On the Pointwise Behavior of Semi-Classical Measures

In this paper we concern ourselves with the small h asymptotics of the inner products of the eigenfunctions of a Schr6dinger-type operator with a coherent state. More precisely, let 0~ and E) denote the eigenfunctions and eigenvalues of a Schr6dinger-type operator H~ with discrete spectrum. Let 0(x,~) be a coherent state centered at the point (x, ~) in phase space. We estimate as h --+ 0 the averages of the squares o f the inner products (o(a,~,OJ~) over an energy interval of size h around a fixed energy, E. This follows from asymptotic expansions o f the form for certain test functions (p and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, 4) is periodic or not. In the periodic case the iterates o f the trajectory contribute to the leading coefficient. We also discuss the case o f the Laplacian on a compact Riernannian manifold.