On the mean-field and semiclassical limit from quantum $N$ -body dynamics

Xuwen Chen Department of Mathematics, University of Rochester, Rochester, NY 14627, USA [email protected] , Shunlin Shen School of Mathematical Sciences, Peking University, Beijing, 100871, China [email protected] and Zhifei Zhang School of Mathematical Sciences, Peking University, Beijing, 100871, China [email protected]

Abstract.

We study the mean-field and semiclassical limit of the quantum many-body dynamics with a repulsive $\delta$ -type potential $N^{3\beta}V(N^{\beta}x)$ and a Coulomb potential, which leads to a macroscopic fluid equation, the Euler-Poisson equation with pressure. We prove quantitative strong convergence of the quantum mass and momentum densities up to the first blow up time of the limiting equation. The main ingredient is a functional inequality on the $\delta$ -type potential for the almost optimal case $\beta\in(0,1)$ , for which we give an analysis of the singular correlation structure between particles.

Key words and phrases:

Mean-field Limit, Semiclassical Limit, Compressible Euler Equation, Quantum Many-body Dynamics, Modulated Energy.

2010 Mathematics Subject Classification:

Primary 35Q31, 76N10, 81V70; Secondary 35Q55, 81Q05.

1. Introduction

1.1. Background and Problems

The foundations of microscopic physics are Newton’s and Schrödinger equations in the classical and the quantum case respectively. By the first principle of quantum mechanics, a quantum system of $N$ particles is described by a wave function satisfying a linear $N$ -body Schrödinger equation. In realistic systems like fluids, the particle number is so large that these $N$ -body equations are almost impossible to solve. The macroscopic dynamics are therefore modeled by phenomenological equations such as the Euler or the Navier-Stokes equations, which are an important part of many areas of pure and applied mathematics, science, and engineering. These macroscopic equations are usually derived from continuum under ideal assumptions, but they are, in principle, consequences of the microscopic physical laws of Newton or Schrödinger. A key goal of statistical mechanics is to justify these macroscopic equations from microscopic theories in appropriate limit regimes. It is thus of fundamental interest to establish macroscopic equations from the microscopic level.

In the current paper, we start from the bosonic¹¹1N₂ and O₂ molecules are bosons (99.03% of air) and 99.05% H₂O molecules are bosons. quantum many-body dynamics with $\delta$ -type and Coulomb potentials, and study the mean-field and semiclassical limit which would lead to macroscopic fluid equations as particle number $N$ tends to infinity and Planck’s constant $\hbar$ tends to zero. The dynamics of $N$ quantum particles in 3D are governed by, according to the superposition principle, the linear $N$ -body Schrödinger equation:

(1.1)

\displaystyle i\hbar\partial_{t}\psi_{N,\hbar}=H_{N,\hbar}\psi_{N,\hbar}.

Our Hamiltonian $H_{N,\hbar}$ is

(1.2)

\displaystyle H_{N,\hbar}=\sum_{j=1}^{N}-\frac{1}{2}\hbar^{2}\Delta_{x_{j}}+\frac{1}{N}\sum_{1\leq j<k\leq N}V_{N}(x_{j}-x_{k})+\frac{\kappa}{N}\sum_{1\leq j<k\leq N}V_{c}(x_{j}-x_{k}),

where the factor $1/N$ is the mean-field averaging factor. The $\delta$ -type and Coulomb potentials are

(1.3)

\begin{cases}&V_{N}(x)=N^{3\beta}V(N^{\beta}x),\\ &V_{c}(x)=\frac{1}{|x|},\end{cases}

in which, the parameter $\beta\in[0,1]$ characterises different density regimes which correspond to different physical situations.

To have a fixed number of variables in the $N\to\infty$ process, we define the marginal densities $\gamma_{N,\hbar}^{(k)}(t)$ associated with $\psi_{N,\hbar}(t)=e^{itH_{N,\hbar}}\psi_{N,\hbar}(0)$ in kernel form by

(1.4)

\displaystyle\gamma_{N,\hbar}^{(k)}(t,X_{k},X_{k}^{\prime})=\int\psi_{N,\hbar}(t,X_{k},X_{N-k})\overline{\psi_{N,\hbar}}(t,X_{k}^{\prime},X_{N-k})dX_{N-k},

where $X_{k}=(x_{1},...,x_{k})\in\mathbb{R}^{3k}$ and $X_{N-k}=(x_{k+1},...,x_{N})\in\mathbb{R}^{3(N-k)}$ . It is believed that nonlinear Schrödinger equations (NLS) is the mean-field limit equation for these quantum $N$ -body dynamics, that is,

(1.5)

\displaystyle\gamma_{N,\hbar}^{(1)}(t,x_{1},x_{1}^{\prime})\sim|\phi(t)\rangle\langle\phi(t)|,

where $\phi(t)$ solves NLS.

There is a large amount of literature devoted to the mean-field theory from quantum many-body dynamics, such as [1, 2, 3, 4, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 18, 16, 17, 19, 20, 21, 22, 25, 26, 32, 28, 29, 30, 31, 33, 39, 40, 37, 38, 36, 41, 42, 43, 44, 48, 46, 47, 55, 58, 59, 60, 61, 62]. In particular, for the case of defocusing $\delta$ -type potential, it was Erdös, Schlein, and Yau who first rigorously derived the 3D cubic defocusing NLS from quantum many-body dynamics in their groundbreaking papers [28, 29, 30, 31].²²2Around the same time, see also [1] for 1D case.In their analysis, apart from the uniqueness of the infinite hierarchy which was widely regarded as the most involved part, understanding the singular correlation structure generated by the $\delta$ -type potential was one of the main challenges.

In the mean-field limit as the particle number $N$ tends to infinity, the potential $V_{N}$ converges formally to the Dirac-delta interaction $(\int V)\delta$ , also called the Fermi potential. For $\beta<\frac{1}{3}$ , the average distance between the particles, which is $O(N^{-\frac{1}{3}})$ , is much less than the range of the interaction potential, which is $O(N^{-\beta})$ , and there are many but weak correlations. For $\beta>\frac{1}{3}$ , then the analysis is much more involved because of the strong correlations between particles. For $\beta$ close or equal to $1$ , as the scaling is starting to match the Laplacian operator, it is expected that the $\delta$ -type potential generates an interparticle singular correlation structure, closely related to the zero-energy scattering equation

(1.6)

\left\{\begin{aligned} &\left(-\hbar^{2}\Delta+\frac{1}{N}V_{N}(x)\right)f_{N,\hbar}(x)=0,\\ &\lim_{|x|\to\infty}f_{N,\hbar}(x)=1.\end{aligned}\right.

The scattering function $(1-f_{N,\hbar}(x))$ varies effectively on the short scale for $|x|\lesssim N^{-\beta}$ and has the same singularity as the Coulomb potential at infinity. It is believed that³³3See for example [49] for the static case and [28, 30, 31] for the time-dependent case., instead of the factorization property, that is,

\displaystyle\gamma_{N,\hbar}^{(2)}(t,x_{1},x_{2};x_{1}^{\prime},x_{2}^{\prime})\sim\gamma_{N,\hbar}^{(1)}(t,x_{1};x_{1}^{\prime})\gamma_{N,\hbar}^{(1)}(t,x_{2};x_{2}^{\prime}),

the marginal densities should be considered as

\displaystyle\gamma_{N,\hbar}^{(2)}(t,x_{1},x_{2};x_{1}^{\prime},x_{2}^{\prime})\sim f_{N,\hbar}(x_{1}-x_{2})f_{N,\hbar}(x_{1}^{\prime}-x_{2}^{\prime})\gamma_{N,\hbar}^{(1)}(t,x_{1};x_{1}^{\prime})\gamma_{N,\hbar}^{(1)}(t,x_{2};x_{2}^{\prime}).

The singular correlation structure is very subtle and plays a crucial role in the mean-field limit from quantum $N$ -body dynamics, as it gives an $O(N^{\beta})$ correction to the $N$ -body energy.

For the semiclassical limit, the connection between Schrödinger-type equations and the classical fluid mechanics was already noted in 1927 by Madelung [51]. Starting from a single NLS, the asymptotic behavior of the wave function as the Planck’s constant goes to zero is studied by many authors using various approaches based on Madelung’s fluid mechanical formulation. See, for example, [35, 45, 50, 64]. For a more detailed survey related to semiclassical limits, see [5, 65] and references within. There are many deep problems on the study of classical limiting dynamics from quantum equations.

The joint mean-field and semiclassical limit from quantum $N$ -body dynamics formally gives a direct connection between quantum microscopic systems and classical macroscopic fluid equations. Providing a rigorous proof is certainly a challenging problem. For the repulsive Coulomb potential, Golse and Paul [34], based on Serfaty’s inequality [57, Proposition 1.1], justified the weak convergence to pressureless Euler-Poisson in the mean-field and semiclassical limit. For the case of the $\delta$ -type potential, in our previous work [23], we derived the compressible Euler equations with strong and quantitative convergence rate from quantum many-body dynamics by a new strategy of combining the accuracy of the hierarchy method and the flexibility of the modulated energy method. Subsequently, such a scheme was adopted in [24] to obtain the quantitative convergence rate from quantum many-body dynamics to the pressureless Euler-Poisson equation.

Despite a series of progress on the mean-field and semiclassical limit from the quantum $N$ -body dynamics with singular potentials, a number of challenges remain open:

(1)

The derivation of the full Euler-Poisson equation with pressure from the quantum many-body dynamics. In [24, 34], the limiting Euler-Poisson equation is pressureless. However, the pressure is a fluid defining feature and essential for the macroscopic fluid equation. It is thus a fundamental question to understand the emergence of pressure from the microscopic level.
(2)

The large $\beta$ problem is known to be difficult in the mean-field and semiclassical limit due to the strong correlations between particles. The main challenge lies in the analysis of the singular correlation structure generated by the $\delta$ -type potential.
(3)

To obtain the quantitative strong convergence rate, a double-exponential restriction between $N$ and $\hbar$ was needed in [23], which is of course not optimal. From the perspective of energy, the restriction should be at least polynomial. To relax the double-exponential restriction, it requires new and finer techniques.
(4)

The scheme in [23] currently cannot deal with the $\mathbb{T}^{3}$ case, since its proof highly relies on a key collapsing estimate, which fails in the $H^{1}$ energy space for the $\mathbb{T}^{3}$ case as proven in [36]. Thus, the $\mathbb{T}^{3}$ case requires new ideas. The torus case is the beginning to understand other related and important problems, such as the microscopic descriptions of the Mach number and Knudsen number and their limit to incompressible fluids.

In this paper, our goal is to settle the above open problems.

1.2. Statement of the Main Theorem

Starting from the quantum $N$ -body dynamics (1.1), we take the normalization that $\|\psi_{N,\hbar}(t)\|_{L_{X_{N}}^{2}}=1$ , and define the quantum mass density and momentum density by

(1.7)

\displaystyle\rho_{N,\hbar}^{(k)}(t,X_{k})=\gamma_{N,\hbar}^{(k)}(t,X_{k};X_{k}),\quad J_{N,h}^{(1)}(t,x)=\operatorname{Im}\left(\hbar\nabla_{x_{1}}\gamma_{N,\hbar}^{(1)}\right)(t,x;x).

The limiting macroscopic equation would be the compressible Euler-Poisson equation with a pressure term $P=\frac{b_{0}}{2}\rho^{2}$ , which is (in velocity form)

(1.8)

\begin{cases}&\partial_{t}\rho+\nabla\cdot\left(\rho u\right)=0,\\ &\partial_{t}u+(u\cdot\nabla)u+b_{0}\nabla_{x}\rho+\kappa\nabla_{x}(V_{c}*\rho)=0,\\ &(\rho,u)|_{t=0}=(\rho^{in},u^{in}),\end{cases}

or (in momentum form)

(1.9)

\begin{cases}&\partial_{t}\rho+\operatorname{div}J=0,\\ &\partial_{t}J+\operatorname{div}\left(\frac{J\otimes J}{\rho}\right)+\frac{1}{2}\nabla\left(b_{0}\rho^{2}\right)+\kappa\rho\nabla_{x}(V_{c}*\rho)=0,\\ &(\rho,J)|_{t=0}=(\rho^{in},J^{in}).\end{cases}

Here, as usual,

	$\displaystyle\rho(t,x):\mathbb{R}\times\mathbb{R}^{3}\mapsto\mathbb{R}$
	$\displaystyle u(t,x)=(u^{1}(t,x),u^{2}(t,x),u^{3}(t,x)):\mathbb{R}\times\mathbb{R}^{3}\mapsto\mathbb{R}^{3}$
	$\displaystyle J(t,x)=\left(\rho u\right)(t,x):\mathbb{R}\times\mathbb{R}^{3}\mapsto\mathbb{R}^{3}$

are respectively the mass density, the velocity, and the momentum of the fluid. The coupling constant $b_{0}=\int V$ is the macroscopic effect of the microscopic interaction $V$ . When the coefficient $\kappa=0$ , the system $(\ref{equ:euler equation})$ is reduced to a compressible Euler equation. Specifically, we consider the initial data satisfying the condition

(1.10)

\left\{\begin{aligned} \rho^{in}\in H^{s-1}(\mathbb{R}^{3}),\quad u^{in}\in H^{s}(\mathbb{R}^{3}),\\ \rho^{in}(x)\geq 0,\quad\int_{\mathbb{R}^{3}}\rho^{in}(x)dx=1,\end{aligned}\right.

with $s>\frac{9}{2}$ and $s\in\mathbb{N}$ , so that the Euler-Poisson system $(\ref{equ:euler equation})$ has a unique solution $(\rho,u)$ up to some time $T_{0}$ such that⁴⁴4We are not dealing with sharp well-posedness of (1.8) here. The local well-posedness of the Euler system here is known by the standard theory on hyperbolic systems,see [52, 53, 54].

(1.11)

\left\{\begin{aligned} &\rho\in C([0,T_{0}];H^{s-1}(\mathbb{R}^{3})),\quad u\in C([0,T_{0}];H^{s}(\mathbb{R}^{3})),\\ &\rho(t,x)\geq 0,\quad\int_{\mathbb{R}^{3}}\rho(t,x)dx=1.\end{aligned}\right.

Theorem 1.1.

Let $\beta\in(0,1)$ , $\kappa\geq 0$ and the marginal densities $\Gamma_{N,\hbar}=\left\{\gamma_{N,\hbar}^{(k)}\right\}$ associated with $\psi_{N,\hbar}$ be the solution to the $N$ -body dynamics (1.1) with a smooth compactly supported, spherically symmetric nonnegative potential $V$ and a repulsive Coulomb potential $V_{c}$ . Assume the initial data satisfy the following conditions:

$(a)$ $\psi_{N,\hbar}(0)$ is normalized and the $N$ -body energy bound holds:

(1.12)

\displaystyle\langle\psi_{N,\hbar}(0),(H_{N,\hbar}/N+1)^{2}\psi_{N,\hbar}(0)\rangle\leq(E_{0})^{2},

for some $E_{0}>0$ .

$(b)$ The initial data $(\rho^{in},u^{in})$ to $(\ref{equ:euler equation})$ satisfy condition $(\ref{equ:initial condition,euler})$ with $s=5$ , and the modulated energy between (1.1) and (1.8) at initial time tends to zero, that is, $\mathcal{M}(0)\to 0$ where

		$\displaystyle\mathcal{M}(0)$
	$\displaystyle:=$	$\displaystyle\int_{\mathbb{R}^{3N}}\|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(0,X_{N})\|^{2}dX_{N}$
		$\displaystyle+\frac{N-1}{N}\int V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(0,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle+b_{0}\int_{\mathbb{R}^{3}}\rho^{in}(x_{1})\rho^{in}(x_{1})dx_{1}-2b_{0}\int_{\mathbb{R}^{3}}\rho^{in}(x_{1})\rho_{N,\hbar}^{(1)}(0,x_{1})dx_{1}$
		$\displaystyle+\int V_{c}(x_{1}-x_{2})\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(0,x_{1},x_{2})+\rho^{in}(x_{1})\rho^{in}(x_{2})-2\rho^{in}(x_{1})\rho_{N,\hbar}^{(1)}(0,x_{2})\right]dx_{1}dx_{2}.$

Then under the polynomial restriction⁵⁵5(1.13) is in fact a rational restriction. We say polynomial to avoid confusing “rational” and “reasonable”.

(1.13)

\displaystyle r(N,\hbar)=C(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4})\to 0,

we have the quantitative estimates on the strong convergence of the mass density

(1.14)

\displaystyle\|\rho_{N,\hbar}^{(1)}(t,x)-\rho(t,x)\|_{L_{t}^{\infty}[0,T_{0}]L_{x}^{2}(\mathbb{R}^{3})}^{2}\lesssim\mathcal{M}(0)+r(N,\hbar)+\hbar^{2},

and on the convergence of the momentum density

(1.15)

\displaystyle\Big{\|}J_{N,\hbar}^{(1)}(t,x)-(\rho u)(t,x)\Big{\|}_{L_{t}^{\infty}[0,T_{0}]L_{x}^{1}(\mathbb{R}^{3})}^{2}\lesssim\mathcal{M}(0)+r(N,\hbar)+\hbar^{2}.

When $\kappa>0$ , Theorem 1.1 is the first result which simultaneously deals with the $\delta$ -type and Coulomb potentials and establishes the quantitative strong convergence to the full Euler-Poisson equation with pressure. Compared to [34, 24], the emergence of the pressure term is the main novelty. We point out that, it is not clear if the scheme in [23, 24] can handle the $\delta$ -type and Coulomb potentials simultaneously, since the energy estimates and collapsing estimates are totally different in the $\delta$ -type and Coulomb potentials. Therefore, it requires completely new ideas for a simultaneous consideration of $\delta$ -type and Coulomb potentials.

