Skip to main content
SpringerLink
Log in

Classical and quantum resonances for hyperbolic surfaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

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

Similar content being viewed by others

Notes

  1. The extension in a strip has been proved in [31, 32] before.

  2. Note that this was proved in the setting of hyperfunctions by Helgason [24].

  3. The case \(\lambda =-1/2\) is not really studied in [6, Lemma 6.8] but the analysis done there for \(\lambda \in -1/2+\mathbb {N}\) applies as well for \(\lambda =-1/2\) by using the explicit expression of the modified Bessel function \(K_{0}(z)\) as a converging series.

  4. The evenness of the expansion at \(t=0\) comes directly from the proof in [6, Lemma 6.8] when acting on functions, since the special functions appearing in the argument are Bessel functions that have even expansions.

References

  1. Borthwick, D.: Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, vol. 256, p. xii+355. Birkhäuser Boston, Inc., Boston (2007)

    MATH  Google Scholar 

  2. Borthwick, D., Judge, C., Perry, P.: Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Comment. Math. Helv. 80(3), 483–515 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2), 301–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bunke, U., Olbrich, M.: Fuchsian groups of the second kind and representations carried by the limit set. Invent. Math. 127(1), 127–154 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bunke, U., Olbrich, M.: Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. Math. 149, 627–689 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dyatlov, S., Faure, F., Guillarmou, C.: Power spectrum of the geodesic flow on hyperbolic manifolds. Anal. PDE 8, 923–1000 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dyatlov, S., Guillarmou, C.: Pollicott–Ruelle resonances for open systems. Ann. Henri Poincaré 17, 3089–3146 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Ec. Norm. Supér. 49, 543–577 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308(2), 325–364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Forster, O.: Lectures on Riemann Surfaces, GTM, vol. 81. Springer, Berlin (1981)

    Book  Google Scholar 

  11. Flaminio, L., Forni, G.: Invariant distributions and the averages for horocylic flows. Duke Math. J. 119(3), 465–526 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. Rend. Circ. Mat. Palermo Ser. II(63), 31–42 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Graham, C.R., Jenne, R., Mason, L., Sparling, G.A.: onformally invariant powers of the Laplacian. I. Existence. J. Lond. Math. Soc. (2) 46(3), 557–565 (1992)

    Article  MATH  Google Scholar 

  14. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Ann. Math. (2) 178(2), 687–773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129(1), 1–37 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guillarmou, C.: Resonances and scattering poles on asymptotically hyperbolic manifolds. Math. Res. Lett. 12(1), 103–119 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guillarmou, C., Guillopé, L.: The determinant of the Dirichlet-to-Neumann map for surfaces with boundary. Int. Math. Res. Not. IMRN 22, 26 (2007). (Art. ID rnm099)

  19. Guillarmou, C., Qing, J.: Spectral characterization of Poincaré–Einstein manifolds with infinity of positive Yamabe type. Int. Math. Res. Not. IMRN 2010(9), 1720–1740 (2009)

    MATH  Google Scholar 

  20. Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved \(2\)-manifolds. Topology 19, 301–312 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guillopé, L., Zworski, M.: Upper bounds on the number of resonances for non-compact Riemann surfaces. J. Funct. Anal. 129(2), 364–389 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guillopé, L., Zworski, M.: Scattering asymptotics for Riemann surfaces. Ann. Math. (2) 145(3), 597–660 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guillopé, L., Zworski, M.: The wave trace for Riemann surfaces. Geom. Funct. Anal. 9(6), 1156–1168 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Helgason, S.: Eigenspaces of the Laplacian; integral representations and irreducibility. J. Funct. Anal. 17(3), 328–353 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Joshi, M.S., Sá Barreto, A.: Inverse scattering on asymptotically hyperbolic manifolds. Acta Math. 184(1), 41–86 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mazzeo, R.: Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds. Am. J. Math. 113(1), 25–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mazzeo, R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260–310 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oshima, T., Sekiguchi, J.: Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math. 57(1), 1–81 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 136(3–4), 241–273 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  30. Patterson, S.J., Perry, P.A.: The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J. 106(2), 321–390 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pollicott, M.: On the rate of mixing of Axiom A flows. Invent. math. 81(3), 413–426 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ruelle, D.: Resonances of chaotic dynamical systems Phys. Rev. Lett. 56(5), 405–407 (1986)

    Article  MathSciNet  Google Scholar 

  33. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)

    MathSciNet  MATH  Google Scholar 

  34. van den Ban, E., Schlichtkrull, H.: Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces. J. Reine Angew. Math. 380, 108–165 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

C.G. is partially supported by ANR-13-BS01-0007-01 and ANR-13-JS01-0006. J.H. and T.W. acknowledge financial support by the DFG Grant DFG HI-412-12/1. We thank S. Dyatlov, F.Faure and M. Zworski for useful discussions, and Viet for suggesting the title.

Author information

Authors and Affiliations

  1. DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 rue d’Ulm, 75230, Paris cedex 05, France

    Colin Guillarmou

  2. Universität Paderborn, Warburgerstr. 100, 33098, Paderborn, Germany

    Joachim Hilgert & Tobias Weich

Authors
  1. Colin Guillarmou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joachim Hilgert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tobias Weich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tobias Weich.

Additional information

Communicated by Nalini Anantharaman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guillarmou, C., Hilgert, J. & Weich, T. Classical and quantum resonances for hyperbolic surfaces. Math. Ann. 370, 1231–1275 (2018). https://doi.org/10.1007/s00208-017-1576-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1576-5

Search

Navigation