# Structural aspects of the anti-de Sitter black hole pseudospectrum

Valentin Boyanov1, Vitor Cardoso1,2, Kyriakos Destounis3,4, José Luis
Jaramillo5 and Rodrigo Panosso Macedo2 1CENTRA, Departamento de Física,
Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco
Pais 1, 1049 Lisboa, Portugal  2 Niels Bohr International Academy, Niels Bohr
Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark  3Dipartimento di Fisica,
Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy  4INFN,
Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy  5Institut de
Mathématiques de Bourgogne (IMB), UMR 5584, CNRS, Université de Bourgogne,
F-21000 Dijon, France

###### Abstract

Black holes in anti-de Sitter spacetime provide an important testing ground
for both gravitational and field-theoretic phenomena. In particular, the study
of perturbations can be useful to further our understanding regarding certain
physical processes, such as superradiance, or the dynamics of strongly coupled
conformal field theories through the holographic principle. In this work we
continue our systematic study of the ultraviolet instabilities of black-hole
quasinormal modes, built on the characterization of the latter as eigenvalues
of a (spectrally unstable) non-selfadjoint operator and using the
pseudospectrum as a main analysis tool, extending our previous studies in the
asymptotically flat setting to Anti-de Sitter asymptotics. Very importantly,
this step provides a singularly well-suited probe into some of the key
structural aspects of the pseudospectrum. This is a consequence of the
specific features of the Schwarzschild-anti-de Sitter geometry, together with
the existence of a sound characterization by Warnick of quasinormal modes as
eigenvalues, that is still absent in asymptotic flatness. This work focuses on
such structural aspects, with an emphasis on the convergence issues of the
pseudospectrum and, in particular, the comparison between the hyperboloidal
and null slicing cases. As a physical by-product of this structural analysis
we assess, in particular, the spectral stability of purely imaginary
“hydrodynamic” modes, which appear for axial gravitational perturbations, that
become dominant when the black-hole horizon is larger than the anti-de Sitter
radius. We find that their spectral stability, under perturbations, depends on
how close they are to the real axis, or conversely how distant they are from
the first oscillatory overtone.

##  I Introduction

Quasinormal mode (QNM) frequencies of black hole (BH) spacetimes, namely the
complex frequencies capturing the linear response to external perturbations
and encoding intrinsic geometric information of the background, play a major
role in different aspects of gravitational physics [1, 2, 3, 4, 5], from
fundamental stability results in mathematical relativity, applications to the
late-time ringdown phase of dynamical BH spacetimes in gravitational-wave (GW)
astronomy, high-energy astrophysical phenomena, such as superradiance and
superradiant instabilities [6, 7, 8, 9], or in the bulk-boundary duality in
the anti-de Sitter/conformal field theory (AdS/CFT) setting [10, 11].
Therefore, the study of their structural stability aspects is of major
importance. Recent studies have put a focus on the QNM spectral stability
under small perturbations of the environment [12, 13, 14, 15], raising
fundamental questions that remain open. In this work we focus on this problem
dwelling in the setting of asymptotically AdS spacetimes, since they provide a
particularly well controlled mathematical environment for the proof-of-
principle study that we aim at here.

###  I.1 QNM spectral instability in BH spacetimes

The sensitivity of BH QNM frequencies to small perturbations of the
environment was early identified in the works by Nollert & Price and
Aguirregabiria & Vishveshwara [16, 17, 18, 19], receiving since then some
attention in the literature, mainly in the GW setting [20, 12, 21, 22].

Recently, an approach to the BH QNM stability problem based on the discussion
of the eigenvalue stability in non-selfadjoint spectral problems has been
proposed111The setting is more general than that of BH QNMs, actually
extending to QNM stability in general linear wave equations, in particular
under perturbations in generic potentials but also the permittivity function
in the optical setting [23]. in [24, 25, 23, 26], in particular introducing
the notion of pseudospectrum [27, 28, 29, 30, 31, 32] both into the QNM
problem and in gravitational physics. This scheme has been then applied to a
variety of BH settings, proving to be robust222A key aspect of this BH
instability is its ultraviolet (high-wavenumber/low-regularity [33, 34, 35]
and Refs. in [26]) character, reflected in particular in the logarithmic
universality of large overtones probing small scales that make that, as long
as asymptotics/boundary conditions are leaky, their detailed form seems
important for the qualitative QNM instability effect. under various spacetime
asymptotics (asymptotically flatness, de Sitter (dS) and AdS) and boundary
conditions [24, 36, 37, 38, 39, 40]. Of particular importance in the present
work is the systematic study in Arean et al. [39], further extended333In the
very late stage of the writing this manuscript, we became aware of the work
[40]. Results in [40] and the ones here presented, in particular regarding the
discussion of the pseudospectrum in a null slicing, are independent and
complementary. by Cownden et al. [40], that discuss the asymptotically AdS
case providing, in particular, an excellent motivation for the study of the BH
QNM spectral instability in the AdS/CFT context. Beyond QNM spectral
instability, other aspects of the pseudospectrum in the (non-selfadjoint) BH
problem, namely transients and pseudo-resonances, have been explored in [41,
37] (cf. also [40]).

These BH QNM instability studies have prompted other related works exploring
associated aspects [42, 43, 44, 45], not specifically using a non-selfadjoint
spectral problem framework, but confirming the qualitative instability
picture, deepening in questions such as the instability of the fundamental QNM
and opening new complementary problems. A recent review, focused on the GW BH
spectroscopy problem, can be found in [46].

####  I.1.1 Pseudospectrum and BH QNMs: open problems

Given a selfadjoint operator $L$ (more generally a so-called ‘normal operator’
[29], namely commuting with its adjoint $[L,L^{\dagger}]=0$), its spectrum
$\sigma(L)$ encodes fundamental information of $L$ that is intrinsic to the
operator, not depending on any other object. However, when the operator is
non-normal ($[L,L^{\dagger}]\neq 0$) the spectrum $\sigma(L)$ is no longer
necessarily the good notion to consider.

In such a non-normal setting, a more robust notion that partially substitutes
the spectrum, in particular regarding spectral stability and evolution aspects
when $L$ is considered as an infinitesimal time generator, is the notion of
the pseudospectrum (or more properly the $\epsilon$-pseudospectrum set) that
we briefly revisit in Sec. III. However, there is a price to pay: the
pseudospectrum is not a notion intrinsic to $L$, but depends on the choice of
a norm. This point has been addressed in detail in [26]. Specifically, the
pseudospectrum sets can be characterized in terms of the norm of the resolvent
$R_{L}(\omega)=(L-\omega\mathrm{Id})^{-1}$, that is, in terms of
$\|R_{L}(\omega)\|$. As a consequence, the choice of norm (and not only the
structure of the operator $L$) plays a critical role in the discussion. In
[24, 25, 23, 26] and the following articles, the chosen norm has been the
‘energy norm’ $\|\cdot\||_{{{}_{E}}}$ defined in terms of the energy of the
propagating linear field. This is a natural norm in many partial differential
equation contexts, in particular to control initial data in second-order
Cauchy problems (see also [28] in a pseudospectrum related context). In
particular, in our BH QNM stability context it seems a very natural norm in
the spirit of estimating the ‘size’ of operator perturbations $\delta L$ in
terms of the injected energy in the system (although such relation is far from
straightforward, cf. [26]). However, the fact that the energy norm, in
particular the associated ‘energy scalar product’, might be an appropriate one
to characterize the size of perturbations from a physical perspective in a ‘BH
QNM stability problem’ does not mean that it is also the good one to ‘define’
the studied BH QNMs. This ‘stability versus definition’ problem will be
addressed in more detail in [47]. Here we mention that other norms might, and
actually are, key in the BH QNM problem.

Let us mention the following issues regarding the use of the pseudospectrum in
the BH QNM stability problem:

  * i)

Choice of norm. As discussed above, different problems (e.g. definition vs.
stability) may demand different norms leading to distinct pseudospectra.

  * ii)

Choice of spacetime slicing. The geometric character of the BH QNM
pseudospectrum, namely the assessment of its independence of the slicing, is
still an open problem, in spite of first explorations [36].

  * iii)

Matrix approximation to the differential operator. We compute the
pseudospectrum of a matrix approximant $L^{N}$, not of the actual differential
operator $L$, and this can be the source of issues, e.g. when the actual
spectrum of $L$ contains a continuous part in addition to eigenvalues (point
spectrum).

  * iv)

Eigenvalues that are not QNMs. There may be eigenvalues of $L$ not admitting
an interpretation as QNMs [48]. Then, specific assessments are needed.

  * v)

Instability of spurious eigenvalues. The use of the approximant $L^{N}$ may
contaminate the pseudospectrum by the instability of eigenvalues of $L^{N}$,
that are not present in the actual spectrum of $L$.

  * vi)

Convergence of the pseudospectrum. The status of the convergence of the
pseudospectrun of $L^{N}$ as $N\to\infty$ is an open issue. This is a critical
point.

###  I.2 Some structural aspects of Schwarzschild-anti-de Sitter geometry

####  I.2.1 Timelike null infinity in AdS

In contrast with the asymptotically flat and dS cases, where the outer
(asymptotic) boundary can be naturally chosen as a null hypersurface (null
infinity in the former and the cosmological horizon in the latter), null
infinity in the asymptotically AdS case is a timelike surface (cf. e.g. Fig.
1). This makes natural to consider, for asymptotically AdS BHs444Actually,
null foliations can also be used for QNM calculations in the asymptotically
flat and dS cases, by imposing vanishing boundary conditions at past null
infinity or the past cosmological horizon, respectively [49, 50, 51, 52].
However, in those cases an actual hyperboloidal slicing is possible and
appears as a privileged one., two different types of ‘regular foliations’ (not
intersecting the horizon bifurcation surface), namely spacelike regular
foliations [48, 39] and null slicings (e.g. [53, 54]). This provides an
assessment to caveat (i) in section VII.A.1 of [24], namely the dependence of
the pseudopectrum on the foliation and therefore its geometric content, in a
richer setting than the restricted one implemented in Reissner-Nordström BHs
[36].

In the AdS setting, boundary conditions must be chosen to be imposed at
timelike null infinity, with the crucial requirement of rendering the wave
evolution into a well-posed problem. Specifically, the allowed boundary
conditions depend (in a given dimension) on the mass of the field. In
particular, for the massless (scalar) field that we will focus on here, only a
homogeneous Dirichlet condition is consistent with smoothness [55, 56].

