UNIVERSITY OF CALIFORNIA,
IRVINE
On the Mathematics of Slow Light
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Mathematics
by
Aaron Thomas Welters
Dissertation Committee:
Professor Aleksandr Figotin, Chair
Professor Svetlana Jitomirskaya
Professor Abel Klein
2011
c 2011 Society for Industrial and Applied Mathematics. Reprinted with
Chapter 2 ⃝
permission. All rights reserved.
c 2011 Aaron Thomas Welters.
All other materials ⃝
DEDICATION
To Jayme
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
v
CURRICULUM VITAE
vi
ABSTRACT OF THE DISSERTATION
ix
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Electrodynamics of Lossless One-Dimensional Photonic Crystals
1.2.1 Time-Harmonic Maxwell’s Equations . . . . . . . . . . .
1.2.2 Lossless 1-D Photonic Crystals . . . . . . . . . . . . . .
1.2.3 Maxwell’s Equations as Canonical Equations . . . . . . .
1.2.4 Definition of Slow Light . . . . . . . . . . . . . . . . . .
1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Perturbation Analysis of Degenerate Eigenvalues from a Jordan block1
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Generic Condition and its Consequences . . . . . . . . . . . . . . . . . .
2.3 Explicit Recursive Formulas for Calculating the Perturbed Spectrum . . . .
2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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41
3 Spectral Perturbation Theory for Holomorphic Matrix Functions
3.1 On Holomorphic Matrix-Valued Functions of One Variable . . . . . .
3.1.1 Local Spectral Theory of Holomorphic Matrix Functions . . .
3.2 On Holomorphic Matrix-Valued Functions of Two Variables . . . . .
3.3 On the Perturbation Theory for Holomorphic Matrix Functions . . .
3.3.1 Eigenvalue Perturbations . . . . . . . . . . . . . . . . . . . . .
3.3.2 Eigenvector Perturbations . . . . . . . . . . . . . . . . . . . .
3.3.3 Analytic Eigenvalues and Eigenvectors . . . . . . . . . . . . .
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c 2011 Society for Industrial and Applied
The contents of this chapter also appear in [61]. Copyright ⃝
Mathematics. Reprinted with permission. All rights reserved.
iii
4 Canonical Equations: A Model for Studying Slow Light
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Canonical ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Energy Flux, Energy Density, and their Averages . . . . . . . . . .
4.2.3 On Points of Definite Type for Canonical ODEs . . . . . . . . . . .
4.2.4 Perturbation Theory for Canonical ODEs . . . . . . . . . . . . . . .
4.3 Canonical DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The Correspondence between Canonical DAEs and Canonical ODEs
4.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Auxiliary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
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103
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ACKNOWLEDGMENTS
I would like to begin by thanking my advisor, Professor Aleksandr Figotin. His insight,
guidance, and support have been invaluable. The efforts he has put in helping me develop as
a mathematician and a researcher went above and beyond the usual duties of an advisor. The
depth and breadth of his knowledge in mathematics and electromagnetism and his passion
for research is a continual source of inspiration for me. He has profoundly influenced my life
and my research and for all of this I am eternally grateful.
To the members of my thesis and advancement committees, Professor Svetlana Jitomirskaya,
Professor Abel Klein, Professor Ozdal Boyraz, and Dr. Ilya Vitebskiy, I would like to thank
them for their time and thoughtful input.
Thank you also to Adam Larios, Jeff Matayoshi, and Mike Rael, for many useful and enjoyable conversations.
A special thanks to Donna McConnell, Siran Kousherian, Tricia Le, Radmila Milosavljevic,
and Casey Sakasegawa, and the other staff at the UCI math department, for handling the
massive amount of paperwork and administrative details that allow me the time to concentrate on research. Their hard work behind the scenes helps make the UCI Mathematics
Department the excellent place that it is.
I would also like to thank my family. My parents, Tom and Kathy Welters, and my brothers,
Matt and Andy Welters, for being great role models on how to work hard and achieve
while still balancing ones life with respect to family, fun, and career and for instilled in
me the importance and joys of family. Thank you so much for all the love, support, and
encouragement over the years. I would like to thank my son, Ashton Welters, for making my
life happier than I thought possible. I would especially like to thank my wife and best friend,
Jayme Welters. I could not have made it without her encouragement, support, commitment,
friendship, love, and laughter. Thank you all that you have done for me over the years, it is
appreciated more than I can possibly say in words.
The work on the explicit recursive formulas in the spectral perturbation analysis of a Jordan
block was first published in SIAM Journal of Matrix Analysis and Applications, Volume 32,
no. 1, (2011) 1–22 and permission to use the copyrighted material in this thesis has been
granted.
I am thankful to the anonymous referees for helpful suggestions and insightful comments on
parts of this research. I am thankful for the University of California, Irvine, where this work
was completed.
This work was supported in part by the Air Force Office of Scientific Research (AFOSR)
under the grant FA9550-08-1-0103.
v
CURRICULUM VITAE
Aaron Thomas Welters
PARTICULARS
EDUCATION
University of California Irvine
Ph. D. in Mathematics
Advisor: Dr. Aleksandr Figotin
Irvine, CA
June 2011
St. Cloud State University
B. A. in Mathematics
Magna Cum Laude
Advisor: Dr. J. -P. Jeff Chen
St. Cloud, MN
May 2004
RESEARCH INTERESTS
My research interests are in mathematical physics, material science, electromagnetics, wave
propagation in periodic media (e.g., photonic crystals), spectral and scattering theory. I specialize in the spectral theory of periodic differential operators, nonlinear eigenvalue problems,
perturbation theory for non-self-adjoint matrices and operators, linear differential-algebraic
equations (DAEs), block operator matrices, boundary value problems, and meromorphic
Fredholm-valued operators. I apply the mathematical methods from these areas to study
spectral and scattering problems of electromagnetic waves in complex and periodic structures.
ACADEMIC HONORS
∙ Von Neumann Award for Outstanding Performance as a Graduate Student, UC Irvine, 2010
∙ Mathematics Department Scholarship, SCSU, 2003
PUBLICATIONS
2. A. Welters, Perturbation Analysis of Slow Waves for Periodic Differential-Algebraic of Definite Type. (in preparation)
1. A. Welters, On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan
Block, SIAM J. Matrix Anal. Appl., 32.1 (2011), pp. 1–22.
vi
TALKS
9. UCLA PDE Seminar, University of California, UCLA, CA, February 2011.
8. 2011 AMS/MAA Joint Mathematics Meetings, New Orleans, LA, January 2011.
7. MGSC (Mathematics Graduate Student Colloquium), University of California, Irvine, CA,
November 2010.
6. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2010.
5. MAA Southern California-Nevada Section Fall 2010 Meeting, University of California, Irvine,
CA, October 2010.
4. Arizona School of Analysis with Applications, University of Arizona, Tucson, AZ, March
2010.
3. 2010 AMS/MAA Joint Mathematics Meetings, San Francisco, CA, January 2010.
2. UCI Mathematical Physics Seminar, University of California, Irvine, CA, November 2009.
1. SCSU Mathematics Colloquium, St. Cloud State University, St. Cloud, MN, November 2009.
RESEARCH EXPERIENCE
∙ Graduate Student Researcher (GSR), UCI, September, 2007 - April, 2011. The GSR
was supported by the AFOSR grant FA9550-08-1-0103 entitled: High-Q Photonic-Crystal
Cavities for Light Amplification.
TEACHING EXPERIENCE
TEACHING ASSISTANT
∙ Math2D: Multivariable Calculus. University of California, Irvine, Summer 2006
∙ Math184: History of Mathematics. University of California, Irvine, Spring 2006
∙ Math146: Fourier Analysis. University of California, Irvine, Spring 2006
∙ Math2B: Single Variable Calculus. University of California, Irvine, Fall 2005
∙ Math2B: Single Variable Calculus. University of California, Irvine, Summer 2005
∙ Math2A: Single Variable Calculus. University of California, Irvine, Summer 2005
∙ Math2J: Infinite Series and Linear Algebra. University of California, Irvine, Spring
2005
∙ Math2B: Single Variable Calculus. University of California, Irvine, Winter 2005
∙ Math2A: Single Variable Calculus. University of California, Irvine, Fall 2004
vii
INSTITUTIONAL SERVICE
∙ Co-Organizer. “Math Graduate Student Colloquium,” University of California, Irvine;
2010-2011
∙ Graduate Recruitment Speaker. “Mathematics Graduate Recruitment Day Event,”
University of California, Irvine; April 2010.
COMPUTATIONAL SKILLS
Programming: C/C++, MATLAB, and Maple.
Markup Languages: LaTeX and Beamer.
Operating Systems: Linux/Unix based systems and Windows.
viii
ABSTRACT OF THE DISSERTATION
On the Mathematics of Slow Light
By
Aaron Thomas Welters
Doctor of Philosophy in Mathematics
University of California, Irvine, 2011
Professor Aleksandr Figotin, Chair
This thesis develops a mathematical framework based on spectral perturbation theory for
the analysis of slow light and the slow wave regime for lossless one-dimensional photonic
crystals.
Electrodynamics of lossless one-dimensional photonic crystals incorporating general bianisotropic and dispersive materials are considered. The time-harmonic Maxwell’s equations
governing the electromagnetic wave propagation through such periodic structures is shown
to reduce to a canonical system of period differential-algebraic equations (DAEs) depending
holomorphically on frequency which we call Maxwell’s DAEs. In this context, a definition
of slow light is given.
We give a detailed perturbation analysis for degenerate eigenvalues of non-self-adjoint matrices. A generic condition is considered and its consequences are studied. We prove the generic
condition implies the degenerate eigenvalue of the unperturbed matrix under consideration
has a single Jordan block in its Jordan normal form corresponding to that eigenvalue. We
find explicit recursive formulas to calculate the perturbation expansions of the splitting eigenvalues and their eigenvectors. The coefficients up to the second order for these expansions
are given.
ix
An exposition on the spectral theory and spectral perturbation theory for holomorphic matrix
functions is given. This serves as a mathematical background for the tools and theory used
in this thesis.
We give a model for studying slow light based on canonical equations which are any canonical
system of periodic differential (canonical ODEs) or differential-algebraic (canonical DAEs)
equations depending holomorphically on a spectral parameter referred to as the frequency.
For canonical ODEs, we prove formulas connecting the dispersion relation, energy flux, and
energy density in our model to the monodromy matrix of the canonical ODEs. We prove one
of the main results of this thesis relating the existence of (non-Bloch) Floquet solutions of the
canonical ODEs and the occurrence of degenerate eigenvalues with nondiagonalizable Jordan
normal form of the monodromy matrix to the band structure of the dispersion relation near
spectral stationary points. For canonical DAEs, of which it is shown Maxwell’s DAEs are
an example, a theorem is given which shows that the model for canonical DAEs, including
the Floquet, spectral, and perturbation theory, is reduced to the model for canonical ODEs.
x
Chapter 1
Introduction
1.1
Motivation
Technology has progressed to the point at which we can now fabricate materials with dimensions on the nanoscale. To give you a perspective on how small this is consider that
one nanometer (nm) is equal to one billionth of a meter (m), the width of a human hair is
about 105 nm, and the wavelength of visible light lies in the approximate range of 400–700
nm. As a result, there has emerged a new class of materials (such as metamaterials [11, 54]),
which allow the possibility of novel electromagnetic properties and effects. For example,
bianisotropic crystals can be used to observe optical singularities [6], photonic crystals [27]
can be used to slow down light [16], negative index materials can be used to create a perfect
lens [50], and metametarials derived from transformation optics can be used to induce electromagnetic invisibility [21, 22, 40, 51]. The novel electromagnetic properties and effects that
metamaterials such as photonic crystals have, is a fascinating and productive area of research
for not only physicists but for mathematicians as well (see for example [32]). In particular,
research on slow light has received considerable attention recently. With its numerous appli-
1
cations such as solar cell efficiency enhancement [10] or antenna miniaturization [49, 59, 62],
there is a growing need for applied and theoretical studies on slow light phenomena.
Although there are several methods for slowing down light including electromagnetically induced transparency (EIT) and photonic crystals, it is the latter that is becoming of particular
interest to mathematicians. A major reason for this is the mathematics of photonic crystals
has been developed significantly over the last 20 years and this gives us a rigorous mathematical setting in which to begin studying problems involving slow light. Moreover, many
of the results on slow wave phenomenon in photonic crystals are expected to have analogies with wave propagation in general linear dispersive media. Thus in order to facilitate
further applied and theoretical studies on slow light, to better understand slow light from a
mathematical perspective, and to explore analogies with slow waves in periodic media, this
thesis considers a general nondissipative but dispersive model for wave propagation using
an important class of periodic differential and differential-algebraic equations (DAEs) called
canonical equations [31]. As we will show in the next section, this model is general enough
to include electromagnetic waves governed by the time-harmonic Maxwell’s equations for
lossless one-dimensional photonic crystals whose constituent layers can be any combination
of isotropic, anisotropic, or bianisotropic materials with or without material dispersion (i.e.,
frequency-dependent response of materials). This makes our work particularly significant
in the study of slow light since metamaterials are widening the range of potential photonic
crystals that can be fabricated and so a model like ours that has the ability to analysis slow
light phenomena for a broad range of photonic crystals is in need.
2
1.2
Electrodynamics of Lossless One-Dimensional Photonic Crystals
1.2.1
Time-Harmonic Maxwell’s Equations
Electromagnetic waves will be governed by the time-harmonic Maxwell’s equations (𝑒−𝑖𝜔𝑡
convention, 𝜔 ∕= 0) without sources in Gaussian units and Cartesian coordinates (with
respect to the standard orthonormal basis vectors e1 , e2 , e3 of ℝ3 ). These equations may be
written in the 2 × 2 block operator matrix form (see [5])
⎡
⎤⎡
⎡
⎤
⎤
∇× ⎥ ⎢ E(r) ⎥
𝜔 ⎢ D(r) ⎥
⎦⎣
⎦ = −𝑖 ⎣
⎦
𝑐 B(r)
−∇× 0
H(r)
0
⎢
⎣
(1.1)
where 𝑐 is the speed of light in a vacuum, r := (𝑥1 , 𝑥2 , 𝑥3 ) are the spatial variables and
the electric field E, magnetic field H, electric induction D, and magnetic induction B take
values in ℂ3 . Here ∇× denotes the curl operator on these fields and it is given by the matrix
operator
⎡
− ∂𝑥∂ 3
∂
∂𝑥2
∂
∂𝑥3
0
− ∂𝑥∂ 1
− ∂𝑥∂ 2
∂
∂𝑥1
0
0
⎢
⎢
∇× := ⎢
⎢
⎣
⎤
⎥
⎥
⎥.
⎥
⎦
(1.2)
The linear constitutive relations in 2 × 2 block matrix form are
⎡
⎤
⎡
⎤
⎡
⎤
⎢ D(r) ⎥
⎢ E(r) ⎥
⎢ 𝜀(𝑥3 , 𝜔) 𝜉(𝑥3 , 𝜔) ⎥
⎣
⎦ = C(𝑥3 , 𝜔) ⎣
⎦ , C(𝑥3 , 𝜔) = ⎣
⎦,
B(r)
H(r)
𝜁(𝑥3 , 𝜔) 𝜇(𝑥3 , 𝜔)
(1.3)
where 𝜀, 𝜇, 𝜉, 𝜁 are 3 × 3 matrix-valued functions representing the electric permittivity, magnetic permeability, and magnetoelectric coupling tensors, respectively. Here we have tacitly
3
assumed materials which are plane parallel layers normal to the 𝑥3 -axis with each layer
consisting of a homogeneous material whose frequency dependency is implicitly indicated.
1.2.2
Lossless 1-D Photonic Crystals
We are considering lossless one-dimensional photonic crystals with dispersive materials and
hence there exists a 𝑑 > 0 and an open connected set Ω ⊆ ℂ, the frequency domain, with
Ωℝ := Ω ∩ ℝ ∕= ∅ such that
(i) C(𝑥3 , 𝜔) = C(𝑥3 + 𝑑, 𝜔), for every 𝜔 ∈ Ω and a.e. 𝑥3 .
(ii) C(𝑥3 , 𝜔)∗ = C(𝑥3 , 𝜔), for every 𝜔 ∈ Ωℝ and a.e. 𝑥3 .
(iii) C(⋅, 𝜔) ∈ 𝑀6 (𝐿2 (𝕋)), C ∈ 𝒪(Ω, 𝑀6 (𝐿2 (𝕋)))1 .
1.2.3
Maxwell’s Equations as Canonical Equations
As in [16], we seek field solutions of the form
⎡
⎡
⎤
⎤
E(𝑥3 ) ⎥
⎢ E(r) ⎥
𝑖k ⋅r ⎢
⎦,
⎣
⎦=𝑒 ⊥ ⊥⎣
H(r)
H(𝑥3 )
(1.4)
where r⊥ = (𝑥1 , 𝑥2 , 0), k⊥ = (𝑘1 , 𝑘2 , 0), and 𝑘1 , 𝑘2 ∈ ℝ. Hence solutions to the time-harmonic
Maxwell’s equations in (1.1) with the field representations (1.4) are solutions to the canonical
equations, which we call Maxwell’s DAEs,
G 𝑦 ′ (𝑥3 ) = 𝑉 (𝑥3 , 𝜔)𝑦(𝑥3 ),
1
See section 4.5 for notation.
4
(1.5)
where 𝑦(𝑥3 ) is a 6 × 1 column vector, G is a 6 × 6 singular matrix, and 𝑉 , the Hamiltonian,
is a 6 × 6 matrix-valued function having the block representations
⎡
⎤
⎡
⎡
⎤
⎤
k⊥ × ⎥
e3 × ⎥
𝜔
⎢
⎢ E(𝑥3 ) ⎥
⎢
𝑦(𝑥3 ) = ⎣
⎦.
⎦, G = 𝑖⎣
⎦ , 𝑉 (𝑥3 , 𝜔) = C(𝑥3 , 𝜔) + ⎣
𝑐
−k⊥ ×
H(𝑥3 )
−e3 ×
(1.6)
Here e3 × and k⊥ × are the matrices
⎡
⎤
⎡
⎤
0 𝑘2 ⎥
⎢ 0 −1 0 ⎥
⎢ 0
⎢
⎥
⎢
⎥
⎥ , k⊥ × := ⎢ 0
⎥
e3 × := ⎢
1
0
0
0
−𝑘
1 ⎥.
⎢
⎥
⎢
⎣
⎦
⎣
⎦
0
0 0
−𝑘2 𝑘1 0
(1.7)
In particular, they are a canonical system of differential-algebraic equations (DAEs) with
periodic coefficients that depend holomorphically on the frequency 𝜔 where the leading
matrix coefficient G ∈ 𝑀6 (ℂ) and the matrix-valued function 𝑉 : ℝ × Ω → 𝑀6 (ℂ) have the
properties:
(i) det(G ) = 0, G ∗ = −G
(ii) 𝑉 (𝑥3 , 𝜔)∗ = 𝑉 (𝑥3 , 𝜔), for each 𝜔 ∈ Ωℝ and a.e. 𝑥3 ∈ ℝ
(iii) 𝑉 (𝑥3 + 𝑑, 𝜔) = 𝑉 (𝑥3 , 𝜔), for each 𝜔 ∈ Ω and a.e. 𝑥3 ∈ ℝ
(iv) 𝑉 ∈ 𝒪(Ω, 𝑀6 (𝐿2 (𝕋))) as a function of frequency.
In Chapter 4 we will consider these types of equations further.
5
1.2.4
Definition of Slow Light
In this context, Bloch solutions of Maxwell’s DAEs, i.e. solutions of the canonical equations
in (1.5) satisfying
𝑦(𝑥3 + 𝑑) = 𝑒𝑖𝑘𝑑 𝑦(𝑥3 ),
(1.8)
for some 𝑘 ∈ ℂ, give rise to the (axial) dispersion relation
𝜔 = 𝜔(𝑘).
(1.9)
For points (𝑘0 , 𝜔 0 ) ∈ ℝ2 on the graph of this dispersion relation, i.e., the Bloch variety ℬ, a
Bloch solution 𝑦 with wavenumber-frequency pair (𝑘0 , 𝜔 0 ) is said to be propagating with its
group velocity
𝑑𝜔 .
𝑑𝑘 (𝑘0 ,𝜔0 )
(1.10)
Under certain reservations [7, 8, 27, 63], its group velocity equals its energy velocity, i.e., the
ratio of its averaged energy flux to its energy density,
𝑑𝜔 =
𝑑𝑘 (𝑘0 ,𝜔0 )
∫
1 𝑑
⟨𝑖G 𝑦(𝑥3 ), 𝑦(𝑥3 )⟩𝑑𝑥3
,
∫ 𝑑 0
1 𝑑
⟨𝑉𝜔 (𝑥3 , 𝜔 0 )𝑦(𝑥3 ), 𝑦(𝑥3 )⟩𝑑𝑥3
𝑑 0
(1.11)
with its energy flux and its energy density are (up to multiplication by a constant) the
functions of the space variable 𝑥3 given by
𝑆 = ⟨𝑖G 𝑦, 𝑦⟩,
(1.12)
𝑈 = ⟨𝑉𝜔 (⋅, 𝜔 0 )𝑦, 𝑦⟩,
(1.13)
6
respectively, where 𝑉𝜔 denotes the derivative of the Hamiltonian 𝑉 with respect to frequency
in the 𝑀𝑁 (𝐿2 (𝕋)) norm and ⟨ , ⟩ is the standard inner product for ℂ6 .
We now give an intuitive definition of slow light for lossless one-dimensional photonic crystals.
In Chapter 4 we give a more precise definition using canonical equations. We will treat the
(axial) dispersion relation 𝜔 = 𝜔(𝑘) as a multi-valued function of the wavenumber 𝑘. We
will use the notation
𝑑𝜔
∣
𝑑𝑘 (𝑘0 ,𝜔0 )
= 0 to mean (𝑘0 , 𝜔 0 ) is a stationary point on the graph of the
dispersion relation 𝜔 = 𝜔(𝑘).
Definition 1 (Slow Light) If (𝑘0 , 𝜔 0 ) ∈ ℝ2 is a stationary point on the graph of the dispersion relation 𝜔 = 𝜔(𝑘), i.e.,
𝑑𝜔 = 0,
𝑑𝑘 (𝑘0 ,𝜔0 )
(1.14)
then any propagating Bloch solution of Maxwell’s DAEs (1.5) with wavenumber-frequency
pair (𝑘, 𝜔) satisfying 0 < ∣∣(𝑘, 𝜔) − (𝑘0 , 𝜔 0 )∣∣ ≪ 1 with its group velocity
𝑑𝜔
∣
𝑑𝑘 (𝑘,𝜔)
satisfying
∣
∣ ≪ 𝑐 is called a slow wave or slow light.
∣ 𝑑𝜔
𝑑𝑘 (𝑘,𝜔)
If (𝑘0 , 𝜔 0 ) ∈ ℝ2 is a stationary point on the graph of the dispersion relation 𝜔 = 𝜔(𝑘) then
an open ball 𝐵((𝑘0 , 𝜔 0 ), 𝑟) in ℂ2 with 0 < 𝑟 ≪ 1 is called the slow wave regime.
1.3
Main Results
The objective of this dissertation is to begin developing a mathematical framework based
on spectral perturbation theory for the analysis of slow light and the slow wave regime for
lossless one-dimensional photonic crystals.
In the following we describe the main contributions of this thesis which are contained in
7
Chapters 2 and 4.
Perturbation Analysis of Degenerate Eigenvalues from a Jordan
block
A fundamental problem in the perturbation theory for non-self-adjoint matrices with a degenerate spectrum is the determination of the perturbed eigenvalues and eigenvectors. Formulas
for the higher order terms of these perturbation expansions are often needed in problems
which require an accurate asymptotic analysis.
For example, my advisor A. Figotin and his colleague, I. Vitebskiy, considered scattering
problems involving slow light in one-dimensional semi-infinite photonic crystals [3, 12–16].
They found that only in the case of the frozen mode regime could incident light enter a
photonic crystal with little reflection and be converted into a slow wave. This frozen mode
regime was found to correspond to a stationary inflection point of the dispersion relation
and a 3 × 3 Jordan block in the Jordan normal form of the unit cell transfer matrix – the
monodromy matrix of the reduced Maxwell’s equations given in [16, §5] or [16, p. 332, (180)]
which are canonical ODEs although not in canonical form. In this setting, the eigenpairs of
the monodromy matrix corresponded to Bloch waves and their Floquet multipliers. Thus in
order for them to rigorously prove the physical results and provide a better understanding of
the very essence of the frozen mode regime, they needed an asymptotic analytic description
of the frozen mode regime which required a sophisticated mathematical framework based on
the spectral perturbation theory of a Jordan block. Unfortunately, at the time when [16] was
written such a theory did not exist and hence this was a big motivating factor for Chapter
22 of this thesis.
2
c 2011 Society for Industrial and Applied
The contents of this chapter also appear in [61]. Copyright ⃝
Mathematics. Reprinted with permission. All rights reserved.
8
In Chapter 2 of this thesis we to develop the spectral perturbation theory of a Jordan block
and address the following:
1. Determine a generic condition that allows a constructive spectral perturbation analysis
for non-self-adjoint matrices with degenerate eigenvalues.
2. What connection is there between that condition and the local Jordan normal form of
the matrix perturbation corresponding to a perturbed eigenvalue?
3. Does there exist explicit recursive formulas to determine the perturbed eigenvalues and
eigenvectors for non-selfadjoint perturbations of matrices with degenerate eigenvalues?
The statement of my main results regarding those three issues are contained in Theorem 1
and Theorem 2 of this thesis which are Theorem 2.1 and Theorem 3.1 from [61]. In particular, I have developed a constructive perturbation theory for non-self-adjoint matrices with
degenerate eigenvalues and found explicit recursive formulas to calculate the perturbation
expansions of the splitting eigenvalues and their eigenvectors, under a generic condition.
Canonical Equations: A Model for Studying Slow Light
We established above that the study of slow light, i.e., slow waves for lossless one-dimensional
photonic crystals is reduced to the study of Maxwell’s DAEs (1.5) near stationary points
of the dispersion relation, i.e., in the slow wave regime. As Maxwell’s DAEs are canonical
equations, it will be beneficial to use canonical equations to study slow wave propagation. By
analogy with that physical model, we get a general mathematical model for wave propagation
in periodic structures in which we can study the dispersion relation, band structure, spectral
stationary points, slow waves, and the slow wave regime. This is exactly what we do in
Chapter 4. The contents of this chapter is part of the paper [60] currently in preparation.
9
One of the major contribution in Chapter 4 of this thesis is Theorem 48. Combined with
Theorem 35, Theorem 47, and Theorem 50, it is our answer to the question:
1. How are the analytic properties of the dispersion relation and the degree of flatness of
spectral bands near stationary points related to (non-Bloch) Floquet solutions and the
occurrence of degenerate eigenvalues and a nondiagonalizable Jordan normal form for
the monodromy matrix of the canonical equations.
To answer this question and to proof Theorem 48 we need some deep results in the spectral
perturbation theory for holomorphic matrix functions. We give an exposition of this theory
in Chapter 3.
1.4
Overview
This thesis is organized in the following manner.
Chapter 2 concerns the perturbation analysis of non-self-adjoint matrices with degenerate
eigenvalues. A generic condition is considered and its consequences are studied. It is shown
that the generic condition implies the degenerate eigenvalue of the unperturbed matrix under
consideration has a single Jordan block in its Jordan normal form corresponding to the
degenerate eigenvalue. Explicit recursive formulas are given to calculate the perturbation
expansions of the splitting eigenvalues and their eigenvectors. The coefficients up to the
second order for these expansions are conveniently listed for quick reference.
Chapter 3 is an exposition on the spectral theory and spectral perturbation theory for holomorphic matrix functions. Its content is the required background material needed in Chapter
4 to prove the main result for that chapter, namely, Theorem 48.
Chapter 4 formulates and studies a model for slow wave propagation in period structures
10
using canonical equations. Both canonical ODEs and canonical DAEs are considered. Electromagnetic wave propagation in lossless one dimensional photonic crystals was discussed in
section 1.2 as a example for the need to include DAEs into the model. Motivation for the
definitions in the model was given by a consideration of their relevance in the example.
Next, in section 4.2, canonical ODEs are considered. Results are given on energy flux and
energy density. Then in section 4.2.3 an important theorem, Theorem 47, is given on points
of definite type for canonical ODEs. In particular, the result justifies the use of the term
definite type and states that to each point of definite type of the canonical ODEs there exists
a neighborhood of such points. This is followed up immediately in the next section, section
4.2.4, with the main result of the chapter, Theorem 48, pertaining to the perturbation theory
for canonical ODEs near points of definite type. As a corollary we give a result that connects
the generic condition of Chapter 2 to points of definite type and the Jordan normal form of
the monodromy matrix of the canonical ODEs.
Next, in section 4.3, canonical DAEs are considered. A theorem is stated that tells us the
theory of canonical DAEs is reduced to the study of canonical ODEs including the Floquet,
spectral, and perturbation theory.
Finally, section 4.4 gives the proofs of all the statements in this chapter. The use of the
spectral perturbation theory of holomorphic matrix functions is use to prove the main result
of the chapter, Theorem 48.
All of the work done here was completed under the guidance and supervision of Professor
Aleksandr Figotin.
11
Chapter 2
Perturbation Analysis of Degenerate
Eigenvalues from a Jordan block1
2.1
Introduction
Consider an analytic square matrix 𝐴 (𝜀) and its unperturbed matrix 𝐴 (0) with a degenerate
eigenvalue 𝜆0 . A fundamental problem in the analytic perturbation theory of non-selfadjoint
matrices is the determination of the perturbed eigenvalues near 𝜆0 along with their corresponding eigenvectors of the matrix 𝐴 (𝜀) near 𝜀 = 0. More specifically, let 𝐴 (𝜀) be a
matrix-valued function having a range in 𝑀𝑛 (ℂ), the set of 𝑛 × 𝑛 matrices with complex
entries, such that its matrix elements are analytic functions of 𝜀 in a neighborhood of the
origin. Let 𝜆0 be an eigenvalue of the matrix 𝐴 (0) with algebraic multiplicity 𝑚 ≥ 1. Then
in this situation, it is well known [4, §6.1.7], [28, §II.1.8] that for sufficiently small 𝜀 all the
perturbed eigenvalues near 𝜆0 , called the 𝜆0 -group, and their corresponding eigenvectors may
be represented as a collection of convergent Puiseux series, i.e., convergent Taylor series in
1
c 2011 Society for Industrial and Applied
The contents of this chapter also appear in [61]. Copyright ⃝
Mathematics. Reprinted with permission. All rights reserved.
12
a fractional power of 𝜀. What is not well known, however, is how we compute these Puiseux
series when 𝐴 (𝜀) is a non-selfadjoint analytic perturbation and 𝜆0 is a defective eigenvalue
of 𝐴 (0). There are sources on the subject like [4, §7.4], [41], [47], [57, §32], and [58] but it
was found that there lacked explicit formulas, recursive or otherwise, to compute the series
coefficients beyond the first order terms. Thus the fundamental problem that this paper
addresses is to find explicit recursive formulas to determine the Puiseux series coefficients
for the 𝜆0 -group and their eigenvectors.
This problem is of applied and theoretic importance, for example, in studying the spectral
properties of dispersive media such as photonic crystals. In particular, this is especially true
in the study of slow light [16, 49, 62], where the characteristic equation, det (𝜆𝐼 − 𝐴 (𝜀)) = 0,
represents implicitly the dispersion relation for Bloch waves in the periodic crystal. In this
setting 𝜀 represents a small change in frequency, 𝐴(𝜀) is the Transfer matrix of a unit cell, and
its eigenpairs, (𝜆(𝜀), 𝑥(𝜀)), correspond to the Bloch waves. From a practical and theoretical
point of view, condition (2.1) on the dispersion relation or its equivalent formulation in
Theorem 1.(i ) of this paper regarding the group velocity for this setting, arises naturally in
the study of slow light where the Jordan normal form of the unperturbed Transfer matrix,
𝐴(0), and the perturbation expansions of the eigenpairs of the Transfer matrix play a central
role in the analysis of slow light waves.
Main Results
In this paper under the generic condition,
∂
det (𝜆𝐼 − 𝐴 (𝜀)) (𝜀,𝜆)=(0,𝜆0 ) ∕= 0,
∂𝜀
(2.1)
we show that 𝜆0 is a non-derogatory eigenvalue of 𝐴(0) and the fundamental problem mentioned above can be solved. In particular, we prove Theorem 1 and Theorem 2 which together
13
state that when condition (2.1) is true then the Jordan normal form of 𝐴 (0) corresponding
to the eigenvalue 𝜆0 consists of a single 𝑚 × 𝑚 Jordan block, the 𝜆0 -group and their corresponding eigenvectors can each be represented by a single convergent Puiseux series whose
branches are given by
𝜆ℎ (𝜀) = 𝜆0 +
∞
∑
)𝑘
(
1
𝛼𝑘 𝜁 ℎ 𝜀 𝑚
𝑘=1
𝑥ℎ (𝜀) = 𝛽 0 +
∞
∑
)𝑘
(
1
𝛽 𝑘 𝜁 ℎ𝜀 𝑚
𝑘=1
1
2𝜋
∞
for ℎ = 0, . . . , 𝑚 − 1 and any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 , {𝛼𝑘 }∞
𝑘=1 ⊆ ℂ, {𝛽 𝑘 }𝑘=0 ⊆
ℂ𝑛 , 𝛼1 ∕= 0, and 𝛽 0 is an eigenvector of 𝐴 (0) corresponding to the eigenvalue 𝜆0 . More
importantly though, Theorem 2 gives explicit recursive formulas that allows us to determine
∞
the Puiseux series coefficients, {𝛼𝑘 }∞
𝑘=1 and {𝛽 𝑘 }𝑘=0 , from just the derivatives of 𝐴 (𝜀) at
𝜀 = 0. Using these recursive formulas, we compute the leading Puiseux series coefficients up
to the second order and list them in Corollary 4.
The key to all of our results is the study of the characteristic equation for the analytic matrix
𝐴 (𝜀) under the generic condition (2.1). By an application of the implicit function theorem,
we are able to derive the functional relation between the eigenvalues and the perturbation
parameter. This leads to the implication that the Jordan normal form of the unperturbed
matrix 𝐴 (0) corresponding to the eigenvalue 𝜆0 is a single 𝑚 × 𝑚 Jordan block. From this,
we are able to use the method of undetermined coefficients along with a careful combinatorial
analysis to get explicit recursive formulas for determining the Puiseux series coefficients.
We want to take a moment here to show how the results of this paper can be used to
determine the Puiseux series coefficients up to the second order for the case in which the
non-derogatory eigenvalue 𝜆0 has algebraic multiplicity 𝑚 ≥ 2. We start by putting 𝐴 (0)
14
into the Jordan normal form [35, §6.5: The Jordan Theorem]
⎤
⎡
⎢ 𝐽𝑚 (𝜆0 )
𝑈 −1 𝐴 (0) 𝑈 = ⎣
𝑊0
⎥
⎦,
(2.2)
where (see notations at end of §1) 𝐽𝑚 (𝜆0 ) is an 𝑚 × 𝑚 Jordan block corresponding to the
eigenvalue 𝜆0 and 𝑊0 is the Jordan normal form for the rest of the spectrum. Next, define
the vectors 𝑢1 ,. . . , 𝑢𝑚 , as the first 𝑚 columns of the matrix 𝑈 ,
𝑢𝑖 := 𝑈 𝑒𝑖 , 1 ≤ 𝑖 ≤ 𝑚
(2.3)
(These vectors have the properties that 𝑢1 is an eigenvector of 𝐴 (0) corresponding to the
eigenvalue 𝜆0 , they form a Jordan chain with generator 𝑢𝑚 , and are a basis for the algebraic eigenspace of 𝐴 (0) corresponding to the eigenvalue 𝜆0 ). We then partition the matrix
𝑈 −1 𝐴′ (0)𝑈 conformally to the blocks 𝐽𝑚 (𝜆0 ) and 𝑊0 of the matrix 𝑈 −1 𝐴 (0) 𝑈 as such
⎡
⎤
∗
⎢ ∗
⎢
..
..
⎢
.
.
⎢
⎢
⎢
⎢ ∗
∗
⎢
⎢
⎢ 𝑎𝑚−1,1 ∗
⎢
𝑈 −1 𝐴′ (0)𝑈 = ⎢
⎢ 𝑎
⎢ 𝑚,1 𝑎𝑚,2
⎢
⎢
⎢ ∗
∗
⎢
⎢
..
..
⎢
.
.
⎢
⎣
∗
∗
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗ ⎥
.. . . .. .. . . .. ⎥
. . .
. . ⎥
.
⎥
⎥
⎥
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗ ⎥
⎥
⎥
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗ ⎥
⎥
⎥.
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗ ⎥
⎥
⎥
⎥
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗ ⎥
⎥
.. . . .. .. . . .. ⎥
. . ⎥
. . .
.
⎥
⎦
∗ ⋅⋅⋅ ∗ ∗ ⋅⋅⋅ ∗
(2.4)
Now, by Theorem 1 and Theorem 2, it follows that
∂
∂𝜀
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 )
𝑎𝑚,1 = − ( ∂ 𝑚
) .
∂𝜆𝑚
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
𝑚!
15
(2.5)
And hence the generic condition is true if and only if 𝑎𝑚,1 ∕= 0. This gives us an alternative
method to determine whether the generic condition (2.1) is true or not.
Lets now assume that 𝑎𝑚,1 ∕= 0 and hence that the generic condition is true. Define 𝑓 (𝜀, 𝜆) :=
det (𝜆𝐼 − 𝐴 (𝜀)). Then by Theorem 2 and Corollary 4 there is exactly one convergent Puiseux
series for the perturbed eigenvalues near 𝜆0 and one for their corresponding eigenvectors
whose branches are given by
∞
)
(
)2 ∑
)𝑘
(
(
1
1
1
𝜆ℎ (𝜀) = 𝜆0 + 𝛼1 𝜁 ℎ 𝜀 𝑚 + 𝛼2 𝜁 ℎ 𝜀 𝑚 +
𝛼𝑘 𝜁 ℎ 𝜀 𝑚
(2.6)
𝑘=3
)
1
(
)2
1
(
𝑥ℎ (𝜀) = 𝑥0 + 𝛽 1 𝜁 ℎ 𝜀 𝑚 + 𝛽 2 𝜁 ℎ 𝜀 𝑚
+
∞
∑
(
)𝑘
1
𝛽 𝑘 𝜁 ℎ𝜀 𝑚
(2.7)
𝑘=3
2𝜋
1
for ℎ = 0, . . . , 𝑚 − 1 and any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 . Furthermore, the series
coefficients up to second order may be given by
(
1/𝑚
𝛼1 = 𝑎𝑚,1 =
𝛼2 =
∂𝑓
(0, 𝜆0 )
− 1 ∂𝜀
𝑚
∂ 𝑓
(0, 𝜆0 )
𝑚! ∂𝜆𝑚
−
𝑎𝑚−1,1 + 𝑎𝑚,2
=
𝑚𝛼𝑚−2
1
(
)1/𝑚
∕= 0,
(2.8)
∂ 𝑚+1 𝑓
1
𝛼𝑚+1
1
(𝑚+1)! ∂𝜆𝑚+1
(
𝑚−1
𝑚𝛼1
(0, 𝜆0 ) +
∂𝑚𝑓
𝑚
1
𝑚! ∂𝜆
∂2𝑓
𝛼1 ∂𝜆∂𝜀
)
(0, 𝜆0 )
⎧

⎨ −Λ𝐴′ (0)𝑢1 + 𝛼2 𝑢2 , if 𝑚 = 2
𝛽 0 = 𝑢1 , 𝛽 1 = 𝛼1 𝑢2 , 𝛽 2 =

⎩
𝛼 𝑢 + 𝛼2 𝑢 , if 𝑚 > 2
2 2
(0, 𝜆0 )
)
,
(2.9)
(2.10)
1 3
for any choice of the 𝑚th root of 𝑎𝑚,1 and where Λ is given in (2.15).
The explicit recursive formulas for computing higher order terms, 𝛼𝑘 , 𝛽 𝑘 , are given by (2.24)
and (2.25) in Theorem 2. The steps which should be used to determine these higher order
terms are discussed in Remark 1 and an example showing how to calculating 𝛼3 , 𝛽 3 using
these steps, when 𝑚 ≥ 3, is provided.
16
Example
The following example may help to give a better idea of these results. Consider
⎡
− 21
1
2
⎤
⎡
⎤
⎥
⎢ 2 0 −1 ⎥
⎢
⎥
⎥
⎥.
1 ⎥ + 𝜀⎢
1
0
−
2
0
−1
⎢
⎥
2
2 ⎥
⎦
⎣
⎦
−1 1
1
1 0
0
⎢
⎢
𝐴 (𝜀) := ⎢
⎢
⎣
1
(2.11)
Here 𝜆0 = 0 is a non-derogatory eigenvalue of 𝐴 (0) of algebraic multiplicity 𝑚 = 2. We put
𝐴 (0) into the Jordan normal form
⎡
⎤
⎡
⎤
⎡
0
⎢ 0 1 0 ⎥
⎢1 1 1⎥
⎢ 1 −1
⎢
⎥
⎢
⎥
⎢
−1
⎥
⎢
⎥
⎢
𝑈 −1 𝐴 (0) 𝑈 = ⎢
1
1
⎢ 0 0 0 ⎥ , 𝑈 = ⎢0 1 1⎥ , 𝑈 = ⎢ −1
⎣
⎦
⎣
⎦
⎣
1 1 0
1
0 −1
0 0 1/2
⎤
⎥
⎥
⎥,
⎥
⎦
so that 𝑊0 = 1/2. We next define the vectors 𝑢1 , 𝑢2 , as the first two columns of the matrix
𝑈,
⎡ ⎤
⎡ ⎤
⎢1⎥
⎢1⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎥
𝑢1 := ⎢
⎢0⎥ , 𝑢2 := ⎢1⎥ .
⎣ ⎦
⎣ ⎦
1
1
Next we partition the matrix 𝑈 −1 𝐴′ (0)𝑈 conformally to the blocks 𝐽𝑚 (𝜆0 ) and 𝑊0 of the
matrix 𝑈 −1 𝐴 (0) 𝑈 as such
⎡
⎤⎡
0
⎢ 1 −1
⎢
𝑈 −1 𝐴′ (0)𝑈 = ⎢
1
1
⎢ −1
⎣
1
0 −1
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
⎤ ⎡
⎤⎡
2 0 −1 ⎥ ⎢1 1 1⎥ ⎢ 0 ∗ ∗
⎥⎢
⎥ ⎢
⎢0 1 1⎥ = ⎢ 1 1 ∗
2 0 −1 ⎥
⎥⎢
⎥ ⎢
⎦ ⎣
⎦⎣
1 1 0
1 0
0
∗ ∗ ∗
17
⎤
⎥
⎥
⎥.
⎥
⎦
Here 𝑎2,1 = 1, 𝑎1,1 = 0, and 𝑎2,2 = 1. Then
∂
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 )
) ,
1 = 𝑎2,1 = − ∂𝜀( ∂ 2
∂𝜆2
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
2!
implying that the generic condition (2.1) is true. Define 𝑓 (𝜀, 𝜆) := det (𝜆𝐼 − 𝐴 (𝜀)) =
𝜆3 − 2𝜆2 𝜀 − 12 𝜆2 + 𝜆𝜀2 − 12 𝜆𝜀 + 𝜀2 + 12 𝜀. Then there is exactly one convergent Puiseux series for
the perturbed eigenvalues near 𝜆0 = 0 and one for their corresponding eigenvectors whose
branches are given by
(
ℎ
𝜆ℎ (𝜀) = 𝜆0 + 𝛼1 (−1) 𝜀
1
2
)
(
ℎ
+ 𝛼2 (−1) 𝜀
1
2
)2
+
(
(
)
)
1
1 2
𝑥ℎ (𝜀) = 𝛽 0 + 𝛽 1 (−1)ℎ 𝜀 2 + 𝛽 2 (−1)ℎ 𝜀 2 +
∞
∑
𝑘=3
∞
∑
(
ℎ
𝛼𝑘 (−1) 𝜀
1
2
)𝑘
(
)
1 𝑘
𝛽 𝑘 (−1)ℎ 𝜀 2
𝑘=3
1
for ℎ = 0, 1 and any fixed branch of 𝜀 2 . Furthermore, the series coefficients up to second
order may be given by
v(
)
u
∂𝑓
u
√
(0,
𝜆
)
√
0
𝛼1 = 1 = 1 = 𝑎2,1 = ⎷ − 1 ∂𝜀
∕= 0,
∂2𝑓
(0,
𝜆
)
2
0
2! ∂𝜆
(
)
∂2𝑓
3 1 ∂3𝑓
−
𝛼
(0,
𝜆
)
+
𝛼
(0,
𝜆
)
0
1 ∂𝜆∂𝜀
0
1 3! ∂𝜆3
1
𝑎1,1 + 𝑎2,2
( 2
)
𝛼2 = =
=
,
2
2
𝛼1 2!1 ∂∂𝜆𝑓2 (0, 𝜆0 )
⎡ ⎤
⎡ ⎤
⎢1⎥
⎢1⎥
⎢ ⎥
⎢ ⎥
′
⎥
⎢ ⎥
𝛽0 = ⎢
⎢0⎥ , 𝛽 1 = ⎢1⎥ , 𝛽 2 = −Λ𝐴 (0)𝑢1 + 𝛼2 𝑢2
⎣ ⎦
⎣ ⎦
1
1
by choosing the positive square root of 𝑎2,1 = 1 and where Λ is given in (2.15). Here
⎡
⎤
∗
⎢ 𝐽𝑚 (0)
Λ=𝑈⎣
(𝑊0 − 𝜆0 𝐼𝑛−𝑚 )−1
⎥ −1
⎦𝑈
18
⎡
⎤⎡
0
⎢1 1 1⎥ ⎢ 0 0
⎢
⎥⎢
⎥⎢
=⎢
0
⎢0 1 1⎥ ⎢ 1 0
⎣
⎦⎣
1 1 0
0 0 (1/2)−1
⎤⎡
⎤
0
⎥ ⎢ 1 −1
⎥⎢
⎥ ⎢ −1
1
1
⎥⎢
⎦⎣
1
0 −1
⎥ ⎢ 3 −1 −2 ⎥
⎥ ⎢
⎥
⎥ = ⎢ 3 −1 −2 ⎥
⎥ ⎢
⎥
⎦ ⎣
⎦
1 −1
0
⎡
⎤
𝛽 2 = −Λ𝐴′ (0)𝑢1 + 𝛼2 𝑢2
⎡ ⎤
⎡ ⎤
⎡
⎤⎡
⎤⎡ ⎤
⎢1⎥
⎢1⎥
⎢ 3 −1 −2 ⎥ ⎢ 2 0 −1 ⎥ ⎢1⎥
⎥⎢
⎥⎢ ⎥ 1 ⎢ ⎥ 1 ⎢ ⎥
⎢
⎢ ⎥
⎢ ⎥
⎥⎢
⎥⎢ ⎥
= −⎢
⎢ 3 −1 −2 ⎥ ⎢ 2 0 −1 ⎥ ⎢0⎥ + 2 ⎢1⎥ = 2 ⎢1⎥ .
⎣ ⎦
⎣ ⎦
⎣
⎦⎣
⎦⎣ ⎦
1
1
1 −1
0
1 0
0
1
Now compare this to the actual perturbed eigenvalues of our example (2.11) near 𝜆0 = 0
and their corresponding eigenvectors
1
1 1
1
𝜆ℎ (𝜀) = 𝜀 + (−1)ℎ 𝜀 2 (𝜀 + 4) 2
2
2
∞
(
(
)
)2 ∑
)𝑘
1(
ℎ 1
ℎ 1
ℎ 1
2
2
2
= (−1) 𝜀 +
(−1) 𝜀
𝛼𝑘 (−1) 𝜀
+
2
𝑘=3
⎡ ⎤ ⎡ ⎤
⎢1⎥ ⎢1⎥
⎢ ⎥ ⎢ ⎥
⎥ ⎢ ⎥
𝑥ℎ (𝜀) = ⎢
⎢0⎥ + ⎢1⎥ 𝜆ℎ (𝜀)
⎣ ⎦ ⎣ ⎦
1
1
⎡ ⎤
⎡ ⎤ ⎡ ⎤
1⎥
⎢1⎥ ⎢1⎥ (
∞
(
) 1⎢
)2 ∑
)
⎢
⎢ ⎥ ⎢ ⎥
⎥(
1
1 𝑘
ℎ
ℎ 1
⎢
⎢
⎥
⎢
⎥
⎥
𝛽 𝑘 (−1)ℎ 𝜀 2
= ⎢0⎥ + ⎢1⎥ (−1) 𝜀 2 + ⎢1⎥ (−1) 𝜀 2 +
2⎣ ⎦
⎣ ⎦ ⎣ ⎦
𝑘=3
1
1
1
1
for ℎ = 0, 1 and any fixed branch of 𝜀 2 . We see that indeed our formulas for the Puiseux
series coefficients are correct up to the second order.
19
Comparison to Known Results
There is a fairly large amount of literature on eigenpair perturbation expansions for analytic perturbations of non-selfadjoint matrices with degenerate eigenvalues (e.g. [1, 2, 4,
9, 23, 26, 28, 33, 34, 37, 41, 43, 46–48, 53, 55–58]). However, most of the literature (e.g.
[1, 2, 23, 33, 34, 37, 41, 43, 46–48, 55, 56]) contains results only on the first order expansions
of the Puiseux series or considers higher order terms only in the case of simple or semisimple
eigenvalues. For those works that do address higher order terms for defective eigenvalues
(e.g. [4, 9, 26, 28, 53, 57, 58]), it was found that there did not exist explicit recursive formulas
for all the Puiseux coefficients when the matrix perturbations were non-linear. One of the
purposes and achievements of this paper are the explicit recursive formulas (2.23)–(2.25)
in Theorem 2 which give all the higher order terms in the important case of degenerate
eigenvalues which are non-derogatory, that is, the case in which a degenerate eigenvalue of
the unperturbed matrix has a single Jordan block for its corresponding Jordan structure.
Our theorem generalizes and extends the results of [4, pp. 315–317, (4.96) & (4.97)], [57, pp.
415–418], and [58, pp. 17–20] to non-linear analytic matrix perturbations and makes explicit
the recursive formulas for calculating the perturbed eigenpair Puiseux expansions. Furthermore, in Proposition 8 we give an explicit recursive formula for calculating the polynomials
{𝑟𝑙 }𝑙∈ℕ . These polynomials must be calculated in order to determine the higher order terms
in the eigenpair Puiseux series expansions (see (2.25) in Theorem 2 and Remark 1). These
polynomials appear in [4, p. 315, (4.95)], [57, p. 414, (32.24)], and [58, p. 19, (34)] under different notation (compare with Proposition 2.41.ii) but no method is given to calculate them.
As such, Proposition 8 is an important contribution in the explicit recursive calculation of
the higher order terms in the eigenpair Puiseux series expansions.
Another purpose of this paper is to give, in the case of degenerate non-derogatory eigenvalues, an easily accessible and quickly referenced list of first and second order terms for the
Puiseux series expansions of the perturbed eigenpairs. When the generic condition (2.1) is
20
satisfied, Corollary 4 gives this list. Now for first order terms there are quite a few papers on
formulas for determining them, see for example [48] which gives a good survey of first order
perturbation theory. But for second order terms, it was difficult to find any results in the
literature similar to and as explicit as Corollary 4 for the case of degenerate non-derogatory
eigenvalues with arbitrary algebraic multiplicity and non-linear analytic perturbations. Results comparable to ours can be found in [4, p. 316], [57, pp. 415–418], [58, pp. 17-20], and
[53, pp. 37–38, 50–54, 125–128], although it should be noted that in [57, p. 417] the formula
for the second order term of the perturbed eigenvalues contains a misprint.
Overview
Section 2 deals with the generic condition (2.1). We give conditions that are equivalent to the
generic condition in Theorem 1. In §3 we give the main results of this paper in Theorem 2,
on the determination of the Puiseux series with the explicit recursive formulas for calculating
the series coefficients. As a corollary we give the exact leading order terms, up to the second
order, for the Puiseux series coefficients. Section 4 contains the proofs of the results in §2
and §3.
Notation
Let 𝑀𝑛 (ℂ) be the set of all 𝑛 × 𝑛 matrices with complex entries and ℂ𝑛 the set of all 𝑛 × 1
column vectors with complex entries. For 𝑎 ∈ ℂ, 𝐴 ∈ 𝑀𝑛 (ℂ), and 𝑥 = [𝑎𝑖,1 ]𝑛𝑖=1 ∈ ℂ𝑛 we
denote by 𝑎∗ , 𝐴∗ , and 𝑥∗ , the complex conjugate of 𝑎, the conjugate transpose of 𝐴, and the
1 × 𝑛 row vector 𝑥∗ := [ 𝑎∗1,1 ⋅ ⋅ ⋅
𝑛
∗
𝑎∗𝑛,1 ]. For 𝑥, 𝑦 ∈ ℂ we let ⟨𝑥, 𝑦⟩ := 𝑥 𝑦 be the standard
inner product. The matrix 𝐼 ∈ 𝑀𝑛 (ℂ) is the identity matrix and its 𝑗th column is 𝑒𝑗 ∈ ℂ𝑛 .
The matrix 𝐼𝑛−𝑚 is the (𝑛 − 𝑚) × (𝑛 − 𝑚) identity matrix. Define an 𝑚 × 𝑚 Jordan block
21
with eigenvalue 𝜆 to be
⎡
⎤
𝜆 1
⎢
⎥
⎢
⎥
⎢
⎥
.
.
⎢
⎥
⎢
⎥
⎢
𝐽𝑚 (𝜆) := ⎢
. . ⎥
⎥.
⎢
⎥
⎢
⎥
. 1⎥
⎢
⎣
⎦
𝜆
When the matrix 𝐴 (𝜀) ∈ 𝑀𝑛 (ℂ) is analytic at 𝜀 = 0 we define 𝐴′ (0) :=
1 𝑑𝑘 𝐴
(0).
𝑘! 𝑑𝜀𝑘
2.2
𝑑𝐴
(0)
𝑑𝜀
and 𝐴𝑘 :=
2𝜋
Let 𝜁 := 𝑒𝑖 𝑚 .
The Generic Condition and its Consequences
The following theorem, which is proved in §4, gives conditions which are equivalent to the
generic one (2.1).
Theorem 1 Let 𝐴 (𝜀) be a matrix-valued function having a range in 𝑀𝑛 (ℂ) such that its
matrix elements are analytic functions of 𝜀 in a neighborhood of the origin. Let 𝜆0 be an
eigenvalue of the unperturbed matrix 𝐴 (0) and denote by 𝑚 its algebraic multiplicity. Then
the following statements are equivalent:
( i) The characteristic polynomial det (𝜆𝐼 − 𝐴 (𝜀)) has a simple zero with respect to 𝜀 at
𝜆 = 𝜆0 and 𝜀 = 0, i.e.,
∂
det (𝜆𝐼 − 𝐴 (𝜀)) (𝜀,𝜆)=(0,𝜆0 ) ∕= 0.
∂𝜀
( ii) The characteristic equation, det(𝜆𝐼 − 𝐴 (𝜀)) = 0, has a unique solution, 𝜀 (𝜆), in a
neighborhood of 𝜆 = 𝜆0 with 𝜀 (𝜆0 ) = 0. This solution is an analytic function with a
22
zero of order 𝑚 at 𝜆 = 𝜆0 , i.e.,
𝑑𝑚−1 𝜀 (𝜆) 𝑑𝑚 𝜀 (𝜆) 𝑑0 𝜀 (𝜆) =
⋅
⋅
⋅
=
=
0,
∕= 0.
𝑑𝜆𝑚 𝜆=𝜆0
𝑑𝜆0 𝜆=𝜆0
𝑑𝜆𝑚−1 𝜆=𝜆0
( iii) There exists a convergent Puiseux series whose branches are given by
ℎ
𝜆ℎ (𝜀) = 𝜆0 + 𝛼1 𝜁 𝜀
1
𝑚
+
∞
∑
(
ℎ
𝛼𝑘 𝜁 𝜀
1
𝑚
)𝑘
,
𝑘=2
1
2𝜋
for ℎ = 0, . . . , 𝑚−1 and any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 , such that the values of
the branches give all the solutions of the characteristic equation, for sufficiently small
𝜀 and 𝜆 sufficiently near 𝜆0 . Furthermore, the first order term is nonzero, i.e.,
𝛼1 ∕= 0.
( iv) The Jordan normal form of 𝐴 (0) corresponding to the eigenvalue 𝜆0 consists of a single
𝑚 × 𝑚 Jordan block and there exists an eigenvector 𝑢0 of 𝐴 (0) corresponding to the
eigenvalue 𝜆0 and an eigenvector 𝑣0 of 𝐴 (0)∗ corresponding to the eigenvalue 𝜆∗0 such
that
⟨𝑣0 , 𝐴′ (0)𝑢0 ⟩ =
∕ 0.
2.3
Explicit Recursive Formulas for Calculating the
Perturbed Spectrum
This section contains the main results of this paper presented below in Theorem 2. To begin
we give some preliminaries that are needed to set up the theorem. Suppose that 𝐴 (𝜀) is
a matrix-valued function having a range in 𝑀𝑛 (ℂ) with matrix elements that are analytic
23
functions of 𝜀 in a neighborhood of the origin and 𝜆0 is an eigenvalue of the unperturbed
matrix 𝐴 (0) with algebraic multiplicity 𝑚. Assume that the generic condition
∂
det (𝜆𝐼 − 𝐴 (𝜀)) (𝜀,𝜆)=(0,𝜆0 ) ∕= 0,
∂𝜀
is true.
Now, by these assumptions, we may appeal to Theorem 1.(iv ) and conclude that the Jordan
canonical form of 𝐴(0) has only one 𝑚 × 𝑚 Jordan block associated with 𝜆0 . Hence there
exists a invertible matrix 𝑈 ∈ ℂ𝑛×𝑛 such that
⎡
⎢ 𝐽𝑚 (𝜆0 )
𝑈 −1 𝐴 (0) 𝑈 = ⎣
⎤
𝑊0
⎥
⎦,
(2.12)
where 𝑊0 is a (𝑛 − 𝑚) × (𝑛 − 𝑚) matrix such that 𝜆0 is not one of its eigenvalues [35, §6.5:
The Jordan Theorem].
We define the vectors 𝑢1 , . . . , 𝑢𝑚 , 𝑣1 , . . . , 𝑣𝑚 ∈ ℂ𝑛 as the first 𝑚 columns of the matrix 𝑈
∗
and (𝑈 −1 ) , respectively, i.e.,
𝑢𝑖 := 𝑈 𝑒𝑖 , 1 ≤ 𝑖 ≤ 𝑚,
(
)∗
𝑣𝑖 := 𝑈 −1 𝑒𝑖 , 1 ≤ 𝑖 ≤ 𝑚.
(2.13)
(2.14)
And define the matrix Λ ∈ 𝑀𝑛 (ℂ) by
⎡
⎤
∗
⎢ 𝐽𝑚 (0)
Λ := 𝑈 ⎣
(𝑊0 − 𝜆0 𝐼𝑛−𝑚 )−1
⎥ −1
⎦𝑈 ,
(2.15)
where (𝑊0 − 𝜆0 𝐼𝑛−𝑚 )−1 exists since 𝜆0 is not an eigenvalue of 𝑊0 (for the important properties
of the matrix Λ see Section 2.5).
24
Next, we introduce the polynomials 𝑝𝑗,𝑖 = 𝑝𝑗,𝑖 (𝛼1 , . . . , 𝛼𝑗−𝑖+1 ) in 𝛼1 ,. . . , 𝛼𝑗−𝑖+1 , for 𝑗 ≥ 𝑖 ≥
0, as the expressions
⎫




⎬
𝑝0,0 := 1, 𝑝𝑗,0 := 0, for 𝑗 > 0,
𝑖
∑
∏
𝑝𝑗,𝑖 (𝛼1 , . . . , 𝛼𝑗−𝑖+1 ) :=
𝛼𝑠𝜚 , for 𝑗 ≥ 𝑖 > 0 



𝑠1 +⋅⋅⋅+𝑠𝑖 =𝑗 𝜚=1
⎭
(2.16)
1≤𝑠𝜚 ≤𝑗−𝑖+1
and the polynomials 𝑟𝑙 = 𝑟𝑙 (𝛼1 , . . . , 𝛼𝑙 ) in 𝛼1 ,. . . , 𝛼𝑙 , for 𝑙 ≥ 1, as the expressions
∑
𝑟1 := 0, 𝑟𝑙 (𝛼1 , . . . , 𝛼𝑙 ) :=
𝑚
∏
𝛼𝑠𝜚 , for 𝑙 > 1
(2.17)
𝑠1 +⋅⋅⋅+𝑠𝑚 =𝑚+𝑙 𝜚=1
1≤𝑠𝜚 ≤𝑙
(see Section 2.5 for more details on these polynomials including recursive formulas for their
calculation).
With these preliminaries we can now state the main results of this paper. Proofs of these
results are contained in the next section.
Theorem 2 Let 𝐴 (𝜀) be a matrix-valued function having a range in 𝑀𝑛 (ℂ) such that its
matrix elements are analytic functions of 𝜀 in a neighborhood of the origin. Let 𝜆0 be an
eigenvalue of the unperturbed matrix 𝐴 (0) and denote by 𝑚 its algebraic multiplicity. Suppose
that the generic condition
∂
det (𝜆𝐼 − 𝐴 (𝜀)) (𝜀,𝜆)=(0,𝜆0 ) ∕= 0,
∂𝜀
(2.18)
is true. Then there is exactly one convergent Puiseux series for the 𝜆0 -group and one for
their corresponding eigenvectors whose branches are given by
𝜆ℎ (𝜀) = 𝜆0 +
∞
∑
𝑘=1
25
(
)𝑘
1
𝛼𝑘 𝜁 ℎ 𝜀 𝑚
(2.19)
𝑥ℎ (𝜀) = 𝛽 0 +
∞
∑
)𝑘
(
1
𝛽 𝑘 𝜁 ℎ𝜀 𝑚
(2.20)
𝑘=1
1
2𝜋
for ℎ = 0, . . . , 𝑚 − 1 and any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 with
∂
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 )
∂𝜀
𝛼𝑚
=
⟨𝑣
,
𝐴
𝑢
⟩
=
−
) ∕= 0
( ∂𝑚
𝑚
1 1
1
∂𝜆𝑚
(Here 𝐴1 denotes
𝑑𝐴
(0)
𝑑𝜀
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
𝑚!
and the vectors 𝑢1 and 𝑣𝑚 are defined in (2.13) and (2.14)). Fur-
thermore, we can choose
𝛼1 = ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩1/𝑚 ,
(2.21)
for any fixed 𝑚th root of ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩ and the eigenvectors to satisfy the normalization conditions
⟨𝑣1 , 𝑥ℎ (𝜀)⟩ = 1, ℎ = 0, ..., 𝑚 − 1.
(2.22)
Consequently, under these conditions 𝛼1 , 𝛼2 ,. . . and 𝛽 0 , 𝛽 1 , . . . are uniquely determined
and are given by the recursive formulas
⎞1/𝑚
⎛
∂
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 ) ⎟
⎜
𝛼1 = ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩1/𝑚 = ⎝− ∂𝜀( ∂ 𝑚
) ⎠
∂𝜆𝑚
−𝑟𝑠−1 +
𝑖=0
𝛼𝑠 =
𝛽𝑠 =
min{𝑠,𝑚}−1
𝑠−1
∑
∑
⟨
𝑝𝑗,𝑖
(2.23)
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
𝑚!
𝑣𝑚−𝑖 ,
⌋
⌊ 𝑚+𝑠−1−𝑗
𝑚
∑
𝑗=𝑖
⟩
𝐴𝑘 𝛽 𝑚+𝑠−1−𝑗−𝑘𝑚
𝑘=1
𝑚𝛼𝑚−1
1
⎧



⎨
𝑠
∑
𝑝𝑠,𝑖 𝑢𝑖+1 , if 0 ≤ 𝑠 ≤ 𝑚 − 1
𝑖=0
𝑗 ⌊∑
𝑚−1
𝑠−𝑚
𝑚 ⌋
∑
∑ ∑



𝑝
𝑢
−
𝑝𝑗,𝑘 Λ𝑘+1 𝐴𝑙 𝛽 𝑠−𝑗−𝑙𝑚 , if 𝑠 ≥ 𝑚
𝑠,𝑖 𝑖+1
⎩
𝑠−𝑗
𝑖=0
(2.24)
𝑗=0 𝑘=0
(2.25)
𝑙=1
where 𝑢𝑖 and 𝑣𝑖 are the vectors defined in (2.13) and (2.14), 𝑝𝑗,𝑖 and 𝑟𝑙 are the polynomials
defined in (2.16) and (2.17), ⌊⌋ denotes the floor function, 𝐴𝑘 denotes the matrix
and Λ is the matrix defined in (2.15).
26
1 𝑑𝑘 𝐴
(0),
𝑘! 𝑑𝜀𝑘
Corollary 3 The calculation of the 𝑘th order terms, 𝛼𝑘 and 𝛽 𝑘 , requires only the matrices
𝐴0 , . . . , 𝐴⌊ 𝑚+𝑘−1 ⌋ .
𝑚
Corollary 4 The coefficients of those Puiseux series up to second order are given by
)1/𝑚
∂𝑓
(0,
𝜆
)
0
− 1 ∂𝜀
= ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩1/𝑚 ,
∂𝑚𝑓
(0, 𝜆0 )
∂𝜆𝑚
⎧ 𝑚!
(
)
∂2𝑓
∂ 𝑚+1 𝑓
𝑚+1
1
1 ∂2𝑓

(0,𝜆
)+𝛼
(0,𝜆
)
(0,𝜆
)+
−
𝛼
0
1
0
0
𝑚+1
2
1

(𝑚+1)!
∂𝜆∂𝜀
2
(
𝛼1 =
∂𝜆
𝛼2 =

⎨
1
𝑚𝛼𝑚−1
( 𝑚!
1
(
1
− 𝛼𝑚+1
1
(𝑚+1)!
∂𝜀
∂𝑚𝑓
(0,𝜆0 )
∂𝜆𝑚
)
, if 𝑚 = 1
)
∂ 𝑚+1 𝑓
∂2𝑓

(0,𝜆0 )+𝛼1 ∂𝜆∂𝜀
(0,𝜆0 )

∂𝜆𝑚+1

⎩
, if 𝑚 > 1
𝑚−1 1 ∂ 𝑚 𝑓
𝑚𝛼1
( 𝑚! ∂𝜆𝑚 (0,𝜆0 ))
⎧

⎨ ⟨𝑣1 , (𝐴2 − 𝐴1 Λ𝐴1 ) 𝑢1 ⟩, if 𝑚 = 1
=
,

𝑚 ,𝐴1 𝑢2 )
⎩ (𝑣𝑚−1 ,𝐴1 𝑢1 )+(𝑣
, if 𝑚 > 1
𝑚𝛼𝑚−2
1
𝛽 0 = 𝑢1 ,
⎧

⎨ −Λ𝐴1 𝑢1 , if 𝑚 = 1
𝛽1 =
,

⎩ 𝛼1 𝑢2 , if 𝑚 > 1
⎧
(
)

