1 Introduction

Analysis of linear perturbation of black holes plays an important role in many topics, such as the (in)stability of black hole solutions, the black hole ringdown phase after binary merger and astrophysics [1,2,3]. Among various liner perturbation modes of black holes, superradiance mode is an interesting one, which can extract energy from the black holes [4,5,6,7]. When a charged bosonic wave is scattering off a charged rotating black hole, the wave is amplified by the black hole if the angular frequency \(\omega \) of the wave satisfies

$$\begin{aligned} \omega < \text {m}\Omega _H + e\Phi _H, \end{aligned}$$
(1.1)

where e and \(\text {m}\) are the charge and azimuthal number of the bosonic wave mode, \(\Omega _H\) is the angular velocity of the black hole horizon and \(\Phi _H\) is the electromagnetic potential of the black hole horizon. This superradiant scattering was studied long time ago [8,9,10,11,12,13,14], and has broad applications in various areas of physics(for a recent comprehensive review, see [4]).

For a superradiant black hole and perturbation system, if a mirror-like mechanism is introduced between the black hole horizon and spatial infinity, the amplified perturbation will be scattered back and forth between the “mirror” and black hole horizon, and this will lead to the superradiant instability of the system. This is dubbed black hole bomb mechanism [15,16,17,18]. Superradiant (in)stability of various charged and rotating black holes has been studied extensively in the literature. The superradiant (in)stability of four-dimensional rotating Kerr black holes under massive scalar or vector perturbation has been studied in [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Rotating or charged black holes with certain asymptotically curved space are proved to be superradiantly unstable under massless or massive bosonic perturbation [34,35,36,37,38,39,40,41,42,43,44,45], where the asymptotically curved geometries provide natural mirror-like boundary conditions.

For asymptotically flat black holes, the four-dimensional extremal or non-extremal Reissner–Nordstrom (RN) black hole has been proved superradiantly stable against charged massive scalar perturbation in the full parameter space of the black hole and scalar perturbation system [46,47,48,49,50,51]. The argument in the proof is that the two conditions for the possible superradiant instability of the system, (i) existence of a trapping potential well outside the black hole horizon and (ii) superradiant amplification of the trapped modes, can’t be satisfied simultaneously in the RN black hole and scalar perturbation system [46, 48].

For various higher dimensional black holes, the linear stability analysis has also been studied in the literature (for an incomplete list, see [52,53,54,55,56,57,58,59,60,61,62]). In Ref. [54], the asymptotically flat RN black holes in \(D=5,6,..,11\) are shown to be stable by studying the time-domain evolution of the massless scalar perturbation with a numerical characteristic integration method. In Ref. [55], the authors have provided numerical evidence that asymptotically flat extremal RN black holes are stable for arbitrary D under massless perturbation.

It is known that the mass term of a scalar perturbation may provide a natural mirror-like boundary condition for low frequency perturbation. Recently, an analytical method based on the Descartes’ rule of signs has been developed by the author to study the superradiant stability of higher dimensional RN black holes under charged massive scalar perturbation [63, 64]. Explicitly, the superradiant stability of five and six dimensional extremal RN black holes and five dimensional non-extremal RN black holes has been studied and it is proved that there is no black hole bomb for each case.

In this work, we will go a step further and apply the above mentioned analytical method to study the superradiant stability of arbitrary D-dimensional (\(D\ge 7\)) extremal RN black hole under charged massive scalar perturbation. The effect on the dynamics of the scalar perturbation, which originates from the curved RN black hole, can be described by an effective potential. We will show that there is no potential well for the effective potential experienced by the scalar perturbation. The two conditions for the possible superradiant instability can not be satisfied simultaneously, so there is no black hole bomb for D-dimensional extremal RN black hole under charged massive scalar perturbation and the system is superradiantly stable.

The organization of this paper is as follows: In Sect. 2, we present a general description of the model and the asymptotic analysis of boundary conditions. In Sect. 3, the effective potential of the radial equation of motion is given and the asymptotic behaviors of the effective potential at the horizon and spatial infinity are discussed. In Sect. 4, we present a general description of the proof that there is no potential well outside the black hole horizon for the superradiant modes. Most of the details of the proof for \(D=7\) case and D-dimensional case are in the appendix. The final Section is devoted to the summary.

