Note on the distribution of poles of the scattering matrix (Q1056832)

Let \(K\subset R^ n\) be a bounded domain with a smooth boundary. Let \(\lambda_ j\) be the poles of the resolvent kernel of the Dirichlet Laplacian in the exterior to K domain. Assume that K has at least one isolated multiple reflecting periodic ray. Under this assumption it is proved that for every \(\epsilon >0\) there exist \(\lambda_ j\) such that \(0<Im \lambda_ j<\epsilon \log | \lambda_ j|.\)