# Tracer particles coupled to an interacting boson gas

Esteban Cárdenas  Department of Mathematics, University of Texas at Austin,
2515 Speedway, Austin TX, 78712, USA<EMAIL_ADDRESS>
###### Abstract.

In this work, we investigate the mean-field limit of a model consisting of
$m\geqslant 1$ tracer particles, coupled to an interacting boson field. We
assume the mass of the tracer particles and the expected number of bosons to
be of the same order of magnitude $N\geqslant 1$ and we investigate the
$N\rightarrow\infty$ limit. In particular, we show that the limiting system
can be effectively described by a pair of variables
$(\textbf{X}_{t},\varphi_{t})\in\mathbb{R}^{3m}\times H^{1}(\mathbb{R}^{3})$
that solve a mean-field equation. Our methods are based on proving estimates
for the number of bosonic particles in a suitable fluctuation state
$\Omega_{N,t}$. The main diffculty of the problem comes from the fact that the
interaction with the tracer particles can create or destroy bosons for states
close to the vacuum.

###### Contents

  1. 1 Introduction
    1. 1.1 The Hamiltonian
    2. 1.2 The Mean-Field Equations
  2. 2 Main Results
    1. 2.1 Discussion of Main Results
  3. 3 Fluctuation States
    1. 3.1 Regularization of the Problem
    2. 3.2 The Truncated Dynamics
    3. 3.3 Proof of Theorem 3.1
  4. 4 Proof of our Main Results
  5. 5 Particle Number Estimates of the Truncated Dynamics
    1. 5.1 Dynamical Estimates
    2. 5.2 Proof of Lemmata
  6. 6 Well-Posedness of the Mean-Field Equations
  7. 7 Well-posedness of the Regularized, Truncated Dynamics
  8. A The Regularization Lemmata
  9. B The Interpolation Lemma

##  1\. Introduction

In this article, we consider $m\geqslant 1$ heavy quantum particles (tracer
particles) that are coupled to a weakly interacting gas of non-relativistic
scalar bosons. We are interested in the regime for which the average number of
bosons and the mass of the tracer particles are, in appropiate physical units,
large and of the same order of magnitude $N$. The microscopic states of this
system are described by wave functions that belong to the Hilbert space

$\mathscr{H}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=L^{2}_{\bm{X}}\otimes\mathcal{F}_{b}$
(1.1)

where $L^{2}_{\bm{X}}=L^{2}(\mathbb{R}^{3m},\mathrm{d}{\bm{X}})$ is the state
space for the tracer particles, with positions labeled by
${\bm{X}}=(X^{(1)},\ldots,X^{(m)})\in\mathbb{R}^{3m}$, and $\mathcal{F}_{b}$
is the spinless bosonic Fock space

$\textstyle\mathcal{F}_{b}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathbb{C}\oplus\bigoplus_{n\geqslant
1}\mathcal{F}_{n}\qquad\textnormal{where}\qquad\mathcal{F}_{n}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\,L^{2}(\mathbb{R}^{3},\mathrm{d}x)^{\otimes_{\mathrm{sym}}n}\,.$
(1.2)

We will denote by $\Omega\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(1,0,\ldots)$
the vacuum vector in $\mathcal{F}_{b}$ and make extensive use of the operator-
valued distributions $a_{x}$ and $a_{x}^{*}$ satisfying the canonical
commutation relations (CCR)

$[a_{x},a_{y}^{*}]=\delta(x-y)\,,\qquad[a_{x},a_{y}]=[a_{x}^{*},a_{y}^{*}]=0\,,\,\,\qquad\quad
x,y\in\mathbb{R}^{3}\,.$ (1.3)

The reader is refered to [33, Section 2] for a nice account on creation and
annihilation operators, and to [32, Section X.7] for a thorough mathematical
treatment.

If $\Psi_{N,t}\in\mathscr{H}$ denotes the wave function of the above system at
time $t\geqslant 0$, the main goal of the present article is to derive an
effective description of this object, in the large $N$ limit. From a heuristic
point of view, one expects that as the mass of the tracer particles gets
larger, classical behaviour should be dominant. On the other hand, for low
enough temperatures, one expects a boson gas to experience condensation–most
of the particles will be in the same quantum state. This leads us to introduce
a pair of interacting mean-field variables

$t\in\mathbb{R}\longmapsto({\bm{X}}_{t},\varphi_{t})\in\mathbb{R}^{3m}\times
L^{2}(\mathbb{R}^{3})$ (1.4)

where ${\bm{X}}_{t}$ describes the classical trajectories followed by the
tracer particles and $\varphi_{t}$ is a self-interacting wave function
describing the spatial probability distribution of the condensate.

###  1.1. The Hamiltonian

Throughout this article, we assume that the boson-tracer and boson-boson
interactions are mediated by even, real-valued functions
$w:\mathbb{R}^{3}\rightarrow\mathbb{R}$ and
$v:\mathbb{R}^{3}\rightarrow\mathbb{R}$. We will always assume enough
regularity so that the $(n+m)$-particle Hamiltonian

$H_{N,n}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=-\frac{\Delta_{{\bm{X}}}}{2N}\otimes\mathds{1}-\sum_{i=1}^{n}\mathds{1}\otimes{\Delta_{x_{i}}}+\mathds{1}\otimes\frac{1}{N}\sum_{i<j}^{n}v(x_{i}-x_{j})+\sum_{i=1}^{n}\sum_{\ell=1}^{m}w(x_{i}-X^{(\ell)})\,,\quad
n\geqslant 1$ (1.5)

is self-adjoint in its natural domain, with the obvious modification for
$n=0$. The dynamics of our combined system will be then described by the
second-quantized Hamiltonian in $\mathscr{H}$

$\mathcal{H}_{N}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=-\frac{\Delta_{{\bm{X}}}}{2N}\otimes\mathds{1}+\mathds{1}\otimes
T_{b}+\mathds{1}\otimes\frac{1}{2N}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)a_{x}^{*}a_{y}^{*}a_{x}a_{y}\,\mathrm{d}x\mathrm{d}y+\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\,\underline{w}(x,{\bm{X}})\otimes
a_{x}^{*}a_{x}\,\mathrm{d}x\,,$ (1.6)

equivalently defined as the direct sum $\bigoplus_{n\geqslant 0}H_{N,n}$. Here
and in the sequel, $T_{b}$ denotes the kinetic energy of the bosons

$\textstyle T_{b}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}a_{x}^{*}(-\Delta_{x})a_{x}\mathrm{d}x=(0)\oplus\bigoplus_{n\geqslant
1}\sum_{i=1}^{n}\big{(}-\Delta_{x_{i}}\big{)}$ (1.7)

and we will be using the following short-hand notation for the total boson-
tracer interaction

$\textstyle\underline{w}(x,{\bm{X}})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sum_{\ell=1}^{m}w(x-X^{(\ell)})\,,\qquad
x\in\mathbb{R}^{3}\,,\ {\bm{X}}=(X^{(1)},\ldots,X^{(m)})\in\mathbb{R}^{3m}\,.$
(1.8)

The Spectral Theorem then gives meaning to the evolution associated to the
Schrödinger equation–from now on refered to as the microscopic dynamics of the
system–

$\begin{cases}&\mathrm{i}\partial_{t}\Psi_{N,t}=\mathcal{H}_{N}\Psi_{N,t}\\\
&\Psi_{N,t}\big{|}_{t=0}=\Psi_{N,0}\end{cases}$ (1.9)

by means of exponentiation
$\Psi_{N,t}=\exp(-\mathrm{i}t\mathcal{H}_{N})\Psi_{N,0}$. A few comments are
in order.

######  Remark 1.1.

Since the Hamiltonian $\mathcal{H}_{N}$ is diagonal with respect to Fock space
$\mathcal{F}_{b}$, it commutes with the particle number operator

$\textstyle N_{b}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}a_{x}^{*}a_{x}\mathrm{d}x=\bigoplus_{n\geqslant
0}n\,.$ (1.10)

Therefore, if $\Psi_{N,0}$ contains in quantum-mechanical average $N$ bosons,
so will $\Psi_{N,t}$ for all later times. In other words, it holds that

$\left\langle\Psi_{N,t},\mathds{1}\otimes
N_{b}\Psi_{N,t}\right\rangle_{\mathscr{H}}=\left\langle\Psi_{N,0},\mathds{1}\otimes
N_{b}\Psi_{N,0}\right\rangle_{\mathscr{H}}=N\,,\qquad\forall t\geqslant 0\,.$
(1.11)

The initial data that we work with will always satisfy this condition.

######  Remark 1.2.

Equation (1.9) is equivalent to an infinite system of equations: one for each
component of $\Psi_{N,t}$ in its direct sum decomposition. Thus, an effective
description of the microscopic dynamics in terms of the $(3m+1)$ mean-field
variables drastically reduces the number of unknowns one should solve for.
Describing such an approximation is the content of the present paper.

######  Remark 1.3.

On a formal level, each term present in the definition of $\mathcal{H}_{N}$ is
of order $N$. Indeed, for the tracer particles, we expect position and
velocities to be of order $1$, so that their kinetic energy is of order $N$
due to their heavy mass. For the boson field, we expect its kinetic term to be
of order $N$ since there should be in average $N$ of them, each of which has
order $1$ kinetic energy. Similary, the interaction terms are scaled so that
they remain of the same order of magnitude in $N$; one can directly count the
powers on creation and annhilation operators present in each term. This is the
so-called weak coupling scaling, or mean-field regime.

######  Remark 1.4.

If the potentials $v$ and $w$ satisfy Condition 2–stated below–it is well-
known that the finite particle Hamiltonian $H_{N,n}$ is self-adjoint in
$H^{2}_{\bm{X}}\otimes H^{2}(\mathbb{R}^{3})^{\otimes_{sym}n}$. Further, for
smooth $\Phi\in\mathcal{F}_{b}$ with finitely many non-zero entries, one may
verify that the boson-boson interaction satisfies the estimate

$\displaystyle\frac{1}{N}\big{\|}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)a_{x}^{*}a_{y}^{*}a_{x}a_{y}\mathrm{d}x\mathrm{d}y\,\Phi\big{\|}_{\mathcal{F}_{b}}\,\lesssim\,\|T_{b}\Phi\|_{\mathcal{F}_{b}}+\frac{1}{N^{2}}\|(\mathds{1}+N_{b})^{3}\Phi\|_{\mathcal{F}_{b}}\,.$
(1.12)

Similarly, the boson-tracer interaction is controlled by $\mathds{1}\otimes
N_{b}$. Consequently, we have the following characterization for the domain of
$\mathcal{H}_{N}$:

$\mathcal{D}(\mathcal{H}_{N})\cap\mathcal{D}(\mathds{1}\otimes
N_{b}^{3})=H_{\bm{X}}^{2}\otimes\mathcal{D}(T_{b})\cap\mathcal{D}(\mathds{1}\otimes
N_{b}^{3})\,,\qquad\forall N\geqslant 1\,.$ (1.13)

###  1.2. The Mean-Field Equations

Let us now introduce the equations satisfied by the mean-field variables.
Heuristically, the classical trajectory of a tracer particle should obey
Newton’s equation, with a time-dependent force depending on the location of
the bosons. Similarly, the quantum boson condensate should feel a time-
dependent potential, related to the positions of the tracer particles, plus
its usual self-interactions as is well-known from the realm of Hartree’s
equation. Thus, we introduce the following system of equations, from now on
refered to as the mean-field equations

$\displaystyle\begin{cases}&\ddot{\bm{X}}_{t}=-\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\,\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})\,|\varphi_{t}(x)|^{2}\,\mathrm{d}x\\\
&\mathrm{i}\partial_{t}\varphi_{t}=-\Delta\varphi_{t}+\underline{w}(x,{\bm{X}}_{t})\varphi_{t}+(v*|\varphi_{t}|^{2}\,)\,\varphi_{t}\\\
&({\bm{X}}_{t},\dot{\bm{X}}_{t},\varphi_{t})\big{|}_{t=0}=({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\end{cases}$
(1.14)

provided some initial data $({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})$ has been
specified at time zero. Global well-posedness in $\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$, for the class of potentials we are going to be working
with, is proven in Section 6. In particular, the $L^{2}$ norm of the bosons
and the energy of the system are conserved quantities.

##  2\. Main Results

###  2.1. Discussion of Main Results

In this section we state our main theorems, which connect the microscopic
dynamics (1.9) with the mean-field equations (1.14). First, let us introduce
the following objects; we will make extensive use of them throughout this
article.

  * •

We denote by ${\bm{X}}$ and ${\bm{P}}=-\mathrm{i}\nabla_{\bm{X}}$ the tracer
particle’s position and momentum observables in $L^{2}_{\bm{X}}$, with domains
$\mathcal{D}({\bm{X}})$ and $\mathcal{D}({\bm{P}})$, respectively.

  * •

For $f\in L^{2}(\mathbb{R}^{3})$ we introduce the Weyl operator on
$\mathcal{F}_{b}$, defined as the unitary transformation

$\mathcal{W}(f)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(a^{*}(f)-a(f))\,,$
(2.1)

where

$a(f)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}f(x)a_{x}\mathrm{d}x\,\quad\textnormal{and}\quad
a^{*}(f)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\widebar{f(x)}a_{x}^{*}\mathrm{d}x\,.$
(2.2)

See [33, Lemma 2.2] for a list of the properties satisfied by Weyl operators.

The following Condition contains the mathematical properties satisfied by the
initial data $\Psi_{N,0}$ that we work with.

######  Condition 1 (Initial Data).

We assume that the initial state of the combined system is a tensor product of
the form

$\Psi_{N,0}=u_{N,0}\otimes\Phi_{N,0}\,,\qquad N\geqslant 1$ (2.3)

where $u_{N,0}\in L^{2}_{\bm{X}}$ and $\Phi_{N,0}\in\mathcal{F}_{b}$ satisfy
the additional requirements:

  1. (1)

$u_{N,0}\in\mathcal{D}({\bm{P}}^{3})\cap\mathcal{D}({\bm{X}}^{3})$ has unit
$L^{2}$-norm. Further, we assume that there exists
$({\bm{X}}_{0},{\bm{V}}_{0})\in\mathbb{R}^{3m}\times\mathbb{R}^{3m}$ and
$C_{0}>0$ such that for any $1\leqslant p\leqslant 3$ and all $N\geqslant 1$
it holds

$\displaystyle\|\,|{\bm{X}}-{\bm{X}}_{0}|^{p}u_{N,0}\,\|_{L^{2}_{\bm{X}}}\,+\,\|\,|N^{-1}{\bm{P}}-{\bm{V}}_{0}|^{p}u_{N,0}\,\|_{L^{2}_{\bm{X}}}\
\leqslant\ \frac{C_{0}}{N^{p/2}}\ .$

  2. (2)

There exists $\varphi_{0}\in H^{1}(\mathbb{R}^{3})$, of unit $L^{2}$-norm,
such that

$\Phi_{N,0}=\mathcal{W}(\sqrt{N}\varphi_{0})\Omega\,.$

######  Remark 2.1.

Given $u\in\mathcal{D}({\bm{P}}^{3})\cap\mathcal{D}({\bm{X}}^{3})$, we may
construct an initial condition that satisfies Condition 1 by appropiate re-
scaling, that is, by setting

$u_{N}({\bm{X}})=N^{3m/4}\exp(\mathrm{i}N\,{\bm{X}}\cdot{\bm{V}}_{0})\,u(N^{1/2}({\bm{X}}-{\bm{X}}_{0}))\,,\qquad{\bm{X}}\in\mathbb{R}^{3m}\,,\
N\geqslant 1\,.$

In particular, $\|u_{N}\|_{L^{2}}=\|u\|_{L^{2}}$ for each $N$.

In $(1)$–besides regularity requirements–we ask the tracer particle’s wave
function to be such that its position and velocity distribution converge to a
Dirac delta, centered at the phase space point
$({\bm{X}}_{0},{\bm{V}}_{0})\in\mathbb{R}^{3m}\times\mathbb{R}^{3m}$. In
particular, we choose the fastest rate of convergence that is in agreement
with Heisenberg’s uncertainty principle:
$\Delta{\bm{X}}\Delta{\bm{V}}\geqslant N^{-1}$. Note that convergence is
rather strong, since it includes up to six moments in the position and
momentum observables. Our aim is to understand how these two conditions are
propagated in time, provided one replaces $({\bm{X}}_{0},{\bm{V}}_{0})$ with
$({\bm{X}}_{t},\dot{\bm{X}}_{t})$.

As for the bosons, in (2) we ask the initial datum to be a coherent state;
each $n$-particle entry may be described as a symmetrization of the boson
field $\varphi_{0}$, in the sense that

$\mathcal{W}(\sqrt{N}\varphi_{0})\Omega=\exp(-N/2)\sum_{n=0}^{\infty}\frac{N^{n/2}}{\sqrt{n!}}\varphi_{0}^{\otimes
n}\,.$ (2.4)

The time evolution of coherent states has been widely studied in the
literature, and the following approach is now standard. We introduce the one-
particle density

$\Gamma_{N,t}(x,y)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{N}\left\langle\Psi_{N,t},\mathds{1}\otimes
a_{x}^{*}a_{y}\Psi_{N,t}\right\rangle_{\mathscr{H}}\,,\qquad(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\,$
(2.5)

and we let $\Gamma_{N,t}$ be the linear operator in $L^{2}(\mathbb{R}^{3})$
whose integral kernel is $\Gamma_{N,t}(x,y)$. In particular, this operator is
positive, self-adjoint and trace-class. Similarly, for the solution of the
mean-field equations $\varphi_{t}\in L^{2}(\mathbb{R}^{3})$, we let
$\ket{\varphi_{t}}\bra{\varphi_{t}}$ be the orthogonal projection onto the
subspace spanned by $\varphi_{t}$. One is then interested in the convergence
of $\Gamma_{N,t}$ towards $\ket{\varphi_{t}}\bra{\varphi_{t}}$.

The class of potentials that we work with is described in the following
Condition. Most notably, we are able to include Coulomb potentials in the
boson-boson interaction. On the other hand, requiring three derivatives in the
boson-tracer interaction is, almost certainly, not optimal.

######  Condition 2 (Potentials).

We assume $w:\mathbb{R}^{3}\rightarrow\mathbb{R}$ and
$v:\mathbb{R}^{3}\rightarrow\mathbb{R}$ are even, real-valued functions that
satisfy

  1. (1)

$w\in C_{b}^{3}(\mathbb{R}^{3})$ .

  2. (2)

$v=v_{1}+v_{2}$ where $v_{1}=\lambda|x|^{-1}$ for some $\lambda\in\mathbb{R}$
and $v_{2}\in L^{\infty}(\mathbb{R}^{3})$ .

Our main results are the following two theorems.

######  Theorem 2.1.

Let $(\Psi_{N,0})_{N\geqslant 1}$ satisfy Condition 1 with respect to
$({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$ and assume the potentials $v$ and $w$ satisfy Condition
2. Let $({\bm{X}}_{t},\varphi_{t})$ solve the mean-field equations (1.14),
with the same initial data, and let
$\Psi_{N,t}=\exp(-\mathrm{i}t\mathcal{H}_{N})\Psi_{N,0}$ solve the microscopic
dynamics (1.9). Then, there exists $C>0$ such that for all $t\geqslant 0$ and
$N\geqslant 1$ it holds that

$\displaystyle\big{|}\left\langle\Psi_{N,t},\,{\bm{X}}\otimes\mathds{1}\
\Psi_{N,t}\right\rangle_{\mathscr{H}}-{\bm{X}}_{t}\big{|}$ $\displaystyle\
\leqslant\ \frac{C\,e^{Ct}}{N^{1/4}}$ (2.6)
$\displaystyle\big{|}N^{-1}\left\langle\Psi_{N,t},\,{\bm{P}}\otimes\mathds{1}\Psi_{N,t}\right\rangle_{\mathscr{H}}-\dot{\bm{X}}_{t}\big{|}$
$\displaystyle\ \leqslant\ \frac{C\,e^{Ct}}{N^{1/4}}\,.$ (2.7)

######  Theorem 2.2.

Under the same conditions of Theorem 2.1, there exists $C>0$ such that for all
$t\geqslant 0$ and $N\geqslant 1$ it holds that

$\mathrm{Tr}\,|\,\Gamma_{N,t}-\ket{\varphi_{t}}\bra{\varphi_{t}}\,|\
\leqslant\ \frac{C\,e^{Ct}}{N^{1/4}}\,.$ (2.8)

######  Remark 2.2.

The conclusion of Theorem 2.1 characterizes the limits of the first moments of
the probability measures

$\mu^{{\bm{X}}}_{N,t}(E)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\|\mathds{1}({\bm{X}}\in
E)\Psi_{N,t}\|^{2}_{\mathscr{H}}\qquad\textnormal{and}\qquad\mu^{{\bm{V}}}_{N,t}(E)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\|\mathds{1}(N^{-1}{\bm{P}}\in
E)\Psi_{N,t}\|^{2}_{\mathscr{H}}$ (2.9)

where $E\in\mathcal{B}(\mathbb{R}^{3m})$ is a Borel set. Since we are actually
able to control their variance in Lemma 4.1, it follows easily that there is
weak-* convergence as $N\rightarrow\infty$:

$\mu^{{\bm{X}}}_{N,t}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\delta_{{\bm{X}}_{t}}\qquad\textnormal{and}\qquad\mu^{{\bm{V}}}_{N,t}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\delta_{\dot{\bm{X}}_{t}}\,.$
(2.10)

######  Remark 2.3.

In the particular situation of Theorem 2.2, the Hilbert-Schmidt norm controls
the trace norm. Namely, for the linear operator $\mathcal{S}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma_{N,t}-\ket{\varphi_{t}}\bra{\varphi_{t}}$
it holds that

$\mathrm{Tr}\,|\mathcal{S}|\leqslant 2\|\mathcal{S}\|_{HS}\,.$ (2.11)

Therefore, we will actually prove convergence in Hilbert-Schmidt norm and then
apply (2.11).

