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The behaviour of eigenstates of arithmetic hyperbolic manifolds. (English) Zbl 0836.58043

Summary: We study some problems arising from the theory of quantum chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
81Q50 Quantum chaos
11F32 Modular correspondences, etc.
11F37 Forms of half-integer weight; nonholomorphic modular forms
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[1] Arnold, V., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968 · Zbl 0167.22901
[2] Aurich, R., Steiner, F.: Energy level statistics of the Hadamard-Gutzwiller ensemble. Physica D43, 155–180 (1990) · Zbl 0704.58039
[3] Aurich, R., Steiner, F.: Statistical properties of highly excited quantum eigenstates of a strongly chaotic system. Preprint DESY 92-091, June 1992 · Zbl 0773.58020
[4] Bogomolny, E.B., Georgeot, B., Giannoni, M., Schmidt, C.: Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett.69, 1477–1480 (1992) · Zbl 0968.81514
[5] Cassels, J.W.: Rational quadratic forms. New York: Academic Press (1978) · Zbl 0395.10029
[6] Colin de Verdiere, Y.: Ergodicité et functions propre du laplacien. Commun. Math. Phys.102, 497–502 (1985) · Zbl 0592.58050
[7] Conway, J., Sloane, N.: Sphere packings, lattices and groups. Berlin, Heidelberg, New York: Springer 1988 · Zbl 0634.52002
[8] Eichler, M.: Lectures on modular correspondences. Tata Institute9, 1955 · Zbl 0068.06601
[9] Hejhal, D., Rackner, B.: On the topography of Maass waveforms forPSL(2,Z); Experiments and heuristics. Experimental Math.1, 275–306 (1992) · Zbl 0813.11035
[10] Helgason, S.: Groups and Geometric Analysis. New York: Academic Press (1984) · Zbl 0543.58001
[11] Heller, E.J.: In: Chaos and Quantum Phyiscs, Les Houches 1989 (ed. by M.J. Giannoni, A. Voros, and J. Zinn-Justin), Amsterdam: North-Holland, 1991, pp. 549–661
[12] Hormander, L.: The Analysis of Linear Partial Differential Operators, Vol.I–IV, Berlin, Heidelberg, New York: Springer-Verlag, 1985
[13] Howe, R., Piatetski-Shapiro, I.: A counter-example to the ”generalized Ramanujan conjecture” for (quasi-) split groups. Proc. Symp. in Pure Math. vol.33, Amer. Math. Soc. 315–322 (1979) · Zbl 0423.22018
[14] Iwaniec, H., Sarnak, P.:L norms of eigenfunctions of arithmetic surfaces. Preprint · Zbl 0833.11019
[15] Landau, E.: Elementary Number Theory. New York: Chelsea Pub. Co., 1958 · Zbl 0079.06201
[16] Luo, W., Sarnak, P.: Number variance for arithmetic hyperbolic surfaces. To appear in Commun. Math. Phys. · Zbl 0797.58069
[17] Maass, H.: Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Annalen138, 287–315 (1959) · Zbl 0089.06102
[18] Sarnak, P.: Arithmetic Quantum Chaos. Schur Lectures, Tel Aviv 1992, preprint
[19] Schnirelman, A.: Usp. Mat. Nauk29, 181–182 (1974)
[20] Selberg, A.: Gottingen Lectures. In: Collected Works, vol.1, Berlin, Heidelberg, New York: Springer-Verlag
[21] Shintani, T.: On construction of holomorphic forms of half integral weight. Nagoya Math. J.58, 83–126 (1975) · Zbl 0316.10016
[22] Seeger, A., Sogge, C.D.: Bounds for eigenfunctions of differential operators. Indiana University Math. J.38, 669–682 (1989) · Zbl 0703.35133
[23] Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J.55, 919–941 (1987) · Zbl 0643.58029
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