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

  • 147 Accesses


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 to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

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


  1. 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, 1922, 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.


  1. 1.

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

  2. 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)

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 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)

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 10.

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

  11. 11.

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

  12. 12.

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

  13. 13.

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

  14. 14.

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

  15. 15.

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

  16. 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)

  17. 17.

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

  18. 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. 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)

  20. 20.

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

  21. 21.

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

  22. 22.

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

  23. 23.

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

  24. 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)

  25. 25.

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

  26. 26.

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

  27. 27.

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

  28. 28.

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

  29. 29.

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

  30. 30.

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

  31. 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)

  32. 32.

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

  33. 33.

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

  34. 34.

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

  35. 35.

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

  36. 36.

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

  37. 37.

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

  38. 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)

  39. 39.

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

  40. 40.

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

  41. 41.

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

  42. 42.

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

  43. 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. 44.

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

  45. 45.

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

  46. 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)

  47. 47.

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

  48. 48.

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

  49. 49.

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

  50. 50.

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

  51. 51.

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

  52. 52.

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

  53. 53.

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

  54. 54.

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

  55. 55.

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

  56. 56.

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

Download references


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

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)

Supplementary material 1 (mp4 62 KB)

Supplementary material 2 (mp4 40 KB)

Rights and permissions

Reprints and Permissions

About this article

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).

Download citation


  • Schrödinger–Newton Equation
  • Semi-classical gravity
  • Critical mass
  • Wave function confinement
  • Gravitational inhibition