Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 248740, 7 pages
http://dx.doi.org/10.1155/2013/248740
Research Article
Dissipative Sturm-Liouville Operators
with Transmission Conditions
Hüseyin Tuna1 and Aytekin EryJlmaz2
1
2
Department of Mathematics, Mehmet Akif Ersoy University, 15100 Burdur, Turkey
Department of Mathematics, Nevsehir University, 50300 Nevsehir, Turkey
Correspondence should be addressed to Aytekin Eryılmaz; eryilmazaytekin@gmail.com
Received 7 December 2012; Accepted 11 February 2013
Academic Editor: Lucas Jódar
Copyright © 2013 H. Tuna and A. Eryılmaz. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
In this paper we study dissipative Sturm-Liouville operators with transmission conditions. By using Pavlov’s method (Pavlov 1947,
Pavlov 1981, Pavlov 1975, and Pavlov 1977), we proved a theorem on completeness of the system of eigenvectors and associated
vectors of the dissipative Sturm-Liouville operators with transmission conditions.
1. Introduction
Spectral theory is one of the main branches of modern functional analysis and it has many applications in mathematics
and applied sciences. There has recently been great interest in
spectral analysis of Sturm-Liouville boundary value problems
with eigenparameter-dependent boundary conditions (see
[1–14]). Furthermore, many researchers have studied some
boundary value problems that may have discontinuities in the
solution or its derivative at an interior point 𝑐 [15–19]. Such
conditions which include left and right limits of solutions
and their derivatives at 𝑐 are often called “transmission
conditions” or “interface conditions.” These problems often
arise in varied assortment of physical transfer problems
[20].
The spectral analysis of non-self-adjoint (dissipative)
operators is based on ideas of the functional model and
dilation theory rather than the method of contour integration
of resolvent which is studied by Naimark [21], but this
method is not effective in studying the spectral analysis of
boundary value problem. The functional model technique
acts a part on the fundamental theorem of Nagy-Foiaş. In
1960s independently from Nagy-Foiaş [22], Lax and Phillips
[23] developed abstract scattering programme that is very
important in scattering theory. Pavlov’s functional model
[24–28] has been extended to dissipative operators which
are finite dimensional extensions of a symmetric operator,
and the corresponding dissipative and Lax-Phillips scattering
matrix was investigated in some detail [5–14, 22–27, 29, 30].
This theory is based on the notion of incoming and outgoing
subspaces to obtain information about analytical properties
of scattering matrix by utilizing properties of original unitary
group. By combining the results of Nagy-Foiaş and LaxPhillips, characteristic function is expressed with scattering
matrix and the dilation of dissipative operator is set up. By
means of different spectral representation of dilation, given
operator can be written very simply and functional models
are obtained. The eigenvalues, eigenvectors and spectral
projection of model operator are expressed obviously by
characteristic function. The problem of completeness of the
system of eigenvectors is solved by writing characteristic
function as factorization.
The purpose of this paper is to study non-self-adjoint
Sturm-Liouville operators with transmission conditions. To
do this, we constructed a functional model of dissipative
operator by means of the incoming and outgoing spectral representations and defined its characteristic function,
because this makes it possible to determine the scattering matrix of dilation according to the Lax and Phillips
scheme [23]. Finally, we proved a theorem on completeness of the system of eigenvectors and associated vectors
of dissipative operators which is based on the method of
Pavlov. While proving our results, we use the machinery of
[5, 7–10].
2
Abstract and Applied Analysis
2. Self-Adjoint Dilation of Dissipative
Sturm-Liouville Operator
and transmission conditions
Consider the differential expression
𝑙 (𝑦) = −𝑦󸀠󸀠 + 𝑞 (𝑥) 𝑦,
𝑥 ∈ [𝑎, 𝑐) ∪ (𝑐, 𝑏) ,
(1)
where 𝐼1 := [𝑎, 𝑐), 𝐼2 := (𝑐, 𝑏) and 𝐼 = 𝐼1 ∪ 𝐼2 , 𝑞(𝑥) is realvalued function on 𝐼, and 𝑞 ∈ 𝐿1loc (𝐼). The points 𝑎 and 𝑐 are
regular and 𝑏 is singular for the differential expression 𝑙(𝑦).
Moreover 𝑞(𝑐±) := lim𝑥 → 𝑐± 𝑞(𝑥) one-sided limits exist and
are finite.
To pass from the differential expression 𝑙(𝑦) to operators,
we introduce the Hilbert space 𝐻 = 𝐿2 (𝐼1 ) ⊕ 𝐿2 (𝐼2 ) with the
inner product
𝑐
𝑏
𝑎
𝑐
⟨𝑓, 𝑔⟩𝐻 := 𝛾1 𝛾2 ∫ 𝑓1 𝑔1 𝑑𝑥 + 𝛿1 𝛿2 ∫ 𝑓2 𝑔2 𝑑𝑥,
𝑓1 (𝑥) ,
𝑓2 (𝑥) ,
𝑥 ∈ 𝐼1
∈ 𝐻,
𝑥 ∈ 𝐼2
𝑔 (𝑥) ,
𝑔 (𝑥) = { 1
𝑔2 (𝑥) ,
𝑥 ∈ 𝐼1
∈ 𝐻,
𝑥 ∈ 𝐼2
𝛾1
𝜃 (𝑐−, 𝜆) ,
𝛿1 1
𝜃2󸀠 (𝑐+, 𝜆) =
𝛾2 󸀠
𝜃 (𝑐−, 𝜆) ,
𝛿2 1
𝜑2 (𝑐+, 𝜆) =
𝛾1
𝜑 (𝑐−, 𝜆) ,
𝛿1 1
𝜑2󸀠 (𝑐+, 𝜆) =
𝛾2 󸀠
𝜑 (𝑐−, 𝜆) ,
𝛿2 1
(7)
where 𝛼, 𝛾1 , 𝛾2 , 𝛿1 , and 𝛿2 ∈ R with 𝛾1 𝛾2 > 0 and 𝛿1 𝛿2 >
0. The solutions 𝜃(𝑥, 𝜆) and 𝜑(𝑥, 𝜆) belong to 𝐻. Let
𝑢 (𝑥) = 𝜃1 (𝑥, 0) , 𝑥 ∈ 𝐼1
𝑢 (𝑥) = { 1
𝑢2 (𝑥) = 𝜃2 (𝑥, 0) , 𝑥 ∈ 𝐼2 ,
V (𝑥) = 𝜑1 (𝑥, 0) , 𝑥 ∈ 𝐼1 ,
V (𝑥) = { 1
V2 (𝑥) = 𝜑2 (𝑥, 0) , 𝑥 ∈ 𝐼2 ,
(2)
(3)
𝑢1 (𝑎) = cos 𝛼,
𝑢1󸀠 (𝑎) = sin 𝛼,
V1 (𝑎) = − sin 𝛼,
V1󸀠 (𝑎) = cos 𝛼
𝑢2 (𝑐+) =
𝛾1
𝑢 (𝑐−) ,
𝛿1 1
𝑢2󸀠 (𝑐+) =
𝛾2 󸀠
𝑢 (𝑐−) ,
𝛿2 1
V2 (𝑐+) =
𝛾1
V (𝑐−) ,
𝛿1 1
V2󸀠 (𝑐+) =
𝛾2 󸀠
V (𝑐−) .
