Pseudospectra of Semiclassical Boundary Value Problems

TL;DR

This paper investigates the spectral instability of semiclassical boundary value problems, characterizing their pseudospectra and quasimode concentration, with applications to diffusion processes and nonlinear evolution equations.

Contribution

It provides a detailed analysis of spectral instability and pseudospectra for semiclassical elliptic boundary value problems, including boundary effects and applications.

Findings

Spectral instability occurs for small semiclassical parameter h.

Pseudospectrum and quasimode concentration are characterized.

Instability extends to certain nonlinear evolution problems.

Abstract

We consider operators $-\Delta + X$ where $X$ is a constant vector field, in a bounded domain and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter h, which should be thought of as the reciprocal of the Peclet constant. This instability is due to the presence of the boundary: just as in the case of $-\Delta + X$, some of our operators are normal when considered on R^d. We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated to these elliptic operators.