We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.
Abstract. We consider a one-dimensional, trapped, focusing Bose gas where [Formula: see text] bosons interact with each other via both a two-body interaction potential of the form [Formula: see text] and an attractive three-body interaction potential of the form [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The system is stable either for any [Formula: see text] as long as [Formula: see text] —the critical strength of the one-dimensional focusing quintic nonlinear Schrödinger (NLS) equation— or for [Formula: see text] when [Formula: see text]. In the former case, fixing [Formula: see text], we prove that in the mean-field limit the many-body system exhibits the Bose–Einstein condensation on the cubic-quintic NLS ground states. When assuming [Formula: see text] and [Formula: see text] as [Formula: see text], with the former convergence being slow enough and “not faster” than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case, [Formula: see text] fixed, we obtain the convergence of many-body energy for small [Formula: see text] when [Formula: see text] is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence [Formula: see text] is “faster” than the slow enough convergence [Formula: see text].
We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.
We consider a system of
N interacting bosons in a rotating harmonic trap in
R2, where the two-body interaction is attractive and scaled as
N2αU(Nαx) with
0<α<1/12, and the three-body interaction is repulsive and scaled as
N4βW(Nβx,Nβy) with
0<β<1/24. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling
a approaches the critical value
a∗ and the three-body coupling
b tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies
ENLS=(1−ζ/4+o(1))Qpotℓn2 with
ζ⩾0 determined by the relative rates of
an→a∗ and
bn↘0. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.