@article{8130,
abstract = {We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N, while its range is proportional to (ε/N)β with scaling parameter β∈(0,1]. We consider the simultaneous limit (N,ε)→(∞,0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈(0,1), we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.},
author = {Bossmann, Lea},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {11},
pages = {541--606},
publisher = {Springer Nature},
title = {{Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons}},
doi = {10.1007/s00205-020-01548-w},
volume = {238},
year = {2020},
}
@article{7650,
abstract = {We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ20). Here ϱ denotes the density of the system and ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.},
author = {Deuchert, Andreas and Seiringer, Robert},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {6},
pages = {1217--1271},
publisher = {Springer Nature},
title = {{Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature}},
doi = {10.1007/s00205-020-01489-4},
volume = {236},
year = {2020},
}
@article{8418,
abstract = {For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set U in phase space of fixed measure, where the set of initial points which lead to collision is O(μ120) dense as μ→0.},
author = {Guardia, Marcel and Kaloshin, Vadim and Zhang, Jianlu},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis},
number = {2},
pages = {799--836},
publisher = {Springer Nature},
title = {{Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem}},
doi = {10.1007/s00205-019-01368-7},
volume = {233},
year = {2019},
}
@article{6002,
abstract = {The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.},
author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan Philip},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {3},
pages = {1037--1090},
publisher = {Springer Nature},
title = {{The Bogoliubov free energy functional I: Existence of minimizers and phase diagram}},
doi = {10.1007/s00205-018-1232-6},
volume = {229},
year = {2018},
}