Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

Abstract

Let 𝐿 denote the operator generated in 𝐿₂(𝑅₊) by Sturm-Liouville equation −𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦, 𝑥∈𝑅₊=[0,∞), 𝑦^′(0)/𝑦(0)=𝛼₀+𝛼₁𝜆+𝛼₂𝜆², where 𝑞 is a complex-valued function and 𝛼_𝑖∈ℂ, 𝑖=0,1,2 with 𝛼₂≠0. In this article, we investigate the eigenvalues and the spectral singularities of 𝐿 and obtain analogs of Naimark and Pavlov conditions for 𝐿.

1. Introduction

Let 𝐿₀ denote Sturm-Liouville operator generated in 𝐿₂(ℝ₊) by the differential expression 𝑙₀(𝑦)∶=−𝑦^′′+𝑞(𝑥)𝑦,𝑥∈ℝ₊,(1.1) and the boundary condition 𝑦(0)=0, where 𝑞∶ℝ₊→ℂ. Since𝑞is a complex-valued function, the operator 𝐿₀ is a non-selfadjoint. The spectral analysis of 𝐿₀ has been investigated byNaĭmark[1]. He proved that some of the poles of the kernel of resolvent of 𝐿₀ are not the eigenvalues of the operator. He also showed that those poles (which are called spectral singularities by Schwartz [2]) are on the continuous spectrum. Moreover, he has shown the spectral singularities play an important role in the spectral analysis of 𝐿₀ , and if ^∞₀𝑒^𝜀𝑥|𝑞(𝑥)|𝑑𝑥<∞,𝜀>0,(N) then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

One very important step in the spectral analysis of 𝐿₀ was taken by Pavlov [3]. He studied the dependence of the structure of the eigenvalues and the spectral singularities of 𝐿₀ on the behavior of potential function at infinity. He also proved that if sup_𝑥∈𝑅+[𝑒^𝜀√𝑥|𝑞(𝑥)|]<∞,𝜀>0,(P) then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

Conditions (N) and (P) are called Naimark and Pavlov conditions for 𝐿₀, respectively.

Lyance showed that the spectral singularities play an important role in the spectral analysis of 𝐿₀ [4, 5]. He also investigated the effect of the spectral singularities in the spectral expansion.

The spectral singularities of non-selfadjoint operator generated in 𝐿₂(ℝ₊) by (1.1) and the boundary condition ^∞₀𝐾(𝑥)𝑦(𝑥)𝑑𝑥+𝛼𝑦^′(0)−𝛽𝑦(0)=0(1.2) was investigated in detail by Krall [6, 7].

Some problems of spectral theory of differential operator and some other types of operators with spectral singularities were studied by some authors [8–14]. Note that in all papers the boundary conditions are not depending on the spectral parameter.

In a recent series of papers, Bindinget al.and Browne[15–18] have studied the spectral theory of regular Sturm-Liouville operators with boundary conditions depending on the spectral parameter.

Let 𝐿 denote the operator generated in 𝐿₂(ℝ₊) by −𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦,𝑥∈𝑅₊,𝑦(1.3)_′(0)𝑦(0)=𝛼₀+𝛼₁𝜆+𝛼₂𝜆²,(1.4) where𝑞is a complex-valued function, 𝛼_𝑖∈ℂ,𝑖=0,1,2, with 𝛼₂≠0. In this paper, we investigate the eigenvalues and the spectral singularities of 𝐿. In particular, we show that the analogs of Naimark and Pavlov conditions for 𝐿 are 𝑞∈𝐴𝐶(ℝ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥|𝑞^′(𝑥)|𝑑𝑥<∞,𝜀>0,𝑞∈𝐴𝐶(ℝ₊),lim_𝑥→∞𝑞(𝑥)=0,sup_𝑥∈𝑅+[𝑒^𝜀√𝑥|𝑞^′(𝑥)|]<∞,𝜀>0,(1.5) respectively, where 𝐴𝐶(ℝ₊)denotes the class of complex-valued absolutely continuous functions on ℝ₊.

