Theories of electrons in one-dimensional disordered systems

A critical survey of the literature on the theory of electronic states in, and electron transmission through, models of one-dimensional disordered solids and liquids is given in the first part. Reference to work on three-dimensional systems is included, especially where exact results have been obtained. The relationship between this subject and the problem of elastic vibrations in disordered solids is pointed out. A complete exposition of the authors' work on one-dimensional conductivity is then presented. It provides a rigorous solution of the problem of average resistance, and of the variance (fluctuations) of resistance, for important classes of disorder which are carefully and precisely defined. Conclusions regarding the role of disorder with respect to the transmission properties are presented and discussed. It is also pointed out that, with appropriate modifications, the results apply generally to wave propagation in inhomogeneous media.