When $\kappa=0$ , it reduces to the sole $\delta$ -type potential case. Compared with our previous work [23], we here list the breakthroughs.

(1)

The parameter $\beta$ is extended to the full range of $(0,1)$ , which is almost optimal in the dilute regime.
(2)

The previous double-exponential restriction between $N$ and $\hbar$ is relaxed to be polynomial, which is a tremendous improvement.
(3)

Our new approach also works for the $\mathbb{T}^{3}$ case with slight modifications, as the proof is independent of the hardcore harmonic analysis on $\mathbb{T}^{3}$ .

Additionally, the convergence rate $\hbar^{2}$ should be optimal since the convergence rate of the modulated kinetic energy part at initial time is at most the order of $\hbar^{2}$ . Besides, this can be achieved with WKB type initial data.

1.3. Outline of the Proof

The proof is based on a modulated energy method.⁶⁶6A closely related method is the relative entropy method, see for example, [63]. The modulated energy we use includes three parts

(1.16)

\displaystyle\mathcal{M}(t)=\mathcal{M}_{K}(t)+\mathcal{F}_{\delta}(t)+\mathcal{F}_{c}(t),

where the kinetic energy part is

(1.17)

\displaystyle\mathcal{M}_{K}(t)=\int_{\mathbb{R}^{3N}}|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(t,X_{N})|^{2}dX_{N},

the $\delta$ -type potential part is

(1.18)		$\displaystyle\mathcal{F}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle+b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho(t,x_{1})dx_{1}-2b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1},$

and the Coulomb potential part is

(1.19)		$\displaystyle\mathcal{F}_{c}(t)=$	$\displaystyle\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}V_{c}(x_{1}-x_{2})\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})\right.$
		$\displaystyle\left.\quad\quad\quad+\rho(t,x_{1})\rho(t,x_{2})-2\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{2})\right]dx_{1}dx_{2}.$

In Section 2, we first derive the time evolution of the modulated energy

(1.20)

\displaystyle\frac{d}{dt}\mathcal{M}(t)=\widetilde{\mathcal{M}}_{K}(t)+\widetilde{\mathcal{F}}_{\delta}(t)+\widetilde{\mathcal{F}}_{c}(t),

where the kinetic energy contribution part is

	$\displaystyle\widetilde{\mathcal{M}}_{K}(t)=$	$\displaystyle-\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)(-i\hbar\partial_{j}\psi_{N,\hbar}-u^{j}\psi_{N,\hbar})\overline{(-i\hbar\partial_{k}\psi_{N,\hbar}-u^{k}\psi_{N,\hbar})}dX_{N}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int_{\mathbb{R}^{3}}\Delta(\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1},$

with the notations $u=(u^{1},u^{2},u^{3})$ , $x_{1}=(x_{1}^{1},x_{1}^{2},x_{1}^{3})\in\mathbb{R}^{3}$ and $\partial_{j}=\partial_{x_{1}^{j}}$ , the $\delta$ -type potential contribution part is

(1.21)		$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-b_{0}\int\operatorname{div}u(t,x_{1})\rho(t,x_{1})\left[\rho(t,x_{1})-2\rho_{N,\hbar}^{(1)}(t,x_{1})\right]dx_{1},$

and the Coulomb potential contribution part is

(1.22)		$\displaystyle\widetilde{\mathcal{F}}_{c}(t)=$	$\displaystyle\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{c}(x_{1}-x_{2})$
		$\displaystyle\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})+\rho(t,x_{1})\rho(t,x_{2})-2\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{2})\right]dx_{1}dx_{2}.$

It is easy to control the kinetic energy contribution part

(1.23)

\displaystyle\widetilde{\mathcal{M}}_{K}(t)\lesssim\mathcal{M}_{K}(t)+\hbar^{2}.

The toughest part in the modulated energy method is to control the potential contribution part both in the classical and quantum setting. See, for example, [27, 34, 50, 56, 57, 64]. In the classical mean-field limit with Coulomb potential, Serfaty in [57, Proposition 1.1] establishes a crucial functional inequality to solve this challenging problem. Then for the quantum many-body systems with Coulomb potential, based on Serfaty’s inequality, Golse and Paul in [34] managed to control the Coulomb potential contribution part $\widetilde{\mathcal{F}}_{c}(t)$ as follows

(1.24)		$\displaystyle\widetilde{\mathcal{F}}_{c}(t)\lesssim$	$\displaystyle\mathcal{F}_{c}(t)+CN^{-\frac{1}{3}},$
(1.25)		$\displaystyle 0\leq$	$\displaystyle\mathcal{F}_{c}(t)+CN^{-\frac{2}{3}}.$

Serfaty’s inequality is a special and impressive tool based on deep observations of the structure of Coulomb potential. It is limited to a special class of singular potentials, as its proof highly relies on the structure and the profile of the potentials, such as the Coulomb characteristic that $-\Delta V_{c}=c_{0}\delta$ . Therefore, it is quite difficult to establish a Serfaty’s inequality for the $\delta$ -type potential case, because of the general profile and sharp singularity of the $\delta$ -type potential. In fact, due to the presence of the singular correlation structure caused by the $\delta$ -type potential, the analysis would be totally different and is expected to be rather intricate.

In this paper, we develop a new scheme without using Serfaty’s inequality to control the $\delta$ -type potential parts $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ and establish

(1.26)		$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)\lesssim$	$\displaystyle\mathcal{F}_{\delta}(t)+r(N,\hbar),$
(1.27)		$\displaystyle 0\leq$	$\displaystyle\mathcal{F}_{\delta}(t)+r(N,\hbar).$

The proof is divided in several steps.

Step 1. Preliminary reduction. Applying the approximation of identity to the one-body term of $\mathcal{F}_{\delta}(t)$ , we have the approximation

(1.28)		$\displaystyle\mathcal{F}_{\delta}(t)\sim$	$\displaystyle\int V_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
		$\displaystyle\quad\quad\left.-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

To have a closed estimate, namely, letting $\widetilde{\mathcal{F}}_{\delta}(t)$ match the approximation of $\mathcal{F}_{\delta}(t)$ , we get by integration by parts for the two-body term that

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\int\operatorname{div}u(t,x_{1})V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-\int(u(t,x)-u(t,y))V_{N}(x-y)\nabla_{x}\rho_{N,\hbar}^{(2)}(t,x,y)dxdy$
		$\displaystyle-b_{0}\int\operatorname{div}u(t,x_{1})\rho(t,x_{1})\left[\rho(t,x_{1})-2\rho_{N,\hbar}^{(1)}(t,x_{1})\right]dx_{1}.$

Using the approximation of identity to the one-body term again, we decompose $\widetilde{\mathcal{F}}_{\delta}(t)$ into the main part and error part

(1.29)

\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=MP+EP,

where

(1.30)	$\displaystyle MP\sim$	$\displaystyle-\int\operatorname{div}u(t,x)V_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
	$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy,$
(1.31)	$\displaystyle EP=$	$\displaystyle-\int(u(t,x)-u(t,y))V_{N}(x-y)\nabla_{x}\rho_{N,\hbar}^{(2)}(t,x,y)dxdy.$

Such a decomposition is based on the key observation that the difference coupled with the $\delta$ -type potential

(1.32)

\displaystyle(u(t,x)-u(t,y))V_{N}(x-y),

when it is tested against a regular function, would vanish in the $N\to\infty$ limit. Such a structure is notably special for the $\delta$ -type potential, since the difference coupled with a common potential including the Coulomb case cannot provide any smallness.

To prove that the error part (1.31) is indeed a small term, it requires the regularity of the two-body density function. Therefore, we delve into the analysis of two-body energy estimates, then deal with the error part and the main part in the Step 3 and 4 respectively.

Step 2. Two-body energy estimate. As usual, a-priori estimates are one of the toughest parts in the study of many-body dynamics as one must seek a regularity high enough for the limiting argument and at the same time low enough that it is provable. In Section 3, we prove that the wave function with added the singular correlation structure satisfies the two-body $H^{1}$ energy bound

(1.33)

\Big{\langle}(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\frac{\psi_{N,\hbar}(t,X_{N})}{1-w_{N,\hbar}(x_{1}-x_{2})},\frac{\psi_{N,\hbar}(t,X_{N})}{1-w_{N,\hbar}(x_{1}-x_{2})}\Big{\rangle}\leq C,

where $w_{N,\hbar}(x)$ satisfies the zero-energy scattering equation

(1.34)

\left\{\begin{aligned} &\left(-\hbar^{2}\Delta+\frac{1}{N}V_{N}(x)\right)(1-w_{N,\hbar}(x))=0,\\ &\lim_{|x|\to\infty}w_{N,\hbar}(x)=0.\end{aligned}\right.

The singular correlation function $w_{N,\hbar}(x)$ varies effectively on the short scale for $|x|\lesssim N^{-\beta}$ and has the same singularity as the Coulomb potential at infinity.

One of the main difficulties here is to understand the interparticle singular correlation structure generated by the $\delta$ -type potential. See, for example, [49] for the study of the static case of Bose gas. For the time-dependent systems, Erdös, Schlein, and Yau [28, 30, 31] first introduced the two-body energy estimate which plays a central role in the derivation of Gross-Pitaevskii equation with the nonlinear interaction given by a scattering length. However, instead of showing the emergence of the scattering length, our purpose here is proving the functional inequalities (1.26) and (1.27).

Another difficulty lies in the Coulomb singularity. The Coulomb potential, if taken to high powers, results in singularities which cannot be controlled by derivatives. The $(H_{N,\hbar})^{2}$ energy estimate (1.33) we prove (and require here) is at the borderline case. Indeed, the square of the Coulomb potential is bounded with respect to the kinetic energy in the sense that as operators $|V_{c}(x)|^{2}\leq C(1-\Delta_{x})$ . However, no such estimates hold for $|V_{c}(x)|^{3}$ due to the singularity of the origin.

Step 3. Analysis of the Error Part. After setting up the energy estimates, we begin to analyze the error part (1.31). Because of the presence of the singular correlation structure, the two-body density function lacks the a-priori energy bound but can be decomposed into the singular and regular (relatively speaking) parts

(1.35)

\displaystyle\rho_{N,\hbar}^{(2)}(t,x,y)=(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}.

Hence, we need to rewrite the error part (1.31) as

		$\displaystyle\int(u(t,x)-u(t,y))\cdot V_{N}(x-y)\nabla_{x}\left[(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right]dxdy$
	$\displaystyle=$	$\displaystyle\int(u(t,x)-u(t,y))\cdot V_{N}(x-y)\left(\nabla_{x}(1-w_{N,\hbar}(x-y))^{2}\right)\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}dxdy$
		$\displaystyle+\int(u(t,x)-u(t,y))\cdot V_{N}(x-y)(1-w_{N,\hbar}(x-y))^{2}\nabla_{x}\left[\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right]dxdy.$

When the derivative hits the singular correlation function, it produces singularities by the defining feature of the singular correlation function, which would give a rise of $O(N^{\beta})$ . On the other hand, when the derivative hits the (relatively) regular part, it still requires a careful analysis as we have limited regularity as discussed before on the modified two-body density function.

In Section 4.1, we prove that, the cancellation structure (1.32) indeed dominates the singularity generated by the $\delta$ -type potential and singular correlation function, and obtain the error estimate

(1.36)

\displaystyle EP\lesssim N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{2}}\hbar^{-4}.

Step 4. Analysis of the Main Part. One difficulty of the analysis of the main part (1.30) is the sharp singularity and the unknown profile of $V_{N}(x)$ . To overcome it, our strategy is to replace $V_{N}(x)$ with a slowly varying potential $G_{N}(x)$ which enjoys a number of good properties, but it comes at a price of the integrand’s regularity. Thus, for the main part (1.30), we again need to decompose the two-body density function into the singular part and relatively regular part as follows

(1.37)		$\displaystyle MP=$	$\displaystyle\int\operatorname{div}u(t,x)V_{N}(x-y)\left[\frac{N-1}{N}\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}(1-w_{N,\hbar}(x-y))^{2}\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

Note that $(1-w_{N,\hbar}(x-y))^{2}\sim 1+O(w_{N,\hbar}(x-y))$ . Then by the two-body energy bound and the property for the scattering function $w_{N,\hbar}(x-y)$ , we can prove that

(1.38)		$\displaystyle MP\sim$	$\displaystyle-\int\operatorname{div}u(t,x)V_{N}(x-y)\left[\frac{N-1}{N}\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

Since the integrand now enjoys the energy bound, we are able to replace $V_{N}$ by $G_{N}$ and get

(1.39)		$\displaystyle MP\sim$	$\displaystyle-b_{0}\int\operatorname{div}u(t,x)G_{N}(x-y)\left[\frac{N-1}{N}\frac{\rho_{N,\hbar}^{(2)}(t,x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy,$

where $G_{N}(x)=N^{3\eta}G(N^{\eta}x)$ with $\eta<\frac{1}{3}$ .

In Section 4.2, we will give a detailed proof of the above analysis and and arrive at the approximations of $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ given by

(1.40)		$\displaystyle\mathcal{F}_{\delta}(t)\sim$	$\displaystyle b_{0}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
		$\displaystyle\quad\quad\left.-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy,$

and

(1.41)		$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)\sim$	$\displaystyle-b_{0}\int\operatorname{div}u(t,x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

Now, from the approximations of $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ , we are left to prove a reduced form of the functional inequality

		$\displaystyle\int\operatorname{div}u(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\lesssim$	$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy+o(1),$

which looks more concise and tractable than the original functional inequality (1.26). But, it is unknown if the integrand

(1.42)

\displaystyle\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)

is non-negative. We cannot simply rule out the term $\operatorname{div}u(x)$ either. Thus, it is still non-trivial to deduce the inequality. In fact, as we will see in Section 4.3, the special structure (1.42) with a slowly varying potential $G_{N}(x)$ plays a crucial role in establishing the reduced version of functional inequality. Then, at the end of Section 4.3, we conclude the functional inequalities (1.26) and (1.27).

Finally in Section 5, by using functional inequalities on $\widetilde{\mathcal{F}}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{c}(t)$ , we prove the Gronwall’s inequality for the positive modulated energy

\displaystyle\frac{d}{dt}\mathcal{M}^{+}(t)\lesssim\mathcal{M}^{+}(t)+\hbar^{2},

where $\mathcal{M}^{+}(t)=\mathcal{M}(t)+2r(N,\hbar)$ . Subsequently, with the quantitative convergence rate of the positive modulated energy, we further conclude the quantitative strong convergence rate of quantum mass and momentum densities, in which the $\delta$ -type potential part plays an indispensable role in upgrading to the quantitative strong convergence.

2. The Time Evolution of the Modulated Energy

We consider the modulated energy in the quantum $N$ -body dynamics corresponding to the $\delta$ -type and Coulomb potentials

\displaystyle\mathcal{M}(t):=

\displaystyle\int_{\mathbb{R}^{3N}}|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(t,X_{N})|^{2}dX_{N}+\mathcal{F}_{\delta}(t)+\mathcal{F}_{c}(t),

where the $\delta$ -type potential part is

(2.1)		$\displaystyle\mathcal{F}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle+b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho(t,x_{1})dx_{1}-2b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1},$

and the Coulomb potential part is

(2.2)		$\displaystyle\mathcal{F}_{c}(t)=$	$\displaystyle\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}V_{c}(x_{1}-x_{2})\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})\right.$
		$\displaystyle\left.\quad\quad\quad+\rho(t,x_{1})\rho(t,x_{2})-2\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{2})\right]dx_{1}dx_{2}.$

Here, we might as well assume that the coefficient $\kappa=1$ , as the proof works the same for $\kappa\geq 0$ .

First, we need to derive a time evolution equation for $\mathcal{M}(t)$ . The related quantities for $\psi_{N,\hbar}$ are given as the following.

Lemma 2.1.

We have the following computations regarding $\psi_{N,\hbar}$ :

(2.3)		$\displaystyle\partial_{t}\rho_{N,\hbar}^{(1)}+\operatorname{div}J_{N,\hbar}^{(1)}=0,$
(2.4)		$\displaystyle\partial_{t}J_{N,\hbar}^{(1)}=\frac{\hbar^{2}}{2}\int\operatorname{Re}\left((-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{2,N}$
	$\displaystyle\quad\quad\quad\quad-\frac{N-1}{N}\int\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{2},$
(2.5)		$\displaystyle E_{N,\hbar}(t)=E_{N,\hbar}(0)\leq E_{0},$

where $X_{2,N}=(x_{2},...,x_{N})$ and the momentum density $J_{N,\hbar}^{(1)}(t,x_{1})$ and the energy $E_{N,\hbar}(t)$ are defined by

(2.6)		$\displaystyle J_{N,\hbar}^{(1)}(t,x_{1})=$	$\displaystyle\operatorname{Im}\left(\hbar\nabla_{x_{1}}\gamma_{N,\hbar}^{(1)}\right)(t,x_{1};x_{1})=\hbar\int\operatorname{Im}(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar})(t,X_{N})dX_{2,N},$
(2.7)		$\displaystyle E_{N,\hbar}(t)=$	$\displaystyle\frac{1}{N}\langle(H_{N,\hbar}+N)\psi_{N,\hbar}(t),\psi_{N,\hbar}(t)\rangle.$

Proof.

