Spectrum of non-planar traveling waves

TL;DR

This paper proves that certain non-self-adjoint differential operators in cylindrical domains have only real discrete spectra located to the right of the essential spectrum, using spatial dynamics and bi-semigroups.

Contribution

It establishes the reality and location of the discrete spectrum for a class of non-self-adjoint operators in cylindrical domains, extending spectral theory results.

Findings

Discrete spectrum is real and located to the right of the essential spectrum.

Essential spectrum described via potential's limiting properties.

Decay rates of eigenfunctions estimated using spatial dynamics and bi-semigroups.

Abstract

In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders $\Omega\times\mathbb R\subseteq\mathbb R^{d+1}$ have only real discrete spectrum located to the right of the right most point of the essential spectrum. We describe the essential spectrum using the limiting properties of the potential. To track the discrete spectrum we use spatial dynamics and bi-semigroups of linear operators to estimate the decay rate of eigenfunctions associated to isolated eigenvalues.