Quantum decay rates appear as imaginary parts of resonances, or poles of the meromorphic continuation of the resolvent of the Laplacian. As energy grows, decay rates are related to properties of geodesic flow and to the structure at infinity. In this thesis we study resonances for manifolds with infinity given by hyperbolic funnels, and establish relationships between trapping of geodesic trajectories and distribution and density of resonances.

We find the largest resonance free regions, corresponding to the fastest decay, in the nontrapping case, that is the case in which all geodesics escape to infinity. When the manifold is trapping, but the trapping is hyperbolic and sufficiently mild, we find resonance free strips with the width of the strip given by the topological pressure. For manifolds with hyperbolic dynamics, but possibly more substantial trapping, we find upper bounds on the rate of accumulation of resonances to the essential spectrum.

We use the method of complex scaling to holomorphically deform our operator and to realize the resonances as eigenvalues of the deformed, nonselfadjoint operator, which allows us to study them directly using microlocal estimates. The principal contribution of this thesis is the adaptation of the method of complex scaling to manifolds with hyperbolic ends, and the ``gluing'' of the estimates it provides near infinity to estimates near the trapped set which we take from other authors.