An efficient spectral method for the fractional Schr\"{o}dinger equation on the real line

TL;DR

This paper introduces a new spectral discretization scheme using Malmquist-Takenaka functions for the fractional Schrödinger equation on the real line, improving computational efficiency and accuracy especially for cases involving the square root of the Laplacian.

Contribution

The paper presents a novel spectral method based on Malmquist-Takenaka functions that outperforms existing schemes in simulating the fractional Schrödinger equation, particularly for the square root Laplacian case.

Findings

The new scheme achieves better performance than existing methods.

Numerical experiments confirm the effectiveness of the proposed discretization.

The method is versatile across different fractional orders of the Schrödinger equation.

Abstract

The fractional Schr\"{o}dinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable or even better performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method.