A new variable-order fractional momentum operator for wave absorption when solving Schrödinger equations

Abstract

Absorbing boundary conditions are often needed for solving Schrödinger equations when the analytical solution is unknown. This article presents a new wave-absorption method by unifying the Schrödinger equations with associated one-way wave equations via a fractional-order momentum operator pˆα+1 where 0≤α≤1. By gradually varying the fractional order α, the computational boundary is made to be transparent. This method allows for incident waves to propagate freely and forces reflected waves to decay exponentially with time. The unified equation is then solved using the Finite-Difference Time-Domain (FDTD) method so that the scheme is iteratively explicit and can be parallelized easily. Finally, the obtained fractional FDTD scheme is tested to simulate soliton and particle propagations and is compared with the existing Perfectly Matched Layer (PML) method. Results show that the new method is promising.