Collocation method for fractional quantum mechanics

TL;DR

This paper introduces a collocation method using Little Sinc Functions to numerically solve fractional quantum mechanics problems involving the fractional Laplacian, demonstrating good convergence and comparing with WKB analysis.

Contribution

It presents a novel collocation approach for fractional quantum mechanics using Little Sinc Functions, enabling efficient numerical solutions with natural boundary condition implementation.

Findings

The method achieves good convergence properties.

It successfully discretizes the Schrödinger equation with fractional Laplacian.

Results are consistent with WKB analysis.

Abstract

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on Little Sinc Functions (LSF), which discretizes the Schr\"odinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a WKB analysis is performed.