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

Abstract

We prove a sharp dispersive estimate|Pacu(t,x)|≤C|t|-1/2{dot operator}{norm of matrix}u(0){norm of matrix}L1(R) for the one dimensional Schrödinger equationiut-uxx+V(x)u+V0(x)u=0, where (1+x2)V∈L1(R) and V0 is a step function, real valued and constant on the positive and negative real axes. © 2011 Elsevier Inc.