© Wuhan University 2024

0 Introduction

This paper focuses on the inverse spectral theory of Atkinson-type Sturm-Liouville problems (SLPs) with a class of non-self-adjoint boundary conditions containing the spectral parameter (also called eigenparameter-dependent boundary conditions). Since such problems have a finite number of eigenvalues, we will take advantage of the associated finite dimensional inverse eigenvalue problems of the form

$\mathit{V}\mathit{Z}=\lambda \mathit{W}\mathit{Z},$(1)

where $\mathit{V}$ and $\mathit{W}$ are square matrices over the complex $\mathbf{ℂ}$ and $\mathit{V}$ is an "almost" Jacobi matrix (in fact, a kind of generalized pseudo-Jacobi matrix), and $\mathit{W}$ is 'almost' diagonal.

Consider the eigenvalue problem consisting of the Sturm-Liouville equation

$-(pf\text{'})\text{'}+qf=\nu wf,\mathrm{o}\mathrm{n}{J}=[\alpha ,\beta ],-\mathrm{\infty }<\alpha <\beta <+\mathrm{\infty },$(2)

subject to the boundary conditions (BCs)

${\mathit{A}}_{\nu }\mathit{F}(\alpha )+{\mathit{B}}_{\nu }\mathit{F}(\beta )=\mathrm{0},\mathit{F}={[f,pf\text{'}]}^{\mathrm{T}},$(3)

where ${\mathit{A}}_{\nu }=\left(\genfrac{}{}{0pt}{}{\nu {\xi }_{\mathrm{1}}\text{'}+{\xi }_{\mathrm{1}}}{\mathrm{0}}\genfrac{}{}{0pt}{}{\nu {\xi }_{\mathrm{2}}\text{'}+{\xi }_{\mathrm{2}}}{\mathrm{0}}\right)$,${\mathit{B}}_{\nu }=\left(\genfrac{}{}{0pt}{}{\mathrm{0}}{\nu {\varsigma }_{\mathrm{1}}\text{'}+{\varsigma }_{\mathrm{1}}}\genfrac{}{}{0pt}{}{\mathrm{0}}{\nu {\varsigma }_{\mathrm{2}}\text{'}+{\varsigma }_{\mathrm{2}}}\right)$,with ${\xi }_j$,${\xi }_j\text{'}$,${\varsigma }_j$,${\varsigma }_j\text{'}\in \mathbb{R},j=\mathrm{1,2},$satisfying ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}>\mathrm{0}$or ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}>\mathrm{0},{\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}<\mathrm{0}.$$\nu$ is the spectral parameter.

In the case of both${\gamma }_{\mathrm{1}}>\mathrm{0}$and ${\gamma }_{\mathrm{2}}>\mathrm{0}$, the corresponding inverse spectral problems have been investigated in Ref. [1], in which the problems are self-adjoint and hence with real eigenvalues. However, it is well known that the conditions as mentioned above mean that the boundary conditions (3) are non-self-adjoint; thus, the eigenvalues of this problem may not be real, and the distribution of eigenvalues is more complex than the self-adjoint cases. The inverse Atkinson-type SLPs with coupled self-adjoint boundary conditions containing the spectral parameter can also be found in Ref. [2].

Similar with the definition in Ref. [1], the Sturm-Liouville equation (2) is said to be of Atkinson-type if there exists a partition of the domain interval $[\alpha ,\beta ]$,

$\alpha ={\alpha }_{\mathrm{0}}<{\beta }_{\mathrm{0}}<{\alpha }_{\mathrm{1}}<{\beta }_{\mathrm{1}}<\cdots <{\alpha }_n<{\beta }_n=\beta ,$(4)

for some positive integer$n>\mathrm{1},$ such that

$r=\frac{\mathrm{1}}{p}=\mathrm{0},\mathrm{o}\mathrm{n}[{\alpha }_j,{\beta }_j],j=\mathrm{0,1},\cdots ,n,{\int }_{{\beta }_j}^{{\alpha }_j}rdx>\mathrm{0},j=\mathrm{1,2},\cdots ,n;$(5)

$q=\mathrm{0},\mathrm{o}\mathrm{n}[{\beta }_{j-\mathrm{1}},{\alpha }_j],j=\mathrm{1,2},\cdots ,n;$(6)

and

$w=\mathrm{0},\mathrm{o}\mathrm{n}[{\beta }_{j-\mathrm{1}},{\alpha }_j],j=\mathrm{1,2},\cdots ,n,{\int }_{{\alpha }_j}^{{\beta }_j}w\mathrm{d}x>\mathrm{0},j=\mathrm{0,1},\cdots ,n.$(7)

SLPs (2) and (3) are said to be of Atkinson-type if (2) is of Atkinson-type, and BCs (3) are self-adjoint or non-self-adjoint.

The problems of Atkinson-type have some significant backgrounds, for instance, frequencies of vibrating strings and diffusion operators^[3]. Following the statement in Ref. [4], in the present work, we still call such problems the Atkinson-type because such problems are initiated by Atkinson and Kong et al^[5,6], though the BCs are non-self-adjoint. The problems that can be transferred into the problems of Atkinson-type also can be found in recent studies in Ref. [7] for discrete Sturm-Liouville problems.