######  Remark 2.4 (Convergence Rates).

If one compares Condition 1 for $p=1$, with the result of Theorem 2.1, it is
evident that there is a loss of a factor $N^{-1/4}$ in the rate of
convergence. For $v=0$, it follows easily from ours methods that optimal
convergence rate is achieved and we only need
$u_{N,0}\in\mathcal{D}({\bm{X}})\cap\mathcal{D}({\bm{P}})$ with (1) in
Condition 1 valid only up to $p=1$. For $v\neq 0$, we can achieve optimal
convergence rate provided we take the following stronger assumptions:  (i)
$u_{N,0}\in\mathcal{D}({\bm{X}}^{4})\cap\mathcal{D}({\bm{P}}^{4})$ with (1) in
Condition 1 valid for $p=4$ and (ii) $w\in C_{b}^{4}(\mathbb{R}^{3})$. This
last statement, however, requires more work as one needs to deal with higher
powers of the observables ${\bm{X}}$ and ${\bm{P}}$.

The author did not investigate the optimality of the rate of convergence in
Theorem 2.2. We believe, however, the optimal rate to be $N^{-1}$ provided one
assumes $p$ to be large enough in Condition 1, and $w$ to be regular enough in
Condition 2.

######  Remark 2.5 (Tracer-tracer Interactions).

With our methods, it would be possible to include interactions between the
tracer particles $V(X^{(i)}-X^{(\ell)})$, provided the potential function is
regular enough. We decide to omit such a term in order to highlight the more
interesting boson-tracer interaction $\underline{w}(x,{\bm{X}})$ and the more
technical boson-boson term $v(x-y)$.

#### Our contribution

There have been several studies in the literature that consider derivation of
effective equations for many-particle systems of interacting bosons. The
classical references are [22, 34, 16, 17]. More recently, [9, 10, 11, 12]
together with [26, 33] have motivated much of the activity in the field; for a
non-exhaustive list we refer to [29, 30, 7, 2, 5, 6, 3, 4, 18, 21, 20, 19, 25,
31].

On the other hand, there have been only few studies of systems that include
interactions between bosons and one or more tracer particle. Most remarkably:

  1. (1)

In [8], Deckert, Fröhlich, Pizzo and Pickl consider the $(N+1)$-particle
Hilbert space $L^{2}(\mathbb{R}^{3(N+1)})$ with the $(N+1)$-particle
Hamiltonian $H_{N,N}$ for $v=0$, i.e. an ideal (non-interacting) Bose gas.
They study the case for which $\varphi_{0}$ is localized in a region that
expands as $N\rightarrow\infty$ and derive a slighly different system of mean-
field equations. We note that they consider only $C_{c}^{\infty}$ potentials
and initial data, but include a smooth tracer-tracer interaction term.

  2. (2)

In [7], Chen and Soffer consider our same model for the case in which there is
only one tracer particle, i.e. $m=1$. The main advantage of this model comes
from translation invariance of the system: one can decompose the Hilbert space
into blocks of constant total momentum $P\in\mathbb{R}^{3}$, i.e. there is a
direct integral representation

$\mathscr{H}=\int_{\mathbb{R}^{3}}^{\oplus}\mathscr{H}(P)\mathrm{d}P$ (2.12)

that allows to mod out the tracer particle dependence in the diagonal term
$\int_{\mathbb{R}^{3}}w(x-X)a_{x}^{*}a_{x}\mathrm{d}x$, see their equation
(1.11). Notably, they are able to give a mean-field description that converges
in Fock space norm once the intertwining between the bosons and the tracer
particle’s momenta has been understood.

  3. (3)

Very recently, Lampart and Pickl in [27] studied the mean-field limit of a
boson gas coupled to a tracer particle, whose mass remains finite. They are
able to derive a norm approximation of the microscopic dynamics via Bogoliubov
theory of the excitations of the condensate.

  4. (4)

In the context of quantum friction/Cherenkov radiation, interactions between
tracer particles and ideal gases of bosons have been studied by Fröhlich and
Gang in [15, 13, 14]. Their starting point are the effective equations derived
in [8].

Our main contribution in this paper can be summarized as follows. Once
translational symmetry is broken, the argument of [7] breaks down and a new
approach towards finding a mean-field description is needed; we develop this
approach while including, at the same time, the possiblity for the Bose gas to
be non-ideal, including the case of Coulomb interactions.

####  2.1.1. Idea of the proof

The proof of our main results is heavily inspired by [8] (to control
expectations of tracer particle’s observables) and by [33] (to control vacuum
fluctuations) and the idea is as follows. First, we introduce fluctuation
states $\Omega_{N,t}\in\mathscr{H}$ that show up naturally when trying to
replace $a_{x}$ with $\sqrt{N}\varphi_{t}(x)$. However, a major difficulty
arises when one writes down its infinitesimal generator. Namely, there is a
(non-diagonal) term of the form

$\sqrt{N}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\otimes\big{(}a_{x}\overline{\varphi_{t}}(x)+a_{x}^{*}\varphi_{t}(x)\big{)}\,\mathrm{d}x$
(2.13)

which has an apparent growth of $\sqrt{N}$. In other words, the presence of
tracer particles can potentially create or destroy a lot of bosons for states
close to the vacuum. In order to control this term, we study simultaneuosly
the difference ${\bm{X}}-{\bm{X}}_{t}$ and apply a Taylor estimate on this
interaction $w(x-X)$. Note that, at the same time, this requires control of
the difference $N^{-1}{\bm{P}}-\dot{\bm{X}}_{t}$ thanks to Ehrenfest’s
Theorem. Subsequently, once that control of $\Omega_{N,t}$ has been
established, important remainder terms can be estimated and the proof of our
main results follows in a few pages.

####  2.1.2. Organization of the paper

In Section 3 we introduce the fundamental notion of a fluctuation state,
together with its regularized version. In Section 4 we prove our main results,
based on particle number estimates satisfied by fluctuation states. In Section
5 we complete the proof of these particle number estimates. In Sections 6 and
7 we prove some required well-posedness results.

####  2.1.3. Notation

  * -

$C_{b}^{k}(\mathbb{R}^{3})$ is the class of $k\in\mathbb{N}$ times
continuously differentiable functions, with all derivatives bounded. We equip
it with the norm $\|f\|_{C_{b}^{k}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=m\sum_{0\leqslant|\alpha|\leqslant
k}\|\partial^{\alpha}f\|_{L^{\infty}}$.

  * -

Given two quantities $A(N,t)$ and $B(N,t)$, we say that $A\lesssim B$ if there
exists a constant $C>0$, depending only on $w$, $v$, $\varphi_{0}$ and
$C_{0}$, such that

$\textstyle A(N,t)\leqslant C\,B(N,t)\,,\qquad\forall N\geqslant 1\,,\ \forall
t\geqslant 0\,.$ (2.14)

  * -

We drop the tensor product symbol $\otimes$ for products of operators.

  * -

We drop the subscript $\mathscr{H}$ associated to the inner product
$\left\langle\cdot,\cdot\right\rangle_{\mathscr{H}}$, whenever there is no
risk of confusion.

  * -

$\chi(A)$ stands for the characteristic function of the measurable set
$A\subset\mathbb{R}$.

  * -

Given a Banach space $E$, we denote by $\mathcal{B}(E)$ the Banach space of
bounded, linear maps $E\rightarrow E$, equipped with its usual norm
$\|\cdot\|_{\mathcal{B}(E)}.$

####  2.1.4. Fundamental estimates

Let us give a list of estimates that will be used throughout this work,
together with close variations of them; since they are well-known, we omit
their proof. In the sequel, we shall refer to them in italics, i.e. as
Particle Number Estimates, Potential Estimates, etc.

Taylor Estimates. For all $x,y\in\mathbb{R}^{3}$ it holds that

$|w(x)-w(y)|+|\nabla w(x)-\nabla w(y)|\lesssim|x-y|\,.$ (2.15)

Particle Number Estimates. For all $\Phi\in\mathscr{H}$ and $f=f(x,{\bm{X}})$
it holds

$\displaystyle\big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}f(x,{\bm{X}})a_{x}^{*}\mathrm{d}x\,\Phi\big{\|}$
$\displaystyle\lesssim\sup_{{\bm{X}}\in\mathbb{R}^{3m}}\|f(\cdot,{\bm{X}})\|_{L^{2}}\|(N_{b}+1)^{1/2}\Phi\|$
(2.16)
$\displaystyle\big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}f(x,{\bm{X}})a_{x}^{*}a_{x}\mathrm{d}x\Phi\big{\|}$
$\displaystyle\lesssim\|f\|_{L^{\infty}}\|N_{b}\Phi\big{\|}\ .$ (2.17)

In addition, for all $\Phi\in\mathscr{H}$ and $\phi_{1}=\phi_{1}(x)$ and
$\phi_{2}=\phi_{2}(y)$ it holds

$\displaystyle\big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)\phi_{1}(x)\phi_{2}(y)a_{y}^{*}a_{x}\mathrm{d}x\mathrm{d}y\,\Phi\big{\|}$
$\displaystyle\lesssim\big{(}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)^{2}|\phi_{1}(x)|^{2}|\phi_{2}(y)|^{2}\mathrm{d}x\mathrm{d}y\big{)}^{1/2}\|(N_{b}+1)\Phi\|$
$\displaystyle\big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)\phi_{1}(y)a_{x}^{*}a_{y}^{*}a_{x}\mathrm{d}x\mathrm{d}y\,\Phi\big{\|}$
$\displaystyle\lesssim\big{(}\sup_{x\in\mathbb{R}^{3}}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}v(x-y)^{2}|\phi_{1}(y)|^{2}\mathrm{d}y\big{)}^{1/2}\|(N_{b}+1)^{3/2}\,\Phi\|\,.$
(2.18)

Potential Estimates. Let $v$ satisfy Condition 2. Then, the following operator
inequality holds

$\left\langle\varphi,v^{2}\varphi\right\rangle\leqslant|v|^{2}\|\varphi\|_{H^{1}}^{2}\,,\qquad\varphi\in
H^{1}(\mathbb{R}^{3})$ (2.19)

for some $|v|>0$. Consequently, for any $\varphi\in H^{1}(\mathbb{R}^{3})$ it
holds

$\displaystyle\sup_{x\in\mathbb{R}^{3}}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}v(x-y)^{2}|\varphi(y)|^{2}\mathrm{d}y$
$\displaystyle\ \leqslant\ |v|^{2}\|\varphi\|_{H^{1}}^{2}$ (2.20)
$\displaystyle\sup_{x\in\mathbb{R}^{3}}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}v(x-y)|\varphi(y)|^{2}\mathrm{d}y$
$\displaystyle\ \leqslant\ |v|\|\varphi\|_{L^{2}}\|\varphi\|_{H^{1}}$ (2.21)
$\displaystyle\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)^{2}|\varphi(x)|^{2}|\varphi(y)|^{2}\mathrm{d}y\mathrm{d}x$
$\displaystyle\ \leqslant\
|v|^{2}\|\varphi\|_{L^{2}}^{2}\|\varphi\|_{H^{1}}^{2}\,.$ (2.22)

Local Lipschitz Property. Let $v$ satisfy Condition 2 and define the map
$J(\varphi)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(v*|\varphi|^{2})\varphi$
for $\varphi\in H^{1}(\mathbb{R}^{3})$. Then, the Leibniz rule and (2.21)
imply that for all $\varphi,\psi\in H^{1}(\mathbb{R}^{3})$

$\|J(\varphi)-J(\psi)\|_{H^{1}}\leqslant
L(v)\big{(}\,\|\varphi\|^{2}_{H^{1}}+\|\psi\|_{H^{1}}^{2}\,\big{)}\|\varphi-\psi\|_{H^{1}}\,.$
(2.23)

where $L(v)=C|v|$. For a detailed account on the Lipschitz property for the
case $v=\lambda|x|^{-\gamma}$ with $\gamma\in(0,3/2)$, we refer the reader to
[23, Lemma A.2], whose analysis follows [28, Lemma 1].

##  3\. Fluctuation States

Let $\Psi_{N,t}=\exp(-\mathrm{i}t\mathcal{H}_{N})\Psi_{N,0}$ and
$({\bm{X}}_{t},\varphi_{t})$ be the solutions of (1.9) and (1.14),
respectively. In this section, we study the number of particles present in the
fluctuation state

$\Omega_{N,t}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{-iS(t)}\mathcal{W}^{*}(\sqrt{N}\varphi_{t})\exp(-\mathrm{i}t\mathcal{H}_{N})\mathcal{W}(\sqrt{N}\varphi_{0})(u_{N,0}\otimes\Omega)\,,\qquad
t\in\mathbb{R}\,,\ N\geqslant 1\,.$ (3.1)

Here, $S(t)$ is a scalar term that modifies $\Omega_{N,t}$ only by an overall
phase; it is not a physically measurable quantity. We choose it to be

$S(t)\,\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\,N\int_{0}^{t}\int_{\mathbb{R}^{3}}\Big{(}\underline{w}\big{(}x,{\bm{X}}_{s}\big{)}\,|\varphi_{s}(x)|^{2}\,+\,\frac{1}{2}\big{(}v*|\varphi_{s}|^{2}\big{)}\\!(x)\,|\varphi_{s}(x)|^{2}\Big{)}\mathrm{d}x\,\mathrm{d}s$
(3.2)

The dynamics that drive states similar to $\Omega_{N,t}$ has been studied
extensively, with the first rigorous study being carried out in [16] and more
recently re-activated in [33]. When no tracer particle is present, the usual
infinitesimal generator consists of kinetic terms, plus second quantized
operators that are polynomials of order bigger than or equal to two in the
creation and annihilation operators $a_{x}$ and $a_{x}^{*}\,$; the scalar term
$S(t)$ and the boson field $\varphi_{t}$ can be chosen so that no zero or
first order terms are present. However, when interacting with tracer
particles, the linear terms that arise from
$\int\underline{w}(x,{\bm{X}})a_{x}^{*}a_{x}\mathrm{d}x$111 In [7], the
${\bm{X}}$ dependence is cleverly eliminated from the very beginning, thanks
to the translational symmetry of the model.  cannot be removed unless one is
willing to introduce a ${\bm{X}}$-dependent boson field
$\varphi_{t}(x,{\bm{X}})$. Note that doing so would have the effect of making
$-\Delta_{{\bm{X}}}$ not commute with the Weyl operator
$\mathcal{W}(\sqrt{N}\varphi_{t}({\bm{X}}))$. We believe that the consequences
outweight the benefits and shall not take this approach.

Our main result concerning particle number estimates for fluctuation states is
the following theorem.

######  Theorem 3.1.

Let $\Omega_{N,t}$ be the fluctuation state defined in (3.1), with initial
data satisfying Condition 1. Then, there is a constant $C>0$ such that for all
$t\geqslant 0$ and $N\geqslant 1$ the following estimates hold

$\displaystyle\|N_{b}^{1/2}\Omega_{N,t}\|^{2}$ $\displaystyle\ \leqslant\
Ce^{Ct}N^{1/2}\,,$ (3.3) $\displaystyle\|N_{b}\,\Omega_{N,t}\|^{2}$
$\displaystyle\ \leqslant\ Ce^{Ct}N^{3/2}\,.$ (3.4)

######  Remark 3.1.

Even though we are not able to show that the fluctuation state $\Omega_{N,t}$
has particle number moments that are uniformly bounded in $N\geqslant 1$, we
are able to reduce by a factor $N^{1/2}$ the apriori growth of these moments.
This in turn is enough to prove our main theorems, the price to pay being
having a worse convergence rate with respect to the initial data.

The proof of Theorem 3.1 consists in a detailed study of the linear generator
that drives the time evolution of $\Omega_{N,t}$; let us then describe it in
more detail. The reader should be aware that we will regularize the problem in
a way that the following formal calculations are properly justified–for the
moment we focus on the algebraic aspects of the proof.

First, note that $\Omega_{N,t}=\mathcal{U}_{N}(t,0)(u_{N,0}\otimes\Omega)$,
where we introduce on $\mathscr{H}$ the following two-parameter family of
unitary transformations

$\mathcal{U}_{N}(t,s)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{-\mathrm{i}S(t)}\mathcal{W}^{*}(\sqrt{N}\varphi_{t})\exp(-\mathrm{i}(t-s)\mathcal{H}_{N})e^{\mathrm{i}S(s)}\mathcal{W}(\sqrt{N}\varphi_{s})\,,\quad
t,s\in\mathbb{R}\,,\,N\geqslant 1\,.$ (3.5)

In particular, straightforward differentiation with respect to time $t$ gives

$\mathrm{i}\partial_{t}\mathcal{U}_{N}(t,s)=\big{(}\,\mathcal{W}_{N,t}^{*}\mathcal{H}_{N}\mathcal{W}_{N,t}\,+\,\mathrm{i}\partial_{t}\mathcal{W}_{N,t}^{*}\,\mathcal{W}_{N,t}-\mathrm{i}\,\dot{S}(t)\,\big{)}\,\mathcal{U}_{N}(t,s)=:\mathcal{L}(t)\,\mathcal{U}_{N}(t,s)$
(3.6)

where we have introduced the short-hand notation
$\mathcal{W}_{N,t}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{W}(\sqrt{N}\varphi_{t}).$
In order to get a more explicit representation of the linear generator
$\mathcal{L}(t)$–defined in (3.6)–we proceed as follows. First, since
$({\bm{X}}_{t},\varphi_{t})$ satisfies the mean field equations (1.14), the
time derivative of the Weyl operator may be computed as

$\displaystyle\mathrm{i}\partial_{t}\mathcal{W}_{N,t}$ $\displaystyle\ =\
[\mathcal{H}_{{\bm{X}},\varphi}(t),\mathcal{W}_{N,t}]$ (3.7)
$\displaystyle\mathcal{H}_{{\bm{X}},\varphi}(t)$ $\displaystyle\ =\
T_{b}+\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\underline{w}\big{(}x,{\bm{X}}_{t}\big{)}\,a_{x}^{*}a_{x}\,\mathrm{d}x+\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}v*|\varphi_{t}|^{2}\big{)}(x)\,a_{x}^{*}a_{x}\,\mathrm{d}x\,,$
(3.8)

see for reference [16, Lemma 3.1]. Moreover, a straightforward, although
tedious, application of the translation properties

$\mathcal{W}_{N,t}^{*}a_{x}\mathcal{W}_{N,t}=a_{x}+\sqrt{N}\,\,\varphi_{t}(x)\quad\mathrm{and}\quad\mathcal{W}_{N,t}^{*}a_{x}^{*}\mathcal{W}_{N,t}=a_{x}^{*}+\sqrt{N}\,\,\overline{\varphi_{t}}(x)$
(3.9)

in the second-quantized operators of (3.6), let us cast the infinitesimal
generator of the dynamics in the form

$\displaystyle\mathcal{L}(t)$ $\displaystyle\ =\
-\frac{\Delta_{\bm{X}}}{2N}\,+T_{b}\,+\,\mathcal{I}(t)+\,\mathcal{I}_{4}\,.$
(3.10)

Here, the first two terms correspond to the kinetic energy of the system. The
third term corresponds to the time-dependent operator

$\mathcal{I}(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{L}^{(d)}(t)+\mathcal{L}^{(od)}(t)$
(3.11)

that contains the diagonal and off-diagonal interaction terms, given by

$\displaystyle\mathcal{L}^{(d)}(t)\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\ \
$
$\displaystyle\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,(N|\varphi_{t}(x)|^{2}-a_{x}^{*}a_{x})\,\,\,\mathrm{d}x$
$\displaystyle+\ \
\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\,\big{(}v*|\varphi_{t}|^{2}\big{)}(x)a_{x}^{*}a_{x}\mathrm{d}x\
\ +\ \
\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}\,v(x-y)\overline{\varphi_{t}}(x)\varphi_{t}(y)a_{y}^{*}a_{x}\,\mathrm{d}x\mathrm{d}y\,,$
(3.12) $\displaystyle\mathcal{L}^{(od)}(t)\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\ \
$
$\displaystyle\sqrt{N}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\,\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,\big{(}a_{x}\overline{\varphi_{t}}(x)+a_{x}^{*}\varphi_{t}(x)\big{)}\,\mathrm{d}x\,$
$\displaystyle+\ \
\frac{1}{2}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}\,v(x-y)\big{(}\,\varphi_{t}(x)\varphi_{t}(y)a_{x}^{*}a_{y}^{*}\,+\,\overline{\varphi_{t}}(x)\,\overline{\varphi_{t}}(y)a_{x}a_{y}\,\big{)}\,\mathrm{d}x\mathrm{d}y$
$\displaystyle+\ \
\frac{1}{\sqrt{N}}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}\,v(x-y)\,a_{x}^{*}\,\big{(}\,\varphi_{t}(y)a_{y}^{*}\
+\
\overline{\varphi_{t}}(y)\,a_{y}\,\big{)}\,a_{x}\,\mathrm{d}x\mathrm{d}y\,.$
(3.13)

The fourth term in (3.10) is the time-independent, diagonal quartic term

$\mathcal{I}_{4}\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\frac{1}{2N}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}\,v(x-y)a_{x}^{*}a_{y}^{*}\,a_{x}a_{y}\,\mathrm{d}x\mathrm{d}y\,.$
(3.14)

Note that both the kinetic terms, $\mathcal{L}^{(d)}(t)$ and $\mathcal{I}_{4}$
commute with the particle number operator $N_{b}$, but $\mathcal{L}^{(od)}(t)$
does not. In other words,
$[\mathcal{L}(t),N_{b}]=[\mathcal{L}^{(od)}(t),N_{b}]$.

