#### Abstract

Let denote the operator generated in by the Sturm-Liouville problem: , , , where is a complex valued function and , with In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of . In particular, we obtain the conditions on under which the operator has a finite number of the eigenvalues and the spectral singularities.

#### 1. Introduction

Let denote the non-self-adjoint Sturm-Liouville operator generated in by the differential expression

and the boundary condition , where is a complex valued function. The spectral analysis of with continuous and discrete spectrum was studied by Naĭmark [1]. In this article, the spectrum of was investigated and shown that it is composed of the eigenvalues, the continuous spectrum and the spectral singularities.The spectral singularities of are poles of the resolvent which are imbedded in the continuous spectrum and are not the eigenvalues. If the function satisfies the Naĭmark condition, that is,

for some , then has a finite number of the eigenvalues and spectral singularities with finite multiplicities.

The results of Naĭmark were extended to the Sturm-Liouville operators on the entire real axis by Kemp [2] and to the differential operators with a singularity at the zero point by Gasymov [3]. The spectral analysis of dissipative Sturm-Liouville operators with spectral singularities was considered by Pavlov [4]. A very important development in the spectral analysis of was made by Lyance [5, 6]. He showed that the spectral singularities play an important role in the spectral theory of . He also investigated the effect of the spectral singularities in the spectral expansion. The spectral singularities of the non-self-adjoint Sturm-Liouville operator generated in by (1.1) and the boundary condition

in which is a complex valued function and was studied in detail by Krall [7–9].

Some problems of spectral theory of differential and difference operators with spectral singularities were also investigated in [10–16]. Note that, the boundary conditions used in [1–17] are independent of spectral parameter. In recent years, various problems of the spectral theory of regular Sturm-Liouville problem whose boundary conditions depend on spectral parameter have been examined in [18–22].

Let us consider the boundary value problem

where is a complex valued function and are complex numbers such that . By we will denote the operator generated in by (1.4) and (1.5). In this paper we discuss the discrete spectrum of and prove that the operator has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity if

for some and . We also show that the analogue of the Naĭmark condition for is the form

for some

#### 2. Jost Solution of (1.4)

We will denote the solution of (1.4) satisfying the condition

by . The solution is called the Jost solution of (1.4). Under the condition

the Jost solution has a representation

for where the kernel satisfies

Moreover, is continuously differentiable with respect to its arguments and

where is a constant [23, Chapter 3].

The solution is analytic with respect to in and continuous on the real axis.

Let denote the class of complex valued absolutely continuous functions in In the sequel we will need the following.

Lemma 2.1. *If
**
then exists and
*

The proof of the lemma is the direct consequence of (2.4).

where is a constant.

#### 3. The Green Function and the Continuous Spectrum

Let denote the solution of (1.4) subject to the initial conditions . Therefore is an entire function of

Let us define the following functions:

where It is obvious that the functions and are analytic in and respectively and continuous on the real axis.

Let

be the Green function of (obtained by the standard techniques), where

We will denote the continuous spectrum of by Using (3.1)–(3.3) in a way similar to Theorem 2 [17, page 303], we get the following:

#### 4. The Discrete Spectrum of the Operator

Let us denote the eigenvalues and the spectral singularities of the operator by and respectively. From (2.3) and (3.1)–(3.4) it follows that

where

*Definition 4.1. *The multiplicity of a zero of (or ) in (or ) is defined as the multiplicity of the corresponding eigenvalue or spectral singularity of .

In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of we need to discuss the quantative properties of the zeros of and in and , respectively. For the sake of simplicity we will consider only the zeros of in A similar procedure may also be employed for zeros of in

Let us define

So we have, by (4.1), that

Theorem 4.2. *Under the conditions in (2.7):*(i)*the discrete spectrum is a bounded, at most countable set and its limit points lie on the bounded subinterval of the real axis;*(ii)*the set is a bounded and its linear Lebesgue measure is zero.*

*Proof. * From (2.3) and (3.1) we obtain that is analytic in , continuous on the real axis and has the form
where
Using (2.5), (2.6), and (2.9) we get that So
From (4.3), (4.6) and uniqueness theorem for analytic functions [24], we get (i) and (ii)*. *

Theorem 4.3. *If
**
then
**
where is the lengths of the boundary complementary intervals of *

*Proof. * From (2.5), (2.6), (2.9), (4.4) and (4.7) we see that is continuously differentiable on Since the function is not identically equal to zero, by Beurling's theorem we obtain (4.8) [25].

Theorem 4.4. *Under the conditions
**
the operator has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity.*

*Proof. *(2.5), (2.7), (2.9), (4.4) and (4.9) imply that the function has an analytic continuation to the half-plane Im. Hence the limit points of its zeros on cannot lie in . Therefore using Theorem 4.2, we have the finiteness of zeros of in . Similarly we find that the function has a finite number of zeros with finite multiplicity in . Then the proof of the theorem is the direct consequence of (4.3).

Note that the conditions in (4.9) are analogous to the Naĭmark condition (1.2) for the operator

It is clear that the condition (4.9) guarantees the analytic continuation of and from the real axis to the lower and the upper half-planes respectively. So the finiteness of the eigenvalues and the spectral singularities of are obtained as a result of these analytic continuations.

Now let suppose that
for some and which is weaker than (4.9). It is obvious that under the condition (4.10) the function is analytic in and infinitely differentiable on the real axis. But does not have analytic continuation from the real axis to the lower half-plane. Similarly, does not have analytic continuation from the real axis to the upper half-plane either. Consequently, under the conditions in (4.10) the finiteness of the eigenvalues and the spectral singularities of cannot be shown in a way similar to Theorem 4.4.

Let us denote the sets of limit points of and by and respectively and the set of all zeros of with infinite multiplicity in by Analogously define the sets and

It is clear from the boundary uniqueness theorem of analytic functions that [24]
and where denote the Lebesgue measure on the real axis.

Theorem 4.5. *If (4.10) holds, then *

*Proof. * We will prove that The case is similar. Under the condition (4.10) is analytic in all of its derivatives are continuous on the real axis and there exists such that
From Theorem 4.2, we get that
Let us define the function
Since the function is not equal to zero identically, by Pavlov's theorem [4],
holds, where is a constant and is the Lebesgue measure of -neighborhood of Using (2.5), (2.6), (2.9) and (4.4) we obtain that
where and are constants depending on and Substituting (4.16) in the definition of we get
Now (4.15) and (4.17) imply that
Since , consequently (4.18) holds for arbitrary if and only if or

Theorem 4.6. *Under the condition (4.10) the operator has a finite number of the eigenvalues and the 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 finite number of zeros with finite multiplicities in and respectively. We will prove it only for The case of is similar.

It follows from (4.11) that So the bounded sets and have no limit points, that is, the has only a finite number of zeros in Since these zeros are of a finite multiplicity.

Theorem 4.7. *If the condition (2.7) is satisfied then the set is of the first category.*

*Proof. *From the continuity of it is clear that the set is closed and is a set of Lebesgue measure zero which is of type . According to Martin's theorem [26] there is measurable set whose metric density exists and is different from and at every point of So, is of the first category from the theorem due to Goffman [27]. We also have obviously same things for Consequently is of the first category by (4.3).