Energy spectrum of quasiparticles in one-dimensional disordered systems (Review)

A review is presented of papers published over the last decade in which a study was made of the behavior of the density of states in one-dimensional disordered systems. The methods used in these papers seem fairly varied at first glance. Nevertheless, they are bare, simple and general ideas, and this allows currently available results to be stated within the framework of a unified approach. The one-dimensional nature of the problem allows closed dynamic equations to be written for any of the quantities which determine the spectral properties. The structure of these equations is always such that it provides the possibility of using them as the basis for obtaining other nonrandom equations for the probability densities of the corresponding quantities. The behavior of the solutions of these equations in all cases of interest is determined by the dynamics of the system, which leads to the derivation of asymptotic forms for the density of states in the investigated ranges of the spectrum. The density of one-electron states in a random field and the density of the squares of the vibration frequencies of a disordered chain are considered. Principal attention is devoted to the behavior of the density of states near singularities—in the neighborhood of the boundaries of the spectrum and in the impurity band.

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