We now calculate

\begin{align*}
\big(\frac{\pa\Delta_b}{\pa r}\rho_b^2-4r\Delta_b\big)A^2+4a r\Delta_bAB& <-2r(r^2-3\Bm r+(a^2+2\BQ^2))A^2+4\abs{a}r\Delta_b\abs{AB}\\
&<-2r(r^2-3\Bm r+(a^2+2\BQ^2))A^2+4\Bm r\Delta_b^{1/2}A^2\\
&<-2r(r^2-5\Bm r+(a^2+2\BQ^2))A^2
\end{align*}

where we use $\abs{a}<\Bm$ and $A^2\geq \Delta_bB^2$ in the second step and $\Delta_b< r^2$ for $r\geq \ehKN$ in the third step. As a consequence, $H^2_{p_{h,z}}f<0$ when $r>5\Bm$.

We now assume that $\ehKN<r\leq 5\Bm$ and $f(r)<\delta'$. Then we have

\begin{align*}
\Delta_b(r)&=r^4f(r)\leq (5\Bm)^4f(r)<(5\Bm)^4
\delta'\\
\pa_r\Delta_b&=2r-2\Bm> 2\sqrt{\Bm^2-a^2-\BQ^2}.
\end{align*}

By choosing $\delta'>0$ sufficiently small such that

\[
2\Bm\sqrt{\Bm^2-a^2-\BQ^2}-4(5\Bm)^4\delta'-4\Bm(5\Bm)^2\sqrt{\delta'}\geq 0,
\]

we conclude that on $\{\ehKN<r\leq 5\Bm\}\cap\{f(r)<\delta'\}$

\begin{align*}
r^5f'(r)&=r\pa_r\Delta_b-4\Delta_b>2\Bm\sqrt{\Bm^2-a^2-\BQ^2}-4(5\Bm)^4\delta'\geq 0,\\
\big(\frac{\pa\Delta_b}{\pa r}\rho_b^2-4r\Delta_b\big)A^2+4a r\Delta_bAB&\geq r\big(\frac{\pa\Delta_b}{\pa r}r-4\Delta_b-4\Bm\Delta_b^{1/2}\big)A^2\\
&>\Bm\big(2\Bm\sqrt{\Bm^2-a^2-\BQ^2}-4(5\Bm)^4\delta'-4\Bm(5\Bm)^2\sqrt{\delta'}\big)A^2\geq 0,
\end{align*}

and thus $H^2_{p_{h,z}}f<0$. Then letting $\delta_0=\min\{f(5\Bm), \delta'\}$ finishes the proof.

We now define the outgoing/incoming tails and the trapped set (see <cit.>,<cit.>). Recall that
\[{}^\scop\!H^{2,0}_{p_{h,z}}=\rho^{-1}\jb{\xi}^{-1}{}^\scop\!H_{p_{h,z}}, \quad\rho=r^{-1}.
\]

Let $f$ be the function defined as in Proposition <ref>. Let $(r,\theta,\varphi,\xi)\in\cscform$ and $\phi(s)=\exp(s{}^\scop\!H^{2,0}_{p_{h,z}})(r,\theta,\varphi,\xi)$ be a maximally extended integral curve with domain of definition $I\subset\BR$ on $\Sigma_h=\{\jb{\xi}^{-2}p_{h,z}=0\}\subset\cscform$. Let $s_-=\inf I, s_+=\sup I$. We say that $\phi(s)$ is trapped as $s\to\pm\infty$, if there exists $\delta>0$ and $T>0$ such that $f(\phi(s))>\delta$ for all $\pm s\geq T$ (as a consequence, $s_\pm=\pm\infty$). We define the incoming tail $\Gamma_-$ and the outgoing tail $\Gamma_+$ as

\[
\Gamma_\mp=\{(r,\theta,\varphi,\xi)\in\cscform\mid \phi(s)=\exp(s{}^\scop\!H^{2,0}_{p_{h,z}})(r,\theta,\varphi,\xi) \mbox{ is trapped as } s\to\pm\infty\}.
\]

Then we define the trapped set as $K:=\Gamma_-\cap\Gamma_+$.

It follows from the definition that $\Gamma_\pm, K$ are invariant under the flow $\exp(s{}^\scop\!H^{2,0}_{p_{h,z}})$. We next study the properties of $\Gamma_{\pm}$ and $K$. We need the following lemmas.

For $z\in\BR\setminus 0$, we have

\begin{equation}
\pm\jb{\xi}^{-1}H_{p_{h,z}}r>0\quad \mbox{on}\quad (\Sigma^\pm_h\cap\{r_-\leq r\leq \ehKN\})\setminus L_\pm.
\end{equation}

We first consider the case $r_-\leq r<\ehKN$ and finite $\xi$. Since $\chi=1, c'(r)=0$ on $\{r_-\leq r<\ehKN\}$, we write $p_{h,z}=0$ as

\[
p_{h,z}=-\rho_b^{-2}\Big(\Delta_b\xi_r^2-2(r^2+a^2)\xi_rz+2a\xi_r\xi_\varphi+\tilde{p}_{h,z}\Big)=0\quad\mbox{where}\quad \tilde{p}_{h,z}=\xi_\theta^2+\frac{1}{\sin^2\theta}(\xi_\varphi-a\sin^2\theta z)^2.
\]

This is a quadratic equation in $\xi_r$ with positive discriminant on $\{r_-\leq r<\ehKN\}$ since $\Delta_b<0$ there. As a consequence, it has two distinct roots $\xi_r^-<0<\xi_r^+$ and $\pm\pa_{\xi_r}p_{h,z}>0$ at $\xi_r^\pm$.
By Lemma <ref>, $\Sigma_h^\pm$ is characterized by the conditions $\pm g^{-1}(-zdt_{\chi}+\xi, d\mathfrak{t})>0$ (because $t_{b,*}=t_{\chi}$ in the region $r_-\leq r\leq \ehKN$), that is, with $\mathfrak{t}=t_\chi+b(r)$ (according to the definition of $\mathfrak{t}$ in (<ref>), we see that $b'(r)\leq 0$ in the region $r_-\leq r\leq\ehKN$), we have

\begin{equation}\label{eq:valueofxir}
\pm\xi_r>\pm\frac{b'(r)\big((r^2+a^2)z-a\xi_\varphi\big)+a\big(a\sin^2\theta z-\xi_\varphi\big)}{r^2+ a^2+b'(r)\Delta_b(r)}.
\end{equation}

If $g^{-1}(-zdt_{\chi}+\xi, d\mathfrak{t})=0$, then $-zdt_{\chi}+\xi$ is a nonzero spacelike vector and thus $p_{h,z}(\xi)=-g^{-1}(-zdt_{\chi}+\xi, -zdt_{\chi}+\xi)<0$. That is, substituting the right-hand side of (<ref>) into $p_{h,z}$ yields a negative number. Therefore,

\[
(r, \theta, \varphi, \xi_r^\pm, \xi_\theta, \xi_\varphi)\in\Sigma_h^\pm\quad \mbox{and}\quad \pm\jb{\xi}^{-1}H_{p_{h,z}}r=\pm\jb{\xi}^{-1}\pa_{\xi_r}p_{h,z}>0\quad \mbox{at}\quad \xi_r^\pm.
\]

In case of fiber infinity $\fiscform\cap\{r_-\leq r<\ehKN\}$, using the coordinates (<ref>), we write

\[
\rho_\xi^2p_{h,z}=-\rho_b^{-2}\Big(\Delta_b\hat{\xi}_r^2+2a\hat{\xi}_r\hat{\xi}_\varphi+\hat{\xi}_\theta^2+\frac{1}{\sin^2\theta}\hat{\xi}_\varphi^2\Big)
\]

\[
\rho_\xi g^{-1}(-zdt_\chi+\xi, d\mathfrak{t})=\rho_b^{-2}\Big(\big(r^2+a^2+b'(r)\Delta_b(r)\big)\hat{\xi}_r+\big(a(1+b'(r))\big)\hat{\xi}_\varphi\Big).
\]

Then the proof proceeds as in the finite $\xi$ case.

It remains to consider the case $r=\ehKN$ where $\Delta_b=0$. For finite $\xi$, we write

\[
p_{h,z}=-\rho_b^{-2}\Big(-2(r^2+a^2)\xi_rz+2a\xi_r\xi_\varphi+\tilde{p}_{h,z}\Big)=0\quad\mbox{where}\quad \tilde{p}_{h,z}=\xi_\theta^2+\frac{1}{\sin^2\theta}(\xi_\varphi-a\sin^2\theta z)^2.
\]

Since $a\xi_\varphi-(r^2+a^2)z\neq 0$ on $\Sigma_h$ (otherwise $a^2\sin^2\theta=r^2+a^2$ at $\ehKN$, which is impossible), the right-hand side of $p_{h,z}$ is a linear equation in $\xi_r$ with nonzero leading coefficient and thus has only one root. Again, by Lemma <ref>, $\Sigma_h^\pm$ is characterized by the conditions $\pm g^{-1}(-zdt_{\chi}+\xi, d\mathfrak{t})>0$, i.e.,

\begin{equation}\label{eq:valueofxirlinear}
\pm\xi_r>\pm\frac{-\big(a(1+b'(r))\big)\xi_\varphi+z\big(a^2\sin^2\theta+(r^2+a^2)b'(r)\big)}{r^2+a^2}.
\end{equation}

Similarly, substituting the right-hand side of (<ref>) into $p_{h,z}$ yields a negative number. Therefore, we have either

\[
(r, \theta, \varphi, \xi_r, \xi_\theta, \xi_\varphi)\in\Sigma_h^+\quad \mbox{and}\quad \jb{\xi}^{-1}H_{p_{h,z}}r=\jb{\xi}^{-1}\pa_{\xi_r}p_{h,z}>0\quad \mbox{at}\quad \xi_r,
\]

\[
(r, \theta, \varphi, \xi_r, \xi_\theta, \xi_\varphi)\in\Sigma_h^-\quad \mbox{and}\quad \jb{\xi}^{-1}H_{p_{h,z}}r=\jb{\xi}^{-1}\pa_{\xi_r}p_{h,z}<0\quad \mbox{at}\quad \xi_r.
\]

In case of fiber infinity $\fiscform\cap\{r=\ehKN\}$, using the coordinates (<ref>), we write

\[
\rho_\xi^2p_{h,z}=-\rho_b^{-2}\Big(2a\hat{\xi}_r\hat{\xi}_\varphi+\hat{\xi}_\theta^2+\frac{1}{\sin^2\theta}\hat{\xi}_\varphi^2\Big).
\]

Since $\hat{\xi}_\varphi\neq 0$ on $(\{\jb{\xi}^{-2}p_{h,z}=0, r=\ehKN\}\cap\fiscform)\setminus L_\pm$, the right-hand side of $\rho_\xi p_{h,z}$ s a linear equation in $\hat{\xi}_r$ with nonzero leading coefficient and thus has only one root.
Then the proof proceeds as in the finite $\xi$ case.

Let $z\in\BR\setminus 0$. Let $\delta_0>0$ and $f(r)$ be defined as in Proposition <ref>. Let $(r,\theta,\varphi,\xi)\in\cscform\setminus(L_\pm\cap \pa_+\CX)$ and $\phi(s)=\exp(s{}^\scop\!H^{2,0}_{p_{h,z}})(r,\theta,\varphi,\xi)$ be a maximally extended flow on $\Sigma_h=\{\jb{\xi}^{-2}p_{h,z}=0\}$.

* If $f(\phi)(s_0)<\delta_0$ and $\pm\pa_sf(\phi(s))\leq0$ for some $\pm s_0\geq0$, then $f(\phi(s))<\delta_0$ and $\pm\pa_sf(\phi(s))<0$ for  $\pm s> s_0$ and $\phi(s)$ is not trapped as $\pm s\to\infty$.

* If $\phi(s)$ is not trapped as $\pm s\to\infty$, then either $f(\phi(s))<\delta_0$ and $\pm\pa_sf(\phi(s))\leq0$ for all sufficiently large $\pm s$, or $\phi(s)\to L_\pm$ (i.e. $f(s)\to0$) as $\pm s\to\infty$.

(3) If $\phi(s)$ is not trapped as $s\to\infty$, then for $\phi(s)\subset \Sigma^-_h$, $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of decreasing $r$ at some finite time $s_0<\infty$ or $\rho(\phi(s))\to0$ as $s\to\infty$, while for $\phi(s)\subset\Sigma^+_h$, $\phi(s)\to L_+$ or $\rho(\phi(s))\to0$ as $s\to\infty$.

(4) If $\phi(s)$ is not trapped as $s\to-\infty$, then for $\phi(s)\subset \Sigma^-_h$, $\phi(s)\to L_-$ or $\rho(\phi(s))\to0$ as $s\to-\infty$, while for $\phi(s)\subset\Sigma_h^+$, $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of decreasing $r$ at some finite time $s_0>-\infty$ or $\rho(\phi(s))\to0$ as $s\to-\infty$.

We only prove the case $\pa_sf(\phi(s))\leq0$ as the other case $-\pa_sf(\phi(s))\leq0$ can be handled similarly. We put
\[
f(s)=f(\phi(s)),\quad \dot{f}(s)=\pa_sf(\phi(s))={}^\scop\!H^{2,0}_{p_{h,z}}f(\phi(s)),\quad \ddot{f}(s)=\pa^2_sf(\phi(s))=({}^\scop\!H^{2,0}_{p_{h,z}})^2f(\phi(s)).
\]

Let $s>s_0$. If $f(s)$ attains its minimum of on the interval $[s_0, s]$ at some point $s_1\in(s_0, s)$, then
\[f(s_1)<\delta_0,\quad \dot{f}(s_1)=0, \quad \ddot{f}(s_1)\geq 0,
\]

which contradicts (<ref>). If $f(s)$ attains its minimum value at $s_0$, then $\dot{f}(s_0)\geq 0$, and thus $\dot{f}(s_0)=0$. Again, it follows from (<ref>) that $s_0$ is a strict local maximum point, which is a contradiction. Therefore, $f(s)$ must attain its minimum value at $s$ and $\dot{f}(s)<0$ (otherwise $\dot{f}(s)=0$ and then by (<ref>), $s$ is a strict local maximum point, which is a contradiction), showing that
\[f(s)<\delta_0,\quad \dot{f}(s)<0\quad \mbox{for all}\quad s>s_0.
\]

Suppose $\phi(s)$ is trapped as $s\to\infty$. Then there exists $\delta>0$ and $T>0$ such that $f(s)>\delta$ for all $s\geq T$. Since $\{f(r)\geq \delta\}\cap\cscform$ is a compact set, we can take a sequence $s_j\to\infty$ such that $\phi(s_j)$ converges to some $(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\in \{\delta\leq f(r)\leq f(s_0)<\delta_0\}$. Since $f(s)$ is non increasing  and bounded for all $s\geq s_0$, $\lim_{s\to\infty}f(s)$ exists. Since $\ddot{f}(s), \dddot{f}(s)$ are bounded for all $s\geq \max\{s_0, T\}$, by Barbălat's Lemma [8], we have ${}^\scop\!H^{1,0}_{p_{h,z}}f((r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty))=({}^\scop\!H^{1,0}_{p_{h,z}})^2f((r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty))=0$. This contradicts (<ref>).

We now prove the statement (2), (3) and (4). We only prove the case $\phi(s)$ is not trapped as $s\to\infty$ in detail as the other case can be handled similarly. If $\phi(s)\subset\{f(r)\leq 0\}$ for all $s\geq0$, then $\phi(s)\subset\{r_-\leq r\leq \ehKN\}$ since $\phi(0)\notin\pa_+\CX$. For the case $\phi(s)\subset\Sigma_h^-$, since $\phi(0)\notin L_-$, it follows from Lemma <ref> that $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of decreasing $r$ at finite time and thus there must exist $s_0>0$ such that $f(s_0)<\delta_0, \dot{f}(s_0)<0$. Proceeding as in the proof of (1), we see that $f(s)<\delta_0, \dot{f}(s)\leq0$ for all $s\geq s_0$. While for the case $\phi(s)\subset\Sigma_h^+$, we claim that $\phi(s)\to L_+$ as $s\to\infty$. Otherwise, there exists a sufficiently small neighborhood $U_+$ of $L_+$ such that $\phi(s)\subset \Sigma_h^+\setminus U_+$ for all $s>0$. By Lemma <ref>, there exists $\delta>0$ such that

\begin{equation}\label{eq:signofderiofrext}
\jb{\xi}^{-1}H_{p_{h,z}}r\geq \delta \quad \mbox{on}\quad (\Sigma_h^+\cap\{r_-\leq r\leq \ehKN+\delta\})\setminus U_+
\end{equation}

(Otherwise, if (<ref>) does not hold for any positive $\delta$, then we can find a sequence $\{(r_j, \theta_j, \varphi_j, \xi_j)\}$ such that $r_-\leq r_j\leq \ehKN+\frac{1}{j}$ and $\jb{\xi}^{-1}H_{p_{h,z}}r((r_j, \theta_j, \varphi_j, \xi_j))< \frac{1}{j}$. Let $j\to\infty$, and then we obtain a limit point
\[(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\in (\Sigma^+_h\cap\{r_-\leq r\leq \ehKN\})\setminus L_+
\]

such that $\jb{\xi}^{-1}H_{p_{h,z}}r((r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty))\leq 0$, which contradicts Lemma <ref>). It follows from (<ref>) that $\phi(s)\subset \{r\geq \ehKN+\delta\}$ for all large enough $s$, which contradicts the assumption that $\phi(s)\subset\{r_-\leq r\leq \ehKN\}$.

On the other hand, suppose there exists $s_0\geq 0$ such that $f(s_0)>0$. Since $\phi(s)$ is not trapped as $s\to\infty$, we can find $s_2>s_1>0$ such that $f(s_2)<f(s_1)<\delta_0$. Then arguing as in the proof of (1), we see that $f(s)$ attains its minimum value on in the interval $[s_1, s_2]$ at the point $s_2$, and thus $\dot{f}(s_2)\leq 0$. Proceeding again as in the proof of (1), we see that $f(s)<\delta_0, \dot{f}(s)\leq0$ for all $s\geq s_2$.

Also, since $\phi(s)$ is not trapped as $s\to\infty$, it follows that either $\rho(\phi(s))\to0$ as $s\to\infty$, or $\phi(s)$ enters the region $\{r\leq r_b+\delta\}$ for any $\delta>0$ in finite time.  Using (<ref>), we find that $\phi(s)\to L_+$ as $s\to\infty$ if $\phi(s)\subset \Sigma^+_h$. If $\phi(s)\subset\Sigma_h^-$, suppose that $\phi(s)$ never crosses $\pa_-\CX$ for $s\geq 0$. Then $\phi(s)$ is well defined for all $s\geq 0$. Since $(\rho,\theta,\varphi,\xi)\notin L_-$, then there exists a neighborhood $V_-$ of $L_-$ such that $ (\rho,\theta,\varphi,\xi)\notin V_-$. Since $L_-$ is a radial source, there exists a neighborhood $U_-$ of $L_-$ and $T>0$ such that

\[
\phi(s)(U_-)\subset V_-\quad \mbox{for all}\quad s\leq-T.
\]

Since $(\rho,\theta,\varphi,\xi)\notin V_-$, it follows that $\phi(s)\notin U_-$ for all $s\geq T$. Proceeding as in the proof of (<ref>) yields

\[
-\jb{\xi}^{-1}H_{p_{h,z}}r\geq \delta \quad \mbox{on}\quad (\Sigma_h^-\cap\{r_-\leq r\leq \ehKN+\delta\})\setminus U_-
\]

for some $\delta>0$. This forces $r(\phi(s))>r_b+\delta$ for all $s>T$ and thus $\rho(\phi(s))\to0$ as $s\to\infty$.

According to Lemma <ref>, it is clear that $K\subset \{f(r)\geq \delta_0\}\subset\{r>\ehKN\}$. Therefore, we now restrict our analysis in the region $r>\ehKN$. Recall the calculation in the proof of Proposition <ref>

\[
\Delta_b>0,\ H_{p_{h,z}}r=0\ \Longrightarrow\  H^2_{p_{h,z}}r=2\rho_b^{-4}\Delta_b\pa_r\Big(\frac{(a\xi_\varphi-(r^2+a^2)z)^2}{\Delta_b(r)}\Big)
\]

We define

\begin{equation}
\end{equation}

The key property of $F(r)$ is given by

Let $z\in\BR\setminus 0$. For each $r\in(\ehKN, \infty)$, we have
\begin{equation}\label{eq:newescapefcn}
p_{h,z}=0,\ \pa_r F(r)=0\ \Longrightarrow\  \pa_r^2F(r)>0.
\end{equation}

Moreover, for each fixed $\xi_\varphi$, there exists a unique $r_{\xi_\varphi}\in(\ehKN, \infty)$ such that $\pa_rF(r_{\xi_\varphi})=0$.

