The inverse discrete transmission eigenvalue problem for absorbing media

Abstract

The inverse problem for the discrete analogue of the transmission eigenvalue problem for absorbing media with a spherically symmetric index of refraction $\mathit{\varrho }$ is considered. Some uniqueness results are provided which imply that $\mathit{\varrho }$ can be recovered uniquely if only the all transmission eigenvalues (counting with their multiples) are given together with partial information on the entries of $\mathit{\varrho }$.

Keywords:

• Absorbing media
• transmission eigenvalue
• inverse spectral problem
• the extended Euclidean algorithm
• discrete version

AMS Subject Classifications:

• Primary: 15A29
• Secondary: 34A55

1. Introduction

Transmission eigenvalue problems have received widespread attention and have become an important area of research in inverse scattering theory, especially in the case of acoustic waves and the electromagnetic waves (see [1–8] and the references therein). The main interest is motivated by the fact that transmission eigenvalues carry information about the material properties of the scattering object and that these eigenvalues can in principle be determined from the scattering data (cf. [3,9]).

The interior transmission problem is a non-selfadjoint boundary-value problem for a pair of fields $\mathit{ϵ}(\mathbf{x})$ and $\mathit{\varrho }(\mathbf{x})$ in a bounded and simply connected domain $\mathrm{\Omega }$ of ${\mathbf{R}}^n$ with the sufficiently smooth boundary $\mathit{\partial }\mathrm{\Omega }.$ Let $\mathrm{\Omega }={\mathrm{\Omega }}_b$ be a ball of radius b in ${\mathbf{R}}^3$ and $\mathit{\varrho }(\mathbf{x})=\widehat{\mathit{\rho }}(x)+i\frac{\mathit{\rho }(x)}{\mathit{\lambda }}$ be a spherically symmetric function corresponding to the square of the refractive index of the medium and $\mathit{ϵ}(\mathbf{x})=\widehat{\mathit{\gamma }}(x)+i\frac{\mathit{\gamma }(x)}{\mathit{\lambda }}$ be another spherically symmetric function i.e. the medium with support ${\mathrm{\Omega }}_b$ and index of refraction $\mathit{\varrho }(x)$ is embedded in a background with index of refraction $\mathit{ϵ}(x)$ at location $\mathbf{x}$ in the electromagnetic case of the reciprocal or the sound speed $v(\mathbf{x})$ in the acoustic case [7].

It is straightforward to see that the corresponding boundary value problem is [10]:(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\omega }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(x)\mathit{\omega }+i\mathit{\lambda }\mathit{\rho }(x)\mathit{\omega }=0,\hfill & 0<x<a,\hfill \\ {\mathit{\upsilon }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(x)\mathit{\upsilon }+i\mathit{\lambda }\mathit{\gamma }(x)\mathit{\upsilon }=0,\hfill & 0<x<a,\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill & \\ \mathit{\omega }(a)=\mathit{\upsilon }(a)\mathrm{a}nd{\mathit{\omega }}^{\prime }(a)={\mathit{\upsilon }}^{\prime }(a).\hfill \end{array}\right.\end{array}$$(1.1)

Here $\widehat{\mathit{\rho }}(x)>0,\mathit{\rho }(x)\ge 0$ and $\widehat{\mathit{\gamma }}(x)>0,\mathit{\gamma }(x)\ge 0$ for $0<x<a$. When $\mathit{\rho }(x)=0$ and $\mathit{\gamma }(x)=0,$ absorption is not present in either the background or inhomogeneity.

This paper is concerned with the inverse discrete transmission eigenvalue problem for absorbing media, which is defined as follows. Let $\mathit{\omega }(n)$ be a complex-valued function of the discrete variable n and $A_K$ be the discrete version of the operator $-{\mathrm{d}}^2/\mathrm{d}{x}^2$, defined by(1.2) $$\begin{array}{c}\hfill (A_K\mathit{\omega })(n)=:-\mathit{\omega }(n+1)-\mathit{\omega }(n-1)+2\mathit{\omega }(n),\end{array}$$(1.2)

where $n=1,2,\dots ,K.$ The discrete transmission eigenvalue problem for absorbing media can be formulated as(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3)

with the transmission conditions(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4)

Here both the natural numbers M, N are given, $\mathit{\omega }(n),\mathit{\upsilon }(n)$ denote the wave-function $\mathit{\omega }(\mathit{\lambda },n),\mathit{\upsilon }(\mathit{\lambda },n),$$\mathit{\rho }(n)=:{\mathit{\rho }}_n\ge 0$, $\widehat{\mathit{\rho }}(n)=:{\widehat{\mathit{\rho }}}_n>0$ for all $n=1,2,\dots ,N$, and $\mathit{\gamma }(n)=:{\mathit{\gamma }}_n\ge 0$, $\widehat{\mathit{\gamma }}(n)=:{\widehat{\mathit{\gamma }}}_n>0$ for all $n=1,2,\dots ,M,$ denote the potential at n, are the discrete indexes of refractions [11]. The $\mathit{\lambda }\in \mathbb{C}$ in (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) ) is the spectral parameter as usual. Those $\mathit{\lambda }$-values for which there exists a nontrivial solution pair $(\overrightarrow{\mathit{\omega }},\overrightarrow{\mathit{\upsilon }})$ to (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are called the transmissions eigenvalues. It will be shown in Section 2 below that the number of the transmission eigenvalues (repeated eigenvalues are allowed) is at most $(2M+2N-2)$. The eigenvalue problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) can be viewed as the discrete version of the transmission eigenvalue problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\omega }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(x)\mathit{\omega }+i\mathit{\lambda }\mathit{\rho }(x)\mathit{\omega }=0,\hfill & 0<x<a,\hfill \\ {\mathit{\upsilon }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(x)\mathit{\upsilon }+i\mathit{\lambda }\mathit{\gamma }(x)\mathit{\upsilon }=0,\hfill & 0<x<a,\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill & \\ \mathit{\omega }(a)=\mathit{\upsilon }(a)\mathrm{a}nd{\mathit{\omega }}^{\prime }(a)={\mathit{\upsilon }}^{\prime }(a).\hfill \end{array}\right.\end{array}$$(1.1) ) with N difference nodes for the first equation and M difference nodes for the second equation.

Let us mention that, for problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ), if M and N become large enough, then the difference operators $A_M$ and $A_N$ defined by (1.2(1.2) $$\begin{array}{c}\hfill (A_K\mathit{\omega })(n)=:-\mathit{\omega }(n+1)-\mathit{\omega }(n-1)+2\mathit{\omega }(n),\end{array}$$(1.2) ) converges to the differential operator $-{\mathrm{d}}^2/\mathrm{d}{x}^2.$ Together with the accurate description of Borcea et al. (for details, see [12,13]) for the connection between the discrete problems and the continuous boundary problems, one knows that the transmission eigenvalues of the discrete problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) converge to the eigenvalues of (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\omega }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(x)\mathit{\omega }+i\mathit{\lambda }\mathit{\rho }(x)\mathit{\omega }=0,\hfill & 0<x<a,\hfill \\ {\mathit{\upsilon }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(x)\mathit{\upsilon }+i\mathit{\lambda }\mathit{\gamma }(x)\mathit{\upsilon }=0,\hfill & 0<x<a,\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill & \\ \mathit{\omega }(a)=\mathit{\upsilon }(a)\mathrm{a}nd{\mathit{\omega }}^{\prime }(a)={\mathit{\upsilon }}^{\prime }(a).\hfill \end{array}\right.\end{array}$$(1.1) ).

In recent years, all of the research on transmission eigenvalue and its inverse problem concentrates mainly on considering the case that $\mathit{\rho }(x)\equiv 0$ and $\mathit{\gamma }(x)\equiv 0,$ i.e. without absorbing media(see [1,4,8,11,14,17] and the references therein). For (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\omega }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(x)\mathit{\omega }+i\mathit{\lambda }\mathit{\rho }(x)\mathit{\omega }=0,\hfill & 0<x<a,\hfill \\ {\mathit{\upsilon }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(x)\mathit{\upsilon }+i\mathit{\lambda }\mathit{\gamma }(x)\mathit{\upsilon }=0,\hfill & 0<x<a,\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill & \\ \mathit{\omega }(a)=\mathit{\upsilon }(a)\mathrm{a}nd{\mathit{\omega }}^{\prime }(a)={\mathit{\upsilon }}^{\prime }(a).\hfill \end{array}\right.\end{array}$$(1.1) ), Aktosun et al. [1] and Wei and Xu [17] gave the conditions to determine uniquely $\widehat{\mathit{\rho }}(x)$ from the transmission eigenvalues; On the other hand, for the inverse discrete transmission eigenvalue problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ), Papanicolaou and Doumas [11] and the author [16] provided the uniqueness theorem in the situation that $M=N$ and all $\widehat{\mathit{\gamma }}(n)\equiv 1$.

The transmission eigenvalue problem for absorbing media was studied by [5,18–20], Particularly, Cakoni et al. [5] showed that the transmission eigenvalues of (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\omega }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(x)\mathit{\omega }+i\mathit{\lambda }\mathit{\rho }(x)\mathit{\omega }=0,\hfill & 0<x<a,\hfill \\ {\mathit{\upsilon }}^{″}+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(x)\mathit{\upsilon }+i\mathit{\lambda }\mathit{\gamma }(x)\mathit{\upsilon }=0,\hfill & 0<x<a,\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill & \\ \mathit{\omega }(a)=\mathit{\upsilon }(a)\mathrm{a}nd{\mathit{\omega }}^{\prime }(a)={\mathit{\upsilon }}^{\prime }(a).\hfill \end{array}\right.\end{array}$$(1.1) ) exist and form at most a discrete set. Though nowadays there are only a rather limited number results on the inverse transmission eigenvalue problem for absorbing media in our knowledge (see [10]).

The main purpose of this paper is, for the cases $M\ge N$ and $M<N$ in Problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ), to study the problem whether ${\mathit{\rho }}_n,{\widehat{\mathit{\rho }}}_n$ ($n=1,\dots ,N$) can be recovered by all of the transmission eigenvalues provided that ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) are known a priori. More precisely, we proved that, when $M>N$, ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined by all transmission eigenvalues; When $M=N$, ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined by all transmission eigenvalues as well as one of ${\mathit{\rho }}_n$, ${\widehat{\mathit{\rho }}}_n$ or constant $\mathit{\delta }$ known a priori; And when $M<N$, all transmission eigenvalues together with some partial information on the entries of ${\mathit{\rho }}_n$ and ${\widehat{\mathit{\rho }}}_n$ can uniquely determine all of ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$

The technique we use to prove our uniqueness theorems is the extended Euclidean algorithm for polynomial. We should like to express our thanks to the anonymous referee for his/her suggestion to use this technique, which have improved the manuscript better than those originally submitted. In fact, the process to prove our results provides a reconstruction procedure of functions ${\mathit{\rho }}_n$ and ${\widehat{\mathit{\rho }}}_n$.

Our paper proceeds as follows. In the next section, we provide some preparatory material, which contains some polynomials and their properties associated with our inverse problem. In Sections 3 and 4, we establish our main results.

2. Preliminaries

In this section, we provide some polynomials and their properties associated with Problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ), which will be used later to prove our main results.

We begin by defining polynomials of a complex variable $\mathit{\lambda }$, say ${\{U(n,\mathit{\lambda })\}}_{n=0}^{N+1},$ which is the unique solution of equation(2.1) $$\begin{array}{c}\hfill (A_N\mathit{\omega })(n,\mathit{\lambda })=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n)\end{array}$$(2.1)

with initial conditions(2.2) $$\begin{array}{c}\hfill U(0,\mathit{\lambda })=0,U(1,\mathit{\lambda })=1.\end{array}$$(2.2)

By straightforward induction, $U(n,\mathit{\lambda })$ is of degree $2n-2$ and formulated as(2.3) $$\begin{array}{c}\hfill U(n,\mathit{\lambda })={(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2n-2}+i{(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\rho }}}_j\sum _{j=1}^{n-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}{\mathit{\lambda }}^{2n-3}+\cdots +n\end{array}$$(2.3)

for $n=2,\dots ,N+1$. Here, $i^2=-1$ and the coefficients can be obtained inductively by using the following recursive formula(2.4) $$\begin{array}{c}\hfill U(n+1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n-1,\mathit{\lambda }).\end{array}$$(2.4)

Thus $U(n,\mathit{\lambda })$ has the form(2.5) $$\begin{array}{c}\hfill U(n,\mathit{\lambda })=\sum _{j=0}^{2n-2}{c}_{2n-1-j}(n){\mathit{\lambda }}^j,n=2,\dots ,N+1.\end{array}$$(2.5)

It should be noted that the coefficients of the first and second terms of $U(n,\mathit{\lambda })$ are(2.6) $$\begin{array}{c}\hfill {(-1)}^{n-1}{\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\cdots {\widehat{\mathit{\rho }}}_{n-1}\end{array}$$(2.6)

and(2.7) $$\begin{array}{c}\hfill i{(-1)}^{n-1}{\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\dots {\widehat{\mathit{\rho }}}_{n-1}\sum _{j=1}^{n-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j},\end{array}$$(2.7)

respectively. Similarly, for the second equation of (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) ), we will refer $P_n(\mathit{\lambda })$ is the solution of(2.8) $$\begin{array}{c}\hfill -P_{n+1}(\mathit{\lambda })-P_{n-1}(\mathit{\lambda })+2P_n(\mathit{\lambda })=(i\mathit{\lambda }{\mathit{\gamma }}_n+{\mathit{\lambda }}^2{\widehat{\mathit{\gamma }}}_n)P_n(\mathit{\lambda }),\end{array}$$(2.8)

with the initial conditions(2.9) $$\begin{array}{c}\hfill P_0(\mathit{\lambda })=0,P_1(\mathit{\lambda })=1.\end{array}$$(2.9)

Thus(2.10) $$\begin{array}{c}\hfill P_n(\mathit{\lambda })={(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\gamma }}}_j{\mathit{\lambda }}^{2n-2}+i{(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\gamma }}}_j\sum _{j=1}^{n-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}{\mathit{\lambda }}^{2n-3}+\cdots +n\end{array}$$(2.10)

for $n=2,\dots ,M+1$.

Lemma 2.1:

The polynomials (in $\mathit{\lambda }$) $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ ($P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$) are relatively prime.

Proof If we suppose $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ are not relatively prime, that is, there exists ${\mathit{\lambda }}_0\in \mathbb{C}$ satisfying $U(N,{\mathit{\lambda }}_0)=U(N+1,{\mathit{\lambda }}_0)=0$, in virtue of (2.4(2.4) $$\begin{array}{c}\hfill U(n+1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n-1,\mathit{\lambda }).\end{array}$$(2.4) ), then $U(N-1,{\mathit{\lambda }}_0)=0$, it follows that $U(n,{\mathit{\lambda }}_0)=0$ for all n, which contradicts $U(1,\mathit{\lambda })=1$ in (2.2(2.2) $$\begin{array}{c}\hfill U(0,\mathit{\lambda })=0,U(1,\mathit{\lambda })=1.\end{array}$$(2.2) ). Similarly, we can prove the polynomials $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are relatively prime.

Let us consider the transmission eigenvalues of Problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ). If $(\overrightarrow{\mathit{\omega }},\overrightarrow{\mathit{\upsilon }})$ is a nontrivial solution, then (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) ) implies that$$\mathit{\omega }(n)=cU(n,\mathit{\lambda }),\mathit{\upsilon }(n)=c^{\prime }{P}_n(\mathit{\lambda }),$$

where $c,c^{\prime }$ are nonzero constants. Thus (1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) can be written as$$\frac{U(N+1,\mathit{\lambda })}{P_{M+1}(\mathit{\lambda })}=\frac{U(N,\mathit{\lambda })}{P_M(\mathit{\lambda })},$$

which is equivalent to(2.11) $$\begin{array}{c}\hfill P_M(\mathit{\lambda })U(N+1,\mathit{\lambda })-P_{M+1}(\mathit{\lambda })U(N,\mathit{\lambda })=0.\end{array}$$(2.11)

If we set(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12)

then the transmission eigenvalues of (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) and the zeros of the polynomial ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ all coincide. It should be noted that the multiplicity of some zeros of ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ may be greater than 1. We refer to the multiplicity of a zero ${\mathit{\lambda }}_j$ of ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ also as the multiplicity of the transmission eigenvalue ${\mathit{\lambda }}_j$ and call ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ the characteristic polynomial of (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ).

Substituting (2.3(2.3) $$\begin{array}{c}\hfill U(n,\mathit{\lambda })={(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2n-2}+i{(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\rho }}}_j\sum _{j=1}^{n-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}{\mathit{\lambda }}^{2n-3}+\cdots +n\end{array}$$(2.3) ) and (2.10(2.10) $$\begin{array}{c}\hfill P_n(\mathit{\lambda })={(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\gamma }}}_j{\mathit{\lambda }}^{2n-2}+i{(-1)}^{n-1}\prod _{j=1}^{n-1}{\widehat{\mathit{\gamma }}}_j\sum _{j=1}^{n-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}{\mathit{\lambda }}^{2n-3}+\cdots +n\end{array}$$(2.10) ) into (2.12(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12) ), we get(2.13) $$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })& ={(-1)}^{M+N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-2}\hfill \\ \hfill & +i{(-1)}^{M+N-1}\left[({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\left(\sum _{j=1}^{N-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}+\sum _{j=1}^{M-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}\right)+({\mathit{\rho }}_N-{\mathit{\gamma }}_M)\right]\hfill \\ \hfill & \times \prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-3}+\cdots +(M-N).\hfill \end{array}$$(2.13)

It is clear that the degree of ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ is at most $2M+2N-2$, and if ${\widehat{\mathit{\rho }}}_N={\widehat{\mathit{\gamma }}}_M$, then the leading coefficient of ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ vanishes. When in addition ${\mathit{\rho }}_N={\mathit{\gamma }}_M$, together with (2.4(2.4) $$\begin{array}{c}\hfill U(n+1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n-1,\mathit{\lambda }).\end{array}$$(2.4) ) and (2.8(2.8) $$\begin{array}{c}\hfill -P_{n+1}(\mathit{\lambda })-P_{n-1}(\mathit{\lambda })+2P_n(\mathit{\lambda })=(i\mathit{\lambda }{\mathit{\gamma }}_n+{\mathit{\lambda }}^2{\widehat{\mathit{\gamma }}}_n)P_n(\mathit{\lambda }),\end{array}$$(2.8) ), we infer that ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })={\mathrm{\Delta }}_{M-1,N-1}(\mathit{\lambda })$. Moreover, if $M=N$, (2.13(2.13) $$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })& ={(-1)}^{M+N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-2}\hfill \\ \hfill & +i{(-1)}^{M+N-1}\left[({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\left(\sum _{j=1}^{N-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}+\sum _{j=1}^{M-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}\right)+({\mathit{\rho }}_N-{\mathit{\gamma }}_M)\right]\hfill \\ \hfill & \times \prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-3}+\cdots +(M-N).\hfill \end{array}$$(2.13) ) implies that ${\mathrm{\Delta }}_{N,N}(0)=0$, which yields that 0 is always a transmission eigenvalue in the case $M=N$.

Assume ${\widehat{\mathit{\rho }}}_N\ne {\widehat{\mathit{\gamma }}}_M,{\mathit{\rho }}_N\ne {\mathit{\gamma }}_M,$ and let ${\mathit{\lambda }}_1,\dots ,{\mathit{\lambda }}_{2M+2N-2}$ be the zeros of ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ (repeated roots are allowed), we have(2.14) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\mathit{\delta }\prod _{j=1}^{2M+2N-2}(\mathit{\lambda }-{\mathit{\lambda }}_j)=:\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda }),\end{array}$$(2.14)

where(2.15) $$\begin{array}{c}\hfill \mathit{\delta }={(-1)}^{M+N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j.\end{array}$$(2.15)

In particular, when $M\ne N,$ we also have from (2.13(2.13) $$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })& ={(-1)}^{M+N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-2}\hfill \\ \hfill & +i{(-1)}^{M+N-1}\left[({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_M)\left(\sum _{j=1}^{N-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}+\sum _{j=1}^{M-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}\right)+({\mathit{\rho }}_N-{\mathit{\gamma }}_M)\right]\hfill \\ \hfill & \times \prod _{j=1}^{M-1}{\widehat{\mathit{\gamma }}}_j\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j{\mathit{\lambda }}^{2M+2N-3}+\cdots +(M-N).\hfill \end{array}$$(2.13) ) and (2.14(2.14) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\mathit{\delta }\prod _{j=1}^{2M+2N-2}(\mathit{\lambda }-{\mathit{\lambda }}_j)=:\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda }),\end{array}$$(2.14) )(2.16) $$\begin{array}{c}\hfill \mathit{\delta }=\frac{M-N}{{\prod }_{j=1}^{2M+2N-2}{\mathit{\lambda }}_j}.\end{array}$$(2.16)

3. The case $M\ge N$

In this section, we will consider the following inverse problem for $M\ge N$ : Given all transmission eigenvalues, reconstruct ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$.

Note that if ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) are known a priori, then $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are known. The first result in this section is presented as the following.

