Dispersive estimate for the 1D Schrödinger equation with a steplike potential

Abstract

We prove a sharp dispersive estimate | P a c u ( t , x ) | ⩽ C | t | − 1 / 2 ⋅ ‖ u ( 0 ) ‖ L 1 ( R ) for the one dimensional Schrödinger equation i u t − u x x + V ( x ) u + V 0 ( x ) u = 0 , where ( 1 + x 2 ) V ∈ L 1 ( R ) and V 0 is a step function, real valued and constant on the positive and negative real axes.