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Abstract

In [13], Guillopé and Zworski prove a Poisson summation formula for the scattering resonances of a hyperbolic surface X with finite geometry, that is, X consisting of a compact manifold with boundary together with a finite number of cusps and funnels. This Poisson summation formula implies a polynomial lower bound on the counting function for scattering resonances and lower bounds for the counting function of scattering resonances in strips. Our purpose here is to generalize both the Poisson summation formula and the lower bounds on resonances to convex cocompact hyperbolic manifolds in any dimension. The result of [13] relies on the analysis of [12], which depends crucially on the special geometry of two-dimensional hyperbolic surfaces; our analysis uses the results of Bunke and Olbrich [4] and Patterson and Perry [26] on the divisor of Selberg's zeta function, which are valid in any dimension, but only for hyperbolic manifolds without cusps. The Poisson summation formula will be shown to be a simple consequence of the main result in [26], and the lower bounds on resonances will follow precisely as in [12, 13] once the Poisson formula is established. To state our results, we recall some basic facts about the spectral and scattering theory of convex cocompact hyperbolic manifolds and the main result of [26] about the zeta function. If Γ is a discrete group of isometries acting on H n+1 , the group Γ is called convex cocompact if there is a finite-sided fundamental domain for Γ which does

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“…( 82) and (80) show up in the scattering problem of Ref. [24] occurring in connection with the trace formula for the Euclidean form of the BTZ black hole metric.…”

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