On the distribution of resonances for some asymptotically hyperbolic manifolds

Froese, R. G.; Hislop, Peter D.

Journées équations aux dérivées partielles (2000), article no. 7, 16 p.

We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute $S$-matrix that is unitary for real values of the energy. This paramatrix is the $S$-matrix for a model laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance counting function requires estimates on the growth of the relative scattering phase, and singular values of a family of integral operators.

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