On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra

Abstract

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

1. Introduction and Main Result

In this paper, we test another formulation of the inverse spectral problem for Sturm–Liouville-type operators on star-shaped graphs with global delay, which were introduced in [1]. This formulation considerably reduces the input data compared with those used for recovering classical Sturm–Liouville operators on trees. Moreover, the new formulation enables gathering all necessary spectral information by altering boundary conditions at only one boundary vertex, which becomes a new quality in the inverse spectral theory on geometrical graphs.

Purely differential (local) operators on graphs, often called quantum graphs, model various processes in networks and branching structures appearing in organic chemistry, mesoscopic physics, quantum mechanics, nanotechnology, hydrodynamics, waveguide theory, and other applications [2,3,4,5,6,7,8,9,10,11]. The theory of quantum graphs includes numerous studies related to inverse problems of recovering operators from their spectral characteristics (see [12,13,14,15,16,17,18,19,20] and references therein). In particular, it was established in [15] that, for the unique determination of the Sturm–Liouville potentials on all edges of an arbitrary compact tree with m boundary vertices, it is sufficient to specify the spectra of precisely m especially chosen boundary value problems. The way how those spectra are chosen formally generalizes the classical inverse Sturm–Liouville problem on an interval due to Borg [21,22,23]. Moreover, such input data remain minimal among other formulations of inverse spectral problems for purely differential operators on trees.

Concerning functional-differential as well as other classes of nonlocal operators on graphs, they were mostly studied in the locally nonlocal case when the corresponding nonlocal equation on each edge can be treated independently of the other edges [24,25,26,27,28,29,30]. This limitation has left uncertain how the nonlocalities could coexist with the internal vertices of the graph.

In [1], however, we introduced Sturm–Liouville-type operators on geometrical graphs with a global delay, which naturally extends through the internal vertices. On a star-shaped graph consisting of m equal edges, this idea can be illustrated by the boundary value problem

$$-y_j^{\prime \prime }\left(x\right)+q_j\left(x\right)y_j(x-a)=\lambda y_j\left(x\right),0<x<1,j=\overline{1,m},$$

(1)

$$y_j(x-a)=y_1(x-a+1),max\{0,a-1\}<x<min\{a,1\},j=\overline{2,m},$$

(2)

$$y_1\left(1\right)=y_j\left(0\right),j=\overline{2,m},y_1^{\prime }\left(1\right)=\sum _{j=2}^m{y}_j^{\prime }\left(0\right),$$

(3)

$$y_1^{\left({\nu }_1\right)}\left(0\right)+{\nu }_1{H}_1{y}_1\left(0\right)=0,V_j\left(y_j\right):=y_j^{\left({\nu }_j\right)}\left(1\right)+{\nu }_j{H}_j{y}_j\left(1\right)=0,j=\overline{2,m},$$

(4)

where $\lambda$ is the spectral parameter and $a\in (0,2),$ while ${\nu }_j\in \{0,1\}$ and $H_j\in \mathbb{C}$ for $j=\overline{1,m}.$ It is assumed that $q_j\left(x\right)$ are complex-valued functions in $L_2(0,1)$ that obey the conditions

$$q_1\left(x\right)=0,x\in (0,min\{a,1\}),q_j\left(x\right)=0,x\in (0,max\{0,a-1\}),j=\overline{2,m},$$

(5)

making the equations in (1) well defined. Supposing the j-th equation in (1) to be defined on the edge $e_j$ of the graph ${\Gamma }_m$ illustrated on Figure 1, one can observe that relations (3) specify the standard matching conditions at the internal vertex $v_1,$ while (4) becomes Dirichlet and/or Robin boundary conditions at the boundary vertices $v_0$ and $v_j$ for $j=\overline{2,m}.$

Relations (1)–(5) generate a nonlocal operator with the global delay a measured in the direction of the boundary vertex $v_0,$ which we refer to as root. Conditions (2) specify an initial function for all equations in (1) except the first one. In other words, those equations always involve the unknown function $y_1\left(x\right)$ on edge $e_1.$ This means that the delay “extends” or “propagates” through the internal vertex $v_1.$ In [1], this new feature was also generalized in a natural way to operators on an arbitrary tree with variable edge lengths.