It is well known that matrix eigenvalue problems, such as the eigenvalue problems of Jacobi matrices and pseudo-Jacobi matrices, have wide applications^[8]. On the other hand, the spectral theory of Sturm-Liouville problems also arises from many practical problems and has been widely studied in various aspects^[9,10]. It is well known that the inverse matrix eigenvalue problems and the inverse Sturm-Liouville problems are important research topics and have been intensely investigated in mathematics, physics, and some other fields in engineering, such as the problems associated with vibrating systems^[11], classical moment problems^[12] and quantum mechanics^[13]. There have been numerous studies on inverse matrix eigenvalue problems in the last decades^[8,14-18]. The inverse Sturm-Liouville problems are initiated by the well-known Ambarzumian's^[19], Borg's^[20], and Levinson's^[21] works, and the extending results can be found in some literatures such as Refs. [9,22-25].

Sturm-Liouville problems with boundary conditions containing the spectral parameter have appeared in some physical problems and engineering problems, such as string vibration and heat transfer problems^[26,27]. Hence, it is an important research topic in mathematical physics^[26-31].

Based on the above, in the present paper, a class of inverse Atkinson-type Sturm-Liouville problems with non-self-adjoint boundary conditions containing the spectral parameter (2),(3) are investigated by using the corresponding inverse matrix eigenvalue problem of (1), which can be seen as a generalization of Ref. [32] to such problems.

The inverse Atkinson-type Sturm-Liouville problems were initiated by Kong and Volkmer et al^[32,33]. In 2012, Kong and Zettl^[32] gave an investigation on the inverse Atkinson-type Sturm-Liouville problems by taking advantage of the so-called matrix representations of these problems presented in Ref. [4] and a generalized method of the inverse matrix eigenvalue problems from Xu^[8]. In Ref. [32], the authors also compared the classical inverse Sturm-Liouville problems with the inverse Atkinson-type Sturm-Liouville problems and showed their differences. In recent years, several works on such problems have been made by some researchers^[1,2,34,35]. However, as far as we know, there is no such result for non-self-adjoint cases.

Some basic notations are needed to be introduced as follows. Let the equation be given as in (2) and the coefficients satisfy the conditions (5)-(7), and let $\sigma ({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}\text{'},{\xi }_{\mathrm{2}}\text{'},{\varsigma }_{\mathrm{1}},{\varsigma }_{\mathrm{2}},{\varsigma }_{\mathrm{1}}\text{'},{\varsigma }_{\mathrm{2}}\text{'})$denote the spectrum of the problem (2), (3) to highlight the dependence of the spectrum on the parameters. Assume that $k\in {\mathbb{N}}^{+}$and $k>\mathrm{4},$and we use ${{M}}_k$to represent the set of complex $k\times k$matrices. For any$\mathit{C}\in {{M}}_k$, $\sigma (\mathit{C})$denotes all the eigenvalues of$\mathit{C}$. Furthermore, let${\mathit{C}}_{\mathrm{1}}$and${\mathit{C}}^{\mathrm{1}}$be the principal submatrices, which are obtained by removing the first row and column and the last row and column from$\mathit{C}$, respectively. For a given$e\in \mathit{C}$, let${\mathit{C}}_{\mathrm{1}}[e]$ denote the matrix, where we add $e$to the $(\mathrm{1,1})$ entry of${\mathit{C}}_{\mathrm{1}}$ and let${\mathit{C}}^{\mathrm{1}}[e]$ denote the matrix, where we add $e$ to the $(k-\mathrm{1},k-\mathrm{1})$entry of${\mathit{C}}^{\mathrm{1}}$. For any$\mathit{C},\mathit{D}\in {{M}}_k$, if there exists a nontrivial vector$\mathit{u}\in {\mathbf{ℂ}}^k$ such that$(\mathit{C}-{\nu }^{*}\mathit{D})\mathit{u}=\mathrm{0}$, then we say that ${\nu }^{*}$is an eigenvalue of the matrix-pair$(\mathit{C},\mathit{D})$. Similarly, let$\sigma (\mathit{C},\mathit{D})$ denote all the eigenvalues of$(\mathit{C},\mathit{D})$. Then it is easy to see,${\nu }^{*}\in \sigma (\mathit{C})$if and only if${\nu }^{*}\in \sigma (\mathit{C},{\mathit{I}}_k)$, where${\mathit{I}}_k$ is the identity matrix in${{M}}_k$.

The arrangement of this paper is as follows. Section 1 contains the equivalences between the matrix problem and the problem studied here. As a main bridge, the inverse matrix eigenvalue problems related to the considered problems are stated in Section 2. The main result and its proof are given in Section 3. A numerical algorithm and related examples are posted in Section 4.

1 Equivalence Matrix Representations of the Problems

To present our result, let us introduce the following lemma firstly, which implies the equivalence between the Atkinson-type SLPs and the SLPs with piecewise constant coefficients.

Lemma 1^[4] Assume $r,q,w\in L({J},\mathbf{ℂ})$ and satisfy the conditions (5)-(7), where $L({J},\mathbf{ℂ})$ denotes the complex-valued functions which are lebesque inteqrable on $J$. Let

$\{\begin{array}{c}{p}_j={({\int }_{{\beta }_{j-\mathrm{1}}}^{{\alpha }_j}r\mathrm{d}x)}^{-\mathrm{1}},j=\mathrm{1,2},\cdots ,n,\\ q_j={\int }_{{\alpha }_j}^{{\beta }_j}q\mathrm{d}x,w_j={\int }_{{\alpha }_j}^{{\beta }_j}w\mathrm{d}x,j=\mathrm{0,1},\cdots ,n.\end{array}$(8)

If we define the piecewise constant functions $\tilde{p}(x)=\frac{\mathrm{1}}{\tilde{r}(x)},\tilde{q}(x)$ and $\tilde{w}(x)$ on $[\alpha ,\beta ]$ by

