In this section, we include the discussion of the subprincipal operator of $\mathcal{P}_{b_0,\gamma}, \mathcal{W}_{b_0,\gamma}$ and $L	_{b_0,\gamma}$ at the trapped sets $K$ for the sake of completeness. The arguments are based on [59], [60] and [61]. In the subsequent discussion, we will list (without proof) the relevant results for the invariant formalism of the subprincipal operator with references (mostly from [59]), and then carry out all the relevant computations.

According to <cit.> and <cit.> (the microlocalized version) (see also the discussion in <cit.>), a sufficient condition for high energy estimates at the trapped set for a semiclassical operator $P_h(z)$ acting on the covariant rank $k$ tensor bundle $\mathcal{T}_k$ to hold is

\begin{equation}
\label{EqBoundAtTrapping}
\sigma_{1,h}\Bigl(\frac{1}{2ih}(P_h(z)-P_h(z)^*)\Bigr)<\nu_{\min}/2
\end{equation}

at the trapped set $K$, where $\nu_{\min}>0$ is the minimal normal expansion rate of the Hamilton flow at the trapping, see Proposition <ref>. Here, the adjoint is taken with respect to a positive definite inner product on $\mathcal{T}_k$. We note that the natural inner product induced by $g$, with respect to which $\Box_{g}$ is symmetric, is not positive definite, except when $k=0$, i.e. for the scalar wave equation. One can introduce a pseudodifferential inner product such that (<ref>) can be arranged, with the adjoint taken with respect to this pseudodifferential inner product.
We work with classical, i.e. one-step polyhomogeneous, symbols and operators, and denote by $S^m_{\mathrm{hom}}(T^*X\setminus 0)$ symbols which are homogeneous of degree $m$ with respect to dilations in the fibers of $T^*X\setminus 0$.

A pseudodifferential inner product (or $\Psi$-inner product) on the vector bundle $\mathcal{E}\to X$ is a pseudodifferential operator $B\in\Psi^0(X;\mathcal{E}\otimes\Omega^\frac12,\overline{\mathcal{E}}^*\otimes\Omega^\frac12)$ satisfying $B=B^*$, and such that moreover the principal symbol $\sigma^0(B)=b\in S^0_{\mathrm{hom}}(T^*X\setminus 0;\pi^*\mathrm{Hom}(\mathcal{E},\overline{\mathcal{E}}^*))$ of $B$ satisfies

\begin{equation}
\label{EqPsdoInnerPosDef}
\langle b(x,\xi)u,\iota(u)\rangle > 0
\end{equation}

for all non-zero $u\in\mathcal{E}_x$, where $x\in X$, $\xi\in T^*_x X\setminus 0$. If the context is clear, we will also call the sesquilinear pairing

\[
C^\infty(X,\mathcal{E}\otimes\Omega^\frac12)\times C^\infty(X,\mathcal{E}\otimes\Omega^\frac12) \ni (u,v) \mapsto \int_X \langle B(x,D)u,\iota(v)\rangle
\]

the pseudodifferential inner product associated with $B$.

The use of a pseudodifferential inner product is equivalent to considering a conjugated operator $QP_h(z)Q^-$, where $Q\in\Psi_{h}^0(X,\mathcal{T}_k)$ is a carefully chosen elliptic operator with parametrix $Q^-$. For any $\epsilon>0$, we can arrange $\sigma_{1,h}(\frac{1}{2ih}(QP_h(z)Q^--(QP_h(z)Q^-)^*))<\epsilon$ (with the adjoint taken relative to an ordinary positive definite inner product on $\mathcal{T}_k$), thus (<ref>) holds for $P_h(z)$ replaced by $QP_h(z)Q^-$.

Let $\pi:T^*X\setminus 0\to X$ be the projection. We will work in standard pseudodifferential operator setting. We specialize to the case that $P\in\Psi^m(X,\mathcal{E}\otimes\Omega^\frac12)$ has a real, scalar principal symbol, which is the case of interest in our application. Fix a coordinate system of $X$ and a local trivialization of $\mathcal{E}$, then the full symbol of $P$ is a sum of homogeneous symbols $p\sim p_m+p_{m-1}+\ldots$, with $p_j$ homogeneous of degree $j$ and valued in complex $N\times N$ matrices. According to <cit.>, the subprincipal symbol

\begin{equation}
\label{EqScalarSubprincipal}
\sigma_\mathrm{sub}(P)=p_{m-1}(x,\xi)-\frac{1}{2i}\sum_j\pa_{x_j\xi_j}p_m(x,\xi) \in S^{m-1}_{\mathrm{hom}}(T^*X\setminus 0,\BC^{N\times N})
\end{equation}

is well-defined under changes of coordinates; however, it does depend on the choice of local trivialization of $\mathcal{E}$. We would like a frame-independent notion of the subprincipal symbol since this provides us with the freedom to choose particularly convenient local frames in concrete computations.

§.§ Invariant formalism for the subprincipal symbols of operators acting on bundles

For $P\in\Psi^m(X,\mathcal{E}\otimes\Omega^\frac12)$ with scalar principal symbol $p$, there is a well-defined subprincipal operator $S_\mathrm{sub}(P)\in\mathrm{Diff}^1(T^*X\setminus 0,\pi^*\mathcal{E})$, homogeneous of degree $m-1$ with respect to dilations in the fibers of $T^*X\setminus 0$, defined as follows: if $\{e_1(x),\ldots,e_N(x)\}$ is a local frame of $\mathcal{E}$, define the operators $P_{jk}\in\Psi^m(X,\Omega^\frac12)$ by $P(\sum_k u_k(x)e_k(x))=\sum_{jk} P_{jk}(u_k)e_j(x)$, $u_k\in C^\infty(X,\Omega^\frac12)$. Then

\[
S_\mathrm{sub}(P)\Bigl(\sum_k q_k(x,\xi)e_k(x)\Bigr) := \sum_{jk} (\sigma_\mathrm{sub}(P_{jk})q_k)e_j - i\sum_k(H_p q_k)e_k.
\]

The symbols of commutators and imaginary parts can be expressed in a completely invariant fashion.

Let $P\in\Psi^m(X,\mathcal{E}\otimes\Omega^\frac12)$ be a pseudodifferential operator with scalar principal symbol $p$.

