Skip to main content
SpringerLink
Log in

Higher Order Deformations of Hyperbolic Spectra

  • Conference paper
  • First Online:
Schrödinger Operators, Spectral Analysis and Number Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 348))

Abstract

This is an expanded writeup of a talk given by the second author at Erik Balslev’s 75th birthday conference on October 1–2, 2010 at Aarhus University. We summarize our work on Fermi’s golden rule and higher order phenomena for hyperbolic manifolds. A topic which occupied the last part of Erik Balslev’s research.

In memory of Erik Balslev

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H. Avelin, Deformation of \(\Gamma _0\)(5)-cusp forms. Math. Comp. 76(257), 361–384 (2007). MR2261026

    Google Scholar 

  2. E. Balslev, Spectral deformation of Laplacians on hyperbolic manifolds. Comm. Anal. Geom. 5(2), 213–247 (1997). MR1483980 (99j:58213)

    Google Scholar 

  3. E. Balslev, J.-M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions. Commun. Math. Phys. 22, 280–294 (1971) (English)

    Google Scholar 

  4. R. Bruggeman, N. Diamantis, Higher-order Maass forms. Alge. Num. Theory 6(7), 1409–1458 (2012). MR3007154

    Google Scholar 

  5. P. Buser, Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106 (Birkhäuser Boston Inc., Boston, MA, 1992). MR1183224 (93g:58149)

    Google Scholar 

  6. E. Balslev, A. Venkov, Selberg’s eigenvalue conjecture and the Siegel zeros for Hecke L-series, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups (Okayama-Kyoto, 1997) (2000), pp. 19–32

    Google Scholar 

  7. E. Balslev, A. Venkov, Spectral theory of Laplacians for Hecke groups with primitive character. Acta Math. 186(2), 155–217 (2001). MR1846029 (2002f:11057)

    Google Scholar 

  8. E. Balslev, A. Venkov, On the relative distribution of eigenvalues of exceptional Hecke operators and automorphic Laplacians. Algebra i Analiz 17(1), 5–52 (2005). MR2140673 (2007b:11071)

    Google Scholar 

  9. E. Balslev, A.B. Venkov, Perturbation of embedded eigenvalues of Laplacians, in Traces in Number Theory, Geometry and Quantum Fields (2007), pp. ix, 223 pp

    Google Scholar 

  10. E. Balslev, A.B. Venkov, The Weyl law for subgroups of the modular group. Geom. Funct. Anal. 8(3), 437–465 (1998). MR1631251 (99g:11069)

    Google Scholar 

  11. J.-M. Deshouillers, H. Iwaniec, R.S. Phillips, P. Sarnak, Maass cusp forms. Proc. Nat. Acad. Sci. 82(11), 3533–3534 (1985)

    Article  MathSciNet  Google Scholar 

  12. D.W. Farmer, S. Lemurell, Deformations of Maass forms. Math. Comput. 74, 1967–1982 (2005)

    Google Scholar 

  13. D. Goldfeld, The distribution of modular symbols, in Number Theory in Progress, vol. 2 (Zakopane-Kościelisko, 1997, 1999), pp. 849–865. MR2000g:11040

    Google Scholar 

  14. D. Goldfeld, Zeta functions formed with modular symbols, in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996, 1999), pp. 111–121. MR2000g:11039

    Google Scholar 

  15. E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung. Math. Ann. 112(1), 664–699 (1936). MR1513069

    Google Scholar 

  16. D.A. Hejhal, The Selberg Trace Formula for PSL(2, R). Vol. 2. Lecture Notes in Mathematics, vol. 1001 (Springer, Berlin, 1983). MR711197 (86e:11040)

    Google Scholar 

  17. V. Ivrii, 100 years of Weyl’s law. Bull. Math. Sci. 6(3), 379–452 (2016). MR3556544

    Google Scholar 

  18. H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd edn. Graduate Studies in Mathematics, vol. 53 (American Mathematical Society, Providence, RI, 2002). MR1942691 (2003k:11085)

    Google Scholar 

  19. J. Jorgenson, C. O’Sullivan, Unipotent vector bundles and higher-order non-holomorphic Eisenstein series. J. Théor. Nombres Bordeaux 20(1), 131–163 (2008). MR2434161

    Google Scholar 

  20. T. Kato, Perturbation theory for linear operators, 2nd edn. (Springer, Berlin, 1976). Grundlehren der Mathematischen Wissenschaften, Band 132. MR0407617 (53,11389)

    Google Scholar 

  21. P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions (Princeton Univ. Press, Princeton, NJ, 1976). Annals of Mathematics Studies, No. 87. MR0562288 (58,27768)

    Google Scholar 

  22. W. Luo, Nonvanishing of L-values and the Weyl law. Ann. of Math. (2) 154(2), 477–502 (2001). MR1865978 (2002i:11084)

    Google Scholar 

  23. H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121, 141–183 (1949). MR0031519 (11,163c)

    Google Scholar 

  24. S. Minakshisundaram, Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math. 1(3), 242–256 (1949) (en)

    Google Scholar 

  25. W. Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109(2), 265–305 (1992). MR93g:58151

    Google Scholar 

  26. Y.N. Petridis, Spectral deformations and Eisenstein series associated with modular symbols. Int. Math. Res. Not. 19, 991–1006 (2002). MR1903 327

    Google Scholar 

  27. Y.N. Petridis, On the singular set, the resolvent and Fermi’s golden rule for finite volume hyperbolic surfaces. Manuscripta Math. 82(3–4), 331–347 (1994). MR1265004 (95b:11053)

    Google Scholar 

  28. Y.N. Petridis, Spectral data for finite volume hyperbolic surfaces at the bottom of the continuous spectrum. J. Funct. Anal. 124(1), 61–94 (1994). MR1284603 (95g:11045)

    Google Scholar 

  29. Y.N. Petridis, M.S. Risager, Modular symbols have a normal distribution. Geom. Funct. Anal. 14(5), 1013–1043 (2004). MR2105951 (2005h:11101)

    Google Scholar 

  30. Y.N. Petridis, M.S. Risager, Dissolving of cusp forms: higher order Fermi’s golden rule. Mathematika 59(2), 269–301 (2013)

    Google Scholar 

  31. R.S. Phillips, P. Sarnak, On cusp forms for co-finite subgroups of PSL(2, R). Invent. Math. 80(2), 339–364 (1985). MR788414 (86m:11037)

    Google Scholar 

  32. R.S. Phillips, P. Sarnak, The Weyl theorem and the deformation of discrete groups. Comm. Pure Appl. Math. 38(6), 853–866 (1985). MR812352 (87f:11035)

    Google Scholar 

  33. R.S. Phillips, P. Sarnak, Perturbation theory for the Laplacian on automorphic functions. J. Amer. Math. Soc. 5(1), 1–32 (1992). MR1127079 (92g:11056)

    Google Scholar 

  34. W. Roelcke, Über die Wellengleichung bei Grenzkreisgruppen erster Art, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1953/1955, 159–267 (1953, 1956). MR18,476d

    Google Scholar 

  35. W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II (1966)

    Google Scholar 

  36. P. Sarnak, On cusp forms. II, Festschrift in honor of I. I. Piatetski- Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), 1990, pp. 237–250. MR1159118 (93e:11068)

    Google Scholar 

  37. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20, 47–87 (1956). MR0088511 (19,531g)

    Google Scholar 

  38. A. Selberg, Göttingen Lecture Notes in Collected Papers, vol. I (Springer, Berlin, 1989). With a foreword by K. Chandrasekharan. MR1117906 (92h:01083)

    Google Scholar 

  39. B. Simon, Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, 247–274 (1973). MR0353896 (50,6378)

    Google Scholar 

  40. A.B. Venkov, Artin-Takagi formula for the Selberg zeta function and the Roelcke hypothesis. Doklady Akademii Nauk SSSR 247(3), 540–543 (1979)

    Google Scholar 

  41. A.B. Venkov, A remark on the discrete spectrum of the automorphic laplacian for a generalized cycloidal subgroup of a general fuchsian group. J. Math. Sci. 52(3), 3016–3021 (1990)

    Article  MathSciNet  Google Scholar 

  42. A.B. Venkov, Spectral theory of automorphic functions and its applications, in Mathematics and Its Applications (Soviet Series), vol. 51 (Kluwer Academic Publishers Group, Dordrecht, 1990). Translated from the Russian by N. B. Lebedinskaya. MR1135112 (93a:11046)

    Google Scholar 

  43. A.B. Venkov, On a multidimensional variant of the Roelcke-Selberg conjecture. Rossiĭskaya Akademiya Nauk. Algebra i Analiz 4(3), 145–158 (1992)

    Google Scholar 

  44. H. Weyl, Ueber die asymptotische Verteilung der Eigenwerte, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1911, 110–117 (1911)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Yiannis N. Petridis

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Morten S. Risager

Authors
  1. Yiannis N. Petridis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morten S. Risager
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morten S. Risager .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Bonn, Germany

    Sergio Albeverio

  2. Hoejbjerg, Denmark

    Anindita Balslev

  3. Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Ricardo Weder

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Petridis, Y.N., Risager, M.S. (2021). Higher Order Deformations of Hyperbolic Spectra. In: Albeverio, S., Balslev, A., Weder, R. (eds) Schrödinger Operators, Spectral Analysis and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-030-68490-7_11

Download citation

Publish with us

Policies and ethics

Search

Navigation