Moreover, in [1], an inverse problem was studied for the particular case when $m=3$ and $a=1$, which consisted in recovering the remaining functions $q_2\left(x\right)$ and $q_3\left(x\right)$ from the spectra corresponding to the two sets of parameters: ${\nu }_1={\nu }_2={\nu }_3-1=0$ and ${\nu }_1={\nu }_2-1={\nu }_3=0,$ while $H_2=H_3=0.$ That statement resembles the inverse problem posed in [15] for the classical Sturm–Liouville operator $(a=0)$ on a tree, when the input spectral data are gathered by altering the boundary conditions at all boundary vertices except the root. The statement in [15], however, would also involve the third spectrum corresponding to ${\nu }_1={\nu }_2={\nu }_3=0.$

In the present paper, we suggest another statement of the inverse problem for (1)–(5) that is assumed to always involve only two spectra obtained by altering the boundary conditions namely at the root, while at the rest boundary vertices, the corresponding conditions remain fixed. This new statement will be tested below for arbitrary m and $a\in [1,2).$

For $\nu =0,1,$ denote by ${\left\{{\lambda }_{n,\nu }\right\}}_{n\in \mathbb{N}}$ the spectrum of the eigenvalue problem ${\mathcal{B}}_{\nu }:={\mathcal{B}}_{\nu }(q,H)$ consisting of relations (1)–(5) under the settings:

$${\nu }_1=\nu ,H_1=0,{\nu }_j=1,j=\overline{2,m},$$

and depending on $q:=[q_1,\dots ,q_m]$ and $H:=[H_2,\dots ,H_m].$

Consider the following inverse problem.

Inverse Problem 1.

Given ${\left\{{\lambda }_{n,0}\right\}}_{n\in \mathbb{N}}$ and ${\left\{{\lambda }_{n,1}\right\}}_{n\in \mathbb{N}}$ as well as the value of $a,$ find $q_j\left(x\right)$ for $j=\overline{1,m}$ provided that $H_j,j=\overline{2,m},$ are known a priori and distinct.

We note that the distinction as well as the knowledge of the coefficients $H_j$ are, obviously, vital whenever one aims to recover each $q_j\left(x\right)$ with reference to the corresponding edge $e_j.$

In the present study, we restrict ourselves to the case $a\in [1,2),$ when the dependence of the characteristic determinants of the problems ${\mathcal{B}}_{\nu }(q,H)$ on q is linear (see the next section for more details). The nonlinear case $a\in (0,1)$ will be discussed at the bottom of this section.

Note that admitted values of $a,$ according to (5), automatically imply $q_1=0,$ while all remaining $q_j\left(x\right)$ vanish a.e. on $(0,a-1).$ The following uniqueness theorem holds.

Theorem 1.

Fix $a\in [1,2)$ and let the coefficients $H_j,j=\overline{2,m},$ be known and distinct. Then the specification of ${\left\{{\lambda }_{n,0}\right\}}_{n\in \mathbb{N}}$ and ${\left\{{\lambda }_{n,1}\right\}}_{n\in \mathbb{N}}$ uniquely determines $q_j\left(x\right),j=\overline{2,m}.$

Similarly to the inverse problem studied in [1], one can expect that an analogous assertion will also hold after reducing the input data to appropriate subspectra of the problems ${\mathcal{B}}_0$ and ${\mathcal{B}}_1,$ but any such refinement is beyond the goals of the present paper.

As was indicated in [1], the problem ${\mathcal{B}}_{\nu }$ in the case $m=2$ becomes a usual Sturm–Liouville-type problem with constant delay on an interval:

$$-y^{\prime \prime }\left(x\right)+q\left(x\right)y(x-a)=\lambda y\left(x\right),0<x<2,y^{\left(\nu \right)}\left(0\right)=y^{\prime }\left(2\right)+H_{2}y\left(2\right)=0,$$

(6)

where

$$y\left(x\right)=\left\{\begin{array}{cc}{y}_1\left(x\right),& 0\le x\le 1,\\ y_2(x-1),& 1<x\le 2,\end{array}\right.q\left(x\right)=\left\{\begin{array}{cc}{q}_1\left(x\right),& 0<x<1,\\ q_2(x-1),& 1<x<2.\end{array}\right.$$

Although the inverse problems for functional-differential operators with constant delay on an interval have been quite extensively studied (see, e.g., [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]), Inverse Problem 1 in the case $m=2$ does not have any direct analog in the literature. Indeed, all studies of recovering $q\left(x\right)$ in the equation in (6) address the case of a common boundary condition at the left-hand rather than the right-hand end-point of the interval as in (6). In other words, the spectra usually used for recovering $q\left(x\right)$ actually corresponded to the boundary conditions

$$y^{\left({\nu }_1\right)}\left(0\right)+{\nu }_1{H}_{1}y\left(0\right)=y^{\left(\nu \right)}\left(2\right)=0,\nu =0,1,$$

where ${\nu }_1\in \{0,1\}$ is fixed. While $q\left(x\right)$ is uniquely determined by those spectra whenever $a\in [4/5,2)$ (see [35] for ${\nu }_1=0$ and [37] for ${\nu }_1=1),$ the recent series of papers [42,43,44] establishes its possible non-uniqueness for any $a\in (0,4/5)$ (also see the brief survey in [45]). One can easily show that these uniqueness and non-uniqueness results remain true with the same ranges of $a,$ respectively, including for the boundary conditions in (6). Therefore, Theorem 1, generally speaking, does not hold for any $a\in (0,4/5)$ but it remains true for all $a\in [4/5,1)$ as soon as $m=2.$ However, the case $m>2$ for such a requires a separate investigation.

