# Monodromy Approach to Pair Production of Charged Black Holes and Electric
Fields

Chiang-Mei Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Chungli 32001, Taiwan  Center for High Energy and High Field
Physics (CHiP), National Central University, Chungli 32001, Taiwan  Toshimasa
Ishige<EMAIL_ADDRESS>Graduate School of Science and Engineering, Chiba
University, Chiba-shi 263-8522, Japan  Sang Pyo Kim<EMAIL_ADDRESS>Department of Physics, Kunsan National University, Kunsan 54150, Korea  Asia
Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea  Akitoshi
Takayasu<EMAIL_ADDRESS>Faculty of Engineering, Information and
Systems, University of Tsukuba, Ibaraki 305-8573, Japan  Chun-Yu Wei
<EMAIL_ADDRESS>Department of Physics, National Central University,
Chungli 32001, Taiwan

###### Abstract

We advance the monodromy method that makes use of the regular singular points
of wave equations to find the pair production, absorption cross section and
quasi-normal modes in background fields. We find the mean number of pairs
produced in background fields whose mode equations belong to the Riemann
differential equation and apply the method to the three particular cases: (i)
charges near the horizon of near-extremal black holes, (ii) charges with
minimal energy under the static balance in nonextremal charged black holes,
and (iii) charges in the Sauter-type electric fields. We then compare the
results from the monodromy with those from the exact wave functions in terms
of the hypergeometric functions with three regular singular points. The
explicit elaboration of monodromy and the model calculations seem to reveal
evidences that the monodromy may provide a practical technique to study the
spontaneous pair production in general black holes and electromagnetic fields.

##  I Introduction

One of nonperturbative aspects of quantum field theory is spontaneous particle
production from background fields, two of whose most prominent phenomena are
the (Sauter-)Schwinger mechanism in electromagnetic fields [1, 2] and the
Hawking radiation in black holes [3]. The physical concept behind the particle
production is that the background fields change the vacuum such that the out-
vacuum is superposed of multiparticle states of the in-vacuum and vice versa
[4]. The Klein-Gordon equation for a scalar field, for instance, has been
solved in a few background fields [5], though linear equations. Indeed it has
been a challenge for a long time either to directly find the exact wave
functions in terms of the special functions or to develop some approximation
schemes for the wave functions, such as the WKB method [6] or Borel-summed WKB
method [7].

Particle production has been an interesting topic in cosmology, in particular,
in expanding universes [8] (for a recent review, see [9]). Recently the
Schwinger pair production of charged particles and antiparticles by a strong
electric field has attracted much attention because ultra-strong lasers have
been proposed to reach field strengths near the Schwinger field in the near
future and spontaneous production of electrons and positrons will be a direct
test of QED in strong field region (for a review on astrophysical
applications, see [10] and for a recent review, see [11]). Charged black holes
are an arena in which both the Schwinger mechanism and the Hawking radiation
get intertwined to spontaneously emit charges.

The field equation for a charged scalar in charged black holes, such as the
Reissner-Nordström (RN) and Kerr-Newman (KN) black holes in an asymptotically
flat or (anti-)de Sitter space (A)dS, has not been exactly solved yet in terms
of special functions in the global covering space. A conventional wisdom has
been to solve the field equation in the near-horizon region and the asymptotic
region and to connect those wave functions. Another method is to use the
enhanced symmetry of background geometry in some limits. The (near-)extremal
black holes have a near-horizon geometry whose enhanced symmetry allows us to
exactly solve wave functions. Two of us (CMC, SPK) have studied spontaneous
production of charged particles from (near-)extremal RN or KN black holes in
the asymptotically flat or (A)dS spaces [12, 13, 14, 15, 16, 17, 18].

To understand the emission of charges from charged black holes, one has to
solve the Klein-Gordon or Dirac equation in the RN or KN black holes. However,
the Klein-Gordon equation in nonextremal charged black holes is a confluent
Heun equation, with three poles including a double pole at infinity, which
generally cannot be analytically solved [19]. In this paper we will study an
alternative technique, the so-called monodromy that makes us directly compute
the scattering coefficients from the information of poles without knowing the
analytic solutions [20, 21]. This technique may pass over the formidable task
of finding the wave functions in terms of the special functions but give us
useful information necessary for the mean number or absorption cross section
of emitted particles from black holes. Moreover, it can also be applied to
calculate the quasi-normal modes by imposing a proper boundary condition.

As the first step in this direction, we find the mean number of pairs produced
in background fields whose mode equations belong to the Riemann differential
equation. And then we consider the three special cases of the Riemann
equation: (i) charges in near-extremal charged black holes, (ii) charges with
minimal energy under a static balance in nonextremal black holes, and (iii)
charges in electric fields that belong to a generalized Pöschl-Teller
potential. All the three cases have wave functions in terms of the
hypergeometric functions with three regular singular points and transformation
formulas between different regions and will provide one with models to test
the monodromy approach. The phase-integral method that also makes use of the
regular singular points [22] gives the leading term for the pair production
from near-extremal charged black holes [12, 13, 14] and explains the Stokes
phenomenon [23]. On the other hand, the monodromy method recovers the exact
results for those models studied in this paper. Thus we argue that the
monodromy seems a completion of the phase-integral method and may work for
more general cases.

In section II, we introduce the definition and properties of monodromy and
impose the boundary conditions for pair production and absorption cross
section. And we find the mean number of pair production in background fields
whose mode equations are described by the Riemann differential function. In
section III, we derive, case by case, the pair production from monodromy when
the governing equations are the hypergeometric function with three regular
singular points. One specific case is pair production of scalar charges from
the near-horizon region of near-extremal RN black holes [12], and the other
case is the scalar charges with minimal energy under a static balance in
nonextremal RN black holes [19], which is also given by the Riemann
differential equations. In section IV, we apply the monodromy to charges in
electric fields of the generalized Pöschl-Teller potential, one of which is
the Sauter-type electric field. The monodromy confirms the pair production
formula from the exact solutions of the wave equation. In section V we discuss
the features and applications of the monodromy in more general cases.

##  II Monodromy and Boundary Conditions

In this section, we are going to give a concise introduction to monodromy and
how to apply this technique to study pair production mechanisms. Such ideas
had been discussed in [21], but the explanations here are more rigorous and
complete, in particular several obscure issues during the physical
applications are clarified. Thus the application turns out to be just a
straightforward process.

###  II.1 Monodromy Representation of Fundamental Group

Consider a second order ordinary differential equation (ODE) of the form

$\frac{d^{2}}{dr^{2}}R(r)+U(r)\frac{d}{dr}R(r)+V(r)R(r)=0,$ (1)

where $U(r)$ and $V(r)$ are meromorphic functions in $\mathbb{C}$ with finite
poles. Let $r_{i}$ ($i=1,2,\dots,m$) be the whole set of poles of either
$U(r)$ or $V(r)$. If both $(r-r_{i})U(r)$ and $(r-r_{i})^{2}V(r)$ are
holomorphic at $r_{i}$, then the singularity $r_{i}$ is called regular,
otherwise irregular.

The ODE (1) can be extended to the Riemann sphere, i.e.
$\mathbb{P}=\mathbb{C}\cup\\{\infty\\}$. Setting $s=1/r$ and $S(s)=R(r)$, the
ODE (1) is transformed into

$\begin{split}&\frac{d^{2}}{ds^{2}}S(s)+U_{\infty}(s)\frac{d}{ds}S(s)+V_{\infty}(s)S(s)=0,\\\
U_{\infty}(s)&=2s^{-1}-s^{-2}U(1/s),\qquad
V_{\infty}(s)=s^{-4}V(1/s).\end{split}$ (2)

Note that $r_{i}\in\mathbb{C}\setminus\\{0\\}$ is a pole of $U(r)$ or $V(r)$
if and only if $s_{i}=1/r_{i}\in\mathbb{C}\setminus\\{0\\}$ is a pole of
$U_{\infty}(s)$ or $V_{\infty}(s)$. If $s=0$ is a pole, then $r_{m+1}=\infty$
is a singularity of (1) and the number of singularities in $\mathbb{P}$
becomes $n=m+1$, otherwise $n=m$. If all of the singularities
$r_{1},\dots,r_{n}$ are regular, then the ODE (1) is called a _Fuchsian
differential equation_. Let
$\mathbb{X}=\mathbb{P}\setminus\\{r_{1},r_{2},\dots,r_{n}\\}$. It is clear
that the solutions of the ODE (1) are holomorphic in $\mathbb{X}$, since they
are differentiable everywhere in $\mathbb{X}$. However, since $\mathbb{X}$ is
not simply connected, a solution of the ODE (1) may be a multivalued
holomorphic function in $\mathbb{X}$. Such multivaluedness of the solutions is
characterized by the _monodromy_ which reflects the topological property of
$\mathbb{X}$.

Figure 1: A sketch of the loops on the Riemann sphere $\mathbb{P}$: The loops
$\Gamma_{1},\Gamma_{2}$, and $\Gamma_{3}$ from the base point $r_{0}$ are
enclosing the singular points $r_{1},r_{2}$, and $r_{3}$, respectively.
$\Gamma_{1}\cdot\Gamma_{2}\cdot\Gamma_{3}$ continuously deforms to $r_{0}$ and
yields the defining relation (3).

Fix a base point $r_{0}\in\mathbb{X}$. Let $\Gamma_{i}$ ($i=1,2,\dots,n$) be a
loop in $\mathbb{X}$ initiated at $r_{0}$ enclosing only $r_{i}$
counterclockwise. The product of loops $\Gamma_{j}\cdot\Gamma_{i}$ is defined
to be a loop going firstly along $\Gamma_{i}$ then $\Gamma_{j}$. Moreover,
$\pi_{1}(\mathbb{X},r_{0})$, the fundamental group of $\mathbb{X}$ with the
base point $r_{0}$, is generated by $[\Gamma_{1}],\dots,[\Gamma_{n}]$ where
$[\Gamma_{i}]$ denotes the homotopy class of loops for $r_{0}$ containing
$\Gamma_{i}$. Since the base point $r_{0}$ is a constant loop (i.e., the image
of the constant map $[0,1]\to\\{r_{0}\\}$), $[r_{0}]$ is the identity element
of $\pi_{1}(\mathbb{X},r_{0})$ denoted by $1$. We can construct
$\Gamma_{1},\dots,\Gamma_{n}$ to be arranged clockwise in this order. Then,
the composite loop $\Gamma_{1}\cdot\Gamma_{2}\cdots\Gamma_{n}$, which is
homotopic to $r_{0}$ in $\mathbb{X}$ (i.e., it continuously deforms to $r_{0}$
in $\mathbb{X}$ as illustrated in Figure 1 for the case of $n=3$), gives the
following _defining relation_ of $\pi_{1}(\mathbb{X},r_{0})$:

$[\Gamma_{1}]\cdot[\Gamma_{2}]\cdots[\Gamma_{n}]=1.$ (3)

Let $\mathbb{D}$ be a simply connected open set in $\mathbb{X}$ with
$r_{0}\in\mathbb{D}$. Since $\mathbb{D}$ is simply connected, the _monodromy
theorem_ in the complex analysis implies that all of the solutions of the ODE
(1) are single-valued in $\mathbb{D}$. In virtue of the existence and
uniqueness of solutions for the initial value problem, the map from the space
of initial values at $r_{0}$, say
$\left[R(r_{0}),R^{\prime}(r_{0})\right]^{T}=\mathbf{R}_{0}$ where
$R^{\prime}=dR/dr$, to the solution space of the ODE (1) over $\mathbb{D}$

$\Phi:\mathbb{C}^{2}\to\mathrm{Sol}(\mathbb{D}),\qquad\mathbf{R}_{0}\mapsto
R(r),$ (4)

is a $\mathbb{C}$-linear isomorphism. Let $\mathbf{R}_{0}^{1}$ and
$\mathbf{R}_{0}^{2}$ be $\mathbb{C}$-linearly independent vectors in
$\mathbb{C}^{2}$, then $\Phi(\mathbf{R}_{0}^{1})$ and
$\Phi(\mathbf{R}_{0}^{2})$ are $\mathbb{C}$-linearly independent solutions in
$\mathrm{Sol}(\mathbb{D})$. Suppose that $\Gamma$ is the image of the smooth
map

$r_{\Gamma}:[0,1]\to\mathbb{X},\qquad t\mapsto
r_{\Gamma}(t)\quad\text{with}\quad r_{\Gamma}(0)=r_{\Gamma}(1)=r_{0}.$ (5)

There exist $0<t_{1}<t_{2}<1$ such that
$r_{\Gamma}\left([0,t_{1}]\right)\subset\mathbb{D}$ and
$r_{\Gamma}\left([t_{2},1]\right)\subset\mathbb{D}$. As for
$r_{\Gamma}([t_{1},t_{2}])$, it is assumed to be any path in $\mathbb{X}$
connecting $r_{\Gamma}(t_{1})$ and $r_{\Gamma}(t_{2})$. In the course of the
analytic continuation along $\Gamma$, the solutions $\Phi(\mathbf{R}_{0}^{1})$
and $\Phi(\mathbf{R}_{0}^{2})$ stay as they are while in
$r_{\Gamma}\left([0,t_{1}]\right)$, and after the analytic continuation along
$r_{\Gamma}\left([t_{1},t_{2}]\right)$ when finally going into
$r_{\Gamma}\left([t_{2},1]\right)$, they consequently become respective
$\mathbb{C}$-linear sums of $\Phi(\mathbf{R}_{0}^{1})$ and
$\Phi(\mathbf{R}_{0}^{2})$. Thus, the _monodromy matrix_ $\mathbf{M}_{\Gamma}$
along the loop $\Gamma$ with respect to
$[\Phi(\mathbf{R}^{1}_{0}),\Phi(\mathbf{R}^{2}_{0})]$ is defined by a
nonsingular matrix such that

$\left[\Phi(\mathbf{R}_{0}^{1}),\Phi(\mathbf{R}_{0}^{2})\right]\,\mathbf{M}_{\Gamma}=\Gamma_{\ast}\left[\Phi(\mathbf{R}^{1}_{0}),\Phi(\mathbf{R}^{2}_{0})\right],$
(6)

where $\Gamma_{\ast}$ denotes the analytic continuation along $\Gamma$.

