Skip to main content
SpringerLink
Log in

Initial condition dependence and wave function confinement in the Schrödinger–Newton equation

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

In this work we study the dynamics of the Schrödinger–Newton (SN) equation upon different choices of initial conditions. Setting up superpositions of Gaussian-like wave packages, a very rich behavior for the critical mass as a function of the parameters of the problem is observed. We find that, for certain values of the parameters, the critical mass is smaller than the critical mass for the system whose initial condition is a single Gaussian wave package, which was the situation previously investigated in the literature. This opens a possibility that more complex initial conditions could in fact produce a significant decrease in the value of the critical mass, which could imply that the SN approach could be tested experimentally. Our conclusions rely on both numerical and analytic estimates. Furthermore, a detailed numerical study is carried out in order to investigate finite-size effects on the simulations, refining earlier results already published. In order to facilitate the reproducibility of our results, a detailed description of our numerical methods has been included in the presentation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Any list of references on this is doomed to be very incomplete, but we mention the following works. For standard treatments of String Theory and its relation to the quantization of the gravitational field, see [30, 45], or [5] for a more recent monograph. Attempts at constructing semi-realistic models out of String Theory and the related problems of stabilization and de Sitter vacua can be found in [2, 9, 16, 19–22, 29, 33, 53] and references therein, while connections with cosmology are explored in [4] and their references. For approaches based on Loop Quantum Gravity, see [48, 49], or the recent survey [13], and references therein. For approaches based on Twistors, see [40, 41, 44] and references therein.

References

  1. Anastopoulos, C., Hu, B.L.: Problems with the Newton–Schrödinger equations? New J. Phys. 16, 085007 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  2. Andriot, D., Goi, E., Minasian, R., Petrini, M.: Supersymmetry breaking branes on solvmanifolds and de sitter vacua in string theory. J. High Energy Phys. 2011(5), 1–65 (2011)

    Article  MathSciNet  Google Scholar 

  3. Arndt, M., Hornberger, K., Zeilinger, A.: Probing the Limits of the Quantum World. Phys. World. 18, 35–40 (2005)

  4. Baumann, D., McAllister, L.: Inflation and String Theory. arXiv preprint arXiv: arXiv:1404.2601

  5. Becker, K., Becker, M., Schwarz, J.H.: String Theory and M-Theory: A Modern Introduction. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  6. Benguria, R., Brézis, H., Lieb, E.H.: The Thomas–Fermi–von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79(2), 167–180 (1981)

    Article  MATH  ADS  Google Scholar 

  7. Blau, M., Theisen, S.: String theory as a theory of quantum gravity: a status report. Gen. Relativ. Gravit. 41(4), 743–755 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Burrage, C., Copeland, E.J., Hinds, E.A.: Probing Dark Energy with Atom Interferometry. arXiv preprint arXiv:1408.1409 (2014)

  9. Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations fr superstrings. Nucl. Phys. B 258, 46–74 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  10. Carlip, S.: Quantum Gravity in 2+1 Dimensions. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  11. Carlip, S.: Is quantum gravity necessary? Class. Quant. Gravit. 25, 154010 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  12. Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Södinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Chiou, D.-W.: Loop quantum gravity. Int. J. Mod. Phys. D 24(1), 1530005 (2015)

  14. Cingolani, S., Clapp, M., Secchi, S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63(2), 233–248 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Clapp, M., Salazar, D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407(1), 1–15 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dabholkar, S.P., Disconzi, M.M., Pingali, V.P.: Remarks on positive energy vacua via effective potentials in string theory. Lett. Math. Phys. 104(7), 893–910 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Disconzi, M.M.: A note on quantization in the presence of gravitational shock waves. Mod. Phys. Lett. A 28(31), 1350111 (2013)

  18. Disconzi, M.M.: Some a priori estimates for a critical Schrodinger–Newton equation. In: Electronic Journal of Differential Equations, Ninth MSU-UAB Conference, vol. 20, pp. 39–51 (2013)

  19. Disconzi, M.M., Douglas, M.R., Pingali, V.: On the boundedness of effective potentials arising from string compactifications. Commun. Math. Phys. 325(3), 847–878 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Douglas, M.R.: Effective potential and warp factor dynamics. J. High Energy Phys. 2010(3), 1–32 (2010)

    Article  Google Scholar 

  21. Douglas, M.R., Kachru, S.: Flux compactification. Rev. Mod. Phys. 79, 733–796 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Douglas, M.R., Kallosh, R.: Compactification on negatively curved manifolds. J. High Energy Phys. 2010(6), 1–18 (2010)

    Article  MathSciNet  Google Scholar 

  23. Fulling, S.A.: Aspects of Quantum Field Theory in Curved Spacetime, vol. 17. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  24. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in rn. Adv. Math. Suppl. Stud. A 7, 369–402 (1981)

