Stationary Schr\"odinger equation in the semi-classical limit: numerical coupling of oscillatory and evanescent regions

TL;DR

This paper develops a hybrid numerical method combining WKB and FEM techniques to efficiently solve 1D Schrödinger equations with mixed oscillatory and evanescent regions, avoiding turning points.

Contribution

It introduces a domain decomposition approach and a hybrid WKB-FEM method with comprehensive error analysis for semi-classical Schrödinger problems.

Findings

The hybrid method achieves high accuracy in semi-classical regimes.

Numerical tests confirm the convergence and efficiency of the proposed approach.

The approach effectively handles discontinuities in the potential without turning points.

Abstract

This paper is concerned with a 1D Schr\"odinger 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.