As the mass and energy conservation laws are well-known, we omit the proof of (2.3) and (2.5). We provide the proof of the evolution (2.4) of the momentum density. From (2.6), we can write out

	$\displaystyle\partial_{t}J_{N,\hbar}^{(1)}=$	$\displaystyle\hbar\int\operatorname{Im}\left(\overline{\partial_{t}\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\partial_{t}\psi_{N,\hbar}\right)dX_{2,N}$
	$\displaystyle=$	$\displaystyle\int\operatorname{Im}\left(i\overline{H_{N,\hbar}\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}-i\overline{\psi_{N,\hbar}}\nabla_{x_{1}}H_{N,\hbar}\psi_{N,\hbar}\right)dX_{2,N}$
	$\displaystyle=$	$\displaystyle\int\operatorname{Re}\left(\overline{H_{N,\hbar}\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}-\overline{\psi_{N,\hbar}}\nabla_{x_{1}}H_{N,\hbar}\psi_{N,\hbar}\right)dX_{2,N}$
	$\displaystyle=$	$\displaystyle I_{K}+I_{V},$

where

\displaystyle I_{K}=

\displaystyle\frac{\hbar^{2}}{2}\int\operatorname{Re}\left(\sum_{i=1}^{N}(-\Delta_{x_{i}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}-\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\sum_{i=1}^{N}-\Delta_{x_{i}}\psi_{N,\hbar}\right)dX_{2,N},

and

	$\displaystyle I_{V}=$	$\displaystyle\int\operatorname{Re}\left(\frac{1}{N}\sum_{i<j}^{N}(V_{N}+V_{c})(x_{i}-x_{j})\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}\right)dX_{2,N}$
		$\displaystyle-\int\operatorname{Re}\left(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\frac{1}{N}\sum_{i<j}^{N}(V_{N}+V_{c})(x_{i}-x_{j})\psi_{N,\hbar}\right)dX_{2,N}.$

For $I_{K}$ , we use integration by parts with $\Delta_{x_{i}}$ to obtain

\displaystyle I_{K}=\frac{\hbar^{2}}{2}\int\operatorname{Re}\left((-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{2,N},

where the other $i$ -summands vanish when $i\geq 2$ .

For $I_{V}$ , we note that the $i$ -summands also vanish when $i\geq 2$ and hence have

	$\displaystyle I_{V}=$	$\displaystyle\int\operatorname{Re}\left(\frac{1}{N}\sum_{j=2}^{N}(V_{N}+V_{c})(x_{1}-x_{j})\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}\right)dX_{2,N}$
		$\displaystyle-\int\operatorname{Re}\left(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\frac{1}{N}\sum_{j=2}^{N}(V_{N}+V_{c})(x_{1}-x_{j})\psi_{N,\hbar}\right)dX_{2,N}$
	$\displaystyle=$	$\displaystyle-\frac{N-1}{N}\int\|\psi_{N,\hbar}\|^{2}\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})dX_{2,N}$
	$\displaystyle=$	$\displaystyle-\frac{N-1}{N}\int\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{2}.$

This completes the proof of (2.4). ∎

Now, we derive the time evolution of $\mathcal{M}(t)$ .

Proposition 2.2.

Let $\mathcal{M}(t)$ be defined in $\eqref{equ:modulated energy}$ , there holds

(2.8)			$\displaystyle\frac{d}{dt}\mathcal{M}(t)$
	$\displaystyle=$	$\displaystyle-\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)(-i\hbar\partial_{j}\psi_{N,\hbar}-u^{j}\psi_{N,\hbar})\overline{(-i\hbar\partial_{k}\psi_{N,\hbar}-u^{k}\psi_{N,\hbar})}dX_{N}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int_{\mathbb{R}^{3}}\Delta(\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}+\widetilde{\mathcal{F}}_{\delta}(t)+\widetilde{\mathcal{F}}_{c}(t),$

where we used the notations $u=(u^{1},u^{2},u^{3})$ , $x_{1}=(x_{1}^{1},x_{1}^{2},x_{1}^{3})\in\mathbb{R}^{3}$ and $\partial_{j}=\partial_{x_{1}^{j}}$ . Here, the $\delta$ -type potential contribution part is

(2.9)		$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-b_{0}\int\operatorname{div}u(t,x_{1})\rho(t,x_{1})\left[\rho(t,x_{1})-2\rho_{N,\hbar}^{(1)}(t,x_{1})\right]dx_{1},$

and the Coulomb potential contribution part is

(2.10)		$\displaystyle\widetilde{\mathcal{F}}_{c}(t)=$	$\displaystyle\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{c}(x_{1}-x_{2})$
		$\displaystyle\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})+\rho(t,x_{1})\rho(t,x_{2})-2\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{2})\right]dx_{1}dx_{2}.$

Proof.

We decompose the modulated energy into five parts to do the calculation.

	$\displaystyle\mathcal{M}_{1}(t)$	$\displaystyle=\int_{\mathbb{R}^{3N}}\|i\hbar\nabla_{x_{1}}\psi_{N,\hbar}(t,X_{N})\|^{2}dX_{N}$
		$\displaystyle+\frac{N-1}{N}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2},$
	$\displaystyle\mathcal{M}_{2}(t)$	$\displaystyle=i\hbar\int_{\mathbb{R}^{3N}}u(t,x_{1})(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}-\psi_{N,\hbar}\nabla_{x_{1}}\overline{\psi_{N,\hbar}})(t,X_{N})dX_{N},$
	$\displaystyle\mathcal{M}_{3}(t)$	$\displaystyle=\int_{\mathbb{R}^{3N}}\|u(t,x_{1})\psi_{N,\hbar}(t,X_{N})\|^{2}dX_{N},$
	$\displaystyle\mathcal{M}_{4}(t)$	$\displaystyle=b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho(t,x_{1})dx_{1}+\int_{\mathbb{R}^{3}}\rho(t,x_{1})(V_{c}*\rho)(t,x_{1})dx_{1},$
	$\displaystyle\mathcal{M}_{5}(t)$	$\displaystyle=-2b_{0}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}-2\int(V_{c}*\rho)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.$

For $\mathcal{M}_{1}(t)$ , by the symmetry of the wave function $\psi_{N,\hbar}(t)$ , we obtain

	$\displaystyle\mathcal{M}_{1}(t)=$	$\displaystyle\int_{\mathbb{R}^{3N}}\left(-\hbar^{2}\Delta_{x_{1}}\psi_{N,\hbar}+\frac{N-1}{N}(V_{N}+V_{c})(x_{1}-x_{2})\psi_{N,\hbar}\right)\overline{\psi_{N,\hbar}}dX_{N}$
	$\displaystyle=$	$\displaystyle\frac{2}{N}\langle H_{N,\hbar}\psi_{N,\hbar}(t),\psi_{N,\hbar}(t)\rangle$
	$\displaystyle=$	$\displaystyle\frac{2}{N}\langle H_{N,\hbar}\psi_{N,\hbar}(0),\psi_{N,\hbar}(0)\rangle,$

where in the last equality we have used the conservation of energy. Therefore, we have that

\frac{d}{dt}\mathcal{M}_{1}(t)=0.

For $\mathcal{M}_{2}(t)$ , from the definition of $J_{N,\hbar}^{(1)}(t,x)$ in (2.6), we note that

	$\displaystyle\mathcal{M}_{2}(t)=$	$\displaystyle i\hbar\int_{\mathbb{R}^{3N}}u(t,x_{1})(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar}-\psi_{N,\hbar}\nabla_{x_{1}}\overline{\psi_{N,\hbar}})(t,X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle-2\int u(t,x_{1})\hbar\int\operatorname{Im}(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar})(t,X_{N})dX_{2,N}dx_{1}$
	$\displaystyle=$	$\displaystyle-2\int u(t,x_{1})J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.$

Thus, we have

(2.11)

\displaystyle\frac{d}{dt}\mathcal{M}_{2}(t)=

\displaystyle-2\int\partial_{t}u(t,x_{1})J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}-2\int u(t,x_{1})\partial_{t}J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.

For the second term on the r.h.s of (2.11), by (2.4) we obtain

		$\displaystyle-2\int u(t,x_{1})\partial_{t}J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
	$\displaystyle=$	$\displaystyle-\hbar^{2}\int u(t,x_{1})\operatorname{Re}\left((-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{N}$
		$\displaystyle+\frac{2(N-1)}{N}\int u(t,x_{1})\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
(2.12)		$\displaystyle=$	$\displaystyle-\hbar^{2}\int u(t,x_{1})\operatorname{Re}\left((-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{N}$
		$\displaystyle+\frac{(N-1)}{N}\int(u(t,x_{1})-u(t,x_{2}))\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2},$

where in the last equality we used the antisymmetry of $\nabla(V_{N}+V_{c})$ .

Next, we deal with (2.12). By integration by parts, we obtain

		$\displaystyle-\hbar^{2}\int u(t,x_{1})\operatorname{Re}\left((-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}+\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{N}$
	$\displaystyle=$	$\displaystyle-\hbar^{2}\operatorname{Re}\int 2u(t,x_{1})(-\Delta_{x_{1}}\overline{\psi_{N,\hbar}})\nabla_{x_{1}}\psi_{N,\hbar}-(\operatorname{div}u)\overline{\psi_{N,\hbar}}\Delta_{x_{1}}\psi_{N,\hbar}dX_{N}$
	$\displaystyle=$	$\displaystyle-\hbar^{2}\sum_{j,k=1}^{3}\operatorname{Re}\int 2\partial_{k}u^{j}(\partial_{k}\overline{\psi_{N,\hbar}})\partial_{j}\psi_{N,\hbar}+2u^{j}(\partial_{k}\overline{\psi_{N,\hbar}})\partial_{k}\partial_{j}\psi_{N,\hbar}dX_{N}$
		$\displaystyle+\hbar^{2}\operatorname{Re}\int(\operatorname{div}u)\overline{\psi_{N,\hbar}}\Delta_{x_{1}}\psi_{N,\hbar}dX_{N}$
	$\displaystyle=$	$\displaystyle\hbar^{2}\sum_{j,k=1}^{3}\int\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)\partial_{j}\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}dX_{N}$
(2.13)			$\displaystyle-\hbar^{2}\sum_{j,k=1}^{3}\operatorname{Re}\int 2u^{j}(\partial_{k}\overline{\psi_{N,\hbar}})\partial_{k}\partial_{j}\psi_{N,\hbar}dX_{N}+\hbar^{2}\operatorname{Re}\int(\operatorname{div}u)\overline{\psi_{N,\hbar}}\Delta_{x_{1}}\psi_{N,\hbar}dX_{N},$

where we used the notations $u=(u^{1},u^{2},u^{3})$ , $x_{1}=(x_{1}^{1},x_{1}^{2},x_{1}^{3})\in\mathbb{R}^{3}$ and $\partial_{j}=\partial_{x_{1}^{j}}$ .

Using again integration by parts on the two terms of (2.13) gives

(2.14)			$\displaystyle-\hbar^{2}\sum_{j,k=1}^{3}\operatorname{Re}\int 2u^{j}(\partial_{k}\overline{\psi_{N,\hbar}})\partial_{k}\partial_{j}\psi_{N,\hbar}dX_{N}+\hbar^{2}\operatorname{Re}\int(\operatorname{div}u)\overline{\psi_{N,\hbar}}\Delta_{x_{1}}\psi_{N,\hbar}dX_{N}$
	$\displaystyle=$	$\displaystyle\hbar^{2}\operatorname{Re}\int\operatorname{div}u\left(\|\nabla_{x_{1}}\psi_{N,\hbar}\|^{2}+\overline{\psi_{N,\hbar}}\Delta_{x_{1}}\psi_{N,\hbar}\right)dX_{N}$
	$\displaystyle=$	$\displaystyle\frac{\hbar^{2}}{2}\operatorname{Re}\int\operatorname{div}u(\Delta_{x_{1}}\|\psi_{N,\hbar}\|^{2})dX_{N}$
	$\displaystyle=$	$\displaystyle\frac{\hbar^{2}}{2}\int(\Delta\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.$

Combining estimates (2.11)–(2.14), we provide

		$\displaystyle\frac{d}{dt}\mathcal{M}_{2}(t)$
	$\displaystyle=$	$\displaystyle-2\langle\partial_{t}u,J_{N,\hbar}^{(1)}\rangle+\hbar^{2}\sum_{j,k=1}^{3}\int\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)\partial_{j}\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}dX_{N}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int(\Delta\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
		$\displaystyle+\frac{(N-1)}{N}\int(u(t,x_{1})-u(t,x_{2}))\cdot\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}.$

For $\mathcal{M}_{3}(t)$ , by the Euler-Poisson equation (1.8) and the mass conservation law (2.3), we obtain

		$\displaystyle\frac{d}{dt}\mathcal{M}_{3}(t)$
	$\displaystyle=$	$\displaystyle\frac{d}{dt}\int\|u(t,x_{1})\|^{2}\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
	$\displaystyle=$	$\displaystyle\int 2u(t,x_{1})\cdot\partial_{t}u(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}+\int\|u(t,x_{1})\|^{2}\partial_{t}\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
	$\displaystyle=$	$\displaystyle-2\int u(t,x_{1})\cdot\left(u\cdot\nabla u+b_{0}\nabla\rho+\nabla V_{c}*\rho\right)\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
		$\displaystyle+\int\nabla\left(\|u(t,x_{1})\|^{2}\right)J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.$

Expanding it gives

	$\displaystyle\frac{d}{dt}\mathcal{M}_{3}(t)=$	$\displaystyle-2\sum_{j,k=1}^{3}\int u^{k}u^{j}\partial_{j}u^{k}\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}-2b_{0}\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla\rho\rangle-2\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla V_{c}*\rho\rangle$
		$\displaystyle+2\sum_{j,k=1}^{3}\int u^{k}\partial_{j}u^{k}J_{N,\hbar}^{(1)}(t,x_{1})dx_{1}.$

For $\mathcal{M}_{4}(t)$ , plugging in the Euler-Poisson equation (1.8), we have

	$\displaystyle\frac{d}{dt}\mathcal{M}_{4}(t)=$	$\displaystyle b_{0}\frac{d}{dt}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho(t,x_{1})dx_{1}+\frac{d}{dt}\int_{\mathbb{R}^{3}}\rho(t,x_{1})(V_{c}*\rho)(t,x_{1})dx_{1},$
	$\displaystyle=$	$\displaystyle 2b_{0}\langle\rho,u\cdot\nabla\rho\rangle+2\langle\rho,u\cdot\nabla V_{c}*\rho\rangle.$

For $\mathcal{M}_{5}(t)$ , similarly we get to

		$\displaystyle\frac{d}{dt}\mathcal{M}_{5}(t)$
	$\displaystyle=$	$\displaystyle-2b_{0}\frac{d}{dt}\int_{\mathbb{R}^{3}}\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}-2\frac{d}{dt}\int(V_{c}*\rho)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
	$\displaystyle=$	$\displaystyle-2b_{0}\left(\langle\partial_{t}\rho,\rho_{N,\hbar}^{(1)}\rangle+\langle\rho,\partial_{t}\rho_{N,\hbar}^{(1)}\rangle\right)-2\left(\langle\partial_{t}\rho,V_{c}\rho_{N,\hbar}^{(1)}\rangle+\langle V_{c}\rho,\partial_{t}\rho_{N,\hbar}^{(1)}\rangle\right).$

Plugging in the Euler-Poisson equation (1.8) and the mass conservation law (2.3), we have

		$\displaystyle\frac{d}{dt}\mathcal{M}_{5}(t)$
	$\displaystyle=$	$\displaystyle 2b_{0}\left(\langle\operatorname{div}(\rho u),\rho_{N,\hbar}^{(1)}\rangle+\langle\rho,\operatorname{div}J_{N,\hbar}^{(1)}\rangle\right)+2\left(\langle\operatorname{div}(\rho u),V_{c}\rho_{N,\hbar}^{(1)}\rangle+\langle V_{c}\rho,\operatorname{div}J_{N,\hbar}^{(1)}\rangle\right)$
	$\displaystyle=$	$\displaystyle-2b_{0}\langle\rho,u\cdot\nabla\rho_{N,\hbar}^{(1)}\rangle-2\langle\rho,u\cdot\nabla V_{c}\rho_{N,\hbar}^{(1)}\rangle-2b_{0}\langle\nabla\rho,J_{N,\hbar}^{(1)}\rangle-2\langle\nabla V_{c}\rho,J_{N,\hbar}^{(1)}\rangle.$

Therefore, putting the five terms together, we reach

	$\displaystyle\frac{d}{dt}\mathcal{M}(t)=$	$\displaystyle\frac{d}{dt}\mathcal{M}_{1}(t)+\frac{d}{dt}\mathcal{M}_{2}(t)+\frac{d}{dt}\mathcal{M}_{3}(t)+\frac{d}{dt}\mathcal{M}_{4}(t)+\frac{d}{dt}\mathcal{M}_{5}(t)$
(2.15)		$\displaystyle=$	$\displaystyle-2\langle\partial_{t}u,J_{N,\hbar}^{(1)}\rangle-\hbar^{2}\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)\partial_{j}\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}dX_{N}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int_{\mathbb{R}^{3}}\Delta(\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
		$\displaystyle+\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\cdot\nabla_{x_{1}}(V_{N}+V_{c})(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
(2.16)			$\displaystyle-2\sum_{j,k=1}^{3}\int u^{k}u^{j}\partial_{j}u^{k}\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}-2b_{0}\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla\rho\rangle-2\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla V_{c}*\rho\rangle$
		$\displaystyle+\langle\nabla(\|u\|^{2}),J_{N,\hbar}^{(1)}\rangle+2b_{0}\langle\rho,u\cdot\nabla\rho\rangle+2\langle\rho,u\cdot\nabla V_{c}*\rho\rangle$
		$\displaystyle-2b_{0}\langle\rho,u\cdot\nabla\rho_{N,\hbar}^{(1)}\rangle-2\langle\rho,u\cdot\nabla V_{c}\rho_{N,\hbar}^{(1)}\rangle-2b_{0}\langle\nabla\rho,J_{N,\hbar}^{(1)}\rangle-2\langle\nabla V_{c}\rho,J_{N,\hbar}^{(1)}\rangle.$

From the above equation, we collect the $\delta$ -type potential contribution part $\widetilde{\mathcal{F}}_{\delta}(t)$ in (2.8) from

		$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\cdot\nabla_{x_{1}}V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-2b_{0}\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla\rho\rangle-2b_{0}\langle\rho,u\cdot\nabla\rho_{N,\hbar}^{(1)}\rangle+2b_{0}\langle\rho,u\cdot\nabla\rho\rangle$
	$\displaystyle=$	$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\cdot\nabla_{x_{1}}V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-b_{0}\int\operatorname{div}u(t,x_{1})\rho(t,x_{1})\left[\rho(t,x_{1})-2\rho_{N,\hbar}^{(1)}(t,x_{1})\right]dx_{1}.$

and the Coulomb potential contribution part $\widetilde{\mathcal{F}}_{c}(t)$ in (2.8) from

		$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\cdot\nabla_{x_{1}}V_{c}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-2\langle\rho_{N,\hbar}^{(1)},u\cdot\nabla V_{c}\rho\rangle-2\langle\rho,u\cdot\nabla V_{c}\rho_{N,\hbar}^{(1)}\rangle+2\langle\rho,u\cdot\nabla V_{c}*\rho\rangle$
	$\displaystyle=$	$\displaystyle\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{c}(x_{1}-x_{2})$
		$\displaystyle\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})+\rho(t,x_{1})\rho(t,x_{2})-2\rho(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{2})\right]dx_{1}dx_{2},$

where in the last equality we used the antisymmetry of $\nabla V_{c}$ .

