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

We consider the mutual distribution of two linearly independent solutions y1(x) and y2(x) of the 1D Schrödinger equation with a random potential. Since individual distributions of y1 and y2 are log-normal, it is reasonable to suggest that their mutual distribution is also log-normal. Such hypothesis is confirmed in the depth of the allowed and forbidden bands, but fails 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 one to demonstrate the 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(ψ), where ψ is a combined phase entering the evolution equations. According to the previous publications, this transition is related to the appearance of the imaginary part of ψ at a certain energy E0, and is not accompanied by singularities in the system resistance. The real meaning of this transition is a change in the configuration of four Lyapunov exponents, which determine the general solution: there are two pairs of complex conjugate exponents for E>E0, while for E<E0 all exponents become real. Realization of two different configurations is confirmed for energies in the depth of the allowed and forbidden bands; it proves the existence of the singular point E0 at the formal level. The phase transition can be observed in optical systems, tracing the sign of the field in a wave, under the variation of the coordinate.