Improved nonrelativistic limit for ground states of the Schrödinger and Hartree equations

Woocheol Choi, Younghun Hong, Jinmyoung Seok

Published 2016-10-19Version 1

Lenzmann \cite{L2} and Choi-Seok \cite{CS} proved that the ground states of the pseudo-relativistic Schr\"odinger equation with Hartree nonlinearity or power type nonlinearity \[ \big(\sqrt{-c^2 \Delta +\tfrac{c^4}{4}} - \tfrac{c^2}{2} \big) u + \mu u = \mathcal{N}(u) \] converges in $H^1 (\mathbb{R}^n)$ to the ground state $u_{\infty}$ of the nonrelativistic limit equation \[ -\Delta u + \mu u = \mathcal{N}(u). \] In this paper, we improve these results by showing the convergence in higher order Sobolev spaces with an explicit convergence rate given by $1/c^2$, which turns out to be optimal.