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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 358912, 12 pages
http://dx.doi.org/10.1155/2011/358912
Research Article

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

Department of Mathematics, Karamanoğlu Mehmetbey University, 70100 Karaman, Turkey

Received 8 March 2011; Accepted 5 April 2011

Academic Editor: Narcisa C.Β Apreutesei

Copyright © 2011 Nihal Yokuş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original paper is properly cited.

Abstract

We consider the operator 𝐿 generated in 𝐿2(ℝ+) by the differential expression 𝑙(𝑦)=βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦, π‘₯βˆˆβ„+∢=[0,∞) and the boundary condition π‘¦ξ…ž(0)/𝑦(0)=𝛼0+𝛼1πœ†+𝛼2πœ†2, where π‘ž is a complex-valued function and π›Όπ‘–βˆˆβ„‚, 𝑖=0,1,2 with 𝛼2β‰ 0. In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of 𝐿.

1. 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 πœŽπ‘(𝑇) and πœŽπ‘‘(𝑇), respectively. Let us assume that πœŽπ‘(𝑇)β‰ βˆ….

Definition 1.1. If πœ†=πœ†0 is a pole of the resolvent of 𝑇 and πœ†0βˆˆπœŽπ‘(𝑇), but πœ†0βˆ‰πœŽπ‘‘(𝑇), then πœ†0 is called a spectral singularity of 𝑇.

Let us consider the nonselfadjoint operator 𝐿0 generated in 𝐿2(ℝ+) by the differential expression𝑙0(𝑦)=βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦,π‘₯βˆˆβ„+,(1.1) and the boundary condition 𝑦(0)=0, where π‘ž is a complex-valued function. The spectrum and spectral expansion of 𝐿0 were investigated by NaΔ­mark [1]. He proved that the spectrum of 𝐿0 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 𝐿0 [2, 3]. He also showed that the spectral singularities play an important role in the spectral analysis of 𝐿0.

The spectral analysis of the non-self-adjoint operator 𝐿1 generated in 𝐿2(ℝ+) by (1.1) and the boundary conditionξ€œβˆž0𝐾(π‘₯)𝑦(π‘₯)𝑑π‘₯+π›Όπ‘¦ξ…ž(0)βˆ’π›½π‘¦(0)=0,(1.2) in which 𝐾∈𝐿2(ℝ+) is a complex valued function and 𝛼,π›½βˆˆβ„‚, was investigated in detail by Krall [4–8] In [4] he obtained the adjoint πΏβˆ—1 of the operator 𝐿1. Note that πΏβˆ—1 deserves special interest, since it is not a purely differential operator. The eigenfunction expansions of 𝐿1 and πΏβˆ—1 were investigated in [5].

In [9] the results of Naimark were extended to the three-dimensional SchrΓΆdinger 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 SchrΓΆdinger, Klein-Gordon, discrete Dirac, and discrete SchrΓΆdinger 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 SchrΓΆdinger operators was obtained in [13]. In [19, 20] the spectral expansion of the discrete Dirac and SchrΓΆdinger 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 𝐿2(ℝ+) by the differential expression𝑙(𝑦)=βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦,π‘₯βˆˆβ„+(1.3) and the boundary conditionπ‘¦ξ…ž(0)𝑦(0)=𝛼0+𝛼1πœ†+𝛼2πœ†2,(1.4) where π‘ž is a complex-valued function and π›Όπ‘–βˆˆβ„‚, 𝑖=0,1,2 with 𝛼2β‰ 0. In this work we obtain the properties of the principal functions corresponding to the spectral singularities of 𝐿.

2. The Jost Solution and Jost Function

We consider the equationβˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†2𝑦,π‘₯βˆˆβ„+(2.1) related to the operator 𝐿.

Now we will assume that the complex valued function π‘ž is almost everywhere continuous in ℝ+ and satisfies the following: ξ€œβˆž0π‘₯||||π‘ž(π‘₯)𝑑π‘₯<∞.(2.2) Let πœ‘(π‘₯,πœ†) and 𝑒(π‘₯,πœ†) denote the solutions of (2.1) satisfying the conditionsπœ‘(0,πœ†)=1,πœ‘ξ…ž(0,πœ†)=𝛼0+𝛼1πœ†+𝛼2πœ†2,limπ‘₯β†’βˆžπ‘’(π‘₯,πœ†)π‘’βˆ’π‘–πœ†π‘₯=1,πœ†βˆˆβ„‚+,(2.3) respectively. The solution 𝑒(π‘₯,πœ†) is called the Jost solution of (2.1). Note that, under the condition (2.2), the solution πœ‘(π‘₯,πœ†) is an entire function of πœ† and the Jost solution is an analytic function of πœ† in β„‚+∢={πœ†βˆΆπœ†βˆˆβ„‚,Imπœ†>0} and continuous in β„‚+={πœ†βˆΆπœ†βˆˆβ„‚,Imπœ†β‰₯0}.

