Quantum particle in a random potential: Exact solution and its implications

Abstract

I accurately solve the Schrödinger equation in a magnetic field B in an arbitrary set of two-dimensional point potentials. When B=0, they yield a mobility edge. When B≠0, all states are localized below a certain energy ${\mathit{E}}_{\mathit{c}}$(B). Above ${\mathit{E}}_{\mathit{c}}$(B), they are extended at the Landau energies. At other energies the localization length is a discontinuous function of B at every rational value of ${\mathit{eBd}}^2$/ch, where d is an average interpotential distance.