$\tilde{p}(x)=\frac{\mathrm{1}}{\tilde{r}(x)}=\{\begin{array}{c}{p}_j({\alpha }_j-{\beta }_{j-\mathrm{1}}),x\in ({\beta }_{j-\mathrm{1}},{\alpha }_j),j=\mathrm{1,2},\cdots ,n,\\ \mathrm{\infty },x\in [{\alpha }_j,{\beta }_j],j=\mathrm{0,1},\cdots ,n;\end{array}$

$\tilde{q}(x)=\{\begin{array}{c}\frac{q_j}{({\beta }_j-{\alpha }_j)},x\in [{\alpha }_j,{\beta }_j],j=\mathrm{0,1},\cdots ,n,\\ \mathrm{0},x\in ({\beta }_{j-\mathrm{1}},{\alpha }_j),j=\mathrm{1,2},\cdots ,n;\end{array}$

$\tilde{w}(x)=\{\begin{array}{c}\frac{w_j}{({\beta }_j-{\alpha }_j)},x\in [{\alpha }_j,{\beta }_j],j=\mathrm{0,1},\cdots ,n,\\ \mathrm{0},x\in ({\beta }_{j-\mathrm{1}},{\alpha }_j),j=\mathrm{1,2},\cdots ,n.\end{array}$

Note that $\tilde{p}(x)=\mathrm{\infty }$ on $[{\alpha }_j,{\beta }_j]$ means that $\tilde{r}(x)=\mathrm{0}$ on $[{\alpha }_j,{\beta }_j],j=\mathrm{0,1},\cdots ,n$.

Then, the eigenvalues of problems (2), (3) and the problem consisting of the equation

$-(\tilde{p}f\text{'})\text{'}+\tilde{q}f=\nu \tilde{w}f,\mathrm{o}\mathrm{n}{J}=[\alpha ,\beta ],$(9)

and the same BCs (3) are exactly the same eigenvalues.

Lemma 1   implies that for a fixed BCs (3) and a given partition of the interval $[\alpha ,\beta ]$, there exists a family of Atkinson-type SLPs such as they have exactly the same eigenvalues as problem (9), (3). Such a family is called the equivalent family of problem (9), (3). Indeed, every problem in this equivalent family must have the same $p_j$, $q_j$ and $w_j$ defined by (8).

We will use the following lemmas and Theorem 2 to prove our main theorem (Theorem 3), which indicates the equivalences between Atkinson-type SLPs with non-self-adjoint boundary conditions containing the spectral parameter and the matrix eigenvalue problems given in Ref. [36].

Lemma 2^[36] Assume that in BCs (3), the parameters satisfy the conditions ${\xi }_j$, ${\xi }_j\text{'}$, ${\varsigma }_j$, ${\varsigma }_j\text{'}\in \mathbb{R},j=\mathrm{1,2},$and ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}>\mathrm{0}$or ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}>\mathrm{0},{\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}<\mathrm{0}$. Let us define an$(n+\mathrm{3})\times (n+\mathrm{3})$ generalized pseudo-Jacobi matrix

$\mathit{P}=\left[\begin{array}{ccc}\begin{array}{ccc}{\xi }_{\mathrm{2}}& {\xi }_{\mathrm{1}}& \\ \mathrm{1}& p_{\mathrm{1}}& -p_{\mathrm{1}}\\ & -p_{\mathrm{1}}& p_{\mathrm{1}}+p_{\mathrm{2}}\end{array}& \begin{array}{ccc}& & \\ & & \\ -p_{\mathrm{2}}& & \end{array}& \\ & \begin{array}{ccc}\cdots & \cdots & \cdots \\ & -p_{n-\mathrm{1}}& p_{n-\mathrm{1}}+p_n\\ & & -p_n\end{array}& \begin{array}{cc}& \\ -p_n& \\ p_n& -\mathrm{1}\end{array}\\ & & \begin{array}{cc}{\varsigma }_{\mathrm{1}}& {\varsigma }_{\mathrm{2}}\end{array}\end{array}\right]$(10)

and a diagonal matrix

$\mathit{Q}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{0},q_{\mathrm{0}},q_{\mathrm{1}},\cdots ,q_n,\mathrm{0})$(11)

as well as an "almost" diagonal matrix

$\mathit{W}=\left[\begin{array}{ccc}\begin{array}{cc}-{\xi }_{\mathrm{2}}\text{'}& -{\xi }_{\mathrm{1}}\text{'}\\ & w_{\mathrm{0}}\end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{l}\\ \end{array}\\ \begin{array}{cc}& \\ & \\ & \end{array}& \begin{array}{cc}\cdots & \\ & w_n\\ & -{\varsigma }_{\mathrm{1}}\text{'}\end{array}& \begin{array}{l}\\ \\ -{\varsigma }_{\mathrm{2}}\text{'}\end{array}\end{array}\right]$(12)

Here the "almost" diagonal matrix means that except for the $(\mathrm{1,2})$ and$(n+\mathrm{3},n+\mathrm{2})$ entries, it is diagonal.

Then

$\sigma ({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}\text{'},{\xi }_{\mathrm{2}}\text{'},{\varsigma }_{\mathrm{1}},{\varsigma }_{\mathrm{2}},{\varsigma }_{\mathrm{1}}\text{'},{\varsigma }_{\mathrm{2}}\text{'})=\sigma (\mathit{P}+\mathit{Q},\mathit{W})$

i.e. the spectrum of the problem (9), (3), and the spectrum of the matrix-pair $(\mathit{P}+\mathit{Q},\mathit{W})$ are the same.

