The generalized Lyapunov exponent for the one-dimensional Schr\"odinger equation with Cauchy disorder: some exact results

TL;DR

This paper derives exact analytical results for the generalized Lyapunov exponent and cumulants of the wave function in a one-dimensional Schrödinger equation with Cauchy disorder, revealing universal behaviors and conductance distribution characteristics.

Contribution

It provides the first exact secular equation and explicit cumulant formulas for the generalized Lyapunov exponent with Cauchy disorder, advancing understanding of wave localization statistics.

Findings

Derived secular equation for the generalized Lyapunov exponent.

Obtained explicit formulas for the first four cumulants.

Identified power-law behavior of conductance distribution at small values.

Abstract

We consider the one-dimensional Schr\"odinger equation with a random potential and study the cumulant generating function of the logarithm of the wave function $\psi(x)$, known in the literature as the "generalized Lyapunov exponent"; this is tantamount to studying the statistics of the so-called "finite size Lyapunov exponent". The problem reduces to that of finding the leading eigenvalue of a certain non-random non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus on the case of Cauchy disorder, for which we derive a secular equation for the generalized Lyapunov exponent. Analytical expressions for the first four cumulants of $\ln|\psi(x)|$ for arbitrary energy and disorder are deduced. In the universal (weak-disorder/high-energy) regime, we obtain simple asymptotic expressions for the generalized Lyapunov exponent and for all the cumulants. The large deviation function controlling the distribution of $\ln|\psi(x)|$ is also obtained in several limits. As an application, we show that, for a disordered region of size $L$, the distribution $\mathcal{W}_L$ of the conductance $g$ exhibits the power law behaviour $\mathcal{W}_L(g)\sim g^{-1/2}$ as $g\to0$.