𝛿2 1
𝑅1 (𝑦) : 𝛾1 𝑦 (𝑐−) − 𝛿1 𝑦 (𝑐+) = 0,
𝑏
𝑏
𝑅2 (𝑦) : 𝛾2 𝑦󸀠 (𝑐−) − 𝛿2 𝑦󸀠 (𝑐+) = 0,
𝑎
𝑎
𝑅3 (𝑦) : 𝑦 (𝑎) cos 𝛼 + 𝑦󸀠 (𝑎) sin 𝛼 = 0,
+ 𝛿1 𝛿2 [𝑦, 𝑧]𝑏 − 𝛿1 𝛿2 [𝑦, 𝑧]𝑐+ ,
𝑅4 (𝑦) : [𝑦, V]𝑏 − 𝐺[𝑦, 𝑢]𝑏 = 0,
(4)
where [𝑦, 𝑧]𝑥 = 𝑦(𝑥)𝑧󸀠 (𝑥) − 𝑦󸀠 (𝑥)𝑧(𝑥) (𝑥 ∈ 𝐼), [𝑦, 𝑧]𝑏 =
lim𝑥 → 𝑏 [𝑦, 𝑧]𝑥 exists and is finite.
Suppose that Weyl’s limit circle case holds for the differential expression 𝑙(𝑦) on 𝐼. There are several sufficient
conditions in which Weyl’s limit circle case holds for a
differential expression [21]. Denote by
𝜑 (𝑥) , 𝑥 ∈ 𝐼1
𝜑 (𝑥) = { 1
𝜑2 (𝑥) , 𝑥 ∈ 𝐼2 ,
(5)
the solutions of the equation 𝑙(𝑦) = 𝜆𝑦, 𝑥 ∈ 𝐼, satisfying the
initial conditions
𝜃1 (𝑎, 𝜆) = cos 𝛼,
𝜑1 (𝑎, 𝜆) = − sin 𝛼,
𝜃1󸀠 (𝑎, 𝜆) = sin 𝛼,
𝜑1󸀠
(𝑎, 𝜆) = cos 𝛼
(6)
(10)
All the maximal dissipative extensions L𝐺 of the operator
𝐿 0 are described by the following conditions (see [4, 15, 18]):
∫ 𝑙 (𝑦) 𝑧 𝑑𝑥 − ∫ 𝑦𝑙 (𝑧) 𝑑𝑥 = 𝛾1 𝛾2 [𝑦, 𝑧]𝑐− − 𝛾1 𝛾2 [𝑦, 𝑧]𝑎
𝑥 ∈ 𝐼1
𝑥 ∈ 𝐼2 ,
(9)
and transmission conditions
and 𝛾1 , 𝛾2 , 𝛿1 , and 𝛿2 are some real numbers with 𝛾1 𝛾2 > 0 and
𝛿1 𝛿2 > 0.
Let 𝐿 0 denote the closure of the minimal operator generated by (1) and by 𝐷0 its domain. Besides, we denote
the set of all functions 𝑓(𝑥) from 𝐻 such that 𝑓, 𝑓󸀠 ∈
𝐴𝐶loc (𝐼), 𝑓(𝑐±), 𝑓󸀠 (𝑐±) one-sided limits exist and are finite
and 𝑙(𝑦) ∈ 𝐻; 𝐷 is the domain of the maximal operator 𝐿.
Furthermore, 𝐿 = 𝐿∗0 [21].
For two arbitrary functions 𝑦(𝑥), 𝑧(𝑥) ∈ 𝐷, we have
Green’s formula
𝜃 (𝑥) ,
𝜃 (𝑥) = { 1
𝜃2 (𝑥) ,
(8)
𝑢(𝑥), and V(𝑥) be solutions of the equation 𝑙(𝑦) = 0, 𝑥 ∈ 𝐼,
satisfying the initial conditions
where
𝑓 (𝑥) = {
𝜃2 (𝑐+, 𝜆) =
(11)
Im 𝐺 > 0.
Let us add the “incoming” and “outgoing” subspaces 𝐷− =
𝐿 (−∞, 0) and 𝐷+ = 𝐿2 (0, ∞) to 𝐻. The orthogonal sum 𝐻 =
𝐷− ⊕ 𝐻 ⊕ 𝐷+ is called main Hilbert space of the dilation.
