Spectral Analysis of q-Sturm-Liouville Problem with the Spectral Parameter in the Boundary Condition

This paper is concerned with q-Sturm-Liouville boundary value problem in the Hilbert space with a spectral parameter in the boundary condition. We construct a self-adjoint dilation of the maximal dissipative q-difference operator and its incoming and outcoming spectral representations, which make it possible to determine the scattering matrix of the dilation. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of operator generated by boundary value problem.


Introduction
Spectral analysis of Sturm-Liouville and Schr ödinger differential equations with a spectral parameter in the boundary conditions has been analyzed intensively see 1-16 .Then spectral analysis of discrete equations became an interesting subject in this field.So there is a substantial literature on this subject see 10, 17-19 .There has recently been great interest in quantum calculus and many works have been devoted to some problems of q-difference equation.In particular, we refer the reader to consult the reference 20 for some definitions and theorems on q-derivative, q-integration, q-exponential function, q-trigonometric function, q-Taylor formula, q-Beta and Gamma functions, Euler-Maclaurin formula, anf so forth.In 21 , Adıvar and Bohner investigated the eigenvalues and the spectral singularities of non-selfa-djoint q-difference equations of second order with spectral singularities.In 12 , Huseynov and Bairamov examined the properties of eigenvalues and eigenvectors of a quadratic pencil of q-difference equations.In 22 , Agarwal examined spectral analysis of self-adjoint equations.In 23 , Shi and Wu presented several classes of explicit self-adjoint Sturm-Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a nondefinite weight function, or a non-self-adjoint boundary condition.In 24 , Annaby and Mansour studied a q-analogue of Sturm-Liouville eigenvalue problems and formulated a self-adjoint q-difference operator in a Hilbert space.They also discussed properties of the eigenvalues and the eigenfunctions.
In this paper, we consider q-Sturm-Liouville Problem and define an adequate Hilbert space.Our main target of the present paper is to study q-Sturm-Liouville boundary value problem in case of dissipation at the right endpoint of 0, a and with the spectral parameter at zero.The maximal dissipative q-Sturm-Liouville operator is constructed using 25, 26 and Lax-Phillips scattering theory in 27 .Then we constructed a functional model of dissipative operator by means of the incoming and outcoming spectral representations and defined its characteristic function in terms of the solutions of the corresponding q-Sturm-Liouville equation.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.Finally, we give theorems on completeness of the system of eigenvectors and associated vectors of the dissipative q-difference operator.
Let q be a positive number with 0 < q < 1, A ⊂ R, and a ∈ C. A q-difference equation is an equation that contains q-derivatives of a function defined on A. Let y x be a complexvalued function on x ∈ A. The q-difference operator D q is defined by where μ x q − 1 x.The q-derivative at zero is defined by if the limit exists and does not depend on x.A right inverse to Dq, the Jackson q-integration, is given by provided that the series converges, and Let L 2 q 0, a be the space of all complex-valued functions defined on 0, a such that f : The space L 2 q 0, a is a separable Hilbert space with the inner product f, g : We will consider the basic Sturm-Liouville equation where v x is defined on 0, a and continuous at zero.The q-Wronskian of y 1 x , y 2 x is defined to be Let L 0 denote the closure of the minimal operator generated by 1.7 and by D 0 its domain.Besides, we denote by D the set of all functions y x from L 2 q 0, a such that y x and D q y x are continuous in 0, a and l y ∈ L 2 q 0, a ; D is the domain of the maximal operator L. Furthermore, L L * 0 2, 4, 13 .Suppose that the operator L 0 has defect index 2, 2 .For every y, z ∈ D we have q-Lagrange's identity 24 Ly, z − y, Lz y, z a − y, z 0 , 1.9 where y, z : y x D q −1 z x − D q −1 y x z x .

Construction of the Dissipative Operator
Consider boundary value problem governed by ly λy, y ∈ D, 2.1 subject to the boundary conditions where λ is spectral parameter and α 1 , α 2 , α 1 , α 2 ∈ R and α is defined by For convenience we assume

2.5
Lemma 2.1.For arbitrary y, z ∈ D, let one suppose that R 0 z R 0 z , R 0 z R 0 z , then one has the following.Proof.

2.7
Let θ 1 , θ 2 denote the solutions of 2.1 satisfying the conditions Then from 2.3 we have We let

2.11
Thus R λ is a Hilbert Schmidt operator on space L 2 q 0, a .The spectrum of the boundary value problem coincides with the roots of the equation Δ λ 0. Since Δ is analytic and not identical to zero, it means that the function Δ has at most a countable number of isolated zeros with finite multiplicity and possible limit points at infinity.Suppose that f 1 ∈ L 2 0, a , f 2 ∈ C, then we denote linear space H L 2 q 0, a ⊕ C with two component of elements of 12 defines an inner product in Hilbert space H. Let us define operator of A h : H → H with equalities suitable for boundary value problem

2.13
Remind that a linear operator A h with domain D A h in Hilbert space H is called dissipative if Im A h f, f ≥ 0 for all f ∈ D A h and maximal dissipative if it does not have a proper extension.
Definition 2.2.If the system of vectors of y 0 , y 1 , y 2 , . . ., y n corresponding to the eigenvalue λ 0 is 2.14 then the system of vectors of y 0 , y 1 , y 2 , . . ., y n corresponding to the eigenvalue λ 0 is called a chain of eigenvectors and associated vectors of boundary value problem 2.2 -2.12 .
Since the operator A h is dissipative in H and from Definition 2.2, we have the following.

Lemma 2.3. The eigenvalue of boundary value problem 2.1 -2.3 coincides with the eigenvalue of dissipative
A h operator.Additionally each chain of eigenvectors and associated vectors y 0 , y 1 , y 2 , . . ., y n corresponding to the eigenvalue λ 0 corresponds to the chain eigenvectors and associated vectors y 0 , y 1 , y 2 , . . ., y n corresponding to the same eigenvalue λ 0 of dissipative A h operator.In this case, the equality Proof.y 0 ∈ D A h and A h y 0 λ 0 y 0 , then the equality l y 0 λ 0 y 0 , R 0 y 0 − λR 0 y 0 0, R 1 y 0 R 2 y 0 0 takes place; that is, y 0 is an eigenfunction of the problem.Conversely, if conditions 2.14 are realized, then y 0 R 0 y 0 y 0 ∈ D A h and A h y 0 λ 0 y 0 , y 0 is an eigenvector of the operator A h .If y 0 , y 1 , y 2 , . . ., y n are a chain of the eigenvectors and associated vectors of the operator A h corresponding to the eigenvalue λ 0 , then by implementing the conditions y k ∈ D A h k 0, 1, 2, . . ., n and equality A h y 0 λ 0 y 0 , A h y s λ 0 y s y s−1 , s 1, 2, . . ., n, we get the equality 2.15 , where y 0 , y 1 , y 2 , . . ., y n are the first components of the vectors y 0 , y 1 , y 2 , . . ., y n .On the contrary, on the basis of the elements y 0 , y 1 , y 2 , . . ., y n corresponding to 2.1 -2.3 , one can construct the vectors y k

2.16
It follows from that Im A h y, y Im h D q −1 y 1 a 2 ≥ 0, A h is a dissipative operator in H. Let us prove that A h is maximal dissipative operator in the space H.It is sufficient to check that To prove 2.17 , let F ∈ H, Im λ < 0 and put where

2.19
The function x → G x, ξ, λ , F 1 satisfies the equation l y − λy F 1 0 ≤ x < ∞ and the boundary conditions 2.1 -2.3 .Moreover, for all F ∈ H and for Im λ < 0, we arrive at Γ ∈ D A h .For each F ∈ H and for Im λ < 0, we have A h − λI Γ F. Consequently, in the case of Im λ < 0, the result is A h − λI D A h H. Hence, Theorem 2.4 is proved.

