Dispersion for Schr\"odinger Operators with One-gap Periodic Potentials on R^1

Abstract

We prove $t^{- \frac 14}-$decay for the solutions of the 1-dim Schrodinger equation with a one-gap periodic potential as $t \to +\infty $. Generically, one has $t^{- \frac 13}$-decay and this decay is optimal. Our approach is to analyze the stationary phase in the Schr\"odinger evolution as an integral operator.