A spectral projection method for transmission eigenvalues

TL;DR

This paper introduces a spectral projection method for efficiently identifying transmission eigenvalues in inverse scattering problems, utilizing contour integrals to detect eigenvalues within specific regions.

Contribution

A novel spectral projection approach for transmission eigenvalues that improves detection efficiency using contour integrals and boundary element discretization.

Findings

Method effectively detects transmission eigenvalues in numerical tests.

Spectral projection accurately identifies zero eigenvalues indicating transmission eigenvalues.

Approach is computationally efficient for complex eigenvalue regions.

Abstract

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.