Orthonormal Strichartz estimate for dispersive equations with potentials

Abstract

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations.