Near-Horizon Modes and Self-adjoint Extensions of the Schrödinger Operator

  • A. P. Balachandran
  • A. R. de Queiroz
  • Alberto SaaEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 229)


We investigate the dynamics of scalar fields in the near-horizon exterior region of a Schwarzschild black hole. We show that  low-energy modes are typically long-living and might be considered as being confined near the black hole horizon. Such dynamics are effectively governed by a Schrödinger operator with infinitely many self-adjoint extensions parameterized by U(1), a situation closely resembling the case of an ordinary free particle moving on a semiaxis. Even though these different self-adjoint extensions lead to equivalent scattering and thermal processes, a comparison with a simplified model suggests a physical prescription to chose the pertinent self-adjoint extensions. However, since all extensions are in principle physically equivalent, they might be considered in equal footing for statistical analyses of near-horizon modes around black holes. Analogous results hold for any non-extremal, spherically symmetric, asymptotically flat black hole.



ARQ and AS thank the University of Zaragoza, where part of this work was carried on, for the warm hospitality. The authors acknowledge the financial support of CNPq and CAPES (ARQ and AS) and FAPESP (AS, Grant 2013/09357-9).


  1. 1.
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory. arXiv:1703.05448
  2. 2.
    E. Berti, V. Cardoso, A.O. Starinets, Quasinormal modes of black holes and black branes. Class. Quantum Grav. 26, 163001 (2009). [arXiv:0905.2975]ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    R.A. Konoplya, A. Zhidenko, Quasinormal modes of black holes: from astrophysics to string theory. Rev. Mod. Phys. 83, 793 (2011). [arXiv:1102.4014]ADSCrossRefGoogle Scholar
  4. 4.
    G. ’t Hooft, On the quantum structure of a black hole. Nucl. Phys. B256, 727 (1985)Google Scholar
  5. 5.
    G. Bonneau, J. Faraut, G. Valent, Self-adjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322 (2001). arXiv:quant-ph/0103153ADSCrossRefGoogle Scholar
  6. 6.
    D.M. Gitman, I.V. Tyutin, B.L. Voronov, Self-adjoint Extensions in Quantum Mechanics: General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials (Birkhuser, 2012)Google Scholar
  7. 7.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, 1974)Google Scholar
  8. 8.
    R.C. Tolman, The Principles of Statistical Mechanics (Dover, 2010)Google Scholar
  9. 9.
    C. Chirenti, A. Saa, J. Skakala, No asymptotically highly damped quasi-normal modes without horizons?. Phys. Rev. D 87, 044034 (2013). arXiv:1211.1046

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of PhysicsSyracuse UniversitySyracuseUSA
  2. 2.Instituto de FísicaUniversidade de BrasíliaBrasiliaBrazil
  3. 3.Department of Applied MathematicsUniversity of CampinasCampinasBrazil

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