Phase-averaged transport for quasi-periodic Hamiltonians

TL;DR

This paper establishes a lower bound on phase-averaged transport for certain quasi-periodic Hamiltonians, linking it to multifractal dimensions of the density of states, under Diophantine conditions, with implications for the Harper operator.

Contribution

It provides a novel lower bound on transport in quasi-periodic Hamiltonians based on multifractal analysis, including the critical Harper operator, under Diophantine conditions.

Findings

Lower bound on phase-averaged transport proven

Connection between transport and multifractal dimensions established

Includes a new solution to the frame problem for Weyl-Heisenberg-Gabor lattices

Abstract

For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heisenberg-Gabor lattices of coherent states is given.