The Semi-Classical Trace Formula and Propagation of Wave Packets

Abstract

We study spectral and propagation properties of operators of the form Sh = ∑Nj=0hjPj where ∀ 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 {φt} of the classical Hamiltonian H(x, p) = ∑Nj=0 σPj(x, p) on T*M, where σPj, is the principal symbol of P j. We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of Sh in terms of periodic trajectors of H. (II) Associated to certain isotropic submanifolds Λ ⊂ T*M we define families of functions {ψh} and prove that ∀t {exp(− ithSħ)(ψħ)} is a family of the same kind associated to φt(Λ).