As for the first term in (2.8), we use the Euler-Poisson equation (1.8) to combine the terms taking the form of $\langle\bullet,J_{N,\hbar}^{(1)}\rangle$

		$\displaystyle-2\langle\partial_{t}u,J_{N,\hbar}^{(1)}\rangle-2b_{0}\langle\nabla\rho,J_{N,\hbar}^{(1)}\rangle-2\langle\nabla V_{c}*\rho,J_{N,\hbar}^{(1)}\rangle+\langle\nabla(\|u\|^{2}),J_{N,\hbar}^{(1)}\rangle$
	$\displaystyle=$	$\displaystyle 2\langle u\cdot\nabla u,J_{N,\hbar}^{(1)}\rangle+2\langle\nabla\cdot(u\otimes u),J_{N,\hbar}^{(1)}\rangle$
(2.17)		$\displaystyle=$	$\displaystyle i\hbar\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)u^{j}(\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}-\overline{\psi_{N,\hbar}}\partial_{k}\psi_{N,\hbar})dX_{N}.$

If we rewrite the first term on the right hand side of (2.8)

		$\displaystyle-\sum_{j,k=1}^{3}\int\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)(-i\hbar\partial_{j}\psi_{N,\hbar}-u^{j}\psi_{N,\hbar})\overline{(-i\hbar\partial_{k}\psi_{N,\hbar}-u^{k}\psi_{N,\hbar})}dX_{N}$
	$\displaystyle=$	$\displaystyle-\hbar^{2}\sum_{j,k=1}^{3}\int\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)\partial_{j}\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}dX_{N}$
		$\displaystyle-2\sum_{j,k=1}^{3}\int u^{k}u^{j}\partial_{j}u^{k}\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}$
		$\displaystyle+i\hbar\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)u^{j}(\psi_{N,\hbar}\partial_{k}\overline{\psi_{N,\hbar}}-\overline{\psi_{N,\hbar}}\partial_{k}\psi_{N,\hbar})dX_{N},$

these are the $\sum_{j,k}^{3}$ terms in (2.15),(2.16) and (2.17). Therefore, we arrive at equation (2.8) and complete the proof.

∎

3. $(H_{N,\hbar})^{2}$ Energy Estimate Using Singular Correlation Structure

As mentioned in the preliminary reduction step in the outline, Section 1.3, the two-body energy estimate is crucial for the analysis of the $\delta$ -type potential parts $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ . The main difficulty is the singularities simultaneously from the Coulomb potential, from the direct $\delta$ -potential in the $N\to\infty$ limit, and from the interparticle singular correlation structure.

Recall the zero-energy scattering equation

(3.1)

\left\{\begin{aligned} &\left(-\hbar^{2}\Delta+\frac{1}{N}V_{N}(x)\right)(1-w_{N,\hbar}(x))=0,\\ &\lim_{|x|\to\infty}w_{N,\hbar}(x)=0.\end{aligned}\right.

and our target estimate

(3.2)

\displaystyle\Big{\langle}(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\frac{\psi_{N,\hbar}(t,X_{N})}{1-w_{N,\hbar}(x_{1}-x_{2})},\frac{\psi_{N,\hbar}(t,X_{N})}{1-w_{N,\hbar}(x_{1}-x_{2})}\Big{\rangle}\leq C.

We first give the properties of the scattering function.

Lemma 3.1.

Suppose that $V\geq 0$ is smooth, spherical symmetric with compact support and $1-w_{N,\hbar}(x)$ satisfies the scattering equation (3.1). Then there exists $C$ , depending on $V$ , such that

(3.3)		$\displaystyle 0\leq w_{N,\hbar}(x)\leq\frac{C}{N\hbar^{2}(\|x\|+N^{-\beta})},$
(3.4)		$\displaystyle\|\nabla w_{N,\hbar}(x)\|\leq\frac{C}{N\hbar^{2}(\|x\|^{2}+N^{-2\beta})},$

for all $x\in\mathbb{R}^{3}$ .

Proof.

The properties of scattering function have been studied by many authors, see, for example, [3, 28, 49]. Here, we include a proof for completeness. First, by the maximum principle, it follows that $(1-w_{N,\hbar}(x))\leq 1$ . From the scattering equation (1.6), we can rewrite

(3.5)

\displaystyle w_{N,\hbar}(x)=\frac{c_{0}}{2N\hbar^{2}}\int\frac{1}{|x-y|}V_{N}(y)(1-w_{N,\hbar}(y))dy,

where $c_{0}$ is the renormalized constant.

Then the Hardy-Littlewood-Sobolev inequality implies that

	$\displaystyle(\|x\|+N^{-\beta})w_{N,\hbar}(x)=$	$\displaystyle\frac{c_{0}}{N\hbar^{2}}\int\frac{\|x\|+N^{-\beta}}{\|x-y\|}V_{N}(y)(1-w_{N,\hbar}(y))dy$
	$\displaystyle\leq$	$\displaystyle\frac{c_{0}}{N\hbar^{2}}\int\frac{\|x-y\|+\|y\|+N^{-\beta}}{\|x-y\|}V_{N}(y)dy$
	$\displaystyle=$	$\displaystyle\frac{c_{0}\\|V\\|_{L^{1}}}{N\hbar^{2}}+\frac{c_{0}}{N^{1+\beta}\hbar^{2}}\int\frac{1+N^{\beta}\|y\|V_{N}(y)}{\|x-y\|}dy$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{N\hbar^{2}}\left(\\|V\\|_{L^{1}}+\\|\langle y\rangle V(y)\\|_{L^{\frac{3}{2}}}\right).$

For (3.4), by taking the gradient of (3.5), we also have

		$\displaystyle\|(\|x\|^{2}+N^{-2\beta})\nabla_{x}w_{N,\hbar}(x)\|$
	$\displaystyle\leq$	$\displaystyle\frac{1}{2N\hbar^{2}}(\|x\|^{2}+N^{-2\beta})\Big{\|}\int\nabla_{x}\frac{1}{\|x-y\|}V_{N}(y)(1-w_{N,\hbar}(y))dy\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{N\hbar^{2}}\int\frac{\|x-y\|^{2}+\|y\|^{2}+N^{-2\beta}}{\|x-y\|^{2}}V_{N}(y)dy$
	$\displaystyle\leq$	$\displaystyle\frac{\\|V\\|_{L^{1}}}{N\hbar^{2}}+\frac{1}{N\hbar^{2}}\int\frac{1+N^{2\beta}\|y\|^{2}V_{N}(y)}{\|x-y\|^{2}}dy$
	$\displaystyle\leq$	$\displaystyle\frac{1}{N\hbar^{2}}\left(\\|V\\|_{L^{1}}+\\|\langle y\rangle^{2}V(y)\\|_{L^{3}}\right).$

∎

For simplicity, we adopt the shorthands

(3.6)

\displaystyle w_{12}=w_{N,\hbar}(x_{1}-x_{2}),\quad\nabla w_{12}=(\nabla w_{N,\hbar})(x_{1}-x_{2}),

and start the proof of (3.2).

Lemma 3.2.

Let $\beta\in(0,1)$ and $N^{\beta-1}\hbar^{-2}\ll 1$ . Then we have

(3.7)

\displaystyle\langle\psi,(H_{N,\hbar}+N)^{2}\psi\rangle\geq\frac{N(N-1)}{16}\Big{\langle}(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\frac{\psi}{1-w_{12}},\frac{\psi}{1-w_{12}}\Big{\rangle}

for $\psi\in L_{s}^{2}(\mathbb{R}^{3N})$ .

Proof.

Let

(3.8)

\displaystyle T_{i}:=1-\frac{\hbar^{2}}{2}\Delta_{i}+\frac{1}{2N}\sum_{j:j\neq i}V_{N}(x_{i}-x_{j})+\frac{1}{2N}\sum_{j:j\neq i}V_{c}(x_{i}-x_{j}),

we rewrite the Hamiltonian (1.2)

\displaystyle H_{N,\hbar}+N=\sum_{i=1}^{N}T_{i}.

By the symmetry of $\psi$ , we have

(3.9)		$\displaystyle\langle\psi,(H_{N,\hbar}+N)^{2}\psi\rangle=$	$\displaystyle\sum_{i,j}^{N}\langle\psi,T_{i}T_{j}\psi\rangle$
	$\displaystyle=$	$\displaystyle N(N-1)\langle\psi,T_{1}T_{2}\psi\rangle+N\langle\psi,T_{1}^{2}\psi\rangle$
	$\displaystyle\geq$	$\displaystyle N(N-1)\langle\psi,T_{1}T_{2}\psi\rangle.$

Note that $\psi=(1-w_{12})\phi_{12}$ and we have

	$\displaystyle-\hbar^{2}\Delta_{1}\psi=$	$\displaystyle-\hbar^{2}\Delta_{1}[(1-w_{12})\phi_{12}]$
	$\displaystyle=$	$\displaystyle(1-w_{12})(-\hbar^{2}\Delta_{1}\phi_{12})+2\hbar\nabla_{1}w_{12}\hbar\nabla_{1}\phi_{12}+\hbar^{2}\Delta_{1}w_{12}\phi_{12}.$

Thus, together with the scattering equation (3.1), we arrive at

(3.10)		$\displaystyle T_{1}\psi=$	$\displaystyle T_{1}[(1-w_{12})\phi_{12}]$
	$\displaystyle=$	$\displaystyle(1-w_{12})(\phi_{12}-\frac{\hbar^{2}}{2}\Delta_{1}\phi_{12})+\hbar\nabla_{1}w_{12}\hbar\nabla_{1}\phi_{12}+\frac{\hbar^{2}}{2}\Delta_{1}w_{12}\phi_{12}$
		$\displaystyle+(1-w_{12})\left[\frac{1}{2N}\sum_{j\geq 2}V_{N}(x_{1}-x_{j})\phi_{12}+\frac{1}{2N}\sum_{j\geq 2}V_{c}(x_{1}-x_{j})\phi_{12}\right]$
	$\displaystyle=$	$\displaystyle(1-w_{12})\left[\phi_{12}-\frac{\hbar^{2}}{2}\Delta_{1}\phi_{12}+\frac{\hbar\nabla_{1}w_{12}}{1-w_{12}}\hbar\nabla_{1}\phi_{12}\right]$
		$\displaystyle+(1-w_{12})\left[\frac{1}{2N}\sum_{j\geq 3}V_{N}(x_{1}-x_{j})\phi_{12}+\frac{1}{2N}\sum_{j\geq 2}V_{c}(x_{1}-x_{j})\phi_{12}\right].$

Similarly, we also have

(3.11)		$\displaystyle T_{2}\psi=$	$\displaystyle(1-w_{12})\left[\phi_{12}-\frac{\hbar^{2}}{2}\Delta_{2}\phi_{12}+\frac{\hbar\nabla w_{12}}{1-w_{12}}\hbar\nabla_{2}\phi_{12}\right]$
		$\displaystyle+(1-w_{12})\left[\frac{1}{2N}\sum_{j\geq 3}V_{N}(x_{2}-x_{j})\phi_{12}+\frac{1}{2N}\sum_{j\neq 2}V_{c}(x_{2}-x_{j})\phi_{12}\right].$

Further define the shorthands

(3.12)		$\displaystyle L_{1}:=1-\frac{\hbar^{2}}{2}\Delta_{1}+\frac{\hbar\nabla_{1}w_{12}}{1-w_{12}}\hbar\nabla_{1},$
(3.13)		$\displaystyle L_{2}:=1-\frac{\hbar^{2}}{2}\Delta_{2}+\frac{\hbar\nabla_{2}w_{12}}{1-w_{12}}\hbar\nabla_{2},$

which are symmetric with respect to the measure $(1-w_{12})^{2}dx$ , that is,

(3.14)		$\displaystyle\int(1-w_{12})^{2}\overline{f}(L_{1}g)=$	$\displaystyle\int(1-w_{12})^{2}(\overline{L_{1}f})g$
	$\displaystyle=$	$\displaystyle\int(1-w_{12})^{2}\left[\overline{f}g+\frac{\hbar^{2}}{2}\nabla_{1}\overline{f}\nabla_{1}g\right].$

Therefore, from (3.10) and (3.11) we obtain

		$\displaystyle\langle T_{1}\psi,T_{2}\psi\rangle$
	$\displaystyle=$	$\displaystyle\int(1-w_{12})^{2}\left(L_{1}+\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{1}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right)\overline{\phi}_{12}$
		$\displaystyle\cdot\left(L_{2}+\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{2}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right)\phi_{12}.$

Expanding it gives

(3.15)			$\displaystyle\langle T_{1}\psi,T_{2}\psi\rangle$
	$\displaystyle=$	$\displaystyle\int(1-w_{12})^{2}L_{1}\overline{\phi}_{12}L_{2}\phi_{12}$
		$\displaystyle+\int(1-w_{12})^{2}(L_{1}\overline{\phi}_{12})\left[\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{2}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right]\phi_{12}$
		$\displaystyle+\int(1-w_{12})^{2}\left[\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{1}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right]\overline{\phi}_{12}L_{2}\phi_{12}$
		$\displaystyle+\int(1-w_{12})^{2}\left[\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{1}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right]\overline{\phi}_{12}$
		$\displaystyle\quad\quad\quad\quad\quad\quad\quad\cdot\left[\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{1}-x_{j})+\frac{1}{2N}V_{c}(x_{1}-x_{2})\right]\overline{\phi}_{12}.$

By the nonnegativity of the potentials, we can discard the last term on the r.h.s of (3.15). The symmetry property (3.14) of the operators $L_{1}$ and $L_{2}$ then yields

		$\displaystyle\langle T_{1}\psi,T_{2}\psi\rangle$
	$\displaystyle\geq$	$\displaystyle\int(1-w_{12})^{2}L_{1}\overline{\phi}_{12}L_{2}\phi_{12}$
		$\displaystyle+\int(1-w_{12})^{2}\left(\|\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{1}\phi_{12}\|^{2}\right)\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{2}-x_{j})$
		$\displaystyle+\int(1-w_{12})^{2}\|\phi_{12}\|^{2}\frac{1}{2N}V_{c}(x_{1}-x_{2})+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}\nabla_{1}\left(\frac{1}{2N}V_{c}(x_{1}-x_{2})\phi_{12}\right)$
		$\displaystyle+\int(1-w_{12})^{2}\left(\|\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{2}\phi_{12}\|^{2}\right)\frac{1}{2N}\sum_{j\geq 3}(V_{N}+V_{c})(x_{2}-x_{j})$
		$\displaystyle+\int(1-w_{12})^{2}\|\phi_{12}\|^{2}\frac{1}{2N}V_{c}(x_{1}-x_{2})+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{2}\left(\frac{1}{2N}V_{c}(x_{1}-x_{2})\overline{\phi}_{12}\right)\nabla_{2}\phi_{12}.$

Again using the nonnegativity of the potentials, we reach

(3.16)		$\displaystyle\langle T_{1}\psi,T_{2}\psi\rangle\geq$	$\displaystyle\int(1-w_{12})^{2}L_{1}\overline{\phi}_{12}L_{2}\phi_{12}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}\nabla_{1}\left(\frac{1}{2N}V_{c}(x_{1}-x_{2})\phi_{12}\right)$
		$\displaystyle+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{2}\left(\frac{1}{2N}V_{c}(x_{1}-x_{2})\overline{\phi}_{12}\right)\nabla_{2}\phi_{12}$
	$\displaystyle=$	$\displaystyle I+II+III.$

For the first term $I$ on the r.h.s of (3.16), by (3.14), we have

(3.17)		$\displaystyle I=$	$\displaystyle\int(1-w_{12})^{2}\left[\overline{\phi}_{12}L_{2}\phi_{12}+\frac{\hbar^{2}}{2}\nabla_{1}\overline{\phi}_{12}\nabla_{1}L_{2}\phi_{12}\right]$
	$\displaystyle=$	$\displaystyle\int(1-w_{12})^{2}\left[\|\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{2}\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{1}\phi_{12}\|^{2}+\frac{\hbar^{4}}{4}\|\nabla_{1}\nabla_{2}\phi_{12}\|^{2}\right]$
		$\displaystyle+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}[\nabla_{1},L_{2}]\phi_{12}$
	$\displaystyle\geq$	$\displaystyle\frac{1}{2}\int\left[\|\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{2}\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{1}\phi_{12}\|^{2}+\frac{\hbar^{4}}{4}\|\nabla_{1}\nabla_{2}\phi_{12}\|^{2}\right]$
		$\displaystyle+\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}[\nabla_{1},L_{2}]\phi_{12},$

where in the last inequality we have used Lemma 3.1 that $(1-w_{12})^{2}\geq\frac{1}{2}$ . To control the last term on the r.h.s of (3.17), we note that

\displaystyle|[\nabla_{1},L_{2}]|=\hbar^{2}\Big{|}\left[\nabla_{1},\frac{\nabla_{2}w_{12}}{1-w_{12}}\right]\Big{|}\leq\hbar^{2}\left[\frac{|\nabla^{2}w_{12}|}{1-w_{12}}+\left(\frac{\nabla w_{12}}{1-w_{12}}\right)^{2}\right].

