The aim of this thesis was to elaborate and extend the results proved in the paper "Convergence of a split-step Hermite method for the Gross-Pitaevskii equation" by L. Gauckler (2010). There, one can find convergence analyses for time and space semi-discretisations and full discretisations applied to the cubic nonlinear Schrödinger equation with a harmonic oscillator potential. The methods used in this treatise include Hermite quadrature and an operator splitting of second order.
The author was able to generalise all these results for Schrödinger equations with a scaled harmonic oscillator potential and a sum of power-nonlinearities up to an arbitrary degree, and was also capable of showing an existence and uniqueness result for equations of this type.
Furthermore, using additionally the formal calculus of Lie derivatives, the author could prove convergence of arbitrary order of the time semi-discretised equation when using an appropriate higher order splitting scheme.
Proving a higher rate of convergence for the fully discretised scheme, however, turned out to be impossible without additional tools, and remains a challenge for future studies.