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Wave propagation in a one-dimensional randomly perturbed periodic medium

We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrödinger operator after propagation through N periods. It is shown that if the frequency of propagation lies inside the band, then the total variance is proportional to Nσ2, where σ is the intensity of the white noise. However, if the wave frequency is close to the band edge (where the transfer matrix has a Jordan block structure), the resulting variance is proportional to Nσ2/3. Thus, propagation becomes highly sensitive to random perturbations. Numerical simulations reveal that even low noise in a periodic potential can suppress transmission near the band edges and make it strongly irregular inside the band. Further increase of the noise amplitude leads to intermittent behaviour of the transmission coefficient, and makes transmission possible only for a few random frequencies in the band.