Advertisement

Communications in Mathematical Physics

, Volume 143, Issue 3, pp 527–544 | Cite as

A generalized spectral duality theorem

  • Wojciech Chojnacki
Article

Abstract

We establish a version of the spectral duality theorem relating the point spectrum of a family of*-representations of a certain covariance algebra to the continuous spectrum of an associated family of*-representations. Using that version, we prove that almost all the images of any element of a certain space of fixed points of some*-automorphism of an irrational rotation algebra via standard*-representations of the algebra inl2ℤ do not have pure point spectrum over any non-empty open subset of the common spectrum of those images. As another application of the spectral duality theorem, we prove that if almost all the Bloch operators associated with a real almost periodic function on ℝ have pure point spectrum over a Borel subset of ℝ, then almost all the Schrödinger operators with potentials belonging to the compact hull of the translates of this function have, over the same set, purely continuous spectrum.

Keywords

Neural Network Covariance Complex System Hull Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avron, J., v. Mouche, P.H.M., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys.132, 103–118 (1990)Google Scholar
  2. 2.
    Bellissard, J.: Schrödinger operators with almost periodic potential: an overview. In: Schrader, R., Seiler, R., Uhlenbrok, D.A. (eds.) Mathematical problems in theoretical physics (Berlin, 1981), pp. 356–363. Lecture Notes in Phys. vol. 153. Berlin, New York: Springer 1982Google Scholar
  3. 3.
    Bellissard, J., Lima, D., Testard, D.: Almost periodic Schrödinger operators. In: Streit, L. (ed.) Mathematics and Physics, Lectures on Recent Results, vol. 1, pp. 1–64. Singapore, Philadelphia: World Scientific 1985Google Scholar
  4. 4.
    Bellissard, J., Testard, D.: Quasi-periodic Hamiltonians. A mathematical approach. In: Kadison, R.V. (ed.) Operator algebras and applications, Part 2 (Kingston, Ontario, 1980), pp. 297–299. Proc. Sympos. Pure Math.38, Providence, R.I. Am. Math. Soc. 1982Google Scholar
  5. 5.
    Bellissard, J., Testard, D.: Almost periodic hamiltonians: An algebraic approach, Preprint CPT-81/P. 1311, Université de Provence, MarseilleGoogle Scholar
  6. 6.
    Bratelli, G.:C *-algebras and their automorphism groups. London, New York, San Francisco: Academic Press 1979Google Scholar
  7. 7.
    Brenken, B.A.: Representations and automorphisms of the irrational rotation algebra. Pacific J. Math.111, 257–282 (1984)Google Scholar
  8. 8.
    Burnat, M.: Die Spektraldarstellung einiger Differentialoperatoren mit periodischen Koeffizienten im Raume der fastperiodischen Funktionen. Studia Math.25, 33–64 (1964)Google Scholar
  9. 9.
    Burnat, M.: The spectral properties of the Schrödinger operator in nonseparable Hilbert spaces. Banach Center Publ.8, 49–56 (1982)Google Scholar
  10. 10.
    Chojnacki, W.: Spectral analysis of Schrödinger operators in non-separable Hilbert spaces. Functional integration with emphasis on the Feynman integral (Sherbrooke, PQ, 1986). Rend. Circ. Mat. Palermo (2) [Suppl.]17, 135–151 (1987)Google Scholar
  11. 11.
    Chojnacki, W.: Some non-trivial cocycles. J. Funct. Anal.77, 9–31 (1988)CrossRefGoogle Scholar
  12. 12.
    Chojnacki, W.: Eigenvalues of almost periodic Schrödinger operator inL 2(bℝ) are at most double. Lett. Math. Phys.22, 7–10 (1991)CrossRefGoogle Scholar
  13. 13.
    Delyon, F.: Absence of localisation in the almost Mathieu equation. J. Phys. A20, L21-L23 (1987)CrossRefGoogle Scholar
  14. 14.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l'équation de Harper. Mém. Soc. Math. France (N.S.)34, 1–113 (1988)Google Scholar
  15. 15.
    Helffer, B., Sjöstrand, J.: Semi-classical analysis for Harper's equation. III. Mém. Soc. Math. France (N.S.)39, 1–124 (1989)Google Scholar
  16. 16.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l'équation de Harper. II. Mém. Soc. Math. France (N.S.)40, 1–139 (1990)Google Scholar
  17. 17.
    Helffer, B., Kerdelhué, P., Sjöstrand, J.: Le papillon de Hofstadter revisité. Mém. Soc. Math. France (N.S.)43, 1–87 (1990)Google Scholar
  18. 18.
    Herczyński, J.: Schrödinger operators with almost periodic potentials in nonseparable Hilbert spaces. Banach Center Publ.19, 121–142 (1987)Google Scholar
  19. 19.
    Kaminker, J., Xia, J.: The spectrum of operators elliptic along the orbits of ℝn actions. Commun. Math. Phys.110, 427–438 (1987)CrossRefGoogle Scholar
  20. 20.
    Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math.334, 141–156 (1982)Google Scholar
  21. 21.
    Krupa, A., Zawisza, B.: Ultrapowers of unbounded selfadjoint operators. Studia Math.85, 107–123 (1987)Google Scholar
  22. 22.
    Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux differences finies aléatoires. Commun. Math. Phys.78, 201–246 (1980)CrossRefGoogle Scholar
  23. 23.
    Morris, S.: Pontryagin duality and the structure of locally compact abelian groups. Cambridge: Cambridge University Press 1977Google Scholar
  24. 24.
    Rudin, W.: Fourier analysis on groups. New York: Interscience 1962Google Scholar
  25. 25.
    Semadeni, Z.: Banach spaces of continuous spaces, vol. 1. Warszawa: PWN 1971Google Scholar
  26. 26.
    Tomiyama, J.: Invitation toC *-algebras and topological dynamics. Singapore, New Jersey, Hong Kong: World Scientific 1987Google Scholar
  27. 27.
    Wilkinson, M.: Critical properties of electron eigenstates in incommensurate systems. Proc. R. Soc. London Ser. A391, 305–350 (1984)Google Scholar
  28. 28.
    Yosida, K.: Functional analysis. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  29. 29.
    Żelazko, W.: Banach algebras. Amsterdam, London, New York: Elsevier, Warszawa: PWN 1973Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Wojciech Chojnacki
    • 1
    • 2
  1. 1.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarszawaPoland
  2. 2.School of Information Science and TechnologyFlinders University of South AustraliaAdelaideAustralia

Personalised recommendations