Volume 2013, Article ID 248740,7pages http://dx.doi.org/10.1155/2013/248740
Research Article
Dissipative Sturm-Liouville Operators with Transmission Conditions
Hüseyin Tuna
1and Aytekin Ery J lmaz
21Department of Mathematics, Mehmet Akif Ersoy University, 15100 Burdur, Turkey
2Department of Mathematics, Nevsehir University, 50300 Nevsehir, Turkey
Correspondence should be addressed to Aytekin Eryılmaz; [email protected] Received 7 December 2012; Accepted 11 February 2013
Academic Editor: Lucas J´odar
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 func- tional 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 point𝑐[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-Foias¸. In 1960s independently from Nagy-Foias¸ [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-Foias¸ and Lax- Phillips, 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 spec- tral representations and defined its characteristic function, because this makes it possible to determine the scatter- ing matrix of dilation according to the Lax and Phillips scheme [23]. Finally, we proved a theorem on complete- ness 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. Self-Adjoint Dilation of Dissipative Sturm-Liouville Operator
Consider the differential expression
𝑙 (𝑦) = −𝑦+ 𝑞 (𝑥) 𝑦, 𝑥 ∈ [𝑎, 𝑐) ∪ (𝑐, 𝑏) , (1) where𝐼1 := [𝑎, 𝑐), 𝐼2 := (𝑐, 𝑏)and𝐼 = 𝐼1∪ 𝐼2,𝑞(𝑥)is real- valued function on𝐼, and𝑞 ∈ 𝐿1loc(𝐼). The points𝑎and𝑐are regular and𝑏is singular for the differential expression𝑙(𝑦).
Moreover𝑞(𝑐±) := lim𝑥 → 𝑐±𝑞(𝑥)one-sided limits exist and are finite.
To pass from the differential expression𝑙(𝑦)to operators, we introduce the Hilbert space𝐻 = 𝐿2(𝐼1) ⊕ 𝐿2(𝐼2)with the inner product
⟨𝑓, 𝑔⟩𝐻:= 𝛾1𝛾2∫𝑐
𝑎𝑓1𝑔1𝑑𝑥 + 𝛿1𝛿2∫𝑏
𝑐 𝑓2𝑔2𝑑𝑥, (2) where
𝑓 (𝑥) = { 𝑓1(𝑥) ,
𝑓2(𝑥) , 𝑥 ∈ 𝐼1 𝑥 ∈ 𝐼2 ∈ 𝐻, 𝑔 (𝑥) = { 𝑔1(𝑥) ,
𝑔2(𝑥) , 𝑥 ∈ 𝐼1 𝑥 ∈ 𝐼2 ∈ 𝐻,
(3)
and𝛾1,𝛾2,𝛿1, and𝛿2are some real numbers with𝛾1𝛾2> 0 and 𝛿1𝛿2> 0.
Let𝐿0 denote the closure of the minimal operator gen- erated by (1) and by 𝐷0 its domain. Besides, we denote the set of all functions 𝑓(𝑥) from 𝐻such that 𝑓, 𝑓 ∈ 𝐴𝐶loc(𝐼), 𝑓(𝑐±), 𝑓(𝑐±)one-sided limits exist and are finite and𝑙(𝑦) ∈ 𝐻; 𝐷is the domain of the maximal operator𝐿.
Furthermore,𝐿 = 𝐿∗0[21].
For two arbitrary functions 𝑦(𝑥),𝑧(𝑥) ∈ 𝐷, we have Green’s formula
∫𝑏
𝑎 𝑙 (𝑦) 𝑧 𝑑𝑥 − ∫𝑏
𝑎 𝑦𝑙 (𝑧) 𝑑𝑥 = 𝛾1𝛾2[𝑦, 𝑧]𝑐−− 𝛾1𝛾2[𝑦, 𝑧]𝑎 + 𝛿1𝛿2[𝑦, 𝑧]𝑏− 𝛿1𝛿2[𝑦, 𝑧]𝑐+,
(4) where[𝑦, 𝑧]𝑥 = 𝑦(𝑥)𝑧(𝑥) − 𝑦(𝑥)𝑧(𝑥) (𝑥 ∈ 𝐼), [𝑦, 𝑧]𝑏 = lim𝑥 → 𝑏[𝑦, 𝑧]𝑥exists and is finite.
