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.
Categories: math.AP math-ph math.MP