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.<br />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.<br />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.<br />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.