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arXiv:1907.03388 · Journal of Mathematical Physics

Rate of convergence toward Hartree type equations for mixture condensates with factorized initial data

Jinyeop Lee

Published Sep 1, 2021• 7 citations
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Abstract

We consider a system of p species of bosons, each of which consists of N1, N2, …, Np particles. The bosons are in three dimensions with interactions via an interaction potential V such that V ≤ D(1 − Δ), which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding p particle dynamics due to a system of coupled Hartree equations is O(N−1), where N=∑q=1pNq.

DOI
10.1063/5.0019679

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We consider a system of p species of bosons, each of which consists of N1, N2, …, Np particles. The bosons are in three dimensions with interactions via an interaction potential V such that V ≤ D(1 − Δ), which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding p particle dynamics due to a system of coupled Hartree equations is O(N−1), where N=∑q=1pNq.
We consider a mean-field model to describe the dynamics of [Formula: see text] bosons of species one and [Formula: see text] bosons of species two in the limit as [Formula: see text] and [Formula: see text] go to infinity. We embed this model into Fock space and use it to describe the time evolution of coherent states which represent two-component condensates. Following this approach, we obtain a microscopic quantum description for the dynamics of such systems, determined by the Schrödinger equation. Associated to the solution to the Schrödinger equation, we have a reduced density operator for one particle in the first component of the condensate and one particle in the second component. In this paper, we estimate the difference between this operator and the projection onto the tensor product of two functions that are solutions of a system of equations of Hartree type. Our results show that this difference goes to zero as [Formula: see text] and [Formula: see text] go to infinity.
The dynamics of a quantum system with a large number NNN of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit N→∞N \rightarrow \inftyN→∞. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum NNN-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.
In this paper, we investigate the dynamics of a system of NNN weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data ΨN,0=φ0⊗NΨ_{N,0}=φ_{0}^{\otimes N}ΨN,0​=φ0⊗N​ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of O(N−(1+4a)/(3+2a))O(N^{-(1+4a)/(3+2a)})O(N−(1+4a)/(3+2a)) for φ0∈H(3/2)+a(R3)φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})φ0​∈H(3/2)+a(R3) where 0≤a<1/20\leq a<1/20≤a<1/2 and O(N−1)O(N^{-1})O(N−1) for a>1a>1a>1. Our proof is based on and extends the Fock space approach.