EPSRC Spectral Theory Network Conference II
27-28th June, 2002
Cardiff University
The Queen’s Buildings
The Parade
Cardiff
EPSRC
Cardiff University
EPSRC Spectral Theory Network Conference II
Cardiff University, 27-28th June, 2002
L IST OF PARTICIPANTS
M van den Berg University of Bristol M.vandenBerg@bristol.ac.uk
B M Brown Cardiff University Malcolm.Brown@cs.cardiff.ac.uk
A W Craig University of Durham Alan.Craig@durham.ac.uk
E B Davies Kings College, University of London E.B.Davies@kcl.ac.uk
M S P Eastham Cardiff University mandh@chesilhay.fsnet.co.uk
W D Evans Cardiff University EvansWD@cardiff.ac.uk
W N Everitt University of Birmingham w.n.everitt@bham.ac.uk
L Greenberg University of Maryland lng@math.umd.edu
Peter K Jimack University of Leeds pkj@comp.leeds.ac.uk
H Kalf Universität München Hubert.Kalf@mathematik.uni-muenchen.de
A Kirpichnikova Loughborough University
A V Kiselev Dublin Institute of Technology akiselev@maths1.kst.dit.ie
Y V Kurylev Loughborough University Y.V.Kurylev@lboro.ac.uk
J Lang The Ohio State University langjan@yahoo.com, lang@math.ohio-state.edu
H Langer Technical University of Vienna hlanger@mail.zserv.tuwien.ac.at
M Langer Universität Bremen mlanger@math.uni-bremen.de
M Levitin Heriot-Watt University m.levitin@ma.hw.ac.uk
L Markus University of Minnesota markus@math.umn.edu
M Marletta University of Leicester mm7@le.ac.uk
C Martinez Kings College, University of London martinez@mth.kcl.ac.uk
V Moroz University of Bristol V.Moroz@bristol.ac.uk
S Naboko St Petersburg State University naboko@felix.math.uab.edu
L Parnovsky University College London leonid@math.ucl.ac.uk
D B Pearson University of Hull d.b.pearson@hull.ac.uk
A Pushnitski Loughborough University A.B.Pushnitski@lboro.ac.uk
R Romanov Cardiff University R.Romanov@cs.cardiff.ac.uk
Y Safarov King’s College, University of London ysafarov@mth.kcl.ac.uk
K M Schmidt Cardiff University SchmidtKM@cardiff.ac.uk
E Shargorodsky University of Sussex E.Shargorodsky@sussex.ac.uk
V A Sloushch Loughborough University V.V.Sloushch@lboro.ac.uk
Ian Sorell Loughborough University i.sorell@lboro.ac.uk
C Tretter Universität Bremen ctretter@math.uni-bremen.de
V A Virtanen University of Sussex J.A.Virtanen@sussex.ac.uk
A Watson Cardiff University WatsonA1@cardiff.ac.uk
R Weikard University of Alabama at Birmingham rudi@math.uab.edu
R Williams Cardiff University Robin.Williams@astro.cf.ac.uk
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W ORKSHOP T IMETABLE
T HURSDAY J UNE 27 TH , 2002
Chair W D Evans
0900 Michael S P Eastham Spectral concentration and resonances
1000 Y V Kurylev Boundary spectral and dynamic inverse problems in electromagnetism
1100 Coffee
1130 Peter K Jimack On the use of finite element multigrid for the optimal solution of differential eigenvalue problems
1230 Lunch
Chair A B Pushnitski
1330 L Parnovsky Integrated density of states and the Bethe-Sommerfeld conjecture
1430 E Shargorodsky Eigenvalues of Toeplitz operators embedded in the essential spectra
1530 Tea
1600 S Naboko Functional model approach for spectral analysis of dissipative
operators – applications to differential operators
1700 Christiane Tretter A new concept for block operator matrices: the Quadratic
Numerical Range (QNR)
1830 Dinner
F RIDAY J UNE 28 TH , 2002
Chair D B Pearson
0900 M van den Berg Subexponential decay of the Dirichlet heat kernel
1000 R Weikard On the inverse resonance problem
1100 Coffee
1130 H Langer Some inverse spectral problems
1230 Lunch
Chair M Marletta
1330 K M Schmidt Critical values and spectral asymptotics of singularly perturbed periodic differential operators
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1430 L Markus Harmonic operators: a new spectral problem
1530 Tea
1600 E B Davies Spectral properties of some non-self-adjoint differential systems
Depart
iv
A BSTRACTS
S UBEXPONENTIAL DECAY OF THE D IRICHLET HEAT
KERNEL
M VAN DEN B ERG
M.vandenBerg@bristol.ac.uk
School of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW, UK
We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for
large time in terms of the survival probability of a Brownian motion, under the
assumption that the latter decays subexponentially for large time.We also obtain a
lower bound for the Dirichlet heat kernel for arbitrary open and connected sets in
Euclidean space.