Theorem 3.1:

Let $M=N$. If ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) and all transmission eigenvalues of the problem defined by (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are known a priori, as well as ${\mathit{\rho }}_N$ or ${\widehat{\mathit{\rho }}}_N$ is known, then ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined.

Proof From (2.12(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12) ) and (2.14(2.14) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\mathit{\delta }\prod _{j=1}^{2M+2N-2}(\mathit{\lambda }-{\mathit{\lambda }}_j)=:\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda }),\end{array}$$(2.14) ), we have(3.1) $$\begin{array}{c}\hfill \left|\begin{array}{cc}{P}_N(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{N+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|=\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda }).\end{array}$$(3.1)

Firstly we prove $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ are known and are determined uniquely by using the Euclidean algorithm for polynomial.

Note that $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are relatively prime, there exist polynomials $A(\mathit{\lambda }),B(\mathit{\lambda })\in \mathbf{C}(X),$ such that(3.2) $$\begin{array}{c}\hfill A(\mathit{\lambda })P_M(\mathit{\lambda })-B(\mathit{\lambda })P_{M+1}(\mathit{\lambda })=1,\end{array}$$(3.2)

and $A(\mathit{\lambda }),B(\mathit{\lambda })$ can be computed by the extended Euclidean algorithm for polynomial. Then doing the Euclidean division of polynomial $A(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })$ by $P_{M+1}(\mathit{\lambda }),$ we have(3.3) $$\begin{array}{c}\hfill A(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })=Q(\mathit{\lambda })P_{M+1}(\mathit{\lambda })+\tilde{A}(\mathit{\lambda }),\end{array}$$(3.3)

where $\mathrm{d}eg\tilde{A}(\mathit{\lambda })<2M.$ Multiply (3.2(3.2) $$\begin{array}{c}\hfill A(\mathit{\lambda })P_M(\mathit{\lambda })-B(\mathit{\lambda })P_{M+1}(\mathit{\lambda })=1,\end{array}$$(3.2) ) by $\mathrm{\Delta }(\mathit{\lambda })$ and multiply (3.3(3.3) $$\begin{array}{c}\hfill A(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })=Q(\mathit{\lambda })P_{M+1}(\mathit{\lambda })+\tilde{A}(\mathit{\lambda }),\end{array}$$(3.3) ) by $P_M(\mathit{\lambda }),$ respectively, and add the two results up, we obtain that(3.4) $$\begin{array}{c}\hfill \tilde{A}(\mathit{\lambda })P_M(\mathit{\lambda })-\tilde{B}(\mathit{\lambda })P_{M+1}(\mathit{\lambda })=\mathrm{\Delta }(\mathit{\lambda }).\end{array}$$(3.4)

Here $\tilde{B}(\mathit{\lambda })=B(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })-Q(\mathit{\lambda })P_M(\mathit{\lambda }).$ Since $\mathrm{d}eg\tilde{A}(\mathit{\lambda })<2M,\mathrm{d}eg{P}_M(\mathit{\lambda })=2M-2,\mathrm{d}eg{P}_{M+1}(\mathit{\lambda })=2M$ and $\mathrm{d}eg\mathrm{\Delta }(\mathit{\lambda })=2M+2N-2\le 4M-2$ one knows that $\mathrm{d}eg\tilde{B}(\mathit{\lambda })\le 2M-2,$ we then have $\mathit{\delta }\tilde{A}(\mathit{\lambda })P_M(\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })P_{M+1}(\mathit{\lambda })={\mathrm{\Delta }}_{M,N}(\mathit{\lambda }).$ Which together with (2.12(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12) ), yields(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5)

Since $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are relatively prime, it implies that $P_M(\mathit{\lambda })|U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })|U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }).$ In addition $\mathrm{d}eg(U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))\le 2M$ and $\mathrm{d}eg(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))\le 2M-2,$ if $M=N,$ one has$$U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda })=a{P}_{N+1}(\mathit{\lambda })\mathrm{a}ndU(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })=b{P}_N(\mathit{\lambda }),a,b\in \mathbf{C}.$$

The values at 0 of the polynomials give$$a=1-\mathit{\delta }\frac{\tilde{A}(0)}{N+1}\mathrm{a}ndb=1-\mathit{\delta }\frac{\tilde{B}(0)}{N}.$$

Which imply that $U(N,\mathit{\lambda }),U(N+1,\mathit{\lambda })$ are known and are determined uniquely by(3.6) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })=P_N(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_N(\mathit{\lambda })\end{array}$$(3.6)

and(3.7) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })=P_{N+1}(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_{N+1}(\mathit{\lambda }),\end{array}$$(3.7)

respectively, where polynomials(3.8) $$\begin{array}{c}\hfill {\mathrm{\Phi }}_N(\mathit{\lambda })=\tilde{B}(\mathit{\lambda })-\frac{\tilde{B}(0)}{N}{P}_N(\mathit{\lambda })=\sum _{k=1}^{2N-2}{a}_{2N-1-k}(N){\mathit{\lambda }}^k\end{array}$$(3.8)

and(3.9) $$\begin{array}{c}\hfill {\mathrm{\Phi }}_{N+1}(\mathit{\lambda })=\tilde{A}(\mathit{\lambda })-\frac{\tilde{A}(0)}{N+1}{P}_{N+1}(\mathit{\lambda })=\sum _{k=1}^{2N}{a}_{2N+1-k}(N+1){\mathit{\lambda }}^k\end{array}$$(3.9)

are known a priori since $\tilde{A}(\mathit{\lambda }),\tilde{B}(\mathit{\lambda })$ are computed by the above process. Note that $\mathrm{d}eg{\mathrm{\Phi }}_N(\mathit{\lambda })=2N-2$, $\mathrm{d}eg{\mathrm{\Phi }}_{N+1}(\mathit{\lambda })=2N$ and ${\mathrm{\Phi }}_N(0)=0={\mathrm{\Phi }}_{N+1}(0).$ Now, both sides of (3.6(3.6) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })=P_N(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_N(\mathit{\lambda })\end{array}$$(3.6) ) are polynomials of degree $2N-2$. Equating the coefficients of the powers ${\mathit{\lambda }}^{2N-2}$ and ${\mathit{\lambda }}^{2N-3}$, we obtain(3.10) $$\begin{array}{c}\hfill {(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j={(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\gamma }}}_j+\mathit{\delta }{a}_1(N)\end{array}$$(3.10)

and(3.11) $$\begin{array}{c}\hfill i{(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j\sum _{j=1}^{N-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}=i{(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\gamma }}}_j\sum _{j=1}^{N-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}+\mathit{\delta }{a}_2(N).\end{array}$$(3.11)

Similarly, equating the coefficients of the powers ${\mathit{\lambda }}^{2N}$ and ${\mathit{\lambda }}^{2N-1}$ in (3.7(3.7) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })=P_{N+1}(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_{N+1}(\mathit{\lambda }),\end{array}$$(3.7) ), we have(3.12) $$\begin{array}{c}\hfill {(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\rho }}}_j={(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\gamma }}}_j+\mathit{\delta }{a}_1(N+1)\end{array}$$(3.12)

and(3.13) $$\begin{array}{c}\hfill i{(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\rho }}}_j\sum _{j=1}^N\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}=i{(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\gamma }}}_j\sum _{j=1}^N\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}+\mathit{\delta }{a}_2(N+1).\end{array}$$(3.13)

For notational convenience, we set(3.14) $$\begin{array}{c}\hfill x={\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\cdots {\widehat{\mathit{\rho }}}_{N-1},\end{array}$$(3.14)

and(3.15) $$\begin{array}{c}\hfill y=\sum _{j=1}^{N-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}.\end{array}$$(3.15)

Furthermore, we denote$${\widehat{\mathit{\gamma }}}_1{\widehat{\mathit{\gamma }}}_2\cdots {\widehat{\mathit{\gamma }}}_{N-1}=d_1,\sum _{j=1}^{N-1}\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}=d_2.$$

Thus the unknown $\mathit{\delta }$ can be expressed as(3.16) $$\begin{array}{c}\hfill \mathit{\delta }=-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1.\end{array}$$(3.16)

Using the new notation, (3.10(3.10) $$\begin{array}{c}\hfill {(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\rho }}}_j={(-1)}^{N-1}\prod _{j=1}^{N-1}{\widehat{\mathit{\gamma }}}_j+\mathit{\delta }{a}_1(N)\end{array}$$(3.10) )–(3.13(3.13) $$\begin{array}{c}\hfill i{(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\rho }}}_j\sum _{j=1}^N\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}=i{(-1)}^N\prod _{j=1}^N{\widehat{\mathit{\gamma }}}_j\sum _{j=1}^N\frac{{\mathit{\gamma }}_j}{{\widehat{\mathit{\gamma }}}_j}+\mathit{\delta }{a}_2(N+1).\end{array}$$(3.13) ) can be rewritten as(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) (3.18) $$\begin{array}{cc}\hfill i{(-1)}^{N-1}xy& =i{(-1)}^{N-1}{d}_1{d}_2-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_2(N),\hfill \end{array}$$(3.18) (3.19) $$\begin{array}{cc}\hfill {(-1)}^{N}x{\widehat{\mathit{\rho }}}_N& ={(-1)}^N{d}_1{\widehat{\mathit{\gamma }}}_N-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N+1),\hfill \end{array}$$(3.19) (3.20) $$\begin{array}{cc}\hfill i{(-1)}^{N}x({\widehat{\mathit{\rho }}}_{N}y+{\mathit{\rho }}_N)& =i{(-1)}^N{d}_1({\widehat{\mathit{\gamma }}}_N{d}_2+{\mathit{\gamma }}_N)-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_2(N+1).\hfill \end{array}$$(3.20)

Multiplying (3.17(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) ) by ${\widehat{\mathit{\gamma }}}_N$ and adding up (3.19(3.19) $$\begin{array}{cc}\hfill {(-1)}^{N}x{\widehat{\mathit{\rho }}}_N& ={(-1)}^N{d}_1{\widehat{\mathit{\gamma }}}_N-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N+1),\hfill \end{array}$$(3.19) ), we obtain(3.21) $$\begin{array}{c}\hfill {(-1)}^{N-1}=({\widehat{\mathit{\gamma }}}_N{a}_1(N)+a_1(N+1))d_1.\end{array}$$(3.21)

The fact that (3.21(3.21) $$\begin{array}{c}\hfill {(-1)}^{N-1}=({\widehat{\mathit{\gamma }}}_N{a}_1(N)+a_1(N+1))d_1.\end{array}$$(3.21) ) is always true implies that equations (3.17(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) ) and (3.19(3.19) $$\begin{array}{cc}\hfill {(-1)}^{N}x{\widehat{\mathit{\rho }}}_N& ={(-1)}^N{d}_1{\widehat{\mathit{\gamma }}}_N-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N+1),\hfill \end{array}$$(3.19) ) are not independent. In the following we discard (3.19(3.19) $$\begin{array}{cc}\hfill {(-1)}^{N}x{\widehat{\mathit{\rho }}}_N& ={(-1)}^N{d}_1{\widehat{\mathit{\gamma }}}_N-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N+1),\hfill \end{array}$$(3.19) ) and work with (3.17(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) ).

