# Parametrix construction and numerical approximation of resonances of
azimuthal harmonics of the charged Klein-Gordon operator on general
cosmological slowly accelerating and rotating charged black hole type
spacetimes

Nicolas Besset

###### Abstract.

We show the index 0 Fredholm property of the spectral family of the azimuthal
harmonics of the charged Klein-Gordon operator on general cosmological slowly
accelerating and rotating charged black hole type spacetimes, including the De
Sitter-Kerr-Newman family, using a parametrix construction for abstract
totally characteristic operators. We then present a numerical scheme to
compute the meromorphic poles of the inverse of the spectral family and
provide an explicit estimate of the numerical error.

Université Paris Saclay, Département de Mathématiques d’Orsay, F-91405 Orsay
Cedex, France. E–mail<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. Resonances and decay of the local energy

Long time behaviour of waves in curved spacetimes is an important first step
in the analysis of stability of the underlying metric (_cf._ [HV] for the non-
linear stability of De Sitter-Kerr metric and [HHV] for the linear stability
of Kerr spacetime). It is also a prerequisite in proving asymptotic
completeness for wave type equations (_cf._ [HN] for massless Dirac fields on
Kerr metric, [GGH] for scalar waves on De Sitter-Kerr type metrics and [B2]
for charged Klein-Gordon (KG) fields on the De Sitter-Reissner-Nordström
metric when the charged product is sufficiently small).

Similarly to solutions of the wave equation on a compact spatial domain expend
as a sum over the (discrete set of) eigenvalues of the spatial Laplacian, we
know since the work of Bony-Häfner [BoHa] that scalar waves on cosmological
uncharged spherical black hole metric with compactly supported inital data
also have a local expansion of the form

(1) $\displaystyle\chi u(t,\bullet)$
$\displaystyle=\sum_{\begin{subarray}{c}{\sigma_{j}\in\mathrm{Res}}\\\
{\mathrm{Im}(\sigma_{j})>-C}\end{subarray}}\sum_{k=0}^{m(\sigma_{j})}t^{k}\mathrm{e}^{-\mathrm{i}\sigma_{j}t}u_{j,k}+\widetilde{u},\qquad\|\widetilde{u}(t,\bullet)\|\lesssim\mathrm{e}^{-Ct}.$

Above $t>0$ is the time variable, $C>0$ and $\chi$ is a smooth spatial cut-
off; the equation holds in some energy space which the norm $\|\bullet\|$ is
associated to. The special frequencies appearing in the expansion are called
quasinormal modes: they are complex frequencies, proper to the black hole and
the ambient spacetime, that describe late time oscillation and damping of
waves. They are not the eigenvalues of the spatial part of the wave operator
as the associated eigenvectors fail in general to be in the appropriate
$L^{2}$ space (whence the appellation _quasi_ normal modes).

Theory of resonances provides a spectral definition of these quasinormal modes
(and then gives a precise description of the long time behaviour of wave type
fields) when the resolvent of the operator describing the underlying dynamics
has sufficiently nice decay at high frequencies. Resonances are defined as
those frequencies $\sigma\in{\mathbb{C}}$ such that a related operator
$P_{\sigma}$ (the spectral family, see Sect. 3.3 below) is invertible between
appropriate functional spaces. Thus resonances and quasinormal modes coincide
provided $\|P_{\sigma}^{-1}\|$ (when it exists) decays sufficiently fast as
$\mathrm{Re}(\sigma)\to\pm\infty$, as we can then shift contour integration
and use the residue theorem to obtain the expansion (1) with resonances (for a
very nice and complete presentation of the mathematical theory of resonances,
we refer the reader to [DZ]). Non trivial elements in the kernel of
$P_{\sigma}$ are called resonant states and those of the dual operator
$P_{\sigma}^{*}$ are called dual states; the term $u_{jk}$ in (1) is the
projection on the $(k+1)$ dimensional vector space spanned by the resonant
states associated to the resonance $\sigma_{j}$ with $k\leq m(\sigma_{j})$,
$m(\sigma_{j})$ being the multiplicity of the resonance $\sigma_{j}$ (see
[BoHa, eq. (1.10) & eq. (1.11)] for details for the wave equation on the De
Sitter-Schwarzschild metric). Resonance expansions have first been obtained
for (massive) waves on De Sitter-Kerr spacetimes by Dyatlov [D]; we refer to
[H3, Sect. 1.1] for more references in this context; see also [B1] for charged
KG fields on De Sitter-Reissner-Nordström spacetimes. Notice from (1) that the
presence of resonances in the upper half complex plane entails exponential
growth in time of the solution (this is the case for small charge $q$ and mass
$m$ of the field with $m\lesssim|q|$, and small rotation of the black hole,
see [BeHa]).

Existence of resonances for (massive) scalar waves in perturbations of De
Sitter-Kerr spacetimes has been proved by the seminal work of Vasy [V] who
cast the theory of resonances in the robust framework of analytic Fredholm
theory for non elliptic operators. Notice however that the operator
$P_{\sigma}$ in [V] is defined on an enlarged spatial domain and a complex
potential supported beyond the horizons is added, so that resonances are
created beyond the horizons; however, in a cut-off sense, that is in the
physical outer domain of the spacetime, they do not depend on the choice of
the extended domain nor on the absorbing potential (as this potential is
microlocally supported above sets lying beyond the horizons of the spacetime).
By the index 0 analytic Fredholm theory, they are thus characterized as the
poles of the meromorphic extension from $\\{\sigma\in{\mathbb{C}}\ |\
\mathrm{Im}(\sigma)\gg 0\\}$ to some larger domain in ${\mathbb{C}}$ of the
cut-off resolvent $\chi P_{\sigma}^{-1}\chi$ (here $\chi$ is a radial cut-
off). This definition coincide with that provided by Mazzeo-Melrose work [MM]
(used by Sá Barreto-Zworski [SZ] then Bony-Häfner [BoHa] to define
resonances).

In this paper, we propose to define resonances of the azimuthal harmonics of
charged KG fields on general cosmological spacetimes, including the De Sitter-
Kerr-Newman family, using a parametrix construction. We will not deal with
semiclassical asymptotics in very high energy regimes. As in [V], resonances
are the poles of the meromorphic extension of the inverse of the spectral
family associated to the charged KG operator, the meromorphic extension being
constructed by analytic Fredholm theory. We do not only extend [V, Thm. 1.1]
(without the trapping statement) to more general settings, but we also make
numerical computations of resonances possible; on the other hand, we only
consider harmonics of the charged KG operators and work on Sobolev spaces with
integer orders (in particular, we do not discuss how and why the order $1/2$
is critical for the Fredholm theory, we refer the reader to [V]). The
parametrix construction and the underlying analytic Fredholm theory are
presented in an abstract setting in Sect. 2 below. As for numerical
approximation of resonances, there exists several methods in the litterature
(we can mention _e.g._ [CCDHJ] on De Sitter-Reissner-Nordström spacetimes and,
more recently, [HX1] on De Sitter and [HX2] on De Sitter-Schwarzschild
spacetimes); to our knowledge, the approach in the present paper provides for
the very first time a scheme with an explicit estimate of the numerical error
for a very large class of cosmological spacetimes. We will only present this
scheme and derive the numerical error in a purely theoretical point of view;
we plan to explicitly apply it in a future work.

###  1.2. Main results

We consider a cosmological, accelerating and rotating charged black hole
spacetime solution to the Einstein-Maxwell equation, _cf._ Sect. 3.1. Let
$(\boldsymbol{P}_{\sigma},\mathcal{X}^{k})$,
$k\in{\mathbb{N}}\setminus\\{0\\}$, be the harmonic111That is, we consider
restriction of $\boldsymbol{P}_{\sigma}$ to $\mathop{\rm
Ker}\nolimits(D_{\varphi}+\ell)$ where $\varphi$ is the azimuth and
$\ell\in{\mathbb{Z}}$. spectral family associated to the charged KG operator
realized on is natural domain $\mathcal{X}^{k}:=\\{u\in H^{k}\ |\
\boldsymbol{P}_{\sigma}u\in H^{k-1}\\}$ ($H^{k}$ are standard Sobolev spaces),
_cf._ Sect. 3.3. Under the Assumptions 19, the following result holds (_cf._
Sect. 3.5 for the proof)

######  Theorem 1.

For all $k\in{\mathbb{N}}\setminus\\{0\\}$, let

$\displaystyle{\mathbb{C}}_{k}$
$\displaystyle:=\big{\\{}\sigma\in{\mathbb{C}}\ \big{|}\
\mathrm{Im}(\sigma)\notin\\{-(k-1/2)\boldsymbol{\kappa}_{-},-(k-1/2)|\boldsymbol{\kappa}_{+}|\\}\big{\\}}$

where
$\boldsymbol{\kappa}_{\pm}:=\frac{(\partial_{r}\mu)(r_{\pm})}{2(1+\lambda)(r_{\pm}^{2}+a^{2})}$.
Then
${\mathbb{C}}_{k}\ni\sigma\mapsto(\boldsymbol{P}_{\sigma},\mathcal{X}^{k})$ is
a analytic family of index 0 Fredholm operators. For non rotating black holes,
this results holds true for the full operator (_i.e._ we do not need any
harmonical restriction).

By analytic Fredholm theory (_cf._ _e.g._ [DZ, Thm. C.8]),
${\mathbb{C}}_{k}\ni\sigma\mapsto(\boldsymbol{P}_{\sigma}^{-1},\mathcal{B}(H^{k-1},\mathcal{X}^{k}))$
is a meromorphic family provided that $\boldsymbol{P}_{\sigma_{0}}^{-1}$
exists for some $\sigma_{0}\in{\mathbb{C}}_{k}$ (in the De Sitter-Kerr
setting, Vasy proved this last point from a semiclassical estimate _cf._ [V,
Sect. 6.4]). The poles of $\boldsymbol{P}_{\sigma}^{-1}$ are by definition the
resonances (of $\boldsymbol{P}_{\sigma}$).

######  Remark 2.

The critical strip ${\mathbb{R}}-\mathrm{i}(k-1/2)|\boldsymbol{\kappa}_{\pm}|$
is associated to the indicial family defined in (7) near the corresponding
horizon ($\rho$ there has to be understood as $|r-r_{\pm}|$, with ’$-$’ near
the event horizon and ’$+$’ near the cosmological horizon), and coincides of
course to that found in [V, Thm. 1.1]; whenever $\sigma$ lies on that strip,
the indicial family has a real root and the Fredholm property is not ensured
anymore. We refer to Sect. 2.3 for details.

