Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity

Article type: Research Article

Authors: Guillopé, Laurent^{; ;} | Zworski, Maciej^;

Affiliations: Institut Fourier, URA 188 C.N.R.S., BP 74, 38402 Saint Martin d'Hères Cedex, France | Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 2I218, USA

Note: [] Correspondence to: L. Guillopé, Institut Fourier, URA 188 C.N.R.S., BP 74, 38402 Saint Martin d'Hères Cedex, France.

Note: [] Partially supported by the European Union under Programme G.AD.G.E.T. SCI-0105C.

Note: [] Partially supported by the National Science Foundation under Grant DMS-9202344.

Abstract: Let X be a conformally compact n-dimensional manifold with constant negative curvature −1 near infinity. The resolvent (Δ−s(n−1−s))−1, Res>n−1, of the Laplacian on X extends to a meromorphic family of operators on C and its poles are called resonances or scattering poles. If NX(r) is the number of resonances in a disc of radius r we prove the following upper bound: NX(r)≤Crn+1+C.

DOI: 10.3233/ASY-1995-11101

Journal: Asymptotic Analysis, vol. 11, no. 1, pp. 1-22, 1995