In addition, Jost solution has a representation ([22])𝑒(π‘₯,πœ†)=π‘’π‘–πœ†π‘₯+ξ€œβˆžπ‘₯𝐾(π‘₯,𝑑)π‘’π‘–πœ†π‘‘π‘‘π‘‘,πœ†βˆˆβ„‚+,(2.4) where the kernel 𝐾(π‘₯,𝑑) satisfies1𝐾(π‘₯,𝑑)=2ξ€œβˆž(π‘₯+𝑑)/21π‘ž(𝑠)𝑑𝑠+2ξ€œπ‘₯(π‘₯+𝑑)/2ξ€œπ‘‘+π‘ βˆ’π‘₯𝑑+π‘₯βˆ’π‘ +1π‘ž(𝑠)𝐾(𝑠,𝑒)𝑑𝑒𝑑𝑠2ξ€œβˆž(π‘₯+𝑑)/2ξ€œπ‘ π‘‘+π‘ βˆ’π‘₯π‘ž(𝑠)𝐾(𝑠,𝑒)𝑑𝑒𝑑𝑠(2.5) and 𝐾(π‘₯,𝑑) is continuously differentiable with respect to its arguments. We also have||||𝐾(π‘₯,𝑑)≀𝑐𝑀π‘₯+𝑑2,(2.6)||𝐾π‘₯||,||𝐾(π‘₯,𝑑)𝑑||≀1(π‘₯,𝑑)4|||π‘žξ‚€π‘₯+𝑑2|||ξ‚€+𝑐𝑀π‘₯+𝑑2,(2.7) where βˆ«π‘€(π‘₯)=∞π‘₯|π‘ž(𝑠)|𝑑𝑠 and 𝑐>0 is a constant.

Let ̂𝑒±(π‘₯,πœ†) denote the solutions of (2.1) subject to the conditionslimπ‘₯β†’βˆžπ‘’Β±π‘–πœ†π‘₯̂𝑒±(π‘₯,πœ†)=1,limπ‘₯β†’βˆžπ‘’Β±π‘–πœ†π‘₯̂𝑒±π‘₯(π‘₯,πœ†)=Β±π‘–πœ†,πœ†βˆˆβ„‚Β±.(2.8) Thenπ‘Šξ€Ίπ‘’(π‘₯,πœ†),̂𝑒±(π‘₯,πœ†)=βˆ“2π‘–πœ†,πœ†βˆˆβ„‚Β±,π‘Š[]𝑒(π‘₯,πœ†),𝑒(π‘₯,βˆ’πœ†)=βˆ’2π‘–πœ†,πœ†βˆˆβ„=(βˆ’βˆž,∞),(2.9) where π‘Š[𝑓1,𝑓2] is the Wronskian of 𝑓1 and 𝑓2, ([23]).

We will denote the Wronskian of the solutions πœ‘(π‘₯,πœ†) with 𝑒(π‘₯,πœ†) and 𝑒(π‘₯,βˆ’πœ†) by 𝐸+(πœ†) and πΈβˆ’(πœ†), respectively, where𝐸+(πœ†)∢=π‘’ξ…žξ€·π›Ό(0,πœ†)βˆ’0+𝛼1πœ†+𝛼2πœ†2𝑒(0,πœ†),πœ†βˆˆβ„‚+,πΈβˆ’(πœ†)∢=π‘’ξ…žξ€·π›Ό(0,βˆ’πœ†)βˆ’0+𝛼1πœ†+𝛼2πœ†2𝑒(0,βˆ’πœ†),πœ†βˆˆβ„‚βˆ’,(2.10) and β„‚βˆ’={πœ†βˆΆπœ†βˆˆβ„‚,Imπœ†β‰€0}. Therefore 𝐸+ and πΈβˆ’ are analytic in β„‚+ and β„‚βˆ’={πœ†βˆΆπœ†βˆˆβ„‚,Imπœ†<0}, respectively, and continuous up to real axis.