This paper is organized as follows. In the next section, we construct the characteristic determinants of the problems ${\mathcal{B}}_{\nu }.$ The proof of Theorem 1 along with a constructive procedure for solving the inverse problem (see Algorithm 1) is given in Section 3.

2. Characteristic Determinants

Denote

$$S_0(x,\lambda ):=\frac{sin\rho x}{\rho },S_1(x,\lambda ):=cos\rho x,{\rho }^2=\lambda .$$

(7)

For $\nu =0,1$ and $j=\overline{2,m},$ we also introduce the designations

$$v_{\nu ,j}\left(\lambda \right):=V_j\left(S_{\nu }(·,\lambda )\right),Q_{\nu ,j}(x,\lambda )={\int }_{a-1}^x\frac{sin\rho (x-t)}{\rho }{q}_j\left(t\right)S_{\nu }(t-a+1,\lambda )dt,$$

(8)

where $V_j(·)$ are the linear forms defined in (4) for ${\nu }_j=1.$ Consider the entire functions

$${\Delta }_{\nu }^0\left(\lambda \right)=S_{\nu }^{\prime }(1,\lambda )\prod _{j=2}^m{v}_{0,j}\left(\lambda \right)+S_{\nu }(1,\lambda )\sum _{j=2}^m{v}_{1,j}\left(\lambda \right)\prod _{\genfrac{}{}{0pt}{}{l\ne j}{l=2}}^m{v}_{0,l}\left(\lambda \right),\nu =0,1,$$

(9)

and

$${\Delta }_{\nu }\left(\lambda \right)={\Delta }_{\nu }^0\left(\lambda \right)+\sum _{j=2}^m{V}_j\left(Q_{\nu ,j}(·,\lambda )\right)\prod _{\genfrac{}{}{0pt}{}{l\ne j}{l=2}}^m{v}_{0,l}\left(\lambda \right),\nu =0,1.$$

(10)

Here and below, the prime always means the differentiation with respect to the first argument.

The function ${\Delta }_{\nu }\left(\lambda \right)$ is called characteristic determinant or characteristic function of the problem ${\mathcal{B}}_{\nu }.$ The following lemma holds.

Lemma 1.

For $\nu =0,1,$ the eigenvalues of the problem ${\mathcal{B}}_{\nu }$ coincide with zeros of ${\Delta }_{\nu }\left(\lambda \right).$

Proof. 

In accordance with [1], the functions

$$y_1\left(x\right)=\sum _{l=0}^1{C}_{l,1}{S}_l(x,\lambda ),y_j\left(x\right)=\sum _{l=0}^1{C}_{l,j}{S}_l(x,\lambda )+\sum _{l=0}^1{C}_{l,1}{Q}_{l,j}(x,\lambda ),j=\overline{2,m},$$

(11)

where $C_{l,j}$ are indefinite constants, form the so-called global general solution of the system of equations (1) subject to the initial-function conditions (2). Substituting (11) into the matching conditions (3) as well as the boundary conditions of ${\mathcal{B}}_{\nu },$ we arrive at a system of linear algebraic equations with respect to the vector ${[C_{\nu ,1},C_{0,2},C_{1,2},\dots ,C_{0,m},C_{1,m}]}^T$ having the determinant

$$D_{\nu }\left(\lambda \right):=\left|\begin{array}{cccccccc}{S}_{\nu }(1,\lambda )& 0& -1& 0& 0& \dots & 0& 0\\ S_{\nu }(1,\lambda )& 0& 0& 0& -1& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ S_{\nu }(1,\lambda )& 0& 0& 0& 0& \dots & 0& -1\\ S_{\nu }^{\prime }(1,\lambda )& -1& 0& -1& 0& \dots & -1& 0\\ V_2\left(Q_{\nu ,2}(·,\lambda )\right)& v_{0,2}\left(\lambda \right)& v_{1,2}\left(\lambda \right)& 0& 0& \dots & 0& 0\\ V_3\left(Q_{\nu ,3}(·,\lambda )\right)& 0& 0& v_{0,3}\left(\lambda \right)& v_{1,3}\left(\lambda \right)& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ V_m\left(Q_{\nu ,m}(·,\lambda )\right)& 0& 0& 0& 0& \dots & v_{0,m}\left(\lambda \right)& v_{1,m}\left(\lambda \right)\end{array}\right|$$

of order $2m-1$ (the m-th entry of the first column is $S_{\nu }^{\prime }(1,\lambda )).$

