Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations.

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using fast Fourier transforms. Splitting the system into the quantum harmonic-oscillator problem and the remaining part allows us to get higher accuracies in many cases, but it requires us to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build methods that combine the advantages of using Fourier methods while solving the time-dependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).