Let us transform (1) into the _Pfaffian form_

$\frac{d}{dr}\begin{bmatrix}R\\\ R^{\prime}\end{bmatrix}=\begin{bmatrix}0&1\\\
-V&-U\end{bmatrix}\begin{bmatrix}R\\\
R^{\prime}\end{bmatrix}\quad\iff\quad\frac{d}{dr}\mathbf{R}=\mathbf{A}\mathbf{R}.$
(7)

Using the discretization of (7), the _path-ordered-exponential_ of
$\mathbf{A}$ along $\Gamma$ is defined by

$\begin{split}\mathbf{L}_{\Gamma}&=\mathcal{P}\exp\left(\int_{\Gamma}\mathbf{A}(r)dr\right)=\mathcal{P}\exp\left(\int_{0}^{1}\mathbf{A}(r_{\Gamma}(t))r_{\Gamma}^{\prime}(t)dt\right)\\\
&=\lim_{N\to\infty}\left(\mathbf{I}+\mathbf{A}\left(r_{\Gamma}(t_{N-1})\right)r_{\Gamma}^{\prime}(t_{N-1})\Delta
t\right)\cdots\left(\mathbf{I}+\mathbf{A}\left(r_{\Gamma}(t_{0})\right)r_{\Gamma}^{\prime}(t_{0})\Delta
t\right),\end{split}$ (8)

where $\Delta t=1/N$, $t_{k}=k/N$. It is clear that

$\mathbf{L}_{\Gamma^{\prime}\cdot\Gamma}=\mathbf{L}_{\Gamma^{\prime}}\mathbf{L}_{\Gamma}$
(9)

holds. Applying $\Phi^{-1}$ on the both sides of (6), we have

$\left[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}\right]\mathbf{M}_{\Gamma}=\mathbf{L}_{\Gamma}\left[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}\right]\quad\iff\quad\mathbf{M}_{\Gamma}=\left[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}\right]^{-1}\mathbf{L}_{\Gamma}\left[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}\right].$
(10)

Thus, the monodromy matrix is similar to the path-ordered-exponential, which
is denoted by $\mathbf{M}_{\Gamma}\sim\mathbf{L}_{\Gamma}$.

The _monodromy representation_ of the fundamental group is given by

$\rho:\pi_{1}(\mathbb{X},r_{0})\to\mathrm{GL}(2,\mathbb{C}),\qquad[\Gamma]\mapsto\mathbf{M}_{\Gamma}.$
(11)

From (9) and (10), it holds that

$\rho([\Gamma^{\prime}]\cdot[\Gamma])=\mathbf{M}_{\Gamma^{\prime}\cdot\Gamma}=\mathbf{M}_{\Gamma^{\prime}}\mathbf{M}_{\Gamma}=\rho([\Gamma^{\prime}])\rho([\Gamma]).$
(12)

Therefore, $\rho$ is a homomorphism of groups. Here the image of $\rho$ is
called the _monodromy group_. From the defining relation of the fundamental
group (3), we obtain the global relation of the monodromy group such that

$\mathbf{M}_{\Gamma_{1}}\mathbf{M}_{\Gamma_{2}}\cdots\mathbf{M}_{\Gamma_{n}}=\rho([\Gamma_{1}])\rho([\Gamma_{2}])\cdots\rho([\Gamma_{n}])=\rho([\Gamma_{1}]\cdot[\Gamma_{2}]\cdots[\Gamma_{n}])=\rho(1)=\mathbf{I}.$
(13)

In particular,
$\mathbf{L}_{\Gamma_{1}}\mathbf{L}_{\Gamma_{2}}\cdots\mathbf{L}_{\Gamma_{n}}=\mathbf{I}$
holds with respect to $[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]=\mathbf{I}$.

Set $\hat{U}_{i}(r)=(r-r_{i})U(r)$ and $\hat{V_{i}}(r)=(r-r_{i})^{2}V(r)$ for
$i=1,2,\dots,m$. If $r_{m+1}=\infty$ is a singularity, set
$\hat{U}_{\infty}(r)=2-rU(r)$ and $\hat{V}_{\infty}(r)=r^{2}V(r)$. Then the
ODE (1) is transformed as follows;

$\begin{split}\text{ODE~{}\eqref{eq:ODE}}&\quad\iff\quad\frac{d^{2}R}{dr^{2}}+\frac{\hat{U}_{i}}{(r-r_{i})}\frac{dR}{dr}+\frac{\hat{V}_{i}}{(r-r_{i})^{2}}R=0\\\
&\quad\iff\quad(r-r_{i})^{2}\frac{d^{2}R}{dr^{2}}+\hat{U}_{i}(r-r_{i})\frac{dR}{dr}+\hat{V}_{i}R=0.\end{split}$
(14)

We introduce the Euler operator $\delta_{i}=(r-r_{i})d/dr$. Using
$(r-r_{i})^{2}d^{2}/dr^{2}=\delta_{i}^{2}-\delta_{i}$, as a sequel to the
transformation in (14) we obtain

$\delta_{i}^{2}R+(\hat{U}_{i}-1)\delta_{i}R+\hat{V}R=0.$ (15)

Thus, we have another Pfaffian form:

$\delta_{i}\begin{bmatrix}R\\\ \delta_{i}R\end{bmatrix}=\begin{bmatrix}0&1\\\
-\hat{V}_{i}&1-\hat{U}_{i}\end{bmatrix}\begin{bmatrix}R\\\
\delta_{i}R\end{bmatrix}\quad\iff\quad\delta_{i}\hat{\mathbf{R}}=\hat{\mathbf{A}}_{i}\hat{\mathbf{R}}.$
(16)

Using the discretization of (16), the path-ordered-exponential of
$\hat{\mathbf{A}}_{i}$ along $\Gamma$ is defined by

$\begin{split}(\hat{\mathbf{L}}_{i})_{\Gamma}&=\mathcal{P}\exp\left(\int_{\Gamma}\hat{\mathbf{A}}_{i}(r)d\log(r-r_{i})\right)=\mathcal{P}\exp\left(\int_{0}^{1}\hat{\mathbf{A}}_{i}(r_{\Gamma}(t))\frac{r_{\Gamma}^{\prime}(t)}{r_{\Gamma}(t)-r_{i}}dt\right)\\\
&=\lim_{N\to\infty}\left(\mathbf{I}+\hat{\mathbf{A}}_{i}\left(r_{\Gamma}(t_{N-1})\right)\frac{r_{\Gamma}^{\prime}(t_{N-1})\Delta
t}{r_{\Gamma}(t_{N-1})-r_{i}}\right)\cdots\left(\mathbf{I}+\hat{\mathbf{A}}_{i}\left(r_{\Gamma}(t_{0})\right)\frac{r_{\Gamma}^{\prime}(t_{0})\Delta
t}{r_{\Gamma}(t_{0})-r_{i}}\right),\end{split}$ (17)

where $\Delta t=1/N,$ and $t_{k}=k/N$.

In virtue of the existence and uniqueness of the solutions for the initial
value problem with the initial value
$(R,\delta_{i}R)^{T}=\hat{\mathbf{R}}_{0}$ for (15), the map

$\hat{\Phi}_{i}:\mathbb{C}^{2}\to\mathrm{Sol}(\mathbb{D}),\qquad\hat{\mathbf{R}}_{0}\mapsto
R(r),$ (18)

is a $\mathbb{C}$-linear isomorphism. Here we suppose
$[\Phi(\mathbf{R}_{0}^{1}),\Phi(\mathbf{R}_{0}^{2})]=[\hat{\Phi}_{i}(\hat{\mathbf{R}}_{0}^{1}),\hat{\Phi}_{i}(\hat{\mathbf{R}}_{0}^{2})]$.
Then, applying $\hat{\Phi}_{i}^{-1}$ on the both sides of (6), we have

$\left[\hat{\mathbf{R}}_{0}^{1},\hat{\mathbf{R}}_{0}^{2}\right]\mathbf{M}_{\Gamma}=(\hat{\mathbf{L}}_{i})_{\Gamma}\left[\hat{\mathbf{R}}_{0}^{1},\hat{\mathbf{R}}_{0}^{2}\right]\quad\iff\quad\mathbf{M}_{\Gamma}=\left[\hat{\mathbf{R}}_{0}^{1},\hat{\mathbf{R}}_{0}^{2}\right]^{-1}(\hat{\mathbf{L}}_{i})_{\Gamma}\left[\hat{\mathbf{R}}_{0}^{1},\hat{\mathbf{R}}_{0}^{2}\right].$
(19)

For a sufficiently small number $\epsilon>0$, one may let $\Gamma_{i}$
continuously deform to
$\Lambda_{0i}(\epsilon)^{-1}\cdot\Lambda_{i}(\epsilon)\cdot\Lambda_{0i}(\epsilon)$
where $\Lambda_{0i}(\epsilon)$ is a path connecting $r_{0}$ to
$r_{i}+\epsilon$, and $\Lambda_{i}(\epsilon)$ is a circle initiated at
$r_{i}+\epsilon$ encircling $r_{i}$ counterclockwise. Thus

$(\hat{\mathbf{L}}_{i})_{\Gamma_{i}}=(\hat{\mathbf{L}}_{i})_{\Lambda_{0i}(\epsilon)}^{-1}(\hat{\mathbf{L}}_{i})_{\Lambda_{i}(\epsilon)}(\hat{\mathbf{L}}_{i})_{\Lambda_{0i}(\epsilon)}.$
(20)

If $r_{i}$ is a regular singularity, the entries of
$\hat{\mathbf{A}}_{i}(r_{i})$ are finite and it follows from (17) that

$\lim_{\epsilon\to
0}\left(\hat{\mathbf{L}}_{i}\right)_{\Lambda_{i}(\epsilon)}=\lim_{N\to\infty}\left(\mathbf{I}+\frac{2\pi
i\hat{\mathbf{A}}_{i}(r_{i})}{N}\right)^{N}=\mathrm{e}^{2\pi
i\hat{\mathbf{A}}_{i}(r_{i})}.$ (21)

In summary, we obtain from (10), (19), (20) and (21) that

$\mathbf{M}_{\Gamma_{i}}\sim\mathbf{L}_{\Gamma_{i}}\sim\left(\hat{\mathbf{L}}_{i}\right)_{\Gamma_{i}}\sim\mathrm{e}^{2\pi
i\hat{\mathbf{A}}_{i}(r_{i})}.$ (22)

Suppose the ODE (1) is a Fuchsian differential equation and the number of
singularities is three, i.e., $n=3$. Then it is known that if
$\mathbf{M}_{1},\mathbf{M}_{2},\mathbf{M}_{3}\in\mathrm{GL}(2,\mathbb{C})$
satisfy

$\mathbf{M}_{1}\sim\mathrm{e}^{2\pi
i\hat{\mathbf{A}}_{1}(r_{1})},\quad\mathbf{M}_{2}\sim\mathrm{e}^{2\pi
i\hat{\mathbf{A}}_{2}(r_{2})},\quad\mathbf{M}_{3}\sim\mathrm{e}^{2\pi
i\hat{\mathbf{A}}_{3}(r_{3})},\quad\text{and}\quad\mathbf{M}_{1}\mathbf{M}_{2}\mathbf{M}_{3}=\mathbf{I},$
(23)

then
$\mathbf{M}_{1}=\mathbf{M}_{\Gamma_{1}},\mathbf{M}_{2}=\mathbf{M}_{\Gamma_{2}}$
and $\mathbf{M}_{3}=\mathbf{M}_{\Gamma_{3}}$ for some initial value
$[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]$. It is worth noting that this is no
longer true for $n>3$. The characteristic equation of
$\hat{\mathbf{A}}_{i}(r_{i})$:

$\det\left(x\mathbf{I}-\hat{\mathbf{A}}_{i}(r_{i})\right)=x^{2}-\mathrm{tr}\hat{\mathbf{A}}_{i}(r_{i})x+\det\hat{\mathbf{A}}_{i}(r_{i})=x^{2}-\left(1-\hat{U}_{i}(r_{i})\right)x+\hat{V}_{i}(r_{i})=0,$
(24)

and its roots, i.e., the eigenvalues are respectively called the _indicial
equation_ and the _characteristic exponents_ of $r_{i}$ for the ODE (1). Let
$\mathbf{\rho}=(\rho^{1},\rho^{2})^{T}$,
$\mathbf{\sigma}=(\sigma^{1},\sigma^{2})^{T}$ and
$\mathbf{\tau}=(\tau^{1},\tau^{2})^{T}$ be the characteristic exponents of
$r_{1},r_{2}$ and $r_{3}$ respectively. It is known that they satisfy the
_Fuchsian relation_

$\rho^{1}+\rho^{2}+\sigma^{1}+\sigma^{2}+\tau^{1}+\tau^{2}=n-2=1.$ (25)

Conversely, the characteristic exponents $\rho,\sigma,\tau\in\mathbb{C}^{2}$
of the singularities $r_{1},r_{2},r_{3}$ satisfying the Fuchsian relation (25)
exactly determine the ODE (1): if $r_{3}=\infty$,

$\begin{split}\frac{d^{2}R}{dr^{2}}&+\left(\frac{1-\rho^{1}-\rho^{2}}{r-r_{1}}+\frac{1-\sigma^{1}-\sigma^{2}}{r-r_{2}}\right)\frac{dR}{dr}\\\
&+\frac{1}{(r-r_{1})(r-r_{2})}\left(\frac{\rho^{1}\rho^{2}(r_{1}-r_{2})}{r-r_{1}}+\frac{\sigma^{1}\sigma^{2}(r_{2}-r_{1})}{r-r_{2}}+\tau^{1}\tau^{2}\right)R=0,\end{split}$
(26)

which is called the _Riemann differential equation_ and whose solution is the
Riemann P-function

$P\begin{pmatrix}r_{1}&r_{2}&\infty&\\\ \rho^{1}&\sigma^{1}&\tau^{1}&;r\\\
\rho^{2}&\sigma^{2}&\tau^{2}&\end{pmatrix}.$ (27)

The monodromy of the Riemann differential equation (26) is given in Theorem
4.3.2 of [20]. Namely, if
$\rho^{i}+\sigma^{j}+\tau^{k}\not\in\mathbb{Z}\quad(i,j,k=1,2)$, then the
monodromy representation $\rho$ is expressed up to conjugacy by the following
matrices:

$\rho([\Gamma_{1}])=\begin{bmatrix}\mathrm{e}^{2\pi i\rho^{1}}&1\\\
0&\mathrm{e}^{2\pi
i\rho^{2}}\end{bmatrix},\quad\rho([\Gamma_{2}])=\begin{bmatrix}\mathrm{e}^{2\pi
i\sigma^{1}}&0\\\ b&\mathrm{e}^{2\pi
i\sigma^{2}}\end{bmatrix},\quad\rho([\Gamma_{3}])=\left(\rho([\Gamma_{1}])\rho([\Gamma_{2}])\right)^{-1},$
(28)

where

$b=\mathrm{e}^{-2\pi i\tau^{1}}+\mathrm{e}^{-2\pi i\tau^{2}}-\mathrm{e}^{2\pi
i(\rho^{1}+\sigma^{1})}-\mathrm{e}^{2\pi i(\rho^{2}+\sigma^{2})}.$ (29)

The monodromy group is generated by $\rho([\Gamma_{1}])$ and
$\rho([\Gamma_{2}])$. In particular, $\rho([\Gamma_{3}])$ is determined by the
global relation
$\rho([\Gamma_{1}])\rho([\Gamma_{2}])\rho([\Gamma_{3}])=\mathrm{I}$. Since
$\rho$ is up to conjugacy, $\mathbf{C}^{-1}\rho(\cdot)\mathbf{C}$ is another
monodromy representation of (26) for any
$\mathbf{C}\in\mathrm{GL}(2,\mathbb{C})$. If
$[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]$ is the initial value at $r_{0}$ for
$\rho$, then $[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]\mathbf{C}$ is the
initial value at $r_{0}$ for $\mathbf{C}^{-1}\rho(\cdot)\mathbf{C}$. In
general, the conjugacy class of the monodromy representation is uniquely
determined by the ODE (1), and called the _monodromy_ of the ODE (1).

###  II.2 Connection Matrix of Pair Production and Quasi-normal Modes

Let $[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]$ be a pair of eigenvectors for
eigenvalues $\alpha_{i}^{1}$ and $\alpha_{i}^{2}$ of
$\mathbf{L}_{\Gamma_{i}}$. We assume $\alpha_{i}^{1}\neq\alpha_{i}^{2}$. Then
$\Phi(\mathbf{F}_{i}^{1})$ and $\Phi(\mathbf{F}_{i}^{2})$ span
$\mathrm{Sol}(\mathbb{D})$. It follows from (10) that

$\begin{split}&\mathbf{L}_{\Gamma_{i}}[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]=[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]\begin{bmatrix}\alpha_{i}^{1}&\\\
&\alpha_{i}^{2}\\\ \end{bmatrix}\\\
\iff\quad&[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]\mathbf{M}_{\Gamma_{i}}[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]^{-1}[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]=[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]\begin{bmatrix}\alpha_{i}^{1}&\\\
&\alpha_{i}^{2}\\\ \end{bmatrix}\\\
\iff\quad&\mathbf{M}_{\Gamma_{i}}[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]^{-1}[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]=[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]^{-1}[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]\begin{bmatrix}\alpha_{i}^{1}&\\\
&\alpha_{i}^{2}\\\ \end{bmatrix}.\end{split}$ (30)

Thus it holds that
$[\mathbf{E}_{i}^{1},\mathbf{E}_{i}^{2}]=[\mathbf{R}_{0}^{1},\mathbf{R}_{0}^{2}]^{-1}[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]$
is a pair of eigenvectors for eigenvalues $\alpha_{i}^{1}$ and
$\alpha_{i}^{2}$ of $\mathbf{M}_{\Gamma_{i}}$ as well.

Furthermore, the _connection matrix_ $\mathbf{P}_{i}^{j}$ from
$[\Phi(\mathbf{F}_{i}^{1}),\Phi(\mathbf{F}_{i}^{2})]$ to
$[\Phi(\mathbf{F}_{j}^{1}),\Phi(\mathbf{F}_{j}^{2})]$ is defined by

$\begin{split}&\left[\Phi(\mathbf{F}_{j}^{1}),\Phi(\mathbf{F}_{j}^{2})\right]=\left[\Phi(\mathbf{F}_{i}^{1}),\Phi(\mathbf{F}_{i}^{2})\right]\mathbf{P}_{i}^{j}\\\
\iff\quad&[\mathbf{F}_{j}^{1},\mathbf{F}_{j}^{2}]=[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]\,\mathbf{P}_{i}^{j}\\\
\iff\quad&\mathbf{P}_{i}^{j}=[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]^{-1}[\mathbf{F}_{j}^{1},\mathbf{F}_{j}^{2}]=[\mathbf{E}_{i}^{1},\mathbf{E}_{i}^{2}]^{-1}[\mathbf{E}_{j}^{1},\mathbf{E}_{j}^{2}].\end{split}$
(31)

For any $\mathbf{P}=\left[\begin{smallmatrix}p_{11}&p_{12}\\\
p_{21}&p_{22}\end{smallmatrix}\right]\in\mathrm{GL}(2,\mathbb{C})$, let us
define

$\begin{split}&\mu(\mathbf{P})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}|p_{11}p_{22}|-|p_{12}p_{21}|,\\\
&\nu(\mathbf{P})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}p_{11}p_{22}p_{12}^{\ast}p_{21}^{\ast}-p_{11}^{\ast}p_{22}^{\ast}p_{12}p_{21}.\end{split}$
(32)

If and only if $\mu(\mathbf{P}_{i}^{j})>0$ and $\nu(\mathbf{P}_{i}^{j})=0$,
one can normalize $\mathbf{P}_{i}^{j}$ to be in
$\mathrm{SU}(1,1)=\\{\mathbf{P}=\left[\begin{smallmatrix}p_{11}&p_{12}\\\
p_{21}&p_{22}\end{smallmatrix}\right]\in\mathrm{GL}(2,\mathbb{C})\mid
p_{11}=p_{22}^{\ast},p_{12}=p_{21}^{\ast},\det\mathbf{P}=1\\}$ by rescaling
the eigenvectors with factors $d_{1},d_{2},d_{3},d_{4}\in\mathbb{C}$, i.e.,

$\hat{\mathbf{P}}_{i}^{j}=\begin{bmatrix}d_{1}&\\\ &d_{2}\\\
\end{bmatrix}\mathbf{P}_{i}^{j}\begin{bmatrix}d_{3}&\\\ &d_{4}\\\
\end{bmatrix}\in\mathrm{SU}(1,1).$ (33)

If $\mu(\mathbf{P}_{i}^{j})<0$, then we can redefine
$\mathbf{P}_{i}^{j}=[\mathbf{F}_{i}^{2},\mathbf{F}_{i}^{1}]^{-1}[\mathbf{F}_{j}^{1},\mathbf{F}_{j}^{2}]$
or
$[\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]^{-1}[\mathbf{F}_{j}^{2},\mathbf{F}_{j}^{1}]$
so that $\mu(\mathbf{P}_{i}^{j})>0$, since

$\begin{split}\mu\left([\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]^{-1}[\mathbf{F}_{j}^{1},\mathbf{F}_{j}^{2}]\right)&=\mu\left([\mathbf{F}_{i}^{2},\mathbf{F}_{i}^{1}]^{-1}[\mathbf{F}_{j}^{2},\mathbf{F}_{j}^{1}]\right)=-\mu\left([\mathbf{F}_{i}^{2},\mathbf{F}_{i}^{1}]^{-1}[\mathbf{F}_{j}^{1},\mathbf{F}_{j}^{2}]\right)\\\
&=-\mu\left([\mathbf{F}_{i}^{1},\mathbf{F}_{i}^{2}]^{-1}[\mathbf{F}_{j}^{2},\mathbf{F}_{j}^{1}]\right).\end{split}$
(34)

As for the condition $\nu(P_{i}^{j})=0$, we will examine below under the
physical assumptions. After the normalization, the following ambiguity of
phases $\theta_{1},\theta_{2},\theta_{3},\theta_{4}\in\mathbb{R}$ with
$\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=0$ remains;

$\hat{\mathbf{P}}_{i}^{j}\in\mathrm{SU}(1,1)\quad\iff\quad\begin{bmatrix}\mathrm{e}^{i\theta_{1}}&\\\
&\mathrm{e}^{i\theta_{2}}\\\
\end{bmatrix}\hat{\mathbf{P}}_{i}^{j}\begin{bmatrix}\mathrm{e}^{i\theta_{3}}&\\\
&\mathrm{e}^{i\theta_{4}}\\\ \end{bmatrix}\in\mathrm{SU}(1,1).$ (35)

Now, we are going to impose physical boundary conditions associated with pair
production. For the pair production created in the region $r_{i}<r<r_{j}$
which is to be observed at $r_{j}$, see the left panel of Fig. 2, one imposes
the vanishing in-going (left-moving) modes at $r_{j}$. If the “incident” flux
(right-moving modes at $r_{i}$) corresponding to virtual pairs of quantum
fluctuation is normalized to unity, the “reflected” flux (left-moving modes at
$r_{i}$), $\mathcal{R}$, represents the reannihilation of virtual pairs and
then the “transmitted” flux (right-moving modes at $r_{j}$), $\mathcal{T}$,
manifests the particle production. The fluxes are conserved, namely
$|\mathcal{R}|^{2}+|\mathcal{T}|^{2}=1$.