The functions 𝐸+ and πΈβˆ’ are called Jost functions of 𝐿.

3. Eigenvalues and Spectral Singularities of 𝐿

Let𝐺𝐺(π‘₯,𝑑;πœ†)=+(π‘₯,𝑑;πœ†),πœ†βˆˆβ„‚+,πΊβˆ’(π‘₯,𝑑;πœ†),πœ†βˆˆβ„‚βˆ’(3.1) be the Green function of 𝐿 (obtained by the standard techniques), where𝐺+⎧βŽͺ⎨βŽͺβŽ©βˆ’(π‘₯,𝑑;πœ†)=πœ‘(𝑑,πœ†)𝑒(π‘₯,πœ†)𝐸+βˆ’(πœ†),0≀𝑑≀π‘₯πœ‘(π‘₯,πœ†)𝑒(𝑑,πœ†)𝐸+𝐺(πœ†),π‘₯≀𝑑<βˆžβˆ’(⎧βŽͺ⎨βŽͺβŽ©βˆ’π‘₯,𝑑;πœ†)=πœ‘(𝑑,πœ†)𝑒(π‘₯,βˆ’πœ†)πΈβˆ’βˆ’(πœ†),0≀𝑑≀π‘₯πœ‘(π‘₯,πœ†)𝑒(𝑑,βˆ’πœ†)πΈβˆ’(πœ†),π‘₯≀𝑑<∞.(3.2) We will denote the set of eigenvalues and spectral singularities of 𝐿 by πœŽπ‘‘(𝐿) and 𝜎ss(𝐿), respectively. From (3.1)–(3.2)πœŽπ‘‘ξ€½(𝐿)=πœ†βˆΆπœ†βˆˆβ„‚+,𝐸+ξ€Ύβˆͺξ€½(πœ†)=0πœ†βˆΆπœ†βˆˆβ„‚βˆ’,πΈβˆ’ξ€Ύ,𝜎(πœ†)=0ss(𝐿)=πœ†βˆΆπœ†βˆˆβ„βˆ—,𝐸+(ξ€Ύβˆͺξ€½πœ†)=0πœ†βˆΆπœ†βˆˆβ„βˆ—,πΈβˆ’(ξ€Ύ,πœ†)=0(3.3) where β„βˆ—=ℝ⧡{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 πΈβˆ’ in β„‚+ and β„‚βˆ’, respectively.

Definition 3.1. The multiplicity of zero of the function 𝐸+(or πΈβˆ’) in β„‚+(or β„‚βˆ’) is called the multiplicity of the corresponding eigenvalue and spectral singularity of 𝐿.
We see from (2.9) that the functionsπœ“+𝐸(π‘₯,πœ†)=+(πœ†)𝐸2π‘–πœ†π‘’(π‘₯,πœ†)βˆ’+(πœ†)2π‘–πœ†Μ‚π‘’+(π‘₯,πœ†),πœ†βˆˆβ„‚+,πœ“βˆ’ξπΈ(π‘₯,πœ†)=βˆ’(πœ†)𝐸2π‘–πœ†π‘’(π‘₯,βˆ’πœ†)βˆ’βˆ’(πœ†)2π‘–πœ†Μ‚π‘’βˆ’(π‘₯,πœ†),πœ†βˆˆβ„‚βˆ’,πΈπœ“(π‘₯,πœ†)=+(πœ†)𝐸2π‘–πœ†π‘’(π‘₯,βˆ’πœ†)βˆ’βˆ’(πœ†)2π‘–πœ†π‘’(π‘₯,πœ†),πœ†βˆˆβ„βˆ—(3.4) are the solutions of the boundary value problem βˆ’π‘¦ξ…žξ…ž+π‘ž(π‘₯)𝑦=πœ†2𝑦,π‘₯βˆˆβ„+,π‘¦ξ…ž(0)𝑦(0)=𝛼0+𝛼1πœ†+𝛼2πœ†2,(3.5) where 𝐸±(πœ†)=̂𝑒±′𝛼(0,πœ†)βˆ’0+𝛼1πœ†+𝛼2πœ†2̂𝑒±(0,πœ†).(3.6) Now let us assume that ξ€·β„π‘žβˆˆπ΄πΆ+ξ€Έ,limπ‘₯β†’βˆžπ‘ž(π‘₯)=0,supπ‘₯βˆˆβ„+ξ‚ƒπ‘’πœ€βˆšπ‘₯||π‘žξ…ž||ξ‚„(π‘₯)<∞,πœ€>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 πœ†1,…,πœ†π‘— and πœ†π‘—+1,…,πœ†π‘˜ denote the zeros of 𝐸+ in β„‚+ and πΈβˆ’ in β„‚βˆ’ (which are the eigenvalues of 𝐿) with multiplicities π‘š1,…,π‘šπ‘— and π‘šπ‘—+1,…,π‘šπ‘˜, respectively. It is obvious that from definition of the Wronskianξ‚»π‘‘π‘›π‘‘πœ†π‘›π‘Šξ€Ίπœ“+ξ€»ξ‚Ό(π‘₯,πœ†),𝑒(π‘₯,πœ†)πœ†=πœ†π‘=ξ‚»π‘‘π‘›π‘‘πœ†π‘›πΈ+ξ‚Ό(πœ†)πœ†=πœ†π‘=0(4.1) for 𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=1,2,…,𝑗, andξ‚»π‘‘π‘›π‘‘πœ†π‘›π‘Š[πœ“βˆ’]ξ‚Ό(π‘₯,πœ†),𝑒(π‘₯,βˆ’πœ†)πœ†=πœ†π‘=ξ‚»π‘‘π‘›π‘‘πœ†π‘›πΈβˆ’ξ‚Ό(πœ†)πœ†=πœ†π‘=0(4.2) for 𝑛=0,1,...,π‘šπ‘βˆ’1,𝑝=𝑗+1,...,π‘˜.

