1 Introduction

Inverse problems of spectral analysis recover operators by their spectral data. Fundamental and vast studies about the classical Sturm-Liouville, Dirac operators, Schrödinger equation, and hyperbolic equations are well studied (see [1–9] and references therein).

Studies of eigenvalue dependence appearing not only in the differential equation but also in the boundary conditions have increased in recent years (see [10–18] and corresponding bibliography). Moreover, boundary conditions which depend linearly and nonlinearly on the spectral parameter are considered in [10, 18–23] and [24–30], respectively. Furthermore, boundary value problems with transmission conditions are also increasingly studied. These types of studies introduce qualitative changes in the exploration. Direct and inverse problems for Sturm-Liouville and Dirac operators with transmission conditions are investigated in some papers (see [7, 31–34] and the corresponding bibliography). Then differential equations with the spectral parameter and transmission conditions arise in heat, mechanics, mass transfer problems, in diffraction problems, and in various physical transfer problems (see [20, 31, 35–42] and corresponding bibliography).

More recently, some boundary value problems with eigenparameter in boundary and transmission conditions were extended to the case of two, more than two or a finite number of transmissions in [43–47] and the references therein.

The present paper deals with the discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions. The aim of the present paper is to obtain the asymptotic formulas of the eigenvalues and eigenfunctions, to construct the Green’s function and the resolvent operator, and to prove some uniqueness theorems. Especially, some parameters of the considered problem can be determined by the Weyl function and some spectral data.

We consider a discontinuous boundary value problem L with function \(\rho (x)\);

$$ ly:=\rho(x)By^{\prime}(x)+\Omega(x)y(x)=\lambda y(x),\quad x\in[ a,\xi _{1} ) \cup( \xi_{1},\xi_{2} ) \cup( \xi_{2},b ] =\Lambda, $$
(1)

where

$$\rho(x)= \textstyle\begin{cases} \rho_{1}^{-1}, &a\leq x< \xi_{1},\\ \rho_{2}^{-1},&\xi_{1}< x< \xi_{2},\\ \rho_{3}^{-1}, &\xi_{2}< x\leq b, \end{cases} $$

and \(\rho_{1}\), \(\rho_{2}\), and \(\rho_{3}\) are given positive real numbers;

$$\begin{aligned} &\Omega(x)= \begin{pmatrix} p ( x ) & q(x)\\ q(x) & r ( x ) \end{pmatrix} ,\quad p ( x ) ,q(x),r ( x ) \in L_{2} [ \Lambda,\mathbb{R} ] ; \\ & B= \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} ,\qquad y ( x ) = \begin{pmatrix} y_{1} ( x ) \\ y_{2} ( x ) \end{pmatrix}, \end{aligned}$$

\(\lambda\in \mathbb{C} \) is a complex spectral parameter; we have boundary conditions at the endpoints,

$$\begin{aligned} &l_{1}y:=\lambda \bigl( \alpha_{1}^{\prime}y_{1}(a)- \alpha_{2}^{\prime}y_{2}(a) \bigr) - \bigl( \alpha_{1}y_{1}(a)-\alpha_{2}y_{2}(a) \bigr) =0, \end{aligned}$$
(2)
$$\begin{aligned} & l_{2}y:=\lambda \bigl( \gamma_{1}^{\prime}y_{1}(b)- \gamma_{2}^{\prime}y_{2}(b) \bigr) + \bigl( \gamma_{1}y_{1}(b)-\gamma_{2}y_{2}(b) \bigr) =0, \end{aligned}$$
(3)

with transmission conditions at the two points \(x=\xi_{1}\), \(x=\xi_{2}\),

$$\begin{aligned} &l_{3}y :=y_{1}(\xi_{1}+0)- \alpha_{3}y_{1}(\xi_{1}-0)=0, \end{aligned}$$
(4)
$$\begin{aligned} &l_{4}y :=y_{2}(\xi_{1}+0)- ( \alpha_{4}+\lambda) y_{1}(\xi_{1}-0)- \alpha_{3}^{-1}y_{2}(\xi_{1}-0)=0, \end{aligned}$$
(5)
$$\begin{aligned} &l_{5}y :=y_{1}(\xi_{2}+0)- \alpha_{5}y_{1}(\xi_{2}-0)=0, \end{aligned}$$
(6)
$$\begin{aligned} &l_{6}y :=y_{2}(\xi_{2}+0)- ( \alpha_{6}+\lambda) y_{1}(\xi_{2}-0)- \alpha_{5}^{-1}y_{2}(\xi_{2}-0)=0, \end{aligned}$$
(7)

where \(\alpha_{i}\), and \(\alpha_{j}^{\prime}\), \(\gamma_{j}^{\prime}\) (\(i=\overline{1,6}\), \(j=1,2\)) are real numbers; \(\alpha_{3}>0\), \(\alpha_{5}>0\), and

$$d_{1}= \biggl\vert \begin{matrix} \alpha_{1} & \alpha_{1}^{\prime}\\ \alpha_{2} & \alpha_{2}^{\prime}\end{matrix} \biggr\vert >0,\qquad d_{2}= \biggl\vert \begin{matrix} \gamma_{1} & \gamma_{1}^{\prime}\\ \gamma_{2} & \gamma_{2}^{\prime}\end{matrix} \biggr\vert >0. $$

2 Operator formulation and properties of spectrum

In this section, we present the inner product in the Hilbert space \(H:=L_{2}(\Lambda)\oplus L_{2}(\Lambda)\oplus \mathbb{C} ^{4}\) and the operator T defined on H such that (1)-(7) can be regarded as the eigenvalue problem of operator T. We define an inner product in H by

$$\begin{aligned} \langle F,G\rangle :={}&\rho^{-1} ( x ) { \int_{a}^{b}} \bigl( f_{1}(x) \overline{g}_{1}(x)+f_{2}(x) \overline{g}_{2}(x) \bigr) \,dx+\alpha_{3}f_{1}( \xi_{1}-0) \overline{g}_{1}(\xi_{1}-0) \\ &{}+\alpha_{5}f_{1}(\xi_{2}-0) \overline{g}_{1}(\xi_{2}-0)+\frac{1}{d_{1}}r \overline{r_{1}}+\frac{1}{d_{2}}s\overline{s_{1}} \end{aligned}$$
(8)

for

$$\begin{aligned} &F= \begin{pmatrix} f(x)\\ r\\ s\\ f_{1} ( \xi_{1}-0 ) \\ f_{1} ( \xi_{2}-0 ) \end{pmatrix} \in H,\qquad G= \begin{pmatrix} g(x)\\ r_{1}\\ s_{1}\\ g_{1} ( \xi_{1}-0 ) \\ g_{1} ( \xi_{2}-0 ) \end{pmatrix} \in H,\qquad f(x)= \begin{pmatrix} f_{1} ( x ) \\ f_{2} ( x ) \end{pmatrix}, \\ &g(x)= \begin{pmatrix} g_{1} ( x ) \\ g_{2} ( x ) \end{pmatrix},\qquad r=\alpha_{1}^{\prime}f_{1}(a)- \alpha_{2}^{\prime}f_{2}(a),\qquad s= \gamma_{1}^{\prime}f_{1}(b)-\gamma_{2}^{\prime}f_{2}(b), \\ &r_{1}=\alpha_{1}^{\prime}g_{1}(a)- \alpha_{2}^{\prime}g_{2}(a),\qquad s_{1}= \gamma_{1}^{\prime}g_{1}(b)-\gamma_{2}^{\prime}g_{2}(b). \end{aligned}$$

Consider the operator T defined in the domain

$$D(T)= \bigl\{ F\in H:f(x)\in AC\bigl( [ a,\xi_{1} ) \cup( \xi _{1},\xi_{2} ) \cup( \xi_{2},b ] \bigr),lf\in L_{2}(\Lambda)\oplus L_{2}( \Lambda),l_{3}f=l_{5}f=0 \bigr\} $$

such that

$$\begin{aligned} TF:={}&\bigl(lf,\alpha_{1}f_{1}(a)-\alpha_{2}f_{2}(a),- \bigl( \gamma_{1}f_{1}(b)-\gamma_{2}f_{2}(b) \bigr) , \\ &f_{2}(\xi_{1}+0)-\alpha_{4}f_{1}( \xi_{1}-0)-\alpha_{3}^{-1}f_{2} ( \xi_{1}-0 ) ,f_{2}(\xi_{2}+0)- \alpha_{6}f_{1}(\xi_{2}-0)-\alpha_{5}^{-1}f_{2} ( \xi_{2}-0 ) \bigr)^{T} \end{aligned}$$

for

$$F=\bigl(f,\alpha_{1}^{\prime}f_{1}(a)- \alpha_{2}^{\prime}f_{2}(a),\gamma _{1}^{\prime}f_{1}(b)- \gamma_{2}^{\prime}f_{2}(b),f_{1} ( \xi _{1}-0 ) ,f_{1} ( \xi_{2}-0 ) \bigr)^{T}\in D(T). $$

Thus, we can rewrite the considered problem (1)-(7) in the operator form as \(TF=\lambda F\), i.e., the problem (1)-(7) can be considered as an eigenvalue problem of the operator T.

We define the solutions

$$\begin{aligned} &\varphi(x,\lambda)= \textstyle\begin{cases} \varphi_{1}(x,\lambda),&x\in[ a,\xi_{1} ), \\ \varphi_{2}(x,\lambda),&x\in( \xi_{1},\xi_{2} ), \\ \varphi_{3}(x,\lambda),&x\in( \xi_{2},b ], \end{cases}\displaystyle \qquad \psi(x,\lambda)= \textstyle\begin{cases} \psi_{1}(x,\lambda),&x\in[ a,\xi_{1} ) ,\\ \psi_{2}(x,\lambda),&x\in( \xi_{1},\xi_{2} ), \\ \psi_{3}(x,\lambda),&x\in( \xi_{2},b ], \end{cases}\displaystyle \\ &\varphi_{1}(x,\lambda)= \bigl( \varphi_{11}(x,\lambda), \varphi_{12}(x,\lambda) \bigr) ^{T}, \\ & \varphi_{2}(x,\lambda)= \bigl( \varphi _{21}(x,\lambda), \varphi_{22}(x,\lambda) \bigr) ^{T}, \\ &\varphi_{3}(x,\lambda)= \bigl( \varphi_{31}(x,\lambda), \varphi_{32}(x,\lambda) \bigr) ^{T}, \end{aligned}$$

and

$$\begin{aligned} &\psi_{1} (x,\lambda)= \bigl( \psi_{11}(x,\lambda), \psi_{12}(x,\lambda) \bigr) ^{T},\\ & \psi_{2}(x, \lambda)= \bigl( \psi_{21}(x,\lambda),\psi_{22}(x,\lambda) \bigr) ^{T}, \\ &\psi_{3}(x,\lambda)= \bigl( \psi_{31}(x,\lambda ), \psi_{32}(x,\lambda) \bigr) ^{T}, \end{aligned}$$

of equation (1) satisfying the initial conditions

$$\begin{aligned} &\varphi_{11}(a,\lambda) =\lambda\alpha_{2}^{\prime}- \alpha_{2},\qquad \varphi_{12}(a,\lambda)=\lambda \alpha_{1}^{\prime}-\alpha_{1}, \\ &\varphi_{21}(\xi_{1},\lambda) =\alpha_{3} \varphi_{11}(\xi_{1},\lambda),\qquad \varphi_{22}(\xi_{1},\lambda)= ( \alpha_{4}+\lambda ) \varphi_{11}(\xi_{1},\lambda)+\alpha_{3}^{-1} \varphi_{12}(\xi_{1},\lambda), \\ &\varphi_{31}(\xi_{2},\lambda) =\alpha_{5} \varphi_{21}(\xi_{2},\lambda),\qquad \varphi_{32}(\xi_{2},\lambda)= ( \alpha_{6}+ \lambda) \varphi_{21}(\xi_{2},\lambda)+\alpha_{5}^{-1} \varphi_{22}(\xi_{2},\lambda), \end{aligned}$$
(9)

and similarly

$$\begin{aligned} &\psi_{31}(b,\lambda) =\lambda\gamma_{2}^{\prime}+ \gamma_{2}, \qquad\psi_{32}(b,\lambda)=\lambda \gamma_{1}^{\prime}+\gamma_{1}, \\ &\psi_{21}(\xi_{2},\lambda) =\frac{\psi_{31}(\xi_{2},\lambda)}{\alpha _{5}},\qquad \psi_{22}(\xi_{2},\lambda)=\alpha_{5} \psi_{32}(\xi_{2},\lambda)- ( \alpha_{6}+\lambda) \psi_{31}(\xi_{2},\lambda), \\ &\psi_{11}(\xi_{2},\lambda) =\frac{\psi_{21}(\xi_{1},\lambda)}{\alpha _{3}},\qquad \psi_{12}(\xi_{2},\lambda)=\alpha_{3} \psi_{22}(\xi_{1},\lambda)- ( \alpha_{4}+\lambda) \psi_{21}(\xi_{1},\lambda), \end{aligned}$$
(10)

respectively.