2
2

−Λ𝐴
+
(Λ𝐴
)
−
𝛼
Λ
𝐴
𝑢1 , if 𝑚 = 1

2
1
1
1

⎨
𝛽2 =
,
−Λ𝐴1 𝑢1 + 𝛼2 𝑢2 , if 𝑚 = 2




⎩
𝛼2 𝑢2 + 𝛼21 𝑢3 , if 𝑚 > 2
where 𝑓 (𝜀, 𝜆) := det (𝜆𝐼 − 𝐴 (𝜀)).
Remark 1 Suppose we want to calculate the terms 𝛼𝑘+1 , 𝛽 𝑘+1 , where 𝑘 ≥ 2, using the
explicit recursive formulas given in the theorem. We may assume we already known or have
calculated
𝑘−1
𝐴0 , . . . , 𝐴⌊ 𝑚+𝑘 ⌋ , {𝑟𝑗 }𝑗=1
, {𝛼𝑗 }𝑘𝑗=1 , {𝛽 𝑗 }𝑘𝑗=0 , {{𝑝𝑗,𝑖 }𝑘𝑗=𝑖 }𝑘𝑖=0 .
(2.26)
𝑚
We need these to calculate 𝛼𝑘+1 , 𝛽 𝑘+1 and the steps to do this are indicated by the following
27
arrow diagram:
(2.45)
(2.24)
(2.44)
(2.25)
(2.26) → 𝑟𝑘 → 𝛼𝑘+1 → {𝑝𝑘+1,𝑖 }𝑘+1
𝑖=0 → 𝛽 𝑘+1 .
(2.27)
After we have followed these steps we not only will have calculated 𝛼𝑘+1 , 𝛽 𝑘+1 but we will
now know
𝑘+1
𝑘+1 𝑘+1
𝐴0 , . . . , 𝐴⌊ 𝑚+𝑘+1 ⌋ , {𝑟𝑗 }𝑘𝑗=1 , {𝛼𝑗 }𝑘+1
𝑗=1 , {𝛽 𝑗 }𝑗=0 , {{𝑝𝑗,𝑖 }𝑗=𝑖 }𝑖=0
(2.28)
𝑚
as well. But these are the terms in (2.26) for 𝑘 + 1 and so we may repeat the steps indicated
above to calculate 𝛼𝑘+2 , 𝛽 𝑘+2 .
It is in this way we see how all the higher order terms can be calculated using the results of
this paper.
Example
In order to illustrate these steps we give the following example which recursively calculates
the third order terms for 𝑚 ≥ 3.
The goal is to determine 𝛼3 , 𝛽 3 . To do this we follow the steps indicated in the above remark
with 𝑘 = 2. The first step is to collect the terms in (2.26). Assuming 𝐴0 , 𝐴1 are known then
by (2.16), (2.17), Corollary 4, and Proposition 7 we have
𝐴0 , 𝐴1 , 𝑟1 = 0, 𝛼1 = ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩1/𝑚 , 𝛼2 =
⟨𝑣𝑚−1 ,𝐴1 𝑢1 ⟩+⟨𝑣𝑚 ,𝐴1 𝑢2 ⟩
,
𝑚𝛼𝑚−2
1
𝛽 0 = 𝑢1 , 𝛽 1 = 𝛼1 𝑢2 , 𝛽 2 = 𝛼2 𝑢2 + 𝛼21 𝑢3 ,
𝑝0,0 = 1, 𝑝1,0 = 0, 𝑝1,1 = 𝛼1 , 𝑝2,0 = 0, 𝑝2,1 = 𝛼2 , 𝑝2,2 = 𝛼21 .
The next step is to determine 𝑟2 using the recursive formula for the 𝑟𝑙 ’s given in (2.45). We
28
find that
1
1
∑
1 ∑
𝑚
[(3 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼3−𝑗 𝛼𝑗+1
[(3 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼3−𝑗 𝑟𝑗 + 𝛼𝑚−2
1
2𝛼1 𝑗=1
2
𝑗=1
𝑚(𝑚 − 1) 𝑚−2 2
=
𝛼1 𝛼2 .
2
𝑟2 =
Now, since 𝑟2 is determined, we can use the recursive formula in (2.24) for the 𝛼𝑠 ’s to
calculate 𝛼3 . In doing so we find that
−𝑟2 +
𝛼3 =
min{3,𝑚}−1
2
∑
∑
𝑖=0
⟨
𝑝𝑗,𝑖
𝑣𝑚−𝑖 ,
𝑗=𝑖
⌊ 𝑚+2−𝑗
⌋
𝑚
∑
⟩
𝐴𝑘 𝛽 𝑚+2−𝑗−𝑘𝑚
𝑘=1
𝑚𝛼𝑚−1
1
−𝑟2 + 𝑝2,1 ⟨𝑣𝑚−1 , 𝐴1 𝛽 0 ⟩ + 𝑝0,0 ⟨𝑣𝑚 , 𝐴1 𝛽 2 ⟩
=
+
𝑚𝛼𝑚−1
1
𝑝2,2 ⟨𝑣𝑚−2 , 𝐴1 𝛽 0 ⟩ + 𝑝1,1 ⟨𝑣𝑚−1 , 𝐴1 𝛽 1 ⟩
𝑚𝛼𝑚−1
1
− 𝑚(𝑚−1)
𝛼𝑚−2
𝛼22 + 𝛼2 ⟨𝑣𝑚−1 , 𝐴1 𝑢1 ⟩ + ⟨𝑣𝑚 , 𝐴1 (𝛼2 𝑢2 + 𝛼21 𝑢3 )⟩
1
2
=
+
𝑚𝛼𝑚−1
1
𝛼21 ⟨𝑣𝑚−2 , 𝐴1 𝑢1 ⟩ + 𝛼1 ⟨𝑣𝑚−1 , 𝐴1 𝛼1 𝑢2 ⟩
𝑚𝛼𝑚−1
1
(
)
3−𝑚
⟨𝑣𝑚−2 , 𝐴1 𝑢1 ⟩ + ⟨𝑣𝑚−1 , 𝐴1 𝑢2 ⟩ + ⟨𝑣𝑚 , 𝐴1 𝑢3 ⟩
2
=
𝛼−1
.
1 𝛼2 +
2
𝑚𝛼𝑚−3
1
Next, since 𝛼3 is determined, we can use (2.44) to calculate {𝑝3,𝑖 }3𝑖=0 . In this case though it
suffices to use Proposition 7 and in doing so we find that
𝑝3,0 = 0, 𝑝3,1 = 𝛼3 , 𝑝3,2 = 2𝛼1 𝛼2 , 𝑝3,3 = 𝛼31 .
Finally, we can compute 𝛽 3 using the recursive formula in (2.25) for the 𝛽 𝑠 ’s. In doing so we
29
find that
𝛽3 =
⎧



⎨
3
∑
𝑝3,𝑖 𝑢𝑖+1 , if 𝑚 > 3
𝑖=0
3−𝑗
𝑗 ⌊∑
𝑚−1
3−𝑚
𝑚 ⌋

∑
∑ ∑


𝑝3,𝑖 𝑢𝑖+1 −
𝑝𝑗,𝑘 Λ𝑘+1 𝐴𝑙 𝛽 3−𝑗−𝑙𝑚 , if 𝑚 = 3
⎩
𝑖=0
𝑗=0 𝑘=0 𝑙=1
⎧

⎨ 𝑝3,1 𝑢2 + 𝑝3,2 𝑢3 + 𝑝3,3 𝑢4 , if 𝑚 > 3
=
2
∑

⎩
𝑝3,𝑖 𝑢𝑖+1 − Λ𝐴1 𝛽 0 , if 𝑚 = 3
𝑖=0
⎧

⎨ 𝛼3 𝑢2 + 2𝛼1 𝛼2 𝑢3 + 𝛼3 𝑢4 , if 𝑚 > 3
1
=

⎩ 𝛼3 𝑢2 + 2𝛼1 𝛼2 𝑢3 − Λ𝐴1 𝑢1 , if 𝑚 = 3.
This completes the calculation of the third order terms, 𝛼3 , 𝛽 3 , when 𝑚 ≥ 3.
2.4
Proofs
This section contains the proofs of the results of this paper. We begin by proving Theorem
1 of §2 on conditions equivalent to the generic condition. We next follow this up with the
proof of the main result of this paper Theorem 2. We finish by proving the Corollaries 3 and
4.
Proof of Theorem 1
To prove this theorem we will prove the chain of statements (i)⇒(ii)⇒(iii)⇒(iv)⇒(i).
We begin by proving (i)⇒(ii). Define 𝑓 (𝜀, 𝜆) := det (𝜆𝐼 − 𝐴 (𝜀)) and suppose (i) is true.
Then 𝑓 is an analytic function of (𝜀, 𝜆) near (0, 𝜆0 ) since the matrix elements of 𝐴 (𝜀) are
analytic functions of 𝜀 in a neighborhood of the origin and the determinant of a matrix
is a polynomial in its matrix elements. Also we have 𝑓 (0, 𝜆0 ) = 0 and
30
∂𝑓
∂𝜀
(0, 𝜆0 ) ∕= 0.
Hence by the holomorphic implicit function theorem [30, §1.4 Theorem 1.4.11] there exists
a unique solution, 𝜀 (𝜆), in a neighborhood of 𝜆 = 𝜆0 with 𝜀 (𝜆0 ) = 0 to the equation
𝑓 (𝜀, 𝜆) = 0, which is analytic at 𝜆 = 𝜆0 . We now show that 𝜀 (𝜆) has a zero there of order
𝑚 at 𝜆 = 𝜆0 . First, the properties of 𝜀(𝜆) imply there exists 𝜀𝑞 ∕= 0 and 𝑞 ∈ ℕ such that
(
)
𝜀 (𝜆) = 𝜀𝑞 (𝜆 − 𝜆0 )𝑞 + 𝑂 (𝜆 − 𝜆0 )𝑞+1 , for ∣𝜆 − 𝜆0 ∣ << 1. Next, by hypothesis 𝜆0 is an
eigenvalue of 𝐴 (0) of algebraic multiplicity 𝑚 hence ∂ 𝑖 𝑓 ∖∂𝜆𝑖 (0, 𝜆0 ) = 0 for 0 ≤ 𝑖 ≤ 𝑚 − 1
but ∂ 𝑚 𝑓 ∖∂𝜆𝑚 (0, 𝜆0 ) ∕= 0. Combining this with the fact that 𝑓 (0, 𝜆0 ) = 0 and
∂𝑓
∂𝜀
(0, 𝜆0 ) ∕= 0
we have
∑
𝑓 (𝜀, 𝜆) = 𝑎10 𝜀 + 𝑎0𝑚 (𝜆 − 𝜆0 )𝑚 +
𝑎𝑖𝑗 𝜀𝑖 (𝜆 − 𝜆0 )𝑗
(2.29)
𝑖+𝑗≥2, 𝑖,𝑗∈ℕ
(𝑖,𝑗)∕∈{(0,𝑗):𝑗≤𝑚}
for ∣𝜀∣ + ∣𝜆 − 𝜆0 ∣ << 1, where 𝑎10 =
∂𝑓
∂𝜀
(0, 𝜆0 ) ∕= 0 and 𝑎0𝑚 =
1 ∂𝑚𝑓
𝑚! ∂𝜆𝑚
(0, 𝜆0 ) ∕= 0. Then
using the expansions of 𝑓 (𝜀, 𝜆) and 𝜀 (𝜆) together with the identity 𝑓 (𝜀 (𝜆) , 𝜆) = 0 for
∣𝜆 − 𝜆0 ∣ << 1, we find that 𝑞 = 𝑚 and
𝜀𝑚 = −
1 ∂ 𝑚 det(𝜆𝐼−𝐴(𝜀)) 𝑚!
∂𝜆𝑚
𝑎0𝑚
(𝜆,𝜀)=(𝜆0 ,0)
=− ∂
.
𝑎10
det
(𝜆𝐼
−
𝐴
(𝜀))
∂𝜀
(𝜆,𝜀)=(𝜆0 ,0)
(2.30)
Therefore we conclude that 𝜀 (𝜆) has a zero of order 𝑚 at 𝜆 = 𝜆0 , which proves (ii).
Next, we prove (ii)⇒(iii). Suppose (ii) is true. The first part of proving (iii) involves inverting
𝜀 (𝜆) near 𝜀 = 0 and 𝜆 = 𝜆0 . To do this we expand 𝜀 (𝜆) in a power series about 𝜆 = 𝜆0 and
find that 𝜀 (𝜆) = 𝑔(𝜆)𝑚 where
(
𝑔(𝜆) = (𝜆 − 𝜆0 ) 𝜀𝑚 +
∞
∑
)1/𝑚
𝜀𝑘 (𝜆 − 𝜆0 )𝑘−𝑚
𝑘=𝑚+1
and we are taking any fixed branch of the 𝑚th root that is analytic at 𝜀𝑚 . Notice that,
for 𝜆 in a small enough neighborhood of 𝜆0 , 𝑔 is an analytic function, 𝑔 (𝜆0 ) = 0, and
𝑑𝑔
(𝜆0 )
𝑑𝜆
1/𝑚
= 𝜀𝑚
∕= 0. This implies, by the inverse function theorem for analytic functions,
that for 𝜆 in a small enough neighborhood of 𝜆0 the analytic function 𝑔 (𝜆) has an analytic
31
inverse 𝑔 −1 (𝜀) in a neighborhood of 𝜀 = 0 with 𝑔 −1 (0) = 𝜆0 . Define a multivalued function
( 1)
1
𝜆 (𝜀), for sufficiently small 𝜀, by 𝜆 (𝜀) := 𝑔 −1 𝜀 𝑚 where by 𝜀 𝑚 we mean all branches of
the 𝑚th root of 𝜀. We know that 𝑔 −1 is analytic at 𝜀 = 0 so that for sufficiently small 𝜀 the
[ 𝑑𝑔
]−1
−1
(𝜆0 )
∕= 0 we have
multivalued function 𝜆 (𝜀) is a Puiseux series. And since 𝑑𝑔𝑑𝜀 (0) = 𝑑𝜆
an expansion
∞
( 1)
( 1 )𝑘
∑
1
𝜆 (𝜀) = 𝑔 −1 𝜀 𝑚 = 𝜆0 + 𝛼1 𝜀 𝑚 +
𝛼𝑘 𝜀 𝑚 .
𝑘=2
Now suppose for fixed 𝜆 sufficiently near 𝜆0 and for sufficiently small 𝜀 that det (𝜆𝐼 − 𝐴 (𝜀)) =
0. We want to show this implies 𝜆 = 𝜆 (𝜀) for one of the branches of the 𝑚th root. We know
by hypothesis we must have 𝜀 = 𝜀 (𝜆). But as we know this implies that 𝜀 = 𝜀 (𝜆) = 𝑔(𝜆)𝑚
hence for some branch of the 𝑚th root, 𝑏𝑚 (⋅ ), we have 𝑏𝑚 (𝜀) = 𝑏𝑚 (𝑔(𝜆)𝑚 ) = 𝑔(𝜆). But
𝜆 is near enough to 𝜆0 and 𝜀 is sufficiently small that we may apply the 𝑔 −1 to both sides
yielding 𝜆 = 𝑔 −1 (𝑔(𝜆)) = 𝑔 −1 (𝑏𝑚 (𝜀)) = 𝜆 (𝜀), as desired. Furthermore, all the 𝑚 branches
𝜆ℎ (𝜀), ℎ = 0, . . . , 𝑚 − 1 of 𝜆 (𝜀) are given by taking all branches of the 𝑚th root of 𝜀 so that
ℎ
𝜆ℎ (𝜀) = 𝜆0 + 𝛼1 𝜁 𝜀
1
𝑚
+
∞
∑
(
)𝑘
1
𝛼𝑘 𝜁 ℎ 𝜀 𝑚
𝑘=2
1
2𝜋
for any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 and
[
]−1
𝑑𝑔
𝑑𝑔 −1
−1/𝑚
(0) =
(𝜆0 )
= 𝜀𝑚
∕= 0,
𝛼1 =
𝑑𝜀
𝑑𝜆
(2.31)
which proves (iii).
Next, we prove (iii)⇒(iv). Suppose (iii) is true. Define the function 𝑦 (𝜀) := 𝜆0 (𝜀𝑚 ).
Then 𝑦 is analytic at 𝜀 = 0 and
𝑑𝑦
𝑑𝜀
(0) = 𝜆1 ∕= 0. Also we have for 𝜀 sufficiently small
det (𝑦 (𝜀) 𝐼 − 𝐴 (𝜀𝑚 )) = 0. Consider the inverse of 𝑦 (𝜀), 𝑦 −1 (𝜆). It has the property
(
(
(
(
𝑚 ))
𝑚 ))
that 0 = det 𝑦 (𝑦 −1 (𝜆)) 𝐼 − 𝐴 [𝑦 −1 (𝜆)]
= det 𝜆𝐼 − 𝐴 [𝑦 −1 (𝜆)]
with 𝑦 −1 (𝜆0 ) = 0,
𝑑𝑦 −1
𝑑𝜆
𝑚
−1
(𝜆0 ) = 𝛼−1
(𝜆)] . Then 𝑔 has a zero of order 𝑚 at 𝜆0 and
1 . Define 𝑔 (𝜆) := [𝑦
det (𝜆𝐼 − 𝐴 (𝑔 (𝜆))) = 0 for 𝜆 in a neighborhood of 𝜆0 .
32
Now we consider the analytic matrix 𝐴 (𝑔 (𝜆)) − 𝜆𝐼 in a neighborhood of 𝜆 = 𝜆0 with the
constant eigenvalue 0. Because 0 is an analytic eigenvalue of it then there exists an analytic
eigenvector, 𝑥 (𝜆), of 𝐴 (𝑔 (𝜆)) − 𝜆𝐼 corresponding to the eigenvalue 0 in a neighborhood of
𝜆0 such that 𝑥 (𝜆0 ) ∕= 0. Hence for 𝜆 near 𝜆0 we have
0 = (𝐴 (𝑔 (𝜆)) − 𝜆𝐼) 𝑥 (𝜆)
(
))
)
( (
(𝜆 − 𝜆0 )𝑚 + 𝑂 (𝜆 − 𝜆0 )𝑚+1 − (𝜆 − 𝜆0 ) 𝐼 − 𝜆0 𝐼 𝑥 (𝜆)
= 𝐴 𝛼−𝑚
1
)
(
𝑑𝑥
(𝜆0 ) − 𝑥 (𝜆0 ) (𝜆 − 𝜆0 ) + ⋅ ⋅ ⋅
= (𝐴 (0) − 𝜆0 𝐼) 𝑥 (𝜆0 ) + (𝐴 (0) − 𝜆0 𝐼)
𝑑𝜆
(
)
𝑑𝑚−1 𝑥
𝑑𝑚−2 𝑥
+ (𝐴 (0) − 𝜆0 𝐼) 𝑚−1 (𝜆0 ) − 𝑚−2 (𝜆0 ) (𝜆 − 𝜆0 )𝑚−1
𝑑𝜆
𝑑𝜆
(
)
𝑚
𝑚−1
𝑑 𝑥
𝑑
𝑥
−𝑚 ′
+ (𝐴 (0) − 𝜆0 𝐼) 𝑚 (𝜆0 ) − 𝑚−1 (𝜆0 ) + 𝛼1 𝐴 (0)𝑥 (𝜆0 ) (𝜆 − 𝜆0 )𝑚
𝑑𝜆
𝑑𝜆
(
)
+ 𝑂 (𝜆 − 𝜆0 )𝑚+1 .
This implies that
𝑑𝑗−1 𝑥
𝑑𝑗 𝑥
(𝐴 (0) − 𝜆0 𝐼) 𝑥 (𝜆0 ) = 0, (𝐴 (0) − 𝜆0 𝐼) 𝑗 (𝜆0 ) = 𝑗−1 (𝜆0 ) , for 𝑗 = 1, . . . , 𝑚 − 1,
𝑑𝜆
𝑑𝜆
𝑑𝑚 𝑥
𝑑𝑚−1 𝑥
′
(𝐴 (0) − 𝜆0 𝐼) 𝑚 (𝜆0 ) = 𝑚−1 (𝜆0 ) − 𝛼−𝑚
(2.32)
1 𝐴 (0)𝑥 (𝜆0 ) .
𝑑𝜆
𝑑𝜆
The first 𝑚 equations imply that 𝑥 (𝜆0 ),
𝑚 generated by
𝑑𝑚−1 𝑥
𝑑𝜆𝑚−1
𝑑𝑥
𝑑𝜆
(𝜆0 ), . . .,
𝑑𝑚−1 𝑥
𝑑𝜆𝑚−1
(𝜆0 ) is a Jordan chain of length
(𝜆0 ). Since the algebraic multiplicity of 𝜆0 for 𝐴 (0) is 𝑚 this implies
that the there is a single 𝑚 × 𝑚 Jordan block corresponding to the eigenvalue 𝜆0 where we
𝑚−1
𝑑𝑥
𝑑
𝑥
can take 𝑥 (𝜆0 ) , 𝑑𝜆
(𝜆0 ) , ..., 𝑑𝜆
𝑚−1 (𝜆0 ) as a Jordan basis. It follows from basic properties of
Jordan chains that there exists an eigenvector 𝑣 of 𝐴(0)∗ corresponding to the eigenvalue 𝜆∗0
𝑚−1
𝑑
𝑥
such that ⟨𝑣, 𝑑𝜆
𝑚−1 (𝜆0 )⟩ = 1. Hence
⟨
0=
⟩
𝑑𝑚 𝑥
(2.32)
′
(𝐴(0) − 𝜆0 𝐼) 𝑣, 𝑚 (𝜆0 )
= 1 − 𝛼−𝑚
1 ⟨𝑣, 𝐴 (0)𝑥 (𝜆0 )⟩
𝑑𝜆
∗
33
implying that ⟨𝑣, 𝑑𝐴
(0) 𝑥 (𝜆0 )⟩ = 𝛼𝑚
1 ∕= 0. Therefore we have shown that the Jordan normal
𝑑𝜀
form of 𝐴 (0) corresponding to the eigenvalue 𝜆0 consists of a single 𝑚 × 𝑚 Jordan block and
there exists an eigenvector 𝑢 of 𝐴 (0) corresponding to the eigenvalue 𝜆0 and an eigenvector
∕ 0. This proves (iv).
𝑣 of 𝐴 (0)∗ corresponding to the eigenvalue 𝜆∗0 such that ⟨𝑣, 𝐴′ (0)𝑢⟩ =
Finally, we show (iv)⇒(i). Suppose (iv) is true. We begin by noting that since
det (𝜆0 𝐼 − 𝐴 (𝜀)) = (−1)𝑛 det ((𝐴 (0) − 𝜆0 𝐼) + 𝐴′ (0)𝜀) + 𝑜 (𝜀)
it suffices to show that
𝑆𝑛−1 :=
𝑑
det ((𝐴 (0) − 𝜆0 𝐼) + 𝐴′ (0)𝜀) 𝜀=0 ∕= 0.
𝑑𝜀
(2.33)
We will use the result from [24, Theorem 2.16] to prove (2.33). Let 𝐴 (0) − 𝜆0 𝐼 = 𝑌 Σ𝑋 ∗ be
a singular-value decomposition of the matrix 𝐴 (0) − 𝜆0 𝐼 where 𝑋, 𝑌 are unitary matrices
and Σ = diag(𝜎 1 , . . . , 𝜎 𝑛−1 , 𝜎 𝑛 ) with 𝜎 1 ≥ . . . ≥ 𝜎 𝑛−1 ≥ 𝜎 𝑛 ≥ 0 (see [35, §5.7, Theorem
2]). Now since the Jordan normal form of 𝐴 (0) corresponding to the eigenvalue 𝜆0 consists
of a single Jordan block this implies that rank of 𝐴 (0) − 𝜆0 𝐼 is 𝑛 − 1. This implies that
𝜎 1 ≥ . . . ≥ 𝜎 𝑛−1 > 𝜎 𝑛 = 0, 𝑢 = 𝑋𝑒𝑛 is an eigenvalue of 𝐴 (0) corresponding to the eigenvalue
𝜆0 , 𝑣 = 𝑌 𝑒𝑛 is an eigenvalue of 𝐴 (0) corresponding to the eigenvalue 𝜆∗0 , and there exist
nonzero constants 𝑐1 , 𝑐2 such that 𝑢 = 𝑐1 𝑢0 and 𝑣 = 𝑐2 𝑣0 .
Now using the result of [24, Theorem 2.16] for (2.33) we find that
∗
𝑆𝑛−1 = det (𝑌 𝑋 )
∑
(
)
∗ ′
𝜎 𝑖1 ⋅ ⋅ ⋅ 𝜎 𝑖𝑛−1 det (𝑌 𝐴 (0)𝑋)𝑖1 ...𝑖𝑛−1 ,
1≤𝑖1 <⋅⋅⋅<𝑖𝑛−1 ≤𝑛
where (𝑌 ∗ 𝐴′ (0)𝑋)𝑖1 ...𝑖𝑛−1 is the matrix obtained from 𝑌 ∗ 𝐴′ (0)𝑋 by removing rows and
34
columns 𝑖1 . . . 𝑖𝑛−1 . But since 𝜎 𝑛 = 0 and
(𝑌 ∗ 𝐴′ (0)𝑋)1...(𝑛−1) = 𝑒∗𝑛 𝑌 ∗ 𝐴′ (0)𝑋𝑒𝑛 = (𝑣, 𝐴′ (0)𝑢) = 𝑐∗2 𝑐1 (𝑣0 , 𝐴′ (0)𝑢0 ) ∕= 0
then 𝑆𝑛−1 = det (𝑌 𝑋 ∗ )
𝑛−1
∏
𝜎 𝑗 𝑐∗2 𝑐1 (𝑣0 , 𝐴′ (0)𝑢0 ) ∕= 0. This completes the proof.
𝑗=1
Proof of Theorem 2
We begin by noting that our hypotheses imply that statements (ii), (iii), and (iv) of Theorem
1 are true. In particular, statement (iii) implies that there is exactly one convergent Puiseux
series for the 𝜆0 -group whose branches are given by
ℎ
𝜆ℎ (𝜀) = 𝜆0 + 𝛼1 𝜁 𝜀
1
𝑚
+
∞
∑
(
)𝑘
1
𝛼𝑘 𝜁 ℎ 𝜀 𝑚 ,
𝑘=2
1
2𝜋
for ℎ = 0, . . . , 𝑚 − 1 and any fixed branch of 𝜀 𝑚 , where 𝜁 = 𝑒 𝑚 𝑖 and 𝛼1 ∕= 0. Then by well
known results [4, §6.1.7, Theorem 2], [28, §II.1.8] there exists a convergent Puiseux series for
the corresponding eigenvectors whose branches are given by
𝑥ℎ (𝜀) = 𝛽 0 +
∞
∑
(
)𝑘
1
𝛽 𝑘 𝜁 ℎ𝜀 𝑚 ,
𝑘=1
for ℎ = 0, . . . , 𝑚−1, where 𝛽 0 is an eigenvector of 𝐴0 = 𝐴 (0) corresponding to the eigenvalue
𝜆0 . Now if we examine the proof of (ii)⇒(iii) in Theorem 1 we see by equation (2.31) that
−1
𝛼𝑚
1 = 𝜀𝑚 , where 𝜀𝑚 is given in equation (2.30) in the proof of (i)⇒(iii) for Theorem 1. Thus
we can conclude that
∂
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 )
∂𝜀
𝛼𝑚
=
−
( ∂𝑚
) ∕= 0.
1
∂𝜆𝑚
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
𝑚!
35
(2.34)
Choose any 𝑚th root of ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩ and denote it by ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩1/𝑚 . By (2.34) we can just
reindexing the Puiseux series (2.19) and (2.20) and assume that
⎞1/𝑚
⎛
∂
det (𝜆𝐼 − 𝐴 (𝜀)) ∣(𝜀,𝜆)=(0,𝜆0 ) ⎟
⎜
𝛼1 = ⎝− ∂𝜀( ∂ 𝑚
) ⎠
∂𝜆𝑚
.
det(𝜆𝐼−𝐴(𝜀))∣(𝜀,𝜆)=(0,𝜆0 )
𝑚!
Next, we wish to prove that we can choose the perturbed eigenvectors (2.20) to satisfy the
normalization conditions (2.22). But this follows by Theorem 1 (iv) and the fact 𝛽 0 is an
eigenvector of 𝐴(0) corresponding to the eigenvalue 𝜆0 since then ⟨𝑣1 , 𝛽 0 ⟩ =
∕ 0 and so we may
take
𝑥ℎ (𝜀)
,
(𝑣1 ,𝑥ℎ (𝜀))
for ℎ = 0, . . . , 𝑚 − 1, to be the perturbed eigenvectors in (2.20) that satisfy
the normalization conditions (2.22).
∞
Now we are ready to begin showing that {𝛼𝑠 }∞
𝑠=1 , {𝛽 𝑠 }𝑠=0 are given by the recursive formulas
(2.23)-(2.25). The first key step is proving the following:
(𝐴0 − 𝜆0 𝐼) 𝛽 𝑠 = −
𝑠 (
∑
𝐴 𝑘 − 𝛼𝑘 𝐼 𝛽 𝑠−𝑘 , for 𝑠 ≥ 1,
(2.35)
𝛽 0 = 𝑢1 , 𝛽 𝑠 = Λ (𝐴0 − 𝜆0 𝐼) 𝛽 𝑠 , for 𝑠 ≥ 1,
(2.36)
𝑘=1
where we define 𝐴 𝑘 := 0, if
𝑚
𝑘
𝑚
)
𝑚
∕∈ ℕ.
The first equality holds since in a neighborhood of the origin
0 = (𝐴 (𝜀) − 𝜆0 (𝜀) 𝐼) 𝑥0 (𝜀) =
∞
∑
𝑠=0
(
𝑠 (
∑
𝑘=0
)
)
𝑠
𝐴 𝑘 − 𝛼𝑘 𝐼 𝛽 𝑠−𝑘 𝜀 𝑚 .
𝑚
The second equality will be proven once we show 𝛽 0 = 𝑢1 and 𝛽 𝑠 ∈ 𝑆 := span{𝑈 𝑒𝑖 ∣2 ≤
𝑖 ≤ 𝑛}, for 𝑠 ≥ 1, where 𝑈 is the matrix from (2.12). This will prove (2.36) because
Λ (𝐴0 − 𝜆0 𝐼) acts as the identity on 𝑆 by Proposition 6.i. But these follow from the facts that
36
𝑆 = {𝑥 ∈ ℂ𝑛 ∣⟨𝑣1 , 𝑥⟩ = 0} and the normalization conditions (2.22) imply that ⟨𝑣1 , 𝛽 0 ⟩ = 1
and ⟨𝑣1 , 𝛽 𝑠 ⟩ = 0, for 𝑠 ≥ 1.
The next key step in this proof is the following lemma:
Lemma 5 For all 𝑠 ≥ 0 the following identity holds
(𝐴0 − 𝜆0 𝐼) 𝛽 𝑠 =
⎧



⎨
𝑠
∑
𝑝𝑠,𝑖 𝑢𝑖 , for 0 ≤ 𝑠 ≤ 𝑚 − 1
𝑖=0
𝑗 ⌊∑
𝑚
𝑠−𝑚
𝑚 ⌋
∑
∑ ∑



𝑝𝑠,𝑖 𝑢𝑖 −
𝑝𝑗,𝑘 Λ𝑘 𝐴𝑙 𝛽 𝑠−𝑗−𝑙𝑚 , for 𝑠 ≥ 𝑚
⎩
𝑠−𝑗
𝑖=0
𝑗=0 𝑘=0
(2.37)
𝑙=1
where we define 𝑢0 := 0.
Proof.
The proof is by induction on 𝑠. The statement is true for 𝑠 = 0 since 𝑝0,0 𝑢0 =
0 = (𝐴0 − 𝜆0 𝐼) 𝛽 0 . Now suppose it was true for all 𝑟 with 0 ≤ 𝑟 ≤ 𝑠 for some nonnegative
integer 𝑠. We will show the statement is true for 𝑠 + 1 as well.
∑
Suppose 𝑠 + 1 ≤ 𝑚 − 1 then (𝐴0 − 𝜆0 𝐼) 𝛽 𝑟 = 𝑟𝑖=0 𝑝𝑟,𝑖 𝑢𝑖 for 0 ≤ 𝑟 ≤ 𝑠 and we must show
∑
that (𝐴0 − 𝜆0 𝐼) 𝛽 𝑠+1 = 𝑠+1
𝑖=0 𝑝𝑠+1,𝑖 𝑢𝑖 . Well, for 1 ≤ 𝑟 ≤ 𝑠,
(2.36)
𝛽 𝑟 = Λ (𝐴0 − 𝜆0 𝐼) 𝛽 𝑟 =
𝑟
∑
(2.40)
𝑝𝑟,𝑖 Λ𝑢𝑖 =
𝑖=0
𝑟
∑
𝑝𝑟,𝑖 𝑢𝑖+1 .
(2.38)
𝑖=1
Hence the statement is true if 𝑠 + 1 ≤ 𝑚 − 1 since
)
𝑠+1
∑(
∑ 𝑠+1−𝑘
∑
(2.35)
(2.38) 𝑠+1
(𝐴0 − 𝜆0 𝐼) 𝛽 𝑠+1 = −
𝐴 𝑘 − 𝛼𝑘 𝐼 𝛽 𝑠+1−𝑘 =
𝛼𝑘 𝑝𝑠+1−𝑘,𝑖 𝑢𝑖+1
𝑚
𝑘=1
𝑘=1 𝑖=0
(𝑠+1−𝑖
)
𝑠
∑
∑
(2.46) ∑
(2.43) 𝑠+1
=
𝛼𝑘 𝑝𝑠+1−𝑘,𝑖 𝑢𝑖+1 =
𝑝𝑠+1,𝑖 𝑢𝑖 .
𝑖=0
𝑘=1
𝑖=0
Now suppose that 𝑠 + 1 ≥ 𝑚. The proof is similar to what we just proved. By the induction
37
(2.36)
hypothesis (2.37) is true for 1 ≤ 𝑟 ≤ 𝑠 and 𝛽 𝑟 = Λ (𝐴0 − 𝜆0 𝐼) 𝛽 𝑟 thus
(2.40)
𝛽𝑟 =
⎧



⎨
𝑟
∑
𝑝𝑟,𝑖 𝑢𝑖+1 , for 0 ≤ 𝑟 ≤ 𝑚 − 1
𝑖=0
(2.39)
𝑗 ⌊∑
𝑚−1
𝑟−𝑚
𝑚 ⌋
∑
∑ ∑



𝑝
𝑢
−
𝑝𝑗,𝑘 Λ𝑘+1 𝐴𝑙 𝛽 𝑟−𝑗−𝑙𝑚 , for 𝑟 ≥ 𝑚.
𝑟,𝑖
𝑖+1
⎩
𝑟−𝑗
𝑗=0 𝑘=0
𝑖=0
𝑙=1
Hence we have
(2.35)
(𝐴0 − 𝜆0 𝐼) 𝛽 𝑠+1 = −
𝑠+1
∑(
)
𝐴 𝑘 − 𝛼𝑘 𝐼 𝛽 𝑠+1−𝑘
𝑚
𝑘=1
(2.39)
= −
⌊ 𝑠+1
𝑚 ⌋
∑
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑠+1−𝑚
∑ 𝑚−1
∑
𝑙=1
−
𝑘=1
𝑗
𝑠+1−𝑚
∑ 𝑠+1−𝑘−𝑚
∑ ∑
𝑗=0
𝑘=1
𝑠+1
∑
𝑠+1−𝑘
∑
𝑘>𝑠+1−𝑚
𝑖=0
+
(2.46)
= −
⌊ 𝑠+1
𝑚 ⌋
∑
⌋
⌊ 𝑠+1−𝑘−𝑗
𝑚
∑
𝑖=0
𝛼𝑘 𝑝𝑠+1−𝑘,𝑖 𝑢𝑖+1
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑗=0
⌊∑⌋
⌊ 𝑠+1−𝑘−𝑗
⌋
𝑚
∑
𝑖=0
= −
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑗
𝑠+1−𝑚
∑ 𝑠+1−𝑘−𝑚
∑ ∑
𝑘=1
Now let 𝑎𝑘,𝑗,𝑖 :=
⌊ 𝑠+1−𝑘−𝑗
⌋
𝑚
∑
)
𝛼𝑘 𝑝𝑠+1−𝑘,𝑖 𝑢𝑖+1
𝑘=1
𝛼𝑘 𝑝𝑗,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑘−𝑗−𝑙𝑚
𝑚
∑
𝑝𝑠+1,𝑖 𝑢𝑖
𝑖=0
𝑙=1
−
(𝑠+1−𝑖
∑
𝑙=1
𝑠+1
𝑚
(2.43)
𝑚−1
∑
𝑖=0
𝑗
𝑠+1−𝑚
∑ 𝑠+1−𝑘−𝑚
∑ ∑
𝑘=1
𝛼𝑘 𝑝𝑗,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑘−𝑗−𝑙𝑚
𝑙=1
𝑙=1
−
𝛼𝑘 𝑝𝑠+1−𝑘,𝑖 𝑢𝑖+1
𝑖=0
𝑗=0
⌋
⌊ 𝑠+1−𝑘−𝑗
𝑚
∑
𝑖=0
𝛼𝑘 𝑝𝑗,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑘−𝑗−𝑙𝑚 .
𝑙=1
𝛼𝑘 𝑝𝑗,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑘−𝑗−𝑙𝑚 . Then using the sum identity
𝑙=1
𝑗
𝑠+1−𝑚
∑ 𝑠+1−𝑘−𝑚
∑ ∑
𝑘=1
𝑗=0
𝑖=0
∑ 𝑠−𝑚
∑ 𝑠+1−𝑗−𝑚
∑
(2.48) 𝑠−𝑚
=
𝑖=0 𝑗=𝑖
𝑗
∑ 𝑠+1−𝑗−𝑚
∑ ∑
(2.46) 𝑠−𝑚
𝑎𝑘,𝑗,𝑖 =
𝑘=1
𝑗=0
𝑘=1
𝑎𝑘,𝑗,𝑖 =
𝑖=0
∑ 𝑠+1−𝑚
∑ 𝑞−𝑖
∑
(2.49) 𝑠−𝑚
𝑎𝑘,𝑗,𝑖 =
𝑖=0 𝑞=𝑖+1 𝑘=1
38
𝑗 𝑠+1−𝑗−𝑚
𝑠−𝑚
∑ ∑
∑
𝑗=0 𝑖=0
𝑎𝑘,𝑞−𝑘,𝑖
𝑘=1
𝑎𝑘,𝑗,𝑖
we can concluded that
(𝐴0 − 𝜆0 𝐼) 𝛽 𝑠+1 = −
⌊ 𝑠+1
𝑚 ⌋
∑
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑠−𝑚
∑ 𝑠+1−𝑚
∑ 𝑞−𝑖
∑
⌊ 𝑠+1−𝑞
⌋
𝑚
∑
𝑖=0 𝑞=𝑖+1 𝑘=1
𝑙=1
⌊∑⌋
𝑠+1
𝑚
=−
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝛼𝑘 𝑝𝑞−𝑘,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑞−𝑙𝑚
𝑚
∑
𝑠−𝑚
∑ 𝑠+1−𝑚
∑ ⌊
𝑖=0 𝑞=𝑖+1
⌊∑⌋
(
∑
∑ ⌋ 𝑞−𝑖
𝑠+1−𝑞
𝑚
= −
𝑘=1
𝑙=1
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑠−𝑚
∑ 𝑠+1−𝑚
∑ ⌊
𝑖=0 𝑞=𝑖+1
⌊∑⌋
∑ ⌋
𝑠+1−𝑞
𝑚
= −
𝑙=1
𝐴𝑙 𝛽 𝑠+1−𝑙𝑚 +
𝑠+1−𝑚
∑⌊
∑ 𝑞−1
𝑞=1
𝑚
(2.16) ∑
=
𝑖=0
𝑚
∑
𝑝𝑠+1,𝑖 𝑢𝑖
𝑖=0
𝑙=1
−
𝑝𝑠+1,𝑖 𝑢𝑖
𝑝𝑞,𝑖+1 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑞−𝑙𝑚
𝑠+1
𝑚
(2.47)
𝑚
∑
𝑖=0
𝑙=1
−
)
𝛼𝑘 𝑝𝑞−𝑘,𝑖 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑞−𝑙𝑚
𝑠+1
𝑚
(2.43)
𝑝𝑠+1,𝑖 𝑢𝑖
𝑖=0
𝑙=1
−
𝑝𝑠+1,𝑖 𝑢𝑖
𝑖=0
𝑙=1
−
𝑚
∑
∑ ⌋
𝑠+1−𝑞
𝑚
𝑝𝑞,𝑖+1 Λ𝑖+1 𝐴𝑙 𝛽 𝑠+1−𝑞−𝑙𝑚
𝑙=1
𝑝𝑠+1,𝑖 𝑢𝑖 −
𝑖=0
𝑗
𝑠+1−𝑚
∑ ∑
𝑗=0
𝑘=0
⌊ 𝑠+1−𝑗
𝑚 ⌋
∑
𝑝𝑗,𝑘 Λ𝑘 𝐴𝑙 𝛽 𝑠+1−𝑗−𝑙𝑚 .
𝑙=1
But this is the statement we needed to prove for 𝑠 + 1 ≥ 𝑚. Therefore by induction the
statement (2.37) is true for all 𝑠 ≥ 0 and the lemma is proved.
The lemma above is the key to prove the recursive formulas for 𝛼𝑠 and 𝛽 𝑠 as given by
(2.23)-(2.25). First we prove that 𝛽 𝑠 is given by (2.25). For 𝑠 = 0 we have already shown
𝛽 0 = 𝑢1 = 𝑝0,0 𝑢1 . So suppose 𝑠 ≥ 1. Then by (2.36) and (2.37) we find that
(2.40)
𝛽𝑠 =
⎧



⎨
𝑠
∑
𝑝𝑠,𝑖 𝑢𝑖+1 , if 0 ≤ 𝑠 ≤ 𝑚 − 1
𝑖=0
𝑗 ⌊∑
𝑚−1
𝑠−𝑚
𝑚 ⌋
∑
∑ ∑



𝑝𝑠,𝑖 𝑢𝑖+1 −
𝑝𝑗,𝑘 Λ𝑘+1 𝐴𝑙 𝛽 𝑠−𝑗−𝑙𝑚 , if 𝑠 ≥ 𝑚.
⎩
𝑠−𝑗
𝑖=0
𝑗=0 𝑘=0
𝑙=1
This proves that 𝛽 𝑠 is given by (2.25).
39
Next we will prove that 𝛼𝑠 is given by (2.23) and (2.24). We start with 𝑠 = 1 and prove 𝛼1
is given by (2.23). First, (𝐴0 − 𝜆0 𝐼)∗ 𝑣𝑚 = 0 and ⟨𝑣𝑚 , 𝑢𝑖 ⟩ = 𝛿 𝑚,𝑖 hence
⟨
⟩
𝑚
∑
(2.42) 𝑚
0 = ⟨𝑣𝑚 , (𝐴0 − 𝜆0 𝐼)𝛽 𝑚 ⟩ =
𝑣𝑚 , 𝑝𝑚,𝑖 𝑢𝑖 − 𝐴1 𝑢1
= 𝛼1 − ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩
(2.37)
𝑖=0
so that 𝛼𝑚
1 = ⟨𝑣𝑚 , 𝐴1 𝑢1 ⟩. This and identity (2.34) imply that formula (2.23) is true.
Finally, suppose that 𝑠 ≥ 2. Then (𝐴0 − 𝜆0 𝐼)∗ 𝑣𝑚 = 0 and ⟨𝑣𝑚 , 𝑢𝑖 ⟩ = 𝛿 𝑚,𝑖 implies
0 = ⟨𝑣𝑚 , (𝐴0 − 𝜆0 𝐼)𝛽 𝑚+𝑠−1 ⟩
⟨
⟩
𝑚+𝑠−1−𝑗
⌋
𝑗 ⌊
𝑚
𝑠−1
𝑚
∑
∑∑
∑
(2.37)
=
𝑣𝑚 , 𝑝𝑚+𝑠−1,𝑖 𝑢𝑖 −
𝑝𝑗,𝑘 Λ𝑘 𝐴𝑙 𝛽 𝑚+𝑠−1−𝑗−𝑙𝑚
𝑖=0
(2.48)
= 𝑝𝑚+𝑠−1,𝑚 −
𝑗=0 𝑘=0
𝑠−1
∑ 𝑠−1
∑
𝑙=1
⟨
𝑝𝑗,𝑘
(Λ∗ )𝑘 𝑣𝑚 ,
𝑘=0 𝑗=𝑘
(2.41)
= 𝑟𝑠−1 + 𝑚𝛼𝑚−1
𝛼𝑠 −
1
⌊ 𝑚+𝑠−1−𝑗
⌋
𝑚
∑
⟩
𝐴𝑙 𝛽 𝑚+𝑠−1−𝑗−𝑙𝑚
𝑙=1
𝑠−1
∑ 𝑠−1
∑
⟨
𝑝𝑗,𝑖
(Λ∗ )𝑖 𝑣𝑚 ,
⌊ 𝑚+𝑠−1−𝑗
⌋
𝑚
∑
𝑖=0 𝑗=𝑖
⟩
𝐴𝑘 𝛽 𝑚+𝑠−1−𝑗−𝑘𝑚
𝑘=1
Therefore with this equality, the fact 𝛼1 ∕= 0, and Proposition 6.iii, we can solve for 𝛼𝑠 and
we will find that it is given by (2.24). This completes the proof.
Proof of Corollaries 3 and 4
Both corollaries follow almost trivially now. To prove Corollary 3, we just examine the recursive formulas (2.23)-(2.25) in Theorem 2 to see that 𝛼𝑘 , 𝛽 𝑘 requires only 𝐴0 , . . . , 𝐴⌊ 𝑚+𝑘−1 ⌋ .
𝑚
To prove Corollary 4, we use Proposition 7 to show that
𝑝0,0 = 1, 𝑝1,0 = 𝑝2,0 = 0, 𝑝1,1 = 𝛼1 , 𝑝2,1 = 𝛼2 , 𝑝2,2 = 𝛼21
40
and then from this and (2.23)-(2.25) we get the desired result for 𝛼1 , 𝛼2 , 𝛽 0 , 𝛽 1 , 𝛽 2 in terms
of 𝐴0 , 𝐴1 , 𝐴2 . The last part to prove is the formula for 𝛼2 in terms of 𝑓 (𝜀, 𝜆) and its partial
derivatives. But the formula follows from the series representation of 𝑓 (𝜀, 𝜆) in (2.29) and
𝜆0 (𝜀) in (2.19) since, for 𝜀 in a neighborhood of the origin,
0 = 𝑓 (𝜀, 𝜆0 (𝜀))
= (𝑎10 + 𝑎0𝑚 𝑝𝑚.𝑚 ) 𝜀
⎧

⎨
(𝑎01 𝑝2,1 + 𝑎02 𝑝2,2 + 𝑎11 𝑝1,1 + 𝑎20 𝑝00 ) 𝜀2 , for 𝑚 = 1
+

⎩ (𝑎0𝑚 𝑝𝑚+1,𝑚 + 𝑎0𝑚+1 𝑝𝑚+1,𝑚+1 + 𝑎11 𝑝1,1 ) 𝜀 𝑚+1
𝑚 , for 𝑚 > 1
( 𝑚+2 )
+𝑂 𝜀 𝑚
which together with Proposition 7 implies the formula for 𝛼2 .
2.5
Auxiliary Results
Properties of the Matrix Λ
The fundamental properties of the matrix Λ defined in (2.15) which are needed in this paper
are given in the following proposition:
Proposition 6
(i) We have Λ (𝐴0 − 𝜆0 𝐼) 𝑈 𝑒1 = 0, Λ (𝐴0 − 𝜆0 𝐼) 𝑈 𝑒𝑖 = 𝑈 𝑒𝑖 , for 2 ≤ 𝑖 ≤
𝑛,
(ii) For 1 ≤ 𝑖 ≤ 𝑚 − 1 we have
Λ𝑢𝑚 = 0, Λ𝑢𝑖 = 𝑢𝑖+1
(iii) Λ∗ 𝑣1 = 0, and Λ∗ 𝑣𝑖 = 𝑣𝑖−1 , for 2 ≤ 𝑖 ≤ 𝑚.
41
(2.40)
Proof. i. Using the fact 𝐽𝑚 (0)∗ 𝐽𝑚 (0) = diag[0, 𝐼𝑚−1 ], (2.12), and (2.15) we find by block
multiplication that 𝑈 −1 Λ (𝐴0 − 𝜆0 𝐼) 𝑈 = diag[0, 𝐼𝑛−1 ]. This implies the result.
ii. & iii. The results follow from the definition of 𝑢𝑖 , 𝑣𝑖 in (2.13), (2.14) and the fact
⎡
⎤
)∗ ⎢ 𝐽𝑚 (0)
(
⎥
(𝑈 −1 Λ𝑈 )∗ = 𝑈 ∗ Λ∗ 𝑈 −1 = ⎣
⎦.
[
−1 ]∗
(𝑊0 − 𝜆0 𝐼𝑛−𝑚 )
Properties of the Polynomials in the Recursive Formulas
This appendix contains two propositions. The first proposition gives fundamental identities
that help to characterize the polynomials {𝑝𝑗,𝑖 }∞
𝑗=𝑖 and {𝑟𝑙 }𝑙∈ℕ in (2.16) and (2.17). The
second proposition gives explicit recursive formulas to calculate these polynomials.
We may assume
∑∞
𝑗=1
𝛼𝑗 𝑧 𝑗 is a convergent Taylor series and 𝛼1 ∕= 0.
Proposition 7 The polynomials {𝑝𝑗,𝑖 }∞
𝑗=𝑖 and {𝑟𝑙 }𝑙∈ℕ have the following properties:
(i)
∞
∑
𝑗=𝑖
(
𝑝𝑗,𝑖 𝑧 𝑗 =
∞
∑
)𝑖
𝛼𝑗 𝑧 𝑗
, for 𝑗 ≥ 𝑖 ≥ 0.
𝑗=1
(ii) For 𝑙 ≥ 1 we have
𝑟𝑙 = 𝑝𝑚+𝑙,𝑚 − 𝑚𝛼𝑚−1
𝛼𝑙+1 .
1
(2.41)
𝑝𝑗,𝑗 = 𝛼𝑗1 .
(2.42)
(iii) 𝑝𝑗,1 = 𝛼𝑗 , for 𝑗 ≥ 1.
(iv) For 𝑗 ≥ 0 we have
42
(v) 𝑝𝑗+1,𝑗 = 𝑗𝛼𝑗−1
1 𝛼2 , for 𝑗 > 0.
(vi) For 𝑗 ≥ 𝑖 > 0 we have
𝑗−𝑖+1
∑
𝛼𝑞 𝑝𝑗−𝑞,𝑖−1 = 𝑝𝑗,𝑖 .
(2.43)
𝑞=1
(
Proof. i. For 𝑖 ≥ 0,
∞
∑
)𝑖
𝛼𝑗 𝑧
𝑗
∞
∑
=
⋅⋅⋅
𝑠1 =1
𝑗=1
∞
∑
(
𝑠𝑖 =1
)
𝑖
∏
𝛼𝑠𝜚 𝑧 𝑠1 +⋅⋅⋅+𝑠𝑖 =
𝜚=1
∞
∑
𝑝𝑗,𝑖 𝑧 𝑗 .
𝑗=𝑖
ii. Let 𝑙 ≥ 1. Then by definition of 𝑝𝑚+𝑙,𝑚 we have
𝑚
∏
∑
𝑝𝑚+𝑙,𝑚 =
𝛼 𝑠𝜚 +
𝜚=1
𝑠1 +⋅⋅⋅+𝑠𝑚 =𝑚+𝑙
1≤𝑠𝜚 ≤𝑙+1
∃𝜚∈{1,...,𝑚} such that 𝑠𝜚 =𝑙+1
𝑚
∏
∑
𝛼 𝑠𝜚
𝜚=1
𝑠1 +⋅⋅⋅+𝑠𝑚 =𝑚+𝑙
1≤𝑠𝜚 ≤𝑙+1
∃/𝜚∈{1,...,𝑚} such that 𝑠𝜚 =𝑙+1
= 𝑚𝛼𝑚−1
𝛼𝑙+1 + 𝑟𝑙 .
1
iii. For 𝑗 ≥ 1 we have 𝑝𝑗,1 =
∑
𝑠1 =𝑗
1≤𝑠𝜚 ≤𝑗
iv. For 𝑗 ≥ 0, 𝑝0,0 = 1 and 𝑝𝑗,𝑗 =
v. For 𝑗 > 0, 𝑝𝑗+1,𝑗 =
∑
∏1
𝜚=1
𝛼 𝑠𝜚 = 𝛼 𝑗 .
∑
𝑠1 +⋅⋅⋅+𝑠𝑗 =𝑗
1≤𝑠𝜚 ≤1
𝑠1 +⋅⋅⋅+𝑠𝑗 =𝑗+1
1≤𝑠𝜚 ≤2
∏𝑗
𝜚=1
∏𝑗
𝛼 𝑠𝜚 =
𝜚=1
∑𝑗
𝛼 𝑠𝜚 =
𝜚=1
∏𝑗
𝜚=1
𝛼1 = 𝛼𝑗1 .
𝑗−1
𝛼𝑗−1
1 𝛼2 = 𝑗𝛼1 𝛼2 .
vi. It follows by
∞
∑
𝑗=𝑖
𝑗 (i)
𝑝𝑗,𝑖 𝑧 =
∞
∑
𝑗=𝑖−1
𝑝𝑗,𝑖−1 𝑧 𝑗
∞
∑
𝑝𝑗,1 𝑧 𝑗 =
𝑗=1
∞
∑
𝑗=𝑖
(𝑗−𝑖+1
∑
)
𝑝𝑗−𝑞,𝑖−1 𝑝𝑞,1
𝑧𝑗 .
𝑞=1
This next proposition gives explicit recursive formulas to calculate the polynomials {𝑝𝑗,𝑖 }∞
𝑗=𝑖
and {𝑟𝑙 }𝑙∈ℕ .
Proposition 8 For each 𝑖 ≥ 0, the sequence of polynomials, {𝑝𝑗,𝑖 }∞
𝑗=𝑖 , is given by the recur43
sive formula
𝑝𝑖,𝑖 =
𝛼𝑖1 ,
𝑝𝑗,𝑖
𝑗−1
∑
1
=
[(𝑗 + 1 − 𝑘)𝑖 − 𝑘]𝛼𝑗+1−𝑘 𝑝𝑘,𝑖 , for 𝑗 > 𝑖.
(𝑗 − 𝑖)𝛼1 𝑘=𝑖
(2.44)
Furthermore, the polynomials {𝑟𝑙 }𝑙∈ℕ are given by the recursive formula:
𝑙−1
1 ∑
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝑟𝑗 +
𝑟1 = 0, 𝑟𝑙 =
𝑙𝛼1 𝑗=1
(2.45)
𝑙−1
𝑚 𝑚−2 ∑
𝛼
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝛼𝑗+1 , for 𝑙 > 1.
𝑙 1 𝑗=1
Proof.
We begin by showing (2.44) is true. For 𝑖 = 0, (2.44) follows from the definition
of the 𝑝𝑗,0 . If 𝑖 > 0 then by (2.42) and [20, (1.1) & (3.2)] it follows that (2.44) is true. Lets
now prove (2.45). From (2.41) and (2.44), we have
𝑟𝑙 = 𝑝𝑚+𝑙,𝑚 − 𝑚𝛼𝑚−1
𝛼𝑙+1
1
𝑚+𝑙−1
1 ∑
=
[(𝑚 + 𝑙 + 1 − 𝑘)𝑚 − 𝑘]𝛼𝑚+𝑙+1−𝑘 𝑝𝑘,𝑚 − 𝑚𝛼𝑚−1
𝛼𝑙+1
1
𝑙𝛼1 𝑘=𝑚
𝑚+𝑙−1
1 ∑
[(𝑚 + 𝑙 + 1 − 𝑘)𝑚 − 𝑘]𝛼𝑚+𝑙+1−𝑘 𝑝𝑘,𝑚
=
𝑙𝛼1 𝑘=𝑚+1
(2.42)
𝑙−1
1 ∑
=
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝑝𝑚+𝑗,𝑚
𝑙𝛼1 𝑗=1
𝑙−1
(
)
1 ∑
=
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝑟𝑗 + 𝑚𝛼𝑚−1
𝛼𝑗+1
1
𝑙𝛼1 𝑗=1
(2.41)
𝑙−1
1 ∑
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝑟𝑗 +
=
𝑙𝛼1 𝑗=1
𝑙−1
𝑚 𝑚−2 ∑
𝛼
[(𝑙 + 1 − 𝑗)𝑚 − (𝑚 + 𝑗)]𝛼𝑙+1−𝑗 𝛼𝑗+1 ,
𝑙 1 𝑗=1
for 𝑙 > 1. This completes the proof.
44
Sum Identities
These double sum identities are used in the proof of Theorem 3.1
𝑑 ∑
𝑑−𝑥
∑
𝑎𝑥,𝑦 =
𝑥=𝑐 𝑦=0
𝑑−1 ∑
𝑑
∑
𝑑−𝑦
𝑑−𝑐 ∑
∑
𝑎𝑥,𝑦 ,
(2.46)
𝑎𝑥,𝑦 ,
(2.47)
𝑎𝑥,𝑦 ,
(2.48)
𝑎𝑥,𝑞−𝑥 .
(2.49)
𝑦=0 𝑥=𝑐
𝑎𝑥,𝑦 =
𝑦−1
𝑑 ∑
∑
𝑥=0 𝑦=𝑥+1
𝑦=1 𝑥=0
𝑑 ∑
𝑥
∑
𝑑 ∑
𝑑
∑
𝑎𝑥,𝑦 =
𝑥=0 𝑦=0
𝑑−𝑦
𝑑−1 ∑
∑
𝑦=𝑐 𝑥=1
𝑎𝑥,𝑦 =
𝑦=0 𝑥=𝑦
𝑞−𝑐
𝑑
∑
∑
𝑞=𝑐+1 𝑥=1
45
Chapter 3
Spectral Perturbation Theory for
Holomorphic Matrix Functions
In this section we consider holomorphic matrix-valued functions of one and two variables.
A holomorphic matrix-valued function of one variable will be referred to as a holomorphic
matrix function. Our goal is to give some basic facts on the spectral theory and spectral
perturbation theory for holomorphic matrix functions that will be needed in this thesis. The
exposition we give here is drawn from many sources, in particular, [44], [29, Appendix A], and
[19] for the spectral theory and [4, Appendix A], [23], and [34] for the spectral perturbation
theory.
3.1
On Holomorphic Matrix-Valued Functions of One
Variable
We introduce the following notation and convention for this section:
46
1. 𝒪(𝜆0 ) :=
∪
𝒪(𝐵(𝜆0 , 𝑟), ℂ) – the set of scalar functions holomorphic at 𝜆0 ∈ ℂ.
𝑟>0
2. 𝒪𝑚×𝑛 (𝜆0 ) :=
∪
𝒪(𝐵(𝜆0 , 𝑟), 𝑀𝑚,𝑛 (ℂ)) – the set of 𝑚 × 𝑛 matrix-valued functions
𝑟>0
holomorphic at 𝜆0 ∈ ℂ.
3. As convention we identify 𝒪1×1 (𝜆0 ) with 𝒪(𝜆0 ).
4. If 𝑓1 , 𝑓2 ∈ 𝒪𝑚×𝑛 (𝜆0 ) then we use the notation 𝑓1 = 𝑓2 to mean there exists an 𝑟 > 0
such that 𝑓1 , 𝑓2 ∈ 𝒪(𝐵(𝜆0 , 𝑟), 𝑀𝑚,𝑛 (ℂ)) and 𝑓1 (𝜆) = 𝑓2 (𝜆) for ∣𝜆 − 𝜆0 ∣ < 𝑟.
5. As is known, if 𝑓 ∈ 𝒪𝑚×𝑛 (𝜆0 ) then 𝑓 (𝜆) is analytic at 𝜆 = 𝜆0 , i.e., 𝑓 (𝜆) is infinitely
differentiable at 𝜆 = 𝜆0 and, denoting its 𝑗th derivative at 𝜆 = 𝜆0 by 𝑓 (𝑗) (𝜆0 ), there
exists an 𝑟 > 0 such that
∞
∑
1 (𝑗)
𝑓 (𝜆0 )(𝜆 − 𝜆0 )𝑗 , ∣𝜆 − 𝜆0 ∣ < 𝑟,
𝑓 (𝜆) =
𝑗!
𝑗=0
where the power series on the right converges absolutely in the 𝑀𝑚,𝑛 (ℂ) norm to 𝑓 (𝜆)
for ∣𝜆 − 𝜆0 ∣ < 𝑟.
We begin with the definition of the spectrum of a holomorphic matrix function and the
important notion of spectral equivalence for holomorphic matrix functions.
Definition 2 If 𝐹 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)), where 𝑈 is an open connected set in ℂ, then the spectrum of 𝐹 , denoted by 𝜎(𝐹 ), is the subset of 𝑈 such that the matrix 𝐹 (𝜆) is not invertible,
i.e.,
𝜎(𝐹 ) := {𝜆 ∈ 𝑈 ∣ det 𝐹 (𝜆) = 0}.
(3.1)
A point 𝜆0 ∈ 𝑈 is called an eigenvalue of 𝐹 provided 𝜆0 ∈ 𝜎(𝐹 ).
Definition 3 Two holomorphic matrix functions 𝐹, 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 ) are called equivalent
47
at 𝜆0 provided there exists 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0 such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are
invertible matrices and
𝐹 (𝜆) = 𝑁 (𝜆)𝐺(𝜆)𝑀 (𝜆),
(3.2)
for every 𝜆 ∈ 𝐵(𝜆0 , 𝑟).
For the local spectral theory of holomorphic matrix functions the follows theorem plays a
central role:
Theorem 9 (Local Smith Form) Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) with 𝜆0 ∈ 𝜎(𝐹 ). Then 𝐹 is equivalent at 𝜆0 to a unique diagonal matrix 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 ) with
𝐺(𝜆) = diag{0, . . . , 0, (𝜆 − 𝜆0 )𝑚𝜇+1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1},
(3.3)
where 0 ≤ 𝜇 ≤ 𝑔 ≤ 𝑛 and the number of zeros down the diagonal is 𝜇, when 𝜇 = 𝑔 there are
no (𝜆 − 𝜆0 ) terms appearing in the diagonal, when 𝜇 < 𝑔 we have 𝑚𝜇+1 ≥ . . . ≥ 𝑚𝑔 ≥ 1 with
{𝑚𝜇+𝑗 }𝑔𝑗=1 ⊆ ℕ, and when 𝑔 = 𝑛 there are no ones appearing in the diagonal.
Proof. This existence is proven in [29, pp. 414–415, Theorem A.6.1] and uniqueness follows
from [19, p. 106, Theorem 4.3.1].
Definition 4 The matrix 𝐺 in (3.3) is called the local Smith form of 𝐹 corresponding to
the eigenvalue 𝜆0 .
Now its obvious from the theorem that
𝑔 = dim(ker(𝐹 (𝜆0 ))),
(3.4)
𝜇 = dim(ker(𝐹 (𝜆))), 0 < ∣𝜆 − 𝜆0 ∣ ≪ 1,
(3.5)
48
and, since det(𝐹 ) ∈ 𝒪(𝜆0 ),
𝜇 = 0 ⇔ det(𝐹 ) ∕= 0.
(3.6)
But an interpretation of the numbers {𝑚𝜇+𝑗 }𝑔𝑗=1 is not as straight forward. Thus we now
give an important example showing how to interpret this theorem in a special case to the
structure of the Jordan normal form of a matrix corresponding to an eigenvalue. Moreover,
the following example will help to motivate the discussion in the next section.
Example 1. Let 𝐴 ∈ 𝑀𝑛 (ℂ) and define 𝐹 (𝜆) := 𝐴 − 𝜆𝐼𝑛 for 𝜆 ∈ ℂ. Let 𝜆0 be an eigenvalue
of 𝐴. Then the hypotheses of Theorem 9 are satisfied and so 𝐹 is equivalent at 𝜆0 to a
diagonal matrix
𝐺(𝜆) = diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}.
(3.7)
where 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1 and {𝑚𝑗 }𝑔𝑗=1 ⊆ ℕ. The numbers 𝑔, 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 :=
𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 are called the geometric multiplicity, partial multiplicities, and algebraic
multiplicity, respectively, of the eigenvalue 𝜆0 of the matrix 𝐴. It follows from [18, §2.2] and
[18, p. 657, §A.3, Theorem A.3.4] that the Jordan normal form of the matrix 𝐴 corresponding
to the eigenvalue 𝜆0 consists of 𝑔 Jordan blocks of order 𝑚1 , . . . , 𝑚𝑔 , respectively, with
eigenvalue 𝜆0 , that is, there exists matrices 𝑆 ∈ 𝑀𝑛 (ℂ) and 𝑊0 ∈ 𝑀𝑛−𝑚 (ℂ) such that 𝑆
is an invertible matrix, 𝜆0 is not an eigenvalue of the matrix 𝑊0 , and 𝑆 −1 𝐴𝑆 is the block
diagonal matrix
𝑆 −1 𝐴𝑆 = diag{𝐽𝑚1 (𝜆0 ), . . . , 𝐽𝑚𝑔 (𝜆0 ), 𝑊0 },
49
(3.8)
where 𝐽𝑚𝑗 (𝜆0 ) is the 𝑚𝑗 × 𝑚𝑗 Jordan block given by
⎡
𝜆 1
⎢ 0
⎢
⎢
. .
⎢
⎢
𝐽𝑚𝑗 (𝜆0 ) = ⎢
. .
⎢
⎢
⎢
.
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥ , for 𝑗 = 1, . . . , 𝑔,
⎥
⎥
⎥
1⎥
⎦
𝜆0
(3.9)
where all unmarked elements are zeros.
In the next section we use Theorem 9 and this example to analogously define for an eigenvalue of a holomorphic matrix function, its geometric multiplicity, partial multiplicities, and
algebraic multiplicity. We also introduce the notion of a Jordan chain for a holomorphic
matrix function and again use this example to compare it with the standard definition of a
Jordan chain of a matrix (see [35, §6.3] for the standard definitions in the matrix case).
3.1.1
Local Spectral Theory of Holomorphic Matrix Functions
For holomorphic matrix functions, a local spectral theory can be given based on the local
Smith form. We now give a brief exposition of this theory.
Definition 5 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ). Then ker(𝐹 (𝜆0 )) is called the eigenspace of 𝐹 at 𝜆0 .
If 𝜑 ∈ ker(𝐹 (𝜆0 )) and 𝜑 ∕= 0 then 𝜑 is called an eigenvector of 𝐹 corresponding to the
eigenvalue 𝜆0 .
Definition 6 If 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝜆0 is an eigenvalue 𝐹 then we say 𝜆0 is an eigenvalue
of 𝐹 of finite (infinite) algebraic multiplicity if det(𝐹 ) ∕= 0 (det(𝐹 ) = 0).
Definition 7 Let 𝐹 be any holomorphic matrix function such that 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝜆0 an
50
eigenvalue of 𝐹 of finite algebraic multiplicity. By Theorem 9 and (3.6) there exists unique
numbers 𝑔, 𝑚1 , . . . , 𝑚𝑔 ∈ ℕ satisfying 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 such that the local Smith form of 𝐹
corresponding to the eigenvalue 𝜆0 is
𝐺(𝜆) = diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}.
(3.10)
The positive integers 𝑔, 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 := 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 are called the geometric
multiplicity, partial multiplicities, and algebraic multiplicity, respectively, of the
eigenvalue 𝜆0 of 𝐹 . We say 𝜆0 is a semisimple eigenvalue of 𝐹 if 𝑚1 = ⋅ ⋅ ⋅ = 𝑚𝑔 = 1.
The following proposition gives a simple but important characterization of the geometric
and algebraic multiplicity.
Proposition 10 Suppose 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝜆0 is an eigenvalue of 𝐹 of finite algebraic
multiplicity. Let 𝑔 and 𝑚 be the geometric multiplicity and algebraic multiplicity of the
eigenvalue 𝜆0 of 𝐹 . Then 𝑔 is the dimension of ker(𝐹 (𝜆0 )) and 𝑚 is the order of the zero of
the function det(𝐹 (𝜆)) at 𝜆 = 𝜆0 .
Proof.
Let 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 be the partial multiplicities of the eigenvalue 𝜆0 of 𝐹 so
that 𝑚 = 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 . By Theorem 9 we know that 𝐹 is equivalent at 𝜆0 to its local
Smith form. Thus there exists 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0 such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are
invertible matrices and
𝐹 (𝜆) = 𝑁 (𝜆) diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}𝑀 (𝜆),
for every 𝜆 ∈ 𝐵(𝜆0 , 𝑟). It follows from this representation that the first 𝑔 columns of the
matrix 𝑀 (𝜆0 )−1 are a basis for ker(𝐹 (𝜆0 )) so that 𝑔 = dim(ker(𝐹 (𝜆0 ))). Moreover, since
51
det(𝑁 ), det(𝑀 ) ∈ 𝒪(𝜆0 ) with det(𝑁 (𝜆0 )), det(𝑀 (𝜆0 )) ∕= 0, we have
det(𝐹 (𝜆)) = det(𝑁 (𝜆) diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}𝑀 (𝜆))
= (𝜆 − 𝜆0 )𝑚 det(𝑁 (𝜆) det(𝑀 (𝜆))
= (𝜆 − 𝜆0 )𝑚 det(𝑁 (𝜆0 )) det(𝑀 (𝜆0 )) + 𝑂((𝜆 − 𝜆0 )𝑚+1 ), as 𝜆 → 𝜆0 .
Hence 𝑚 is the order of the zero of the function det(𝐹 (𝜆)) at 𝜆 = 𝜆0 . This completes the
proof.
Definition 8 Let 𝐹 be any holomorphic matrix function such that 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝜆0 an
𝑛
eigenvalue of 𝐹 . A sequence {𝜑𝑗 }𝑙−1
𝑗=0 ⊆ ℂ , where 𝑙 ∈ ℕ, is called a Jordan chain of 𝐹 of
length 𝑙 corresponding to the eigenvalue 𝜆0 (associated with the eigenvector 𝜑0 ) if
𝜑0 ∕= 0 and
𝑗
∑
1 (ℎ)
𝐹 (𝜆0 )𝜑𝑗−ℎ = 0,
ℎ!
ℎ=0
𝑗 = 0, . . . , 𝑙 − 1.
(3.11)
The maximal length of a Jordan chain of 𝐹 corresponding to the eigenvalue 𝜆0 and associated
with 𝜑0 is called the multiplicity of the eigenvector 𝜑0 and is denoted by 𝑚(𝜑0 ), i.e.,
𝑚(𝜑0 ) := sup{𝑙 ∈ ℕ : {𝜑𝑗 }𝑙−1
𝑗=0 satisfies (3.11)}.
(3.12)
Example 2. Let 𝐴 ∈ 𝑀𝑛 (ℂ) and define 𝐹 (𝜆) := 𝐴 − 𝜆𝐼𝑛 for 𝜆 ∈ ℂ. Then by Definition
5, 𝜆0 is an eigenvalue of 𝐴 if and only if 𝜆0 is an eigenvalue of 𝐹 . We will now show that
𝑛
a sequence of vectors {𝜑𝑗 }𝑙−1
𝑗=0 ⊆ ℂ is a Jordan chain of 𝐴 of length 𝑙 corresponding to the
eigenvalue 𝜆0 (see [35, p. 230, §6.3] for definition) if and only if {𝜑𝑗 }𝑙−1
𝑗=0 is a Jordan chain of
𝐹 of length 𝑙 corresponding to the eigenvalue 𝜆0 .
52
Suppose 𝜆0 is an eigenvalue of 𝐴 and {𝜑𝑗 }𝑙−1
𝑗=0 is a Jordan chain of 𝐴 of length 𝑙 corresponding
to the eigenvalue 𝜆0 , i.e.,
𝜑0 ∕= 0, (𝐴 − 𝜆0 𝐼𝑛 )𝜑𝑗 = 𝜑𝑗−1 ,
𝑗 = 0, . . . , 𝑙 − 1
where 𝜑−1 := 0. Then 𝜑0 ∕= 0 and
𝑗
∑
1 (ℎ)
𝐹 (𝜆0 )𝜑𝑗−ℎ = 𝐹 (0) (𝜆0 )𝜑𝑗 + 𝐹 (1) (𝜆0 )𝜑𝑗−1 = (𝐴 − 𝜆0 𝐼𝑛 )𝜑𝑗 − 𝜑𝑗−1 = 0
ℎ!
ℎ=0
for 𝑗 = 0, . . . , 𝑙 − 1. This implies by Definition 8 that the sequence {𝜑𝑗 }𝑙−1
𝑗=0 is a Jordan chain
of 𝐹 of length 𝑙 corresponding to the eigenvalue 𝜆0 .
Conversely, if {𝜑𝑗 }𝑙−1
𝑗=0 is a Jordan chain of 𝐹 of length 𝑙 corresponding to the eigenvalue 𝜆0
then by Definition 8 this sequence satisfies
𝑗
∑
1 (ℎ)
𝜑0 ∕= 0 and 0 =
𝐹 (𝜆0 )𝜑𝑗−ℎ = (𝐴 − 𝜆0 𝐼𝑛 )𝜑𝑗 − 𝜑𝑗−1 ,
ℎ!
ℎ=0
𝑗 = 0, . . . , 𝑙 − 1,
𝑙−1
where 𝜑−1 := 0. This implies {𝜑𝑗 }𝑗=0
⊆ ℂ𝑛 is a Jordan chain of 𝐴 of length 𝑙 corresponding
to the eigenvalue 𝜆0 .
Definition 9 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ). A vector-valued function 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) is called a generating function of order 𝑙 for 𝐹 at the eigenvalue 𝜆0 , where 𝑙 ∈ ℕ, if 𝜑(𝜆0 ) ∕= 0 and the
function 𝐹 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) has a zero of order ≥ 𝑙 at 𝜆0 , i.e.,
𝜑(𝜆0 ) ∕= 0 𝑎𝑛𝑑 𝐹 (𝜆)𝜑(𝜆) = 𝑂((𝜆 − 𝜆0 )𝑙 ), as 𝜆 → 𝜆0
(3.13)
Proposition 11 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ). Then 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) is a generating function of order 𝑙
53
for 𝐹 at the eigenvalue 𝜆0 if and only if 𝜆0 is an eigenvalue 𝐹 and the sequence
𝜑𝑗 :=
1 (𝑗)
𝜑 (𝜆0 ),
𝑗!
𝑗 = 0, . . . , 𝑙 − 1
(3.14)
is a Jordan chain of F of length 𝑙 corresponding to the eigenvalue 𝜆0 .
Proof. This statement and its proof can be found in [44, p. 57, §II.11, Lemma 11.3]. The
result follows from the fact if 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) then 𝐹 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) and
hence
𝐹 (𝜆)𝜑(𝜆) =
∞
∞
∑
∑
1 (𝑗)
1 (𝑗)
𝐹 (𝜆0 )(𝜆 − 𝜆0 )𝑗
𝜑 (𝜆0 )(𝜆 − 𝜆0 )𝑗
𝑗!
𝑗!
𝑗=0
𝑗=0
𝑗
∞ ∑
∑
1
1 (ℎ)
𝐹 (𝜆0 )
𝜑(𝑗−ℎ) (𝜆0 )
=
ℎ!
(𝑗
−
ℎ)!
𝑗=0 ℎ=0
𝑗
𝑙−1 ∑
∑
1 (ℎ)
1
=
𝐹 (𝜆0 )
𝜑(𝑗−ℎ) (𝜆0 ) + 𝑂((𝜆 − 𝜆0 )𝑙 ), as 𝜆 → 𝜆0 .
ℎ!
(𝑗 − ℎ)!
𝑗=0 ℎ=0
Definition 10 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) such that 𝜆0 an eigenvalue of 𝐹 of finite algebraic mul𝑚 −1
𝑗
tiplicity. We say {𝜑𝑖,𝑗 }𝑖=0
, 𝑗 = 1, . . . , 𝑔 is a canonical system of Jordan chains of 𝐹
corresponding to the eigenvalue 𝜆0 if the following three conditions are satisfied:
1. The numbers 𝑔, 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 := 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 are the geometric multiplicity,
partial multiplicities, and algebraic multiplicity, respectively, of the eigenvalue 𝜆0 of 𝐹 .
𝑚 −1
𝑗
2. The sequence {𝜑𝑖,𝑗 }𝑖=0
is a Jordan chain of 𝐹 of length 𝑚𝑗 , for 𝑗 = 1, . . . , 𝑔 and
𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1.
3. The vectors 𝜑0,𝑗 , 𝑗 = 1, . . . , 𝑔 form a basis for the eigenspace ker(𝐹 (𝜆0 )).
54
Example 3. Let 𝐴 ∈ 𝑀𝑛 (ℂ) and define 𝐹 (𝜆) := 𝐴 − 𝜆𝐼𝑛 for 𝜆 ∈ ℂ. Then it can be shown
𝑚 −1
𝑗
that {𝜑𝑖,𝑗 }𝑖=0
, 𝑗 = 1, . . . , 𝑔 is a canonical system of Jordan chains of 𝐹 corresponding to
∪
𝑚𝑗 −1
the eigenvalue 𝜆0 if and only if the set of vectors 𝑔𝑗=1 {𝜑𝑖,𝑗 }𝑖=0
form a Jordan basis for the
generalized eigenspace of the matrix 𝐴 corresponding to the eigenvalue 𝜆0 (see [35, p. 232,
𝑚 −1
𝑗
§6.4] for the definition) where {𝜑𝑖,𝑗 }𝑖=0
is a Jordan chain of 𝐴 of length 𝑚𝑗 corresponding
to the eigenvalue 𝜆0 and 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1.
Lemma 12 The relation ∼ defined by 𝐹 ∼ 𝐺 if 𝐹, 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 ) are equivalent at 𝜆0 , is
an equivalence relation on 𝒪𝑛×𝑛 (𝜆0 ).
Proof.
First, its obvious the relation ∼ is reflexive since the identity matrix 𝐼𝑛 as a
constant function belongs in 𝒪𝑛×𝑛 (𝜆0 ). Now lets show the relation is symmetric. Suppose
𝐹, 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝐹 ∼ 𝐺. This implies there exists 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0
such that 𝐹 (𝜆) = 𝑁 (𝜆)𝐺(𝜆)𝑀 (𝜆) for ∣𝜆 − 𝜆0 ∣ < 𝑟 and 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are invertible matrices.
It follows from [45, p. 7, §1.2, Proposition 1.2.5] that there exists a possibly smaller 𝑟 > 0
such that 𝑁 −1 (𝜆) := 𝑁 (𝜆)−1 and 𝑀 −1 (𝜆) := 𝑀 (𝜆)−1 exist for ∣𝜆 − 𝜆0 ∣ < 𝑟 and 𝑁 −1 , 𝑀 −1 ∈
𝒪(𝐵(𝜆0 , 𝑟), 𝑀𝑛 (ℂ)). But then 𝑁 −1 , 𝑀 −1 ∈ 𝒪𝑛×𝑛 (𝜆0 ), 𝑁 −1 (𝜆), 𝑀 −1 (𝜆0 ) are invertible, and
𝐺(𝜆) = 𝑁 −1 (𝜆)𝐹 (𝜆)𝑀 −1 (𝜆) for ∣𝜆 − 𝜆0 ∣ < 𝑟. This proves 𝐺 ∼ 𝐹 and so ∼ is symmetric. We
now prove ∼ is transitive. Indeed, let 𝐹 ∼ 𝐺 and 𝐺 ∼ 𝐻. Then there exists 𝑁1 , 𝑁2 , 𝑀1 , 𝑀2 ∈
𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0 such that 𝐹 (𝜆) = 𝑁1 (𝜆)𝐺(𝜆)𝑀1 (𝜆) = 𝑁1 (𝜆)𝑁2 (𝜆)𝐻(𝜆)𝑀1 (𝜆)𝑀2 (𝜆)
for ∣𝜆 − 𝜆0 ∣ < 𝑟 but the products 𝑁1 𝑁2 and 𝑀1 𝑀2 are in 𝒪𝑛×𝑛 (𝜆0 ) and are invertible at 𝜆0 .
Thus 𝐹 ∼ 𝐻 and so ∼ is transitive. This proves the relation ∼ is an equivalence relation on
𝒪𝑛×𝑛 (𝜆0 ).
Proposition 13 Let 𝐹, 𝐺 be any holomorphic matrix function such that 𝐹, 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 )
are equivalent at 𝜆0 . Then
1. 𝜆0 is an eigenvalue of 𝐹 of finite algebraic multiplicity if and only if 𝜆0 is an eigenvalue
55
of 𝐺 of finite algebraic multiplicity.
2. Let 𝜆0 be an eigenvalue of 𝐹 of finite algebraic multiplicity. Then the local Smith
forms of 𝐹 and 𝐺 corresponding to the eigenvalue 𝜆0 are equal and, in particular, the
geometric, partial, and algebraic multiplicities of the eigenvalue 𝜆0 of 𝐹 and 𝐺 are the
same.
3. Let 𝜆0 be an eigenvalue of 𝐹 of finite algebraic multiplicity. Let 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) be
such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are invertible matrices and
𝐹 (𝜆) = 𝑁 (𝜆)𝐺(𝜆)𝑀 (𝜆)
(3.15)
for every ∣𝜆 − 𝜆0 ∣ < 𝑟. If {𝜑𝑗 }𝑙𝑗=0 is a Jordan chain of 𝐹 of length 𝑙 corresponding to
the eigenvalue 𝜆0 then {˜
𝜑𝑗 }𝑙𝑗=0 is a Jordan chain of 𝐺 of length 𝑙 corresponding to the
eigenvalue 𝜆0 where
𝑗
𝜑
˜ 𝑗 :=
Proof.
1 ∑ 1 (ℎ)
𝑀 (𝜆0 )𝜑𝑗−ℎ ,
𝑗! ℎ=0 ℎ!
𝑗 = 0, . . . , 𝑙 − 1.
(3.16)
Let 𝐹, 𝐺 be any holomorphic matrix function such that 𝐹, 𝐺 ∈ 𝒪𝑛×𝑛 (𝜆0 ) are
equivalent at 𝜆0 . It follows from Lemma 12 and Theorem 9 that the local Smith forms of
𝐹 and 𝐺 are equal. This implies that 𝜆0 is an eigenvalue of 𝐹 finite algebraic multiplicity
if and only if 𝜆0 is an eigenvalue of 𝐺 finite algebraic multiplicity. It also implies that if 𝜆0
is an eigenvalue of 𝐹 finite algebraic multiplicity then the geometric, partial, and algebraic
multiplicities of the eigenvalue 𝜆0 of 𝐹 and 𝐺 are the same.
Suppose that 𝜆0 is an eigenvalue of 𝐹 finite algebraic multiplicity. Let 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) be
such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are invertible matrices and
𝐹 (𝜆) = 𝑁 (𝜆)𝐺(𝜆)𝑀 (𝜆)
56
(3.17)
for every ∣𝜆 − 𝜆0 ∣ < 𝑟. Suppose {𝜑𝑗 }𝑙𝑗=0 is a Jordan chain of 𝐹 of length 𝑙 corresponding to
the eigenvalue 𝜆0 . Define the vector-valued function 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) by
𝜑(𝜆) :=
𝑙−1
∑
𝜑𝑗 (𝜆 − 𝜆0 )𝑗 ,
𝜆 ∈ ℂ.
𝑗=0
It follows from Proposition 11 that 𝜑 is a generating function of order 𝑙 for 𝐹 at the eigenvalue
𝜆0 . This implies
𝜑(𝜆0 ) ∕= 0,
𝐹 (𝜆)𝜑(𝜆) = 𝑂((𝜆 − 𝜆0 )𝑙 ), as 𝜆 → 𝜆0 .
But since 𝑁 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝑁 (𝜆0 ) is invertible this implies there exists a possibly smaller
𝑟 > 0 such that 𝑁 −1 (𝜆) := 𝑁 (𝜆)−1 exists for ∣𝜆 − 𝜆0 ∣ < 𝑟 and 𝑁 −1 ∈ 𝒪(𝐵(𝜆0 , 𝑟), 𝑀𝑛 (ℂ)).
This implies
𝐺(𝜆)𝑀 (𝜆)𝜑(𝜆) = 𝑁 −1 (𝜆)𝐹 (𝜆)𝜑(𝜆) = 𝑂((𝜆 − 𝜆0 )𝑙 ), as 𝜆 → 𝜆0 .
But since 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and 𝑀 (𝜆0 ) is invertible then 𝑀 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) and
𝑀 (𝜆0 )𝜑(𝜆0 ) ∕= 0,
𝐺(𝜆)𝑀 (𝜆)𝜑(𝜆) = 𝑂((𝜆 − 𝜆0 )𝑙 ), as 𝜆 → 𝜆0 .
This implies by Definition 9 that 𝑀 𝜑 is a generating function of order 𝑙 for 𝐹 at the eigenvalue
𝜆0 and so by Proposition 11, {˜
𝜑𝑗 }𝑙𝑗=0 is a Jordan chain of 𝐺 of length 𝑙 corresponding to the
eigenvalue 𝜆0 where
𝑗
1 ∑ 1 (ℎ)
1 𝑑𝑗
(𝑀 (𝜆)𝜑(𝜆)) 𝜆=𝜆0 =
𝑀 (𝜆0 )𝜑𝑗−ℎ ,
𝜑
˜ 𝑗 :=
𝑗! 𝑑𝜆𝑗
𝑗! ℎ=0 ℎ!
𝑗 = 0, . . . , 𝑙 − 1.
This completes the proof.
Theorem 14 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) such that 𝜆0 an eigenvalue of 𝐹 of finite algebraic mul57
tiplicity. Then there exists a canonical system of Jordan chains of 𝐹 corresponding to the
eigenvalue 𝜆0 .
Proof.
Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) such that 𝜆0 an eigenvalue of 𝐹 of finite algebraic multiplicity.
Denote by 𝑔, 𝑚1 , . . . , 𝑚𝑔 , 𝑚 the geometric, partial, and algebraic multiplicities, respectively,
of the eigenvalue 𝜆0 of 𝐹 where 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1 and 𝑚 = 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 . By Theorem
9 and Lemma 12 there exists 𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0 such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are
invertible matrices and
𝐺(𝜆) = 𝑁 (𝜆)𝐹 (𝜆)𝑀 (𝜆),
for every 𝜆 ∈ 𝐵(𝜆0 , 𝑟) where
𝐺(𝜆) = diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}.
Let 𝑒1 , . . . , 𝑒𝑛 denote the standard orthonormal basis vectors in ℂ𝑛 . Define vector-valued
functions 𝜑𝑗 ∈ 𝒪𝑛×1 (𝜆0 ), 𝑗 = 1, . . . , 𝑔 by
𝜑𝑗 (𝜆) = 𝑒𝑗 ,
𝜆 ∈ ℂ,
𝑗 = 1, . . . , 𝑔.
(3.18)
Then since
𝜑𝑗 (𝜆0 ) = 𝑒𝑗 ∕= 0,
𝐺(𝜆)𝜑𝑖 (𝜆) = (𝜆 − 𝜆0 )𝑚𝑗 𝑒𝑗 = 𝑂((𝜆 − 𝜆0 )𝑚𝑖 ), as 𝜆 → 𝜆0 ,
(3.19)
each 𝜑𝑗 is a generating function of order 𝑚𝑗 for 𝐺 at the eigenvalue 𝜆0 and so by Proposition
𝑚 −1
𝑗
11 the sequence {𝜑𝑖,𝑗 }𝑖=0
defined by
𝜑𝑖,𝑗 :=
1 (𝑖)
𝜑 (𝜆0 ) = 𝛿 0,𝑖 𝑒𝑗 ,
𝑖! 𝑗
58
𝑖 = 0, . . . , 𝑚𝑗 − 1,
(3.20)
where 𝛿 0,𝑖 is the Kronecker delta symbol, is a Jordan chain of 𝐺 of length 𝑚𝑗 corresponding to
𝑚 −1
𝑗
the eigenvalue 𝜆0 , for 𝑗 = 1, . . . , 𝑔. It follows that {𝜑𝑖,𝑗 }𝑖=0
, 𝑗 = 1, . . . , 𝑔 is canonical system
of Jordan chains for 𝐺 corresponding to the eigenvalue 𝜆0 , that is, we have constructed
a canonical system of Jordan chains for the local Smith form of 𝐹 corresponding to the
𝑚 −1
𝑗
eigenvalue 𝜆0 . By Proposition 13 the sequence {˜
𝜑𝑖,𝑗 }𝑖=0
defined by
𝑖
𝜑
˜ 𝑖,𝑗
1 ∑ 1 (ℎ)
𝑀 (𝜆0 )𝜑𝑖−ℎ,𝑗 ,
:=
𝑖! ℎ=0 ℎ!
𝑖 = 0, . . . , 𝑚𝑗 − 1
(3.21)
is a Jordan chain of 𝐹 of length 𝑚𝑗 corresponding to the eigenvalue 𝜆0 , for 𝑗 = 1, . . . , 𝑔. Moreover, since 𝑒1 , . . . , 𝑒𝑔 is a basis for ker 𝐺(𝜆0 ), 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are invertible, and 𝑁 (𝜆0 )𝐺(𝜆0 ) =
𝐹 (𝜆0 )𝑀 (𝜆0 ) then 𝜑
˜ 0,𝑗 = 𝑀 (𝜆0 )𝑒𝑗 , 𝑗 = 1, . . . , 𝑔 is a basis for ker 𝐹 (𝜆0 ). Thus we have shown
the numbers 𝑔, 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 = 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 are the geometric multiplicity, partial
multiplicities, and algebraic multiplicity, respectively, of the eigenvalue 𝜆0 of 𝐹 , the sequence
𝑚 −1
𝑗
{𝜑𝑖,𝑗 }𝑖=0
is a Jordan chain of 𝐹 of length 𝑚𝑗 , for 𝑗 = 1, . . . , 𝑔 and 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1, and
the vectors 𝜑0,𝑗 , 𝑗 = 1, . . . , 𝑔 form a basis for the eigenspace ker(𝐹 (𝜆0 )). Therefore, by Defi𝑚 −1
𝑗
nition 10, {𝜑𝑖,𝑗 }𝑖=0
, 𝑗 = 1, . . . , 𝑔 is a canonical system of Jordan chains of 𝐹 corresponding
to the eigenvalue 𝜆0 . This completes the proof.
It will be important in this thesis to know when an eigenvalue of a holomorphic matrix
function is both of finite algebraic multiplicity and semisimple. The next proposition gives
us a simple sufficient condition for this to be true.
Proposition 15 If 𝜆0 is an eigenvalue of 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and the equation
𝐹 (1) (𝜆0 )𝜑0 = −𝐹 (𝜆0 )𝜑1
(3.22)
has no solution for 𝜑0 ∈ ker(𝐹 (𝜆0 ))/{0} and 𝜑1 ∈ ℂ𝑛 then 𝜆0 is an eigenvalue of 𝐹 of finite
algebraic multiplicity and a semisimple eigenvalue of 𝐹 .
59
Proof.
Let the hypotheses of this proposition be true for some 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ). Then the
hypothesis that (3.22) has no solution for 𝜑0 ∈ ker(𝐹 (𝜆0 ))/{0} and 𝜑1 ∈ ℂ𝑛 is equivalent
by Definition 8 to there are no Jordan chains of 𝐹 of length 𝑙 ≥ 2 corresponding to the
eigenvalue 𝜆0 . Thus if 𝜆0 was an eigenvalue of 𝐹 but not of finite algebraic multiplicity then
by Theorem 9 we could find a vector-valued function 𝜑 ∈ 𝒪𝑛×1 (𝜆0 ) and an 𝑟 > 0 such that
𝐹 (𝜆)𝜑(𝜆) = 0 for ∣𝜆 − 𝜆0 ∣ < 𝑟. By Proposition 11 we could find a Jordan chain of order 𝑙
for any 𝑙 ∈ ℕ and in particular, for 𝑙 ≥ 2, a contradiction. Hence 𝜆0 must be an eigenvalue
of 𝐹 of finite algebraic multiplicity. If 𝜆0 was an eigenvalue of 𝐹 but was not a semisimple
eigenvalue of 𝐹 then by Theorem 14 there would exist a Jordan chain of 𝐹 of length ≥ 2,
a contradiction. Therefore, 𝜆0 is an eigenvalue of 𝐹 of finite algebraic multiplicity and a
semisimple eigenvalue of 𝐹 . This completes the proof.
Another important result needed in this thesis is given by the next proposition.
Proposition 16 Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and let 𝑓 ∈ 𝒪𝑛×𝑛 (𝑧0 ) be such that 𝑓 (𝑧0 ) = 𝜆0 . Define
𝐺(𝑧) := 𝐹 (𝑓 (𝑧)) for all 𝑧 such that the composition is well-defined. Then 𝐺 ∈ 𝒪𝑛×𝑛 (𝑧0 )
and 𝜆0 is an eigenvalue of 𝐹 if and only if 𝑧0 is an eigenvalue of 𝐺. Moreover, if 𝜆0 is an
eigenvalue of 𝐹 and 𝑓 ′ (𝑧0 ) ∕= 0 then
1. 𝜆0 is an eigenvalue of 𝐹 of finite algebraic multiplicity if and only if 𝑧0 is an eigenvalue
of 𝐺 of finite algebraic multiplicity.
2. Let 𝜆0 be an eigenvalue of 𝐹 of finite algebraic multiplicity. Then the geometric, partial,
and algebraic multiplicities of the eigenvalue 𝜆0 of 𝐹 and of the eigenvalue 𝑧0 of 𝐺 are
in one-to-one correspondence.
Proof.
Let 𝐹 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and let 𝑓 ∈ 𝒪𝑛×𝑛 (𝑧0 ) such that 𝑓 (𝑧0 ) = 𝜆0 . It follows there
exists an 𝜖 > 0 such that 𝐹 ∈ 𝒪(𝐵(𝜆0 , 𝜖), 𝑀𝑛 (ℂ)). Thus there exists a 𝛿 > 0 such that
𝑓 ∈ 𝒪(𝐵(𝜆0 , 𝛿), ℂ)) and if 𝑧 ∈ 𝐵(𝜆0 , 𝛿) then 𝑓 (𝑧) ∈ 𝐵(𝜆0 , 𝜖). It then follows 𝐺(𝑧) = 𝐹 (𝑓 (𝑧))
60
is well-defined on 𝐵(𝜆0 , 𝛿) and 𝐺′ (𝑧) = 𝑓 ′ (𝑧)𝐹 ′ (𝑓 (𝑧)) for every 𝑧 ∈ 𝐵(𝜆0 , 𝛿). Thus 𝐺 ∈
𝒪(𝐵(𝑧0 , 𝛿), 𝑀𝑛 (ℂ)) and so 𝐺 ∈ 𝒪𝑛×𝑛 (𝑧0 ). It follows from the fact det 𝐹 (𝜆0 ) = det 𝐺(𝑧0 )
that 𝜆0 is an eigenvalue of 𝐹 if and only if 𝑧0 is an eigenvalue of 𝐺.
Suppose 𝜆0 is an eigenvalue of 𝐹 . Then by the local Smith form, Theorem 9, there exists
𝑁, 𝑀 ∈ 𝒪𝑛×𝑛 (𝜆0 ) and an 𝑟 > 0 with 𝑟 ≤ 𝜖 such that 𝑁 (𝜆0 ), 𝑀 (𝜆0 ) are invertible matrices
and
𝐹 (𝜆) = 𝑁 (𝜆) diag{0𝜇 , (𝜆 − 𝜆0 )𝑚𝜇+1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 𝐼𝑛−𝑔 }𝑀 (𝜆),
for every 𝜆 ∈ 𝐵(𝜆0 , 𝑟), where 0 ≤ 𝜇 ≤ 𝑔 ≤ 𝑛, 0𝜇 is the 𝜇 × 𝜇 zero matrix, 𝐼𝑛−𝑔 is the
(𝑛−𝑔)×(𝑛−𝑔) identity matrix, the number of zeros down the diagonal is 𝜇, when 𝜇 = 𝑔 there
are no (𝜆 − 𝜆0 ) terms appearing in the diagonal, when 𝜇 < 𝑔 we have 𝑚𝜇+1 ≥ . . . ≥ 𝑚𝑔 ≥ 1
with {𝑚𝜇+𝑗 }𝑔𝑗=1 ⊆ ℕ, and when 𝑔 = 𝑛 there are no ones appearing in the diagonal.
Again we can find a 𝛿 1 ≤ 𝛿 such that 𝑓 (𝑧) ∈ 𝐵(𝜆0 , 𝑟) for 𝑧 ∈ 𝐵(𝑧0 , 𝛿 1 ). But from what we
˜ (𝑧) := 𝑁 (𝑓 (𝑧)) and 𝑀
˜ (𝑧) := 𝑀 (𝑓 (𝑧)), for all 𝑧 such that the
just proved this implies for 𝑁
˜, 𝑀
˜ ∈ 𝒪𝑛×𝑛 (𝑧0 ) with 𝑁
˜ (𝑧0 ) = 𝑁 (𝜆0 ), 𝑀
˜ (𝑧0 ) = 𝑀 (𝜆0 )
composition is well-defined, that 𝑁
invertible matrices such that
𝐺(𝑧) = 𝐹 (𝑓 (𝑧))
˜ (𝑧) diag{0𝜇 , (𝑓 (𝑧) − 𝜆0 )𝑚𝜇+1 , . . . , (𝑓 (𝑧) − 𝜆0 )𝑚𝑔 , 𝐼𝑛−𝑔 }𝑀
˜ (𝑧)
=𝑁
for every 𝑧 ∈ 𝐵(𝑧0 , 𝛿 1 ).
Now since 𝑓 ′ (𝑧0 ) ∕= 0 then for the function 𝑓0 (𝑧) :=
𝑓 (𝑧)
𝑧−𝑧0
for 𝑧 ∈ 𝐵(𝑧0 , 𝛿 1 ) we have
𝒪(𝐵(𝑧0 , 𝛿 1 ), ℂ) with 𝑓0 (𝑧0 ) = 𝑓 ′ (𝑧0 ) ∕= 0. Thus there exists a 𝛿 2 > 0 with 𝛿 2 ≤ 𝛿 1 such
that 𝑓0−1 (𝑧) :=
𝑧−𝑧0
𝑓 (𝑧)
with 𝑧 ∈ 𝐵(𝑧0 , 𝛿 2 ) is well-defined and 𝑓0−1 ∈ 𝒪(𝐵(𝑧0 , 𝛿 2 ), ℂ).
61
ˆ (𝑧) for 𝑧 ∈ 𝐵(𝑧0 , 𝛿 2 ) by
Define the matrix 𝑀
˜ (𝑧).
ˆ (𝑧) = diag{𝐼𝜇 , 𝑓0 (𝑧)𝑚𝜇+1 , . . . , 𝑓0 (𝑧)𝑚𝑔 , 𝐼𝑛−𝑔 }𝑀
𝑀
ˆ ∈ 𝒪(𝐵(𝑧0 , 𝛿 2 ), 𝑀𝑛 (ℂ)) with 𝑀
ˆ (𝑧0 ) an invertible matrix and 𝑀
ˆ (𝑧)
It follows from this that 𝑀
is invertible for every 𝑧 ∈ 𝐵(𝑧0 , 𝛿 2 ). Moreover, since we have (𝑓 (𝑧) − 𝜆0 )𝑗 = (𝑧 − 𝑧0 )𝑗 𝑓0 (𝑧)𝑗 ,
it follows that
𝐺(𝑧) = 𝐹 (𝑓 (𝑧))
˜ (𝑧) diag{0𝜇 , (𝑓 (𝑧) − 𝜆0 )𝑚𝜇+1 , . . . , (𝑓 (𝑧) − 𝜆0 )𝑚𝑔 , 𝐼𝑛−𝑔 }𝑀
˜ (𝑧)
=𝑁
˜ (𝑧) diag{0𝜇 , (𝑧 − 𝑧0 )𝑚𝜇+1 , . . . , (𝑧 − 𝑧0 )𝑚𝑔 , 𝐼𝑛−𝑔 }𝑀
ˆ (𝑧)
=𝑁
for every 𝑧 ∈ 𝐵(𝑧0 , 𝛿 2 ).
Therefore by the uniqueness portion of Theorem 9 it follows that the local Smith form of 𝐹
corresponding to the eigenvalue 𝜆0 is
diag{0𝜇 , (𝜆 − 𝜆0 )𝑚𝜇+1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 𝐼𝑛−𝑔 }
and the local Smith form of 𝐺 corresponding to the eigenvalue 𝑧0 is
diag{0𝜇 , (𝑧 − 𝑧0 )𝑚𝜇+1 , . . . , (𝑧 − 𝑧0 )𝑚𝑔 , 𝐼𝑛−𝑔 }.
The proof of the proposition now follows from the uniqueness of those local Smith forms.
The following example arises in our thesis in Floquet theory when studying the Jordan normal
form of the monodromy matrix for canonical systems of periodic differential equations with
period 𝑑.
62
Example 4. Let 𝐴 ∈ 𝑀𝑛 (ℂ) and define 𝐹 (𝜆) := 𝐴 − 𝜆𝐼𝑛 for 𝜆 ∈ 𝑀𝑛 (ℂ). Define 𝐺(𝑘) :=
𝐹 (𝑒𝑖𝑘𝑑 ) = 𝐴−𝑒𝑖𝑘𝑑 𝐼𝑛 , for 𝑘 ∈ ℂ. Let 𝜆0 be an eigenvalue of the matrix 𝐴. Then we know from
example 1 that if we let 𝑔, 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 , 𝑚 := 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 denote the geometric, partial,
and algebraic multiplicities, respectively, of the eigenvalue 𝜆0 of 𝐹 then the Jordan normal
form of the matrix 𝐴 corresponding to the eigenvalue 𝜆0 has exactly 𝑔 Jordan blocks of order
𝑚1 , . . . , 𝑚𝑔 , respectively, with eigenvalue 𝜆0 . Now let 𝑘0 ∈ ℂ such that 𝑒𝑖𝑘0 𝑑 = 𝜆0 . Then,
since
𝑑 𝑖𝑘𝑑
𝑒
𝑑𝑘
= 𝑖𝑑𝑒𝑖𝑘𝑑 ∕= 0 for any 𝑘, it follows from Proposition 16 that 𝑘0 is an eigenvalue of 𝐺
with finite algebraic multiplicity such that its geometric, partial, and algebraic multiplicities
are 𝑔, 𝑚1 , . . . , 𝑚𝑔 and 𝑚, respectively. By the same proposition the converse is also true.
We conclude by noting that it can be helpful sometimes to study the spectrum of the matrix
𝐴 by studying the spectrum of the holomorphic matrix function 𝐺(𝑘) := 𝐴 − 𝑒𝑖𝑘𝑑 𝐼𝑛 . This
perspective is found useful when considering perturbation problems such as when 𝐴 := 𝐴(𝜔)
depends on a perturbation parameter 𝜔.
3.2
On Holomorphic Matrix-Valued Functions of Two
Variables
We introduce the following notation and convention for this section:
(i) 𝒪(𝑈, 𝐸) := {𝑓 : 𝑈 → 𝐸) ∣ 𝑓 is holomorphic on 𝑈 } where 𝑈 is an open connected set
in ℂ2 , (𝐸, ∣∣ ⋅ ∣∣) is a Banach space, and a function 𝑓 : 𝑈 → 𝐸 is said to be holomorphic
on 𝑈 provided the partial derivatives of 𝑓 exist at each point in 𝑈 , i.e., the limits
∂𝑓 := 𝑓𝜆 (𝜆0 , 𝑧0 ) := lim (𝜆 − 𝜆0 )−1 (𝑓 (𝜆, 𝑧0 ) − 𝑓 (𝜆0 , 𝑧0 ))
𝜆→𝜆0
∂𝜆 (𝜆,𝑧)=(𝜆0 ,𝑧0 )
∂𝑓 := 𝑓𝑧 (𝜆0 , 𝑧0 ) := lim (𝑧 − 𝑧0 )−1 (𝑓 (𝜆0 , 𝑧) − 𝑓 (𝜆0 , 𝑧0 ))
𝑧→𝑧0
∂𝑧 (𝜆,𝑧)=(𝜆0 ,𝑧0 )
63
exist for every (𝜆0 , 𝑧0 ) ∈ 𝑈 .
(ii) 𝒪((𝜆0 , 𝑧0 )) :=
∪
𝒪(𝐵((𝜆0 , 𝑧0 ), 𝑟), ℂ) – the set of scalar functions holomorphic at
𝑟>0
(𝜆0 , 𝑧0 ) ∈ ℂ2 .
(iii) 𝒪𝑚×𝑛 ((𝜆0 , 𝑧0 )) :=
∪
𝒪(𝐵((𝜆0 , 𝑧0 ), 𝑟), 𝑀𝑚,𝑛 (ℂ)) – the set of 𝑚 × 𝑛 matrix-valued func-
𝑟>0
tions holomorphic at (𝜆0 , 𝑧0 ) ∈ ℂ2 .
(iv) As convention we identify 𝒪1×1 ((𝜆0 , 𝑧0 )) with 𝒪((𝜆0 , 𝑧0 )).
(v) If 𝑓1 , 𝑓2 ∈ 𝒪𝑚×𝑛 ((𝜆0 , 𝑧0 )) then we use the notation 𝑓1 = 𝑓2 to mean there exists
an 𝑟 > 0 such that 𝑓1 , 𝑓2 ∈ 𝒪(𝐵((𝜆0 , 𝑧0 ), 𝑟), 𝑀𝑚,𝑛 (ℂ)) and 𝑓1 (𝜆, 𝑧) = 𝑓2 (𝜆, 𝑧) for
∣∣(𝜆, 𝑧) − (𝜆0 , 𝑧0 )∣∣ℂ < 𝑟.
The following is an important characterization of holomorphic functions as analytic functions:
Lemma 17 If 𝑓 ∈ 𝒪𝑚×𝑛 ((𝜆0 , 𝑧0 )) then 𝑓 (𝜆, 𝑧) is analytic at (𝜆, 𝑧) = (𝜆0 , 𝑧0 ), that is,
𝑗1 +𝑗2
the partial and mixed partial derivatives 𝑓 (𝑗1 ,𝑗2 ) (𝜆0 , 𝑧0 ) := ∂∂𝑗1 𝜆∂ 𝑗2𝑓𝑧 (𝜆,𝑧)=(𝜆0 ,𝑧0 ) exist for every 𝑗1 , 𝑗2 ∈ ℕ ∪ {0} and there exists an 𝑟 > 0 such that
𝑓 (𝜆, 𝑧) =
∞
∑
𝑗1 ,𝑗2
1
𝑓 (𝑗1 ,𝑗2 ) (𝜆0 , 𝑧0 )(𝜆 − 𝜆0 )𝑗1 (𝑧 − 𝑧0 )𝑗2 , ∣𝜆 − 𝜆0 ∣ < 𝑟, ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑗
!𝑗
!
=0 1 2
where the power series on the right converges absolutely in the 𝑀𝑚,𝑛 (ℂ) norm to 𝑓 (𝜆, 𝑧) for
(𝜆, 𝑧) ∈ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟).
Proof. The space 𝑀𝑚,𝑛 (ℂ) is finite dimensional and since all norms on a finite dimensional
space are equivalent, we need only show that the statement is true for the norm given by
∣∣[𝑎𝑖,𝑗 ]𝑚,𝑛
[𝑎𝑖,𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∣∣∞ := max ∣𝑎𝑖,𝑗 ∣,
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (ℂ).
𝑖,𝑗
64
Thus it follows from this that 𝒪𝑚×𝑛 ((𝜆0 , 𝑧0 )) = 𝑀𝑚,𝑛 (𝒪((𝜆0 , 𝑧0 ))) and the statement will be
true if we can show that its true for every 𝑓 ∈ 𝒪((𝜆0 , 𝑧0 )). But this statement for functions
in 𝒪((𝜆0 , 𝑧0 )) is a well known fact from the theory of functions of several complex variables
[17, p. 39, Theorem 3.2] and follows from a deep theorem of Hartogs (see [17, p. 21, Theorem
1.6]).
In the spectral perturbation theory of holomorphic matrix functions, the following theorem
plays a central role:
Theorem 18 (Weierstrass Preparation Theorem) If 𝑓 ∈ 𝒪((𝜆0 , 𝑧0 )), 𝑓 (𝜆0 , 𝑧0 ) = 0,
and 𝑓 ∕= 0, then there exists an 𝑟 > 0 such that
𝑓 (𝜆, 𝑧) = (𝑧 − 𝑧0 )𝜇 𝑓0 (𝜆, 𝑧)𝑓1 (𝜆, 𝑧),
𝑓1 (𝜆, 𝑧) ∕= 0, and
𝑓0 (𝜆, 𝑧) = (𝜆 − 𝜆0 )𝑚 + 𝑎1 (𝑧)(𝜆 − 𝜆0 )𝑚−1 + ⋅ ⋅ ⋅ + 𝑎𝑚−1 (𝑧)(𝜆 − 𝜆0 ) + 𝑎𝑚 (𝑧)
(3.23)
(3.24)
for every (𝜆, 𝑧) ∈ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟). Here 𝜇, 𝑚 ∈ ℕ ∪ {0}, 𝑎𝑗 ∈ 𝒪(𝐵(𝑧0 , 𝑟), ℂ) with
𝑎𝑗 (𝑧0 ) = 0 for 𝑗 = 1, . . . , 𝑚 − 1, and 𝑓1 ∈ 𝒪(𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟), ℂ).
Proof. The proof of this statement follows from [17, p. 55, §I.4, Theorem 4.1].
It follows from this that to study the zero set of the function 𝑓 ∈ 𝒪((𝜆0 , 𝑧0 )) near (𝜆0 , 𝑧0 )
we need to study the zero set of the monic polynomial 𝑓0 in 𝜆 (if 𝑚 ≥ 1) given by (3.24),
whose coefficients 𝑎1 , . . . , 𝑎𝑚 are in 𝒪(𝑧0 ) and satisfy 𝑎1 (𝑧0 ) = ⋅ ⋅ ⋅ = 𝑎𝑚 (𝑧0 ) = 0.
Hence we define the set
𝒪0 ((𝜆0 , 𝑧0 )) := {𝑓 ∈ 𝒪((𝜆0 , 𝑧0 )) ∣ 𝑓 has the form in (3.24)}.
65
(3.25)
Let 𝑓 ∈ 𝒪0 ((𝜆0 , 𝑧0 )). Then 𝑓 (𝜆, 𝑧0 ) = (𝜆 − 𝜆0 )𝑚 for some 𝑚 ∈ ℕ ∪ {0}. We call deg 𝑓 := 𝑚
the degree of 𝑓 .
A function 𝑓 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) with 𝑓 ∕= 1 is said to be irreducible if whenever 𝑓 = 𝑝1 𝑝2 with
𝑝1 , 𝑝2 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) then 𝑝1 = 1 or 𝑝2 = 1.
Theorem 19 Every 𝑓 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) with 𝑓 ∕= 1, factors uniquely into a finite product of
irreducibles, i.e., there exists unique irreducibles 𝑝1 , . . . , 𝑝𝑙 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) (not necessarily
distinct) such that
𝑓=
𝑙
∏
𝑝𝑗 .
(3.26)
𝑗=1
Furthermore, deg 𝑓 = deg 𝑝1 + ⋅ ⋅ ⋅ + deg 𝑝𝑙 . Moreover, there exists an 𝑟 > 0 such that for
each 𝑗 = 1, . . . , 𝑙 and for each fixed 𝑧 satisfying 0 < ∣𝑧 − 𝑧0 ∣ < 𝑟, the polynomial 𝑝𝑗 (𝜆, 𝑧) has
only simple roots.
Proof.
The uniqueness of a factorization into irreducibles follows from [4, p. 397, §A.2.6,
Theorem 1] and our choice of notation in A.2.(v). From the definition of the degree of an
element from 𝒪0 ((𝜆0 , 𝑧0 )) we have
(𝜆 − 𝜆0 )
deg 𝑓
= 𝑓 (𝜆, 𝑧0 ) =
𝑙
∏
𝑙
∏
𝑝𝑗 (𝜆, 𝑧0 ) =
(𝜆 − 𝜆0 )deg 𝑝𝑗 = (𝜆 − 𝜆0 )deg 𝑝1 +⋅⋅⋅+deg 𝑝𝑙
𝑗=1
𝑗=1
implying deg 𝑓 = deg 𝑝1 + ⋅ ⋅ ⋅ + deg 𝑝𝑙 . The latter part of this theorem follows from [4, p.
399, §A.3.2, Theorem 2].
Definition 11 Let 𝑞 ∈ ℕ and 𝑟 > 0. If 𝑣 ∈ 𝒪(𝐵(0,
√
𝑞
𝑟), 𝑀𝑚,𝑛 (ℂ)), where
√
𝑞
𝑟 denotes the
real 𝑞th root of 𝑟, and is given by the convergent power series
𝑣(𝜀) =
∞
∑
𝑐𝑗 𝜀𝑗 , ∣𝜀∣ <
𝑗=0
66
√
𝑞
𝑟
(3.27)
then the multi-valued function
1
𝑢(𝑧) := 𝑣((𝑧 − 𝑧0 ) 𝑞 ) =
∞
∑
𝑗
𝑐𝑗 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
(3.28)
𝑗=0
is called a convergent Puiseux series expanded about 𝑧0 with limit point 𝑐0 , domain
𝐵(𝑧0 , 𝑟), and period 𝑞.
Fix any branch of the 𝑞th root function and denote its evaluation at (𝑧 − 𝑧0 ) by (𝑧 − 𝑧0 )1/𝑞
1
(e.g., using the principal branch of the 𝑞th root (𝑧 − 𝑧0 )1/𝑞 = ∣𝑧 − 𝑧0 ∣1/𝑞 𝑒 𝑞 𝑖arg(𝑧−𝑧0 ) where
−𝜋 < arg(𝑧 −𝑧0 ) ≤ 𝜋). Then the single-valued functions 𝑢0 , . . . , 𝑢𝑞−1 given by the convergent
series
ℎ
𝑢ℎ (𝑧) := 𝑣(𝜁 (𝑧 − 𝑧0 )
1/𝑞
)=
∞
∑
𝑐𝑗 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑗 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞 − 1, (3.29)
𝑗=0
where 𝜁 is any primitive 𝑞th root of unity (e.g., 𝜁 = 𝑒𝑖2𝜋/𝑞 ), are called the branches of the
Puiseux series 𝑢. The values of the Puiseux series 𝑢 at a point 𝑧 ∈ 𝐵(𝑧0 , 𝑟) are the values
of its branches at this point, namely, 𝑢0 (𝑧), . . . , 𝑢𝑞−1 (𝑧). The graph of the Puiseux series 𝑢
∪
is the union of the graphs of its branches, namely, 𝑞−1
ℎ=0 {(𝑧, 𝑢ℎ (𝑧)) : ∣𝑧 − 𝑧0 ∣ < 𝑟}.
Theorem 20 Let 𝑓 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) with 𝑓 ∕= 1. Let 𝑓 =
𝑙
∏
𝑝𝑗 be a factorization of 𝑓 into
𝑗=1
unique irreducibles 𝑝1 , . . . , 𝑝𝑙 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) (not necessarily distinct) with deg 𝑓 = deg 𝑝1 +
⋅ ⋅ ⋅ + deg 𝑝𝑙 and 𝑟 > 0 be such that for each 𝑗 = 1, . . . , 𝑙 and for each fixed 𝑧 satisfying
0 < ∣𝑧 − 𝑧0 ∣ < 𝑟, the polynomial 𝑝𝑗 (𝜆, 𝑧) has only simple roots. Then there exists 𝑙 convergent
Puiseux series 𝜆1 (𝑧), . . . , 𝜆𝑙 (𝑧) expanded about 𝑧0 with limit point 𝜆0 , domain 𝐵(𝑧0 , 𝑟), and
periods deg 𝑝1 , . . . , deg 𝑝𝑙 , respectively, such that for every (𝜆, 𝑧) ∈ 𝐵(𝑧0 , 𝑟) × ℂ,
deg 𝑝𝑗 −1
𝑝𝑗 (𝜆, 𝑧) =
∏
(𝜆 − 𝜆𝑗,ℎ (𝑧)),
ℎ=0
67
𝑗 = 1, . . . , 𝑙
(3.30)
and
𝑝𝑗 −1
𝑙 deg∏
∏
𝑓 (𝜆, 𝑧) =
(𝜆 − 𝜆𝑗,ℎ (𝑧)),
𝑗=1
(3.31)
ℎ=0
where 𝜆𝑗,ℎ (𝑧), ℎ = 0, . . . , deg 𝑝𝑗 − 1 denotes all the branches of the Puiseux series 𝜆𝑗 (𝑧), for
𝑗 = 1, . . . , 𝑙.
Proof. This statement follows from [4, p. 406, §A.5.4, Theorem 3].
3.3
On the Perturbation Theory for Holomorphic Matrix Functions
In this section, we consider a holomorphic matrix-valued function of two variables 𝐿 ∈
𝒪𝑛×𝑛 ((𝜆0 , 𝑧0 )) which satisfies det(𝐿(𝜆0 , 𝑧0 )) = 0. For such a function, we can consider
𝐿(𝜆, 𝑧) as a family, indexed by a perturbation parameter 𝑧, of holomorphic matrix functions
in the variable 𝜆. We will give basic facts regarding the dependency on 𝑧 of the eigenvalues
𝜆(𝑧) and corresponding eigenvectors 𝜑(𝑧) of the holomorphic matrix function 𝐿(𝜆, 𝑧) when
(𝜆, 𝑧) is near (𝜆0 , 𝑧0 ). In order to avoid confusion we denote by 𝐿(⋅, 𝑧), the holomorphic
matrix function 𝐿(𝜆, 𝑧) in the variable 𝜆 for a fix 𝑧.
3.3.1
Eigenvalue Perturbations
We begin by defining the spectrum for a holomorphic matrix-valued function of two variables
in a manner similar to the one variable case.
Definition 12 If 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) then the spectrum of 𝐿, denoted by 𝜎(𝐿), is the subset
68
of 𝑈 such that the matrix 𝐿(𝜆, 𝑧) is not invertible, i.e.,
𝜎(𝐿) := {(𝜆, 𝑧) ∈ 𝑈 ∣ det 𝐿(𝜆, 𝑧) = 0}.
(3.32)
Notice that (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿) if and only if 𝜆0 ∈ 𝜎(𝐿(⋅, 𝑧0 )), that is, 𝜆0 is an eigenvalue of the
holomorphic matrix function 𝐿(⋅, 𝑧0 ) ∈ 𝒪(𝜆0 ). Thus the spectral theory developed in the
previous sections for holomorphic matrix functions applies.
Conceptually, we can consider the eigenvalues 𝜆(𝑧) of 𝐿(⋅, 𝑧) as a multi-valued function of 𝑧
implicitly defined by the equation det 𝐿(𝜆, 𝑧) = 0. Thus it is useful to introduce the notation
𝜎(𝐿)−1 for its graph, i.e.,
𝜎(𝐿)−1 := {(𝑧, 𝜆) ∣ (𝜆, 𝑧) ∈ 𝜎(𝐿)}.
(3.33)
With this in mind, the next theorem can be interpreted as a description of the local analytic
properties of this multi-valued function.
Theorem 21 Suppose 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) such that (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿). If 𝜆0 an eigenvalue
of 𝐿(⋅, 𝑧0 ) with algebraic multiplicity 𝑚 then for any 𝜀 > 0 there exists an 𝛿 > 0 such
that 𝜎(𝐿)−1 ∩ 𝐵(𝑧0 , 𝛿) × 𝐵(𝜆0 , 𝜀) is the union of the graphs of 𝑙 convergent Puiseux series
𝜆1 (𝑧), . . . , 𝜆𝑙 (𝑧) expanded about 𝑧0 with limit point 𝜆0 , domain 𝐵(𝑧0 , 𝛿), and periods 𝑞1 , . . . , 𝑞𝑙 ,
respectively, which satisfy 𝑚 = 𝑞1 + ⋅ ⋅ ⋅ + 𝑞𝑙 .
Proof.
First, 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) implies 𝑓 := det(𝐿) ∈ 𝒪(𝑈, ℂ) since the determinant of
a matrix is just a polynomial in its entries. Hence 𝜎(𝐿) is the zero set of the holomorphic
function 𝑓 : 𝑈 → ℂ and since (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿) then 𝑓 (𝜆0 , 𝑧0 ) = 0. Suppose 𝜆0 an eigenvalue of 𝐿(⋅, 𝑧0 ) with algebraic multiplicity 𝑚. Then by Proposition 10 the order of the
zero of 𝑓 (𝜆, 𝑧0 ) = det(𝐿(𝜆, 𝑧0 )) at 𝜆 = 𝜆0 is 𝑚 and so 𝑓 (⋅, 𝑧0 ) ∕= 0. By the Weierstrass
69
Preparation Theorem, Theorem 18 above, there exists 𝑓0 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) with degree 𝑚 and
𝑓1 ∈ 𝒪((𝜆0 , 𝑧0 )) with 𝑓1 (𝜆0 , 𝑧0 ) ∕= 0 such that 𝑓 = 𝑓0 𝑓1 in 𝒪((𝜆0 , 𝑧0 )). By Theorem 19,
there exists unique irreducibles 𝑝1 , . . . , 𝑝𝑙 ∈ 𝒪0 ((𝜆0 , 𝑧0 )) (not necessarily distinct) such that
in 𝒪((𝜆0 , 𝑧0 )) we have
𝑓 = 𝑓1 𝑓0 = 𝑓1
𝑙
∏
𝑝𝑗 .
𝑗=1
Moreover, by the same theorem, deg 𝑓0 = deg 𝑝1 + ⋅ ⋅ ⋅ + deg 𝑝𝑙 and there exists an 𝑟 > 0 such
that for each 𝑗 = 1, . . . , 𝑙 and for each fixed 𝑧 satisfying 0 < ∣𝑧 − 𝑧0 ∣ < 𝑟, the polynomial
𝑝𝑗 (𝜆, 𝑧) has only simple roots. By taking a smaller 𝑟, we may also assume without loss of
generality that 𝑓, 𝑓1 , 𝑝1 , . . . , 𝑝𝑙 ∈ 𝒪(𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟), ℂ) and
𝑓 (𝜆, 𝑧) = 𝑓1 (𝜆, 𝑧)
𝑙
∏
𝑝𝑗 (𝜆, 𝑧),
𝑓1 (𝜆, 𝑧) ∕= 0,
𝑗=1
for every (𝜆, 𝑧) ∈ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟). We now define the numbers 𝑞𝑗 := deg 𝑝𝑗 , 𝑗 = 1, . . . , 𝑙.
Thus 𝑚 = deg 𝑓0 = deg 𝑝1 + ⋅ ⋅ ⋅ + deg 𝑝𝑙 = 𝑞1 + ⋅ ⋅ ⋅ + 𝑞𝑙 . By Theorem 20 there exists 𝑙
convergent Puiseux series 𝜆1 (𝑧), . . . , 𝜆𝑙 (𝑧) expanded about 𝑧0 with limit point 𝜆0 , domain
𝐵(𝑧0 , 𝑟), and periods 𝑞1 , . . . , 𝑞𝑙 , respectively, such that
𝑞𝑗 −1
𝑝𝑗 (𝜆, 𝑧) =
∏
(𝜆 − 𝜆𝑗,ℎ (𝑧)),
𝑗 = 1, . . . , 𝑙
ℎ=0
where 𝜆𝑗,ℎ (𝑧), ℎ = 0, . . . , 𝑞𝑗 − 1 denotes all the branches of the Puiseux series 𝜆𝑗 (𝑧), for
𝑗 = 1, . . . , 𝑙. This implies
𝑗 −1
𝑙 𝑞∏
∏
det(𝐿(𝜆, 𝑧)) = 𝑓1 (𝜆, 𝑧)
(𝜆 − 𝜆𝑗,ℎ (𝑧)),
𝑗=1 ℎ=0
for every (𝜆, 𝑧) ∈ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟).
70
𝑓1 (𝜆, 𝑧) ∕= 0,
Let 𝜀 > 0 be given. Then since
lim 𝜆𝑗,ℎ (𝑧) = 𝜆𝑗,ℎ (𝑧0 ) = 𝜆0 ,
𝑧→𝑧0
ℎ = 0, . . . , 𝑞𝑗 − 1,
𝑗 = 1, . . . , 𝑙,
ℎ = 0, . . . , 𝑞𝑗 − 1,
𝑗 = 1, . . . , 𝑙,
there exists 𝛿 > 0 with 𝛿 ≤ 𝑟 such that
∣𝜆𝑗,ℎ (𝑧) − 𝜆0 ∣ < min{𝜀, 𝑟},
if ∣𝑧 − 𝑧0 ∣ < 𝛿. Then, since 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝛿) ⊆ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟), we have
𝜎(𝐿) ∩ 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝛿) = 𝜎(𝐿) ∩ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟) ∩ 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝛿)
𝑗 −1
𝑙 𝑞∪
∪
=
{(𝜆𝑗,ℎ (𝑧), 𝑧) : (𝜆𝑗,ℎ (𝑧), 𝑧) ∈ 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟)} ∩ 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝛿)
𝑗=1 ℎ=0
𝑗 −1
𝑙 𝑞∪
∪
{(𝜆𝑗,ℎ (𝑧), 𝑧) ∣ 𝑧 ∈ 𝐵(𝑧0 , 𝛿) }
=
𝑗=1 ℎ=0
It follows from this that if we just take the convergent Puiseux series 𝜆1 (𝑧), . . . , 𝜆𝑙 (𝑧) to have
the domain 𝐵(𝑧0 , 𝛿) instead of 𝐵(𝑧0 , 𝑟) then we have 𝑙 convergent Puiseux series expanded
about 𝑧0 with limit point 𝜆0 , domain 𝐵(𝑧0 , 𝛿), and periods 𝑞1 , . . . , 𝑞𝑙 , respectively, which
satisfy 𝑚 = deg 𝑞1 + ⋅ ⋅ ⋅ + deg 𝑞𝑙 and such that the union of their graphs is
𝑗 −1
𝑙 𝑞∪
∪
{(𝑧, 𝜆𝑗,ℎ (𝑧)) ∣ 𝑧 ∈ 𝐵(𝑧0 , 𝛿) } = {(𝑧, 𝜆) ∣ (𝜆, 𝑧) ∈ 𝜎(𝐿) ∩ 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝛿)}
𝑗=1 ℎ=0
= 𝜎(𝐿)−1 ∩ 𝐵(𝑧0 , 𝛿) × 𝐵(𝜆0 , 𝜀).
This completes the proof.
Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and assume that (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿) and 𝜆0 an eigenvalue of 𝐿(⋅, 𝑧0 ) with
finite algebraic multiplicity. Denote by 𝑔, 𝑚1 ≥ . . . ≥ 𝑚𝑔 , and 𝑚 = 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 the
geometric multiplicity, partial multiplicities, and algebraic multiplicity, respectively of the
71
eigenvalue 𝜆0 of 𝐿(⋅, 𝑧0 ). Then it follows Theorem 21 that given an 𝜀 > 0 there exists a 𝛿 > 0
such that, for ∣𝑧 −𝑧0 ∣ < 𝛿, the spectrum of 𝐿(𝜆, 𝑧) in ∣𝜆−𝜆0 ∣ < 𝜀, that is, 𝜎(𝐿(⋅, 𝑧))∩𝐵(𝜆0 , 𝜀),
consists of 𝑚 eigenvalues and they are the values of 𝑙 convergent Puiseux series of the form
𝜆𝑗 (𝑧) = 𝜆0 +
∞
∑
𝑠
𝑐𝑗,𝑠 (𝑧 − 𝑧0 ) 𝑞𝑗 , ∣𝑧 − 𝑧0 ∣ < 𝛿,
𝑗 = 1, . . . , 𝑙
(3.34)
𝑠=1
where there periods satisfy 𝑚 = 𝑞1 + . . . + 𝑞𝑙 . Specifically, letting 𝜆𝑗,ℎ (𝑧), ℎ = 0, . . . , 𝑞𝑗 − 1
denote all the branches of the Puiseux series 𝜆𝑗 (𝑧), for 𝑗 = 1, . . . , 𝑙 then
𝑙
∪
𝜎(𝐿(⋅, 𝑧)) ∩ 𝐵(𝜆0 , 𝜀) =
{𝜆𝑗,ℎ (𝑧) : ℎ = 0, . . . , 𝑞𝑗 − 1}
(3.35)
𝑗=1
for each 𝑧 ∈ 𝐵(𝑧0 , 𝛿).
In general the only connection between the numbers 𝑙 and 𝑞1 , . . . , 𝑞𝑙 with the geometric mul∑𝑔
∑𝑙
tiplicity 𝑔 and partial multiplicities 𝑚1 , . . . , 𝑚𝑔 is the equality 𝑚 =
𝑗=1 𝑚𝑗 .
𝑗=1 𝑞𝑗 =
Because of this two important problems arise in the spectral perturbation theory of holomorphic matrix functions:
1. Find general conditions involving just the partial derivatives of 𝐿 at (𝜆0 , 𝑧0 ) up to first
order, namely, 𝐿(𝜆0 , 𝑧0 ), 𝐿𝜆 (𝜆0 , 𝑧0 ), and 𝐿𝑧 (𝜆0 , 𝑧0 ) such that we may determine the
values 𝑙, 𝑞1 , . . . , 𝑞𝑙 and, in particular, those conditions which imply 𝑙 = 𝑔 and 𝑞𝑗 = 𝑚𝑗 ,
for 𝑗 = 1, . . . , 𝑔.
2. Find formulas for the first order coefficients of those Puiseux series, i.e., formulas for
coefficient 𝑐𝑗,1 , 𝑗 = 1, . . . , 𝑙 in (3.34).
Several papers have addressed these problems (see [23, 34, 37–39]) and significant progress
was made towards their resolution.
72
3.3.2
Eigenvector Perturbations
We will now proceed to give a useful description for the dependency on 𝑧 of the perturbed
eigenvectors 𝜑(𝑧) of the holomorphic matrix function 𝐿(𝜆, 𝑧) corresponding to perturbed
eigenvalue 𝜆(𝑧) when (𝜆, 𝑧) is near a point (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿).
Definition 13 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)). We say 𝜆(𝑧) is a eigenvalue Puiseux series of
𝐿(⋅, 𝑧) provided it is a convergent Puiseux series whose values at each point 𝑧 in its domain
are eigenvalues of 𝐿(⋅, 𝑧). If
𝜆(𝑧) = 𝜆0 +
∞
∑
𝑠
𝑐𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝛿
(3.36)
𝑠=1
is a eigenvalue Puiseux series of 𝐿(⋅, 𝑧) then we say 𝜑(𝑧) is an eigenvector Puiseux series
of 𝐿(⋅, 𝑧) corresponding to the eigenvalue Puiseux series 𝜆(𝑧) provided it is a convergent
Puiseux series with
𝜑(𝑧) = 𝛽 0 +
∞
∑
𝑠
𝛽 𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟
(3.37)
𝑠=1
𝑛
where 0 < 𝑟 ≤ 𝛿, {𝛽 𝑠 }∞
𝑠=0 ⊆ ℂ , 𝛽 0 ∕= 0, and such that if we fix any branch of the 𝑞th root
function and denote its evaluation at (𝑧 − 𝑧0 ) by (𝑧 − 𝑧0 )1/𝑞 then their branches
𝜆ℎ (𝑧) = 𝜆0 +
𝜑ℎ (𝑧) = 𝛽 0 +
∞
∑
𝑠=1
∞
∑
𝑐𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞,
(3.38)
𝛽 𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞,
(3.39)
𝑠=1
where 𝜁 is any primitive 𝑞th root of unity, satisfy
𝐿(𝜆ℎ (𝑧), 𝑧)𝜑ℎ (𝑧) = 0,
𝜑ℎ (𝑧) ∕= 0,
73
ℎ = 0, . . . , 𝑞 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟.
(3.40)
We call the pair (𝜆(𝑧), 𝜑(𝑧)) an eigenpair Puiseux series of 𝐿(⋅, 𝑧).
The following lemma tells us that to every eigenvalue Puiseux series there exists a corresponding eigenvector Puiseux series.
Lemma 22 Suppose 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)). If 𝜆(𝑧) is a eigenvalue Puiseux series of 𝐿(⋅, 𝑧) then
there exists an eigenvector Puiseux series 𝜑(𝑧) of 𝐿(⋅, 𝑧) corresponding to the eigenvector
Puiseux series 𝜆(𝑧).
Proof.
Assume 𝜆(𝑧) be an eigenvalue Puiseux series of 𝐿(⋅, 𝑧). This implies for some
√
𝑞 ∈ ℕ, 𝛿 > 0, 𝑣 ∈ 𝒪(𝐵(0, 𝑞 𝛿), ℂ), and some (𝜆0 , 𝑧0 ) ∈ 𝑈 , 𝜆(𝑧) is a convergent Puiseux series
with
1
𝑞
𝜆(𝑧) = 𝑣((𝑧 − 𝑧0 ) ) = 𝜆0 +
∞
∑
𝑠
𝑐𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝛿
(3.41)
𝑠=1
and taking any fixed branch of the 𝑞th root and denoting its evaluation at 𝑧−𝑧0 by (𝑧−𝑧0 )1/𝑞 ,
its branches
ℎ
1/𝑞
𝜆ℎ (𝑧) = 𝑣(𝜁 (𝑧 − 𝑧0 )
) = 𝜆0 +
∞
∑
𝑐𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞, (3.42)
𝑠=1
where 𝜁 is any primitive 𝑞th root of unity, satisfy (𝜆ℎ (𝑧), 𝑧) ∈ 𝑈 for ∣𝑧 − 𝑧0 ∣ < 𝛿 and
det 𝐿(𝜆ℎ (𝑧), 𝑧) = 0, ∣𝑧 − 𝑧0 ∣ < 𝛿,
ℎ = 0, . . . , 𝑞 − 1,
(3.43)
since the values 𝜆ℎ (𝑧), ℎ = 0, . . . , 𝑞−1 for ∣𝑧−𝑧0 ∣ < 𝛿 are eigenvalues of 𝐿(⋅, 𝑧) by assumption.
√
We introduce a new variable 𝜀 ∈ 𝐵(0, 𝑞 𝛿). Then 𝑧(𝜀) := 𝜀𝑞 + 𝑧0 ∈ 𝐵(𝑧0 , 𝛿) is an analytic
√
function. Moreover, for each 𝜀 ∈ 𝐵(0, 𝑞 𝛿) there exists an ℎ ∈ {0, . . . , 𝑞 − 1} such that 𝜀 =
𝜁 ℎ (𝑧(𝜀)−𝑧0 )1/𝑞 . This implies (𝑣(𝜀), 𝑧(𝜀)) = (𝜆ℎ (𝑧(𝜀)), 𝑧(𝜀)) ∈ 𝑈 and hence det 𝐿(𝑣(𝜀), 𝑧(𝜀)) =
74
det 𝐿(𝜆ℎ (𝑧), 𝑧) = 0. Thus we have shown for every 𝜀 ∈ 𝐵(0,
√
𝑞
𝑟), the analytic functions 𝑣(𝜀)
and 𝑧(𝜀) satisfy (𝑣(𝜀), 𝑧(𝜀)) ∈ 𝑈 and so have ranges in the domain of the analytic function
𝐿 and satisfy
det 𝐿(𝑣(𝜀), 𝑧(𝜀)) = 0.
(3.44)
√
𝑞
𝛿) → 𝑀𝑛 (ℂ) defined by 𝐹 (𝜀) := 𝐿(𝑣(𝜀), 𝑧(𝜀)) is an
√
√
analytic function as well, i.e., 𝐹 ∈ 𝒪(𝐵(0, 𝑞 𝛿), 𝑀𝑛 (ℂ)), and hence det 𝐹 ∈ 𝒪(𝐵(0, 𝑞 𝛿), ℂ)
But this implies the function 𝐹 : 𝐵(0,
with
det 𝐹 = 0.
(3.45)
But this implies that 0 is an eigenvalue of the holomorphic matrix function 𝐹 of infinite
algebraic multiplicity. Consider the local Smith form 𝐺 of 𝐹 corresponding to the eigenvalue
0 of infinite algebraic multiplicity. By Theorem 9 it must be of the form
′
′
𝐺(𝜀) = diag{0, . . . , 0, 𝜀𝑚𝜇+1 , . . . , 𝜀𝑚𝑔 , 1, . . . , 1}
(3.46)
where 1 ≤ 𝜇 ≤ 𝑔 = dim ker 𝐹 (0) = dim ker 𝐿(𝜆0 , 𝑧0 ) ≤ 𝑛, the number of zeros down the
diagonal is 1 ≤ 𝜇 = dim ker 𝐹 (𝜀) = dim ker 𝐿(𝑣(𝜀), 𝑧(𝜀)) for 0 < 𝜀 ≪ 1, when 𝜇 = 𝑔
there are no 𝜀 terms down the diagonal, when 𝜇 < 𝑔 we have 𝑚′𝜇+1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚′𝑔 ≥ 1 with
{𝑚′𝜇+𝑗 }𝑔𝑗=1 ⊆ ℕ, and when 𝑔 = 𝑛 there are no ones appearing in the diagonal. Hence by
√
√
Theorem 9 and Lemma 12 there exists an 𝑟 > 0 with 0 < 𝑞 𝑟 ≤ 𝑞 𝛿 and matrix functions
√
𝑀, 𝑁 ∈ 𝒪(𝐵(0, 𝑞 𝑟), 𝑀𝑛 (ℂ)) such that 𝑀 (𝜀), 𝑁 (𝜀) are invertible matrices for every 𝜀 ∈
√
𝐵(0, 𝑞 𝑟) and satisfy
′
′
diag{0, . . . , 0, 𝜀𝑚𝜇+1 , . . . , 𝜀𝑚𝑔 , 1, . . . , 1} = 𝑁 (𝜀)𝐹 (𝜀)𝑀 (𝜀)
= 𝑁 (𝜀)𝐿(𝑣(𝜀), 𝑧(𝜀))𝑀 (𝜀), ∣𝜀∣ <
75
√
𝑞
𝑟.
(3.47)
Denote the standard orthonormal basis vectors in ℂ𝑛 by 𝑒1 , . . . , 𝑒𝑛 . Then
𝜙𝑗 (𝜀) := 𝑀 (𝜀)𝑒𝑗 ,
𝑗 = 1, . . . , 𝜇
(3.48)
√
are the first 𝜇 columns of 𝑀 (𝜀) and so are a basis for ker 𝐿(𝑣(𝜀), 𝑧(𝜀)) for every 𝜀 ∈ 𝐵(0, 𝑞 𝑟).
√
√
Hence for 𝑗 = 1, . . . , 𝜇 we have 𝜙𝑗 ∈ 𝒪(𝐵(0, 𝑞 𝑟), ℂ𝑛 ) and 𝜙𝑗 (𝜀) ∕= 0 for every 𝜀 ∈ 𝐵(0, 𝑞 𝑟).
In particular, 𝜙1 ∈ 𝒪(𝐵(0,
√
𝑞
𝑟), ℂ𝑛 ) and satisfies
𝜑1 (𝜀) ∕= 0, ∣𝜀∣ <
𝐿(𝑣(𝜀), 𝑧(𝜀))𝜑1 (𝜀) = 0,
√
𝑞
𝑟.
(3.49)
√
𝑞
𝑟), ℂ𝑛 ) it follows that it has a power series expansion about the
√
point 𝑧0 which converges for ∣𝜀∣ < 𝑞 𝑟. Thus we have
Now since 𝜙1 ∈ 𝒪(𝐵(0,
𝜙1 (𝑧) = 𝛽 0 +
∞
∑
𝛽 𝑠 𝜀𝑠 , ∣𝜀∣ <
√
𝑞
𝑟.
(3.50)
𝑠=1
𝑛
𝑛
where {𝛽 𝑠 }∞
𝑠=0 ⊆ ℂ , 𝛽 0 ∕= 0, and the series converges absolutely in the ℂ norm for every
√
𝜀 ∈ 𝐵(0, 𝑞 𝑟). It follows that a convergent Puiseux series 𝜑(𝑧) is given by
1
𝑞
𝜑(𝑧) = 𝜙1 ((𝑧 − 𝑧0 ) ) = 𝛽 0 +
∞
∑
𝑠
𝛽 𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟.
(3.51)
𝑠=1
We will now show that 𝜑(𝑧) is an eigenvector Puiseux series of 𝐿(⋅, 𝑧) corresponding to
the eigenvalue Puiseux series 𝜆(𝑧). Fix any branch of the 𝑞th root function denoting its
evaluation at (𝑧 − 𝑧0 ) by (𝑧 − 𝑧0 )1/𝑞 and let 𝜁 be any primitive 𝑞th root of unity. Now if
√
√
∣𝑧 − 𝑧0 ∣ < 𝑟 then 𝜀ℎ (𝑧) := 𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 ∈ 𝐵(0, 𝑞 𝑟) ⊆ 𝐵(0, 𝑞 𝛿) for ℎ = 0, . . . , 𝑞 and so it
follows that the branches of 𝜆(𝑧) and 𝜑(𝑧) are given by the convergent series
𝜆ℎ (𝑧) = 𝜆0 +
∞
∑
𝑐𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
𝑠=1
76
ℎ = 0, . . . , 𝑞,
(3.52)
𝜑ℎ (𝑧) = 𝛽 0 +
∞
∑
𝛽 𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞.
(3.53)
𝑠=1
Moreover, if ∣𝑧 − 𝑧0 ∣ < 𝑟 then we have 𝜀ℎ (𝑧) = 𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 ∈ 𝐵(0,
√
𝑞
𝑟) ⊆ 𝐵(0,
√
𝑞
𝛿) implies
𝑧 = 𝑧(𝜀ℎ (𝑧)), 𝜆ℎ (𝑧) = 𝑣(𝜀ℎ (𝑧)), (𝑣(𝜀ℎ (𝑧)), 𝑧(𝜀ℎ (𝑧))) = (𝜆ℎ (𝑧), 𝑧) ∈ 𝑈 , 𝜑ℎ (𝑧) = 𝜙1 (𝜀ℎ (𝑧)) ∕= 0,
and
𝐿(𝜆ℎ (𝑧), 𝑧)𝜑ℎ (𝑧) = 𝐿(𝑣(𝜀ℎ (𝑧)), 𝑧(𝜀ℎ (𝑧)))𝜙1 (𝜀ℎ (𝑧)) = 0
(3.54)
for ℎ = 0, . . . , 𝑞 − 1. Therefore 𝜑(𝑧) is an eigenvector Puiseux series of 𝐿(⋅, 𝑧) corresponding
to the eigenvalue Puiseux series 𝜆(𝑧). This completes the proof.
Theorem 23 Suppose 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) such that (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿). If 𝜆0 an eigenvalue of
𝐿(⋅, 𝑧0 ) with algebraic multiplicity 𝑚 then for any 𝜀 > 0 there exists a 𝑟 > 0 and 𝑙 eigenvalue
Puiseux series (𝜆1 (𝑧), 𝜑1 (𝑧)), . . . , (𝜆𝑙 (𝑧), 𝜑𝑙 (𝑧)) expanded about 𝑧0 with domain 𝐵(𝑧0 , 𝑟) and
periods 𝑞1 , . . . , 𝑞𝑙 , respectively, satisfying 𝑚 = 𝑞1 +⋅ ⋅ ⋅+𝑞𝑙 such that if we fix any branch of the
𝑞th root function, for 𝑞 ∈ {𝑞1 , . . . , 𝑞𝑙 }, denoting its evaluation at (𝑧 − 𝑧0 ) by (𝑧 − 𝑧0 )1/𝑞 and
let 𝜁 𝑞 be any primitive 𝑞th root of unity then their branches given by the convergent series
𝜆𝑗,ℎ (𝑧) = 𝜆0 +
∞
∑
𝑐𝑗,𝑠 (𝜁 ℎ𝑞𝑗 (𝑧 − 𝑧0 )1/𝑞𝑗 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟, ℎ = 0, . . . , 𝑞𝑗 − 1,
(3.55)
𝛽 𝑗,ℎ (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞𝑗 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟,
(3.56)
𝑠=1
𝜑𝑗,ℎ (𝑧) = 𝛽 𝑗,0 +
∞
∑
ℎ = 0, . . . , 𝑞𝑗 − 1.
𝑠=1
satisfy
det 𝐿(𝜆𝑗,ℎ (𝑧), 𝑧) = 0,
ℎ = 0, . . . , 𝑞𝑗 − 1,
𝜎(𝐿) ∩ 𝐵(𝜆0 , 𝜀) × 𝐵(𝑧0 , 𝑟) =
𝑗 −1
𝑙 𝑞∪
∪
𝑗 = 1, . . . , 𝑙,
∣𝑧 − 𝑧0 ∣ < 𝑟
{(𝜆𝑗,ℎ (𝑧), 𝑧) : ∣𝑧 − 𝑧0 ∣ < 𝑟}
(3.57)
(3.58)
𝑗=1 ℎ=1
𝜎(𝐿(⋅, 𝑧)) ∩ 𝐵(𝜆0 , 𝜀) =
𝑗 −1
𝑙 𝑞∪
∪
{𝜆𝑗,ℎ (𝑧)}, ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑗=1 ℎ=1
77
(3.59)
𝐿(𝜆ℎ (𝑧), 𝑧)𝜑ℎ (𝑧) = 0,
𝜑ℎ (𝑧) ∕= 0,
ℎ = 0, . . . , 𝑞𝑗 − 1,
𝑗 = 1, . . . , 𝑙,
∣𝑧 − 𝑧0 ∣ < 𝑟. (3.60)
Proof. This theorem follows directly from Theorem 21 and Lemma 22.
Now in general the zero-order eigenvectors {𝛽 𝑗,0 }𝑔𝑗=1 ⊆ ker 𝐿(𝜆0 , 𝑧0 ) need not be a basis for
ker 𝐿(𝜆0 , 𝑧0 ), the eigenspace of 𝐿(⋅, 𝑧0 ) corresponding to the eigenvalue 𝜆0 . Because of this
two important problems arise in the spectral perturbation theory of holomorphic matrix
functions:
1. Find general conditions involving just the partial derivatives of 𝐿 at (𝜆0 , 𝑧0 ) up to first
order, namely, 𝐿(𝜆0 , 𝑧0 ), 𝐿𝜆 (𝜆0 , 𝑧0 ), and 𝐿𝑧 (𝜆0 , 𝑧0 ) such that we may determine whether
or not we can choose eigenpair Puiseux series (𝜆1 (𝑧), 𝜑1 (𝑧)), . . . , (𝜆𝑙 (𝑧), 𝜑𝑙 (𝑧)) of 𝐿(⋅, 𝑧)
as given in the above theorem so that their zero-order eigenvectors {𝜑𝑗 (𝑧0 )}𝑔𝑗=1 span
the whole eigenspace ker 𝐿(𝜆0 , 𝑧0 ).
2. Find formulas to determine the zero-order eigenvectors {𝛽 𝑗,0 }𝑔𝑗=1 .
The literature which addresses these problems is scarce and this author is only aware of
[23, 34] which addresses these problems for general perturbations of holomorphic matrix
functions. Even in the special case 𝐿(𝜆, 𝑧) = 𝐴(𝑧) − 𝜆𝐼, few results exist with the exception
of [4, p. 270, Theorem 4; pp. 310-311, Theorem 8, Corollary 2, & Corollary 3], [41, Theorem
2], [58, p. 64, Chap. 2, §7.10, Theorem 6], and [47, Theorem 2.2]. In particular, the second
problem remains open even in this case, when total splitting at the first-order approximation
(see [4, p. 311] for definition) does not occur, and seems a difficult problem to resolve.
78
3.3.3
Analytic Eigenvalues and Eigenvectors
In this section we give a survey of the main results from [23, 34] that we will need in our
thesis.
Preliminaries. We first begin with some preliminaries.
Definition 14 Let 𝐹 ∈ 𝒪(𝑈, 𝑀𝑚,𝑛 (ℂ)) where 𝑈 ⊆ ℂ is an open connected set. Define
𝑈 ∗ := {𝜆 ∣ 𝜆 ∈ 𝑈 }. The matrix function 𝐹 ∗ : 𝑈 ∗ → 𝑀𝑚,𝑛 (ℂ) defined by 𝐹 ∗ (𝜆) = 𝐹 (𝜆)∗ is
called the adjoint of the holomorphic matrix function 𝐹 .
Lemma 24 If 𝐹 ∈ 𝒪(𝑈, 𝑀𝑚,𝑛 (ℂ)) where 𝑈 ⊆ ℂ is an open connected set then 𝑈 ∗ is an
open connected set and 𝐹 ∗ ∈ 𝒪(𝑈 ∗ , 𝑀𝑚,𝑛 (ℂ)).
Proof.
If 𝐹 ∈ 𝒪(𝑈, 𝑀𝑚,𝑛 (ℂ)) then 𝐹 = [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 where 𝑎𝑖𝑗 ∈ 𝒪(𝑈, ℂ). Now obviously
𝑈 ∗ is an open connected set and from complex analysis we know 𝑎∗𝑖𝑗 (𝜆) := 𝑎𝑖𝑗 (𝜆) belongs to
∗
𝒪(𝑈 ∗ , ℂ). But this implies 𝐹 ∗ = [𝑎∗𝑗𝑖 ]𝑛,𝑚
𝑖=1,𝑗=1 ∈ 𝒪(𝑈 , 𝑀𝑚,𝑛 (ℂ)). This completes the proof.
Proposition 25 Let 𝐹 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)). Then
1. 𝜆0 ∈ ℂ is an eigenvalue of 𝐹 if and only if 𝜆0 is an eigenvalue of 𝐹 ∗ .
2. 𝜆0 is an eigenvalue of 𝐹 of finite algebraic multiplicity if and only if 𝜆0 is an eigenvalue
of 𝐹 ∗ of finite algebraic multiplicity.
3. If 𝜆0 is an eigenvalue of 𝐹 of finite algebraic multiplicity then the geometric, partial,
and algebraic multiplicities of 𝜆0 coincide with the geometric, partial, and algebraic
multiplicities, respectively, of the eigenvalue 𝜆0 of 𝐹 ∗ .
79
Proof. The first and second properties follows from the equality det 𝐹 (𝜆0 ) = det 𝐹 (𝜆0 )∗ =
det 𝐹 ∗ (𝜆0 ) for any 𝜆0 ∈ 𝑈 . To prove the third part let 𝜆0 be an eigenvalue of 𝐹 of finite
algebraic multiplicity and let 𝑔, 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 , 𝑚 := 𝑚1 + . . . + 𝑚𝑔 be its geometric, partial,
and algebraic multiplicities, respectively. Then by Theorem 9 there exists 𝑁, 𝑀 ∈ 𝒪(𝜆0 )
and an 𝑟 > 0 such that 𝑀 (𝜆0 ), 𝑁 (𝜆0 ) are invertible and for every 𝜆 ∈ 𝐵(𝜆0 , 𝑟) we have
𝐹 (𝜆) = 𝑁 (𝜆) diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}𝑀 (𝜆).
where if 𝑔 = 𝑛 there are no ones appearing in the diagonal. But this implies for any
𝜆 ∈ 𝐵(𝜆0 , 𝑟) we have
𝐹 ∗ (𝜆) = 𝑀 ∗ (𝜆) diag{(𝜆 − 𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1}𝑁 ∗ (𝜆).
where if 𝑔 = 𝑛 there are no ones appearing in the diagonal. And since 𝑀 ∗ , 𝑁 ∗ ∈ 𝒪(𝜆0 ) with
𝑀 ∗ (𝜆0 )) = 𝑀 (𝜆0 )∗ , 𝑁 ∗ (𝜆0 )) = 𝑁 (𝜆0 )∗ are invertible, this implies that 𝐺∗ (𝜆) = diag{(𝜆 −
𝜆0 )𝑚1 , . . . , (𝜆 − 𝜆0 )𝑚𝑔 , 1, . . . , 1} is the local Smith form of 𝐹 ∗ corresponding to the eigenvalue
𝜆0 . Therefore, 𝑔, 𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 , 𝑚 are the geometric, partial, and algebraic multiplicities,
respectively, of the eigenvalue 𝜆0 of 𝐹 ∗ . This completes the proof.
Definition 15 Let 𝑈 ⊆ ℂ2 . Define 𝑈 ∗ := {(𝜉, 𝜂) ∣ (𝜉, 𝜂) ∈ 𝑈 }. If 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ))
where 𝑈 is an open connected set then the matrix function 𝐿∗ : 𝑈 ∗ → 𝑀𝑛 (ℂ) defined by
𝐿∗ (𝜉, 𝜂) = 𝐿(𝜉, 𝜂)∗ is called the adjoint of the holomorphic matrix function 𝐿.
Lemma 26 If 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) where 𝑈 ⊆ ℂ2 is an open connected set then 𝑈 ∗ is an open
connected set and 𝐿∗ ∈ 𝒪(𝑈 ∗ , 𝑀𝑛 (ℂ)).
Proof.
Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) where 𝑈 ⊆ ℂ2 is an open connected set. Obviously 𝑈 ∗ is
an open connected set. The adjoint of 𝐿 is 𝐿∗ (𝜉, 𝜂) := 𝐿(𝜉, 𝜈). Now for any (𝜉 0 , 𝜂 0 ) ∈ 𝑈 ∗
80
there exist (𝜆0 , 𝑧0 ) ∈ 𝑈 such that (𝜉 0 , 𝜂 0 ) = (𝜆0 , 𝑧0 ). We define 𝐹1 (𝜉) := 𝐿(𝜉, 𝑧0 ) = 𝐿∗ (𝜉, 𝜂 0 )
and 𝐹2 (𝜂) := 𝐿(𝜆0 , 𝜂) = 𝐿∗ (𝜉 0 , 𝜂). It follows from the one variable case that 𝐹1 ∈ 𝒪(𝜉 0 ) and
𝐹2 ∈ 𝒪(𝜂 0 ). This implies the partial derivatives of 𝐿∗ (𝜉, 𝜂) exist at (𝜉 0 , 𝜂 0 ). But since this
is true for every (𝜉 0 , 𝜂 0 ) ∈ 𝑈 ∗ then 𝐿∗ ∈ 𝒪(𝑈 ∗ , ℂ). This completes the proof.
Lemma 27 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)). Then 𝜆(𝑧) ∈ 𝜎(𝐿(⋅, 𝑧)) if and only if 𝜆(𝜂) ∈ 𝜎(𝐿∗ (⋅, 𝜂)),
where 𝜂 = 𝑧. Moreover,
𝜎(𝐿∗ ) = 𝜎(𝐿)∗ .
Proof.
(3.61)
If 𝜆(𝑧) ∈ 𝜎(𝐿(⋅, 𝑧)) then 0 = det 𝐿(𝜆(𝑧), 𝑧) = det 𝐿(𝜆(𝑧), 𝑧)∗ = det 𝐿∗ (𝜆(𝑧), 𝑧) =
det 𝐿∗ (𝜆(𝜂), 𝜂) where 𝜂 = 𝑧. Hence 𝜆(𝜂) ∈ 𝜎(𝐿∗ (⋅, 𝜂)). Conversely, if 𝜆(𝜂) ∈ 𝜎(𝐿∗ (⋅, 𝜂))
then letting 𝜂 = 𝑧 we have 0 = det 𝐿∗ (𝜆(𝜂), 𝜂) = det 𝐿∗ (𝜆(𝑧), 𝑧) = det 𝐿(𝜆(𝑧), 𝑧)∗ =
det 𝐿(𝜆(𝑧), 𝑧). Hence 𝜆(𝑧) ∈ 𝜎(𝐿(⋅, 𝑧)). The latter part of this statement follows from
the identity det 𝐿∗ (𝜉, 𝜂) = det 𝐿(𝜉, 𝜂)∗ = det 𝐿(𝜉, 𝜂) for any (𝜉, 𝜂) ∈ 𝑈 ∗ . This completes the
proof.
Definition 16 Suppose that 𝜆(𝑧) is a convergent Puiseux series given by
𝜆(𝑧) =
∞
∑
𝑠
𝑐𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟
(3.62)
𝑠=0
where {𝑐𝑠 }∞
𝑠=0 ⊆ ℂ. Then a convergent Puiseux series is given by
𝜆∗ (𝜂) :=
∞
∑
𝑠
𝑐𝑠 (𝜂 − 𝑧0 ) 𝑞 , ∣𝜂 − 𝑧0 ∣ < 𝑟.
(3.63)
𝑠=0
We call the Puiseux series 𝜆∗ (𝜂) the adjoint of the Puiseux series 𝜆(𝑧).
Lemma 28 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)). Then 𝜆(𝑧) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑧) if
81
and only if 𝜆∗ (𝜂) is an eigenvalue Puiseux series of 𝐿∗ (⋅, 𝜂), where 𝜂 = 𝑧. Moreover, if 𝜆ℎ (𝑧),
ℎ = 0, . . . , 𝑞 − 1 are all the branches of 𝜆(𝑧) and 𝜆∗ℎ (𝜂), ℎ = 0, . . . , 𝑞 − 1 are all the branches
of 𝜆∗ (𝜂) then for 𝜂 = 𝑧 we have
𝑞−1
𝑞−1
∪
{𝜆∗ℎ (𝜂)}
ℎ=0
𝑞−1
∪
=
∪
{𝜆ℎ (𝜂)},
(3.64)
ℎ=0
𝑞−1
{𝜆ℎ (𝑧)} =
ℎ=0
∪
{𝜆∗ℎ (𝑧)}.
(3.65)
ℎ=0
Proof. Its obvious that
𝜆(𝑧) = 𝜆0 +
∞
∑
𝑠
𝑐𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟
(3.66)
𝑠=1
is a convergent Puiseux series if and only if its adjoint
∗
𝜆 (𝜂) := 𝜆0 +
∞
∑
𝑠
𝑐𝑠 (𝜂 − 𝑧0 ) 𝑞 , ∣𝜂 − 𝑧0 ∣ < 𝑟
(3.67)
𝑠=1
is a convergent Puiseux series since (𝜆∗ )∗ (𝑧) = 𝜆(𝑧). Denote the principal branch of the 𝑞th
√
1
√
root by 𝑞 𝑧 − 𝑧0 := 𝑞 ∣𝑧 − 𝑧0 ∣𝑒 𝑞 𝑖arg(𝑧−𝑧0 ) for 0 ≤ arg(𝑧 − 𝑧0 ) ≤ 2𝜋 and let 𝜁 := 𝑒𝑖2𝜋/𝑞 . Denote
√
1
another branch of the 𝑞th root by (𝑧 −𝑧0 )1/𝑞 := 𝑞 ∣𝑧 − 𝑧0 ∣𝑒 𝑞 𝑖arg(𝑧−𝑧0 ) for −𝜋 ≤ arg(𝑧 −𝑧0 ) < 𝜋
and notice 𝜁 = 𝑒−𝑖2𝜋/𝑞 is a primitive 𝑞th root of unity. A quick calculation shows
√
𝑞
𝑧 − 𝑧0 = (𝑧 − 𝑧0 )1/𝑞 ,
√
(𝑧 − 𝑧0 )1/𝑞 = 𝑞 𝑧 − 𝑧0 ,
for every 𝑧 ∈ ℂ.
82
(3.68)
(3.69)
Now all the branches of 𝜆(𝑧) and 𝜆∗ (𝜂) are given by the convergent series
𝜆ℎ (𝑧) = 𝜆0 +
∞
∑
√
𝑐𝑠 (𝜁 ℎ 𝑞 𝑧 − 𝑧0 )𝑠 ,
ℎ = 0, . . . , 𝑞 − 1 ∣𝑧 − 𝑧0 ∣ < 𝑟,
𝑠=1
𝜆∗ℎ (𝜂) = 𝜆0 +
∞
∑
ℎ
𝑐𝑠 (𝜁 (𝜂 − 𝑧0 )1/𝑞 )𝑠 ,
ℎ = 0, . . . , 𝑞 − 1, ∣𝜂 − 𝑧0 ∣ < 𝑟.
𝑠=1
Making the substituitions 𝜂 = 𝑧 and 𝑧 = 𝜂 as input to those branches and then using (3.68)
and (3.69), we prove the identities
𝜆ℎ (𝑧) = 𝜆∗ℎ (𝑧),
ℎ = 0, . . . , 𝑞 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟,
(3.70)
𝜆∗ℎ (𝜂) = 𝜆ℎ (𝜂),
ℎ = 0, . . . , 𝑞 − 1, ∣𝜂 − 𝑧0 ∣ < 𝑟.
(3.71)
This proves
𝑞−1
𝑞−1
𝑞−1
∪
∪
∪
{𝜆∗ℎ (𝜂)}
ℎ=0
=
{𝜆ℎ (𝜂)},
ℎ=0
𝑞−1
{𝜆ℎ (𝑧)} =
ℎ=0
∪
{𝜆∗ℎ (𝑧)}
ℎ=0
which is just the set of all values of the Puiseux series 𝜆∗ (𝜂) and 𝜆(𝑧) at 𝜂 and 𝑧, respectively,
where 𝜂 = 𝑧. And hence, since for any convergent Puiseux series with say period 𝑞, the set
of all values at a point in its domain is independent of our choice of the branch of the 𝑞th
root function and primitive 𝑞th root of unity that we use, this proves the latter part of the
statement of this lemma. It now follows this and Lemma 27 that the values of the branches
of 𝜆(𝑧) are eigenvalues of 𝐿(⋅, 𝑧) if and only if the branches of its adjoint 𝜆∗ (𝜂) are eigenvalues
of 𝐿∗ (⋅, 𝜂), where 𝜂 = 𝑧. And hence 𝜆(𝑧) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑧) if and only
if 𝜆∗ (𝜂) is an eigenvalue Puiseux series of 𝐿∗ (⋅, 𝜂), where 𝜂 = 𝑧. This completes the proof.
On Results from the Spectral Perturbation Theory. We are now ready to begin
giving some of the major results from [23, 34] that we will need in our thesis. The following
definition comes from [34].
83
Definition 17 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and (𝜆0 , 𝑧0 ) ∈ 𝑈 . Let (𝜆(𝑧), 𝜑(𝑧)) be an eigenpair
Puiseux series of 𝐿(⋅, 𝑧) such that 𝜆(𝑧) and 𝜑(𝑧) are expanded about 𝑧0 with limit point 𝜆0
and 𝛽 0 , respectively. The vector 𝛽 0 is called a generating eigenvector of 𝐿(⋅, 𝑧) (at the
point (𝜆0 , 𝑧0 ) and associated with 𝜆(𝑧)).
Remark. It should be noted that in the case 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) with (𝜆0 , 𝑧0 ) ∈ 𝑈 and 𝜆0 is
an eigenvalue of 𝐿(⋅, 𝑧0 ) of finite algebraic multiplicity, that by Theorem 23 there exists an
eigenpair Puiseux series (𝜆(𝑧), 𝜑(𝑧)) of 𝐿(⋅, 𝑧) such that 𝜆(𝑧) and 𝜑(𝑧) are expanded about
𝑧0 with limit point 𝜆0 and 𝛽 0 , respectively, for some 𝛽 0 . And hence a generating eigenvector
𝛽 0 of 𝐿(⋅, 𝑧) at the point (𝜆0 , 𝑧0 ) and associated with 𝜆(𝑧) exists. And, in particular, we
have 𝛽 0 ∕= 0 and 𝛽 0 ∈ ker(𝐿(𝜆0 , 𝑧0 )). Moreover, by Lemma 28 and Lemma 22 it follows
there exists an eigenpair Puiseux series (𝜆∗ (𝜂), 𝛾(𝜂)) of 𝐿∗ (⋅, 𝜂) such that 𝜆∗ (𝜂) and 𝛾(𝜂)
are expanded about 𝑧0 with limit point 𝜆0 and 𝛾 0 , respectively, for some 𝛾 0 . Thus 𝛾 0 is
a generating eigenvector of 𝐿∗ (⋅, 𝜂) at the point (𝜆0 , 𝑧0 ) and associated with 𝜆∗ (𝜂), where
𝜆∗ (𝜂) is the adjoint of the Puiseux series 𝜆(𝑧) and 𝐿∗ is the adjoint of 𝐿. And, in particular,
𝛾 0 ∕= 0 and 𝛾 0 ∈ ker(𝐿∗ (𝜆0 , 𝑧0 )) = ker 𝐿(𝜆0 , 𝑧0 )∗ .
The next proposition is equivalent to Lemma 7 from [34] and its proof comes from [23].
As we will shall soon see this proposition is actually one of the key results in the spectral
perturbation theory of holomorphic matrix functions.
Proposition 29 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and (𝜆0 , 𝑧0 ) ∈ 𝑈 . Let 𝜆1 (𝑧), 𝜆2 (𝑧) be any two eigenvalue Puiseux series of 𝐿 expanded about 𝑧0 with common limit point 𝜆0 . Suppose that for
some branch 𝜆1,ℎ1 (𝑧) of 𝜆1 (𝑧) and some branch 𝜆2,ℎ2 (𝑧) of 𝜆2 (𝑧) there existed an 𝑟 > 0 such
that 𝜆1,ℎ1 (𝑧) ∕= 𝜆2,ℎ2 (𝑧) for ∣𝑧 − 𝑧0 ∣ < 𝑟. Then for any generating eigenvector 𝛽 0 of 𝐿(⋅, 𝑧) at
the point (𝜆0 , 𝑧0 ) and associated with 𝜆1 (𝑧) and for any generating eigenvector 𝛾 0 of 𝐿∗ (⋅, 𝜂)
84
at the point (𝜆0 , 𝑧0 ) and associated with 𝜆∗2 (𝜂) we have
⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 , 𝛾 0 ⟩ℂ = 0.
(3.72)
Proof. By assumption there exists eigenvalue Puiseux series
𝜆1 (𝑧) =
∞
∑
𝑠
𝑐1,𝑠 (𝑧 − 𝑧0 ) 𝑞1 , ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑠=0
𝜆2 (𝑧) =
∞
∑
𝑠
𝑐2,𝑠 (𝑧 − 𝑧0 ) 𝑞2 , ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑠=0
of 𝐿(⋅, 𝑧) such that for some fixed branch of the 𝑞1 th and 𝑞2 th root functions say 𝑓1 and 𝑓2
with its evaluation at (𝑧 − 𝑧0 ) given by 𝑓1 (𝑧 − 𝑧0 ) and 𝑓2 (𝑧 − 𝑧0 ), respectively, and some fixed
primitive 𝑞1 th and 𝑞2 th root of unity say 𝜁 1 and 𝜁 2 , respectively, we have all their branches
given by the convergent series
𝜆1,ℎ (𝑧) = 𝜆0 +
𝜆2,ℎ (𝑧) = 𝜆0 +
∞
∑
𝑠=1
∞
∑
𝑐1,𝑠 (𝜁 ℎ1 𝑓1 (𝑧 − 𝑧0 ))𝑠 ,
ℎ = 0, . . . , 𝑞1 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑐2,𝑠 (𝜁 ℎ2 𝑓2 (𝑧 − 𝑧0 ))𝑠 ,
ℎ = 0, . . . , 𝑞2 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟
𝑠=1
and for some ℎ1 ∈ {0, . . . , 𝑞1 − 1} and ℎ2 ∈ {0, . . . , 𝑞2 − 1},
𝜆1,ℎ1 (𝑧) ∕= 𝜆2,ℎ2 (𝑧), ∣𝑧 − 𝑧0 ∣ < 𝑟.
Now let 𝛽 0 be a generating eigenvector of 𝐿(⋅, 𝑧) at the point (𝜆0 , 𝑧0 ) and associated with
𝜆1 (𝑧) and 𝛾 0 be a generating eigenvector of 𝐿∗ (⋅, 𝜂) at the point (𝜆0 , 𝑧0 ) and associated
with 𝜆∗2 (𝜂). By definition of generating eigenvector, there exists an eigenpair Puiseux series
(𝜆(𝑧), 𝜑(𝑧)) of 𝐿(⋅, 𝑧) such that 𝜆(𝑧) and 𝜑(𝑧) are expanded about 𝑧0 with limit point 𝜆0 and
𝛽 0 , respectively, and there exists an eigenpair Puiseux series (𝜆∗2 (𝜂), 𝛾(𝜂)) of 𝐿∗ (⋅, 𝜂) such
85
that 𝜆(𝑧) and 𝜑(𝑧) are expanded about 𝑧0 with limit point 𝜆0 and 𝛾 0 , respectively.
Now because of the definition of eigenvector Puiseux series and the definition of adjoint of
a Puiseux series and because none of the properties just mentioned change by going to a
smaller domain for 𝑧, i.e., taking 𝑟 smaller, we can assume without loss of generality that all
the branches of 𝜑(𝑧), 𝜆∗2 (𝜂), 𝛾(𝜂) are given by the convergent series
∞
∑
𝜑ℎ (𝑧) = 𝛽 0 +
𝛽 1,𝑠 (𝜁 ℎ1 𝑓1 (𝑧 − 𝑧0 ))𝑠 ,
𝑠=1
∞
∑
𝜆∗2,ℎ (𝜂) = 𝜆0 +
𝛾 ℎ (𝜂) = 𝜆0 +
𝑐2,𝑠 (𝜁 ℎ2 𝑓2 (𝜂 − 𝑧0 ))𝑠 ,
𝑠=1
∞
∑
𝜒𝑠 (𝜁 ℎ2 𝑓2 (𝜂 − 𝑧0 ))𝑠 ,
ℎ = 0, . . . , 𝑞1 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞2 − 1, ∣𝜂 − 𝑧0 ∣ < 𝑟,
ℎ = 0, . . . , 𝑞2 − 1, ∣𝜂 − 𝑧0 ∣ < 𝑟,
𝑠=1
respectively, and satisfy
ℎ = 0, . . . , 𝑞1 − 1, ∣𝑧 − 𝑧0 ∣ < 𝑟,
𝐿(𝜆1,ℎ (𝑧), 𝑧)𝜑ℎ (𝑧) = 0,
𝐿∗ (𝜆∗2,ℎ (𝜂), 𝜂)𝛾 ℎ (𝜂) = 0,
ℎ = 0, . . . , 𝑞2 − 1, ∣𝜂 − 𝑧0 ∣ < 𝑟.
Now it follows from Lemma 28 that
𝑞2 −1
∪
ℎ=0
𝑞2 −1
{𝜆∗2,ℎ (𝜂)}
=
∪
{𝜆2,ℎ (𝜂)}, ∣𝜂 − 𝑧0 ∣ < 𝑟.
ℎ=0
Thus for every 𝜂 ∈ 𝐵(𝑧0 , 𝑟) there exists ℎ(𝜂) ∈ {0, . . . , 𝑞2 − 1} such that
𝜆∗2,ℎ(𝜂) (𝜂) = 𝜆2,ℎ2 (𝜂).
Hence if ∣𝑧 − 𝑧0 ∣ < 𝑟 then for 𝜂 = 𝑧 we have
0 = 𝐿∗ (𝜆∗2,ℎ(𝜂) (𝜂), 𝜂)𝛾 ℎ(𝜂) (𝜂) = 𝐿(𝜆2,ℎ2 (𝜂), 𝜂)∗ 𝛾 ℎ(𝜂) (𝜂) = 𝐿(𝜆2,ℎ2 (𝑧), 𝑧)∗ 𝛾 ℎ(𝑧) (𝑧)
86
and thus
⟨𝛾 ℎ(𝑧) (𝑧), 𝐿(𝜆2,ℎ2 (𝑧), 𝑧)𝜑ℎ1 (𝑧)⟩ℂ = ⟨𝐿(𝜆2,ℎ2 (𝑧), 𝑧)∗ 𝛾 ℎ(𝑧) (𝑧), 𝜑ℎ1 (𝑧)⟩ℂ = 0.
Now it follows from Lemma 17, since 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and (𝜆0 , 𝑧0 ) ∈ 𝑈 , that 𝐿 is analytic at (𝜆, 𝑧) = (𝜆0 , 𝑧0 ), i.e., the partial and mixed partial derivatives 𝐿(𝑗1 ,𝑗2 ) (𝜆0 , 𝑧0 ) :=
∂ 𝑗1 +𝑗2 𝐿 exist for every 𝑗1 , 𝑗2 ∈ ℕ ∪ {0} and there exists an 𝑟0 > 0 such that
∂ 𝑗1 𝜆∂ 𝑗2 𝑧 (𝜆,𝑧)=(𝜆0 ,𝑧0 )
𝐿(𝜆, 𝑧) =
∞
∑
𝑗1 ,𝑗2
1
𝐿(𝑗1 ,𝑗2 ) (𝜆0 , 𝑧0 )(𝜆 − 𝜆0 )𝑗1 (𝑧 − 𝑧0 )𝑗2 , ∣𝜆 − 𝜆0 ∣ < 𝑟0 , ∣𝑧 − 𝑧0 ∣ < 𝑟0
𝑗
!𝑗
!
=0 1 2
where the power series on the right converges absolutely in the 𝑀𝑛 (ℂ) norm to 𝐿(𝜆, 𝑧) for
(𝜆, 𝑧) ∈ 𝐵(𝜆0 , 𝑟0 ) × 𝐵(𝑧0 , 𝑟0 ). We may assume without loss of generality that 𝑟0 ≤ 𝑟.
Thus from the facts above and the facts
𝑥𝑗 − 𝑦 𝑗 = (𝑥 − 𝑦)
𝑗−1
∑
𝑥𝑙 𝑦 𝑗−1−𝑙 ,
𝑥, 𝑦 ∈ ℂ,
𝑗∈ℕ
𝑙=0
lim𝑧→𝑧0 𝜆1,ℎ1 (𝑧) = lim 𝜆1,ℎ2 (𝑧) = 𝜆0
𝑧→𝑧0
lim𝑧→𝑧0 𝛾 ℎ(𝑧) (𝑧) = lim 𝛾 ℎ (𝑧) = 𝛾 0 ,
𝑧→𝑧0
ℎ = 0, . . . , 𝑞2 − 1
it follows that
(𝜆1,ℎ1 (𝑧) − 𝜆0 )𝑗 − (𝜆2,ℎ2 (𝑧) − 𝜆0 )𝑗
= 𝑜(1), as 𝑧 → 𝑧0 , if 𝑗 ≥ 2
𝜆1,ℎ1 (𝑧) − 𝜆2,ℎ2 (𝑧)
and so
𝐿(𝜆1,ℎ1 (𝑧), 𝑧) − 𝐿(𝜆2,ℎ2 (𝑧), 𝑧)
= 𝐿(1,0) (𝜆0 , 𝑧0 ) + 𝑜(1), as 𝑧 → 𝑧0
𝜆1,ℎ1 (𝑧) − 𝜆2,ℎ2 (𝑧)
87
which implies
⟨𝛾 ℎ(𝑧) (𝑧), 𝐿(𝜆1,ℎ1 (𝑧), 𝑧)𝜑ℎ1 (𝑧)⟩ℂ − ⟨𝛾 ℎ(𝑧) (𝑧), 𝐿(𝜆2,ℎ2 (𝑧), 𝑧)𝜑ℎ1 (𝑧)⟩ℂ
0=
𝜆1,ℎ1 (𝑧) − 𝜆2,ℎ2 (𝑧)
⟩
⟨
𝐿(𝜆1,ℎ1 (𝑧), 𝑧) − 𝐿(𝜆2,ℎ2 (𝑧), 𝑧)
𝜑ℎ1 (𝑧)
= 𝛾 ℎ(𝑧) (𝑧),
𝜆1,ℎ1 (𝑧) − 𝜆2,ℎ2 (𝑧)
ℂ
= ⟨𝛾 0 , 𝐿(1,0) (𝜆0 , 𝑧0 )𝛽 0 ⟩ℂ + 𝑜(1), as 𝑧 → 𝑧0 .
Hence we must have
0 = ⟨𝛾 0 , 𝐿(1,0) (𝜆0 , 𝑧0 )𝛽 0 ⟩ℂ = ⟨𝛾 0 , 𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 ⟩ℂ = ⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 , 𝛾 0 ⟩ℂ ,
where by our notation 𝐿𝜆 (𝜆0 , 𝑧0 ) := 𝐿(1,0) (𝜆0 , 𝑧0 ). Therefore ⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 , 𝛾 0 ⟩ℂ = 0. This
completes the proof.
Lemma 30 Let 𝜆(𝑧) be a convergent Puiseux series expanded about 𝑧0 with domain 𝐵(𝑧0 , 𝑟0 )
and period 𝑞. Suppose 𝜆(𝑧) is not a single-valued analytic function of 𝑧 in 𝐵(𝑧0 , 𝑟0 ). Then
if we let 𝜆1 (𝑧), . . . , 𝜆𝑞−1 (𝑧) be all the branches of 𝜆(𝑧) with respect to a fix branch of the 𝑞th
root function and a fix primitive 𝑞th root of unity, there exists ℎ1 , ℎ2 ∈ {0, . . . , 𝑞 − 1} and an
𝑟 > 0 with 𝑟 ≤ 𝑟0 such that
𝜆ℎ1 (𝑧) ∕= 𝜆ℎ2 (𝑧), 0 < ∣𝑧 − 𝑧0 ∣ < 𝑟.
Proof.
(3.73)
Suppose 𝜆(𝑧) is a convergent Puiseux series expanded about 𝑧0 with domain
𝐵(𝑧0 , 𝑟0 ) and period 𝑞, where
𝜆(𝑧) =
∞
∑
𝑠
𝑐𝑠 (𝑧 − 𝑧0 ) 𝑞 , ∣𝑧 − 𝑧0 ∣ < 𝑟0 ,
𝑠=0
and suppose that 𝜆(𝑧) is not a single-valued analytic function of 𝑧 in 𝐵(𝑧0 , 𝑟0 ), i.e., there
exists 𝑠 ∈ ℕ such that
𝑠
𝑞
∕∈ ℕ and 𝑐𝑠 ∕= 0. We will denote by 𝑠0 the smallest such 𝑠 with that
88
property. Note that for the Puiseux series 𝜆(𝑧) with the series representation give above we
must have the period 𝑞 ≥ 2.
Now fix a branch of the 𝑞th root and denote its evaluation at 𝑧 − 𝑧0 by (𝑧 − 𝑧0 )1/𝑞 and fix
some primitive 𝑞th root of unity denoting it by 𝜁. Then the branches of 𝜆(𝑧) are given by
the convergent series
𝜆ℎ (𝑧) =
∞
∑
𝑐𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 , ∣𝑧 − 𝑧0 ∣ < 𝑟0 ,
ℎ = 0, . . . , 𝑞 − 1.
𝑠=0
By the definition of 𝑠0 there exists an 𝑙 ∈ ℕ ∪ {0} such that
𝜆ℎ (𝑧) =
𝑙
∑
𝑗
ℎ
1/𝑞 𝑠0
𝑐𝑗𝑞 (𝑧 − 𝑧0 ) + 𝑐𝑠0 (𝜁 (𝑧 − 𝑧0 )
) +
𝑠0 −1
𝑞
∞
∑
= 𝑙, 𝑐𝑠0 ∕= 0, and
𝑐𝑠 (𝜁 ℎ (𝑧 − 𝑧0 )1/𝑞 )𝑠 ,
𝑠=𝑠0 +1
𝑗=0
for ∣𝑧 − 𝑧0 ∣ < 𝑟0 and ℎ = 0, . . . , 𝑞 − 1. It follows from this that for any ℎ1 , ℎ2 ∈ {0, . . . , 𝑞 − 1}
we have
𝑠0
∣𝜆ℎ1 (𝑧) − 𝜆ℎ2 (𝑧)∣ = ∣𝑐𝑠0 ∣∣𝜁 ℎ1 𝑠0 − 𝜁 ℎ2 𝑠0 ∣∣𝑧 − 𝑧0 ∣ 𝑞 + 𝑂(∣𝑧 − 𝑧0 ∣
𝑠0 +1
𝑞
), as 𝑧 → 𝑧0 .
The leading order term here can only be zero if ∣𝜁 ℎ1 𝑠0 − 𝜁 ℎ2 𝑠0 ∣ = 0 which would imply
𝜁 (ℎ1 −ℎ2 )𝑠0 = 1 and since 𝜁 is a primitive 𝑞th root of unity this implies
letting ℎ1 := 1, ℎ2 := 0 ∈ {0, . . . , 𝑞 − 1} (since 𝑞 ≥ 2) then, because
(ℎ1 −ℎ2 )𝑠0
𝑞
(ℎ1 −ℎ2 )𝑠0
𝑞
=
∈ ℤ. Thus
𝑠0
𝑞
∕∈ ℤ, we
have ∣𝜁 ℎ1 𝑠0 − 𝜁 ℎ2 𝑠0 ∣ =
∕ 0 and so
∣𝜆ℎ1 (𝑧) − 𝜆ℎ2 (𝑧)∣
𝑠0
∣𝑐𝑠0 ∣∣𝜁 ℎ1 𝑠0 − 𝜁 ℎ2 𝑠0 ∣∣𝑧 − 𝑧0 ∣ 𝑞
= 1 + 𝑜(1),
𝑧 ∕= 𝑧0 , as 𝑧 → 𝑧0 .
Therefore, since lim 𝜆ℎ (𝑧) = 𝑐0 for ℎ = 0, . . . , 𝑞 − 1, it follows there exists 𝑟 > 0 with 𝑟 ≤ 𝑟0
𝑧→𝑧0
such that
𝜆ℎ1 (𝑧) ∕= 𝜆ℎ2 (𝑧), 0 < ∣𝑧 − 𝑧0 ∣ < 𝑟.
89
This completes the proof.
The following theorem is Theorem 9 from [34].
Theorem 31 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and suppose (𝜆0 , 𝑧0 ) ∈ 𝑈 . Let 𝜆(𝑧) be an eigenvalue
Puiseux series of 𝐿(⋅, 𝑧) expanded about 𝑧0 with limit point 𝜆0 . Assume that for every generating eigenvector 𝛽 of 𝐿(⋅, 𝑧) at the point (𝜆0 , 𝑧0 ) and associated with 𝜆(𝑧) there exists a
generating eigenvector 𝛾 of 𝐿∗ (⋅, 𝜂) at the point (𝜆0 , 𝑧0 ) and associated with 𝜆∗ (𝜂) such that
⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽, 𝛾⟩ℂ ∕= 0,
(3.74)
where 𝜆∗ (𝜂) denotes the adjoint of 𝜆(𝑧) and 𝐿∗ denotes the adjoint of 𝐿. Then 𝜆(𝑧) is a
single-valued analytic function of 𝑧 and there exists an eigenvector Puiseux series 𝜑(𝑧) of
𝐿(⋅, 𝑧) corresponding to 𝜆(𝑧) which is also a single-valued analytic function of 𝑧.
Proof.
Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and suppose (𝜆0 , 𝑧0 ) ∈ 𝑈 . Let 𝜆(𝑧) be an eigenvalue
Puiseux series of 𝐿(⋅, 𝑧) expanded about 𝑧0 , domain 𝐵(𝑧0 , 𝑟0 ), with limit point 𝜆0 , and
period 𝑞. Suppose the hypotheses of this theorem are true for 𝜆(𝑧). But suppose that 𝜆(𝑧)
was not a single-valued analytic function of 𝑧 ∈ 𝐵(𝑧0 , 𝑟0 ). Then by Lemma 30, if we let
𝜆1 (𝑧), . . . , 𝜆𝑞−1 (𝑧) be all the branches of 𝜆(𝑧) with respect to a fix branch of the 𝑞th root
function and a fix primitive 𝑞th root of unity, there exists ℎ1 , ℎ2 ∈ {0, . . . , 𝑞 − 1} and an
𝑟 > 0 with 𝑟 ≤ 𝑟0 such that
𝜆ℎ1 (𝑧) ∕= 𝜆ℎ2 (𝑧), 0 < ∣𝑧 − 𝑧0 ∣ < 𝑟.
(3.75)
Now by Lemma 22 there exists an eigenvector Puiseux series 𝜑(𝑧) of 𝐿(⋅, 𝑧) corresponding
to the eigenvalue Puiseux series 𝜆(𝑧). Let 𝛽 0 denote the limit point of 𝜑(𝑧) when expanded
about 𝑧0 . By definition 𝛽 0 is a generating eigenvector of 𝐿(⋅, 𝑧) at the point (𝜆0 , 𝑧0 ) and
90
associated with 𝜆(𝑧) and so by our hypotheses their exists a eigenvalue Puiseux series 𝛾(𝜂),
expanded about 𝑧0 with limit point 𝛾 0 , of 𝐿∗ (⋅, 𝜂) corresponding to the eigenvalue Puiseux
series 𝜆∗ (𝜂) such that
⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 , 𝛾 0 ⟩ℂ ∕= 0.
(3.76)
But according to Proposition 29 with 𝜆1 (𝑧) := 𝜆2 (𝑧) = 𝜆(𝑧) we must have
⟨𝐿𝜆 (𝜆0 , 𝑧0 )𝛽 0 , 𝛾 0 ⟩ℂ = 0.
(3.77)
This is a contradiction. Therefore, 𝜆(𝑧) is single-valued analytic function of 𝑧 ∈ 𝐵(𝑧0 , 𝑟0 )
and so by Lemma 22 there exists an eigenvector Puiseux series 𝜑(𝑧) of 𝐿(⋅, 𝑧) corresponding
to 𝜆(𝑧) which is also a single-valued analytic function of 𝑧. This completes the proof.
The following proposition comes from [23, Lemma 4.1].
Proposition 32 Let 𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and suppose (𝜆0 , 𝑧0 ) ∈ 𝜎(𝐿). Assume (𝜆(𝑧), 𝜑(𝑧)) is
an eigenpair Puiseux series of 𝐿(⋅, 𝑧) such that both 𝜆(𝑧) and 𝜑(𝑧) are analytic functions at
𝑧0 with 𝜆(𝑧0 ) = 𝜆0 . Denote the order of the zero of the function 𝜆(𝑧) − 𝜆0 at 𝑧0 by 𝑞. Then
𝜑(𝑧) is a generating function of order 𝑞 for 𝐿(𝜆0 , ⋅) at the eigenvalue 𝑧0 .
Proof.
By hypotheses there exists an 𝑟 > 0 such that 𝐵(𝜆0 , 𝑟) × 𝐵(𝑧0 , 𝑟) ⊆ 𝑈 , 𝜆(⋅) ∈
𝒪(𝐵(𝑧0 , 𝑟), ℂ), and 𝜑(⋅) ∈ 𝒪(𝐵(𝑧0 , 𝑟), ℂ𝑛 ) such that 𝜑(𝑧0 ) ∕= 0 and 𝐿(𝜆(𝑧), 𝑧)𝜑(𝑧) = 0 for
every 𝑧 ∈ 𝐵(𝑧0 , 𝑟). Thus it follows from this by using the power series expansion of 𝐿 at
(𝜆0 , 𝑧0 ) from Lemma 17, that
𝜑(𝑧0 ) ∕= 0,
𝐿(𝜆0 , 𝑧)𝜑(𝑧) = 𝐿(𝜆0 , 𝑧)𝜑(𝑧) − 𝐿(𝜆(𝑧), 𝑧)𝜑(𝑧)
= (𝐿(𝜆0 , 𝑧) − 𝐿(𝜆(𝑧), 𝑧))𝜑(𝑧)
= −(𝜆(𝑧) − 𝜆0 )𝐿𝜔 (𝜆0 , 𝑧0 )𝜑(𝑧) + 𝑜(𝜆(𝑧) − 𝜆0 )
91
= −𝜆(𝑞) (𝑧0 )(𝑧 − 𝑧0 )𝑞 𝐿𝜔 (𝜆0 , 𝑧0 )𝜑(𝑧0 ) + 𝑜((𝑧 − 𝑧0 )𝑞 )
= 𝑂((𝑧 − 𝑧0 )𝑞 ), as 𝑧 → 𝑧0 .
By definition of generating function this implies 𝜑(𝑧) is a generating function of order 𝑞 for
𝐿(𝜆0 , ⋅) at the eigenvalue 𝑧0 .
92
Chapter 4
Canonical Equations: A Model for
Studying Slow Light
4.1
Introduction
In this chapter we consider a general nondissipative but dispersive model for wave propagation using an important class of periodic differential and differential-algebraic equations
(DAEs) called canonical equations [31]. This model is general enough to include electromagnetic waves governed by the time-harmonic Maxwell’s equations for lossless one-dimensional
photonic crystals whose constituent layers can be any combination of isotropic, anisotropic,
or bianisotropic materials with or without material dispersion (i.e., frequency-dependent response of materials). This makes our work particularly significant in the study of slow light
since metamaterials are widening the range of potential photonic crystals that can be fabricated and so a model like ours that has the ability to analysis slow light phenomena for a
broad range of photonic crystals is in need.
93
4.1.1
Model Formulation
In this chapter1 , we will denoted a wave by 𝑦 and its evaluation at a position 𝑥 by 𝑦(𝑥),
where 𝑥 ∈ ℝ, 𝑦(𝑥) ∈ ℂ𝑁 , and 𝑦 ∈ (𝐿2𝑙𝑜𝑐 (ℝ))𝑁 . The frequency domain Ω is an open connected
set in ℂ and the real frequency domain Ωℝ := Ω ∩ ℝ is nonempty.
Wave propagation will be governed by canonical systems of differential equations with periodic coefficients that depend holomorphically on the frequency parameter 𝜔, i.e., differential
equations of the form
G 𝑦 ′ (𝑥) = 𝑉 (𝑥, 𝜔)𝑦(𝑥),
(4.1)
where the leading matrix coefficient G ∈ 𝑀𝑁 (ℂ) and the matrix-valued function 𝑉 : ℝ×Ω →
𝑀𝑁 (ℂ), the Hamiltonian, have the properties:
(i) G ∗ = −G
(ii) 𝑉 (𝑥, 𝜔)∗ = 𝑉 (𝑥, 𝜔), for each 𝜔 ∈ Ωℝ and almost every 𝑥 ∈ ℝ
(iii) 𝑉 (𝑥 + 𝑑, 𝜔) = 𝑉 (𝑥, 𝜔), for each 𝜔 ∈ Ω and almost every 𝑥 ∈ ℝ
(iv) 𝑉 ∈ 𝒪(Ω, 𝑀𝑁 (𝐿𝑝 (𝕋))) as a function of frequency where
⎧

