Hidden spectral symmetries and mode stability of subextremal Kerr(-dS) black holes

We uncover hidden spectral symmetries of the Teukolsky equation in Kerr(-de Sitter) black holes, recently conjectured by Aminov, Grassi and Hatsuda (arXiv: 2006.06111 and arXiv: 2007.07906, 2020) in the zero cosmological constant case. Using these symmetries, we provide a new, simpler proof of mode stability for subextremal Kerr black holes. We also present a partial mode stability result for Kerr-de Sitter black holes.


Introduction
In General Relativity, a vacuum spacetime is a (1 + 3)-dimensional Lorentzian manifold solving the Einstein equations Ric(g) = Λg , (1.1) where g is the Lorentzian metric and Λ is the cosmological constant. Here, we will focus especially on the Λ ≥ 0 case. Of paramount importance are the Kerr and Kerr-de Sitter, black hole families of solutions to (1.1) with Λ = 0 and Λ > 0, respectively. These are parametrized by their mass M > 0 and a specific angular momentum a ∈ R which is constrained in terms of M and Λ; for instance, in the case Λ = 0, Kerr black holes verify the bound |a| ≤ M . As Kerr(-de Sitter) black holes are stationary spacetimes, they correspond to equilibrium states for (1.1), and one would like to determine whether they are stable or unstable equilibria [RW57]. A great deal of progress on the problem has been achieved for the spherically symmetric non-rotating (a = 0) subfamily and perturbations thereof. In the Λ > 0 case, Hintz and Vasy showed a full nonlinear stability statement when the black hole parameters are such that |a| M, Λ. In the more nuanced Λ = 0 setting, a complete picture of the nonlinear stability of the a = 0 subfamily was only very recently established by Dafermos, Holzegel, Rodnianski and Taylor in [DHRT21]; see also [KS21] for some progress in the direction of an extension to |a| M black holes. Outside these special classes, stability of Kerr(-de Sitter) black holes has remained an open question. Nevertheless, the previous works lay out a clear roadmap for investigating it. The key step in the program is to understand the so-called Teukolsky equation with s = ±2, as it describes the dynamics of some gauge-invariant curvature components in the linearized Einstein equations around Kerr(-de Sitter) black holes. The Teukolsky equation was first obtained in the Λ = 0 case by Teukolsky [Teu73], and later for Λ > 0 in [Kha83]. For Λ ≥ 0, writing Ξ = 1 + a 2 Λ/3, this equation takes the form g α [ in Boyer-Lindquist-type coordinates, where ∆ is a function of r given by ∆ = (r 2 + a 2 ) 1 − Λr 2 3 − 2M r ; (1.3) see already Sections 2.1 and 3.1 for the definitions of ∆ θ and ρ. In (1.2), g is the covariant wave operator on the fixed Kerr(-de Sitter) metric. The parameter s is called a spin-weight taking values in 1 2 Z. Aside from the important case s = ±2 concerning gravitational perturbations, (1.2) gives the dynamics of some gauge-invariant electromagnetic components in the linearized Maxwell equations if s = ±1, and describes perturbations by Dirac fields if s = ± 1 2 and by conformal scalar fields if s = 0. Note that a conformal scalar field is massless in the Λ = 0 setting, but has a specific Klein-Gordon mass in the Λ = 0 case, see Section 3.2.2 for further insight regarding our choice of mass.
If Kerr-(de Sitter) black holes are nonlinearly stable, the most basic statement we can hope to prove for (1.2) is that it is modally stable, i.e. that there are no separable solutions to (1.2) which are exponentially growing or bounded but non-decaying in time. By separable solution we mean a solution of the form where S and α satisfy, respectively, an angular ODE and a radial ODE with suitable boundary conditions; note that α [s] is exponentially growing if Im ω > 0 and bounded, non-decaying in time if ω ∈ R. Our motivation for studying solutions as in (1.4) comes from Carter's result [Car68] that g , and hence (1.2), is separable. The only systematic way of establishing mode stability for a PDE is to find a conserved coercive energy. In a stationary spacetime such as Kerr(-de Sitter) we have an obvious candidate for such an energy: the conserved quantity associated to the stationary Killing field. This energy is coercive for non-rotating, i.e. a = 0, black holes, thus mode stability follows at once in such spacetimes. Perturbative arguments, relying on the celebrated redshift effect of Dafermos and Rodnianski [DR09], allow us to extend the a = 0 mode stability result to the larger class of black holes which are very slowly rotating, i.e. where |a| M [DR10] and, if Λ > 0, |a| M, Λ [Dya11b]. In the general a = 0 case, this approach breaks down completely. The conserved energy associated to the Killing field is generally non-coercive. In the cases s = 0, ±1, ±2 the non-coercivity goes by the name of superradiance. From the point of view of the black hole geometry, superradiance is a consequence of the fact that rotating black holes have ergoregions where the stationary field becomes spacelike. Under the separable ansatz (1.4), superradiance is captured by a simple condition on the frequency parameters ω and m. For instance, take ω ∈ R: if Λ = 0, the superradiant condition reads m = 0 , 0 < ω m < a r 2 + + a 2 , where r + is the largest root of (1.3); if Λ > 0, it reads m = 0 , a r 2 2 + a 2 < ω m < a r 2 1 + a 2 , (1.6) where r 2 , r 1 are, respectively, the largest and second largest roots of (1.3). Furthermore, the stabilizing effect of redshift is generally not strong enough to overcome superradiance. Superradiance, therefore, emerges as an important obstacle in establishing mode stability and, indeed, black hole stability for general back hole parameters: there are several examples where superradiance leads to mode instability. Massive scalar fields on Kerr can produce a black hole bomb [SR15a], and even milder modifications of the scalar potential can lead to non-decaying modes [Mos17] in Kerr. For Kerr's Λ < 0 cousin, Kerr-Anti de Sitter, superradiance may be even more damaging to its stability 1 : massive and massless scalar fields also admit exponentially growing modes [Dol17] (see also [CD04]) on black holes which lie below the Hawking-Reall [HR00] bound.
In light of all these mode instabilities, it is remarkable that, in the Λ = 0 case, mode stability holds for the entire Kerr black hole family, even in the endpoint case |a| = M . This is mostly to the credit of the pioneering work of Whiting [Whi89]. In 1989, Whiting proved mode stability for Im ω > 0 and |a| < M by demonstrating that such mode solutions (1.4) to (1.2) can be injectively mapped into mode solutions of a scalar wave equation on a new spacetime in which the energy associated to the stationary Killing field is coercive. Whiting's map consists of taking appropriate integral and differential transformations of the radial and angular functions, respectively, in (1.4). It turns out, as Shlapentokh-Rothman showed in 2015 [SR15b], that Whiting's integral radial transformation suffices to prove mode stability in the upper half-plane and even on the real axis for |a| < M , see also a different extension to the real axis in [AMPW17]. However, as the transformation is very sensible to the change in the nature of singularities in the radial ODE, it breaks down at the endpoint case |a| = M . There, Whiting's method requires a very different integral transformation, which was only very recently found by the second author [TdC20]. We emphasize that while Whiting's approach to mode stability may be bypassed when considering |a| M black holes, by the reasons described above, it was absolutely crucial to the characterization of solutions to the Teukolsky equation (2.1) in the full subextremal range of parameters |a| < M in the works [DRSR16,SRTdC20,SRTdC21], as no other approach to mode stability for general a was known.
Turning to the Kerr-de Sitter Λ > 0 case, the picture is much less complete, and it remains an open problem to determine whether (1.2) is modally stable for general black hole parameters. The lack of a mode stability statement is one of the reasons why Hintz and Vasy's proof [HV18] of nonlinear stability of the Kerr-de Sitter family cannot be extended past the very slowly rotating |a| M, Λ setting. In fact, in the linear setting of the scalar wave equation, recent work of Petersen and Vasy [PV21] has singled out mode stability as the only obstruction to showing decay in the full subextremal range. This state of affairs is also somewhat surprising: the Λ > 0 case in spherical symmetry and perturbations thereof is much better behaved than the Λ = 0 case (compare, for instance, [DR10] and [Dya11a]), so naively one would expect mode stability to be easier to show in the former case than in the latter. Yet, to employ Whiting's method in Kerr-de Sitter one requires, much like in the |a| = M, Λ = 0 case, a new radial integral transformation, as the nature of the singular points of the radial ODE is very different from the Λ = 0 case. Attempts at finding such a transformation have been unsuccessful, see [Ume00] for a discussion, and no other mechanism of establishing mode stability has been put forth.
