- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Operators with spectral density increasing as Ω(E)∼eConst.E2

Erik Aurell1,* and Satya N. Majumdar2,†

  • *Contact author: eaurell@kth.se
  • †Contact author: satyanarayan.majumdar@cnrs.fr
PDF

Phys. Rev. Research 7, 043165 – Published 13 November, 2025

DOI: https://doi.org/10.1103/5bxn-pt83

Abstract

Motivated by a putative model of black holes as quantum objects, we consider what types of operators would have a corresponding spectrum. We find that there are indeed such operators, but of a rather unusual type, and for which the wave functions are only barely localized. We point out a tension between such almost delocalized states and black holes as compact objects.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (76)

  1. S. W. Hawking, Black hole explosions? Nature (London) 248, 30 (1974).
  2. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  3. R. M. Wald, On particle creation by black holes, Commun. Math. Phys. 45, 9 (1975).
  4. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, Chicago, Illinois, 1994).
  5. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  6. V. F. Mukhanov, On the origin of black-hole entropy, Found. Phys. 33, 271 (2003).
  7. G. ’t Hooft, Dimensional reduction in quantum gravity, arXiv:gr-qc/9310026.
  8. L. Susskind, The world as a hologram, J. Math. Phys. 36, 6377 (1995).
  9. R. Bousso, The holographic principle, Rev. Mod. Phys. 74, 825 (2002).
  10. C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77, 3288 (1996).
  11. A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80, 904 (1998).
  12. K. A. Meissner, Black-hole entropy in loop quantum gravity, Classical Quantum Gravity 21, 5245 (2004).
  13. S. D. Mathur, A proposal to resolve the black hole information paradox, Int. J. Mod. Phys. D 11, 1537 (2002).
  14. K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rep. 467, 117 (2008).
  15. M. Maggiore, Physical interpretation of the spectrum of black hole quasinormal modes, Phys. Rev. Lett. 100, 141301 (2008).
  16. E. Berti, V. Cardoso, and A. O Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  17. R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83, 793 (2011).
  18. J. L. Jaramillo, R. P. Macedo, and L. A. Sheikh, Pseudospectrum and black hole quasinormal mode instability, Phys. Rev. X 11, 031003 (2021).
  19. C. G. Callan and J. M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472, 591 (1996).
  20. J. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475, 679 (1996).
  21. G. Dvali and C. Gomez, Black hole’s quantum N-portrait arXiv:1112.3359.
  22. G. Dvali, Non-thermal corrections to Hawking radiation versus the information paradox, Fortschr. Phys. 64, 106 (2016).
  23. J. D. Bekenstein, Bekenstein-Hawking entropy, Scholarpedia 3, 7375 (2008).
  24. D. N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7, 203 (2005).
  25. S. Hossenfelder and L. Smolin, Conservative solutions to the black hole information problem, Phys. Rev. D 81, 064009 (2010).
  26. D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88, 015002 (2016).
  27. D. Marolf, The black hole information problem: Past, present and future, Rep. Prog. Phys. 80, 092001 (2017).
  28. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. 12 (2019) 063.
  29. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013.
  30. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  31. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  32. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  33. V. Balasubramanian, A. Lawrence, J. M. Magán, and M. Sasieta, Microscopic origin of the entropy of astrophysical black holes, Phys. Rev. Lett. 132, 141501 (2024).
  34. V. Balasubramanian, A. Lawrence, J. M. Magán, and M. Sasieta, Microscopic origin of the entropy of black holes in general relativity, Phys. Rev. X 14, 011024 (2024).
  35. R. Bousso and G. Penington, Islands far outside the horizon, J. High Energy Phys. 11 (2024) 164.
  36. Y. Matsuo, Universal structure of islands in evaporating black holes, J. High Energy Phys. 03 (2025) 068.
  37. A. Climent, R. Emparan, J. M. Magán, M. Sasieta, and A. Vilar López, Universal construction of black hole microstates, Phys. Rev. D 109, 086024 (2024).
  38. W. Mück, Black holes and Marchenko-Pastur distribution, Phys. Rev. D 109, 126001 (2024).
  39. C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, The black hole interior from non-isometric codes and complexity, J. High Energy Phys. 06 (2024) 155.
  40. J. van der Heijden and E. Verlinde, An operator algebraic approach to black hole information, J. High Energy Phys. 02 (2025) 207.
  41. V. Mukhanov, Are black holes quantised? Pis. Eksp. Teor. Fiz. 44, 50 (1986).
  42. J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, Phys. Lett. B 360, 7 (1995).
