The surfboard Schrödinger equations

Abstract

We study the large time behavior of solutions of time dependent Schrödinger equationsi∂u/∂t=−(1/2)Δu+t ^α V(x/t)u with bounded potentialV(x). We show that (1) ifα>−1, all solutions are asymptotically free ast→∞, (2) ifα≦−1 a solution becomes asymptotically free if and only if it has the momentum support outside of suppV for large time, (3) if −1 ≦α<0 all solutions are still asymptotically “modified free” ast→∞ and that (4) if 0 ≦α<2, for each local minimumx ₀ ofV(x), there exist solutions which are asymptotically Gaussians centered atx=tx ₀ and spreading slowly ast→∞.