## The fractional Laplacian and its application to the fractional Schrödinger equation

- Sept. 25, 2017
- 2:15 p.m.
- LeConte 312

## Abstract

Recently, the fractional Laplacian has been received much attention in modeling different complex phenomena that involve the long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis. In this talk, I present a novel and accurate finite difference method -- the weighted trapezoidal method, to discretize the fractional Laplacian. Compared to the current state-of-the-art methods, our method has higher accuracy but less computational complexity. As an application, we studied the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well, which is one of the fundamental problem in the fractional quantum mechanics. Our results show that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well are different from those of the standard case, and this provides insights into one open problem in this field.