Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-dxfhg Total loading time: 0.286 Render date: 2021-03-06T08:56:11.480Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Canadian Mathematical Bulletin Canadian Mathematical Bulletin

Article contents

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
Corresponding
E-mail address:
i.petridis@ucl.ac.uk
e-mail: nicole.raulf@math.univ-lille1.fr
risager@math.ku.dk
Save pdf (0.28 mb)
Rights & Permissions[Opens in a new window]

Abstract.

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

11F72 58G25 35P25 quantum limits Eisenstein series scattering 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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 30 *
View data table for this chart

* Views captured on Cambridge Core between 20th November 2018 - 6th March 2021. This data will be updated every 24 hours.

Access
3 Cited by

Linked content

Please note a has been issued for this article.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Quantum Limits of Eisenstein Series and Scattering States
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Quantum Limits of Eisenstein Series and Scattering States
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Quantum Limits of Eisenstein Series and Scattering States
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *