Matrix elements of Fourier Integral Operators

Abstract

This article is concerned with the semi-classical limits of matrix elements $<F \phi_j, \phi_j>$ of eigenfunctions of the Laplacian $\Delta_g$ of a compact Riemannian manifold $(M, g)$ with respect to a Fourier integral operator $F$ on $L^2(M)$. Many results exist for the case where $F$ is a pseudo-differential operator, but matrix elements of Fourier integral operators involve new considerations. The limits reflect the extent to which the canonical relation of $F$ is invariant under the geodesic flow of $(M, g)$. When the canonical relation is almost nowhere invariant, a density one subsequence of the matrix elements tends to zero (related results arose first in the study of quantum ergodic restriction theorems). The limit states are invariant measures on the canonical relation of $F$ and their invariance properties are explained. The invariance properties in the case of Hecke operators answers an old question raised by the author.