####  I.2.2 Absence of tails, spectral instability and hydrodynamic modes in
AdS

Another crucial aspect of AdS, as it is also the case for dS, is the absence
of Price-law tails in the late-time behavior of BH perturbations,
corresponding to the absence of a branch cut (or the continuous part of the
spectrum in our setting) present in the asymptotically flat case (see also
Ref. [57] for a description on how AdS dynamics connects smoothly to that of
asymptotic flat spacetime in the limit of small cosmological constant). Note
that the Pöschl-Teller case, studied in detail in [24] as a test-bed for BH
QNM (and actually corresponding to scalar field perturbations in dS [24, 58]),
falls also in this category of potentials without Price-law tails555See also
accelerating Schwarschild and Reissner-Nordström BHs, described by the
$C$-metric, where perturbations decay at late times in an exponential manner
[59, 60]..

The use of matrix approximants $L^{N}$ for the operator $L$, introduce
spurious eigenvalues in the spectrum, corresponding to a discretization of the
‘branch cut’, whose potential instability contaminates the pseudospectrum. In
this context, the AdS case (as well as Pöschl-Teller and dS) provides a much
cleaner BH setting than Schwarzschild or Reissner-Nordström. This has
important consequences at a technical and at a physical level.

At the technical level, in addition to the absence of contamination of the
spectra and pseudospectra by spurious eigenvalues in the imaginary axis, the
complementary approach tool to explore the spectral instability by using
random perturbations (‘filling’ pseudospectra sets, cf. the Bauer-Fike theorem
in [29, 24]) can be fully explored in this setting. This was initiated for the
Pöschl-Teller potential in [24] (see also [38]) and further developed in [39]
(also [40]). Crucially, (random) perturbation analyses are not spoiled by
eigenvalues in the ‘branch cut’.

From the physical point of view, the absence of a non-convergent continuous
spectrum along the imaginary axis, permits to study in a clean manner the so-
called ‘hydrodynamic’ modes of AdS BHs [3], namely QNMs placed also along the
imaginary axis corresponding to actual (convergent) eigenvalues, and not to
points in the continuum spectrum that manifest as (non-convergent) eigenvalues
upon discretisation of the operator. Hydrodynamic modes are relics of
holography that connect perturbations of the bulk BH spacetime to
corresponding hydrodynamic quantities in the dual CFT at the boundary of AdS
[61, 62, 63, 64, 57, 3]. These modes can be complex or purely imaginary
depending on the perturbation channel of the CFT, are non-trivially connected
to the properties of hydrodynamics, and can become long-lived for AdS BHs when
the event horizon radius is larger than the AdS length scale [65, 3]. The
absence of numerically non-convergent but unstable eigenvalues corresponding
to the ‘branch-cut’ in the continuum limit, will allow us to study the
stability of these hydrodynamic modes, both when they are fundamental, i.e.
for ‘large’ Schwarzschild-AdS (SAdS) BHs or when they are overtones, for
‘small’ SAdS BHs. The study of their spectral stability will be a physical
aspect on which we will focus in this work.

####  I.2.3 QNMs as eigenvalues of a non-selfdjoint operator: the special
status of AdS

Independently of the spacetime asymptotics, adopting a spacetime foliation
that intersects the BH horizon and $\mathscr{I}^{+}$ in regular sections
permits to cast the QNM calculation as an eigenvalue problem. For
concreteness, let us focus on spacelike regular foliations, leaving the
discussion of the setting of QNMs in the null slicing approach for section
II.1.2. Such regular spacelike foliations are referred to as ‘hyperboloidal
slicings’ and have been used in the BH QNM setting in the asymptotically flat
case (e.g. [66, 67, 68, 24, 23]) with dS asymptotics (e.g. [69, 38]), and the
AdS case (e.g. [48, 39], where they are referred as spacelike ‘regular
slicings’). Considering the wave equation for a scalar field (that can be the
one in the master equation of, say, gravitational perturbations) in such
hyperboloidal setting, and upon a first-order reduction in time followed by a
Fourier transform, the QNM problem can be cast (see section II.1.1 below) as

$\displaystyle Lu_{n}=\omega_{n}u_{n}\ ,$ (1)

where

$\displaystyle u=\begin{pmatrix}\phi_{n}\\\ \psi_{n}\end{pmatrix}\ \ ,\ \
L=\frac{1}{i}\\!\left(\begin{array}[]{c|c}0&1\\\ \hline\cr
L_{1}&L_{2}\end{array}\right)\ ,$ (4)

and operators $L_{1}$ and $L_{2}$ are given in Eq. (21) below.

As formulated in Eq. (1), the QNM system looks as a relatively simple and
harmless eigenvalue problem, in which even outgoing boundary conditions are no
longer explicit, having being incorporated into the bulk of the operator. In
addition, the success in [67, 68] where (1) is combined with accurate spectral
Chebyshev methods, supports such a vision. However, such apparent simplicity
is misleading. In fact, the trade-off of eliminating outgoing boundary
conditions in favor of conditions on the regularity of the eigenfunctions
introduces a key subtlety in the very definition of QNMs as eigenvalues.

A clear manifestation of such regularity issues is the following key remark in
[67]: considering the convention in which QNM frequencies $\omega_{n}$ are in
the upper-half plane $\mathbb{C}$, it holds that Eq. (1) admits a smooth
($C^{\infty}$) solution not only for that discrete set $\\{\omega_{n}\\}$, but
actually for any complex number $\omega$ in the upper-half plane666The
discussion in Sec. IV.A and Appendix B of Ref. [67] is presented in terms of
the Laplace-transform parameter $s$, rather than the Fourier $\omega$ (related
by $s=i\omega$) and without prior first-order reduction in time. Specifically,
our Eq. (1) corresponds to Eq. (27) in [67], in the particular case of the
Schwarzschild BH.. In other words, any $\omega$ is an eigenvalue for an
appropriate smooth eigenfunction. This has the remarkable consequence that
initial data can be chosen such that the evolved fields decays as an
exponentially damped sinusoidal time-signal, in which the time decay and the
oscillating frequency can be chosen freely (cf. Fig. 3 in [67] and discussion
therein). That is, the field decays exactly in the way QNMs do (and without
tails), but for an arbitrary complex frequency. What actually singles out the
discrete set of QNM frequencies $\omega_{n}$ is that the corresponding
eigenfunctions possess an enhanced regularity.As discussed in Ref. [67],
withing the approach originally put forward by Leaver [70], this property is
cast in terms of the behaviour of the coefficients $H_{k}$ of a Taylor
expansion of the eigenfunction $\phi(\omega)$ around the horizon (a regular
singular point): QNMs correspond to $\omega_{n}$ whose corresponding
$\phi_{n}=\phi(\omega_{n})$ present decreasing coefficients $H_{k}$ as
$k\to\infty$, whereas for an arbitrary point $\omega\notin\\{\omega_{n}\\}$
coefficients $H_{k}$ diverge (and still $\phi(\omega)$ is smooth, cf. Appendix
B in [67]).

A systematic manner of dealing with these regularity requirements consists of
enforcing QNM eigenfunctions to belong to a Hilbert space where the scalar
product is sufficiently stringent to enforce the appropriate regularity. This
is the approach taken by Warnick [48] for the characterization of QNMs as
‘proper’ eigenvalues in asymptotically AdS (as well as dS) BH spacetimes by
using Sobolev’s spaces $H^{k}$ (see also [71, 58, 72]) and then extended to
some BH models in the asymtotically flat case in [73, 74, 75], in terms of
Gevrey-2 classes of functions.

The key point in the setting of the present work is that Warnick’s discussion
of the asymptotically AdS (and dS) case, in terms of Sobolev’s spaces $H^{k}$
controlling the norm of the first $k$ derivatives777In this context of control
of derivatives, a manifestation of the underlying regularity issues is the
observed spectral instability in [24, 25, 23, 26], where the use of the energy
norm (essentially an $H^{1}$ norm) is not sufficiently stringent to control
QNM overtones., offers a particularly complete and robust characterization of
QNMs as eigenvalues. Unfortunately, such a level of mathematical control has
not been yet attained in the generic asymptotically flat case (in terms of so-
called Gevrey-2 classes of functions, needing the control of all derivatives).
It is for this reason that the AdS case offers a particularly well-suited
test-bed for our exploration of the pseudospectrum and related tools, given
the existence of a well-understood mathematical counterpart. The basic
elements of QNMs as eigenvalues in AdS will be discussed in Sec. III.

##  II Perturbations of Schwarzschild-Anti de Sitter black holes

The SAdS geometry [76] is given by the line element

${\rm d}s^{2}=-f(r){\rm d}t^{2}+f^{-1}(r){\rm d}r^{2}+r^{2}{\rm d}\Omega^{2},$
(5)

where ${\rm d}\Omega^{2}$ is the line element of the unit sphere, and the
redshift function reads

$f(r)=1-\frac{r_{s}}{r}+\frac{r^{2}}{R^{2}},$ (6)

with $r_{s}$ and $R$ positive constants. This redshift function has only one
zero at a positive value of the radial coordinate, which corresponds to the
position of the BH event horizon, designated as $r=r_{h}$ from here on. With
this last parameter, the function (6) can be recast into the form

$f(r)=\left(1-\frac{r_{h}}{r}\right)\left[1+\alpha^{2}\left(1+\frac{r}{r_{h}}+\frac{r^{2}}{r_{h}^{2}}\right)\right],$
(7)

where $\alpha=r_{h}/R$ is a dimensionless constant which relates the size of
the BH to the AdS length scale $R$. The form (6) can be recovered with the
relation

$r_{s}=r_{h}(1+\alpha^{2}).$

The form (7) will be used throughout this work, since it allows us to simplify
the geometric approach of imposing QNM boundary conditions at the horizon.

Assuming linear-field (e.g. scalar or gravitational) perturbations on SAdS
spacetime, and imposing appropriate transformations of the degrees of freedom
together with a decomposition of the angular sector into spherical harmonics,
the radial-temporal part of each angular mode of different types of
fluctuations satisfies a wave equation of the form

$-\Psi_{,tt}+\Psi_{,r^{*}r^{*}}-V_{s}\Psi=0,$ (8)

where a comma in the subscript denotes partial differentiation, $r^{*}$ is the
tortoise coordinate, defined as ${\rm d}r^{*}={\rm d}r/f(r)$, and the
potential $V_{s}$ depends on the angular multipole $\ell$ and the nature of
the perturbations. In this work we will mainly focus on scalar $s=0$ and axial
gravitational $s=2$ waves, described, respectively, by the potentials [3]

$\displaystyle V_{0}$
$\displaystyle=\frac{f}{r^{2}}\left[\ell(\ell+1)+rf^{\prime}\right],$ (9)
$\displaystyle V_{2}$
$\displaystyle=\frac{f}{r^{2}}\left[\ell(\ell+1)-\frac{3r_{s}}{r}\right],$
(10)

respectively, where prime denotes differentiation with respect to the
function’s argument. For the sake of brevity, in what follows we will present
results only for these two potentials. The electromagnetic and polar
gravitational cases lead to qualitatively similar results, as we will briefly
discuss in the concluding remarks.

