Complex scaling and the distribution of scattering poles

The purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self-adjoint perturbations of the Laplacian in R', n odd. We also consider more general types of bounds that give sharper estimates in certain situations. The general conclusion can be stated as follows: The order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems. From the few known cases, however, the exact asymptotics are expected to be different. The scattering poles for compactly supported perturbations were rigorously defined by Lax and Phillips [13]. In a more general setting they correspond to resonances, the study of which has a long tradition in mathematical physics. In the Lax-Phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator. Because of the latter characterization, they can be considered as the analogue of the discrete spectral data for problems on noncompact domains. The problem of estimating the counting function