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 inl 2ℤ 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.
Similar content being viewed by others
References
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)
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 1982
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 1985
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. 1982
Bellissard, J., Testard, D.: Almost periodic hamiltonians: An algebraic approach, Preprint CPT-81/P. 1311, Université de Provence, Marseille
Bratelli, G.:C *-algebras and their automorphism groups. London, New York, San Francisco: Academic Press 1979
Brenken, B.A.: Representations and automorphisms of the irrational rotation algebra. Pacific J. Math.111, 257–282 (1984)
Burnat, M.: Die Spektraldarstellung einiger Differentialoperatoren mit periodischen Koeffizienten im Raume der fastperiodischen Funktionen. Studia Math.25, 33–64 (1964)
Burnat, M.: The spectral properties of the Schrödinger operator in nonseparable Hilbert spaces. Banach Center Publ.8, 49–56 (1982)
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)
Chojnacki, W.: Some non-trivial cocycles. J. Funct. Anal.77, 9–31 (1988)
Chojnacki, W.: Eigenvalues of almost periodic Schrödinger operator inL 2(bℝ) are at most double. Lett. Math. Phys.22, 7–10 (1991)
Delyon, F.: Absence of localisation in the almost Mathieu equation. J. Phys. A20, L21-L23 (1987)
Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l'équation de Harper. Mém. Soc. Math. France (N.S.)34, 1–113 (1988)
Helffer, B., Sjöstrand, J.: Semi-classical analysis for Harper's equation. III. Mém. Soc. Math. France (N.S.)39, 1–124 (1989)
Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l'équation de Harper. II. Mém. Soc. Math. France (N.S.)40, 1–139 (1990)
Helffer, B., Kerdelhué, P., Sjöstrand, J.: Le papillon de Hofstadter revisité. Mém. Soc. Math. France (N.S.)43, 1–87 (1990)
Herczyński, J.: Schrödinger operators with almost periodic potentials in nonseparable Hilbert spaces. Banach Center Publ.19, 121–142 (1987)
Kaminker, J., Xia, J.: The spectrum of operators elliptic along the orbits of ℝn actions. Commun. Math. Phys.110, 427–438 (1987)
Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math.334, 141–156 (1982)
Krupa, A., Zawisza, B.: Ultrapowers of unbounded selfadjoint operators. Studia Math.85, 107–123 (1987)
Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux differences finies aléatoires. Commun. Math. Phys.78, 201–246 (1980)
Morris, S.: Pontryagin duality and the structure of locally compact abelian groups. Cambridge: Cambridge University Press 1977
Rudin, W.: Fourier analysis on groups. New York: Interscience 1962
Semadeni, Z.: Banach spaces of continuous spaces, vol. 1. Warszawa: PWN 1971
Tomiyama, J.: Invitation toC *-algebras and topological dynamics. Singapore, New Jersey, Hong Kong: World Scientific 1987
Wilkinson, M.: Critical properties of electron eigenstates in incommensurate systems. Proc. R. Soc. London Ser. A391, 305–350 (1984)
Yosida, K.: Functional analysis. Berlin, Heidelberg, New York: Springer 1980
Żelazko, W.: Banach algebras. Amsterdam, London, New York: Elsevier, Warszawa: PWN 1973
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Dedicated to Professor Marek Burnat
Rights and permissions
About this article
Cite this article
Chojnacki, W. A generalized spectral duality theorem. Commun.Math. Phys. 143, 527–544 (1992). https://doi.org/10.1007/BF02099263
Issue Date:
DOI: https://doi.org/10.1007/BF02099263