Inverse problems for discontinuous Sturm-Liouville operators with mixed spectral data

Abstract

In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators with boundary conditions linearly dependent on the spectral parameter and a finite number of interior discontinuities and show that if $q$ is given a priori on the interval $[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$, then the potential $q$ on the whole interval $[0,1]$ can be uniquely determined either by parts of a finite number of spectra, or by a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. We still establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuous conditions from the above two spectral data.

Keywords:

• inverse problem
• Sturm–Liouville operator
• potential
• spectral parameter
• spectrum

1 Introduction

Consider the following Sturm–Liouville operator $L:=L(q,h_l,H_l,d_m)$ defined by(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) with boundary conditions(1.2) $$\begin{array}{cc}\hfill U_0(u)& :=(\mathit{\lambda }-h_1)u^{\prime }(0,\mathit{\lambda })+(h_2\mathit{\lambda }-h_3)u(0,\mathit{\lambda })=0,\hfill \end{array}$$(1.2) (1.3) $$\begin{array}{cc}\hfill U_1(u)& :=(\mathit{\lambda }-H_1)u^{\prime }(1,\mathit{\lambda })-(H_2\mathit{\lambda }-H_3)u(1,\mathit{\lambda })=0,\hfill \end{array}$$(1.3) and interior discontinuities(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) where $h_l,H_l,a_m,b_m,d_m\in \mathbb{R}$, $a_m>0,0<d_m<d_{m+1}<1,l=\overline{1,3},m=\overline{1,m_0},m_0\in \mathbb{N},$ such that$$r_0=h_3-h_1{h}_2>0\mathrm{and}{r}_1=H_1{H}_2-H_3>0,$$$\mathit{\lambda }$ is the spectral parameter, $q(x)$ is a real-valued function and $q\in L^2[0,1]$.

For (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ), Yao and Sun [1] showed that all eigenvalues are real and simple and all eigenfunctions of the Sturm–Liouville operator completes in $L^2[0,1]$. Later, Wang [2] discussed two inverse problems for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) either from partial information on the potential and parts of one spectrum, or from a set of values of eigenfunctions at some interior point and parts of two spectra. More related results for Sturm–Liouville operators with boundary conditions dependent on the spectral parameter were studied by many authors (see [1–14]). In particular, Freiling and Yurko [7] further explored three inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter either from Weyl function, or from discrete spectral data, or from two spectra and provided procedures for reconstructing this differential operator from the above spectral data. Mclaughlin and Polyakov [8] addressed the inverse problem for Sturm–Liouville operators with Dirichlet boundary condition at $x=0$ and the boundary condition having transcendental functions on the spectral parameter at $x=1$ and established an interesting uniqueness theorem on the potential $q$, which is a generalization of Hochstadt–Lieberman theorem.[15] Hald [16] first discussed the half-inverse problem for the Sturm–Liouville operator with one discontinuous condition. More results or generalizations for differential operators with interior discontinuities were studied by many authors (see [1, 2, 9, 10, 13, 14, 16–24]).

Borg [25], or Levinson [26], or Levitan [27], respectively, considered the inverse problem for Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) with Robin boundary conditions and showed that the specification of two spectra can uniquely determine the potential $q$ and coefficients $h_0,h_1$ of the boundary conditions. Hochstadt and Lieberman [15] first addressed the half-inverse problem for Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) with Robin boundary conditions and proved that if $q$ is prescribed on $[0,1/2]$ and coefficients $h_0,h_1$ of the boundary condition is given a priori, then the potential $q$ on the interval $[0,1]$ can be uniquely determined by one spectrum. Castillo [28], or Suzuki [29] independently showed that the fixed boundary condition at $x=0$ is necessary for Hochstadt–Lieberman theorem by an example. Hochstadt–Lieberman type theorem for differential operators was established by many authors (see [2, 8, 10, 11, 15–17, 23, 28–36]). Marchenko [37] first adopted an alternative approach to the inverse spectral problem and showed that the Weyl function of Sturm–Liouville operators uniquely determined the potential $q$ and coefficients $h_0,h_1$ of the boundary conditions. Numerous authors discussed the inverse spectral problems for differential operators by the Weyl function of this operator (see [2, 5, 7, 9, 11, 19–21, 30, 31, 35–41]). Gesztesy and Simon [31] (see [31, Theorem 1.3]) used the Weyl function to study the inverse spectral problem for Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) with Robin boundary conditions and showed that if potential $q$ on the interval $\left[0,,,\frac{1+\mathit{\alpha }}{2}\right]$ for some $0\le \mathit{\alpha }<1$ is given a priori, then parts of one spectrum is sufficient to determine the potential $q$ on the whole interval. Suzuki [29] showed that if $q$ is prescribed on $\left[0,,,\frac{1}{2},-,\mathit{\epsilon }\right]$ for $0<\mathit{\epsilon }<\frac{1}{2}$, then one spectrum cannot uniquely determine the potential $q$ by a counterexample. Later, using the Weyl function and methods developed in [31], Wei and Xu [36] proved that if $q$ is given a priori on the interval $[0,\mathit{\alpha }]$ for some $0\le \mathit{\alpha }<1$, then parts of pairs of eigenvalues and the norming constants of the corresponding eigenvalues is sufficient to determine the potential $q$ on the whole interval $[0,1]$. Hence, the following inverse problem is of interest:

Inverse Problem-1: For (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ), if $q$ is prescribed on $[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$ and coefficients $h_l,l=\overline{1,3}$ of the boundary condition are given a priori, what extra conditions can ensure the unique determination of the potential $q$ on the interval $[0,1]$?

The purpose of this article is to solve Inverse Problem-1 given by (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). At first, we discuss two inverse problems for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) either from partial information on the potential and parts of a finite number of spectra, or from partial information on the potential and parts of a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. Then we establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and arbitrary finite number of interior discontinuous conditions from the above two spectral data. The techniques used here are based on the Weyl function and methods developed in [7, 31, 36]. Borg type theorem, or Gesztesy–Simon theorem, or Hochstadt–Lieberman type theorem for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) is one of Theorem 3.1 under special cases (see below).

This article is organized as follows. In Section 2, we present preliminaries. In Section 3, we discuss the inverse spectral problems for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) from the above two spectral data. In Section 4, we establish several uniqueness theorems for Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) with Robin boundary conditions and interior discontinuous conditions with the above spectral data.

2 Preliminaries

Let $S_1(x,\mathit{\lambda }),S_2(x,\mathit{\lambda }),u_{-}(x,\mathit{\lambda })$ and $u_{+}(x,\mathit{\lambda })$ be solutions of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) under the initial conditions(2.1) $$\begin{array}{cc}& S_1(0,\mathit{\lambda })=S_2^{\prime }(0,\mathit{\lambda })=0,S_1^{\prime }(0,\mathit{\lambda })=S_2(0,\mathit{\lambda })=1\hfill \\ \hfill & u_{-}(0,\mathit{\lambda })=\mathit{\lambda }-h_1,u_{-}^{\prime }(0,\mathit{\lambda })=h_3-h_2\mathit{\lambda },\hfill \\ \hfill & u_{+}(1,\mathit{\lambda })=\mathit{\lambda }-H_1,u_{+}^{\prime }(1,\mathit{\lambda })=H_2\mathit{\lambda }-H_3.\hfill \end{array}$$(2.1) and satisfying the jump conditions (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ), respectively.

Denote ${\mathrm{\Delta }}_k(\mathit{\lambda })=U_1(S_k)$. Clearly, $U_0(u_{-})=U_1(u_{+})=0$, and(2.2) $$\begin{array}{cc}\hfill u_{-}(x,\mathit{\lambda })& =(\mathit{\lambda }-h_1)S_2(x,\mathit{\lambda })+(h_3-h_2\mathit{\lambda })S_1(x,\mathit{\lambda }),\hfill \end{array}$$(2.2) (2.3) $$\begin{array}{cc}\hfill u_{+}(x,\mathit{\lambda })=& (\mathit{\lambda }-H_1)S_2(1-x,\mathit{\lambda })-(H_2\mathit{\lambda }-H_3)S_1(1-x,\mathit{\lambda })\hfill \\ \hfill =& {\mathrm{\Delta }}_1(\mathit{\lambda })S_2(x,\mathit{\lambda })-{\mathrm{\Delta }}_2(\mathit{\lambda })S_1(x,\mathit{\lambda }).\hfill \end{array}$$(2.3) The following formula is called as the Green’s formula:(2.4) $$\begin{array}{c}\hfill {\int }_0^1(yL(z)-zL(y))=[y,z](1)-[y,z](0),\end{array}$$(2.4) where $[y,z](x):=y(x)z^{\prime }(x)-y^{\prime }(x)z(x)$ is called the Wronskian of $y$ and $z$.

Let(2.5) $$\begin{array}{c}\hfill \mathrm{\Delta }(\mathit{\lambda }):=[u_{+},u_{-}](x,\mathit{\lambda }).\end{array}$$(2.5) By calculation, we have(2.6) $$\begin{array}{c}\hfill [u_{+},u_{-}](x+,\mathit{\lambda })=[u_{+},u_{-}](x-,\mathit{\lambda }),\forall x\in [0,1].\end{array}$$(2.6) Therefore, $\mathrm{\Delta }(\mathit{\lambda })$ does not depend on $x$ and(2.7) $$\begin{array}{cc}\hfill \mathrm{\Delta }(\mathit{\lambda })& =(\mathit{\lambda }-h_1){\mathrm{\Delta }}_2(\mathit{\lambda })+(h_3-h_2\mathit{\lambda }){\mathrm{\Delta }}_1(\mathit{\lambda })\hfill \\ \hfill & =U_1(u_{-})=-U_0(u_{+}),\hfill \end{array}$$(2.7) which is called the characteristic function of $L$.

Let $\mathit{\sigma }(L):={\{{\mathit{\lambda }}_n\}}_{n=0}^{\infty }$ be the set of all eigenvalues of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). From [1], we see that all zeros ${\mathit{\lambda }}_n$ of $\mathrm{\Delta }(\mathit{\lambda })$ are real and simple. Suppose that $u_{-}(x,{\mathit{\lambda }}_n)$ and $u_{+}(x,{\mathit{\lambda }}_n)$ are eigenfunctions of the corresponding eigenvalue ${\mathit{\lambda }}_n$, then there exists ${\mathit{\kappa }}_n$ such that$$\begin{array}{c}\hfill u_{+}(x,{\mathit{\lambda }}_n)={\mathit{\kappa }}_n{u}_{-}(x,{\mathit{\lambda }}_n),\end{array}$$where ${\mathit{\kappa }}_n$ is called the norming constant of the corresponding eigenvalue ${\mathit{\lambda }}_n$. Consequently,(2.8) $$\begin{array}{c}\hfill u_{+}(0,{\mathit{\lambda }}_n)={\mathit{\kappa }}_n({\mathit{\lambda }}_n-h_1),\end{array}$$(2.8) and ${\mathit{\kappa }}_n\ne 0,\infty$.

Denote $\mathit{\lambda }={\mathit{\rho }}^2$ and $\mathit{\tau }=|Im\mathit{\rho }|$. Analogous to [13, 14, 20], we have the following asymptotic formulae of $u_{-}(x,\mathit{\lambda })$ and $u_{+}(x,\mathit{\lambda })$(2.9) $$\begin{array}{cc}\hfill \left|u_{-}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }x}\right)0\le x\le 1,\mathit{\nu }=0,1,\hfill \end{array}$$(2.9) (2.10) $$\begin{array}{cc}\hfill \left|u_{+}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }(1-x)}\right),0\le x\le 1,\mathit{\nu }=0,1.\hfill \end{array}$$(2.10) Therefore, for sufficiently large $|\mathit{\rho }|$, we have the asymptotic formula of $\mathrm{\Delta }(\mathit{\lambda })$(2.11) $$\begin{array}{c}\hfill \mathrm{\Delta }(\mathit{\lambda })={\mathrm{\Delta }}_0(\mathit{\lambda })+O\left({\mathit{\rho }}^4,e^{\mathit{\tau }}\right),\end{array}$$(2.11) where(2.12) $$\begin{array}{cc}\hfill {\mathrm{\Delta }}_0(\mathit{\lambda })=& {\mathit{\rho }}^5\left[{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,\cdots ,{\mathit{\alpha }}_{m_0},sin,\mathit{\rho },+,\sum _{k=1}^{m_0},{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,\cdots ,{\mathit{\alpha }}_k^{\prime },\cdots ,{\mathit{\alpha }}_{m_0},sin,(\mathit{\rho }-2d_k)\right.\hfill \\ \hfill & +\sum _{1\le k<j\le m_0}{\mathit{\alpha }}_1\cdots {\mathit{\alpha }}_k^{\prime }\cdots {\mathit{\alpha }}_j^{\prime }\cdots {\mathit{\alpha }}_{m_0}sin(\mathit{\rho }+2d_k-2d_j)\hfill \\ \hfill & \left.+,\cdots ,+,{\mathit{\alpha }}_1^{\prime },{\mathit{\alpha }}_2^{\prime },\cdots ,{\mathit{\alpha }}_{m_0}^{\prime },sin,\left(\mathit{\rho },+,2,\sum _{k=1}^{m_0},{(-1)}^{m_0-k},d_k\right)\right],\hfill \end{array}$$(2.12) where ${\mathit{\alpha }}_k=\frac{a_k+b_k}{2}$ and ${\mathit{\alpha }}_k^{\prime }=\frac{a_k-b_k}{2}$.

Let ${\mathit{\lambda }}_n^0={({\mathit{\rho }}_n^0)}^2$ be zeros of ${\mathrm{\Delta }}_0(\mathit{\lambda })$. Denote $G_{\mathit{\delta }}:=\{\mathit{\rho }||\mathit{\rho }-{\mathit{\rho }}_n^0|>\mathit{\delta },n\in \mathbb{Z}\}$, where $\mathit{\delta }$ sufficiently small, then there exists a constant $C_{\mathit{\delta }}$ (see [20, 40]), such that for sufficiently large $|\mathit{\lambda }|$(2.13) $$\begin{array}{c}\hfill |\mathrm{\Delta }(\mathit{\lambda })|\ge C_{\mathit{\delta }}{|\mathit{\rho }|}^5{e}^{\mathit{\tau }},\forall \mathit{\rho }\in G_{\mathit{\delta }}.\end{array}$$(2.13) Hence, ${\mathit{\lambda }}_n$ satisfy(2.14) $$\begin{array}{c}\hfill {\mathit{\lambda }}_0<{\mathit{\lambda }}_1<{\mathit{\lambda }}_2<\cdots \to +\infty \end{array}$$(2.14) and(2.15) $$\begin{array}{c}\hfill {\mathit{\lambda }}_n={\mathit{\lambda }}_n^0+o(1),\end{array}$$(2.15) where ${\mathit{\lambda }}_n^0=n^2{\mathit{\pi }}^2+O(1)$.

Let $\mathrm{\Phi }(x,\mathit{\lambda })$ be the solution of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) under the conditions $U_1(\mathrm{\Phi })=1$, $U_0(\mathrm{\Phi })=0$ and the jump conditions (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). Then(2.16) $$\begin{array}{c}\hfill \mathrm{\Phi }(x,\mathit{\lambda })=-\frac{u_{+}(x,\mathit{\lambda })}{\mathrm{\Delta }(\mathit{\lambda })}=S_1(x,\mathit{\lambda })+M(\mathit{\lambda })u_{-}(x,\mathit{\lambda }),\end{array}$$(2.16) where(2.17) $$\begin{array}{c}\hfill M(\mathit{\lambda }):=-\frac{u_{+}^{\prime }(0,\mathit{\lambda })}{\mathrm{\Delta }(\mathit{\lambda })},\end{array}$$(2.17) which is called the Weyl function of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ).

The following two lemmas are important for proofs of our main results.

Lemma 2.1

([7, 14]) Let $M(\mathit{\lambda })$ be the Weyl function of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) and coefficients $h_l,l=\overline{1,3}$, of the boundary condition be given a priori. Then $M(\mathit{\lambda })$ uniquely determines the coefficients $H_l,l=\overline{1,3}$, of the boundary condition and $a_m,b_m,d_m,m=\overline{1,m_0}$, of the jump conditions as well as $q(x)$ (a.e.) on the interval $[0,1]$.

Lemma 2.2

([31, Proposition B.6]) Let $f(z)$ be an entire function such that

(1)	${sup}_{|z|=R_k}|f(z)|\le C_{1}exp(C_2{R}_k^{\mathit{\alpha }})$ for some $0<\mathit{\alpha }<1$, some sequence $R_k\to \infty$ as $k\to \infty$ and $C_1,C_2>0$.
(2)	${lim}_{|x|\to \infty }|f(ix)|=0,x\in \mathbb{R}$. Then $f\equiv 0.$

3 Inverse spectral problems

In this section, we discuss two inverse problems for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) and show that if the potential $q$ is prescribed on the interval $[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$, then the potential $q$ on the whole interval $[0,1]$ can be uniquely determined by parts of a finite number of spectra or by parts of a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues.

We agree that together with $L$ we consider in this section a boundary value problem $L_{kj}=L(q_k,h_{lj},H_{lk},d_{km}),k=1,2,l=\overline{1,3},j=\overline{1,j_0},m=\overline{1,m_0},j_0,m_0\in \mathbb{N},$ of the same form but with other coefficients. That is, we use $(q_k,h_{lj},H_{lk},d_{km})$ instead of $(q,h_l,H_l,d_m)$ of the operator $L$, respectively, such that$$\frac{h_{2j_1}\mathit{\lambda }-h_{3j_1}}{\mathit{\lambda }-h_{1j_1}}\ne \frac{h_{2j_2}\mathit{\lambda }-h_{3j_2}}{\mathit{\lambda }-h_{1j_2}},\forall \mathit{\lambda }\in \mathbb{C}.$$for all $j_1\ne j_2,j_1,j_2\in \{1,2,\cdots ,j_0\}$.

Let $\mathit{\sigma }(L_{kj})={\{{\mathit{\lambda }}_{kjn}\}}_{n=0}^{\infty },k=1,2,j=\overline{1,j_0},$ be the spectrum of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_l,H_{lk},d_{km})$. Since $\frac{h_{2j_1}\mathit{\lambda }-h_{3j_1}}{\mathit{\lambda }-h_{1j_1}}\ne \frac{h_{2j_2}\mathit{\lambda }-h_{3j_2}}{\mathit{\lambda }-h_{1j_2}},\forall \mathit{\lambda }\in \mathbb{C}.$ It is easy to prove $\mathit{\sigma }(L_{k{j}_1})\cap \mathit{\sigma }(L_{k{j}_2})=\mathrm{\varnothing }$ for fixed $k=1$ or $2$.

For any $S={\{{\mathit{\lambda }}_n\}}_{n=0}^{\infty },{\mathit{\lambda }}_n\in \mathbb{R}$, we denote$$\begin{array}{c}\hfill N_S=♯\{n\in {\mathbb{N}}_0:{\mathit{\lambda }}_n\le t\}\end{array}$$for all sufficiently large $t\in {\mathbb{R}}^{+}$.

We establish the following uniqueness theorem.