######  Remark 3.

When the cosmological constant $\Lambda$ is $0$, the manifold has a hyperbolic
and an Euclidean ends. There is then an accumulation of resonances at the 0
energy (_cf._ [H3, Thm. 1.1]) so that 0 is not itself a resonance and the
resolvent is not meromorphic in any neighborhood of 0. In this situation, only
polynomial decay of waves is expected (_cf._ _e.g._ [H2] for a sharp rate of
decay on Kerr metric).

While there is seemingly no continuity in the limit $\Lambda\to 0$ in the
expansion (1), we can take this limit on the spectral side in Thm 1: in the
case of a non accelerating black hole with sufficiently small charge and
angular momentum, we can check that $r_{-}\to M+\sqrt{M^{2}-(Q^{2}+a^{2})}$
and $r_{+}=\mathcal{O}(\Lambda^{-1/2})$ as $\Lambda\to 0$, so that
$\boldsymbol{\kappa}_{-}\to\frac{\sqrt{M^{2}-(Q^{2}+a^{2})}}{2M(M+\sqrt{M^{2}-(Q^{2}+a^{2})})-Q^{2}}>0$
while $\boldsymbol{\kappa}_{+}=\mathcal{O}(\Lambda^{1/2})\to 0$. As a result,
the highest critical strip in the Fredholm theory is simply ${\mathbb{R}}$; it
is then no longer possible to use a simple contour deformation to get a long
time behaviour for solutions to the charged KG equation as in [BoHa] or [V]
(but we can use a limiting absorption principle as in [H2]).

######  Remark 4.

In the first version of this paper [B3], we considered other realizations of
$\boldsymbol{P}_{\sigma}$ on De Sitter-Reissner-Nordström metrics. It turned
out that the index 0 Fredholm property holds for
$\mathrm{Im}(\sigma)>-\min\\{\boldsymbol{\kappa}_{-},|\boldsymbol{\kappa}_{+}|\\}$
independently of the value of $k$. This rigid restriction also applies when
using the Regge-Wheeler, or tortoise, coordinate (_cf._ [B1, Thm. 3.8]) –
using this coordinate pushes the horizons (the boundaries) at infinity and we
do not really see them. Thus, it is crucial to take into account the
boundaries of the spacetime in the analysis as we do in the present paper or
as Vasy did in [V] in order to lower the critical strips as we increase
Sobolev regularity.

As explained in Sect. 1.1 above, the proof of Thm. 1 relies on a parametrix
construction. The main point using such an approach is that it allows
numerical computations of resonances. This is the point of our second main
result (see Sect. 4.2 for the proof):

######  Theorem 5.

We assume that $\boldsymbol{P}_{\sigma_{0}}^{-1}$ exists for some
$\sigma_{0}\in{\mathbb{C}}_{k}$ (it is the case if all the parameters of the
problem (charges, mass of the field, angular momentum and acceleration of the
black hole) are sufficiently small in order to enter the framework of small
perturbations of De Sitter-Kerr spacetimes treated in [V]). For all
$k,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$, let

$\displaystyle{\mathbb{C}}_{k,N^{\prime}}$
$\displaystyle:=\big{\\{}\sigma\in{\mathbb{C}}\ \big{|}\
\mathrm{Im}(\sigma)\notin\\{-(k-1/2+j-1)\boldsymbol{\kappa}_{-},-(k-1/2+j-1)|\boldsymbol{\kappa}_{+}|\\},\
\forall j\in\\{1,\ldots,N^{\prime}\\}\big{\\}}\subset{\mathbb{C}}_{k}.$

There exists an analytic function ${\mathbb{C}}_{k,N^{\prime}}\ni\sigma\mapsto
D(\sigma)\in{\mathbb{C}}$ whose zeros are exactly the poles (counted with
multiplicity) of
$(\boldsymbol{P}_{\sigma}^{-1},\mathcal{B}(H^{k-1},\mathcal{X}^{k}))$.

Furthermore, for all familly of projectors
$(\Pi_{R})_{R\in{\mathbb{N}}\setminus\\{0\\}}$ acting on $H^{k-1}$, there
exists a family of analytic functions
${\mathbb{C}}_{k,N^{\prime}}\ni\sigma\mapsto(D_{R}(\sigma))_{R\in{\mathbb{N}}\setminus\\{0\\}}$
(which are the determinants of matrices acting on the range of the $\Pi_{R}$)
with the following property: for all positively oriented contour
$\Gamma\in{\mathbb{C}}_{k,N^{\prime}}$, let $N(\Gamma)$ and $S(\Gamma,n)$
($n\in{\mathbb{N}}\setminus\\{0\\}$) be respectively the number of zeros of
$D$ lying inside $\Gamma$ and the sum of the $n$-th power of these zeros
(counted with their multiplicity), then define similarly $N_{R}(\Gamma)$ and
$S_{R}(\Gamma,n)$; then

$\displaystyle|N(\Gamma)-N_{R}(\Gamma)|$ $\displaystyle\leq
C_{R}(\Gamma),\qquad|S(\Gamma,n)-S_{R}(\Gamma,n)|\leq\widetilde{C}_{R}(\Gamma,n)$

where $C_{R}(\Gamma),\widetilde{C}_{R}(\Gamma,n)$ are explicitly given in (20)
and decay as
$\mathcal{O}(\|\mathds{1}_{H^{k-1}}-\Pi_{R}\|_{\mathcal{B}(H^{k-1})})$ as
$R\to+\infty$.

######  Remark 6.

Since $N(\Gamma)\in{\mathbb{N}}$, the _exact_ number of resonances (counted
with their multiplicity) inside $\Gamma$ is given by $N_{R}(\Gamma)$ when
$R\gg 0$ so that $C_{R}(\Gamma)<1$. Besides, if we do know that only one
resonance lies inside $\Gamma$, then $S_{R}(\Gamma,1)$ gives an approximative
value of that resonance.

######  Remark 7.

The farther below the real line we want to seek resonances, the larger we have
to take $k\in{\mathbb{N}}\setminus\\{0\\}$: this increases the numerical error
as it uses the Sobolev norm $\|\bullet\|_{H^{k-1}}$ (for simple spherically
symmetric problems, we expect that resonances associated to large values of
the angular momentum of the KG field are localized far below the real line,
see _e.g._ [BoHa, Thm. 2.1], [B1, Thm. 5.1] and [DZ, Fig. 1.7]). Increasing
$N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$ offers better rate of convergence
of the scheme at the cost of adding much more terms in the parametrix.

In practice, we first look for resonances starting with very simple black
holes (_e.g._ De Sitter-Reissner-Nordström black holes) for which we do know
from [BoHa] that all the resonances for the wave equation lie below
${\mathbb{R}}$ except one which is at 0, and from [B1] and [BeHa] that that
resonance 0 for waves is expelled from 0 for charged KG fields and moves above
or below the real axis depending on the respective size of the charge and the
mass of the field (when the mass controls the charge then the resonance moves
below and _vice versa_). Then, we increase the perturbative parameters (such
as the angular momentum of the black hole or the product of the black hole
charge with the KG field charge) to guess where the resonances move (by
analyticity of the spectral family in these parameters, the path followed by
the resonances under these perturbations is smooth).

As we expect from the expansion (1) that resonances are multiple of the (tiny)
characteristic length $\sqrt{\Lambda}$ (which is of order $10^{-26}$ according
to Planck’s latest measure _cf._ [PC]), we can use the following rescaling
(see Sect. 3.1 for the notations):

$\displaystyle\tilde{t}:=\sqrt{\Lambda}t,\qquad$
$\displaystyle\tilde{r}:=\sqrt{\Lambda}r,\qquad\tilde{M}:=\sqrt{\Lambda}M,\qquad\tilde{Q}:=\sqrt{\Lambda}Q,\qquad\tilde{\Lambda}:=1,$
$\displaystyle\tilde{a}:=\sqrt{\Lambda}a,\qquad$
$\displaystyle\tilde{\alpha}:=\frac{\alpha}{\sqrt{\Lambda}},\qquad\tilde{q}:=\frac{q}{\sqrt{\Lambda}},\qquad\tilde{m}:=\frac{m}{\sqrt{\Lambda}},\qquad\tilde{\sigma}:=\frac{\sigma}{\sqrt{\Lambda}}.$

In practice, this amounts to ”zooming” on resonances and changing the
characteristic distance from $\sqrt{\Lambda}$ to $1$. Observe that the charge
product $\tilde{q}\tilde{Q}=qQ$ is _invariant under the rescaling_.

######  Remark 8.

It is possible to define resonances as the zeros of some Wronskian $W$ for
numerical approximation (see [B-th, Sect. 5.4]; note that this approach uses
the Regge-Wheeler coordinate $x\in{\mathbb{R}}$ so that only resonances above
the upper critical strip
${\mathbb{R}}-\mathrm{i}\min\\{\boldsymbol{\kappa}_{-},|\boldsymbol{\kappa}_{+}|\\}$
can be approximated222Moreover, the error estimate of the numerical
approximation worsens below ${\mathbb{R}}$ in the following sense: if $h>0$ is
the numerical step (that is the scheme converges as $h\to 0$), then the error
of approximation of a resonance $\sigma$ lying below ${\mathbb{R}}$ is
proportional to $h^{1-|\mathrm{Im}(\sigma)|/\kappa}$ where $\kappa>0$ is such
that the upper critical strip is $\\{z\in{\mathbb{C}}\ |\
\mathrm{Im}(z)=-\mathrm{i}\kappa\\}$., accordingly to Rmk. 4). This Wronskian
involves Jost solutions $e_{\pm}$ (_cf._ [B1, Sect. 3.3]) which are outgoing
solutions to $\boldsymbol{P}_{\sigma}u=0$; recall that outgoing means that
$e_{\pm}(x)\sim\mathrm{e}^{\pm\mathrm{i}\sigma x}$ as $x\to\pm\infty$, that
is, they behave as free linear waves near the appropriate horizon. Thus, above
the upper critical strip, $W$ and $D$ have the same zeros (counted with their
multiplicity); for one dimensional Schrödinger type operator with suitable
potential (_i.e._ when $\boldsymbol{P}_{\sigma}=-\partial_{x}^{2}+V(x)$), [S,
Prop. 5.7] shows that, in fact, $W=D$.

###  1.3. Outline of the method

While some parts in this paper use tedious computations, the overall method is
simple.

In the quite general setting of Sect. 3.1, the restriction to azimuthal
harmonics of the charged KG operator is totally characteristic. An operator is
said to be totally characteristic in some domain if its Fourier principal
symbol vanishes at the boundary of that domain – in other words, the operator
is elliptic inside the domain but its characteristic set is not empty on the
boundary. The lack of ellipticity prevents us from using the standard
construction of parametrices; however, in the Mellin quantization sense, the
operator becomes uniformly elliptic and an explicit (thus workable for
numerics) construction is now possible. This construction amounts to first
produce a decay in the fiber direction, then in the basis direction of the
cotangent bundle in order to obtain compact remainder.

We then deduce the Fredholm property of the spectral family
$\boldsymbol{P}_{\sigma}$, $\sigma\in{\mathbb{C}}$, associated to the
(harmonic) charged KG operator outside critical strips of frequencies in
${\mathbb{C}}$; this restriction comes from the existence of real roots of
indicial polynomials associated to $\boldsymbol{P}_{\sigma}$. This amounts to
working separately near each radial boundary of the spacetime in a Mellin
setting, construct local parametrices there, then glue them together. As a
rough summary:

$\displaystyle\text{Fourier ellipticity}\qquad\implies\qquad\text{Fredholm
property},$ $\displaystyle\text{Mellin ellipticity }+\text{ restriction
outside critical strips}\qquad\implies\qquad\text{Fredholm property}.$

In the last case, we will see that we need to lose one derivative with respect
to the standard Fourier ellipticity; another noticeable difference is the role
of the subprincipal part of the spectral family in the frequency restriction
(_cf._ Rmk. 12 below where $b(0)$ belongs to the subprincipal part of
$\boldsymbol{P}_{\sigma}$).

Having established the equation
$\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma}=\mathds{1}+\boldsymbol{K}_{\sigma}$
for some suitable $\boldsymbol{Q}_{\sigma}$ and $\boldsymbol{K}_{\sigma}$, we
can push the spectral analysis of the latter further and show its trace class
property. As a result, the Fredholm determinant
$\det(\mathds{1}+\boldsymbol{K}_{\sigma})$ is well-defined and can be
approximated by $\det(\Pi_{R}(\mathds{1}+\boldsymbol{K}_{\sigma}))$ where
$\Pi_{R}$ projects onto a $R$ dimensional linear space; furthermore, the
determinant cancels if and only if $\boldsymbol{K}_{\sigma}$ has a non trivial
kernel333The power of the Fredholm theory is to allow one to apply the
standard linear algebra results in finite dimension to problems in infinite
dimension.. If we manage to show that $\boldsymbol{P}_{\sigma}$ has index 0
(that is both the dimensions of its kernel and cokernel coincide), then

$\displaystyle\sigma\text{ is a resonance}\
\qquad\Longleftrightarrow\qquad\det(\mathds{1}+\boldsymbol{K}_{\sigma})=0.$

As $\sigma\mapsto\det(\mathds{1}+\boldsymbol{K}_{\sigma})$ is analytic for
$\sigma$ outside the critical strips, we can finally use complex integrals of
$\det(\Pi_{R}(\mathds{1}+\boldsymbol{K}_{\sigma}))$ (which is the determinant
of a matrix) to approximate or count the number of resonances inside a fixed
domain in ${\mathbb{C}}$.

###  1.4. Quantizations

We introduce in this section the two types of quantization that we will use in
this paper.

The Fourier transform
$\mathcal{F}\in\mathcal{B}(L^{2}({\mathbb{R}},\mathrm{d}x))$ and its inverse
are defined by:

$\displaystyle\mathcal{F}[u](\xi)$
$\displaystyle:=\int_{{\mathbb{R}}}\mathrm{e}^{-\mathrm{i}x\xi}u(x)\mathrm{d}x,\qquad\qquad\mathcal{F}^{-1}[v](x)=\frac{1}{2\pi}\int_{{\mathbb{R}}}\mathrm{e}^{\mathrm{i}x\xi}v(\xi)\mathrm{d}\xi.$

The Mellin transform
$\mathcal{M}\in\mathcal{B}\left(L^{2}\left((0,+\infty),\frac{\mathrm{d}\rho}{\rho}\right),L^{2}({\mathbb{R}},\mathrm{d}\xi)\right)$
and its inverse are defined by:

$\displaystyle\mathcal{M}[u](\xi)$
$\displaystyle:=\int_{{\mathbb{R}}_{+}}\rho^{-\mathrm{i}\xi}u(\rho)\frac{\mathrm{d}\rho}{\rho},\qquad\qquad\mathcal{M}^{-1}[v](\rho)=\frac{1}{2\pi}\int_{\mathbb{R}}\rho^{\mathrm{i}\xi}v(\xi)\mathrm{d}\xi.$

The change of variable $\rho:=\mathrm{e}^{x}$, $x\in{\mathbb{R}}$, allows us
to transpose properties of Fourier type quantization to Mellin type
quantization444In particular, that
$\mathcal{M}\in\mathcal{B}\left(L^{2}\left((0,+\infty),\frac{\mathrm{d}\rho}{\rho}\right),L^{2}({\mathbb{R}},\mathrm{d}\xi)\right)$
is simply a consequence of Plancherel formula. We should by the way talk
respectively about the Fourier-Plancherel and Mellin-Plancherel extensions of
the standard Fourier and Mellin transforms.. Setting $v=u\circ\exp$, we have
$\mathcal{M}[u]=\mathcal{F}[v]$ and
$(\rho\partial_{\rho}u)(\rho)=(\partial_{x}v)(x)$555From the spectral
perspective, Mellin type quantization associates $\rho D_{\rho}$ to $\xi$,
_i.e._ totally characteristic operators near a boundary are in fact Mellin
elliptic.; in particular,

$\displaystyle u\in
L^{2}\left((0,+\infty),\frac{\mathrm{d}\rho}{\rho}\right)\qquad$
$\displaystyle\Longleftrightarrow\qquad v\in L^{2}({\mathbb{R}},\mathrm{d}x).$

Note however that this equivalence fails to hold true between Sobolev spaces,
as the above change of variable modifies the differential structures (in
short, $\partial_{\rho}$ is not equivalent to $\partial_{x}$). Thus, when
dealing with Mellin type pseudo-differential operators
$H^{k^{\prime}}\left((0,+\infty),\frac{\mathrm{d}\rho}{\rho}\right)\to
H^{k}\left((0,+\infty),\frac{\mathrm{d}\rho}{\rho}\right)$ with
$k,k^{\prime}\in{\mathbb{N}}$ such that $k^{\prime}\geq k$, we will first take
$(k^{\prime}-k)$ derivatives (with respect to the variable
$\rho\in(0,+\infty)$) then we will change the variable to work with a Fourier
type pseudo-differential operator acting on $L^{2}({\mathbb{R}},\mathrm{d}x)$
(this will be used in the proof of Lem. 18 below).

Let
$a\in\mathcal{C}^{\infty}((0,+\infty)_{\rho}\times{\mathbb{R}}^{n}_{\omega},{\mathbb{C}})$.
The mixed Mellin-Fourier quantization of $a$ is by definition the following
operator:

$\displaystyle(\mathrm{Op}_{\mathcal{M},\mathcal{F}}[a]u)(\rho,\omega)$
$\displaystyle:=\frac{1}{(2\pi)^{n+1}}\int_{{\mathbb{R}}_{\xi}}\int_{{\mathbb{R}}^{n}_{\eta}}\rho^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}a(\rho,\omega,\xi,\eta)\left(\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}y^{-\mathrm{i}\xi}\mathrm{e}^{-\mathrm{i}\zeta\cdot\eta}u(y,\zeta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}\right)\mathrm{d}\eta\mathrm{d}\xi.$

As explained above, we can see this operator as a full Fourier type
quantization on ${\mathbb{R}}_{x}\times{\mathbb{R}}^{n}_{\omega}$. We refer
the reader to _e.g._ [Hö, Chap. XVIII] for the basic properties (depending on
$a$) of these pseudo-differential operators acting on Sobolev spaces.

###  1.5. Plan of the paper and notations

The paper is organized as follows: Sect. 2 describes the construction in an
abstract setting of a parametrix for radially totally characteristic operators
and establishes the associated local Fredholm theory, Sect. 3 presents and
puts the charged KG operator defined in some cosmological black hole type
spacetimes into the general setting of the previous section and Sect. 4
presents a numerical scheme to compute resonances. We give in App. 5 a
detailed derivation of the spectral family of Sect. 3.3. in coordinates
defined near the poles of $\mathbb{S}^{2}$.

Throughout the paper, we will use the following notations:
$[\bullet\,,\bullet]_{+}$ is the positive commutator ($[a,b]_{+}:=ab+ba$),
$\mathcal{C}^{\omega}(X,{\mathbb{C}})$ denotes the set of all the analytic
functions defined on a vector space $X$, $a\lesssim b$ means that there exists
a universal constant $C>0$ such that $a\leq Cb$, and
$D_{\bullet}:=-\mathrm{i}\partial_{\bullet}$.

##  2\. General boundary parametrix construction

We construct in this section local (_i.e._ near a boundary) parametrices for
abstract totally characteristic operators in the case where the domain has a
radial symmetry, using Mellin type quantization in the radial direction.

###  2.1. Definition of the functional framework and main results of the
section

Let $(k,l,\sigma)\in\mathbb{N}\times{\mathbb{R}}\times{\mathbb{C}}$ and let
$\mathcal{M}$ be a $n$ dimensional compact smooth manifold. Let
$(\phi_{\bullet})_{\bullet}$ be a smooth partition of unity of $\mathcal{M}$
then choose
$\widetilde{\phi}_{\bullet}\in\mathcal{C}^{\infty}(\mathcal{M},[0,1])$ such
that $\widetilde{\phi}_{\bullet}\phi_{\bullet}=\phi_{\bullet}$ on
$\mathrm{Supp}(\phi_{\bullet})$; in the sequel, we will identify
$\mathcal{M}\cap\mathrm{Supp}(\widetilde{\phi}_{\bullet})$ with an open subset
of ${\mathbb{R}}^{n}$. Let also
$\chi\in\mathcal{C}^{\infty}([0,+\infty),[0,1])$ such that

(2) $\displaystyle\chi(\rho)$ $\displaystyle=\begin{cases}1&\text{if
$\rho\leq\rho_{0}$},\\\ 0&\text{if $\rho\geq\rho_{0}^{\prime}$}\end{cases}$

for some $0<\rho_{0}<\rho_{0}^{\prime}$. In applications, $\sigma$ is a
spectral parameter and $\mathcal{M}$ is the unit sphere $\mathbb{S}^{n}$.

Let $X:=(0,+\infty)\times\mathcal{M}$. On
$\mathrm{Supp}(\chi)\times\mathrm{Supp}(\phi_{\bullet})$, we consider

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle:=\rho^{-1}P_{\sigma}+\cancel{P}_{\\!\sigma},$ $\displaystyle
P_{\sigma}$ $\displaystyle:=a(\rho,\omega,\sigma)(\rho
D_{\rho})^{2}+b(\rho,\omega,\sigma)\rho
D_{\rho}+c(\rho,\omega,\sigma),\qquad\cancel{P}_{\\!\sigma}:=\sum_{j=1}^{n}\left(\cancel{a}_{j}(\rho,\omega,\sigma)D_{\omega_{j}}^{2}+\cancel{b}_{j}(\rho,\omega,\sigma)D_{\omega_{j}}\right)$

where
$a,b,c,\cancel{a}_{j},\cancel{b}_{j}\in\mathcal{C}^{\infty}(\overline{\mathrm{Supp}(\chi)}\times\mathcal{M}\times{\mathbb{C}})$
– these functions as well as the coordinate $\omega$ of course depend on the
cut-off $\phi_{\bullet}$ and we should write
$a_{\bullet}(\rho,\omega_{\bullet},\sigma)$,
$\cancel{a}_{j,\bullet}(\rho,\omega_{\bullet},\sigma)$ and so on; we will
however not emphasize this dependence in notations below in order to not
overload them.

We define the following weighted boundary Sobolev spaces:

$\displaystyle\mathcal{H}$
$\displaystyle:=\mathcal{H}^{0,0}:=L^{2}\Big{(}X,\frac{\mathrm{d}\rho}{\rho}\mathrm{d}\omega\Big{)},$
$\displaystyle\mathcal{H}^{k,l}$ $\displaystyle:=\big{\\{}u:X\to{\mathbb{C}}\
\big{|}\ \rho^{-l}u\in\mathcal{H},\
\rho^{-l}(\partial_{\rho}^{k_{1}}\partial_{\omega}^{k_{2}}u)\in\mathcal{H}\;\text{for
all }k_{1}+|k_{2}|=k\big{\\}}.$

We equip $\mathcal{H}$ with its standard scalar product and define:

$\displaystyle\|u\|_{\mathcal{H}^{k,l}}^{2}$
$\displaystyle:=\|\rho^{-l}u\|_{\mathcal{H}}^{2}+\|\rho^{-l}(\partial_{\rho}^{k}u)\|_{\mathcal{H}}^{2}+\sum_{j=1}^{n}\|\rho^{-l}(\partial_{\omega_{j}}^{k}u)\|_{\mathcal{H}}^{2}.$

We will use the realization $(\boldsymbol{P}_{\sigma},\mathcal{X}^{k+1,l})$
with

$\displaystyle\mathcal{X}^{k+1,l}$ $\displaystyle:=\\{u\in\mathcal{H}^{k+1,l}\
|\
\boldsymbol{P}_{\sigma}u\in\mathcal{H}^{k,l}\\},\qquad\|u\|_{\mathcal{X}^{k+1,l}}^{2}:=\|u\|_{\mathcal{H}^{k+1,l}}^{2}+\|\boldsymbol{P}_{\sigma}u\|_{\mathcal{H}^{k,l}}^{2}.$

We now make some assumptions. Given
$(\rho,\omega,\xi,\eta,\sigma)\in[0,+\infty)\times\mathcal{M}\times{\mathbb{R}}\times{\mathbb{R}}^{n}\times{\mathbb{C}}$,
we let

$\displaystyle\xi_{k,l}$
$\displaystyle:=\xi-\mathrm{i}(k+l),\qquad|\eta|^{2}_{\cancel{a}}:=\sum_{j=1}^{n}\cancel{a}_{j}(\rho,\omega,\sigma)\eta_{j}^{2}.$

######  Assumptions 9.

We assume that $k+l>0$. We also assume that for all
$(\xi,\eta,\sigma)\in\mathrm{Supp}(\chi)\times\mathcal{M}\times{\mathbb{R}}\times{\mathbb{R}}^{n}\times{\mathbb{C}}$,

$\displaystyle a(\rho,\omega,\sigma)\neq 0\ \mathrm{on}\
\overline{\mathrm{Supp}(\chi)},\qquad\cancel{a}_{j}(\rho,\omega,\sigma)\neq 0\
\mathrm{on}\ \overline{\mathrm{Supp}(\phi_{\bullet})}$

and if $(\rho,\omega)\in(0,+\infty)\times\mathcal{M}$, then

$\displaystyle|\cancel{a}_{j}(\rho,\omega)\eta_{j}|\lesssim\big{|}|\eta|_{\cancel{a}}\big{|},\qquad
a(\rho,\omega,\sigma)\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}\neq 0,$
$\displaystyle\lim_{\|(\xi,\eta)\|\to+\infty}|a(\rho,\omega,\sigma)\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}|=+\infty.$

######  Remark 10.

Assumptions 9 are verified for example if $a$ and $\cancel{a}$ are positive
valued, except the condition
$a(\rho,\omega,\sigma)\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}\neq 0$ which
may fail to be true if $\xi=0$ and
$\rho|\eta|_{\cancel{a}}^{2}=a(\rho,\omega,\sigma)(k+l)^{2}$. In that case, we
can replace it by
$a(\rho,\omega,\sigma)(\xi_{k,l}^{2}+(k+l+\epsilon)^{2})+\rho|\eta|_{\cancel{a}}^{2}$,
$\epsilon>0$, and modify accordingly the denominator of
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$ in Sect. 2.2 below (and replace $c$
with $c-a(k+l+\epsilon)^{2}$) to proceed, as we will do in Sect. 3.4.

Under Assumptions 9, we will establish the following result (_cf._ Sect. 2.4
for the proof):

######  Proposition 11.

We defined the indicial polynomials associated to $\boldsymbol{P}_{\sigma}$:

$\displaystyle\mathfrak{p}_{\sigma,k,l}(\xi)$
$\displaystyle:=a(0,\omega,\sigma)\xi_{k,l}^{2}+(b(0,\omega,\sigma)-2\mathrm{i}a(0,\omega,\sigma))\xi_{k,l}-\mathrm{i}b(0,\omega,\sigma)-a(0,\omega,\sigma)+c(0,\omega,\sigma).$

For all $N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$, let

(3) $\displaystyle{\mathbb{C}}_{k,l,N^{\prime}}$
$\displaystyle:=\bigg{\\{}\sigma\in{\mathbb{C}}\ \bigg{|}\
0\notin\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}})\bigg{\\}}.$

If $N>2(N^{\prime}-1)$, then there exist a right local parametrix
$\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{H}^{,k,l},\mathcal{X}^{k+1,l})$
as well as a left local parametrix
$\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{H}^{,k,l},\mathcal{X}^{k+1,l})$
such that:

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k,l}$
$\displaystyle=\chi\mathds{1}_{\mathcal{H}^{,k,l}}+\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k,l},$
$\displaystyle\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k,l}\widetilde{\boldsymbol{P}}_{\sigma}$
$\displaystyle=\chi\mathds{1}_{\mathcal{X}^{k+1,l}}+\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k,l}.$

The remainders
$\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{H}^{k,l})$
and
$\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{X}^{k+1,l})$
are compact. If
$\left(\sigma\mapsto\boldsymbol{P}_{\sigma}\right)\in\mathcal{C}^{p}({\mathbb{C}},\mathcal{B}(\mathcal{X}^{k+1,l},\mathcal{H}^{k,l}))$
for some $p\in{\mathbb{N}}\cup\\{+\infty,\omega\\}$, then
$\Big{(}\sigma\mapsto\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k,l}\Big{)}\in\mathcal{C}^{p}({\mathbb{C}}_{k,l,N^{\prime}},\mathcal{B}(\mathcal{H}^{k,l}))$
and
$\Big{(}\sigma\mapsto\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k,l}\Big{)}\in\mathcal{C}^{p}({\mathbb{C}}_{k,l,N^{\prime}},\mathcal{B}(\mathcal{X}^{k+1,l}))$.
Furthermore, if $N\geq 3N^{\prime}-2$, then
$\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{H}^{k+N-2(N^{\prime}-1),l})$
and if $0\notin\bigcup_{0\leq\alpha\leq
N^{\prime}}\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}}-\mathrm{i}\alpha)$,
then
$\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{X}^{k+1+N-2(N^{\prime}-1),l})$.

######  Remark 12.

If $c(0,\omega,\sigma)=0$ then

$\displaystyle\mathfrak{p}_{\sigma,k+j-1,l}(\xi)$
$\displaystyle=a(0,\omega,\sigma)\xi_{k+l+j}\left(\xi_{k+l+j}+\frac{b(0,\omega,\sigma)}{a(0,\omega,\sigma)}\right).$

and $\mathfrak{p}_{\sigma,k,l,j}$ has a real root if and only if
$\mathrm{Im}(b(0,\omega,\sigma))\mathrm{Re}(a(0,\omega,\sigma))-\mathrm{Re}(b(0,\omega,\sigma))\mathrm{Im}(a(0,\omega,\sigma))=k+l+j$.
Otherwise, notice that all the complex roots of $\mathfrak{p}_{\sigma,k,l,j}$
lie in the upper half plane whenever
$\mathrm{Im}(b(0,\omega,\sigma))\mathrm{Re}(a(0,\omega,\sigma))-\mathrm{Re}(b(0,\omega,\sigma))\mathrm{Im}(a(0,\omega,\sigma))\leq
k+l+j$.

