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In the framework of real hyperbolic geometry, this review note begins with the Helgason correspondence induced by the Poisson transform between eigenfunctions of the Laplace–Beltrami operator on the hyperbolic space and hyperfunctions on its boundary at infinity . Focused on the scattering operator for real hyperbolic manifolds of finite geometry, discussion is given on the two different constructions (pseudo-differential calculus for degenerate operators and harmonic analysis for the conformal group) and some applications (Selberg zeta functions, resonances and scattering poles).

Following a brief presentation of the database activities of the Cellule MathDoc (CMD), the paper focuses on the NUMDAM project lead by CMD on behalf of the CNRS. The purpose of the project is the retro-digitization of mathematical materials back to the 19th Century in order to contribute to the preservation of the world's cultural heritage in mathematics and improve mathematicians' access to the literature. The basic elements underlying NUMDAM are non-proprietary standards, such as XML and Unicode, a highly structured hyperlink model and collaboration with the various players in the scientific publishing community. A preliminary version of the NUMDAM site, conforming to the basic tenets of an OAI environment, was opened at the end of 2002. This paper also covers the major issues of metadata, intellectual property rights, copyright, and economic models. Examples of NUMDAM-related projects are given. In conclusion, some good practices on the part of researchers are discussed.

Malgre la proliferation de journaux dits electroniques sur la Toile, la lecture de documents sur ecran reste un probleme ouvert. Souvent cet affichage n’est qu’une etape indiquant une transmission reussie, permettant un apercu sur le document, preludant a une impression qui sera le support de la lecture et du travail sur le texte. La lecture-ecran reste, malgre les avancees technologiques, eloignee de la lecture-livre.

Guillope,L. and M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Analysis 11 (1995) 1-22. Let X be a conforrnally compact n-dimensional manifold with constant negative curvature -1 near infinity. The resolvent (ilsCn 1 s»-I, Re s > n 1, of the Laplacian on X extends to a meromorphic family of operators on C and its poles are called resonances or scattering poles. If NxCr) is the number of resonances in a disc of radius r we prove the following upper bound: NxCr) :( Crn+! + C.

Resume On dkrit la structure des codes A barre EAN (un code EAN13 orne souvent la demiere de couverture d’un livre) et un ensemble de macros QX utilist pour les realiser. On termine par quelques remarques sur la rhlisation effective. Abstract We dercribe the atrncture of the batcodes of the EAN family (a EAN13 code is often printcd nowadags on the lest covcr-page of a book) and a set of l&Y-macros ued to realize them. We end with *orne nmarks on thc eflective realisction.

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., 16 pages

International audience; We give a new upper bound on the Selberg zeta function for a convexco-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in stripsparallel to the imaginary axis the zeta function is bounded by $ \exp ( C|s|^\delta ) $ where $ \delta $ is the dimension of the limit set of thegroup. 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 proofof this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transferoperators and on the quasi-self-similarity of limit sets. We also studythis problem numerically and provide evidence that the bound may beoptimal. Our motivation comes from molecular dynamics and we consider $\Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantumchaotic scattering. The proof of this result is based on the applicationof holomorphic $L^2$-techniques to the study of the determinants of theRuelle transfer operators and on the quasi-self-similarity of limitsets.

Soit $M$ une surface riemannienne a courbure $-1$ et a bord compact totalement geodesique. Il lui est attachee une fonction zeta $Z_M$, generalisant la fonction que Selberg introduisit pour les surfaces d'aire finie. L'expression de cette fonction en termes d'invariants spectraux (valeurs propres et resonances) etablit son prolongement meromorphe au plan complexe. La preuve combine deux types de formules de trace, a la Selberg et a la Birman-Krein, en s'appuyant sur l'analyse du prolongement meromophe des resolvantes de divers laplaciens et des resultats de type diffusion (stationnaire et non stationnaire) qui en resultent.

In this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends. We also give Weyl asymptotics for the relative scattering phase of such a surface. The proofs are based on trace formulae analogous to those of the Euclidean odd-dimensional scattering. The main technical ingredient is a new proof of the Poisson formula (Theorem 5.7) which is applicable in the Euclidean case as well. Our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries. The previous general lower bounds or asymptotics were either nonoptimal ([25], [58], [9]), one-dimensional or radial ([65], [67] and [54], [41]1) or they required some degeneracy of the

On etudie la theorie spectrale d'operateurs type laplacien et operateurs de Dirac sur une variete de Riemann complete a bouts (cylindriques ou isometriques a des pointes localement symetriques). La resolvante de tels operateurs admet un prolongement meromorphe a un revetement du plan spectral C, avec des ramifications d'ordre 2 au dessus d'un ensemble discret. On precise l'etude de la partie absolument continue grâce a des fonctions dites ondes planes, qui satisfont un ensemble d'equations fonctionnelles

Let X be a Riemannian surface of finite geometric type and with hyperbolic ends. The resolvent (ΔX−s(1−s))−1, Re s > 1 of the Laplacian on X extends to a meromorphic family of operators on C and its poles are called resonances. We prove an optimal polynomial bound for their counting function.

In this PhD thesis we consider the finite meromorphic continuation of the resolvent of the free Laplacian on complete manifolds with dimension n greater than 2. The poles of this continuation are called resonances. We assume that the manifold has some symmetries like S^1, (S^1)^m or SO(n). With this condition, we construct potentials V which are isoresonant i.e. such that the Laplacian plus V has the same resonances as the free Laplacian with the same multiplicities. During this construction we had to find an estimate of the first term of the spectrum of the Laplacian acting on S^1 homogeneous functions with compact support. We also show that these isoresonant potentials can change the order of the resonances. Finally, sometimes, resonances are defined as the poles of the scattering operator : we prove that in this framework we also have the isoresonance of our potentials.; Dans cette thèse on considère le prolongement méromorphe fini de la résolvante du laplacien libre sur une variété riemannienne connexe non compacte de dimension supérieure ou égale à 2. Ses pôles sont appelés résonances. On suppose que la variété possède certaines symétries comme S^1, (S^1)^m ou encore SO(n). Avec cette hypothèse, on construit des potentiels V dits isorésonants c'est-à-dire tels que le laplacien plus V ait les mêmes résonances que le laplacien libre avec les mêmes multiplicités. Au passage on est amené à estimer le bas du spectre du laplacien agissant sur les fonctions S^1 homogènes à support compact. On montre également que ces potentiels isorésonants peuvent modifier l'ordre des résonances. Enfin, les résonances sont parfois définies comme pôles de l'opérateur de diffusion : on montre que dans ce cadre on a aussi l'isorésonance de nos potentiels.