Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Note on quantum unique ergodicity

Author: Steve Zelditch
Journal: Proc. Amer. Math. Soc. 132 (2004), 1869-1872
MSC (2000): Primary 58J50, 58J40, 35P99, 81S10
Published electronically: November 21, 2003
MathSciNet review: 2051153
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that (near) off-diagonal matrix elements $\langle A \varphi_i, \varphi_j \rangle$ ($i \not= j$) of pseudodifferential operators relative to eigenfunctions of quantum unique- ly ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.

References [Enhancements On Off] (What's this?)

  • [BSS] A. Backer, R. Schubert, and P. Stifter, On the number of bouncing ball modes in billiards, J. Phys. A 30 (1997), no. 19, 6783-6795.
  • [BL] J. Bourgain and E. Lindenstrauss, Entropy of Quantum Limits, Commun. Math. Phys. 233 (2003), 153-171.
  • [BZ1] N. Burq and M. Zworski, Control in the presence of a black box, arxiv preprint math.AP/0304184 (2003).
  • [BZ2] N. Burq and M. Zworski, Bouncing ball modes and quantum chaos, arxiv preprint math.AP/0306278 (2003).
  • [CdV] Y. Colin de Verdière, Quasi-modes sur les variétés Riemanniennes, Invent. Math. 43 (1977), no. 1, 15-52. MR 58:18615
  • [D] H. G. Donnelly, Quantum unique ergodicity, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2945-2951.
  • [FN] F. Faure and S. Nonnenmacher, On the maximal scarring for quantum cat map eigenstates, arxiv preprint nlin.CD/0304031 (2003).
  • [FND] F. Faure, S. Nonnenmacher, and S. De Bievre, Scarred eigenstates for quantum cat maps of minimal periods, arxiv preprint nlin.CD/0207060 (2003), Comm. Math. Phys. 239 (2003), 449-492.
  • [H] E. J. Heller, Wavepacket dynamics and quantum chaology. Chaos et physique quantique (Les Houches, 1989), 547-664, North-Holland, Amsterdam, 1991. MR 94i:81031
  • [HO] E. J. Heller and P. W. O'Connor, Quantum localization for a strongly classically chaotic system, Phys. Rev. Lett. 61 (20) (1988), 2288-2291. MR 89j:81069
  • [L] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, preprint, 2003.
  • [RS] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195-213. MR 95m:11052
  • [S] P. Sarnak, Arithmetic quantum chaos, The Schur lectures (1992) (Tel Aviv), 183-236, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. MR 96d:11059
  • [W] S. A. Wolpert, The modulus of continuity for $\Gamma\sb 0(m)\backslash{\mathbb H}$ semi-classical limits, Comm. Math. Phys. 216 (2001), no. 2, 313-323. MR 2002f:11059
  • [Z] S. Zelditch, Quantum transition amplitudes for ergodic and for completely integrable systems, J. Funct. Anal. 94 (1990), no. 2, 415-436. MR 92b:58181

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J50, 58J40, 35P99, 81S10

Retrieve articles in all journals with MSC (2000): 58J50, 58J40, 35P99, 81S10

Additional Information

Steve Zelditch
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Received by editor(s): January 28, 2003
Received by editor(s) in revised form: March 10, 2003
Published electronically: November 21, 2003
Additional Notes: This research was partially supported by NSF grant DMS-0071358 and by the Clay Mathematics Institute
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society