Pointwise decay for radial solutions of the Schr\"odinger equation with a repulsive Coulomb potential

TL;DR

This paper analyzes the long-time decay of radial solutions to the Schrödinger equation with a repulsive Coulomb potential, providing explicit decay rates and a detailed Fourier transform analysis.

Contribution

It introduces a method to explicitly compute the distorted Fourier transform for the Coulomb potential, enabling precise decay estimates for solutions.

Findings

Established decay rate of $t^{-3/2}$ for solutions in $L^{ ext{infinity}}$

Derived explicit formulas for the distorted Fourier transform

Analyzed the kernel to understand long-time behavior

Abstract

We study the long-time behavior of solutions to the Schr\"odinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-\Delta+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.