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Blow-up profile of 2D focusing mixture Bose gases

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Abstract

We study the collapse of a many-body system which is used to model two-component Bose–Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for N identical bosons in \({\mathbb {R}}^2\), interacting with potentials rescaled in the mean-field manner \(-N^{2\beta -1}w^{(\sigma )}(N^{\beta }x)\) with \(\int \limits _{{\mathbb {R}}^{2}}w^{(\sigma )}(x)\mathrm{d}x=1\). Assuming that \(0<\beta <1/2\), we first show that the leading order of the quantum energy is captured correctly by the Gross–Pitaevskii energy. Secondly, we investigate the blow-up behavior of the quantum energy as well as the ground states when \(N\rightarrow \infty \), and either the total interaction strength of intra-species and inter-species or the strengths of intra-species interactions of each component approach sufficiently slowly a critical value, which is the critical strength for the focusing Gross–Pitaevskii functional. We prove that the many-body ground states fully condensate on the (unique) Gagliardo–Nirenberg solution.

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References

  1. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science, pp. 198–201 (1995)

  2. Bao, W., Cai, Y.: Ground states of two-component Bose–Einstein condensates with an internal atomic Josephson junction. East Asian J. Appl. Math. 1, 49–81 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Rel. Models 6, 1–135 (2013)

    Article  MathSciNet  Google Scholar 

  4. Baym, G., Pethick, C.: Ground-state properties of magnetically trapped Bose-condensed rubidium gas. Phys. Rev. Lett. 76, 6 (1996)

    Article  Google Scholar 

  5. Bradley, C.C., Sackett, C., Tollett, J., Hulet, R.G.: Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 75, 1687 (1995)

    Article  Google Scholar 

  6. Cornell, E.A., Wieman, C.E.: Nobel lecture: Bose–Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Rev. Mod. Phys. 74, 875 (2002)

    Article  Google Scholar 

  7. Dalfovo, F., Stringari, S.: Bosons in anisotropic traps: ground state and vortices. Phys. Rev. A 53, 2477 (1996)

    Article  Google Scholar 

  8. Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D., Ketterle, W.: Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969 (1995)

    Article  Google Scholar 

  9. Donley, E.A., Claussen, N.R., Cornish, S.L., Roberts, J.L., Cornell, E.A., Wieman, C.E.: Dynamics of collapsing and exploding Bose–Einstein condensates. Nature 412, 295 (2001)

    Article  Google Scholar 

  10. Gerton, J.M., Strekalov, D., Prodan, I., Hulet, R.G.: Direct observation of growth and collapse of a Bose–Einstein condensate with attractive interactions. Nature 408, 692 (2000)

    Article  Google Scholar 

  11. Guo, Y., Li, S., Wei, J., Zeng, X.: Ground states of two-component attractive Bose–Einstein condensates I: existence and uniqueness. J. Funct. Anal. 276, 183–230 (2019)

    Article  MathSciNet  Google Scholar 

  12. Guo, Y., Li, S., Wei, J., Zeng, X.: Ground states of two-component attractive Bose–Einstein condensates II: Semi-trivial limit behavior. Trans. Am. Math. Soc. 371, 6903–6948 (2019)

    Article  MathSciNet  Google Scholar 

  13. Guo, Y., Seiringer, R.: On the mass concentration for Bose–Einstein condensates with attractive interactions. Lett. Math. Phys. 104, 141–156 (2014)

    Article  MathSciNet  Google Scholar 

  14. Guo, Y., Zeng, X., Zhou, H.-S.: Blow-up solutions for two coupled Gross–Pitaevskii equations with attractive interactions. Discret. Cont. Dyn. Syst.-A 37, 3749–3786 (2017)

    Article  MathSciNet  Google Scholar 

  15. Guo, Y., Zeng, X., Zhou, H.-S.: Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions. J. Differ. Equ. 264, 1411–1441 (2018)

    Article  Google Scholar 

  16. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16, 1782–1785 (1977)

    Article  MathSciNet  Google Scholar 

  17. Kagan, Y., Muryshev, A., Shlyapnikov, G.: Collapse and Bose–Einstein condensation in a trapped Bose gas with negative scattering length. Phys. Rev. Lett. 81, 933 (1998)

