Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter

and Applied Analysis 3 2. Jost Solution of 1.4 We will denote the solution of 1.4 satisfying the condition lim x→∞ y x, λ e−iλx 1, λ ∈ C : {λ : λ ∈ C, Imλ ≥ 0}, 2.1 by e x, λ . The solution e x, λ is called the Jost solution of 1.4 . Under the condition ∫∞ 0 x ∣ ∣q x ∣ ∣dx < ∞, 2.2 the Jost solution has a representation


Introduction
Let L denote the non-self-adjoint Sturm-Liouville operator generated in L 2 R by the differential expression l y −y q x y, x ∈ R 1.1 and the boundary condition y 0 0, where q is a complex valued function.The spectral analysis of L with continuous and discrete spectrum was studied by Naȋmark 1 .In this article, the spectrum of L was investigated and shown that it is composed of the eigenvalues, the continuous spectrum and the spectral singularities.The spectral singularities of L are poles of the resolvent which are imbedded in the continuous spectrum and are not the eigenvalues.
If the function q satisfies the Naȋmark condition, that is, ∞ 0 e εx q x dx < ∞ 1.2 for some ε > 0, then L has a finite number of the eigenvalues and spectral singularities with finite multiplicities.The results of Naȋmark 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 L was made by Lyance 5,6 .He showed that the spectral singularities play an important role in the spectral theory of L. 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 L 2 R by 1.1 and the boundary condition is a complex valued function and α, β ∈ C, was studied in detail by  .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 −y q x y λ 2 y, x ∈ R , 1.4 where q is a complex valued function and α 0 , α 1 , β 0 , β 1 are complex numbers such that α 0 β 1 − α 1 β 0 / 0. By A we will denote the operator generated in L 2 R by 1.4 and 1.5 .In this paper we discuss the discrete spectrum of A and prove that the operator A has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity if for some ε > 0 and 1/2 ≤ δ < 1.We also show that the analogue of the Naȋmark condition for A is the form for some ε > 0.

Jost Solution of 1.4
We will denote the solution of 1.4 satisfying the condition lim 1 by e x, λ .The solution e x, λ is called the Jost solution of 1.4 .Under the condition the Jost solution has a representation

2.4
Moreover, K x, t is continuously differentiable with respect to its arguments and where c > 0 is a constant 23, Chapter 3 .
The solution e x, λ is analytic with respect to λ in C : {λ : λ ∈ C, Im λ > 0} and continuous on the real axis.
Let AC R denote the class of complex valued absolutely continuous functions in R .In the sequel we will need the following.

2.8
The proof of the lemma is the direct consequence of 2.4 .From 2.5 -2.8 we find that where c > 0 is a constant.

The Green Function and the Continuous Spectrum
Let ϕ x, λ denote the solution of 1.4 subject to the initial conditions ϕ 0, λ α 0 α 1 λ, ϕ 0, λ β 0 β 1 λ.Therefore ϕ x, λ is an entire function of λ.Let us define the following functions: where C ± {λ : λ ∈ C, ± Im λ ≥ 0}.It is obvious that the functions D and D − are analytic in C and C − : {λ : λ ∈ C, Im λ < 0}, respectively and continuous on the real axis.Let be the Green function of A obtained by the standard techniques , where

3.3
We will denote the continuous spectrum of A by σ c .Using 3.1 -3.3 in a way similar to Theorem 2 17, page 303 , we get the following: 3.4

The Discrete Spectrum of the Operator A
Let us denote the eigenvalues and the spectral singularities of the operator A by σ d and σ ss respectively.From 2.3 and 3.1 -3.4 it follows that where R * R − {0}.In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of A we need to discuss the quantative properties of the zeros of D and D − in C and C − , respectively.For the sake of simplicity we will consider only the zeros of D in C .A similar procedure may also be employed for zeros of D − in C − .
Let us define So we have, by 4.1 , that Theorem 4.2.Under the conditions in 2.7 : i the discrete spectrum σ d is a bounded, at most countable set and its limit points lie on the bounded subinterval of the real axis; ii the set σ ss is a bounded and its linear Lebesgue measure is zero.
Proof.From 2.3 and 3.1 we obtain that D is analytic in C , continuous on the real axis and has the form where

4.5
Using 2.5 , 2.6 , and 2.9 we get that From 4.3 , 4.6 and uniqueness theorem for analytic functions 24 , we get i and ii .
where |l ν | is the lengths of the boundary complementary intervals of σ ss .
Proof.From 2.5 , 2.6 , 2.9 , 4.4 and 4.7 we see that D is continuously differentiable on R. Since the function D is not identically equal to zero, by Beurling's theorem we obtain 4.8 25 .
Theorem 4.4.Under the conditions the operator A 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 D has an analytic continuation to the half-plane Imλ > −ε/2.Hence the limit points of its zeros on C cannot lie in R. Therefore using Theorem 4.2, we have the finiteness of zeros of D in C .Similarly we find that the function D − has a finite number of zeros with finite multiplicity in C − .Then the proof of the theorem is the direct consequence of 4.3 .Note that the conditions in 4.9 are analogous to the Naȋmark condition 1.2 for the operator A.
It is clear that the condition 4.9 guarantees the analytic continuation of D and D − from the real axis to the lower and the upper half-planes respectively.So the finiteness of the eigenvalues and the spectral singularities of A are obtained as a result of these analytic continuations.Now let suppose that for some ε > 0 and 1/2 ≤ δ < 1, which is weaker than 4.9 .It is obvious that under the condition 4.10 the function D is analytic in C and infinitely differentiable on the real axis.But D does not have analytic continuation from the real axis to the lower half-plane.Similarly, D − 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 A cannot be shown in a way similar to Theorem 4.4.
Let us denote the sets of limit points of M 1 and M 2 by M 3 and M 4 respectively and the set of all zeros of D with infinite multiplicity in C by M ∞ .Analogously define the sets M − 3 , M − 4 and M − ∞ .It is clear from the boundary uniqueness theorem of analytic functions that 24 Proof.We will prove that M ∞ ∅.The case M − ∞ ∅ is similar.Under the condition 4.10 D is analytic in C all of its derivatives are continuous on the real axis and there exists N > 0 such that

4.12
From Theorem 4.2, we get that  Proof.To be able to prove the theorem we have to show that the functions D and D − have finite number of zeros with finite multiplicities in C and C − , respectively.We will prove it only for D .The case of D − is similar.
It follows from 4.11 that M 3 M 4 ∅.So the bounded sets M 1 and M 2 have no limit points, that is, the D has only a finite number of zeros in C .Since M ∞ ∅ these zeros are of a finite multiplicity.Theorem 4.7.If the condition 2.7 is satisfied then the set σ ss is of the first category.

Definition 4 . 1 .
The multiplicity of a zero of D or D − in C or C − is defined as the multiplicity of the corresponding eigenvalue or spectral singularity of A.

Theorem 4 . 6 .
Under the condition 4.10 the operator A has a finite number of the eigenvalues and the spectral singularities and each of them is of a finite multiplicity.