Periodic-orbit theory of Anderson localization on graphs

We present the first quantum system where Anderson localization is complete ly described within periodic-orbit theory. The model isa quantum graph anal ogous to an aperiodic Kronig-Penney model in one dimension. The exact expre ssion for the probability to return to an initially localized state is comp uted in terms of classical trajectories. It saturates to a finite value due to localization, while the diagonal approximation decays diffusively. Our theory is based on the identification of families of isometric orbits. The coherent periodic-orbit sums within these families, and the summation over ail families, are performed analytically using advanced combinatorial metho ds.