Abstract
We consider a gas of N bosons in a box with volume one interacting through a two-body potential with scattering length of order \({N^{-1}}\) (Gross–Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently weak, we prove complete Bose–Einstein condensation for the ground state and for many-body states with finite excitation energy in the limit of large N with a uniform (N-independent) bound on the number of excitations.
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de Benedikter N., Oliveira G., Schlein B.: Quantitative derivation of the Gross–Pitaevskii equation. Commun. Pure Appl. Math. 68(8), 1399–1482 (2014)
Bogoliubov, N.N.: On the theory of superfluidity. Izv. Akad. Nauk. USSR 11 (1947), 77. Engl. Transl. J. Phys. (USSR) 11, 23 (1947)
Brennecke, C., Schlein, B.: Gross–Pitaevskii dynamics for Bose–Einstein condensates. Preprint arXiv:1702.05625
Chen X., Holmer J.: Correlation structures, many-body scattering processes and the derivation of the Gross–Pitaevskii hierarchy. Int. Math. Res. Not. 10, 3051–3110 (2016)
Dereziński J., Napiórkowski M.: Excitation spectrum of interacting bosons in the mean-field infinite-volume limit. Ann. Henri Poincaré 15, 2409–2439 (2014)
Erdős L., Michelangeli A., Schlein B.: Dynamical formation of correlations in a Bose–Einstein condensate. Commun. Math. Phys. 289(3), 1171–1210 (2009)
Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006)
Erdős L., Schlein B., Yau H.-T.: Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems. Inv. Math. 167, 515–614 (2006)
Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. Math. (2) 172(1), 291–370 (2010)
Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation. Phys. Rev. Lett. 98((4), 040404 (2007)
Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. 22, 1099–1156 (2009)
Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322((2), 559–591 (2013)
Lewin M., Nam P.T., Serfaty S., Solovej J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. 68(3), 413–471 (2015)
Lieb E.H., Seiringer R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)
Lieb E.H., Seiringer R.: Derivation of the Gross–Pitaevskii equation for rotating Bose gases. Commun. Math. Phys. 264, 505–537 (2006)
Lieb E.H., Seiringer R., Yngvason J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A. 61, 043602 (2000)
Lieb E.H., Yngvason J.: Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998)
Nam P.T., Rougerie N., Seiringer R.: Ground states of large bosonic systems: the Gross–Pitaevskii limit revisited. Anal. PDE 9(2), 459–485 (2016)
Pickl P.: Derivation of the time dependent Gross–Pitaevskii equation with external fields. Rev. Math. Phys. 27, 1550003 (2015)
Pizzo, A.: Bose particles in a box III. A convergent expansion of the ground state of the Hamiltonian in the mean field limiting regime. Preprint arXiv:1511.07026
Seiringer R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)
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Boccato, C., Brennecke, C., Cenatiempo, S. et al. Complete Bose–Einstein Condensation in the Gross–Pitaevskii Regime. Commun. Math. Phys. 359, 975–1026 (2018). https://doi.org/10.1007/s00220-017-3016-5
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DOI: https://doi.org/10.1007/s00220-017-3016-5