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

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Guillope,L. and M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Analysis 11 (1995) 1-22. Let X be a conforrnally compact n-dimensional manifold with constant negative curvature -1 near infinity. The resolvent (ilsCn 1 s»-I, Re s > 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 NxCr) is the number of resonances in a disc of radius r we prove the following upper bound: NxCr) :( Crn+! + C.