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

Authors: I. Dehghani Tazehkand; A. Jodayree Akbarfam
Journal: ISRN Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 [1–7] and the bibliographies thereof). In recent years there has been considerable interest in the spectral theory of Sturm-Liouville operators on graphs (see [8–10]). The direct spectral and scattering problems on compact and noncompact graphs, respectively, were considered in many publications (see, e.g., [11–15]). 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 graphG 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 ej∈E is viewed as an interval [0,a]. Parametrizing ej∈E 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:(1.1)-yj′′(x)+qj(x)yj(x)=λ2yj(x),x∈[0,a],j=1,2,3,
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:(1.2)yi(a)=yj(a)fori,j=1,2,3,(continuitycondition),∑j=13yj′(a)+βy1(a,λ)=0(Kirchhoff′scondition),
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:(1.3)y1(0)=y2(0)=y3′(0)-hy3(0)=0,
where h 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 potentialq(x)=[qj(x)]j=1,2,3  and the real numbers h 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:(1.4)-yj′′(x)+qj(x)yj(x)=λ2yj(x),x∈[0,a],yj(0)=yj(a)=0,j=1,2,-y3′′(x)+q3(x)y3(x)=λ2y3(x),x∈[0,a],y3′(0)-hy3(0)=y3(a)=0,
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 h 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 byLj′, j=1,2,3 the following boundary-value problems:(1.5)-yj′′(x)(x)+qj(x)yj(x)=λ2yj(x),x∈[0,a],yj(0)=y′j(a)=0,j=1,2,-y3′′(x)+q3(x)y3(x)=λ2y3(x),x∈[0,a],y3′(0)-hy3(0)=y3′(a)=0.
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 h 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 spaceH=L2(0,a)⊕L2(0,a)⊕L2(0,a) with standard inner product (·,·)H, according to the formulas(1.6)AY=A(y1(x)y2(x)y3(x))=(-y1′′(x)+q1(x)y1(x)-y2′′(x)+q2(x)y2(x)-y3′′(x)+q3(x)y3(x)),D(A)={(y1(x)y2(x)y3(x))|yj(x)∈W22(0,a)forj=1,2,3,yi(a)=yj(a)fori,j=1,2,3,∑j=13y′j(a)+βy1(a)=0,y1(0)=y2(0)=y3′(0)-hy3(0)=0},
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 (A≫0). Furthermore, integrating by parts, we obtain the following equality for any vector function Y=(y1(x),y2(x),y3(x))t∈D(A) (t denotes the transpose of a matrix):(1.7)(AY,Y)H=∑j=13∫0a(|y′j(x)|2+qj(x)|yj(x)|2)dx+β|y1(a)|2+h|y3(0)|2.
Relation (1.7) yields the following simple sufficient condition for the strict positivity of the operator A:(1.8)qj(x)≥ϵ>0a.e.on⁡[0,a],j=1,2,3,β≥0,h≥0.
On the other hand, if A≫0, then setting in turn Y=(y1(x),0,0)t∈D(A),Y=(0,y2(x),0)t∈D(A) and Y=(0,0,y3(x))t∈D(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 λ2-q0,q0>0, in (1.1). For this reason, we assume in what follows without loss of generality that A≫0. 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 Section2 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 problemsL0 and Lj,j=1,2,3 that are necessary for what follows.Let us denote bycj(x,λ),sj(x,λ),j=1,2,3 the solutions of (1.1) on the edge ej which satisfy the initial conditions(2.1)c′j(0,λ)=cj(0,λ)-1=0,sj(0,λ)=sj′(0,λ)-1=0.
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), are(2.2)yj(x,λ)=Cjuj(x,λ),j=1,2,3,
where Cj, j=1,2,3 are constants and(2.3)uj(x,λ)={sj(x,λ),j=1,2,c3(x,λ)+hs3(x,λ),j=3.
Substituting (2.2) into (1.2), we establish that the eigenvalues of the boundary-value problem L0 are zeros of the entire function(2.4)Φ(λ)∶=|u1(a,λ)-u2(a,λ)0u1(a,λ)0-u3(a,λ)u1′(a,λ)+βu1(a,λ)u2′(a,λ)u3′(a,λ)|or(2.5)Φ(λ)=∑i=13(u′i(a,λ)∏j=1j≠i3uj(a,λ))+β∏j=13uj(a,λ).
For what follows, we need the definition presented below.Definition 2.1 (see [21]).
Let{zk}-∞∞({zk}-∞,k≠0∞) 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)
z-k=-zk¯(Rezk≠0);(ii)
Rezk≤Rezk+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 Section2, denote(2.6)Bj={12∫0aqj(x)dx,j=1,2,h+12∫0aq3(x)dx,j=3.
We introduce the entire function(2.7)Ψ(λ)=∏j=13uj(a,λ).
Let us denote by {λk}-∞,k≠0∞ the set of zeros of Φ(λ) and by {κk}-∞,k≠0∞ the set of zeros of the function Ψ(λ). Denote by {νk(j)}-∞,k≠0∞, 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}-∞,k≠0∞  is the union of the sets ⋃j=13{νk(j)}-∞,k≠0∞, 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}-∞,k≠0∞,{νk(j)}-∞,k≠0∞,j=1,2,3 and {κk}-∞,k≠0∞ in the proper way (λ-k=-λk, λk≤λk+1, ν-k(j)=-νk(j), νk(j)<νk+1(j)  for j=1,2,3 and κ-k=-κk, κk≤κk+1). Note that the sets of eigenvalues {νk(j)}-∞,k≠0∞, j=1,2,3 behave asymptotically as follows (see [20, Section 1.5]):(2.8)νk(j)=kπa+Bjπk+δk(j)k,j=1,2,νk(3)=π(k-(1/2))a+B3π(k-(1/2))+δk(3)k,
where {δk(j)}-∞k≠0∞∈l2 for j=1,2,3.Let us denote byLd, 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 functionsΦ(λ) and Ψ(λ) can be represented as follows:
(2.9)Φ(λ)=2sin⁡λa-3sin⁡3λaλ+(2B1+2B2+3B3+β)sin⁡2λacos⁡λaλ2-(B1+B2)cos⁡3λaλ2+ω1(λ)λ2,(2.10)Ψ(λ)=sin⁡2λacos⁡λaλ2-(B1+B2)cos⁡2λasin⁡λaλ3+B3sin⁡3λaλ3+ω2(λ)λ3,
where ω1(λ),ω2(λ)∈L3a.Proof.
Using the formulas of [19, page 18], [20, page 9] and taking into account that
(2.11)∫0af(t)cos⁡λtdt∈La,∫0af(t)sin⁡λtdt∈La,
whenever f∈L2(0,a) by the Paley-Wiener theorem [23, page 103], we obtain
(2.12)uj(a,λ)=sin⁡λaλ+ϱj1(λ)λ=sin⁡λaλ-Bjcos⁡λaλ2+ϱj2(λ)λ2,j=1,2,u3(a,λ)=cos⁡λa+ϱ31(λ)=cos⁡λa+B3sin⁡λaλ+ϱ32(λ)λ,uj′(a,λ)=cos⁡λa+Bjsin⁡λaλ+σj(λ)λ,j=1,2,u3′(a,λ)=-λsin⁡λa+B3cos⁡λa+σ3(λ),
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}-∞,k≠0∞ of zeros of  Φ(λ) can be represented as the union of three pairwise disjoint subsequences  ⋃j=13{λk(j)}-∞,k≠0∞which, being enumerated in the following way:  λ-k(1)=-λk(1),λ-k(2)=-λk(3),λ-k(3)=-λk(2), and  λk(j)≤λk+1(j)  for  j=1,2,3, behave asymptotically as follows:
(2.13)λk(1)=kπa+B1+B22kπ+γk(1)k,(2.14)λk(j)=kπ+(-1)jsin⁡-12/3a+3B1+3B2+6B3+2β12kπ+γk(j)k,j=2,3,
where {γk(j)}-∞,k≠0∞∈l2 for j=1,2,3.Proof.
In the same way as [16, Lemma 1.3], we can show that the set of zeros {λk}-∞,k≠0∞  can be arranged into three pairwise disjoint subsequences {λk(j)}-∞,k≠0∞, j=1,2,3 enumerated in the following way: λ-k(1)=-λk(1), λ-k(2)=-λk(3), λ-k(3)=-λk(2), and λk(j)≤λk+1(j) for j=1,2,3, such that {λk}-∞,k≠0∞=⋃j=13{λk(j)}-∞,k≠0∞, and
(2.15)λk(1)=kπa+εk(1),(2.16)λk(j)=kπ+(-1)jsin⁡-12/3a+εk(j),j=2,3,
where εk(j)=o(1), as k→∞ for j=1,2,3. It is not difficult to see that
(2.17)εk(j)=O(1k),k⟶∞,j=1,2,3.
