nonnegative. The nonnegativity of $a_{\beta}$ follows by the arbitrariness of
$h$. ∎

##  4\. Degenerate case: Gaussian equilibrium

This section is devoted to the case of a Gaussian mean-field equilibrium
$\mu_{\beta}$, that is, we assume that potentials $V,W$ satisfy for some
$R\in(0,\infty)$,

$(V+W\ast\mu_{\beta})(x)=\tfrac{1}{2}R|x|^{2},\qquad\mu_{\beta}(x)=\tfrac{\beta
R}{2\pi}e^{-\frac{1}{2}\beta R|x|^{2}},$ (4.1)

and we shall then prove Theorem 1.2.

###  4.1. Preliminary BBGKY analysis

We start by noting that the BBGKY hierarchy for correlations as derived in
Lemma 2.1 simplifies drastically in the Gaussian setting.

######  Lemma 4.1 (BBGKY hierarchy for correlations).

In the Gaussian setting (4.1), the correlation functions satisfy the BBGKY
hierarchy (2.7) where the defining operators are now given for all $1\leq
m\leq N$ by

$\displaystyle iL_{N,\beta}^{m}g_{N}^{m}$ $\displaystyle:=$
$\displaystyle-\tfrac{N-m}{N}\sum_{j=2}^{m}\beta
Rx_{j}\cdot\int_{\mathbb{R}^{2}}K(x_{j}-x_{*})\,g_{N}^{m}(x_{[m]\setminus\\{j\\}},x_{*})\,\mu_{\beta}(x_{*})\,dx_{*},$
$\displaystyle iS_{N,\beta}^{m,+}g_{N}^{m+1}$ $\displaystyle:=$
$\displaystyle-\tfrac{N-m}{N}\sum_{j=1}^{m}\nabla_{j;\beta}\cdot\int_{\mathbb{R}^{2}}K(x_{j}-x_{*})\,g_{N}^{m+1}(x_{[m]},x_{*})\,\mu_{\beta}(x_{*})\,dx_{*},$
$\displaystyle iS_{N,\beta}^{m,\circ}g_{N}^{m}$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{m}\sum_{2\leq j\leq m\atop i\neq
j}\nabla_{i;\beta}\cdot\int_{\mathbb{R}^{2}}K(x_{i}-x_{*})\,g_{N}^{m}(x_{[m]\setminus\\{j\\}},x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}$
$\displaystyle-\sum_{i,j=1}^{m}\Big{(}K(x_{i}-x_{j})-K\ast\mu_{\beta}(x_{i})\Big{)}\cdot\nabla_{i;\beta}g_{N}^{m},$
$\displaystyle iS_{N,\beta}^{m,-}g_{N}^{m-1}$ $\displaystyle:=$
$\displaystyle-\sum_{2\leq i,j\leq m}^{\neq}\beta
Rx_{i}\cdot\int_{\mathbb{R}^{2}}K(x_{i}-x_{*})\,g_{N}^{m-1}(x_{[m]\setminus\\{i,j\\}},x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}$
$\displaystyle-\sum_{i=1}^{m}\sum_{2\leq j\leq m\atop i\neq
j}\Big{(}K(x_{i}-x_{j})-(K\ast\mu_{\beta})(x_{i})\Big{)}\cdot\nabla_{i;\beta}g_{N}^{m-1}(x_{[m]\setminus\\{j\\}})$
$\displaystyle+\sum_{i=2}^{m}\sum_{j=1}^{m}\beta Rx_{i}\cdot
K(x_{i}-x_{j})\,g_{N}^{m-1}(x_{[m]\setminus\\{i\\}}),$ $\displaystyle
iS_{N,\beta}^{m,=}g_{N}^{m-2}$ $\displaystyle:=$ $\displaystyle 0,$

with the short-hand notation $\nabla_{i;\beta}=\nabla_{i}-\beta Rx_{i}$.

###### Proof.

As $V,W,f^{\circ}$ are radial, we first note that the initial data (1.11)
satisfies

$f_{N}^{\circ}(Ox_{1},\ldots,Ox_{N})\,=\,f_{N}^{\circ}(x_{1},\ldots,x_{N}),\qquad\text{for
all $O\in O(2)$,}$

which remains true over time by the Liouville equation. In particular, via
(2.6), we deduce

$g_{N}^{m}(Ox_{1},\ldots,Ox_{m})\,=\,g_{N}^{m}(x_{1},\ldots,x_{m}),\qquad\text{for
all $1\leq m\leq N$ and $O\in O(2)$,}$

which implies the differential identity

$\sum_{j=1}^{m}x_{j}^{\bot}\cdot\nabla_{j}g_{N}^{m}\,=\,0.$

In the Gaussian setting (4.1), as we have $F+K\ast\mu_{\beta}=-Rx^{\bot}$, we
deduce

$\sum_{j=1}^{m}(K\ast\mu_{\beta}+F)(x_{j})\cdot\nabla_{j}g_{N}^{m}\,=\,0,$

which yields the different simplifications in the definition of the relevant
operators compared to Lemma 2.1. ∎

The above shows that the linearized mean-field operators
$\\{L_{N}^{m}\\}_{1\leq m\leq N}$ can now be written as Kronecker sums

$L_{N,\beta}^{m}\,=\,\tfrac{N-m}{N}\sum_{j=2}^{m}(\operatorname{Id}_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})})^{\otimes
j-1}\otimes
T_{\beta}\otimes(\operatorname{Id}_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})})^{\otimes(m-j)},$
(4.2)

involving the following single-particle operator on
$\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$,

$(iT_{\beta}h)(x)\,:=\,-\beta
Rx\cdot\int_{\mathbb{R}^{2}}K(x-x_{*})\,h(x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}.$
(4.3)

The following result states that this operator is compact, and identifies its
kernel.

######  Lemma 4.2.

Consider the Gaussian setting (4.1) and assume that $W$ does not vanish
identically. Then, for all $\beta>0$, the above-defined operator $T_{\beta}$
is compact and self-adjoint on $\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$.
Moreover, its kernel coincides with the set of radial functions.

###### Proof.

We split the proof into three steps.

Step 1\. Proof that $T_{\beta}$ is compact.
Given a weakly converging sequence $h_{n}\rightharpoonup h$ in
$\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$, we find $T_{\beta}h_{n}\to
T_{\beta}h$ a.e. By the Cauchy–Schwarz inequality, the definition (4.3) of
$T_{\beta}$ can be bounded by

$|T_{\beta}h_{n}(x)|\,\leq\,|\beta
Rx|\,\|K\|_{\operatorname{L}^{\infty}(\mathbb{R}^{2})}\big{(}\textstyle\sup_{n}\|h_{n}\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}\big{)}\leavevmode\nobreak\
\,\in\,\leavevmode\nobreak\ \operatorname{L}^{2}_{\beta}(\mathbb{R}^{2}),$

so that the dominated convergence theorem entails $T_{\beta}h_{n}\to
T_{\beta}h$ strongly in $\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$. This
proves that $T_{\beta}$ is compact.

Step 2\. Proof that $T_{\beta}$ is self-adjoint.
By Step 1, in order to show that $T_{\beta}$ is self-adjoint on
$\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$, it remains to check that it is
symmetric. Recalling that $x\cdot K(x)=0$, we can write

$\displaystyle\langle
h^{\prime},T_{\beta}h\rangle_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}$
$\displaystyle=$ $\displaystyle i\beta
R\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}}x\cdot
K(x-x_{*})\,\overline{h^{\prime}(x)}\,h(x_{*})\,\mu_{\beta}(x)\,\mu_{\beta}(x_{*})\,dxdx_{*}$
$\displaystyle=$ $\displaystyle\tfrac{i}{2}\beta
R\iint_{\mathbb{R}^{2}\times\mathbb{R}^{2}}(x+x_{*})\cdot
K(x-x_{*})\,\overline{h^{\prime}(x)}\,h(x_{*})\,\mu_{\beta}(x)\,\mu_{\beta}(x_{*})\,dxdx_{*}.$

As $K$ satisfies $K(-x)=-K(x)$, this proves that $T_{\beta}$ is symmetric.

Step 3\. Identification of $\ker(T_{\beta})$.
Recalling again that $x\cdot K(x)=0$, as well as $\nabla\mu_{\beta}=-\beta
Rx\mu_{\beta}$ and $K=-\nabla^{\bot}W$, and integrating by parts, we can
rewrite the definition (4.3) of $T_{\beta}$ as

$\displaystyle(iT_{\beta}h)(x)$ $\displaystyle=$ $\displaystyle-\beta
R\int_{\mathbb{R}^{2}}K(x-x_{*})\cdot
x_{*}\,h(x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}$ $\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}^{2}}K(x-x_{*})\cdot(h\sqrt{\mu_{\beta}})(x_{*})\nabla\sqrt{\mu_{\beta}}(x_{*})\,dx_{*}$
$\displaystyle=$
$\displaystyle-2\int_{\mathbb{R}^{2}}W(x-x_{*})\nabla^{\bot}(h\sqrt{\mu_{\beta}})(x_{*})\cdot\nabla\sqrt{\mu_{\beta}}(x_{*})\,dx_{*}.$

If $h\in\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$ belongs to
$\ker(T_{\beta})$, we deduce from this identity that

$H:=F\big{\\{}\nabla^{\bot}(h\sqrt{\mu_{\beta}})\cdot\nabla\sqrt{\mu_{\beta}}\big{\\}}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e. on the support of
$F\\{W\\}$,}$

where we use here the notation $F\\{g\\}$ for the Fourier transform of a
function $g$ on $\mathbb{R}^{2}$. As $\mu_{\beta}$ is Gaussian and as
$h\sqrt{\mu_{\beta}}\in\operatorname{L}^{2}(\mathbb{R}^{d})$, we note that $H$
is real analytic. Given that it vanishes on the support of $F\\{W\\}$, and
noting that the latter contains an open set as $W$ is integrable and does not
vanish identically, we deduce that $H$ vanishes everywhere on
$\mathbb{R}^{2}$. Inverting the Fourier transform, this means

$\nabla^{\bot}(h\sqrt{\mu_{\beta}})\cdot\nabla\sqrt{\mu_{\beta}}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{a.e.},$

or equivalently $x^{\bot}\cdot\nabla h(x)=0$ a.e., which precisely means that
$h$ is radial. ∎

Next, we establish the following useful preliminary estimates on BBGKY
operators in the weighted negative Sobolev spaces defined in (2.18). We
emphasize that item (ii) establishes that linearized mean-field evolutions are
almost uniformly bounded in negative Sobolev spaces in sharp contrast with the
non-Gaussian case of Lemma 3.5(ii).

######  Lemma 4.3.

  1. (i)

_Weak bounds on BBGKY operators:_
For all $1\leq m\leq N$, $s\geq 0$, and $h^{m+r}\in
C^{\infty}_{c}((\mathbb{R}^{2})^{m+r})$ for $r\in\\{-1,0,1\\}$, we have

$\displaystyle\|L^{m}_{N,\beta}h^{m}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m})}$
$\displaystyle\lesssim_{m,s}$
$\displaystyle\|h^{m}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m})},$
$\displaystyle\|S^{m,\circ}_{N,\beta}h^{m}\|_{H^{-s-1}_{\beta}((\mathbb{R}^{2})^{m})}$
$\displaystyle\lesssim_{m,s}$
$\displaystyle\|h^{m}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m})},$
$\displaystyle\|S^{m,+}_{N,\beta}h^{m+1}\|_{H^{-s-1}_{\beta}((\mathbb{R}^{2})^{m})}$
$\displaystyle\lesssim_{m,s}$
$\displaystyle\|h^{m+1}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m+1})},$
$\displaystyle\|S^{m,-}_{N,\beta}h^{m-1}\|_{H^{-s-1}_{\beta}((\mathbb{R}^{2})^{m})}$
$\displaystyle\lesssim_{m,s}$
$\displaystyle\|h^{m-1}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m-1})}.$

  2. (ii)

_Weak bounds on linearized mean-field evolutions:_
For all $1\leq m\leq N$, $s\geq 0$, $\delta>0$, and $h^{m}\in
C^{\infty}_{c}((\mathbb{R}^{2})^{m})$, we have

$\|e^{iL_{N,\beta}^{m}t}h^{m}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m})}\,\lesssim_{m,s,\delta}\,\langle
t\rangle^{\delta}\|h^{m}\|_{H^{-s}_{\beta}((\mathbb{R}^{2})^{m})}.$

###### Proof.

Item (i) is obtained by duality, as in the proof of Lemma 3.5, and we skip the
detail. We turn to the proof of item (ii). By duality, recalling that the
operator $L_{N,\beta}^{m}$ in (4.2) is self-adjoint by Lemma 4.2, it suffices
to prove for all $s\geq 0$, $\delta>0$, and $h^{m}\in
C^{\infty}_{c}((\mathbb{R}^{2})^{m})$,

$\|e^{iL_{N,\beta}^{m}t}h^{m}\|_{H_{\beta}^{s}((\mathbb{R}^{2})^{m})}\,\lesssim_{m,s,\delta}\,\langle
t\rangle^{\delta}\|h^{m}\|_{H^{s}_{\beta}((\mathbb{R}^{2})^{m})}.$

As by definition $L_{N,\beta}^{m}$ is (a multiple of) the Kronecker sum of
$T_{\beta}$ over the last $m-1$ directions on
$\operatorname{L}^{2}_{\beta}((\mathbb{R}^{2})^{m})$, cf. (4.2), it suffices
to show, for all $s\geq 0$, $\delta>0$, and $h\in
C^{\infty}_{c}(\mathbb{R}^{2})$,

$\|e^{iT_{\beta}t}h\|_{H_{\beta}^{s}(\mathbb{R}^{2})}\,\lesssim_{s,\delta}\,\langle
t\rangle^{\delta}\|h\|_{H^{s}_{\beta}(\mathbb{R}^{2})}.$ (4.4)

For that purpose, we first note that by definition (4.3) the operator
$T_{\beta}$ satisfies the following regularizing property: for all $r\geq 0$,

$\|T_{\beta}h\|_{H^{r}_{\beta}(\mathbb{R}^{2})}\,\lesssim_{r}\,\|h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}.$

Writing
$e^{iT_{\beta}t}h-h=\int_{0}^{t}iT_{\beta}(e^{iT_{\beta}t^{\prime}}h)\,dt^{\prime}$
and using this regularizing property, we then get

$\|e^{iT_{\beta}t}h-h\|_{H^{r}_{\beta}(\mathbb{R}^{2})}\,\leq\,\int_{0}^{t}\|T_{\beta}(e^{iT_{\beta}t^{\prime}}h)\|_{H^{r}_{\beta}(\mathbb{R}^{2})}\,dt^{\prime}\,\lesssim_{r}\,\int_{0}^{t}\|e^{iT_{\beta}t^{\prime}}h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}\,dt^{\prime}.$

Now we combine this with the interpolation inequality of Lemma 2.3(ii): for
all $r\geq s\geq 0$, we get

$\displaystyle\|e^{iT_{\beta}t}h-h\|_{H_{\beta}^{s}(\mathbb{R}^{2})}$
$\displaystyle\lesssim_{s,r}$
$\displaystyle\|e^{iT_{\beta}t}h-h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}^{1-\frac{s}{r}}\|e^{iT_{\beta}t}h-h\|_{H_{\beta}^{r}(\mathbb{R}^{2})}^{\frac{s}{r}}$
$\displaystyle\lesssim_{r}$
$\displaystyle\|e^{iT_{\beta}t}h-h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}^{1-\frac{s}{r}}\Big{(}\int_{0}^{t}\|e^{iT_{\beta}t^{\prime}}h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})}\,dt^{\prime}\Big{)}^{\frac{s}{r}}.$

By the triangle inequality and the self-adjointness of $T_{\beta}$ on
$\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$, cf. Lemma 4.2, we conclude
that

$\|e^{iT_{\beta}t}h\|_{H_{\beta}^{s}(\mathbb{R}^{2})}\,\lesssim_{s,r}\,\|h\|_{H_{\beta}^{s}(\mathbb{R}^{2})}+\langle
t\rangle^{\frac{s}{r}}\|h\|_{\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})},$

and the claim (4.4) follows by choosing $r\geq\frac{s}{\delta}$. ∎

###  4.2. Proof of Theorem 1.2

We start by using the cumulant estimates of Lemma 2.2 to truncate the BBGKY
hierarchy and get a closed description of the tagged particle density. We
manage to get arbitrarily close to the critical timescale $t=O(N^{1/2})$ by
considering multiple iterations of the hierarchy. This iteration procedure
works precisely thanks to the almost uniform bound on linearized mean-field
evolutions that we proved above, cf. Lemma 4.3(ii), in contrast with the non-
Gaussian case where we could not avoid a strong time restriction, cf. Lemma
3.6.

######  Lemma 4.4.

Consider the Gaussian setting (4.1). For all $\sigma<\frac{1}{2}$, there exist
$\delta,s>0$ (with $\delta\simeq\tfrac{1}{2}-\sigma$ and $s\simeq\delta^{-2}$)
such that

$\sup_{0\leq t\leq
N^{\sigma}}\Big{\|}N\partial_{t}^{2}g_{N}^{1;t}-iS_{N,\beta}^{1,+}e^{-it(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})\Big{\|}_{H^{-s}_{\beta}(\mathbb{R}^{2})}\,\lesssim_{\sigma}\,N^{-\delta}.$

###### Proof.

Recall that in the Gaussian setting the BBGKY hierarchy (2.7) holds with
$L_{N,\beta}^{1}=0$, with $S_{N,\beta}^{m,=}=0$ for all $m\geq 1$, and with
$L_{N,\beta}^{2}=\frac{N-2}{N}(\operatorname{Id}\otimes T_{\beta})$, where
$T_{\beta}$ is defined in (4.3), cf. Lemma 4.1. For the tagged particle
density, we then find

$\partial_{t}g_{N}^{1}\,=\,iS_{N,\beta}^{1,+}g_{N}^{2}+\tfrac{1}{N}iS_{N,\beta}^{1,\circ}g_{N}^{1},$
(4.5)

and thus, taking another time derivative and iterating,

$N\partial_{t}^{2}g_{N}^{1}-iS_{N,\beta}^{1,+}N\partial_{t}g_{N}^{2}\,=\,-S_{N,\beta}^{1,\circ}S_{N,\beta}^{1,+}g_{N}^{2}-\tfrac{1}{N}S_{N,\beta}^{1,\circ}S_{N,\beta}^{1,\circ}g_{N}^{1}.$
(4.6)

Next, we further appeal to the BBGKY hierarchy to express the correlation
function $N\partial_{t}g_{N}^{2}$ in the left-hand side. For that purpose, we
shall use the following version of the Duhamel formula: if $g,h$ satisfy an
equation of the form

$\partial_{t}g+iLg=h,\qquad g|_{t=0}=0,$

for a self-adjoint operator $L$, then we can write the solution as
$g^{t}=\int_{0}^{t}e^{-iL(t-s)}h^{s}\,ds$, from which we can deduce for
instance, taking a time derivative and integrating by parts,

$\displaystyle\partial_{t}g^{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\big{(}\delta(s-t)-e^{-iL(t-s)}iL\big{)}h^{s}\,ds$
(4.7) $\displaystyle=$ $\displaystyle
e^{-iLt}h^{\circ}+\int_{0}^{t}e^{-iL(t-s)}\partial_{s}h^{s}\,ds.$

Now, as $L^{2}_{N,\beta}=\frac{N-2}{N}(\operatorname{Id}\otimes T_{\beta})$,
the BBGKY equation (2.7) for the correlation function $g_{N}^{2}$ takes the
form

$\partial_{t}g^{2}_{N}+i(\operatorname{Id}\otimes
T_{\beta})g^{2}_{N}=iS^{2,+}_{N,\beta}g_{N}^{3}+\tfrac{1}{N}\Big{(}iS^{2,\circ}_{N,\beta}g_{N}^{2}+iS_{N,\beta}^{2,-}g^{1}_{N}+2i(\operatorname{Id}\otimes
T_{\beta})g_{N}^{2}\Big{)},$

hence, using $g_{N}^{1}|_{t=0}=\tfrac{1}{\mu_{\beta}}f^{\circ}$ and
$g_{N}^{m}|_{t=0}=0$ for all $m\geq 2$, the above Duhamel formula (4.7) yields

$N\partial_{t}g_{N}^{2;t}\,=\,e^{-it(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})+\int_{0}^{t}e^{-i(t-s)(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,+}N\partial_{s}g_{N}^{3;s}\,ds\\\
+\int_{0}^{t}e^{-i(t-s)(\operatorname{Id}\otimes
T_{\beta})}\Big{(}iS_{N,\beta}^{2,\circ}\partial_{s}g_{N}^{2;s}+iS_{N,\beta}^{2,-}\partial_{s}g_{N}^{1;s}+2i(\operatorname{Id}\otimes
T_{\beta})\partial_{s}g_{N}^{2;s}\Big{)}ds.$

Further replacing $\partial_{s}g_{N}^{1}$ by (4.5) in the right-hand side, and
integrating by parts as in (4.7) to remove the time derivative on $g_{N}^{2}$,
we get

$N\partial_{t}g_{N}^{2;t}\,=\,e^{-it(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})+\int_{0}^{t}e^{-i(t-s)(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,+}N\partial_{s}g_{N}^{3;s}\,ds\\\
+\int_{0}^{t}e^{-i(t-s)(\operatorname{Id}\otimes
T_{\beta})}\Big{(}iS_{N,\beta}^{2,-}iS_{N,\beta}^{1,+}g_{N}^{2;s}+\tfrac{1}{N}iS_{N,\beta}^{2,-}iS_{N,\beta}^{1,\circ}g_{N}^{1;s}\Big{)}ds\\\
+\int_{0}^{t}\Big{(}\delta(s-t)-e^{-i(t-s)(\operatorname{Id}\otimes
T_{\beta})}i(\operatorname{Id}\otimes
T_{\beta})\Big{)}\big{(}iS_{N,\beta}^{2,\circ}+2i(\operatorname{Id}\otimes
T_{\beta})\big{)}g_{N}^{2;s}\,ds.$

Appealing to Lemmas 2.2 and 4.3 to estimate the different right-hand side
terms, we deduce for all $s\geq 1$ and $T,\delta>0$,

$\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{2;t}\|_{H^{-s-1}_{\beta}(\mathbb{R}^{2})}\,\lesssim_{s,\delta}\,1+N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta}+\langle T\rangle^{1+\delta}\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{3;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})},$ (4.8)

and similarly, further combining with (4.6),

$\sup_{0\leq t\leq
T}\Big{\|}N\partial_{t}^{2}g_{N}^{1;t}-iS_{N,\beta}^{1,+}e^{-it(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})\Big{\|}_{H^{-s-2}_{\beta}(\mathbb{R}^{2})}\\\
\,\lesssim_{s,\delta}\,N^{-\frac{1}{2}}\langle T\rangle^{1+\delta}+\langle
T\rangle^{1+\delta}\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{3;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}.$ (4.9)

It remains to estimate the $3$-particle correlation function
$\partial_{t}g_{N}^{3}$. In order to get an estimate valid up to the critical
timescale $t=O(N^{1/2})$, some special care is needed and we shall argue by
iterations on the whole hierarchy. For any $m\geq 3$, applying the Duhamel
formula (4.7) to the BBGKY equation (2.7) for $g_{N}^{m}$, we get

$N\partial_{t}g_{N}^{m;t}=\int_{0}^{t}e^{-iL_{N}^{n}(t-s)}iS_{N,\beta}^{m,+}N\partial_{s}g_{N}^{m+1;s}\,ds\\\
+\int_{0}^{t}e^{-iL_{N}^{n}(t-s)}\Big{(}iS_{N,\beta}^{m,\circ}\partial_{s}g_{N}^{m;s}+iS_{N,\beta}^{m,-}\partial_{s}g_{N}^{m-1;s}\Big{)}\,ds,$

and therefore, by Lemmas 2.2 and 4.3, we get for any $s\geq 0$ and
$T,\delta>0$,

$\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{m;t}\|_{H^{-s-1}_{\beta}(\mathbb{R}^{2})}\,\lesssim_{s,\delta}\,N^{-1}\langle
T\rangle^{1+\delta}\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{m;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}\\\ +\langle
T\rangle^{1+\delta}\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{m+1;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}+N^{-1}\langle
T\rangle^{1+\delta}\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{m-1;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}.$

Denoting

$A_{s}^{m}(T)\,:=\,\sup_{0\leq t\leq
T}\|N^{\frac{m-1}{2}}\partial_{t}g_{N}^{m;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})},$
(4.10)

the above means for all $m\geq 3$, $s\geq 0$, and $T,\delta>0$,

$A_{s+1}^{m}(T)\,\lesssim_{s,\delta}\,N^{-1}\langle
T\rangle^{1+\delta}A_{s}^{m}(T)+N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta}\big{(}A_{s}^{m+1}(T)+A_{s}^{m-1}(T)\big{)}.$ (4.11)

Further note that the BBGKY equation (2.7) for $\partial_{t}g_{N}^{m}$,
combined with Lemmas 2.2 and 4.3, yields the following a priori estimates, for
all $m\geq 2$, $s\geq 1$, and $T>0$,

$\displaystyle A_{s}^{m}(T)$ $\displaystyle\leq$ $\displaystyle\sup_{0\leq
t\leq
T}N^{\frac{m-1}{2}}\Big{(}\|iL_{N,\beta}^{m}g^{m;t}_{N}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}+\|iS_{N,\beta}^{m,+}g_{N}^{m+1;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}$
(4.12) $\displaystyle\hskip
56.9055pt+\tfrac{1}{N}\|iS^{m,\circ}_{N,\beta}g_{N}^{m;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}+\tfrac{1}{N}\|iS_{N,\beta}^{m,-}g_{N}^{m-1;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}\Big{)}$
$\displaystyle\lesssim$ $\displaystyle 1.$

Now combining (4.11) and (4.12), we deduce by direct iteration, for all $s\geq
1$, $k\geq 0$, and $T,\delta>0$, provided that $N^{-1/2}\langle
T\rangle^{1+\delta}\leq 1$,

$A_{s+k}^{3}(T)\,\lesssim_{s,k,\delta}\,N^{-1}\langle
T\rangle^{1+\delta}+(N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta})^{k}+N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta}A_{s}^{2}(T).$

Noting that (4.8) means, with the notation (4.10), for $s\geq 1$,

$A_{s+1}^{2}(T)\,\lesssim_{s,\delta}\,N^{-\frac{1}{2}}+N^{-1}\langle
T\rangle^{1+\delta}+N^{-\frac{1}{2}}\langle T\rangle^{1+\delta}A_{s}^{3}(T),$
(4.13)

and inserting this into the above, we deduce for all $s\geq 1$, $k\geq 0$, and
$T,\delta>0$, provided that $N^{-1/2}\langle T\rangle^{1+\delta}\leq 1$,

$A_{s+k+1}^{3}(T)\,\lesssim_{s,k,\delta}\,N^{-1}\langle
T\rangle^{1+\delta}+(N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta})^{k}+(N^{-\frac{1}{2}}\langle
T\rangle^{1+\delta})^{2}A_{s}^{3}(T).$

Now further iterating this inequality for $A^{3}$, together with the a priori
estimate (4.12), we deduce for all $k\geq 1$, $s\geq k^{2}$, and $T>0$,
provided that $N^{-1/2}\langle T\rangle^{1+\delta}\leq 1$,

$\sup_{0\leq t\leq
T}\|N\partial_{t}g_{N}^{3;t}\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}\,=\,A_{s}^{3}(T)\,\lesssim_{s,k,\delta}\,N^{-1}\langle
T\rangle^{1+\delta}+(N^{-\frac{1}{2}}\langle T\rangle^{1+\delta})^{k}.$

Combining this with (4.9) and optimizing in $k$ yield the conclusion. ∎

In view of the compact nature of the linearized mean-field operator
$T_{\beta}$, as formulated in Lemma 4.2, we may now pass to the limit in the
above closed description of the tagged particle density and conclude the proof
of Theorem 1.2.

###### Proof of Theorem 1.2.

By definition of BBGKY operators in Lemma 2.1, we have that

$iS_{N,\beta}^{1,+}e^{-it(\operatorname{Id}\otimes
T_{\beta})}iS_{N,\beta}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})\\\
\,=\,-\tfrac{N-1}{N}(\nabla-\beta
Rx)\cdot\int_{\mathbb{R}^{2}}K(x-x_{*})\,(e^{-it(\operatorname{Id}\otimes
T_{\beta})}H)(x,x_{*})\,\mu_{\beta}(x_{*})\,dx_{*},$ (4.14)

where $H\in\operatorname{L}^{2}_{\beta}((\mathbb{R}^{2})^{2})$ is given by

$\displaystyle H(x,x_{*})$ $\displaystyle=$
$\displaystyle-\Big{(}K(x-x_{*})-(K\ast\mu_{\beta})(x)\Big{)}\cdot(\nabla-\beta
Rx)(\tfrac{1}{\mu_{\beta}}f^{\circ})(x)$ $\displaystyle=$
$\displaystyle-K(x-x_{*})\cdot(\tfrac{1}{\mu_{\beta}}\nabla f^{\circ})(x).$

Since $T_{\beta}$ is compact, cf. Lemma 4.2, we find as in (1.5), for all
$h\in\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$ and $\sigma>0$,

$e^{-i\tau N^{\sigma}T_{\beta}}h\,\xrightarrow{N\uparrow\infty}\,\pi_{0}h,$

in the weak-* sense of
$\operatorname{L}^{\infty}(\mathbb{R}^{+};\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2}))$,
where $\pi_{0}$ stands for the orthogonal projection of
$\operatorname{L}^{2}_{\beta}(\mathbb{R}^{2})$ onto the kernel of $T_{\beta}$,
that is, by Lemma 4.2, onto the subspace of radial functions. Using this in
(4.14) with $t=N^{\sigma}\tau$, we deduce the following weak-* convergence in
$\operatorname{L}^{\infty}(\mathbb{R}^{+};H^{-1}_{\beta}(\mathbb{R}^{2}))$,

$iS_{N}^{1,+}e^{-i\tau N^{\sigma}(\operatorname{Id}\otimes
T_{\beta})}iS_{N}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})\\\
\,\xrightarrow{N\uparrow\infty}\,-(\nabla-\beta
Rx)\cdot\int_{\mathbb{R}^{2}}K(x-x_{*})\,((\operatorname{Id}\otimes\pi_{0})H)(x,x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}.$

By definition of $H$, the weak-* limit can be rewritten as

$iS_{N}^{1,+}e^{-i\tau N^{\sigma}(\operatorname{Id}\otimes
T_{\beta})}iS_{N}^{2,-}(\tfrac{1}{\mu_{\beta}}f^{\circ})\\\
\,\xrightarrow{N\uparrow\infty}\,\tfrac{1}{\mu_{\beta}}{\operatorname{div}}(A\nabla
f^{\circ}),\qquad\text{in
$\operatorname{L}^{\infty}(\mathbb{R}^{+};H^{-1}_{\beta}(\mathbb{R}^{2}))$},$

where the coefficient field
$A\in\operatorname{L}^{\infty}(\mathbb{R}^{2})^{2\times 2}$ is given by

$A(x)\,:=\,\int_{\mathbb{R}^{2}}K(x-x_{*})\otimes\Big{(}(\operatorname{Id}\otimes\pi_{0})\big{[}(x_{1},x_{2})\mapsto
K(x_{1}-x_{2})\big{]}\Big{)}(x,x_{*})\,\mu_{\beta}(x_{*})\,dx_{*}.$

The projection $\pi_{0}$ onto the subspace of radial functions can be
explicitly computed and we recover formula (1.16) for $A$. Combining this
convergence with the estimate of Lemma 4.4, and using (2.19) in form of
$\|h\|_{H^{-s}(\mathbb{R}^{2})}\lesssim\|\frac{1}{\mu_{\beta}}h\|_{H^{-s}_{\beta}(\mathbb{R}^{2})}$,
the conclusion follows. ∎

##  5\. Special degenerate case: uniform equilibrium

This section is devoted to the special degenerate case of a tagged particle in
a uniformly distributed background on the torus $\mathbb{T}^{d}$, say in
arbitrary spatial dimension $d\geq 1$. More precisely, given a smooth force
kernel $K\in C^{\infty}(\mathbb{T}^{d})^{d}$ that satisfies the
incompressibility condition ${\operatorname{div}}(K)=0$ and the action-
reaction condition $K(-x)=-K(x)$, we now consider the associated point-vortex
dynamics

$\partial_{t}x_{i}=\tfrac{1}{N}\sum_{j=1}^{N}K(x_{i}-x_{j}),\qquad
x_{i}|_{t=0}=x_{i}^{\circ},\qquad 1\leq i\leq N.$

The Liouville equation for the $N$-point density
$f_{N}\in\mathcal{P}((\mathbb{T}^{d})^{N})$ then reads

$\partial_{t}f_{N}+\tfrac{1}{N}\sum_{i,j=1}^{N}K(x_{i}-x_{j})\cdot\nabla_{i}f_{N}=0,\qquad
f_{N}|_{t=0}=f_{N}^{\circ}.$ (5.1)

Consider a tagged particle (labeled ‘1’) in a uniform equilibrium background:
in other words, we assume that initially the $N$-point density
$f_{N}|_{t=0}=f_{N}^{\circ}$ takes the form

$f_{N}^{\circ}(x_{1},\ldots,x_{N})\,=\,f^{\circ}(x_{1}),$ (5.2)

for some $f^{\circ}\in\mathcal{P}\cap C^{\infty}(\mathbb{T}^{d})$. At later
times, the tagged particle density is given by the first marginal

$f_{N}^{1}(t,x_{1})\,:=\,\int_{(\mathbb{R}^{d})^{N-1}}f_{N}(t,x_{1},x_{2},\ldots,x_{N})\,dx_{2}\ldots
dx_{N}.$

As linearized mean-field operators actually vanish at uniform equilibrium,
Conjecture 1 leads us to expect a nontrivial slow conservative dynamics for
the tagged particle on the timescale $t=O(N^{1/2})$, displaying no
thermalization in the strict sense. In the spirit of (1.7), we start by
showing that the tagged particle density satisfies a linear wave equation to
leading order for relatively short times $t\ll N^{1/2}$ (that is, $\tau\ll
1$). This is analogous to Theorem 1.2 in the Gaussian degenerate setting. The
proof is displayed in Section 5.1.

######  Proposition 5.1.

The critically-rescaled tagged particle density
$\bar{f}_{N}^{1}(\tau):=f_{N}^{1}(N^{\frac{1}{2}}\tau)$ satisfies

$\|\partial_{\tau}^{2}\bar{f}_{N}^{1}(\tau)-{\operatorname{div}}(A\nabla
f^{\circ})\|_{\operatorname{L}^{2}(\mathbb{T}^{d})}\,\lesssim_{f^{\circ}}\,\tau^{2}+\tfrac{1}{N},$

with constant diffusion matrix $A:=\int_{\mathbb{T}^{d}}K\otimes K$.

For a description of the dynamics on the critical timescale $t=O(N^{1/2})$, we
need to consider the tagged particle density together with the whole family of
its correlation functions with respect to background particles. We first
recall some standard definitions. As in (2.1), we denote by
$\\{f_{N}^{m}\\}_{1\leq m\leq N}$ the marginals of $f_{N}$,

$f_{N}^{m}(t,x_{1},\ldots,x_{m})\,:=\,\int_{(\mathbb{T}^{d})^{N-m}}f_{N}(t,x_{1},\ldots,x_{m},x_{m+1},\ldots,x_{N})\,dx_{m+1}\ldots
dx_{N}.$ (5.3)

As background particles with labels $2,\ldots,N$ are exchangeable initially,
cf. (5.2), they remain so over time, hence $f_{N}^{m}$ is symmetric in its
last $m-1$ variables. The correlation functions $\\{g_{N}^{m}\\}_{1\leq m\leq
N}$ for the tagged particle with respect to the initially uniform background
are defined so as to satisfy the following cluster expansions,

$f_{N}^{m}(t,x_{1},\ldots,x_{m})\,=\,\sum_{n=1}^{m}\sum_{\sigma\in
P_{n-1}^{m-1}}g_{N}^{n}(t,x_{1},x_{\sigma}),\qquad 1\leq m\leq N.$ (5.4)

For all $m$, the correlation function $g_{N}^{m}$ is uniquely chosen to be
symmetric in its last $m-1$ variables and to satisfy
$\int_{\mathbb{T}^{d}}g_{N}^{m}(x_{1},\ldots,x_{m})\,dx_{j}=0$ for all $2\leq
j\leq m$. More explicitly, the above relations can be inverted and correlation
functions are given by

$g_{N}^{m}(t,x_{1},\ldots,x_{m}):=\sum_{n=1}^{m}(-1)^{m-n}\sum_{\sigma\in
P_{n-1}^{m-1}}f_{N}^{n}(t,x_{1},x_{\sigma}).$ (5.5)

In these terms, we can now state the following result. In accordance with
(1.6), the tagged particle density does not satisfy a closed equation on the
critical timescale $t=O(N^{1/2})$, but its dynamics takes the form of a
(formally) unitary evolution for an infinite hierarchy of coupled equations
describing limits of all rescaled correlation functions.