These solutions are entire functions of λ for each fixed \(x\in[ a,b ] \) and satisfy the relation \(\psi(x,\lambda_{n})=\kappa_{n}\varphi(x,\lambda_{n})\) for each eigenvalue \(\lambda_{n}\), where

$$\kappa_{n}=\frac{\alpha_{1}^{\prime}\psi_{11}(a,\lambda_{n})-\alpha _{2}^{\prime}\psi_{12}(a,\lambda_{n})}{d_{1}}. $$

Lemma 1

T is a self-adjoint operator. Therefore, all eigenvalues and eigenfunctions of the problem (1)-(7) are real and the two eigenfunctions \(\varphi(x,\lambda _{1})= ( \varphi_{1}(x,\lambda_{1}),\varphi_{2}(x,\lambda_{1}) ) ^{T}\) and \(\varphi(x,\lambda_{2})= ( \varphi_{1}(x,\lambda_{2}),\varphi_{2}(x,\lambda_{2}) ) ^{T}\) corresponding to different eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) are orthogonal in the sense of

$$\begin{aligned} &\rho^{-1} ( x ) { \int_{a}^{b}} \bigl[ \varphi_{1}(x,\lambda_{1})\varphi_{1}(x, \lambda_{2})+\varphi _{2}(x,\lambda_{1}) \varphi_{2}(x,\lambda_{2}) \bigr] \,dx \\ &\quad{}+\alpha_{3}\varphi_{1}(\xi_{1}-0, \lambda_{1})\varphi_{1}(\xi_{1}-0, \lambda_{2})+\alpha_{5}\varphi_{1}( \xi_{2}-0,\lambda_{1})\varphi_{1}( \xi_{2}-0,\lambda_{2}) \\ &\quad{} +\frac{1}{d_{1}} \bigl( \alpha _{1}^{\prime} \varphi_{11}(a,\lambda_{1})-\alpha_{2}^{\prime} \varphi _{12}(a,\lambda_{1}) \bigr) \bigl( \alpha_{1}^{\prime}\varphi_{11}(a, \lambda_{2})-\alpha_{2}^{\prime}\varphi_{12}(a, \lambda_{2}) \bigr) \\ &\quad{} +\frac{1}{d_{2}} \bigl( \gamma_{1}^{\prime}\varphi _{31}(b,\lambda_{1})-\gamma_{2}^{\prime} \varphi_{32}(b,\lambda_{1}) \bigr) \bigl( \gamma_{1}^{\prime}\varphi_{31}(b,\lambda_{2})- \gamma_{2}^{\prime }\varphi_{32}(b,\lambda_{2}) \bigr) =0. \end{aligned}$$

By the method of variation of parameters, integral equations in Lemmas 2, 3 can be obtained and with the help of these integral equations, we also have their asymptotic behaviors.

Lemma 2

The following integral equations and asymptotic behaviors hold:

$$\begin{aligned} &\varphi_{11}(x,\lambda)=- \bigl( \lambda\alpha_{1}^{\prime}- \alpha _{1} \bigr) \sin\lambda\rho_{1}(x-a)+ \bigl( \lambda \alpha_{2}^{\prime }-\alpha_{2} \bigr) \cos\lambda \rho_{1}(x-a) \\ &\phantom{\varphi_{11}(x,\lambda)=}{} +{ \int_{a}^{x}} \bigl[ p ( t ) \sin\lambda \rho_{1}(x-t)+q(t)\cos\lambda\rho _{1}(x-t) \bigr] \rho_{1}\varphi_{11}(t,\lambda)\,dt \\ &\phantom{\varphi_{11}(x,\lambda)=}{} +{ \int_{a}^{x}} \bigl[ q ( t ) \sin\lambda \rho_{1}(x-t)+r(t)\cos\lambda\rho _{1}(x-t) \bigr] \rho_{1}\varphi_{12}(t,\lambda)\,dt \\ &\phantom{\varphi_{11}(x,\lambda)}=- \bigl( \lambda\alpha_{1}^{\prime}-\alpha_{1} \bigr) \sin\lambda\rho_{1}(x-a)+ \bigl( \lambda\alpha_{2}^{\prime}- \alpha_{2} \bigr) \cos\lambda\rho_{1}(x-a)+o \bigl(\vert \lambda\vert e^{\vert \operatorname{Im}\lambda\vert (x-a)\rho_{1}}\bigr), \\ &\varphi_{12}(x,\lambda)= \bigl( \lambda \alpha_{1}^{\prime}-\alpha_{1} \bigr) \cos \lambda \rho_{1}(x-a)+ \bigl( \lambda\alpha_{2}^{\prime}- \alpha_{2} \bigr) \sin\lambda\rho_{1}(x-a) \\ &\phantom{\varphi_{12}(x,\lambda)=}{}+{ \int_{a}^{x}} \bigl[ -p ( t ) \cos\lambda \rho_{1}(x-t)+q(t)\sin\lambda\rho _{1}(x-t) \bigr] \rho_{1}\varphi_{11}(t,\lambda)\,dt \\ &\phantom{\varphi_{12}(x,\lambda)=}{}+{ \int_{a}^{x}} \bigl[ -q ( t ) \cos\lambda \rho_{1}(x-t)+r(t)\sin\lambda\rho _{1}(x-t) \bigr] \rho_{1}\varphi_{12}(t,\lambda)\,dt \\ &\phantom{\varphi_{12}(x,\lambda)}= \bigl( \lambda\alpha_{1}^{\prime}-\alpha_{1} \bigr) \cos\lambda\rho_{1}(x-a)+ \bigl( \lambda\alpha_{2}^{\prime}- \alpha_{2} \bigr) \sin\lambda\rho_{1}(x-a)+o \bigl(\vert \lambda\vert e^{\vert \operatorname{Im}\lambda\vert (x-a)\rho_{1}}\bigr), \\ &\varphi_{21}(x,\lambda)=\alpha_{3} \varphi_{11}(\xi_{1},\lambda)\cos\lambda\rho_{2}(x- \xi_{1}) \\ &\phantom{\varphi_{21}(x,\lambda)=}{}- \biggl( ( \alpha_{4}+\lambda) \varphi _{11}( \xi_{1},\lambda)+\frac{1}{\alpha_{3}}\varphi_{12}( \xi_{1},\lambda) \biggr) \sin\lambda\rho_{2}(x- \xi_{1}) \\ &\phantom{\varphi_{21}(x,\lambda)=}{} +{ \int_{\xi_{1}}^{x}} \bigl[ p ( t ) \sin\lambda \rho_{2}(x-t)+q(t)\cos\lambda\rho _{2}(x-t) \bigr] \rho_{2}\varphi_{21}(t,\lambda)\,dt \\ &\phantom{\varphi_{21}(x,\lambda)=}{}+{ \int_{\xi_{1}}^{x}} \bigl[ q ( t ) \sin\lambda \rho_{2}(x-t)+r(t)\cos\lambda\rho _{2}(x-t) \bigr] \rho_{2}\varphi_{22}(t,\lambda)\,dt \\ &\phantom{\varphi_{21}(x,\lambda)}= ( \alpha_{4}+\lambda) \bigl[ \bigl( \lambda\alpha_{1}^{\prime}- \alpha_{1} \bigr) \sin\lambda\rho_{1} ( \xi_{1}-a ) \sin \lambda\rho_{2} ( x-\xi_{1} ) \\ &\phantom{\varphi_{21}(x,\lambda)=}{} - \bigl( \lambda\alpha_{2}^{\prime}-\alpha_{2} \bigr) \cos\lambda\rho_{1} ( \xi_{1}-a ) \sin\lambda \rho_{2} ( x-\xi_{1} ) \bigr] +o\bigl(\vert \lambda\vert ^{2}e^{\vert \operatorname{Im}\lambda\vert ( (\xi_{1}-a)\rho_{1}+(x-\xi_{1})\rho_{2} ) }\bigr), \\ &\varphi_{22}(x,\lambda)=\alpha_{3}\varphi_{11}( \xi _{1},\lambda)\sin\lambda\rho_{2}(x-\xi_{1}) \\ &\phantom{\varphi_{22}(x,\lambda)=}{}+ \biggl( ( \alpha_{4}+\lambda) \varphi_{11}( \xi_{1},\lambda)+\frac {1}{\alpha_{3}}\varphi_{12}( \xi_{1},\lambda) \biggr) \cos\lambda\rho_{2}(x- \xi_{1}) \\ &\phantom{\varphi_{22}(x,\lambda)=}{} +{ \int_{\xi_{1}}^{x}} \bigl[ -p ( t ) \cos\lambda \rho_{2}(x-t)+q(t)\sin\lambda\rho _{2}(x-t) \bigr] \rho_{2}\varphi_{21}(t,\lambda)\,dt \\ &\phantom{\varphi_{22}(x,\lambda)=}{}+{ \int_{\xi_{1}}^{x}} \bigl[ -q ( t ) \cos\lambda \rho_{2}(x-t)+r(t)\sin\lambda\rho _{2}(x-t) \bigr] \rho_{2}\varphi_{22}(t,\lambda)\,dt \\ &\phantom{\varphi_{22}(x,\lambda)}=- ( \alpha_{4}+\lambda) \bigl[ \bigl( \lambda\alpha_{1}^{\prime}- \alpha_{1} \bigr) \sin\lambda\rho_{1} ( \xi_{1}-a ) \cos \lambda\rho_{2} ( x-\xi_{1} ) \\ &\phantom{\varphi_{22}(x,\lambda)=}{} - \bigl( \lambda\alpha_{2}^{\prime}-\alpha_{2} \bigr) \cos\lambda\rho_{1} ( \xi_{1}-a ) \cos\lambda \rho_{2} ( x-\xi_{1} ) \bigr] +o\bigl(\vert \lambda\vert ^{2}e^{\vert \operatorname{Im}\lambda\vert ( (\xi_{1}-a)\rho_{1}+(x-\xi_{1})\rho_{2} ) }\bigr), \\ & \varphi_{31}(x,\lambda)=\alpha_{5}\varphi_{21}( \xi _{2},\lambda)\cos\lambda\rho_{3} ( x-\xi_{2} ) \\ &\phantom{\varphi_{31}(x,\lambda)=}{}- \biggl( \frac{1}{\alpha_{5}}\varphi_{22}(\xi_{2}, \lambda)+ ( \alpha_{6}+\lambda) \varphi_{21}( \xi_{2},\lambda) \biggr) \sin\lambda\rho _{3} ( x- \xi_{2} ) \\ &\phantom{\varphi_{31}(x,\lambda)=}{}+{ \int_{\xi_{2}}^{x}} \bigl[ p ( t ) \sin\lambda \rho_{3}(x-t)+q(t)\cos\lambda\rho _{3}(x-t) \bigr] \rho_{3}\varphi_{31}(t,\lambda)\,dt \\ &\phantom{\varphi_{31}(x,\lambda)=}{}+{ \int_{\xi_{2}}^{x}} \bigl[ q ( t ) \sin\lambda \rho_{3}(x-t)+r(t)\cos\lambda\rho _{3}(x-t) \bigr] \rho_{3}\varphi_{32}(t,\lambda)\,dt \\ &\phantom{\varphi_{31}(x,\lambda)}= ( \alpha_{4}+\lambda) ( \alpha_{6}+\lambda) \bigl[ - \bigl( \lambda\alpha_{1}^{\prime}-\alpha_{1} \bigr) \sin\lambda\rho_{1} ( \xi_{1}-a ) \sin\lambda \rho_{2} ( \xi_{2}-\xi_{1} ) \\ &\phantom{\varphi_{31}(x,\lambda)=}{} + \bigl( \lambda\alpha_{2}^{\prime}-\alpha _{2} \bigr) \cos\lambda\rho_{1} ( \xi_{1}-a ) \sin\lambda\rho _{2} ( \xi_{2}-\xi_{1} ) \bigr] \sin\lambda \rho_{3} ( x-\xi_{2} ) \\ &\phantom{\varphi_{31}(x,\lambda)=}{}+o\bigl(\vert \lambda\vert ^{3}e^{\vert \operatorname{Im}\lambda\vert ( (\xi_{1}-a)\rho_{1}+ ( \xi_{2}-\xi_{1} ) \rho_{2}+(x-\xi_{2})\rho _{3} ) }\bigr), \\ &\varphi_{32}(x,\lambda)=\alpha_{5}\varphi_{21}(\xi_{2},\lambda)\sin\lambda\rho_{3} ( x-\xi_{2} ) \\ &\phantom{\varphi_{32}(x,\lambda)=}{}+ \biggl( \frac{1}{\alpha_{5}}\varphi_{22}(\xi_{2}, \lambda)+ ( \alpha_{6}+\lambda) \varphi_{21}( \xi_{2},\lambda) \biggr) \cos\lambda\rho_{3} ( x- \xi_{2} ) \\ &\phantom{\varphi_{32}(x,\lambda)=}{}+{ \int_{\xi_{2}}^{x}} \bigl[ -p ( t ) \cos\lambda \rho_{3}(x-t)+q(t)\sin\lambda\rho _{3}(x-t) \bigr] \rho_{3}\varphi_{31}(t,\lambda)\,dt \\ &\phantom{\varphi_{32}(x,\lambda)=}{} +{ \int_{\xi_{2}}^{x}} \bigl[ -q ( t ) \cos\lambda \rho_{3}(x-t)+r(t)\sin\lambda\rho _{3}(x-t) \bigr] \rho_{3}\varphi_{32}(t,\lambda)\,dt \\ &\phantom{\varphi_{32}(x,\lambda)}{} =- ( \alpha_{4}+\lambda) ( \alpha_{6}+\lambda) \bigl[ - \bigl( \lambda\alpha_{1}^{\prime}-\alpha_{1} \bigr) \sin\lambda\rho_{1} ( \xi_{1}-a ) \sin\lambda \rho_{2} ( \xi_{2}-\xi_{1} ) \\ &\phantom{\varphi_{32}(x,\lambda)=}{} + \bigl( \lambda\alpha_{2}^{\prime}-\alpha _{2} \bigr) \cos\lambda\rho_{1} ( \xi_{1}-a ) \sin\lambda\rho _{2} ( \xi_{2}-\xi_{1} ) \bigr] \cos\lambda \rho_{3} ( x-\xi_{2} ) \\ &\phantom{\varphi_{32}(x,\lambda)=}{}+o\bigl(\vert \lambda\vert ^{3}e^{\vert \operatorname{Im}\lambda\vert ( (\xi_{1}-a)\rho_{1}+ ( \xi_{2}-\xi_{1} ) \rho_{2}+(x-\xi_{2})\rho _{3} ) }\bigr). \end{aligned}$$