Since $	p_{h,z}=0$, by the discussion around (<ref>), we see that $a\xi_\varphi-(r^2+a^2)z\neq 0$. Then

\[
\pa_rF(r)=-\frac{(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b^2}\Big(\frac{\pa\Delta_b}{\pa r}(a\xi_\varphi-(r^2+a^2)z)+4zr\Delta_b\Big)=0
\]

\begin{align*}
\pa_r^2F(r)&=-\frac{(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b^2}\Big(\frac{\pa^2\Delta_b}{\pa r^2}(a\xi_\varphi-(r^2+a^2)z)+2zr\frac{\pa\Delta_b}{\pa r}+4z\Delta_b\Big)\\
&=-\frac{8z^2r}{\Delta_b(\pa_r \Delta_b)^2}\Big(2\frac{\pa^2\Delta_b}{\pa r^2}r\Delta_b-r(\pa_r\Delta_b)^2-2\Delta_b\pa_r\Delta_b\Big)\\
&=\frac{32z^2r}{\Delta_b(\pa_r \Delta_b)^2}\Big(r^3-3\Bm r^2+3\Bm^2r-\Bm(a^2+\BQ^2)\Big)>0\quad \mbox{on}\quad \ehKN<r<\infty.
\end{align*}

Since $F(r)\to \infty$ as $r\to \ehKN$ and $r\to\infty$, $F(r)$ must attains its global minimum value at $r_{\xi_\varphi}\in(\ehKN, \infty)$ and $\pa_rF(r_{\xi_\varphi})=0$. Moreover, the existence of the critical point of $F$ is unique (otherwise, suppose that there are two points $r_1, r_2\in(\ehKN, \infty)$ with $r_1<r_2$ such that $\pa_r F(r_1)=\pa_rF(r_2)=0$. Then let $D=\{r_1<r<r_2\mid \pa_rF(r)>0\}$. Since $\pa_rF(r_1)=0, \pa_r^2F(r_1)>0$, it follows that $D$ is nonempty and we let $d:=\sup D$. Since $\pa_rF(r_2)=0, \pa^2_rF(r_2)>0$, we see that $d\in(r_1,r_2)$. By continuity, we have $\pa_rF(d)=0$. By the definition of $d$, we have $\pa_r^2F(d)\leq 0$, which is a contradiction).

Let $(r^0,\theta^0, \varphi^0,\xi^0)\in\Sigma_h$ and $(r(s),\theta(s),\varphi(s),\xi(s))=\exp(sH_{p_{h,z}})((r^0,\theta^0, \varphi^0,\xi^0))$. Then we define

\begin{equation}
\Phi^0(r):=\Phi(r;\theta^0, \varphi^0,\xi_\theta^0,\xi^0_\varphi)=F(r;\xi_\varphi^0)-\tilde{p}_{h,z}((\theta^0, \varphi^0,\xi_\theta^0,\xi_\varphi^0)).
\end{equation}

Since $\tilde{p}_{h,z}$ and $\xi_\varphi$ are constants along the Hamiltonian flows $\exp(sH_{p_{h,z}})$ on $\Sigma_h$, it follows that $(r(s), \xi_r(s))$ is a Hamiltonian flow trajectory of

\[
H^0(r, \xi_r):=\Delta_b\big(\xi_r-zc'(r)+\frac{\chi(a\xi^0_\varphi-(r^2+a^2)z)}{\Delta_b}\big)^2-\Phi^0(r)
\]

In particular, $(r(s), \xi_r(s))$ solves the equation $H^0(r,\xi_r)=0$.

In the region $r>\ehKN$, the outgoing tail $\Gamma^+$ and incoming tail $\Gamma^-$ are given by

\begin{equation}\label{eq:defoftails}
\Gamma^{\pm}=\Big\{(r,\theta,\varphi,\xi)\!\mid \!p_{h,z}=0,\  \xi_r-zc'(r)+\frac{\chi(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b}=\mp\sgn(r-r_{\xi_\varphi})\sqrt{\frac{\Phi(r;\theta,\varphi,\xi_\theta,\xi_\varphi)}{\Delta_b}}\Big\}
\end{equation}

\[
\Phi(r;\theta,\varphi,\xi_\theta,\xi_\varphi)=F(r;\xi_\varphi)-\tilde{p}_{h,z}(\theta,\varphi, \xi_\theta,\xi_\varphi)
\]

and $r_{\xi_\varphi}$ is the only solution to the equations $\Phi(r;\theta,\varphi,\xi_\theta,\xi_\varphi)=0$ and $\pa_r\Phi(r;\theta,\varphi,\xi_\theta,\xi_\varphi)=0$. Moreover, $\Gamma^\pm$ are smooth codimension $1$ submanifolds of $\Sigma_h$ intersecting transversely, and their intersection is the trapped set

\begin{equation}\label{eq:exoftrapping}
K=\Big\{(r,\theta,\varphi,\xi)\!\mid \!p_{h,z}=0,\  r=r_{\xi_\varphi},\ \xi_r-zc'(r)+\frac{\chi(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b}=0\Big\},
\end{equation}

which is a smooth codimension $2$ submanifold of $\Sigma_h$

Let $(r^0,\theta^0, \varphi^0,\xi^0)\in\Sigma_h$ and $(r(s),\theta(s),\varphi(s),\xi(s))=\exp(sH_{p_{h,z}})((r^0,\theta^0, \varphi^0,\xi^0))$. By Proposition <ref>, $\pa_r\Phi^0(r_{\xi^0_\varphi})=0$. If $\Phi^0(r_{\xi^0_\varphi})\neq 0$, we have $\abs{\Phi^0(r(s))}+\abs{\pa_r\Phi^0(r(s))}>0$. Suppose that $(r(s),\theta(s),\varphi(s),\xi(s))$ is  trapped as $s\to\infty$. Since $(r(s),\xi_r(s))$ solves the equation $H^0(r,\xi_r)=0$, arguing as in the proof of the statement (1) of Lemma <ref>, we can find a sequence $s_j\to\infty$ such that $(r(s_j),\xi_r(s_j))$ converges to some $(r_\infty, (\xi_r)_\infty)$ which satisfies $H^0(r_\infty, (\xi_r)_\infty)=0, H_{H^0}r((r_\infty, (\xi_r)_\infty))=H_{H^0}\xi_r((r_\infty, (\xi_r)_\infty))=0$. This contradicts the fact that the Hamiltonian vector field associated to $H^0$ is nonzero on the set $H^0=0$. This proves that $(r(s),\theta(s),\varphi(s),\xi(s))$ is  not trapped as $s\to\infty$. Similarly, $(r(s),\theta(s),\varphi(s),\xi(s))$ escapes as $s\to-\infty$ as well. As a result, $(r^0,\theta^0, \varphi^0,\xi^0)\notin \Gamma^{\pm}$.

On the other hand, we consider the case $\Phi^0(r_{\xi^0_\varphi})=0$. We note that the set of solutions to the equation $H^0(r,\xi_r)=0$ is given by the union $\Gamma^{+,0}\cup \Gamma^{-,0}$ where

\[		\Gamma^{\pm,0}=\Big\{\xi_r-zc'(r)+\frac{\chi(a\xi^0_\varphi-(r^2+a^2)z)}{\Delta_b}=\mp\sgn(r-r_{\xi^0_\varphi})\sqrt{\frac{\Phi(r;\theta^0,\varphi^0,\xi^0_\theta,\xi^0_\varphi)}{\Delta_b}}\Big\}.
\]

First, $(r^0,\theta^0,\varphi^0,\xi^0)\in\Sigma_h$ is equivalent to $H^0(r^0, \xi_r^0)=0$, i.e., $(r^0,\theta^0,\varphi^0,\xi^0)\in\Sigma_h$ implies $(r^0,\xi_r^0)\in \Gamma^{+,0}\cup \Gamma^{-,0}$. Next, if $(r^0, \xi_r^0)\in \Gamma^{\mp,0}$, we see that $(r(s),\theta(s),\varphi(s),\xi(s))$ is trapped as $s\to\pm\infty$. Therefore, we conclude that $(r(s),\theta(s),\varphi(s),\xi(s))$ is trapped as $s\to\pm\infty$ if and only if $(r^0, \xi_r^0)\in \Gamma^{\mp,0}$. Then we obtain

\begin{align*}
K=\Gamma^+\cap\Gamma^-&=\Big\{(r,\theta,\varphi,\xi)\!\mid \!p_{h,z}=0,\  r=r_{\xi_\varphi},\ \xi_r-zc'(r)+\frac{\chi(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b}=0\Big\}\\&=\Big\{(r,\theta,\varphi,\xi)\!\mid \!p_{h,z}=0,\  \pa_r F=0,\ H_{p_{h,z}}r=0\Big\}.
\end{align*}

We note that $d(H_{p_{h,z}}r)\in\{dr,d\xi_r, d\xi_\varphi\}$ and the coefficient of $d\xi_r$ is $1$. In view of (<ref>), $d(\pa_rF)\in\{dr, d\xi_\varphi\}$ and the coefficient of $dr$ is positive on $K$. Since $H_{p_{h,z}}r=0$ and $\pa_rF=0$ imply $\pa_{\xi_r}p_{h,z}=0$ and $\pa_r p_{h,z}=0$ respectively on $K$, we have $dp_{h,z}\in\{d\theta, d\xi_\theta, d\xi_\varphi\}$. Therefore, $dp_{h,z}, d(\pa_rF)$ and $d(H_{p_{h,z}}r)$ are linearly independent on $K$ and $K$ is a smooth codimension $2$ submanifold of $\Sigma_h$.

For any $(r^0,\theta^0, \varphi^0,\xi^0)\in K$, since  $\Phi^0(r)=\frac{1}{2}\abs{\pa_r^2\Phi^0(r_{\xi_\varphi^0})}(r-r_{\xi_\varphi^0})^2+\mathcal{O}((r-r_{\xi_\varphi^0})^3)$, the defining functions of $\Gamma^{\pm,0}$ are smooth functions in $r, \xi_r$ and thus $\Gamma^{\pm,0}$ is a smooth codimension $1$ submanifold of $T^*(\ehKN,\infty)$. If $(r,\theta,\varphi,\xi)\in\Gamma^\pm$, then its projection onto $(\theta, \varphi, \xi_\theta,\xi_\phi)$ must lie in the set of projection of $K$ onto $(\theta, \varphi, \xi_\theta,\xi_\phi)$. Therefore, $\Gamma^\pm$ are smooth codimension $1$ submanifolds of $\Sigma_h$ intersecting transversely at $K$.

Since $TK$ is in the kernel of $d(H_{p_{h,z}}r)$ and  $d(\pa_r F)$, we see that the symplectic complement of $TK$ in $\Sigma_h$ is $(TK)^\perp=\{H_{H_{p_{h,z}}r}, H_{\pa_rF}\}$. By a direct calculation, we see that $H_{\pa_r F}(H_{p,z}r)=-2\Delta_b\pa_r^2F\neq 0$ on $K$, which implies $TK\cap (TK)^\perp=0$ and thus $K$ is symplectic.

Now we verify that the trapped set $K$ is normally hyperbolic whose definition will be given below (see <cit.>).

Let $K$ be given as above, i.e., $K$ is symplectic. Let $\phi_s=\exp(sH_{p_{h,z}}):\fscform\to\fscform$ be the Hamiltonian flow and $\varphi_\pm$ be the defining functions of $\Gamma^\pm$. We say $K$ is normally hyperbolic if there exist $C, \nu>0$ such that for all $(r,\theta,\varphi,\xi)\in K$

\begin{equation}\label{eq:defofnormalhypertra}
\frac{d(\varphi_\pm\circ\phi_{\pm s})(r,\theta,\varphi,\xi)}{d\varphi_\pm(r,\theta,\varphi,\xi)}\leq Ce^{-\nu s},\quad  s\geq0
\end{equation}

Define the minimal expansion rate $\nu_{\min}>0$ as the supremum of all values of $\nu$ for which there exists a constant $C$ such that (<ref>) holds.

Let $K$ be defined as in (<ref>). Then $K$ is normally hyperbolic.

It suffices to prove that the minimal expansion $\nu_{\min}$ rate defined above is positive. By (<ref>), we have

\[
\Gamma^{\pm}=\{\varphi_{\pm}=\xi_r-zc'(r)+\frac{\chi(a\xi_\varphi-(r^2+a^2)z)}{\Delta_b}\pm\sgn(r-r_{\xi_\varphi})\sqrt{\frac{\Phi(r;\theta,\varphi,\xi_\theta,\xi_\varphi)}{\Delta_b}}=0\}
\]

where $(\theta,\varphi,\xi_\theta,\xi_\varphi)\in K$. That is $\varphi_\pm$ are defining function of $\Gamma^\pm$. Since $p_{h,z}=\rho_b^{-2}\Delta_b\varphi_+\varphi_-$, it follows that

\[
H_{p_{h,z}}\varphi_\pm=\mp \rho_b^{-2}\Delta_b(H_{\varphi_+}\varphi_-)\varphi_\pm+(H_{\rho_b^{-2}\Delta_b}
\varphi_\pm)\varphi_\mp\varphi_\pm:=\mp\nu_\pm\varphi_{\pm}
\]

and $\nu_+|_K=\nu_-|_K=\rho_b^{-2}\Delta_b(H_{\varphi_+}\varphi_-)|_K=\nu$. Since

\[
\pa_{\xi_r}(H_{p_{h,z}}\varphi_\pm)=\pa_{\xi_r}(\mp\nu_\pm\varphi_{\pm})=\mp\big((\pa_{\xi_r}\nu_\pm)\varphi_\pm+\nu_\pm\big),
\]

using Proposition <ref> we find that

\begin{align*}
\nu&=-\pa_{\xi_r}(H_{p_{h,z}}\varphi_+)|_K=\rho_b^{-2}\sqrt{2\Delta_b\pa_r^2F(r_{\xi_\varphi})}>0.
\end{align*}

For $(r,\theta,\varphi,\xi)\in K$, we calculate

\[
\pa_sd(\varphi_\pm\circ\phi_{\pm s})=\mp d((\nu_\pm\varphi_\pm)\circ\phi_{\pm s})
=\mp(\nu_\pm\circ\phi_{\pm s}) d(\varphi_\pm\circ\phi_{\pm s}).
\]

Therefore, for all $T>0$ and $(r,\theta,\varphi,\xi)\in K$, we have

\[
\frac{d(\varphi_\pm\circ\phi_{\pm T})(r,\theta,\varphi,\xi)}{d\varphi_\pm(r,\theta,\varphi,\xi)}=e^{-\jb{\nu}^\pm_T T},\quad \jb{\nu}^\pm_T=\frac{1}{T}\int_0^T\nu\circ\phi_{\pm s}(r,\theta,\varphi,\xi)\,ds.
\]

As a result, the minimal expansion rate is given by

\begin{equation}\label{eq:calofmer}
\nu_{\min}=\min\{\ \liminf_{T\to\infty}\inf_{(r,\theta,\varphi,\xi)\in K}\jb{\nu}^+_T,\quad \liminf_{T\to\infty}\inf_{(r,\theta, \varphi,\xi)\in K} \jb{\nu}^-_T\ \}>0.
\end{equation}

Finally, we further determine the position of the tapped set $K$.

For $z\in\BR\setminus 0$, let $\Sigma_h^{\pm}$ be defined as in Lemma <ref>. Then $K\subset \Sigma_h^\pm$ for $\pm z>0$. Moreover, $\Gamma^+\cup\Gamma^-\subset\Sigma^\pm_h$ for $\pm z>0$.

First, according to (<ref>) in Lemma <ref>, we have
\[
\Sigma_h^{\mp}\subset\{r\geq r_-\mid\Delta_b\leq a^2\sin^2\theta\}\subset \{r_-\leq r\leq \Bm+\sqrt{\Bm^2-\BQ^2}\}\quad \mbox{for}\quad \pm z>0
\]

and thus

\[
K\cap\{r> \Bm+\sqrt{\Bm^2-\BQ^2}\}\subset \Sigma^\pm_h\quad \mbox{for}\quad \pm z>0.
\]

Since $c'(r)=0, \chi=1$ on $r_-\leq r\leq \Bm+\sqrt{\Bm^2-\BQ^2}$, then for

\[
(r,\theta,\varphi,\xi)\in K\cap\{r_-\leq r\leq \Bm+\sqrt{\Bm^2-\BQ^2}\}=K\cap\{r_b< r\leq \Bm+\sqrt{\Bm^2-\BQ^2}\},
\]

we have

\[
a\xi_\varphi-(r^2+a^2)z=-\frac{4zr\Delta_b}{\pa_r\Delta_b}\quad\mbox{and}\quad \xi_r=\frac{4zr}{\pa_r\Delta_b},
\]

and thus

\begin{align*}
\pm g^{-1}(-zdt_{b,*}+\xi, d\mathfrak{t})&=\pm\rho_b^{-2}\Big(\xi_r(r^2+a^2+b'(r)\Delta_b)+b'(r)(a\xi_\varphi-(r^2+a^2)z)+(a\xi_\varphi-a^2\sin^2\theta z)\Big)\\
&=\pm\frac{4zr}{\rho_b^2\pa_r\Delta_b}\Big(r^2+a^2-\Delta_b\Big)\pm z=\pm\frac{4zr}{\rho_b^2\pa_r\Delta_b}\Big(2\Bm r-\BQ^2\Big)\pm z>0\quad \mbox{for}\quad \pm z>0.
\end{align*}

This implies that

\[
K\cap\{r_-\leq r\leq \Bm+\sqrt{\Bm^2-\BQ^2}\}\subset \Sigma^\pm_h\quad \mbox{for}\quad \pm z>0,
\]

and thus proves $K\subset\Sigma^\pm_h$ for $\pm z>0$.

In order to prove $\Gamma^+\cup\Gamma^-\subset \Sigma^\pm_h$ for $\pm z>0$, it suffices to show that if $(r,\theta,\varphi,\xi)\in\Gamma^{\pm}$, then $\phi(s)=\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})(r,\theta, \varphi,\xi)\to K$ as $s\to\mp\infty$. We only prove the case of $(r,\theta,\varphi,\xi)\in\Gamma^-$ as the other case can be handled in a similar manner. Suppose that $\phi(s)\nrightarrow K$ as $s\to\infty$. Then there exists a sequence $s_j\to\infty$ and a neighborhood $U$ of $K$ such that $\phi(s_j)\notin U$ for all $j$. Since $\phi(s)$ is trapped as $s\to\infty$ (i.e., there exist $\delta, T>0$ such that $f(\phi(s))>\delta$ for all $s>T$), passing to a subsequence we have

\[
\phi(s_j)\to(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\quad \mbox{for some}\quad (r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\notin K=\Gamma^+\cap\Gamma^-.
\]

By Lemma <ref>, we find that $\Gamma^-$ is closed and thus $(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\in \Gamma^-$. It follows that $(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)\notin \Gamma^+$, which means that $\phi(s)$ is not trapped as $s\to-\infty$. Then there exists $T_1>0$ such that $f(\phi_\infty(-T_1))<\delta$ where $\phi_\infty(s)=\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})(r_\infty,\theta_\infty, \varphi_\infty,\xi_\infty)$. Since $	\phi(s_j)\to(r_\infty, \theta_\infty, \varphi_\infty, \xi_\infty)$, it follows that
\[\exp(-T_1{}^{\scop}\!H_{p_{h,z}}^{2,0})(\phi(s_j))\to\phi_\infty(-T_1).
\]

Therefore, $f(\phi(s_j-T_1))<\delta$ for all sufficiently large $j$. But this contradicts the fact that $\phi(s)$ is trapped as $s\to\infty$.

§.§.§ Global dynamics of the Hamiltonian flow

Now we shall analyze the global behavior of the flow of $	{}^{\scop}\!H_{p_{h,z}}^{2,0}:=\rho^{-1}\jb{\xi}^{-1}{}^{\scop}\!H_{p_{h,z}}$ on the semiclassical characteristic set $\Sigma_h=\{\jb{\xi}^{-2}p_{h,z}=0\}$. We need the following technical lemma.

For $z\in\BR\setminus 0$, let $(\rho,\theta, \varphi,\xi)\in\Sigma_h$ and we put $\phi(s)=\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})(\rho,\theta, \varphi,\xi)$. If $\rho(\phi( s))\to 0$ as $s\to\pm\infty$, then one has

\[
\phi(s)\to R(\frac{z}{2}\pm\frac{\abs{z}}{2})=\{\rho=0, \xi=(z\pm\abs{z})\frac{d\rho}{\rho^2}\}\quad \mbox{as}\quad s\to \pm\infty.
\]

Recall that

\[
\jb{\xi}{}^{\scop}\!H_{p_{h,z}}^{2,0}=(-2\xi_\rho+2z)\rho\pa_\rho+2\tilde{p}\pa_{\xi_\rho}+(-2\xi_\rho+2z)\sum_{\mu=\theta,\varphi}\xi_\mu\pa_{\xi_\mu}-H_{\tilde{p}}+\rho V, \quad V\in\mathcal{V}_\bop(\fscform).
\]

Since $p_{h,z}=0$ along the integral curves of ${}^{\scop}\!H_{p_{h,z}}^{2,0}$, we calculate

\begin{align*}
\jb{\xi}{}^{\scop}\!H_{p_{h,z}}^{2,0}\Big(\frac{\xi_\rho-z}{\rho}\Big)=\jb{\xi}\frac{2(\xi_\rho-z)^2+2\tilde{p}+\rho a_1}{\rho}=\jb{\xi}\frac{z^2+\rho a_2}{\rho},\quad a_1, a_2\in\mathcal{A}^0(\CX).
\end{align*}

Since $\rho(\phi( s))\to 0$ as $s\to\pm\infty$, then for any $\epsilon>0$, there exists $T>0$ such that $\rho(\phi( s))<\epsilon$ for all $\pm s>T$. Picking $\epsilon$ sufficiently small, we see that ${}^{\scop}\!H_{p_{h,z}}^{2,0}\Big(\frac{\xi_\rho-z}{\rho}\Big)\geq c_1>0$ for all $\pm s>T$. This implies that $(\xi_\rho-z)/\rho\to\pm\infty$ as $s\to\pm\infty$; in particular $\pm(\xi_\rho-z)\geq 0$ for all $\pm s>T_1$ where $T_1>0$ is a sufficiently large constant.