###  II.1 Boundary conditions

Asymptotically AdS spacetimes have a timelike boundary at $r\to\infty$ (see
Fig. 1), where not just spacelike, but also null and timelike curves can begin
and end. Therefore, in order for the evolution of fields on this spacetime to
be determined unambiguously from initial data on a spacelike hypersurface, a
boundary condition that fits the physical properties of the geometry at
$r\to\infty$ must be imposed. A typical choice is the reflective (Dirichlet)
boundary condition [77]

$\Psi|_{r=\infty}=0,$ (11)

which we will adopt in this work. For scalar perturbations $\Psi_{sc}$, the
divergent potential (9) at this boundary along with the condition (11) result
in a falloff at large $r$ as

$\Psi_{sc}\sim\frac{1}{r^{2}}.$ (12)

For the other types of perturbations, the potentials are finite at the
boundary, and imposing the behaviour $1/r$ is sufficient for solutions to
satisfy the boundary condition.

At the event horizon, all potentials vanish and Eq. (8) reduces to a free wave
equation. The boundary condition there which defines QNMs is that of an
ingoing wave,

$\left(\Psi_{,t}-\Psi_{,r^{*}}\right)|_{r=r_{h}}=0.$ (13)

A Fourier decomposition on $\Psi(t,r_{*})$ translates this behavior to the
frequency domain as

$\Psi\sim
e^{i\omega(r^{*}+t)}\sim\left(1-\frac{r_{h}}{r}\right)^{\frac{i\omega
r_{h}}{1+3\alpha^{2}}}e^{i\omega t}.$ (14)

From the temporal part we observe that in order for the solution to decay in
time, i.e. for the spacetime to be linearly stable, the QNM frequencies
$\omega_{n}$ must have a positive imaginary part. This, however, makes the
spatial part having a divergent, oscillatory behavior. This issue is typical
of QNMs when working in Schwarzschild-like coordinates, and we will deal with
it geometrically in what follows.

Figure 1: SAdS conformal diagram and different types of hypersurface slicing
families on which the evolution of a decaying field outside the horizon can be
analyzed.

####  II.1.1 Regular (“hyperboloidal”) slicing

Much like how the analytical extension toward the past of BH spacetimes leads
to non-physical white hole configurations, the extension of QNMs toward the
past describes oscillations with exponentially increasing amplitude, which are
equally non-physical. At the horizon bifurcation surface (and at spacelike
infinity, if the spacetime were asymptotically flat) this issue is still
partially present, leading to the singular oscillatory behavior observed at
the boundary in Eq. (14). In other words, this singular tendency can be seen
as a consequence of evolving the wave equation along $t=\text{const.}$
hypersurfaces. These span from the horizon bifurcation surface to the AdS
boundary, as shown in Fig. 1, where they are labelled as Cauchy slices.

A geometric method of working around this issue is to choose a set of
hypersurfaces which intersect the future event horizon $\mathscr{H}^{+}$, as
shown in Fig. 1. One possibility is to use a (half-)hyperboloidal888We say
‘half’ because the standard hyperboloidal scheme also bends the hypersurfaces
toward future null infinity, when this boundary is null. scheme [67, 68, 78],
which keeps constant-time hypersurfaces spacelike but bends them toward the
future horizon. Another is to directly use a foliation of ingoing null
hypersurfaces. Both of these transformations, when applied to Eq. (8), change
the problem in a way that not only makes the QNM solutions regular at the
horizon, but also makes regularity equivalent to the ingoing boundary
condition (13).

Aside from this change in the spacetime foliation, in order to evaluate the
problem numerically, we also need to compactify the infinite span of the
radial (tortoise) coordinate to a finite domain. We thus define a new
coordinate system $\\{\tau,\chi\\}$ for the radial-temporal sector as

$\begin{split}t(\tau,\chi)&=\tau-h(\chi),\\\
r^{*}(\tau,\chi)&=g(\chi).\end{split}$ (15)

The function $g$ ensures that the $r^{*}\in(-\infty,r_{\infty}^{*})$ interval,
where $r^{*}_{\infty}$ is an integration constant corresponding to
$r\to\infty$, is compactified into a finite domain $\chi\in[a,b]$. It is also
chosen such that the potential in the wave equations can be written explicitly
in terms of $\chi$. A standard choice is the one defined by the relation
$\chi=r_{h}/r$, formally given by

$g(\chi)=\int^{r_{h}/\chi}\frac{{\rm d}r}{f(r)}.$ (16)

As for the height function $h$, its purpose is to bend the spacetime foliation
toward the future horizon $\mathscr{H}^{+}$, such that the ingoing wave
condition is satisfied geometrically, as described above. This is achieved as
long as the condition

$h(\chi)\sim
g(\chi)\sim\frac{1}{1+3\alpha^{2}}\log\left(1-\frac{r_{h}}{r(\chi)}\right),$
(17)

is satisfied when approaching the event horizon. In other words, $\tau$ must
behave like the Eddington-Finkelstein ingoing null coordinate, $v=t+r^{*}$,
around the horizon. When (17) is satisfied only in the vicinity of $r_{h}$,
while in the rest of the BH exterior the $\tau=\text{const.}$ hypersurfaces
are spacelike ($h^{\prime}<g^{\prime}$), the slicing is hyperboloidal. In
particular, we will work with a slicing

$h(\chi)=\frac{1}{1+3\alpha^{2}}\log(1-\chi),$ (18)

obtained by integrating only the dominant term of the redshift function around
the horizon in Eq. (16), and subsequently checking that the condition
$h^{\prime}<g^{\prime}$ is satisfied for $r>r_{\rm h}$. Note that for any
$\tau$ defined through (15), the Fourier frequency variable $\omega$ is the
same as the one corresponding to $t$, since $\partial_{\tau}=\partial_{t}$ for
these transformations.

After this transformation, and a reduction of order in time through

$\phi=\partial_{\tau}\psi,$ (19)

the wave equation (8) can be written as

$L_{1}\psi+L_{2}\phi=\partial_{\tau}\phi,$ (20)

where

$\begin{split}L_{1}&=\frac{p}{w}\partial_{\chi}^{2}+\frac{p^{\prime}}{w}\partial_{\chi}-\frac{q}{w}=\frac{1}{w}\left(\partial_{\chi}(p\;\partial_{\chi})-q\right),\\\
L_{2}&=2\frac{\gamma}{w}\partial_{\chi}+\frac{\gamma^{\prime}}{w}=\frac{1}{w}\left(\gamma\partial_{\chi}+\partial_{\chi}(\gamma\cdot)\right),\end{split}$
(21)

with a prime denoting a derivative with respect to $\chi$, and the functions

$w=\frac{g^{\prime 2}-h^{\prime 2}}{|g^{\prime}|},\quad
p=\frac{1}{|g^{\prime}|},\quad\gamma=\frac{h^{\prime}}{|g^{\prime}|},\quad
q=|g^{\prime}|V.$ (22)

The first-order problem can be obtained from combining Eqs. (19) and (20),

$iL\,u=\partial_{\tau}u,$ (23)

where we define the vector and block operator (cf. Eq. (4)) as

$u=\left(\begin{array}[]{c}\psi\\\ \phi\end{array}\right),\quad
L=\frac{1}{i}\left(\begin{array}[]{c|c}0&1\\\ \hline\cr
L_{1}&L_{2}\end{array}\right).$ (24)

####  II.1.2 Null slicing

The third possibility depicted in Fig. 1 is to analyse the wave equation in an
ingoing null slicing, with a time parameter given directly by the advanced
Eddington-Finkelstein coordinate $v=t+r^{*}$ (equivalent to a transformation
(15) with $h=g$). This choice in fact turns out to be the most efficient
method for the numerical computations presented in the following sections. We
note that a similar approach has been recently pursued in [40]. That said, all
computations have been performed with both the hyperboloidal and null
foliations, for the sake of comparison (as we will see, some of the results
differ between the two). In this sense, our work encompasses both the
hyperboloidal treatment in Arean et al. [39] and the null slicing approach in
Cownden et al. [40], allowing us to compare and assess both approches to BH
QNM instability in an AdS setting.

The wave equation (8) cast in the ingoing null coordinate system acquires the
form

$iM\psi=B\partial_{v}\psi,$ (25)

where

$\begin{split}M&=\frac{1}{2i}\left[(\partial_{\chi}p\;\partial_{\chi})-q\right]=\frac{1}{2i}\left(p\partial_{\chi}^{2}+p^{\prime}\partial_{\chi}-q\right),\\\
B&=-\partial_{\chi},\end{split}$ (26)

with $p$ and $q$ given again by (22).

####  II.1.3 AdS boundary

The ingoing condition at the horizon (13) is satisfied automatically for
regular solutions in both the hyperboloidal and null coordinates, as can be
seen by applying the transformation (15) to (13). On the other hand, the
Dirichlet condition, which we have chosen for the AdS boundary, must be
imposed by hand. For gravitational (and electromagnetic) perturbations, this
condition can be applied directly in the numerical implementation of the
problem999The same method has been used in [37] to apply Dirichlet boundary
conditions at the perfectly reflective surface of horizonless compact
objects., while for scalar perturbations, due to the requirement for a sharper
decay in $r$ (see Eq. (12)), an additional transformation must be applied to
the equation through the rescaling

$\Psi_{sc}(\tau,\chi)=\chi\,\xi(\tau,\chi).$ (27)

With this redefinition, Eqs. (25) and (23) (appropriately regularized with a
multiplication by $\chi$ on both sides) automatically impose that $\xi$ must
tend to zero at the timelike AdS boundary, which, together with smoothness in
$\chi$, completely imposes the required asymptotic behavior (12).

However, one important change is worth noting. Eq. (23), after the rescaling
(27), takes on the form

$i\tilde{L}u_{\xi}=\chi^{2}\partial_{\tau}u_{\xi},$ (28)

with $u_{\xi}$ a vector of components $\xi$ and $\partial_{\tau}\xi$, and
$\tilde{L}$ obtained by direct substitution of (27) into (23), followed by a
multiplication by $\chi$. The key feature is that the $\chi^{2}$ multiplying
the right-hand side cannot be absorbed into $\tilde{L}$ without making this
operator singular at the AdS boundary, due to the very divergence of the
potential this transformation is meant to regularise. This makes the problem
in the hyperboloidal framework more similar to the one in the null case (25),
with operators on both sides of the equation. After the Fourier transform
discussed below, this equation turns into a _generalized_ eigenvalue problem.
As for the wave equation in the null case (25), after the rescaling (27) and
the multiplication by $\chi$, the right-hand side operator simply turns into
$\tilde{B}=-\chi^{2}\partial_{\chi}-\chi$.

##  III QNMs as eigenvalues, spectral instability and norm

The discussion in the previous Sec. II leads naturally to casting the
calculation of QNMs as an eigenvalue problem in two distinct but (in
principle) formally equivalent formulations corresponding, respectively, to
the (regular) hyperboloidal and the null slicings.

For the gravitational perturbation problem expressed in the hyperboloidal
frame, described in Sec. II.1.1, taking a Fourier transform in Eq. (23) with
respect to $\tau$ leads to the (“frequency domain”) eigenvalue problem for the
infinitesimal time generator $L$ in Eq. (24)

$L\,u=\omega u.$ (29)

On the other hand, when we consider the same problem in the null slicing
(II.1.2), we obtain QNMs characterized in terms of a _generalised_ eigenvalue
problem101010These generalised eigenvalue problems have been studied in the
QNM instability context in [23] in optical cavity scenarios. (see also [40]).
Specifically, taking the Fourier transform in the (advanced) time $v$, the
wave equation (25) becomes

$M\psi=\omega B\psi\ .$ (30)

Although the analysis of such a generalized eigenvalue problem requires some
additional care, a key difference with Eq. (29) is that such an approach does
not involve a previous first-order reduction in time, as in the hyperboloidal
approach. This may prove to be relevant in the analysis of the pseudospectrum,
as we comment below.

For scalar perturbations, as noted above, the Fourier-transformed equation
turns into a generalised eigenvalue problem in both the hyperboloidal and null
slicings. In the hyperboloidal one, this is in addition to the time-order
reduction, while for the null one the problem remains structurally the same as
the gravitational perturbation case (30). In the following discussion, the two
spectral problems (29) and (30) will be referenced, and the latter should be
understood to encompass the scalar field case for both slicings.

The QNM frequencies, being eigenvalues in spectral problems of the type (29)
and (30), can be defined as $\omega\in\mathbb{C}$ for which the operators
$(L-\omega\mathrm{Id})$ and $(M-\omega B)$, respectively, are not invertible.
In other words, frequencies $\omega$ for which there exist eigenfunctions
$u(\omega)$ and $\psi(\omega)$ (respectively for Eq. (29) and (30)) in the
appropriate Hilbert space. Indeed, as discussed in Sec. I.2.3, such
$u(\omega)$ and $\psi(\omega)$ QNM eigenfunctions must be defined in the
appropriate Hilbert space for enforcing the regularity required to correctly
characterize the QNMs. The choice of Hilbert space is key. In terms of the
corresponding resolvent operators, namely

$\displaystyle R_{L}(\omega)=(L-\omega\mathrm{Id})^{-1}\ ,$ (31)

for the hyperboloidal case and [40, 23, 29]

$\displaystyle R_{M,B}(\omega)$ $\displaystyle=$ $\displaystyle(M-\omega
B)^{-1}\ ,$ (32)

for the null slicing, QNM frequencies are those $\omega\in\mathbb{C}$ for
which the resolvents $R_{L}(\omega)$ and $R_{M,B}(\omega)$, defined in the
corresponding appropriate Hilbert space, do not exist. In particular, when
considering the operator norm $\|\cdot\|$ induced from the Hilbert space norm,
these resolvent norms diverge as $\omega$ tends to one of these eigenvalues.

###  III.1 QNMs as eigenvalues in AdS: $H^{k}$ regularity

As discussed in Sec. I.2.3, the question about the appropriate regularity
required to characterize QNMs in asymptotically AdS BHs has been fully
addressed by Warnick in [48]. As a result of this analysis, it follows that
the upper-complex-half plane $\mathrm{Im}(\omega)>0$ presents a structure in
horizontal bands of width given by the BH horizon surface gravity $\kappa$,
where increasing regularity of the QNM eigenfunctions is required to identify
QNMs as they become more damped. More specifically, focusing on the
hyperboloidal slicing eigenvalue problem (29):

  * i)

Given $k\in\mathbb{N}$, $H^{k}$-QNMs are introduced as eigenfunctions of the
spectral problem (29) with $H^{k}$-regularity, i.e. whose first $k$
derivatives are $L^{2}$-integrable (namely, they are in the Sobolev space
$H^{k}$). Specifically, defining the $H^{k}$-regular $L_{k}$ operator

$\displaystyle\begin{array}[]{ cccc}L_{k}:&H^{k}\times H^{k-1}&\to&H^{k}\times
H^{k-1}\\\ &(\phi\;\;,\;\psi)&\mapsto&L(\phi,\psi)=\\\
&&&(\psi,L_{1}\phi+L_{2}\psi)\end{array}\ ,$ (36)

$H^{k}$-QNM frequencies are proper eigenvalues of $L_{k}$.

  * ii)

$H^{k}$-QNM frequencies in the horizontal band above the real axis,
characterized by

$\displaystyle\mathrm{Im}(\omega)<a+\kappa(k-\frac{1}{2})\ ,$ (37)

form a discrete set (here $a$ is a fixed constant depending on the spacetime
and not on $k$).

  * iii)

The $H^{k}$-resolvent $R_{L_{k}}(\omega)=(L_{k}-\omega\mathrm{Id})^{-1}$,
where

$\displaystyle R_{L_{k}}(\omega):H^{k}\times H^{k-1}$ $\displaystyle\to$
$\displaystyle H^{k}\times H^{k-1}\ ,$ (38)

exists and is a $H^{k}$-bounded operator in the region defined by inequality
(37), of course except in the discrete set given by $H^{k}$-QNM frequencies.

  * iv)

However, for every $k\in\mathbb{N}$, there exists a constant $c_{k}$ such that
$R_{L_{k}}(\omega)$ does not exist for $\omega\in\mathbb{C}$ with

$\displaystyle\mathrm{Im}(\omega)>c_{k}\ .$ (39)

In other words, all complex numbers $\omega$ in the half-plane defined by (39)
are proper $H^{k}$-QNM frequencies, namely actual eigenvalues of $L_{k}$.

In more simple terms, if we want to characterize and locate the QNMs in
asymptotically AdS BHs as eigenvalues in the spectral problem (29), we must
proceed band by band, of width $\kappa$, starting from the real
axis111111Actually, the starting point depends on the constant $a$ in (37),
but this is just a technical point. by imposing more and more control on the
derivatives of the corresponding eigenfunctions. In order to find the discrete
QNMs in the first $\kappa$-band we impose the first derivative to be in
$L^{2}$ (so the eigenfunction is in $H^{1}$). All frequencies above this band
are valid QNMs at this regularity level. Then, to access the discrete QNMs in
the second $\kappa$-band we impose second derivatives to be in $L^{2}$
(eigenfunctions in $H^{2}$) and so on and so forth: for each $\kappa$-band
where we want to locate discrete eigenvalues we must add control on one more
derivative. QNMs become more regular as they are more damped, thus requiring
more regularity to ‘pick up’ the appropriate discrete QNM frequencies in each
new band. But, for any finite $k$, there always remain a continuous set of
QNMs with that $H^{k}$-regularity above the $k$-th $\kappa$-band121212This
formalizes in terms of $H^{k}$ Hilbert spaces, and for asymptotically AdS BHs,
the remark in [67] commented in Sec. I.2.3, according to which all frequencies
in the upper complex half-plane are QNMs of the Schwarzschild BH if no
appropriate enhanced regularity is enforced on the corresponding
eigenfunctions..

###  III.2 Spectral stability of QNMs: pseudospectrum and choice of norm

In the previous section we have sketched what in Sec. I.1.1 we have referred
to as the ‘BH QNM definition problem’, in the specific context of
asymptotically AdS BHs, where it is fully addressed. We have seen that the
choice of the proper Hilbert space, with its associated norm, is key to
control the required regularity. The focus of the present manuscript is
however placed in the related, but different, ‘BH QNM stability problem’,
where the choice of norm is also a key part of the analysis.

####  III.2.1 Characterizations of the pseudospectrum

Non-selfadjoint operators $L$ (and, more generally non-normal operators, for
which $[L,L^{\dagger}]\neq 0$), can potentially suffer from spectral
instability, as a consequence of the general non-orthogonality of the
eigenfunctions of $L$ and its adjoint $L^{\dagger}$. In this setting, the
spectrum $\sigma(L)$ is not necessarily the proper object to consider, and the
notion of $\epsilon$-pseudospectrum $\sigma^{\epsilon}(L)$ becomes relevant.
The latter is defined as the set, for a given $\epsilon>0$, of all complex
numbers $\omega\in\mathbb{C}$ that are eigenvalues for ‘some’ perturbed
version $L+\delta L$ of the operator $L$, with perturbations $\delta L$ of
‘size’ $\|\delta L\|<\epsilon$ in some appropriate norm. In other words [29,
31, 32],

$\displaystyle\sigma^{\epsilon}(L)=\\{\omega\in\mathbb{C},\exists\delta
L,\|\delta L\|<\epsilon:\omega\in\sigma(L+\delta L)\\}\ ,$ (40)

Spectrally stable operators are such that the perturbed eigenvalues remain at
a distance of order $\epsilon$ under perturbations with $\|\delta
L\|\sim\epsilon$, giving rise to $\epsilon$-pseudospectral levels structured
as concentric circles (more specifically ‘tubular neighborhoods’) of radii
$\epsilon$. On the contrary, spectrally unstable operators present
$\epsilon$-pseudospectra extending over much larger regions in the complex
plane. Clearly, such a notion of spectral stability depends on what identifies
as a “big” or “small” perturbation, and therefore on the choice of norm (see
discussion in [24, 26]). Moreover, the natural choice of norm in this
“stability” problem need not coincide with the appropriate one in the
“definition” problem [47]: whereas the latter is dictated by the mathematical
structure of the operator, the former may be determined on physical grounds.

