`Abstract and Applied AnalysisVolumeΒ 2011, Article IDΒ 358912, 12 pageshttp://dx.doi.org/10.1155/2011/358912`
Research Article

## Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

Department of Mathematics, KaramanoΔlu Mehmetbey University, 70100 Karaman, Turkey

Received 8 March 2011; Accepted 5 April 2011

Copyright Β© 2011 Nihal YokuΕ. 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 paper is properly cited.

#### Abstract

We consider the operator generated in by the differential expression , and the boundary condition , where is a complex-valued function and , with . In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of .

#### 1. Introduction

Let be a nonselfadjoint, closed operator in a Hilbert space . We will denote the continuous spectrum and the set of all eigenvalues of by and , respectively. Let us assume that .

Definition 1.1. If is a pole of the resolvent of and , but , then is called a spectral singularity of .

Let us consider the nonselfadjoint operator generated in by the differential expression and the boundary condition , where is a complex-valued function. The spectrum and spectral expansion of were investigated by NaΔ­mark [1]. He proved that the spectrum of is composed of continuous spectrum, eigenvalues, and spectral singularities. He showed that spectral singularities are on the continuous spectrum and are the poles of the resolvent kernel, which are not eigenvalues.

Lyance investigated the effect of the spectral singularities in the spectral expansion in terms of the principal functions of [2, 3]. He also showed that the spectral singularities play an important role in the spectral analysis of .

The spectral analysis of the non-self-adjoint operator generated in by (1.1) and the boundary condition in which is a complex valued function and , was investigated in detail by Krall [4β8] In [4] he obtained the adjoint of the operator . Note that deserves special interest, since it is not a purely differential operator. The eigenfunction expansions of and were investigated in [5].

In [9] the results of Naimark were extended to the three-dimensional SchrΓΆdinger operators.

The Laurent expansion of the resolvents of the abstract non-self-adjoint operators in the neighborhood of the spectral singularities was studied in [10].

Using the boundary uniqueness theorems of analytic functions, the structure of the eigenvalues and the spectral singularities of a quadratic pencil of SchrΓΆdinger, Klein-Gordon, discrete Dirac, and discrete SchrΓΆdinger operators was investigated in [11β20]. By regularization of a divergent integral, the effect of the spectral singularities in the spectral expansion of a quadratic pencil of SchrΓΆdinger operators was obtained in [13]. In [19, 20] the spectral expansion of the discrete Dirac and SchrΓΆdinger operators with spectral singularities was derived using the generalized spectral function (in the sense of Marchenko [21]) and the analytical properties of the Weyl function.

Let denote the operator generated in by the differential expression and the boundary condition where is a complex-valued function and , with . In this work we obtain the properties of the principal functions corresponding to the spectral singularities of .

#### 2. The Jost Solution and Jost Function

We consider the equation related to the operator .

Now we will assume that the complex valued function is almost everywhere continuous in and satisfies the following: Let and denote the solutions of (2.1) satisfying the conditions respectively. The solution is called the Jost solution of (2.1). Note that, under the condition (2.2), the solution is an entire function of and the Jost solution is an analytic function of in and continuous in .

In addition, Jost solution has a representation ([22]) where the kernel satisfies and is continuously differentiable with respect to its arguments. We also have where and is a constant.

Let denote the solutions of (2.1) subject to the conditions Then where is the Wronskian of and , ([23]).

We will denote the Wronskian of the solutions with and by and , respectively, where and . Therefore and are analytic in and , respectively, and continuous up to real axis.

The functions and are called Jost functions of .

#### 3. Eigenvalues and Spectral Singularities of πΏ

Let be the Green function of (obtained by the standard techniques), where We will denote the set of eigenvalues and spectral singularities of by and , respectively. From (3.1)β(3.2) where .

From (3.3) we obtain that to investigate the structure of the eigenvalues and the spectral singularities of , we need to discuss the structure of the zeros of the functions and in and , respectively.

Definition 3.1. The multiplicity of zero of the function or in or is called the multiplicity of the corresponding eigenvalue and spectral singularity of .
We see from (2.9) that the functions are the solutions of the boundary value problem where Now let us assume that

Theorem 3.2 (see [24]). Under the condition (3.7) the operator has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

#### 4. Principal Functions of πΏ

In this section we assume that (3.7) holds. Let and denote the zeros of in and in (which are the eigenvalues of ) with multiplicities and , respectively. It is obvious that from definition of the Wronskian for , and for .

Theorem 4.1. The fallowing formulae: , where , where holds.

Proof. We will proceed by mathematical induction, we prove first (4.3). Let . From (4.1) we get where . Let us assume that for , (4.3) holds; that is, Now we will prove that (4.3) holds for . If is a solution of (2.1), then satisfies Writing (4.9) for and , and using (4.8), we find where From (4.1) we have Hence there exists a constant such that This shows that (4.3) holds for .
Similarly we can prove that (4.5) holds.

Definition 4.2. Let be an eigenvalue of L. If the functions satisfy the equations then the function is called the eigenfunction corresponding to the eigenvalue of . The functions are called the associated functions corresponding . The eigenfunctions and the associated functions corresponding to are called the principal functions of the eigenvalue .
The principal functions of the spectral singularities of are defined similarly.

Now using (4.3) and (4.5) define the functions

and.

Then for ,,

hold, where and denotes the differential expressions whose coefficients are the m-th derivatives with respect to of the corresponding coefficients of the differential expression . Equation (4.18) shows that is the eigenfunction corresponding to the eigenvalue are the associated functions of ([25, 26]).

are called the principal functions corresponding to the eigenvalue of .

