Local Solvability and Stability of an Inverse Spectral Problem for Higher-Order Differential Operators

Abstract

In this paper, we, for the first time, prove the local solvability and stability of an inverse spectral problem for higher-order ($n>3$) differential operators with distribution coefficients. The inverse problem consists of the recovery of differential equation coefficients from $(n-1)$ spectra and the corresponding weight numbers. The proof method is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation remains uniquely solvable. Furthermore, we estimate the differences of the coefficients in the corresponding functional spaces.

1. Introduction

This paper deals with the differential equation:

$$y^{\left(n\right)}+p_{n-2}\left(x\right)y^{(n-2)}+p_{n-3}\left(x\right)y^{(n-3)}+\cdots +p_1\left(x\right)y^{\prime }+p_0\left(x\right)y=\lambda y,x\in (0,1),$$

(1)

where $n\ge 2$, $p_k$ are complex-valued functions, $p_k\in W_2^{k-1}[0,1]$, $k=\overline{0,n-2}$, and $\lambda$ is the spectral parameter. Recall that:

• For $s\ge 1$, $W_2^s[0,1]$ is the space of functions $f\left(x\right)$ whose derivatives $f^{\left(j\right)}\left(x\right)$ are absolutely continuous on $[0,1]$ for $j=\overline{0,s-1}$ and $f^{\left(s\right)}\in L_2[0,1]$.
• $W_2^0[0,1]=L_2[0,1]$.
• $W_2^{-1}[0,1]$ is the space of generalised functions (distributions) $f\left(x\right)$ whose antiderivatives $f^{(-1)}\left(x\right)$ belong to $L_2[0,1]$.

We study the inverse spectral problem that consists of the recovery of the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ from the eigenvalues ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}$ and the weight numbers ${\left\{{\beta }_{l,k}\right\}}_{l\ge 1}$ of the boundary value problems ${\mathcal{L}}_k$, $k=\overline{1,n-1}$, for Equation (1) with the corresponding boundary conditions:

$$y^{\left(j\right)}\left(0\right)=0,j=\overline{0,k-1},y^{\left(s\right)}\left(1\right)=0,s=\overline{0,n-k-1}.$$

(2)

The main goal of this paper is to prove the local solvability and stability of the inverse problem. This is the first result of such a kind for arbitrary-order differential operators with distribution coefficients.

1.1. Historical Background

The spectral theory of linear ordinary differential operators has a fundamental significance for mathematics and applications. For $n=2$, Equation (1) turns into the Sturm–Liouville (one-dimensional Schrödinger) equation $y^{″}+p_{0}y=\lambda y$, which models various processes in quantum and classical mechanics, materials science, astrophysics, acoustics, and electronics. The third-order differential equations are applied to describing the thin membrane flow of a viscous liquid [1] and elastic beam vibrations [2]. The third-order spectral problems also arise in the integration of the nonlinear Boussinesq equation by the inverse scattering transform [3]. The fourth-order and the sixth-order linear differential operators appear in geophysics [4] and in vibration theory [5,6]. Therefore, the development of general mathematical methods for the investigation of spectral problems for arbitrary-order linear differential operators is fundamentally important.

The inverse problems of spectral analysis consist of the reconstruction of differential operators by their spectral characteristics. Such problems have been studied fairly completely for Sturm–Liouville operators $-y^{″}+q\left(x\right)y$ with regular (integrable) potentials $q\left(x\right)$ (see the monographs [7,8,9,10,11] and the references therein), as well as with the distribution potentials of class $W_2^{-1}$ (see, e.g., the papers [12,13,14,15,16,17,18] and a more-extensive bibliography in [19]). The basic results for the inverse Sturm–Liouville problem were obtained by the method of Gelfand and Levitan [20], which is based on transformation operators. However, inverse problems for differential operators of higher orders $n>2$ are significantly more difficult for investigation, since the Gelfand–Levitan method does not work for them. Therefore, Yurko [21,22,23] has developed the method of spectral mappings, which is based on the theory of analytic functions. The central place in this method is taken by the contour integration in the complex plane of the spectral parameter $\lambda$ of some meromorphic functions (spectral mappings), which were introduced by Leibenson [24,25]. Applying the method of spectral mappings, Yurko created the theory of inverse spectral problems for arbitrary-order differential operators with regular coefficients and also with the Bessel-type singularities on a finite interval and on the half-line (see [21,22,23,26,27]). Another approach was developed by Beals et al. [28,29] for inverse scattering problems on the line, which are essentially different from the spectral problems on a finite interval.

In [30], Mirzoev and Shkalikov proposed a regularisation approach for higher-order differential equations with distribution coefficients. This has motivated a number of researchers to study the solutions and spectral theory for such equations (see the bibliography in the recent papers [31,32]). Inverse spectral problems for higher-order differential operators with distribution coefficients were investigated in [32,33,34,35,36]. In particular, the uniqueness theorems were proven in [32,33,34]. The paper [35] is concerned with a reconstruction approach, based on developing the ideas of the method of spectral mappings. In [36], the necessary and sufficient conditions for the inverse problem solvability were obtained for a third-order differential equation.

In this paper, we focus on the local solvability and stability of the inverse problem. These aspects for the Sturm–Liouville operators were studied in [8,9,10,12,15,16,18,37,38,39,40,41,42,43,44,45,46] and many other papers. Local solvability has a fundamental significance in inverse problem theory, especially for such problems, for which global solvability theorems are absent or contain hard-to-verify conditions. Stability is important for the justification of numerical methods. For the higher-order differential Equation (1) on a finite interval, the stability of the inverse problem in the uniform norm was proven by Yurko for regular coefficients $p_k\in W_1^k[0,1]$ (see Theorem 2.3.2 in [23]). Furthermore, local solvability and stability in the $L_2$-norm were formulated without proofs as Theorems 2.3.4 and 2.5.3 in [23]. For the distribution coefficients, the local solvability and stability theorems were proven in [12] for $n=2$ and in [36] for $n=3$. However, to the best of the author’s knowledge, there were no results in this direction for $n\ge 4$.

It is worth mentioning that, for the Sturm–Liouville operators with distribution potentials of the classes $W_2^{\alpha }[0,1]$, $\alpha >-1$, Savchuk and Shkalikov [15] obtained the uniform stability of the inverse spectral problem. This result was extended by Hryniv [16] to $\alpha =-1$ by another method. Recently, the uniform stability of inverse problems was also proven for some nonlocal operators (see [47,48,49]). However, the approaches of the mentioned studies do not work for higher-order differential operators. Thus, the uniform stability of them is an open problem, which is not considered in this paper.

1.2. Main Results

In this paper, we, for the first time, proved the local solvability and stability of an inverse problem for Equation (1) and, in addition, obtained some sufficient conditions for global solvability. To the best of the author’s knowledge, these aspects were not considered in previous studies for arbitrary-order differential equations with distribution coefficients, so our results are fundamentally novel. In order to prove the main theorems, we derived reconstruction formulas for coefficients ${\left(p_k\right)}_{k=0}^{n-2}$, which are also novel for the considered class of operators.

Let us formulate the inverse problem for Equation (1) and the corresponding local solvability and stability theorem.

Consider the problems ${\mathcal{L}}_k$ given by (1) and (2). For each $k\in \{1,2,\cdots ,n-1\}$, the spectrum of the problem ${\mathcal{L}}_k$ is a countable set of eigenvalues ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}$. They are supposed to be numbered, counting with multiplicities according to the asymptotics obtained in [50]:

$${\lambda }_{l,k}={(-1)}^{n-k}{\left(\frac{\pi }{sin\frac{\pi k}{n}}(l+{\theta }_k+{\varkappa }_{l,k})\right)}^n,l\ge 1,k=\overline{1,n-1},$$

(3)

where ${\theta }_k$ are some constants independent of ${\left(p_s\right)}_{s=0}^{n-2}$ and $\left\{{\varkappa }_{l,k}\right\}\in l_2$.

The weight numbers ${\left\{{\beta }_{l,k}\right\}}_{l\ge 1k=\overline{1,n-1}}$ will be defined in Section 3. Thus, the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ are generated by the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$ of Equation (1). In this paper, we confined ourselves to the p of some class W with certain restrictions on the spectra.

Definition 1.

We say that $p={\left(p_k\right)}_{k=0}^{n-2}$ belongs to the class W if:

(W-1) $p_k\in W_2^{k-1}[0,1]$, $k=\overline{0,n-2}$.

(W-2) For each $k\in \{1,2,\cdots ,n-1\}$, the eigenvalues ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}$ are simple.

(W-3) ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}\cap {\left\{{\lambda }_{l,k+1}\right\}}_{l\ge 1}=⌀$ for $k=\overline{1,n-2}$.

Thus, we study the following inverse problem.

Problem 1.

Given the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$, find the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}\in W$.

Problem 1 generalises the classical inverse Sturm–Liouville problem studied by Marchenko [10] and Gelfand and Levitan [20] (see Example 1). The uniqueness for the solution of Problem 1 follows from the results of [34,35] (see Section 3 for details). Note that, if the conditions (W-2) and (W-3) are violated, then the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ do not uniquely specify the coefficients p and additional spectral characteristics are needed. In particular, for multiple eigenvalues in the case $n=2$, the generalised weight numbers were defined in [51,52]. For $n=3$ and ${\lambda }_{l,1}={\lambda }_{l,2}$, the additional weight numbers ${\gamma }_l$ were used in [36]. For higher orders n, the situation becomes much more complicated, so in this paper, we confined ourselves to the class W. Anyway, in view of the eigenvalues’ asymptotics (3), the assumptions (W-2) and (W-3) hold for all sufficiently large indices l.

Along with p, we considered an analogous vector $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}\in W$. We agree that, if a symbol $\alpha$ denotes an object related to p, then the symbol $\tilde{\alpha }$ with the tilde will denote the analogous object related to $\tilde{p}$. The main result of this paper is the following theorem on the local solvability and stability of Problem 1.

Theorem 1.

Let $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}\in W$ be fixed. Then, there exists $\delta >0$ (which depends on $\tilde{p}$) such that, for any complex numbers ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ satisfying the inequality:

$$\mathsf{\Omega }:={\left(\sum _{l=1}^{\infty }{\left(\sum _{k=1}^{n-1}\left(l^{-1}|{\lambda }_{l,k}-{\tilde{\lambda }}_{l,k}|+l^{-2}|{\beta }_{l,k}-{\tilde{\beta }}_{l,k}|\right)\right)}^2\right)}^{1/2}\le \delta ,$$

(4)

there exists a unique $p={\left(p_k\right)}_{k=0}^{n-2}$ with the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$. Moreover,

$$\parallel p_k-{\tilde{p}}_k{\parallel }_{W_2^{k-1}[0,1]}\le C\mathsf{\Omega },k=\overline{0,n-2},$$

(5)

where the constant C depends only on $\tilde{p}$ and δ.

Theorem 1 generalises the previous results of [12] for $n=2$ and of [36] for $n=3$. However, for $n\ge 4$, to the best of the author’s knowledge, Theorem 1 is the first existence result for the inverse problem solution in the case of distribution coefficients.

The proof of Theorem 1 is based on the constructive approach of [35]. Namely, we reduced the nonlinear inverse problem to the so-called main equation, which is a linear equation in the Banach space of bounded infinite sequences. The unique solvability of the main equation follows from the smallness of $\delta$. Furthermore, we derived reconstruction formulas for the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ in the form of infinite series. The crucial step in the proof is establishing the convergence of those series in the corresponding spaces $W_2^{k-1}[0,1]$ (including the space of generalised functions $W_2^{-1}[0,1]$). In order to prove the convergence, we rigorously analysed the solution of the main equation and obtained the precise estimates for the Weyl solutions. Along with Theorem 1, we also proved Theorem 2 on the global solvability of the inverse problem under several requirements on an auxiliary model problem.

The paper is organised as follows. In Section 2, we discuss the regularisation of Equation (1) and provide other preliminaries. In Section 3, the weight numbers are defined and the properties of the spectral data are described. In Section 4, we derive the main equation based on the results of [35]. In Section 5, the reconstruction formulas for the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ are obtained. In Section 6, we prove the solvability and stability of the inverse problem. Section 7 contains concluding remarks.

