Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T03:24:51.596Z Has data issue: false hasContentIssue false
  • English
  • Français

Group C*-Algebras and the Spectrum of a Periodic Schrödinger Operator on a Manifold

Published online by Cambridge University Press:  20 November 2018

Toshikazu Sunada
Toshikazu Sunada*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The spectrum of the Laplacian or more generally of a Schrödinger operator on an open manifold may have possibly a complicated aspect. For example, a Cantor set in the real axis may appear as the spectrum even for an innocent looking potential on a standard Riemannian manifold (see J. Moser [10]). The fundamental result of the spectral theory of periodic Schrödinger operators, however, says that the picture of the spectrum of a Schrödinger operator on ℝn with a periodic potential is simple; indeed the spectrum consists of a series of closed intervals of the real axis without accumulation, separated in general by gaps outside the spectrum (see M. Reed and B. Simon [13] or M. M. Skriganov [15] for instance).

Keywords

58G2547C15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 1 , 01 December 1991 , pp. 180 - 193
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Arveson, W., An invitation to C*-algebra. Springer-Verlag, New York, 1976.Google Scholar
2. Atiyah, M.F., Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32/ 33(1976), 43–72.Google Scholar
3. Bellissard, J., Lima, R. and Testard, D., Almost periodic Schrödinger operators. Mathematics + Physics, Lectures on Recent Results, 1, World Scientific, Singapore, 1985. 1–64.Google Scholar
4. Diximier, J., C* -algebras. North-Holland, Amsterdam, 1977.Google Scholar
5. Donnelly, H., On the L2-Betti numbers for abelian groups, Canad. Math. Bull. 24(1981), 91–95.Google Scholar
6. P, M.S.. Eastham, The spectral theory of periodic differential equations. Scottish Academic Press, Edinburgh-London, 1973.Google Scholar
7. Effros, E.G., Why the circle is connected: an introduction to quantized topology, Math. Intell. 11(1989), 27–34.Google Scholar
8. Kobayashi, T., Ono, K. and Sunada, T., Periodic Schrôdinger operators on a manifold, Forum Math. 1(1989), 69–79.Google Scholar
9. Lance, C., K-theoryfor certain group C*-algebras, Acta Math. 151(1983), 209–230.Google Scholar
10. Moser, J., An example of a Schrödinger equation with an almost periodic potential and nowhere dense spectrum, Comment. Math. Helv. 56(1981), 198–224.Google Scholar
11. Pimsner, M. and Voiculescu, D., K-groups ofreduced crossed products by the free groups, J. Operator Theory 8(1982), 131–156.Google Scholar
12. Pimsner, M., KK-groups of crossed products by groups acting on trees, Invent. Math. 86(1986), 603–634.Google Scholar
13. Reed, M. and Simon, B., Methods of modern mathematical physics, IV. Analysis of operators. Academic Press, London, 1978.Google Scholar
14. Serre, J.P., Trees. Springer-Verlag, 1980.Google Scholar
15. Skriganov, M.M., Geometric and arithmetic methods in the spectral theory of multidimentional periodic operators, Proc. of the Steklov Inst, of Math. 171(1985), 1–117.Google Scholar
16. Sunada, T., Trace formula for Hill's operators, Duke Math. J. 47(1980), 529–546.Google Scholar
17. Sunada, T. ,Trace formulas, Wiener integrals and asymptotics. Proc. Spectra of Riemannian Manifolds, Kaigai Publ., Tokyo, 1983. 103–113.Google Scholar
18. Sunada, T. , Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28(1989), 125–132.Google Scholar
19. Sunada, T., Fundamental groups and Laplacians. Proceedings of the Taniguchi Symposium on Geometry and Analysis on Manifolds, 1987. Springer Lecture Notes in Mathematics 1339(1988), 248–277.Google Scholar
20. Taylor, M.E., Pseudodifferential operators. Princeton University Press, Princeton, New Jersey, 1981..Google Scholar
21. Feller, W., An Introduction to the Theory of Probability, Vol. II. Wiley, New York, 1971.Google Scholar