TY - JOUR
T1 - Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well
AU - Siwei Duo & Yanzhi Zhang
JO - Communications in Computational Physics
VL - 2
SP - 321
EP - 350
PY - 2018
DA - 2018/04
SN - 18
DO - http://doi.org/10.4208/cicp.300414.120215a
UR - https://global-sci.org/intro/article_detail/cicp/11030.html
KW - 
AB - <p style="text-align: justify;">In this paper, we numerically study the ground and first excited states of the
fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of
the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions
of the fractional Schrödinger equation analytically. We first introduce a normalized
fractional gradient flow and then discretize it by a quadrature rule method in space
and the semi-implicit Euler method in time. Our numerical results suggest that the
eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ
from those of the standard (non-fractional) Schrödinger equation. We find that the
strong nonlocal interactions represented by the fractional Laplacian can lead to a large
scattering of particles inside of the potential well. Compared to the ground states,
the scattering of particles in the first excited states is larger. Furthermore, boundary
layers emerge in the ground states and additionally inner layers exist in the first excited
states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are
consistent with the lower and upper bound estimates in the literature.</p>