Lemma 3   Assume that in BCs (3), the parameters satisfy the conditions ${\xi }_{\mathrm{1}}={\xi }_{\mathrm{2}}=\mathrm{0},{\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\varsigma }_j,{\varsigma }_j\in \mathbb{R},j=\mathrm{1,2}$, and ${\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}>\mathrm{0}$. Let us define an$(n+\mathrm{2})\times (n+\mathrm{2})$generalized pseudo-Jacobi matrix

${\mathit{P}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]=\left[\begin{array}{ccc}\begin{array}{ccc}{p}_{\mathrm{1}}-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}& -p_{\mathrm{1}}& \\ -p_{\mathrm{1}}& p_{\mathrm{1}}+p_{\mathrm{2}}& -p_{\mathrm{2}}\\ & \cdots & \cdots \end{array}& & \\ & \begin{array}{ccc}\cdots & \cdots & \cdots \\ & -p_{n-\mathrm{1}}& p_{n-\mathrm{1}}+p_n\\ & & -p_n\end{array}& \begin{array}{cc}& \\ -p_n& \\ p_n& -\mathrm{1}\end{array}\\ & & \begin{array}{cc}{\varsigma }_{\mathrm{1}}& {\varsigma }_{\mathrm{2}}\end{array}\end{array}\right]$(13)

and a diagonal matrix

${\mathit{Q}}_{\mathrm{1}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(q_{\mathrm{0}},q_{\mathrm{1}},\cdots ,q_n,\mathrm{0}),$(14)

as well as an "almost" diagonal matrix

${\mathit{W}}_{\mathrm{1}}=\left[\begin{array}{ccc}\begin{array}{cc}{w}_{\mathrm{0}}& \\ & w_{\mathrm{1}}\end{array}& & \\ & \begin{array}{cc}\cdots & \\ & w_n\\ & -{\varsigma }_{\mathrm{1}}\text{'}\end{array}& \begin{array}{l}\\ \\ -{\varsigma }_{\mathrm{2}}\text{'}\end{array}\end{array}\right].$(15)

Then

$\sigma (\mathrm{0,0},{\xi }_{\mathrm{1}}\text{'},{\xi }_{\mathrm{2}}\text{'},{\varsigma }_{\mathrm{1}},{\varsigma }_{\mathrm{2}},{\varsigma }_{\mathrm{1}}\text{'},{\varsigma }_{\mathrm{2}}\text{'})=\sigma ({\mathit{P}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{Q}}_{\mathrm{1}},{\mathit{W}}_{\mathrm{1}}).$

Proof   The proof is similar to the proof of Lemma 2 in Ref. [36]; since it is routine, we omit the details.

Lemma 4   Assume that in BCs (3), the parameters satisfy the conditions ${\varsigma }_{\mathrm{1}}={\varsigma }_{\mathrm{2}}=\mathrm{0},{\varsigma }_{\mathrm{2}}\text{'}<\mathrm{0},{\xi }_j,{\xi }_j\in \mathbb{R},j=\mathrm{1,2}$, and ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}>\mathrm{0}$. Let us define an $(n+\mathrm{2})\times (n+\mathrm{2})$ generalized pseudo-Jacobi matrix

${\mathit{P}}^{\mathrm{1}}[\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}]=\left[\begin{array}{ccc}\begin{array}{ccc}{\xi }_{\mathrm{2}}& {\xi }_{\mathrm{1}}& \\ \mathrm{1}& p_{\mathrm{1}}& -p_{\mathrm{1}}\\ & -p_{\mathrm{1}}& p_{\mathrm{1}}+p_{\mathrm{2}}\end{array}& \begin{array}{ccc}& & \\ & & \\ -p_{\mathrm{2}}& & \end{array}& \\ & \begin{array}{ccc}\cdots & \cdots & \cdots \\ & -p_{n-\mathrm{1}}& p_{n-\mathrm{1}}+p_n\\ & & -p_n\end{array}& \begin{array}{l}\\ -p_n\\ p_n+\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}\end{array}\end{array}\right]$(16)

and a diagonal matrix

${\mathit{Q}}^{\mathrm{1}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{0},q_{\mathrm{0}},q_{\mathrm{1}},\cdots ,q_n),$(17)

as well as an "almost" diagonal matrix

${\mathit{W}}^{\mathrm{1}}=\left[\begin{array}{ccc}\begin{array}{cc}-{\xi }_{\mathrm{2}}\text{'}& -{\xi }_{\mathrm{1}}\text{'}\\ & w_{\mathrm{0}}\end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{l}\\ \end{array}\\ \begin{array}{cc}& \\ & \\ & \end{array}& \begin{array}{cc}\cdots & \\ & w_{n-\mathrm{1}}\\ & \end{array}& \begin{array}{l}\\ \\ w_n\end{array}\end{array}\right].$(18)

Then

$\sigma ({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}\text{'},{\xi }_{\mathrm{2}}\text{'},\mathrm{0,0},{\varsigma }_{\mathrm{1}}\text{'},{\varsigma }_{\mathrm{2}}\text{'})=\sigma ({\mathit{P}}^{\mathrm{1}}[\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}]+{\mathit{Q}}^{\mathrm{1}},{\mathit{W}}^{\mathrm{1}}).$

Proof   The proof is similar with those in Lemma 3.

