Dispersive and Strichartz estimates for 3D wave equation with a class of many-electric potentials

Abstract

We prove the dispersive and Strichartz estimates for solutions to the wave equation with a class of many-electric potentials in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schr\"odinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schr\"odinger semigroup, obtain a Gaussian type upper bound, establish Bernstein type inequalities, and finally pass to the M\"uller-Seeger's subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel-Tao.