In the space H, we consider the operator L𝐺 on the set
̂ 𝜑+ ⟩,
𝐷(L𝐺), its elements consisting of vectors 𝑤 = ⟨𝜑− , 𝑦,
generated by the expression
2
̂ 𝜑+ ⟩ = ⟨𝑖
L𝐺 ⟨𝜑− , 𝑦,
𝑑𝜑−
𝑑𝜑
̂ ,𝑖 +⟩
, 𝑙 (𝑦)
𝑑𝜉
𝑑𝜉
(12)
satisfying the conditions 𝜑− ∈ 𝑊21 (−∞, 0), 𝜑+ ∈ 𝑊21 (0, ∞),
̂ = 0, 𝑅2 (𝑦)
̂ = 0, 𝑅3 (𝑦)
̂ = 0, [𝑦,
̂ V]𝑏 − 𝐺[𝑦,
̂ 𝑢]𝑏 =
𝑦̂ ∈ 𝐻, 𝑅1 (𝑦)
̂ V]𝑏 −𝐺[𝑦,
̂ 𝑢]𝑏 = (𝐶/√𝛿1 𝛿2 )𝜑+ (0), where
(𝐶/√𝛿1 𝛿2 )𝜑− (0), [𝑦,
𝑊21 are Sobolev spaces and 𝐶2 := 2 Im 𝐺, 𝐶 > 0.
Theorem 1. The operator L𝐺 is self-adjoint in H and it is a
̃ 𝐺(= 𝐿 𝐾 ).
self-adjoint dilation of the operator 𝐿
Abstract and Applied Analysis
3
Proof. We first prove that L𝐺 is symmetric in H. Namely,
(L𝐺𝑓, 𝑔)H − (𝑓, L𝐺𝑔)H = 0. Let 𝑓, 𝑔 ∈ 𝐷(L𝐺), 𝑓 =
̂ 𝜑+ ⟩ and 𝑔 = ⟨𝜓− , 𝑧̂, 𝜓+ ⟩. Then we have
⟨𝜑− , 𝑦,
(L𝐺𝑓, 𝑔)H − (𝑓, L𝐺𝑔)H
̂ 𝜑+ ⟩ , ⟨𝜓− , 𝑧̂, 𝜓+ ⟩)
= (L𝐺 ⟨𝜑− , 𝑦,
̂ 𝜑+ ⟩ , L𝐺 ⟨𝜓− , 𝑧̂, 𝜓+ ⟩)
− (⟨𝜑− , 𝑦,
=∫
0
∞
−∞
̂ , 𝑧̂)𝐻 + ∫ 𝑖𝜑+󸀠 𝜓+ 𝑑𝜉
𝑖𝜑−󸀠 𝜓− 𝑑𝜉 + (𝑙 (𝑦)
0
0
−∫
−∞
=∫
0
∞
̂ 𝑙 (̂𝑧))𝐻 − ∫ 𝑖𝜓+󸀠 𝜑+ 𝑑𝜉
𝑖𝜓−󸀠 𝜑− 𝑑𝜉 − (𝑦,
(13)
0
∞
−∞
̂ 𝑧̂]𝑏 + ∫ 𝑖𝜑+󸀠 𝜓+ 𝑑𝜉
𝑖𝜑−󸀠 𝜓− 𝑑𝜉 + 𝛿1 𝛿2 [𝑦,
0
−∫
−∞
∞
for 𝑦̂ and Im 𝜆 < 0. By applying onto the mapping 𝑃, we
obtain (17), and
We obtain by direct computation
(14)
Thus, L𝐺 is a symmetric operator. To prove that L𝐺 is
self-adjoint, we need to show that L𝐺 ⊆ L∗𝐺. Take 𝑔 =
⟨𝜓− , 𝑧̂, 𝜓+ ⟩ ∈ 𝐷(L∗𝐺). Let L∗𝐺𝑔 = 𝑔∗ = ⟨𝜓−∗ , 𝑧̂∗ , 𝜓+∗ ⟩ ∈ H,
so that
(15)
By choosing elements with suitable components as the
𝑓 ∈ 𝐷(L𝐺) in (15), it is not difficult to show that 𝜓− ∈
𝑊21 (−∞, 0), 𝜓+ ∈ 𝑊21 (0, ∞), 𝑔 ∈ 𝐷(L), and 𝑔∗ =
L𝐺𝑔; the operator L𝐺 is defined by (12). Therefore (15)
is obtained from (L𝐺𝑓, 𝑔)H = (𝑓, L𝐺𝑔)H for all 𝑓 ∈
𝐷(L∗𝐺). Furthermore, 𝑔 ∈ 𝐷(L∗𝐺) satisfies the conditions
̂ V]𝑏 + 𝐺[𝑦,
̂ 𝑢]𝑏 =
[𝑦,
𝐶
𝜑+ (0) .
√𝛿1 𝛿2
√𝛿1 𝛿2
̂ 𝑢]𝑏 ) 𝑒−𝑖𝜆𝜉 ⟩
̂ V]𝑏 + 𝐺[𝑦,
([𝑦,
𝐶
(18)
−1
̂ 𝑧̂]𝑏 .
= 𝑖𝜓− (0) 𝜑− (0) − 𝑖𝜑+ (0) 𝜓+ (0) + 𝛿1 𝛿2 [𝑦,
𝐶
𝜑− (0) ,
√𝛿1 𝛿2
For this purpose, we set (L𝐺 − 𝜆𝐼)−1 𝑃1 𝑦̂ = 𝑔 = ⟨𝜓− , 𝑧̂, 𝜓+ ⟩
̂ and hence ̃𝑙(̂𝑧) − 𝜆̂𝑧 =
which implies that (L𝐺 − 𝜆𝐼)𝑔 = 𝑃1 𝑦,
−𝑖𝜆𝜉
̂ 𝜓− (𝜉) = 𝜓− (0)𝑒
and 𝜓+ (𝜉) = 𝜓+ (0)𝑒−𝑖𝜆𝜉 . Since 𝑔 ∈
𝑦,
1
𝐷(L𝐺), then 𝜓− ∈ 𝑊2 (−∞, 0); it follows that 𝜓− (0) = 0,
̂ V]𝑏 −
and consequently 𝑧 satisfies the boundary condition [𝑦,
̃ 𝐺), and since point 𝜆 with
̂ 𝑢]𝑏 = 0. Therefore 𝑧̂ ∈ 𝐷(𝐿
𝐺[𝑦,
Im 𝜆 < 0 cannot be an eigenvalue of dissipative operator,
̃ 𝐺 − 𝜆𝐼)−1 𝑦.