Lemma 3

The following integral equations and asymptotic behaviors hold:

$$\begin{aligned} &\psi_{31}(x,\lambda)= \bigl( \lambda\gamma_{2}^{\prime}+ \gamma_{2} \bigr) \cos\lambda\rho_{3}(x-b)-\bigl(\lambda \gamma_{1}^{\prime}+\gamma_{1}\bigr)\sin \lambda \rho_{3}(x-b) \\ &\phantom{\psi_{31}(x,\lambda)=}{}-{ \int_{x}^{b}} \bigl[ p(t)\sin\lambda \rho_{3}(x-t)+q(t)\cos\lambda\rho_{3}(x-t) \bigr] \rho_{3}\psi_{31}(t,\lambda)\,dt \\ &\phantom{\psi_{31}(x,\lambda)=}{}-{ \int_{x}^{b}} \bigl[ q(t)\sin\lambda \rho_{3}(x-t)+r(t)\cos\lambda\rho_{3}(x-t) \bigr] \rho_{3}\psi_{32}(t,\lambda)\,dt \\ &\phantom{\psi_{31}(x,\lambda)} = \bigl( \lambda\gamma_{2}^{\prime }+\gamma_{2} \bigr) \cos\lambda\rho_{3}(x-b)-\bigl(\lambda\gamma_{1}^{\prime }+ \gamma_{1}\bigr)\sin\lambda\rho_{3}(x-b)+o\bigl(\vert \lambda\vert e^{\vert \operatorname{Im}\lambda\vert (b-x)\rho_{3}}\bigr),\\ &\psi_{32}(x,\lambda)= \bigl( \lambda\gamma_{2}^{\prime }+ \gamma_{2} \bigr) \sin\lambda\rho_{3}(x-b)+\bigl(\lambda \gamma_{1}^{\prime }+\gamma_{1}\bigr)\cos\lambda \rho_{3}(x-b) \\ &\phantom{\psi_{32}(x,\lambda)=}{}+{ \int_{x}^{b}} \bigl[ p(t)\cos\lambda \rho_{3}(x-t)-q(t)\sin\lambda\rho_{3}(x-t) \bigr] \rho_{3}\psi_{31}(t,\lambda)\,dt \\ &\phantom{\psi_{32}(x,\lambda)=}{} +{ \int_{x}^{b}} \bigl[ q(t)\cos\lambda \rho_{3}(x-t)-r(t)\sin\lambda\rho_{3}(x-t) \bigr] \rho_{3}\psi_{32}(t,\lambda)\,dt \\ &\phantom{\psi_{32}(x,\lambda)} = \bigl( \lambda\gamma_{2}^{\prime }+\gamma_{2} \bigr) \sin\lambda\rho_{3}(x-b)+\bigl(\lambda\gamma_{1}^{\prime }+ \gamma_{1}\bigr)\cos\lambda\rho_{3}(x-b)+o\bigl(\vert \lambda\vert e^{\vert \operatorname{Im}\lambda\vert (b-x)\rho_{3}}\bigr),\\ &\psi_{21}(x,\lambda)= \bigl[ ( \alpha_{6}+ \lambda) \psi_{31}(\xi_{2},\lambda)-\alpha_{5} \psi_{32}(\xi _{2},\lambda) \bigr] \sin\lambda \rho_{2}(x-\xi_{2}) \\ &\phantom{\psi_{21}(x,\lambda)=}{}+\frac {1}{\alpha_{5}}\psi_{31}(\xi_{2},\lambda)\cos\lambda \rho_{2}(x-\xi_{2})\\ &\phantom{\psi_{21}(x,\lambda)=}{}- { \int_{x}^{\xi_{2}}} \bigl[ p(t)\sin\lambda \rho_{2}(x-t)+q(t)\cos\lambda\rho_{2}(x-t) \bigr] \rho_{2}\psi_{21}(t,\lambda)\,dt \\ &\phantom{\psi_{21}(x,\lambda)=}{}-{ \int_{x}^{\xi_{2}}} \bigl[ q(t)\sin\lambda \rho_{2}(x-t)+r(t)\cos\lambda\rho_{2}(x-t) \bigr] \rho_{2}\psi_{22}(t,\lambda)\,dt \\ &\phantom{\psi_{21}(x,\lambda)}= ( \alpha_{6}+\lambda) \bigl[ \bigl( \lambda\gamma_{2}^{\prime}+ \gamma_{2} \bigr) \cos\lambda \rho_{3} ( \xi_{2}-b ) \sin\lambda\rho_{2} ( x-\xi_{2} ) \\ &\phantom{\psi_{21}(x,\lambda)=}{} -\bigl(\lambda\gamma_{1}^{\prime}+\gamma _{1} \bigr)\sin\lambda\rho_{3} ( \xi_{2}-b ) \sin\lambda \rho_{2} ( x-\xi_{2} ) \bigr] +o\bigl(\vert \lambda\vert ^{2}e^{\vert \operatorname{Im}\lambda\vert ( ( b-\xi_{2} ) \rho _{3}+(\xi_{2}-x)\rho_{2} ) }\bigr), \\ & \psi_{22}(x,\lambda )= \bigl[ - ( \alpha_{6}+\lambda) \psi_{31}(\xi_{2},\lambda )+\alpha_{5} \psi_{32}(\xi_{2},\lambda) \bigr] \cos\lambda \rho_{2}(x-\xi_{2}) \\ &\phantom{\psi_{22}(x,\lambda )=}{} +\frac{1}{\alpha_{5}}\psi_{31}(\xi_{2},\lambda)\sin \lambda \rho _{2}(x-\xi_{2})\\ &\phantom{\psi_{22}(x,\lambda )=}{}+{ \int_{x}^{\xi_{2}}} \bigl[ p(t)\cos\lambda \rho_{2}(x-t)-q(t)\sin\lambda\rho_{2}(x-t) \bigr] \rho_{2}\psi_{21}(t,\lambda)\,dt \\ &\phantom{\psi_{22}(x,\lambda )=}{}+{ \int_{x}^{\xi_{2}}} \bigl[ q(t)\cos\lambda \rho_{2}(x-t)-r(t)\sin\lambda\rho_{2}(x-t) \bigr] \rho_{2}\psi_{22}(t,\lambda)\,dt \\ &\phantom{\psi_{22}(x,\lambda )}{} =- ( \alpha_{6}+\lambda) \bigl[ \bigl( \lambda \gamma_{2}^{\prime}+\gamma_{2} \bigr) \cos\lambda \rho_{3} ( \xi_{2}-b ) \cos\lambda\rho_{2} ( x- \xi_{2} ) \\ &\phantom{\psi_{22}(x,\lambda )=}{} -\bigl(\lambda\gamma_{1}^{\prime}+\gamma _{1} \bigr)\sin\lambda\rho_{3} ( \xi_{2}-b ) \cos\lambda \rho_{2} ( x-\xi_{2} ) \bigr] +o\bigl(\vert \lambda\vert ^{2}e^{\vert \operatorname{Im}\lambda\vert ( ( b-\xi_{2} ) \rho _{3}+(\xi_{2}-x)\rho_{2} ) }\bigr), \\ & \psi_{11}(x,\lambda )= \bigl( \alpha_{3}\psi_{22}(\xi_{1},\lambda)- ( \alpha_{4}+\lambda) \psi_{21}(\xi_{1}, \lambda) \bigr) \sin\lambda\rho_{1}(x-\xi_{1}) \\ &\phantom{\psi_{11}(x,\lambda )=}{} -\frac{1}{\alpha_{3}}\psi_{21}(\xi_{1},\lambda )\cos \lambda\rho_{1}(x-\xi_{1})\\ &\phantom{\psi_{22}(x,\lambda )=}{}-{ \int_{x}^{\xi_{1}}} \bigl[ p(t)\sin\lambda \rho_{1}(x-t)+q(t)\cos\lambda\rho_{1}(x-t) \bigr] \rho_{2}\psi_{11}(t,\lambda)\,dt \\ &\phantom{\psi_{11}(x,\lambda )=}{}-{ \int_{x}^{\xi_{1}}} \bigl[ q(t)\sin\lambda \rho_{1}(x-t)+r(t)\cos\lambda\rho_{1}(x-t) \bigr] \rho_{2}\psi_{12}(t,\lambda)\,dt \\ &\phantom{\psi_{11}(x,\lambda )}=- ( \alpha_{4}+\lambda) ( \alpha_{6}+\lambda) \bigl[ \bigl( \lambda\gamma_{2}^{\prime }+\gamma_{2} \bigr) \cos\lambda\rho_{3} ( \xi_{2}-b ) \\ &\phantom{\psi_{11}(x,\lambda )=}{} -\bigl(\lambda\gamma_{1}^{\prime}+\gamma_{1}\bigr)\sin\lambda\rho_{3} ( \xi_{2}-b ) \bigr] \sin\lambda \rho _{2} ( \xi_{1}-\xi_{2} ) \sin\lambda \rho_{1} ( x-\xi _{1} ) \\ &\phantom{\psi_{11}(x,\lambda )=}{}+o\bigl(\vert \lambda\vert ^{3}e^{\vert \operatorname{Im}\lambda\vert ( ( b-\xi_{2} ) \rho _{3}+ ( \xi_{2}-\xi_{1} ) \rho_{2}+(\xi_{1}-x)\rho_{1} ) }\bigr), \\ &\psi_{12}(x,\lambda)= \bigl( ( \alpha_{4}+ \lambda) \psi_{21}(\xi_{1},\lambda)-\alpha_{3} \psi_{22}(\xi _{1},\lambda) \bigr) \cos\lambda \rho_{1}(x-\xi_{1}) \\ &\phantom{\psi_{12}(x,\lambda)=}{}-\frac {1}{\alpha_{3}}\psi_{21}(\xi_{1},\lambda)\sin\lambda \rho_{1}(x-\xi_{1})\\ &\phantom{\psi_{12}(x,\lambda)=}{}+ { \int_{x}^{\xi_{1}}} \bigl[ p(t)\cos\lambda \rho_{1}(x-t)-q(t)\sin\lambda\rho_{1}(x-t) \bigr] \rho_{1}\psi_{11}(t,\lambda)\,dt \\ &\phantom{\psi_{12}(x,\lambda)=}{}+{ \int_{x}^{\xi_{1}}} \bigl[ q(t)\cos\lambda \rho_{1}(x-t)-r(t)\sin\lambda\rho_{1}(x-t) \bigr] \rho_{1}\psi_{12}(t,\lambda)\,dt \\ &\phantom{\psi_{12}(x,\lambda)}= ( \alpha_{4}+\lambda) ( \alpha_{6}+\lambda) \bigl[ \bigl( \lambda\gamma_{2}^{\prime }+\gamma_{2} \bigr) \cos\lambda\rho_{3} ( \xi_{2}-b ) \\ &\phantom{\psi_{12}(x,\lambda)=}{} -\bigl(\lambda\gamma_{1}^{\prime}+\gamma_{1}\bigr)\sin\lambda\rho_{3} ( \xi_{2}-b ) \bigr] \sin\lambda \rho _{2} ( \xi_{1}-\xi_{2} ) \cos\lambda \rho_{1} ( x-\xi _{1} ) \\ &\phantom{\psi_{12}(x,\lambda)=}{}+o\bigl(\vert \lambda\vert ^{3}e^{\vert \operatorname{Im}\lambda\vert ( ( b-\xi_{2} ) \rho_{3}+ ( \xi_{2}-\xi_{1} ) \rho_{2}+(\xi _{1}-x)\rho_{1} ) }\bigr). \end{aligned}$$

