A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

We develop a scattering theory for the linear wave equation □gψ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Box _g \psi = 0 $$\end{document} on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} and ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}. This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior.


Introduction
One of the most stunning predictions of general relativity is the formation of black holes, defined by the property that information cannot propagate from their interior region to outside far-away observers. Fortunately, we can count ourselves among the latter; nevertheless, if a group of physicists were so courageous as to cross the event horizon and enter a black hole, they could still very well perform experiments and compare the outcomes among themselves. Indeed, the problem of determining the fate of these black hole explorers (and their laboratories) has led to some of the most central conceptual puzzles in gravitational physics. In view of the above, there has been a lot of recent activity analyzing the Cauchy problem on black hole interiors, e.g., [15][16][17]31,47]. However, for certain physical processes it is more natural to consider the scattering problem (see [18] for scattering on the exterior of black holes). With this paper, we initiate the mathematical study of the finite energy scattering problem on black hole interiors. Specifically, we will consider solutions of the wave equation on what can be viewed as the most elementary interior, that of Reissner-Nordström. The Reissner-Nordström metrics constitute a family of spacetimes, parameterized by mass M and charge Q, which satisfy the Einstein-Maxwell system in spherical symmetry [41,45] and admit an additional Killing vector field T . For vanishing charge Q = 0, the family reduces to Schwarzschild. We shall moreover restrict in the following to the subextremal case where 0 < |Q| < M. In addition to the bifurcate event horizon, these black hole interiors then admit an additional bifurcate inner horizon, the so-called Cauchy horizon. Our past and future scattering states will be defined as suitable traces of the solution on the bifurcate event horizon and bifurcate Cauchy horizon, respectively, restricted to have finite T energy flux on each component of the horizons.
In the rest of the introduction we will state our main results for the scattering problem on the interior of Reissner-Nordström (Theorems 1-5), relate them to existing literature in fixed frequency scattering, and draw links to various recent results in the interior and exterior of black holes. Finally, we will see that the existence of a bounded scattering map for the wave equation on Reissner-Nordström turns out to be a very fragile property; we shall show that there does not exist an analogous scattering theory in the presence of a cosmological constant (Theorem 6) or Klein-Gordon mass (Theorem 7).
The scattering problem on Reissner-Nordström interior. In this paper, we will establish a scattering theory for finite energy solutions of the linear wave equation,  Fig. 1. The first main result of our paper is Theorem 1 (see Sect. 3.1) in which we will show existence, uniqueness and asymptotic completeness of finite energy scattering states. In this context, existence and uniqueness mean that for given finite energy data ψ 0 on the event horizon H, there exist unique finite energy data on the Cauchy horizon CH arising from ψ 0 as the evolution of (1.1). With asymptotic completeness we denote the property that all finite energy data on the Cauchy horizon CH can indeed be achieved from finite energy data on the event horizon H. This provides a way to construct solutions with desired asymptotic properties which is a Figure 1. Penrose diagram of the interior of the Reissner-Nordström black hole and visualization of the scattering map necessary first step to properly understand quantum theories in the interior of a Reissner-Nordström black hole (cf. [14,23,51]). The energy spaces on the event and Cauchy horizon are associated with the Killing field and generator of the time translation T . Indeed, T is null on the horizons and, in particular, is the generator of the event and Cauchy horizon H and CH. Because of the sign-indefiniteness of the energy flux of the vector field T on the bifurcate event (resp. Cauchy) horizon [see already (1.4)], we define our energy space by requiring the finiteness of the T energy on both components separately of the event (resp. Cauchy) horizon. These define Hilbert spaces with respect to which the scattering map is proven to be bounded.
Finally, it is instructive to draw a comparison between the interior of Reissner-Nordström and the interior of Schwarzschild (Q = 0). As opposed to Reissner-Nordström discussed above, the Schwarzschild interior terminates at a singular boundary at which solutions to (1.1) generically blow up (see [15]). In contrast, the non-singular and, moreover, Killing, Cauchy horizons (see Fig. 1) of Reissner-Nordström immediately yield natural Hilbert spaces of finite energy data to consider. In view of this, Reissner-Nordström with Q = 0 can be considered the most elementary interior on which to study the scattering problem. Furthermore, in view of the recent work [7], we have that the causal structure of Reissner-Nordström is stable in a weak sense (see the discussion below about related works in the interior). with potential V [see already (2.37)], where ω ∈ R is the time frequency and ∈ N 0 is the angular parameter. Indeed, most of the existing literature concerning scattering of waves in the interior of Reissner-Nordström mainly considers fixed frequency solutions, e.g., [5,21,22,[33][34][35]52]. For a purely incoming (i.e., supported only on H A ) fixed frequency solution with parameters (ω, ), we can associate transmission and reflection coefficients T(ω, ) and R(ω, ). The transmission coefficient T(ω, ) measures what proportion of the incoming wave is transmitted to CH B , whereas the reflection coefficient specifies the proportion reflected to CH A . An essential step to go from fixed frequency scattering to physical space scattering is to prove uniform boundedness of T(ω, ) and R(ω, ). This is non-trivial in view of the discussion of the energy identity (1.4). In this paper, we indeed obtain this uniform bound in Theorem 2 (see Sect. 3.2). In particular, the regime ω → 0, → ∞ is the most involved frequency range to prove uniform boundedness. As we shall see, the proof relies on an explicit computation at ω = 0 (see [21]) together with a careful analysis of special functions and perturbations thereof.
The uniform boundedness of the scattering coefficients is the main ingredient to prove the boundedness of the scattering map in Theorem 1. Moreover, it allows us to connect the separated picture to the physical space picture by means of a Fourier representation formula. This is stated as Theorem 3 (see Sect. 3.3). A somewhat surprising, direct consequence of the Fourier representation of the scattered data on the Cauchy horizon is that purely incoming compactly supported data lead to a solution which vanishes at the future bifurcation sphere B + . This is moreover shown to be a necessary condition for the existence of a bounded scattering map (Corollary 3.1).
Comparison to scattering on the exterior of black holes. On the exterior of black holes, the scattering problem has been studied more extensively; see the pioneering works [2,3,[11][12][13], the book [18] and related results on conformal scattering in [32,37,40,49]. Note that for the exterior of a Schwarzschild or Reissner-Nordström black hole, the uniform boundedness of the scattering coefficients or equivalently, the boundedness of the scattering map, can be viewed a posteriori 1 as a consequence of the global T energy identity Considering only incoming radiation from I − , this statement translates into |R| 2 + |T| 2 = 1 for the reflection coefficient R and transmission coefficients T. In the interior, however, due to the different orientations of the T vector field on the horizons (cf. Fig. 2), boundedness of the scattering map is not at all manifest. In particular, the global T energy identity on the interior of a Reissner-Nordström black hole reads Figure 2. Interior of Reissner-Nordström (left) and exterior of Schwarzschild or Reissner-Nordström (right). In both diagrams, the arrows denote the direction of the T Killing vector field. Note that we have the identifications H A = H + and B − = B from which we cannot deduce boundedness of the scattering map even a posteriori. (Indeed, identity (1.4) corresponds only to the "pseudo-unitarity" statement of Theorem 1.) Again, considering only ingoing radiation from H A , this translates to for the reflection coefficient R and the transmission coefficient T. Hence, while for fixed |ω| > 0 and , it is straightforward to show that T and R are finite, there is no a priori obvious obstruction from (1.5) for these scattering coefficients to blow up in the limits ω → 0, ±∞ and → ∞. Moreover, note that in the exterior, the Killing field T is timelike, so the radial o.d.e. (1.2) should be considered as an equation for a fixed time frequency wave on a constant time slice. In the interior, however, the Killing field T is spacelike, so the radial o.d.e. (1.2) is rather an evolution equation for a constant spatial frequency.
The Schwarzschild family can be viewed as a special case (a = 0) of the two-parameter Kerr family, describing rotating black holes with mass parameter M and rotation parameter a [26]. 2 On the exterior of Kerr many other difficulties arise: superradiance, intricate trapping, presence of ergoregion, etc. [8]. Nevertheless, using the decay results in [8], a definitive physical space scattering theory for Kerr black holes has been established in [9] (see also the earlier [19]). The proof on the exterior of Kerr involved first establishing a scattering map from past null infinity I − to a constant time slice Σ and then concatenating that map with a second scattering map from Σ to the future event horizon H + and future null infinity I + . In the interior, however, we will directly show the existence of a "global" scattering map from the event horizon H to the Cauchy horizon CH. Indeed, due to blueshift instabilities (see [10]), we do not expect that the analogous concatenation of scattering maps (event horizon H to spacelike hypersurface Σ and then from Σ to the Cauchy horizon CH) as in the Kerr exterior, to be bounded in the interior of Reissner-Nordström.

Injectivity of the reflection map and blueshift instabilities.
We can also conclude from our work that there is always non-vanishing reflection to the Cauchy horizon CH A arising from non-vanishing purely ingoing radiation at H A . This follows from the fact that in the separated picture and for fixed , the reflection coefficient R(ω, ) can be analytically continued to the strip | Im(ω)| < κ + and hence, only vanishes on a discrete set of points on the real axis. This is shown in Theorem 4 (see Sect. 3.4).
We will also deduce from the Fourier representation of the scattered data on the Cauchy horizon CH, and a suitable meromorphic continuation of the transmission coefficient, that there exist purely incoming compactly supported data on the event horizon H leading to solutions which fail to be C 1 on the Cauchy horizon CH. This C 1 -blowup of linear waves puts on a completely rigorous footing the pioneering work of Chandrasekhar and Hartle [5]. We state this as Theorem 5 (see Sect. 3.5).
For generic solutions arising from localized data on an asymptotically flat hypersurface, one expects a stronger instability, namely non-degenerate energy blowup at the Cauchy horizon CH. Such non-degenerate energy blowup was proven in [27] for generic compactly supported data on an asymptotically flat Cauchy hypersurface. Later, for the more difficult Kerr interior, non-degenerate energy blowup was proven in [31] assuming certain polynomial lower bounds on the tail of incoming data on the event horizon H and in [10] for solutions arising from generic initial data along past null infinity I − with polynomial tails.
Finally, we mention the forthcoming work [30] which studies the problem of non-degenerate energy blowup from a scattering theory perspective and also uses the non-triviality of reflection to establish results related to mass inflation for the spherically symmetric Einstein-Maxwell-scalar field system (cf. [28,29]).
Related results on the interior. There has been a lot of recent progress studying the interior of black holes. In particular, new insights were gained concerning the stability of the Cauchy horizon and the strong cosmic censorship conjecture.
For the Cauchy problem for (1.1) on the interior of both a fixed Kerr and a Reissner-Nordström black hole, the works [16,17,24] establish uniform boundedness (in L ∞ ) and continuity up to and including the Cauchy horizon for solutions arising from smooth and compactly supported data on an asymptotically flat spacelike hypersurface. Such data in particular give rise to solutions with polynomial decay along the event horizon.
In contrast, for the scattering problem considered in the present paper, we are required to work with spaces which are symmetric with respect to the event and Cauchy horizons. This naturally leads to the rougher class of finite T energy data in the statement of Theorem 1. Note that for such data on the Cauchy horizon, continuity or boundedness (in L ∞ ) does not necessarily hold true.
Turning finally to the full nonlinear dynamics of the Einstein equations, it is shown in [7] that the Kerr Cauchy horizon is C 0 -stable. Thus, the existence of a Cauchy horizon, a very natural setting parameterizing scattering data in the interior, is not a pure artifact of symmetry but rather a stable property at least in a weak sense. On the other hand, in [28,29,50] it is proven that for a suitable Einstein-matter system under spherical symmetry, the Cauchy horizon, while C 0 -stable, is generically C 2 -unstable. Finally, we mention that for the Schwarzschild black hole (Q = 0), which does not admit a Cauchy horizon, it is shown in [15] that solutions to (1.1) generically blow up at the spacelike singularity {r = 0}.
Breakdown of T energy scattering for Λ = 0 or μ = 0. If a cosmological constant Λ ∈ R is added to the Einstein-Maxwell system, we can consider the analogous (anti-) de Sitter-Reissner-Nordström family of solutions whose interiors have the same Penrose diagram as depicted in Fig. 1. For very slowly rotating Kerr-de Sitter and Reissner-Nordström-de Sitter spacetimes, boundedness, continuity, and regularity up to and including the Cauchy horizon has been shown for solutions to (1.1) arising from smooth and compactly supported data on a Cauchy hypersurface [25]. However, remarkably, there is no analogous T energy scattering theory for either the linear wave equation (1.1) or the Klein-Gordon equation with conformal mass. This is the statement of Theorem 6 (see Sect. 3.6). The reason for this failure is the unboundedness of the transmission coefficient T and reflection coefficients R in the vanishing frequency limit ω → 0. To be more precise, we will prove that there does not exist a T energy scattering theory of the wave or Klein-Gordon equation in the interior of a (anti-) de Sitter-Reissner-Nordström black hole for generic subextremal black hole parameters (M, Q, Λ). In particular, for fixed 0 < |Q| < M, there is an > 0 such that there does not exist a T energy scattering theory for all 0 = |Λ| < .
Similarly, we prove in Theorem 7 (see Sect. 3.7) that there does not exist a T energy scattering theory for the Klein-Gordon equation g ψ − μψ = 0 on the Reissner-Nordström interior for a generic set of masses μ. This is in contrast to the exterior, where T energy scattering theories were established for massive fields in [3,36].
It remains an open problem to formulate an appropriate scattering theory in the cosmological setting and for the Klein-Gordon equation as well as for the interior of Kerr.
Outline. This paper is organized as follows. In Sect. 2, we shall set up the spacetime, introduce the relevant energy spaces, and define the scattering coefficients in the separated picture. In Sect. 3, we state the main results of this paper, Theorems 1-7. Section 4 is devoted to the proof of Theorem 2. Then, the statement of Theorem 2 allows us to prove Theorem 1 in Sect. 5. Finally, in the last two sections are show our non-existence results: In Sect. 6, we prove Theorem 6 and in Sect. 7, we give the proof of Theorem 7.

