Hostname: page-component-7dd5485656-jtdwj Total loading time: 0 Render date: 2025-10-29T07:27:52.292Z Has data issue: false hasContentIssue false
  • English
  • Français

An Exposition of the Connection between Limit-PeriodicPotentials and Profinite Groups

Published online by Cambridge University Press:  12 May 2010

Z. Gan
Z. Gan*
Affiliation:
Department of Mathematics, Rice University, 77005 Houston, USA
*
* E-mail: zheng.gan@rice.edu
Get access

Abstract

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

Keywords

profinite grouplimit-periodic potentialSchrödinger operator

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Dedicated to the memory of M. S. Birman

References

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