An equivalent characterization of the $\epsilon$-pseudospectrum providing a
direct link to the resolvent, as well as an efficient manner of determining
the $\epsilon$-pseudospectral sets131313Alternatively, the characterization
(40) of the $\epsilon$-pseudospectrum can be employed to determine the
$\epsilon$-pseudospectra sets by using random perturbations $\delta L$ of norm
$\|\delta L\|\leq\epsilon$, cf. [29]. This has been systematically used in
[39] in the AdS black brane setting. is given in terms of the norm of the
resolvent. In the hyperboloidal case (31), this is given by

$\sigma^{\epsilon}(L)=\\{\omega\in\mathbb{C}:\|R_{L}(\omega)\|=\|(L-\omega\mathrm{Id})^{-1}\|>1/\epsilon\\},$
(41)

whereas in the generalised eigenvalue case (32) we have141414See Chapter 45 in
[29] for the specifities of pseudospectra in this generalized eigenvalue
formulation.

$\sigma^{\epsilon}(M,B)=\\{\omega\in\mathbb{C}:\|R_{M,B}(\omega)\|=\|(M-\omega
B)^{-1}\|>1/\epsilon\\}.$ (42)

In the case that the norm $\|\cdot\|$ is induced from a scalar product, the
pseudospectrum calculation can be reduced to the calculation of the minimum of
the (generalized) singular values $s^{\mathrm{min}}(A)$ of the appropriate
operator $A$ (see details in [24], in particular in its Appendix B). In the
hyperboloidal case we have the characterization

$\displaystyle\sigma^{\epsilon}(L)=\\{\omega\in\mathbb{C}:s^{\mathrm{min}}(L-\omega\mathrm{Id})<\epsilon\\}\
,$ (43)

whereas in the null slicing case, the expression is

$\displaystyle\sigma^{\epsilon}(M,B)=\\{\omega\in\mathbb{C}:s^{\mathrm{min}}(M-\omega
B)<\epsilon\\}\ .$ (44)

####  III.2.2 Energy and $H^{k}$ Sobolev norms

As we have mentioned above, the natural norm to assess QNM stability is not
necessarily the same as the one needed in order to control the appropriate
regularity of the QNMs. Whereas in the asymptotically AdS (and dS) case the
latter is given by $H^{k}$ norms controlling the ‘size’ of the first $k$
derivatives (cf. Sec. III.1), from a physical perspective a natural norm
concerning the stability issue is the one related to the energy introduced in
the system by the perturbation $\delta L$. Such an energy norm is essentially
an $H^{1}$ norm, which is natural in the control of the well posedness of the
initial value problem of the second-order wave equation (8), but it is clearly
insufficient to characterize BH QNMs beyond the first $\kappa$-band in the
upper complex plane (namely the fundamental or slowest decaying QNM
frequency).

Starting from the expression of the energy, in a stationary spacetime,
associated with a field $\phi$ in a given hypersurface, we can write
explicitly the energy scalar product in the hyperboloidal case, as (cf. [24,
26])

$\\!\\!\\!\langle
u_{1},\\!u_{2}\rangle_{{}_{E}}\\!\\!=\\!\frac{1}{2}\\!\\!\int_{a}^{b}\\!\\!\\!\left(\\!w(\chi)\bar{\phi}_{1}\phi_{2}+p(\chi)\partial_{\chi}\bar{\psi}_{1}\\!\partial_{\chi}\psi_{2}+q(\chi)\bar{\psi}_{1}\psi_{2}\\!\right)\\!d\chi.$
(45)

In the null slicing case, the product is essentially the same, keeping in mind
that there is no reduction of order in time and, correspondingly, $w(\chi)=0$,
leading to

$\langle\psi_{1},\\!\psi_{2}\rangle_{{}_{E}}=\frac{1}{2}\int_{a}^{b}\\!\left(p(\chi)\partial_{\chi}\bar{\psi}_{1}\\!\partial_{\chi}\psi_{2}+q(\chi)\bar{\psi}_{1}\psi_{2}\\!\right)\\!d\chi$
(46)

For the case of a scalar field, where the function $\psi$ is rescaled
according to (27), the above expressions apply to the quantity
$\chi\xi(\tau,\chi)$. They can be further simplified to remove the cross
products between differentiated and non-differentiated functions using
integration by parts. Defining the order-reduced rescaled function
$\eta(\tau,\chi)=\partial_{\tau}\xi(\tau,\chi)=\phi(\tau,\chi)/\chi$, and the
vector $u_{\xi}$ with components $\xi$ and $\eta$, the resulting product reads

$\begin{split}\langle u_{\xi 1},\\!u_{\xi
2}\rangle_{{}_{E}}&=\frac{1}{2}\int_{a}^{b}\left(\chi^{2}w(\chi)\bar{\eta}_{1}\eta_{2}+\chi^{2}p(\chi)\partial_{\chi}\bar{\xi}_{1}\partial_{\chi}\xi_{2}\right.\\\
&\left.+(\chi^{2}q(\chi)-\chi
p^{\prime}(\chi))\bar{\xi}_{1}\xi_{2}\right)d\chi\end{split}$ (47)

For the null slicing, the above expression once again applies, setting
$w(\chi)=0$ and considering the appropriate functions $p(\chi)$ and $q(\chi)$
corresponding to the coordinate transformation.

For higher-order $H^{k}$ Sobolev norms, we consider an extension to the above
expression through the direct addition of a term

$\chi(1-\chi)\partial_{\chi}^{k}\bar{\psi}\partial_{\chi}^{k}\psi$ (48)

to the integrand. The weight $\chi(1-\chi)$ has been chosen simply in order to
avoid adding non-trivial terms at the boundaries, as such terms are directly
related to the non-selfadjoint nature of the operator $L$ in the energy norm
(see [24]). In practice, any simple weight function, even one which does not
vanish at the boundaries, appears to lead to the same qualitative results
described below.

Since, both the energy and the $H^{k}$ norms are induced by the scalar product
in the appropriate Hilbert space, we can use the characterizations (43) and
(44) in the standard and the generalised eigenvalue problems, respectively.

###  III.3 Numerical implementation

To obtain the QNM spectrum, and subsequently the pseudospectrum, of the simple
and generalised eigenvalue problems discussed above, we follow the method
employed in [24, 36, 37] and use a Chebyshev-Lobatto grid discretisation in
the $\chi$ variable with $N$ grid points. The discretised expressions for both
the derivative and integral operators defined above are obtained by expanding
the solution in $N$ Chebyshev polynomials, and expressing the
derivative/integral of each of them in the same basis, with the operators
making the corresponding connection.

The ingoing boundary condition at the horizon is automatically satisfied in
the coordinate system used for solutions regular at $r=r_{h}$, which are the
only ones the numerical method can approximate. As for the reflective boundary
condition at $r\to\infty$, for the axial gravitational perturbations it is
imposed by removing a row and column from the discretized matrix operators
corresponding to the position of $\chi=0$ in the grid; for scalar
perturbations, as mentioned above, the rescaling (27) and the regularisation
of the resulting equation already impose the necessary condition for the
solutions at the boundary, so no additional steps need to be taken.

In what follows, we use several different values of $N$ for different
calculations. For calculating the QNM spectrum of the BHs, we find that
$N=100$ is more than sufficient to resolve at least the first 6-7 modes
closest to the origin to within several decimal places of accuracy, for all
cases studied. For the contour plots of the pseudospectrum, we therefore use
the same $N$, finding the result to be qualitatively well behaved. For the QNM
spectra of the perturbed operators, we increase the resolution to $N=150$ for
the purpose of numerically resolving the highly oscillating perturbation well.
Lastly, for the convergence tests we use values in the range $N\in[50,800]$,
with jumps of $25$.

For the calculation of the pseudospectrum, we use the generalised singular
value decomposition from Eqs. (43) and (44), with

$s^{\rm min}_{E}\left(R^{-1}(\omega)\right)={\rm
min}\\{\sqrt{c}:c\in\sigma((R^{-1}(\omega))^{\dagger}R^{-1}(\omega))\\},$ (49)

where $R(\omega)$ stands for the resolvent $R_{L}(\omega)$ or $R_{M,B}$,
defined above for the eigenvalue problems studied, and
$(R^{-1}(\omega))^{\dagger}$ is the adjoint of its inverse. The calculation of
the adjoint is performed with respect to the norms descibed above for each
case, with its numerical implementation given by

$A^{\dagger}=(G)^{-1}\cdot(A^{t})^{*}\cdot G,$ (50)

where $A$ is any operator, with its conjugate transpose denoted as
$(A^{t})^{*}$, and $G$ is the Gram matrix corresponding to each norm, as
defined in Appendix C of [24].

##  IV Results

In this section we present the results concerning the stability of SAdS QNMs,
focusing on their generic structural aspects, rather than systematically
exploring the parameter space of this problem. We start by presenting the
results in the the hyperboloidal slicing setting, and then in the null
slicing. After this we discuss and interpret these results in light of
Warnick’s analytical results in [48]. Finally, this analysis allows us to
focus on the physical problem concerning the stability of hydrodynamic modes.

###  IV.1 Pseudospectra: hyperboloidal slicing

Figures 2 and 3 illustrate, respectively, the stability properties of QNMs for
a scalar field and the axial gravitational perturbation (both for the $\ell=2$
mode) in SAdS BHs for given choices of the parameter $\alpha=r_{h}/R$. These
and subsequent figures include two independent classes of calculations:

  * i)

QNM calculations: they corresponds to the resolution (through an appropriate
numerical algorithm) of the eigenvalue problem (29), both for the exact SAdS
spacetime (red points) and for effective potentials subject to perturbations
$\delta V$ (differently colored and shaped points). These perturbations are of
the form

$\delta V=\varepsilon\sin(2\pi k\chi),$ (51)

with the wavenumber $k$ generally chosen to be as high as possible while still
safely allowing for convergent results with the numerical resolution used
(since high wavenumber perturbations lead to the strongest destabilisation, as
shown in [24]), and $\varepsilon$ is chosen in a way that fixes the energy
norm of the perturbation operator ($\delta L$ or $\delta M$) to the specified
values.

  * ii)

Pseudospectrum calculation: this corresponds to the contour lines map of the
function $f_{L}:\mathbb{C}\to\mathbb{R}^{+}$, with
$f_{L}(\omega)=\|R_{L}(\omega)\|$ (actually we plot the inverse
$1/f_{L}(\omega)$) where $R_{L}(\omega)$ is given in (31). Pseudospectral sets
$\sigma^{\epsilon}(L)$ in (41) are given by the regions bounded below by the
$\epsilon$-contour lines.

These numerical results recover the qualitative features and contour-line
patterns previously found when using the hyperboloidal slicing to study BH QNM
stability in the asymptotically flat [24, 25, 23, 36, 37, 46], asymptotically
dS [38] and asymptotically AdS [39] configurations.

Figure 2: Pseudospectrum in the energy norm of $\ell=2$ scalar perturbations
on SAdS with $\alpha=1$, in the hyperboloidal slicing. The spectrum of the
unperturbed operator is shown with red circles while the resulting spectra of
perturbed versions, where perturbations of the type (51) with $k=20$ have been
added to the discretized potential of the operator, are shown with differently
shaped and colored points according to their energy norm, indicated in the
legend. The grid-point resolution used is $N=100$ for the pseudospectrum and
$N=150$ for the perturbed spectra.

Figure 3: Pseudospectrum of $\ell=2$ axial gravitational perturbations on SAdS
with $\alpha=1/2$ (top panel), $\alpha=1$ (middle panel) and $\alpha=10$
(bottom panel), for the hyperboloidal slicing and the energy norm. Again, the
spectrum of the unperturbed operator is shown with red circles while the
resulting spectrum of perturbed versions of the effective potential under
perturbations (51) with $k=20$ is shown with differently shaped and colored
points. Once again, the grid-point number used is $N=100$ for the
pseudospectrum and $N=150$ for the perturbed spectra.

####  IV.1.1 Pseudospectra

Concerning the (numerical approximation to the) $\epsilon$-pseudospectral
sets, they indicate a strong spectral instability of the BH QNM spectral
problem, with increasing instability as $\mathrm{Im}(\omega)$ grows, that is,
for higher overtones of the non-perturbed spectrum. The fundamental QNM is the
most stable of the QNMs but, at the pseudospectrum level, the transition to
instability in the overtones is a smooth one, without any drastic qualitative
difference in the transition to the first overtone, as compared to the
transition between the first and second overtones.

Although the pseudospectrum analysis does not allow us to identify the origin
of the QNM instability, an interesting point concerns the asymptotic structure
of the contour lines for large $\mathrm{Re}(\omega)$. They present
qualitatively the same open structure found in the asymptotically flat case
(and also in dS). More specifically, we find that the contour lines are
consistent with an asymptotic logarithm structure $\mathrm{Im}(\omega)\sim
C_{1}+C_{2}\ln(\mathrm{Re}(\omega)+C_{3})$, for $\mathrm{Re}(\omega)\gg 1$.
This structure was identified [25, 23, 36] as universal, in the sense of not
depending on the details of the potential in asymptotic flatness. Results in
Figs. 2 and 3 (together, with [38, 39]) suggest that such universality extends
to different asymptotic boundary conditions.151515 This is in tension with the
results in [40], suggesting a power-law for large $|\mathrm{Re}(\omega)|$
asymptotics, namely $\mathrm{Im}(\omega)\sim
C_{1}+C_{2}|\mathrm{Re}(\omega)|^{\alpha}$. Results in [40] are obtained in a
null slicing. Elucidating this point, and in particular the possible
dependence of the asymptotics on the slicing requires a devoted study.. This
would reflect the underlying ‘ultraviolet’ origin of such asymptotic
logarithmic branches, where large frequency $\mathrm{Re}(\omega)$ values
explore the small-scale structures and are therefore insensitive to the
asymptotics and large scale features of the effective potentials.

####  IV.1.2 BH QNM perturbations

Regarding the calculation of QNMs as eigenvalues in (29), QNMs corresponding
to the non-perturbed BH potential correctly recover the values found by other
methods. In particular, the convergence of such eigenvalues of the $L^{N}$
finite-rank approximant, with the grid resolution $N$, demonstrates their
convergence to the actual QNMs of the differential operator $L$.

Regarding QNM instability, the implementation of perturbations $\delta V$ to
$\delta L$ allows us to probe the origin of the instabilities encoded in the
pseudospectra figures. In particular, the use of ‘physical’ perturbations
$\delta V$ of the effective potential permits us to conclude that high-
wavenumber perturbations do trigger QNM instabilities, whereas low-wavenumber
perturbations leave the QNM spectrum stable. A crucial point in the
asymptotically AdS setting is that the absence of a continuous part of the
spectrum along the upper imaginary axis (corresponding to the branch cut in
the scattering resonance approach to QNMs) permits us to use both
deterministic and random perturbations of the potentials, as in the Pöschl-
Teller case in [24]. This is crucial since, on the one hand, this feature
permits us to soundly implement a convergence analysis of the perturbed QNMs,
namely under deterministic perturbations. This will be crucial in the
convergence issues discussed below. On the other hand, the possibility of
enforcing random perturbations on $V$ (that were not available in the
asymptotically flat BH cases due to the contamination from the ‘branch cut’)
allows us to push the study of the instability analysis, since they maximize
the migration of QNMs to the new Nollert-Price perturbed QNM branches, in
particular saturating the pseudospectrum and probing the Regge QNM conjecture
under ‘ultraviolet’ perturbations [25, 26]. While the plots in Figs. 2 and 3
(and subsequently in Figs. 4 and 5) only present the QNMs after convergent
deterministic perturbations, the same qualitative behaviour has been observed
for the case of random perturbations.

The resulting qualitative picture recovers the generic picture of the
asymptotically flat case, but in a much cleaner setting due to the absence of
‘branch cut’. In particular, the fundamental QNM remains much more
stable161616Perturbations non-vanishing at the horizon, not necessarily of
‘ultraviolet’ nature do indeed perturb the fundamental QNM, as stressed in
[44] and shown in the AdS context in [39]. In our present context, and in
contrast to Ref. [39] we are not considering an actual potential in the
original geometric wave equation, and the effective one coming from the
D’Alembertian do not allow such non-vanishing perturbations at the horizon if
we want to preserve the (classical) BH horizon structure. under such
ultraviolet perturbations (but less stable than in asymptotically flat case,
as in the dS case[38]). Concerning SAdS QNM overtones, they become unstable
under high-wavenumber perturbations of the underlying geometry, migrating to
perturbed open Nollert-Price-like QNM branches. QNMs in such perturbed
Nollert-Price-like branches are stable under further high-wavenumber
perturbations. A crucial point to underline here, in view of the latter
discussion, is that the exact SAdS QNM overtones are spectrally unstable, and
this conclusion can be reached independently of the study of the
pseudospectrum. This will prove to crucial in the discussion below.

###  IV.2 Pseudospectra: null slicing

We consider now the BH QNM spectral instability in the setting of null
slicings. In particular, together with [40], we present the first BH QNM
pseudospectra constructed in a null slicing, since hyperboloidal slicings had
been used in the calculation of all previous BH QNM pseudospectra,
independently of the $\mathscr{I}^{+}$ spacetime null asymptotics. In
accordance with point (ii) in Sec. I.1.1, the calculation of the
pseudospectrum in different slicings has the interest of providing a test on
the geometric nature (slicing independence) of the pseudospectrum. As we will
see below, this comparison problem turns out to be a subtle one, in particular
reinforcing the interest of the null slicing pseudospectrum by itself.

The spectral stability properties of QNMs of SAdS in the null slicing are
illustrated in Figs. 4 and 5, respectively corresponding to a scalar field and
an axial gravitational perturbation (all of the with $\ell=2$), and with the
same choices for the parameter $\alpha$ as in Figs. 2 and 3. As in the
hyperboloidal case, such figures present two independent sets of calculations:

  * i)

QNMs: now calculated from the generalised eigenvalue problem (30), for the
exact effective potential $V_{\rm eff}=V_{s}/(\chi^{2}f)$ of SAdS spacetime
(red points) and for perturbed potentials (differently shaped and colored
points).

  * ii)

Pseudospectrum: now giving the ‘topographic map’ of (the inverse of) function
$f_{M,B}:\mathbb{C}\to\mathbb{R}^{+}$, where
$f_{M,B}(\omega)=\|R_{M,B}(\omega)\|$ with $R_{L}(\omega)$ in (31).

Figure 4: Pseudospectrum in the energy norm of $\ell=2$ scalar perturbations
on SAdS with $\alpha=1$, in the null slicing. The parameters and perturbations
used are the same as those of Fig. 2.

Figure 5: Pseudospectrum of $\ell=2$ axial gravitational perturbations on SAdS
with $\alpha=1/2$ (top panel), $\alpha=1$ (middle panel) and $\alpha=10$
(bottom panel). The parameters and perturbations used are the same as those of
Fig. 3.

When considering the qualitative picture of BH QNM instability for overtones
in the null slicing, both from the pseudospectra and QNM perturbation
perspectives (and with a criterion for ‘big/small’ perturbations given by the
energy norm) it coincides with that obtained in the hyperboloidal slicing.
Specifically, if we consider the pseudospectrum far away from the fundamental
QNM, we observe in the null slicing (as we did in the hyperboloidal case) an
increment of the spectral instability for higher values of
$\mathrm{Im}(\omega)$ in the upper-half complex plane, with $\epsilon$-contour
lines opening for large $\mathrm{Re}(\omega)$ and presenting the same (but cf.
the caveats in footnote 15) qualitative logarithmic asymptotic behavior.
Likewise, regarding QNM perturbations and as in the hyperboloidal case, QNM
overtones migrate to open Nollert-Price-like branches, triggered by high-
wavenumber perturbations in the effective potential, whereas the fundamental
mode shows a spectrally stable behavior. This provides an overall consistency
in that region away the fundamental QNM.

However, discrepancies appear in both the $\epsilon$-pseudospectra contour
lines and the perturbed QNMs as we compare more closely the region around the
fundamental QNM. We comment on this below.

####  IV.2.1 Issues in the comparison of hyperboloidal and null slicing
pseudospectra near the fundamental QNM

We comment on two differences in the BH QNM instability in the hyperboloidal
and the null slicings:

  * i)

Pseudospectrum contour lines crossing into the unstable complex half-plane.
The most apparent difference between the hyperboloidal and null slicing
pseudospectra is the fact that, in the null slicing case, pseudospectrum
contour lines near the fundamental QNM cross into the lower half of the
complex plane. Such a behaviour is a potential indication of a transient
dynamical phenomenon (see e.g. [27, 29, 41, 37, 40]). However this qualitative
behaviour of pseudopectrum contour lines is absent in the hyperboloidal case.
Actually, this points out a structural difference between the AdS BH QNM
pseudospectra in Arean et al. [39] (hyperboloidal slicing case) and in Cownden
et al. [40] (null slicing case). A careful assessment is then needed.