S PECTRAL PROPERTIES OF SOME
NON - SELF - ADJOINT DIFFERENTIAL SYSTEMS
E B DAVIES
E.B.Davies@kcl.ac.uk
Department of Mathematics
Kings College, University of London
The Strand
London WC2R 2LS, UK
We obtain an asymptotic estimate of the spectral counting function for certain
non-self-adjoint ordinary differential systems. It appears that the leading term
depends upon the boundary conditions, unlike the self-adjoint case.
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S PECTRAL CONCENTRATION AND RESONANCES
M S P E ASTHAM
mandh@chesilhay.fsnet.co.uk
Department of Computer Science
Cardiff University
PO Box 916, Cardiff CF24 3XF, UK
The lecture discusses several outstanding problems in the spectral theory of the
Sturm-Liouville equation under the following headings:
1. Spectral concentration and perturbed discrete spectra.
2. Asymptotic distribution of resonances for super-exponentially decaying potentials.
3. Analytic continuation into the unphysical sheet for exponentially and power
decaying potentials.
4. The location of resonances.
5. Antibound states and a varying coupling constant.
6. Hurwitz-type results on the absence of antibound states.
O N THE USE OF FINITE ELEMENT MULTIGRID FOR
THE OPTIMAL SOLUTION OF DIFFERENTIAL
EIGENVALUE PROBLEMS
P ETER K J IMACK
pkj@comp.leeds.ac.uk
Computational PDE Unit
School of Computing
University of Leeds
Leeds LS2 9JT, UK
Multigrid is a widely used tool for the finite element solution of many elliptic
PDEs due to the fact that the overall solution time grows only linearly with the
number of degrees of freedom in the finite element grid. The first part of this talk
will introduce the fundamentals of multigrid and explain how it can also be used
in the solution of differential eigenvalue problems.
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The second half of the talk will describe research in progress on the development
of a new, more direct, approach to the application of the multigrid philosophy to
the solution of eigenvalue problems. The idea is based upon considering multigrid in terms of energy minimization when applied to an elliptic PDE which is the
Euler-Lagrange equation for some energy functional. It is then possible to extend
this consideration to the direct minimization of the Rayleigh quotient when seeking the smallest eigenvalue for a linear differential eigenvalue problem. When
only a small number of extreme eigenvalues and eigenfunctions are required it
appears that this approach has some promise.
B OUNDARY SPECTRAL AND DYNAMIC INVERSE
PROBLEMS IN ELECTROMAGNETISM
Y V K URYLEV
Y.V.Kurylev@lboro.ac.uk
Department of Mathematical Sciences
Loughborough University
Loughborough, Leicestershire LE11 3TU, UK
We consider inverse problems of determination of electric and magnetic parameters, (x) and µ(x) from boundary measurements. We prove uniqueness for
isotropic and some anisotropic and µ.
S OME INVERSE SPECTRAL PROBLEMS
H L ANGER
hlanger@mail.zserv.tuwien.ac.at
Institute for Analysis and Technical Mathematics
Technical University of Vienna
Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria
A general result of L. de Branges states that every Nevanlinna function (this is a
function which is analytic in the upper half plane and has a non-negative imaginary part there or, equivalently, for which the kernel
NQ (z, ζ) =
Q(z) − Q(ζ)∗
,
z − ζ∗
=z, =ζ > 0,
is positive definite), is the Titchmarsh–Weyl coefficient of a 2 × 2 canonical system with a positive definite Hamiltonian on an interval [0, L). If Q is a generalized
Nevanlinna function (this means that Q is meromorphic in the upper half plane
and that the kernel NQ has a finite number of negative squares) a corresponding general result is not yet known. However, it is known and will be explained
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in the lecture that under some additional assumptions such a function Q is the
Titchmarsh–Weyl coefficient of a differential equation which is at certain points
nonlinear in the spectral parameter or for which the Hamiltonian is not positive
definite. Connections with a continuous fraction expansion of the Titchmarsh–
Weyl coefficient or the generalized Schur algorithm are explained. The general
feature of these problems is that they are not self-adjoint in a Hilbert space but
they are self-adjoint with respect to an indefinite inner product.
H ARMONIC OPERATORS : A NEW SPECTRAL
PROBLEM
L M ARKUS
markus@math.umn.edu
University of Minnesota
School of Mathematics
127 Vincent Hall
206 Church St. S.E.
Minneapolis, MN 55455, USA
This report, representing joint work of W.N. Everitt, L. Markus and M. Plum,
analyses some self-adjoint operators that are generated through boundary value
problems for certain elliptic partial differential expressions. In particular, harmonic operators, which arose in studies of differential operators and symplectic
algebra by the first two mentioned authors, are illustrated for the classical Laplacian in the unit open disk Ω of the complex plane.
Here the harmonic operator
o
∆
THar : f → −∆f on D(THar ) = W 2 (Ω) u L2 (Ω)
is defined on the direct sum of two submanifolds of L2 (Ω), namely:
o
∂f
2
2
= 0 on ∂Ω
W (Ω) = f ∈ W (Ω) : f =
∂n
and
∆
L2 (Ω) = {f ∈ L2 (Ω) : f is harmonic in Ω}.
The spectrum of the self-adjoint operator THar consists of the eigenvalue 0, of infinite multiplicity, and countably many real non-zero eigenvalues, each of finite
multiplicity. New methods of spectral analysis are introduced to define eigenfunctions modulo a subspace.
Examples of eigenfunctions, which have singular (or no) boundary values are
described within Bergman and Hardy function spaces.
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F UNCTIONAL MODEL APPROACH FOR SPECTRAL
ANALYSIS OF DISSIPATIVE OPERATORS –
APPLICATIONS TO DIFFERENTIAL OPERATORS
S NABOKO
naboko@felix.math.uab.edu
Department of Mathematical Physics
Institute for Physics
St Petersburg State University
Ulyanovskaya 1
St Petersburg 198904, Russia
The spectral analysis of dissipative operators from the point of view of the Szokefalvi–
Nagy–Foias functional model will be considered. It is based on the consideration
of the so-called characteristic function of the operator which is a contractive analytic operator-valued function in the upper half-plane.The characteristic function plays the role of a “matrix” characteristic polynomial. It contains the whole
spectral information about the so-called completely nonselfadjoint part of the dissipative operator. This part can be reconstructed via the characterictic function
uniquely up to the unitarily equivalence. Generally speaking the spectral analysis
of the operator can be reduced to the factorization of its characteristic function
and therefore to the problems related to the analytic matrix-valued functions. We
plan to consider the application to the Friedrichs model operators, Schrödinger
operators with complex potentials and the Boltzmann transport operators
I NTEGRATED DENSITY OF STATES AND THE
B ETHE -S OMMERFELD CONJECTURE
L PARNOVSKY
leonid@math.ucl.ac.uk
University College London
Department of Mathematics
Gower Street
London WC1E 6BT, UK
I will discuss recent progress in understanding the asymptotic behaviour of the integrated density of states of Schrödinger operators with periodic potentials. I also
explain the relationship between these results and the Bethe-Sommerfeld conjecture. This is a joint work with A. Sobolev.
