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 C 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. 1 Introduction and Main Results Let P be a compactly supported perturbation of the Laplacian in Rn, n odd, defined by the “black box scattering” formalism, i.e., P D outside the ball B.0;R0 / and P satisfies the hypotheses in [SjZ2], see section 2. As usual, P] denotes a reference operator on the “perturbed torus ” TR, R> R0, see next section. Let N.r / be the number o