  In this paper, we numerically study the ground and first excited states of
the fractional Schrodinger equation in an infinite potential well. Due to the
non-locality of the fractional Laplacian, it is challenging to find the
eigenvalues and eigenfunctions of the fractional Schrodinger equation either
analytically or numerically. We first introduce a fractional gradient flow with
discrete normalization and then discretize it by using the trapezoidal type
quadrature rule in space and the semi-implicit Euler method in time. Our method
can be used to compute the ground and first excited states not only in the
linear cases but also in the nonlinear cases. Moreover, it can be generalized
to solve the fractional partial differential equations (PDEs) with Riesz
fractional derivatives in space. Our numerical results suggest that the
eigenfunctions of the fractional Schrodinger equation in an infinite potential
well are significantly different from those of the standard (non-fractional)
Schrodinger equation. In addition, 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
Schrodinger equation.
