The Spectrum of Schr\"odinger Operators with Randomly Perturbed Ergodic Potentials
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Publication:6416118
DOI10.1007/S00039-023-00632-ZarXiv2211.02173MaRDI QIDQ6416118
Artur Avila, A. S. Gorodetskii, David Damanik
Publication date: 3 November 2022
Abstract: We consider Schr"odinger operators in $ell^2(mathbb{Z})$ whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a connected compact metric space and a continuous sampling function, we show that the almost sure spectrum arises in an explicitly described way from the unperturbed spectrum and the topological support of the single-site distribution. In particular, assuming that the latter is compact and contains at least two points, this explicit description of the almost sure spectrum shows that it will always be given by a finite union of non-degenerate compact intervals. The result can be viewed as a far reaching generalization of the well known formula for the spectrum of the classical Anderson model.
Special classes of linear operators (47Bxx) General theory of linear operators (47Axx) General mathematical topics and methods in quantum theory (81Qxx)
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