Approximation of wave packets on the real line

Abstract

In this paper we compare three different orthogonal systems in L2(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 L2(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.

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

“…Remark 2.3. Recently, Iserles, Luong and Webb in [19] compared the approximation power of the Malmquist-Takenaka, Hermite and stretched Fourier functions for Gaussian wave packet functions of the form f (x) = exp(−β(x − x 0 ) 2 ) cos(ωx), where β > 0 and x 0 , ω ∈ R. After some lengthy algebra, they derived the decay rates of the coefficients with respect to these three orthogonal systems, respectively, and concluded that the MTFs are superior to the other two functions. Note that all those three functions have banded skew-Hermitian differentiation matrices.…”

“…where δ n,m is the Kronecker delta. Theoretical aspects of MTFs as well as their applications in designing algorithms for Fourier and Hilbert transforms have been investigated during the past few decades (see, e.g., [8,15,18,19,35,36]). We list below some of theoretical properties of MTFs that will be used for the construction of spectral method.…”