Strong rates of convergence of a splitting scheme for Schr\"{o}dinger equations with nonlocal interaction cubic nonlinearity and white noise dispersion

Abstract

We analyse a splitting integrator for the time discretization of the Schr\"odinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has order of convergence one in the $p$-th mean sense, for any $p\geq1$ in some Sobolev spaces. We prove that the splitting schemes preserves the $L^2$-norm, which is a crucial property for the proof of the strong convergence result. Finally, numerical experiments illustrate the performance of the proposed numerical scheme.