z,\>|\gamma_{t}-t|\leq\delta t\right)+\mathbb{P}\left(|\gamma_{t}-t|>\delta
t\right),$

and on the other hand, we have

$\mathbb{P}\left(t^{-1/\alpha}Z_{t}\geq
z\right)\geq\mathbb{P}\left(|\gamma_{t}-t|\leq\delta
t\right)-\mathbb{P}\left(t^{-1/\alpha}Z_{t}<z,\>|\gamma_{t}-t|\leq\delta
t\right).$

It follows from the strong law of large number that
$\mathbb{P}(|\gamma_{t}-t|>\delta t)$ converges to $0$ as $t\to\infty$. Now on
the event $\\{|\gamma_{t}-t|\leq\delta t\\}$, we have
$\gamma_{t}\in[(1-\delta)t,(1+\delta)t]$, and thus, reminding that
$Z_{t}=X_{\gamma_{t}}$, we have

$\mathbb{P}\left(t^{-1/\alpha}Z_{t}\geq z,\>|\gamma_{t}-t|\leq\delta
t\right)\leq\mathbb{P}\left(\sup_{s\in[1-\delta,1+\delta]}t^{-1/\alpha}X_{st}\geq
z\right),$

and

$\mathbb{P}\left(t^{-1/\alpha}Z_{t}<z,\>|\gamma_{t}-t|\leq\delta
t\right)\leq\mathbb{P}\left(\inf_{s\in[1-\delta,1+\delta]}t^{-1/\alpha}X_{st}<z\right).$

The two quantities on the right-hand-side of the above equations converge by
Proposition 29 to $\mathbb{P}(\sup_{s\in[1-\delta,1+\delta]}Z_{s}^{\alpha}\geq
z)$ and $\mathbb{P}(\inf_{s\in[1-\delta,1+\delta]}Z_{s}^{\alpha}<z)$ as $t$
tends to infinity. Putting the pieces together, we have

$\limsup_{t\to\infty}\mathbb{P}\left(t^{-1/\alpha}Z_{t}\geq
z\right)\leq\mathbb{P}\left(\sup_{s\in[1-\delta,1+\delta]}Z_{s}^{\alpha}\geq
z\right).$

and

$\liminf_{t\to\infty}\mathbb{P}\left(t^{-1/\alpha}Z_{t}\geq
z\right)\geq\mathbb{P}\left(\inf_{s\in[1-\delta,1+\delta]}Z_{s}^{\alpha}\geq
z\right),$

Since almost surely, $1$ is not a jumping time of $(Z_{t}^{\alpha})_{t\geq
0}$, it should be clear that by letting $\delta\to 0$, we can conclude that
$\lim_{t\to\infty}\mathbb{P}(t^{-1/\alpha}Z_{t}\geq
z)=\mathbb{P}(Z_{1}^{\alpha}\geq z)$. ∎

###  5.3 Proof of Theorem 7

###### Proof of Theorem 7.

Step 1: As explained above, we start by showing that Theorem 7 holds for a
solution $(X_{t},V_{t})_{t\geq 0}$ starting at $(0,0)$. We need to show that
(33) holds. Let $(Z_{t})_{t\geq 0}=(X_{\gamma_{t}})_{t\geq 0}$ as in the
previous subsection, where $(\gamma_{t})_{t\geq 0}$ is the inverse of the
local time $(L_{t})_{t\geq 0}$ at $0$ of $(V_{t})_{t\geq 0}$. Since the
$\bm{\mathrm{M}}_{1}$-topology is weaker than the
$\bm{\mathrm{J}}_{1}$-topology, by Proposition 30 and Theorem 6, we get, for
any $T>0$, for any $\eta>0$,

$\lim_{\delta\to 0}\limsup_{\epsilon\to
0}\mathbb{P}(w(\epsilon^{1/\alpha}Z_{t/\epsilon},T,\delta)>\eta)=0.$

Now we show that for any $T>0$, for any $\eta>0$ and any $\delta\in(0,1)$,

$\mathbb{P}\left(w(\epsilon^{1/\alpha}X_{t/\epsilon},T,\delta)>\eta\right)\leq\mathbb{P}\left(w(\epsilon^{1/\alpha}Z_{t/\epsilon},T+1,2\delta)>\eta\right)+\mathbb{P}\left(\sup_{t\in[0,T]}|\epsilon
L_{t/\epsilon}-t|\geq\delta\right)$ (35)

This will achieve the first step by (34). We first introduce for $t\geq 0$,

$g_{t}=\sup\\{s\leq t,V_{s}=0\\}\qquad\text{and}\qquad d_{t}=\inf\\{s\geq
t,V_{s}=0\\}.$

Note that $g_{t}$ and $d_{t}$ can be expressed in terms of the local time and
its inverse, i.e. $g_{t}=\gamma_{L_{t}-}$ and $d_{t}=\gamma_{L_{t}}$. We also
introduce the random function $\nu(t)$ such that $\nu(t)=1$ if the excursion
straddling the time $t$ is positive and $\nu(t)=-1$ if it is negative, i.e.
$\nu(t)=\bm{1}_{\\{V_{t}>0\\}}-\bm{1}_{\\{V_{t}<0\\}}$.

Let $T>0$ and $\delta\in(0,1)$, we first place ourselves on the event
$A_{T,\delta}=\\{\sup_{t\in[0,T]}|\epsilon L_{t/\epsilon}-t|<\delta\\}$. Let
$t\in[0,T]$ and $t_{\delta-}\leq t_{1}<t_{2}<t_{3}\leq t_{\delta+}$, where
$t_{\delta-}=0\vee(t-\delta)$ and $t_{\delta+}=T\wedge(t+\delta)$. We also
introduce $t_{2\delta-}=0\vee(t-2\delta)$ and
$t_{2\delta+}=(T+1)\wedge(t+2\delta)$. We emphasize that, since we are on the
event $A_{T,\delta}$,
$d_{t/\epsilon}=\gamma_{L_{t/\epsilon}}\leq\gamma_{(t+\delta)/\epsilon}$ for
every $t\in[0,T]$. We first bound the distance
$d(\epsilon^{1/\alpha}X_{t_{2}/\epsilon},[\epsilon^{1/\alpha}X_{t_{1}/\epsilon},\epsilon^{1/\alpha}X_{t_{3}/\epsilon}])=\epsilon^{1/\alpha}d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])$.
Without loss of generality, we will assume that $X_{t_{1}/\epsilon}\leq
X_{t_{3}/\epsilon}$.

  * •

First case: $X_{t_{1}/\epsilon}\leq X_{t_{2}/\epsilon}\leq
X_{t_{3}/\epsilon}$. Then we have

$d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])=0\leq\sup_{t_{2\delta-}\leq
t_{1}<t_{2}<t_{3}\leq
t_{2\delta+}}d(Z_{t_{2}/\epsilon},[Z_{t_{1}/\epsilon},Z_{t_{3}/\epsilon}]).$

  * •

Second case: $X_{t_{2}/\epsilon}<X_{t_{1}/\epsilon}\leq X_{t_{3}/\epsilon}$.
In this case,
$d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])=X_{t_{1}/\epsilon}-X_{t_{2}/\epsilon}$.
Let us note that, since $(X_{t})_{t\geq 0}$ is monotonic on every excursion of
$(V_{t})_{t\geq 0}$, $t_{1}/\epsilon$ and $t_{3}/\epsilon$ can not belong to
the same excursion, i.e. $d_{t_{1}/\epsilon}\leq g_{t_{3}/\epsilon}$. We
define, for $i\in\\{1,2,3\\}$ and $\epsilon>0$, the positive real numbers
$u_{i,\epsilon}$ defined as follows

    1. 1.

$u_{2,\epsilon}=g_{t_{2}/\epsilon}$ if $\nu(t_{2}/\epsilon)=1$,
$u_{2,\epsilon}=d_{t_{2}/\epsilon}$ if $\nu(t_{2}/\epsilon)=-1$ and
$u_{2,\epsilon}=t_{2}/\epsilon$ if $\nu(t_{2}/\epsilon)=0$. Since
$(X_{t})_{t\geq 0}$ is monotonic on every excursion of $(V_{t})_{t\geq 0}$, we
have $X_{t_{2}/\epsilon}\geq X_{u_{2,\epsilon}}$.

    2. 2.

For $i\in\\{1,3\\}$, $u_{i,\epsilon}=d_{t_{i}/\epsilon}$ if
$\nu(t_{i}/\epsilon)=1$, $u_{i,\epsilon}=g_{t_{i}/\epsilon}$ if
$\nu(t_{i}/\epsilon)=-1$ and $u_{i,\epsilon}=t_{i}/\epsilon$ if
$\nu(t_{i}/\epsilon)=-0$. We have $X_{t_{i}/\epsilon}\leq X_{u_{i,\epsilon}}$.

Therefore, we have $X_{u_{2,\epsilon}}<X_{u_{1,\epsilon}}$ and
$X_{u_{2,\epsilon}}<X_{u_{3,\epsilon}}$ so that necessarily,
$u_{1,\epsilon}<u_{2,\epsilon}<u_{3,\epsilon}$. Now we stress that, if $r\geq
0$ is such that $V_{r}=0$ (i.e. such that $\nu(r)=0$), then since the zero set
of $(V_{t})_{t\geq 0}$ has no isolated points, either $r=d_{r}$ or $r=g_{r}$.
In any case, we always have for every $i\in\\{1,2,3\\}$,
$u_{i,\epsilon}=\gamma_{L_{t_{i}/\epsilon}}$ or
$u_{i,\epsilon}=\gamma_{L_{t_{i}/\epsilon}-}$. Let $\theta>0$ and remember
that on the event $A_{T,\delta}$, we have
$L_{t_{i}/\epsilon}\in((t_{i}-\delta)/\epsilon,(t_{i}+\delta)/\epsilon)$ for
every $i\in\\{1,2,3\\}$. Using the fact that $(\gamma_{t})_{t\geq 0}$ is
increasing and that $(X_{t})_{t\geq 0}$ is continuous, we can always find
$s_{1}<s_{2}<s_{3}\in[t_{2\delta-},t_{2\delta+}]$ such that

$X_{t_{2}/\epsilon}\geq X_{\gamma_{s_{2}/\epsilon}}-\frac{\theta}{2},\quad
X_{t_{1}/\epsilon}\leq
X_{\gamma_{s_{1}/\epsilon}}+\frac{\theta}{2}\quad\text{and}\quad
X_{t_{3}/\epsilon}\leq X_{\gamma_{s_{3}/\epsilon}}+\frac{\theta}{2},$

which leads to

$d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])\leq(Z_{s_{1}/\epsilon}-Z_{s_{2}/\epsilon})\wedge(Z_{s_{3}/\epsilon}-Z_{s_{2}/\epsilon})+\theta=d(Z_{s_{2}/\epsilon},[Z_{s_{1}/\epsilon},Z_{s_{3}/\epsilon}])+\theta.$

Since this holds for every $\theta>0$, we deduce the following bound

$d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])\leq\sup_{t_{2\delta-}\leq
t_{1}<t_{2}<t_{3}\leq
t_{2\delta+}}d(Z_{t_{2}/\epsilon},[Z_{t_{1}/\epsilon},Z_{t_{3}/\epsilon}]).$

  * •

Third case: $X_{t_{1}/\epsilon}\leq X_{t_{3}/\epsilon}<X_{t_{2}/\epsilon}$. We
can adapt the previous case to deduce that

$d(X_{t_{2}/\epsilon},[X_{t_{1}/\epsilon},X_{t_{3}/\epsilon}])\leq\sup_{t_{2\delta-}\leq
t_{1}<t_{2}<t_{3}\leq
t_{2\delta+}}d(Z_{t_{2}/\epsilon},[Z_{t_{1}/\epsilon},Z_{t_{3}/\epsilon}]).$

To summarize, we proved that, on the event $A_{T,\delta}$, the following bound
holds

$w(\epsilon^{1/\alpha}X_{t/\epsilon},T,\delta)\leq
w(\epsilon^{1/\alpha}Z_{t/\epsilon},T+1,2\delta).$

This implies (35). We proved Theorem 7 in the case $v_{0}=0$.

