L² Solutions to the Schrödinger–Poisson System: Existence, Uniqueness, Time Behaviour, and Smoothing Effects

Abstract

We study a system of infinitely many coupled Schrödinger equations with self-consistent Coulomb potential as the initial data has only a regularity of L 2 -type. We first establish Strichartz' inequalities in the framework of vector-valued wave functions (density matrices). This allows us to prove a well-posedness result, and strong smoothing effects. Also, we state blow-up (resp. decay) estimates for the solution as time goes to zero (resp. infinity).