Self-Adjoint Dilation of Dissipative Operator
We first construct the self-adjoint dilation of the operator A h .Let us add the "incoming" and "outgoing" subspaces called main Hilbert space of the dilation.In the space H we consider the operator L h on the set D L h , its elements consisting of vectors w ϕ − , y, ϕ , generated by the expression where W 1 2 •, • are Sobolev spaces and β 2 : 2 Im h, β > 0. Then we have the following.
Theorem 3.1.The operator L h is self-adjoint in H and it is a self-adjoint dilation of the operator A h .
Proof.We first prove that L h is symmetric in H.

3.3
On the other hand, y a z a .

3.4
By 3.3 , we have From equalities 3.3 and 3.5 , we have L h f, g H − f, L h g H 0. Thus, L h is a symmetric operator.To prove that L h is self-adjoint, we need to show that L h ⊆ L * h .We consider the bilinear form Integrating by parts, we get L * h g i dψ − /dξ , z * ,i dψ /dξ , where , where the operator L is defined by 3.1 .Therefore, the sum of the integrated terms in the bilinear form L h f, g H must be equal to zero:

3.7
Then by 2.6 , we get From the boundary conditions for L h , we have 3.9 Afterwards, by 3.8 we get

3.10
Comparing the coefficients of ϕ − 0 in 3.10 , we obtain Similarly, comparing the coefficients of ϕ 0 in 3.10 we get z a − hD q −1 z a βψ 0 .

3.13
Therefore conditions 3.12 and 3.13 imply D L * h ⊆ D L h , hence L h L * h .The self-adjoint operator L h generates on H a unitary group U t exp iL h t t ∈ R 0, ∞ .Let us denote by P : H → H and P 1 : H → H the mapping acting according to the formulae P : ϕ − , y, ϕ → y and P 1 : y → 0, y, 0 .Let Z t : PU t P 1 ,t ≥ 0, by using U t .The family {Z t } t ≥ 0 of operators is a strongly continuous semigroup of completely nonunitary contraction on H. Let us denote by B h the generator of this semigroup: B h y lim t → 0 it −1 Z t y − y .The domain of B h consists of all the vectors for which the limit exists.The operator B h is dissipative.The operator L h is called the self-adjoint dilation of B h see 2, 9, 18 .We show that B h A h , hence L h is self-adjoint dilation of B h .To show this, it is sufficient to verify the equality For this purpose, we set L h − λI −1 P 1 y g ψ − , z, ψ which implies that L h − λI g P 1 y, and hence l z − λ z y, ψ − ξ ψ − 0 e −iλξ and ψ ξ ψ 0 e −iλξ .Since g ∈ D L h , then ψ − ∈ L 2 −∞, 0 , and it follows that ψ − 0 0, and consequently z satisfies the boundary condition z a − hD q −1 z a 0. Therefore, z ∈ D A h , and since point λ with Im λ < 0 cannot be an eigenvalue of dissipative operator, it follows that ψ 0 is obtained from the formula ψ 0 β −1 z a − hD q −1 z a .Thus for y and Im λ < 0. On applying the mapping P , we obtain 3.14 , and

so this clearly shows that A h B h .
The unitary group {U t } has an important property which makes it possible to apply it to the Lax-Phillips 27 , that is, it has orthogonal incoming and outcoming subspaces D − L 2 −∞, 0 , 0, 0 and D 0, 0, L 2 0, ∞ having the following properties: To be able to prove property 1 for D the proof for D − is similar , we set R λ L h − λI −1 .For all λ, with Im λ < 0 and for any f 0, 0, ϕ ∈ D , we have which implies that U t f, g H 0 for all t ≥ 0. Hence, for t ≥ 0, U t D ⊂ D , and property 1 has been proved.In order to prove property 2 , we define the mappings P : H → L 2 0, ∞ and P 1 : L 2 0, ∞ → D as follows: P : ϕ − , y, ϕ → ϕ and P 1 : ϕ → 0, 0, ϕ , respectively.We take into consideration that the semigroup of isometries U t : P U t P 1 t ≥ 0 is a one-sided shift in L 2 0, ∞ .Indeed, the generator of the semigroup of the one-sided shift V t in L 2 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 U t t ≥ 0 is the operator Sϕ P L h P 1 ϕ P L h 0, 0, ϕ P 0, 0, i d/dξ ϕ i d/dξ ϕ, where ϕ ∈ W 1 2 0, ∞ and ϕ 0 0. Since a semigroup is uniquely determined by its generator, it follows that U t V t , and hence so, the proof of property 2 is completed.
Definition 3.2.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 self-adjoint.
To prove property 3 of the incoming and outcoming subspaces, let us prove following lemma.

