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We consider all of the transmission eigenvalues for one-dimensional media. We give some conditions under which complex eigenvalues exist. In the case when the index of refraction is constant, it is shown that all the transmission eigenvalues are real if and only if the index of refraction is an odd number or reciprocal of an odd number.

The transmission eigenvalue problem appears in the inverse scattering theory for acoustic and electromagnetic waves [

Since only real eigenvalues can be determined from the scattering data and the physical properties of the scattering object can be obtained from the transmission eigenvalues, it is of interest to find the existence conditions of the complex transmission eigenvalues and to research the conditions under which there are no complex eigenvalues at all. For the existence of the transmission eigenvalues, to the author’s knowledge, only the sufficient conditions are given.

The plan of our paper is as follows. In Section

Our research methods rely on transforming the first Helmholtz equation in (

Suppose that the fundamental solutions

The value

The boundary conditions in (

The determinant condition

With the help of the Liouville transformation (

Assume that

So we have the following asymptotic expansions.

Assume that

After substituting the asymptotic expansions of

According to the fundamental inequality

Our aim here is to find conditions under which

Let

Then we have the following result.

Assume that

Based on Cohn’s theorem, our first aim is to look for conditions of

The following proof for the existence of complex zeros for

Next, inspired by the results and methods in [

Let

Assume that

From the identities

Furthermore, at

Suppose that

We consider the eigenvalue problem (

In the symmetric domain, we can separate (

In the case when

By the fact that

In the case when

From now on, we further assume that

Our goal is to determine under what conditions there exist complex eigenvalues

Let

Let

Let

In order to find the transmission eigenvalues, we study the roots of

First, when

As we can see from the above examples, this problem may only have real transmission eigenvalues under some conditions. The following theorem presents a sufficient and necessary condition for the nonexistence of complex transmission eigenvalues in the case when

Let

We see that if

For the second part of this theorem, we only need to show that

From this point on, take

In the case when

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China under Grant no. 61170019.