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Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

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Abstract

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

  2. Department of Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, NJ, 08544, USA

    Robert Seiringer

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  1. Elliott H. Lieb
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  2. Robert Seiringer
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Correspondence to Robert Seiringer.

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Communicated by H.-T. Yau

© 2006 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Work partially supported by U.S. National Science Foundation grant PHY 01 39984.

Work partially supported by U.S. National Science Foundation grant PHY 03 53181, and by an A.P. Sloan Fellowship

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Lieb, E., Seiringer, R. Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases. Commun. Math. Phys. 264, 505–537 (2006). https://doi.org/10.1007/s00220-006-1524-9

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