Therefore, we have

	$\displaystyle\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}[\nabla_{1},L_{2}]\phi_{12}\leq$	$\displaystyle\frac{\hbar^{4}}{2}\int\left(\|\nabla^{2}w_{12}\|+\|\nabla w_{12}\|^{2}\right)\|\nabla_{1}\phi_{12}\|\|\phi_{12}\|$
	$\displaystyle=$	$\displaystyle I_{1}+I_{2}.$

For $I_{1}$ , by Hölder and Sobolev inequalities we have

	$\displaystyle I_{1}\leq$	$\displaystyle\hbar^{4}\\|\nabla^{2}w_{12}\\|_{L_{x_{2}}^{\frac{3}{2}}}\\|\nabla_{1}\phi_{12}\\|_{L^{2}L_{x_{2}}^{6}}\\|\phi_{12}\\|_{L^{2}L_{x_{2}}^{6}}$
	$\displaystyle\lesssim$	$\displaystyle\hbar^{4}\\|\nabla^{2}w_{12}\\|_{L_{x_{2}}^{\frac{3}{2}}}\\|\nabla_{1}\nabla_{2}\phi_{12}\\|_{L^{2}L_{x_{2}}^{2}}\\|\nabla_{2}\phi_{12}\\|_{L^{2}L_{x_{2}}^{2}}.$

By the Calderón-Zygmund theory which implies that $\|\nabla^{2}f\|_{L^{p}}\lesssim\|\Delta f\|_{L^{p}}$ for $1<p<\infty$ and the scattering equation (3.1), we get

\displaystyle\hbar^{2}\|\nabla^{2}w_{12}\|_{L_{x_{2}}^{\frac{3}{2}}}\lesssim\hbar^{2}\|\Delta w_{12}\|_{L_{x_{2}}^{\frac{3}{2}}}\leq\frac{1}{N}\|V_{N}(x_{1}-x_{2})\|_{L_{x_{2}}^{\frac{3}{2}}}\lesssim\frac{N^{\beta}}{N}.

Thus, we arrive at

(3.18)

\displaystyle I_{1}\lesssim\frac{N^{\beta}}{N}\hbar^{2}\left(\|\nabla_{1}\nabla_{2}\phi_{12}\|_{L^{2}}^{2}+\|\nabla_{1}\phi_{12}\|_{L^{2}}^{2}\right).

For $I_{2}$ , by the properties of the scattering function in Lemma 3.1, we have

\displaystyle|\nabla w_{12}|\leq\frac{C}{N\hbar^{2}(|x_{1}-x_{2}|^{2}+N^{-2\beta})},

which implies that

\displaystyle|\nabla w_{12}|^{2}\lesssim\frac{N^{2\beta}}{N\hbar^{2}}\frac{1}{N\hbar^{2}|x_{1}-x_{2}|^{2}}=\frac{N^{2\beta}}{N^{2}\hbar^{4}|x_{1}-x_{2}|^{2}}.

Then by Cauchy-Schwarz and Hardy’s inequalities, we get

(3.19)		$\displaystyle I_{2}\leq$	$\displaystyle\hbar^{4}\int\|\nabla w_{12}\|^{2}(\|\nabla_{1}\phi_{12}\|^{2}+\|\phi_{12}\|^{2})$
	$\displaystyle\lesssim$	$\displaystyle\frac{N^{2\beta}}{N^{2}}\left(\\|\nabla_{1}\nabla_{2}\phi_{12}\\|_{L^{2}}^{2}+\\|\nabla_{2}\phi_{12}\\|_{L^{2}}^{2}\right).$

Next, we deal with the terms $II$ and $III$ in (3.17). For $II$ , we have

	$\displaystyle II=$	$\displaystyle\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\nabla_{1}\overline{\phi}_{12}\nabla_{1}\left(\frac{1}{2N}V_{c}(x_{1}-x_{2})\phi_{12}\right)$
	$\displaystyle=$	$\displaystyle\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}\left[\|\nabla_{1}\phi_{12}\|^{2}\frac{1}{2N}V_{c}(x_{1}-x_{2})+(\nabla_{1}\overline{\phi}_{12})\phi_{12}\nabla_{1}\frac{1}{2N}V_{c}(x_{1}-x_{2})\right]$
	$\displaystyle\geq$	$\displaystyle\frac{\hbar^{2}}{2}\int(1-w_{12})^{2}(\nabla_{1}\overline{\phi}_{12})\phi_{12}\nabla_{1}\frac{1}{2N}V_{c}(x_{1}-x_{2}),$

where in the last inequality we used the positivity of the Coulomb potential. Noting that $|\nabla V_{c}(x)|\lesssim|x|^{-2}$ , we can use Cauchy-Schwarz and Hardy’s inequalities to obtain

(3.20)		$\displaystyle II\geq$	$\displaystyle-\frac{\hbar^{2}}{2N}\int\left(\|\nabla_{1}\phi_{12}\|^{2}+\|\phi_{12}\|^{2}\right)\frac{1}{\|x_{1}-x_{2}\|^{2}}$
	$\displaystyle\gtrsim$	$\displaystyle-\frac{\hbar^{2}}{N}\left(\\|\nabla_{1}\nabla_{2}\phi_{12}\\|_{L^{2}}^{2}+\\|\nabla_{2}\phi_{12}\\|_{L^{2}}^{2}\right).$

As the term $III$ can be estimated in the same way as $II$ , we also have

(3.21)

\displaystyle III\gtrsim

\displaystyle-\frac{\hbar^{2}}{N}\left(\|\nabla_{1}\nabla_{2}\phi_{12}\|_{L^{2}}^{2}+\|\nabla_{2}\phi_{12}\|_{L^{2}}^{2}\right).

Together with estimates (3.16)–(3.21), we arrive at

	$\displaystyle\langle T_{1}\psi,T_{2}\psi\rangle\geq$	$\displaystyle\frac{1}{2}\int\left[\|\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{2}\phi_{12}\|^{2}+\frac{\hbar^{2}}{2}\|\nabla_{1}\phi_{12}\|^{2}+\frac{\hbar^{4}}{4}\|\nabla_{1}\nabla_{2}\phi_{12}\|^{2}\right]$
		$\displaystyle-C(\frac{N^{\beta}}{N}\hbar^{2}+\frac{N^{2\beta}}{N^{2}})\left(\\|\nabla_{1}\nabla_{2}\phi_{12}\\|_{L^{2}}^{2}+\\|\nabla_{1}\phi_{12}\\|_{L^{2}}^{2}\right)$
	$\displaystyle\geq$	$\displaystyle\frac{1}{16}\int\|\phi_{12}\|^{2}+\hbar^{2}\|\nabla_{2}\phi_{12}\|^{2}+\hbar^{2}\|\nabla_{1}\phi_{12}\|^{2}+\hbar^{4}\|\nabla_{1}\nabla_{2}\phi_{12}\|^{2}$
	$\displaystyle=$	$\displaystyle\frac{1}{16}\langle(1-\hbar^{2}\Delta_{1})(1-\hbar^{2}\Delta_{2})\phi_{12},\phi_{12}\rangle,$

where in the second-to-last inequality we have used that $N^{\beta-1}\hbar^{-2}\ll 1$ . With (3.9), we complete the proof of the estimate (3.7). ∎

Proposition 3.3.

Let $\beta\in(0,1)$ and $N^{\beta-1}\hbar^{2}\ll 1$ . Define

\displaystyle\phi_{N,\hbar,12}(t,X_{N})=\frac{\psi_{N,\hbar}(t,X_{N})}{1-w_{N,\hbar}(x_{1}-x_{2})}.

There exists a constant $C>0$ such that

(3.22)

\displaystyle\langle(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\phi_{N,\hbar,12}(t),\phi_{N,\hbar,12}(t)\rangle\leq C

for all $t\in\mathbb{R}$ .

Proof.

By the $(H_{N,\hbar})^{2}$ energy estimate in Lemma 3.2, we have

		$\displaystyle\langle(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\phi_{N,\hbar,12}(t),\phi_{N,\hbar,12}(t)\rangle$
	$\displaystyle\leq$	$\displaystyle\frac{16}{N(N-1)}\langle\psi_{N,\hbar}(t),(N+H_{N,\hbar})^{2}\psi_{N,\hbar}(t)\rangle$
	$\displaystyle=$	$\displaystyle\frac{16}{N(N-1)}\langle\psi_{N,\hbar}(0),(N+H_{N,\hbar})^{2}\psi_{N,\hbar}(0)\rangle$
	$\displaystyle\leq$	$\displaystyle 16(E_{0})^{2},$

where we have used the conservation of $(H_{N,\hbar})^{2}$ in the second-to-last equality, and the initial energy condition (1.12) in the last inequality. ∎

4. Functional Inequalities

In the section, with the a-priori energy bound established in Proposition 3.3, we control the $\delta$ -type potential parts $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ and establish the functional inequalities (1.26) and (1.27). In Section 4.1, we deal with the error analysis of the two-body term of $\widetilde{\mathcal{F}}_{\delta}(t)$ and then find the main part of $\widetilde{\mathcal{F}}_{\delta}(t)$ . In Section 4.2, we estimate the main part. By a replacement argument, we find proper approximations of $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ , and hence arrive at a reduction of the functional inequality. We then complete the proof of the reduced version of functional inequalities in Section 4.3.

The main goal of the section is the following proposition which is the precise form of (1.26) and (1.27).

Proposition 4.1.

Let $\beta\in(0,1)$ , we have the estimate

(4.1)

\widetilde{\mathcal{F}}_{\delta}(t)\lesssim\mathcal{F}_{\delta}(t)+O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4})

and a lower bound of $\mathcal{F}_{\delta}(t)$

(4.2)

\displaystyle 0\leq\mathcal{F}_{\delta}(t)+O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4}).

Here, the notation $O(a+b)$ is a shorthand for $O(a)+O(b)$ and the notation $O(N^{-\alpha_{1}}\hbar^{-\alpha_{2}})$ denotes the same order of $N^{-\alpha_{1}}\hbar^{-\alpha_{2}}$ up to an unimportant constant⁷⁷7The constant could depend on the usual Sobolev constants and the fixed parameters such as the time $T_{0}$ , the energy bound $E_{0}$ , and the Sobolev norms of $(\rho,u)$ but the constant is independent of $(N,\hbar)$ . $C$ .

Proof.

We postpone the proof of Proposition 4.1 to the end of the Section 4.3. ∎

4.1. Error Analysis of Two-Body Term

From the expression of $\widetilde{\mathcal{F}}_{\delta}(t)$

(4.3)		$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=$	$\displaystyle\frac{N-1}{N}\int(u(t,x_{1})-u(t,x_{2}))\nabla V_{N}(x_{1}-x_{2})\rho_{N,\hbar}^{(2)}(t,x_{1},x_{2})dx_{1}dx_{2}$
		$\displaystyle-b_{0}\int\operatorname{div}u(t,x_{1})\rho(t,x_{1})\left[\rho(t,x_{1})-2\rho_{N,\hbar}^{(1)}(t,x_{1})\right]dx_{1},$

the difficult part is the two-body term

\displaystyle\int(u(t,x)-u(t,y))\cdot\nabla V_{N}(x-y)\rho_{N,\hbar}^{(2)}(t,x,y)dxdy.

At first sight, the lack of a uniform regularity estimate for the two-body density function $\rho_{N,\hbar}^{(2)}(x,y)$ makes further analysis difficult. With the singular correlation structure in mind, we decompose the two-body density function into the singular and regular parts

\displaystyle\rho_{N,\hbar}^{(2)}(x,y)=(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}},

and rewrite the two-body term as

(4.4)

\displaystyle\int(u(x)-u(y))\cdot\nabla V_{N}(x-y)(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}}dxdy.

That is, the singularities come from the potential $\nabla V_{N}$ and the singular correlation function $(1-w_{N,\hbar}(x-y))^{2}$ . As mentioned in (1.32) at the outline, a key observation to beat the singularities is a cancellation structure from the difference coupled with the $\delta$ -type potential

(4.5)

\displaystyle(u(x)-u(y))V_{N}(x-y),

which would vanish as $N$ tends to the infinity. Such a structure is special for the $\delta$ -type potential. Many common potentials including the Coulomb do not carry such a property.

We will prove that, based on (3.22), the cancellation structure (4.5) dominates the singularities generated by the delta-potential and singular correlation function, which allows us extract the main term from the two-body term.

Lemma 4.2.

Let $\beta\in(0,1)$ , we have

(4.6)			$\displaystyle\int(u(x)-u(y))\cdot\nabla V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
	$\displaystyle=$	$\displaystyle-\int\operatorname{div}u(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{2}}\hbar^{-4}),$

where the notation $f=g\pm O(N,\hbar)$ means $|f-g|\leq O(N,\hbar)$ .

Proof.

First, due to the singular correlation structure, we rewrite the two-body term as (4.4). To employ the cancellation structure (4.5), we take the derivative off $V_{N}$ by integrating by parts

		$\displaystyle\int(u(x)-u(y))\cdot\nabla V_{N}(x-y)(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}}dxdy$
	$\displaystyle=$	$\displaystyle-\int\operatorname{div}u(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
(4.7)			$\displaystyle-\int(u(x)-u(y))V_{N}(x-y)\nabla_{x}\left[(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right]dxdy.$

It remains to show the term (4.7) is indeed an error term. For simplicity, we set

w_{12}=w_{N,\hbar}(x_{1}-x_{2}),\quad\nabla w_{12}=(\nabla w_{N,\hbar})(x_{1}-x_{2}),\quad\phi_{N,\hbar,12}=(1-w_{12})\psi_{N,\hbar}.

Then we have

(4.8)			$\displaystyle\int(u(x)-u(y))V_{N}(x-y)\nabla_{x}\left[(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right]dxdy$
	$\displaystyle=$	$\displaystyle\int(u(x_{1})-u(x_{2}))V_{N}(x_{1}-x_{2})\nabla_{x_{1}}\left[(1-w_{12})\phi_{N,\hbar,12}\right]^{2}dX_{N}$
	$\displaystyle\leq$	$\displaystyle 2\\|\nabla u\\|_{L^{\infty}}\int\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})\|\nabla w_{12}\|(1-w_{12})\|\phi_{N,\hbar,12}\|^{2}dX_{N}$
		$\displaystyle+2\\|\nabla u\\|_{L^{\infty}}\int\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})(1-w_{12})^{2}\|\nabla_{x_{1}}\phi_{N,\hbar,12}\|\|\phi_{N,\hbar,12}\|dX_{N}$
	$\displaystyle=:$	$\displaystyle 2(A+B).$

We bound $A$ and $B$ , using the properties of the scattering function, the two-body energy estimate (3.22) and the operator inequalities in Lemma A.2.

For the term $A$ , by Lemma 3.1, we have the upper bound estimate

(1-w_{12})\leq 1,\quad|\nabla w_{12}|\lesssim\frac{N^{2\beta}}{N\hbar^{2}}.

Therefore, we arrive at

(4.9)		$\displaystyle A=$	$\displaystyle\int\\|\nabla u\\|_{L^{\infty}}\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})\|\nabla w_{12}\|(1-w_{12})\|\phi_{N,\hbar,12}\|^{2}dX_{N}$
	$\displaystyle\lesssim$	$\displaystyle\frac{N^{\beta}}{N\hbar^{2}}\\|\nabla u\\|_{L^{\infty}}\int N^{\beta}\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})\|\phi_{N,\hbar,12}\|^{2}dX_{N}$
	$\displaystyle\lesssim$	$\displaystyle\frac{N^{\beta}}{N\hbar^{2}}\\|\nabla u\\|_{L^{\infty}}\\|\|x\|V(x)\\|_{L^{1}}\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle,$

where in the last inequality we used the operator inequality (A.2).

