An Exposition of the Connection between Limit-Periodic Potentials and Profinite Groups

Z. Gan

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 158-174
  • ISSN: 0973-5348

Abstract

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We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some recent results on Schrödinger operators obtained in this context.

How to cite

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Gan, Z.. "An Exposition of the Connection between Limit-Periodic Potentials and Profinite Groups." Mathematical Modelling of Natural Phenomena 5.4 (2010): 158-174. <http://eudml.org/doc/197717>.

@article{Gan2010,
abstract = {We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some recent results on Schrödinger operators obtained in this context.},
author = {Gan, Z.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {profinite group; limit-periodic potential; Schrödinger operator},
language = {eng},
month = {5},
number = {4},
pages = {158-174},
publisher = {EDP Sciences},
title = {An Exposition of the Connection between Limit-Periodic Potentials and Profinite Groups},
url = {http://eudml.org/doc/197717},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Gan, Z.
TI - An Exposition of the Connection between Limit-Periodic Potentials and Profinite Groups
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 158
EP - 174
AB - We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some recent results on Schrödinger operators obtained in this context.
LA - eng
KW - profinite group; limit-periodic potential; Schrödinger operator
UR - http://eudml.org/doc/197717
ER -

References

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  1. A. Avila. On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators. Commun. Math. Phys., 288 (2009), 907–918. Zbl1188.47023
  2. J. Avron, B. Simon. Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys., 82 (1981), 101–120. Zbl0484.35069
  3. W. Craig, B. Simon. Subharmonicity of the Lyaponov index. Duke Math. J., 50:2 (1983), 551–560.  Zbl0518.35027
  4. D. Damanik, Z. Gan. Spectral properties of limit-periodic Schrödinger operators. To appear in to appear in Discrete Contin. Dyn. Syst. Ser. S.  
  5. D. Damanik, Z. Gan. Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents. J. Funct. Anal.258:12 (2010), 4010–4025 Zbl1191.47058
  6. D. Damanik, Z. Gan. Limit-periodic Schrödinger operators with uniformly localized eigenfunctions. Preprint, (arXiv:1003.1695).  Zbl1314.47053
  7. D. Damanik, A. Gorodetski. The spectrum of the weakly coupled Fibonacci Hamiltonian. Electron. Res. Announc. Math. Sci., 16 (2009), 23–29. Zbl1169.82009
  8. A. Figotin, L. Pastur. An exactly solvable model of a multidimensional incommensurate structure. Commun. Math. Phys., 95 (1984), 401–425. Zbl0582.35101
  9. S. Fishman, D. Grempel, R. Prange. Localization in a d-dimensional incommensurate structure. Phys. Rev. B, 29 (1984), 4272–4276. 
  10. Z. Gan, H. Krüger. Optimality of log Hölder continuity of the integrated density of states. To appear in Math. Nachr.  Zbl1238.47021
  11. S. Jitomirskaya. Continuous spectrum and uniform localization for ergodic Schrödinger operators. J. Funct. Anal., 145 (1997), 312–322. Zbl0883.47044
  12. S. Jitomirskaya, B. Simon. Operators with singular continuous spectrum, III. Alomost periodic Schrödinger operators. Commun. Math. Phys., 165 (1994), 201–205. Zbl0830.34074
  13. J. Pöschel. Examples of discrete Schrödinger operators with pure point spectrum. Commun. Math. Phys., 88 (1983), 447–463. Zbl0532.35070
  14. R. Prange, D. Grempel, S. Fishman. A solvable model of quantum motion in an incommensurate potential. Phys. Rev. B, 29 (1984), 6500–6512. 
  15. L. Ribes, P. Zalesskii. Profinite Groups. Springer-Verlag, Berlin, 2000.  Zbl0949.20017
  16. B. Simon. Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging, 1 (2007), No. 4, 713–772. Zbl1149.31004
  17. J. Wilson. Profinite Groups. Oxford University Press, New York, 1998.  Zbl0909.20001

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