Theorem 4.3. One has

Proof. Let and . Using (2.6) and (3.7) we obtain that From (2.4) we get where is a constant. Since for the eigenvalues , of , (4.21) implies that The proof of theorem is obtained from (4.16) and (4.22). In a similar way using (4.17) we may also prove the results for and .

Let , and be the zeros of and in (which are the spectral singularities of ) with multiplicities and , respectively.

Similar to (4.3) and (4.5) we can show the following:

,

where

,

where Now define the generalized eigenfunctions and generalized associated functions corresponding to the spectral singularities of by the following:, .

Then , also satisfy the equations analogous to (4.18).

are called the principal functions corresponding to the spectral singularities of .

Theorem 4.4. One has

Proof. If we consider (4.21) for the principal functions corresponding to the spectral singularities , of and consider that for the spectral singularities, then we have (4.28), by (4.26) and (4.27).

Now introduce the Hilbert spaces with respectively. Then and is isomorphic to the dual of .

Theorem 4.5. One has

Proof. From (2.4) we have Using (4.26), (4.33) we obtain In a similar way, we find

Let us choose so that

By Theorem 4.5 and (4.31) we get following theorem

Theorem 4.6. One has

#### Acknowledgment

The author would like to thank Professor E. Bairamov for his helpful suggestions during the preparation of this work.

#### References

1. M. A. Naĭmark, βInvestigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,β American Mathematical Society Translations, vol. 16, pp. 103β193, 1960.
2. V. E. Lyance, βA differential operator with spectral singularities I,β American Mathematical Society Transactions Series 2, vol. 60, pp. 185β225, 1967.
3. V. E. Lyance, βA differential operator with spectral singularities II,β American Mathematical Society Transactions Series 2, vol. 60, pp. 227β283, 1967.
4. A. M. Krall, βThe adjoint of a differential operator with integral boundary conditions,β Proceedings of the American Mathematical Society, vol. 16, pp. 738β742, 1965.
5. A. M. Krall, βA nonhomogeneous eigenfunction expansion,β Transactions of the American Mathematical Society, vol. 117, pp. 352β361, 1965.
6. A. M. Krall, βSecond order ordinary differential operators with general boundary conditions,β Duke Mathematical Journal, vol. 32, pp. 617β625, 1965.
7. A. M. Krall, βNonhomogeneous differential operators,β The Michigan Mathematical Journal, vol. 12, pp. 247β255, 1965.
8. A. M. Krall, βOn non-self-adjoint ordinary differential operators of the second order,β Doklady Akademii Nauk SSSR, vol. 165, pp. 1235β1237, 1965.
9. M. G. Gasymov, βExpansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrödinger equation,β Soviet Mathematics—Doklady, vol. 9, pp. 390β393, 1968.
10. M. G. Gasymov and F. G. Maksudov, βThe principal part of the resolvent of nonselfadjoint operators in the neighborhood of spectral singularities,β Functional Analysis and Its Applications, vol. 6, no. 3, pp. 185β192, 1972.
11. M. Adıvar and E. Bairamov, βSpectral properties of non-selfadjoint difference operators,β Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461β478, 2001.
12. M. Adıvar and E. Bairamov, βDifference equations of second order with spectral singularities,β Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 714β721, 2003.
13. E. Bairamov, Ö. Çakar, and A. M. Krall, βAn eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities,β Journal of Differential Equations, vol. 151, no. 2, pp. 268β289, 1999.
14. E. Bairamov, Ö. Çakar, and A. M. Krall, βNon-selfadjoint difference operators and Jacobi matrices with spectral singularities,β Mathematische Nachrichten, vol. 229, pp. 5β14, 2001.
15. E. Bairamov, Ö. Çakar, and A. M. Krall, βSpectral properties, including spectral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis,β Quaestiones Mathematicae, vol. 26, no. 1, pp. 15β30, 2003.
16. E. Bairamov, Ö. Cakar, and C. Yanik, βSpectral singularities of the Klein-Gordon s-wave equation,β Indian Journal of Pure and Applied Mathematics, vol. 32, no. 6, pp. 851β857, 2001.
17. E. Bairamov and A. O. Çelebi, βSpectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,β The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 50, no. 200, pp. 371β384, 1999.
18. E. Bairamov and Ö. Karaman, βSpectral singularities of Klein-Gordon s-wave equations with an integral boundary condition,β Acta Mathematica Hungarica, vol. 97, no. 1-2, pp. 121β131, 2002.
19. A. M. Krall, E. Bairamov, and Ö. Çakar, βSpectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition,β Journal of Differential Equations, vol. 151, no. 2, pp. 252β267, 1999.
20. A. M. Krall, E. Bairamov, and Ö. Çakar, βSpectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities,β Mathematische Nachrichten, vol. 231, pp. 89β104, 2001.
21. V. A. Marchenko, βExpansion in eigenfunctions of non-selfadjoint singular second order differential operators,β American Mathematical Society Transactions Series 2, vol. 25, pp. 77β130, 1963.
22. V. A. Marchenko, Sturm-Liouville Operators and Applications, vol. 22, Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1986.
23. M. Jaulent and C. Jean, βThe inverse s-wave scattering problem for a class of potentials depending on energy,β Communications in Mathematical Physics, vol. 28, pp. 177β220, 1972.
24. E. Bairamov and N. Yokus, βSpectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions,β Abstract and Applied Analysis, p. Art. ID 289596, 8, 2009.
25. M. V. Keldysh, βOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,β Soviet Mathematics—Doklady, vol. 77, no. 4, pp. 11β14, 1951.
26. M. V. Keldysh, βOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,β Russian Mathematical Surveys, vol. 26, no. 4, pp. 15β41, 1971.