Preliminaries
In this section, we will define the background differentiable structure and metric for the Reissner-Nordström spacetime and introduce some convenient coordinate systems.

Interior of the Subextremal Reissner-Nordström Black Hole
The global geometry of Reissner-Nordström was first described in [20]. For completeness, we will explicitly construct in this section the coordinates for the region considered. We start, in Sect. 2.1.1, by defining a Lorentzian manifold corresponding to the interior of the Reissner-Nordström black hole without the horizons. Then, in Sect. 2.1.2, we will attach the boundaries which will correspond to the event horizon and Cauchy horizon.

The Interior Without Boundary.
We will now give an explicit description of the differential structure and metric. The Reissner-Nordström solutions [41,45] are a two-parameter family of spherically symmetric spacetimes with mass parameter M ∈ R and electric charge parameter Q ∈ R solving the Einstein-Maxwell system In this paper, we consider the subextremal family of black holes with parameter range 0 < |Q| < M. Define the manifold M by where r ± = M ± M 2 − Q 2 > 0. The manifold is parameterized by the standard coordinates t ∈ R, r ∈ (r − , r + ), and a choice of spherical coordinates (θ, φ) on the sphere S 2 . We denote the global coordinate vector field ∂ t by T : Using the above coordinates, we equip M with the Lorentzian metric where / g S 2 is the round metric on the 2-sphere. We also define the quantities

4)
Furthermore, define r * by where we choose r * ( r++r− 2 ) = 0 for definiteness. Thus, for a constant C only depending on the black hole parameters and (2.8) We also introduce null coordinates v = r * + t and u = r * − t (2.9) on M. The Penrose diagram of M is depicted in Fig. 3.

Attaching the Event and Cauchy
Horizon. Now, we will attach the Cauchy and event horizon to the manifold. The Cauchy horizon characterizes the future boundary up to which the spacetime is uniquely determined as a solution to the Einstein-Maxwell system arising from data on the event horizon. Although the metric is smoothly extendible beyond the Cauchy horizon, any such extension fails to be uniquely determined from initial data, leading to a severe failure of determinism.
Attaching the event and Cauchy horizon gives rise to a manifold with corners. To do so, we first define the following two pairs of null coordinates.
Let U H : R → (0, ∞) and V H : R → (0, ∞) be smooth and strictly increasing functions such that Similarly, let U CH : R → (−∞, 0) and V CH : R → (−∞, 0) be smooth and strictly increasing functions such that To define the event horizon, we consider the coordinate chart (U H , V H , θ, φ). We now define the event horizon without the bifurcation sphere as the union Analogously, we also define the Cauchy horizon without the bifurcation sphere in the coordinate chart (U CH , V CH , θ, φ) as the union (2.14) The Lorentzian metric on M extends smoothly toM. In particular, the boundary ofM consists of four disconnected null hypersurfaces. In Fig. 4, we have depicted its Penrose diagram. In mild abuse of notation, we shall also use the coordinate systems (2.18) In particular, we can write (2.22) Note also that the vector field T , initially defined on M in (2.3), extends to a smooth vector field onM with  sphere B + to the Cauchy horizon which will be denoted by CH. We call the resulting manifold M RN . Further details are not given since the precise construction is straightforward and the metric extends smoothly. Moreover, the T vector field extends smoothly to B + and B − and vanishes there. Its Penrose diagram is depicted in Fig. 5.
Further details about the coordinate systems can be found in [42]. From a dynamical point of view, we can also consider the spacetimes (M RN , g M,Q ) as being contained in the Cauchy development of a spacelike hypersurface with two asymptotically flat ends solving the Einstein-Maxwell system in spherical symmetry.

The Characteristic Initial Value Problem for the Wave Equation
In the context of scattering theory, we will be interested in solutions to the wave equation (1.1) arising from suitable characteristic initial data. Recall the following well-posedness result for (1.1) with characteristic initial data. The previous proposition is well known, see [38,46]. Analogously, we have the following for the backward evolution.

Proposition 2.2.
Let Ψ ∈ C ∞ c (CH) be smooth compactly supported data on the Cauchy horizon CH. Then, there exists a unique smooth solution ψ to (1.1) on M RN \H such that ψ CH = Ψ.

Hilbert Spaces of Finite T Energy on Both Horizon Components
Now, we are in the position to define the Hilbert spaces on the event H = H A ∪ H B ∪ B − and Cauchy horizon CH = CH A ∪ CH B ∪ B + , respectively.
We will start with constructing the Hilbert space on the event horizon. Roughly speaking, it will be defined by requiring finiteness of the T energy flux on H A minus the T energy flux on H B . More precisely, let C ∞ c (H) be the vector space of smooth compactly supported functions on H. Recall that the Killing vector field T is also a null generator of H and vanishes at the past bifurcation sphere B − . This allows us to define the norm · 2 where ψ ∈ C ∞ c (H), T[ψ] is the energy momentum tensor The norm (2.27) defines an inner product, hence its completion is a Hilbert space. Analogously, we can consider the vector space C ∞ c (CH) and define the norm · 2 E T CH as the T energy flux on the component CH B minus the T energy flux on the component CH A : Again, in view of the orientation of the T vector field (cf. Fig. 2 (2.31) Definition 2.2. We define the Hilbert space of finite T energy E T CH on both components of the Cauchy horizon as the completion of smooth and compactly supported functions C ∞ c (CH) the Cauchy horizon CH = CH A ∪CH B ∪B + with respect to the norm (2.30).

Separation of Variables
In this section, we show how the radial o.d.e. (1.2) arises from decomposing a general solution in spherical harmonics and Fourier modes. For concreteness, let ψ be a smooth solution to g ψ = 0 such that on each {r = const.} slice, ψ is compactly supported in the t variable. 4 Then, we can define its Fourier transform in the t variable and also decompose ψ in spherical harmonics to end up witĥ Due to the compact support on constant r slices, the wave equation g ψ = 0 implies thatψ In Sect. 4, we will analyze solutions to (2.34) and denote a solution thereof with R(r). Moreover, it is useful to introduce the function u defined as u(r) := rR(r) (2.35) and consider u = u(r(r * )) as a function of r * , which is defined in (2.7). Using the r * variable, the o.d.e. (2.34) finally reduces to on the real line with potential In Lemma A.3 in Appendix, it is proven that, as a function of r * , the potential V decays exponentially as r * → ±∞. In particular, this indicates that we have asymptotic free waves (asymptotic states) near the event and Cauchy horizon of the form e ±iωr * as |r * | → ∞. In order to construct these solutions, we use the following proposition for Volterra integral equations (see Lemma 2.4 of [48]).
Then, the Volterra integral equation has a unique solution f satisfying

If in addition K is smooth in both variables and
for all k ∈ N, then the solution f is smooth on (− ∞, x 0 ) and the derivatives can be computed by formal differentiation of (2.39).
Remark 2.1. Analogous results as in Proposition 2.3 also hold true for Volterra integral equations on intervals of the form ( This allows us to define the following fundamental pairs of solutions to the o.d.e. (2.36). In view of the exponential decay of the potential, it is straightforward to check that the assumptions of Proposition 2.3 are satisfied.
Analogously, define v 1 and v 2 as the unique solutions to the Volterra integral equations For ω = 0, we set sin(ω(r * −y)) ω ω=0 = r * − y in the integral kernel in which case u 1 and u 2 coincide. We definẽ and similarly,ṽ Since u 1 (0, r * ) = u 2 (0, r * ) for ω = 0, there exists another linearly independent fundamental solutionũ 2 solving the Volterra integral equatioñ Similarly, we also have another fundamental solution, which is linearly independent fromṽ 1 , solving Since r * is not uniformly bounded, we cannot apply Proposition 2.3 to constructũ 2 andṽ 2 . Nevertheless, after switching to coordinates which are regular at H or CH, the existence of the desired solutions follows immediately from the usual local theory of regular singularities (see [44]).
Remark 2.2. Due to the exponential decay of the potential V (see Lemma A.3 in Appendix), it follows from standard theory that the solutions u 1 (ω, r * ), u 2 (ω, r * ), v 1 (ω, r * ) and v 2 (ω, r * ) can be continued to holomorphic functions of ω in the strip | Im(ω)| < κ + for fixed r * ∈ R. Indeed, in [5] it is shown that u 1 (ω, r * ) is analytic in C\{imκ + : m ∈ N} with possible poles at {imκ + : m ∈ N} and similarly for u 2 , v 1 , and v 2 . See also the proof of Proposition A.2 in Appendix.
This allows us now to define the reflection and transmission coefficients R and T.

Definition 2.4.
Let ω = 0. Then, we define the transmission coefficient T(ω, ) and reflection coefficient R(ω, ) as the unique coefficients such that (2.50) Using the fact that the Wronskian of two solutions f and g is independent of r * , we can equivalently define the scattering coefficients as and The transmission and reflection coefficients satisfy a pseudo-unitarity property proven in the following. Applying this to the solution u 1 = Tv 1 + Rv 2 shows the claim.

Proposition 2.4 (Pseudo-unitarity in the separated picture). The transmission and reflection coefficients satisfy
In the following, we shall see that the reflection and transmission coefficients are regular at ω = 0. Proposition 2.5. Let ∈ N 0 be fixed. Then, the scattering coefficients R(ω, ) and T(ω, ) are analytic functions of ω in the strip {ω ∈ C : | Im(ω)| < κ + } with values for ω = 0 given by In particular, the reflection coefficient R(ω, ) only vanishes on a discrete set of points ω.

Conventions
Let X be a point set with a limit point c (e.g., X = R, [a, b], C). Throughout this paper, we will use the symbols and , where the implicit constants might depend on the black hole parameters M and Q. In particular, for functions (or constants) a(x), b(x) > 0 the notation a b means that there is a constant for all x ∈ X. We will also make use of the notation or which means that the constant may additionally also depend on . We also write a We shall also make use of the standard Landau notation O and o [39,44].
We will also use the notation O if the constant C in (2.61) may additionally depend on .

Main Theorems
In this section, we will formulate our main theorems. Theorem 1, which we state in Sect. 3.1, establishes the existence of a scattering map S T of the form which is a Hilbert space isomorphism, i.e., a bounded and invertible map with bounded inverse. Theorem 1 will be proven in Sect. 5. In the separated picture, the boundedness of S T corresponds to the uniform boundedness of the transmission and reflection coefficients which is stated as Theorem 2 in Sect. 3.2. Theorem 2 will be proven in Sect. 4 (and later used in the proof of Theorem 1). Section 3.3 is then devoted to Theorem 3, which connects our physical space scattering theory to the fixed frequency scattering theory. (We will infer Theorem 3 as a consequence of Theorem 1). In Theorem 3.4, this connection allows us to prove that the reflection map is injective, which is the content of Theorem 4. In Theorem 5, which is stated and proven in Sect. 3.5, we construct data which are incoming and compactly supported but, nevertheless, lead to a solution which fails to be in C 1 on the Cauchy horizon.
We end this section with the statement of our two non-existence results. In Sect. 3.6, we formulate Theorem 6, the non-existence of the T energy scattering theory for the Klein-Gordon equation with conformal mass on the interior of (anti-) de Sitter-Reissner-Nordström black holes. The proof of Theorem 6 is given in Sect. 6. Finally, in Theorem 7, stated in Sect. 3.7, we show the non-existence of the T energy scattering map for the Klein-Gordon equation on the interior of Reissner-Nordström. The proof of Theorem 7 is given in Sect. 7.

Existence and Boundedness of the T Energy Scattering Map
First, we define the forward (resp. backward) evolution on a dense domain.
Definition 3.1. The domains of the forward and backward evolution are defined as the Cauchy evolution of ψ has × compact support on constant r = const. hypersurfaces} (3.2) and respectively. Here, we consider r − < r < r + and note that if ψ is compactly supported on one {r = const.} slice, then, as a direct consequence of the domain of dependence, its evolution will be compactly supported on all other {r = const.} hypersurfaces for r − < r < r + . We will prove in Lemma 5.
These definitions of the domains are motivated by the following observation.
Remark 3.1. Suppose we are given data in D T H on the event horizon H. Consider now the unique Cauchy development (cf. Proposition 2.1) and observe that its restriction to the Cauchy horizon CH will lie in D T CH . This holds true since we can first smoothly extend the metric beyond the Cauchy horizon CH and then use the compact support on a constant r * hypersurface to solve an equivalent Cauchy problem in an appropriate region which extends the Cauchy horizon CH, includes the support of the solution, but does not include i + . The smoothness of the solution up to and including the Cauchy horizon CH follows now from Cauchy stability.
In view of Remark 3.1, we can define the forward and backward map on the domains D T H and D T CH , respectively.
as the unique forward evolution from data on the event horizon to data on the Cauchy horizon. More precisely, let ψ be the solution to (1.1) arising from initial data Now, we are in the position to state our main theorem.

bounded and uniquely extends to
called the "scattering map". The scattering map S T is a Hilbert space isomorphism, i.e., a bounded and invertible linear map with bounded inverse Here, In more traditional language, Theorem 1 yields existence, uniqueness, and asymptotic completeness of scattering states.
The proof of Theorem 1 is given in Sect. 5. Let us note already that Theorem 1 is a posteriori the physical space equivalent of the uniform boundedness of the scattering coefficients proven in Theorem 2 (see Sect. 3.2). This equivalence is made precise in Theorem 3 (see Sect. 3.3).
Remark 3.3. Note that in general, neither initial data nor scattered data have to be bounded in L ∞ or continuous. Indeed, we have that H such that its image under the scattering map is not in L ∞ and not continuous. We emphasize the contrast with the estimates from [17] for which more regularity and decay along the event horizon H are necessary.

Uniform Boundedness of the Transmission and Reflection Coefficients
On the level of the o.d.e. (2.36) in the separated picture, the problem of boundedness of the scattering map reduces to proving that the transmission coefficient T and the reflection coefficient R are uniformly bounded over all parameter ranges of ω ∈ R and ∈ N 0 . This is stated as Theorem 2.
Theorem 2. The reflection and transmission coefficients R(ω, ) and T(ω, ) are uniformly bounded, i.e., they satisfy Theorem 2 is proved in Sect. 4. As discussed in the introduction, the proof relies on an explicit calculation for ω = 0 together with a careful analysis of the radial o.d.e. (2.36), involving properties of special functions and perturbations thereof.
Let us note that, given Theorem 1, we could infer Theorem 2 as a corollary (using the theory to be described in Sect. 3.3). We emphasize, however, that in the present paper we use Theorem 2 to prove Theorem 1 in Sect. 5.

Connection Between the Separated and the Physical Space Picture
In this section, we will make the connection of the separated and physical space picture precise.
First, let us note that we have natural Hilbert space decompositions E T Proof. Clearly, the embedding i : as → 0. Analogously, we can do this for H B from which the claim follows.
We will use this identification to represent the scattering map also in the Fourier picture and show how these pictures connect. To do so, we define the following.
and analogously for Ψ B . Hence, in mild abuse of notation, we can define the Fourier and spherical harmonics coefficients F HA (Ψ A ) and F HB (Ψ B ) as (3.14) Also, recall the Hilbert space decomposition E T Thus, the scattering matrix can be also decomposed as is a bounded linear map for i, j ∈ {A, B}. 5

Definition 3.4. Define the Hilbert spaceŝ
The Hilbert spaces defined in Definition 3.4 are unitary isomorphic to their corresponding physical energy spaces. This is captured in Proof. This follows from the fact that the Fourier transform and the decomposition into spherical harmonics are unitary maps.
Now, we will define the scattering map in the separated picture and show that it is bounded.

Proposition 3.3. The scattering map in the separated picturê
defined as the multiplication operator is bounded. Moreover, the mapŜ T is invertible with bounded inverse given bŷ Proof. Indeed,Ŝ T is bounded in view of the uniform boundedness of the transmission and reflection coefficients T and R (cf. Theorem 2). Also note that |T| 2 = 1 + |R| 2 implies that det Ŝ T = 1 (3.22) which shows (3.21). The boundedness ofŜ T −1 is again immediate since the scattering coefficients are uniformly bounded.
Using the previous definitions, we obtain the following connection for the scattering map between the physical space and the separated picture. Theorem 3. The following diagram commutes and each arrow is a Hilbert space isomorphism: Moreover, the maps S T andŜ T are pseudo-unitary satisfying (3.6) and (2.54), respectively. More concretely, for represented by regular distributions as
From the previous representation of the scattered solution, we can draw a link between the boundedness of the scattering map and the fact that compactly supported incoming data will lead to solutions which vanish on the future bifurcation sphere B + . This is the content of the following HA ⊕ E T HB be purely incoming smooth data. Assume further that Ψ A is supported away from the past bifurcation sphere B − and future timelike infinity i + .
Then, the Cauchy evolution ψ arising from Ψ A vanishes at the future bifurcation sphere B + .
On the other hand, if Ψ, as above, led to a solution which does not vanish at the future bifurcation sphere B + , then the scattering map S T : Proof. The first claim is a direct consequence of (3.27) in Theorem 3. For the second statement, let Ψ A be compactly supported data on the event horizon and assume that its Cauchy evolution ψ does not vanish at the future bifurcation sphere B + . Now take dataΨ A which is supported away from the past bifurcation sphere B − and satisfies TΨ A = Ψ A . Then,Ψ A ∈ E T but its Cauchy evolutionψ satisfiesψ CH / ∈ E T CH since  Fig. 5.

Injectivity of the Reflection Map
In this section, we define the reflection operator of purely incoming radiation (cf. Fig. 6) and prove that it is has trivial kernel as an operator from E T HA → E T CHA . Figure 6. Reflection R of purely incoming radiation for all m, , and a.e. ω ∈ R. Moreover, since R(ω, ) only vanishes on a discrete set (cf. Proposition 2.5), we obtain that F HA (Ψ A )(ω, m, ) = 0 for all m, , and a.e. ω ∈ R. Again, in view of Theorem 3, we conclude Ψ A = 0 as an element of E T HA .

C 1 -Blowup on the Cauchy Horizon
In this section, we shall revisit and discuss the seminal work [5] of Chandrasekhar and Hartle. The Fourier representation of the scattered data on the Cauchy horizon in Theorem 3 serves as a good framework to provide a completely rigorous framework for the C 1 -blowup at the Cauchy horizon studied in [5].
Theorem 5 (C 1 -blowup on the Cauchy horizon [5]). There exist smooth, compactly supported, and purely incoming data Ψ A on the event horizon H A for which the Cauchy evolution of (1.1) fails to be C 1 at the Cauchy horizon CH. More precisely, the solution ψ arising from Ψ A fails to be C 1 at every point on the Cauchy horizon CH A ∪ B + . Moreover, the incoming radiation can be chosen to be only supported on any angular parameter 0 which satisfies 0 ( 0 + 1) = r 2 + (r + − 3r − ). Proof. Let 0 be fixed and satisfy 0 ( 0 + 1) = r 2 is smooth and compactly supported. Moreover, assume that the entire function f satisfiesf (iκ + ) = 0. In view of the representation formula (3.27) from Theorem 3, the degenerate derivative of its Cauchy evolution Φ B on the Cauchy horizon CH B reads Since T(ω, ) has a simple pole at ω = iκ + (cf. Proposition A.2 in Appendix), we pick up the residue at iκ + when we deform the contour of integration in (3.32) from the real line to the line Im(ω) = κ + + δ for some κ + > δ > 0. Here, we use that the compact support of f (v) implies the bound |f (ω)| ≤ e | Im(ω)| sup | supp(f )|f (Re(ω)) and that, in addition, by Proposition A.2, the transmission coefficient T remains bounded as | Re(ω)| → ∞. This ensures that the deformation of the integration contour is valid. Hence, by construction. Thus, Φ B is not in C 1 at the future bifurcation sphere as the non-degenerate derivative diverges as v → ∞: where we recall that κ − < −κ + < 0. Finally, propagation of regularity gives that the solution is not in C 1 at each point on the Cauchy horizon CH A . More precisely, expressing (1.1) is (u, v) coordinates gives where Δ is as in (2.5) and where we have used that Δ S 2 ψ = − 0 ( 0 + 1)ψ. Now, note that |ψ|, |∂ u ψ| and |∂ v ψ| are uniformly bounded in the interior by a higher order norm of Ψ A . This follows from [17], commuting with T and angular momentum operators as well as elliptic estimates. Finally, integrating (3.36) in u, using the estimate |Δ| e κ−(u+v) for r * ≥ 0 [see (A.7)] and using the non-degenerate coordinate V CH gives the C 1 blowup also everywhere on CH A .