Theorem 4.1. The fallowing formulae: ξ‚»πœ•π‘›πœ•πœ†π‘›πœ“+ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘=π‘›ξ“π‘š=0π΄π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘,(4.3)𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=1,2,…,𝑗, where π΄π‘šξ€·πœ†π‘ξ€Έ=ξ‚΅π‘›π‘šπœ•ξ‚Άξ‚»π‘›βˆ’π‘šπœ•πœ†π‘›βˆ’π‘šξπΈ+ξ‚Ό(πœ†)πœ†=πœ†π‘,(4.4)ξ‚»πœ•π‘›πœ•πœ†π‘›πœ“βˆ’ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘=π‘›ξ“π‘š=0π΅π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,βˆ’πœ†)πœ†=πœ†π‘,(4.5)𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=𝑗+1,…,π‘˜, where π΅π‘šξ€·πœ†π‘ξ€Έ=ξ‚΅π‘›π‘šπœ•ξ‚Άξ‚»π‘›βˆ’π‘šπœ•πœ†π‘›βˆ’π‘šξπΈβˆ’ξ‚Ό(πœ†)πœ†=πœ†π‘(4.6) holds.

Proof. We will proceed by mathematical induction, we prove first (4.3). Let 𝑛=0. From (4.1) we get πœ“+ξ€·π‘₯,πœ†π‘ξ€Έ=π‘Ž0ξ€·πœ†π‘ξ€Έξ€·β‹…π‘’π‘₯,πœ†π‘ξ€Έ,(4.7) where π‘Ž0(πœ†π‘)β‰ 0. Let us assume that for 1≀𝑛0β‰€π‘šπ‘βˆ’2, (4.3) holds; that is, ξ‚»πœ•π‘›0πœ•πœ†π‘›0πœ“+ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘=𝑛0ξ“π‘š=0π΄π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘.(4.8) Now we will prove that (4.3) holds for 𝑛0+1. If 𝑦(π‘₯,πœ†) is a solution of (2.1), then (πœ•π‘›/πœ•πœ†π‘›)𝑦(π‘₯,πœ†) satisfies ξ‚Έβˆ’π‘‘2𝑑π‘₯2+π‘ž(π‘₯)βˆ’πœ†2ξ‚Ήπœ•π‘›πœ•πœ†π‘›πœ•π‘¦(π‘₯,πœ†)=2πœ†π‘›π‘›βˆ’1πœ•πœ†π‘›βˆ’1πœ•π‘¦(π‘₯,πœ†)+𝑛(π‘›βˆ’1)π‘›βˆ’2πœ•πœ†π‘›βˆ’2𝑦(π‘₯,πœ†).(4.9) Writing (4.9) for πœ“+(π‘₯,πœ†) and 𝑒(π‘₯,πœ†), and using (4.8), we find ξ‚Έβˆ’π‘‘2𝑑π‘₯2+π‘ž(π‘₯)βˆ’πœ†2𝑓𝑛0+1ξ€·π‘₯,πœ†π‘ξ€Έ=0,(4.10) where 𝑓𝑛0+1ξ€·π‘₯,πœ†π‘ξ€Έ=ξ‚»πœ•π‘›0+1πœ•πœ†π‘›0+1πœ“+ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘βˆ’π‘›0+1ξ“π‘š=0π΄π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘.(4.11) From (4.1) we have π‘Šξ€Ίπ‘“π‘›0+1ξ€·π‘₯,πœ†π‘ξ€Έξ€·,𝑒π‘₯,πœ†π‘=𝑑𝑛0+1π‘‘πœ†π‘›0+1π‘Šξ€Ίπœ“+ξ€»ξ‚Ό(π‘₯,πœ†),𝑒(π‘₯,πœ†)πœ†=πœ†π‘=0.(4.12) Hence there exists a constant π‘Žπ‘›0+1(πœ†π‘) such that 𝑓𝑛0+1ξ€·π‘₯,πœ†π‘ξ€Έ=π‘Žπ‘›0+1ξ€·πœ†π‘ξ€Έπ‘’ξ€·π‘₯,πœ†π‘ξ€Έ.(4.13) This shows that (4.3) holds for 𝑛=𝑛0+1.
Similarly we can prove that (4.5) holds.