Denote by $D_{\nu }^0\left(\lambda \right)$ the determinant that is obtained from $D_{\nu }\left(\lambda \right)$ after making all $q_j\left(x\right)$ to equal zero, i.e. by zeroing the last $m-1$ entries of the first column. Then we have

$$D_{\nu }\left(\lambda \right)=D_{\nu }^0\left(\lambda \right)+\sum _{j=m+1}^{2m-1}{V}_{j-m+1}\left(Q_{\nu ,j-m+1}(·,\lambda )\right)A_j\left(\lambda \right),\nu =0,1,$$

(12)

where $A_j\left(\lambda \right)$ is the cofactor of the j-th entry of the first column. In particular, we have

$$A_m\left(\lambda \right)={(-1)}^{m+1}\left|\begin{array}{ccccccc}0& -1& 0& 0& \dots & 0& 0\\ 0& 0& 0& -1& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& 0& 0& 0& \dots & 0& -1\\ v_{0,2}\left(\lambda \right)& 0& 0& 0& \dots & 0& 0\\ 0& 0& v_{0,3}\left(\lambda \right)& 0& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& 0& 0& 0& \dots & v_{0,m}\left(\lambda \right)& 0\end{array}\right|={\sigma }_m\prod _{l=2}^m{v}_{0,l}\left(\lambda \right),$$

(13)

where ${\sigma }_m={(-1)}^{\frac{(m-1)(m+2)}{2}}.$ For $j=\overline{m+1,2m-1},$ we have

$$A_j\left(\lambda \right)={(-1)}^{j+1}\left|\begin{array}{ccccccc}0& -1& 0& 0& \dots & 0& 0\\ 0& 0& 0& -1& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& 0& 0& 0& \dots & 0& -1\\ -1& 0& -1& 0& \dots & -1& 0\\ v_{0,2}\left(\lambda \right)& 0& 0& 0& \dots & 0& 0\\ 0& 0& v_{0,3}\left(\lambda \right)& 0& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ 0& 0& 0& 0& \dots & v_{0,m}\left(\lambda \right)& 0\end{array}\right|,$$

where the row possessing $v_{0,j-m+1}\left(\lambda \right)$ is absent. Thus, we obtain the representation

$$A_j\left(\lambda \right)={\sigma }_m\prod _{\genfrac{}{}{0pt}{}{l\ne j-m+1}{l=2}}^m{v}_{0,l}\left(\lambda \right),j=\overline{m+1,2m-1},$$

(14)

which also fits (13) for $j=m.$ Substituting (14) into (12), we obtain

$$D_{\nu }\left(\lambda \right)=D_{\nu }^0\left(\lambda \right)+{\sigma }_m\sum _{j=m+1}^{2m-1}{V}_{j-m+1}\left(Q_{\nu ,j-m+1}(·,\lambda )\right)\prod _{\genfrac{}{}{0pt}{}{l\ne j-m+1}{l=2}}^m{v}_{0,l}\left(\lambda \right),\nu =0,1.$$

(15)

For calculating $D_{\nu }^0\left(\lambda \right),$ it is convenient to write it in the form

$$D_{\nu }^0\left(\lambda \right)=\left|\begin{array}{cccccccc}{S}_{\nu }(1,\lambda )& 0& -1& 0& 0& \dots & 0& 0\\ S_{\nu }(1,\lambda )& 0& 0& 0& -1& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ S_{\nu }(1,\lambda )& 0& 0& 0& 0& \dots & 0& -1\\ S_{\nu }^{\prime }(1,\lambda )& -1& 0& -1& 0& \dots & -1& 0\\ v_{1,2}\left(\lambda \right)S_{\nu }(1,\lambda )& v_{0,2}\left(\lambda \right)& 0& 0& 0& \dots & 0& 0\\ v_{1,3}\left(\lambda \right)S_{\nu }(1,\lambda )& 0& 0& v_{0,3}\left(\lambda \right)& 0& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& & ⋮& ⋮\\ v_{1,m}\left(\lambda \right)S_{\nu }(1,\lambda )& 0& 0& 0& 0& \dots & v_{0,m}\left(\lambda \right)& 0\end{array}\right|,$$

which immediately implies

$$D_{\nu }^0\left(\lambda \right)=S_{\nu }^{\prime }(1,\lambda )A_m\left(\lambda \right)+S_{\nu }(1,\lambda )\sum _{j=m+1}^{2m-1}{v}_{1,j-m+1}\left(\lambda \right)A_j\left(\lambda \right).$$