Theorem 3.1

Let $\mathit{\sigma }(L_{kj})$ be as that defined above, $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(L_{1j})\bigcap \mathit{\sigma }(L_{2j})$, ${\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, for each $j=\overline{1,j_0}$ and coefficients $h_{lj},l=\overline{1,3}$, of the boundary conditions be given a priori. If

(1)	$q_1=q_{2}on[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1).$
(2)	$2\left({\sum }_{j=1}^{j_0},{\mathit{\alpha }}_j,-,{\mathit{\alpha }}_0\right)=j_0-2$, ${\sum }_{j=1}^{j_0}{\mathit{\beta }}_j=5j_0-4$, and ${\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j>0$ for some ${\mathit{\alpha }}_j\in \left[0,,,\frac{1}{2}\right]$, ${\mathit{\beta }}_j\in {\mathbb{N}}_0$ and ${\mathit{\epsilon }}_j\ge 0$.
(3)	For each $j=\overline{1,j_0},$ the inequality(3.1) $$\begin{array}{c}\hfill N_{S_j}\ge (1-2{\mathit{\alpha }}_j)N_{\mathit{\sigma }(L_{1j})}+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4}\end{array}$$(3.1) holds for all sufficiently large $t\in {\mathbb{R}}^{+}$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Let $j_0=2,{\mathit{\alpha }}_0={\mathit{\alpha }}_1={\mathit{\alpha }}_2=0,{\mathit{\beta }}_1=2{\mathit{\epsilon }}_1=2{\mathit{\epsilon }}_2=1,{\mathit{\beta }}_2=5$, we have the Borg type theorem except for one eigenvalue.

Theorem 3.2

Let $\mathit{\sigma }(L_{kj}),k,j=1,2$, be as that defined above and coefficients $h_{lj},l=\overline{1,3}$, of the boundary condition be given a priori. If one of the following conditions hold

(1)	${\mathit{\lambda }}_{11n}={\mathit{\lambda }}_{21n},\forall n\in {\mathbb{N}}_{0}and{\mathit{\lambda }}_{12n}={\mathit{\lambda }}_{22n},\forall n\in {\mathbb{N}}_0\backslash \{n_1\};$
(2)	${\mathit{\lambda }}_{11n}={\mathit{\lambda }}_{21n},\forall n\in {\mathbb{N}}_0\backslash \{n_2\}and{\mathit{\lambda }}_{12n}={\mathit{\lambda }}_{22n},\forall n\in {\mathbb{N}}_0,$

then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill \&a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\end{array}\end{array}$$

If $j_0=1,$${\mathit{\alpha }}_0=\frac{1+\mathit{\alpha }}{2},{\mathit{\alpha }}_1=\frac{\mathit{\alpha }}{2},{\mathit{\beta }}_1=1,{\mathit{\epsilon }}_1>0$, Theorem 3.1 leads to the Gesztesy–Simon type theorem for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). i.e.:

Theorem 3.3

Let $\mathit{\sigma }(L_k)={\{{\mathit{\lambda }}_{kn}\}}_{n=0}^{\infty }$ be the spectrum of  (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_l,H_{lk},d_{km})$, $S={\{{\mathit{\lambda }}_{1n}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(L_1)\bigcap \mathit{\sigma }(L_2)$$,{\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, and coefficients $h_l,l=\overline{1,3}$, of the boundary condition be given a priori. For some $\mathit{\alpha }\in [0,1)$, $\mathit{\epsilon }$ sufficiently small positive number, if $q_1=q_2$ on the interval $\left[0,,,\frac{1+\mathit{\alpha }}{2}\right]$ and(3.2) $$\begin{array}{c}\hfill N_S\ge (1-\mathit{\alpha })N_{\mathit{\sigma }(L_1)}+\frac{5\mathit{\alpha }-1+2\mathit{\epsilon }}{4}\end{array}$$(3.2) is satisfied for all sufficiently large $t\in {\mathbb{R}}^{+}$, then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

In particular, let $\mathit{\alpha }=0$ and $\mathit{\epsilon }=\frac{1}{2}$ in Theorem 3.3, we have the Hochstadt–Lieberman type theorem for the problem (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). i.e.

Theorem 3.4

Let $q$ be prescribed on the interval $\left[0,,,\frac{1}{2}\right]$ and coefficients $h_l,l=\overline{1,3}$, of the boundary condition be given a priori. Then one spectrum is sufficient to determine the potential $q$ on the whole interval $[0,1]$ and coefficients $H_l,l=\overline{1,3}$, of the boundary condition and coefficients $a_m$, $b_m$ and $d_m,m=\overline{1,m_0},$ of the jump conditions.

Let $j_0=3,{\mathit{\alpha }}_0=0$, ${\mathit{\alpha }}_j=\frac{1}{6},j=\overline{1,3}$, ${\mathit{\beta }}_1={\mathit{\beta }}_2=4,{\mathit{\beta }}_3=3$, ${\mathit{\epsilon }}_1={\mathit{\epsilon }}_2=\frac{7}{3},{\mathit{\epsilon }}_3=\frac{4}{3}$, we get the following uniqueness theorem for the problem (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_{lj},H_{lk},d_{km})$ from two-thirds of three spectra. i.e.

Theorem 3.5

Let $\mathit{\sigma }(L_{kj})$, $k=1,2,j=\overline{1,3}$, be as that defined above, $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(L_{1j})\bigcap \mathit{\sigma }(L_{2j})$ and coefficients $h_{lj},l,j=\overline{1,3}$, of the boundary condition be given a priori. If(3.3) $$\begin{array}{c}\hfill N_{S_j}=\frac{2}{3}{N}_{\mathit{\sigma }(L_{1j})},j=\overline{1,3},\end{array}$$(3.3) then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Next, we present the proof of Theorem 3.1.

Proof of Theorem 3.1

Let $u_{k+}(x,\mathit{\lambda }),k=1,2$, be the solution of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) for $q_k$ under the terminal conditions $u_{k+}(1,\mathit{\lambda })=\mathit{\lambda }-H_{1k}$ and $u_{k+}^{\prime }(1,\mathit{\lambda })=H_{2k}\mathit{\lambda }-H_{3k}$ and the interior discontinuities (1.4) for $d_{km}$. By the Green’s formula, we have(3.4) $$\begin{array}{cc}\hfill {\int }_0^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })\mathrm{d}x=& [u_{1+},u_{2+}](1,\mathit{\lambda })-[u_{1+},u_{2+}](0,\mathit{\lambda })\hfill \\ \hfill =& F_1(1,\mathit{\lambda })-F_1(0,\mathit{\lambda }),\hfill \end{array}$$(3.4) where $Q(x)=q_2(x)-q_1(x)$ and(3.5) $$\begin{array}{c}\hfill F_1(x,\mathit{\lambda })=[u_{1+},u_{2+}](x,\mathit{\lambda }).\end{array}$$(3.5) From $Q(x)=0$ on $[0,{\mathit{\alpha }}_0]$ together with the terminal conditions $u_{k+}(1,\mathit{\lambda })$ and $u_{k+}^{\prime }(1,\mathit{\lambda })$, we get(3.6) $$\begin{array}{c}\hfill F_1(0,\mathit{\lambda })=F_1({\mathit{\alpha }}_0,\mathit{\lambda })=F_1(1,\mathit{\lambda })-{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx.\end{array}$$(3.6) In addition, for each $j=\overline{1,j_0}$, we obtain(3.7) $$\begin{array}{cc}\hfill F_1(0,\mathit{\lambda })& =u_{1+}(0,\mathit{\lambda })u_{2+}^{\prime }(0,\mathit{\lambda })-u_{1+}^{\prime }(0,\mathit{\lambda })u_{2+}(0,\mathit{\lambda })\hfill \\ \hfill & =\frac{1}{\mathit{\lambda }-h_{1j}}[u_{1+}(0,\mathit{\lambda })U_{0j}(u_{2+})-u_{2+}(0,\mathit{\lambda })U_{0j}(u_{1+})].\hfill \end{array}$$(3.7) This implies(3.8) $$\begin{array}{c}\hfill F_1(0,{\mathit{\lambda }}_{1jn})=0,\forall {\mathit{\lambda }}_{1jn}\in S_j\subseteq \mathit{\sigma }(L_{1j})\bigcap \mathit{\sigma }(L_{2j}),j=\overline{1,j_0}.\end{array}$$(3.8) Therefore(3.9) $$\begin{array}{c}\hfill F_1({\mathit{\alpha }}_0,{\mathit{\lambda }}_{1jn})=0,\forall {\mathit{\lambda }}_{1jn}\in S_j\subseteq \mathit{\sigma }(L_{1j})\bigcap \mathit{\sigma }(L_{2j}),j=\overline{1,j_0}.\end{array}$$(3.9) Denote(3.10) $$\begin{array}{c}\hfill {\mathit{\omega }}_{kj}(\mathit{\lambda }):=(\mathit{\lambda }-h_{1j})u_{k+}(0,\mathit{\lambda })+(h_{2j}\mathit{\lambda }-h_{3j})u_{k+}^{\prime }(0,\mathit{\lambda }),k=1,2,j=\overline{1,j_0}.\end{array}$$(3.10) Without loss of generality, let’s assume all eigenvalues ${\mathit{\lambda }}_{1jn}>1,n\in {\mathbb{N}}_0,j=\overline{1,j_0}$, of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $q_1,h_{lj},H_{l1},d_{1m}$. By virtue of [1], we get(3.11) $$\begin{array}{c}\hfill {\mathit{\omega }}_{1j}(\mathit{\lambda })=c_{1j}\prod _{n\in {\mathbb{N}}_0}\left(1,-,\frac{\mathit{\lambda }}{{\mathit{\lambda }}_{1jn}}\right),j=\overline{1,j_0},\end{array}$$(3.11) where $c_{1j}$ is constant.

By virtue of (3.6(3.6) $$\begin{array}{c}\hfill F_1(0,\mathit{\lambda })=F_1({\mathit{\alpha }}_0,\mathit{\lambda })=F_1(1,\mathit{\lambda })-{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx.\end{array}$$(3.6) ) and (2.10(2.10) $$\begin{array}{cc}\hfill \left|u_{+}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }(1-x)}\right),0\le x\le 1,\mathit{\nu }=0,1.\hfill \end{array}$$(2.10) ) together with Schwarz inequality, this yields(3.12) $$\begin{array}{cc}& |F_1(0,\mathit{\lambda })|=|F_1({\mathit{\alpha }}_0,\mathit{\lambda })|\hfill \\ \hfill & \le |F_1(1,\mathit{\lambda })|+|{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx|\hfill \\ \hfill & \le |F_1{(1,\mathit{\lambda })|+||Q||}_2{\left({\int }_{{\mathit{\alpha }}_0}^1,|,u_{1+}(x,\mathit{\lambda })u_{2+}{(x,\mathit{\lambda })|}^{2}dx\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le c_1{|\mathit{\rho }|}^4+c_2{|\mathit{\rho }|}^4{||Q||}_2\sqrt{1-{\mathit{\alpha }}_0}{e}^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\hfill \\ \hfill & =O\left({\mathit{\rho }}^4,e^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\right),\hfill \end{array}$$(3.12) where $c_1,c_2$ are constant.

