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.
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For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of particular values of dynamical zeta functions by using natural uniformizations, one due to Mazzeo-Taylor, the other to Osgood-Phillips-Sarnak. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization.
L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
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