2. Jost Functions of (1.3)-(1.4)

Under the condition ^∞₀𝑥|𝑞(𝑥)|𝑑𝑥<∞,(2.1) (1.3) has a solution 𝑒(𝑥,𝜆)satisfying lim_𝑥→∞𝑒(𝑥,𝜆)𝑒^{−𝑖𝜆𝑥}=1,𝜆∈ℂ₊,(2.2) where ℂ₊={𝜆∶𝜆∈ℂ,Im𝜆≥0}. The solution 𝑒(𝑥,𝜆) is called Jost solution of (1.3). Note that Jost solution has a representation [19] 𝑒(𝑥,𝜆)=𝑒^𝑖𝜆𝑥+^∞_𝑥𝐾(𝑥,𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡,𝜆∈ℂ₊,(2.3) where 𝐾(𝑥,𝑡) is the solution of the integral equation 1𝐾(𝑥,𝑡)=2^∞_{(𝑥+𝑡)/2}1𝑞(𝑠)𝑑𝑠+2_𝑥^{(𝑥+𝑡)/2𝑡+𝑠−𝑥}_{𝑡+𝑥−𝑠}1𝑞(𝑠)𝐾(𝑠,𝑢)𝑑𝑢𝑑𝑠+2^∞_{(𝑥+𝑡)/2𝑠}^{𝑡+𝑠−𝑥}𝑞(𝑠)𝐾(𝑠,𝑢)𝑑𝑢𝑑𝑠,(2.4) and 𝐾(𝑥,𝑡) are continuously differentiable with respect to their arguments. We also have |𝐾(𝑥,𝑡)|⩽𝑐𝑤(𝑥+𝑡2),|𝐾_𝑥(𝑥,𝑡)|,|𝐾_𝑡1(𝑥,𝑡)|≤4|𝑞(𝑥+𝑡2)|+𝑐𝑤(𝑥+𝑡2),(2.5) where 𝑤(𝑥)=∫^∞_𝑥|𝑞(𝑠)|𝑑𝑠 and 𝑐>0 is a constant.

Let 𝐸⁺(𝜆)∶=𝑒^′(0,𝜆)−(𝛼₀+𝛼₁𝜆+𝛼₂𝜆²)𝑒(0,𝜆),𝜆∈ℂ₊,𝐸⁻(𝜆)∶=𝑒^′(0,−𝜆)−(𝛼₀+𝛼₁𝜆+𝛼₂𝜆²)𝑒(0,−𝜆),𝜆∈ℂ₋,(2.6) where ℂ₋={𝜆∶𝜆∈ℂ,Im𝜆≤0}. Therefore, 𝐸⁺and 𝐸⁻ are analytic in ℂ₊={𝜆∶𝜆∈ℂ,Im𝜆>0} and ℂ₋={𝜆∶𝜆∈ℂ,Im𝜆<0}, respectively, and continuous up to real axis. The functions 𝐸⁺ and 𝐸⁻ are called Jost functions of 𝐿.

Let us denote the eigenvalues and the spectral singularities of 𝐿 by 𝜎_𝑑(𝐿) and 𝜎_𝑠𝑠(𝐿), respectively. It is evident that 𝜎_𝑑(𝐿)={𝜆∶𝜆∈ℂ₊,𝐸⁺(𝜆)=0}∪{𝜆∶𝜆∈ℂ₋,𝐸⁻𝜎(𝜆)=0},_𝑠𝑠(𝐿)={𝜆∶𝜆∈ℝ^∗,𝐸⁺(𝜆)=0}∪{𝜆∶𝜆∈ℝ^∗,𝐸⁻(𝜆)=0},(2.7){𝜆∶𝜆∈ℝ^∗,𝐸⁺(𝜆)=0}∩{𝜆∶𝜆∈ℝ^∗,𝐸⁻(𝜆)=0}=∅,(2.8) where ℝ^∗=ℝ⧵{0}.

Definition 2.1. The multiplicity of a zero 𝐸⁺(or 𝐸⁻) in ℂ₊ (or ℂ₋) is defined as the multiplicity of the corresponding eigenvalue and spectral singularity of 𝐿.

In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of 𝐿, we need to discuss the quantitative properties of the zeros of 𝐸⁺ and 𝐸⁻ in ℂ₊and ℂ₋, respectively.

