Uniqueness of Solutions to Schrodinger Equations on Complex Semi-simple Lie Groups

We consider the time dependent Schrodinger equation on a complex semi-simple Lie group. We consider initial data a bi-invariant function. We prove that if the initial data decays fast enough, and the solution decays fast enough at one time slice, then the solution hs to vanish identically for all time. The hypothesis for decay we impose is shown to be optimal, uniqueness fails otherwise. We also consider the Heisenberg roup. There we show that the presence of closed loops that arise as the projection of geodesics to the contact plane at the origin precludes writing down a solution operator.