Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schr'odinger equations

摘要

We consider the numerical integration of the Gross-Pitaevskii equation with apotential trap given by a time-dependent harmonic potential or a smallperturbation thereof. Splitting methods are frequently used with Fouriertechniques since the system can be split into the kinetic and remaining part,and each part can be solved efficiently using Fast Fourier Transforms. To splitthe system into the quantum harmonic oscillator problem and the remaining partallows to get higher accuracies in many cases, but it requires to changebetween Hermite basis functions and the coordinate space, and this is notefficient for time-dependent frequencies or strong nonlinearities. We show howto build new methods which combine the advantages of using Fourier methodswhile solving the timedependent harmonic oscillator exactly (or with a highaccuracy by using a Magnus integrator and an appropriate decomposition).