Definition 4.2. Let πœ†=πœ†0 be an eigenvalue of L. If the functions 𝑦0ξ€·π‘₯,πœ†0ξ€Έ,𝑦1ξ€·π‘₯,πœ†0ξ€Έ,…,𝑦𝑠π‘₯,πœ†0ξ€Έ(4.14) satisfy the equations 𝑙𝑦0ξ€Έβˆ’πœ†0𝑦0𝑦=0,π‘™π‘—ξ€Έβˆ’πœ†0π‘¦π‘—βˆ’π‘¦π‘—βˆ’1=0,𝑗=1,2,…,𝑠,(4.15) then the function 𝑦0(π‘₯,πœ†0) is called the eigenfunction corresponding to the eigenvalue πœ†=πœ†0 of 𝐿. The functions 𝑦1(π‘₯,πœ†0),…,𝑦𝑠(π‘₯,πœ†0) are called the associated functions corresponding πœ†=πœ†0. The eigenfunctions and the associated functions corresponding to πœ†=πœ†0 are called the principal functions of the eigenvalue πœ†=πœ†0.
The principal functions of the spectral singularities of 𝐿 are defined similarly.

Now using (4.3) and (4.5) define the functionsπ‘ˆπ‘›,π‘ξ‚»πœ•(π‘₯)=π‘›πœ•πœ†π‘›πœ“+ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘=π‘›ξ“π‘š=0π΄π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘,(4.16)𝑛=0,1,…,π‘šπ‘βˆ’1,…𝑝=1,2,…,𝑗

andπ‘ˆπ‘›,π‘ξ‚»πœ•(π‘₯)=π‘›πœ•πœ†π‘›πœ“βˆ’ξ‚Ό(π‘₯,πœ†)πœ†=πœ†π‘=π‘›ξ“π‘š=0π΅π‘šξ€·πœ†π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,βˆ’πœ†)πœ†=πœ†π‘,(4.17)𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=𝑗+1,…,π‘˜.

Then for πœ†=πœ†π‘,𝑝=1,2,…,𝑗,𝑗+1,…,π‘˜,π‘™ξ€·π‘ˆ0,π‘ξ€Έπ‘™ξ€·π‘ˆ=0,1,𝑝+1πœ•1!π‘™ξ€·π‘ˆπœ•πœ†0,π‘ξ€Έπ‘™ξ€·π‘ˆ=0,𝑛,𝑝+1πœ•1!π‘™ξ€·π‘ˆπœ•πœ†π‘›βˆ’1,𝑝+1πœ•2!2πœ•πœ†2π‘™ξ€·π‘ˆπ‘›βˆ’2,𝑝=0,(4.18)𝑛=2,3,…,π‘šπ‘βˆ’1,

hold, where 𝑙(𝑒)=βˆ’π‘’ξ…žξ…ž+π‘ž(π‘₯)π‘’βˆ’πœ†2𝑒 and (πœ•π‘š/πœ•πœ†π‘š)𝑙(𝑒) denotes the differential expressions whose coefficients are the m-th derivatives with respect to πœ† of the corresponding coefficients of the differential expression 𝑙(𝑒). Equation (4.18) shows that π‘ˆ0,𝑝 is the eigenfunction corresponding to the eigenvalue πœ†=πœ†π‘;π‘ˆ1,𝑝,π‘ˆ2,𝑝,…,π‘ˆπ‘šπ‘βˆ’1,𝑝 are the associated functions of π‘ˆ0,𝑝 ([25, 26]).

