Microlocal limits of plane waves and Eisenstein functions

Semyon Dyatlov (1) , Colin Guillarmou (2)

1 Department of Mathematics [Berkeley]
2 DMA - Département de Mathématiques et Applications - ENS Paris

We study microlocal limits of plane waves on noncompact Riemannian manifolds $(M,g)$ which are either Euclidean or asymptotically hyperbolic with curvature $-1$ near infinity. The plane waves $E(z,\xi)$ are functions on $M$ parametrized by the square root of energy $z$ and the direction of the wave, $\xi$, interpreted as a point at infinity. If the trapped set $K$ for the geodesic flow has Liouville measure zero, we show that, as $z\to +\infty$, $E(z,\xi)$ microlocally converges to a measure $\mu_\xi$, in average on energy intervals of fixed size, $[z,z+1]$, and in $\xi$. We express the rate of convergence to the limit in terms of the classical escape rate of the geodesic flow and its maximal expansion rate~--- when the flow is Axiom~A on the trapped set, this yields a negative power of $z$. As an application, we obtain Weyl type asymptotic expansions for local traces of spectral projectors with a remainder controlled in terms of the classical escape rate.

Mots clés: semiclassical limits; plane waves

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Semyon Dyatlov, Colin Guillarmou. Microlocal limits of plane waves and Eisenstein functions. Annales Scientifiques de l'École Normale Supérieure, 2014, 47 (2), pp.371--448. ⟨10.24033/asens.2217⟩. ⟨hal-00685779⟩