zbMATH — the first resource for mathematics

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.

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
Full Text: DOI