Probabilités, Physique Mathématique et Analyse (2PMA)

The objective of this work is to study how wave propagates in the frequency regime in quasi-periodic media. Quasi-periodic media are characterized by quasi-periodic coefficients, i.e. functions that are the value in a certain (irrational) direction of periodic functions in a higher dimension. Quasi-periodic media are somehow between periodic media and random media: they have a structure without being periodic and they exhibit properties well known in random media such as ergodicity or localization. I want to explain in this talk how we can take advantage of their structure to define, when it is possible, the outgoing/physical wave and a notion of group velocity.

TBA

This talk explores the intriguing realm of scattering resonances within two-dimensional transparent cavities, which arose in the modeling of micro-resonators constructed from dielectric materials (with positive permittivity) or metallic nanoparticles (with negative permittivity). Specifically, our investigation is focused on resonances that closely align with the real axis, characterized by highly oscillatory behavior and localization along the interface separating the cavity from its external environment. Notable exemplars of such resonances include whispering-gallery modes observed in dielectric cavities and surface plasmon waves associated with metallic particles.

The asymptotic behaviour of the partition function is one of the central questions of statistical mechanics. The asymptotic expansion of this partition function can be regarded as an infinite dimensional version of the Laplace method, since the number of integrations is also growing with N. By analogy, if the potential is complex one needs an infinite dimensional version of the steepest descent (Saddle point) method. We address this problem in the context of Beta ensembles. This is joint work with A. Guionnet and K. Kozlowski.

Dans cet exposé, nous nous intéressons à la théorie de la diffusion pour un modèle abstrait d'opérateurs non-auto-adjoints agissant sur un espace de Hilbert. L'opérateur non-auto-adjoint H est donné par une perturbation relativement compacte V d'un opérateur auto-adjoint H_0. Sous des hypothèses de principe d'absorption limite, nous expliquerons comment les opérateurs d'ondes non-unitaires associés à H et H_0 peuvent être définis et présenterons leurs propriétés. Finalement nous définirons la notion de complétude asymptotique pour ces opérateurs d'ondes et la relierons à la notion de singularité spectrale. Nos résultats s'appliquent à des opérateurs de Schrödinger avec des potentiels à valeurs complexes.