Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well (Communications in Computational Physics, 2018)
In this paper, we numerically study the ground and first excited states of thefractional Schrödinger equation in an infinite potential well. Due to the nonlocality ofthe fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctionsof the fractional Schrödinger equation analytically. We first introduce a normalizedfractional gradient flow and then discretize it by a quadrature rule method in spaceand the semi-implicit Euler method in time. Our numerical results suggest that theeigenfunctions of the fractional Schrödinger equation in an infinite potential well differfrom those of the standard (non-fractional) Schrödinger equation. We find that thestrong nonlocal interactions represented by the fractional Laplacian can lead to a largescattering of particles inside of the potential well. Compared to the ground states,the scattering of particles in the first excited states is larger. Furthermore, boundarylayers emerge in the ground states and additionally inner layers exist in the first excitedstates of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues areconsistent with the lower and upper bound estimates in the literature.