Bose–Einstein Condensation in the Luttinger–Sy Model with Contact Interaction

  • Joachim KernerEmail author
  • Maximilian Pechmann
  • Wolfgang Spitzer


We study bosons on the real line in a Poisson random potential (Luttinger–Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose–Einstein condensation (BEC) occurs almost surely if the intensity \(\nu _N\) of the Poisson potential satisfies \([\ln (N)]^4/N^{1 - 2\eta } \ll \nu _N\lesssim 1\) for arbitrary \(0 < \eta \le 1/3\). We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. Furthermore, for sufficiently strong contact interactions and \(0< \eta < 1/6\), we prove that the mean particle density in the largest interval is almost surely bounded asymptotically by \(\nu _NN^{3/5+\delta }\) for \(\delta > 0\).


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  1. 1.
    Bolte, J., Kerner, J.: Many-particle quantum graphs and Bose–Einstein condensation. J. Math. Phys. 55(6), 061901 (2014)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bolte, J., Kerner, J.: Instability of Bose–Einstein condensation into the one-particle ground state on quantum graphs under repulsive perturbations. J. Math. Phys. 57(4), 043301 (2016)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Casimir, H.: On Bose–Einstein condensation. Fundam. Probl. Stat. Mech. 3, 188–196 (1968)zbMATHGoogle Scholar
  4. 4.
    Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27, 642–669 (1956)MathSciNetzbMATHGoogle Scholar
  5. 5.
    de Smedt, P.: The effect of repulsive interactions on Bose–Einstein condensation. J. Stat. Phys. 45, 201–213 (1986)ADSMathSciNetGoogle Scholar
  6. 6.
    Einstein, A.: Quantentheorie des einatomigen idealen Gases. Sitzber. Kgl. Preuss. Akad. Wiss. 261–267 (1924)Google Scholar
  7. 7.
    Einstein, A.: Quantentheorie des einatomigen idealen Gases, II. Abhandlung, Sitzber. Kgl. Preuss. Akad. Wiss. pp. 3–14 (1925)Google Scholar
  8. 8.
    Girardeau, M.: Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1(6), 516–523 (1960)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967)ADSGoogle Scholar
  10. 10.
    Jaeck, T., Pulé, J.V., Zagrebnov, V.A.: On the nature of Bose–Einstein condensation enhanced by localization. J. Math. Phys. 51(10), 103302 (2010)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Kingman, J.: Poisson Processes. Clarendon Press, New York (1993)zbMATHGoogle Scholar
  12. 12.
    Kennedy, T., Lieb, E.H., Shastry, B.S.: The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61, 2582–2584 (1988)ADSGoogle Scholar
  13. 13.
    Kirsch, W., Simon, B.: Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators. Commun. Math. Phys. 97(3), 453–460 (1985)ADSzbMATHGoogle Scholar
  14. 14.
    Lieb, E.H., Liniger, W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963)ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Lenoble, O., Pastur, L.A., Zagrebnov, V.A.: Bose–Einstein condensation in random potentials. C. R. Phys. 5(1), 129–142 (2004)ADSGoogle Scholar
  16. 16.
    Luttinger, J.M., Sy, H.K.: Bose–Einstein condensation in a one-dimensional model with random impurities. Phys. Rev. A 7(2), 712 (1973)ADSGoogle Scholar
  17. 17.
    Luttinger, J.M., Sy, H.K.: Low-lying energy spectrum of a one-dimensional disordered system. Phys. Rev. A 7, 701–712 (1973)ADSGoogle Scholar
  18. 18.
    Lieb, E.H., Seiringer, R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)ADSGoogle Scholar
  19. 19.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and Its Condensation, Oberwolfach Seminars, vol. 34. Birkhäuser Verlag, Basel (2005)Google Scholar
  20. 20.
    Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61(4), 043602 (2000)ADSGoogle Scholar
  21. 21.
    Lieb, E.H., Seiringer, R., Yngvason, J.: One-dimensional behavior of dilute, trapped Bose gases. Commun. Math. Phys. 244(2), 347–393 (2004)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Lauwers, J., Verbeure, A., Zagrebnov, V.A.: Proof of Bose–Einstein condensation for interacting gases with a one-particle gap. J. Phys. A 36, 169–174 (2003)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Landau, L.J., Wilde, I.F.: On the Bose–Einstein condensation of an ideal gas. Commun. Math. Phys. 70(1), 43–51 (1979)ADSMathSciNetGoogle Scholar
  24. 24.
    Massart, P.: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18(3), 1269–1283 (1990)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Michelangeli, A.: Reduced density matrices and Bose–Einstein condensation. SISSA 39 (2007)Google Scholar
  26. 26.
    Pastur, A.L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)zbMATHGoogle Scholar
  27. 27.
    Penrose, O., Onsager, L.: Bose–Einstein condensation and liquid helium. Phys. Rev. 104, 576–584 (1956)ADSzbMATHGoogle Scholar
  28. 28.
    Seiringer, R., Yngvason, J., Zagrebnov, V.A.: Disordered Bose–Einstein condensates with interaction in one dimension. J. Stat. Mech. Theory Exp. 2012(11), P11007 (2012)Google Scholar
  29. 29.
    Seiringer, R., Yngvason, J., Zagrebnov, V.A.: Disordered Bose–Einstein condensates with interaction. In: Proceedings of XVIIth International Congress on Mathematical Physics. World Scientific, pp. 610–619 (2013)Google Scholar
  30. 30.
    van den Berg, M.: On condensation in the free-boson gas and the spectrum of the Laplacian. J. Stat. Phys. 31, 623–637 (1983)ADSMathSciNetGoogle Scholar
  31. 31.
    van den Berg, M., Lewis, J.T.: On generalized condensation in the free boson gas. Phys. A Stat. Mech. Appl. 110(3), 550–564 (1982)MathSciNetGoogle Scholar
  32. 32.
    van den Berg, M., Lewis, J.T., Lunn, M.: On the general theory of Bose–Einstein condensation and the state of the free boson gas. Helv. Phys. Acta 59(8), 1289–1310 (1986)MathSciNetGoogle Scholar
  33. 33.
    van den Berg, M., Lewis, J.T., Pulé, J.V.: A general theory of Bose–Einstein condensation. Helv. Phys. Acta 59(8), 1271–1288 (1986)MathSciNetGoogle Scholar
  34. 34.
    Zagrebnov, V.A.: Bose–Einstein condensation in a random media. J. Phys. Stud. 11(1), 108–121 (2007)Google Scholar
  35. 35.
    Zagrebnov, V.A., Bru, J.B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350(5–6), 291–434 (2001)ADSMathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Joachim Kerner
    • 1
    Email author
  • Maximilian Pechmann
    • 1
  • Wolfgang Spitzer
    • 1
  1. 1.Department of Mathematics and Computer ScienceFernUniversität in HagenHagenGermany

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