The Gaussian Wave Packet Transform for the Semi-Classical Schr??dinger Equation with Vector Potentials

In this paper, we reformulate the semi-classical Schrodinger equation in the presence of electromagnetic field by the Gaussian wave packet transform. With this approach, the highly oscillatory Schrodinger equation is equivalently transformed into another Schrodinger type wave equation, the omega equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the w equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.