Suppose that Weyl’s limit circle case holds for the dif- ferential 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
𝜃 (𝑥) = {𝜃1(𝑥) , 𝑥 ∈ 𝐼1
𝜃2(𝑥) , 𝑥 ∈ 𝐼2, 𝜑 (𝑥) = {𝜑1(𝑥) , 𝑥 ∈ 𝐼1 𝜑2(𝑥) , 𝑥 ∈ 𝐼2,
(5) the solutions of the equation𝑙(𝑦) = 𝜆𝑦, 𝑥 ∈ 𝐼, satisfying the initial conditions
𝜃1(𝑎, 𝜆) =cos𝛼, 𝜃1(𝑎, 𝜆) =sin𝛼,
𝜑1(𝑎, 𝜆) = −sin𝛼, 𝜑1(𝑎, 𝜆) =cos𝛼 (6)
and transmission conditions 𝜃2(𝑐+, 𝜆) = 𝛾1
𝛿1𝜃1(𝑐−, 𝜆) , 𝜃2(𝑐+, 𝜆) = 𝛾2
𝛿2𝜃1(𝑐−, 𝜆) , 𝜑2(𝑐+, 𝜆) = 𝛾1
𝛿1𝜑1(𝑐−, 𝜆) , 𝜑2(𝑐+, 𝜆) = 𝛾2
𝛿2𝜑1(𝑐−, 𝜆) , (7) where𝛼, 𝛾1, 𝛾2, 𝛿1, and𝛿2 ∈ Rwith𝛾1𝛾2 > 0and𝛿1𝛿2 >
0.The solutions𝜃(𝑥, 𝜆)and𝜑(𝑥, 𝜆)belong to𝐻. Let 𝑢 (𝑥) = {𝑢1(𝑥) = 𝜃1(𝑥, 0) , 𝑥 ∈ 𝐼1
𝑢2(𝑥) = 𝜃2(𝑥, 0) , 𝑥 ∈ 𝐼2, V(𝑥) = {V1(𝑥) = 𝜑1(𝑥, 0) , 𝑥 ∈ 𝐼1, V2(𝑥) = 𝜑2(𝑥, 0) , 𝑥 ∈ 𝐼2,
(8)
𝑢(𝑥), andV(𝑥)be solutions of the equation𝑙(𝑦) = 0, 𝑥 ∈ 𝐼, satisfying the initial conditions
𝑢1(𝑎) =cos𝛼, 𝑢1(𝑎) =sin𝛼,
V1(𝑎) = −sin𝛼, V1(𝑎) =cos𝛼 (9) and transmission conditions
𝑢2(𝑐+) = 𝛾1
𝛿1𝑢1(𝑐−) , 𝑢2(𝑐+) = 𝛾2
𝛿2𝑢1(𝑐−) , V2(𝑐+) = 𝛾1
𝛿1V1(𝑐−) , V2(𝑐+) = 𝛾2 𝛿2V1(𝑐−) .
(10)
All the maximal dissipative extensionsL𝐺of the operator 𝐿0are described by the following conditions (see [4,15,18]):
𝑅1(𝑦) : 𝛾1𝑦 (𝑐−) − 𝛿1𝑦 (𝑐+) = 0, 𝑅2(𝑦) : 𝛾2𝑦(𝑐−) − 𝛿2𝑦(𝑐+) = 0, 𝑅3(𝑦) : 𝑦 (𝑎)cos 𝛼 + 𝑦(𝑎)sin𝛼 = 0, 𝑅4(𝑦) : [𝑦,V]𝑏− 𝐺[𝑦, 𝑢]𝑏= 0, Im𝐺 > 0.
(11)
Let us add the “incoming” and “outgoing” subspaces𝐷−= 𝐿2(−∞, 0)and𝐷+ = 𝐿2(0, ∞)to𝐻. The orthogonal sum𝐻 = 𝐷−⊕ 𝐻 ⊕ 𝐷+is calledmain Hilbert space of the dilation.
In the spaceH, we consider the operatorL𝐺on the set 𝐷(L𝐺), its elements consisting of vectors𝑤 = ⟨𝜑−, ̂𝑦, 𝜑+⟩, generated by the expression
L𝐺⟨𝜑−, ̂𝑦, 𝜑+⟩ = ⟨𝑖𝑑𝜑−
𝑑𝜉 , 𝑙 ( ̂𝑦) , 𝑖𝑑𝜑+
𝑑𝜉 ⟩ (12)
satisfying the conditions𝜑− ∈ 𝑊21(−∞, 0),𝜑+ ∈ 𝑊21(0, ∞),
̂𝑦 ∈ 𝐻,𝑅1( ̂𝑦) = 0, 𝑅2( ̂𝑦) = 0, 𝑅3( ̂𝑦) = 0,[ ̂𝑦,V]𝑏− 𝐺[ ̂𝑦, 𝑢]𝑏= (𝐶/√𝛿1𝛿2)𝜑−(0), [ ̂𝑦,V]𝑏−𝐺[ ̂𝑦, 𝑢]𝑏= (𝐶/√𝛿1𝛿2)𝜑+(0), where 𝑊21are Sobolev spaces and𝐶2:= 2Im𝐺,𝐶 > 0.
Theorem 1. The operatorL𝐺is self-adjoint inHand it is a self-adjoint dilation of the operator̃𝐿𝐺(= 𝐿𝐾).