⎨ 1 if det(G ) ∕= 0,
𝑝=

⎩ 2 if det(G ) = 0.
In the case det(G ) = 0, the domain of equation (4.1) and definition of its leading term are
1,1
(ℝ))𝑁 }, G 𝑦 ′ := G (𝑃⊥ 𝑦)′ ,
𝒟 := {𝑦 ∈ (𝐿2𝑙𝑜𝑐 (ℝ))𝑁 : 𝑃⊥ 𝑦 ∈ (𝑊𝑙𝑜𝑐
1
See section 4.5 for notation and convention.
94
(4.2)
where 𝑃 , 𝑃⊥ ∈ 𝑀𝑁 (ℂ) denote the projections onto the kernel and range of G , respectively.
1,1
In the case det(G ) ∕= 0, we take the standard domain 𝒟 := (𝑊𝑙𝑜𝑐
(ℝ))𝑁 and definition of 𝑦 ′
as the weak derivative of the entries of 𝑦.
The equations of the form (4.1) with det(G ) = 0 are differential-algebraic equations (DAEs)
and are not just ODEs. Their solutions must satisfy an algebraic equation as well, namely,
𝑃 𝑉 (𝑥, 𝜔)𝑦(𝑥) = 0 for a.e. 𝑥 ∈ ℝ. Thus, in order to guarantee the existence of a nontrivial
solution for each frequency and a solution space which depends holomorphically on frequency,
we impose an additional hypothesis for equations of the form (4.1) whenever det(G ) = 0,
namely, (G + 𝑃 𝑉 (𝑥, 𝜔)𝑃 )−1 exists for each 𝜔 ∈ Ω and a.e. 𝑥 ∈ ℝ and as a function of
frequency
(G + 𝑃 𝑉 𝑃 )−1 ∈ 𝒪(Ω, 𝑀𝑁 (𝐿∞ (0, 𝑑))).
(4.3)
We will refer to this as the index-1 hypothesis.
Differential-algebraic equations of the form given by (4.1), for real frequencies, are known as
canonical equations [31] and, in the case det(G ) ∕= 0, they are known as canonical differential
equations, linear canonical systems, linear Hamiltonian systems, or Hamiltonian equations.
We distinguish between the two classes of canonical equations of the form (4.1) by referring
to those with det(G ) = 0 as canonical DAEs and those with det(G ) ∕= 0 as canonical ODEs.
We introduce the following definitions into our model:
Definition 18 We say 𝑦 is a solution of equation (4.1) provided 𝑦 ∈ 𝒟 and equation (4.1)
is satisfied for some 𝜔 ∈ Ω and for a.e. 𝑥 in ℝ. In which case we call 𝜔 its frequency.
Definition 19 We say 𝑦 is a Floquet solution provided for some 𝜔 ∈ Ω, it is a solution
95
of equation (4.1) with 𝜔 as its frequency and has the Floquet representation
𝑦(𝑥) = 𝑒𝑖𝑘𝑥
𝑚
∑
𝑥𝑗 𝑢𝑗 (𝑥),
(4.4)
𝑗=0
where 𝑚 ∈ ℕ ∪ {0} and 𝑢𝑗 ∈ 𝒟 ∩ (𝐿𝑝 (𝕋))𝑁 , for 𝑗 = 1, . . . , 𝑚. We call 𝑘, (𝑘, 𝜔), and 𝜆 := 𝑒𝑖𝑘𝑑
its wavenumber, wavenumber-frequency pair, and Floquet multiplier, respectively.
If 𝑢𝑚 ∕= 0 then we call 𝑚 its order. We call 𝑦 a Bloch solution if it is a Floquet solution
with 𝑚 = 0. If, in addition, 𝑦 is a Bloch solution with real frequency then we call it a
propagating wave if it has a real wavenumber and an evanescent wave otherwise.
Definition 20 We define the Bloch variety to be the set
ℬ := {(𝑘, 𝜔) ∈ ℂ × Ω ∣ (𝑘, 𝜔) is the wavenumber-frequency pair of some
(4.5)
nontrivial Bloch solution of equation (4.1)}
and the real Bloch variety to be the set
ℬℝ := ℬ ∩ ℝ2 .
(4.6)
Definition 21 We define the dispersion relation denoted by 𝜔 = 𝜔(𝑘) as the multivalued function whose graph is the Bloch variety. We say a point (𝑘0 , 𝜔 0 ) ∈ ℬℝ is amicable
if for every 𝜖 > 0 there exists an 𝛿 > 0 such that ℬ ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) is the union
of the graphs of a finite number of real analytic functions, say 𝜔 1 (𝑘), . . . , 𝜔 𝑔 (𝑘), with the
property 𝜔 1 (𝑘0 ) = ⋅ ⋅ ⋅ = 𝜔 𝑔 (𝑘0 ) = 𝜔 0 . We call these functions the band functions and
their derivatives
𝑔
𝑑𝜔 1
, . . . , 𝑑𝜔
𝑑𝑘
𝑑𝑘
the group velocities. The intersection of the graph of a band
function with ℝ2 is called a band.
Definition 22 For any solution 𝑦 of equation (4.1) with real frequency 𝜔 0 we define its
96
energy flux and energy density to be the functions of position given by
⟨𝑖G 𝑦, 𝑦⟩ℂ
(4.7)
⟨𝑉𝜔 (⋅, 𝜔 0 )𝑦, 𝑦⟩ℂ
(4.8)
respectively, where 𝑉𝜔 denotes the derivative of the Hamiltonian 𝑉 with respect to frequency in
the 𝑀𝑁 (𝐿𝑝 (𝕋)) norm. We define its average energy flux and average energy density
to be the averages
∫
1 𝑑
⟨𝑖G 𝑦(𝑥), 𝑦(𝑥)⟩ℂ 𝑑𝑥
𝑑 0
∫
1 𝑑
⟨𝑉𝜔 (𝑥, 𝜔 0 )𝑦(𝑥), 𝑦(𝑥)⟩ℂ 𝑑𝑥
𝑑 0
(4.9)
(4.10)
respectively. We also define its energy velocity as the ratio of its average energy flux to
its average energy density provided the latter is nonzero, i.e.,
∫
1 𝑑
⟨𝑖G 𝑦(𝑥), 𝑦(𝑥)⟩ℂ 𝑑𝑥
𝑑 0
.
∫
𝑑
1
⟨𝑉
(𝑥,
𝜔
)𝑦(𝑥),
𝑦(𝑥)⟩
𝑑𝑥
𝜔
0
ℂ
𝑑 0
(4.11)
Remark: The integrals (4.9) and (4.10) are well-defined because Corollary 40, Corollary 43,
and Theorem 50 show the functions (4.7) and (4.8) are in 𝐿1loc (ℝ).
Definition 23 We say (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the canonical equations
in (4.1) and say these equations are of definite type at this point provided the average energy
density of any nontrivial Bloch solution 𝑦 with (𝑘0 , 𝜔 0 ) as its wavenumber-frequency pair is
nonzero, i.e.,
1
𝑑
∫
𝑑
⟨𝑉𝜔 (𝑥, 𝜔 0 )𝑦(𝑥), 𝑦(𝑥)⟩ℂ 𝑑𝑥 ∕= 0.
0
97
(4.12)
The canonical equations in (4.1) along with the definitions just given constitute the model
of wave propagation that we will analyze in this thesis. Moreover, based on our studies
on electromagnetic wave propagation in one-dimensional photonic crystals, it is found by
physical considerations that a point of definite type is the right assumption in which to
begin a perturbation analysis from stationary points of the dispersion relation.
We are particularly interested in the perturbation analysis of canonical equations in the slow
wave regime in which points in the Bloch variety (𝑘, 𝜔) belong to a small neighborhood of
a point (𝑘0 , 𝜔 0 ) in the real Bloch variety where (𝑘0 , 𝜔 0 ) is both a point of definite type for
the canonical equations (such a point is amicable (see Theorem 48 and Theorem 50)) and a
stationary point of the dispersion relation, i.e., at (𝑘0 , 𝜔 0 ) one of the bands has a stationary
point. For this model and with this regime in mind, we give the following definition for a
slow wave:
Definition 24 Let (𝑘0 , 𝜔 0 ) be any point which is amicable and 0 < 𝑟 ≪ 1. Then any
propagating Bloch wave 𝑦 with wavenumber-frequency pair (𝑘, 𝜔) belonging to a band with
a stationary point at (𝑘0 , 𝜔 0 ) and satisfying 0 < ∣∣(𝑘, 𝜔) − (𝑘0 , 𝜔 0 )∣∣ℂ < 𝑟 is called a slow
wave.
The main purpose of this chapter is to begin developing a mathematical framework for this
model, including the Floquet, spectral, and perturbation theory, and to use this framework
to analyze in detail the slow wave regime.
4.2
Canonical ODEs
In this section we consider the canonical ODEs
J 𝜓 ′ (𝑥) = 𝐴(𝑥, 𝜔)𝜓(𝑥),
98
1,1
𝜓 ∈ (𝑊loc
(ℝ))𝑛
(4.13)
where, as discussed in section 4.1.1, the leading matrix coefficient J ∈ 𝑀𝑛 (ℂ) and the
Hamiltonian 𝐴 : ℝ × Ω → 𝑀𝑛 (ℂ) have the properties:
(i) det(J ) ∕= 0, J ∗ = −J ,
(ii) 𝐴(𝑥, 𝜔)∗ = 𝐴(𝑥, 𝜔), for each 𝜔 ∈ Ωℝ and a.e. 𝑥 ∈ ℝ,
(iii) 𝐴(𝑥 + 𝑑, 𝜔) = 𝐴(𝑥, 𝜔), for each 𝜔 ∈ Ω and a.e. 𝑥 ∈ ℝ,
(iv) 𝐴 ∈ 𝒪(Ω, 𝑀𝑁 (𝐿1 (𝕋))) as a function of frequency.
Denote by 𝐴𝜔 the derivative of the Hamiltonian 𝐴 with respect to frequency in the 𝑀𝑛 (𝐿1 (𝕋))
norm.
4.2.1
Preliminaries
We begin with some preliminary results on the solutions of the canonical ODEs in (4.13)
and the Floquet theory relating to the matricant and monodromy matrix of these canonical ODEs. The dependency of the matricant and monodromy matrix on frequency is also
discussed. We will hold off on proving the results of this subsection until section 4.4. The
statements in this subsection are known, but we state them here in order to setup the main
results of this chapter. We also will give proofs of these statements for completeness in
section 4.4.
Proposition 33 For each 𝜔 ∈ Ω, let Ψ(⋅, 𝜔) denote the fundamental matrix solution of
the canonical ODEs in (4.13) satisfying Ψ(0, 𝜔) = 𝐼𝑛 , i.e., the unique function Ψ(⋅, 𝜔) ∈
1,1
𝑀𝑛 (𝑊loc
(ℝ)) satisfying a.e. the matrix differential equation with initial condition
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
99
(4.14)
Then the matrix-valued function Ψ : ℝ × Ω → 𝑀𝑛 (ℂ) has the following properties:
(i) 𝜓 is a solution of the canonical ODEs in (4.13) at the frequency 𝜔 ∈ Ω if and only if
𝜓 = Ψ(⋅, 𝜔)𝛾 for some 𝛾 ∈ ℂ𝑛 . Moreover, 𝛾 is unique.
(ii) For every (𝑥, 𝜔) ∈ ℝ × Ω,
∫
Ψ(𝑥, 𝜔) = 𝐼𝑛 +
𝑥
J −1 𝐴(𝑡, 𝜔)Ψ(𝑡, 𝜔)𝑑𝑡.
(4.15)
0
(iii) For every 𝑥 ∈ ℝ, Ψ(𝑥, ⋅) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)). We will denote the partial derivative of the
function Ψ with respect to frequency in the 𝑀𝑛 (ℂ) norm by Ψ𝜔 .
1,1
(iv) For every (𝑥, 𝜔) ∈ ℝ × Ω, Ψ𝜔 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) and
∫
Ψ𝜔 (𝑥, 𝜔) =
𝑥
J −1 𝐴𝜔 (𝑡, 𝜔)Ψ(𝑡, 𝜔) + J −1 𝐴(𝑡, 𝜔)Ψ𝜔 (𝑡, 𝜔)𝑑𝑡
(4.16)
0
1,1
(v) For every (𝑥, 𝜔) ∈ ℝ × Ω, Ψ−1 (𝑥, 𝜔) := Ψ(𝑥, 𝜔)−1 exists and Ψ−1 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
(vi) For every (𝑥, 𝜔) ∈ ℝ × Ω,
Ψ(𝑥 + 𝑑, 𝜔) = Ψ(𝑥, 𝜔)Ψ(𝑑, 𝜔).
(4.17)
Now for each fixed 𝜔 ∈ Ω, Ψ(⋅, 𝜔) and Ψ(𝑑, 𝜔) are the matricant and monodromy matrix,
respectively, of the canonical ODEs in (4.13). Thus the previous proposition implies the
following:
Theorem 34 (Floquet-Lyapunov) Let 𝜔 ∈ Ω. Then there exists a matrix 𝐾(𝜔) ∈ 𝑀𝑛 (ℂ)
1,1
and a function 𝐹 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) such that for every 𝑥 ∈ ℝ,
Ψ(𝑥, 𝜔) = 𝐹 (𝑥, 𝜔)𝑒𝑖𝑥𝐾(𝜔) ,
100
𝐹 (𝑥 + 𝑑, 𝜔) = 𝐹 (𝑥, 𝜔), 𝐹 (0, 𝜔) = 𝐼𝑛 , 𝐹 −1 (𝑥, 𝜔) := 𝐹 (𝑥, 𝜔)−1 exists, and 𝐹 −1 (⋅, 𝜔) ∈
1,1
𝑀𝑛 (𝑊loc
(ℝ)).
Theorem 35 If 𝜓 is a Floquet solution of the canonical ODEs in (4.13) with wavenumberfrequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚 then 𝜓 = Ψ(⋅, 𝜔)𝛾 where
𝛾 is a generalized eigenvector of Ψ(𝑑, 𝜔) of order 𝑚 + 1 corresponding to the eigenvalue 𝜆.
Conversely, if 𝛾 is a generalized eigenvector of order 𝑚 + 1 of the monodromy matrix Ψ(𝑑, 𝜔)
corresponding to the eigenvalue 𝜆 then for any 𝑘 ∈ ℂ such that 𝜆 = 𝑒𝑖𝑘𝑑 , 𝜓 = Ψ(⋅, 𝜔)𝛾 is
a Floquet solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (𝑘, 𝜔),
Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚.
Corollary 36 Let 𝜔 ∈ Ω. Let {𝛾 𝑗 }𝑛𝑗=1 be a Jordan basis for the matrix Ψ(𝑑, 𝜔). For
𝑗 = 1, . . . , 𝑛, let 𝑙𝑗 , 𝜆𝑗 , and 𝑘𝑗 be such that 𝛾 𝑗 is a generalized eigenvalue of Ψ(𝑑, 𝜔) of
order 𝑙𝑗 corresponding to the eigenvalue 𝜆𝑗 = 𝑒𝑖𝑘𝑗 𝑑 . Define {𝜓 𝑗 }𝑛𝑗=1 by 𝜓 𝑗 := Ψ(⋅, 𝜔)𝛾 𝑗 for
𝑗 = 1, . . . , 𝑛. Then the following statements are true:
(i) The set of solutions of the canonical ODEs in (4.13) at the frequency 𝜔 is a vector
space over ℂ and {𝜓 𝑗 }𝑛𝑗=1 is a basis for this space.
(ii) For 𝑗 = 1, . . . , 𝑛, 𝜓 𝑗 is a Floquet solution of the canonical ODEs in (4.13) with
wavenumber-frequency pair (𝑘𝑗 , 𝜔), Floquet multiplier 𝜆𝑗 = 𝑒𝑖𝑘𝑗 𝑑 , and order 𝑙𝑗 − 1.
Corollary 37 Let ℬ denote the Bloch variety of the canonical ODEs in (4.13). Then, for
any 𝑙 ∈ ℤ,
ℬ = (2𝜋𝑙/𝑑, 0) + ℬ.
(4.18)
Corollary 38 Define the function 𝐷 : ℂ × Ω → ℂ by
(
)
𝐷(𝑘, 𝜔) := det 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔) .
101
(4.19)
Then 𝐷 is a nonconstant holomorphic function and its zero set is the Bloch variety of the
canonical ODEs in (4.13), i.e.,
ℬ = {(𝑘, 𝜔) ∈ ℂ × Ω ∣ 𝐷(𝑘, 𝜔) = 0}.
4.2.2
(4.20)
Energy Flux, Energy Density, and their Averages
The following results describe how the energy flux, energy density, their averages, and points
of definite type of the canonical ODEs in (4.13) are related to the leading matrix coefficient,
the matricant, and the monodromy matrix.
We start by considering the energy flux and its average. The following proposition is the key
result, as we will see in this chapter, in the study of energy flux for canonical equations.
1,1
Proposition 39 For each 𝜔 ∈ Ωℝ we have Ψ(⋅, 𝜔)∗ 𝑖J Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) and
Ψ(⋅, 𝜔)∗ 𝑖J Ψ(⋅, 𝜔) = 𝑖J .
(4.21)
This proposition is important because it tells us the energy flux of any solution of canonical
ODEs with real frequency is a conserved quantity. A more precise statement of this is found
in the next corollary.
Corollary 40 For each 𝜔 ∈ Ωℝ and every 𝛾 1 , 𝛾 2 ∈ ℂ𝑛 , if 𝜓 1 = Ψ(⋅, 𝜔)𝛾 1 and 𝜓 2 = Ψ(⋅, 𝜔)𝛾 2
1,1
then ⟨𝑖J 𝜓 1 , 𝜓 2 ⟩ℂ ∈ 𝑊loc
(ℝ) and
1
𝑑
∫
⟨𝑖J 𝜓 1 , 𝜓 2 ⟩ℂ = ⟨𝑖J 𝛾 1 , 𝛾 2 ⟩ℂ ,
(4.22)
⟨𝑖J 𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥 = ⟨𝑖J 𝛾 1 , 𝛾 2 ⟩ℂ .
(4.23)
𝑑
0
102
We will now consider the energy density and its average.
Lemma 41 For each 𝜔 ∈ Ωℝ we have 𝐴𝜔 (⋅, 𝜔) ∈ 𝑀𝑛 (𝐿1 (𝕋)) and
𝐴𝜔 (⋅, 𝜔)∗ = 𝐴𝜔 (⋅, 𝜔).
(4.24)
Theorem 42 For each 𝜔 ∈ Ωℝ we have
1,1
(ℝ)),
Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(4.25)
Ψ(⋅, 𝜔)∗ 𝐴𝜔 (⋅, 𝜔)Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝐿1loc (ℝ)),
(4.26)
Ψ(⋅, 𝜔)∗ 𝐴𝜔 (⋅, 𝜔)Ψ(⋅, 𝜔) = (Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔))′ ,
∫
1 𝑑
1
Ψ(𝑡, 𝜔)∗ 𝐴𝜔 (𝑡, 𝜔)Ψ(𝑡, 𝜔)𝑑𝑡 = Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔).
𝑑 0
𝑑
(4.27)
(4.28)
Corollary 43 For each 𝜔 ∈ Ωℝ and every 𝛾 1 , 𝛾 2 ∈ ℂ𝑛 , if 𝜓 1 = Ψ(⋅, 𝜔)𝛾 1 and 𝜓 2 = Ψ(⋅, 𝜔)𝛾 2
1,1
then ⟨Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ ∈ 𝑊loc
(ℝ), ⟨𝐴𝜔 (⋅, 𝜔)𝜓 1 , 𝜓 2 ⟩ℂ ∈ 𝐿1loc (ℝ) and
⟨𝐴𝜔 (⋅, 𝜔)𝜓 1 , 𝜓 2 ⟩ℂ = ⟨Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ ′ ,
1
𝑑
4.2.3
∫
0
𝑑
1
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥 = ⟨Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ .
𝑑
(4.29)
(4.30)
On Points of Definite Type for Canonical ODEs
Let (𝑘, 𝜔) ∈ ℬℝ and on the eigenspace of the monodromy matrix Ψ(𝑑, 𝜔) corresponding to
the eigenvalue 𝑒𝑖𝑘𝑑 we define the sesquilinear form 𝑞(𝑘,𝜔) by
𝑞(𝑘,𝜔) (𝛾 1 , 𝛾 2 ) :=
1
𝑑
∫𝑑
0
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥 (= 𝑑1 ⟨Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ ),
(4.31)
𝛾 1 , 𝛾 2 ∈ ker(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)),
𝜓 1 = Ψ(⋅, 𝜔)𝛾 1 ,
𝜓 2 = Ψ(⋅, 𝜔)𝛾 2 .
It follows from Corollary 43 that 𝑞(𝑘,𝜔) is well-defined.
103
Lemma 44 For each (𝑘, 𝜔) ∈ ℬℝ , 𝑞(𝑘,𝜔) is a Hermitian form, that is, 𝑞(𝑘,𝜔) satisfies
𝑞(𝑘,𝜔) (𝛾 1 , 𝛾 2 ) = 𝑞(𝑘,𝜔) (𝛾 2 , 𝛾 1 )
(4.32)
for every 𝛾 1 , 𝛾 2 ∈ ker(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)). In particular, 𝑞(𝑘,𝜔) (𝛾, 𝛾) ∈ ℝ for every 𝛾 ∈
ker(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)).
Denote the sign function by sgn(𝑥) :=
𝑥
∣𝑥∣
∈ {−1, 1}, for 𝑥 ∈ ℝ/{0}.
Proposition 45 The canonical ODEs in (4.13) are of definite type at a point (𝑘0 , 𝜔 0 ) ∈ ℬℝ
if and only if 𝑞(𝑘0 ,𝜔0 ) is a definite sesquilinear form which is bounded, i.e., the following
properties hold:
(i) For every nonzero 𝛾 ∈ ker(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)) we have
𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾) ∈ ℝ/{0}
(4.33)
and its sign sgn(𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾)) =: sgn(𝑞(𝑘0 ,𝜔0 ) ) is independent of the choice of 𝛾.
(ii) The sesquilinear form
⟨𝛾 1 , 𝛾 2 ⟩(𝑘0 ,𝜔0 ) := sgn(𝑞(𝑘0 ,𝜔0 ) )𝑞(𝑘0 ,𝜔0 ) (𝛾 1 , 𝛾 2 ),
(4.34)
𝛾 1 , 𝛾 2 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 ))
is inner product on ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
(iii) There exists constants 𝐶1 , 𝐶2 > 0 such that
𝐶1 ∣∣𝛾∣∣2ℂ ≤ ∣𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾)∣ ≤ 𝐶2 ∣∣𝛾∣∣2ℂ , for all 𝛾 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
104
(4.35)
Lemma 46 The family of sesquilinear forms {𝑞(𝑘,𝜔) }(𝑘,𝜔)∈ℬℝ depends continuously on the
indexing parameter (𝑘, 𝜔) in the following sense:
If {(𝑘𝑗 , 𝜔 𝑗 )}𝑗∈ℕ ⊆ ℬℝ , {𝛾 𝑗 }𝑗∈ℕ ⊆ ℂ𝑛 such that 𝛾 𝑗 ∈ ker(𝑒𝑖𝑘𝑗 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 𝑗 )) with ∣∣𝛾 𝑗 ∣∣ℂ = 1
for all 𝑗 ∈ ℕ and
∣∣⋅∣∣ℂ
∣∣⋅∣∣ℂ
(𝑘𝑗 , 𝜔 𝑗 ) −−→ (𝑘0 , 𝜔 0 ) and 𝛾 𝑗 −−→ 𝛾 0 , as 𝑗 → ∞
then (𝑘0 , 𝜔 0 ) ∈ ℬℝ , 𝛾 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) with ∣∣𝛾 0 ∣∣ℂ = 1 and
∣⋅∣
𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛾 𝑗 , 𝛾 𝑗 ) −
→ 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 ) as 𝑗 → ∞.
(4.36)
Theorem 47 If (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the canonical ODEs in (4.13)
then there exists an 𝑟 > 0 such that every (𝑘, 𝜔) ∈ 𝐵((𝑘0 , 𝜔 0 ), 𝑟) ∩ ℬℝ is a point of definite
type and
sgn(𝑞(𝑘,𝜔) ) = sgn(𝑞(𝑘0 ,𝜔0 ) ).
4.2.4
(4.37)
Perturbation Theory for Canonical ODEs
This section contains the main results of this chapter on the perturbation theory for canonical
ODEs.
Theorem 48 Suppose the canonical ODEs in (4.13) are of definite type at (𝑘0 , 𝜔 0 ) ∈ ℬℝ .
Let 𝑔 be the number of Jordan blocks (geometric multiplicity) in the Jordan form of the
monodromy matrix Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 and 𝑚1 ≥ ⋅ ⋅ ⋅ ≥
𝑚𝑔 ≥ 1 the dimensions of each of those Jordan blocks (partial multiplicities). Define 𝑚 :=
𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 (algebraic multiplicity). Let 𝜖 > 0 be given. Then there exists an 𝛿 > 0 such
105
that
(i) The order of the zero of 𝐷(𝑘0 , 𝜔) at 𝜔 = 𝜔 0 is 𝑔 and the order of the zero of 𝐷(𝑘, 𝜔 0 )
at 𝑘 = 𝑘0 is 𝑚.
(ii) The set ℬ ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) is the union of the graphs of 𝑔 nonconstant real analytic
functions 𝜔 1 (𝑘), . . . , 𝜔 𝑔 (𝑘) given by the convergent power series
𝜔 𝑗 (𝑘) = 𝜔 0 + 𝜈 𝑗,𝑚𝑗 (𝑘 − 𝑘0 )𝑚𝑗 +
∞
∑
𝜈 𝑗,𝑙 (𝑘 − 𝑘0 )𝑙 , ∣𝑘 − 𝑘0 ∣ < 𝛿
(4.38)
𝑙=𝑚𝑗 +1
where
𝜈 𝑗,𝑚𝑗 ∕= 0,
(4.39)
for 𝑗 = 1, . . . , 𝑔. Moreover, there exists analytic functions 𝜑1 (𝑘), . . . , 𝜑𝑔 (𝑘) belonging
to 𝒪(𝐵(𝑘0 , 𝛿), ℂ𝑛 ) such that
𝜑𝑗 (𝑘) ∕= 0, Ψ(𝑑, 𝜔 𝑗 (𝑘))𝜑𝑗 (𝑘) = 𝑒𝑖𝑘𝑑 𝜑𝑗 (𝑘), ∣𝑘 − 𝑘0 ∣ < 𝛿,
𝑗 = 1, . . . , 𝑔,
(4.40)
the vectors 𝜑1 (𝑘), . . . , 𝜑𝑔 (𝑘) are linearly independent for ∣𝑘−𝑘0 ∣ < 𝛿, and, in particular,
the vectors 𝜑1 (𝑘0 ), . . . , 𝜑𝑔 (𝑘0 ) form a basis for ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
Corollary 49 The conditions
𝐷(𝑘0 , 𝜔 0 ) = 0,
∂𝐷
(𝑘0 , 𝜔 0 ) ∕= 0, (𝑘0 , 𝜔 0 ) ∈ ℝ × Ωℝ
∂𝜔
(4.41)
are satisfied if and only if (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the canonical ODEs in
(4.13) and the Jordan normal form of the monodromy matrix Ψ(𝑑, 𝜔 0 ) corresponding to the
eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 consists of a single Jordan block, i.e., dim ker(𝜆0 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) = 1.
106
Remark. In this case, by denoting 𝜆0 := 𝑒𝑖𝑘0 𝑑 , the conditions of the previous corollary are
det(𝜆0 𝐼𝑛 − Ψ(𝑑, 𝜔)) = 𝐷(𝑘0 , 𝜔) = 0 and
∂
∂𝜔
det(𝜆𝐼𝑛 − Ψ(𝑑, 𝜔))∣(𝜔0 ,𝜆0 ) =
∂𝐷
(𝑘0 , 𝜔 0 )
∂𝜔
∕= 0,
but this is just the generic condition (2.1) in chapter 2 (see also [61, See (1.1)]). This
condition was throughly investigated in chapter 2 (see also [61, See (1.1)]) for general matrix
perturbations. And, in particular, in Theorem 2 of chapter 2 (see also [61, Theorem 3.1])
explicit recursive formulas are given for the unique eigenvalue Puiseux series 𝜆(𝜔) which
satisfies 𝜆(𝜔 0 ) = 𝜆0 and for its associated eigenvector Puiseux series of the monodromy
matrix Ψ(𝑑, 𝜔) near the frequency 𝜔 = 𝜔 0 . Moreover, their Puiseux series coefficients up to
the second order are conveniently listed in Corollary 4 of chapter 2 (see also [61, Corollary
3.3]).
4.3
Canonical DAEs
In this section we consider the canonical DAEs
G 𝜓 ′ (𝑥) = 𝑉 (𝑥, 𝜔)𝑦(𝑥),
𝑦∈𝒟
(4.42)
where, as discussed in §1, the leading matrix coefficient G ∈ 𝑀𝑛 (ℂ) and the Hamiltonian
𝑉 : ℝ × Ω → 𝑀𝑁 (ℂ) have the properties:
(i) det(G ) = 0, G ∗ = −G ,
(ii) 𝑉 (𝑥, 𝜔)∗ = 𝑉 (𝑥, 𝜔), for each 𝜔 ∈ Ωℝ and a.e. 𝑥 ∈ ℝ,
(iii) 𝑉 (𝑥 + 𝑑, 𝜔) = 𝑉 (𝑥, 𝜔), for each 𝜔 ∈ Ω and a.e. 𝑥 ∈ ℝ,
(iv) 𝑉 ∈ 𝒪(Ω, 𝑀𝑁 (𝐿2 (𝕋))) as a function of frequency.
107
Denote by 𝑉𝜔 the derivative of the Hamiltonian 𝑉 with respect to frequency in the 𝑀𝑛 (𝐿2 (𝕋))
norm.
The domain 𝒟 of the canonical DAEs in (4.42) and definition of its leading term are
1,1
𝒟 := {𝑦 ∈ (𝐿2𝑙𝑜𝑐 (ℝ))𝑁 : 𝑃⊥ 𝑦 ∈ (𝑊𝑙𝑜𝑐
(ℝ))𝑁 }, G 𝑦 ′ := G (𝑃⊥ 𝑦)′ ,
(4.43)
where 𝑃 , 𝑃⊥ ∈ 𝑀𝑁 (ℂ) denote the projections onto the kernel and range of G , respectively.
We shall assume that the canonical DAEs in (4.42) satisfy the index-1 hypothesis, namely,
(G + 𝑃 𝑉 𝑃 )−1 ∈ 𝒪(Ω, 𝑀𝑁 (𝐿∞ (0, 𝑑))).
(4.44)
The goal of this section is to describe the theory of canonical DAEs including the Floquet
and spectral perturbation theory in terms of the theory developed in the previous section
for canonical ODEs. We will only present the statement of our main result, the proof and
consequences will appear in [60].
4.3.1
The Correspondence between Canonical DAEs and Canonical ODEs
The following represents our main result in the study of canonical DAEs:
Theorem 50 Let 𝑛 := dim ran(G ). Then there exists an 𝑛 × 𝑛 system of canonical ODEs
J 𝜓 ′ (𝑥) = 𝐴(𝑥, 𝜔)𝜓(𝑥),
1,1
𝜓 ∈ (𝑊loc
(ℝ))𝑛
(4.45)
with J ∈ 𝑀𝑛 (ℂ), 𝐴 ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝕋))), and a function 𝑄 ∈ 𝒪(Ω, 𝑀𝑁 ×𝑛 (𝐿2 (𝕋))) such that
1,1
the multiplication map 𝑄(⋅, 𝜔) : (𝑊loc
(ℝ))𝑛 → 𝒟 is injective and the following properties
108
hold:
(i) The solutions of the canonical DAEs in (4.42) with frequency 𝜔 are given by 𝑦 =
𝑄(⋅, 𝜔)𝜓 where 𝜓 is a solution of the canonical ODEs in (4.45) with frequency 𝜔.
(ii) If 𝑦 = 𝑄(⋅, 𝜔)𝜓 then 𝑦 is Floquet solution of the canonical DAEs with wavenumberfrequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚 if and only if 𝜓 is a
Floquet solution of the canonical ODEs with wavenumber-frequency pair (𝑘, 𝜔), Floquet
multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚.
(iii) If 𝑦 = 𝑄(⋅, 𝜔 0 )𝜓 is a solution of the canonical DAEs with frequency 𝜔 0 ∈ Ωℝ then its
energy flux and energy density are in 𝐿1loc (ℝ) and
⟨𝑖G 𝑦, 𝑦⟩ℂ = ⟨𝑖J 𝜓, 𝜓⟩ℂ , ⟨𝑉𝜔 (⋅, 𝜔 0 )𝑦, 𝑦⟩ℂ = ⟨𝐴𝜔 (⋅, 𝜔 0 )𝜓, 𝜓⟩ℂ .
(4.46)
To summarize this theorem, the study of canonical DAEs (with index-1 hypothesis), including the Floquet, spectral, and perturbation theory, is reduced to the study of canonical
ODEs and the theory already developed in this chapter. In [60] we will elaborate on this
correspondence in more detail and discuss the spectral perturbation theory as well.
4.4
Proofs
Proofs for Section 4.2.1
We begin by introducing a differential operator associated to the canonical ODEs in (4.13).
We consider the frequency dependent operator
𝑇 (𝜔)𝜓 := 𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓,
109
1,1
𝜓 ∈ (𝑊loc
(ℝ))𝑛 ,
𝜔 ∈ Ω.
(4.47)
1,1
Lemma 51 For each 𝜔 ∈ Ω, 𝑇 (𝜔) is a linear operator with 𝑇 (𝜔) : (𝑊loc
(ℝ))𝑛 → (𝐿1loc (ℝ))𝑛 .
Proof.
1,1
(ℝ))𝑛 → (𝐿1loc (ℝ))𝑛 given by
We begin by proving the operator 𝑇 (𝜔) : (𝑊loc
1,1
1,1
(ℝ))𝑛 we
(ℝ))𝑛 . Then by definition of the space (𝑊loc
(4.47) is well-defined. Let 𝜓 ∈ (𝑊loc
have 𝜓 ′ ∈ (𝐿1loc (ℝ))𝑛 . We now show J −1 𝐴(⋅, 𝜔)𝜓 ∈ (𝐿1loc (ℝ))𝑛 . To do this let (𝑎, 𝑏) ⊆
ℝ be any bounded interval. It follows from Lemma 88 that 𝐴(⋅, 𝜔)∣(𝑎,𝑏) ∈ 𝑀𝑛 (𝐿1 (𝑎, 𝑏))
and from Lemma 69 that 𝜓∣(𝑎,𝑏) ∈ (𝑊 1,1 (𝑎, 𝑏))𝑛 . It then follows from Lemma 83 that
𝐴(⋅, 𝜔)∣(𝑎,𝑏) 𝜓∣(𝑎,𝑏) ∈ (𝐿1 (𝑎, 𝑏))𝑛 and hence J −1 𝐴(⋅, 𝜔)∣(𝑎,𝑏) 𝜓∣(𝑎,𝑏) ∈ (𝐿1 (𝑎, 𝑏))𝑛 . From this
it follows that (J −1 𝐴(⋅, 𝜔)𝜓)∣(𝑎,𝑏) = J −1 𝐴(⋅, 𝜔)∣(𝑎,𝑏) 𝜓∣(𝑎,𝑏) ∈ (𝐿1 (𝑎, 𝑏))𝑛 for every bounded
interval (𝑎, 𝑏) ⊆ ℝ, this implies J −1 𝐴(⋅, 𝜔)𝜓 ∈ (𝐿1loc (ℝ))𝑛 . And hence 𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓 ∈
1,1
(𝐿1loc (ℝ))𝑛 . This proves that the operator 𝑇 (𝜔) : (𝑊loc
(ℝ))𝑛 → (𝐿1loc (ℝ))𝑛 given by (4.47) is
well-defined. Therefore, since its obvious the operator 𝑇 (𝜔) is linear, the proof is complete.
Lemma 52 𝜓 is a solution of the canonical ODEs in (4.13) with frequency 𝜔 ∈ Ω if and
only if 𝜓 ∈ ker(𝑇 (𝜔)).
Proof.
Let 𝜓 is a solution of the canonical ODEs in (4.13) with frequency 𝜔 ∈ Ω. This
1,1
means by Definition 18 that 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and for a.e. 𝑥 ∈ ℝ 𝜓 satisfies the equation
J 𝜓 ′ (𝑥) = 𝐴(𝑥, 𝜔)𝜓(𝑥).
Hence for a.e. 𝑥 ∈ ℝ satisfies the equation
𝜓 ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)𝜓(𝑥).
110
But this implies, since as we showed in the proof of Lemma 4.47 that J −1 𝐴(⋅, 𝜔)𝜓 ∈
(𝐿1loc (ℝ))𝑛 , that 𝑇 (𝜔)𝜓 = 𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓 = 0 in (𝐿1loc (ℝ))𝑛 . This proves 𝜓 ∈ ker(𝑇 (𝜔)).
1,1
Conversely, if 𝜓 ∈ ker(𝑇 (𝜔)) then 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and 0 = 𝑇 (𝜔)𝜓 = 𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓 in
(𝐿1loc (ℝ))𝑛 . This implies that for a.e. 𝑥 ∈ ℝ, 𝜓 satisfies the equation
𝜓 ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)𝜓(𝑥),
or equivalently, the equation
J 𝜓 ′ (𝑥) = 𝐴(𝑥, 𝜔)𝜓(𝑥).
Therefore by Definition 18 this means 𝜓 is a solution of the canonical ODEs in (4.13) with
frequency 𝜔 ∈ Ω. This completes the proof.
Let (𝑎, 𝑏) ⊆ ℝ be a bounded interval. Then we can consider (𝑊 1,1 (𝑎, 𝑏))𝑛 and (𝐿1 (𝑎, 𝑏))𝑛 as
1,1
a subspace of (𝑊loc
(ℝ))𝑛 and (𝐿1loc (ℝ))𝑛 , respectively, by identifying them with the image
of those latter spaces under the surjective map 𝑓 7→ 𝑓 ∣(𝑎,𝑏) . Then we can restrict the domain
of 𝑇 (𝜔) to the subspace (𝑊 1,1 (𝑎, 𝑏))𝑛 and it becomes a differential operator on (𝑊 1,1 (𝑎, 𝑏))𝑛
with range in (𝐿1 (𝑎, 𝑏))𝑛 . It is with this perspective in mind that we introduce and consider
the following frequency dependent operator
𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 := 𝜓 ′ − J −1 𝐴(⋅, 𝜔)∣(𝑎,𝑏) 𝜓,
𝜓 ∈ (𝑊 1,1 (𝑎, 𝑏))𝑛 ,
𝜔 ∈ Ω.
(4.48)
Proposition 53 For each 𝜔 ∈ Ω we have
𝑇 ∣(𝑎,𝑏) (𝜔) : (𝑊 1,1 (𝑎, 𝑏))𝑛 → (𝐿1 (𝑎, 𝑏))𝑛
111
(4.49)
𝑇 ∣(𝑎,𝑏) (𝜔) ∈ L ((𝑊 1,1 (𝑎, 𝑏))𝑛 , (𝐿1 (𝑎, 𝑏))𝑛 )
(4.50)
𝑇 ∣(𝑎,𝑏) ∈ 𝒪(Ω, L ((𝑊 1,1 (𝑎, 𝑏))𝑛 , (𝐿1 (𝑎, 𝑏))𝑛 )).
(4.51)
(𝑇 (𝜔)𝜓)∣(𝑎,𝑏) = 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) ,
(4.52)
Moreover,
1,1
for every 𝜔 ∈ Ω, every 𝜓 ∈ (𝑊loc
(ℝ))𝑛 , and every bounded interval (𝑎, 𝑏) ⊆ ℝ.
Proof.
To be begin we note that by hypothesis 𝐴 ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝕋))). Let (𝑎, 𝑏) ⊆ ℝ)
be any bounded interval. It follows from Lemma 88 that for each 𝜔 ∈ Ω, 𝐴∣(𝑎,𝑏) (⋅, 𝜔) :=
𝐴(⋅, 𝜔)∣(𝑎,𝑏) ∈ 𝑀𝑛 (𝐿1 (𝑎, 𝑏)). It then follows from Lemma 93 that 𝐴∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝑎, 𝑏))).
It then follows from Lemma 89 with 𝑝 = ∞, 𝑞 = 𝑠 = 1 that we have 𝐴˜ := J −1 𝐴∣(𝑎,𝑏) ∈
𝒪(Ω, 𝑀𝑛 (𝐿1 (𝑎, 𝑏))). But according to our definition we have
˜ 𝜔)𝜓,
𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 𝜓 ′ − 𝐴(⋅,
𝜓 ∈ (𝑊 1,1 (𝑎, 𝑏))𝑛 ,
𝜔 ∈ Ω.
with 𝐴˜ ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝑎, 𝑏))). It now follows from the fact ()′ ∈ L ((𝑊 1,1 (𝑎, 𝑏))𝑛 , (𝐿1 (𝑎, 𝑏))𝑛 )
and Lemma 83 that 𝑇 ∣(𝑎,𝑏) (𝜔) : (𝑊 1,1 (𝑎, 𝑏))𝑛 → (𝐿1 (𝑎, 𝑏))𝑛 is a well-defined linear operator
and 𝑇 ∣(𝑎,𝑏) (𝜔) ∈ L ((𝑊 1,1 (𝑎, 𝑏))𝑛 , (𝐿1 (𝑎, 𝑏))𝑛 ), for each 𝜔 ∈ Ω. It follows now from both
Lemma 79 and Lemma 90 that 𝑇 ∣(𝑎,𝑏) ∈ 𝒪(Ω, L ((𝑊 1,1 (𝑎, 𝑏))𝑛 , (𝐿1 (𝑎, 𝑏))𝑛 )).
1,1
Now let 𝜔 ∈ Ω, 𝜓 ∈ (𝑊loc
(ℝ))𝑛 , and (𝑎, 𝑏) ⊆ ℝ) be a bounded interval. Then by Lemma
69 we have 𝜓∣(𝑎,𝑏) ∈ (𝑊 1,1 (𝑎, 𝑏))𝑛 and so 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) ∈ (𝐿1 (𝑎, 𝑏))𝑛 . But (𝑇 (𝜔)𝜓)∣(𝑎,𝑏) ∈
(𝐿1 (𝑎, 𝑏))𝑛 since 𝑇 (𝜔)𝜓 ∈ (𝐿1loc (ℝ))𝑛 and by Lemma 69 we have 𝜓∣′(𝑎,𝑏) = 𝜓 ′ ∣(𝑎,𝑏) . Thus we
conclude that
(𝑇 (𝜔)𝜓)∣(𝑎,𝑏) = (𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓)∣(𝑎,𝑏) = 𝜓∣′(𝑎,𝑏) − J −1 𝐴(⋅, 𝜔)∣(𝑎,𝑏) 𝜓∣(𝑎,𝑏)
112
= 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) .
And hence we have shown that (𝑇 (𝜔)𝜓)∣(𝑎,𝑏) = 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) for every 𝜔 ∈ Ω, every
1,1
𝜓 ∈ (𝑊loc
(ℝ))𝑛 , and every bounded interval (𝑎, 𝑏) ⊆ ℝ. This completes the proof.
Corollary 54 Let 𝜔 ∈ Ω. Then the following statements are equivalent:
(i) 𝜓 is a solution of the canonical ODEs in (4.13) with frequency 𝜔.
(ii) 𝜓 ∈ ker(𝑇 (𝜔)).
1,1
(iii) 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and 𝜓∣(𝑎,𝑏) ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)) every bounded interval (𝑎, 𝑏) ⊆ ℝ.
Proof. We have already proven in Lemma 52 that statement (i) is equivalent to statement
(ii). We now complete the proof of this corollary by proving statement (ii) is equivalent to
1,1
statement (iii). Suppose 𝜓 ∈ ker(𝑇 (𝜔)) for some 𝜔 ∈ Ω. Then 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and for any
bounded interval (𝑎, 𝑏) ⊆ ℝ by Proposition 53 we have 0 = (𝑇 (𝜔)𝜓)∣(𝑎,𝑏) = 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) im1,1
plying 𝜓∣(𝑎,𝑏) ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)). Conversely, if 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and 𝜓∣(𝑎,𝑏) ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)) for
every bounded interval (𝑎, 𝑏) ⊆ ℝ then by Proposition 53 this implies 0 = 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) =
(𝑇 (𝜔)𝜓)∣(𝑎,𝑏) for every bounded interval (𝑎, 𝑏) ⊆ ℝ. But since 𝑇 (𝜔)𝜓 ∈ (𝐿1loc (ℝ))𝑛 this
implies 𝑇 (𝜔)𝜓 = 0 in (𝐿1loc (ℝ))𝑛 and therefore 𝜓 ∈ ker(𝑇 (𝜔)). This completes the proof.
In the following proposition, we collect together some facts from [45, §II.2.5] that will be
need in this chapter.
Proposition 55 For every bounded interval (𝑎, 𝑏) ⊆ ℝ with 0 ∈ (𝑎, 𝑏) and for each 𝜔 ∈ Ω,
there exists a unique function Ψ∣(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) satisfying for a.e. 𝑥 ∈ (𝑎, 𝑏) the
113
matrix differential equation with initial condition
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
(4.53)
Moreover, the following statements are true:
(i) If 𝜓 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)) then there exists a unique 𝛾 ∈ ℂ𝑛 such that 𝜓 = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾.
Conversely, if 𝛾 ∈ ℂ𝑛 and we define 𝜓 := Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾 then 𝜓 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)).
(ii) For every (𝑥, 𝜔) ∈ [𝑎, 𝑏] × Ω,
∫
Ψ∣(𝑎,𝑏) (𝑥, 𝜔) = 𝐼𝑛 +
𝑥
J −1 𝐴(𝑡, 𝜔)Ψ∣(𝑎,𝑏) (𝑡, 𝜔)𝑑𝑡.
(4.54)
0
(iii) As a function of frequency, Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))).
−1
(iv) For every (𝑥, 𝜔) ∈ [𝑎, 𝑏] × Ω, Ψ∣−1
exists and as function of
(𝑎,𝑏) (𝑥, 𝜔) := Ψ∣(𝑎,𝑏) (𝑥, 𝜔)
1,1
position Ψ∣−1
(𝑎, 𝑏)).
(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊
Proof.
Let us fix a bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0. We proceed by proving the
existence and uniqueness portions of this proposition first and then we will prove statements
(i)–(iv).
We begin by reminding the reader that, by the proof of Proposition 53, we have
˜ 𝜔)𝜓,
𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 𝜓 ′ − 𝐴(⋅,
𝜓 ∈ (𝑊 1,1 (𝑎, 𝑏))𝑛 ,
𝜔 ∈ Ω.
˜ 𝜔) := J −1 𝐴(⋅, 𝜔)∣(𝑎,𝑏) for 𝜔 ∈ Ω. Following [45, p.
with 𝐴˜ ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝑎, 𝑏))) where 𝐴(⋅,
69, §II.2.5, Definition 2.5.2], we introduce two new definitions.
Definition 25 Let 𝜔 0 ∈ Ω. A matrix 𝑌0 ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) is called a fundamental matrix
of 𝑇 ∣(𝑎,𝑏) (𝜔 0 )𝜓 = 0 if for each 𝜓 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔 0 )) there exists a 𝛾 ∈ ℂ𝑛 such that 𝜓 = 𝑌0 𝛾.
114
A matrix function 𝑌 : Ω → 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) is called a fundamental matrix function of
𝑇 ∣(𝑎,𝑏) 𝜓 = 0 if 𝑌 (𝜔) is a fundamental matrix of 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 for each 𝜔 ∈ Ω.
We now prove the existence portion of this proposition. By [45, p. 69, §II.2.5, Theorem 2.5.3]
there exists a fundamental matrix function 𝑌 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))) of 𝑇 ∣(𝑎,𝑏) 𝜓 = 0 with
the property that for all 𝜔 ∈ Ω, 𝑌 (𝑎, 𝜔) = 𝐼𝑛 and 𝑌 (⋅, 𝜔) is invertible in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)).
Hence for each 𝜔 ∈ Ω, there exists 𝑌 −1 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) such that 𝑌 −1 (⋅, 𝜔)𝑌 (⋅, 𝜔) =
𝑌 (⋅, 𝜔)𝑌 −1 (⋅, 𝜔) = 𝐼𝑛 . But the functions 𝑌 (⋅, 𝜔), 𝑌 −1 (⋅, 𝜔) : [𝑎, 𝑏] → 𝑀𝑛 (ℂ) are continuous
implying 𝑌 (𝑥, 𝜔)𝑌 −1 (𝑥, 𝜔) = 𝐼𝑛 for all 𝑥 ∈ [𝑎, 𝑏]. Thus 𝑌 (𝑥, 𝜔) is invertible for all 𝑥 ∈ [𝑎, 𝑏]
and 𝑌 −1 (𝑥, 𝜔) = 𝑌 (𝑥, 𝜔)−1 .
Now for each 𝜔 ∈ Ω, we define Ψ∣(𝑎,𝑏) (⋅, 𝜔) := 𝑌 (⋅, 𝜔)𝑌 −1 (0, 𝜔). It follows by [45, p. 71,
§II.2.5, Proposition 2.5.4] that Ψ∣(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) is a fundamental matrix of
𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 0 for each 𝜔 ∈ Ω. It then follows from [45, p. 72, §II.2.5, Corollary 2.5.5] that
Ψ∣(𝑎,𝑏) (⋅, 𝜔) is a solution of the equation
˜ 𝜔)Ψ = 0
Ψ′ − 𝐴(⋅,
in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) for each 𝜔 ∈ Ω. But this means for each 𝜔 ∈ Ω, Ψ∣(𝑎,𝑏) (⋅, 𝜔) satisfies for
a.e. 𝑥 ∈ (𝑎, 𝑏) the matrix differential equation with initial condition
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
(4.55)
This proves existence.
We now prove the uniqueness portion of this proposition. Let 𝜔 ∈ Ω. Suppose Φ ∈
𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) satisfies for a.e. 𝑥 ∈ (𝑎, 𝑏) the matrix differential equation with initial condi-
115
tion
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
(4.56)
It follows that Φ is a solution of the equation
˜ 𝜔)Ψ = 0
Ψ′ − 𝐴(⋅,
in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) with 0 ∈ [𝑎, 𝑏] and Φ(0) = 𝐼𝑛 . This implies by [45, p. 73, §II.2.5, Proposition 2.5.9] that Φ is a fundamental matrix of 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 0. But from the existence
portion of this proof, 𝑌 (⋅, 𝜔) is also a fundamental matrix of 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 0 and so by [45,
p. 71, §II.2.5, Proposition 2.5.4] there exists an invertible matrix 𝐶 ∈ 𝑀𝑛 (ℂ) such that Φ =
𝑌 (⋅, 𝜔)𝐶. Thus since 𝐼𝑛 = Φ(0) = 𝑌 (0, 𝜔)𝐶 this implies Φ = 𝑌 (⋅, 𝜔)𝑌 −1 (0, 𝜔) = Ψ∣(𝑎,𝑏) (⋅, 𝜔).
This proves uniqueness.
Now we will prove statements (i)–(iv) of this proposition. We begin with statement (i). First,
it follows from what we just proved that for each 𝜔 ∈ Ω, Ψ∣(𝑎,𝑏) (⋅, 𝜔) = 𝑌 (⋅, 𝜔)𝑌 −1 (0, 𝜔) ∈
𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) is a fundamental matrix of 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓 = 0. By Definition 25 this means
if 𝜓 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)) then there exists 𝛾 ∈ (ℂ)𝑛 such that 𝜓 = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾. But as
discussed 𝑌 (⋅, 𝜔) is invertible with inverse 𝑌 −1 (⋅, 𝜔)−1 ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)). Uniqueness of the
representation 𝜓 = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾 follows since 𝛾 = 𝑌 (0, 𝜔)𝑌 −1 (⋅, 𝜔)𝜓 in (𝑊 1,1 (𝑎, 𝑏))𝑛 .
Let 𝜔 ∈ Ω, let 𝛾 ∈ (ℂ)𝑛 , and define 𝜓 := Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾. Now because 𝜓 = 𝑌 (⋅, 𝜔)(𝑌 −1 (0, 𝜔)𝛾)
and 𝑌 is a fundamental matrix function of 𝑇 ∣(𝑎,𝑏) 𝜓 = 0, it follows by Definition 25 that
𝜓 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)). This completes the proof of statement (i).
116
The proof of statement (ii) is straightforward. Let (𝑥, 𝜔) ∈ [𝑎, 𝑏] × Ω. Then Ψ := Ψ∣(𝑎,𝑏) (⋅, 𝜔)
satisfies (4.53) and so upon integrating and applying Lemma 70 we arrive at
𝑥
∫
∫
′
𝑥
Ψ (𝑡, 𝜔)𝑑𝑡 = 𝐼𝑛 +
Ψ∣(𝑎,𝑏) (𝑥, 𝜔) = 𝐼𝑛 +
∫0 𝑥
= 𝐼𝑛 +
J −1 𝐴(𝑡, 𝜔)Ψ(𝑡, 𝜔)𝑑𝑡
0
J −1 𝐴(𝑡, 𝜔)Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝑑𝑡.
0
This proves statement (ii).
We complete the proof of this proposition now by proving statements (iii) and (iv). We have
already shown that there exists 𝑌 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))) such that for each 𝜔 ∈ Ω, 𝑌 (⋅, 𝜔)
is invertible in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) and, denoting this inverse in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) by 𝑌 −1 (⋅, 𝜔), we
have for every 𝑥 ∈ [𝑎, 𝑏], 𝑌 (𝑥, 𝜔) is invertible with 𝑌 −1 (𝑥, 𝜔) = 𝑌 (𝑥, 𝜔)−1 and Ψ∣(𝑎,𝑏) (⋅, 𝜔) =
𝑌 (⋅, 𝜔)𝑌 −1 (0, 𝜔). This proves statement (iv) since for every (𝑥, 𝜔) ∈ [𝑎, 𝑏] × Ω, we have
−1
Ψ∣(𝑎,𝑏) (𝑥, 𝜔) is invertible with inverse Ψ∣−1
= 𝑌 (0, 𝜔)𝑌 −1 (𝑥, 𝜔) and
(𝑎,𝑏) (𝑥, 𝜔) := Ψ∣(𝑎,𝑏) (𝑥, 𝜔)
−1
Ψ∣−1
(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) by Lemma 72.
(𝑎,𝑏) (⋅, 𝜔) = 𝑌 (0, 𝜔)𝑌
We will now show Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))). Since 𝑌 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))) and
𝑌 (⋅, 𝜔) is invertible in 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) with the inverse 𝑌 −1 (⋅, 𝜔) for every 𝜔 ∈ Ω, it follows
from [45, p. 66, §II.2.3, Proposition 2.3.3] and [45, p. 7, §I.1.2, Proposition 1.2.5] that
𝑌 −1 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))). It follows from this and Lemma 91 that if we let 𝛿 0 denote the
evaluation map at 𝑥 = 0 then 𝑌 −1 (0, ⋅) = 𝛿 0 𝑌 −1 ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)). But the map 𝜄 : 𝑀𝑛 (ℂ) →
𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) given by (𝜄𝐵)(⋅) := 𝐵 is a continuous linear map and so by Lemma 78 we
have 𝑌 −1 (0, ⋅) = 𝜄𝛿 0 𝑌 −1 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))). Therefore by Lemma 78 and Lemma 72
we have Ψ∣(𝑎,𝑏) = 𝑌 −1 𝜄𝛿 0 𝑌 −1 ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))). This proves statement (iii) and hence
the proof of the proposition is complete.
From the functions Ψ∣(𝑎,𝑏) (⋅, 𝜔) we define a matrix-valued function Ψ : ℝ × Ω → 𝑀𝑛 (ℂ) by
Ψ(𝑥, 𝜔) := Ψ∣(𝑎,𝑏) (𝑥, 𝜔)
117
(4.57)
for each (𝑥, 𝜔) ∈ ℝ × Ω and for any interval (𝑎, 𝑏) containing 0 and 𝑥.
Lemma 56 The matrix-valued function Ψ : ℝ × Ω → 𝑀𝑛 (ℂ) is well-defined. Moreover, for
1,1
(ℝ)) and for any 𝑎, 𝑏 ∈ ℝ with 𝑎 < 0 < 𝑏 we have
each 𝜔 ∈ Ω, Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
Ψ(⋅, 𝜔)∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (⋅, 𝜔).
Proof.
(4.58)
First, fix 𝜔 ∈ Ω and let (𝑢, 𝑣) be an interval such that 0 ∈ (𝑢, 𝑣) ⊆ (𝑎, 𝑏). We will
begin by showing
Ψ∣(𝑎,𝑏) (⋅, 𝜔)∣(𝑢,𝑣) = Ψ∣(𝑢,𝑣) (⋅, 𝜔).
(4.59)
We define 𝑌 := Ψ∣(𝑎,𝑏) (⋅, 𝜔)∣(𝑢,𝑣) so that 𝑌 (0) = 𝐼𝑛 . Then by [45, §II.2.2, Proposition 2.2.1]
we have 𝑌 ∈ 𝑀𝑛 (𝑊 1,1 (𝑢, 𝑣)) and 𝑌 ′ = Ψ∣(𝑎,𝑏) (⋅, 𝜔)′ ∣(𝑢,𝑣) . This implies by the definition of
Ψ∣(𝑎,𝑏) (⋅, 𝜔) in Proposition 55 that 𝑌 is a function in 𝑀𝑛 (𝑊 1,1 (𝑢, 𝑣)) satisfying the for a.e.
𝑥 ∈ (𝑢, 𝑣) the matrix differential equation with initial condition
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
By the uniqueness portion of Proposition 55 we conclude that 𝑌 := Ψ∣(𝑢,𝑣) (⋅, 𝜔). Thus we
have shown that Ψ∣(𝑎,𝑏) (⋅, 𝜔)∣(𝑢,𝑣) = Ψ∣(𝑢,𝑣) (⋅, 𝜔).
We now prove the function Ψ defined by (4.57) is well-defined. Let (𝑥, 𝜔) ∈ ℝ × Ω. Let
(𝑎1 , 𝑏1 ) and (𝑎2 , 𝑏2 ) be any intervals containing 0 and 𝑥. Then there exists 𝑢, 𝑣 such that
(𝑢, 𝑣) = (𝑎1 , 𝑏1 ) ∩ (𝑎2 , 𝑏2 ). Now since 0 and 𝑥 are in the interval (𝑢, 𝑣) then Ψ∣(𝑢,𝑣) (⋅, 𝜔)
and Ψ∣(𝑢,𝑣) (𝑥, 𝜔) are well-defined by Proposition 55. And thus since 0 ∈ (𝑢, 𝑣) ⊆ (𝑎𝑗 , 𝑏𝑗 ) for
𝑗 = 1, 2, we get from (4.59) that
Ψ∣(𝑎1 ,𝑏1 ) (𝑥, 𝜔) = Ψ∣(𝑢,𝑣) (𝑥, 𝜔) = Ψ∣(𝑎2 ,𝑏2 ) (𝑥, 𝜔).
118
This proves the function Ψ is well-defined.
1,1
(ℝ)) for any 𝜔 ∈ Ω its enough to show, by Lemma 71,
Next, to show that Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
that Ψ(⋅, 𝜔)∣(𝑎,𝑏) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) for any interval (𝑎, 𝑏) containing 0. But this follows from
the definition (4.57) since for any 𝑥 ∈ (𝑎, 𝑏)
Ψ(𝑥, 𝜔) = Ψ∣(𝑎,𝑏) (𝑥, 𝜔)
which implies
Ψ(⋅, 𝜔)∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)).
This completes the proof of the lemma.
We are in a position now to prove Proposition 33.
Proof. [Proposition 33] To begin this proof we need to show that the matrix-valued
function Ψ : ℝ × Ω → 𝑀𝑛 (ℂ), as defined by (4.57), for a fixed 𝜔 ∈ Ω satisfies a.e. 𝑥 ∈ ℝ the
matrix differential equation with initial condition
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 ,
1,1
and Ψ(⋅, 𝜔) is the only function in 𝑀𝑛 (𝑊loc
(ℝ)) that does so. After which we will prove that
it has the properties described in Proposition 33.
This first part follows now from the previous lemma since for any 𝜔 ∈ Ω and interval (𝑎, 𝑏)
containing 0 we have Ψ(𝑥, 𝜔)∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (𝑥, 𝜔) for all 𝑥 ∈ (𝑎, 𝑏) and so by Proposition
55 the function Ψ(⋅, 𝜔) satisfies a.e. 𝑥 ∈ (𝑎, 𝑏) the matrix differential equation with initial
119
condition
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
Hence this implies Ψ(⋅, 𝜔) satisfies a.e. 𝑥 ∈ ℝ the matrix differential equation with initial
condition
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
1,1
Lets now prove uniqueness. Fix 𝜔 ∈ Ω. Suppose Ψ1 ∈ 𝑀𝑛 (𝑊loc
(ℝ)) satisfied a.e. 𝑥 ∈ ℝ the
matrix differential equation with initial condition
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
Then for any interval (𝑎, 𝑏) containing 0, Ψ1 ∣(𝑎,𝑏) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) and satisfies a.e. 𝑥 ∈ (𝑎, 𝑏)
the matrix differential equation with initial condition
Ψ′ (𝑥) = J −1 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ(0) = 𝐼𝑛 .
and so by the uniqueness part of Proposition 55 this implies Ψ1 (⋅)∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (⋅, 𝜔). By
definition (4.57) this implies Ψ1 (⋅)∣(𝑎,𝑏) = Ψ(⋅, 𝜔)∣(𝑎,𝑏) . Since this is true for any interval (𝑎, 𝑏)
containing 0 we must have Ψ1 (⋅) = Ψ(⋅, 𝜔). This proves uniqueness.
Now we will prove Propositions 33.(i)–(vi). We start with property (i). Suppose that 𝜓 is
a solution of the canonical ODEs in (4.13) at the frequency 𝜔 ∈ Ω. Then we must prove
1,1
there exists a unique 𝛾 ∈ ℂ𝑛 such that, as elements of (𝑊loc
(ℝ))𝑛 , 𝜓 = Ψ(⋅, 𝜔)𝛾. We begin
by proving the uniqueness statement. Suppose that there did exist 𝛾 ∈ ℂ𝑛 such that 𝜓 =
1,1
1,1
Ψ(⋅, 𝜔)𝛾. It follows from this, since 𝜓 ∈ (𝑊loc
(ℝ))𝑛 and Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) by Lemma
120
70 they have unique locally absolutely continuous representative functions 𝜓(⋅) : ℝ → ℂ𝑛
and Ψ(⋅, 𝜔) : ℝ → 𝑀𝑛 (ℂ) which are continuous, that 𝜓(𝑥) = Ψ(𝑥, 𝜔)𝛾 for all 𝑥 ∈ ℝ. In
particular, since Ψ(0, 𝜔) = 𝐼𝑛 this means 𝜓(0) = Ψ(0, 𝜔)𝛾 = 𝛾. This proves uniqueness of 𝛾.
Lets now prove existence. By Lemma 52 we have 𝜓 ∈ ker(𝑇 (𝜔)). By Corollary 54 we have
𝜓∣(𝑎,𝑏) ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)) for any bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0 and so by Proposition
55.((i)) there exists a 𝛾∣(𝑎,𝑏) ∈ ℂ𝑛 such that 𝜓∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾∣(𝑎,𝑏) . But 0 ∈ (𝑎, 𝑏) and so
by (4.57) we have 𝛾 := 𝜓(0) = 𝜓(0)∣(𝑎,𝑏) = 𝜓∣(𝑎,𝑏) (0) = Ψ∣(𝑎,𝑏) (0, 𝜔)𝛾∣(𝑎,𝑏) = Ψ(0, 𝜔)𝛾∣(𝑎,𝑏) =
𝛾∣(𝑎,𝑏) . Thus from this and Lemma 56 we have 𝜓∣(𝑎,𝑏) = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾 = Ψ(⋅, 𝜔)∣(𝑎,𝑏) 𝛾 =
Ψ(⋅, 𝜔)𝛾∣(𝑎,𝑏) . But since this is true for every bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0 this
1,1
implies as elements of (𝑊loc
(ℝ))𝑛 we have 𝜓 = Ψ(⋅, 𝜔)𝛾. This proves existence.
1,1
Conversely, let 𝛾 ∈ (ℝ)𝑛 and define 𝜓 := Ψ(⋅, 𝜔)𝛾. By Lemma 73 we have 𝜓 ∈ (𝑊loc
(ℝ))𝑛 .
Then for any bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0 we have by Lemma 56 and Proposition
55.(i) that 𝜓∣(𝑎,𝑏) = Ψ(⋅, 𝜔)∣(𝑎,𝑏) 𝛾 = Ψ∣(𝑎,𝑏) (⋅, 𝜔)𝛾 ∈ ker(𝑇 ∣(𝑎,𝑏) (𝜔)). It follows from this and
Proposition 53 that we have (𝑇 (𝜔)𝜓)∣(𝑎,𝑏) = 𝑇 ∣(𝑎,𝑏) (𝜔)𝜓∣(𝑎,𝑏) = 0 in (𝐿1 (𝑎, 𝑏))𝑛 . But this is
