# Pseudospectra of complex momentum modes

David García Fariña1,2, Karl Landsteiner1,
Pau G. Romeu1,2 and Pablo Saura-Bastida1,3

###### Abstract

We initiate the study of stability and pseudospectra of complex momentum modes
of asymptotically anti-de Sitter black holes. Similar to quasinormal modes,
these can be defined as the poles of the holographic Green’s function, albeit
for real frequency and complex momentum. Their pseudospectra are in stark
contrast to the pseudospectra of quasinormal modes of AdS black holes.
Contrary to the case of quasinormal mode pseudospectra, the resolvent is well-
defined, and the numerical approximation shows fast convergence. At zero
frequency, complex momentum modes are stable normal modes of a Hermitian
operator. Even for large frequencies, they show only comparatively mild
spectral instability. We also find that local potential perturbations cannot
destabilize the lowest complex momentum mode.

1Instituto de Física Teórica UAM/CSIC, c/Nicolás Cabrera 13-15, Universidad
Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

2Departamento de Física Teórica, Universidad Autónoma de Madrid, Campus de
Cantoblanco, 28049 Madrid, Spain

3 Departamento de Automática, Ingeniería Eléctrica y Tecnología Electrónica,
Universidad Politécnica de Cartagena, Calle Dr. Fleming, S/N, 30202 Cartagena,
Murcia

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>IFT-UAM/CSIC-24-100

###### Contents

  1. 1 Introduction
  2. 2 Pseudospectra and Stability
  3. 3 Convergence of Pseudospectrum of Quasinormal Modes
  4. 4 Complex Momentum Modes
  5. 5 Holographic model
    1. 5.1 Energy Norm and Function Space
    2. 5.2 Construction of $\mathcal{L}$ and $\mathcal{L}^{\dagger}$
  6. 6 Numerical method
  7. 7 Results
  8. 8 Summary and Discussion
  9. A Non-normality in AdS
  10. B Numerical Values of the Complex Momenta
  11. C Pseudospectrum plots
  12. D Comparison with quasinormal frequencies at complex momentum

##  1 Introduction

Eigenvalue problems for linear operators are fundamental to all of physics,
from classical mechanics to quantum mechanics and general relativity. The
eigenvalues are of course determined by the particular boundary conditions.
Closed systems lead to the emblematic Sturm-Liouville type problems for self-
adjoint differential operators. Open systems on the other hand are
characterized by non-self-adjoint and potentially non-normal operators. Non-
normality implies that under a small perturbation to the operator, the
eigenvalues might suffer a displacement much larger than the size of the
perturbation. This phenomenon, labelled spectral instability, is a fundamental
feature of non-normal operators. For normal operators, such as the ones
appearing in conservative systems, the spectral theorem ensures that the
eigenvalues cannot be displaced more than the size of the perturbation, and
thus the spectrum is stable.

At a practical level, spectral instability has significant implications. It
tells us that if we use some non-normal operator to model a non-conservative
system and assume a small error in its construction (e.g. disregarding some
interactions to simplify computations), we cannot ensure that the spectrum of
the operator should match the actual spectrum of the real system. In short,
the presence of spectral instability warns us that to make predictions to a
given accuracy, we will need to construct a model with a much higher one. For
that reason, when dealing with non-conservative systems, one should complement
the standard eigenvalue analysis with a discussion on the stability of the
eigenvalues, which typically is approached through the study of the
pseudospectrum [1, 2, 3].

In the context of gravitational physics, non-self-adjoint operators
generically appear when studying quasinormal modes (QNMs) of black holes [4,
5, 6]. The non-conservative nature of black hole geometries, derived from the
existence of an event horizon which acts as a perfectly absorbing membrane,
results in the time evolution of linearized fluctuations being governed by a
non-self-adjoint differential operator, whose eigenvalues are the quasinormal
frequencies (QNFs). In asymptotically flat spacetimes, fluctuations are also
outgoing at asymptotic infinity further contributing to the non-self-
adjointness.

The study of the spectral stability of quasinormal frequencies using a
pseudospectrum analysis has been recently pioneered in [7]. Further aspects of
the pseudospectrum of quasinormal modes of asymptotically flat black holes
have been investigated in [8, 9, 10, 11, 12, 13] (see [14] also for
horizonless compact objects and [15] for quantum corrected black holes), as
well as in de Sitter (dS) [16, 17] and anti-de Sitter (AdS) [18, 19, 20]
geometries. All these studies found the spectrum of quasinormal frequencies to
be unstable due to the non-conservative nature of the geometry. Further
studies have also covered the implications of this spectral instability on the
strong cosmic censorship [21] and on the gravitational waves emitted in black
hole mergers [22, 23].

In the case of asymptotically AdS spacetimes, the observation of spectral
instability is of particular interest in view of gauge/gravity duality [24,
25, 26, 27, 28] which postulates that black hole geometries are dual to
thermal states in a strongly coupled quantum field theory [29]. In this
context, black hole quasinormal frequencies are interpreted as poles of
retarded propagators of the thermal field theory [30, 31, 32]. Consequently,
the study of quasinormal frequencies in AdS black holes has led to important
insights into hydrodynamics and transport theory in the relativistic regime
[33, 34, 35], with some remarkable results being the extremely low specific
shear viscosity of holographic models of the quark-gluon plasma [36], phase
transitions towards superconducting states [37, 38, 39, 40] and strongly
coupled quantum critical phases [41, 42, 43]. To better understand the
physical meaning of this spectral instability is therefore clearly of utmost
importance for gauge/gravity duality.

Quasinormal frequencies determine the time evolution of an open system that is
slightly perturbed away from its ground state. Figuratively one hits the
system with a “hammer” and observes the damped oscillations that return the
system to its ground state. This is however not the only way of probing an
open or dissipative system. Another way is to couple it locally to an
oscillating source (“antenna”) and observe the wavelength and absorption
length of the forced oscillation. This type of response is also determined by
the poles of the retarded Green’s function, this time however one fixes the
frequency to be real and searches for poles in the complexified momentum
plane. In gauge/gravity duality this means that one solves the same system of
equations as for the quasinormal modes but considers now the frequency as
given and solves for the complex eigenvalues of the momentum [44]. These
complex momentum modes are the AdS analogues of the complex angular momentum
modes in asymptotically flat spacetimes [45]. We denote complex momentum modes
by $\mathbb{C}$MMs and their corresponding eigenvalues, the complex (linear)
momenta by $\mathbb{C}$LMs.

In this paper, we therefore depart from the study of the stability of QNFs to
instead consider the stability of $\mathbb{C}$LMs of planar AdS black holes.
While the former are eigenvalues of the generator of time translations, the
latter correspond to eigenvalues of the generator of spatial translations
along some direction parallel to the brane. QNFs and $\mathbb{C}$LMs are dual
to poles of the retarded Green’s function at fixed momentum (relaxation times)
and fixed frequency (absorption lengths), respectively. $\mathbb{C}$LMs also
play an essential role in studying causality of the boundary field theory [46,
47]. At zero frequency, they are dual to the glueball masses of a
dimensionally compactified toy model for QCD [29, 48, 49, 50, 51].
$\mathbb{C}$MMs have also appeared in the holographic context in [52, 53, 54,
55]. It is also worth noting that in the context of asymptotically flat
spacetimes [56, 57, 58] studied the stability of Regge poles (complex angular
momenta).

We choose to study the stability of complex momenta for two main reasons

  * •

Complex momenta offer a new window to probe the spectral stability of the
theory. Although both $\mathbb{C}$LMs and QNFs are poles of the retarded
Green’s function, they appear in qualitatively very different settings, and
thus, their spectral properties need not be the same. Hence, we could find a
theory whose QNFs are stable while the $\mathbb{C}$LMs are not and vice versa.

  * •

The pseudospectra of complex momenta computed at real frequencies is
convergent. When computing the pseudospectra of QNFs, one finds that it does
not converge for the overtones [20]. This issue seems to be related to the
definition of size for the perturbations and, as we shall see in greater
detail, hints at the need to introduce a cutoff to how localized the
perturbations can be. Notably, this prevents us from making quantitative
statements about the stability, although it is believed that the qualitative
picture is correct. On the other hand, working at fixed real frequency solves
these issues and thus allows us to reach quantitative conclusions about the
full pseudospectrum.

We compute the $\mathbb{C}$LM pseudospectrum for a real scalar field in a
Schwarzschild AdS4+1 black brane. Although this setup lacks some interesting
phenomenology associated with the existence of hydrodynamic modes, we favour
its simplicity and lack of gauge symmetries, which would introduce some extra
subtleties. We leave a more complete study for fields with gauge symmetries to
a follow-up work.

The paper is structured as follows. In section 2 we offer a short review of
pseudospectra in the context of stability analysis. Here, we follow closely
[18] and draw heavily from [1].

In section 3 we discuss the lack of convergence of the pseudospectrum of QNFs.
Our presentation is not in-depth; instead, we try to offer a more qualitative
picture. We refer the interested reader looking for a more detailed discussion
to [20].

In section 4, we define $\mathbb{C}$MMs and showcase, from the point of view
of the dual field theory, in what situations they appear. We discuss their
differences with respect to QNFs and conclude by arguing some of the stability
properties complex momenta should have. In particular, in the zero frequency
limit, we argue that as $\mathbb{C}$LMs are dual to the glueball spectrum of
the effective theory for the lowest Kaluza-Klein modes of the compactification
on a thermal circle, they should be stable.

In section 5, we introduce the holographic model and construct a well-
motivated notion of size for the $\mathbb{C}$MMs. We follow the usual
prescription introduced in [7, 59] and more generally argued for in [1], and
use the energy as a guide to construct a norm in subsection 5.1. In subsection
5.2, we define the relevant differential operator whose eigenvalues are the
$\mathbb{C}$LMs, and construct its adjoint. We pay special attention to how
our norm manages to reproduce the expected stability properties discussed in
section 4.