For the term $B$ , we first discard $(1-w_{12})^{2}$ and then use Cauchy-Schwarz to get

	$\displaystyle B=$	$\displaystyle\\|\nabla u\\|_{L^{\infty}}\int\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})(1-w_{12})^{2}\|\nabla_{x_{1}}\phi_{N,\hbar,12}\|\|\phi_{N,\hbar,12}\|dX_{N}$
(4.10)		$\displaystyle\leq$	$\displaystyle\frac{\\|\nabla u\\|_{L^{\infty}}}{N^{\beta}}\left[\alpha\langle N^{\beta}\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\right.$
		$\displaystyle\left.+\alpha^{-1}\langle N^{\beta}\|x_{1}-x_{2}\|V_{N}(x_{1}-x_{2})\nabla_{x_{1}}\phi_{N,\hbar,12},\nabla_{x_{1}}\phi_{N,\hbar,12}\rangle\right].$

By applying the operator inequality (A.2) to the first term and the operator inequality (A.3) to the second term on the r.h.s of (4.1), we obtain

(4.11)		$\displaystyle B\leq$	$\displaystyle\frac{\\|\nabla u\\|_{L^{\infty}}}{N^{\beta}}\left(\alpha\\|\|x\|V(x)\\|_{L^{1}}\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\right.$
		$\displaystyle\left.+\alpha^{-1}N^{\beta}\\|\|x\|V(x)\\|_{L^{\frac{3}{2}}}\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\right)$
	$\displaystyle\lesssim$	$\displaystyle N^{-\frac{\beta}{2}}\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle,$

where in the last inequality we optimized the choice of $\alpha$ .

Together with (4.8) and estimates for the terms $A$ and $B$ , we reach

		$\displaystyle\int(u(x)-u(y))V_{N}(x-y)\nabla_{x}\left[(1-w_{N,\hbar}(x-y))^{2}\frac{\rho_{N,\hbar}^{(2)}(x,y)}{(1-w_{N,\hbar}(x-y))^{2}}\right]dxdy$
	$\displaystyle\lesssim$	$\displaystyle\left(N^{\beta-1}\hbar^{-2}+N^{-\frac{\beta}{2}}\right)\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle\left(N^{\beta-1}\hbar^{-2}+N^{-\frac{\beta}{2}}\right)\hbar^{-4}\langle(1-\hbar^{2}\Delta_{x_{1}})(1-\hbar^{2}\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{2}}\hbar^{-4},$

where in the last inequality we used the two-body $H^{1}$ energy bound (3.22). This completes the proof of (4.6).

∎

4.2. Tamed Singularities

As a result of the error analysis of the two-body term, we are able to capture the main term

\displaystyle\int\operatorname{div}u(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy.

Using the identity approximation to the one-body term of $\widetilde{\mathcal{F}}_{\delta}(t)$ , we arrive at

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)\sim$	$\displaystyle-\int\operatorname{div}u(t,x)V_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

By the identity approximation again, we also have the approximation of $\mathcal{F}_{\delta}(t)$ that

	$\displaystyle\mathcal{F}_{\delta}(t)\sim$	$\displaystyle\int V_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(t,x,y)\right.$
		$\displaystyle\quad\quad\left.-\rho_{N,\hbar}^{(1)}(t,x)\rho(t,y)-\rho(t,x)\rho_{N,\hbar}^{(1)}(t,y)+\rho(t,x)\rho(t,y)\right]dxdy.$

We now need to deal with the sharp singularity of $V_{N}(x)$ . We tame the singularity by replacing $V_{N}(x)$ with a slowly varying potential with a number of good properties. However, the replacement relies on the regularity of the integrand. Therefore, we again need to decompose the two-body density function as the singular and relatively regular parts. We obtain proper approximations of $\mathcal{F}_{\delta}(t)$ and $\widetilde{\mathcal{F}}_{\delta}(t)$ and arrive at a reduced version of functional inequalities via a careful analysis.

The following is the main lemma of the section.

Lemma 4.3.

Let

G(x)=\left(\frac{1}{\pi}\right)^{\frac{3}{2}}e^{-|x|^{2}},\quad G_{N}(x)=N^{3\eta}G(N^{\eta}x).

Then for the two-body term we have

(4.12)			$\displaystyle\int F(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
	$\displaystyle=$	$\displaystyle b_{0}\int F(x)G_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}),$

and for the one-body term we have

(4.13)			$\displaystyle b_{0}\int F(x)\rho(x)\left[\rho(x)-2\rho_{N,\hbar}^{(1)}(x)\right]dx$
	$\displaystyle=$	$\displaystyle b_{0}\int F(x)G_{N}(x-y)\left[-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy\pm O(N^{-\eta}).$

In particular, given $F(x)=1$ , we have the approximation of $\mathcal{F}_{\delta}(t)$

(4.14)	$\displaystyle\mathcal{F}_{\delta}(t)=$	$\displaystyle b_{0}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)\right.$
	$\displaystyle\quad\quad\left.-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}),$

and given $F(x)=\operatorname{div}u(x)$ , we have the approximation of $\widetilde{\mathcal{F}}_{\delta}(t)$

(4.15)	$\displaystyle\widetilde{\mathcal{F}}_{\delta}(t)=$	$\displaystyle-b_{0}\int\operatorname{div}u(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)\right.$
	$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}).$

Proof.

For (4.12), we recall $\phi_{N,\hbar,12}=(1-w_{12})\psi_{N,\hbar}$ and rewrite

		$\displaystyle\int F(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
	$\displaystyle=$	$\displaystyle\langle F(x)V_{N}(x-y)(1-w_{12})^{2}\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle=$	$\displaystyle\langle F(x)V_{N}(x-y)\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle+\langle F(x)V_{N}(x-y)(-2w_{12}+(w_{12})^{2})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle=$	$\displaystyle I+II.$

For the term $I$ , we use the Poincar $\acute{e}$ type inequality with $\theta=\frac{1}{3}$ in Lemma A.3 to obtain

		$\displaystyle\|\langle F(x)(V_{N}(x-y)-b_{0}\delta(x-y))\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\|$
	$\displaystyle\lesssim$	$\displaystyle N^{-\frac{\beta}{3}}\\|\langle\nabla_{x_{1}}\rangle\langle\nabla_{x_{2}}\rangle F(x_{1})\phi_{N,\hbar,12}\\|_{L^{2}}\\|\langle\nabla_{x_{1}}\rangle\langle\nabla_{x_{2}}\rangle\phi_{N,\hbar,12}\\|_{L^{2}}$
	$\displaystyle\lesssim$	$\displaystyle N^{-\frac{\beta}{3}}(\\|F\\|_{L^{\infty}}+\\|\nabla F\\|_{L^{\infty}})\\|\langle\nabla_{x_{1}}\rangle\langle\nabla_{x_{2}}\rangle\phi_{N,\hbar,12}\\|_{L^{2}}^{2}\lesssim N^{-\frac{\beta}{3}}\hbar^{-4},$

where in the last inequality we used the two-body energy bound (3.22).

For the term $II$ , by Lemma 3.1, we have $|w_{12}|\lesssim N^{\beta-1}\hbar^{-2}$ . Therefore, we get

	$\displaystyle II\lesssim$	$\displaystyle\\|F\\|_{L^{\infty}}\langle V_{N}(x-y)\|w_{12}\|\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle N^{\beta-1}\hbar^{-2}\langle V_{N}(x-y)\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle N^{\beta-1}\hbar^{-2}\langle(1-\Delta_{x_{1}})(1-\Delta_{x_{2}})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\lesssim N^{\beta-1}\hbar^{-6},$

where we used the operator inequality (A.2) in the second-to-last inequality and the two-body energy bound (3.22) in the last inequality.

In the same way, we also obtain

		$\displaystyle b_{0}\int F(x)G_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
	$\displaystyle=$	$\displaystyle b_{0}\int F(x)\delta(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy\pm O(N^{-\frac{\eta}{3}}\hbar^{-4}+N^{\beta-1}\hbar^{-6}).$

Then by the triangle inequality, we arrive at

		$\displaystyle\Big{\|}\int F(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy-b_{0}\int F(x)G_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy\Big{\|}$
	$\displaystyle\lesssim$	$\displaystyle N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4},$

which completes the proof of (4.12).

For (4.13), we rewrite

		$\displaystyle\int F(x)\rho(x)\left[\rho(x)-2\rho_{N,\hbar}^{(1)}(x)\right]dx$
	$\displaystyle=$	$\displaystyle\int F(x)\delta(x-y)\left[-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle\int F(x)G_{N}(x-y)\left[-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
(4.16)			$\displaystyle-\langle F\rho_{N,\hbar}^{(1)},(G_{N}-\delta)\rho\rangle-\langle(G_{N}-\delta)(F\rho),\rho_{N,\hbar}^{(1)}\rangle+\langle F\rho,(G_{N}-\delta)*\rho\rangle.$

For the error terms in (4.16), we use Hölder and Sobolev inequalities to get

		$\displaystyle\|\langle F\rho_{N,\hbar}^{(1)},(G_{N}-\delta)\rho\rangle\|+\|\langle(G_{N}-\delta)(F\rho),\rho_{N,\hbar}^{(1)}\rangle\|+\|\langle F\rho,(G_{N}-\delta)*\rho\rangle\|$
	$\displaystyle\leq$	$\displaystyle\\|F\\|_{L^{\infty}}(\\|\rho_{N,\hbar}^{(1)}\\|_{L^{1}}+\\|\rho\\|_{L^{1}})\\|(G_{N}-\delta)\rho\\|_{L^{\infty}}+\\|\rho_{N,\hbar}^{(1)}\\|_{L^{1}}\\|(G_{N}-\delta)(F\rho)\\|_{L^{\infty}}$
	$\displaystyle\lesssim$	$\displaystyle\\|F\\|_{L^{\infty}}(\\|\rho_{N,\hbar}^{(1)}\\|_{L^{1}}+\\|\rho\\|_{L^{1}})\\|(G_{N}-\delta)\langle\nabla\rangle^{2}\rho\\|_{L^{2}}+\\|\rho_{N,\hbar}^{(1)}\\|_{L^{1}}\\|(G_{N}-\delta)\langle\nabla\rangle^{2}(F\rho)\\|_{L^{2}}$
	$\displaystyle\lesssim$	$\displaystyle N^{-\eta}\left(\\|F\\|_{L^{\infty}}\\|\rho\\|_{H^{3}}+\\|F\rho\\|_{H^{3}}\right)$
	$\displaystyle\lesssim$	$\displaystyle N^{-\eta}\\|F\\|_{H^{3}}\\|\rho\\|_{H^{3}},$

where in the second-to-last inequality we used Lemma A.1 and the mass conservation, and in the last inequality we used Leibniz rule and Sobolev inequality. Therefore, we complete the proof of (4.13).

For (4.14), by taking $F(x)=1$ in (4.12) and (4.13), we arrive at the approximation of $\mathcal{F}_{\delta}(t)$ .

For (4.15), by the error analysis (4.6) in Lemma 4.2 we get

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}=$	$\displaystyle-b_{0}\int\operatorname{div}u(x)V_{N}(x-y)\rho_{N,\hbar}^{(2)}(x,y)dxdy$
		$\displaystyle-b_{0}\int\operatorname{div}u(x)\rho(x)\left[\rho(x)-2\rho_{N,\hbar}^{(1)}(x)\right]dx$
		$\displaystyle\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{2}}\hbar^{-4}).$

Then by taking $F(x)=\operatorname{div}u(x)$ in (4.12) and (4.13), we get the approximation (4.15) of $\widetilde{\mathcal{F}}_{\delta}(t)$ . ∎

4.3. Reduced Version of Functional Inequality

After the analysis of error terms and simplification, we now work with a reduced form of functional inequality

(4.17)			$\displaystyle\int\operatorname{div}u(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\lesssim$	$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy+o(1),$

which is more concise than the original functional inequality. However, it is unknown whether or not the integrand

(4.18)

\displaystyle\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)

is non-negative, so we cannot directly bound the term $\operatorname{div}u(x)$ in (4.17). We prove that, if integrated against $G_{N}(x-y)$ , (4.18) provides a non-negative contribution up to a small correction and use that to prove the lower bound of $\mathcal{F}_{\delta}(t)$ . The special structure of a relatively slowly varying and explicit potential $G_{N}(x)$ plays a critical role in establishing the reduced version of functional inequality. We then complete the proof of Proposition 4.1.

Lemma 4.4 (Reduced Version of Functional Inequality).

Let $\eta<\frac{1}{3}$ to be determined, we have

(4.19)			$\displaystyle\int F(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\leq$	$\displaystyle\\|F\\|_{L^{\infty}}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle+O(N^{-\eta}\hbar^{-2}+N^{3\eta-1}).$

Proof.

For simplicity, set $\rho_{N,\hbar}(X_{N})=|\psi_{N,\hbar}(X_{N})|^{2}$ . By the symmetry of $\rho_{N,\hbar}(X_{N})$ , we can write

		$\displaystyle\int F(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle\frac{1}{N^{2}}\sum_{i\neq j}^{N}\int F(x_{i})G_{N}(x_{i}-x_{j})\rho_{N,\hbar}(X_{N})dX_{N}+\int F(x)G_{N}(x-y)\rho(x)\rho(y)dxdy$
		$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\int\int F(x_{i})G_{N}(x_{i}-y)\rho(y)dy\rho_{N,\hbar}(X_{N})dX_{N}$
		$\displaystyle-\frac{1}{N}\sum_{j=1}^{N}\int\int F(x)G_{N}(x-x_{j})\rho(x)dx\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\int F(x)G_{N}(x-y)\left[\frac{1}{N^{2}}\sum_{i\neq j}^{N}\delta_{x_{i}}(x)\delta_{x_{j}}(y)+\rho(x)\rho(y)\right.$
		$\displaystyle\left.-\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}(x)\rho(y)-\rho(x)\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{j}}(y)\right]dxdy\rho_{N,\hbar}(X_{N})dX_{N}.$

To simplify, we define the measure

(4.20)

\displaystyle\nu_{X_{N}}(dx)=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}(dx)-\rho(x)dx.

We rewrite

(4.21)			$\displaystyle\int F(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle\int F(x)G_{N}(x-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}-\frac{G_{N}(0)}{N}\int F(x)\rho_{N,\hbar}^{(1)}(x)dx.$

where the last term on the r.h.s of (4.21) comes from the diagonal summation. In particular, if we take $F(x)=1$ , we also have

(4.22)			$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle\int G_{N}(x-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}-\frac{G_{N}(0)}{N}\int\rho_{N,\hbar}^{(1)}(x)dx.$

Note that

(4.23)

\displaystyle\frac{G_{N}(0)}{N}\int F(x)\rho_{N,\hbar}^{(1)}(x)dx\leq N^{3\eta-1}\|F\|_{L^{\infty}},

which is a smallness term as long as $\eta<\frac{1}{3}$ .

Next, we get into the analysis of the main term. Note that the convolution property of the Gaussian function $G$ , which is

(4.24)

\displaystyle G_{N}(x-y)=\int G_{0,N}(x-z)G_{0,N}(z-y)dz,

where $G_{0,N}(x)=N^{3\eta}G_{0}(N^{\eta}x)$ and $G_{0}(x)=\left(\frac{2}{\pi}\right)^{\frac{3}{2}}e^{-2|x|^{2}}$ . Putting (4.24) into the main term of (4.22) gives

(4.25)			$\displaystyle\int F(x)G_{N}(x-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\int F(x)G_{0,N}(x-z)G_{0,N}(z-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle=$	$\displaystyle A+B,$

where

(4.26)		$\displaystyle A=$	$\displaystyle\int(F(x)-F(z))G_{0,N}(x-z)G_{0,N}(z-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dzdX_{N},$
(4.27)		$\displaystyle B=$	$\displaystyle\int F(z)G_{0,N}(x-z)G_{0,N}(z-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dzdX_{N}.$

Thus, we are left to bound the terms $A$ and $B$ .

For the term $A$ , we use Cauchy-Schwarz inequality to get

	$\displaystyle A^{2}\leq$	$\displaystyle\int\left[\int(F(x)-F(z))G_{0,N}(x-z)\nu_{X_{N}}(dx)\right]^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}$
		$\displaystyle\cdot\int\left[\int G_{0,N}(z-y)\nu_{X_{N}}(dy)\right]^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle\leq$	$\displaystyle 2(A_{1}+A_{2})\int\left[\int G_{0,N}(z-y)\nu_{X_{N}}(dy)\right]^{2}\rho_{N,\hbar}(X_{N})dzdX_{N},$

where

	$\displaystyle A_{1}=$	$\displaystyle\int\left[\int(F(x)-F(z))G_{0,N}(x-z)\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}(dx)\right]^{2}\rho_{N,\hbar}(X_{N})dzdX_{N},$
	$\displaystyle A_{2}=$	$\displaystyle\int\left[\int(F(x)-F(z))G_{0,N}(x-z)\rho(x)dx\right]^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}.$

For $A_{1}$ , we further decompose it into two parts $A_{1}=A_{11}+A_{12}$ , where the diagonal part is

\displaystyle A_{11}=

\displaystyle\frac{1}{N^{2}}\sum_{i=1}^{N}\int(F(x_{i})-F(z))G_{0,N}(x_{i}-z)(F(x_{i})-F(z))G_{0,N}(x_{i}-z)\rho_{N,\hbar}(X_{N})dzdX_{N},

and the off-diagonal part is

\displaystyle A_{12}=

\displaystyle\frac{1}{N^{2}}\sum_{i\neq j}^{N}\int(F(x_{i})-F(z))G_{0,N}(x_{i}-z)(F(x_{j})-F(z))G_{0,N}(x_{j}-z)\rho_{N,\hbar}(X_{N})dzdX_{N}.