The main difficulty of the present article is understanding the first term of
$\mathcal{L}^{(od)}(t)$ as it involves a $\sqrt{N}$ factor. To control the
other two terms, we proceed very similarly as in [33]. Namely, we introduce a
truncated dynamics whose generator is $\mathcal{L}(t)$ but with a cut-off in
the interaction term; this is described in Subsection 3.2. First, we
regularize the problem at hand.

###  3.1. Regularization of the Problem

The purpose of regularizing the problem is twofold: (1) all of the upcoming
calculations are rigorously justified, and (2) the proof of well-posedness for
the (auxiliary) truncated dynamics is short. In this subsection, we show how
to regularize both the microscopic dynamics (1.9) and the mean-field equations
(1.14). This in turn is enough to regularize the fluctuation state
$\Omega_{N,t}$

Let us be more precise. Consider some initial datum
$\Psi_{N,0}=u_{N,0}\otimes\mathcal{W}(\sqrt{N}\varphi_{0})$ satisfying
Condition 1, and let $v=\lambda|x|^{-1}+v_{2}$ and $w$ be potentials
satisfying Condition 2. We will regularize these objects according to the
following definition.

######  Definition 3.2.

In what follows, we always assume that $\varepsilon\in(0,1)$.

  1. (1)

We let $w^{\varepsilon}\in\mathcal{S}(\mathbb{R}^{3},\mathbb{R})$ be a
Schwartz class function such that

$\lim_{\varepsilon\downarrow 0}\|w-w^{\varepsilon}\|_{C_{b}^{3}}=0\,.$ (3.15)

  2. (2)

We define the bounded potential $v^{\varepsilon}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\lambda(\varepsilon^{2}+|x|^{2})^{-1/2}+v_{2}\in
L^{\infty}(\mathbb{R}^{3},\mathbb{R})$; it satisfies

$\|v-v^{\varepsilon}\|_{L^{p}}\leqslant C(p)\,\varepsilon^{(3-p)/p}\,,\qquad
p\in[1,3)\,.$ (3.16)

  3. (3)

We let $\varphi^{\varepsilon}_{0}\in H^{2}(\mathbb{R}^{3})$ be an initial
datum for the boson field that satisfies

$\lim_{\varepsilon\downarrow
0}\|\varphi_{0}-\varphi_{0}^{\varepsilon}\|_{H^{1}}=0\,.$ (3.17)

  4. (4)

For each $N\geqslant 1$, we let
$u_{N,0}^{\varepsilon}\in\mathcal{S}(\mathbb{R}^{3})$ be an initial datum that
satisfies

$\lim_{\varepsilon\downarrow
0}\|(|{\bm{X}}|^{3}+|{\bm{P}}|^{3})(u_{N,0}-u_{N,0}^{\varepsilon})\|_{L^{2}}=0\,.$
(3.18)

######  Remark 3.2.

For the boson-boson interaction, the pointwise bound
$(\varepsilon^{2}+x^{2})^{-1/2}\leqslant|x|^{-1}$ implies that the Potential
Estimates and the Local Lipschitz Property are both satisfied by
$v^{\varepsilon}$, uniformly in $\varepsilon$. On the other hand, since giving
explicit formulas for the other regularized objects is not enlightening, we
refrain to do so.

####  3.1.1. Regularization of the Microscopic Dynamics

In the sequel, we refer to the solution
$\Psi_{N,t}^{\varepsilon}=\exp(-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon})\Psi_{N,0}^{\varepsilon}$
of the Schrödinger equation

$\begin{cases}&\mathrm{i}\partial_{t}\Psi_{N,t}^{\varepsilon}=\mathcal{H}_{N}^{\varepsilon}\Psi_{N,t}^{\varepsilon}\\\
&\Psi_{N,t}^{\varepsilon}\big{|}_{t=0}=u_{N,0}^{\varepsilon}\otimes\mathcal{W}(\sqrt{N}\varphi_{0}^{\varepsilon})\Omega\end{cases}$
(3.19)

as the regularized microscopic dynamics, where the Hamiltonian
$\mathcal{H}_{N}^{\varepsilon}$ is given by (1.6) but with
$(v^{\varepsilon},w^{\varepsilon})$ replacing the original potentials $(v,w)$.
In particular,
$\Psi_{N,t}^{\varepsilon}\in\mathcal{D}(\mathcal{H}_{N}^{\varepsilon})$ for
all $t\geqslant 0$ and (3.19) holds in the strong sense.

The next result establishes the validity of the approximation when
$\varepsilon\downarrow 0$. Since the proof is of rather technical character,
we postpone it to Appendix A.

######  Lemma 3.1.

Let $\Psi_{N,t}$ and $\Psi_{N,t}^{\varepsilon}$ be the microscopic dynamics
and the regularized microscopic dynamics, respectively. Then, for all
$N\geqslant 1$ and $t\geqslant 0$ it holds that

$\lim_{\varepsilon\downarrow
0}\|\Psi_{N,t}-\Psi_{N,t}^{\varepsilon}\|_{\mathscr{H}}=0\,.$ (3.20)

####  3.1.2. Regularization of the Mean-field Equations

In the sequel, we consider a regularized pair of mean-field variables
$({\bm{X}}_{t}^{\varepsilon},\varphi_{t}^{\varepsilon})$ satisfying the
coupled equations

$\displaystyle\begin{cases}\ddot{\bm{X}}^{\varepsilon}_{t}=-\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\,\nabla_{{\bm{X}}}\underline{w}^{\varepsilon}\big{(}x,{\bm{X}}^{\varepsilon}_{t}\big{)}|\varphi^{\varepsilon}_{t}(x)|^{2}\,\mathrm{d}x\\\
\mathrm{i}\partial_{t}\varphi_{t}^{\varepsilon}=-\Delta\varphi_{t}^{\varepsilon}+\underline{w}^{\varepsilon}\big{(}x,{\bm{X}}^{\varepsilon}_{t}\big{)}\varphi_{t}^{\varepsilon}+\big{(}v^{\varepsilon}*|\varphi^{\varepsilon}_{t}|^{2}\,\big{)}\,\varphi^{\varepsilon}_{t}\\\
({\bm{X}}^{\varepsilon}_{t=0},\dot{\bm{X}}^{\varepsilon}_{t=0},\varphi^{\varepsilon}_{0})=({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0}^{\varepsilon})\,,\end{cases}$
(3.21)

from now on refered to as the regularized mean-field equations. In particular,
as proven in Section 6, the regularized boson field
$\varphi_{t}^{\varepsilon}$ remains in $H^{2}(\mathbb{R}^{3})$ for all later
times and (3.21) holds in the strong sense.

The following result is the analogous of Lemma 3.1. For the same reasons, its
proof gets postponed to Appendix A.

######  Lemma 3.2.

Let $({\bm{X}}_{t},\varphi_{t})$ and
$({\bm{X}}_{t}^{\varepsilon},\varphi_{t}^{\varepsilon})$ be the mean-field
variables and regularized mean-field variables, respectively. Then, it holds
that for all $t\geqslant 0$

$\lim_{\varepsilon\downarrow
0}\big{(}|{\bm{X}}_{t}-{\bm{X}}_{t}^{\varepsilon}|+|\dot{\bm{X}}_{t}-\dot{\bm{X}}_{t}^{\varepsilon}|+\|\varphi_{t}-\varphi_{t}^{\varepsilon}\|_{H^{1}}\big{)}=0\,.$
(3.22)

####  3.1.3. Regularization of the Fluctuation State

In terms of the regularized microscopic dynamics $\Psi_{N,t}^{\varepsilon}$
and the regularized mean-field variables
$({\bm{X}}_{t}^{\varepsilon},\varphi_{t}^{\varepsilon})$, we define the
regularized fluctuation state as

$\Omega_{N,t}^{\varepsilon}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{-iS^{\varepsilon}(t)}\mathcal{W}^{*}(\sqrt{N}\varphi_{t}^{\varepsilon})\exp(-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon})\mathcal{W}(\sqrt{N}\varphi_{0}^{\varepsilon})(u_{N,0}^{\varepsilon}\otimes\Omega)\,,\quad
N\geqslant 1\,,\ t\in\mathbb{R}\,.$ (3.23)

Here, $S^{\varepsilon}(t)$ is the regularized version of the scalar term
$S(t)$ defined in (3.2). The following result establishes the validity of the
approximation $\varepsilon\downarrow 0$

######  Lemma 3.3.

Let $t\geqslant 0$, $N\geqslant 1$ and $k\in\mathbb{N}$. Then, the following
holds:

(1) $\ \lim_{\varepsilon\downarrow
0}\|\Omega_{N,t}-\Omega_{N,t}^{\varepsilon}\|_{\mathscr{H}}=0\,.$

(2) $\|N_{b}^{k}\Omega_{N,t}^{\varepsilon}\|+\|N_{b}^{k}\Omega_{N,t}\|\lesssim
N^{k}$ uniformly in $t\in\mathbb{R}$ and $\varepsilon\in(0,1)$ .

(3) $\lim_{\varepsilon\downarrow
0}\|N_{b}^{k/2}(\Omega_{N,t}-\Omega_{N,t}^{\varepsilon})\|_{\mathscr{H}}=0\,.$

###### Proof.

(1) Since $\|u_{N,0}\|_{L_{\bm{X}}^{2}}$ and
$\|\mathcal{W}(\sqrt{N}\varphi_{0}^{\varepsilon})\Omega\|_{\mathcal{F}_{b}}$
are uniformly bounded in $N\geqslant$ and $\varepsilon\in(0,1)$ one obtains
that

$\|\Omega_{N,t}-\Omega_{N,t}^{\varepsilon}\|_{\mathscr{H}}\lesssim\|\big{(}\mathcal{W}(\sqrt{N}\varphi_{t})-\mathcal{W}(\sqrt{N}\varphi_{t}^{\varepsilon})\big{)}\Psi_{N,t}\|_{\mathscr{H}}+\|\Psi_{N,t}-\Psi_{N,t}^{\varepsilon}\|_{\mathscr{H}}+|S(t)-S^{\varepsilon}(t)|\,.$
(3.24)

It is known [16, Lemma 3.1] that the map $f\in L^{2}\mapsto\mathcal{W}(f)\Phi$
is continuous for every $\Phi\in\mathcal{F}_{b}$. Since
$\varphi_{t}^{\varepsilon}\rightarrow\varphi_{t}$ in $L^{2}$, this implies
that the first term vanishes in the limit. The $\varepsilon\downarrow 0$ limit
of the second term is zero, thanks to Lemma 3.1. Finally, since
$\varphi_{t}^{\varepsilon}\rightarrow\varphi_{t}$ in $H^{1}$, one may easily
adapt the proof of Lemma 3.2 to show that $S(t)=\lim_{\varepsilon\downarrow
0}S^{\varepsilon}(t)$. Thus,
$\Omega_{N,t}=\mathscr{H}-\lim_{\varepsilon\downarrow
0}\Omega_{N,t}^{\varepsilon}$.

(2) The proof may be adapted from the weak bounds presented in [33, Lemma
3.6]; one only needs to use the translation property (3.9) of the Weyl
operator $\mathcal{W}(\sqrt{N}\varphi_{t})$ , combined with the uniform-in-
time growth $\|N_{b}^{k}\Psi_{N,t}\|\lesssim N^{k}$.

(3) The Cauchy-Schwarz inequality implies

$\|N_{b}^{k/2}(\Omega_{N,t}-\Omega_{N,t}^{\varepsilon})\|^{2}_{\mathscr{H}}\leqslant\big{(}\|N_{b}^{k}\Omega_{N,t}\|_{\mathscr{H}}+\|N_{b}^{k}\Omega_{N,t}^{\varepsilon}\|_{\mathscr{H}}\big{)}\|\Omega_{N,t}-\Omega_{N,t}^{\varepsilon}\|_{\mathscr{H}}\,.$
(3.25)

It suffices to apply the first two parts of the lemma. ∎

###  3.2. The Truncated Dynamics

Let $M\geqslant 1$, from now on refered to as the cut-off parameter. Given a
solution $({\bm{X}}_{t}^{\varepsilon},\varphi_{t}^{\varepsilon})$ of the mean-
field equations, we introduce on $\mathscr{H}$ the auxiliary time-dependent
operator

$\displaystyle\mathcal{L}_{M}^{\varepsilon}(t)$ $\displaystyle\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
-\frac{\Delta_{\bm{X}}}{2N}\,+\,T_{b}\,+\,\chi(N_{b}\leqslant
M)\mathcal{I}^{\varepsilon}(t)+\mathcal{I}_{4}^{\varepsilon}\,.$ (3.26)

Here, the new $\varepsilon$-dependent operators are defined according to the
formulae (3.10)–(3.14), but with the regularized potentials and mean-field
variables replacing the old ones, i.e. we change

$(v,w,{\bm{X}}_{t},\varphi_{t})\mapsto(v^{\varepsilon},w^{\varepsilon},{\bm{X}}_{t}^{\varepsilon},\varphi_{t}^{\varepsilon})\,.$

We shall refer to $\mathcal{I}_{M}^{\varepsilon}(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\chi(N_{b}\leqslant
M)\mathcal{I}^{\varepsilon}(t)$ as the truncated interaction term; we also
refer to $\mathcal{L}_{M}^{\varepsilon}(t)$ as the truncated generator. As
will become clear later on, the particle number cut-off will let us “close
equations” when trying to run a Grönwall-type argument. In particular,
estimates will be given in terms of both $M$ and $N$ and we will choose $M=N$
at the end of the calculations. Let us note that besides
$\mathcal{L}_{M}^{\varepsilon}(t)$ one could have chosen different auxiliary
operators. For instance, one could have considered introducing more or less
terms with a cut-off in particle number. In particular, we decide not to
introduce a cut-off in the quartic term because we do not have good operator
bounds on $\mathcal{I}_{4}$ ($v^{\varepsilon}$ is not bounded uniformly in
$\varepsilon$)–it is, however, diagonal and time independent.

In Section 7, we prove that to the truncated generator we can associate a
unique unitary propagator222A strongly continuous, two-parameter family of
unitary transformations $(t,s)\in\mathbb{R}^{2}\mapsto
U(t,s)\in\mathcal{B}(\mathscr{H})$ is said to be a unitary propagator, if
$U(t,s)=U(t,r)U(r,s)$ and $U(t,t)=\mathds{1}$. that satisfies

$\displaystyle\mathrm{i}\partial_{t}\,\mathcal{U}_{N}^{(\varepsilon,M)}(t,s)=\mathcal{L}_{M}^{\varepsilon}(t)\,\mathcal{U}_{N}^{(\varepsilon,M)}(t,s)\quad\textnormal{and}\quad\mathcal{U}_{N}^{(\varepsilon,M)}(t,t)=\mathds{1}\,,\quad\forall
t,s\in\mathbb{R}\,.$ (3.27)

We show that this equation is well-posed and establish basic results related
to propagation of regularity–see Propositon 7.1 and 7.2. Our main ingredients
are the uniform boundedness of the interaction term, together with the map
$t\in\mathbb{R}\mapsto\mathcal{I}_{M}^{\varepsilon}(t)\in\mathcal{B}(\mathscr{H})$
being locally Lipschitz; see Lemma 7.1.

For the rest of the section, we focus on the case where the inital condition
that is being evolved satisfies (up to regularization) Condition 1, i.e. we
introduce

$\Omega_{N,t}^{(\varepsilon,M)}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{U}_{N}^{(\varepsilon,M)}(t,0)(u_{N,0}^{\varepsilon}\otimes\Omega)\,,\qquad
t\geqslant 0\,,\ N,M\geqslant 1\,,\ \varepsilon\in(0,1)\,.$ (3.28)

One of our most important estimates is contained in the following proposition;
it is concerned with particle number estimates for
$\Omega_{N,t}^{(\varepsilon,M)}$. Note that, if one chooses $M=N$, we obtain
estimates for the third particle number moment that are uniform in $N\geqslant
1$.

######  Proposition 3.1.

Let $\Omega_{N,t}^{(\varepsilon,M)}$ be as above, with
$(u_{N,0}^{\varepsilon})_{N\geqslant 1}$ being regularized initial data,
satisfying Condition 1. Then, there is $C>0$ such that for all $t\geqslant 0$
and $M,N\geqslant 1$ it holds that

$\displaystyle\limsup_{\varepsilon\downarrow
0}\|N_{b}^{3/2}\Omega_{N,t}^{(\varepsilon,M)}\|$ $\displaystyle\ \leqslant\
C\exp\big{(}C\,t\,(1+M/N)^{2}\,\big{)}\,,$ (3.29)
$\displaystyle\limsup_{\varepsilon\downarrow
0}\|N_{b}^{2}\Omega_{N,t}^{(\varepsilon,M)}\|$ $\displaystyle\ \leqslant\
C\,\sqrt{N}\,\exp\big{(}C\,t\,(1+M/N)^{2}\,\big{)}\,.$ (3.30)

######  Remark 3.3.

The proof of Proposition 3.1 contains the most intricate technical work of
this article. This is due to the fact that the time evolution of powers of the
particle number operator $N_{b}$ get intertwined with powers of the tracer
particle observables ${\bm{X}}$ and ${\bm{P}}$; we postpone its proof and
dedicate Section 5 to it.

One of the main consequences of Proposition 3.1 is that it allows to control
the difference between the original and the truncated evolution. The following
result is the first step in this direction.

######  Corollary 1.

Let $\Omega_{N,t}^{(\varepsilon,M)}$ be as in Proposition 3.1. Then, there is
$C>0$ such that for all $t\geqslant 0$, and $N=M\geqslant 1$ it holds that

$\limsup_{\varepsilon\downarrow
0}\|\chi(N_{b}>N)\,\mathcal{I}^{\varepsilon}(t)\,\Omega_{N,t}^{(\varepsilon,N)}\|\
\leqslant\ \frac{C\,e^{Ct}}{\sqrt{N}}\,.$ (3.31)

###### Proof.

For simplicity, let us introduce the following notation
$\chi_{N}\equiv\chi(N_{b}>N)$ for the projection operator. Without loss of
generality we take $N\geqslant 2$ and $t\geqslant 0$; they shall remain fixed.
Moreover, since the upcoming estimates are uniform in $\varepsilon\in(0,1)$,
we shall not write it explicitly. Finally, we let $\Phi$ denote a generic
element of $L_{\bm{X}}^{2}\otimes\mathcal{D}(N_{b}^{2})$ and we let $C>0$
denote a constant, independent of these quantities, whose value may change
from line to line.

