Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Massive waves gravitationally bound to static bodies
HTML articles powered by AMS MathViewer

by Ethan Sussman;
Proc. Amer. Math. Soc. 152 (2024), 3319-3337
DOI: https://doi.org/10.1090/proc/16761
Published electronically: June 5, 2024

Abstract:

We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the Klein–Gordon equation having Schwartz initial data. In fact, there exist infinitely many independent such solutions. The proof is a variational argument based on the long range nature of the effective potential. We give two sets of test functions which serve to verify the hypothesis of the variational argument. One set consists of cutoff versions of the hydrogen bound states and is used to prove the existence of eigenvalues near the hydrogen spectrum.
References
  • Y. Angelopoulos, S. Aretakis, and D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes, Adv. Math. 323 (2018), 529–621. MR 3725885, DOI 10.1016/j.aim.2017.10.027
  • Y. Angelopoulos, S. Aretakis, and D. Gajic, A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes, Ann. PDE 4 (2018), no. 2, Paper No. 15, 120. MR 3859608, DOI 10.1007/s40818-018-0051-2
  • Juan Barranco et al., Schwarzschild scalar wigs: spectral analysis and late time behavior, Phys. Rev. D 89 (2014). DOI 10.1103/PhysRevD.89.083006.
  • Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
  • Lior M. Burko and Gaurav Khanna, Universality of massive scalar field late-time tails in black-hole spacetimes, Phys. Rev. D (3) 70 (2004), no. 4, 044018, 8. MR 2114399, DOI 10.1103/PhysRevD.70.044018
  • Roland Donninger, Wilhelm Schlag, and Avy Soffer, A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta, Adv. Math. 226 (2011), no. 1, 484–540. MR 2735767, DOI 10.1016/j.aim.2010.06.026
  • Roland Donninger, Wilhelm Schlag, and Avy Soffer, On pointwise decay of linear waves on a Schwarzschild black hole background, Comm. Math. Phys. 309 (2012), no. 1, 51–86. MR 2864787, DOI 10.1007/s00220-011-1393-8
  • A. Erdélyi, Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. (N.S.) 24 (1960), 235–250 (1961). MR 123751
  • A. Erdélyi, Asymptotic solutions of differential equations with transition points or singularities, J. Mathematical Phys. 1 (1960), 16–26. MR 111915, DOI 10.1063/1.1703631
  • Brian C. Hall, Quantum theory for mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, New York, 2013. MR 3112817, DOI 10.1007/978-1-4614-7116-5
  • Peter Hintz, A sharp version of Price’s law for wave decay on asymptotically flat spacetimes, Comm. Math. Phys. 389 (2022), no. 1, 491–542. MR 4365146, DOI 10.1007/s00220-021-04276-8
  • Peter Hintz, Linear waves on asymptotically flat spacetimes. I, arXiv:2302.14647 [math.AP], (2023).
  • Shahar Hod and Tsvi Piran, Late-time tails in gravitational collapse of a self-interacting (massive) scalar-field and decay of a self-interacting scalar hair, Phys. Rev. D 58 (1998). DOI 10.1103/PhysRevD.58.044018.
  • H. A. Kramers, Quantum mechanics, North-Holland Publishing Co., Amsterdam; Interscience Publishers, Inc., New York, 1957. MR 87487
  • Hiroko Koyama and Akira Tomimatsu, Asymptotic tails of massive scalar fields in a Schwarzschild background, Phys. Rev. D (3) 64 (2001), no. 4, 044014, 8. MR 1853983, DOI 10.1103/PhysRevD.64.044014
  • Hiroko Koyama and Akira Tomimatsu, Slowly decaying tails of massive scalar fields in spherically symmetric spacetimes, Phys. Rev. D (3) 65 (2002), no. 8, 084031, 8. MR 1899216, DOI 10.1103/PhysRevD.65.084031
  • Roman A. Konoplya, Alexander Zhidenko, and Carlos Molina, Late time tails of the massive vector field in a black hole background, Phys. Rev. D 75 (2007). DOI 10.1103/PhysRevD.75.084004.
  • L. D. Landau and E. M. Lifshitz, Quantum mechanics: non-relativistic theory. Course of Theoretical Physics, Vol. 3, Addison-Wesley Series in Advanced Physics, Pergamon Press, Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, MA, 1958. Translated from the Russian by J. B. Sykes and J. S. Bell. MR 93319
  • Shi-Zhuo Looi, Pointwise decay for the wave equation on nonstationary spacetimes, J. Math. Anal. Appl. 527 (2023), no. 1, Paper No. 126939, 44. MR 4598501, DOI 10.1016/j.jmaa.2022.126939
  • Richard B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 85–130. MR 1291640
  • Richard B. Melrose, Geometric scattering theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995. MR 1350074
  • Katrina Morgan, Wave Decay in the Asymptotically Flat Stationary Setting, ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. MR 4035543
  • Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu, Price’s law on nonstationary space-times, Adv. Math. 230 (2012), no. 3, 995–1028. MR 2921169, DOI 10.1016/j.aim.2012.03.010
  • Katrina Morgan and Jared Wunsch, Generalized Price’s law on fractional-order asymptotically flat stationary spacetimes, arXiv:2105.02305 [math.AP] 2021.
  • Simon Pasternack, On the mean value of $r^s$ for Keplerian systems, Proc. Natl. Acad. Sci. USA 23 (1937), no. 2, 91–94.
  • Richard H. Price, Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations, Phys. Rev. D (3) 5 (1972), 2419–2438. MR 376103, DOI 10.1103/PhysRevD.5.2419
  • Richard H. Price, Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D (3) 5 (1972), 2439–2454. MR 376104, DOI 10.1103/PhysRevD.5.2439
  • Federico Pasqualotto, Yakov Shlapentokh-Rothman, and Maxime Van de Moortel, The asymptotics of massive fields on stationary spherically symmetric black holes for all angular momenta, arXiv:2303.17767 [gr-qc], (2023).
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
  • Michael Reed and Barry Simon, Methods of modern mathematical physics, vol. 1, Academic Press, 1980, Revised and enlarged edition.
  • Yakov Shlapentokh-Rothman, Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes, Comm. Math. Phys. 329 (2014), no. 3, 859–891. MR 3212872, DOI 10.1007/s00220-014-2033-x
  • Ethan Sussman, Hydrogen-like Schrödinger operators at low energies, arXiv:2204.08355 [math.AP], (2022).
  • Daniel Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math. 135 (2013), no. 2, 361–401. MR 3038715, DOI 10.1353/ajm.2013.0012
  • András Vasy, A minicourse on microlocal analysis for wave propagation, Asymptotic analysis in general relativity, London Math. Soc. Lecture Note Ser., vol. 443, Cambridge Univ. Press, Cambridge, 2018, pp. 219–374. MR 3792086
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2020): 35P05, 35P15, 81Q05
  • Retrieve articles in all journals with MSC (2020): 35P05, 35P15, 81Q05
Bibliographic Information
  • Ethan Sussman
  • Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
  • MR Author ID: 1511280
  • Email: ethanws@stanford.edu
  • Received by editor(s): November 15, 2022
  • Received by editor(s) in revised form: May 18, 2023, December 1, 2023, and December 8, 2023
  • Published electronically: June 5, 2024
  • Communicated by: Tanya Christiansen
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3319-3337
  • MSC (2020): Primary 35P05; Secondary 35P15, 81Q05
  • DOI: https://doi.org/10.1090/proc/16761
  • MathSciNet review: 4767265