Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension

We consider the Schrödinger equation with a random potential of the form V(x) = w^2(x)/4-w'(x)/2 where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent Ω = γ - iπN where N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω 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 Lévy process. We review the known solvable cases—where Ω can be expressed in terms of special functions—and discover a new one.