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An efficient spectral method for the fractional Schrödinger equation on the real line

Mengxia Shen, Haiyong Wang

TL;DR

This work tackles the fractional Schrödinger equation on the real line, where nonlocal fractional Laplacians and power-law decay pose significant numerical challenges. It introduces a spectral Galerkin method based on Malmquist-Takenaka functions, enabling highly accurate spatial discretization on unbounded domains, and combines it with suitable time integrators (splitting for linear parts and exponential time-differencing for nonlinearities). A key finding is the exponential convergence of the MT-based method when $\alpha=1$, with comparable or superior performance for other $\alpha$ values, along with efficient $O(N\log N)$ coefficient computations via FFTs. The approach is demonstrated on linear and nonlinear FSEs, showing mass conservation in the nonlinear case and robust accuracy, thereby offering a powerful tool for simulations of nonlocal PDEs on unbounded domains. This method broadens the applicability of spectral techniques to problems with slow decay and nonlocal operators, and suggests avenues for higher-dimensional extensions and rigorous conservation analysis.

Abstract

The fractional Schrödinger 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.

An efficient spectral method for the fractional Schrödinger equation on the real line

TL;DR

This work tackles the fractional Schrödinger equation on the real line, where nonlocal fractional Laplacians and power-law decay pose significant numerical challenges. It introduces a spectral Galerkin method based on Malmquist-Takenaka functions, enabling highly accurate spatial discretization on unbounded domains, and combines it with suitable time integrators (splitting for linear parts and exponential time-differencing for nonlinearities). A key finding is the exponential convergence of the MT-based method when , with comparable or superior performance for other values, along with efficient coefficient computations via FFTs. The approach is demonstrated on linear and nonlinear FSEs, showing mass conservation in the nonlinear case and robust accuracy, thereby offering a powerful tool for simulations of nonlocal PDEs on unbounded domains. This method broadens the applicability of spectral techniques to problems with slow decay and nonlocal operators, and suggests avenues for higher-dimensional extensions and rigorous conservation analysis.

Abstract

The fractional Schrödinger 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.
Paper Structure (9 sections, 3 theorems, 55 equations, 12 figures)

This paper contains 9 sections, 3 theorems, 55 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Malmquist-Takenaka functions
  3. Malmquist-Takenaka spectral method
  4. Spatial discretization
  5. Temporal discretization
  6. The linear case
  7. The nonlinear case
  8. Numerical examples
  9. Conclusions

Key Result

Theorem 2.1

Let $\Pi_Nf$ denote the spectral approximation defined in eq:MTExp and let $\mathcal{A}_{\rho}$ denote the annulus defined by $\mathcal{A}_{\rho}=\{z\in\mathbb{C}:\rho^{-1}<|z|<\rho\}$. Moreover, let If $\hat{f}(z)$ is analytic in $\mathcal{A}_{\rho}$ for some $\rho>1$. Then for $x\in\mathbb{R}$,

Figures (12)

  • Figure 1: The magnitude of the Malmquist-Takenaka coefficients of $\{a_k\}_{k\in\mathbb{Z}}$. Left: $f(x)=\exp(-x^2)$ and $\mathrm{sech}(x)$. Right: $f(x)=1/(1+x^2)$ and $1/(1+x^4)$.
  • Figure 2: Maximum errors of MTF and MCF approximations for $f(x)=\exp(-x^2)/(1+\mathrm{i}x)$, $1/(x^2+4)$, $1/(x^4+1)$ (left column) and $f(x)=\exp(-x^2)$, $\mathrm{sech}(x)$, $1/(x^4+1)^{1.2}$ (right column).
  • Figure 3: The logarithm of the magnitude of the elements of $C$ for $\alpha=0.6$ (left) and $\alpha=1.4$ (right).
  • Figure 4: The logarithm of the magnitude of the elements of $M$ for $V(x)=1/(1+x^2)$ (left) and $V(x)=\exp(-x^2)$ (right).
  • Figure 5: Stability regions of the Krogstad-P22 scheme for several values of $y$.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 6 more