For $A_{11}$ , by the symmetry of $\rho_{N,\hbar}(X_{N})$ , we have

	$\displaystyle A_{11}\leq$	$\displaystyle\frac{1}{N}\int(F(x_{1})-F(z))G_{0,N}(x_{1}-z)(F(x_{1})-F(z))G_{0,N}(x_{1}-z)\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle\leq$	$\displaystyle\frac{\\|\nabla F\\|_{L^{\infty}}^{2}}{N}\int\left(\|x_{1}-z\|G_{0,N}(x_{1}-z)\right)^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle=$	$\displaystyle\frac{\\|\nabla F\\|_{L^{\infty}}^{2}}{N}\\|\|x\|G_{0,N}(x)\\|_{L^{2}}^{2}\int\rho_{N,\hbar}(X_{N})dX_{N}\lesssim N^{\eta-1},$

where in the last inequality we used that $\||x|G_{0,N}(x)\|_{L^{2}}^{2}\lesssim N^{\eta}$ and the mass conservation for $\rho_{N,\hbar}(X_{N})$ .

For $A_{12}$ , by the symmetry of $\rho_{N,\hbar}(X_{N})$ , we also have

	$\displaystyle A_{12}\leq$	$\displaystyle\int\|(F(x_{1})-F(z))G_{0,N}(x_{1}-z)(F(x_{2})-F(z))G_{0,N}(x_{2}-z)\|\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle\leq$	$\displaystyle\\|\nabla F\\|_{L^{\infty}}^{2}\int\|x_{1}-z\|G_{0,N}(x_{1}-z)\|x_{2}-z\|G_{0,N}(x_{2}-z)\|\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle\leq$	$\displaystyle\frac{\\|\nabla F\\|_{L^{\infty}}^{2}}{N^{2\eta}}\int G_{1,N}(x_{1}-x_{2})\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\frac{\\|\nabla F\\|_{L^{\infty}}^{2}}{N^{2\eta}}\langle G_{1,N}(x_{1}-x_{2})\psi_{N,\hbar},\psi_{N,\hbar}\rangle,$

where

\displaystyle G_{1,N}(x_{1}-x_{2})=\int N^{\eta}|x_{1}-z|G_{0,N}(x_{1}-z)N^{\eta}|x_{2}-z|G_{0,N}(x_{2}-z)|dz.

To bound $A_{12}$ , we recall $\phi_{N,\hbar,12}=(1-w_{12})\psi_{N,\hbar}$ then get

	$\displaystyle A_{12}\lesssim$	$\displaystyle N^{-2\eta}\langle G_{1,N}(x_{1}-x_{2})(1-w_{12})^{2}\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle N^{-2\eta}\langle G_{1,N}(x_{1}-x_{2})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle$
	$\displaystyle\lesssim$	$\displaystyle N^{-2\eta}\\|G_{1,N}\\|_{L^{1}}\langle(1-\Delta_{1})(1-\Delta_{2})\phi_{N,\hbar,12},\phi_{N,\hbar,12}\rangle\lesssim N^{-2\eta}\hbar^{-4},$

where we discarded $(1-w_{12})^{2}$ in the second line and used the operator inequality (A.2) in the second-to-last inequality, and the two-body $H^{1}$ energy bound (3.22) in the last inequality.

For $A_{2}$ , we rewrite

	$\displaystyle A_{2}=$	$\displaystyle\\|F(G_{0,N}\rho)-G_{0,N}(F\rho)\\|_{L^{2}}^{2}\int\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\\|F(G_{0,N}\rho)-G_{0,N}(F\rho)\\|_{L^{2}}^{2},$

where in the last inequality we used the mass conservation for $\rho_{N,\hbar}(X_{N})$ . By the triangle, Hölder inequalities and Lemma A.1 we get

	$\displaystyle A_{2}\leq$	$\displaystyle 2\\|F(G_{0,N}\rho)-F\rho\\|_{L^{2}}^{2}+2\\|F\rho-G_{0,N}(F\rho)\\|_{L^{2}}^{2}$
	$\displaystyle\lesssim$	$\displaystyle\\|F\\|_{L^{\infty}}^{2}\\|(G_{0,N}-\delta)\rho\\|_{L^{2}}^{2}+\\|(G_{0,N}-\delta)(F\rho)\\|_{L^{2}}^{2}$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{N^{2\eta}}\\|F\\|_{L^{\infty}}^{2}\\|\langle\nabla\rangle\rho\\|_{L^{2}}^{2}+\frac{1}{N^{2\eta}}\\|\langle\nabla\rangle(F\rho)\\|_{L^{2}}^{2}\lesssim N^{-2\eta}.$

To sum up, we complete the estimates for the term $A$ and reach

(4.28)

\displaystyle A\leq\sqrt{A_{11}}+\sqrt{A_{12}}+\sqrt{A_{2}}\lesssim N^{\frac{\eta-1}{2}}+N^{-\eta}\hbar^{-2}\lesssim N^{-\eta}\hbar^{-2},

where in the last inequality we used that $N^{\frac{\eta-1}{2}}\leq N^{-\eta}$ for $\eta<\frac{1}{3}$ .

For the term $B$ , we rewrite

	$\displaystyle B=$	$\displaystyle\int F(z)G_{0,N}(x-z)G_{0,N}(z-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)dz\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\int F(z)\|G_{0,N}*\nu_{X_{N}}(z)\|^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}.$

Observe that

\displaystyle|G_{0,N}*\nu_{X_{N}}(z)|^{2}\rho_{N,\hbar}(X_{N})\geq 0

for a.e. $(z,X_{N})\in\mathbb{R}^{3}\times\mathbb{R}^{3N}$ . Therefore, we can directly bound $F(z)$ and get

(4.29)		$\displaystyle B\leq$	$\displaystyle\\|F\\|_{L^{\infty}}\int\|G_{0,N}*\nu_{X_{N}}(z)\|^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}$
	$\displaystyle=$	$\displaystyle\\|F\\|_{L^{\infty}}\int\int G_{0,N}(x-z)G_{0,N}(z-y)dz\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\\|F\\|_{L^{\infty}}\int G_{N}(x-y)\nu_{X_{N}}(dx)\nu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\\|F\\|_{L^{\infty}}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle+\\|F\\|_{L^{\infty}}\frac{G_{N}(0)}{N}\int\rho_{N,\hbar}^{(1)}(x)dx,$

where in the second-to-last equality we used the property (4.24), and in the last equality we used the equation (4.22).

With the approximation forms (4.21) and (4.25), we use estimates (4.23), (4.28) for $A$ and (4.29) for $B$ to arrive at

		$\displaystyle\int F(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle A+B+O(N^{3\eta-1})$
	$\displaystyle\leq$	$\displaystyle\\|F\\|_{L^{\infty}}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle+O(N^{-\eta}\hbar^{-2}+N^{3\eta-1}),$

which is the desired estimate (4.19). ∎

To prove the lower bound estimate (4.2) for $\mathcal{F}_{\delta}(t)$ , we give the following estimate.

Lemma 4.5.

Let $\eta<\frac{1}{3}$ to be determined, we have

(4.30)			$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle\geq$	$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(x)-\rho(x))(\rho_{N,\hbar}^{(1)}(y)-\rho(y))dxdy-O(N^{3\eta-1}).$

Proof.

We decompose

		$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle I+II,$

where

(4.31)		$\displaystyle I=$	$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(x)-\rho(x))(\rho_{N,\hbar}^{(1)}(y)-\rho(y))dxdy,$
(4.32)		$\displaystyle II=$	$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho_{N,\hbar}^{(1)}(y)\right]dxdy.$

It suffices to prove a lower bound of the term $II$ . By the symmetry of the density function $\rho_{N,\hbar}(X_{N})$ , we rewrite

	$\displaystyle II=$	$\displaystyle\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)-\rho_{N,\hbar}^{(1)}(x)\rho_{N,\hbar}^{(1)}(y)\right]dxdy$
	$\displaystyle=$	$\displaystyle\int G_{N}(x-y)\left[\frac{1}{N^{2}}\sum_{i\neq j}^{N}\delta_{x_{i}}(x)\delta_{x_{j}}(y)+\rho_{N,\hbar}^{(1)}(x)\rho_{N,\hbar}^{(1)}(y)\right.$
		$\displaystyle\left.-\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}(x)\rho_{N,\hbar}^{(1)}(y)-\rho_{N,\hbar}^{(1)}(x)\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{j}}(y)\right]dxdy\rho_{N,\hbar}(X_{N})dX_{N}$
	$\displaystyle=$	$\displaystyle\int G_{N}(x-y)\mu_{X_{N}}(dx)\mu_{X_{N}}(dy)\rho_{N,\hbar}(X_{N})dX_{N}-\frac{G_{N}(0)}{N}\int\rho_{N,\hbar}^{(1)}(x)dx,$

where

\displaystyle\mu_{X_{N}}(dx)=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}(dx)-\rho_{N,\hbar}^{(1)}(x)dx.

Then by (4.24) and (4.23), we obtain

\displaystyle II=

\displaystyle\int|G_{0,N}*\mu_{X_{N}}(z)|^{2}\rho_{N,\hbar}(X_{N})dzdX_{N}-\frac{G_{N}(0)}{N}\int\rho_{N,\hbar}^{(1)}(x)dx\gtrsim-N^{3\eta-1},

which completes the proof of estimate (4.30). ∎

To the end, we get into the proof of Proposition 4.1.

Proof of Proposition 4.1.

For estimate (4.1), the approximation (4.15) of $\widetilde{\mathcal{F}}_{\delta}(t)$ in Lemma 4.3 gives

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}=$	$\displaystyle-b_{0}\int\operatorname{div}u(x)G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)\right.$
		$\displaystyle\left.\quad\quad-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}).$

Then by the functional inequality (4.19) in Lemma 4.4, we get

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}\leq$	$\displaystyle\\|\operatorname{div}u\\|_{L^{\infty}}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)\right.$
		$\displaystyle\left.-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle+O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}+N^{-\eta}\hbar^{-2}+N^{3\eta-1}).$

Using the approximation (4.14) of $\mathcal{F}_{\delta}$ in Lemma 4.3, we arrive at

	$\displaystyle\widetilde{\mathcal{F}}_{\delta}\leq$	$\displaystyle\\|\operatorname{div}u\\|_{L^{\infty}}\mathcal{F}_{\delta}+O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}+N^{-\eta}\hbar^{-2}+N^{3\eta-1})$
	$\displaystyle\lesssim$	$\displaystyle\mathcal{F}_{\delta}+O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4}),$

where in the last inequality we took $\eta=\frac{3}{10}$ . Therefore, we complete the proof of the estimate (4.1).

For the lower bound estimate (4.2) on $\mathcal{F}_{\delta}$ , we use the approximation (4.14) of $\mathcal{F}_{\delta}$ and estimate (4.30) to obtain

	$\displaystyle\mathcal{F}_{\delta}=$	$\displaystyle b_{0}\int G_{N}(x-y)\left[\frac{N-1}{N}\rho_{N,\hbar}^{(2)}(x,y)\right.$
		$\displaystyle\quad\quad\left.-\rho_{N,\hbar}^{(1)}(x)\rho(y)-\rho(x)\rho_{N,\hbar}^{(1)}(y)+\rho(x)\rho(y)\right]dxdy$
		$\displaystyle\pm O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4})$
(4.33)		$\displaystyle\geq$	$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(x)-\rho(x))(\rho_{N,\hbar}^{(1)}(y)-\rho(y))dxdy$
		$\displaystyle-O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{\eta}{3}}\hbar^{-4}+N^{3\eta-1}).$

By (4.24), we observe that the term on the r.h.s of (4.33)

		$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(x)-\rho(x))(\rho_{N,\hbar}^{(1)}(y)-\rho(y))dxdy$
	$\displaystyle=$	$\displaystyle\int\|G_{N,0}*(\rho_{N,\hbar}^{(1)}-\rho)(z)\|^{2}dz\geq 0.$

Thus, we can discard this positive term and then take $\eta=\frac{3}{10}$ to get

\displaystyle\mathcal{F}_{\delta}\geq-O(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4}),

which is the lower bound estimate (4.2). ∎

5. Quantitative Strong Convergence of Quantum Densities

In the section, using functional inequalities, we prove the Gronwall’s inequality for the modulated energy. Subsequently, with the quantitative convergence rate of the modulated energy, we further conclude the quantitative strong convergence of quantum mass and momentum densities. Notably, the $\delta$ -type potential part is crucial in upgrading to the quantitative strong convergence, that is, in the case of only the Coulomb potential, one cannot deduce the strong convergence here.

Recall the modulated energy

(5.1)

\displaystyle\mathcal{M}(t)=\mathcal{M}_{K}(t)+\mathcal{M}_{P}(t),

where the kinetic energy part is

(5.2)

\displaystyle\mathcal{M}_{K}(t)=\int_{\mathbb{R}^{3N}}|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(t,X_{N})|^{2}dX_{N},

and the potential energy part is

(5.3)

\displaystyle\mathcal{M}_{P}(t)=\mathcal{F}_{\delta}(t)+\mathcal{F}_{c}(t).

From lower bound estimates (4.2) on $\mathcal{F}_{\delta}(t)$ and (1.25) on $\mathcal{F}_{c}(t)$ , we can add a small compensation such that

(5.4)

\displaystyle\mathcal{F}_{\delta}(t)+\mathcal{F}_{c}(t)+r(N,\hbar)\geq 0,

where $r(N,\hbar)=C(N^{\beta-1}\hbar^{-6}+N^{-\frac{\beta}{3}}\hbar^{-4}+N^{-\frac{1}{10}}\hbar^{-4}).$ Thus, we introduce the positive modulated energy

(5.5)

\displaystyle\mathcal{M}^{+}(t)=

\displaystyle\mathcal{M}(t)+2r(N,\hbar)\geq r(N,\hbar)\geq 0.

We now provide a closed estimate for the positive modulated energy.

Proposition 5.1.

For $t\in[0,T_{0}]$ , we have the differential inequality

(5.6)

\displaystyle\frac{d}{dt}\mathcal{M}^{+}(t)\lesssim\mathcal{M}^{+}(t)+\hbar^{2}.

Moreover, we conclude

(5.7)

\int_{\mathbb{R}^{3N}}|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(t,X_{N})|^{2}dX_{N}\lesssim\mathcal{M}^{+}(0)+\hbar^{2},

and

(5.8)

\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(x)-\rho(x))(\rho_{N,\hbar}^{(1)}(y)-\rho(y))dxdy\lesssim\mathcal{M}^{+}(0)+\hbar^{2}.

Proof.

From the evolution of the modulated energy (2.8), we find that

		$\displaystyle\frac{d}{dt}\mathcal{M}^{+}(t)$
	$\displaystyle=$	$\displaystyle-\sum_{j,k=1}^{3}\int_{\mathbb{R}^{3N}}\left(\partial_{j}u^{k}+\partial_{k}u^{j}\right)(-i\hbar\partial_{j}\psi_{N,\hbar}-u^{j}\psi_{N,\hbar})\overline{(-i\hbar\partial_{k}\psi_{N,\hbar}-u^{k}\psi_{N,\hbar})}dX_{N}$
		$\displaystyle+\frac{\hbar^{2}}{2}\int_{\mathbb{R}^{3}}\Delta(\operatorname{div}u)(t,x_{1})\rho_{N,\hbar}^{(1)}(t,x_{1})dx_{1}+\widetilde{\mathcal{F}}_{\delta}(t)+\widetilde{\mathcal{F}}_{c}(t)$
	$\displaystyle\lesssim$	$\displaystyle\\|\nabla u\\|_{L^{\infty}}\int_{\mathbb{R}^{3N}}\|(i\hbar\nabla_{x_{1}}-u)\psi_{N,\hbar}\|^{2}dX_{N}+\hbar^{2}\\|\psi_{N,\hbar}\\|_{L^{2}}^{2}\\|\Delta\operatorname{div}u\\|_{L^{\infty}}+\widetilde{\mathcal{F}}_{\delta}(t)+\widetilde{\mathcal{F}}_{c}(t).$

By the functional inequalities $(\ref{equ:functional inequality,fdelta})$ on $\widetilde{\mathcal{F}}_{\delta}(t)$ and (1.24) on $\widetilde{\mathcal{F}}_{c}(t)$ , we get

(5.9)

\displaystyle\frac{d}{dt}\mathcal{M}^{+}(t)\lesssim\mathcal{M}^{+}(t)+\hbar^{2}.

Then by Gronwall’s inequality, we arrive at

(5.10)

\displaystyle\mathcal{M}^{+}(t)\leq\exp(CT_{0})\left(\mathcal{M}^{+}(0)+\hbar^{2}t\right)\lesssim

\displaystyle\mathcal{M}^{+}(0)+\hbar^{2}

for $t\in[0,T_{0}]$ .

For the kinetic energy estimate (5.7), by (5.5) and (5.10) we have that

\displaystyle\int_{\mathbb{R}^{3N}}|\left(i\hbar\nabla_{x_{1}}-u(t,x_{1})\right)\psi_{N,\hbar}(t,X_{N})|^{2}dX_{N}\leq

\displaystyle\mathcal{M}^{+}(t)\lesssim\mathcal{M}^{+}(0)+\hbar^{2},

which completes the proof of (5.7).

For the potential energy estimate (5.8), by (4.30) in Lemma 4.5 and (4.14) in Lemma 4.3, we have

	$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(t,x)-\rho(t,x))(\rho_{N,\hbar}^{(1)}(t,y)-\rho(t,y))dxdy\leq$	$\displaystyle\widetilde{\mathcal{F}}_{c}(t)+r(N,\hbar)$
	$\displaystyle\leq$	$\displaystyle\mathcal{M}^{+}(t)+r(N,\hbar).$

Again by (5.10), we arrive at (5.8). ∎

To the end, we get into the proof of Theorem 1.1.

