Produzione Accademica

We study a system of $N$ trapped bosons in the Thomas–Fermi regime with an interacting pair potential of the form $g_N{N}^{3\beta -1}V(N^{\beta }x)$, for some $\beta \in (0,1/3)$ and $g_N$diverging as $N\to \mathrm{\infty }$. We prove that there is complete Bose–Einstein condensation at the level of the ground state and, furthermore, that, if $\beta \in (0,1/6)$, condensation is preserved by the time evolution.

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In this paper we discuss some aspects of fragmented condensation from a mathematical perspective. We first propose a simple way of characterizing finite fragmentation. Then, inspired by recent results of semiclassical analysis applied to bosonic systems with infinitely many degrees of freedom, we address the problem of persistence of fragmented condensation. We show that the latter occurs in interacting systems, in the mean-field regime, and in the limit of large gap of the one-body Hamiltonian.

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We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε⁻⁴, with ε⁻² denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ω_c such that if Ω = Ω₀/ε⁴ with Ω₀ > Ω_c, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.

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