According to Lemma <ref>, there exist neighborhoods $V^{\epsilon'}_\pm=\{(\xi_\rho-(z\pm(\sgn z)z)))^2+\tilde{p}+\rho^2\leq \epsilon'\}$ of $R(\frac{z}{2}\pm\frac{\abs{z}}{2})$ such that once $\phi(s)$ enters $V^{\epsilon'}_\pm$ at some time $\pm s_0>0$, it will converge to $R(\frac{z}{2}\pm\frac{\abs{z}}{2})$ as $\pm s\to\infty$. Suppose that $\phi(s)\notin V^{\epsilon'}_\pm$ for all $\pm s\geq 0$. Since $p_{h,z}=0$ along the integral curves of ${}^{\scop}\!H_{p_{h,z}}^{2,0}$, it follows that

\begin{equation}\label{eq:awayfromradialpt}
\abs{\xi_\rho-z}\leq \delta_1\quad \tilde{p}>\delta_2\quad \mbox{for all} \quad \pm s>T_2
\end{equation}

for some $\delta_1, \delta_2>0,T_2>T_1$. Then for all $\pm s>T_2$, we have

\[
{}^{\scop}\!H_{p_{h,z}}^{2,0}(\xi_\rho-z)=\jb{\xi}^{-1}(2\tilde{p}+\rho a_2)>c_2>0,\quad a_2\in\mathcal{A}^0(\CX).
\]

which contradicts (<ref>). Therefore, $\phi(s)$ must enter $V^{\epsilon'}_\pm$ at some time $\pm s_0>0$ and this finishes the proof of the lemma.

Now we are at the position to describe the the global behavior of the flow of $	{}^{\scop}\!H_{p_{h,z}}^{2,0}:=\rho^{-1}\jb{\xi}^{-1}{}^{\scop}\!H_{p_{h,z}}$ on the semiclassical characteristic set $\Sigma_h=\{\jb{\xi}^{-2}p_{h,z}=0\}$.

[color=gray!50] (-3, 0.2) to [out=-20, in=180] (3,-1)–(3,2) to [out=180, in=-90] (-1,3)–(-3,3)–cycle;
[color=gray!50] (-1, -3) – (-3,-3) –(-3,-0.5) to [out=0, in=90] (-1,-3);

[dotted, semithick](-2,3)–(-2,-3);

[circle,inner sep=1pt,fill=black,label=above:$L_+$] at (-2,3) ;
[circle,inner sep=1pt,fill=black,label=below:$L_-$] at (-2,-3) ;
[circle,inner sep=1pt,fill=black,label=right:$R(z)$] at (3,-1) ;
[circle,inner sep=1pt,fill=black,label=right:$R(0)$] at (3,2) ;
[circle,inner sep=1pt,fill=black,label=right:$K$] at (0.5,0.5) ;

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(-2, -3)–(-3,-3);

decoration=markings, mark=at position 0.5 with stealth,

[ decoration=markings, mark=at position 0.5 with stealth,

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] (-2, -3) to [out=35, in=-10] (-3,-0.8);
decoration=markings, mark=at position 0.75 with stealth,
] (-2, -3) to [out=90+10, in=-10]  (-3, -1.2);

[ decoration=markings, mark=at position 0.5 with stealth,
(3,2) to [out=-90-30, in=90+30](3,-1);

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[ decoration=markings, mark=at position 0.5 with stealth,
(-3,0.5) to [out=-10, in=180-10]  (3,-1);

[label= above right:$\Gamma^-$] at (0.8, 1) ;
[label= below right:$\Gamma^+$] at (1.3,0.3) ;
[label= above right:$\Sigma_h^-$] at (-3, -3) ;
[label= below right:$\Sigma_h^+$] at (-3, 3) ;
at (0, -3.75) $z>0$;

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[dotted, semithick](-2,3)–(-2,-3);

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[circle,inner sep=1pt,fill=black,label=right:$R(z)$] at (3,1) ;
[circle,inner sep=1pt,fill=black,label=right:$R(0)$] at (3,-2) ;
[circle,inner sep=1pt,fill=black,label=right:$K$] at (0.5,-0.5) ;

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[label= below right:$\Gamma^+$] at (0.6, -1) ;
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[label= above right:$\Sigma_h^-$] at (-3, -3) ;
[label= below right:$\Sigma_h^+$] at (-3, 3) ;
at (0, -3.75) $z<0$;

The flow of $\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})$, which is projected to the coordinates $(r, \xi_r)$ and drawn in a fiber-radially compactified view. The shaded region is the semiclassical characteristic set $\Sigma_h=\Sigma^+_h\sqcup\Sigma^-_h=\{\jb{\xi}^{-2}p_{h,z}=0\}$. The horizontal coordinate is $r$ and the rightmost vertical line corresponds to $\pa_+\CX$. The dotted line is the even horizon $r=r_b$ and the vertical line in the middle is $\{r=r_{\mathrm{ergo}}\}$ where $r_{\mathrm{ergo}}$ is the equatorial radius (i.e. greatest radius) of the ergosphere. The vertical coordinate is $\xi_r/\jb{\xi_r}$, so the top and bottom lines stand for the fiber infinity.

Let $\pm z>0$. Let $\phi(s)=\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})(\rho,\theta, \varphi,\xi)$ be the maximally extended integral curve of ${}^{\scop}\!H_{p_{h,z}}^{2,0}$ on $\Sigma_h$ with the domain of definition $s\in I\subset\BR$. Let $s_-=\inf I, s_+=\sup I$. (see Figure <ref>).

* Let $z>0$.

If $\phi(s)\subset\Sigma^-_h$, then either $\phi(s)\subset L_-$, or $\phi(s)\to L_-$ as $s\to s_-=-\infty$ and $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at finite time $s_+<\infty$.

(2) If $\phi(s)\subset\Sigma^+_h$, then either $\phi(s)\subset L_+\cup R(z)\cup R(0)\cup K$, or $\phi(s)\to L_+\cup R(z)\cap K$ as $s\to s_+=\infty$ and $\phi(s)\to R(0)\cup K$ as $s\to s_-=-\infty$ or $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at finite time $s_->-\infty$. Moreover, if $(\rho,\theta,\varphi,\xi)\notin K$, then $\phi(s)$ cannot converge to $K$ in both the forward and backward direction.

* Let $z<0$.

If $\phi(s)\subset\Sigma^+_h$, then either $\phi(s)\subset L_+$, or $\phi(s)\to L_+$ as $s\to s_+=\infty$ and $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at finite time $s_->-\infty$.

(2) If $\phi(s)\subset\Sigma^-_h$, then either $\phi(s)\subset L_-\cup R(z)\cup R(0)\cup K$, or $\phi(s)\to L_-\cup R(z)\cup K$ as $s\to s_-=-\infty$ and $\phi(s)\to R(0)\cup K$ as $s\to s_+=\infty$ or $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at finite time $s_+<\infty$. Moreover, if $(\rho,\theta,\varphi,\xi)\notin K$, then $\phi(s)$ cannot converge to $K$ in both the forward and backward direction.

We only consider the case $z>0$ as the other case $z<0$ can be proved similarly. First, we note that $L_+, L_-, R(0), R(z), K$ are invariant under the flow $\phi(s)$ and if $(\rho,\theta,\varphi,\xi)\in \Gamma^\pm$, then $\phi(s)\to K$ as $\mp s\to\infty$. If $(\rho,\theta,\varphi,\xi)\notin L_+\cup L_-\cup R(0)\cup R(z)\cup K$, then either $(\rho,\theta,\varphi,\xi)\notin\Gamma^-$ or $(\rho,\theta,\varphi,\xi)\notin\Gamma^+$.

If $(\rho,\theta,\varphi,\xi)\notin\Gamma^-$, by Lemma <ref>, for $\phi(s)\subset \Sigma_h^+$, either $\phi(s)\to L_+$ or $\rho(\phi(s))\to 0$ as $s\to\infty$, while for $\phi(s)\subset \Sigma_h^-$, $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at some finite time $s_0<\infty$. If $\rho(\phi(s))\to 0$ as $s\to\infty$, according to Lemma <ref>, we have $\phi(s)\to R(z)$ as $s\to\infty$.

If $(\rho,\theta,\varphi,\xi)\notin\Gamma^+$, by Lemma <ref>, for $\phi(s)\subset \Sigma_h^+$, either $\rho(\phi(s))\to 0$ as $s\to\infty$ or $\phi(s)$ crosses $\pa_-\CX$ into the inward direction of deceasing $r$ at some finite time $s_0>-\infty$, while for $\phi(s)\subset \Sigma_h^-$, $\phi(s)\to L_-$ as $s\to-\infty$. If $\rho(\phi(s))\to 0$ as $s\to-\infty$, according to Lemma <ref>, we have $\phi(s)\to R(0)$ as $s\to-\infty$.

§.§ High energy estimates

In this section, we prove the high energy estimates first for operators $h^2\widehat{\Box_{g_{b}}}(h^{-1}z)$ acting on scalar functions, then for $h^2\widehat{\mathcal{P}_{b, \gamma}}(h^{-1}z), h^2\widehat{\mathcal{W}_{b, \gamma}}(h^{-1}z)$ acting on scattering $1$-forms and the linearized gauge-fixed Einstein-Maxwell operator $h^2\widehat{L_{b,\gamma}}(h^{-1}z)$, as well as their formal adjoints.

To this end, we combine the global dynamics of of the Hamiltonian flow of the semiclassical principal symbol $p_{h,z}$ established in <ref> with the elliptic estimate, propagation of singularities estimate, the radial point estimate at event horizon, the scattering radial point estimate at spatial infinity $\pa_+\CX$ and hyperbolic estimate, all of which are in the semiclassical version, together with the estimate at normally hyperbolic trapping.

§.§.§ 	High energy estimates for scalar wave operators

We first prove high energy estimates for the operator $h^2\widehat{\Box_{g_{b}}}(h^{-1}z)$ acting on scalar functions, as well as its formal adjoint with respect to volume density $L^2(\CX;\sqrt{\det\abs{g_b}}drd\theta d\varphi)$.

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A phase space picture of the proof of the estimate (<ref>), on the left, and (<ref>), on the right. The coordinates and notations $L_\pm, R(0), R(z),\Gamma^\pm,K$ are the same as in Figure <ref>. For (<ref>), we use semiclassical hyperbolic estimates to control $u$ via $\chi u$; $\chi$ is controlled (modulo elliptic estimates) by $B_\pm, B_1, B_0, B_z$; $B_0$ is controlled by $E_0$ using low b-decay radial point estimates and $E_0$ is again controlled by $B_+, B_1, B_z$; $B_1$ is controlled by $B_2$ by using normally hyperbolic trapping estimate and $B_2$ is controlled by $B_+,B_z$; finally $B_\pm, B_z$ are controlled using high regularity radial points estimates at $L_\pm$ and high sc-decay radial point estimates at $R(z)$. For (<ref>), we use semiclassical hyperbolic estimates to bound $(1-\chi)u$; $\chi$ is controlled (modulo elliptic estimates) by $B'_\pm, B'_1, B'_0, B'_z$; $B'_\pm$ is controlled by $E'_\pm$ using low regularity radial point estimates and $E'_\pm$ is controlled by $B'_1, B_0', 1-\chi$; $B'_z$ is controlled by $E'_z$ using low sc-decay radial points estimates and $E'_z$ is controlled by $B_0', B_1', 1-\chi$; $B'_1$ is controlled by $B'_2$ by using normally hyperbolic trapping estimate and $B'_2$ is controlled by $B'_+, 1-\chi$;  finally $B'_0$ is controlled using high b-decay radial points estimates at $R(0)$.

Let $b_0=(\Bm_0, \Ba_0, \BQ_0)$ with $\abs{\BQ_0}+\abs{\Ba_0}<\Bm_0$. Then there exists $\epsilon>0$ such that for $b=(\Bm, \Ba,\BQ)$ with $\abs{b-b_0}<\epsilon$ and for $s>\frac{1}{2}, \ell<-\frac{1}{2}$ with $s+\ell>-\frac12$, the following holds. For any fixed $C_1>0$, there exist $C_2>1, C>0$ which are independent of $b$, such that for $\sigma\in\BC, \IM\sigma\in[0, C_1], \abs{\RE\sigma}>C_2$, we have

\begin{align}\label{eq:highenergy1}
\norm{u}_{\bar{H}_{\bop,h}^{s,\ell}(\CX)}&\leq C\norm{\widehat{\Box_{g_{b}}}(\sigma)u}_{\bar{H}_{\bop,h}^{s,\ell+1}(\CX)},\quad\norm{u}_{\dot{H}_{\bop,h}^{-s,-\ell-1}(\CX)}\leq C\norm{\widehat{\Box_{g_{b}}}(\sigma)^*u}_{\dot{H}_{\bop,h}^{-s,-\ell}(\CX)}; \\\label{eq:highenergy2}
\norm{u}_{\bar{H}_{\bop,h}^{s,\ell}(\CX)}&\leq C\norm{\widehat{\Box_{g_{b}}}(\sigma)u}_{\bar{H}_{\bop,h}^{s-1,\ell+2}(\CX)},\quad \norm{u}_{\dot{H}_{\bop,h}^{-s+1,-\ell-2}(\CX)}\leq C\norm{\widehat{\Box_{g_{b}}}(\sigma)^*u}_{\dot{H}_{\bop,h}^{-s,-\ell}(\CX)},
\end{align}

where $h=\abs{\sigma}^{-1}$ and $C$ only depends on $b_0, s,\ell, C_1$.

\[
\widehat{\Box_{g_{b}}}:\{u\in\bar{H}_{\bop,h}^{s,\ell}(\CX)\mid\bar{H}_{\bop,h}^{s,\ell+1}(\CX)\}\to\bar{H}_{\bop,h}^{s,\ell+1}(\CX)
\quad\mbox{and}\quad
\widehat{\Box_{g_{b}}}:\{u\in\bar{H}_{\bop,h}^{s,\ell}(\CX)\mid\bar{H}_{\bop,h}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop,h}^{s-1,\ell+2}(\CX)
\]

are invertible for $\sigma\in\BC, \IM\sigma\in[0, C_1], \abs{\RE\sigma}>C_2$.

For the simplicity of notations, we let $P_h(z):=h^2\widehat{\Box_{g_{b}}}(h^{-1}z)$.	We first claim that it suffices to prove the following estimates for $s>\frac 12, r>-\frac12, \ell<-\frac 12$

\begin{equation}\label{eq:highenergysecondmic}
\norm{u}_{\bar{H}_{\scop,\bop,h}^{s,r,\ell}(\CX)}\leq Ch^{-2}	\norm{P_h(z)u}_{\bar{H}_{\scop,\bop,h}^{s-1,r+1,\ell+1}(\CX)}
\end{equation}

\begin{equation}\label{eq:highenergysecondmicdual}
\norm{u}_{\dot{H}_{\scop,\bop,h}^{-s+1,-r-1,-\ell-1}(\CX)}\leq Ch^{-2}	\norm{P_h(z)^*u}_{\dot{H}_{\scop,\bop,h}^{-s,-r,-\ell}(\CX)}
\end{equation}

Since $\bar{H}_{\scop,\bop,h}^{s,s+\ell,\ell}=\bar{H}_{\bop,h}^{s,l}$ and $\dot{H}_{\scop,\bop,h}^{s,s+\ell,\ell}=\dot{H}_{\bop,h}^{s,l}$, letting $r=s+\ell>-\frac 12$ we have

\[
\norm{u}_{\bar{H}_{\scop,\bop,h}^{s,s+\ell,\ell}(\CX)}\leq Ch^{-2}	\norm{P_h(z)u}_{\bar{H}_{\scop,\bop,h}^{s-1,s+\ell+1,\ell+1}(\CX)}
\]

\[
\norm{u}_{\dot{H}_{\scop,\bop,h}^{-s+1,-s-r-1,-\ell-1}(\CX)}\leq Ch^{-2}	\norm{P_h(z)^*u}_{\dot{H}_{\scop,\bop,h}^{-s,-s-r,-\ell}(\CX)}.
\]

Then using the facts that

\begin{gather*} \bar{H}_{\bop,h}^{s,\ell+1}=\bar{H}^{s,s+\ell+1, \ell+1}_{\scop,\bop,h}\subset \bar{H}^{s-1,s+\ell+1,\ell+1}_{\scop,\bop,h},\quad\bar{H}_{\bop,h}^{s-1,\ell+2}=\bar{H}^{s-1,s+\ell+1, \ell+2}_{\scop,\bop,h}\subset \bar{H}^{s-1,s+\ell+1,\ell+1}_{\scop,\bop,h},\\
\dot{H}_{\bop,h}^{-s,-\ell-1}=\dot{H}^{-s,-s-\ell-1, -\ell-1}_{\scop,\bop,h}\supset \dot{H}^{-s+1,-s-\ell-1,-\ell-1}_{\scop,\bop,h},\\ \dot{H}_{\bop,h}^{-s+1,-\ell-2}=\dot{H}^{-s+1,-s-\ell-1,-\ell-2}_{\scop,\bop,h}\supset \dot{H}^{-s+1,-s-\ell-1,-\ell-1}_{\scop,\bop,h},
\end{gather*}

we obtain (<ref>) and (<ref>).  Now we will show (<ref>) and (<ref>). Here we only discuss the case $\RE z>0$ (see Figure <ref> for a phase space illustration of the proof of estimates (<ref>) and (<ref>)), as the case $\RE z<0$ can be handled in a similar manner.

* Proof of the estimate (<ref>). For $s>s_0>\frac 12\geq\frac{1}{2}-\IM(h^{-1}z)\frac{\ehKN^2+a^2}{\ehKN^2-\Bm}$ (as calculated in (<ref>) and (<ref>)), using the semiclassical high regularity radial point estimate (see <cit.>, <cit.>), there exist $B_\pm, G_\pm, S_\pm \in\Psi^0_h(\CX)$ microlocally supported near $L_\pm$ with $L_\pm\subset \mathrm{ell}_h(B_\pm), L_\pm\subset \mathrm{ell}_h(S_\pm)$ and satisfying that all the forward (backward) null characteristics from $\mathrm{WF}_h(B_\pm)$ tend to $L_\pm$, while remaining in the elliptic set of $G_\pm$, such that

\begin{equation}\label{eq:highregradiales}
\norm{B_\pm u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{G_\pm P_h(z)u}_{H^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+Ch^{s-s_0}\norm{S_\pm u}_{H^{s_0,r_0,\ell_0}_{\scop,\bop,h}(\CX)}
\end{equation}

where $r_0<r, \ell_0<\ell$ and the sc-decay order $r$ and b-decay order $\ell$ are actually irrelevant here.

For $r>r_0>-\frac 12$ (as calculated in (<ref>)), using the semiclassical high sc-decay radial point estimates at the non-zero scattering section $R(z)$ (see [111]. The proof follows from a positive commutator estimate

\[
i(P_h(z)^*A-AP_h(z))=\IM P_h(z)A+A\IM P_h(z)+i[\RE P_h(z), A]\quad
\]

\[
\RE P_h(z)=\frac{P_h(z)+P_h(z)^*}{2},\quad\IM P_h(z)=\frac{P_h(z)-P_h(z)^*}{2i}.
\]

Let $A\in\Psi_{\scop,\bop,h}^{2s-1,2r+1,2l+1}(\CX)$ with semiclassical principal symbol
\[
a=\chi_0(\xi_\theta^2+\frac{1}{\sin^2\theta}\xi_\varphi^2)\chi_1((\xi_\rho-2\RE z)^2)\rho^{-2r+1}(\xi_\theta^2+\frac{1}{\sin^2\theta}\xi_\varphi^2+\xi_\rho^2)^{s-\frac{1}{2}}
\]

where $\chi_0, \chi_1$ are identically 0 near $0$ and have compact support sufficiently close to $0$, and $\chi_1$ has relatively large support such that $\mbox{supp}\chi_0\cap \mbox{supp}\chi_1'$ is disjoint from the characteristic set of $\RE P_h(z)$. Then using Lemma 4.1 and the calculation before Lemma 4.8 in [111], it follows that $h^{-1}(\IM P_h(z)A+A\IM P_h(z)+i[\RE P_h(z), A])\in \Psi_{\scop,\bop,h}^{2s,2r,2l}(\CX)$ whose semiclassical principal symbol is positive definite elliptic near $R(z)$ if $r>-\frac{1}{2}$. This, together with a regularization argument,  proves semiclassical high sc-decay radial point estimates at $R(z)$), there exist $B_z, G_z, S_z \in\Psi^{0,0}_{\scop,h}(\CX)$ microlocally supported near $R(z)$ with $R(z)\subset \mathrm{ell}_{\scop,h}(B_z), R(z)\subset\mathrm{ell}_{\scop,h}(S_z)$ and satisfying that all the forward null characteristics from $\mathrm{WF}_{\scop,h}(B_z)$ tend to $R(z)$, while remaining in the scattering elliptic set of $G_z$, such that

\begin{equation}\label{eq:highscdecayradiales}
\norm{B_z u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{G_zP_h(z)u}_{H^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+Ch^{s-s_0}\norm{S_z u}_{H^{s_0,r_0,\ell_0}_{\scop,\bop,h}(\CX)}
\end{equation}

where $s_0<s, \ell_0<\ell$ and the differential order $s$ and b-decay order $\ell$ are actually irrelevant here.

