Solving the Pertubed Quantum Harmonic Oscillator in Imaginary Time Using Splitting Methods with Complex Coefficients

Abstract

[EN] Efficient splitting algorithms for the Schrödinger eigenvalue problem with perturbed harmonic oscillator potentials in higher dimensions are considered. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. Using algebraic techniques, we show how to apply Fourier spectral methods to propagate higher dimensional quantum harmonic oscillators, thus retaining the near integrable structure and fast computability. This methods is then used to solve the eigenvalue problem by imaginary time propagation. High order fractional time steps of order greater than two necessarily have negative steps and can not be used for this class of diffusive problems. However, the use of fractional complex time steps with positive real parts does not negatively impact on stability and only moderately increases the computational cost. We analyze the performance of this class of schemes and propose new highly optimized sixth-order schemes for near integrable systems which outperform the existing ones in most cases.