Approximation of wave packets on the real line

TL;DR

This paper compares three orthogonal systems for spectral methods to solve the Schrödinger equation on the real line, highlighting the advantages of Malmquist–Takenaka functions in approximating wave packets accurately.

Contribution

It introduces and analyzes the use of Malmquist–Takenaka functions as a superior basis for spectral methods in quantum wave packet approximation.

Findings

Malmquist–Takenaka basis outperforms Hermite and Fourier bases in high-frequency wave packet approximation.

All three bases have banded skew-Hermitian differentiation matrices, ensuring stability and unitarity.

Asymptotic coefficient approximations are highly accurate in the high-frequency regime.

Abstract

In this paper we compare three different orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schr\"odinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist--Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary -- this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the $\mathrm{L}_2(\mathbb{R})$ norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist--Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets