Semiclassical limits for the hyperbolic plane

Duke Mathematical Journal

Volumn 108, Issue 3, 2001, Pages 449-509

  Wolpert, Scott A ^a  

^a NONE

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[No Author keywords available]

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EID: 0742317389     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-01-10833-8     Document Type: Article

References (53)

1. A. Alvarez-Parrilla, Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series, Trans. Amer. Math. Soc. 352 (2000), 5563-5582. MR CMP 1 675 233
2. R. Aurich, E. B. Bogomolny, and F. Steiner, Periodic orbits on the regular hyperbolic octagon, Phys. D 48 (1991), 91-101. MR 92b:58173
3. R. Aurich and F. Steiner, From classical periodic orbits to the quantization of chaos, Proc. Roy. Soc. London Ser. A 437 (1992), 693-714. MR 93k:81056
4. _, Statistical properties of highly excited quantum eigenstates of a strongly chaotic system, Phys. D 64 (1993), 185-214. MR 94e:81058
5. N. L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rep. 143 (1986), 109-240. MR 88h:58070
6. M. V. Berry, Quantum scars of classical closed orbits in phase space, Proc. Roy. Soc. London Ser. A 423 (1989), 219-231. MR 92c:81075
7. A. Borel, Automorphic Forms on $\SL\sb 2(\textbfR)$, Cambridge Tracts in Math. 130, Cambridge Univ. Press, Cambridge, 1997. MR 98j:11028
8. R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), 1-18. MR 57:12394
9. Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), 497-502. MR 87d:58145
10. J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), 219-288. MR 84m: 10015
11. _, "The nonvanishing of Rankin-Selberg zeta-functions at special points" in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984), Contemp. Math. 53, Amer. Math. Soc., Providence, 1986, 51-95. MR 88d:11047
12. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. I (based, in part, on notes left by Harry Bateman), McGraw-Hill, New York, 1954. MR 15:868a
13. _, Tables of Integral Transforms, Vol. II (based, in part, on notes left by Harry Bateman), McGraw-Hill, New York, 1954. MR 16:468c
14. A. Good, On various means involving the Fourier coefficients of cusp forms, Math. Z. 183 (1983), 95-129. MR 84g:10052
15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., Academic Press, Boston, 1994. MR 94g:00008
16. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, New York, 1979. MR 81i:10002
17. D. A. Hejhal, The Selberg Trace Formula for $\PSL(2, \textbfR)$, Vol. 2, Lecture Notes in Math. 1001, Springer, Berlin, 1983. MR 86e:11040
18. _, "Eigenfunctions of the Laplacian, quantum chaos, and computation" in Journées "Équations aux Dérivées Partielles" (Saint-Jean-de-Monts, France, 1995), École Polytech., Palaiseau, 1995, exp. no. 7. MR CMP 1 360 476
19. _, "On eigenfunctions of the Laplacian for Hecke triangle groups" in Emerging Applications of Number Theory (Minneapolis, 1996), IMA Vol. Math. Appl. 109, Springer, New York, 1999, 291-315. MR 2000f: 11063
20. D. A. Hejhal and B. N. Rackner, On the topography of Maass waveforms for $\PSL(2,\mathbfZ) $, Experiment. Math. 1 (1992), 275-305. MR 95f:11037
21. S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progr. Math. 13, Birkhäuser, Boston, 1981. MR 83g:43009
22. _, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr. 39, Amer. Math. Soc., Providence, 1994. MR 96h:43009
23. J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), 161-181. MR 95m:11048
24. A. E. Ingham, Some asymptotic formulae in the theory of numbers, J. London Math. Soc. 2 (1927), 202-208.
25. H. Iwaniec, Small eigenvalues of Laplacian for $\Gamma\sb 0(N)$, Acta Arith. 56 (1990), 65-82. MR 92h: 11045
26. _, The spectral growth of automorphic $L$-functions, J. Reine Angew. Math. 428 (1992), 139-159. MR 93g:11049
27. _, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoam., Madrid, 1995. MR 96f:11078
28. D. Jakobson, Equidistribution of cusp forms on $\PSL\sb 2(\mathbfZ)\backslash\PSL\sb 2 (\mathbfR)$, Ann. Inst. Fourier (Grenoble) 47 (1997), 967-984. MR 99c:11063
29. Y. Katznelson, An Introduction to Harmonic Analysis, 2d ed., Dover, New York, 1976. MR 54:10976
30. N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture: Sums of Kloosterman sums (in Russian), Mat. Sb. (N.S.) 111 (153), no. 3 (1980), 334-383., 479. MR 81m: 10053
31. S. Lang, S\SL \sb 2(\textbfR)S, Grad. Texts in Math. 105, Springer, New York, 1985. MR 86j:22018
32. N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972. MR 50:2568
33. W. Z. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on $\PSL \sb 2(\mathbfZ) \backslash \mathbfH\sp 2S, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207-237. MR 97f:11037
34. H. M. Macdonald, Zeroes of the Bessel functions, Proc. London Math. Soc. 30 (1899), 165-179.
35. T. Meurman, "On the order of the Maass $L$-function on the critical line" in Number Theory (Budapest, 1987), Vol. I, Colloq. Math. Soc. János Bolyai 51, North- Holland, Amsterdam, 1990, 325-354. MR 91m:11037
36. J.-P. Otal, Sur les fonctions propres du laplacien du disque hyperbolique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 161-166. MR 99e:35161
37. S. Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Math. 45 (1915), 81-84.
38. Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195-213. MR 95m: 11052
39. P. Sarnak, "Arithmetic quantum chaos" in The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, 1995, 183-236. MR 96d:11059
40. C. Schmit, "Quantum and classical properties of some billiards on the hyperbolic plane" in Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, 1991, 331-370. MR 94m:58176
41. A. I. Šnirelman, Ergodic properties of eigenfunctions (in Russian), Uspekhi. Mat. Nauk 29, no. 6 (1974), 181-182. MR 53:6648
42. A. Selberg, "On the estimation of Fourier coefficients of modular forms" in Theory of Numbers (Pasadena, Calif., 1963), Proc. Sympos. Pure Math. 8, Amer. Math. Soc., Providence, 1965, 1-15. MR 32:93
43. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, I, Springer, New York, 1985. MR 87f:22010
44. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2d ed., Oxford Univ. Press, New York, 1986. MR 88c: 11049
45. F. G. Tricomi, Integral Equations, Dover, New York, 1985. MR 86k:45001
46. A. B. Venkov, Spectral Theory of Automorphic Functions and Its Applications, Math. Appl. (Soviet Ser.) 51, Kluwer, Dordrecht, 1990. MR 93a:11046
47. D. V. Widder, The Laplace Transform, Princeton Math. Ser. 6, Princeton Univ. Press, Princeton, 1941. MR 3:232d
48. S. A. Wolpert, Asymptotic relations among Fourier coefficients of automorphic eigenfunctions, to appear in Trans. Amer. Math. Soc.
49. S. Zelditch, Pseudodifferential analysis on hyperbolic surfaces, J. Funct. Anal. 68 (1986), 72-105. MR 87j:58092
50. _, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919-941. MR 89d:58129
51. _, The averaging method and ergodic theory for pseudo-differential operators on compact hyperbolic surfaces, J. Funct. Anal. 82 (1989), 38-68. MR 91e:58194
52. -, Trace formula for compact $\Gamma \backslash \PSL \sb 2(\mathbbR)$ and the equidistribution theory of closed geodesics, Duke Math. J. 59 (1989), 27-81. MR 91 d: 11056
53. _, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, J. Funct. Anal. 97(1991), 1-49. MR 92h: 11046