true for every bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0 and so it follows this and Lemma 4.47
that 𝑇 (𝜔)𝜓 = 0 in (𝐿1loc (ℝ))𝑛 which means 𝜓 ∈ ker(𝑇 (𝜔)). Therefore by Corollary 54 this
implies 𝜓 is a solution of the canonical ODEs in (4.13) with frequency 𝜔. This completes
the proof of property (i).
Next we will prove property (ii). Let (𝑥, 𝜔) ∈ ℝ × Ω. Take any interval (𝑎, 𝑏) containing 0
and 𝑥. Then by Proposition 55 and the previous lemma we have
∫
𝑥
Ψ(𝑥, 𝜔) = Ψ∣(𝑎,𝑏) (𝑥, 𝜔) = 𝐼𝑛 +
J −1 𝐴(𝑡, 𝜔)Ψ∣(𝑎,𝑏) (𝑡, 𝜔)𝑑𝑡
0
∫ 𝑥
= 𝐼𝑛 +
J −1 𝐴(𝑡, 𝜔)Ψ(𝑡, 𝜔)𝑑𝑡.
0
This proves property (ii).
121
Next we will prove property (iii). That is, we must show for each fixed 𝑥 ∈ ℝ, that Ψ(𝑥, ⋅) ∈
𝒪(Ω, 𝑀𝑛 (ℂ)). To do this we fix 𝑥 ∈ ℝ and choose any bounded interval (𝑎, 𝑏) ⊆ ℝ with
0, 𝑥 ∈ (𝑎, 𝑏). Then by (4.57) we have Ψ(𝑥, ⋅) = Ψ∣(𝑎,𝑏) (𝑥, ⋅). Let 𝛿 𝑥 : 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑛 (ℂ)
be the evaluation map at 𝑥 defined by 𝛿 𝑥 𝐵 := 𝐵(𝑥) – the value of the unique absolutely
continuous representative of 𝐵 ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) at 𝑥. Then by Proposition 55.(iii) we
have Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))) and so by Lemma 91 we have 𝛿 𝑥 Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)).
And thus we have Ψ(𝑥, ⋅) = Ψ∣(𝑎,𝑏) (𝑥, ⋅) = 𝛿 𝑥 Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)) as desired. This proves
property (iii).
Next we prove property (iv). To start, since Ψ(𝑥, ⋅) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)) for each 𝑥 ∈ ℝ, we can
define Ψ𝜔 : ℝ × Ω → 𝑀𝑛 (ℂ) to be the partial derivative with respect to frequency in the
𝑀𝑛 (ℂ) norm of the function Ψ : ℝ × Ω → 𝑀𝑛 (ℂ), i.e.,
Ψ𝜔 (𝑥, 𝜔 0 ) := lim (𝜔 − 𝜔 0 )−1 (Ψ(𝑥, 𝜔) − Ψ(𝑥, 𝜔 0 )), ∀(𝑥, 𝜔 0 ) ∈ ℝ × Ω.
𝜔→𝜔 0
1,1
Our goal is to prove that for any (𝑥, 𝜔 0 ) ∈ ℝ × Ω, Ψ𝜔 (⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) and
∫
Ψ𝜔 (𝑥, 𝜔 0 ) =
𝑥
J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 ) + J −1 𝐴(𝑡, 𝜔 0 )Ψ𝜔 (𝑡, 𝜔 0 )𝑑𝑡.
0
To do this we begin by choosing any bounded interval (𝑎, 𝑏) ⊆ ℝ with 0 ∈ (𝑎, 𝑏). Next, let
ℐ : 𝑀𝑛 (𝐿1 (𝑎, 𝑏)) → 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)) be the integral map defined by
∫
(ℐ𝐵)(𝑥) :=
𝑥
𝐵(𝑡)𝑑𝑡,
𝐵 ∈ 𝑀𝑛 (𝐿1 (𝑎, 𝑏)),
𝑥 ∈ (𝑎, 𝑏).
0
Now by hypothesis we have 𝐴 ∈ 𝒪(Ω, 𝐿1 (𝕋)) and so by Lemma 93 we have 𝐴∣(𝑎,𝑏) ∈
𝒪(Ω, 𝐿1 (𝑎, 𝑏)) and (𝐴∣(𝑎,𝑏) )𝜔 = 𝐴𝜔 ∣(𝑎,𝑏) . It then follows from this and Lemma 89 with
𝑝 = ∞, 𝑞 = 𝑠 = 1 that J −1 𝐴∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝐿1 (𝑎, 𝑏)) and (J −1 𝐴∣(𝑎,𝑏) )𝜔 = J −1 𝐴𝜔 ∣(𝑎,𝑏) .
From this and the fact by Proposition 55.(iii) we have Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))), it follows by Lemma 90 with 𝑝 = 1 that J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑛 (𝐿1 (𝑎, 𝑏))) whose derivative
122
is (J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) )𝜔 = J −1 𝐴𝜔 ∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) + J −1 𝐴∣(𝑎,𝑏) (Ψ∣(𝑎,𝑏) )𝜔 . It follows from this and
Lemma 92 that we must have ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ) ∈ 𝒪(Ω, 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏))) and
(ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ))𝜔 = ℐ(J −1 𝐴𝜔 ∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) + J −1 𝐴∣(𝑎,𝑏) (Ψ∣(𝑎,𝑏) )𝜔 ).
This and Lemma 91 imply that for any 𝑥 ∈ (𝑎, 𝑏), 𝛿 𝑥 ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ))
and
(𝛿 𝑥 ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ))𝜔 = 𝛿 𝑥 ℐ(J −1 𝐴𝜔 ∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) + J −1 𝐴∣(𝑎,𝑏) (Ψ∣(𝑎,𝑏) )𝜔 ).
(4.60)
Now from Proposition 55.(ii) it follows that we have the identity
Ψ(𝑥, ⋅) = Ψ∣(𝑎,𝑏) (𝑥, ⋅) = 𝛿 𝑥 Ψ∣(𝑎,𝑏) = 𝐼𝑛 + 𝛿 𝑥 ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ), ∀𝑥 ∈ (𝑎, 𝑏).
From this and (4.60) it follows that
Ψ𝜔 (𝑥, ⋅) = 𝛿 𝑥 (Ψ∣(𝑎,𝑏) )𝜔 = (𝐼𝑛 + 𝛿 𝑥 ℐ(J −1 𝐴∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) ))𝜔
= 𝛿 𝑥 ℐ(J −1 𝐴𝜔 ∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) + J −1 𝐴∣(𝑎,𝑏) (Ψ∣(𝑎,𝑏) )𝜔 ), ∀𝑥 ∈ (𝑎, 𝑏).
(4.61)
But this implies for any (𝑥, 𝜔 0 ) ∈ (𝑎, 𝑏) × Ω we have Ψ𝜔 (𝑥, 𝜔 0 ) = 𝛿 𝑥 (Ψ∣(𝑎,𝑏) )𝜔 (𝜔 0 ) =
(Ψ∣(𝑎,𝑏) )𝜔 (𝑥, 𝜔 0 ). And hence this shows that for each 𝜔 0 ∈ Ω,
Ψ𝜔 (⋅, 𝜔 0 )∣(𝑎,𝑏) = (Ψ∣(𝑎,𝑏) )𝜔 (⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝑊 1,1 (𝑎, 𝑏)).
(4.62)
But since this is true for any bounded interval (𝑎, 𝑏) ⊆ ℝ with 0 ∈ (𝑎, 𝑏), then by Lemma
1,1
71 for any 𝜔 0 ∈ Ω we have Ψ𝜔 (⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝑊loc
(ℝ)). Moreover, it follows that for any
(𝑥, 𝜔 0 ) ∈ ℝ × Ω we can choose a bounded interval (𝑎, 𝑏) ⊆ ℝ with 0, 𝑥 ∈ (𝑎, 𝑏) so that by
123
(4.57), (4.61), and (4.62) we have
Ψ𝜔 (𝑥, 𝜔 0 ) = 𝛿 𝑥 ℐ(J −1 𝐴𝜔 ∣(𝑎,𝑏) Ψ∣(𝑎,𝑏) + J −1 𝐴∣(𝑎,𝑏) (Ψ∣(𝑎,𝑏) )𝜔 )(𝜔 0 )
∫ 𝑥
J −1 𝐴𝜔 ∣(𝑎,𝑏) (𝑡, 𝜔 0 )Ψ∣(𝑎,𝑏) (𝑡, 𝜔 0 ) + J −1 𝐴∣(𝑎,𝑏) (𝑡, 𝜔 0 )(Ψ∣(𝑎,𝑏) )𝜔 (𝑡, 𝜔 0 )𝑑𝑡
=
∫0 𝑥
=
J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 ) + J −1 𝐴(𝑡, 𝜔 0 )Ψ𝜔 (𝑡, 𝜔 0 )𝑑𝑡.
0
This completes the proof of Proposition 33.(iv).
Next we prove property (v) of Proposition 33. The goal is to prove for every (𝑥, 𝜔) ∈ ℝ × Ω
the matrix Ψ(𝑥, 𝜔) is invertible and, denoting Ψ−1 (𝑥, 𝜔) := Ψ(𝑥, 𝜔)−1 , that Ψ−1 (⋅, 𝜔) ∈
1,1
𝑀𝑛 (𝑊loc
(ℝ)). Well, by Proposition 55.(iv) for any bounded interval (𝑎, 𝑏) ⊆ ℝ containing
0 and any (𝑥, 𝜔) ∈ (𝑎, 𝑏) × Ω we have Ψ∣(𝑎,𝑏) (𝑥, 𝜔) is invertible and denoting Ψ∣−1
(𝑎,𝑏) (𝑥, 𝜔) :=
1,1
Ψ∣(𝑎,𝑏) (𝑥, 𝜔)−1 we have Ψ∣−1
(𝑎, 𝑏)). But by (4.57) and since Ψ(𝑥, 𝜔) =
(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊
Ψ∣(𝑎,𝑏) (𝑥, 𝜔) this implies Ψ−1 (𝑥, 𝜔) = Ψ∣−1
(𝑎,𝑏) (𝑥, 𝜔) for any (𝑥, 𝜔) ∈ (𝑎, 𝑏) × Ω. This proves for
every (𝑥, 𝜔) ∈ ℝ × Ω the matrix Ψ(𝑥, 𝜔) is invertible and for any 𝜔 ∈ Ω, Ψ−1 (⋅, 𝜔)∣(𝑎,𝑏) =
1,1
Ψ∣−1
(𝑎, 𝑏)) for any bounded interval (𝑎, 𝑏) ⊆ ℝ containing 0. And therefore
(𝑎,𝑏) (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊
1,1
by Lemma 71 this proves Ψ−1 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)) for any 𝜔 ∈ Ω. This completes the proof
of property (v) of Proposition 33.
Finally, we will complete the proof of Proposition 33 by proving property (vi). To do this
we must show that for every (𝑥, 𝜔) ∈ ℝ × Ω we have Ψ(𝑥 + 𝑑, 𝜔) = Ψ(𝑥, 𝜔)Ψ(𝑑, 𝜔). Fix
1,1
𝜔 ∈ Ω and define Φ := Ψ(⋅ + 𝑑, 𝜔)Ψ(𝑑, 𝜔)−1 . We now show that Φ ∈ 𝑀𝑛 (𝑊loc
(ℝ)). First,
1,1
1,1
since Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)), it follows from Lemma 74 that Ψ(⋅ + 𝑑, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
1,1
(ℝ)). It now
Second, since Ψ(𝑑, 𝜔)−1 is just a constant matrix then Ψ(𝑑, 𝜔)−1 ∈ 𝑀𝑛 (𝑊loc
1,1
follows from Lemma 73 that the product Φ = Ψ(⋅ + 𝑑, 𝜔)Ψ(𝑑, 𝜔)−1 ∈ 𝑀𝑛 (𝑊loc
(ℝ)). We will
1,1
now show that Φ ∈ 𝑀𝑛 (𝑊loc
(ℝ)) satisfies a.e. the matrix differential equation with initial
124
condition
1,1
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ ∈ 𝑀𝑛 (𝑊loc
(ℝ)), Ψ(0) = 𝐼𝑛 .
1,1
1,1
(ℝ)) denote the translation operator as
(ℝ)) → 𝑀𝑛 (𝑊loc
To do this we let ℒ𝑑 : 𝑀𝑛 (𝑊loc
defined in Lemma 74. Then Φ = ℒ𝑑 (Ψ(⋅, 𝜔)Ψ(𝑑, 𝜔)−1 ). Then by Lemma 74 and Lemma 73
we have
J Φ′ = J (ℒ𝑑 (Ψ(⋅, 𝜔)Ψ(𝑑, 𝜔)−1 ))′ = J ℒ𝑑 (Ψ(⋅, 𝜔)Ψ(𝑑, 𝜔)−1 )′
= J ℒ𝑑 (Ψ(⋅, 𝜔)′ Ψ(𝑑, 𝜔)−1 ) = ℒ𝑑 (J Ψ(⋅, 𝜔)′ )Ψ(𝑑, 𝜔)−1 .
But Proposition 33 implies J Ψ(⋅, 𝜔)′ = 𝐴(⋅, 𝜔)Ψ(⋅, 𝜔) and since by hypothesis 𝐴(⋅ + 𝑑, 𝜔) =
𝐴(⋅, 𝜔) this implies ℒ𝑑 (𝐴(⋅, 𝜔)Ψ(⋅, 𝜔)) = 𝐴(⋅ + 𝑑, 𝜔)Ψ(⋅ + 𝑑, 𝜔) = 𝐴(⋅, 𝜔)ℒ𝑑 (Ψ(⋅, 𝜔)). Hence
these facts imply
J Φ′ = ℒ𝑑 (J Ψ(⋅, 𝜔)′ )Ψ(𝑑, 𝜔)−1 = ℒ𝑑 (𝐴(⋅, 𝜔)Ψ(⋅, 𝜔))Ψ(𝑑, 𝜔)−1
= 𝐴(⋅, 𝜔)ℒ𝑑 (Ψ(⋅, 𝜔))Ψ(𝑑, 𝜔)−1 = 𝐴(⋅, 𝜔)Φ.
But this is an equality of elements in 𝑀𝑛 (𝐿1loc (ℝ)) and so together with the fact that Φ(0) =
1,1
Ψ(0 + 𝑑, 𝜔)Ψ(𝑑, 𝜔)−1 = 𝐼𝑛 this implies that Φ ∈ 𝑀𝑛 (𝑊loc
(ℝ)) satisfies a.e. the matrix
differential equation with initial condition
1,1
J Ψ′ (𝑥) = 𝐴(𝑥, 𝜔)Ψ(𝑥), Ψ ∈ 𝑀𝑛 (𝑊loc
(ℝ)), Ψ(0) = 𝐼𝑛 .
Now though the uniqueness portion of Proposition 33 implies that we must have Ψ(⋅, 𝜔) = Φ.
Thus Ψ(𝑥 + 𝑑, 𝜔) = Ψ(𝑥, 𝜔)Ψ(𝑑, 𝜔) for a.e. 𝑥 ∈ ℝ and so, since Proposition 33.(ii) implies
the function Ψ(𝑥, 𝜔) is continuous as a function of 𝑥 ∈ ℝ, we must have Ψ(𝑥 + 𝑑, 𝜔) =
Ψ(𝑥, 𝜔)Ψ(𝑑, 𝜔) for all 𝑥 ∈ ℝ. This proves Proposition 33.(vi) and therefore completes the
125
proof of Proposition 33.
We will use Proposition 33 now to prove the Floquet-Lyapunov Theorem.
Proof. [Theorem 34] We fix 𝜔 ∈ Ω. To simplify notation we will denote the matricant
by Φ := Ψ(⋅, 𝜔) and we will agree to treat, in the rest of this proof, all objects as depending
implicitly on the frequency 𝜔. Our goal here is to show that there exists a matrix 𝐾 ∈ 𝑀𝑛 (ℂ)
1,1
and a function 𝐹 ∈ 𝑀𝑛 (𝑊loc
(ℝ)) such that for every 𝑥 ∈ ℝ,
Φ(𝑥) = 𝐹 (𝑥)𝑒𝑖𝑥𝐾 ,
1,1
where 𝐹 (𝑥 + 𝑑) = 𝐹 (𝑥), 𝐹 (0) = 𝐼𝑛 , 𝐹 −1 (𝑥) := 𝐹 (𝑥)−1 exist, and 𝐹 −1 ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
To begin, let 𝜆1 , . . . , 𝜆𝑠 denote the distinct eigenvalues of the monodromy matrix Φ(𝑑). Let
log(𝑧) be any branch of the logarithm which is analytic at each 𝜆𝑗 , for 𝑗 = 1, . . . , 𝑠. Using
the Riesz-Dunford functional calculus as described for matrices in [35, pp. 304–334, §9], it
follows that the matrix 𝐾 :=
1
𝑖𝑑
log(Φ(𝑑)) satisfies
𝑒𝑖𝑑𝐾 = Φ(𝑑),
where 𝑒𝑧 is the exponential function. From the functional calculus it follows that the matrix
1,1
function 𝐵(𝑥) := 𝑒𝑖𝑥𝐾 , for 𝑥 ∈ ℝ has the properties 𝐵 ∈ 𝑀𝑛 (𝑊loc
(ℝ)), 𝐵(𝑥) is invertible
1,1
with inverse 𝐵 −1 (𝑥) := 𝑒−𝑖𝑥𝐾 for 𝑥 ∈ ℝ, and 𝐵 −1 ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
We define the function 𝐹 by 𝐹 (𝑥) := Φ(𝑥)𝑒−𝑖𝑥𝐾 , 𝑥 ∈ ℝ. It then follows from Lemma
1,1
73 that 𝐹 ∈ 𝑀𝑛 (𝑊loc
(ℝ)). We also find that for every 𝑥 ∈ ℝ, 𝐹 (𝑥) is invertible with
inverse 𝐹 −1 (𝑥) := 𝑒𝑖𝑥𝐾 Φ(𝑥)−1 . It follows from Proposition 33.(v) and Lemma 73 that 𝐹 −1 ∈
1,1
𝑀𝑛 (𝑊loc
(ℝ)). Furthermore, since Φ(0) = 𝐼𝑛 = 𝑒−𝑖0𝐾 then 𝐹 (0) = 𝐼𝑛 . Moreover, since
𝑒𝑖𝑑𝐾 = Φ(𝑑), 𝑒−𝑖𝑑𝐾 = Φ(𝑑)−1 , and, by Proposition 33.(vi), Φ(𝑥 + 𝑑) = Φ(𝑥)Φ(𝑑) for every
126
𝑥 ∈ ℝ, we have
𝐹 (𝑥 + 𝑑) = Φ(𝑥 + 𝑑)𝑒−𝑖(𝑥+𝑑)𝐾 = Φ(𝑥)Φ(𝑑)𝑒−𝑖𝑑𝐾 𝑒−𝑖𝑥𝐾 = Φ(𝑥)Φ(𝑑)Φ(𝑑)−1 𝑒−𝑖𝑥𝐾
= Φ(𝑥)𝑒−𝑖𝑥𝐾 = 𝐹 (𝑥), ∀𝑥 ∈ ℝ.
This completes the proof of Theorem 34.
Using the Floquet-Lyapunov Theorem and its proof we will now establish the validity of
Theorem 35.
Proof. [Theorem 35] We first prove the statement: if 𝜓 is a nontrivial Floquet solution
of the canonical ODEs in (4.13) with wavenumber-frequency pair (𝑘, 𝜔), Floquet multiplier
𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚 ∈ ℕ ∪ {0} then 𝜓 = Ψ(⋅, 𝜔)𝛾 where 𝛾 is a generalized eigenvector of
the monodromy matrix Ψ(𝑑, 𝜔) of order 𝑚 + 1 corresponding to the eigenvalue 𝜆. We prove
this by induction on the order 𝑚 of the Floquet solution. Before we begin we will prove the
following lemma:
1,1
𝑛
1
𝑛
Lemma 57 If {𝑢𝑗 }𝑚
𝑗=0 ⊆ (𝑊loc (ℝ)) ∩ (𝐿 (𝕋)) and for a.e. 𝑥 ∈ ℝ,
𝑚
∑
𝑥𝑗 𝑢𝑗 (𝑥) = 0
𝑗=0
then 𝑢1 = ⋅ ⋅ ⋅ = 𝑢𝑚 = 0.
Proof. We prove the statement by induction. The statement is obviously true for 𝑚 = 0.
Suppose the statement is true for 𝑚 ∈∈ ℕ ∪ {0}. Lets show its true for 𝑚 + 1. Suppose
1,1
𝑛
1
𝑛
{𝑢𝑗 }𝑚+1
𝑗=0 ⊆ (𝑊loc (ℝ)) ∩ (𝐿 (𝕋)) and satisfies for a.e. 𝑥 ∈ ℝ,
𝑚+1
∑
𝑥𝑗 𝑢𝑗 (𝑥) = 0
𝑗=0
127
𝑛
By Lemma 70 we can assume {𝑢𝑗 }𝑚+1
𝑗=0 ⊆ (𝐴𝐶loc (ℝ)) and satisfy 𝑢𝑗 (𝑥 + 𝑑) = 𝑢𝑗 (𝑥) for all
𝑥 ∈ ℝ and for 𝑗 = 0, . . . , 𝑚 + 1. Hence letting 𝑢𝑗,𝑖 denote the 𝑖th row entry of 𝑢𝑗 we have
𝑢𝑗,𝑖 ∈ 𝐴𝐶loc (ℝ) is a 𝑑-periodic function and hence is bounded on ℝ, for 𝑗 = 0, . . . , 𝑚 + 1,
𝑖 = 0, . . . , 𝑛. This implies
𝑥𝑗 𝑢𝑗,𝑖 (𝑥)
=0
𝑥→∞ 𝑥𝑚+1
lim
for 𝑗 = 0, . . . , 𝑚, 𝑖 = 0, . . . , 𝑛 and there exists 𝑥𝑖 ∈ [0, 𝑑) such that
𝑢𝑚+1,𝑖 (𝑥𝑖 ) = sup ∣𝑢𝑚+1,𝑖 (𝑥)∣
𝑥∈ℝ
for 𝑖 = 0, . . . , 𝑛. Now it follows from our hypothesis that
𝑚+1
∑
𝑥𝑗 𝑢𝑗,𝑖 (𝑥) = 0, ∀𝑥 ∈ ℝ,
𝑖 = 1, . . . , 𝑛.
𝑗=0
Thus we conclude for 𝑙 ∈ ℕ,
sup ∣𝑢𝑚+1,𝑖 (𝑥)∣ = 𝑢𝑚+1,𝑖 (𝑥𝑖 ) = lim 𝑢𝑚+1,𝑖 (𝑥𝑖 + 𝑙𝑑)
𝑙→∞
𝑥∈ℝ
= lim −
𝑙→∞
𝑚
∑
𝑗=0
(𝑥𝑖 + 𝑙𝑑)𝑗 𝑢𝑗,𝑖 (𝑥𝑖 + 𝑙𝑑)
=0
(𝑥𝑖 + 𝑙𝑑)𝑚+1
for 𝑖 = 0, . . . , 𝑛. This implies 𝑢𝑚+1 = 0. But now the sequence {𝑢𝑗 }𝑚
𝑗=0 satisfies the hypotheses of this lemma and so by the induction hypotheses we conclude that 𝑢𝑚 = 𝑢𝑚 − 1 = ⋅ ⋅ ⋅ =
𝑢0 = 0. Therefore by induction the statement is true for all 𝑚 ∈ ℕ ∪ {0}. This completes
the proof.
We now begin the proof of the above statement. Let us show the base case 𝑚 = 0 is
true. Suppose 𝜓 is a Floquet solution of the canonical ODEs in (4.13) with wavenumberfrequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚 = 0. Then by Definition
128
1,1
(ℝ))𝑛 ∩ (𝐿1 (𝕋))𝑛 with 𝑢0 ∕= 0. By
19, 𝜓(𝑥) = 𝑒𝑖𝑘𝑥 𝑢0 (𝑥) for a.e. 𝑥 ∈ ℝ, for some 𝑢0 ∈ (𝑊loc
Lemma 57 we conclude 𝜓 ∕= 0 and so by Proposition 33.(i), 𝜓 = Ψ(⋅, 𝜔)𝛾 for some nonzero
𝛾 ∈ ℂ𝑛 . These facts and Proposition 33.(vi) imply for a.e. 𝑥 ∈ ℝ, Ψ(𝑥, 𝜔)Ψ(𝑑, 𝜔)𝛾 =
𝜓(𝑥 + 𝑑) = 𝑒𝑖𝑘𝑥 𝑒𝑖𝑘𝑑 𝑢0 (𝑥 + 𝑑) = 𝜆𝑒𝑖𝑘𝑑 𝑢0 (𝑥) = 𝜆𝜓(𝑥) = Ψ(𝑥, 𝜔)𝜆𝛾. Hence Proposition 33.(v)
implies Ψ(𝑑, 𝜔)𝛾 = 𝜆𝛾. But this implies, since 𝛾 ∕= 0, that 𝛾 is a generalized eigenvector of
Ψ(𝑑, 𝜔) of order 1 corresponding to the eigenvalue 𝜆. This proves the base case.
Suppose now the statement is true for 𝑚 ∈ ℕ ∪ {0}. We will now show its true for 𝑚 + 1.
Let 𝜓 be a Floquet solution of the canonical ODEs in (4.13) with wavenumber-frequency
pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚 + 1. Then by Definition 19, 𝜓(𝑥) =
∑
1,1
𝑚+1
𝑗
𝑛
1
𝑛
𝑒𝑖𝑘𝑥 𝑚+1
𝑗=0 𝑥 𝑢𝑗 (𝑥) for a.e. 𝑥 ∈ ℝ, for some {𝑢𝑗 }𝑗=0 ⊆ (𝑊loc (ℝ)) ∩ (𝐿 (𝕋)) with 𝑢𝑚+1 ∕= 0.
By Lemma 57 we conclude 𝜓 ∕= 0 and so by Proposition 33.(i), 𝜓 = Ψ(⋅, 𝜔)𝛾 for some nonzero
𝛾 ∈ ℂ𝑛 . We define
𝛾˜ := (Ψ(𝑑, 𝜔) − 𝜆𝐼𝑛 )𝛾,
˜ := Ψ(⋅, 𝜔)˜
𝜓
𝛾.
˜ is a Floquet solution of the canonical ODEs in (4.13) with wavenumberWe will now prove 𝜓
frequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚. First, it follows by Proposi˜ is a solution of the canonical ODEs in (4.13) at the frequency 𝜔 ∈ Ω. Next,
tion 33.(i) that 𝜓
∑ ()
by Proposition 33.(vi), the Floquet representation of 𝜓, and the fact (𝑥+𝑑)𝑗 = 𝑗𝑙=0 𝑗𝑙 𝑑𝑗−𝑙 𝑥𝑙 ,
it follows for a.e. 𝑥 ∈ ℝ that
˜
𝜓(𝑥)
= Ψ(𝑥, 𝜔)(Ψ(𝑑, 𝜔) − 𝑒𝑖𝑘𝑑 𝐼𝑛 )𝛾 = 𝜓(𝑥 + 𝑑) − 𝑒𝑖𝑘𝑑 𝜓(𝑥)
=𝑒
𝑖(𝑘+𝑑)𝑥
𝑚+1
∑
𝑗
(𝑥 + 𝑑) 𝑢𝑗 (𝑥 + 𝑑) − 𝑒
𝑗=0
𝑖𝑘𝑑 𝑖𝑘𝑥
𝑒
𝑚+1
∑
𝑥𝑗 𝑢𝑗 (𝑥)
𝑗=0
𝑗−1 ( )
𝑚+1
∑∑
(
) 𝑖𝑘𝑑
𝑗 𝑗−𝑙 𝑙 𝑖𝑘𝑑
𝑗
𝑗
𝑖𝑘𝑥
=𝑒
(𝑥 + 𝑑) − 𝑥 𝑒 𝑢𝑗 (𝑥) = 𝑒
𝑑 𝑥 𝑒 𝑢𝑗 (𝑥)
𝑙
𝑗=0
𝑗=1 𝑙=0
( 𝑚 ( )
)
𝑚
𝑚
∑
∑ 𝑗
∑
𝑖𝑘𝑥
𝑙
𝑗+1−𝑙 𝑖𝑘𝑑
𝑖𝑘𝑥
=𝑒
𝑥
𝑑
𝑒 𝑢𝑗+1 (𝑥) = 𝑒
𝑥𝑙 𝑢˜𝑗 (𝑥),
𝑙
𝑙=0
𝑗=𝑙
𝑙=0
𝑖𝑘𝑥
𝑚+1
∑
129
where 𝑢˜𝑗 =
∑𝑚 (𝑗 ) 𝑗+1−𝑙 𝑖𝑘𝑑
1,1
(ℝ))𝑛 ∩ (𝐿1 (𝕋))𝑛 for 𝑗 = 0, . . . , 𝑚 and 𝑢˜𝑚 =
𝑒 𝑢𝑗+1 ∈ (𝑊loc
𝑗=𝑙 𝑙 𝑑
˜ is a Floquet solution of the canonical ODEs in (4.13) with
𝑑𝑒𝑖𝑘𝑑 𝑢𝑚+1 ∕= 0. But this proves 𝜓
wavenumber-frequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚. Hence by the
induction hypothesis and the uniqueness portion of Proposition 33.(i) we conclude that 𝛾˜ is
a generalized eigenvector of Ψ(𝑑, 𝜔) of order 𝑚 + 1 corresponding to the eigenvalue 𝜆. But
since 𝛾˜ = (Ψ(𝑑, 𝜔) − 𝜆𝐼𝑛 )𝛾 this implies 𝛾 is a generalized eigenvector of Ψ(𝑑, 𝜔) of order
𝑚 + 2 corresponding to the eigenvalue 𝜆. This proves the statement for 𝑚 + 1. Therefore by
induction the statement is true for all 𝑚 ∈ ℕ ∪ {0}.
To complete the proof of this theorem we must show that if 𝛾 is a generalized eigenvector
of Ψ(𝑑, 𝜔) of order 𝑚 + 1 corresponding to the eigenvalue 𝜆 then for any 𝑘 ∈ ℂ such
that 𝜆 = 𝑒𝑖𝑘𝑑 , 𝜓 = Ψ(⋅, 𝜔)𝛾 is a Floquet solution of the canonical ODEs in (4.13) with
wavenumber-frequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚.
Let 𝛾 be a generalized eigenvector of Ψ(𝑑, 𝜔) of order 𝑚 + 1 corresponding to the eigenvalue
𝜆 where 𝑚 ∈ ℕ ∪ {0}. Define 𝜓 := Ψ(⋅, 𝜔)𝛾. By Proposition 33.(i), 𝜓 is a nontrivial solution
of the canonical ODEs in (4.13) at the frequency 𝜔 ∈ Ω. We will now show that 𝜓 is a
Floquet solution with the desired properties.
To simplify notation we will denote the matricant by Φ := Ψ(⋅, 𝜔). Then by the Floquet1,1
Lyapunov Theorem there exists a matrix 𝐾 ∈ 𝑀𝑛 (ℂ) and a function 𝐹 ∈ 𝑀𝑛 (𝑊loc
(ℝ)) such
that for every 𝑥 ∈ ℝ,
Φ(𝑥) = 𝐹 (𝑥)𝑒𝑖𝑥𝐾 ,
1,1
where 𝐹 (𝑥 + 𝑑) = 𝐹 (𝑥), 𝐹 (0) = 𝐼𝑛 , 𝐹 −1 (𝑥) := 𝐹 (𝑥)−1 exist, and 𝐹 −1 ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
Moreover, by the proof of the Floquet-Lyapunov Theorem we can assume that
𝐾=
1
log(Φ(𝑑))
𝑖𝑑
130
where, letting 𝜆1 , . . . , 𝜆𝑠 denote the distinct eigenvalues of the monodromy matrix Φ(𝑑),
log(𝑧) is a branch of the logarithm which is analytic at each 𝜆𝑙 , for 𝑙 = 1, . . . , 𝑠. Fix an
𝑥 ∈ ℝ. We define the function
𝑥
𝑓 (𝑧) := 𝑒 𝑑 𝑙𝑜𝑔(𝑧)
with the same domain as the function log(𝑧). It follows that 𝑓 is analytic at eigenvalues of
the monodromy matrix Φ(𝑑) as well. By the Riesz-Dunford functional calculus as described
1
for matrices in [35, pp. 304–334, §9] we can define the matrix 𝑓 (Φ(𝑑)). But 𝑓 (𝑧) = 𝑒𝑖𝑥( 𝑖𝑑 𝑙𝑜𝑔(𝑧))
and so it follows by [35, pp. 324, §9.7, Theorem 2] that
𝑓 (Φ(𝑑)) = 𝑒𝑖𝑥𝐾 .
For each 𝑙 = 1, . . . , 𝑠, let 𝑃𝑙 ∈ 𝑀𝑛 (ℂ) denote the projection onto the generalized eigenspace of
the monodromy matrix Φ(𝑑) corresponding to the eigenvalue 𝜆𝑙 . Without loss of generality
we may assume 𝜆1 = 𝜆. Let 𝑚𝑙 denote the index of the eigenvalue 𝜆𝑙 for 𝑙 = 1, . . . , 𝑠. Let
𝑓𝑙,𝑗 denote the value of the 𝑗th derivative of 𝑓 (𝑧) at the eigenvalue 𝜆𝑙 for 𝑙 = 1, . . . , 𝑠 and
𝑗 = 0, . . . , 𝑚𝑘 − 1. By spectral resolution of 𝑓 (Φ(𝑑)) (see [35, p. 314, §9.5, Theorem 1], [35,
p. 319, §9.5, Theorem 3], and [35, p. 321, §9.6, Theorem 1]) we have
𝑓 (Φ(𝑑)) =
𝑠 𝑚
𝑙 −1
∑
∑
𝑓𝑙,𝑗
𝑙=1 𝑗=0
𝑗!
(Φ(𝑑) − 𝜆𝑙 𝐼𝑛 )𝑗 𝑃𝑙
Now since by hypothesis 𝛾 is a generalized eigenvector of Φ(𝑑) of order 𝑚 + 1 corresponding
to the eigenvalue 𝜆1 this implies
𝑃1 𝛾 = 𝛾,
𝑃𝑙 𝛾 = 0, for 𝑙 ∕= 1,
(Φ(𝑑) − 𝜆1 𝐼𝑛 )𝑚 𝛾 ∕= 0, (Φ(𝑑) − 𝜆1 𝐼𝑛 )𝑗 𝛾 = 0, for 𝑗 > 𝑚.
131
Thus we have
𝑓 (Φ(𝑑))𝛾 =
𝑚
∑
𝑓1,𝑗
𝑗=0
𝑗!
(Φ(𝑑) − 𝜆1 𝐼𝑛 )𝑗 𝛾,
𝑓1,𝑗 =
𝑑𝑗 𝑓 .
𝑑𝑧 𝑗 𝑧=𝜆1
But we have
)
(𝑥
) 𝑥
(𝑥) (𝑥
𝑑𝑗 𝑓 − 1 ⋅⋅⋅
− (𝑗 − 1) 𝑒( 𝑑 −(𝑗−1)) log(𝜆1 ) ,
=
𝑗
𝑑𝑧 𝑧=𝜆1
𝑑
𝑑
𝑑
𝑗 = 1, . . . , 𝑚.
We define the polynomials in the variable 𝑥 by
𝑝0 (𝑥) := 1,
𝑝𝑗 (𝑥) :=
(𝑥) (𝑥
𝑑
)
(𝑥
)
− 1 ⋅⋅⋅
− (𝑗 − 1) 𝑒−(𝑗−1) log(𝜆1 ) .
𝑑
𝑑
Choose any 𝑘 ∈ ℂ such that
𝜆1 = 𝑒𝑖𝑘𝑑 .
Then, since 𝜆1 = 𝑒log(𝜆1 ) which follows from the fact log(𝑧) is a branch of the logarithm
analytic at 𝜆1 , there exist 𝑞 ∈ ℤ such that
𝑖𝑘𝑑 = log(𝜆1 ) + 𝑖2𝜋𝑞.
From these facts it follows that
𝑒
𝑖𝑥𝐾
𝛾 = 𝑓 (Φ(𝑑))𝛾 =
𝑚
∑
𝑓1,𝑗
𝑗=0
= 𝑒𝑖𝑘𝑥
𝑚
∑
𝑝𝑗 (𝑥)
𝑗=0
𝑗!
𝑗!
𝑗
(Φ(𝑑) − 𝜆1 𝐼𝑛 ) 𝛾 =
𝑚
∑
𝑗=0
𝑒−𝑖𝑥2𝜋𝑞/𝑑 (Φ(𝑑) − 𝜆1 𝐼𝑛 )𝑗 𝛾.
132
𝑥
𝑒 𝑑 log(𝜆1 )
𝑝𝑗 (𝑥)
(Φ(𝑑) − 𝜆1 𝐼𝑛 )𝑗 𝛾
𝑗!
And from this we conclude
𝜓(𝑥) = Φ(𝑥)𝛾 = 𝐹 (𝑥)𝑒𝑖𝑥𝐾 𝛾 = 𝑒𝑖𝑘𝑥
𝑚
∑
𝑝𝑗 (𝑥)
𝑗=0
𝑗!
𝑒−𝑖𝑥2𝜋𝑞/𝑑 𝐹 (𝑥)(Φ(𝑑) − 𝜆𝐼𝑛 )𝑗 𝛾,
(4.63)
1,1
for every 𝑥 ∈ ℝ. But 𝑣𝑗 (⋅) := 𝑗!1 𝑒−𝑖(⋅)2𝜋𝑞/𝑑 𝐹 (⋅)(Φ(𝑑) − 𝜆𝐼𝑛 )𝑗 𝛾 ∈ (𝑊loc
(ℝ))𝑛 ∩ (𝐿1 (𝕋))𝑛 and the
polynomial 𝑝𝑗 (⋅) has degree 𝑗, for 𝑗 = 0, . . . , 𝑚. Moreover, 𝑣𝑚 (0) =
1
(Φ(𝑑)
𝑚!
− 𝜆𝐼𝑛 )𝑚 𝛾 ∕= 0
and 𝜆 = 𝑒𝑖𝑘𝑑 . From these facts and the representation (4.63) we conclude that for any 𝑘 ∈ ℂ
such that 𝜆 = 𝑒𝑖𝑘𝑑 , 𝜓 is Floquet solution of the canonical ODEs in (4.13) with wavenumberfrequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 𝑚. This completes the proof of
Proposition 35.
Proof. [Corollary 36] Fix an 𝜔 ∈ Ω. Let {𝛾 𝑗 }𝑛𝑗=1 be a Jordan basis for the matrix Ψ(𝑑, 𝜔).
Then for 𝑗 = 1, . . . , 𝑛, we can define 𝑙𝑗 , 𝜆𝑗 , and 𝑘𝑗 to be such that 𝛾 𝑗 is a generalized
eigenvalue of Ψ(𝑑, 𝜔) of order 𝑙𝑗 corresponding to the eigenvalue 𝜆𝑗 = 𝑒𝑖𝑘𝑗 𝑑 . We define
{𝜓 𝑗 }𝑛𝑗=1 by 𝜓 𝑗 := Ψ(⋅, 𝜔)𝛾 𝑗 for 𝑗 = 1, . . . , 𝑛. Then since {𝛾 𝑗 }𝑛𝑗=1 is a basis for (ℂ)𝑛 it follows
by 33.(i) and 33.(v) that the set of solutions of the canonical ODEs in (4.13) at the frequency
𝜔 is a vector space over ℂ and {𝜓 𝑗 }𝑛𝑗=1 is a basis for that space. Moreover, by Theorem 35,
𝜓 𝑗 is a Floquet solution of the canonical ODEs in (4.13)with wavenumber-frequency pair
(𝑘𝑗 , 𝜔), Floquet multiplier 𝜆𝑗 = 𝑒𝑖𝑘𝑗 𝑑 , and order 𝑙𝑗 − 1, for 𝑗 = 1, . . . , 𝑛. This completes the
proof.
Proof. [Corollary 37 & Corollary 38] Let ℬ denote the Bloch variety of the canonical
ODEs in (4.13), i.e.,
ℬ := {(𝑘, 𝜔) ∈ ℂ × Ω ∣ (𝑘, 𝜔) is the wavenumber-frequency pair of some nontrivial
Bloch solution of the canonical ODEs in (4.13)}.
133
Define the function 𝐷 : ℂ × Ω → ℂ by
(
)
𝐷(𝑘, 𝜔) := det 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)
and the set 𝒞 ⊆ ℂ × Ω by
𝒞 := {(𝑘, 𝜔) ∈ ℂ × Ω ∣ 𝐷(𝑘, 𝜔) = 0}.
Now Corollary 37 follows from Corollary 38 since 𝐷(𝑘 + 2𝜋/𝑑, 𝜔) = 𝐷(𝑘, 𝜔). Hence to prove
these corollaries we need only show the function 𝐷 is a nonconstant holomorphic function
and ℬ = 𝒞.
We will now prove ℬ = 𝒞. Let (𝑘, 𝜔) ∈ ℬ. Then there exists a 𝜓 ∕= 0 which is a Bloch
solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (𝑘, 𝜔). But this
implies by Definition 19 that 𝜓 is a Floquet solution of the canonical ODEs in (4.13) with
wavenumber-frequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 1. By Theorem 35
this implies 𝜓 = Ψ(⋅, 𝜔)𝛾 where 𝛾 is an eigenvector of the monodromy matrix Ψ(⋅, 𝜔) with
(
)
𝜆 = 𝑒𝑖𝑘𝑑 as its corresponding eigenvalue. Thus we must have det 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔) = 0 and
hence (𝑘, 𝜔) ∈ 𝒞. This proves ℬ ⊆ 𝒞.
(
)
To prove the reverse inclusion, let (𝑘, 𝜔) ∈ 𝒞. Then det 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔) = 0 and this
implies 𝜆 = 𝑒𝑖𝑘𝑑 is an eigenvalue of the monodromy matrix Ψ(𝑑, 𝜔). Let 𝛾 be an eigenvector of
the monodromy matrix Ψ(𝑑, 𝜔) corresponding to the eigenvalue 𝜆 = 𝑒𝑖𝑘𝑑 . Then Theorem 35
implies 𝜓 := Ψ(⋅, 𝜔)𝛾 is a Floquet solution of the canonical ODEs in (4.13) with wavenumberfrequency pair (𝑘, 𝜔), Floquet multiplier 𝜆 = 𝑒𝑖𝑘𝑑 , and order 1. This and Lemma 57 imply 𝜓
is a nontrivial Bloch solution of the canonical ODEs in (4.13) with wavenumber-frequency
pair (𝑘, 𝜔) and hence (𝑘, 𝜔) ∈ ℬ. This proves 𝒞 ⊆ ℬ. Therefore ℬ = 𝒞.
134
We will now prove the function 𝐷 : ℂ × Ω → ℂ is a nonconstant function. First, 𝑓 (𝜆, 𝜔) :=
det (𝜆𝐼𝑛 − Ψ(𝑑, 𝜔)) is a monic polynomial of degree 𝑛 in the variable 𝜆 for each 𝜔 ∈ ℂ. Thus
𝑓 : ℂ × Ω → ℂ is a nonconstant function. Now 𝐷(𝑘, 𝜔) = 𝑓 (𝑒𝑖𝑘𝑑 , 𝜔) and this implies 𝐷 must
be a nonconstant function for otherwise we could fix an 𝜔 0 ∈ Ω and then there would exist
𝑐 ∈ ℂ such that 𝑓 (𝑒𝑖𝑘𝑑 , 𝜔 0 ) − 𝑐 = 0 for every 𝑘 ∈ ℂ implying, since 𝑓 (𝜆, 𝜔 0 ) − 𝑐 is a monic
polynomial of degree 𝑛 in the variable 𝜆, the function 𝑔(𝑘) := 𝑒𝑖𝑘𝑑 maps into a finite set of
values in ℂ which is a contradiction. Thus we have proven the function 𝐷 is a nonconstant
function.
We complete the proof of these corollaries now by proving the function 𝐷 : ℂ × Ω → ℂ
is a holomorphic function. By Lemma 80 we have 𝒪(Ω, 𝑀𝑛 (ℂ)) = 𝑀𝑛 (𝒪(Ω, ℂ)). Thus
by Proposition 33.(iii) we have Ψ(𝑑, ⋅) ∈ 𝑀𝑛 (𝒪(Ω, ℂ)), i.e., the entries of the monodromy
matrix function Ψ(𝑑, ⋅) are holomorphic functions of frequency. This implies the entries of
the matrix 𝑀 (𝑘, 𝜔) := 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔) are holomorphic functions of the variable (𝑘, 𝜔) in
the domain ℂ × Ω. And since the function 𝐹 (𝐵) := det(𝐵) is a polynomial in the entries of
the matrix 𝐵 ∈ 𝑀𝑛 (ℂ) this implies 𝐷(𝑘, 𝜔) = 𝐹 (𝑀 (𝑘, 𝜔)) is a holomorphic functions of the
variable (𝑘, 𝜔) in the domain ℂ × Ω. This completes the proof.
Proofs for Section 4.2.2
We begin with proving a key result in the study of the energy flux for canonical ODEs.
Proof. [Proposition 39] Fix an 𝜔 ∈ Ωℝ . Then by Proposition 33 we have Ψ(⋅, 𝜔) ∈
1,1
1,1
𝑀𝑛 (𝑊loc
(ℝ)), by Lemma 75 we have Ψ(⋅, 𝜔)∗ ∈ 𝑀𝑛 (𝑊loc
(ℝ)), and hence by Lemma 73 we
1,1
(ℝ)). For simplification of notation let Φ := Ψ(⋅, 𝜔).
have Ψ(⋅, 𝜔)∗ 𝑖J Ψ(⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
1,1
To complete the proof we must show Φ∗ 𝑖J Φ = 𝑖J in 𝑀𝑛 (𝑊loc
(ℝ)). First, it follows from
Proposition 33 that Φ′ = J −1 𝐴(⋅, 𝜔)Φ in 𝑀𝑛 (𝐿1loc (ℝ)). Second, by our assumptions we
135
have (J −1 )∗ = −J −1 and 𝐴(⋅, 𝜔)∗ = 𝐴(⋅, 𝜔) in 𝑀𝑛 (𝐿1loc (ℝ)). Thus by these facts and by
Lemmas 75 and 73 we have
(Φ∗ 𝑖J Φ)′ = (Φ′ )∗ 𝑖J Φ + Φ∗ 𝑖J Φ′ = (J −1 𝐴(⋅, 𝜔)Φ)∗ 𝑖J Φ + Φ∗ 𝑖𝐴(⋅, 𝜔)Φ
= −𝑖Φ∗ 𝐴(⋅, 𝜔)Φ + 𝑖Φ∗ 𝐴(⋅, 𝜔)Φ = 0.
It follows from this that there exists a constant matrix 𝐶 ∈ 𝑀𝑛 (ℂ) such that
Φ∗ 𝑖J Φ = 𝐶.
But since Φ(0) = Ψ(0, 𝜔) = 𝐼𝑛 this implies 𝐶 = 𝑖J and hence
Φ∗ 𝑖J Φ = 𝑖J .
This completes the proof.
Proof.
[Lemma 41] Let 𝜔 0 ∈ Ωℝ . By Lemma 76 and the hypotheses the Hamilto-
nian satisfies 𝐴(⋅, 𝜔)∗ = 𝐴(⋅, 𝜔) as an element of 𝑀𝑛 (𝐿1 (𝕋))) for every 𝜔 ∈ Ωℝ and 𝐴 ∈
𝒪(Ω, 𝑀𝑛 (𝐿1 (𝕋))) as a function of frequency. By definition since 𝐴𝜔 is derivative of the 𝐴
with respect to frequency in the Banach space 𝑀𝑛 (𝐿1 (𝕋)) we have 𝐴𝜔 (⋅, 𝜔) ∈ 𝑀𝑛 (𝐿1 (𝕋)) for
every 𝜔 ∈ Ω and
lim ∥(𝜔 − 𝜔 0 )−1 (𝐴(⋅, 𝜔) − 𝐴(⋅, 𝜔 0 )) − 𝐴𝜔 (⋅, 𝜔 0 )∥𝐿1 (𝕋) = 0.
𝜔→𝜔 0
It then follows from these facts and Lemma 76 that
0 = lim ∥(𝜔 − 𝜔 0 )−1 (𝐴(⋅, 𝜔) − 𝐴(⋅, 𝜔 0 )) − 𝐴𝜔 (⋅, 𝜔 0 )∥𝐿1 (𝕋)
𝜔→𝜔 0
𝜔∈Ωℝ
(
)∗
= lim ∥ (𝜔 − 𝜔 0 )−1 (𝐴(⋅, 𝜔) − 𝐴(⋅, 𝜔 0 )) − 𝐴𝜔 (⋅, 𝜔 0 ) ∥𝐿1 (𝕋)
𝜔→𝜔 0
𝜔∈Ωℝ
136
= lim ∥(𝜔 − 𝜔 0 )−1 (𝐴(⋅, 𝜔) − 𝐴(⋅, 𝜔 0 )) − 𝐴𝜔 (⋅, 𝜔 0 )∗ ∥𝐿1 (𝕋) .
𝜔→𝜔 0
𝜔∈Ωℝ
From which it follows that 𝐴𝜔 (⋅, 𝜔 0 )∗ = 𝐴𝜔 (⋅, 𝜔 0 ) in 𝑀𝑛 (𝐿1 (𝕋))). This completes the proof.
Proof. [Theorem 42] Let 𝜔 0 ∈ Ωℝ . Then Proposition 33 implies Ψ(⋅, 𝜔 0 ), Ψ𝜔 (⋅, 𝜔 0 ) ∈
1,1
1,1
(ℝ)). Thus by Lemma 73 we
(ℝ)) and by Lemma 75 have Ψ(⋅, 𝜔 0 )∗ ∈ 𝑀𝑛 (𝑊loc
𝑀𝑛 (𝑊loc
1,1
have Ψ(⋅, 𝜔 0 )∗ J Ψ𝜔 (⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝑊loc
(ℝ)).
Thus by notation 4.5.(xxix) we must have
(Ψ(⋅, 𝜔 0 )∗ J Ψ𝜔 (⋅, 𝜔 0 ))′ ∈ 𝑀𝑛 (𝐿1loc (ℝ)). Now since 𝐴𝜔 (⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝐿1 (𝕋))) ⊆ 𝑀𝑛 (𝐿1loc (ℝ))
then it follows from Lemma 84 that we also have Ψ(⋅, 𝜔 0 )∗ 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 ) ∈ 𝑀𝑛 (𝐿1loc (ℝ)).
Our next goal is to show that as elements of 𝑀𝑛 (𝐿1loc (ℝ)),
Ψ(⋅, 𝜔 0 )∗ 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 ) = (Ψ(⋅, 𝜔 0 )∗ J Ψ𝜔 (⋅, 𝜔 0 ))′ .
We begin by defining two operators that will play a key role in our proof. The first is the
integral map defined by
∫
𝑥
(ℐ𝐵)(𝑥) :=
𝐵(𝑡)𝑑𝑡,
𝐵 ∈ (𝐿1loc (ℝ))𝑛 ,
𝑥 ∈ ℝ.
0
1,1
By Lemma 87, the map ℐ : (𝐿1loc (ℝ))𝑛 → (𝑊loc
(ℝ))𝑛 is well-defined and
(ℐ𝑓 )′ = 𝑓,
1,1
for every 𝑓 ∈ (𝑊loc
(ℝ))𝑛 . The second map is defined for each 𝜔 ∈ Ω by
∫
(𝑆(𝜔)𝑓 )(𝑥) := Ψ(𝑥, 𝜔)
𝑥
Ψ(𝑡, 𝜔)−1 𝑓 (𝑡)𝑑𝑡,
0
137
𝑓 ∈ (𝐿1loc (ℝ))𝑛 ,
𝑥 ∈ ℝ.
1,1
(ℝ))𝑛 is a right inverse of
Lemma 58 For each 𝜔 ∈ Ω, the map 𝑆(𝜔) : (𝐿1loc (ℝ))𝑛 → (𝑊loc
1,1
the map 𝑇 (𝜔) : (𝑊loc
(ℝ))𝑛 → (𝐿1loc (ℝ))𝑛 defined in (4.47), i.e., for every 𝑓 ∈ (𝐿1loc (ℝ))𝑛 we
have
𝑇 (𝜔)𝑆(𝜔)𝑓 = 𝑓.
Proof.
1,1
(ℝ))𝑛 is wellLet 𝜔 ∈ Ω. Lets first prove the map 𝑆(𝜔) : (𝐿1loc (ℝ))𝑛 → (𝑊loc
defined. By Proposition 33.(v) it follows that Ψ−1 (𝑥, 𝜔) := Ψ(𝑥, 𝜔)−1 exists for every
1,1
𝑥 ∈ ℝ and Ψ−1 (⋅, 𝜔) ∈ 𝑀𝑛 (𝑊loc
(ℝ)). Let 𝑓 ∈ (𝐿1loc (ℝ))𝑛 . Then it follows by Lemma
84 that Ψ−1 (⋅, 𝜔)𝑓 ∈ (𝐿1loc (ℝ))𝑛 and applying the integral map to this yields ℐ(Ψ−1 (⋅, 𝜔)𝑓 ) ∈
1,1
(𝑊loc
(ℝ))𝑛 . By Lemma 73 and the definition of 𝑆(𝜔) we find that
1,1
𝑆(𝜔)𝑓 = Ψ(⋅, 𝜔)ℐ(Ψ−1 (⋅, 𝜔)𝑓 ) ∈ (𝑊loc
(ℝ))𝑛 .
Thus the map is well-defined.
1,1
Next, recall that the map 𝑇 (𝜔) : (𝑊loc
(ℝ))𝑛 → (𝐿1loc (ℝ))𝑛 in (4.47) is
𝑇 (𝜔)𝜓 := 𝜓 ′ − J −1 𝐴(⋅, 𝜔)𝜓,
1,1
𝜓 ∈ (𝑊loc
(ℝ))𝑛 .
We now show 𝑆(𝜔) is a right inverse of 𝑇 (𝜔). Let 𝑓 ∈ (𝐿1loc (ℝ))𝑛 then by Lemma 73 we have
(𝑆(𝜔)𝑓 )′ = (Ψ(⋅, 𝜔)ℐ(Ψ−1 (⋅, 𝜔)𝑓 ))′ = Ψ(⋅, 𝜔)′ ℐ(Ψ−1 (⋅, 𝜔)𝑓 ) + Ψ(⋅, 𝜔)(ℐ(Ψ−1 (⋅, 𝜔)𝑓 ))′
= J −1 𝐴(⋅, 𝜔)Ψ(⋅, 𝜔)ℐ(Ψ−1 (⋅, 𝜔)𝑓 ) + Ψ(⋅, 𝜔)Ψ−1 (⋅, 𝜔)𝑓 = J −1 𝐴(⋅, 𝜔)𝑆(𝜔)𝑓 + 𝑓.
Hence it follows that
𝑇 (𝜔)𝑆(𝜔)𝑓 = (𝑆(𝜔)𝑓 )′ − J −1 𝐴(⋅, 𝜔)𝑆(𝜔)𝑓 = 𝑓.
138
Thus 𝑆(𝜔) is a right inverse of 𝑇 (𝜔). This completes the proof.
Let 𝛾 ∈ ℂ𝑛 . Define 𝜙 := Ψ𝜔 (⋅, 𝜔 0 )𝛾 and 𝑓 := J −1 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 )𝛾. We will now show
that 𝜙 = 𝑆(𝜔 0 )𝑓 . It follows from Lemma 84 that 𝑓 ∈ (𝐿1loc (ℝ))𝑛 . It follows by Proposition
1,1
33.(iv) that 𝜙 ∈ (𝑊loc
(ℝ))𝑛 and satisfies
(
)
𝜙 = Ψ𝜔 (⋅, 𝜔 0 )𝛾 = ℐ J −1 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 )𝛾 + J −1 𝐴(⋅, 𝜔 0 )Ψ𝜔 (⋅, 𝜔 0 )𝛾
(
)
= ℐ 𝑓 + J −1 𝐴(⋅, 𝜔 0 )𝜙
since J −1 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 )𝛾 ∈ 𝑀𝑛 (𝐿1loc (ℝ)) by Lemma 84. Hence by Lemma 87 it follows
that
𝑇 (𝜔 0 )𝜙 = 𝜙′ − J −1 𝐴(⋅, 𝜔 0 )𝜙 = 𝑓.
1,1
But by Lemma 58 we also have 𝑇 (𝜔 0 )𝑆(𝜔 0 )𝑓 = 𝑓 with 𝑆(𝜔 0 )𝑓 ∈ (𝑊loc
(ℝ))𝑛 . Hence
1,1
𝜓 := 𝜙 − 𝑆(𝜔 0 )𝑓 ∈ (𝑊loc
(ℝ))𝑛 and satisfies
𝑇 (𝜔 0 )𝜓 = 0,
that is, 𝜓 ∈ ker(𝑇 (𝜔 0 )). Thus by Corollary 54 and Proposition 33.(i) there exists a unique
𝛾˜ ∈ ℂ𝑛 such that 𝜓 = Ψ(⋅, 𝜔 0 )˜
𝛾 . In fact, 𝛾˜ = 𝜓(0) = 𝜙(0)−(𝑆(𝜔 0 )𝑓 )(0). But (𝑆(𝜔)𝑓 )(0) = 0
and 𝜙(0) = Ψ𝜔 (0, 𝜔 0 )𝛾 = 0. Thus we have 𝜓 = 0 and so 𝜙 = 𝑆(𝜔 0 )𝑓 as desired.
Now it follows from what we have just shown that for every 𝛾 ∈ ℂ𝑛 ,
Ψ𝜔 (⋅, 𝜔 0 )𝛾 = 𝑆(𝜔 0 )(J −1 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 )𝛾)
139
from which it follows by the definition of 𝑆(𝜔 0 ) and Lemma 87 that
∫
𝑥
Ψ(𝑡, 𝜔 0 )−1 J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝛾𝑑𝑡
Ψ𝜔 (𝑥, 𝜔 0 )𝛾 = Ψ(𝑥, 𝜔 0 )
∫0 𝑥
= Ψ(𝑥, 𝜔 0 )
Ψ(𝑡, 𝜔 0 )−1 J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡𝛾,
∀𝑥 ∈ ℝ.
0
But since this is true for every 𝛾 ∈ ℂ𝑛 this implies
∫
Ψ𝜔 (𝑥, 𝜔 0 ) = Ψ(𝑥, 𝜔 0 )
𝑥
Ψ(𝑡, 𝜔 0 )−1 J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡, ∀𝑥 ∈ ℝ.
0
Now if we multiply both sides by Ψ(𝑥, 𝜔)∗ J and apply Proposition 39 we get
∫
𝑥
Ψ(𝑡, 𝜔 0 )−1 J −1 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡
Ψ(𝑥, 𝜔 0 ) J Ψ𝜔 (𝑥, 𝜔 0 ) = Ψ(𝑥, 𝜔) J Ψ(𝑥, 𝜔 0 )
0
∫ 𝑥
∫ 𝑥
−1
−1
=J
Ψ(𝑡, 𝜔 0 ) J 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡 =
Ψ(𝑡, 𝜔 0 )∗ 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡
∗
∗
0
0
for every 𝑥 ∈ ℝ. It follows from this and Lemma 87 that
(Ψ(⋅, 𝜔 0 )∗ J Ψ𝜔 (⋅, 𝜔 0 ))′ = Ψ(⋅, 𝜔 0 )∗ 𝐴𝜔 (⋅, 𝜔 0 )Ψ(⋅, 𝜔 0 )
and
1
𝑑
∫
0
𝑑
1
Ψ(𝑡, 𝜔 0 )∗ 𝐴𝜔 (𝑡, 𝜔 0 )Ψ(𝑡, 𝜔 0 )𝑑𝑡 = Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔 0 ).
𝑑
This completes the proof.
Proof. [Corollary 43] Let 𝜔 ∈ Ωℝ and 𝛾 1 , 𝛾 2 ∈ ℂ𝑛 . Then by Theorem 42 we have
1,1
Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 ∈ (𝑊loc
(ℝ))𝑛
Ψ(⋅, 𝜔)∗ 𝐴𝜔 (⋅, 𝜔)Ψ(⋅, 𝜔)𝛾 1 ∈ (𝐿1loc (ℝ))𝑛
(Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 )′ = Ψ(⋅, 𝜔)∗ 𝐴𝜔 (⋅, 𝜔)Ψ(⋅, 𝜔)𝛾 1
140
1,1
(ℝ),
Hence it follows from this and Lemma 75 that ⟨Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ ∈ 𝑊loc
⟨𝐴𝜔 (⋅, 𝜔)𝜓 1 , 𝜓 2 ⟩ℂ ∈ 𝐿1loc (ℝ), and ⟨Ψ(⋅, 𝜔)∗ J Ψ𝜔 (⋅, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ ′ = ⟨𝐴𝜔 (⋅, 𝜔)𝜓 1 , 𝜓 2 ⟩ℂ . And by
Theorem 42 we conclude that
1
𝑑
∫
0
𝑑
1
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥 = ⟨Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)𝛾 1 , 𝛾 2 ⟩ℂ .
𝑑
This completes the proof.
Proofs for Section 4.2.3
We begin by proving the sesquilinear form defined in (4.31) is a Hermitian form.
Proof. [Lemma 44] Let (𝑘, 𝜔) ∈ ℬℝ . By Lemma 41, for any 𝛾 1 , 𝛾 2 ∈ ker (𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)),
if we let 𝜓 1 := Ψ(⋅, 𝜔)𝛾 1 , 𝜓 2 := Ψ(⋅, 𝜔)𝛾 2 then we have
∫
∫
1 𝑑
1 𝑑
𝑞(𝑘,𝜔) (𝛾 1 , 𝛾 2 ) =
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥 =
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥), 𝜓 2 (𝑥)⟩ℂ 𝑑𝑥
𝑑 0
𝑑 0
∫
∫
1 𝑑
1 𝑑
=
⟨𝜓 (𝑥), 𝐴𝜔 (𝑥, 𝜔)𝜓 1 (𝑥)⟩ℂ 𝑑𝑥 =
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 2 (𝑥), 𝜓 1 (𝑥)⟩ℂ 𝑑𝑥 = 𝑞(𝑘,𝜔) (𝛾 2 , 𝛾 1 ),
𝑑 0 2
𝑑 0
implying 𝑞(𝑘,𝜔) (𝛾 1 , 𝛾 1 ) ∈ ℝ. This completes the proof.
Proof. [Proposition 45] By Definition 23, the canonical ODEs in (4.13) are of definite
type at a point (𝑘0 , 𝜔 0 ) ∈ ℬℝ if and only if
1
𝑑
∫
𝑑
⟨𝐴𝜔 (𝑥, 𝜔 0 )𝜓(𝑥), 𝜓(𝑥)⟩ℂ 𝑑𝑥 ∕= 0
0
for any nontrivial Bloch solution 𝜓 of the canonical ODEs in (4.13) with wavenumberfrequency (𝑘0 , 𝜔 0 ). But by Theorem 35, 𝜓 is a nontrivial Bloch solution 𝜓 of the canonical
ODEs in (4.13) with wavenumber-frequency (𝑘0 , 𝜔 0 ) if and only if 𝜓 = Ψ(⋅, 𝜔 0 )𝛾 where 𝛾
141
is an eigenvector of the monodromy matrix Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 .
And hence the canonical ODEs in (4.13) are of definite type at a point (𝑘0 , 𝜔 0 ) ∈ ℬℝ if and
only if ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) ∕= {0} and
𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾) ∕= 0, for every nonzero 𝛾 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
(4.64)
Thus to complete the proof we need only show that (4.64) implies 𝑞(𝑘0 ,𝜔0 ) , as defined in (4.31),
is a definite sesquilinear form which is bounded.
Suppose that (4.64) is true. Then it follows from Lemma 44 that 𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾) ∈ ℝ/{0}
for every nonzero 𝛾 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )). Suppose there existed nonzero 𝛾 − , 𝛾 + ∈
ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) such that 𝑞(𝑘0 ,𝜔0 ) (𝛾 − 𝛾 − ) < 0 < 𝑞(𝑘0 ,𝜔0 ) (𝛾 + , 𝛾 + ). Then we can define a
continuous real-valued function 𝑓 : [0, 1] → ℝ by
𝑓 (𝑥) := 𝑞(𝑘0 ,𝜔0 ) (𝑥𝛾 + + (1 − 𝑥)𝛾 − , 𝑥𝛾 + + (1 − 𝑥)𝛾 − ).
Now 𝑓 (0) < 0 < 𝑓 (1) and so by the intermediate value theorem that there exists 𝑥0 ∈
(0, 1) such that 𝑓 (𝑥0 ) = 0 implying by hypothesis that 𝑥0 𝛾 + + (1 − 𝑥0 )𝛾 − = 0. But this
implies 0 < 𝑥20 𝑞(𝑘0 ,𝜔0 ) (𝛾 + , 𝛾 + ) = (1 − 𝑥0 )2 𝑞(𝑘0 ,𝜔0 ) (𝛾 − , 𝛾 − ) < 0, a contradiction. Thus we
conclude that sgn(𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾)) =: sgn(𝑞(𝑘0 ,𝜔0 ) ) is independent of the choice of nonzero 𝛾 in
ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
We define a sesquilinear form by
⟨𝛾 1 , 𝛾 2 ⟩(𝑘0 ,𝜔0 ) := sgn(𝑞(𝑘0 ,𝜔0 ) )𝑞(𝑘0 ,𝜔0 ) (𝛾 1 , 𝛾 2 ),
𝛾 1 , 𝛾 2 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
It follows from what we have just shown that ⟨ , ⟩(𝑘0 ,𝜔0 ) is an inner product on the finite-
142
dimensional vector space ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) and so
1
1
2
∣∣𝛾∣∣(𝑘0 ,𝜔0 ) := ⟨𝛾, 𝛾⟩(𝑘
= ∣𝑞(𝑘0 ,𝜔0 ) (𝛾, 𝛾)∣ 2 ,
0 ,𝜔 0 )
𝛾 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )),
defines a norm on this space as does the Euclidean norm ∣∣ ⋅ ∣∣ℂ . But since any two norms
on a finite dimensional vector space are equivalent this implies there exists 𝐶1 , 𝐶2 > 0 such
that
𝐶1 ∣∣𝛾∣∣2ℂ ≤ ∣∣𝛾∣∣2(𝑘0 ,𝜔0 ) ≤ 𝐶2 ∣∣𝛾∣∣2ℂ
for every 𝛾 ∈ ker (𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )). This completes the proof.
Proof. [Lemma 46] Let {(𝑘𝑗 , 𝜔 𝑗 )}𝑗∈ℕ be a sequence in ℬℝ and {𝛾 𝑗 }𝑗∈ℕ a sequence in ℂ𝑛
such that 𝛾 𝑗 ∈ ker(𝑒𝑖𝑘𝑗 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 𝑗 )), ∣∣𝛾 𝑗 ∣∣ℂ = 1 for all 𝑗 ∈ ℕ and
∣∣⋅∣∣ℂ
∣∣⋅∣∣ℂ
(𝑘𝑗 , 𝜔 𝑗 ) −−→ (𝑘0 , 𝜔 0 ) and 𝛾 𝑗 −−→ 𝛾 0 , as 𝑗 → ∞,
for some 𝑘0 , 𝜔 0 ∈ ℂ and 𝛾 0 ∈ ℂ𝑛 . We will now show that ∣∣𝛾 0 ∣∣ℂ = 1, 𝛾 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 −
Ψ(𝑑, 𝜔 0 )), and (𝑘0 , 𝜔 0 ) ∈ ℬℝ .
∣∣⋅∣∣ℂ
First, since 𝛾 𝑗 −−→ 𝛾 0 as 𝑗 → ∞, then
1 = lim ∣∣𝛾 𝑗 ∣∣ℂ = ∣∣𝛾 0 ∣∣ℂ .
𝑗→∞
Second, if we define a matrix-valued function by 𝑀 (𝑘, 𝜔) := 𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔), (𝑘, 𝜔) ∈ ℂ × Ω
then this function is continuous since
∣∣𝑀 (𝑘, 𝜔) − 𝑀 (𝑘0 , 𝜔 0 )∣∣ℂ ≤ ∣∣𝑀 (𝑘, 𝜔) − 𝑀 (𝑘0 , 𝜔)∣∣ℂ + ∣∣𝑀 (𝑘0 , 𝜔) − 𝑀 (𝑘0 , 𝜔 0 )∣∣ℂ
1
= ∣𝑒𝑖𝑘𝑑 − 𝑒𝑖𝑘0 𝑑 ∣𝑛 2 + ∣∣Ψ(𝑑, 𝜔) − Ψ(𝑑, 𝜔 0 )∣∣ℂ
143
and by Proposition 33.(iii) we have Ψ(𝑑, ⋅) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ) implying ∣∣Ψ(𝑑, 𝜔) − Ψ(𝑑, 𝜔 0 )∣∣ℂ →
0 as 𝜔 → 𝜔 0 . Thus we conclude from this that
∣∣𝑀 (𝑘0 , 𝜔 0 )𝛾 0 ∣∣ℂ = lim ∣∣𝑀 (𝑘𝑗 , 𝜔 𝑗 )𝛾 𝑗 − 𝑀 (𝑘0 , 𝜔 0 )𝛾 0 ∣∣ℂ
𝑗→∞
≤ lim ∣∣𝑀 (𝑘𝑗 , 𝜔 𝑗 )𝛾 𝑗 − 𝑀 (𝑘0 , 𝜔 0 )𝛾 𝑗 ∣∣ℂ + lim ∣∣𝑀 (𝑘0 , 𝜔 0 )𝛾 𝑗 − 𝑀 (𝑘0 , 𝜔 0 )𝛾 0 ∣∣ℂ
𝑗→∞
𝑗→∞
≤ lim ∣∣𝑀 (𝑘𝑗 , 𝜔 𝑗 ) − 𝑀 (𝑘0 , 𝜔 0 )∣∣ℂ ∣∣𝛾 𝑗 ∣∣ℂ + lim ∣∣𝑀 (𝑘0 , 𝜔 0 )∣∣ℂ ∣∣𝛾 𝑗 − 𝛾 0 ∣∣ℂ
𝑗→∞
𝑗→∞
=0
and hence 𝛾 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )). It now follows from this and Corollary 38 that
(𝑘0 , 𝜔 0 ) ∈ ℬℝ .
Now we complete the proof by showing
∣⋅∣
𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛾 𝑗 , 𝛾 𝑗 ) −
→ 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 ), as 𝑗 → ∞.
(4.65)
First, it follows from Corollary 43 that for any (𝑘, 𝜔) ∈ ℬℝ and for any 𝛽 1 , 𝛽 2 ∈ ker(𝑒𝑖𝑘𝑑 𝐼𝑛 −
Ψ(𝑑, 𝜔)) we have
1
𝑞(𝑘,𝜔) (𝛽 1 , 𝛽 2 ) = ⟨Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)𝛽 1 , 𝛽 2 ⟩ℂ .
𝑑
Thus we extend the definition of 𝑞(𝑘,𝜔) to all of ℂ𝑛 × ℂ𝑛 in the obvious way by
1
𝑞(𝑘,𝜔) (𝛽 1 , 𝛽 2 ) := ⟨Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)𝛽 1 , 𝛽 2 ⟩ℂ ,
𝑑
𝛽 1 , 𝛽 2 ∈ ℂ𝑛 .
It follows from this definition that 𝑞(𝑘,𝜔) : ℂ𝑛 × ℂ𝑛 → ℂ is a continuous sesquilinear form
since
∣𝑞(𝑘,𝜔) (𝛽 1 , 𝛽 2 )∣ ≤ 𝑑−1 ∣∣Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)∣∣ℂ ∣∣𝛽 1 ∣∣ℂ ∣∣𝛽 2 ∣∣ℂ , ∀𝛽 1 , 𝛽 2 ∈ ℂ𝑛 .
144
Hence for any (𝑘, 𝜔) ∈ ℬℝ and any 𝛽 ∈ ℂ𝑛 we have the inequality
∣𝑞(𝑘,𝜔) (𝛽, 𝛽) − 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 )∣ ≤ ∣𝑞(𝑘,𝜔) (𝛽, 𝛽) − 𝑞(𝑘,𝜔) (𝛽, 𝛾 0 )∣
+ ∣𝑞(𝑘,𝜔) (𝛽, 𝛾 0 ) − 𝑞(𝑘0 ,𝜔0 ) (𝛽, 𝛾 0 )∣
+ ∣𝑞(𝑘0 ,𝜔0 ) (𝛽, 𝛾 0 ) − 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 )∣
and the inequalities
∣𝑞(𝑘,𝜔) (𝛽, 𝛽) − 𝑞(𝑘,𝜔) (𝛽, 𝛾 0 )∣ = ∣𝑞(𝑘,𝜔) (𝛽, 𝛽 − 𝛾 0 )∣
≤ 𝑑−1 ∣∣Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)∣∣ℂ ∣∣𝛽∣∣ℂ ∣∣𝛽 − 𝛾 0 ∣∣ℂ ,
∣𝑞(𝑘0 ,𝜔0 ) (𝛽, 𝛾 0 ) − 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 )∣ = ∣𝑞(𝑘0 ,𝜔0 ) (𝛽 − 𝛾 0 , 𝛾 0 )∣
≤ 𝑑−1 ∣∣Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)∣∣ℂ ∣∣𝛽 − 𝛾 0 ∣∣ℂ ∣∣𝛾 0 ∣∣ℂ ,
∣𝑞(𝑘,𝜔) (𝛽, 𝛾 0 ) − 𝑞(𝑘0 ,𝜔0 ) (𝛽, 𝛾 0 )∣ = 𝑑−1 ∣∣Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔)
− Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 )∣∣ℂ ∣∣𝛽∣∣ℂ ∣∣𝛾 0 ∣∣ℂ
And thus it follows from these inequalities that (4.65) is true if
∣∣⋅∣∣ℂ
∣⋅∣
Ψ(𝑑, 𝜔)∗ J Ψ𝜔 (𝑑, 𝜔) −−→ Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 ), as 𝜔 −
→ 𝜔0.
But by Lemma 77, multiplication of matrices in 𝑀𝑛 (ℂ) is a continuous bilinear map and so
in order to prove the latter statement we need only show
∣∣⋅∣∣ℂ
∣∣⋅∣∣ℂ
∣⋅∣
Ψ(𝑑, 𝜔)∗ −−→ Ψ(𝑑, 𝜔 0 )∗ and Ψ𝜔 (𝑑, 𝜔) −−→ Ψ𝜔 (𝑑, 𝜔 0 ), as 𝜔 −
→ 𝜔0.
145
Convergence of the first limit follows from the fact that the adjoint map ()∗ : 𝑀𝑛 (ℂ) →
𝑀𝑛 (ℂ) is a norm preserving map, i.e., ∣∣𝐵 ∗ ∣∣ℂ = ∣∣𝐵∣∣ℂ for every 𝐵 ∈ 𝑀𝑛 (ℂ), since ∣∣Ψ(𝑑, 𝜔)∗ −
Ψ(𝑑, 𝜔 0 )∗ ∣∣ℂ = ∣∣(Ψ(𝑑, 𝜔)−Ψ(𝑑, 𝜔 0 ))∗ ∣∣ℂ = ∣∣Ψ(𝑑, 𝜔)−Ψ(𝑑, 𝜔 0 )∣∣ℂ → 0 as 𝜔 → 𝜔 0 . Now for the
other limit, by Lemma 81 since Ψ(𝑑, ⋅) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)) and, by Proposition 33.(iii), Ψ𝜔 (𝑑, ⋅)
is its derivative in the ∣∣ ⋅ ∣∣ℂ norm then Ψ𝜔 (𝑑, ⋅) ∈ 𝒪(Ω, 𝑀𝑛 (ℂ)). But this implies Ψ𝜔 (𝑑, ⋅) is
∣∣⋅∣∣ℂ
a continuous matrix-valued function on Ω in the ∣∣ ⋅ ∣∣ℂ norm and so Ψ𝜔 (𝑑, 𝜔) −−→ Ψ𝜔 (𝑑, 𝜔 0 )
∣⋅∣
as 𝜔 −
→ 𝜔 0 . Therefore (4.65) is true and this completes the proof.
Proof. [Theorem 47] Suppose (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the canonical
ODEs in (4.13). We will first show that there exists an 𝑟 > 0 such that every (𝑘, 𝜔) ∈
𝐵((𝑘0 , 𝜔 0 ), 𝑟) ∩ ℬℝ is a point of definite type for the canonical ODEs in (4.13). We will prove
this by contradiction. Suppose the statement was not true. Then we can find a sequence of
1,1
∞
𝑛
points {(𝑘𝑗 , 𝜔 𝑗 )}∞
𝑗=1 ⊆ ℬℝ and a sequence {𝜓 𝑗 }𝑗=1 ⊆ (𝑊loc (ℝ)) such that 𝜓 𝑗 is a nontrivial
Bloch solution of the canonical ODEs in (4.13) with wavenumber-frequency pair (𝑘𝑗 , 𝜔 𝑗 ) and
1
𝑑
∫
𝑑
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 𝑗 (𝑥), 𝜓 𝑗 (𝑥)⟩ℂ 𝑑𝑥 = 0,
0
for every 𝑗 ∈ ℕ. By Theorem 35 there exists a nonzero 𝛾 𝑗 ∈ ker(𝑒𝑖𝑘𝑗 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 𝑗 )) such that
𝜓 𝑗 = Ψ(𝑑, 𝜔 𝑗 )𝛾 𝑗 , for every 𝑗 ∈ ℕ. Let 𝛽 𝑗 :=
𝛾𝑗
,
∣∣𝛾 𝑗 ∣∣ℂ
for 𝑗 ∈ ℕ. Then {𝛽 𝑗 }∞
𝑗=1 is a sequence
of vectors with unit norm in ℂ𝑛 and hence since {𝛽 ∈ ℂ𝑛 : ∣∣𝛽∣∣ℂ = 1} is a compact set
∣∣⋅∣∣ℂ
this implies there exists a convergent subsequence {𝛽 𝑗𝑙 }∞
𝑙=1 such that 𝛽 𝑗𝑙 −−→ 𝛽 0 as 𝑙 → ∞
for some 𝛽 0 ∈ ℂ𝑛 with ∣∣𝛽 0 ∣∣ℂ = 1. Thus since {(𝑘𝑗𝑙 , 𝜔 𝑗𝑙 )}𝑙∈ℕ ⊆ ℬℝ , {𝛽 𝑗𝑙 }𝑗∈ℕ ⊆ ℂ𝑛 such that
𝛽 𝑗𝑙 ∈ ker(𝑒𝑖𝑘𝑗𝑙 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 𝑗𝑙 )) with ∣∣𝛽 𝑗𝑙 ∣∣ℂ = 1 for all 𝑙 ∈ ℕ and
∣∣⋅∣∣ℂ
∣∣⋅∣∣ℂ
(𝑘𝑗𝑙 , 𝜔 𝑗𝑙 ) −−→ (𝑘0 , 𝜔 0 ) and 𝛽 𝑗𝑙 −−→ 𝛽 0 , as 𝑙 → ∞
146
then by Lemma 46, 𝛽 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) with ∣∣𝛽 0 ∣∣ℂ = 1 and
∣⋅∣
→ 𝑞(𝑘0 ,𝜔0 ) (𝛽 0 , 𝛽 0 ) as 𝑙 → ∞.
𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛽 𝑗𝑙 , 𝛽 𝑗𝑙 ) −
𝑙
𝑙
But for every 𝑙 ∈ ℕ we have
𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛽 𝑗𝑙 , 𝛽 𝑗𝑙 ) = ∣∣𝛾 𝑗𝑙 ∣∣−2
ℂ 𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛾 𝑗𝑙 , 𝛾 𝑗𝑙 )
𝑙
𝑙
𝑙
𝑙
∫ 𝑑
1
= ∣∣𝛾 𝑗𝑙 ∣∣−2
⟨𝐴𝜔 (𝑥, 𝜔)𝜓 𝑗𝑙 (𝑥), 𝜓 𝑗𝑙 (𝑥)⟩ℂ 𝑑𝑥 = 0
ℂ
𝑑 0
implying
𝑞(𝑘0 ,𝜔0 ) (𝛽 0 , 𝛽 0 ) = 0, ∣∣𝛽 0 ∣∣ℂ = 1,
𝛽 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )).
By Proposition 45 this contradicts that (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the
canonical ODEs in (4.13). Thus we conclude that there exists an 𝑟 > 0 such that every
(𝑘, 𝜔) ∈ 𝐵((𝑘0 , 𝜔 0 ), 𝑟) ∩ ℬℝ is a point of definite type for the canonical ODEs in (4.13).
Now let {(𝑘𝑗 , 𝜔 𝑗 )}∞
𝑗=1 ⊆ ℬℝ be any sequence converging to (𝑘0 , 𝜔 0 ) in the Euclidean norm
∣∣ ⋅ ∣∣ℂ . Then since the set {𝛽 ∈ ℂ𝑛 : ∣∣𝛽∣∣ℂ = 1} is compact we can find a sequence {𝛾 𝑗 }∞
𝑗=1
such that 𝛾 𝑗 ∈ ker(𝑒𝑖𝑘𝑗 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 𝑗 )) with ∣∣𝛾 𝑗 ∣∣ℂ = 1 for all 𝑗 ∈ ℕ and
∣∣⋅∣∣ℂ
𝛾 𝑗 −−→ 𝛾 0 , as 𝑗 → ∞
for some 𝛾 0 ∈ {𝛽 ∈ ℂ𝑛 : ∣∣𝛽∣∣ℂ = 1}. From what we just proved there exists a 𝐽 ∈ ℕ such
that if 𝑗 ≥ 𝐽 then (𝑘𝑗 , 𝜔 𝑗 ) is a point of definite type for the canonical ODEs in (4.13). Thus
by Proposition 45 and Lemma 46 it follows that 𝛾 0 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) with ∣∣𝛾 0 ∣∣ℂ = 1
and
sgn(𝑞(𝑘𝑗 ,𝜔𝑗 ) ) =
𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛾 𝑗 , 𝛾 𝑗 )
∣𝑞(𝑘𝑗 ,𝜔𝑗 ) (𝛾 𝑗 , 𝛾 𝑗 )∣
∣⋅∣
−
→
𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 )
∣𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 )∣
147
= sgn(𝑞(𝑘0 ,𝜔0 ) ), as 𝑗 → ∞.
From which we conclude that for 𝑗 sufficiently large we must have sgn(𝑞(𝑘𝑗 ,𝜔𝑗 ) ) = sgn(𝑞(𝑘0 ,𝜔0 ) ).
Therefore, since this is true for an arbitrary sequence in ℬℝ , we conclude that there exists an
𝑟 > 0 such that every (𝑘, 𝜔) ∈ 𝐵((𝑘0 , 𝜔 0 ), 𝑟) ∩ ℬℝ is a point of definite type for the canonical
ODEs in (4.13) and sgn(𝑞(𝑘,𝜔) ) = sgn(𝑞(𝑘0 ,𝜔0 ) ). This completes the proof.
Proofs for Section 4.2.4
We now begin proving our main results on the perturbation theory for canonical ODEs. Our
proof relies heavily on the spectral perturbation theory for holomorphic matrix functions
and the reader may wish to review chapter 3 before preceding.
Proof. [Theorem 48] Let (𝑘0 , 𝜔 0 ) ∈ ℬℝ be a point of definite type for the canonical ODEs
in (4.13). Let 𝑔 be the number of Jordan blocks (geometric multiplicity) in the Jordan
form of the monodromy matrix Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 and
𝑚1 ≥ ⋅ ⋅ ⋅ ≥ 𝑚𝑔 ≥ 1 the dimensions of each of those Jordan blocks (partial multiplicities).
Define 𝑚 := 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 (algebraic multiplicity). Let 𝜖 > 0 be given.
Choose an 𝑟0 > 0 such that 0 < 𝑟0 ≤ 𝜖 and 𝐵(𝜔 0 , 𝑟0 ) ∈ Ω. Define the function 𝐿 : 𝑈 →
𝑀𝑛 (ℂ) by
𝐿(𝜔, 𝑘) := Ψ(𝑑, 𝜔) − 𝑒𝑖𝑘𝑑 𝐼𝑛 .
(4.66)
where 𝑈 := 𝐵(𝜔 0 , 𝑟0 ) × 𝐵(𝑘0 , 𝑟0 ). By definition of a holomorphic matrix-valued function of
two variables, see 3.2.(i), and the fact Ψ(𝑑, ⋅) ∈ 𝒪(𝐵(𝜔 0 , 𝑟0 ), 𝑀𝑛 (ℂ)), it follows that
𝐿 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)).
(4.67)
We will now use the results from section 3.2 and give a spectral perturbation analysis of the
148
holomorphic matrix function 𝐿(⋅, 𝑘) near the point (𝜔 0 , 𝑘0 ) ∈ 𝜎(𝐿) = ℬ −1 .
First, since 𝐵(𝜔 0 , 𝑟0 )∗ = 𝐵(𝜔 0 , 𝑟0 ), the adjoint of the holomorphic function Ψ(𝑑, 𝜔), see
Definition 14, is
Ψ∗ (𝑑, 𝜔) = Ψ(𝑑, 𝜔)∗ ,
𝜔 ∈ 𝐵(𝜔 0 , 𝑟0 ),
(4.68)
and by Lemma 24 we have
Ψ∗ (𝑑, ⋅) ∈ 𝒪(𝐵(𝜔 0 , 𝑟0 ), 𝑀𝑛 (ℂ)).
(4.69)
Next, since 𝑈 ∗ = 𝑈 , the adjoint 𝐿∗ of the holomorphic function of two variables 𝐿, see
Definition 15, is
𝐿∗ (𝜔, 𝑘) = 𝐿(𝜔, 𝑘)∗ , (𝜔, 𝑘) ∈ 𝑈,
(4.70)
𝐿∗ ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)).
(4.71)
and by Lemma 26 we have
We now prove series of lemmas which relate the spectrum of 𝐿 to the spectrum of its adjoint
𝐿∗ .
Lemma 59 For every (𝜔, 𝑘) ∈ 𝑈 we have
𝐿∗ (𝜔, 𝑘)J = −𝑒−𝑖𝑘𝑑 Ψ∗ (𝑑, 𝜔)J 𝐿(𝜔, 𝑘).
149
(4.72)
Proof. By Proposition 39 we have
Ψ∗ (𝑑, 𝜔)J Ψ(𝑑, 𝜔) = Ψ(𝑑, 𝜔)∗ J Ψ(𝑑, 𝜔) = J
for every 𝐵(𝜔 0 , 𝑟0 ) ∩ ℝ ⊆ Ωℝ . Thus if (𝜔, 𝑘) ∈ 𝑈 ∩ ℝ2 then
− 𝑒−𝑖𝑘𝑑 Ψ∗ (𝑑, 𝜔)J 𝐿(𝜔, 𝑘) = −𝑒−𝑖𝑘𝑑 Ψ(𝑑, 𝜔)∗ J (Ψ(𝑑, 𝜔) − 𝑒𝑖𝑘𝑑 𝐼𝑛 )
= Ψ(𝑑, 𝜔)∗ J − 𝑒−𝑖𝑘𝑑 J = (Ψ(𝑑, 𝜔)∗ − 𝑒−𝑖𝑘𝑑 )J
= 𝐿∗ (𝜔, 𝑘)J .
But for the matrix-valued function 𝑓 (𝜔, 𝑘) := −𝑒−𝑖𝑘𝑑 Ψ∗ (𝑑, 𝜔)J 𝐿(𝜔, 𝑘)−𝐿∗ (𝜔, 𝑘)J we have
𝑓 ∈ 𝒪(𝑈, 𝑀𝑛 (ℂ)) and its identically equal to zero on 𝑈 ∩ℝ2 where 𝑈 = 𝐵(𝜔 0 , 𝑟0 )×𝐵(𝑘0 , 𝑟0 ).
By Lemma 17, 𝑓 is an analytic function of two variables identically equal to zero on 𝑈 ∩ ℝ2
hence using the power series expansion of 𝑓 at the point (𝜔 0 , 𝑘0 ) and the fact 𝑓 = 0 on
𝑈 ∩ ℝ2 we can conclude 𝑓 = 0 in an open connected set of ℂ2 in 𝑈 which contains (𝜔 0 , 𝑘0 ).
Thus, since 𝑈 is an open connected set on which 𝑓 is analytic, by analytic continuation we
conclude that 𝑓 = 0 on 𝑈 . This completes the proof of the lemma.
The following two lemmas are the key to the proof of this theorem.
Lemma 60 If 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit
point 𝜔 0 then 𝜔(𝑘) and 𝜔 ∗ (𝑘), the adjoint of the Puiseux series 𝜔(𝑘), both are eigenvalue
Puiseux series of 𝐿∗ (⋅, 𝑘) and 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit point 𝜔 0 .
Proof.
By Lemma 59, the fact J and Ψ(𝑑, 𝜔) are invertible matrices for every 𝜔 ∈ Ω,
implies the holomorphic matrix function 𝐿(⋅, 𝑘) and its adjoint 𝐿∗ (⋅, 𝑘) are equivalent, i.e,
𝐿(⋅, 𝑘) ∼ 𝐿∗ (⋅, 𝑘), at 𝜔 in the sense of holomorphic matrix functions from chapter 3 for every
(𝜔, 𝑘) ∈ 𝑈 . In particular, 𝐿(⋅, 𝑘) and its adjoint 𝐿∗ (⋅, 𝑘) have the same eigenvalues for every
150
𝑘 ∈ 𝐵(𝑘0 , 𝑟0 ), i.e., for every 𝑘 ∈ 𝐵(𝑘0 , 𝑟0 )
𝜎(𝐿∗ (⋅, 𝑘)) = 𝜎(𝐿(⋅, 𝑘)).
This implies if 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit
point 𝜔 0 then by Lemma 28 its adjoint 𝜔 ∗ (𝑘) is an eigenvalue Puiseux series of 𝐿∗ (⋅, 𝑘)
expanded about 𝑘0 = 𝑘0 with limit point 𝜔 0 = 𝜔 0 . But since 𝜎(𝐿∗ (⋅, 𝑘)) = 𝜎(𝐿(⋅, 𝑘)) then
all the values of the branches of the Puiseux series 𝜔 ∗ (𝑘) and 𝜔(𝑘) are eigenvalues of both
𝐿∗ (⋅, 𝑘) and 𝐿(⋅, 𝑘) which, by the definition of eigenvalue Puiseux series, means 𝜔 ∗ (𝑘) and
𝜔(𝑘) are also eigenvalue Puiseux series of 𝐿∗ (⋅, 𝑘) expanded about 𝑘0 with limit point 𝜔 0 .
This completes the proof.
Lemma 61 If 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit
point 𝜔 0 and 𝛽 is generating eigenvector of 𝐿(⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and associated with
𝜔(𝑘) then J 𝛽 is a generating eigenvector of 𝐿∗ (⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and associated with
𝜔(𝑘). Moreover,
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛽, J 𝛽⟩ℂ ∕= 0.
Proof.
(4.73)
Let 𝜔(𝑘) be an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit
point 𝜔 0 and 𝛽 is generating eigenvector of 𝐿(⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and associated
with 𝜔(𝑘). By definition of generating eigenvector, there exists a eigenpair Puiseux series
(𝜔(𝑘), 𝛾(𝑘)) of 𝐿(⋅, 𝑘) expanded about 𝑘0 such that 𝜔(𝑘) has limit point 𝜔 0 and 𝛾(𝑘) has
limit point 𝛽. By Lemma 59 and the fact det(J ) ∕= 0 it follows that (𝜔(𝑘), J 𝛾(𝑘)) is
an eigenpair Puiseux series of 𝐿∗ (⋅, 𝑘) expanded about 𝑘0 . Hence J 𝛾(𝑘0 ) = J 𝛽 0 is the
limit point of the eigenvector Puiseux series J 𝛾(𝑘). By definition of generating eigenvector
this means J 𝛽 is a generating eigenvector of 𝐿∗ (⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and associated
with 𝜔(𝑘). Moreover, since 𝛽 is a generating eigenvector of 𝐿(⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) this
151
means 𝛽 ∕= 0 and 𝛽 ∈ ker 𝐿(𝜔 0 , 𝑘0 ) = ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )). This means by Lemma 45 and
Corollary 43 that
1
1
0 ∕= 𝑞( 𝑘0 ,𝜔0 ) (𝛽, 𝛽) = ⟨Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 )𝛽, 𝛽⟩ℂ = ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛽, Ψ(𝑑, 𝜔 0 )𝛽⟩ℂ
𝑑
𝑑
𝑖𝑘0 𝑑
1
𝑒
= ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛽, 𝑒𝑖𝑘0 𝑑 𝛽⟩ℂ = −
⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛽, J 𝛽⟩ℂ .
𝑑
𝑑
Therefore,
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 , J 𝛽 0 ⟩ℂ = ⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛽, J 𝛽⟩ℂ ∕= 0.
This completes the proof of the lemma.
Proposition 62 If 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with
limit point 𝜔 0 then 𝜔(𝑘) is a single-valued real analytic nonconstant function of 𝑘.
Proof. Let 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit point
𝜔 0 . Then by Lemma 60 we know that 𝜔 ∗ (𝑘) is also an eigenvalue Puiseux series of 𝐿(⋅, 𝑘)
expanded about 𝑘0 with limit point 𝜔 0 . Let 𝜔 1 (𝑘) := 𝜔(𝑘) and 𝜔 2 (𝑘) := 𝜔 ∗ (𝑘). By definition
of the adjoint of a convergent Puiseux series we have 𝜔 ∗2 (𝑘) = (𝜔 ∗ )∗ (𝑘) = 𝜔(𝑘) = 𝜔 1 (𝑘).
Suppose that for some branch 𝜔 1,ℎ1 (𝑘) of 𝜔 1 (𝑘) and some branch 𝜔 2,ℎ2 (𝑘) of 𝜔 2 (𝑘) there
existed an 𝑟 > 0 such that 𝜔 1,ℎ1 (𝑘) ∕= 𝜔 2,ℎ2 (𝑘) for ∣𝑘 − 𝑘0 ∣ < 𝑟. Then this implies by
Proposition 29 that for every generating eigenvector 𝛽 of 𝐿(⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and
associated with 𝜔 1 (𝑘) and for any generating eigenvector 𝛾 of 𝐿∗ (⋅, 𝑘) at the point (𝜔 0 , 𝑘0 )(=
(𝜔 0 , 𝑘0 )) and associated with 𝜔 ∗2 (𝑘)(= 𝜔 1 (𝑘)) we have
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛽, 𝛾⟩ℂ = 0.
But by Lemma 61 we know that J 𝛽 is a generating eigenvector of 𝐿∗ (⋅, 𝑘) at the point
152
(𝜔 0 , 𝑘0 ) = (𝜔 0 , 𝑘0 ) and associated with 𝜔 ∗2 (𝑘) = 𝜔 1 (𝑘) and
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛽, J 𝛽⟩ℂ ∕= 0.
Thus we have a contradiction. This contradiction implies that there must exist an 𝑟 > 0
such that if 𝜔 1,ℎ1 (𝑘) is any branch of 𝜔 1 (𝑘) and if 𝜔 2,ℎ2 (𝑘) is any branch of 𝜔 2 (𝑘) then
𝜔 1,ℎ1 (𝑘) = 𝜔 2,ℎ2 (𝑘) for ∣𝑘 − 𝑘0 ∣ < 𝑟. Thus if we fix the branch 𝜔 2,ℎ2 (𝑘) then all branches of
𝜔 1 (𝑘) are equal to 𝜔 2,ℎ2 (𝑘) for ∣𝑘 −𝑘0 ∣ < 𝑟. By Lemma 30 this implies 𝜔 1 (𝑘) is a single-valued
analytic function of 𝑘 for 𝑘 ∈ 𝐵(𝑘0 , 𝑟). For the exact same reasoning this implies 𝜔 2 (𝑘) is
a single-valued analytic function of 𝑘 and we have 𝜔 1 (𝑘) = 𝜔 2 (𝑘) for all 𝑘 ∈ 𝐵(𝑘0 , 𝑟). But
since 𝜔(𝑘) = 𝜔 1 (𝑘) = 𝜔 2 (𝑘) = 𝜔 ∗ (𝑘) where 𝜔 ∗ (𝑘) is the adjoint of the analytic function 𝜔(𝑘),
this implies by definition of adjoint of a Puiseux series, that 𝜔(𝑘) = 𝜔 ∗ (𝑘) = 𝜔(𝑘) for all
𝑘 ∈ 𝐵(𝑘0 , 𝑟). And so 𝜔(𝑘) is a single-valued analytic function in 𝐵(𝑘0 , 𝑟) and is real-valued
for real 𝑘, i.e., is a real analytic function of 𝑘 in 𝐵(𝑘0 , 𝑟).
To complete the proof we need only show that 𝜔(𝑘) is a nonconstant function of 𝑘, i.e.,
𝜔(𝑘) = 𝜔(𝑘0 ) for every 𝑘 in 𝐵(𝑘0 , 𝑟) is a contradiction. By hypothesis we have 𝜔(𝑘0 ) = 𝜔 0 and
since 𝜔(𝑘) is an eigenvalue Puiseux series of 𝐿(⋅, 𝑘) this means 0 = det 𝐿(𝜔(𝑘), 𝑘) for every
𝑘 in 𝐵(𝑘0 , 𝑟). If 𝜔(𝑘) was a constant function then this would mean 0 = det 𝐿(𝜔(𝑘), 𝑘) =
det 𝐿(𝜔 0 , 𝑘) = det(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) for every 𝑘 in 𝐵(𝑘0 , 𝑟). But det(𝜆𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) is a
monic polynomial in 𝜆 and so has a nonzero but finite number of zeros where as 𝑓 (𝑘) := 𝑒𝑖𝑘𝑑
being a nonconstant analytic function is an open map and hence 𝑓 (𝐵(𝑘0 , 𝑟)) is a nonempty
open set in ℂ and hence contains an uncountable number of elements. But then because
0 = det(𝑓 (𝑘)𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) for every 𝑘 in 𝐵(𝑘0 , 𝑟) this implies 𝑓 (𝐵(𝑘0 , 𝑟)) has only a finite
number of elements since they are all roots of det(𝜆𝐼𝑛 −Ψ(𝑑, 𝜔 0 )). A contradiction. Therefore
𝜔(𝑘) is a nonconstant function. This completes the proof.
Lemma 63 Let 𝜔 1 (𝑘) and 𝜔 2 (𝑘) are two different eigenvalue Puiseux series of 𝐿(⋅, 𝑘) ex-
153
panded about 𝑘0 with limit point 𝜔 0 . If 𝛽 1 , 𝛽 2 are generating eigenvectors of 𝐿(⋅, 𝑘) at the
point (𝜔 0 , 𝑘0 ) corresponding to 𝜔 1 (𝑘) and 𝜔 2 (𝑘), respectively, then
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛽 1 , J 𝛽 2 ⟩ℂ = 0.
(4.74)
Proof. Let 𝜔 1 (𝑘) and 𝜔 2 (𝑘) are two different eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded
about 𝑘0 with limit point 𝜔 0 . Let 𝛽 1 , 𝛽 2 are generating eigenvectors of 𝐿(⋅, 𝑘) at the point
(𝜔 0 , 𝑘0 ) corresponding to 𝜔 1 (𝑘) and 𝜔 2 (𝑘), respectively. Then by Proposition 62 𝜔 1 (𝑘) and
𝜔 2 (𝑘) are single-valued real analytic functions and hence there adjoints satisfy 𝜔 ∗𝑗 (𝑘) =
𝜔 𝑗 (𝑘) = 𝜔 𝑗 (𝑘) for 𝑗 = 1, 2. By Lemma 61 we know J 𝛽 2 is a generating eigenvector of 𝐿∗ (⋅, 𝑘)
at the point (𝜔 0 , 𝑘0 ) = (𝜔 0 , 𝑘0 ) and associated with 𝜔 2 (𝑘). This implies by Proposition 29,
since 𝜔 1 (𝑘) and 𝜔 ∗2 (𝑘) = 𝜔 2 (𝑘) are different analytic functions, that
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛽 1 , J 𝛽 2 ⟩ℂ = 0.
(4.75)
This completes the proof.
Proposition 64 The holomorphic matrix function 𝐿(⋅, 𝑘0 ) has 𝜔 0 an eigenvalue of finite
algebraic multiplicity. Furthermore, it is a semisimple eigenvalue of 𝐿(⋅, 𝑘0 ). Moreover, its
geometric multiplicity as an eigenvalue of 𝐿(⋅, 𝑘0 ) is 𝑔 and this is the order of the analytic
function det 𝐿(𝜔, 𝑘0 ) = 𝐷(𝑘0 , 𝜔) at 𝜔 = 𝜔 0 .
Proof.
First, since (𝑘0 , 𝜔 0 ) ∈ ℬℝ then 0 = 𝐷(𝑘0 , 𝜔 0 ) = det(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) =
(−1)𝑛 det(𝐿(𝜔 0 , 𝑘0 )) so that 𝜔 0 ∈ 𝜎(𝐿(⋅, 𝑘0 )) and hence by definition 𝜔 0 an eigenvalue of
the holomorphic matrix function 𝐿(⋅, 𝑘0 ). To show that it is an eigenvalue of finite algebraic
multiplicity and a semisimple eigenvalue, it suffices to show, by Proposition 15, that
𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 = −𝐿(𝜔 0 , 𝑘0 )𝛾 1
154
has no solution for 𝛾 0 ∈ ker(𝐿(𝜔 0 , 𝑘0 ))/{0} and 𝛾 1 ∈ ℂ𝑛 .
Suppose there existed vectors 𝛾 0 ∈ ker(𝐿(𝜔 0 , 𝑘0 ))/{0}, 𝛾 1 ∈ ℂ𝑛 such that 𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 =
−𝐿(𝜔 0 , 𝑘0 )𝛾 1 . It follows from this, the fact ker(𝐿(𝜔 0 , 𝑘0 )) = ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )), Proposition 45 along with the fact (𝑘0 , 𝜔 0 ) is a point of definite type, that we have
1
0 ∕= 𝑞(𝑘0 ,𝜔0 ) (𝛾 0 , 𝛾 0 ) = ⟨Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 )𝛾 0 , 𝛾 0 ⟩ℂ
𝑑
1
1
= ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛾 0 , Ψ(𝑑, 𝜔 0 )𝛾 0 ⟩ℂ = ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛾 0 , 𝑒𝑖𝑘0 𝑑 𝛾 0 ⟩ℂ
𝑑
𝑑
𝑒𝑖𝑘0 𝑑
𝑒𝑖𝑘0 𝑑
=−
⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛾 0 , J 𝛾 0 ⟩ℂ = −
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 , J 𝛾 0 ⟩ℂ .
𝑑
𝑑
But 𝐿(𝜔 0 , 𝑘0 )𝛾 0 = 0 and so by Lemma 59 this implies 0 = 𝐿∗ (𝜔 0 , 𝑘0 )J 𝛾 0 = 𝐿(𝜔 0 , 𝑘0 )∗ J 𝛾 0
which implies
⟨𝐿(𝜔 0 , 𝑘0 )𝛾 1 , J 𝛾 0 ⟩ℂ = ⟨𝛾 1 , 𝐿(𝜔 0 , 𝑘0 )J 𝛾 0 ⟩ℂ = 0
which implies
0 ∕= −
𝑒𝑖𝑘0 𝑑
𝑒𝑖𝑘0 𝑑
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 , J 𝛾 0 ⟩ℂ = −
⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝛾 0 + 𝐿(𝜔 0 , 𝑘0 )𝛾 1 , J 𝛾 0 ⟩ℂ = 0.
𝑑
𝑑
This is a contradiction. Thus 𝜔 0 is an eigenvalue of 𝐿(⋅, 𝑘0 ) of finite algebraic multiplicity
and a semisimple eigenvalue. And so its geometric multiplicity equals its algebraic multiplicity and hence by Proposition 10, its geometric multiplicity is dim ker(𝐿(𝜔 0 , 𝑘0 )) =
dim ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) = 𝑔 and its algebraic multiplicity again 𝑔 is equal to the order of
the analytic function det(𝐿(𝜔, 𝑘0 )) at 𝜔 = 𝜔 0 . And since det(𝐿(𝜔, 𝑘0 )) = (−1)𝑛 det(𝑒𝑖𝑘0 𝑑 𝐼𝑛 −
Ψ(𝑑, 𝜔 0 )) = 𝐷(𝑘0 , 𝜔) for 𝜔 ∈ 𝐵(𝜔 0 , 𝑟0 ) this implies 𝑔 is the order of the zero of the analytic
function 𝐷(𝑘0 , 𝜔) at 𝜔 = 𝜔 0 . This completes the proof of the proposition.
Proposition 65 The holomorphic matrix function 𝐿(𝜔 0 , ⋅) has 𝑘0 an eigenvalue of finite
algebraic multiplicity. Moreover, 𝑘0 as an eigenvalue of 𝐿(𝜔 0 , ⋅) has 𝑔, 𝑚1 ≥ . . . ≥ 𝑚𝑔 , 𝑚
155
as its geometric, partial, and algebraic multiplicities, respectively. In particular, the order of
the zero of the analytic function det 𝐿(𝜔 0 , 𝑘) = 𝐷(𝑘, 𝜔 0 ) at 𝑘 = 𝑘0 is 𝑚.
Proof. We have 𝐿(𝜔 0 , 𝑘) = Ψ(𝑑, 𝜔 0 )−𝑒𝑖𝑘𝑑 𝐼𝑛 . If we define the holomorphic matrix functions
𝐹 (𝜆) := Ψ(𝑑, 𝜔 0 ) − 𝜆𝐼𝑛 and 𝐺(𝑘) := Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘𝑑 𝐼𝑛 then it follows that 𝐺(𝑘) = 𝐹 (𝑒𝑖𝑘𝑑 ). It
thus follows that 𝑘0 is an eigenvalue of 𝐺, 𝜆0 := 𝑒𝑖𝑘0 𝑑 is an eigenvalue of 𝐹 , and there is a oneto-one correspondence with their geometric, partial, and algebraic multiplicities. But as we
know by Example 1 in chapter 3 and by definition of the numbers 𝑔, 𝑚1 ≥ . . . ≥ 𝑚𝑔 , and 𝑚,
it follows that the geometric, partial, and algebraic multiplicities of the eigenvalue 𝜆0 of 𝐹 are
just, in the Jordan normal form of the matrix Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 , 𝑔
the number of Jordan blocks, their orders 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 = 𝑚1 + ⋅ ⋅ ⋅ + 𝑚𝑔 , respectively.
Thus 𝑔, 𝑚1 , . . . , 𝑚𝑔 , and 𝑚 are the geometric, partial, and algebraic multiplicities of the
eigenvalue 𝑘0 of 𝐿(𝜔 0 , 𝑘). By Proposition 10, since 𝑚 being the algebraic multiplicity of the
eigenvalue 𝑘0 of 𝐿(𝜔 0 , ⋅), we know that 𝑚 is the order of the zero of the function det 𝐿(𝜔 0 , 𝑘)
at 𝑘 = 𝑘0 . And since det 𝐿(𝜔 0 , 𝑘) = 𝐷(𝑘, 𝜔 0 ), the proof is complete.
Theorem 66 There exists a 𝛿 > 0 such that 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) ⊆ 𝑈 and ℬ ∩ 𝐵(𝑘0 , 𝛿) ×
𝐵(𝜔 0 , 𝜖) is the union of the graphs of 𝑔 nonconstant real analytic functions 𝜔 1 (𝑘), . . . , 𝜔 𝑔 (𝑘)
given by convergent power series
𝑚𝑗
𝜔 𝑗 (𝑘) = 𝜔 0 + 𝜈 𝑗,𝑚𝑗 (𝑘 − 𝑘0 )
+
∞
∑
𝜈 𝑗,𝑙 (𝑘 − 𝑘0 )𝑙 , ∣𝑘 − 𝑘0 ∣ < 𝛿
(4.76)
𝑙=𝑚𝑗 +1
where
𝜈 𝑗,𝑚𝑗 ∕= 0,
(4.77)
for 𝑗 = 1, . . . , 𝑔. Moreover, there exists analytic functions 𝜑1 (𝑘), . . . , 𝜑𝑔 (𝑘) belonging to
156
𝒪(𝐵(𝑘0 , 𝛿), ℂ𝑛 ) such that
𝜑𝑗 (𝑘) ∕= 0,
𝐿(𝜔 𝑗 (𝑘), 𝑘)𝜑𝑗 (𝑘) = 0, ∣𝑘 − 𝑘0 ∣ < 𝛿,
𝑗 = 1, . . . , 𝑔
(4.78)
and the vectors 𝜑1 (𝑘), . . . , 𝜑𝑔 (𝑘) are linearly independent for ∣𝑘 − 𝑘0 ∣ < 𝛿.
Proof. The spectrum 𝜎(𝐿) of the holomorphic matrix-valued function 𝐿 is given by
𝜎(𝐿) = {(𝜔, 𝑘) ∈ 𝑈 ∣ det 𝐿(𝜔, 𝑘) = 0}.
It follows from the definition of 𝐿 and the set 𝑈 = 𝐵(𝜔 0 , 𝑟0 ) × 𝐵(𝑘0 , 𝑟0 ) ⊆ Ω × ℂ that the
inverse set 𝜎(𝐿)−1 is a subset of the Bloch variety ℬ since
𝜎(𝐿)−1 = {(𝑘, 𝜔) ∈ 𝐵(𝑘0 , 𝑟0 ) × 𝐵(𝜔 0 , 𝑟0 ) ∣ det 𝐿(𝜔, 𝑘) = 0}
= {(𝑘, 𝜔) ∈ 𝐵(𝑘0 , 𝑟0 ) × 𝐵(𝜔 0 , 𝑟0 ) ∣ det 𝐷(𝑘, 𝜔) = 0}
= ℬ ∩ 𝐵(𝑘0 , 𝑟0 ) × 𝐵(𝜔 0 , 𝑟0 ).
Now by Proposition 64 the algebraic multiplicity of the eigenvalue 𝜔 0 of 𝐿(⋅, 𝑘0 ) is 𝑔 and so
by Theorem 21, there exists an 𝛿 > 0, which we may assume without loss of generality 𝛿 ≤ 𝑟0 ,
such that 𝜎(𝐿)−1 ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) is the union of the graphs of 𝑙 convergent Puiseux
series 𝜔 1 (𝑘), . . . , 𝜔 𝑙 (𝑘) expanded about 𝑘0 with limit point 𝜔 0 , domain 𝐵(𝑘0 , 𝛿), and periods
𝑞1 , . . . , 𝑞𝑙 , respectively, which satisfy 𝑔 = 𝑞1 +⋅ ⋅ ⋅+𝑞𝑙 . This implies each 𝜔 𝑗 (𝑘) is an eigenvalue
Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 with limit point 𝜔 0 and so by Proposition 62,
𝜔 𝑗 (𝑘) is a single-valued nonconstant real analytic function of 𝑘 in the domain 𝐵(𝑘0 , 𝛿), for
𝑗 = 1, . . . , 𝑙. Then, since 𝛿 ≤ 𝑟0 ≤ 𝜖 and 𝜎(𝐿)−1 = ℬ ∩ 𝐵(𝑘0 , 𝑟0 ) × 𝐵(𝜔 0 , 𝑟0 ), we have
𝜎(𝐿)−1 ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) = ℬ ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) and it is the union of the graphs of 𝑔
convergent Puiseux series 𝜔 1 (𝑘), . . . , 𝜔 𝑙 (𝑘) single-valued nonconstant real analytic function
of 𝑘 in the domain 𝐵(𝑘0 , 𝛿) with 𝜔 𝑗 (𝑘0 ) = 𝜔 0 , for 𝑗 = 1, . . . , 𝑙.
157
Let 𝜔 1 (𝑘), . . . , 𝜔 𝑔ˆ(𝑘) denote all those analytic functions which are not identically equal on
𝐵(𝑘0 , 𝛿).
This implies now that they have a convergent series expansion
𝜔 𝑗 (𝑘) = 𝜔 0 + 𝜈 𝑗,𝑞𝑗 (𝑘 − 𝑘0 )𝑞𝑗 +
∞
∑
𝜈 𝑗,𝑙 (𝑘 − 𝑘0 )𝑙 , ∣𝑘 − 𝑘0 ∣ < 𝛿
𝑙=𝑞𝑗 +1
where
𝜈 𝑗,𝑞𝑗 ∕= 0,
for 𝑗 = 1, . . . , 𝑔ˆ. Without loss of generality we may assume 𝑞1 ≥ ⋅ ⋅ ⋅ ≥ 𝑞𝑔ˆ.
Now by the Weierstrass Preparation Theorem, see Theorem 18 in the chapter 3, since det 𝐿 ∈
𝒪((𝜔 0 , 𝑘0 )), det 𝐿(𝜔 0 , 𝑘0 ) = 0, det 𝐿(⋅, 𝑘0 ) ∕= 0 which follows since 𝐿(⋅, 𝑘0 ) has 𝜔 0 as a
semisimple eigenvalue, then
det 𝐿 = 𝑓0 𝑓1 ,
𝑓0 ∈ 𝒪0 ((𝜔 0 , 𝑘0 )),
𝑓1 ∈ 𝒪((𝜔 0 , 𝑘0 ))
where 𝑓1 (𝜔, 𝑘) ∕= 0 for (𝜔, 𝑘) ∈ 𝐵(𝜔 0 , 𝑟) × 𝐵(𝑘0 , 𝑟) with 0 < 𝑟 ≪ 1 and deg 𝑓0 = 𝑔, since 𝑔 is
the order of the zero of det 𝐿(𝜔, 𝑘0 ) = 𝑓0 (𝜔, 𝑘0 )𝑓1 (𝜔, 𝑘0 ) because by Proposition 64 we know
𝜔 0 is a semisimple eigenvalue of 𝐿(⋅, 𝑘0 ), which means its geometric multiplicity 𝑔 equals
its algebraic multiplicity equals the order of the zero of det 𝐿(𝜔, 𝑘0 ) at 𝜔 = 𝜔 0 which thus
equals the order of the zero of 𝑓0 (𝜔, 𝑘0 ) at 𝜔 = 𝜔 0 which is by definition the deg 𝑓0 . Now by
Theorem 19 since 𝑓0 ∕= 1 because deg 𝑓0 = 𝑔 > 1 there exists unique and distinct irreducibles
𝑝1 , . . . , 𝑝𝑠 ∈ 𝒪0 ((𝜔 0 , 𝑘0 )) such that
𝑓0 =
𝑠
∏
𝑗=1
158
𝑚(𝑗)
𝑝𝑗
where 𝑚(𝑗) is the unique number representing the multiplicity of the irreducible factor 𝑝𝑗 ,
∑𝑠
for 𝑗 = 1, . . . , 𝑠 and 𝑔 = deg 𝑓0 =
ˆ𝑗 (𝜔, 𝑘) := 𝜔 − 𝜔 𝑗 (𝑘) =
𝑗=1 𝑚(𝑗) deg 𝑝𝑗 . But for 𝑝
(𝜔 − 𝜔 0 ) − (𝜔 𝑗 (𝑘) − 𝜔 0 ) we have 𝑝ˆ𝑗 ∈ 𝒪0 ((𝜔 0 , 𝑘0 )) and it irreducible, for 𝑗 = 1, . . . , 𝑔ˆ. Now
though as we have shown 𝜎(𝐿)−1 ∩ 𝐵(𝑘0 , 𝛿) × 𝐵(𝜔 0 , 𝜖) is the graph of the 𝜔 1 (𝑘), . . . , 𝜔 𝑔ˆ(𝑘)
and hence since det 𝐿 = 𝑓0 𝑓1 then the zeros of 𝑓0 are exactly the zeros of 𝑝ˆ𝑗 ∈ 𝒪0 ((𝜔 0 , 𝑘0 ))
implying by Theorem 20 and the uniqueness of the factorization into irreducibles we must
have after possible reordering of the indices, 𝑔ˆ = 𝑠, 𝑝𝑗 = 𝑝ˆ𝑗 for 𝑗 = 1, . . . , 𝑔ˆ, and
𝑔ˆ
∏
𝑓0 =
𝑚(𝑗)
𝑝ˆ𝑗
.
𝑗=1
This of course implies, since deg 𝑝ˆ𝑗 = 1 for each 𝑗, that
𝑔 = deg 𝑓0 =
𝑔ˆ
∑
𝑚(𝑗)
deg 𝑝ˆ𝑗
=
𝑗=1
𝑔ˆ
∑
𝑚(𝑗) deg 𝑝ˆ𝑗 =
𝑗=1
𝑔ˆ
∑
𝑚(𝑗).
𝑗=1
Now it follows from what we have just shown that
det 𝐿(𝜔 0 , 𝑘) = 𝑓0 (𝜔 0 , 𝑘)𝑓1 (𝜔 0 , 𝑘) = 𝑓1 (𝜔 0 , 𝑘)
𝑔ˆ
∏
𝑚(𝑗)
𝑝ˆ𝑗
(𝜔 0 , 𝑘)
𝑗=1
𝑔ˆ
𝑔ˆ
∏
∏
𝑚(𝑗)
= 𝑓1 (𝜔 0 , 𝑘) (𝜔 0 − 𝜔 𝑗 (𝑘))
= 𝑓1 (𝜔 0 , 𝑘) (−𝜈 𝑗,𝑞𝑗 (𝑘 − 𝑘0 )𝑞𝑗 + 𝑜((𝑘 − 𝑘0 )𝑞𝑗 ))𝑚(𝑗)
𝑗=1
(
=
𝑓1 (𝜔 0 , 𝑘)
𝑔ˆ
𝑗=1
)
∏
(−𝜈 𝑗,𝑞𝑗 )𝑚(𝑗)
∑𝑔ˆ
𝑗=1 𝑞𝑗 𝑚(𝑗)
(𝑘 − 𝑘0 )
(
)
∑𝑔ˆ
+ 𝑜 (𝑘 − 𝑘0 ) 𝑗=1 𝑞𝑗 𝑚(𝑗)
𝑗=1
𝑔ˆ
∏
𝑓1 (𝜔 0 , 𝑘0 ) (−𝜈 𝑗,𝑞𝑗 )𝑚(𝑗)
(
=
)
∑𝑔ˆ
(𝑘 − 𝑘0 )
𝑗=1 𝑞𝑗 𝑚(𝑗)
(
∑𝑔ˆ
+ 𝑜 (𝑘 − 𝑘0 )
𝑗=1 𝑞𝑗 𝑚(𝑗)
)
𝑗=1
𝑔ˆ
∏
as 𝑘 → 𝑘0 . The leading term 𝑓1 (𝜔 0 , 𝑘0 ) (−𝜈 𝑗,𝑞𝑗 )𝑚(𝑗) ∕= 0 implying the order of the zero
𝑗=1
∑𝑔ˆ
of det 𝐿(𝜔 0 , 𝑘) at 𝑘 = 𝑘0 is 𝑗=1 𝑞𝑗 𝑚(𝑗), but by Proposition 65 the order of the zero of
159
det 𝐿(𝜔 0 , 𝑘) at 𝑘 = 𝑘0 is 𝑚. This implies
𝑚=
𝑔ˆ
∑
𝑞𝑗 𝑚(𝑗).
𝑗=1
Lemma 67 For 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1, 𝜔 𝑗 (𝑘) is a semisimple eigenvalue of 𝐿(⋅, 𝑘) and
𝑚(𝑗) = dim ker 𝐿(𝜔 𝑗 (𝑘), 𝑘),
𝑗 = 1, . . . , 𝑔ˆ.
(4.79)
Proof. For 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1 we have 𝜔 𝑗1 (𝑘) ∕= 𝜔 𝑗2 (𝑘) for 𝑗1 ∕= 𝑗2 with 1 ≤ 𝑗1 , 𝑗2 ≤ 𝑔ˆ, since
by assumption the functions 𝜔 1 (𝑘), . . . , 𝜔 𝑔ˆ(𝑘) where not identically equal on 𝐵(𝑘0 , 𝛿). And
since
𝑔ˆ
∏
det 𝐿(𝜔, 𝑘) = 𝑓0 (𝜔, 𝑘)𝑓1 (𝜔, 𝑘) = 𝑓1 (𝜔, 𝑘) (𝜔 − 𝜔 𝑗 (𝑘))𝑚(𝑗) ,
𝑓1 (𝜔, 𝑘) ∕= 0
𝑗=1
for all (𝜔, 𝑘) ∈ 𝐵(𝜔 0 , 𝑟) × 𝐵(𝑘0 , 𝑟) for 0 < 𝑟 ≪ 1, these fact together with 𝜔 𝑗 (𝑘0 ) = 𝜔 0
imply that if 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1 then the order of the zero of the function det 𝐿(⋅, 𝑘) at
𝜔 = 𝜔 𝑗 (𝑘) is 𝑚(𝑗) which by 10 this is the algebraic multiplicity of the eigenvalue 𝜔 𝑗 (𝑘) of
the holomorphic matrix function 𝐿(⋅, 𝑘), for 𝑗 = 1, . . . , 𝑔ˆ. Now it follows by the local Smith
form (see Theorem 9) of the holomorphic matrix function 𝐿(𝜔 𝑗 (𝑘), 𝑘) at the eigenvalue 𝑘0
that
𝜇(𝑗) := dim ker 𝐿(𝜔 𝑗 (𝑘), 𝑘),
𝑗 = 1, . . . , 𝑔ˆ, 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1,
are constant. Our goal is to now show that 𝜇(𝑗) = 𝑚(𝑗), for 𝑗 = 1, . . . , 𝑔ˆ. Notice first that by
definition of 𝜇(𝑗), it is the geometric multiplicity of the eigenvalue of 𝜔 𝑗 (𝑘) of the holomorphic
matrix function 𝐿(⋅, 𝑘) and since 𝑚(𝑗) is the algebraic multiplicity of the eigenvalue 𝜔 𝑗 (𝑘) of
160
the holomorphic matrix function 𝐿(⋅, 𝑘) for 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1. But it follows from Theorem
47 that if ∣𝑘 − 𝑘0 ∣ ≪ 1 and 𝑘 is real then (𝑘, 𝜔 𝑗 (𝑘)) is point of definite type for the canonical
ODEs in (4.13) and hence by Proposition 64 we know 𝜔 𝑗 (𝑘) is a semisimple eigenvalue of
the holomorphic matrix function 𝐿(⋅, 𝑘) which by definition of semisimple eigenvalue means
its geometric multiplicity 𝜇(𝑗) equals its algebraic multiplicity 𝑚(𝑗), i.e., 𝜇(𝑗) ≤ 𝑚(𝑗), for
𝑗 = 1, . . . , 𝑔ˆ. And therefore this implies for all 𝑘 satisfying 0 < ∣𝑘 − 𝑘0 ∣ ≪ 1 we have
𝑚(𝑗) = 𝜇(𝑗) = dim ker 𝐿(𝜔 𝑗 (𝑘), 𝑘),
𝑗 = 1, . . . , 𝑔ˆ,
which means 𝜔 𝑗 (𝑘) is a semisimple eigenvalue of 𝐿(⋅, 𝑘). This completes the proof.
Now it follows this lemma and the local Smith form, Theorem 9, of the holomorphic matrix
function 𝐿(𝜔 𝑗 (𝑘), 𝑘) in the variable 𝑘 with eigenvalue 𝑘0 , that there exists an 𝑟 > 0 and there
exists function 𝑁𝑗 , 𝑀𝑗 ∈ 𝒪(𝐵(𝑘0 , 𝑟), 𝑀𝑛 (ℂ)) with 𝑁𝑗 (𝑘), 𝑀𝑗 (𝑘) invertible matrices and
𝑁𝑗 (𝑘)𝐿(𝜔 𝑗 (𝑘), 𝑘)𝑀𝑗 (𝑘) = diag{0, . . . , 0, (𝑘 − 𝑘0 )𝑚𝑗,𝑚(𝑗)+1 , . . . , (𝑘 − 𝑘0 )𝑚𝑗,𝑛 }
(4.80)
for every 𝑘 ∈ 𝐵(𝑘0 , 𝑟), where the number of zeros down the diagonal is given by 𝑚(𝑗) and
{𝑚𝑗,𝑖 }𝑛𝑖=𝑚(𝑗)+1 is a decreasing sequence of nonnegative integers, for 𝑗 = 1, . . . , 𝑔ˆ.
Let 𝑒1 , . . . , 𝑒𝑛 denote the standard orthonormal vectors from ℂ𝑛 . We now define the vectorvalued functions
𝜑𝑗,𝑖 (𝑘) := 𝑀𝑗 (𝑘)𝑒𝑖 ,
𝑖 = 1, . . . , 𝑚(𝑗),
𝑗 = 1, . . . , 𝑔ˆ, ∣𝑘 − 𝑘0 ∣ < 𝑟.
It follows, since 𝑀𝑗 ∈ 𝒪(𝐵(𝑘0 , 𝑟), 𝑀𝑛 (ℂ)), that 𝜑𝑗,𝑖 ∈ 𝒪(𝐵(𝑘0 , 𝑟), ℂ𝑛 ) for all 𝑖, 𝑗 and
𝐿(𝜔 𝑗 (𝑘), 𝑘)𝜑𝑗,𝑖 (𝑘) = 0,
𝑖 = 1, . . . , 𝑚(𝑗),
161
𝑗 = 1, . . . , 𝑔ˆ, ∣𝑘 − 𝑘0 ∣ < 𝑟.
Lemma 68 For every 𝑘 with ∣𝑘 − 𝑘0 ∣ ≪ 1, the vectors 𝜑𝑗,𝑖 (𝑘), 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ
𝑚(𝑖)
are linear independent, {𝜑𝑗,𝑖 (𝑘)}𝑖=1 is a basis for ker 𝐿(𝜔 𝑗 (𝑘), 𝑘), for 𝑗 = 1, . . . , 𝑔ˆ, and
∪𝑔ˆ
𝑚(𝑖)
𝑗=1 {𝜑𝑗,𝑖 (𝑘0 )}𝑖=1 is a basis for ker 𝐿(𝜔 0 , 𝑘0 ).
Proof. For the first part of the theorem it suffices to prove that their limit points 𝜑𝑗,𝑖 (𝑘0 ),
𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ are linearly independent. Suppose they were not. Then there
∑ˆ ∑𝑚(𝑗)
exists constants 𝑐𝑗,𝑖 , 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ with 𝑔𝑗=1
𝑖=1 ∣𝑐𝑗,𝑖 ∣ > 0 such that
0=
𝑔ˆ 𝑚(𝑗)
∑
∑
𝑐𝑗,𝑖 𝜑𝑗,𝑖 (𝑘0 ).
𝑗=1 𝑖=1
But if 𝑗1 , 𝑗2 ∈ {1, . . . , 𝑔ˆ, 𝑖1 ∈ {1, . . . , 𝑚(𝑗1 ), and 𝑖2 ∈ {1, . . . , 𝑚(𝑗2 )} then (𝜔 𝑗1 (𝑘), 𝜑𝑖1 ,𝑗1 (𝑘))
and (𝜔 𝑗2 (𝑘), 𝜑𝑖2 ,𝑗2 (𝑘)) are eigenpair Puiseux series of 𝐿(⋅, 𝑘) expanded about 𝑘0 . By the
definition of generating eigenvector, 𝜑𝑗1 ,𝑖1 (𝑘0 ) and 𝜑𝑗2 ,𝑖2 (𝑘0 ) are generating eigenvectors of
𝐿(⋅, 𝑘) at the point (𝜔 0 , 𝑘0 ) and associated with 𝜔 1 (𝑘) and 𝜔 2 (𝑘), respectively. Now if 𝑗1 ∕= 𝑗2
then 𝜔 𝑗1 (𝑘) and 𝜔 𝑗2 (𝑘) are two different eigenvalue Puiseux series of 𝐿(⋅, 𝑘) expanded about
𝑘0 with the same limit point 𝜔 0 . This implies by Lemma 63 that
0 = ⟨𝐿𝜔 (𝜔 0 , 𝑘0 )𝜑𝑗1 ,𝑖1 (𝑘0 ), J 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩ℂ = −⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩ℂ
= −𝑒𝑖𝑘0 𝑑 𝑒−𝑖𝑘0 𝑑 ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩ℂ
= −𝑒−𝑖𝑘0 𝑑 ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝜑𝑗1 ,𝑖1 (𝑘0 ), Ψ(𝑑, 𝜔 0 )𝜑𝑗2 ,𝑖2 (𝑘0 )⟩ℂ
= −𝑒−𝑖𝑘0 𝑑 ⟨Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 )𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩ℂ
= −𝑑𝑒−𝑖𝑘0 𝑑 𝑞(𝑘0 ,𝜔0 ) (𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 ))
= −𝑑𝑒−𝑖𝑘0 𝑑 sgn(𝑞(𝑘0 ,𝑑) )⟨𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩(𝑘0 ,𝜔0 ) .
162
Thus we have the orthogonality relations
⟨𝜑𝑗1 ,𝑖1 (𝑘0 ), 𝜑𝑗2 ,𝑖2 (𝑘0 )⟩(𝑘0 ,𝜔0 ) = 0,
Now if we let 𝜓 𝑗 :=
∑𝑚(𝑗)
𝑖=1
𝑐𝑗,𝑖 𝜑𝑗,𝑖 (𝑘0 ) then it follows from these orthogonality relations that
⟨𝜓 𝑗1 , 𝜓 𝑗2 ⟩(𝑘0 ,𝜔0 ) = 0,
And hence since 0 =
∑𝑔ˆ
𝑗=1
𝑗1 ∕= 𝑗2 .
∑𝑚(𝑗)
0=
𝑖=1
𝑐𝑗,𝑖 𝜑𝑗,𝑖 (𝑘0 ) =
⟨ 𝑔ˆ
∑
𝑗=1
∑𝑔ˆ
𝑗=1
𝑗1 ∕= 𝑗2 .
𝜓 𝑗 it follows that
⟩
= ⟨𝜓 𝑗1 , 𝜓 𝑗1 ⟩(𝑘0 ,𝜔0 ) .
𝜓 𝑗 , 𝜓 𝑗1
(𝑘0 ,𝜔 0 )
But by Proposition 45 we know that ⟨ , ⟩(𝑘0 ,𝜔0 ) is an inner product on ker 𝐿(𝜔 0 , 𝑘0 ) and
so we conclude 𝜓 𝑗 = 0 for every 𝑗. But 𝑀𝑗 (𝑘0 ) is an invertible matrix and hence 0 =
∑𝑚(𝑗)
∑
𝑀𝑗 (𝑘0 )−1 𝜓 𝑗 = 𝑚(𝑗)
𝑖=1 ∣𝑐𝑗,𝑖 ∣ = 0 for every 𝑗. Thus we conclude
𝑖=1 𝑐𝑗,𝑖 𝑒𝑖 which implies that
∑𝑔ˆ ∑𝑚(𝑗)
𝑖=1 ∣𝑐𝑗,𝑖 ∣ = 0 a contradiction. Therefore the limit points 𝜑𝑗,𝑖 (𝑘0 ), 𝑖 = 1, . . . , 𝑚(𝑗),
𝑗=1
𝑗 = 1, . . . , 𝑔ˆ are linearly independent. This completes the proof.
Without loss of generality we may take 0 < 𝑟 ≤ 𝛿 such that the consequences of Lemma 68 is
true for the functions 𝜑𝑗,𝑖 (𝑘)), 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ. Define the functions 𝜔 𝑗,𝑖 (𝑘) :=
∑ˆ ∑𝑚(𝑗)
𝜔 𝑗 (𝑘), 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ on 𝐵(𝑘0 , 𝑟). It follows that there is 𝑔𝑗=1
𝑖=1 = 𝑔 of
them, they are nonconstant real analytic functions with the union of there graphs being ℬ ∩
𝐵(𝑘0 , 𝑟)×𝐵(𝜔 0 , 𝜀). The order of the zero of 𝜔 𝑗 (𝑘)−𝜔 0 at 𝑘 = 𝑘0 is 𝑞𝑗,𝑖 := 𝑞𝑗 , 𝑖 = 1, . . . , 𝑚(𝑗),
∑ˆ ∑𝑚(𝑗)
∑𝑔ˆ
𝑗 = 1, . . . , 𝑔ˆ such that 𝑔𝑗=1
𝑖=1 𝑞𝑖,𝑗 =
𝑗=1 𝑚(𝑗)𝑞𝑖,𝑗 = 𝑚. Moreover, the functions 𝜑𝑗,𝑖 (𝑘),
𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ are analytic functions belonging to 𝒪(𝐵(𝑘0 , 𝑟), ℂ𝑛 ) such that
for each 𝑘 ∈ 𝐵(𝑘0 , 𝑟) the vectors 𝜑𝑗,𝑖 (𝑘), 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ are linear independent,
∪ˆ
𝑚(𝑖)
𝑚(𝑖)
{𝜑𝑗,𝑖 (𝑘)}𝑖=1 are a basis for ker 𝐿(𝜔 𝑗,𝑖 (𝑘), 𝑘), for 𝑗 = 1, . . . , 𝑔ˆ, and 𝑔𝑗=1
{𝜑𝑗,𝑖 (𝑘0 )}𝑖=1 is a
basis for ker 𝐿(𝜔 0 , 𝑘0 ). Thus the proof this theorem will be complete if we can show that 𝑞𝑗,𝑖 ,
163
𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ are the dimensions of the Jordan blocks in the Jordan normal
form of Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 .
It follows from what we just stated that (𝜔 𝑖,𝑗 (𝑘), 𝜑𝑗,𝑖 (𝑘)), 𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ are
eigenpair Puiseux series of 𝐿(⋅, 𝑘). And the order of the zero of 𝜔 𝑗,𝑖 (𝑘) − 𝜔 0 at 𝑘 = 𝑘0 is 𝑞𝑗,𝑖 ,
𝑖 = 1, . . . , 𝑚(𝑗), 𝑗 = 1, . . . , 𝑔ˆ. We denote any one of them by (𝜔(𝑘), 𝜑(𝑘)). Then both 𝜔(𝑘)
and 𝜑(𝑘) are analytic at 𝑘 = 𝑘0 with 𝜔(𝑘0 ) = 𝜔 0 . Let 𝑞 be the order of the zero of 𝜔 𝑗 (𝑘) − 𝜔 0
at 𝑘 = 𝑘0 . It follows from Proposition 32 that 𝜑(𝑘) is a generating function of order 𝑞 for
𝐿(𝜔 0 , ⋅) at the eigenvalue 𝑘0 . It follows by Proposition 11 that
𝜑𝑙 :=
1 (𝑙)
𝜑 (𝜆0 ),
𝑗!
𝑙 = 0, . . . , 𝑞 − 1
is a Jordan chain of 𝐿(𝜔 0 , ⋅) of length 𝑞 corresponding to the eigenvalue 𝑘0 . By definition of
a Jordan chain for a holomorphic function we have
𝜑0 ∕= 0 and
𝑙
∑
1 (0,ℎ)
𝐿
(𝜔 0 , 𝑘0 )𝜑𝑙−ℎ = 0,
ℎ!
ℎ=0
𝑙 = 0, . . . , 𝑞 − 1.
But 𝐿(0,0) (𝜔 0 , 𝑘0 ) = (Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘0 𝑑 𝐼𝑛 ) and 𝐿(0,ℎ) (𝜔 0 , 𝑘0 ) = −(𝑖𝑑)ℎ 𝑒𝑖𝑘0 𝑑 𝐼𝑛 for ℎ ≥ 0. Hence
for 𝑙 = 0, . . . , 𝑞 − 1 we have
0 = (Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘0 𝑑 𝐼𝑛 )𝜑0 ,
𝜑0 ∕= 0,
𝑙
𝑙
∑
∑
1 (0,ℎ)
𝑖𝑘0 𝑑
0=
𝐿
(𝜔 0 , 𝑘0 )𝜑𝑙−ℎ = (Ψ(𝑑, 𝜔 0 ) − 𝑒 𝐼𝑛 )𝜑𝑙 +
−(𝑖𝑑)ℎ 𝑒𝑖𝑘0 𝑑 𝜑𝑙−ℎ ,
ℎ!
ℎ=0
ℎ=1
Thus it follow that
(Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘0 𝑑 𝐼𝑛 )𝜑0 = 0,
𝜑0 ∕= 0,
(Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘0 𝑑 𝐼𝑛 )𝑙 𝜑𝑙 = (𝑖𝑑𝑒𝑖𝑘0 𝑑 )𝑙 𝜑0 , 1 ≤ 𝑙 ≤ 𝑞 − 1
164
𝑙≥1
This implies that 𝛾 𝑞−1 :=
𝜑𝑞−1
(𝑖𝑑𝑒𝑖𝑘0 𝑑 )𝑞−1
is a generalized eigenvector of Ψ(𝑑, 𝜔) of order 𝑞 corre-
sponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 . Thus if we define
𝛾 𝑙 := (Ψ(𝑑, 𝜔 0 ) − 𝑒𝑖𝑘0 𝑑 𝐼𝑛 )𝑞−1−𝑙 𝛾 𝑞−1 ,
𝑙 = 0, . . . , 𝑞 − 1
𝑞−1
is a Jordan chain of Ψ(𝑑, 𝜔 0 ) of length 𝑞 corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑
then {𝛾 𝑙 }𝑙=0
with 𝛾 0 = 𝜑0 the eigenvector.
𝑞
−1
𝑗,𝑖
Thus to each pair (𝜔 𝑖,𝑗 (𝑘), 𝜑𝑗,𝑖 (𝑘)) we get a Jordan chain, {𝛾 𝑗,𝑖,𝑙 }𝑙=0
, of Ψ(𝑑, 𝜔 0 ) of length
𝑞𝑗,𝑖 − 1 corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 with 𝛾 𝑗,𝑖,0 = 𝜑𝑗,𝑖 (𝑘0 ) the eigenvector. Thus since
∪ˆ
𝑚(𝑖)
{𝜑𝑗,𝑖 (𝑘0 )}𝑖=1
the elements in a Jordan chain are always linearly independent and, since 𝑔𝑗=1
is a basis for ker 𝐿(𝜔 0 , 𝑘0 ), the eigenvectors of these chains 𝛾 𝑗,𝑖,0 = 𝜑𝑗,𝑖 (𝑘0 ), 𝑖 = 1, . . . , 𝑚(𝑗),
𝑗 = 1, . . . , 𝑔ˆ are linearly independent and this implies the collection of Jordan chains
∪𝑔ˆ
𝑚(𝑖)
𝑗=1 {𝜑𝑗,𝑖 (𝑘0 )}𝑖=1 are linearly independent. But the dimension of the generalized eigenspace
for Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 is equal to its algebraic multiplicity which is
∪ˆ
𝑚(𝑖)
{𝜑𝑗,𝑖 (𝑘0 )}𝑖=1 are linearly independent and there
𝑚. And the collection of Jordan chains 𝑔𝑗=1
∑ˆ ∑𝑚(𝑗)
are a total of 𝑔𝑗=1
𝑖=1 𝑞𝑖,𝑗 = 𝑚 of them and they belong to the generalized eigenspace
for Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 . Therefore the collection of Jordan chains
∪𝑔ˆ
𝑚(𝑖)
𝑗=1 {𝜑𝑗,𝑖 (𝑘0 )}𝑖=1 are a basis for the generalized eigenspace for Ψ(𝑑, 𝜔 0 ) corresponding to
the eigenvalue 𝑒𝑖𝑘0 𝑑 , and in particular, they are a Jordan basis for that space for matrix
Ψ(𝑑, 𝜔 0 ). This implies in the Jordan normal form of Ψ(𝑑, 𝜔 0 ) corresponding to the eigen∑ˆ ∑𝑚(𝑗)
value 𝑒𝑖𝑘0 𝑑 it has 𝑔𝑗=1
𝑖=1 = 𝑔 Jordan blocks whose orders are exactly 𝑞𝑖,𝑗 , 𝑖 = 1, . . . , 𝑚(𝑗),
𝑗 = 1, . . . , 𝑔ˆ, but by hypothesis these orders where 𝑚1 , . . . , 𝑚𝑔 implying by the uniqueness
of the orders of the Jordan blocks these lists have the same numbers in them. This therefore
completes the proof.
The proof of Theorem 48.(i) now follows from Proposition 64 and Proposition 65. The
proof of Theorem 48.(ii) follows from Theorem 66. This completes the proof of Theorem 48.
165
Proof. [Corollary 49] Suppose the conditions
𝐷(𝑘0 , 𝜔 0 ) = 0,
∂𝐷
(𝑘0 , 𝜔 0 ) ∕= 0, (𝑘0 , 𝜔 0 ) ∈ ℝ × Ωℝ
∂𝜔
are satisfied where 𝐷(𝑘, 𝜔) = det(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔)), defined in Corollary 38. By Corollary
38 the first condition implies (𝑘0 , 𝜔 0 ) ∈ ℬℝ . Choose 𝑟 > 0 such that 𝐵(𝜔 0 , 𝑟) ⊆ Ω. Let
𝜆0 := 𝑒𝑖𝑘0 𝑑 and 𝑀 (𝜀) := Ψ(𝑑, 𝜀 + 𝜔 0 ) for 𝜀 ∈ 𝐵(0, 𝑟). By Proposition 33.(iii) we have
Ψ(𝑑, ⋅) ∈ 𝒪(𝐵(𝜔 0 , 𝑟), 𝑀𝑛 (ℂ)) and hence 𝑀 (⋅) ∈ 𝒪(𝐵(0, 𝑟), 𝑀𝑛 (ℂ)) which implies 𝑀 (𝜀) is
an analytic matrix function of the perturbation parameter 𝜀. Furthermore, 𝜆0 is an eigenvalue
of the unperturbed matrix 𝑀 (0) = Ψ(𝑑, 𝜔 0 ) since 0 = 𝐷(𝜔 0 , 𝑘0 ) = det(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) =
det(𝜆0 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )). Moreover, we have
∂
∂
det(𝜆𝐼𝑛 − 𝑀 (𝜀))∣(0,𝜆0 ) =
det(𝜆𝐼𝑛 − Ψ(𝑑, 𝜀 + 𝜔 0 ))∣(0,𝜆0 )
∂𝜀
∂𝜀
∂
∂
=
det(𝜆𝐼𝑛 − Ψ(𝑑, 𝜔))∣(𝜔0 ,𝜆0 ) =
det(𝑒𝑖𝑘𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔))∣(𝜔0 ,𝑘0 )
∂𝜔
∂𝜔
∂𝐷
=
(𝑘0 , 𝜔 0 ) ∕= 0
∂𝜔
Thus the generic condition, defined in [61, see (1.1)], is satisfied for the matrix function
𝑀 (𝜀) for the unperturbed matrix 𝑀 (0) with the eigenvalue 𝜆0 . By [61, Theorem 2.1.(iv)]
the Jordan normal form of 𝑀 (0) = Ψ(𝑑, 𝜔 0 )) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑
consists of a single 𝑚 × 𝑚 Jordan block and there exists an eigenvector 𝛾 0 of Ψ(𝑑, 𝜔 0 ) and
an eigenvector 𝛽 0 of Ψ(𝑑, 𝜔 0 )∗ corresponding to the eigenvalue 𝜆0 = 𝑒−𝑖𝑘0 𝑑 such that
⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛾 0 , 𝛽 0 ⟩ℂ ∕= 0.
But since the Jordan normal form of Ψ(𝑑, 𝜔 0 )) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑
consists of a single 𝑚 × 𝑚 Jordan block, this implies that dim ker(𝜆0 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) = 1.
166
But because Ψ(𝑑, 𝜔 0 )∗ is a adjoint of Ψ(𝑑, 𝜔 0 ) this implies the Jordan normal form corresponding to the eigenvalue 𝜆0 = 𝑒−𝑖𝑘0 𝑑 also consists of a single 𝑚 × 𝑚 Jordan block so that
dim ker(𝜆0 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) = 1 as well. And therefore, since any eigenvector 𝛾 of Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 must be a scalar multiple of 𝛾 0 and similarly any eigenvector
𝛽 of Ψ(𝑑, 𝜔 0 )∗ corresponding to the eigenvalue 𝜆0 must be a scalar multiple of 𝛽 0 , we must
have
⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛾, 𝛽⟩ℂ ∕= 0
for any eigenvector 𝛾 of Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 and for any
eigenvector 𝛽 of Ψ(𝑑, 𝜔 0 )∗ corresponding to the eigenvalue 𝜆0 = 𝑒−𝑖𝑘0 𝑑 . That is the key
statement.
Indeed, Proposition 39 implies
Ψ(𝑑, 𝜔 0 )∗ J Ψ(𝑑, 𝜔 0 ) = J
and by Proposition 33.(v) we know that Ψ(𝑑, 𝜔 0 ) is an invertible matrix and hence for any
eigenvector 𝛾 of Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 we have
Ψ(𝑑, 𝜔 0 )∗ J 𝛾 = J Ψ(𝑑, 𝜔 0 )−1 𝛾 = 𝑒−𝑖𝑘0 𝑑 J 𝛾.
Then since J is invertible this implies J 𝛾 is an eigenvector of Ψ(𝑑, 𝜔 0 )∗ corresponding to
the eigenvalue 𝑒−𝑖𝑘0 𝑑 = 𝜆0 . But this means
0 ∕= ⟨Ψ𝜔 (𝑑, 𝜔 0 )𝛾, J 𝛾⟩ℂ = −𝑒𝑖𝑘0 𝑑 𝑒−𝑖𝑘0 𝑑 ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛾, 𝛾⟩ℂ
= −𝑒−𝑖𝑘0 𝑑 ⟨J Ψ𝜔 (𝑑, 𝜔 0 )𝛾, Ψ(𝑑, 𝜔 0 )𝛾⟩ℂ
= −𝑒−𝑖𝑘0 𝑑 ⟨Ψ(𝑑, 𝜔 0 )∗ J Ψ𝜔 (𝑑, 𝜔 0 )𝛾, 𝛾⟩ℂ
167
−𝑖𝑘0 𝑑 1
= −𝑑𝑒
𝑑
∫
𝑑
⟨𝐴𝜔 (𝑥, 𝜔)𝜓(𝑥), 𝜓(𝑥)⟩ℂ 𝑑𝑥,
𝜓 := Ψ(⋅, 𝜔 0 )𝛾,
0
where the last equality follows from Corollary 43.
Thus we have shown for any 𝛾 ∈
ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) with 𝛾 ∕= 0 we have
1
𝑑
∫
𝑑
⟨𝐴𝜔 (𝑥, 𝜔)𝜓(𝑥), 𝜓(𝑥)⟩ℂ 𝑑𝑥 ∕= 0,
𝜓 := Ψ(⋅, 𝜔 0 )𝛾.
0
Therefore if 𝜓 is any nontrivial Bloch solution of the canonical ODEs in 4.13 with (𝑘0 , 𝜔 0 )
as its wavenumber-frequency pair, the by Theorem 35 there exists an eigenvector 𝛾 of the
monodromy matrix Ψ(𝑑, 𝜔 0 ) corresponding to the eigenvalue 𝑒𝑖𝑘0 𝑑 such that 𝜓 = Ψ(⋅, 𝜔 0 )𝛾,
i.e., 𝛾 ∈ ker(𝑒𝑖𝑘0 𝑑 𝐼𝑛 − Ψ(𝑑, 𝜔 0 )) with 𝛾 ∕= 0, and thus from what we just proved we must have
1
𝑑
∫
𝑑
⟨𝐴𝜔 (𝑥, 𝜔)𝜓(𝑥), 𝜓(𝑥)⟩ℂ 𝑑𝑥 ∕= 0.
0
This proves (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the canonical ODEs in 4.13. Moreover, from what we have shown the Jordan normal form of Ψ(𝑑, 𝜔 0 )) corresponding to the
eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 consists of a single 𝑚 × 𝑚 Jordan block.
We will now prove the converse. Suppose that (𝑘0 , 𝜔 0 ) ∈ ℬℝ is a point of definite type for the
canonical ODEs in (4.13) such that the Jordan normal form of Ψ(𝑑, 𝜔 0 ) corresponding to the
eigenvalue 𝜆0 = 𝑒𝑖𝑘0 𝑑 consists of a single 𝑚 × 𝑚 Jordan block, for some 𝑚 ∈ ℕ. By Theorem
48.(i), since 𝑔 = 1 in this case, we know that the order of the zero of the function 𝐷(𝑘0 , 𝜔)
at 𝜔 = 𝜔 0 is 1. But this is equivalent to 𝐷(𝑘0 , 𝜔 0 ) = 0 and
converse is true. This completes the proof.
168
∂𝐷
(𝑘0 , 𝜔 0 )
∂𝜔
∕= 0. Therefore the
4.5
Auxiliary Material
In this section we give some notation, convention, and background material frequently used
throughout this chapter.
Notation and Convention
(i) 𝑀𝑚,𝑛 (𝐸) := the set of all 𝑚 × 𝑛 matrices with entries in a space 𝐸.
(ii) 𝑀𝑚 (𝐸) := 𝑀𝑚,𝑚 (𝐸).
(iii) 𝐸 𝑚 := 𝑀𝑚,1 (𝐸).
(iv) As convention we identify 𝐸 with 𝐸 1 and the set of 𝑚-tuples whose entries are in 𝐸
with 𝑀1,𝑚 (𝐸).
(v) 𝐴𝑇 :=the transpose of the matrix 𝐴 ∈ 𝑀𝑚,𝑛 (𝐸).
(vi) (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) ) := the Banach space with norm
(
∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸) :=
𝑚 ∑
𝑛
∑
) 21
∣∣𝑎𝑖𝑗 ∣∣2
, for any 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐸)
(4.81)
𝑖=1 𝑖=1
where (𝐸, ∣∣ ⋅ ∣∣) is a Banach space.
(vii) L (𝐸, 𝐹 ) := the Banach space of all continuous linear operators from a Banach space
(𝐸, ∣∣ ⋅ ∣∣𝐸 ) to a Banach space (𝐹, ∣∣ ⋅ ∣∣𝐹 ) with the operator norm
∣∣𝑇 𝑒∣∣𝐹
𝑒∈𝐸, ∣∣𝑒∣∣𝐸
∣∣𝑇 ∣∣L (𝐸,𝐹 ) := sup
𝑒∕=0
(viii) 𝐵(𝑎, 𝑟) := {𝑒 ∈ 𝐸 : ∣∣𝑒 − 𝑎∣∣ < 𝑟}, where (𝐸, ∣∣ ⋅ ∣∣) is a Banach space, 𝑎 ∈ 𝐸, and
𝑟 > 0.
169
(ix) 𝒪(Ω, 𝐸) := {𝑓 : Ω → 𝐸 ∣ 𝑓 is holomorphic on Ω} where (𝐸, ∣∣ ⋅ ∣∣) is a Banach space, Ω
an open connected set in ℂ, and 𝑓 : Ω → 𝐸 is said to be holomorphic on Ω provided
𝑓 is differentiable at each point in Ω, i.e., the limit
𝑓𝜔 (𝜔 0 ) := lim (𝜔 − 𝜔 0 )−1 (𝑓 (𝜔) − 𝑓 (𝜔 0 ))
𝜔→𝜔 0
exists for every 𝜔 0 ∈ Ω. The function 𝑓𝜔 : Ω → 𝐸 is called the derivative of 𝑓 .
(x) (ℂ, ∣ ⋅ ∣):= complex numbers with the standard norm ∣𝑎 + 𝑖𝑏∣ :=
√
𝑎2 + 𝑏2 and complex
conjugation 𝑎 + 𝑖𝑏 := 𝑎 − 𝑖𝑏, for 𝑎, 𝑏 ∈ ℝ.
(xi) ∣∣ ⋅ ∣∣ℂ := ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (ℂ)
(xii) 𝐼𝑚 := the identity matrix in 𝑀𝑚 (ℂ).
(xiii) 𝐴 := the matrix formed from the matrix 𝐴 ∈ 𝑀𝑚,𝑛 (ℂ) by replacing each entry by its
complex conjugate.
𝑇
(xiv) 𝐴∗ := 𝐴 , the adjoint of the matrix 𝐴 ∈ 𝑀𝑚,𝑛 (ℂ).
(xv) ⟨𝑢, 𝑣⟩ℂ := 𝑢∗ 𝑣, the standard inner product on ℂ𝑚 .
(xvi) (𝐿𝑝 (𝑎, 𝑏), ∣∣ ⋅ ∣∣𝑝 ) := the Banach space of measurable functions 𝑓 on (𝑎, 𝑏) (modulo
functions which vanish almost everywhere (a.e.)) such that
(∫
∣∣𝑓 ∣∣𝑝 :=
𝑏
) 𝑝1
∣𝑓 (𝑥)∣𝑝 𝑑𝑥
< ∞ (1 ≤ 𝑝 < ∞),
𝑎
∣∣𝑓 ∣∣∞ := ess sup𝑥∈(𝑎,𝑏) ∣𝑓 (𝑥)∣ < ∞ (𝑝 = ∞).
(xvii) ∣∣ ⋅ ∣∣𝑝 := ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐿𝑝 (𝑎,𝑏))
(xviii) 𝐿𝑝loc (ℝ) := the space of all measurable functions 𝑓 on ℝ (modulo functions which vanish
a.e.) such that 𝑓 ∣(𝑎,𝑏) ∈ 𝐿𝑝 (𝑎, 𝑏) for all 𝑎 < 𝑏.
170
(xix) 𝐿𝑝 (𝕋) := {𝑓 ∈ 𝐿𝑝loc (ℝ) : 𝑓 (𝑥 + 𝑑) = 𝑓 (𝑥) for a.e. 𝑥 ∈ ℝ}, the Banach space with norm
∣∣𝑓 ∣∣𝐿𝑝 (𝕋)
( ∫ 𝑑
) 𝑝1
1
:=
∣𝑓 (𝑥)∣𝑝 𝑑𝑥
< ∞ (1 ≤ 𝑝 < ∞),
𝑑 0
∣∣𝑓 ∣∣𝐿∞ (𝕋) := ess sup𝑥∈(0,𝑑) ∣𝑓 (𝑥)∣ < ∞ (𝑝 = ∞).
(xx) ∣∣ ⋅ ∣∣𝐿𝑝 (𝕋) := ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))
(xxi) As convention we identify an element 𝐴 of 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) or 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) with any
one of its representative functions 𝐴(⋅) (and visa versa). And hence any identities,
inequalities, etc. throughout this chapter are to be understood as being true almost
everywhere unless otherwise stated.
(xxii) As convention we identify an element 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))) with the function
𝐴 : (𝑎, 𝑏) × Ω → 𝑀𝑚,𝑛 (ℂ) defined by 𝐴(⋅, 𝜔) := 𝐴(𝜔). Similarly, a function 𝐴 :
(𝑎, 𝑏) × Ω → 𝑀𝑚,𝑛 (ℂ) is said to be in 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))) as a function of frequency
and we write 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))) provided 𝐴(𝜔) := 𝐴(⋅, 𝜔) ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) for
each 𝜔 ∈ Ω and 𝐴(⋅) ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))).
(xxiii) As convention we identify an element 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))) with the function 𝐴 :
ℝ × Ω → 𝑀𝑚,𝑛 (ℂ) defined by 𝐴(⋅, 𝜔) := 𝐴(𝜔). Similarly, a function 𝐴 : ℝ × Ω →
𝑀𝑚,𝑛 (ℂ) is said to be in 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))) as a function of frequency and we write
𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))) provided 𝐴(𝜔) := 𝐴(⋅, 𝜔) ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))) for each 𝜔 ∈ Ω and
𝐴(⋅) ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)))).
(xxiv)
∫
𝑈
𝐴(𝑥)𝑑𝑥 :=
[∫
𝑈
𝑎𝑖𝑗 (𝑥)𝑑𝑥
]𝑚,𝑛
𝑖=1,𝑗=1
, for any measurable set 𝑈 contained in [𝑎, 𝑏] and any
𝑝
𝑝
𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐿 (𝑎, 𝑏)) ∪ 𝑀𝑚,𝑛 (𝐿loc (ℝ)).
(xxv) As convention we set
∫𝑥
𝑐
𝐴(𝑡)𝑑𝑡 :=
∫
[𝑐,𝑥]
𝐴(𝑡)𝑑𝑡 if 𝑥 ≤ 𝑐 and
∫𝑥
𝑐
𝐴(𝑡)𝑑𝑡 := −
𝑐 < 𝑎, whenever 𝑐, 𝑥 ∈ [𝑎, 𝑏] and 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) ∪ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)).
171
∫𝑐
𝑥
𝐴(𝑡)𝑑𝑡 if
(xxvi) 𝑊 1,1 (𝑎, 𝑏) := {𝑓 ∈ 𝐿1 (𝑎, 𝑏) ∣ 𝑓 ′ ∈ 𝐿1 (𝑎, 𝑏)}, the standard Sobolev space with norm
∣∣𝑓 ∣∣1,1 := ∣∣𝑓 ∣∣1 + ∣∣𝑓 ′ ∣∣1
where 𝑓 ′ denotes the weak derivative of 𝑓 .
(xxvii) ∣∣ ⋅ ∣∣1,1 := ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝑊 1,1 (𝑎,𝑏))
1,1
(ℝ) := {𝑓 ∈ 𝐿1loc (ℝ) ∣ 𝑓 ′ ∈ 𝐿1loc (ℝ)}.
(xxviii) 𝑊loc
[ ]𝑚,𝑛
1,1
1,1
(𝑎, 𝑏)) or 𝑀𝑚,𝑛 (𝑊loc
(ℝ)).
(xxix) 𝐴′ := 𝑎′𝑖𝑗 𝑖=1,𝑗=1 , for any 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 in 𝑀𝑚,𝑛 (𝑊
1,1
(xxx) As convention we always identify an element 𝐴 of 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) or 𝑀𝑚,𝑛 (𝑊loc
(ℝ))
with its unique absolutely continuous or unique locally absolutely continuous representative, respectively, (see Lemma 70 below) so that, according to this identification,
∫𝑥
𝐴(𝑥) = 𝐴(𝑢) + 𝑢 𝐴′ (𝑡)𝑑𝑡 for all 𝑢, 𝑥 ∈ [𝑎, 𝑏], if 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)), and for all
1,1
𝑢, 𝑥 ∈ ℝ, if 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)).
(xxxi) As convention we identify an element 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) with the function
𝐴 : [𝑎, 𝑏] × Ω → 𝑀𝑚,𝑛 (ℂ) defined by 𝐴(𝑥, 𝜔) := 𝐴(𝜔)(𝑥) – the value of the function
𝐴 : Ω → 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) at 𝜔 ∈ Ω and the value of the unique absolutely continuous
representative of 𝐴(𝜔) at 𝑥 ∈ [𝑎, 𝑏]. Similarly, a function 𝐴 : 𝐼 × Ω → 𝑀𝑚,𝑛 (ℂ),
where 𝐼 := (𝑎, 𝑏) or [𝑎, 𝑏], is said to be in 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) and we write 𝐴 ∈
𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) provided 𝐴(𝜔) := 𝐴(⋅, 𝜔) ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) for each 𝜔 ∈ Ω
and 𝐴(⋅) ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))).
Background Material
The following is background material for this chapter.
In what follows we will need notation for the derivative of a function in the classical sense. Let
172
𝑓 : 𝐼 → ℂ be a complex-valued function defined on an interval 𝐼 ⊆ ℝ. If 𝑓 is differentiable
(in the classical sense) everywhere on 𝐼 we denote its derivative by 𝑑𝑓 /𝑑𝑥, otherwise we
define the function 𝑑𝑓 /𝑑𝑥 : 𝐼 → ℂ by
⎧

⎨ lim𝑥→𝑡
𝑑𝑓 :=

𝑑𝑥 𝑥=𝑡
⎩
𝑓 (𝑥)−𝑓 (𝑡)
,
𝑥−𝑡
if the limit exists
0,
else.
1,1
Lemma 69 If 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) then 𝐴∣(𝑎,𝑏) ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) and 𝐴∣′(𝑎,𝑏) = 𝐴′ ∣(𝑎,𝑏) for
any bounded interval (𝑎, 𝑏) ⊆ ℝ.
Proof.
1,1
We first prove the statement in the case 𝑚 = 𝑛 = 1. Suppose 𝑓 ∈ 𝑊loc
(ℝ). Then
by the definition of the weak derivative ()′ since 𝑓 ′ ∈ 𝐿1loc (ℝ) we have
∫
∫
′
𝜑(𝑥)𝑓 ′ (𝑥)𝑑𝑥, ∀𝜑 ∈ 𝐶0∞ (ℝ),
𝜑 (𝑥)𝑓 (𝑥)𝑑𝑥 = −
ℝ
ℝ
where 𝐶0∞ (ℝ) is the set of all infinitely differentiable function on ℝ with compact support
and 𝜑′ = 𝑑𝜙/𝑑𝑥 for 𝜑 ∈ 𝐶0∞ (ℝ). This implies that for any (𝑎, 𝑏) open interval and any
𝜙 ∈ 𝐶0∞ (𝑎, 𝑏), the set of infinitely differentiable functions on (𝑎, 𝑏) with compact support, we
can take the zero extension of 𝜙, i.e.,
⎧

⎨ 𝜙(𝑥) if 𝑥 ∈ (𝑎, 𝑏)
𝜙𝑒 (𝑥) :=

⎩ 0
else.
so that 𝜙𝑒 ∈ 𝐶0∞ (ℝ). And hence
∫
𝑏
′
∫
𝜑′𝑒 (𝑥)𝑓 (𝑥)𝑑𝑥
𝜑 (𝑥)𝑓 (𝑥)𝑑𝑥 =
𝑎
ℝ
∫
=−
𝜑𝑒 (𝑥)𝑓 ′ (𝑥)𝑑𝑥
ℝ
∫
=−
𝑏
𝜑(𝑥)𝑓 ′ (𝑥)𝑑𝑥, ∀𝜑 ∈ 𝐶0∞ (𝑎, 𝑏).
𝑎
173
This implies by the definition of the weak derivative that 𝑓 ∣(𝑎,𝑏) ∈ 𝑊 1,1 (𝑎, 𝑏) and 𝑓 ∣′(𝑎,𝑏) =
𝑓 ′ ∣(𝑎,𝑏) . This proves the statement for 𝑚 = 𝑛 = 1. For the case 𝑚, 𝑛 ≥ 1, it follows
𝑚,𝑛
now trivially from the notation 4.5.(xxix) and the fact [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∣(𝑎,𝑏) = [𝑎𝑖𝑗 ∣(𝑎,𝑏) ]𝑖=1,𝑗=1 for
1
1
[𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐿 (𝑎, 𝑏)) ∪ 𝑀𝑚,𝑛 (𝐿loc (ℝ)). This completes the proof.
Definition 26 For any compact interval [𝑎, 𝑏] ⊆ ℝ, denote by 𝐴𝐶[𝑎, 𝑏] the space of absolutely
continuous functions on [𝑎, 𝑏], i.e.,
𝐴𝐶[𝑎, 𝑏] := {𝑓 : [𝑎, 𝑏] → ℂ ∣ 𝑓 is differentiable a.e., 𝑑𝑓 /𝑑𝑥 ∈ 𝐿1 (𝑎, 𝑏), and
∫ 𝑢 𝑑𝑓 𝑓 (𝑢) = 𝑓 (𝑣) +
𝑑𝑡, ∀𝑢, 𝑣 ∈ [𝑎, 𝑏] }.
𝑣 𝑑𝑥 𝑥=𝑡
We denote by 𝐴𝐶loc (ℝ) the space of locally absolutely continuous functions on ℝ, i.e.,
𝐴𝐶loc (ℝ) := {𝑓 : ℝ → ℂ ∣ 𝑓 is differentiable a.e., 𝑑𝑓 /𝑑𝑥 ∈ 𝐿1loc (ℝ), and
∫ 𝑢 𝑑𝑓 𝑓 (𝑢) = 𝑓 (𝑣) +
𝑑𝑡, ∀𝑢, 𝑣 ∈ ℝ }.
𝑣 𝑑𝑥 𝑥=𝑡
For 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐴𝐶[𝑎, 𝑏]) we define
𝑑𝐴
𝑑𝑥
:= [
𝑑𝑎𝑖𝑗 𝑚,𝑛
]
𝑑𝑥 𝑖=1,𝑗=1
∈ 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)). Sim-
ilarly, for 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐴𝐶loc (ℝ)) we define the matrix
𝑑𝐴
𝑑𝑥
:= [
𝑑𝑎𝑖𝑗 𝑚,𝑛
]
𝑑𝑥 𝑖=1,𝑗=1
∈
𝑀𝑚,𝑛 (𝐿1loc (ℝ)).
Lemma 70 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) if and only if 𝐴(⋅) ∈ 𝑀𝑚,𝑛 (𝐴𝐶[𝑎, 𝑏]) for some representative function 𝐴(⋅) of 𝐴. Furthermore, this representative function is unique. Moreover, for
the absolutely continuous representative 𝐴(⋅) of 𝐴 we have 𝐴′ =
∫
𝐴(𝑥) = 𝐴(𝑢) +
𝑑𝐴(⋅)
𝑑𝑥
and
𝑥
𝐴′ (𝑡)𝑑𝑡, ∀𝑢, 𝑥 ∈ [𝑎, 𝑏].
𝑢
1,1
Similarly, 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) if and only if 𝐴(⋅) ∈ 𝑀𝑚,𝑛 (𝐴𝐶loc (ℝ))) for some representative
function 𝐴(⋅) of 𝐴. Furthermore, this representative function is unique. Moreover, for the
174
locally absolutely continuous representative 𝐴(⋅) of 𝐴 we have 𝐴′ =
𝑑𝐴(⋅)
𝑑𝑥
and
𝑥
∫
𝐴′ (𝑡)𝑑𝑡, ∀𝑢, 𝑥 ∈ ℝ.
𝐴(𝑥) = 𝐴(𝑢) +
𝑢
Proof.
We first prove the statement for 𝑚 = 𝑛 = 1. By [45, p. 56, §II.2.1, Proposition
2.1.5], if 𝑓 ∈ 𝑊 1,1 (𝑎, 𝑏) then 𝑓 has a representative function 𝑓 (⋅) : [𝑎, 𝑏] → ℂ such that
∫
𝑓 (𝑥) = 𝑓 (𝑢) +
𝑥
𝑓 ′ (𝑡)𝑑𝑡, ∀𝑢, 𝑥 ∈ [𝑎, 𝑏].
𝑢
This implies 𝑓 (⋅) ∈ 𝐴𝐶[𝑎, 𝑏] with 𝑑𝑓 (⋅)/𝑑𝑥 = 𝑓 ′ . To prove uniqueness we note that if
𝑔 ∈ 𝐴𝐶[𝑎, 𝑏] is another representative function of 𝑓 then since 𝑓 (⋅) is as well we have
𝑔(𝑥) = 𝑓 (𝑥) for a.e. 𝑥 ∈ (𝑎, 𝑏). But then by the fact absolutely continuous functions on [𝑎, 𝑏]
are continuous functions on [𝑎, 𝑏] this implies 𝑔(𝑥) = 𝑓 (𝑥) for every 𝑥 ∈ [𝑎, 𝑏]. This proves
uniqueness. On the other hand, if 𝑓 (⋅) ∈ 𝐴𝐶[𝑎, 𝑏] then by [45, p. 55, §II.2.1, Proposition
2.1.3] we have 𝑓 ∈ 𝑊 1,1 (𝑎, 𝑏).
1,1
Now suppose 𝑓 ∈ 𝑊loc
(ℝ). Then by Lemma 69 we have 𝑓 ∣(𝑎,𝑏) ∈ 𝑊 1,1 (𝑎, 𝑏) for any bounded
open interval (𝑎, 𝑏) and 𝑓 ∣′(𝑎,𝑏) = 𝑓 ′ ∣(𝑎,𝑏) . Hence from what we just proved 𝑓 ∣(𝑎,𝑏) has an
absolutely continuous representative function 𝑓 ∣(𝑎,𝑏) (⋅) : [𝑎, 𝑏] → ℂ such that
∫
𝑥
𝑓 ∣(𝑎,𝑏) (𝑥) = 𝑓 ∣(𝑎,𝑏) (𝑢) +
∫𝑢 𝑥
= 𝑓 ∣(𝑎,𝑏) (𝑢) +
𝑓 ∣′(𝑎,𝑏) (𝑡)𝑑𝑡
∫
= 𝑓 ∣(𝑎,𝑏) (𝑢) +
𝑥
𝑓 ′ ∣(𝑎,𝑏) (𝑡)𝑑𝑡
𝑢
𝑓 ′ (𝑡)𝑑𝑡, ∀𝑢, 𝑥 ∈ [𝑎, 𝑏].
𝑢
Define a function 𝑔(⋅) : ℝ → ℂ by
𝑔(𝑥) := 𝑓 ∣(𝑎,𝑏) (𝑥), for any bounded interval (𝑎, 𝑏) with 𝑥 ∈ (𝑎, 𝑏).
We first show this function is well-defined. Let (𝑎, 𝑏) and (𝑢, 𝑣) be any two bounded open
175
interval such that 𝐼 := (𝑎, 𝑏) ∩ (𝑢, 𝑣) ∕= ∅ and let 𝑓 ∣(𝑎,𝑏) (⋅) ∈ 𝐴𝐶[𝑎, 𝑏] be an absolutely continuous representative function of 𝑓 ∣(𝑎,𝑏) ∈ 𝑊 1,1 (𝑎, 𝑏) and 𝑓 ∣(𝑢,𝑣) (⋅) ∈ 𝐴𝐶[𝑢, 𝑣] be an absolutely
continuous representative function of 𝑓 ∣(𝑢,𝑣) ∈ 𝑊 1,1 (𝑢, 𝑣) . Now fix any representative function 𝑓˜ : ℝ → ℂ of 𝑓 . Then 𝑓 ∣(𝑎,𝑏) (𝑥) = 𝑓˜(𝑥) for a.e. 𝑥 ∈ [𝑎, 𝑏] and 𝑓 ∣(𝑢,𝑣) (𝑥) = 𝑓˜(𝑥) for
a.e. 𝑥 ∈ [𝑢, 𝑣] implying 𝑓 ∣(𝑎,𝑏) (𝑥) = 𝑓˜(𝑥) = 𝑓 ∣(𝑢,𝑣) (𝑥) for a.e. 𝑥 ∈ 𝐼. Hence since 𝑓 ∣(𝑎,𝑏) (⋅)
and 𝑓 ∣(𝑢,𝑣) (⋅) are continuous on 𝐼 this implies 𝑓 ∣(𝑎,𝑏) (𝑥) = 𝑓 ∣(𝑢,𝑣) (𝑥) for all 𝑥 ∈ 𝐼. This
proves 𝑔(⋅) : ℝ → ℂ is a well-defined function. From the definition of 𝑔 it also proves
1,1
𝑔∣(𝑎,𝑏) = 𝑓 ∣(𝑎,𝑏) ∈ 𝑊 1,1 (𝑎, 𝑏) for any bounded open interval (𝑎, 𝑏) and hence 𝑔 = 𝑓 ∈ 𝑊loc
(ℝ).
It also implies that for any 𝑢, 𝑥 ∈ ℝ, if we take any bounded open interval (𝑎, 𝑏) containing
𝑢, 𝑥, then we have
∫
𝑔(𝑥) = 𝑓 ∣(𝑎,𝑏) (𝑥) = 𝑓 ∣(𝑎,𝑏) (𝑢) +
𝑥
′
∫
𝑓 (𝑡)𝑑𝑡 = 𝑔(𝑢) +
𝑢
𝑥
𝑓 ′ (𝑡)𝑑𝑡.
𝑢
Thus implying 𝑔(⋅) ∈ 𝐴𝐶loc (ℝ) with 𝑑𝑔(⋅)/𝑑𝑥 = 𝑓 ′ . Hence we have proven 𝑓 has a locally
absolutely continuous representative 𝑓 (⋅) := 𝑔(⋅) ∈ 𝐴𝐶loc (ℝ) with 𝑑𝑓 (⋅)/𝑑𝑥 = 𝑓 ′ and such
that
∫
𝑓 (𝑥) = 𝑓 (𝑢) +
𝑥
𝑓 ′ (𝑡)𝑑𝑡, ∀𝑢, 𝑥 ∈ ℝ.
𝑢
To prove uniqueness we note that if 𝑓1 , 𝑓2 ∈ 𝐴𝐶loc (ℝ) are two representative functions of 𝑓
then 𝑓1 and 𝑓2 are equal a.e. on ℝ and so since functions in 𝐴𝐶loc (ℝ) are continuous this
implies 𝑓1 and 𝑓2 are equal everywhere on ℝ. This proves uniqueness.
Suppose now that 𝑓 ∈ 𝐴𝐶loc (ℝ). Then we have 𝑓 ∈ 𝐿1loc (ℝ) and 𝑑𝑓 /𝑑𝑥 ∈ 𝐿1loc (ℝ). We need
only show now that 𝑓 has 𝑑𝑓 /𝑑𝑥 as its weak derivative, i.e., 𝑓 ′ = 𝑑𝑓 /𝑑𝑥. To do this we need
to show
∫
∫
𝑑𝑓 𝜑 (𝑡)𝑓 (𝑡)𝑑𝑡 = − 𝜑(𝑡) 𝑑𝑡, ∀𝜑 ∈ 𝐶0∞ (ℝ),
𝑑𝑥 𝑥=𝑡
ℝ
ℝ
′
176
where 𝐶0∞ (ℝ) is the set of all infinitely differentiable function on ℝ with compact support.
Let 𝜑 ∈ 𝐶0∞ (ℝ) and let [𝑎, 𝑏] be any nonempty compact interval such that supp(𝜙) ⊆ [𝑎, 𝑏].
Then by integration by parts we have
∫
∫
′
𝑏
∫
′
𝜑 (𝑡)𝑓 (𝑡)𝑑𝑡 = 𝑓 (𝑏)𝜑(𝑏) − 𝑓 (𝑎)𝜑(𝑎) −
𝑎
∫
∫ 𝑏
𝑑𝑓 𝑑𝑓 𝜑(𝑡) 𝑑𝑡 = − 𝜑(𝑡) 𝑑𝑡.
=−
𝑑𝑥 𝑥=𝑡
𝑑𝑥 𝑥=𝑡
ℝ
𝑎
𝜑 (𝑡)𝑓 (𝑡)𝑑𝑡 =
ℝ
𝑏
𝜑(𝑡)
𝑎
𝑑𝑓 𝑑𝑡
𝑑𝑥 𝑥=𝑡
And therefore we have shown
∫
∫
𝑑𝑓 𝜑 (𝑡)𝑓 (𝑡)𝑑𝑡 = − 𝜑(𝑡) 𝑑𝑡, ∀𝜑 ∈ 𝐶0∞ (ℝ),
𝑑𝑥 𝑥=𝑡
ℝ
ℝ
′
1,1
which proves 𝑓 ′ = 𝑑𝑓 /𝑑𝑥 ∈ 𝐿1loc (ℝ) and hence 𝑓 ∈ 𝑊loc
(ℝ).
This completes the prove in the case 𝑚 = 𝑛 = 1. The case 𝑚, 𝑛 ≥ 1 follows now trivially
from the case 𝑚 = 𝑛 = 1 by the notation 4.5.(xxiv) and 4.5.(xxix). This completes the
proof.
The following lemma, along with Lemma 69 and Lemma 70, completes our characterization
1,1
of spaces 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) and 𝑀𝑚,𝑛 (𝑊loc
(ℝ)).
Lemma 71 Let 𝐴 : ℝ → 𝑀𝑚,𝑛 (ℂ) be a function such that there exists an increasing se∪∞
quence of intervals {(𝑎𝑗 , 𝑏𝑗 )}∞
𝑗=1 with ℝ =
𝑗=1 (𝑎𝑗 , 𝑏𝑗 ) such that for every 𝑗, 𝐴∣(𝑎𝑗 ,𝑏𝑗 ) ∈
1,1
𝑀𝑚,𝑛 (𝑊 1,1 (𝑎𝑗 , 𝑏𝑗 )). Then 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)).
We first prove the statement for 𝑚 = 𝑛 = 1. Let 𝑓 : ℝ → ℂ be a function
∪∞
such that there exists an increasing sequence of intervals {(𝑎𝑗 , 𝑏𝑗 )}∞
𝑗=1 with ℝ =
𝑗=1 (𝑎𝑗 , 𝑏𝑗 )
Proof.
such that for every 𝑗, 𝑓 ∣(𝑎𝑗 ,𝑏𝑗 ) ∈ 𝑊 1,1 (𝑎𝑗 , 𝑏𝑗 ). It follows from this that on every interval
(𝑎𝑗 , 𝑏𝑗 ) the function 𝑓 on this interval is equal a.e. to a function which is measurable and
integrable on (𝑎𝑗 , 𝑏𝑗 ) and this implies the function 𝑓 restricted to (𝑎𝑗 , 𝑏𝑗 ) is a measurable and
177
integrable function. Thus since ℝ =
∪∞
and the restriction of 𝑓 : ℝ → ℂ to (𝑎𝑗 , 𝑏𝑗 )
𝑗=1 (𝑎𝑗 , 𝑏𝑗 )
is measurable and integrable for every 𝑗, this implies 𝑓 is a measurable function which is
locally integrable and so 𝑓 ∈ 𝐿1loc (ℝ).
Now 𝑓 ∣(𝑎𝑗 ,𝑏𝑗 ) ∈ 𝑊 1,1 (𝑎𝑗 , 𝑏𝑗 ) for every 𝑗 and so by Lemma 70 there exists an absolutely
continuous function 𝑔𝑗 ∈ 𝐴𝐶[𝑎𝑗 , 𝑏𝑗 ] such that 𝑓 (𝑥) = 𝑔𝑗 (𝑥) for a.e. 𝑥 ∈ (𝑎𝑗 , 𝑏𝑗 ) for each 𝑗.
But this implies for any fix 𝑗0 we must have 𝑔𝑗 (𝑥) = 𝑓 (𝑥) = 𝑔𝑗0 (𝑥) for a.e. 𝑥 ∈ (𝑎𝑗0 , 𝑏𝑗0 ) for any
fixed 𝑗 ≥ 𝑗0 . But the functions 𝑔𝑗 , 𝑔𝑗0 are continuous on [𝑎𝑗0 , 𝑏𝑗0 ] and so this implies 𝑔𝑗 (𝑥) =
𝑔𝑗0 (𝑥) for all 𝑥 ∈ [𝑎𝑗0 , 𝑏𝑗0 ] and every 𝑗 ≥ 𝑗0 . But this implies 𝑔(𝑥) := lim𝑗→∞ 𝑔𝑗 (𝑥) exists for
every 𝑥 ∈ ℝ and so defines a function 𝑔 : ℝ → ℂ. Moreover, from what we proved we have
𝑔(𝑥) = 𝑔𝑗 (𝑥) for all 𝑥 ∈ [𝑎𝑗0 , 𝑏𝑗0 ] and any 𝑗. Thus for every 𝑗 we have 𝑔∣[𝑎𝑗 ,𝑏𝑗 ] = 𝑔𝑗 ∈ 𝐴𝐶[𝑎𝑗 , 𝑏𝑗 ]
implying 𝑔 is differentiable a.e. on [𝑎𝑗 , 𝑏𝑗 ] with 𝑑𝑔/𝑑𝑥∣[𝑎𝑗 ,𝑏𝑗 ] = 𝑑𝑔𝑗 /𝑑𝑥 = 𝑔𝑗′ ∈ 𝐿1 (𝑎𝑗 , 𝑏𝑗 ). But
∪
1
since ℝ = ∞
𝑗=1 (𝑎𝑗 , 𝑏𝑗 ) this implies 𝑔 is differentiable a.e. on ℝ with 𝑑𝑔/𝑑𝑥 ∈ 𝐿loc (ℝ). Hence
if 𝜑 ∈ 𝐶0∞ (ℝ) then we can find 𝑗 sufficiently large so that supp(𝜑) ⊆ [𝑎𝑗 , 𝑏𝑗 ] and so
∫
∫
′
𝑏𝑗
∫
𝑏𝑗
𝑎𝑗
𝑏𝑗
∫
=
𝑎𝑗
∫
′
𝑏𝑗
𝜑 (𝑡)𝑔𝑗 (𝑡)𝑑𝑡 =
𝜑 (𝑡)𝑔(𝑡)𝑑𝑡 =
𝜑 (𝑡)𝑔(𝑡)𝑑𝑡 =
ℝ
′
𝜑(𝑡)𝑔𝑗′ (𝑡)𝑑𝑡
𝑎𝑗
𝑎𝑗
∫
𝑑𝑔 𝑑𝑔 𝜑(𝑡) 𝑑𝑡.
𝜑(𝑡) 𝑑𝑡 =
𝑑𝑥 𝑥=𝑡
𝑑𝑥 𝑥=𝑡
ℝ
1,1
This implies that 𝑔 ′ = 𝑑𝑔/𝑑𝑥 ∈ 𝐿1loc (ℝ) and therefore 𝑔 ∈ 𝑊loc
(ℝ). But 𝑓 = 𝑔 a.e. on ℝ and
1,1
so 𝑓 ∈ 𝑊loc
(ℝ). This proves the statement for 𝑛 = 𝑚 = 1. The case where 𝑚, 𝑛 ≥ 1 follows
trivially now from the case 𝑛 = 𝑚 = 1. This completes the proof.
Lemma 72 Let (𝑎, 𝑏) ⊆ ℝ be a bounded interval.
Then the matrix multiplication ⋅ :
𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) × 𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝑊 1,1 (𝑎, 𝑏)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈
𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)), 𝐵 ∈ 𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)), is a continuous bilinear map and
(𝐴𝐵)′ = 𝐴′ 𝐵 + 𝐴𝐵 ′ .
178
We first prove the statement for 𝑚 = 𝑛 = 𝑟 = 1. To begin, let (𝑎, 𝑏) ⊆ ℝ
Proof.
be a bounded interval. Then by [45, p. 65, §II.2.3, Proposition 2.3.1], multiplication ⋅ :
𝑊 1,1 (𝑎, 𝑏) × 𝑊 1,1 (𝑎, 𝑏) → 𝑊 1,1 (𝑎, 𝑏), where 𝑓 ⋅ 𝑔 := 𝑓 𝑔 for 𝑓, 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏), is a continuous
bilinear map and
(𝑓 𝑔)′ = 𝑓 ′ 𝑔 + 𝑓 𝑔 ′ .
This proves the statement for 𝑚 = 𝑛 = 𝑟 = 1. The statement for 𝑚, 𝑛, 𝑟 ≥ 1 now follows
from this, Lemma 77, and notation 4.5.(xxix). This completes the proof.
1,1
1,1
1,1
Lemma 73 Matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) × 𝑀𝑛,𝑟 (𝑊loc
(ℝ)) → 𝑀𝑚,𝑟 (𝑊loc
(ℝ)),
1,1
1,1
where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)), 𝐵 ∈ 𝑀𝑛,𝑟 (𝑊loc
(ℝ)), is a well-defined map and
(𝐴𝐵)′ = 𝐴′ 𝐵 + 𝐴𝐵 ′ .
Proof.
We first prove the statement for 𝑚 = 𝑛 = 𝑟 = 1. To begin, let (𝑎, 𝑏) ⊆ ℝ be any
1,1
bounded interval. It follows from Lemma 69 and Lemma 72 that if 𝑓, 𝑔 ∈ 𝑊loc
(ℝ) then we
have (𝑓 𝑔)∣(𝑎,𝑏) = 𝑓 ∣(𝑎,𝑏) 𝑔∣(𝑎,𝑏) ∈ 𝑊 1,1 (𝑎, 𝑏). Now since this is true for every bounded interval
1,1
(𝑎, 𝑏) ⊆ ℝ, this implies by Lemma 71 that 𝑓 𝑔 ∈ 𝑊loc
(ℝ). Moreover, by Lemma 69 and
Lemma 72 we have for every bounded interval (𝑎, 𝑏) ⊆ ℝ,
((𝑓 𝑔)∣(𝑎,𝑏) )′ = (𝑓 ∣(𝑎,𝑏) 𝑔∣(𝑎,𝑏) )′ = (𝑓 ∣(𝑎,𝑏) )′ 𝑔∣(𝑎,𝑏) + 𝑓 ∣(𝑎,𝑏) (𝑔∣(𝑎,𝑏) )′
= 𝑓 ′ ∣(𝑎,𝑏) 𝑔∣(𝑎,𝑏) + 𝑓 ∣(𝑎,𝑏) 𝑔 ′ ∣(𝑎,𝑏) .
It follows from this that 𝑓 ′ 𝑔 + 𝑓 𝑔 ′ ∈ 𝐿1loc (ℝ) and for any 𝜑 ∈ 𝐶0∞ (ℝ) we can find a bounded
interval (𝑎, 𝑏) ⊆ ℝ such that supp(𝜑) ⊆ (𝑎, 𝑏) implying
∫
′
∫
𝜑 (𝑥)(𝑓 𝑔)(𝑥)𝑑𝑥 =
ℝ
𝑏
𝜑′ (𝑥)(𝑓 𝑔)∣(𝑎,𝑏) (𝑥)𝑑𝑥
𝑎
179
∫
𝑏
𝜑(𝑥)(𝑓 ′ ∣(𝑎,𝑏) (𝑥)𝑔∣(𝑎,𝑏) (𝑥) + 𝑓 ∣(𝑎,𝑏) (𝑥)𝑔 ′ ∣(𝑎,𝑏) (𝑥))𝑑𝑥
=−
𝑎
∫
𝑏
𝜑(𝑥)(𝑓 ′ (𝑥)𝑔(𝑥) + 𝑓 (𝑥)𝑔 ′ (𝑥))𝑑𝑥
=−
𝑎
∫
=−
𝑏
𝜑(𝑥)(𝑓 ′ 𝑔 + 𝑓 𝑔 ′ )(𝑥)𝑑𝑥.
𝑎
And because this is true for every 𝜑 ∈ 𝐶0∞ (ℝ) this means the weak derivative of 𝑓 𝑔 is
𝑓 ′ 𝑔 + 𝑓 𝑔 ′ , i.e.,
(𝑓 𝑔)′ = 𝑓 ′ 𝑔 + 𝑓 𝑔 ′ .
This proves the statement for 𝑚 = 𝑛 = 𝑟 = 1. The statement for 𝑚, 𝑛, 𝑟 ≥ 1 now follows
trivially from this one by notation 4.5.(xxix). This completes the proof.
1,1
Lemma 74 Let 𝐸 = 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) or 𝑀𝑚,𝑛 (𝑊loc
(ℝ)), where 1 ≤ 𝑝 ≤ ∞. The translation
operator ℒ𝑑 : 𝐸 → 𝐸, where (ℒ𝑑 𝐴)(⋅) = 𝐴(⋅+𝑑) for 𝐴 ∈ 𝐸, is a well-defined map. Moreover,
1,1
for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) we have
(ℒ𝑑 𝐴)′ = ℒ𝑑 𝐴′ .
Proof. We first prove the statement for 𝑛 = 𝑚 = 1. Let 𝑓 ∈ 𝐿𝑝loc (ℝ). Then (ℒ𝑑 𝑓 )(⋅) : ℝ →
ℂ given by (ℒ𝑑 𝑓 )(𝑥) := 𝑓 (𝑥 + 𝑑), 𝑥 ∈ ℝ is a measurable function. And for any bounded
interval (𝑎, 𝑏) ⊆ ℝ, since ∣∣(ℒ𝑑 𝑓 )∣(𝑎,𝑏) ∣∣𝑝 = ∣∣𝑓 ∣(𝑎+𝑑,𝑏+𝑑) ∣∣𝑝 , we have (ℒ𝑑 𝑓 )∣(𝑎,𝑏) ∈ 𝐿𝑝 (𝑎, 𝑏). This
1,1
(ℝ) then 𝑓 ′ ∈ 𝐿1loc (ℝ) and from what we have just
proves ℒ𝑑 𝑓 ∈ 𝐿𝑝loc (ℝ). Now if 𝑓 ∈ 𝑊loc
1,1
proved, ℒ𝑑 𝑓 ′ ∈ 𝐿1loc (ℝ). We will now show ℒ𝑑 𝑓 ∈ 𝑊loc
(ℝ) with (ℒ𝑑 𝑓 )′ = ℒ𝑑 𝑓 ′ . To start, we
have (𝜑(⋅ − 𝑑))′ = 𝜑′ (⋅ − 𝑑) ∈ 𝐶0∞ (ℝ) for all 𝜑 ∈ 𝐶0∞ (ℝ) since ()′ = 𝑑/𝑑𝑥 is just the derivative
in the classical sense on smooth functions. From this and the fact the Lebesgue integral on
180
ℝ is translation invariant it follows that
∫
∫
′
𝜙′ (𝑥 − 𝑑)(ℒ𝑑 𝑓 )(𝑥 − 𝑑)𝑑𝑥
∫ℝ
∫
′
=
𝜙 (𝑥 − 𝑑)𝑓 (𝑥)𝑑𝑥 =
𝜙′ (𝑥 − 𝑑)𝑓 (𝑥)𝑑𝑥
ℝ
∫ℝ
∫
′
= (𝜙(⋅ − 𝑑)) (𝑥)𝑓 (𝑥)𝑑𝑥 =
𝜙(𝑥 − 𝑑)𝑓 ′ (𝑥)𝑑𝑥
ℝ
∫ℝ
∫
=
𝜙(𝑥)𝑓 ′ (𝑥 + 𝑑)𝑑𝑥 =
𝜙(𝑥)(ℒ𝑑 𝑓 ′ )(𝑥)𝑑𝑥, 𝜑 ∈ 𝐶0∞ (ℝ).
𝜙 (𝑥)(ℒ𝑑 𝑓 )(𝑥)𝑑𝑥 =
ℝ
ℝ
ℝ
But this implies (ℒ𝑑 𝑓 )′ = ℒ𝑑 𝑓 ′ and, since we have already shown ℒ𝑑 𝑓 ′ ∈ 𝐿1loc (ℝ), this proves
1,1
ℒ𝑑 𝑓 ∈ 𝑊loc
(ℝ). This proves the statement for 𝑚 = 𝑛 = 1. The statement for 𝑚, 𝑛, 𝑟 ≥ 1
now follows trivially from this one by notation 4.5.(xxix). This completes the proof.
Lemma 75 For 1 ≤ 𝑝 ≤ ∞ and 𝑚, 𝑛 ∈ ℕ, the adjoint extends via 𝐴∗ (⋅) := 𝐴(⋅)∗ for
𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)), to a well-defined map
∗
: 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) → 𝑀𝑛,𝑚 (𝐿𝑝loc (ℝ)). Moreover, if
1,1
1,1
𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) then 𝐴∗ ∈ 𝑀𝑛,𝑚 (𝑊loc
(ℝ)) and
(𝐴∗ )′ = (𝐴′ )∗ .
Proof.
In the case 𝑚 = 𝑛 = 1, it just follows from the facts that 𝑓 ∈ 𝐿1loc (ℝ) if and only
1,1
1,1
(ℝ) if and only if Re(𝑓 ), Im(𝑓 ) ∈ 𝑊loc
(ℝ), Re(𝑓 ′ ) =
if Re(𝑓 ), Im(𝑓 ) ∈ 𝐿1loc (ℝ) and 𝑓 ∈ 𝑊loc
′
(Re(𝑓 ))′ , Im(𝑓 ′ ) = (Im(𝑓 ))′ , and 𝑓 ′ = 𝑓 . In the case 𝑚, 𝑛 ≥ 1, if 𝐴 := {𝑎𝑖𝑗 }𝑚,𝑛
𝑖=1,𝑗=1 ∈
1
𝑀𝑚,𝑛 (𝐿1loc (ℝ)) then, by the case 𝑚 = 𝑛 = 1, we have 𝐴∗ = {𝑎∗𝑖𝑗 }𝑛,𝑚
𝑗=1,𝑖=1 ∈ 𝑀𝑛,𝑚 (𝐿loc (ℝ)) and
1,1
1,1
if 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) then 𝐴∗ ∈ 𝑀𝑛,𝑚 (𝑊loc
(ℝ)) with
{ }𝑛,𝑚
𝑛,𝑚
(𝐴∗ )′ = {𝑎𝑖𝑗 ′ }𝑗=1,𝑖=1 = 𝑎𝑖𝑗 ′ 𝑗=1,𝑖=1 = (𝐴′ )∗ .
This completes the proof.
181
Lemma 76 If 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)) and 1 ≤ 𝑝 ≤ ∞ then 𝐴∗ ∈ 𝑀𝑛,𝑚 (𝐿𝑝 (𝕋)) with ∣∣𝐴∗ ∣∣𝐿𝑝 (𝕋) =
∣∣𝐴∣∣𝐿𝑝 (𝕋) .
Proof. By Lemma 75 the map ∗ : 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) → 𝑀𝑛,𝑚 (𝐿𝑝loc (ℝ)) extending the adjoint by
𝐴∗ (⋅) = 𝐴(⋅)∗ for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) is well-defined. By Lemma 74, the translation operator
ℒ𝑑 : 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) → 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) given by (L𝑑 𝐴)(⋅) = 𝐴(⋅ + 𝑑) for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ))
is well-defined. Thus for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)), since 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)) ⊆ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)), then
𝐴∗ , ℒ𝑑 𝐴∗ , (ℒ𝑑 𝐴)∗ are all well-defined and belong to 𝑀𝑛,𝑚 (𝐿𝑝loc (ℝ)). Also, 𝐵 ∈ 𝑀𝑛,𝑚 (𝐿𝑝 (𝕋))
if and only if 𝑀𝑛,𝑚 (𝐿𝑝loc (ℝ)) and ℒ𝑑 𝐵 = 𝐵. Thus since 𝐴∗ ∈ 𝑀𝑛,𝑚 (𝐿𝑝loc (ℝ)) and
ℒ𝑑 𝐴∗ = 𝐴∗ (⋅ + 𝑑) = 𝐴(⋅ + 𝑑)∗ = (ℒ𝑑 𝐴)∗ = 𝐴∗
then 𝐴∗ ∈ 𝑀𝑛,𝑚 (𝐿𝑝 (𝕋)).
Now we prove ∣∣𝐴∗ ∣∣𝐿𝑝 (𝕋) = ∣∣𝐴∣∣𝐿𝑝 (𝕋) for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)). We start with the case 𝑚 = 𝑛 =
1. If 𝑓 ∈ 𝐿𝑝 (𝕋) then as we just proved 𝑓 ∈ 𝐿𝑝 (𝕋) and
∣∣𝑓 ∣∣𝐿𝑝 (𝕋)
) 𝑝1
( ∫ 𝑑
) 𝑝1 ( ∫ 𝑑
1
1
𝑝
𝑝
∣𝑓 (𝑥)∣ 𝑑𝑥
=
∣𝑓 (𝑥)∣ 𝑑𝑥
= ∣∣𝑓 ∣∣𝐿𝑝 (𝕋) (1 ≤ 𝑝 < ∞),
=
𝑑 0
𝑑 0
∣∣𝑓 ∣∣𝐿∞ (𝕋) = ∣∣𝑓 ∣∣∞ = ∣∣𝑓 ∣∣∞ = ∣∣𝑓 ∣∣𝐿∞ (𝕋) (𝑝 = ∞).
This proves the statement for 𝑚 = 𝑛 = 1. Consider now the case 𝑚, 𝑛 ≥ 1. If 𝐴 :=
𝑛,𝑚
𝑝
𝑝
∗
{𝑎𝑖𝑗 }𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐿 (𝕋)) then 𝐴 = {𝑎𝑖𝑗 }𝑗=1,𝑖=1 ∈ 𝑀𝑛,𝑚 (𝐿loc (ℝ)) and so
(
∣∣𝐴∗ ∣∣𝐿𝑝 (𝕋) =
𝑚
𝑛 ∑
∑
) 21
∣∣𝑎𝑖𝑗 ∣∣2𝐿𝑝 (𝕋)
(
=
𝑗=1 𝑖=1
𝑛
𝑚 ∑
∑
) 12
∣∣𝑎𝑖𝑗 ∣∣2𝐿𝑝 (𝕋)
= ∣∣𝐴∣∣𝐿𝑝 (𝕋) .
𝑖=1 𝑗=1
And hence the statement is true for 𝑚, 𝑛 ≥ 1. This completes the proof.
182
General Statement
In this section we give the statements we need for general Banach spaces. We split this
section into statements that do not involve the frequency 𝜔 and those that do.
Parameter Independent
This is the collection of statements which do not involve the frequency parameter.
Definition 27 Let 𝐸1 , 𝐸2 , and 𝐸3 be Banach spaces. A function ⋅ : 𝐸1 × 𝐸2 → 𝐸3 is called
a continuous bilinear map provided it satisfies the following three properties:
(i) For any 𝑎1 ∈ 𝐸1 , the map 𝑎2 7→ 𝑎1 ⋅ 𝑎2 is a linear map from 𝐸2 to 𝐸3 .
(ii) For any 𝑎2 ∈ 𝐸2 , the map 𝑎1 7→ 𝑎1 ⋅ 𝑎2 is a linear map from 𝐸1 to 𝐸3 .
(iii) There exists a constant 𝐶 > 0 such that ∣∣𝑎1 ⋅ 𝑎2 ∣∣𝐸3 ≤ 𝐶∣∣𝑎1 ∣∣𝐸1 ∣∣𝑎2 ∣∣𝐸2 for all 𝑎1 ∈ 𝐸1 ,
𝑎2 ∈ 𝐸2 .
Lemma 77 Let 𝐸1 , 𝐸2 , and 𝐸3 be Banach spaces and ⋅ : 𝐸1 × 𝐸2 → 𝐸3 be a bilinear map.
Define the map ⋅ : 𝑀𝑚,𝑛 (𝐸1 ) × 𝑀𝑛,𝑟 (𝐸2 ) → 𝑀𝑚,𝑟 (𝐸3 ) defined by
[
𝐴 ⋅ 𝐵 :=
𝑛
∑
]𝑚,𝑟
𝑎𝑖𝑘 ⋅ 𝑏𝑘𝑗
𝑘=1
𝑖=1,𝑗=1
𝑛,𝑟
for any 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐸1 ), 𝐵 := [𝑏𝑖𝑗 ]𝑖=1,𝑗=1 ∈ 𝑀𝑛,𝑟 (𝐸2 ). Then ⋅ : 𝑀𝑚,𝑛 (𝐸1 ) ×
𝑀𝑛,𝑟 (𝐸2 ) → 𝑀𝑚,𝑟 (𝐸3 ) is a continuous bilinear map.
Proof. The map is obviously linear in both of its arguments and so properties (i) and (ii)
are satisfied of Definition 27. Now choose any 𝐶 > 0 such that ∣∣𝑎1 ⋅ 𝑎2 ∣∣𝐸3 ≤ 𝐶∣∣𝑎1 ∣∣𝐸1 ∣∣𝑎2 ∣∣𝐸2
183
for all 𝑎1 ∈ 𝐸1 , 𝑎2 ∈ 𝐸2 . Then
⎛
∣∣𝐴 ⋅ 𝐵∣∣𝐸3 = ⎝
𝑖=1 𝑗=1
(
≤
𝑚 ∑
𝑟 ∑
𝑛
∑
𝑚 ∑
𝑟 ∑
𝑛
∑
𝑘=1
2 ⎞ 12 (
) 12
𝑚 ∑
𝑟 ∑
𝑛
∑
𝑎𝑖𝑘 ⋅ 𝑏𝑘𝑗 ⎠ ≤
∣∣𝑎𝑖𝑘 ⋅ 𝑏𝑘𝑗 ∣∣2𝐸3
𝑖=1 𝑗=1 𝑘=1
𝐸3
) 21
𝐶 2 ∣∣𝑎𝑖𝑘 ∣∣2𝐸1 ∣∣𝑏𝑘𝑗 ∣∣2𝐸3
𝑖=1 𝑗=1 𝑘=1
≤𝐶
(( 𝑚 𝑛
∑∑
𝑖=1 𝑘=1
∣∣𝑎𝑖𝑘 ∣∣2𝐸1
)( 𝑛 𝑟
∑∑
)) 21
∣∣𝑏𝑘𝑗 ∣∣2𝐸3
𝑘=1 𝑗=1
= 𝐶∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸1 ) ∣∣𝐵∣∣𝑀𝑛,𝑟 (𝐸2 ) .
This proves property (iii) of Definition 27 and thus we have proved the lemma.
Parameter Dependent
This is the collection of statements which involve the frequency parameter.
Lemma 78 Let 𝐸1 , 𝐸2 , and 𝐸3 be Banach spaces and ⋅ : 𝐸1 × 𝐸2 → 𝐸3 be a bilinear map.
If 𝑢 ∈ 𝒪(Ω, 𝐸1 ) and 𝑣 ∈ 𝒪(Ω, 𝐸2 ) then 𝑢 ⋅ 𝑣 ∈ 𝒪(Ω, 𝐸3 ) and
(𝑢 ⋅ 𝑣)𝜔 = 𝑢𝜔 ⋅ 𝑣 + 𝑢 ⋅ 𝑣𝜔 .
Proof. First we choose any 𝐶 > 0 such that ∣∣𝑎1 ⋅ 𝑎2 ∣∣𝐸3 ≤ 𝐶∣∣𝑎1 ∣∣𝐸1 ∣∣𝑎2 ∣∣𝐸2 for all 𝑎1 ∈ 𝐸1 ,
𝑎2 ∈ 𝐸2 . Now let 𝑢 ∈ 𝒪(Ω, 𝐸1 ) and 𝑣 ∈ 𝒪(Ω, 𝐸2 ). Let 𝜔 0 ∈ Ω. Then
(𝜔 − 𝜔 0 )−1 (𝑢(𝜔) ⋅ 𝑣(𝜔) − 𝑢(𝜔 0 ) ⋅ 𝑣(𝜔 0 )) − (𝑢𝜔 (𝜔 0 ) ⋅ 𝑣(𝜔 0 ) + 𝑢(𝜔 0 ) ⋅ 𝑣𝜔 (𝜔 0 ))
𝐸3
= [(𝜔 − 𝜔 0 )−1 (𝑢(𝜔) − 𝑢𝜔 (𝜔 0 ))] ⋅ 𝑣(𝜔) + 𝑢(𝜔 0 ) ⋅ [(𝜔 − 𝜔 0 )−1 (𝑣(𝜔) − 𝑣𝜔 (𝜔 0 ))]𝐸3
≤ 𝐶∣∣(𝜔 − 𝜔 0 )−1 (𝑢(𝜔) − 𝑢𝜔 (𝜔 0 ))∣∣𝐸1 ∣∣(𝜔 − 𝜔 0 )−1 (𝑣(𝜔) − 𝑣𝜔 (𝜔 0 ))∣∣𝐸2
184
Thus as 𝜔 → 𝜔 0 the RHS of that inequality goes to zero implying the LHS of that inequality
does as well which implies the function (𝑢 ⋅ 𝑣)(𝜔) := 𝑢(𝜔) ⋅ 𝑣(𝜔) is differentiable at 𝜔 0 and
(𝑢 ⋅ 𝑣)𝜔 (𝜔 0 ) = (𝑢𝜔 ⋅ 𝑣)(𝜔 0 ) + (𝑢 ⋅ 𝑣𝜔 )(𝜔 0 ).
Therefore 𝑢 ⋅ 𝑣 ∈ 𝒪(Ω, 𝐸3 ) and (𝑢 ⋅ 𝑣)𝜔 = 𝑢𝜔 ⋅ 𝑣 + 𝑢 ⋅ 𝑣𝜔 .
Lemma 79 Let 𝐸1 and 𝐸2 be Banach spaces. If ℒ ∈ 𝒪(Ω, L (𝐸1 , 𝐸2 )) and 𝑢 ∈ 𝒪(Ω, 𝐸1 )
then ℒ𝑢 ∈ 𝒪(Ω, 𝐸2 ) and
(ℒ𝑢)𝜔 = ℒ𝜔 𝑢 + ℒ𝑢𝜔 .
Proof. From the Banach spaces L (𝐸1 , 𝐸2 ) and 𝐸1 , we define the map ⋅ : L (𝐸1 , 𝐸2 )×𝐸1 →
𝐸2 by ℒ ⋅ 𝑢 := ℒ𝑢. Then it is a continuous bilinear map with ∣∣ℒ ⋅ 𝑢∣∣𝐸2 ≤ ∣∣ℒ∣∣L (𝐸1 ,𝐸2 ) ∣∣𝑢∣∣𝐸1 .
Thus it follows from Lemma 78 above that if ℒ ∈ 𝒪(Ω, L (𝐸1 , 𝐸2 )) and 𝑢 ∈ 𝒪(Ω, 𝐸1 ) then
ℒ𝑢 ∈ 𝒪(Ω, 𝐸2 ) and (ℒ𝑢)𝜔 = ℒ𝜔 𝑢 + ℒ𝑢𝜔 . This completes the proof.
Lemma 80 Let 𝐸 be a Banach space. Then 𝒪(Ω, 𝑀𝑚,𝑛 (𝐸)) = 𝑀𝑚,𝑛 (𝒪(Ω, 𝐸)) and for any
𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐸)),
𝐴𝜔 = [(𝑎𝑖𝑗 )𝜔 ]𝑚,𝑛
𝑖=1,𝑗=1 .
Proof. Let (𝐸, ∣∣⋅∣∣) be a Banach space and (𝑀𝑚,𝑛 (𝐸), ∣∣⋅∣∣𝑀𝑚,𝑛 (𝐸) ) the Banach space whose
norm is defined in (4.81). To begin, we know that any two norms on a finite dimensional
vector space over ℂ are equivalent so that there exists constants 𝐶1 , 𝐶2 > 0 such that
𝐶1 ∣∣𝐵∣∣𝑀𝑚,𝑛 (ℂ),∞ ≤ ∣∣𝐵∣∣𝑀𝑚,𝑛 (ℂ),2 ≤ 𝐶2 ∣∣𝐵∣∣𝑀𝑚,𝑛 (ℂ),∞ , ∀𝐵 ∈ 𝑀𝑚,𝑛 (ℂ),
185
where ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (ℂ),∞ , ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (ℂ),2 : 𝑀𝑚,𝑛 (ℂ) → [0, ∞) are the norms defined by
(
∣∣𝐵∣∣𝑀𝑚,𝑛 (ℂ),∞ :=
max
1≤𝑖≤𝑚,1≤𝑗≤𝑛
∣𝑏𝑖𝑗 ∣, ∣∣𝐵∣∣𝑀𝑚,𝑛 (ℂ),2 :=
𝑚 ∑
𝑛
∑
) 21
∣𝑏𝑖𝑗 ∣2
,
𝑖=1 𝑖=1
for any 𝐵 := [𝑏𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (ℂ).
Now we define a function ∣ ⋅ ∣ : 𝑀𝑚,𝑛 (𝐸) → [0, ∞) by
∣𝐴∣ := [∣∣𝑎𝑖𝑗 ∣∣]𝑚,𝑛
∀𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ,
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐸).
We define two norms ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ , ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),2 : 𝑀𝑚,𝑛 (𝐸) → [0, ∞) by
∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸),∞ := ∣𝐴∣𝑀𝑚,𝑛 (ℂ),∞ , ∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸),2 := ∣𝐴∣𝑀𝑚,𝑛 (ℂ),2
for any 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐸). But since ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),2 = ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) it follows that
𝐶1 ∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸),∞ ≤ ∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸) ≤ 𝐶2 ∣∣𝐴∣∣𝑀𝑚,𝑛 (𝐸),∞ , ∀𝐴 ∈ 𝑀𝑚,𝑛 (𝐸).
From this it follows that the identity map 𝜄 : (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) ) → (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅
∣∣𝑀𝑚,𝑛 (𝐸),∞ ) belongs to L ((𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) ), (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ )) and has a continuous inverse. These facts and Lemma 79 imply that 𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) )) =
𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ )) and (𝜄𝐴)𝜔 = 𝜄𝐴𝜔 = 𝐴𝜔 for all 𝐴 ∈ 𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅
∣∣𝑀𝑚,𝑛 (𝐸) )).
Now it follows from the definition of the norm ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ that the statement of this
lemma is true for 𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ )) and hence 𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) )) =
𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸),∞ )) = 𝑀𝑚,𝑛 (𝒪(Ω, 𝐸)) with
[(𝑎𝑖𝑗 )𝜔 ]𝑚,𝑛
𝑖=1,𝑗=1 = (𝜄𝐴)𝜔 = 𝜄𝐴𝜔 = 𝐴𝜔
186
for every 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝒪(Ω, (𝑀𝑚,𝑛 (𝐸), ∣∣ ⋅ ∣∣𝑀𝑚,𝑛 (𝐸) )). This completes the proof of the
lemma.
Lemma 81 If 𝑓 ∈ 𝒪(Ω, 𝐸) then 𝑓𝜔 ∈ 𝒪(Ω, 𝐸).
Proof.
This statement is proven in [19, pp. 21–22, Theorem 1.8.4 & Theorem 1.8.5].
Specific Statements
In the second section we give the statements we need for the specific Banach spaces used in
this chapter. We split this section into statements that do not involve the frequency 𝜔 and
those that do.
Parameter Independent
This is the collection of statements which do not involve the frequency parameter.
Lemma 82 If 𝑝, 𝑞, 𝑠 ∈ ℝ with 1 ≤ 𝑝, 𝑞, 𝑠 ≤ ∞ and
1
𝑝
+
1
𝑞
=
1
𝑠
then matrix multiplication ⋅ :
𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))×𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝐿𝑠 (𝑎, 𝑏)), where 𝐴⋅𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)),
𝐵 ∈ 𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)), is a continuous bilinear map.
Proof.
Multiplication ⋅ : 𝐿𝑝 (𝑎, 𝑏) × 𝐿𝑞 (𝑎, 𝑏) → 𝐿𝑠 (𝑎, 𝑏), where 𝑓 ⋅ 𝑔 := 𝑓 𝑔 for 𝑓 ∈ 𝐿𝑝 (𝑎, 𝑏),
𝑔 ∈ 𝐿𝑠 (𝑎, 𝑏), is a bilinear map since it follows from Holder’s inequality that
∣∣𝑓 𝑔∣∣𝑠 ≤ ∣∣𝑓 ∣∣𝑝 ∣∣𝑔∣∣𝑞 .
187
Thus by Lemma 77 it follows that matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))×𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)) →
𝑀𝑚,𝑟 (𝐿𝑠 (𝑎, 𝑏)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) and 𝐵 ∈ 𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)), is also
continuous bilinear map. This completes the proof.
Lemma 83 Let 1 ≤ 𝑝 ≤ ∞. Then the map of matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) ×
𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝐿𝑝 (𝑎, 𝑏)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) and 𝐵 ∈
𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)), is a continuous bilinear map.
Proof.
Let (𝐶[𝑎, 𝑏], ∣∣ ⋅ ∣∣∞ ) denote the Banach space of all continuous functions on [𝑎, 𝑏]
with norm ∣∣𝑔∣∣∞ := sup𝑥∈[𝑎,𝑏] ∣𝑔(𝑥)∣. By our convention, if 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏) then 𝑔 is absolutely
continuous on [𝑎, 𝑏] and hence 𝑔 ∈ 𝐶[𝑎, 𝑏]. Thus we have 𝑊 1,1 (𝑎, 𝑏) ⊆ 𝐶[𝑎, 𝑏]. Now by [45,
p. 57, §II.2.1, Proposition 2.1.7] we have for the inclusion map 𝜄 : 𝑊 1,1 (𝑎, 𝑏) → 𝐶[𝑎, 𝑏] given
by 𝜄𝑔 = 𝑔 that 𝜄 ∈ L (𝑊 1,1 (𝑎, 𝑏), 𝐶[𝑎, 𝑏]). This implies there exists a 𝐶 > 0 such that
sup ∣𝑔(𝑥)∣ = ∣∣𝑔∣∣∞ ≤ 𝐶∣∣𝑔∣∣1,1 , for every 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏).
𝑥∈[𝑎,𝑏]
It follows from this that for all 𝑓 ∈ 𝐿𝑝 (𝑎, 𝑏), 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏) we have
∣∣𝑓 𝑔∣∣𝑝 ≤ ∣∣𝑓 ∣∣𝑝 ∣∣𝑔∣∣∞ ≤ 𝐶∣∣𝑓 ∣∣𝑝 ∣∣𝑔∣∣1,1 .
This implies that multiplication ⋅ : 𝐿𝑝 (𝑎, 𝑏) × 𝑊 1,1 (𝑎, 𝑏) → 𝐿𝑝 (𝑎, 𝑏), where 𝑓 ⋅ 𝑔 := 𝑓 𝑔 for
𝑓 ∈ 𝐿𝑝 (𝑎, 𝑏), 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏), is a continuous bilinear map. Therefore it follows by Lemma 77
that the matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) × 𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝐿𝑝 (𝑎, 𝑏)), where
𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)), 𝐵 ∈ 𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)), is also a continuous bilinear map.
This completes the proof.
Lemma 84 Let 1 ≤ 𝑝 ≤ ∞. Then the map of matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) ×
1,1
𝑀𝑛,𝑟 (𝑊loc
(ℝ)) → 𝑀𝑚,𝑟 (𝐿𝑝loc (ℝ)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) and 𝐵 ∈
1,1
𝑀𝑛,𝑟 (𝑊loc
(ℝ)), is a well-defined map.
188
Proof.
1,1
(ℝ)). Then, for any bounded interval
Let 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝loc (ℝ)) and 𝐵 ∈ 𝑀𝑛,𝑟 (𝑊loc
(𝑎, 𝑏) ⊆ ℝ by Lemma 83, (𝐴𝐵)∣(𝑎,𝑏) = 𝐴∣(𝑎,𝑏) 𝐵∣(𝑎,𝑏) ∈ 𝑀𝑚,𝑟 (𝐿𝑝 (𝑎, 𝑏)). But this implies
𝐴𝐵 ∈ 𝑀𝑚,𝑟 (𝐿𝑝loc (ℝ)). This completes the proof.
Lemma 85 For each 𝑥 ∈ [𝑎, 𝑏], the evaluation map at 𝑥 defined by 𝛿 𝑥 : 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) →
𝑀𝑚,𝑛 (ℂ) where 𝛿 𝑥 𝐴 := 𝐴(𝑥) for 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)). Then
𝛿 𝑥 ∈ L (𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)), 𝑀𝑚,𝑛 (ℂ)).
(4.82)
Proof. It follows from the proof of Lemma 83 that there exists a constant 𝐶 > 0 such that
∣𝛿 𝑥 𝑔∣ ≤ 𝐶∣∣𝑔∣∣1,1 , for every 𝑔 ∈ 𝑊 1,1 (𝑎, 𝑏).
Now the evaluation map is linear so we need only prove its bounded. If 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈
𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) we have
(
∣∣𝛿 𝑥 𝐴∣∣ℂ = ∣∣𝐴(𝑥)∣∣ℂ =
𝑚 ∑
𝑛
∑
) 21
∣𝑎𝑖𝑗 (𝑥)∣2
1
1
≤ 𝑚 2 𝑛 2 max ∣𝑎𝑖𝑗 (𝑥)∣
1≤𝑖≤𝑚
1≤𝑗≤𝑛
𝑖=1 𝑗=1
1
2
1
2
1
2
≤ 𝐶𝑚 𝑛 max ∣∣𝑎𝑖𝑗 ∣∣1,1 ≤ 𝐶𝑚 𝑛
1≤𝑖≤𝑚
1≤𝑗≤𝑛
1
2
( 𝑚 𝑛
∑∑
) 12
∣∣𝑎𝑖𝑗 ∣∣21,1
1
1
= 𝐶𝑚 2 𝑛 2 ∣∣𝐴∣∣1,1 .
𝑖=1 𝑗=1
1
1
This implies that ∣∣𝛿 𝑥 ∣∣L (𝑀𝑚,𝑛 (𝑊 1,1 (𝑎,𝑏)),𝑀𝑚,𝑛 (ℂ)) ≤ 𝐶𝑚 2 𝑛 2 and hence it follows that 𝛿 𝑥 ∈
L (𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)), ℂ). This completes the proof.
Lemma 86 Let 𝑎, 𝑏 ∈ ℝ with 𝑎 < 𝑏 and let 𝑢 ∈ [𝑎, 𝑏]. We define the integral map by
∫
(ℐ𝐴)(𝑥) :=
𝑥
𝐴(𝑡)𝑑𝑡,
𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)),
𝑢
189
𝑥 ∈ (𝑎, 𝑏).
Then ℐ ∈ L (𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)), 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) and
(ℐ𝐴)′ = 𝐴,
for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)).
Proof. By [45, p. 56, §II.2.1, Proposition 2.1.5] it follows that the map ℐ : 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)) →
𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) is a well-defined map which is linear and (ℐ𝐴)′ = 𝐴. Thus we need only
prove that it is bounded to complete the proof. Now if 𝑓 ∈ 𝐿1 (𝑎, 𝑏) then it follows from the
definition of the norm ∣∣ ⋅ ∣∣1,1 on 𝑊 1,1 (𝑎, 𝑏) that
∣∣ℐ𝑓 ∣∣1,1 = ∣∣(ℐ𝑓 )′ ∣∣1 + ∣∣ℐ𝑓 ∣∣1 = ∣∣𝑓 ∣∣1 + ∣∣ℐ𝑓 ∣∣1 ≤ (𝑏 − 𝑎 + 1)∣∣𝑓 ∣∣1 .
1
Hence it follows that if 𝐴 := [𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐿 (𝑎, 𝑏)) then
(
∣∣ℐ𝐴∣∣1,1 =
𝑚 ∑
𝑛
∑
) 21
∣∣ℐ𝑎𝑖𝑗 ∣∣21,1
1
1
≤ 𝑚 2 𝑛 2 max ∣∣ℐ𝑎𝑖𝑗 ∣∣1,1
1≤𝑖≤𝑚
1≤𝑗≤𝑛
𝑖=1 𝑗=1
1
2
1
2
1
2
≤ (𝑏 − 𝑎 + 1)𝑚 𝑛 max ∣∣𝑎𝑖𝑗 ∣∣1 ≤ (𝑏 − 𝑎 + 1)𝑚 𝑛
1
2
( 𝑚 𝑛
∑∑
1≤𝑖≤𝑚
1≤𝑗≤𝑛
1
) 12
∣∣𝑎𝑖𝑗 ∣∣21
𝑖=1 𝑗=1
1
= (𝑏 − 𝑎 + 1)𝑚 2 𝑛 2 ∣∣𝐴∣∣1 .
1
1
This implies ∣∣ℐ∣∣L (𝑀𝑚,𝑛 (𝐿1 (𝑎,𝑏)),𝑀𝑚,𝑛 (𝑊 1,1 (𝑎,𝑏))) ≤ (𝑏 − 𝑎 + 1)𝑚 2 𝑛 2 and hence it follows that
ℐ ∈ L (𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)), 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))). This completes the proof.
1,1
(ℝ)) defined by
Lemma 87 The integral map ℐ : 𝑀𝑚,𝑛 (𝐿1loc (ℝ)) → 𝑀𝑚,𝑛 (𝑊loc
∫
(ℐ𝐴)(𝑥) :=
𝑥
𝐴(𝑡)𝑑𝑡,
𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1loc (ℝ)),
0
190
𝑥 ∈ ℝ,
is well-defined and satisfies
(ℐ𝐴)′ = 𝐴,
for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1loc (ℝ)).
Proof.
Let 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1loc (ℝ)). Then it follows that the function 𝐵𝐴 (⋅) : ℝ → 𝑀𝑚,𝑛 (ℂ),
defined by
∫
𝑥
𝐴(𝑡)𝑑𝑡,
𝐵𝐴 (𝑥) :=
𝑥 ∈ ℝ,
0
1,1
is well-defined. Moreover, by Lemma 86 we have that 𝐵𝐴 ∣(𝑎,𝑏) ∈ 𝑀𝑚,𝑛 (𝑊loc
(𝑎, 𝑏)) for any
bounded interval (𝑎, 𝑏) in ℝ containing 0 and 𝐵𝐴 ∣′(𝑎,𝑏) = 𝐴∣(𝑎,𝑏) . By Lemma 71 it follows
1,1
that 𝐵𝐴 ∈ 𝑀𝑚,𝑛 (𝑊loc
(ℝ)) and 𝐵𝐴 ′ ∣(𝑎,𝑏) = 𝐵𝐴 ∣′(𝑎,𝑏) = 𝐴∣(𝑎,𝑏) for any bounded interval (𝑎, 𝑏)
in ℝ containing 0. This proves 𝐵𝐴′ = 𝐴. But by the definition of the integral map we have
1,1
(ℝ)) is
ℐ𝐴 = 𝐵𝐴 for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1loc (ℝ)). Therefore ℐ : 𝑀𝑚,𝑛 (𝐿1loc (ℝ)) → 𝑀𝑚,𝑛 (𝑊loc
well-defined and (ℐ𝐴)′ = 𝐴 for any 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1loc (ℝ)). This completes the proof.
Lemma 88 Let 1 ≤ 𝑝 ≤ ∞ and (𝑎, 𝑏) ⊆ ℝ be a bounded open interval. Define the restriction
map by 𝐴 7→ 𝐴∣(𝑎,𝑏) where 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)). Then ∣(𝑎,𝑏) ∈ L (𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)), 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)).
Proof.
Fix 𝑝 ∈ [1, ∞] and a bounded open interval (𝑎, 𝑏). Pick 𝑞 ∈ ℕ such that (𝑎, 𝑏) ⊆
[−𝑞𝑑, 𝑞𝑑]. Let 𝑓 ∈ 𝐿𝑝 (𝕋). Then 𝑓 ∈ 𝐿𝑝loc (ℝ) and 𝑓 (𝑥 + 𝑑) = 𝑓 (𝑥) for a.e. 𝑥 ∈ ℝ. If 𝑝 = ∞
then
∣∣𝑓 ∣(𝑎,𝑏) ∣∣∞ ≤ ∣∣𝑓 ∣(−𝑞𝑑,𝑞𝑑) ∣∣∞ = ∣∣𝑓 ∣(0,𝑑) ∣∣∞ = ∣∣𝑓 ∣∣𝐿∞ (𝕋) .
191
If 𝑝 ∈ [0, ∞) then by periodicity we have
∫ 𝑗𝑑
∣𝑓 (𝑥)∣𝑝 𝑑𝑥 =
(𝑗−1)𝑑
∫𝑑
0
∣𝑓 (𝑥)∣𝑝 𝑑𝑥 for any 𝑗 ∈ ℤ.
Hence implying
∣∣𝑓 ∣(𝑎,𝑏) ∣∣𝑝𝑝
𝑏
∫
∣𝑓 (𝑥)∣𝑝 𝑑𝑥
−𝑞𝑑
𝑎
𝑞
𝑗𝑑
∑∫
𝑝
∣𝑓 (𝑥)∣ 𝑑𝑥 +
𝑞 ∫
∑
(𝑗−1)𝑑
𝑗=1
𝑞
=
𝑞𝑑
∣𝑓 (𝑥)∣ 𝑑𝑥 ≤
=
=
∫
𝑝
𝑝
∣𝑓 (𝑥)∣ 𝑑𝑥 +
0
𝑗=1
𝑞 ∫
∑
∣𝑓 (𝑥)∣𝑝 𝑑𝑥
−𝑗𝑑
𝑗=1
𝑑
∑∫
−(𝑗−1)𝑑
𝑑
∣𝑓 (𝑥)∣𝑝 𝑑𝑥
0
𝑗=1
𝑑
∫
= 2𝑞
0
∣𝑓 (𝑥)∣𝑝 𝑑𝑥 = 2𝑞𝑑∣∣𝑓 ∣∣𝑝𝐿𝑝 (𝕋) .
1
Thus for any 𝑝 ∈ [1, ∞] we have ∣∣𝑓 ∣(𝑎,𝑏) ∣∣𝑝 ≤ (2𝑞𝑑) 𝑝 ∣∣𝑓 ∣∣𝐿𝑝 (𝕋) . But then for any 𝐴 :=
𝑝
[𝑎𝑖𝑗 ]𝑚,𝑛
𝑖=1,𝑗=1 ∈ 𝑀𝑚,𝑛 (𝐿 (𝕋)) we have
(
∣∣𝐴∣(𝑎,𝑏) ∣∣𝑝 =
𝑚 ∑
𝑛
∑
) 12
∣∣𝑎𝑖𝑗 ∣(𝑎,𝑏) ∣∣2𝑝
1
1
≤ 𝑚 2 𝑛 2 max ∣∣𝑎𝑖𝑗 ∣(𝑎,𝑏) ∣∣𝑝
1≤𝑖≤𝑚
1≤𝑗≤𝑛
𝑖=1 𝑗=1
1
1
1
≤ (2𝑞𝑑) 𝑝 𝑚 2 𝑛 2 max ∣∣𝑎𝑖𝑗 ∣∣𝐿𝑝 (𝕋)
1≤𝑖≤𝑚
1≤𝑗≤𝑛
1
𝑝
1
2
≤ (2𝑞𝑑) 𝑚 𝑛
1
2
( 𝑚 𝑛
∑∑
) 21
∣∣𝑎𝑖𝑗 ∣∣2𝐿𝑝 (𝕋)
1
1
1
= (2𝑞𝑑) 𝑝 𝑚 2 𝑛 2 ∣∣𝐴∣∣𝐿𝑝 (𝕋) .
𝑖=1 𝑗=1
And hence we conclude the restriction map ∣(𝑎,𝑏) : 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)) → 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) is a
1
1
1
bounded linear operator with norm bounded above by (2𝑞𝑑) 𝑝 𝑚 2 𝑛 2 . This proves the statement for 𝑝 ∈ [1, ∞].
Parameter Dependent
This is the collection of statements which involve the frequency parameter.
Lemma 89 Let 𝑝, 𝑞, 𝑠 ∈ ℝ with 1 ≤ 𝑝, 𝑞, 𝑠 ≤ ∞ and
192
1
𝑝
+ 1𝑞 = 1𝑠 . If 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)))
and 𝐵 ∈ 𝒪(Ω, 𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏))) then 𝐴𝐵 ∈ 𝒪(Ω, 𝑀𝑚,𝑟 (𝐿𝑠 (𝑎, 𝑏))) and
(𝐴𝐵)𝜔 = 𝐴𝜔 𝐵 + 𝐴𝐵𝜔 .
Proof. By Lemma 82 we know that the map of matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) ×
𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝐿𝑠 (𝑎, 𝑏)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) and 𝐵 ∈
𝑀𝑛,𝑟 (𝐿𝑞 (𝑎, 𝑏)), is a continuous bilinear map. Thus by Lemma 78 we must have 𝐴𝐵 ∈
𝒪(Ω, 𝑀𝑚,𝑟 (𝐿𝑠 (𝑎, 𝑏))) and
(𝐴𝐵)𝜔 = 𝐴𝜔 𝐵 + 𝐴𝐵𝜔 .
This completes the proof.
Lemma 90 Let 1 ≤ 𝑝 ≤ ∞. If 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏))) and 𝐵 ∈ 𝒪(Ω, 𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)))
then 𝐴𝐵 ∈ 𝒪(Ω, 𝑀𝑛,𝑟 (𝐿𝑝 (𝑎, 𝑏))) and
(𝐴𝐵)𝜔 = 𝐴𝜔 𝐵 + 𝐴𝐵𝜔 .
Proof. By Lemma 83 we know that the map of matrix multiplication ⋅ : 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) ×
𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑚,𝑟 (𝐿𝑝 (𝑎, 𝑏)), where 𝐴 ⋅ 𝐵 := 𝐴𝐵 for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)), 𝐵 ∈
𝑀𝑛,𝑟 (𝑊 1,1 (𝑎, 𝑏)), is a continuous bilinear map. Thus by Lemma 78 we must have 𝐴𝐵 ∈
𝒪(Ω, 𝑀𝑚,𝑟 (𝐿𝑝 (𝑎, 𝑏))) and
(𝐴𝐵)𝜔 = 𝐴𝜔 𝐵 + 𝐴𝐵𝜔 .
This completes the proof.
Lemma 91 Let 𝑥 ∈ [𝑎, 𝑏] and 𝛿 𝑥 : 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)) → 𝑀𝑚,𝑛 (ℂ) be the evaluation map
193
at 𝑥, i.e., 𝛿 𝑥 𝐴 = 𝐴(𝑥) for all 𝐴 ∈ 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)). If 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) then
𝛿 𝑥 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (ℂ)) and
(𝛿 𝑥 𝐴)𝜔 = 𝛿 𝑥 𝐴𝜔 .
Proof.
By Lemma 85 we know 𝛿 𝑥 ∈ L (𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏)), 𝑀𝑚,𝑛 (ℂ)). Thus by Lemma 79,
if 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) then 𝛿 𝑥 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (ℂ)) and
(𝛿 𝑥 𝐴)𝜔 = 𝛿 𝑥 𝐴𝜔 .
This completes the proof.
Lemma 92 Let 𝑎, 𝑏 ∈ ℝ, 𝑎 < 𝑏, and 𝑢 ∈ [𝑎, 𝑏]. Let ℐ : 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)) → 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))
be the integral map, i.e.,
∫
𝑥
𝐴(𝑡)𝑑𝑡,
(ℐ𝐴)(𝑥) :=
𝐴 ∈ 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)),
𝑥 ∈ (𝑎, 𝑏).
𝑢
If 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏))) then ℐ𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) and
(ℐ𝐴)𝜔 = ℐ𝐴𝜔 .
Proof.
By Lemma 86 we know that ℐ ∈ L (𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏)), 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))). Thus by
Lemma 79, if 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿1 (𝑎, 𝑏))) then ℐ𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑊 1,1 (𝑎, 𝑏))) and
(ℐ𝐴)𝜔 = ℐ𝐴𝜔 .
This completes the proof.
194
Lemma 93 Let 𝑝 ∈ [1, ∞) and (𝑎, 𝑏) be any bounded open interval in ℝ. Let ∣(𝑎,𝑏) :
𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)) → 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)) be the restriction map, i.e., the domain restriction map 𝐴 7→
𝐴∣(𝑎,𝑏) for 𝐴 ∈ 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)). If 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝕋))) then 𝐴∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)))
and
(𝐴∣(𝑎,𝑏) )𝜔 = 𝐴𝜔 ∣(𝑎,𝑏) .
Proof.
By Lemma 88 we know that ∣(𝑎,𝑏) ∈ L (𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)), 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)). Thus by
Lemma 79, if 𝐴 ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝑀𝑚,𝑛 (𝐿𝑝 (𝕋)))) then we have 𝐴∣(𝑎,𝑏) ∈ 𝒪(Ω, 𝑀𝑚,𝑛 (𝐿𝑝 (𝑎, 𝑏)))
and
(𝐴∣(𝑎,𝑏) )𝜔 = 𝐴𝜔 ∣(𝑎,𝑏) .
This completes the proof.
195
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