The goal of this paper is precisely to revisit mode stability for Kerr(-de Sitter) black holes. In the Λ = 0 setting, we provide a new proof of the classical mode stability result Theorem 1. Fix M > 0 and |a| < M , and let s ∈ 1 2 Z. Then there are no non-trivial mode solutions (1.4) to (1.2) for ω = 0 with Im ω ≥ 0.
Our proof makes use of previously unknown symmetries of the point spectrum of the radial Teukolsky equation. Such symmetries were conjectured to exist by Aminov, Grassi and Hatsuda [AGH21,Hat21a] by comparing the Teukolsky equation with quantization conditions for some supersymmetric gauge theories [IKO17], see also [BCGM21,BILT21]. To establish their conjecture, we rely on a Jaffé expansion for the radial Teukolsky ODE, see [MDW + 95, Part B], a method usually attributed in the black hole community to Leaver's seminal work on quasinormal modes [Lea85]. We also sketch an alternative proof via the so-called MST method of Mano, Suzuki and Tagasuki [MST96], see also [ST03]. Finally, we emphasize that our results hold for Im ω ≥ 0, and we refer the reader to the previous references and the more recent [GW21,GW20] for results concerning the case Im ω < 0.
In the Λ > 0 or Kerr-de Sitter setting, we show that symmetries analogous to those for Λ = 0 hold for the point spectrum of the radial Teukolsky ODE. These are once again inspired by the supersymmetric gauge theories of [IKO17], though to the best of our knowledge have not been conjectured or shown earlier. To prove their existence, rather than a Jaffé expansion, we rely on an expansion in hypergeometric polynomials which, despite being well-known in the classical texts on special ODEs, is to our knowledge new in the General Relativity literature. An alternative proof based on a variant of the MST method introduced in [STU99, STU00] is also sketched briefly. As for Kerr, making use of these novel symmetries, we are able to establish a partial mode stability result for Kerr-de Sitter: Theorem 2. Fix Λ > 0, M > 0 and |a| < 3/Λ so that (1.3) has four distinct real roots, labeled r 3 < r 0 < r 1 < r 2 , and let s ∈ 1 2 Z. Then there are no non-trivial mode solutions (1.4) to (1.2) with ω such that nor with ω such that Im ω > 0 and |ω| ∈ |m| 0, 2a a 2 + 3/Λ − (r 0 + r 2 ) 2 .
Note that no smallness assumptions are made on the black hole specific angular momentum a nor on how close ω is to the endpoints of the superradiant regime, given in (1.6) for real ω. We also remark that the Λ → 0 limit of the proof of Theorem 2 yields precisely Theorem 1.
Finally, let us comment on the distinction between cases |s| = 1 2 , 3 2 and the rest. Similarly to Kerr, if |s| = 1 2 , 3 2 then the conservation law associated to the stationary Killing is coercive for ω ∈ R, and mode stability then follows. In fact, superradiance does not occur for half-integer s in general: the only obstacle to coercivity of the conservation law in this setting is the possible negativity of the so-called Teukolsky-Starobinsky constants for |s| > 2, which, see our previous work [CTdC21], may occur outside the superradiant set (1.6). Thus, the case s ∈ Z is where Theorem 2 is most useful: it rules out some of the modes in the superradiant range (1.5), see Figure 1. The remainder of this paper is organized into two sections. Section 2 addresses the case Λ = 0 and gives a new proof of Theorem 1. Section 3 addresses the case Λ > 0 and contains the proof of Theorem 2. 314824/2020-0. R.TdC. acknowledges support from EPSRC (United Kingdom) grant EP/L016516/1 and from NSF (United States) award DMS-2103173, and thanks André Guerra for his enthusiasm for this project and many useful discussions, as well as Mihalis Dafermos for useful suggestions and Igor Rodnianski for insightful comments. Both authors thank one of the anonymous referees for a very careful reading of this manuscript, and many useful suggestions.
Data availability statement. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Subextremal Kerr black holes 2.1 Geometry of the exterior
In this section, we recall, for the benefit of the reader, some of the basic geometric properties of subextremal Kerr black holes, see for instance [Cha83] or [DR13, Section 5.1] for more details. Fix M > 0, |a| < M , and let The subextremal Kerr black hole exterior is a manifold covered globally (modulo the usual degeneration of polar coordinates) by so-called Boyer-Lindquist coordinates (t, r, θ, φ) ∈ R × (r + , ∞) × S 2 [BL67], and endowed with the Lorentzian metric where we have ρ 2 := r 2 + a 2 cos 2 θ , ∆ := r 2 − 2M r + a 2 = (r − r + )(r − r − ) .
Finally, we will also find it convenient to work with a rescaling of the Boyer-Lindquist r, the tortoise coordinate r * = r * (r) defined by We remark that, throughout this section, we take to denote a derivative with respect to r * .

The Teukolsky equation and its separability
Fix M > 0, |a| ≤ M and s ∈ 1 2 Z. The Teukolsky equation [Teu73] is for ω ∈ C, m − s ∈ Z and a separation constant λ. Plugging (2.2) into (2.1), we find that S each satisfy ODEs, which are introduced in the next two subsections. (2.4)

The angular ODE and its eigenvalues
The reader will find a proof of (2.4) in the positive cosmological setting in Lemma 3.1 below.

The radial ODE and its boundary conditions
Let us introduce the notation (2.6) For j = ±, the quantities ω j and κ j are, respectively, the angular velocity and the surface gravity of the horizon at r = r j . As before, we are interested in studying (2.5) under boundary conditions which ensure that (2.2) can arise from suitably regular initial data for the Teukolsky equation (2.1). Based on the classical theory of regular and irregular singularities for ODEs, see [Olv73, Chapters 5 and 7], we will consider the following boundary conditions: , a,ω m,λ (r)e −iωr r −2iM ω+s admits an asymptotic series as r → ∞ in powers of r −1 .

Precise definition of mode solution
We are finally ready to define mode solution precisely:

Some hidden spectral symmetries
Consider the radial ODE where m 1 , m 2 , m 3 , E, p ∈ C. It is easy to see that (2.5) may be cast in this form: (2.8) Proof. We note the identities