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C RITICAL VALUES AND SPECTRAL ASYMPTOTICS OF
SINGULARLY PERTURBED PERIODIC DIFFERENTIAL
OPERATORS
K M S CHMIDT
SchmidtKM@cardiff.ac.uk
School of Mathematics
Cardiff University
Senghennydd Road, Cardiff CF24 4YH, UK
The remarkable spectral properties of radially periodic Schrödinger and Dirac operators are taken as a motivation to study periodic ordinary differential operators
of Sturm-Liouville and Dirac type with a singular perturbation of the type of the
angular momentum term.
This talk presents results on critical phenomena in connection with the number of
eigenvalues in spectral gaps, and the eigenvalue asymptotics both near the band
edge and in the large-coupling limit.
E IGENVALUES OF T OEPLITZ OPERATORS EMBEDDED
IN THE ESSENTIAL SPECTRA
E S HARGORODSKY
E.Shargorodsky@sussex.ac.uk
School of Mathematical Sciences
University of Sussex
Falmer, Brighton BN1 9QH, UK
This is a joint work with my PhD student Jani Virtanen.
Let T (a) be a Toeplitz operator with a continuous coefficient a. It is well known
that the essential spectrum of T (a) coincides with the set of values of a. Suppose
λ does not belong to the essential spectrum of T (a), i.e. the function λ − a has no
zeros. Then T (a) − λI is invertible iff the winding number of λ − a is zero, and λ
is an eigenvalue iff this winding number is positive, in which case the multiplicity
of λ equals the winding number. It is much more difficult to decide whether or not
λ is an eigenvalue and find its multiplicity if λ belongs to the essential spectrum
of T (a), i.e. λ − a has zeros.
We study the latter case and show the effect of (the discontinuities of) the argument of λ − a on the (non)existence of Lp –eigenfunctions corresponding to λ.
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A NEW CONCEPT FOR BLOCK OPERATOR MATRICES :
THE Q UADRATIC N UMERICAL R ANGE (QNR)
C T RETTER
ctretter@math.uni-bremen.de
FB 3 - Mathematik
Universität Bremen
Bibliothekstrasse 1
D-28359 Bremen, Germany
In this talk a new concept for 2 × 2 block operator matrices – the quadratic numerical range – is presented. The main results are a spectral inclusion theorem, a
classification of the corners of the quadratic numerical range, an estimate of the
resolvent in terms of the quadratic numerical range, factorization theorems for
the Schur complements, and a theorem about angular operator representations of
spectral invariant subspaces which implies, e.g., the existence of solutions of the
corresponding Riccati equations and a block diagonalization. All results are new
in the operator as well as in the matrix case.
Joint work with H. Langer, A. Markus, V. Matsaev
O N THE INVERSE RESONANCE PROBLEM
R W EIKARD
rudi@math.uab.edu
Department of Mathematics
University of Alabama at Birmingham
University Station
Birmingham AL 35294, USA
Consider the Sturm-Liouville problem given by the equation −y 00 + qy = λy
on [0, ∞) and the boundary condition y(0) = 0. It was shown by Korotyaev
and by Zworski that eigenvalues and resonances determine uniquely a compactly
supported real potential.
We give sufficient (albeit somewhat implicit) conditions on q for which the statement remains to hold. Our approach allows the potential to be complex-valued.
This is joint work with B. M. Brown and I. W. Knowles.
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