First, we recall that
$\mathcal{I}(t)=\mathcal{L}^{(d)}(t)+\mathcal{L}^{(od)}(t)$. For the diagonal
part, we easily find thanks to our Particle Number Estimates and the Potential
Estimates that

$\displaystyle\|\chi_{N}\,\mathcal{L}^{(d)}(t)\Phi\|$
$\displaystyle=\|\mathcal{L}^{(d)}(t)\chi_{N}\Phi\|$ $\displaystyle\
\leqslant\ C\Big{(}N\|\chi_{N}\Phi\|\,+\,\|(N_{b}+1)\chi_{N}\Phi\|\Big{)}\,.$
(3.32)

Next, for the non-diagonal term we use the relations
$\chi_{N}a_{x}=a_{x}\chi_{N-1}$ and $\chi_{N}a_{x}^{*}=a_{x}^{*}\chi_{N-1}$ to
find, thanks again to our Particle Number Estimates and the Potential
Estimates, that

$\displaystyle\|\chi_{N}\,\mathcal{L}^{(od)}(t)\Phi\|\ \leqslant\ $
$\displaystyle
C\sqrt{N}\,\Big{(}\|(N_{b}+1)^{1/2}\chi_{N-1}\Phi\|+\|(N_{b}+1)^{1/2}\chi_{N+1}\Phi\|\Big{)}$
$\displaystyle+\
C\,\Big{(}\|(N_{b}+1)\chi_{N-2}\Phi\|+\|(N_{b}+1)\chi_{N+2}\Phi\|\Big{)}$
$\displaystyle+\
\frac{C}{\sqrt{N}}\Big{(}\|(N_{b}+1)^{3/2}\chi_{N-1}\Phi\|+\|(N_{b}+1)^{3/2}\chi_{N+1}\Phi\|\Big{)}\,.$
(3.33)

In order to control the intermediate terms involving $N_{b}^{j}$ for
$0\leqslant j\leqslant 3/2$ we use the following inequalities:

$\displaystyle\|(N_{b}+1)^{j}\,\chi(N_{b}>M)\,\Phi\|\leqslant\frac{1}{(M+1)^{J-j}}\|(N_{b}+1)^{J}\Phi\|\,,\qquad
0\leqslant j\leqslant J\,$ (3.34)

for $M=N-2\,,\cdots\,,N+2$ and $J=3/2$. Gathering estimates we find that

$\|\chi(N_{b}>N)\mathcal{I}(t)\Phi\|\
\leqslant\frac{C}{\sqrt{N}}\|(N_{b}+1)^{3/2}\Phi\|\,.$ (3.35)

The last inequality for $\Phi=\Omega_{N,t}^{(N)}$, combined with Proposition
3.1, finishes the proof. ∎

###  3.3. Proof of Theorem 3.1

Starting from Propositon 3.1 (particle number estimates for the truncated
dynamics), we can sacrifice growth in $N$ and prove Theorem 3.1 (particle
number estimates for the original dynamics).

###### Proof of Theorem 3.1.

Let $N\geqslant 1$, $\varepsilon\in(0,1)$ and $t\geqslant 0$. The following
proof is heavily inspired by that of [33, Lemma 3.7]; the idea is as follows:
pick $k\in[0,2]$ and write

$\displaystyle\|N_{b}^{k/2}\Omega_{N,t}^{\varepsilon}\|^{2}$
$\displaystyle\,=\,\|N_{b}^{k/2}\Omega_{N,t}^{(\varepsilon,N)}\|^{2}+\big{\langle}N_{b}^{k}\Omega_{N,t}^{\varepsilon},\big{(}\Omega_{N,t}^{\varepsilon}-\Omega_{N,t}^{(\varepsilon,N)}\big{)}\big{\rangle}+\big{\langle}N_{b}^{k}\Omega_{N,t}^{(\varepsilon,N)},\big{(}\Omega_{N,t}^{\varepsilon}-\Omega_{N,t}^{(\varepsilon,N)}\big{)}\big{\rangle}\,,$
$\displaystyle\,\lesssim\,\|N_{b}^{k/2}\Omega_{N,t}^{(\varepsilon,N)}\|^{2}+\big{(}\|N_{b}^{k}\Omega_{N,t}^{(\varepsilon,N)}\|+\|N_{b}^{k}\Omega_{N,t}^{\varepsilon}\|\big{)}\,\|\Omega_{N,t}^{\varepsilon}-\Omega_{N,t}^{(\varepsilon,N)}\|\,.$
(3.36)

We control each term separately. Since $k\leqslant 2$, the first one may be
controlled using Proposition 3.1. For the second one, we use Lemma 3.3 and
Proposition 3.1 to obtain

$\|N_{b}^{k}\Omega_{N,t}^{\varepsilon}\|+\|N_{b}^{k}\Omega_{N,t}^{(N,\varepsilon)}\|\lesssim\exp(Ct)(N^{k}+N^{1/2})\,,\quad\
t\in\mathbb{R}\,,\ N\geqslant 1\,,\ \varepsilon\in(0,1)\,.$ (3.37)

In addition, we may use Duhamel’s formula to find that

$\|\Omega_{N,t}^{\varepsilon}-\Omega_{N,t}^{(\varepsilon,N)}\|\leqslant\int_{0}^{t}\|\big{(}\mathcal{L}^{\varepsilon}(s)-\mathcal{L}_{N}^{\varepsilon}(s)\big{)}\Omega_{N,s}^{(\varepsilon,N)}\|\,\mathrm{d}s\leqslant\int_{0}^{t}\|\chi(N_{b}>N)\mathcal{I}^{\varepsilon}(s)\Omega_{N,s}^{(\varepsilon,N)}\|\,\mathrm{d}s\,,$
(3.38)

which, in combination with Corollary 1, controls the last term in (3.36).
Thus, we gather estimates and find that

$\limsup_{\varepsilon\downarrow
0}\|N_{b}^{k/2}\Omega_{N,t}^{\varepsilon}\|^{2}\,\lesssim\,\exp(Ct)\Big{(}1+N^{k-1/2}\Big{)}\,.$
(3.39)

Thus, the proof of the theorem is complete after we use Lemma 3.3 and select
$k=1$ and $k=2$ in the last estimate. ∎

##  4\. Proof of our Main Results

Throughout this section, the assumptions made in Theorem 2.1 hold and we keep
on using the same notation from the previous section. In particular,
$\Omega_{N,t}$ denotes the fluctuation state defined in (3.1) and, thanks to
Theorem 3.1, the following particle number estimates hold

$\displaystyle\frac{\|N_{b}\Omega_{N,t}\|}{N}\ +\
\frac{\|N_{b}^{1/2}\Omega_{N,t}\|}{N^{1/2}}$ $\displaystyle\ \leqslant\
\frac{C\,e^{Ct}}{N^{1/4}}$ (4.1)

for some constant $C>0$, independent of $N\geqslant 1$ and $t\geqslant 0$.

For $t\geqslant 0$, let us introduce the following operators to take into
account the classical-quantum error associated to the tracer particle
variables

$\Delta{\bm{X}}(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\bm{X}}-{\bm{X}}_{t}\quad\text{
and }\quad\Delta{\bm{V}}(t)=N^{-1}{\bm{P}}-\dot{\bm{X}}_{t}\ .$ (4.2)

We note that propagation of regularity for the microscopic state
$\Psi_{N,t}=e^{-\mathrm{i}t\mathcal{H}_{N}}\Psi_{N,0}$ follows from standard
techniques. In particular,
$\Psi_{N,t}\in\mathcal{D}({\bm{X}}^{3})\cap\mathcal{D}({\bm{P}}^{3})$ for all
$t\geqslant 0$.

Let us start with the following estimate, from which Theorem 2.1 follows
easily after elementary manipulations.

######  Lemma 4.1.

There exists $C>0$ such that for all $t\geqslant 0$ and $N\geqslant 1$ the
following inequality holds

$\textstyle\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}+\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}\
\leqslant\ C\exp(Ct)N^{-1/2}\,.$ (4.3)

###### Proof.

Let us first assume that $(t\mapsto\Psi_{N,t})\in
C^{1}(\mathbb{R},\mathscr{H})\cap C(\mathbb{R},\mathcal{D}(\mathcal{H}_{N}))$.
We start with the difference in position; straightforward differentiation and
the Cauchy-Schwarz inequality gives

$\displaystyle\frac{d}{dt}\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}\,=\,2\left\langle\Psi_{N,t},\Delta{\bm{X}}(t)\cdot\Delta{\bm{V}}(t)\Psi_{N,t}\right\rangle\,\leqslant\,\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}+\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}\,.$
(4.4)

The difference in velocity is a little more intricate. First, differentiation
with respect to time and the Cauchy-Schwarz inequality gives

$\displaystyle\frac{d}{dt}\|\Delta$ $\displaystyle{\bm{V}}(t)\Psi_{N,t}\|^{2}$
(4.5)
$\displaystyle\,=\,2\,\big{\langle}\Psi_{N,t},\Delta{\bm{V}}(t)\cdot\Big{(}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})\frac{a_{x}^{*}a_{x}}{N}\,\mathrm{d}x-\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})|\varphi_{t}(x)|^{2}\,\mathrm{d}x\Big{)}\Psi_{N,t}\big{\rangle}$
$\displaystyle\,\leqslant\,\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}+\Big{\|}\Big{(}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})\frac{a_{x}^{*}a_{x}}{N}\,\mathrm{d}x-\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})|\varphi_{t}(x)|^{2}\,\mathrm{d}x\Big{)}\Psi_{N,t}\Big{\|}^{2}.$

Here, we have used the commutation relation
$\mathrm{i}[{\bm{P}},\mathcal{H}_{N}]=-\int_{\mathbb{R}^{3}}\nabla_{{\bm{X}}}\underline{w}(x,{\bm{X}})a_{x}^{*}a_{x}\mathrm{d}x$
together with the equation of motion
$\ddot{\bm{X}}_{t}=-\int\nabla_{{\bm{X}}}\underline{w}(x,{\bm{X}})|\varphi_{t}(x)|^{2}\mathrm{d}x$.
Next, we control the second term in the right hand side; the triangle
inequality gives

$\displaystyle\Big{\|}\bigg{(}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}w(x,{\bm{X}})\frac{a_{x}^{*}a_{x}}{N}\,\mathrm{d}x\,-\,$
$\displaystyle\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})|\varphi_{t}(x)|\,\mathrm{d}x\Big{)}\Psi_{N,t}\bigg{\|}$
(4.6) $\displaystyle\ \leqslant\ $
$\displaystyle\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})\Big{(}\frac{a_{x}^{*}a_{x}}{N}-|\varphi_{t}(x)|^{2}\Big{)}\mathrm{d}x\Psi_{N,t}\Big{\|}$
$\displaystyle+\,\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\Big{(}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})-\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})\Big{)}|\varphi_{t}(x)|^{2}\mathrm{d}x\,\Psi_{N,t}\Big{\|}\
.$

The first term on the right hand side can be estimated using the translation
property (3.9) and the Particle Number Estimates. We obtain

$\displaystyle\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})\Big{(}\frac{a_{x}^{*}a_{x}}{N}\,-\,$
$\displaystyle|\varphi_{t}(x)|^{2}\Big{)}\mathrm{d}x\Psi_{N,t}\Big{\|}$
$\displaystyle=\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})\Big{(}\frac{a_{x}^{*}a_{x}}{N}+\frac{1}{\sqrt{N}}(a_{x}^{*}\varphi_{t}(x)+\textnormal{h.c})\Big{)}\mathrm{d}x\,\Omega_{N,t}\Big{\|}$
$\displaystyle\leqslant 2\,\|\nabla w\|_{L^{\infty}}\,\|\varphi_{0}\|_{L^{2}}$
$\displaystyle\qquad\times\Big{(}\frac{1}{N}\|(N_{b}+1)\Omega_{N,t}\|+\frac{1}{N^{1/2}}\|(N_{b}+1)^{1/2}\Omega_{N,t}\|\Big{)}\,.$
(4.7)

Similarly, we use a first order Taylor estimate to find that

$\displaystyle\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\Big{(}\nabla_{\bm{X}}\underline{w}(x,{\bm{X}})-\nabla_{\bm{X}}\underline{w}(x,{\bm{X}}_{t})\Big{)}|\varphi_{t}(x)|^{2}\mathrm{d}x\,\Psi_{N,t}\Big{\|}$
$\displaystyle\ \leqslant\
\|w\|_{C_{b}^{2}}\|\varphi_{0}\|_{L^{2}}\|\Delta{\bm{X}}(t)\Psi_{N,t}\|\,.$
(4.8)

We put the last four estimates together to find that for some $C>0$ it holds
that

$\frac{d}{dt}\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}\lesssim\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}+\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}+\frac{\exp{(Ct)}}{N^{1/2}}\,,\qquad
t\geqslant 0\,,N\geqslant 1\,,$ (4.9)

where we have used the number estimates (4.1). We combine (4.4) and (4.9) to
find that, thanks to the Grönwall inequality, the following estimate holds for
some $C>0$

$\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}+\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}\lesssim
e^{Ct}\Big{(}\|\Delta{\bm{X}}(0)\,u_{N,0}\|^{2}_{L^{2}}+\|\Delta{\bm{V}}(0)\,u_{N,0}\|^{2}_{L^{2}}+\frac{1}{N^{1/2}}\Big{)}\,.$
(4.10)

Since $u_{N,0}$ satisfies Condition 1, the right hand side is of order
$\exp(Ct)N^{-1/2}$. Thus, the proof of the lemma is complete under our time
differentiability assumption. To remove it, one may re-do all the calculations
for $\Psi_{N,t}^{\varepsilon}$ and ${\bm{X}}_{t}^{\varepsilon}$, introduced in
Section 3. Moreover, thanks to Lemma 3.1 and 3.2, we may adapt the argument
presented in Lemma 3.3 to show that

$\textstyle\|\Delta{\bm{X}}(t)\Psi_{N,t}\|^{2}+\|\Delta{\bm{V}}(t)\Psi_{N,t}\|^{2}\leqslant\limsup_{\varepsilon\downarrow
0}\Big{(}\|\Delta{\bm{X}}^{\varepsilon}(t)\Psi^{\varepsilon}_{N,t}\|^{2}+\|\Delta{\bm{V}}^{\varepsilon}(t)\Psi^{\varepsilon}_{N,t}\|^{2}\Big{)}\,.$
(4.11)

Further, all we used in the proof of (4.10) were standard bounds, and the
particle number estimates (4.1). Thus, our claim follows in view of (3.39). ∎

The proof of our first main theorem now becomes a straightforward corollary of
the last result.

###### Proof of Theorem 2.1.

Let us fix $t\geqslant 0$ and $N\geqslant 1$. Since $\Psi_{N,t}$ has unit
norm, one easily finds thanks to the Cauchy-Schwarz inequality that

$|\left\langle\Psi_{N,t},{\bm{X}}\Psi_{N,t}\right\rangle-{\bm{X}}_{t}|\leqslant\left\langle\Psi_{N,t},|{\bm{X}}-{\bm{X}}_{t}|\Psi_{N,t}\right\rangle\leqslant\|\Delta{\bm{X}}(t)\Psi_{N,t}\|\,.$
(4.12)

Thus, it suffices to apply Lemma 4.1. The argument for the difference in
velocity remains unchanged. ∎

###### Proof of Theorem 2.2.

Let $\varphi_{t}$ be the boson field solving the system (1.14). Then, the
translation property (3.9) satisfied by the Weyl operator implies that

$\frac{1}{N}\langle\Psi_{N,t},a_{x}^{*}a_{y}\Psi_{N,t}\rangle=\overline{\varphi_{t}}(x)\varphi_{t}(y)\\!+\\!\frac{1}{N}\langle\Omega_{N,t},\big{(}a_{x}^{*}a_{y}\\!+\\!\sqrt{N}\varphi_{t}(y)a_{x}^{*}\\!+\\!\sqrt{N}\,\overline{\varphi_{t}}(x)\,a_{y}\big{)}\Omega_{N,t}\rangle.$
(4.13)

In particular, we have the following expression for the difference of the
integral kernels:

$\displaystyle\Gamma_{N,t}(x,y)-\overline{\varphi_{t}}(x)\varphi_{t}(y)=\frac{1}{N}\left\langle
a_{x}\Omega_{N,t},a_{y}\Omega_{N,t}\right\rangle+\frac{\varphi_{t}(y)}{\sqrt{N}}\left\langle
a_{x}\Omega_{N,t},\Omega_{N,t}\right\rangle+\frac{\overline{\varphi_{t}}(x)}{\sqrt{N}}\left\langle\Omega_{N,t},a_{y}\Omega_{N,t}\right\rangle\,.$
(4.14)

Next, let $K(x,y)\in L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3})$ be the
integral kernel of the Hilbert-Schmidt operator $\mathcal{K}$. We integrate
both sides of the last identity against $K(x,y)$ and use the Cauchy-Schwarz
inequality to find

$\displaystyle\big{|}\mathrm{Tr}\,\,\mathcal{K}\big{(}\Gamma_{N,t}-$
$\displaystyle\ket{\varphi_{t}}\bra{\varphi_{t}}\big{)}\,\big{|}$
$\displaystyle\leqslant\frac{\|K\|_{L^{2}}}{N}\left(\,\scaleobj{1.2}{\textstyle\int_{\mathbb{R}^{6}}}|\left\langle
a_{x}\Omega_{N,t},a_{y}\Omega_{N,t}\right\rangle|^{2}\,\right)^{1/2}+\frac{2\|K\|_{L^{2}}}{\sqrt{N}}\left(\scaleobj{1.2}{\textstyle\int_{\mathbb{R}^{6}}}|\varphi_{t}(y)|^{2}\|a_{x}\Omega_{N,t}\|^{2}\,\right)^{1/2}$
$\displaystyle\leqslant\frac{\|\mathcal{K}\|_{HS}}{N}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\|a_{x}\Omega_{N,t}\|^{2}\mathrm{d}x+\,\frac{2\,\|\mathcal{K}\|_{HS}}{\sqrt{N}}\left(\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\|a_{x}\Omega_{N,t}\|^{2}\mathrm{d}x\right)^{1/2}$
$\displaystyle=\frac{\|\mathcal{K}\|_{HS}}{N}\|N_{b}^{1/2}\Omega_{N,t}\|^{2}+\frac{2\,\|\mathcal{K}\|_{HS}}{\sqrt{N}}\|N_{b}^{1/2}\Omega_{N,t}\|\,,$
(4.15)

where we have also used the fact that $\Omega_{N,t}$ and $\varphi_{t}$ have
unit norms. It follows now from the number estimates (4.1) that there is $C>0$
such that for all $t\geqslant 0$ and $N\geqslant 1$

$\displaystyle\big{|}\mathrm{Tr}\,\,\mathcal{K}\big{(}\Gamma_{N,t}-\ket{\varphi_{t}}\bra{\varphi_{t}}\big{)}\,\big{|}$
$\displaystyle\ \leqslant\
C\|\mathcal{K}\|_{HS}\Big{(}\frac{e^{Ct}}{N^{1/2}}+\frac{e^{Ct}}{N^{1/4}}\Big{)}\,.$
(4.16)

Therefore, from the previous estimates we conclude that

$\|\Gamma_{N,t}-\ket{\varphi_{t}}\bra{\varphi_{t}}\|_{HS}\
=\sup_{\|\mathcal{K}\|_{HS}\leqslant
1}\big{|}\mathrm{Tr}\,\mathcal{K}\big{(}\Gamma_{N,t}-\ket{\varphi_{t}}\bra{\varphi_{t}}\big{)}\big{|}\
\leqslant\ \frac{C\,e^{Ct}}{N^{1/4}}\,,$ (4.17)

for all $t\geqslant 0$ and $N\geqslant 1$. In view of (2.11), this completes
the proof. ∎

##  5\. Particle Number Estimates of the Truncated Dynamics

The main goal of this section is proving Proposition 3.1, which contains the
key estimate used to prove Theorem 3.1. Let us then fix $N,M\geqslant 1$.

