Critical-Point-Based Stability Analyses of Finite-Difference Time-Domain Methods for Schrödinger Equation Incorporating Vector and Scalar Potentials

Abstract

This paper presents the critical-point-based stability analyses of finite-difference time-domain (FDTD) methods for Schrödinger equation incorporating vector and scalar potentials. Most previous FDTD formulations and stability analyses for the Schrödinger equation involve only the scalar potentials. On the other hand, the existing stability conditions that include both vector and scalar potentials were not thoroughly nor rigorously analyzed, hence they are inadequate for general cases. In this paper, rigorous stability analyses of the FDTD methods will be performed for Schrödinger equation in full 3D incorporating both vector and scalar potentials. New stability conditions are derived rigorously based on the critical points within the interior and boundary regions, while considering the local and global extrema across all variables. Two FDTD schemes are considered, of which one is updated entirely in complex form, and the other is decomposed into real and imaginary parts and updated in a leapfrog manner. Comparisons of the new stability conditions are made against those of prior works, highlighting the thoroughness, completeness and adequacy. Numerical experiments further validate the derived stability conditions and demonstrate their applicability in FDTD methods. Using these stability conditions, the FDTD methods are useful for simulations of quantum-electromagnetic interactions involving vector and scalar potentials.