  * ii)

Calibration of perturbation ‘sizes’ and $\epsilon$-contour values. When
comparing the perturbation of the fundamental QNM in Figs. 2 and 3
(hyperboloidal slicing) with Figs. 4 and 5 (null slicing), we observe that the
fundamental QNM seems much more stable in the null slicing case. When trying
to assess this issue by using the values in the pseudospectrum contour lines
another difficulty arises, namely assigning a neat correspondence between
contour lines with the same $\epsilon$ in both sets of pseudospectra.

The origin of these difficulties is two-fold. A first problem concerns the
relation between the ‘size’ of the operator perturbation, on the one hand, and
the resulting perturbation of the eigenvalue, on the other side. A neat
relation between the module $|\delta\omega|$ of the eigenvalue in the complex
plane and the size $\|\delta L\|$ of the operator perturbation holds for the
standard eigenvalue problem (29) and the corresponding
$\epsilon$-pseudospectra in Eq. (41) (cf. [29], in particular the discussion
around the Bauer-Fike theorem). However, such a relation is more difficult to
establish for the generalized eigenvalue problem (30). If $B$ were invertible,
the proper comparison would be between $|\delta\omega|$ and $\|B^{-1}\delta
M\|$ (cf. Chapter 45 in [29]). However, in our problem $B$ is not invertible
(actually, the corresponding matrices are very ill-posed for inversion).
Avenues to circumvent this issue can be explored, but the second problem is
more serious: the pseudospectrum in the hyperboloidal slicing does not
converge when $N\to\infty$ in the finite-rank approximants $L^{N}$ to $L$.

In the following subsection we discuss this pseudospectrum convergence issue.
The AdS structure is key now, since in this case we have a theorem to compare
with.

###  IV.3 Pseudospectra convergence properties: comparison with Warnick’s
characterization

The non-convergence of the hyperboloidal pseudospectrum with the size $N\times
N$ of the matrices $L^{N}$, signalled at the end of the last subsection, is a
major issue in the scheme. Indeed, whereas perturbed eigenvalues in both the
hyperboloidal and the null slicing approaches do converge as the numerical
grid gets finer, at the pseudospectrum level the situation is less clear, with
a convergent behavior in the null slicing case and a non-convergent one in the
hyperboloidal setting. QNM spectral instability is not under question, since
the convergence of the perturbed eigenvalues soundly demonstrates the
ultraviolet instability of QNMs. What is under question is the actual
significance of the numerically calculated pseudospectra in the hyperboloidal
slicing and, in particular, their capability to ‘predict’ the displacement in
the complex plane of QNMs under operator perturbations of ‘size’ $\epsilon$.
We address below this pseudospectrum convergence issue, both in the
hyperboloidal and the null slicing settings.

####  IV.3.1 Hyperboloidal slicing

In order to properly characterize the non-convergence of the pseudospectrum we
proceed as follows: (i) we take an arbitrary complex number $\omega$ in the
upper-half plane, (ii) we consider a finite rank approximant $L^{N}$ of the
operator $L$ and calculate the norm of its resolvent $\|R_{L^{N}}\|$, (iii) we
take the limit of $\|R_{L^{N}}\|$ as $N\to\infty$. The statement is then that
this limit diverges. In other words, according to our numerical scheme for the
pseudospectrum, all points in the upper complex plane are in the spectrum
$\sigma(L)$ of $L$.

The last remark is consistent with the statement in [67], formulated in the
asymptotically flat context, according to which all points in the upper
complex plane are eigenvalues of $L$ if we allow their eigenfunctions to be of
sufficiently low regularity. In the asymptotically AdS case, however, the
results by Warnick [48] provide a sound characterization of the appropriate
regularity and the manner to control it in terms of the corresponding Sobolev
norm. Indeed, as discussed in Sec. III.1, if we consider a band of width $\sim
k\cdot\kappa$, the $H^{k}$-QNM spectrum is a discrete set. In particular, for
points $\omega$ in this band that are not in the $H^{k}$-spectrum, the point
(iii) in III.1 guarantees that the resolvent has finite $H^{k}$ norm.

We use the energy norm in our analysis of QNM stability, namely an $H^{1}$
norm. Therefore, if the resolvent $R_{L}(\omega)$ of the differential operator
$L$ is well attained as the limit of the resolvents $R_{L^{N}}(\omega)$ of the
matrix approximants $L^{N}$, as $N\to\infty$, then Warnick’s theorem implies
that we should find convergence in an horizontal band above the real axis with
width of the order of the surface gravity $\kappa$, except for the fundamental
QNM frequency. However, this is not the case. Indeed, the left panel in Fig. 6
demonstrates a convergence test of the (inverse of the) energy/$H^{1}$ norm of
the resolvent at an illustrative point $\omega=2+i0.5$, providing strong
support for a (power-law) divergence. This behaviour is generic for any point
in the upper half plane, no matter how close to the real axis. Even worse, the
right panel of Fig. 6 shows the convergence test of the norm of the resolvent
for a point in the lower-half plane, showing also non-convergence.

Figure 6: Norm of (inverse of) $R_{L}(\omega)$ as function of the numerical
resolution $N$, for $\ell=2$ axial gravitational perturbations on an SAdS BH
with $\alpha=1$. Left figure: $H^{k},k=1$ (energy norm); $r_{h}\omega=2+i0.5$.
Right figure: $H^{k},k=1$ (energy norm); $r_{h}\omega=2-i0.5$. (These points
are not part of the discrete spectrum.) The linear fit in the Log-Log scale
shows whether there is an inverse-polynomial approach to zero.

In spite of their unexpected character, these negative results on the
convergence of the pseudospectrum (namely of the norm of the resolvent
evaluated any point $\omega$) are not in contradiction with Warnick’s theorem.
The reason is that the resolvent $R_{L}(\omega)$ is a non-compact operator. As
pointed out in (iii) of Sec. III.1, in [48] it has been proven that the
resolvent $R_{L}(\omega)$ exists and it is $H^{k}$-bounded in the band
$\mathrm{Im}(\omega)<a+\kappa(k-1/2)$ (cf. inequality 37), except in a
discrete set of points, namely the QNM frequencies. But it is also shown that
$R_{L}(\omega)$ is not a compact operator. This means that $R_{L}(\omega)$
cannot be obtained as the limit of a sequence of matrices; in other words, it
cannot be approximated (as a whole) by the resolvent of our finite-rank
approximants $L^{N}$.

Assessing the previous point is a subtle issue. On the one hand, we know that
partial features of other non-compact operators, such as $L$, admit good
approximations by matrices. If this were not the case, we could not
approximate the QNMs of $L$ from the eigenvalues of $L^{N}$. But such a
calculation works, because we retain only a subset of all the eigenvalues of
$L^{N}$. However, when calculating the pseudospectrum, the procedure simply
fails, and there is no inconsistency in it. On the other hand, not everything
is spurious in the resolvent of finite-rank approximations $L^{N}$: indeed the
logarithmic contour-lines of the pseudospectrum of $L^{N}$ matrices do capture
the qualitative distribution of perturbed operators, that we know to be
correct since such perturbed eigenvalues do converge as $N\to\infty$. What
fails is the assignment of a particular value $\epsilon$ to a given contour
line. In other words, the finite-rank approximants of the non-compact operator
$R_{L}(\omega)$ do capture the qualitative aspects of the pseudospectrum, but
fail in the quantitative ones.

####  IV.3.2 Null slicing

The situation changes when considering the null slicing. In this case the
pseudospectrum presents much better convergence properties, in particular
consistent with the horizontal $\kappa$-band structure discussed in section
III.1. This is in spite of the fact that, a priori, the comparison with the
results in [48], which are presented in the hyperboloidal setting, is now less
straightforward (in particular regarding the compactness of $R_{MB}(\omega)$).
In the following we present results, at an exploratory stage, demonstrating
such convergence (for a more detailed study, cf. [47]).

As commented above, since the energy norm is an $H^{1}$ one, we should expect
convergence in a horizontal band of approximate width given by the surface
gravity $\kappa$ and, then, convergence in this energy norm should fail in all
$\omega$’s points above such a band. In order to extend by $\kappa$ the width
of the convergent horizontal band in the upper-half plane, we need to control
also the $L^{2}$ size of the second derivatives, i.e. we must use an $H^{2}$
norm. Convergence in $H^{2}$ is then expected in a band of approximate width
$2\cdot\kappa$ and, above this band the pseudospectrum in the $H^{2}$ should
not converge.

Figure 7: Norm of (inverse of) $R_{M,B}(\omega)$ as function of the numerical
resolution $N$, for $\ell=2$ axial gravitational perturbations on an SAdS BH
with $\alpha=1$. The surface gravity of this BH is $\kappa=2/r_{h}$. Left
panel: first row $H^{k},k=1;r_{h}\omega=2+i1.5$, second row
$H^{k},k=2;r_{h}\omega=8+i3$. Right panel: first row $H^{k},k=1$ (energy
norm); $r_{h}\omega=2+i2.6$, second row $H^{k},k=2$; $r_{h}\omega=8+i5$.
(These points are not part of the discrete spectrum.) The linear fit in the
Log-Log scale shows whether there is an inverse-polynomial approach to zero.

In Fig. 7 we test this picture (in our setting, we have $\kappa=2/M$). In the
upper panel of the left row, we present the convergence test for the energy
norm ($H^{1}$) resolvent at a point in the first $\kappa$-band, indicating a
tendency to convergence in stark contrast with the hyperboloidal foliation. In
the upper panel of this first row in Fig. 7 we repeat the convergence test in
the energy norm, but for a point above the first $\kappa$-band, finding a non-
convergent behavior. This is consistent with Warnick’s notion of $H^{k}$-QNMs.
In order to further test the latter, in the lower panel of the left row we
consider the convergence test in an $H^{2}$ norm for a point in the second
$\kappa$-band, finding a clearly convergent tendency. Finally, keeping the
$H^{2}$ norm but considering a point in the third $\kappa$-band, the lower
panel of the right row shows again a non-convergent behaviour. These tests
provide a non-trivial confirmation of the $\kappa$-band structure of the SAdS
BH, consistent with Warnick’s theorem, and illustrate the good convergence
properties of the resolvent, and therefore the pseudospectrum, in the null
slicing.