Denote

$$\Delta_{i} ( \lambda) :=W(\varphi_{i}, \psi_{i},x):=\varphi _{i1}\psi_{i2}- \varphi_{i2}\psi_{i1},\quad x\in\Lambda_{i}\ ( i= \overline {1,3} ) , $$

which are independent of \(x\in\Lambda_{i}\) and are entire functions such that \(\Lambda_{1}= [ a,\xi_{1} ) \), \(\Lambda_{2}= ( \xi_{1},\xi_{2} ) \), \(\Lambda_{3}= ( \xi _{2},b ] \).

Let

$$ \Delta_{3} ( \lambda) =\Delta(\lambda)=W ( \varphi,\psi,b ) = \bigl( \lambda\gamma_{1}^{\prime}+\gamma_{1} \bigr) \varphi_{31}(b,\lambda)- \bigl( \lambda\gamma_{2}^{\prime}+ \gamma_{2} \bigr) \varphi_{32}(b,\lambda) $$
(11)

and

$$\begin{aligned} \mu_{n} :={}&\rho^{-1} ( x ) { \int_{a}^{b}} \bigl[ \varphi_{1}^{2}(x, \lambda_{n})+ \varphi_{2}^{2}(x, \lambda_{n}) \bigr] \,dx \\ &{} +\alpha_{3}\varphi_{1}^{2}( \xi_{1}-0,\lambda_{n})+\alpha_{5} \varphi_{1}^{2}(\xi_{2}-0, \lambda_{n})+\frac{1}{d_{1}} \bigl( \alpha_{1}^{\prime}\varphi_{11}(a,\lambda_{n})-\alpha_{2}^{\prime} \varphi_{12}(a,\lambda_{n}) \bigr) ^{2} \\ &{ }+\frac{1}{d_{2}} \bigl( \gamma_{1}^{\prime} \varphi_{31}(b,\lambda_{n})-\gamma_{2}^{\prime} \varphi_{32}(b,\lambda_{n}) \bigr) ^{2}. \end{aligned}$$
(12)

The function \(\Delta(\lambda)\) is called the characteristic function and numbers \(\{ \mu_{n} \} _{n\in \mathbb{Z} }\) are called the normalizing constants of the problem (1)-(7).

Lemma 4

The following equality holds for each eigenvalue \(\lambda_{n}\):

$$\dot{\Delta}(\lambda_{n})=-\kappa_{n}\mu_{n}. $$

Proof

Since

$$\begin{aligned} &\rho(x)\varphi_{2}^{\prime}(x,\lambda_{n})+p ( x ) \varphi_{1} ( x,\lambda_{n} ) +q ( x ) \varphi _{2} ( x,\lambda_{n} ) =\lambda_{n} \varphi_{1} ( x,\lambda _{n} ), \\ &\rho(x)\psi_{2}^{\prime}(x,\lambda)+p ( x ) \psi_{1} ( x,\lambda) +q ( x ) \psi _{2} ( x,\lambda) =\lambda \psi_{1} ( x,\lambda), \end{aligned}$$

and

$$\begin{aligned} &{-}\rho(x)\varphi_{1}^{\prime}(x,\lambda_{n})+q ( x ) \varphi_{1} ( x,\lambda_{n} ) +r ( x ) \varphi_{2} ( x,\lambda_{n} ) =\lambda_{n}\varphi_{2} ( x, \lambda_{n} ), \\ &{-}\rho(x)\psi_{1}^{\prime}(x,\lambda)+q ( x ) \psi_{1} ( x,\lambda) +r ( x ) \psi_{2} ( x,\lambda) = \lambda\psi_{2} ( x,\lambda), \end{aligned}$$

we obtain

$$\begin{aligned} &\varphi_{1}(x,\lambda_{n}) \psi_{2}(x,\lambda)-\varphi_{2}(x,\lambda_{n}) \psi _{1}(x,\lambda) \bigl( \big\vert _{a}^{\xi_{1}}+\big\vert _{\xi_{1}}^{\xi_{2}} +\big\vert _{\xi_{2}}^{b} \bigr) \\ &\quad=(\lambda-\lambda_{n})\rho_{1}{ \int_{a}^{\xi_{1}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx \\ &\qquad{}+(\lambda-\lambda_{n})\rho_{2} { \int_{\xi_{1}}^{\xi_{2}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx \\ &\qquad{}+(\lambda-\lambda_{n})\rho_{3} { \int_{\xi_{2}}^{b}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx. \end{aligned}$$

After adding and subtracting \(\Delta(\lambda)\) on the left-hand side of the last equality and by using the conditions (2)-(7) one can obtain

$$\begin{aligned} &\Delta(\lambda)-(\lambda-\lambda_{n}) \bigl( \alpha_{2}^{\prime}\psi_{2}(a,\lambda)-\alpha_{1}^{\prime} \psi_{1}(a,\lambda) \bigr) +(\lambda-\lambda_{n}) \bigl( \gamma_{2}^{\prime}\varphi_{2}(b,\lambda _{n})-\gamma_{1}^{\prime}\varphi_{1}(b, \lambda_{n}) \bigr) \\ &\qquad{}+\alpha_{3} ( \lambda-\lambda_{n} ) \varphi_{1}(\xi _{1}-0,\lambda_{n}) \psi_{1}(\xi_{1}-0,\lambda)+\alpha_{5} ( \lambda- \lambda_{n} ) \varphi_{1}(\xi_{2}-0, \lambda_{n})\psi_{1}(\xi _{2}-0,\lambda) \\ &\quad=( \lambda-\lambda_{n})\rho_{1}{ \int_{a}^{\xi_{1}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx \\ &\qquad{}+(\lambda-\lambda_{n})\rho_{2} { \int_{\xi_{1}}^{\xi_{2}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx \\ &\qquad{}+(\lambda-\lambda_{n})\rho_{3} { \int_{\delta_{2}}^{b}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx, \end{aligned}$$

or

$$\begin{aligned} \frac {\Delta(\lambda)}{\lambda-\lambda_{n}}={}&\rho_{1}{ \int_{a}^{\xi_{1}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx \\ &{}+\rho_{2}{ \int_{\xi_{1}}^{\xi_{2}}} \bigl[ \psi_{1}(x,\lambda)\varphi_{1}(x,\lambda_{n})+ \psi_{2}(x,\lambda )\varphi_{2}(x,\lambda_{n}) \bigr] \,dx\\ &{}+\rho_{3} \bigl[ \psi_{1}(x,\lambda ) \varphi_{1}(x,\lambda_{n})+\psi_{2}(x,\lambda) \varphi_{2}(x,\lambda _{n}) \bigr] \,dx \\ &{}+\frac{ ( \alpha_{1}^{\prime}\psi_{1}(a,\lambda)-\alpha_{2}^{\prime}\psi_{2}(a,\lambda) ) ( \alpha_{1}^{\prime}\varphi_{1}(a,\lambda_{n})-\alpha_{2}^{\prime}\varphi _{2}(a,\lambda_{n}) ) }{d_{1}} \\ &{}+\frac{ ( \gamma_{2}^{\prime}\psi_{2}(b,\lambda)-\gamma_{1}^{\prime}\psi_{1}(b,\lambda) ) ( \gamma_{2}^{\prime}\varphi_{2}(b,\lambda _{n})-\gamma_{1}^{\prime}\varphi_{1}(b,\lambda_{n}) ) }{d_{2}}\\ &{}+\alpha_{3}\varphi_{1}( \xi_{1}-0,\lambda_{n})\psi_{1}( \xi_{1}-0,\lambda ) \\ &{}+\alpha_{5}\varphi_{1}(\xi_{2}-0, \lambda_{n})\psi_{1}(\xi_{2}-0, \lambda). \end{aligned}$$

For \(\lambda\rightarrow\lambda_{n}\), \(-\dot{\Delta}(\lambda_{n})=\kappa_{n}\mu_{n}\) is obtained by using the equality \(\psi(x,\lambda_{n})=\kappa_{n}\varphi(x,\lambda _{n})\) and (12). □

From Lemma 4, we see that \(\dot{\Delta}(\lambda_{n})\neq0\). Thus, the eigenvalues of problem L are simple.

Lemma 5

(cf. [48])

Let \(\{ \alpha_{i} \} _{i=1}^{p}\) be the set of real numbers satisfying the inequalities \(\alpha_{0}>\cdots >\alpha_{p-1}>0\) and \(\{ a_{i} \} _{i=1}^{p}\) be the set of complex numbers. If \(a_{p}\neq0\) then the roots of the equation \(e^{\alpha_{0}\lambda}+a_{1}e^{\alpha_{1}\lambda}+\cdots +a_{p-1}e^{\alpha _{p-1}\lambda}+a_{p}=0\) have the form

$$\lambda_{n}=\frac{2\pi ni}{\alpha_{0}}+\Psi(n)\quad (n=0,\pm1,\ldots), $$

where \(\Psi(n)\) is a bounded sequence.

