Stationary Schrödinger equation in the semi-classical limit: numerical coupling of oscillatory and evanescent regions
This work addresses the efficient and accurate numerical solution of Schrödinger equations in the semi-classical limit for problems with discontinuous potentials, which is relevant for quantum mechanics simulations.
The paper tackles the 1D Schrödinger scattering problem in the semi-classical limit with oscillatory and evanescent regions separated by potential jumps. They derive a domain decomposition method and a hybrid WKB-based numerical method, providing a complete error analysis and numerical validation.
This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid "turning points". We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests.