On the wave equation with Coulomb potential

Wave/Schr\"{o}dinger equations with potentials naturally originates from both the quantum physics and the study of nonlinear equations. The distractive Coulomb potential is a quantum mechanical description of distractive Coulomb force between two particles with the same charge. The spectrum of the operator \(-\Delta +1/|x|\) is well known and there are also a few results on the Strichartz estimates, local and global well-posedness and scattering result about the nonlinear Schr\"{o}dinger equation with a distractive Coulomb potential. In the contrast, much less is known for the global and asymptotic behaviour of solutions to the corresponding wave equations with a Coulomb potential. In this work we consider the wave equation with a distractive Coulomb potential in dimensions \(d\geq 3\). We first describe the asymptotic behaviour of the solutions to the linear homogeneous Coulomb wave equation, especially their energy distribution property and scattering profiles, then show that the radial finite-energy solutions to suitable defocusing Coulomb wave equation are defined for all time and scatter in both two time directions, by establishing a family of radial Strichartz estimates and combining them with the decay of the potential energy.