Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rn, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rn) 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)⩽Anrn+o(rn) with an explicit An. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.