In section 6 we discuss the numerical implementation based on pseudospectral
methods and introduce the selective pseudospectrum used to test the stability
under random local potential perturbations. We also construct some specific
potentials to test the nature of the (in)stability.

Section 7 is the core of the paper and contains the results of our
pseudospectrum computations. We find that at non-zero frequency, the spectrum
becomes rapidly unstable. Remarkably, we observe that the effect of local
potential perturbations seems to be very mild.

In section 8, we summarize our findings and present our conclusions.

In appendix B we present the numerical values of the complex momenta. Relevant
figures are collected in appendix C. A comparison between the stability
properties of $\mathbb{C}$LMs and QNFs can be found in appendix D.

##  2 Pseudospectra and Stability

In this section, we summarize some important definitions and results
concerning pseudospectra and condition numbers and their relevance for the
study of the spectral stability of non-normal operators. We follow closely the
exposition of [18] and refer the interested reader to [1] for further details.

Given a closed linear operator $\mathcal{L}$ acting on a Hilbert space
$\mathcal{H}$ with domain $\mathcal{D}(\mathcal{L})$, its spectrum
$\sigma(\mathcal{L})$ is defined as the set of points $\lambda$ in the complex
plane where the resolvent
$\mathcal{R}(\mathcal{L};\lambda)=(\mathcal{L}-\lambda)^{-1}$ is not defined.
An eigenvalue $\lambda\in\sigma(\mathcal{L})$ is defined as a solution to the
eigenvalue equation

$(\mathcal{L}-\lambda)u_{\lambda}=0\,,$ (1)

where $u_{\lambda}\in\mathcal{D}(\mathcal{L})$ is the corresponding
eigenvector.

An important property of eigenvalues is that for self-adjoint adjoint
operators (or, in more physical terms, conservative systems), the spectral
theorem ensures that if we perturb the system with a bounded operator of size
$\varepsilon$ the eigenvalues of the perturbed operator cannot suffer a
displacement greater than $\varepsilon$ [60, 61]. This property holds in
general for any normal operator $A$ satisfying $\left[A,A^{\dagger}\right]=0$,
with $A^{\dagger}$ the adjoint of $A$.

However, in non-conservative systems normality of the relevant operators is
not guaranteed and small perturbations could potentially alter the spectrum in
a significant manner. For that reason, one concludes [1] that in non-
conservative systems eigenvalue analysis alone is insufficient as the spectrum
might be unstable.

To characterize the stability of eigenvalues one introduces the notion of
$\varepsilon$-pseudospectrum, which can be defined in three mathematically
equivalent ways [1]:

######  Def. 2.1 (Resolvent norm approach).

Given a closed linear operator $\mathcal{L}$ acting on a Hilbert space
$\mathcal{H}$ with domain $\mathcal{D}(\mathcal{L})$, and $\varepsilon>0$, the
$\varepsilon$-pseudospectrum $\sigma_{\varepsilon}(\mathcal{L})$ is

$\sigma_{\varepsilon}(\mathcal{L})=\\{z\in\mathbb{C}:\lVert\mathcal{R}(z;\mathcal{L})\rVert>1/\varepsilon\\}\,,$
(2)

with the convention $\lVert\mathcal{R}(z;\mathcal{L})\rVert=\infty$ for
$z\in\sigma(\mathcal{L})$.

######  Def. 2.2 (Perturbative approach).

Given a closed linear operator $\mathcal{L}$ acting on a Hilbert space
$\mathcal{H}$ with domain $\mathcal{D}(\mathcal{L})$, and $\varepsilon>0$, the
$\varepsilon$-pseudospectrum $\sigma_{\varepsilon}(\mathcal{L})$ is

$\sigma_{\varepsilon}(\mathcal{L})=\\{z\in\mathbb{C},\exists V,\rVert
V\lVert<\varepsilon:z\in\sigma(\mathcal{L}+V)\\}\,.$ (3)

######  Def. 2.3 (Pseudoeigenvalue approach).

Given a closed linear operator $\mathcal{L}$ acting on a Hilbert space
$\mathcal{H}$ with domain $\mathcal{D}(\mathcal{L})$, and $\varepsilon>0$, the
$\varepsilon$-pseudospectrum $\sigma_{\varepsilon}(\mathcal{L})$ is

$\sigma_{\varepsilon}(\mathcal{L})=\\{z\in\mathbb{C},\exists
u^{\varepsilon}\in\mathcal{D}(\mathcal{L}):\;\rVert(\mathcal{L}-z)u^{\varepsilon}\lVert<\varepsilon\lVert
u^{\varepsilon}\rVert\\}\,,$ (4)

where $u^{\varepsilon}$ is a $\varepsilon$-pseudoeigenvector with
$\varepsilon$-pseudoeigenvalue $z$.

Note that, contrary to the spectrum, the pseudospectrum depends on the
operator norm

$\rVert V\lVert=\max_{u\in\mathcal{H}}\frac{\rVert Vu\lVert}{\rVert
u\lVert}\,,$ (5)

as, in order to quantify stability, it needs a notion of what constitutes a
small perturbation.

Remarkably, definition 2.2 corresponds to the physical intuition we were
seeking: the $\varepsilon$-pseudospectrum constitutes the maximal region
containing all possible displacements of the eigenvalues under perturbations
of size $\varepsilon$. It is then quite natural to represent the
pseudospectrum as a contour map indicating the boundaries of these regions for
multiple values of $\varepsilon$. However, computationally definition 2.2 is
very inefficient as one should compute the spectra for all possible bounded
perturbations. For that reason, when studying general instability properties
one instead uses definition 2.1 which, despite lacking such a clear physical
interpretation, is much more manageable to compute in a finite-dimensional
setting such as the ones arising when employing numerical approximations to
the operators. When constructing a selective pseudospectrum where we only
analyze the stability under a certain restricted type of perturbations (e.g.
local potentials), we will still use definition 2.2 with a finite set of
randomly generated perturbations.

Another useful tool for studying the stability of eigenvalues is the set of
condition numbers $\\{\kappa_{i}\\}$ defined as

$\kappa_{i}=\frac{\norm{v_{i}}\norm{u_{i}}}{|\expectationvalue{v_{i},u_{i}}|}\,,$
(6)

with $\expectationvalue{\cdot,\cdot}$ the inner product associated with the
norm $\norm{\cdot}$, $u_{i}$ the right-eigenvector satisfying
$\mathcal{L}u_{i}=\lambda_{i}u_{i}$ and $v_{i}$ the left-eigenvector
satisfying $\mathcal{L}^{\dagger}v_{i}=\bar{\lambda}_{i}v_{i}$.111We denote
the complex conjugate with a bar and complex conjugate transpose with an
asterisk. Remarkably, condition numbers manage to quantify the effect of
perturbations of size $\varepsilon$ through only the knowledge of the
orthogonality between the eigenvectors of the unperturbed operator and its
adjoint. Explicitly, for a bounded perturbation $V$ of size $\varepsilon$ we
have:

$|\lambda_{i}(\varepsilon)-\lambda_{i}|\leq\varepsilon\kappa_{i}\,,$ (7)

where $\\{\lambda_{i}(\varepsilon)\\}$ are the eigenvalues of the perturbed
operator $\mathcal{L}(\varepsilon)=\mathcal{L}+V$ and $||V||=\varepsilon$. For
normal operators all eigenvalues are stable and have condition number 1.

As in this work, we limit ourselves to studying pseudospectra of matrices
arising from the discretization of differential operators, we collect below a
relevant theorem specialized to matrices, which allows to efficiently compute
the pseudospectrum numerically. Again further details can be found on [1].

######  Thm. 2.1.

Given the $\ell^{2}_{N}$-inner product $\expectationvalue{\cdot,\cdot}_{2}$

$\expectationvalue{v,u}_{2}=\bar{v}^{i}u^{i}\,,$ (8)

and generic $G$-inner product $\expectationvalue{\cdot,\cdot}_{G}$ such that:

$\expectationvalue{v,u}_{G}=G_{ij}\bar{v}^{i}u^{j}\,,$ (9)

with $G=F^{*}F$ a symmetric positive definite $N\times N$ matrix.

  * •

The $\varepsilon$-pseudospectrum of a matrix $M$ in the $G$-norm
$\sigma_{\varepsilon}^{G}(M)$ satisfies

$\sigma_{\varepsilon}^{G}(M)=\sigma_{\varepsilon}^{\ell^{2}_{N}}\left(FMF^{-1}\right)\,,$
(10)

with $\sigma_{\varepsilon}^{\ell^{2}_{N}}(M)$ the pseudospectrum in the
$\ell^{2}_{N}$-norm.

  * •

The condition number of the eigenvalue $\lambda_{i}$ of a matrix $M$ in the
G-norm $\kappa_{i}^{G}$ satisfies:

$\kappa_{i}^{G}=\frac{\norm{\tilde{v}_{i}}_{2}\norm{\tilde{u}_{i}}_{2}}{|\expectationvalue{\tilde{v}_{i},\tilde{u}_{i}}_{2}|}\,,$
(11)

where $\tilde{u}_{i}$ and $\tilde{v}_{i}$ fulfill:

$FMF^{-1}\tilde{u}_{i}=\lambda_{i}\tilde{u}_{i}\,,\qquad\left(FMF^{-1}\right)^{*}\tilde{v}_{i}=\bar{\lambda}_{i}\tilde{v}_{i}\,.$
(12)

Lastly, we note that throughout this section we have assumed $\varepsilon$ to
be small. Formally this corresponds to saying that $\varepsilon$ is much
smaller than the minimum distance $d_{\min}$ between disconnected regions of
the spectrum, i.e., $\varepsilon/d_{\min}\ll 1$.

##  3 Convergence of Pseudospectrum of Quasinormal Modes

In this section we offer a short qualitative discussion on the lack of
convergence of the pseudospectrum of quasinormal modes reported in [20].

As pointed out in [62], when defining QNMs in regular coordinates one needs to
demand analyticity on the event horizon. Otherwise, outgoing and ingoing modes
would be indistinguishable and any frequency would then be a quasinormal
frequency. This implies that if we define QNFs as eigenvalues of a non-normal
operator acting on some Hilbert space, we should ensure that the norm of the
said Hilbert space does indeed eliminate all non-analytic modes, i.e. that
their norm is infinite.

