An Alternative Numerical Method for Initial Value Problems Involving the Contact Nonlinear Hamiltonians

We suggest a new difference scheme for dealing with contact nonlinear Hamiltonians. The scheme has two parts. First, the system is transformed to the interaction picture of quantum mechanics using the time-independent Hamiltonian H0. This reduces the problem to a system of ordinary differential equations in time. Subsequently, the system is integrated in time for a time step Δt and then transformed back to the initial representation. Standard time integration schemes make it possible to eliminate explicit use of transformation operators, thus significantly reducing the number of calculations. We give explicit expressions for integration using the Runge–Kutta scheme. We consider three applications of the method and illustrate the behavior of the norms of the resulting wave functions after many time steps. The method is compared to the standard split-step method, and we show that our method has five N(u0(τ)) more calculations in a single step of the scheme, for the simplest case of one time and one spatial dimension. Here N(u0(τ)) is the number of calculations needed to apply the evolution operator u0(τ) to the wave function, where u0 is defined in terms of the (time-independent) Hamiltonian. This increase in the number of steps is offset by at least one order higher accuracy of the method. Its implementation is straightforward. It uses a unique arrangement of the steps of the split-step method.