Abstract
We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct super-scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds. We show that these super-scars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we compute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions.
-
[BD1]
F. Bonechi and S. De Bièvre, "Controlling strong scarring for quantized ergodic toral automorphisms," Duke Math. J., vol. 117, iss. 3, pp. 571-587, 2003.@article {BD1, MRKEY = {1979054},
AUTHOR = {Bonechi, Francesco and De Bi{è}vre, Stephan},
TITLE = {Controlling strong scarring for quantized ergodic toral automorphisms},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {117},
YEAR = {2003},
NUMBER = {3},
PAGES = {571--587},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {81Q50 (37A25 37D20 37N20 81Q20)},
MRNUMBER = {2004i:81079},
MRREVIEWER = {Jens Marklof},
DOI = {10.1215/S0012-7094-03-11736-6},
ZBLNUMBER = {1049.81028},
} -
[BD] A. Bouzouina and S. De Bièvre, "Equipartition of the eigenfunctions of quantized ergodic maps on the torus," Comm. Math. Phys., vol. 178, iss. 1, pp. 83-105, 1996.
@article {BD, MRKEY = {1387942},
AUTHOR = {Bouzouina, A. and De Bi{è}vre, S.},
TITLE = {Equipartition of the eigenfunctions of quantized ergodic maps on the torus},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {178},
YEAR = {1996},
NUMBER = {1},
PAGES = {83--105},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {81Q50 (58F19 82B03)},
MRNUMBER = {97b:81023},
MRREVIEWER = {N. L. Balazs},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104286555},
ZBLNUMBER = {0876.58041},
} -
[BoS] E. Bogomolny and C. Schmit, "Multiplicities of periodic orbit lengths for non-arithmetic models," J. Phys. A, vol. 37, iss. 16, pp. 4501-4526, 2004.
@article {BoS, MRKEY = {2065941},
AUTHOR = {Bogomolny, E. and Schmit, C.},
TITLE = {Multiplicities of periodic orbit lengths for non-arithmetic models},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics. A. Mathematical and General},
VOLUME = {37},
YEAR = {2004},
NUMBER = {16},
PAGES = {4501--4526},
ISSN = {0305-4470},
CODEN = {JPHAC5},
MRCLASS = {81Q50 (20F05 37C27 37D40)},
MRNUMBER = {2005f:81073},
MRREVIEWER = {Marco Lenci},
DOI = {10.1088/0305-4470/37/16/003},
ZBLNUMBER = {1050.37008},
} -
[Bo] E. Bombieri, "On exponential sums in finite fields," Amer. J. Math., vol. 88, pp. 71-105, 1966.
@article {Bo, MRKEY = {0200267},
AUTHOR = {Bombieri, Enrico},
TITLE = {On exponential sums in finite fields},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {88},
YEAR = {1966},
PAGES = {71--105},
ISSN = {0002-9327},
MRCLASS = {14.48 (10.41)},
MRNUMBER = {34 \#166},
MRREVIEWER = {D. J. Lewis},
DOI = {10.2307/2373048},
ZBLNUMBER = {0171.41504},
} -
[Cohn] H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, New York: Springer-Verlag, 1978.
@book {Cohn, MRKEY = {506156},
AUTHOR = {Cohn, Harvey},
TITLE = {A Classical Invitation to Algebraic Numbers and Class Fields},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1978},
PAGES = {xiii+328},
ISBN = {0-387-90345-3},
MRCLASS = {12-01 (12Axx)},
MRNUMBER = {80c:12001},
MRREVIEWER = {Ezra Brown},
ZBLNUMBER = {0395.12001},
} -
[DD] S. De Bièvre and M. Degli Esposti, "Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and baker maps," Ann. Inst. H. Poincaré Phys. Théor., vol. 69, iss. 1, pp. 1-30, 1998.
@article {DD, MRKEY = {1635807},
AUTHOR = {De Bi{è}vre, S. and Degli Esposti, M.},
TITLE = {Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and baker maps},
JOURNAL = {Ann. Inst. H. Poincaré Phys. Théor.},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Physique Théorique},
VOLUME = {69},
YEAR = {1998},
NUMBER = {1},
PAGES = {1--30},
ISSN = {0246-0211},
CODEN = {AHPAAO},
MRCLASS = {58G15 (58F17)},
MRNUMBER = {99g:58119},
MRREVIEWER = {{É}ric Leichtnam},
URL = {http://www.numdam.org/item?id=AIHPA_1998__69_1_1_0},
ZBLNUMBER = {0922.58074},
} -
[DEG] M. Degli Esposti and S. Graffi, "Mathematical aspects of quantum maps," in The Mathematical Aspects of Quantum Maps, New York: Springer-Verlag, 2003, pp. 49-90.
@incollection {DEG, MRKEY = {2159323},
AUTHOR = {Degli Esposti, Mirko and Graffi, Sandro},
TITLE = {Mathematical aspects of quantum maps},
BOOKTITLE = {The Mathematical Aspects of Quantum Maps},
SERIES = {Lecture Notes in Phys.},
NUMBER = {618},
PAGES = {49--90},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2003},
MRCLASS = {81Q50 (11K55 37E99 37N20)},
MRNUMBER = {2159323},
ZBLNUMBER = {1058.81542},
} -
[DGI] M. Degli Esposti, S. Graffi, and S. Isola, "Classical limit of the quantized hyperbolic toral automorphisms," Comm. Math. Phys., vol. 167, iss. 3, pp. 471-507, 1995.
@article {DGI, MRKEY = {1316757},
AUTHOR = {Degli Esposti, Mirko and Graffi, Sandro and Isola, Stefano},
TITLE = {Classical limit of the quantized hyperbolic toral automorphisms},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {167},
YEAR = {1995},
NUMBER = {3},
PAGES = {471--507},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {81Q50 (11L05 11Z05 58F11 81S30)},
MRNUMBER = {96c:81054},
MRREVIEWER = {Jens Bolte},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104272155},
ZBLNUMBER = {0822.58022},
} -
[EFK1] B. Eckhardt, S. Fishman, J. Keating, O. Agam, J. Main, and K. Müller, "Approach to ergodicity in quantum wave functions," Phys. Rev. E, vol. 52, pp. 5893-5903, 1995.
@article{EFK1,
author={Eckhardt, B. and Fishman, S. and Keating, J. and Agam, O. and Main, J. and Müller, K.},
TITLE={Approach to ergodicity in quantum wave functions},
JOURNAL={Phys. Rev. E},
VOLUME={52},
YEAR={1995},
PAGES={5893--5903},
} -
[FP] M. Feingold and A. Peres, "Distribution of matrix elements of chaotic systems," Phys. Rev. A, vol. 34, iss. 1, pp. 591-595, 1986.
@article {FP, MRKEY = {848319},
AUTHOR = {Feingold, Mario and Peres, Asher},
TITLE = {Distribution of matrix elements of chaotic systems},
JOURNAL = {Phys. Rev. A},
FJOURNAL = {Physical Review. A. Third Series},
VOLUME = {34},
YEAR = {1986},
NUMBER = {1},
PAGES = {591--595},
ISSN = {1050-2947},
CODEN = {PLRAAN},
MRCLASS = {81C15 (58F13)},
MRNUMBER = {87i:81055},
DOI = {10.1103/PhysRevA.34.591},
} -
[FND] F. Faure, S. Nonnenmacher, and S. De Bièvre, "Scarred eigenstates for quantum cat maps of minimal periods," Comm. Math. Phys., vol. 239, iss. 3, pp. 449-492, 2003.
@article {FND, MRKEY = {2000926},
AUTHOR = {Faure, Fr{é}d{é}ric and Nonnenmacher, St{é}phane and De Bi{è}vre, Stephan},
TITLE = {Scarred eigenstates for quantum cat maps of minimal periods},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {239},
YEAR = {2003},
NUMBER = {3},
PAGES = {449--492},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {81Q50 (37N20)},
MRNUMBER = {2005a:81076},
DOI = {10.1007/s00220-003-0888-3},
ZBLNUMBER = {1033.81024},
} -
[FJ] M. D. Fried and M. Jarden, Field Arithmetic, Second ed., New York: Springer-Verlag, 2005.
@book {FJ, MRKEY = {2102046},
AUTHOR = {Fried, Michael D. and Jarden, Moshe},
TITLE = {Field Arithmetic},
SERIES = {Ergeb. Math. Grenzgeb.},
NUMBER = {11},
EDITION = {Second},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2005},
PAGES = {xxiv+780},
ISBN = {3-540-22811-X},
MRCLASS = {12E30 (03B25 03C10 03C60 03H05)},
MRNUMBER = {2005k:12003},
MRREVIEWER = {Ido Efrat},
ZBLNUMBER = {1055.12003},
} -
[Ge] P. Gérardin, "Weil representations associated to finite fields," J. Algebra, vol. 46, iss. 1, pp. 54-101, 1977.
@article {Ge, MRKEY = {0460477},
AUTHOR = {G{é}rardin, Paul},
TITLE = {Weil representations associated to finite fields},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {46},
YEAR = {1977},
NUMBER = {1},
PAGES = {54--101},
ISSN = {0021-8693},
MRCLASS = {20G05},
MRNUMBER = {57 \#470},
MRREVIEWER = {G. I. Lehrer},
DOI = {10.1016/0021-8693(77)90394-5},
ZBLNUMBER = {0359.20008},
} -
[G] S. Gurevich, "Weil representation, Deligne sheaf, and proof of the Kurlberg-Rudnick conjecture," PhD Thesis , Tel-Aviv University, 2005.
@phdthesis{G,
author={Gurevich, Shagmar},
TITLE={Weil representation, Deligne sheaf, and proof of the Kurlberg-Rudnick conjecture},
SCHOOL={Tel-Aviv University},
YEAR={2005},
} -
[GH] S. Gurevich and R. Hadani, "Proof of the Kurlberg-Rudnick rate conjecture," in $p$-adic Mathematical Physics, Melville, NY: Amer. Inst. Phys., 2006, pp. 74-80.
@incollection {GH, MRKEY = {2258675},
AUTHOR = {Gurevich, Shagmar and Hadani, Ronny},
TITLE = {Proof of the {K}urlberg-{R}udnick rate conjecture},
BOOKTITLE = {{$p$}-adic Mathematical Physics},
SERIES = {{\rm AIP} Conf. Proc.},
NUMBER = {826},
PAGES = {74--80},
PUBLISHER = {Amer. Inst. Phys.},
ADDRESS = {Melville, NY},
YEAR = {2006},
MRCLASS = {58J50 (11L07 81Q50)},
MRNUMBER = {2008f:58033},
ZBLNUMBER = {1152.58310},
} -
[GH2] S. Gurevich and R. Hadani, The higher-dimensional Rudnick-Kurlberg conjecture, 2004.
@misc {GH2,
author = {Gurevich, Shagmar and Hadani, Ronny},
TITLE = {The higher-dimensional Rudnick-Kurlberg conjecture},
NOTE={preprint},
YEAR={2004},
ARXIV={math-ph/0409031},
} -
[HB] J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating," Phys. D, vol. 1, iss. 3, pp. 267-290, 1980.
@article {HB, MRKEY = {602111},
AUTHOR = {Hannay, J. H. and Berry, M. V.},
TITLE = {Quantization of linear maps on a torus-{F}resnel diffraction by a periodic grating},
JOURNAL = {Phys. D},
FJOURNAL = {Physica D. Nonlinear Phenomena},
VOLUME = {1},
YEAR = {1980},
NUMBER = {3},
PAGES = {267--290},
ISSN = {0167-2789},
CODEN = {PHYSA},
MRCLASS = {81B05 (58F06 58G15)},
MRNUMBER = {82g:81005},
DOI = {10.1016/0167-2789(80)90026-3},
} -
[Hua] L. K. Hua and I. Reiner, "On the generators of the symplectic modular group," Trans. Amer. Math. Soc., vol. 65, pp. 415-426, 1949.
@article {Hua, MRKEY = {0029942},
AUTHOR = {Hua, L. K. and Reiner, I.},
TITLE = {On the generators of the symplectic modular group},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {65},
YEAR = {1949},
PAGES = {415--426},
ISSN = {0002-9947},
MRCLASS = {10.0X},
MRNUMBER = {10,684d},
MRREVIEWER = {W. Givens},
DOI = {10.2307/1990620},
ZBLNUMBER = {0034.30503},
} -
[Iw] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002.
@book {Iw, MRKEY = {1942691},
AUTHOR = {Iwaniec, Henryk},
TITLE = {Spectral Methods of Automorphic Forms},
SERIES = {Grad. Studi. Math.},
NUMBER = {53},
EDITION = {Second},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2002},
PAGES = {xii+220},
ISBN = {0-8218-3160-7},
MRCLASS = {11F72 (11F12 11F37)},
MRNUMBER = {2003k:11085},
ZBLNUMBER = {1006.11024},
} -
[K1] D. Kelmer, "On the quantum variance of matrix elements for the cat map on the 4-dimensional torus," Int. Math. Res. Not., iss. 36, pp. 2223-2236, 2005.
