Abstract
We consider Schrödinger 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.
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Notes
Here, \(\tilde{g}{(x)}\) is defined for the cocycle (T, A) in a way analogous to our definition of \(\tilde{g}_E{(x)}\) associated with the cocycle \((T,A_E)\) above.
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Acknowledgements
We would like to thank Jake Fillman and the anonymous referee for several helpful comments that have resulted in an improvement of the presentation.
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D. D. was supported in part by NSF Grants DMS–1700131 and DMS–2054752 A. G. was supported in part by NSF Grant DMS–1855541.
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Avila, A., Damanik, D. & Gorodetski, A. The Spectrum of Schrödinger Operators with Randomly Perturbed Ergodic Potentials. Geom. Funct. Anal. 33, 364–375 (2023). https://doi.org/10.1007/s00039-023-00632-z
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DOI: https://doi.org/10.1007/s00039-023-00632-z