Microlocal limits of plane waves and Eisenstein functions

Abstract

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.