Research Article | Open Access

# Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

**Academic Editor:**Ağacik Zafer

#### Abstract

Let denote the operator generated in by Sturm-Liouville equation , , , where is a complex-valued function and , with . In this article, we investigate the eigenvalues and the spectral singularities of and obtain analogs of Naimark and Pavlov conditions for .

#### 1. Introduction

Let denote Sturm-Liouville operator generated in by the differential expression and the boundary condition where Sinceis a complex-valued function, the operator is a non-selfadjoint. The spectral analysis of has been investigated byNaĭmark[1]. He proved that some of the poles of the kernel of resolvent of are not the eigenvalues of the operator. He also showed that those poles (which are called spectral singularities by Schwartz [2]) are on the continuous spectrum. Moreover, he has shown the spectral singularities play an important role in the spectral analysis of , and if then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

One very important step in the spectral analysis of was taken by Pavlov [3]. He studied the dependence of the structure of the eigenvalues and the spectral singularities of on the behavior of potential function at infinity. He also proved that if then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

Conditions (N) and (P) are called Naimark and Pavlov conditions for , respectively.

Lyance showed that the spectral singularities play an important role in the spectral analysis of [4, 5]. He also investigated the effect of the spectral singularities in the spectral expansion.

The spectral singularities of non-selfadjoint operator generated in by (1.1) and the boundary condition was investigated in detail by Krall [6, 7].

Some problems of spectral theory of differential operator and some other types of operators with spectral singularities were studied by some authors [8–14]. Note that in all papers the boundary conditions are not depending on the spectral parameter.

In a recent series of papers, Bindinget al.and Browne[15–18] have studied the spectral theory of regular Sturm-Liouville operators with boundary conditions depending on the spectral parameter.

Let denote the operator generated in by whereis a complex-valued function, with In this paper, we investigate the eigenvalues and the spectral singularities of . In particular, we show that the analogs of Naimark and Pavlov conditions for are respectively, where denotes the class of complex-valued absolutely continuous functions on .

#### 2. Jost Functions of (1.3)-(1.4)

Under the condition (1.3) has a solution satisfying where The solution is called Jost solution of (1.3). Note that Jost solution has a representation [19] where is the solution of the integral equation and are continuously differentiable with respect to their arguments. We also have where and is a constant.

Let where Therefore, and are analytic in and respectively, and continuous up to real axis. The functions and are called Jost functions of

Let us denote the eigenvalues and the spectral singularities of by and , respectively. It is evident that where

*Definition 2.1. *The multiplicity of a zero or in (or is defined as the multiplicity of the corresponding eigenvalue and spectral singularity of

In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of , we need to discuss the quantitative properties of the zeros of and in and , respectively.

Define then by (2.7), we have

Now, let us assume that

Theorem 2.2. *Under condition (2.11), the functions and have the representations
**
where and *

*Proof. *Using (2.3),(2.4), and (2.6), we get (2.12), where
From (2.4), we see that
holds, where and is a constant. It follows from (2.5), (2.14), and (2.15) that In a similar way, we obtain (2.13).

Theorem 2.3. *Under condition (2.11), we have the following.*(i)*The set of is bounded and has at most a countable number of elements, and its limit points can lie only in a bounded subinterval of the real axis.*(ii)*The set of is bounded and its linear Lebesgue measure is zero.*

*Proof. *From (2.14) and (2.15), we see that
Using (2.10), (2.16), and the uniqueness theorem of analytic functions [20], we get (i) and (ii).

#### 3. Naĭmark and Pavlov Conditions for

We will denote the set of all limit points of and by and , respectively, and the set of all zeros of and with infinity multiplicity in and , by and , respectively. It is obvious that and the linear Lebesgue measures of and are zero.

Theorem 3.1. *If
**
then the operator has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.*

*Proof. *From (2.5), (2.14), (2.15), and (3.2), we find that
where is a constant. By (2.12) and (3.3), we observe that the function has an anlytic continuation to the half-plane So we get that It follows from (3.1) that Therefore the sets and have a finite number of elements with a finite multiplicity. We obtain similar results for the sets and . By (2.10) we have the proof of the theorem.

Now let us assume that

Hence, we have the following lemma.