2 Scalar field in D-dimensional RN black holes

We first present our model with non-extremal D-dimensional RN black hole and then take the extremal limit for further discussion. The metric of the D-dimensional non-extremal RN black hole [63,64,65] is

$$\begin{aligned} ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega _{D-2}^2. \end{aligned}$$
(2.1)

The function f(r) reads

$$\begin{aligned} f(r)=1-\frac{2m}{r^{D-3}}+\frac{q^2}{r^{2(D-3)}}, \end{aligned}$$
(2.2)

where the parameters m and q are related with the ADM mass M and electric charge Q of the RN black hole,

$$\begin{aligned} m= & {} \frac{8\pi }{(D-2)Vol(S^{D-2})}M,~~ \nonumber \\ q= & {} \frac{8\pi }{\sqrt{2(D-2)(D-3)}Vol(S^{D-2})}Q. \end{aligned}$$
(2.3)

Here \(Vol(S^{D-2})=2\pi ^{\frac{D-1}{2}}/\Gamma (\frac{D-1}{2})\) is the volume of unit (\(D-2\))-sphere. \(d\Omega _{D-2}^2\) is the common line element of a (\(D-2\))-dimensional unit sphere \( S^{D-2}\) and can be written as

$$\begin{aligned} d\Omega _{D-2}^2=d\theta _{D-2}^2+\sum ^{D-3}_{i=1} \prod _{j=i+1}^{D-2}\sin ^2(\theta _{j})d\theta _i^2, \end{aligned}$$
(2.4)

where the ranges of the angular coordinates are taken as \(\theta _i\in [0,\pi ](i=2,..,D-2), \theta _1\in [0,2\pi ]\). The inner and outer horizons of this RN black hole are

$$\begin{aligned} r_\pm =(m\pm \sqrt{m^2-q^2})^{1/(D-3)}. \end{aligned}$$
(2.5)

It is obvious that we have the following two equalities

$$\begin{aligned} r_+^{D-3}+r_-^{D-3}=2m,~r_+^{D-3}r_-^{D-3}=q^2. \end{aligned}$$
(2.6)

The electromagnetic field outside the black hole horizon is described by the following 1-form vector potential

$$\begin{aligned} A=-\sqrt{\frac{D-2}{2(D-3)}}\frac{q}{r^{D-3}} dt=-c_D\frac{q}{r^{D-3}} dt. \end{aligned}$$
(2.7)

The equation of motion for a charged massive scalar perturbation in this D-dimensional non-extremal black hole background is governed by the covariant Klein–Gordon equation

$$\begin{aligned} (D_\nu D^\nu -\mu ^2)\phi =0, \end{aligned}$$
(2.8)

where \(D_\nu =\nabla _\nu -ie A_\nu \) is the covariant derivative and \(\mu ,~e\) are the mass and charge of the scalar field respectively. The solution of this equation with definite angular frequency can be decomposed as

$$\begin{aligned} \phi (t,r,\theta _i)=e^{-i\omega t}R(r)\Theta (\theta _i). \end{aligned}$$
(2.9)

The angular eigenfunctions \(\Theta (\theta _i)\) are \((D-2)\)-dimensional scalar spherical harmonics and the corresponding eigenvalues are given by \(-l(l+D-3), (l=0,1,2,..)\) [66,67,68,69,70].

The radial equation of motion is described by

$$\begin{aligned} \Delta \frac{d}{dr}\left( \Delta \frac{d R}{dr}\right) +U R=0, \end{aligned}$$
(2.10)

where

$$\begin{aligned} \Delta= & {} r^{D-2}f(r), \nonumber \\ U&{=}&(\omega {+}e A_t)^2 r^{2(D{-}2)}{-}l(l{+}D{-}3) r^{D{-}4}\Delta {-}\mu ^2 r^{D-2}\Delta .\nonumber \\ \end{aligned}$$
(2.11)

In order to analyze the physical boundary conditions needed here at the horizon and spatial infinity, we define the tortoise coordinate y by \(dy=\frac{r^{D-2}}{\Delta }dr\) and a new radial function \(\tilde{R}=r^{\frac{D-2}{2}}R\), then the radial equation (2.10) can be rewritten as

$$\begin{aligned} \frac{d^2\tilde{R}}{dy^2}+\tilde{U} \tilde{R}=0, \end{aligned}$$
(2.12)

where

$$\begin{aligned} \tilde{U}=\frac{U}{r^{2(D-2)}}-\frac{(D-2)f(r)[(D-4)f(r)+ 2 r f'(r)]}{4r^2}\nonumber \\ \end{aligned}$$
(2.13)

The asymptotic behaviors of \(\tilde{U}\) at the spatial infinity and the outer horizon are

$$\begin{aligned} \lim _{r\rightarrow +\infty }\tilde{U}= & {} \omega ^2-\mu ^2, \nonumber \\ \lim _{r\rightarrow r_+} {\tilde{U}}= & {} \left( \omega -c_D\frac{e q}{r_+^{D-3}}\right) ^2=(\omega -e\Phi _h)^2, \end{aligned}$$
(2.14)

where \(\Phi _h\) is the electric potential of the outer horizon of the RN black hole. Here we need purely ingoing wave condition at the horizon and bound state condition at spatial infinity, which leads to the following two conditions

$$\begin{aligned} \omega< & {} e\Phi _h=c_D\frac{e q}{r_+^{D-3}}, \end{aligned}$$
(2.15)
$$\begin{aligned} \omega< & {} \mu . \end{aligned}$$
(2.16)

The first inequality is the superradiance condition and the second inequality gives the bound state condition.

3 Effective potential and its asymptotic behaviors

In order to analyze the superradiant stability of the RN black hole and scalar perturbation system, we define a new radial function \(\psi =\Delta ^{1/2} R\), then the radial equation of motion (2.10) can be written as a Schrodinger-like equation

$$\begin{aligned} \frac{d^2\psi }{dr^2}+(\omega ^2-V)\psi =0, \end{aligned}$$
(3.1)

where V is the effective potential, which is the main object we will discuss. The explicit expression for the effective potential V is \(V=\omega ^2+\frac{B_1}{A_1}\) and

$$\begin{aligned} A_1= & {} 4r^{2}(r^{2D-6}-2 m r^{D-3}+q^2)^2, \end{aligned}$$
(3.2)
$$\begin{aligned} B_1= & {} 4(\mu ^2-\omega ^2)r^{4D-10}\nonumber \\&+\,(2l+D-2)(2l+D-4)r^{4D-12}\nonumber \\&-\,8(m\mu ^2-c_D eq\omega )r^{3D-7}\nonumber \\&-\,4m(2\lambda _l+(D-4)(D-2))r^{3D-9}\nonumber \\&+\,4q^2(\mu ^2-c_D^2 e^2)r^{2D-4}\nonumber \\&-\,2(2m^2-q^2(2 \lambda _l+3(D-4)(D-2)+2))r^{2D-6}\nonumber \\&-\,4m q^2(D-4)(D-2)r^{D-3}\nonumber \\&+\,q^4(D-4)(D-2), \end{aligned}$$
(3.3)

where \(\lambda _l=l(l+D-3)\).

3.1 Extremal limit

Now we consider the extremal limit by taking \(m=q\). In this limit, the superradiance condition becomes into

$$\begin{aligned} \omega <c_D e. \end{aligned}$$
(3.4)

The expression of the effective potential V becomes into

$$\begin{aligned} V=\omega ^2+\frac{B}{A}, \end{aligned}$$
(3.5)

where A and B read

$$\begin{aligned} A= & {} 4r^{2}(r^{2D-6}-2 m r^{D-3}+m^2)^2\nonumber \\= & {} 4r^2(r^{D-3}-m)^4, \end{aligned}$$
(3.6)
$$\begin{aligned} B= & {} 4(\mu ^2-\omega ^2)r^{4D-10}+(2l+D-2)(2l+D-4)r^{4D-12}\nonumber \\&-\,8(m\mu ^2-c_D e m \omega )r^{3D-7}\nonumber \\&-\,4m(2\lambda _l+(D-4)(D-2))r^{3D-9}\nonumber \\&+\,4m^2(\mu ^2-c_D^2 e^2)r^{2D-4}\nonumber \\&+\,2m^2(2 \lambda _l+3(D-4)(D-2))r^{2D-6}\nonumber \\&-\,4m^3(D-4)(D-2)r^{D-3}\nonumber \\&+\,m^4(D-4)(D-2). \end{aligned}$$
(3.7)

In the extremal limit, the asymptotic behaviors of V at the horizon and spatial infinity are

$$\begin{aligned} r\rightarrow r_h,~~ V\rightarrow -\infty ; \end{aligned}$$
(3.8)
$$\begin{aligned} r\rightarrow +\infty ,~~ V\rightarrow \mu ^2. \end{aligned}$$
(3.9)

At the spatial infinity, the asymptotic behavior of the derivative of the effective potential, \(V'(r)\), is

$$\begin{aligned} V'(r)\rightarrow \left\{ \begin{array}{ll} \frac{-(D-2)(D-4)-4\lambda _l-8m(\mu ^2+c_D e \omega -2\omega ^2)}{2r^3}, &{} {D=5;} \\ \frac{-(D-2)(D-4)-4\lambda _l}{2r^3}, &{} {D\geqslant 6.} \end{array} \right. \nonumber \\ \end{aligned}$$
(3.10)

Given the superradiance condition (3.4) and bound state condition (2.16), we can prove \(V'(r)<0\) at spatial infinity when \(D=5\). It is also obvious that \(V'(r)<0\) at spatial infinity when \(D\geqslant 6\). This means that there is no potential well near the spatial infinity and one maximum exists for the effective potential V(r) outside the black hole horizon.

In the next section, we will prove that there is only one extreme (it is just the maximum mentioned above) outside the event horizon \(r_h\) for the effective potential in the D-dimensional extremal RN black hole case, no potential well exists outside the event horizon for the superradiance modes. So there is no black hole bomb and D-dimensional extremal RN black holes are superradiantly stable under charged massive scalar perturbation. In our proof, the mathematical theorem Descartes’ rule of signs plays an important role, which asserts that the number of positive roots of a polynomial equation with real coefficients is at most the number of sign changes in the sequence of the polynomial’s coefficients.

4 Analysis of the potential wells of V

In this section, we show that there is only one extreme for the effective potential outside the RN black hole horizon by analyzing the derivative of the effective potential \(V'(r)\). Explicitly, it is shown that only one real root exists for the following equation

$$\begin{aligned} V'(r)=0, \end{aligned}$$
(4.1)

when \(r>r_h\).

In the extremal RN black hole case, the derivative of effective potential (3.5) can be expressed as \(V'(r)=\frac{E(r)}{F(r)}\). E(r) and F(r) are polynomials of r, which read as follows

$$\begin{aligned} F(r)= & {} 2 r^3(r^{D-3}-m)^5, \end{aligned}$$
(4.2)
$$\begin{aligned} E(r)= & {} - (D_1 + 4 \lambda _l)r^{5(D-3)}\nonumber \\&+\,m(5D_1+(24-4D)\lambda _l)r^{4(D-3)}\nonumber \\&-\,2m^2(5D_1+(18-4D)\lambda _l)r^{3(D-3)}\nonumber \\&+\,2m^3(5D_1+(8-2D)\lambda _l)r^{2(D-3)}\nonumber \\&-\,4(D-3)m(\mu ^2+\omega (c_D e-2\omega ))r^{4D-10}\nonumber \\&+\,4(D-3)m^2(c_D^2 e^2+2\mu ^2-3c_D e \omega )r^{3D-7}\nonumber \\&+\,4(D-3)m^3(c_D^2 e^2-\mu ^2)r^{2D-4}\nonumber \\&-\,5m^4D_1r^{D-3}+m^5D_1\nonumber \\= & {} a_0+a_1 r^{D-3}+a_2 r^{2D-6}+a'_2 r^{2D-4}+a_3 r^{3D-9}\nonumber \\&+\,a'_3 r^{3D-7}+a_4 r^{4D-12}\nonumber \\&+\,a'_4 r^{4D-10}+a_5 r^{5D-15},\nonumber \\ \end{aligned}$$
(4.3)

where

$$\begin{aligned} a_5= & {} - (D_1 + 4 \lambda _l),~a_4=m(5D_1+(24-4D)\lambda _l),\nonumber \\ a_3= & {} -2m^2(5D_1+(18-4D)\lambda _l),\nonumber \\ a_2= & {} 2m^3(5D_1+(8-2D)\lambda _l),~a_1=-5m^4D_1,\nonumber \\ a_0= & {} m^5D_1,a'_4=-4(D-3)m(\mu ^2+\omega (c_D e-2\omega ))\nonumber \\ a'_3= & {} 4(D-3)m^2(c_D^2 e^2+2\mu ^2-3c_D e \omega ),\nonumber \\ a'_2= & {} 4(D-3)m^3(c_D^2 e^2-\mu ^2). \end{aligned}$$
(4.4)

and \(D_1=D^2-6D+8=(D-2)(D-4)\).

