Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

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.

LaTeX2e, 34 pages; minor modifications, dedication added. To appear in Commun. Math. Phys

References in corpus (3)

• Justification of c-Number Substitutions in Bosonic Hamiltonians
• Ground State Energy of the Low Density Fermi Gas
• Ground State Asymptotics of a Dilute, Rotating Gas

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