Location and Weyl formula for the eigenvalues of some non self-adjoint operators

TL;DR

This paper surveys recent results on the distribution and Weyl asymptotics of complex eigenvalues for certain non self-adjoint operators, including those related to wave equations and interior transmission problems, using semi-classical methods.

Contribution

It provides a comprehensive overview of the location and Weyl formula for eigenvalues of non self-adjoint operators in wave and transmission problems, with new results on Weyl asymptotics.

Findings

Weyl formula with remainder for interior transmission eigenvalues

Location results for eigenvalues of the generator of contraction semigroups

Semi-classical analysis applied to non self-adjoint operators

Abstract

We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator $G$ of the contraction semigroup $e^{tG}, \: t \geq 0,$ related to the wave equation in an unbounded domain $\Omega$ with dissipative boundary conditions on $\partial \Omega$. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain $K$ obtaining a Weyl formula with remainder for the counting function $N(r)$ of complex (ITE). The analysis is based on a semi-classical approach.