×

zbMATH — the first resource for mathematics

On the topography of Maass waveforms for \(\text{PSL}(2,\mathbb Z)\). (English) Zbl 0813.11035
The paper provides “a glimpse into arithmetic quantum chaos”. In particular, one finds some beautiful color contour maps for eigenfunctions \(\phi_ n\) of the Laplace operator on the fundamental domain of the modular group \(\text{SL}(2, \mathbb Z)\). Nodal lines \((\phi_ n = 0)\) are also plotted. Physicists have asked whether we can see ridges or scars in contour maps like these as the eigenvalue goes to infinity. And the question is whether the scarring is along geodesics. See, for example, M. C. Gutzwiller [Chaos in classical and quantum mechanics. New York etc.: Springer-Verlag (1990; Zbl 0727.70029)]. Recently Sarnak and others have addressed this question [see P. Sarnak, Arithmetic quantum chaos, The Schur lectures (1992). Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 183–236 (1995; Zbl 0831.58045)]. In studying the pictures the authors find, for example, that there do seem to be ridges but they do not seem to lie along geodesics.
Other questions are also asked. Look at the \(n\)-th eigenfunction \(\phi_ n\) of \(\Delta\) on \(L^ 2 (\text{SL}(2, \mathbb Z) \backslash H)\) and set \[ \nu_{n,A} (E) = \mu (A)^{-1} \mu \left\{ z \in A \mid \phi_ n (z) \in E \right\}. \] Does this converge nicely to a Gaussian distribution? The authors find that this seems to be true by studying histograms of the value distribution of \(\phi_ n\) for large \(n\).

MSC:
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F11 Holomorphic modular forms of integral weight
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML
References:
[1] Aurich R., Physica 48 pp 445– (1991)
[2] Balazs N. L., Phys. Rep. 143 pp 109– (1986)
[3] Balogh C., SIAM J. Appl. Math. 15 pp 1315– (1967) · Zbl 0157.12303
[4] Berry M., J. Phys. 10 pp 2083– (1977)
[5] Berry M., Proc. Royal Soc. London 423 pp 219– (1989)
[6] Berry M., Chaos et Physique Quantique, Les Houches 1989 pp 251– (1991)
[7] Billingsley P., Probability and Measure,, 2. ed. (1986) · Zbl 0649.60001
[8] Bogomolny E. B., Physica 31 pp 169– (1988)
[9] Bombieri E., C. R. Acad. Sci. Paris 304 pp 213– (1987)
[10] Brüning J., Math. Zeit. 158 pp 15– · Zbl 0349.58012
[11] Bump D., Number Theory, Trace Formulas, and Discrete Groups pp 49– (1989)
[12] Colin de Verdiere Y., Comm. Math. Phys. 102 pp 497– (1985) · Zbl 0592.58050
[13] Courant R., Methods of Mathematical Physics 1 (1953) · Zbl 0051.28802
[14] Courant R., Methods of Mathematical Physics 2 (1962) · Zbl 0099.29504
[15] Delande D., Comments on Atom. Molec. Phys. 25 pp 281– (1991)
[16] Deshouillers J.-M., Invent. Math. 70 pp 219– (1982) · Zbl 0502.10021
[17] Donnelly H., Invent. Math. 93 pp 161– (1988) · Zbl 0659.58047
[18] Donnelly H., J. Amer. Math. Soc. pp 333– (1990)
[19] Epstein C., Math. Zeit. 190 pp 113– (1985) · Zbl 0565.10026
[20] Erdélyi A., Higher Transcendental Functions (1953) · Zbl 0051.30303
[21] Esseen C. G., Acta Math. 77 (1945) · Zbl 0060.28705
[22] Feller W., An Introduction to Probabilility Theory and Its Applications,, 2. ed. (1971)
[23] Gelbart S., Analytic Properties of Automorphic L-functions (1988) · Zbl 0654.10028
[24] Ghosh A., J. Number Th. 17 pp 93– (1983) · Zbl 0511.10030
[25] Goldfeld D., Adv. in Math. 39 pp 240– (1981) · Zbl 0458.10022
[26] Good A., Math. Ann. 255 pp 523– (1981) · Zbl 0439.30031
[27] Good A., Math. Zeit. 183 pp 95– (1983) · Zbl 0497.10015
[28] Gutzwiller M. C., Chaos in Classical and Quantum Mechanics (1990) · Zbl 0727.70029
[29] Hedlund G. A., Amer. J. Math. 57 pp 668– (1937) · Zbl 0012.20301
[30] Hedlund G. A., Bull. Amer. Math. Soc. 45 pp 241– (1939) · Zbl 0020.40303
[31] Hejhal D. A., The Selberg Trace Formula for (1976) · Zbl 0347.10018
[32] Hejhal D. A., C. R. Acad. Sci. Paris 294 pp 637– (1982)
[33] Hejhal D. A., The Selberg Trace Formula for (1983) · Zbl 0543.10020
[34] Hejhal D. A., ”On polynomial approximations to Z(t) · Zbl 0790.11062
[35] Hejhal D. A., J. d’Analyse Math. 55 pp 59– (1990) · Zbl 0723.11039
[36] Hejhal, D. A. ”Eigenvalues of the Laplacian for PSL(2, Z): some new results and computational techniques”. International Symposium in Memory of Hua Loo-Keng. Edited by: Gong, S. vol. 1, pp.59–102. Beijing: Science Press. [Hejhal 1991], and Springer-Verlag, New York, Reprinted with [Hejhal 1992b] · Zbl 0805.11044
[37] Hejhal D. A., Internat. Math. Res. Notices, Duke Math. J. 66 pp 83– (1992) · Zbl 0761.11036
[38] Hejhal D. A., Memoirs Amer. Math. Soc. 469 (1992)
[39] Hejhal D. A., On Fourier coefficients of Maass waveforms for PSL(2, Z) (1992) · Zbl 0781.11020
[40] Heller E. J., Phys. Rev. Lett. 53 pp 1515– (1984)
[41] Heller E. J., Physica Scripta 40 pp 354– (1989) · Zbl 1063.81538
[42] Hobson E. W., The Theory of Spherical and Ellipsoidal Harmonics (1931) · Zbl 0004.21001
[43] Hoffstein J., ”Coefficients of Maass forms and the Siegel zero” (1992)
[44] Hopf E., Ergodentheorie (1937)
[45] Hörmander L., Acta Math. 121 pp 193– (1968) · Zbl 0164.13201
[46] Huntebrinker W., Ph.D. dissertation, Univ. Bonn, Bonner Math. Schriften 225 (1991)
[47] Iwaniec H., Modular Forms pp 157– (1984)
[48] Iwaniec H., Topics in Analytic Number Theory pp 221– (1985)
[49] Iwaniec H., Acta Arith. 56 pp 65– (1990)
[50] Kac M., Statistical Independence in Probability, Analysis, and Number Theory (1959) · Zbl 0088.10303
[51] Kahane J. P., Some Random Series of Functions,, 2. ed. (1985) · Zbl 0571.60002
[52] Kuznecov N. V., Math. USSR Sbornik 39 pp 299– (1981) · Zbl 0461.10017
[53] Kuznecov N. V., J. Soviet Math. 18 pp 398– (1982) · Zbl 0476.10040
[54] Kuznecov N. V., J. Soviet Math. 29 pp 1131– (1985) · Zbl 0564.10027
[55] Linnik Ju. V., The Dispersion Method in Binary Additive Problems (1963)
