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Ground-state solution of Bose–Einstein condensate by directly minimizing the energy functional. (English) Zbl 1028.82500

Summary: In this paper, we propose a new numerical method to compute the ground-state solution of trapped interacting Bose-Einstein condensation at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground-state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground-state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground-state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented.

MSC:

82-08 Computational methods (statistical mechanics) (MSC2010)
81-08 Computational methods for problems pertaining to quantum theory
81V80 Quantum optics
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
49K35 Optimality conditions for minimax problems
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B10 Quantum equilibrium statistical mechanics (general)
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