Spectral theory of elliptic differential operators with indefinite weights

TL;DR

This paper investigates the spectral properties of non-selfadjoint elliptic operators with indefinite weights, showing that their nonreal spectrum is bounded and can be characterized explicitly in certain decomposable domains.

Contribution

It provides new insights into the spectral behavior of indefinite-weight elliptic operators, including boundedness of nonreal spectrum and eigenvalue characterization via Dirichlet-to-Neumann maps.

Findings

Nonreal spectrum of such operators is bounded.

In decomposable domains, nonreal spectrum consists only of normal eigenvalues.

Eigenvalues can be characterized using Dirichlet-to-Neumann maps.

Abstract

The spectral properties of a class of non-selfadjoint second order elliptic operators with indefinite weight functions on unbounded domains $\Omega$ are investigated. It is shown that under an abstract regularity assumption the nonreal spectrum of the associated elliptic operator in $L^2(\Omega)$ is bounded. In the special case that $\Omega=R^n $decomposes into subdomains $\Omega_+$ and $\Omega_-$ with smooth compact boundaries and the weight function is positive on $\Omega_+$ and negative on $\Omega_-$, it turns out that the nonreal spectrum consists only of normal eigenvalues which can be characterized with a Dirichlet-to-Neumann map.