Substituting (13) and (14) into the last formula, we arrive at

$$D_{\nu }^0\left(\lambda \right)={\sigma }_m{S}_{\nu }^{\prime }(1,\lambda )\prod _{l=2}^m{v}_{0,l}\left(\lambda \right)+{\sigma }_m{S}_{\nu }(1,\lambda )\sum _{j=2}^m{v}_{1,j}\left(\lambda \right)\prod _{\genfrac{}{}{0pt}{}{l\ne j}{l=2}}^m{v}_{0,l}\left(\lambda \right),\nu =0,1.$$

(16)

Comparing (16) and (15) with (9) and (10), respectively, we observe that the functions $D_{\nu }^0\left(\lambda \right)$ and ${\Delta }_{\nu }^0\left(\lambda \right)$ as well as $D_{\nu }\left(\lambda \right)$ and ${\Delta }_{\nu }\left(\lambda \right)$ differ only by one and the same sign ${\sigma }_m,$ which finishes the proof.    □

In particular, this lemma implies that the function ${\Delta }_{\nu }^0\left(\lambda \right)$ determined by Formula (9) is the characteristic determinant of the problem ${\mathcal{B}}_{\nu }(0,H),$ while the function

$${\Delta }_{\nu }^{00}\left(\lambda \right)=S_{\nu }^{\prime }(1,\lambda ){(cos\rho )}^{m-1}+(1-m)S_{\nu }(1,\lambda ){(cos\rho )}^{m-2}\rho sin\rho$$

is the characteristic determinant of ${\mathcal{B}}_{\nu }(0,0).$ Moreover, we have the asymptotics

$${\Delta }_{\nu }\left(\lambda \right)={\Delta }_{\nu }^{00}\left(\lambda \right)+O\left({\rho }^{\nu -1}{e}^{m\left|\mathrm{Im}\rho \right|}\right),\lambda \to \infty ,\nu =0,1,$$

while ${\rho }^{-\nu }{\Delta }_{\nu }^{00}\left({\rho }^2\right)$ are sine-type functions of the exponential type m in the $\rho$-plane. Hence, using Rouché’s theorem as, e.g., in the proof of Theorem 4 in [50], one can immediately conclude that zeros of ${\Delta }_{\nu }^{00}\left(\lambda \right)$ form a principal part of the eigenvalues ${\lambda }_{n,\nu }$ as $n\to \infty .$

Thus, analogously to Theorem 1.1.4 in [22], one can prove the following assertion.

Lemma 2.

The functions ${\Delta }_0\left(\lambda \right)$ and ${\Delta }_1\left(\lambda \right)$ are determined by their zeros uniquely. Specifically, the representations

$${\Delta }_0\left(\lambda \right)=\prod _{n=1}^{\infty }\frac{{\lambda }_{n,0}-\lambda }{{\lambda }_{n,0}^0},{\Delta }_1\left(\lambda \right)=m({\lambda }_{1,1}-\lambda )\prod _{n=2}^{\infty }\frac{{\lambda }_{n,1}-\lambda }{{\lambda }_{n,1}^0}$$

(17)

take place, where ${\left\{{\lambda }_{n,\nu }^0\right\}}_{n\in \mathbb{N}}$ are zeros of the function ${\Delta }_{\nu }^{00}\left(\lambda \right).$

3. Solution of the Inverse Problem

Before directly proceeding to the proof of Theorem 1, we fulfill some preparatory work. Using Rouché’s theorem, one can prove the following assertion.

Lemma 3.

For $j=\overline{2,m},$ zeros ${\left\{{\xi }_{n,j}\right\}}_{n\in \mathbb{N}}$ of the function $v_{0,j}\left(\lambda \right)$ have the asymptotics

$${\xi }_{n,j}={\eta }_{n,j}^2,{\eta }_{n,j}=\pi \left(n-\frac{1}{2}\right)+\frac{H_j}{\pi n}+O\left(\frac{1}{n^2}\right),n\to \infty .$$

(18)

Without loss of generally, we assume that all multiple zeros ${\xi }_{n,j}$ are neighboring, i.e.,

$${\xi }_{n,j}={\xi }_{n+1,j}=\dots ={\xi }_{n+m_{n,j}-1,j},$$

where $m_{n,j}$ is the multiplicity of ${\xi }_{n,j}.$ Thus, the set

$${\mathcal{S}}_j:=\{n:{\xi }_{n,j}\ne {\xi }_{n-1,j},n-1\in \mathbb{N}\}\cup \left\{1\right\}$$

indexes the zeros ${\xi }_{n,j}$ ignoring their multiplicities.

