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Internal Lifshits tails for random magnetic Schrödinger operators. (English) Zbl 1121.82022

Loosely speaking, Lifshits tails refer to the feature (discovered by this author) that, at the bottom of the spectrum of a random Schrödinger operator, the density of states decays exponentially fast. This phenomenon is particularly relevant to the quantum Hall effect. Here one studies the Lifshits tails for a random magnetic Schrödinger operator in which the potential is the sum of two terms. A potential \(V\) which is periodic, and a potential \(w_{ii}(x-\gamma)\) where the \(w_{ii}\)’s are i.i.d. bounded random variables. The paper gives exact conditions under which there exists a Lifshits tail.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80 Random linear operators
47N55 Applications of operator theory in statistical physics (MSC2000)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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