Computing the ground and first excited states of the fractional Schrodinger equation in an infinite potential well

Abstract

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.

References in corpus (9)

• Mathematical theory and numerical methods for Bose-Einstein condensation
• Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$
• Fractional Laplacian in Bounded Domains
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well
• Comment on: "On the consistency of solutions of the space fractional Schrödinger equation"
• Levy path integral approach to the solution of the fractional Schrödinger equation with infinite square well
• The fractional Schrödinger equation and the infinite potential well - numerical results using the Riesz derivative
• Leaky-box approximation to the fractional diffusion model of cosmic rays

Cited by in corpus (15)

• Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
• A unified meshfree pseudospectral method for solving both classical and fractional PDEs
• Eigenvalues of the fractional Laplace operator in the unit ball
• Numerical approximations for the tempered fractional Laplacian: Error analysis and applications
• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• A novel and simple spectral method for nonlocal PDEs with the fractional Laplacian
• A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
• Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• Ultrarelativistic bound states in the shallow spherical well
• Fractional Laplacians and Levy flights in bounded domains
• Existence and non-existence of normalized solutions for a nonlinear fractional Schrödinger system
• A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations
• Approximating and preconditioning the stiffness matrix in the GoFD approximation of the fractional Laplacian