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We discuss the localization of a particle in a random potential in terms of a probability of return and a sojourn time. At the second order in the potential, we find qualitatively different behaviours in dimension one, two and three. In particular for dimension two, we find states neither localized, nor truly extended. These results hold also for systems with off-diagonal disorder. In the presence of a magnetic field, our results suggest complete localization at weak concentration of impurities or at large field.
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Kunz et al. (1982) studied this question.
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