From (2.10(2.10) $$\begin{array}{cc}\hfill \left|u_{+}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }(1-x)}\right),0\le x\le 1,\mathit{\nu }=0,1.\hfill \end{array}$$(2.10) ), we have(3.13) $$\begin{array}{c}\hfill |{\mathit{\omega }}_{1j}(\mathit{\lambda })|=O\left({\mathit{\rho }}^5,e^{\mathit{\tau }}\right),j=\overline{1,j_0}.\end{array}$$(3.13) Define the functions $G_{1j}(\mathit{\lambda }),j=\overline{1,j_0}$, and $K_1(\mathit{\lambda })$ by(3.14) $$\begin{array}{c}\hfill G_{1j}(\mathit{\lambda })=\prod _{{\mathit{\lambda }}_{1jn}\in S_j}\left(1,-,\frac{\mathit{\lambda }}{{\mathit{\lambda }}_{1jn}}\right)\end{array}$$(3.14) and(3.15) $$\begin{array}{c}\hfill K_1(\mathit{\lambda })=\frac{F_1({\mathit{\alpha }}_0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{1j}(\mathit{\lambda })}.\end{array}$$(3.15) Then, $K_1(\mathit{\lambda })$ is an entire function in $\mathit{\lambda }$.

By virtue of (3.1(3.1) $$\begin{array}{c}\hfill N_{S_j}\ge (1-2{\mathit{\alpha }}_j)N_{\mathit{\sigma }(L_{1j})}+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4}\end{array}$$(3.1) ), this yields(3.16) $$\begin{array}{c}\hfill N_{G_{1j}}(t)\ge (1-2{\mathit{\alpha }}_j)N_{{\mathit{\omega }}_{1j}}(t)+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4}.\end{array}$$(3.16) Since ${\mathit{\omega }}_{1j}(\mathit{\lambda })$ is an entire function in $\mathit{\lambda }$ of order $\frac{1}{2}$, there exists a positive constant $C$ such that(3.17) $$\begin{array}{c}\hfill N_{G_{1j}}(t)\le N_{{\mathit{\omega }}_{1j}}(t)\le C{t}^{\frac{1}{2}}.\end{array}$$(3.17) Therefore, $N_{G_{1j}}(1)=N_{{\mathit{\omega }}_{1j}}(1)=0$. For fixed $y\in \mathbb{R},$ and $|y|$ sufficiently large, we have(3.18) $$\begin{array}{ccc}\hfill ln|G_{1j}(\mathrm{i}y)|& =\frac{1}{2}ln{G}_{1j}(\mathrm{i}y)\overline{G_{1j}(\mathrm{i}y)}\hfill & \hfill \\ \hfill & =\frac{1}{2}\sum _{{\mathit{\lambda }}_{1jn}\in S_{1j}}ln\left(1,+,\frac{y^2}{{({\mathit{\lambda }}_{1jn})}^2}\right)=\frac{1}{2}{\int }_1^{\infty }ln\left(1,+,\frac{y^2}{t^2}\right)\mathrm{d}{N}_{G_{1j}}(t)\hfill \\ \hfill & =\frac{1}{2}ln\left(1,+,\frac{y^2}{t^2}\right)N_{G_{1j}}{(t)|}_1^{\infty }-\frac{1}{2}{\int }_1^{\infty }{N}_{G_{1j}}(t)\mathrm{d}\left[ln,\left(1,+,\frac{y^2}{t^2}\right)\right].\hfill \end{array}$$(3.18) For sufficiently large $t$, since$$ln\left(1,+,\frac{y^2}{t^2}\right)=O\left(\frac{1}{t^2}\right),$$then$$\underset{t\to \infty }{lim}ln\left(1,+,\frac{y^2}{t^2}\right)N_{G_{1j}}(t)=0\mathrm{and}\underset{t\to \infty }{lim}ln\left(1,+,\frac{y^2}{t^2}\right)N_{{\mathit{\omega }}_{1j}}(t)=0.$$By the assumption (3.3(3.3) $$\begin{array}{c}\hfill N_{S_j}=\frac{2}{3}{N}_{\mathit{\sigma }(L_{1j})},j=\overline{1,3},\end{array}$$(3.3) ) on $G_{1j}$ of Theorem 3.1, there exist constants $t_j\ge 1$ and $C_{1j}$ such that$$N_{G_{1j}}=\left\{\begin{array}{c}{N}_{G_{1j}}(t)\ge (1-2{\mathit{\alpha }}_j)N_{{\mathit{\omega }}_{1j}}(t)+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4},t\ge t_j,\hfill \\ N_{G_{1j}}(t)\ge (1-2{\mathit{\alpha }}_j)N_{{\mathit{\omega }}_{1j}}(t)-C_{1j},t<t_j.\hfill \end{array}\right.$$By virtue of (3.18(3.18) $$\begin{array}{ccc}\hfill ln|G_{1j}(\mathrm{i}y)|& =\frac{1}{2}ln{G}_{1j}(\mathrm{i}y)\overline{G_{1j}(\mathrm{i}y)}\hfill & \hfill \\ \hfill & =\frac{1}{2}\sum _{{\mathit{\lambda }}_{1jn}\in S_{1j}}ln\left(1,+,\frac{y^2}{{({\mathit{\lambda }}_{1jn})}^2}\right)=\frac{1}{2}{\int }_1^{\infty }ln\left(1,+,\frac{y^2}{t^2}\right)\mathrm{d}{N}_{G_{1j}}(t)\hfill \\ \hfill & =\frac{1}{2}ln\left(1,+,\frac{y^2}{t^2}\right)N_{G_{1j}}{(t)|}_1^{\infty }-\frac{1}{2}{\int }_1^{\infty }{N}_{G_{1j}}(t)\mathrm{d}\left[ln,\left(1,+,\frac{y^2}{t^2}\right)\right].\hfill \end{array}$$(3.18) ) together with the following relation$$\frac{y^2}{t^3+t{y}^2}=-\frac{d}{dt}\left(\frac{1}{2},ln,\left(1,+,\frac{y^2}{t^2}\right)\right),$$we have(3.19) $$\begin{array}{cc}& ln|G_{1j}(\mathrm{i}y)|={\int }_1^{\infty }\frac{y^2}{{}^{t^3+t{y}^2}}{N}_{G_{1j}}(t)\mathrm{d}t\hfill \\ \hfill & ={\int }_1^{t_j}\frac{y^2}{{}^{t^3+t{y}^2}}{N}_{G_{1j}}(t)\mathrm{d}t+{\int }_{t_j}^{\infty }\frac{y^2}{{}^{t^3+t{y}^2}}{N}_{G_{1j}}(t)\mathrm{d}t\hfill \\ \hfill & \ge (1-2{\mathit{\alpha }}_j){\int }_1^{\infty }\frac{y^2}{{}^{t^3+t{y}^2}}{N}_{{\mathit{\omega }}_{1j}}(t)\mathrm{d}t+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4}{\int }_1^{\infty }\frac{y^2}{{}^{t^3+t{y}^2}}\mathrm{d}t\hfill \\ \hfill & -\left(\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4},+,C_{1j}\right){\int }_1^{t_j}\frac{y^2}{{}^{t^3+t{y}^2}}dt\hfill \\ \hfill & =(1-2{\mathit{\alpha }}_j)ln|{\mathit{\omega }}_{1j}(\mathrm{i}y)|+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4}ln(1+y^2)MYAMP]+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j+2C_{1j}}{4}ln\left(\frac{1+y^2}{t_j^2+y^2}\right)+\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j+2C_{1j}}{4}ln{t}_j.\hfill \end{array}$$(3.19) This implies(3.20) $$\begin{array}{c}\hfill |G_j(\mathrm{i}y)|\ge C_{0j}|{\mathit{\omega }}_{1j}{(iy)|}^{1-2{\mathit{\alpha }}_j}{|y|}^{\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}},y\in \mathbb{R},j=\overline{1,j_0},\end{array}$$(3.20) where $C_{0j}$ is constant.