We will assume that the ODE (1) is a Fuchsian differential equation, and for
physical applications, all of the coefficients in $U(r)$ and $V(r)$ and the
singularities $r_{1}<\dots<r_{n-1}$ are real, and $r_{n-1}>0,r_{n}=\infty$. We
will further assume that the discriminant of the indicial equation for
$r_{i}(i=1,2,\dots,n)$ is negative, i.e.,
$\mathrm{tr}\hat{\mathbf{A}}_{i}(r_{i})^{2}-4\det\hat{\mathbf{A}}_{i}(r_{i})<0$.
Then the characteristic exponents are $s_{i}^{\pm}=\beta_{i}\pm i\alpha_{i}$
with $\beta_{i},\alpha_{i}\in\mathbb{R}$ and $\alpha_{i}\neq 0$. Consider the
Frobenius solutions
$F_{i}^{\pm}=c_{i}^{\pm}(r-r_{i})^{s_{i}^{\pm}}(\sum_{k\geq
0}c_{i,k}^{\pm}(r-r_{i})^{k})$ at the singularity $r_{i}\neq\infty$, and
$F_{n}^{\pm}=c_{n}^{\pm}r^{-s_{n}^{\pm}}(\sum_{k\geq 0}c_{n,k}^{\pm}r^{-k})$
at $r_{n}=\infty$. From the hypothesis that all of the coefficients of $U(r)$
and $V(r)$ are real, we may assume all $c_{i,k}$ are real. It is known that
for $r_{i}\neq\infty$ the radius of convergence for $\sum_{k\geq
0}c_{i,k}^{\pm}(r-r_{i})^{k}$ is $R_{i}=\min_{k\neq i}|r_{k}-r_{i}|$, and
therefore the disc of convergence is
$\mathbb{D}_{i}=\\{r\in\mathbb{C}\mid|r-r_{i}|<R_{i}\\}$. As for
$r_{n}=\infty$, the radius of convergence for $\sum_{k\geq
0}c_{n,k}^{\pm}r^{-k}$ is $R_{n}=\max_{k\neq n}|r_{k}|$, and the disc of
convergence is $\mathbb{D}_{n}=\\{r\in\mathbb{C}\mid|r-r_{i}|>R_{n}\\}$. For
adjacent singularities $r_{i},r_{j}$ with $j=i+1$, we fix the base point
$r_{0}$ such that $r_{i}<r_{0}<r_{j}$. We set $c_{i}^{\pm}=1$, and
$c_{j}^{\pm}=1$ if $j=n$ and $c_{j}^{\pm}=(-1)^{-s_{i}^{\pm}}$ if $j<n$. Then
if $r_{0}\in\mathbb{D}_{i}\cap\mathbb{D}_{j}$ we immediately have
$F_{i}^{+}=(F_{i}^{-})^{*}$ and $F_{j}^{+}=(F_{j}^{-})^{*}$ at $r=r_{0}$,
otherwise analytically continue $F_{i}^{\pm}$ and $F_{j}^{\pm}$ to $r_{0}$
along the real axis using (8) with the integrands being real. Consequently,
$F_{i}^{\pm}$ and $F_{j}^{\pm}$ have the respective initial values
$[(\mathbf{F}_{i}^{+})_{0},(\mathbf{F}_{i}^{-})_{0}]$ with
$(\mathbf{F}_{i}^{+})_{0}=(\mathbf{F}_{i}^{-})_{0}^{\ast}$ and
$[(\mathbf{F}_{j}^{+})_{0},(\mathbf{F}_{j}^{-})_{0}]$ with
$(\mathbf{F}_{j}^{+})_{0}=(\mathbf{F}_{j}^{-})_{0}^{\ast}$ at $r_{0}$, which
are the respective pairs of eingenvectors of $\mathbf{L}_{\Gamma_{i}}$ and
$\mathbf{L}_{\Gamma_{j}}$. Thus, the connection matrix is obtained by

$\mathbf{P}_{i}^{j}=[(\mathbf{F}_{i}^{+})_{0},(\mathbf{F}_{i}^{-})_{0}]^{-1}[(\mathbf{F}_{j}^{+})_{0},(\mathbf{F}_{j}^{-})_{0}].$
(36)

If $\mu(\mathbf{P}_{i}^{j})<0$, we redefine $\mathbf{P}_{i}^{j}$ so that
$\mu(\mathbf{P}_{i}^{j})>0$ as mentioned above. On the other hand,

$(\mathbf{P}_{i}^{j})^{\ast}=[(\mathbf{F}_{i}^{-})_{0},(\mathbf{F}_{i}^{+})_{0}]^{-1}[(\mathbf{F}_{j}^{-})_{0},(\mathbf{F}_{j}^{+})_{0}]=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\mathbf{P}_{i}^{j}\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\quad\Longrightarrow\quad\nu(\mathbf{P}_{i}^{j})=0.$ (37)

Thus $\mathbf{P}_{i}^{j}$ can be normalized to be in $\mathrm{SU}(1,1)$.

Therefore the connection matrix $\mathbf{P}_{i}^{j}$ for pair production can
be expressed as

$\mathbf{P}_{i}^{j}=\begin{bmatrix}1/\mathcal{T}&\mathcal{R}/\mathcal{T}\\\
\mathcal{R}^{*}/\mathcal{T}^{*}&1/\mathcal{T}^{*}\end{bmatrix},$ (38)

which had been derived in [21]. The conservation of flux implies the
determinant of the connection matrix $\mathbf{P}_{+}^{\infty}$, up to an
irrelevant phase, should be one. Fortunately, the choice of the permutation of
the eigenvectors when forcing $\mu(\mathbf{P}_{i}^{j})>0$ and the ambiguity of
phases in (33) become irrelevant when computing $|\mathcal{R}|$ and
$|\mathcal{T}|$.

Additionally, we can also derive the connection matrix for the quasi-normal
mode (QNM) with the boundary condition of only ingoing flux
$\mathcal{T}^{\prime}$ at $r_{i}$ (generally horizon) and outgoing flux
$\mathcal{T}$ at $r_{j}$ (infinity), see the right panel of Fig. 2. In others
words, we only need to replace $\mathcal{R}$ with $\mathcal{T}^{\prime}$ and
take the infinity limit on $\mathcal{T}$ with finite ratio
$\mathcal{T}^{\prime}/\mathcal{T}$. Hence, the connection matrix of QNM will
be

$\mathbf{P}_{i}^{j}=\begin{bmatrix}0&\mathcal{T^{\prime}}/\mathcal{T}\\\
\mathcal{T^{\prime}}^{*}/\mathcal{T}^{*}&0\end{bmatrix},$ (39)

which has vanishing diagonal components and $\det(\mathbf{P}_{i}^{j})=-1$. A
similar approach had been discussed in [21].

In the following sections we will apply this method to study the pair
production in a few physical backgrounds.

Figure 2: The boundary condition for pair production [left] and quasi-normal
mode [right] that are created in the region $r_{i}<r<r_{j}$ and to be observed
at $r_{j}$.

###  II.3 Algorithm of Monodromy Approach

Here we summarize the algorithm for applying monodromy method to study pair
production and quasi-normal modes in problems whose wave functions are given
by the Riemann differential equation:

  * •

determine the characteristic exponents: (26)

  * •

construct monodromy matrices: (28)

  * •

compute connection matrix from eigenvectors of monodromy matrices: (31)

  * •

normalize connection matrix: (33)

  * •

read out physical quantities: pair production (38) and QNM (39).

###  II.4 Mean Number of Pair Production in the Riemann Differential Equation

We now consider physical problems whose mode equations are described by the
Riemann differential equation (26) with the Riemann scheme (27) and give a
general formula for pair production. To be specific, let us consider pair
production from $r=r_{2}$ to $r=r_{\infty}$, in which the imaginary part of
$\sigma^{1}$ and $\tau^{1}$ (similarly the sign of $\sigma^{2}$ and
$\tau^{2}$) should have the same sign. The associated monodromy matrices are

$\displaystyle\mathbf{M}_{1}=\begin{bmatrix}\mathrm{e}^{2\pi i\rho^{1}}&1\\\
0&\mathrm{e}^{2\pi
i\rho^{2}}\end{bmatrix},\qquad\mathbf{M}_{2}=\begin{bmatrix}\mathrm{e}^{2\pi
i\sigma^{1}}&0\\\ b&\mathrm{e}^{2\pi i\sigma^{2}}\end{bmatrix},$
$\displaystyle\mathbf{M}_{\infty}=\begin{bmatrix}\mathrm{e}^{-2\pi
i(\rho^{1}+\sigma^{1})}&-\mathrm{e}^{-2\pi i(\rho^{1}+\sigma^{1}+\rho^{2})}\\\
-b\,\mathrm{e}^{-2\pi i(\rho^{1}+\sigma^{1}+\sigma^{2})}&\mathrm{e}^{-2\pi
i(\rho^{2}+\sigma^{2})}+b\,\mathrm{e}^{-2\pi
i(\rho^{1}+\sigma^{1}+\rho^{2}+\sigma^{2})}\end{bmatrix},$ (40)

where

$b=\mathrm{e}^{-2\pi i\tau^{1}}+\mathrm{e}^{-2\pi i\tau^{2}}-\mathrm{e}^{2\pi
i(\rho^{1}+\sigma^{1})}-\mathrm{e}^{2\pi i(\rho^{2}+\sigma^{2})}.$ (41)

The eigenvalues of $\mathbf{M}_{2}$ and $\mathbf{M}_{\infty}$ are
$(\mathrm{e}^{2\pi i\sigma^{1}},\mathrm{e}^{2\pi i\sigma^{2}})$ and
$(\mathrm{e}^{2\pi i\tau^{2}},\mathrm{e}^{2\pi i\tau^{1}})$, respectively, and
the corresponding eigenvectors are

$\mathbf{E}_{2}=\begin{bmatrix}\mathrm{e}^{2\pi i\sigma^{1}}-\mathrm{e}^{2\pi
i\sigma^{2}}&0\\\
b&1\end{bmatrix},\qquad\mathbf{E}_{\infty}=\begin{bmatrix}\mathrm{e}^{2\pi
i\sigma^{2}}&\mathrm{e}^{2\pi i\sigma^{2}}\\\ \mathrm{e}^{2\pi
i(\rho^{2}+\sigma^{2})}-\mathrm{e}^{-2\pi i\tau^{1}}&\mathrm{e}^{2\pi
i(\rho^{2}+\sigma^{2})}-\mathrm{e}^{-2\pi i\tau^{2}}\end{bmatrix}.$ (42)

Hence, the connection matrix relating $r=r_{2}$ to $r=\infty$ is

$\displaystyle\mathbf{P}_{2}^{\infty}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}d_{1}&0\\\
0&d_{2}\end{bmatrix}(\mathbf{E}_{2})^{-1}\mathbf{E}_{\infty}\begin{bmatrix}d_{3}&0\\\
0&d_{4}\end{bmatrix}=\frac{1}{\mathrm{e}^{2\pi i\sigma^{1}}-\mathrm{e}^{2\pi
i\sigma^{2}}}\begin{bmatrix}d_{1}d_{3}\,\mathrm{e}^{2\pi
i\sigma^{2}}&d_{1}d_{4}\,\mathrm{e}^{2\pi i\sigma^{2}}\\\
-d_{2}d_{3}\,\Xi_{1}&-d_{2}d_{4}\,\Xi_{2}\end{bmatrix},$ (44)
$\displaystyle\Xi_{1}=\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{1})}+\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{2})}-\mathrm{e}^{2\pi
i(\rho^{1}+\sigma^{1}+\sigma^{2})}-\mathrm{e}^{2\pi
i(\rho^{2}+\sigma^{2}+\sigma^{1})},$ $\displaystyle\Xi_{2}=\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{2})}+\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{1})}-\mathrm{e}^{2\pi
i(\rho^{1}+\sigma^{1}+\sigma^{2})}-\mathrm{e}^{2\pi
i(\rho^{2}+\sigma^{2}+\sigma^{1})}.$

To normalize the connection matrix $\mathbf{P}_{2}^{\infty}$ to have the unit
determinant, one should impose the constraint

$d_{1}d_{2}d_{3}d_{4}=-\frac{\left(\mathrm{e}^{2\pi
i\sigma^{1}}-\mathrm{e}^{2\pi i\sigma^{2}}\right)^{2}}{\mathrm{e}^{2\pi
i\sigma^{2}}\left[\mathrm{e}^{2\pi i(\sigma^{1}-\tau^{2})}-\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{1})}+\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{1})}-\mathrm{e}^{2\pi i(\sigma^{2}-\tau^{2})}\right]},$
(45)

which has the following relation to the fluxes for the pair production
boundary condition

$\mathbf{P}_{2}^{\infty}=\begin{bmatrix}1/\mathcal{T}&\mathcal{R}/\mathcal{T}\\\
\mathcal{R}^{*}/\mathcal{T}^{*}&1/\mathcal{T}^{*}\end{bmatrix}.$ (46)

Finally, the mean number of pair production via the “tunneling” process is

${\cal
N}=\frac{|\mathcal{T}|^{2}}{|\mathcal{R}|^{2}}=-\frac{\left(\mathrm{e}^{2\pi
i\sigma^{1}}-\mathrm{e}^{2\pi
i\sigma^{2}}\right)^{2}}{d_{1}d_{2}d_{3}d_{4}\,\mathrm{e}^{2\pi
i\sigma^{2}}\,\Xi_{1}}=\frac{\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{2})}-\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{1})}+\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{1})}-\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{2})}}{\mathrm{e}^{2\pi
i(\sigma^{1}-\tau^{1})}+\mathrm{e}^{2\pi
i(\sigma^{2}-\tau^{2})}-\mathrm{e}^{2\pi
i(\rho^{1}+\sigma^{1}+\sigma^{2})}-\mathrm{e}^{2\pi
i(\rho^{2}+\sigma^{2}+\sigma^{1})}}.$ (47)

The mean number can be written as

${\cal
N}=\frac{\cos\pi(\sigma^{1}-\sigma^{2}+\tau^{1}-\tau^{2})-\cos\pi(\sigma^{1}-\sigma^{2}-\tau^{1}+\tau^{2})}{\cos\pi(\sigma^{1}-\sigma^{2}-\tau^{1}+\tau^{2})+\cos\pi(\rho^{1}-\rho^{2})}.$
(48)

##  III Pair Production from Charged Black Holes: Two Special Cases

In order to investigate the Hawking radiation and the Schwinger effect, we
consider a massive charged scalar field $\Phi$ carrying charge $q$ and mass
$m$ which satisfies the Klein-Gordon (KG) equation

$\left(D_{\nu}D^{\nu}-m^{2}\right)\Phi=0,$ (49)

where the covariant derivative is defined as $D_{\mu}=\nabla_{\mu}-iqA_{\mu}$.
This field propagates in the background of the RN black holes with charge $Q$
and mass $M$

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{2}^{2},\qquad
A_{\mu}dx^{\mu}=-\frac{Q}{r}dt,$ (50)

where $f(r)=(r-r_{+})(r-r_{-})/r^{2}$ with $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$.
By separating the Fourier modes

$\Phi(t,r,\theta,\phi)=\int
d\omega\sum_{ln}\mathrm{e}^{in\phi}S_{ln}(\theta)R_{ln}(\omega,r)\mathrm{e}^{-i\omega
t},$ (51)

where $\mathrm{e}^{in\phi}S_{ln}(\theta)$ are the spherical harmonics, the KG
equation (49) reduces to an ODE

$f\frac{d}{dr}\left(r^{2}f\frac{d}{dr}R\right)+\left[(\omega
r-qQ)^{2}-(m^{2}r^{2}+\lambda_{l})f\right]R=0,$ (52)

where $R=R_{ln}(\omega,r)$ and $\lambda_{l}$ is the separation constant
$\lambda_{l}=l(l+1)$ with an integer $l$ ($l=0,1,\cdots$).

The Eq. (52) is a confluent Heun equation with three poles including a
degenerate pole at spatial infinity. There are some technical difficulties in
applying the monodromy approach. Here we consider two special cases such that
the associated Klein-Gordon equation reduces to the Riemann differential
equations which can be directly analyzed by the monodromy technique.

###  III.1 Near Horizon of Near Extremal Black Holes

Let’s firstly focus on pair production in the near-horizon region of near-
extremal RN black holes [12]. By performing the following coordinate re-
scaling, as shown in Fig. 3, and taking $\epsilon\to 0$

$r\rightarrow Q+\epsilon\rho,\qquad M\rightarrow
Q+\frac{\epsilon^{2}B^{2}}{2Q},\qquad t\rightarrow\frac{\tau}{\epsilon},$ (53)

the metric and the electromagnetic potential take the form in the
$\mathrm{AdS}_{2}\times S^{2}$ space

$ds^{2}=-\frac{\rho^{2}-B^{2}}{Q^{2}}d\tau^{2}+\frac{Q^{2}}{\rho^{2}-B^{2}}d\rho^{2}+Q^{2}d\Omega_{2}^{2},\qquad
A_{\mu}dx^{\mu}=-\frac{\rho}{Q}d\tau,$ (54)

which $B$ is the black holes horizon and $-B$ the inner horizon in the new
scale. Hence, the radial part of the KG equation in the near-horizon region
which can be rewritten as

$\frac{d^{2}R}{d\rho^{2}}+\frac{2\rho}{\rho^{2}-B^{2}}\frac{dR}{d\rho}+\left[\frac{Q^{2}(\omega
Q-q\rho)^{2}}{(\rho^{2}-B^{2})^{2}}-\frac{m^{2}Q^{2}+\lambda_{l}}{\rho^{2}-B^{2}}\right]R=0,$
(55)

is described by a Riemann differential equation (26) with three regular
singular points at $\rho_{\pm}=\pm B$ and infinity $\rho_{\infty}=\infty$.

Figure 3: Re-scaling of the coordinates near the horizon of near-extremal
charged black holes.

To construct the monodromy matrices $\mathbf{M}_{i}$ we should firstly compute
the characteristic exponents $s_{\rightarrow}^{i}$ and $s_{\leftarrow}^{i}$ at
every singular point, $\rho_{i},i=\pm,\infty$. By the indicial equation (24),
we can straightforwardly figure out the exponents and find the solution in the
Riemann P-function

$P\begin{pmatrix}\rho_{-}&\rho_{+}&\infty&\\\
s_{\rightarrow}^{-}&s_{\rightarrow}^{+}&s_{\rightarrow}^{\infty}&;\rho\\\
s_{\leftarrow}^{-}&s_{\leftarrow}^{+}&s_{\leftarrow}^{\infty}&\end{pmatrix}=P\begin{pmatrix}-B&B&\infty&\\\
i\alpha_{-}&i\alpha_{+}&\frac{1}{2}+i\alpha_{\infty}&;\rho\\\
-i\alpha_{-}&-i\alpha_{+}&\frac{1}{2}-i\alpha_{\infty}&\end{pmatrix},$ (56)

where

$2\alpha_{\pm}=Q^{2}\frac{\omega}{B}\mp
qQ,\qquad\alpha_{\infty}=\sqrt{(q^{2}-m^{2})Q^{2}-\lambda_{l}-1/4}.$ (57)

Moreover, the property (23) requires that the monodromy matrices
$\mathbf{M}_{i}$ should hold two invariants

$\det\mathbf{M}_{i}=\mathrm{e}^{i2\pi\left(s_{\rightarrow}^{i}+s_{\leftarrow}^{i}\right)}=1,\quad\text{for}\quad
i=\pm,\,\infty,$ (58)

and

$\mathrm{tr}\,\mathbf{M}_{i}=\mathrm{e}^{i2\pi
s_{\rightarrow}^{i}}+\mathrm{e}^{i2\pi s_{\leftarrow}^{i}}=\begin{cases}\quad
2\cosh(2\pi\alpha_{i})&\mathrm{for}\;i=\pm,\\\
-2\cosh(2\pi\alpha_{\infty})&\mathrm{for}\;i=\infty.\end{cases}$ (59)

According to (28) and (23), we assume the monodromy matrices as the following

$\begin{split}&\mathbf{M}_{-}=\begin{bmatrix}\mathrm{e}^{-2\pi\alpha_{-}}&1\\\
0&\mathrm{e}^{2\pi\alpha_{-}}\end{bmatrix},\qquad\mathbf{M}_{+}=\begin{bmatrix}\mathrm{e}^{-2\pi\alpha_{+}}&0\\\
b&\mathrm{e}^{2\pi\alpha_{+}}\end{bmatrix},\\\
&\mathbf{M}_{\infty}=\begin{bmatrix}\mathrm{e}^{2\pi(\alpha_{+}+\alpha_{-})}&-\mathrm{e}^{2\pi\alpha_{+}}\\\
-b\,\mathrm{e}^{2\pi\alpha_{-}}&b+\mathrm{e}^{-2\pi(\alpha_{+}+\alpha_{-})}\end{bmatrix},\end{split}$
(60)

with

$b=-\mathrm{e}^{2\pi\alpha_{\infty}}-\mathrm{e}^{-2\pi\alpha_{\infty}}-\mathrm{e}^{2\pi(\alpha_{+}+\alpha_{-})}-\mathrm{e}^{-2\pi(\alpha_{+}+\alpha_{-})}.$
(61)

After then, it is easy to figure out the eigenvectors with eigenvalues
$\left\\{\mathrm{e}^{2\pi\alpha_{+}},\mathrm{e}^{-2\pi\alpha_{+}}\right\\}$
for the monodromy matrix $\mathbf{M}_{+}$

$\mathbf{E}_{+}=\begin{bmatrix}\mathbf{E}_{+}^{1},\mathbf{E}_{+}^{2}\end{bmatrix}=\begin{bmatrix}0&\mathrm{e}^{-2\pi\alpha_{+}}-\mathrm{e}^{2\pi\alpha_{+}}\\\
1&b\end{bmatrix},$ (62)

and
$\left\\{-\mathrm{e}^{-2\pi\alpha_{\infty}},-\mathrm{e}^{2\pi\alpha_{\infty}}\right\\}$
for the monodromy matrix $\mathbf{M}_{\infty}$

$\mathbf{E}_{\infty}=\begin{bmatrix}\mathrm{e}^{2\pi\alpha_{+}}&\mathrm{e}^{2\pi\alpha_{+}}\\\
\mathrm{e}^{-2\pi\alpha_{\infty}}+\mathrm{e}^{2\pi(\alpha_{+}+\alpha_{-})}&\mathrm{e}^{2\pi\alpha_{\infty}}+\mathrm{e}^{2\pi(\alpha_{+}+\alpha_{-})}\end{bmatrix}.$
(63)

Following (31) and (33) it is straightforward to compute the connection matrix
$\mathbf{P}_{+}^{\infty}$

$\begin{split}\mathbf{P}_{+}^{\infty}&=\begin{bmatrix}d_{1}&0\\\
0&d_{2}\end{bmatrix}(\mathbf{E}_{+})^{-1}\mathbf{E}_{\infty}\begin{bmatrix}d_{3}&0\\\
0&d_{4}\end{bmatrix}\\\
&=\begin{bmatrix}-d_{1}d_{3}\frac{\cosh(2\pi\alpha_{-})+\cosh(2\pi\alpha_{+}+2\pi\alpha_{\infty})}{\sinh(2\pi\alpha_{+})}&-d_{1}d_{4}\frac{\cosh(2\pi\alpha_{-})+\cosh(2\pi\alpha_{+}-2\pi\alpha_{\infty})}{\sinh(2\pi\alpha_{+})}\\\
-d_{2}d_{3}\frac{2\mathrm{e}^{2\pi\alpha_{+}}}{\sinh(2\pi\alpha_{+})}&-d_{2}d_{4}\frac{2\mathrm{e}^{2\pi\alpha_{+}}}{\sinh(2\pi\alpha_{+})}\end{bmatrix}.\end{split}$
(64)

To normalize the connection matrix $\mathbf{P}_{+}^{\infty}$ with unit
determinant, one should impose the constraint

$d_{1}d_{2}d_{3}d_{4}=\frac{\sinh^{2}(2\pi\alpha_{+})}{4\mathrm{e}^{2\pi\alpha_{+}}\sinh(2\pi\alpha_{+})\sinh(2\pi\alpha_{\infty})}.$
(65)

Furthermore, from Eq. (38) the absorption cross section ratio and the inverse
of product of diagonal elements, are independent of $d_{i}$ as shown

$\sigma_{\mathrm{Near}}=\left|\mathcal{T}_{\mathrm{Near}}\right|^{2}=\frac{\sinh(2\pi\alpha_{+})\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\alpha_{+}+\pi\alpha_{-}+\pi\alpha_{\infty})\cosh(\pi\alpha_{+}-\pi\alpha_{-}+\pi\alpha_{\infty})},$
(66)

and, similarly, the mean particle number is

$\mathcal{N}_{\mathrm{Near}}=\frac{\left|\mathcal{T}_{\mathrm{Near}}\right|^{2}}{\left|\mathcal{R}_{\mathrm{Near}}\right|^{2}}=\frac{\sinh(2\pi\alpha_{+})\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\alpha_{+}+\pi\alpha_{-}-\pi\alpha_{\infty})\cosh(\pi\alpha_{+}-\pi\alpha_{-}-\pi\alpha_{\infty})},$
(67)

for the pair production from the near-horizon region of near-extremal black
holes. Indeed, the mean number can be obtained directly from general formula
(48) with
$\rho^{1}=i\alpha_{-},\rho^{2}=-i\alpha_{-},\sigma^{1}=i\alpha_{+},\sigma^{2}=-i\alpha_{+}$
and $\tau^{1}=1/2+i\alpha_{\infty},\tau^{2}=1/2-i\alpha_{\infty}$. These
results are identical with those obtained in [12] by a different approach
which used the transformation formulas of the Whittaker and hypergeometric
functions.

