Pair excitations and the mean field approximation of interacting Bosons, II

Abstract

We consider a large number of Bosons with interaction potential $v_N(x)=N^{3\beta }v(N^{\beta }x)$. In earlier papers we considered a set of equations for the condensate $\varphi$ and pair excitation function $k$ and proved that they provide a Fock space approximation to the exact evolution of the condensate for $\beta <\frac{1}{3}$. This result was extended to the case $\beta <\frac{1}{2}$ by E. Kuz, where it was also argued informally that the equations of our earlier work do not provide an approximation for $\beta >\frac{1}{2}$. In 2013, we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range $0\le \beta <1$. In the current paper we prove that this is indeed the case for $\beta <\frac{2}{3}$, at least locally in time. In order to do that, we re-formulate the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of X. Chen and J. Holmer to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function $k$.

References in corpus (5)

• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• The Second Order Upper Bound for the Ground Energy of a Bose Gas
• The ground state energy of a low density Bose gas: a second order upper bound
• Beyond mean field: On the role of pair excitations in the evolution of condensates

Cited by in corpus (21)

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• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Central limit theorem for Bose gases interacting through singular potentials
• The rigorous derivation of the ${\mathbb{T}}^2$ focusing cubic NLS from 3D
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data
• Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• On the time dependence of the rate of convergence towards Hartree dynamics for interacting Bosons
• A note on dependent random variables in quantum dynamics
• Uniform in $N$ Global Well-posedness of the Time-Dependent Hartree-Fock-Bogoliubov Equations in ${\mathbb{R}}^{1+1}$
• Analysis of fluctuations around non linear effective dynamics
• Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential
• On the mean-field and semiclassical limit from quantum $N$-body dynamics
• Differential Equations of Quantum Mechanics
• The Second Order 2D Behaviors of a 3D Bose Gases in the Gross-Pitaevskii Regime