Now, from Lemma 2 and (11), we can note that

$$\Delta(\lambda)-\Delta_{0}(\lambda)=o\bigl(\vert \lambda\vert ^{4}e^{\vert \operatorname{Im}\lambda\vert ( (\xi_{1}-a)\rho_{1}+(\xi_{2}-\xi_{1})\rho_{2}+(b-\xi_{2})\rho_{3} ) }\bigr), $$

where

$$\begin{aligned} \Delta_{0}(\lambda)={}&\lambda^{4}\sin\lambda \rho_{2} ( \xi_{2}-\xi_{1} ) \bigl[ \gamma_{1}^{\prime}\sin\lambda\rho_{3} ( b- \xi_{2} ) +\gamma_{2}^{\prime}\cos\lambda \rho_{3} ( b-\xi _{2} ) \bigr] \\ &{}\times\bigl[ \alpha_{2}^{\prime }\cos\lambda \rho_{1} ( \xi_{1}-a ) -\alpha_{1}^{\prime} \sin \lambda\rho_{1} ( \xi_{1}-a ) \bigr] . \end{aligned}$$

On the other hand, we can see non-zero roots, namely the \(\lambda _{n}^{0}\) of the equation \(\Delta_{0}(\lambda)=0\) are real and simple.

Furthermore, it can be proved by using Lemma 5 that

$$ \lambda_{n}^{0}=\frac{n\pi}{(\xi_{1}-a)\rho_{1}+(\xi_{2}-\xi_{1})\rho _{2}+(b-\xi_{2})\rho_{3}}+\Psi_{n}, \quad \sup_{n}\vert\Psi_{n}\vert< \infty,n=0, \mp1,\mp2,\ldots. $$
(13)

Theorem 1

The eigenvalues \(\{ \lambda_{n} \} \) which are located on the positive side of the real axis satisfy the following asymptotic behavior:

$$ \lambda_{n}=\lambda_{n-4}^{0}+o(1) , \quad n \rightarrow\infty. $$
(14)

Proof

Denote

$$G_{n}:= \bigl\{ \lambda:0\leq\operatorname{Re}\lambda\leq \lambda_{n}^{0}-\delta, \vert \operatorname{Im} \lambda\vert \leq \lambda_{n}^{0}-\delta, n=0,1,2,\ldots \bigr\} \cup\bigl\{ \lambda:\vert \lambda\vert < \delta\bigr\} , $$

where δ is a sufficiently small number. The relations

$$\bigl\vert \Delta_{0}(\lambda)\bigr\vert \geq C\vert \lambda\vert ^{4}e^{\vert \operatorname{Im}\lambda\vert ((\xi_{1}-a)\rho _{1}+(\xi_{2}-\xi_{1})\rho_{2}+(b-\xi_{2})\rho_{3})} $$

and

$$\Delta(\lambda)-\Delta_{0}(\lambda)=o\bigl(\vert \lambda\vert ^{4}e^{\vert \operatorname{Im}k\vert ((\xi_{1}-a)\rho_{1}+(\xi _{2}-\xi_{1})\rho_{2}+(b-\xi_{2})\rho_{3})}\bigr) $$

are valid for \(\lambda\in\partial G_{n}\). Then, by Rouche’s theorem, we see that the number of zeros of \(\Delta_{0}(\lambda)\) coincides with the number of zeros of \(\Delta (\lambda)\) in \(G_{n}\), namely \(n+4\) zeros, \(\lambda_{0},\lambda_{1},\lambda_{2},\ldots,\lambda_{n+3}\). In the annulus between \(G_{n}\) and \(G_{n+1}\), \(\Delta(\lambda)\) has accurately one zero, namely \(k_{n}:k_{n}=\lambda_{n}^{0}+\delta_{n}\), for \(n\geq1\). So, it follows that \(\lambda_{n+4}=k_{n}\). Applying Rouche’s theorem in \(\eta_{\varepsilon }= \{ \lambda:\vert \lambda-\lambda_{n}^{0}\vert \leq\varepsilon\} \) for sufficiently small ε and sufficiently large n, we get \(\delta_{n}=o(1)\). Finally, we obtain the asymptotic formula \(\lambda_{n}=\lambda_{n-4}^{0}+o(1)\). □

3 Construction of Green’s function

In this section, we get the resolvent of the boundary value problem (1)-(7) for λ, not an eigenvalue. Hence, we find the solution of the non-homogeneous differential equation

$$ \rho(x)By^{\prime}(x)+\Omega(x)y(x)=\lambda y(x)+f ( x ) ,\quad x\in \Lambda, $$
(15)

which satisfies the conditions (2)-(7).

We can find the general solution of homogeneous differential equation

$$\rho(x)By^{\prime}(x)+\Omega(x)y(x)=\lambda y(x), \quad x\in\Lambda, $$

in the form

$$\begin{aligned} &U_{1} ( x,\lambda) = \begin{pmatrix} c_{1}\varphi_{11} ( x,\lambda) +c_{2}\chi_{11} ( x,\lambda) \\ c_{1}\varphi_{12} ( x,\lambda) +c_{2}\chi_{12} ( x,\lambda) \end{pmatrix} ,\quad [ a,\xi_{1} ) , \\ &U_{2} ( x,\lambda) = \begin{pmatrix} c_{3}\varphi_{21} ( x,\lambda) +c_{4}\chi_{21} ( x,\lambda) \\ c_{3}\varphi_{22} ( x,\lambda) +c_{4}\chi_{22} ( x,\lambda) \end{pmatrix} , \quad ( \xi_{1},\xi_{2} ) , \\ &U_{3} ( x,\lambda) = \begin{pmatrix} c_{5}\varphi_{31} ( x,\lambda) +c_{6}\chi_{31} ( x,\lambda) \\ c_{5}\varphi_{32} ( x,\lambda) +c_{6}\chi_{32} ( x,\lambda) \end{pmatrix} , \quad( \xi_{2},b ] , \end{aligned}$$

where \(c_{i}\), \(i=\overline{1,6}\) are arbitrary constants. By using the method of variation of parameters, we shall investigate the general solution of the non-homogeneous linear differential equation (15) in the form

$$\begin{aligned} &U_{1} ( x,\lambda) = \begin{pmatrix} c_{1} ( x,\lambda) \varphi_{11} ( x,\lambda) +c_{2} ( x,\lambda) \chi_{11} ( x,\lambda) \\ c_{1} ( x,\lambda) \varphi_{12} ( x,\lambda) +c_{2} ( x,\lambda) \chi_{12} ( x,\lambda) \end{pmatrix} , \quad\text{for }x\in[ a, \xi_{1} ), \\ &U_{2} ( x,\lambda) = \begin{pmatrix} c_{3} ( x,\lambda) \varphi_{21} ( x,\lambda) +c_{4} ( x,\lambda) \chi_{21} ( x,\lambda) \\ c_{3} ( x,\lambda) \varphi_{22} ( x,\lambda) +c_{4} ( x,\lambda) \chi_{22} ( x,\lambda) \end{pmatrix} ,\quad \text{for }x \in( \xi_{1},\xi_{2} ) , \\ &U_{3} ( x,\lambda) = \begin{pmatrix} c_{5} ( x,\lambda) \varphi_{31} ( x,\lambda) +c_{6} ( x,\lambda) \chi_{31} ( x,\lambda) \\ c_{5} ( x,\lambda) \varphi_{32} ( x,\lambda) +c_{6} ( x,\lambda) \chi_{32} ( x,\lambda) \end{pmatrix} , \quad\text{for }x \in( \xi_{2},b ] , \end{aligned}$$
(16)

where the functions \(c_{i} ( x,\lambda) \) (\(i=\overline{1\text{-}6}\)) satisfy the following linear system of equations:

$$\begin{aligned} & \begin{pmatrix} c_{1}^{\prime} ( x,\lambda) \varphi_{11} ( x,\lambda) +c_{2}^{\prime} ( x,\lambda) \chi_{11} ( x,\lambda) =f_{1} ( x ) \\ c_{1}^{\prime} ( x,\lambda) \varphi_{12} ( x,\lambda) +c_{2}^{\prime} ( x,\lambda) \chi_{12} ( x,\lambda) =f_{2} ( x ) \end{pmatrix}, \quad \mbox{for } x\in[ a,\xi_{1} ) , \\ & \begin{pmatrix} c_{3}^{\prime} ( x,\lambda) \varphi_{21} ( x,\lambda) +c_{4}^{\prime} ( x,\lambda) \chi_{21} ( x,\lambda) =f_{1} ( x ) \\ c_{3}^{\prime} ( x,\lambda) \varphi_{22} ( x,\lambda) +c_{4}^{\prime} ( x,\lambda) \chi_{22} ( x,\lambda) =f_{2} ( x ) \end{pmatrix}, \quad \mbox{for } x\in( \xi_{1}, \xi_{2} ), \\ & \begin{pmatrix} c_{5}^{\prime} ( x,\lambda) \varphi_{31} ( x,\lambda) +c_{6}^{\prime} ( x,\lambda) \chi_{31} ( x,\lambda) =f_{1} ( x ) \\ c_{5}^{\prime} ( x,\lambda) \varphi_{32} ( x,\lambda) +c_{6}^{\prime} ( x,\lambda) \chi_{32} ( x,\lambda) =f_{2} ( x ) \end{pmatrix}, \quad\mbox{for } x\in( \xi_{2},b] . \end{aligned}$$

Since λ is not an eigenvalue, each of the linear system of equations has a unique solution. Thus,

$$\begin{aligned} & \biggl\vert \begin{matrix} \varphi_{11} ( x,\lambda) & \chi_{11} ( x,\lambda) \\ \varphi_{12} ( x,\lambda) & \chi_{12} ( x,\lambda) \end{matrix} \biggr\vert \neq0,\qquad \biggl\vert \begin{matrix} \varphi_{21} ( x,\lambda) & \chi_{21} ( x,\lambda) \\ \varphi_{22} ( x,\lambda) & \chi_{22} ( x,\lambda) \end{matrix} \biggr\vert \neq0, \end{aligned}$$

and

$$\biggl\vert \begin{matrix} \varphi_{31} ( x,\lambda) & \chi_{31} ( x,\lambda) \\ \varphi_{32} ( x,\lambda) & \chi_{32} ( x,\lambda) \end{matrix} \biggr\vert \neq0. $$

It is obvious that

$$\begin{aligned} &c_{1} ( x,\lambda) =\frac{1}{\Delta_{1} ( \lambda) }{ \int_{x}^{\xi_{1}}} \bigl( \chi_{11} ( t,\lambda) f_{2} ( t ) -\chi _{12} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{1}, \\ &c_{2} ( x,\lambda) =\frac{1}{\Delta_{1} ( \lambda) }{ \int_{a}^{x}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{2}, \\ &c_{3} ( x,\lambda) =\frac{1}{\Delta_{2} ( \lambda) }{ \int_{x}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi _{22} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{3}, \\ &c_{4} ( x,\lambda) =\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{x}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{4}, \\ &c_{5} ( x,\lambda) =\frac{1}{\Delta_{3} ( \lambda) }{ \int_{x}^{b}} \bigl( \chi_{31} ( t,\lambda) f_{2} ( t ) -\chi _{32} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{5}, \\ &c_{6} ( x,\lambda) =\frac{1}{\Delta_{3} ( \lambda) }{ \int_{\xi_{2}}^{x}} \bigl( \varphi_{31} ( t,\lambda) f_{2} ( t ) -\varphi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \,dt+c_{6}, \end{aligned}$$

where \(c_{i}\), \(i=\overline{1,6}\) are arbitrary constants. Substituting these above expressions in (16), then we obtain the general solution of non-homogeneous linear differential equation (15) in the form

$$\begin{aligned} &\text{for }x \in[ a,\xi_{1} ) ,\quad U_{1} ( x,\lambda) = \frac{1}{\Delta_{1} ( \lambda) }{ \int_{x}^{\xi_{1}}} \bigl( \chi_{11} ( t,\lambda) f_{2} ( t ) -\chi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{11} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in[ a,\xi_{1} ) ,\quad U_{1} ( x,\lambda) =}{ }+\frac{1}{\Delta _{1} ( \lambda) }{ \int_{x}^{\xi_{1}}} \bigl( \chi_{11} ( t,\lambda) f_{2} ( t ) -\chi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{12} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in[ a,\xi_{1} ) ,\quad U_{1} ( x,\lambda) =}{ } +\frac{1}{\Delta _{1} ( \lambda) }{ \int_{a}^{x}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{11} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in[ a,\xi_{1} ) ,\quad U_{1} ( x,\lambda) =}{ }+\frac{1}{\Delta _{1} ( \lambda) }{ \int_{a}^{x}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{12} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in[ a,\xi_{1} ) ,\quad U_{1} ( x,\lambda) =} { }+c_{1}\varphi_{11} ( x,\lambda) +c_{2} \chi_{11} ( x,\lambda) +c_{1} \varphi_{12} ( x, \lambda) +c_{2}\chi_{12} ( x, \lambda) , \\ &\text{for }x \in( \xi_{1},\xi_{2} ) ,\quad U_{2} ( x,\lambda) =\frac{1}{\Delta_{2} ( \lambda) }{ \int_{x}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{21} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{1},\xi_{2} ) ,\quad U_{2} ( x,\lambda) =} { }+\frac{1}{\Delta _{2} ( \lambda) }{ \int_{x}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{22} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{1},\xi_{2} ) ,\quad U_{2} ( x,\lambda) =}{ }+\frac{1}{\Delta _{2} ( \lambda) }{ \int_{\xi_{1}}^{x}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{21} ( x,\lambda) \,dt \\ & \phantom{\text{for }x \in( \xi_{1},\xi_{2} ) ,\quad U_{2} ( x,\lambda) =}{ }+\frac{1}{\Delta _{2} ( \lambda) }{ \int_{\xi_{1}}^{x}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{22} ( x,\lambda) \,dt \\ & \phantom{\text{for }x \in( \xi_{1},\xi_{2} ) ,\quad U_{2} ( x,\lambda) =}{ }+c_{3}\varphi_{21} ( x,\lambda) +c_{4} \chi_{21} ( x,\lambda) +c_{3} \varphi_{22} ( x, \lambda) +c_{4}\chi_{22} ( x, \lambda) , \\ &\text{for }x \in( \xi_{2},b ] ,\quad U_{3} ( x,\lambda) = \frac{1}{\Delta_{3} ( \lambda) }{ \int_{x}^{b}} \bigl( \chi_{31} ( t, \lambda) f_{2} ( t ) -\chi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{31} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{2},b ] ,\quad U_{3} ( x,\lambda) =}{} +\frac{1}{\Delta _{3} ( \lambda) }{ \int_{x}^{b}} \bigl( \chi_{31} ( t, \lambda) f_{2} ( t ) -\chi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{32} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{2},b ] ,\quad U_{3} ( x,\lambda) =}{} +\frac{1}{\Delta _{3} ( \lambda) }{ \int_{\xi_{2}}^{x}} \bigl( \varphi_{31} ( t,\lambda) f_{2} ( t ) -\varphi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{31} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{2},b ] ,\quad U_{3} ( x,\lambda) =}{} +\frac{1}{\Delta _{3} ( \lambda) }{ \int_{\xi_{2}}^{x}} \bigl( \varphi_{31} ( t,\lambda) f_{2} ( t ) -\varphi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{32} ( x,\lambda) \,dt \\ &\phantom{\text{for }x \in( \xi_{2},b ] ,\quad U_{3} ( x,\lambda)=} {}+c_{5}\varphi_{31} ( x,\lambda) +c_{6} \chi_{31} ( x,\lambda) +c_{5}\varphi_{32} ( x, \lambda) +c_{6}\chi_{32} ( x,\lambda) . \end{aligned}$$
(17)

Therefore, we easily see that \(l_{1} ( U_{1} ) =-c_{2}\Delta_{1} ( \lambda) \), \(l_{2} ( U_{3} ) =c_{5}\Delta_{3} ( \lambda) \). Since \(\Delta_{1} ( \lambda) \neq0\), \(\Delta_{2} ( \lambda) \neq0\), and from the boundary conditions (2)-(3), \(c_{2}=0\), and \(c_{5}=0\).

On the other hand, considering these results and transmission conditions (4)-(7), the following linear equation system according to the variables \(c_{1}\), \(c_{3}\), \(c_{4}\), and \(c_{6}\) is acquired:

$$\begin{aligned} & {-}c_{1}\varphi_{21} ( \xi_{1}+0 ) +c_{3}\varphi_{21} ( \xi_{1}+0 ) +c_{4} \chi_{21} ( \xi_{1}+0 ) \\ &\quad =-\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi_{22} ( t, \lambda) f_{1} ( t ) \bigr) \varphi_{21} ( \xi_{1}+0, \lambda) \,dt \\ &\qquad{} +\frac{1}{\Delta_{1} ( \lambda) }{ \int_{a}^{\xi_{1}}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{21} ( \xi_{1}+0,\lambda) \,dt, \\ & {-}c_{1}\varphi_{22} ( \xi_{1}+0 ) +c_{3}\varphi_{22} ( \xi_{1}+0 ) +c_{4} \chi_{22} ( \xi_{1}+0 ) \\ &\quad =-\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi_{22} ( t, \lambda) f_{1} ( t ) \bigr) \varphi_{22} ( \xi_{1}+0, \lambda) \,dt \\ &\qquad{} +\frac{1}{\Delta_{1} ( \lambda) }{ \int_{a}^{\xi_{1}}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{22} ( \xi_{1}+0,\lambda) \,dt, \\ & {-}c_{3}\varphi_{31} ( \xi_{2}+0 ) -c_{4}\chi_{31} ( \xi_{2}+0 ) +c_{6} \chi_{31} ( \xi_{2}+0 ) \\ &\quad =-\frac{1}{\Delta_{3} ( \lambda) }{ \int_{\xi_{2}}^{b}} \bigl( \chi_{31} ( t,\lambda) f_{2} ( t ) -\chi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{31} ( \xi_{2}+0, \lambda) \,dt \\ &\qquad{} +\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{31} ( \xi_{2}+0,\lambda) \,dt, \\ & {-}c_{3}\varphi_{32} ( \xi_{2}+0 ) -c_{4}\chi_{32} ( \xi_{2}+0 ) +c_{6} \chi_{32} ( \xi_{2}+0 ) \\ & \quad=-\frac{1}{\Delta_{3} ( \lambda) }{ \int_{\xi_{2}}^{b}} \bigl( \chi_{31} ( t,\lambda) f_{2} ( t ) -\chi_{32} ( t,\lambda) f_{1} ( t ) \bigr) \varphi_{32} ( \xi_{2}+0, \lambda) \,dt \\ &\qquad{} +\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \chi_{32} ( \xi_{2}+0,\lambda) \,dt. \end{aligned}$$
(18)

Recalling the definitions of the solutions \(\varphi_{ij} ( x,\lambda) \) and \(\chi_{ij} ( x,\lambda)\) (\(i=2,3\), \(j=1,2\)), the following relation is gotten for the determinant of this linear equation system:

$$\left\vert \begin{matrix} -\varphi_{21} ( \xi_{1}+0 ) & \varphi_{21} ( \xi_{1}+0 ) & \chi_{21} ( \xi_{1}+0 ) & 0\\ -\varphi_{22} ( \xi_{1}+0 ) & \varphi_{22} ( \xi_{1}+0 ) & \chi_{22} ( \xi_{1}+0 ) & 0\\ 0 & -\varphi_{31} ( \xi_{2}+0 ) & -\chi_{31} ( \xi _{2}+0 ) & \chi_{31} ( \xi_{2}+0 ) \\ 0 & -\varphi_{32} ( \xi_{2}+0 ) & -\chi_{32} ( \xi _{2}+0 ) & \chi_{32} ( \xi_{2}+0 ) \end{matrix} \right\vert =-\Delta_{2} ( \lambda) \Delta_{3} ( \lambda) . $$

Since the above determinant is different from zero, the solution of (18) is unique. When we solve system (18), we obtain the following equality for the coefficients \(c_{1}\), \(c_{3}\), \(c_{4}\), and \(c_{6}\):

$$\begin{aligned} c_{1}={}&\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \chi_{21} ( t,\lambda) f_{2} ( t ) -\chi _{22} ( t,\lambda) f_{1} ( t ) \bigr) \,dt \\ &{}+\frac{1}{\Delta_{3} ( \lambda) }{ \int_{\xi_{2}}^{b}} \bigl( \chi_{31} ( t,\lambda) f_{2} ( t ) -\chi _{32} ( t,\lambda) f_{1} ( t ) \bigr) \,dt, \\ c_{3}={}&\frac{1}{\Delta_{3} ( \lambda) }{ \int_{\xi_{2}}^{b}} \bigl( \chi_{31} ( t,\lambda) f_{2} ( t ) -\chi _{32} ( t,\lambda) f_{1} ( t ) \bigr) \,dt, \\ c_{4}={}&\frac{1}{\Delta_{1} ( \lambda) }{ \int_{a}^{\xi_{1}}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \,dt, \\ c_{6}={}&\frac{1}{\Delta_{1} ( \lambda) }{ \int_{a}^{\xi_{1}}} \bigl( \varphi_{11} ( t,\lambda) f_{2} ( t ) -\varphi_{12} ( t,\lambda) f_{1} ( t ) \bigr) \,dt \\ &{} +\frac{1}{\Delta_{2} ( \lambda) }{ \int_{\xi_{1}}^{\xi_{2}}} \bigl( \varphi_{21} ( t,\lambda) f_{2} ( t ) -\varphi_{22} ( t,\lambda) f_{1} ( t ) \bigr) \,dt. \end{aligned}$$

As a result, if we substitute the coefficients \(c_{i}\) (\(i=1,3,4,6\)) in (17), then we get the resolvent \(U ( x,\lambda) \) as follows:

$$ U ( x,\lambda) =\frac{\chi( x,\lambda) }{\Delta _{i} ( \lambda) }{ \int_{a}^{x}} ( f_{2} \varphi_{i1}-f_{1}\varphi_{i2} ) \,dt+ \frac{\varphi( x,\lambda) }{\Delta_{i} ( \lambda) }{ \int_{x}^{b}} ( f_{2} \chi_{i1}-f_{1}\chi_{i2} ) \,dt, \quad i= \overline{1,3} $$
(19)

such that

$$\begin{aligned} &\varphi( x,\lambda) = \textstyle\begin{cases} \bigl({\scriptsize\begin{matrix}{}\varphi_{11} ( x,\lambda) \cr \varphi_{12} ( x,\lambda)\end{matrix}}\bigr) ,& x\in[ a,\xi_{1} ), \\ \bigl({\scriptsize\begin{matrix}{} \varphi_{21} ( x,\lambda) \\ \varphi_{22} ( x,\lambda) \end{matrix}}\bigr) ,&x\in( \xi_{1},\xi_{2} ), \\ \bigl({\scriptsize\begin{matrix}{} \varphi_{31} ( x,\lambda) \\ \varphi_{32} ( x,\lambda) \end{matrix}}\bigr) ,&x\in( \xi_{2},b ], \end{cases}\displaystyle \\ &\chi( x,\lambda) = \textstyle\begin{cases} \bigl({\scriptsize\begin{matrix}{} \chi_{11} ( x,\lambda) \\ \chi_{12} ( x,\lambda) \end{matrix}}\bigr) ,& x\in[ a,\xi_{1} ) , \\ \bigl({\scriptsize\begin{matrix}{} \chi_{21} ( x,\lambda) \\ \chi_{22} ( x,\lambda) \end{matrix}}\bigr),&x\in( \xi_{1},\xi_{2} ) , \\ \bigl({\scriptsize\begin{matrix}{} \chi_{31} ( x,\lambda) \\ \chi_{32} ( x,\lambda) \end{matrix}}\bigr) ,&x\in( \xi_{2},b ]. \end{cases}\displaystyle \end{aligned}$$

We can easily find the Green’s function from the resolvent (19) as follows:

$$G ( x,t;\lambda) = \textstyle\begin{cases} \frac{\chi( x,\lambda) }{\Delta_{i} ( \lambda) } \bigl({\scriptsize\begin{matrix}{} -\varphi_{i2} ( x,\lambda) \\ \varphi_{i1} ( x,\lambda) \end{matrix}}\bigr) ^{T},&a\leq t\leq x\leq b,x\neq\xi_{1},\xi_{2}, t\neq\xi_{1},\xi_{2}, \\ \frac{\varphi( x,\lambda) }{\Delta_{i} ( \lambda) } \bigl({\scriptsize\begin{matrix}{} -\chi_{i2} ( x,\lambda) \\ \chi_{i1} ( x,\lambda) \end{matrix}}\bigr) ^{T},& a\leq t\leq x\leq b,x\neq\xi_{1},\xi_{2}, t\neq\xi_{1},\xi_{2}. \end{cases} $$

We can rewrite equation (19) in the following form:

$$U ( x,\lambda) ={ \int_{a}^{b}} G ( x,t;\lambda) f ( t ) \,dt\quad\mbox{such that }f ( t ) = \begin{pmatrix} f_{1} ( t ) \\ f_{2} ( t ) \end{pmatrix} . $$

Now, we define the resolvent operator

$$R ( T,\lambda) := ( T-\lambda I ) ^{-1}:H_{1}\rightarrow H_{1}. $$

It is easy to see that the operator equation \(( T-\lambda I ) Y=F\), \(F\in H_{1}\) is equivalent to the boundary value problem (15), (2)-(7) where λ is not an eigenvalue,

$$\begin{aligned} &Y= \bigl( y_{1} ( x ) ,y_{2} ( x ) ,y_{3},y_{4},y_{5},y_{6} \bigr) ^{T}\quad\mbox{such that }y_{3}=\alpha_{1}^{\prime}y_{1} ( a ) -\alpha_{2}^{\prime }y_{2} ( a ) , \\ &y_{4}=\gamma_{1}^{\prime}y_{1} ( b ) -\gamma_{2}^{\prime}y_{2} ( b ) ,\qquad y_{5}=y_{1} ( \xi_{1}-0 ) ,\qquad y_{6}=y_{1} ( \xi_{2}-0 )\quad \mbox{and} \\ &F= \bigl( f_{1} ( x ) ,f_{2} ( x ) ,z_{3},z_{4},z_{5},z_{6} \bigr) ^{T} \quad\mbox{where } z_{3}=z_{4}=z_{5}=z_{6}=0. \end{aligned}$$

4 Inverse problems

In this section, we study the inverse problems for the reconstruction of boundary value problem (1)-(7) by the Weyl function and spectral data.

We consider the boundary value problem L̃ which has the same form as L but with different coefficients \(\tilde{\Omega}(x)\), \(\tilde {\alpha }_{j}\), \(\tilde{\gamma}_{j}\), \(\tilde{\alpha}_{j}^{\prime}\), \(\tilde{\gamma}_{j}^{\prime}\), \(j=1,2\), such that

$$\tilde{\Omega}(x)= \begin{pmatrix} \tilde{p} ( x ) & q(x)\\ q(x) & \tilde{r} ( x ) \end{pmatrix}. $$

If a certain symbol σ denotes an object related to L, then the symbol σ̃ denotes the corresponding object related to L̃.

Let \(\Phi(x,\lambda)\) be a solution of equation (1) which satisfies the condition \(( \lambda\alpha_{2}^{\prime}-\alpha _{2} ) \Phi_{2}(a,\lambda)- ( \lambda\alpha_{1}^{\prime}-\alpha _{1} ) \Phi_{1}(a,\lambda)=1\) and the transmissions (4)-(7).

Assume that the function \(\phi(x,\lambda)= ( \phi_{1}(x,\lambda),\phi _{2}(x,\lambda) ) ^{T}\) is the solution of equation (1) that satisfies the conditions \(\phi_{1}(a,\lambda)=d_{1}^{-1}\alpha _{2}^{\prime}\), \(\phi_{2}(a,\lambda)=d_{1}^{-1}\alpha_{1}^{\prime}\) and the transmission conditions (4)-(7).

Since \(W [ \varphi,\phi] =1\), the functions ϕ and φ are linearly independent. Therefore, the function \(\psi(x,\lambda)\) can be represented by

$$\psi(x,\lambda)=d_{1}^{-1} \bigl( \alpha_{1}^{\prime} \psi_{1}(a,\lambda)-\alpha_{2}^{\prime} \psi_{2}(a,\lambda) \bigr) \varphi(x,\lambda)+\Delta( \lambda) \phi(x, \lambda) $$

or

$$ \Phi(x,\lambda)=\frac{\psi(x,\lambda)}{\Delta( \lambda) }=\phi(x,\lambda)+ \frac{\alpha_{1}^{\prime}\psi_{1}(a,\lambda)-\alpha _{2}^{\prime}\psi_{2}(a,\lambda)}{d_{1}\Delta( \lambda) }\varphi(x,\lambda), $$
(20)

which is called the Weyl solution, and

$$ \frac{\alpha_{1}^{\prime}\psi_{1}(a,\lambda)-\alpha_{2}^{\prime}\psi _{2}(a,\lambda)}{d_{1}\Delta(\lambda)}=M(\lambda)=d_{1}^{-1} \bigl( \alpha _{1}^{\prime}\Phi_{1}(a,\lambda)-\alpha_{2}^{\prime} \Phi_{2}(a,\lambda) \bigr) $$
(21)

is called the Weyl function.

Lemma 6

The following representation is true:

$$M(\lambda)={ \sum_{n=-\infty}^{\infty}} \frac{1}{\mu_{n} ( \lambda_{n}-\lambda) }. $$

Proof

The Weyl function \(M(\lambda)\) is a meromorphic function with respect to λ, which has simple poles at \(\lambda_{n}\). Therefore, we calculate

$$\operatorname{Re}s_{\lambda=\lambda_{n}}M(\lambda)=\frac {\alpha_{1}^{\prime}\psi_{1}(a,\lambda_{n})-\alpha_{2}^{\prime}\psi _{2}(a,\lambda_{n})}{d_{1}\dot{\Delta}(\lambda_{n})}. $$

Since

$$\kappa _{n}=\frac{\alpha_{1}^{\prime}\psi_{1}(a,\lambda_{n})-\alpha_{2}^{\prime }\psi_{2}(a,\lambda_{n})}{d_{1}} $$

and \(\dot{\Delta}(\lambda_{n})=-\kappa _{n}\mu_{n}\),

$$ \operatorname{Re}s_{\lambda=\lambda_{n}} M(\lambda)=- \frac{1}{\mu_{n}}. $$
(22)

Let \(\Gamma_{n}= \{ \lambda:\vert \lambda\vert \leq\vert \lambda_{n}^{o}\vert +\varepsilon\} \), where ε is a sufficiently small number. Consider the contour integral \(I_{n} ( \lambda) =\frac{1}{2\pi i}{ \int_{\Gamma_{n}}} \frac{M ( \mu) }{\mu-\lambda}\,d\mu\), \(\lambda\in \operatorname{int}\Gamma_{n}\).

Since \(\Delta( \lambda) \geq C_{\delta}\lambda ^{4}e^{\vert \operatorname{Im}\lambda\vert ( ( \xi _{1}-a ) \rho_{1}+ ( \xi_{2}-\xi_{1} ) \rho_{2}+ ( b-\xi_{2} ) \rho_{3} ) }\) and \(M(\lambda)=\frac{\alpha _{1}^{\prime}\psi_{1}(a,\lambda)-\alpha_{2}^{\prime}\psi_{2}(a,\lambda)}{d_{1}\Delta(\lambda)}\), \(\vert M ( \lambda) \vert \leq\frac{C_{\delta}}{\vert \lambda\vert }\) for \(\lambda\in F_{\delta}= \{ \lambda:\vert \lambda-\lambda_{n}\vert \geq\delta, n=0,\pm1,\ldots\} \), where δ is a sufficiently small number. Thus, \(\lim_{n\rightarrow\infty}I_{n} ( \lambda) =0\). Then the residue theorem yields

$$M(\lambda)={ \sum_{n=-\infty}^{\infty}} \frac{1}{\mu_{n} ( \lambda_{n}-\lambda) }. $$

 □

Theorem 2

If \(M(\lambda)=\tilde{M}(\lambda)\), then \(L=\tilde{L}\), i.e., \(\Omega (x)=\tilde{\Omega}(x)\), a.e. and \(\alpha_{j}=\tilde{\alpha}_{j}\), \(\gamma _{j}=\tilde{\gamma}_{j}\), \(\alpha_{j}^{\prime}=\tilde{\alpha }_{j}^{\prime}\), \(\gamma_{j}^{\prime}=\tilde{\gamma}_{j}^{\prime}\), \(j=1,2\).

Proof

We introduce a matrix \(P(x,\lambda)= [ P_{kj}(x,\lambda) ] _{k,j=1,2}\) by the formula

$$ P(x,\lambda) \begin{pmatrix} \tilde{\varphi}_{1} & \tilde{\Phi}_{1}\\ \tilde{\varphi}_{2} & \tilde{\Phi}_{2}\end{pmatrix} = \begin{pmatrix} \varphi_{1} & \Phi_{1}\\ \varphi_{2} & \Phi_{2}\end{pmatrix} $$
(23)

or

$$ \begin{pmatrix} P_{11}(x,\lambda) & P_{12}(x,\lambda)\\ P_{21}(x,\lambda) & P_{22}(x,\lambda) \end{pmatrix} = \begin{pmatrix} \varphi_{1}\tilde{\Phi}_{2}-\Phi_{1}\tilde{\varphi}_{2} & -\varphi_{1}\tilde{\Phi}_{1}+\Phi_{1}\tilde{\varphi}_{1}\\ \varphi_{2}\tilde{\Phi}_{2}-\tilde{\varphi}_{2}\Phi_{2} & -\varphi_{2}\tilde{\Phi}_{1}+\tilde{\varphi}_{1}\Phi_{2}\end{pmatrix} , $$
(24)

where \(\Phi(x,\lambda)=\frac{\psi(x,\lambda)}{\Delta(\lambda)}\) and \(W ( \tilde{\Phi},\tilde{\varphi} ) =1\). Thus, we find

$$\begin{aligned} \begin{aligned} &P_{11}(x,\lambda) =\varphi_{1}(x,\lambda) \frac{\tilde{\psi}_{2}(x,\lambda)}{\tilde{\Delta}(x,\lambda)}-\frac{\psi_{1}(x,\lambda)}{\Delta(\lambda)}\tilde{\varphi}_{2}(x,\lambda), \\ &P_{12}(x,\lambda) =-\varphi_{1}(x,\lambda) \frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta}(\lambda)}+\frac{\psi_{1}(x,\lambda)}{\Delta(\lambda)}\tilde{\varphi}_{1}(x,\lambda), \\ &P_{21}(x,\lambda) =\varphi_{2}(x,\lambda) \frac{\tilde{\psi}_{2}(x,\lambda)}{\tilde{\Delta}(\lambda)}-\frac{\psi_{2}(x,\lambda)}{\Delta(\lambda)}\tilde{\varphi}_{2}(x,\lambda), \\ &P_{22}(x,\lambda) =-\varphi_{2}(x,\lambda) \frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta}(\lambda)}+\frac{\psi_{2}(x,\lambda)}{\Delta(\lambda)}\tilde{\varphi}_{1}(x,\lambda). \end{aligned} \end{aligned}$$
(25)