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
(2.18)λk(1)Φ(λk(1))=(-1)k(2sin⁡εk(1)a-3sin⁡3εk(1)a)+(-1)ka(2B1+2B2+3B3+β)sin⁡2εk(1)acos⁡εk(1)akπ-(-1)ka(B1+B2)cos⁡3εk(1)akπ+aω1(λk(1))kπ+O(1k2)=(-1)k2sin⁡εk(1)a+O(1k)=0,k⟶∞.
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))}-∞,k≠0∞∈l2 (see [20, Lemma 1.4.3]), we obtain
(2.19)2εk(1)a-a(B1+B2)kπ+τkk=0,
where {τk}-∞,k≠0∞∈l2. 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 problemL0 with the sufficient condition which will be obtained in Section 3, we need more precise asymptotics.Theorem 2.4.
Letqj(x)∈W21(0,a) for j=1,2,3.  Then the subsequences of Theorem 2.3 behave asymptotically as follows:
(2.20)λk(1)=kπa+B1+B22kπ+γk(1)k2,(2.21)λk(j)=kπ+(-1)jsin⁡-12/3a+3B1+3B2+6B3+2β12kπ+γk(j)k2,j=2,3,
where {γk(j)}-∞,k≠0∞∈l2 for j=1,2,3.Proof.
Ifqj(x)∈W21(0,a), twice integrating by parts the formulas of [20, page 9] and [19, page 18], we obtain
(2.22)uj(a,λ)=sin⁡λaλ-Bjcos⁡λaλ2+Djsin⁡λaλ3+ϱj(λ)λ3,j=1,2,u3(a,λ)=cos⁡λa+B3sin⁡λaλ+D3cos⁡λaλ2+ϱ3(λ)λ2,uj′(a,λ)=cos⁡λa+Bjsin⁡λaλ+Dj′cos⁡λaλ2+σj(λ)λ2,j=1,2,u3′(a,λ)=-λsin⁡λa+B3cos⁡λa+D3′sin⁡λaλ+σ3(λ)λ,
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
(2.23)Φ(λ)=2sin⁡λa-3sin⁡3λaλ+(2B1+2B2+3B3+β)sin⁡2λacos⁡λaλ2-(B1+B2)cos⁡3λaλ2+E1sin⁡3λaλ3+E2cos⁡2λasin⁡λaλ3+ω3(λ)λ3,
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 Theorem2.4, the spectra {νk(j)}-∞,k≠0∞  of the boundary-value problems Lj  for j=1,2,3 behave asymptotically as follows (see [20, page 75]):
(2.24)νk(j)=kπa+Bjπk+δk(j)k2,j=1,2,νk(3)=π(k-(1/2))a+B3π(k-(1/2))+δk(3)k2,
where {δk(j)}-∞,k≠0∞∈l2 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.IfΩ⊆ℂ is an open set, we denote by ℋ(Ω) the set of all functions which are analytic in Ω and by ℳ(Ω) the set of all functions meromorphic in Ω.Definition 2.6 (see [24]).
Let𝒦⊆ℳ(ℂ) and let φ,ψ∈ℋ(ℂ).(i)
The pair(φ,ψ) is called a 1-𝒦-pair, if ψ-1φ∈𝒦 and φ and ψ have no common zeros.(ii)
Letn∈ℕ and n≥2. The pair (φ,ψ) is called an n-𝒦-pair, if ψ-1φ∈𝒦, and there exist 1-𝒦-pairs (φ1,ψ1),…,(φn,ψn) such that(2.25)ψ=∏i=1nψi,φ=∑i=1n(φi∏j=1j≠inψj),
and no representation of this kind is possible with less than n many 1-𝒦-pairs.Definition 2.7 (see [24]).
A functionf∈ℋ(ℂ∖ℝ) is said to be of Nevanlinna class 𝒩 if(i)
f(z)=f(z¯)¯forz∈ℂ∖ℝ;(ii)
Imf(z)≥0 for Imz>0.Definition 2.8 (see [24]).
The class𝒩ep of essentially positive Nenalinna functions is defined as the set of all functions f∈𝒩 which are analytic in ℂ∖[0,∞) with possible exception of finitely many poles. Moreover, the class 𝒩-ep is defined as the set of all functions f∈𝒩 such that for some b∈ℝ we have f∈ℋ(ℂ∖[b,∞)) and f(z)≤0 for z∈(-∞,b).It is easy to check that𝒩-ep⊆𝒩ep.Definition 2.9 (see [25]).
An entire functionω(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 ofω(z) lie in a strip |Imz|<h<∞;(ii)
for someh1 and all z∈{λ:Imz=h1}, the following inequalities hold:
(2.26)0<m≤|ω(z)|≤M<∞;(iii)
the type ofω(z) in the lower half-plane coincides with that in the upper half-plane.Let us introduce the entire functions(2.27)φj(z)=-uj′(a,z)-β3uj(a,z),j=1,2,3,(2.28)ψj(z)=uj(a,z),j=1,2,3,(2.29)φ(z)=-Φ(z),ψ(z)=Ψ(z).Using (2.5) and (2.7), we obtain(2.30)φ(z)=∑i=13(φi(z)∏j=1j≠i3ψj(z)),ψ(z)=∏j=13ψj(z)
and consequently(2.31)φ(z)ψ(z)=∑j=13φj(z)ψj(z).Lemma 2.10.
(1) the zeros of the functionsφj(z) and ψj(z)(j=1,2,3) are real;
(2) the functionsφj(z) and ψj(z)(j=1,2,3) have no common zeros.Proof.
The zeros ofφj(z), j=1,2,3 coincide with the squares of the eigenvalues of the boundary-value problems
(2.32)-yj′′(x)+qj(x)yj(x)=λ2yj(x),x∈[0,a],yj(0)=yj′(a)+β3yj(a)=0,j=1,2,-y3′′(x)+q3(x)y3(x)=λ2y3(x),x∈[0,a],y3′(0)-hy3(0)=y3′(a)+β3y3(a)=0,
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
(2.33)(z-z0)∫0auj(x,z)uj(x,z0)dx=(uj(x,z)uj′(x,z0)-uj′(x,z)uj(x,z0))|0a=φj(z)ψj(z0)-φj(z0)ψj(z).
For z→z0 we get
(2.34)∫0auj2(x,z0)dx=φ̇j(z0)ψj(z0)-φj(z0)ψ̇j(z0)=0,
where φ̇j(z)=(d/dz)φj(z) and ψ̇j(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 functionsφj(z)/ψj(z),j=1,2,3 and φ(z)/ψ(z) are of the Nevanlinna class 𝒩.Proof.
Letj∈{1,2,3}. Using the Lagrange identity for the solution uj(a,z) of (1.1), we have
(2.35)(uj′(x,z)¯uj(x,z)-uj′(x,z)uj(x,z)¯)|0a=2iIm⁡z∫0a|uj(x,z)|2dx.
Since
(2.36)Im⁡(uj′(a,z)¯uj(a,z)-uj′(a,z)uj(a,z)¯)=-2|uj(a,z)|2Im⁡uj′(a,z)uj(a,z),
then (2.35) yields
(2.37)-Im⁡uj′(a,z)uj(a,z)=Im⁡z∫0a|uj(x,z)|2dx|uj(a,z)|2,Im⁡z≠0.
Thus,
(2.38)Im⁡(-uj′(a,z)uj(a,z))≥0forIm⁡z>0
and consequently
(2.39)Im⁡φj(z)ψj(z)=Im⁡(-uj′(a,z)uj(a,z)-β3)≥0forIm⁡z>0.
Also, according to Lemma 2.10 the zeros of φj(z) and ψj(z) are real and hence φj(z)/ψj(z)∈ℋ(ℂ∖ℝ). Therefore, φj(z)/ψj(z)∈𝒩. Now it follows from (2.31) and (2.39) that φ(z)/ψ(z)∈ℋ(ℂ∖ℝ) and
(2.40)Im⁡φ(z)ψ(z)=∑j=13Im⁡φj(z)ψj(z)≥0forIm⁡z>0.
Consequently φ(z)/ψ(z)∈𝒩. Lemma 2.11 is proved.Lemma 2.12.
The functionsφj(z)/ψj(z),j=1,2,3 and φ(z)/ψ(z) are of the class 𝒩-ep.Proof.
By virtue of the formulas (2.12) we get
(2.41)uj(a,z)=e|z|a2|z|(1+o(1)),z⟶-∞,j=1,2,u3(a,z)=e|z|a2(1+o(1)),z⟶-∞,uj′(a,z)=e|z|a2(1+o(1)),z⟶-∞,j=1,2,u3′(a,z)=|z|e|z|a2(1+o(1)),z⟶-∞.
Using these asymptotics we obtain from (2.27) and (2.28)
(2.42)φj(z)ψj(z)=-|z|(1+o(1)),z⟶-∞,j=1,2,3,
and consequently
(2.43)lim⁡z→-∞φj(z)ψj(z)=-∞,j=1,2,3.
It follows form Lemmas 2.10 and 2.11, and (2.43) that there exist real numbers bj∈ℝ,j=1,2,3 such that
(2.44)φj(z)ψj(z)∈N∩H(C∖[bj,∞)),φj(z)ψj(z)<0forz∈(-∞,bj).
Therefore, φj(z)/ψj(z)∈𝒩-ep for j=1,2,3.
Now using (2.31) and (2.44) and Lemma 2.11, we conclude that
(2.45)φ(z)ψ(z)∈N∩H(C∖[b,∞)),φ(z)ψ(z)=∑j=13φj(z)ψj(z)<0forz∈(-∞,b),
where b=min{b1,b2,b3}. Thus, φ(z)/ψ(z)∈𝒩-ep. Lemma 2.12 is proved.Theorem 2.13.
The sequences{λk}-∞,k≠0∞ and {κk}-∞,k≠0∞ satisfy the following conditions:(1)
0<λ1<κ1≤λ2≤κ2≤⋯≤λk≤κk≤⋯(λ-k=-λk,κ-k=-κk);(2)
κk=λk+1 if and only if λk+1=κk+1  for k∈ℕ;(3)
the maximal multiplicity ofκk is 3.Proof.
Denote𝒩-ep°:=ℳ(ℂ)∩𝒩-ep. 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)∈𝒩-ep° for j=1,2,3 and φ(z)/ψ(z)∈𝒩-ep°. 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-𝒩-ep°-pairs and consequently, in view of (2.30), the pair (φ,ψ) is an m-𝒩-ep°-pair with some m≤3 (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 potentialq(x)=[qj(x)]j=1,2,3 and the real numbers h, β from the given spectral characteristics. Let us denote by Q the class of sets {[qj(x)]j=1,2,3,h,β} which satisfy the following conditions:(i)
qj(x), j=1,2,3 are real-valued functions from L2(0,a);(ii)
h,β∈ℝ;(iii)
the operatorA constructed via (1.6) is strictly positive.Theorem 3.1.
Let the following conditions be satisfied.(1)
Three sequences{νk(j)}-∞,k≠0∞, j=1,2,3 of real numbers are such that
(i)
ν-k(j)=-νk(j),νk(j)<νk+1(j),νk(j)≠0 for all k∈ℕ and j=1,2,3;(ii)
{νk(i)}-∞,k≠0∞⋂{νk(j)}-∞,k≠0∞=∅ for i≠j,i,j=1,2,3;(iii)
one has(3.1)νk(j)=πka+Bjπk+δk(j)k2,j=1,2,(3.2)νk(3)=π(k-(1/2))a+B3π(k-(1/2))+δk(3)k2,
where Bj are real constants, Bi≠Bj for i≠j and {δk(j)}-∞,k≠0∞∈l2 for j=1,2,3.(2)
A sequence{λk}-∞,k≠0∞ of real numbers (λ-k=-λk, λk≤λk+1, λk≠0 for all k∈ℕ) can be represented as the union of three pairwise disjoint subsequences {λk}-∞,k≠0∞=⋃j=13{λk(j)}-∞,k≠0∞(λ-k(1)=-λk(1),λ-k(2)=-λk(3), λ-k(3)=-λk(2) and λk(j)≤λk+1(j)  for j=1,2,3) which behave asymptotically as follows:(3.3)λk(1)=kπa+B1+B22kπ+γk(1)k2,(3.4)λk(j)=kπ+(-1)jsin⁡-12/3a+B0kπ+γk(j)k2,j=2,3,
where B0 is a real constant and {γk(j)}-∞,k≠0∞∈l2  for j=1,2,3.(3)
The sequences{λk}-∞,k≠0∞ and {κk}-∞∞:=⋃j=13{νk(j)}-∞,k≠0∞⋃{0}(κ-k=-κk,κk<κk+1) interlace in the following strict sense:(3.5)⋯<κ-2<λ-2<κ-1<λ-1<κ0=0<λ1<κ1<λ2<κ2<⋯.
Then there exists a unique set {[qj(x)]j=1,2,3,h,β}∈Q such that the sequence {λk}-∞,k≠0∞ coincide with the spectrum of the boundary-value problem L0, where β=6B0-(3/2)B1-(3/2)B2-3B3, h=B3-(1/2)∫0aq3(x)dx and the sequences {νk(j)}-∞,k≠0∞,j=1,2,3 coincides with the spectra of the boundary-value problems Lj, j=1,2,3, respectively.Proof.
Denote by(3.6){ρk(0)}-∞,k≠0∞∶={πk-ξa}-∞,k≠0∞⋃{πk+ξa}-∞,k≠0∞,ξ:=sin⁡-123,{ρk}-∞,k≠0∞∶={λk(2)}-∞,k≠0∞⋃{λk(3)}-∞,k≠0∞.
