Bounds on positive interior transmission eigenvalues

TL;DR

This paper establishes bounds on the positive interior transmission eigenvalues, proving their infinitude and providing asymptotic estimates, which are crucial for understanding wave propagation in inhomogeneous media.

Contribution

It introduces new lower and upper bounds on the counting function of interior transmission eigenvalues, including cases with obstacles, advancing spectral theory in this area.

Findings

Proves the existence of infinitely many interior transmission eigenvalues.

Provides asymptotic upper bounds for large wave numbers.

Derives upper estimates for the first few eigenvalues.

Abstract

The paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.