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Localization for some continuous random Schrödinger operators. (English) Zbl 0820.60044

Summary: We study the spectrum of random Schrödinger operators acting on \(L^ 2(\mathbb R^ d)\) of the following type \(H = -\Delta + W + \sum_{x \in \mathbb Z^ d} t_ x V_ x\). The \((t_ x)_{x\in \mathbb Z^ d}\) are i.i.d. random variables. Under weak assumptions on \(V\), we prove exponential localization for \(H\) at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions on \(V\).

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80 Random linear operators
47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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