######  Theorem 5.2.

The critically-rescaled correlation functions

$\bar{g}_{N}^{m}(\tau):=N^{\frac{m-1}{2}}g_{N}^{m}(N^{\frac{1}{2}}\tau),\qquad
1\leq m\leq N,$ (5.6)

converge weakly-* in
$W^{1,\infty}(\mathbb{R}^{+};\operatorname{L}^{2}((\mathbb{T}^{d})^{m}))$ as
$N\uparrow\infty$,

$\bar{g}_{N}^{m}\overset{*}{\rightharpoonup}\bar{g}^{m},\qquad m\geq 1,$

where the limit $\\{\bar{g}^{m}\\}_{m}$ is the unique weak solution of the
limit hierarchy

$\left\\{\begin{array}[]{lll}\partial_{\tau}\bar{g}^{m}=iS^{m,+}\bar{g}^{m+1}+iS^{m,-}\bar{g}^{m-1}&:&m\geq
1,\\\ \bar{g}^{1}|_{\tau=0}=f^{\circ},\\\
\bar{g}^{m}|_{\tau=0}=0&:&m>1,\end{array}\right.$ (5.7)

such that

$\sup_{\tau\geq
0}\sum_{m=1}^{\infty}\tfrac{1}{(m-1)!}\|\partial_{\tau}^{k}\bar{g}^{m}\|^{2}_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}\,<\,\infty,\qquad\text{for
all $k\geq 0$},$ (5.8)

and where the operators $S^{m,\pm}$ are explicitly defined as follows,

$\displaystyle iS^{m,+}\bar{g}^{m+1}$ $\displaystyle:=$
$\displaystyle-\sum_{i=1}^{m}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}\bar{g}^{m+1}(x_{[m]},x_{*})\,dx_{*},$
$\displaystyle iS^{m,-}\bar{g}^{m-1}$ $\displaystyle:=$
$\displaystyle-\sum_{i=1}^{m}\sum_{j=2}^{m}K(x_{i}-x_{j})\cdot\nabla_{i}\bar{g}^{m-1}(x_{[m]\setminus\\{j\\}}).$

In addition, the limit hierarchy (5.7) has the following (formal) unitary
structure: defining $\mathcal{H}^{m}$ as the Hilbert space of functions
$h^{m}\in\operatorname{L}^{2}((\mathbb{T}^{d})^{m})$ that are symmetric in
their last $m-1$ entries, and endowing this space with the norm

$\|h^{m}\|_{\mathcal{H}^{m}}^{2}:=\tfrac{1}{(m-1)!}\|h^{m}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}^{2},$

the operators $S^{m,+}:\mathcal{H}^{m+1}\to\mathcal{H}^{m}$ and
$S^{m+1,-}:\mathcal{H}^{m}\to\mathcal{H}^{m+1}$ satisfy

$(S^{m,+})^{*}=S^{m+1,-},\qquad(S^{m+1,-})^{*}=S^{m,+}.$ (5.9)

Let us further describe the structure of the limit hierarchy (5.7)–(5.8) and
investigate its actual unitary structure. For that purpose, consider the
Hilbert space

$\mathcal{H}\,:=\,\overline{\textstyle\bigoplus_{m\geq 1}\mathcal{H}^{m}},$
(5.10)

that is, the Hilbert closure of the algebraic direct sum $\bigoplus_{m\geq
1}\mathcal{H}^{m}$ with respect to the norm

$\|h\|_{\mathcal{H}}^{2}\,:=\,\sum_{m=1}^{\infty}\|h^{m}\|_{\mathcal{H}^{m}}^{2}\,=\,\sum_{m=1}^{\infty}\tfrac{1}{(m-1)!}\|h^{m}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}^{2}.$

In other words, this means
$\mathcal{H}=\operatorname{L}^{2}(\mathbb{T}^{d})\otimes\mathcal{F}_{+}(\operatorname{L}^{2}(\mathbb{T}^{d}))$,
where the bosonic Fock space
$\mathcal{F}_{+}(\operatorname{L}^{2}(\mathbb{T}^{d}))$ is viewed as the state
space for background correlations. In this setting, we consider the operator
$S:=S^{+}+S^{-}$ on $\mathcal{H}$ given by

$(Sh)^{m}\,:=\,S^{m,+}h^{m+1}+S^{m,-}h^{m-1},\qquad m\geq 1,$ (5.11)

which is well-defined for all $h$ in the dense subset

$\textstyle\mathcal{C}\,:=\,\bigoplus_{m\geq
1}C^{\infty}_{b}((\mathbb{T}^{d})^{m})\,\subset\,\mathcal{H}.$ (5.12)

(Note that the direct sum is understood in the algebraic sense.) The symmetry
relations (5.9) precisely mean that this densely-defined operator $S$ is
symmetric on $\mathcal{C}$. In these terms, the fact that the limit
$\bar{g}=\\{\bar{g}^{m}\\}_{m}$ is a weak solution of the limit hierarchy
(5.7) and that it satisfies the a priori estimates (5.8) is equivalent to the
following: the limit $\bar{g}=\\{\bar{g}^{m}\\}_{m}$ belongs to
$C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ and satisfies

$\langle h,\partial_{\tau}\bar{g}\rangle_{\mathcal{H}}=-\langle
iSh,\bar{g}\rangle_{\mathcal{H}},\qquad\text{for all $h\in\mathcal{C}$,}$

with initial condition $\bar{g}|_{\tau=0}=\bar{g}^{\circ}\in\mathcal{C}$ given
by

$(\bar{g}^{\circ})^{m}\,:=\,\left\\{\begin{array}[]{ll}f^{\circ},&\text{for
$m=1$},\\\ 0,&\text{for $m>1$}.\end{array}\right.$ (5.13)

This actually means that $\bar{g}$ is a strong solution of

$\partial_{\tau}\bar{g}=iS^{*}\bar{g},\qquad\bar{g}|_{\tau=0}=\bar{g}^{\circ},$
(5.14)

in terms of the adjoint $S^{*}$ of $S$ on $\mathcal{H}$. The uniqueness of the
solution to this limit equation and the unitarity of the so-defined semigroup
would amount to the essential self-adjointness of the operator $S$ on
$\mathcal{H}$, or equivalently to the symmetry of its adjoint $S^{*}$.
However, we do not expect $S$ to be essentially self-adjoint: this operator
can indeed be viewed as a cubic expression in terms of canonical creation and
annihilation operators on the Fock space, and in dimension $1$ nontrivial
symmetric polynomials of order $3$ in creation and annihilation operators are
known to be non-self-adjoint [25, 23]. This expected lack of self-adjointness,
leading to a lack of unitarity, is related to statistical closure problems in
link with turbulence [19]. In fact, it is not even clear whether $S^{*}$
generates a semigroup.

Leaving these delicate issues aside, we can at least show that the limit
equation (5.14) is well-posed in some sense. This explains the uniqueness of
the limit in Theorem 5.2. The proof is postponed to Section 5.4. Note that the
proof of uniqueness only relies on the symmetry of $S$ on its core
$\mathcal{C}$ and on the observation that $S\mathcal{C}\subset\mathcal{C}$.

######  Proposition 5.3 (Well-posedness of limiting hierarchy).

For all $g^{\circ}\in\mathcal{C}$, there is a unique strong solution $g\in
C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ of the limit equation

$\partial_{\tau}g=iS^{*}g,\qquad g|_{\tau=0}=g^{\circ}.$ (5.15)

Moreover, it satisfies the following properties:

  1. (i)

_Contraction:_ for all $\tau\geq 0$ and $k\geq 0$,

$\|\partial_{\tau}^{k}g(\tau)\|_{\mathcal{H}}\,\leq\,\|S^{k}g^{\circ}\|_{\mathcal{H}}.$

  2. (ii)

_Approximate isometricity up to $O(\tau^{\infty})$:_ for all $k\geq 0$,

$-\tfrac{1}{k!}\tau^{k}\sum_{j=0}^{k}\binom{k}{j}\|S^{j}g^{\circ}\|_{\mathcal{H}}^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak\
\|g(\tau)\|_{\mathcal{H}}^{2}-\|g^{\circ}\|_{\mathcal{H}}^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0.$

Finally, we establish the following RAGE theorem, which further describes the
structure and the mixing behavior of the limit dynamics. The proof is
postponed to Section 5.5.

######  Proposition 5.4 (RAGE theorem).

Let $\\{\lambda_{k}\\}_{k}$ be the set of real eigenvalues of $S^{*}$ (if
any). There exists a family of positive contractions $\\{P_{k}\\}_{k}$ on
$\mathcal{H}$ such that for all $k$ the image $\operatorname{ran}(P_{k})$ is a
subset of the eigenspace of $S^{*}$ associated with $\lambda_{k}$, such that
the orthogonality condition
$\operatorname{ran}(P_{k})\bot\operatorname{ran}(P_{l})$ holds for
$|\lambda_{k}|\neq|\lambda_{l}|$, and such that the following RAGE theorem
holds: for all $g^{\circ}\in\mathcal{C}$, the unique strong solution $g$ of
the limit equation (5.15) as given by Proposition 5.3 can be decomposed as

$g(\tau)\,=\,\sum_{k}e^{i\tau\lambda_{k}}P_{k}g^{\circ}+R(\tau),$

where the remainder $R$ satisfies for all $h\in\mathcal{H}$,

$\lim_{T\uparrow\infty}\frac{1}{T}\int_{0}^{T}|\langle
h,R(\tau)\rangle_{\mathcal{H}}|^{2}d\tau\,=\,0.$

###  5.1. Proof of Proposition 5.1

Taking time derivatives in the Liouville equation (5.1), and using that
$\operatorname{L}^{2}$ norms are conserved, we find for all $k\geq 0$,

$\|\partial_{t}^{k}f_{N}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}\,=\,\|\partial_{t}^{k}f_{N}|_{t=0}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}.$
(5.16)

From the Liouville equation (5.1) and the choice (5.2) of initial data, we
find

$\partial_{t}^{k}f_{N}|_{t=0}\,=\,\bigg{[}-\tfrac{1}{N}\sum_{i,j=1}^{N}K(x_{i}-x_{j})\cdot\nabla_{i}\bigg{]}^{k}f^{\circ}(x_{1}).$
(5.17)

By integration by parts, the norm of this quantity can be written as

$\|\partial_{t}^{k}f_{N}|_{t=0}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}^{2}\,=\,(-1)^{k}N^{-2k}\sum_{i_{1},j_{1},\ldots,i_{2k},j_{2k}=1}^{N}\int_{(\mathbb{T}^{d})^{N}}f^{\circ}(x_{1})\,K_{\alpha_{1}}(x_{i_{1}}-x_{j_{1}})(\nabla_{i_{1}})_{\alpha_{1}}\\\
\ldots
K_{\alpha_{2k}}(x_{i_{2k}}-x_{j_{2k}})(\nabla_{i_{2k}})_{\alpha_{2k}}f^{\circ}(x_{1})\,dx_{1}\ldots
dx_{N}.$

Taking advantage of cancellations when computing the derivatives, and
recalling that ${\operatorname{div}}(K)=0$, we find that the only non-
vanishing contributions in this sum are those such that for all $1\leq l\leq
2k$ the index $i_{l}$ belongs both to $\\{1,j_{1},\ldots,j_{l-1}\\}$ and to
$\\{j_{l+1},\ldots,j_{2k},1\\}$. Moreover, recalling that
$\int_{\mathbb{T}^{d}}K=0$, we find that each value in
$\\{i_{1},j_{2},\ldots,i_{2k},j_{2k}\\}\setminus\\{1\\}$ must be taken by at
least two different indices. From these two restrictions in the above
summation, we can immediately conclude that

$\|\partial_{t}^{k}f_{N}|_{t=0}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}\,\lesssim_{k,f^{\circ}}\,N^{-\frac{k}{2}}.$

Combining this with (5.16), we deduce for all $k\geq 0$,

$\|\partial_{t}^{k}f_{N}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}\,\lesssim_{k,f^{\circ}}\,N^{-\frac{k}{2}}.$
(5.18)

A second-order Taylor expansion then yields in particular,

${\big{\|}\partial_{\tau}^{2}(f_{N}(N^{\frac{1}{2}}\tau))-N(\partial_{t}^{2}f_{N}|_{t=0})-N^{\frac{3}{2}}\tau(\partial_{t}^{3}f_{N}|_{t=0})\big{\|}_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}}\\\
\,=\,\Big{\|}N\int_{0}^{N^{\frac{1}{2}}\tau}(N^{\frac{1}{2}}\tau-s)\,(\partial_{t}^{4}f_{N})(s)\,ds\Big{\|}_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}\,\lesssim_{f^{\circ}}\,\tau^{2}.$

Averaging over variables $x_{2},\ldots,x_{N}$, this entails

$\big{\|}\partial_{\tau}^{2}(f_{N}^{1}(N^{\frac{1}{2}}\tau))-N(\partial_{t}^{2}f_{N}^{1}|_{t=0})-N^{\frac{3}{2}}\tau(\partial_{t}^{3}f_{N}^{1}|_{t=0})\big{\|}_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}\,\lesssim_{f^{\circ}}\,\tau^{2}.$
(5.19)

Starting from (5.17), direct computations yield

$\displaystyle\partial_{t}^{2}f_{N}^{1}|_{t=0}$ $\displaystyle=$
$\displaystyle\int_{(\mathbb{T}^{d})^{N-1}}\bigg{[}-\tfrac{1}{N}\sum_{i,j=1}^{N}K(x_{i}-x_{j})\cdot\nabla_{i}\bigg{]}^{2}f^{\circ}(x_{1})\,dx_{2}\ldots
dx_{N}$ $\displaystyle=$
$\displaystyle\tfrac{1}{N^{2}}\sum_{i,j,k=1}^{N}\int_{(\mathbb{T}^{d})^{N-1}}K(x_{i}-x_{j})\cdot\nabla_{i}K(x_{1}-x_{k})\cdot\nabla_{1}f^{\circ}(x_{1})\,dx_{2}\ldots
dx_{N}$ $\displaystyle=$
$\displaystyle\tfrac{N-1}{N^{2}}\Big{(}\int_{\mathbb{T}^{d}}K\otimes
K\Big{)}:\nabla^{2}f^{\circ},$

and

$\partial_{t}^{3}f_{N}^{1}|_{t=0}\,=\,0,$

which concludes the proof of Proposition 5.1.∎

######  Remark 5.5.

The above proof can be pursued to capture higher-order corrections in a
similar perturbative way up to $O(\tau^{\infty}+\frac{1}{N})$. For instance,
the next-order Taylor expansion yields, instead of (5.19),

