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

We prove a sharp dispersive estimate|Pacu(t,x)|⩽C|t|−1/2⋅‖u(0)‖L1(R)|Pacu(t,x)|⩽C|t|−1/2⋅‖u(0)‖L1(R) for the one dimensional Schrödinger equationiut−uxx+V(x)u+V0(x)u=0,iut−uxx+V(x)u+V0(x)u=0, where (1+x2)V∈L1(R)(1+x2)V∈L1(R) and V0V0 is a step function, real valued and constant on the positive and negative real axes.