Strenghtening Assumptions 9, we can obtain the following Fredholm property for
$\boldsymbol{P}_{\sigma}$ (_cf._ Sect. 2.5 for the proof):

######  Proposition 13.

Let
$(k,l,N^{\prime})\in{\mathbb{N}}\times{\mathbb{R}}\times{\mathbb{N}}\setminus\\{0\\}$
and $X^{\prime}\Subset X$ such that
$\chi_{|X^{\prime}}=\mathds{1}_{X^{\prime}}$. Denote by $r_{X^{\prime}}$ the
restriction to $X^{\prime}$ and $\mathcal{H}^{\prime k,l},\mathcal{X}^{\prime
k+1,l}$ the functional spaces defined over $X^{\prime}\times\mathcal{M}$.
Assume that $\pm a(\rho,\omega,\sigma)>0$ and
$\pm\cancel{a}_{j}(\rho,\omega,\sigma)>0$ on
$\overline{X^{\prime}}\times\mathcal{M}$ (so that Assumptions 9 are
satisfied). Then the restriction of
${\mathbb{C}}_{k,l,N^{\prime}}\ni\sigma\mapsto
r_{X^{\prime}}\circ\boldsymbol{P}_{\sigma}\in\mathcal{B}(\mathcal{X}^{\prime
k+1,l},\mathcal{H}^{\prime k,l})$ is an analytic family of index 0 Fredholm
operators.

In proving Prop. 11, we will use the following function:

$\displaystyle F_{k,l}(\rho,\xi)$
$\displaystyle:=\frac{\rho^{k+l}}{(\mathrm{i}\xi+k+l)_{(k)}}.$

Above $(z)_{(w)}:=\frac{\Gamma(z+1)}{\Gamma(z-w+1)}$ denotes the falling
factorial ($\Gamma$ is the Gamma function).

######  Lemma 14.

Let $u\in H^{k,l}(X,\frac{\mathrm{d}\rho}{\rho}\mathrm{d}\omega)$ and put
$v(\rho,\omega):=\rho^{-l}(\partial_{\rho}^{k}u)(\rho,\omega)$. Then
$\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}]v=u$.

###### Proof.

As $F_{k,l}$ is purely radial, we can directly integrate with respect to the
angular variable $\omega$ and its dual variable $\eta$. Next, shifting the
contour integral from ${\mathbb{R}}$ to ${\mathbb{R}}+\mathrm{i}k$ then
performing $k$ integrations by parts, we compute:

$\displaystyle(\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}]v)(\rho,\omega)$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\rho^{\mathrm{i}\xi+k+l}\frac{1}{(\mathrm{i}\xi+k+l)_{(k)}}\left(\int_{(0,+\infty)}y^{-\mathrm{i}\xi}(y^{-l}\partial_{y}^{k}u)(y,\omega)\frac{\mathrm{d}y}{y}\right)\mathrm{d}\xi$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\rho^{\mathrm{i}\xi+l}\frac{1}{(\mathrm{i}\xi+l)_{(k)}}\left(\int_{(0,+\infty)}y^{-\mathrm{i}\xi+k}(y^{-l}\partial_{y}^{k}u)(y,\omega)\frac{\mathrm{d}y}{y}\right)\mathrm{d}\xi$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\rho^{\mathrm{i}\xi+l}\frac{1}{(\mathrm{i}\xi+l)_{(k)}}\left(\int_{(0,+\infty)}(-1)^{k}(\partial_{y}^{k}y^{-\mathrm{i}\xi+k-l-1})u(y,\omega)\mathrm{d}y\right)\mathrm{d}\xi$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\rho^{\mathrm{i}\xi+l}\frac{1}{(\mathrm{i}\xi+l)_{(k)}}\left(\int_{(0,+\infty)}(\mathrm{i}\xi+l)_{(k)}y^{-\mathrm{i}\xi}y^{-l}u(y,\omega)\frac{\mathrm{d}y}{y}\right)\mathrm{d}\xi$
$\displaystyle=\rho^{l}\mathrm{Op}_{\mathcal{M}}[1]((\rho,\omega)\mapsto\rho^{-l}u(\rho,\omega))$
$\displaystyle=u(\rho,\omega).$

The contour shift uses Cauchy-Riemann equations: to check them, we first need
to consider $u\in\mathcal{C}^{\infty}_{0}((0,+\infty),{\mathbb{C}})$ (the
space of smooth functions vanishing at $+\infty$ together with all their
derivatives) and integrate by parts once to change
$y^{-\mathrm{i}\xi-l-1}\partial_{y}^{k}u$ to
$y^{-\mathrm{i}\xi-l}\partial_{y}^{k+1}u$, then we use the density of
$\mathcal{C}^{\infty}_{0}((0,+\infty),{\mathbb{C}})$ into
$H^{k}((0,+\infty),\mathrm{d}\rho)$ (_cf._ [AF, Thm. 3.22]). This completes
the proof. ∎

###  2.2. Decay in the fiber

Let $u\in H^{k,l}(X,\frac{\mathrm{d}\rho}{\rho}\mathrm{d}\omega)$ and set

$\displaystyle v_{k,l,N/S}(\rho,\omega)$
$\displaystyle:=\widetilde{\phi}_{N/S}(\omega)\rho^{-l}(\partial_{\rho}^{k}u)(\rho,\omega).$

We define

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},1}u$
$\displaystyle:=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},1}]v_{k,l,\bullet}$

with

$\displaystyle\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},1}$
$\displaystyle:=\chi\phi_{\bullet}\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1},\qquad\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}:=\frac{\rho}{a\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}}.$

In order to lighten notations below, we will just write
$\mathfrak{q}_{\bullet}^{\mathrm{f},1}$ instead of
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},1}$ (and similarly for
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$). We will also use the notation
$\widetilde{w}:=\rho^{-1}w$. Let us compute:

$\displaystyle\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}_{\bullet}^{\mathrm{f},1}$
$\displaystyle=\chi\phi_{\bullet}\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{f},1}+[\boldsymbol{P}_{\sigma},\chi\phi_{\bullet}]\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{f},1},$
$\displaystyle\quad\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{f},1}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\omega\cdot\eta}P_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathfrak{q}^{\mathrm{f},1}+\rho^{\mathrm{i}\xi+k+l}\cancel{P}_{\\!\sigma}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{f},1}$
$\displaystyle=\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\Bigg{\\{}\\!\left(\frac{a\xi_{k,l}^{2}+b\xi_{k,l}+c}{\rho}\right)\mathfrak{q}^{\mathrm{f},1}-(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+a)(\partial_{\rho}\mathfrak{q}^{\mathrm{f},1})-a\rho(\partial_{\rho}^{2}\mathfrak{q}^{\mathrm{f},1})$
$\displaystyle\qquad\qquad\qquad\quad+\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})\mathfrak{q}^{\mathrm{f},1}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\mathfrak{q}^{\mathrm{f},1})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\mathfrak{q}^{\mathrm{f},1})\right)\\!\Bigg{\\}}$
$\displaystyle=\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\\!\left(a\xi_{k,l}^{2}+(b-2\mathrm{i}a)\xi_{k,l}-\mathrm{i}b-a+c\right)\widetilde{\mathfrak{q}}^{\mathrm{f},1}-\rho(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+3a)(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{f},1})-a\rho^{2}(\partial_{\rho}^{2}\widetilde{\mathfrak{q}}^{\mathrm{f},1})$
$\displaystyle\qquad\qquad\qquad\quad+\rho\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})\widetilde{\mathfrak{q}}^{\mathrm{f},1}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{f},1})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)\\!\bigg{\\}},$
$\displaystyle\quad[\boldsymbol{P}_{\sigma},\chi\phi_{\bullet}]\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{f},1}$
$\displaystyle=\left(\phi_{\bullet}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}[P_{\sigma},\chi]\rho^{\mathrm{i}\xi+k+l}+\chi\rho^{\mathrm{i}\xi+k+l}[\cancel{P}_{\sigma},\phi_{\bullet}]\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\right)\mathfrak{q}^{\mathrm{f},1}$
$\displaystyle=-\left(\phi_{\bullet}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\left(\chi^{\prime}(2a\rho\partial_{\rho}+\mathrm{i}b+a)+a\rho\chi^{\prime\prime}\right)\rho^{\mathrm{i}\xi+k+l}+\chi\rho^{\mathrm{i}\xi+k+l}\sum_{j=1}^{n}\left((\partial_{\omega_{j}}\phi_{\bullet})(2\cancel{a}_{j}\partial_{\omega_{j}}+\mathrm{i}\cancel{b}_{j})+\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\phi_{\bullet})\right)\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\right)\mathfrak{q}^{\mathrm{f},1}$
$\displaystyle=-\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\phi_{\bullet}\left((2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+a)\chi^{\prime}\mathfrak{q}^{\mathrm{f},1}+a\rho\chi^{\prime\prime}\mathfrak{q}^{\mathrm{f},1}+2a\rho\chi^{\prime}(\partial_{\rho}\mathfrak{q}^{\mathrm{f},1})\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\chi\sum_{j=1}^{n}\left((2\mathrm{i}\cancel{a}_{j}\eta_{j}+\mathrm{i}\cancel{b}_{j})(\partial_{\omega_{j}}\phi_{\bullet})\mathfrak{q}^{\mathrm{f},1}+\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\phi_{\bullet})\mathfrak{q}^{\mathrm{f},1}+2\cancel{a}_{j}(\partial_{\omega_{j}}\phi_{\bullet})(\partial_{\omega_{j}}\mathfrak{q}^{\mathrm{f},1})\right)\bigg{\\}}$
$\displaystyle=-\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\phi_{\bullet}\rho\left(\big{(}(2(1+\mathrm{i}\xi_{k,l})a+\mathrm{i}b+a)\chi^{\prime}+a\rho\chi^{\prime\prime}\big{)}\widetilde{\mathfrak{q}}^{\mathrm{f},1}+2a\rho\chi^{\prime}(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)$
$\displaystyle\qquad\qquad\qquad\qquad+\chi\rho\sum_{j=1}^{n}\left(\big{(}\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\phi_{\bullet})+\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\phi_{\bullet})\big{)}\widetilde{\mathfrak{q}}^{\mathrm{f},1}+2\cancel{a}_{j}(\partial_{\omega_{j}}\phi_{\bullet})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)\bigg{\\}}.$

Above ′ denotes the derivative with respect to $\rho$. It follows:

$\displaystyle\boldsymbol{P}_{\sigma}\mathrm{Op}$
${}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\bullet}^{\mathrm{f},1}]=\chi\phi_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}]$
$\displaystyle+\chi\phi_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}\bigg{[}F_{k,l}\bigg{\\{}\\!\left((b-2\mathrm{i}a)\xi_{k,l}-\mathrm{i}b-a+c\right)\widetilde{\mathfrak{q}}^{\mathrm{f},1}-\rho(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+3a)(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{f},1})-a\rho^{2}(\partial_{\rho}^{2}\widetilde{\mathfrak{q}}^{\mathrm{f},1})$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\rho\sum_{j=1}^{n}\left(\cancel{b}_{j}\eta_{j}\widetilde{\mathfrak{q}}^{\mathrm{f},1}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{f},1})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)\\!\bigg{\\}}\bigg{]}$
$\displaystyle+\mathrm{Op}_{\mathcal{M},\mathcal{F}}\bigg{[}F_{k,l}\bigg{\\{}\phi_{\bullet}\rho\left(\big{(}(2(1+\mathrm{i}\xi_{k,l})a+\mathrm{i}b+a)\chi^{\prime}+a\rho\chi^{\prime\prime}\big{)}\widetilde{\mathfrak{q}}^{\mathrm{f},1}+2a\rho\chi^{\prime}(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)$
(4) $\displaystyle\qquad\qquad\qquad\quad\
+\chi\rho\sum_{j=1}^{n}\left(\big{(}\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\phi_{\bullet})+\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\phi_{\bullet})\big{)}\widetilde{\mathfrak{q}}^{\mathrm{f},1}+2\cancel{a}_{j}(\partial_{\omega_{j}}\phi_{\bullet})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\right)\\!\bigg{\\}}\bigg{]}.$

By Lem. 14, we have

$\displaystyle\chi\phi_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}]v_{k,l,\bullet}$
$\displaystyle=\chi\phi_{\bullet}(\widetilde{\phi}_{\bullet}u)=\chi\phi_{\bullet}u;$

Denoting by $\mathfrak{N}_{d}\subset{\mathbb{N}}^{d}$ the subset of multi-
indices with all coefficients zero but one which lies in $\\{0,1,2\\}$ (so
$(0,2,0,\ldots,0)\in\mathfrak{N}_{d}$ but
$(1,1,0,\ldots,0)\notin\mathfrak{N}_{d}$), we get

(5)
$\displaystyle\boldsymbol{P}_{\sigma}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\bullet}^{\mathrm{f},1}]v_{k,l,\bullet}$
$\displaystyle=\chi\phi_{\bullet}u+\sum_{\alpha\in\mathfrak{N}_{n+1}}\partial^{\alpha}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha,\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},1})]v_{k,l,\bullet}$

where the multi-index $\beta$ indicates the powers of $\xi_{k,l}$ or
$\rho\eta_{j}$ _i.e._
$A_{\alpha,\beta,\gamma}=\mathcal{O}\big{(}\xi_{k,l}^{\beta_{1}}+(\rho\eta)^{\beta_{2}}\big{)}$
with $|\beta|=\beta_{1}+|\beta_{2}|$ (so
$A_{(0,\ldots,0),(0,\ldots,0),(0,\ldots,0)}=-\mathrm{i}b-a+c$,
$A_{(0,\ldots,0),(1,\ldots,0),(0,\ldots,0)}=(b-2\mathrm{i}a)\xi_{k,l}$, etc.).
Notice that $\eta_{j}$ in $A_{\alpha,\beta,\gamma}$ is always multiplied by
$\rho$. Notice also that $A_{\alpha,\beta,\gamma}=0$ whether $|\alpha|\neq 0$
and $|\gamma|=2$, or $|\beta|\neq 0$ and $|\gamma|=2$.

To lower the total order in $\xi$ and $\eta_{j}$ of the symbol of the
remainder, we set

$\displaystyle\mathfrak{q}^{\mathrm{f},2}_{\bullet}$
$\displaystyle:=\sum_{\alpha\in\mathfrak{N}_{n+1}}\partial^{\alpha}(\chi\phi_{\bullet})\mathfrak{q}^{\mathrm{f},2,\alpha},\qquad\mathfrak{q}^{\mathrm{f},2,\alpha}:=-\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=1\end{subarray}}\sum_{\begin{subarray}{c}\gamma\in{\mathbb{N}}^{2n+3}\\\
|\gamma|\leq
1\end{subarray}}A_{\alpha,\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},1})\mathfrak{q}^{\mathrm{f},1};$

then using (5), we compute

$\displaystyle\boldsymbol{P}_{\sigma}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\bullet}^{\mathrm{f},2}]$
$\displaystyle=-\sum_{\alpha\in\mathfrak{N}_{n+1}}\partial^{\alpha}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha,\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},1})]$
$\displaystyle\quad+\sum_{\alpha_{1}\in\mathfrak{N}_{n+1}}\sum_{\alpha_{2}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\alpha_{2}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha_{2},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},2,\alpha_{1}})]$

_i.e._

$\displaystyle\boldsymbol{P}_{\sigma}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}(\mathfrak{q}_{\bullet}^{\mathrm{f},1}+\mathfrak{q}_{\bullet}^{\mathrm{f},2})$
$\displaystyle]v_{k,l,\bullet}=\chi\phi_{\bullet}u+\sum_{\alpha\in\mathfrak{N}_{n+1}}\partial^{\alpha}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha,\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},1})]v_{k,l,\bullet}$
$\displaystyle+\sum_{\alpha_{1}\in\mathfrak{N}_{n+1}}\sum_{\alpha_{2}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\alpha_{2}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha_{2},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{f},2,\alpha_{1}})]v_{k,l,\bullet}.$

By induction over $N\in{\mathbb{N}}\setminus\\{0,1,2\\}$, we show that

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$
$\displaystyle=\chi u+\boldsymbol{K}_{\sigma,k,l}^{\mathrm{f},N}u$

where

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$
$\displaystyle=\sum_{j=1}^{N}\left(\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}]\right)v_{k,l,\bullet},\qquad\boldsymbol{K}_{\sigma,k,l}^{\mathrm{f},N}u=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}]v_{k,l,\bullet}$

with666By convention,
$\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-2,\alpha_{1},\ldots,\alpha_{\lfloor
j/2\rfloor-1}}:=\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},1}$ for
$j=3$.

$\displaystyle\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}$
$\displaystyle=\sum_{i=\lfloor
j/2\rfloor}^{j-1}\sum_{\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\ldots+\alpha_{i}}(\chi\phi_{\bullet})\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j,\alpha_{1},\ldots,\alpha_{i}},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j,\alpha_{1},\ldots,\alpha_{\lfloor
j/2\rfloor}}$
$\displaystyle=-\sum_{\gamma\in\mathfrak{N}_{2n+3}}\left(\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}A_{\alpha_{\lfloor
j/2\rfloor},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-2,\alpha_{1},\ldots,\alpha_{\lfloor
j/2\rfloor-1}})\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+\delta_{j\in
2{\mathbb{N}}}\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=1\end{subarray}}A_{\alpha_{\lfloor
j/2\rfloor},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-1,\alpha_{1},\ldots,\alpha_{\lfloor
j/2\rfloor-1}})\right)\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j,\alpha_{1},\ldots,\alpha_{i}}$
$\displaystyle=-\sum_{\gamma\in\mathfrak{N}_{2n+3}}\left(\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-2,\alpha_{1},\ldots,\alpha_{i-1}})+\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=1\end{subarray}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-1,\alpha_{1},\ldots,\alpha_{i-1}})\right)\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$

for $\lfloor j/2\rfloor<i<j-1$,

$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j,\alpha_{1},\ldots,\alpha_{j-1}}$
$\displaystyle=-\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=1\end{subarray}}A_{\alpha_{j-1},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},j-1,\alpha_{1},\ldots,\alpha_{j-2}})\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$

and

$\displaystyle\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$
$\displaystyle=\sum_{i=\lfloor(N+1)/2\rfloor}^{N-1}\sum_{\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\ldots+\alpha_{i}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N-1,\alpha_{1},\ldots,\alpha_{i-1}})$
$\displaystyle\qquad+\sum_{i=\lceil(N+1)/2\rceil}^{N}\sum_{\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\ldots+\alpha_{i}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N,\alpha_{1},\ldots,\alpha_{i-1}}).$

Indeed, defining $\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},N+1}$
according to the above formula, we get with (5):

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}(\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},1}+\ldots+\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},N+1})]v_{k,l,\bullet}=\chi\phi_{\bullet}u$
$\displaystyle+\sum_{i=\lceil(N+1)/2\rceil}^{N}\sum_{\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\ldots+\alpha_{i}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N,\alpha_{1},\ldots,\alpha_{i-1}})]v_{k,l,\bullet}$
$\displaystyle+\sum_{i=\lfloor(N+1)/2\rfloor+1}^{N+1}\sum_{\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{n+1}}\partial^{\alpha_{1}+\ldots+\alpha_{i}}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N+1,\alpha_{1},\ldots,\alpha_{i-1}})]v_{k,l,\bullet}.$

There has been
$2N-3-\lfloor(N+1)/2\rfloor-\lceil(N+1)/2\rceil=2N-4+\delta_{N+1\in
2{\mathbb{N}}}-2\lfloor(N+1)/2\rfloor$ terms cancelled in the remainder
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$; it remains to notice
that $\lceil(N+1)/2\rceil=\lfloor(N+2)/2\rfloor$ and
$\lfloor(N+1)/2\rfloor+1=\lceil(N+2)/2\rceil$ to complete the induction proof
of the above formula.

######  Lemma 15.

For all $N\in{\mathbb{N}}\setminus\\{0\\}$,
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l})$.

