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

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.