Numerical simulation of the two-dimensional fractional Schrödinger equation for describing the quantum dynamics on a comb with the absorbing boundary conditions

Abstract

This paper mainly contributes to investigating a distinctive fractional Schrödinger equation (FSE) for describing the quantum dynamics on a comb. The special characteristic of the comb is that the diffusion versus the $x$-direction only occurs on the $x$-axis. We replace the infinite boundary with the absorbing boundary conditions (ABCs), which can be obtained using the Laplace transform. We utilize the integration property to deal with the singularity function in the FSE and propose the finite difference method. The stability and convergence of this scheme are investigated. The $\mathrm{L1}$ scheme to discretize the Caputo fractional derivative needs a particularly expensive computational and storage cost. To increase the calculation speed, a fast algorithm is used to improve the operation efficiency. By introducing a source term, the exact solution and the numerical solution are compared. The comparison of the CPU time for the $\mathrm{L1}$ scheme and fast scheme is given. The comparison between the FSE on a comb and the classical one is also analyzed and discussed. The transport process versus the $x$-axis and the $y$-axis with various parameters are shown. Finally, the mean square displacement (MSD) is investigated. We find that the solution with the ABCs has a better agreement with the exact expression compared to the solution with truncated zero boundary conditions.