̂ Thus we have
then 𝑧̂ = (𝐿
̃ 𝐺 − 𝜆𝐼) 𝑦,
̂
= ⟨0, (𝐿
0
̂ 𝑢]𝑏 =
̂ V]𝑏 + 𝐺[𝑦,
[𝑦,
𝑦̂ ∈ 𝐻, Im ℎ < 0.
(17)
−1
𝑖𝜓−󸀠 𝜑− 𝑑𝜉 − ∫ 𝑖𝜓+󸀠 𝜑+ 𝑑𝜉
(L𝐺𝑓, 𝑔)H = (𝑓, L∗𝐺𝑔)H = (𝑓, 𝑔∗ )H .
−1
−1
̃ 𝐺 − 𝜆𝐼) 𝑦,
̂
𝑃(L𝐺 − 𝜆𝐼) 𝑃1 𝑦̂ = (𝐿
(L𝐺 − 𝜆𝐼) 𝑃1 𝑦̂
0
̂ 𝑧̂]𝑏 = 0.
𝑖𝜓− (0) 𝜑− (0) − 𝑖𝜑+ (0) 𝜓+ (0) + 𝛿1 𝛿2 [𝑦,
is self-adjoint dilation of 𝐵𝐺. To show this, it is sufficient to
verify the equality
(16)
Hence, 𝐷(L∗𝐺) ⊆ 𝐷(L𝐺); that is, L𝐺 = L∗𝐺.
The self-adjoint operator L𝐺 generates on H a unitary
group 𝑈𝑡 = exp (𝑖L𝐺𝑡) (𝑡 ∈ R+ = (0, ∞)). Let us denote
by 𝑃 : H → 𝐻 and 𝑃1 : 𝐻 → H the mapping acting
̂ 𝜑+ ⟩ → 𝑦̂ and 𝑃1 :
according to the formulae 𝑃 : ⟨𝜑− , 𝑦,
̂ 0⟩. Let 𝑍𝑡 := 𝑃𝑈𝑡 𝑃1 , 𝑡 ≥ 0, by using 𝑈𝑡 . The
𝑦̂ → ⟨0, 𝑦,
family {𝑍𝑡 } (𝑡 ≥ 0) of operators is a strongly continuous
semigroup of completely nonunitary contraction on 𝐻. Let
us denote by 𝐵𝐺 the generator of this semigroup: 𝐵𝐺𝑦̂ =
̂ The domain of 𝐵𝐺 consists of all
lim𝑡 → +0 (𝑖𝑡)−1 (𝑍𝑡 𝑦̂ − 𝑦).
the vectors for which the limit exists. The operator 𝐵𝐺 is
dissipative. The operator L𝐺 is called the self-adjoint dilation
̃ 𝐺; hence L𝐺
of 𝐵𝐺 (see [10, 21, 30]). We show that 𝐵𝐺 = 𝐿
̃ 𝐺 − 𝜆𝐼)
(𝐿
−1
∞
−1
= 𝑃(L𝐺 − 𝜆𝐼) 𝑃1 = −𝑖𝑃 ∫ 𝑈𝑡 𝑒−𝑖𝜆𝑡 𝑑𝑡𝑃1
0
∞
−1
= − 𝑖 ∫ 𝑍𝑡 𝑒−𝑖𝜆𝑡 𝑑𝑡 = (𝐵𝐺 − 𝜆𝐼) ,
Im 𝜆 < 0,
0
(19)
̃ 𝐺 = 𝐵𝐺.
so this clearly shows that 𝐿
3. Functional Model of Dissipative
Sturm-Liouville Operator
The unitary group {𝑈𝑡 } has an important property which
makes it possible to apply it to the Lax-Phillips [23]. It
has orthogonal incoming and outgoing subspaces 𝐷− =
⟨𝐿2 (−∞, 0), 0, 0⟩ and 𝐷+ = ⟨0, 0, 𝐿2 (0, ∞)⟩ having the
following properties:
(1) 𝑈𝑡 𝐷− ⊂ 𝐷− , 𝑡 ≤ 0 and 𝑈𝑡 𝐷+ ⊂ 𝐷+ , 𝑡 ≥ 0,
(2) ∩𝑡≤0 𝑈𝑡 𝐷− = ∩𝑡≥0 𝑈𝑡 𝐷+ = {0},
(3) ∪𝑡≥0 𝑈𝑡 𝐷− = ∪𝑡≤0 𝑈𝑡 𝐷+ = H,
(4) 𝐷− ⊥ 𝐷+ .
Property (4) is clear. To be able to prove property (1) for
𝐷+ (the proof for 𝐷− is similar), we set R𝜆 = (L𝐺 − 𝜆𝐼)−1 .
For all 𝜆, with Im 𝜆 < 0 and for any 𝑓 = ⟨0, 0, 𝜑+ ⟩ ∈ 𝐷+ , we
have
𝜉
R𝜆 𝑓 = ⟨0, 0, −𝑖𝑒−𝑖𝜆𝜉 ∫ 𝑒𝑖𝜆𝑠 𝜑+ (𝑠) 𝑑𝑠⟩ ,
0
(20)
as R𝜆 𝑓 ∈ 𝐷+ . Therefore, if 𝑔 ⊥ 𝐷+ , then
∞
0 = (R𝜆 𝑓, 𝑔)H = −𝑖 ∫ 𝑒−𝑖𝜆𝑡 (𝑈𝑡 𝑓, 𝑔)H 𝑑𝑡,
0
Im 𝜆 < 0,
(21)
4
Abstract and Applied Analysis
which implies that (𝑈𝑡 𝑓, 𝑔)H = 0 for all 𝑡 ≥ 0. Hence, for
𝑡 ≥ 0, 𝑈𝑡 𝐷+ ⊂ 𝐷+ , and property (1) has been proved.
