Decay of scattering solutions to one-dimensional free Schr\"{o}dinger equation

TL;DR

This paper studies how scattering solutions to the one-dimensional free Schrödinger equation decay over time, showing that decay rates depend on the initial data's Fourier transform near zero, with proofs based on basic calculus.

Contribution

It provides a simple proof linking decay rates to the Fourier transform behavior near zero, clarifying the decay mechanism for solutions.

Findings

Decay rate depends on Fourier transform near zero

Proof uses basic calculus techniques

Results clarify the decay behavior of solutions

Abstract

In this paper, we investigate the decay property of scattering solutions to the initial value problem for the free Schr\"{o}dinger equation in $\mathbb{R}$. It becomes clear that the rate of time decay is essentially determined by the behavior of the Fourier transform of initial data near the origin. The proof is described by basic calculus.