The distributions of some transmission eigenvalues in one dimension

作者：Zhang, Yalin; Shi, Guoliang^*

来源：IMA Journal of Applied Mathematics, 2017, 82(3): 601-628.

DOI：10.1093/imamat/hxx005

• transmission eigenvalue problem
• real transmission eigenvalues
• distributions of transmission eigenvalues

This work focuses on two main topics about the transmission eigenvalue problem defined on the unit interval [0, 1]: (i) the distributions of transmission eigenvalues in a region Sigma(e) which is symmetric about the axes; (ii) the existence of real transmission eigenvalues under some conditions on acoustic profiles n(1) (x) and n(2) (x). These subjects are central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Specifically, in the case where n(1)(x) = n(2)(x) for x = 0, 1, we show how to locate the transmission eigenvalues in Sigma(e). The existence of infinitely many real transmission eigenvalues is proven in the following cases: (i) n(2) (0) /n(1) (0) not equal n(2) (1) /n(1) (1) and n(1) (0) n(1) (1) not equal n(2) (0) n(2) (1); (ii) n(1) (0) = n(2) (0), n(1) (1) = n(2) (1) and n(1)' (1) - n(2)' (1) not equal n (1)' (0) - n(2)' (0). The asymptotics of the real eigenvalues, as the byproduct of their existence, is obtained in both cases. All the results are obtained under the assumptions on the values of n(i) (x) and n(i)' (x) for i = 1, 2, x = 0, 1. There are no more restrictions on the values of n(1) (x), n(2) (x) in the interval (0, 1).

• 出版日期2017-6
• 单位天津大学; 天津理工大学