Suppression of quantum-mechanical collapse in bosonic gases with intrinsic repulsion: A brief review

On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions

“…where C 1 is the corresponding Gagliardo-Nirenberg constant (see Theorem 2.8). From (26), one directly obtains that ψ(t) 4 is uniformly bounded in time by some positive constant C depending only on λ 1 , λ 2 , λ 3 , ψ 0 and p. Now we also obtain that…”

“…We consider only the case g p > 0. For p = 5 we obtain the Lee-Huang-Yang correction; the Lee-Huang-Yang coefficient g 5 is then always positive (see [26]). The case p = 6 corresponds to the energy critical case and, with a positive coefficient g 6 , may describe short-range conservative threebody interactions (see [10]); the case g 6 < 0 models three-body losses ( [28]) and due to its high complexity lies out of the scope of this paper.…”

“…For a proof of the unique local solution, we refer to [13,Theorem 9.2.6]. In order to show that the solution is global we estimate like in (26) and (27), by replacing ∇ψ 2 2 to ∇ψ 2 2 + V ext |u| 2 1 .…”

“…Thus it was suggested to modify (1) by adding a nonlocal (convolution integral) term and a nonlinear higher order term, respectively modelling the long range dipole-dipole interactions and the beyond mean field quantum fluctuations (the so-called Lee-Huang-Yang correction); see also [9,26] and references therein.…”