Acta mathematica scientia, Series B >
MEAN-FIELD LIMIT OF BOSE-EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS IN R2
Received date: 2014-08-06
Revised date: 2015-10-12
Online published: 2016-04-25
Supported by
This work is partially supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China, and National Center for Mathematics and Interdisciplinary Sciences in China.
Starting with the many-body Schrödinger Hamiltonian in R2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the mean-field regime, as the particle number N→∞ and however the scattering length κ→0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R2.
Yujin GUO , Lu LU . MEAN-FIELD LIMIT OF BOSE-EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS IN R2[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 317 -324 . DOI: 10.1016/S0252-9602(16)30001-7
[1] Anderson M H, Ensher J R, Matthews M R, et al. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 1995, 269:198-201
[2] Baumgartner B, Solovej J P, Yngvason J. Atoms in strong magnetic fields:the high field limit at fixed nuclear charge. Comm Math Phys, 2000, 212:703-724
[3] Benguria R, Lieb E H. Proof of the stability of highly negative Ions in the absence of the pauli principle. Phys Rev Lett, 1983, 50:1771-1774
[4] Bloch I, Dalibard J, ZwergerW. Many-body physics with ultracold gases. Rev Mod Phys, 2008, 80:885-964
[5] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York:Springer, 2001
[6] Cazenave T. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics 10. New York:Courant Institute of Mathematical Science/AMS, 2003
[7] Cooper N R. Rapidly rotating atomic gases. Adv Phys, 2008, 57:539-616
[8] Dalfovo F, Giorgini S, Pitaevskii L P, et al. Theory of Bose-Einstein condensation in trapped gases. Rev Mod Phys, 1999, 71:463-512
[9] Davis K B, Mewes M O, Andrews M R, et al. Bose-Einstein condensation in a gas of sodium atoms. Phys Rev Lett, 1995, 75:3969-3973
[10] Erdös L, Schlein B, Yau H T. Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J Amer Math Soc, 2009, 22:1099-1156
[11] Fetter A L. Rotating trapped Bose-Einstein condensates. Rev Mod Phys, 2009, 81:647
[12] Fröhlich J, Lenzmann E. Mean-field limit of quantum Bose gases and nonlinear Hartree equation. Séminaire:Équations aux Dérivées Partielles, 2004:1-26
[13] Gidas B, Ni W M, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical analysis and applications Part A. Adv in Math Suppl Stud, 7a. New York-London:Academic Press, 1981:369-402
[14] Grech P, Seiringer R. The excitation spectrum for weakly interacting bosons in a trap. Comm Math Phys, 2013, 322:559-591
[15] Guo Y J, Seiringer R. On the mass concentration for Bose-Einstein condensates with attactive interactions. Lett Math Phys, 2014, 104:141-156
[16] Guo Y J, Zeng X Y, Zhou H S. Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials[J/OL]. Ann I H Poincaré-AN, 2015, http://dx.doi.org/10.1016/j.anihpc. 2015.01.005
[17] Huepe C, Metens S, Dewel G, et al. Decay rates in attractive Bose-Einstein condensates. Phys Rev Lett, 1999, 82:1616-1619
[18] Kagan Y, Muryshev A E, Shlyapnikov G V. Collapse and Bose-Einstein condensation in a trapped Bose gas with nagative scattering length. Phys Rev Lett, 1998, 81:933-937
[19] Kwong M K. Uniqueness of positive solutions of Δu-u+up=0 in RN. Arch Rational Mech Anal, 1989, 105:243-266
[20] Li Y, Ni W M. Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm Partial Differential Equations, 1993, 18:1043-1054
[21] Lieb E H, Seiringer R. Proof of Bose-Einstein condensation for dilute trapped gases. Phys Rev Lett, 2002, 88:170409-1-4
[22] Lieb E H, Seiringer R. Derivation of the Gross-Pitaevskii equation for rotating Bose gases. Comm Math Phys, 2006, 264:505-537
[23] Lieb E H, Seiringer R, Solovej J P, et al. The mathematics of the Bose gas and its condensation. Oberwolfach Seminars 34. Basel:Birkhäuser Verlag, 2005
[24] Lieb E H, Seiringer R, Yngvason J. Bosons in a trap:A rigorous derivation of the Gross-Pitaevskii energy functional. Phys Rev A, 2000, 61:043602-1-13
[25] Lieb E H, Seiringer R, Yngvason J. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Comm Math Phys, 2001, 224:17-31
[26] Lieb E H, Yau H T. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm Math Phys, 1987, 112:147-174
[27] Seiringer R. The excitation spectrum for weakly interacting bosons. Comm Math Phys, 2011, 306:565-578
[28] Seiringer R, Yngvason J, Zagrebnov V A. Disordered Bose-Einstein condensates with interaction in one dimension. J Stat Mech, 2012, 2012:P11007
[29] Weinstein M I. Nonlinear Schrödinger equations and sharp interpolations estimates. Comm Math Phys, 1983, 87:567-576
[30] Zhang J. Stability of attractive Bose-Einstein condensates. J Statist Phys, 2000, 101:731-746
/
| 〈 |
|
〉 |