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Quantum Limits of Eisenstein Series and Scattering States

Published online by Cambridge University Press:  20 November 2018

Yiannis N. Petridis ,
Nicole Raulf  and
Morten S. Risager
Yiannis N. Petridis
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom e-mail: i.petridis@ucl.ac.uk
Nicole Raulf
Affiliation:
Laboratoire Paul Painlevé, U.F.R. de Mathematiques, Université Lille 1 Sciences et Technologies, 59 655 Villeneuve d’Ascq Cédex, France e-mail: e-mail: nicole.raulf@math.univ-lille1.fr
Morten S. Risager
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Denmark e-mail: risager@math.ku.dk
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Abstract.

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We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Keywords

11F7258G2535P25quantum limitsEisenstein seriesscattering poles

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 814 - 826
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Bui, H., Conrey, J. B., and Young, M., More than 41 of the zeros of the zeta function are on the critical line. arxiv:1002.4127v2.Google Scholar
[2] Colin de Verdiére, Y., Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102 (1985), no. 3, 497–502. http://dx.doi.org/10.1007/BF01209296 Google Scholar
[3] Dyatlov, S., Quantum ergodicity of Eisenstein functions at complex energies. arxiv:1109.3338v1.Google Scholar
[4] Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products. Fifth ed., Academic Press, Inc., San Diego, CA, 1994.Google Scholar
[5] Guillarmou, C. and Naud, F., Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds. arxiv:1107.2655v1.Google Scholar
[6] Holowinsky, R. and Soundararajan, K., Mass equidistribution for Hecke eigenforms. Ann. of Math. (2) 172 (2010), no. 2, 1517–1528.Google Scholar
[7] Iwaniec, H.. Spectral methods of automorphic forms. Second ed., Graduate Studies in Mathematics, 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002.Google Scholar
[8] Iwaniec, H. and Kowalski, E., Analytic number theory. American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
[9] Jakobson, D., Quantum unique ergodicity for Eisenstein series on PSL2(Z)nPSL2(R). Ann. Inst. Fourier (Grenoble) 44 (1994), no. 5, 1477–1504. http://dx.doi.org/10.5802/aif.1442 Google Scholar
[10] Koyama, S., Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Comm. Math. Phys. 215 (2000), no. 2, 477–486. http://dx.doi.org/10.1007/s002200000317 Google Scholar
[11] Lindenstrauss, E., Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163 (2006), no. 1, 165–219. http://dx.doi.org/10.4007/annals.2006.163.165 Google Scholar
[12] Luo, W. and Sarnak, P., Quantum ergodicity of eigenfunctions on PSL2(Z)nH2. Inst. Hautes E´ tudes Sci. Publ. Math. 81 (1995), 207–237.Google Scholar
[13] Michel, P. and Venkatesh, A., The subconvexity problem for GL2. Publ. Math. Inst. Hautes E´ tudes Sci. 111 (2010), 171–271.Google Scholar
[14] Meurman, T., On the order of the Maass L-function on the critical line. In: Number theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, 51, North-Holland, Amsterdan, 1990, pp. 325–354.Google Scholar
[15] Petridis, Y. and Sarnak, P., Quantum unique ergodicity for SL2(O)nH3 and estimates for L-functions. J. Evol. Equ. 1 (2001), no. 3, 277–290. http://dx.doi.org/10.1007/PL00001371 Google Scholar
[16] Selberg, A., Harmonic analysis, Göttingen lecture notes. In: Collected papers, Vol I, Springer Verlag, 1989, pp. 626–674.Google Scholar
[17] Shnirelman, A., Ergodic properties of eigenfunctions. (Russian) Uspehi Mat. Nauk 29 (1974), no. 6 (180), 181–182.Google Scholar
[18] Soundararajan, K., Quantum unique ergodicity for SL2(Z)nH. Ann. of Math. (2) 172 (2010), no. 2, 1529–1538.Google Scholar
[19] Titchmarsh, E., The theory of the Riemann zeta-function. Second Ed., The Clarendon Press, Oxford University Press, New York, 1986.Google Scholar
[20] Truelsen, J. L., Quantum unique ergodicity of Eisenstein series on the Hilbert modular group over a totally real field. Forum Mathematicum, 23, no. 5, 891–931.Google Scholar
[21] Zelditch, S., Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55 (1987), no. 4, 919–941. http://dx.doi.org/10.1215/S0012-7094-87-05546-3 Google Scholar
[22] Zelditch, S., Selberg trace formulae and equidistribution theorems for closed geodesics and Laplace eigenfunctions: finite area surfaces. Mem. Amer. Math. Soc. 96 (1992), no. 465.Google Scholar