On the pointwise behavior of semi-classical measures

Abstract

In this paper we concern ourselves with the small ħ asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let ψ j ħ and E j ħ denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H ħ with discrete spectrum. Let ψ ( x ,ξ) be a coherent state centered at the point ( x , ξ) in phase space. We estimate as ħ→0 the averages of the squares of the inner products (ψ a ( x ,ξ) ,ψ j ħ ) over an energy interval of size ħ around a fixed energy, E . This follows from asymptotic expansions of the form for certain test function φ 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 , ξ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.