A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space (Communications in Computational Physics, 2020)
We propose a hybrid method which combines the Bloch decomposition-basedtime splitting (BDTS) method and the Gaussian beam method to simulate theSchrödinger equation with periodic potentials in the case of band-crossings. With thehelp of the Bloch transformation, we develop a Bloch decomposition-based Gaussianbeam (BDGB) approximation in the momentum space to solve the Schrödinger equation.Around the band-crossing a BDTS method is used to capture the inter-band transitions,and away from the crossing, a BDGB method is applied in order to improve theefficiency. Numerical results show that this method can capture the inter-band transitionsaccurately with a computational cost much lower than the direct solver. Wealso compare the Schrödinger equation with its Dirac approximation, and numericallyshow that, as the rescaled Planck number ε→0, the Schrödinger equation converges tothe Dirac equations when the external potential is zero or small, but for general externalpotentials there is an O(1) difference between the solutions of the Schrödingerequation and its Dirac approximation.
Keywords: Schrödinger equation; band-crossing; Dirac point; Bloch decomposition; time-splitting spectral method; Gaussian beam method.