Gaussian beam methods for the Schrödinger equation with discontinuous potentials

We propose Eulerian and Lagrangian Gaussian beam methods for the Schrödinger equation with discontinuous potentials. At the quantum barriers where the potential is discontinuous, we derive suitable interface conditions to account for quantum scattering information. These scattering interface conditions are then built into the numerical fluxes in the Eulerian level set formulation of the Gaussian beam methods, and are also used in the Lagrangian formulation, including an interface condition for the Hessian matrix. We carry both 1D and 2D numerical examples to verify the accuracy of the method.