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

Abstract

Let 𝐴 denote the operator generated in 𝐿₂(ℛ₊) by the Sturm-Liouville problem: −𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦, 𝑥∈ℛ₊=[0,∞), (𝑦^′/𝑦)(0)=(𝛽₁𝜆+𝛽₀)/(𝛼₁𝜆+𝛼₀), where 𝑞 is a complex valued function and 𝛼₀,𝛼₁,𝛽₀,𝛽₁∈𝒞, with 𝛼₀𝛽₁−𝛼₁𝛽₀≠0. In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of 𝐴. In particular, we obtain the conditions on 𝑞 under which the operator 𝐴 has a finite number of the eigenvalues and the spectral singularities.

1. Introduction

Let 𝐿 denote the non-self-adjoint Sturm-Liouville operator generated in 𝐿₂(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 Naĭmark [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 Naĭmark condition, that is,

^∞₀𝑒^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 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 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 𝐿₂(ℛ₊) by (1.1) and the boundary condition

^∞₀𝐾(𝑥)𝑦(𝑥)𝑑𝑥+𝛼𝑦^′(0)−𝛽𝑦(0)=0, (1.3) in which 𝐾∈𝐿₂(ℛ₊) is a complex valued function and 𝛼,𝛽∈𝒞, 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

−𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦,𝑥∈ℛ₊,𝑦^′𝑦𝛽(0)=¹𝜆+𝛽₀𝛼₁𝜆+𝛼₀, (1.4)(1.5) where 𝑞 is a complex valued function and 𝛼₀,𝛼₁,𝛽₀,𝛽₁ are complex numbers such that 𝛼₀𝛽₁−𝛼₁𝛽₀≠0. By 𝐴 we will denote the operator generated in 𝐿₂(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

lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥𝛿|𝑞^′(𝑥)|𝑑𝑥<∞ (1.6) for some 𝜀>0 and 1/2≤𝛿<1. We also show that the analogue of the Naĭmark condition for 𝐴 is the form

lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥|𝑞^′(𝑥)|𝑑𝑥<∞ (1.7) for some 𝜀>0.

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

^∞₀𝑥|𝑞(𝑥)|𝑑𝑥<∞, (2.2) the Jost solution has a representation

𝑒(𝑥,𝜆)=𝑒^𝑖𝜆𝑥+^∞_𝑥𝐾(𝑥,𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡 (2.3) for 𝜆∈𝒞₊, where the kernel 𝐾(𝑥,𝑡) satisfies

1𝐾(𝑥,𝑡)=2^∞_{(𝑥+𝑡)/2}1𝑞(𝜉)𝑑𝜉+2_𝑥^{(𝑥+𝑡)/2𝑡+𝜉−𝑥}_{𝑡+𝑥−𝜉}+1𝐾(𝜉,𝜂)𝑞(𝜉)𝑑𝜂𝑑𝜉2^∞_{(𝑥+𝑡)/2𝜉}^{𝑡+𝜉−𝑥}𝐾(𝜉,𝜂)𝑞(𝜉)𝑑𝜂𝑑𝜉. (2.4) Moreover, 𝐾(𝑥,𝑡) is continuously differentiable with respect to its arguments and

|𝐾(𝑥,𝑡)|≤𝑐^∞_{(𝑥+𝑡)/2}|𝑞(𝜉)|𝑑𝜉,|𝐾_𝑥(𝑥,𝑡)|,|𝐾_𝑡1(𝑥,𝑡)|≤4|𝑞(𝑥+𝑡2)|+𝑐^∞_{(𝑥+𝑡)/2}|𝑞(𝜉)|𝑑𝜉, (2.5)(2.6) 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,Im𝜆>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.

Lemma 2.1. If 𝑞∈𝒜𝒞(ℛ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑥²|𝑞^′(𝑥)|𝑑𝑥<∞, (2.7) then 𝐾_𝑥𝑡(𝑥,𝑡)∶=(𝜕²/𝜕𝑡𝜕𝑥)𝐾(𝑥,𝑡) exists and 𝐾_𝑥𝑡1(𝑥,𝑡)=−8𝑞^′(𝑥+𝑡21)−4𝐾(𝑥+𝑡2,𝑥+𝑡2)𝑞(𝑥+𝑡2)−12_𝑥^{(𝑥+𝑡)/2}[𝐾_𝑡(𝜉,𝑡+𝑥−𝜉)+𝐾_𝑡−1(𝜉,𝑡−𝑥+𝜉)]𝑞(𝜉)𝑑𝜉2^∞_{(𝑥+𝑡)/2}𝐾_𝑡(𝜉,𝑡−𝑥+𝜉)𝑞(𝜉)𝑑𝜉. (2.8)

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

From (2.5)–(2.8) we find that

|𝐾_𝑥𝑡𝑡(0,𝑡)|≤𝑐[|𝑞(2)|+|𝑞^′(𝑡2)|+^∞_𝑡/2|𝑞(𝜉)|𝑑𝜉], (2.9) where 𝑐>0 is a constant.

3. 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)=𝛼₀+𝛼₁𝜆,𝜑^′(0,𝜆)=𝛽₀+𝛽₁𝜆. Therefore 𝜑(𝑥,𝜆) is an entire function of 𝜆.

Let us define the following functions:

𝐷_±(𝜆)=(𝛼₀+𝛼₁𝜆)𝑒_𝑥(0,±𝜆)−(𝛽₀+𝛽₁𝜆)𝑒(0,±𝜆)𝜆∈𝒞_±, (3.1) where 𝒞_±={𝜆∶𝜆∈𝒞,±Im𝜆≥0}. It is obvious that the functions 𝐷₊ and 𝐷₋ are analytic in 𝒞₊ and 𝒞₋∶={𝜆∶𝜆∈𝒞,Im𝜆<0}, respectively and continuous on the real axis.

Let

𝐺𝐺(𝑥,𝑡;𝜆)=⁺(𝑥,𝑡;𝜆),𝜆∈𝒞₊,𝐺₋(𝑥,𝑡;𝜆),𝜆∈𝒞₋ (3.2) be the Green function of 𝐴 (obtained by the standard techniques), where

𝐺_±−(𝑥,𝑡;𝜆)=𝑒(𝑥,±𝜆)𝜑(𝑡,𝜆)𝐷_±−(𝜆),0≤𝑡≤𝑥,𝑒(𝑡,±𝜆)𝜑(𝑥,𝜆)𝐷_±(𝜆),𝑥≤𝑡<∞. (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:

𝜎_𝑐=ℛ. (3.4)

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

𝜎_𝑑={𝜆∶𝜆∈𝒞₊,𝐷₊(𝜆)=0}∪{𝜆∶𝜆∈𝒞₋,𝐷₋𝜎(𝜆)=0},_𝑠𝑠={𝜆∶𝜆∈ℛ^∗,𝐷₊(𝜆)=0}∪{𝜆∶𝜆∈ℛ^∗,𝐷₋(𝜆)=0}, (4.1) where ℛ^∗=ℛ−{0}.

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.

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

𝑀^±₁={𝜆∶𝜆∈𝒞_±,𝐷_±(𝜆)=0},𝑀^±₂={𝜆∶𝜆∈ℛ,𝐷_±(𝜆)=0}. (4.2) So we have, by (4.1), that

𝜎_𝑑=𝑀⁺₁∪𝑀⁻₁,𝜎_𝑠𝑠=𝑀⁺₂∪𝑀⁻₂−{0}. (4.3)

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 𝐷₊(𝜆)=𝑖𝛼₁𝜆²+𝑎𝜆+𝑏+^∞₀𝑓(𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡, (4.4) where 𝑎=𝑖𝛼₀−𝛼₁𝐾(0,0)−𝛽₁,𝑏=−(𝛼₀+𝑖𝛽₁)𝐾(0,0)−𝛽₀+𝑖𝛼₁𝐾_𝑥(0,0),𝑓(𝑡)=−𝛽₀𝐾(0,𝑡)−𝑖𝛽₁𝐾_𝑡(0,𝑡)+𝛼₀𝐾_𝑥(0,𝑡)+𝑖𝛼₁𝐾_𝑥𝑡(0,𝑡). (4.5) Using (2.5), (2.6), and (2.9) we get that 𝑓∈𝐿₁(ℛ₊). So 𝐷₊(𝜆)=𝑖𝛼₁𝜆²+𝑎𝜆+𝑏+𝑜(1),𝜆∈𝒞₊,|𝜆|→∞. (4.6) From (4.3), (4.6) and uniqueness theorem for analytic functions [24], we get (i) and (ii).

Theorem 4.3. If 𝑞∈𝒜𝒞(ℛ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑥³|𝑞^′(𝑥)|𝑑𝑥<∞, (4.7) then _𝜈|𝑙_𝜈1|ln|𝑙_𝜈|<∞, (4.8) 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 𝑞∈𝒜𝒞(ℛ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥|𝑞^′(𝑥)|𝑑𝑥<∞,𝜀>0, (4.9) 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 Naĭmark 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 𝑞∈𝒜𝒞(ℛ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥𝛿|𝑞^′(𝑥)|𝑑𝑥<∞, (4.10) 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] 𝑀^±₁∩𝑀^±_∞=∅,𝑀^±₃⊂𝑀^±₂,𝑀^±₄⊂𝑀^±₂,𝑀^±_∞⊂𝑀^±₂,𝑀^±₃⊂𝑀^±_∞,𝑀^±₄⊂𝑀^±_∞, (4.11) and 𝜇(𝑀^±₃)=𝜇(𝑀^±₄)=𝜇(𝑀^±_∞)=0, where 𝜇 denote the Lebesgue measure on the real axis.

Theorem 4.5. If (4.10) holds, then 𝑀⁺_∞=𝑀⁻_∞=∅.

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 |𝑑^𝑛𝑑𝜆^𝑛𝐷₊(𝜆)|≤𝐵_𝑛,𝑛=0,1,2,…,𝜆∈𝒞₊𝐵,|𝜆|<2𝑁,₀=4|𝛼₁|𝑁²+2|𝑎|𝑁+|𝑏|+^∞₀𝐵|𝑓(𝑡)|𝑑𝑡,₁=4|𝛼₁|𝑁+|𝑎|+^∞₀𝐵𝑡|𝑓(𝑡)|𝑑𝑡,₂=2|𝛼₁|+^∞₀𝑡²𝐵|𝑓(𝑡)|𝑑𝑡,_𝑛=^∞₀𝑡^𝑛|𝑓(𝑡)|𝑑𝑡,𝑛≥3. (4.12) From Theorem 4.2, we get that |^−𝑁_−∞ln|𝐷₊(𝜆)|1+𝜆²𝑑𝜆|<∞,|^∞_𝑁ln|𝐷₊(𝜆)|1+𝜆²𝑑𝜆|<∞. (4.13) Let us define the function 𝑇(𝑠)=inf_𝑛𝐵_𝑛𝑠^𝑛.𝑛! (4.14) Since the function 𝐷₊ is not equal to zero identically, by Pavlov's theorem [4], ^ℎ₀ln𝑇(𝑠)𝑑𝜇(𝑀⁺_∞,𝑠)>−∞ (4.15) holds, where ℎ>0 is a constant and 𝜇(𝑀⁺_∞,𝑠) is the Lebesgue measure of 𝑠-neighborhood of 𝑀⁺_∞. Using (2.5), (2.6), (2.9) and (4.4) we obtain that 𝐵_𝑛≤𝐵𝑑^𝑛𝑛!𝑛^{𝑛(1/𝛿−1)}, (4.16) where 𝐵 and 𝑑 are constants depending on 𝜀 and 𝛿. Substituting (4.16) in the definition of 𝑇(𝑠) we get 1𝑇(𝑠)≤𝐵exp{−(𝛿−1)𝑒⁻¹𝑑^{−𝛿/(1−𝛿)}𝑠^{−𝛿/(1−𝛿)}}. (4.17) Now (4.15) and (4.17) imply that ^ℎ₀𝑠^{−𝛿/(1−𝛿)}𝑑𝜇(𝑀⁺_∞,𝑠)<∞. (4.18) Since 𝛿/(1−𝛿)≥1, consequently (4.18) holds for arbitrary 𝑠 if and only if 𝜇(𝑀⁺_∞,𝑠)=0 or 𝑀⁺_∞=∅.

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.

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.

Proof. From the continuity of 𝐷₊ it is clear that the set 𝑀⁺₂ is closed and is a set of Lebesgue measure zero which is of type 𝐹_𝜎. According to Martin's theorem [26] there is measurable set whose metric density exists and is different from 0 and 1 at every point of 𝑀⁺₂. So, 𝑀⁺₂ is of the first category from the theorem due to Goffman [27]. We also have obviously same things for 𝑀⁻₂. Consequently 𝜎_𝑠𝑠 is of the first category by (4.3).