Breakdown of T Energy Scattering for Cosmological Constants Λ = 0
Interestingly, the analogous result to Theorem 1 on the interior of a subextremal (anti-) de Sitter-Reissner-Nordström black hole does not hold for almost all cosmological constants Λ. In the presence of a cosmological constant, it is also natural to consider the Klein-Gordon equation with conformal mass μ = 3 2 Λ. We will consider in fact a general mass term of the form μ = νΛ, where ν ∈ R. Note that ν = 3 2 corresponds to the conformal invariant Klein-Gordon equation.
To be more precise, we prove that for generic subextremal black hole parameters (M, Q, Λ), there exists a normalized (in E T H -norm) sequence of Schwartz initial data on the event horizon for which the E T CH -norm of the evolution restricted to the Cauchy horizon blows up.
We arising from Ψ n has unbounded T energy at the Cauchy horizon Proof. See Sect. 6.
Remark 3.5. Note that from Theorem 6 it also follows that for fixed 0 < |Q| < M, the T energy scattering breaks down (in sense of Theorem 6) for all cosmological constants 0 < |Λ| < , where = (M, Q) > 0 is small enough.

Breakdown of T Energy Scattering for the Klein-Gordon Equation
Finally, we will also prove that the T energy scattering theory does not hold for the Klein-Gordon equation for a generic set of masses μ, even in the case of vanishing cosmological constant Λ = 0. Proof. See Theorem 7.
The Theorem 6 and Theorem 7 show that the existence of a T energy scattering theory for the wave equation (1.1) on the interior of Reissner-Nordström is in retrospect a surprising property. Implications of the non-existence of a T energy scattering map and, in particular the unboundedness of the scattering map in the cosmological setting Λ = 0, are yet to be understood. (|R(ω, )| + |T(ω, )|) 1.

Proof of Theorem 2: Uniform Boundedness of the Transmission and Reflection Coefficients
(3.7) The proof of Theorem 2 will involve different arguments for different regimes of parameters. Also, note that in view of (2.56) and (2.57) it is enough to assume ω = 0.
The first regime for bounded frequencies (|ω| ≤ ω 0 , arbitrary) requires the most work. One of its main difficulties is to obtain estimates which are uniform in the limit → ∞. We shall use that the o. To prove this, we will essentially divide the real line R r * into three regions.
The first region will be near the event horizon (r * = −∞), where we will consider the potential V as a perturbation. The second region will be the intermediate region, where we will consider the term involving ω as a perturbation. Finally, in the third region near the Cauchy horizon (r * = +∞), we consider the potential V as a perturbation again. This eventually allows us to prove the uniform boundedness of the reflection and transmission coefficients R and T in the bounded frequency regime |ω| < ω 0 .
The second regime will be bounded angular momenta and ω-frequencies bounded from below (|ω| ≥ ω 0 , ≤ 0 ). For this parameter range, we will consider V as a perturbation of the o.d.e. since V might only grow with , which is, however, bounded in that range. Again, this allows us to show uniform boundedness for the transmission and reflection coefficients T and R.
The third regime will be angular momenta and frequencies both bounded from below (|ω| ≥ ω 0 , ≥ 0 ). To prove boundedness of reflection and transmission coefficients R and T, we will consider 1 as a small parameter to perform a WKB approximation.

Low Frequencies (|ω| ≤ ω 0 )
We first analyze the o.d.e. for the special case of vanishing frequency. Then, we will summarize properties of special functions, which we will need to finally prove the boundedness of reflection and transmission coefficients in the low frequency regime. Let be a fixed constant. We define the coordinate x(r) as

Explicit Solution for Vanishing Frequency (ω = 0).
or equivalently, Then, we can write We will denote by P (x) and Q (x) the two independent solutions, the Legendre polynomials and the Legendre functions of second kind, respectively [39,44]. We will prove later in Proposition 4.2 thatũ 1 andũ 2 from Definition 2.3 satisfỹ u 1 (r * ) = w 1 (r * ) := (−1) r(r * ) r + P (x(r * )), (4.8) These are a fundamental pair of solutions for the o.d.e. in the case ω = 0. We will perturb these explicit solutions for the regime of low frequencies (|ω| ≤ ω 0 ). To do so, we will need properties about special functions which will be considered first.
In view of the fact that ω 0 is fixed, constants appearing in and may also depend on ω 0 . Before we begin, we shall summarize the special functions we will use and list their relevant properties in the case |ω| ≤ ω 0 .

Special Functions.
Good references for the following discussion are [1,39,44]. First, we shall recall the definition of the Gamma and Digamma function.

Definition 4.1.
For z ∈ C with Re(z) > 0, we denote the Gamma function with Γ(z) and will also make use of the Digamma function (z) defined as where γ is the Euler-Mascheroni constant.
As we mentioned above, we shall use the Legendre polynomials and the Legendre functions of second kind. We will express them in terms of the hypergeometric function F(a, b; c; x) for x ∈ (−1, 1), a, b, c ∈ R as defined in [44,Equation (9.3)].