    MathSciNet  Google Scholar 

  25. Ginibre, J., Velo, G.: On a class of non linear Schrödinger equations with non local interaction. Math. Z. 170(2), 109–136 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Giulini, D., Großardt, A.: Gravitationally induced inhibitions of dispersion according to the Schrödinger–Newton equation. Class. Quantum Gravit. 28(19), 195026 (2011)

    Article  ADS  Google Scholar 

  27. Giulini, D., Großardt, A.: Class. Quantum Gravit. 29, 215010 (2012)

    Article  ADS  Google Scholar 

  28. Giulini, D., Großardt, A.: Centre-of-mass motion in multi-particle Schrödinger–Newton dynamics. New J. Phys. 16(7), 075005 (2014)

    Article  MathSciNet  Google Scholar 

  29. Grana, M.: Flux compactifications in string theory: a comprehensive review. Phys. Rep. 423(3), 91–158 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  30. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, vol. 1, 2. Cambridge University Press (1987)

  31. Hackermüller, L., Uttenthaler, S., Hornberger, K., Reiger, E., Brezger, B., Zeilinger, A., Arndt, M.: Wave nature of biomolecules and fluorofullerenes. Phys. Rev. Lett. 91(9), 090408 (2003)

    Article  ADS  Google Scholar 

  32. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43(3), 199–220 (1975)

  33. Kachru, S., Kallosh, R., Linde, A., Trivedi, S.P.: De sitter vacua in string theory. Phys. Rev. D 68, 046005 (2006)

  34. Lions, P.L.: The Choquard equation and related questions. Nonlinear Anal.: Theory Methods Appl. 4(6), 1063–1072 (1980)

    Article  MATH  Google Scholar 

  35. Lions, P.L.: Compactness and topological methods for some nonlinear variational problems of mathematical physics. N.-Holl. Math. Stud. 61, 17–34 (1982)

    Article  Google Scholar 

  36. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. Manfredi, G.: The Schrödinger–Newton equations beyond Newton. Gen. Relativ. Gravit. 47(2), 1–12 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  38. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Parker, L., Toms, D.: Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  40. Penrose, R.: Twistor algebra. J. Math. Phys. 8(2), 345–366 (1967)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. Penrose, R.: On the origins of twistor theory. Gravit. Geom. 341–361 (1987)

  42. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28(5), 581–600 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. Penrose, R.: Quantum computation, entanglement and state reduction. In: Philosofical Transactions-Royal Society of Londo Series A Mathematical Physical and Engineering Sciences, pp. 1927–1937 (1998)

  44. Penrose, R.: The central programme of twistor theory. Chaos Solitons Fract. 10(2), 581–611 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Polchinski, J.: String theory, vols. 1 and 2, vol. 402, p. 531. Cambridge University Press, Cambridge (1998)

  46. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  47. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  48. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  49. Rovelli, C.: Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  50. Salzman, P.J., Carlip, S.: A Possible Experimental Test of Quantized Gravity. arXiv preprint gr-qc/0606120 (2006)

  51. Shatah, J., Strauss, W.: Instability of nonlinear bound states. Commun. Math. Phys. 100(2), 173–190 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  52. Shomer, A.: A Pedagogical Explanation for the Non-renormalizability of Gravity. arXiv preprint arXiv:0709.3555 (2007)

  53. Silverstein, E.: Simple de sitter solutions. Phys. Rev. D 77(10), 106006 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  54. van Meter, J.R.: Schrödinger–Newton “collapse of the wave function”. Class. Quant. Gravit. 28, 215013 (2011)

    Article  ADS  Google Scholar 

  55. Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago Lectures in Physics). University of Chicago Press (1994)

  56. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (2010)

    Google Scholar 

Download references

Acknowledgments

Marcelo M. Disconzi is partially supported by NSF award 1305705. Marion Silvestrini, leonardo G. Brunnet and Carolina Brito thank the Brazilian funding agencies CNPq, Capes and Fapergs. We thank the supercomputing laboratory at IF-UFRGS and at New York University, where the simulations were run, for computer time.

Author information

Authors and Affiliations

  1. Physics Department, Universidade Federal do Rio Grande do Sul, Av. Bento Gonalves 9500 - Caixa Postal 15051, Porto Alegre, RS, CEP 91501-970, Brazil

    Marion Silvestrini, Leonardo G. Brunnet & Carolina Brito

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Marcelo Disconzi

Authors
  1. Marion Silvestrini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonardo G. Brunnet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcelo Disconzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Carolina Brito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carolina Brito.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (mp4 62 KB)

Supplementary material 2 (mp4 40 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Silvestrini, M., Brunnet, L.G., Disconzi, M. et al. Initial condition dependence and wave function confinement in the Schrödinger–Newton equation. Gen Relativ Gravit 47, 129 (2015). https://doi.org/10.1007/s10714-015-1975-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-015-1975-4

Keywords

Search

Navigation