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arXiv:2210.04515 · SIAM Journal on Mathematical Analysis

On One-Dimensional Bose Gases with Two-Body and (Critical) Attractive Three-Body Interactions

Dinh-Thi Nguyen, Julien Ricaud

Published May 3, 2024• 7 citations
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Abstract

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].

DOI
10.1137/22m1535139

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•Aug 4, 2026•0 citations
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 investigate the behavior of a homogeneous one-dimensional Bose gas composed of NNN identical bosons experiencing attractive two-body and repulsive three-body interactions. Utilizing intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the many-body Hamiltonian. Particular attention is given to the impact of the repulsive three-body term on ground state energy estimates and mass concentration properties
We consider a system of NNN interacting bosons in a rotating harmonic trap in R2\mathbb{R}^2R2, where the two-body interaction is attractive and scaled as N2αU(Nαx)N^{2α}U(N^αx)N2αU(Nαx) with 0<α<1/120<α<1/120<α<1/12, and the three-body interaction is repulsive and scaled as N4βW(Nβx,Nβy)N^{4β}W(N^βx,N^βy)N4βW(Nβx,Nβy) with 0<β<1/240<β<1/240<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 aaa approaches the critical value a∗a_*a∗​ and the three-body coupling bbb 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))Qpotℓn2E^{\mathrm{NLS}} = (1-ζ/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2ENLS=(1−ζ/4+o(1))Qpotℓn2​ with ζ⩾0ζ\geqslant0ζ⩾0 determined by the relative rates of an→a∗a_n\to a_*an​→a∗​ and bn↘0b_n\searrow0bn​↘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.
We consider a trapped dilute gas of NNN bosons in R3\mathbb{R}^3R3 interacting via a three-body interaction potential of the form N V(N1/2(x−y,x−z))N\, V(N^{1/2}(x-y,x-z))NV(N1/2(x−y,x−z)). In the limit N→∞N\to \inftyN→∞, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schrödinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the NNN-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schrödinger minimizer is unique up to a complex phase.
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a3β−1V(aβ⋅)a^{3β-1}V(a^β\cdot)a3β−1V(aβ⋅) where ∫V⩽0\int V\leqslant 0∫V⩽0 and aaa matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D NNN-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty $. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.
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.
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.