2. Preliminaries

In this section, we explain in which sense we understand Equation (1). For this purpose, an associated matrix and quasi-derivatives are introduced. In addition, we provide other preliminaries. We begin with some notations, which are used throughout the paper:

• ${\delta }_{j,k}$ is the Kronecker delta.
• $C_k^j=\frac{k!}{j!(k-j)!}$ are the binomial coefficients.
• In estimates, the same symbol C is used for various positive constants that do not depend on x, l, $\lambda$, etc.
• The spaces $W_2^k[0,1]$ are equipped with the following norms:

  $$\begin{array}{cc}\hfill {\parallel y\parallel }_{W_2^k[0,1]}& ={\left(\sum _{j=0}^k{\parallel y^{\left(j\right)}\parallel }_{L_2[0,1]}^2\right)}^{1/2},k\ge 0,\hfill \\ \hfill {\parallel y\parallel }_{W_2^{-1}[0,1]}& =\underset{c\in \mathbb{C}}{inf}{\parallel (y^{(-1)}+c)\parallel }_{L_2[0,1]}.\hfill \end{array}$$

For $n\ge 3$, consider the differential expression:

$${\ell }_n\left(y\right):=y^{\left(n\right)}+\sum _{k=0}^{n-2}{p}_k\left(x\right)y,x\in (0,1).$$

Fix any function $\sigma \in L_2[0,1]$ such that $p_0={\sigma }^{\prime }$. Define the associated matrix $F\left(x\right)={\left[f_{k,j}\left(x\right)\right]}_{k,j=1}^n$ for the differential expression ${\ell }_n\left(y\right)$ by the formulas:

$$f_{n-1,1}:=-\sigma ,f_{n,2}:=\sigma -p_1,f_{n,k}:=-p_{k-1},k=\overline{3,n-1}.$$

(6)

All the other entries $f_{k,j}$ are assumed to be zero. Clearly, $f_{k,j}\in L_2[0,1]$.

Using the matrix function $F\left(x\right)$, introduce the quasi-derivatives:

$$y^{\left[0\right]}:=y,y^{\left[k\right]}:={\left(y^{[k-1]}\right)}^{\prime }-\sum _{j=1}^k{f}_{k,j}{y}^{[j-1]},k=\overline{1,n},$$

(7)

and the domain:

$${\mathcal{D}}_F:=\{y:y^{\left[k\right]}\in AC[0,1],k=\overline{0,n-1}\}.$$

Due the the special structure of the associated matrix $F\left(x\right)$, we have

$$\begin{array}{c}{y}^{\left[k\right]}=y^{\left(k\right)},k=\overline{0,n-2},y^{[n-1]}=y^{(n-1)}+\sigma y,\end{array}$$

(8)

$$\begin{array}{c}{y}^{\left[n\right]}={\left(y^{[n-1]}\right)}^{\prime }+\sum _{k=1}^{n-2}{p}_k{y}^{\left(k\right)}+\sigma y^{\prime },\end{array}$$

(9)

and so, ${\mathcal{D}}_F\subset W_1^{n-1}[0,1]$.

Note that the differential expression is correctly defined in the sense of generalised functions for any $y\in W_1^{n-1}[0,1]$. However, if $y\in {\mathcal{D}}_F$, then $y^{\left[n\right]}\in L_1[0,1]$ and Relations (8) and (9) directly imply the following lemma.

Lemma 1.

For $y\in {\mathcal{D}}_F$, ${\ell }_n\left(y\right)$ is a regular generalised function and ${\ell }_n\left(y\right)=y^{\left[n\right]}$.

Thus, for $y\in {\mathcal{D}}_F$, ${\ell }_n\left(y\right)$ is a function of $L_1[0,1]$ and the relation ${\ell }_n\left(y\right)=y^{\left[n\right]}$ gives the regularization of this differential expression. We call a matrix function $F\left(x\right)$ an associated matrix of the differential expression ${\ell }_n\left(y\right)$ if $F\left(x\right)$ defines the quasi-derivatives $y^{\left[k\right]}$ and the domain ${\mathcal{D}}_F$, so that the assertion of Lemma 1 holds. A function y is called a solution of Equation (1) if $y\in {\mathcal{D}}_F$ and ${\ell }_n\left(y\right)=\lambda y$ a.e. on $(0,1)$.

Following the technique of ([35], Section 2), we considered along with $F\left(x\right)$ the matrix function $F^{★}\left(x\right)={\left[f_{k,j}^{★}\left(x\right)\right]}_{k,j=1}^n$ such that

$$f_{k,j}^{★}\left(x\right)={(-1)}^{k+j+1}{f}_{n-j+1,n-k+1}\left(x\right),k,j=\overline{1,n}.$$

Using (6), we obtain

$$f_{k,1}^{★}={(-1)}^{k+1}{p}_{n-k},k=\overline{2,n-2},f_{n-1,1}^{★}={(-1)}^n(p_1-\sigma ),f_{n,2}^{★}={(-1)}^n\sigma ,$$

(10)

and all the other entries $f_{k,j}^{★}$ equal zero. For example, for $n=6$, we have

$$F\left(x\right)=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ -\sigma & 0& 0& 0& 0& 0\\ 0& \sigma -p_1& -p_2& -p_3& -p_4& 0\end{array}\right],F^{★}\left(x\right)=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ -p_4& 0& 0& 0& 0& 0\\ p_3& 0& 0& 0& 0& 0\\ -p_2& 0& 0& 0& 0& 0\\ p_1-\sigma & 0& 0& 0& 0& 0\\ 0& \sigma & 0& 0& 0& 0\end{array}\right].$$

Using the matrix function $F^{★}\left(x\right)$, define the quasi-derivatives:

$$z^{\left[0\right]}:=z,z^{\left[k\right]}:={\left(z^{[k-1]}\right)}^{\prime }-\sum _{j=1}^k{f}_{k,j}^{★}{z}^{[j-1]},k=\overline{1,n},$$

(11)

the domain:

$${\mathcal{D}}_{F^{★}}:=\{z:z^{\left[k\right]}\in AC[0,1],k=\overline{0,n-1}\},$$

(12)

and the differential expression:

$${\ell }_n^{★}\left(z\right):={(-1)}^n{z}^{\left[n\right]}.$$

(13)

Note that, in (12) and (13), we used the quasi-derivatives defined by (11). Below, we call a function z a solution of the differential equation:

$${\ell }_n^{★}\left(z\right)=\lambda z,x\in (0,1),$$

(14)

if $z\in {\mathcal{D}}_{F^{★}}$ and the equality (14) holds a.e. on $(0,1)$. Throughout this paper, we always use the quasi-derivatives (7) for functions of ${\mathcal{D}}_F$ and the quasi-derivatives (11) for functions of ${\mathcal{D}}_{F^{★}}$.

For $z\in {\mathcal{D}}_{F^{★}}$, Relations (10) and (11) imply

$$z^{\left[k\right]}={\left(z^{[k-1]}\right)}^{\prime }+{(-1)}^k{p}_{n-k}z,k=\overline{2,n-2}.$$

Therefore, one can show by induction that

$$\begin{array}{cc}\hfill z^{\left[k\right]}& =z^{\left(k\right)}+\sum _{j=0}^{k-2}\left(\sum _{s=j}^{k-2}{(-1)}^{s+k}{C}_s^j{p}_{n-k+s}^{(s-j)}\right)z^{\left(j\right)},k=\overline{0,n-2},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill z^{[n-1]}& =z^{(n-1)}+\sum _{j=0}^{n-3}\left(\sum _{s=j}^{n-3}{(-1)}^{s+n-1}{C}_s^j{p}_{s+1}^{(s-j)}\right)z^{\left(j\right)}+{(-1)}^n\sigma z,\hfill \\ \hfill z^{\left[n\right]}& ={\left(z^{[n-1]}\right)}^{\prime }+{(-1)}^{n+1}\sigma z^{\prime }.\hfill \end{array}$$

(16)

Consequently, induction implies $z^{\left(k\right)}\in AC[0,1]$ for $k=\overline{0,n-2}$. Hence, ${\mathcal{D}}_{F^{★}}\subset W_1^{n-1}[0,1]$.

For $z\in {\mathcal{D}}_{F^{★}}$ and $y\in {\mathcal{D}}_F$, define the Lagrange bracket:

$$\langle z,y\rangle =\sum _{k=0}^{n-1}{(-1)}^k{z}^{\left[k\right]}{y}^{[n-k-1]}.$$

(17)

Then, the Lagrange identity holds (see ([35], Section 2)):

$$\frac{d}{dx}\langle z,y\rangle =z{\ell }_n\left(y\right)-y{\ell }_n^{★}\left(z\right).$$

In particular, if z and y solve the equations ${\ell }^{★}\left(z\right)=\mu z$ and $\ell \left(y\right)=\lambda y$, respectively, then we obtain

$$\frac{d}{dx}\langle z,y\rangle =(\lambda -\mu )yz.$$

(18)

Substituting (15) and (16) into (17), we derive the relation

$$\langle z,y\rangle =\sum _{k=0}^{n-1}{(-1)}^{n-k-1}{z}^{(n-k-1)}{y}^{\left(k\right)}+\sum _{k=0}^{n-3}{y}^{\left(k\right)}\sum _{j=0}^{n-k-3}\left(\sum _{s=j}^{n-k-3}{(-1)}^s{C}_s^j{p}_{s+k+1}^{(s-j)}\right)z^{\left(j\right)},$$

(19)

where all the derivatives are regular, since $z,y\in W_1^{n-1}[0,1]$ and $p_k\in W_2^{k-1}[0,1]$, $k=\overline{1,n-2}$.

Remark 1.

The associated matrix $F\left(x\right)$ given by (6) regularises the differential expression ${\ell }_n\left(y\right)$ only for $n\ge 3$. For $n=2$, the associated matrix is constructed in a different way (see [53]):

$$F=\left[\begin{array}{cc}-\sigma & 0\\ -{\sigma }^2& \sigma \end{array}\right].$$

Nevertheless, the main result of this paper (Theorem 1) holds for $n=2$ and, moreover, was already proven in [12] for real-valued potentials. Therefore, below in the proofs, we confine ourselves to the case $n\ge 3$ and use the associated matrix (6). For $n=2$, the proofs are valid with minor modifications.

Remark 2.

For the regularisation of the differential expression ${\ell }_n\left(y\right)$, different associated matrices can be used (see [30,32,34]). However, it was proven in [34] that the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ do not depend on the choice of the associated matrix.

3. Spectral Data

In this section, we discuss the properties of the spectral characteristics for the boundary value problems ${\mathcal{L}}_k$, $k=\overline{1,n-1}$. In particular, the weight numbers $\left\{{\beta }_{l,k}\right\}$ are defined as the residues of some entries of the Weyl–Yurko matrix.

For $k=\overline{1,n}$, denote by ${\mathcal{C}}_k(x,\lambda )$ and ${\mathsf{\Phi }}_k(x,\lambda )$ the solutions of Equation (1) satisfying the initial conditions:

$${\mathcal{C}}_k^{[j-1]}(0,\lambda )={\delta }_{k,j},j=\overline{1,n},$$

and the boundary conditions:

$${\mathsf{\Phi }}_k^{[j-1]}(0,\lambda )={\delta }_{k,j},j=\overline{1,k},{\mathsf{\Phi }}_k^{[s-1]}(1,\lambda )=0,s=\overline{1,n-k}.$$

respectively. The functions ${\left\{{\mathsf{\Phi }}_k(x,\lambda )\right\}}_{k=1}^n$ are called the Weyl solutions of Equation (1).

Let us summarise the properties of the solutions ${\mathcal{C}}_k(x,\lambda )$ and ${\mathsf{\Phi }}_k(x,\lambda )$. For the details, see [33]. The functions ${\mathcal{C}}_k(x,\lambda )$, $k=\overline{1,n}$, are uniquely defined as solutions of the initial value problems, and they are entire in $\lambda$ for each fixed $x\in [0,1]$ together with their quasi-derivatives ${\mathcal{C}}_k^{\left[j\right]}(x,\lambda )$, $j=\overline{1,n-1}$. The Weyl solutions ${\mathsf{\Phi }}_k(x,\lambda )$, $k=\overline{1,n}$, and their quasi-derivatives are meromorphic in $\lambda$. Furthermore, the fundamental matrices:

$$\mathcal{C}(x,\lambda ):={\left[{\mathcal{C}}_k^{[j-1]}(x,\lambda )\right]}_{j,k=1}^n,\mathsf{\Phi }(x,\lambda ):={\left[{\mathsf{\Phi }}_k^{[j-1]}(x,\lambda )\right]}_{j,k=1}^n$$

are related to each other as follows:

$$\mathsf{\Phi }(x,\lambda )=\mathcal{C}(x,\lambda )M\left(\lambda \right),$$

where $M\left(\lambda \right)={\left[M_{j,k}\left(\lambda \right)\right]}_{j,k=1}^n$ is called the Weyl–Yurko matrix.

