2011/05/31 by Alain Comtet, Christophe Texier, Yves Tourigny
Mathematics · Physics and Astronomy · #Applied mathematics #Dimension (graph theory) #Exponent #Generality #Lyapunov equation #Lyapunov exponent #Lyapunov function #Lévy process #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear system #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Quantum statistical mechanics #Schrödinger equation #Schrödinger's cat #Statistical Mechanics and Entropy #Statistical mechanics #Statistical physics #Superpotential #Supersymmetric quantum mechanics #Supersymmetry #cond-mat.dis-nn #math-ph #math.MP #math.PR #msc:60G51 #msc:82B44
paper · pdf · doi:10.1007/s10955-011-0351-3
published as Journal of Statistical Physics vol: 145 , Issue: 5 , Pages: 1291 - 1323, 2011 · 31 pages, to appear in Journal of Statistical Physics. This arxiv version corrects a minor mistake appearing in the J. Stat. Phys. version: see the equation now labelled (5.11)
openalex publication_date 2011/09/23 · arxiv created 2011/09/25 · arxiv updated 2013/07/02 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are, respectively, the Lyapunov exponent and the integrated density of states of the system. In the case where the Levy process is non-decreasing, we show that the calculation of the complex Lyapunov exponent reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Levy process. We review the known solvable cases, where the complex Lyapunov exponent can be expressed in terms of special functions, and discover a new one.