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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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