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Singular continuous spectrum in a class of random Schrödinger operators. (English) Zbl 0770.60064

Summary: For a class of random Schrödinger operators in \(L^ 2(R^ d)\) \[ H(\omega)=-\Delta+\sum_{j\in\mathbb{Z}^ d}q_ j(\omega)f(x-j), \] where \(q_ j\) are continuous independent identically distributed bounded random variables and \(f\) has a power decay and defined sign, in any energy interval the singular continuous spectrum is either empty or with positive Lebesgue measure. As a consequence, the proof of localization for a class of random but deterministic one-dimensional operators is shifted to showing that the singular continuous spectrum has null Lebesgue measure.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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References:

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