Quantum ergodicity of C*-dynamical systems

Steven Zelditch

Séminaire de théorie spectrale et géométrie (1994-1995)

  • Volume: 13, page 81-95
  • ISSN: 1624-5458

How to cite


Zelditch, Steven. "Quantum ergodicity of C*-dynamical systems." Séminaire de théorie spectrale et géométrie 13 (1994-1995): 81-95. <http://eudml.org/doc/114384>.

author = {Zelditch, Steven},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {operator algebras; pseudodifferential operators; Fourier integral operators; Toeplitz operators; quantized contact transformations; quantized symplectic torus automorphisms; semi-classical quantum ergodicity; -dynamical systems; quantum ergodicity},
language = {eng},
pages = {81-95},
publisher = {Institut Fourier},
title = {Quantum ergodicity of C*-dynamical systems},
url = {http://eudml.org/doc/114384},
volume = {13},
year = {1994-1995},

AU - Zelditch, Steven
TI - Quantum ergodicity of C*-dynamical systems
JO - Séminaire de théorie spectrale et géométrie
PY - 1994-1995
PB - Institut Fourier
VL - 13
SP - 81
EP - 95
LA - eng
KW - operator algebras; pseudodifferential operators; Fourier integral operators; Toeplitz operators; quantized contact transformations; quantized symplectic torus automorphisms; semi-classical quantum ergodicity; -dynamical systems; quantum ergodicity
UR - http://eudml.org/doc/114384
ER -


