JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 4, Number 4, October 1991 COMPLEX SCALING AND THE DISTRIBUTION OF SCATTERING POLES JOHANNES SJOSTRAND AND MACIEJ ZWORSKI 1. INTRODUCTION AND STATEMENT OF RESULTS The purpose of this paper is to establish sharp polynomial bounds on the num- ber of scattering poles for a general class of compactly supported self-adjoint perturbations of the Laplacian in ]Rn, 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 prob- lems. 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 continua- tion 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 (1.1) N(r) = #{Aj : I)) ::; r, Aj is a scattering pole}, where the poles are included according to their multiplicities, was introduced by Melrose [17], who after obtaining a polynomial bound for a general class of problems [18, 19] established a sharp bound ( 1.2) in the case of obstacle scattering [20]. The bound (1.2) was then obtained in the case of scattering by compactly supported bounded potentials by the second al,lthor [29] (the exponent n + 1 was obtained earlier by Melrose). Vodev [26] has recently extended that result to compactly supported metric perturbations. Received by the editors February 25, 1991. 1991 Mathematics Subject Classification. Primary 35P25, 47 A40. The second author's research was partially supported by NSF Grant 8922720. 729 © 1991 American Mathematical Society 0894-0347/91 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use