Remark 1   It can be noted that in Lemmas 2-4, the following equalities hold

$\begin{array}{l}{(\mathit{P}+\mathit{Q})}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]={\mathit{P}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{Q}}_{\mathrm{1}},\\ {(\mathit{P}+\mathit{Q})}^{\mathrm{1}}[\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}]={\mathit{P}}^{\mathrm{1}}[\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}]+{\mathit{Q}}^{\mathrm{1}}.\end{array}$

2 Inverse Matrix Eigenvalue Problems

In this section, we discuss the isospetral inverse matrix eigenvalue problems. Compared with the matrices given in Lemmas 2-4, we first consider the so-called generalized pseudo-Jacobi matrix in ${{M}}_k$, which takes the form as

${\mathit{J}}_{gp}=\left[\begin{array}{cc}\begin{array}{ccc}{\eta }_{\mathrm{1}}& {ϵ}_{\mathrm{1}}{l}_{\mathrm{1}}{\zeta }_{\mathrm{1}}& \\ {\zeta }_{\mathrm{1}}& {\eta }_{\mathrm{2}}& {\zeta }_{\mathrm{2}}\\ & \cdots & \cdots \end{array}& \\ & \begin{array}{ccc}\cdots & & \\ {\zeta }_{k-\mathrm{2}}& {\eta }_{k-\mathrm{1}}& {\zeta }_{k-\mathrm{1}}\\ & {ϵ}_{\mathrm{2}}{l}_{\mathrm{2}}{\zeta }_{k-\mathrm{1}}& {\eta }_k\end{array}\end{array}\right].$(19)

Definition 1   If a matrix ${\mathit{J}}_{gp}\in {{M}}_k$ takes the form of (19) and satisfies with the conditions ${\zeta }_{\mathrm{1}}>\mathrm{0},{\zeta }_j<\mathrm{0}$ for all$j=\mathrm{2},\cdots ,k-\mathrm{1},{ϵ}_{\mathrm{1}}={ϵ}_{\mathrm{2}}=\pm \mathrm{1},l_{\mathrm{1}},l_{\mathrm{2}}>\mathrm{0}$, then it is called a generalized pseudo-Jacobi matrix.

Definition 2   If a matrix ${\mathit{J}}_g\in {{M}}_k$ takes the form as

${\mathit{J}}_g=\left[\begin{array}{cc}\begin{array}{ccc}{\widehat{\eta }}_{\mathrm{1}}& {\widehat{l}}_{\mathrm{1}}{\widehat{\zeta }}_{\mathrm{1}}& \\ {\widehat{\zeta }}_{\mathrm{1}}& {\widehat{\eta }}_{\mathrm{2}}& {\widehat{\zeta }}_{\mathrm{2}}\\ & \cdots & \cdots \end{array}& \\ & \begin{array}{ccc}\cdots & & \\ {\widehat{\zeta }}_{k-\mathrm{2}}& {\widehat{\eta }}_{k-\mathrm{1}}& {\widehat{\zeta }}_{k-\mathrm{1}}\\ & {\widehat{l}}_{\mathrm{2}}{\widehat{\zeta }}_{k-\mathrm{1}}& {\widehat{\eta }}_k\end{array}\end{array}\right],$(20)

and satisfies with the conditions ${\widehat{\zeta }}_j>\mathrm{0},j=\mathrm{1},\cdots ,k-\mathrm{1}$, and ${\widehat{l}}_{\mathrm{1}}<\mathrm{0},{\widehat{l}}_{\mathrm{2}}<\mathrm{0}$, then it is called a generalized positive pseudo-Jacobi matrix.

If we set$\widehat{\mathit{L}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{1},\sqrt[]{-{\widehat{l}}_{\mathrm{1}}},\cdots ,\sqrt[]{-{\widehat{l}}_{\mathrm{1}}},\sqrt[]{\frac{-{\widehat{l}}_{\mathrm{1}}{\widehat{l}}_{\mathrm{2}}}{{\widehat{l}}_{\mathrm{2}}}})$and let$\mathit{J}=\widehat{\mathit{L}}{\mathit{J}}_g{\widehat{\mathit{L}}}^{-\mathrm{1}}$, then the matrix ${\mathit{J}}_g$ can be transformed to a positive pseudo-Jacobi matrix $\mathit{J}$ of the form

$\mathit{J}=\left[\begin{array}{cc}\begin{array}{ccc}{\widehat{\eta }}_{\mathrm{1}}& -\sqrt[]{-{\widehat{l}}_{\mathrm{1}}}{\widehat{\zeta }}_{\mathrm{1}}& \\ \sqrt[]{-{\widehat{l}}_{\mathrm{1}}}{\widehat{\zeta }}_{\mathrm{1}}& {\widehat{\eta }}_{\mathrm{2}}& {\widehat{\zeta }}_{\mathrm{2}}\\ & \cdots & \cdots \end{array}& \\ & \begin{array}{ccc}\cdots & & \\ {\widehat{\zeta }}_{k-\mathrm{2}}& {\widehat{\eta }}_{k-\mathrm{1}}& \sqrt[]{{\widehat{l}}_{\mathrm{2}}}{\widehat{\zeta }}_{k-\mathrm{1}}\\ & \sqrt[]{{\widehat{l}}_{\mathrm{2}}}{\widehat{\zeta }}_{k-\mathrm{1}}& {\widehat{\eta }}_k\end{array}\end{array}\right].$

Since it is a similarity transformation, hence the eigenvalues of matrices $\mathit{J}$ and ${\mathit{J}}_g$ are the same.