The entries $M_{j,k}\left(\lambda \right)$ satisfy the relations:

$$\begin{array}{cc}\hfill M_{j,k}\left(\lambda \right)& ={\delta }_{j,k},j\le k,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill M_{j,k}\left(\lambda \right)& =-\frac{{\Delta }_{j,k}\left(\lambda \right)}{{\Delta }_{k,k}\left(\lambda \right)},k=\overline{1,n-1},j=\overline{k+1,n},\hfill \end{array}$$

(21)

where ${\Delta }_{k,k}\left(\lambda \right):=det\left({\left[{\mathcal{C}}_j^{[n-s]}(1,\lambda )\right]}_{s,j=k+1}^n\right)$, $k=\overline{1,n-1}$, and ${\Delta }_{j,k}\left(\lambda \right)$ is obtained from ${\Delta }_{k,k}\left(\lambda \right)$ by replacing ${\mathcal{C}}_j$ by ${\mathcal{C}}_k$. Clearly, the functions ${\Delta }_{j,k}\left(\lambda \right)$ for $j\ge k$ are entire in $\lambda$. Thus, $M\left(\lambda \right)$ is a unit lower-triangular matrix, whose entries under the main diagonal are meromorphic in $\lambda$, and the poles of the k-th column coincide with the zeros of ${\Delta }_{k,k}\left(\lambda \right)$.

On the other hand, the zeros of ${\Delta }_{k,k}\left(\lambda \right)$ coincide with the eigenvalues of the problem ${\mathcal{L}}_k$ for Equation (1) with the boundary conditions (2) for each $k=\overline{1,n-1}$. Therefore, under the assumption (W-2) of Definition 1, all the poles of $M\left(\lambda \right)$ are simple. The Laurent series at $\lambda ={\lambda }_{l,k}$ has the form:

$$M\left(\lambda \right)=\frac{M_{\langle -1\rangle }\left({\lambda }_{l,k}\right)}{\lambda -{\lambda }_{l,k}}+M_{\langle 0\rangle }\left({\lambda }_{l,k}\right)+M_{\langle 1\rangle }\left({\lambda }_{l,k}\right)(\lambda -{\lambda }_{l,k})+\cdots ,$$

where $M_{\langle j\rangle }\left({\lambda }_{l,k}\right)$ are the corresponding $(n\times n)$-matrix coefficients.

Define the weight matrices:

$$\mathcal{N}\left({\lambda }_{l,k}\right):={\left(M_{\langle 0\rangle }\left({\lambda }_{l,k}\right)\right)}^{-1}{M}_{\langle -1\rangle }\left({\lambda }_{l,k}\right).$$

Theorem 4.4 of [34] implies the following uniqueness result.

Proposition 1

([34]). The spectral data ${\{{\lambda }_{l,k},\mathcal{N}\left({\lambda }_{l,k}\right)\}}_{l\ge 1,k=\overline{1,n-1}}$ uniquely specify the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$ satisfying the conditions (W-1) and (W-2) of Definition 1.

The structural properties of the weight matrices $\mathcal{N}\left({\lambda }_{l,k}\right)={\left[{\mathcal{N}}_{s,j}\left({\lambda }_{l,k}\right)\right]}_{s,j=1}^n$ are similar to the ones for the case of regular coefficients (see [23,35]). In view of (20), ${\mathcal{N}}_{s,j}\left({\lambda }_{l,k}\right)=0$ for $s\le j$. Moreover, under the condition (W-3), ${\mathcal{N}}_{s,j}\left({\lambda }_{l,k}\right)=0$ for $s>j+1$. Thus, the only non-zero entries of $\mathcal{N}\left({\lambda }_{l,k}\right)$ are ${\mathcal{N}}_{j+1,j}\left({\lambda }_{l,k}\right)$ for such j that ${\Delta }_{j,j}\left({\lambda }_{l,k}\right)=0$. Therefore, instead of the weight matrices $\mathcal{N}\left({\lambda }_{l,k}\right)$, it is sufficient to use the weight numbers:

$${\beta }_{l,k}:={\mathcal{N}}_{k+1,k}\left({\lambda }_{l,k}\right)=-\frac{{\Delta }_{k+1,k}\left({\lambda }_{l,k}\right)}{\frac{d}{d\lambda }{\Delta }_{k,k}\left({\lambda }_{l,k}\right)}.$$

Indeed, if ${\lambda }_{l_1,k_1}={\lambda }_{l_2,k_2}=\cdots ={\lambda }_{l_r,k_r}$ is a group of equal eigenvalues (of different problems ${\mathcal{L}}_{k_j}$), which is maximal by inclusion, we have

$$\mathcal{N}\left({\lambda }_{l_1,k_1}\right)=\mathcal{N}\left({\lambda }_{l_2,k_2}\right)=\cdots =\mathcal{N}\left({\lambda }_{l_r,k_r}\right)=\sum _{s=1}^r{\beta }_{l_s,k_s}{E}_{k_s+1,k_s},$$

where $E_{i,j}$ denotes the matrix with the unit entry at the position $(i,j)$, and all the other entries equal zero. Hence, Proposition 1 implies the following corollary.

Corollary 1.

The spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ uniquely specify the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}\in W$.

By Theorem 6.2 of [50], the weight numbers have the asymptotics:

$${\beta }_{l,k}=l^n({\beta }_k+{\varkappa }_{l,k}^0),l\ge 1,k=\overline{1,n-1},$$

(22)

where ${\beta }_k\ne 0$ and $\left\{{\varkappa }_{l,k}^0\right\}\in l_2$.

Example 1.

For $n=2$, ${\left\{{\lambda }_{l,1}\right\}}_{l\ge 1}$ are the Dirichlet eigenvalues of the Sturm–Liouville equation $y^{″}+p_{0}y=\lambda y$ and

$$\mathcal{N}\left({\lambda }_{l,1}\right)=\left[\begin{array}{cc}0& 0\\ {\beta }_{l,1}& 0\end{array}\right].$$

One can easily check that ${\beta }_{l,1}={\alpha }_l^{-1}$, where ${\alpha }_l:={\int }_0^1{y}_l^2\left(x\right)dx$ and $y_l\left(x\right)$ is the eigenfunction of ${\lambda }_{l,1}$ such that $y_l^{\left[1\right]}\left(0\right)=1$ for a distributional potential $p_0$ or $y^{\prime }\left(0\right)=1$ for an integrable potential $p_0$. Therefore, ${\{{\lambda }_{l,1},{\beta }_{l,1}\}}_{l\ge 1}$ are equivalent to the spectral data ${\{{\lambda }_{l,1},{\alpha }_l\}}_{l\ge 1}$ of the classical inverse Sturm–Liouville problem studied by Marchenko [10], Gelfand and Levitan [20], etc.

Example 2.

For $n=3$, (W-3) implies $\left\{{\lambda }_{l,1}\right\}\cap \left\{{\lambda }_{l,2}\right\}=⌀$. Hence,

$$\mathcal{N}\left({\lambda }_{l,1}\right)=\left[\begin{array}{ccc}0& 0& 0\\ {\beta }_{l,1}& 0& 0\\ 0& 0& 0\end{array}\right],\mathcal{N}\left({\lambda }_{l,2}\right)=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& {\beta }_{l,2}& 0\end{array}\right].$$

The recovery of the coefficients $p=(p_0,p_1)\in W$ from the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=1,2}$ was investigated in [36].

4. Main Equation

In this section, we reduce Problem 1 to a linear equation in the Banach space m of bounded infinite sequences. First, we deduced an infinite system of linear equations. Second, this system was transformed to achieve the absolute convergence of the series by the method of [23]. Although we relied on the general approach of [35], the construction of the main equation was simplified because of the separation condition (W-3).

Consider the two coefficient vectors $p={\left(p_k\right)}_{k=0}^{n-2}$ and $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}$ of the class W. Note that the differential expression ${\tilde{\ell }}_n\left(y\right)$ with the coefficients $\tilde{p}$ has the associated matrix $\tilde{F}\left(x\right)$, which can be different from $F\left(x\right)$, so the corresponding quasi-derivatives differ. The matrix ${\tilde{F}}^{★}\left(x\right)$ and the corresponding quasi-derivatives are defined analogously to $F^{★}\left(x\right)$ and (11), respectively.

For $k=\overline{1,n}$, denote by ${\tilde{\mathsf{\Phi }}}_k^{★}(x,\lambda )$ the solution of the boundary value problem:

$$\begin{array}{c}{\tilde{\ell }}_n^{★}\left({\tilde{\mathsf{\Phi }}}_k^{★}\right)=\lambda {\tilde{\mathsf{\Phi }}}_k^{★},x\in (0,1),\\ {\tilde{\mathsf{\Phi }}}_k^{★[j-1]}(0,\lambda )={\delta }_{k,j},j=\overline{1,k},{\tilde{\mathsf{\Phi }}}_k^{★[s-1]}(1,\lambda )=0,s=\overline{1,n-k}.\end{array}$$

Introduce the notations:

$$\begin{array}{c}V:=\{(l,k,\epsilon ):l\in \mathbb{N},k=\overline{1,n-1}.\epsilon =0,1\},\\ {\lambda }_{l,k,0}:={\lambda }_{l,k},{\lambda }_{l,k,1}:={\tilde{\lambda }}_{l,k},{\beta }_{l,k,0}:={\beta }_{l,k},{\beta }_{l,k,1}:={\tilde{\beta }}_{l,k},\end{array}$$

$$\begin{array}{c}{\phi }_{l,k,\epsilon }\left(x\right):={\mathsf{\Phi }}_{k+1}(x,{\lambda }_{l,k,\epsilon }),{\tilde{\phi }}_{l,k,\epsilon }\left(x\right):={\tilde{\mathsf{\Phi }}}_{k+1}(x,{\lambda }_{l,k,\epsilon }),(l,k,\epsilon )\in V,\end{array}$$

(23)

$$\begin{array}{c}{\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}\left(x\right):=\left\{\begin{array}{cc}{(-1)}^{n-k}{\beta }_{l,k,\epsilon }{\int }_0^x{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(t,{\lambda }_{l,k,\epsilon }){\tilde{\mathsf{\Phi }}}_{k_0+1}(t,{\lambda }_{l_0,k_0,{\epsilon }_0})dt,\hfill & (l,k,\epsilon )=(l_0,k_0,{\epsilon }_0),\hfill \\ {(-1)}^{n-k}{\beta }_{l,k,\epsilon }{\displaystyle \frac{\langle {\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon }),{\tilde{\mathsf{\Phi }}}_{k_0+1}(x,{\lambda }_{l_0,k_0,{\epsilon }_0})\rangle }{{\lambda }_{l_0,k_0,{\epsilon }_0}-{\lambda }_{l,k,\epsilon }}},\hfill & (l,k,\epsilon )\ne (l_0,k_0,{\epsilon }_0).\hfill \end{array}\right.\end{array}$$

(24)

Note that, for each fixed $k\in \{1,2,\cdots ,n-2\}$, the Weyl solution ${\mathsf{\Phi }}_{k+1}(x,\lambda )$ has the poles ${\left\{{\lambda }_{l,k+1,0}\right\}}_{l\ge 1}$, which do not coincide with ${\left\{{\lambda }_{l,k,0}\right\}}_{l\ge 1}$ because of (W-3). Furthermore, the solution ${\mathsf{\Phi }}_n(x,\lambda )\equiv {\mathcal{C}}_n(x,\lambda )$ is entire in $\lambda$. For technical convenience, assume that ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1,k=\overline{1,n-1}}\cap {\left\{{\tilde{\lambda }}_{l,k}\right\}}_{l\ge 1,k=\overline{1,n-1}}=⌀$. The opposite case requires minor changes. Hence, the functions ${\phi }_{l,k,\epsilon }\left(x\right)$ are correctly defined by (23) and so are ${\tilde{\phi }}_{l,k,\epsilon }\left(x\right)$, $(l,k,\epsilon )\in V$.

