Mutual distribution of two partial solutions in 1D localization: new information on the phase transition

Abstract

We consider the mutual distribution of two linearly independent solutions y_1(x) and y_2(x) of the 1D Schroedinger equation with a random potential. Since individual distributions of $y_1$ and $y_2$ are log-normal, it is naturally to suggest that their mutual distribution is also log-normal. Such hypothesis is confirmed in the deep of the allowed and forbidden bands, but failed near the initial band edge. The mechanism of deviations from the log-normal form is elucidated, and the first correction to it is calculated. The latter allows to demonstrate broadening of the spectral lines in the universal conductance fluctuations. A lot of new information is obtained on the phase transition in the distribution P(\psi), where \psi is a combined phase entering the evolution equations. According to the previous publications, this transition is related with appearance of the imaginary part of \psi at a certain energy E_0, and is not accompanied by singularities in the system resistance. The real sense of this transition consists in the change of configuration of four Lyapunov exponents, which determine the general solution: there are two pairs of complex-conjugated exponents for E>E_0, while for E<E_0 all exponents become real. Realization of two different configurations is confirmed for energies in the deep of the allowed and forbidden bands; it proves the existence of the singular point E_0 at the formal level. The phase transition can be observed in optical systems, tracing the sign of the field in a wave, when the coordinate is changed.