To best understand the theorems below, we still need to introduce a kind of generalized pseudo-Jacobi matrix $\mathit{M}\in {{M}}_k$, which takes the form

$\mathit{M}=\left[\begin{array}{cc}\begin{array}{ccc}{\check{\eta }}_{\mathrm{1}}& {\check{l}}_{\mathrm{1}}{\check{\zeta }}_{\mathrm{1}}& \\ {\check{\zeta }}_{\mathrm{1}}& {\check{\eta }}_{\mathrm{2}}& {\check{\zeta }}_{\mathrm{2}}\\ & \cdots & \cdots \end{array}& \\ & \begin{array}{ccc}\cdots & & \\ {\check{\zeta }}_{k-\mathrm{2}}& {\check{\eta }}_{k-\mathrm{1}}& {\check{\zeta }}_{k-\mathrm{1}}\\ & -{\check{l}}_{\mathrm{2}}{\check{\zeta }}_{k-\mathrm{1}}& {\check{\eta }}_k\end{array}\end{array}\right],$(21)

where ${\check{\zeta }}_{\mathrm{1}}>\mathrm{0},{\check{\zeta }}_j<\mathrm{0}$ for all $j=\mathrm{2},\cdots ,k-\mathrm{1},{\check{l}}_{\mathrm{1}},{\check{l}}_{\mathrm{2}}\ne \mathrm{0}$.

Lemma 5   (see Ref. [14]) Let $\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$and$\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$ be two sets of numbers satisfying one of the following four possibilities:

$\begin{array}{l}\mathrm{1}){\lambda }_{\mathrm{1}}>{\nu }_{\mathrm{1}}>{\lambda }_{\mathrm{2}}>{\nu }_{\mathrm{2}}>\cdots >{\nu }_{k-\mathrm{2}}>{\lambda }_{k-\mathrm{1}}>{\nu }_{k-\mathrm{1}}>{\nu }_k;\\ \mathrm{2}){\nu }_k>{\nu }_{\mathrm{1}}>{\lambda }_{\mathrm{1}}>{\nu }_{\mathrm{2}}>{\lambda }_{\mathrm{2}}>\cdots >{\nu }_{k-\mathrm{1}}>{\lambda }_{k-\mathrm{1}};\\ \mathrm{3})\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}[h\in \{\mathrm{1},\cdots ,k-\mathrm{2}\}]\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}:{\lambda }_{\mathrm{1}}>{\nu }_{\mathrm{1}}>\\ \cdots >{\lambda }_h>{\nu }_h>{\nu }_k>{\nu }_{h+\mathrm{1}}>{\lambda }_{h+\mathrm{1}}>\cdots >{\nu }_{k-\mathrm{1}}>{\lambda }_{k-\mathrm{1}};\\ \mathrm{4}){\lambda }_{\mathrm{1}}>{\nu }_{\mathrm{1}}>{\lambda }_{\mathrm{2}}>{\nu }_{\mathrm{2}}>\cdots >{\nu }_{k-\mathrm{2}}>{\lambda }_{k-\mathrm{1}},\\ \mathrm{a}\mathrm{n}\mathrm{d}{\nu }_k=\stackrel{\_\_\_\_\_}{{\nu }_{k-\mathrm{1}}}.\end{array}\}$(22)

Then, there exists a pseudo-Jacobi matrix $\mathit{J}\in {{M}}_k$ (may not be unique) such that $\sigma (\mathit{J})=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\sigma ({\mathit{J}}_{\mathrm{1}})=\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$.

Theorem 1   Let $\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$ be two sets of given numbers satisfying one of the four possibilities in (22). Then for arbitrary ${\widehat{l}}_{\mathrm{1}}<\mathrm{0},{\widehat{l}}_{\mathrm{2}}>\mathrm{0}$, there exists a generalized positive pseudo-Jacobi matrix ${\mathit{J}}_g\in {{M}}_k$ such that $\sigma ({\mathit{J}}_g)=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\sigma (({\mathit{J}}_g{)}_{\mathrm{1}})=$$\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$.

Proof   Let $\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$ be two sets of numbers satisfying one of the four possibilities in (22). Then by Lemma 5, there exists a positive pseudo-Jacobi matrix $\mathit{J}\in {{M}}_k$ such that

$\sigma (\mathit{J})=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}\mathrm{a}\mathrm{n}\mathrm{d}\sigma ({\mathit{J}}_{\mathrm{1}})=\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}.$

Now for arbitrary ${\widehat{l}}_{\mathrm{1}}<\mathrm{0},{\widehat{l}}_{\mathrm{2}}>\mathrm{0}$, there exists a generalized positive pseudo-Jacobi matrix ${\mathit{J}}_g\in {{M}}_k$ such that ${\mathit{J}}_g={\widehat{\mathit{L}}}^{-\mathrm{1}}\mathit{J}\widehat{\mathit{L}},({\mathit{J}}_g{)}_{\mathrm{1}}={\widehat{\mathit{L}}}^{-\mathrm{1}}{\mathit{J}}_{\mathrm{1}}\mathit{L}$,where $\widehat{\mathit{L}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{1},\sqrt[]{-{\widehat{l}}_{\mathrm{1}}},\cdots ,\sqrt[]{-{\widehat{l}}_{\mathrm{1}}},\sqrt[]{\frac{-{\widehat{l}}_{\mathrm{1}}{\widehat{l}}_{\mathrm{2}}}{{\widehat{l}}_{\mathrm{2}}}})$. This implies that the eigenvalues of matrices $\mathit{J}$ and ${\mathit{J}}_g$ are the same. Thus, $\sigma ({\mathit{J}}_g)=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\sigma (({\mathit{J}}_g{)}_{\mathrm{1}})=\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$.

The proof is finished.