By virtue of the assumption of Theorem 3.1 together with (3.20(3.20) $$\begin{array}{c}\hfill |G_j(\mathrm{i}y)|\ge C_{0j}|{\mathit{\omega }}_{1j}{(iy)|}^{1-2{\mathit{\alpha }}_j}{|y|}^{\frac{10{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}},y\in \mathbb{R},j=\overline{1,j_0},\end{array}$$(3.20) ), we get(3.21) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{1j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{1j}{|y|}^{2+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{2Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.21) By virtue of (3.12(3.12) $$\begin{array}{cc}& |F_1(0,\mathit{\lambda })|=|F_1({\mathit{\alpha }}_0,\mathit{\lambda })|\hfill \\ \hfill & \le |F_1(1,\mathit{\lambda })|+|{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx|\hfill \\ \hfill & \le |F_1{(1,\mathit{\lambda })|+||Q||}_2{\left({\int }_{{\mathit{\alpha }}_0}^1,|,u_{1+}(x,\mathit{\lambda })u_{2+}{(x,\mathit{\lambda })|}^{2}dx\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le c_1{|\mathit{\rho }|}^4+c_2{|\mathit{\rho }|}^4{||Q||}_2\sqrt{1-{\mathit{\alpha }}_0}{e}^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\hfill \\ \hfill & =O\left({\mathit{\rho }}^4,e^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\right),\hfill \end{array}$$(3.12) ), (3.15(3.15) $$\begin{array}{c}\hfill K_1(\mathit{\lambda })=\frac{F_1({\mathit{\alpha }}_0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{1j}(\mathit{\lambda })}.\end{array}$$(3.15) ) and (3.21(3.21) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{1j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{1j}{|y|}^{2+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{2Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.21) ), for $|y|$ sufficiently large, we obtain(3.22) $$\begin{array}{c}\hfill |K_1(iy)|=\frac{F_1({\mathit{\alpha }}_0,iy)}{{\prod }_{j=1}^{j_0}{G}_{1j}(iy)}=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right).\end{array}$$(3.22) From Lemma 2.2 together with (3.22(3.22) $$\begin{array}{c}\hfill |K_1(iy)|=\frac{F_1({\mathit{\alpha }}_0,iy)}{{\prod }_{j=1}^{j_0}{G}_{1j}(iy)}=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right).\end{array}$$(3.22) ), we have$$\begin{array}{c}\hfill K_1(\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$Hence,(3.23) $$\begin{array}{c}\hfill F_1({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.23) By virtue of (3.6(3.6) $$\begin{array}{c}\hfill F_1(0,\mathit{\lambda })=F_1({\mathit{\alpha }}_0,\mathit{\lambda })=F_1(1,\mathit{\lambda })-{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx.\end{array}$$(3.6) ) and (3.23(3.23) $$\begin{array}{c}\hfill F_1({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.23) ), this yields$$\begin{array}{c}\hfill u_{1+}(0,\mathit{\lambda })u_{2+}^{\prime }(0,\mathit{\lambda })-u_{1+}^{\prime }(0,\mathit{\lambda })u_{2+}(0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$This implies$$\begin{array}{cc}& \frac{u_{1+}^{\prime }(0,\mathit{\lambda })}{(h_{21}\mathit{\lambda }-h_{31})u_{1+}(0,\mathit{\lambda })-(\mathit{\lambda }-h_{11})u_{1+}^{\prime }(0,\mathit{\lambda })}\hfill \\ \hfill & =\frac{u_{2+}^{\prime }(0,\mathit{\lambda })}{(h_{21}\mathit{\lambda }-h_{31})u_{2+}(0,\mathit{\lambda })-(\mathit{\lambda }-h_{11})u_{2+}^{\prime }(0,\mathit{\lambda })}.\hfill \end{array}$$Consequently,(3.24) $$\begin{array}{c}\hfill M_1(\mathit{\lambda })=M_2(\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.24) By virtue of Lemma 2.1 together with (3.24(3.24) $$\begin{array}{c}\hfill M_1(\mathit{\lambda })=M_2(\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.24) ), we obtain$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}\mathrm{and}{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$This completes the proof of Theorem 3.1.

Instead of partial information on the potential and parts of a finite number of spectra, one can use partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues to establish the following uniqueness theorem for (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ). The techniques used here are based on the methods in [20, 36].$\square$

Theorem 3.6

Let $\mathit{\sigma }(L_{kj})$ be as that defined above, $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(L_{1j})\bigcap \mathit{\sigma }(L_{2j})$, ${\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, for $j=\overline{1,j_0}$, and coefficients $h_{lj},l=\overline{1,3}$, of the boundary conditions be given a priori. If

(1)	$q_1=q_{2}on[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$.
(2)	${\sum }_{j=1}^{j_0}{\mathit{\alpha }}_j-{\mathit{\alpha }}_0=j_0-1$, ${\sum }_{j=1}^{j_0}{\mathit{\beta }}_j=5j_0-2$ and ${\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j>0$ for some ${\mathit{\alpha }}_j\in [0,1]$, ${\mathit{\beta }}_j\in {\mathbb{N}}_0$ and ${\mathit{\epsilon }}_j\ge 0$.
(3)	${\mathit{\kappa }}_{1n}={\mathit{\kappa }}_{2n},\forall {\mathit{\lambda }}_{1jn}\in S_j$.
(4)	For each $j=\overline{1,j_0}$, the inequality(3.25) $$\begin{array}{c}\hfill N_{S_j}\ge (1-{\mathit{\alpha }}_j)N_{\mathit{\sigma }(L_{1j})}+\frac{5{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{4},j=\overline{1,j_0},\end{array}$$(3.25) holds for all sufficiently large $t\in {\mathbb{R}}^{+}$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Clearly, Theorem 3.6 leads to the following Corollary 3.7, which is a generalization of Theorem 4.1 in [36].

Corollary 3.7

Let $\mathit{\sigma }(L_k)={\{{\mathit{\lambda }}_{kn}\}}_{n=0}^{\infty }(k=1,2)$ be the spectrum of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_l,H_{lk},d_{km})$, $l=\overline{1,3}$ and coefficients $h_l$ of the boundary condition be given a priori. Denote the subset $S={\{{\mathit{\lambda }}_{1n}\}}_{n\in {\mathrm{\Lambda }}_1}\subseteq \mathit{\sigma }(L_1)\bigcap \mathit{\sigma }(L_2)$. If

(1)	$q_1=q_2$ a.e. on $[0,\mathit{\alpha }]$ for some $\mathit{\alpha }\in [0,1)$.
(2)	${\mathit{\kappa }}_{1n}={\mathit{\kappa }}_{2n},\forall {\mathit{\lambda }}_{1n}\in S.$
(3)	The subset $S$ satisfies(3.26) $$\begin{array}{c}\hfill N_S\ge (1-\mathit{\alpha })N_{\mathit{\sigma }(L_1)}+\frac{5\mathit{\alpha }-3+2\mathit{\epsilon }}{4}\end{array}$$(3.26) for all sufficiently large $t\in {\mathbb{R}}^{+}$ and $\mathit{\epsilon }>0$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Let $\mathit{\alpha }=\frac{1}{2}$ and $\mathit{\epsilon }=\frac{1}{4}$, we have the following corollary.

Corollary 3.8

Let $\mathit{\sigma }(L)={\{{\mathit{\lambda }}_n\}}_{n=0}^{\infty }$ be the spectrum of (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) )–(1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) and coefficients $h_l,l=\overline{1,3}$, of the boundary condition be given a priori. If $q$ is prescribed on the interval $[0,\frac{1}{2}]$, then the even spectral data ${\{({\mathit{\lambda }}_{2n},{\mathit{\kappa }}_{2n})\}}_{j=0}^{\infty }$, or the odd spectral data ${\{({\mathit{\lambda }}_{2n-1},{\mathit{\kappa }}_{2n-1})\}}_{n=1}^{\infty }$ is sufficient to determine the potential $q$ on the whole interval $[0,1]$ and coefficients $H_l,l=\overline{1,3}$, of the boundary condition and coefficients $a_m,b_m$ and $d_m,m=\overline{1,m_0},m_0\in \mathbb{N}$, of the jump conditions.

In the rest parts of this section, we prove Theorem 3.6.

Proof of Theorem 3.6

We use the same symbols as these in Theorem 3.1. Let $v_2(x,\mathit{\lambda })$ be the solution of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) for $q_2$.

Denote(3.27) $$\begin{array}{c}\hfill F_v(x,\mathit{\lambda }):=[u_{1+}-u_{2+},v_2](x,\mathit{\lambda })\end{array}$$(3.27) By virtue of (3.27(3.27) $$\begin{array}{c}\hfill F_v(x,\mathit{\lambda }):=[u_{1+}-u_{2+},v_2](x,\mathit{\lambda })\end{array}$$(3.27) ), for each $j=\overline{1,j_0}$, we have(3.28) $$\begin{array}{cc}& F_v(0,\mathit{\lambda })\hfill \\ \hfill & =(u_{1+}^{\prime }(1,\mathit{\lambda })-u_{2+}^{\prime }(1,\mathit{\lambda }))v_2(1,\mathit{\lambda })-{\int }_0^1(q_1-q_2)u_{1+}{v}_{2}dx\hfill \\ \hfill & =[u_{1+}-u_{2+},v_2](0,\mathit{\lambda })\hfill \\ \hfill & =\frac{1}{\mathit{\lambda }-h_{1j}}\left|\begin{array}{cc}{u}_{1+}(0,\mathit{\lambda })-u_{2+}(0,\mathit{\lambda })& v_2(0,\mathit{\lambda })\\ {\mathit{\omega }}_{2j}(\mathit{\lambda })-{\mathit{\omega }}_{1j}(\mathit{\lambda })& (\mathit{\lambda }-h_{1j})v_2^{\prime }(0,\mathit{\lambda })+(h_{3j}-h_{2j}\mathit{\lambda })v_2(0,\mathit{\lambda })\end{array}\right|.\hfill \end{array}$$(3.28) Without loss of generality, we assume that ${\mathit{\lambda }}_{1jn}-h_{1j}\ne 0$ for all ${\mathit{\lambda }}_{1jn}\in S_j$. This implies(3.29) $$\begin{array}{c}\hfill {\mathit{\omega }}_{2j}({\mathit{\lambda }}_{1jn})-{\mathit{\omega }}_{1j}({\mathit{\lambda }}_{1jn})=0\mathrm{and}{u}_{1+}(0,{\mathit{\lambda }}_{1jn})=u_{2+}(0,{\mathit{\lambda }}_{1jn}),\forall {\mathit{\lambda }}_{1jn}\in S_j.\end{array}$$(3.29) From (3.28(3.28) $$\begin{array}{cc}& F_v(0,\mathit{\lambda })\hfill \\ \hfill & =(u_{1+}^{\prime }(1,\mathit{\lambda })-u_{2+}^{\prime }(1,\mathit{\lambda }))v_2(1,\mathit{\lambda })-{\int }_0^1(q_1-q_2)u_{1+}{v}_{2}dx\hfill \\ \hfill & =[u_{1+}-u_{2+},v_2](0,\mathit{\lambda })\hfill \\ \hfill & =\frac{1}{\mathit{\lambda }-h_{1j}}\left|\begin{array}{cc}{u}_{1+}(0,\mathit{\lambda })-u_{2+}(0,\mathit{\lambda })& v_2(0,\mathit{\lambda })\\ {\mathit{\omega }}_{2j}(\mathit{\lambda })-{\mathit{\omega }}_{1j}(\mathit{\lambda })& (\mathit{\lambda }-h_{1j})v_2^{\prime }(0,\mathit{\lambda })+(h_{3j}-h_{2j}\mathit{\lambda })v_2(0,\mathit{\lambda })\end{array}\right|.\hfill \end{array}$$(3.28) ) and (3.29(3.29) $$\begin{array}{c}\hfill {\mathit{\omega }}_{2j}({\mathit{\lambda }}_{1jn})-{\mathit{\omega }}_{1j}({\mathit{\lambda }}_{1jn})=0\mathrm{and}{u}_{1+}(0,{\mathit{\lambda }}_{1jn})=u_{2+}(0,{\mathit{\lambda }}_{1jn}),\forall {\mathit{\lambda }}_{1jn}\in S_j.\end{array}$$(3.29) ), we get(3.30) $$\begin{array}{c}\hfill F_v(0,{\mathit{\lambda }}_{1jn})=0,\forall {\mathit{\lambda }}_{1jn}\in S_j.\end{array}$$(3.30) By virtue of the assumption $q_1=q_2$ on $[0,{\mathit{\alpha }}_0]$ of Theorem 3.6 together with the Green’s formula, we obtain(3.31) $$\begin{array}{cc}\hfill F_v(0,\mathit{\lambda })& =(u_{1+}^{\prime }(1,\mathit{\lambda })-u_{2+}^{\prime }(1,\mathit{\lambda }))v_2(1,\mathit{\lambda })-{\int }_0^1(q_1-q_2)u_{1+}{v}_{2}dx\hfill \\ \hfill & =(u_{1+}^{\prime }(1,\mathit{\lambda })-u_{2+}^{\prime }(1,\mathit{\lambda }))v_2(1,\mathit{\lambda })-{\int }_{{\mathit{\alpha }}_0}^1(q_1-q_2)u_{1+}{v}_{2}dx\hfill \\ \hfill & =[u_{1+}-u_{2+},v_2]({\mathit{\alpha }}_0,\mathit{\lambda }).\hfill \end{array}$$(3.31) Next, we will prove that(3.32) $$\begin{array}{c}\hfill u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })=u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })\mathrm{and}{u}_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.32) First, we show that $u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })=u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}$, holds.

Let $v_2(x,\mathit{\lambda })=:v_D(x,\mathit{\lambda })$ be the solution of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) for $q_2$ with the initial conditions $v_D({\mathit{\alpha }}_0,\mathit{\lambda })=0$ and $v_D^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=1$. Consequently,(3.33) $$\begin{array}{c}\hfill F_{v_D}(0,\mathit{\lambda })=u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })-u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda }).\end{array}$$(3.33) Define the entire functions $G_{S_j}(\mathit{\lambda })$ and $H_D(\mathit{\lambda })$ by(3.34) $$\begin{array}{c}\hfill G_{S_j}(\mathit{\lambda })=\prod _{{\mathit{\lambda }}_{1jn}\in S_j}\left(1,-,\frac{\mathit{\lambda }}{{\mathit{\lambda }}_{1jn}}\right)\mathrm{and}{H}_D(\mathit{\lambda })=\frac{F_{v_D}(0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{S_j}(\mathit{\lambda })}.\end{array}$$(3.34) By virtue of (2.10(2.10) $$\begin{array}{cc}\hfill \left|u_{+}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }(1-x)}\right),0\le x\le 1,\mathit{\nu }=0,1.\hfill \end{array}$$(2.10) ), for sufficiently large $y\in \mathbb{R}$, this yields(3.35) $$\begin{array}{c}\hfill |u_{1+}({\mathit{\alpha }}_0,iy)-u_{2+}({\mathit{\alpha }}_0,iy)|=O\left(|y|,e^{(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\right).\end{array}$$(3.35) Similar to (3.21(3.21) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{1j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{1j}{|y|}^{2+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{2Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.21) ), by calculation, we have(3.36) $$\begin{array}{c}\hfill |G_{S_j}(\mathrm{i}y)|\ge C_{S_j}|{\mathit{\omega }}_{1j}{(iy)|}^{1-{\mathit{\alpha }}_j}{|y|}^{\frac{5{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}}\ge C_{S_j}{|y|}^{\frac{5{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}}{e}^{\sqrt{i}(1-{\mathit{\alpha }}_j){|y|}^{\frac{1}{2}}}\end{array}$$(3.36) for each $j=\overline{1,j_0}$, where $C_{S_j}$ is a positive constant.

From (3.36(3.36) $$\begin{array}{c}\hfill |G_{S_j}(\mathrm{i}y)|\ge C_{S_j}|{\mathit{\omega }}_{1j}{(iy)|}^{1-{\mathit{\alpha }}_j}{|y|}^{\frac{5{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}}\ge C_{S_j}{|y|}^{\frac{5{\mathit{\alpha }}_j-{\mathit{\beta }}_j+2{\mathit{\epsilon }}_j}{2}}{e}^{\sqrt{i}(1-{\mathit{\alpha }}_j){|y|}^{\frac{1}{2}}}\end{array}$$(3.36) ) together with the assumption of Theorem 3.6, we get(3.37) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{S_j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{S_j}{|y|}^{1+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.37) By virtue of (3.34(3.34) $$\begin{array}{c}\hfill G_{S_j}(\mathit{\lambda })=\prod _{{\mathit{\lambda }}_{1jn}\in S_j}\left(1,-,\frac{\mathit{\lambda }}{{\mathit{\lambda }}_{1jn}}\right)\mathrm{and}{H}_D(\mathit{\lambda })=\frac{F_{v_D}(0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{S_j}(\mathit{\lambda })}.\end{array}$$(3.34) ), (3.35(3.35) $$\begin{array}{c}\hfill |u_{1+}({\mathit{\alpha }}_0,iy)-u_{2+}({\mathit{\alpha }}_0,iy)|=O\left(|y|,e^{(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\right).\end{array}$$(3.35) ) and (3.37(3.37) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{S_j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{S_j}{|y|}^{1+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.37) ), this yields(3.38) $$\begin{array}{c}\hfill |H_D(iy)|=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right),y\in \mathbb{R}.\end{array}$$(3.38) By Lemma 2.2 together with (3.38(3.38) $$\begin{array}{c}\hfill |H_D(iy)|=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right),y\in \mathbb{R}.\end{array}$$(3.38) ), we obtain(3.39) $$\begin{array}{c}\hfill H_D(\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.39) Therefore,(3.40) $$\begin{array}{c}\hfill u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })-u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.40) Second, we prove $u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}$.

Let $v_2(x,\mathit{\lambda })=:v_N(x,\mathit{\lambda })$ be the solution of Equation (1.1(1.1) $$\begin{array}{c}\hfill Lu=-u^{″}+q(x)u=\mathit{\lambda }u,x\in (0,1)\end{array}$$(1.1) ) for $q_2$ with the initial conditions $v_N({\mathit{\alpha }}_0,\mathit{\lambda })=1$ and $v_N^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=0$. Therefore,(3.41) $$\begin{array}{c}\hfill F_{v_N}(0,\mathit{\lambda })=u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda }).\end{array}$$(3.41) Define the entire function $H_N(\mathit{\lambda })$ by(3.42) $$\begin{array}{c}\hfill H_N(\mathit{\lambda })=\frac{F_{v_N}(0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{S_j}(\mathit{\lambda })}.\end{array}$$(3.42) Since $u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })=u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })$, we get(3.43) $$\begin{array}{cc}\hfill F_1({\mathit{\alpha }}_0,\mathit{\lambda })& =u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })\hfill \\ \hfill & =(u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda }))u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda }).\hfill \end{array}$$(3.43) From (2.10(2.10) $$\begin{array}{cc}\hfill \left|u_{+}^{(\mathit{\nu })},(x,\mathit{\lambda })\right|& =O\left({|\mathit{\rho }|}^{\mathit{\nu }+2},e^{\mathit{\tau }(1-x)}\right),0\le x\le 1,\mathit{\nu }=0,1.\hfill \end{array}$$(2.10) ), (3.12(3.12) $$\begin{array}{cc}& |F_1(0,\mathit{\lambda })|=|F_1({\mathit{\alpha }}_0,\mathit{\lambda })|\hfill \\ \hfill & \le |F_1(1,\mathit{\lambda })|+|{\int }_{{\mathit{\alpha }}_0}^{1}Q(x)u_{1+}(x,\mathit{\lambda })u_{2+}(x,\mathit{\lambda })dx|\hfill \\ \hfill & \le |F_1{(1,\mathit{\lambda })|+||Q||}_2{\left({\int }_{{\mathit{\alpha }}_0}^1,|,u_{1+}(x,\mathit{\lambda })u_{2+}{(x,\mathit{\lambda })|}^{2}dx\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le c_1{|\mathit{\rho }|}^4+c_2{|\mathit{\rho }|}^4{||Q||}_2\sqrt{1-{\mathit{\alpha }}_0}{e}^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\hfill \\ \hfill & =O\left({\mathit{\rho }}^4,e^{2\mathit{\tau }(1-{\mathit{\alpha }}_0)}\right),\hfill \end{array}$$(3.12) ) and (3.43(3.43) $$\begin{array}{cc}\hfill F_1({\mathit{\alpha }}_0,\mathit{\lambda })& =u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })\hfill \\ \hfill & =(u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda }))u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda }).\hfill \end{array}$$(3.43) ), we obtain the following asymptotic formula(3.44) $$\begin{array}{c}\hfill |u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })|=O({\mathit{\rho }}^2{e}^{\mathit{\tau }(1-{\mathit{\alpha }}_0)}).\end{array}$$(3.44) This implies(3.45) $$\begin{array}{c}\hfill |F_{v_N}(0,iy)|=|u_{2+}^{\prime }({\mathit{\alpha }}_0,iy)-u_{1+}^{\prime }({\mathit{\alpha }}_0,iy)|=O(|y|e^{(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}})\end{array}$$(3.45) for sufficiently large $y\in \mathbb{R}$.