π‘ˆ0,𝑝,π‘ˆ1,𝑝,…,π‘ˆπ‘šπ‘βˆ’1,𝑝,𝑝=1,2,…,𝑗,𝑗+1,…,π‘˜ are called the principal functions corresponding to the eigenvalue πœ†=πœ†π‘,𝑝=1,2,…,𝑗,𝑗+1,…,π‘˜ of 𝐿.

Theorem 4.3. One has π‘ˆπ‘›,π‘βˆˆπΏ2ℝ+ξ€Έ,𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=1,2,…,𝑗,𝑗+1,…,π‘˜.(4.19)

Proof. Let 0β‰€π‘›β‰€π‘šπ‘βˆ’1 and 1≀𝑝≀𝑗. Using (2.6) and (3.7) we obtain that ||||𝐾(π‘₯,𝑑)β‰€π‘π‘’βˆšβˆ’πœ€(π‘₯+𝑑)/2.(4.20) From (2.4) we get |||||ξ‚»πœ•π‘›πœ•πœ†π‘›ξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘|||||≀π‘₯π‘›π‘’βˆ’π‘₯Imπœ†π‘ξ€œ+π‘βˆžπ‘₯π‘‘π‘›π‘’βˆšβˆ’πœ€(π‘₯+𝑑)/2π‘’βˆ’π‘‘Imπœ†π‘π‘‘π‘‘,(4.21) where 𝑐>0 is a constant. Since Imπœ†π‘>0 for the eigenvalues πœ†π‘,𝑝=1,…,𝑗, of 𝐿, (4.21) implies that ξ‚»πœ•π‘›πœ•πœ†π‘›ξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ†π‘βˆˆπΏ2ℝ+ξ€Έ,𝑛=0,1,…,π‘šπ‘βˆ’1,𝑝=1,2,…,𝑗.(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 0β‰€π‘›β‰€π‘šπ‘βˆ’1 and 𝑗+1β‰€π‘β‰€π‘˜.

Let πœ‡1,…,πœ‡π‘£, and πœ‡π‘£+1,…,πœ‡π‘™ be the zeros of 𝐸+ and πΈβˆ’ in β„βˆ—=ℝ⧡{0} (which are the spectral singularities of 𝐿) with multiplicities 𝑛1,…,𝑛𝑣 and 𝑛𝑣+1,…,𝑛𝑙, respectively.

Similar to (4.3) and (4.5) we can show the following:ξ‚»πœ•π‘›πœ•πœ†π‘›ξ‚Όπœ“(π‘₯,πœ†)πœ†=πœ‡π‘=π‘›ξ“π‘š=0πΆπ‘šξ€·πœ‡π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ‡π‘,(4.23)

𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣,

whereπΆπ‘šξ€·πœ‡π‘ξ€Έξ‚΅π‘›π‘šπœ•=βˆ’ξ‚Άξ‚»π‘›βˆ’π‘šπœ•πœ†π‘›βˆ’π‘šπΈβˆ’ξ‚Ό(πœ†)πœ†=πœ‡π‘,ξ‚»πœ•π‘›πœ•πœ†π‘›ξ‚Όπœ“(π‘₯,πœ†)πœ†=πœ‡π‘=π‘›ξ“π‘š=0π·π‘šξ€·πœ‡π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,βˆ’πœ†)πœ†=πœ‡π‘,(4.24)

𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=𝑣+1,…,𝑙,

whereπ·π‘šξ€·πœ‡π‘ξ€Έ=ξ‚΅π‘›π‘šπœ•ξ‚Άξ‚»π‘›βˆ’π‘šπœ•πœ†π‘›βˆ’π‘šπΈ+ξ‚Ό(πœ†)πœ†=πœ‡π‘.(4.25) Now define the generalized eigenfunctions and generalized associated functions corresponding to the spectral singularities of 𝐿 by the following:πœπ‘›,π‘ξ‚»πœ•(π‘₯)=π‘›πœ•πœ†π‘›ξ‚Όπœ“(π‘₯,πœ†)πœ†=πœ‡π‘=π‘›ξ“π‘š=0πΆπ‘šξ€·πœ‡π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,πœ†)πœ†=πœ‡π‘,(4.26)𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣, πœπ‘›,π‘ξ‚»πœ•(π‘₯)=π‘›πœ•πœ†π‘›ξ‚Όπœ“(π‘₯,πœ†)πœ†=πœ‡π‘=π‘›ξ“π‘š=0π·π‘šξ€·πœ‡π‘ξ€Έξ‚»πœ•π‘šπœ•πœ†π‘šξ‚Όπ‘’(π‘₯,βˆ’πœ†)πœ†=πœ‡π‘,(4.27)𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=𝑣+1,…,𝑙.

Then πœπ‘›,𝑝,𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙, also satisfy the equations analogous to (4.18).

𝜐0,𝑝,𝜐1,𝑝,…,πœπ‘›π‘βˆ’1,𝑝,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙 are called the principal functions corresponding to the spectral singularities πœ†=πœ‡π‘,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙 of 𝐿.

Theorem 4.4. One has πœπ‘›,π‘βˆ‰πΏ2ℝ+ξ€Έ,𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙.(4.28)

Proof. If we consider (4.21) for the principal functions corresponding to the spectral singularities πœ†=πœ‡π‘,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙, of 𝐿 and consider that Imπœ†π‘=0 for the spectral singularities, then we have (4.28), by (4.26) and (4.27).

Now introduce the Hilbert spaces𝐻𝑛=ξ‚»ξ€œπ‘“βˆΆβˆž0(1+π‘₯)2𝑛||||𝑓(π‘₯)2𝐻𝑑π‘₯<∞,𝑛=1,2,…,βˆ’π‘›=ξ‚»ξ€œπ‘”βˆΆβˆž0(1+π‘₯)βˆ’2𝑛||||𝑔(π‘₯)2𝑑π‘₯<∞,𝑛=1,2,…,(4.29) with‖𝑓‖2𝑛=ξ€œβˆž0(1+π‘₯)2𝑛||||𝑓(π‘₯)2𝑑π‘₯;‖𝑔‖2βˆ’π‘›=ξ€œβˆž0(1+π‘₯)βˆ’2𝑛||||𝑔(π‘₯)2𝑑π‘₯,(4.30) respectively. Then𝐻𝑛+1⫋𝐻𝑛⫋𝐿2ℝ+ξ€Έβ«‹π»βˆ’π‘›β«‹π»βˆ’(𝑛+1),𝑛=1,2,…,(4.31) and π»βˆ’π‘› is isomorphic to the dual of 𝐻𝑛.

Theorem 4.5. One has πœπ‘›,π‘βˆˆπ»βˆ’(𝑛+1),𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙.(4.32)

Proof. From (2.4) we have ξ€œβˆž0(1+π‘₯)βˆ’2(𝑛+1)||(𝑖π‘₯)π‘›π‘’π‘–πœ‡π‘π‘₯||2ξ€œπ‘‘π‘₯<∞,∞0(1+π‘₯)βˆ’2(𝑛+1)||||ξ€œβˆžπ‘₯(𝑖𝑑)𝑛𝐾(π‘₯,𝑑)π‘’π‘–πœ‡π‘π‘‘||||𝑑𝑑2𝑑π‘₯<∞.(4.33) Using (4.26), (4.33) we obtain πœπ‘›,π‘βˆˆπ»βˆ’(𝑛+1),𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣.(4.34) In a similar way, we find πœπ‘›,π‘βˆˆπ»βˆ’(𝑛+1),𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=𝑣+1,…,𝑙.(4.35)

Let us choose 𝑛0 so that𝑛0𝑛=max1,𝑛2…,𝑛𝑣,𝑛𝑣+1,…,𝑛𝑙.(4.36)

By Theorem 4.5 and (4.31) we get following theorem

Theorem 4.6. One has πœπ‘›,π‘βˆˆπ»βˆ’π‘›0,𝑛=0,1,…,π‘›π‘βˆ’1,𝑝=1,2,…,𝑣,𝑣+1,…,𝑙.(4.37)

Acknowledgment

The author would like to thank Professor E. Bairamov for his helpful suggestions during the preparation of this work.

References

  1. M. A. Naĭmark, β€œInvestigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,” American Mathematical Society Translations, vol. 16, pp. 103–193, 1960. View at Google Scholar
  2. V. E. Lyance, β€œA differential operator with spectral singularities I,” American Mathematical Society Transactions Series 2, vol. 60, pp. 185–225, 1967. View at Google Scholar
  3. V. E. Lyance, β€œA differential operator with spectral singularities II,” American Mathematical Society Transactions Series 2, vol. 60, pp. 227–283, 1967. View at Google Scholar
  4. A. M. Krall, β€œThe adjoint of a differential operator with integral boundary conditions,” Proceedings of the American Mathematical Society, vol. 16, pp. 738–742, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. A. M. Krall, β€œA nonhomogeneous eigenfunction expansion,” Transactions of the American Mathematical Society, vol. 117, pp. 352–361, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. A. M. Krall, β€œSecond order ordinary differential operators with general boundary conditions,” Duke Mathematical Journal, vol. 32, pp. 617–625, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. A. M. Krall, β€œNonhomogeneous differential operators,” The Michigan Mathematical Journal, vol. 12, pp. 247–255, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. A. M. Krall, β€œOn non-self-adjoint ordinary differential operators of the second order,” Doklady Akademii Nauk SSSR, vol. 165, pp. 1235–1237, 1965. View at Google Scholar
  9. M. G. Gasymov, β€œExpansion in terms of the solutions of a scattering theory problem for the non-selfadjoint Schrödinger equation,” Soviet Mathematics—Doklady, vol. 9, pp. 390–393, 1968. View at Google Scholar
  10. M. G. Gasymov and F. G. Maksudov, β€œThe principal part of the resolvent of nonselfadjoint operators in the neighborhood of spectral singularities,” Functional Analysis and Its Applications, vol. 6, no. 3, pp. 185–192, 1972. View at Google Scholar
  11. M. Adıvar and E. Bairamov, β€œSpectral properties of non-selfadjoint difference operators,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461–478, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  12. M. Adıvar and E. Bairamov, β€œDifference equations of second order with spectral singularities,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 714–721, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. E. Bairamov, Ö. Çakar, and A. M. Krall, β€œAn eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities,” Journal of Differential Equations, vol. 151, no. 2, pp. 268–289, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  14. E. Bairamov, Ö. Çakar, and A. M. Krall, β€œNon-selfadjoint difference operators and Jacobi matrices with spectral singularities,” Mathematische Nachrichten, vol. 229, pp. 5–14, 2001. View at Google Scholar Β· View at Zentralblatt MATH
  15. E. Bairamov, Ö. Çakar, and A. M. Krall, β€œSpectral properties, including spectral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis,” Quaestiones Mathematicae, vol. 26, no. 1, pp. 15–30, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. E. Bairamov, Ö. Cakar, and C. Yanik, β€œSpectral singularities of the Klein-Gordon s-wave equation,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 6, pp. 851–857, 2001. View at Google Scholar
  17. E. Bairamov and A. O. Çelebi, β€œSpectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,” The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 50, no. 200, pp. 371–384, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  18. E. Bairamov and Ö. Karaman, β€œSpectral singularities of Klein-Gordon s-wave equations with an integral boundary condition,” Acta Mathematica Hungarica, vol. 97, no. 1-2, pp. 121–131, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  19. A. M. Krall, E. Bairamov, and Ö. Çakar, β€œSpectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition,” Journal of Differential Equations, vol. 151, no. 2, pp. 252–267, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  20. A. M. Krall, E. Bairamov, and Ö. Çakar, β€œSpectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities,” Mathematische Nachrichten, vol. 231, pp. 89–104, 2001. View at Google Scholar Β· View at Zentralblatt MATH
  21. V. A. Marchenko, β€œExpansion in eigenfunctions of non-selfadjoint singular second order differential operators,” American Mathematical Society Transactions Series 2, vol. 25, pp. 77–130, 1963. View at Google Scholar
  22. V. A. Marchenko, Sturm-Liouville Operators and Applications, vol. 22, Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1986.
  23. M. Jaulent and C. Jean, β€œThe inverse s-wave scattering problem for a class of potentials depending on energy,” Communications in Mathematical Physics, vol. 28, pp. 177–220, 1972. View at Publisher Β· View at Google Scholar
  24. E. Bairamov and N. Yokus, β€œSpectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions,” Abstract and Applied Analysis, p. Art. ID 289596, 8, 2009. View at Google Scholar
  25. M. V. Keldysh, β€œOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,” Soviet Mathematics—Doklady, vol. 77, no. 4, pp. 11–14, 1951. View at Google Scholar
  26. M. V. Keldysh, β€œOn the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,” Russian Mathematical Surveys, vol. 26, no. 4, pp. 15–41, 1971. View at Google Scholar