For $\ell<-\frac 12$ (as calculated in (<ref>)), using the semiclassical low b-decay radial point estimates at the zero scattering section $R(0)$ (see <cit.>), there exist $B_0, G_0\in\Psi^{0,0,0}_{\scop,\bop,h}(\CX)$ microlocally supported near $R(0)$ (in fact near the blown up $R(0)$ in the second microlocalized space introduced in <ref>) with $R(0)\subset \mathrm{ell}_{\scop,\bop,h}(B_0)$ and $E_0\in\Psi_{\scop,\bop,h}^{0,0,0}(\CX), \mathrm{WF}_{\scop,\bop,h}(E_0)\cap R(0)=\emptyset$, and satisfying that all the forward null characteristics from $\mathrm{WF}_{\scop,\bop,h}(B_0)\setminus R(0)$ enter $\mathrm{ell}_{\scop,\bop,h}(E_0)$, while remaining in the second microlocalized scattering elliptic set of $G_0$, such that for any sufficiently large $N>0$

\begin{equation}\label{eq:lowbdecayradiales}
\norm{B_0 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{G_0P_h(z)u}_{H^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+C\norm{E_0 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\bar{H}^{-N,-N,\ell}_{\scop,\bop,h}(\CX)}
\end{equation}

where the differential order $s$ is actually irrelevant here.

Since at the tapped set $K$

\begin{align*}
\sigma_h(h^{-1}\IM P_{h}(z))&=\sigma_h(\frac{P_h(z)-P_h(z)^*}{2ih})=\sigma_h(\frac{P_h(z)-P_h(\bar{z})}{2ih})\\
&=-\rho_b^{-2}\IM\sigma\bigg(-\Big(\xi_r- c'(r)\RE z+\frac{\chi(a \xi_\varphi-(r^2+\Ba^2)\RE z)}{\Delta_b}\Big)\Big(c'(r)+\frac{\chi(r^2+a^2)}{\Delta_b}\Big)\\
&\quad\quad+\frac{2}{\Delta_b}\Big(a\xi_\varphi-(r^2+a^2)\RE z\Big)\Big(r^2+a^2\Big)-2a\Big(\xi_\varphi-a\sin^2\theta \RE z\Big)\bigg)\\
&=-\rho_b^{-2}\IM\sigma\bigg(\frac{2}{\Delta_b}\Big(a\xi_\varphi-(r^2+a^2)\RE z\Big)\Big(r^2+a^2\Big)-2a\Big(\xi_\varphi-a\sin^2\theta \RE z\Big)\bigg)>0
\end{align*}

where we use $\xi_r- c'(r)\RE z+\frac{\chi(a \xi_\varphi-(r^2+a^2)\RE z)}{\Delta_b}=0$ at $K$ in the last equality and
\[a\xi_\varphi-(r^2+a^2)\RE z=-\frac{-4 r\Delta_b\RE z}{\pa_r\Delta_b}<0, \quad (a\xi_\varphi-(r^2+a^2)\RE z)^2\geq \Delta_b(\xi_\varphi-a\sin^2\theta\RE z)^2\quad \mbox{at }K,
\]

which follow from Proposition <ref> and (<ref>), in the last step, it follows that

\[
\sigma_h(h^{-1}\IM -P_{h}(z))=\sigma_h(\frac{(-P_h(z))-(-P_h(z))^*}{2ih})<0<\frac{\nu_{\min}}{2}\quad \mbox{at}\quad K.
\]

Using the hyperbolic trapping estimates (see [39] and <cit.>), there exist $B_1, G_1\in\Psi^{0}_{h}(\CX)$ microlocally supported near $K$ with $K\subset \mathrm{ell}_{h}(B_1)$ and $B_2\in\Psi_{h}^{0}(\CX), \mathrm{WF}_{h}(B_2)\cap \Gamma^-=\emptyset$, and satisfying that $
\mathrm{WF}_h(B_1)\cup\mathrm{WF}_h(B_2)$ is contained in the elliptic set of $G_1$, such that for any $N, s',r',\ell'\in\BR$

\begin{equation}\label{eq:normallytrapes1}
\norm{B_1 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-2}\norm{G_1P_h(z)u}_{H^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+C\norm{B_2 u}_{H^{s,r,\ell}_{,h\scop,\bop}(\CX)}+Ch^{N}\norm{u}_{\bar{H}^{s',r',\ell'}_{\scop,\bop,h}(\CX)}
\end{equation}

where the differential order $s'$, the sc-decay order $r'$ and b-decay order $\ell'$ are actually irrelevant here.

Let $r_-<r_0<\ehKN$ and $\chi\in C_c^\infty(\CX)$ with $\chi=1$ near $r\geq\ehKN$ and $\mathrm{supp}\chi\subset\{r\geq r_0\}$. By part (1) in Proposition <ref>, semiclassical propagation of singularities estimates and semiclassical elliptic estimates (Concretely, when $(\rho,\theta, \varphi, \xi)\in \mathrm{WF}_h(\chi)\cap\{\jb{\xi}^{-2}p_{h,z}(\xi)\neq0\}$, we use semiclassical elliptic estimates. If $(\rho,\theta, \varphi, \xi)\in \mathrm{WF}_h(\chi)\cap\{\jb{\xi}^{-2}p_{h,z}(\xi)=0\}$, then there exists $s\in\BR$ with $\exp(s{}^{\scop}\!H_{p_{h,z}}^{2,0})(\rho,\theta, \varphi,\xi)\in \mathrm{ell}_h(B_\pm)\cup\mathrm{ell}_h(B_1)\cup\mathrm{ell}_h(B_0)\cup\mathrm{ell}_h(B_z)$, and thus we use semiclassical propagation of singularities estimates. We point out that in the set $\{\RE p_{h}(z)=0\}\cap\pa_+\CX$, the semiclassical symbol has a nonnegative imaginary part, so one can propagate estimates towards $R(0)$. Finally, by using a pseudodifferential partition of unity, $\chi$ can be written as s sum of operators falling into the above two case), we have

\begin{equation}
\begin{split}
\norm{\chi u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}&\leq C h^{-1}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+C\norm{B_+ u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+C\norm{B_- u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\\
&+C\norm{B_1 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+C\norm{B_0 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+C\norm{B_z u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\bar{H}^{-N,-N,\ell}_{\scop,\bop,h}(\CX)}.
\end{split}
\end{equation}

By the same reasoning as above in the control of $\chi u$, it follows that

\begin{equation}
\begin{split}
\norm{E_0 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}&\leq C h^{-1}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+C\norm{B_+ u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+C\norm{B_1 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\\
&\quad+C\norm{B_z u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\bar{H}^{-N,-N,\ell}_{\scop,\bop,h}(\CX)}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\norm{B_2 u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}&\leq C h^{-1}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+C\norm{B_+ u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\\
&\quad+C\norm{B_z u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\bar{H}^{-N,-N,\ell}_{\scop,\bop,h}(\CX)}.
\end{split}
\end{equation}

Putting all the above estimates together yields for $s>s_0>\frac 12, r>r_0>-\frac 12, \ell<-\frac 12$

\begin{equation}\label{eq:semiglobales1}
\norm{\chi u}_{H^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-2}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+Ch^{s-s_0}\norm{u}_{\bar{H}^{s_0,r_0,\ell}_{\scop,\bop,h}(\CX)}.
\end{equation}

\[
\]

\[ \tilde{p}_{h,z}=\xi_\theta^2+\frac{1}{\sin^2\theta}(\xi_\varphi-a\sin^2\theta z)^2
\]

is semiclassical hyperbolic with respect to $r$ in the region $\{r_-\leq r<\ehKN\}$ (see <cit.>), using the semiclassical hyperbolic estimates (see <cit.>), we have for all $s,r,\ell\in \BR$

\begin{equation}\label{eq:semihyperesext}
\norm{(1-\chi) u}_{\bar{H}^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+\norm{\chi u}_{\bar{H}^{s,r,\ell}_{\scop,\bop,h}(\CX)}
\end{equation}

where the sc-decay order $r$ and b-decay order $\ell$ are actually relevant here. Combining (<ref>) with (<ref>) yields for $s>s_0>\frac 12, r>r_0>-\frac 12, \ell<-\frac12$

\begin{equation}
\norm{u}_{\bar{H}^{s,r,\ell}_{\scop,\bop,h}(\CX)}\leq C h^{-2}\norm{P_h(z)u}_{\bar{H}^{s-1, r+1,\ell+1}_{\scop,\bop,h}(\CX)}+Ch^{s-s_0}\norm{u}_{\bar{H}^{s_0,r_0,\ell}_{\scop,\bop,h}(\CX)}.
\end{equation}

Taking $h$ small enough, the term $Ch^{s-s_0}\norm{u}_{\bar{H}^{s_0,r_0,\ell}_{h,\scop,\bop}(\CX)}$ can be absorbed into the left-hand side and this proves the estimate (<ref>).

* Proof of the estimate (<ref>). For $s>\frac 12\geq\frac{1}{2}-\IM(h^{-1}z)\frac{\ehKN^2+a^2}{\ehKN^2-\Bm}$, we have $1-s\leq\frac{1}{2}-\IM(h^{-1}\overline{z})\frac{\ehKN^2+a^2}{\ehKN^2-\Bm}$. Then using the semiclassical low regularity radial point estimates (see <cit.>, <cit.>), there exist $B'_\pm, G'_\pm\in\Psi^0_h(\CX)$ microlocally supported near $L_\pm$ with $L_\pm\subset \mathrm{ell}_h(B'_\pm)$ and $E'_\pm\in\Psi^0_h(\CX), \mathrm{WF}_{h}(E'_\pm)\cap L_\pm=\emptyset$, and satisfying that all the backward (forward) null characteristics from $\mathrm{WF}_h(B'_\pm)\setminus L_\pm$ reach $\mathrm{ell}_h(E'_\pm)$, while remaining in the elliptic set of $G'_\pm$, such that for any $N\in\BR$

\begin{equation}\label{eq:highregradialesdual}
\begin{split}
&\norm{B'_\pm u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\leq C h^{-1}\norm{G'P_h(z)^*u}_{H^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{E'_\pm u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^N\norm{u}_{\dot{H}^{-N,-N,-N}_{\scop,\bop,h}(\CX)}
\end{split}
\end{equation}

where the sc-decay order $r$ and b-decay order $\ell$ are actually irrelevant here.

For $r>-\frac 12$, we have $-r-1<-\frac 12$. Then using the semiclassical low sc-decay radial point estimates at the non-zero scattering section $R(z)$ (see [111] and the above discussion about the proof of high sc-decay radial point estimates at $R(z)$. We note that the only difference is that now the term involving $\chi_0'$ does not have the correct sign and needs to be treated as an error term $E_z'u$), there exist $B'_z, G'_z \in\Psi^{0,0}_{\scop,h}(\CX)$ microlocally supported near $R(z)$ with $R(z)\subset \mathrm{ell}_{\scop,h}(B_z)$ and $E'_z \in\Psi^{0,0}_{\scop,h}(\CX), \mathrm{WF}_{\scop,h}(E'_z)\cap R(z)=\emptyset$, and satisfying that all the backward null characteristics from $\mathrm{WF}_{\scop,h}(B'_z)\setminus R(z)$ reach $\mathrm{ell}_{\scop,h}(E'_z)$, while remaining in the scattering elliptic set of $G'_z$, such that

\begin{equation}\label{eq:lowscdecayradialesdual}
\begin{split}
&\norm{B'_z u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\leq C h^{-1}\norm{G'_zP_h(z)^*u}_{H^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{E'_z u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^N\norm{u}_{\dot{H}^{-N,-N,-N}_{\scop,\bop,h}(\CX)}
\end{split}
\end{equation}

where the differential order $s$ and b-decay order $\ell$ are actually irrelevant here.

For $\ell<-\frac 12$, we have $-\ell-1>-\frac 12$. Then using the semiclassical high b-decay radial point estimates at the zero scattering section $R(0)$ (see <cit.>), there exist $B'_0, G'_0\in\Psi^{0,0,0}_{\scop,\bop,h}(\CX)$ microlocally supported near $R(0)$ (in fact near the blown up $R(0)$ in the second microlocalized space introduced in <ref>) with $R(0)\subset \mathrm{ell}_{\scop,\bop,h}(B'_0)$, and satisfying all the backward null characteristics from $\mathrm{WF}_{\scop,\bop,h}(B'_0)$ tend to $R(0)$, while remaining in the second microlocalized scattering elliptic set of $G'_0$, such that for any $N\in\BR$

\begin{equation}\label{eq:highbdecayradialesdual}
\norm{B'_0 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{G'_0P_h(z)^*u}_{H^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{-N,-N,-\ell-1}_{\scop,\bop,h}(\CX)}
\end{equation}

where the differential order $s$ is actually irrelevant here.

We again calculate

\[
\sigma_h(h^{-1}\IM P_{h}(z)^*)=\sigma_h(\frac{P_h(\bar{z})-P_h(z)}{2ih})<0<\frac{\nu_{\min}}{2}\quad \mbox{at}\quad K.
\]

Then using the hyperbolic trapping estimates (see [39] and <cit.>), there exist $B'_1, G'_1,\in\Psi^{0}_{h}(\CX)$ microlocally supported near $K$ with $K\subset \mathrm{ell}_{h}(B'_1)$ and $B'_2\in\Psi_{h}^{0}(\CX), \mathrm{WF}_{h}(B'_2)\cap \Gamma^+=\emptyset$, and satisfying that $
\mathrm{WF}_h(B'_1)\cup\mathrm{WF}_h(B'_2)$ is contained in the elliptic set of $G'_1$, one has for any $N, s',r',\ell'\in\BR$

\begin{equation}\label{eq:normallytrapes2}
\begin{split}
&\norm{B'_1 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\leq C h^{-2}\norm{G'_1P_h(z)^*u}_{H^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{B'_2 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{s',r',\ell'}_{\scop,\bop,h}(\CX)}
\end{split}
\end{equation}

where the differential order $s'$, the sc-decay order $r'$ and b-decay order $\ell'$ are actually irrelevant here.

By the same reasoning as in the proof of the estimate (<ref>), we have

\begin{equation}
\begin{split}
&\norm{\chi u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\leq C h^{-1}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{B_+ u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\quad+C\norm{B'_- u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}
+C\norm{B'_1 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\quad+C\norm{B'_0 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+C\norm{B'_z u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{-N,-N,-\ell-1}_{\scop,\bop,h}(\CX)},
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&\norm{E'_\pm u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+\norm{E'_z u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad\leq C h^{-1}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{(1-\chi) u}_{\dot{H}^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\&\quad\quad+C\norm{B'_1 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+C\norm{B'_0 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{-N,-N,-\ell-1}_{\scop,\bop,h}(\CX)}
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\norm{B'_2 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}&\leq C h^{-1}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{(1-\chi) u}_{\dot{H}^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
&\quad+C\norm{B'_0 u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{-N,-N,-\ell-1}_{\scop,\bop,h}(\CX)}.
\end{split}
\end{equation}

Putting all the above estimates together yields for $s>\frac 12, r>-\frac 12, \ell<-\frac 12$

\begin{equation}\label{eq:semiglobales2}
\begin{split}
\norm{\chi u}_{H^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}&\leq C h^{-2}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+C\norm{(1-\chi) u}_{\dot{H}^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\\
\end{split}
\end{equation}

Again using the semiclassical hyperbolic estimates (see <cit.>), we have for all $s,r,\ell\in \BR$

\begin{equation}\label{eq:semihyperessupp}
\norm{(1-\chi) u}_{\dot{H}^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\leq C h^{-1}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop}(\CX)}
\end{equation}

where the sc-decay order $r$ and b-decay order $\ell$ are actually relevant here. Combining (<ref>) with (<ref>) yields for $s>\frac 12, r>-\frac 12, \ell<-\frac12$

\begin{equation}
\norm{u}_{\dot{H}^{1-s,-r-1,-\ell-1}_{\scop,\bop,h}(\CX)}\leq C h^{-2}\norm{P_h(z)^*u}_{\dot{H}^{-s, -r,-\ell}_{\scop,\bop,h}(\CX)}+Ch^{N}\norm{u}_{\dot{H}^{-N,-N,-\ell-1}_{\scop,\bop,h}(\CX)}.
\end{equation}

Taking $h$ small enough, the term $Ch^{N}\norm{u}_{\dot{H}^{-N,-N,\ell}_{h,\scop,\bop}(\CX)}$ can be absorbed into the left-hand side and this proves the estimate (<ref>).

* Invertibility of $\widehat{\Box_{g_{b}}}(\sigma)$ for $0\leq\IM\sigma\leq C_1, \RE\sigma\geq C_2$. We only discuss the proof of the invertibility of the operator $\widehat{\Box_{g_{b}}}:\{u\in\bar{H}_{\bop,h}^{s,\ell}(\CX)\mid\bar{H}_{\bop,h}^{s,\ell+1}(\CX)\}\to\bar{H}_{\bop,h}^{s,\ell+1}(\CX)$ in detail as the operator $
\widehat{\Box_{g_{b}}}:\{u\in\bar{H}_{\bop,h}^{s,\ell}(\CX)\mid\bar{H}_{\bop,h}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop,h}^{s-1,\ell+2}(\CX)$ can be handled in a completely analogous manner. We define the space
\[
\mathcal{X}(\sigma):=\{u\in\bar{H}_\bop^{s,\ell}(\CX)\mid \widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_\bop^{s,\ell+1}(\CX)\},
\]

which is a Hilbert space endowed with the norm $\norm{u}_{\mathcal{X}(\sigma)}:=\norm{u}_{\bar{H}_\bop^{s,\ell}}+\norm{\widehat{\Box_{g_{b}}}(\sigma)u}_{\bar{H}_\bop^{s,\ell+1}}$. First, the injectivity of $\widehat{\Box_{g_{b}}}(\sigma):\mathcal{X}(\sigma)\to\bar{H}^{s,\ell}_\bop$ immediately follows from (<ref>). As for the surjectivity, we need to show that for any $f\in\bar{H}_\bop^{s,\ell+1}$, there exists $u\in\bar{H}^{s,\ell}_\bop$ such that $\widehat{\Box_{g_{b}}}(\sigma)u=f$. We define

\[
\mathcal{Y}(\sigma):=\{v\in\dot{H}_\bop^{-s,-\ell-1}(\CX)\mid \widehat{\Box_{g_{b}}}(\sigma)^*v\in\dot{H}_\bop^{-s,-\ell}(\CX)\}.
\]

According to (<ref>), we have

\[
\abs{\angles{f}{v}}\leq C\norm{f}_{\bar{H}_\bop^{s,\ell+1}}\norm{\widehat{\Box_{g_{b}}}(\sigma)^*v}_{\dot{H}_\bop^{-s,-\ell}},\quad v\in\mathcal{Y}(\sigma).
\]

By Hahn-Banach Theorem, the anti-linear form $\widehat{\Box_{g_{b}}}(\sigma)^*v\to \angles{f}{v}$ with $v\in\mathcal{Y}(\sigma)$ can be extended to a continuous anti-linear form on $\dot{H}^{-s,-\ell}_\bop$. Since $(\dot{H}^{-s,-\ell}_\bop)^*=\bar{H}^{s,\ell}_\bop$, there exists $u\in\bar{H}^{s,\ell}_\bop$ such that $\angles{u}{\widehat{\Box_{g_{b}}}(\sigma)^*v}=\angles{f}{v}$ for $v\in\mathcal{Y}(\sigma)$. In particular, for $v\in C_c^\infty(X^\circ)$

\[
\angles{\widehat{\Box_{g_{b}}}(\sigma)u}{v}=\angles{u}{\widehat{\Box_{g_{b}}}(\sigma)^*v}=\angles{f}{v}.
\]

This proves that $\widehat{\Box_{g_{b}}}(\sigma)u=f$.

Now we further determine the index of the Fredholm operators $\widehat{\Box_{g_{b}}}(\sigma)$ on suitable function spaces for $\sigma\in\BC, \IM\sigma\geq 0$.

Let $s>\frac12, \ell<-\frac12, s+\ell>-\frac12$. Then the following holds.

* For $\sigma\in\BC, \IM\sigma\geq 0, \sigma\neq 0$, the operators

\begin{align}\label{eq:Fredholmindex1}
&\widehat{\Box_{g_{b}}}(\sigma):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_{\bop}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop}^{s-1,\ell+2}(\CX)\\\label{eq:Fredholmindex2}
&\widehat{\Box_{g_{b}}}(\sigma):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_{\bop}^{s,\ell+1}(\CX)\}\to\bar{H}_{\bop}^{s,\ell+1}(\CX)
\end{align}

are Fredholm of index $0$.

* Let $-\frac32<\ell<-\frac12$. Then

\begin{equation}\label{eq:Fredholmindex0}
\widehat{\Box_{g_{b}}}(0):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(0)u\in\bar{H}_{\bop}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop}^{s-1,\ell+2}(\CX)
\end{equation}

is Fredholm of index $0$.

We now prove that (<ref>) and (<ref>) have index $0$, and the index $0$ property of (<ref>) follows in a similar manner.

