Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 736437, 17 pages

doi:10.1155/2012/736437

Research Article

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

Aytekin Eryılmaz

Department of Mathematics, Nevsehir University, Nevsehir 50300, Turkey

Received 20 April 2012; Accepted 2 August 2012

Academic Editor: Feliz Minhós

Copyright © 2012 Aytekin 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.

Abstract.

This paper is concerned with -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 -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.

1. Introduction

Spectral analysis of Sturm-Liouville and Schrödinger 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 -difference equation. In particular, we refer the reader to consult the reference [20] for some definitions and theorems on -derivative, -integration, -exponential function, -trigonometric function, -Taylor formula, -Beta and Gamma functions, Euler-Maclaurin formula, anf so forth. In [21], Advar and Bohner investigated the eigenvalues and the spectral singularities of non-selfa-djoint -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 -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 -analogue of Sturm-Liouville eigenvalue problems and formulated a self-adjoint -difference operator in a Hilbert space. They also discussed properties of the eigenvalues and the eigenfunctions.

In this paper, we consider -Sturm-Liouville Problem and define an adequate Hilbert space. Our main target of the present paper is to study -Sturm-Liouville boundary value problem in case of dissipation at the right endpoint of and with the spectral parameter at zero. The maximal dissipative -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 -Sturm-Liouville equation. By combining the results of Nagy-Foiaş 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 -difference operator.

Let be a positive number with , and . A -difference equation is an equation that contains -derivatives of a function defined on . Let be a complex-valued function on . The -difference operator is defined by

where . The -derivative at zero is defined by

if the limit exists and does not depend on . A right inverse to , the Jackson -integration, is given by

provided that the series converges, and

Let be the space of all complex-valued functions defined on such that

The space is a separable Hilbert space with the inner product

We will consider the basic Sturm-Liouville equation

where is defined on and continuous at zero. The -Wronskian of is defined to be

Let denote the closure of the minimal operator generated by (1.7) and by its domain. Besides, we denote by the set of all functions from such that and are continuous in and is the domain of the maximal operator . Furthermore, [2, 4, 13]. Suppose that the operator has defect index

For every we have Lagrange’s identity [24]

where .

2. Construction of the Dissipative Operator

Consider boundary value problem governed by

subject to the boundary conditions

where is spectral parameter and and is defined by

For convenience we assume

Lemma 2.1. For arbitrary , let one suppose that , then one has the following.

Proof.

Let ,   denote the solutions of (2.1) satisfying the conditions

Then from (2.3) we have

We let

It can be shown that satisfies (2.1) and boundary conditions (2.2)–(2.3). is a Green function of the boundary value problem (2.1)–(2.3). Thus, we obtain that the is a Hilbert-Schmidt kernel and the solution of the boundary value problem can be expressed by

Thus is a Hilbert Schmidt operator on space . The spectrum of the boundary value problem coincides with the roots of the equation . Since is analytic and not identical to zero, it means that the function has at most a countable number of isolated zeros with finite multiplicity and possible limit points at infinity.

Suppose that ,  , then we denote linear space with two component of elements of . If and ,  , then the formula

defines an inner product in Hilbert space . Let us define operator of with equalities suitable for boundary value problem

Remind that a linear operator with domain in Hilbert space is called dissipative if for all and maximal dissipative if it does not have a proper extension.

Definition 2.2. If the system of vectors of corresponding to the eigenvalue is

then the system of vectors of corresponding to the eigenvalue is called a chain of eigenvectors and associated vectors of boundary value problem (2.2)–(2.12).

Since the operator is dissipative in 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 operator. Additionally each chain of eigenvectors and associated vectors corresponding to the eigenvalue corresponds to the chain eigenvectors and associated vectors corresponding to the same eigenvalue of dissipative operator. In this case, the equality

holds.

Proof. and , then the equality ,  , takes place; that is, is an eigenfunction of the problem. Conversely, if conditions (2.14) are realized, then and is an eigenvector of the operator . If are a chain of the eigenvectors and associated vectors of the operator corresponding to the eigenvalue , then by implementing the conditions and equality ,  ,  we get the equality (2.15), where are the first components of the vectors . On the contrary, on the basis of the elements corresponding to (2.1)–(2.3), one can construct the vectors for which and ,  .

Theorem 2.4. The operator is maximal dissipative in the space .

Proof. Let . From (2.6), we have

It follows from that ,   is a dissipative operator in . Let us prove that is maximal dissipative operator in the space . It is sufficient to check that

To prove (2.17), let ,   and put

where

The function satisfies the equation and the boundary conditions (2.1)–(2.3). Moreover, for all and for , we arrive at . For each and for , we have . Consequently, in the case of , the result is . Hence, Theorem 2.4 is proved.

3. Self-Adjoint Dilation of Dissipative Operator

We first construct the self-adjoint dilation of the operator . Let us add the “incoming” and “outgoing” subspaces and to . The orthogonal sum is called main Hilbert space of the dilation. In the space we consider the operator on the set , its elements consisting of vectors , generated by the expression

satisfying the conditions: ,  ,  ,  ,  ,  , and

where are Sobolev spaces and ,  . Then we have the following.

Theorem 3.1. The operator is self-adjoint in and it is a self-adjoint dilation of the operator .

Proof. We first prove that is symmetric in . Namely . Let ,,   and . Then we have

On the other hand,

By (3.3), we have

From equalities (3.3) and (3.5), we have . Thus, is a symmetric operator. To prove that is self-adjoint, we need to show that . We consider the bilinear form on elements , where , ,. Integrating by parts, we get ,, where , . Similarly, if , then integrating by parts in , we obtain

Consequently, we have , for each by (3.6), where the operator is defined by (3.1). Therefore, the sum of the integrated terms in the bilinear form must be equal to zero:

Then by (2.6), we get

From the boundary conditions for , we have

Afterwards, by (3.8) we get

Comparing the coefficients of in (3.10), we obtain

or

Similarly, comparing the coefficients of in (3.10) we get

Therefore conditions (3.12) and (3.13) imply , hence .

The self-adjoint operator generates on a unitary group (). Let us denote by and the mapping acting according to the formulae and . Let ,, by using . The family of operators is a strongly continuous semigroup of completely nonunitary contraction on . Let us denote by the generator of this semigroup. The domain of consists of all the vectors for which the limit exists. The operator is dissipative. The operator is called the self-adjoint dilation of (see [2, 9, 18]). We show that , hence is self-adjoint dilation of . To show this, it is sufficient to verify the equality

For this purpose, we set which implies that , and hence and . Since , then , and it follows that , and consequently satisfies the boundary condition . Therefore, , and since point with cannot be an eigenvalue of dissipative operator, it follows that is obtained from the formula . Thus

for and . On applying the mapping , we obtain (3.14), and

so this clearly shows that .

The unitary group 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 and having the following properties:

(1),   and ,  ;(2); (3); (4).

To be able to prove property for (the proof for is similar), we set . For all , with and for any , we have

as . Therefore, if , then

which implies that for all . Hence, for ,  , and property has been proved.

In order to prove property , we define the mappings and as follows: and , respectively. We take into consideration that the semigroup of isometries is a one-sided shift in . Indeed, the generator of the semigroup of the one-sided shift in is the differential operator with the boundary condition . On the other hand, the generator of the semigroup of isometries is the operator , where and . Since a semigroup is uniquely determined by its generator, it follows that , and hence

so, the proof of property (2) is completed.

Definition 3.2. The linear operator with domain acting in the Hilbert space is called completely non-self-adjoint (or  simple) if there is no invariant subspace of the operator on which the restriction to is self-adjoint.

To prove property of the incoming and outcoming subspaces, let us prove following lemma.

Lemma 3.3. The operator is completely non-self-adjoint (simple).

Proof. Let be a nontrivial subspace in which induces a self-adjoint operator with domain . If , then and

and taking , we have

Since ,   holds condition above. Moreover, eigenvectors of the operator should also hold this condition. Therefore, for the eigenvectors of the operator acting in and the eigenvectors of the operator , we have . From the boundary conditions, we get and . Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator , we obtain . Hence the operator is simple. The proof is completed.

Let us define ,  .

Lemma 3.4. The equality holds.

Proof. Considering property of the subspace , it is easy to show that the subspace is invariant relative to the group and has the form , where is a subspace in . Therefore, if the subspace (and hence also ) was nontrivial, then the unitary group restricted to this subspace would be a unitary part of the group , and hence the restriction of to would be a self-adjoint operator in . Since the operator is simple, it follows that . The lemma is proved.

Assume that and are solutions of satisfying the conditions

The Titchmarsh-Weyl function is a meromorphic function on the complex plane with a countable number of poles on the real axis. Further, it is possible to show that the function possesses the following properties: for all , and for all , except the real poles . We set

We note that the vectors for real do not belong to the space . However, satisfies the equation and the corresponding boundary conditions for the operator . By means of vector , we define the transformation by

on the vectors in which ,  ,   are smooth, compactly supported functions.

Lemma 3.5. The transformation isometrically maps onto . For all vectors the Parseval equality and the inversion formulae hold:

where and .

Proof. For ,  ,  , with Paley-Wiener theorem, we have

and by using usual Parseval equality for Fourier integrals

Here, denote the Hardy classes in 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 . We consider in the dense set of of the vectors obtained as follows from the smooth, compactly supported functions in if ,  ,  , where is a nonnegative number depending on . If , then for and we have ; moreover, the first components of these vectors belong to . Therefore, since the operators are unitary, by the equality

we have

By taking the closure (3.30), we obtain the Parseval equality for the space . 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 , that is, maps onto the whole of . The lemma is proved.

We set

We note that the vectors for real do not belong to the space . However, satisfies the equation and the corresponding boundary conditions for the operator . With the help of vector , we define the transformation by on the vectors in which ,   and are smooth, compactly supported functions.

Lemma 3.6. The transformation isometrically maps onto . For all vectors the Parseval equality and the inversion formula hold:

where   and   .

Proof. The proof is analogous to Lemma 3.5.

It is obvious that the matrix-valued function is meromorphic in and all poles are in the lower half-plane. From (3.23), for ; and is the unitary matrix for all . Therefore, it explicitly follows from the formulae for the vectors and that

It follows from Lemmas 3.5 and 3.6 that . Together with Lemma 3.4, this shows that ; therefore, property above has been proved for the incoming and outcoming subspaces. Finally property is clear.

Thus, the transformation isometrically maps onto with the subspace mapped onto and the operators are transformed into the operators of multiplication by . This means that is the incoming spectral representation for the group . Similarly, is the outgoing spectral representation for the group . It follows from (3.33) that the passage from the representation of an element to its representation is accomplished as . Consequently, according to [27] we have proved the following.

Theorem 3.7. The function is the scattering matrix of the group (of  the  self-adjoint  operator  ).

Let be an arbitrary nonconstant inner function on the upper half-plane (the analytic function on the upper half-plane is called inner function on if for all and for almost all . Define . Then is a subspace of the Hilbert space . We consider the semigroup of operators acting in according to the formula ,, where is the orthogonal projection from onto . The generator of the semigroup is denoted by

in which is a maximal dissipative operator acting in and with the domain consisting of all functions , such that the limit exists. The operator 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 Foiaş [26]). The basic assertion is that is the characteristic function of the operator .

Let , so that . It follows from the explicit form of the unitary transformation under the mapping

The formulae (3.35) show that operator is a unitarily equivalent to the model dissipative operator with the characteristic function . 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 coincides with the function defined in (3.23).

Using characteristic function, the spectral properties of the maximal dissipative operator can be investigated. The characteristic function of the maximal dissipative operator is known to lead to information of completeness about the spectral properties of this operator. For instance, the absence of a singular factor of the characteristic function in the factorization , where is a Blaschke product, ensures completeness of the system of eigenvectors and associated vectors of the operator in the space (see [25]).

Theorem 3.9. For all the values of with , except possibly for a single value , the characteristic function of the maximal dissipative operator is a Blaschke product. The spectrum of is purely discrete and belongs to the open upper half-plane. The operator 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 is complete in the space .

Proof. From (3.23), it is clear that is an inner function in the upper half-plane, and it is meromorphic in the whole complex -plane. Therefore, it can be factored in the form

where is a Blaschke product. It follows from (3.36) that

Further, for in terms of , we find from (3.23) that

If for a given value , then (3.37) implies that , and then (3.24) gives us that . Since does not depend on , this implies that can be nonzero at not more than a single point (and further ). 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 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 with , except possible for a single value , the boundary value problem (2.1)–(2.3) has a countable number of isolated eigenvalues with finite multiplicity and limit points and infinity. The system of the eigenfunctions and associated functions of this problem is complete in the space .

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