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

and Applied Analysis 3 Now we will assume that the complex valued function q is almost everywhere continuous in and satisfies the following: ∫∞ 0 x ∣q x ∣dx <∞. 2.2 Let φ x, λ and e x, λ denote the solutions of 2.1 satisfying the conditions φ 0, λ 1, φ′ 0, λ α0 α1λ α2λ, lim x→∞ e x, λ e−iλx 1, λ ∈ , 2.3 respectively. The solution e x, λ is called the Jost solution of 2.1 . Note that, under the condition 2.2 , the solution φ x, λ is an entire function of λ and the Jost solution is an analytic function of λ in : {λ : λ ∈ , Im λ > 0} and continuous in {λ : λ ∈ , Im λ ≥ 0}. In addition, Jost solution has a representation 22 e x, λ e ∫∞ x K x, t edt, λ ∈ , 2.4 where the kernelK x, t satisfies K x, t 1 2 ∫∞ x t /2 q s ds 1 2 ∫ x t /2 x ∫ t s−x t x−s q s K s, u duds 1 2 ∫∞ x t /2 ∫ t s−x s q s K s, u duds 2.5 and K x, t is continuously differentiable with respect to its arguments. We also have |K x, t | ≤ cw ( x t 2 ) , 2.6 |Kx x, t |, |Kt x, t | ≤ 14 ∣∣ ∣∣q ( x t 2 )∣∣ ∣∣ cw ( x t 2 ) , 2.7 where w x ∫∞ x |q s |ds and c > 0 is a constant. Let ê± x, λ denote the solutions of 2.1 subject to the conditions lim x→∞ e±iλxê± x, λ 1, lim x→∞ e±iλxê± x x, λ ±iλ, λ ∈ ±. 2.8 Then W [ e x, λ , ê± x, λ ] ∓2iλ, λ ∈ ± , W e x, λ , e x,−λ −2iλ, λ ∈ −∞,∞ , 2.9 whereW f1, f2 is the Wronskian of f1 and f2, 23 . 4 Abstract and Applied Analysis We will denote the Wronskian of the solutions φ x, λ with e x, λ and e x,−λ by E λ and E− λ , respectively, where E λ : e′ 0, λ − ( α0 α1λ α2λ ) e 0, λ , λ ∈ , E− λ : e′ 0,−λ − ( α0 α1λ α2λ ) e 0,−λ , λ ∈ −, 2.10 and − {λ : λ ∈ , Im λ ≤ 0}. Therefore E and E− are analytic in and − {λ : λ ∈ , Im λ < 0}, respectively, and continuous up to real axis. The functions E and E− are called Jost functions of L. 3. Eigenvalues and Spectral Singularities of L Let G x, t; λ ⎧ ⎨ ⎩ G x, t; λ , λ ∈ , G− x, t; λ , λ ∈ − 3.1 be the Green function of L obtained by the standard techniques , where G x, t; λ ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ − t, λ e x, λ E λ , 0 ≤ t ≤ x − x, λ e t, λ E λ , x ≤ t <∞ G− x, t; λ ⎧ ⎪⎨ ⎪ ⎪⎩ − t, λ e x,−λ E− λ , 0 ≤ t ≤ x − x, λ e t,−λ E− λ , x ≤ t < ∞. 3.2 We will denote the set of eigenvalues and spectral singularities of L by σd L and σss L , respectively. From 3.1 – 3.2 σd L {λ : λ ∈ , E λ 0} ∪ { λ : λ ∈ − , E− λ 0 } , σss L {λ : λ ∈ ∗ , E λ 0} ∪ { λ : λ ∈ ∗ , E− λ 0, 3.3 where ∗ \ {0}. From 3.3 we obtain that to investigate the structure of the eigenvalues and the spectral singularities of L, we need to discuss the structure of the zeros of the functions E and E− in and −, respectively. Definition 3.1. The multiplicity of zero of the function E or E− in or − is called the multiplicity of the corresponding eigenvalue and spectral singularity of L. Abstract and Applied Analysis 5 We see from 2.9 that the functionsand Applied Analysis 5 We see from 2.9 that the functions ψ x, λ Ê λ 2iλ e x, λ − E λ 2iλ ê x, λ , λ ∈ , ψ− x, λ Ê− λ 2iλ e x,−λ − E − λ 2iλ ê− x, λ , λ ∈ − , ψ x, λ E λ 2iλ e x,−λ − E − λ 2iλ e x, λ , λ ∈ ∗ 3.4 are the solutions of the boundary value problem −y′′ q x y λ2y, x ∈ , y′ 0 y 0 α0 α1λ α2λ, 3.5 where Ê± λ ê± ′ 0, λ − ( α0 α1λ α2λ ) ê± 0, λ . 3.6 Now let us assume that q ∈ AC , lim x→∞ q x 0, sup x∈ [ e √ x ∣q′ x ∣∣ ] <∞, ε > 0. 3.7 Theorem 3.2 see 24 . Under the condition 3.7 the operator L has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity. 4. Principal Functions of L In this section we assume that 3.7 holds. Let λ1, . . . , λj and λj 1, . . . , λk denote the zeros of E in and E− in − which are the eigenvalues of L with multiplicities m1, . . . , mj and mj 1, . . . , mk, respectively. It is obvious that from definition of the Wronskian { d dλn W [ ψ x, λ , e x, λ ]} λ λp { d dλn E λ } λ λp 0 4.1 for n 0, 1, . . . , mp − 1, p 1, 2, . . . , j, and { d dλn W [ ψ− x, λ , e x,−λ ] }


Introduction
Let T be a nonselfadjoint, closed operator in a Hilbert space H.We will denote the continuous spectrum and the set of all eigenvalues of T by σ c T and σ d T , respectively.Let us assume that σ c T / ∅.Definition 1.1.If λ λ 0 is a pole of the resolvent of T and λ 0 ∈ σ c T , but λ 0 / ∈ σ d T , then λ 0 is called a spectral singularity of T.
Let us consider the nonselfadjoint operator L 0 generated in L 2 Ê by the differential expression l 0 y −y q x y, x ∈ Ê , 1.1 and the boundary condition y 0 0, where q is a complex-valued function.The spectrum and spectral expansion of L 0 were investigated by Naȋmark 1 .He proved that the spectrum of L 0 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 L 0 2, 3 .He also showed that the spectral singularities play an important role in the spectral analysis of L 0 .
The spectral analysis of the non-self-adjoint operator L 1 generated in L 2 Ê by 1.1 and the boundary condition in which K ∈ L 2 Ê is a complex valued function and α, β ∈ , was investigated in detail by  In 4 he obtained the adjoint L * 1 of the operator L 1 .Note that L * 1 deserves special interest, since it is not a purely differential operator.The eigenfunction expansions of L 1 and L * 1 were investigated in 5 .
In 9 the results of Naimark were extended to the three-dimensional Schr ödinger 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 Schr ödinger, Klein-Gordon, discrete Dirac, and discrete Schr ödinger operators was investigated in 11-20 .By regularization of a divergent integral, the effect of the spectral singularities in the spectral expansion of a quadratic pencil of Schr ödinger operators was obtained in 13 .In 19, 20 the spectral expansion of the discrete Dirac and Schr ödinger 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 L denote the operator generated in L 2 Ê by the differential expression l y −y q x y, x ∈ Ê 1.3 and the boundary condition where q is a complex-valued function and α i ∈ , i 0, 1, 2 with α 2 / 0. In this work we obtain the properties of the principal functions corresponding to the spectral singularities of L.

The Jost Solution and Jost Function
We consider the equation −y q x y λ 2 y, x ∈ Ê 2.1 related to the operator L. Now we will assume that the complex valued function q is almost everywhere continuous in Ê and satisfies the following: Let ϕ x, λ and e x, λ denote the solutions of 2.1 satisfying the conditions where the kernel K x, t satisfies and K x, t is continuously differentiable with respect to its arguments.We also have where Then where W f 1 , f 2 is the Wronskian of f 1 and f 2 , 23 .
We will denote the Wronskian of the solutions ϕ x, λ with e x, λ and e x, −λ by E λ and E − λ , respectively, where The functions E and E − are called Jost functions of L.

Eigenvalues and Spectral Singularities of
be the Green function of L obtained by the standard techniques , where

3.2
We will denote the set of eigenvalues and spectral singularities of L by σ d L and σ ss L , respectively.From 3.1 -3.2 where Ê * Ê \ {0}.
From 3.3 we obtain that to investigate the structure of the eigenvalues and the spectral singularities of L, we need to discuss the structure of the zeros of the functions E and E − in and − , respectively.
Definition 3.1.The multiplicity of zero of the function E or E − in or − is called the multiplicity of the corresponding eigenvalue and spectral singularity of L.
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 L has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

Principal Functions of L
In this section we assume that 3.7 holds.Let λ 1 , . . ., λ j and λ j 1 , . . ., λ k denote the zeros of E in and E − in − which are the eigenvalues of L with multiplicities m 1 , . . ., m j and m j 1 , . . ., m k , respectively.It is obvious that from definition of the Wronskian Writing 4.9 for ψ x, λ and e x, λ , and using 4.8 , we find where

4.11
From 4.1 we have Hence there exists a constant a n 0 1 λ p such that f n 0 1 x, λ p a n 0 1 λ p e x, λ p .

4.13
This shows that 4.3 holds for n n 0 1.Similarly we can prove that 4.5 holds.
Definition 4.2.Let λ λ 0 be an eigenvalue of L. If the functions y 0 x, λ 0 , y 1 x, λ 0 , . . ., y s x, λ 0 4.14 satisfy the equations l y 0 − λ 0 y 0 0, l y j − λ 0 y j − y j−1 0, j 1, 2, . . ., s, 4.15 then the function y 0 x, λ 0 is called the eigenfunction corresponding to the eigenvalue λ λ 0 of L. The functions y 1 x, λ 0 , . . ., y s x, λ 0 are called the associated functions corresponding λ λ 0 .The eigenfunctions and the associated functions corresponding to λ λ 0 are called the principal functions of the eigenvalue λ λ 0 .The principal functions of the spectral singularities of L are defined similarly.Now using 4.3 and 4.5 define the functions Then for λ λ p , p 1, 2, . . ., j, j 1, . . ., k, l U 0,p 0, hold, where l u −u q x u − λ 2 u and ∂ m /∂λ m l u denotes the differential expressions whose coefficients are the m-th derivatives with respect to λ of the corresponding coefficients of the differential expression l u .Equation 4.18 shows that U 0,p is the eigenfunction corresponding to the eigenvalue λ λ p ; U 1,p , U

4.22
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 0 ≤ n ≤ m p − 1 and j 1 ≤ p ≤ k.