Recurrence of singularities for second order isotropic pseudodifferential operators

Abstract

Let H be a self‐adjoint isotropic elliptic pseudodifferential operator of order 2. Denote by u(t) the solution of the Schrödinger equation (i∂t−H)u=0 with initial data ufalse(0false)=u0. If u0 is compactly supported the solution u(t) is smooth for small t>0, but not for all t. We determine the wavefront set of u(t) in terms of the wavefront set of u0 and the principal and subprincipal symbol of H.