It was shown in [35] that ${\tilde{\mathsf{\Phi }}}_j^{★}(x,\lambda )$ has the poles ${\tilde{\lambda }}_{l,j}^{★}={\tilde{\lambda }}_{l,n-j-1}$, $l\ge 1$, for $j=\overline{1,n-1}$, and ${\tilde{\mathsf{\Phi }}}_n^{★}(x,\lambda )$ is entire in $\lambda$. Therefore, ${\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon })$ is correctly defined, and so is ${\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}\left(x\right)$ for $(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)\in V$.

In [35], the following infinite linear system was obtained:

$${\phi }_{l_0,k_0,{\epsilon }_0}\left(x\right)={\tilde{\phi }}_{l_0,k_0,{\epsilon }_0}\left(x\right)+\sum _{(l,k,\epsilon )\in V}{(-1)}^{\epsilon }{\phi }_{l,k,\epsilon }\left(x\right){\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}\left(x\right),(l_0,k_0,{\epsilon }_0)\in V.$$

(25)

Our next goal is to combine the terms in (25) to achieve the absolute convergence of the series. Introduce the numbers:

$${\xi }_l:=\sum _{k=1}^{n-1}\left(l^{-(n-1)}|{\lambda }_{l,k}-{\tilde{\lambda }}_{l,k}|+l^{-n}|{\beta }_{l,k}-{\tilde{\beta }}_{l,k}|\right),l\ge 1,$$

(26)

which characterise the “distance” between the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ and ${\{{\tilde{\lambda }}_{l,k},{\tilde{\beta }}_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ of p and $\tilde{p}$, respectively. The asymptotics (3) and (22) imply that $\left\{{\xi }_l\right\}\in l_2$. In addition, define the functions:

$$w_{l,k}\left(x\right):=l^{-k}exp(-xlcot(k\pi /n)),$$

which characterise the growth of ${\phi }_{l,k,\epsilon }\left(x\right)$: the estimate $|{\phi }_{l,k,\epsilon }\left(x\right)|\le C{w}_{l,k}\left(x\right)$ holds by Lemma 7 in [35].

Pass to the new variables:

$$\left[\begin{array}{c}{\psi }_{l,k,0}\left(x\right)\\ {\psi }_{l,k,1}\left(x\right)\end{array}\right]:=w_{l,k}^{-1}\left(x\right)\left[\begin{array}{cc}{\xi }_l^{-1}& -{\xi }_l^{-1}\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\phi }_{l,k,0}\left(x\right)\\ {\phi }_{l,k,1}\left(x\right)\end{array}\right],$$

(27)

$$\begin{array}{c}\left[\begin{array}{cc}{\tilde{R}}_{(l_0,k_0,0),(l,k,0)}\left(x\right)& {\tilde{R}}_{(l_0,k_0,0),(l,k,1)}\left(x\right)\\ {\tilde{R}}_{(l_0,k_0,1),(l,k,0)}\left(x\right)& {\tilde{R}}_{(l_0,k_0,1),(l,k,1)}\left(x\right)\end{array}\right]:=\hfill \\ \hfill \frac{w_{l,k}\left(x\right)}{w_{l_0,k_0}\left(x\right)}\left[\begin{array}{cc}{\xi }_{l_0}^{-1}& -{\xi }_{l_0}^{-1}\\ 0& 1\end{array}\right]\left[\begin{array}{cc}{\tilde{G}}_{(l,k,0),(l_0,k_0,0)}\left(x\right)& {\tilde{G}}_{(l,k,1),(l_0,k_0,0)}\left(x\right)\\ {\tilde{G}}_{(l,k,0),(l_0,k_0,1)}\left(x\right)& {\tilde{G}}_{(l,k,1),(l_0,k_0,1)}\left(x\right)\end{array}\right]\left[\begin{array}{cc}{\xi }_l& 1\\ 0& -1\end{array}\right].\end{array}$$

(28)

Analogous to ${\psi }_{l,k,\epsilon }\left(x\right)$, define ${\tilde{\psi }}_{l,k,\epsilon }\left(x\right)$. For brevity, denote $v=(l,k,\epsilon )$, $v_0=(l_0,k_0,{\epsilon }_0)$, $v,v_0\in V$. Then, Relation (25) is transformed into

$${\psi }_{v_0}\left(x\right)={\tilde{\psi }}_{v_0}\left(x\right)+\sum _{v\in V}{\tilde{R}}_{v_0,v}\left(x\right){\psi }_v\left(x\right),v_0\in V,$$

(29)

where

$$|{\psi }_v\left(x\right)|,|{\tilde{\psi }}_v\left(x\right)|\le C,|{\tilde{R}}_{v_0,v}\left(x\right)|\le \frac{C{\xi }_l}{|l-l_0|+1},v_0,v\in V.$$

(30)

Consider the Banach space m of bounded infinite sequences $a={\left[a_v\right]}_{v\in V}$ with the norm ${\parallel a\parallel }_m:={sup}_{v\in V}\left|a_v\right|$. For each fixed $x\in [0,1]$, define the linear operator $\tilde{R}\left(x\right):m\to m$ as follows:

$${\left(\tilde{R}\left(x\right)a\right)}_{v_0}=\sum _{v\in V}{\tilde{R}}_{v_0,v}\left(x\right)a_v,v_0\in V.$$

In view of the estimates (30), $\psi \left(x\right),\tilde{\psi }\left(x\right)\in m$, and the operator $\tilde{R}\left(x\right)$ is bounded in m for each fixed $x\in [0,1]$. Denote by I the identity operator in m. Then, the system (29) can be represented as a linear equation in the Banach space m:

$$(I-\tilde{R}\left(x\right))\psi \left(x\right)=\tilde{\psi }\left(x\right),x\in [0,1].$$

(31)

Equation (31) is called the main equation of Problem 1. We derived (31) under the assumption that $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ and $\{{\tilde{\lambda }}_{l,k},{\tilde{\beta }}_{l,k}\}$ are the spectral data of the two problems with the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$ and $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}$, respectively. Anyway, the main Equation (31) can be used for the reconstruction of p by $\{{\lambda }_{l,k},{\beta }_{l,k}\}$. Indeed, one can choose an arbitrary $\tilde{p}\in W$, find $\tilde{\psi }\left(x\right)$ and $\tilde{R}\left(x\right)$ by using $\tilde{p}$, $\{{\tilde{\lambda }}_{l,k},{\tilde{\beta }}_{l,k}\}$, and $\{{\lambda }_{l,k},{\beta }_{l,k}\}$, then find $\psi \left(x\right)$ by solving the main equation. In order to find p from $\psi \left(x\right)$, we need reconstruction formulas, which are obtained in the next section.

5. Reconstruction Formulas

In this section, we derive formulas for recovering the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ from the solution $\psi \left(x\right)$ of the main Equation (31). For the derivation, we used the special structure of the associated matrices $F\left(x\right)$ and $F^{★}\left(x\right)$. The arguments of this section are based on formal calculations with infinite series. The convergence of those series will be rigorously studied in the next section.

Find ${\left\{{\phi }_{l,k,\epsilon }\left(x\right)\right\}}_{(l,k,\epsilon )\in V}$ from (27):

$$\left[\begin{array}{c}{\phi }_{l,k,0}\left(x\right)\\ {\phi }_{l,k,1}\left(x\right)\end{array}\right]=w_{l,k}\left(x\right)\left[\begin{array}{cc}{\xi }_l& 1\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\psi }_{l,k,0}\left(x\right)\\ {\psi }_{l,k,1}\left(x\right)\end{array}\right].$$

(32)

Then, we can recover the Weyl solutions for $k_0=\overline{1,n}$:

$${\mathsf{\Phi }}_{k_0}(x,\lambda ):={\tilde{\mathsf{\Phi }}}_{k_0}(x,\lambda )+\sum _{(l,k,\epsilon )\in V}{(-1)}^{\epsilon +n-k}{\beta }_{l,k,\epsilon }{\phi }_{l,k,\epsilon }\left(x\right)\frac{\langle {\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon }),{\tilde{\mathsf{\Phi }}}_{k_0+1}(x,\lambda )\rangle }{\lambda -{\lambda }_{l,k,\epsilon }}.$$

(33)

Formally applying the differential expression ${\ell }_n\left(y\right)$ to the left- and the right-hand sides of (33), after some transforms (see ([35], Section 4.1)), we arrive at the relation:

$$\begin{array}{cc}\hfill \sum _{(l,k,\epsilon )\in V}{(-1)}^{\epsilon }{\phi }_{l,k,\epsilon }\left(x\right)\langle {\tilde{\eta }}_{l,k,\epsilon }\left(x\right),{\tilde{\mathsf{\Phi }}}_{k_0}(x,\lambda )\rangle =& \sum _{s=0}^{n-2}{\widehat{p}}_s\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )+\sum _{s=0}^{n-1}{t}_{n,s}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )\hfill \\ \hfill & +\sum _{s=0}^{n-3}\sum _{r=s+1}^{n-2}{p}_r\left(x\right)t_{r,s}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda ),\hfill \end{array}$$

(34)

where ${\widehat{p}}_s:=p_s-{\tilde{p}}_s$,

$$\begin{array}{cc}\hfill {\tilde{\eta }}_{l,k,\epsilon }\left(x\right)& :={(-1)}^{n-k}{\beta }_{l,k,\epsilon }{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon }),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill t_{r,s}\left(x\right)& :=\sum _{u=s}^{r-1}{C}_r^{u+1}{C}_u^s{T}_{r-u-1,u-s}\left(x\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill T_{j_1,j_2}\left(x\right)& :=\sum _{(l,k,\epsilon )\in V}{(-1)}^{\epsilon }{\phi }_{l,k,\epsilon }^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,\epsilon }^{\left(j_2\right)}\left(x\right).\hfill \end{array}$$

(37)

Note that, for $(l,k,\epsilon )\in V$, the functions ${\tilde{\eta }}_{l,k,\epsilon }\left(x\right)$ are solutions of the equation ${\tilde{\ell }}^{★}\left(z\right)={\lambda }_{l,k,\epsilon }z$, so their quasi-derivatives are generated by the matrix function ${\tilde{F}}^{★}\left(x\right)$. Thus, Relation (19) for the Lagrange bracket in (34) implies

$$\begin{array}{cc}\hfill \langle {\tilde{\eta }}_{l,k,\epsilon }\left(x\right),{\tilde{\mathsf{\Phi }}}_{k_0}(x,\lambda )\rangle =& \sum _{s=0}^{n-1}{(-1)}^{n-s-1}{\tilde{\eta }}_{l,k,\epsilon }^{(n-s-1)}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )\hfill \\ \hfill & +\sum _{s=0}^{n-3}\left(\sum _{j=0}^{n-s-3}\sum _{r=j}^{n-s-3}{(-1)}^r{C}_r^j{\tilde{p}}_{s+r+1}^{(r-j)}\left(x\right){\tilde{\eta }}_{l,k,\epsilon }^{\left(j\right)}\left(x\right)\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda ).\hfill \end{array}$$

(38)

Substituting (38) into (34) and using (37), we obtain

$$\begin{array}{cc}\hfill & \sum _{s=0}^{n-1}{(-1)}^{n-s-1}{T}_{0,n-s-1}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )+\sum _{s=0}^{n-3}\left(\sum _{j=0}^{n-s-3}\sum _{r=j}^{n-s-3}{(-1)}^r{C}_r^j{\tilde{p}}_{s+r+1}^{(r-j)}\left(x\right)T_{0,j}\left(x\right)\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )\hfill \\ & =\sum _{s=0}^{n-2}{\widehat{p}}_s\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )+\sum _{s=0}^{n-1}{t}_{n,s}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda )+\sum _{s=0}^{n-3}\sum _{r=s+1}^{n-2}{p}_r\left(x\right)t_{r,s}\left(x\right){\tilde{\mathsf{\Phi }}}_{k_0}^{\left(s\right)}(x,\lambda ).\hfill \end{array}$$

(39)

Let us group the terms at ${\mathsf{\Phi }}_{k_0}^{\left(s\right)}(x,\lambda )$ and assume that the corresponding left- and right-hand sides are equal to each other:

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{k_0}^{(n-1)}(x,\lambda ):& T_{0,0}\left(x\right)=T_{0,0}\left(x\right),\hfill \\ \hfill {\mathsf{\Phi }}_{k_0}^{(n-2)}(x,\lambda ):& -T_{0,1}\left(x\right)={\widehat{p}}_{n-2}\left(x\right)+t_{n,n-2}\left(x\right),\hfill \\ \hfill {\mathsf{\Phi }}_{k_0}^{\left(s\right)}(x,\lambda ):& {(-1)}^{n-s-1}{T}_{0,n-s-1}\left(x\right)+\sum _{j=0}^{n-s-3}\sum _{r=j}^{n-s-3}{(-1)}^r{C}_r^j{\tilde{p}}_{s+r+1}^{(r-j)}\left(x\right)T_{0,j}\left(x\right)\hfill \\ \hfill & ={\widehat{p}}_s\left(x\right)+t_{n,s}\left(x\right)+\sum _{r=s+1}^{n-2}{p}_r\left(x\right)t_{r,s}\left(x\right),s=\overline{0,n-3}.\hfill \end{array}$$

From here, we obtain the reconstruction formulas:

$$\begin{array}{cc}\hfill p_s\left(x\right)=& {\tilde{p}}_s\left(x\right)-\left(t_{n,s}\left(x\right)+{(-1)}^{n-s}{T}_{0,n-s-1}\left(x\right)\right)\hfill \\ \hfill & +\sum _{j=0}^{n-s-3}\sum _{r=j}^{n-s-3}{(-1)}^r{C}_r^j{\tilde{p}}_{r+s+1}^{(r-j)}\left(x\right)T_{0,j}\left(x\right)-\sum _{r=s+1}^{n-2}{p}_r\left(x\right)t_{r,s}\left(x\right),\hfill \end{array}$$

(40)

for $s=n-2,n-3,\cdots ,1,0$.