Definition 4.2 (Legendre functions of first and second kind ).
We use the standard conventions which are used in [39,44].
For x ∈ (−1, 1), we define the associated Legendre polynomials by and the associated Legendre functions of second kind by (4.14) Here, (4.16) We shall also use the convention P = P 0 and Q m = Q 0 . Also, recall the symmetry In the asymptotic expansion in the parameter for the Legendre polynomials and functions, we will make use of Bessel functions which we define in the following. (4.20) and the Bessel functions of second kind

Definition 4.3 (Bessel functions of first and second kind ). Recall the Bessel functions of first kind
where H k = k n=1 n −1 is the k-the harmonic number. We have the asymptotic expansions Note that bounds deduced from (4.23)-(4.26) hold uniformly on any interval (0, a] of finite length. We shall also use the bounds for 0 < x ≤ 1 and In the proof, we will also use the following asymptotic formulae for P and Q for large in terms of Bessel functions.
where the error terms can be estimated by for θ ∈ (0, π − δ) and for any fixed δ > 0. In particular, this holds uniformly as θ → 0.
We shall use the following asymptotic formulae for the Legendre functions at the singular endpoints.
Lemma 4.2 [39,Sect. 14.8]. For 0 < x < 1, we have x). Moreover, analogous results hold true for −1 < x < 0 due to symmetry. Now, we will estimate the derivatives of the Legendre polynomials and Legendre functions of second kind.
We will consider both summands separately.
Finally, we prove asymptotics for the derivatives of the Legendre of functions of second kind near the singular points.
By symmetry, this also yields for −1 < x < 0 and x → −1 Having reviewed the required facts about special functions, we shall now proceed to prove the uniform boundedness of the reflection and transmission coefficients.

Boundedness of the Reflection and Transmission Coefficients.
As mentioned before, we will consider three different regions: a region near the event horizon, an intermediate region, and a region near the Cauchy horizon. In r * coordinates, we separate these regions at and for 0 < |ω| < ω 0 and ∈ N 0 . Note that −∞ < R * 1 (ω, ) < 0 < R * 2 (ω, ) < ∞.

Region near the event horizon
Proposition 4.1. Let 0 < |ω| < ω 0 and ∈ N 0 . Then, we have |ω|, (4.54) 1. where P and Q are the Legendre polynomials and Legendre functions of second kind, respectively. Our first claim is that we have constructed this fundamental pair (w 1 , w 2 ) to have unit Wronskian and moreoverũ 1 = w 1 and u 2 = w 2 holds true.
For the intermediate region, we will need the following result in order to get uniform bounds for the Volterra iteration. Lemma 4.5. Let 0 < |ω| < ω 0 and ∈ N 0 , then Proof. We first prove (4.93). We shall split the integral in two regions. The first region is from r * = R * 1 to r * = 0. In that region, we define θ ∈ (0, π 2 ] such that cos(θ) = −x(r * ). Using also Lemma 4.1, we obtain The last term shall be treated as an error term. Thus, Here, C and C 1 are positive constants only depending on the black hole parameters. We further estimate using equation (4.95) where we will take care of the term later. First, we look at the term where we have used equation (4.27) and (4.28). Now, we are left with the error term Completely analogously, we can compute The proof of equation (4.93) is completely similar up to a term which involves appearing in the estimate analogous to (4.99).
Region near the Cauchy horizon. Completely analogously to Proposition 4.1, we have Proposition 4.5. Let 0 < |ω| < ω 0 and ∈ N 0 . Then, we have Boundedness of the scattering coefficients. Finally, we conclude that the reflection and transmission coefficients are uniformly bounded for parameters 0 < |ω| < ω 0 and ∈ N 0 . and (4.124)

Frequencies Bounded from Below and Bounded Angular
Momenta (|ω| ≥ ω 0 , ≤ 0 ) Now, we will consider parameters of the form |ω| ≥ ω 0 and ≤ 0 , where ω 0 is small and determined from Sect. 4.1. Also, the upper bound on the angular momentum 0 will be determined from Sect. 4.3. As before, constants appearing in and may depend on ω 0 . In this regime, we assume ω ≥ ω 0 and ≥ 0 , where we choose 0 large enough such that V < 0 everywhere. Note that such an 0 can be chosen only depending on the black hole parameters. We write the o.d.e. as and will represent the solution of the o.d.e. via a WKB approximation. For concreteness, we will use the following theorem which is a slight modification of [43,Theorem 4].
Lemma 4.6 (Theorem 4 of [43]). Let p ∈ C 2 (R) be a positive function such that satisfies sup x∈R F (x) < ∞. Then, the differential equation has conjugate solutions u andū such that  Now we have to show that F is uniformly bounded on the real line. Note that we have the following bounds on the potential and its derivatives |V (r * )|, |V (r * )|, |V (r * )| 2 e 2κ+r * and 2 e 2κ+r * |V (r * )| for r * ≤ 0, (4.147) |V (r * )|, |V (r * )|, |V (r * )| 2 e 2κ−r * and 2 e 2κ−r * |V (r * )| for r * ≥ 0. (4.148) Here, we might have to choose 0 (M, Q) even larger (r 2 + (r + −3r − )+ ( +1) > 0, cf. (A.16)) in order to assure the lower bounds on the potential. Finally, we can estimate F by  The bound on the reflection and transmission coefficients follows now from and analogously for T. Finally, A can be determined from the asymptotic behavior u → e iωr * as r * → −∞ and it is given by which converges since V tends to zero exponentially fast. In particular, this also shows that |A| = 1.
Finally, Theorem 2 is a consequence of Propositions 4.6, 4.7 and 4.8.

Proof of Theorem 1: Existence and Boundedness of the T Energy Scattering Map
Having performed the analysis of the radial o.d.e. and having in particular proven uniform boundedness of the transmission coefficient T and the reflection coefficients R, we shall prove Theorem 1 in this section.

CH
In the following proposition, we shall lift the boundedness of the transmission and reflection coefficients (Theorem 2) to the physical space picture on the dense domains D T H and D T CH . Proposition 5.1. Let ψ be a smooth solution to (1.1) on M RN such that ψ H ∈ D T H (or equivalently, ψ CH ∈ D T CH ). Then, and ψ HA for constants B andB only depending on the black hole parameters.
Proof. Set φ := T ψ and note that φ H ∈ D T H and φ also solves (1.1). Since ψ ∈ D T H ⊂ E T H , we have that φ HA = T ψ HA ∈ L 2 (H A ) with respect to the unique volume form induced by the normal vector field T . Analogously, we also have φ HB = T ψ HB ∈ L 2 (H B ). Thus, we can define the Fourier transform on the event horizon with the charts (2.15) and (2.16) as From Plancherel's theorem, we obtain We can further decompose the Fourier coefficients in spherical harmonics to obtain Again, in view of Plancherel's theorem (5.14) and similarly for CH B . We shall also decompose φ on a constant r slice. Fix r ∈ (r − , r + ), then set This is well defined since φ(t, r, θ, φ) is compactly supported on each r = const. slice.
Since φ is smooth, we also know thatφ m satisfies the radial o.d.e. (2.34) and can be expanded aŝ where |u 1 − e iωr * | e 2κ+r * ∼ (r + − r), (5.18) |u 2 − e −iωr * | e 2κ+r * ∼ (r + − r) (5.19) for r * ≤ 0. Note that this holds uniformly in ω. We shall show in the following that indeed α ,m HA = a ,m HA and α ,m HB = a ,m HB . To do so, note that for r(r * ) with r * ≤ 0 we have for fixed (m, ) that (5.20) We want to interchange the limit r → r + with the integral. In order to use Lebesgue's dominated convergence theorem we will estimate α ,m HA and α ,m HB . Note that which is independent of r(r * ) and integrable since ω →ψ m (ω, r * ) is a Schwartz function. Now, we shall fix v = r * + t and let r → r + such that r * → −∞.  (5.32) show the claim in view of (5.8), (5.9), (5.13), and (5.14). Finally, in view of the Fourier inversion theorem, note that the previous also justifies the Fourier representation of scattering map (3.20), and the Fourier representations (3.24) and (3.25).

