Solve Time-dependent Schr(?)dinger Equations With Perturbed Collocation Method

In this paper, we mainly discuss the perturbed collocation method. We apply the method to the Schrodinger equation which is a very importent model in quantum mechanics. Perturbed collocation method added a perturbation operator to the classical collocation method, so it generalizes the classical collocation method. A perturbed collocation method is equivalent to a Runge-Kutta method theoretically. This paper suggests the symplecticity condition for the perturbed collocation method from the equivalent symplectic Runge-Kutta method and prove the order condition for this method in detail.We construct a symplectic s-stage perturbed collocation method of order 2s-2 and apply it to the Schrodinger equation. Especially we implement the numerical experiments for s=2 and s=3. It is well known, the equivalence theoretically does not mean the same numerically. In comparison with the symplectic perturbed collocation method, we give the numerical experiments for the equivalent symplectic Runge-Kutta method and another two non-symplectic methods. It shows that a remarkable advantage of the symplectic methods applied to the Schrodinger equation is the precise preservation of charge conservation law. In order to study the numerical stability, we also adopt the different time step sizes. In our numerical experiments, the numerical results show that our algorithms here are stable.