Define 𝑀^±₁={𝜆∶𝜆∈ℂ_±,𝐸^±(𝜆)=0},𝑀^±₂={𝜆∶𝜆∈ℝ^∗,𝐸^±(𝜆)=0},(2.9) then by (2.7), we have 𝜎_𝑑(𝐿)=𝑀⁺₁∪𝑀⁻₁,𝜎_𝑠𝑠(𝐿)=𝑀⁺₂∪𝑀⁻₂.(2.10)

Now, let us assume that 𝑞∈𝐴𝐶(𝑅₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑥³|𝑞^′(𝑥)|𝑑𝑥<∞.(2.11)

Theorem 2.2. Under condition (2.11), the functions 𝐸⁺ and 𝐸⁻ have the representations 𝐸⁺(𝜆)=−𝛼₂𝜆²+𝛽⁺𝜆+𝛿⁺+^∞₀𝑓⁺(𝑡)𝑒^𝑖𝜆𝑡𝑑𝑡,𝜆∈ℂ₊,𝐸(2.12)₋(𝜆)=−𝛼₂𝜆²+𝛽⁻𝜆+𝛿⁻+^∞₀𝑓⁻(𝑡)𝑒^{−𝑖𝜆𝑡}𝑑𝑡,𝜆∈ℂ₋,(2.13) where 𝛽^±,𝛿^±∈ℂ, and 𝑓^±∈𝐿₁(ℝ₊).

Proof. Using (2.3),(2.4), and (2.6), we get (2.12), where 𝛽⁺=𝑖−𝛼₁−𝑖𝛼₂𝛿𝐾(0,0),₊=−𝐾(0,0)−𝛼₀−𝑖𝛼₁𝐾(0,0)+𝛼₂𝐾_𝑡𝑓(0,0),₊(𝑡)=𝐾_𝑥(0,𝑡)−𝛼₀𝐾(0,𝑡)−𝑖𝛼₁𝐾_𝑡(0,𝑡)+𝛼₂𝐾_𝑡𝑡(0,𝑡).(2.14) From (2.4), we see that |𝐾_𝑡𝑡𝑡(0,𝑡)|≤𝑐[𝑡|𝑞(2)|+|𝑞^′(𝑡2𝑡)|+𝑡𝑤(2)+𝑤₁(𝑡2)](2.15) holds, where 𝑤₁(𝑡)=∫^∞_𝑡𝑤(𝑠)𝑑𝑠 and 𝑐>0 is a constant. It follows from (2.5), (2.14), and (2.15) that 𝑓⁺∈𝐿₁(ℝ₊). In a similar way, we obtain (2.13).

Theorem 2.3. Under condition (2.11), we have the following.(i)The set of 𝜎_𝑑(𝐿) is bounded and has at most a countable number of elements, and its limit points can lie only in a bounded subinterval of the real axis.(ii)The set of 𝜎_𝑠𝑠(𝐿) is bounded and its linear Lebesgue measure is zero.

Proof. From (2.14) and (2.15), we see that 𝐸⁺(𝜆)=−𝛼₂𝜆²+𝛽⁺𝜆+𝛿⁺+𝑜(1),𝜆∈ℂ₊𝐸,|𝜆|→∞,₋(𝜆)=−𝛼₂𝜆²+𝛽⁻𝜆+𝛿⁻+𝑜(1),𝜆∈ℂ₋,|𝜆|→∞.(2.16) Using (2.10), (2.16), and the uniqueness theorem of analytic functions [20], we get (i) and (ii).

3. Naĭmark and Pavlov Conditions for 𝐋

We will denote the set of all limit points of 𝑀⁺₁ and 𝑀⁻₁ by 𝑀⁺₃ and 𝑀⁻₃, respectively, and the set of all zeros of 𝐸⁺ and 𝐸⁻ with infinity multiplicity in ℂ₊ and ℂ₋, by 𝑀⁺₄ and 𝑀⁻₄, respectively. It is obvious that 𝑀^±₃⊂𝑀^±₂,𝑀^±₄⊂𝑀^±₂,𝑀^±₃⊂𝑀^±₄,(3.1) and the linear Lebesgue measures of 𝑀^±₃ and 𝑀^±₄ are zero.