Step 2: We consider the solution $(X_{t},V_{t})_{t\geq 0}$ of (1) starting at
$(0,v_{0})$ where $v_{0}\in\mathbb{R}$. We show that there exists a constant
$C>0$ such that for any $T>0$,

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|\Big{]}\leq C(1+T^{1/2}).$ (36)

To this aim, we start by studying $\sup_{t\in[0,T]}|V_{t}|^{\beta+1}$ and we
introduce, recall Assumption 1, the even function $\ell$ defined as

$\ell(v)=2\int_{0}^{v}\Theta^{-\beta}(u)\int_{0}^{u}\Theta^{\beta}(w)\mathrm{d}w\mathrm{d}u,$

which solves the Poisson equation
$\ell^{\prime}\mathrm{F}+\frac{1}{2}\ell^{\prime\prime}=1$. Then by the Itô
formula, we have

$\ell(V_{t})=\ell(v_{0})+t+\int_{0}^{t}\ell^{\prime}(V_{s})\mathrm{d}B_{s}.$

Moreover, remember that $|v|\Theta(v)\to 1$ as $v\to\pm\infty$ and that
$\beta>1$. As a consequence, there exist positive constants $c,c^{\prime}>0$
such that

$\ell(v)\sim c|v|^{\beta+1}\quad\text{and}\quad\ell^{\prime}(v)\sim
c^{\prime}\text{sgn}(v)|v|^{\beta}\quad\text{as }v\to\pm\infty.$

Therefore, there exist positive constants $M,M^{\prime}>0$ such that for all
$v\in\mathbb{R}$,

$|v|^{\beta+1}\leq M(1+\ell(v))\quad\text{and}\quad[\ell^{\prime}(v)]^{2}\leq
M^{\prime}(1+v^{2\beta}).$ (37)

Using (37) and Doob’s inequality, we get

$\displaystyle\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta+1}\Big{]}$
$\displaystyle\leq
M\left[1+\ell(v_{0})+T+4\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}[\ell^{\prime}(V_{s})]^{2}\mathrm{d}s\Big{)}^{1/2}\Big{]}\right]$
$\displaystyle\leq
M\left[1+\ell(v_{0})+T+4(M^{\prime}T)^{1/2}\right]+4M(M^{\prime}T)^{1/2}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta}\Big{]}.$
(38)

Finally, using Young’s inequality with $p=\beta+1$ and $q=(\beta+1)/\beta$, we
find some $C>0$ such that

$\displaystyle
4M(M^{\prime}T)^{1/2}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta}\Big{]}\leq$
$\displaystyle
CT^{(\beta+1)/2}+\frac{1}{2}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta}\Big{]}^{(\beta+1)/\beta}$
$\displaystyle\leq$ $\displaystyle
CT^{(\beta+1)/2}+\frac{1}{2}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta+1}\Big{]},$

which, inserted in (38), implies that there exists a constant $K>0$ such that

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}|V_{t}|^{\beta+1}\Big{]}\leq
K(1+T^{(\beta+1)/2}).$

We deduce (36) by Hölder’s inequality again.

Step 3: We finally show the result for any solution $(X_{t},V_{t})_{t\geq 0}$
starting at $(0,v_{0})$, where $v_{0}\in\mathbb{R}$, and we extend the
technique used in [25, page 21]. If we set $T_{0}=\inf\\{t\geq 0,V_{t}=0\\}$,
then the process

$(\bar{X}_{t},\bar{V}_{t})_{t\geq
0}=(X_{t+T_{0}}-X_{T_{0}},V_{t+T_{0}})_{t\geq 0}$

is a solution of (1) starting at $(0,0)$. Therefore, by the first step, the
process $(\epsilon^{1/\alpha}\bar{X}_{t/\epsilon})_{t\geq 0}$ converges to
$(Z_{t}^{\alpha})_{t\geq 0}$ in the space $\mathcal{D}$ endowed with the
$\bm{\mathrm{M}}_{1}$-topology. Then, by a version of the Slutsky lemma, see
for instance [11, Section 3, Theorem 3.1], it is enough to show that for any
$T>0$,

$\sup_{t\in[0,T]}\epsilon^{1/\alpha}\left|X_{t/\epsilon}-\bar{X}_{t/\epsilon}\right|\overset{\mathbb{P}}{\longrightarrow}0\quad\hbox{as
$\epsilon\to 0$}.$ (39)

Indeed, the $\bm{\mathrm{M}}_{1}$-topology is weaker than the topology induced
by the uniform convergence on compact time-intervals. We distinguish two
cases. First,

$\bm{1}_{\\{T_{0}\geq
t/\epsilon\\}}\left|X_{t/\epsilon}-\bar{X}_{t/\epsilon}\right|\leq\bm{1}_{\\{T_{0}\geq
t/\epsilon\\}}\left|X_{t/\epsilon}\right|+\bm{1}_{\\{T_{0}\geq
t/\epsilon\\}}\left|X_{t/\epsilon+T_{0}}-X_{T_{0}}\right|\leq\int_{0}^{2T_{0}}|V_{s}|\mathrm{d}s=D_{1}.$

Second,

$\bm{1}_{\\{T_{0}<t/\epsilon\\}}\left|X_{t/\epsilon}-\bar{X}_{t/\epsilon}\right|\leq\left|X_{T_{0}}\right|+\bm{1}_{\\{T_{0}<t/\epsilon\\}}\left|X_{t/\epsilon+T_{0}}-X_{t/\epsilon}\right|\leq
D_{1}+\bm{1}_{\\{T_{0}<t/\epsilon\\}}\int_{t/\epsilon}^{t/\epsilon+T_{0}}|V_{s}|\mathrm{d}s.$

Hence, if we set
$D_{t,\epsilon}^{2}=\bm{1}_{\\{T_{0}<t/\epsilon\\}}\int_{t/\epsilon}^{t/\epsilon+T_{0}}|V_{s}|\mathrm{d}s$,
we get

$\sup_{t\in[0,T]}\epsilon^{1/\alpha}\left|X_{t/\epsilon}-\bar{X}_{t/\epsilon}\right|\leq\epsilon^{1/\alpha}D_{1}+\sup_{t\in[0,T]}\epsilon^{1/\alpha}D_{t,\epsilon}^{2}.$

The first term converges almost surely to $0$ as $\epsilon\to 0$. Regarding
the second one, we have

$\displaystyle\mathbb{E}\Big{[}\sup_{t\in[0,T]}\epsilon^{1/\alpha}D_{t,\epsilon}^{2}\Big{|}\mathcal{F}_{T_{0}}\Big{]}$
$\displaystyle\leq\epsilon^{1/\alpha}T_{0}\mathbb{E}\Big{[}\sup_{t\in[0,T]}\bm{1}_{\\{T_{0}<t/\epsilon\\}}\sup_{s\in[t/\epsilon,t/\epsilon+T_{0}]}|V_{s}|\Big{|}\mathcal{F}_{T_{0}}\Big{]}$
$\displaystyle\leq\epsilon^{1/\alpha}T_{0}\bm{1}_{\\{T_{0}<T/\epsilon\\}}\mathbb{E}\Big{[}\sup_{t\in[0,2T/\epsilon]}|V_{s}|\Big{]}$
$\displaystyle\leq\epsilon^{1/\alpha}T_{0}C(1+(2T)^{1/2}\epsilon^{-1/2})$

by Step 2. This last quantity almost surely goes to $0$ as $\alpha\in(0,2)$.
This implies the convergence in probability of
$\sup_{t\in[0,T]}\epsilon^{1/\alpha}D_{t,\epsilon}^{2}$ to $0$, so that (39)
holds. ∎

## Appendix

##  Appendix A Some useful results

######  Lemma 31.

Let $X$ and $Y$ be two positive random variables and let $C>0$. Assume that we
have $\mathbb{P}(X>t)\sim Ct^{-1/2}$ as $t\to\infty$. Then the following
assertions are equivalent.

  1. (i)

$\lim_{t\to\infty}t^{1/2}\mathbb{P}(Y>t)=0$.

  2. (ii)

$\mathbb{P}(X+Y>t)\sim Ct^{-1/2}$ as $t\to\infty$.

###### Proof.

Let us first show that (i) implies (ii). Let $\delta\in(0,1)$ and write

$\displaystyle\mathbb{P}(X+Y>t)=$ $\displaystyle\>\mathbb{P}(X+Y>t,\>X>\delta
t)+\mathbb{P}(X+Y>t,\>X\leq\delta t)$ $\displaystyle\leq$
$\displaystyle\>\mathbb{P}(X>\delta t)+\mathbb{P}(Y>(1-\delta)t).$

We deduce that

$C=\liminf_{t\to\infty}t^{1/2}\mathbb{P}(X>t)\leq\liminf_{t\to\infty}t^{1/2}\mathbb{P}(X+Y>t)\leq\limsup_{t\to\infty}t^{1/2}\mathbb{P}(X+Y>t)\leq
C\delta^{-1/2}.$

Letting $\delta\to 1$ completes the first step.

We now show that (ii) implies (i). We first remark that

$\\{Y>3t\\}\subset\bigcup_{n\in\mathbb{N}}\\{X+Y>(n+1)t\\}\cap\\{X\leq nt\\}.$

Indeed, if $Y>3t$, we set $n+2=\lfloor\frac{X+Y}{t}\rfloor\geq 3$, and get
$\frac{X}{t}\leq\frac{X+Y}{t}-3\leq n$ and $\frac{X+Y}{t}>n+1$. Now let
$N\in\mathbb{N}$, then we have $(\cup_{n>N}\\{X+Y>(n+1)t\\}\cap\\{X\leq
nt\\})\subset\\{X+Y>(N+2)t\\}$ and therefore

$\mathbb{P}(Y>3t)\leq\mathbb{P}(X+Y>(N+2)t)+\sum_{n=1}^{N}\mathbb{P}(X+Y>(n+1)t,\>X\leq
nt).$

For any $n\geq 1$, we have

$\displaystyle\mathbb{P}(X+Y>(n+1)t,\>X\leq nt)=$
$\displaystyle\>\mathbb{P}(X+Y>(n+1)t)-\mathbb{P}(X+Y>(n+1)t,\>X>nt)$
$\displaystyle\leq\mathbb{P}(X+Y>(n+1)t)-\mathbb{P}(X>(n+1)t),$

and thus $\lim_{t\to\infty}t^{1/2}\mathbb{P}(X+Y>(n+1)t,\>X\leq nt)=0$, from
which we deduce that for any $N\in\mathbb{N}$,

$\limsup_{t\to\infty}t^{1/2}\mathbb{P}(Y>3t)\leq C(N+2)^{-1/2}.$

Letting $N\to\infty$ completes the proof. ∎

######  Proposition 32.

Let $(X_{n})_{n\in\mathbb{N}}$ be i.i.d positive random variables and let
$\alpha\in(0,1)$ such that $\lim_{t\to\infty}t^{\alpha}\mathbb{P}(X_{1}>t)=0$.
If $S_{n}=\sum_{k=1}^{n}X_{k}$, then $n^{-1/\alpha}S_{n}$ converges to $0$ in
probability as $n\to\infty$.

