On the Bandwidths of Periodic Approximations to Discrete Schr\"odinger Operators

TL;DR

This paper investigates the relationship between spectral properties of ergodic Schrödinger operators and their periodic approximations, establishing bounds on bandwidths and conditions for exponential localization of eigenvectors.

Contribution

It provides new bounds on the bandwidths of periodic approximations in terms of the Lyapunov exponent and identifies conditions for exponential localization of eigenvectors.

Findings

Bandwidths can be bounded from below by the Lyapunov exponent.

Under certain conditions, an upper bound on bandwidths is established.

Eigenvectors exhibit exponential localization with a Floquet-independent center.

Abstract

We study how the spectral properties of ergodic Schr\"odinger operators are reflected in the asymptotic properties of its periodic approximation as the period tends to infinity. The first property we address is the asymptotics of the bandwidths on the logarithmic scale, which quantifies the sensitivity of the finite volume restriction to the boundary conditions. We show that the bandwidths can always be bounded from below in terms of the Lyapunov exponent. Under an additional assumption satisfied by i.i.d potentials, we also prove a matching upper bound. Finally, we provide an additional assumption which is also satisfied in the i.i.d case, under which the corresponding eigenvectors are exponentially localised with a localisation centre independent of the Floquet number.