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On ground state of spinor Bose-Einstein condensates. (English) Zbl 1228.35218

Summary: We prove the existence of the ground state for the spinor Bose-Einstein condensates in the one-dimensional case.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
82B10 Quantum equilibrium statistical mechanics (general)
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[1] Anderson, M.H.; Ensher, J.R.; Mathews, M.R.; Wieman, C.E.; Cornell, E.A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 198, (1995)
[2] Ambrosetti, A.; Colorado, E., Bound and ground states of coupled nonlinear schrodinger equations, C. R. Math. Acad. Sci. Paris, 342, 453-458, (2006) · Zbl 1094.35112
[3] Barrett, M.D.; Sauer, J.A.; Chapman, M.S., All-optical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett., 87, 010404, (2001)
[4] Bartsch, T.; Wang, Z.-Q.; Wei, J., Bound states for a coupled Schrödinger system, J. Fixed. Point. Theory. Appl., 2, 353-367, (2007) · Zbl 1153.35390
[5] Bradley, C.C.; Sackett, C.A.; Tollett, J.J.; Hulet, R.G., Evidence of Bose-Einstein condensation in an atomic gas with attractive interaction, Phys. Rev. Lett., 75, 1687, (1995)
[6] Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463-512, (1999)
[7] Davis, K.B.; Mewes, M.-O.; Andrews, M.R.; Druten, N.J.; Durfee, D.D.; Kurn, D.M.; Ketterle, W., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75, 3969, (1995)
[8] Esry, B.D.; Greene, C.H.; Burke, J.P.; Bohn, J.L., Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594-3597, (1997)
[9] Gerdjikov, V.S., Kostov, N.A., Valchev, T.I.: Solutions of multi-component NLS models and spinor Bose-Einstein condensates. Physica D. doi:10.1016/j.physd.2008.06.007 · Zbl 1167.82326
[10] Görlitz, A.; Gustavson, T.L.; Leanhardt, A.E.; Löw, E.; Chikkatur, A.P.; Gupta, S.; Inouye, S.; Pritchard, D.E.; Katterle, W., Sodium Bose-Einstein condensates in the \(F\) = 2 state in a large-volume optical trap, Phys. Rev. Lett., 90, 090401, (2003)
[11] Gross, E.P., Nuovo, structure of a quantized vortex in boson systems, Nuovo Cimento, 20, 454-477, (1961) · Zbl 0100.42403
[12] Ho, T.L., Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81, 742-745, (1998)
[13] Law, C.K.; Pu, H.; Bigelow, N.P., Quantum spins mixing in spinor Bose-Einstein condensates, Phys. Rev. Lett., 81, 5257-5261, (1998)
[14] Lieb, E., Loss, M.: Analysis. Amer. Math. Soc., Providence (1996) · Zbl 0527.35023
[15] Lin, T.-C.; Wei, J.-C., Ground state of \(N\) coupled nonlinear Schrödinger equations in \(R\)\^{}{\(n\)}, \(n\) ≤ 3, Commun. Math. Phys., 255, 629-653, (2005) · Zbl 1119.35087
[16] Lin, T.-C.; Wei, J.-C., Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Poincare Anal. Non-Lin., 22, 403-439, (2005) · Zbl 1080.35143
[17] Maia, L.A.; Montefusco, E.; Pellacci, B., Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 229, 743-767, (2006) · Zbl 1104.35053
[18] Miesner, H.J.; Stamper-Kurn, D.M.; Stenger, J.; Inouye, S.; Chikkatur, A.P.; Ketterle, W., Observation of metastable states in spinor Bose-Einstein condensates, Phys. Rev. Lett., 82, 2228-2231, (1999)
[19] Myatt, C.J.; Burt, E.A.; Ghrist, R.W.; Cornell, E.A.; Wieman, C.E., Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78, 586-589, (1997)
[20] Ohmi, T.; Machida, K., Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, J. Phys. Soc. Jpn., 67, 1822-1825, (1998)
[21] Pitaevskii, L.P., Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13, 451-454, (1961)
[22] Stamper-Kurn, D.M.; Andrews, M.R.; Chikkatur, A.P.; Inouye, S.; Miesner, H.-J.; Stenger, J.; Ketterle, W., Optical confinement of a Bose-Einstein condensate, Phys. Rev. Lett., 80, 2027-2030, (1998)
[23] Stenger, J.; Inouye, S.; Stamper-Kurn, D.M.; Miesner, H.-J.; Chikkatur, A.P.; Ketterle, W., Spin domains in ground state Bose-Einstein condensates, Nature, 396, 345-348, (1998)
[24] Sirakov, B., Least energy solitary waves for a system of nonlinear schrodinger equations, Commun. Math. Phys., 271, 199-221, (2007) · Zbl 1147.35098
[25] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87, 567-576, (1983) · Zbl 0527.35023
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