Supersymmetric Quantum Mechanics with L,vy Disorder in One Dimension

We consider the Schrodinger equation with a random potential of the form V(x) = w(2)(x)/4 - w'(x)/2 where w is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent Omega := gamma - i pi N where N is the integrated density of states of the system, and. is the Lyapunov exponent. In the case where the Levy process is non-decreasing, we show that the calculation of Omega 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 Omega can be expressed in terms of special functions-and discover a new one.