From (3.17(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) ), we obtain(3.22) $$\begin{array}{c}\hfill x=\frac{d_1}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}.\end{array}$$(3.22)

Next we multiply (3.17(3.17) $$\begin{array}{cc}\hfill {(-1)}^{N-1}x& ={(-1)}^{N-1}{d}_1-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N),\hfill \end{array}$$(3.17) ) by $i{\mathit{\rho }}_N$, (3.18(3.18) $$\begin{array}{cc}\hfill i{(-1)}^{N-1}xy& =i{(-1)}^{N-1}{d}_1{d}_2-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_2(N),\hfill \end{array}$$(3.18) ) by ${\widehat{\mathit{\rho }}}_N$. Adding up the two results gives(3.23) $$\begin{array}{c}\hfill i{(-1)}^{N-1}x({\widehat{\mathit{\rho }}}_{N}y+{\mathit{\rho }}_N)=i{(-1)}^{N-1}{d}_1(d_2{\widehat{\mathit{\rho }}}_N+{\mathit{\rho }}_N)-x{d}_1({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)({\widehat{\mathit{\rho }}}_N{a}_2(N)+i{\mathit{\rho }}_N{a}_1(N)).\end{array}$$(3.23)

Adding up (3.23(3.23) $$\begin{array}{c}\hfill i{(-1)}^{N-1}x({\widehat{\mathit{\rho }}}_{N}y+{\mathit{\rho }}_N)=i{(-1)}^{N-1}{d}_1(d_2{\widehat{\mathit{\rho }}}_N+{\mathit{\rho }}_N)-x{d}_1({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)({\widehat{\mathit{\rho }}}_N{a}_2(N)+i{\mathit{\rho }}_N{a}_1(N)).\end{array}$$(3.23) ) and (3.20(3.20) $$\begin{array}{cc}\hfill i{(-1)}^{N}x({\widehat{\mathit{\rho }}}_{N}y+{\mathit{\rho }}_N)& =i{(-1)}^N{d}_1({\widehat{\mathit{\gamma }}}_N{d}_2+{\mathit{\gamma }}_N)-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_2(N+1).\hfill \end{array}$$(3.20) ) yields(3.24) $$\begin{array}{c}\hfill x=\frac{{(-1)}^{N-1}[{\mathit{\rho }}_N-{\mathit{\gamma }}_N+d_2({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)]}{({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)[{\mathit{\rho }}_N{a}_1(N)-i({\widehat{\mathit{\rho }}}_N{a}_2(N)+a_2(N+1))]}.\end{array}$$(3.24)

From (3.22(3.22) $$\begin{array}{c}\hfill x=\frac{d_1}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}.\end{array}$$(3.22) ) and (3.24(3.24) $$\begin{array}{c}\hfill x=\frac{{(-1)}^{N-1}[{\mathit{\rho }}_N-{\mathit{\gamma }}_N+d_2({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)]}{({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)[{\mathit{\rho }}_N{a}_1(N)-i({\widehat{\mathit{\rho }}}_N{a}_2(N)+a_2(N+1))]}.\end{array}$$(3.24) ) we obtain(3.25) $$\begin{array}{c}\hfill \frac{{(-1)}^{N-1}{d}_1({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}=\frac{{\mathit{\rho }}_N-{\mathit{\gamma }}_N+d_2({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{{\mathit{\rho }}_N{a}_1(N)-i({\widehat{\mathit{\rho }}}_N{a}_2(N)+a_2(N+1))}.\end{array}$$(3.25)

If ${\widehat{\mathit{\rho }}}_N$ or ${\mathit{\rho }}_N$ is known a priori, then we can use (3.25(3.25) $$\begin{array}{c}\hfill \frac{{(-1)}^{N-1}{d}_1({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}=\frac{{\mathit{\rho }}_N-{\mathit{\gamma }}_N+d_2({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{{\mathit{\rho }}_N{a}_1(N)-i({\widehat{\mathit{\rho }}}_N{a}_2(N)+a_2(N+1))}.\end{array}$$(3.25) ) to compute the other. It follows that x is known from (3.22(3.22) $$\begin{array}{c}\hfill x=\frac{d_1}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}.\end{array}$$(3.22) ), and $\mathit{\delta }$ is also known from (3.16(3.16) $$\begin{array}{c}\hfill \mathit{\delta }=-x({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1.\end{array}$$(3.16) ). Now (3.6(3.6) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })=P_N(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_N(\mathit{\lambda })\end{array}$$(3.6) ) and (3.7(3.7) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })=P_{N+1}(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_{N+1}(\mathit{\lambda }),\end{array}$$(3.7) ) imply $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ are completely known. We continue by observing that (2.4(2.4) $$\begin{array}{c}\hfill U(n+1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n-1,\mathit{\lambda }).\end{array}$$(2.4) ) can be written as(3.26) $$\begin{array}{c}\hfill U(n-1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n+1,\mathit{\lambda }).\end{array}$$(3.26)

From the above discussion it follows that $U(N-1,\mathit{\lambda })$ is completely known. We have from (2.6(2.6) $$\begin{array}{c}\hfill {(-1)}^{n-1}{\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\cdots {\widehat{\mathit{\rho }}}_{n-1}\end{array}$$(2.6) ) and (2.7(2.7) $$\begin{array}{c}\hfill i{(-1)}^{n-1}{\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\dots {\widehat{\mathit{\rho }}}_{n-1}\sum _{j=1}^{n-1}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j},\end{array}$$(2.7) ),(3.27) $$\begin{array}{c}\hfill {\widehat{\mathit{\rho }}}_{N-1}=-\frac{c_1(N)}{c_1(N-1)},{\mathit{\rho }}_{N-1}=i\frac{c_2(N)c_1(N-1)-c_2(N-1)c_1(N)}{c_1^2(N-1)}\end{array}$$(3.27)

are also known. So $U(N-2,\mathit{\lambda })$ is completely known, continuing in the same way, ${\widehat{\mathit{\rho }}}_{N-2},{\mathit{\rho }}_{N-2}$ are known, and so on. Thus, eventually, all ${\widehat{\mathit{\rho }}}_k,{\mathit{\rho }}_k,k=1,2,\dots ,N,$ are determined.

Moreover, if $\mathit{\delta }$ and the transmission eigenvalues ${\mathit{\lambda }}_1,\dots ,{\mathit{\lambda }}_{4N-2}$ are known, in virtue of (3.6(3.6) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })=P_N(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_N(\mathit{\lambda })\end{array}$$(3.6) ) and (3.7(3.7) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })=P_{N+1}(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_{N+1}(\mathit{\lambda }),\end{array}$$(3.7) ), $U(N,\mathit{\lambda }),U(N+1,\mathit{\lambda })$ are also known. By a similar argument as Theorem 3.1, we have:

Theorem 3.2:

Let $M=N$. If ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) and all transmission eigenvalues of the problem defined by (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are known a priori, as well as $\mathit{\delta }$ is known, then ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined.

In the following, we consider the case $M>N$. It should be noted that, in this case, $\mathrm{d}eg{P}_M(\mathit{\lambda })>\mathrm{d}egU(N,\mathit{\lambda })$ and $\mathrm{d}eg{P}_{M+1}(\mathit{\lambda })>\mathrm{d}egU(N+1,\mathit{\lambda })$. In virtue of (3.5(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5) ), one has(3.28) $$\begin{array}{cc}\hfill P_{M+1}(\mathit{\lambda })& =a(\mathit{\lambda })(U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda })),a(\mathit{\lambda })\in \mathbf{C}(X),\hfill \\ \hfill P_M(\mathit{\lambda })& =b(\mathit{\lambda })(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })),b(\mathit{\lambda })\in \mathbf{C}(X).\hfill \end{array}$$(3.28)

Which together with (3.5(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5) ) yields $a(\mathit{\lambda })=b(\mathit{\lambda }).$ Since $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are relatively prime, we get$$U(N+1,\mathit{\lambda })=\mathit{\delta }\tilde{A}(\mathit{\lambda })$$

and$$U(N,\mathit{\lambda })=\mathit{\delta }\tilde{B}(\mathit{\lambda }).$$

Thus $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ are completely known, then we have

Corollary 3.3:

Let $M>N$. If ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) and all transmission eigenvalues of the problem defined by (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are known a priori, then ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined.

4. The case $M<N$

In this section, we consider the case $M<N$. On the condition that we set $Q_M(\mathit{\lambda })=P_M(\mathit{\lambda }),$ we begin by defining polynomials $Q_{M+m}(\mathit{\lambda })$ for $m=1,2,\dots ,N$ as(4.1) $$\begin{array}{cc}\hfill Q_{M+1}(\mathit{\lambda })& =-P_{M+1}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_N-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_N)Q_M(\mathit{\lambda }),\hfill \end{array}$$(4.1) (4.2) $$\begin{array}{cc}\hfill Q_{M+k}(\mathit{\lambda })& =-Q_{M+k-2}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_{N-(k-1)}-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_{N-(k-1)})Q_{M+k-1}(\mathit{\lambda })\hfill \end{array}$$(4.2)

for $k=2,3,\dots ,N.$ It is not hard to see that $Q_{M+k}$ is of degree $2(M+k-1)$.

Lemma 4.1:

For $1\le m<N$, the polynomials $Q_{M+m}(\mathit{\lambda })$ and $Q_{M+m-1}(\mathit{\lambda })$ are relatively prime.

Proof This can be justified as follows. If the polynomials $Q_{M+m}(\mathit{\lambda })$ and $Q_{M+m-1}(\mathit{\lambda })$ are not relatively prime, then there exists ${\mathit{\lambda }}_0$ such that $Q_{M+m}({\mathit{\lambda }}_0)=0=Q_{M+m-1}({\mathit{\lambda }}_0)$, it follows from (4.1(4.1) $$\begin{array}{cc}\hfill Q_{M+1}(\mathit{\lambda })& =-P_{M+1}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_N-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_N)Q_M(\mathit{\lambda }),\hfill \end{array}$$(4.1) ) and (4.2(4.2) $$\begin{array}{cc}\hfill Q_{M+k}(\mathit{\lambda })& =-Q_{M+k-2}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_{N-(k-1)}-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_{N-(k-1)})Q_{M+k-1}(\mathit{\lambda })\hfill \end{array}$$(4.2) ) that $P_M({\mathit{\lambda }}_0)=0=P_{M+1}({\mathit{\lambda }}_0)$, which contradicts Lemma 2.1.

