One-Dimensional Disordered Supersymmetric Quantum Mechanics: A Brief Survey

We consider a one-dimensional model of localization based on the Witten Hamiltonian of supersymmetric quantum mechanics. The low energy spectral properties are reviewed and compared with those of other models with off-diagonal disorder. Using recent results on exponential functionals of a Brownian motion we discuss the statistical properties of the ground state wave function and their multifractal behaviour.