Skip to main content
Book cover

Spectral Analysis of Quantum Hamiltonians pp 67–105Cite as

Birkhäuser

The Topological Bloch-Floquet Transform and Some Applications

Part of the Operator Theory: Advances and Applications book series (OT,volume 224)

Abstract

We investigate the relation between the symmetries of a SchrÖdinger operator and the related topological quantum numbers.W e show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch- Floquet transform induces a direct integral decomposition of the algebra of observables.More relevantly, we prove that the generalized transform selects uniquely the set of “continuous sections” in the direct integral decomposition, thus yielding a Hilbert bundle.T he proof is constructive and provides an explicit description of the fibers.Th e emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere [DFP12].T wo running examples provide an Ariadne’s thread through the paper.F or the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.

Keywords

  • Topological quantum numbers
  • spectral decomposition
  • Bloch-Floquet transform
  • Hilbert bundle

Mathematics Subject Classification (2000). 81Q70, 46L08, 46L45, 57R22.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-0348-0414-1_5
  • Chapter length: 39 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-0348-0414-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Hardcover Book
USD   109.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.E. Avron. Colored Hofstadter butterflies. Multiscale Methods in Quantum Mechanics: Theory and Experiments. Birkhäuser, 2004.

    Google Scholar 

  2. F.P. Boca. Rotations C *-algebras and almost Mathieu operators. Theta Foundation, 2001.

    Google Scholar 

  3. O. Bratteli and D.W. Robinson. C * - and W *-Algebras, Symmetry Groups, Decomposition of States, volume I of Operator Algebras and Quantum Statistical Mechanics. Springer-Verlag, 1987.

    Google Scholar 

  4. H. Brézis. Analyse fonctionnelle, Théorie et Application. Masson, 1987.

    Google Scholar 

  5. J.V. Bellissard, H. Schulz-Baldes, and A. van Elst. The Non Commutative Geometry of the Quantum Hall Effect. J. Math. Phys., 35: 5373-5471, 1994.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. G. Choquet. Topology. Academic Press, 1966.

    Google Scholar 

  7. K.R. Davidson. C *-Algebras by Example. American Mathematical Society, 1996.

    Google Scholar 

  8. J. Dixmier and A. Douady. Champs continus d’espaces hilbertiens et de C * - algebres. Bull. Soc. math. France, 91: 227-284, 1963.

    MathSciNet  MATH  Google Scholar 

  9. G. De Nittis, F. Faure, and G. Panati. Colored Hofstadter butterflies and duality of vector bundles. Available as preprint at www.arxiv.org , 2012.

  10. J. Dixmier. von Neumann Algebras. North-Holland, 1981.

    Google Scholar 

  11. J. Dixmier. C*-Algebras. North-Holland, 1982.

    Google Scholar 

  12. G. De Nittis and G. Landi. Generalized tknn equations. Available as preprint at http://arxiv.org/abs/1104.1214 , 2011.

  13. G. De Nittis and G. Panati. Effective models for conductance in magnetic fields: derivation of Harper and Hofstadter models. Available as preprint at http://arxiv.org/abs/1007.4786 , 2010.

  14. J.M.G. Fell and R.S. Doran. Basic Representation Theory of Groups and Algebras, volume 1 of Representation of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles. Academic Press Inc., 1988.

    Google Scholar 

  15. M.J. Gruber. Non-commutative Bloch theory. J. Math. Phys., 42: 2438-2465, 2001.

    MATH  Google Scholar 

  16. J.M. Gracia-Bondia, J.C. Várilly, and H. Figueroa. Elements of Noncommutative Geometry. Birkhäuser, 2001.

    Google Scholar 

  17. L. Hörmander. Complex Analysis in Several Variables. North-Holland, 1990.

    Google Scholar 

  18. P. Kuchment. Floquet Theory for Partial Differential Equations. Operator Theory: Advances and Applications. Birkhäuser, 1993.

    Google Scholar 

  19. S. Lang. Differential Manifolds. Springer, 1985.

    Google Scholar 

  20. G. Landi. An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Springer, 1997.

    MATH  Google Scholar 

  21. K. Maurin. General Eigenfunction Expansions and Unitary Representations of Topological Groups. PWN, 1968.

    Google Scholar 

  22. B. Sz. Nagy and C. Foias. Harmonic Analysis of Operators on Hilbert Space. American Elsevier. North-Holland, 1970.

    Google Scholar 

  23. D. Osadchy and J.E. Avron. Hofstadter butterfly as quantum phase diagram. J. Math. Phys., 42: 5665-5671, 2001.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. G. Panati. Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré, 8: 995-1011, 2007.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. W. Rudin. Fourier Analysis on Groups. Number 12 in Interscience Tracts in Pure and Applied Mathematics. Interscience, 1962.

    Google Scholar 

  26. [Rud87] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1987.

    Google Scholar 

  27. Y.S. Samoilenko. Spectral Theory of Families of Self-Adjoint Operators. Mathematics and Its Applications. Kluwer, 1991.

    Google Scholar 

  28. F. Treves. Topological vector spaces, distributions and kernels. Academic Press, 1967.

    Google Scholar 

  29. K. von Klitzing, G. Dorda, and M. Pepper. New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance. Phys. Rev. Lett., 45: 494-497, 1980.

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Analyse, Géométrie, et Modélisation, Université de Cergy-Pontoise, Cergy-Pontoise, France

    G. De Nittis

  2. Dipartimento di Matematica, Università di Roma “La Sapienza”, Roma, Italy

    G. Panati

Authors
  1. G. De Nittis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Panati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. De Nittis .

Editor information

Editors and Affiliations

  1. , Departamento de Física, P. Universidad Católica de Chile, Santiago, 22, Chile

    Rafael Benguria

  2. , Departamento de Matemáticas, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Eduardo Friedman

  3. , Departamento de Matemática, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Marius Mantoiu

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer Basel

About this paper

Cite this paper

De Nittis, G., Panati, G. (2012). The Topological Bloch-Floquet Transform and Some Applications. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_5

Download citation