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September  2016, 36(9): 5097-5118. doi: 10.3934/dcds.2016021

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

Florian Méhats 1, and  Christof Sparber 2,

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

Received  July 2015 Revised  February 2016 Published  May 2016

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement.
Keywords: Gross-Pitaevskii equation, angular momentum operator., averaging, dimension reduction, Bose-Einstein condensation.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B2.
Citation: Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
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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