Proof. We first prove thatL𝐺is symmetric inH. Namely, (L𝐺𝑓, 𝑔)H − (𝑓,L𝐺𝑔)H = 0. Let 𝑓, 𝑔 ∈ 𝐷(L𝐺), 𝑓 =
⟨𝜑−, ̂𝑦, 𝜑+⟩and𝑔 = ⟨𝜓−, ̂𝑧, 𝜓+⟩. Then we have (L𝐺𝑓, 𝑔)H− (𝑓,L𝐺𝑔)H
= (L𝐺⟨𝜑−, ̂𝑦, 𝜑+⟩ , ⟨𝜓−, ̂𝑧, 𝜓+⟩)
− (⟨𝜑−, ̂𝑦, 𝜑+⟩ ,L𝐺⟨𝜓−, ̂𝑧, 𝜓+⟩)
= ∫0
−∞𝑖𝜑−𝜓−𝑑𝜉 + (𝑙 ( ̂𝑦) , ̂𝑧)𝐻+ ∫∞
0 𝑖𝜑+𝜓+𝑑𝜉
− ∫0
−∞𝑖𝜓−𝜑−𝑑𝜉 − ( ̂𝑦, 𝑙 (̂𝑧))𝐻− ∫∞
0 𝑖𝜓+𝜑+𝑑𝜉
= ∫0
−∞𝑖𝜑−𝜓−𝑑𝜉 + 𝛿1𝛿2[ ̂𝑦, ̂𝑧]𝑏+ ∫∞
0 𝑖𝜑+𝜓+𝑑𝜉
− ∫0
−∞𝑖𝜓−𝜑−𝑑𝜉 − ∫∞
0 𝑖𝜓+𝜑+𝑑𝜉
= 𝑖𝜓−(0) 𝜑−(0) − 𝑖𝜑+(0) 𝜓+(0) + 𝛿1𝛿2[ ̂𝑦, ̂𝑧]𝑏. (13)
We obtain by direct computation
𝑖𝜓−(0) 𝜑−(0) − 𝑖𝜑+(0) 𝜓+(0) + 𝛿1𝛿2[ ̂𝑦, ̂𝑧]𝑏= 0. (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−, ̂𝑧, 𝜓+⟩ ∈ 𝐷(L∗𝐺). LetL∗𝐺𝑔 = 𝑔∗ = ⟨𝜓−∗, ̂𝑧∗, 𝜓+∗⟩ ∈ H, so that
(L𝐺𝑓, 𝑔)H = (𝑓,L∗𝐺𝑔)H = (𝑓, 𝑔∗)H. (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
[ ̂𝑦,V]𝑏+ 𝐺[ ̂𝑦, 𝑢]𝑏= 𝐶
√𝛿1𝛿2𝜑−(0) , [ ̂𝑦,V]𝑏+ 𝐺[ ̂𝑦, 𝑢]𝑏= 𝐶
√𝛿1𝛿2𝜑+(0) .
(16)
Hence,𝐷(L∗𝐺) ⊆ 𝐷(L𝐺); that is,L𝐺=L∗𝐺.
The self-adjoint operatorL𝐺generates onHa unitary group𝑈𝑡 = exp(𝑖L𝐺𝑡)(𝑡 ∈ R+ = (0, ∞)). Let us denote by𝑃 : H → 𝐻and𝑃1 : 𝐻 → Hthe mapping acting according to the formulae𝑃 : ⟨𝜑−, ̂𝑦, 𝜑+⟩ → ̂𝑦and 𝑃1 :
̂𝑦 → ⟨0, ̂𝑦, 0⟩. Let𝑍𝑡 := 𝑃𝑈𝑡𝑃1, 𝑡 ≥ 0, by using𝑈𝑡. The family{𝑍𝑡} (𝑡 ≥ 0) of operators is a strongly continuous semigroup of completely nonunitary contraction on𝐻. Let us denote by 𝐵𝐺the generator of this semigroup: 𝐵𝐺̂𝑦 = lim𝑡 → +0(𝑖𝑡)−1(𝑍𝑡̂𝑦 − ̂𝑦). The domain of 𝐵𝐺 consists of all the vectors for which the limit exists. The operator 𝐵𝐺 is dissipative. The operatorL𝐺is called the self-adjoint dilation of𝐵𝐺(see [10,21,30]). We show that𝐵𝐺 = ̃𝐿𝐺; henceL𝐺
is self-adjoint dilation of𝐵𝐺. To show this, it is sufficient to verify the equality
𝑃(L𝐺− 𝜆𝐼)−1𝑃1̂𝑦 = (̃𝐿𝐺− 𝜆𝐼)−1̂𝑦, ̂𝑦 ∈ 𝐻, Imℎ < 0.
(17) For this purpose, we set(L𝐺− 𝜆𝐼)−1𝑃1̂𝑦 = 𝑔 = ⟨𝜓−, ̂𝑧, 𝜓+⟩ which implies that(L𝐺− 𝜆𝐼)𝑔 = 𝑃1̂𝑦, and hencẽ𝑙(̂𝑧) − 𝜆̂𝑧 =
̂𝑦, 𝜓−(𝜉) = 𝜓−(0)𝑒−𝑖𝜆𝜉 and𝜓+(𝜉) = 𝜓+(0)𝑒−𝑖𝜆𝜉. Since 𝑔 ∈ 𝐷(L𝐺), then𝜓− ∈ 𝑊21(−∞, 0); it follows that𝜓−(0) = 0, and consequently𝑧satisfies the boundary condition[ ̂𝑦,V]𝑏− 𝐺[ ̂𝑦, 𝑢]𝑏 = 0. Thereforê𝑧 ∈ 𝐷(̃𝐿𝐺), and since point𝜆with Im𝜆 < 0 cannot be an eigenvalue of dissipative operator, then̂𝑧 = (̃𝐿𝐺− 𝜆𝐼)−1̂𝑦. Thus we have
(L𝐺− 𝜆𝐼)−1𝑃1̂𝑦
= ⟨0, (̃𝐿𝐺− 𝜆𝐼)−1̂𝑦, √𝛿1𝛿2
𝐶 ([ ̂𝑦,V]𝑏+ 𝐺[ ̂𝑦, 𝑢]𝑏) 𝑒−𝑖𝜆𝜉⟩ (18) for ̂𝑦and Im𝜆 < 0. By applying onto the mapping𝑃, we obtain (17), and
(̃𝐿𝐺− 𝜆𝐼)−1= 𝑃(L𝐺− 𝜆𝐼)−1𝑃1= −𝑖𝑃 ∫∞
0 𝑈𝑡𝑒−𝑖𝜆𝑡𝑑𝑡𝑃1
= − 𝑖 ∫∞
0 𝑍𝑡𝑒−𝑖𝜆𝑡𝑑𝑡 = (𝐵𝐺− 𝜆𝐼)−1, Im𝜆 < 0, (19) so this clearly shows that̃𝐿𝐺= 𝐵𝐺.
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)𝑈𝑡𝐷− ⊂ 𝐷−, 𝑡 ≤ 0and𝑈𝑡𝐷+⊂ 𝐷+, 𝑡 ≥ 0, (2)∩𝑡≤0𝑈𝑡𝐷−= ∩𝑡≥0𝑈𝑡𝐷+= {0},
(3)∪𝑡≥0𝑈𝑡𝐷−= ∪𝑡≤0𝑈𝑡𝐷+=H, (4)𝐷−⊥ 𝐷+.
Property (4) is clear. To be able to prove property (1) for 𝐷+ (the proof for𝐷−is similar), we setR𝜆 = (L𝐺− 𝜆𝐼)−1. For all𝜆, with Im𝜆 < 0and for any𝑓 = ⟨0, 0, 𝜑+⟩ ∈ 𝐷+, we have
R𝜆𝑓 = ⟨0, 0, −𝑖𝑒−𝑖𝜆𝜉∫𝜉
0 𝑒𝑖𝜆𝑠𝜑+(𝑠) 𝑑𝑠⟩ , (20) asR𝜆𝑓 ∈ 𝐷+. Therefore, if𝑔 ⊥ 𝐷+, then
0 = (R𝜆𝑓, 𝑔)H= −𝑖 ∫∞
0 𝑒−𝑖𝜆𝑡(𝑈𝑡𝑓, 𝑔)H𝑑𝑡, Im𝜆 < 0, (21)
which implies that(𝑈𝑡𝑓, 𝑔)H = 0for all𝑡 ≥ 0. Hence, for 𝑡 ≥ 0, 𝑈𝑡𝐷+⊂ 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 follows:𝑃+ : ⟨𝜑−, ̂𝑦, 𝜑+⟩ → 𝜑+ and𝑃1+ : 𝜑 → ⟨0, 0, 𝜑⟩, respectively. We take into consider that the semigroup of isometries𝑈𝑡+ := 𝑃+𝑈𝑡𝑃1+ (𝑡 ≥ 0) is a one-sided shift in 𝐿2(0, ∞). Indeed, the generator of the semigroup of the one- sided shift𝑉𝑡in𝐿2(0, ∞)is the differential operator𝑖(𝑑/𝑑𝜉) with the boundary condition𝜑(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 that𝑈𝑡+= 𝑉𝑡, and, hence,
⋂
𝑡≥0𝑈𝑡𝐷+= ⟨0, 0, ⋂
𝑡≤0𝑉𝑡𝐿2(0, ∞)⟩ = {0} , (22) so the proof is completed.