Similarly to Proposition 1 in [23], one can prove the following assertion, which remains valid for arbitrary sequences ${\left\{{\xi }_{n,j}\right\}}_{n\in \mathbb{N}}$ of the form ${\xi }_{n,j}={\pi }^2{(n-1/2+{\varkappa }_n)}^2$ with $\left\{{\varkappa }_n\right\}\in l_2.$

Lemma 4.

For any fixed $\nu \in \{0,1\}$ and $j=\overline{2,m},$ the functional system ${\left\{s_{n,j,\nu }\left(x\right)\right\}}_{n\in \mathbb{N}}$ forms a Riesz basis in $L_2(0,1),$ where

$$s_{k+s,j,\nu }\left(x\right)=k^{1-\nu }\frac{d^s}{d{\lambda }^s}{S}_{\nu }(x,\lambda ){|}_{\lambda ={\xi }_{k,j}},k\in {\mathcal{S}}_j,s=\overline{0,m_{k,j}-1},$$

(19)

while $S_{\nu }(x,\lambda )$ are determined in (7).

For each $j=\overline{2,m}$ and $k\in {\mathcal{S}}_j,$ differentiating representation (10) $p=\overline{0,m_{k,j}-1}$ times and substituting $\lambda ={\xi }_{k,j}$ into the obtained relations, we arrive at the linear systems

$${\gamma }_{k+p,j,\nu }=\sum _{s=0}^p\left(\genfrac{}{}{0pt}{}{p}{s}\right){\alpha }_{k+p-s,j,\nu }{\beta }_{k+s,j,\nu },p=\overline{0,m_{k,j}-1},k\in {\mathcal{S}}_j,\nu =0,1,$$

(20)

where

$${\gamma }_{k+s,j,\nu }:=k^{1-\nu }{({\Delta }_{\nu }-{\Delta }_{\nu }^0)}^{\left(s\right)}\left({\xi }_{k,j}\right),{\alpha }_{k+s,j,\nu }:=\frac{d^s}{d{\lambda }^s}\prod _{\genfrac{}{}{0pt}{}{l\ne j}{l=2}}^m{v}_{0,l}\left(\lambda \right){|}_{\lambda ={\xi }_{k,j}},$$

(21)

$${\beta }_{k+s,j,\nu }:=k^{1-\nu }\frac{d^s}{d{\lambda }^s}{V}_j\left(Q_{\nu ,j}(·,\lambda )\right){|}_{\lambda ={\xi }_{k,j}},k\in {\mathcal{S}}_j,s=\overline{0,m_{k,j}-1}.$$

(22)

Lemma 5.

For $j=\overline{2,m},$ the following representations hold:

$$V_j\left(Q_{0,j}(·,\lambda )\right)=\frac{{\omega }_j}{2\rho }sin\rho (2-a)+{\int }_0^{2-a}{u}_{0,j}\left(x\right)\frac{sin\rho x}{\rho }dx,$$

(23)

$$V_j\left(Q_{1,j}(·,\lambda )\right)=\frac{{\omega }_j}{2}cos\rho (2-a)+H_j{\omega }_j\frac{sin\rho (2-a)}{\rho }+{\int }_0^{2-a}{u}_{1,j}\left(x\right)cos\rho xdx,$$

(24)

where

$${\omega }_j={\int }_{a-1}^1{q}_j\left(x\right)dx,j=\overline{2,m},$$

(25)

and

$$u_{\nu ,j}\left(x\right)=\frac{1}{4}\left(p_j\left(\frac{a+x}{2}\right)-{(-1)}^{\nu }{p}_j\left(\frac{a-x}{2}\right)\right),0<x<2-a,\nu =0,1,$$

(26)

while

$$p_j\left(x\right)=q_j\left(x\right)-2H_j{\int }_x^1{q}_j\left(t\right)dt,a-1<x<1.$$

(27)

Proof. 

In accordance with (7) and (8), we have

$$Q_{0,j}(x,\lambda )={\int }_{a-1}^x\frac{sin\rho (x-t)}{\rho }{q}_j\left(t\right)\frac{sin\rho (t-a+1)}{\rho }dt$$

$$=\frac{1}{2{\rho }^2}{\int }_{a-1}^x{q}_j\left(t\right)\left(cos\rho (x-2t+a-1)-cos\rho (x-a+1)\right)dt,$$

(28)

$$Q_{1,j}(x,\lambda )={\int }_{a-1}^x\frac{sin\rho (x-t)}{\rho }{q}_j\left(t\right)cos\rho (t-a+1)dt$$

$$=\frac{1}{2\rho }{\int }_{a-1}^x{q}_j\left(t\right)\left(sin\rho (x-2t+a-1)+sin\rho (x-a+1)\right)dt.$$

(29)

By the differentiation, we obtain

$$Q_{0,j}^{\prime }(x,\lambda )=\frac{1}{2\rho }{\int }_{a-1}^x{q}_j\left(t\right)\left(sin\rho (x-a+1)-sin\rho (x-2t+a-1)\right)dt,$$

$$Q_{1,j}^{\prime }(x,\lambda )=\frac{1}{2}{\int }_{a-1}^x{q}_j\left(t\right)\left(cos\rho (x-a+1)+cos\rho (x-2t+a-1)\right)dt.$$

Integrating by parts in (28) and (29), we arrive at

$$Q_{0,j}(x,\lambda )=\frac{1}{\rho }{\int }_{a-1}^{x}sin\rho (x-2t+a-1)dt{\int }_t^x{q}_j\left(\tau \right)d\tau ,$$

$$Q_{1,j}(x,\lambda )=\frac{sin\rho (x-a+1)}{\rho }{\int }_{a-1}^x{q}_j\left(t\right)dt-{\int }_{a-1}^{x}cos\rho (x-2t+a-1)dt{\int }_t^x{q}_j\left(\tau \right)d\tau .$$

Substituting $x=1$ into the last four formulae and using the designation (25), we obtain

$$Q_{0,j}^{\prime }(1,\lambda )=\frac{{\omega }_j}{2\rho }sin\rho (2-a)-\frac{1}{2\rho }{\int }_{a-1}^1{q}_j\left(x\right)sin\rho (a-2x)dx,$$

$$Q_{1,j}^{\prime }(1,\lambda )=\frac{{\omega }_j}{2}cos\rho (2-a)+\frac{1}{2}{\int }_{a-1}^1{q}_j\left(x\right)cos\rho (a-2x)dx,$$

$$Q_{0,j}(1,\lambda )=\frac{1}{\rho }{\int }_{a-1}^{1}sin\rho (a-2x)dx{\int }_x^1{q}_j\left(t\right)dt,$$

$$Q_{1,j}(1,\lambda )=\frac{{\omega }_j}{\rho }sin\rho (2-a)-{\int }_{a-1}^{1}cos\rho (a-2x)dx{\int }_x^1{q}_j\left(t\right)dt,$$

which, along with (4) for ${\nu }_j=1$ and (27), leads to the representations

$$V_j\left(Q_{0,j}(·,\lambda )\right)=\frac{{\omega }_j}{2\rho }sin\rho (2-a)-\frac{1}{2\rho }{\int }_{a-1}^1{p}_j\left(x\right)sin\rho (a-2x)dx,$$

$$V_j\left(Q_{1,j}(·,\lambda )\right)=\frac{{\omega }_j}{2}cos\rho (2-a)+H_j{\omega }_j\frac{sin\rho (2-a)}{\rho }+\frac{1}{2}{\int }_{a-1}^1{p}_j\left(x\right)cos\rho (a-2x)dx.$$

Changing the integration variable and taking (26) into account, we arrive at (23) and (24).    □

By virtue of (23) and (24) along with (7) and (19), formula (22) takes the form

$${\beta }_{n,j,\nu }=\frac{{\omega }_j}{2}{s}_{n,j,\nu }(2-a)+{\delta }_{\nu ,1}{H}_j{\omega }_j{\tilde{s}}_{n,j,0}(2-a)+{\int }_0^{2-a}{u}_{\nu ,j}\left(x\right)s_{n,j,\nu }\left(x\right)dx,n\in \mathbb{N},$$

(30)

where ${\delta }_{\nu ,1}$ is the Kronecker delta and

$${\tilde{s}}_{k+s,j,0}\left(x\right)=\frac{1}{k}{s}_{k+s,j,0}\left(x\right),k\in {\mathcal{S}}_0,s=\overline{0,m_{k,0}-1}.$$

Proof of Theorem 1 

By virtue of Lemma 2, the specification of the spectra along with the coefficients $H_j$ uniquely determines the numbers ${\alpha }_{n,j,\nu }$ and ${\gamma }_{n,j,\nu }$ defined in (21). Furthermore, since all $H_j$ are distinct, any two functions $v_{0,j}\left(\lambda \right)$ and $v_{0,l}\left(\lambda \right)$ defined in (8) for $j,l=\overline{2,m}$ have no common zeros whenever $j\ne l,$ i.e., ${\left\{{\xi }_{n,j}\right\}}_{n\in \mathbb{N}}\cap {\left\{{\xi }_{n,l}\right\}}_{n\in \mathbb{N}}=\varnothing .$ Hence, ${\alpha }_{k,j,\nu }\ne 0$ for all $k\in {\mathcal{S}}_j,$$j=\overline{2,m}$ and $\nu =0,1.$ Therefore, each triangular system of linear algebraic equations (20) always has the unique solution ${\beta }_{k+s,j,\nu },$ $s=\overline{0,m_{k,j}-1}.$ Thus, all sequences ${\left\{{\beta }_{n,j,\nu }\right\}}_{n\in \mathbb{N}}$ are uniquely determined by specifying the spectra of the problems ${\mathcal{B}}_0$ and ${\mathcal{B}}_1.$

