Supersymmetric quantum mechanics with Levy disorder in one dimension

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.