Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole. I. Meromorphic Continuations

Abstract

This paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider the eigenvalue problem for a self-adjoint elliptic differential operator in a multidimensional tube-like domain which is infinite along one dimension and can be bounded or unbounded in other dimensions. This self-adjoint eigenvalue problem is perturbed by a small hole cut out of the domain. The boundary of the hole is described by a non-Hermitian Robin-type boundary condition. The main result of the present paper states the existence and describes the properties of local meromorphic continuations of the resolvent of the operator in question through the essential spectrum. The continuations are constructed near the edge of the spectrum and in the vicinity of certain internal threshold points of the spectrum. Then we define the eigenvalues and resonances of the operator as the poles of these continuations and prove that both the edge and the internal thresholds bifurcate into eigenvalues and/or resonances. The total multiplicity of the eigenvalues and resonances bifurcating from internal thresholds can be up to twice larger than the multiplicity of the thresholds. In other words, the perturbation can increase the total multiplicity.