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We study how the spectral properties of ergodic Schr\"odinger operators arereflected in the asymptotic properties of its periodic approximation as theperiod tends to infinity. The first property we address is the asymptotics ofthe bandwidths on the logarithmic scale, which quantifies the sensitivity ofthe finite volume restriction to the boundary conditions. We show that thebandwidths 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 amatching upper bound. Finally, we provide an additional assumption which isalso satisfied in the i.i.d case, under which the corresponding eigenvectorsare exponentially localised with a localisation centre independent of theFloquet number.

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