Viscosity limits for 0th order pseudodifferential operators

TL;DR

This paper studies the spectral behavior of 0th order pseudodifferential operators on tori under viscosity limits, showing that eigenvalues of the regularized operators converge as viscosity vanishes, thus supporting physics claims.

Contribution

It adapts Helffer–Sj"ostrand theory to analyze viscosity limits of 0th order pseudodifferential operators on tori, providing rigorous justification for physics-based assertions.

Findings

Eigenvalues of regularized operators have well-defined limits as viscosity approaches zero.

The results justify previous physics claims about spectral properties in viscosity limits.

The approach extends scattering resonance theory to the setting of pseudodifferential operators on tori.

Abstract

Motivated by the work of Colin de Verdi\`ere and Saint-Raymond on spectral theory 0th order pseudodifferential operators on tori we consider viscosity limits in which 0th order operators $ P $ are replaced by $ P + i \nu \Delta $, $ \nu > 0 $. By adapting the Helffer--Sj\"ostrand theory of scattering resonances we show that in a complex neighbourhood of the continuous spectrum eigenvalues of $ P + i \nu \Delta $ have limits as viscosity $ \nu $ goes to 0. In the simplified setting of tori this justifies claims made in the physics literature.