From quasimodes to resonances: exponentially decaying perturbations

TL;DR

This paper establishes exponential bounds on the resolvent for certain self-adjoint operators and demonstrates how clusters of quasimodes lead to resonances, advancing understanding of spectral properties in quantum mechanics.

Contribution

It introduces new exponential resolvent bounds for operators close to the Laplacian and links quasimodes to resonances with a local maximum principle approach.

Findings

Exponential resolvent bounds away from resonances.

Clusters of quasimodes generate resonances with multiplicity.

Rapid convergence of resonances to quasimodes.

Abstract

We consider self-adjoint operators of black-box type which are exponentially close to the free Laplacian near infinity, and prove an exponential bound for the resolvent in a strip away from resonances. Here the resonances are defined as poles of the meromorphic continuation of the resolvent between appropriate exponentially weighted spaces. We then use a local version of the maximum principle to prove that any cluster of real quasimodes generates at least as many resonances, with multiplicity, rapidly converging to the quasimodes.