Note on quantum unique ergodicity

Author:
Steve Zelditch

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1869-1872

MSC (2000):
Primary 58J50, 58J40, 35P99, 81S10

DOI:
https://doi.org/10.1090/S0002-9939-03-07298-8

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 () 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.

**[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 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**

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

Email:
zelditch@math.jhu.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07298-8

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