Hidden Symmetry in 1D Localization

Resistance \rho of an one-dimensional disordered system of length l has the log-normal distribution in the limit of large l. Parameters of this distribution depend on the Fermi level position, but are independent on the boundary conditions. However, the boundary conditions essentially affect the distribution of phases entering the transfer matrix, and generally change the parameters of the evolution equation for the distribution P(\rho). This visible contradiction is resolved by existence of the hidden symmetry, whose nature is revealed by derivation of the equation for the stationary phase distribution and by analysis of its transformation properties.