Furthermore, we can recast the ODE boundary conditions in terms of the new parameters:
The potential in (2.5) can thus be written as from where we may now read off the appropriate values of m 1 , m 2 , m 3 , p, E.
Remark 2.7 (Connection with SQCD). In [AGH21] (see also [Hat21a], Aminov Next, we give a characterization of the point spectrum of (2.7), in the space of solutions with suitable boundary conditions, by a Jaffé expansion [Jaf34], also known in the black hole community as Leaver's method [Lea85].
where i 1 , i 2 and i 3 are distinct natural numbers between 1 and 3, of complex numbers verifying three conditions: Let R E,p (mi 1 ,mi 2 ,mi 3 ) be the unique, up to rescaling, solution of the differential equation (c.f. (2.7)) • R(z)e −pz z −mi 3 admits an asymptotic series as z → ∞ in powers of z −1 . Then is nontrivial if and only if the continued fraction condition Proof. From the classical theory of ODEs, see [Olv73, Chapter 7], the second boundary condition is verified if and only if 2 we have that R E,p (mi 1 ,mi 2 ,mi 3 ) (z)e −pz z −mi 3 has a limit as z → ∞. We thus conclude that g defined through is the unique (up to rescaling) nontrivial solution to the ODE with the boundary conditions that g is smooth at z = 1 and has a limit as (z − 1)/z → 1. Here, the coefficients B 1 , . . . , B 5 are given by In light of the strong rigidity in terms of holomorphicity of g around z = 1 afforded by the classical theory of ODEs, see [Olv73,Chapter 5], it is natural to consider a series expansion in known special functions with such properties at z = 1. Following [MDW + 95, Part B, Chapter 3.2], we consider the ansatz for a solution to (2.12) given by (2.14) for some coefficients {b n } n≥0 . Notice that, as long as y can be shown to converge in a neighborhood of z = 1, this ansatz is without loss of generality: by uniqueness of solutions to ODEs with prescribed boundary conditions, any solution of (2.12) which is regular at z = 1 must be a multiple of the ansatz (2.14), so g = Cy for some C ∈ C\{0} in the domain where y is defined. Let us turn to determining this domain. From the identities we deduce that the coefficients {b n } n≥0 in (2.14) satisfy a three-term recurrence formula with respect to the recurrence coefficients The case b n ≡ 0 is trivial. If b n ≡ 0, a large-n asymptotic analysis of (2.15) shows that there are two linearly independent behaviors: writing In either case, lim n→∞ |b n+1 /b n | = 1, and hence (2.14) converges, and defines a holomorphic function, at least for |(z − 1)/z| < 1. Hence, we deduce that g ≡ y up to rescaling for |(z − 1)/z| < 1. At this point, let us note that the continued fraction (2.9) naturally enters the problem because it is a formal solvability condition for (2.15): which holds rigorously as long as the right hand side of (2.9) converges. By [Gau67, Theorem 1.1], convergence occurs if and only if the recurrence relation (2.15) possesses a so-called "minimal solution". By our assumptions on p we have Re C 0 > 0, and hence by comparing (2.16) and (2.17), we find that the latter defines a minimal solution to (2.15).
From the point of view of the Teukolsky equation, Proposition 2.8 is the statement that its point spectrum, under suitable boundary conditions, is invariant under exchanges m i ↔ m j . The precise statement we will use is: (iii) (m 2 ↔ m 3 symmetry) there is a nontrivial solution to the radial ODE Proof. The conclusion follows from Proposition 2.8 after setting z(r + − r − ) = r − r − and rewriting the ODE (2.18) and the boundary conditions in terms of (m 1 , m 2 , m 3 , E, p).
A few remarks concerning the above symmetries are in order.
(2.21) as noted by Hatsuda [Hat21a]. Under the assumption |a| < M , it has been known for some time that this ODE has the same (empty) point spectrum as (2.5) for Im ω ≥ 0 and ω = 0: indeed, this is exactly the ODE obtained by Whiting's (injective) radial transformation in the seminal mode stability paper [Whi89] for Im ω > 0 which is also used to show mode stability for ω ∈ C [SR15b], see also [AMPW17] and [TdC20, Section 3.2].
Remark 2.12 (The extremal limit). Lemma 2.6 clearly fails to hold for |a| = M , as the variable z and the parameter m 2 are not well-defined in this case. The latter degeneration is remedied if we exchange m 2 ↔ m 3 , as from (2.8) we obtain The former degeneration is also cured if we take z = z−r+ r+−M for somer + > r + = r − = M . LetR [s], a,ω m,λ be a solution to (2.7) under these conditions and let denote a derivative with respect to a modified r * variable defined by and an initial condition. Then, the rescaling solves the ODE given by u +Ṽũ = 0, wherẽ

Proof of mode stability
In this section, we give a proof of Theorem 1, alternative to those of the literature [Whi89, SR15b, AMPW17, TdC20], which relies on the hidden symmetries uncovered in Proposition 2.8 and Corollary 2.9. To be precise, we prove Theorem 2.13. Fix M > 0, |a| < M , s ∈ 1 2 Z, m − s ∈ Z and (ω, λ) such that we have either Im ω > 0 and Im(λω) ≤ 0 or ω ∈ R\{0} and λ ∈ R .
If R where prime again denotes a derivative with respect to r * . Note that (2.21) is real if ω is real, that the coefficient of λ in (2.21) and the (ω, m, λ)-independent part ofṼ are non-positive. Thus, from (2.24) and our assumptions, we obtain

