Author:
Iserles Arieh,Luong Karen,Webb Marcus
Abstract
AbstractIn this paper we compare three different orthogonal systems in $$\textrm{L}_2({\mathbb {R}})$$
L
2
(
R
)
which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger 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 $$\textrm{L}_2({\mathbb {R}})$$
L
2
(
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.
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,General Mathematics,Analysis
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