On quantum unique ergodicity for locally symmetric spaces

Geometric and Functional Analysis

Volumn 17, Issue 3, 2007, Pages 960-998

  Silberman, Lior ^a,b   Venkatesh, Akshay ^c,d,e  

^a PRINCETON UNIVERSITY   (United States)

^b HARVARD UNIVERSITY   (United States)

^c MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^d INSTITUTE FOR ADVANCED STUDY   (United States)

^e NEW YORK UNIVERSITY   (United States)

Author keywords

Automorphic forms; Invariant measures; Lie groups quantum chaos; Locally symmetric spaces; Microlocal lift; Quantum unique ergodicity

Indexed keywords

EID: 35148900927     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-007-0611-1     Document Type: Article

References (28)

1. J. BOURGAIN, E. LINDENSTRAUSS, Entropy of quantum limits, Comm. Math. Phys. 233:1 (2003), 153-171.
2. Y. COLIN DE VERDIÈRE, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102:3 (1985), 497-502.
3. J. DIXMIER, Enveloping Algebras, Graduate Studies in Mathematics, 11, American Mathematical Society, Providence, RI, 1996 (Revised reprint of the 1977 translation).
4. J.J. DUISTERMAAT, J.A.C. KOLK, V.S. VARADARAJAN, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52:1 (1979), 27-93.
5. J.J. DUISTERMAAT, J.A.C. KOLK, V.S. VARADARAJAN, Erratum: "Spectra of compact locally symmetric manifolds of negative curvature", Invent. Math. 54:1 (1979), 101.
6. M. EINSIEDLER AND A. KATOK, Invariant measures on G/Γ for split simple Lie groups G, Comm. Pure Appl. Math. 56:8 (2003), 1184-1221.
7. R.P. FEYNMAN, Statistical Mechanics: A Set of Lectures, Frontiers in Physics, W. A. Benjamin, Inc., Reading, Massachusetts, 1972.
8. R. GANGOLLI, V.S. VARADARAJAN, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 101, Springer-Verlag, Berlin (1988).
9. S. HELGASON, Groups and Geometrie Analysis, Mathematical Surveys and Monographs 83, American Mathematical Society, Providence, RI, 2000.
10. E. HOPF, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sachs. Akad. Wiss. Leipzig 91 (1939), 261-304.
11. L. HÖRMANDER, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218.
12. L. HÖRMANDER, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 274, Springer-Verlag, Berlin (1994).
13. ∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141:2 (1995), 301-320.
14. A.W. KNAPP, Representation Theory of Semisimple Groups, Princeton Mathematical Series 36, Princeton University Press, Princeton, NJ, (1986).
15. E. LINDENSTRAUSS, On quantum unique ergodicity for Γ\H×H, Internat. Math. Res. Notices 17 (2001), 913-933.
16. E. LINDENSTRAUSS, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. 163:1 (2006), 165-219.
17. E. LINDENSTRAUSS, A. VENKATESH, Existence and Weyl's law for spherical cusp forms, GAFA, Geom. funct. anal. 17:1 (2007), 220-251.
18. 3(ℤ), J. reine angew. Math. 533 (2001), 127-169.
19. G.D. MOSTOW, Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies 78, Princeton University Press, Princeton, NJ (1973),
20. Z. RUDNICK, P. SARNAK, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161:1 (1994), 195-213.
21. A.I. SHNIRELMAN, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk 29:6(180) (1974), 181-182.
22. p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77:1 (1988), 123-138.
23. p norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4:2 (2003), 343-368.
24. E. WIGNER, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40:5 (1932), 749-759.
25. S.A. WOLPERT, Semiclassical limits for the hyperbolic plane, Duke Math. J. 108:3 (2001), 449-509.
26. S. ZELDITCH, Pseudodifferential analysis on hyperbolic surfaces, J. Funct. Anal. 68:1 (1986), 72-105.
27. S. ZELDITCH, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55:4 (1987), 919-941.
28. S. ZELDITCH, The averaging method and ergodic theory for pseudo-differential operators on compact hyperbolic surfaces, J. Funct. Anal. 82:1 (1989), 38-68.