Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations

TL;DR

This paper extends Sylvester's approach to prove the discreteness of transmission eigenvalues for complex higher-order operators with specific smoothness and boundary conditions, advancing spectral theory in PDEs.

Contribution

It introduces a novel method for establishing eigenvalue discreteness for higher-order transmission problems with boundary vanishing and smoothness conditions.

Findings

Transmission eigenvalues are discrete under specified conditions.

Higher-order perturbations require smoothness and boundary vanishing.

Zeroeth order term must satisfy coercivity near boundary.

Abstract

In this paper we extend Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher-order main terms and higher-order perturbations. The coefficients of the perturbations must be sufficiently smooth and the coefficients of the higher-order terms of the perturbation must vanish in a neighbourhood of the boundary of the underlying domain. The zeroeth order term must satisfy a suitable coercivity condition in a neighbourhood of the boundary.