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

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.

I. Introduction

The Schrödinger equation is crucial for applications involving nanostructures, semiconductor and quantum devices, etc. The finite-difference time-domain (FDTD) method is one of the popular methods for solving the Schrödinger equation due to its robustness and flexibility. In the FDTD method, the Schrödinger equation can be discretized and updated in its complex form entirely [1], [2], [3], [4], [5], [6], [7]. Alternatively, the Schrödinger equation can first be decomposed into real and imaginary parts, and they are discretized and updated in a leapfrog or synchronous manner [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. The scheme could be more efficient as there are no longer operations involving complex numbers. It has been shown that decomposing the Schrödinger equation into real and imaginary parts can achieve higher efficiency than the complex one in the implicit FDTD method [18]. In [19], [20], the Schrödinger equation is transformed into diffusion equation via the imaginary time transformation before discretization, which yields a dissipative scheme. Other higher order schemes have also been proposed in [21], [22], [23], [24]. In electromagnetics (EM), the dynamic fields involve both magnetic vector potential and electric scalar potential . Hence, the EM-quantum interactions generally would require both potentials to be incorporated in the Schrödinger equation. Nevertheless, most FDTD formulations [1], [2], [5], [6], [7], [8], [9], [10], [11], [14], [15], [18], [19], [20] for the Schrödinger equation involve only the scalar potential (). Most are also only in one (1D) or two dimension (2D) and not in full three dimension (3D). Henceforth, our analyses shall focus on the nondissipative, second-order FDTD methods for Schrödinger equation in full 3D incorporating vector and scalar potentials.