The determinant of the Dirichlet-to-Neumann map for surfaces with boundary

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.Comment: 16 page

The Selberg zeta function for convex co-compact Schottky groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on ${\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $\exp (C |s|^\delta)$ where $\delta$ 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 $\Gamma \backslash {\mathbb H}^{n+1}$. 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 $\Gamma \backslash {\mathbb H}^{n+1}$ 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

Mathematics & database (open) access

The textual version of this presentation at the Conference "Open Access to Scientific and Technical Information: State of the Art and Future Trends" was published with the title 'Mathematics and databases: Open Access' in "Information Services and Use", vol. 23 (2003), issue 2-3, p. 127-131

Statique et dynamique de documents mathématiques

National audienceDivers prototypes visant à permettre l’exploration d’un ensemble critique en mathématiques sont décrits. Si aucun n’arrive à répondre aux contraintes contradictoires de ce type de montage numérique, des progrès notables sont néanmoins constatés

Statique et dynamique de documents mathématiques

National audienceDivers prototypes visant à permettre l’exploration d’un ensemble critique en mathématiques sont décrits. Si aucun n’arrive à répondre aux contraintes contradictoires de ce type de montage numérique, des progrès notables sont néanmoins constatés