By proceeding inductively, in terms of (2.12(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12) ), (4.1(4.1) $$\begin{array}{cc}\hfill Q_{M+1}(\mathit{\lambda })& =-P_{M+1}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_N-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_N)Q_M(\mathit{\lambda }),\hfill \end{array}$$(4.1) ) and (4.2(4.2) $$\begin{array}{cc}\hfill Q_{M+k}(\mathit{\lambda })& =-Q_{M+k-2}(\mathit{\lambda })+(2-i\mathit{\lambda }{\mathit{\rho }}_{N-(k-1)}-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_{N-(k-1)})Q_{M+k-1}(\mathit{\lambda })\hfill \end{array}$$(4.2) ), it is easy to see that

Lemma 4.2:

If $1\le m<N$, then(4.3) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{Q}_{M+m}(\mathit{\lambda })& U(N-m,\mathit{\lambda })\\ Q_{M+m-1}(\mathit{\lambda })& U(N-m+1,\mathit{\lambda })\end{array}\right|.\end{array}$$(4.3)

Theorem 4.3:

Let $M<N$. If ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) and all transmission eigenvalues of the problem defined by (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are known, as well as ${\mathit{\rho }}_{N-\mathit{\theta }},\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_{N-\mathit{\theta }},\dots ,{\widehat{\mathit{\rho }}}_N$ are known, where $\mathit{\theta }=\frac{1}{2}(N-M-1)$ for $M+N$ is odd and $\mathit{\theta }=\frac{1}{2}(N-M)$ for $M+N$ is even, then ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined.

Proof From Lemma 4.2, we obtain(4.4) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda })=\left|\begin{array}{cc}{Q}_{M+\mathit{\theta }+1}(\mathit{\lambda })& U(N-\mathit{\theta }-1,\mathit{\lambda })\\ Q_{M+\mathit{\theta }}(\mathit{\lambda })& U(N-\mathit{\theta },\mathit{\lambda })\end{array}\right|.\end{array}$$(4.4)

Here the polynomials $Q_{M+\mathit{\theta }+1}(\mathit{\lambda })$ and $Q_{M+\mathit{\theta }}(\mathit{\lambda })$ are known if ${\mathit{\rho }}_{N-\mathit{\theta }},\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_{N-\mathit{\theta }},\dots ,{\widehat{\mathit{\rho }}}_N$ are known a priori.

Note that $Q_{M+\mathit{\theta }+1}(\mathit{\lambda })$ and $Q_{M+\mathit{\theta }}(\mathit{\lambda })$ are relatively prime, there exist polynomials $G(\mathit{\lambda }),H(\mathit{\lambda })\in \mathbf{C}(X),$ such that(4.5) $$\begin{array}{c}\hfill G(\mathit{\lambda })Q_{M+\mathit{\theta }+1}(\mathit{\lambda })-H(\mathit{\lambda })Q_{M+\mathit{\theta }}(\mathit{\lambda })=1,\end{array}$$(4.5)

where $G(\mathit{\lambda }),H(\mathit{\lambda })$ can be computed by the extended Euclidean algorithm for polynomial. Notice that $\mathit{\delta }$ is known from (2.16(2.16) $$\begin{array}{c}\hfill \mathit{\delta }=\frac{M-N}{{\prod }_{j=1}^{2M+2N-2}{\mathit{\lambda }}_j}.\end{array}$$(2.16) ), that is ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ is known. Then doing the Euclidean division of polynomial $H(\mathit{\lambda }){\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ by $Q_{M+\mathit{\theta }+1}(\mathit{\lambda }),$ we have(4.6) $$\begin{array}{c}\hfill H(\mathit{\lambda }){\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=R(\mathit{\lambda })Q_{M+\mathit{\theta }+1}(\mathit{\lambda })+\tilde{H}(\mathit{\lambda }),\end{array}$$(4.6)

where $\mathrm{d}eg\tilde{H}(\mathit{\lambda })<2(M+\mathit{\theta }).$ Multiply (4.5(4.5) $$\begin{array}{c}\hfill G(\mathit{\lambda })Q_{M+\mathit{\theta }+1}(\mathit{\lambda })-H(\mathit{\lambda })Q_{M+\mathit{\theta }}(\mathit{\lambda })=1,\end{array}$$(4.5) ) by ${\mathrm{\Delta }}_{M,N}(\mathit{\lambda })$ and multiply (4.6(4.6) $$\begin{array}{c}\hfill H(\mathit{\lambda }){\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=R(\mathit{\lambda })Q_{M+\mathit{\theta }+1}(\mathit{\lambda })+\tilde{H}(\mathit{\lambda }),\end{array}$$(4.6) ) by $Q_{M+\mathit{\theta }}(\mathit{\lambda }),$ respectively, and add the two results up, we have(4.7) $$\begin{array}{c}\hfill \tilde{G}(\mathit{\lambda })Q_{M+\mathit{\theta }+1}(\mathit{\lambda })-\tilde{H}(\mathit{\lambda })Q_{M+\mathit{\theta }}(\mathit{\lambda })={\mathrm{\Delta }}_{M,N}(\mathit{\lambda }).\end{array}$$(4.7)

Here $\tilde{G}(\mathit{\lambda })=G(\mathit{\lambda }){\mathrm{\Delta }}_{M,N}(\mathit{\lambda })-R(\mathit{\lambda })Q_{M+\mathit{\theta }}(\mathit{\lambda }),$ and $\mathrm{d}eg\tilde{G}(\mathit{\lambda })\le 2(N-\mathit{\theta }-1)$. In virtue of (4.3(4.3) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{Q}_{M+m}(\mathit{\lambda })& U(N-m,\mathit{\lambda })\\ Q_{M+m-1}(\mathit{\lambda })& U(N-m+1,\mathit{\lambda })\end{array}\right|.\end{array}$$(4.3) ), thus we have(4.8) $$\begin{array}{c}\hfill (U(N-\mathit{\theta },\mathit{\lambda })-\tilde{G}(\mathit{\lambda }))Q_{M+\mathit{\theta }+1}(\mathit{\lambda })=(U(N-\mathit{\theta }-1,\mathit{\lambda })-\tilde{H}(\mathit{\lambda }))Q_{M+\mathit{\theta }}(\mathit{\lambda }).\end{array}$$(4.8)

Since $Q_{M+\mathit{\theta }+1}(\mathit{\lambda })$ and $Q_{M+\mathit{\theta }}(\mathit{\lambda })$ are relatively prime, together with $\mathrm{d}egU(N-\mathit{\theta }-1,\mathit{\lambda })-\tilde{H}(\mathit{\lambda })<\mathrm{d}eg{Q}_{M+\mathit{\theta }+1}(\mathit{\lambda })$ and $\mathrm{d}egU(N-\mathit{\theta },\mathit{\lambda })-\tilde{G}(\mathit{\lambda })<\mathrm{d}eg{Q}_{M+\mathit{\theta }}(\mathit{\lambda }),$ it implies that $Q_{M+\mathit{\theta }}(\mathit{\lambda })|U(N-\mathit{\theta },\mathit{\lambda })-\tilde{G}(\mathit{\lambda })$ and $Q_{M+\mathit{\theta }+1}(\mathit{\lambda })|U(N-\mathit{\theta }-1,\mathit{\lambda })-\tilde{H}(\mathit{\lambda }).$ Set(4.9) $$\begin{array}{cc}\hfill Q_{M+\mathit{\theta }}(\mathit{\lambda })& =c(\mathit{\lambda })(U(N-\mathit{\theta },\mathit{\lambda })-\tilde{G}(\mathit{\lambda })),c(\mathit{\lambda })\in \mathbf{C}(X),\hfill \\ \hfill Q_{M+\mathit{\theta }+1}(\mathit{\lambda })& =d(\mathit{\lambda })(U(N-\mathit{\theta }-1,\mathit{\lambda })-\tilde{H}(\mathit{\lambda })),d(\mathit{\lambda })\in \mathbf{C}(X).\hfill \end{array}$$(4.9)

Which together with (4.8(4.8) $$\begin{array}{c}\hfill (U(N-\mathit{\theta },\mathit{\lambda })-\tilde{G}(\mathit{\lambda }))Q_{M+\mathit{\theta }+1}(\mathit{\lambda })=(U(N-\mathit{\theta }-1,\mathit{\lambda })-\tilde{H}(\mathit{\lambda }))Q_{M+\mathit{\theta }}(\mathit{\lambda }).\end{array}$$(4.8) ) yields $c(\mathit{\lambda })=d(\mathit{\lambda }).$ Since $Q_{M+\mathit{\theta }+1}(\mathit{\lambda })$ and $Q_{M+\mathit{\theta }}(\mathit{\lambda })$ are relatively prime, we get$$U(N-\mathit{\theta },\mathit{\lambda })=\tilde{G}(\mathit{\lambda })$$

and$$U(N-\mathit{\theta }-1,\mathit{\lambda })=\tilde{H}(\mathit{\lambda }).$$

Thus $U(N-\mathit{\theta },\mathit{\lambda })$ and $U(N-\mathit{\theta }-1,\mathit{\lambda })$ are completely known, and can be written as(4.10) $$\begin{array}{c}\hfill U(N-\mathit{\theta }-1,\mathit{\lambda })=\sum _{j=0}^{2(N-\mathit{\theta })-4}{c}_{2(N-\mathit{\theta })-3-j}(N-\mathit{\theta }-1){\mathit{\lambda }}^j,\end{array}$$(4.10)

and(4.11) $$\begin{array}{c}\hfill U(N-\mathit{\theta },\mathit{\lambda })=\sum _{j=0}^{2(N-\mathit{\theta })-2}{b}_{2(N-\mathit{\theta })-1-j}(N-\mathit{\theta }){\mathit{\lambda }}^j,\end{array}$$(4.11)

where the coefficients $c_{2(N-\mathit{\theta })-3-j}$${}^{,}s$ and $b_{2(N-\mathit{\theta })-1-j}$${}^{,}s$ are known. Denote(4.12) $$\begin{array}{c}\hfill x={\widehat{\mathit{\rho }}}_1{\widehat{\mathit{\rho }}}_2\cdots {\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-2},\end{array}$$(4.12)

and(4.13) $$\begin{array}{c}\hfill y=\sum _{j=1}^{N-\mathit{\theta }-2}\frac{{\mathit{\rho }}_j}{{\widehat{\mathit{\rho }}}_j}.\end{array}$$(4.13)

Together with (4.10(4.10) $$\begin{array}{c}\hfill U(N-\mathit{\theta }-1,\mathit{\lambda })=\sum _{j=0}^{2(N-\mathit{\theta })-4}{c}_{2(N-\mathit{\theta })-3-j}(N-\mathit{\theta }-1){\mathit{\lambda }}^j,\end{array}$$(4.10) ), equating the coefficients of the powers ${\mathit{\lambda }}^{2(N-\mathit{\theta })-4}$ and ${\mathit{\lambda }}^{2(N-\mathit{\theta })-5}$ of $U(N-\mathit{\theta }-1,\mathit{\lambda })$ , we obtain(4.14) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-2}x=c_1(N-\mathit{\theta }-1)\end{array}$$(4.14)

and(4.15) $$\begin{array}{c}\hfill i{(-1)}^{N-\mathit{\theta }-2}xy=c_2(N-\mathit{\theta }-1).\end{array}$$(4.15)

Similarly, equating the coefficients of the powers ${\mathit{\lambda }}^{2(N-\mathit{\theta })-2}$ and ${\mathit{\lambda }}^{2(N-\mathit{\theta })-3}$ of $U(N-\mathit{\theta },\mathit{\lambda })$ and by using (4.11(4.11) $$\begin{array}{c}\hfill U(N-\mathit{\theta },\mathit{\lambda })=\sum _{j=0}^{2(N-\mathit{\theta })-2}{b}_{2(N-\mathit{\theta })-1-j}(N-\mathit{\theta }){\mathit{\lambda }}^j,\end{array}$$(4.11) ), one has(4.16) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-1}x{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}=b_1(N-\mathit{\theta })\end{array}$$(4.16)

and(4.17) $$\begin{array}{c}\hfill i{(-1)}^{N-\mathit{\theta }-1}x{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}\left(y+\frac{{\mathit{\rho }}_{N-\mathit{\theta }-1}}{{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}}\right)=b_2(N-\mathit{\theta }).\end{array}$$(4.17)

From (4.14(4.14) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-2}x=c_1(N-\mathit{\theta }-1)\end{array}$$(4.14) ) and (4.16(4.16) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-1}x{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}=b_1(N-\mathit{\theta })\end{array}$$(4.16) ), we have(4.18) $$\begin{array}{c}\hfill {\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}=-\frac{b_1(N-\mathit{\theta })}{c_1(N-\mathit{\theta }-1)}.\end{array}$$(4.18)

Furthermore, (4.14(4.14) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-2}x=c_1(N-\mathit{\theta }-1)\end{array}$$(4.14) ) and (4.15(4.15) $$\begin{array}{c}\hfill i{(-1)}^{N-\mathit{\theta }-2}xy=c_2(N-\mathit{\theta }-1).\end{array}$$(4.15) ) yield(4.19) $$\begin{array}{c}\hfill y=-i\frac{c_2(N-\mathit{\theta }-1)}{c_1(N-\mathit{\theta }-1)}.\end{array}$$(4.19)

In virtue of (4.16(4.16) $$\begin{array}{c}\hfill {(-1)}^{N-\mathit{\theta }-1}x{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}=b_1(N-\mathit{\theta })\end{array}$$(4.16) ) and (4.17(4.17) $$\begin{array}{c}\hfill i{(-1)}^{N-\mathit{\theta }-1}x{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}\left(y+\frac{{\mathit{\rho }}_{N-\mathit{\theta }-1}}{{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}}\right)=b_2(N-\mathit{\theta }).\end{array}$$(4.17) ), one has$$y+\frac{{\mathit{\rho }}_{N-\mathit{\theta }-1}}{{\widehat{\mathit{\rho }}}_{N-\mathit{\theta }-1}}=-i\frac{b_2(N-\mathit{\theta })}{b_1(N-\mathit{\theta })},$$

which implies$${\mathit{\rho }}_{N-\mathit{\theta }-1}=i\frac{b_2(N-\mathit{\theta })c_1(N-\mathit{\theta }-1)-b_1(N-\mathit{\theta })c_2(N-\mathit{\theta }-1)}{c_1^2(N-\mathit{\theta }-1)}.$$

By using (3.28(3.28) $$\begin{array}{cc}\hfill P_{M+1}(\mathit{\lambda })& =a(\mathit{\lambda })(U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda })),a(\mathit{\lambda })\in \mathbf{C}(X),\hfill \\ \hfill P_M(\mathit{\lambda })& =b(\mathit{\lambda })(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })),b(\mathit{\lambda })\in \mathbf{C}(X).\hfill \end{array}$$(3.28) ), $U(N-\mathit{\theta }-2,\mathit{\lambda })$ is completely known. Continuing in the same way as in the proof of Theorem 3.1, all ${\widehat{\mathit{\rho }}}_k,{\mathit{\rho }}_k,k=1,2,\dots ,N,$ can be determined.

We proceed to denote the transmission eigenvalues of the problem (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) ) with the transmission conditions(4.20) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M),\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1)+a_0\mathit{\upsilon }(M),\end{array}$$(4.20)

by ${\widehat{\mathit{\lambda }}}_1,\dots ,{\widehat{\mathit{\lambda }}}_{2M+2N-2}$ (repeated roots are allowed, of course). which coincide with the zeros of the polynomial(4.21) $$\begin{array}{c}\hfill {\widehat{\mathrm{\Delta }}}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })+a_0{P}_M(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|.\end{array}$$(4.21)

In virtue of (2.12(2.12) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{M,N}(\mathit{\lambda })=\left|\begin{array}{cc}{P}_M(\mathit{\lambda })& U(N,\mathit{\lambda })\\ P_{M+1}(\mathit{\lambda })& U(N+1,\mathit{\lambda })\end{array}\right|,\end{array}$$(2.12) ), we have(4.22) $$\begin{array}{c}\hfill {\widehat{\mathrm{\Delta }}}_{M,N}(\mathit{\lambda })=\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda })-a_0{P}_M(\mathit{\lambda })U(N,\mathit{\lambda }).\end{array}$$(4.22)

If $P_M({\widehat{\mathit{\lambda }}}_j)\ne 0,$ (4.22(4.22) $$\begin{array}{c}\hfill {\widehat{\mathrm{\Delta }}}_{M,N}(\mathit{\lambda })=\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda })-a_0{P}_M(\mathit{\lambda })U(N,\mathit{\lambda }).\end{array}$$(4.22) ) yields that$$\begin{array}{c}\hfill U(N,{\widehat{\mathit{\lambda }}}_j)=\frac{\mathit{\delta }\mathrm{\Delta }({\widehat{\mathit{\lambda }}}_j)}{a_0{P}_M({\widehat{\mathit{\lambda }}}_j)}=:\mathit{\delta }{H}_0(N,{\widehat{\mathit{\lambda }}}_j).\end{array}$$

Furthermore, if the multiplicity of the root ${\widehat{\mathit{\lambda }}}_j$ is $r_j$ , that is ${\widehat{\mathrm{\Delta }}}_{M,N}^{(k)}({\widehat{\mathit{\lambda }}}_j)=0$ for $j=0,1,\dots ,r_j-1$, then$$\begin{array}{c}\hfill U^{\prime }(N,{\widehat{\mathit{\lambda }}}_j)=\frac{\mathit{\delta }{\mathrm{\Delta }}^{\prime }({\widehat{\mathit{\lambda }}}_j)P_M({\widehat{\mathit{\lambda }}}_j)-\mathit{\delta }\mathrm{\Delta }({\widehat{\mathit{\lambda }}}_j)P_M^{\prime }({\widehat{\mathit{\lambda }}}_j)}{a_0{P}_M^2({\widehat{\mathit{\lambda }}}_j)}=:\mathit{\delta }{H}_1(N,{\widehat{\mathit{\lambda }}}_j).\end{array}$$

Proceeding by induction by differentiating (4.22(4.22) $$\begin{array}{c}\hfill {\widehat{\mathrm{\Delta }}}_{M,N}(\mathit{\lambda })=\mathit{\delta }\mathrm{\Delta }(\mathit{\lambda })-a_0{P}_M(\mathit{\lambda })U(N,\mathit{\lambda }).\end{array}$$(4.22) ) for k times, we obtain(4.23) $$\begin{array}{c}\hfill U^{(k)}(N,{\widehat{\mathit{\lambda }}}_j)=:\mathit{\delta }{H}_k(N,{\widehat{\mathit{\lambda }}}_j)\end{array}$$(4.23)

for $k=1,2,\dots ,r_j-1$, and $H_k(N,{\widehat{\mathit{\lambda }}}_j)$ is known. We give the following result:

Theorem 4.4:

Let $M<N$. Suppose $P_M({\widehat{\mathit{\lambda }}}_j)\ne 0$ for $j=1,2,\dots ,m$. If ${\mathit{\gamma }}_n,{\widehat{\mathit{\gamma }}}_n$ ($n=1,\dots ,M$) and all transmission eigenvalues of the problem defined by (1.3(1.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}(A_N\mathit{\omega })(n)=i\mathit{\lambda }\mathit{\rho }(n)\mathit{\omega }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\rho }}(n)\mathit{\omega }(n),\hfill \\ (A_M\mathit{\upsilon })(n)=i\mathit{\lambda }\mathit{\gamma }(n)\mathit{\upsilon }(n)+{\mathit{\lambda }}^2\widehat{\mathit{\gamma }}(n)\mathit{\upsilon }(n),\hfill \\ \mathit{\omega }(0)=\mathit{\upsilon }(0)=0,\hfill \end{array}\right.\end{array}$$(1.3) )–(1.4(1.4) $$\begin{array}{c}\hfill \mathit{\omega }(N)=\mathit{\upsilon }(M)\mathrm{a}nd\mathit{\omega }(N+1)=\mathit{\upsilon }(M+1).\end{array}$$(1.4) ) are known, as well as m different zeros of ${\widehat{\mathrm{\Delta }}}_{M,N}(\mathit{\lambda })$, denoted by ${\widehat{\mathit{\lambda }}}_1,\dots ,{\widehat{\mathit{\lambda }}}_m,$ and their corresponding multiplicities $r_1,\dots ,r_m$, ,satisfying $r_1+\cdots +r_m=2N-2M$, are known a priori, then ${\mathit{\rho }}_1,\dots ,{\mathit{\rho }}_N$ and ${\widehat{\mathit{\rho }}}_1,\dots ,{\widehat{\mathit{\rho }}}_N$ are uniquely determined.