* Suppose $Q\in\Psi^{m'}(X,\mathcal{E}\otimes\Omega^\frac12)$ is an operator acting on $\mathcal{E}$-valued half-densities, with principal symbol $q$. Then

\begin{equation}
\label{EqSubprImagPart}
\sigma_{m+m'-1}([P,Q]) = [S_\mathrm{sub}(P),q].
\end{equation}

If $Q$ is elliptic with parametrix $Q^-$, then

\begin{equation}
\label{EqSubprChangeOfBasis}
S_\mathrm{sub}(QPQ^-)=q S_\mathrm{sub}(P)q^{-1}.
\end{equation}

* Suppose in addition that $p$ is real. Let $B$ be a $\Psi$-inner product (pseudodifferential inner product) on $\mathcal{E}$ with principal symbol $b$, then

\begin{equation}
\label{EqInvImagSymbol}
\sigma_{m-1}(\IM^B P) = \IM^b S_\mathrm{sub}(P),
\end{equation}

where $\IM^BP=\frac{1}{2i}(P-P^{*B})$ and $\IM^b S_\mathrm{sub}(P)=\frac{1}{2i}\bigl(S_\mathrm{sub}(P)-S_\mathrm{sub}(P)^{*b}\bigr)$; we take the adjoint of $P$ with respect to the $\Psi$-inner product $B$ and the adjoint of the differential operator $S_\mathrm{sub}(P)$ with respect to the inner product $b$ on $\pi^*\mathcal{E}$ and the symplectic volume density on $T^*X$.

The imaginary part $\IM^B P$ of an operator $P$ with respect to a $\Psi$-inner product $B$ can be interpreted in terms of the imaginary part of a conjugated version of $P$ with respect to a standard inner product.

Let $B$ be a $\Psi$-inner product on $\mathcal{E}$. Then for any positive definite Hermitian inner product $B_0\in C^\infty(X,\mathrm{Hom}(\mathcal{E}\otimes\Omega^\frac12,\overline{\mathcal{E}}^*\otimes\Omega^\frac12))$ on $\mathcal{E}$, there exists an elliptic operator $Q\in\Psi^0(X,\mathrm{End}(\mathcal{E}\otimes\Omega^\frac12))$ such that $B-Q^*B_0Q\in\Psi^{-\infty}(X,\mathrm{Hom}(\mathcal{E}\otimes\Omega^\frac12,\overline{\mathcal{E}}^*\otimes\Omega^\frac12))$.

In particular, denoting by $Q^-\in\Psi^0(X,\mathrm{End}(\mathcal{E}\otimes\Omega^\frac12))$ a parametrix of $Q$, we have for any $P\in\Psi^m(X,\mathcal{E}\otimes\Omega^\frac12)$ with real and scalar principal symbol.

\begin{equation}
\label{EqPsdoInnerToNormal}
Q(\IM^B P)Q^- = \IM^{B_0}(QPQ^-),
\end{equation}

and $\sigma_{m-1}(\IM^B P)$ and $\sigma_{m-1}(\IM^{B_0}(QPQ^-))$ (which are self-adjoint with respect to $\sigma^0(B)$ and $B_0$, respectively, hence diagonalizable) have the same eigenvalues.

Therefore, the analysis of  $\sigma_{m-1}(\IM^{B_0}(QPQ^-)))$ is reduced to finding an inner product $b$ such that $(S_\mathrm{sub}(L)-S_\mathrm{sub}(L)^{*b})$ has a desired form. (In our applications, the choice of $b$ will be clear from an inspection of the form of $S_\mathrm{sub}(L)$).

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$ of arbitrary signature. Denote by $\mathcal{T}_k M=\bigotimes^k T^*M$, $k\geq 1$, the bundle of (covariant) tensors of rank $k$ on $M$. The metric $g$ induces a metric (which we also call $g$) on $\mathcal{T}_k M$. The subprincipal operator of $\Delta_k=\tr\nabla^2\in\mathrm{Diff}^2(M,\mathcal{T}_k M)$ acting on the bundle $\mathcal{T}_k M$ has a nice form.

The subprincipal operator of $\Delta_k$ is

\begin{equation}
\label{EqLaplaceSubpr}
S_\mathrm{sub}(\Delta_k)(x,\xi) = i\nabla^{\pi^*\mathcal{T}_k M}_{H_G} \in \mathrm{Diff}^1(T^*M\setminus 0,\pi^*\mathcal{T}_k M),
\end{equation}

where $\nabla^{\pi^*\mathcal{T}_k M}$ is the pullback connection, with $\pi\colon T^*M\setminus 0\to M$ being the projection, and $G=|\xi|^2_{g^{-1}(x)}$.

§.§ Calculation  of subprincipal operators

We notice that modulo terms of order $0$, which are sub-subprincipal and thus irrelevant in the calculation of $S_{\mathrm{sub}}(\mathcal{P}_{b_0,\gamma}), S_{\mathrm{sub}}(\mathcal{W}_{b_0,\gamma})$ and $S_{\mathrm{sub}}(L_{b_0,\gamma})$ (for simplicity of notation, from now on, we drop the notations $b_0, \gamma$, i.e., $g=g_{b_0}$),
we have

\begin{equation}
\end{equation}

\begin{equation}
\label{EqESGL}
-L(\dg,\dA) \equiv \Bigl( \Box_g\dg- 2(\tilde{\delta}_g^*-\delta_g^*)\delta_g G_g\dg -2\delta^*_g(\tilde{\delta}_g-\delta_g)G_g\dg+ L_{12}(\dA),\  \Box_g\dA + L_{21}(\dg) \Bigr),
\end{equation}

\begin{gather*}
L_{12}(\dA) = 8\tr_g^{24}(F\otimes_s d\dA) - 2 g^{-1}(F,d\dA)\,g, \quad F=dA=d(\frac{\BQ_0}{r}dt)=\frac{\BQ_0}{r^2}dt\wedge dr,\\
L_{21}(\dg) =  \tr_g^{12}(\delta_g G_g\dg\otimes F) - \frac{1}{2}\tr_g^{24}\tr_g^{35}(d^\nabla\dg\otimes F),\quad (d^\nabla\dg)_{\nu}^{\ \mu\kappa}=\nabla^{\mu}\dg_{\nu}^{\ \kappa}-\nabla_\nu^{\kappa}\dg^{\ \mu}.
\end{gather*}

Since $\tilde{\delta}_g^*-\delta_g^*$ and $\tilde{\delta}_g-\delta_g$ are compactly supported away from the trapping for Reissner-Nordström metric, we further have that modulo terms of order $0$

\begin{equation}
\label{EqESGLTrapping}
\begin{gathered}
-L(\dg,\dA) \equiv \Bigl( \Box_g\dg+L_{12}(\dA),\  \Box_g\dA + L_{21}(\dg) \Bigr)
\end{gathered}
\end{equation}

at the trapping $K$.

Let $g=g_{b_0}$ be the Reissner-Nördstrom metric with parameter $b_0=(\Bm_0, 0,\BQ_0)$

\begin{equation}
g=-\alpha^2dt^2+\alpha^{-2}dr^2+r^2\sg,\quad \alpha=\sqrt{\mu_{b_0}}=\sqrt{1-\frac{2\Bm_0}{r}+\frac{\BQ_0^2}{r^2}}.
\end{equation}

That is, we work in the coordinates $(t,r,\omega),\omega\in\BS^2$ at the trapping $K$. Let $(\sigma, \xi, \eta)$ be the dual variables to $(t, r, \omega)$. We define

\begin{equation}
e^0:=\alpha dt,\quad e^1:=\alpha^{-1}dr
\end{equation}

and use the following splittings of $T^*\mathcal{M}$ and $S^2\,T^*\mathcal{M}$

\begin{equation}\label{Eqsplittingtrap}
\begin{split}
T^*\mathcal{M}&=\langle e^0\rangle\oplus\langle e^1\rangle \oplus T^*\BS^2,\\
S^2\,T^*\mathcal{M}&=\langle e^0e^0\rangle\oplus\langle2e^0\otimes_s e^1\rangle\oplus (2e^0\otimes_s T^*\BS^2)\oplus \langle e^1e^1\rangle\oplus (2e^1\otimes_s T^*\BS^2)\oplus S^2\,T^*\BS^2.
\end{split}
\end{equation}

In this section, we are interested in the subprincipal operators of $\Box_{g,1}$ and $ L$ at the trapped set

\begin{equation}
K=\{r=r_p, \xi=0, \sigma^2=\mu_{b_0}r^{-2}\abs{\eta}^2\},\quad \mbox{where}\quad \pa_r(r\alpha^{-1})(r_p)=0,\quad \abs{\eta}^2=\abs{\eta}^2_{\sg^{-1}}.
\end{equation}

Therefore, at the trapping $K$, we have

\[
\]

\[
\sigma^2H_{\alpha^{-2}}-	H_{\alpha^{2}\xi^2+r^{-2}\abs{\eta}^2}=-r^{-2}H_{\abs{\eta}^2}.
\]

Let $x^0=t$, and $x'=(r,\omega)$ be coordinates on $X$ (independent of $t$). We let Greek indices $\mu,\nu,\lambda,\ldots$ run from $0$ to $3$. Moreover, the canonical dual variables $\xi_0=:\sigma$ and $\xi'=(\xi,\eta)$ on the fibers of $T^*\mathcal{M}$ are indexed by decorated Greek indices $\wt\mu$ running from $0$ to $3$. For a section $u$ of $\pi^*T^*\mathcal{M}$, we have

\[
\nabla^{\pi^*T^*M}_\mu u_\nu = \nabla^{M}_\mu u_\nu, \quad \nabla^{\pi^*T^*M}_{\wt\mu} u_\nu = \pa_{\wt\mu}u_\nu.
\]

§.§.§ Calculation of the subprincipal operator of $\Box_{g,1}$

According to Proposition <ref> and using the fact that $G=-\alpha^{-2}\sigma^2+\alpha^2\xi^2+r^{-2}\abs{\eta}^2$, a direct calculation implies that at trapping $K=\{r=r_p,\xi=0, \sigma^2=\mu_{b_0}r^{-2}\abs{\eta}^2\}$

\begin{equation}
\label{EqSDSSubpr}
i S_\mathrm{sub}(\Box_{g,1})
2\alpha^{-2}\sigma\pa_t - r^{-2}H_{|\eta|^2} & 0 & 0\\
0	& 2\alpha^{-2}\sigma\pa_t - r^{-2}H_{|\eta|^2} &  0 \\
0 & 0 & 2\alpha^{-2}\sigma\pa_t - r^{-2}\nabla^{\pi_{\BS^2}^*T^*\BS^2}_{H_{|\eta|^2}}
\end{pmatrix}+S_{(1)}
\end{equation}

\begin{equation}
\label{EqSDSSubprzero}
0 & -2r^{-1}\sigma & 0\\
-2r^{-1}\sigma	& 0&  2\alpha r^{-3}\iota_\eta \\
0 & -2\alpha r^{-1}\eta & 0
\end{pmatrix}.
\end{equation}

For simplicity of calculation, we further decompose $\pi^*T^*\mathcal{M}\to T^*\mathcal{M}$ over the trapping $K$. For $\eta\in T^*\BS^2\setminus o$, we define $\eta^\perp:=\sstar\eta$ and

\[
\widehat{\eta}  :=\alpha\sigma^{-1}\eta ,\quad\widehat{\eta}^\perp=\alpha\sigma^{-1}\eta^\perp.
\]

Then we write

\begin{equation}
\label{EqESGTrapMlSplitSph}
\begin{gathered}
\pi_{\BS^2}^*T^*\BS^2= \langle \widehat\eta \rangle \oplus \langle\widehat\eta^\perp\rangle,
%\quad \pi_{\BS^2}^*\Lambda^2 T^*\BS^2 = \langle\widehat\eta\wedge\widehat\eta^\perp\rangle,
\\
\pi_{\BS^2}^*S^2T^*\BS^2 = \langle \widehat\eta\widehat\eta \rangle \oplus \langle 2\widehat\eta\widehat\eta^\perp\rangle \oplus \langle\widehat\eta^\perp\widehat\eta^\perp\rangle,
\end{gathered}
\end{equation}

and this induces the following decompositions

\begin{equation}
\label{EqESGTrapMlSplit}
\begin{split}
\pi^* T^*M &= \langle e^0\rangle \oplus \Bigl(\langle e^1\rangle \oplus \bigl(\langle \widehat\eta \rangle \oplus \langle\widehat\eta^\perp\rangle\bigr)\Bigr), \\
%	\pi^*\Lambda^2 T^*M &= \Bigl(\langle e^0\wedge e^1\rangle \oplus\bigl(\langle e^0\wedge\widehat\eta \rangle \oplus\langle e^0\wedge\widehat\eta^\perp\rangle\bigr)\Bigr) \\
%	&\hspace{6em} \oplus\bigl(\langle e^1\wedge\widehat\eta\rangle \oplus \langle e^1\wedge\widehat\eta^\perp\rangle \bigr) \oplus \langle\widehat\eta\wedge\widehat\eta^\perp\rangle, \\
\pi^*S^2 T^*M &= \Bigl(\langle e^0e^0\rangle \oplus \bigl(\langle 2e^0e^1\rangle \oplus \bigl( \langle 2e^0\widehat\eta\rangle \oplus \langle 2e^0\widehat\eta^\perp\rangle \bigr)\bigr)\Bigr) \\
&\hspace{6em} \oplus \Bigl(\langle e^1e^1\rangle \oplus \bigl(\langle 2e^1\widehat\eta\rangle \oplus \langle 2e^1\widehat\eta^\perp\rangle \bigr)\Bigr) \\
&\hspace{6em} \oplus \bigl(\langle\widehat\eta\widehat\eta\rangle \oplus \langle 2\widehat\eta\widehat\eta^\perp\rangle \oplus \langle\widehat\eta^\perp\widehat\eta^\perp\rangle \bigr),
\end{split}
\end{equation}

In terms of (<ref>), we have

\[
\eta=\begin{pmatrix}\alpha^{-1}\sigma\\0\end{pmatrix},\quad r^{-2}\iota_\eta=\begin{pmatrix} \alpha^{-1}\sigma& 0 \end{pmatrix}\quad \mbox{at}\quad K.
\]