It is possible to enumerate {ρk(0)}-∞,k≠0∞ and {ρk}-∞,k≠0∞ in the proper way (ρ-k(0)=-ρk(0), ρk(0)<ρk+1(0) and ρ-k=-ρk, ρk≤ρk+1). Let us construct the following entire functions:
(3.7)uj(λ)=a∏1∞(a2π2k2(νk(j)2-λ2)),j=1,2,u3(λ)=∏1∞(a2π2(k-(1/2))2(νk(3)2-λ2)),ϕ1(λ)=a∏1∞(a2π2k2(λk(1)2-λ2)),ϕ2(λ)=2∏1∞(1ρk(0)2(ρk2-λ2)).
Using [27, Lemma 2.1], we obtain
(3.8)uj(λ)=sin⁡λaλ-Bjcos⁡λaλ2+Fjsin⁡λaλ3+fj(λ)λ3,j=1,2,
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
(3.9)u3(λ)=cos⁡λa+B3sin⁡λaλ+F3cos⁡λaλ2+f3(λ)λ2,ϕ1(λ)=sin⁡λaλ-(B1+B22)cos⁡λaλ2+G1sin⁡λaλ3+g1(λ)λ3,ϕ2(λ)=2-3sin⁡2λa+3B0sin⁡2λaλ+G22-3sin⁡2λaλ2+g2(λ)λ2,
where F3, Gj, j=1,2 are constants and f3(λ),g1(λ)∈La and g2(λ)∈L2a.
Let us set(3.10)Xk(j)∶=νk(j)(ϕ1(νk(j))ϕ2(νk(j))ui(νk(j))u3(νk(j))-cos⁡νk(j)a-Bjsin⁡νk(j)νk(j)),i,j=1,2,i≠j,(3.11)Xk(3)∶=(ϕ1(νk(3))ϕ2(νk(3))u1(νk(3))u2(νk(3))+νk(3)sin⁡νk(3)a-B3cos⁡νk(3)a)
where the numbers Bj,j=1,2,3 can be determined by
(3.12)Bj=lim⁡k→∞kπ(νk(j)-πka),j=1,2,B3=lim⁡k→∞π(k-12)(νk(3)-π(k-(1/2))a).
It is clear that X-k(j)=-Xk(j) for j=1,2 and X-k(3)=Xk(3). To complete the proof we need the following lemma.
Lemma 3.2.
One has(3.13){Xk(j)}-∞,k≠0∞∈l2forj=1,2,3.Proof.
Substituting (3.1) into (3.8) and (3.9), we obtain
(3.14)u2(νk(1))=(-1)ka2(B1-B2)π2k2+ζk(1)k3,u3(νk(1))=(-1)k(1-a2B122π2k2+a2B1B3π2k2+a2F3π2k2)+ζk(2)k2,ϕ1(νk(1))=(-1)ka2(B1-B2)π2k2+ζk(3)k3,ϕ2(νk(1))=2+ζk(4)k,
where {ζk(j)}-∞,k≠0∞∈l2 for j=1,4¯. Also, using (3.1), we obtain the asymptotic relation
(3.15)cos⁡νk(1)a+B1sin⁡νk(1)νk(1)=(-1)k(1+ηkk),
where {ηk}-∞,k≠0∞∈l2. If we substitute (3.14) and (3.15) into (3.10), then we conclude that {Xk(1)}-∞,k≠0∞∈l2. Analogously we can show that {Xk(2)}-∞,k≠0∞∈l2. We show that {Xk(3)}-∞,k≠0∞∈l2. Let us substitute (3.2) into (3.8) and (3.9). We obtain
(3.16)uj(νk(3))=(-1)k+1νk(3)(1+ζ′k(j)k),j=1,2,ϕ1(νk(3))=(-1)k+1νk(3)(1+ζ′k(3)k),ϕ2(νk(3))=-1+ζ′k(4)k,
where {ζk′(j)}-∞,k≠0∞∈l2 for j=1,4¯. Furthermore, taking into account (3.2), we have
(3.17)νk(3)sin⁡νk(3)a-B3cos⁡νk(3)a=(-1)k+1νk(3)(1+ηk′k),
where {ηk′}-∞,k≠0∞∈l2. Using (3.16) and (3.17) in (3.11), the assertion of Lemma 3.2 for j=3 follows.