Proof of Theorem 1.1.

Convergence of the mass density $\rho_{N,\hbar}^{(1)}(t)$ .

We decompose

		$\displaystyle\int G_{N}(x-y)(\rho_{N,\hbar}^{(1)}(t,x)-\rho(t,x))(\rho_{N,\hbar}^{(1)}(t,y)-\rho(t,y))dxdy$
(5.11)		$\displaystyle=$	$\displaystyle\langle(G_{N}-\delta)*(\rho_{N,\hbar}^{(1)}(t)-\rho(t)),\rho_{N,\hbar}^{(1)}(t)-\rho(t)\rangle+\\|\rho_{N,\hbar}^{(1)}(t)-\rho(t)\\|_{L^{2}}^{2}.$

For the first term on the r.h.s of (5.11), we use Hölder inequality and Lemma A.1 to obtain

		$\displaystyle\langle(G_{N}-\delta)*(\rho_{N,\hbar}^{(1)}(t)-\rho(t)),\rho_{N,\hbar}^{(1)}(t)-\rho(t)\rangle$
	$\displaystyle\leq$	$\displaystyle\\|(G_{N}-\delta)*(\rho_{N,\hbar}^{(1)}(t)-\rho(t))\\|_{L^{\frac{3}{2}}}\\|\rho_{N,\hbar}^{(1)}(t)-\rho(t)\\|_{L^{3}}$
	$\displaystyle\lesssim$	$\displaystyle N^{-\eta}\left(\\|\langle\nabla\rangle\rho_{N,\hbar}^{(1)}(t)\\|_{L^{\frac{3}{2}}}+\\|\rho(t)\\|_{L^{\frac{3}{2}}}\right)\left(\\|\rho_{N,\hbar}^{(1)}(t)\\|_{L^{3}}+\\|\rho(t)\\|_{L^{3}}\right).$

Next, we estimate the terms $\|\langle\nabla\rangle\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}$ and $\|\rho_{N,\hbar}^{(1)}(t)\|_{L^{3}}$ . By the Calderón-Zygmund theory which implies that $\|\langle\nabla\rangle f\|_{L^{p}}\lesssim\|\nabla f\|_{L^{p}}+\|f\|_{L^{p}}$ for $1<p<\infty$ , we get

\displaystyle\|\langle\nabla\rangle\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}\lesssim\|\nabla\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}+\|\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}.

By the Leibniz rule, Minkowski, Hölder, and Sobolev inequalities, we then obtain

\displaystyle\|\nabla\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}\lesssim\|\nabla_{x_{1}}\psi_{N,\hbar}(t)\|_{L^{2}}\|\psi_{N,\hbar}(t)\|_{L^{2}L_{x_{1}}^{6}}\lesssim\|\langle\nabla_{x_{1}}\rangle\psi_{N,\hbar}(t)\|_{L^{2}}^{2}\lesssim\hbar^{-2},

where in the last inequality we have used the $H^{1}$ energy bound (2.5) for $\psi_{N,\hbar}$ . Similarly, we also have

\displaystyle\|\rho_{N,\hbar}^{(1)}(t)\|_{L^{\frac{3}{2}}}\lesssim\|\psi_{N,\hbar}(t)\|_{L^{2}}\|\psi_{N,\hbar}(t)\|_{L^{2}L_{x_{1}}^{6}}\lesssim\|\psi_{N,\hbar}(t)\|_{L^{2}}\|\langle\nabla_{x_{1}}\rangle\psi_{N,\hbar}(t)\|_{L^{2}}\lesssim\hbar^{-1},

and

\displaystyle\|\rho_{N,\hbar}^{(1)}(t)\|_{L^{3}}\leq\|\psi_{N,\hbar}(t)\|_{L^{2}L_{x_{1}}^{6}}^{2}\lesssim\|\langle\nabla_{x_{1}}\rangle\psi_{N,\hbar}(t)\|_{L^{2}}^{2}\lesssim\hbar^{-2}.

With $\eta=\frac{3}{10}$ , these bounds give that

(5.12)

\displaystyle\langle(G_{N}-\delta)*(\rho_{N,\hbar}^{(1)}(t)-\rho(t)),\rho_{N,\hbar}^{(1)}(t)-\rho(t)\rangle\lesssim N^{-\frac{3}{10}}\hbar^{-4}\lesssim r(N,\hbar).

Thus, combining (5.11), (5.12) with (5.8), we arrive at

(5.13)

\displaystyle\|\rho_{N,\hbar}^{(1)}(t)-\rho(t)\|_{L^{2}}^{2}\lesssim

\displaystyle r(N,\hbar)+\mathcal{M}^{+}(0)+\hbar^{2}\lesssim\mathcal{M}^{+}(0)+\hbar^{2},

where in the last inequality we have used that $r(N,\hbar)\leq\mathcal{M}^{+}(0)$ .

Convergence of the momentum density $J_{N,\hbar}^{(1)}(t)$ .

Recall the momentum density

\displaystyle J_{N,\hbar}^{(1)}(t,x_{1})=\hbar\int\operatorname{Im}(\overline{\psi_{N,\hbar}}\nabla_{x_{1}}\psi_{N,\hbar})(t,X_{N})dx_{2}\cdot\cdot\cdot dx_{N}.

Then by the triangle and Hölder’s inequalities, we have

		$\displaystyle\\|J_{N,\hbar}^{(1)}(t)-(\rho u)(t)\\|_{L^{1}}$
	$\displaystyle\leq$	$\displaystyle\\|J_{N,\hbar}^{(1)}(t)-(\rho_{N,\hbar}^{(1)}u)(t)\\|_{L^{1}}+\\|(\rho_{N,\hbar}^{(1)}u)(t)-(\rho u)(t)\\|_{L^{1}}$
	$\displaystyle=$	$\displaystyle\\|\operatorname{Im}\left(\overline{\psi_{N,\hbar}(t)}\left(\hbar\nabla_{x_{1}}-iu(t)\right)\psi_{N,\hbar}(t)\right)\\|_{L^{1}}+\\|(\rho_{N,\hbar}^{(1)}u)(t)-(\rho u)(t)\\|_{L^{1}}$
	$\displaystyle\leq$	$\displaystyle\\|\psi_{N,\hbar}(t)\\|_{L^{2}}\\|(i\hbar\nabla_{x_{1}}-u(t))\psi_{N,\hbar}(t)\\|_{L^{2}}+\\|u(t)\\|_{L^{2}}\\|\rho_{N,\hbar}^{(1)}(t)-\rho(t)\\|_{L^{2}}$
	$\displaystyle\lesssim$	$\displaystyle\mathcal{M}^{+}(0)+\hbar^{2},$

where in the last inequality we used the mass conservation, estimates $(\ref{equ:convergence of kinetic energy})$ and $(\ref{equ:convergence of mass density,L2})$ . ∎

Acknowledgements. X. Chen was supported in part by NSF grant DMS-2005469 and a Simons fellowship numbered 916862, S. Shen was supported in part by the Postdoctoral Science Foundation of China under Grant 2022M720263, and Z. Zhang was supported in part by NSF of China under Grant 12171010 and 12288101.

Appendix A Sobolev Type Estimates

Lemma A.1 ([21], Lemma A.5).

Let $d=3$ and $W_{N}(x)=N^{3\beta}V(N^{\beta}x)-b_{0}\delta$ , where $b_{0}=\int V(x)dx$ . For any $0\leq s\leq 1$ ,

(A.1)

\displaystyle\|W_{N}*f\|_{L^{p}}\leq C\|\langle x\rangle V(x)\|_{L^{1}}N^{-\beta s}\|\langle\nabla\rangle^{s}f\|_{L^{p}}

for any $1<p<\infty$ .

Lemma A.2 ([29], Lemma A.3).

Let $d=3$ and $V_{N}(x)=N^{3\beta}V(N^{\beta}x)$ . Then

(A.2)		$\displaystyle V_{N}(x_{1}-x_{2})\leq C\\|V\\|_{L^{1}}(1-\Delta_{x_{1}})(1-\Delta_{x_{2}}),$
(A.3)		$\displaystyle V_{N}(x_{1}-x_{2})\leq CN^{\beta}\\|V\\|_{L^{\frac{3}{2}}}(1-\Delta_{x_{1}}),$
(A.4)		$\displaystyle V_{N}(x_{1}-x_{2})\leq CN^{3\beta}\\|V\\|_{L^{\infty}}.$

Lemma A.3.

Suppose that $f\in L^{1}$ such that

\int|f(x)||x|^{\frac{1}{2}}dx<\infty.

Let $f_{\varepsilon}(x)=\varepsilon^{3}f(\varepsilon x)$ and $d_{0}=\int fdx$ , then we have

		$\displaystyle\|\langle(f_{\varepsilon}(x-y)-d_{0}\delta(x-y))\varphi,\psi\rangle\|$
	$\displaystyle\lesssim$	$\displaystyle\varepsilon^{\theta}\langle(1-\Delta_{x})(1-\Delta_{y})\varphi,\varphi\rangle^{\frac{1}{2}}\langle(1-\Delta_{x})(1-\Delta_{y})\psi,\psi\rangle^{\frac{1}{2}}$

for $\theta\in(0,\frac{1}{2})$ .

Proof.

For the derivation of NLS, this Poincar $\acute{e}$ type inequality is usually used in the convergence part of the hierarchy method. See, for example, [29, 30, 31, 46]. For completeness, we here include a proof. Without loss of generality, we might as well assume that $d_{0}=\int fdx=1$ . Switching to Fourier space, we observe that

		$\displaystyle\langle\varphi,(f_{\varepsilon}(x-y)-\delta(x-y))\psi\rangle$
	$\displaystyle=$	$\displaystyle\int dxdpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)f(x)(e^{i\varepsilon p\cdot x}-1).$

By using $|e^{ia}-1|\leq\min\left\{a,2\right\}\leq 2a^{\theta}$ for $\theta\in(0,1)$ and $|p|^{\theta}\leq\langle\xi_{1}\rangle^{\theta}+\langle\xi_{1}+p\rangle^{\theta}$ , we have

		$\displaystyle\langle\varphi,(f_{\varepsilon}(x-y)-\delta(x-y))\psi\rangle$
	$\displaystyle\leq$	$\displaystyle 2\varepsilon^{\theta}\int f(x)\|x\|^{\theta}dx\int dpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)\|p\|^{\theta}$
	$\displaystyle\leq$	$\displaystyle 2\varepsilon^{\theta}\int f(x)\|x\|^{\theta}dx\int dpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)\left(\langle\xi_{1}\rangle^{\theta}+\langle\xi_{1}+p\rangle^{\theta}\right).$

It suffices to bound the term containing $\langle\xi_{1}\rangle^{\theta}$ , as the term containing $\langle\xi_{1}-p\rangle^{\theta}$ can be estimated similarly. We rewrite

		$\displaystyle\int dpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)\langle\xi_{1}\rangle^{\theta}$
	$\displaystyle=$	$\displaystyle\int dpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)\frac{\langle\xi_{1}\rangle\langle\xi_{2}\rangle\langle\xi_{1}+p\rangle\langle\xi_{2}-p\rangle}{\langle\xi_{1}\rangle^{1-\theta}\langle\xi_{2}\rangle\langle\xi_{1}+p\rangle\langle\xi_{2}-p\rangle}$

By Cauchy-Schwarz inequality,

		$\displaystyle\int dpd\xi_{1}d\xi_{2}\widehat{\varphi}(\xi_{1},\xi_{2})\overline{\widehat{\psi}}(\xi_{1}+p,\xi_{2}-p)\langle\xi_{1}\rangle^{\theta}$
	$\displaystyle\leq$	$\displaystyle\left[\int dpd\xi_{1}d\xi_{2}\frac{\langle\xi_{1}\rangle^{2}\langle\xi_{2}\rangle^{2}}{\langle\xi_{1}+p\rangle^{2}\langle\xi_{2}-p\rangle^{2}}\|\widehat{\varphi}(\xi_{1},\xi_{2})\|^{2}\right]^{\frac{1}{2}}$
		$\displaystyle\cdot\left[\int dpd\xi_{1}d\xi_{2}\frac{\langle\xi_{1}+p\rangle^{2}\langle\xi_{2}-p\rangle^{2}}{\langle\xi_{1}\rangle^{2-2\theta}\langle\xi_{2}\rangle^{2}}\|\widehat{\psi}(\xi_{1}+p,\xi_{2}-p)\|^{2}\right]^{\frac{1}{2}}$
	$\displaystyle=$	$\displaystyle\left[\int dpd\xi_{1}d\xi_{2}\frac{\langle\xi_{1}\rangle^{2}\langle\xi_{2}\rangle^{2}}{\langle\xi_{1}+p\rangle^{2}\langle\xi_{2}-p\rangle^{2}}\|\widehat{\varphi}(\xi_{1},\xi_{2})\|^{2}\right]^{\frac{1}{2}}$
		$\displaystyle\cdot\left[\int dpd\xi_{1}d\xi_{2}\frac{\langle\xi_{1}\rangle^{2}\langle\xi_{2}\rangle^{2}}{\langle\xi_{1}+p\rangle^{2-2\theta}\langle\xi_{2}-p\rangle^{2}}\|\widehat{\psi}(\xi_{1},\xi_{2})\|^{2}\right]^{\frac{1}{2}}$
	$\displaystyle\lesssim$	$\displaystyle\langle(1-\Delta_{1})(1-\Delta_{2})\varphi,\varphi\rangle^{\frac{1}{2}}\langle(1-\Delta_{1})(1-\Delta_{2})\psi,\psi\rangle^{\frac{1}{2}}$

where in the last inequality we used that

\displaystyle\sup_{\xi_{1},\xi_{2}}\int\frac{1}{\langle\xi_{1}-p\rangle^{2-2\theta}\langle\xi_{2}-p\rangle^{2}}dp<\infty

for all $0\leq\theta<\frac{1}{2}$ . ∎

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	$\displaystyle(\|x\|+N^{-\beta})w_{N,\hbar}(x)=$	$\displaystyle\frac{c_{0}}{N\hbar^{2}}\int\frac{\|x\|+N^{-\beta}}{\|x-y\|}V_{N}(y)(1-w_{N,\hbar}(y))dy$
	$\displaystyle\leq$	$\displaystyle\frac{c_{0}}{N\hbar^{2}}\int\frac{\|x-y\|+\|y\|+N^{-\beta}}{\|x-y\|}V_{N}(y)dy$
	$\displaystyle=$	$\displaystyle\frac{c_{0}\\|V\\|_{L^{1}}}{N\hbar^{2}}+\frac{c_{0}}{N^{1+\beta}\hbar^{2}}\int\frac{1+N^{\beta}\|y\|V_{N}(y)}{\|x-y\|}dy$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{N\hbar^{2}}\left(\\|V\\|_{L^{1}}+\\|\langle y\rangle V(y)\\|_{L^{\frac{3}{2}}}\right).$

		$\displaystyle\|(\|x\|^{2}+N^{-2\beta})\nabla_{x}w_{N,\hbar}(x)\|$
	$\displaystyle\leq$	$\displaystyle\frac{1}{2N\hbar^{2}}(\|x\|^{2}+N^{-2\beta})\Big{\|}\int\nabla_{x}\frac{1}{\|x-y\|}V_{N}(y)(1-w_{N,\hbar}(y))dy\Big{\|}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{N\hbar^{2}}\int\frac{\|x-y\|^{2}+\|y\|^{2}+N^{-2\beta}}{\|x-y\|^{2}}V_{N}(y)dy$
	$\displaystyle\leq$	$\displaystyle\frac{\\|V\\|_{L^{1}}}{N\hbar^{2}}+\frac{1}{N\hbar^{2}}\int\frac{1+N^{2\beta}\|y\|^{2}V_{N}(y)}{\|x-y\|^{2}}dy$
	$\displaystyle\leq$	$\displaystyle\frac{1}{N\hbar^{2}}\left(\\|V\\|_{L^{1}}+\\|\langle y\rangle^{2}V(y)\\|_{L^{3}}\right).$

On the mean-field and semiclassical limit from quantum NN-body dynamics

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

1.1. Background and Problems

1.2. Statement of the Main Theorem

Theorem 1.1.

1.3. Outline of the Proof

2. The Time Evolution of the Modulated Energy

Lemma 2.1.

Proof.

Proposition 2.2.

Proof.

3. (HN,ℏ)2(H_{N,\hbar})^{2} Energy Estimate Using Singular Correlation Structure

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Proposition 3.3.

Proof.

4. Functional Inequalities

Proposition 4.1.

Proof.

4.1. Error Analysis of Two-Body Term

Lemma 4.2.

Proof.

4.2. Tamed Singularities

Lemma 4.3.

Proof.

4.3. Reduced Version of Functional Inequality

Lemma 4.4 (Reduced Version of Functional Inequality).

Proof.

Lemma 4.5.

Proof.

Proof of Proposition 4.1.

5. Quantitative Strong Convergence of Quantum Densities

Proposition 5.1.

Proof.

Proof of Theorem 1.1.

Appendix A Sobolev Type Estimates

Lemma A.1 ([21], Lemma A.5).

Lemma A.2 ([29], Lemma A.3).

Lemma A.3.

Proof.

References

On the mean-field and semiclassical limit from quantum $N$ -body dynamics

3. $(H_{N,\hbar})^{2}$ Energy Estimate Using Singular Correlation Structure