* Proof of the index $0$ property of (<ref>). In view of the high energy estimates in Theorem <ref>,
\[\widehat{\Box_{g_b}}(\sigma):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_{\bop}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop}^{s-1,\ell+2}(\CX)
\]

is invertible and thus has index $0$ for fixed $\IM\sigma=C\geq 0$ when $\abs{\RE\sigma}\gg1$. We claim that the index of $\widehat{\Box_{g_b}}(\sigma)$ with $\IM \sigma\geq 0,\sigma\neq 0$ is a constant on the horizontal line $\IM\sigma=C$ and thus is $0$ for all $\sigma$ with $\IM\sigma\geq 0,\sigma\neq0$. Let

\[
\mathcal{X}(\sigma):=
\{u\in\eHb^{s,\ell}(\CX):\widehat{\Box_{g_b}}(\sigma)u\in \eHb^{s-1,\ell+2}(\CX)\}.
\]

For any fixed $\sigma_0\neq 0$ with $\IM\sigma_0=C\geq0$, without loss of generality, we may assume $\RE \sigma_0\geq 0$. First, there exists a $C'$ such that $\widehat{\Box_{g_b}}(\sigma)$ is invertible and thus has index $0$ for $\sigma\in[C'+iC, \infty+iC)$. We next prove that for each $\sigma'\in[\sigma_0=\RE\sigma_0+iC, C'+iC]$, there exists $\epsilon>0$ such that the index of $\widehat{\Box_{g_b}}(\sigma):\mathcal{X}(\sigma)\to\bar{H}^{s-1,\ell+2}_\bop(\CX)$ is a constant for $\sigma$ with $\IM\sigma\geq 0,\sigma\neq 0,\abs{\sigma-\sigma'}<\epsilon$. Then by compactness of $[\sigma_0=\RE\sigma_0+iC, C'+iC]$ we conclude that the index of $\widehat{\Box_{g_b}}(\sigma_0)$ is the same as that of $\widehat{\Box_{g_b}}(C'+iC)$ and thus is $0$. Suppose $\widehat{\Box_{g_b}}(\sigma')$ has kernel and cokernel of dimension $k_+$ and $k_-$, respectively, then we establish a Grushin problem <cit.>. More specifically, we define the following operator

\[
\widehat{\Box_{g_b}}(\sigma)&L_-\\
\end{pmatrix}: \mathcal{X}(\sigma)\oplus\BC^{k_-}\to \eHb^{s-1,\ell+2}(\CX)\oplus \BC^{k_+}
\]

where $L_+, L_-$ are defined as follows. If $\ker\widehat{\Box_{g_b}}(\sigma')$ is spanned by $x_j\in \mathcal{X}(\sigma'), j=1,\cdots, k_+$, then by Hahn-Banach Theorem  we can find $x_j^*\in \sHb^{-s,-\ell}(\CX)$ with $\norm{x_j^*}\leq 1$ such that $\angles{x_i}{x_j^*}=\delta_{ij}$. It follows that

\[
L_+: \eHb^{s,\ell}(\CX)\to C^{k_+},\quad L_+(f):=(\angles{f}{x_1^*},\cdots,\angles{f}{x_{k_+}^*})
\]

restricts to an isomorphism $\ker\widehat{\Box_{g_b}}(\sigma')\to \BC^{k_+}$. As for the construction of $L_-$, we choose $y_j\in \eHb^{s-1,\ell+2}(\CX)$ with $\norm{y_j}\leq 1$, $ j=1, \cdots, k_-$ so that $y_j+\Ran\widehat{\Box_{g_b}}(\sigma')$ form a basis of $\eHb^{s-1,\ell+2}/\Ran\widehat{\Box_{g_b}}(\sigma')$. Then we define

\[
L_-:\BC^{k_-}\to \eHb^{s-1,\ell+2}(\CX),\quad L_-((a_1, \cdots,a_{k_-})):=\sum_{j=1}^{k_-}a_jy_j.
\]

The operator $L_-$ has maximal rank and $\Ran L_-\cap \Ran\widehat{\Box_{g_b}}(\sigma')=\emptyset$. The above construction of $L_+,L_-$ implies $L(\sigma')$ has a trivial kernel and onto, and thus is invertible. Since $\widehat{\Box_{g_b}}(\sigma)$ satisfy the uniform Fredholm estimates (<ref>) for all $\sigma\in\BC, \IM\sigma\in[0,2C]$ satisfying $\abs{\sigma_0}/2\leq \abs{\sigma}\leq 2(C+C') $, so do $L(\sigma)$. That is, there exists a constant $\tilde{C}$ (independent of $b$ and $\sigma$) such that

\[
\norm{(u, u^-)}_{\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}}\leq \tilde{C}\Bigl(\norm{L(\sigma)(u, u^-)}_{\eHb^{s-1,\ell+2}(\CX)\oplus \BC^{k_+}}+\norm{(u,u^-)}_{\eHb^{s_0,\ell_0}(\CX)\oplus \BC^{k_-}}\Bigr)
\]

for $\sigma\in\BC, \IM\sigma\in[0,2C]$ satisfying $\abs{\sigma_0}/2\leq \abs{\sigma}\leq 2(C+C')$, and $s_0<s, \ell_0<\ell$. Then following the perturbation arguments in <cit.>, we prove the invertibility of $L(\sigma)$ when $\sigma$ is close to $\sigma'$. Concretely, suppose there exists a sequence $\delta_j\to\delta'$ such that $L(\delta_j)$ is not invertible, so either $\ker L(\delta_j) $ on $\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}$ or $\ker L(\delta_j)^* $ on $\sHb^{-s+1,-\ell-2}(\CX)\oplus \BC^{k_+}$ is non-trivial. By passing to a subsequence, we may assume that the former possibility holds for all $j$, as the case of the adjoint is completely analogous. Now, if $\norm{(u_j, u^-_j)}_{\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}}=1$ and $L(\delta_j)(u_j, u^-_j)=0$, then the above uniform Fredholm estimate gives $\tilde{C}^{-1}\leq \norm{(u_j,u_j^-)}_{\eHb^{s_0,\ell_0}(\CX)\oplus \BC^{k_-}}$. Using the sequential compactness of the unit ball in $\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}$ in the weak topology, and the compactness of the inclusion $\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}\to\eHb^{s_0,\ell_0}(\CX)\oplus \BC^{k_-}, s_0<s,\ell_0<\ell$,  it follows that $(u_j, u^-_j)$ has a weakly convergent subsequence in $\eHb^{s,\ell}(\CX)\oplus \BC^{k_-}$ to some $(u_0, u^+_0) \in \eHb^{s,\ell}(\CX)\oplus \BC^{k_-}$ which is  norm-convergent in $\eHb^{s_0,\ell_0}(\CX)\oplus \BC^{k_-}$. We also have

\[
\qquad \qquad L(\delta')(u_0, u^+_0)=L(\delta')\left((u_0, u^+_0)-(u_j, u^-_j)\right)+(L(\delta')-L(\delta_j))(u_j, u^-_j)\to 0\quad\mbox{in}\quad \eHb^{s_0-2, \ell_0}(\CX)\oplus\BC^{k_-}
\]

since $L(\delta_j)\to L(\delta')$ as bounded operators in $\mathcal{L}(\eHb^{s_0, \ell_0}(\CX)\oplus\BC^{k_-}, \eHb^{s_0-2, \ell_0}(\CX)\oplus\BC^{k_-})$ and $(u_j, u^-_j)\to(u_0, u^-_0)$ in $\eHb^{s_0, \ell_0}(\CX)\oplus\BC^{k_-}$. Therefore, $L(\delta')(u_0, u^-_0)=0$. Since $\tilde{C}^{-1}\leq \norm{(u,u^-)}_{\eHb^{s_0,\ell_0}(\CX)\oplus \BC^{k_-}}$, $(u_0, u^-_0)\neq 0$ and thus $\ker L(\sigma')$ is non-trivial, which leads to a contradiction. This proves that $L(\sigma)$ is invertible for $\sigma$ close to $\sigma'$, which implies $\widehat{\Box_{g_b}}(\sigma)$ has index $k_+-k_-$ (see <cit.>) for for $\sigma$ close to $\sigma'$.

* Proof of $0$ index property of (<ref>). Suppose that $-\frac32<\ell<-\frac 12$. Then using the uniform Fredholm estimates (<ref>) (up to $\sigma=0$) and the argument as above yields the index $0$ property of (<ref>).

§.§.§ High energy estimates for tensor wave operators

Now we prove an analogy to Theorem <ref> for the semicalssical operators  $h^2\widehat{\mathcal{P}_{b,\gamma}}(h^{-1}z),\  h^2\widehat{\mathcal{W}_{b,\gamma}}(h^{-1}z)$ and $h^2\widehat{L_{b, \gamma}}(h^{-1}z)$ acting on bundles.

Let $b_0=(\Bm_0, \Ba_0, \BQ_0)$ with $\abs{\BQ_0}+\abs{\Ba_0}<\Bm_0$ and $\abs{\Ba_0}\ll\abs{\Bm_0}+\abs{\BQ_0}$. Then there exists $\epsilon>0$ such that for $b=(\Bm, \Ba,\BQ)$ with $\abs{b-b_0}<\epsilon$ and for $s>s_0>2$ (resp. $s>s_0>3$), $\ell<-\frac{1}{2}$ with $s+\ell>-\frac12$, the conclusions in Theorem <ref> hold for $h^2\widehat{\mathcal{P}_{b,\gamma}}(h^{-1}z)$ and $h^2\widehat{\mathcal{W}_{b,\gamma}}(h^{-1}z)$ (resp. $h^2\widehat{L_{b,\gamma}}(h^{-1}z)$).

The proof is analogous to that of Theorem <ref>, except for the computation of threshold regularity in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at spatial infinity $\pa_+\CX$, and the verification of

\begin{equation}
\label{eq:verificationatK}
\sigma_{1,h}\Big(\frac{1}{2ih}\big(h^2\widehat{\bullet}(h^{-1}z)-(h^2\widehat{\bullet}(h^{-1}z))^*\big)\Big)<\frac{\nu_{\min}}{2},\quad \bullet=\mathcal{P}_{b,\gamma},\ \mathcal{W}_{b,\gamma},\ L_{b,\gamma}
\end{equation}

at trapping $K$.

The computation of threshold regularity in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at spatial infinity $\pa_+\CX$ is the same as that in the fixed $\sigma$ case in Theorem <ref>. The verification of (<ref>) at trapping $K$ follows from the discussion in appendix <ref> (see Theorems <ref>, <ref> and Remark <ref>) and the relevant calculation in the proof of Theorem <ref>.

Correspondingly, we have the following analogy to Lemma <ref> for $\widehat{\mathcal{P}_{b,\gamma}},\ \widehat{\mathcal{W}_{b,\gamma}}$ and $\widehat{L_{b,\gamma}}$.

Let $b=(\Bm, \Ba, \BQ)$ with $\abs{\BQ}+\abs{\Ba}<\Bm$ and $\abs{\Ba}\ll\abs{\Bm}+\abs{\BQ}$. Let $s>2$(resp. $s>3$), $\ell<-\frac12$ and $s+\ell>-\frac12$. Then the conclusions in Lemma <ref> also hold for $\widehat{\mathcal{P}_{b,\gamma}}(\sigma)$ and $\widehat{\mathcal{W}_{b,\gamma}}(\sigma)$ (resp. $\widehat{L_{b,\gamma}}(\sigma)$).

The proof is completely analogous to that of Lemma <ref>.

§.§ Energy estimates

In this section, we shall establish the energy estimates for the solutions to various wave type equations on slowly rotating Kerr-Newman metrics.

First, we introduce another modification of the Boyer-Lindquist coordinates, which is defined by

\[
\bar{t}=t+F_t(r),\quad \bar{\varphi}=\varphi+F_\varphi(r).
\]

In the new coordinates $(\bar{t}, r, \theta, \bar{\varphi})$, the Kerr-Newman metric takes the following form

\begin{equation}\label{eq:extendedmetric}
\begin{split}
g_b&=-\frac{\Delta_b}{\rho_b^2}\Big( d\bar{t}-a\sin^2\theta d\bar{\varphi}+\big(a\sin^2\theta F_\varphi'(r)-F_t'(r)\big)dr\Big)^2\\
&\quad+\frac{\sin^2\theta}{\rho_b^2}\Big(a d\bar{t}-(r^2+a^2)d\bar{\varphi}+\big((r^2+a^2)F_\varphi'(r)-a F_t'(r)\big)dr\Big)^2+\rho_b^2(\frac{dr^2}{\Delta_b}+d\theta^2)
\end{split}
\end{equation}

and its inverse is

\begin{equation}\label{eq:extendedinversemetric}
\begin{split}
\end{split}
\end{equation}

In order for $g_b, g_b^{-1}$ to be smooth at event horizon, $F_t(r)$ and $F_\varphi(r)$ are required to satisfy that $F'_t(r)=\pm(r^2+a^2)/\Delta_b, F'_\varphi(r)=\pm a/\Delta_b$ (up to a smooth modification) at event horizon. Now we shall prove

There exist $F_t(r), F_\varphi(r)$ such that they satisfy the following conditions

(i) $F'_\varphi(r)=\frac{a}{\Delta_b}+\tilde{F}_\varphi(r)$ where $\tilde{F}_\varphi(r)$ is smooth on $[r_-, \infty)_r$, and $F_\varphi(r)=0$ near $r=\infty$.

(ii) $F'_t(r)=\frac{r^2+a^2}{\Delta_b}+\tilde{F}_t(r)$ where $\tilde{F}_t(r)$ is smooth on $[r_-,\infty)$, and $F_t(r)=0$ near $r=\infty$.

(iii)The hypersurfaces $\bar{t}=const$ are spacelike, i.e., $d\bar{t}$ is timelike. More precisely, $g_b^{-1}(d\bar{t}, d\bar{t})\leq-1$.

With this choice of $F_t(r), F_\varphi(r)$, $g_b$ in the coordinate $(\bar{t}, r, \theta, \bar{\varphi})$ are smooth and non degenerate on the extended manifold $\BR_{\bar{t}}\times [r_-, \infty)_r\times\BS^2$.

Let $\chi(r)\in C_c^\infty(\BR)$ such that $\chi(r)=1$ when $r\leq 3\Bm$ and $\chi=0$ when $r\geq 4\Bm$. We first define $F_\varphi(r)=\chi \int a/\Delta_b\,dr$ which satisfies (i).
Next, we expect to construct $F_t(r)$ such that

\[
g_b^{-1}(d\bar{t}, d\bar{t})=-\frac{1}{\rho_b^2}\Big(\frac{(r^2+\Ba^2)^2}{\Delta_b}-\Delta_bF_t'(r)^2-\Ba^2\sin^2\theta\Big)=\begin{cases}
-1,\quad &r\leq 3\Bm\\
\leq -1,\quad &3\Bm\leq r\leq 4\Bm\\
g_b^{-1}(dt, dt),\quad &r\geq 4\Bm
\end{cases}.
\]

Since we also expect

\begin{equation}\label{eq:requireforhyperblodal}
F'_t(r)=\frac{r^2+a^2}{\Delta_b}+\mu_1(r)\quad \mbox{near}\quad r=\ehKN
%;\quad 	F'_t(r)=-\frac{r^2+\Ba^2}{\Delta_b}+\mu_2(r)\quad \mbox{near}\quad r=\infty
\end{equation}

where $\mu_1(r)$ is smooth near event horizon, it follows that we can define $F_t(r)$ as follows

\[
\sqrt{\Big(\frac{r^2+a^2}{\Delta_b}\Big)^2-\frac{r^2+a^2}{\Delta_b}}=\frac{r^2+a^2}{\Delta_b}\sqrt{1-\frac{\Delta_b}{r^2+a^2}},\quad & r\leq 3\Bm\\
0,\quad & r \geq 4\Bm
\end{cases}.
\]

To define a global $F_t(r)$, we let

\[
\]

With this choice of $F_t'(r)$, we have $
g_b^{-1}(d\bar{t},d\bar{t})=-g_b^{-1}(dt,dt)\leq -1$ for $ r\geq 4\Bm$ and $g_b^{-1}(d\bar{t},d\bar{t})\leq -1$ for $ r\leq  4\Bm$, and thus the requirement (iii) is satisfied. By choosing a proper integration constant we have $F_t(r)=0$ for $r\geq 4\Bm$, and thus the requirement (ii) is satisfied

There exist $F_t(r), F_\varphi(r)$ such that they satisfy the following conditions

* $F'_\varphi(r)=\frac{a}{\Delta_b}+\tilde{F}_\varphi(r)$ where $\tilde{F}_\varphi(r)$ is smooth on $[r_-, \infty)_r$, and $F_\varphi(r)=0$ near $r=\infty$.

* $F'_t(r)=\frac{r^2+a^2}{\Delta_b}+\tilde{F}_t(r)$ where $\tilde{F}_t(r)$ is smooth on $[r_-,\infty)$, and $F_t(r)=-r_{b,*}+\mathcal{O}(r^{-1})$ near $r=\infty$ where $\frac{dr_{b,*}}{dr}=\frac{1}{\Delta_b}$.

* The hypersurfaces $\bar{t}=const$ are spacelike, i.e., $d\bar{t}$ is timelike. More precisely,
\[g_b^{-1}(d\bar{t}, d\bar{t})\leq-\Bm^2\rho_b^{-2}<0.
\]

With this choice of $F_t(r), F_\varphi(r)$, $g_b$ in the coordinates $(\bar{t}, r, \theta, \bar{\varphi})$ is smooth and non degenerate on the extended manifold $\BR_{\bar{t}}\times [r_-, \infty)_r\times\BS^2$.

Let $\chi(r)\in C_c^\infty(\BR)$ such that $\chi(r)=1$ when $r\leq 3\Bm$ and $\chi=0$ when $r\geq 4\Bm$. We first define $F_\varphi(r)=\chi \int a/\Delta_b\,dr$ which satisfies statement ($1$). Next, we expect

\begin{equation}\label{eq:requireforhyperblodal}
F'_t(r)=\frac{r^2+a^2}{\Delta_b}+\mu_1(r)\quad \mbox{near}\quad r=\ehKN;\quad 	F'_t(r)=-\frac{r^2+a^2}{\Delta_b}+\mu_2(r)\quad \mbox{near}\quad r=\infty
\end{equation}

where $\mu_1(r)$ is smooth near event horizon and $\mu_2(r)\sim r^{-2}$ near $r=\infty$. We let

\[
g_b^{-1}(d\bar{t}, d\bar{t})=-\frac{1}{\rho_b^2}\Big(\frac{(r^2+a^2)^2}{\Delta_b}-\Delta_bF_t'(r)^2-a^2\sin^2\theta\Big)=-\frac{\Bm^2}{\rho_b^2}<0,
\]

In order to arrange (<ref>), we need

\[
\sqrt{\Big(\frac{r^2+a^2}{\Delta_b}\Big)^2-\frac{\Bm^2+a^2\sin^2\theta}{\Delta_b}}=\frac{r^2+a^2}{\Delta_b}\sqrt{1-\frac{\Delta_b(\Bm^2+a^2\sin^2\theta)}{(r^2+a^2)^2}},\quad & r \mbox{ is near } \ehKN\\
-\sqrt{\Big(\frac{r^2+a^2}{\Delta_b}\Big)^2-\frac{1+a^2\sin^2\theta}{\Delta_b}}=-\frac{r^2+a^2}{\Delta_b}\sqrt{1-\frac{\Delta_b(\Bm^2+a^2\sin^2\theta)}{(r^2+a^2)^2}},\quad & r \mbox{ is near } \infty
\end{cases}.
\]

Therefore, we define

\[
\]

With this choice of $F_t'(r)$, we have $
g_b^{-1}(d\bar{t},d\bar{t})=-\Bm^2\rho_b^{-2}$ for $r\leq 3\Bm, r\geq 4\Bm$ and $
g_b^{-1}(d\bar{t},d\bar{t})\leq -\Bm^2\rho_b^{-2}$ for $3\Bm\leq r\leq  4\Bm$, and thus the requirement statement ($3$) is satisfied. By choosing a proper integration constant we have $\bar{t}=r_{b,*}+\mathcal{O}(r^{-1})$ near $r=\infty$, and thus the requirement statement ($2$) is satisfied

We now establish the energy estimates for the solutions to the equation $\Box_{g_b} u=f$ on the extended Kerr-Newman manifold $M=\BR_{\bar{t}}\times[r_-,\infty)_r\times \BS^2$. We denote by $\Sigma_{\bar{t}_0}$ the spacelike hypersurfaces $\{\bar{t}=\bar{t}_0\}
\cap M$. Here, the integral on $M$ is taken with respect to the volume form $dV=dvol_{g_b}=\rho_b^2\sin\theta d\bar{t}drd\theta d\varphi$.

We begin with the
energy-momentum tensor

\[
T_{\alpha\beta}[g]=\partial_\alpha u \partial_\beta u -
\frac{1}{2}g_{\alpha\beta}\partial^\gamma u \partial_\gamma u
\]

Its contraction with respect to a vector field $X$ is denoted by

\[
\]

and its divergence is

\[
\nabla^\alpha P_\alpha[g,X] = \Box_g u \cdot Xu + \frac{1}{2}T_{\alpha\beta}[g]\pi_X^{\alpha \beta}
\]

and $\pi_X^{\alpha \beta} $ is the deformation tensor of $X$, which is given in terms of the Lie derivative by

\[
\pi_{\alpha\beta}^X=\nabla_\alpha X_\beta + \nabla_\beta X_\alpha=({\mathcal L}_X g)_{\alpha\beta}.
\]

A special role is played by the Killing vector field whose deformation tensor is zero

\[
\]

In general for a vector field $X$ independent of the metric we compute

\begin{equation*}
\pi^{\alpha\beta}_X=-{\mathcal L}_X g^{\alpha\beta}=-X(g^{\alpha\beta})
+g^{\alpha\gamma} \partial_\gamma X^\beta+g^{\beta\gamma}\partial_\gamma X^\alpha.
\end{equation*}

Since $\frac{1}{\sqrt{|g|}}X(\sqrt{|g|})=\frac{1}{2}g^{\alpha\beta}X(g_{\alpha\beta})=-\frac{1}{2}g_{\alpha\beta}X(g^{\alpha\beta})$, we can write

\begin{equation}\label{eq:deformationtensor}
Q[g, X] =-\,\frac{1}{\!\!\!\!\sqrt{|g|}}(X(\sqrt{|g|}g^{\alpha\beta}))
\partial_\alpha u\,\partial_\beta u \\
+(g^{\alpha\gamma}\partial_\gamma X^\beta +g^{\beta\gamma}\partial_\gamma X^\alpha)\partial_\alpha u\partial_\beta u- \partial_\gamma X^\gamma \, g^{\alpha\beta}\partial_\alpha u\,\partial_\beta u.
\end{equation}

Integrating the above divergence relation for a suitable choice of $X$
does not suffice in order to prove the local energy estimates, as in
general the deformation tensor can only be made positive modulo a
Lagrangian term $q(r) \partial^\alpha u \partial_\alpha u $.
Hence some lower order corrections are required.  For a vector field
$X$, a scalar function $q(r)$ and a 1-form $m$ we define

\[
P_\alpha[g,X,q,m] = P_\alpha[g,X] + q u \partial_\alpha u - \frac12
(\partial_\alpha q(r)) u^2 + \frac{1}{2}m_{\alpha}u^2.
\]

The divergence formula gives

\begin{equation}
\nabla^\alpha P_\alpha[g,X,q,m] =  \Box_g u \big(Xu +
q(r) u\big)+ Q[g,X,q,m],
\label{div}\end{equation}

\[
Q[g,X,q,m] =
\frac{1}{2}Q[g, X] + q\,
\partial^\alpha u\, \partial_\alpha u + m_\alpha u\,
\partial^\alpha u +\frac{1}{2} \big(\nabla^\alpha m_\alpha -
\nabla^\alpha \partial_\alpha q(r)\big) \, u^2.
\]

Since $\Sigma_{\bar{t}}$ is “asymptotically null”, we shall define the (degenerate) energy at $\Sigma_{\bar{t}_0}$ as

\[
E[u](\bar{t}_0):=\int_{r_-}^\infty\int_{\BS^2}\frac{1}{r^2}\abs{\pa_{\bar{t}}u(\bar{t}_0,\cdot)}^2+\abs{\pa_ru(\bar{t}_0,\cdot)}^2+\frac{1}{r^2}\Big(\abs{\pa_\theta u(\bar{t}_0, \cdot)}^2+\frac{1}{\sin^2\theta}\abs{\pa_\varphi u(\bar{t}_0, \cdot)}^2\Big) r^2\sin\theta\,dr\,d\theta\,d\varphi.
\]

Now we establish the energy estimate.

Let $b=(\Bm,\Ba, \BQ)$ with $\abs{\Ba}\ll\abs{\BQ}+\Bm$. Let $u$ be the solution to the following initial value problem

\begin{equation}\label{eq:ivp}
\begin{split}
\Box_{g_b}u=f, \quad u|_{\bar{t}=0}=u_0,\quad \pa_{\bar{t}}u|_{\bar{t}=0}=u_1
\end{split}
\end{equation}

where $rf\in L^1([0, \infty)_{\bar{t}}; L^2(\Sigma_{\bar{t}}))$ and
$\frac{u_0}{r}, \frac{u_1}{r}, \pa u_0\in L^2(\Sigma_0; r^2\sin\theta\,dr\,d\theta\,d\varphi)$. Then there exists a constant $C(b)$ which depends on the parameter $b=(\Bm, \Ba,\BQ)$ such that

\begin{equation}\label{eq:energyestimate}
E[u](\bar{t})\leq C(b)\Big(\norm{\frac{u_0}{r}}_{L^2(\Sigma_0)}+\norm{\pa u_0}_{L^2(\Sigma_0)}+\norm{\frac{u_1}{r}}_{L^2(\Sigma_0)}+\norm{rf}^2_{L^1([0, \bar{t}); L^2(\Sigma_{\bar{t}}))}\Big)e^{C(b)\bar{t}}.
\end{equation}

where the integral is taken with respect the volume form $r^2\,dr\,d\theta\,d\varphi$ and $\pa\in\{\pa_r,\frac{1}{r}\pa_\theta, \frac{1}{r\sin\theta}\pa_\varphi\}$.

Moreover, the above statements also hold for the wave type operators $\mathcal{P}_{b,\gamma}, \mathcal{W}_{b,\gamma}$ and $L_{b,\gamma}$.

We only do the analysis for the case $\Ba=0$ here and the same argument will also work for small $\Ba$ since all the calculation below near event horizon depends continuously on $b=(\Bm, \Ba,\BQ)$ (away from the event the horizon, we make use of the timelike property of $d\bar{t}$ and $\pa_{\bar{t}}$). Therefore, we let $b=(\Bm, 0,\BQ)$ and then we have

\begin{align*}
g=g_b&=-\frac{\Delta_b}{r^2} d\bar{t}^2+2\frac{\Delta_b}{r^2}F_t'(r)d\bar{t}dr+\Big(\frac{r^2}{\Delta_b}-\frac{\Delta_b}{r^2}F'_t(r)^2\Big)dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2,\\
\end{align*}

By a density statement, we may assume that $u$ is supported away from spatial infinity.

* Killing multiplier: $X=\pa_{\bar{t}}$.
We compute

\begin{equation}\label{eq:boundaryterm1}
\begin{split}
g^{-1}(d\bar{t}, P[g,\pa_{\bar{t}}])&=\Big(\frac{\Delta_b}{r^2}F_t'(r)^2-\frac{r^2}{\Delta_b}\Big)T(\partial_{\bar{t}}, \partial_{\bar{t}})+\frac{\Delta_b}{r^2}F'_t(r)T(\partial_r, \partial_{\bar{t}})\\
&=-\frac 12\bigg(\Big(\frac{r^2}{\Delta_b}-\frac{\Delta_b}{r^2}F_t'(r)^2\Big)\lvert\partial_{\bar{t}}u\rvert^2+\frac{\Delta_b}{r^2}\lvert\partial_r u\rvert^2+\abs{\pa_\theta u}^2+\frac{1}{\sin^2\theta}\abs{\pa_\varphi}^2\bigg)\\
&\sim  -\Big(\frac{1}{r^2}\lvert\partial_{\bar{t}}u\rvert^2+\lvert\partial_r u\rvert^2+\frac{1}{r^2}\abs{\pa_\theta u}^2+\frac{1}{r^2\sin^2\theta}\abs{\pa_\varphi u}^2\Big)
\quad \mbox{on}
\quad [r_-,\infty),\\
g^{-1}(dr, P[g,\pa_{\bar{t}}])&=\frac{\Delta_b}{r^2}F_t'(r)T(\partial_{\bar{t}}, \partial_{\bar{t}})+\frac{\Delta_b}{r^2}T(\partial_r, \partial_{\bar{t}})\\
&=\frac{\Delta_b(r_-)}{r_-^2}F_t'(r_-)\lvert\partial_{\bar{t}}u\rvert^2+\frac{\Delta_b(r_-)}{r_-^2}\partial_{\bar{t}}u\partial_r u	\quad\mbox{at}\quad r=r_-.
\end{split}
\end{equation}

We see that $g^{-1}(d\bar{t}, P[g,\pa_{\bar{t}}])$ gives control of $\abs{\frac{1}{r}\partial_{\bar{t}}u}$ and angular derivative but not $\partial_ru$ near event horizon because $\Delta_b(r)$ is nonpositive at and beyond event horizon. We also note that the first term of $g^{-1}(dr, P[g,\pa_{\bar{t}}])$ is positive if $r_-$ is sufficiently close to event horizon $r=\ehKN$ while the second term does not have a definite sign. So we need to introduce a suitable multiplier which allows us to deal with the above issues at event horizon.

* Energy estimates near event horizon. Following [87], We  introduce $X=b(r)\pa_r$ which is expressed in terms of the coordinates $(\bar{t}, r,\theta,\varphi)$ with $b(r)=-\chi(r)$ where $\chi(r)=1$ when $r\leq 3\Bm$ and $\chi=0$ when $r\geq 4\Bm$.

First we compute the boundary terms at the spacelike hypersurfaces $\bar{t}=const$

\begin{align*}
g^{-1}(d\bar{t}, P[g,X])&=\Big(\frac{\Delta_b}{r^2}F_t'(r)^2-\frac{r^2}{\Delta_b}\Big)T(\partial_{\bar{t}}, b(r)\partial_r)+\frac{\Delta_b}{r^2}F_t'(r)T(\partial_r, b(r)\partial_r)\\
&=-\chi(r)\bigg(-\Big(\frac{\Delta_b}{r^2}F_t'(r)^2-\frac{r^2}{\Delta_b}\Big)\partial_{\bar{t}}u\partial_r u+\frac{\Delta_b}{r^2}F_t'(r)\lvert\partial_r u\rvert^2\bigg).
\end{align*}

Since $\frac{\Delta_b}{r^2}F_t'(r)\sim 1$ near event horizon $r=\ehKN$, it follows that on $r\in[r_-,\infty)$

\begin{equation}\label{eq:boundaryt}
g^{-1}(d\bar{t}, P[g,C\pa_{\bar{t}}+X])\sim
- E[u](\bar{t})
%\Big(\lvert\partial_{\bar{t}}u\rvert^2+\lvert\partial_r u\rvert^2+\frac{1}{r^2}\abs{\pa_\theta^2 u}^2+\frac{1}{r^2\sin^2\theta}\abs{\pa_\varphi u}^2\Big)\
\end{equation}

provided $C$ is large enough.
At the hypersurface $r=r_-$, we have

\begin{align*}
g^{-1}(dr, P[g,X])&=\frac{\Delta_b}{r^2}F_t'(r)T(\partial_{\bar{t}}, b(r)\partial_r)+\frac{\Delta_b}{r^2}T(\partial_r, b(r)\partial_r)\\
&=-\frac 12\chi(r)\bigg(\Big(\frac{r^2}{\Delta_b}-\frac{\Delta_b}{r^2}F_t'(r)^2\Big)\lvert\partial_{\bar{t}}u\rvert^2+\frac{\Delta_b}{r^2}\lvert\partial_r u\rvert^2-\frac{1}{r^2}\abs{\pa_\theta^2 u}^2-\frac{1}{r^2\sin^2\theta}\abs{\pa_\varphi u}^2\bigg).
\end{align*}

Therefore, combining the calculation of $	g^{-1}(dr, P[g,X])$ with (<ref>) yields

\begin{equation}\label{eq:boundaryr}
\begin{split}
&g^{-1}(dr, C\pa_{\bar{t}}+X)\\
\Big(C\frac{\Delta_b(r_-)}{r_-^2}F_t'(r_-)+\frac12\big(\frac{r^2}{\Delta_b}-\frac{\Delta_b}{r^2}F_t'(r)^2\big)-\epsilon\Big)\lvert\partial_{\bar{t}}u\rvert^2+\epsilon(\partial_{\bar{t}}u+\frac{C\Delta_b(r_-)}{2\epsilon r_-^2}\partial_ru\Big)^2\\
&\quad+\Big(-\frac{\Delta_b(r_-)}{2r_-^2}-\frac{C^2\Delta_b^2(r_-)}{4\epsilon r_-^4}\Big)\lvert\partial_r u\rvert^2+\frac{1}{2r_-^2}\abs{\pa_\theta^2 u}^2+\frac{1}{2r_-^2\sin^2\theta}\abs{\pa_\varphi u}^2\geq 0\quad\mbox{at}\quad r=r_-.
\end{split}
\end{equation}

provided $C$ is sufficiently large, $\epsilon$ is small enough and $r_-$ is sufficiently close to $\ehKN$.

Then we integrate $\nabla^\alpha P_\alpha[g,	C\pa_{\bar{t}}+X]=\Box_gu\cdot (\pa_{\bar{t}}u+u)+\frac 12 T_{\alpha\beta}[g]\pi_X^{\alpha \beta}$ over the region
\[\mathcal{D}=\{0\leq\bar{t}\leq\bar{t}_1, r\geq r_-\}
\]

with respect to the volume form $dV_{g_b}=r^2\sin\theta d\bar{t}drd\theta d\varphi$ and use Divergence Theorem

\[
\int_\mathcal{D}\nabla^\alpha P_\alpha\,dV=\int_{\pa\mathcal{D}}\iota_{P}dV
\]

to obtain

\begin{align*}
&\int_{\mathcal{D}}\Box_gu\cdot (C\pa_{\bar{t}}u+Xu) \,r^2\sin\theta d\bar{t}drd\theta d\varphi+\frac 12\int_{\mathcal{D}}T_{\alpha\beta}[g]\pi_X^{\alpha \beta} \,r^2\sin\theta d\bar{t}drd\theta d\varphi\\
&\quad=\int_{r_-}^\infty\int_{\BS^2} g^{-1}(d\bar{t}, P[g,C\pa_{\bar{t}}+X])\, r^2\sin\theta drd\theta d\varphi|_{\bar{t}=0}^{\bar{t}=\bar{t}_1}\\
&\quad\quad-\int_{0}^{\bar{t}_1}\int_{\BS^2}g^{-1}(dr, P[g,C\pa_{\bar{t}}+X]) r^2\sin\theta d\bar{t}d\theta d\varphi|_{r=r_-}.
\end{align*}

Since $g^{-1}(dr, P[g,C\pa_{\bar{t}}+X])|_{r=r_-}\geq0$, and $T_{\alpha\beta}[g]\pi_X^{\alpha \beta}$ is quadratic in $\partial u$,  we find that

\begin{align*}
\sup_{0\leq\bar{t}\leq\bar{t}_1}E[u](\bar{t})
%&\lesssim E[u](\bar{t}_1)+\int_{\mathcal{D}}\lvert \Box_{g}u\rvert\cdot\lvert C\pa_{\bar{t}}u+Xu\rvert\,dV+\frac 12\int_{\mathcal{D}}\lvert Q[g, X]\rvert\,dV\\
&\leq C'\Big(\norm{\pa u_0}_{L^2(\Sigma_0)}+\norm{\frac{u_1}{r}}_{L^2(\Sigma_0)}+\norm{rf}^2_{L^1([0, \bar{t}); L^2(\Sigma_{\bar{t}}))}\Big)\\
&\quad +C'\int_{0}^{\bar{t}_1}E[u](\bar{t})\,d\bar{t}.
\end{align*}

where $\pa\in\{\pa_r, \frac{1}{r}\pa_\theta, \frac{1}{r\sin\theta}\pa_\varphi\}$ and $L^2(\Sigma_0)=L^2(\Sigma_0;r^2\sin\theta\,dr\,d\theta\,d\varphi)$.

As for the $L^2$ norm of $u$, we have

\[
\norm{\frac{u}{r}}_{L^2(\Sigma_{\bar{t}})}\lesssim\norm{\frac{u_0}{r}}_{L^2(\Sigma_{0})}+\int_0^{\bar{t}_1}\norm{\frac{\pa_{\bar{t}}u}{r}}_{L^2(\Sigma_{\bar{t}})}\,d\bar{t}
\]

where $L^2(\Sigma_{\bar{t}})=L^2(\Sigma_{\bar{t}};r^2\sin\theta\,dr\,d\theta\,d\varphi)$.
Therefore, we have

\begin{align*}
\sup_{0\leq\bar{t}\leq \bar{t}_1} E[u](\bar{t})	&\leq C\Big(\norm{\frac{ u_0}{r}}_{L^2(\Sigma_0)}+\norm{\pa u_0}_{L^2(\Sigma_0)}+\norm{\frac{u_1}{r}}_{L^2(\Sigma_0)}+\norm{rf}^2_{L^1([0, \bar{t}); L^2(\Sigma_{\bar{t}}))}\Big)\\
&\quad +C\int_{0}^{\bar{t}_1}E[u](\bar{t})\,d\bar{t}.
\end{align*}

Finally by Grönwall inequality we obtain the exponential growth of the energy

\[
E[u](\bar{t})\leq C\Big(\norm{\frac{u_0}{r}}_{H^1(\Sigma_0)}+\norm{\pa u_0}_{L^2(\Sigma_0)}+\norm{\frac{u_1}{r}}_{L^2(\Sigma_0)}+\norm{rf}^2_{L^1([0, \bar{t}); L^2(\Sigma_{\bar{t}}))}\Big)e^{C\bar{t}}.
\]

* Proof for $\mathcal{P}_{b,\gamma}, \mathcal{W}_{b,\gamma}$ and $L_{b,\gamma}$. According to Lemma <ref>, $-2\mathcal{P}_{b,\gamma}, -2\mathcal{W}_{b,\gamma}$ (resp. $-L_{b,\gamma}$) are $\Box_{g_b}$ tensored with $4\times 4$ (resp. $14\times 14$) identity matrix plus a a matrix whose entries are first order differential operators with coefficients decaying like $r^{-2}$, and then the above arguments in the proof for $\Box_{g_b}$ still apply here.

We next show that if $\IM\sigma$ is sufficiently large, then $\widehat{\Box_{g_{b}}}(\sigma), \widehat{\mathcal{P}_{b, \gamma}}(\sigma),\widehat{\mathcal{W}_{b,\gamma}}(\sigma), \widehat{L_{b,\gamma}}(\sigma)$ are invertible on suitable function spaces.

Let $b_0=(\Bm_0, 0,\BQ_0)$ with $\abs{\BQ_0}<\Bm_0$. Then there exists $\epsilon>0$ such that for $b=(\Bm, \BQ,\Ba)$ with $\abs{b-b_0}<\epsilon$ and for $s>\frac{1}{2}, \ell<-\frac{1}{2}$ with $s+\ell>-\frac12$, the operators

\begin{align}\label{eq:largeiminvertible1}
&\widehat{\Box_{g_{b}}}(\sigma):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_{\bop}^{s,\ell+1}(\CX)\}\to\bar{H}_{\bop}^{s,\ell+1}(\CX)\\\label{eq:largeiminvertible2}
&\widehat{\Box_{g_{b}}}(\sigma):\{u\in\bar{H}_{\bop}^{s,\ell}(\CX)\mid	\widehat{\Box_{g_{b}}}(\sigma)u\in\bar{H}_{\bop}^{s-1,\ell+2}(\CX)\}\to\bar{H}_{\bop}^{s-1,\ell+2}(\CX)
\end{align}

are invertible for $\IM\sigma\geq C'(b_0)$ where $C'(b_0)>0$ is a sufficiently large constant only depending on $b_0$.

Moreover, the above statements also hold for the wave type operators $\mathcal{P}_{b,\gamma}, \mathcal{W}_{b,\gamma}$ and $L_{b,\gamma}$.

According to Proposition <ref>, there exist $\epsilon>0, \tilde{C}(b_0)>0$ such that for $\abs{b-b_0}<\epsilon$, the solution $u$ to the initial value problem (<ref>) satisfies

\[
E[u](\bar{t})\leq \Big(\tilde{C}(b_0)E[u](0)+\tilde{C}(b_0)\big(\int_{0}^{\bar{t}}\lVert rf\rVert_{L^2(\Sigma_{\bar{t}})}\,d\bar{t}\big)^2\Big)e^{\tilde{C}(b_0)\bar{t}}.
\]

We now show that if $\IM\sigma>\tilde{C}(b_0)$, then (<ref>) and (<ref>) are invertible. According to Lemma <ref>, it suffices to prove that (<ref>) and (<ref>) are injective. Assume the contrary, by Proposition <ref>, there exists a nontrivial solution $w=\rho C^\infty(\pa_+\CX)+\mathcal{A}^{2-}(\CX)$ to $\widehat{\Box_{g_{b}}}(\sigma)w=0$. Then, $u(t_{b,*}, r,\theta, \varphi)=e^{-it_{b,*}\sigma}w$ is a solution to $\Box_{g_b}u=0$. Let $\chi(\bar{t})\in C^\infty(\BR)$ be such that $\mbox{supp}\chi\subset[1,\infty)$ and $\mbox{supp}(1-\chi)\subset(-\infty, 2]$. Then $\chi u$ is the unique solution which is supported in $(1,\infty)$ to the inhomogeneous equation $\Box\tilde{u}=[\Box_{g_b},\chi]u$. According to Lemma <ref> and $\bar{t}-t_{b,*}=\mathcal{O}(\frac{1}{r})$, we have

\[
\Box\tilde{u}=[\Box_{g_b},\chi]u\sim\frac{1}{r^3-}
\]

and thus $\chi(\bar{t})u$ satisfies $E(\chi u)\lesssim e^{\tilde{C}(b_0)\bar{t}}$. But a direct calculation implies $E(\chi u)\sim e^{\IM\sigma t_{b,*}}\sim e^{\IM\sigma\bar{t}}$, which leads to a contradiction.

The proof for $\widehat{\mathcal{P}_{b, \gamma}}(\sigma), \widehat{\mathcal{W}_{b, \gamma}}(\sigma), \widehat{L_{b,\gamma}}(\sigma)$ is completely analogous.

§ SPHERICAL HARMONIC DECOMPOSITIONS

In this section, we shall introduce the spherical harmonic decompositions, which will be used in the proof of the geometric mode stability of Reissner-Nordström spacetime (see <ref>).

Let $\slashed{g}$ be the standard metric on the unit sphere $\BS^2$. We denote the operators on $\BS^2$ using a slash, e.g. $\slashed{\tr}=\tr_{\sg}, \slashed{\delta}=\delta_{\slashed{g}}, \slashed{\delta}^*=\delta_{\slashed{g}}^*$, etc. We denote by $Y_l^m$ with $l\in\BN, m\in\BZ, \abs{m}\leq l$ the spherical harmonic function of degree $l$ and order $m$ which satisfies $\slashed{\Delta}_{H}Y_l^m=l(l+1)Y_l^m$ where $\slashed{\Delta}_{H}=\slashed{d}\slashed{\delta}+\slashed{\delta}\slashed{d}$ is the Hodge Laplacian. We denote by $\BFS_l=\{Y_l^m: \abs{m}\leq l\}$ the space of degree $l$ spherical harmonic functions. Then one can conclude that $\{\BFS_l: l\in\BN\}$ form an orthogonal basis of $L^2(\BS^2)$.

According to Hodge decomposition Theorem and the fact that the de Rham cohomology group $H^1_{dR}(\BS^2)=0$, we see that any $1$-form $\omega\in C^{\infty}(\BS^2, T^*\BS^2)$ can be uniquely decomposed as $\omega=\slashed{d}\phi+\slashed{\delta} \eta$ where $\phi\in C^{\infty}(\BS^2), \eta\in C^\infty(\BS^2; \Lambda^2T^*\BS^2)$. Since $\sdelta=-\slashed{*}\sd \sstar$ where $\sstar$ is the Hodge star operator, we can rewrite $\omega=\sd\phi+\sstar\sd\psi$ where $\phi, \psi\in C^\infty(\BS^2)$. Therefore, one can conclude that an orthogonal basis of $L^2(\BS^2, T^*\BS^2)$ is given by $\{\sd\BFS_l, \BFV_l=\sstar\sd\BFS_l: l\in\BN\}$ where $\BFS_l, \BFV_l$ are the eigen-$1$-forms of the Hodge Laplacian $\Delta_H$ with eigenvalue $l(l+1)$. We note that $\sdelta\BFV_l=0$ and $\sd\BFS_0=\BFV_0=0$. In other words, a spectral decomposition of the negative tensor Laplacian $-\slashed{\Delta}=-\str\slashed{\nabla}_a\slashed{\nabla}_b$ satisfying $-\slashed{\Delta}=\slashed{\Delta}_H-1$ is given by the scalar part

\begin{equation}
\sd\BFS_l\in\ker(-\slashed{\Delta}-\left(l(l+1)-1\right))\quad\mbox{with}\quad l\geq 1
\end{equation}

and the vector part

\begin{equation}
\BFV_l\in\ker(-\slashed{\Delta}-\left(l(l+1)-1\right))\quad\mbox{with}\quad l\geq 1.
\end{equation}

As for the symmetric $2$-tensors $h\in C^\infty(\BS^2; S^2T^*\BS^2)$, we have the transverse-traceless decomposition [120] $h=\frac12(\str h)\sg+\sdelta_0^*\omega+h_{TT}$ where $\sdelta_0^*=\sdelta^*+\frac 12\sg\sdelta$ is the traceless symmetric gradient, $\omega\in C^\infty(\BS^2; T^*\BS^2)$ and $h_{TT}$ is both traceless and divergence free. According to [58], we know that $h_{TT}=0$. We then decompose $\omega$ further into eigen-$1$-forms (we also call them spherical harmonic $1$-forms) of $-\slashed{\Delta}$ of scalar type $\sd\BFS_l$ and vector type $\BFV_l$, and $\str h$ into scalar spherical harmonic functions $\BFS_l$. In the spherical coordinates $(\theta, \varphi)$, since
\[\sd Y_1^0=-\sin\theta d\theta, \quad\sd Y_1^1=\cos\theta \cos\varphi d\theta-\sin\theta\sin\varphi d\varphi, \quad\sd Y_1^{-1}=\cos\theta\sin\varphi d\theta+\sin\theta\cos\varphi d\varphi,
\]

we see that $\sd\BFS_1$ is conformal killing and thus $\sdelta^*_0\sd\BFS_1=0$. We also have
\begin{equation}\label{eq:Vkilling}
\sstar\sd Y_1^0=-\sin^2\theta d\varphi, \quad\sstar\sd Y_1^1=\sin\theta\cos\theta \cos\varphi d\varphi+\sin\varphi d\theta, \quad\sstar\sd Y_1^{-1}=\sin\theta\cos\theta \sin\varphi d\varphi-\cos\varphi d\theta.
\end{equation}

and find that $\BFV_1^{\sharp}$ are rotations, and thus killing, i.e. $\sdelta^*\BFV_1=0$. Then any symmetric $2$-tensors on sphere can be decomposed into the following spherical harmonic symmetric $2$-tensors of the scalar type

\begin{equation}
\BFS_l\sg~(l\geq0),\quad \sdelta^*_0\sd\BFS_l~(l\geq 2)
\end{equation}

and the vector type

\begin{equation}
\sdelta_0^*\BFV_l=	\sdelta^*\BFV_l~(l\geq2).
\end{equation}

We claim that the following geometric operators on sphere $\BS^2$

\[\sg, \str, \sd, \sdelta, \sdelta^*, -\slashed{\Delta} \mbox{ and their compositions}
\]

preserve the the scalar and vector type of the spherical harmonics. More precisely, these geometric operators map the scalar type functions/$1$-forms/symmetric $2$-tensors built out of a particular $\mathsf{S}\in \BFS_l$ into another scalar type tensor with the same $\mathsf{S}$, likewise for the vector type $1$-forms/symmetric $2$-tensors. This is obviously true for $\sg$ acting on functions, $\str$ on symmetric $2$-tensors, $\sd$ on functions, $\sdelta$ on $1$-forms, and $-\slashed{\Delta}$ on functions and $1$-forms. For the verification of $\sdelta, \sdelta^*, -\slashed{\Delta}$, we make use of the identities $\sdelta\sdelta^*\omega=-\frac 12 \slashed{\Delta}\omega+\frac 12\sd\sdelta\omega-\frac 12\omega$ and $-\slashed{\Delta}\sdelta^*\omega=\sdelta^*(-\slashed{\Delta}\omega)-3\sdelta^*\omega-2\sg\sdelta\omega$ to obtain

\begin{equation}
\begin{split}
\sdelta(\mathsf{S}\sg)&=-\sd \mathsf{S},~\sdelta \sdelta_0^*\sd \mathsf{S}=\frac{l(l+1)-2}{2}\sd\mathsf{S},~ \sdelta\sdelta^*\!\!\sstar\sd \mathsf{S}=\frac{l(l+1)-2}{2}\sstar\sd \mathsf{S}; ~\sdelta^*\!\!\sd \mathsf{S}=\sdelta_0^*\sd \mathsf{S}\!-\!\!\frac{l(l+1)}{2}\mathsf{S}\sg;\\
-\slashed{\Delta}(\mathsf{S}\sg)&=l(l+1)\mathsf{S}\sg,\quad-\slashed{\Delta}(\sdelta_0^*\sd \mathsf{S})=\left(l(l+1)-4\right)(\sdelta_0^*\sd \mathsf{S}),\quad -\slashed{\Delta}(\delta^*\sstar\sd \mathsf{S})=\left(l(l+1)-4\right)	(\delta^*\sstar\sd \mathsf{S}).
\end{split}
\end{equation}

We now decompose the spacetime $M$ into the aspherical $\hX=\BR_{t_*}\times [r_-, \infty)_r$ and spherical part $\BS^2$, i.e.

\begin{equation}\label{eq:as_sdecom}
M=\hX\times \BS^2, \quad \hpi: M\to\hX,\quad \slashed{\pi}: M\to \BS^2.
\end{equation}

We then call the functions aspherical if they are only functions of $(t_*, r)$, and spherical if they are only functions on $\BS^2$. Alternatively, we can identify the aspherical and spherical functions as subspaces of $C^\infty(M)$ as follows:

\begin{equation}
C^\infty(\hX)\cong\hpi^*C^\infty(\hX)\subset C^\infty(M),\quad C^\infty(\BS^2)\cong\slashed{\pi}^*C^\infty(\BS^2)\subset C^\infty(M).
\end{equation}

Then functions on $M$, say $L^2(M)$, can be decomposed into an infinite sum of products of aspherical functions and spherical harmonic functions.

We can also split the cotangent bundle $T^*M$ into aspherical and spherical part as follows:

\begin{equation}\label{eq:splitof1}
T^*M=T^*_{AS}\oplus T^*_{S}\quad \mbox{where}\quad T^*_{AS}=\hpi^* T^*\hX\quad\mbox{and}\quad T^*_{S}=\slashed{\pi}^*T^*\BS^2.
\end{equation}

We can further write $\omega\in T^*_S$ as an infinite sum of the spherical harmonic $1$-forms $\sd\BFS_l, \BFV_l$ of $-\slashed{\Delta}$. Therefore, a $1$-form $\omega\in C^\infty(M; T^*M)$ can be expressed as an infinite sum of products of aspherical functions/$1$-forms on $\hX$ and spherical harmonic functions/$1$-forms.

The above splitting of $T^*M$ induces a natural splitting of the symmetric $2$-tensors,into aspherical part $S^2T^*_{AS}$, mixed part $T^*_{AS}\otimes T^*_S$ and spherical part $S^2T^*_S$ as follows:

\begin{equation}\label{eq:splitofsym2}
S^2T^*M=S^2T^*_{AS}\oplus(T^*_{AS}\otimes T^*_S)\oplus S^2T^*_S
\end{equation}

where we identify $a\otimes b\in T^*_{AS}\otimes T^*_S$ as $2a\otimes_s b=a\otimes b+b\otimes a$.
Therefore, a symmetric $2$-tensor $h\in C^\infty(M; S^2T^*M)$ can be expressed as an infinite sum of products of aspherical functions/$1$-forms/symmetric $2$-tensors on $\hX$ and spherical harmonic functions/$1$-forms/symmetric $2$-tensors.

The above decomposition for functions/$1$-form/symmetric $2$-tensors on $M$ implies that the perturbation $(\dot{g}, \dot{A})$ can be divided into two categories: the first consists of scalar perturbations (also called even parity modes in [102] and closed portions in [71]):

\begin{equation}\label{eq:scalarper}
\begin{split}
&	\mbox{scalar}~l\geq 2:\quad\begin{cases}
\dot{g}=\widetilde{f}\sfS+(f\otimes_s\sd\sfS)+(H_L\sfS\sg+H_T\sdelta_0^*\sd\sfS)\\
\dot{A}=\widetilde{K}\sfS+K\sd\sfS
\end{cases}\quad\mbox{with}\quad \sfS\in\BFS_l\\
&	\mbox{scalar}~l=1:\quad\begin{cases}
\dot{g}=\widetilde{f}\sfS+(f\otimes_s\sd\sfS)+H_L\sfS\sg\\
\dot{A}=\widetilde{K}\sfS+K\sd\sfS
\end{cases}\quad\mbox{with}\quad \sfS\in\BFS_1\\
&	\mbox{scalar}~l=0:\quad\begin{cases}
\dot{g}=\widetilde{f}+H_L\sg\\
\dot{A}=\widetilde{K}
\end{cases}
\end{split}
\end{equation}

\[
H_L, H_T, K\in C^\infty(\hX), \quad f, \widetilde{K}\in C^\infty(\hX; T^*\hX), \quad\widetilde{f}\in C^\infty(\hX; S^2T^*\hX).
\]

The second comprises of the vector perturbations (also called odd parity modes in [102] and co-closed portions in [71]):

\begin{equation}\label{eq:vectorper}
\begin{split}
&	\mbox{vector}~l\geq 2:\quad\begin{cases}
\dot{g}=f\otimes_s\sfV+H_T\sdelta^*\sfV\\
\dot{A}=K\sfV
\end{cases}\quad\mbox{with}\quad \sfV\in\BFV_l\\
&	\mbox{vector}~l=1:\quad\begin{cases}
\dot{g}=f\otimes_s\sfV\\
\dot{A}=K\sfV
\end{cases}\quad\mbox{with}\quad \sfV\in\BFV_1
\end{split}
\end{equation}

where $
H_T,K\in C^\infty(\hX)$ and $f\in C^\infty(\hX; T^*\hX)$.

We decompose the Reissner-Nordström metric $g$ into aspherical and spherical part as in (<ref>), which takes the form in static coordinates $(t,r, \omega)$ with $\omega\in\BS^2$ as follows

\begin{equation}
\end{equation}

where $\hg$ is a Lorentzian metric on $\hX=\BR_{t_*}\times [r_-, \infty)$ and $\sg$ denote the standard metric on the unit sphere $\BS^2$. In static coordinates $(t,r)$, $\hg$ takes the form

\begin{equation}
\hg=-\weight dt^2+\weight^{-1}dr^2\quad\mbox{with}\quad\weight=1-\frac{2\Bm}{r}+\frac{\BQ^2}{r^2},
\end{equation}

and it can be extended beyond the event horizon by introducing $(t_*, r)$ coordinates. We denote the operators associated with $\hg$ and $\sg$ by rings and slashes respectively. Here we use the Einstein summation convention with Greek indices $\mu, \nu,\dots$ for coordinates on $M$, lower case Roman indices $i,j,k,m,n$ for coordinates on $\hX$ and $a,b,c,d,e$ for coordinates on $\BS^2$. Indices are raised and lowered using the Reissner-Nordström metric $g$, except that we write $\sg^{ab}=(\sg^{-1})_{ab}$ for the inverse metric to $\sg$ on $\BS^2$.

§ MODE STABILITY OF THE REISSNER-NORDSTRÖM SPACETIME

In this section we shall characterize the non-decaying (generalized) modes of the linearized Einstein-Maxwell system around the Reissner-Nordström spacetime $(M, g_{b_0}, A_{b_0})$ with subextremal charge, following the method of Kodama-Ishibashi [79, 80] for the study of perturbations of black holes without (or with) charge in high dimensions. Throughout this section, we drop the subscript $b_0$, i.e., we take $(g, A)=(g_{b_0}, A_{b_0})$ and $\Bm=\Bm_0, \BQ=\BQ_0$. Here we adopt some notations from [80, 60, 57].

Let $\sigma\in \BC$ with $\IM\sigma\geq0$, suppose $(\dot{g}, \dot{A})$ is an outgoing mode solution to the linearized Einstein-Maxwell system, i.e., $(\dot{g}, \dot{A})=(e^{-i\sigma t_*}\dot{g}_0, e^{-i\sigma t_*}\dot{A}_0)$ with $(\dot{g}_0, \dot{A}_0)\in \eHb^{\infty,
\ell}(\CX; \scsym\oplus \scform)$ for some $\ell\in\BR$ solves

\begin{equation}
\begin{split}
\mathscr{L}_1(\dot{g}, \dot{A})&=-\frac 12\Box_{g}\dot{g}-\delta^*_g\delta_gG_g\dot{g}+\mathscr{R}_g\dot{g}-2D_{(g, dA)}T(\dot{g}, d\dot{A})=0\\
\mathscr{L}_2(\dot{g}, \dot{A})&=D_{(g, A)}\left(\delta_{g}dA\right)(\dot{g}, \dot{A})=0.
\end{split}
\end{equation}

Then there exist parameters $\dot{\Bm}\in \BR, \dot{\Ba}\in \BR^3, \dot{\BQ}\in\BR$ and an outgoing $1$-form $\omega$ and a function $\phi$ on $\CM$, i.e. $(\omega, \phi)\in e^{-i\sigma t_*}\eHb^{\infty,
\ell'}(\CX; \scform \oplus \BC)$ for some $\ell'\in\BR$, such that

\begin{equation}\label{eq:modesoln}
(\dot{g}, \dot{A})=\left(\dot{g}_{(\Bm, 0,\BQ)}(\dot{\Bm}, \dot{\Ba},\dot{\BQ})+2\delta_g^*\omega, ~\dot{A}_{(\Bm, 0,\BQ)}(\dot{\Bm}, \dot{\Ba},\dot{\BQ})+\mathcal{L}_{\omega^\#}A+d\phi\right).
\end{equation}

More specifically

* If $\sigma\neq 0$, then

\[
(\dot{g}, \dot{A})=\left(2\delta_g^*\omega, ~\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in e^{-i\sigma t_*}\eHb^{\infty,
\ell'}(\CX; \scform\oplus \BC)$ for some $\ell'\in\BR$.

* If $\sigma=0$ and $-\frac 32<\ell<-\frac 12$, we decompose the perturbation $(\dot{g}, \dot{A})$ into spherical harmonics whose detail is given in <ref>. Then

(1) If $(\dot{g}, \dot{A})$ is of scalar type with $l\geq 1$ or vector type $l\geq 2$, then

\[
(\dot{g}, \dot{A})=\left(2\delta_g^*\omega, ~\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in \eHb^{\infty,
\ell-1}(\CX; \scform\oplus \BC)$ is of the same type as $(\dot{g}, \dot{A})$.

(2) If $(\dot{g}, \dot{A})$ is of scalar type with $l=0$, i.e. spherically symmetric, then

\[
(\dot{g}, \dot{A})=\left(\dot{g}_{(\Bm, 0,\BQ)}(\dot{\Bm}, 0, \dot{\BQ})+2\delta_g^*\omega, ~\dot{A}_{(\Bm, 0,\BQ)}(0, 0,\dot{\BQ})+\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in\eHb^{\infty,
\ell-1}(\CX; \scform\oplus \BC)$ is spherically symmetric.

(3) If $(\dot{g}, \dot{A})$ is of vector type $l=1$, then

\[
(\dot{g}, \dot{A})=\left(\dot{g}_{(\Bm, 0,\BQ)}(0, \dot{\Ba},0)+2\delta_g^*\omega, ~\dot{A}_{(\Bm, 0,\BQ)}(0, \dot{\Ba},0)+\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in\eHb^{\infty,
\ell-1}(\CX; \scform\oplus \BC)$ is of vector type with $l=1$.

The statement of stationary perturbation $(\dot{g}, \dot{A})$ ($\sigma=0$) of scalar type with $l=0,1$ can be extended to the following cases:

(4) If $(\dot{g}, \dot{A})\in \eHb^{\infty,
\ell}(\CX; \scsym\oplus \scform)$ with $-\frac 52<\ell<-\frac 32$ is a stationary perturbation of scalar type $l=1$, then

\[
(\dot{g}, \dot{A})=\left(2\delta_g^*\omega, ~\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in\eHb^{\infty,
\ell-1}(\CX; \scform\oplus \BC)$ is of the scalar type $l=1$.

(5) If $(\dot{g}, \dot{A})\in \mbox{Poly}(t_*)^k\eHb^{\infty,
\ell}(\CX; \scsym\oplus \scform)$ is of scalar type $l=0$, then

\[
(\dot{g}, \dot{A})=\left(\dot{g}_{(\Bm, 0,\BQ)}(\dot{\Bm}, 0, \dot{\BQ})+2\delta_g^*\omega, ~\dot{A}_{(\Bm, 0,\BQ)}(0, 0,\dot{\BQ})+\mathcal{L}_{\omega^\sharp}A+d\phi\right)
\]

where $(\omega, \phi)\in\mbox{Poly}(t_*)^{k+1}\eHb^{\infty,
\ell'}(\CX; \scform\oplus \BC)$ is of scalar type $l=0$ for some $\ell'<\ell$.

Our discussion corresponds to the case $n=2,K=1, \kappa^2=2, E_0=r^{-2}\BQ, Q^2=\BQ^2, \Lambda=0$ (and thus $\lambda=0$), in [80]. We note that [80] discusses the scalar and vector type perturbations for non-stationary modes ($\sigma\neq0$) with $l\geq2$ in detail, but the corresponding stationary perturbations ($\sigma=0$) are only treated in the uncharged case in[79]. [80] also studies the scalar (Appendix D) and vector (4) perturbations for modes $l=1$ (which is called exceptional modes there), however, they do not give a full description of the scalar $l=0$ (spherically symmetric) perturbations.