Lemma 3.2. *It holds that *

*Proof. *From (2.12) and (3.4), we find that the function is analytic in , and all of its derivatives are continuous in . For a sufficiently large we have
where
and is a constant. Since the function is not equal to zero identically, then by Pavlov's theorem, satisfies
where is the linear Lebesgue measure of -neighborhood of [3]. Now, we obtain the following estimates for
where and are constants depending on and From (3.8), we get that
Now, (3.7) yields that
However, (3.10) holds for an arbitrary , if and only if or . In a similar way we can prove that

Theorem 3.3. *Under condition (3.4), the operator has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.*

*Proof. *To be able to prove the theorem, we have to show that the functions and have a finite number of zeros with finite multiplicities in and , respectively. We give the proof for

From Lemma 3.2 and (3.1), we find that So the bounded sets and have no limit points, that is, the function has only a finite number of zeros in . Since these zeros are of finite multiplicity.

#### Acknowledgment

This work was supported by TUBITAK.

#### References

- M. A. Naĭmark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,”
*American Mathematical Society Translations Series 2*, vol. 16, pp. 103–193, 1960. View at: Google Scholar | MathSciNet - J. Schwartz, “Some non-selfadjoint operators,”
*Communications on Pure and Applied Mathematics*, vol. 13, pp. 609–639, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. S. Pavlov, “The non-selfadjoint Schrödinger operator,”
*Topics in Mathematical Physics*, vol. 1, pp. 87–110, 1967. View at: Google Scholar - V. E. Lyance, “A differential operator with spectral singularities. I,”
*American Mathematical Society Translations Series 2*, vol. 60, pp. 185–225, 1967. View at: Google Scholar - V. E. Lyance, “A differential operator with spectral singularities. II,”
*American Mathematical Society Translations Series 2*, vol. 60, pp. 227–283, 1967. View at: Google Scholar - A. M. Krall, “A nonhomogeneous eigenfunction expansion,”
*Transactions of the American Mathematical Society*, vol. 117, pp. 352–361, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Krall, “The adjoint of a differential operator with integral boundary conditions,”
*Proceedings of the American Mathematical Society*, vol. 16, pp. 738–742, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Adıvar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,”
*Journal of Mathematical Analysis and Applications*, vol. 261, no. 2, pp. 461–478, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Adıvar and E. Bairamov, “Difference equations of second order with spectral singularities,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 714–721, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Bairamov, Ö. Çakar, and A. M. Krall, “An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities,”
*Journal of Differential Equations*, vol. 151, no. 2, pp. 268–289, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Bairamov, Ö. Çakar, and A. M. Krall, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,”
*Mathematische Nachrichten*, vol. 229, pp. 5–14, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Bairamov and A. O. Çelebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,”
*The Quarterly Journal of Mathematics. Oxford. Second Series*, vol. 50, no. 200, pp. 371–384, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Krall, E. Bairamov, and Ö. Çakar, “Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition,”
*Journal of Differential Equations*, vol. 151, no. 2, pp. 252–267, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Krall, E. Bairamov, and Ö. Çakar, “Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities,”
*Mathematische Nachrichten*, vol. 231, pp. 89–104, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. A. Binding, P. J. Browne, and B. A. Watson, “Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions,”
*The Bulletin of the London Mathematical Society*, vol. 33, no. 6, pp. 749–757, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I,”
*Proceedings of the Edinburgh Mathematical Society*, vol. 33, pp. 749–757, 2001. View at: Google Scholar - P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II,”
*Journal of Computational and Applied Mathematics*, vol. 148, no. 1, pp. 147–168, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. J. Code and P. J. Browne, “Sturm-Liouville problems with boundary conditions depending quadratically on the eigenparameter,”
*Journal of Mathematical Analysis and Applications*, vol. 309, no. 2, pp. 729–742, 2005. View at: Publisher Site | Google Scholar | MathSciNet - V. A. Marchenko,
*Sturm-Liouville Operators and Applications*, vol. 22 of*Operator Theory: Advances and Applications*, Birkhäuser, Basel, Switzerland, 1986. View at: MathSciNet - E. P. Dolzhenko, “Boundary value uniqueness theorems for analytic functions,”
*Mathematical Notes*, vol. 26, pp. 437–442, 1979. View at: Google Scholar

#### Copyright

Copyright © 2009 Elgiz Bairamov and Nihal Yokus. 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.