Epilogue: mass symmetries within the MST method
In this section, we provide an alternative proof of Corollary 2.9 which is based not on Jaffé expansions of Proposition 2.8 but on the matching of (confluent) hypergeometric expansions. In the study of quasinormal modes, this method was first introduced by Mano, Suzuki and Tagasuki [MST96], and is thus known as the MST method. In the exposition below we follow the review [ST03].
Let us fix M > 0, |a| < M , s ∈ 1 2 Z, m − s ∈ Z, ω ∈ C\{0} with Re ω ≥ 0 and Im ω ≥ 0, and λ ∈ C, assuming additionally that s ≤ 0 if Re ω = mω + . To aid the reader, we write the MST quantities , τ and κ in [ST03] in terms of the quantities identified in (2.8) and vice-versa: Within the MST formalism, existence of a non-trivial solution to (2.5) with outgoing boundary conditions at I + and ingoing boundary conditions at H + is, by [ST03,Equations (165), (167) and (168)] and the properties of the Gamma function, equivalent to the condition 3 B Γ (ν + 1 + m 3 ) where we have used the shorthand notation cannot be a negative integer under our assumptions, so that |Γ(1 − m 1 − m 2 )| < ∞.
Furthermore, the parameter ν is the so-called renormalized angular momentum parameter and its value is chosen to ensure that the continued fraction equation

26) is equivalent to
Clearly, α ν n , β ν n and γ ν n are all separately invariant under the map (m 1 , m 2 , m 3 ) → (m i , m j , m k ) with i = j = k, and so b ν n and ν must also be preserved. Consequently, the same is true for A ν , C ν and D ν . Hence, we conclude that the condition (2.28) is invariant under the map (m 1 , m 2 , m 3 ) → (m i1 , m i2 , m i3 ) with i 1 = i 2 = i 3 = i 1 . By using the fact that (2.5) and the boundary conditions are invariant under taking at once Re ω → − Re ω, m → −m and complex conjugation, we arrive at the same conclusion for Re ω ≤ 0. The statement of Corollary 2.9 then follows easily from exploiting these symmetries.

Geometry of the exterior
In this section, we recall for the benefit of the reader some of the basic geometric properties of subextremal Kerr-de Sitter black holes, see for instance [Dya11b, Section 1] for more details.
Finally, we will also find it convenient to work with a rescaling of the Chambers-Moss r, the tortoise coordinate r * = r * (r) defined by and an initial condition. Note that we have Furthermore, we remark that in this section derivatives denoted by will be assumed to be taken with respect to r * , unless otherwise stated.

The Teukolsky equation and its separability
Fix M > 0, and |a|, L > 0 subextremal Kerr-de Sitter parameters. For s ∈ 1 2 Z, the Teukolsky equation in Kerr-de Sitter [Kha83] is

The angular ODE and its eigenvalues
Let s ∈ Z be fixed. Consider (3.5) and replace aω by a parameter ν ∈ C. The angular ODE verified by S (3.6) We are interested in solutions of (3.6) with boundary conditions which ensure that, when ν is taken to be aω, (3.5) is a smooth s-spin weighted function on R, which can be defined similarly to the Kerr case of the previous section. By analogy with the Kerr case of Lemma 2.1, we have: ∈ R can be analytically continued to ν ∈ C except for finitely many branch points (with no finite accumulation point), located away from the real axis, and branch cuts emanating from these. We define λ , integrating by parts and taking the real part: where the right hand side is clearly non-negative if Im ν > 0. In fact, it is only zero if S [s], Ξ,ν m,λ ≡ 0. As trivial functions are not in the scope of the lemma, we obtain (3.7).
Remark 3.2. We note that, as before, λ denotes a complex number with no restrictions whereas λ denotes one of the eigenvalues identified in Lemma 3.1.

