Generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: Some exact results

We consider the one-dimensional Schrödinger equation with a random potential and study the cumulant generating function of the logarithm of the wave function 𝜓⁡(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 nonrandom 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⁡|𝜓⁡(𝑥)| for arbitrary energy and disorder are deduced. In the universal (weak-disorder and 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⁡|𝜓⁡(𝑥)| is also obtained in several limits. As an application, we show that, for a disordered region of size 𝐿, the distribution 𝒲𝐿 of the conductance 𝑔 exhibits the power-law behavior 𝒲𝐿⁡(𝑔)∼𝑔−1/2 as 𝑔→0.