Table of Contents
Retracted

This article has been retracted as it is essentially identical in content with the published article “Determination of Sturm-Liouville operator on a three-star graph from four spectra,” by I. Dehghani Tazehkand and A. Jodayree Akbarfam and published in Acta Universitatis Apulensis No. 32/2012, pp.147-172.

View the full Retraction here.

References

  1. I. Dehghani Tazehkand and A. J. Akbarfam, “An inverse spectral problem for the Sturm-Liouville operator on a three-star graph,” ISRN Applied Mathematics, vol. 2012, Article ID 132842, 23 pages, 2012.
ISRN Applied Mathematics
Volume 2012, Article ID 132842, 23 pages
http://dx.doi.org/10.5402/2012/132842
Research Article

An Inverse Spectral Problem for the Sturm-Liouville Operator on a Three-Star Graph

Faculty of Mathematical Sciences, University of Tabriz, 29 Bahman Boulevard, Tabriz, Iran

Received 10 January 2012; Accepted 15 March 2012

Academic Editor: D. Georges

Copyright © 2012 I. Dehghani Tazehkand and A. Jodayree Akbarfam. 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 work is properly cited.

Abstract

We study an inverse spectral problem for the Sturm-Liouville operator on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. As spectral characteristics, we consider the spectrum of the main problem together with the spectra of two Dirichlet-Dirichlet problems and one Robin-Dirichlet problem on the edges of the graph and investigate their properties and asymptotic behavior. We prove that if these four spectra do not intersect, then the inverse problem of recovering the operator is uniquely solvable. We give an algorithm for the solution of the inverse problem with respect to this quadruple of spectra.

1. Introduction

This paper is devoted to the study of the inverse spectral problem for Sturm-Liouville operators on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. The considered inverse problem consists of recovering the Sturm-Liouville operator on a graph from the given spectral characteristics. Differential operators on graphs (networks, trees) often appear in mathematics, mechanics, physics, geophysics, physical chemistry, electronics, nanoscale technology and branches of natural sciences and engineering (see [17] and the bibliographies thereof). In recent years there has been considerable interest in the spectral theory of Sturm-Liouville operators on graphs (see [810]). The direct spectral and scattering problems on compact and noncompact graphs, respectively, were considered in many publications (see, e.g., [1115]). The considered inverse spectral problem is not studied yet. However, inverse spectral problems of recovering differential operators on star-type graphs with the boundary conditions other than considered here were studied in [16, 17] and other papers. Hochstadt-Liberman type inverse problems on star-type graphs were investigated in [16, 18].

We consider a three-star graph 𝐺 with vertex set 𝑉={𝑣0,𝑣1,𝑣2,𝑣3} and edge set 𝐸={𝑒1,𝑒2,𝑒3}, where 𝑣1,𝑣2,𝑣3 are the boundary vertices, 𝑣0 is the internal vertex, and 𝑒𝑗=[𝑣𝑗,𝑣0] for 𝑗=1,2,3. We assume that the length of every edge is equal to 𝑎, 𝑎>0. Every edge 𝑒𝑗𝐸 is viewed as an interval [0,𝑎]. Parametrizing 𝑒𝑗𝐸 by 𝑥[0,𝑎], the following choice of orientation is convenient for us: 𝑥=0 corresponds to the boundary vertices 𝑣1,𝑣2,𝑣3 and 𝑥=𝑎 corresponds to the internal vertex 𝑣0. A function 𝑌 on 𝐺 may be represented as a vector 𝑌(𝑥)=[𝑦𝑗(𝑥)]𝑗=1,2,3,𝑥[0,𝑎] and the function 𝑦𝑗(𝑥) is defined on the edge 𝑒𝑗. Let 𝑞(𝑥)=[𝑞𝑗(𝑥)]𝑗=1,2,3 be a function on 𝐺 which is called the potential and 𝑞𝑗(𝑥)𝐿2(0,𝑎) is a real-valued function defined on the edge 𝑒𝑗. Let us consider the following Sturm-Liouville equations on 𝐺:𝑦𝑗(𝑥)+𝑞𝑗(𝑥)𝑦𝑗(𝑥)=𝜆2𝑦𝑗[](𝑥),𝑥0,𝑎,𝑗=1,2,3,(1.1) where 𝜆 is the spectral parameter. The functions 𝑦𝑗(𝑥) and 𝑦𝑗(𝑥) are absolutely continuous and satisfy the following matching conditions in the internal vertex 𝑣0:𝑦𝑖(𝑎)=𝑦𝑗(𝑎)for𝑖,𝑗=1,2,3,(continuitycondition),3𝑗=1𝑦𝑗(𝑎)+𝛽𝑦1(𝑎,𝜆)=0Kirchho,scondition(1.2) where 𝛽 is a real number. In electrical circuits, (1.2) expresses Kirchhof’s law; in an elastic string network, it expresses the balance of tension and so on. Let us denote by 𝐿0 the boundary-value problem for (1.1) with the matching conditions (1.2) and the following boundary conditions at the boundary vertices 𝑣1,𝑣2,𝑣3:𝑦1(0)=𝑦2(0)=𝑦3(0)𝑦3(0)=0,(1.3) where is a real number.

The problem of small transverse vibrations of a three-star graph consisting of three inhomogeneous smooth strings joined at the internal vertex with two pendent ends fixed and one pendent end which can move without friction in the directions orthogonal to their respective equilibrium positions can be reduced to this problem by the Liouville transformation. This problem occurs also in quantum mechanics when one considers a quantum particle subject to the Shrödinger equation moving in a quasi-one-dimensional graph domain.

In this paper, we study the inverse problem of recovering the potential 𝑞(𝑥)=[𝑞𝑗(𝑥)]𝑗=1,2,3  and the real numbers and 𝛽 from the given spectral characteristics. Similar inverse spectral problems on star-type graphs with three and arbitrary number of edges but only with the Dirichlet conditions at the boundary vertices were considered in [16, 17]. As spectral characteristics, we consider the set of eigenvalues of problem 𝐿0 together with the sets of eigenvalues of the following two Dirichlet-Dirichlet problems and one Robin-Dirichlet problem on the edges of the graph 𝐺:𝑦𝑗(𝑥)+𝑞𝑗(𝑥)𝑦𝑗(𝑥)=𝜆2𝑦𝑗[],𝑦(𝑥),𝑥0,𝑎𝑗(0)=𝑦𝑗(𝑎)=0,𝑗=1,2,𝑦3(𝑥)+𝑞3(𝑥)𝑦3(𝑥)=𝜆2𝑦3[],𝑦(𝑥),𝑥0,𝑎3(0)𝑦3(0)=𝑦3(𝑎)=0,(1.4) through which we denote these problems by 𝐿𝑗,𝑗=1,2,3. We obtain conditions for four sequences of real numbers that enable one to reconstruct the potential 𝑞(𝑥)=[𝑞𝑗(𝑥)]𝑗=1,2,3  and the real numbers and 𝛽 so that one of the sequences describes the spectrum of the boundary-value problem 𝐿0 and other three sequences coincide with the spectra of the problems 𝐿𝑗, 𝑗=1,2,3. We give an algorithm for the construction of the potential and the coefficients of the boundary and matching conditions corresponding to these four sequences.