  43. V. Mukhanov, Quantum black holes, Jacob Bekenstein (World Scientific, Singapore, 2018), pp. 99–108.
  44. J. D. Bekenstein and A. Meisels, Einstein A and B coefficients for a black hole, Phys. Rev. D 15, 2775 (1977).
  45. G. H Hardy and S. Ramanujan, Asymptotic formulæ in combinatory analysis, Proc. London Math. Soc. s2-17, 75 (1918).
  46. P. Erdös and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8, 335 (1941).
  47. F. C. Auluck and D. S. Kothari, Statistical mechanics and the partitions of numbers, Math. Proc. Cambridge Philos. Soc. 42, 272 (1946).
  48. B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Math. Proc. Cambridge Philos. Soc. 47, 207 (1951).
  49. C. B. Haselgrove and H. N. V. Temperley, Asymptotic formulæ in the theory of partitions, Math. Proc. Cambridge Philos. Soc. 50, 225 (1954).
  50. M. N. Tran, M. V. N. Murthy, and R. K. Bhaduri, On the quantum density of states and partitioning an integer, Ann. Phys. 311, 204 (2004).
  51. A. Comtet, P. Leboeuf, and S. N. Majumdar, Level density of a Bose gas and extreme value statistics, Phys. Rev. Lett. 98, 070404 (2007).
  52. A. Comtet, S. N. Majumdar, and S. Ouvry, Integer partitions and exclusion statistics, J. Phys. A: Math. Theor. 40, 11255 (2007).
  53. A. Comtet, S. N. Majumdar, S. Ouvry, and S. Sabhapandit, Integer partitions and exclusion statistics: Limit shapes and the largest parts of Young diagrams, J. Stat. Mech.: Theory Exp. (2007) P10001.
  54. C. Echter, G. Maier, J.-D. Urbina, C. Lewenkopf, and K. Richter, Many-body density of states of bosonic and fermionic gases: A combinatorial approach, J. Phys. A: Math. Theor. 58, 065002 (2025).
  55. J. C. Peyen, L. V. Bogachev, and P. P. Martin, Boltzmann distribution on “short” integer partitions with power parts: Limit laws and sampling, Adv. Appl. Math. 159, 102739 (2024).
  56. H. H. Rugh, Dynamical approach to temperature, Phys. Rev. Lett. 78, 772 (1997).
  57. R. H. Swendsen and J.-S. Wang, Negative temperatures and the definition of entropy, Physica A 453, 24 (2016).
  58. P. Buonsante, R. Franzosi, and A. Smerzi, On the dispute between Boltzmann and Gibbs entropy, Ann. Phys. 375, 414 (2016).
  59. A. Puglisi, A. Sarracino, and A. Vulpiani, Temperature in and out of equilibrium: A review of concepts, tools and attempts, Phys. Rep. 709-710, 1 (2017).
  60. G. Gradenigo, S. Iubini, R. Livi, and S. N. Majumdar, Localization transition in the discrete nonlinear Schrödinger equation: Ensembles inequivalence and negative temperatures, J. Stat. Mech. (2021) 023201.
  61. E. Poisson and W. Israel, Inner-horizon instability and mass inflation in black holes, Phys. Rev. Lett. 63, 1663 (1989).
  62. A. Bonanno, S. Droz, W. Israel, and S. M. Morsink, Structure of the charged spherical black hole interior, Proc. R. Soc. London, Ser. A 450, 553 (1995).
  63. D. Christodoulou, On the global initial value problem and the issue of singularities, Classical Quantum Gravity 16, A23 (1999).
  64. M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Commun. Pure Appl. Math. 58, 445 (2005).
  65. M. Dafermos and J. Luk, The interior of dynamical vacuum black holes I: The c0-stability of the Kerr Cauchy horizon, Ann. of Math. 202, 309 (2025).
  66. M. De Clerck, S. A. Hartnoll, and J. E. Santos, Mixmaster chaos in an ADS black hole interior, J. High Energy Phys. 07 (2024) 202.
  67. P. Bueno, P. A. Cano, and R. A. Hennigar, Kasner epochs, eras and eons, Phys. Rev. D 110, L041503 (2024).
  68. E. Cáceres, Á. J. Murcia, A. K. Patra, and J. F. Pedraza, Kasner eons with matter: Holographic excursions to the black hole singularity, J. High Energy Phys. 12 (2024) 077.
  69. N. Zilberman, M. Casals, A. Ori, and A. C. Ottewill, Quantum fluxes at the inner horizon of a spinning black hole, Phys. Rev. Lett. 129, 261102 (2022).
  70. R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, Mass inflation without Cauchy horizons, Phys. Rev. Lett. 133, 181402 (2024).
  71. J. Wheeler, Geometrodynamics and the issue of the final state, in 1963 Houches Lectures, edited by B. S. DeWitt and C. M. DeWitt (Gordon and Breach, London, UK, 1963).
  72. J. Chakravarty, Overcounting of interior excitations: A resolution to the bags of gold paradox in ADS, J. High Energy Phys. 02 (2021) 027.
  73. M. Mézard and A. Montanari, Information, Physics, and Computation (Oxford University Press, Oxford, UK, 2009).
  74. P. Charbonneau, E. Marinari, M. Mézard, G. Parisi, F. Ricci-Tersenghi, G. Sicuro, and F. Zamponi, Spin Gass Theory and Far Beyond (World scientific, Singapore, 2023).
  75. M. Mézard and G. Parisi, The cavity method at zero temperature, J. Stat. Phys. 111, 1 (2003).
  76. M. Kac, Can one hear the shape of a drum? Am. Math. Mon. 73, 1 (1966).

Outline

Information

Phys. Rev. Research 7, 043165– Published 13 November, 2025
  • Received 10 April 2025
  • Accepted 12 October 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation