一维局域化中两个部分解的联合分布：关于相变的新信息
Mutual distribution of two partial solutions in 1D localization: new information on the phase transition
AI总结：
研究了一维随机势薛定谔方程两个独立解的联合分布，发现其偏离对数正态的机制与修正，揭示了由 Lyapunov 指数构型变化导致的相变，并在光学系统中可观测。
摘要：
我们研究了具有随机势的一维薛定谔方程的两个线性独立解 y_1(x) 和 y_2(x) 的联合分布。由于 y_1 和 y_2 的个体分布是对数正态的，因此自然假设它们的联合分布也是对数正态的。这一假设在允带和禁带的深处得到证实，但在初始能带边缘附近失效。我们阐明了偏离对数正态形式的机制，并计算了其第一阶修正。后者允许我们证明在普适电导涨落中谱线的展宽。关于分布 P(ψ) 的相变获得了大量新信息，其中 ψ 是进入演化方程的组合相位。根据先前的出版物，该相变与在特定能量 E_0 处 ψ 虚部的出现有关，且不伴随系统电阻的奇异性。该相变的真实含义在于四个 Lyapunov 指数构型的变化，这些指数决定了通解：对于 E>E_0 存在两对共轭复指数，而对于 E<E_0 所有指数变为实数。在允带和禁带深处的能量证实了两种不同构型的实现；这在形式层面上证明了奇点 E_0 的存在。通过在光学系统中追踪坐标变化时波场符号的变化，可以观察到该相变。
英文摘要：
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(ψ), where ψis a combined phase entering the evolution equations. According to the previous publications, this transition is related with appearance of the imaginary part of ψat 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.
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