Splitting of the series of resonances on the nonphysical sheet

Abstract

The paper is concerned with the behavior of the poles (resonances) of the analytic continuation to the “nonphysical sheet” of the resolvent of the Dirichlet problem for the operator\( - \frac{\partial }{{\partial x_i }}a_{ij} (x)\frac{\partial }{{\partial xj}} + q(x)\) in the exterior of a trapping domain as the channel connecting the cavity with the exterior of the trap closes up. It is proved that the set of poles decomposes into two series one of which approaches the eigenvalues of the Dirichlet problem in the cavity and the other the resonances of the exterior of the closed trap.