Random discrete Schrödinger operators from Random Matrix Theory

Abstract

We investigate random, discrete Schrödiner operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature $\beta$. They belong to the class of "critical" random Schrödiner operators with random potentials which diminish as $|x{|}^{-1/2}$. We show that as a function of $\beta$ their eigenstates undergo a transition from extended ($\beta \ge 2$) to power-law localized ($0<\beta <2$).

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