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 operato r 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 TM, where sigma(p), is the princi pal symbol of P-j. We present two sets of results. (1) The ''semiclass ical trace formula'', on the asymptotic behavior of eigenvalues and ei genfunctions of S-h in terms of periodic trajectors of H. (II) Associa ted to certain isotropic submanifolds Lambda subset of TM we define f amilies 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.