Spectral degeneracy in the one-dimensional Anderson problem: A uniform expansion in the energy band

Statistics of finite-time Lyapunov exponents in a random time-dependent potential

“…This situation appears to be related to the band-center case of one-dimensional localization in the Anderson model [41,42] (where space is discretized on the lattice), since at the band-center the effective mass of the particle diverges (and hence the mean of K 12 vanishes). b) Chaotic dynamics with an isotropic phase space may be modeled by independent fluctuations of all four matrix elements K ij with identical amplitude and vanishing mean.…”

Scaling and the Center-of-Band Anomaly in a One-Dimensional Anderson Model with Diagonal Disorder

“…The new description of the band structure can be viewed as a transition to the reduced zone representation, where the wave number of the original dispersion equation [Eq. (6)] is restricted to the interval 0 ka =2. This operation is equivalent to doubling of the elementary cell of the original periodic chain.…”

“…(3) from this phase randomization hypothesis for several different models. The phase randomization itself was rigorously proven only for a few particular models: banded matrices [9], the Anderson tight-binding model with diagonal disorder [6], and a continuous model with a whitenoise random potential [10] for certain spectral regions. At the same time, it was shown that for some values of energies, such as E 0 or E 2 for the model P H Y S I C A L R E V I E W L E T T E R S week ending 29 AUGUST 2003 VOLUME 91, NUMBER 9 096601-1 0031-9007=03=91(9)=096601(4)$20.00  2003 The American Physical Society 096601-1 described by Eq.…”

“…What actually happens is that the uniform contribution to the stationary phase distribution vanishes in the vicinity of these special energies. As a result, the nonuniform part, which always exists but is usually small, becomes dominant [5][6][7]. Thus, the phase randomization length, as it was introduced in Ref.…”

Scaling in the One-Dimensional Anderson Localization Problem in the Region of Fluctuation States

“…The phase randomization was proven rigorously for in-band states (those belonging to the spectrum of underlying ordered systems) for some one-dimensional models (Anderson model [6] and a continuous model with a white-noise random potential [7]) as well as for some quasi-one-dimensional models [8]. For the Lloyd model, the authors of Ref.…”