From (3.37(3.37) $$\begin{array}{c}\hfill \left|\prod _{j=1}^{j_0},G_{S_j},(iy)\right|\ge \prod _{j=1}^{j_0}{C}_{S_j}{|y|}^{1+{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}{e}^{Im\sqrt{i}(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}}\end{array}$$(3.37) ), (3.42(3.42) $$\begin{array}{c}\hfill H_N(\mathit{\lambda })=\frac{F_{v_N}(0,\mathit{\lambda })}{{\prod }_{j=1}^{j_0}{G}_{S_j}(\mathit{\lambda })}.\end{array}$$(3.42) ) and (3.45(3.45) $$\begin{array}{c}\hfill |F_{v_N}(0,iy)|=|u_{2+}^{\prime }({\mathit{\alpha }}_0,iy)-u_{1+}^{\prime }({\mathit{\alpha }}_0,iy)|=O(|y|e^{(1-{\mathit{\alpha }}_0){|y|}^{\frac{1}{2}}})\end{array}$$(3.45) ), we have(3.46) $$\begin{array}{c}\hfill |H_N(iy)|=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right),y\in \mathbb{R}.\end{array}$$(3.46) By Lemma 2.2 together with (3.46(3.46) $$\begin{array}{c}\hfill |H_N(iy)|=O\left(\frac{1}{{|y|}^{{\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j}}\right),y\in \mathbb{R}.\end{array}$$(3.46) ), we get(3.47) $$\begin{array}{c}\hfill H_N(\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.47) Therefore,(3.48) $$\begin{array}{c}\hfill u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.48) From (3.40(3.40) $$\begin{array}{c}\hfill u_{1+}({\mathit{\alpha }}_0,\mathit{\lambda })-u_{2+}({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.40) ) and (3.48(3.48) $$\begin{array}{c}\hfill u_{2+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })-u_{1+}^{\prime }({\mathit{\alpha }}_0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.48) ), we get(3.49) $$\begin{array}{c}\hfill [u_{1+},u_{2+}]({\mathit{\alpha }}_0,\mathit{\lambda })=0.\end{array}$$(3.49) In virtue of $q_1=q_2$ on $[0,{\mathit{\alpha }}_0]$ and (3.49(3.49) $$\begin{array}{c}\hfill [u_{1+},u_{2+}]({\mathit{\alpha }}_0,\mathit{\lambda })=0.\end{array}$$(3.49) ), this yields(3.50) $$\begin{array}{c}\hfill u_{1+}(0,\mathit{\lambda })u_{2+}^{\prime }(0,\mathit{\lambda })-u_{1+}^{\prime }(0,\mathit{\lambda })u_{2+}(0,\mathit{\lambda })=0,\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.50) Consequently,(3.51) $$\begin{array}{c}\hfill M_1(\mathit{\lambda })=M_2(\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.51) From Lemma 2.1 together with (3.51(3.51) $$\begin{array}{c}\hfill M_1(\mathit{\lambda })=M_2(\mathit{\lambda }),\forall \mathit{\lambda }\in \mathbb{C}.\end{array}$$(3.51) ), we obtain$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.\mathrm{on}[0,1],H_{l1}=H_{l2},l=\overline{1,3},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}\mathrm{and}{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$By now, the proof of Theorem 3.6 is completed.$\square$

4 Uniqueness theorems

In this section, we present several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuities from partial information on the potential and parts of a finite number of spectra or from partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues.

Denote the Sturm–Liouville operators $l_{kj}:=l(q_k,h_{0kj},h_{1k},d_{km})$, $k=1,2,j=\overline{1,j_0},j_0\in \mathbb{N}$, of the form(4.1) $$\begin{array}{c}\hfill l{u}_k=-u_k^{″}+q_k(x)u_k=\mathit{\lambda }{u}_k,x\in (0,1)\end{array}$$(4.1) with boundary conditions(4.2) $$\begin{array}{cc}\hfill U_{0kj}(u_k)& :=u_k^{\prime }(0,\mathit{\lambda })-h_{0kj}{u}_k(0,\mathit{\lambda })=0,\hfill \end{array}$$(4.2) (4.3) $$\begin{array}{cc}\hfill U_{1k}(u_k)& :=u_k^{\prime }(1,\mathit{\lambda })-h_{1k}{u}_k(1,\mathit{\lambda })=0,\hfill \end{array}$$(4.3) and interior discontinuities (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ), where $h_{0kj},h_{1k}\in \mathbb{R}$, $h_{0k{j}_1}\ne h_{0k{j}_2},j_1\ne j_2\in \{1,2,\cdots ,j_0\}$, and $q_k(x)$ are real-valued functions and $q_k\in L^2[0,1]$.

Let $\mathit{\sigma }(l_{kj})={\{{\mathit{\lambda }}_{kjn}\}}_{n=0}^{\infty },k=1,2,j=\overline{1,j_0}$, be the spectrum of (4.1(4.1) $$\begin{array}{c}\hfill l{u}_k=-u_k^{″}+q_k(x)u_k=\mathit{\lambda }{u}_k,x\in (0,1)\end{array}$$(4.1) )–(4.3(4.3) $$\begin{array}{cc}\hfill U_{1k}(u_k)& :=u_k^{\prime }(1,\mathit{\lambda })-h_{1k}{u}_k(1,\mathit{\lambda })=0,\hfill \end{array}$$(4.3) ) and (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_{0kj},h_{1k},d_{km})$. Since $h_{0k{j}_1}\ne h_{0k{j}_2},j_1\ne j_2\in \{1,2,\cdots ,j_0\}$, it is easy to prove that $\mathit{\sigma }(l_{k{j}_1})\cap \mathit{\sigma }(l_{k{j}_2})=\mathrm{\varnothing }$ for fixed $k=1$ or $2$.

We obtain the following uniqueness theorem for the discontinuities Sturm–Liouville operator.

Theorem 4.1

Let $\mathit{\sigma }(l_{kj})$ be as that defined above, $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(l_{1j})\bigcap \mathit{\sigma }(l_{2j})$, ${\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, for each $j=\overline{1,j_0}$ and coefficients $h_{01j}=h_{02j}=h_{0j}$, of the boundary condition be given a priori. If

(1)	$q_1=q_{2}on[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$.
(2)	$2\left({\sum }_{j=1}^{j_0},{\mathit{\alpha }}_j,-,{\mathit{\alpha }}_0\right)=j_0-2$, ${\sum }_{j=1}^{j_0}{\mathit{\beta }}_j=j_0$ and ${\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j>0$ for some ${\mathit{\alpha }}_j\in [0,\frac{1}{2}]$, ${\mathit{\beta }}_j\in {\mathbb{N}}_0$ and ${\mathit{\epsilon }}_j\ge 0$.
(3)	For each $j=\overline{1,j_0},$ the inequality$$\begin{array}{c}\hfill N_{S_j}\ge (1-2{\mathit{\alpha }}_j)N_{\mathit{\sigma }(L_{1j})}+\frac{2{\mathit{\alpha }}_j-{\mathit{\beta }}_j+{\mathit{\epsilon }}_j}{4}\end{array}$$ holds for all sufficiently large $t\in {\mathbb{R}}^{+}$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],h_{11}=h_{12},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Proof

Applying the same arguments as that in the proof of Theorem 3.1, we can prove Theorem 4.1 and omit the proof of Theorem 4.1 here.$\square$

Remark 1

   

(1)	Let $j_0=2,{\mathit{\alpha }}_0={\mathit{\alpha }}_j=0,{\mathit{\beta }}_1+{\mathit{\beta }}_2=2,{\mathit{\epsilon }}_j={\mathit{\beta }}_j(j=1,2),$$a_{km}=1,b_{km}=0,m=\overline{1,m_0}$, then Theorem 4.1 leads to the Borg theorem.[25]
(2)	If $j_0=1,{\mathit{\alpha }}_0=\frac{1+\mathit{\alpha }}{2},{\mathit{\alpha }}_1=\frac{\mathit{\alpha }}{2},{\mathit{\beta }}_1=1,$${\mathit{\epsilon }}_1=1-\mathit{\alpha }$, $a_{km}=1,b_{km}=0,m=\overline{1,m_0}$, then Theorem 4.1 leads to the Gesztesy–Simon theorem.[31]
(3)	Let $j_0=1,{\mathit{\alpha }}_0=\frac{1}{2},{\mathit{\alpha }}_1=0,{\mathit{\beta }}_1={\mathit{\epsilon }}_1=1,$$a_{km}=1,b_{km}=0,m=\overline{1,m_0}$, then Theorem 4.1 leads to the Hochstadt–Lieberman theorem.[15]
(4)	Let $j_0=3,{\mathit{\alpha }}_0=0,{\mathit{\alpha }}_j=\frac{1}{6},{\mathit{\beta }}_j=1,{\mathit{\epsilon }}_j=\frac{4}{3},a_{km}=1,b_{km}=0,m=\overline{1,m_0}$, then Theorem 4.1 leads to Theorem 4.1 in [30].

Remark 2

If $0\le {\mathit{\alpha }}_j<\frac{1}{2}$ for each $j=\overline{1,j_0}$, then $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(l_{1j})\bigcap \mathit{\sigma }(l_{2j})$, ${\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, for each $j$ in Theorem 4.1 is an infinite set. Using the condition ‘coefficient $h_{01j_1}=h_{02j_1}=h_{0j_1},j_1\in \{1,2,\dots ,j_0\}$, of the boundary condition is given a priori’ instead of ‘coefficients $h_{01j}=h_{02j}=h_{0j}$, for each $j=\overline{1,j_0}$ of the boundary conditions are given a priori’ in Theorem 4.1 and other conditions remain unchanged, we can also prove $h_{01j}=h_{02j}$ for each $j\ne j_1.$ Therefore, Theorem 4.1 still holds.