Therefore, in terms of the splitting  (<ref>), the operator $S_{(1)}$  in (<ref>) at trapping $K$ is given as

\begin{equation}
\label{EqL22microK}
0 & -2 & 0&0\\
0& -2& 0&0\\
\end{pmatrix}.
\end{equation}

Now, for any fixed $\epsilon>0$, we define the change of basis matrix

\[
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & \frac{1}{\epsilon} & 0 \\
0 & -\frac{2}{\epsilon^2} & 0 & 0 \\
-\frac{4}{\epsilon^3} & 0 & -\frac{4}{\epsilon^3} & 0 \\
\end{array}
\right),
\]

and then we have

\[
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & \epsilon& 0 \\
0 & 0 & 0 & \epsilon \\
0 & 0 & 0 & 0 \\
\end{array}
\right).
\]

Since $q_1$ is a constant coefficient operator in the region where the splitting (<ref>) is well defined, in particular, in a neighborhood of the trapping $K$, it commutes with the diagonal part $iS_{\mathrm{sub}} (\Box_{g,1})-S$ of $S_{\mathrm{sub}} (\Box_{g,1})$. Equipping the 1-form bundle over $M$ with the Hermitian inner product $B_0^1$ which is given by an identity matrix in the splitting (<ref>), $q_1S_\mathrm{sub}(\Box_{g,1})q_1^{-1}$ has imaginary part (with respect to $B_0$) of size $\mathcal{O}(\epsilon)$. Put differently, $S_{\mathrm{sub}}(\Box_{g,1})$ has imaginary part of size $\mathcal(\epsilon)$ relative to the Hermitian inner product $b:=B^1_0(q\cdot,q\cdot)$, which is the symbol of a pseudodifferential inner product on $\pi^*T^*M$. To summarize,

For any $\epsilon>0$, there exists a (positive definite) $t$-independent pseudodifferential inner product $B=b(x,D)$ on $T^*\mathcal{M}$ (thus, $b$ is an inner product on $\pi^*T^*\mathcal{M}$, homogeneous of degree $0$ with respect to dilations in the base $T^*\mathcal{M}\setminus 0$), such that

\[
\sup_K |\sigma|^{-1}\left\|\frac{1}{2i}(S_\mathrm{sub}(\Box_{g,1})-S_\mathrm{sub}(\Box_{g,1})^{*b})\right\|_b \leq \epsilon,
\]

where $K$ is the trapped set. Put differently, there is an elliptic pseudodifferential operator $Q$, invariant under $t$-translations, acting on sections of $T^*\mathcal{M}$, with parametrix $Q^-$, such that relative to the ordinary positive definite inner product $B_0^1$ we have

\[
\sup_\Gamma |\sigma|^{-1}\left\|\sigma_1\left(\frac{1}{2i}(Q\Box_{g,1}Q^- - (Q\Box_{g,1} Q^-)^{*B^1_0})\right)\right\|_{B^1_0} \leq \epsilon.
\]

§.§.§ Calculation of the subprincipal operator of $L$

Using Proposition <ref> again, we see that  $S_{\mathrm{sub}}(\Box_{g,2})=i\nabla^{\pi^*S^2T^*\mathcal{M}}_{H_G}$. Since $\pi^*S^2T^*\mathcal{M}$ is simply the restriction of the product connection $\nabla^{\pi^*T^*\mathcal{M}}\otimes\nabla^{\pi^*T^*\mathcal{M}}$ to $\pi^*S^2T^*\mathcal{M}$, it follows that

\[
S_\mathrm{sub}(\Box_{g,2})(w_1 w_2) = S_\mathrm{sub}(\Box_{g,1}) w_1\cdot w_2 + w_1\cdot S_\mathrm{sub}(\Box_{g,1}) w_2
\]

for $w_1,w_2\in C^\infty(T^*\mathcal{M},\pi^*T^*\mathcal{M})$, under the splitting (<ref>), $S_\mathrm{sub}(\Box_{g,2})$ has a canonical first order part, induced by the canonical first order part of $S_\mathrm{sub}(\Box_{g,1})$ which is given in (<ref>). The $0$-order part $S_{(2)}$ of $S_\mathrm{sub}(\Box_{g,2})$ is equal to the second symmetric tensor power of the $0$-th order part of $S_\mathrm{sub}(\Box_{g,1})$ in (<ref>). To summarize,

\begin{equation}
\label{EqESGWaveSubpr}
H_{|\eta|^2} &0&0&0&0&0&\\
0&	 H_{|\eta|^2} &0&0&0&0\\
0&0& H^{\pi^*_{\BS^2}T^*\BS^2}_{|\eta|^2} &0&0&0\\
0&0&0&	 H_{|\eta|^2} &0&0\\
0&0&0&0& H^{\pi^*_{\BS^2}T^*\BS^2}_{|\eta|^2}&0\\
\end{pmatrix}+S_{(2)}
\end{equation}

\begin{equation}
\label{EqESGWaveSubprzero}
0              & -4r^{-1}\sigma & 0                 & 0              & 0                 & 0                 \\
-2r^{-1}\sigma & 0              & 2 \alpha r^{-3}i_\eta & -2r^{-1}\sigma & 0                 & 0                 \\
0              & -2 \alpha r^{-1}\eta & 0                 & 0              & -2r^{-1}\sigma    & 0                 \\
0              & -4r^{-1}\sigma & 0                 & 0              & 4 \alpha r^{-3}i_\eta & 0                 \\
0              & 0              & -2r^{-1}\sigma    & -2 \alpha r^{-1}\eta & 0                 & 2 \alpha r^{-3}i_\eta \\
0              & 0              & 0                 & 0              & -4 \alpha r^{-1}\eta    & 0
\end{pmatrix}.
\end{equation}

Again, we turn to the `microlocal splitting' (<ref>), under which we write

\begin{gather*}
\eta
\alpha^{-1}\sigma & 0 \\
0            & \frac{1}{2}\alpha^{-1}\sigma \\
0            & 0
\end{pmatrix}
\colon T^*\BS^2\to S^2T^*\BS^2, \\
\alpha^{-1}\sigma & 0               & 0 \\
0               & \alpha^{-1}\sigma & 0
\end{pmatrix}
\colon S^2T^*\BS^2\to T^*\BS^2
\end{gather*}

