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A note on 2D focusing many-boson systems


Authors: Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie
Journal: Proc. Amer. Math. Soc. 145 (2017), 2441-2454
MSC (2010): Primary 35Q40, 81V70
DOI: https://doi.org/10.1090/proc/13468
Published electronically: February 10, 2017
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Abstract: We consider a 2D quantum system of $ N$ bosons in a trapping potential $ \vert x\vert^s$, interacting via a pair potential of the form $ N^{2\beta -1} w(N^\beta x)$. We show that for all $ 0<\beta <(s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $ N$ limit by the corresponding cubic nonlinear Schrödinger energy functional. Our result covers the focusing case ($ w<0$) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X. Chen and J. Holmer for harmonic traps ($ s=2$). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all $ 0<\beta <3/4$.


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  • [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science, 269 (5221) (1995), pp. 198-201.
  • [2] W. H. Aschbacher, J. Fröhlich, G. M. Graf, K. Schnee, and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys. 43 (2002), no. 8, 3879-3891. MR 1915631, https://doi.org/10.1063/1.1488673
  • [3] Niels Benedikter, Marcello Porta, and Benjamin Schlein, Effective evolution equations from quantum dynamics, SpringerBriefs in Mathematical Physics, vol. 7, Springer, Cham, 2016. MR 3382225
  • [4] C. Boccato, S. Cenatiempo, and B. Schlein, Quantum many-body fluctuations around nonlinear Schrödinger dynamics, Preprint (2015) arXiv:1509.03837.
  • [5] X. Chen and J. Holmer, The rigorous derivation of the 2d cubic focusing NLS from quantum many-body evolution, Preprint (2015) arXiv:1508.07675.
  • [6] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium Atoms, Phys. Rev. Lett., 75 (1995), pp. 3969-3973.
  • [7] László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515-614. MR 2276262, https://doi.org/10.1007/s00222-006-0022-1
  • [8] height 2pt depth -1.6pt width 23pt, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2), 172 (2010), pp. 291-370.
  • [9] László Erdős and Horng-Tzer Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169-1205. MR 1926667, https://doi.org/10.4310/ATMP.2001.v5.n6.a6
  • [10] R. L. Frank, Ground states of semi-linear PDE.
    Lecture notes from the ``Summer School on Current Topics in Mathematical Physics'', CIRM Marseille, Sept. 2013.
  • [11] Yujin Guo and Robert Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), no. 2, 141-156. MR 3152151, https://doi.org/10.1007/s11005-013-0667-9
  • [12] M. Lewin, Mean-Field limit of Bose systems: rigorous results, Preprint (2015) arXiv:1510.04407.
  • [13] Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math. 254 (2014), 570-621. MR 3161107, https://doi.org/10.1016/j.aim.2013.12.010
  • [14] height 2pt depth -1.6pt width 23pt, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc, in press. arXiv:1405.3220 (2014).
  • [15] Mathieu Lewin, Phan Thành Nam, and Benjamin Schlein, Fluctuations around Hartree states in the mean-field regime, Amer. J. Math. 137 (2015), no. 6, 1613-1650. MR 3432269, https://doi.org/10.1353/ajm.2015.0040
  • [16] Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, vol. 34, Birkhäuser Verlag, Basel, 2005. MR 2143817
  • [17] Elliott H. Lieb, Robert Seiringer, and Jakob Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys. 224 (2001), no. 1, 17-31. Dedicated to Joel L. Lebowitz. MR 1868990, https://doi.org/10.1007/s002200100533
  • [18] Masaya Maeda, On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud. 10 (2010), no. 4, 895-925. MR 2683688
  • [19] P. Nam and M. Napiórkowski, Bogoliubov correction to the mean-field dynamics of interacting bosons, Preprint (2015) arXiv:1509.04631.
  • [20] P. Nam, N. Rougerie, and R. Seiringer, Ground states of large Bose systems: The Gross-Pitaevskii limit revisited, arXiv:1503.07061.
  • [21] N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation, Lecture Notes for a course at LMU, Munich (2014). arXiv:1506.05263.
  • [22] Robert Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Comm. Math. Phys. 229 (2002), no. 3, 491-509. MR 1924365, https://doi.org/10.1007/s00220-002-0695-2
  • [23] Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567-576. MR 691044
  • [24] Jian Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys. 101 (2000), no. 3-4, 731-746. MR 1804895, https://doi.org/10.1023/A:1026437923987

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Additional Information

Mathieu Lewin
Affiliation: CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534), Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
Email: mathieu.lewin@math.cnrs.fr

Phan Thành Nam
Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: pnam@ist.ac.at

Nicolas Rougerie
Affiliation: Université Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France
Email: nicolas.rougerie@grenoble.cnrs.fr

DOI: https://doi.org/10.1090/proc/13468
Received by editor(s): October 21, 2015
Published electronically: February 10, 2017
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 by the authors

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