Denote by 𝐿𝑗, 𝑗=1,2,3 the following boundary-value problems:𝑦𝑗(𝑥)(𝑥)+𝑞𝑗(𝑥)𝑦𝑗(𝑥)=𝜆2𝑦𝑗[],𝑦(𝑥),𝑥0,𝑎𝑗(0)=𝑦𝑗(𝑎)=0,𝑗=1,2,𝑦3(𝑥)+𝑞3(𝑥)𝑦3(𝑥)=𝜆2𝑦3[],𝑦(𝑥),𝑥0,𝑎3(0)𝑦3(0)=𝑦3(𝑎)=0.(1.5) The main idea of the solution of the inverse problem for the considered system is its reduction to three independent inverse problems of reconstruction of the functions 𝑞𝑗(𝑥)𝐿2(0,𝑎),𝑗=1,2,3 and on the basis of two spectra, namely, the spectrum of the problem 𝐿𝑗 and the spectrum of the problem 𝐿𝑗. Since the solutions of the later inverse problems are known (see [19, Section 1.5], [20, Section 3.4]), this reduction gives an algorithm for the reconstruction of the potential and coefficients of the boundary-value problem 𝐿0.

Let us consider the operator-theoretical interpretation of our problem. Denote by A the operator acting in the Hilbert space 𝐻=𝐿2(0,𝑎)𝐿2(0,𝑎)𝐿2(0,𝑎) with standard inner product (,)𝐻, according to the formulas𝑦𝐴𝑌=𝐴1𝑦(𝑥)2(𝑦𝑥)3=(𝑥)𝑦1(𝑥)+𝑞1(𝑥)𝑦1(𝑥)𝑦2(𝑥)+𝑞2(𝑥)𝑦2(𝑥)𝑦3(𝑥)+𝑞3(𝑥)𝑦3,𝐷𝑦(𝑥)(𝐴)=1𝑦(𝑥)2(𝑦𝑥)3||||||||||||||𝑦(𝑥)𝑗(𝑥)𝑊22(𝑦0,𝑎)for𝑗=1,2,3,𝑖(𝑎)=𝑦𝑗(𝑎)for𝑖,𝑗=1,2,3,3𝑗=1𝑦𝑗(𝑎)+𝛽𝑦1𝑦(𝑎)=0,1(0)=𝑦2(0)=𝑦3(0)𝑦3(,0)=0(1.6) where 𝑊22(0,𝑎) is a Sobolev space. By constructing the adjoint operator 𝐴, it is easy to show that 𝐴 is self-adjoint. The operator 𝐴 has a discrete spectrum and its eigenvalues coincide with the squares of the eigenvalues of the boundary-value problem 𝐿0. Thus, for all eigenvalues of the boundary-value problem 𝐿0 to be real and nonzero, it is necessary and sufficient that the operator 𝐴 be strictly positive (𝐴0). Furthermore, integrating by parts, we obtain the following equality for any vector function 𝑌=(𝑦1(𝑥),𝑦2(𝑥),𝑦3(𝑥))𝑡𝐷(𝐴) (𝑡 denotes the transpose of a matrix):(𝐴𝑌,𝑌)𝐻=3𝑗=1𝑎0||𝑦𝑗||(𝑥)2+𝑞𝑗||𝑦(𝑥)𝑗||(𝑥)2||𝑦𝑑𝑥+𝛽1||(𝑎)2||𝑦+3||(0)2.(1.7) Relation (1.7) yields the following simple sufficient condition for the strict positivity of the operator 𝐴:𝑞𝑗[](𝑥)𝜖>0a.e.on0,𝑎,𝑗=1,2,3,𝛽0,0.(1.8) On the other hand, if 𝐴0, then setting in turn 𝑌=(𝑦1(𝑥),0,0)𝑡𝐷(𝐴),𝑌=(0,𝑦2(𝑥),0)𝑡𝐷(𝐴) and 𝑌=(0,0,𝑦3(𝑥))𝑡𝐷(𝐴) in (1.7), we establish that the eigenvalues of the problems 𝐿𝑗, 𝑗=1,2,3 are also real and nonzero. The strict positivity of the operator 𝐴 can be realized by shifting the spectral parameter 𝜆2𝑞0,𝑞0>0, in (1.1). For this reason, we assume in what follows without loss of generality that 𝐴0. Thus, the eigenvalues of the boundary-value problems 𝐿0 and 𝐿𝑗,𝑗=1,2,3 are nonzero real numbers.

This paper has the following structure: in Section 2 the direct problem is considered. Aspects of the theory of entire and meromorphic functions are used as tools for a description of the set of eigenvalues of the boundary-value problem 𝐿0 and the spectra of the auxiliary problems 𝐿𝑗,𝑗=1,2,3 associated with this system. As a consequence we prove that the eigenvalues of the main problem and the spectra of the auxiliary problems interlace in some sense. In Section 3 we solve the inverse spectral problem for 𝐿0 within the framework of the statement indicated above.

2. Direct Problem

In this section, we describe the properties of sequences of eigenvalues of the boundary-value problems 𝐿0 and 𝐿𝑗,𝑗=1,2,3 that are necessary for what follows.

Let us denote by 𝑐𝑗(𝑥,𝜆),𝑠𝑗(𝑥,𝜆),𝑗=1,2,3 the solutions of (1.1) on the edge 𝑒𝑗 which satisfy the initial conditions𝑐𝑗(0,𝜆)=𝑐𝑗(0,𝜆)1=0,𝑠𝑗(0,𝜆)=𝑠𝑗(0,𝜆)1=0.(2.1) For each fixed 𝑥[0,𝑎], the functions 𝑐𝑗(𝜈)(𝑥,𝜆) and 𝑠𝑗(𝜈)(𝑥,𝜆),𝜈=0,1,𝑗=1,2,3 are entire in 𝜆. Since {𝑐𝑗(𝑥,𝜆),𝑠𝑗(𝑥,𝜆)} is a fundamental system of solutions of (1.1) on the edge 𝑒𝑗, then the solutions of (1.1), which satisfy the conditions (1.3), are𝑦𝑗(𝑥,𝜆)=𝐶𝑗𝑢𝑗(𝑥,𝜆),𝑗=1,2,3,(2.2) where 𝐶𝑗, 𝑗=1,2,3 are constants and𝑢𝑗𝑠(𝑥,𝜆)=𝑗𝑐(𝑥,𝜆),𝑗=1,2,3(𝑥,𝜆)+𝑠3(𝑥,𝜆),𝑗=3.(2.3) Substituting (2.2) into (1.2), we establish that the eigenvalues of the boundary-value problem 𝐿0 are zeros of the entire function|||||||||𝑢Φ(𝜆)=1(𝑎,𝜆)𝑢2𝑢(𝑎,𝜆)01(𝑎,𝜆)0𝑢3𝑢(𝑎,𝜆)1(𝑎,𝜆)+𝛽𝑢1(𝑎,𝜆)𝑢2(𝑎,𝜆)𝑢3|||||||||(𝑎,𝜆)(2.4)

orΦ(𝜆)=3𝑖=1𝑢𝑖(𝑎,𝜆)3𝑗=1𝑗𝑖𝑢𝑗(𝑎,𝜆)+𝛽3𝑗=1𝑢𝑗(𝑎,𝜆).(2.5) For what follows, we need the definition presented below.

Definition 2.1 (see [21]). Let {𝑧𝑘}({𝑧𝑘},𝑘0) be a sequence of complex numbers of finite multiplicities which satisfy the following conditions: (1) the sequence is symmetric with respect to the imaginary axis and symmetrically located numbers possess the same multiplicities; (2) any strip |Re𝑧|𝑝< contains not more than a finite number of 𝑧𝑘. Then, the following way of enumeration is called proper:(i)𝑧𝑘=𝑧𝑘(Re𝑧𝑘0);(ii)Re𝑧𝑘Re𝑧𝑘+1;(iii)the multiplicities are taken into account.

If a sequence has even number of pure imaginary elements, we exclude the index zero from enumeration to make it proper.

Throughout Section 2, denote𝐵𝑗=12𝑎0𝑞𝑗1(𝑥)𝑑𝑥,𝑗=1,2,+2𝑎0𝑞3(𝑥)𝑑𝑥,𝑗=3.(2.6) We introduce the entire functionΨ(𝜆)=3𝑗=1𝑢𝑗(𝑎,𝜆).(2.7) Let us denote by {𝜆𝑘},𝑘0 the set of zeros of Φ(𝜆) and by {𝜅𝑘},𝑘0 the set of zeros of the function Ψ(𝜆). Denote by {𝜈𝑘(𝑗)},𝑘0, 𝑗=1,2,3 the sets of zeros of the functions 𝑢𝑗(𝑎,𝜆), 𝑗=1,2,3, respectively. It is clear from (2.7) that the set {𝜅𝑘},𝑘0  is the union of the sets 3𝑗=1{𝜈𝑘(𝑗)},𝑘0, that is, the spectra of the auxiliary problems 𝐿𝑗,𝑗=1,2,3. According to the remark presented in Section 1, all numbers 𝜆𝑘,𝜈𝑘(𝑗),𝑗=1,2,3 and 𝜅𝑘 are real and nonzero. We enumerate the sets {𝜆𝑘},𝑘0,{𝜈𝑘(𝑗)},𝑘0,𝑗=1,2,3 and {𝜅𝑘},𝑘0 in the proper way (𝜆𝑘=𝜆𝑘, 𝜆𝑘𝜆𝑘+1, 𝜈(𝑗)𝑘=𝜈𝑘(𝑗), 𝜈𝑘(𝑗)<𝜈(𝑗)𝑘+1  for 𝑗=1,2,3 and 𝜅𝑘=𝜅𝑘, 𝜅𝑘𝜅𝑘+1). Note that the sets of eigenvalues {𝜈𝑘(𝑗)},𝑘0, 𝑗=1,2,3 behave asymptotically as follows (see [20, Section 1.5]):𝜈𝑘(𝑗)=𝑘𝜋𝑎+𝐵𝑗+𝛿𝜋𝑘𝑘(𝑗)𝑘𝜈,𝑗=1,2,𝑘(3)=𝜋(𝑘(1/2))𝑎+𝐵3+𝛿𝜋(𝑘(1/2))𝑘(3)𝑘,(2.8) where {𝛿𝑘(𝑗)}𝑘0𝑙2 for 𝑗=1,2,3.

Let us denote by 𝐿𝑑, 𝑑>0 the class (introduced in [22, page 149]) of entire functions of exponential type ≤𝑑 whose restrictions on the real line belong to 𝐿2(,).

Lemma 2.2. The functions Φ(𝜆) and Ψ(𝜆) can be represented as follows: Φ(𝜆)=2sin𝜆𝑎3sin3𝜆𝑎𝜆+2𝐵1+2𝐵2+3𝐵3+𝛽sin2𝜆𝑎cos𝜆𝑎𝜆2𝐵1+𝐵2cos3𝜆𝑎𝜆2+𝜔1(𝜆)𝜆2,(2.9)Ψ(𝜆)=sin2𝜆𝑎cos𝜆𝑎𝜆2𝐵1+𝐵2cos2𝜆𝑎sin𝜆𝑎𝜆3+𝐵3sin3𝜆𝑎𝜆3+𝜔2(𝜆)𝜆3,(2.10) where 𝜔1(𝜆),𝜔2(𝜆)𝐿3𝑎.

Proof. Using the formulas of [19, page 18], [20, page 9] and taking into account that 𝑎0𝑓(𝑡)cos𝜆𝑡𝑑𝑡𝐿𝑎,𝑎0𝑓(𝑡)sin𝜆𝑡𝑑𝑡𝐿𝑎,(2.11) whenever 𝑓𝐿2(0,𝑎) by the Paley-Wiener theorem [23, page 103], we obtain 𝑢𝑗(𝑎,𝜆)=sin𝜆𝑎𝜆+𝜚𝑗1(𝜆)𝜆=sin𝜆𝑎𝜆𝐵𝑗cos𝜆𝑎𝜆2+𝜚𝑗2(𝜆)𝜆2𝑢,𝑗=1,2,3(𝑎,𝜆)=cos𝜆𝑎+𝜚31(𝜆)=cos𝜆𝑎+𝐵3sin𝜆𝑎𝜆+𝜚32(𝜆)𝜆,𝑢𝑗(𝑎,𝜆)=cos𝜆𝑎+𝐵𝑗sin𝜆𝑎𝜆+𝜎𝑗(𝜆)𝜆𝑢,𝑗=1,2,3(𝑎,𝜆)=𝜆sin𝜆𝑎+𝐵3cos𝜆𝑎+𝜎3(𝜆),(2.12) where 𝜚𝑗1(𝜆),𝜚𝑗2(𝜆),𝜎𝑗(𝜆),𝑗=1,2,3, are entire functions of class 𝐿𝑎. Substituting (2.12) into (2.5) and (2.7), we get (2.9) and (2.10).

Theorem 2.3. The set {𝜆𝑘},𝑘0 of zeros of Φ(𝜆) can be represented as the union of three pairwise disjoint subsequences 3𝑗=1{𝜆𝑘(𝑗)},𝑘0which, being enumerated in the following way: 𝜆(1)𝑘=𝜆𝑘(1),𝜆(2)𝑘=𝜆𝑘(3),𝜆(3)𝑘=𝜆𝑘(2), and 𝜆𝑘(𝑗)𝜆(𝑗)𝑘+1  for 𝑗=1,2,3, behave asymptotically as follows: 𝜆𝑘(1)=𝑘𝜋𝑎+𝐵1+𝐵2+𝛾2𝑘𝜋𝑘(1)𝑘,𝜆(2.13)𝑘(𝑗)=𝑘𝜋+(1)𝑗sin12/3𝑎+3𝐵1+3𝐵2+6𝐵3+2𝛽+𝛾12𝑘𝜋𝑘(𝑗)𝑘,𝑗=2,3,(2.14) where {𝛾𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,2,3.

Proof. In the same way as [16, Lemma 1.3], we can show that the set of zeros {𝜆𝑘},𝑘0  can be arranged into three pairwise disjoint subsequences {𝜆𝑘(𝑗)},𝑘0, 𝑗=1,2,3 enumerated in the following way: 𝜆(1)𝑘=𝜆𝑘(1), 𝜆(2)𝑘=𝜆𝑘(3), 𝜆(3)𝑘=𝜆𝑘(2), and 𝜆𝑘(𝑗)𝜆(𝑗)𝑘+1 for 𝑗=1,2,3, such that {𝜆𝑘},𝑘0=3𝑗=1{𝜆𝑘(𝑗)},𝑘0, and 𝜆𝑘(1)=𝑘𝜋𝑎+𝜀𝑘(1),𝜆(2.15)𝑘(𝑗)=𝑘𝜋+(1)𝑗sin12/3𝑎+𝜀𝑘(𝑗),𝑗=2,3,(2.16) where 𝜀𝑘(𝑗)=𝑜(1), as 𝑘 for 𝑗=1,2,3. It is not difficult to see that 𝜀𝑘(𝑗)1=𝑂𝑘,𝑘,𝑗=1,2,3.(2.17) Substituting (2.15) into 𝜆𝑘(1)Φ(𝜆𝑘(1))=0, then from (2.9) and taking into account that the function 𝜔1(𝜆) is bounded on the real axis by the Paley-Wiener theorem, we obtain 𝜆𝑘(1)Φ𝜆𝑘(1)=(1)𝑘2sin𝜀𝑘(1)𝑎3sin3𝜀𝑘(1)𝑎+(1)𝑘𝑎2𝐵1+2𝐵2+3𝐵3+𝛽sin2𝜀𝑘(1)𝑎cos𝜀𝑘(1)𝑎𝑘𝜋(1)𝑘𝑎𝐵1+𝐵2cos3𝜀𝑘(1)𝑎+𝑘𝜋𝑎𝜔1𝜆𝑘(1)1𝑘𝜋+𝑂𝑘2=(1)𝑘2sin𝜀𝑘(1)1𝑎+𝑂𝑘=0,𝑘.(2.18) This yields sin𝜀𝑘(1)𝑎=𝑂(1/𝑘). Thus, 𝜀𝑘(1)=𝑂(1/𝑘). Similarly, we can show that 𝜀𝑘(𝑗)=𝑂(1/𝑘) for 𝑗=2,3. Substituting (2.15) into the equation 𝜆𝑘(1)Φ(𝜆𝑘(1))=0, where Φ(𝜆) is given by (2.9), by expanding the left-hand side of resulting equation in power series and taking into account (2.17) and {𝜔1(𝜆𝑘(1))},𝑘0𝑙2 (see [20, Lemma 1.4.3]), we obtain 2𝜀𝑘(1)𝑎𝐵𝑎1+𝐵2+𝜏𝑘𝜋𝑘𝑘=0,(2.19) where {𝜏𝑘},𝑘0𝑙2. Solving this equation we get (2.13). In the same way, we get (2.14).

To compare necessary conditions on a sequence to be the spectrum of the boundary-value problem 𝐿0 with the sufficient condition which will be obtained in Section 3, we need more precise asymptotics.

Theorem 2.4. Let 𝑞𝑗(𝑥)𝑊12(0,𝑎) for 𝑗=1,2,3.  Then the subsequences of Theorem 2.3 behave asymptotically as follows: 𝜆𝑘(1)=𝑘𝜋𝑎+𝐵1+𝐵2+𝛾2𝑘𝜋𝑘(1)𝑘2,𝜆(2.20)𝑘(𝑗)=𝑘𝜋+(1)𝑗sin12/3𝑎+3𝐵1+3𝐵2+6𝐵3+2𝛽+𝛾12𝑘𝜋𝑘(𝑗)𝑘2,𝑗=2,3,(2.21) where {𝛾𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,2,3.

Proof. If 𝑞𝑗(𝑥)𝑊12(0,𝑎), twice integrating by parts the formulas of [20, page 9] and [19, page 18], we obtain 𝑢𝑗(𝑎,𝜆)=sin𝜆𝑎𝜆𝐵𝑗cos𝜆𝑎𝜆2+𝐷𝑗sin𝜆𝑎𝜆3+𝜚𝑗(𝜆)𝜆3𝑢,𝑗=1,2,3(𝑎,𝜆)=cos𝜆𝑎+𝐵3sin𝜆𝑎𝜆+𝐷3cos𝜆𝑎𝜆2+𝜚3(𝜆)𝜆2,𝑢𝑗(𝑎,𝜆)=cos𝜆𝑎+𝐵𝑗sin𝜆𝑎𝜆+𝐷𝑗cos𝜆𝑎𝜆2+𝜎𝑗(𝜆)𝜆2𝑢,𝑗=1,2,3(𝑎,𝜆)=𝜆sin𝜆𝑎+𝐵3cos𝜆𝑎+𝐷3sin𝜆𝑎𝜆+𝜎3(𝜆)𝜆,(2.22) where 𝐷𝑗,𝐷𝑗,𝑗=1,2,3 are constants and 𝜚𝑗(𝜆),𝜎𝑗(𝜆),𝑗=1,2,3 are entire functions of class 𝐿𝑎. Substituting (2.22) into (2.5), we obtain Φ(𝜆)=2sin𝜆𝑎3sin3𝜆𝑎𝜆+2𝐵1+2𝐵2+3𝐵3+𝛽sin2𝜆𝑎cos𝜆𝑎𝜆2𝐵1+𝐵2cos3𝜆𝑎𝜆2+𝐸1sin3𝜆𝑎𝜆3+𝐸2cos2𝜆𝑎sin𝜆𝑎𝜆3+𝜔3(𝜆)𝜆3,(2.23) where 𝐸1, 𝐸2 are constants and 𝜔3(𝜆)𝐿3𝑎. Substituting (2.13) into the equation 𝜆𝑘(1)Φ(𝜆𝑘(1))=0, where Φ(𝜆) is given by (2.23) and by expanding the left-hand side of resulting equation in power series, we get (2.20). Analogously, we obtain (2.21). Theorem 2.4 is proved.

Remark 2.5. Under the conditions of Theorem 2.4, the spectra {𝜈𝑘(𝑗)},𝑘0  of the boundary-value problems 𝐿𝑗  for 𝑗=1,2,3 behave asymptotically as follows (see [20, page 75]): 𝜈𝑘(𝑗)=𝑘𝜋𝑎+𝐵𝑗+𝛿𝜋𝑘𝑘(𝑗)𝑘2𝜈,𝑗=1,2,𝑘(3)=𝜋(𝑘(1/2))𝑎+𝐵3+𝛿𝜋(𝑘(1/2))𝑘(3)𝑘2,(2.24) where {𝛿𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,2,3.

For investigation of direct and inverse spectral problems, methods of the theory of entire and meromorphic functions are widely used. For this reason, we give several notation and definitions for what follows.

If Ω is an open set, we denote by (Ω) the set of all functions which are analytic in Ω and by (Ω) the set of all functions meromorphic in Ω.

Definition 2.6 (see [24]). Let 𝒦() and let 𝜑,𝜓(). (i)The pair (𝜑,𝜓) is called a 1-𝒦-pair, if 𝜓1𝜑𝒦 and 𝜑 and 𝜓 have no common zeros.(ii)Let 𝑛 and 𝑛2. The pair (𝜑,𝜓) is called an 𝑛-𝒦-pair, if 𝜓1𝜑𝒦, and there exist 1-𝒦-pairs (𝜑1,𝜓1),,(𝜑𝑛,𝜓𝑛) such that𝜓=𝑛𝑖=1𝜓𝑖,𝜑=𝑛𝑖=1𝜑𝑖𝑛𝑗=1𝑗𝑖𝜓𝑗,(2.25) and no representation of this kind is possible with less than 𝑛 many 1-𝒦-pairs.

Definition 2.7 (see [24]). A function 𝑓() is said to be of Nevanlinna class 𝒩 if(i)𝑓(𝑧)=𝑓(𝑧)for𝑧;(ii)Im𝑓(𝑧)0 for Im𝑧>0.

Definition 2.8 (see [24]). The class 𝒩ep of essentially positive Nenalinna functions is defined as the set of all functions 𝑓𝒩 which are analytic in [0,) with possible exception of finitely many poles. Moreover, the class 𝒩ep is defined as the set of all functions 𝑓𝒩 such that for some 𝑏 we have 𝑓([𝑏,)) and 𝑓(𝑧)0 for 𝑧(,𝑏).

It is easy to check that 𝒩ep𝒩ep.

Definition 2.9 (see [25]). An entire function 𝜔(𝑧) of exponential type 𝜎>0 is said to be a function of sine-type if it satisfies the following conditions:(i)all the zeros of 𝜔(𝑧) lie in a strip |Im𝑧|<<;(ii)for some 1 and all 𝑧{𝜆Im𝑧=1}, the following inequalities hold: ||||0<𝑚𝜔(𝑧)𝑀<;(2.26)(iii)the type of 𝜔(𝑧) in the lower half-plane coincides with that in the upper half-plane.

Let us introduce the entire functions𝜑𝑗(𝑧)=𝑢𝑗𝑎,𝑧𝛽3𝑢𝑗𝑎,𝑧𝜓,𝑗=1,2,3,(2.27)𝑗(𝑧)=𝑢𝑗𝑎,𝑧,𝑗=1,2,3,(2.28)𝜑(𝑧)=Φ𝑧,𝜓(𝑧)=Ψ𝑧.(2.29)

Using (2.5) and (2.7), we obtain𝜑(𝑧)=3𝑖=1𝜑𝑖(𝑧)3𝑗=1𝑗𝑖𝜓𝑗(𝑧),𝜓(𝑧)=3𝑗=1𝜓𝑗(𝑧)(2.30) and consequently𝜑(𝑧)𝜓=(𝑧)3𝑗=1𝜑𝑗(𝑧)𝜓𝑗(𝑧).(2.31)

Lemma 2.10. (1) the zeros of the functions 𝜑𝑗(𝑧) and 𝜓𝑗(𝑧)(𝑗=1,2,3) are real;
(2) the functions 𝜑𝑗(𝑧) and 𝜓𝑗(𝑧)(𝑗=1,2,3) have no common zeros.

Proof. The zeros of 𝜑𝑗(𝑧), 𝑗=1,2,3 coincide with the squares of the eigenvalues of the boundary-value problems 𝑦𝑗(𝑥)+𝑞𝑗(𝑥)𝑦𝑗(𝑥)=𝜆2𝑦𝑗[],𝑦(𝑥),𝑥0,𝑎𝑗(0)=𝑦𝑗𝛽(𝑎)+3𝑦𝑗(𝑎)=0,𝑗=1,2,𝑦3(𝑥)+𝑞3(𝑥)𝑦3(𝑥)=𝜆2𝑦3[],𝑦(𝑥),𝑥0,𝑎3(0)𝑦3(0)=𝑦3𝛽(𝑎)+3𝑦3(𝑎)=0,(2.32) respectively, and the zeros of 𝜓𝑗(𝑧) coincide with the squares of the eigenvalues of the boundary-value problems 𝐿𝑗,𝑗=1,2,3, respectively. These problems are self-adjoint and it follows from [26, Part I, Theorem 3] that the squares of their eigenvalues are real. Assertion 1 is proved. To prove assertion 2, let 𝑧0 be a common zero of 𝜑𝑗(𝑧) and 𝜓𝑗(𝑧). Using the Lagrange identity(see [26, Part II, page 50]) for solutions 𝑢𝑗(𝑎,𝑧) and 𝑢𝑗(𝑎,𝑧0) of (1.1), we obtain 𝑧𝑧0𝑎0𝑢𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧0𝑢𝑑𝑥=𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧0𝑢𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧0|||𝑎0=𝜑𝑗(𝑧)𝜓𝑗𝑧0𝜑𝑗𝑧0𝜓𝑗(𝑧).(2.33) For 𝑧𝑧0 we get 𝑎0𝑢2𝑗𝑥,𝑧0𝑑𝑥=̇𝜑𝑗𝑧0𝜓𝑗𝑧0𝜑𝑗𝑧0̇𝜓𝑗𝑧0=0,(2.34) where ̇𝜑𝑗(𝑧)=(𝑑/𝑑𝑧)𝜑𝑗(𝑧) and ̇𝜓𝑗(𝑧)=(𝑑/𝑑𝑧)𝜓𝑗(𝑧). This implies that 𝑢𝑗(𝑥,𝑧0)0 which is a contradiction. Therefore, 𝜑𝑗(𝑧) and 𝜓𝑗(𝑧) have no common zeros.

Lemma 2.11. The functions 𝜑𝑗(𝑧)/𝜓𝑗(𝑧),𝑗=1,2,3 and 𝜑(𝑧)/𝜓(𝑧) are of the Nevanlinna class 𝒩.

Proof. Let 𝑗{1,2,3}. Using the Lagrange identity for the solution 𝑢𝑗(𝑎,𝑧) of (1.1), we have 𝑢𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧𝑢𝑗𝑥,𝑧||||𝑎0=2𝑖Im𝑧𝑎0|||𝑢𝑗𝑥,𝑧|||2𝑑𝑥.(2.35) Since Im𝑢𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧|||𝑢=2𝑗𝑎,𝑧|||2𝑢Im𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧,(2.36) then (2.35) yields 𝑢Im𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧=Im𝑧𝑎0|||𝑢𝑗𝑥,𝑧|||2𝑑𝑥|||𝑢𝑗𝑎,𝑧|||2,Im𝑧0.(2.37) Thus, 𝑢Im𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧0forIm𝑧>0(2.38) and consequently 𝜑Im𝑗(𝑧)𝜓𝑗𝑢(𝑧)=Im𝑗𝑎,𝑧𝑢𝑗𝑎,𝑧𝛽30forIm𝑧>0.(2.39) Also, according to Lemma 2.10 the zeros of 𝜑𝑗(𝑧) and 𝜓𝑗(𝑧) are real and hence 𝜑𝑗(𝑧)/𝜓𝑗(𝑧)(). Therefore, 𝜑𝑗(𝑧)/𝜓𝑗(𝑧)𝒩. Now it follows from (2.31) and (2.39) that 𝜑(𝑧)/𝜓(𝑧)() and Im𝜑(𝑧)𝜓=(𝑧)3𝑗=1𝜑Im𝑗(𝑧)𝜓𝑗(𝑧)0forIm𝑧>0.(2.40) Consequently 𝜑(𝑧)/𝜓(𝑧)𝒩. Lemma 2.11 is proved.

Lemma 2.12. The functions 𝜑𝑗(𝑧)/𝜓𝑗(𝑧),𝑗=1,2,3 and 𝜑(𝑧)/𝜓(𝑧) are of the class 𝒩ep.

Proof. By virtue of the formulas (2.12) we get 𝑢𝑗𝑎,𝑧=𝑒|𝑧|𝑎2𝑢|𝑧|(1+𝑜(1)),𝑧,𝑗=1,2,3𝑎,𝑧=𝑒|𝑧|𝑎2𝑢(1+𝑜(1)),𝑧,𝑗𝑎,𝑧=𝑒|𝑧|𝑎2𝑢(1+𝑜(1)),𝑧,𝑗=1,2,3𝑎,𝑧=|𝑧|𝑒|𝑧|𝑎2(1+𝑜(1)),𝑧.(2.41) Using these asymptotics we obtain from (2.27) and (2.28) 𝜑𝑗(𝑧)𝜓𝑗(𝑧)=|𝑧|(1+𝑜(1)),𝑧,𝑗=1,2,3,(2.42) and consequently lim𝑧𝜑𝑗(𝑧)𝜓𝑗(𝑧)=,𝑗=1,2,3.(2.43) It follows form Lemmas 2.10 and 2.11, and (2.43) that there exist real numbers 𝑏𝑗,𝑗=1,2,3 such that 𝜑𝑗(𝑧)𝜓𝑗𝑏(𝑧)𝒩𝐻𝑗,𝜑,𝑗(𝑧)𝜓𝑗(𝑧)<0for𝑧,𝑏𝑗.(2.44) Therefore, 𝜑𝑗(𝑧)/𝜓𝑗(𝑧)𝒩ep for 𝑗=1,2,3.
Now using (2.31) and (2.44) and Lemma 2.11, we conclude that 𝜑(𝑧)𝜓[(𝑧)𝒩(𝑏,)),𝜑(𝑧)𝜓=(𝑧)3𝑗=1𝜑𝑗(𝑧)𝜓𝑗(𝑧)<0for𝑧(,𝑏),(2.45) where 𝑏=min{𝑏1,𝑏2,𝑏3}. Thus, 𝜑(𝑧)/𝜓(𝑧)𝒩ep. Lemma 2.12 is proved.