In order to prove property (2), we define the mappings
+
𝑃 : H → 𝐿2 (0, ∞) and 𝑃1+ : 𝐿2 (0, ∞) → 𝐷+ as
̂ 𝜑+ ⟩ → 𝜑+ and 𝑃1+ : 𝜑 → ⟨0, 0, 𝜑⟩,
follows: 𝑃+ : ⟨𝜑− , 𝑦,
respectively. We take into consider that the semigroup of
isometries 𝑈𝑡+ := 𝑃+ 𝑈𝑡 𝑃1+ (𝑡 ≥ 0) is a one-sided shift in
𝐿2 (0, ∞). Indeed, the generator of the semigroup of the onesided shift 𝑉𝑡 in 𝐿2 (0, ∞) is the differential operator 𝑖(𝑑/𝑑𝜉)
with the boundary condition 𝜑(0) = 0. On the other hand,
the generator 𝑆 of the semigroup of isometries 𝑈𝑡+ (𝑡 ≥
0) is the operator 𝑆𝜑 = 𝑃+ L𝐺𝑃1+ 𝜑 = 𝑃+ L𝐺⟨0, 0, 𝜑⟩ =
𝑃+ ⟨0, 0, 𝑖(𝑑/𝑑𝜉)𝜑⟩ = 𝑖(𝑑/𝑑𝜉)𝜑, where 𝜑 ∈ 𝑊21 (0, ∞) and
𝜑(0) = 0. Since a semigroup is uniquely determined by its
generator, it follows that 𝑈𝑡+ = 𝑉𝑡 , and, hence,
2
⋂ 𝑈𝑡 𝐷+ = ⟨0, 0, ⋂ 𝑉𝑡 𝐿 (0, ∞)⟩ = {0} ,
𝑡≥0
(22)
𝑡≤0
Definition 2. The linear operator 𝐴 with domain 𝐷(𝐴) acting
in the Hilbert space 𝐻 is called completely non-self-adjoint (or
simple) if there is no invariant subspace 𝑀 ⊆ 𝐷(𝐴)(𝑀 ≠ {0})
of the operator 𝐴 on which the restriction 𝐴 to 𝑀 is selfadjoint.
To prove property (3) of the incoming and outgoing
subspaces, let us prove following lemma.
̃ 𝐺 is completely non-self-adjoint
Lemma 3. The operator 𝐿
(simple).
̃𝐺
Proof. Let 𝐻󸀠 ⊂ 𝐻 be a nontrivial subspace in which 𝐿
󸀠
󸀠
̃ with domain 𝐷(𝐿
̃ ) =
induces a self-adjoint operator 𝐿
𝐺
𝐺
󸀠
󸀠
∗
̂
̂
̃
̃
̃
𝐻 ∩ 𝐷(𝐿 𝐺). If 𝑓 ∈ 𝐷(𝐿 𝐺), then 𝑓 ∈ 𝐷(𝐿 𝐺) and
𝑑 󵄩󵄩󵄩 𝑖𝐿̃󸀠𝐺𝑡 ̂󵄩󵄩󵄩2
𝑑 𝑖𝐿̃󸀠 𝑡 ̂ 𝑖𝐿̃󸀠𝐺𝑡 ̂
𝑒 𝑓)
󵄩𝑒 𝑓󵄩󵄩 = (𝑒 𝐺 𝑓,
󵄩𝐻 𝑑𝑡
𝐻
𝑑𝑡 󵄩󵄩
2
̃ 󸀠𝐺 𝑡
𝑖𝐿
= − 𝐶 ([𝑒
Assume that
𝜑 (𝑥, 𝜆) := {
𝜑1 (𝑥, 𝜆) , 𝑥 ∈ 𝐼1
𝜑2 (𝑥, 𝜆) , 𝑥 ∈ 𝐼2 ,
2
(24)
𝜓 (𝑥, 𝜆) , 𝑥 ∈ 𝐼1
𝜓 (𝑥, 𝜆) := { 1
𝜓2 (𝑥, 𝜆) , 𝑥 ∈ 𝐼2
are solutions of 𝑙(𝑦) = 𝜆𝑦 satisfying the conditions
𝜑1 (𝑎, 𝜆) = cos 𝛼,
𝜑1󸀠 (𝑎, 𝜆) = sin 𝛼,
𝜓1󸀠 (𝑎, 𝜆) = cos 𝛼,
𝜓1 (𝑎, 𝜆) = − sin 𝛼,
so the proof is completed.
0=
relative to the group {𝑈𝑡 } and has the form H󸀠 = ⟨0, 𝐻󸀠 , 0⟩,
where 𝐻󸀠 is a subspace in 𝐻. Therefore, if the subspace
H󸀠 (and hence also 𝐻󸀠 ) was nontrivial, then the unitary group
{𝑈𝑡󸀠 } restricted to this subspace would be a unitary part of the
̃ 󸀠 of 𝐿
̃ 𝐺 to 𝐻󸀠 would be
group {𝑈𝑡 }, and hence, the restriction 𝐿
𝐺
̃ 𝐺 is simple,
a self-adjoint operator in 𝐻󸀠 . Since the operator 𝐿
󸀠
it follows that 𝐻 = {0}. The lemma is proved.
𝜑2 (𝑐+, 𝜆) =
𝛾1
𝜑 (𝑐−, 𝜆) ,
𝛿1 1
𝜑2󸀠 (𝑐+, 𝜆) =
𝛾2 󸀠
𝜑 (𝑐−, 𝜆) .
𝛿2 1
(25)
Let us adopt the following notations:
𝐾 (𝜆) =
[𝜑, V]𝑏
[𝜓, 𝑢]𝑏
,
𝑀 (𝜆) = −
[𝜓, 𝑢]𝑏
[𝜑, 𝑢]𝑏
,
−1
𝑆𝐺 (𝜆) = (𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) (𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) ,
(26)
(27)
where 𝑀(𝜆) is a meromorphic function on the complex plane
C with a countable number of poles on the real axis. Further,
it is possible to show that the function 𝑀(𝜆) possesses the
following properties: Im 𝑀(𝜆) ≤ 0 for all Im 𝜆 ≠ 0, and
𝑀(𝜆) = 𝑀(𝜆) for all 𝜆 ∈ C, except the real poles 𝑀(𝜆).
We set
𝑈𝜆− (𝑥, 𝜉, 𝜁)
(23)
̂ 𝑢] ) .