Thus, we arrive at the following constructive procedure for solving Problem 1.

Procedure 1.

Suppose that the spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ are given. We have to find the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$:

1. 

Choose $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}\in W$, and find the spectral data ${\{{\tilde{\lambda }}_{l,k},{\tilde{\beta }}_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$.

2. 

Find the Weyl solutions ${\tilde{\phi }}_{l,k,\epsilon }\left(x\right)={\tilde{\mathsf{\Phi }}}_{k+1}(x,{\lambda }_{l,k,\epsilon })$ and ${\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon })$ for $(l,k,\epsilon )\in V$, and construct ${\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}\left(x\right)$ by (24).

3. 

Construct ${\tilde{\psi }}_v\left(x\right)$ for $v\in V$ by (27) and ${\tilde{R}}_{v_0,v}\left(x\right)$ for $v_0,v\in V$ by (28).

4. 

Find ${\psi }_v\left(x\right)$, $v\in V$, by solving the main Equation (31).

5. 

For $(l,k,\epsilon )\in V$, determine ${\phi }_{l,k,\epsilon }\left(x\right)$ by (32) and ${\tilde{\eta }}_{l,k,\epsilon }\left(x\right)$ by (35).

6. 

For $s=n-2,n-3,\cdots ,1,0$, find $p_s\left(x\right)$ by Formula (40), in which $t_{r,s}\left(x\right)$ and $T_{j_1,j_2}\left(x\right)$ are defined by (36) and (37), respectively.

Procedure 1 will be used in the next section for proving Theorem 1. In general, there is a challenge to choose a model problem $\tilde{p}$ so that the series for $p_s$ converge in the corresponding spaces. Note that Steps 1–5 work for any $\tilde{p}\in W$, since for them, the estimate $\left\{{\xi }_l\right\}\in l_2$ is sufficient. However, the situation differs for Step 6. In Section 6, we prove the validity of Step 6 in the case $\left\{l^{n-2}{\xi }_l\right\}\in l_2$.

6. Solvability and Stability

In this section, we prove the following theorem on the solvability of Problem 1.

Theorem 2.

Let complex numbers ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ and coefficients $\tilde{p}={\left({\tilde{p}}_k\right)}_{k=0}^{n-2}\in W$ be such that:

(S-1) For each $k=\overline{1,n-1}$, the numbers ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}$ are distinct.

(S-2) ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}\cap {\left\{{\lambda }_{l,k+1}\right\}}_{l\ge 1}=⌀$ for $k=\overline{1,n-2}$.

(S-3) ${\beta }_{l,k}\ne 0$ for all $l\ge 1$, $k=\overline{1,n-1}$.

(S-4) ${\left\{l^{n-2}{\xi }_l\right\}}_{l\ge 1}\in l_2$, where the numbers ${\xi }_l$ are defined in (26).

(S-5) The operator $(I-\tilde{R}\left(x\right))$, which is constructed by using $\{{\lambda }_{l,k},{\beta }_{l,k}\}$, and $\tilde{p}$ according to Section 4, has a bounded inverse operator for each fixed $x\in [0,1]$.

Then, ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ are the spectral data of some (unique) $p={\left(p_k\right)}_{k=0}^{n-2}\in W$.

Theorem 2 provides sufficient conditions for the global solvability of the inverse problem. Theorem 1 on local solvability and stability will be obtained as a corollary of Theorem 2. Thus, Theorem 2 plays an auxiliary role in this paper, but also has a separate significance. The proof of Theorem 2 is based on Procedure 1. We investigated the properties of the solution $\psi \left(x\right)$ of the main equation and proved the convergence of the series in (37) and (40) in the corresponding spaces of regular and generalised functions. This part of the proofs is the most-difficult one, since the series converge in different spaces and precise estimates for the Weyl solutions are needed. Finally, we show that the numbers $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ satisfying the conditions of Theorem 2 are the spectral data of the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$ reconstructed by Formula (40). In the end of this section, we prove Theorem 1.

Proceed to the proof of Theorem 2. Let ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ and $\tilde{p}$ satisfy the hypotheses (S-1)–(S-5). We emphasise that $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ are not necessarily the spectral data corresponding to some p. We have to prove this.

By virtue of (S-5), the operator $(I-\tilde{R}\left(x\right))$ has a bounded inverse. Therefore, the main Equation (31) is uniquely solvable in m for each fixed $x\in [0,1]$. Consider its solution $\psi \left(x\right)={\left[{\psi }_v\left(x\right)\right]}_{v\in V}$. Recover the functions ${\phi }_{l,k,\epsilon }\left(x\right)$ for $(l,k,\epsilon )\in V$ by (32). Let us study their properties. For this purpose, we need the auxiliary estimates for ${\tilde{\mathsf{\Phi }}}_k(x,\lambda )$ and ${\tilde{\mathsf{\Phi }}}_k^{★}(x,\lambda )$, which were deduced from the results of [33] and used in Section 4.2 of [35].

Proposition 2

([33,35]). For $(l,k,\epsilon )\in V$, $x\in [0,1]$, and $\nu =\overline{0,n-1}$, the following estimates hold:

$$\begin{array}{c}|{\tilde{\mathsf{\Phi }}}_{k+1}^{\left[\nu \right]}(x,{\lambda }_{l,k,\epsilon })|\le C{l}^{\nu }{w}_{l,k}\left(x\right),|{\tilde{\mathsf{\Phi }}}_{k+1}^{\left[\nu \right]}(x,{\lambda }_{l,k,0})-{\tilde{\mathsf{\Phi }}}_{k+1}^{\left[\nu \right]}(x,{\lambda }_{l,k,1})|\le C{l}^{\nu }{w}_{l,k}\left(x\right){\xi }_l,\\ |{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,{\lambda }_{l,k,\epsilon })|\le C{l}^{\nu -n}{w}_{l,k}^{-1}\left(x\right),|{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,{\lambda }_{l,k,0})-{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,{\lambda }_{l,k,1})|\le C{l}^{\nu -n}{w}_{l,k}^{-1}\left(x\right){\xi }_l,\end{array}$$

where $w_{l,k}\left(x\right):=l^{-k}exp(-xlcot(k\pi /n))$.

Corollary 2.

The functions ${\tilde{\mathsf{\Phi }}}_k(.,\lambda )$ and ${\tilde{\mathsf{\Phi }}}_k^{★}(.,\lambda )$ belong to $C^{n-2}[0,1]$ for $k=\overline{1,n}$. Moreover, for $\nu =\overline{0,n-2}$, the estimates of Proposition 2 are valid for the quasi-derivatives replaced with the classical derivatives:

$$\begin{array}{c}|{\tilde{\mathsf{\Phi }}}_{k+1}^{\left(\nu \right)}(x,{\lambda }_{l,k,\epsilon })|\le C{l}^{\nu }{w}_{l,k}\left(x\right),|{\tilde{\mathsf{\Phi }}}_{k+1}^{\left(\nu \right)}(x,{\lambda }_{l,k,0})-{\tilde{\mathsf{\Phi }}}_{k+1}^{\left(\nu \right)}(x,{\lambda }_{l,k,1})|\le C{l}^{\nu }{w}_{l,k}\left(x\right){\xi }_l,\end{array}$$

(41)

$$\begin{array}{c}|{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,{\lambda }_{l,k,\epsilon })|\le C{l}^{\nu -n}{w}_{l,k}^{-1}\left(x\right),|{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,{\lambda }_{l,k,0})-{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,{\lambda }_{l,k,1})|\le C{l}^{\nu -n}{w}_{l,k}^{-1}\left(x\right){\xi }_l,\end{array}$$

(42)

where $(l,k,\epsilon )\in V$, $x\in [0,1]$.

Proof. 

Recall that ${\left\{{\tilde{\mathsf{\Phi }}}_k(x,\lambda )\right\}}_{k=1}^n$ and ${\left\{{\tilde{\mathsf{\Phi }}}_k^{★}(x,\lambda )\right\}}_{k=1}^n$ are solutions of the equations $\tilde{\ell }\left(y\right)=\lambda y$ and ${\tilde{\ell }}^{★}\left(z\right)=\lambda z$, respectively. Hence,

$${\tilde{\mathsf{\Phi }}}_k(.,\lambda )\in {\mathcal{D}}_{\tilde{F}}\subset W_1^{n-1}[0,1],{\tilde{\mathsf{\Phi }}}_k^{★}(.,\lambda )\in {\mathcal{D}}_{{\tilde{F}}^{★}}\subset W_1^{n-1}[0,1].$$

This implies ${\tilde{\mathsf{\Phi }}}_k(.,\lambda ),{\tilde{\mathsf{\Phi }}}_k^{★}(.,\lambda )\in C^{n-2}[0,1]$.

In view of (8), we have ${\tilde{\mathsf{\Phi }}}_k^{\left[\nu \right]}(x,\lambda )={\tilde{\mathsf{\Phi }}}_k^{\left(\nu \right)}(x,\lambda )$ for $\nu =\overline{0,n-2}$, so the estimates (41) readily follow from Proposition 2.

The quasi-derivatives for ${\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,\lambda )$ are generated by the matrix function ${\tilde{F}}^{★}\left(x\right)$. Hence, a relation similar to (15) is valid for them:

$${\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,\lambda )={\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,\lambda )+\sum _{j=0}^{\nu -2}\left(\sum _{s=j}^{\nu -2}{(-1)}^{s+\nu }{C}_s^j{\tilde{p}}_{n-\nu +s}^{(s-j)}\left(x\right)\right){\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(j\right)}(x,\lambda ),$$

(43)

where $\nu =\overline{0,n-2}$, $k=\overline{1,n}$.

Since ${\tilde{p}}_k\in W_2^{k-1}[0,1]$ for $k=\overline{0,n-2}$, we obtain that all the derivatives ${\tilde{p}}_{n-\nu +s}^{(s-j)}$ in (43) belong to $W_1^1[0,1]$, so they are bounded. Let us prove the estimates (42) by induction. For $\nu =0,1$, we have ${\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,\lambda )={\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,\lambda )$, so the estimates (42) directly follow from Proposition 2. Next, consider $\nu \ge 2$, and assume that the estimates (42) were already proven for $0,1,\cdots ,\nu -1$. Then, using (43), Proposition 2, and the induction hypothesis, we obtain

$$|{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(\nu \right)}(x,{\lambda }_{l,k,\epsilon })|\le |{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left[\nu \right]}(x,{\lambda }_{l,k,\epsilon })|+C\sum _{j=0}^{\nu -2}\left|{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★\left(j\right)}(x,{\lambda }_{l,k,\epsilon })\right|\le C{l}^{\nu -n}{w}_{l,k}^{-1}\left(x\right).$$

The second estimate in (42) is obtained similarly. □

Comparing (4) to (26) and taking (S-4) into account, we conclude that

$$\mathsf{\Omega }={\left(\sum _{l=1}^{\infty }{\left(l^{n-2}{\xi }_l\right)}^2\right)}^{1/2}<\infty .$$

(44)

Lemma 2.

For $(l,k,\epsilon )\in V$, we have ${\phi }_{l,k,\epsilon }\in C^{n-2}[0,1]$ and

$$\begin{array}{c}|{\phi }_{l,k,\epsilon }^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}\left(x\right),|{\phi }_{l,k,0}^{\left(\nu \right)}\left(x\right)-{\phi }_{l,k,1}^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}\left(x\right){\xi }_l,\nu =\overline{0,n-2},\\ |{\phi }_{l,k,\epsilon }\left(x\right)-{\tilde{\phi }}_{l,k,\epsilon }\left(x\right)|\le C\mathsf{\Omega }{w}_{l,k}\left(x\right){\chi }_l,\\ |{\phi }_{l,k,0}\left(x\right)-{\phi }_{l,k,1}\left(x\right)-{\tilde{\phi }}_{l,k,0}\left(x\right)+{\tilde{\phi }}_{l,k,1}\left(x\right)|\le C\mathsf{\Omega }{w}_{l,k}\left(x\right){\chi }_l{\xi }_l,\\ \left.\begin{array}{c}|{\phi }_{l,k,\epsilon }^{\left(\nu \right)}\left(x\right)-{\tilde{\phi }}_{l,k,\epsilon }^{\left(\nu \right)}\left(x\right)|\le C\mathsf{\Omega }{l}^{\nu -1}{w}_{l,k}\left(x\right),\\ |{\phi }_{l,k,0}^{\left(\nu \right)}\left(x\right)-{\phi }_{l,k,1}^{\left(\nu \right)}-{\tilde{\phi }}_{l,k,0}^{\left(\nu \right)}\left(x\right)+{\tilde{\phi }}_{l,k,1}^{\left(\nu \right)}\left(x\right)|\le C\mathsf{\Omega }{l}^{\nu -1}{w}_{l,k}\left(x\right){\xi }_l,\end{array}\right\}\nu =\overline{1,n-2},\end{array}$$

where $x\in [0,1]$ and

$${\chi }_l:={\left(\sum _{k=1}^{\infty }\frac{1}{k^2{\left(\right|l-k|+1)}^2}\right)}^{1/2},{\left\{{\chi }_l\right\}}_{l\ge 1}\in l_2.$$

Proof. 

First, let us investigate the smoothness of the functions ${\tilde{\psi }}_v\left(x\right)$ and ${\tilde{R}}_{v_0,v}\left(x\right)$ in the main equation. Due to (23), (27), and Corollary 2, we obtain ${\tilde{\phi }}_{l,k,\epsilon }\in C^{n-2}[0,1]$ for $(l,k,\epsilon )\in V$, and so, ${\tilde{\psi }}_v\in C^{n-2}[0,1]$ for $v\in V$. Next, applying (18) to (24), we obtain

$${\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)={(-1)}^{n-k}{\beta }_{l,k,\epsilon }{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,\epsilon }){\tilde{\mathsf{\Phi }}}_{k_0+1}(x,{\lambda }_{l_0,k_0,{\epsilon }_0})\in C^{n-2}[0,1].$$

(45)

Using (45) together with (28), we conclude that ${\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}\in C^{n-1}[0,1]$, and so, ${\tilde{R}}_{v_0,v}\in C^{n-1}[0,1]$. Moreover, using (23) and Corollary 2, we obtain

$$|{\tilde{\phi }}_{l,k,\epsilon }^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}\left(x\right),|{\tilde{\phi }}_{l,k,0}^{\left(\nu \right)}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}\left(x\right){\xi }_l,$$

(46)

for $\nu =\overline{0,n-2}$, $(l,k,\epsilon )\in V$.

The asymptotics (22) and (26) imply the following estimates for the weight numbers:

$$|{\beta }_{l,k,\epsilon }|\le C{l}^n,|{\beta }_{l,k,0}-{\beta }_{l,k,1}|\le C{l}^n{\xi }_l,(l,k,\epsilon )\in V.$$

(47)

Using (45), (47) and Corollary 2, we obtain

$$|{\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)|\le C{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right).$$

Furthermore,

$$\begin{array}{c}{\tilde{G}}_{(l,k,0),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)-{\tilde{G}}_{(l,k,1),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)={(-1)}^{n-k}({\beta }_{l,k,0}-{\beta }_{l,k,1}){\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,0}){\tilde{\mathsf{\Phi }}}_{k_0+1}(x,{\lambda }_{l_0,k_0,{\epsilon }_0})\hfill \\ \hfill +{(-1)}^{n-k}{\beta }_{l,k,1}({\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,0})-{\tilde{\mathsf{\Phi }}}_{n-k+1}^{★}(x,{\lambda }_{l,k,1})){\tilde{\mathsf{\Phi }}}_{k_0+1}(x,{\lambda }_{l_0,k_0,{\epsilon }_0}).\end{array}$$

Applying the estimates of Corollary 2 and (47), we obtain

$$|{\tilde{G}}_{(l,k,0),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)-{\tilde{G}}_{(l,k,1),(l_0,k_0,{\epsilon }_0)}^{\prime }\left(x\right)|\le C{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right){\xi }_l.$$

Calculating the derivatives of higher-orders for (45), we similarly obtain

$$\begin{array}{c}|{\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}^{\left(\nu \right)}\left(x\right)|\le C{(l+l_0)}^{\nu -1}{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right),\\ |{\tilde{G}}_{(l,k,0),(l_0,k_0,{\epsilon }_0)}^{\left(\nu \right)}\left(x\right)-{\tilde{G}}_{(l,k,1),(l_0,k_0,{\epsilon }_0)}^{\left(\nu \right)}\left(x\right)|\le C{(l+l_0)}^{\nu -1}{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right){\xi }_l\end{array}$$

for $\nu =\overline{2,n-2}$. Analogously, we deduce

$$\begin{array}{c}|{\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,0)}^{\left(\nu \right)}\left(x\right)-{\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,1)}^{\left(\nu \right)}\left(x\right)|\le C{(l+l_0)}^{\nu -1}{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right){\xi }_{l_0},\\ |{\tilde{G}}_{(l,k,0),(l_0,k_0,0)}^{\left(\nu \right)}\left(x\right)-{\tilde{G}}_{(l,k,1),(l_0,k_0,0)}^{\left(\nu \right)}\left(x\right)-{\tilde{G}}_{(l,k,0),(l_0,k_0,1)}^{\left(\nu \right)}\left(x\right)+{\tilde{G}}_{(l,k,1),(l_0,k_0,1)}^{\left(\nu \right)}\left(x\right)|\\ \le C{(l+l_0)}^{\nu -1}{w}_{l_0,k_0}\left(x\right)w_{l,k}^{-1}\left(x\right){\xi }_{l_0}{\xi }_l,\end{array}$$

for $\nu =\overline{1,n-2}$.

Then, using (27), (28), (46), and the above estimates for ${\tilde{G}}_{(l,k,\epsilon ),(l_0,k_0,{\epsilon }_0)}^{\left(\nu \right)}\left(x\right)$, we obtain

$$|{\tilde{\psi }}_v^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu },\left|{\tilde{R}}_{v_0,v}^{\left(\nu \right)}\left(x\right)\right|\le C{(l+l_0)}^{\nu -1}{\xi }_l,v_0,v\in V,\nu =\overline{1,n-2}.$$

(48)

In addition, we have Formula (30) for ${\tilde{\psi }}_v\left(x\right)$, ${\tilde{R}}_{v_0,v}\left(x\right)$, $\nu =0$. The obtained estimates coincide with the ones for the case of regular coefficients (see the formulas (2.3.40) in [23]). Therefore, the remaining part of the proof almost repeats the proof of Lemma 1.6.7 in [23], so we omit the technical details. By differentiating the relation $\psi \left(x\right)={(I-\tilde{R}\left(x\right))}^{-1}\tilde{\psi }\left(x\right)$ and analysing the convergence of the obtained series, we prove the following properties of $\psi \left(x\right)={\left[{\psi }_v\left(x\right)\right]}_{v\in V}$:

$$\begin{array}{c}{\psi }_v\in C^{n-2}[0,1],\left|{\psi }_v^{\left(\nu \right)}\left(x\right)\right|\le C{l}^{\nu },\nu =\overline{0,n-2},\\ |{\psi }_v\left(x\right)-{\tilde{\psi }}_v\left(x\right)|\le C\mathsf{\Omega }{\chi }_l,|{\psi }_v^{\left(\nu \right)}\left(x\right)-{\tilde{\psi }}_v^{\left(\nu \right)}\left(x\right)|\le C\mathsf{\Omega }{l}^{\nu -1},\nu =\overline{1,n-2}.\end{array}$$

Using the latter estimates together with (32), we readily arrive at the claimed estimates for ${\phi }_{l,k,\epsilon }\left(x\right)$. □

Analogous estimates can be obtained for ${\tilde{\eta }}_{l,k,\epsilon }\left(x\right)$ defined by (35).

Lemma 3.

For $(l,k,\epsilon )\in V$, we have ${\eta }_{l,k,\epsilon }\in C^{n-2}[0,1]$ and

$$|{\tilde{\eta }}_{l,k,\epsilon }^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}^{-1}\left(x\right),|{\tilde{\eta }}_{l,k,0}^{\left(\nu \right)}\left(x\right)-{\tilde{\eta }}_{l,k,1}^{\left(\nu \right)}\left(x\right)|\le C{l}^{\nu }{w}_{l,k}^{-1}\left(x\right){\xi }_l,\nu =\overline{0,n-2}.$$

Proof. 

The assertion of the lemma immediately follows from the definition (35), the estimates (47) for the weight numbers, and Corollary 2. □

Proceed to the investigation of the convergence for the series $t_{r,s}\left(x\right)$ and $T_{j_1,j_2}\left(x\right)$ in the reconstruction Formula (40). We relied on Proposition 3, which (due to our notations) readily follows from Lemma 8 in [35] and its proof.

Proposition 3

([35]). The following statements hold:

1. 

If $j_1+j_2=n-2$, then there exist regularisation constants $a_{j_1,j_2,l,k}$ such that the series:

$${\mathcal{T}}_{j_1,j_2}^{reg}\left(x\right):=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}\left({\tilde{\phi }}_{l,k,0}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left[j_2\right]}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left[j_2\right]}\left(x\right)-a_{j_1,j_2,l,k}\right)$$

converges in $L_2[0,1]$, $\parallel {\mathcal{T}}_{j_1,j_2}^{reg}{\parallel }_{L_2[0,1]}\le C\mathsf{\Omega }$, and $a_{j_1,j_2,l,k}+a_{j_1+1,j_2-1,l,k}=0$, so the series:

$$\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}\left({\tilde{\phi }}_{l,k,0}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left[j_2\right]}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left[j_2\right]}\left(x\right)+{\tilde{\phi }}_{l,k,0}^{[j_1+1]}\left(x\right){\tilde{\eta }}_{l,k,0}^{[j_2-1]}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{[j_1+1]}\left(x\right){\tilde{\eta }}_{l,k,1}^{[j_2-1]}\left(x\right)\right)$$

converges in $L_2[0,1]$ without regularisation.

2. 

If $j_1+j_2<n-2$, then the series:

$${\mathcal{T}}_{j_1,j_2}\left(x\right):=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}\left({\tilde{\phi }}_{l,k,0}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left[j_2\right]}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left[j_1\right]}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left[j_2\right]}\left(x\right)\right)$$

converges absolutely and uniformly on $[0,1]$. Moreover, ${max}_{x\in [0,1]}\left|{\mathcal{T}}_{j_1,j_2}\left(x\right)\right|\le C\mathsf{\Omega }$.

The constants $a_{j_1,j_2,l,k}$ are explicitly found in the proof of Lemma 8 in [35]. However, we do not provide them here in order not to introduce many additional notations. Moreover, explicit formulas for $a_{j_1,j_2,l,k}$ are not needed in the proofs.