Theorem 2.13. The sequences {𝜆𝑘},𝑘0 and {𝜅𝑘},𝑘0 satisfy the following conditions:(1)0<𝜆1<𝜅1𝜆2𝜅2𝜆𝑘𝜅𝑘(𝜆𝑘=𝜆𝑘,𝜅𝑘=𝜅𝑘);(2)𝜅𝑘=𝜆𝑘+1 if and only if 𝜆𝑘+1=𝜅𝑘+1  for 𝑘;(3)the maximal multiplicity of 𝜅𝑘 is 3.

Proof. Denote 𝒩ep=()𝒩ep. The functions 𝑢𝑗(𝑎,𝑧) and 𝑢𝑗(𝑎,𝑧) are entire in 𝑧 and hence in view of Lemma 2.12, 𝜑𝑗(𝑧)/𝜓𝑗(𝑧)𝒩ep for 𝑗=1,2,3 and 𝜑(𝑧)/𝜓(𝑧)𝒩ep. Also, by Lemma 2.10 the functions 𝜑𝑗(𝑧),𝜓𝑗(𝑧) have no common zeros. Therefore, the pairs (𝜑𝑗,𝜓𝑗),𝑗=1,2,3 are 1-𝒩ep-pairs and consequently, in view of (2.30), the pair (𝜑,𝜓) is an 𝑚-𝒩ep-pair with some 𝑚3 (see Definition 2.6). On the other hand by virtue of (2.29), the squares of the zeros of Φ(𝜆) and Ψ(𝜆) coincide with the zeros of 𝜑(𝑧) and 𝜓(𝑧), respectively. Now the assertions of Theorem 2.13 immediately follows from [24, Corollary 4.6].

3. Inverse Problem

In the present section, we study the problem of reconstruction of the potential 𝑞(𝑥)=[𝑞𝑗(𝑥)]𝑗=1,2,3 and the real numbers , 𝛽 from the given spectral characteristics. Let us denote by 𝑄 the class of sets {[𝑞𝑗(𝑥)]𝑗=1,2,3,,𝛽} which satisfy the following conditions:(i)𝑞𝑗(𝑥), 𝑗=1,2,3 are real-valued functions from 𝐿2(0,𝑎);(ii),𝛽;(iii)the operator 𝐴 constructed via (1.6) is strictly positive.

Theorem 3.1. Let the following conditions be satisfied.(1)Three sequences {𝜈𝑘(𝑗)},𝑘0, 𝑗=1,2,3 of real numbers are such that(i)𝜈(𝑗)𝑘=𝜈𝑘(𝑗),𝜈𝑘(𝑗)<𝜈(𝑗)𝑘+1,𝜈𝑘(𝑗)0 for all 𝑘 and 𝑗=1,2,3;(ii){𝜈𝑘(𝑖)},𝑘0{𝜈𝑘(𝑗)},𝑘0= for 𝑖𝑗,𝑖,𝑗=1,2,3;(iii)one has 𝜈𝑘(𝑗)=𝜋𝑘𝑎+𝐵𝑗+𝛿𝜋𝑘𝑘(𝑗)𝑘2𝜈,𝑗=1,2,(3.1)𝑘(3)=𝜋(𝑘(1/2))𝑎+𝐵3+𝛿𝜋(𝑘(1/2))𝑘(3)𝑘2,(3.2) where 𝐵𝑗 are real constants, 𝐵𝑖𝐵𝑗 for 𝑖𝑗 and {𝛿𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,2,3.(2)A sequence {𝜆𝑘},𝑘0 of real numbers (𝜆𝑘=𝜆𝑘, 𝜆𝑘𝜆𝑘+1, 𝜆𝑘0 for all 𝑘) can be represented as the union of three pairwise disjoint subsequences {𝜆𝑘},𝑘0=3𝑗=1{𝜆𝑘(𝑗)},𝑘0(𝜆(1)𝑘=𝜆𝑘(1),𝜆(2)𝑘=𝜆𝑘(3), 𝜆(3)𝑘=𝜆𝑘(2) and 𝜆𝑘(𝑗)𝜆(𝑗)𝑘+1  for 𝑗=1,2,3) which behave asymptotically as follows:𝜆𝑘(1)=𝑘𝜋𝑎+𝐵1+𝐵2+𝛾2𝑘𝜋𝑘(1)𝑘2,𝜆(3.3)𝑘(𝑗)=𝑘𝜋+(1)𝑗sin12/3𝑎+𝐵0+𝛾𝑘𝜋𝑘(𝑗)𝑘2,𝑗=2,3,(3.4) where 𝐵0 is a real constant and {𝛾𝑘(𝑗)},𝑘0𝑙2  for 𝑗=1,2,3.(3)The sequences {𝜆𝑘},𝑘0 and {𝜅𝑘}=3𝑗=1{𝜈𝑘(𝑗)},𝑘0{0}(𝜅𝑘=𝜅𝑘,𝜅𝑘<𝜅𝑘+1) interlace in the following strict sense:<𝜅2<𝜆2<𝜅1<𝜆1<𝜅0=0<𝜆1<𝜅1<𝜆2<𝜅2<.(3.5) Then there exists a unique set {[𝑞𝑗(𝑥)]𝑗=1,2,3,,𝛽}𝑄 such that the sequence {𝜆𝑘},𝑘0 coincide with the spectrum of the boundary-value problem 𝐿0, where 𝛽=6𝐵0(3/2)𝐵1(3/2)𝐵23𝐵3, =𝐵3(1/2)𝑎0𝑞3(𝑥)𝑑𝑥 and the sequences {𝜈𝑘(𝑗)},𝑘0,𝑗=1,2,3 coincides with the spectra of the boundary-value problems 𝐿𝑗, 𝑗=1,2,3, respectively.

Proof. Denote by 𝜌𝑘(0),𝑘0=𝜋𝑘𝜉𝑎,𝑘0𝜋𝑘+𝜉𝑎,𝑘0,𝜉=sin123,𝜌𝑘,𝑘0𝜆=𝑘(2),𝑘0𝜆𝑘(3),𝑘0.(3.6) It is possible to enumerate {𝜌𝑘(0)},𝑘0 and {𝜌𝑘},𝑘0 in the proper way (𝜌(0)𝑘=𝜌𝑘(0), 𝜌𝑘(0)<𝜌(0)𝑘+1 and 𝜌𝑘=𝜌𝑘, 𝜌𝑘𝜌𝑘+1). Let us construct the following entire functions: 𝑢𝑗(𝜆)=𝑎1𝑎2𝜋2𝑘2𝜈𝑘(𝑗)2𝜆2𝑢,𝑗=1,2,3(𝜆)=1𝑎2𝜋2(𝑘(1/2))2𝜈𝑘(3)2𝜆2,𝜙1(𝜆)=𝑎1𝑎2𝜋2𝑘2𝜆𝑘(1)2𝜆2,𝜙2(𝜆)=211𝜌𝑘(0)2𝜌2𝑘𝜆2.(3.7) Using [27, Lemma 2.1], we obtain 𝑢𝑗(𝜆)=sin𝜆𝑎𝜆𝐵𝑗cos𝜆𝑎𝜆2+𝐹𝑗sin𝜆𝑎𝜆3+𝑓𝑗(𝜆)𝜆3,𝑗=1,2,(3.8) where 𝐹𝑗, 𝑗=1,2 are constants and 𝑓𝑗(𝜆)𝐿𝑎 for 𝑗=1,2. In the same way as [27, Lemma 2.1] we can prove that 𝑢3(𝜆)=cos𝜆𝑎+𝐵3sin𝜆𝑎𝜆+𝐹3cos𝜆𝑎𝜆2+𝑓3(𝜆)𝜆2,𝜙1(𝜆)=sin𝜆𝑎𝜆𝐵1+𝐵22cos𝜆𝑎𝜆2+𝐺1sin𝜆𝑎𝜆3+𝑔1(𝜆)𝜆3,𝜙2(𝜆)=23sin2𝜆𝑎+3𝐵0sin2𝜆𝑎𝜆+𝐺223sin2𝜆𝑎𝜆2+𝑔2(𝜆)𝜆2,(3.9) where 𝐹3, 𝐺𝑗, 𝑗=1,2 are constants and 𝑓3(𝜆),𝑔1(𝜆)𝐿𝑎 and 𝑔2(𝜆)𝐿2𝑎.
Let us set 𝑋𝑘(𝑗)=𝜈𝑘(𝑗)𝜙1𝜈𝑘(𝑗)𝜙2𝜈𝑘(𝑗)𝑢𝑖𝜈𝑘(𝑗)𝑢3𝜈𝑘(𝑗)cos𝜈𝑘(𝑗)𝑎𝐵𝑗sin𝜈𝑘(𝑗)𝜈𝑘(𝑗)𝑋,𝑖,𝑗=1,2,𝑖𝑗,(3.10)𝑘(3)𝜙=1𝜈𝑘(3)𝜙2𝜈𝑘(3)𝑢1𝜈𝑘(3)𝑢2𝜈𝑘(3)+𝜈𝑘(3)sin𝜈𝑘(3)𝑎𝐵3cos𝜈𝑘(3)𝑎(3.11) where the numbers 𝐵𝑗,𝑗=1,2,3 can be determined by 𝐵𝑗=lim𝑘𝜈𝑘𝜋𝑘(𝑗)𝜋𝑘𝑎𝐵,𝑗=1,2,3=lim𝑘𝜋1𝑘2𝜈𝑘(3)𝜋(𝑘(1/2))𝑎.(3.12) It is clear that 𝑋(𝑗)𝑘=𝑋𝑘(𝑗) for 𝑗=1,2 and 𝑋(3)𝑘=𝑋𝑘(3). To complete the proof we need the following lemma.
Lemma 3.2. One has 𝑋𝑘(𝑗),𝑘0𝑙2for𝑗=1,2,3.(3.13)Proof. Substituting (3.1) into (3.8) and (3.9), we obtain 𝑢2𝜈𝑘(1)=(1)𝑘𝑎2𝐵1𝐵2𝜋2𝑘2+𝜁𝑘(1)𝑘3,𝑢3𝜈𝑘(1)=(1)𝑘𝑎12𝐵212𝜋2𝑘2+𝑎2𝐵1𝐵3𝜋2𝑘2+𝑎2𝐹3𝜋2𝑘2+𝜁𝑘(2)𝑘2,𝜙1𝜈𝑘(1)=(1)𝑘𝑎2𝐵1𝐵2𝜋2𝑘2+𝜁𝑘(3)𝑘3,𝜙2𝜈𝑘(1)𝜁=2+𝑘(4)𝑘,(3.14) where {𝜁𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,4. Also, using (3.1), we obtain the asymptotic relation cos𝜈𝑘(1)𝑎+𝐵1sin𝜈𝑘(1)𝜈𝑘(1)=(1)𝑘𝜂1+𝑘𝑘,(3.15) where {𝜂𝑘},𝑘0𝑙2. If we substitute (3.14) and (3.15) into (3.10), then we conclude that {𝑋𝑘(1)},𝑘0𝑙2. Analogously we can show that {𝑋𝑘(2)},𝑘0𝑙2. We show that {𝑋𝑘(3)},𝑘0𝑙2. Let us substitute (3.2) into (3.8) and (3.9). We obtain 𝑢𝑗𝜈𝑘(3)=(1)𝑘+1𝜈𝑘(3)1+𝜁𝑘(𝑗)𝑘𝜙,𝑗=1,2,1𝜈𝑘(3)=(1)𝑘+1𝜈𝑘(3)1+𝜁𝑘(3)𝑘,𝜙2𝜈𝑘(3)=1+𝜁𝑘(4)𝑘,(3.16) where {𝜁𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,4. Furthermore, taking into account (3.2), we have 𝜈𝑘(3)sin𝜈𝑘(3)𝑎𝐵3cos𝜈𝑘(3)𝑎=(1)𝑘+1𝜈𝑘(3)𝜂1+𝑘𝑘,(3.17) where {𝜂𝑘},𝑘0𝑙2. Using (3.16) and (3.17) in (3.11), the assertion of Lemma 3.2 for 𝑗=3 follows.
Now since the functions 𝜆𝑢𝑗(𝜆), 𝑗=1,2 and 𝑢3(𝜆) are sine-type functions (see Definition 2.9) and by virtue of (3.1), (3.2), and (3.5), inf𝑘𝑝|𝜈𝑘(𝑗)𝜈𝑝(𝑗)|>0 for 𝑗=1,2,3 (and hence the zeros of 𝜆𝑢𝑗(𝜆),𝑗=1,2 and 𝑢3(𝜆) are simple), the Lagrange interpolation series 𝜆𝑢𝑗(𝜆)𝑘0𝑋𝑘(𝑗)𝑑𝜆𝑢𝑗||(𝜆)/𝑑𝜆𝜆=𝜈𝑘(𝑗)𝜆𝜈𝑘(𝑗)𝑢,𝑗=1,2,(3.18)3(𝜆)𝑘0𝑋𝑘(3)𝑑𝑢3||(𝜆)/𝑑𝜆𝜆=𝜈𝑘(3)𝜆𝜈𝑘(3),(3.19) constructed on the basis of the sequences {𝑋𝑘(𝑗)},𝑘0, define functions 𝜀𝑗(𝜆)𝐿𝑎, 𝑗=1,2,3, respectively (see [28, Theorem A]). Using these functions, we define the even entire functions 𝑣𝑗(𝜆)=cos𝜆𝑎+𝐵𝑗sin𝜆𝑎𝜆+𝜀𝑗(𝜆)𝜆𝑣,𝑗=1,2,3(𝜆)=𝜆sin𝜆𝑎+𝐵3cos𝜆𝑎+𝜀3(𝜆).(3.20) It follows directly from (3.18) and (3.19) that 𝜀𝑗(𝜈𝑘(𝑗))=𝑋𝑘(𝑗) for 𝑗=1,2,3 and hence 𝑣𝑗𝜈𝑘(𝑗)=𝜙1𝜈𝑘(𝑗)𝜙2𝜈𝑘(𝑗)𝑢𝑖𝜈𝑘(𝑗)𝑢3𝜈𝑘(𝑗)𝑣,𝑖,𝑗=1,2,𝑖𝑗,3𝜈𝑘(3)=𝜙1𝜈𝑘(3)𝜙2𝜈𝑘(3)𝑢1𝜈𝑘(3)𝑢2𝜈𝑘(3).(3.21) Let us denote by {𝜇𝑘(𝑗)},𝑘0 the set of zeros of the functions 𝑣𝑗(𝜆), 𝑗=1,2,3, respectively. These sets are symmetric with respect to the real axis and to the imaginary axis. Hence, we number the zeros in the proper way: 𝜇(𝑗)𝑘=𝜇𝑘(𝑗), Re𝜇𝑘(𝑗)Re𝜇(𝑗)𝑘+1 for all 𝑘 and the multiplicities are taken into account (we will prove that all 𝜇𝑘(𝑗)2 are real and all 𝜇𝑘(𝑗) are simple except for 𝜇1(𝑗), if 𝜇1(𝑗)=𝜇(𝑗)1=0). It follows from (3.20) that 𝜇𝑘(𝑗)=𝜋(𝑘(1/2))𝑎+𝐵𝑗+𝜃𝜋(𝑘(1/2))𝑘(𝑗)𝑘𝜇,𝑗=1,2,(3.22)𝑘(3)=(𝑘1)𝜋𝑎+𝐵3+𝜃𝑘𝜋𝑘(3)𝑘,(3.23) where {𝜃𝑘(𝑗)},𝑘0𝑙2 for 𝑗=1,2,3.
