Supersymmetric quantum mechanics with Levy disorder in one dimension

Abstract

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.

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)

References in corpus (3)

• Products of random matrices and generalised quantum point scatterers
• On an imaginary exponential functional of Brownian motion
• Localization for one-dimensional random potentials with large local fluctuations

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