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

Mathematical Modelling of Natural Phenomena (2010)

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

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topGan, 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

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

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