Because we are interested in the real roots of the equation \(V'(r)=0\), only the numerator E(r) of \(V'(r)\) is important for our analysis. It is equivalent to consider the real roots of the equation \(E(r)=0\). After changing the variable r to \(z=r-r_h\), E(r) can be rewritten as a polynomial of z, E(z). A real root of \(E(r)=0\) when \(r>r_h\) is equivalent to a positive root of \(E(z)=0\). The polynomial E(z) can be expanded as

$$\begin{aligned} E(z)=\sum _{i=0}^{5D-15} b_i z^i. \end{aligned}$$
(4.5)

In the following of this section, we will prove that there is only one positive real root for the equation \(E(z)=0\), i.e., only one maximum outside the event horizon \(r_h\) for the effective potential in the D-dimensional extremal RN black hole case and no potential well exists outside the event horizon for the superradiance modes. This is achieved by showing

$$\begin{aligned} \text {sign}(b_{p+1})\leqslant \text {sign}(b_{p}),\quad 0\leqslant p <5D-15. \end{aligned}$$
(4.6)

Then, for the sequence of the real coefficients \((b_{5D-15},b_{5D-16},\ldots ,b_0)\) in the polynomial E(z), the sign change is always 1 and according to Descartes’ rule of signs, the equation \(E(z)=0\) has at most one positive real root.

The constant term \(b_0\) in E(z) is

$$\begin{aligned} b_0= & {} a_0+a_1 r_h^{D-3}+a_2 r_h^{2D-6}+a'_2 r_h^{2D-4}+a_3 r_h^{3D-9}\nonumber \\&+\,a'_3 r_h^{3D-7}+a_4 r_h^{4D-12}+a'_4 r_h^{4D-10}+a_5 r_h^{5D-15}.\nonumber \\ \end{aligned}$$
(4.7)

Plugging (4.4) into the above equation and after a straightforward calculation, we can obtain

$$\begin{aligned} b_0=8(D-3)m^5 r_h^{2}(\omega -c_D e)^2>0. \end{aligned}$$
(4.8)

where we use the equation \(r_h^{D-3}=m\).

It is easy to see that \(a_5=- (D_1 + 4 \lambda _l)<0\). After considering the superradiance condition (3.4) and bound state condition (2.16), it is also easy to verify that

$$\begin{aligned} a'_4= & {} -4(D-3)m(\mu ^2+\omega (c_D e-2\omega ))\nonumber \\= & {} -4(D-3)m(\mu ^2-\omega ^2+\omega (c_D e-\omega ))<0. \end{aligned}$$
(4.9)

So we can immediately find that

$$\begin{aligned} b_{5D-15}= & {} a_{5}=- (D_1 + 4 \lambda _l)<0,\nonumber \\ b_{5D-16}= & {} a_5 C_{5D-15}^1 r_h<0,\nonumber \\&...,\nonumber \\ b_{4D-9}= & {} a_5 C_{5D-15}^{4D-9} r_h^{D-6}<0,\nonumber \\ b_{4D-10}= & {} a_5 C_{5D-15}^{4D-10}r_h^{D-5}+a'_4<0,\nonumber \\ b_{4D-11}= & {} a_5 C_{5D-15}^{4D-11}r_h^{D-4}+a'_4C_{4D-10}^{4D-11}r_h<0. \end{aligned}$$
(4.10)

Then let us consider the coefficient \(b_{4D-12}\) of \(z^{4D-12}\), which is

$$\begin{aligned} b_{4D-12}=a_5 C_{5D-15}^{4D-12}r_h^{D-3}+a'_4 C_{4D-10}^{4D-12}r_h^{2}+a_4. \end{aligned}$$
(4.11)

The term involving \(a'_4\) is negative. Now, we show that the sum of the left two terms, \(a_5 C_{5D-15}^{4D-12}r_h^{D-3}+a_4\), is also negative. It is easy to check that when \(D\ge 7\), \(C_{5D-15}^{4D-12}>5D-15>15\). Then

$$\begin{aligned} a_5 C_{5D-15}^{4D-12}r_h^{D-3}+a_4= & {} -\, m D_1 (C_{5D-15}^{4D-12}-5)\\&-\,4m \lambda _l (C_{5D-15}^{4D-12}-6+D), \end{aligned}$$

The first and second terms on the right of the above equation are both negative, so we have

$$\begin{aligned} b_{4D-12}<0. \end{aligned}$$
(4.12)

4.1 Coefficients of \(z^p\), \(3D-7<p<4D-12\)

For \(3D-7<p<4D-12\), the coefficient of \(z^p\) can be written as follows,

$$\begin{aligned} b_p= & {} a_5 C_{5D-15}^{p}r_h^{5D-15-p}+a'_4 C_{4D-10}^{p}r_h^{4D-10-p}\nonumber \\&+a_4C_{4D-12}^p r_h^{4D-12-p}\nonumber \\= & {} r_h^{4D-12-p}(a_5 C_{5D-15}^{p}m+a'_4C_{4D-10}^{p}r_h^{2}+a_4 C_{4D-12}^p ).\nonumber \\ \end{aligned}$$
(4.13)

On the right of the above equation, the term involving \(a'_4\) is negative because \(a'_4<0\) . Now, let’s prove the sum of the left two terms is also negative in the following and we will neglect the positive factor \(r_h^{4D-12-p}\) for simplicity,

$$\begin{aligned}&a_5 C_{5D-15}^{p}m+a_4 C_{4D-12}^p\nonumber \\&=m C_{5D-15}^{p}(- D_1-4\lambda _l)\nonumber \\&\quad +m C_{4D-12}^p(5D_1+4(6-D)\lambda _l)\nonumber \\&=m D_1(-C_{5D-15}^{p}+5C_{4D-12}^p)\nonumber \\&\quad -4m\lambda _l (C_{5D-15}^{p}+(D-6) C_{4D-12}^p)\nonumber \\&=m D_1C_{4D-12}^p(-\frac{5D-15}{4D-12}\cdot \frac{5D-16}{4D-13}\nonumber \\&\quad \cdots \frac{5D-15-p+1}{4D-12-p+1}+5)\nonumber \\&\quad -4m\lambda _l (C_{5D-15}^{p}+(D-6) C_{4D-12}^p). \end{aligned}$$
(4.14)

The \(\lambda _l\) term in the above equation is obviously negative when \(D\ge 7\). Because

$$\begin{aligned} -\frac{5D-15}{4D-12}\cdot \frac{5D-16}{4D-13}\cdots \frac{5D-15-p+1}{4D-12-p+1}<-\left( \frac{5}{4}\right) ^p,\nonumber \\ \end{aligned}$$
(4.15)

and \((\frac{5}{4})^p>5\) when \(p>3D-7>11\), the \(D_1\) term in (4.14) is also negative. So (4.14) is negative.

We finally obtain that

$$\begin{aligned} b_p<0,\quad 3D-7<p<4D-12. \end{aligned}$$
(4.16)

4.2 Coefficients of \(z^p\), \(p=3D-7,3D-8,3D-9\)

In this subsection, we prove the sign relations between pairs of adjacent coefficients of \(z^p\), \(p=3D-7,3D-8,3D-9\). The three coefficients are listed as following

$$\begin{aligned} b_{3D-7}= & {} a_5 C_{5D-15}^{3D-7}r_h^{2D-8}+a'_4 C_{4D-10}^{3D-7}r_h^{D-3}\nonumber \\&+a_4C_{4D-12}^{3D-7}r_h^{D-5}+a'_3,\nonumber \\ b_{3D-8}= & {} a_5 C_{5D-15}^{3D-8}r_h^{2D-7}+a'_4 C_{4D-10}^{3D-8}r_h^{D-2}\nonumber \\&+a_4C_{4D-12}^{3D-8}r_h^{D-4}+a'_3 C_{3D-7}^{3D-8}r_h,\nonumber \\ b_{3D-9}= & {} a_5 C_{5D-15}^{3D-9}r_h^{2D-6}+a'_4 C_{4D-10}^{3D-9}r_h^{D-1}\nonumber \\&+a_4C_{4D-12}^{3D-9}r_h^{D-3}+a'_3 C_{3D-7}^{3D-9}r_h^2+a_3. \end{aligned}$$
(4.17)

Plugging (4.4) into the above equations and after a straightforward calculation, we can obtain

$$\begin{aligned} b_{3D-7}= & {} r_h^{D-5}(a_5 m C_{5D-15}^{3D-7}+a_4C_{4D-12}^{3D-7})\nonumber \\&+a'_4 m C_{4D-10}^{3D-7}+a'_3\nonumber \\= & {} r_h^{D-5}(-D_1 m(C_{5D-15}^{3D-7}-5C_{4D-12}^{3D-7})\nonumber \\&-4m\lambda _l(C_{5D-15}^{3D-7}+(D-6)C_{4D-12}^{3D-7}))\nonumber \\&+4(D-3)m^2(2C_{3D-7}^{3D-7}+C_{4D-10}^{3D-7})(c_D e-\omega )\nonumber \\&\times \left( \frac{C_{3D-7}^{3D-7} c_D e}{2C_{3D-7}^{3D-7}+C_{4D-10}^{3D-7}}-\omega \right) \nonumber \\&+4(D-3)m^2(\mu ^2-\omega ^2)(2C_{3D-7}^{3D-7}-C_{4D-10}^{3D-7}), \end{aligned}$$
(4.18)
$$\begin{aligned} b_{3D-8}= & {} r_h^{D-4}(a_5 m C_{5D-15}^{3D-8}+a_4C_{4D-12}^{3D-8})+r_h(a'_4 m C_{4D-10}^{3D-8}\nonumber \\&+a'_3C_{3D-7}^{3D-8})\nonumber \\= & {} r_h^{D-4}(-D_1 m(C_{5D-15}^{3D-8}-5C_{4D-12}^{3D-8})\nonumber \\&-4m\lambda _l(C_{5D-15}^{3D-8}+(D-6)C_{4D-12}^{3D-8}))\nonumber \\&+4(D-3)m^2r_h(2C_{3D-7}^{3D-8}+C_{4D-10}^{3D-8})(c_D e-\omega )\nonumber \\&\times \left( \frac{ C_{3D-7}^{3D-8}c_D e}{2C_{3D-7}^{3D-8}+C_{4D-10}^{3D-8}}-\omega \right) \nonumber \\&+4(D-3)m^2r_h(\mu ^2-\omega ^2)(2C_{3D-7}^{3D-8}-C_{4D-10}^{3D-8}), \end{aligned}$$
(4.19)
$$\begin{aligned} b_{3D-9}= & {} r_h^{D-3}(a_5 m C_{5D-15}^{3D-9}+a_4C_{4D-12}^{3D-9})+r_h^2(a'_4 m C_{4D-10}^{3D-9}\nonumber \\&+a'_3C_{3D-7}^{3D-9})+a_3\nonumber \\= & {} r_h^{D-3}(-D_1 m(C_{5D-15}^{3D-9}-5C_{4D-12}^{3D-9}+10)\nonumber \\&-4m\lambda _l(C_{5D-15}^{3D-9}+(D-6)C_{4D-12}^{3D-9}+9-2D))\nonumber \\&+4(D-3)m^2r_h^2(2C_{3D-7}^{3D-9}+C_{4D-10}^{3D-9})(c_D e-\omega )\nonumber \\&\times \left( \frac{ C_{3D-7}^{3D-9} c_D e}{2C_{3D-7}^{3D-9}+C_{4D-10}^{3D-9}}-\omega \right) \nonumber \\&+4(D-3)m^2r_h^2(\mu ^2-\omega ^2)(2C_{3D-7}^{3D-9}-C_{4D-10}^{3D-9}).\nonumber \\ \end{aligned}$$
(4.20)

Here it is not easy to fix the signs of the three coefficients. Now we define the following three normalized coefficients

$$\begin{aligned} \tilde{b}_{3D-7}= & {} \frac{r_h^{3D-7}}{2C_{3D-7}^{3D-7}+C_{4D-10}^{3D-7}}b_{3D-7},\nonumber \\ \tilde{b}_{3D-8}= & {} \frac{r_h^{3D-8}}{2C_{3D-7}^{3D-8}+C_{4D-10}^{3D-8}}b_{3D-8},\nonumber \\ \tilde{b}_{3D-9}= & {} \frac{r_h^{3D-9}}{2C_{3D-7}^{3D-9}+C_{4D-10}^{3D-9}}b_{3D-9}. \end{aligned}$$
(4.21)

