Sharp upper bounds on the number of the scattering poles

摘要

We study the scattering poles of a compactly supported "black box" perturbations of the Laplacian in R-n, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(r(n)) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N (r) < A(n)r(n) + o(r(n)) with an explicit A(n). We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula. (C) 2005 Elsevier Inc. All rights reserved.