Ground States for a Superlinear Fractional Schrödinger Equation with Potentials

2021 ◽  
pp. 145-194
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Ground States ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

scholarly journals Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

2015 ◽  
Vol 18 (2) ◽  
pp. 321-350 ◽  
Author(s):  
Siwei Duo ◽  
Yanzhi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Euler Method ◽  
Potential Well ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation ◽  
Infinite Potential Well

AbstractIn 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.

The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain

2018 ◽  
Vol 18 (1) ◽  
pp. 77-94
Author(s):  
Dan Li ◽  
Jiwei Zhang ◽  
Zhimin Zhang
Keyword(s):  
Schrödinger Equation ◽  
Fractional Derivative ◽  
Unbounded Domain ◽  
Caputo Fractional Derivative ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Main Idea ◽  
Fractional Schrödinger Equation ◽  
Direct Scheme ◽  
Fractional Schrodinger Equation

AbstractA fast and accurate numerical scheme is presented for the computation of the time fractional Schrödinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.

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