Below, similar to the series in Proposition 3, we considered the series $T_{j_1,j_2}\left(x\right)$ with the brackets:

$$T_{j_1,j_2}\left(x\right)=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}\left({\phi }_{l,k,0}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)-{\phi }_{l,k,1}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left(j_2\right)}\left(x\right)\right).$$

Moreover, we agree that we understand the summation of several series $T_{j_1,j_2}\left(x\right)$ (in particular, in (36) and in (40)) in the sense that:

$$\sum _{(l,k,\epsilon )\in V}{b}_{l,k,\epsilon }+\sum _{(l,k,\epsilon )\in V}{c}_{l,k,\epsilon }=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}(b_{l,k,0}+b_{l,k,1}+c_{l,k,0}+c_{l,k,1}).$$

(49)

Using Lemmas 2, 3, and Proposition 3, we obtain the lemma on the convergence of $T_{j_1,j_2}\left(x\right)$.

Lemma 4.

The following statements hold:

1. 

If $j_1+j_2=n-2$, then the series $T_{j_1,j_2}\left(x\right)$ converges in $L_2[0,1]$ with the regularisation constants $a_{j_1,j_2,l,k}$ from Proposition 3, that is the series:

$$T_{j_1,j_2}^{reg}\left(x\right):=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}\left({\phi }_{l,k,0}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)-{\phi }_{l,k,1}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left(j_2\right)}\left(x\right)-a_{j_1,j_2,l,k}\right)$$

converges in $L_2[0,1]$. Moreover,

$$\parallel T_{j_1,j_2}^{reg}{\left(x\right)\parallel }_{L_2[0,1]}\le C\mathsf{\Omega }.$$

(50)

2. 

If $j_1+j_2=n-2-s$ and $s\in \{1,2,\cdots ,n-2\}$, then $T_{j_1,j_2}\left(x\right)$ converges in $W_2^s[0,1]$ and $\parallel T_{j_1,j_2}{\left(x\right)\parallel }_{W_2^s[0,1]}\le C\mathsf{\Omega }$.

Proof. 

Observe that, for $j_1+j_2\le n-2$, we have ${\tilde{\phi }}_{l,k,\epsilon }^{\left[j_1\right]}\left(x\right)={\tilde{\phi }}_{l,k,\epsilon }^{\left(j_1\right)}\left(x\right)$ and ${\tilde{\eta }}_{l,k,\epsilon }^{\left[j_2\right]}\left(x\right)$ satisfies the relation analogous to (43):

$${\tilde{\eta }}_{l,k,\epsilon }^{\left[j_2\right]}\left(x\right)={\tilde{\eta }}_{l,k,\epsilon }^{\left(j_2\right)}\left(x\right)+\sum _{j=0}^{\nu -2}\left(\sum _{s=j}^{\nu -2}{(-1)}^{s+\nu }{C}_s^j{\tilde{p}}_{n-\nu +s}^{(s-j)}\left(x\right)\right){\tilde{\eta }}_{l,k,\epsilon }^{\left(j\right)}\left(x\right).$$

Therefore, using the induction by $j_1+j_2=0,1,\cdots ,n-2$, we obtain from Proposition 3 that, for $j_1+j_2<n-2$, the series:

$${\tilde{T}}_{j_1,j_2}\left(x\right):=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}({\tilde{\phi }}_{l,k,0}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left(j_1\right)}{\tilde{\eta }}_{l,k,1}^{\left(j_2\right)}\left(x\right))$$

converges absolutely and uniformly on $[0,1]$, and for $j_1+j_2=n-2$, it converges in $L_2[0,1]$ with regularisation; in other words, the series

$${\tilde{T}}_{j_1,j_2}^{reg}\left(x\right):=\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}({\tilde{\phi }}_{l,k,0}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left(j_1\right)}\left(x\right){\tilde{\eta }}_{l,k,1}^{\left(j_2\right)}\left(x\right)-a_{j_1,j_2,l,k})$$

converges in $L_2[0,1]$. The regularisation constants $a_{j_1,j_2,l,k}$ are the same as in Proposition 3, because the difference ${\mathcal{T}}_{j_1,j_2}^{reg}\left(x\right)-{\tilde{T}}_{j_1,j_2}^{reg}\left(x\right)$ for $j_1+j_2=n-2$ can be represented as a linear combination of several series $T_{i_1,i_2}\left(x\right)$ with lower powers ($i_1+i_2<n-2$), which converge absolutely and uniformly on $[0,1]$. The addition and subtraction of series throughout this proof are understood in the sense of (49). Moreover, the corresponding estimates hold:

$$\begin{array}{cc}\hfill \underset{x\in [0,1]}{max}\left|{\tilde{T}}_{j_1,j_2}\left(x\right)\right|\le C\mathsf{\Omega },& j_1+j_2<n-2,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill \parallel {\tilde{T}}_{j_1,j_2}^{reg}{\left(x\right)\parallel }_{L_2[0,1]}\le C\mathsf{\Omega },& j_1+j_2=n-2.\hfill \end{array}$$

(52)

Consider the series:

$$\begin{array}{cc}\hfill T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right)=& \sum _{(l,k,\epsilon )\in V}({\phi }_{l,k,\epsilon }^{\left(j_1\right)}\left(x\right)-{\tilde{\phi }}_{l,k,\epsilon }^{\left(j_1\right)}\left(x\right)){\tilde{\eta }}_{l,k,\epsilon }^{\left(j_2\right)}\left(x\right)\hfill \\ \hfill =& \sum _{l=1}^{\infty }\sum _{k=1}^{n-1}({\phi }_{l,k,0}^{\left(j_1\right)}\left(x\right)-{\phi }_{l,k,1}^{\left(j_1\right)}\left(x\right)-{\tilde{\phi }}_{l,k,0}^{\left(j_1\right)}\left(x\right)+{\tilde{\phi }}_{l,k,1}^{\left(j_1\right)}\left(x\right)){\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)\hfill \\ \hfill & +\sum _{l=1}^{\infty }\sum _{k=1}^{n-1}({\phi }_{l,k,1}^{\left(j_1\right)}\left(x\right)-{\tilde{\phi }}_{l,k,1}^{\left(j_1\right)}\left(x\right))({\tilde{\eta }}_{l,k,0}^{\left(j_2\right)}\left(x\right)-{\tilde{\eta }}_{l,k,1}^{\left(j_2\right)}\left(x\right)).\hfill \end{array}$$

Let us apply the estimates of Lemmas 2 and 3 for ${\phi }_{l,k,\epsilon }^{\left(j_1\right)}\left(x\right)$ and ${\tilde{\eta }}_{l,k,\epsilon }^{\left(j_2\right)}\left(x\right)$, respectively. We have the two cases:

$$\begin{array}{cc}\hfill j_1=0:& |T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right)|\le C\mathsf{\Omega }\sum _{l=1}^{\infty }{l}^{j_2}{\xi }_l{\chi }_l,\hfill \\ \hfill j_1>0:& |T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right)|\le C\mathsf{\Omega }\sum _{l=1}^{\infty }{l}^{j_1+j_2-1}{\xi }_l.\hfill \end{array}$$

Since $\left\{l^{n-2}{\xi }_l\right\}\in l_2$ and $\left\{{\chi }_l\right\}\in l_2$, we conclude that the series $T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right)$ in both cases converges absolutely and uniformly on $[0,1]$ for $j_1+j_2\le n-2$. Moreover, for the difference $T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right)$, the estimates similar to (51) and (52) hold. For $j_1+j_2=n-2$, this readily implies the convergences of the series:

$$T_{j_1,j_2}^{reg}\left(x\right)={\tilde{T}}_{j_1,j_2}^{reg}\left(x\right)+(T_{j_1,j_2}\left(x\right)-{\tilde{T}}_{j_1,j_2}\left(x\right))$$

in $L_2[0,1]$ and the estimate (50).

Now, let $j_1+j_2=n-3$. Formal differentiation together with the summation rule (49) imply

$$T_{j_1,j_2}^{\prime }\left(x\right)=T_{j_1+1,j_2}\left(x\right)+T_{j_1,j_2+1}\left(x\right).$$

As we have already shown, the series $T_{j_1+1,j_2}\left(x\right)$ and $T_{j_1,j_2+1}\left(x\right)$ converge in $L_2[0,1]$ with regularisation and satisfy the estimate (50). Moreover, for their sum, the regularisation constants vanish: $a_{j_1+1,j_2,l,k}+a_{j_1,j_2+1,l,k}=0$, so the series $T_{j_1,j_2}^{\prime }\left(x\right)$ converges in $L_2[0,1]$ without regularisation. Consequently, $T_{j_1,j_2}\in W_2^1[0,1]$ and

$$\begin{array}{cc}\hfill \parallel T_{j_1,j_2}{\left(x\right)\parallel }_{W_2^1[0,1]}& \le \parallel T_{j_1,j_2}{\left(x\right)\parallel }_{L_2[0,1]}+{\parallel T_{j_1,j_2}^{\prime }\left(x\right)\parallel }_{L_2[0,1]}\hfill \\ & \le \underset{x\in [0,1]}{max}|T_{j_1,j_2}\left(x\right)|+\parallel T_{j_1+1,j_2}^{reg}{\left(x\right)\parallel }_{L_2[0,1]}+{\parallel T_{j_1,j_2+1}^{reg}\left(x\right)\parallel }_{L_2[0,1]}\le C\mathsf{\Omega }.\hfill \end{array}$$

Thus, the lemma was already proven for $j_1+j_2=n-2$ and $j_1+j_2=n-3$. By induction, we complete the proof for $j_1+j_2=n-4,\cdots ,1,0$. □

Thus, we are ready to investigate the convergence of the series in the reconstruction Formula (40).

Lemma 5.

The reconstruction Formula (40) define the functions $p_s$ of the corresponding spaces $W_2^{s-1}[0,1]$ for $s=\overline{0,n-2}$ and $\parallel p_s\left(x\right)-{\tilde{p}}_s{\left(x\right)\parallel }_{W_2^{s-1}[0,1]}\le C\mathsf{\Omega }$.

Proof. 

We prove the lemma by induction. Fix $s\in \{0,1,\cdots ,n-2\}$. Suppose that the assertion of the lemma has already been proven for all $s_1>s$. Due to (36), we have

$$\begin{array}{cc}\hfill {\mathcal{S}}_1\left(x\right)& :=t_{n,s}\left(x\right)+{(-1)}^{n-s}{T}_{0,n-s-1}\left(x\right)=\sum _{j=0}^{n-s-1}{b}_j{T}_{n-s-1-j,j}\left(x\right)\hfill \\ & =\frac{d}{dx}\left(\sum _{j=0}^{n-s-2}{d}_j{T}_{n-s-2-j,j}\left(x\right)\right),\hfill \end{array}$$

where

$$\begin{array}{c}{b}_j:=C_n^{j+s+1}{C}_{j+s}^s+{(-1)}^{n-s}{\delta }_{j,n-s-1},\sum _{j=0}^{n-s-1}{b}_j=0,\\ d_j:=\sum _{i=0}^j{(-1)}^{j-i}{b}_i,j=\overline{0,n-s-2}.\end{array}$$

By virtue of Lemma 4, the series $T_{n-s-2-j,j}$ for $j=\overline{0,n-s-2}$ belong to $W_2^s[0,1]$ for $s\ge 1$ and converge with regularisation in $L_2[0,1]$ for $s=0$. In both cases, we conclude that ${\mathcal{S}}_1\in W_2^{s-1}[0,1]$. Furthermore, $\parallel {\mathcal{S}}_1{\left(x\right)\parallel }_{W_2^{s-1}[0,1]}\le C\mathsf{\Omega }$.

Consider the next term in (40):

$${\mathcal{S}}_2\left(x\right):=\sum _{j=0}^{n-s-3}\sum _{r=j}^{n-s-3}{(-1)}^r{C}_r^j{\tilde{p}}_{r+s+1}^{(r-j)}\left(x\right)T_{0,j}\left(x\right).$$

In view of $p_k\in W_2^{k-1}[0,1]$ and Lemma 4, we have

$$\begin{array}{cc}\hfill & {\tilde{p}}_{r+s+1}^{(r-j)}\in W_2^{s+j}[0,1]\subseteq W_2^s[0,1],\hfill \\ \hfill & T_{0,j}\in W_2^{n-j-2}[0,1]\subseteq W_2^{s+1}[0,1].\hfill \end{array}$$

Hence, ${\mathcal{S}}_2\in W_2^s[0,1]$.

