The Selberg Zeta Function for Convex Co-Compact Schottky Groups

By Laurent Guillopé, Kevin K. Lin, Maciej Zworski. 2002

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We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn+1: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|n+1) , and it gives new bounds on the number of resonances (scattering poles) of Γ\ℍn+1 . The proof of this result is based on the application of holomorphic L2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\ℍn+1 as the simplest model of quantum chaotic scattering.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eefbe870f31be70fac8a3c6e86893897424378f0

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Citations in the corpus

Resonance projectors and asymptotics for r -normally hyperbolic trapped sets (2013, background): “or the number of resonances in strips near the real axis have been proved in dierent situations, both for normally hyperbolic and for hyperbolic trapping, by Sjostrand [Sj90], Guillope{Lin{Zworski [GuLiZw], Sjostrand{Zworski [SjZw07], Nonnenmacher{Sjostrand{Zworski [NoSjZw1,NoSjZw2], Faure{Sjostrand”
Fractal upper bounds on the density of semiclassical resonances (2005, background): “The proof of (3.5) follows from writing B(t) = V (t)BU(t), so that, in view of the properties of V (t), hDtB(t) ≡ [A(t), B(t)] mod Ψ−∞,−∞h , B(0) = B0 .”