Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms

TL;DR

This paper extends convergence results of Schrödinger equations to fractional Hartree equations with singular potentials, under more natural and weaker assumptions, including cases with less regular initial data.

Contribution

It generalizes previous convergence results to fractional and more singular potentials, and weakens initial data regularity requirements for deriving the Hartree equation.

Findings

Convergence established for fractional Hartree equations with singular potentials.

Results hold under weaker assumptions on initial data and potential regularity.

Extension to both defocusing and focusing cases in semi-relativistic and non-relativistic regimes.

Abstract

This is a work extending the results of \cite{AH} and \cite{AHH}. We want to show convergence of the Schr\"odinger equation towards the Hartree equation with more natural assumptions. We first consider both the defocusing and the focusing semi-relativistic Hartree equation. We show that the tools of \cite{P} are essentially sufficient for deriving the Hartree equation in those cases. Next, we extend this result to the case of fractional Hartree equations with possibly more singular potentials than the Coulomb potential. Finally, we show that, in the non-relativistic case, one can derive the Hartree equation assuming only $L^2$-data in the case of potentials that are more than or as regular as the Coulomb potential. We also derive the Hartree equation for more singular potentials in this case. This work is inspired by talks given at the conference 'MCQM 2018' at Sapienza/Rome.