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

Abstract

This is a work extending the results of \cite{AH} and \cite{AHH}. We want to show convergence of the Schrödinger 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.

38 pages

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Cited by in corpus (3)

• Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data
• Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate