A. A. Abrikosov, On the Magnetic properties of superconductors of the second group, Sov. Phys. JETP, vol.5, pp.1174-1182, 1957.

A. Aftalion, Vortices in Bose Einstein condensates, Progress in Nonlinear Differential Equations and Their Applications, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00017351

A. Aftalion, X. Blanc, and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose???Einstein condensates, Journal of Functional Analysis, vol.241, issue.2, pp.661-702, 2006.
DOI : 10.1016/j.jfa.2006.04.027
URL : https://hal.archives-ouvertes.fr/hal-00447032

A. Aftalion and X. Blanc, Reduced energy functionals for a three-dimensional fast rotating Bose Einstein condensates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.25, issue.2, pp.339-355, 2008.
DOI : 10.1016/j.anihpc.2006.11.011

V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Fast Rotation of a Bose-Einstein Condensate, Physical Review Letters, vol.92, issue.5, pp.92-050403, 2004.
DOI : 10.1103/PhysRevLett.92.050403

V. S. Buslaev, Quantization and the WKB method, Trudy Mat. Inst. Steklov, vol.110, pp.5-28, 1970.

E. A. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, Journal of Functional Analysis, vol.97, issue.1, pp.231-249, 1991.
DOI : 10.1016/0022-1236(91)90022-W

A. L. Fetter, Lowest-Landau-level description of a Bose-Einstein condensate in a rapidly rotating anisotropic trap, Physical Review A, vol.75, issue.1, p.13620, 2007.
DOI : 10.1103/PhysRevA.75.013620

T. L. Ho, Bose-Einstein Condensates with Large Number of Vortices, Physical Review Letters, vol.87, issue.6, p.60403, 2001.
DOI : 10.1103/PhysRevLett.87.060403

L. Hörmander, The analysis of linear partial differential operators. III Pseudo-differential operators, Reprint of the 1994 edition, Classics in Mathematics, p.35006, 2007.

J. Leray, Lagrangian analysis and quantum mechanics A mathematical structure related to asymptotic expansions and the Maslov index, Translated from the French by Carolyn Schroeder, p.64463358081, 1981.

N. Lerner, The Wick calculus of pseudo-differential operators and some of its applications, Cubo Mat. Educ, vol.5, issue.1, pp.213-236, 2003.

H. Elliott and . Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys, issue.3, pp.31-594, 1990.

K. Madison, F. Chevy, V. Bretin, and J. Dalibard, Vortex Formation in a Stirred Bose-Einstein Condensate, Physical Review Letters, vol.84, issue.5, p.806, 2000.
DOI : 10.1103/PhysRevLett.84.806

D. Mcduff and D. Salamon, Introduction to symplectic topology Vortex lattice of a bose-einstein condensate in a rotating anisotropic trap, Oxford Mathematical Monographs, 1998.

C. J. Pethick and H. Smith, Bose Einstein condensation in dilute gases, 2002.
DOI : 10.1017/cbo9780511755583
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.382.7192

L. Pitaevskii and S. Stringari, Bose einstein condensation, International series of monographs on physics, 2003.

P. Sanchez-lotero and J. J. Palacios, Vortices in a rotating Bose-Einstein condensate under extreme elongation, Physical Review A, vol.72, issue.4, p.43613, 2005.
DOI : 10.1103/PhysRevA.72.043613

S. Sinha and G. V. Shlyapnikov, Two-Dimensional Bose-Einstein Condensate under Extreme Rotation, Physical Review Letters, vol.94, issue.15, 2005.
DOI : 10.1103/PhysRevLett.94.150401
URL : https://hal.archives-ouvertes.fr/hal-00004895

G. Watanabe, G. Baym, and C. J. Pethick, Landau Levels and the Thomas-Fermi Structure of Rapidly Rotating Bose-Einstein Condensates, Physical Review Letters, vol.93, issue.19, p.93, 2004.
DOI : 10.1103/PhysRevLett.93.190401

A. Aftalion, C. , and E. Polytechnique, CNRS, 91128 Palaiseau cedex, France E-mail address: amandine.aftalion@polytechnique

X. Blanc, U. Pierre, and M. Curie-paris6, UMR 7598, laboratoire Jacques-Louis Lions, 175 rue du Chevaleret, Paris F-75013 France E-mail address: blanc@ann, 175 rue du Chevaleret, p.75013

F. Paris, E-mail address: lerner@math.jussieu.fr URL