The radial ODE and its boundary conditions
Our starting point is the radial Teukolsky ODE. In the literature, this ODE is usually presented in terms of α for r ∈ (r 1 , r 2 ). This equation has a regular singularity at r = ∞ and singularities at each of the roots of ∆, which are also regular if the Kerr-de Sitter parameters are subextremal. In this paper, we will consider the radial ODE in terms of the variable R (3.9) As noted in [STU98], if µ = 1, equation (3.9) has singularities only at the roots of ∆: r = ∞ is now a regular point of the ODE. Thus, from this point onwards, we take µ ≡ 1. Let us here introduce the notation for j ∈ {0, 1, 2, 3}. Note that we have 3 j=0 η j = 0. The quantities ω j and κ j are, respectively, the angular velocity and the surface gravity of the horizon at r = r j .
Based on the classical theory of regular singularities for ODEs, see [Olv73, Chapter 5], we can consider the following boundary conditions for (3.9): Definition 3.3. We say that a solution, R if Re ω = mω 2 and s ≤ 0.

R
[s], Ξ,a,ω m,λ (r) solves the radial ODE (3.9) with ingoing boundary conditions at H + and outgoing boundary conditions at H + c . We note that Remark 2.5 still applies here, mutatis mutandis.

Some hidden spectral symmetries
For z 2 > 1, consider the radial ODE where m 1 , m 2 , m 3 , m 4 , E ∈ C. The radial ODE (3.9) may also be cast in this form: Lemma 3.5. The radial ODE (3.9) may be written as (3.11) if we let

Furthermore, the boundary conditions in Definition 3.3 can be recast in terms of the new parameters, noting that
The choices of m 1 and m 2 , and of m 3 and m 4 are not canonical: these boundary conditions and the ODE (3.11) are invariant by the exchange of m 1 and m 2 , and by the exchange of m 3 and m 4 . Proof. Under the Möbius transformation (3.12), the singularities of (3.8) transform as follows Note that z 2 < z ∞ as r 3 < r 0 , and furthermore z 2 > 1, since The result then follows by direct computations, which have been independently obtained in the literature in [STU98,STU99], see also [Hat21b].
Remark 3.6. To our knowledge, Lemma 3.5 has not appeared elsewhere in the literature. However, as in the case Lemma 2.6 in the Kerr case, the observation in Lemma 3.5 comes from comparing the Teukolsky equation in subextremal Kerr-de Sitter black holes and the SU (2) Seiberg-Witten theory, but now with four fundamental hypermultiplets, in supersymmetric quantum chromodynamics [IKO17]. We carry out this comparison in analogy to those drawn throughout Aminov, Grassi and Hatsuda's paper [AGH21], and present here the final outcome as Lemma 3.5.
Remark 3.7. Taking the L → ∞ limit of Lemma 3.5 yields Lemma 2.6. For further details about the confluence process which turns Heun equations into confluent Heun equations, we refer the reader to the classical books [MDW + 95, Sla00] and references therein.
Next, we give a characterization of the point spectrum of (3.11), in the space of solutions with suitable boundary conditions. Though other methods such as Jaffé-type expansions are sometimes considered in the literature, see [YUF10], we find it convenient to replace the Jaffé polynomials of Proposition 2.8 with hypergeometric polynomials.
Step 1: a formal expansion. As in Proposition 2.9, it is convenient to identify a set of adequate special functions in which to expand solutions to (3.16). Our construction is loosely based on a small modification of [MDW + 95, Part A, Chapter 4] (see also [Sva39] and [Erd44]). Let us introduce the operators where t := z−1 z2−1 . Then, L may be written as Let us introduce the hypergeometric functions which satisfy where the coefficientsÃ ν are given bỹ .