It is easy to see that if ${\widehat{l}}_{\mathrm{1}}=-\mathrm{1},{\widehat{l}}_{\mathrm{2}}=\mathrm{1}$in the matrix ${\mathit{J}}_g$, then the generalized positive pseudo-Jacobi matrix ${\mathit{J}}_g$ will reduce to the positive pseudo-Jacobi matrix $\mathit{J}$. We now state a theorem on the inverse eigenvalue problem for a generalized pseudo-Jacobi matrix, which can be seen as an extension of Theorem 1.

Theorem 2   Let $\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$ be two sets of numbers satisfying one of the four possibilities in (22).

Let $\mathit{W}=\left[\begin{array}{ccc}\begin{array}{cc}-{\xi }_{\mathrm{2}}\text{'}& -{\xi }_{\mathrm{1}}\text{'}\\ & w_{\mathrm{0}}\end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{l}\\ \end{array}\\ \begin{array}{cc}& \\ & \\ & \end{array}& \begin{array}{cc}\cdots & \\ & w_n\\ & -{\varsigma }_{\mathrm{1}}\text{'}\end{array}& \begin{array}{l}\\ \\ -{\varsigma }_{\mathrm{2}}\text{'}\end{array}\end{array}\right]$be an "almost" diagonal matrix in which $w_j>\mathrm{0}$ for $j=\mathrm{0,1},\cdots ,k-\mathrm{3}$, and ${\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\varsigma }_{\mathrm{2}}\text{'}<\mathrm{0}$. Then for arbitrary ${\widehat{l}}_{\mathrm{1}}<\mathrm{0},{\widehat{l}}_{\mathrm{2}}>\mathrm{0}$, there exists a generalized pseudo-Jacobi matrix $\mathit{M}\in {{M}}_k$ such that

$\sigma (\mathit{M},\mathit{W})=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}and\sigma ({\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{W}}_{\mathrm{1}})=\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}.$(23)

Proof   First, from Theorem 1, we know that there exists a generalized positive pseudo-Jacobi matrix ${\mathit{J}}_g\in {{M}}_k$ such that $\sigma ({\mathit{J}}_g)=\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\sigma (({\mathit{J}}_g{)}_{\mathrm{1}})=$$\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$.

Now, for each $\nu ={\nu }_j,j=\mathrm{1},\cdots ,k,$ there exists a nontrivial $\mathit{u}\in {\mathbf{ℂ}}^k$ such that $({\mathit{J}}_g-\nu {\mathit{I}}_k)\mathit{u}=\mathrm{0}$.

Let us set $\mathit{H}=\left[\begin{array}{ccc}\begin{array}{cc}\mathrm{1}& -\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}\\ & \mathrm{1}\end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{l}\\ \end{array}\\ \begin{array}{cc}& \\ & \\ & \end{array}& \begin{array}{cc}\cdots & \\ & \mathrm{1}\\ & -\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}\end{array}& \begin{array}{l}\\ \\ \mathrm{1}\end{array}\end{array}\right]$ , and $\mathit{W}\mathit{H}={\mathit{R}}^{\mathrm{2}}$. Then we have $\mathit{R}=\sqrt[]{\mathit{W}\mathit{H}}:=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sqrt[]{-{\xi }_{\mathrm{2}}\text{'}},\sqrt[]{w_{\mathrm{0}}},$

$-\sqrt[]{w_{\mathrm{1}}},\cdots ,{(-\mathrm{1})}^{k-\mathrm{3}}\sqrt[]{w_{k-\mathrm{3}}},{(-\mathrm{1})}^{k-\mathrm{2}}\sqrt[]{-{\varsigma }_{\mathrm{2}}\text{'}}).$ If we let $\mathit{u}=$$\mathit{R}{\mathit{H}}^{-\mathrm{1}}\tilde{\mathit{u}}$, and left multiplying both sides of the equation (${\mathit{J}}_g-\nu {\mathit{I}}_k)\mathit{u}=\mathrm{0}$ by $\mathit{R}$ we obtain

$(\mathit{R}{\mathit{J}}_g\mathit{R}{\mathit{H}}^{-\mathrm{1}}-\nu {\mathit{R}}^{\mathrm{2}}{\mathit{H}}^{-\mathrm{1}})\tilde{\mathit{u}}=\mathrm{0}.$

Now by setting $\mathit{M}=\mathit{R}{\mathit{J}}_g\mathit{R}{\mathit{H}}^{-\mathrm{1}}$, we arrive at $(\mathit{M}-\nu \mathit{W})\tilde{\mathit{u}}=\mathrm{0}$. Therefore,

$\{\begin{array}{l}{\check{\zeta }}_{\mathrm{1}}=\sqrt[]{-{\xi }_{\mathrm{2}}\text{'}{w}_{\mathrm{0}}}{\widehat{\zeta }}_{\mathrm{1}},\\ {\check{\zeta }}_{\mathrm{2}}=\sqrt[]{w_{\mathrm{0}}{w}_{\mathrm{1}}}{\widehat{\zeta }}_{\mathrm{2}},\\ ⋮\\ {\check{\zeta }}_{k-\mathrm{2}}={(-\mathrm{1})}^{\mathrm{2}k-\mathrm{7}}\sqrt[]{w_{k-\mathrm{4}}{w}_{k-\mathrm{3}}}{\widehat{\zeta }}_{k-\mathrm{2}},\\ {\check{\zeta }}_{k-\mathrm{1}}={(-\mathrm{1})}^{\mathrm{2}k-\mathrm{5}}\sqrt[]{-{\varsigma }_{\mathrm{2}}\text{'}{w}_{k-\mathrm{3}}}{\widehat{\zeta }}_{k-\mathrm{1}};\end{array}\{\begin{array}{l}{\check{\eta }}_{\mathrm{1}}=-{\xi }_{\mathrm{2}}\text{'}{\widehat{\eta }}_{\mathrm{1}},\\ {\check{\eta }}_{\mathrm{2}}=w_{\mathrm{0}}{\widehat{\eta }}_{\mathrm{2}}+\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}{\check{\zeta }}_{\mathrm{1}},\\ ⋮\\ {\check{\eta }}_{k-\mathrm{1}}=w_{k-\mathrm{3}}{\widehat{\eta }}_{k-\mathrm{1}}+\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}{\check{\zeta }}_{k-\mathrm{1}},\\ {\check{\eta }}_k=-{\varsigma }_{\mathrm{2}}\text{'}{\widehat{\eta }}_k;\end{array}$