On the other hand, from (20), we get

$$\begin{aligned} \begin{aligned} &P_{11}(x,\lambda) =\varphi_{1}(x,\lambda)\tilde{ \phi}_{2} ( x,\lambda) -\tilde{\varphi}_{2}(x,\lambda) \phi_{1} ( x,\lambda) + \bigl( \tilde{M} ( \lambda) -M ( \lambda) \bigr) \varphi_{1}(x,\lambda)\tilde{\varphi}_{2}(x, \lambda), \\ &P_{12}(x,\lambda) =-\varphi_{1}(x,\lambda)\tilde{ \phi}_{1} ( x,\lambda) +\tilde{\varphi}_{1}(x,\lambda) \phi_{1} ( x,\lambda) - \bigl( \tilde{M} ( \lambda) -M ( \lambda) \bigr) \varphi_{1}(x,\lambda)\tilde{\varphi}_{1}(x, \lambda), \\ &P_{21}(x,\lambda) =\varphi_{2}(x,\lambda)\tilde{ \phi}_{2} ( x,\lambda) -\tilde{\varphi}_{2}(x,\lambda) \phi_{2} ( x,\lambda) + \bigl( \tilde{M} ( \lambda) -M ( \lambda) \bigr) \varphi_{2}(x,\lambda)\tilde{\varphi}_{2}(x, \lambda), \\ &P_{22}(x,\lambda) =-\varphi_{2}(x,\lambda)\tilde{ \phi}_{1} ( x,\lambda) +\tilde{\varphi}_{1}(x,\lambda) \phi_{2} ( x,\lambda) - \bigl( \tilde{M} ( \lambda) -M ( \lambda) \bigr) \tilde{\varphi}_{1}(x,\lambda)\varphi_{2}(x, \lambda). \end{aligned} \end{aligned}$$
(26)

Thus, if \(M(\lambda)\equiv\tilde{M}(\lambda)\) then the functions \(P_{ij}(x,\lambda)\) (\(i,j=1,2\)) are entire in λ for each fixed x. Moreover, since asymptotic behaviors of \(\varphi_{i}(x,\lambda)\), \(\tilde{\varphi}_{i}(x,\lambda)\), \(\psi _{i}(x,\lambda)\), \(\tilde{\psi}_{i}(x,\lambda)\), and \(\vert \Delta (\lambda)\vert \geq C_{\delta}\vert \lambda\vert ^{4}e^{\vert \operatorname{Im}\lambda\vert ((\xi_{1}-a)\rho _{1}+(\xi_{2}-\xi_{1})\rho_{2}+(b-\xi_{2})\rho_{3})}\) in \(F_{\delta}\cap \tilde{F}_{\delta}\), we can easily see that the functions \(P_{ij}(x,\lambda)\) are bounded with respect to λ. As a result, these functions do not depend on λ. Here, we denote \(\tilde{F}_{\delta}= \{ \lambda:\vert \lambda-\tilde{\lambda}_{n}\vert \geq\delta, n=0,\pm1,\pm2,\ldots\} \) where n is sufficiently small number, \(\tilde{\lambda}_{n}\) are eigenvalues of the problem L̃.

From (25), we get

$$\begin{aligned} &P_{11}(x,\lambda)-1=\frac{\tilde{\psi}_{2}(x,\lambda) ( \varphi _{1}(x,\lambda)-\tilde{\varphi}_{1}(x,\lambda) ) }{\tilde{\Delta} ( \lambda) }-\tilde{\varphi}_{2}(x, \lambda) \biggl( \frac{\psi _{1}(x,\lambda)}{\Delta(\lambda)}-\frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta}(\lambda)} \biggr) , \\ &P_{12}(x,\lambda)=\frac{\psi_{1}(x,\lambda) ( \tilde{\varphi}_{1}(x,\lambda)-\varphi_{1}(x,\lambda) ) }{\Delta( \lambda) }+\varphi_{1}(x, \lambda) \biggl( \frac{\psi_{1}(x,\lambda)}{\Delta(\lambda )}-\frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta}(\lambda)} \biggr) , \\ &P_{21}(x,\lambda)=\varphi_{2}(x,\lambda) \biggl( \frac{\tilde{\psi}_{2}(x,\lambda)}{\tilde{\Delta}(\lambda)}-\frac{\psi_{2}(x,\lambda)}{\Delta( \lambda) } \biggr) +\psi_{2}(x,\lambda) \biggl( \frac{\varphi_{2}(x,\lambda)-\tilde{\varphi}_{2}(x,\lambda)}{\Delta (\lambda )} \biggr), \\ & P_{22}(x,\lambda)-1=\frac{\psi_{2}(x,\lambda) ( \tilde{\varphi}_{1}(x,\lambda)-\varphi_{1}(x,\lambda) ) }{\Delta( \lambda) }+\varphi_{2}(x, \lambda) \biggl( \frac{\psi_{1}(x,\lambda)}{\Delta(\lambda )}-\frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta}(\lambda)} \biggr) . \end{aligned}$$

It follows from the representations of the solutions \(\varphi(x,\lambda )\) and \(\psi(x,\lambda)\), that

$$\lim_{\lambda\rightarrow \infty}\frac{\tilde{\psi}_{2}(x,\lambda) ( \varphi_{1}(x,\lambda)-\tilde{\varphi}_{1}(x,\lambda) ) }{\tilde{\Delta} ( \lambda) }=0 \quad\mbox{and} \quad \lim_{\lambda\rightarrow\infty} \tilde{ \varphi}_{2}(x,\lambda) \biggl( \frac{\psi_{1}(x,\lambda)}{\Delta(\lambda)}-\frac{\tilde{\psi}_{1}(x,\lambda)}{\tilde{\Delta }(\lambda )} \biggr) =0 $$

for all x in Λ. Thus, \(\lim_{\lambda \rightarrow\infty} ( P_{11}(x,\lambda)-1 ) =0\) is valid uniformly with respect to x. So we have \(P_{11}(x,\lambda )\equiv1\) and similarly \(P_{12}(x,\lambda)\equiv0\), \(P_{21}(x,\lambda )\equiv 0\), \(P_{22}(x,\lambda)\equiv1\).

From (23), we obtain \(\varphi_{1}(x,\lambda)\equiv\tilde{\varphi }_{1}(x,\lambda)\), \(\Phi_{1}\equiv\tilde{\Phi}_{1}\), \(\varphi_{2}(x,\lambda)\equiv\tilde{\varphi}_{2}(x,\lambda)\), and \(\Phi_{2}\equiv \tilde{\Phi}_{2}\) for all x and λ. Moreover, from \(\Phi (x,\lambda)=\frac{\psi(x,\lambda)}{\Delta(\lambda)}\), we get \(\frac{\psi_{2} ( x,\lambda) }{\psi_{1} ( x,\lambda) }=\frac{\tilde{\psi}_{2} ( x,\lambda) }{\tilde{\psi}_{1} ( x,\lambda) }\). Hence, \(\Omega(x)=\tilde{\Omega}(x)\), i.e., \(p ( x ) =\tilde{p} ( x ) \), \(r ( x ) =\tilde{r} ( x ) \) almost everywhere. On the other hand, since

$$ \begin{pmatrix} \varphi_{11}(a,\lambda)\\ \varphi_{12}(a,\lambda) \end{pmatrix} = \begin{pmatrix} \lambda\alpha_{2}^{\prime}-\alpha_{2}\\ \lambda\alpha_{1}^{\prime}-\alpha_{1}\end{pmatrix} , \qquad \begin{pmatrix} \psi_{31}(b,\lambda)\\ \psi_{32}(b,\lambda) \end{pmatrix} = \begin{pmatrix} \lambda\gamma_{2}^{\prime}+\gamma_{2}\\ \lambda\gamma_{1}^{\prime}+\gamma_{1}\end{pmatrix}, $$

we easily see that \(\alpha_{2}^{\prime}=\tilde{\alpha}_{2}^{\prime}\), \(\alpha_{2}=\tilde{\alpha}_{2}\), \(\alpha_{1}^{\prime}=\tilde{\alpha}_{1}^{\prime}\), \(\alpha_{1}=\tilde{\alpha}_{1}\), and \(\gamma _{2}^{\prime}=\tilde{\gamma}_{2}^{\prime}\), \(\gamma_{2}=\tilde{\gamma}_{2}\), \(\gamma_{1}^{\prime}=\tilde{\gamma}_{1}^{\prime}\), \(\gamma_{1}=\tilde {\gamma }_{1}\). Therefore, \(L\equiv\tilde{L}\). □

Theorem 3

If \(\lambda_{n}=\tilde{\lambda}_{n}\) and \(\mu_{n}=\tilde{\mu}_{n}\) for all n, then \(L\equiv\tilde{L}\), i.e., \(\Omega(x)=\tilde{\Omega }(x)\), a.e., \(\alpha_{j}=\tilde{\alpha}_{j}\), \(\gamma_{i}=\tilde{\gamma}_{i}\), \(\alpha _{j}^{\prime}=\tilde{\alpha}_{j}^{\prime}\), \(\gamma_{j}^{\prime}=\tilde {\gamma}_{j}^{\prime}\), \(j=1,2\). Hence, the problem (1)-(7) is uniquely determined by the spectral data \(\{ \lambda_{n},\mu_{n} \} \).

Proof

If \(\lambda_{n}=\tilde{\lambda}_{n}\) and \(\mu_{n}=\tilde{\mu}_{n}\) for all n, then \(M(\lambda)=\tilde{M}(\lambda)\) by Lemma 6. Therefore, we get \(L=\tilde{L}\) by Theorem 2. □

Let us consider the boundary value problem \(L_{1}\) with the condition \(\alpha_{1}^{\prime}y_{1}(a,\lambda)-\alpha_{2}^{\prime}y_{2}(a,\lambda)=0\) instead of the condition (2) in L. Let \(\{ \tau_{n} \} _{n\in \mathbb{Z} }\) be the eigenvalues of the problem \(L_{1}\). It is clear that the \(\tau _{n}\) are zeros of \(\Delta_{1}(\tau):=\alpha_{1}^{\prime}\psi_{1}(a,\tau )-\alpha_{2}^{\prime}\psi_{2}(a,\tau)\), which is the characteristic function of \(L_{1}\).

Theorem 4

If \(\lambda_{n}=\tilde{\lambda}_{n}\) and \(\tau_{n}=\tilde{\tau}_{n}\) for all n, then \(L(\Omega,\gamma_{i},\gamma_{j}^{\prime})=L(\tilde{\Omega},\tilde{\gamma}_{i},\tilde{\gamma}_{j}^{\prime})\), \(j=1,2\).

Hence, the problem L is uniquely determined by the sequences \(\{ \lambda_{n} \} \) and \(\{ \tau_{n} \} \) except coefficients \(\alpha_{j}\) and \(\alpha_{j}^{\prime}\).

Proof

Since the characteristic functions \(\Delta(\lambda)\) and \(\Delta _{1}(\tau)\) are entire of order 1, the functions \(\Delta(\lambda)\) and \(\Delta _{1}(\tau)\) are uniquely determined up to a multiplicative constant with their zeros by Hadamard’s factorization theorem [49],

$$\begin{aligned} &\Delta( \lambda) =C{ \prod _{n=-\infty}^{\infty}} \biggl( 1-\frac{\lambda}{\lambda_{n}} \biggr) , \\ & \Delta_{1}(\tau)=C_{1}{ \prod_{n=-\infty}^{\infty}} \biggl( 1- \frac{\tau}{\tau_{n}} \biggr) , \end{aligned}$$

where C and \(C_{1}\) are constants depending on \(\{ \lambda_{n} \} \) and \(\{ \tau _{n} \} \), respectively. When \(\lambda_{n}=\tilde{\lambda}_{n}\) and \(\tau_{n}=\tilde{\tau}_{n}\) for all n, \(\Delta(\lambda)\equiv\tilde{\Delta}(\lambda)\) and \(\Delta_{1}(\tau )\equiv\tilde{\Delta}_{1}(\tau)\). Hence, \(\alpha_{1}^{\prime}\psi_{1}(a,\tau)-\alpha_{2}^{\prime}\psi_{2}(a,\tau)=\tilde{\alpha}_{1}^{\prime}\tilde{\psi}_{1}(a,\tau)-\tilde{\alpha}_{2}^{\prime}\tilde {\psi }_{2}(a,\tau)\). As a result, we get \(M(\lambda)=\tilde{M}(\lambda)\) by (21). So, the proof is completed by Theorem 2. □