𝑓,
𝑏 𝐻
̂ 𝑢]𝑏 = 0. Using this result with
Consequently, we have [𝑦,
̂ 𝑢]𝑏 = 0, we have [𝑦,
̂ 𝑢]𝑏 =
̂ V]𝑏 + 𝐺[𝑦,
boundary condition [𝑦,
̂
0; that is, 𝑦(𝜆)
= 0. Since all solutions of 𝑙(𝑦) = 𝜆𝑦 belong
to 𝐿2 (0, ∞), from this it can be concluded that the resolvent
̃ 𝐺) is a compact operator, and the spectrum of 𝐿
̃ 𝐺 is
𝑅𝜆 (𝐿
purely discrete. Consequently, by the theorem on expansion
̃ 󸀠 we obtain
in the eigenvectors of the self-adjoint operator 𝐿
𝐺
󸀠
̃
𝐻 = {0}. Hence the operator 𝐿 𝐺 is simple. The proof is
completed.
Let us define 𝐻− = ∪𝑡≥0 𝑈𝑡 𝐷− , 𝐻+ = ∪𝑡≤0 𝑈𝑡 𝐷+ .
Lemma 4. The equality 𝐻− + 𝐻+ = H holds.
Proof. Considering property (1) of the subspace 𝐷+ , it is easy
to show that the subspace H󸀠 = H ⊝ (𝐻− + 𝐻+ ) is invariant
= ⟨𝑒−𝑖𝜆𝜉 ,
𝐶
−1
𝑀 (𝜆) [(𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) [𝜓, 𝑢]𝑏 ] 𝜑 (𝑥, 𝜆) ,
√𝛿1 𝛿2
𝑆𝐺 (𝜆) 𝑒−𝑖𝜆𝜁 ⟩ .
(28)
We note that the vectors 𝑈𝜆− (𝑥, 𝜉, 𝜁) for real 𝜆 do not belong
to the space H. However, 𝑈𝜆− (𝑥, 𝜉, 𝜁) satisfies the equation
L𝑈 = 𝜆𝑈 and the corresponding boundary conditions for
the operator Lℎ .
By means of vector 𝑈𝜆− (𝑥, 𝜉, 𝜁), we define the transformã (𝜆) by
tion 𝐹− : 𝑓 → 𝑓
−
̃ (𝜆) :=
(𝐹− 𝑓) (𝜆) := 𝑓
−
1
(𝑓, 𝑈𝜆)H
√2𝜋
(29)
Abstract and Applied Analysis
5
̂ 𝜑+ ⟩ in which 𝜑− (𝜉), 𝜑+ (𝜁), 𝑦(𝑥)
on the vectors 𝑓 = ⟨𝜑− , 𝑦,
are smooth, compactly supported functions.
Lemma 5. The transformation 𝐹− isometrically maps 𝐻− onto
𝐿2 (R). For all vectors 𝑓, 𝑔 ∈ 𝐻− the Parseval equality and the
inversion formulae hold:
∞
̃ , 𝑔̃ ) = ∫
(𝑓, 𝑔)H = (𝑓
− − 𝐿2
−∞
(30)
Proof. For 𝑓, 𝑔 ∈ 𝐷− , 𝑓 = ⟨𝜑− , 0, 0⟩, 𝑔 = ⟨𝜓− , 0, 0⟩, with
Paley-Wiener theorem, we have
0
1
1
(𝑓, 𝑈𝜆 )H =
∫ 𝜑− (𝜉) 𝑒−𝑖𝜆𝜉 𝑑𝜉 ∈ 𝐻−2 ,
√2𝜋
√2𝜋 −∞
(31)
and by using usual Parseval equality for Fourier integrals,
∞
(𝑓, 𝑔)H = ∫
−∞
∞
= ∫
−∞
𝜑− (𝜉) 𝜓− (𝜉)𝑑𝜉
(32)
̃ (𝜆) 𝑔̃ (𝜆)𝑑𝜆 = (𝐹 𝑓, 𝐹 𝑔) 2 .
𝑓
−
−
−
− 𝐿
Here, 𝐻±2 denote the Hardy classes in 𝐿2 (R) consisting of the
functions analytically extendible to the upper and lower halfplanes, respectively.
We now extend the Parseval equality to the whole of 𝐻− .
We consider in 𝐻− the dense set of 𝐻−󸀠 of the vectors obtained
as follows from the smooth, compactly supported functions
in 𝐷− : 𝑓 ∈ 𝐻−󸀠 if 𝑓 = 𝑈𝑇 𝑓0 , 𝑓0 = ⟨𝜑− , 0, 0⟩, 𝜑− ∈ 𝐶0∞ (−∞, 0),
where 𝑇 = 𝑇𝑓 is a nonnegative number depending on 𝑓. If
𝑓, 𝑔 ∈ 𝐻−󸀠 , then for 𝑇 > 𝑇𝑓 and 𝑇 > 𝑇𝑔 we have 𝑈−𝑇 𝑓, 𝑈−𝑇 𝑔 ∈
𝐷− ; moreover, the first components of these vectors belong
to 𝐶0∞ (−∞, 0). Therefore, since the operators 𝑈𝑡 (𝑡 ∈ R) are
unitary, by the equality
𝐹− 𝑈𝑡 𝑓 =
(𝑈𝑡 𝑓, 𝑈𝜆− )H
𝑖𝜆𝑡
=𝑒
(𝑓, 𝑈𝜆− )H
= ⟨𝑆𝐺 (𝜆) 𝑒−𝑖𝜆𝜉 ,
𝑒−𝑖𝜆𝜁 ⟩ .
(36)
̃ (𝜆) = (𝐹 𝑓)(𝜆) and 𝑔̃ (𝜆) = (𝐹 𝑔)(𝜆).
where 𝑓
−
−
−
−
̃ (𝜆) =
𝑓
−
(𝑥, 𝜉, 𝜁)
−1
𝐶
𝑀 (𝜆) [(𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) [𝜓, 𝑢]𝑏 ] 𝜑 (𝑥, 𝜆) ,
√𝛿1 𝛿2
̃ (𝜆) 𝑔̃ (𝜆) 𝑑𝜆,
𝑓
−
−
∞
1
̃ (𝜆) 𝑈 𝑑𝜆,
𝑓=
∫ 𝑓
𝜆
√2𝜋 −∞ −
We set
𝑈𝜆+
𝑖𝜆𝑡
= 𝑒 𝐹− 𝑓,
(33)
we have
(𝑓, 𝑔)H = (𝑈−𝑇 𝑓, 𝑈−𝑇 𝑔)H = (𝐹− 𝑈−𝑇 𝑓, 𝐹− 𝑈−𝑇 𝑔)𝐿2 ,
̃ 𝑔)
̃ 𝐿2 .