3.21
Since f ∈ D A h , A h holds condition above.Moreover, eigenvectors of the operator A h should also hold this condition.Therefore, for the eigenvectors y λ of the operator A h acting in H and the eigenvectors of the operator A h , we have D q −1 y 1 a 0. From the boundary conditions, we get y 1 a 0 and y x, λ 0. Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator A h , we obtain H {0}. Hence the operator A h is simple.The proof is completed.

Let us define H
Lemma 3.4.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 relative to the group {U t } 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 {U t } restricted to this subspace would be a unitary part of the group {U t }, and hence the restriction B h of B h to H would be a self-adjoint operator in H . Since the operator B h is simple, it follows that H {0}. The lemma is proved.
Assume that ϕ λ and ψ λ are solutions of l y λy satisfying the conditions

3.22
The Titchmarsh-Weyl function m a 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 a λ possesses the following properties: Im m a λ ≥ 0 for all Im λ > 0, and m a λ m a λ for all λ ∈ C, except the real poles m a λ .We set

3.24
We note that the vectors U − λ x, ξ, ζ for real λ do not belong to the space H.However, U − λ x, ξ, ζ satisfies the equation LU λU and the corresponding boundary conditions for the operator L H .By means of vector U − λ x, ξ, ζ , we define the transformation on the vectors f ϕ − , y, ϕ in which ϕ − ξ , ϕ ζ , y x are smooth, compactly supported functions.
Lemma 3.5.The transformation F − isometrically maps H − onto L 2 R .For all vectors f, g ∈ H − the Parseval equality and the inversion formulae hold: Proof.For f, g ∈ D − , f ϕ − , 0, 0 , g ψ , 0, 0 , with Paley-Wiener theorem, we have and by using usual Parseval equality for Fourier integrals

3.28
Here, H 2 ± denote the Hardy classes in L 2 R consisting of the functions analytically extendible to the upper and lower half-planes, respectively.
We now extend to 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 ; moreover, the first components of these vectors belong to C ∞ 0 −∞, 0 .Therefore, since the operators U t t ∈ R are unitary, by the equality

3.30
By taking the closure 3.30 , 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 of integrals over finite intervals.Finally where f λ F f λ and g λ F g λ .
Proof.The proof is analogous to Lemma 3.5.
It is obvious that the matrix-valued function S h λ is meromorphic in C and all poles are in the lower half-plane.From 3.23 , |S h λ | ≤ 1 for Im λ > 0; and S h λ is the unitary matrix for all λ ∈ R. Therefore, it explicitly follows from the formulae for the vectors U − λ and U λ that

3.35
The formulae 3.35 show that operator A h is a unitarily equivalent to the model dissipative operator with the characteristic function S h λ .Since the characteristic functions of unitary equivalent dissipative operator coincide see 26 , we have thus proved the following theorem.
Theorem 3.8.The characteristic function of the maximal dissipative operator A h coincides with the function S h λ defined in 3.23 .
Using characteristic function, the spectral properties of the maximal dissipative operator A h can be investigated.The characteristic function of the maximal dissipative operator A h is known to lead to information of completeness about the spectral properties of this operator.For instance, the absence of a singular factor s λ of the characteristic function S h λ in the factorization det S h λ s λ B λ , where B λ is a Blaschke product, ensures completeness of the system of eigenvectors and associated vectors of the operator A h in the space L 2 q 0, a see 25 .
Theorem 3.9.For all the values of h with Im h > 0, except possibly for a single value h h 0 , the characteristic function S h λ of the maximal dissipative operator A h is a Blaschke product.The spectrum of A h is purely discrete and belongs to the open upper half-plane.The operator A h has a countable number of isolated eigenvalues with finite multiplicity and limit points at infinity.The system of all eigenvectors and associated vectors of the operator A h is complete in the space H.
Proof.From 3.23 , it is clear that S h 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

3.37
Further, for m a λ in terms of S h λ , we find from 3.23 that m a λ h − hS h λ S h λ − 1 .

3.38
If c h > 0 for a given value h Im h > 0 , then 3.37 implies that lim t → ∞ S h it 0, and then 3.24 gives us that lim t → ∞ m a it −G.Since m a λ does not depend on h, this implies that c h can be nonzero at not more than a single point h h 0 and further h 0 −lim t → ∞ m a it .The theorem is proved.
Due to Theorem 2.4, since the eigenvalues of the boundary value problem 2.1 -2.3 and eigenvalues of the operator A h coincide, including their multiplicity and, furthermore, for the eigenfunctions and associated functions the boundary problems 2.1 -2.3 , then theorem is interpreted as follows.
Corollary 3.10.The spectrum of the boundary value problem 2.1 -2.3 is purely discrete and belongs to the open upper half-plane.For all the values of h with Im λ > 0, except possible for a single value h h 0 , the boundary value problem 2.1 -2.3 h / h 0 has a countable number of isolated Journal of Function Spaces and Applications eigenvalues with finite multiplicity and limit points and infinity.The system of the eigenfunctions and associated functions of this problem h / h 0 is complete in the space L 2 q 0, a .
S h λ e iλc B h λ , c c h ≥ 0, 3.36 where B h λ is a Blaschke product.It follows from 3.36 that |S h λ | e iλc |B h λ | ≤ e −b h Im λ , Im λ ≥ 0.
, that is, F − maps H − onto the whole of L 2 R .The lemma is proved.We note that the vectors U λ x, ξ, ζ for real λ do not belong to the space H.However, U λ x, ξ, ζ satisfies the equation LU λU and the corresponding boundary conditions for the operator L H .With the help of vector U λ x, ξ, ζ , we define the transformationF : f → f λ by F f λ : f λ : 1/ √ 2π f, U λ H on the vectors f ϕ − , y, ϕ in which ϕ − ξ , ϕ ζand yx are smooth, compactly supported functions.The transformation F isometrically maps H onto L 2 R .For all vectors f, g ∈ H the Parseval equality and the inversion formula hold: 33It follows from Lemmas 3.5 and 3.6 that H − H . Together withLemma 3.4, this shows that H − H H; therefore, property 3 above has been proved for the incoming and outcoming subspaces.Finally property 4 is clear.Thus, the transformation F − isometrically maps H − onto L 2 R with the subspace D − mapped onto H 2 − and the operators U t are transformed into the operators of multiplication by e iλt .This means that F − is the incoming spectral representation for the group {U t }.Similarly, F is the outgoing spectral representation for the group {U t }.It follows from 3.33 that the passage from the F − representation of an element f ∈ H to its F representation is accomplished as f λ S −1 h λ f − λ .Consequently, according to 27 we have proved the following.The function S −1 h λ is the scattering matrix of the group {U t } (of the selfadjoint operator L H ). S λ be an arbitrary nonconstant inner function on the upper half-plane the analytic function S λ on the upper half-plane C is called inner function on C if |S h λ | ≤ 1 for all λ ∈ C and |S h 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 Z t t ≥ 0 acting in K according to the formula Z t ϕ P e iλt ϕ ,ϕ ϕ λ ∈ K, where P is the orthogonal projection from H 2 onto K.The generator of the semigroup {Z t } is denoted by 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 we remark that this model dissipative operator, which is associated with the names of Lax-Phillips 27 , is a special case of a more general model dissipative operator constructed by Nagy and Foias ¸ 26 .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