To compute the QNM connection matrix (39), we should firstly use
$\det(\mathbf{P}_{i}^{j})=-1$ to determine $d_{1}d_{2}d_{3}d_{4}$ and then the
property of vanishing diagonal components gives the condition, which is
equivalent to $|{\cal T}|^{2}\to\infty$,

$\frac{\cosh(\pi\alpha_{-}+\pi\alpha_{+}+\pi\alpha_{\infty})\cosh(\pi\alpha_{-}-\pi\alpha_{+}-\pi\alpha_{\infty})}{\sinh(2\pi\alpha_{+})\sinh(2\pi\alpha_{\infty})}=0,$
(68)

where $\alpha_{\pm}$ in $\omega$ (57) are function of frequency. Therefore,
the condition for the QNM frequencies is

$\cosh(\pi\alpha_{-}+\pi\alpha_{+}+\pi\alpha_{\infty})=0\quad\Rightarrow\quad\alpha_{-}+\alpha_{+}+\alpha_{\infty}=\pm
i(N+1/2),\quad N=0,1,\cdots,$ (69)

which leads to

$\omega=-\frac{B}{Q^{2}}\alpha_{\infty}\pm
i\left(N+\frac{1}{2}\right)\frac{B}{Q^{2}},\qquad N=0,1,\cdots.$ (70)

The extremal limit can be fulfilled by taking limit $B=0$ and the
corresponding KG equations are confluent hypergeometric equations with a
degenerate pole at $\rho=0$. Consequently, the cross section ratio
$\sigma_{\mathrm{Ex}}$ is

$\sigma_{\mathrm{Ex}}=\left|\mathcal{T}_{\mathrm{Ex}}\right|^{2}=\mathrm{e}^{-\pi(\alpha_{\infty}+\tilde{\alpha})}\frac{\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\tilde{\alpha}-\pi\alpha_{\infty})},$
(71)

and the mean particle number $\mathcal{N}_{\mathrm{Ex}}$ is

$\mathcal{N}_{\mathrm{Ex}}=\frac{\left|\mathcal{T}_{\mathrm{Ex}}\right|^{2}}{\left|\mathcal{R}_{\mathrm{Ex}}\right|^{2}}=\mathrm{e}^{\pi(\alpha_{\infty}-\tilde{\alpha})}\frac{\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\tilde{\alpha}+\pi\alpha_{\infty})},$
(72)

with $\tilde{\alpha}=qQ$ and again agreeing with the results in [12].

###  III.2 Production of minimal energy particle under static balance in
nonextremal black hole

In Ref. [19], pair production was studied under a special condition:
$\omega=m$ and $mM=qQ$. The condition $mM=qQ$ is the static balance
(gravitational and electric forces) of a charge with $m$ and $q$ in the
background of nonextremal RN black hole up to the ppN approximation [24]. The
other condition $\omega=m$ is the minimal energy of the particle in the
asymptotic region.111The Hamiltonian for the charge in the metric (50),
$H=-m^{2}=-\frac{1}{f(r)}(\pi_{t}-qA_{t}(r))^{2}+f(r)\pi_{r}^{2}+\frac{L^{2}}{r^{2}}$,
gives the minimal energy when $\pi_{t}=\omega=m$ at $r=\infty$. Under the
specific condition above, the radial part of the KG equation (52) reduces to

$\frac{d^{2}}{dr^{2}}R+\left(\frac{1}{r-r_{+}}+\frac{1}{r-r_{-}}\right)\frac{d}{dr}R+\left[\left(\frac{\omega
r(r_{+}-r_{-})}{2(r-r_{-})(r-r_{+})}\right)^{2}-\frac{\lambda_{l}}{(r-r_{-})(r-r_{+})}\right]R=0.$
(73)

Certainly, Eq. (73) belongs to a Riemann differential equation (26) and,
according to the procedure in the previous subsection, we write down the
Riemann P-function as

$P\begin{pmatrix}r_{-}&r_{+}&\infty&\\\ iA_{-}&iA_{+}&1/2+iA_{\infty}&;r\\\
-iA_{-}&-iA_{+}&1/2-iA_{\infty}&\end{pmatrix},$ (74)

with

$2A_{\pm}=\omega r_{\pm},\qquad
A_{\infty}=\sqrt{\omega^{2}(r_{+}-r_{-})^{2}/4-\lambda_{l}-1/4}.$ (75)

Repeating the same computation in the previous subsection by replacing
$\alpha_{i}$ by $A_{i}$, the mean particle number $\mathcal{N}_{\mathrm{noF}}$
and the absorption cross section ratio $\sigma_{\mathrm{noF}}$ are

$\displaystyle\mathcal{N}_{\mathrm{noF}}=\frac{\left|\mathcal{T}_{\mathrm{noF}}\right|^{2}}{\left|\mathcal{R}_{\mathrm{noF}}\right|^{2}}=\frac{\sinh(2\pi
A_{+})\sinh(2\pi A_{\infty})}{\cosh(\pi A_{+}+\pi A_{-}-\pi
A_{\infty})\cosh(\pi A_{+}-\pi A_{-}-\pi A_{\infty})},$ (76)
$\displaystyle\sigma_{\mathrm{noF}}=\left|\mathcal{T}_{\mathrm{noF}}\right|^{2}=\frac{\sinh(2\pi
A_{+})\sinh(2\pi A_{\infty})}{\cosh(\pi A_{+}+\pi A_{-}+\pi
A_{\infty})\cosh(\pi A_{+}-\pi A_{-}+\pi A_{\infty})}.$ (77)

Eqs. (67) and (68) agree with the results obtained in [19], which follows the
method in [12]. Note that these results apply to pair production of specific
(no-force) scalar particles in the full spacetime outside the horizon of RN
black holes. This a new result without any direct comparison with literature.
Moreover, this particular case may not be physically interesting due to the
strong specific constraints. However, the result may shed light on
understanding pair production in non-extremal black holes, which requires a
further study.

##  IV Pair Production from Generalized Pöschl-Teller Potential

As a model with exact wave functions known in terms of special functions, we
consider the generalized Pöschl-Teller potential, which was introduced by
Rosen and Morse, and apply the monodromy method to find the pair production as
well as the absorption cross section in Sec. III. This potential occurs for a
scalar field in the global de Sitter space and a charged scalar field in a
Sauter-type electric field. The wave equation takes the form

$\psi^{\prime\prime}(x)+\left(A+B\tanh x+C\tanh^{2}x\right)\psi(x)=0,$ (78)

where $\psi$ is a scalar field in the background of curved spaces or electric
fields and $x$ denotes a dimensionless space variable. The equation can be
transformed, by using the new coordinate $z\equiv 2/(1-\tanh x)$, to

$\psi^{\prime\prime}(z)+\frac{1}{z-1}\psi^{\prime}(z)+\left(\frac{C}{z^{2}}+\frac{A-B+C}{4(z-1)^{2}}+\frac{2C-B}{2z}-\frac{2C-B}{2(z-1)}\right)\psi(z)=0,$
(79)

which is a Riemann differential equation (26) with three simple poles:
$z=0\,(x=i\pi/2)$, $z=1\,(x=-\infty)$, $z=\infty\,(x=\infty)$, and the
associated Riemann P-function is

$P\begin{pmatrix}0&1&\infty&\\\
1/2+i\alpha_{0}&i\alpha_{1}&i\alpha_{\infty}&;z\\\
1/2-i\alpha_{0}&-i\alpha_{1}&-i\alpha_{\infty}&\end{pmatrix},$ (80)

where

$\alpha_{0}=\sqrt{C-1/4},\qquad 2\alpha_{1}=\sqrt{A-B+C},\qquad
2\alpha_{\infty}=\sqrt{A+B+C}.$ (81)

Analogous to the previous section, the monodromy matrices in this case are
chosen as

$\begin{split}&\mathbf{M}_{1}=\begin{bmatrix}\mathrm{e}^{-2\pi\alpha_{1}}&1\\\
0&\mathrm{e}^{2\pi\alpha_{1}}\end{bmatrix},\qquad\mathbf{M}_{\infty}=\begin{bmatrix}\mathrm{e}^{-2\pi\alpha_{\infty}}&0\\\
b&\mathrm{e}^{2\pi\alpha_{\infty}}\end{bmatrix},\\\
&\mathbf{M}_{0}=\begin{bmatrix}\mathrm{e}^{2\pi(\alpha_{1}+\alpha_{\infty})}&-\mathrm{e}^{2\pi\alpha_{\infty}}\\\
-b\,\mathrm{e}^{2\pi\alpha_{1}}&b+\mathrm{e}^{-2\pi(\alpha_{1}+\alpha_{\infty})}\end{bmatrix},\end{split}$
(82)

with

$b=-\mathrm{e}^{2\pi\alpha_{0}}-\mathrm{e}^{-2\pi\alpha_{0}}-\mathrm{e}^{2\pi(\alpha_{1}+\alpha_{\infty})}-\mathrm{e}^{-2\pi(\alpha_{1}+\alpha_{\infty})}.$
(83)

The eigenvectors of $\mathbf{M}_{1}$ and $\mathbf{M}_{\infty}$ are

$\mathbf{E}_{1}=\begin{bmatrix}1&1\\\
\mathrm{e}^{2\pi\alpha_{1}}-\mathrm{e}^{-2\pi\alpha_{1}}&0\end{bmatrix},\qquad\mathbf{E}_{\infty}=\begin{bmatrix}\mathrm{e}^{-2\pi\alpha_{\infty}}-\mathrm{e}^{2\pi\alpha_{\infty}}&0\\\
b&1\end{bmatrix},$ (84)

from which we get the associated connection matrix

$\begin{split}\mathbf{P}_{1}^{\infty}&=\begin{bmatrix}d_{1}&0\\\
0&d_{2}\end{bmatrix}(\mathbf{E}_{1})^{-1}\mathbf{E}_{\infty}\begin{bmatrix}d_{3}&0\\\
0&d_{4}\end{bmatrix}\\\
&=\frac{1}{\sinh(2\pi\alpha_{1})}\begin{bmatrix}-d_{1}d_{3}\left[\cosh(2\pi\alpha_{0})+\cosh(2\pi\alpha_{1}+2\pi\alpha_{\infty})\right]&2d_{1}d_{4}\\\
d_{2}d_{3}\left[\cosh(2\pi\alpha_{0})+\cosh(2\pi\alpha_{1}-2\pi\alpha_{\infty})\right]&-2d_{2}d_{4}\end{bmatrix}.\end{split}$
(85)

Here, the normalization gives the condition

$d_{1}d_{2}d_{3}d_{4}=\frac{\sinh^{2}(2\pi\alpha_{0})}{4\sinh(2\pi\alpha_{1})\sinh(2\pi\alpha_{\infty})}.$
(86)

Finally, the absorption cross section can be read out as

$\sigma=\frac{\sinh(2\pi\alpha_{1})\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\alpha_{0}-\pi\alpha_{1}-\pi\alpha_{\infty})\cosh(\pi\alpha_{0}+\pi\alpha_{1}+\pi\alpha_{\infty})},$
(87)

as well as the mean particle number

$\mathcal{N}=\frac{\sinh(2\pi\alpha_{1})\sinh(2\pi\alpha_{\infty})}{\cosh(\pi\alpha_{0}+\pi\alpha_{1}-\pi\alpha_{\infty})\cosh(\pi\alpha_{0}-\pi\alpha_{1}+\pi\alpha_{\infty})}.$
(88)

Actually, from the general formula (48) we can easily compute the mean number
by substituting
$\rho^{1}=1/2+i\alpha_{0},\rho^{2}=1/2-i\alpha_{0},\sigma^{1}=i\alpha_{1},\sigma^{2}=-i\alpha_{1}$
and $\tau^{1}=i\alpha_{\infty},\tau^{2}=-i\alpha_{\infty}$.

We now compare the monodromy result with that of the generalized Pösch-Teller
potential. The model explains the Schwinger pair production in a Sauter-type
electric field. To do so, we first consider the component equation of the
scalar field in a background field [5]

$\psi^{\prime\prime}(x)+k^{2}(x)\psi(x)=0,$ (89)

where the momentum square is

$k^{2}(x)=k_{-}^{2}+\frac{1}{2}(k_{+}^{2}-k_{-}^{2})\left(1+\tanh\frac{x}{L}\right)+\frac{V}{4}\left(\tanh^{2}\frac{x}{L}-1\right).$
(90)

Here, $L$ is a characteristic length scale to make $x/L$ dimensionless, and an
effective length scale for the Sauter-type electric field
$E(x)=E_{0}/\cosh^{2}(x/L)$.

The momentum has two limits in the far left and right: $k(-\infty)=k_{-}$ and
$k(+\infty)=k_{+}$. The solution that has a unit incident flux $x=-\infty$ and
only out-going flux at $x=\infty$ is given by

$\psi(x)\\!=\\!\left(1\\!+\\!\tanh\frac{x}{L}\right)^{-ik_{-}L/2}\left(1\\!-\\!\tanh\frac{x}{L}\right)^{-ik_{+}L/2}F\left(\frac{\mu}{2},\frac{\nu}{2},1\\!-\\!ik_{+}L;\frac{1\\!-\\!\tanh\frac{x}{L}}{2}\right),$
(91)

where

$\mu=1-iL(k_{-}+k_{+})-i\Delta,\qquad\nu=1-iL(k_{-}+k_{+})+i\Delta.$ (92)

Here, $\Delta=\sqrt{VL^{2}-1}$. Using the transformation formula 9.131.2 of
[29], we find the asymptotic forms of the wave function

$\psi(x\rightarrow-\infty)=\mathrm{e}^{ik_{-}x}+R\,\mathrm{e}^{-ik_{-}x},\qquad\psi(x\rightarrow\infty)=T\,\mathrm{e}^{ik_{+}x},$
(93)

where

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\Gamma\bigl{(}ik_{-}L\bigr{)}\,\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L+k_{-}L+\Delta)\Bigr{)}\,\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L+k_{-}L-\Delta)\Bigr{)}}{\Gamma\bigl{(}-ik_{-}L\bigr{)}\,\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L-k_{-}L+\Delta)\Bigr{)}\,\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L-k_{-}L+\Delta)\Bigr{)}},$
(94) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L+k_{-}L+\Delta)\Bigr{)}\,\Gamma\Bigl{(}\frac{1}{2}-\frac{i}{2}(k_{+}L+k_{-}L-\Delta)\Bigr{)}}{\Gamma\bigl{(}1-ik_{+}L\bigr{)}\,\Gamma\bigl{(}-ik_{-}L\bigr{)}},$
(95)

The Bogoliubov coefficients related to fluxes are defined as [12]
$|\alpha|^{2}=\mathrm{(incident~{}flux)}/\mathrm{(reflection~{}flux)}=1/|R|^{2}$
and
$|\beta|^{2}=\mathrm{(transmission~{}flux)}/\mathrm{(reflection~{}flux)}=|T|^{2}/|R|^{2}$
which lead to

$\displaystyle|\alpha|^{2}=\frac{\cosh\left[\frac{\pi}{2}(k_{+}L+k_{-}L+\Delta)\right]\cosh\left[\frac{\pi}{2}(k_{+}L+k_{-}L-\Delta)\right]}{\cosh\left[\frac{\pi}{2}(k_{+}L-k_{-}L+\Delta)\right]\cosh\left[\frac{\pi}{2}(k_{+}L-k_{-}L-\Delta)\right]},$
(96) $\displaystyle|\beta|^{2}=\frac{\sinh(\pi k_{-}L)\sinh(\pi
k_{+}L)}{\cosh\left[\frac{\pi}{2}(k_{+}L-k_{-}L+\Delta)\right]\cosh\left[\frac{\pi}{2}(k_{+}L-k_{-}L-\Delta)\right]}.$
(97)

The Bogoliubov coefficients satisfy the relation $|\alpha|^{2}-|\beta|^{2}=1$.
The absorption cross section defined as
$\sigma=(\mathrm{tranmission~{}flux})/(\mathrm{incident~{}flux})=|T|^{2}=|\beta|^{2}/|\alpha|^{2}$
is

$\displaystyle\sigma=\frac{\sinh(\pi k_{-}L)\sinh(\pi
k_{+}L)}{\cosh\left[\frac{\pi}{2}(k_{+}L+k_{-}L+\Delta)\right]\cosh\left[\frac{\pi}{2}(k_{+}L+k_{-}L-\Delta)\right]}.$
(98)

By comparing equation (89) with the equations (78), we can identify

$A=L^{2}[(k_{+}^{2}+k_{-}^{2})/2-V/4],\qquad
B=-L^{2}(k_{+}^{2}-k_{-}^{2})/2,\qquad C=VL^{2}/4,$ (99)

and thereby the corresponding parameters

$\begin{split}&\alpha_{0}=\sqrt{C-1/4}=\frac{1}{2}\sqrt{VL^{2}-1}=\Delta/2,\qquad
2\alpha_{1}=\sqrt{A-B+C}=k_{+}L,\\\
&2\alpha_{\infty}=\sqrt{A+B+C}=k_{-}L.\end{split}$ (100)

Remarkably, substituting these parameters into (87, 88), the result from the
monodromy is identical to (98, 97) and consistent with the result of [25, 26]
for scalar pair production in the Sauter-type electric field with
$V/4=(qE_{0}L)^{2}$. Note that for spin-1/2 fermions, setting
$V/4=(qE_{0}L)^{2}\pm iqE_{0}$ in (87) gives the correct result for the mean
number of fermions in the Sauter-type electric field [26, 27, 28]. A passing
remark is the phase-integral method that also makes use of the regular
singular points gives the leading Boltzmann factor for the pair production in
the Sauter-type electric field [22]. We argue that the monodromy may complete
the phase-integral method.

##  V Conclusion

In this paper we have advanced the monodromy approach to find the exact mean
number of the pair production in background fields whose mode equations belong
to the Riemann diffrential equation. Further, we have investigated the pair
production of charged scalar in two special cases of RN black holes and in the
generalized Pöschl-Teller potential. In these three cases the mode equations
after separating conserved quantum numbers are (confluent) hypergeometric
functions, the specific case of the Riemann differential equation, from which
scattering coefficients and therefrom the associated Bogolubov coefficients
have been directly computed. The monodromy technique provides a practical and
powerful method to compute the mean number and absorption cross section while
it avoids cumbersome and subtle transformation formulas of the hypergeometric
functions that connect different regions related to the singular points of the
underlying equation. Instead, we only need to compute the exponents of the
Frobenius solutions at relevant singular points, which significantly
simplifies computation.

For the next step, we will study the Schwinger pair production and Hawking
radiation in the general RN black holes, in which the relevant mode equation
is a confluent Heun equation with an irregular singularity at infinity.
Resolving the irregular singularity and regular poles more than three is a
challenging task in applying the monodromy approach. A naive idea is firstly
to apply the monodromy technique for the Heun equation with four simple poles,
and then take a proper limit to get the results for confluent Heun equations,
analogously from Eqs. (66), (67) to Eqs. (71), (72). Of course, we have to
deal with some new subtleties, such as appearance of accessary parameters and
derivation of monodromy matrices etc. Hopefully the results of the no-force
case in Sec. III.2 may provide in part some clues for possible ambiguities.
Nevertheless, even without closed forms, developing rigorous numerics will
contribute numerical results for physical problems under study.

###### Acknowledgements.

C.M.C. would like to appreciate the hospitality at Kunsan National University
and CQUeST, Sogang University, where part of the revision was made. The work
of C.M.C. was supported by the National Science and Technology Council of the
R.O.C. (Taiwan) under the grants MOST 111-2112-M-008-012 and
110-2923-M-002-016-MY3. The work of S.P.K. was supported in part by National
Research Foundation of Korea (NRF) funded by the Ministry of Education
(2019R1I1A3A01063183). AT was supported by JSPS KAKENHI Grant Numbers
JP22K03411 and JP21H01001.

## References

  * [1] J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82 (1951), 664-679.
  * [2] F. Sauter, “Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69 (1931), 742-764.
  * [3] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206]
  * [4] B. S. DeWitt, The global approach to quantum field theory, (Oxford University Press, USA, 2003).
  * [5] V. G. Bagrov and D. Gitman, Exact solutions of relativistic wave equations, (Springer Science & Business Media, 1990).
  * [6] C. M. Bender, S. Orszag and S. A. Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory, (Springer Science & Business Media, 1999).
  * [7] N. Honda, T. Kawai and Y. Takei, Virtual turning points, (Springer, Tokyo, 2015).
  * [8] L. Parker, “Particle creation in expanding universes,” Phys. Rev. Lett. 21 (1968), 562-564.
  * [9] L. H. Ford, “Cosmological particle production: a review,” Rept. Prog. Phys. 84 (2021) no.11, 116901 [arXiv:2112.02444 [gr-qc]].
  * [10] R. Ruffini, G. Vereshchagin and S. S. Xue, “Electron-positron pairs in physics and astrophysics: from heavy nuclei to black holes,” Phys. Rept. 487 (2010), 1-140 [arXiv:0910.0974 [astro-ph.HE]].
  * [11] A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya and G. Torgrimsson, “Advances in QED with intense background fields,” Phys. Rept. 1010, 1-138 (2023) [arXiv:2203.00019 [hep-ph]].
  * [12] C.-M. Chen, S. P. Kim, I.-C. Lin, J.-R. Sun and M.-F. Wu, “Spontaneous Pair Production in Reissner-Nordstrom Black Holes,” Phys. Rev. D 85 (2012), 124041 [arXiv:1202.3224 [hep-th]].
  * [13] C.-M. Chen, S. P. Kim, J.-R. Sun and F.-Y. Tang, “Pair Production in Near Extremal Kerr-Newman Black Holes,” Phys. Rev. D 95 (2017) no.4, 044043 [arXiv:1607.02610 [hep-th]].
  * [14] C.-M. Chen, S. P. Kim, J.-R. Sun and F.-Y. Tang, “Pair production of scalar dyons in Kerr–Newman black holes,” Phys. Lett. B 781 (2018), 129-138 [arXiv:1705.10629 [hep-th]].
  * [15] C.-M. Chen and S. P. Kim, “Schwinger Effect from Near-extremal Black Holes in (A)dS Space,” Phys. Rev. D 101 (2020) no.8, 085014 [arXiv:2002.00394 [hep-th]].
  * [16] J. Zhang, Y.-Y. Lin, H.-C. Liang, K.-J. Chi, C.-M. Chen, S. P. Kim and J.-R. Sun, “Pair production in Reissner-Nordström-Anti de Sitter black holes,” Chin. Phys. C 45 (2021) no.6, 065105 [arXiv:2003.06398 [hep-th]].
  * [17] R.-G. Cai, C.-M. Chen, S. P. Kim and J.-R. Sun, “Schwinger effect in near-extremal charged black holes in high dimensions,” Phys. Rev. D 101 (2020) no.10, 105015 [arXiv:2004.00735 [hep-th]].
  * [18] C.-M. Chen and S. P. Kim, “Dyon Production from Near-Extremal Kerr-Newman-(Anti-)de Sitter Black Holes,” [arXiv:2111.14124 [hep-th]].
  * [19] B. Hsiao, “Pair Production in Charged Black Holes Admitting Heun-to-Hypergeometric Reduction,” MSc. Thesis, National Central University, 2019
  * [20] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé: a modern theory of special functions, (Braunschweig: Vieweg, 2012).
  * [21] A. Castro, J. M. Lapan, A. Maloney and M. J. Rodriguez, “Black Hole Scattering from Monodromy,” Class. Quant. Grav. 30 (2013), 165005 [arXiv:1304.3781 [hep-th]].
  * [22] S. P. Kim and D. N. Page, “Improved Approximations for Fermion Pair Production in Inhomogeneous Electric Fields,” Phys. Rev. D 75 (2007), 045013 [arXiv:hep-th/0701047 [hep-th]].
  * [23] S. P. Kim, “Geometric Origin of Stokes Phenomenon for de Sitter Radiation,” Phys. Rev. D 88, no.4, 044027 (2013) [arXiv:1307.0590 [hep-th]].
  * [24] W. B. Bonnor, “The equilibrium of two charged masses in general relativity,” Phys. Lett. A 83 (1981), 414.
  * [25] A. Chervyakov and H. Kleinert, “Exact Pair Production Rate for a Smooth Potential Step,” Phys. Rev. D 80 (2009), 065010 [arXiv:0906.1422 [hep-th]].
  * [26] S. P. Kim, H. K. Lee and Y. Yoon, “Effective Action of QED in Electric Field Backgrounds II. Spatially Localized Fields,” Phys. Rev. D 82 (2010), 025015 [arXiv:0910.3363 [hep-th]].
  * [27] A. I. Nikishov, “Barrier scattering in field theory removal of klein paradox,” Nucl. Phys. B 21 (1970), 346-358.
  * [28] A. Hansen and F. Ravndal, “Klein’s Paradox and Its Resolution,” Phys. Scripta 23 (1981), 1036
  * [29] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, (Academic press, 2014)


As for $(b)$ and $(c)$, I only thoroughly derive the asymptotic distribution
of $(b)$, because that of $(c)$ comes from a similar argument. I begin by
considering the distribution of $(b)$ given $X$ and $D$. Notice that under
Assumption 1, iid data, and condition (i) for the theorem,

$\displaystyle\operatorname{var}\biggr{(}(b)\ \biggr{|}\
X,D\biggr{)}=\operatorname{var}(\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}\epsilon_{i}\
|\ X,D)=\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$ (85)

Additionally, under conditions (iii) and (v) for the theorem,
$\exists\delta>0$ such that

$\displaystyle\frac{\sum_{i:D_{i}=0}\mathbbm{E}(|\frac{w_{i}}{n_{c}}\epsilon_{i}|^{2+\delta}\
|\ X,D)}{\operatorname{var}(\sum_{i:D_{i}=0}\frac{w_{i}}{n_{c}}\epsilon_{i}\
|\ X,D)^{\frac{2+\delta}{2}}}\rightarrow 0$ (86)

