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

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity(−ħ2c2Δ+m2c4−mc2)u+μu=N(u), where c denotes the speed of light. We prove that as c→∞ the ground states of this equation converges to the ground state of its nonrelativistic counterpart−ħ22mΔu+μu=N(u) with convergence rate 1/c2 in every Hs norms. Moreover, we show that this rate is optimal.