Perturbation of resonances

Abstract

Resonances are often treated under the assumption that they are simple poles of the resolvent kernel (Green's function). There are no physical or mathematical reasons which exclude the existence of multiple poles of Green's functions even in the case when the potential is real valued. In this case the poles in the physical sheet are simple since the Schrödinger operator is selfadjoint but the poles on the unphysical sheet are not necessarily simple. We prove that sufficiently small perturbations of the potential (or boundary in diffraction problems) do not increase the multiplicity of the resonances on any fixed compact domain of the complex plane. We also observe that a knowledge of the set of the Green's function poles does not determine the potential uniquely.