###  5.1. Dynamical Estimates

We will be working with the following space of smooth functions

$\mathscr{D}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathscr{D}_{\infty}\cap\mathcal{D}(T_{b})\,,\quad\textnormal{where}\quad\mathscr{D}_{\infty}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\bigcap_{k=1}^{\infty}\mathcal{D}(|{\bm{X}}|^{k})\cap\mathcal{D}(|{\bm{P}}|^{k})\cap\mathcal{D}(N_{b}^{k})\,.$
(5.1)

In particular, it follows from our results in Section 7 that the regularized,
truncated dynamics propagates smoothness in the sense that

$\mathcal{U}_{N}^{(M,\varepsilon)}(t,s)\mathscr{D}\subset\mathscr{D}\,,\qquad\forall
t,s\in\mathbb{R}\,.$ (5.2)

All of the upcoming calculations are justified in this space.

We will be mostly concerned in proving estimates for
$\|N_{b}^{3/2}\,\mathcal{U}_{N}^{(M,\varepsilon)}(t,0)\,\Phi\|$, when
$\Phi\in\mathscr{D}$ and $t\geqslant 0$. Let us note that thanks to the non-
diagonal term

$\sqrt{N}\int_{\mathbb{R}^{3}}\big{(}\underline{w}^{\varepsilon}(x,{\bm{X}})-\underline{w}^{\varepsilon}(x,{\bm{X}}^{\varepsilon}_{t})\big{)}\big{(}a_{x}\overline{\varphi_{t}}^{\varepsilon}(x)+a_{x}^{*}\varphi_{t}^{\varepsilon}(x)\big{)}\mathrm{d}x$
(5.3)

present in the generator of the (regularized) truncated dynamics, there is a
link between the production of particles and the interaction between bosons
and tracer particles. In order to account for this phenomenon, we introduce
the time-dependent operators

$\displaystyle\Delta{\bm{X}}(t,\varepsilon)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\bm{X}}-{\bm{X}}^{\varepsilon}_{t}$
(5.4) $\displaystyle\Delta{\bm{V}}(t,\varepsilon)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=N^{-1}{\bm{P}}-\dot{\bm{X}}^{\varepsilon}_{t}\,.$
(5.5)

where ${\bm{X}}$ and ${\bm{P}}$ are the tracer particle’s position and
momentum observables. It will be convenient to work with their Euclidean
components. Namely, if $i\in\\{1,\ldots,m\\}$ labels the different tracer
particles and $k\in\\{1,2,3\\}$ labels their components, we write

$\displaystyle\Delta{\bm{X}}(t,\varepsilon)=\big{(}\Delta
X_{k}^{(i)}(t,\varepsilon)\big{)}_{i=1,k=1}^{m,3}\quad\textnormal{ and
}\quad\Delta{\bm{V}}(t,\varepsilon)=\big{(}\Delta
V_{k}^{(i)}(t,\varepsilon)\big{)}_{i=1,k=1}^{m,3}\ .$ (5.6)

We will be using the following quadratic form to quantify the production of
bosonic particles associated to the interaction with the tracer particles.
Here and in the sequel, we let
$\mathcal{U}_{N,t}^{(M,\varepsilon)}\equiv\mathcal{U}_{N}^{(M,\varepsilon)}(t,0)$
unless stated otherwise.

######  Definition 5.1.

For $\Phi\in\mathscr{D}$, $t\geqslant 0$ and $\varepsilon\in(0,1)$ we
introduce the quantity

$\displaystyle\mathcal{G}_{\Phi}(t,\varepsilon)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=$
$\displaystyle\sum_{i=1}^{m}\sum_{k=1}^{3}\bigg{(}\|\,\Delta
X_{k}^{(i)}(t,\varepsilon)^{3}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}+\|\,\Delta
V_{k}^{(i)}(t,\varepsilon)^{3}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\bigg{)}$
$\displaystyle+\frac{1}{N^{2}}\sum_{i=1}^{m}\sum_{k=1}^{3}\bigg{(}\|\,\Delta
X_{k}^{(i)}(t,\varepsilon)\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}+\|\,\Delta
V_{k}^{(i)}(t,\varepsilon)\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\bigg{)}$
$\displaystyle+\frac{1}{N^{3}}\bigg{(}\|(N_{b}+1)^{3/2}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}+\|(N_{b}+1)^{1/2}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\bigg{)}\,.$ (5.7)

######  Remark 5.1.

When we evaluate at our (regularized) initial condition
$\Phi_{N}^{\varepsilon}=u_{N,0}^{\varepsilon}\otimes\Omega$, the following
order-of-magnitude estimate holds at time $t=0:$

$\limsup_{\varepsilon\downarrow
0}\mathcal{G}_{\Phi_{N}^{\varepsilon}}(0,\varepsilon)\lesssim N^{-3}\,.$ (5.8)

Thus, our goal will be to show that there is $C$, independent of
$N,M,t,\varepsilon$ and $\Phi$, such that

$\partial_{t}\,\mathcal{G}_{\Phi}(t,\varepsilon)\ \leqslant\
C(1+M/N)^{2}\,\mathcal{G}_{\Phi}(t,\varepsilon)$ (5.9)

holds. We then apply the Grönwall inequality.

The proof of (5.9) will be split into three parts, contained in the following
three lemmas.

######  Lemma 5.1.

There is $C>0$, independent of $N$ and $M$, such that for all
$\Phi\in\mathscr{D}$ the following holds

  1. ($a_{1}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|(N_{b}+1)^{3/2}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
C\big{(}1+\sqrt{M/N}\big{)}N^{3}\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

  2. ($a_{2}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|(N_{b}+1)^{1/2}\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\
\leqslant\
C\big{(}1+\sqrt{M/N}\big{)}N^{3}\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

  3. ($a_{3}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\|(N_{b}+1)^{2}\ \mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\
\leqslant\ C\exp\big{(}Ct(1+\sqrt{M/N})\big{)}\,N\,\|(N_{b}+1)^{3/2}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\,.$

######  Lemma 5.2.

Let $i\in\\{1,\ldots,m\\}$ and $k\in\\{1,2,3\\}$. There is $C>0$, independent
of $N$ and $M$, such that for all $\Phi\in\mathscr{D}$ the following holds

  1. ($b_{1}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|\,\Delta X_{k}^{(i)}(t,\varepsilon)\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
CN^{2}\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

  2. ($b_{2}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|\,\Delta V_{k}^{(i)}(t,\varepsilon)\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
C(1+M/N)N^{2}\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

######  Lemma 5.3.

Let $i\in\\{1,\ldots,m\\}$ and $k\in\\{1,2,3\\}$. There is $C>0$, independent
of $N$ and $M$, such that for all $\Phi\in\mathscr{D}$ the following holds

  1. ($c_{1}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|\,\Delta X_{k}^{(i)}(t,\varepsilon)^{3}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
C\,\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

  2. ($c_{2}$)

For any $t\geqslant 0$ and $\varepsilon\in(0,1)$

$\displaystyle\frac{d}{dt}\|\,\Delta V_{k}^{(i)}(t,\varepsilon)^{3}\
\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
C(1+M/N)^{2}\,\mathcal{G}_{\Phi}(t,\varepsilon)\,.$

We postpone the proof of Lemma 5.1, 5.2 and 5.3 to the next subsection; let us
first prove the very important Proposition 3.1.

###### Proof of Proposition 3.1 .

We will show in full detail the proof of (3.29) and omit that of (3.30); the
proof of the latter follows directly from (3.29) and Lemma 5.1. Indeed, we put
Lemma 5.1, 5.2 and 5.3 together and we conclude that there is $C>0$ such that
for all $t\geqslant 0$, $\varepsilon\in(0,1)$, $N,M\geqslant 1$ and
$\Phi\in\mathscr{D}$ it holds that

$\partial_{t}\mathcal{G}_{\Phi}(t,\varepsilon)\ \leqslant\
C(1+M/N)^{2}\mathcal{G}_{\Phi}(0,\varepsilon)\,.$ (5.10)

Therefore, an application of the Grönwall inequality yields

$\mathcal{G}_{\Phi}(t,\varepsilon)\ \leqslant\
\exp\big{(}C\,t\,(1+M/N)^{2}\,\big{)}\mathcal{G}_{\Phi}(0,\varepsilon)\,.$
(5.11)

In particular, it follows from the definition of $\mathcal{G}_{\Phi}$ that

$\|(N_{b}+1)^{3/2}\ \mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi\|^{2}\ \leqslant\
N^{3}\exp\big{(}C\,t\,(1+M/N)^{2}\big{)}\
\mathcal{G}_{\Phi}(0,\varepsilon)\,.$ (5.12)

Let us now take $\Phi\equiv\Phi_{N}^{\varepsilon}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=u_{N,0}^{\varepsilon}\otimes\Omega\in\mathscr{D}$,
where $u_{N,0}^{\varepsilon}$ is the regularized version of the initial data
$u_{N,0}$ satisfying Condition 1. Thanks to (3.18), it follows that

$\limsup_{\varepsilon\downarrow
0}\mathcal{G}_{\Phi_{N}^{\varepsilon}}(0,\varepsilon)\lesssim N^{-3}\,.$
(5.13)

The proof is complete once we put the last two estimates together for
$\Omega_{N,t}^{(\varepsilon,M)}=\mathcal{U}_{N,t}^{(M,\varepsilon)}\Phi_{N}^{\varepsilon}$.
∎

###  5.2. Proof of Lemmata

Let us now turn to the proof of Lemma 5.1, 5.2 and 5.3. Since one should think
about the following calculations as a regularized version of estimates done
for the original dynamics, we will drop the $\varepsilon\in(0,1)$ superscript
and assume instead that all of the involved quantities are smooth; since our
estimates are uniform in $\varepsilon$, there is no risk in doing so. We also
denote $\mathcal{G}_{\Phi}(t,\varepsilon)\equiv\mathcal{G}_{\Phi}(t)$.

The following (formal) Interpolation Lemma will be helpful. Its use will only
be put into use when all the manipulations find a rigorous meaning; for a
proof see Appendix B.

######  Lemma 5.4 (Interpolation).

Let $A$ and $B$ be self-adjoint operators such that
$\mathrm{i}[A,B]=\lambda\mathds{1}$ for some $\lambda\in\mathbb{C}$. Then,
there is $c>0$ such that

$\|A^{2}B\Psi\|^{2}+\|ABA\Psi\|^{2}+\|BA^{2}\Psi\|^{2}\leqslant
c\Big{(}\|A^{3}\Psi\|^{2}+\|B^{3}\Psi\|^{2}+\lambda^{2}\|A\Psi\|^{2}+\lambda^{2}\|B\Psi\|^{2}\Big{)}$
(5.14)

for all $\Psi$.

###### Proof of Lemma 5.1.

Let us fix $N,M\geqslant 1$ and $\Phi\in\mathscr{D}$. We start by recalling
the interaction terms to be $\mathcal{I}_{M}(t)=\chi(N_{b}\leqslant
M)(\mathcal{L}^{(od)}(t)+\mathcal{L}^{(d)}(t))$. In particular, we may write
the truncated, off-diagonal term as

$\chi(N_{b}\leqslant
M)\mathcal{L}^{(od)}(t)=\mathcal{L}_{1}(t)+\mathcal{L}_{2}(t)+\mathcal{L}_{3}(t)$
(5.15)

where we introduce the notation

$\displaystyle\mathcal{L}_{1}(t)=\chi(N_{b}\leqslant
M)\sqrt{N}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,\big{(}a_{x}\overline{\varphi_{t}(x)}+a_{x}^{*}\varphi_{t}(x)\big{)}\,\mathrm{d}x\,$
$\displaystyle\mathcal{L}_{2}(t)=\chi(N_{b}\leqslant
M)\frac{1}{2}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)\bigg{(}\varphi_{t}(x)\varphi_{t}(y)a_{x}^{*}a_{y}^{*}\,+\,\overline{\varphi_{t}}(x)\,\overline{\varphi_{t}}(y)a_{x}a_{y}\bigg{)}\mathrm{d}x\mathrm{d}y$
$\displaystyle\mathcal{L}_{3}(t)=\chi(N_{b}\leqslant
M)\frac{1}{\sqrt{N}}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-y)\,a_{x}^{*}\,\bigg{(}\varphi_{t}(y)a_{y}^{*}\
+\ \overline{\varphi_{t}}(y)\,a_{y}\bigg{)}\,a_{x}\,\mathrm{d}x\mathrm{d}y\,.$
(5.16)

The labeling has been chosen so that each $\mathcal{L}_{i}(t)$ depends on a
$i$-th power of the creation and annihilation operators.

The terms $\mathcal{L}_{2}(t)$ and $\mathcal{L}_{3}(t)$ have already been
analyzed in the literature; we will take some commutator estimates from [33].
More precisely, it is known that for all $j\in\mathbb{Z}$ there is $C(j)>0$
such that for all smooth $\Psi\in\mathscr{D}$ and $t\geqslant 0$ it holds that

$\big{|}\left\langle\Psi,[\mathcal{L}_{2}(t)+\mathcal{L}_{3}(t),(N_{b}+1)^{j}]\Psi\right\rangle\big{|}\leqslant
C(j)\big{(}1+\sqrt{M/N}\big{)}\left\langle\Psi,(N_{b}+1)^{j}\Psi\right\rangle\,.$
(5.17)

The constant $C(j)$ depends on the potentials only through $|v|$, as the above
estimates are a consequence of Particle Number Estimates and the Potential
Estimates. Therefore, it suffices to focus on deriving similar commutator
estimates for $\mathcal{L}_{1}(t)$ for the cases $j=1,2,3\,.$

Proof of $(a_{1})$. Let $\Psi\in\mathscr{D}$ be smooth but arbitrary. Starting
from the definition of $\mathcal{L}_{1}(t)$ one easily finds that

$\displaystyle\big{|}\langle\Psi,[\mathcal{L}_{1}(t),(N_{b}+$ $\displaystyle
1)^{3}]\Psi\rangle\big{|}$ (5.18)
$\displaystyle\leqslant\sqrt{N}\,\big{|}\langle\Psi,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}[a_{x}\overline{\varphi_{t}}(x)+a_{x}^{*}\varphi_{t}(x),(N_{b}+1)^{3}]\Psi\,\mathrm{d}x\rangle\big{|}$
$\displaystyle\lesssim\sqrt{N}\,\big{|}\langle\Psi,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,\overline{\varphi_{t}}(x)\,[a_{x},(N_{b}+1)^{3}]\Psi\,\mathrm{d}x\rangle\big{|}\,.$

The remaining commutator can be controlled as follows: by means of the
relation $a_{x}N_{b}=(N_{b}+1)a_{x}$ one finds that for any
$\ell\in\mathbb{N}$ it holds that

$[a_{x},(N_{b}+1)^{\ell}]=\sum_{k=0}^{\ell-1}\binom{\ell}{k}(N_{b}+1)^{k}a_{x}\,.$
(5.19)

We plug this relationship back in (5.18) and use the Cauchy-Schwarz inequality
to find that

$\displaystyle\big{|}\langle\Psi,[\mathcal{L}_{1}(t),(N_{b}$
$\displaystyle+1)^{3}]\Psi\rangle\big{|}$
$\displaystyle\lesssim\sqrt{N}\,\sum_{k=0}^{2}\,\binom{3}{k}\big{|}\big{\langle}\Psi,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,\overline{\varphi_{t}}(x)\,(N_{b}+1)^{k}a_{x}\Psi\,\mathrm{d}x\big{\rangle}\big{|}$
$\displaystyle\lesssim\sqrt{N}\,\|(N_{b}+1)^{3/2}\Psi\|\,\big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\,\overline{\varphi_{t}}(x)\,(N_{b}+1)^{1/2}a_{x}\mathrm{d}x\,\Psi\big{\|}$
$\displaystyle\lesssim\sqrt{N}\,\|(N_{b}+1)^{3/2}\Psi\|\,\|\Delta{\bm{X}}(t)(N_{b}+1)\Psi\|$
$\displaystyle\lesssim\|(N_{b}+1)^{3/2}\Psi\|^{2}\ +\
N\,\|\Delta{\bm{X}}(t)(N_{b}+1)\Psi\|^{2}$ (5.20)

where, in the third line, we have used a Taylor estimate together with the
Particle Number Estimates. In order to control the second term that appears in
(5.20) we use the Interpolation Lemma for $A=|\Delta{\bm{X}}(t)|$,
$B=N^{-1/2}(N_{b}+1)^{1/2}$ and $\lambda=0$ to find that

$N\,\|\Delta{\bm{X}}(t)\,(N_{b}+1)\Psi\|^{2}\ \lesssim\
\|(N_{b}+1)^{3/2}\Psi\|^{2}+N^{3}\||\Delta{\bm{X}}(t)|^{3}\Psi\|^{2}\,.$
(5.21)

By putting together estimates (5.17), (5.20) and (5.21) for
$\Psi=\mathcal{U}_{N,t}^{(M)}\Phi$ we find

$\displaystyle\frac{d}{dt}\|(N_{b}+1)^{3/2}\mathcal{U}_{N,t}^{(M)}\Phi\|^{2}$
$\displaystyle\ \leqslant\
\big{|}\left\langle\mathcal{U}_{N,t}^{(M)}\Phi,[\mathcal{L}_{1}(t)+\mathcal{L}_{2}(t)+\mathcal{L}_{3}(t),(N_{b}+1)^{3}]\
\mathcal{U}_{N,t}^{(M)}\Phi\right\rangle\big{|}$ $\displaystyle\ \lesssim\
\big{(}1+\sqrt{M/N}\big{)}N^{3}\,\mathcal{G}_{\Phi}(t)\,.$ (5.22)

Proof of $(a_{2})$. We repeat the argument presented above; the proof is
actually shorter. In particular, the estimate

$\displaystyle\big{|}\left\langle\Psi,[\mathcal{L}_{1}(t),N_{b}+1]\Psi\right\rangle\big{|}\lesssim\|(N_{b}+1)^{1/2}\Psi\|^{2}\
+\ N\|\,\Delta{\bm{X}}(t)\,\Psi\|^{2}$ (5.23)

is the analogous of (5.20) but the operators $N_{b}$ and $\Delta X(t)$ are
already decoupled. For $\Psi=\mathcal{U}_{N,t}^{(M)}\Phi$ the right hand side
may be immediately bounded above by $N^{3}\,\mathcal{G}_{\Phi}(t)$; an
application of (5.17) for $j=1$ then finishes the proof.

Proof of $(a_{3})$. Similarly as in the proof of $(a_{1})$, we estimate the
following commutator as

$\displaystyle\big{|}\langle\Psi,[\mathcal{L}_{1}(t),(N_{b}+1)^{4}]$
$\displaystyle\Psi\rangle\big{|}$
$\displaystyle\lesssim\sqrt{N}\|(N_{b}+1)^{2}\Psi\|\Big{\|}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\underline{w}(x,{\bm{X}})-\underline{w}(x,{\bm{X}}_{t})\big{)}\overline{\varphi}_{t}(x)(N_{b}+1)a_{x}\Psi\mathrm{d}x\Big{\|}$
$\displaystyle\lesssim\sqrt{N}\|(N_{b}+1)^{2}\Psi\|\|(N_{b}+1)^{3/2}\Psi\|$
$\displaystyle\lesssim\|(N_{b}+1)^{2}\Psi\|^{2}+N\|(N_{b}+1)^{3/2}\Psi\|^{2}\,,$
(5.24)

where we have used boundedness of $w$ instead of a Taylor estimate to control
the ${\bm{X}}$-dependent term. We put together estimates (5.24) and (5.17) for
$j=4$, to find that for $\Psi=\mathcal{U}_{N,t}^{(M)}\Phi$ we have

$\frac{d}{dt}\|(N_{b}+1)^{2}\mathcal{U}_{N,t}^{(M)}\Phi\|^{2}\lesssim(1+\sqrt{M/N})\|(N_{b}+1)^{2}\mathcal{U}_{N,t}^{(M)}\Phi\|^{2}+N\|(N_{b}+1)^{3/2}\mathcal{U}_{N,t}^{(M)}\Phi\|^{2}\,.$
(5.25)

The proof of the lemma is finished after we apply the Grönwall inequality. ∎

###### Proof of Lemma 5.2.

Let us introduce simplifications in the notation:

  1. (i)

Since the following proof is conceptually independent of the number of tracer
particles $m$, we shall assume for simplicity that $m=1$. In particular, we
drop the boldface notation for ${\bm{X}}$ and ${\bm{V}}$; we write instead

${\bm{X}}=X=(X_{1},X_{2},X_{3})\in\mathbb{R}^{3}\,,\quad\textnormal{and }\quad
w(x-X)=\underline{w}(x,{\bm{X}})$ (5.26)

and similarly for velocity.

