Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T03:29:58.297Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
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