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arXiv · 2309.01313

Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential

Adam Black·Ebru Toprak·Bruno Vergara·Jiahua Zou
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Abstract

We study the long-time behavior of solutions to the Schrödinger 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 $H=-Δ+\frac{q}{|x|}$ 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.

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Adam Black, Ebru Toprak, Bruno Vergara, Jiahua Zou. 2024-02-05. Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential. https://arxiv.org/abs/2309.01313

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