###### Proof.

We will show that the Laplace transform of $n^{-1/\alpha}S_{n}$ converges to 1
as $n\to\infty$. We have for any $\lambda>0$,

$\mathbb{E}\left[e^{-\lambda
n^{-1/\alpha}S_{n}}\right]=\mathbb{E}\left[e^{-\lambda
n^{-1/\alpha}X_{1}}\right]^{n},$

and thus

$\log\mathbb{E}\left[e^{-\lambda
n^{-1/\alpha}S_{n}}\right]\underset{n\to\infty}{\sim}n\mathbb{E}\left[e^{-\lambda
n^{-1/\alpha}X_{1}}-1\right]=\frac{-\lambda}{n^{1/\alpha-1}}\int_{0}^{\infty}e^{-\lambda
u/n^{1/\alpha}}\mathbb{P}(X_{1}>u)\mathrm{d}u.$

Then we use that for $u\geq n^{1/\alpha}$,
$\mathbb{P}(X_{1}>u)\leq\mathbb{P}(X_{1}>n^{1/\alpha})$ and we get

$\frac{\lambda}{n^{1/\alpha-1}}\int_{0}^{\infty}e^{-\lambda
u/n^{1/\alpha}}\mathbb{P}(X_{1}>u)\mathrm{d}u\leq\frac{\lambda}{n^{1/\alpha-1}}\int_{0}^{n^{1/\alpha}}\mathbb{P}(X_{1}>u)\mathrm{d}u+e^{-\lambda}n\mathbb{P}(X_{1}>n^{1/\alpha}).$

The second term on the right-hand-side converges to $0$ by assumption.
Regarding the first term, since $\mathbb{P}(X_{1}>u)=o(u^{-\alpha})$ as
$u\to\infty$ and since
$\lim_{x\to\infty}\int_{1}^{x}u^{-\alpha}\mathrm{d}=\infty$, we have
$\int_{0}^{n^{1/\alpha}}\mathbb{P}(X_{1}>u)\mathrm{d}u=o(\int_{1}^{n^{1/\alpha}}u^{-\alpha}\mathrm{d}u)=o(n^{1/\alpha-1})$
as $n\to\infty$, which completes the proof. ∎

##  Appendix B On the PDE result

In this subsection, we formalize the P.D.E. result briefly exposed in the
introduction. We first explain how the law of $(\bm{X}_{t},\bm{V}_{t})_{t\geq
0}$ is linked with a kinetic Fokker-Planck equation with diffusive boundary
conditions. For every $\varphi\in C^{2}_{b}(\mathbb{R})$, we define
$\mathcal{L}\varphi=\mathrm{F}\varphi^{\prime}+\frac{1}{2}\varphi^{\prime\prime}$.
Then $\mathcal{L}$ is the infinitesimal generator of the (free) speed process
$(V_{t})_{t\geq 0}$. We also denote by $\mathcal{L}^{*}$ its adjoint operator
which is such that
$\mathcal{L}^{*}\varphi=\frac{1}{2}\varphi^{\prime\prime}-[F\varphi]^{\prime}$.

######  Proposition 33.

Let $(\bm{X}_{t},\bm{V}_{t})_{t\geq 0}$ be a solution to (3) starting at
$(x_{0},v_{0})\in((0,\infty)\times\mathbb{R})\cup(\\{0\\}\times(0,\infty))$.
Let us denote by
$f(\mathrm{d}t,\mathrm{d}x,\mathrm{d}v)=\mathbb{P}(\bm{X}_{t}\in\mathrm{d}x,\bm{V}_{t}\in\mathrm{d}v)\mathrm{d}t$
which is a measure on $\mathbb{R}_{+}^{2}\times\mathbb{R}$. There exist two
measures $\nu_{-}\in\mathcal{M}(\mathbb{R}_{+}\times\mathbb{R}_{-})$ and
$\nu_{+}\in\mathcal{M}(\mathbb{R}_{+}^{2})$ such that for every $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+}^{2}\times\mathbb{R})$, we have

$\displaystyle\varphi(0,x_{0},v_{0})+\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}$
$\displaystyle\left[\partial_{t}\varphi+v\partial_{x}\varphi+\mathcal{L}\varphi\right]f(\mathrm{d}s,\mathrm{d}x,\mathrm{d}v)$
$\displaystyle+\int_{\mathbb{R}_{+}^{2}}\varphi(s,0,v)\nu_{+}(\mathrm{d}s,\mathrm{d}v)-\int_{\mathbb{R}_{+}\times\mathbb{R}_{-}}\varphi(s,0,v)\nu_{-}(\mathrm{d}s,\mathrm{d}v)=0.$
(40)

Moreover the measures $\nu_{-}$ and $\nu_{+}$ satisfy
$\nu_{+}(\mathrm{d}t,\mathrm{d}v)=\mu(\mathrm{d}v)\int_{w\in\mathbb{R}_{-}}\nu_{-}(\mathrm{d}t,\mathrm{d}w)$.

###### Proof.

Let $\varphi\in C^{\infty}_{c}(\mathbb{R}_{+}^{2}\times\mathbb{R})$, then by
Itô’s formula, and passing to the expectation, we get for every $T>0$ that

$\displaystyle\mathbb{E}\left[\varphi(T,\bm{X}_{T},\bm{V}_{T})\right]=$
$\displaystyle\>\varphi(0,x_{0},v_{0})+\int_{0}^{T}\mathbb{E}\left[(\partial_{t}\varphi+v\partial_{x}\varphi+\mathcal{L}\varphi)(s,\bm{X}_{s},\bm{V}_{s})\right]\mathrm{d}s$
$\displaystyle+\sum_{n\in\mathbb{N}}\mathbb{E}\left[(\varphi(\bm{\tau}_{n},0,M_{n})-\varphi(\bm{\tau}_{n},0,\bm{V}_{\tau_{n}-}))\bm{1}_{\\{\bm{\tau}_{n}\leq
t\\}}\right].$

The local martingale part
$\int_{0}^{T}\partial_{v}\varphi(s,\bm{X}_{s},\bm{V}_{s})\mathrm{d}B_{s}$ is
indeed a true martingale since $\partial_{v}\varphi$ is bounded. Let us now
define the measures
$\nu_{-}\in\mathcal{M}(\mathbb{R}_{+}\times\mathbb{R}_{-})$ and
$\nu_{+}\in\mathcal{M}(\mathbb{R}_{+}^{2})$ by

$\nu_{-}(\mathrm{d}t,\mathrm{d}v)=\sum_{n\in\mathbb{N}}\mathbb{P}(\bm{\tau}_{n}\in\mathrm{d}t,\bm{V}_{\tau_{n}-}\in\mathrm{d}v)\quad\text{and}\quad\nu_{+}(\mathrm{d}t,\mathrm{d}v)=\sum_{n\in\mathbb{N}}\mathbb{P}(\bm{\tau}_{n}\in\mathrm{d}t,M_{n}\in\mathrm{d}v).$

Notice that for any $T>0$, we have
$\nu_{-}([0,T]\times\mathbb{R}_{-})=\nu_{+}([0,T]\times\mathbb{R}_{+})=\mathbb{E}[\sum_{n\in\mathbb{N}}\bm{1}_{\\{\bm{\tau}_{n}\leq
T\\}}]$, which is finite since $(\bm{\tau}_{n+1}-\bm{\tau}_{n})_{n\geq 1}$ is
an i.i.d. sequence of positive random variables. This also justifies the above
exchange between $\mathbb{E}$ and $\sum$. Therefore we have for any $T>0$

$\displaystyle\mathbb{E}\left[\varphi(T,\bm{X}_{T},\bm{V}_{T})\right]=$
$\displaystyle\>\varphi(0,x_{0},v_{0})+\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}\left[\partial_{t}\varphi+v\partial_{x}\varphi+\mathcal{L}\varphi\right]\bm{1}_{\\{s\leq
T\\}}f(\mathrm{d}s,\mathrm{d}x,\mathrm{d}v)$
$\displaystyle+\int_{\mathbb{R}_{+}^{2}}\varphi(s,0,v)\bm{1}_{\\{s\leq
T\\}}\nu_{+}(\mathrm{d}s,\mathrm{d}v)-\int_{\mathbb{R}_{+}\times\mathbb{R}_{-}}\varphi(s,0,v)\bm{1}_{\\{s\leq
T\\}}\nu_{-}(\mathrm{d}s,\mathrm{d}v).$

Choosing $T$ large enough so that the support of $\varphi$ is included in
$[0,T]\times\mathbb{R}_{+}\times\mathbb{R}$, we get the desired identity.

The relation between the two measures comes from the fact that for every
$n\in\mathbb{N}$, $\bm{\tau}_{n}$ and $M_{n}$ are independent and that $M_{n}$
is $\mu$-distributed. Indeed it is clear that, since
$(M_{n})_{n\in\mathbb{N}}$ is i.i.d. and also independent from the driving
Brownian moton, $M_{n}$ is independent from $(\bm{X}_{t},\bm{V}_{t})_{0\leq
t<\bm{\tau}_{n}}$ for every $n\in\mathbb{N}$. Hence we have

$\nu_{+}(\mathrm{d}t,\mathrm{d}v)=\mu(\mathrm{d}v)\sum_{n\in\mathbb{N}}\mathbb{P}(\bm{\tau}_{n}\in\mathrm{d}t)=\mu(\mathrm{d}v)\int_{w\in\mathbb{R}_{-}}\nu_{-}(\mathrm{d}t,\mathrm{d}w),$

which achieves the proof. ∎

######  Remark 34.

Assume for simplicity that $\mu(\mathrm{d}v)=\mu(v)\mathrm{d}v$. The preceding
proposition shows that $f$ is a weak solution of

$\left\\{\begin{aligned}
\partial_{t}&f+v\partial_{x}f=\mathcal{L}^{*}f\quad&\text{for}\>\>(t,x,v)\in(0,\infty)^{2}\times\mathbb{R}\\\
v&f(t,0,v)=-\mu(v)\int_{(-\infty,0)}wf(t,0,w)\mathrm{d}w&\text{for}\>\>(t,v)\in(0,\infty)^{2}\\\
\>\>f&(0,\cdot,\cdot)=\delta_{(0,v_{0})}\end{aligned}\right.$

Informally, it automatically holds that
$\nu_{+}(\mathrm{d}s,\mathrm{d}v)=vf(s,0,v)\bm{1}_{\\{v>0\\}}\mathrm{d}s\mathrm{d}v$
and
$\nu_{-}(\mathrm{d}s,\mathrm{d}v)=-vf(s,0,v)\bm{1}_{\\{v<0\\}}\mathrm{d}s\mathrm{d}v$.

For similar notions of weak solutions associated to closely related equations,
we refer to Jabir-Profeta [26, Theorem 4.2.1] and Bernou-Fournier [4,
Definition 4].

###### Proof.

We assume that
$f(\mathrm{d}t,\mathrm{d}x,\mathrm{d}v)=f(t,x,v)\mathrm{d}x\mathrm{d}v\mathrm{d}t$
with $f$ smooth enough. Let $\varphi$ be a function belonging to
$C^{\infty}_{c}(\mathbb{R}_{+}^{2}\times\mathbb{R})$. We perform some
integrations by parts. We have

$\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}f\partial_{t}\varphi=-\varphi(0,x_{0},v_{0})-\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}\varphi\partial_{t}f\quad\text{and}\quad\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}f\mathcal{L}\varphi=\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}\varphi\mathcal{L}^{*}f.$

Regarding the integration by part in $x$, we have

$\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}vf\partial_{x}\varphi=-\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}\varphi
v\partial_{x}f-\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}v\varphi(s,0,v)f(s,0,v)\mathrm{d}v\mathrm{d}s.$

Inserting the previous identities in (33), it comes that

$\displaystyle\int_{\mathbb{R}_{+}^{2}\times\mathbb{R}}\left[\partial_{t}f+v\partial_{x}f-\mathcal{L}^{*}f\right]\varphi+$
$\displaystyle\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}v\varphi(s,0,v)f(s,0,v)\mathrm{d}v\mathrm{d}s$
$\displaystyle=\int_{\mathbb{R}_{+}^{2}}\varphi(s,0,v)\nu_{+}(\mathrm{d}s,\mathrm{d}v)-\int_{\mathbb{R}_{+}\times\mathbb{R}_{-}}\varphi(s,0,v)\nu_{-}(\mathrm{d}s,\mathrm{d}v).$

Since this holds for every $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+}\times(0,\infty)\times\mathbb{R})$, we first
conclude that $\partial_{t}f+v\partial_{x}f=\mathcal{L}^{*}f$ for
$(t,x,v)\in(0,\infty)^{2}\times\mathbb{R}$. Then in a second time, we see that

$vf(s,0,v)\mathrm{d}s\mathrm{d}v=\nu_{+}(\mathrm{d}s,\mathrm{d}v)-\nu_{-}(\mathrm{d}s,\mathrm{d}v),$

i.e.
$\nu_{+}(\mathrm{d}s,\mathrm{d}v)=vf(s,0,v)\bm{1}_{\\{v>0\\}}\mathrm{d}s\mathrm{d}v$
and
$\nu_{-}(\mathrm{d}s,\mathrm{d}v)=-vf(s,0,v)\bm{1}_{\\{v<0\\}}\mathrm{d}s\mathrm{d}v$.
Then, since
$\nu_{+}(\mathrm{d}s,\mathrm{d}v)=\mu(\mathrm{d}v)\int_{w\in\mathbb{R}_{-}}\nu_{-}(\mathrm{d}s,\mathrm{d}w)$,
we conclude that $vf(t,0,v)=-\mu(v)\int_{(-\infty,0)}wf(t,0,w)\mathrm{d}w$ for
$(t,v)\in(0,\infty)^{2}$. ∎

We finally study the limiting fractional diffusion equation. We have the
following result.

######  Proposition 35.

Let $(R_{t}^{\alpha})_{t\geq
0}=(Z_{t}^{\alpha}-\inf_{s\in[0,t]}Z_{s}^{\alpha})_{t\geq 0}$ where
$(Z_{t}^{\alpha})_{t\geq 0}$ is the stable process from Theorem 3. Let us
denote by $\rho_{t}(\mathrm{d}x)=\mathbb{P}(R_{t}^{\alpha}\in\mathrm{d}x)$.
The following assertions hold.

  1. (i)

If $\alpha\in(0,1)$, then for every $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+})$, we have

$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}(\mathrm{d}x)=\varphi(0)+\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s,$

where

$\mathcal{L}^{\alpha}\varphi(x)=\frac{\sigma_{\alpha}}{2}\int_{\mathbb{R}}\frac{\varphi((x+z)_{+})-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z=\frac{\sigma_{\alpha}}{2\alpha}\int_{0}^{\infty}\frac{\varphi^{\prime}(y)(y-x)}{|y-x|^{\alpha+1}}\mathrm{d}y.$

  2. (ii)

If $\alpha\in[1,2)$, then for every $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+})$ such that $\varphi^{\prime}(0)=0$, we have

$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}(\mathrm{d}x)=\varphi(0)+\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s,$

where

$\mathcal{L}^{\alpha}\varphi(x)=\frac{\sigma_{\alpha}}{2}\mathrm{P.V.}\int_{\mathbb{R}}\frac{\varphi((x+z)_{+})-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z=\frac{\sigma_{\alpha}}{2\alpha}\mathrm{P.V.}\int_{0}^{\infty}\frac{\varphi^{\prime}(y)(y-x)}{|y-x|^{\alpha+1}}\mathrm{d}y,$

where $\mathrm{P.V.}$ stands for principal values.

###### Proof.

We will denote by
$\nu(\mathrm{d}z)=\frac{\sigma_{\alpha}}{2}|z|^{-\alpha-1}\mathrm{d}z$ the
Lévy measure of the symmetric stable process $(Z_{t}^{\alpha})_{t\geq 0}$ and
we set $I_{t}^{\alpha}=\inf_{s\in[0,t]}Z_{s}^{\alpha}$.

Item (i): Let us denote by $\Pi(\mathrm{d}t,\mathrm{d}z)$ the random Poisson
measure on $\mathbb{R}_{+}\times\mathbb{R}$ with intensity
$\mathrm{d}t\otimes\nu(\mathrm{d}z)$, associated with $(Z_{t}^{\alpha})_{t\geq
0}$. Since $\alpha\in(0,1)$, $(Z_{t}^{\alpha})_{t\geq 0}$ has finite
variations and we have
$Z_{t}^{\alpha}=\int_{0}^{t}\int_{\mathbb{R}}z\Pi(\mathrm{d}s,\mathrm{d}z)$.
The process $(R_{t}^{\alpha})_{t\geq 0}$ is also a pure jump Markov process
and we check that $\Delta R_{t}^{\alpha}=(R_{t-}^{\alpha}+\Delta
Z_{t}^{\alpha})_{+}-R_{t-}^{\alpha}$. If first $\Delta
Z_{t}^{\alpha}>-R_{t-}^{\alpha}=-Z_{t-}^{\alpha}+I_{t-}^{\alpha}$, then
$Z_{t}^{\alpha}>I_{t-}^{\alpha}$ so that $I_{t}^{\alpha}=I_{t-}^{\alpha}$ and
$\Delta R_{t}^{\alpha}=\Delta Z_{t}^{\alpha}$. If next $\Delta
Z_{t}^{\alpha}\leq-R_{t-}^{\alpha}$, then $Z_{t}^{\alpha}\leq I_{t-}^{\alpha}$
so that $I_{t}^{\alpha}=Z_{t}^{\alpha}$ and thus $R_{t}^{\alpha}=0$ i.e.
$\Delta R_{t}^{\alpha}=-R_{t-}^{\alpha}$. Therefore, we have
$R_{t}^{\alpha}=\int_{0}^{t}\int_{\mathbb{R}}[(R_{s-}^{\alpha}+z)_{+}-R_{s-}^{\alpha}]\Pi(\mathrm{d}s,\mathrm{d}z)$.
By Itô’s formula, we get that for any $\varphi\in
C^{\infty}_{c}(\mathbb{R}_{+})$,

$\mathbb{E}\left[\varphi(R_{t}^{\alpha})\right]=\varphi(0)+\int_{0}^{t}\int_{\mathbb{R}}\mathbb{E}\left[\varphi((R_{s}^{\alpha}+z)_{+})-\varphi(R_{s}^{\alpha})\right]\nu(\mathrm{d}x)\mathrm{d}s,$

which exactly means
$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}(\mathrm{d}x)=\varphi(0)+\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s$,
where the operator $\mathcal{L}^{\alpha}$ is defined as
$\mathcal{L}^{\alpha}\varphi(x)=\frac{\sigma_{\alpha}}{2}\int_{\mathbb{R}}[\varphi((x+z)_{+})-\varphi(x)]|z|^{-1-\alpha}\mathrm{d}z$.
It only remains to prove the second identity for $\mathcal{L}^{\alpha}$. We
assume that $x>0$, the proof for $x=0$ being similar. We have

$\frac{2}{\sigma_{\alpha}}\mathcal{L}^{\alpha}\varphi(x)=\frac{\varphi(0)-\varphi(x)}{\alpha}x^{-\alpha}+\int_{-x}^{0}\frac{\varphi(x+z)-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z+\int_{0}^{\infty}\frac{\varphi(x+z)-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z$

Then, performing carefully two integration by parts, we see that

$\int_{-x}^{0}\frac{\varphi(x+z)-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z=-\frac{\varphi(0)-\varphi(x)}{\alpha}x^{-\alpha}+\frac{1}{\alpha}\int_{-x}^{0}\frac{\varphi^{\prime}(x+z)z}{|z|^{1+\alpha}}\mathrm{d}z$

and

$\int_{0}^{\infty}\frac{\varphi(x+z)-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z=\frac{1}{\alpha}\int_{0}^{\infty}\frac{\varphi^{\prime}(x+z)z}{|z|^{1+\alpha}}\mathrm{d}z.$

Putting the pieces together, we get

$\frac{2}{\sigma_{\alpha}}\mathcal{L}^{\alpha}\varphi(x)=\frac{1}{\alpha}\int_{-x}^{\infty}\frac{\varphi^{\prime}(x+z)z}{|z|^{1+\alpha}}\mathrm{d}z=\frac{1}{\alpha}\int_{0}^{\infty}\frac{\varphi^{\prime}(y)(y-x)}{|y-x|^{1+\alpha}}\mathrm{d}y.$

Item (ii): When $\alpha\in[1,2)$, the second expression of
$\mathcal{L}^{\alpha}$ is obtained as in the previous case. We need to remove
the small jumps and work with the pure jump Lévy process
$(Z_{t}^{\alpha,\delta})_{t\geq 0}$ with Lévy measure
$\nu_{\delta}(\mathrm{d}z)=\bm{1}_{\\{|z|>\delta\\}}\nu(\mathrm{d}z)$. Since
$\nu$ is symmetric, $Z_{t}^{\alpha,\delta}\to Z_{t}^{\alpha}$ in law as
$\delta\to 0$ for every $t\geq 0$. This implies, see [29, Chapter VII,
Corollary 3.6], that $(Z_{t}^{\alpha,\delta})_{t\geq 0}$ converges in law to
$(Z_{t}^{\alpha})_{t\geq 0}$ in the space of càdlàg functions endowed with the
$\bm{\mathrm{J}}_{1}$-topology as $\delta\to 0$. But since the reflection map
is continuous with respect to this topology, see [44, Chapter 13, Theorem
13.5.1], the continuous mapping theorem implies that
$(R_{t}^{\alpha,\delta})_{t\geq
0}=(Z_{t}^{\alpha,\delta}-\inf_{s\in[0,t]}Z_{s}^{\alpha,\delta})_{t\geq 0}$
converges weakly to $(R_{t}^{\alpha})_{t\geq 0}$ as $\delta\to 0$. In
particular, if we set
$\rho_{t}^{\delta}(\mathrm{d}x)=\mathbb{P}(R_{t}^{\alpha,\delta}\in\mathrm{d}x)$,
then the probability measure $\rho_{t}^{\delta}$ converges weakly to
$\rho_{t}$ as $\delta\to 0$.

But since $(R_{t}^{\alpha,\delta})_{t\geq 0}$ has finite variations, we can
use the very same argument as in the first step to see that for every function
$\varphi\in C^{\infty}_{c}(\mathbb{R}_{+})$, we have

$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}^{\delta}(\mathrm{d}x)=\varphi(0)+\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha,\delta}\varphi(x)\rho_{s}^{\delta}(\mathrm{d}x)\mathrm{d}s,$

where $\mathcal{L}^{\alpha,\delta}$ is such that for every $x\geq 0$

$\mathcal{L}^{\alpha,\delta}\varphi(x)=\frac{\sigma_{\alpha}}{2}\int_{|z|>\delta}\frac{\varphi((x+z)_{+})-\varphi(x)}{|z|^{1+\alpha}}\mathrm{d}z.$

First, it is clear that
$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}^{\delta}(\mathrm{d}x)$ converges to
$\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}(\mathrm{d}x)$ as $\delta\to 0$. We
will now conclude by showing that, if $\varphi^{\prime}(0)=0$,

$\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha,\delta}\varphi(x)\rho_{s}^{\delta}(\mathrm{d}x)\mathrm{d}s\longrightarrow\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s\quad\text{as
}\delta\to 0.$ (41)

Step 1: To do so, we show in Step 2 that there exists a positive constant
$C_{\varphi}$ such that

$||\mathcal{L}^{\alpha}\varphi-\mathcal{L}^{\alpha,\delta}\varphi||_{\infty}\leq
C_{\varphi}\delta^{2-\alpha}.$ (42)

This shows that $\mathcal{L}^{\alpha}\varphi$ is a continuous and bounded
function. Moreover, since

$\displaystyle\Big{|}\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s-$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha,\delta}\varphi(x)\rho_{s}^{\delta}(\mathrm{d}x)\mathrm{d}s\Big{|}$
$\displaystyle\leq\int_{0}^{t}\int_{\mathbb{R}_{+}}|\mathcal{L}^{\alpha}\varphi(x)-\mathcal{L}^{\alpha,\delta}\varphi(x)|\rho_{s}^{\delta}(\mathrm{d}x)\mathrm{d}s$
$\displaystyle+\Big{|}\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}^{\delta}(\mathrm{d}x)\mathrm{d}s-\int_{0}^{t}\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)\mathrm{d}s\Big{|},$

it is clear that (42) implies (41). Indeed the first term clearly goes to $0$
as $\delta\to 0$ and the second term converges since for every $s\in[0,t]$,
$\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}^{\delta}(\mathrm{d}x)\to\int_{\mathbb{R}_{+}}\mathcal{L}^{\alpha}\varphi(x)\rho_{s}(\mathrm{d}x)$
as $\mathcal{L}^{\alpha}\varphi$ is a continuous and bounded function and we
can thus apply the dominated convergence theorem.

