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ON THE KLAINERMAN-MACHEDON CONJECTURE OF THE QUANTUM BBGKY HIERARCHY WITH SELF-INTERACTION
"... Abstract. We consider the 3D quantum BBGKY hierarchy which corresponds to the N-particle Schrödinger equation. We assume the pair interaction is N 3β−1V (N β•). For interaction parameter β ∈ (0, 2 3), we prove that, as N → ∞, the limit points of the solutions to the BBGKY hierarchy satisfy the space ..."
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Abstract. We consider the 3D quantum BBGKY hierarchy which corresponds to the N-particle Schrödinger equation. We assume the pair interaction is N 3β−1V (N β•). For interaction parameter β ∈ (0, 2 3), we prove that, as N → ∞, the limit points of the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman-Machedon [37] in 2008. This allows for the application of the Klainerman-Machedon uniqueness theorem, and hence implies that the limit is uniquely determined as a tensor product of solutions to the Gross-Pitaevski equation when the N-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović [11] for β ∈ (0, 1 4) and subsequently by X. Chen [15] for β ∈ (0, 2 7]. We build upon the approach of X. Chen but apply frequency localized Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate to extend the range to β ∈ (0, 2 3). Overall, this provides an alternative approach
FOCUSING QUANTUM MANY-BODY DYNAMICS: THE RIGOROUS DERIVATION OF THE 1D FOCUSING CUBIC NONLINEAR SCHRÖDINGER EQUATION
"... We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N β−1 V (N β ·) where R V 6 0. We develop new techniques in treating the N−body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establis ..."
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Cited by 9 (4 self)
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We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N β−1 V (N β ·) where R V 6 0. We develop new techniques in treating the N−body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N → ∞ limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.
ON THE RIGOROUS DERIVATION OF THE 2D CUBIC NONLINEAR SCHRÖDINGER EQUATION FROM 3D QUANTUM MANY-BODY DYNAMICS
"... Abstract. We consider the 3D quantum many-body dynamics describing a dilute bose gas with strong confining in one direction. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to ∞. We find that this diverging coefficien ..."
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Cited by 6 (5 self)
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Abstract. We consider the 3D quantum many-body dynamics describing a dilute bose gas with strong confining in one direction. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to ∞. We find that this diverging coefficient is counterbalanced by the limiting structure of the density matrices and establish the convergence of the BBGKY hierarchy. Moreover, we prove that the limit is fully described by a 2D cubic NLS and obtain the exact 3D to 2D coupling constant.
Mean-field Evolution of Fermionic Systems
, 2013
"... The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one ..."
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The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection ωN with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree-Fock equation with initial data ωN. Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree-Fock dynamics. 1
A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on T³ from the dynamics of many-body quantum systems
, 2014
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Positive Semidefiniteness and Global Well-Posedness of Solutions to the Gross-Pitaevskii Hierarchy
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Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy. arXiv:1409.1425
"... Abstract. We consider the dynamics of N bosons in three dimensions. We assume the pair interaction is given by N31V (N ). By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by t ..."
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Abstract. We consider the dynamics of N bosons in three dimensions. We assume the pair interaction is given by N31V (N ). By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the N-body wave function, for 2 (0; 1], we derive the Gross-Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the N!1 limit, all k-body scattering processes vanishes if k> 3 and thus provide a direct answer to a question raised by Erdös, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for 2 (0; 1) and weakly for = 1. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve < 1, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time
FOCUSING QUANTUM MANY-BODY DYNAMICS II: THE RIGOROUS DERIVATION OF THE 1D FOCUSING CUBIC NONLINEAR SCHRÖDINGER EQUATION FROM 3D
"... 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. ..."
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Cited by 2 (2 self)
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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
4 ON THE UNCONDITIONAL UNIQUENESS OF SOLUTIONS TO THE INFINITE RADIAL CHERN-SIMONS-SCHRÖDINGER HIERARCHY
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