Periodic-Orbit Theory of Anderson Localization on Graphs

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Abstract
We present the first quantum system where Anderson localization is completely described within periodic-orbit theory. The model is a quantum graph analogous to an aperiodic Kronig-Penney model in one dimension. The exact expression for the probability to return to an initially localized state is computed 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 all families, are performed analytically using advanced combinatorial methods.