From quasimodes to resonances: exponentially decaying perturbations

Understand

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.