Proposition 3.3. The following inequalities are valid: 𝜇1(𝑗)2<𝜈1(𝑗)2<𝜇2(𝑗)2<𝜈2(𝑗)2<,𝑗=1,2,3.(3.24)Proof. In the same way as proof of [16, Proposition 2.3], we can show that (1)𝑘𝜙1𝜈𝑘(𝑗)𝜙2𝜈𝑘(𝑗)𝑢𝑖𝜈𝑘(𝑗)𝑢3𝜈𝑘(𝑗)>0,𝑖,𝑗=1,2,𝑖𝑗,(1)𝑘𝜙1𝜈𝑘(3)𝜙2𝜈𝑘(3)𝑢1𝜈𝑘(3)𝑢2𝜈𝑘(3)>0.(3.25) From these inequalities and (3.21), it follows that (1)𝑘𝑣𝑗𝜈𝑘(𝑗)>0,𝑗=1,2,3.(3.26) Let 𝑗{1,2,3} be fixed. It follows from (3.26) that between consecutive 𝜈𝑘(𝑗)’s there is an odd number (with account of multiplicities) of 𝜇𝑘(𝑗)’s. Suppose that there are three or more of them between 𝜈𝑘(𝑗) and 𝜈(𝑗)𝑘+1. Then comparing (3.22) and (3.23) with (3.1) and (3.2), we conclude that there are no 𝜇𝑝(𝑗)’s between some 𝜈(𝑗)𝑘 and 𝜈(𝑗)𝑘+1, where 𝑘𝑘, a contradiction. Thus, 𝜈1(𝑗)2<𝜇2(𝑗)2<𝜈2(𝑗)2<. If 𝑣𝑗(0)>0, then 0<𝜇1(𝑗)<𝜈1(𝑗). If 𝑣𝑗(0)=0, then 𝜇1(𝑗)=0. If 𝑣𝑗(0)<0, then 𝜇1(𝑗) is a pure imaginary number and hence 𝜇1(𝑗)2<𝜈1(𝑗)2. Proposition 3.3 is proved.
Let 𝑗{1,2}. It follows from (3.24) and the asymptotic relations (3.1) and (3.22) that the sequences {𝜈𝑘(𝑗)},𝑘0 and {𝜇𝑘(𝑗)},𝑘0 satisfy the conditions of [20, Theorem 3.4.1]. Thus, there exists a unique real-valued function 𝑞𝑗(𝑥)𝐿2(0,𝑎) such that {𝜈𝑘(𝑗)},𝑘0 and {𝜇𝑘(𝑗)},𝑘0 are the spectra of the boundary-value problems 𝐿𝑗 and 𝐿𝑗, respectively. An algorithm for the reconstruction of this potential 𝑞𝑗(𝑥) is as follows (see [20, Section 3.4]): without loss of generality, let us assume that 𝜇1(𝑗)2>0, otherwise we apply a shift. The function 𝑒𝑗(𝜆)=𝑒𝑖𝜆𝑎𝑣𝑗(𝜆)+𝑖𝜆𝑢𝑗(𝜆)(3.27) is the so-called Jost function of the corresponding prolonged Sturm-Liouville problem on the semiaxis: 𝑦𝑗(𝑥)+̃𝑞𝑗(𝑥)𝑦𝑗(𝑥)=𝜆2𝑦𝑗[(𝑥),𝑥0,),𝑦𝑗(0)=0,(3.28) where ̃𝑞𝑗𝑞(𝑥)=𝑗[],(𝑥)if𝑥0,𝑎0if𝑥(𝑎,)(3.29) Then we construct the 𝑆-function of the problem on the semi-axis: 𝑆𝑗𝑒(𝜆)=𝑗(𝜆)𝑒𝑗(𝜆)(3.30) and the function 𝐹𝑗1(𝑥)=2𝜋1𝑆𝑗𝑒(𝜆)𝑖𝜆𝑥𝑑𝜆.(3.31) Solving the Marchenko integral equation 𝐾𝑗(𝑥,𝑡)+𝐹𝑗(𝑥+𝑡)+𝑥𝐾𝑗(𝑥,𝑠)𝐹𝑗(𝑥+𝑠)𝑑𝑠=0,𝑡>𝑥,(3.32) we find the unique solution 𝐾𝑗(𝑥,𝑡) and 𝑞𝑗(𝑥)=2𝑑𝐾𝑗(𝑥,𝑥)[]𝑑𝑥,𝑥0,𝑎.(3.33) The two sequences {𝜈𝑘(3)},𝑘0 and {𝜇𝑘(3)},𝑘0 satisfy (due to (3.2), (3.23) and (3.24)) the conditions of [19, Theorem 1.5.4]. Thus, there exists a unique real-valued function 𝑞3(𝑥)𝐿2(0,𝑎) and a unique real number such that {𝜈𝑘(3)},𝑘0 and {𝜇𝑘(3)},𝑘0 are the spectra of the boundary-value problems 𝐿3 and 𝐿3, respectively. Below we give the algorithm of recovering of 𝑞3(𝑥) as it is described in [19, Section 1.5]. Calculate the so-called weight numbers {𝛼𝑘}1 of the problem 𝐿3 by 𝛼𝑘=12𝜇𝑘(3)̇𝑣3𝜇𝑘(3)𝑢3𝜇𝑘(3),(3.34) where ̇𝑣3(𝜆)=(d/𝑑𝜆)𝑣3(𝜆). If 𝜇1(3)=0, then ̇𝑣3(𝜇1(3))=0 and we set 𝛼1̈𝑣=(1/2)3(0)𝑢3(0) where ̈𝑣(𝜆)=(𝑑2/𝑑𝜆2)𝑣3(𝜆). Construct the function 𝐹(𝑥,𝑡)=𝑘=1cos𝜇𝑘(3)2𝑥cos𝜇𝑘(3)2𝑡𝛼𝑘cos(𝑘1)𝑥cos(𝑘1)𝑡𝛼0𝑘,(3.35) where 𝛼0𝑘=𝑎2,𝑘>1,𝑎,𝑘=1.(3.36) Then using the unique solution of the Gel’fand-Levitan integral equation 𝐾3(𝑥,𝑡)+𝐹(𝑥,𝑡)+𝑥0𝐾3(𝑥,𝑠)𝐹(𝑠,𝑡)𝑑