Definition 2. The linear operator𝐴with domain𝐷(𝐴)acting in the Hilbert space𝐻is calledcompletely non-self-adjoint(or simple) if there is no invariant subspace𝑀 ⊆ 𝐷(𝐴)(𝑀 ̸= {0}) of the operator𝐴on which the restriction𝐴to𝑀is self- adjoint.
To prove property (3) of the incoming and outgoing subspaces, let us prove following lemma.
Lemma 3. The operator ̃𝐿𝐺 is completely non-self-adjoint (simple).
Proof. Let𝐻 ⊂ 𝐻be a nontrivial subspace in which ̃𝐿𝐺 induces a self-adjoint operator ̃𝐿𝐺with domain 𝐷(̃𝐿𝐺) = 𝐻∩ 𝐷(̃𝐿𝐺). If𝑓 ∈ 𝐷(̃𝐿̂ 𝐺), then𝑓 ∈ 𝐷(̃𝐿̂ ∗𝐺)and
0 = 𝑑
𝑑𝑡𝑒𝑖̃𝐿𝐺𝑡𝑓̂2𝐻= 𝑑
𝑑𝑡(𝑒𝑖̃𝐿𝐺𝑡𝑓, 𝑒̂ 𝑖̃𝐿𝐺𝑡𝑓)̂
𝐻
= − 𝐶2([𝑒𝑖̃𝐿𝐺𝑡𝑓, 𝑢]̂
𝑏)2
𝐻.
(23)
Consequently, we have[ ̂𝑦, 𝑢]𝑏 = 0.Using this result with boundary condition[ ̂𝑦,V]𝑏+ 𝐺[ ̂𝑦, 𝑢]𝑏= 0,we have[ ̂𝑦, 𝑢]𝑏= 0; that is, ̂𝑦(𝜆) = 0. Since all solutions of𝑙(𝑦) = 𝜆𝑦belong to𝐿2(0, ∞), from this it can be concluded that the resolvent 𝑅𝜆(̃𝐿𝐺) is a compact operator, and the spectrum of ̃𝐿𝐺 is purely discrete. Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator̃𝐿𝐺we obtain 𝐻 = {0}. Hence the operator ̃𝐿𝐺 is simple. The proof is completed.
Let us define𝐻− = ∪𝑡≥0𝑈𝑡𝐷−, 𝐻+ = ∪𝑡≤0𝑈𝑡𝐷+. Lemma 4. The equality𝐻−+ 𝐻+=Hholds.
Proof. Considering property (1) of the subspace𝐷+, it is easy to show that the subspaceH=H⊝ (𝐻−+ 𝐻+)is invariant
relative to the group{𝑈𝑡}and has the formH = ⟨0, 𝐻, 0⟩, where 𝐻 is a subspace in 𝐻. Therefore, if the subspace H(and hence also𝐻)was nontrivial, then the unitary group {𝑈𝑡}restricted to this subspace would be a unitary part of the group{𝑈𝑡}, and hence, the restrictioñ𝐿𝐺of̃𝐿𝐺to𝐻would be a self-adjoint operator in𝐻. Since the operator̃𝐿𝐺is simple, it follows that𝐻= {0}. The lemma is proved.
Assume that
𝜑 (𝑥, 𝜆) := {𝜑1(𝑥, 𝜆) , 𝑥 ∈ 𝐼1 𝜑2(𝑥, 𝜆) , 𝑥 ∈ 𝐼2, 𝜓 (𝑥, 𝜆) := {𝜓1(𝑥, 𝜆) , 𝑥 ∈ 𝐼1 𝜓2(𝑥, 𝜆) , 𝑥 ∈ 𝐼2
(24)
are solutions of𝑙(𝑦) = 𝜆𝑦satisfying the conditions 𝜑1(𝑎, 𝜆) =cos𝛼, 𝜑1(𝑎, 𝜆) =sin𝛼, 𝜓1(𝑎, 𝜆) = −sin𝛼, 𝜓1(𝑎, 𝜆) =cos𝛼, 𝜑2(𝑐+, 𝜆) = 𝛾1
𝛿1𝜑1(𝑐−, 𝜆) , 𝜑2(𝑐+, 𝜆) = 𝛾2
𝛿2𝜑1(𝑐−, 𝜆) . (25) Let us adopt the following notations:
𝐾 (𝜆) = [𝜑,V]𝑏
[𝜓, 𝑢]𝑏, 𝑀 (𝜆) = −[𝜓, 𝑢]𝑏
[𝜑, 𝑢]𝑏, (26) 𝑆𝐺(𝜆) = (𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) (𝑀 (𝜆) 𝐾 (𝜆) + 𝐺)−1, (27) where𝑀(𝜆)is a meromorphic function on the complex plane Cwith a countable number of poles on the real axis. Further, it is possible to show that the function𝑀(𝜆)possesses the following properties: Im𝑀(𝜆) ≤ 0 for all Im𝜆 ̸= 0, and 𝑀(𝜆) = 𝑀(𝜆)for all𝜆 ∈C, except the real poles𝑀(𝜆).
We set 𝑈𝜆−(𝑥, 𝜉, 𝜁)
= ⟨𝑒−𝑖𝜆𝜉, 𝐶
√𝛿1𝛿2𝑀 (𝜆) [(𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) [𝜓, 𝑢]𝑏]−1𝜑 (𝑥, 𝜆) , 𝑆𝐺(𝜆) 𝑒−𝑖𝜆𝜁⟩ .
(28) We note that the vectors𝑈𝜆−(𝑥, 𝜉, 𝜁)for real𝜆do not belong to the space H. However,𝑈𝜆−(𝑥, 𝜉, 𝜁)satisfies the equation L𝑈 = 𝜆𝑈and the corresponding boundary conditions for the operatorLℎ.
By means of vector𝑈𝜆−(𝑥, 𝜉, 𝜁), we define the transforma- tion𝐹− : 𝑓 → ̃𝑓−(𝜆)by
(𝐹−𝑓) (𝜆) := ̃𝑓−(𝜆) := 1
√2𝜋(𝑓, 𝑈𝜆)H (29)
on the vectors𝑓 = ⟨𝜑−, ̂𝑦, 𝜑+⟩in which𝜑−(𝜉), 𝜑+(𝜁), 𝑦(𝑥) are smooth, compactly supported functions.
Lemma 5. The transformation𝐹−isometrically maps𝐻−onto 𝐿2(R). For all vectors𝑓, 𝑔 ∈ 𝐻−the Parseval equality and the inversion formulae hold:
(𝑓, 𝑔)H = (̃𝑓−, ̃𝑔−)𝐿2= ∫∞
−∞
𝑓̃−(𝜆) ̃𝑔−(𝜆) 𝑑𝜆,
𝑓 = 1
√2𝜋∫∞
−∞
𝑓̃−(𝜆) 𝑈𝜆𝑑𝜆,
(30)
where𝑓̃−(𝜆) = (𝐹−𝑓)(𝜆)and𝑔̃−(𝜆) = (𝐹−𝑔)(𝜆).
Proof. For𝑓, 𝑔 ∈ 𝐷−, 𝑓 = ⟨𝜑−, 0, 0⟩, 𝑔 = ⟨𝜓−, 0, 0⟩, with Paley-Wiener theorem, we have
𝑓̃−(𝜆) = 1
√2𝜋(𝑓, 𝑈𝜆)H= 1
√2𝜋∫0
−∞𝜑−(𝜉) 𝑒−𝑖𝜆𝜉𝑑𝜉 ∈ 𝐻−2, (31) and by using usual Parseval equality for Fourier integrals,
(𝑓, 𝑔)H= ∫∞
−∞𝜑−(𝜉) 𝜓−(𝜉)𝑑𝜉
= ∫∞
−∞
𝑓̃−(𝜆) ̃𝑔−(𝜆)𝑑𝜆 = (𝐹−𝑓, 𝐹−𝑔)𝐿2.
(32)
Here,𝐻±2denote the Hardy classes in𝐿2(R)consisting of the functions analytically extendible to the upper and lower half- planes, respectively.
We now extend the Parseval equality to the whole of𝐻−. We consider in𝐻−the dense set of𝐻−of the vectors obtained as follows from the smooth, compactly supported functions in𝐷−: 𝑓 ∈ 𝐻−if𝑓 = 𝑈𝑇𝑓0,𝑓0= ⟨𝜑−, 0, 0⟩,𝜑− ∈ 𝐶0∞(−∞, 0), where𝑇 = 𝑇𝑓is a nonnegative number depending on𝑓. If 𝑓, 𝑔 ∈ 𝐻−, then for𝑇 > 𝑇𝑓and𝑇 > 𝑇𝑔we have𝑈−𝑇𝑓, 𝑈−𝑇𝑔 ∈ 𝐷−; moreover, the first components of these vectors belong to𝐶∞0 (−∞, 0).Therefore, since the operators𝑈𝑡(𝑡 ∈ R)are unitary, by the equality
𝐹−𝑈𝑡𝑓 = (𝑈𝑡𝑓, 𝑈𝜆−)H= 𝑒𝑖𝜆𝑡(𝑓, 𝑈𝜆−)H= 𝑒𝑖𝜆𝑡𝐹−𝑓, (33) we have
(𝑓, 𝑔)H= (𝑈−𝑇𝑓, 𝑈−𝑇𝑔)H= (𝐹−𝑈−𝑇𝑓, 𝐹−𝑈−𝑇𝑔)𝐿2, (34) (𝑒𝑖𝜆𝑇𝐹−𝑓, 𝑒𝑖𝜆𝑇𝐹−𝑔)𝐿2 = ( ̃𝑓, ̃𝑔)𝐿2. (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. Finally𝐹−𝐻− =
∪𝑡≥0𝐹−𝑈𝑡𝐷− = ∪𝑡≥0𝑒𝑖𝜆𝑡𝐻−2= 𝐿2(R); that is,𝐹−maps𝐻−onto the whole of𝐿2(R). The lemma is proved.