  2. (ii)

In order to avoid double subscripts, we write

${\bm{X}}_{t}=X_{t}=\big{(}X_{1}(t),X_{2}(t),X_{3}(t)\big{)}$ (5.27)

to denote time dependence of the components of the tracer’s particle position,
and similarly for velocity.

  3. (iii)

Whenever there is no risk of confusion, we write

$\Phi(t)\equiv\Phi_{t}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{U}_{N,t}^{(M)}\Phi\,.$
(5.28)

Let us then fix $k\in\\{1,2,3\\}$, together with $N,M\geqslant 1$ and
$\Phi\in\mathscr{D}.$

Proof of $(b_{1})$. Let us first calculate that

$\displaystyle\frac{d}{dt}\|\Delta X_{k}(t)\ \Phi(t)\|^{2}$
$\displaystyle=\Big{\langle}\Phi(t),\Big{(}\mathrm{i}[-\frac{\Delta_{X}}{2N},\Delta
X_{k}(t)^{2}]-2\,\Delta
X_{k}(t)\dot{X}_{k}(t)\Big{)}\,\Phi(t)\Big{\rangle}\,.$ (5.29)

Moreover, the commutation relations
$\mathrm{i}[P_{\ell},X_{k}]=\delta_{k,\ell}$ imply that the following identity
holds

$\mathrm{i}[P^{2},\,\Delta X_{k}(t)^{2}]=P_{k}\Delta X_{k}(t)+\Delta
X_{k}(t)P_{k}\,.$ (5.30)

Thus, since $\Delta V_{k}(t)=N^{-1}P_{k}-\dot{X}_{k}(t)$, we find thanks to
the Cauchy-Schwarz inequality that

$\displaystyle\frac{d}{dt}\|\Delta X_{k}(t)\Phi(t)\|^{2}$ $\displaystyle\ =\
2\mathrm{Re}\left\langle\Phi(t),\Delta V_{k}(t)\Delta
X_{k}(t)\Phi(t)\right\rangle$ $\displaystyle\ \leqslant\ \|\Delta V_{k}(t)\
\Phi(t)\|^{2}+\|\Delta X_{k}(t)\ \Phi(t)\|^{2}$ (5.31)

from which our claim follows easily.

Proof of $(b_{2})$. First, let us identify the term cointained in the
generator $\mathcal{L}_{M}(t)$ that does not commute with
$P_{k}=-\mathrm{i}\partial_{k}$; it is given by

$\mathcal{F}_{M}(t)=\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}w(x-X)\,\big{(}N|\varphi_{t}(x)|^{2}+\sqrt{N}(a_{x}\overline{\varphi_{t}}(x)+h.c)-a_{x}^{*}a_{x}\big{)}\,\mathrm{d}x\,.$
(5.32)

Moreover, we recall that the trajectory $X(t)$ satisfies the mean-field
equations (1.14). Therefore, similarly as we calculated for $(b_{1}),$ we find

$\displaystyle\frac{d}{dt}\|\Delta V_{k}(t)\Phi(t)\|^{2}$ $\displaystyle\ =\
2\,\mathrm{Re}\Big{\langle}\Phi(t),\Big{(}\mathrm{i}\,[\mathcal{F}_{M}(t),N^{-1}\\!P_{k}]-\ddot{X}_{k}(t)\Big{)}\Delta
V_{k}(t)\mathcal{U}_{N,t}^{(M)}\Phi\Big{\rangle}$ $\displaystyle\ \leqslant\
\|\Delta
V_{k}(t)\Phi(t)\|^{2}+\big{\|}\big{(}\mathrm{i}\,[\mathcal{F}_{M}(t),N^{-1}P_{k}]\,-\ddot{X}_{k}(t)\big{)}\Phi(t)\big{\|}^{2}\,.$
(5.33)

The first term that shows up in (5.33) is clearly bounded above by
$N^{2}\mathcal{G}_{\Phi}(t)$. Therefore, it remains to estimate the first one
in terms of $N^{2}\mathcal{G}_{\Phi}(t)$. To this end, we write

$\Delta F_{k}(t)\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\mathrm{i}\,[\mathcal{F}_{M}(t),N^{-1}P_{k}]\,-\ddot{X}_{k}(t)$ (5.34)

to denote the error in the force term exherted on the tracer particle.
Further, we decompose it as $\Delta F_{k}(t)=\Delta F_{k}^{(1)}(t)\,+\,\Delta
F_{k}^{(2)}(t)$, where

$\displaystyle\Delta F_{k}^{(1)}(t)\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\big{(}\partial_{k}w(x-X)-\partial_{k}w(x-X_{t})\big{)}|\varphi_{t}(x)|^{2}\,\mathrm{d}x$
(5.35) $\displaystyle\Delta F_{k}^{(2)}(t)\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\partial_{k}w(x-X)\bigg{(}\frac{1}{\sqrt{N}}(a_{x}\overline{\varphi_{t}}(x)+h.c)-\frac{1}{N}a_{x}^{*}a_{x}\bigg{)}\,\mathrm{d}x\,.$
(5.36)

Since $w\in C_{b}^{2}(\mathbb{R}^{3},\mathbb{R})$, we may use a Taylor
Estimate and the Particle Number Estimates to find that the following
inequalities

$\displaystyle\|\Delta
F_{k}^{(1)}(t)\Psi\|\leqslant\|\nabla^{2}w\|_{L^{\infty}}\||\Delta
X(t)|\Psi\|$ (5.37) $\displaystyle\|\Delta
F_{k}^{(2)}(t)\Psi\|\leqslant\frac{\|\nabla
w\|_{L^{\infty}}}{\sqrt{N}}\big{(}1+\sqrt{M/N}\big{)}\|(N_{b}+1)^{1/2}\Psi\|\,.$
(5.38)

hold, for all $\Psi\in\mathscr{D}.$ By plugging these estimates back in (5.33)
for $\Psi=\Phi(t)$ one easily finds that

$\frac{d}{dt}\|\Delta V_{k}(t)\Phi(t)\|^{2}\ \lesssim\
(1+M/N)N^{2}\,\mathcal{G}_{\Phi}(t)\,.$

This finishes the proof of the lemma. ∎

###### Proof of Lemma 5.3.

We will keep on using the same simplifications and notation introduced at the
beginning of the proof of lemma 5.2. Let us fix $k\in\\{1,2,3\\}$, together
with $N,M\geqslant 1$ and $\Phi\in\mathscr{D}.$

The following general general calculation will turn out to be useful. Let
$A(t)$ be a time-dependent, self-adjoint operator. As long as everything is
well-defined, one formally has that

$\displaystyle\frac{d}{dt}\|A(t)^{3}\Phi(t)\|^{2}$ $\displaystyle\ =\
\left\langle\Phi(t),\Big{(}\mathrm{i}[\mathcal{L}_{M}(t),A(t)^{6}]-6A(t)^{5}\dot{A}(t)\Big{)}\Phi(t)\right\rangle$
$\displaystyle\ \lesssim\
\|A(t)^{3}\Phi(t)\|^{2}+\|A(t)^{2}\Big{(}\mathrm{i}[\mathcal{L}_{M}(t),A]-\dot{A}(t)\Big{)}\Phi(t)\|^{2}$
$\displaystyle\
\quad+\|A(t)\Big{(}\mathrm{i}[\mathcal{L}_{M}(t),A]-\dot{A}(t)\Big{)}A(t)\Phi(t)\|^{2}$
$\displaystyle\
\quad+\|\Big{(}\mathrm{i}[\mathcal{L}_{M}(t),A]-\dot{A}(t)\Big{)}A(t)^{2}\Phi(t)\|^{2}\,.$
(5.39)

Proof of $(c_{1})$. First, we specialize to $A(t)=\Delta X_{k}(t)$ for which
the above estimate turns into

$\displaystyle\frac{d}{dt}\|\Delta X_{k}(t)^{3}\Phi_{t}\|^{2}\ \lesssim\ $
$\displaystyle\|\Delta X_{k}(t)^{3}\Phi_{t}\|^{2}+\|\Delta X_{k}(t)^{2}\Delta
V_{k}(t)\Phi_{t}\|^{2}$ $\displaystyle+\|\Delta X_{k}(t)\Delta V_{k}(t)\Delta
X_{k}(t)\Phi_{t}\|^{2}+\|\Delta V_{k}(t)\Delta X_{k}(t)^{2}\Phi_{t}\|^{2}\,.$
(5.40)

It suffices to show that every term contained in the right hand side above may
be bounded in terms of $\mathcal{G}_{\Phi}(t).$ The first one is trivial and
follows from the definition, so we omit it. The other three may be controlled
using the Interpolation Lemma with $A=\Delta X_{k}(t)$, $B=V_{k}(t)$ and
$\lambda=-1$: we find that for all $t\geqslant 0$

$\|\Delta X_{k}(t)^{2}\Delta V_{k}(t)\Phi_{t}\|^{2}+\|\Delta X_{k}(t)\Delta
V_{k}(t)\Delta X_{k}(t)\Phi_{t}\|^{2}+\|\Delta V_{k}(t)\Delta
X_{k}(t)^{2}\Phi_{t}\|^{2}\,\,\lesssim\,\mathcal{G}_{\Phi}(t)\,.$ (5.41)

Proof of $(c_{2})$. Let us now specialize our general calculation to
$A(t)=\Delta V_{k}(t)$. We find that

$\displaystyle\frac{d}{dt}\|\Delta V_{k}(t)^{3}\Phi_{t}\|^{2}\ \lesssim\ $
$\displaystyle\|\Delta V_{k}(t)^{3}\Phi_{t}\|^{2}+\|\Delta V_{k}(t)^{2}\Delta
F_{k}(t)\Phi_{t}\|^{2}$ $\displaystyle+\|\Delta V_{k}(t)\Delta F_{k}(t)\Delta
V_{k}(t)\Phi_{t}\|^{2}\,+\|\Delta F_{k}(t)\Delta
V_{k}(t)^{2}\Phi_{t}\|^{2}\,,$ (5.42)

where we recall that $\Delta F_{k}(t)$ was defined in (5.34). Let us introduce
some notation for the different terms that we found above:

$\displaystyle T_{1}\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\|\Delta F_{k}(t)\Delta V_{k}(t)^{2}\Phi(t)\|^{2}\,$ (5.43) $\displaystyle
T_{2}\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\|\Delta V_{k}(t)\Delta F_{k}(t)\Delta V_{k}(t)\Phi(t)\|^{2}\,$ (5.44)
$\displaystyle T_{3}\ :=\ \|\Delta V_{k}(t)^{2}\Delta
F_{k}(t)\Phi(t)\|^{2}\,.$ (5.45)

Our goal is to show that $T_{i}\lesssim(1+M/N)^{2}\mathcal{G}_{\Phi}(t)$ for
$i\in\\{1,2,3\\}$.

Bounds on $T_{1}:$

Thanks to the force estimates (5.37), (5.38) one easily finds that

$\displaystyle T_{1}$ $\displaystyle\ \lesssim\ \||\Delta X|\,\Delta
V_{k}(t)^{2}\Phi(t)\|^{2}+(1+M/N)\,\|N^{-1/2}(N_{b}+1)^{1/2}\Delta
V_{k}(t)^{2}\Phi(t)\|^{2}$ $\displaystyle\ \lesssim\ \sum_{\ell=1}^{3}\|\Delta
X_{\ell}(t)\,\Delta
V_{k}(t)^{2}\Phi(t)\|^{2}+(1+M/N)\,\|N^{-1/2}(N_{b}+1)^{1/2}\Delta
V_{k}(t)^{2}\Phi(t)\|^{2}$ (5.46)

Both the first and the second term that appear in the right hand side above
can be controlled by $\mathcal{G}_{\Phi}(t)$ using the Interpolation Lemma:

(i) For the the first term choose: $A_{\ell}=\Delta X_{\ell}(t)$, $B=\Delta
V_{k}(t)$ with $\ell\in\\{1,2,3\\}$ and $\lambda_{\ell}\in\\{-1,0\\}$.

(ii) For the second term choose: $A=N^{-1/2}(N_{b}+1)^{1/2}$, $B=\Delta
V_{k}(t)$ and $\lambda=0$.
We conclude that $T_{1}\lesssim(1+M/N)\mathcal{G}_{\Phi}(t)$.

Bounds on $T_{2}:$

The idea will be to give an upper bound for $T_{2}$ in terms on $T_{1}$, plus
a commutator term arising from $[\Delta F_{k}(t),\Delta
V_{k}(t)]=\frac{1}{N}[\Delta F_{k}(t),P_{k}]$. For the sake of the upcoming
calculations, let us briefly denote by

$\mathcal{A}_{N,t}(x)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{N}}\big{(}a_{x}\overline{\varphi_{t}}(x)+h.c\big{)}+\frac{1}{N}a_{x}^{*}a_{x}\,,\qquad
x\in\mathbb{R}^{3}$ (5.47)

the operator-valued distribution, found in the definition of the force term
$\Delta F_{k}(t)$. We find that

$\displaystyle T_{2}\ \lesssim\
T_{1}\,+\,\frac{1}{N^{2}}\big{\|}\,\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\partial_{k}^{2}w(x-X)\Big{(}\,|\varphi_{t}(x)|^{2}+\mathcal{A}_{N,t}(x)\,\Big{)}\mathrm{d}x\Delta
V_{k}(t)\Phi(t)\big{\|}^{2}\,.$ (5.48)

Since $w\in C_{b}^{2}(\mathbb{R}^{3})$ and
$\|\varphi_{t}\|_{L^{2}}=\|\varphi_{0}\|_{L^{2}}$ we find, thanks to the cut-
off term $\chi(N_{b}\leqslant M)$ and the Particle Number Estimates, that

$\displaystyle T_{2}\ \lesssim\
T_{1}\,+\,\frac{1}{N^{2}}\big{(}1+\sqrt{{M}/{N}}+{M}/{N}\big{)}^{2}\|\Delta
V_{k}(t)\Phi(t)\|^{2}\,\ \lesssim\
T_{1}+\big{(}1+{M}/{N}\big{)}^{2}\mathcal{G}_{\Phi}(t)\,.$ (5.49)

Thanks to the bound $T_{1}\lesssim(1+M/N)\mathcal{G}_{\Phi}(t)$ we conclude
that $T_{2}\lesssim(1+M/N)^{2}\mathcal{G}_{\Phi}(t)$.

Bounds on $T_{3}:$

Similarly as before, we give an upper bound for $T_{3}$ in terms of $T_{2}$,
plus commutator terms. In particular, the following estimate is the only one
in this article that requires three derivatives of the boson-tracer
interaction $w$. We find

$\displaystyle T_{3}\ \lesssim\ $ $\displaystyle
T_{2}\,+\,\frac{1}{N^{2}}\big{\|}\Delta V_{k}(t)\,\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\partial_{k}^{2}w(x-X)\big{(}|\varphi_{t}(x)|^{2}+\mathcal{A}_{N,t}(x)\big{)}\mathrm{d}x\,\Phi(t)\big{\|}^{2}$
$\displaystyle\ \lesssim\ $ $\displaystyle
T_{2}\,+\,\frac{1}{N^{2}}\big{\|}\,\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\partial_{k}^{2}w(x-X)\big{(}|\varphi_{t}(x)|^{2}+\mathcal{A}_{N,t}(x)\big{)}\mathrm{d}x\
\Delta V_{k}(t)\,\Phi(t)\big{\|}^{2}$
$\displaystyle+\frac{1}{N^{4}}\big{\|}\,\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\partial_{k}^{3}w(x-X)\big{(}|\varphi_{t}(x)|^{2}+\mathcal{A}_{N,t}(x)\big{)}\mathrm{d}x\,\Phi(t)\big{\|}^{2}$
$\displaystyle\ \lesssim\ $ $\displaystyle
T_{2}+\frac{1}{N^{2}}\big{(}1\,+\,\sqrt{{M}/{N}}+{M}/{N}\big{)}^{2}\|\Delta
V_{k}(t)\,\Phi(t)\|^{2}\,$
$\displaystyle+\frac{1}{N^{4}}\big{(}1+\sqrt{{M}/{N}}+{M}/{N}\big{)}^{2}\|\Phi(t)\|^{2}\,$
$\displaystyle\ \lesssim\ $ $\displaystyle
T_{2}\,+\,(1+M/N)^{2}\,\mathcal{G}_{\Phi}(t)+N^{-4}(1+M/N)^{2}\,.$ (5.50)

Thanks to the upper bound $T_{2}\lesssim(1+M/N)^{2}\mathcal{G}_{\Phi}(t)$ and
$N^{-4}\leqslant\mathcal{G}_{\Phi}(t)$ we conclude that
$T_{3}\lesssim(1+M/N)^{2}\mathcal{G}_{\Phi}(t)$. This finishes the proof of
the lemma. ∎

##  6\. Well-Posedness of the Mean-Field Equations

In this section, we establish well-posedness of the mean-field equations,
introduced in (1.14). The energy of the system will be described by the
functional $E:\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})\rightarrow\mathbb{R}$ defined as

$E({\bm{X}},{\bm{V}},\varphi)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{2}{\bm{V}}^{2}+\|\varphi\|_{H^{1}}^{2}+\,\frac{1}{2}\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}|\varphi(x)|^{2}v(x-y)|\varphi(y)|^{2}\mathrm{d}x\mathrm{d}y+\,\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\underline{w}(x,{\bm{X}})|\varphi(x)|^{2}\mathrm{d}x\,.$
(6.1)

The fundamental estimate satisfied by the energy functional $E$ is the lower
bound

${\bm{V}}^{2}+\|\varphi\|_{H^{1}}^{2}\,\leqslant\,\,\mu\,\big{(}E({\bm{X}},{\bm{V}},\varphi)+\|\varphi\|_{L^{2}}^{2}+\|\varphi\|_{L^{2}}^{6}\big{)}\,,\qquad({\bm{X}},{\bm{V}},\varphi)\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$ (6.2)

where $\mu=\mu(v,w)>0$ is a constant depending only on the potentials. Its
proof follows from the operator inequality (2.19) for $v$, and the boundedness
of $w$.

For notational simplicity, given a solution $({\bm{X}}_{t},\varphi_{t})$ of
the mean-field equations, we shall simply write
$E(t)=E({\bm{X}}_{t},\dot{\bm{X}}_{t},\varphi_{t})$. In addition, we denote by
${\bm{V}}_{t}=\dot{\bm{X}}_{t}$ the velocity of the tracer particles. The main
result of this section is

######  Theorem 6.1.

Assume that $v$ and $w$ satisfy Condition 2. Then, for any initial condition
$({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$, there is a unique global solution

$({\bm{X}},{\bm{V}},\varphi)\in
C\big{(}\mathbb{R};\,\mathbb{R}^{6m}\\!\times\\!H^{1}(\mathbb{R}^{3})\big{)}$

to the mean-field equations (1.14) in mild form; it preserves the $L^{2}$-norm
of the boson field and the total energy of the system. In particular, for all
$t\in\mathbb{R}$ it holds that

${\bm{V}}_{t}^{2}\,+\,\|\varphi_{t}\|_{H^{1}}^{2}\,\leqslant\,\mu\,\big{(}E(0)+\|\varphi_{0}\|_{L^{2}}^{2}+\|\varphi_{0}\|_{L^{2}}^{6}\big{)}\,.$
(6.3)

Further, if $\varphi_{0}\in H^{2}(\mathbb{R}^{3})$, it holds that $\varphi\in
C^{1}\big{(}\mathbb{R};L^{2}(\mathbb{R}^{3})\big{)}\cap
C\big{(}\mathbb{R};H^{2}(\mathbb{R}^{3})\big{)}$ and (1.14) holds in the
strong sense.

Its proof is divided into three steps: local well-posedness, propagation of
$H^{2}$ regularity, and conservation laws. These steps are given as the
following three lemmas.

