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

@article{Comtet2022GeneralizedLE,
  title={Generalized Lyapunov exponent for the one-dimensional Schr{\"o}dinger equation with Cauchy disorder: Some exact results},
  author={Alain Comtet and Christophe Texier and Yves Tourigny},
  journal={Physical Review E},
  year={2022}
}
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 non-random non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus… 

Figures from this paper

References

SHOWING 1-10 OF 90 REFERENCES
Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation
Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schr\"odinger equation with a random potential $V$, are studied. I
Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson localization models
For the 2D matrix Langevin dynamics that correspond to the continuous-time limit of the products of some 2 × 2 random matrices, the finite-time Lyapunov exponent can be written as an additive
Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension
AbstractWe consider the Schrödinger equation with a random potential of the form where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent where N is
Statistics of finite-time Lyapunov exponents in a random time-dependent potential.
  • H. Schomerus, M. Titov
  • Mathematics, Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
TLDR
This work compute for a particle moving in a randomly time-dependent, one-dimensional potential how the distribution function P(lambda;t) approaches the limiting distribution P( lambda; infinity)=delta(lambda-lambda(infinity)).
The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity
We study products of arbitrary random real 2×2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,ℝ), we identify a continuum regime where the mean values and the
Lyapunov exponents, one-dimensional Anderson localization and products of random matrices
The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the
Random walkers in one-dimensional random environments: exact renormalization group analysis.
TLDR
Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group, which yields exact results at long times and rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are characterized in detail.
Breaking supersymmetry in a one-dimensional random Hamiltonian
The one-dimensional supersymmetric random Hamiltonian , where (x) is a Gaussian white noise of zero mean and variance g, presents particular spectral and localization properties at low energy: a
Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent
I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\Pi_n=M_nM_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following
...
...