To construct the pseudospectrum, we would like to have a physically motivated
norm. For QNMs, as argued in [7, 59], such norm is given by their energy which
typically contains at most two derivatives. However we should check whether
such norm does discard all non-analytic modes. Consider the non-analytic
function $\psi(x)=x^{3/2}$. Then for a norm containing only two derivatives
(such as the energy)

$\norm{\psi}^{2}=\int_{0}^{1}dx\,\left[a\partial_{x}\psi\partial_{x}\bar{\psi}+b\psi\bar{\psi}\right]=\int
dx\,\left[a\frac{9}{4}\left(x^{1/2}\right)^{2}+b\left(x^{3/2}\right)^{2}\right]=\frac{9}{8}a+\frac{1}{4}b<\infty\,.$
(13)

Since $\norm{\psi}<\infty$ our construction fails to remove this non-analytic
mode. On the other hand, if we consider a norm with four derivatives such as

$\norm{\psi}^{2}=\int_{0}^{1}dx\,\left[a\partial_{x}\psi\partial_{x}\bar{\psi}+b\psi\bar{\psi}+c\partial_{x}^{2}\psi\partial_{x}^{2}\bar{\psi}\right]=\int
dx\,\left[a\frac{9}{4}\left(x^{1/2}\right)^{2}+b\left(x^{3/2}\right)^{2}+c\frac{9}{16}\left(x^{-1/2}\right)^{2}\right]=\infty\,,$
(14)

we see that $\norm{\psi}=\infty$; and the mode is indeed discarded.
Nonetheless a four-derivative norm will still fail to eliminate other non-
analytic modes. To ensure that one properly removes all non-analytic modes one
is forced to consider a Sobolev norm containing an infinite number of
derivatives [62].

In summary, the problem of computing the pseudospectrum of the QNMs seems to
be ill-posed. To define the quasinormal modes we need to consider a Sobolev
norm with infinite derivatives while the physically relevant norm is the
energy which contains only two derivatives [20]. More concretely, the energy
norm fails to differentiate between modes entering and exiting the horizon for
large enough $-\imaginary(\omega)$, i.e. for the overtones (in our convention
exponential decay in time means $\imaginary(\omega)<0)$.

This interpretation then sheds light on the apparent reason as to why the
pseudospectrum computed numerically fails to converge. In the numerical
approach, the pseudospectrum is computed for a discretized version of the
system using a Chebysev grid. This in turn implies that only analytic
functions enter the computation, since non-analytic functions cannot be
represented to infinite accuracy in the grid. However, as we go to the
continuum limit where the grid is removed, the non-analytic functions can be
better and better approximated in the grid. Hence, we tend to the
pseudospectrum expected for the continuum limit where every point in the
complex frequency plane satisfying $\Im{\omega}<a$, with $a$ a constant
dependent on the number of derivatives in the norm [62], becomes a quasinormal
frequency and thus the norm of the resolvent tends to infinity.

This seems to be indicating a very interesting physical picture. As we allow
the perturbations entering in the pseudospectrum to be more and more localized
in the radial direction (by increasing the number of grid points) the size of
the perturbation needed to displace a given QNF to any arbitrary point in the
complex plane becomes smaller and smaller (recall definition 2.2 of
$\varepsilon$-pseudospectrum). This seems to agree with the results of [20],
where the authors showed that for a fixed perturbation the effect on the
spectrum was independent of the grid size. Consequently, the continuum limit
corresponds to allowing more and more localized perturbations. However, one
expects general relativity to break down at small scales. Thus, in this
picture the grid size acts as a cutoff and to go to the continuum limit we
expect to need to include higher derivative corrections to the theory and,
consequently, to the energy norm; thus obtaining a Sobolev norm. A similar
conclusion was reached in [63] in terms of the response of fluctuations to
sources localized on the horizon. Further aspects of the instability of the
pseudospectrum were also discussed in [64].

##  4 Complex Momentum Modes

Similar to QNMs, complex momentum modes are defined as solutions to the
linearized equations of motion in a fixed black brane background satisfying
infalling boundary conditions on the event horizon and normalizable boundary
conditions on the AdS boundary. However, while QNMs are eigenvalues of the
killing vector associated with time translations, complex $\mathbb{C}$MMs are
eigenvalues of the generator of spatial translations along an arbitrary
spatial direction parallel to the brane which we denote $x^{3}$.

We consider a Schwarzschild anti-de Sitter black brane background with metric

$ds^{2}=\frac{r^{2}}{l^{2}}(-f(r)d\hat{t}^{2}+d\mathbb{x}^{2})+\frac{l^{2}}{r^{2}f(r)}dr^{2}\,,\qquad
f=1-\frac{r_{h}^{4}}{r^{4}}\,,$ (15)

where $f$ is the blackening factor which has a zero at $r=r_{h}$ and $l$ the
AdS scale. We introduce regular coordinates [62, 18] by the following
coordinate transformation

$t=\hat{t}-\frac{l^{2}}{r_{h}}\left(1-\frac{r_{h}}{r}\right)+\int\frac{dr}{f(r)}\left(\frac{l}{r}\right)^{2}\,,\qquad\rho=1-\frac{r_{h}}{r}\,,$
(16)

where we also have compactified the radius so that the AdS boundary is at
$\rho=1$ and the horizon at $\rho=0$. Remarkably, on the boundary $t=\hat{t}$,
which can be identified with the time in the dual field theory. The metric in
regular coordinates takes the form

$ds^{2}=\frac{l^{2}}{z_{h}^{2}(1-\rho)^{2}}\left\\{-f(\rho)dt^{2}+(dx^{3})^{2}+d\mathbb{x}_{\perp}^{2}+2(1-f(\rho))z_{h}dtd\rho+(2-f(\rho))z_{h}^{2}d\rho^{2}\right\\}\,,$
(17)

Here $z_{h}=l^{2}/r_{h}$, the Hawking temperature is given by $T=(\pi
z_{h})^{-1}$ and $\mathbb{x}_{\perp}$ denotes the spatial directions
perpendicular to $x^{3}$ and $\rho$. We note that regularity of the
$\mathbb{C}$MMs is equivalent to demanding infalling boundary
conditions.222Regular coordinates are the AdS analogue to the hyperboloidal
ones used in asymptotically flat spacetimes [65, 66, 67, 68, 69, 70].
Alternatively one could have also used infalling Eddington-Finkelstein
coordinates as in [19].

For a $\mathbb{C}$MM $\Phi(t,x^{3},\mathbb{x}_{\perp},z)$, after Fourier
transforming along the time direction $t$ and the spatial coordinates
$\mathbb{x}_{\perp}$

$\Phi(t,x^{3},\mathbb{x}_{\perp},z)=\int\frac{d\omega\,d^{\text{d}-2}k_{\perp}}{(2\pi)^{d-1}}\Phi(\omega,x^{3},\mathbb{x}_{\perp},z)\exp{-i\omega
t+i\mathbb{k}_{\perp}\mathbb{x}_{\perp}}\,,$ (18)

the corresponding $\mathbb{C}$LM $k$ is defined through the following equation

$\partial_{3}\Phi(\omega,x^{3},\mathbb{x}_{\perp},z)=ik\Phi(\omega,x^{3},\mathbb{x}_{\perp},z)\,,$
(19)

where note that $k$ is a function of $\omega$ and $\mathbb{k}_{\perp}$.
Henceforth, we always take $\mathbb{k}_{\perp}=0$ as we can use the rotational
invariance of the background (17) to align the momentum with the $x^{3}$
direction.

QNFs and $\mathbb{C}$LMs are just two different real sections of the complex
lines that are the poles of the holographic retarded Green’s function
$G_{R}(\omega,k)$ in the two dimensional complex space spanned by
$(\omega,k)$. QNFs are the sections $\imaginary(k)=0$ whereas $\mathbb{C}$LMs
are the sections $\imaginary(\omega)=0$. Generically these complex lines
intersect in complicated ways. That can make the properties of the modes very
different when viewed as quasinormal modes or as complex momentum modes (see
for example the discussion of the diffusive mode in [44] and its relevance for
causality [46]). It also gives rise to a rich and complex life story [71].

$\mathbb{C}$LMs and QNFs arise naturally in very different physical settings.
Let us illustrate this from the point of view of the dual CFT. We consider a
simple example within linear response theory and follow closely the discussion
of [44]. In linear response theory the expectation value of an operator
$\mathcal{O}$ is given by

$\expectationvalue{\mathcal{O}(t,\mathbb{x})}=-\int
dt^{\prime}\,d^{\text{d}-1}x^{\prime}\,\,G_{R}(t-t^{\prime},\mathbb{x}-\mathbb{x}^{\prime})j_{\mathcal{O}}(t^{\prime},x^{\prime})\,,$
(20)

where $G_{R}$ is the retarded propagator and $j_{\mathcal{O}}$ the source of
the said operator. If we now take the source to be an antenna of the form
$j_{\mathcal{O}}=\chi(x^{3})\exp(-i\omega t)$ with $\chi(x^{3})$ an arbitrary
function with analytic Fourier transform, then we get

$\displaystyle\expectationvalue{\mathcal{O}(t,\mathbb{x})}$
$\displaystyle=-\int
dt^{\prime}\,d^{\text{d}-1}x^{\prime}\,\,\int\frac{d\nu\,d^{d-1}q}{(2\pi)^{\text{d}}}\tilde{G}_{R}(\nu,\mathbb{q})\int\frac{dk}{2\pi}\tilde{\chi}(k)\exp{ik(x^{3})^{\prime}+iq(x-x^{\prime})-i\omega
t-i\nu(t-t^{\prime}))}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dk}{2\pi}\tilde{\chi}(k)\tilde{G}_{R}(\omega,k,\mathbb{k}_{\perp}=0)\exp(ikx^{3}-i\omega
t)$ $\displaystyle=-i\,\text{sign}(x^{3})e^{-i\omega
t}\sum_{k_{n}:\text{poles}}e^{ik_{n}x^{3}}\text{Res}\left[\tilde{\chi}(k)\tilde{G}_{R}(\omega,k,\mathbb{k}_{\perp}=0)\right]\,,$
(21)

where in the last equality we have used Cauchy’s theorem (see figure 1) and we
have denoted the Fourier transforms with a tilde for clarity. Exponential
decay away from the origin of the perturbation happens as long as

$\text{sign}\left(\frac{\omega}{\real(k)}\right)=\text{sign}\left(\imaginary(k)\right)\,,$
(22)

which means that for positive frequency the poles are located in the first and
third quadrant [44].

$\text{Re}(k)$$\text{Im}(k)$$\Gamma_{x^{3}<0}$ (a)

$\text{Re}(k)$$\text{Im}(k)$$\Gamma_{x^{3}>0}$ (b)

Figure 1: The relevant integration contours for the poles in the complexified
momentum-plane. Figure 1(a) shows the contour for $x<0$, and figure 1(b) shows
the contour for $x>0$. In order to obtain exponentially decaying waves
travelling away from the origin of the perturbation, it is necessary that the
poles lie in the 1st and 3rd quadrants for positive frequency (as assumed in
the figure).

Alternatively, we could take the source to be a hammer of the form
$j_{\mathcal{O}}=\xi(t)\exp(ikx^{3})$, with $\xi(t)$ a function with analytic
Fourier transform. Then

$\displaystyle\expectationvalue{\mathcal{O}(t,\mathbb{x})}$
$\displaystyle=-\int
dt^{\prime}\,d^{\text{d}-1}x^{\prime}\,\,\int\frac{d\nu\,d^{d-1}q}{(2\pi)^{\text{d}}}\tilde{G}_{R}(\nu,\mathbb{q})\int\frac{d\omega}{2\pi}\tilde{\xi}(\omega)\exp{-i\omega
t^{\prime}+ik(x^{3})^{\prime}+iq(x-x^{\prime})-i\nu(t-t^{\prime}))}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\tilde{\xi}(\omega)\tilde{G}_{R}(\omega,k,\mathbb{k}_{\perp}=0)\exp{ikx^{3}-i\omega
t}$
$\displaystyle=i\Theta(t)\,e^{ikx^{3}}\sum_{\omega_{n}:\text{poles}}e^{-i\omega_{n}t}\text{Res}\left[\tilde{\xi}(\omega)\tilde{G}_{R}(\omega,k,\mathbb{k}_{\perp}=0)\right]\,,$
(23)

Thus, we conclude that $\mathbb{C}$LMs arise naturally when considering
sources that behave as antennas, while QNFs appear when studying sources that
behave as hammers. This has many deep fundamental implications as both setups
are qualitatively very different. In particular, in the former case we always
necessarily have a non-conservative system, even in the absence of
thermalization, as the source introduces/extracts energy for any $t$; while in
the latter setup we need not expect such behaviour as sources should vanish
for large enough times $|t|>|t_{*}|$.

It is also worth stressing that when studying $\mathbb{C}$LMs $k(\omega)$ we
are probing the stability of a sector with a given fixed frequency. This is
particularly important in the $\omega=0$ limit. In this limit the
$\mathbb{C}$MMs at $\omega=0$ are dual to the glueball masses of the effective
theory of the lowest lying Kaluza-Klein modes of the theory compactifid on the
thermal circle [29, 72]. As this effective theory is conservative, we expect
the spectrum of glueball masses to be spectrally stable. Hence, the
$\mathbb{C}$LMs $k(\omega=0)$ should be stable provided we choose the norm
adequately. On the other hand, a QNF $\omega=0$ need not be stable as we work
in the fixed $k$ sector instead.

We briefly review the idea behind the interpretation of complex momenta as
glueball masses in a dimensionally reduced theory. In order to do so we go
momentarily back to the coordinate system (15) but note that the time like
Killing vectors of this metric and of (17) are the same
$\frac{\partial}{\partial\hat{t}}=\frac{\partial}{\partial t}$. In particular
they have the same zero modes. In the black brane metric (15) we can now do a
double Wick rotation $\hat{t}\rightarrow i\sigma$ and $x^{3}\rightarrow
ix^{0}$ The dual field theory lives then on the geometry $S^{1}\times M_{3}$
with $M_{3}$ being three dimensional Minkowski space. Zero modes on the
$S^{1}$ are the zero modes of $\frac{\partial}{\partial_{\sigma}}$ which
coincide with the zero modes of $\frac{\partial}{\partial\hat{t}}$. These zero
modes live in three dimensional Minkowski space. The eigenvalues of the three
dimensional momenta $k^{2}$ can be interpreted as masses of excitation in the
dimensionally reduced theory (glueballs). Note that now $k^{2}<0$ simply means
that $k$ is timelike in the reduced theory. Thus in the original theory,
$\sqrt{-k^{2}}$ is interpreted as an absorption coefficient or inverse
screening length, whereas in the double Wick rotated dimensionally reduced
theory $\sqrt{-k^{2}}=M$ is interpreted as the mass of a propagating
excitation of a (Hermitian) dual quantum field theory.

Lastly we note that, the behaviour of $\mathbb{C}$MMs near the event horizon
depends only on the frequency $\omega$. Hence, following the discussion on the
previous section, we expect the pseudospectrum of $\mathbb{C}$MMs to converge
as modes exiting the brane with $\imaginary(\omega)=0$ are discarded as we see
in section 5.2. We also explicitly see in section 7 that our numerics do
converge.

##  5 Holographic model

We want to study the spectral stability of the $\mathbb{C}$LMs of a real
scalar field $\phi$ in a SAdS4+1 background with metric (17). The
corresponding action is given by

$S[\phi]=-\frac{1}{2}\int
d^{5}x\left[(\partial\phi)^{2}+m^{2}\phi^{2}\right]\,,$ (24)

where $m$ is the mass, which we take to be above the Breitenlohner-Freedman
(BF) bound ($m^{2}l^{2}>-4$). We further assume standard quantization of the
scalar, such that the leading mode in the asymptotic expansion around the AdS
boundary $\rho=1$ is always the source term.

Our stability analysis is based on the study of pseudospectra as discussed in
section 2. In subsection 5.1 we construct the energy norm and define the
relevant function space. Following that, in subsection 5.2 we construct the
eigenvalue problem and identify the relevant operator $\mathcal{L}$ and its
adjoint $\mathcal{L}^{\dagger}$ with respect to the energy norm.

###  5.1 Energy Norm and Function Space

Following [7] we try to define a function space which automatically imposes
the adequate boundary conditions for the $\mathbb{C}$MMs and equip it with a
physically-motivated norm based on the energy, which we label the energy norm.

We begin by first constructing the energy. To do so, recall that the energy
momentum tensor for a real scalar field $\phi$ with action (24) is given by

$T_{MN}[\phi]=\partial_{M}\phi\partial_{N}\phi-\frac{1}{2}g_{MN}\left[(\partial\phi)^{2}+m^{2}\phi^{2}\right]\,,$
(25)

and that, consequently, the energy along a constant $t$-hypersurface
($\Sigma_{t}$) is

$\displaystyle E[\phi]$
$\displaystyle=\int_{\Sigma_{t}}d^{d-1}x\sqrt{\gamma_{\Sigma}}\>n_{\Sigma}^{M}T_{MN}\mathrm{t}^{N}=-\int
d\rho dx^{1}dx^{2}dx^{3}\sqrt{-g}\>\mathrm{t}^{M}T^{t}\mathstrut_{M}$
$\displaystyle=\int\frac{d\rho}{(1-\rho)^{3}}dx^{1}dx^{2}dx^{3}\left\\{f(\partial_{\rho}\phi)^{2}+z_{h}^{2}(\partial_{a}\phi)^{2}+z_{h}^{2}(\partial_{3}\phi)^{2}+z_{h}^{2}(2-f)(\partial_{t}\phi)^{2}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi^{2}\right\\}\,,$
(26)

where $\mathrm{t}=\partial_{t}$ is the killing vector associated with time
translations, $n_{\Sigma}$ is the normal vector orthogonal to the integration
surface and $\gamma_{\Sigma}$ the induced metric on it.

Now, if we want to promote the expression for the energy above to a norm for
the function space containing the $\mathbb{C}$MMs, we find ourselves facing
two main problems. First, we would like to remove the $x^{3}$ integral so that
the norm would not involve integration over all possible values of $x^{3}$,
i.e., so that we do not need to know the explicit $x^{3}$ dependence of all
$\phi$ in our function space. This is easily solved by considering the energy
density along $x^{3}$

$\displaystyle\varrho_{E}[\phi]$
$\displaystyle=\int_{\Sigma^{\prime}}d^{d-2}x\sqrt{\gamma_{\Sigma^{\prime}}}\>n_{(\Sigma^{\prime})}{}^{MN}\mathrm{k}^{(3)}_{M}T_{NP}\mathrm{t}^{P}$
$\displaystyle=\int\frac{d\rho}{(1-\rho)^{3}}dx^{1}dx^{2}\left\\{f(\partial_{\rho}\phi)^{2}+z_{h}^{2}(\partial_{a}\phi)^{2}+z_{h}^{2}(\partial_{3}\phi)^{2}+z_{h}^{2}(2-f)(\partial_{t}\phi)^{2}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi^{2}\right\\}\,,$
(27)

where $\Sigma^{\prime}$ is a hypersurface with $t$ and $x^{3}$ constant,
$n_{(\Sigma^{\prime})}$ the corresponding normal two-vector and
$\mathrm{k}^{(3)}=k^{(3)\>M}\partial_{M}=\partial_{3}$.

Second, we want to add an integral over time to be able to eliminate the
dependence on time derivatives and focus on sectors with fixed frequency
$\omega$. This can be achieved by considering a time-averaged energy density
$\bar{\varrho}_{E}[\phi]$

$\bar{\varrho}_{E}[\phi]=\underset{s\rightarrow\infty}{\lim}\int_{-s}^{s}\frac{dt}{2s}\int\frac{d\rho
dx^{1}dx^{2}}{(1-\rho)^{3}}\left\\{f(\partial_{\rho}\phi)^{2}+z_{h}^{2}(\partial_{a}\phi)^{2}+z_{h}^{2}(\partial_{3}\phi)^{2}+z_{h}^{2}(2-f)(\partial_{t}\phi)^{2}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi^{2}\right\\}\,.$
(28)

With all this, we can now Fourier transform in the directions
$\mathbb{x}_{\perp}$ and $t$ and rewrite the expression above as

$\bar{\varrho}_{E}[\phi]=\underset{s\rightarrow\infty}{\lim}\int\frac{d\omega
d^{2}k_{a}}{2s(2\pi)^{3}}\frac{d\rho}{(1-\rho)^{3}}\left\\{f\partial_{\rho}\phi\partial_{\rho}\bar{\phi}+z_{h}^{2}(k_{a})^{2}\phi\bar{\phi}+z_{h}^{2}\partial_{3}\phi\partial_{3}\bar{\phi}+z_{h}^{2}(2-f)\omega^{2}\phi\bar{\phi}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi\bar{\phi}\right\\}\,,$
(29)

where we abuse notation and write $\phi=\phi(\omega,k_{a},x^{3},\rho)$ and an
integral in $\omega$ as opposed the sum corresponding to finite $s$. We now
consider a sector with fixed $\\{\omega,\mathbb{k}_{\perp}\\}$ and drop the
integral over $\\{\omega,\mathbb{k}_{\perp}\\}$ as it only contributes with a
prefactor which does not affect the operator norm. We further simplify by
using the rotational invariance to set $\mathbb{k}_{\perp}=0$. Hence, the
final expression for the time-averaged energy density without numerical
prefactors is given by

$\bar{\varrho}_{E}[\phi]=\int\frac{d\rho}{(1-\rho)^{3}}\left\\{f\partial_{\rho}\phi\partial_{\rho}\bar{\phi}+z_{h}^{2}\partial_{3}\phi\partial_{3}\bar{\phi}+z_{h}^{2}(2-f)\omega^{2}\phi\bar{\phi}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi\bar{\phi}\right\\}\,.$
(30)

Now we promote the above expression to a physically-motivated inner product
for studying $\mathbb{C}$MMs. As stated in section 4, these modes are regular
(infalling) functions satisfying normalizable boundary conditions in the
boundary of AdS which are solutions to the following eigenvalue problem

$z_{h}\partial_{3}\phi=iz_{h}k\phi=i\mathfrak{q}\phi\,.$ (31)

where we have defined the dimensionless momentum $\mathfrak{q}$. Thus, we need
to eliminate the explicit dependence on $\partial_{3}\phi$ from the inner-
product to make it well defined. Mathematically, this necessity arises from
the fact that the function space contains more elements than just the
eigenvectors of $\partial_{3}$ (e.g. the sum of two eigenvectors with
different eigenvalues is not an eigenvector). To solve this issue, we
introduce an auxiliary field $\psi=z_{h}\partial_{3}\phi$, and arrive at the
following expression for the energy norm on the space of doublets
$\Psi=(\phi,\psi)$

$||\Psi||_{E}=\int\frac{d\rho}{(1-\rho)^{3}}\left\\{f\partial_{\rho}\bar{\phi}\partial_{\rho}\phi+\bar{\psi}\psi+\mathfrak{w}^{2}(2-f)\bar{\phi}\phi+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\bar{\phi}\phi\right\\}\,,$
(32)

where $\mathfrak{w}=z_{h}\omega$. To ensure convergence at $\rho=1$ we demand
that $\phi=(1-\rho)^{2}\xi(\rho)$ where $\xi(\rho)$ fulfills a Dirichlet
boundary condition $\xi(1)=0$. The second and third terms are obviously
positive definite. Expressing the kinetic and mass terms in the rescaled field
we get

$\displaystyle\int\frac{d\rho}{1-\rho}\left[(4f+m^{2}l^{2})\xi^{2}+2f\partial_{\rho}|\xi^{2}|+(1-\rho)^{2}|\partial_{\rho}\xi|^{2}\right]=$
$\displaystyle=\int\frac{d\rho}{1-\rho}\left[(4+m^{2}l^{2}+4(1-\rho)^{4})\,|\xi|^{2}+(1-\rho)^{2}|\partial_{\rho}\xi|^{2}\right]>0\,,$
(33)

where we have done a partial integration and used that $f\xi^{2}$ vanishes on
the horizon and on the boundary. The remainder is positive definite as long as
$m^{2}l^{2}\geq-4$ and so is the norm.

We can further extend the norm to an inner product

$\expectationvalue{\Psi_{1},\Psi_{2}}_{E}=\int\frac{d\rho}{(1-\rho)^{3}}\left\\{f\partial_{\rho}\bar{\phi}_{1}\partial_{\rho}\phi_{2}+\bar{\psi}_{1}\psi_{2}+\mathfrak{w}^{2}(2-f)\bar{\phi}_{1}\phi_{2}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\bar{\phi}_{1}\phi_{2}\right\\}\,.$
(34)

In the next section we will show that only the $\mathbb{C}$MMs with infalling
boundary conditions and normalizability at the AdS boundary belong to the
Hilbert space defined by (34).

###  5.2 Construction of $\mathcal{L}$ and $\mathcal{L}^{\dagger}$

We now want to use the equation of motion of the scalar field (24)

$\nabla^{2}\phi-m^{2}\phi=0\,,$ (35)

to rewrite the eigenvalue equation

$z_{h}\partial_{3}\Psi=i\mathfrak{q}\Psi\,.$ (36)

in terms of a differential operator
$\mathcal{L}=\mathcal{L}[\mathfrak{w},\rho;\partial_{\rho},\partial_{\rho}^{2}]$.
We achieve this by first explicitly writing the equation of motion (35) as

$z_{h}^{2}\partial_{3}^{2}\phi=\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi-(1-\rho)^{3}\partial_{\rho}\left(\frac{f\partial_{\rho}\phi}{(1-\rho)^{3}}\right)+\mathfrak{w}^{2}(-2+f)\phi+i\mathfrak{w}\left[(1-\rho)^{3}\partial_{\rho}\left(\frac{(1-f)\phi}{(1-\rho)^{3}}\right)+(1-f)\partial_{\rho}\phi\right]\,.$
(37)

and then introducing the auxiliary field $\psi=z_{h}\partial_{3}\phi$ so that
we get

$i\mathfrak{q}\Psi=\begin{pmatrix}0&1\\\
L&0\end{pmatrix}\begin{pmatrix}\phi\\\ \psi\end{pmatrix}=\mathcal{L}\Psi\,,$
(38)

where $L$ is a differential operator whose action on $\phi$ is

$L\phi=\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi-(1-\rho)^{3}\partial_{\rho}\left(\frac{f\partial_{\rho}\phi}{(1-\rho)^{3}}\right)+\mathfrak{w}^{2}(-2+f)\phi+i\mathfrak{w}\left[(1-\rho)^{3}\partial_{\rho}\left(\frac{(1-f)\phi}{(1-\rho)^{3}}\right)+(1-f)\partial_{\rho}\phi\right]\,,$
(39)

and
$\mathcal{L}=\mathcal{L}[\mathfrak{w},\rho;\partial_{\rho},\partial_{\rho}^{2}]$
is the differential operator which we initially wanted to obtain.

We now argue that only the infalling and normalizable solutions belong to the
function space introduced in the previous section. First we note that the
differential operator has regular singular points at $\rho=0$ and $\rho=1$.
The coefficients of the derivatives are analytic in the interior
$\rho\in(0,1)$. We can therefore construct two analytic solutions valid in the
interior by a simple power series ansatz. Therefore solutions are certainly
locally integrable except possibly for the boundary points. At the (conformal)
boundary of AdS $(\rho=1)$ it is well known that the two local solution behave
like

$\phi^{(\mathrm{nn})}\propto(1-\rho)^{\Delta_{-}}\,,\qquad\phi^{(\mathrm{n})}\propto(1-\rho)^{\Delta+}\,,$
(40)

with $\Delta_{\pm}=\left(2\pm\sqrt{4+m^{2}l^{2}}\right)$. Integrability of the
inner product demands $\Delta_{a}+\Delta_{b}>4$ with $a,b\in\\{+,-\\}$. We see
that in the range of masses $-4<m^{2}l^{2}\leq 0$ the inner product (34)
indeed rejects the solution $\phi^{(\mathrm{nn})}$ as non-normalizable.333We
restrict ourselves here to standard quantization.

Now let us proceed to the Horizon at $\rho=0$. There we have to distinguish
between the cases $\omega\neq 0$ and $\omega=0$. For
$\omega\in\mathbb{R}\backslash\\{0\\}$ the two local solutions near the
horizon behave as

$\phi^{(\mathrm{in})}\propto\rho^{0}\,,\qquad\phi^{(\mathrm{out})}\propto\rho^{i\mathfrak{w}/2}\,.$
(41)

The most singular term is the derivative term in the norm. Since the
blackening factor $f\propto\rho$ near the Horizon we find

$\bar{\varrho}_{E}\approx\int
d\rho\,\rho\,|\partial_{\rho}\phi^{(\mathrm{out})}|^{2}\approx\int
d\rho\,\rho\,\frac{1}{\rho^{2}}\rightarrow\infty\,.$ (42)

For $\omega=0$ the local solutions behave as
$\phi^{(\mathrm{in})}\propto\rho^{0}$ and
$\phi^{(\mathrm{out})}\propto\log(\rho)$. Clearly the logarithmic solution
again is rejected by the norm as non-integrable at the horizon. We see
therefore that our choice of norm is such that only solutions to the
eigenvalue problem with the correct boundary conditions (infalling at the
horizon and normalizable at the boundary) are elements of the Hilbert space
defined through the inner product (34). There is a further subtlety concerning
the asymptotic behavior of $\psi$. If we only demand that the norm exists then
it seems enough to demand that $\psi$ vanishes faster than $(1-\rho)$ towards
the boundary. However we also want $\norm{\mathcal{L}\Psi}$ to be well defined
thus $\psi$ has to go faster that $(1-\rho)^{2}$ towards the boundary.

The eigenvalues of $\mathcal{L}$ are the $\mathbb{C}$LMs times the imaginary
unit $i$. In practice, when presenting our results in section 7, we find it
more convenient to work with differential operator
$\tilde{\mathcal{L}}=-i\mathcal{L}$ rather than with $\mathcal{L}$. This
choice ensures that the eigenvalues are the $\mathbb{C}$LMs without any
imaginary prefactor and does not alter any of the relevant conclusions derived
throughout this section in terms of $\mathcal{L}$.

Regarding the adjoint $\mathcal{L}^{\dagger}$, we have the following
expression

$\displaystyle\expectationvalue{\mathcal{L}^{\dagger}\Psi_{1},\Psi_{2}}_{E}-\expectationvalue{\mathcal{L}\Psi_{1},\Psi_{2}}_{E}=$
$\displaystyle=\int\frac{d\rho}{(1-\rho)^{3}}\Biggl{\\{}\left[2\mathfrak{w}^{2}(2-f)\bar{\phi}_{1}\psi_{2}-i\mathfrak{w}(1-f)\phi_{2}\partial_{\rho}\bar{\psi}_{1}-i\mathfrak{w}(1-\rho)^{3}\phi_{2}\partial_{\rho}\left(\frac{(1-f)\bar{\psi}_{1}}{(1-\rho)^{3}}\right)-(\phi\leftrightarrow\psi)\right]$
$\displaystyle+\phi_{2}\left[-2i\mathfrak{w}\bar{\psi}_{1}\delta(\rho)\right]\Biggr{\\}}\,.$
(43)

Thus for generic $\mathfrak{w}^{2}\neq 0$ we find that
$\mathcal{L}^{\dagger}\neq\mathcal{L}$. Moreover, we cannot rewrite
$\mathcal{L}^{\dagger}$ as $\mathcal{L}$ plus some boundary terms; we
necessarily have contributions to the non-normality arising from the bulk.
This nicely matches the discussion of section 4. We expect that the system is
non-conservative even in the absence of temperature, hence there should be
contributions to the non-normality that arise independently of the existence
of an horizon, i.e. contributions arising from the bulk. In fact, as we see in
appendix A, the existence of bulk contributions for $\mathfrak{w}\neq 0$ is
guaranteed even in AdS, further confirming the physical intuition.

Remarkably $\mathcal{L}$ is self-adjoint at $\mathfrak{w}=0$ and has a stable
spectrum. This is in particularly good agreement with the arguments presented
in section 4. At $\mathfrak{w}=0$ we are studying the glueball spectrum of the
dimensionally reduced theory and, as this theory lacks any dissipation
mechanism, a well-defined norm should predict that $\mathcal{L}$ is Hermitian;
which indeed matches our results.

##  6 Numerical method

We follow [7, 18] and perform a numerical stability analysis using
pseudospectra á la [1]. For completeness, here we briefly review the some of
the details and subtleties.

Our numerical approach is based on the discretization of the radial coordinate
$\rho$ on a Chebyshev grid with points:

$\rho_{j}=\frac{1}{2}\left[1-\cos\left(\frac{j\pi}{N}\right)\right]\,,\qquad
j=0,1,...,N\,,$ (44)

and the discretization of the differential operators using the corresponding
Chebyshev differentiation matrices [73].

In order to numerically select the correct function space, we find it
convenient work with the rescaled scalar doublet $u$ defined as

$u=(1-\rho)^{2}\Psi\,,$ (45)

where we impose Dirichlet boundary conditions for $u$ at $\rho=1$. This
boundary condition can be easily implemented by removing the rows and columns
corresponding to the AdS boundary from all discretized operators, including
the discretized energy norm. We use Wolfram Engine to compute condition
numbers and pseudospectra of the resulting matrices according to the
discussion presented in section 2.

To further explore the origin of the (in)stability, we study the selective
pseudospectra associated with local potential perturbations to the original
equation of motion (35). Concretely, we consider perturbed equations of motion
of the form:

$\left[-\nabla_{M}\nabla^{M}+m^{2}+\frac{V(\rho)}{l^{2}}\right]\phi=0\,,$ (46)

where, in order to preserve the asymptotic behavior on the AdS boundary, we
choose potentials with $V(1)=0$. The selective pseudospectrum is then computed
using randomly generated potential perturbations constructed as diagonal
matrices with random entries and normalized to a given size. Moreover, we
complement this analysis with the computation of the $\mathbb{C}$LMs for the
perturbed system with the following deterministic potentials:

$\displaystyle V_{1}(\rho)$ $\displaystyle=A_{1}(1-\rho)\cos(2\pi\rho)\,,$
(47a) $\displaystyle V_{2}(\rho)$
$\displaystyle=A_{2}(1-\rho)\cos(90\pi\rho)\,,$ (47b) $\displaystyle
V_{3}(\rho)$
$\displaystyle=A_{3}(1-\rho)\left\\{1-\tanh\left[20\rho\right]\right\\}\,,$
(47c) $\displaystyle V_{4}(\rho)$
$\displaystyle=A_{4}(1-\rho)\left\\{1-\tanh\left[20(1-\rho)\right]\right\\}\,,$
(47d)

which shed light on a few interesting regimes. With $V_{1}$ and $V_{2}$, we
probe the effect of long and short $\rho$-wavelength (wavelength in the $\rho$
direction) perturbations; while with $V_{3}$ and $V_{4}$, we analyze the
stability under localized perturbations near the horizon (IR of the dual CFT)
and the boundary (UV of the dual CFT). The coefficients $A_{i}$ are
normalization constants that fix the magnitude of the perturbation. We plot
these potentials in figure 2.

Figure 2: Deterministic potentials (47) with $A_{i}=1$. Recall that the
horizon is at $\rho=0$ and the boundary at $\rho=1$.

##  7 Results

Here we present the results of the analysis described in the previous
sections. Our numerical simulations are performed in a grid of 100 points with
a precision of 5$\times$MachinePrecision.

In appendix B we some the values for the $\mathbb{C}$LMs at different masses
$m^{2}l^{2}$ and frequencies $\mathfrak{w}$. All pseudospectrum plots are
collected in appendix C.

We are mainly interested in the stability of the $\mathbb{C}$LMs lying closest
to the real axis as these dominate the long-distance behaviour of the
system.444The system is symmetric under
$\mathfrak{q}\rightarrow-\mathfrak{q}$. Consequently, henceforth when talking
about any given $\mathbb{C}$LM we will be generically referring to the pair
$\\{\mathfrak{q},-\mathfrak{q}\\}$. For this reason our computations are
restricted to the region of the complex $\mathfrak{q}$-plane defined by
$|\imaginary(\mathfrak{q})|<10$ and $|\real(\mathfrak{q})|<10$. To ensure the
convergence of the pseudospectrum in that region, in figure 3 we plot the
value of the inverse of the norm of the resolvent
$\norm{(\mathcal{L}-\mathfrak{q})^{-1}}$ at the edges of the said region as a
function of $N$. Clearly, we can see that in agreement with the discussion
presented in section 3, the pseudospectrum indeed converges for real
$\mathfrak{w}$.

In figures 4 and 5 we plot the full and selective pseudospectra and the
corresponding condition numbers for different values of $m^{2}l^{2}$ and
$\mathfrak{w}$. As expected from the analytic results discussed in section
5.2, at zero frequency the spectrum is indeed stable as the operator becomes
normal. We can easily appreciate this in figures 4(a) and 4(c) where the
$10^{-1}$ pseudospectrum around the first $\mathbb{C}$LM is shown to match the
circle of radius $10^{-1}$ around the said momenta. At non-zero $\mathfrak{w}$
the full pseudospectrum opens up denoting spectral instability. Remarkably, at
$\mathfrak{w}=10$ we find that in order to drive the background unstable by
making one of complex momenta cross into the second or fourth quadrants of the
complex plane [44], we need to consider perturbations of size $0.7$ and $0.5$,
for masses $m^{2}l^{2}=0$ and $m^{2}l^{2}=-3$, respectively. As such
perturbations are of the order of magnitude of the distance between
$\mathbb{C}$LMs, we consider that they are relatively big. Hence we claim that
while the spectrum of $\mathbb{C}$LMs is unstable, this spectral instability
cannot drive the background unstable. We refer the reader to section 3.3 of
[18] for a detailed discussion on the difference between spectral and
background instability. It is also worth noting that, similarly to what
happened for QNFs, higher $\mathbb{C}$MMs are increasingly unstable as
indicated by their condition numbers (see figure 5).

Regarding the selective pseudospectrum of local potential perturbations, we
find a striking decrease in instability compared to the instability found for
generic perturbations. In particular, we have not managed to find any random
local potential perturbation capable of destabilizing the lowest lying
$\mathbb{C}$LM (see figure 4). This is in contrast with the observations for
the lowest QNF in [18]. Furthermore, for higher $\mathbb{C}$LMs, we always
find that the effect of these perturbations is much smaller than that of
generic perturbations (see figure 6). This suggests that in order to cause a
significant impact on the spectrum, one necessarily has to consider non-local
potentials.

To further explore this surprising spectral stability, in figures 7 and 8 we
plot the effect of the deterministic potentials (47) on the lowest lying
$\mathbb{C}$LM and on the full region of the complex plane under
consideration, respectively. We see that as indicated by the selective
pseudospectrum, the first $\mathbb{C}$LM is stable under the deterministic
potentials (47). Moreover, in figure 8 we find that this stability is shared
by all the plotted $\mathbb{C}$LMs; as we observe that the perturbations do
not displace them in a significant manner. This contrasts greatly with the
picture found for the QNFs in [18], thus exemplifying the different stability
properties of complex momenta and quasinormal frequencies in agreement with
the discussion of section 4.

