Microlocal limits of plane waves and Eisenstein functions

TL;DR

This paper investigates the microlocal behavior of plane waves and Eisenstein functions on certain noncompact Riemannian manifolds, establishing convergence results and asymptotic expansions related to the classical escape rate of geodesic flow.

Contribution

It provides new results on the microlocal limits of plane waves on Euclidean and asymptotically hyperbolic manifolds, linking convergence rates to classical dynamical quantities.

Findings

Microlocal limits of plane waves converge to a measure as energy increases.

Convergence rate is expressed in terms of the classical escape rate and maximal expansion rate.

Weyl type asymptotics for spectral projectors with controlled remainder.

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.