Dynamics of resonances for 0th order pseudodifferential operators

TL;DR

This paper investigates the behavior of resonances and eigenvalues in 0th order pseudodifferential operators under perturbations, providing new results on resonance dynamics and eigenvalue convergence relevant to internal wave models.

Contribution

It establishes a Fermi golden rule for resonances at embedded eigenvalues and analyzes eigenvalue dynamics for specific operator perturbations.

Findings

Proves a Fermi golden rule for resonances at embedded eigenvalues.

Studies eigenvalue convergence in perturbed operators.

Provides microlocal models for internal waves.

Abstract

We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators $P(s)$. In particular, we prove a Fermi golden rule for resonances of $P(s)$ at embedded eigenvalues of $P=P(0)$. We also study the dynamics of eigenvalues of $P(t)=P+it\Delta$ as the eigenvalues converge to simple eigenvalues of $P$. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.