Existence and Regularity of Propagators for Multi-Particle Schrödinger Equations in External Fields

Abstract

We consider Schrödinger equations for N number of particles in (classical) electro-magnetic fields that are interacting with each other via time dependent inter-particle potentials. We prove that they uniquely generate unitary propagators ${\{U(t,s), t,s \in \mathbb{R}\}}$ on the state space ${\mathcal{H}}$ under the conditions that fields are spatially smooth and do not grow too rapidly at infinity so that propagators for single particles satisfy Strichartz estimates locally in time, and that local singularities of inter-particle potentials are not too strong that time frozen Hamiltonians define natural selfadjoint realizations in ${\mathcal{H}}$. We also show, under very mild additional assumptions on the time derivative of inter-particle potentials, that propagators possess the domain of definition of the quantum harmonic oscillator ${\Sigma(2)}$ as an invariant subspace such that, for initial states in ${\Sigma(2)}$, solutions are C ¹ functions of the time variable with values in ${\mathcal{H}}$. New estimates of Strichartz type for propagators for N independent particles in the field will be proved and used in the proof.