Therefore, in terms of (<ref>), we see that

\begin{equation}
\label{EqL11microK}
\end{pmatrix}.
\end{equation}

Let $S_L$ be the 0-th order part of $iS_\mathrm{sub}(-L)$ at $K$, with $L$ given in (<ref>). Acting on the bundle $S^2T^*\mathcal{M}\oplus T^*\mathcal{M}$, we can write $S_L$ as a $2\times 2$ block matrix,

\[
S_L = \begin{pmatrix} S_L^{11} & S_L^{12} \\ S_L^{21} & S_L^{22} \end{pmatrix}\quad \mbox{where} \quad S_L^{11}=S_{(2)},\ S^{22}_L=S_{(1)},\ S^{12}_L=i\sigma_1(L_{12}),\ S_L^{21}=i\sigma_1(L_{21})
\]

\begin{gather*}
L_{12}(\dA) = 8\tr_g^{24}(F\otimes_s d\dA) - 2 g^{-1}(F,d\dA)\,g, \quad F=dA=d(\frac{\BQ_0}{r}dt)=\frac{\BQ_0}{r^2}dt\wedge dr,\\
L_{21}(\dg) =  \tr_g^{12}(\delta_g G_g\dg\otimes F) - \frac{1}{2}\tr_g^{24}\tr_g^{35}(d^\nabla\dg\otimes F),\quad (d^\nabla\dg)_{\nu}^{\ \mu\kappa}=\nabla^{\mu}\dg_{\nu}^{\ \kappa}-\nabla_\nu^{\kappa}\dg^{\ \mu}.
\end{gather*}

In the splitting (<ref>) and

\[
\Lambda^2T^*\mathcal{M}=\langle e^0\wedge e^1\rangle\oplus \langle e^0\wedge T^*\BS^2\rangle\oplus \langle e^1\wedge T^*\BS^2 \rangle\oplus \Lambda^2T^*\BS^2,
\]

$d:T^*\mathcal{M}\to\Lambda^2T^*\mathcal{M}$ and $\tr_g^{24}(F\otimes_s (\cdot)):\Lambda^2T^*\mathcal{M}\to S^2T^*\mathcal{M}$ are given by

\[
-\sd& 0&\alpha^{-1}\pa_t\\
\end{pmatrix},\quad 2\tr_g^{24}(F\otimes_s (\cdot))=\frac{\BQ_0}{r^2}\begin{pmatrix}
\end{pmatrix},
\]
and we also have

\[ g^{-1}(F,(\cdot))g=\begin{pmatrix}
\frac{2\BQ_0}{r^2}&0&0&0\\
\end{pmatrix}:\Lambda^2T^*\mathcal{M}\to S^2T^*\mathcal{M}.
\]

Therefore, we obtain

\begin{equation}
\label{symbolofL121}
\sigma_1(L_{12})=i\frac{4\BQ_0}{r^2}
\begin{pmatrix}
0&\eta&-\alpha\xi	\\
\alpha\xi&-\alpha^{-1}\sigma&0\\
\eta&0&-\alpha^{-1}\sigma\\
-\alpha\xi r^2\sg&\alpha^{-1}\sigma r^2\sg&0
\end{pmatrix}.
\end{equation}

Turning to the `microlocal splitting' (<ref>), we write $\sg=
\sg=\abs{\eta}^{-2}\eta\eta+\abs{\eta}^{-2}\eta^\perp\eta^\perp$ and then we have

\[
\sg=\begin{pmatrix}
\end{pmatrix}\quad \mbox{at}\quad K.
\]

Therefore, in terms of the splitting (<ref>),

\begin{equation}
\label{EqL12microK}
i\sigma_1(L_{12})=r^{-1}\sigma\frac{4\BQ_0}{r\alpha}	\begin{pmatrix}
\end{pmatrix}\quad \mbox{at}\quad K.
\end{equation}

So it remains to analyze $L_{21}(\dg)$. We first compute

\begin{equation}
\label{Eqneededlateron1}
\frac12&0&0&\frac12&0&\frac{1}{2r^2}\str\\
\frac12&0&0&\frac12&0&-\frac{1}{2r^2}\str\\
\frac12r^2\sg&0&0&-\frac12r^2\sg&0&G_{\sg}
\end{pmatrix},\quad \tr_g^{12}((\cdot)\otimes F)=\begin{pmatrix}
\end{pmatrix}: T^*\mathcal{M}\to T^*\mathcal{M}
\end{equation}

\begin{equation}
\label{Eqneededlateron2}
\sigma_1(\delta_g)=i\begin{pmatrix}
\alpha^{-1}\sigma&-\alpha\xi&-r^{-2}\iota_\eta&0&0&0\\
\end{pmatrix}.
\end{equation}

Therefore, we find that

\begin{equation}
\label{EqL211}
\sigma_1(\tr_g^{12}(\delta_gG_g(\cdot)\otimes F))=i\frac{\BQ_0}{r^2}\begin{pmatrix}
\frac12\alpha\xi&-\alpha^{-1}\sigma&0&\frac12\alpha\xi&r^{-2}\iota_{\eta}&-\frac{1}{2r^2}
\alpha\xi\str\\
\end{equation}

Next, we calculate

\begin{equation}
\label{EqL212}
\sigma_1(d^{\nabla}(\cdot)\otimes F)=i\frac{2\BQ_0}{r^2}\begin{pmatrix}
\alpha\xi&-\alpha^{-1}\sigma&0&0&0&0&\\
\end{pmatrix}.
\end{equation}

In the `microlocal splitting' (<ref>), we have

\[
\str=\begin{pmatrix}
\end{pmatrix}\quad \mbox{at}\quad K.
\]

and thus we have

\begin{equation}
\label{EqL21microK}
\end{pmatrix}.
\end{equation}

Putting (<ref>), (<ref>), (<ref>) and (<ref>) together yields

\begin{equation}
\begin{array}{cccccccccccccc}
0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 \BQ' & 0 & 0 \\
-2 & 0 & 2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -2 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & -8 \BQ' & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 8 \BQ' & 0 & 0 \\
0 & 0 & -2 & 0 & -2 & 0 & 0 & 2 & 0 & 0 & -8 \BQ'& 0 & 8 \BQ' & 0 \\
0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 8 \BQ' \\
0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & -8 \BQ'& 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 \BQ'& 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 \BQ' & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 \\
\BQ'& 0 & -2 \BQ'& 0 & -\BQ' & 0 & 0 & \BQ' & 0 & \BQ' & -2& 0 & 2& 0 \\
0 & 0 & 0 & 0 & 0 & -2 \BQ' & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -2 \BQ' & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)
\end{equation}

where $\BQ'=\frac{\BQ_0}{2r\alpha}$. Since $S_L$ has the following Jordan form

\[
\left(
\begin{array}{cccccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 i \BQ' & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 i \BQ'& 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 i \BQ'& 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 i \BQ'\\
\end{array}
\right),
\]

one can choose a suitable matrix $q_L$ which has constant coefficients at $K$ such that

\begin{equation}
\begin{array}{cccccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \epsilon  & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \epsilon  & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \epsilon  & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \epsilon  & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 i \BQ'& 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 i \BQ' & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 i \BQ' & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 i \BQ' \\
\end{array}
\right).
\end{equation}

Again, equipping $S^2T^*\mathcal{M}\oplus T^*\mathcal{M}$ over $M$ with the Hermitian inner product $B_0^L$ which is given as an identity matrix in the  splitting (<ref>),

$q_LS_\mathrm{sub}(-L)q_L^{-1}$ has imaginary part (with respect to $B^L_0$) of size $\mathcal{O}(\epsilon)$. Put differently, $S_{\mathrm{sub}}(-L)$ has imaginary part of size $\mathcal(\epsilon)$ relative to the Hermitian inner product $b:=B^L_0(q_L\cdot,q_L\cdot)$, which is the symbol of a pseudodifferential inner product on $\pi^*T^*M$. To summarize,

For any $\epsilon>0$, there exists a (positive definite) $t$-independent pseudodifferential inner product $B=b(x,D)$ on $S^2T^*\mathcal{M}\oplus T^*\mathcal{M}$ (thus, $b$ is an inner product on $\pi^*(S^2T^*\mathcal{M}\oplus T^*\mathcal{M})$, homogeneous of degree $0$ with respect to dilations in the base $T^*\mathcal{M}\setminus 0$), such that

\[
\sup_K |\sigma|^{-1}\left\|\frac{1}{2i}(S_\mathrm{sub}(-L)-S_\mathrm{sub}(-L)^{*b})\right\|_b \leq \epsilon,
\]

where $K$ is the trapped set. Put differently, there is an elliptic pseudodifferential operator $Q$, invariant under $t$-translations, acting on sections of $S^2T^*\mathcal{M}\oplus T^*\mathcal{M}$, with parametrix $Q^-$, such that relative to the ordinary positive definite inner product $B_0^L$ we have

\[
\sup_\Gamma |\sigma|^{-1}\left\|\sigma_1\left(\frac{1}{2i}(Q(-L)Q^- - (Q(-L) Q^-)^{*B^L_0})\right)\right\|_{B^L_0} \leq \epsilon.
\]

Finally, we point out the relation between the operators $\mathcal{P}, \mathcal{W}, L$ and their Fourier transform (semiclassically rescaled) $h^2\widehat{\mathcal{P}}(h^{-1}z),\ h^2\widehat{\mathcal{W}}(h^{-1}z)$ and $h^2\widehat{L}(h^{-1}z)$.

Given a operator $P\in\{\Box_{g,1}, -L\}$, we define $\widehat{P}(\sigma):=e^{i\sigma t_*}Pe^{-i\sigma t_*}$ and its semiclassical rescaling $h^2\widehat{P}(h^{-1}z)$ with $h=\abs{\sigma}^{-1}$. Let $p_2=\sigma_2(P)$ and $p_{2,h}(z)=\sigma_{2,h}(h^2\widehat{P}(h^{-1}z))$. Here we also use $\xi'=(\xi,\eta) $ as the semiclassical dual variables of $x'=(r,\omega)$. Then using the correspondence $\RE z\longleftrightarrow-\sigma$, we have

\[
S_{\mathrm{sub}}(h^2\widehat{P}(h^{-1}z))=S_{\mathrm{sub}}(P)+i\alpha^{-2}\sigma\pa_t+i\IM(hz)\pa_\sigma p_2
\]

and thus

\[
\sigma_{1,h}\Big(\frac{1}{2ih}(h^2\widehat{P}(h^{-1}z)-(h^2\widehat{P}(h^{-1}z))^{*B})\Big)=\sigma_1\left(\frac{1}{2i}(P - P^{*B})\right)+ \IM(hz)\pa_\sigma p_2.
\]

Finally, we note that $\IM(hz)\pa_\sigma p_2=\IM(hz)\pa_zp_{2,h}(\RE z)=\sigma_{1,h}\Big(\frac{1}{2ih}(h^2\widehat{\Box_{g,0}}(h^{-1}z)-(h^2\widehat{\Box_{g,0}}(h^{-1}z))^*)\Big)$.

§ CALCULATION OF THE SUBPRINCIPAL OPERATOR AT RADIAL POINTS AT EVENT HORIZON

In this section, we calculate the threshold regularity at the radial points at event horizon. To this end, we discuss the subprincipal operator of $\mathcal{P}_{b_0,\gamma}, \mathcal{W}_{b_0,\gamma}$ and $L	_{b_0,\gamma}$ at radial points at event horizon. Here we again drop the notations $b_0, \gamma$.

We recall that modulo terms of order $0$, we have

\begin{equation}
\end{equation}

\begin{equation}
\label{EqESGLRecall}
-L(\dg,\dA) \equiv \Bigl( \Box_g\dg- 2(\tilde{\delta}_g^*-\delta_g^*)\delta_g G_g\dg -2\delta^*_g(\tilde{\delta}_g-\delta_g)G_g\dg+ L_{12}(\dA),\  \Box_g\dA + L_{21}(\dg) \Bigr).
\end{equation}

Near the radial points at the event horizon, we use the coordinates $(t_0,r,\omega)$ in (<ref>), in which the Reissner-Nordström metric $g=g_{b_0}, b_0=(\Bm_0,0,\BQ_0)$ takes the form

\[
\]

Writing the covectors as

\[
\sigma\,dt_0 + \xi\,dr + \eta,\quad \eta\in T^*\BS^2,
\]

we have $G=g^{-1}(\sigma\,dt_0 + \xi\,dr + \eta,\sigma\,dt_0 + \xi\,dr + \eta)=2\sigma\xi+\mu_{b_0}\xi^2+r^{-2}\abs{\eta}$ and

\[
\]

The radial points at event horizon is given as

\begin{equation}
L_\pm=\{(t_0, \ehRN, \omega; 0,\xi,0)\mid \pm\xi>0\}.
\end{equation}

Then at $L_\pm$, we have

\begin{equation}
\end{equation}

\begin{equation}
\kappa=\frac12\mu_{b_0}'(\ehRN).
\end{equation}

being the surface gravity. We keep the term $2\mu_{b_0}\xi\pa_r$ here as it is needed in the computation of the skew adjoint part.

The basis $e^0=\alpha dt, e^1=\alpha^{-1}dr$ are not defined at $r=\ehRN$. To deal with this issue, we follow the method in <cit.>. Concretely, we relate the basis $e^0=\alpha dt, e^1=\alpha^{-1}dr$ to a smooth trivialization of $T^*M$ and $S^2 T^*M$ as follows: writing a 1-form $u$ in the static region $\mathcal{M}$ as

\[
u = u_N\,e^0 + u^T_N\,e^1 + u^T_T = \wt u_N\,dt_0 + \wt u^T_N\,dr+\wt u^T_T,
\]

where $u^T_T$ and $\wt u^T_T$ are 1-forms on $\BS^2$, we see that

\begin{equation}
\label{EqESGRadTrans1}
\begin{pmatrix} u_N \\ u^T_N \\ u^T_T \end{pmatrix} = \mathscr{C}^{(1)} \begin{pmatrix} \wt u_N \\ \wt u^T_N \\ \wt u^T_T \end{pmatrix},
\quad
\mathscr{C}^{(1)} = \begin{pmatrix} \alpha^{-1} & 0 & 0 \\ \alpha^{-1} & \alpha & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\end{equation}

The following smooth (at $r=\ehRN$) bundle splitting

\begin{equation}
\label{EqESGRadBundle1}
T^*M = \langle dt_0 \rangle \oplus \langle dr \rangle \oplus T^*\BS^2
\end{equation}

\begin{equation}
\label{EqESGRadBundle2}
S^2T^*M= \langle dt_0^2 \rangle \oplus \langle 2dt_0\,dr\rangle\oplus (2 dt_0\cdot T^*\BS^2) \oplus \langle dr^2 \rangle\oplus (2 dr\cdot T^*\BS^2) \oplus S^2T^*\BS^2.
\end{equation}

Then one can write a section $u$ of $S^2T^*M$ as

\begin{align*}
u &= u_{NN}\,e^0e^0 + 2 u_{NTN}\,e^0e^1 + 2e^0\cdot u_{NTT} + u^T_{NN}\,e^1e^1 + 2e^1\cdot u^T_{NT} + u^T_{TT} \\
&= \wt u_{NN}\,dt_0^2 + 2 \wt u_{NTN}\,dt_0\,dr + 2dt_0\cdot \wt u_{NTT} + \wt u^T_{NN}\,dr^2 + 2dr\cdot \wt u^T_{NT} + \wt u^T_{TT},
\end{align*}

and the transition matrix is defined by

\begin{equation}
\label{EqESGRadTrans2}
\begin{pmatrix} u_{NN} \\ u_{NTN} \\ u_{NTT} \\ u^T_{NN} \\ u^T_{NT} \\ u^T_{TT} \end{pmatrix} = \mathscr{C}^{(2)} \begin{pmatrix} \wt u_{NN} \\ \wt u_{NTN} \\ \wt u_{NTT} \\ \wt u^T_{NN} \\ \wt u^T_{NT} \\ \wt u^T_{TT} \end{pmatrix},
\quad
\mathscr{C}^{(2)} = \begin{pmatrix} \alpha^{-2} & 0 & 0 & 0 & 0 & 0 \\ \alpha^{-2} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha^{-1} & 0 & 0 & 0 \\ \alpha^{-2} & 2 & 0 & \alpha^2 & 0 & 0 \\ 0 & 0 & \alpha^{-1} & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}.
\end{equation}

Then using the coordinates $(t_0,r,\omega)$ and splitting (<ref>), we again use Proposition <ref> to compute

\begin{equation}
\label{EqL22Rad}
\end{pmatrix}\quad \mbox{at} \quad L_\pm
\end{equation}

\begin{equation}
\label{EqL11Rad}
\end{pmatrix}\quad \mbox{at} \quad L_\pm.
\end{equation}

Next, we turn to the calculation of $S_{\mathrm{sub}}(L_{12})$ and $S_{\mathrm{sub}}(L_{21})$ at $L_\pm$. We first calculate the symbols at $N^*\{r=r_0\}=\{t,r_0, \omega, 0,\xi,0\}$, for $r_0\in(\ehRN,\infty)$ close to $\ehRN$, in the static coordinates $(t,r,\omega)$ and bundle splittings (<ref>). Then we conjugate by $\mathscr{C}^{(1)}$ or $\mathscr{C}^{(2)}$ and evaluate at $\mu_{b_0}=0$. Concretely, according to (<ref>), in terms of the splitting (<ref>), we have at $N^*\{r=r_0\}$

\[
\alpha\xi&0&0\\
\alpha\xi r^2\sg&0&0
\end{pmatrix}.
\]

So in the smooth splittings (<ref>) and (<ref>) (multiplying from the left by $(\mathscr{C}^{(2)})^{-1}$ and from the right by $\mathscr{C}^{(1)}$ and evaluating at $\mu=0$),

\begin{equation}
\label{EqESGRadL12}
=\frac{4\BQ_0}{r^2} \xi
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \\
r^2\sg& 0 & 0 \\
\end{array}
\right)\quad \mbox{at}\quad L_\pm.
\end{equation}

Using (<ref>) and (<ref>), in terms of the splitting (<ref>), we have at $N^*\{r=r_0\}$

\[
\frac12\alpha\xi&0&0&-\frac12\alpha\xi&0&\frac{1}{2r^2}\alpha\xi\str\\
\end{pmatrix}.
\]

Again multiplying this from the left by $(\mathscr{C}^{(1)})^{-1}$ and from the right by $\mathscr{C}^{(2)}$ and evaluating at $\mu=0$ give rise to $i\sigma(L_{21})$ in the smooth splittings (<ref>) and (<ref>)

\begin{equation}
\label{EqESGRadL21}
i\sigma(L_{21})	=\frac{\BQ_0}{2r^2}\xi\left(
\begin{array}{cccccc}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & -\frac{1}{r^2}\str \\
0 & 0 & 2 & 0 & 0 & 0 \\
\end{array}
\right)\quad\mbox{at}\quad L_\pm.
\end{equation}

Finally, it remains to calculate the subprincipal operators of $(\tilde{\delta}_g^*-\delta_g^*)\delta_g G_g\dg$ and  $\delta^*_g(\tilde{\delta}_g-\delta_g)G_g\dg$. First, according to (<ref>) and (<ref>), we see that in the static splitting (<ref>)

\[
\frac12\alpha\xi&0&0&\frac12\alpha\xi&0&-\frac{\alpha\xi}{2r^2}\str\\
\end{pmatrix}\]

at $N^*\{r=r_0\}$, and thus

\[
\begin{array}{cccccc}
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0& 0 & -\frac{1}{r^2}\str \\
0 & 0 & 2 & 0 & 0 & 0 \\
\end{array}
\right)\quad \mbox{at}\quad L_\pm.
\]

Also, in the smooth splitting, we have at $L_\pm$

\[
\end{pmatrix}.
\]

Since $\tilde{\delta}_g^*-\delta_g^*= 2\gamma\mathfrak{c}\otimes_s(\cdot)-\frac12\gamma g^{-1}(\mathfrak{c},\cdot)g$ and $\tilde{\delta}_g-\delta_g=2\gamma\iota_{\mathfrak{c}}^g(\cdot)-\frac12\gamma\mathfrak{c}\tr_g(\cdot)$ where $\mathfrak{c}=\tilde{\chi}(r)(dt_0-\mathfrak{b}dr)$ with $\tilde{\chi}(r)=1$ near event horizon and $\mathfrak{b}>2$, it follows that in terms of the smooth splitting (<ref>) and (<ref>)

\[
\tilde{\delta}_g^*-\delta_g^*=\begin{pmatrix}
-\frac12\mathfrak{b}\gamma	&\frac12\gamma&0\\
\frac12\mathfrak{b}\gamma r^2\sg&-\frac12\gamma r^2\sg&0
\end{pmatrix},\quad \tilde{\delta}_g-\delta_g=\begin{pmatrix}
-2\mathfrak{b}\gamma&\gamma&0&0&0&-\frac12\gamma r^{-2}\str\\
0&-\mathfrak{b}\gamma&0&2\gamma&0&\frac12\mathfrak{b}\gamma r^{-2}\str\\
\end{pmatrix}
\]

at $L_\pm$.
Therefore, in terms of the smooth splitting, we have at $L_\pm$

\begin{align}
\label{EqL11sub1}
i\sigma_1((\tilde{\delta}_g^*-\delta_g^*)\delta_g G_g)&=\frac{1}{2}\gamma\xi\begin{pmatrix}
\mathfrak{b}
\end{pmatrix},\\
\label{EqPsub}
i\sigma_1(\delta_g G_g(\tilde{\delta}_g^*-\delta_g^*))&=\frac{1}{2}\gamma\xi\begin{pmatrix}
\end{pmatrix},\\
\label{EqL11sub2}
\end{pmatrix},\\
\label{EqWsub}
\mathfrak{b}&-4&0\\
\end{pmatrix}.
\end{align}

