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Correlation integral and determinism for a family of $2^\infty$ maps
Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement
1. | IRMAR, Université de Rennes 1 and IPSO, INRIA Rennes, 35042 Rennes Cedex |
2. | Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607 |
References:
[1] |
P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334 (2015), 367-396.
doi: 10.1007/s00220-014-2166-y. |
[2] |
P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation, Discrete Contin. Dyn. Syst., 32 (2012), 703-715.
doi: 10.3934/dcds.2012.32.703. |
[3] |
W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741.
doi: 10.1137/110850451. |
[4] |
W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), 767-782.
doi: 10.1142/S0218202505000534. |
[5] |
N. Ben Abdallah, Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinet. Relat. Models, 4 (2011), 831-856.
doi: 10.3934/krm.2011.4.831. |
[6] |
N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differential Equ., 245 (2008), 154-200.
doi: 10.1016/j.jde.2008.02.002. |
[7] |
N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), 189-199.
doi: 10.1137/040614554. |
[8] |
R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete Contin. Dyn. Syst., 13 (2005), 385-398.
doi: 10.3934/dcds.2005.13.385. |
[9] |
R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.
doi: 10.1137/S0036141002416936. |
[10] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, American Mathematical Society, Providence, RI, 2003. |
[11] |
F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field, Comm. Math. Phys., 292 (2009), 829-870.
doi: 10.1007/s00220-009-0868-3. |
[12] |
S. Flügge, Practical Quantum Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1999. |
[13] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988. |
[14] |
B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques, Société mathématique de France, 1984. |
[15] |
H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193-226. |
[16] |
E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars, 34, Birkhäuser Verlag, Basel, 2005. |
[17] |
E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys., 264 (2006), 505-537.
doi: 10.1007/s00220-006-1524-9. |
[18] |
C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. |
[19] |
L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. |
[20] |
N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces, Commun. Partial Differ. Equ., 37 (2012), 125-135.
doi: 10.1080/03605302.2011.574306. |
show all references
References:
[1] |
P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334 (2015), 367-396.
doi: 10.1007/s00220-014-2166-y. |
[2] |
P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation, Discrete Contin. Dyn. Syst., 32 (2012), 703-715.
doi: 10.3934/dcds.2012.32.703. |
[3] |
W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741.
doi: 10.1137/110850451. |
[4] |
W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), 767-782.
doi: 10.1142/S0218202505000534. |
[5] |
N. Ben Abdallah, Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinet. Relat. Models, 4 (2011), 831-856.
doi: 10.3934/krm.2011.4.831. |
[6] |
N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differential Equ., 245 (2008), 154-200.
doi: 10.1016/j.jde.2008.02.002. |
[7] |
N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), 189-199.
doi: 10.1137/040614554. |
[8] |
R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete Contin. Dyn. Syst., 13 (2005), 385-398.
doi: 10.3934/dcds.2005.13.385. |
[9] |
R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.
doi: 10.1137/S0036141002416936. |
[10] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, American Mathematical Society, Providence, RI, 2003. |
[11] |
F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field, Comm. Math. Phys., 292 (2009), 829-870.
doi: 10.1007/s00220-009-0868-3. |
[12] |
S. Flügge, Practical Quantum Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1999. |
[13] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988. |
[14] |
B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques, Société mathématique de France, 1984. |
[15] |
H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193-226. |
[16] |
E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars, 34, Birkhäuser Verlag, Basel, 2005. |
[17] |
E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Comm. Math. Phys., 264 (2006), 505-537.
doi: 10.1007/s00220-006-1524-9. |
[18] |
C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. |
[19] |
L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. |
[20] |
N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces, Commun. Partial Differ. Equ., 37 (2012), 125-135.
doi: 10.1080/03605302.2011.574306. |
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