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Recent progress on the quantum unique ergodicity conjecture


Author: Peter Sarnak
Journal: Bull. Amer. Math. Soc. 48 (2011), 211-228
MSC (2010): Primary 11Fxx, 11Mxx, 35Qxx, 37Axx, 81Sxx
DOI: https://doi.org/10.1090/S0273-0979-2011-01323-4
Published electronically: January 10, 2011
MathSciNet review: 2774090
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Abstract: We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context.


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  • [An] N. Anantharaman, ``Entropy and the localization of eigenfunctions,'' Annals of Math. (2), 168 (2008), 435-475. MR 2434883
  • [A-N] N. Anantharaman and S. Nonnenmacher, ``Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold,'' Ann. Inst. Four. (Grenoble), 57, 6 (2007), 2465-2523. MR 2394549 (2009m:81076)
  • [A-S] R. Aurich and F. Steiner, ``Exact theory for the quantum eigenstates of a strongly chaotic system,'' Phys. D, 48 (1991), 445-470. MR 1102171 (92c:81040)
  • [B-G-H-T] T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, ``A family of Calabi-Yau varieties and potential automorphy II'', preprint (2009).
  • [Ba] A.H. Barnett, ``Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards,'' Comm. Pure Appl. Math., 59 (2006), 1457-1488. MR 2248896 (2007d:81085)
  • [Be] M. Berry, ``Quantum scars of classical closed orbits in phase space,'' Proc. Roy. Soc, London, Ser A, 423 (1989), 219-231. MR 1059234 (92c:81075)
  • [Ber] P. Berard, ``On the wave equation on a compact Riemannian manifold without conjugate points'', Math. Z., 155 (1977), 2083-2091.
  • [Bl] D. Blasius, ``Hilbert modular forms and the Ramanujan Conjecture'', ARXIV (2005). MR 2327298 (2008g:11077)
  • [Bog] E. Bogomolny, ``Smoothed wave functions of chaotic quantum systems,'' Phys. D, 31 (1988), 169-189. MR 955627 (89h:81062)
  • [Bos] R.J. Boscovich, ``The ergodic properties of certain billiards'', Sectionun Conicarum Elementa, (1757), Venice.
  • [B-L] J. Bourgain and E. Lindenstrauss, ``Entropy of quantum limits'', Comm. Math. Phys., 233 (2003), 153-171. MR 1957735 (2004c:11076)
  • [Br] S. Brooks, Entropy bounds for quantum limits, Princeton University Thesis, (2009). MR 2713056
  • [Bu] L. Bunimovich, ``The ergodic properties of certain billiards'', Funct. Anal. and Appl., 8 (1974), 254.
  • [B-Z] N. Burq and M. Zworski, ``Bouncing ball modes and quantum chaos,'' SIAM Rev., 47 (2005), 43-49. MR 2149100 (2006d:81111)
  • [C-M] N. Chernov and R. Markarian, ``Chaotic billiards'', AMS, Surveys 127 (2006).
  • [CdV1] Y. Colin de Verdiere, ``Quasi-modes sur les variétés Riemanniennes,'' Invent. Math., 3 (1977), 15-52. MR 0501196 (58:18615)
  • [CdV2] Y. Colin de Verdiere, ``Ergodicité et fonctions propres du laplacien,'' Comm. Math. Phys., 102 (1985), 497-502. MR 818831 (87d:58145)
  • [CdV3] Y. Colin de Verdiere, ``Hyperbolic geometry in two dimensions and trace formulas'', in Chaos and Quantum Physics, Les-Houches, (1989), 307-329.
  • [De] P. Deligne, ``La conjecture de Weil. I.,'' Inst. Hautes Études Sci. Publ. Math., 43 (1974), 273-307. MR 0340258 (49:5013)
  • [Dona] S. Donaldson, ``Plank's constant in complex and almost complex geometry'', XII ICMP (2000), 63-72, International Press, Boston. MR 1883295 (2003c:53124)
  • [Do] H. Donnelly, ``Quantum unique ergodicity,'' Proc. Amer. Math. Soc., 131 (2003), 2945-2951. MR 1974353 (2005a:58048)
  • [E] Y. Egorov, ``The canonical transformations of pseudo-differential operators'', Uspehi. Mat. Nauk, 24 (1969), 235-236.
  • [E-K-L] M. Einsiedler-A. Katok and E. Lindenstrauss, ``Invariant measures and the set of exceptions to Littlewood's conjecture,'' Annals of Math. (2), 164 (2006), 513-560. MR 2247967 (2007j:22032)
  • [E-M-S] P. D. Elliott, C. J. Moreno and F. Shahidi, ``On the absolute value of Ramanujan's $ \tau$-function,'' Math. Ann., 266 (1984), 507-511. MR 735531 (85f:11030)
  • [F-N-D] A. Faure, S. Nonnenmacher and S. DeBieure, ``Scarred eigenstates for quantum cat maps of minimal periods'', CMP, 293 (2003), 449-492. MR 20009236 (2005a:81076)
  • [Fr] J. Friedlander, ``Bounds for $ L$-functions,'' Proc. ICM (1994), 363-373. MR 1403937 (97e:11108)
  • [G-H-L] D. Goldfeld, J. Hoffstein and D. Lieman, ``Coefficients of Maass forms and the Siegel zero,'' Annals of Math. (2), 140 (1994), 161-181. MR 1289494 (95m:11048)
  • [G-L] P. Gerard and E. Leichtnam, ``Ergodic properties of eigenfunctions for the Dirichlet problem,'' Duke Math. J., 71 (1993), 559-607. MR 1233448 (94i:35146)
  • [G-S] A. Ghosh and P. Sarnak, ``Zeros and nodal domains of modular forms,'' in preparation.
  • [Go] A. Good, ``The square mean of Dirichlet series associated with cusp forms,'' Mathematika, 29 (1982), 278-295. MR 696884 (84f:10036)
  • [G-S] A. Granville and K. Soundararajan, ``The distribution of values of $ L(1,\chi_d)$'', Geom. Funct. Anal., 15 (2003), 992-1028. MR 2024414 (2005d:11129)
  • [Hel] E. Heller, ``Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits,'' Phys. Rev. Lett., 53 (1984), 1515-1518. MR 762412 (85k:81055)
  • [H-O] E. Heller and P. O'Conner, ``Quantum localization for a strongly classically chaotic system,'' Phys. Lett. Rev., 61 (1988), 2288-2291. MR 966831 (89j:81069)
  • [Hi] A. Hilderbrand, ``A note on Burgess' character sum estimate'', C.R. Math. Rep. Acad. Sc., Canada, 8 (1986), 147-152.
  • [Ha] A. Hassell, ``Ergodic billiards that are not quantum unique ergodic,'' Annals of Math. (2) 171 (2010), 605-619. MR 2630052
  • [Ho] R. Holowinsky, ``Sieving for mass equidistribution'' Ann. of Math. (2), 172 (2010), 1499-1516. MR 2680498
  • [H-S] R. Holowinsky and K. Soundararajan, ``Mass equidistribution of Hecke eigenfunctions,'' Ann. of Math. (2), 172 (2010), 1517-1528. MR 2680499
  • [Ich] A. Ichino, ``Trilinear forms and the central values of triple product $ L$-functions,'' Duke Math. J., 145 (2008), 281-307. MR 2449948 (2009i:11066)
  • [I-S] H. Iwaniec and P. Sarnak, ``Perspectives on the analytic theory of $ L$-functions,'' GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal., 2000, Special Volume, Part II, 705-741. MR 1826269 (2002b:11117)
  • [Ja] D. Jakobson, ``Quantum unique ergodicity for Eisenstein series on $ \mathrm{PSL}_2(\mathbf Z) \mathrm{PSL}_2(\mathbf R)$,'' Ann. Inst. Fourier (Grenoble), 44 (1994), 1477-1504. MR 1313792 (96b:11068)
  • [Ka] S. Katok, ``Fuchsian groups'' Chicago Lectures in Mathematics University of Chicago Press, Chicago, 1992. MR 1177168 (93d:20088)
  • [K-S] H. Kim and F. Shahidi, ``Cuspidality of symmetric powers with applications,'' Duke Math. J., 112 (2007), 177-197. MR 1890650 (2003a:11057)
  • [K-H] S. Kudla and M. Harris, ``The central critical value of a triple product $ L$-function,'' Ann. of Math. (2), 133 (1991), 605-672. MR 1109355 (93a:11043)
  • [K-R] P. Kurlberg and Z. Rudnick, ``Hecke theory and equidistribution for the quantization of linear maps of the torus,'' Duke Math. J., 103 (2000), 47-77. MR 1758239 (2001f:11065)
  • [L-L-Y] Y. K. Lau, J. Liu and Y. Ye, ``Subconvexity bounds for Rankin-Selberg $ L$-functions for congruence subgroups,'' J. Number Theory, 121 (2006), 204-223. MR 2274903 (2007h:11106)
  • [La] V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Ergebnisse der Mathamatik und ihrer Grenzgebiete (3), Vol. 24, (1993). MR 1239173 (94m:58069)
  • [Li1] E. Lindenstrauss, ``Invariant measures and arithmetic quantum unique ergodicity,'' Ann. of Math. (2), 163 (2006), 165-219. MR 2195133 (2007b:11072)
  • [Li2] E. Lindenstrauss, ``On quantum unique ergodicity for $ \Gamma \setminus {\mathbb{H}}\times {\mathbb{H}}$,'' Internat. Math. Res. Notices 2001, 913-933. MR 1859345 (2002k:11076)
  • [L-S2] W. Luo and P. Sarnak, ``Mass equidistribution for Hecke eigenforms,'' Comm. Pure Appl. Math. 56 (2003), 874-891. MR 1990480 (2004e:11038)
  • [L-S1] W. Luo and P. Sarnak, ``Quantum ergodicity of eigenfunctions on $ \mathrm{PSL}_2({mathbf Z})\setminus {\mathbf H}^2$,'' Inst. Hautes Études Sci. Publ. Math., 81 (1995), 207-237. MR 1361757 (97f:11037)
  • [Lu] W. Luo, ``Values of symmetric square $ L$-functions at $ 1$,'' J. Reine Agnew. Math., 506 (1999), 215-235. MR 1665705 (2001d:11055)
  • [M-V] P. Michel and A. Venkatesh, ``The subconvexity problem for $ GL(2)$'' arXiv: 0903.3591. MR 2653249
  • [Ma] G. Margulis, ``Problems and conjectures in rigidity theory'', Mathematics: frontiers and perspectives, 161-174, Amer. Math. Soc., Providence, RI, 2000. MR 1754775 (2001d:22008)
  • [Mark] J. Marklof, ``Arithmetic quantum chaos'', Encyc. of Math. Phys., Vol 1 (J.-P. Francoise, G. L. Naber, and S.T. Tsou, editors), Oxford, Elsevier, 2006, pp. 212-220.
  • [Mars] S. Marshall, ``On the cohomology and quantum chaos of the general linear group in two variables'', Thesis, Princeton University (2009).
  • [Mel] R. Melrose, ``Geometric scattering theory,'' Stanford Lectures. Cambridge University Press, Cambridge, 1995. MR 1350074 (96k:35129)
  • [Mu] T. Muerman, ``On the order of the Maass $ L$-function on the critical line'', Colloq. Math. Soc. Janos Bolyai, 51 (1990), 325-354.
  • [Na] M. Nair, ``Multiplicative functions of polynomial values in short intervals,'' Acta Arith., 62 (1992), 257-269. MR 1197420 (94b:11093)
  • [N-V] S. Nonnenmacher and A. Voros, ``Chaotic eigenfunctions in phase space,'' J. Statist. Phys., 92 (1998), 431-518. MR 1649013 (2000c:81079)
  • [P-R] V. Platonov and A. Rapinchuk, ``Algebraic Groups and Number Theory'', A.P. (1991), Chapter 9.
  • [Ri] G. Riviere, ``Entropy of semiclassical measures in dimension 2'', to appear in Duke Math. J.
  • [Ru1] Z. Rudnick, ``On the asymptotic distribution of zeros of modular forms'', IMRN (2005), 2059-2074. MR 2181743 (2006k:11099)
  • [Ru2] Z. Rudnick, ``The arithmetic theory of quantum maps: Equidistribution in number theory, an introduction'' 331-342. in NATO Sci. Ser. II Math. Phys. Chem.,, 237 Springer, Dorcrecht, 2007. MR 2290505 (2007k:81071)
  • [R-S] Z. Rudnick and P. Sarnak, ``The behavior of eigenstates of arithmetic hyperbolic manifolds'', CMP, 161 (1994), 195-213. MR 1266075 (95m:11052)
  • [Ru] D. Rudolph, ``$ \times 2$ and $ \times 3$ invariant measures and entropy'', Erg. Th. and Dyn. Systems, 10 (1990), 395-406. MR 1062766 (91g:28026)
  • [Sa1] P. Sarnak, ``Arithmetic quantum chaos'', Schur Lectures, Israel Math. Conf. Proc. (1995). MR 1321639 (96d:11059)
  • [Sa2] P. Sarnak, ``Spectra of arithmetic manifolds'', BAMS (2003), Vol. 40, 441-478. MR 1997348 (2004f:11107)
  • [Sa3] P. Sarnak, ``Estimates for Rankin-Selberg $ L$-functions and quantum unique ergodicity,'' J. Funct. Anal., 184 (2001), 419-453. MR 1851004 (2003c:11050)
  • [Sa4] P. Sarnak, ``Notes on the generalized Ramanujan conjectures: Harmonic analysis, the trace formula, and Shimura varieties'', 659-685. Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005. MR 2192019 (2007a:11067)
  • [Se1] J.P. Serre, ``A course in arithmetic'' (Translated from the French). Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. MR 0344216 (49:8956)
  • [Se2] J.P. Serre, ``Trees'', (Translated from the French by J. Stillwell.) Springer-Verlag, Berlin-New York, 1980. MR 607504 (82c:20083)
  • [S-Z] B. Shiffmann and S. Zelditch, ``Distribution of zeros of random and quantum chaotic sections of positive line bundles,'' Comm. Math. Phys., 200 (1999), 661-683. MR 1675133 (2001j:32018)
  • [Sh1] A. Shnirelman, ``Ergodic properties of eigenfunctions'', Uspenski Math. Nauk 29/6 (1974), 181-182.
  • [Sh2] A. Shnirelman, Appendix to ``KAM theory and semiclassical approximations to eigenfunctions'' by V. Lazutkin, Ergebnisse der Mathematik, 24, Springer-Verlag, Berlin, 1993.
  • [S-V1] L. Silberman and A. Venkatesh, ``On quantum unique ergodicity for locally symmetric spaces,'' Geom. Funct. Anal., 17 (2007), 960-998. MR 2346281 (2009a:81072)
  • [S-V2] L. Silberman and A. Venkatesh, GAFA, (to appear).
  • [Si1] Y. Sinai, ``Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards'', Russian Math. Surveys, 25 (1970), 137-189.
  • [Si2] Y. Sinai, ``Flows with finite energy'', Doklady of Russian Academy of Sciences (1959), 124, 754-755.
  • [So1] K. Soundararajan, ``Quantum Unique Ergodicity for $ SL_2 ( \mathbb{Z} \backslash \mathbb{H})$'', Ann. of Math. (2), (to appear).
  • [So2] K. Soundararajan, ``Weak Subconvexity and central values of $ L$-functions'', Ann. of Math.(2), (to appear).
  • [Ti] E. Titchmarsh, ``The Theory of the Riemann Zeta-function'', Oxford, at The Clarendon Press, 1956. MR 0046485 (13:741c)
  • [Wa] T. Watson, Princeton University Thesis (2001), ``Rakin triple product and quantum chaos,'' ArXiv: 0810:0425. MR 2703041
  • [We] H. Weyl, ``Über die Gleichverteilung von Zahlen mod. Eins,'' Math. Ann., 77 (1916), 313-352. MR 1511862
  • [Wo] S. Wolpert, ``The modulus of continuity for $ \Gamma_0(m)\setminus{\mathbb{H}}$ semi-classical limits,'' Comm. Math. Phys. 216 (2001), no. 2, 313-323. MR 1814849 (2002f:11059)
  • [Zel1] S. Zelditch, ``Uniform distribution of eigenfunctions on compact hyperbolic surfaces,'' Duke Math. J., 55 (1987), 919-941. MR 916129 (89d:58129)
  • [Zel2] S. Zelditch, ``Note on quantum unique ergodicity,'' Proc. Amer. Math. Soc., 132 (2004), 1869-1872. MR 2051153 (2005i:58039)
  • [Zel3] S. Zelditch, ``Quantum ergodicity and mixing'', Encyc. of Math. Phys., Vol. 1 (J.-P. Francoise, G. L. Naber, and S.T. Tsou, editors), Oxford, Elsevier, 2006, pp. 183-196.
  • [Zel4] S. Zelditch, ``Recent developments in mathematical quantum chaos,'' ArXiv0911.4321.
  • [Zel5] S. Zelditch, ``Index and dynamics of quantized contact transformations,'' Ann. Inst. Fourier (Grenoble) 47 (1997), 305-363. MR 1437187 (99a:58082)
  • [Z-Z] S. Zelditch and M. Zworski, ``Ergodicity of eigenfunctions for ergodic billiards,'' Comm. Math. Phys. 175 (1996), no. 3, 673-682. MR 1372814 (97a:58193)

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Additional Information

Peter Sarnak
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: sarnak@math.princeton.edu

DOI: https://doi.org/10.1090/S0273-0979-2011-01323-4
Received by editor(s): February 8, 2010
Published electronically: January 10, 2011
Article copyright: © Copyright 2011 American Mathematical Society
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