Step 2: We show that (42) holds. Let us first explicit a bit more
$\mathcal{L}^{\alpha}\varphi(x)$. Let us remark that since
$\varphi^{\prime}(0)=0$, there is no need for principal values for
$\mathcal{L}^{\alpha}\varphi(0)=\int_{0}^{\infty}(\varphi(z)-\varphi(0))|z|^{-1-\alpha}\mathrm{d}z$.
If $x>0$, we have for any $\epsilon<x$,
$\int_{-x}^{\epsilon}z|z|^{-1-\alpha}\mathrm{d}z+\int_{\epsilon}^{x}z|z|^{-1-\alpha}=0$

$\mathcal{L}^{\alpha}\varphi(x)=\int_{\mathbb{R}}\frac{\varphi((x+z)_{+})-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z.$
(43)

For the very same reason, the same identity holds for
$\mathcal{L}^{\alpha,\delta}\varphi(x)$ and therefore, for any $x>0$

$\displaystyle\mathcal{L}^{\alpha}\varphi(x)-\mathcal{L}^{\alpha,\delta}\varphi(x)=$
$\displaystyle\int_{|z|<\delta}\frac{\varphi((x+z)_{+})-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z$
$\displaystyle=$
$\displaystyle\frac{\varphi(0)-\varphi(x)}{\alpha}(x^{-\alpha}-\delta^{-\alpha})\bm{1}_{\\{\delta>x\\}}$
$\displaystyle+\int_{-(\delta\wedge
x)}^{\delta}\frac{\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z.$

Since $\varphi\in C^{\infty}_{c}(\mathbb{R}_{+})$ and $\varphi^{\prime}(0)=0$,
it is clear that if we set $D_{\varphi}=||\varphi^{\prime\prime}||_{\infty}$
we have for any $x\geq 0$ and any $z\geq-x$,
$|\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)|\leq D_{\varphi}z^{2}/2$. Since
$\varphi^{\prime}(0)=0$, we also have for any $x\geq 0$,
$|\varphi^{\prime}(x)|\leq D_{\varphi}|x|$, from which we get that
$|\varphi(x+z)-\varphi(x)|\leq D_{\varphi}|x||z|+D_{\varphi}z^{2}$. In any
case, we see that for any $x\geq 0$ and any $z\geq-x$

$\left|\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}\right|\leq
2D_{\varphi}z^{2},$

It comes that

$\left|\int_{-(\delta\wedge
x)}^{\delta}\frac{\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z\right|\leq\frac{4D_{\varphi}}{2-\alpha}\delta^{2-\alpha}.$

We also get

$\Big{|}\frac{\varphi(0)-\varphi(x)}{\alpha}(x^{-\alpha}-\delta^{-\alpha})\Big{|}\bm{1}_{\\{\delta>x\\}}\leq\frac{D_{\varphi}}{\alpha}x^{2-\alpha}\bm{1}_{\\{\delta>x\\}}\leq\frac{D_{\varphi}}{\alpha}\delta^{2-\alpha}.$

All in all, we showed that for any $x>0$,
$|\mathcal{L}^{\alpha}\varphi(x)-\mathcal{L}^{\alpha,\delta}\varphi(x)|\leq
C_{\varphi}\delta^{2-\alpha}$. When $x=0$, we have

$\left|\mathcal{L}^{\alpha}\varphi(0)-\mathcal{L}^{\alpha,\delta}\varphi(0)\right|=\left|\int_{0}^{\delta}\frac{\varphi(z)-\varphi(0)}{|z|^{1+\alpha}}\mathrm{d}z\right|\leq\frac{D_{\varphi}}{2-\alpha}\delta^{2-\alpha}\leq
C_{\varphi}\delta^{2-\alpha}.$

This shows that (42) holds. ∎

######  Remark 36.

The above proposition shows that $\rho$ is a weak solution of

$\left\\{\begin{aligned}
&\partial_{t}\rho_{t}(x)=\frac{\sigma_{\alpha}}{2}\int_{\mathbb{R}}\frac{\rho_{t}(x-z)\bm{1}_{\\{x>z\\}}-\rho_{t}(x)+z\partial_{x}\rho_{t}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z\;\;&\text{for}\>\>(t,x)\in(0,\infty)^{2},\\\
&\int_{0}^{\infty}\rho_{t}(x)\mathrm{d}x=1\quad&\text{for}\>\>t\in(0,\infty),\\\
&\>\rho_{0}(\cdot)=\delta_{0}.\end{aligned}\right.$

The term $z\partial_{x}\rho_{t}(x)\bm{1}_{\\{|z|<x\\}}$ is useless when
$\alpha\in(0,1)$.

###### Proof.

As usual, we do as if $\rho_{t}$ was sufficiently regular so that all the
computations below hold true. By the way, it might actually be the case, see
for instance Chaumont-Malecki [21], but this is not our purpose. Consider a
function $\varphi\in C^{\infty}_{c}((0,\infty))$. Then, by Proposition 35, we
have that

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}_{+}}\varphi(x)\rho_{t}(x)\mathrm{d}x=\int_{\mathbb{R}_{+}}\rho_{t}(x)\mathcal{L}^{\alpha}\varphi(x)\mathrm{d}x=\frac{\sigma_{\alpha}}{2}[I_{t}(\varphi)-J_{t}(\varphi)],$

where, recalling (43),

$I_{t}(\varphi)=\int_{\mathbb{R}_{+}}\int_{-x}^{\infty}\rho_{t}(x)\frac{\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z\mathrm{d}x$

and, since $\varphi(0)=0$,

$J_{t}(\varphi)=\int_{\mathbb{R}_{+}}\frac{\varphi(x)\rho_{t}(x)}{\alpha
x^{\alpha}}.$

We first focus on $I_{t}(\varphi)$. Exchanging integrals, we get

$I_{t}(\varphi)=\int_{\mathbb{R}}|z|^{-1-\alpha}\int_{(-z)\vee
0}^{\infty}\rho_{t}(x)[\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}]\mathrm{d}x\mathrm{d}z.$

We next write, for any $z\in\mathbb{R}$,

$\displaystyle\int_{(-z)\vee 0}^{\infty}$
$\displaystyle\rho_{t}(x)[\varphi(x+z)-\varphi(x)-z\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}]\mathrm{d}x$
$\displaystyle=\int_{z\vee
0}^{\infty}\rho_{t}(x-z)\varphi(x)\mathrm{d}x-\int_{(-z)\vee
0}^{\infty}\rho_{t}(x)\varphi(x)\mathrm{d}x-z\int_{(-z)\vee
0}^{\infty}\rho_{t}(x)\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}\mathrm{d}x.$

Regarding the third term, we have $\int_{(-z)\vee
0}\rho_{t}(x)\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}\mathrm{d}x=\int_{|z|}^{\infty}\rho_{t}(x)\varphi^{\prime}(x)\mathrm{d}x$,
so that, performing an integration by part, we have

$\displaystyle z\int_{(-z)\vee
0}\rho_{t}(x)\varphi^{\prime}(x)\bm{1}_{\\{|z|<x\\}}\mathrm{d}x=z\rho_{t}(|z|)\varphi(|z|)-z\int_{(-z)\vee
0}\varphi(x)\partial_{x}\rho_{t}(x)\bm{1}_{\\{|z|<x\\}}\mathrm{d}x.$

All in all, it holds that

$\displaystyle I_{t}(\varphi)=$
$\displaystyle\int_{\mathbb{R}_{+}}\varphi(x)\int_{\mathbb{R}}\frac{\rho_{t}(x-z)\bm{1}_{\\{x>z\\}}-\rho_{t}(x)\bm{1}_{\\{x>-z\\}}+z\partial_{x}\rho_{t}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z\mathrm{d}x$
$\displaystyle+\int_{\mathbb{R}}\frac{z}{|z|^{1+\alpha}}\rho_{t}(|z|)\varphi(|z|)\mathrm{d}z.$

The last term is equal to zero since the integrand is an odd function.
Finally, we write

$J_{t}(\varphi)=\int_{\mathbb{R}_{+}}\varphi(x)\int_{-\infty}^{-x}\frac{\rho_{t}(x)}{|z|^{1+\alpha}}\mathrm{d}z.$

Recombinig all the terms, we get that

$\int_{\mathbb{R}_{+}}\rho_{t}(x)\mathcal{L}^{\alpha}\varphi(x)\mathrm{d}x=\int_{\mathbb{R}_{+}}\varphi(x)\mathcal{A}\rho_{t}(x)\mathrm{d}x,$

where $\mathcal{A}\rho_{t}$ is defined for every $x>0$ as

$\mathcal{A}\rho_{t}(x)=\frac{\sigma_{\alpha}}{2}\int_{\mathbb{R}}\frac{\rho_{t}(x-z)\bm{1}_{\\{x>z\\}}-\rho_{t}(x)+z\partial_{x}\rho_{t}(x)\bm{1}_{\\{|z|<x\\}}}{|z|^{1+\alpha}}\mathrm{d}z.$

Therefore, we conclude that for any $\varphi\in C^{\infty}_{c}((0,\infty))$,
we have

$\int_{\mathbb{R}_{+}}\varphi(x)[\partial_{t}\rho_{t}(x)-\mathcal{A}\rho_{t}(x)]\mathrm{d}z=0,$

which is enough to deduce that for any $t>0$ and any $x>0$,
$\partial_{t}\rho_{t}(x)=\mathcal{A}\rho_{t}(x)$. ∎

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# Structure and lattice excitations of the copper substituted lead oxyapatite
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3)

Qiang Zhang<EMAIL_ADDRESS>Neutron Scattering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Yingdong Guan  Department of
Physics, Pennsylvania State University, University Park, Pennsylvania 16802,
USA  Yongqiang Cheng  Neutron Scattering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Lujin Min  Department of Physics,
Pennsylvania State University, University Park, Pennsylvania 16802, USA  Jong
K. Keum  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA  Center for Nanophase Materials Sciences, Oak
Ridge national Laboratory, Oak Ridge, Tennessee 37831, USA  Zhiqiang Mao
Department of Physics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA  Matthew B. Stone<EMAIL_ADDRESS>Neutron Scattering
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

The copper substituted lead oxyapatite, Pb10-xCux(PO3.92(4))6O0.96(3)
(x=0.94(6)) was studied using neutron and x-ray diffraction and neutron
spectroscopy techniques. The crystal structure of the main phase of our
sample, which has come to be colloquially known as LK-99, is verified to
possess a hexagonal structure with space group $P6_{3}/m$, alongside the
presence of impurity phases Cu and Cu2S. We determine the primary substitution
location of the Cu as the Pb1 ($6h$) site, with a small substitution at the
Pb2 ($4f$) site. Consequently, no clear Cu-doping-induced structural
distortion was observed in the investigated temperature region between 10 K
and 300 K. Specially, we did not observe a reduction of coordinate number at
the Pb2 site or a clear tilting of PO4 tetrahedron. Magnetic characterization
reveals a diamagnetic signal in the specimen, accompanied by a very weak
ferromagnetic component at 2 K. No long-range magnetic order down to 10 K was
detected by the neutron diffraction. Inelastic neutron scattering measurements
did not show magnetic excitations for energies up to 350 meV. There is no sign
of a superconducting resonance in the excitation spectrum of this material.
The measured phonon density of states compares well with density functional
theory calculations performed for the main LK-99 phase and its impurity
phases. Our study may shed some insight into the role of the favored
substitution site of copper in the absence of structural distortion and
superconductivity in LK-99.

###### pacs:

25.40.Fq,74.25.Kc,61.05.F-,61.05.cp

##  I Introduction

Since claims of the observation of room temperature superconductivity in a
copper-substituted lead apatite compound emerged in 2023 [1], there was a
rapid growth of interest in this compound. The parent compound, lead apatite
Pb10(PO4)6O was proposed to become a superconductor with the substitution of
copper as Cu2+ on the Pb2+ sites in the structure [2]. Spurred by this
announcement, other researchers quickly attempted to reproduce the results.
Synthesis of the compound, which has come to be colloquially known as LK-99,
was found to be accompanied by an impurity phase of Cu2S although no
superconductivity was observed in subsequent magnetization measurements [3].
We note that this impurity phase was observed by the original group proposing
superconductivity in this compound. Additional synthesis by other groups also
observed the Cu2S impurity phase along with electrical transport and
magnetization measurements that are consistent with LK-99 being a
semiconductor and not a superconductor [4]. It was argued that the first-order
structural phase transition in Cu2S is responsible for the signal in the
magnetization, which was misinterpreted as superconductivity [5]. Growths of
LK-99 with improved purity of precursor materials or single crystals were
likewise unable to reproduce the proposed high-temperature superconducting
phase [6, 7]. Prompted by these experimental results, density functional
theory (DFT) calculations of the parent compound and the copper substituted
LK-99 have been performed by several groups [8, 9, 10, 11, 12, 13, 14, 15]. Si
and Held predicted an insulating electronic structure in the parent compound
with a reduction in unit cell volume and flat bands near the Fermi energy
appearing upon substituting Cu for Pb [8]. These flat bands imply a large
electronic density of states near the Fermi level such that correlated
electron behavior may be present in the material. Nonmagnetic DFT predicts the
flat bands to have both hole and electron pockets while spin-polarized DFT
calculations find that Cu substitution results in a ferromagnetic metal with
0.57 $\mu_{B}$ on the Cu site and a small hybridized moment of 0.06 $\mu_{B}$
on the nearest neighbor O site [9]. DFT calculations examining the lattice
excitations of the Cu substituted compound have found that in a hexagonal
structure, bands of phonons occur in the ranges 0 to 7, 11 to 13, 15 to 17 and
27 to 30 THz accompanied by an imaginary phonon mode, while in a triclinic
lattice, the phonons have similar banding but without any imaginary modes
[11].

Although room-temperature superconductivity in LK-99 has not been reproduced
and its original observation is likely a misidentification due to a non-
superconducting transition in an impurity phase, theoretical calculations do
suggest the potential for correlated electron behavior. Such calculations rely
heavily on details of the crystal structure and where the Cu sites are located
in the substituted compound. Griffin et al. [15] predicted that the
substitution of Cu on the Pb1 ($4f$) site induces a structural distortion and
plays an important role in inducing key characteristics for high-$T_{C}$
superconductivity, such as fluctuating magnetism, flat isolated d-manifold,
charge and phonons. This underscores the importance of investigating the
substitution site of the copper, particularly considering the presence of two
nonequivalent Pb sites in LK-99. In addition, as compared to x-ray
diffraction, neutron diffraction is imperative for obtaining more precise
information about oxygen positions and occupancy, which, unfortunately, has
not been pursued. Here, in addition to the x-ray diffraction experiments, we
conducted for the first time a series of neutron scattering measurements
designed to better understand the nature of the crystal structure, investigate
the lattice excitations, and probe for any indications of magnetic long-range
order, magnetic fluctuations, or a superconducting resonance of LK-99.

Figure 1:  (a) Three dimensional and (b) the projection of crystal structure
in the $ab$ plane of Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) obtained from the
Rietveld analysis on the XRD and neutron powder diffraction data. (c).
Comparison of the chemical environments of PO4, Pb1O8, and Pb2O9 in the
Pb10(PO4)6O at RT, Cu-doped LK-99 at RT and 10 K. The bond lengths of P-O in
PO4 tetrahedron and the longest/shortest Pb-O bond lengths have been added for
comparison.

##  II  Experimental and Theoretical Details

###  II.1 Sample preparation and magnetization measurements

We synthesized the Pb10-xCux(PO4)6O compound following the procedures reported
by Lee et al. [2] High purity PbO and PbSO4 powder were thoroughly mixed and
loaded into an alumina crucible. The crucible was then heated to 725 °C and
kept at this temperature for 24 hours in air. The product of Pb2(SO4)O was
verified by powder x-ray diffraction (XRD) analysis. Cu3P was synthesized via
heating the vacuum-sealed mixture of high-purity Cu and P powder. Our XRD
measurements indicated that a tiny amount of Cu impurity remained in the Cu3P
product. Pb2(SO4)O and Cu3P were then mixed with a 1:1 molar ratio and sealed
under vacuum in a quartz tube. The sealed tube was then heated to 925 °C and
kept at this temperature for 15 hours. The formation of the Pb10-xCux(PO4)6O
phase was verified through Rietveld analysis on both XRD and neutron
diffraction patterns. However, the product also contained impurity phases,
including Cu and Cu2S. Temperature and field dependence of the magnetization
of the synthesized material was conducted using a commercial Superconducting
Quantum Interference Device (SQUID, Quantum Design).

###  II.2 x-ray and neutron powder diffraction

XRD measurements were conducted on a PANalytical X’Pert Pro MPD equipped with
an X’Celerator solid-state line detector at room temperature (RT). For the XRD
measurements, the x-ray beam was generated at 45 kV/40 mA, and the x-ray beam
wavelength was set at $\lambda$=1.5418 Å (Cu K$\alpha$ radiation). The step
size ($\Delta 2\theta$) was 0.016o and the exposure time at each step was 200
seconds.

Neutron powder diffraction measurements were conducted at the time-of-flight
neutron powder diffractometer POWGEN, located at the Spallation Neutron Source
(SNS) at Oak Ridge National Laboratory [16, 17]. The POWGEN automatic changer
(PAC) was used to collect data at temperatures $T=10$ and $T=300$ K.
Approximately 0.5 g of powder sample was sealed in a standard vanadium PAC can
with helium exchange gas to ensure thermal contact with the PAC temperature
stage. Two neutron detector banks with central neutron wavelengths of 0.8 Å
and 2.665 Å were used to measure neutron powder diffraction over a wide range
of $d$-spacing. To check the homogeneity of the sample and the substitution
site of the Copper, the entire sample and two portions of the powder were
subsequently used for XRD measurements at RT. The Rietveld analysis on XRD and
neutron diffraction patterns was conducted using GSAS-II software [18].

###  II.3 Inelastic neutron scattering

Inelastic neutron scattering measurements were performed using the SEQUOIA
instrument at the SNS at Oak Ridge National Laboratory [19, 20]. The sample
used was the same sample from the POWGEN measurements but sealed within a 6.35
mm diameter aluminum sample can in one atmosphere of helium gas to ensure good
thermal contact with the sample environment cooling stage. Measurements were
performed using the three-sample changer closed cycle refrigerator (CCR)
environment at the spectrometer. An identical aluminum sample can was sealed
and attached to the three-sample changer to provide a measurement of the
empty-can scattering under the same instrument configurations as the sample
measurements. Unless otherwise noted, presented data have had this empty-can
scattering subtracted. Measurements were performed with the high-flux Fermi
chopper set for incident energies of $E_{i}=2000$, 1000, 504, 249, 124, 59.4,
and 29.5 meV with frequencies of 600, 600, 480, 360, 240, 180, and 120 Hz.
respectively. Measurements were also performed with the high-resolution Fermi
chopper set for $E_{i}=11.4$ meV at 180 Hz. All configurations were measured
at $T=5$ K with sample(empty-can) measurement counting times of 5(0), 5(2.5),
5(2.5), 5(2.5), 7.5(2.5), 9.23(3.87), 7.29(2.53) and 10(2.5) Coulombs of
charge accumulated on the spallation target for the respective neutron
energies listed above (from $E_{i}=2000$ to $11.4$ meV). Such longer counting
times were used to account for the reduction in neutron flux at higher
incident energies and the relatively small sample size. Some of these
configurations were also measured at $T=300$ K. The calculated full width at
half maximum energy resolution for elastically scattered neutrons was
$\delta\hbar\omega=240$, 86, 38, 18, 9.6, 4.3, 2.2, and 0.24 meV for the
incident energies of $E_{i}=2000$, 1000, 504, 249, 124, 59.4, 29.5, and 11.4
meV respectively. Powder inelastic neutron scattering data are binned into
intensity histograms as a function of scattering angle, $2\theta$, and energy
transfer, $\hbar\omega$ using the inelastic algorithms in the MANTID software
package [21]. Inelastic neutron scattering data are further binned as a
function of the magnitude of wave-vector transfer, $Q$, and $\hbar\omega$ for
presentation and analysis using DAVE software [22].

###  II.4 DFT calculations

Density functional theory (DFT) calculations of Cu, Cu2S, and LK-99 were
performed using the Vienna Ab initio Simulation Package (VASP) [23]. The
calculation used the Projector Augmented Wave (PAW) [24, 25] method to
describe the effects of core electrons, and Perdew-Burke-Ernzerhof (PBE) [26]
implementation of the Generalized Gradient Approximation (GGA) for the
exchange-correlation functional. The spin-polarized calculation of LK-99 used
the structure model and the Hubbard U (2.0 eV) as reported by Cabezas-Escares
et al [11]. Phonopy [27] and finite displacement method was used to obtain the
interatomic force constants and subsequently the phonon frequencies and modes.
The OCLIMAX [28] software was used to convert the DFT-calculated phonon
results to the simulated INS spectra.

The structural model used for the DFT calculations is exactly the same as
reported by Cabezas-Escares et al.[11] The unit cell contains 1 Cu, 9 Pb, 6 P,
and 25 O atoms, with the single Cu atom occupying the 4f site. While this does
not fully match the composition and Cu arrangement observed in our structural
refinement, we show that this simpler model properly accounts for most of the
phonon features observed in the inelastic neutron scattering experiment. To
account for the refined structure would require a much larger cell to describe
the partial occupancies and disorder. Barring a fundamental change in the
electronic structure, the relatively minor differences between the model used
for the DFT calculation and the refined structure are unlikely to cause
significant changes in the phonon density of states.

##  III Results and Discussion

###  III.1 Rietveld analysis of the crystal structure

Figure 2:  Rietveld refinements of the (a) x-ray diffraction and (b) neutron
powder diffraction patterns of Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and its
impurity phases at 300 K. The observed data and the fit are indicated by the
open circles and solid lines, respectively. The difference curve is shown at
the bottom. The vertical bars between the data and the difference curve mark
the positions of Bragg reflections for the phases of LK-99 (upper ticks),
impurity phases Cu (middle ticks) and Cu2S (bottom ticks).

To study the details of the crystal structure, we performed x-ray and neutron
diffraction measurements on the same entire sample ($\approx$0.5 grams). The
determined crystal structure is displayed in Fig. 1 (a) and (b). The XRD and
neutron patterns at 300 K and the Rietveld co-refinement analysis of these
patterns are shown in Fig. 2(a) and (b), respectively. In addition to the main
LK-99 phase, there exist two impurity phases: Cu and Cu2S. The crystal
structure of Cu is cubic with space group Fm-3m (No. 225). Cu2S crystallizes
in a monoclinic structure with a space group $P2_{1}/c$ (No. 14), consistent
with previous reports [3, 4, 5]. The weight fractions of LK-99, Cu and Cu2S
are refined to be 41.2 %, 24.1 % and 34.8 %, respectively, based on the co-
refinement on XRD and POWGEN patterns. Rietveld analysis reveals that we have
synthesized the LK-99 phase that crystallizes in the hexagonal structure with
space group $P6_{3}/m$ (No. 176). The lattice constants are refined to be
$a$=9.839(7) Å, $c$ = 7.428(6) Å at $T=300$ K, which are consistent with the
initial reports by Lee et al. on LK-99 ($a$=9.843(4) Å, $c$ = 7.428(3) Å) [2,
1]. The volume of the unit cell of our LK-99 sample is reduced by $\approx$
0.56 % relative to the parent compound ($a$=9.8650 Å, $c$= 7.4306 Å) [29],
which is slightly larger than $\approx$ 0.48% in the previous reports by Lee
et al. [2, 1] due to smaller lattice constant $a$. This indicates that Pb
sites may be substituted by Cu and the smaller atomic radius of Cu than Pb is
responsible for the shrinking of the unit cell. Note that our Rietveld
analysis on both XRD and neutron data excluded superstructures with double
$c$-axis parameter as reported for the parent compound Pb10(PO4)6O very
recently [30].

Since the x-ray scattering cross-section is generally proportional to the
atomic number, Z, the x-ray scattering factor of Pb (Z=82) is much larger than
Cu (Z=29). Consequently, x-ray diffraction proves to be a powerful tool to the
determination of doping concentration of Cu on the Pb1 ($6h$) and/or Pb2
($4f$) sites. Moreover, considering the larger neutron scattering length of Pb
(9.405) in comparison to Cu (7.718), there exists an additional contrast that
aids in elucidating the substitution sites of Cu by neutron diffraction. A
careful Rietveld co-refinement on both XRD and neutron patterns at 300 K has
discerned that the doping concentration of Cu on Pb1 and Pb2 sites is 12.9(8)%
and 3.9(6)%, respectively. This indicates that Cu is primarily substituted at
the Pb1 ($6h$) site rather than the Pb2 ($4f$) site. Note that neutron data
and XRD patterns co-refined were collected on the same 0.5 g of powder sample
in its entirety. To assess sample homogeneity and the substitution site of the
Copper, two additional XRD measurements were conducted on selected partial
samples. The refinement on weight fractions unveiled a non-uniform
distribution of LK-99, Cu, and Cu2S within the sample. However, the weight
fractions derived from the refinement of XRD and POWGEN patterns on the entire
sample remained consistent. Notably, despite variations in weight fractions
among the three samples, Rietveld analysis on their XRD patterns yields
consistent doping concentrations of Cu on the Pb1 and Pb2 sites within the
uncertainties for the main LK-99 phase. This finding aligns seamlessly with
the Rietveld refinement results obtained from the POWGEN data, demonstrating a
robust and coherent result. Furthermore, our findings corroborate theoretical
calculations [15], illustrating that the replacement of Cu on the Pb1 ($6h$)
site is energetically favored by 1.08 eV compared to the Pb2 ($4f$) site (Note
that the labeling of Pb1 and Pb2 in this reference [15] is reversed with our
labeling). In addition, we observed negligible vacancies at P, O1, O2 and O4
sites, whereas there is approximately a 7 % vacancy at the O3 site. The
refined composition of our sample is Pb10-xCux(PO3.92(4))6O0.96(3)
(x=0.94(6)), falling into the regime of the 0.9$<$x$<$1.1 for the LK-99 in the
previous reports [2, 1].

Table 1: Refined atomic positions, isotropic temperature factors, occupancy,
from modeling high-resolution powder neutron diffraction data of
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) at 10 K. Analysis of the nuclear Bragg
reflections lead to the space group selection of $P6_{3}/m$ (No. 176) and
indexed unit cell constants of $a=b=$ 9.807(2) Å and $c=$ 7.399(8) Å. The
refined composition is Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3)

atom | site | x | y | z | $U$ | occ
---|---|---|---|---|---|---
Pb1 | 6h | 0.00210(5) | 0.755(3) | 0.25000 | 0.0070(5) | 0.871(8)
Cu1 | 6h | 0.00210(6) | 0.755(6) | 0.25000 | 0.0070(5) | 0.129(8)
Pb2 | 4f | 0.6667 | 0.3333 | -0.00238(4) | 0.0070(5) | 0.961(6)
Cu2 | 4f | 0.6667 | 0.3333 | -0.00238(4) | 0.0070(5) | 0.039(6)
P1 | 6h | 0.624(4) | 0.597(6) | 0.25000 | 0.0030(7) | 1.021(7)
O1 | 6h | 0.514(7) | 0.664(6) | 0.25000 | 0.0093(8) | 1.032(7)
O2 | 12i | 0.733(5) | 0.651(2) | 0.0842(7) | 0.0093(8) | 0.990(4)
O3 | 6h | 0.530(6) | 0.416(8) | 0.25000 | 0.0093(8) | 0.933(6))
O4 | 4e | 0 | 0 | 0.215(8) | 0.0093(8) | 0.241(8)

It is instructive to compare the refined structural features between our LK-99
sample, the parent compound, and the prior theoretical calculations examining
LK-99. In the parent compound Pb10(PO4)6O [29], three key polyhedra are
noteworthy: tetrahedra PO4, Pb1O8 and Pb2O9 (with the cutoff 3 Å for the Pb-O
bond length). Figure 1 (c) displays the comparison of these polyhedra in the
parent compound at RT, our LK-99 sample at RT and 10 K based on our Rietveld
refinements. The theoretical predictions by Griffin et al. [15] suggest that
the substitution of Cu on the Pb1 ($4f$) site, corresponding to the Pb2 ($4f$)
site here, induces interesting structural distortion with two primary
features. The first involves a reduction in coordination around the Pb2 ($4f$)
site from nine to six, presenting a distorted trigonal prism coordination. The
second entails a modification and tilting of the PO4 tetrahedra. However, in
our Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) compound, the coordination of
Cu/Pb1 is maintained at 9 with the same Pb-O distance cutoff of 3 Å. For
clarity, the longest and shortest Pb2-O bond lengths in Pb2O9 have been
included for the parent compound and our LK-99 in Fig.1 (c). No substantial
changes in the Pb2-O bond lengths or clear distorted trigonal prism were
observed due to Cu doping in our LK-99 compound. Furthermore, the coordination
of Cu/Pb1 remains 8 using the same cutoff of 3 Å. The PO4 tetrahedra in the
parent compound does not form a regular tetrahedron since the O-P-O bond angle
ranges from 106.4o to 111.9o, slightly deviating from the ideal angle 109.47o.
In the Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) compound, we find that the O-P-O
bond angle falls in the range from 106.23o to 111.6 o, showing comparable bond
angles relative to the parent compound at RT. Thus, no clear tetrahedral
tilting is observed in the LK-99 compound. The P-O bond lengths were also
incorporated in Fig.1 (c). Both parameters suggest that the Cu doping does not
induce significant tilting of the PO4 tetrahedron. Therefore, the predicted
structural distortion [15] was not observed in our LK-99 compound.

It comes to our attention that Griffin et al. [15] predicted the structural
distortion under the assumption that Cu substituted to the Pb2 ($4f$) site.
However, our investigation reveals a minimal substitution ratio of Cu on the
Pb2 ($4f$) site, with the majority of Cu predominantly entering the Pb1 ($6h$)
site in the LK-99 sample. This finding may clarify why Cu doping does not
induce the predicted structural distortion in the actual LK-99 sample.
Furthermore, Griffin et al. [15] also predicted that the substitution of Cu on
the Pb2 ($4f$) site plays a pivotal role in inducing key characteristics for
high-$T_{C}$ superconductivity, such as fluctuating magnetism, a flat isolated
d-manifold, charge and phonons. However, the predominant incorporation of Cu
into the Pb1 ($6h$) site in the LK-99 sample, as opposed to the Pb2 ($4f$)
site, may offer insights into the observed absence of room-temperature
superconductivity in recent experiments [11]. The distinct distributions of Cu
between the Pb1 and Pb2 sites necessitate a reevaluation in subsequent
theoretical calculations.

Figure 3:  (a) Temperature dependence of total magnetization after the ZFC and
FC processes in a low field of 10 Oe and 100 Oe of the
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and its impurity phases. The inset
shows the magnetic hysteresis loop with field sweeping from +5000 to 0 to
-5000 to 0 to 5000 Oe consecutively. (b) Comparison of low-Q POWGEN data
between 300 K and 10 K showing the absence of any long-range magnetic order.
The vertical bars between the data and the difference curve mark the positions
of Bragg reflections for the phases of LK-99 (upper ticks), impurity phases Cu
(middle ticks) and Cu2S (bottom ticks).

###  III.2 Magnetic properties

The temperature dependence of the magnetization measured under zero-field-
cooling (ZFC) and field-cooling (FC) histories in a field of 10 and 100 Oe is
presented in Fig. 3(a). The magnetization is negative across the investigated
temperature region of 2 K to 300 K, indicative of an overall diamagnetic
response from the sample. Given that the impurity phases Cu and Cu2S are not
diamagnetic in this temperature region, the diamagnetic behavior arises from
the main LK-99 phase. This is consistent with observations in other
polycrystalline compounds [2, 5, 3] or single crystal [7]. The upturn observed
at low temperatures in an applied field of 100 Oe implies the existence of
paramagnetic impurities, a phenomenon less prominent under an applied field of
10 Oe. Note that such an upturn is much weaker than those in the previous
reports [2, 5, 7], indicating a lower fraction of paramagnetic impurities in
our sample. The magnetic hysteresis loop, illustrated in the inset of Fig. 3
exhibits diamagnetic behavior. A small coercive field $\approx$ 400 Oe at 2 K
indicates a very weak ferromagnetic component. This finding is consistent with
observations in LK-99 single crystals [7], possibly arising from frustrated
exchange interactions within the Cu-rich clusters.

The overplot of the POWGEN data in the low-$Q$ region at 300 K and 10 K is
displayed in Fig. 3(b). No discernible pure magnetic peaks or an increase in
peak intensity attributed to magnetic contributions were observed.
Consequently, there is no indication of long-range magnetic order from 300 K
down to 10 K in LK-99. The POWGEN data at 10 K can be well fitted using these
three phases: LK-99, Cu and Cu2S, as shown in Fig. 4(a-b). Furthermore, there
is no structural transition or clear structural distortion (see Fig. 1(c))
observed down to 10 K. The refined lattice constants, atomic positions,
isotropic temperature factors and occupancy at 10 K are summarized in Table 1.

Figure 4:  Rietveld refinement fit to neutron powder diffraction patterns
collected by (a) neutron bank 3 and (b) neutron bank 1 for
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and its impurity phases at the POWGEN
instrument at 10 K. The observed data and the fit are indicated by the open
circles and solid lines, respectively. The difference curve is shown at the
bottom. The vertical bars mark the positions of Bragg reflections for the
phases of LK-99 (top), impurity phases Cu (middle) and Cu2S (bottom).

###  III.3 Lattice excitations

Figure 5(a) shows the measured scattering intensity as a function of $Q$ and
$\hbar\omega$ for $E_{i}=11.4$ meV neutrons measured at $T=5$ K. There are no
apparent magnetic fluctuations present in the spectrum. Figure 5(b) shows the
integrated scattering intensity for elastically scattered neutrons with
$-0.24<\hbar\omega<0.24$ meV. The reflections at $Q\approx 0.74$ and $1.1$ Å-1
corresponding to d-spacings of $\approx 8.5$ and $5.7$ Å respectively, are in
agreement with the 010 and 011 reflections observed in the POWGEN
measurements. No additional Bragg reflections, beyond those measured at
POWGEN, are discernible in the inelastic measurements. Higher $Q$ peaks are
also structural and consistent with the POWGEN measurements.

Figure 5:  Scattering intensity of Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and
its impurity phases at $T=5$ K measured with $E_{i}=11.4$ meV neutrons. Data
have been background subtracted as described in the text. (a) Scattering
intensity as a function of $Q$ and $\hbar\omega$. (b) Scattering intensity for
$-0.24<\hbar\omega<0.24$ meV as a function of $Q$.

Figure 6 shows the measured scattering intensity as a function of $Q$ and
$\hbar\omega$ for several incident energy measurements at both $T=5$ and 300
K. For the $Ei=504$, 249, 124, and 60 meV measurements, there is no
significant scattering observed at lower values of wave-vector transfer that
would be consistent with magnetic scattering. The proposed superconducting
transition temperature of Pb10-xCux(PO4)6O was $T_{C}\approx 400$ K [1, 2]. If
LK-99 were an unconventional superconductor, one would expect potential
superconducting resonances in the excitation spectrum [31]. Superconducting
resonance energies, $E_{r}$, in inelastic neutron scattering measurements are
observed at temperatures that are linearly proportional to the transition
temperature. For example in the Fe-based superconducting compounds, this
relationship has been found to be $E_{r}=4.9k_{B}T_{C}$ [32], where $K_{B}$ is
Boltzman’s constant, and $E_{r}=5.8k_{B}T_{C}$ for the cuprate superconductors
[33]. Using the cuprate characteristic energy of the resonance for LK-99 would
place the superconducting resonance at approximately 200 meV at low $Q$. No
such feature is observed in our measured spectra. At larger values of wave-
vector transfer, a powder phonon density of states (PDOS) with bands of
phonons extending up to $\hbar\omega\approx 120$ meV is observed.

Figure 6:  Scattering intensity of Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and
its impurity phases at $T=5$ K and $T=300$ K as a function of $Q$ and
$\hbar\omega$ measured with different incident energy, $E_{i}$, neutrons. Data
have been background subtracted as described in the text. (a) Scattering
intensity for $Ei=504$ meV and $T=5$ K. (b) Scattering intensity for $Ei=249$
meV and $T=5$ K. (c)-(d) Scattering intensity for $Ei=124$ meV and $T=5$ K and
$T=300$ K respectively. (e)-(f) Scattering intensity for $Ei=59.4$ meV and
$T=5$ K and $T=300$ K respectively. (g)-(h) Scattering intensity for $Ei=29.5$
meV and $T=5$ K and $T=300$ K respectively. The measured intensity shown in
panels (a) and (b) is multiplied by a factor of 10.

The measured spectra for $E_{i}=29.5$ meV have some scattering intensity
appearing near $\hbar\omega=6$ meV. This is present in both the $T=5$ K and
$T=300$ K data as shown in Figs. 6(g) and (h), respectively. Figure 7 are
plots of scattering intensity as a function of (a) $Q$ and (b)-(c)
$\hbar\omega$ for the $T=5$ and 300 K measurements. The wave-vector
dependence, Fig 7(a) is integrated between 4 and 7.5 meV and shows quadratic
behavior for both temperatures consistent with this mode being associated with
a lattice excitation. Intensity as a function of energy transfer for two
different ranges of integration of wave-vector transfer, Figs. 7(b)-(c), show
that the $T=5$ K mode has a peak in the vicinity of 5.5 meV which softens
slightly with increasing temperature and increases rapidly in intensity as a
function of wave-vector transfer making it further consistent with it being a
lattice excitation.

Figure 7:  Scattering intensity of Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and
its impurity phases at $T=5$ K and $T=300$ K for the $Ei=29.5$ meV
measurement. Data are cut through the spectra shown in Figs. 6(g)-(h) as a
function of $Q$ and $\hbar\omega$. Data have been background subtracted as
described in the text. (a) Scattering intensities as a function of $Q$ for
energy transfers $4<\hbar\omega<7.5$ meV. (b) Scattering intensities for wave-
vectors $2<Q<3.5$ Å-1. (c) Scattering intensities for wave-vectors $0<Q<1.5$
Å-1. Solid lines are fits to the $T=5$ K data as the sum of scattering
contributions from LK-99, Cu2s, and Cu as described in the text. The range of
the solid line along the independent variable indicates the range used for the
fit of the data.

We individually examine the generalized phonon density of states (PDOS) from
the different incident energy neutron spectroscopy measurements. The
scattering intensity was first plotted as a function of wave-vector transfer
for the $E_{i}=29.5$, $59.4$, and $124$ meV configurations integrated over
energy transfer ranges $[4,8]$, $[10,30]$, and $[60,74]$ meV respectively.
These data were individually compared as a function of energy transfer to the
function $I(Q)=A+B*Q^{2}\exp(-|u|^{2}Q^{2})$ where $A$ is an overall constant
background, $B$ is a multiplicative constant, and $u$ quantifies the Debye-
Waller factor. The mean Debye-Waller factor was then used with the DAVE
software to extract the general phonon density of states for the $E_{i}=29.5$,
$59.4$, $124$, and $249$ meV measurements for the integrated ranges of wave-
vector transfer $[0.75,3.1]$,$[2,4.5]$, $[2.5,6.5]$, and $[3,8.5]$ Å-1
respectively [22]. The individual generalized phonon density of states is
shown as a function of energy transfer for four different incident energies in
Fig. 8. The variation in neutron flux as a function of incident energy was not
applied as a normalization for these data. This accounts for the large
variation in the magnitude of the density of states across the measurement
configurations, which span an order of magnitude in neutron incident energy.
Each measurement has characteristic features, which we enumerate as we discuss
the analysis of these data.

Figure 8:  Generalized phonon density of states for
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) and its impurity phases for $T=5$ K.
Data have been extracted into a general phonon density of states as described
in the text. Data were acquired with four different incident energy
configurations as described in the text: (a) $E_{i}=249$ meV, (b) $E_{i}=124$
meV, (c) $E_{i}=59.4$ meV, and (d) $E_{i}=29.5$ meV. The red solid lines are
the calculated phonon density of states of
Pb9.06(7)Cu0.94(6)(PO3.92(4))6O0.96(3) convolved with the respective energy
transfer dependent resolution function for each of the measurement
configurations as described in the text. The dashed blue and green dotted
lines are similar calculations for Cu2S and Cu, respectively, as described in
the text. The heavy solid line is the refined combined phonon density of
states that has been fit simultaneously to all of the measurements shown. This
fit is further described in the text. The range used for the is the range of
the black line. The individual phonon density of states is shown as their
respective contribution to the overall measured phonon density of states but
with no additional flat background included.

DFT calculations of Cu, Cu2S, and LK-99 were performed. The scattering
intensity based upon the lattice excitation spectra calculated for the LK-99,
Cu2S, and Cu was convoluted individually with the instrumental energy
resolution function for the $E_{i}=29.5$ meV configuration. The three spectra
were added according to the percentages of each compound refined in analyzing
the powder diffraction data. We fit the wave-vector dependent scattering
intensity at low energy transfer for the same cut through the data to this
calculated resolution convolved scattering intensity, including an overall
constant background and an overall multiplicative scaling factor. The result
is shown in Fig. 7(a) as a solid line that reproduces the quadratic wave-
vector dependent measured scattering intensity. We also fit separately the
scattering intensity as a function of energy transfer using the same method
for the data in Figs. 7(b) and (c). For this case, we include a Gaussian peak
at zero energy transfer to account for the elastic scattering. The wave-vector
and energy transfer-dependent scattering intensity of the feature at
approximately 5 meV energy transfer is well explained from the calculated
lattice excitations of the LK-99 and its impurity phases.

We perform a similar fitting of the phonon density of states measurements
shown in Fig. 8 to the calculated phonon density of states convolved with the
instrumental energy resolution. We fit simultaneously the four phonon spectra
measurements shown in Fig. 8 to the sum of the calculated phonon density of
states of LK-99, Cu2S, and Cu with a fixed weighted percentage based upon the
refined percentage of these compounds found in the analysis of the diffraction
data. An overall constant background and a multiplicative scaling factor are
also included for each of the incident energies in Fig. 8. The resulting fit
to the phonon density of states is shown as a heavy solid line in the figure.
The individual phonon density of states without the additive background is
also shown in this figure separately for the three compounds. This fit
provides an excellent representation of the measured density of states over
more than two orders of magnitude of energy transfer. The bands of calculated
phonons agree with calculations of phonon frequencies for LK-99. [11, 34]
There is a larger relative background contribution in the phonon density of
states at higher energy transfers. This is likely due to multiple scattering
or multi-phonon effects contributing to these higher energy transfers. Above
$\hbar\omega\approx 40$ meV, the phonon density of states is dominated by the
contribution from LK-99. The LK-99 modes in the vicinity of $70$ meV are due
to the O-P-O bending modes with motions deforming the PO4 tetrahedra. The
highest energy mode at approximately 110 meV is due to the stretching modes of
the P-O bonds. One location with a difference in the calculated and measured
phonon density of states is in the vicinity of $\hbar\omega=44.6$ meV, Fig.
8(c). This mode corresponds to a twisting and scissoring type of motion
resulting in squeezing of the PO4 tetrahedra. The actual energy of this mode
appears to be 4 meV higher at $\hbar\omega=48.8$ meV. We speculate that this
difference may be due to the relatively small unit cell used in the
calculations.

Earlier, we considered where the superconducting resonance would occur in the
spectrum if LK-99 were an unconventional superconductor. If LK-99 were a
normal superconductor, electron-phonon coupling associated with the
superconductivity would produce features in the excitation spectrum in the
vicinity of the superconducting energy gap, $2\Delta$. [35, 36, 37]
Considering a proposed transition temperature of $T_{C}\approx 400$ K and a
range of plus or minus fifty percent deviation from weak,
$2\Delta=3.35k_{B}T_{C}$, or strong, $2\Delta=4k_{B}T_{C}$ coupling limits
results in a range between approximately 70 and 210 meV over which one would
potentially see changes in the phonon spectrum [38]. We observe only modes
that are consistent with the calculated phonon density of states in this
energy range. All of the spectral features are well accounted for by the
lattice excitations calculated for LK-99 and the Cu and Cu2S impurity phases.

##  IV Conclusions

We present a comprehensive analysis of the crystal structure, encompassing
site occupancy, oxygen positions, and polyhedral environments, along with
lattice excitations for the copper-substituted lead oxyapatite,
Pb10-xCux(PO3.92(4))6O0.96(3) (x=0.94(6)). Our structural refinements reveal a
favored substitution site of Copper on Pb1 ($6h$) site, which may provide
insight into the absence of the Cu-doping-induced structural distortion and
superconductivity in LK-99. Our DFT calculations for the Pb10-xCux(PO4)6O,
alongside impurity phases Cu and Cu2S, predict a phonon spectrum entirely
consistent with the measured spectrum. We find no experimental evidence of
static long-range magnetic order, dynamic magnetic fluctuations,
superconducting resonances, or superconductivity-correlated electron behavior
in the LK-99.

###### Acknowledgements.

We acknowledge useful discussions with G. Granroth regarding multiphonon
backgrounds. This research used resources at the Spallation Neutron Source,
DOE Office of Science User Facilities operated by the Oak Ridge National
Laboratory. This research used computing resources made available through the
VirtuES project, funded by the Laboratory Directed Research and Development
program and Compute and Data Environment for Science (CADES) at ORNL, as well
as resources of the National Energy Research Scientific Computing Center
(NERSC), a U.S. Department of Energy Office of Science User Facility located
at Lawrence Berkeley National Laboratory, operated under Contract No. DE-
AC02-05CH11231 using NERSC award ERCAP0024340. This research used the
resources of the Center for Nanophase Materials Sciences (CNMS) and Neutron
Scattering Division (NSD), which are DOE Office of Science User Facilities.
Y.D.G. and Z.Q. M. acknowledge the support by the US Department of Energy
under grants DE-SC0019068. L.J.M. acknowledges the partial support from NSF
through the Materials Research Science and Engineering Center DMR 2011839
(2020–2026).

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