Upper and lower bounds on resonances for manifolds hyperbolic near infinity

Details Upper and lower bounds on resonances for manifolds hyperbolic near infinity Upper and lower bounds on resonances for manifolds hyperbolic near infinity

2007-10-22

arxiv.org/abs/0710.3894v2

Mathematics

Spectral Theory

29 pages, minor corrections, added one figure

Scientific paper

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.

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