(𝑒𝑖𝜆𝑇 𝐹− 𝑓, 𝑒𝑖𝜆𝑇 𝐹− 𝑔)𝐿2 = (𝑓,
(34)
(35)
By taking the closure (35), we obtain the Parseval equality
for the space 𝐻− . The inversion formula is obtained from
the Parseval equality if all integrals in it are considered as
limits in the integrals over finite intervals. Finally 𝐹− 𝐻− =
∪𝑡≥0 𝐹− 𝑈𝑡 𝐷− = ∪𝑡≥0 𝑒𝑖𝜆𝑡 𝐻−2 = 𝐿2 (R); that is, 𝐹− maps 𝐻− onto
the whole of 𝐿2 (R). The lemma is proved.
vectors 𝑈𝜆+ (𝑥, 𝜉, 𝜁) for real 𝜆 do not belong
However, 𝑈𝜆+ (𝑥, 𝜉, 𝜁) satisfies the equation
We note that the
to the space H.
L𝑈 = 𝜆𝑈 and the corresponding boundary conditions for
the operator Lℎ . With the help of vector 𝑈𝜆+ (𝑥, 𝜉, 𝜁), we define
̃ (𝜆) by (𝐹 𝑓)(𝜆) := 𝑓
̃ (𝜆) :=
the transformation 𝐹+ : 𝑓 → 𝑓
+
+
+
+
√
̂
(1/ 2𝜋)(𝑓, 𝑈𝜆 )H on the vectors 𝑓 = ⟨𝜑− , 𝑦, 𝜑+ ⟩ in which
𝜑− (𝜉), 𝜑+ (𝜁), and 𝑦(𝑥) are smooth, compactly supported
functions.
Lemma 6. The transformation 𝐹+ isometrically maps 𝐻+ onto
𝐿2 (R). For all vectors 𝑓, 𝑔 ∈ 𝐻+ the Parseval equality and the
inversion formula hold:
̃ , 𝑔̃ ) = ∫
(𝑓, 𝑔)H = (𝑓
+ + 𝐿2
∞
−∞
̃ (𝜆) 𝑔̃ (𝜆)𝑑𝜆,
𝑓
+
+
∞
1
̃ (𝜆) 𝑈+ 𝑑𝜆,
𝑓=
∫ 𝑓
𝜆
√2𝜋 −∞ +
(37)
̃ (𝜆) = (𝐹 𝑓)(𝜆) and 𝑔̃ (𝜆) = (𝐹 𝑔)(𝜆).
where𝑓
+
+
+
+
Proof. The proof is analogous to Lemma 6.
It is obvious that the matrix-valued function 𝑆𝐺(𝜆) is
meromorphic in C and all poles are in the lower half-plane.
From (27), |𝑆𝐺(𝜆)| ≤ 1 for Im 𝜆 > 0, and 𝑆𝐺(𝜆) is the unitary
matrix for all 𝜆 ∈ R. Therefore, it explicitly follows from the
formulae for the vectors 𝑈𝜆− and 𝑈𝜆+ that
𝑈𝜆+ = 𝑆𝐺 (𝜆) 𝑈𝜆− .
(38)
It follows from Lemmas 6 and 5 that 𝐻− = 𝐻+ . Together with
Lemma 5, this shows that 𝐻− = 𝐻+ = H; therefore property
(3) has been proved for the incoming and outgoing subspaces.
Thus, the transformation 𝐹− isometrically maps 𝐻− onto
𝐿2 (R) with the subspace 𝐷− mapped onto 𝐻−2 and the
operators 𝑈𝑡 are transformed into the operators of multiplication by 𝑒𝑖𝜆𝑡 . This means that 𝐹− is the incoming spectral
representation for the group {𝑈𝑡 }. Similarly, 𝐹+ is the outgoing
spectral representation for the group {𝑈𝑡 }. It follows from (38)
that the passage from the 𝐹− representation of an element
̃ (𝜆) =
𝑓 ∈ H to its 𝐹+ representation is accomplished as 𝑓
+
−1
̃ (𝜆). Consequently, according to [22], we have proved
𝑆𝐺 (𝜆)𝑓
−
the following.
Theorem 7. The function 𝑆𝐺(𝜆) is the scattering matrix of the
group {𝑈𝑡 } (of the self-adjoint operator L𝐺).
6
Abstract and Applied Analysis
Let 𝑆(𝜆) be an arbitrary nonconstant inner function (see
[19]) on the upper half-plane (the analytic function 𝑆(𝜆) and
the upper half-plane C+ is called inner function on C+ if
|𝑆ℎ (𝜆)| ≤ 1 for all 𝜆 ∈ C+ and |𝑆ℎ (𝜆)| = 1 for almost
all 𝜆 ∈ R). Define 𝐾 = 𝐻+2 ⊝ 𝑆𝐻+2 . Then 𝐾 ≠ {0} is a
subspace of the Hilbert space 𝐻+2 . We consider the semigroup
of operators 𝑍𝑡 (𝑡 ≥ 0) acting in 𝐾 according to the formula
𝑍𝑡 𝜑 = 𝑃[𝑒𝑖𝜆𝑡 𝜑], 𝜑 = 𝜑(𝜆) ∈ 𝐾, where 𝑃 is the orthogonal
projection from 𝐻+2 onto 𝐾. The generator of the semigroup
{𝑍𝑡 } is denoted by
−1
𝑇𝜑 = lim (𝑖𝑡) (𝑍𝑡 𝜑 − 𝜑) ,
𝑡 → +0
(39)
where 𝑇 is a maximal dissipative operator acting in 𝐾 and
with the domain 𝐷(𝑇) consisting of all functions 𝜑 ∈
𝐾, such that the limit exists. The operator 𝑇 is called a
model dissipative operator. Recall that this model dissipative
operator, which is associated with the names of Lax-Phillips
[23], is a special case of a more general model dissipative
operator constructed by Nagy and Foiaş [22]. The basic
assertion is that 𝑆(𝜆) is the characteristic function of the
operator 𝑇.
