Spectral theory of damped quantum chaotic systems

Théorie spectrale des systèmes quantiques amortis

Stéphane Nonnenmacher (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.

Mots clés: damped wave equation; Anosov manifold; energy decay; semiclassical spectrum

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Dynamique Chaotique [nlin.CD]; Systèmes dynamiques [math.DS]

Stéphane Nonnenmacher. Spectral theory of damped quantum chaotic systems. Journées Equations aux Dérivées Partielles 2011, Jun 2011, Biarritz, France. ⟨hal-00619146v2⟩