Now since the functionsλuj(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), infk≠p|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
(3.18)λuj(λ)∑-∞k≠0∞Xk(j)dλuj(λ)/dλ|λ=νk(j)(λ-νk(j)),j=1,2,(3.19)u3(λ)∑-∞k≠0∞Xk(3)du3(λ)/dλ|λ=νk(3)(λ-νk(3)),
constructed on the basis of the sequences {Xk(j)}-∞,k≠0∞, define functions εj(λ)∈La, j=1,2,3, respectively (see [28, Theorem A]). Using these functions, we define the even entire functions
(3.20)vj(λ)=cos⁡λa+Bjsin⁡λaλ+εj(λ)λ,j=1,2,v3(λ)=-λsin⁡λa+B3cos⁡λa+ε3(λ).
It follows directly from (3.18) and (3.19) that εj(νk(j))=Xk(j) for j=1,2,3 and hence
(3.21)vj(νk(j))=ϕ1(νk(j))ϕ2(νk(j))ui(νk(j))u3(νk(j)),i,j=1,2,i≠j,v3(νk(3))=ϕ1(νk(3))ϕ2(νk(3))u1(νk(3))u2(νk(3)).
Let us denote by {μk(j)}-∞,k≠0∞ 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-k(j)=-μk(j), Reμk(j)≤Reμk+1(j) for all k∈ℕ 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-1(j)=0). It follows from (3.20) that
(3.22)μk(j)=π(k-(1/2))a+Bjπ(k-(1/2))+θk(j)k,j=1,2,(3.23)μk(3)=(k-1)πa+B3kπ+θk(3)k,
where {θk(j)}-∞,k≠0∞∈l2 for j=1,2,3.
Proposition 3.3.
The following inequalities are valid:(3.24)μ1(j)2<ν1(j)2<μ2(j)2<ν2(j)2<⋯,j=1,2,3.Proof.
In the same way as proof of [16, Proposition 2.3], we can show that
(3.25)(-1)kϕ1(νk(j))ϕ2(νk(j))ui(νk(j))u3(νk(j))>0,i,j=1,2,i≠j,(-1)kϕ1(νk(3))ϕ2(νk(3))u1(νk(3))u2(νk(3))>0.
From these inequalities and (3.21), it follows that
(3.26)(-1)kvj(νk(j))>0,j=1,2,3.
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 νk+1(j). Then comparing (3.22) and (3.23) with (3.1) and (3.2), we conclude that there are no μp(j)’s between some νk′(j) and νk′+1(j), where k≠k′, 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.
Letj∈{1,2}. It follows from (3.24) and the asymptotic relations (3.1) and (3.22) that the sequences {νk(j)}-∞,k≠0∞ and {μk(j)}-∞,k≠0∞ 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)}-∞,k≠0∞ and {μk(j)}-∞,k≠0∞ 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
(3.27)ej(λ)=e-iλa(vj(λ)+iλuj(λ))
is the so-called Jost function of the corresponding prolonged Sturm-Liouville problem on the semiaxis:
(3.28)-yj′′(x)+q̃j(x)yj(x)=λ2yj(x),x∈[0,∞),yj(0)=0,
where
(3.29)q̃j(x)={qj(x)ifx∈[0,a],0ifx∈(a,∞)
Then we construct the S-function of the problem on the semi-axis:
(3.30)Sj(λ)=ej(λ)ej(-λ)
and the function
(3.31)Fj(x)=12π∫-∞∞(1-Sj(λ))eiλxdλ.
Solving the Marchenko integral equation
(3.32)Kj(x,t)+Fj(x+t)+∫x∞Kj(x,s)Fj(x+s)ds=0,t>x,
we find the unique solution Kj(x,t) and
(3.33)qj(x)=-2dKj(x,x)dx,x∈[0,a].
The two sequences {νk(3)}-∞,k≠0∞ and {μk(3)}-∞,k≠0∞ 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 h such that {νk(3)}-∞,k≠0∞ and {μk(3)}-∞,k≠0∞ 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 L′3 by
(3.34)αk=12μk(3)v̇3(μk(3))u3(μk(3)),
where v̇3(λ)=(d/dλ)v3(λ). If μ1(3)=0, then v̇3(μ1(3))=0 and we set α1=(1/2)v̈3(0)u3(0) where v̈(λ)=(d2/dλ2)v3(λ). Construct the function
(3.35)F(x,t)=∑k=1∞(cos⁡μk(3)2xcos⁡μk(3)2tαk-cos⁡(k-1)xcos⁡(k-1)tαk0),
where
(3.36)αk0={a2,k>1,a,k=1.