Let 𝐾 = ⟨0, 𝐻, 0⟩, so that H = 𝐷− ⊕ 𝐾 ⊕ 𝐷+ . It follows
from the explicit form of the unitary transformation 𝐹− under
the mapping 𝐹− that
2
H 󳨀→ 𝐿 (R) ,
𝐷− 󳨀→ 𝐻−2 ,
̃ (𝜆) = (𝐹 𝑓) (𝜆) ,
𝑓 󳨀→ 𝑓
−
−
𝐷+ 󳨀→ 𝑆𝐺𝐻+2 ,
𝐾 󳨀→ 𝐻+2 ⊝ 𝑆𝐺𝐻+2 ,
corresponding singular measure 𝜇 must be contained in the
set of poles 𝑆𝐺(𝜆). But in this case the singular measure 𝜇 is
a simple step function. If we require 𝑆𝐺(𝜆) to have no zeros
of infinite multiplicity, then 𝜇 = 0. So the singular factor
vanishes. The characteristic function 𝑆𝐺(𝜆) of the maximal
̃ 𝐺 has the form
dissipative operator 𝐿
𝑆𝐺 (𝜆) :=
𝑀 (𝜆) 𝐾 (𝜆) + 𝐺
,
𝑀 (𝜆) 𝐾 (𝜆) + 𝐺
(41)
where Im 𝐺 > 0.
Theorem 9. For all the values of 𝐺 with Im 𝐺 > 0, except
possibly for a single value 𝐺 = 𝐺0 , the characteristic function
̃ 𝐺 is a Blaschke
𝑆𝐺(𝜆) of the maximal dissipative operator 𝐿
̃
product. The spectrum of 𝐿 𝐺 is purely discrete and belongs
̃ 𝐺(𝐺 ≠ 𝐺0 ) has a
to the open upper half-plane. The operator 𝐿
countable number of isolated eigenvalues with finite multiplicity and limit points at infinity. The system of all eigenvectors and
̃ 𝐺 is complete in the space 𝐻.
associated vectors of the operator 𝐿
Proof. From (35), it is clear that 𝑆𝐺(𝜆) is an inner function
in the upper half-plane, and it is meromorphic in the whole
complex 𝜆-plane. Therefore, it can be factored in the form
𝑆𝐺 (𝜆) = 𝑒𝑖𝜆𝑐 𝐵𝐺 (𝜆) ,
𝑐 = 𝑐 (𝐺) ≥ 0,
(42)
where 𝐵𝐺(𝜆) is a Blaschke product. It follows from (42) that
(40)
̃ ) (𝜆) = 𝑒𝑖𝜆𝑡 𝑓
̃ (𝜆) .
𝑈𝑡 󳨀→ (𝐹− 𝑈𝑡 𝐹−−1 𝑓
−
−
̃ 𝐺 is unitarily equivaThe formulas (40) show that operator 𝐿
lent to the model dissipative operator with the characteristic
function 𝑆𝐺(𝜆). We have thus proved the following theorem.
Theorem 8. The characteristic function of the maximal dissĩ 𝐺 coincides with the function 𝑆𝐺(𝜆) defined
pative operator 𝐿
by (27).
4. The Spectral Properties of Dissipative
Sturm-Liouville Operators
By using characteristic function, the spectral properties of the
̃ 𝐺(𝐿 𝐾 ) can be investigated. The
maximal dissipative operator 𝐿
characteristic function of the maximal dissipative operator
̃ 𝐺 is known to lead to information of completeness about the
𝐿
spectral properties of this operator. For instance, the absence
of a singular factor 𝑠(𝜆) of the characteristic function 𝑆𝐺(𝜆)
in the factorization det 𝑆𝐺(𝜆) = 𝑠(𝜆)𝐵(𝜆) (𝐵(𝜆) is a Blaschke
product) ensures completeness of the system of eigenvectors
̃ 𝐺(𝐿 𝐾 ) in the space
and associated vectors of the operator 𝐿
𝐿 2 (0, ∞) (see [10, 21, 30]). If the characteristic function 𝑆𝐺(𝜆)
has nontrivial singular factor, the system of eigenvectors
̃ 𝐺(𝐿 𝐾 ) can fail to
and associated vectors of the operator 𝐿
be complete. Because 𝑆𝐺(𝜆) is smooth, the support of the
󵄨 󵄨 𝑖𝜆𝑐 󵄨 󵄨
󵄨
󵄨󵄨
−𝑏(𝐺) Im 𝜆
,
󵄨󵄨𝑆𝐺 (𝜆)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝑒 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝐵𝐺 (𝜆)󵄨󵄨󵄨 ≤ 𝑒
Im 𝜆 ≥ 0.
(43)
Further, expressing 𝑛𝐺(𝜆) := 𝑀(𝜆)𝐾(𝜆) in terms of 𝑆𝐺(𝜆), we
find from (35) that
𝑛𝐺 (𝜆) =
𝐺𝑆𝐺 (𝜆) − 𝐺
.
1 − 𝑆𝐺 (𝜆)
(44)
For a given value 𝐺 (Im 𝐺 > 0), if 𝑐(𝐺) > 0, then (43)
implies that lim𝑡 → +∞ 𝑆𝐺(𝑖𝑡) = 0, and then (44) gives us that
lim𝑡 → +∞ 𝑛𝐺(𝑖𝑡) = 𝐺0 . Since 𝑛𝐺(𝜆) does not depend on 𝐺, this
implies that 𝑐(𝐺) can be nonzero at not more than a single
point 𝐺 = 𝐺0 (and further 𝐺0 = −lim𝑡 → +∞ 𝑛𝐺(𝑖𝑡)). This
completes the proof.
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