$\{\begin{array}{l}{\check{l}}_{\mathrm{1}}{\check{\zeta }}_{\mathrm{1}}=-{\xi }_{\mathrm{1}}\text{'}{\widehat{\zeta }}_{\mathrm{1}}+{\widehat{l}}_{\mathrm{1}}{\check{\zeta }}_{\mathrm{1}},\\ -{\check{l}}_{\mathrm{2}}{\check{\zeta }}_{k-\mathrm{1}}=-{\varsigma }_{\mathrm{1}}\text{'}{\widehat{\zeta }}_{k-\mathrm{1}}+{\widehat{l}}_{\mathrm{2}}{\check{\zeta }}_{k-\mathrm{1}}.\end{array}$

It is clear $\nu \in \sigma (\mathit{M},\mathit{W})$, and $\mathit{M}\in {{M}}_k$ is a generalized pseudo-Jacobi matrix. Similarly, for each $\lambda ={\lambda }_j,j=\mathrm{1},\cdots ,k-\mathrm{1}$, the same argument also applies. This means that ${\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]={\mathit{R}}_{\mathrm{1}}({\mathit{J}}_g{)}_{\mathrm{1}}{\mathit{R}}_{\mathrm{1}}({\mathit{H}}_{\mathrm{1}}{)}^{-\mathrm{1}},{\mathit{W}}_{\mathrm{1}}={\mathit{R}}_{\mathrm{1}}^{\mathrm{2}}({\mathit{H}}_{\mathrm{1}}{)}^{-\mathrm{1}}$ and $\lambda \in \sigma ({\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{W}}_{\mathrm{1}})$. Thus, we can conclude that

$\sigma ({\mathit{J}}_g)\subset \sigma (\mathit{M},\mathit{W})\mathrm{a}\mathrm{n}\mathrm{d}\sigma (({\mathit{J}}_g{)}_{\mathrm{1}})\subset \sigma ({\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{W}}_{\mathrm{1}}).$(24)

On the other hand, it is easy to get

$\sigma ({\mathit{J}}_g)\supset \sigma (\mathit{M},\mathit{W})\mathrm{a}\mathrm{n}\mathrm{d}\sigma (({\mathit{J}}_g{)}_{\mathrm{1}})\supset \sigma ({\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]+{\mathit{W}}_{\mathrm{1}}),$(25)

by reversing the above steps.

The proof is finished following (24) and (25).

Corollary 1   The statement of Theorem 2 still holds true, when ${\mathit{M}}_{\mathrm{1}}[-\frac{{\xi }_{\mathrm{1}}\text{'}}{{\xi }_{\mathrm{2}}\text{'}}]$ and ${\mathit{W}}_{\mathrm{1}}$ are replaced by ${\mathit{M}}^{\mathrm{1}}[\frac{{\varsigma }_{\mathrm{1}}\text{'}}{{\varsigma }_{\mathrm{2}}\text{'}}]$ and ${\mathit{W}}^{\mathrm{1}}$, respectively.

Corollary 2   The statement of Theorem 2 still holds true, when the condition $w_j>\mathrm{0}$ is replaced by $w_j<\mathrm{0}$ for $j=\mathrm{0,1},\cdots ,k-\mathrm{3}$.

3 Main Result and Proof

We now state our result on the inverse problem of SLPs with non-self-adjoint boundary conditions containing the spectral parameters (2), (3).

Theorem 3   Let ${\xi }_j$, ${\xi }_j\text{'}$, ${\varsigma }_j$, ${\varsigma }_j\text{'}\in \mathbb{R},j=\mathrm{1,2}$ be priori given numbers and satisfy the conditions ${\gamma }_{\mathrm{1}}={\xi }_{\mathrm{1}}\text{'}{\xi }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\gamma }_{\mathrm{2}}={\varsigma }_{\mathrm{1}}{\varsigma }_{\mathrm{2}}\text{'}-{\varsigma }_{\mathrm{1}}\text{'}{\varsigma }_{\mathrm{2}}>\mathrm{0}$ and ${\xi }_{\mathrm{2}}\text{'}<\mathrm{0},{\varsigma }_{\mathrm{2}}\text{'}<\mathrm{0}$. Let $\{{\nu }_j:j=\mathrm{1},\cdots ,k\}$ and $\{{\lambda }_j:j=\mathrm{1},\cdots ,k-\mathrm{1}\}$ be two sets of numbers satisfying one of the four possibilities in (22). Let $n=k-\mathrm{3}$. Then we have that for any $-\mathrm{\infty }<\alpha <\beta <+\mathrm{\infty }$, any partition (4) on ${J}$ , and any $w\in L$($\mathit{J},\mathbb{R}$) satisfying (7):

(a) There exist $p,q\in L$($\mathit{J},\mathbb{R}$) satisfying (5) and (6) such that the associated equivalent family of the problem (2), (3) has