From quasimodes to resonances: exponentially decaying perturbations

Oran Gannot

Vol. 277 (2015), No. 1, 77–97

DOI: 10.2140/pjm.2015.277.77

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.

Keywords

scattering resonances, quasimodes, exponentially decaying potentials

Mathematical Subject Classification 2010

Primary: 35P25

Secondary: 47F05, 47A40

Milestones

Received: 6 August 2013

Revised: 10 December 2014

Accepted: 23 December 2014

Published: 6 August 2015