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Note on quantum unique ergodicity
Author:
Steve Zelditch
Journal:
Proc. Amer. Math. Soc. 132 (2004), 1869-1872
MSC (2000):
Primary 58J50, 58J40, 35P99, 81S10
Posted:
November 21, 2003
MathSciNet review:
2051153
Full-text PDF Free Access
Abstract |
References |
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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]
Yves
Colin de Verdière, Quasi-modes sur les
variétés Riemanniennes, Invent. Math.
43 (1977), no. 1, 15–52 (French). MR 0501196
(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]
Eric
J. Heller, Wavepacket dynamics and quantum chaology, Chaos et
physique quantique (Les Houches, 1989) North-Holland, Amsterdam, 1991,
pp. 547–664. MR 1188425
(94i:81031)
- [HO]
Patrick
W. O’Connor and Eric
J. Heller, Quantum localization for a strongly classically chaotic
system, Phys. Rev. Lett. 61 (1988), no. 20,
2288–2291. MR 966831
(89j:81069), http://dx.doi.org/10.1103/PhysRevLett.61.2288
- [L]
E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, preprint, 2003.
- [RS]
Zeév
Rudnick and Peter
Sarnak, The behaviour of eigenstates of arithmetic hyperbolic
manifolds, Comm. Math. Phys. 161 (1994), no. 1,
195–213. MR 1266075
(95m:11052)
- [S]
Peter
Sarnak, Arithmetic quantum chaos, The Schur lectures (1992)
(Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat
Gan, 1995, pp. 183–236. MR 1321639
(96d:11059)
- [W]
Scott
A. Wolpert, The modulus of continuity for
Γ₀(𝑚)\ℍ semi-classical limits, Comm. Math.
Phys. 216 (2001), no. 2, 313–323. MR 1814849
(2002f:11059), http://dx.doi.org/10.1007/PL00005549
- [Z]
Steven
Zelditch, Quantum transition amplitudes for ergodic and for
completely integrable systems, J. Funct. Anal. 94
(1990), no. 2, 415–436. MR 1081652
(92b:58181), http://dx.doi.org/10.1016/0022-1236(90)90021-C
- [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
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Additional Information
Steve Zelditch
Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email:
zelditch@math.jhu.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07298-8
PII:
S 0002-9939(03)07298-8
Received by editor(s):
January 28, 2003
Received by editor(s) in revised form:
March 10, 2003
Posted:
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
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