Advertisement

Annales Henri Poincaré

, Volume 15, Issue 9, pp 1697–1732 | Cite as

Pseudo-Differential Calculus on Homogeneous Trees

  • Etienne Le Masson
Article

Abstract

To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L 2 and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.

Keywords

Regular Graph Symmetry Condition Rapid Decay Homogeneous Tree Hyperbolic Surface 
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.
    Anantharaman, N., Le Masson, E.: Quantum Ergodicity on Large Regular Graphs. arXiv:1304.4343 (2013) (preprint)Google Scholar
  2. 2.
    Berkolaiko G., Keating J.P., Smilansky U.: Quantum ergodicity for graphs related to interval maps. Comm. Math. Phys. 273((1), 137–159 (2007)CrossRefzbMATHMathSciNetADSGoogle Scholar
  3. 3.
    Berkolaiko G., Keating J.P., Winn B.: No quantum ergodicity for star graphs. Comm. Math. Phys. 250(2), 259–285 (2004)CrossRefzbMATHMathSciNetADSGoogle Scholar
  4. 4.
    Brooks S., Lindenstrauss E.: Non-localization of eigenfunctions on large regular graphs. Israel J. Math. 193(1), 1–14 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Colin de Verdière Y.: Ergodicité et fonctions propres du Laplacien. Comm. Math. Phys. 102(3), 497–502 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cowling M., Meda S., Setti A.G.: An overview of harmonic analysis on the group of isometries of a homogeneous tree. Expo. Math. 16(5), 385–423 (1998)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Cowling M., Setti A.G.: The range of the Helgason–Fourier transformation on homogeneous trees. Bull. Aust. Math. Soc. 59(2), 237–246 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Figà-Talamanca, A., Nebbia, C.: Harmonic analysis and representation theory for groups acting on homogeneous trees. In: London Mathematical Society Lecture Note Series, vol. 162. Cambridge University Press, Cambridge (1991)Google Scholar
  9. 9.
    Keating J.P., Marklof J., Winn B.: Value distribution of the eigenfunctions and spectral determinants of quantum star graphs. Comm. Math. Phys. 241(2–3), 421–452 (2003)zbMATHMathSciNetADSGoogle Scholar
  10. 10.
    Kottos T., Smilansky U.: Quantum chaos on graphs. Phys. Rev. Lett. 79, 4794–4797 (1997)CrossRefADSGoogle Scholar
  11. 11.
    Kottos T., Smilansky U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274(1), 76–124 (1999)CrossRefzbMATHMathSciNetADSGoogle Scholar
  12. 12.
    Smilansky U.: Quantum chaos on discrete graphs. J. Phys. A 40(27), F621–F630 (2007)CrossRefzbMATHMathSciNetADSGoogle Scholar
  13. 13.
    Smilansky, U.: Discrete graphs—a paradigm model for quantum chaos. In: Séminaire Poincaré XIV, pp. 89–114. (2010)Google Scholar
  14. 14.
    Šnirel′man A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6(180)), 181–182 (1974)Google Scholar
  15. 15.
    Zelditch S.: Pseudodifferential analysis on hyperbolic surfaces. J. Funct. Anal. 68(1), 72–105 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Université Paris-Sud 11Orsay CedexFrance

Personalised recommendations