######  Lemma 6.1 (Local Well-Posedness).

Let $({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$ be initial conditions for which the system has energy
$E_{0}.$ Then, there exists $T=T(v,w,E_{0},\|\varphi_{0}\|_{L^{2}})>0$ such
that the system of equations

$\displaystyle{\bm{X}}_{t}={\bm{X}}_{0}+\int_{0}^{t}{\bm{V}}_{s}\mathrm{d}s$
(6.4)
$\displaystyle{\bm{V}}_{t}={\bm{V}}_{0}-\int_{0}^{t}\int_{\mathbb{R}^{3}}\nabla_{\bm{X}}\underline{w}\big{(}x,{\bm{X}}_{s}\big{)}\,|\varphi_{s}(x)|^{2}\mathrm{d}x$
(6.5)
$\displaystyle\varphi_{t}=e^{-\mathrm{i}t\Delta}\varphi_{0}-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}(t-s)\Delta}\Big{(}(v*|\varphi_{s}|^{2})\,\varphi_{s}+\underline{w}\big{(}x,{\bm{X}}_{s}\big{)}\varphi_{s}\Big{)}\,\mathrm{d}s\,$
(6.6)

has a unique solution $({\bm{X}},{\bm{V}},\varphi)\in
C\big{(}[0,T];\,\mathbb{R}^{6m}\times H^{1}(\mathbb{R}^{3})\big{)}$. In
addition, the solution depends continuously on the initial data.

######  Remark 6.1.

The map $({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})\mapsto T\in(0,\infty)$ is continuous; see (6.7) and
(6.20).

###### Proof.

Let us fix some initial conditions
$({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})\in\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})$. In particular, it follows from (6.2) that the
inequality

$M^{2}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mu(E_{0}+\|\varphi_{0}\|_{L^{2}}^{2}+\|\varphi_{0}\|_{L^{2}}^{6})\geqslant\|\varphi_{0}\|_{H^{1}}^{2}+|{\bm{V}}_{0}|^{2}\geqslant
0\,,$ (6.7)

holds. For $T=T(M,v,w)>0$, we will set up a fixed-point argument in the Banach
space

$Y_{T}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Big{\\{}({\bm{X}},{\bm{V}},\varphi)\in
C\big{(}[0,T];\,\mathbb{R}^{6m}\\!\times\\!H^{1}(\mathbb{R}^{3})\big{)}\,\big{|}\,\sup_{|t|\leqslant
T}\Big{(}|{\bm{V}}(t)-{\bm{V}}_{0}|+\|\varphi(t)-e^{-\mathrm{i}t\Delta}\varphi_{0}\|_{H^{1}}\Big{)}\leqslant
M\Big{\\}}\,,$ (6.8)

endowed with the norm $\|({\bm{X}},{\bm{V}},\varphi)\|\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sup_{0\leqslant
t\leqslant
T}\big{(}|{\bm{X}}(t)|+|{\bm{V}}(t)|+\|\varphi(t)\|_{H^{1}}\big{)}$. We
consider the map
$\mathcal{M}=(\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3})$ on
$C\big{(}[0,T];\,\mathbb{R}^{3m}\times\mathbb{R}^{3m}\\!\times\\!H^{1}(\mathbb{R}^{3})\big{)}$
defined as

$\displaystyle\mathcal{M}_{1}({\bm{X}},{\bm{V}},\varphi)(t)$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\bm{X}}_{0}+\int_{0}^{t}{\bm{V}}_{s}\mathrm{d}s$
(6.9) $\displaystyle\mathcal{M}_{2}({\bm{X}},{\bm{V}},\varphi)(t)$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}={\bm{V}}_{0}-\int_{0}^{t}\int_{\mathbb{R}^{3}}\nabla_{\bm{X}}\underline{w}\big{(}x,{\bm{X}}_{s}\big{)}\,|\varphi_{s}(x)|^{2}\mathrm{d}x\mathrm{d}s$
(6.10) $\displaystyle\mathcal{M}_{3}({\bm{X}},{\bm{V}},\varphi)(t)$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{-\mathrm{i}t\Delta}\varphi_{0}-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}(t-s)\Delta}\Big{(}(v*|\varphi_{s}|^{2})\,\varphi_{s}+\underline{w}\big{(}x,{\bm{X}}_{s}\big{)}\varphi_{s}\Big{)}\,\mathrm{d}s\,.$
(6.11)

Thanks to Banach’s fixed point theorem, existence and uniqueness of solutions
is established once we show that $\mathcal{M}$ is a contraction from $Y_{T}$
into itself, for $T$ small enough. We will show that such $T$ can be chosen to
depend only on $M$, $v$ and $w$. The details of the proof of continuity with
respect to initial data are left as an exercise to the reader.

Proof of $\mathcal{M}(Y_{T})\subset Y_{T}$. Let
$({\bm{X}},{\bm{V}},\varphi)\in Y_{T}$ and $t\in[0,T]$. Then, we have the
estimates

$\displaystyle|\mathcal{M}_{2}({\bm{X}},{\bm{V}},\varphi)(t)-{\bm{V}}_{0}|$
$\displaystyle\leqslant\,T\,\|w\|_{C_{b}^{1}}\sup_{0\leqslant\tau\leqslant
T}\|\varphi_{\tau}\|_{L^{2}}^{2}$ (6.12)
$\displaystyle\|\mathcal{M}_{3}({\bm{X}},{\bm{V}},\varphi)(t)-e^{-\mathrm{i}t\Delta}\varphi_{0}\|_{H^{1}}$
$\displaystyle\leqslant T\sup_{0\leqslant\tau\leqslant
T}\big{(}\|(v*|\varphi_{\tau}|^{2})\varphi_{\tau}\|_{H^{1}}+\|w(x,{\bm{X}}_{s})\varphi_{s}\|_{H^{1}}\big{)}$
$\displaystyle\leqslant\,T\,(L(v)+2\|w\|_{C_{b}^{1}})\sup_{0\leqslant\tau\leqslant
T}\big{(}\|\varphi_{\tau}\|_{H^{1}}^{3}+\|\varphi_{\tau}\|_{H^{1}}\big{)}$
(6.13)

where, in deriving the last inequality, we have used the Lipschitz property
(2.23) for $\psi=0$. Thanks to the triangle inequality and the definition of
$M$ and $Y_{T}$, it follows that

$\|\varphi_{\tau}\|_{H_{1}}+|{\bm{V}}_{\tau}|\leqslant\|e^{-i\tau\Delta}\varphi_{0}\|_{H^{1}}+|{\bm{V}}_{0}|+\|\varphi_{\tau}-e^{-\mathrm{i}\tau\Delta}\varphi_{0}\|_{H^{1}}+|{\bm{V}}_{\tau}-{\bm{V}}_{0}|\leqslant
3M\,.$ (6.14)

Therefore, we may combine the last three estimates to conclude that there is
$C_{v,w}^{(1)}>0$ such that for all $t\in[0,T]$

$|\mathcal{M}_{2}({\bm{X}},{\bm{V}},\varphi)(t)-{\bm{V}}_{0}|+\|\mathcal{M}_{3}({\bm{X}},{\bm{V}},\varphi)(t)-e^{-\mathrm{i}t\Delta}\varphi_{0}\|_{H^{1}}\leqslant
C_{v,w}^{(1)}\,T\,(M+M^{3}).$ (6.15)

Thus, $\mathcal{M}$ maps $Y_{T}$ into itself provided
$T\leqslant(C_{v,w}^{(1)}(1+M^{2}))^{-1}$.

Proof of Contractivity. Consider $({\bm{X}},{\bm{V}},\varphi)$ and
$(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})$ in $Y_{T}$, and let
$t\in[0,T]$. For the position variables, one simply has that

$\displaystyle|\mathcal{M}_{1}({\bm{X}},{\bm{V}},\varphi)(t)-\mathcal{M}_{1}(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})(t)|$
$\displaystyle\leqslant T\textstyle\sup_{0\leqslant\tau\leqslant
T}|{\bm{V}}_{\tau}-\tilde{\bm{V}}_{\tau}|\,.$ (6.16)

For the velocity variables, we use a Taylor estimate and the triangle
inequality to find that

$\displaystyle|\mathcal{M}_{2}({\bm{X}},{\bm{V}},\varphi$
$\displaystyle)(t)-\mathcal{M}_{2}(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})(t)|$
(6.17) $\displaystyle\leqslant\textstyle T\sup_{0\leqslant\tau\leqslant
T}\Big{(}\|w\|_{C_{b}^{1}}(\|\varphi_{\tau}\|_{L^{2}}+\|\tilde{\varphi}_{\tau}\|_{L^{2}})\|\varphi_{\tau}-\tilde{\varphi}_{\tau}\|_{L^{2}}+\|w\|_{C_{b}^{2}}\|\varphi_{\tau}\|_{L^{2}}^{2}|{\bm{X}}_{\tau}-\tilde{\bm{X}}_{\tau}|\Big{)}.$

For the boson fields, we use the Lipschitz property (2.23), a Taylor estimate
and the triangle inequality to find that

$\displaystyle\|\mathcal{M}_{3}({\bm{X}},{\bm{V}},\varphi)(t)\,-\,\mathcal{M}_{3}$
$\displaystyle(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})(t)\|_{H^{1}}$
$\displaystyle\ \leqslant\ \textstyle T\sup_{0\leqslant\tau\leqslant
T}L(v)(\|\varphi_{\tau}\|_{H^{1}}^{2}+\|\tilde{\varphi}_{\tau}\|_{H^{1}}^{2})\|\varphi_{\tau}-\tilde{\varphi}_{\tau}\|_{H^{1}}$
$\displaystyle\quad\ +\ T\sup_{0\leqslant\tau\leqslant
T}3\|w\|_{C_{b}^{2}}(\|\varphi_{\tau}\|_{H^{1}}|{\bm{X}}_{\tau}-\tilde{\bm{X}}_{\tau}|+\|\varphi_{\tau}-\tilde{\varphi}_{\tau}\|_{H^{1}})\,.$
(6.18)

Since the last three estimates are uniform in $t\in[0,T]$, we put them
together to find that there is $C_{v,w}^{(2)}>0$ such that

$\|\mathcal{M}({\bm{X}},{\bm{V}},\varphi)-\mathcal{M}(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})\|\leqslant
T\,C_{v,w}^{(2)}(M+M^{2})\|({\bm{X}},{\bm{V}},\varphi)-(\tilde{\bm{X}},\tilde{\bm{V}},\tilde{\varphi})\|$
(6.19)

where we have used $\eqref{M bound}$ as well. Since $M+M^{2}\leqslant
2(1+M^{2})$, it follows that by choosing

$T(M,v,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\min\Big{\\{}\frac{1}{C_{v,w}^{(1)}},\frac{1}{4C_{v,w}^{(2)}}\Big{\\}}\frac{1}{1+M^{2}}$
(6.20)

the map $\mathcal{M}$ becomes Lipschitz continuous, with Lipschitz constant
$1/2$. In particular, it becomes a contraction and the proof is complete. ∎

The following proof of propagation of regularity is heavily inspired by [1,
Chapter 4].

######  Lemma 6.2 (Propagation of $H^{2}$ Regularity).

Let $({\bm{X}},{\bm{V}},\varphi)\in
C\big{(}[0,T];\,\mathbb{R}^{6m}\\!\times\\!H^{1}(\mathbb{R}^{3})\big{)}$ be as
in Lemma 6.1 with initial data $\varphi_{0}\in H^{2}(\mathbb{R}^{3})$. Then,
we have that $\varphi\in C^{1}\big{(}[0,T];L^{2}(\mathbb{R}^{3})\big{)}\cap
C\big{(}[0,T];H^{2}(\mathbb{R}^{3})\big{)}$ and the mean-field equation (1.14)
holds in the strong sense.

###### Proof.

In terms of ${\bm{X}}\in C^{1}([0,T],\mathbb{R}^{3m})$, as given by Lemma 6.1,
we define the map

$F(t,\phi)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(v*|\phi|^{2})\phi+w\big{(}x,{\bm{X}}_{t}\big{)}\phi\,,\qquad(t,\phi)\in[0,T]\times
H^{1}(\mathbb{R}^{3})\,.$ (6.21)

In particular, a Taylor estimate combined with the Potential Estimate (2.21),
imply that the following Lipschitz estimates are satisfied

$\displaystyle\|F(t,\phi)-F(s,\phi)\|_{L^{2}}\ \lesssim\
\sup_{0\leqslant\tau\leqslant T}\\!|\dot{\bm{X}}_{\tau}|\ \|\phi\|_{L^{2}}\
|t-s|$ (6.22) $\displaystyle\|F(t,\phi)-F(t,\psi)\|_{L^{2}}\ \lesssim\
(\|\phi\|_{H^{1}}^{2}+\|\psi\|_{H^{1}}^{2})\|\phi-\psi\|_{L^{2}}\,.$ (6.23)

Let $\varphi\in C([0,T],H^{1})$ be the solution constructed in Lemma 6.1. Our
goal will be to show that $t\in[0,T]\mapsto\varphi_{t}\in L^{2}$ is a
Lipschitz map. Indeed, after a change of variables, we are able to write
$\varphi_{t}=e^{-\mathrm{i}t\Delta}\varphi_{0}-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}s\Delta}F(t-s,\varphi_{t-s})\mathrm{d}t$
for all $t\in(0,T)$. Thus, for all $h$ small enough we find thanks to (6.22)
and (6.23)

$\displaystyle\|\varphi_{t+h}-\varphi_{t}\|_{L^{2}}\ \lesssim\ $
$\displaystyle|h|\,\|\Delta\varphi_{0}\|_{L^{2}}+\int_{t}^{t+h}\|F(t+h-s,\varphi_{t+h-s})\|_{L^{2}}\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\|F(t+h-s,\varphi_{t+h-s})-F(t-s,\varphi_{t-s})\|_{L^{2}}\mathrm{d}s\,,$
$\displaystyle\ \lesssim\ $
$\displaystyle\,|h|\,\big{(}\|\Delta\varphi_{0}\|_{L^{2}}+\nu^{3}\big{)}+\int_{0}^{t}\big{(}\nu^{2}|h|+\nu^{2}\|\varphi_{t+h-s}-\varphi_{t-s}\|_{L^{2}}\big{)}\mathrm{d}s\,,$
(6.24)

where $\nu\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sup_{0\leqslant
t\leqslant T}(\,\|\varphi_{t}\|_{H^{1}}+|\dot{\bm{X}}_{t}|\,)<\infty$. A
second change of variables in the last integral let us apply the Grönwall
inequality and conclude that for some $C>0$, independent of $t$ and $h$, it
holds that

$\|\varphi_{t+h}-\varphi_{t}\|_{L^{2}}\leqslant
Ce^{C\nu^{2}t}\big{(}\|\Delta\varphi_{0}\|_{L^{2}}+\nu^{3}+\nu^{2}t\big{)}\,|h|\,.$
(6.25)

Consequently, the map $t\in[0,T]\mapsto f(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=F(t,\varphi_{t})\in
L^{2}(\mathbb{R}^{3}))$ is Lipschitz continuous and, in particular, $f\in
W^{1,1}(0,T;L^{2}(\mathbb{R}^{3}))$. One may then view the mean-field equation
for $\varphi_{t}$ as a semi-linear Schrödinger equation with source term $f$.
In particular $W^{1,1}$ regularity of the source term implies that $H^{2}$
regularity of the solution is propagated in time; see [1, Proposition 4.1.16].
∎

In order to obtain global well-posedness, we prove that both the energy of the
system and the $L^{2}$ mass of the boson field are constant in time.
Consequently, the time $T>0$–introduced in Lemma 6.1–may be used to patch
local solutions and cover the real line.

######  Lemma 6.3 (Conservation Laws).

Let $({\bm{X}},{\bm{V}},\varphi)\in
C\big{(}[0,T];\,\mathbb{R}^{6m}\\!\times\\!H^{1}(\mathbb{R}^{3})\big{)}$ be as
in Lemma 6.1. Then, it holds that for all $t\in[0,T]$

$\|\varphi_{t}\|_{L^{2}}=\|\varphi_{0}\|_{L^{2}}\qquad\text{ and }\qquad
E(t)=E(0)\,.$ (6.26)

where $E(t)$ is the energy of the system defined in (6.1).

###### Proof.

Let us first consider the case $\varphi_{0}\in H^{2}(\mathbb{R}^{3})$ and let
$({\bm{X}},{\bm{V}},\varphi)$ be the local solution of the mean-field
equations, as given by Lemma 6.1. Thanks to Lemma 6.2, the boson field remains
in $H^{2}$ and satisfies $\dot{\varphi}_{t}=-\mathrm{i}\,h(t)\varphi_{t}$,
where we introduce the self-adjoint operator

$h(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=-\Delta+v*|\varphi_{t}|^{2}+\underline{w}\big{(}x,{\bm{X}}_{t}\big{)}\,,\qquad
t\in[0,T]\,.$ (6.27)

Conservation of $L^{2}$-norm. We differentiate at $t\in(0,T)$ to find

$\frac{d}{dt}\|\varphi_{t}\|_{L^{2}}^{2}=2\mathrm{Re}\left\langle\varphi_{t},\mathrm{i}\,h(t)\varphi_{t}\right\rangle_{L^{2}}=0$
(6.28)

thanks to self-adjointness of $h(t)$. It suffices to apply the Fundamental
Theorem of Calculus.

Conservation of energy. Let $DE_{Z}:\mathbb{R}^{6m}\times
H^{1}(\mathbb{R}^{3})\mapsto\mathbb{R}$ be the Fréchet derivative of the
energy functional $E$, evaluated at $Z$. Then, the chain rule gives at
$t\in(0,T)$

$\displaystyle\frac{d}{dt}E(t)$ $\displaystyle\ =\
DE_{({\bm{X}}_{t},{\bm{V}}_{t},\varphi_{t})}(\dot{\bm{X}}_{t},\dot{\bm{V}}_{t},\dot{\varphi}_{t})\,,$
$\displaystyle\ =\
\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\nabla_{\bm{X}}\underline{w}\big{(}x,{\bm{X}}_{t}\big{)}|\varphi_{t}(x)|^{2}\mathrm{d}x\cdot\dot{\bm{X}}_{t}\,+\,{\bm{V}}_{t}\cdot\dot{\bm{V}}_{t}\,+\,2\mathrm{Re}\left\langle
h(t)\varphi_{t},\dot{\varphi}_{t}\right\rangle_{L^{2}}\,.$ (6.29)

Thanks to the mean-field equations, the first and second term cancel out. The
third term vanishes, thanks to self-adjointess of $h(t)$.

Finally, for the case $\varphi_{0}\in H^{1}(\mathbb{R}^{3})$ we take a
sequence $(\varphi_{0,n})_{n\geqslant 1}\subset H^{2}(\mathbb{R}^{3})$ that
converges to $\varphi_{0}$ in $H^{1}(\mathbb{R}^{3})$. In particular, thanks
to continuity with respect to intial data, it holds that

$\lim_{n\rightarrow\infty}\|\varphi_{t,n}-\varphi_{t}\|_{H^{1}}+|\dot{\bm{X}}_{t,n}-\dot{\bm{X}}_{t}|+|{\bm{X}}_{t,n}-{\bm{X}}_{t}|=0\,,\qquad\forall
t\in[0,T]$ (6.30)

where $({\bm{X}}_{t},\varphi_{t})$ solves (1.14) with initial data
$({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0})$ and $({\bm{X}}_{t,n},\varphi_{t,n})$
solves (1.14) with initial data $({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{n,0})$.
Thus, it easily follows that for all $t\in[0,T]$

$E(t)=E({\bm{X}}_{t},\dot{\bm{X}}_{t},\varphi_{t})=\lim_{n\rightarrow\infty}E({\bm{X}}_{t,n},\dot{\bm{X}}_{t,n},\varphi_{t,n})=\lim_{n\rightarrow\infty}E({\bm{X}}_{0},{\bm{V}}_{0},\varphi_{0,n})=E_{0}\,,$
(6.31)

which concludes the proof. ∎

##  7\. Well-posedness of the Regularized, Truncated Dynamics

Let $N,M\geqslant 1$ and $\varepsilon\in(0,1)$. Throughout this section, we
shall drop the regularization paremeter $\varepsilon\in(0,1)$ and assume
instead that the potentials and the initial data that we work with satisfy the
stronger assumptions:

$v\in L^{\infty}(\mathbb{R}^{3})\,,\quad
w\in\mathcal{S}(\mathbb{R}^{3})\,,\quad\textnormal{and}\quad\varphi_{0}\in
H^{2}(\mathbb{R}^{3})\,.$ (7.1)

We will then only apply well-posedness results presented in this section, to
the truncated dynamics. Indeed, we now turn our attention to the study of the
Cauchy problem for solutions of the equation

$\mathrm{i}\partial_{t}\mathcal{U}_{N}^{(M)}(t,s)=\mathcal{L}_{M}(t)\mathcal{U}_{N}^{(M)}(t,s)\,,\quad\textnormal{and}\quad\mathcal{U}_{N}^{(M)}(t,t)=\mathds{1}\,,\qquad
t,s\in\mathbb{R}\,.$ (7.2)

Here, $\mathcal{L}_{M}(t)$ is the regularized, truncated generator introduced
in (3.26). We will decompose it in the form

$\displaystyle\mathcal{L}_{M}(t)\ =\ \mathbb{H}\ +\
\mathcal{I}_{M}(t)\,,\quad\textnormal{where}\quad\mathbb{H}\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
-\frac{1}{2N}\Delta_{{\bm{X}}}\,+\,T_{b}\,+\,\mathcal{I}_{4}\,.$ (7.3)

######  Remark 7.1.

$\mathbb{H}$ is a self-adjoint operator with a domain
$\mathcal{D}(\mathbb{H})$ satisfying the anlogous characterization (1.13), but
with $N_{b}^{2}$ instead of $N_{b}^{3}$; this thanks to boundedness of $v$. In
particular, its domain becomes a Banach space when endowed with the norm

$\|\Psi\|_{\mathbb{H}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\|\Psi\|+\|\mathbb{H}\Psi\|\,,\qquad\Psi\in\mathcal{D}(\mathbb{H})\,.$
(7.4)

Further, note that $\mathbb{H}$ commutes with both the particle number
operator $N_{b}$ and the momentum operator
${\bm{P}}=-\mathrm{i}\nabla_{{\bm{X}}}$.

Thanks to the particle number cut-off, the potentials being bounded, and the
boson field being in $C^{1}(\mathbb{R},L^{2}_{x})$, the interaction term
$\mathcal{I}_{M}(t)$ enjoys the following two regularity properties.

######  Lemma 7.1.

Let $T\geqslant 0$ and $N,M\geqslant 1$. Then, the following holds:

(1)
$\sup_{t\in\mathbb{R}}\|\mathcal{I}_{M}(t)\|_{\mathcal{B}(\mathscr{H})}<\infty.$

(2) There exists $C=C(v,w,T,N,M)$ such that

$\|\mathcal{I}_{M}(t)-\mathcal{I}_{M}(s)\|_{\mathcal{B}(\mathscr{H})}\leqslant
C\,|t-s|\,,\qquad\forall t,s\in[0,T]\,.$ (7.5)

###### Proof.

(1) Using the Particle Number Estimates for $v,w\in
L^{\infty}(\mathbb{R}^{3})$ one finds, thanks to the particle number cut-off,
that for all $t\in\mathbb{R}$

$\|\mathcal{I}_{M}(t)\|_{\mathcal{B}(\mathscr{H})}\,\lesssim\,C(N,M)\big{(}\|v\|_{L^{\infty}}+\|w\|_{L^{\infty}}\big{)}(1+\|\varphi_{t}\|_{L^{2}}^{2})$
(7.6)

from which our claims follows easily.

(2) Let $t,s\in[0,T]$ and write the difference as

$\mathcal{I}_{M}(t)-\mathcal{I}_{M}(s)=\chi(N_{b}\leqslant
M)\big{(}N\Delta\mathcal{I}_{0}(t,s)+N^{1/2}\Delta\mathcal{I}_{1}(t,s)+\Delta\mathcal{I}_{2}(t,s)+N^{-1/2}\Delta\mathcal{I}_{3}(t,s)\big{)}$
(7.7)

where each term $\Delta\mathcal{I}_{i}(t)$ that has beed introduced above,
depends on a $i$-th power on creation and annihilation operator. Since the
estimate for each one of these is very similar, we shall only show one in
detail. Indeed, for the cubic term we have that

$\displaystyle\|\chi(N_{b}\leqslant M)\Delta\mathcal{I}_{3}(t,s)\|$
$\displaystyle\,\lesssim\,\|\chi(N_{b}\leqslant
M)\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{6}}}}v(x-t)a_{x}^{*}\big{(}\varphi_{t}(y)-\varphi_{s}(y)\big{)}a_{y}^{*}a_{x}\mathrm{d}x\mathrm{d}y\|$
$\displaystyle\,\lesssim\,M^{3/2}\big{(}\sup_{x\in\mathbb{R}^{3}}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}v(x-y)^{2}|(\varphi_{t}-\varphi_{s})(y)|^{2}\mathrm{d}y\big{)}^{1/2}$
$\displaystyle\,\lesssim\,\|v\|_{L^{\infty}}M^{3/2}\|\varphi_{t}-\varphi_{s}\|_{L^{2}}\,\lesssim\,\|v\|_{L^{\infty}}M^{3/2}\sup_{\tau\leqslant
T}\|\partial_{\tau}\varphi_{\tau}\|_{L^{2}}|t-s|\,,$ (7.8)

where, in the last line, we have used
$\varphi_{t}-\varphi_{s}=\int_{s}^{t}\partial_{\tau}\varphi_{\tau}\mathrm{d}\tau$.
Since $(t\mapsto\varphi_{t})\in C^{1}(\mathbb{R},L^{2}_{x})$, the Lipschitz
constant of the right hand side of (7.8) is finite. ∎

We combine Lemma 7.1, together with [32, Theorem X.69] and [1, Proposition
4.1.16] to prove existence, uniqueness and basic propagation of regularity for
the propagator of the dynamics defined by (7.2). Note that a similar result
could have been obtained using the general theory of Kato [24].

######  Proposition 7.1.

There exists a unique unitary propagator
$\big{(}\mathcal{U}_{N}^{(M)}(t,s)\big{)}_{t,s\in\mathbb{R}}$ such that for
all $\Psi_{0}\in\mathcal{D}(\mathbb{H})$ and $s\in\mathbb{R}$, the map
$t\in\mathbb{R}\mapsto\Psi(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{U}_{N}^{(M)}(t,s)\Psi_{0}\in\mathscr{H}$
satisfies:

(1) $\Psi\in C^{1}(\mathbb{R},\mathscr{H})\cap
C(\mathbb{R},\mathcal{D}(\mathbb{H}))$

(2) $\mathrm{i}\partial_{t}\Psi(t)=\mathcal{L}_{M}(t)\Psi(t)$ holds in
$\mathscr{H}$ .

######  Remark 7.2.

This result can be extended to the following abstract setting. Consider time-
dependent operators of the form

$\mathbb{H}(t)=\mathbb{H}_{0}+\mathcal{I}(t)\,,\qquad t\in\mathbb{R}\,,$ (7.9)

where $\mathbb{H}_{0}$ is self-adjoint with domain
$\mathcal{D}(\mathbb{H}_{0})$, and $\mathcal{I}(t)$ is bounded, self-adjoint
and satisfies assumptions (1) and (2) of Lemma 7.1. Then, there is a unique
unitary propagator $U(t,s)$ associated to the evolution of $\mathbb{H}(t)$, in
the sense of Proposition 7.1. Note that we do not require $\mathbb{H}_{0}$ to
be bounded from below, nor we require additional assumptions on the derivative
of $\mathcal{I}(t)$ or on the commutator $[\mathbb{H}_{0},\mathcal{I}(t)]$.

###### Proof of Proposition 7.1.

Let us first pass to the interaction picture. Namely, we define

$\widetilde{\mathcal{I}}_{M}(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{\mathrm{i}t\mathbb{H}}\mathcal{I}_{M}(t)e^{-\mathrm{i}t\mathbb{H}}\,,\qquad
t\in\mathbb{R}\,.$ (7.10)

Then, Lemma 7.1 and [32, Theorem X.69] imply the existence of a unitary
propagator $\widetilde{\mathcal{U}}_{N}^{M}(t,s)$–explicitly given by the
absolutely convergent Dyson series–that satisfies the integral equation

$\widetilde{\mathcal{U}}_{N}^{(M)}(t,s)=\mathds{1}-\mathrm{i}\int_{s}^{t}\widetilde{\mathcal{I}}_{M}(r)\widetilde{\mathcal{U}}_{N}^{(M)}(r,s)\mathrm{d}r\,,\qquad
t,s\in\mathbb{R}\,.$ (7.11)

Consequently, for $t,s\in\mathbb{R}$ we let our original propagator be
$\mathcal{U}_{N}^{(M)}(t,s)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e^{-\mathrm{i}t\mathbb{H}}\widetilde{\mathcal{U}}_{N}^{(M)}(t,s)e^{\mathrm{i}s\mathbb{H}}$.
Let now $\Psi(t)$ be as in the statement of the proposition. Then, it is
straightforward to verify that for all $t,s\in\mathbb{R}$ it holds that

$\Psi(t)=e^{-\mathrm{i}(t-s)\mathbb{H}}\Psi_{0}-\mathrm{i}\int_{s}^{t}e^{-\mathrm{i}(t-r)\mathbb{H}}\mathcal{I}_{M}(r)\Psi(r)\,\mathrm{d}r\,.$
(7.12)

Since the interaction term $\mathcal{I}_{M}(t)$ is locally Lipschitz
continuous, we may adapt the argument presented in the proof of Lemma 6.2 to
show that the map $t\in\mathbb{R}\mapsto\Psi(t)\in\mathscr{H}$ is locally
Lipschitz continuous, as well. In particular, the source term $t\mapsto
F(t)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{I}_{M}(t)\Psi(t)\in\mathscr{H}$
becomes locally Lipschitz continuous and, therefore, belongs to
$W^{1,1}_{loc}(\mathbb{R},\mathscr{H})$. We may then apply [1, Proposition
4.1.6] to conclude propagation of regularity, in the sense that (1) and (2)
hold true. Uniqueness follows from standard arguments using the symmetry of
$\mathcal{L}_{M}(t)$ and denseness of
$\mathcal{D}(\mathbb{H})\subset\mathscr{H}$. ∎

Next, we show that the regularized truncated dynamics propagates smoothness
with respect to the tracer particle variables, together with boundedness with
respect to the particle number operator and its powers. We remind the reader
that the space $\mathscr{D}_{\infty}$ has been introduced in (5.1)

######  Proposition 7.2.

In the notation of Proposition 7.1, if $\Psi_{0}\in\mathscr{D}_{\infty}$, then
$\Psi(t)\in\mathscr{D}_{\infty}$ for all $t\in\mathbb{R}$.

###### Proof.

Let $N,M\geqslant 1$ be fixed and assume $s=0$ for simplicity. We will be
using a fixed-point argument. Indeed, let $k\in\mathbb{N}\cup\\{0\\}$ and
consider the following Banach space

$\mathscr{H}_{k}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\\{\Psi\in\mathscr{H}\,|\,\|\Psi\|_{\mathscr{H}_{k}}<\infty\\}$
(7.13)

which we endow with the norm

$\|\Psi\|_{\mathscr{H}_{k}}\textstyle\ \mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\
\|\,|{\bm{X}}|^{k}\Psi\|_{\mathscr{H}}+\|\,|{\bm{P}}|^{k}\Psi\|_{\mathscr{H}}+\|N_{b}^{k}\Psi\|_{\mathscr{H}}+\|\Psi\|_{\mathscr{H}}\,.$
(7.14)

Step 1. First, we show that the interaction term $\mathcal{I}_{M}(t)$ is
continuous with respect to $\mathscr{H}_{k}$, uniformly in $t\in\mathbb{R}$.
To this end, let $\alpha$ be a multi-index with $|\alpha|\leqslant k$. Then,
the terms in $\mathcal{I}_{M}(t)$ that depend on ${\bm{X}}\in\mathbb{R}^{3m}$
satisfy the following estimate

$\displaystyle\|\partial_{\bm{X}}^{\alpha}\Big{(}\scaleobj{1.4}{\textstyle\int_{\scaleobj{.85}{\mathbb{R}^{3}}}}\underline{w}(x,{\bm{X}})\,\chi(N_{b}\leqslant
M)(N|\varphi_{t}|^{2}+$
$\displaystyle\sqrt{N}a_{x}\varphi_{t}(x)+a_{x}^{*}a_{x})\mathrm{d}x\
\Psi\Big{)}\|_{\mathscr{H}}$ $\displaystyle\leqslant
C(k)\|w\|_{C_{b}^{k}}(N+\sqrt{NM}+M)\|\Psi\|_{\mathscr{H}_{k}}$ (7.15)

where $C(k)>0$ depends only on $k$. The other terms that show up in
$\mathcal{I}_{M}(t)$ do not depend on the tracer particle variables
${\bm{X}}\in\mathbb{R}^{3m}$ and can be controlled analogously, thanks to the
particle number cut-off. We put all of the terms together and find that for
all $t\in\mathbb{R}$ it holds that

$\||{\bm{P}}|^{k}\mathcal{I}_{M}(t)\Psi\|_{\mathscr{H}}\ \lesssim\
(N+\sqrt{NM}+M+\sqrt{M^{3}/N})\,\|\Psi\|_{\mathscr{H}_{k}}\,,\qquad\Psi\in\mathscr{H}_{k}\,.$
(7.16)

Next, for the particle number operator we can use the particle number cut-off
to easily find that the following (rough) estimate holds

$\|N_{b}^{k}\mathcal{I}_{M}(t)\Psi\|_{\mathscr{H}}\lesssim
M^{k}\sup_{t\in\mathbb{R}}\|\mathcal{I}_{M}(t)\|_{\mathcal{B}(\mathscr{H})}\|\Psi\|_{\mathscr{H}}\lesssim
M^{k}\sup_{t\in\mathbb{R}}\|\mathcal{I}_{M}(t)\|_{\mathcal{B}(\mathscr{H})}\|\Psi\|_{\mathscr{H}_{k}}\,.$
(7.17)

For the position variables, simply note that ${\bm{X}}$ commutes with
$\mathcal{I}_{M}(t)$. Thus, we put these estimates together to find that for
all $t\in\mathbb{R}$

$\|\mathcal{I}_{M}(t)\Psi\|_{\mathscr{H}_{k}}\lesssim\big{(}N+M+\sqrt{M^{3}/N}+M^{k}\sup_{t\in\mathbb{R}}\|\mathcal{I}_{M}(t)\|_{\mathcal{B}(\mathscr{H})}\big{)}\|\Psi\|_{\mathscr{H}_{k}}\,.$
(7.18)

Step 2. Secondly, we show that the evolution group
$(e^{\mathrm{i}t\mathcal{H}_{N}})_{t\in\mathbb{R}}$ is continuous with respect
to $\mathscr{H}_{k}$, locally uniformly in time. To this end, we note that in
one dimension the following estimate holds for the free Schrödinger propagator

$\|X^{k}e^{-\mathrm{i}tP^{2}}\psi\|_{L^{2}(\mathbb{R})}=\|(X+tP)^{k}\psi\|_{L^{2}(\mathbb{R})}\leqslant
C(k)(\|X^{k}\psi\|_{L^{2}(\mathbb{R})}+t^{k}\|P^{k}\psi\|_{L^{2}(\mathbb{R})}+\|\psi\|_{L^{2}(\mathbb{R})})$
(7.19)

as it can be verified by using the commutation relation $XP=PX+\mathrm{i}$ to
control mixed powers. This in turn easily implies that

$\displaystyle\|e^{-\mathrm{i}t\mathbb{H}}\Psi\|_{\mathscr{H}_{k}}$
$\displaystyle=\||{\bm{X}}|^{k}e^{-\mathrm{i}t\mathbb{H}}\Psi\|_{\mathscr{H}}+\||{\bm{P}}|^{k}\Psi\|_{\mathscr{H}}+\|N_{b}^{k}\Psi\|_{\mathscr{H}}+\|\Psi\|_{\mathscr{H}}\leqslant
C(1+t^{k})\|\Psi\|_{\mathscr{H}_{k}}\,$ (7.20)

for some constant $C=C(k)>0$ and all $t\in\mathbb{R}$.

Step 3. Finally, we use the continuity estimates (7.18) and (7.20) to set up a
fixed-point argument in $C([0,T],\mathscr{H}_{k})$ to show that the equation

$\widetilde{\Psi}(t)=e^{-\mathrm{i}t\mathbb{H}}\Psi_{0}-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}(t-r)\mathbb{H}}\mathcal{I}_{M}(r)\widetilde{\Psi}(r)\
\mathrm{d}r.$

has a unique solution. In particular, if $\Psi(t)$ is the solution constructed
in Proposition 7.1, it holds that
$\Psi(t)=\widetilde{\Psi}(t)\in\mathscr{H}_{k}$ for all $t\in[0,T]$, thanks to
uniquess of solutions in $C([0,T],\mathscr{H})$ (i.e. the $k=0$ case).
Further, $T<1$ can be chosen to depend only on the constants that show up in
the continuity estimates. Thus, we can iterate the argument to cover
$\mathbb{R}$. Since $k\in\mathbb{N}$ was arbitrary, this concludes the proof.
∎

##  Appendix A The Regularization Lemmata

###### Proof of Lemma 3.1 .

Let us fix $N\geqslant 1$ and $t\geqslant 0$. Then, the triangle inequality
and the unitarity of the evolution group
$(e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}})_{t\in\mathbb{R}}$ implies

$\displaystyle\|\Psi_{N,t}-\Psi_{N,t}^{\varepsilon}\|\leqslant\|\Psi_{N,0}-\Psi_{N,0}^{\varepsilon}\|+\|(e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}}-e^{-\mathrm{i}t\mathcal{H}_{N}})\Psi_{N,0}\|\,,\quad\forall\varepsilon\in(0,1)\,.$
(A.1)

We shall estimate these terms separately.

The First Term. The triangle inequality implies that

$\|\Psi_{N,0}-\Psi_{N,0}^{\varepsilon}\|\,\lesssim\,\|\mathcal{W}(\sqrt{N}\varphi_{0})\Omega-\mathcal{W}(\sqrt{N}\varphi_{0}^{\varepsilon})\Omega\|_{\mathcal{F}_{b}}+\|u_{N,0}-u_{N,0}^{\varepsilon}\|_{L_{\bm{X}}^{2}}$
(A.2)

where we used the fact that the norms of $u_{N,0}^{\varepsilon}$ and
$\mathcal{W}(\sqrt{N}\varphi_{0})\Omega$ are uniformly bounded in $N$ and
$\varepsilon$. It is known [16, Lemma 3.1] that the map $f\in
L^{2}\mapsto\mathcal{W}(f)\Phi\in\mathcal{F}_{b}$ is continuous for every
$\Phi\in\mathcal{F}_{b}$. Thus, we apply the definitions of the regularized
quantities $u_{N,0}^{\varepsilon}$ and $\varphi_{0}^{\varepsilon}$ to conclude
that the right hand side of (A.2) vanishes as $\varepsilon\downarrow 0$.

The Second Term. Since the Hamiltonians preserve particle number, for any
$n_{0}\in\mathbb{N}$ we may write

$\|(e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}}-e^{-\mathrm{i}t\mathcal{H}_{N}})\Psi_{N,0}\|\lesssim\|\chi(N_{b}\leqslant
n_{0})(e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}}-e^{-\mathrm{i}t\mathcal{H}_{N}})\Psi_{N,0}\|+\|\chi(N_{b}>n_{0})\Psi_{N,0}\|\,.$
(A.3)

Let us decompose
$e^{-\mathrm{i}t\mathcal{H}_{N}}\Psi_{N,0}=(\psi_{n}(t))_{n\geqslant 0}$ and
$e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}}\Psi_{N,0}=(\psi_{n}^{\varepsilon}(t))_{n\geqslant
0}$ according to their direct sum representation in $\bigoplus_{n\geqslant
0}L^{2}_{{\bm{X}}}\otimes\mathcal{F}_{n}$. Then, we obtain the following
standard estimates

$\displaystyle\|\psi_{n}(t)-\psi_{n}^{\varepsilon}$
$\displaystyle(t)\|^{2}_{L^{2}_{\bm{X}}\otimes\mathcal{F}_{n}}$ (A.4)
$\displaystyle\lesssim\int_{0}^{t}|\left\langle\psi_{n}(s),(H_{N,n}-H_{N,n}^{\varepsilon})\psi_{n}^{\varepsilon}(s)\right\rangle_{L^{2}_{\bm{X}}\otimes\mathcal{F}_{n}}|\,\mathrm{d}s\quad\textnormal{(in
the sense of quad. forms)}$
$\displaystyle\lesssim\int_{0}^{t}|\big{\langle}\psi_{n}(s),\Big{(}\,\frac{n^{2}}{N}(v-v^{\varepsilon})(x_{1}-x_{2})+n(\underline{w}-\underline{w}^{\varepsilon})(x_{1},{\bm{X}})\,\Big{)}\psi_{n}^{\varepsilon}(s)\big{\rangle}_{L^{2}_{\bm{X}}\otimes\mathcal{F}_{n}}|\mathrm{d}s$