To see this, Assumption 1, iid data, and conditions (i) and (iii) give

$\displaystyle\frac{\sum_{i:D_{i}=0}\mathbbm{E}(|\frac{w_{i}}{n_{c}}\epsilon_{i}|^{2+\delta}\
|\ X,D)}{\operatorname{var}(\sum_{i:D_{i}=0}\frac{w_{i}}{n_{c}}\epsilon_{i}\
|\ X,D)^{\frac{2+\delta}{2}}}$
$\displaystyle=\frac{\sum_{i:D_{i}=0}\mathbbm{E}(|w_{i}\epsilon_{i}|^{2+\delta}\
|\ X,D)}{\operatorname{var}(\sum_{i:D_{i}=0}w_{i}\epsilon_{i}\ |\
X,D)^{\frac{2+\delta}{2}}}$
$\displaystyle=\frac{\sum_{i:D_{i}=0}w_{i}^{2+\delta}\mathbbm{E}(|\epsilon_{i}|^{2+\delta}\
|\ X)}{(\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2})^{\frac{2+\delta}{2}}}$
$\displaystyle\leq\biggr{(}\frac{1}{\underline{C}^{\frac{2+\delta}{2}}}\biggr{)}\biggr{(}\frac{\sum_{i:D_{i}=0}w_{i}^{2+\delta}\mathbbm{E}(|\epsilon_{i}|^{2+\delta}\
|\ X)}{(\sum_{i:D_{i}=0}w_{i}^{2})^{\frac{2+\delta}{2}}}\biggr{)}$ (87)

Then, by condition (v) for the theorem,
$\exists\delta,\overline{C}_{\delta}>0$ that allow (B.3) to continue as

$\displaystyle\frac{\sum_{i:D_{i}=0}\mathbbm{E}(|\frac{w_{i}}{n_{c}}\epsilon_{i}|^{2+\delta}\
|\ X,D)}{\operatorname{var}(\sum_{i:D_{i}=0}\frac{w_{i}}{n_{c}}\epsilon_{i}\
|\
X,D)^{\frac{2+\delta}{2}}}\leq\biggr{(}\frac{\overline{C}_{\delta}}{\underline{C}^{\frac{2+\delta}{2}}}\biggr{)}\biggr{(}\frac{\sum_{i:D_{i}=0}w_{i}^{2+\delta}}{(\sum_{i:D_{i}=0}w_{i}^{2})^{\frac{2+\delta}{2}}}\biggr{)}\rightarrow
0$ (88)

Therefore, $(b)$ satisfies the Lyapunov condition conditional on $X$ and $D$,
meaning that, by the Lindberg-Feller theorem,

$\displaystyle\frac{(b)}{\sqrt{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}}\
\biggr{|}\ X,D\overset{d}{\rightarrow}N(0,1)$ (89)

which also implies here that

$\displaystyle\frac{(b)}{\sqrt{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}}\overset{d}{\rightarrow}N(0,1)$
(90)

or, after some manipulation, that

$\displaystyle\sqrt{n}\times(b)\overset{d}{\rightarrow}N\biggr{(}0,\
\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=0)}\biggr{)}$
(91)

where $\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}<\infty$ by condition
(ii).

Moving to $(c)$, the steps to derive its asymptotic distribution are nearly
identical to those for $(b)$, so I omit them here and simply state the final
results. One finds that

$\displaystyle\frac{(c)}{\sqrt{\frac{1}{n_{t}^{2}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}}\
\biggr{|}\ X,D\overset{d}{\rightarrow}N(0,1)$ (92)

which also means that

$\displaystyle\frac{(c)}{\sqrt{\frac{1}{n_{t}^{2}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}}\overset{d}{\rightarrow}N(0,1)$
(93)

and, after some manipulation, that

$\displaystyle\sqrt{n}\times(c)\overset{d}{\rightarrow}N\biggr{(}0,\
\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=1)}\biggr{)}$
(94)

where $\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}<\infty$ by condition
(ii).

Putting these pieces together, $\sqrt{n}\times(a)$, $\sqrt{n}\times(b)$, and
$\sqrt{n}\times(c)$ are asymptotically independent because they are
independent given $X$ and $D$ and (89) and (92) both hold given $X$ and $D$.
Therefore, by (B.3), (84), (91), and (94)

$\displaystyle\sqrt{n}\biggr{(}\hat{\tau}_{\mathrm{wdim}}^{(\mathrm{ATE})}-\mathrm{ATE}-(d)\biggr{)}$
$\displaystyle\ \ \ \ \ =\sqrt{n}\biggr{(}(a)-(b)+(c)\biggr{)}$
$\displaystyle\ \ \ \ \
\overset{d}{\rightarrow}N\biggr{(}0,\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=0)}+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=1)}\biggr{)}$
(95)

Further, $\sqrt{n}\times(d)\overset{p}{\rightarrow}0$ by condition (vi) for
the theorem. Thus, (B.3) can be rewritten as

$\displaystyle\sqrt{n}(\hat{\tau}_{\mathrm{wdim}}^{(\mathrm{ATE})}-\mathrm{ATE})$
$\displaystyle\ \ \ \ \
\overset{d}{\rightarrow}N\biggr{(}0,\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=0)}+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=1)}\biggr{)}$
(96)

Finally, examining $V_{\mathrm{ATE}}$, it is then evident that

$\displaystyle n\times V_{\mathrm{ATE}}$ $\displaystyle\ \ \ \ \
=\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})+\frac{\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}{\frac{n_{c}}{n}}+\frac{\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}{\frac{n_{t}}{n}}$
$\displaystyle\ \ \ \ \
\overset{p}{\rightarrow}\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=0)}+\frac{\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{t}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}}{p(D_{i}=1)}$ (97)

Thus, applying (B.3) to (B.3) completes the proof.

$\square$

###  B.4 Proof of Theorem 6

$\hat{V}_{\mathrm{ATE}}$ can be rewritten as:

$\displaystyle\hat{V}_{\mathrm{ATE}}$
$\displaystyle=\underbrace{\frac{1}{n^{2}}\sum_{i=1}^{n}\biggr{(}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]+[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]+[\hat{f}_{1}(X_{i})-f_{1}(X_{i})]+[\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}]\biggr{)}^{2}}_{(a)}$
$\displaystyle+\underbrace{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\biggr{(}\epsilon_{i}+[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]\biggr{)}^{2}}_{(b)}$
$\displaystyle+\underbrace{\frac{1}{n_{t}^{2}}\sum_{i:D_{i}=1}w_{i}^{2}\biggr{(}\epsilon_{i}+[f_{1}(X_{i})-\hat{f}_{1}(X_{i})]\biggr{)}^{2}}_{(c)}$
(98)

I prove the theorem by showing that
$(a)\overset{p}{\rightarrow}\frac{1}{n}\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})$,
$(b)\overset{p}{\rightarrow}\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$,
and
$(c)\overset{p}{\rightarrow}\frac{1}{n_{t}^{2}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}$.

Starting with $(a)$,

$\displaystyle(a)$
$\displaystyle=\underbrace{\frac{1}{n^{2}}\sum_{i=1}^{n}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2}}_{(a.1)}$
$\displaystyle+\underbrace{\frac{1}{n^{2}}\sum_{i=1}^{n}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]^{2}}_{(a.2)}$
$\displaystyle+\underbrace{\frac{1}{n^{2}}\sum_{i=1}^{n}[f_{1}(X_{i})-\hat{f}_{1}(X_{i})]^{2}}_{(a.3)}$
$\displaystyle+\underbrace{\frac{1}{n^{2}}\sum_{i=1}^{n}[\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}]^{2}}_{(a.4)}$
$\displaystyle+\underbrace{\frac{2}{n^{2}}\sum_{i=1}^{n}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]\biggr{(}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]+[\hat{f}_{1}(X_{i})-f_{1}(X_{i})]\biggr{)}}_{(a.5)}$
$\displaystyle+\underbrace{\frac{2}{n^{2}}\sum_{i=1}^{n}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}][\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}]}_{(a.6)}$
$\displaystyle+\underbrace{\frac{2}{n^{2}}\sum_{i=1}^{n}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})][\hat{f}_{1}(X_{i})-f_{1}(X_{i})]}_{(a.7)}$
$\displaystyle+\underbrace{\frac{2}{n^{2}}\sum_{i=1}^{n}\biggr{(}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]+[\hat{f}_{1}(X_{i})-f_{1}(X_{i})]\biggr{)}[\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}]}_{(a.8)}$
(99)

Given iid data, and thus by the Law of Large Numbers,

$\displaystyle(a.1)\overset{p}{\rightarrow}\frac{1}{n}\mathbbm{E}\biggr{[}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2}\biggr{]}$
(100)

where $\mathbbm{E}([f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2})$ is finite by
condition (iv) for Theorem 5. By condition (vii) of the theorem,

$\displaystyle(a.2)=o_{p}(n^{-1})$ (101) $\displaystyle(a.3)=o_{p}(n^{-1})$
(102)

and, by the Cauchy-Schwartz Inequality, along with (100), (101), and (102)

$\displaystyle|(a.5)|$ $\displaystyle\leq
2\sqrt{(a.1)}\sqrt{(a.2)}+2\sqrt{(a.1)}\sqrt{(a.3)}$
$\displaystyle=2\sqrt{O_{p}(n^{-1})}\sqrt{o_{p}(n^{-1})}+2\sqrt{O_{p}(n^{-1})}\sqrt{o_{p}(n^{-1})}$
$\displaystyle=o_{p}(n^{-1})$ (103) $\displaystyle|(a.7)|$ $\displaystyle\leq
2\sqrt{(a.2)}\sqrt{(a.3)}=2\sqrt{o_{p}(n^{-1})}\sqrt{o_{p}(n^{-1})}=o_{p}(n^{-1})$
(104)

Then, under the conditions for Theorem 5,
$\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}\overset{p}{\rightarrow}\mathrm{ATE}$,
so

$\displaystyle(a.4)=\frac{1}{n}(\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}})^{2}=o_{p}(n^{-1})$
(105)

and, by iid data, and thus by the Law of Large Numbers,

$\displaystyle(a.6)\overset{p}{\rightarrow}2\biggr{(}\frac{\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}}{n}\biggr{)}\mathbbm{E}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]=o_{p}(n^{-1})$
(106)
$\displaystyle(a.8)\overset{p}{\rightarrow}2\biggr{(}\frac{\mathrm{ATE}-\hat{\tau}^{(\mathrm{ATE})}_{\mathrm{wdim}}}{n}\biggr{)}\mathbbm{E}\biggr{(}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]+[\hat{f}_{1}(X_{i})-f_{1}(X_{i})]\biggr{)}=o_{p}(n^{-1})$
(107)

Therefore, applying (100), (101), (102), (B.4), (104), (105), (106), and (107)
to (B.4) yields

$\displaystyle(a)\overset{p}{\rightarrow}\frac{1}{n}\mathbbm{E}\biggr{[}[f_{1}(X_{i})-f_{0}(X_{i})-\mathrm{ATE}]^{2}\biggr{]}$
(108)

Moving to $(b)$ in (B.4),

$\displaystyle(b)$
$\displaystyle=\underbrace{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\epsilon_{i}^{2}}_{(b.1)}$
$\displaystyle+\underbrace{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]^{2}}_{(b.2)}$
$\displaystyle+\underbrace{\frac{2}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\epsilon_{i}[f_{0}(X_{i})-\hat{f}_{0}(X_{i})]}_{(b.3)}$
(109)

By condition (viii) for the theorem,

$\displaystyle(b.1)\overset{p}{\rightarrow}\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$
(110)

Then, by condition (ix) for the theorem,

$\displaystyle(b.2)=o_{p}(n^{-1})$ (111)

and using the Cauchy-Schwartz Inequality and (110),

$\displaystyle|(b.3)|$ $\displaystyle\leq 2\sqrt{(b.1)}\sqrt{(b.2)}$
$\displaystyle\overset{p}{\rightarrow}2\sqrt{\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}}\sqrt{(b.2)}$
(112)

Then, because $\underset{n\rightarrow\infty}{\mathrm{plim}}\
\frac{1}{n_{c}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$ is finite by condition
(ii) for Theorem 5, we find that
$\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$ in (B.4) is
$O_{p}(n^{-1})$. Additionally using (111) allows (B.4) to continue as

$\displaystyle|(b.3)|\leq
2\sqrt{O_{p}(n^{-1})}\sqrt{o_{p}(n^{-1})}=o_{p}(n^{-1})$ (113)

Therefore, applying (110), (111), and (113) to (B.4) gives

$\displaystyle(b)\overset{p}{\rightarrow}\frac{1}{n_{c}^{2}}\sum_{i:D_{i}=0}w_{i}^{2}\sigma_{i}^{2}$
(114)

As for $(c)$, deriving its probability limit follows nearly identical steps as
does deriving the probability limit of $(b)$. Therefore, I omit the derivation
for $(c)$ and simply state the result. One finds that

$\displaystyle(c)\overset{p}{\rightarrow}\frac{1}{n_{t}^{2}}\sum_{i:D_{i}=1}w_{i}^{2}\sigma_{i}^{2}$
(115)

Finally, applying (108), (114), and (115) to (B.4) completes the proof.

$\square$