The results in this section demonstrate the possibility of using the null
slicing for constructing the pseudospectrum of SAdS BHs, in contrast with the
hyperboloidal slicing, where convergence issues must be addressed. For this
reason, we use the null slicing pseudospectrum to assess in the next section
the physical question concerning the spectral stability of SAdS hydrodynamic
QNMs.

###  IV.4 Stability of hydrodynamic modes

We observe a special mode appearing in the complex plane, namely the
hydrodynamic mode. This mode, shown in Fig. 5, has decreasing imaginary part
as the BH radius increases. Approximately when the event horizon radius is
comparable to the AdS scale, and beyond, the hydrodynamic mode become the
dominant one and approaches the real axis. In this section we investigate its
pseudospectral properties, as well as if this mode migrates under
perturbations of the effective potential.

The stability of the hydrodynamic mode under the random perturbations
considered seems to depend on whether it is the fundamental mode or an
overtone. In the case where the hydrodynamic mode dominates the dynamics at
late times, it seems to exhibit spectral stability features in contrast to the
overtones as it shown in the middle and bottom panels of Fig. 5. In contrast,
when the hydrodynamic mode is not fundamental then it becomes spectrally
unstable as shown in the top panel of Fig. 5. Figure 8 zooms in the
pseudospectrum contour lines around this mode for the $\alpha=1$ and $10$
cases, when the mode is spectrally stable, to further enhance our conclusions.
The contour lines of rather large values of $\epsilon$ (up to approximately
$10^{-1.7}$ in the first case, and up to well above $\epsilon=1$ in the latter
case) have a circular shape, which is a direct imprint of spectral stability
(see also Fig. 6 in [24] and Fig. 10 in [36]). Furthermore, the perturbed
versions of these modes migrate less than what the pseudospectrum would
quantitatively suggest, implying an even higher degree of stability, since we
only perturb the effective potential and not the whole operator $L$. To our
knowledge, this is the first BH spacetime in the literature that exhibits
spectrally stable purely imaginary modes when they are fundamental, while the
overtones are still spectrally unstable.

Despite the qualitative evidence for the stability of the hydrodynamic mode, a
quantitative and definitive conclusion relies on a direct identification
between the size $\epsilon$ associated to the norm the perturbed term
$\|\delta M\|$, with the contour line for the $\epsilon$-pseudospectra
$\sigma^{(\epsilon)}$ as defined in Eq. (44), and the circles of radius
$\epsilon$ around the mode. Reference [24], however, points out some
limitations in the direct identification between the size $\epsilon$ for the
operator’s perturbation and the contour line for the $\epsilon$-pseudospectra
$\sigma^{(\epsilon)}$. From the technical perspective, even though the
qualitative behavior for the pseudospectra does not change as one increases
the numerical resolution, the $\epsilon$ value associated with a given
pseudospectrum contour line does. This lack of absolute convergence for the
pseudospectra on the original results raises, at first, a red flag to the
conclusion on the spectral stability for the hydrodynamic mode.

However, we do observe the convergence of the pseudospectra contour lines
around the hydrodynamic mode when it becomes the slowest decaying one of the
system. Indeed, not only does Fig. 8 show that the $\epsilon$-pseudospectral
contour lines retain the same values as we increase the numerical resolution
(e.g. $N=100,150$ and $200$), but also that these lines coincide with the
circles of radius $\epsilon$ around the mode under question. The convergence
becomes even faster as the hydrodynamic mode moves further apart from the
first oscillating mode and approaches the real axis. This result may hint
towards an underlying convergence on the pseudospectrum in a larger region of
the complex plane as well.

Figure 8: Zoom in around the hydrodynamic mode (shown in red), for the middle
case of Fig. 5 (top) with $\alpha=1$ and the bottom case of Fig. 5 (bottom)
with $\alpha=10$ panels. The pseudospectrum contour lines are shown for
$N=100$ (dashed), $N=150$ (dot-dashed) and $N=200$ (continuous). The
differently shaped and colored points designate the perturbed versions of the
hydrodynamic modes resulting from random perturbations added to the potential,
with energy norm $\|\delta L_{1}\|\simeq 0.5$.

##  V Conclusions

This work has focused on assessing some key structural aspects of the spectral
stability of BH QNMs, focusing on the convergence properties of the
pseudospectrum and using asymptotically AdS BHs as a test-bed, thanks to the
existing mathematical characterization by Warnick [48] of QNMs as eigenvalues
of non-selfadjoint operators.

Specifically, we have studied the SAdS BH pseudospectrum in hyperboloidal and
null slicings providing, in particular, a common perspective and synthesis of
the recent studies of this problem, namely [39] in the hyperboloidal slicing
and [40] in the null case. Our main conclusion is that, in their current
(straightforward) numerical implementation, pseudospectrum convergence
properties present a better behaviour in the null case and, crucially and non-
trivially, are consistent with Warnick’s theorems for AdS BH QNMs. It is
indeed the existence of such a neat mathematical characterization of BH QNMs
in the asymptotically AdS case [48] that has provided the key ingredient
allowing us to reach this conclusion on the convergence of the respective
numerical schemes. In particular, it has been crucial for demonstrating the
structure of the frequency $\omega$-complex stable half plane in horizontal
bands of width given by the surface gravity $\kappa$, in such a way that
increasing regularity is required to identify QNM frequencies as we explore
higher overtones.

Regarding the numerical BH QNM pseudospectra calculated in the hyperboloidal
slicing, we observe evidence of non-convergent features in the numerical
scheme. Such non-convergence does not hamper the validity of the (asymptotic)
structure of the pseudospectrum contour lines, but it critically affects the
$\epsilon$ values assigned to such boudaries $\epsilon$-pseudospectra sets.
Although the presented analysis has been focused on SAdS BH, the conclusion
actually extends to pseudospectra calculated in hyperboloidal slicings for all
spacetime asymptotics, namely asymptotically flat, dS and AdS spacetimes.
These poor convergence properties are indeed consistent with the underlying
mathematical structure, the ultimate reason being the non-compact nature of
the infinitesimal time generator in such hyperboloidal slicings. Although
hints for such non-convergent issues had been found in the asymptotically flat
case [24], the contamination by spurious eigenvalues in the continuous ‘branch
cut’ hindered a clear assessment of this problem. Very importantly, we
emphasise that BH QNM spectral instability is not in question, as the
convergence of (deterministically) perturbed QNMs demonstrates. The latter had
been soundly shown in the asymptotically flat and dS cases and has now also
been established for the AdS case in Refs. [39, 40] and the present work. Yet,
the numerical approximations to the hyperboloidal pseudospectrum must be taken
with a grain of salt: although the open contour-lines at a given resolution
$N$ do capture the qualitative spectral instability, the non-convergence with
$N$ of their associated $\epsilon$ values prevents any quantitative
conclusion. In summary, this behaviour signals a fundamental issue of the
numerical scheme for hyperboloidal pseudospectra in its current
implementations but, at the same time, it identifies its underlying cause (the
non-compactness of the operator) and therefore points to a concrete avenue for
its solution. We conclude the urgent need of devoted studies to address,
clarify and, hopefully, cure this pseudospectrum non-convergence problem.

Finally, as a concrete application of the insight gained by the previous
discussion on the structural aspects of BH QNMs as eigenvalues of non-
selfadjoint operators, we have addressed the physical problem concerning the
spectral stability of AdS BH hydrodynamic modes. Specifically, given the good
convergence properties of the pseudospectrum in the null slicing, we have
chosen this foliation (instead of the ‘worse-posed’ hyperboloidal one) to
study the problem. This has permitted us to conclude the convergence of such
BH AdS hydrodynamic modes. As a by-product, the confidence gained in the null
slicing pseudospectrum also provides support for the potential presence of
transients phenomena in the AdS BH setting, suggested by the crossing of
pseudospectra contour lines into the unstable half plane (cf. Fig.5). Both the
hydrodynamic modes and the possible transients have been discussed in [40],
but it is the pseudospectrum convergence analysis here presented and its
consistence with Warnick’s theorems that provides a solid picture.

We plan [47] to further push forward these structural studies, extending the
present analysis to include further physical aspects of the problem, that
remain open.

###### Acknowledgements.

We would like to thank Claude Warnick for generously sharing his insights into
this problem. We also thank Daniel Areán, Piotr Bizoń, Thierry Daudé, Dejan
Gajic, Jeﬀrey Galkowski, David García Fariña, Jason Joykutty, Karl
Landsteiner, Johannes Sjöstrand and Maciej Zworski for discussions. This work
was founded by the VILLUM Foundation (grant no. VIL37766), the DNRF Chair
program (grant no. DNRF162) by the Danish National Research Foundation, and
the European Union’s H2020 ERC Advanced Grant “Black holes: gravitational
engines of discovery” grant agreement no. Gravitas–101052587. Views and
opinions expressed are however those of the author only and do not necessarily
reflect those of the European Union or the European Research Council. Neither
the European Union nor the granting authority can be held responsible for
them. V.B. acknowledges support from the Spanish Government through the
projects PID2020-118159GB-C43, PID2020-118159GB-C44 (with FEDER contribution).
V.C. is a Villum Investigator and a DNRF Chair. This project has received
funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 101007855. K.D.
acknowledges financial support provided under the European Union’s H2020 ERC,
Starting Grant agreement no. DarkGRA–757480 and the MIUR PRIN and FARE
programmes (GW-NEXT, CUP:B84I20000100001). K.D. also acknowledges hospitality
and financial support provided by the Theoretical Astrophysics group of the
University of Tübingen and Instituto Superior Técnico where parts of this
project were worked out. J.L.J. acknowledges the program “Investissements
d’Avenir” through project ISITE-BFC (ANR-15-IDEX-03), the ANR “Quantum Fields
interacting with Geometry” (QFG) project (ANR-20-CE40-0018- 02), the EIPHI
Graduate School (ANR-17- EURE- 0002) and the Spanish FIS2017-86497-C2-1
project (with FEDER contribution). R.P.M. acknowledges the financial support
provided partially by STFC via grant number ST/V000551/1 STFC via grant number
ST/V000551/1. We thank FCT for financial support through Project No.
UIDB/00099/2020. We acknowledge financial support provided by FCT/Portugal
through grants PTDC/MAT-APL/30043/2017 and PTDC/FIS-AST/7002/2020. This
research project was conducted using the computational resources of “Baltasar
Sete-Sois” cluster at Instituto Superior Técnico.

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