Exact solvable family of discrete Schrödinger operators with long-range hoppings

Authors

  • César R. de Oliveira
  • Mariane Pigossi

DOI:

https://doi.org/10.7146/math.scand.a-147791

Abstract

It is shown that one-dimensional discrete Schrödinger operators, with a uniform electric field and long-range hoppings, form an isospectral family with discrete and simple spectrum (and isospectral operators with eigenfunctions having different decay properties). Explicit relations between hopping decay rates and eigenvectors decay rates are obtained. Small perturbations of exponentially decay hopping operators preserve dynamical localization.

References

de Oliveira, C. R., Intermediate spectral theory and quantum dynamics, Progress in Mathematical Physics, 54. Birkhäuser, Basel 2009. https://doi.org/10.1007/978-3-7643-8795-2

de Oliveira, C. R., and Pigossi, M., Point spectrum and SULE for time-periodic perturbations of discrete 1D Schrödinger operators with electric fields, J. Stat. Phys. 173 (2018), no. 1, 140–162. https://doi.org/10.1007/s10955-018-2126-6

de Oliveira, C. R., and Pigossi, M., Proof of dynamical localization for perturbations of discrete 1D Schrödinger operators with uniform electric fields, Math. Z. 291 (2019), no. 3–4, 1525–1541. https://doi.org/10.1007/s00209-018-2103-4

de Oliveira, C. R., and Pigossi, M., Persistence of point spectrum for perturbations of one-dimensional operators with discrete spectra. Spectral theory and mathematical physics, 125–151, Lat. Amer. Math. Ser., Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-55556-6_7

Dinaburg, E. I., Stark effect for a difference Schrödinger operator, Teoret. Mat. Fiz. 78 (1989), no. 1, 70–80; translation in Theoret. and Math. Phys. 78 (1989), no. 1, 50–57. https://doi.org/10.1007/BF01016916

Figotin, A. L., and Pastur, L. A., An exactly solvable model of a multidimensional incomensurate structure, Comm. Math. Phys. 95 (1984), no. 4, 401–425. http://projecteuclid.org/euclid.cmp/1103941644

Jitomirskaya, S., and Liu, W., Upper bounds on transport exponents for long-range operators, J. Math. Phys. 62 (2021), no. 7, Paper No. 073506, 9 pp. https://doi.org/10.1063/5.0054834

Liu, W., Power law logarithmic bounds of moments for long range operators in arbitrary dimension, J. Math. Phys. 64 (2023), no. 3, Paper No. 033508, 11 pp. https://doi.org/10.1063/5.0138325

Shamis, M., and Sodin, S., Upper bounds on quantum dynamics in arbitrary dimension, J. Funct. Anal. 285 (2023), no. 7, Paper No. 110034, 20 pp. https://doi.org/10.1016/j.jfa.2023.110034

Shi, Y., Localization for almost-periodic operators with power-law long-range hopping: A Nash-Moser iteration type reducibility approach, Comm. Math. Phys. 402 (2023), no. 2, 1765–1806. https://doi.org/10.1007/s00220-023-04756-z

Shi, Y., and Wen, L., Localization for a class of discrete long-range quasi-periodic operators, Lett. Math. Phys. 112 (2022), no. 5, Paper No. 86, 18 pp. https://doi.org/10.1007/s11005-022-01581-8

Shi, Y., and Wen, L., Diagonalization in a quantum kicked rotor model with non-analytic potential, J. Differential Equations 355 (2023), 334–368. https://doi.org/10.1016/j.jde.2023.01.033

Published

2024-11-04

How to Cite

de Oliveira, C. R., & Pigossi, M. (2024). Exact solvable family of discrete Schrödinger operators with long-range hoppings. MATHEMATICA SCANDINAVICA, 130(3). https://doi.org/10.7146/math.scand.a-147791

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Articles