On occurrence of resonances from multiple eigenvalues of the Schrödinger operator in a cylinder with scattering perturbations

In this paper, the Schrödinger operator with a localized potential in a multidimensional cylinder is considered. The boundary of the cylinder is split into three parts, two of which are “sleeves” going to infinity, and the third (central) part is located between them. On the sleeves and the central part, respectively, the Neumann and Dirichlet boundary conditions are posed. We examine the situation where the distance between the sleeves increases. We assume that the same Schrödinger operator in the same cylinder endowed with the Dirichlet condition on the whole boundary has an isolated double eigenvalue. We show that for a sufficiently large distance between the sleeves, this double eigenvalue splits into a pair of resonances of the original operator. For these resonances, we explicitly obtain the first terms of their asymptotic expansions and describe the behavior of the imaginary part of the resonances.