[56] Longuet-Higgins M. S., Philos. Trans. Royal Soc. London 249 pp 321– (1957) · Zbl 0077.12707
[57] Longuet-Higgins M. S., Philos. Trans. Royal Soc. London 250 pp 157– (1957) · Zbl 0078.32701
[58] Longuet-Higgins M. S., Proc. Symposia Applied Math. 13 pp 105– (1962)
[59] Maass H., Math. Ann. 121 pp 141– (1949)
[60] Maass H., Lectures on Modular Functions of One Complex Variable, (1983) · Zbl 0539.10021
[61] McDonald S., Phys. Rev. Lett. 42 pp 1189– (1979)
[62] McDonald S., Phys. Rev. 37 pp 3067– (1988)
[63] Mehta M. L., Random Matrices, (1967) · Zbl 0925.60011
[64] Moran P., An Introduction to Probability Theory (1968) · Zbl 0169.48602
[65] Moreno C., Number Theory Day pp 73– (1977)
[66] Moreno C., Math. Ann. 266 pp 233– (1983) · Zbl 0508.10014
[67] Moreno C., Canad. Math. Bull. 28 pp 405– (1985) · Zbl 0577.10027
[68] Ozorio de Almeida A. M., Hamiltonian Systems: Chaos and Quantization (1988)
[69] Petersson H., Modulfunktionen und quadratische Formen (1982)
[70] Phillips R. S., Invent. Math. 80 pp 339– (1985) · Zbl 0558.10017
[71] Rice S. O., Bell Sys. Tech. J. 23 pp 282– (1944) · Zbl 0063.06485
[72] Rudnick Z., ”Notes on arithmetic quantum chaos” (1992)
[73] Salem R., Acta Math. 91 pp 245– (1954) · Zbl 0056.29001
[74] Selberg A., J. Indian Math. Soc. 20 pp 47– (1956)
[75] Selberg A., Proc. Symposia Pure Math. (1965)
[76] Selberg A., Collected Papers 2 pp 47– (1991) · Zbl 0729.11001
[77] Shahidi F., Automorphic Forms and Analytic Number Theory pp 135– (1990)
[78] Shapiro M., Phys. Rev. Lett. 53 pp 1714– (1984)
[79] Shapiro M., Chem. Phys. Lett. 148 pp 177– (1988)
[80] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions (1971) · Zbl 0221.10029
[81] Shnirelman A. I., Uspekhi Mat. Nauk. 29 pp 181– (1974)
[82] Siegel C. L., Advanced Analytic Number Theory (1980) · Zbl 0478.10001
[83] Sinai Ya. G., Introduction to Ergodic Theory (1977)
[84] Smith R. A., Proc. Amer. Math. Soc. 82 pp 179– (1981)
[85] Takeuchi K., J. Fac. Sci. Univ. Tokyo 24 pp 201– (1977)
[86] Takeuchi K., J. Math. Soc. Japan 29 pp 91– (1977) · Zbl 0344.20035
[87] Takeuchi K., J. Math. Soc. Japan 35 pp 381– (1983) · Zbl 0517.20022
[88] Titchmarsh E., The Theory of the Riemann Zeta-Function (1951) · Zbl 0042.07901
[89] Tsang K. M., Ph.D. dissertation, in: The Distribution of the Values of the Riemann Zeta-Function (1984)
[90] Uhlenbeck K., Amer. J. Math. 98 pp 1059– (1976) · Zbl 0355.58017
[91] Venkov A. B., Spectral Theory of Automorphic Functions (1983) · Zbl 0527.10026
[92] Vinogradov A. I., J. Soviet Math. 36 pp 57– (1987) · Zbl 0609.10023
[93] Weyl H., Bull. Amer. Math. Soc. 56 pp 115– (1950) · Zbl 0041.21003
[94] Zelditch S., Duke Math. J. 55 pp 919– (1987) · Zbl 0643.58029
[95] Zelditch S., J. Funct. Anal. 97 (1991) · Zbl 0743.58034
[96] Zygmund A., Trigonometric Series, 1, 2. ed. (1959) · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.