Convergence analysis of a symplectic semi-discretization for stochastic NLS equation with quadratic potential

In this paper, we investigate the convergence in probability of a stochastic symplectic scheme for stochastic nonlinear Schr"{o}dinger equation with quadratic potential and an additive noise. Theoretical analysis shows that our symplectic semi-discretization is of order one in probability under appropriate regularity conditions for the initial value and noise. Numerical experiments are given to simulate the long time behavior of the discrete average charge and energy as well as the influence of the external potential and noise, and to test the convergence order.