Theorem 3.1. If 𝑞∈𝐴𝐶(ℝ₊),lim_𝑥→∞𝑞(𝑥)=0,^∞₀𝑒^𝜀𝑥|𝑞^′(𝑥)|𝑑𝑥<∞,𝜀>0,(3.2) then the operator 𝐿 has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

Proof. From (2.5), (2.14), (2.15), and (3.2), we find that |𝑓⁺(𝑡)|≤𝑐𝑒^{−(𝜀/2)𝑡},(3.3) where 𝑐>0 is a constant. By (2.12) and (3.3), we observe that the function 𝐸⁺ has an anlytic continuation to the half-plane Im𝜆>−𝜀/4. So we get that 𝑀⁺₄=∅. It follows from (3.1) that 𝑀⁺₃=∅. Therefore the sets 𝑀⁺₁ and 𝑀⁺₂ have a finite number of elements with a finite multiplicity. We obtain similar results for the sets 𝑀⁻₁ and 𝑀⁻₂. By (2.10) we have the proof of the theorem.

Now let us assume that 𝑞∈𝐴𝐶(ℝ₊),lim_𝑥→∞𝑞(𝑥)=0,sup_𝑥∈𝑅+[𝑒^𝜀√𝑥|𝑞^′(𝑥)|]<∞,𝜀>0.(3.4)

Hence, we have the following lemma.

Lemma 3.2. It holds that 𝑀⁺₄=𝑀⁻₄=∅.

Proof. From (2.12) and (3.4), we find that the function 𝐸⁺ is analytic in ℂ₊, and all of its derivatives are continuous in ℂ₊. For a sufficiently large 𝑇>0, we have |𝑑^𝑘𝑑𝜆^𝑘𝐸⁺(𝜆)|≤𝐴_𝑘,𝜆∈ℂ₊,|𝜆|≤𝑇,𝑘=0,1,2,…,(3.5) where 𝐴_𝑘=2^𝑘𝑐^∞₀𝑡^𝑘𝑒^{√−(𝜀/2)𝑡}𝑑𝑡,𝑘=0,1,2,…,(3.6) and 𝑐>0 is a constant. Since the function 𝐸⁺ is not equal to zero identically, then by Pavlov's theorem, 𝑀⁺₄ satisfies ^ℎ₀ln𝐴(𝑠)𝑑𝜇(𝑀⁺₄,𝑠)>−∞,(3.7) where 𝐴(𝑠)=inf_𝑘(𝐴_𝑘𝑠^𝑘/𝑘!), 𝜇(𝑀⁺₄,𝑠) is the linear Lebesgue measure of 𝑠-neighborhood of 𝑀⁺₄,[3]. Now, we obtain the following estimates for 𝐴_𝑘∶𝐴_𝑘≤𝐵𝑏^𝑘𝑘^𝑘𝑘!,(3.8) where 𝐵 and 𝑏 are constants depending on 𝑐 and 𝜀. From (3.8), we get that 𝐴(𝑠)≤𝐵inf_𝑘(𝑏^𝑘𝑠^𝑘𝑘^𝑘)≤𝐵exp(−𝑏⁻¹𝑒⁻¹𝑠⁻¹).(3.9) Now, (3.7) yields that ^ℎ₀1𝑠𝑑𝜇(𝑀⁺₄,𝑠)<∞.(3.10) However, (3.10) holds for an arbitrary 𝑠, if and only if 𝜇(𝑀⁺₄,𝑠)=0 or 𝑀⁺₄=∅. In a similar way we can prove that 𝑀⁻₄=∅.

Theorem 3.3. Under condition (3.4), the operator 𝐿 has a finite number of eigenvalues and 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 𝐸⁺ and 𝐸⁻ have a finite number of zeros with finite multiplicities in ℂ₊and ℂ₋, respectively. We give the proof for 𝐸⁺.
From Lemma 3.2 and (3.1), we find that 𝑀⁺₃=∅. So the bounded sets 𝑀⁺₁ and 𝑀⁺₂ have no limit points, that is, the function 𝐸⁺ has only a finite number of zeros in ℂ₊. Since 𝑀⁺₄=∅, these zeros are of finite multiplicity.

Acknowledgment

This work was supported by TUBITAK.