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

2023/09/04 by Adam T. Black, Black, Adam, Ebru Toprak +5
Mathematics · #33C15 #35J10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2309.01313

openalex publication_date 2023/09/04 · openalex created_date 2023/09/08 · openalex updated_date 2026/07/28

Abstract

We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on ℝ³ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian H=-Δ+(q)/(|x|) on radial data f, which allows us to explicitly write the evolution e^itHf. A comprehensive analysis of the kernel is then used to establish that, for large times, ‖e^{i t H}f‖_L^∞ ≤ C t^-(3)/(2)‖f‖_L¹. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

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