ON THE POINTWISE BEHAVIOR OF SEMICLASSICAL MEASURES

In this paper we concern ourselves with the small h asymptotics of the inner products of the eigenfunctions of a Schrodinger-type operator w ith a coherent state. More precisely, let psi(j)(h) and E(j)(h) denote the eigenfunctions and eigenvalues of a Schrodinger-type operator H-h , with discrete spectrum. Let psi((x,zeta)) be a coherent state center ed at the point (x, zeta) in phase space. We estimate as h --> 0 the a verages of the squares of the inner products (psi((x,zeta))(a)(psi(j)( h)) over an energy interval of size fL around a fixed energy, E. This follows from asymptotic expansions of the form [GRAPHICS] for certain test functions phi and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on wh ether the classical trajectory through (x,zeta) is periodic or not. In the periodic case the iterates of the trajectory contribute to the le ading coefficient. We also discuss the case of the Laplacian on a comp act Riemannian manifold.