Analytic Continuation of Resolvents of Elliptic Operators in a Multidimensional Cylinder

Abstract

We consider the scalar second order elliptic differential operator in a multidimensional cylinder with the Dirichlet boundary condition. The coefficients of the operator are periodic on both outlets of the cylinder and are arbitrary in its finite part. We study a local analytic continuation of the bordered resolvent of the operator from the upper half-plane to the lower one with respect to the spectral parameter in a neighborhood of an interior point of the essential spectrum. It is shown that the size of the neighborhood depends only on geometric properties of the cylinder and the behavior of periodic components of coefficients of the operator. We introduce the notion of a resonance and formulate the corresponding boundary value problems. We describe the behavior of the bordered resolvent with respect to the spectral parameter in the neighborhood of the point of the essential spectrum.