@article {K1, MRKEY = {2181455},
AUTHOR = {Kelmer, Dubi},
TITLE = {On the quantum variance of matrix elements for the cat map on the 4-dimensional torus},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2005},
NUMBER = {36},
PAGES = {2223--2236},
ISSN = {1073-7928},
MRCLASS = {81Q50 (11Z05 37D45 58J50)},
MRNUMBER = {2007j:81066},
MRREVIEWER = {Roman Schubert},
ZBLNUMBER = {05004181},
} -
[Knabe] S. Knabe, "On the quantisation of Arnold’s cat," J. Phys. A, vol. 23, iss. 11, pp. 2013-2025, 1990.
@article {Knabe, MRKEY = {1063430},
AUTHOR = {Knabe, Stefan},
TITLE = {On the quantisation of {A}rnold's cat},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics. A. Mathematical and General},
VOLUME = {23},
YEAR = {1990},
NUMBER = {11},
PAGES = {2013--2025},
ISSN = {0305-4470},
CODEN = {JPHAC5},
MRCLASS = {81S99 (58F11 58F13 81Q50)},
MRNUMBER = {91f:81086},
MRREVIEWER = {Vesselin M. Petkov},
URL = {http://stacks.iop.org/0305-4470/23/2013},
ZBLNUMBER = {0715.58014},
} -
[KR2] Pär. Kurlberg and Z. Rudnick, "Hecke theory and equidistribution for the quantization of linear maps of the torus," Duke Math. J., vol. 103, iss. 1, pp. 47-77, 2000.
@article {KR2, MRKEY = {1758239},
AUTHOR = {Kurlberg, P{ä}r and Rudnick, Ze{é}v},
TITLE = {Hecke theory and equidistribution for the quantization of linear maps of the torus},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {103},
YEAR = {2000},
NUMBER = {1},
PAGES = {47--77},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F25 (11G40 11L07 37D40 81Q50)},
MRNUMBER = {2001f:11065},
MRREVIEWER = {Peter Sarnak},
DOI = {10.1215/S0012-7094-00-10314-6},
ZBLNUMBER = {1013.81017},
} -
[KR1] Pär. Kurlberg and Z. Rudnick, "On the distribution of matrix elements for the quantum cat map," Ann. of Math., vol. 161, iss. 1, pp. 489-507, 2005.
@article {KR1, MRKEY = {2150390},
AUTHOR = {Kurlberg, P{ä}r and Rudnick, Ze{é}v},
TITLE = {On the distribution of matrix elements for the quantum cat map},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {161},
YEAR = {2005},
NUMBER = {1},
PAGES = {489--507},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {81Q50 (11F25 11L07 37D20 81Q20)},
MRNUMBER = {2006h:81091},
MRREVIEWER = {Dieter H. Mayer},
DOI = {10.4007/annals.2005.161.489},
ZBLNUMBER = {1082.81054},
} -
[KRR] Pär. Kurlberg, L. Rosenzweig, and Z. Rudnick, "Matrix elements for the quantum cat map: fluctuations in short windows," Nonlinearity, vol. 20, pp. 2289-2304, 2007.
@article{KRR,
author={Kurlberg, P{ä}r and Rosenzweig, L. and Rudnick, Ze{é}v},
TITLE={Matrix elements for the quantum cat map: fluctuations in short windows},
JOURNAL={Nonlinearity},
VOLUME={20},
YEAR={2007},
PAGES={2289--2304},
} -
[Mo] C. Moeglin, M. F. Vignéras, and J. L. Waldspurger, Correspondances de Howe sur un Corps $p$-Adique, New York: Springer-Verlag, 1987, vol. 1291.
@book{Mo,
author={M{\oe}glin, C. and Vign{é}ras, M. F. and Waldspurger, J. L.},
TITLE={Correspondances de {H}owe sur un Corps {$p$}-Adique},
SERIES={Lecture Notes in Math.},
VOLUME={1291},
YEAR={1987},
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
MRNUMBER={91f:11040},
} -
[Ne] M. Neuhauser, "An explicit construction of the metaplectic representation over a finite field," J. Lie Theory, vol. 12, iss. 1, pp. 15-30, 2002.
@article {Ne, MRKEY = {1885034},
AUTHOR = {Neuhauser, Markus},
TITLE = {An explicit construction of the metaplectic representation over a finite field},
JOURNAL = {J. Lie Theory},
FJOURNAL = {Journal of Lie Theory},
VOLUME = {12},
YEAR = {2002},
NUMBER = {1},
PAGES = {15--30},
ISSN = {0949-5932},
MRCLASS = {20C33 (20G05)},
MRNUMBER = {2003e:20014},
MRREVIEWER = {Cornelius Pillen},
ZBLNUMBER = {1026.22018},
} -
[No] S. Nonnenmacher, Private communication, 2003.
