Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 358912, 12 pages

http://dx.doi.org/10.1155/2011/358912

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

Department of Mathematics, KaramanoΔlu Mehmetbey University, 70100 Karaman, Turkey

Received 8 March 2011; Accepted 5 April 2011

Academic Editor: Narcisa C.Β Apreutesei

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 [4β8] 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 [11β20]. 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

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

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

- M. A. Naĭmark, βInvestigation 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. 103β193, 1960. View at Google Scholar - V. E. Lyance, βA differential operator with spectral singularities I,β
*American Mathematical Society Transactions Series 2*, vol. 60, pp. 185β225, 1967. View at Google Scholar - V. E. Lyance, βA differential operator with spectral singularities II,β
*American Mathematical Society Transactions Series 2*, vol. 60, pp. 227β283, 1967. View at Google Scholar - A. M. Krall, βThe adjoint of a differential operator with integral boundary conditions,β
*Proceedings of the American Mathematical Society*, vol. 16, pp. 738β742, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. Krall, βA nonhomogeneous eigenfunction expansion,β
*Transactions of the American Mathematical Society*, vol. 117, pp. 352β361, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. Krall, βSecond order ordinary differential operators with general boundary conditions,β
*Duke Mathematical Journal*, vol. 32, pp. 617β625, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. Krall, βNonhomogeneous differential operators,β
*The Michigan Mathematical Journal*, vol. 12, pp. 247β255, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. Krall, βOn non-self-adjoint ordinary differential operators of the second order,β
*Doklady Akademii Nauk SSSR*, vol. 165, pp. 1235β1237, 1965. View at Google Scholar - M. G. Gasymov, βExpansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrödinger equation,β
*Soviet Mathematics—Doklady*, vol. 9, pp. 390β393, 1968. View at Google Scholar - M. G. Gasymov and F. G. Maksudov, βThe 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. 185β192, 1972. View at Google Scholar - M. Adıvar and E. Bairamov, βSpectral properties of non-selfadjoint difference operators,β
*Journal of Mathematical Analysis and Applications*, vol. 261, no. 2, pp. 461β478, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Adıvar and E. Bairamov, βDifference equations of second order with spectral singularities,β
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 714β721, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. Bairamov, Ö. Çakar, and A. M. Krall, βAn eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities,β
*Journal of Differential Equations*, vol. 151, no. 2, pp. 268β289, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. Bairamov, Ö. Çakar, and A. M. Krall, βNon-selfadjoint difference operators and Jacobi matrices with spectral singularities,β
*Mathematische Nachrichten*, vol. 229, pp. 5β14, 2001. View at Google Scholar Β· View at Zentralblatt MATH - E. Bairamov, Ö. Çakar, and A. M. Krall, βSpectral properties, including spectral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis,β
*Quaestiones Mathematicae*, vol. 26, no. 1, pp. 15β30, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - E. Bairamov, Ö. Cakar, and C. Yanik, βSpectral singularities of the Klein-Gordon
*s*-wave equation,β*Indian Journal of Pure and Applied Mathematics*, vol. 32, no. 6, pp. 851β857, 2001. View at Google Scholar - E. Bairamov and A. O. Çelebi, βSpectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,β
*The Quarterly Journal of Mathematics. Oxford. Second Series*, vol. 50, no. 200, pp. 371β384, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. Bairamov and Ö. Karaman, βSpectral singularities of Klein-Gordon
*s*-wave equations with an integral boundary condition,β*Acta Mathematica Hungarica*, vol. 97, no. 1-2, pp. 121β131, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - A. M. Krall, E. Bairamov, and Ö. Çakar, βSpectrum 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. 252β267, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. M. Krall, E. Bairamov, and Ö. Çakar, βSpectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities,β
*Mathematische Nachrichten*, vol. 231, pp. 89β104, 2001. View at Google Scholar Β· View at Zentralblatt MATH - V. A. Marchenko, βExpansion in eigenfunctions of non-selfadjoint singular second order differential operators,β
*American Mathematical Society Transactions Series 2*, vol. 25, pp. 77β130, 1963. View at Google Scholar - V. A. Marchenko,
*Sturm-Liouville Operators and Applications*, vol. 22, Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1986. - M. Jaulent and C. Jean, βThe inverse
*s*-wave scattering problem for a class of potentials depending on energy,β*Communications in Mathematical Physics*, vol. 28, pp. 177β220, 1972. View at Publisher Β· View at Google Scholar - E. Bairamov and N. Yokus, βSpectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions,β
*Abstract and Applied Analysis*, p. Art. ID 289596, 8, 2009. View at Google Scholar - M. V. Keldysh, βOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,β
*Soviet Mathematics—Doklady*, vol. 77, no. 4, pp. 11β14, 1951. View at Google Scholar - M. V. Keldysh, βOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,β
*Russian Mathematical Surveys*, vol. 26, no. 4, pp. 15β41, 1971. View at Google Scholar