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

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