Further, in accordance with (7), (18) and (19), formula (30) implies

$$\frac{2{\eta }_{n,j}{\beta }_{n,j,0}}{n}={\omega }_{j}sin{\eta }_{n,j}(2-a)+2{\int }_0^{2-a}{u}_{0,j}\left(x\right)sin{\eta }_{n,j}xdx,n\gg 1.$$

Hence, the numbers ${\omega }_j$ are uniquely determined too by the formulae

$${\omega }_j=\underset{k\to \infty }{lim}\frac{2{\eta }_{n_k,j}{\beta }_{n,j,0}}{n_{k}sin{\eta }_{n_k,j}(2-a)},j=\overline{2,m},$$

(31)

where $\left\{n_k\right\}$ is an increasing sequence of natural numbers that are chosen so that

$$|a_{n_k}|\ge c>0,a_n:=sin\pi \left(n-\frac{1}{2}\right)(2-a),$$

whose existence, in turn, follows from the non-convergence of the sequence $\left\{a_n\right\},$ which can be shown as in Lemma 3.3 in [39].

Thus, by virtue of Lemma 4, all functions $u_{\nu ,j}\left(x\right)$ are determined by relations (30) uniquely. Hence, according to (26), the functions $p_j\left(x\right)$ are uniquely determined by the formulae

$$p_j\left(x\right)=\left\{\begin{array}{cc}2(u_{1,j}-u_{0,j})(a-2x),& {\displaystyle x\in \left(a-1,\frac{a}{2}\right),}\hfill \\ 2(u_{1,j}+u_{0,j})(2x-a),& {\displaystyle x\in \left(\frac{a}{2},1\right),}\hfill \end{array}\right.j=\overline{2,m}.$$

(32)

Finally, all $q_j\left(x\right)$ are unique solutions of the Volterra integral Equations (27), i.e.

$$q_j\left(x\right)=p_j\left(x\right)+2H_j{\int }_x^1{e}^{2H_j(t-x)}{p}_j\left(t\right)dt,a-1<x<1,j=\overline{2,m},$$

(33)

which finishes the proof.    □

This proof is constructive and gives the following algorithm for solving the inverse problem.

Algorithm 1 Let the spectra ${\left\{{\lambda }_{n,0}\right\}}_{n\in \mathbb{N}}$ and ${\left\{{\lambda }_{n,1}\right\}}_{n\in \mathbb{N}}$ be given. Then:

(i) Calculate the characteristic functions ${\Delta }_0\left(\lambda \right)$ and ${\Delta }_1\left(\lambda \right)$ by the formulae in (17); (ii) Construct the sequences ${\left\{{\alpha }_{n,j,\nu }\right\}}_{n\in \mathbb{N}}$ and ${\left\{{\gamma }_{n,j,\nu }\right\}}_{n\in \mathbb{N}}$ for $j=\overline{2,m}$ and $\nu =0,1$ by the formulae in (21) and using (7)–(9); (iii) Find the sequences ${\left\{{\beta }_{n,j,\nu }\right\}}_{n\in \mathbb{N}}$ for $j=\overline{2,m}$ and $\nu =0,1$ via relations (20); (iv) Calculate the numbers ${\omega }_j,j=\overline{2,m},$ using (31); (v) In accordance with (30) and Lemma 4, construct the functions $u_{\nu ,j}\left(x\right)\in L_2(0,2-a)$ by the formula $$u_{\nu ,j}\left(x\right)=\sum _{n=1}^{\infty }\left({\beta }_{n,j,\nu }-\frac{{\omega }_j}{2}{s}_{n,j,\nu }(2-a)-{\delta }_{\nu ,1}{H}_j{\omega }_j{\tilde{s}}_{n,j,0}(2-a)\right)s_{n,j,\nu }^{*}\left(x\right),\nu =0,1,j=\overline{2,m},$$ where ${\left\{s_{n,j,\nu }^{*}\left(x\right)\right\}}_{n\in \mathbb{N}}$ is the basis that is biorthogonal to the basis ${\left\{\overline{s_{n,j,\nu }\left(x\right)}\right\}}_{n\in \mathbb{N}};$ (vi) Finally, find the functions $p_j\left(x\right)$ by (32) and calculate $q_j\left(x\right)$ by (33).