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

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 N_X(r) is the number of resonances in a disc of radius r we prove the following upper bound: N_X(r)≤Crⁿ⁺¹+C.