Proof Let(4.24) $$\begin{array}{c}\hfill D(\mathit{\lambda })=\prod _{j=1}^{2N-2M}(\mathit{\lambda }-{\widehat{\mathit{\lambda }}}_j).\end{array}$$(4.24)

It should be noted that $D^{(k)}({\widehat{\mathit{\lambda }}}^j)=0$ for $j=1,2,\dots ,r,k=1,2,\dots ,r_j-1.$ Repeating the proof of Theorem 3.1, we can prove that (3.2(3.2) $$\begin{array}{c}\hfill A(\mathit{\lambda })P_M(\mathit{\lambda })-B(\mathit{\lambda })P_{M+1}(\mathit{\lambda })=1,\end{array}$$(3.2) )–(3.5(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5) ) can be obtained again. We set(4.25) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda })=M_1(\mathit{\lambda })D(\mathit{\lambda })+V_1(\mathit{\lambda }),\end{array}$$(4.25)

and(4.26) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })=M_0(\mathit{\lambda })D(\mathit{\lambda })+V_0(\mathit{\lambda }),\end{array}$$(4.26)

where $\mathrm{d}eg{M}_1(\mathit{\lambda })=2M,\mathrm{d}eg{V}_1(\mathit{\lambda })\le 2N-2M-1$ and $\mathrm{d}eg{M}_0(\mathit{\lambda })=2M-2,\mathrm{d}eg{V}_0(\mathit{\lambda })\le 2N-2M-1.$ Thus, together with (3.5(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5) ), we have(4.27) $$\begin{array}{c}\hfill V_1(\mathit{\lambda })=\frac{P_{M+1}(\mathit{\lambda })(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))}{P_M(\mathit{\lambda })}-M_1(\mathit{\lambda })D(\mathit{\lambda }),\end{array}$$(4.27)

following one has(4.28) $$\begin{array}{c}\hfill V_1({\widehat{\mathit{\lambda }}}^j)=\frac{P_{M+1}({\widehat{\mathit{\lambda }}}^j)(U(N,{\widehat{\mathit{\lambda }}}^j)-\mathit{\delta }\tilde{B}({\widehat{\mathit{\lambda }}}^j))}{P_M({\widehat{\mathit{\lambda }}}^j)}=:\mathit{\delta }{G}_0({\widehat{\mathit{\lambda }}}^j),j=1,2,\dots ,m.\end{array}$$(4.28)

Furthermore, proceeding by induction by differentiating (4.27(4.27) $$\begin{array}{c}\hfill V_1(\mathit{\lambda })=\frac{P_{M+1}(\mathit{\lambda })(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))}{P_M(\mathit{\lambda })}-M_1(\mathit{\lambda })D(\mathit{\lambda }),\end{array}$$(4.27) ) for k times and by using (4.23(4.23) $$\begin{array}{c}\hfill U^{(k)}(N,{\widehat{\mathit{\lambda }}}_j)=:\mathit{\delta }{H}_k(N,{\widehat{\mathit{\lambda }}}_j)\end{array}$$(4.23) ), we obtain(4.29) $$\begin{array}{c}\hfill V_1^{(k)}({\widehat{\mathit{\lambda }}}^j)=:\mathit{\delta }{G}_k({\widehat{\mathit{\lambda }}}^j),j=1,2,\dots ,m\end{array}$$(4.29)

for $k=1,2,r_j-1,$ and $G_k({\widehat{\mathit{\lambda }}}^j)$ is known. Thus, Hermite interpolation yields the polynomial $V_1(\mathit{\lambda })$ is known.

Moreover, (3.5(3.5) $$\begin{array}{c}\hfill (U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda }))P_M(\mathit{\lambda })=(U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }).\end{array}$$(3.5) ) together with (4.25(4.25) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })-\mathit{\delta }\tilde{A}(\mathit{\lambda })=M_1(\mathit{\lambda })D(\mathit{\lambda })+V_1(\mathit{\lambda }),\end{array}$$(4.25) ) and (4.26(4.26) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })-\mathit{\delta }\tilde{B}(\mathit{\lambda })=M_0(\mathit{\lambda })D(\mathit{\lambda })+V_0(\mathit{\lambda }),\end{array}$$(4.26) ) yields(4.30) $$\begin{array}{c}\hfill P_{M+1}(\mathit{\lambda })(M_0(\mathit{\lambda })D(\mathit{\lambda })+V_0(\mathit{\lambda }))=P_M(\mathit{\lambda })(M_1(\mathit{\lambda })D(\mathit{\lambda })+V_1(\mathit{\lambda })),\end{array}$$(4.30)

then we have(4.31) $$\begin{array}{c}\hfill V_0({\widehat{\mathit{\lambda }}}^j)=\frac{P_M({\widehat{\mathit{\lambda }}}^j)V_1({\widehat{\mathit{\lambda }}}^j)}{P_{M+1}({\widehat{\mathit{\lambda }}}^j)}=:\mathit{\delta }{F}_0({\widehat{\mathit{\lambda }}}^j),j=1,2,\dots ,m.\end{array}$$(4.31)

Proceeding by induction by differentiating (4.30(4.30) $$\begin{array}{c}\hfill P_{M+1}(\mathit{\lambda })(M_0(\mathit{\lambda })D(\mathit{\lambda })+V_0(\mathit{\lambda }))=P_M(\mathit{\lambda })(M_1(\mathit{\lambda })D(\mathit{\lambda })+V_1(\mathit{\lambda })),\end{array}$$(4.30) ) for k times, one yields(4.32) $$\begin{array}{c}\hfill V_0^{(k)}({\widehat{\mathit{\lambda }}}^j)=:\mathit{\delta }{F}_k({\widehat{\mathit{\lambda }}}^j),j=1,2,\dots ,m\end{array}$$(4.32)

for $k=1,2,r_j-1,$ and $F_k({\widehat{\mathit{\lambda }}}^j)$ is known. Thus, Hermite interpolation yields the polynomial $V_0(\mathit{\lambda })$ is known. We proceed by using (4.30(4.30) $$\begin{array}{c}\hfill P_{M+1}(\mathit{\lambda })(M_0(\mathit{\lambda })D(\mathit{\lambda })+V_0(\mathit{\lambda }))=P_M(\mathit{\lambda })(M_1(\mathit{\lambda })D(\mathit{\lambda })+V_1(\mathit{\lambda })),\end{array}$$(4.30) ), obtain(4.33) $$\begin{array}{c}\hfill P_{M+1}(\mathit{\lambda })M_0(\mathit{\lambda })-P_M(\mathit{\lambda })M_1(\mathit{\lambda })=S(\mathit{\lambda }),\end{array}$$(4.33)

where $S(\mathit{\lambda })=(P_M(\mathit{\lambda })V_1(\mathit{\lambda })-P_{M+1}(\mathit{\lambda })V_0(\mathit{\lambda }))/D(\mathit{\lambda })$ and $S(\mathit{\lambda })$ is known. Then doing the Euclidean division of polynomial $A(\mathit{\lambda })S(\mathit{\lambda })$ by $P_{M+1}(\mathit{\lambda }),$ we have(4.34) $$\begin{array}{c}\hfill A(\mathit{\lambda })S(\mathit{\lambda })=P_{M+1}(\mathit{\lambda })c(\mathit{\lambda })+d(\mathit{\lambda }).\end{array}$$(4.34)

Which together with (3.2(3.2) $$\begin{array}{c}\hfill A(\mathit{\lambda })P_M(\mathit{\lambda })-B(\mathit{\lambda })P_{M+1}(\mathit{\lambda })=1,\end{array}$$(3.2) ), yields(4.35) $$\begin{array}{c}\hfill (M_1(\mathit{\lambda })+d(\mathit{\lambda }))P_M(\mathit{\lambda })=(M_0(\mathit{\lambda })+\tilde{d}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }),\end{array}$$(4.35)

where $\tilde{d}(\mathit{\lambda })=B(\mathit{\lambda })S(\mathit{\lambda })-c(\mathit{\lambda })P_M(\mathit{\lambda }).$

Since $P_M(\mathit{\lambda })$ and $P_{M+1}(\mathit{\lambda })$ are coprime, similarly to the proof of Theorem 3.2, (4.35(4.35) $$\begin{array}{c}\hfill (M_1(\mathit{\lambda })+d(\mathit{\lambda }))P_M(\mathit{\lambda })=(M_0(\mathit{\lambda })+\tilde{d}(\mathit{\lambda }))P_{M+1}(\mathit{\lambda }),\end{array}$$(4.35) ) implies that$$M_0(\mathit{\lambda })=P_M(\mathit{\lambda })-\tilde{d}(\mathit{\lambda }),M_1(\mathit{\lambda })=P_{M+1}(\mathit{\lambda })-d(\mathit{\lambda }),$$

that is(4.36) $$\begin{array}{c}\hfill U(N,\mathit{\lambda })=(P_M(\mathit{\lambda })-\tilde{d}(\mathit{\lambda }))D(\mathit{\lambda })+V_0(\mathit{\lambda })+\mathit{\delta }\tilde{B}(\mathit{\lambda })=:\sum _{k=1}^{2N-2}{c}_{2N-1-k}(N){\mathit{\lambda }}^k,\end{array}$$(4.36)

and(4.37) $$\begin{array}{c}\hfill U(N+1,\mathit{\lambda })=(P_{M+1}(\mathit{\lambda })-\tilde{d}(\mathit{\lambda }))D(\mathit{\lambda })+V_1(\mathit{\lambda })+\mathit{\delta }\tilde{A}(\mathit{\lambda })=:\sum _{k=1}^{2N}{c}_{2N+1-k}(N+1){\mathit{\lambda }}^k.\end{array}$$(4.37)

Here $c_j(N)$${}^{,}s$ and $c_j(N+1)$${}^{,}s$ are known. Thus the polynomial $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$ are completely known.

Similarly to the proof of Theorem 3.1, by equating the coefficients of the first and second terms of $U(N,\mathit{\lambda })$ and $U(N+1,\mathit{\lambda })$, we obtain$${\widehat{\mathit{\rho }}}_N=-\frac{c_1(N+1)}{c_1(N)},{\mathit{\rho }}_N=i\frac{c_2(N+1)c_1(N)-c_2(N)c_1(N+1)}{c_1^2(N)}.$$

It follows that $U(N-1,\mathit{\lambda })$ is completely known, continuing in the same way, ${\widehat{\mathit{\rho }}}_{N-1},{\mathit{\rho }}_{N-1}$ are known, and so on. Thus, eventually, all ${\widehat{\mathit{\rho }}}_k,{\mathit{\rho }}_k,k=1,2,\dots ,N,$ are determined.

5. One example

The mathematical treatment of the scattering of a time harmonic electromagnetic plane wave by an inhomogeneous infinite cylinder with cross section D such that the electric field E is polarized parallel to the axis of the cylinder leads to an interior transmission eigenvalue problem, that is a boundary value problem for electromagnetic waves (see [6] for details). In this section, we consider the discrete version of the corresponding problem for the case $M=N$.

If we take $M=N=3,$${\widehat{\mathit{\gamma }}}_3={\widehat{\mathit{\gamma }}}_2={\widehat{\mathit{\gamma }}}_1=1,{\mathit{\gamma }}_3={\mathit{\gamma }}_2={\mathit{\gamma }}_1=2,$ and set the transmission eigenvalues are known a priori, which satisfy(5.1) $$\begin{array}{cc}\hfill \mathrm{\Delta }(\mathit{\lambda })& ={\mathit{\lambda }}^{10}-2i{\mathit{\lambda }}^9-9.25{\mathit{\lambda }}^8-11.75i{\mathit{\lambda }}^7+8.25{\mathit{\lambda }}^6-i{\mathit{\lambda }}^5\hfill \\ \hfill & +10.5{\mathit{\lambda }}^4+12.5i{\mathit{\lambda }}^3-7{\mathit{\lambda }}^2-3.25i\mathit{\lambda }.\hfill \end{array}$$(5.1)

From the relations (2.8(2.8) $$\begin{array}{c}\hfill -P_{n+1}(\mathit{\lambda })-P_{n-1}(\mathit{\lambda })+2P_n(\mathit{\lambda })=(i\mathit{\lambda }{\mathit{\gamma }}_n+{\mathit{\lambda }}^2{\widehat{\mathit{\gamma }}}_n)P_n(\mathit{\lambda }),\end{array}$$(2.8) ) and (2.9(2.9) $$\begin{array}{c}\hfill P_0(\mathit{\lambda })=0,P_1(\mathit{\lambda })=1.\end{array}$$(2.9) ), we have(5.2) $$\begin{array}{c}\hfill P_2(\mathit{\lambda })=-{\mathit{\lambda }}^2-2i\mathit{\lambda }+2,P_3(\mathit{\lambda })={\mathit{\lambda }}^4+4i{\mathit{\lambda }}^3-8{\mathit{\lambda }}^2-8i\mathit{\lambda }+3,\end{array}$$(5.2)

and(5.3) $$\begin{array}{c}\hfill P_4(\mathit{\lambda })=-{\mathit{\lambda }}^6-6i{\mathit{\lambda }}^5+18{\mathit{\lambda }}^4+32i{\mathit{\lambda }}^3-34{\mathit{\lambda }}^2-20i\mathit{\lambda }+4.\end{array}$$(5.3)

Since $P_3(\mathit{\lambda })$ and $P_4(\mathit{\lambda })$ are coprime, we have(5.4) $$\begin{array}{c}\hfill P_3(\mathit{\lambda })A(\mathit{\lambda })-P_4(\mathit{\lambda })B(\mathit{\lambda })=1,\end{array}$$(5.4)

by using the extended Euclidean algorithm for polynomial, we obtain(5.5) $$\begin{array}{c}\hfill A(\mathit{\lambda })={\mathit{\lambda }}^4+4i{\mathit{\lambda }}^3-8{\mathit{\lambda }}^2-8i\mathit{\lambda }+3,B(\mathit{\lambda })=-{\mathit{\lambda }}^2-2i\mathit{\lambda }+2.\end{array}$$(5.5)

Then doing the Euclidean division of polynomial $A(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })$ by $P_4(\mathit{\lambda }),$ we have(5.6) $$\begin{array}{c}\hfill A(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })=Q(\mathit{\lambda })P_4(\mathit{\lambda })+\tilde{A}(\mathit{\lambda }),\end{array}$$(5.6)

where$$Q(\mathit{\lambda })=-{\mathit{\lambda }}^8-2.00i{\mathit{\lambda }}^7+3.25{\mathit{\lambda }}^6+1.25i{\mathit{\lambda }}^5-0.25{\mathit{\lambda }}^4+4.00i{\mathit{\lambda }}^3-1.75{\mathit{\lambda }}^2-2.00i\mathit{\lambda }-1.75,$$

and$$\tilde{A}(\mathit{\lambda })=-9.00i{\mathit{\lambda }}^5+28.50{\mathit{\lambda }}^4+57.75i{\mathit{\lambda }}^3-61.50{\mathit{\lambda }}^2-37.50i\mathit{\lambda }+7.00.$$

Thus(5.7) $$\begin{array}{c}\hfill \tilde{B}(\mathit{\lambda })=B(\mathit{\lambda })\mathrm{\Delta }(\mathit{\lambda })-Q(\mathit{\lambda })P_3(\mathit{\lambda })={\mathit{\lambda }}^4+7.00i{\mathit{\lambda }}^3-13.75{\mathit{\lambda }}^2-15.00i\mathit{\lambda }+5.25,\end{array}$$(5.7)

following we have,(5.8) $$\begin{array}{cc}\hfill {\mathrm{\Phi }}_3(\mathit{\lambda })& =\tilde{B}(\mathit{\lambda })-\frac{\tilde{B}(0)}{3}{P}_3(\mathit{\lambda })\hfill \\ \hfill & =-0.75{\mathit{\lambda }}^4+0.25{\mathit{\lambda }}^2-i\mathit{\lambda }\hfill \\ \hfill & =:\hfill & \hfill \sum _{k=1}^4{a}_{5-k}(3){\mathit{\lambda }}^k,\end{array}$$(5.8)

and(5.9) $$\begin{array}{cc}\hfill {\mathrm{\Phi }}_4(\mathit{\lambda })& =\tilde{A}(\mathit{\lambda })-\frac{\tilde{A}(0)}{4}{P}_4(\mathit{\lambda })\hfill \\ \hfill & =1.75{\mathit{\lambda }}^6+1.50i{\mathit{\lambda }}^5-3.00{\mathit{\lambda }}^4+1.75i{\mathit{\lambda }}^3-2.00{\mathit{\lambda }}^2-2.50i\mathit{\lambda }\hfill \\ \hfill & =:\hfill & \hfill \sum _{k=1}^6{a}_{7-k}(4){\mathit{\lambda }}^k,\end{array}$$(5.9)

Note that ${\widehat{\mathit{\gamma }}}_3={\widehat{\mathit{\gamma }}}_2={\widehat{\mathit{\gamma }}}_1=1,{\mathit{\gamma }}_3={\mathit{\gamma }}_2={\mathit{\gamma }}_1=2,$ we know that $d_1={\widehat{\mathit{\gamma }}}_1{\widehat{\mathit{\gamma }}}_2=1,d_2={\mathit{\gamma }}_1/{\widehat{\mathit{\gamma }}}_1+{\mathit{\gamma }}_2/{\widehat{\mathit{\gamma }}}_2=4.$ If ${\widehat{\mathit{\rho }}}_3=2$ is known also, then from the relation (3.25(3.25) $$\begin{array}{c}\hfill \frac{{(-1)}^{N-1}{d}_1({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}=\frac{{\mathit{\rho }}_N-{\mathit{\gamma }}_N+d_2({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)}{{\mathit{\rho }}_N{a}_1(N)-i({\widehat{\mathit{\rho }}}_N{a}_2(N)+a_2(N+1))}.\end{array}$$(3.25) ), one has$$\frac{d_1({\widehat{\mathit{\rho }}}_3-{\widehat{\mathit{\gamma }}}_3)}{1+({\widehat{\mathit{\rho }}}_3-{\widehat{\mathit{\gamma }}}_3)a_1(3)}=\frac{{\mathit{\rho }}_3-{\mathit{\gamma }}_3+d_2({\widehat{\mathit{\rho }}}_3-{\widehat{\mathit{\gamma }}}_3)}{{\mathit{\rho }}_3{a}_1(3)-i({\widehat{\mathit{\rho }}}_3{a}_2(3)+a_2(4))},$$

which yields$${\mathit{\rho }}_3=1.00.$$

In virtue of (3.22(3.22) $$\begin{array}{c}\hfill x=\frac{d_1}{1+{(-1)}^{N-1}({\widehat{\mathit{\rho }}}_N-{\widehat{\mathit{\gamma }}}_N)d_1{a}_1(N)}.\end{array}$$(3.22) ), one obtains$$x=\frac{d_1}{1+({\widehat{\mathit{\rho }}}_3-{\widehat{\mathit{\gamma }}}_3)d_1{a}_1(3)}=4.00,$$

thus$$\mathit{\delta }=-x({\widehat{\mathit{\rho }}}_3-{\widehat{\mathit{\gamma }}}_3)d_1=-4.00.$$

Then one has(5.10) $$\begin{array}{cc}\hfill U(3,\mathit{\lambda })& =P_3(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_3(\mathit{\lambda })\hfill \\ \hfill & =4.00{\mathit{\lambda }}^4+4.00i{\mathit{\lambda }}^3-9.00{\mathit{\lambda }}^2-4.00i\mathit{\lambda }+3.00\hfill \end{array}$$(5.10)

and(5.11) $$\begin{array}{cc}\hfill U(4,\mathit{\lambda })& =P_4(\mathit{\lambda })+\mathit{\delta }{\mathrm{\Phi }}_4(\mathit{\lambda })\hfill \\ \hfill & =-8.00{\mathit{\lambda }}^6-12.00i{\mathit{\lambda }}^5+30.00{\mathit{\lambda }}^4+25.00i{\mathit{\lambda }}^3\hfill \\ \hfill & -26.00{\mathit{\lambda }}^2-10.00i\mathit{\lambda }+4.00.\hfill \end{array}$$(5.11)

By using (3.26(3.26) $$\begin{array}{c}\hfill U(n-1,\mathit{\lambda })=(2-i\mathit{\lambda }{\mathit{\rho }}_n-{\mathit{\lambda }}^2{\widehat{\mathit{\rho }}}_n)U(n,\mathit{\lambda })-U(n+1,\mathit{\lambda }).\end{array}$$(3.26) ), we obtain(5.12) $$\begin{array}{cc}\hfill U(2,\mathit{\lambda })& =(2-i\mathit{\lambda }-2{\mathit{\lambda }}^2)U(3,\mathit{\lambda })-U(4,\mathit{\lambda })=-2{\mathit{\lambda }}^2-i\mathit{\lambda }+2.\hfill \end{array}$$(5.12)

That is ${\widehat{\mathit{\rho }}}_1=2,{\mathit{\rho }}_1=1.$ In virtue of (3.27(3.27) $$\begin{array}{c}\hfill {\widehat{\mathit{\rho }}}_{N-1}=-\frac{c_1(N)}{c_1(N-1)},{\mathit{\rho }}_{N-1}=i\frac{c_2(N)c_1(N-1)-c_2(N-1)c_1(N)}{c_1^2(N-1)}\end{array}$$(3.27) ),then we have ${\widehat{\mathit{\rho }}}_2=2,{\mathit{\rho }}_2=1$.

Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions. Special thanks to the referee who advices us using the extended Euclidean algorithm which led to a much improved manuscript.

Additional information

Funding

The research was supported in part by the National Natural Science Foundation of China [grant number 11571212]; Youth Innovation Team Fund of Xi’an Shiyou University [grant number 2015QNKYCXTD03]; Youth Innovation Fund of Xi’an Shiyou University [grant number Z15135].

Notes

No potential conflict of interest was reported by the authors.