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

A uniform quantitative description of the properties of the one-dimensional Anderson model is obtained by mapping that problem onto an infinitely quasidegenerate master equation. This quasidegeneracy is identified as the source of the small-denominator problem encountered before in investigations of this problem. An appropriate quasidegenerate perturbation theory is developed to obtain a uniform asymptotic expansion, in powers of the strength of the noise, for the probability distribution function of the ratio of the value of the wave function at neighboring sites. Well known results, such as those obtained by Thouless, Kappus and Wegner, and Derrida and co-workers are reproduced and systematic corrections to these results as well as some more results are found. In particular, we find internal layers in the above-mentioned distribution function for values of the energy given by E=2 cosπα with α rational. We also find crossovers in the behavior of the distribution function (and consequently in quantities derived from it) near the band-edge and band-center regions. The properties of the model in the band-edge region were studied by us in detail in a previous publication [Phys. Rev. B 47, 1918 (1992)].