Skip to main content
SpringerLink
Log in

Quantum Unique Ergodicity of Degenerate Eisenstein Series on GL(n)

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In the area of quantum chaos, it is of great interest to study the distribution of the \(L^2\)-mass of eigenfunctions of the Laplacian as eigenvalues tend to infinity. Luo and Sarnak first formulated and proved arithmetic quantum unique ergodicity for the continuous spectrum (spanned by Eisenstein series) of the hyperbolic Laplacian on \(SL(2,\mathbb {Z})\backslash \mathbb {H}\) by utilizing the sub-convexity bounds of L-functions associated to Maass cusp forms. In this paper, we build on Luo and Sarnak’s method and explore the structure properties of the constant terms of GL(n) Eisenstein series and extend their results to higher ranks. We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Buttcane, J.: Higher weight on GL(3), II: the cusp forms. Algebra Number Theory 12(10), 2237–2294 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buttcane, J.: Higher weight on \({\rm GL}(3)\). I: the Eisenstein series. Forum Math. 30(3), 681–722 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. de Verdière, Y.C.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)

    Article  ADS  MATH  Google Scholar 

  4. Goldfeld, D.: Automorphic Forms and L-Functions for the Group \({\rm GL}(n,\rm R)\), Cambridge Studies in Advanced Mathematics, vol. 9. Cambridge University Press, Cambridge (2015). (With an Appendix by Kevin A. Broughan, Paperback edition of the (2006) original)

    Google Scholar 

  5. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007). (Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM. (Windows, Macintosh and UNIX))

    MATH  Google Scholar 

  6. Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)

    Google Scholar 

  7. Jakobson, D.: Quantum unique ergodicity for Eisenstein series on \({\rm PSL}_2( \bf { Z}) \backslash {\rm PSL}_2( { R})\). Ann. Inst. Fourier (Grenoble) 44(5), 1477–1504 (1994)

    Article  MathSciNet  Google Scholar 

  8. Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, vol. 544. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  9. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. (2) 163(1), 165–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Luo, W.Z., Sarnak, P.: Quantum ergodicity of eigenfunctions on \({\rm PSL}_2( { Z}) \backslash \bf { H}^2\). Inst. Hautes Études Sci. Publ. Math. 81, 207–237 (1995)

    Article  MATH  Google Scholar 

  11. Mœglin, C., Waldspurger, J.-L.: Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1995). (Une paraphrase de l’Écriture [A paraphrase of Scripture])

    Book  MATH  Google Scholar 

  12. Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  13. Ramanujan, S.: Some formulae in the analytic theory of numbers. Messenger Math. 45, 81–84 (1916)

    MathSciNet  Google Scholar 

  14. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Silberman, L.: Quantum unique ergodicity on locally symmetric spaces: the degenerate lift. Can. Math. Bull. 58(3), 632–650 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Silberman, L., Venkatesh, A.: Entropy Bounds and Quantum Unique Ergodicity for Hecke Eigenfunctions on Division Algebras (2006). arXiv:1606.02267

  17. Silberman, L., Venkatesh, A.: On quantum unique ergodicity for locally symmetric spaces. Geom. Funct. Anal. 17(3), 960–998 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Soundararajan, K.: Quantum unique ergodicity for \({\rm SL}_2(\mathbb{Z})\backslash \mathbb{H}\). Ann. Math. (2) 172(2), 1529–1538 (2010)

    MathSciNet  Google Scholar 

  19. Šnirel’ man, A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6(180)), 181–182 (1974)

    MathSciNet  Google Scholar 

  20. Venkov, A.B.: The Selberg trace formula for \({\rm SL}(3, Z)\). Dokl. Akad. Nauk SSSR 228(2), 273–276 (1976)

    MathSciNet  Google Scholar 

  21. Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zelditch, S.: Selberg trace formulae and equidistribution theorems for closed geodesics and Laplace eigenfunctions: finite area surfaces. Mem. Am. Math. Soc. 96(465), vi+102 (1992)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Alex Kontorovich for suggesting the problem, many enlightening discussions, and careful readings of the various versions of this paper. I would like to thank Valentin Blomer for valuable comments. I also would like to thank the anonymous referee and editor for providing valuable suggestions and corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics, Maloney Hall, Boston College, Boston, MA, 02467, USA

    Liyang Zhang

Authors
  1. Liyang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liyang Zhang.

Additional information

Communicated by J. Marklof

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was supported by the following grants from Alex Kontorovich: NSF CAREER Grant DMS-1254788 and DMS-1455705.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L. Quantum Unique Ergodicity of Degenerate Eisenstein Series on GL(n). Commun. Math. Phys. 369, 1–48 (2019). https://doi.org/10.1007/s00220-019-03464-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03464-x

Search

Navigation