### The giant vortex state for a Bose-Einstein condensate in a rotating anharmonic trap : extreme rotation regimes.

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**Abstract :**We study the fast rotation limit for a Bose-Einstein condensate in a quadratic plus quartic confining potential within the framework of the two dimensional Gross-Pitaevskii energy functional. As the rotation speed tends to infinity with a proper scaling of the other parameters in the model, a linear limit problem appears for which we are able to derive precise energy estimates. We prove that the energy and density asymptotics of the problem can be obtained by minimizing a simplified one dimensional energy functional. In the case of a fixed coupling constant we also prove that a giant vortex state appears. It is an annulus with pure irrotational fl ow encircling a central low-density hole around which there is a macroscopic phase circulation.

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**Date:** 2008-09-11