Then using the unique solution of the Gel’fand-Levitan integral equation
(3.37)K3(x,t)+F(x,t)+∫0xK3(x,s)F(s,t)ds=0,0≤t≤x≤a,
we find
(3.38)q3(x)=2dK3(x,x)dx,h=K3(0,0)=B3-12∫0aq3(x)dx.
To find β, we compare (2.21) and (3.4) and set
(3.39)β=6B0-32B1-32B2-3B3,
where B0 can be determined by
(3.40)B0=lim⁡k→∞kπ(λk(2)-kπ+sin⁡-12/3a).
Now we prove that the spectrum of the problem L0 which is generated by the obtained [qj(x)]j=1,2,3, h and β coincides with {λk}-∞,k≠0∞. Due to [20, Theorem 3.4.1] and [19, Theorem 1.5.4], the spectra of the problems Lj,j=1,2,3 which are generated by the obtained [qj(x)]j=1,2,3 and h coincide with {νk(j)}-∞,k≠0∞, j=1,2,3, respectively. The functions uj(a,λ)(j=1,2,3) where uj(x,λ) is the solution of (1.1) with obtained [qj(x)]j=1,2,3 and h which satisfy the initial conditions (1.3) coincide with uj(λ), since they have the same zeros and the same asymptotics. Also, since uj′(a,λ) and vj(λ)(j=1,2,3) have the same asymptotics and according to [20, Theorem 3.4.1] and [19, Theorem 1.5.4] have the same zeros, hence they coincide. Thus, the values of the function Φ(λ) (defined by (2.5)) at λ=νk(j) coincide with
(3.41)ϕ1(νk(j))ϕ2(νk(j))
for all k∈ℤ∖{0} and all j=1,2,3, that is, with the corresponding values of the function ϕ1(λ)ϕ2(λ). This implies that the entire function
(3.42)Δ(λ)∶=Φ(λ)-ϕ1(λ)ϕ2(λ)
of exponential type 3a can be represented as follows:
(3.43)Δ(λ)=t(λ)∏j=13uj(a,λ),
where t(λ) is an entire function. Using (2.10) and (3.9) we have
(3.44)Δ(λ)=t(λ)(sin⁡2λacos⁡λaλ2-(B1+B2)cos⁡2λasin⁡λaλ3+B3sin⁡3λaλ3+ω1(λ)λ3),ϕ1(λ)ϕ2(λ)=2sin⁡λa-3sin⁡3λaλ+(6B0+12B1+12B2)sin⁡2λacos⁡λaλ2-(B1+B2)cos⁡3λaλ2+E1′sin⁡3λaλ3+E2′cos⁡2λasin⁡λaλ3+ω2(λ)λ3,
where E1′, E2′ are constants and ω1(λ),ω2(λ)∈L3a. Substituting (2.23) and (3.44) into (3.42) and using (3.39), we obtain
(3.45)t(λ)(λsin⁡2λacos⁡λa-(B1+B2)cos⁡2λasin⁡λa+B3sin⁡3λa+ω1(λ))=(E1-E1′)sin⁡3λa+(E2-E2′)cos⁡2λasin⁡λa+ω3(λ),
where ω3(λ)∈L3a. Since the functions sin3λa, cos2λasinλa, ω1(λ) and ω3(λ) are bounded on the real axis, hence relation (3.45) implies that t(λ)≡0 and from (3.42), it follows that Φ(λ)=ϕ1(λ)ϕ2(λ). Consequently, the sequence {λk}-∞,k≠0∞ coincides with the spectrum of the boundary-value problem L0. The operator A constructed by (1.6) using the obtained [qj(x)]j=1,2,3, h and β is strictly positive, because it is self-adjoint and its spectrum is positive. The uniqueness of the solution of the inverse problem follows from the fact that formulas (3.18) and (3.19) establishes one-to-one correspondence between l2 and La (see [28, Theorem A]). The proof of Theorem 3.1 is finished.Remark 3.4.
If condition 1(ii) of Theorem3.1 fails, that is, the sets {νk(j)}-∞,k≠0∞,j=1,2,3 are not pairwise disjoint (consequently, the condition 3 fails too), either the uniqueness or the existence result of mentioned theorem can also fails, for the same reasons as in the case of three spectra (see [27, 29]). If the sequences {λk}-∞,k≠0∞ and {κk}-∞∞=⋃j=13{νk(j)}-∞,k≠0∞⋃{0} are not pairwise disjoint and satisfy the statements of Theorem 2.13, then the solution of the inverse problem exists but is not unique.



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