  1. [A.dP.W] S. AXELROD, S. DELLA PIETRA and E. WITTEN. _ Geometric quantization of the Chern-Simons gauge theory, J.D.G. 33 ( 1991), 787-902. Zbl0697.53061MR1100212
  2. [Bai] W. BAILY . _ Classical theory of θ-fonctions, in AMS Proc.Symp.Pure. Math. IX AMS ( 1966), 306-311. Zbl0178.55002MR208016
  3. [Be] J. BELLISSARD. _ Non-commutative method in semi-classical analysis, in Transition to Chaos in Classical and Quantum Mechanics, éd. S.Graffi, Lecture Notes in Math. 1589, Springer-Verlag, N.Y., 1994. Zbl0818.46072
  4. [B] F. BENATTI. _ Deterministic Chaos in Infinite Quantum Systems, Trieste, Notes in Physics, Springer-Verlag, New York, 1993. MR1261185
  5. [B.N.S] F. BENATTI, H. NARNHOFER and G.L. SEWELL . _ A non-commutative version of the Arnold cat map, Lett. Math.Phys. 21 ( 1991), 157-172. Zbl0722.46033MR1093527
  6. [B.H] M.V. BERRY and J.H. HANNAY. _ Quantization oflinearmaps on a torus, Physica Dl ( 1980), 267. MR602111
  7. [dB.B] S. DE BIEVRE and A. BOUZOUINA. _ Equipartition of the eigenfùnctions of quantized ergodic maps on the torus, to appear in Comm. Math. Phys. Zbl0876.58041MR1387942
  8. [B] L. BOUTET DE MONVEL. _ Toeplitz operators, an asymptotic quantization of symplectic cones, in: Stochastic Processes and Their Applications, S. Albeverio (Ed.), Kluwer Acad. Pub., Netherlands, 1990. Zbl0735.47014MR1086184
  9. [B.G] L. BOUTET DE MONVEL and V. GUILLEMIN. _ The Spectral Theory of Toeplitz Operators, Ann. Math. Studies 99, Princeton U. Press, 1981. Zbl0469.47021MR620794
  10. [B.S] L. BOUTET DE MONVEL and J. SJÖSTRAND. _ Sur la singularité des noyaux de Bergmann et de Szegö, Asterisque 34-35 ( 1976), 123-164. Zbl0344.32010MR590106
  11. [B.R] O. BRATTELI and D.W. ROBINSON. _ Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlay, 1979. Zbl0421.46048MR611508
  12. [Co] A. CONNES . _ Non commutative Geometry, Acad. Press, N.Y., 1994. 
  13. [dE.G.I] M. D'EGLI ESPOSTI, S. GRAFFI and S. ISOLA. _ Stochastic properties of the quantum Arnold cat in the classical limit, Comm.Math.Phys. 167 ( 1995), 471-509. MR1316757
  14. [D] R.G. DOUGLAS . _ C * - Algebra Extensions and K-Homology, Ann.Math.Studies 95, Princeton U.Press, Princeton, 1980. Zbl0458.46049MR571362
  15. [F] G. FOLLAND. _ Harmonic Analysis in Phase Space, Ann. Math. Studies, 122, Princeton U. Press, 1989. Zbl0682.43001MR983366
  16. [ES] G. FOLLAND and E. STEIN. _ , Comm. P.A.M. 27 ( 1974), 429-522. Zbl0293.35012MR367477
  17. [G.S] A. GRIGIS and J. SJOSTRAND. _ Microlocal Analysis of Spectral Problems, Cambridge Univ. Press, 1994. Zbl0804.35001MR1269107
  18. [G] V. GUILLEMIN. _ Residue traces for certain algebras of Fourier Integral operators, J.Fun.Anal. 115 ( 1993),381-417. Zbl0791.35162MR1234397
  19. [G.1] V. GUILLEMIN. _ Some classical theorems in spectral theory revisited, in Seminar on Singularises, Ann. Math. Studies, 91, Princeton U. Press, 1979. Zbl0452.35093MR547021
  20. [GS] V. GUILLEMIN and S. STERNBERG. _ Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 ( 1979), 915-955. Zbl0446.58019MR536046
  21. [H] W. HELTON . _ An operatoralgebra approach to partial diffevential equations, propagation of singularities and spectral theory, Indiana Univ. Math. Journal, 1976. Zbl0373.35060
  22. [HoIV] L. HÖRMANDER. _ The Analysis of Linear Partial Differential Operators IV, Grundlehren 275, Springer-Verlag, 1985. Zbl0612.35001MR781537
  23. [Herm] C. HERMITE. _ Sur quelques formules relatives a la transformation des fonctions elliptiques, Journal de Liouville III ( 1858), 26. 
  24. [K] V. KAC. _ Infinite Dimensional LieAlgebras, 3rd ed. Cambridge: Cambridge Univ.Press, 1990. Zbl0716.17022MR1104219
  25. [K.P] V. KAC and D.H. PETERSON. _ Infinite dimensional Lie algebras, theta fonctions and modular forms, Adv in Math. 53 ( 1984), 125-264. Zbl0584.17007MR750341
  26. [Ke] J. KEATING. _ The cat maps: quantum mechanics and classical motion, Nonlinearity 4 ( 1991), 309-341. Zbl0726.58037MR1107009
  27. [R] D. RUELLE . _ Statistical Mechanics, Benjamin, 1969. Zbl0177.57301MR289084
  28. [S] P. SARNAK. _ Arithmetic quantum chaos, Tel Aviv lectures, 1993. Zbl0831.58045
  29. [Sn] A.I. SNIRELMAN. _ Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29 ( 1974), 181-182. 
  30. [St] E. STEIN . _ Harmonic Analysis, Princeton: Princeton Univ.Press, 1993. Zbl0821.42001
  31. [Su] T. SUNADA. _ Quanrum ergodicity, (preprint 1994). 
  32. [Tr] F. TREVES. _ Introduction to Pseudodifferential and Fourier Integral Operators II, Plenum, New York, 1980. Zbl0453.47027MR597145
  33. [W] P. WALTERS. _ An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer-Verlag, NY, 1982. Zbl0475.28009MR648108
  34. [Wei] A. WEINSTEIN. _ Fourier Intégral Operators, quantization, and the spectrum of a Riemannian manifold, Colloques Internationaux C.N.R.S. 237, Geometrie Symplectique et Physique, 1976. 
  35. [We] J. WEITSMAN. _ Quantization via real polarization of the moduli space of flat connections and Chern-Simons gauge theory in genus one, Comm. Math. Phys. 137 ( 1991), 175-190. Zbl0717.53065MR1099261
  36. [Z.1] S. ZELDITCH. _ Quantum ergodicity of C* - dynamical systems, (Comm.Math.Phys., to appear). Zbl0856.58019MR1384146
  37. [Z.2] S. ZELDITCH. _ Index and dynamics of quantized contact transformations, (preprint 1995). MR1437187
  38. [Z.3] S. ZELDITCH. _ Quantum Mixing, (J. Fun. Anal., to appear). Zbl0858.58049MR1404574
  39. [Z.4] S. ZELDITCH. _ Quantum transition amplitudes for ergodic and for completely integrable Systems, J. Fun. Anal. 94 ( 1990), 415-436. Zbl0721.58051MR1081652
  40. [Z.Z] S. ZELDITCH, M. ZWORSKI. _ Quantum ergodicity and ergodic billiards, (Comm. Math. Phys., to appear). Zbl0840.58048MR1372814

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.