$\Big{\|}\partial_{\tau}^{2}(f_{N}^{1}(N^{\frac{1}{2}}\tau))-N(\partial_{t}^{2}f_{N}|_{t=0})-\tau
N^{\frac{3}{2}}(\partial_{t}^{3}f_{N}^{1}|_{t=0})\\\
-\tfrac{1}{2}\tau^{2}N^{2}(\partial_{t}^{4}f_{N}|_{t=0})-\tfrac{1}{6}\tau^{3}N^{\frac{5}{2}}(\partial_{t}^{5}f_{N}|_{t=0})\Big{\|}_{\operatorname{L}^{2}(\mathbb{T}^{d})}\,\lesssim\,\tau^{4}.$

Further computing $\partial_{t}^{4}f_{N}|_{t=0}$, starting from (5.17), and
noting that $\partial_{t}^{5}f_{N}|_{t=0}=0$, we are led to the following
next-order version of Proposition 5.1,

$\Big{\|}\partial_{\tau}^{2}(f_{N}^{1}(N^{\frac{1}{2}}\tau))-{\operatorname{div}}(A\nabla
f^{\circ})-\tfrac{3}{2}\tau^{2}({\operatorname{div}}A\nabla)^{2}f^{\circ}-\tfrac{1}{2}\tau^{2}{\operatorname{div}}(B\nabla
f^{\circ})\Big{\|}_{\operatorname{L}^{2}(\mathbb{T}^{d})}\,\lesssim\,\tau^{4}+\tfrac{1}{N},$

where the matrix $B$ is explicitly given by

$\displaystyle B_{\alpha\beta}$
$\displaystyle=\Big{(}\int_{\mathbb{T}^{d}}(\nabla_{\gamma}K_{\delta})(\nabla_{\delta}K_{\gamma})(K_{\alpha}\ast
K_{\beta})\Big{)}-2\Big{(}\int_{\mathbb{T}^{d}}(\nabla_{\delta}K_{\alpha})(\nabla_{\gamma}K_{\beta})\Big{)}\Big{(}\int_{\mathbb{T}^{d}}K_{\delta}K_{\gamma}\Big{)}$
$\displaystyle\qquad-2\Big{(}\int_{\mathbb{T}^{d}}(\nabla_{\delta}K_{\alpha})(\nabla_{\gamma}K_{\beta})(K_{\delta}\ast
K_{\gamma})\Big{)}-2\Big{(}\int_{\mathbb{T}^{d}}(\nabla_{\delta}K_{\alpha})(\nabla_{\gamma}K_{\delta})(K_{\gamma}\ast
K_{\beta})\Big{)},$

where we implicitly sum over indices $\delta,\gamma$.

###  5.2. Rigorous BBGKY analysis

We start by reformulating the BBGKY hierarchy of Lemma 2.1 in the present
uniform setting, noting that it gets drastically simplified and that in
particular linearized mean-field operators vanish.

######  Lemma 5.6 (BBGKY hierarchy for cumulants).

For all $1\leq m\leq N$,

$\partial_{t}g_{N}^{m}\,=\,iS_{N}^{m,+}g_{N}^{m+1}+\frac{1}{N}\Big{(}iS_{N}^{m,\circ}g_{N}^{m}+iS_{N}^{m,-}g_{N}^{m-1}\Big{)},$
(5.20)

where we have set for notational convenience $g^{r}_{N}=0$ for $r<1$ or $r>N$,
and where we have defined the operators

$\displaystyle iS_{N}^{m,+}g_{N}^{m+1}$ $\displaystyle:=$
$\displaystyle-\tfrac{N-m}{N}\sum_{i=1}^{m}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}g_{N}^{m+1}(x_{[m]},x_{*})\,dx_{*},$
$\displaystyle iS_{N}^{m,\circ}g_{N}^{m}$ $\displaystyle:=$
$\displaystyle-\sum_{i,j=1}^{m}K(x_{i}-x_{j})\cdot\nabla_{i}g_{N}^{m}$
$\displaystyle+\sum_{i=1}^{m}\sum_{2\leq j\leq m\atop i\neq
j}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}g_{N}^{m}(x_{[m]\setminus\\{j\\}},x_{*})\,dx_{*},$
$\displaystyle iS_{N}^{m,-}g_{N}^{m-1}$ $\displaystyle:=$
$\displaystyle-\sum_{i=1}^{m}\sum_{j=2}^{m}K(x_{i}-x_{j})\cdot\nabla_{i}g_{N}^{m-1}(x_{[m]\setminus\\{j\\}}),$

where we recall the short-hand notation $[m]=\\{1,\ldots,m\\}$.

###### Proof.

Upon partial integration, the Liouville equation (5.1) yields the following
BBGKY hierarchy of equations for the marginals,

$\partial_{t}f_{N}^{m}+\tfrac{1}{N}\sum_{i,j=1}^{m}K(x_{i}-x_{j})\cdot\nabla_{i}f_{N}^{m}\\\
+\tfrac{N-m}{N}\sum_{i=1}^{m}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}f_{N}^{m+1}(x_{1},\ldots,x_{m},x_{*})\,dx_{*}=0.$
(5.21)

By definition (5.5) of correlation functions, we deduce

$\partial_{t}g_{N}^{m}\,=\,-\tfrac{1}{N}\sum_{j=2}^{m}K(x_{1}-x_{j})\cdot(\nabla_{1}-\nabla_{j})\sum_{n=1}^{m}(-1)^{m-n}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{j\in\sigma}f_{N}^{n}(x_{1},x_{\sigma})\\\
-\tfrac{1}{N}\sum_{i,j=2}^{m}K(x_{i}-x_{j})\cdot\nabla_{i}\sum_{n=1}^{m}(-1)^{m-n}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{i,j\in\sigma}f_{N}^{n}(x_{1},x_{\sigma})\\\
-\sum_{n=1}^{m}(-1)^{m-n}\tfrac{N-n}{N}\sum_{\sigma\in
P_{n-1}^{m-1}}\int_{\mathbb{T}^{d}}K(x_{1}-x_{*})\cdot\nabla_{1}f_{N}^{n+1}(x_{1},x_{\sigma},x_{*})\,dx_{*}\\\
-\sum_{i=2}^{m}\sum_{n=1}^{m}(-1)^{m-n}\tfrac{N-n}{N}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{i\in\sigma}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}f_{N}^{n+1}(x_{1},x_{\sigma},x_{*})\,dx_{*}.$
(5.22)

Replacing marginals in terms of cumulants, cf. (5.4), and arguing as in (2.9),
we get for the first right-hand side term, for all $j\in[m]\setminus\\{1\\}$,

$\sum_{n=1}^{m}(-1)^{m-n}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{j\in\sigma}f_{N}^{n}(x_{1},x_{\sigma})\,=\,g_{N}^{m}(x_{[m]})+g_{N}^{m-1}(x_{[m]\setminus\\{j\\}}).$

Similarly, for the second right-hand side term in (5.22), we find for all
$i,j\in[m]\setminus\\{1\\}$,

$\sum_{n=1}^{m}(-1)^{m-n}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{i,j\in\sigma}f_{N}^{n}(x_{1},x_{\sigma})\\\
\,=\,g_{N}^{m}(x_{[m]})+g_{N}^{m-1}(x_{[m]\setminus\\{i\\}})+g_{N}^{m-1}(x_{[m]\setminus\\{j\\}})+g_{N}^{m-2}(x_{[m]\setminus\\{i,j\\}}).$

For the third right-hand side term in (5.22), arguing as in (2.12), replacing
again marginals by cumulants and using that $\int_{\mathbb{T}^{d}}K=0$, we
find that

$\displaystyle\sum_{n=1}^{m}(-1)^{m-n}\tfrac{N-n}{N}\sum_{\sigma\in
P_{n-1}^{m-1}}\int_{\mathbb{T}^{d}}K(x_{1}-x_{*})\cdot\nabla_{1}f_{N}^{n+1}(x_{1},x_{\sigma},x_{*})\,dx_{*}$
(5.23) $\displaystyle=$
$\displaystyle\sum_{n=1}^{m}(-1)^{m-n}\tfrac{N-n}{N}\sum_{\sigma\in
P_{n-1}^{m-1}}\sum_{r=1}^{n}\sum_{\tau\in
P_{r-1}^{\sigma}}\int_{\mathbb{T}^{d}}K(x_{1}-x_{*})\cdot\nabla_{1}g_{N}^{r+1}(x_{1},x_{\tau},x_{*})\,dx_{*}$
$\displaystyle=$
$\displaystyle\tfrac{N-m}{N}\int_{\mathbb{T}^{d}}K(x_{1}-x_{*})\cdot\nabla_{1}g_{N}^{m+1}(x_{[m]},x_{*})\,dx_{*}$
$\displaystyle\hskip
85.35826pt-\tfrac{1}{N}\sum_{j=2}^{m}\int_{\mathbb{T}^{d}}K(x_{1}-x_{*})\cdot\nabla_{1}g_{N}^{m}(x_{[m]\setminus\\{j\\}},x_{*})\,dx_{*}.$

For the last right-hand side term in (5.22), we similarly get

$\sum_{i=2}^{m}\sum_{n=1}^{m}(-1)^{m-n}\tfrac{N-n}{N}\sum_{\sigma\in
P_{n-1}^{m-1}}\mathds{1}_{i\in\sigma}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}f_{N}^{n+1}(x_{1},x_{\sigma},x_{*})\,dx_{*}\\\
\,=\,\tfrac{N-m}{N}\sum_{i=2}^{m}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}g_{N}^{m+1}(x_{[m]},x_{*})\,dx_{*}\\\
-\tfrac{1}{N}\sum_{2\leq i,j\leq
m}^{\neq}\int_{\mathbb{T}^{d}}K(x_{i}-x_{*})\cdot\nabla_{i}g_{N}^{m}(x_{[m]\setminus\\{j\\}},x_{*})\,dx_{*}.$

Combining the above identities, the conclusion follows. ∎

Next, we prove uniform-in-time propagation-of-chaos estimates for the particle
system in form of a priori estimates on correlation functions. This is deduced
from a straightforward symmetry argument inspired by the work of Bodineau,
Gallagher, and Saint-Raymond [3]. In contrast with the more delicate case of
Lemma 2.2, no large deviation theory is needed here. Note that in the present
situation the $N$-scaling $g_{N}^{m+1}=O(N^{-m/2})$ is actually optimal on
long timescales, cf. Theorem 5.2, whereas the scaling $g_{N}^{m+1}=O(N^{-m})$
that can be heuristically guessed from the BBGKY hierarchy (5.20) is only
valid on short times $t=O(1)$. For later purposes, we also include estimates
on time derivatives of correlation functions.

######  Lemma 5.7 (A priori cumulant estimates).

For all $0\leq m<N$ and $k\geq 0$, we have

$\|\partial_{t}^{k}g_{N}^{m+1}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m+1})}\,\lesssim_{m,k,f^{\circ}}\,N^{-\frac{m+k}{2}}.$

###### Proof.

Recall that correlation functions satisfy
$\int_{\mathbb{T}^{d}}g_{N}^{m}(x_{[m]})\,dx_{j}=0$ for $2\leq j\leq m$.
Computing the $\operatorname{L}^{2}$ norm of the $N$-point density $f_{N}$ and
inserting the cluster expansion (5.4) in terms of correlation functions, we
then get

$\int_{(\mathbb{T}^{d})^{N}}|f_{N}|^{2}\,=\,\sum_{m=1}^{N}\binom{N-1}{m-1}\int_{(\mathbb{T}^{d})^{m}}|g_{N}^{m}|^{2},$
(5.24)

hence in particular, for all $0\leq m<N$,

$\|g_{N}^{m+1}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m+1})}\,\lesssim_{m}\,N^{-\frac{m}{2}}\|f_{N}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}.$

By linearity, we similarly find for all $k\geq 0$,

$\int_{(\mathbb{T}^{d})^{N}}|\partial_{t}^{k}f_{N}|^{2}\,=\,\sum_{m=1}^{N}\binom{N-1}{m-1}\int_{(\mathbb{T}^{d})^{m}}|\partial_{t}^{k}g_{N}^{m}|^{2},$
(5.25)

hence

$\|\partial_{t}^{k}g_{N}^{m+1}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m+1})}\,\lesssim_{m}\,N^{-\frac{m}{2}}\|\partial_{t}^{k}f_{N}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}.$

Combining this with the energy estimate (5.18) for the Liouville equation, the
conclusion follows. ∎

###  5.3. Proof of Theorem 5.2

For all $m\geq 1$, we deduce from Lemma 5.7 that the rescaled correlation
function $\bar{g}_{N}^{m}$ defined in (5.6) is bounded in
$W^{k,\infty}(\mathbb{R}^{+};\operatorname{L}^{2}((\mathbb{T}^{d})^{m}))$, for
any $k\geq 0$, as $N\uparrow\infty$. By weak-* compactness, up to a
subsequence, we deduce that there exists a limiting family
$\bar{g}=\\{\bar{g}^{m}\\}_{m\geq 1}$ such that for all $m,k$ we have

$\bar{g}^{m}_{N}\rightharpoonup\bar{g}^{m},\qquad\text{weakly-* in
$W^{k,\infty}(\mathbb{R}^{+};\operatorname{L}^{2}((\mathbb{T}^{d})^{m}))$}.$

As the choice (5.2) of initial data yields
$\bar{g}_{N}^{1}|_{\tau=0}=f^{\circ}$ and $\bar{g}_{N}^{m}|_{\tau=0}=0$ for
$m>1$, the convergence implies in particular

$\bar{g}^{1}|_{\tau=0}=f^{\circ}\qquad\text{and}\qquad\bar{g}^{m}|_{\tau=0}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for $m>1$.}$ (5.26)

Next, we note that, after rescaling, the BBGKY hierarchy for cumulants in
Lemma 5.6 takes the form

$\partial_{\tau}\bar{g}_{N}^{m}=iS_{N}^{m,+}\bar{g}_{N}^{m+1}+iS_{N}^{m,-}\bar{g}_{N}^{m-1}+N^{-\frac{1}{2}}iS_{N}^{m,\circ}\bar{g}_{N}^{m},\qquad
1\leq m\leq N.$

Passing to the limit in the weak formulation of those equations, we deduce
that the limit $\bar{g}$ satisfies the following hierarchy of equations in the
weak sense,

$\partial_{\tau}\bar{g}^{m}=iS^{m,+}\bar{g}^{m+1}+iS^{m,-}\bar{g}^{m-1},\qquad
m\geq 1,$ (5.27)

where the limiting operators $iS^{m,+},iS^{m,-}$ are defined in the statement.
Moreover, reformulating (5.25) in terms of rescaled correlations, truncating
the sum, and combining with the energy estimate (5.18), we find for any fixed
$M,k$,

$\sum_{m=1}^{M\wedge
N}N^{1-m}\binom{N-1}{m-1}\|\partial_{\tau}^{k}\bar{g}_{N}^{m}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}^{2}\,\leq\,N^{k}\|\partial_{t}^{k}f_{N}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{N})}^{2}\,\lesssim_{k,f^{\circ}}\,1.$

Passing to the limit $N\uparrow\infty$, with
$N^{1-m}\binom{N-1}{m-1}\to\tfrac{1}{(m-1)!}$, we deduce by weak lower
semicontinuity,

$\sum_{m=1}^{M}\tfrac{1}{(m-1)!}\|\partial_{\tau}^{k}\bar{g}^{m}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}^{2}\,\lesssim_{k,f^{\circ}}\,1,$

and thus, letting $M\uparrow\infty$, we infer for all $k\geq 1$,

$\sum_{m=1}^{\infty}\tfrac{1}{(m-1)!}\|\partial_{\tau}^{k}\bar{g}^{m}\|_{\operatorname{L}^{2}((\mathbb{T}^{d})^{m})}^{2}\,\lesssim_{k,f^{\circ}}\,1.$
(5.28)

Finally, in order to get rid of the extraction of a subsequence in the above
argument, it remains to check that there is at most one
$\bar{g}=\\{\bar{g}^{m}\\}_{m}$ that satisfies the limit hierarchy (5.27) in
the weak sense and such that the a priori estimates (5.28) hold. As explained,
this is equivalent to proving that there is at most one strong solution
$\bar{g}\in C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ of the following
equation,

$\partial_{\tau}\bar{g}=iS^{*}\bar{g},\qquad\bar{g}|_{\tau=0}=\bar{g}^{\circ},$

where the Hilbert space $\mathcal{H}$ and the densely-defined symmetric
operator $S$ are defined in (5.10)–(5.11), and where the initial condition
$\bar{g}^{\circ}$ is given by (5.13). Noting that $\bar{g}^{\circ}$ belongs to
the core $\mathcal{C}$ of $S$ and that the latter satisfies
$S\mathcal{C}\subset\mathcal{C}$, the desired uniqueness is a consequence of
Lemma 5.8 below. This ends the proof of Theorem 5.2. ∎

###  5.4. Proof of Proposition 5.3

We start by proving the uniqueness of a solution of the limit hierarchy in
$C^{2}_{b}(\mathbb{R}^{+};\mathcal{H})$. We state it separately in form of the
following general abstract result.

######  Lemma 5.8.

Let $\mathcal{H}$ be a Hilbert space, let $S$ be a densely-defined symmetric
operator, defined on a dense subset $\mathcal{C}\subset\mathcal{H}$, and
assume that $S\mathcal{C}\subset\mathcal{C}$. Then, for all
$g^{\circ}\in\mathcal{H}$, there is at most one solution $g\in
C^{2}_{b}(\mathbb{R}^{+};\mathcal{H})$ of the following equation,

$\partial_{\tau}g=iS^{*}g,\qquad g|_{\tau=0}=g^{\circ}.$ (5.29)

###### Proof.

As $S\mathcal{C}\subset\mathcal{C}$, we note that the squared operator $S^{2}$
is well-defined, symmetric, and positive on $\mathcal{C}$. By Friedrichs’
theorem, it admits a canonical self-adjoint extension $\mathcal{L}_{0}$. Given
a solution $g\in C^{2}(\mathbb{R}^{+};\mathcal{H})$ of equation (5.29), we may
then compute for all $h\in\mathcal{C}$,

$\langle h,\partial_{\tau}^{2}g(\tau)\rangle_{\mathcal{H}}=-\langle
S^{2}h,g(\tau)\rangle_{\mathcal{H}}=-\langle\mathcal{L}_{0}h,g(\tau)\rangle_{\mathcal{H}}.$

As $g$ belongs to $C^{2}(\mathbb{R}^{+};\mathcal{H})$, the left-hand side is
bounded by $\|\partial_{\tau}^{2}g(\tau)\|_{\mathcal{H}}\|h\|_{\mathcal{H}}$,
hence so is the right-hand side, which entails that $g(\tau)$ belongs to the
domain of $\mathcal{L}_{0}$ for all $\tau$. This implies that $g$ is a strong
solution the following equation,

$\partial_{\tau}^{2}g+\mathcal{L}_{0}g=0,\qquad
g|_{\tau=0}=g^{\circ},\qquad\partial_{\tau}g|_{\tau=0}=iS^{*}g^{\circ}.$

By self-adjointness and non-negativity of $\mathcal{L}_{0}$, the strong
solution of this equation is unique. (Note that equation (5.29) and the
condition $g\in C^{2}(\mathbb{R}^{+};\mathcal{H})$ ensure
$g^{\circ}\in\mathcal{D}(S^{*})$, so $iS^{*}g^{\circ}$ is indeed well-
defined.) ∎

######  Remark 5.9.

The proof of Lemma 5.8 is easily adapted to establish the following result:
given a Hilbert space $\mathcal{H}$ and a densely-defined symmetric operator
$S$, defined on a dense subset $\mathcal{C}\subset\mathcal{H}$, if $S$ admits
a self-adjoint extension $S_{0}$, then for all
$g^{\circ}\in\mathcal{D}(S_{0})$ there is a unique strong solution $g\in
C^{1}_{b}(\mathbb{R}^{+};\mathcal{H})$ of the equation

$\partial_{\tau}g=iS^{*}g,\qquad g|_{\tau=0}=g^{\circ},$

and it coincides with the unitary group $g(\tau)=e^{i\tau S_{0}}g^{\circ}$. It
is however not clear to us whether this result applies to our situation as we
do not know how to prove the existence of a self-adjoint extension for the
operator $S$ defined in (5.11).

With the above general uniqueness result at hand, we can now conclude the
proof of Proposition 5.3, establishing the well-posedness, contraction, and
approximate isometricity of solutions of the limit hierarchy.

###### Proof of Proposition 5.3.

Let the Hilbert space $\mathcal{H}$, the densely-defined symmetric operator
$S$, and its core $\mathcal{C}\subset\mathcal{H}$ be defined as in
(5.10)–(5.12). We split the proof into two steps.

Step 1\. Proof that there exists a unique contraction-valued strongly-
continuous map $U:\mathbb{R}^{+}\to L(\mathcal{H})$, which might not be a
semigroup, such that for all $g^{\circ}\in\mathcal{C}$ the evolution
$g(\tau):=U(\tau)g^{\circ}$ is the unique strong solution in
$C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ of the equation

$\partial_{\tau}g=iS^{*}g,\qquad g|_{\tau=0}=g^{\circ}.$ (5.30)

Moreover, we shall show that it satisfies the stability property

$\|\partial_{\tau}^{k}U(\tau)g^{\circ}\|_{\mathcal{H}}\,\leq\,\|S^{k}g^{\circ}\|_{\mathcal{H}},\qquad\text{for
all $k\geq 0$ and $g^{\circ}\in\mathcal{C}$}.$ (5.31)

To prove this result, we start by defining self-adjoint truncations of $S$:
for all $N\geq 1$, we consider the closed subspace $\mathcal{H}_{\leq
N}:=\bigoplus_{1\leq m\leq N}\mathcal{H}^{m}\subset\mathcal{H}$, we let
$\pi_{\leq N}:\mathcal{H}\to\mathcal{H}_{\leq N}$ be the associated orthogonal
projection, and we define the truncated operator $S_{N}:=\pi_{\leq
N}S\pi_{\leq N}$ on $\mathcal{C}$. Note that this truncation could be replaced
by the original hierarchy (5.20) for fixed $N$: it does not change the
argument, but the present truncation is simpler to handle. By definition of
$S$, using the symmetry relations (5.9), we easily check that this truncated
operator $S_{N}$ is essentially self-adjoint on $\mathcal{C}$ (self-
adjointness poses no difficulty here thanks to truncations). We may then
consider the unitary semigroup $U_{N}:\mathbb{R}^{+}\to L(\mathcal{H})$ given
by

$U_{N}(\tau)\,:=\,e^{i\tau S_{N}}.$

Up to extraction of a subsequence, as $N\uparrow\infty$, the semigroup $U_{N}$
converges pointwise in the weak operator topology to some strongly-continuous
map $U:\mathbb{R}^{+}\to L(\mathcal{H})$ with $U(0)=\operatorname{Id}$ and
$\|U(\tau)\|\leq 1$ for all $\tau\geq 0$. Note that this weak limit map $U$
might no longer be a semigroup nor take its values among unitary operators.

Given $g^{\circ}\in\mathcal{C}$, let us examine the properties of the limit
evolution $\tau\mapsto U(\tau)g^{\circ}$. For all $N\geq 1$, as $iS_{N}$ is
the generator of $U_{N}$ and as $S_{N}\mathcal{C}\subset\mathcal{C}$, the flow
$\tau\mapsto U_{N}(\tau)g^{\circ}$ belongs to
$C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ and satisfies

$\|\partial_{\tau}^{k}U_{N}(\tau)g^{\circ}\|_{\mathcal{H}}\,=\,\|S_{N}^{k}g^{\circ}\|_{\mathcal{H}},\qquad\text{for
all $k\geq 0$,}$ (5.32)

and

$\langle h,\partial_{\tau}U_{N}(\tau)g^{\circ}\rangle\,=\,\langle
S_{N}h,iU_{N}(\tau)g^{\circ}\rangle,\qquad\text{for all $h\in\mathcal{C}$}.$

Note that the tridiagonal structure of $S$ yields $S_{N}^{k}h=S^{k}h$ for all
$h\in\pi_{\leq N-k}\mathcal{C}$ and $k\geq 1$, and therefore $S_{N}^{k}h\to
S^{k}h$ strongly as $N\uparrow\infty$ for all $h\in\mathcal{C}$ and $k\geq 1$.
We may then pass to the limit along the extracted subsequence in the above
properties of $U_{N}(\tau)g^{\circ}$: we deduce that the limit evolution
$\tau\mapsto U(\tau)g^{\circ}$ also belongs to
$C^{\infty}_{b}(\mathbb{R}^{+};\mathcal{H})$ and satisfies

$\|\partial_{\tau}^{k}U(\tau)g^{\circ}\|_{\mathcal{H}}\,\leq\,\|S^{k}g^{\circ}\|_{\mathcal{H}},\qquad\text{for
all $k\geq 0$},$ (5.33)

and

$\langle h,\partial_{\tau}U(\tau)g^{\circ}\rangle\,=\,\langle
Sh,iU(\tau)g^{\circ}\rangle,\qquad\text{for all $h\in\mathcal{C}$}.$

The former is the desired contraction property (5.31) and the latter precisely
means that the limit evolution $g(\tau):=U(\tau)g^{\circ}$ satisfies equation
(5.30) in the strong sense.

Now, by Lemma 5.8, the solution of equation (5.30) is necessarily unique in
$C^{2}_{b}(\mathbb{R}^{+};\mathcal{H})$. In particular, the limit evolution
$\tau\mapsto U(\tau)g^{\circ}$ is uniquely determined for all
$g^{\circ}\in\mathcal{C}$. As $U$ is contraction-valued and as $\mathcal{C}$
is dense in $\mathcal{H}$, this entails that $U$ is itself uniquely determined
as a contraction-valued strongly-continuous map $\mathbb{R}^{+}\to
L(\mathcal{H})$.

Step 2\. Approximate unitarity: proof that for all $k\geq 0$ and
$g^{\circ}\in\mathcal{C}$,

$-\tfrac{1}{k!}\tau^{k}\sum_{j=0}^{k}\binom{k}{j}\|S^{j}g^{\circ}\|_{\mathcal{H}}^{2}\,\leq\,\|U(\tau)g^{\circ}\|_{\mathcal{H}}^{2}-\|g^{\circ}\|_{\mathcal{H}}^{2}\,\leq\,0.$
(5.34)

The upper bound follows from (5.33) and it remains to prove the lower bound.
Let $g^{\circ}\in\mathcal{C}$ be fixed. For $M,N\geq 1$, consider

$E_{M,N}(\tau)\,:=\,\|\pi_{\leq
M}U_{N}(\tau)g^{\circ}\|^{2}_{\mathcal{H}}\,=\,\sum_{m=1}^{M}\|U_{N}(\tau)g^{\circ}\|_{\mathcal{H}^{m}}^{2},$

and appeal to Taylor’s expansion

$\Big{|}E_{M,N}(\tau)-\sum_{j=0}^{k-1}\tfrac{1}{j!}\tau^{j}\partial_{\tau}^{j}E_{M,N}(0)\Big{|}\,\leq\,\tfrac{1}{k!}\tau^{k}\|\partial_{\tau}^{k}E_{M,N}\|_{\operatorname{L}^{\infty}(\mathbb{R}^{+})}.$

Using (5.32), time derivatives of $E_{M,N}$ can be estimated as follows, for
all $k\geq 1$,

$\displaystyle|\partial_{\tau}^{k}E_{M,N}(\tau)|$ $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{k}\binom{k}{j}\|\pi_{\leq
M}\partial_{\tau}^{j}U_{N}(\tau)g^{\circ}\|_{\mathcal{H}}\|\pi_{\leq
M}\partial_{\tau}^{k-j}U_{N}(\tau)g^{\circ}\|_{\mathcal{H}}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{k}\binom{k}{j}\|S_{N}^{j}g^{\circ}\|_{\mathcal{H}}\|S_{N}^{k-j}g^{\circ}\|_{\mathcal{H}}.$

By the symmetry in the sum and the trivial bound $2ab\leq a^{2}+b^{2}$, the
above becomes

$\Big{|}E_{M,N}(\tau)-\sum_{j=0}^{k-1}\tfrac{1}{j!}\tau^{j}\partial_{\tau}^{j}E_{M,N}(0)\Big{|}\,\leq\,\tfrac{1}{k!}\tau^{k}\sum_{j=0}^{k}\binom{k}{j}\|S_{N}^{j}g^{\circ}\|_{\mathcal{H}}^{2}.$
(5.35)

Now for $M<N$, recalling $S_{N}=\pi_{\leq N}S\pi_{\leq N}$ and the definition
of $S$, we can compute the first time derivative of $E_{M,N}$ as follows:
setting $g_{N}(\tau):=U_{N}(\tau)g^{\circ}$ and $g_{N}=\\{g_{N}^{m}\\}_{m\geq
1}$,

$\displaystyle\partial_{\tau}E_{M,N}(\tau)$ $\displaystyle=$ $\displaystyle
2\Re\sum_{m=1}^{M}\langle
g_{N}^{m}(\tau),\partial_{\tau}g_{N}^{m}(\tau)\rangle_{\mathcal{H}^{m}}$
$\displaystyle=$ $\displaystyle
2\Re\sum_{m=1}^{M}\big{\langle}g^{m}_{N}(\tau),iS^{m,+}\bar{g}^{m+1}_{N}(\tau)+iS^{m,-}g^{m-1}_{N}(\tau)\big{\rangle}_{\mathcal{H}^{m}},$

and thus, using $(S^{m,-})^{*}=S^{m-1,+}$ and recognizing a telescoping sum,

$\displaystyle\partial_{\tau}E_{M,N}(\tau)$ $\displaystyle=$ $\displaystyle
2\Re\sum_{m=1}^{M}\Big{(}\langle
g^{m}_{N}(\tau),iS^{m,+}g^{m+1}_{N}(\tau)\rangle_{\mathcal{H}_{m}}-\langle
iS^{m-1,+}g^{m}_{N}(\tau),g^{m-1}_{N}(\tau)\rangle_{\mathcal{H}^{m-1}}\Big{)}$
$\displaystyle=$ $\displaystyle 2\Re\langle
g_{N}^{M}(\tau),iS^{M,+}g_{N}^{M+1}(\tau)\rangle_{\mathcal{H}_{M}}.$

Evaluating at $\tau=0$, this means

$\partial_{\tau}E_{M,N}(0)\,=\,2\Re\langle(g^{\circ})^{M},iS^{M,+}(g^{\circ})^{M+1}\rangle_{\mathcal{H}_{M}}.$

For $g^{\circ}\in\mathcal{C}$, there is $M_{0}(g^{\circ})<\infty$ such that
$(g^{\circ})^{M}=0$ for all $M>M_{0}(g^{\circ})$. Hence, we deduce
$\partial_{\tau}E_{M,N}(0)=0$ for $M\geq M_{0}(g^{\circ})$. Taking additional
time derivatives of the above expression and using the tridiagonal structure
of $S$, we find $\partial_{\tau}^{j}E_{M,N}(0)=0$ for all $1\leq j\leq
M+1-M_{0}(g^{\circ})$. By definition of $E_{M,N}$, the estimate (5.35) then
takes on the following guise: for all $M<N$ and $k\leq M+2-M_{0}(g^{\circ})$,

$\Big{|}\|\pi_{\leq M}U_{N}(\tau)g^{\circ}\|^{2}_{\mathcal{H}}-\|\pi_{\leq
M}g^{\circ}\|^{2}_{\mathcal{H}}\Big{|}\,\leq\,\tfrac{1}{k!}\tau^{k}\sum_{j=0}^{k}\binom{k}{j}\|S_{N}^{j}g^{\circ}\|_{\mathcal{H}}^{2}.$

Letting $N\uparrow\infty$ and then $M\uparrow\infty$, using that
$U_{N}(\tau)g^{\circ}$ converges weakly to $U(\tau)g^{\circ}$ in
$\mathcal{H}$, and recalling that $S_{N}^{j}g^{\circ}$ converges strongly to
$S^{j}g^{\circ}$ in $\mathcal{H}$ for all $j\geq 0$, the claimed lower bound
(5.34) follows. ∎

###  5.5. Proof of Proposition 5.4

We show that the RAGE theorem still applies to the weak limit of a sequence of
unitary groups, although the limit might no longer be a semigroup nor take its
values among unitary operators. Note however that we loose (part of) the usual
orthogonality property for periodic solutions. Proposition 5.4 is a direct
consequence of the following general abstract result.

######  Lemma 5.10.

Let $(U_{N})_{N}$ be a sequence of unitary semigroups $U_{N}:\mathbb{R}^{+}\to
L(\mathcal{H})$ on a Hilbert space $\mathcal{H}$, and assume that their
generators $(S_{N})_{N}$ are essentially self-adjoint on a common (dense) core
$\mathcal{C}\subset\mathcal{H}$. As $N\uparrow\infty$, assume that $S_{N}$
converges in the strong operator topology to some operator $S$ on
$\mathcal{C}$ (necessarily symmetric on $\mathcal{C}$, but possibly not
essentially self-adjoint), and that $U_{N}$ converges pointwise in the weak
operator topology to a map $U:\mathbb{R}^{+}\to L(\mathcal{H})$ (necessarily
strongly-continuous and contraction-valued, but possibly not unitary and not a
semigroup). Then the following RAGE theorem holds for $U$: denoting by
$\\{\lambda_{k}\\}_{k}$ the set of real eigenvalues of $S^{*}$, there exists a
family of positive contractions $\\{P_{k}\\}_{k}$ on $\mathcal{H}$ such that
for all $k$ the image $\operatorname{ran}(P_{k})$ is a subset of the
eigenspace of $S^{*}$ associated with $\lambda_{k}$, and such that for all
$g^{\circ}\in\mathcal{H}$ we can decompose the limit evolution as

$U(\tau)g^{\circ}\,=\,\sum_{k}e^{i\tau\lambda_{k}}P_{k}g^{\circ}+R(\tau)g^{\circ},$

where the remainder $R(\tau)g^{\circ}$ satisfies for all $h\in\mathcal{H}$,

$\lim_{T\uparrow\infty}\frac{1}{T}\int_{0}^{T}|\langle
h,R(\tau)g^{\circ}\rangle_{\mathcal{H}}|^{2}d\tau\,=\,0.$

If in addition we have $S_{N}\mathcal{C}\subset\mathcal{C}$,
$S\mathcal{C}\subset\mathcal{C}$, and if the squared operator $S_{N}^{2}$ also
converges in the strong operator topology to $S^{2}$ on $\mathcal{C}$, then we
have the following partial orthogonality property:
$\operatorname{ran}(P_{k})\bot\operatorname{ran}(P_{l})$ for
$|\lambda_{k}|\neq|\lambda_{l}|$.

###### Proof.

For all $N$, we can consider the spectral measure $E_{N}$ of the essentially
self-adjoint generator $S_{N}$ and represent the unitary group $U_{N}$ as

$U_{N}(\tau)\,=\,e^{iS_{N}\tau}\,=\,\int_{\mathbb{R}}e^{i\lambda\tau}dE_{N}(\lambda).$

Under the considered assumptions, the spectral measure $E_{N}$ converges
weakly as $N\uparrow\infty$ to some positive operator-valued measure $E$ such
that

$U(\tau)\,=\,\int_{\mathbb{R}}e^{i\lambda\tau}dE(\lambda),$

and in addition for all $g^{\circ}\in\mathcal{C}$ we find that the limit
evolution $\tau\mapsto U(\tau)g^{\circ}$ belongs to
$C^{1}_{b}(\mathbb{R}^{+};\mathcal{H})$ and is a strong solution of

$\partial_{\tau}U(\tau)g^{\circ}\,=\,iS^{*}U(\tau)g^{\circ}\,=\,iU(\tau)Sg^{\circ},\qquad
U(\tau)g^{\circ}|_{\tau=0}=g^{\circ}.$ (5.36)

Note that the limit measure $E$ is a priori not projection-valued, hence is
not a spectral measure, in link with the fact that the limit operator $S$
might not be essentially self-adjoint and might not even generate a semigroup.
By Naimark’s dilation theorem, see e.g. [24, Theorem 4.6], there exists an
extended Hilbert space $\hat{\mathcal{H}}$, a bounded linear map
$V:\hat{\mathcal{H}}\to\mathcal{H}$ with $VV^{*}=\operatorname{Id}$, and a
spectral measure $\hat{E}$ on $\mathbb{R}$ such that
$dE(\lambda)=Vd\hat{E}(\lambda)V^{*}$. In terms of the self-adjoint operator
$\hat{S}:=\int_{\mathbb{R}}\lambda d\hat{E}(\lambda)$ on $\hat{\mathcal{H}}$,
we then get

$U(\tau)\,=\,Ve^{i\tau\hat{S}}V^{*}\,=\,V\int_{\mathbb{R}}e^{i\tau\lambda}d\hat{E}(\lambda)V^{*}.$

We may now appeal to the standard form of the RAGE theorem for $\hat{S}$, see
e.g. [11, Section 5.4]: denoting by $\\{\lambda_{k}\\}_{k}\subset\mathbb{R}$
the set of eigenvalues of $\hat{S}$ and denoting by
$\hat{P}_{k}:\hat{\mathcal{H}}\to\hat{\mathcal{H}}$ the orthogonal projection
onto the eigenspace of $\hat{S}$ associated with $\lambda_{k}$, we can
decompose for all $g^{\circ}\in\mathcal{H}$,

$U(\tau)g^{\circ}\,=\,\sum_{k}e^{i\tau\lambda_{k}}V\hat{P}_{k}V^{*}g^{\circ}+R(\tau)g^{\circ},$
(5.37)

where the remainder $R(\tau)g^{\circ}$ satisfies for all $h\in\mathcal{H}$,

$\lim_{T\uparrow\infty}\frac{1}{T}\int_{0}^{T}|\langle
h,R(\tau)g^{\circ}\rangle_{\mathcal{H}}|^{2}d\tau\,=\,0.$ (5.38)

Note that by definition the remainder can be written as

$R(\tau)g^{\circ}\,=\,V\Big{(}e^{i\tau\hat{S}}-\sum_{k}e^{i\tau\lambda_{k}}\hat{P}_{k}\Big{)}V^{*}g^{\circ}\,=\,Ve^{i\tau\hat{S}}\hat{\pi}_{c}V^{*}g^{\circ},$
(5.39)

where $\hat{\pi}_{c}:=1-\sum_{k}\hat{P}_{k}$ is the orthogonal projection onto
the continuous subspace of $\hat{S}$.

Next, we show that for all $k$ the value $\lambda_{k}$ is actually also an
eigenvalue of the adjoint $S^{*}$ on $\mathcal{H}$ and that the image of
$P_{k}:=V\hat{P}_{k}V^{*}$ is a subset of the associated eigenspace in
$\mathcal{H}$. For that purpose, applying the operator
$\partial_{\tau}-iS^{*}$ to (5.37) and using the equation (5.36) satisfied by
the limit evolution, we find in the weak sense

$(\partial_{\tau}-iS^{*})R(\tau)g^{\circ}\,=\,-i\sum_{k}e^{i\tau\lambda_{k}}\,(\lambda_{k}-S^{*})\,V\hat{P}_{k}V^{*}g^{\circ}.$

Given some $k_{0}$, multiplying both sides of this identity by
$e^{-i\tau\lambda_{k_{0}}}$ and integrating over some time interval $[0,T]$,
with $T>0$, we deduce in the weak sense

$\frac{1}{iT}\Big{(}e^{-iT\lambda_{k_{0}}}R(T)g^{\circ}-R(0)g^{\circ}\Big{)}+\frac{1}{T}\int_{0}^{T}e^{-i\tau\lambda_{k_{0}}}(\lambda_{k_{0}}-S^{*})R(\tau)g^{\circ}\,d\tau\\\
\,=\,-(\lambda_{k_{0}}-S^{*})\,V\hat{P}_{k_{0}}V^{*}\bar{g}^{\circ}-\sum_{k:k\neq
k_{0}}\frac{e^{iT(\lambda_{k}-\lambda_{k_{0}})}-1}{iT(\lambda_{k}-\lambda_{k_{0}})}\,(\lambda_{k}-S^{*})\,V\hat{P}_{k}V^{*}g^{\circ}.$

Now testing this identity with some $h\in\mathcal{C}$, and singling out the
first right-hand side term, we deduce

$\big{|}\big{\langle}(\lambda_{k_{0}}-S)h,V\hat{P}_{k_{0}}V^{*}g^{\circ}\big{\rangle}_{\mathcal{H}}\big{|}\\\
\,\leq\,\frac{1}{T}\|h\|_{\mathcal{H}}\Big{(}\|R(T)g^{\circ}\|_{\mathcal{H}}+\|R(0)g^{\circ}\|_{\mathcal{H}}\Big{)}+\frac{1}{T}\int_{0}^{T}\big{|}\big{\langle}(\lambda_{k_{0}}-S)h,R(\tau)g^{\circ}\big{\rangle}_{\mathcal{H}}\big{|}\\\
+\sum_{k:k\neq
k_{0}}\bigg{|}\frac{e^{iT(\lambda_{k}-\lambda_{k_{0}})}-1}{iT(\lambda_{k}-\lambda_{k_{0}})}\bigg{|}\|\hat{P}_{k}V^{*}(\lambda_{k}-S)h\|_{\mathcal{H}}\|\hat{P}_{k}V^{*}g^{\circ}\|_{\mathcal{H}}.$
(5.40)

By (5.39), we have
$\|R(\tau)g^{\circ}\|_{\mathcal{H}}\leq\|g^{\circ}\|_{\mathcal{H}}$, and the
first right-hand side term in (5.40) thus converges to $0$ as $T\uparrow 0$.
By (5.38) and Jensen’s inequality, the second right-hand side term also
converges to $0$. For the last right-hand side term, using $|e^{ix}-1|\leq
2\wedge|x|$, and decomposing
$(\lambda_{k}-S)h=(\lambda_{k}-\lambda_{k_{0}})h+(\lambda_{k_{0}}-S)h$, we
find

$\displaystyle\sum_{k:k\neq
k_{0}}\bigg{|}\frac{e^{iT(\lambda_{k}-\lambda_{k_{0}})}-1}{iT(\lambda_{k}-\lambda_{k_{0}})}\bigg{|}\|\hat{P}_{k}V^{*}(\lambda_{k}-S)h\|_{\mathcal{H}}\|\hat{P}_{k}V^{*}g^{\circ}\|_{\mathcal{H}}$
$\displaystyle\leq$
$\displaystyle\tfrac{2}{T}\sum_{k}\|\hat{P}_{k}V^{*}h\|_{\mathcal{H}}\|\hat{P}_{k}V^{*}g^{\circ}\|_{\mathcal{H}}$
$\displaystyle+\sum_{k:k\neq
k_{0}}\big{(}1\wedge\tfrac{2}{T|\lambda_{k}-\lambda_{k_{0}}|}\big{)}\|\hat{P}_{k}V^{*}(\lambda_{k_{0}}-S)h\|_{\mathcal{H}}\|\hat{P}_{k}V^{*}g^{\circ}\|_{\mathcal{H}},$

which converges to $0$ as $T\uparrow\infty$ by dominated convergence. Going
back to (5.40), we conclude for all $h\in\mathcal{C}$,

$\big{\langle}(\lambda_{k_{0}}-S)h,V\hat{P}_{k_{0}}V^{*}g^{\circ}\big{\rangle}_{\mathcal{H}}\,=\,0,$
(5.41)

which precisely means that $V\hat{P}_{k_{0}}V^{*}g^{\circ}$ is an eigenvector
of $S^{*}$ with eigenvalue $\lambda_{k_{0}}$.

Finally, let us further assume $S_{N}\mathcal{C}\subset\mathcal{C}$,
$S\mathcal{C}\subset\mathcal{C}$, and that the squared operator $S_{N}^{2}$
also converges in the strong operator topology to $S^{2}$ on $\mathcal{C}$.
This easily implies that for $g^{\circ}\in\mathcal{C}$ the limit evolution
$\tau\mapsto U(\tau)g^{\circ}$ belongs to
$C^{2}_{b}(\mathbb{R}^{+};\mathcal{H})$. By Lemma 5.8, we then learn that this
limit evolution is the unique strong solution of equation (5.36) in
$C^{2}_{b}(\mathbb{R}^{+};\mathcal{H})$. Moreover, in terms of the canonical
self-adjoint extension $\mathcal{L}_{0}$ of the squared operator $S^{2}$ on
$\mathcal{C}$ as given by Friedrichs’ theorem, the proof of Lemma 5.8 shows
that for $g^{\circ}\in\mathcal{C}$ the limit evolution $\tau\mapsto
U(\tau)g^{\circ}$ coincides with the unique strong solution of

$\partial_{\tau}^{2}g+\mathcal{L}_{0}g=0,\qquad
g|_{\tau=0}=g^{\circ},\qquad\partial_{\tau}g|_{\tau=0}=iS^{*}g^{\circ}.$

For all $k$, a straightforward adaptation of the proof of (5.41) above then
leads us to conclude that $V\hat{P}_{k}V^{*}g^{\circ}$ is also an eigenvector
of $\mathcal{L}_{0}$ with eigenvalue $\lambda_{k}^{2}$. As $\mathcal{L}_{0}$
is self-adjoint, this entails that $V\hat{P}_{k}V^{*}g^{\circ}$ is orthogonal
to $V\hat{P}_{l}V^{*}g^{\circ}$ whenever $\lambda_{k}^{2}\neq\lambda_{l}^{2}$.
This ends the proof of Proposition 5.4. ∎

## Acknowledgements

The authors thank Jean-Baptiste Fouvry and Pierre-Henri Chavanis for
motivating discussions and for pointing out relevant physics references. M.
Duerinckx acknowledges financial support from the F.R.S.-FNRS, as well as from
the European Union (ERC, PASTIS, Grant Agreement n∘101075879).555Views and
opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. P.-E. Jabin was partially
supported by NSF DMS Grants 2205694 and 2219297.

## References

  * [1] R. Balescu.  Irreversible Processes in Ionized Gases.  Phys. Fluids, 3(1):52–63, 1960.
  * [2] G. Ben Arous and M. Brunaud.  Méthode de Laplace: Étude variationnelle des fluctuations de diffusions de type champ moyen.  Stoch. Stoch. Rep., 31(1-4):79–144, 1990.
  * [3] T. Bodineau, I. Gallagher, and L. Saint-Raymond.  From hard sphere dynamics to the Stokes–Fourier equations: an $L^{2}$ analysis of the Boltzmann–Grad limit.  Ann. PDE, 3(1):Art. 2, 118, 2017.
  * [4] M. Bostan.  The Vlasov–Poisson system with strong external magnetic field. Finite Larmor radius regime.  Asymptot. Anal., 61(2):91–123, 2007.
  * [5] P.-H. Chavanis.  Systematic drift experienced by a point vortex in two-dimensional turbulence.  Phys. Rev. E, 58(2):R1199–1202, 1998.
  * [6] P.-H. Chavanis.  Kinetic theory of point vortices: diffusion coefficient and systematic drift.  Phys. Rev. E, 64(2):026309, 2001.
  * [7] P.-H. Chavanis.  Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics.  Physica A, 391(14):3657–3679, 2012.
  * [8] P.-H. Chavanis.  Kinetic theory of two-dimensional point vortices with collective effects.  J. Stat. Mech.: Theory Exp., 2012(2):P02019, 2012.
  * [9] P.-H. Chavanis.  Kinetic theory of two-dimensional point vortices and fluctuation-dissipation theorem.  Eur. Phys. J. Plus, 138(2):1–51, 2023.
  * [10] P.-H. Chavanis and M. Lemou.  Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations.  Eur. Phys. J. B, 59:217–247, 2007.
  * [11] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon.  Schrödinger operators with application to quantum mechanics and global geometry.  Texts and Monographs in Physics. Springer-Verlag, Berlin, 1987.
  * [12] M. Duerinckx.  On the size of chaos via Glauber calculus in the classical mean-field dynamics.  Commun. Math. Phys., 382:613–653, 2021.
  * [13] M. Duerinckx and L. Saint-Raymond.  Lenard–Balescu correction to mean-field theory.  Prob. Math. Phys., 2(1):27–69, 2021.
  * [14] J. B. Fouvry, B. Bar-Or, and P.-H. Chavanis.  Vector resonant relaxation of stars around a massive black hole.  Astrophys. J., 883(2):161, 2019.
  * [15] E. Frénod and E. Sonnendrücker.  The finite Larmor radius approximation.  SIAM J. Math. Anal., 32(6):1227–1247, 2001.
  * [16] R. L. Guernsey.  Kinetic Equation for a Completely Ionized Gas.  Phys. Fluids, 5:322–328, 1962.
  * [17] T. Kato.  Perturbation theory for linear operators.  Classics in Mathematics. Springer-Verlag, Berlin, 1995.
  * [18] B. Kocsis and S. Tremaine.  A numerical study of vector resonant relaxation.  Mon. Notices Royal Astron. Soc., 448(4):3265–3296, 2015.
  * [19] J. A. Krommes.  Fundamental statistical descriptions of plasma turbulence in magnetic fields.  Phys. Rep., 360:1–352, 2002.
  * [20] A. Lenard.  On Bogoliubov’s kinetic equation for a spatially homogeneous plasma.  Ann. Phys., 10:390–400, 1960.
  * [21] C. S. Lin.  Interpolation inequalities with weights.  Comm. Partial Differential Equations, 11(14):1515–1538, 1986.
  * [22] N. Magnan, J. B. Fouvry, and P. H. Chavanis.  Orbital alignment and mass segregation in galactic nuclei via vector resonant relaxation.  Mon. Notices Royal Astron. Soc., 514(3):3452–3465, 2022.
  * [23] P. Otte.  Deficiency indices of operator polynomials in creation and annihilation operators.  J. Math. Phys., 44(12):6209–6222, 2003.
  * [24] V. Paulsen.  Completely bounded maps and operator algebras, volume 78 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2002.
  * [25] S. Rabsztyn.  Deficiency indices for Wick powers in one dimension.  Rep. Math. Phys., 27(2):161–168, 1989.
  * [26] K. P. Rauch and S. Tremaine.  Resonant relaxation in stellar systems.  New Astron., 1(2):149–170, 1996.
  * [27] J. B. Taylor and B. McNamara.  Plasma diffusion in two dimensions.  Phys. Fluids, 14(7):1492–1499, 1971.
  * [28] D. A. Tidman, R. L. Guernsey, and D. Montgomery.  “Test Particle” Problem for an Equilibrium Plasma.  Phys. Fluids, 7:1089–1091, 1964.