Existence and regularity of propagators for multi-particle Schr\"odinger equations in external fields

TL;DR

This paper establishes the existence, uniqueness, and regularity of propagators for multi-particle Schrödinger equations in external electromagnetic fields with specific growth and singularity conditions, using new Strichartz estimates.

Contribution

It introduces new Strichartz estimates for multi-particle Schrödinger propagators and proves their existence and regularity under near-critical conditions.

Findings

Unique unitary propagator exists under specified conditions

Domain of quantum harmonic oscillator is invariant under the propagator

Solutions are continuously differentiable in time for initial states in the domain

Abstract

We prove that the Schr\"odinger equation for N number of particles in the time dependent electro-magnetic field generates a unique unitary propagator on the state space under the condition that the field is smooth and moderately but almost critically increases at the spatial infinity such that propagator for every single particle in the field enjoys the time local Strichartz estimates and that the time dependent inter-particle potentials are almost critically singular for Hamiltonians to have a unique selfadjoint realization at every fixed time. We also show that the domain of definition of the quantum harmonic oscillator is invariant under the propagator and, for initial states in that space, solutions are continuously differentiable function of time variable with values in the state space under the additional assumption that the time derivative of inter-particle potentials exists almost everywhere and it increases the spatial singularities by at most the inverse power of the inter-particle distances. New estimates of Strichartz type for the propagator for N independent particles in the field are proved and used for the proof.