Diffusion on a one-dimensional lattice and quantum localization

The problem of diffusion in modulated and random media as described by a master equation is reduced to a discrete Schrödinger equation, and further transformed into a recursive nonlinear map. Short- and long-time diffusion is then discussed in terms of dynamical properties of such a map. Analytical expressions for scaling behavior of the integrated density of states are derived for the top and bottom of the spectrum. The numerical results allow estimation of the scaling region and the corrections to the leading behavior. © 1988 The American Physical Society.