@misc{No,
author={Nonnenmacher, S.},
TITLE={private communication},
YEAR={2003},
} -
[RSO] A. M. F. Rivas, M. Saraceno, and A. M. Ozorio de Almeida, "Quantization of multidimensional cat maps," Nonlinearity, vol. 13, iss. 2, pp. 341-376, 2000.
@article {RSO, MRKEY = {1745370},
AUTHOR = {Rivas, A. M. F. and Saraceno, M. and Ozorio de Almeida, A. M.},
TITLE = {Quantization of multidimensional cat maps},
JOURNAL = {Nonlinearity},
FJOURNAL = {Nonlinearity},
VOLUME = {13},
YEAR = {2000},
NUMBER = {2},
PAGES = {341--376},
ISSN = {0951-7715},
CODEN = {NONLE5},
MRCLASS = {37J10 (37N20 81S30)},
MRNUMBER = {2002a:37086},
DOI = {10.1088/0951-7715/13/2/302},
ZBLNUMBER = {0952.37016},
} -
[RS] Z. Rudnick and P. Sarnak, "The behaviour of eigenstates of arithmetic hyperbolic manifolds," Comm. Math. Phys., vol. 161, iss. 1, pp. 195-213, 1994.
@article {RS, MRKEY = {1266075},
AUTHOR = {Rudnick, Ze{é}v and Sarnak, Peter},
TITLE = {The behaviour of eigenstates of arithmetic hyperbolic manifolds},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {161},
YEAR = {1994},
NUMBER = {1},
PAGES = {195--213},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {11F72 (11F32 11F37 81Q50)},
MRNUMBER = {95m:11052},
MRREVIEWER = {Jens Bolte},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104269797},
ZBLNUMBER = {0836.58043},
} -
[S] A. I. vSnirelcprimeman, "Ergodic properties of eigenfunctions," Uspehi Mat. Nauk, vol. 29, iss. 6(180), pp. 181-182, 1974.
@article {S, MRKEY = {0402834},
AUTHOR = {{Š}nirel{\cprime}man, A. I.},
TITLE = {Ergodic properties of eigenfunctions},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {29},
YEAR = {1974},
NUMBER = {6(180)},
PAGES = {181--182},
ISSN = {0042-1316},
MRCLASS = {58G99 (35P20)},
MRNUMBER = {53 \#6648},
MRREVIEWER = {D. Newton},
} -
[Winnie] W. W. C. Li, Number Theory with Applications, River Edge, NJ: World Scientific Publishing Co., 1996.
@book {Winnie, MRKEY = {1390759},
AUTHOR = {Li, W. C. Winnie},
TITLE = {Number Theory with Applications},
SERIES = {Series on University Mathematics},
NUMBER = {7},
PUBLISHER = {World Scientific Publishing Co.},
ADDRESS = {River Edge, NJ},
YEAR = {1996},
PAGES = {xii+229},
ISBN = {981-02-2226-2},
MRCLASS = {11-02 (11D79 11Fxx 11G25 11L07 11R52 68R10)},
MRNUMBER = {98b:11001},
MRREVIEWER = {M. Ram Murty},
ZBLNUMBER = {0849.11006},
} -
[Weil] A. Weil, "On some exponential sums," Proc. Nat. Acad. Sci. U. S. A., vol. 34, pp. 204-207, 1948.
@article {Weil, MRKEY = {0027006},
AUTHOR = {Weil, Andr{é}},
TITLE = {On some exponential sums},
JOURNAL = {Proc. Nat. Acad. Sci. U. S. A.},
VOLUME = {34},
YEAR = {1948},
PAGES = {204--207},
MRCLASS = {10.0X},
MRNUMBER = {10,234e},
MRREVIEWER = {H. D. Kloosterman},
ZBLNUMBER = {0032.26102},
} -
[Z] S. Zelditch, "Uniform distribution of eigenfunctions on compact hyperbolic surfaces," Duke Math. J., vol. 55, iss. 4, pp. 919-941, 1987.
@article {Z, MRKEY = {916129},
AUTHOR = {Zelditch, Steven},
TITLE = {Uniform distribution of eigenfunctions on compact hyperbolic surfaces},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {55},
YEAR = {1987},
NUMBER = {4},
PAGES = {919--941},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {58G25 (58G30)},
MRNUMBER = {89d:58129},
MRREVIEWER = {Alejandro Uribe},
DOI = {10.1215/S0012-7094-87-05546-3},
ZBLNUMBER = {0643.58029},
}
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