    Article  Google Scholar 

  18. Ketterle, W.: Nobel lecture: when atoms behave as waves: Bose–Einstein condensation and the atom laser. Rev. Mod. Phys. 74, 1131 (2002)

    Article  Google Scholar 

  19. Lévy-Leblond, J.-M.: Nonsaturation of gravitational forces. J. Math. Phys. 10, 806–812 (1969)

    Article  Google Scholar 

  20. Lewin, M.: Mean-Field limit of Bose systems: rigorous results. In: Proceedings of the International Congress of Mathematical Physics, ICMP (2015)

  21. Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)

    Article  MathSciNet  Google Scholar 

  22. Lewin, M., Nam, P.T., Rougerie, N.: The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Am. Math. Soc. 368, 6131–6157 (2016)

    Article  Google Scholar 

  23. Lewin, M., Nam, P. T., Rougerie, N.: Blow-up profile of rotating 2d focusing Bose gases. In: Workshop on Macroscopic Limits of Quantum Systems, pp. 145–170. Springer (2017)

  24. Lewin, M., Nam, P.T., Rougerie, N.: A note on 2d focusing many-boson systems. Proc. Am. Math. Soc. 145, 2441–2454 (2017)

    Article  MathSciNet  Google Scholar 

  25. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence, RI (2001)

  26. Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  27. Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984)

    Article  MathSciNet  Google Scholar 

  28. Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of Stellar Collapse as the Limit of Quantum Mechanics. Commun. Math. Phys. 112, 147–174 (1987)

    Article  MathSciNet  Google Scholar 

  29. Maeda, M.: On the symmetry of the ground states of nonlinear Schrödinger equations with potential. Advanced Nonlinear Studies 10, 895–925 (2010)

    Article  MathSciNet  Google Scholar 

  30. Michelangeli, A., Nam, P.T., Olgiati, A.: Ground state energy of mixture of bose gases. Reviews in Mathematical Physics 31, 1950005 (2019)

    Article  MathSciNet  Google Scholar 

  31. Michelangeli, A., Olgiati, A.: Mean-field quantum dynamics for a mixture of Bose-Einstein condensates. Analysis and Mathematical Physics 7, 377–416 (2017)

    Article  MathSciNet  Google Scholar 

  32. Mueller, E.J., Baym, G.: Finite-temperature collapse of a Bose gas with attractive interactions. Phys. Rev. A 62, 053605 (2000)

    Article  Google Scholar 

  33. Nam, P.T., Rougerie, N.: Improved stability for 2D attractive Bose gases. Journal of Mathematical Physics 61, 021901 (2020)

    Article  MathSciNet  Google Scholar 

  34. Nguyen, D.-T.: Many-Body Blow-Up Profile of Boson Stars with External Potentials. Review in Mathematical Physics 31, 1950034 (2019)

    Article  MathSciNet  Google Scholar 

  35. Onsager, L.: Electrostatic Interaction of Molecules. J. Phys. Chem. 43, 189–196 (1939)

    Article  Google Scholar 

  36. Sackett, C., Stoof, H., Hulet, R.: Growth and collapse of a Bose-Einstein condensate with attractive interactions. Phys. Rev. Lett. 80, 2031 (1998)

    Article  Google Scholar 

  37. Ueda, M., Leggett, A.J.: Macroscopic quantum tunneling of a Bose-Einstein condensate with attractive interaction. Phys. Rev. Lett. 80, 1576 (1998)

    Article  Google Scholar 

  38. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1983)

    Article  Google Scholar 

  39. Zhang, J.: Stability of Attractive Bose-Einstein Condensates. J. Stat. Phys. 101, 731–746 (2000)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is indebted to the referee for many useful suggestions which improved significantly the presentation of the paper. Also, he is very grateful to T. König for his proofreading of the manuscript. He cordially thanks A. Triay and X. Zeng for some helpful discussions. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2111-390814868.

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  1. Mathematisches Institut, Ludwig–Maximilians–Universität München, Theresienstrasse 39, 80333, Munich, Germany

    Dinh-Thi Nguyen

  2. Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799, Munich, Germany

    Dinh-Thi Nguyen

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Nguyen, DT. Blow-up profile of 2D focusing mixture Bose gases. Z. Angew. Math. Phys. 71, 81 (2020). https://doi.org/10.1007/s00033-020-01302-y

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  • DOI: https://doi.org/10.1007/s00033-020-01302-y

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