Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials

  • Giulia BastiEmail author
  • Serena Cenatiempo
  • Alessandro Teta


We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N2V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.


Quantum Lorentz gas Gross-Pitaevskii potentials Effective behavior Low-energy neutron scattering 

Mathematics Subject Classification (2010)

81Q99 81V35 82C10 



The authors gratefully acknowledge the support from the GNFM Gruppo Nazionale per la Fisica Matematica - INDAM.


  1. 1.
    Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. 2nd edn. With an appendix by P. Exner. AMS Chelsea Publishing (2005)Google Scholar
  2. 2.
    Benedikter, N., de Oliveira, G., Schlein, B.: Quantitative derivation of the Gross-Pitaevskii equation. Commun. Pure Appl. Math. 68(8), 1399–1482 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benedikter, N., Porta, M., Schlein, B.: Effective evolution equations from quantum dynamics. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brasche, J.F., Figari, R., Teta, A.: Singular Schrödinger operators as limits of point interaction Hamiltonians. Potential Anal. 8, 163–178 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brennecke, C., Schlein, B.: Gross-Pitaevskii dynamics for Bose-Einstein condensates, arXiv:1702.05625 1702.05625
  6. 6.
    Erdös, L., Schlein, B., Yau, H.T.: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun. Pure Appl. Math. 59, 1659–1741 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdös, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. 172(1), 291–370 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdös, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. 22, 1099–1156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Figari, R., Holden, H., Teta, A.: A law of large numbers and a central limit for the Schroedinger operator with zero-range potentials. J. Stat. Phys. 51(1/2), 205–214 (1988)CrossRefzbMATHADSGoogle Scholar
  10. 10.
    Figari, R., Orlandi, E., Teta, A.: The laplacian in regions with many small obstacles: fluctuations around the limit operator. J. Stat. Phys. 41, 465–487 (1985)MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. 11.
    Figari, R., Teta, A.: Effective potential and fluctuations for a boundary value problem on a randomly perforated domain. Lett. Math Effective Phys. 4, 295–305 (1992)MathSciNetCrossRefzbMATHADSGoogle Scholar
  12. 12.
    Figari, R., Teta, A.: A boundary value problem of mixed type on perforated domains. Asymptotic Anal. 6(3), 271–284 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jeblick, M., Pickl, P.: Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials. arXiv:1801.04799(2018)
  14. 14.
    Kingman, J.F.C.: Uses of exchangeability. Ann. Probab. 6(2), 183–197 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lamperti, J.W.: Probability: a survey of the mathematical theory. Wiley, Hoboken (2011)zbMATHGoogle Scholar
  16. 16.
    Michelangeli, A., Olgiati, A.: Gross-pitaevskii non-linear dynamics for pseudo-spinor condensates. J. Nonlinear Math. Phys. 24(3), 426–464 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ozawa, S.: On an elaboration of M. Kac’s theorem concerning eigenvalues of the laplacian in a region with randomly distributed small obstacles. Commun. Math. Phys. 91, 473–487 (1983)MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    Papanicolaou, G.C., Varadhan, S.R.S.: Diffusion in regions with many small holes. In: Lecture notes in control and information sciences, vol. 25, pp 190–206. Springer, Berlin (1980)Google Scholar
  19. 19.
    Pickl, P.: Derivation of the time dependent Gross Pitaevskii equation with external fields. Rev. Math. Phys. 27(1), 1550003 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reed, M., Simon, B.: Methods of modern mathematical physics. Vol I: Functional Analysis. Academic Press, Cambridge (1981)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZurichZurichSwitzerland
  2. 2.Gran Sasso Science InstituteL’AquilaItaly
  3. 3.Dipartimento di Matematica G. CastelnuovoSapienza Università di RomaRomaItaly

Personalised recommendations