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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 248740, 7 pages
Dissipative Sturm-Liouville Operators with Transmission Conditions
1Department of Mathematics, Mehmet Akif Ersoy University, 15100 Burdur, Turkey
2Department of Mathematics, Nevsehir University, 50300 Nevsehir, Turkey
Received 7 December 2012; Accepted 11 February 2013
Academic Editor: Lucas Jódar
Copyright © 2013 Hüseyin Tuna and 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.
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.
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 [15–19]. Such conditions which include left and right limits of solutions and their derivatives at are often called “transmission conditions” or “interface conditions.” These problems often arise in varied assortment of physical transfer problems .
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 , 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 , Lax and Phillips  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 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 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 . 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. Self-Adjoint Dilation of Dissipative Sturm-Liouville Operator
Consider the differential expression where and , is real-valued function on , and . The points and are regular and is singular for the differential expression. Moreover one-sided limits exist and are finite.
To pass from the differential expression to operators, we introduce the Hilbert space with the inner product where and , , , and are some real numbers with and .
Let denote the closure of the minimal operator generated by (1) and by its domain. Besides, we denote the set of all functions from such that one-sided limits exist and are finite and is the domain of the maximal operator . Furthermore, .
For two arbitrary functions , , we have Green’s formula where exists and is finite.
Suppose that Weyl’s limit circle case holds for the differential expression on . There are several sufficient conditions in which Weyl’s limit circle case holds for a differential expression . Denote by the solutions of the equation , satisfying the initial conditions and transmission conditions where , and with and The solutions and belong to . Let , and be solutions of the equation , satisfying the initial conditions and transmission conditions
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 , , , , , , where are Sobolev spaces and , .
Theorem 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
We obtain by direct computation
Thus, is a symmetric operator. To prove that is self-adjoint, we need to show that . Take . Let , so that
By choosing elements with suitable components as the in (15), it is not difficult to show that , , and ; the operator is defined by (12). Therefore (15) is obtained from for all Furthermore, satisfies the conditions
Hence,; that is, .
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 [10, 21, 30]). 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 ; it follows that , and consequently satisfies the boundary condition . Therefore , and since point with cannot be an eigenvalue of dissipative operator, then . Thus we have for and . By applying onto the mapping , we obtain (17), and so this clearly shows that .
3. Functional Model of Dissipative Sturm-Liouville Operator
The unitary group has an important property which makes it possible to apply it to the Lax-Phillips . It has orthogonal incoming and outgoing subspaces and having the following properties:(1), and , ,(2),(3),(4).
Property (4) is clear. To be able to prove property (1) 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 (1) has been proved.
In order to prove property (2), we define the mappings and as follows: and , respectively. We take into consider 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 is completed.
Definition 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 (3) of the incoming and outgoing subspaces, let us prove following lemma.
Lemma 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 Consequently, we have Using this result with boundary condition we have ; that is, . Since all solutions of belong to , from this it can be concluded that the resolvent is a compact operator, and the spectrum of is purely discrete. 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 4. The equality holds.
Proof. Considering property (1) 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 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 are solutions of satisfying the conditions Let us adopt the following notations: where 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 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 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 (35), 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 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 6. The transformation isometrically maps onto . For all vectors the Parseval equality and the inversion formula hold: whereand .
Proof. The proof is analogous to Lemma 6.
It is obvious that the matrix-valued function is meromorphic in and all poles are in the lower half-plane. From (27), for , and is the unitary matrix for all . Therefore, it explicitly follows from the formulae for the vectors and that It follows from Lemmas 6 and 5 that . Together with Lemma 5, this shows that ; therefore property (3) has been proved for the incoming and outgoing subspaces.
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 (38) that the passage from the representation of an element to its representation is accomplished as . Consequently, according to , we have proved the following.
Theorem 7. The function is the scattering matrix of the group (of the self-adjoint operator ).
Let be an arbitrary nonconstant inner function (see ) on the upper half-plane (the analytic function and 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 where 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. Recall that this model dissipative operator, which is associated with the names of Lax-Phillips , is a special case of a more general model dissipative operator constructed by Nagy and Foiaş . 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 that The formulas (40) show that operator is unitarily equivalent to the model dissipative operator with the characteristic function . We have thus proved the following theorem.
Theorem 8. The characteristic function of the maximal dissipative operator coincides with the function defined by (27).
4. The Spectral Properties of Dissipative Sturm-Liouville Operators
By 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 ( is a Blaschke product) ensures completeness of the system of eigenvectors and associated vectors of the operator in the space (see [10, 21, 30]). If thecharacteristic function has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator can fail to be complete. Because is smooth, the support of the corresponding singular measure must be contained in the set of poles . But in this case the singular measure is a simple step function. If we require to have no zeros of infinite multiplicity, then So the singular factor vanishes. The characteristic function of the maximal dissipative operator has the form where .
Theorem 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 (35), 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 (42) that Further, expressing in terms of , we find from (35) that For a given value (), if , then (43) implies that , and then (44) gives us that . Since does not depend on , this implies that can be nonzero at not more than a single point (and further ). This completes the proof.
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