THE SEMICLASSICAL TRACE FORMULA AND PROPAGATION OF WAVE-PACKETS

We study spectral and propagation properties of operators of the form S-h = Sigma(j=0)h(j)P(j) where For All j P-j is a differential operator of order j on a manifold M, asymptotically as h --> 0. The estimates are in terms of the flow {phi(t)} of the classical Hamiltonian H(x, p) = Sigma(j=0)(N) sigma(pi)(x,p) on T*M, where sigma(p), is the principal symbol of P-j. We present two sets of results. (1) The ''semiclassical trace formula'', on the asymptotic behavior of eigenvalues and eigenfunctions of S-h in terms of periodic trajectors of H. (II) Associated to certain isotropic submanifolds Lambda subset of T*M we define families of functions {psi(h)} and prove that For All(t){exp(-ithS(h))(psi(h))} is a family of the same kind associated to phi(t)(Lambda). (C) 1995 Academic Press. Inc.