To conclude, in figures 9 and 10 we plot the condition numbers for the first
three $\mathbb{C}$LMs as a function of $m^{2}l^{2}$ and $\mathfrak{w}$,
respectively. As expected at $\mathfrak{w}=0$ the condition numbers are 1, up
to small numerical errors. Remarkably we find that instability increases with
mass and with frequency. From the dual perspective this implies that
absorption lengths of operators with larger conformal dimension at large
frequencies are more unstable.

##  8 Summary and Discussion

This work presents complex linear momenta ($\mathbb{C}$LMs) as a new probe to
explore the spectral instability of holographic theories. From the
gravitational point of view, these are eigenvalues of the translation operator
along a space-like direction parallel to the AdS boundary at fixed, real
frequency, subject to appropriate boundary conditions. From the holographic
perspective, $\mathbb{C}$LMs are dual to the poles of the retarded Green’s
function at fixed real frequency. They contain information about the causal
structure of the dual field theory and, at zero frequency, precisely describe
the masses of the states in the theory compactified on the thermal circle.

The study of Complex Momentum Modes ($\mathbb{C}$MMs), with non-zero frequency
( $\omega\neq 0$), implies pumping energy into the system; making it non-
conservative and thus vulnerable to spectral instabilities. In particular, we
have computed the pseudospectra of $\mathbb{C}$LMs for a massive, real scalar
on a Schwarzschild-AdS4+1 background. In order to measure the displacement of
the momenta, we use the (two derivative) energy norm. Remarkably, we have
found that $\mathbb{C}$LMs are unstable at non-zero frequency. Furthermore, we
have argued that the spectral stability observed at zero frequency is a
physical feature related to the fact that in that case $\mathbb{C}$LMs are
dual to the glueball spectrum of a conservative theory arising from
dimensional reduction on the thermal circle. At non-zero frequency, the
spectral instability increases with mass and frequency. This suggests that in
the dual field theory, the poles of retarded Green’s functions of scalar
operators with large mass dimension at large frequency are more unstable.
Additionally, the pseudospectrum appears to be particularly resistant to local
potential perturbations, indicating that non-local potentials may need to be
considered to produce a significant displacement of the $\mathbb{C}$LM.

As explained in Section 3 the QNF pseudospectrum fails to converge for
negative enough $\imaginary(\omega)$. One of the paper’s main results is the
striking numerical convergence of the $\mathbb{C}$LMs pseudospectra in our
particular setup. We expect this to be a generic property since, for branes
without momenta along the $x^{3}$ direction, the near-horizon behaviour of
$\mathbb{C}$MMs is independent of $k$. Thus, we claim that the pseudospectrum
of $\mathbb{C}$LMs offers a new alternative to assess the spectral stability
of the theory, which allows us to derive quantitative results, which in turn
would facilitate making precise predictions for observables.

Even though QNMs and $\mathbb{C}$MMs seem very similar in flavour, we have
shown that they probe different regimes of the theory. As such, neither the
stability of their pseudospectra nor their numerical convergence are directly
related. It seems odd since both quantities are holographically dual to the
same object, poles in the retarded Green’s functions. We are looking at two
different real sections of the solutions of $G_{R}(k,\omega)^{-1}=0$. In
principle analytic continuation should allow us to relate both sections. If
such an “analytic continuation” of sorts was possible, we could then attempt
to reconstruct a numerically convergent pseudospectra for QNFs; without
needing to employ higher derivative Sobolev norms. This, however, is well
beyond the scope of this work.

## Acknowledgements

We thank D. Areán, V. Boyanov and J.L. Jaramillo for valuable discussions.
This work is supported through the grants CEX2020-001007-S and
PID2021-123017NB-100, PID2021-127726NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”. The work
of D.G.F. is supported by FPI grant PRE2022-101810. The work of PSB is
supported by Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de
Murcia, grant 21609/FPI/21 and Spanish Ministerio de Ciencia e Innovación
PID2021-125700NA-C22.

##  Appendix A Non-normality in AdS

To better understand the bulk contributions to $\mathcal{L}^{\dagger}$ that we
found in 5.2, here we repeat the computations of the preceding sections taking
the AdS limit $f=1$. As we are interested only in the bulk contributions, we
do not keep track of possible boundary terms in the computation of
$\mathcal{L}^{\dagger}$.

In AdS4+1, the operator $\mathcal{L}$ reads

$\mathcal{L}_{0}=\begin{pmatrix}0&1\\\ L_{0}&0\end{pmatrix}\,,$ (48)

where $L_{0}$ is a differential operator whose action on $\phi$ is defined as

$L_{0}\phi=\frac{m^{2}l^{2}}{(1-\rho)^{2}}\phi-(1-\rho)^{3}\partial_{\rho}\left(\frac{\partial_{\rho}\phi}{(1-\rho)^{3}}\right)-\mathfrak{w}^{2}\phi\,,$
(49)

and the energy norm simplifies to

$\expectationvalue{\Psi_{1},\Psi_{2}}_{E}=\int\frac{d\rho}{(1-\rho)^{3}}\left\\{\partial_{\rho}\bar{\phi}_{1}\partial_{\rho}\phi_{2}+\bar{\psi}_{1}\psi_{2}+\mathfrak{w}^{2}\bar{\phi}_{1}\phi_{2}+\frac{m^{2}l^{2}}{(1-\rho)^{2}}\bar{\phi}_{1}\phi_{2}\right\\}\,.$
(50)

Hence, a straightforward computation allows us to conclude the following

$\displaystyle\expectationvalue{\mathcal{L}_{0}^{\dagger}\Psi_{1},\Psi_{2}}_{E}-\expectationvalue{\mathcal{L}_{0}\Psi_{1},\Psi_{2}}_{E}=\int\frac{d\rho}{(1-\rho)^{3}}\left\\{\left[2\mathfrak{w}^{2}\bar{\phi}_{1}\psi_{2}-(\phi\leftrightarrow\psi)\right]\right\\}\,.$
(51)

which, as anticipated, shows that $\mathcal{L}_{0}^{\dagger}$ contains bulk
contributions to the non-normality for $\mathfrak{w}\neq 0$.

Remarkably, the existence of non-normality even in the absence of a black hole
matches the naive physical intuition discussed in section 4. From the CFT
perspective, exciting $\mathbb{C}$MMs corresponds to turning on a source of
the form $j(t,x^{3})=\chi(x^{3})e^{-i\omega t}$ for the dual operator; thus,
even without thermal dissipation, the system has an influx/outflow of matter
associated with the time dependent source, which in turn makes the system non-
conservative. Again, in agreement with the naive physical intuition, these
bulk contributions also vanish in the $\mathfrak{w}=0$ limit, signaling that
the system becomes conservative as the source is no longer time-dependent.

##  Appendix B Numerical Values of the Complex Momenta

In this appendix we provide the numerical values of the first 10
$\mathbb{C}$LMs for the real scalar with action (24). For purposes of
presentation we limit the precision to 15 significant figures.

$n$ | $\real(\mathfrak{q}_{n})$ | $\imaginary(\mathfrak{q}_{n})$
---|---|---
$1$ | $0$ | $\pm 3.40406555853814$
$2$ | $0$ | $\pm 5.87596663030813$
$3$ | $0$ | $\pm 8.30511662249606$
$4$ | $0$ | $\pm 10.7196286254632$
$5$ | $0$ | $\pm 13.1274967745451$
$6$ | $0$ | $\pm 15.5317934290101$
$7$ | $0$ | $\pm 17.9339496075144$
$8$ | $0$ | $\pm 20.3347218217190$
$9$ | $0$ | $\pm 22.7345476095668$
$10$ | $0$ | $\pm 25.1336976323937$
Table 1: $\mathbb{C}$LMs for $m^{2}l^{2}=0$ and $\mathfrak{w}=0$. $n$ | $\real(\mathfrak{q}_{n})$ | $\imaginary(\mathfrak{q}_{n})$
---|---|---
$1$ | $\pm 9.02226257097657$ | $\pm 1.87239874470858$
$2$ | $\pm 8.17892555723022$ | $\pm 3.97449830746049$
$3$ | $\pm 7.50532453968757$ | $\pm 6.40222333417884$
$4$ | $\pm 7.04748494704286$ | $\pm 8.96919759174839$
$5$ | $\pm 6.75242590366829$ | $\pm 11.5596060709257$
$6$ | $\pm 6.55981285259286$ | $\pm 14.1324913026753$
$7$ | $\pm 6.42974689165010$ | $\pm 16.6795521362831$
$8$ | $\pm 6.33868833659767$ | $\pm 19.2024829197097$
$9$ | $\pm 6.27280863252675$ | $\pm 21.7051740861311$
$10$ | $\pm 6.22376285335282$ | $\pm 24.1914304968759$
Table 2: $\mathbb{C}$LMs for $m^{2}l^{2}=0$ and $\mathfrak{w}=10$. $n$ | $\real(\mathfrak{q}_{n})$ | $\imaginary(\mathfrak{q}_{n})$
---|---|---
$1$ | $0$ | $\pm 2.39628046947118$
$2$ | $0$ | $\pm 4.79256093894237$
$3$ | $0$ | $\pm 7.18884140841355$
$4$ | $0$ | $\pm 9.58512187788474$
$5$ | $0$ | $\pm 11.9814023473559$
$6$ | $0$ | $\pm 14.3776828168271$
$7$ | $0$ | $\pm 16.7739632862983$
$8$ | $0$ | $\pm 19.1702437557695$
$9$ | $0$ | $\pm 21.5665242252407$
$10$ | $0$ | $\pm 23.9628046947118$
Table 3: $\mathbb{C}$LMs for $m^{2}l^{2}=-3$ and $\mathfrak{w}=0$. $n$ | $\real(\mathfrak{q}_{n})$ | $\imaginary(\mathfrak{q}_{n})$
---|---|---
$1$ | $\pm 9.38974080989012$ | $\pm 1.10875947814635$
$2$ | $\pm 8.54763762913947$ | $\pm 2.94898874493230$
$3$ | $\pm 7.78565832114999$ | $\pm 5.22055753548535$
$4$ | $\pm 7.23018023647193$ | $\pm 7.72858499087675$
$5$ | $\pm 6.86711966426586$ | $\pm 10.3091717841825$
$6$ | $\pm 6.63309570199478$ | $\pm 12.8879323622292$
$7$ | $\pm 6.47813029086701$ | $\pm 15.4439100354085$
$8$ | $\pm 6.37172016550565$ | $\pm 17.9750953890531$
$9$ | $\pm 6.29604904918147$ | $\pm 20.4845878580473$
$10$ | $\pm 6.24054508579031$ | $\pm 22.9762411905281$
Table 4: $\mathbb{C}$LMs for $m^{2}l^{2}=-3$ and $\mathfrak{w}=10$.

