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Spectral structure of Anderson type Hamiltonians. (English) Zbl 0962.60056
The authors consider random self-adjoint operators of the form \(H_\omega =H_0+\sum \lambda_\omega (n)\langle \delta_n,\cdot \rangle \delta_n \) where \(H_0\) is a bounded non-random operator, \(\{ \delta_n\}\) is a family of orthonormal vectors, and \(\lambda_\omega (n)\) are independent random variables with absolutely continuous distributions. Conditions regarding the boundedness of \(H_0\) and the independence of \(\lambda_\omega (n)\) can be relaxed.
Let \(\mathcal H_{\omega ,n}\) be a closure of the set of vectors \(f(H_\omega)\delta_n\) where \(f\) runs the set of all complex-valued continuous functions vanishing at infinity. It is shown that if almost surely \(\mathcal H_{\omega ,n}\) and \(\mathcal H_{\omega ,m}\) are not orthogonal, then the restrictions of \(H_\omega\) to \(\mathcal H_{\omega ,n}\) and \(\mathcal H_{\omega ,m}\) are unitarily equivalent. This general result is used for finding intervals of purely absolutely continuous spectra for some discrete Schrödinger operators with random potentials.

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
47B80 Random linear operators
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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