Gevrey Smoothing Properties of the Schrödinger Evolution Group in Weighted Sobolev Spaces

The Cauchy problem for the Schrödinger Equation i∂u/∂t = − 12 Δu + Vu is studied. It is found that for initial data decaying sufficiently rapidly at infinity and Gevrey regular potentials V, the solutions are infinitely differentiable functions of x and t (in fact they are in Gevrey classes). Further, for V = V1 + V2, where V1 satisfies certain smoothness conditions and V2 is a rough potential that decays sufficiently rapidly at infinity, the solutions are still Gevrey regular functions of t. Applications to Scattering Theory are discussed.