Using partial information on the potential and a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues as the spectral data, we obtain the following uniqueness theorem for (4.1(4.1) $$\begin{array}{c}\hfill l{u}_k=-u_k^{″}+q_k(x)u_k=\mathit{\lambda }{u}_k,x\in (0,1)\end{array}$$(4.1) )–(4.3(4.3) $$\begin{array}{cc}\hfill U_{1k}(u_k)& :=u_k^{\prime }(1,\mathit{\lambda })-h_{1k}{u}_k(1,\mathit{\lambda })=0,\hfill \end{array}$$(4.3) ) and (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ).

Theorem 4.2

Let $\mathit{\sigma }(l_{kj})$ be as that defined above, $S_j={\{{\mathit{\lambda }}_{1jn}\}}_{n\in {\mathrm{\Lambda }}_j}\subseteq \mathit{\sigma }(l_{1j})\bigcap \mathit{\sigma }(l_{2j})$, ${\mathrm{\Lambda }}_j\subseteq {\mathbb{N}}_0$, for each $j=\overline{1,j_0}$ and coefficients $h_{01j}=h_{02j}=h_{0j}$, of the boundary conditions be given a priori. Suppose that the following conditions hold:

(1)	$q_1=q_2$ a.e. on $[0,{\mathit{\alpha }}_0]$ for some ${\mathit{\alpha }}_0\in [0,1)$.
(2)	${\sum }_{j=1}^{j_0}{\mathit{\alpha }}_j-{\mathit{\alpha }}_0=j_0-1$, ${\sum }_{j=1}^{j_0}{\mathit{\beta }}_j=j_0$ and ${\sum }_{j=1}^{j_0}{\mathit{\epsilon }}_j>0$ for some ${\mathit{\alpha }}_j\in [0,1]$, ${\mathit{\beta }}_j\in {\mathbb{N}}_0$ and ${\mathit{\epsilon }}_j\ge 0$.
(3)	${\mathit{\kappa }}_{1jn}={\mathit{\kappa }}_{2jn}forall{\mathit{\lambda }}_{1jn}\in S_j.$
(4)	For each $j=\overline{1,j_0}$ the inequality$$\begin{array}{c}\hfill N_{S_j}\ge (1-{\mathit{\alpha }}_j)N_{\mathit{\sigma }(L_{1j})}+\frac{{\mathit{\alpha }}_j-{\mathit{\beta }}_j+{\mathit{\epsilon }}_j}{4}\end{array}$$ holds for all sufficiently large $t\in {\mathbb{R}}^{+}$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],h_{11}=h_{12},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Proof

The proof is analogous to that of Theorem 3.6 in Section 3.

Remark 3

Clearly, if $j_0=1,a_{km}=1,b_{km}=0,m=\overline{1,m_0}$, Theorem 4.2 leads to Theorem 4.1 in [36].

Let $j_0=1$, ${\mathit{\alpha }}_0={\mathit{\alpha }}_1={\mathit{\epsilon }}_1=\frac{1}{2},{\mathit{\beta }}_1=1$, we then have the following corollary.

Corollary 4.3

Let $\mathit{\sigma }(l_k)={\{{\mathit{\lambda }}_{k,n}\}}_{n=0}^{\infty }$ be the spectrum of (4.1(4.1) $$\begin{array}{c}\hfill l{u}_k=-u_k^{″}+q_k(x)u_k=\mathit{\lambda }{u}_k,x\in (0,1)\end{array}$$(4.1) )–(4.3(4.3) $$\begin{array}{cc}\hfill U_{1k}(u_k)& :=u_k^{\prime }(1,\mathit{\lambda })-h_{1k}{u}_k(1,\mathit{\lambda })=0,\hfill \end{array}$$(4.3) ) and (1.4(1.4) $$\begin{array}{cc}& V_{1m}(u):=u(d_m+,\mathit{\lambda })-a_{m}u(d_m-,\mathit{\lambda })=0,\hfill \\ \hfill & V_{2m}(u):=u^{\prime }(d_m+,\mathit{\lambda })-a_m^{-1}{u}^{\prime }(d_m-,\mathit{\lambda })-b_{m}u(d_m-,\mathit{\lambda })=0,\hfill \end{array}$$(1.4) ) for $(q_k,h_{0k1},h_{1k},d_{km})$ and coefficient $h_{011}=h_{021}=h_0$ of the boundary condition be given a priori. Suppose that the following conditions hold.

(1)	$q_1=q_2$ a.e. on the interval $[0,\frac{1}{2}]$.
(2)	The even spectral data ${\{({\mathit{\lambda }}_{1,2n},{\mathit{\kappa }}_{1,2n})\}}_{n=0}^{\infty }={\{({\mathit{\lambda }}_{2,2n},{\mathit{\kappa }}_{2,2n})\}}_{n=0}^{\infty }$, or the odd spectral data ${\{({\mathit{\lambda }}_{1,2n-1},{\mathit{\kappa }}_{1,2n-1})\}}_{n=1}^{\infty }={\{({\mathit{\lambda }}_{2,2n-1},{\mathit{\kappa }}_{2,2n-1})\}}_{n=1}^{\infty }$.

Then$$\begin{array}{c}\hfill \begin{array}{cc}& q_1=q_{2}a.e.on[0,1],h_{11}=h_{12},\hfill \\ \hfill & a_{1m}=a_{2m},b_{1m}=b_{2m}and{d}_{1m}=d_{2m},m=\overline{1,m_0}.\hfill \end{array}\end{array}$$

Finally, we present an example for Theorem 4.1 for the case ${\mathit{\alpha }}_0=0,d_1=\frac{1}{2}$, which is given in data [14, 20]. Denote ${\mathit{\gamma }}_n={\int }_0^1{\mathit{\phi }}^2(x,{\mathit{\lambda }}_n)dx$, where $\mathit{\phi }(x,{\mathit{\lambda }}_n)$ is the eigenfunction of the eigenvalue ${\mathit{\lambda }}_n$. It is well known that the spectral data ${\{({\mathit{\lambda }}_n,{\mathit{\kappa }}_n)\}}_{n=0}^{\infty }$ is equivalent to the spectral ${\{({\mathit{\lambda }}_n,{\mathit{\gamma }}_n)\}}_{n=0}^{\infty }$ in the inverse spectral problem. Therefore, we reconstruct the potential $q$ and coefficients by the spectral data ${\{({\mathit{\lambda }}_n,{\mathit{\gamma }}_n)\}}_{n=0}^{\infty }$. Since some symbols are undefined in this paper, the reader might check out these in [14, 20].

Example

[14, 20] Take $\tilde{l}$ such that $\tilde{q}(x)=0$, ${\tilde{h}}_0=0$, ${\tilde{h}}_1=0$ and ${\tilde{b}}_1=0$. Let ${\{({\tilde{\mathit{\lambda }}}_n,{\tilde{\mathit{\gamma }}}_n)\}}_{n=0}^{\infty }$ be the spectral data of $\tilde{l}$. Clearly$$\begin{array}{c}\hfill {\tilde{\mathit{\lambda }}}_0=0,{\tilde{\mathit{\gamma }}}_0=\frac{1}{2}(1+a_1),{\tilde{\mathit{\phi }}}_{00}(x)=1,0<x<\frac{1}{2},{\tilde{\mathit{\phi }}}_{00}(x)=a_1,\frac{1}{2}<x<1.\end{array}$$Let ${\mathit{\lambda }}_n={\tilde{\mathit{\lambda }}}_n,n\ge 0,$ and ${\mathit{\gamma }}_n={\tilde{\mathit{\gamma }}}_n,n\ge 1$, and ${\mathit{\gamma }}_0\ne {\tilde{\mathit{\gamma }}}_0$ be an arbitrary positive number. Denote $A=\frac{1}{{\mathit{\gamma }}_0}-\frac{1}{{\tilde{\mathit{\gamma }}}_0}$. By virtue of (73) and (77) in [14, 20], this yields$$\begin{array}{c}\hfill {\tilde{\mathit{\phi }}}_{00}(x)={\mathit{\phi }}_{00}(x)\left(1,+,A,{\int }_0^x,{\tilde{\mathit{\phi }}}_{00}^2,(t),d,t\right),{\mathit{\epsilon }}_0(x)=A{\tilde{\mathit{\phi }}}_{00}(x){\mathit{\phi }}_{00}(x).\end{array}$$Therefore$$\begin{array}{cc}\hfill {\mathit{\phi }}_{00}(x)& =\left\{\begin{array}{cc}{(1+Ax)}^{-1},\hfill & 0<x<\frac{1}{2},\hfill \\ a_1{(B+A{a}_1^{2}x)}^{-1},\hfill & \frac{1}{2}<x<1,\hfill \end{array}\right.\hfill \\ \hfill {\mathit{\epsilon }}_0(x)& =\left\{\begin{array}{cc}A{(1+Ax)}^{-1},\hfill & 0<x<\frac{1}{2},\hfill \\ A{a}_1^2{(B+A{a}_1^{2}x)}^{-1},\hfill & \frac{1}{2}<x<1,\hfill \end{array}\right.\hfill \end{array}$$where $B=1+\frac{1}{2}A(1-a_1^2)$. Using (78)–(80) in [14, 20], we calculate$$\begin{array}{cc}\hfill q(x)& =\left\{\begin{array}{cc}2A^2{(1+Ax)}^{-2},\hfill & 0<x<\frac{1}{2},\hfill \\ 2A^2{a}_1^4{(B+A{a}_1^{2}x)}^{-2},\hfill & \frac{1}{2}<x<1,\hfill \end{array}\right.\hfill \\ \hfill h_0& =-A,h_1=A{a}_1^2{(B+A{a}_1^2)}^{-1}\text{and}{b}_1=(a_1^{-1}-a_1^3)A{(1+A{a}_1)}^{-1}.\hfill \end{array}$$

Acknowledgements

The author would like to express his gratitude to anonymous referees, editors and Professor D. Lesnic, Department of Applied Mathematics, University of Leeds, UK., for careful examination and valuable suggestions.