Single parameter scaling in one-dimensional localization revisited

The variance of the Lyapunov exponent is calculated exactly in the one-dime nsional Anderson model with random site energies distributed according to t he Cauchy distribution. We find a new significant scaling parameter in the system, and derive an exact analytical criterion for single parameter scali ng which differs from the commonly used condition of phase randomization. T he results obtained are applied to the Kronig-Penney model with the potenti al in the form of periodically positioned delta functions with random stren gth.