What determines the spreading of a wave packet?

TL;DR

This paper investigates how multifractal properties of energy spectra and eigenfunctions influence the long-term spreading behavior of wave packets, revealing scaling laws and spatial decay patterns.

Contribution

It establishes a direct link between multifractal dimensions and wave packet spreading, providing new theoretical insights and numerical validation.

Findings

Wave packet width scales as t^(beta) with beta=D2^mu/D2^psi.

In systems with shape-preserving wave packets, moments grow as t^(k*beta).

Wave packet centers decay spatially as a power law with exponent D2^psi - d.

Abstract

The multifractal dimensions D2^mu and D2^psi of the energy spectrum and eigenfunctions, resp., are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is preserved the k-th moment increases as t^(k*beta) with beta=D2^mu/D2^psi, while in general t^(k*beta) is an optimal lower bound. Furthermore, we show that in d dimensions asymptotically in time the center of any wave packet decreases spatially as a power law with exponent D_2^psi - d and present numerical support for these results.