- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Schrödinger-Newton solitons with axial symmetry

A. Flores, C. Stegner, and S. S. Chabysheva

J. R. Hiller

PDF

Phys. Rev. D 112, 044055 – Published 26 August, 2025

DOI: https://doi.org/10.1103/hjtl-ypxt

Abstract

We solve the Schrödinger-Newton problem of Newtonian gravity coupled to a nonrelativistic scalar particle for solutions with axial symmetry. The gravitational potential is driven by a mass density assumed to be proportional to the probability density of the scalar. Unlike related calculations for condensates of ultralight dark matter or boson stars, no assumption of spherical symmetry is made for the effective gravitational potential. Instead, the potential has only axial symmetry, consistent with the axial symmetry of the particle’s probability density for eigenstates of Lz. With total angular momentum no longer a good quantum number, there are, in general, contributions from a range of partial waves. This permits us to study the partial-wave content of self-consistent solutions of the Schrödinger-Newton system.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. D. A. Taylor, S. S. Chabysheva, and J. R. Hiller, Gravitational soliton solutions to self-coupled Klein-Gordon and Schrödinger equations, Phys. Rev. D 107, 124049 (2023).
  2. R. Ruffini and S. Bonazzola, Systems of self-gravitating particles in general relativity and the concept of an equation of state, Phys. Rev. 187, 1767 (1969).
  3. I. M. Moroz, R. Penrose, and K. P. Tod, Spherically symmetric solutions of the Schrödinger–Newton equations, Classical Quantum Gravity 15, 2733 (1998).
  4. D.H Bernstein, E. Giladi, and K. R. W. Jones, Eigenstates of the gravitational Schrödinger equation, Mod. Phys. Lett. A 13, 2327 (1998).
  5. K. P. Tod, The ground state energy of the Schrödinger–Newton equation, Phys. Lett. A 280, 173 (2001).
  6. R. Harrison, I. M. Moroz, and K. P. Tod, A numerical study of the Schrödinger–Newton equations, Nonlinearity 16, 101 (2002).
  7. J. Luna Zagorac, I. Sands, N. Padmanabhan, and R. Easther, Schrödinger–Poisson solitons: Perturbation theory, Phys. Rev. D 105, 103506 (2022).
  8. I. Álvarez-Rios and F. S. Guzmán, Spherical solutions of the Schrödinger-Poisson system with core-tail structure, Phys. Rev. D 108, 063519 (2023).
  9. B. Schupp and J. J. van der Bij, An axially-symmetric Newtonian boson star, Phys. Lett. B 366, 85 (1996).
  10. V. Silveira and C. M. G. de Sousa, Boson star rotation: A Newtonian approximation, Phys. Rev. D 52, 5724 (1995).
  11. S. Yoshida and Y. Eriguchi, Rotating boson stars in general relativity, Phys. Rev. D 56, 762 (1997); New static axisymmetric and nonvacuum solutions in general relativity: Equilibrium solutions of boson stars, 55, 1994 (1997).
  12. F. E. Schunck and E. W. Mielke, Rotating boson star as an effective mass torus in general relativity, Phys. Lett. A 249, 389 (1998).
  13. R. Harrison, A numerical study of the Schrödinger–Newton equations, Ph.D. Thesis, University of Oxford, 2001 (unpublished).
  14. F. S. Guzmán and L. A. Ureña-López, Evolution of the Schrodinger-Newton system for a selfgravitating scalar field, Phys. Rev. D 69, 124033 (2004).
  15. M. Alcubierre, J. Barranco, A. Bernal, J. C. Degollado, A. Diez-Tejedor, M. Megevand, D. Núñez, and O. Sarbach, ℓ-boson stars, Classical Quantum Gravity 35, 19LT01 (2018); Dynamical evolutions of ℓ-boson stars in spherical symmetry, 36, 215013 (2019); Boson stars and their relatives in semiclassical gravity, Phys. Rev. D 107, 045017 (2023).
  16. F. S. Guzmán and L. A. Ureña-López, Gravitational atoms: General framework for the construction of multistate axially symmetric solutions of the Schrödinger–Poisson system, Phys. Rev. D 101, 081392 (2020).
  17. V. Jaramillo, N. Sanchis-Gual, J. Barranco, A. Bernal, J. C. Degollado, C. Herdeiro, and D. Núñez, Dynamical ℓ-boson stars: Generic stability and evidence for nonspherical solutions, Phys. Rev. D 101, 124020 (2020).
  18. A. S. Dmitriev, D. G. Levkov, A. G. Panin, E. K. Pushnaya, and I. I. Tkachev, Instability of rotating Bose stars, Phys. Rev. D 104, 023504 (2021).
  19. E. C. Nambo, A. A. Roque, and O. Sarbach, Are nonrelativistic ground state l-boson stars only stable for l=0 and l=1?, Phys. Rev. D 108, 124065 (2023).
  20. H. Y. Schive, T. Chiueh, and T. Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nat. Phys. 10, 496 (2014).
  21. A. H. Guth, M. P. Hertzberg, and C. Prescod-Weinstein, Do dark matter axions form a condensate with long-range correlation?, Phys. Rev. D 92, 103513 (2015).
  22. P. H. Chavanis and L. Delfini, Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: II. Numerical results, Phys. Rev. D 84, 043532 (2011); P. H. Chavanis, Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results, 84, 043531 (2011).
  23. D. J. Kaup, Klein-Gordon geon, Phys. Rev. 172, 1331 (1998).
  24. R. Ferrell and M. Gleiser, Gravitational atoms: Gravitational radiation from excited boson stars, Phys. Rev. D 40, 2524 (1989).
  25. L. Diósi, Gravitation and quantum-mechanical localization of macro-objects, Phys. Lett. 105A, 199 (1984).
  26. R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativ. Gravit. 28, 581 (1996).
  27. R. Bahrami, A. Großardt, S. Donadi, and A. Bassi, The Schrödinger–Newton equation and its foundations, New J. Phys. 16, 115007 (2014).
  28. M. Di Mauro, S. Esposito, and A. Naddeo, A road map for Feynman’s adventures in the land of gravitation, Eur. Phys. J. H 46, 22 (2021).
  29. R. D. Lehn, S. S. Chabysheva, and J. R. Hiller, Klein–Gordon equation in curved space-time, Eur. J. Phys. 39, 045405 (2018).
  30. D. Giulini and A. Großardt, The Schrödinger–Newton equation as a nonrelativistic limit of self-gravitating Klein–Gordon and Dirac fields, Classical Quantum Gravity 29, 215010 (2012).
  31. D. Brizuela and A. Duran-Cabacés, Relativistic effects on the Schrödinger–Newton equation, Phys. Rev. D 106, 124038 (2022).
  32. A. Flores, C. Stegner, S. S. Chabysheva, and J. R. Hiller, Data Repository for the University of Minnesota (DRUM), 2025 https://hdl.handle.net/11299/275132.

Outline

Information

Phys. Rev. D 112, 044055– Published 26 August, 2025
  • Received 26 December 2024
  • Accepted 28 July 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 SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation