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Localization for Anderson models on metric and discrete tree graphs

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Abstract

We establish spectral and dynamical localization for several Anderson models on metric and discrete radial trees. The localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies. All results are proved under the minimal hypothesis on the type of disorder: the random variables generating the trees assume at least two distinct values. This level of generality, in particular, allows us to treat radial trees with disordered geometry as well as Schrödinger operators with Bernoulli-type singular potentials. Our methods are based on an interplay between graph-theoretical properties of radial trees and spectral analysis of the associated random differential and difference operators on the half-line.

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Notes

  1. I.e., \(|{{\,\mathrm{tr}\,}}M_j|<2\).

  2. Recall that L and \(\widetilde{L}\) are related via (3.5).

  3. In the sequel \(\zeta \) will be determined by the center of localization.

  4. N will depend on u through \(C_u\). In particular, if all generalized eigenfunctions are uniformly bounded, N is u-independent.

  5. \(\zeta \) from (3.23) is called the center of localization of f.

  6. I.e., \(|{{\,\mathrm{tr}\,}}(A)|>2\).

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Acknowledgements

We thank G. Berkolaiko, M. Lukic, and G. Stolz for helpful discussions, and P. Hislop for bringing our attention to this subject and for motivating discussions.

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Correspondence to Selim Sukhtaiev.

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Communicated by Loukas Grafakos.

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David Damanik was supported in part by NSF Grant DMS–1700131. Jake Fillman was supported in part by an AMS-Simons travel Grant, 2016–2018. Selim Sukhtaiev was supported in part by an AMS-Simons travel Grant, 2017–2019.

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Damanik, D., Fillman, J. & Sukhtaiev, S. Localization for Anderson models on metric and discrete tree graphs. Math. Ann. 376, 1337–1393 (2020). https://doi.org/10.1007/s00208-019-01912-6

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