Abstract
In this paper we study discrete quasi-periodic operators with certain long-range hopping and monotonic meromorphic potentials. The hopping amplitude decays with the interparticle distance \(|\varvec{n}|\) as \(e^{-r\log ^t(1+|\varvec{n}|)}\) (\(t>1, r>0, \varvec{n}\in {\mathbb {Z}}^{v}\)). By employing the KAM method, we prove such operators have pure point spectrum with eigenfunctions \(\left\{ \varphi _{\varvec{i}} \right\} _{\varvec{i}\in {\mathbb {Z}}^v}\) obeying \(\left| \varphi _{\varvec{i}} (\varvec{n})\right| \le 2e^{-\frac{r}{2}\log ^t(1+|\varvec{n}-\varvec{i}|)}\) for all \(\varvec{i}\in {\mathbb {Z}}^{v}, \varvec{n}\in {\mathbb {Z}}^{v}\).
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References
Bellissard, J., Lima, R., Scoppola, E.: Localization in \(v\)-dimensional incommensurate structures. Commun. Math. Phys. 88(4), 465–477 (1983)
Chulaevsky, V.A., Dinaburg, E.I.: Methods of KAM-theory for long-range quasi-periodic operators on \({ Z}^\nu \). Pure point spectrum. Commun. Math. Phys. 153(3), 559–577 (1993)
Craig, W.: Pure point spectrum for discrete almost periodic Schrödinger operators. Commun. Math. Phys. 88(1), 113–131 (1983)
Dinaburg, E.I., Sinai, J.G.: The one-dimensional Schrödinger equation with quasiperiodic potential. Funkc. Anal. i Priložen. 9(4), 8–21 (1975)
Eliasson, L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179(2), 153–196 (1997)
Grempel, D., Fishman, S., Prange, R.: Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49(11), 833 (1982)
Pöschel, J.: Examples of discrete Schrödinger operators with pure point spectrum. Commun. Math. Phys. 88(4), 447–463 (1983)
Pöschel, J.: Small divisors with spatial structure in infinite-dimensional Hamiltonian systems. Commun. Math. Phys. 127(2), 351–393 (1990)
Sarnak, P.: Spectral behavior of quasiperiodic potentials. Commun. Math. Phys. 84(3), 377–401 (1982)
Shi, Y.: Power-Law Localization for Almost Periodic Operators: A Nash-Moser Iteration Approach (2022). arXiv:2201.07405
Sinai, Y.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys. 46(5–6), 861–909 (1987)
Vittot, M.: Théorie Classique des Perturbations et Grand Nombre de Degres de Liberté. Ph.D. Thesis (1985)
Acknowledgements
Our research is supported by the National Key R &D Program of China (No. 2021YFA1001600) and NSFC (No. 11901010). We are very grateful to the referees for helpful suggestions.
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Shi, Y., Wen, L. Localization for a class of discrete long-range quasi-periodic operators. Lett Math Phys 112, 86 (2022). https://doi.org/10.1007/s11005-022-01581-8
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DOI: https://doi.org/10.1007/s11005-022-01581-8