Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations

TL;DR

This paper establishes the optimal convergence rate of ground states from a pseudo-relativistic nonlinear equation to its nonrelativistic limit as the speed of light tends to infinity, with a precise rate of 1/c^2.

Contribution

It proves the convergence rate of ground states in the nonrelativistic limit for pseudo-relativistic equations with explicit optimal rate 1/c^2.

Findings

Ground states converge to nonrelativistic counterparts at rate 1/c^2

The convergence rate is proven to be optimal

Results apply to equations with Hartree or power nonlinearities

Abstract

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 \Delta +m^2c^4} - mc^2 \right) u + \mu u = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}\Delta u + \mu u = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary order as $c \to \infty$. Moreover, we show that this rate is optimal.