The Selberg Zeta Function for Convex Co-Compact Schottky Groups

Abstract

Abstract. We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on H n+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\H n+1 . The proof of this result is based on the application of holomorphic L 2 -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\H n+1 as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic L 2 -techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets.

Lower bounds for resonances of infinite-area Riemann surfaces

“…Guillopé et al [2004] actually prove a stronger statement than (1). Let Ᏸ(z) be the number of resonances in the disc centered at z and radius one:…”

“…Physically, these are the most relevant resonances, because they correspond to metastable states that live the longest (the imaginary part corresponding to the decay rate). In the case of Schottky groups (equivalently, convex cocompact quotients in dimension 2), a "fractal" upper bound was obtained in [Zworski 1999;Guillopé et al 2004], namely…”

Mathematical Physics of Quantum Mechanics

“…Several numerical studies have attempted to confirm the above estimate for a variety of scattering Hamiltonians [10,12,13,14], but with rather inconclusive results. Indeed, it is numerically demanding to compute resonances.…”

“…If K(E) consists in one unstable periodic orbit, the resonances form a (slightly deformed) rectangular lattice of sides ∝ , so each -box contains at most finitely many resonances [22]. For intermediate situations (0 < d E < n − 1), one has only been able to prove one half of the above estimate, namely the upper bound for this resonance counting [21,27,10,24]. The dimension appearing in these upper bounds is the Minkowski dimension defined by measuring ǫ-neighborhoods of K(E).…”

“…In the case of the geodesic flow on a convex co-compact quotient of the Poincaré disk (which has a fractal trapped set), the resonances of the Laplace operator are exactly given by the zeros of Selberg's zeta function. Even in that case, it has been difficult to check the asymptotic Weyl law (2), due to the necessity to reach sufficiently high values of the energy [10].…”