In the splitting (<ref>), combining (<ref>) with (<ref>) and (<ref>) yields

\[

\[
\mathfrak{b}\gamma&-\gamma-2\kappa&0\\
\end{pmatrix},\quad S_{\mathcal{W}}=\xi\begin{pmatrix}
\end{pmatrix}.
\]

In the splitting (<ref>), equipping $T^*M$ with the Hermitian inner product $1\oplus1\oplus\sg^{-1}$,  the first three terms of $S_{\mathrm{sub}}(-2\mathcal{P}), S_{\mathrm{sub}}(-2\mathcal{W})$ are formally self adjoint with respect to the symplectic volume form on $T^*M$, and the last term $S_{\mathcal{P}}$ resp. $S_{\mathcal{W}}$ of $iS_{\mathrm{sub}}(-2\mathcal{P})$ resp. $iS_{\mathrm{sub}}(-2\mathcal{W})$, multiplied by $\xi^{-1}$, has eigenvalues

\begin{equation}
\label{EqeigenPW}
2(\kappa-2\gamma),\ -2\kappa-\gamma,\ -2\gamma,\quad \mbox{resp.}\quad -2(\kappa-2\gamma),\ 2\gamma,\ 2\kappa+\gamma.
\end{equation}

Now we turn to the calculation $L$. Using the following decomposition

\[
S^2 T^*\BS^2 = \langle r^2\sg \rangle \oplus \sg^\perp,
\]

we have

\[
r^2\sg = \begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad r^{-2}\str = \begin{pmatrix} 2 & 0 \end{pmatrix}.
\]

This further induces the splitting

\begin{equation}
\label{EqsplitRad}
\begin{split}
&\langle dt_0^2 \rangle \oplus \langle 2 dt_0\,dr \rangle \oplus (2 dt_0\cdot T^*\BS^2) \oplus \langle dr^2 \rangle \oplus (2 dr\cdot T^*\BS^2) \oplus \langle r^2\sg \rangle \oplus \sg^\perp \\
& \qquad \oplus \langle dt_0\rangle \oplus \langle dr \rangle \oplus T^*\BS^2.
\end{split}
\end{equation}

In the above splitting (<ref>), putting (<ref>),(<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) together yields

\[
\]

\[
\end{pmatrix}
\]

with $\BQ''=\BQ_0r^{-2}$. In the splitting (<ref>), equipping $S^2T^*M\oplus T^*M$ with the Hermitian inner product $1\oplus1\oplus\sg^{-1}\oplus1\oplus\sg^{-1}\oplus 1\oplus1\oplus 1\oplus 1\oplus \sg{-1}$, the first three terms of $S_{\mathrm{sub}}(-L)$ are formally self adjoint with respect to the symplectic volume form on $S^2T^*M\oplus T^*M$, and the last term $S_L$ of $iS_{\mathrm{sub}}(-L)$, multiplied by $\xi^{-1}$ has eigenvalues

\begin{equation}
0,\ 0,\ -2\kappa,\ -2\kappa,\ 2\kappa,\ -4(\kappa-\gamma),\ 2(\kappa-\gamma),\ 4(\kappa-\gamma),\ -\gamma,\ 2\gamma.
\end{equation}

Let $P\in\{-2\mathcal{P}, -2\mathcal{W},-L\}$. Then the quantity $\tilde{\beta}$ for $P$ is defined as

\[
\abs{\xi}^{-1}\frac{1}{2i}\Big(S_{\mathrm{sub}}(P)-\big(S_{\mathrm{sub}}(P)\big)^*\Big)=\mp\tilde{\beta}\beta_0\quad \mbox{at}\quad L_\pm
\]

where $\beta_0=2\kappa$. Recall that $\beta_{\sup}=\sup\tilde{\beta}$ and $\beta_{\inf}=\inf \tilde{\beta}$. Then for sufficiently small $\gamma>0$, we have

\begin{equation}
\label{EqRadReg}
\begin{gathered}
-2\mathcal{P}:\  \beta_{\sup}=1-\frac{2\gamma}{\kappa},\quad \beta_{\inf}=-1-\frac{\gamma}{2\kappa};\\
-2\mathcal{W}:\  \beta_{\sup}=1+\frac{\gamma}{2\kappa},\quad \beta_{\inf}=-1+\frac{2\gamma}{\kappa};\\
-L:\  \beta_{\sup}=2-\frac{2\gamma}{\kappa},\quad \beta_{\inf}=-2+\frac{2\gamma}{\kappa}.
\end{gathered}
\end{equation}

We again point out the relation between the operators $\mathcal{P}, \mathcal{W}, L$ and their Fourier transform $\widehat{\mathcal{P}}(\sigma),\ \widehat{\mathcal{W}}(\sigma)$ and $\widehat{L}(\sigma)$.

Given a operator $P\in\{-2\mathcal{P},-2\mathcal{W}, -L\}$, we define $\widehat{P}(\sigma):=e^{i\sigma t_*}Pe^{-i\sigma t_*}$ with $\sigma\in\BC$. The radial points set of $\widehat{P}$ (which we still denote by $L_\pm$) is given by $L_\pm=\{(\ehRN,\omega; \xi,0)\mid \pm\xi>0\}$. Then we have

\[
\frac{1}{2i}\Big(S_{\mathrm{sub}}(\widehat{P})-\big(S_{\mathrm{sub}}(\widehat{P})\big)^*\Big)=\frac{1}{2i}\Big(S_{\mathrm{sub}}(P)-\big(S_{\mathrm{sub}}(P)\big)^*\Big)+\sigma_1(\frac{\widehat{\Box_{g,0}}(\sigma)-(\widehat{\Box_{g,0}}(\sigma))^*}{2i})	\quad \mbox{at}\quad L_\pm.
\]

Finally, according to the discussion around (<ref>), the calculation of the threshold regularity at radial points at event horizon for the semiclassical operator $h^2\widehat{P}(h^{-1}z)$ is equal to that of $\widehat{P}(\sigma)$.

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