We set 𝑈𝜆+(𝑥, 𝜉, 𝜁)
= ⟨𝑆𝐺(𝜆) 𝑒−𝑖𝜆𝜉, 𝐶
√𝛿1𝛿2𝑀 (𝜆) [(𝑀 (𝜆) 𝐾 (𝜆) + 𝐺) [𝜓, 𝑢]𝑏]−1𝜑 (𝑥, 𝜆) , 𝑒−𝑖𝜆𝜁⟩ .
(36) We note that the vectors𝑈𝜆+(𝑥, 𝜉, 𝜁)for real𝜆do not belong to the space H. However,𝑈𝜆+(𝑥, 𝜉, 𝜁)satisfies the equation L𝑈 = 𝜆𝑈and the corresponding boundary conditions for the operatorLℎ. With the help of vector𝑈𝜆+(𝑥, 𝜉, 𝜁), we define the transformation𝐹+: 𝑓 → ̃𝑓+(𝜆)by(𝐹+𝑓)(𝜆) := ̃𝑓+(𝜆) :=
(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 transformation𝐹+isometrically maps𝐻+onto 𝐿2(R). For all vectors𝑓, 𝑔 ∈ 𝐻+ the Parseval equality and the inversion formula hold:
(𝑓, 𝑔)H= (̃𝑓+, ̃𝑔+)𝐿2 = ∫∞
−∞
𝑓̃+(𝜆) ̃𝑔+(𝜆)𝑑𝜆,
𝑓 = 1
√2𝜋∫∞
−∞
𝑓̃+(𝜆) 𝑈𝜆+𝑑𝜆,
(37)
where𝑓̃+(𝜆) = (𝐹+𝑓)(𝜆) and𝑔̃+(𝜆) = (𝐹+𝑔)(𝜆).
Proof. The proof is analogous to Lemma6.
It is obvious that the matrix-valued function 𝑆𝐺(𝜆) is meromorphic inCand all poles are in the lower half-plane.
From (27),|𝑆𝐺(𝜆)| ≤ 1for Im𝜆 > 0, and𝑆𝐺(𝜆)is the unitary matrix for all𝜆 ∈ R. Therefore, it explicitly follows from the formulae for the vectors𝑈𝜆−and𝑈𝜆+that
𝑈𝜆+= 𝑆𝐺(𝜆) 𝑈𝜆−. (38) It follows from Lemmas6and5that𝐻−= 𝐻+. Together with Lemma5, this shows that𝐻− = 𝐻+=H; therefore property (3) has been proved for the incoming and outgoing subspaces.
Thus, the transformation𝐹−isometrically maps𝐻−onto 𝐿2(R) with the subspace 𝐷− mapped onto 𝐻−2 and the operators𝑈𝑡are transformed into the operators of multipli- cation by𝑒𝑖𝜆𝑡. This means that𝐹− 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 the𝐹− representation of an element 𝑓 ∈ Hto its𝐹+ representation is accomplished as𝑓̃+(𝜆) = 𝑆−1𝐺(𝜆)̃𝑓−(𝜆). Consequently, according to [22], we have proved the following.
Theorem 7. The function𝑆𝐺(𝜆)is the scattering matrix of the group{𝑈𝑡}(of the self-adjoint operatorL𝐺).
Let𝑆(𝜆)be an arbitrary nonconstant inner function (see [19]) on the upper half-plane (the analytic function𝑆(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 space𝐻+2. We consider the semigroup of operators𝑍𝑡(𝑡 ≥ 0)acting in𝐾according to the formula 𝑍𝑡𝜑 = 𝑃[𝑒𝑖𝜆𝑡𝜑],𝜑 = 𝜑(𝜆) ∈ 𝐾, where𝑃is the orthogonal projection from𝐻+2onto𝐾. The generator of the semigroup {𝑍𝑡}is denoted by
𝑇𝜑 = lim
𝑡 → +0(𝑖𝑡)−1(𝑍𝑡𝜑 − 𝜑) , (39) where𝑇is a maximal dissipative operator acting in𝐾 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 Foias¸ [22]. The basic assertion is that 𝑆(𝜆) is the characteristic function of the operator𝑇.
