Upper bounds on wavepacket spreading for random Jacobi matrices

TL;DR

This paper introduces a new method to establish upper bounds on quantum wavepacket spreading in random Jacobi matrices, providing sharp bounds for diffusion exponents and demonstrating logarithmic growth of moments under certain conditions.

Contribution

It presents a novel approach for bounding wavepacket spreading in random Jacobi matrices, applicable to polymer models and Lyapunov exponent conditions, without relying on traditional multiscale analysis.

Findings

Sharp upper bounds on diffusion exponents for random polymer models.

Logarithmic growth of moments under positive Lyapunov exponents.

A new elementary method avoiding multiscale analysis.

Abstract

A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work. The second application is an elementary argument (not using multiscale analysis or the Aizenman-Molchanov method) showing that under the condition of uniformly positive Lyapunov exponents, the moments of the position operator grow at most logarithmically in time.