##  Appendix C Pseudospectrum plots

In this appendix we collect the pseudospectrum plots discussed in section 7.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=0$.

(b) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(c) $m^{2}l^{2}=-3$, $\mathfrak{w}=0$.

(d) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 3: Norm of the inverse of the resolvent ($\sigma_{\epsilon}$) as a
function of the grid size $N$ evaluated on $\mathfrak{q}=\pm(10\pm 10i)$ for
different values of $\mathfrak{w}$ and $m^{2}l^{2}$. Remarkably, the
pseudospectrum converges rapidly and for $N=100$ we can claim that our results
should be quantitatively correct up to corrections of order $1\%$.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=0$.

(b) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(c) $m^{2}l^{2}=-3$, $\mathfrak{w}=0$.

(d) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 4: Close-up of the scalar pseudospectrum in the energy norm around the
first $\mathbb{C}$LM for different values of $\mathfrak{w}$ and $m^{2}l^{2}$.
The red dot corresponds to the $\mathbb{C}$LM, the white lines represent the
boundaries of various full $\varepsilon$-pseudospectra, and the dashed blue
circle symbolizes a circle with a radius of $10^{-1}$ centered on the
$\mathbb{C}$LM. The heat map corresponds to the logarithm in base 10 of the
inverse of the the norm of the resolvent, while the blue dots indicate
selective $\varepsilon$-pseudospectra computed with random local potential
perturbations of size $10^{-1}$. Remarkably, in (a) and (c) we observe
stability as expected from the compactified theory.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=0$.

(b) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(c) $m^{2}l^{2}=-3$, $\mathfrak{w}=0$.

(d) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 5: Scalar pseudospectrum in the energy norm for different values of
$\mathfrak{w}$ and $m^{2}l^{2}$. In the lower panels, we present selective and
full pseudospectra. The red dots represent the (unperturbed) $\mathbb{C}$LMs.
The white lines denote the boundaries of different full
$\varepsilon$-pseudospectra. The heat map corresponds to the logarithm in base
10 of the inverse of the norm of the resolvent, while the blue dots indicate
different selective $\varepsilon$-pseudospectra computed with random local
potential perturbations of size $10^{-1}$. In the upper panels, we represent
the condition numbers. Most notably, at $\mathfrak{w}\neq 0$, for small values
of $\varepsilon$, the full $\varepsilon$-pseudospectra present open regions
containing multiple QNFs, which signals spectral instability.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(b) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 6: Close-up of the scalar pseudospectrum in the energy norm around the
third and fourth $\mathbb{C}$LMs for $\mathfrak{w}=10$ and different values of
$m^{2}l^{2}$. The red dots represent the (unperturbed) $\mathbb{C}$LMs. The
white lines denote the boundaries of different full
$\varepsilon$-pseudospectra. The heat map corresponds to the logarithm in base
10 of the inverse of the norm of the resolvent, while the blue dots indicate
different selective $\varepsilon$-pseudospectra computed with random local
potential perturbations of size $10^{-1}$. Remarkably the instability under
random local potential perturbations probed by the selective pseudospectrum is
orders of magnitude milder than that of generic perturbations probed by the
full pseudospectrum.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=0$.

(b) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(c) $m^{2}l^{2}=-3$, $\mathfrak{w}=0$.

(d) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 7: Effect on $\mathbb{C}$LMs of the deterministic perturbations (47)
with size $\norm{V_{i}}_{E}=10^{-1}$. The unperturbed $\mathbb{C}$LM is shown
in red, while the perturbed $\mathbb{C}$LMs are depicted in blue. The dashed
blue line represents the circle of radius $10^{-1}$ centered in the
unperturbed $\mathbb{C}$LM. Remarkably, even for $\mathfrak{w}\neq 0$ we
observe stability under deterministic perturbations.

(a) $m^{2}l^{2}=0$, $\mathfrak{w}=0$.

(b) $m^{2}l^{2}=0$, $\mathfrak{w}=10$.

(c) $m^{2}l^{2}=-3$, $\mathfrak{w}=0$.

(d) $m^{2}l^{2}=-3$, $\mathfrak{w}=10$.

Figure 8: Effect on the spectrum of the scalar of the deterministic
perturbations (47) with size $\norm{V_{i}}_{E}=10^{-1}$. In the lower panels
we present the spectra and in the upper ones the condition numbers for the
lowest $\mathbb{C}$LMs. The unperturbed $\mathbb{C}$LMs are shown in red,
while the perturbed ones are depicted in blue. In the plotted region of the
spectrum is stable under all perturbations even for $\mathfrak{w}\neq 0$.

(a) $\mathfrak{w}=0$.

(b) $\mathfrak{w}=10$.

Figure 9: Mass dependence of the condition numbers of the complex momenta.
Note that for $\mathfrak{w}=0$ all condition numbers are one.

(a) $m^{2}l^{2}=0$.

(b) $m^{2}l^{2}=-3$.

Figure 10: Frequency dependence of the condition numbers of the
$\mathbb{C}$LMs.

##  Appendix D Comparison with quasinormal frequencies at complex momentum

In this appendix we explicitly show the differences in the stability
properties of QNFs and $\mathbb{C}$LMs. As we argued in the main text, while
related, QNFs and $\mathbb{C}$LMs appear in very different physical settings.
Thus, the stability properties of a mode depend heavily on whether we consider
it a QNM or a $\mathbb{C}$MM.

To explicitly show this we consider the scalar field (24) with $m=0$. This
scalar has a $\mathbb{C}$LM $\mathfrak{q}=-3.40407i$ at $\mathfrak{w}=0$ and,
correspondingly, a QNF $\mathfrak{w}=0$ at $\mathfrak{q}=-3.40407i$. To
analyze the differences in stability we compare the pseudospectra of the
$\mathbb{C}$LM and of the QNF in figure 11. Clearly we can see that the
$\mathbb{C}$LM is stable, while the QNF is not. Thus showcasing their
different stability properties.

(a) QNF $\mathfrak{w}=0$ at $\mathfrak{q}=-3.40407i$.

(b) $\mathbb{C}$LM $\mathfrak{q}=-3.40407i$ at $\mathfrak{w}=0$.

Figure 11: Comparison of the pseudospectra of a QNF and a $\mathbb{C}$LM. The
red dots represent the eigenvalues. The heat map corresponds to the logarithm
in base 10 of the inverse of the norm of the resolvent. The white lines denote
the boundaries $\varepsilon$-pseudospectra and the red (yellow) lines
correspond to the boundaries of the $10^{-3/2}$ ($10^{-7/4}$) pseudospectra.
The dashed red (yellow) circle represents the circle of radius $10^{-3/2}$
($10^{-7/4}$) centered around the eigenvalue. Note that in 11(b) the spectrum
is stable as the dashed red (yellow) circle coincides with the boundary of the
$10^{-3/2}$ ($10^{-7/4}$) pseudospectra. On the other hand, in 11(a) the
spectrum is unstable as the boundary of the $10^{-3/2}$ ($10^{-7/4}$)
pseudospectra is much larger than the dashed red (yellow) circle.

The $\mathbb{C}$LM pseudospectrum is computed following the procedure
discussed throughout this paper. The differential operator is given by (38)
and the inner product by (34). On the other hand, to compute the QNF
pseudospectrum we consider the following eigenvalue problem and norm
introduced in [18]

$\mathfrak{w}\Psi=i\begin{pmatrix}0&1\\\
L_{1}\left[\partial_{\rho}^{2},\partial_{\rho};\bm{\mathfrak{q}},\rho\right]&L_{2}\left[\partial_{\rho};\bm{\mathfrak{q}},\rho\right]\end{pmatrix}\Psi\,,$
(52)
$\expectationvalue{\Psi_{1},\Psi_{2}}_{E}=\int\frac{d\rho}{(1-\rho)^{3}}\,\,\left[\mathfrak{|q|}^{2}\bar{\phi}_{1}\phi_{2}+f\partial_{\rho}\bar{\phi}_{1}\partial_{\rho}\phi_{2}+(2-f)\bar{\psi_{1}}\psi_{2}\right]\,,$
(53)

where $\Psi=(\phi,z_{h}\partial_{t}\phi)=(\phi,\psi)$ and the differential
operators $L_{1}$ and $L_{2}$ take the form:

$\displaystyle
L_{1}\left[\partial_{\rho}^{2},\partial_{\rho};\bm{\mathfrak{q}},\rho\right]$
$\displaystyle=\left[f-2\right]^{-1}\left[\bm{\mathfrak{q}}^{2}-(1-\rho)^{3}\left(\frac{f}{(1-\rho)^{3}}\right)^{\prime}\partial_{\rho}-f\partial_{\rho}^{2}\right]\,,$
(54) $\displaystyle L_{2}\left[\partial_{\rho};\bm{\mathfrak{q}},\rho\right]$
$\displaystyle=\left[f-2\right]^{-1}\left[(1-\rho)^{3}\left(\frac{f-1}{(1-\rho)^{3}}\right)^{\prime}+2\left(f-1\right)\partial_{\rho}\right]\,.$
(55)

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