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arXiv · 2101.02566

Approximation of wave packets on the real line

Arieh Iserles·Karen Luong·Marcus Webb
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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

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Arieh Iserles, Karen Luong, Marcus Webb. 2021-01-07. Approximation of wave packets on the real line. https://arxiv.org/abs/2101.02566

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