Now, suppose that the function
is well defined and its derivatives are obtained by differentiating term by term. If y solves L y = 0, it follows that b n verify the recursive relation Up to this point, ν has remained a free parameter; let us now fix ν to be ν = (ω H − 1)/2. One can then check that in (3.21), we shall have b n = 0 for n < 0. Therefore, our final ansatz for a solution of the ODE (3.16) is where b n satisfy (3.21) and f n (z) = F (ω H −1)/2+n (z) are polynomials of degree n in z − 1: denoting by (·) k the rising factorial, where P (·,·) n denote the usual Jacobi polynomials. To compute the region of convergence of the series (3.22), we need to examine the asymptotics of both f n (z) and b n as n → ∞. We begin with the latter: a large n asymptotic analysis of the recursion relation following [Gau67, Theorem 2.3(b)] shows that, as long as b n ≡ 0, we have either Step 2: from (3.14) to g ≡ 0. If the continued fraction equation (3.14) is satisfied, by classical results in the theory of three-term recurrence relations (see [Gau67, Theorem 1.1]), then there exists a nonzero minimal solution to the recursion (3.21). Thus, there are coefficients b n ≡ 0 such that (3.24) holds, implying that, through (3.22), we may construct a solution, g ≡ 0, to the ODE (3.16) which is holomorphic in E .
Step 3: from g ≡ 0 to (3.14). It is a classical result, see [Sze39, Theorem 9.1.1] and the more recent [Car74], that a function which is analytic in an interval admits an expansion in Jacobi polynomials (in addition to, of course, power series), with large freedom in the choice of Jacobi polynomial parameters, which converges in an ellipse around the interval of analyticity. It is convenient to introduce a coordinate adapted to the Jacobi polynomials, x := 1 − 2(z − 1) z 2 − 1 ; we will always assume x = x(z) in what follows. Under the conditions on m i that we impose, the theory guarantees that g(z) = ∞ n=0 2 n (n + ω H ) n G (n) (n + δ, n + ; −1, +1)P (δ−1, −1) converges and defines a holomorphic function in the ellipse E , yielding a representation of the form (3.22) for g in E : by (3.23) In the previous formulas, the coefficients G (n) are given by the integral where Γ is a closed contour in E which contains the interval z ∈ [1, z 2 ] ⇔ x ∈ [−1, 1], and B denotes the beta function.
It is worth noting that the functions in (3.26) are, unsurprisingly, closely related to Jacobi polynomials and verify many of the same identities. Such identities can be used to show that the term-by-term manipulations in step 1 of the series representing g hold. As an example, in view of the decomposition (3.18), let us try to find a representation for Λ 1 g. By assumption, this is a holomorphic function in the ellipse E and thus, by the aforementioned classical theory, it can be written as with coefficients given bỹ G (n) (n + δ, n + ; −1, +1) := n! 2πi Γ Λ 1 g(z)R −n−1 (n + δ, n + ; For the last equality, we have integrated by parts, keeping in mind that Γ lies in the region of holomorphicity of Λ 1 g and is a closed contour, and we have used the notation From the identity Λ 3 R −n−1 (n + δ, n + ; x + 1, x − 1) = −n(n + ω H )R −n−1 (n + δ, n + ; x + 1, x − 1) , which follows from (3.26), we now deduce that consistently with (3.19) after taking ν → n + (ω H − 1)/2. By arguing similarly for the remaining terms of L in (3.18), we deduce that the coefficients b n must satisfy the recursion relation (3.21). Then, by the convergence of the series in (3.25) and the convergence analysis at the end of step 1, b n also need to satisfy the asymptotic relation (3.24), i.e. they must be a minimal solution to the recursion (3.21). Once again, the classical result [Gau67, Theorem 1.1] then implies that the associated continued fraction equation (3.14) (see Proposition 2.9 for details on this relationship) must be satisfied.
If ω ∈ R we cannot argue just from (3.31). As in the Kerr case, we need to appeal to unique continuation for ODEs such as (3.8) to deduce that, unless the superradiant condition holds, we may infer that u ≡ 0, thus concluding the proof for s = 0. In fact, the conclusion can be shown to hold more generally for s ∈ Z ≤2 by appealing to the Teukolsky-Starobinsky identities, see [TM93]. By the same method, one may deduce that if |s| ≤ 2 is half-integer, the energy identity implies that in fact u ≡ 0 holds independently of ω. However, for if |s| > 2, for integer and half-integer spins alike, the energy identity may fail to be coercive even for frequencies not in (3.32), as our previous work [CTdC21] suggests.