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

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 $H=-\Delta +\frac{q}{|x|}$ 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, $\Vert e^{itH}f{\Vert }_{L^{\infty }}\le C{t}^{-\frac{3}{2}}\Vert f{\Vert }_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

60 pages