Skip to main content
SpringerLink
Log in

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let denote the field of algebraic numbers in A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M d ((G, σ)), regarded as an operator on l 2(G)d, the eigenvalues of A are algebraic numbers, where σ ∈ Z 2(G, ) is an algebraic multiplier, and denotes the unitary elements of . Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σ n=1 for some positive integer n) this property holds for a larger class of groups containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahlfors, L., Sario, L.: Riemann surfaces, Princeton Mathematical Series, No. 26 Princeton, N.J.: Princeton University Press, 1960

  2. Atiyah, M.: Elliptic operators, discrete groups and Von Neumann algebras. Astérisque 32–33, 43–72 (1976)

  3. Bellissard, J.: Gap Labeling Theorems for Schrödinger's Operators. In: From number theory to physics (Les Houches, 1989), Berlin: Springer, 1992, pp. 538–630

  4. Boca, F.: Rotation C*-algebras and almost Mathieu operators. Theta Series in Advanced Mathematics, 1. Bucharest: The Theta Foundation, 2001

  5. Brown, K.: Cohomology of groups. Graduate Texts in Mathematics, 87. New York-Berlin: Springer-Verlag, 1982

  6. Carey, A., Hannabuss, K., Mathai, V., McCann, P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190 (3), 629–673 (1998)

    Article  MathSciNet  Google Scholar 

  7. Carey, A., Hannabuss, K., Mathai, V.: Quantum Hall Effect on the hyperbolic plane in the presence of disorder. Lett. Math. Phys. 47, 215–236 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dodziuk, J.: Rham-Hodge theory for L2-cohomology of infinite coverings. Topology 16, 157–165 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dodziuk, J., Linnell, P., Mathai, V., Schick, T., Yates, S.: Approximating L 2-invariants, and the Atiyah conjecture. Commun. in Pure and Appl. Math. 56(7), 839–873 (2003)

    Article  MathSciNet  Google Scholar 

  10. Dodziuk, J., Mathai, V.: Approximating L 2 invariants of amenable covering spaces: a combinatorial approach. J. Funct. Anal. 154(2), 359–378 (1998)

    Article  MathSciNet  Google Scholar 

  11. Elek, G.: On the analytic zero divisor conjecture of Linnell, Bull. London Math. Soc. 35(2), 236–238 (2003)

    Article  MathSciNet  Google Scholar 

  12. Grigorchuk, R.: On the Milnor problem of group growth. (Russian) Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983)

    MathSciNet  Google Scholar 

  13. Grigorchuk, R., Zuk, A.: The lamplighter group as a group generated by a 2-state automaton and its spectrum. Geom. Dedicata 87, 209–244 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Linnell, P.A.: Division rings and group von Neumann algebras. Forum Math. 5, 561–576 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lück, W.: Approximating L 2 invariants by their finite dimensional analogues. Geom. Func. Anal. 4, 455–481 (1994)

    Article  MATH  Google Scholar 

  16. Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, I: noncommutative Bloch theory. Commun. Contemp. Math. 1(4), 553–587 (1999)

    Article  MathSciNet  Google Scholar 

  17. Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, II: fractional quantum numbers. Commun. Math. Phys. 217(1), 55–87 (2001)

    Article  MathSciNet  Google Scholar 

  18. Mathai, V., Yates, S.: Approximating spectral invariants of Harper operators on graphs. J. Funct. Anal. 188(1), 111–136 (2002)

    Article  MathSciNet  Google Scholar 

  19. Mathai, V., Schick, T., Yates, S.: Approximating spectral invariants of Harper operators on graphs II. Proc. Amer. Math. Soc. 131(6), 1917–1923 (2003)

    Article  MathSciNet  Google Scholar 

  20. Thomas Schick,: L 2-determinant class and approximation of L 2-Betti numbers. Trans. Amer. Math. Soc. 353(8), 3247–3265 (2001)

    Article  MathSciNet  Google Scholar 

  21. Shubin, M.: Discrete Magnetic Schrödinger operators. Commun. Math. Phys. 164(2), 259–275 (1994)

    Article  MathSciNet  Google Scholar 

  22. Shubin, M.: von Neumann algebras and L 2 techniques in geometry and topology. Book in preparation

  23. Sunada, T.: A discrete analogue of periodic magnetic Schrödinger operators. Contemp. Math. 173, 283–299 (1994)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ph.D. Program in Mathematics, Graduate Center of CUNY, New York, NY, 10016, USA

    Józef Dodziuk

  2. Department of Mathematics, University of Adelaide, Adelaide, 5005, Australia

    Varghese Mathai & Stuart Yates

  3. Max Planck Institut für Mathematik, Bonn, Germany

    Stuart Yates

Authors
  1. Józef Dodziuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Varghese Mathai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stuart Yates
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Józef Dodziuk.

Additional information

Communicated by P. Sarnak

The second and third authors acknowledge support from the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dodziuk, J., Mathai, V. & Yates, S. Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs. Commun. Math. Phys. 262, 269–297 (2006). https://doi.org/10.1007/s00220-005-1489-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1489-0

Keywords

Search

Navigation