For the last term:

$${\mathcal{S}}_3\left(x\right):=-\sum _{r=s+1}^{n-2}{p}_r\left(x\right)t_{r,s}\left(x\right),$$

we have $p_r\in W_2^{r-1}[0,1]$ by the induction hypothesis, so $p_r\in W_2^s[0,1]$, and $t_{r,s}\in W_2^{s+1}[0,1]$ according to (36) and Lemma 4. Hence, ${\mathcal{S}}_3\in W_2^s[0,1]$. Lemma 4 also implies $\parallel {\mathcal{S}}_j{\left(x\right)\parallel }_{W_2^s[0,1]}\le C\mathsf{\Omega }$ for $j=2,3$.

Since $p_s\left(x\right)={\tilde{p}}_s\left(x\right)+{\mathcal{S}}_1\left(x\right)+{\mathcal{S}}_2\left(x\right)+{\mathcal{S}}_3\left(x\right)$, we arrive at the assertion of the lemma. □

Thus, we have obtained the vector $p={\left(p_k\right)}_{k=0}^{n-2}$ by the reconstruction formulas. It remains to prove the following lemma.

Lemma 6.

The spectral data of p coincide with ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$.

Proof. 

The proof is based on the approximation approach, which was considered in [36] for $n=3$ in detail. The proof for higher orders n is similar, so we omit the technical details and just outline the main idea.

Along with ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$, consider the “truncated” data

$${\lambda }_{l,k}^N:=\left\{\begin{array}{cc}{\lambda }_{l,k},\hfill & l\le N,\hfill \\ {\tilde{\lambda }}_{l,k},\hfill & l>N,\hfill \end{array}\right.{\beta }_{l,k}^N:=\left\{\begin{array}{cc}{\beta }_{l,k},\hfill & l\le N,\hfill \\ {\tilde{\beta }}_{l,k},\hfill & l>N.\hfill \end{array}\right.$$

Then, by using $\{{\lambda }_{l,k}^N,{\beta }_{l,k}^N\}$ instead of $\{{\lambda }_{l,k},{\beta }_{l,k}\}$, one can construct the main equation:

$$(I-{\tilde{R}}^N\left(x\right)){\psi }^N\left(x\right)={\tilde{\psi }}^N\left(x\right),x\in [0,1],$$

(53)

analogously to (31). It can be shown that Equation (53) is uniquely solvable for sufficiently large values of N. Therefore, one can find the functions ${\phi }_{l,k,\epsilon }^N\left(x\right)$, $(l,k,\epsilon )\in V$, by using the solution ${\psi }^N\left(x\right)$ similarly to (32). Then, construct the functions ${\left\{{\mathsf{\Phi }}_k^N(x,\lambda )\right\}}_{k=1}^n$ and $p^N={\left(p_k^N\right)}_{k=0}^{n-2}$ analogously to (33) and (40), respectively. The advantage of the “truncated” data $\{{\lambda }_{l,k}^N,{\beta }_{l,k}^N\}$ over $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ is that the series in (33) and (40) are finite, so one can show by direct calculations that ${\left\{{\mathsf{\Phi }}_k^N(x,\lambda )\right\}}_{k=1}^n$ are the Weyl solutions of Equation (1) with the coefficients $p^N$ and deduce that $\{{\lambda }_{l,k}^N,{\beta }_{l,k}^N\}$ are the spectral data of $p^N$. At this stage, the assumptions (S-1)–(S-3) of Theorem 2 are crucial. If these assumptions do not hold, then one needs additional data to recover $p^N$. Next, using (40), we show that

$$\underset{N\to \infty }{lim}{\parallel p_k^N-p_k\parallel }_{W_2^{k-1}[0,1]}=0,k=\overline{0,n-2}.$$

Furthermore, it can be shown that the spectral data depend continuously on the coefficients $p={\left(p_k\right)}_{k=0}^{n-2}$. This concludes the proof. □

Let us summarise the arguments of this section in the proof of Theorem 2.

Proof of Theorem 2.

Let $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ and $\tilde{p}$ satisfy the hypothesis of the theorem. Then, the main Equation (31) is uniquely solvable. By using its solution $\psi \left(x\right)$, we find the functions ${\left\{{\phi }_{l,k,\epsilon }\left(x\right)\right\}}_{(l,k,\epsilon )\in V}$ by (32) and reconstruct the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ by (40). By virtue of Lemma 5, $p_k\in W_2^{k-1}[0,1]$ for $k=\overline{0,n-2}$. Lemma 6 implies that $\{{\lambda }_{l,k},{\beta }_{l,k}\}$ are the spectral data of $p={\left(p_k\right)}_{k=0}^{n-2}$. Taking (S-1) and (S-2) into account, we conclude that $p\in W$. The uniqueness of p is given by Corollary 1. □

Proof of Theorem 1.

Let us show that, if data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n}}$ and $\tilde{p}$ satisfy the conditions of Theorem 1 for sufficiently small $\delta >0$, then they also satisfy the hypothesis of Theorem 2.

It follows from (4) that

$$|{\lambda }_{l,k}-{\tilde{\lambda }}_{l,k}|\le \delta ,|{\beta }_{l,k}-{\tilde{\beta }}_{l,k}|\le \delta ,l\ge 1,k=\overline{1,n-1}.$$

On the other hand, Definition 1 and the asymptotics (3) and (22) imply that the eigenvalues ${\left\{{\tilde{\lambda }}_{l,k}\right\}}_{l\ge 1}$ are separated for each fixed $k\in \{1,2,\cdots ,n-1\}$, as well as for neighbouring values of k and the weight numbers $\left\{{\tilde{\beta }}_{l,k}\right\}$ are separated from zero. Rigorously speaking, we have

$$\begin{array}{cc}\hfill & |{\tilde{\lambda }}_{l,k}-{\tilde{\lambda }}_{l_0,k}|\ge {\delta }_0,l_0\ne l,k=\overline{1,n-1},\hfill \\ \hfill & |{\tilde{\lambda }}_{l_0,k}-{\tilde{\lambda }}_{l,k+1}|\ge {\delta }_0,l,l_0\ge 1,k=\overline{1,n-2},\hfill \\ \hfill & |{\tilde{\beta }}_{l,k}|\ge {\delta }_0,l\ge 1,k=\overline{1,n-1}.\hfill \end{array}$$

By choosing $\delta <{\delta }_0/2$, we achieve the conditions (S-1)–(S-3) of Theorem 2 for ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$. Furthermore, (S-4) directly follows from (26) and (4).

Using (30) and (44), we estimate

$$\parallel \tilde{R}{\left(x\right)\parallel }_{m\to m}=\underset{v_0\in V}{sup}\sum _{v\in V}\left|{\tilde{R}}_{v_0,v}\left(x\right)\right|\le C\underset{l_0\ge 1}{sup}\sum _{l=1}^{\infty }\frac{{\xi }_l}{|l-l_0|+1}\le C\mathsf{\Omega },$$

where the constant C depends only on $\tilde{p}$ and $\delta$ if $\mathsf{\Omega }\le \delta$. Therefore, choosing a sufficiently small $\delta$ for the fixed $\tilde{p}$, we achieve $\parallel \tilde{R}\left(x\right)\parallel \le \frac{1}{2}$. Then, the operator $(I-\tilde{R}\left(x\right))$ has a bounded inverse, that is the condition (S-5) of Theorem 2 is fulfilled.

Thus, the numbers ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l\ge 1,k=\overline{1,n-1}}$ satisfy the conditions (S-1)–(S-5) of Theorem 2, which implies the existence of $p={\left(p_k\right)}_{k=0}^{n-2}\in W$ with the spectral data $\{{\lambda }_{l,k},{\beta }_{l,k}\}$. It is worth noting that, if $\mathsf{\Omega }\le \delta$, then the constant C in Lemmas 2–5 depends only on $\tilde{p}$ and $\delta$. Hence, the estimate (5) follows from Lemma 5. This concludes the proof. □

7. Conclusions

In this paper, we proved the local solvability and stability of recovering the coefficients ${\left(p_k\right)}_{k=0}^{n-2}$ of Equation (1) from the spectral data, which consist of the eigenvalues ${\left\{{\lambda }_{l,k}\right\}}_{l\ge 1}$ and the weight numbers ${\left\{{\beta }_{l,k}\right\}}_{l\ge 1}$ of the boundary value problems ${\mathcal{L}}_k$, $k=\overline{1,n-1}$. Moreover, we obtained some sufficient conditions for the global solution of the inverse problem (Theorem 2). The proof method is constructive. It is based on the reduction of the inverse problem to the linear main Equation (31) in the Banach space m and on the reconstruction formulas (40) for the coefficients $p_k$, $k=\overline{0,n-2}$. We proved the convergence of the series from the reconstruction formulas in the corresponding spaces $W_2^{k-1}[0,1]$, $k=\overline{0,n-2}$. Theorem 1 on local solvability and stability generalises the analogous results of [12] for $n=2$ and of [36] for $n=3$.

Our results have the following advantages over the previous studies:

• We, for the first time, proved the local solvability and stability of an inverse spectral problem for differential operators with distribution coefficients of orders $n\ge 4$.
• Our method is constructive. Based on Procedure 1, one can develop a numerical method for solving the inverse problem.
• Our method does not require self-adjointness. All the results of this paper were obtained for the general non-self-adjoint case.

The results and the methods of this paper can be used for the future development of spectral theory for higher-order differential operators with regular and distribution coefficients. Let us mention some open problems in this direction:

• In the method of spectral mappings, the unique solvability of the main equation plays an important role. In this paper, we used the smallness of the spectral data perturbation (i.e., the smallness of $\delta$ in Theorem 1) to guarantee the existence of the main equation solution. It is worth considering other cases, when the unique solvability of the main equation can be proven. In particular, for $n=2$ and $n=3$, the main equation is uniquely solvable in the self-adjoint case. For differential operators on a finite interval for $n\ge 4$, this issue is open.
• Using the technique of [50], the following precise asymptotics can be obtained for the spectral data:

  $$\begin{array}{cc}\hfill {\lambda }_{l,k}& =l^n\left(c_{0,k}+c_{1,k}{l}^{-1}+c_{2,k}{l}^{-2}+\cdots +c_{n-1,k}{l}^{-(n-1)}+l^{-(n-1)}{\varkappa }_{l,k}\right),\hfill \\ \hfill {\beta }_{l,k}& =l^n\left(d_{0,k}+d_{1,k}{l}^{-1}+d_{2,k}{l}^{-2}+\cdots +d_{n-2,k}{l}^{-(n-2)}+l^{-(n-2)}{\varkappa }_{l,k}^0\right),\hfill \end{array}$$

  where $c_{j,k}$, $d_{j,k}$ are constants and $\left\{{\varkappa }_{l,k}\right\}$, $\left\{{\varkappa }_{l,k}^0\right\}$ are $l_2$-sequences. Hence, if $c_{j,k}={\tilde{c}}_{j,k}$ and $d_{j,k}={\tilde{d}}_{j,k}$, then $\left\{l^{n-2}{\xi }_l\right\}\in l_2$, that is the assumption (S-4) of Theorem 2 holds. Thus, in order to achieve (S-4), one has to construct a model problem with the known coefficients $c_{j,k}$ and $d_{j,k}$ in the spectral data asymptotics. The construction of such a problem will help to finalise the global solvability results for higher-order inverse problems.
• Remove the conditions (W-2) and (W-3) of Definition 1 (i.e. the simplicity of the spectra and the separation condition). Then, other spectral data are required, and the problem becomes more technically complicated. In addition, one has to take the splitting of multiple eigenvalues under small perturbations of the spectra into account (see [38] for $n=2$).
• Obtain solvability conditions for differential expressions with coefficients of higher singularity orders than $p_k\in W_2^{k-1}[0,1]$. Although the uniqueness theorems in [32,33,34] and the reconstruction approach [35] are obtained for coefficients of wider distribution spaces, it is a challenge to prove the existence of the inverse problem solution.
• Investigate the uniform stability of inverse problems for higher-order differential operators (see [15,16] for $n=2$).
• Study the reconstruction of higher-order differential operators from the finite spectral data ${\{{\lambda }_{l,k},{\beta }_{l,k}\}}_{l=1}^N$. For $n=2$, see, e.g., [18,42].