Let 𝐾 = ⟨0, 𝐻, 0⟩, so thatH= 𝐷−⊕ 𝐾 ⊕ 𝐷+. It follows from the explicit form of the unitary transformation𝐹−under the mapping𝐹−that
H→ 𝐿2(R) , 𝑓 → ̃𝑓−(𝜆) = (𝐹−𝑓) (𝜆) , 𝐷− → 𝐻−2, 𝐷+→ 𝑆𝐺𝐻+2,
𝐾 → 𝐻+2⊝ 𝑆𝐺𝐻+2,
𝑈𝑡→ (𝐹−𝑈𝑡𝐹−−1𝑓̃−) (𝜆) = 𝑒𝑖𝜆𝑡𝑓̃−(𝜆) .
(40)
The formulas (40) show that operator̃𝐿𝐺is unitarily equiva- lent to the model dissipative operator with the characteristic function𝑆𝐺(𝜆). We have thus proved the following theorem.
Theorem 8. The characteristic function of the maximal dissi- pative operator̃𝐿𝐺coincides with the function𝑆𝐺(𝜆)defined by(27).
4. The Spectral Properties of Dissipative Sturm-Liouville Operators
By using characteristic function, the spectral properties of the maximal dissipative operator̃𝐿𝐺(𝐿𝐾)can be investigated. The characteristic function of the maximal dissipative operator
̃𝐿𝐺is known to lead to information of completeness about the spectral properties of this operator. For instance, the absence of a singular factor𝑠(𝜆)of the characteristic function𝑆𝐺(𝜆) in the factorization det𝑆𝐺(𝜆) = 𝑠(𝜆)𝐵(𝜆)(𝐵(𝜆)is a Blaschke product) ensures completeness of the system of eigenvectors and associated vectors of the operator̃𝐿𝐺(𝐿𝐾)in the space 𝐿2(0, ∞)(see [10,21,30]). If the characteristic function𝑆𝐺(𝜆) has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator ̃𝐿𝐺(𝐿𝐾)can fail to be complete. Because𝑆𝐺(𝜆)is smooth, the support of the
corresponding singular measure𝜇must be contained in the set of poles𝑆𝐺(𝜆). But in this case the singular measure𝜇is a simple step function. If we require𝑆𝐺(𝜆)to have no zeros of infinite multiplicity, then𝜇 = 0. So the singular factor vanishes. The characteristic function𝑆𝐺(𝜆)of the maximal dissipative operator̃𝐿𝐺has the form
𝑆𝐺(𝜆) := 𝑀 (𝜆) 𝐾 (𝜆) + 𝐺
𝑀 (𝜆) 𝐾 (𝜆) + 𝐺, (41) where Im𝐺 > 0.
Theorem 9. For all the values of𝐺withIm𝐺 > 0, except possibly for a single value𝐺 = 𝐺0, the characteristic function 𝑆𝐺(𝜆)of the maximal dissipative operator ̃𝐿𝐺 is a Blaschke product. The spectrum of ̃𝐿𝐺 is purely discrete and belongs to the open upper half-plane. The operator̃𝐿𝐺(𝐺 ̸= 𝐺0)has a countable number of isolated eigenvalues with finite multiplic- ity and limit points at infinity. The system of all eigenvectors and associated vectors of the operator̃𝐿𝐺is complete in the space𝐻.
Proof. From (35), it is clear that𝑆𝐺(𝜆)is an inner function in the upper half-plane, and it is meromorphic in the whole complex𝜆-plane. Therefore, it can be factored in the form
𝑆𝐺(𝜆) = 𝑒𝑖𝜆𝑐𝐵𝐺(𝜆) , 𝑐 = 𝑐 (𝐺) ≥ 0, (42) where𝐵𝐺(𝜆)is a Blaschke product. It follows from (42) that
𝑆𝐺(𝜆) =𝑒𝑖𝜆𝑐𝐵𝐺(𝜆) ≤ 𝑒−𝑏(𝐺)Im𝜆, Im𝜆 ≥ 0. (43) Further, expressing𝑛𝐺(𝜆) := 𝑀(𝜆)𝐾(𝜆)in terms of𝑆𝐺(𝜆), we find from (35) that
𝑛𝐺(𝜆) =𝐺𝑆𝐺(𝜆) − 𝐺
1 − 𝑆𝐺(𝜆) . (44)
For a given value 𝐺 (Im𝐺 > 0), if𝑐(𝐺) > 0, then (43) implies that lim𝑡 → +∞𝑆𝐺(𝑖𝑡) = 0, and then (44) gives us that lim𝑡 → +∞𝑛𝐺(𝑖𝑡) = 𝐺0. Since𝑛𝐺(𝜆)does not depend on𝐺, this implies that𝑐(𝐺)can be nonzero at not more than a single point𝐺 = 𝐺0 (and further𝐺0 = −lim𝑡 → +∞𝑛𝐺(𝑖𝑡)). This completes the proof.
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