Matrix elements of Fourier Integral Operators

TL;DR

This paper investigates the semi-classical limits of matrix elements of eigenfunctions of the Laplacian with respect to Fourier integral operators on compact Riemannian manifolds, revealing how these limits relate to the invariance of the canonical relation under geodesic flow.

Contribution

It extends the analysis of matrix elements from pseudo-differential operators to Fourier integral operators, elucidating the role of canonical relation invariance in semi-classical limits.

Findings

Matrix elements tend to zero when the canonical relation is almost nowhere invariant.

Limit states are invariant measures on the canonical relation.

Invariance properties of limit states are explained, including for Hecke 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.