Communications in Mathematical Physics

, Volume 330, Issue 2, pp 771–799 | Cite as

Quasinormal Modes for Schwarzschild–AdS Black Holes: Exponential Convergence to the Real Axis

  • Oran GannotEmail author


We study quasinormal modes for massive scalar fields in Schwarzschild–anti-de Sitter black holes. When the mass-squared is above the Breitenlohner–Freedman bound, we show that for large angular momenta, , there exist quasinormal modes with imaginary parts of size exp(−/C). We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.


Black Hole Asymptotic Expansion Quasinormal Mode Hardy Inequality Meromorphic Continuation 
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  1. 1.
    Avis S.J., Isham C.J., Storey D.: Quantum field theory in anti-de Sitter space-time. Phys. Rev. D 18, 3565–3576 (1978)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachelot A.: The Dirac system on the anti-de Sitter universe. Commun. Math. Phys. 283, 126–167 (2008)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bailey P.B., Everitt W.N., Zettl A.: The SLEIGN2 Sturm–Liouville Code. ACM Trans. Math. Softw. 21, 143–192 (2001)CrossRefGoogle Scholar
  4. 4.
    Berti E., Cardoso V., Pani P.: Breit–Wigner resonances and the quasinormal modes of anti-de Sitter black holes. Phys. Rev. D 79, 101501 (2009)ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berti E., Cardoso V., Starinets A.: Quasinormal modes of black holes and black branes. Class. Quantum Gravity 26, 163001 (2009)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Breitenlohner P., Freedman D.: Stability in gauged extended supergravity. Ann. Phys. 144, 249 (1982)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dias Ó. et al.: On the nonlinear stability of asymptotically anti-de Sitter solutions. Class. Quantum Gravity 29, 235019 (2012)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Everitt, W.N., Kalf, H.: The Bessel differential equation and the Hankel transform. J. Comput. Appl. Math. 208, 3-19 (2007)Google Scholar
  9. 9.
    Festuccia G., Liu H.: A Bohr–Sommerfeld quantization formula for quasinormal frequencies of AdS black holes. Adv. Sci. Lett. 2, 221 (2009)CrossRefGoogle Scholar
  10. 10.
    Fujiié, S., Ramond, T.: Exact WKB analysis and the Langer modification with application to barrier top resonances. In: Howls, C. (ed.) Toward the exact WKB analysis of differential equation, linear or non-linear, pp. 15–31. Kyoto University Press, Japan (2000)Google Scholar
  11. 11.
    Gannot, O.: From quasimodes to resonances: exponentially decaying perturbations. arXiv:1305.2896
  12. 12.
    Gannot, O.: Quasinormal modes for Schwarzschild–AdS black holes: expansions of scattered waves (in preparation)Google Scholar
  13. 13.
    Grain J., Barrau A.: A WKB approach to scalar fields dynamics in curved space-time. Nucl. Phys. B 742, 253 (2006)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Hall R., Saad N.: Perturbation expansions for the spiked harmonic oscillator and related series involving the gamma function. J. Phys. A Math. Gen. 33, 5531 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Helffer, B., Francis, N.: Hypoelliptic estimates and spectral theory for Fokker–Planck operators and witten laplacians. Lecture Notes in Mathematics 1862. Springer, New York (2005)Google Scholar
  16. 16.
    Holzegel G.: On the massive wave equation on slowly rotating Kerr–AdS spacetimes. Commun. Math. Phys. 294(1), 169–197 (2009)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Holzegel G., Smulevici J.: Decay properties of Klein–Gordon fields on Kerr–AdS spacetimes. Commun. Pure Appl. Math. 66(11), 1751–1802 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Holzegel, G., Warnick, C.: Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes. arXiv:1209.3308
  19. 19.
    Holzegel, G., Smulevici, J.: Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes. arXiv:1303.5944
  20. 20.
    Hull T.E., Infeld L.: The factorization method. Rev. Mod. Phys. 23, 21–68 (1951)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Konoplya R., Zhidenko A.: Quasinormal modes of black holes: from astrophysics to string theory. Rev. Mod. Phys. 83, 793–836 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    Nakamura S., Stefanov P., Zworski M.: Resonance expansions of propagators in the presence of potential barriers. J. Funct. Anal 205, 180–205 (2002)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nedelec L.: Multiplicity of resonances in black box scattering. Can. Math. Bull. 47, 407–416 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ramond T.: Semiclassical study of quantum scattering on the line. Commun. Math. Phys. 177(1), 221–254 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Simon B.: Resonances in one dimension and Fredholm determinants. J. Funct. Anal. 178, 396–420 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Sjöstrand, J.: A trace formula and review of some estimates for resonances. Microlocal Anal. Spectr. Theory NATO ASI Ser. 490, 377–437 (1997)Google Scholar
  27. 27.
    Sjöstrand J., Zworski M.: Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4(4), 729–769 (1991)CrossRefzbMATHGoogle Scholar
  28. 28.
    Stefanov P.: Approximating resonances with the complex absorbing potential method. Commun. Partial Differ. Equ. 30(10–12), 1843–1862 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Tang S.H., Zworski M.: From quasimodes to resonances. Math. Res. Lett. 5(3), 261–272 (1998)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Warnick C.M.: The massive wave equation in asymptotically AdS spacetimes. Commun. Math. Phys. 294, 169–197 (2009)Google Scholar
  31. 31.
    Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics 1258. Springer, New York (1987)Google Scholar
  32. 32.
    Zettl, A.: Sturm–Liouville Theory, Mathematical Surveys and Monographs, vol. 121. AMS, Providence (2000)Google Scholar
  33. 33.
    Zworski M.: Resonances in physics and geometry. Not. Am. Math. Soc. 46(3), 319–328 (1999)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Zworski, M.: Semiclassical Analysis, Graduate Studies in Mathematics, AMS, Providence (2012)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Evans HallUniversity of CaliforniaBerkeleyUSA

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