Transmission eigenvalues for elliptic operators

TL;DR

This paper reduces the transmission eigenvalue problem for elliptic operators with sign-definite perturbations to a non-selfadjoint eigenproblem, providing conditions for eigenvalue existence and completeness, especially in the trace class case.

Contribution

It introduces a novel reduction of the transmission eigenvalue problem to a non-selfadjoint operator framework and establishes existence conditions, including in the trace class scenario.

Findings

Reduction of the problem to a non-selfadjoint eigenproblem

Sufficient conditions for eigenvalue existence and completeness

Generic existence of transmission eigenvalues in the trace class case

Abstract

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.