Eigenfunctions of the laplacian, quantum chaos, and computation
Hejhal, Dennis A.
Journées équations aux dérivées partielles, (1995), p. 1-11 / Harvested from Numdam
Publié le : 1995-01-01
@article{JEDP_1995____A7_0,
     author = {Hejhal, Dennis A.},
     title = {Eigenfunctions of the laplacian, quantum chaos, and computation},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {1995},
     pages = {1-11},
     mrnumber = {1360476},
     zbl = {0948.35507},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_1995____A7_0}
}
Hejhal, Dennis A. Eigenfunctions of the laplacian, quantum chaos, and computation. Journées équations aux dérivées partielles,  (1995), pp. 1-11. http://gdmltest.u-ga.fr/item/JEDP_1995____A7_0/

[B] M. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A10 (1977) 2083-2091. | MR 58 #8961 | Zbl 0377.70014

[C] Y. Colin De Verdiere, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985) 497-502. | MR 87d:58145 | Zbl 0592.58050

[D] P. Deligne, La conjecture de Weil I, IHES Publ. Math. 53 (1974) 273-307. | Numdam | MR 49 #5013 | Zbl 0287.14001

[G] A. Good, Cusp forms and eigenfunctions of the Laplacian, Math. Ann. 255 (1981) 523-548. | MR 82i:10029 | Zbl 0439.30031

[H1] D.A. Hejhal, The Selberg Trace Formula for PSL (2,ℝ), vol. 1, Springer Lecture Notes in Mathematics 548 (1976), 516 pp. | MR 55 #12641 | Zbl 0347.10018

[H2] D.A. Hejhal, The Selberg Trace Formula for PSL (2,ℝ), vol. 2, Springer Lecture Notes in Mathematics 1001 (1983), 806 pp. | MR 86e:11040 | Zbl 0543.10020

[H3] D.A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, American Math. Soc. Memoir 469 (1992), 165 pp. | Zbl 0746.11025

[HA] D.A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for PSL (2,ℤ), Math. of Comp. 61 (1993) 245-267 and S11-S16. | MR 94a:11062 | Zbl 0781.11020

[HR] D.A. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL (2,ℤ), Experimental Math. 1 (1992) 275-305. | MR 95f:11037 | Zbl 0813.11035

[L] D.H. Lehmer, Note on the distribution of Ramanujan's tau function, Math. of Comp. 24 (1970) 741-743. | MR 43 #166 | Zbl 0214.30601

[LS] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL (2,ℤ) \ H1995, 38 pp. | Numdam | MR 97f:11037 | Zbl 0852.11024

[MS] C. Moreno and F. Shahidi, The fourth moment of Ramanujan's τ - function, Math. Ann. 266 (1983) 233-239. | MR 85i:11039 | Zbl 0508.10014

[P] Y. Petridis, On squares of eigenfunctions for the hyperbolic plane and a new bound on certain L-series, International Math. Res. Notices (1995) 111-127. | MR 96d:11058 | Zbl 0833.11021

[PS] R. Phillips and P. Sarnak, On cusp forms for cofinite subgroups of PSL (2,ℝ), Invent. Math. 80 (1985) 339-364. | MR 86m:11037 | Zbl 0558.10017

[R1] R.A. Rankin, Sums of powers of cusp form coefficients II, Math. Ann. 272 (1985) 593-600. | MR 87d:11032 | Zbl 0556.10018

[R2] R.A. Rankin, A family of newforms, Ann. Acad. Sci. Fennicae 10 (1985) 461-467. | MR 87b:11036 | Zbl 0595.10019

[R3] R.A. Rankin, Fourier coefficients of cusp forms, Math. Proc. Cambr. Phil. Soc. 100 (1986) 5-29, especially 26-27. | MR 87i:11060 | Zbl 0616.10020

[Ra] M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Th. and Dyn. Sys. 7 (1987) 267-288. | MR 88j:58103 | Zbl 0623.22008

[Re] F. Reza, An Introduction to Information Theory, Dover Publications, 1994, especially §§8.6 and 8.7. | MR 1298628 | Zbl 0925.94068

[RS] Z. Rudnick and P. Sarnak, The behavior of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994) 195-213. | MR 95m:11052 | Zbl 0836.58043

[S1] P. Sarnak, Asymptotic behavior of periodic orbits of the horocyclic flow and Eisenstein series, Comm. Pure Appl. Math. 34 (1981) 719-739. | MR 83m:58060 | Zbl 0501.58027

[S2] P. Sarnak, On cusp forms, American Math. Soc. Contemp. Math. 53 (1986) 393-407. | MR 87j:11047 | Zbl 0618.10018

[S3] P. Sarnak, Inner products of eigenfunctions, International Math. Res. Notices (1994) 251-260. | MR 95i:11039 | Zbl 0833.11020

[ShW] C. Shannon and W. Weaver, The Mathematical Theory of Communication, Univ. of I11. Press, 1949, especially pp. 55, 56. | MR 11,258e | Zbl 0041.25804

[Sh1] A. Shnirelman, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk. 29 (6) (1974) 181-182 [in Russian]. | MR 53 #6648

[Sh2] A. Shnirelman, On the asymptotic properties of eigenfunctions in the regions of chaotic motion, Addendum in the book : V.F. Lazutkin, KAM Theory and Semiclassical Asymptotics to Eigenfunctions, Springer-Verlag, 1993, pp. 313-337.

[St] G. Steil, Eigenvalues of the Laplacian and of the Hecke operators for PSL (2,ℤ), Hamburg DESY Preprint 94-028, 1994, 25 pp.

[T] J. Thomas, An Introduction to Statistical Communication Theory, Wiley, 1969, especially pages 481 (top), 561, 562. | Zbl 0202.17801

[V] A. Venkov, Spectral Theory of Automorphic Functions, Proc. Steklov Inst. Math. 153 (1982) 163 pp. | MR 85j:11060b | Zbl 0501.10029

[Z1] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987) 919-941. | MR 89d:58129 | Zbl 0643.58029

[Z2] S. Zelditch, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, J. Funct. Anal. 97 (1991) 1-49. | MR 92h:11046 | Zbl 0743.58034