Dispersive estimate for the 1D Schr\"odinger equation with a steplike potential

TL;DR

This paper establishes a precise decay rate for solutions to the 1D Schrödinger equation with a steplike potential, extending dispersive estimates to this class of problems with specific decay conditions on the potential.

Contribution

It provides a sharp dispersive estimate for the 1D Schrödinger equation with a steplike potential, a case not thoroughly analyzed before.

Findings

Proves a decay rate of |t|^{-1/2} for the solution's dispersive behavior.

Demonstrates the estimate holds under specific decay conditions on the potential.

Extends dispersive analysis to steplike potentials with bounded variation.

Abstract

We prove a sharp dispersive estimate $$ |P_{ac}u(t,x)|\le C|t|^{-1/2}\|u(0)\|_{L^1(R)} $$ for the one dimensional Schr\"odinger equation $$ iu_{t}-u_{xx}+V(x)u+V_0 u=0, $$ where $(1+x^2)V\in L^1(R)$ and $V_0$ is a step function, real valued and consant on the positive and negative real axes.