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arXiv:2307.00956 · Analysis & PDE

Focusing dynamics of 2D Bose gases in the instability regime

Lea Boßmann, Charlotte Dietze, Phan Thành Nam

Published Jan 22, 2026• 1 citation
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Abstract

We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2β-1}w(N^β\cdot)$ for $β\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{ö}dinger equation (NLS) may blow up in finite time. We show that to leading order, the $N$-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.

DOI
10.2140/apde.2026.19.281

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We consider the dynamics of a 2D Bose gas with an interaction potential of the form N2β−1w(Nβ⋅)N^{2β-1}w(N^β\cdot)N2β−1w(Nβ⋅) for β∈(0,3/2)β\in (0,3/2)β∈(0,3/2). The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{ö}dinger equation (NLS) may blow up in finite time. We show that to leading order, the NNN-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.
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.
This thesis investigates the rigorous derivation of non-linear effective equations describing the time evolution of Bose-Einstein condensation from the N-particle linear Schrödinger equation, focusing on the one-particle reduced density matrix. The first chapter outlines the mathematical framework of many-body quantum mechanics, including the Fock space formalism. The second chapter studies the mean-field regime, proving that the condensed state converges to the non-linear Hartree equation by analyzing fluctuation dynamics to quantify the convergence rate. The third chapter addresses the physically realistic but mathematically challenging Gross-Pitaevskii regime, characterized by rare but strong interactions. Here, Bogoliubov transformations are used to capture crucial microscopic short-scale correlations, allowing for a quantitative derivation of the cubic non-linear Schrödinger equation. While the preceding chapters focus exclusively on repulsive interactions, the fourth chapter explores the focusing case with attractive potentials, which can lead to finite-time blow-up. Operating in an intermediate scaling regime, the effective focusing cubic non-linear Schrödinger equation is rigorously derived. Since the negative energy cannot bound many-body fluctuations, convergence is proven by approximating the dynamics via a quadratic Hamiltonian, in the spirit of Bogoliubov theory, and avoiding Grönwall estimates through the analysis of a coupled system of one-body linear equations for the one-particle density matrices.
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.
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 derive rigorously, for both R2{\Bbb R}^2R2 and [−L,L]×2[{-}L,L]^{\times 2}[−L,L]×2, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
Abstract. 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 a 3β−1 V (a β ·) where ∫ V � 0 and a 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 N-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 ∞. 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