It is worth emphasizing that all the normalization factors are positive and the signs of \(b_*\) and \(\tilde{b}_*\) are the same. We show that these coefficients satisfy

$$\begin{aligned} \tilde{b}_{3D-7}<\tilde{b}_{3D-8},\quad \tilde{b}_{3D-8}<\tilde{b}_{3D-9}. \end{aligned}$$

When \(\tilde{b}_{3D-7}<\tilde{b}_{3D-8}\), the possible signs of these two coefficients are \((-,-)(-,+),(+,+)\), which can be denoted as \( \text {sign}(\tilde{b}_{3D-7})\leqslant \text {sign}(\tilde{b}_{3D-8}). \) Similarly, we have

$$\begin{aligned} \text {sign}(b_{3D-7})\leqslant \text {sign}(b_{3D-8}), \end{aligned}$$
(4.22)
$$\begin{aligned} \text {sign}(b_{3D-8})\leqslant \text {sign}(b_{3D-9}). \end{aligned}$$
(4.23)

4.3 Coefficients of \(z^p\), \(0<p<3D-9\)

Similarly, we can prove the sign relations between pairs of adjacent coefficients of \(z^p\), \(0<p<3D-9\). After complicated checks on a case by case basis, we show the following sign relations

$$\begin{aligned} \text {sign}(b_{p+1})\leqslant \text {sign}(b_{p}),~0<p<3D-9. \end{aligned}$$
(4.24)

The details of the proofs for \(D=7\) case and D-dimensional case can be found in the Appendix.

5 Summary

In this work, superradiant stability of D-dimensional (\(D\ge 7\)) extremal RN black hole under charged massive scalar perturbation is studied analytically. Based on the asymptotic analysis of the effective potential V(r) experienced by the scalar perturbation, we know there is one maximum for the effective potential outside the black hole horizon. Then we derive the numerator E(z) of the derivative of the effective potential, which is a polynomial of \(z=r-r_h\) with real coefficients. In Sect. 4, we show in (4.8), (4.10), (4.12), (4.16) that

$$\begin{aligned} b_0>0,\quad b_p<0~(3D-7<p<5D-15). \end{aligned}$$
(5.1)

According the results in (4.22), (4.23), (4.24), we obtain

$$\begin{aligned} \text {sign}(b_{p})\geqslant \text {sign}(b_{p+1}),~(0<p<3D-7). \end{aligned}$$
(5.2)

So the sign change in the following sequence of the real coefficients of E(z),

$$\begin{aligned} (b_{5D-15},b_{5D-16},\ldots ,b_{p+1},b_p,..,b_1,b_0), \end{aligned}$$
(5.3)

is always 1. Then according to Descartes’ rule of signs, we know there is at most 1 positive root for the equation \(E(z)=0\)(i.e. \(V'(r)=0\) when \(r>r_h\)). Thus there is only one extreme for the effective potential outside the horizon, which is the maximum obtained by asymptotical analysis and there is no potential well outside the horizon for the superradiance modes. Given the superradiance condition and bound state condition, a typical shape of the effective potential outside the horizon is shown in Fig. 1. The two conditions for the possible superradiant instability can not be satisfied simultaneously, so there is no black hole bomb for D-dimensional extremal RN black hole under charged massive scalar perturbation.

Fig. 1
figure 1

Given the superradiance condition and bound state condition, a typical shape of the effective potential V(r) outside the black hole horizon \(r_h\)

Full size image

The analytical method used in this paper seems to be efficient to analyze the superradiant stability of higher dimensional black holes. We have already applied it to study the superradiant stability of 5-dimensional non-extremal RN black hole under charged massive scalar perturbation in [63]. As a step further, it is interesting to apply it to study other higher dimensional non-extremal RN black hole cases and even the D-dimensional (\(D\geqslant 6\)) non-extremal RN black hole case. It is also interesting to apply it in studying higher dimensional rotating black hole cases.