Completing the Proof
Having proven Lemma 5.1 and Proposition 5.1, we can finally show Theorem 1 in the following.
Proof of Theorem 1.

Proof of Theorem 6: Breakdown of T Energy Scattering for Cosmological Constants Λ = 0
In the presence of a cosmological constant Λ, the situation regarding the T energy scattering problem is changed radically. In this section we will consider the subextremal (anti-) de Sitter-Reissner-Nordström black hole interior (M (a)dSRN , g Q,M,Λ ) which is completely analogous to (M RN , g Q,M ). We will assume that (M, Q, Λ) ∈ P se as defined in Sect. 3.6. Also, recall that in the presence of a cosmological constant it is natural to look at the Klein-Gordon equation with mass μ = 3 2 Λ for the conformal invariant equation or more general μ = νΛ for fixed ν ∈ R.
This section is devoted to prove Theorem 6 which relies on the fact that solutions of the corresponding radial o.d.e. in the vanishing frequency limit ω = 0 generically map bounded solutions at r * = −∞ to unbounded solutions at r * = +∞. More precisely, for Λ = 0 we obtain-after separation of variables for (6.1) and setting dr * = h −1 dr-the o.d.e.
for u(r * ) = r(r * )R(r * ), where Here, consider r(r * ) as a function r * and recall that denotes the derivative with respect to r * . The presence of the mass and the cosmological constant leads to a modification of the potential V ,Λ . Nevertheless, the potential V ,Λ still decays exponentially at ±∞ and we can define asymptotic states u for all Λ such that (M, Q, Λ) ∈ P se .
Step 1: P se ⊂ R 3 is open and has two connected components where either Q > 0 or Q < 0 For the sake of completeness, we will give a proof of Step 1, although this seems a quite well-known fact. Note that P se = P To conclude, note that for every (M 0 , Q 0 , Λ 0 = 0) ∈ P Λ=0 se , there exist paths from (M 0 , Q 0 , Λ 0 ) to both (M 0 , Q 0 , ) ∈ P Λ>0 se and (M 0 , Q 0 , − ) ∈ P Λ<0 se for some (M 0 , Q 0 ) > 0. Together with the fact that P Λ=0 se ∩ {Q > 0} is pathconnected, this shows that {Q > 0} ∩ P se is path-connected and similarly that {Q < 0} ∩ P se is path-connected which proves the claim.

(6.22)
Now, taking ∂ Λ Λ=0 and integrating by parts once again yields Recall that we are in the subextremal range which guarantees that p(r(1)) = 0. We will now distinguish two cases, ν = 0 and ν = 0.
Proof. Fix 0 = 0 (ν) from Proposition 6.1 and (M, Q, Λ) ∈ P se such that B( 0 , Λ, M, Q) = 0. Now, note that the o.d.e. implies that d dr * Im(ūu ) = 0 which shows that 1 = |T| 2 − |R| 2 . In particular, either |T| and |R| are both bounded or both unbounded as ω → 0. Also note that as ω → 0, we have that u Finally, this allows us to prove Theorem 6 which we restate in the following for the convenience of the reader. Proof. Fix 0 = 0 (ν) from Proposition 6.2 such that the reflection and transmission coefficients blow up as ω → 0. Define a sequence of compactly supported functions Ψ n on H A by Ψ n (v, θ, ϕ) = f n (v)Y 0 (θ, ϕ), such that f n ∈ C ∞ c (R), R ω 2 |f n (ω)| 2 dω = 1 and 1 n − 1 n ω 2 |f n (ω)| 2 dω ≥ R ω 2 |f n (ω)| 2 dω = (6.31) for some > 0. 6 Imposing vanishing data on H B , this gives rise to a unique smooth solutions ψ n up to but excluding the Cauchy horizon. Arguments completely analogous to those given in the proof of

Proof of Theorem 7: Breakdown of T Energy Scattering for the Klein-Gordon Equation
In this last section, we will prove that for a generic set of Klein-Gordon masses, there does not exist a T scattering theory on the interior of Reissner-Nordström for the Klein-Gordon equation. For the convenience of the reader, we have restated Theorem 7.  Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Proof. We shall use the vector field S = r −2 ∂ r * . By potentially making r red larger, we can assure that the bulk term K S := ∇ μ J S μ of the vector field S has a fixed negative sign in r 0 ∈ (r red , r + ). This current is analogous to the current introduced in [ This concludes the proof.
Analytic properties of the potential and the scattering coefficients. In the following, we would like to summarize analytic properties of the potential V (r) and u 1 ,u 2 , v 1 and v 2 as functions of ω. This is similar to parts of [5]. First, however, we will show the exponential decay of the potential V as r * → ±∞. for a smooth function f (r * ), which is uniformly bounded below and above for r * ≤ 0. Moreover, we have f (r * ), f (r * ) → 0 exponentially fast as r * → −∞. The estimates (A.8) and (A.9) are now straightforward applications of the chain rule and the fact that dr dr * = Δ r 2 and Δ = (r − r − )(r − r + ). where C > 0 is such that z( r++r− 2 ) = 1. From the inverse function theorem, it follows that V (z) = V (r(z)) can be analytically continued in a neighborhood of z = 0 and thus, there exists a Taylor expansion around z = 0 such that (A.14) Hence, Moreover, T(ω, ) has a pole of order one at ω = iκ + given that ( + 1) = r 2 + (r + − 3r − ). Proof. Recall, that u 1 is the unique solution to In [5], it is shown that the Volterra iteration has the form u 1 (r * ) = e iωr * 1 + has a pole of order one at ω = iκ + since C 1 = 0, see (A.16). Since for n = 1 there is no term of the form e 2κr * in (A.22) as m n ≥ n, the pole at ω = iκ + cannot be canceled by the other terms and must occur in u 1 . Moreover, this pole of u 1 at ω = iκ + is not of higher order that one since d 1 does not occur at higher powers than one in the Volterra iteration. This implies that T(ω, ) has a pole of order one at ω = iκ + .