§.§ Scalar type perturbations

We shall consider the scalar type perturbations of modes $l\geq 2, l=1$ and $l=0$ separately.  Recall that we can write the perturbations under the splitting (<ref>) and (<ref>) as in (<ref>). We further introduce a rescaled version of the traceless part of the spherical harmonic symmetric $2$-tensor $\sdelta_0^*\sd \mathsf{S}_l$ which is built from the spherical harmonic function $\mathsf{S}\in \BFS_l$ with eigenvalues $k^2=l(l+1)$:

\[
\rsH_k\mathsf{S}=k^{-2}\sdelta_0^*\sd\mathsf{S}\quad \mbox{where}\quad \mathsf{S}\in\BFS_l\mbox{ with }l\neq 0.
\]

§.§.§ Modes with $l\geq 2$

Suppose $(\dg, \dA)$ is the scalar perturbations of the following form

\begin{equation}\label{eq:perturbation_l2}
\dg=\begin{pmatrix}
\widetilde{f}\sfS\\
\end{pmatrix},\quad \dA=\begin{pmatrix}
\widetilde{K}\sfS\\
\end{pmatrix}\quad\mbox{with}\quad \sfS\in\BFS_l ~(l\geq2).
\end{equation}

We notice that the pure gauge solutions take the form

\begin{equation}
\bdelta\dg=2\delta_g^*\omega=\begin{pmatrix}
2\hdelta^*T	\\
-\frac{r}{k}(-\frac{k}{r}T+r\hd r^{-1}L)\sd\sfS\\
\end{pmatrix},\quad\!\! \bdelta\dA=\mathcal{L}_{\omega^\sharp}A+d\phi=\begin{pmatrix}
(\hd\iota_AT+\iota_{T}\hd A+\hd P)\sfS\\
\end{pmatrix}
\end{equation}

\begin{equation}
\omega=\begin{pmatrix}
\end{pmatrix},\quad\!\! \phi=P\sfS
\end{equation}

where $T\in C^\infty(\hX; T^*\hX), L, P\in C^\infty({\hX})$. When adding $(\bdelta\dg, \bdelta\dA)$ to $(\dg, \dA)$, the quantities $\tilde{f}, f, H_L, H_T, \tilde{K}, K$ change by

\begin{equation}
\begin{gathered}
\bdelta\widetilde{f}=2\hdelta^*T,\quad\bdelta f=-\frac{k}{r}T+r\hd (r^{-1}L),\quad \bdelta H_L=r^{-1}\iota_{dr}T+\frac{k}{2r}L,\quad \bdelta H_T=-\frac{k}{r}L;\\
\bdelta\widetilde{K}=\hd\iota_AT+\iota_T\hd A+\hd P,\quad \bdelta K=-\frac{k}{r}(\iota_A T+P).
\end{gathered}
\end{equation}

We define $\mathbf{X}:=\frac{r}{k}(f+\frac{r}{k}\hd H_T)$, then $\bdelta\mathbf{X}=\frac{r}{k}(\bdelta f+\frac{r}{k}\hd \bdelta H_T)=-T$ because $\bdelta$ commutes with any geometric operators we use in this manuscript. As a consequence, the following quantities

\begin{equation}
\begin{split}
\widetilde{F}&:=\widetilde{f}+2\hdelta^*\mathbf{X}\in C^{\infty}(\hX; S^2T^*\hX),\\
J&:=H_L+\frac{1}{2}H_T+r^{-1}\iota_{dr}\mathbf{X}\in C^\infty(\hX;T^*\hX),\\
N&:=\widetilde{K}+\hd\left(\frac{r}{k}K\right)+\iota_{\mathbf{X}}\hd A\in C^\infty(\hX; T^*\hX)
\end{split}
\end{equation}

are gauge-invariant, that is, $\bdelta\widetilde{F}=0, \bdelta J=\bdelta N=0$. Conversely, if $\widetilde{F}=0,  J= N=0$, one can verify $(\dg, \dA)$ is a pure gauge solution

\begin{equation}
(\dg, \dA)=(2\delta^*\omega, \mathcal{L}_{\omega^\sharp}A+d\phi) \quad\!\! \mbox{with}\quad\!\! \omega=\begin{pmatrix}
\frac{r^2}{k^2}H_T\sd\sfS
\end{pmatrix},\quad \!\! \phi=-\frac{r}{k}K+\iota_A\mathbf{X}.
\end{equation}

If $(\dg, \dA)$ are outgoing mode solutions with frequency $\sigma\neq 0$, then $\omega$ and $ \phi$ are modes of the same type, with an extra factor $r^2$ and $r$ respectively relative to $\dg$ and $\dA$.  If $\sigma=0$, $(\omega, \phi)$ grow at most by a factor $r$ more than $(\dg, \dA)$.

Since the linearized Einstein-Maxwell system is gauge-invariant, we can express it in terms of the gauge-invariant quantities defined above. Concretely, one can choose a gauge, i.e., add a pure gauge solution $(\bdelta\dg, \bdelta\dA)$ to $(\dg, \dA)$ for suitable $\omega=\omega(T,L), \phi=\phi(P)$ such that the non gauge-invariant quantities $\tilde{f}$ etc. take a simple form in the new gauge. More specifically, we adds $(\bdelta\dg, \bdelta\dA)$ built from $T=\mathbf{X}, L=\frac{r}{k}H_T, P=\frac{r}{k}K-\iota_A\mathbf{X}$, then $\mathbf{X}+\bdelta\mathbf{X}=T-T=0, H_T+\bdelta H_T=\frac{k}{r}L-\frac{k}{r}L=0$ which implies $f+\bdelta f=0, \widetilde{f}+\bdelta\widetilde{f}=\tilde{F}, H_L+\bdelta H_L=J$, and $K+\bdelta K=K-\frac{k}{r}\iota_A\mathbf{X}-\frac{k}{r}(\frac{r}{k}K-\iota_A\mathbf{X})=0$ which implies $\widetilde{K}+\bdelta \widetilde{K}=N$. To summarize, if we replace $(\dg, \dA)$ by $(\dg+\bdelta\dg, \dA+\bdelta\dA)$, in the new gauge we have $(\widetilde{f}, f, H_L, H_T, \widetilde{K}, K)=(\widetilde{F}, 0, J, 0, N, 0)$.
Therefore, using the detailed calculation in <ref> and <ref> we can write the linearized 	Einstein-Maxwell system, acting on the new $(\dg, \dA)$

\[
\dg=\begin{pmatrix}
&	\widetilde{F}\sfS\\
&2r^2 J\sfS\sg
\end{pmatrix}, \quad \dA=\begin{pmatrix}
\end{pmatrix}
\]
in terms of the gauge-invariant quantities $\widetilde{F}, J,N$.

Now we express $2\mathcal{L}_1(\dg, \dA)=0$ and $\mathcal{L}_2(\dg, \dA)=0$ in the form of (<ref>) in terms of $\widetilde{f}^E, f^E, H_L^E, H_T^E$ and $\widetilde{K}^E, K^E$ respectively. Then the linearized Einstein-Maxwell system reads

\begin{align}
\widetilde{f}^E&=\left(-\hBox-2\hdelta^*\hdelta-\hdelta^*\hd \htr\right)\widetilde{F}+2r^{-1}\left(2\hdelta^*\iota_{dr}\widetilde{F}-(\pa^i r)\hnabla_i\widetilde{F}\right)-4\hdelta^*\hd J-8r^{-1}(dr)\otimes_s \hd J\notag \\
&\quad+(-\weight''+k^2r^{-2})\widetilde{F}+(-2\BQ^2 r^{-4}+\weight'')\hg\htr\widetilde{F}-4\BQ r^{-2}\hg\hstar\hd N=0,\label{eq:Etildef}\\
-\frac{r}{k}f^E&=-\hdelta\widetilde{F}-2\hd J-r\hd(r^{-1}\htr\widetilde{F})+4\BQ r^{-2}\hstar N=0,\label{eq:Ef}\\
2r^2H_L^E&=-\hBox(2r^2J)-2r\iota_{dr}\hdelta\widetilde{F}+2\iota_{dr}\iota_{dr}\widetilde{F}-r\iota_{dr}\hd\htr\widetilde{F}+(r\weight'+\frac{k^2}{2}+2\BQ^2 r^{-2})\htr \widetilde{F}\notag\\
&\quad+	(2k^2-4\BQ^2r^{-2})J+4\BQ\hstar\hd N=0,\label{eq:EH_L}\\
\widetilde K^E&=r^{-2}\hdelta(r^2\hd N)+k^2r^{-2}N+\frac{1}{2}\BQ r^{-2}\hstar\hd\htr\widetilde{F}-2\BQ r^{-2}\hstar\hd J=0,\label{eq:EtildeK}\\
-\frac{r}{k}K^E&=-\hdelta N=0.\label{eq:EK}
\end{align}

By (<ref>), all terms containing $\htr \widetilde{F}$ vanish, then plugging $
\hdelta\widetilde{F}=-2\hd J+4\BQ r^{-2}\hstar N$ into (<ref>) yields the cancellation of the term $4\hdelta^*\hd J$ and thus one obtains a wave equation of $\widetilde{F}$ (coupled to $J, N$ only via subprincipal terms). Moreover, by (<ref>), one adds the zero term $\hd\hdelta N$ to (<ref>) and then can obtain a wave equation for $N$ (coupled to $J$ via subprincipal terms). In conclusion, we can rewrite equation (<ref>), (<ref>) and (<ref>) as a principally scalar system of wave equations for $(\widetilde{F}, J, N)$

\begin{equation}
-\hBox B-\mathscr{D}B=0\quad \mbox{where}\quad
\begin{pmatrix}
\widetilde{F}\\
\end{pmatrix}
\end{equation}

where $\mathscr{D}$ is a first order stationary differential operator acting on $C^{\infty}(\hX; S^2T^*\hX\oplus \BR\oplus T^*\hX)$. When $(\dg, \dA)$ and thus $B$ are smooth modes, this system of equations become a system of ODEs on the one dimensional space $t_*^{-1}(0)$ with a regular singular point at the event horizon $r=\ehRN$ whose solution is given by Frobenius series, which means that the vanishing of $B$ in the static region $r>\ehRN$ implies the vanishing of $B$ on $\hX$. As a result, it suffices to prove $B=0$ in the static region $r>\ehRN$. To achieve this, we can work in the static coordinates $(t,r)$.

First, since $\hdelta N=\hstar\hd\hstar N=0$, we have $\hd\hstar N=0$ and then  $\hstar N=\hd\mathcal{N}$ for some $\mathcal{N}\in C^\infty(\hX)$ according to $H_{dR}^1(\hX)=0$, which implies $N=\hstar\hd\mathcal{N}$. Returning to the equation (<ref>), we find

\begin{equation}
\hd J=\hd\left(-r^2\hBox \mathcal{N}+k^2\mathcal{N}-2\BQ J\right)=0
\end{equation}

and thus

\[
-r^2\hBox \mathcal{N}+k^2\mathcal{N}-2\BQ J=C\quad\mbox{for some}\quad\!\!C\in\BR.
\]

Replacing $\mathcal{N}$ by $\mathcal{N}-\frac{C}{k^2}$, we still have $N=\hstar\hd\mathcal{N}$ and meanwhile

\begin{equation}\label{eq:eqnforN}
-\hBox \mathcal{N}+k^2r^{-2}\mathcal{N}-2\BQ r^{-2}J=0.
\end{equation}

We note that since $N$, and thus $\hstar N$ is a mode, $\mathcal{N}$ can also be chosen as a mode. More specifically, suppose $\hstar 	N=e^{-it_*\sigma}(N_{0}dt_*+N_1dr)$ where $N_0, N_1$ are smooth functions of $r$ and $\sigma\neq 0$, we let $\mathcal{N}=e^{-it_*\sigma}\mathcal{N}_0$ with $\mathcal{N}_0=i\sigma^{-1}N_0$. Then $N_1=\pa_r\mathcal{N}_0=i\sigma^{-1}\pa_r N_0$ will be satisfied automatically because $\hd\hstar N=e^{-it_*\sigma}(-\pa_rN_0-i\sigma N_1)dt_*\wedge dr=0$.

Next, according to the second Bianchi identity $\delta_gG_g\Ric(g)=0$ (which holds for any metric $g$), one concludes for all $(g, F)$

\[
\delta_g G_g(\Ric(g)-2T(g,F))=-2\delta_g T(g,F)=-2g^{\al\be}(\delta_gF)_\al F_{\n\be}-g^{\al\kappa}g^{\be\lambda}F_{\al\be}(\nabla_\n F_{\kappa\lambda}+2\nabla_{\kappa}F_{\lambda\n}).
\]

Since $dF_{\n\kappa\lambda}=\nabla_{\n}F_{\kappa\lambda}
-\nabla_{\lambda}F_{\n\kappa}+\nabla_{\kappa}F_{\lambda\n}$ and $g^{\al\kappa}g^{\be\lambda}F_{\al\be}(\nabla_{\lambda}F_{\n\kappa}+\nabla_{\kappa}F_{\lambda\n})=0$, it follows that

\begin{equation}\label{eq:Lsecondbianchi}
\delta_g G_g(\Ric(g)-2T(g,F))=-2\delta_g T(g,F)=-2g^{\al\be}(\delta_gF)_\al F_{\n\be}-g^{\al\kappa}g^{\be\lambda}F_{\al\be}(dF)_{\n\kappa\lambda}.
\end{equation}

Linearizing the above equation around $(g, F)=(g_{b_0}, dA_{b_0})$, if $(\dg, \dA)$ is a solution to the Maxwell part of the linearized Einstein-Maxwell system, i.e., $\mathcal{L}_2(\dg, \dA)=0$, we find by using $\Ric(g)-2T(g,dA)=0, \delta_g dA=0, d^2A=0$ that

\begin{equation}\label{eq:LsecondBianchiM}
\delta_g G_g\mathcal{L}_1(\dg, d\dA)=0,\quad\mbox{i.e., }\quad\!\! \delta_g G_g\begin{pmatrix}
\widetilde{f}^E\sfS\\
-\frac{r}{k}f^E\otimes \sd\sfS\\
\end{pmatrix}=0,
\end{equation}

which gives the following system of equations for $(\widetilde{f}^E, f^E, H_L^E, H_T^E)$

\begin{gather*}
\frac{1}{2}\htr\widetilde{f}^E-\frac{1}{kr^2}\hdelta(r^3f^E)+\frac{k^2-2}{k^2}H_T^E=0.
\end{gather*}

We define

\begin{equation}\label{eq:defoftildeE}
\widetilde{E}=G_g\widetilde{f}^E-2H_L^E\hg,
\end{equation}

it is clear that $\widetilde{E}=0$ implies $(\widetilde{f}^E, H_L^E)=(h\hg, 0)$ for some $h\in C^\infty(\hX)$. If in addition equations (<ref>) and (<ref>) hold (i.e. $f^E=0$ and $H_T^E=0$), one obtains $\htr\widetilde{f}^E=0, \hdelta(r^2\widetilde{E})=0$, and thus $(\widetilde{f}^E, H_L^E)=(0, 0)$. We further write $\hdelta(r^2\widetilde{E})=0$ in $(t,r)$ coordinates and find its $dr$ component

\[
-\frac {\weight'}{2\weight^2}\widetilde{E}_{tt}+\weight'\pa_t\widetilde{E}_{tr}+\weight^{-1/2}\pa_r(\weight^{3/2}\widetilde{E}_{rr})=0.
\]

Then the vanishing of $\widetilde{E}_{tr}$ and $\widetilde{E}_{rr}$ implies $\widetilde{E}_{tt}=0$ because $\weight^{-2}\weight'=\weight^{-2}(2\Bm r^{-2}-2\BQ^2r^{-3})\neq0$ in the static region $r>\ehRN=\Bm+\sqrt{\Bm^2-\BQ^2}>\BQ^2/\Bm$ in the subextremal charge case $\Bm>\BQ$. In summary, $(\widetilde{f}^E, f^E, H_L^E, H_T^E)=0$ is equivalent to $(f^E, H_T^E, \widetilde{E}_{tr}, \widetilde{E}_{rr})=0$. So our goal is to prove $(\widetilde{F}, J, \mathcal{N})=0$, provided that $(f^E, H_T^E, \widetilde{E}_{tr}, \widetilde{E}_{rr}, \widetilde{K}^E,K^E)=0$.

Following <cit.> or <cit.>, we write

\begin{equation}\label{eq:defofXYZ}
\widetilde{F}-2J\hg=\begin{pmatrix}
\weight X\\
-\weight^{-1} Z\\
-\weight^{-1} Y
\end{pmatrix}
\end{equation}

in the splitting (<ref>). From now on, we will use the fact $\htr\widetilde{F}=0$ which is implied by $H_T^E=0$ without specified till the end of this subsubsection. One can recover $\widetilde{F}, J$ as

\begin{equation}\label{eq:recoverofFJ}
\widetilde{F}=\begin{pmatrix}
\frac{\weight}{2}(X-Y)\\
\frac{1}{2\weight}(X-Y)
\end{pmatrix},\quad J=\frac{X+Y}{4}.
\end{equation}

Then the equation (<ref>) $f^E=0$ implies $\hdelta(\widetilde{F}-2J\hg)=4\BQ r^{-2}\hg\hd\mathcal{N}$, which we express in terms of $X,Y,Z, \mathcal{N}$ as

\begin{equation}\label{eq:XYZ1}
\pa_tX+\pa_r Z=4\BQ r^{-2}\pa_t\mathcal{N},\quad -\frac{\weight'}{2\weight}(X-Y)-\weight^{-2}\pa_tZ+\pa_rY=4\BQ r^{-2}\pa_r\mathcal{N}.
\end{equation}

The equation $\widetilde{E}_{tr}=\widetilde{f}^E_{tr}=0$ reads (we use the equation (<ref>) during the calculation)

\begin{equation}\label{eq:XYZ2}
-\pa_t\pa_r X-\pa_t\pa_r Y+\frac{\weight'}{2\weight}
\pa_t X+(\frac{\weight'}{2\weight}-\frac{2}{r})\pa_t Y-\frac{k^2}{r^2\weight}Z=0.
\end{equation}

Finally the equation $\weight\widetilde{E}_{rr}=\frac{1}{2}(\weight^{-1}\widetilde{f}^E_{tt}+\weight\widetilde{f}^E_{rr})-2H_L^E=0$ becomes (we use equation (<ref>) during the calculation)

\begin{equation}\label{eq:XYZ3}
-\weight^{-1}\pa_t^2X-\weight^{-1}\pa_t^2 Y+\frac{\weight'}{2}\pa_rX+(\frac{\weight'}{2}+\frac{2\weight}{r})\pa_r Y-\frac{4}{r\weight}\pa_t Z-\frac{\BQ^2}{r^4}X-\left(\frac{k^2-2}{r^2}+\frac{3\BQ^2}{r^2}\right)Y+\frac{4k^2\BQ}{r^4}\mathcal{N}=0.
\end{equation}

Therefore, it suffices to prove $(X, Y, Z, \mathcal{N})=0$ provided that they satisfy the equations (<ref>) and (<ref>)–(<ref>).

We now discuss the case that $(\dg,\dA)$ is a mode solution with $\sigma\neq 0$. After taking Fourier transform with respect to $t$, setting $\pa_t (X, Y, Z,\mathcal{N})=-i\sigma(X, Y, Z, \mathcal{N})$ and solving for $(X', Y', Z')=\pa_r(X, Y,Z)$, we obtain from equations (<ref>) and (<ref>)

\[
\begin{pmatrix}
\frac{Z'}{i\sigma}
\end{pmatrix}=T\begin{pmatrix}
\frac{Z}{i\sigma}
\end{pmatrix}+f \quad\mbox{where}\quad T=\begin{pmatrix}
\frac{\weight'}{2\weight}&-\frac{\weight'}{2\weight}&\frac{\sigma^2}{\weight^2}\\
\end{pmatrix},\quad f=\frac{4\BQ}{r^2}\begin{pmatrix}
\mathcal{N}'\\
\end{pmatrix},
\]

while the equation (<ref>) implies the following linear constraint on $X,Y,Z$

\begin{equation}\label{eq:constraintonXYZ}
\gamma\begin{pmatrix}
\frac{Z}{i\sigma}
\end{pmatrix}=h
\end{equation}

where $h=-\frac{4\BQ}{r^2}(\frac{2\weight}{r}\mathcal{N}'+\frac{k^2}{r^2}\mathcal{N})$ and

\[
\gamma=(\frac{\sigma^2}{\weight}-\frac{\BQ^2}{r^4}+\frac{(\weight')^2}{4\weight}+\frac{\weight'}{r},\  \frac{\sigma^2}{\weight}-\frac{k^2-2}{r^2}-\frac{3\BQ^2}{r^4}+\frac{(\weight')^2}{4\weight}-\frac{2\weight'}{r},\  -\frac{2\sigma^2}{r\weight}+\frac{k^2\weight'}{2r^2\weight}).
\]

Following [80] and [60], one can find a linear combination $\Phi$ of $X, Y, \frac{Z}{i\sigma}$ which satisfies a second order ODE. Concretely, let

\begin{equation}
\Phi:=\frac{\frac{2Z}{i\sigma}-r(X+Y)}{H}\quad\mbox{with}\quad m:=k^2-2,\quad x:=\frac{2\Bm}{r},\quad z:=\frac{\BQ^2}{r^2},\quad H:=m+3x-4z.
\end{equation}

Then the second order ODE for $\Phi$ is (see <cit.>, <cit.>)

\begin{equation}\label{eq:eqforPhi}
\end{equation}

where $V_\Phi$ and $F_\Phi$ are (see <cit.>, <cit.>)

\begin{equation}\label{eq:VFPhi}
\begin{split}
\end{split}.
\end{equation}

Conversely, $X, Y, \frac{Z}{i\sigma}$ can be expressed in terms of $\Phi$ and $\mathcal{N}$ as

\begin{equation}\label{eq:eqforXYZs}
\begin{split}
X&=(\frac{\sigma^2r}{\weight}-\frac{P_{X_0}}{2rH^2})\Phi+\frac{P_{X_1}}{2H}\Phi'-\frac{2\BQ P_{X\mathcal{N}}}{r^2H^2}\mathcal{N}-\frac{8\BQ\weight}{rH}\mathcal{N}'\\
Y&=(-\frac{\sigma^2r}{\weight}-\frac{P_{Y_0}}{2rH^2})\Phi+\frac{P_{Y_1}}{2H}\Phi'-\frac{2\BQ P_{Y\mathcal{N}}}{r^2H^2}\mathcal{N}+\frac{8\BQ\weight}{rH}\mathcal{N}'\\
\frac{Z}{i\sigma}&=\frac{P_Z}{2H}\Phi-r\weight\Phi'-\frac{8\BQ\weight}{rH}\mathcal{N}'
\end{split}
\end{equation}

where $P_{X_0}, P_{X_1},P_{X\mathcal{N}},P_{Y_0}, P_{Y_1},P_{Y\mathcal{N}}, P_Z$ are functions of $r$ (see <cit.> and <cit.>)

\begin{equation}\label{eq:eqforP_XYZ}
\begin{split}
&\qquad -64z^3+48mz^2-8m(m-2)z+2m^2(m+2),\\
&\qquad -64z^3+16(m-4)z^2-8m(m+2)z,\\
\end{split}
\end{equation}