Supersymmetric quantum mechanics with Levy disorder in one dimension

TL;DR

This paper studies a supersymmetric quantum system with Levy noise, deriving the complex Lyapunov exponent and density of states, revealing new solvable cases and low-energy behavior insights.

Contribution

It introduces a novel approach to compute the complex Lyapunov exponent for Levy-disordered systems and identifies a new solvable case.

Findings

Reduction to a Stieltjes moment problem for non-decreasing Levy processes

Low-energy behavior of the density of states characterized

Discovery of a new solvable case involving special functions

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.