###### Proof.

We can show (_e.g._ using Faà di Bruno formula) that, for all
$N_{1},N_{2}\in{\mathbb{N}}\setminus\\{0\\}$ and $\lceil N_{2}/2\rceil\leq
N_{2}^{\prime}\leq N_{2}$,
$\rho^{N_{2}^{\prime}}\partial_{\omega_{j}}^{N_{2}}\rho^{N_{1}}\partial_{\rho}^{N_{1}}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},1}$
is a sum of terms controlled on the support of $\chi$ by

$\displaystyle f(\rho,\omega,\xi,\eta)$
$\displaystyle:=\left|\frac{\rho^{s+t^{\prime}}(\xi_{k,l}^{2s}+|\eta|_{\cancel{a}}^{2s})(\xi_{k,l}^{2t}+\rho^{t}|\eta|_{\cancel{a}}^{2t})}{(a\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2})^{s+t+1}}\right|$

where $0\leq s\leq N_{1}$, $0\leq t\leq N_{2}$ and $\lceil t/2\rceil\leq
t^{\prime}\leq t$; the $s$ terms come from the derivatives
$\rho^{s}\partial_{\rho}^{s}$ (recall from (2.2) that each derivative
$\partial_{\rho}$ is multiplied on the left by $\rho$ for terms in front of
$\chi$) while the $t,t^{\prime}$ terms come from the derivatives
$\rho^{t^{\prime}}\partial_{\omega_{j}}^{t}$ (as we have
$\rho\partial_{\omega_{j}}$ and $\rho\partial_{\omega_{j}}^{2}$ in (2.2) for
terms in front of $\chi$). Recall that the denominator of $f$ does not cancel
on the support of $\chi$ by hypothesis on $a$ and $\cancel{a}$. If
$|\eta|_{\cancel{a}}\leq\rho^{-1/2}$, then

$\displaystyle f(\rho,\omega,\xi,\eta)$
$\displaystyle\leq\frac{\rho^{t^{\prime}}(\rho^{s+t^{\prime}}|\xi_{k,l}|^{2s}+1)(|\xi_{k,l}|^{2t}+1)}{\big{|}a\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}\big{|}^{s+t+1}}\lesssim
1$

as the total order in $\xi$ and $|\eta|_{\cancel{a}}$ is $-2$ (we use here
that $|\eta_{j}|\lesssim\big{|}|\eta|_{\cancel{a}}\big{|}$ by hypothesis). On
the other hand, if $|\eta|_{\cancel{a}}\geq\rho^{-1/2}$, then

(6) $\displaystyle f(\rho,\omega,\xi,\eta)$
$\displaystyle\leq\frac{\left(\frac{|\xi_{k,l}|^{2s}}{\big{|}|\eta|_{\cancel{a}}\big{|}^{2s}}+1\right)\left(\frac{|\xi_{k,l}|^{2t}}{\rho^{t}\big{|}|\eta|_{\cancel{a}}\big{|}^{2t}}+1\right)}{\left|\frac{a\xi_{k,l}^{2}}{\rho|\eta|_{\cancel{a}}^{2}}+1\right|^{s+t+1}}\frac{\rho^{s+t+t^{\prime}}\big{|}|\eta|_{\cancel{a}}\big{|}^{2(s+t)}}{\rho^{s+t+1}\big{|}|\eta|_{\cancel{a}}\big{|}^{2(s+t+1)}}\lesssim
1$

as again the total order in $\xi$ and $|\eta|_{\cancel{a}}$ is $-2$. The same
conclusion obviously holds for terms on the support of $\chi^{\prime}$ and
$\chi^{\prime\prime}$ (this time derivatives in (2.2) are not necessarily
multiplied on the left by $\rho$, but $\rho>0$ there).

Using the inductive construction of the
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}$, we see that all the
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j,\alpha_{1},\ldots,\alpha_{i}}$ are
written in terms of
$A_{\alpha,\beta,\gamma}(\partial^{\gamma}\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j-2,\alpha_{1},\ldots,\alpha_{i-1}})$
with $|\beta|=0$ (this means no $\xi_{k,l}$ nor $\eta_{1},\ldots,\eta_{n}$ in
$A_{\alpha,\beta,\gamma}$) and of
$A_{\alpha,\beta,\gamma}(\partial^{\gamma}\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},j-1,\alpha_{1},\ldots,\alpha_{i-1}})$
with $|\beta|=1$ (this means one power of $\xi_{k,l}$ or
$\eta_{1},\ldots,\eta_{n}$ in $A_{\alpha,\beta,\gamma}$); the total order in
$\xi$ and $|\eta|_{\cancel{a}}$ of
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}$ is therefore $-(j+1)$. The
derivatives computation above of course applies to
$\mathrm{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},1}$ (the absence of
$t^{\prime}\geq 1$ in (6) is compensated by the extra $\rho$ in
$\mathrm{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},1}$). We thus deduce:

$\displaystyle(|\xi|^{N_{1}}+|\eta_{j}|^{N_{2}})|\partial_{\omega_{j}}^{N_{2}}(\rho\partial_{\rho})^{N_{1}}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}|$
$\displaystyle\lesssim 1.$

Passing to the $x:=\ln\rho$ variable, the mixed quantization
$\mathrm{Op}_{\mathcal{M},\mathcal{F}}$ becomes a standard Fourier type
quantization and
${\mathbb{R}}\times\mathcal{M}\times{\mathbb{R}}\times{\mathbb{R}}^{n}\ni(x,\omega,\xi,\eta)\mapsto\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}(\mathrm{e}^{x},\omega,\xi,\eta)$
is a (Fourier type) order 0 symbol (as $\partial_{x}=\rho\partial_{\rho}$)
applied to $(x,\omega)\mapsto v_{k,l,\bullet}(\mathrm{e}^{x},\omega)\in
L^{2}({\mathbb{R}}\times\mathcal{M},\mathrm{d}x\mathrm{d}\omega)$: the
standard theory of pseudo-differential calculus implies that this term remains
bounded in $L^{2}({\mathbb{R}}\times\mathcal{M},\mathrm{d}x\mathrm{d}\omega)$
(the term $\rho^{l}=\mathrm{e}^{lx}$ being controlled in the support of
$\chi$), which means that, back to the $\rho$ variable,
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u\in\mathcal{H}^{0,l}$ _i.e._
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{H}^{0,l})$.

Next, we take derivatives of $\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$.
First of all,
$\partial_{\rho}^{k+1}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$ involves
the terms

$\displaystyle(\partial_{\rho}^{k+1}\rho^{\mathrm{i}\xi}F_{k,l})(\partial_{\rho}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$
$\displaystyle=\frac{\rho^{\mathrm{i}\xi+k+l-s}(\mathrm{i}\xi+k+l)_{(s)}}{(\mathrm{i}\xi+k+l)_{(k)}}(\partial_{\rho}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$
$\displaystyle=\rho^{l}\rho^{\mathrm{i}\xi}\frac{(\mathrm{i}\xi+k+l)_{(s)}}{(\mathrm{i}\xi+k+l)_{(k)}\langle\xi\rangle}\frac{\langle\xi\rangle}{\rho}(\rho^{k+1-s}\partial_{\rho}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$

for $1\leq j\leq N$ and $0\leq s\leq k+1$; the term $\rho^{l}$ gives the
suitable weight for the target space
$\mathcal{X}^{k+1,l}\subset\mathcal{H}^{k+1,l}$, $\rho^{\mathrm{i}\xi}$ is
used in the inverse Mellin transform,
$\frac{(\mathrm{i}\xi+k+l)_{(s)}}{(\mathrm{i}\xi+k+l)_{(k)}\langle\xi\rangle}$
is bounded uniformly in $\xi\in{\mathbb{R}}$, and the same argument as at the
beginning of the proof sets that
$(\rho^{k+1-s}\partial_{\rho}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$
is bounded uniformly in
$(\rho,\omega,\xi,\eta)\in\mathrm{Supp}(\chi)\times\mathrm{Supp}(\phi_{\bullet})\times{\mathbb{R}}\times{\mathbb{R}}^{n}$
and is of order 1 in $\rho$ (because of the presence of
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$) and of order $-1$ in $\xi$, which
controls the term $\frac{\langle\xi\rangle}{\rho}$. It follows that, after
passing to the $x:=\ln\rho$ variable,
$\partial_{\rho}^{k+1}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$ is the
Fourier type quantization of a order 0 symbol applied to $(x,\omega)\mapsto
v_{k,l,\bullet}(\mathrm{e}^{x},\omega)\in
L^{2}({\mathbb{R}}\times\mathcal{M},\mathrm{d}x\mathrm{d}\omega)$: as above,
we deduce that this term in bounded in $\mathcal{H}^{0,l}$, that is,
$\partial_{\rho}^{k+1}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{H}^{0,l})$.
Next, $\partial_{\omega_{i}}^{k+1}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}u$
involves the terms

$\displaystyle(\partial_{\omega_{i}}^{k+1}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}F_{k,l})(\partial_{\omega_{i}}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$
$\displaystyle=\rho^{l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\frac{(\mathrm{i}\eta_{i})^{s}}{(\mathrm{i}\xi+k+l)_{(k)}\langle\eta\rangle}\rho^{s}\frac{\langle\eta\rangle}{\rho}(\rho^{k+1-s}\partial_{\omega_{i}}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j}).$

for all $1\leq i\leq n$; all the terms $\rho^{l}$,
$\frac{1}{(\mathrm{i}\xi+k+l)_{(k)}}$ and
$\frac{\langle\eta\rangle}{\rho}(\rho^{k+1-s}\partial_{\omega_{i}}^{k+1-s}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},j})$
are treated as for the $\rho$ derivatives above, $\rho^{s}\leq\rho_{0}^{s}$
($\rho_{0}$ as in (2)), while
$\int_{{\mathbb{R}}^{n}}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}(\mathrm{i}\eta_{i})^{s}\langle\eta\rangle^{-1}\mathcal{F}[v_{k,l,\bullet}]\mathrm{d}\eta$
gives an element of $\mathcal{H}$. We conclude that
$\partial_{\omega_{i}}^{k+1}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{H}^{0,l})$
as well. The mixed derivatives
$\partial_{\rho}^{s_{1}}\partial_{\omega}^{s_{2}}$, $s_{1}+|s_{2}|=k+1$ are
dealt with similarly.

At last, it is clear that
$\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l})$
since, by construction,
$\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}=\chi+\boldsymbol{K}_{\sigma,k,l}^{\mathrm{f},N}$
and $\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$ has the same form as
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$ (with more decay in $\xi$
and $\eta$). This proves that
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l})$
and thus completes the proof. ∎

######  Remark 16.

Notice the loss of one derivative with respect to the Fourier elliptic case:
taking $k+2$ derivatives in $\rho$ would produce an extra term $\rho^{-1}$
which would target the wrong weighted Sobolev space.

###  2.3. Decay in the basis

We keep the same notations as in Sect. 2.2. The remainder term is not yet
compact due to the lack of decay in the basis of the cotangent space, so we
add corrections to the boundary parametrix
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}$ in order to fix it.

We set

$\displaystyle\breve{w}_{0}$
$\displaystyle:=\breve{w}(0),\qquad\breve{w}(\rho):=\frac{w(\rho)-w_{0}}{\rho}$

for any function $w\in\mathcal{C}^{1}([0,+\infty),{\mathbb{C}})$, and

$\displaystyle\mathfrak{N}_{d}^{0}$
$\displaystyle:=\big{\\{}\alpha\in\mathfrak{N}_{d}\ \big{|}\ \text{the first
coordinate of $\alpha$ is $0$}\big{\\}},$ $\displaystyle\mathfrak{N}_{d}^{1}$
$\displaystyle:=\big{\\{}\alpha\in\mathfrak{N}_{d}\ \big{|}\ \text{the first
coordinate of $\alpha$ is not $0$}\big{\\}}.$

Given $\alpha_{1},\ldots,\alpha_{i}\in\mathfrak{N}_{d}$, we will write
$\boldsymbol{\alpha}_{i}:=(\alpha_{1},\ldots,\alpha_{i})\in\mathfrak{N}_{id}$
and use the notation $:\boldsymbol{\alpha}_{i}:$ for
$\alpha_{1}+\ldots+\alpha_{i}\in\mathfrak{N}_{d}$.

We define

(7) $\displaystyle\mathfrak{p}_{\sigma,k,l}(\xi)$
$\displaystyle:=a_{0}\xi_{k,l}^{2}+(b_{0}-2\mathrm{i}a_{0})\xi_{k,l}-\mathrm{i}b_{0}-a_{0}+c_{0}$

the indicial family of $P_{\sigma}$ at $0$. It is the indicial polynomial from
Frobenius method applied to $P_{\sigma}$ with the variable
$\rho^{-\mathrm{i}\xi_{k,l}+1}$ instead of $\rho$, where $+1$ compensates the
$\rho^{-1}$ factor of $P_{\sigma}$ and $k$, $l$ are the underlying Sobolev
regularity and weight. In other words, if $\mathrm{Ind}_{\sigma}$ is the
indicial polynomial777Recall that the indicial polynomial associated to
$\rho^{2}\partial_{\rho}^{2}+p(\rho)\partial_{\rho}+q(\rho)$ is
$\xi\mapsto\xi(\xi-1)+p(0)\xi+q(0)$. associated to $P_{\sigma}$ then
$\mathfrak{p}_{\sigma,k,l}(\xi)=\mathrm{Ind}_{\sigma}(-\mathrm{i}\xi_{k,l}+1)$).
We assume that $0\notin\mathfrak{p}_{\sigma,k,l}({\mathbb{R}})$. Next, we
define

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,1}u$
$\displaystyle:=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,1}]v_{k,l,\bullet}$

with888This term will cancel the parts of
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$ containing $\chi$. We
could remove from $\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,1,\bullet}$ all the
terms in $\boldsymbol{k}_{\sigma,k,l}^{\mathrm{f},N,\bullet}$ of order $\rho$;
this however would complicate the correction process below as we would have to
track the dependence in $\rho$ of the coefficients of
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$.

$\displaystyle\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,1}$
$\displaystyle:=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{\lfloor(N+1)/2\rfloor}}$
$\displaystyle:=-\delta_{N\in
2{\mathbb{N}}}\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{\lfloor(N+1)/2\rfloor},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N-1,\alpha_{1},\ldots,\alpha_{\lfloor(N+1)/2\rfloor-1}})\frac{\rho}{\mathfrak{p}_{\sigma,k,l}},$

for all $\lceil(N+1)/2\rceil\leq i\leq N-1$,

$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}}$
$\displaystyle:=-\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N-1,\alpha_{1},\ldots,\alpha_{i-1}})\frac{\rho}{\mathfrak{p}_{\sigma,k,l}}$
$\displaystyle\qquad-\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N,\alpha_{1},\ldots,\alpha_{i-1}})\frac{\rho}{\mathfrak{p}_{\sigma,k,l}},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{N}}$
$\displaystyle:=-\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{N},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N,\alpha_{1},\ldots,\alpha_{N-1}})\frac{\rho}{\mathfrak{p}_{\sigma,k,l}}.$

Note that $\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,1}$ is of order
$\rho$. We again drop the indices $\sigma,k,l$ off the notations in the
following computation to simplify them. Using the beginning of Sect. 2.2, we
have:

$\displaystyle\quad\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{b},N,1}$
$\displaystyle=\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\\!\left(a_{0}\xi_{k,l}^{2}+(b_{0}-2\mathrm{i}a_{0})\xi_{k,l}-\mathrm{i}b_{0}-a_{0}+c_{0}\right)\widetilde{\mathfrak{q}}^{\mathrm{b},N,1}$
$\displaystyle\qquad\qquad\qquad\quad+\rho\left(\breve{a}\xi_{k,l}^{2}+(\breve{b}-2\mathrm{i}\breve{a})\xi_{k,l}-\mathrm{i}\breve{b}-\breve{a}+\breve{c}\right)\widetilde{\mathfrak{q}}^{\mathrm{b},N,1}-\rho(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+3a)(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1})-a\rho^{2}(\partial_{\rho}^{2}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1})$
$\displaystyle\qquad\qquad\qquad\quad+\rho\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})\widetilde{\mathfrak{q}}^{\mathrm{b},N,1}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1})\right)\\!\bigg{\\}}.$

In order to further simplify notations for the next computation, we simply
write $\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}$ for
$\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}}$. It
follows

$\displaystyle\quad\boldsymbol{P}_{\sigma}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathfrak{q}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}]$
$\displaystyle=\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{p}_{\sigma,k,l}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}]$
$\displaystyle\quad+\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathrm{Op}_{\mathcal{M},\mathcal{F}}\bigg{[}F_{k,l}\bigg{\\{}\\!\rho\left(\breve{a}\xi_{k,l}^{2}+(\breve{b}-2\mathrm{i}\breve{a})\xi_{k,l}-\mathrm{i}\breve{b}-\breve{a}+\breve{c}\right)\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\quad-\rho(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+3a)(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})-a\rho^{2}(\partial_{\rho}^{2}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\quad+\rho\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})\right)\\!\bigg{\\}}\bigg{]}$
$\displaystyle\quad+\mathrm{Op}_{\mathcal{M},\mathcal{F}}\bigg{[}F_{k,l}\bigg{\\{}\\!\left(\left((2a\rho+\mathrm{i}b+a)\partial_{\rho}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})(1+\mathrm{i}\xi_{k,l})+a\rho\partial_{\rho}^{2}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\right)\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.+(2a\rho+\mathrm{i}b+a)\partial_{\rho}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})(\partial_{\rho}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\rho\sum_{j=1}^{n}\left((2\cancel{a}_{j}+\mathrm{i}\cancel{b}_{j})\partial_{\omega_{j}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathrm{i}\eta_{j}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}+\cancel{a}_{j}\partial_{\omega_{j}}^{2}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.+(2\cancel{a}_{j}+\mathrm{i}\cancel{b}_{j})\partial_{\omega_{j}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})(\partial_{\omega_{j}}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\boldsymbol{\alpha}_{i}})\right)\\!\bigg{\\}}\bigg{]}$

_i.e._

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,1}]=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{p}_{\sigma,k,l}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}}]$
(8)
$\displaystyle+\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\begin{subarray}{c}\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{0}\\\
\alpha_{i+1}\in\mathfrak{N}_{n+1}\end{subarray}}\partial^{:\boldsymbol{\alpha}_{i+1}:}(\chi\phi_{\bullet})\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{0\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}})]$

where $B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,\delta}$ denotes the
coefficients in (2.3) in front of
$\partial^{:\boldsymbol{\alpha}_{i+1}:}(\chi\phi_{\bullet})(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}})$
containing a term
$\rho^{\delta}\xi_{k,l}^{\beta^{\prime}}\eta_{1}^{\beta_{1}^{\prime\prime}}\ldots\eta_{n}^{\beta_{n}^{\prime\prime}}$
with $|\beta|=\beta^{\prime}+|\beta^{\prime\prime}|$ (note that all the
$\beta^{\prime}$, $\beta^{\prime\prime}_{j}$ are zero but one which can be 0,1
or 2 because $\beta\in\mathfrak{N}_{n+1}$). Notice that
$B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,\delta}$ is either of order
$\rho$, $\rho^{2}$ (when $\alpha_{i+1}\in\mathfrak{N}_{n+1}^{0}$) or contains
$\chi^{\prime}$ or $\chi^{\prime\prime}$ (when $\delta=0$). Note also that the
remainder (the term on the last line) in (2.3) is of order $\rho$.

To increase the total order in $\rho$ of the remainder, we define

$\displaystyle\mathfrak{p}_{\sigma,k,l,2}(\xi)$
$\displaystyle:=a_{0}\xi_{k,l}^{2}+(b_{0}-4\mathrm{i}a_{0})\xi_{k,l}-2\mathrm{i}b_{0}-4a_{0}+c_{0}=\mathfrak{p}_{\sigma,k+1,l}(\xi)$

and we assume that $0\notin\mathfrak{p}_{\sigma,k,l,2}({\mathbb{R}})$. We then
set

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,2}u$
$\displaystyle:=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2}]v_{k,l,\bullet}$

with

$\displaystyle\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2}$
$\displaystyle:=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i+1}\in\mathfrak{N}_{(i+1)(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i+1}:}(\chi\phi_{\bullet})\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,2,\alpha_{1},\ldots,\alpha_{i+1}},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,2,\alpha_{1},\ldots,\alpha_{i+1}}$
$\displaystyle:=-\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{1\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}})]\frac{\rho}{\mathfrak{p}_{\sigma,k,l,2}}.$

Once again, $B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,0}=0$ when
$\alpha_{1},\ldots,\alpha_{i+1}\in\mathfrak{N}_{n+1}^{0}$ so that the above
term will cancel the remainder of order $\rho$. Note that
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2}$ is of order $\rho^{2}$
but is of order $-N+2$ in $|\eta|_{\cancel{a}}$ (contrary to the first
iteration, we had here to cancel terms containing
$\rho|\eta|_{\cancel{a}}^{2}$). If
$\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2}:=\rho^{-2}\mathfrak{q}^{\mathrm{b},N,2}$
then:

$\displaystyle\quad\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{b},N,2}$
$\displaystyle=\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\rho\left(a_{0}\xi_{k,l}^{2}+(b_{0}-4\mathrm{i}a_{0})\xi_{k,l}-2\mathrm{i}b_{0}-4a_{0}+c_{0}\right)\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2}$
$\displaystyle\qquad\qquad\qquad\quad\\!\\!+\rho^{2}\left(\breve{a}\xi_{k,l}^{2}+(\breve{b}-4\mathrm{i}\breve{a})\xi_{k,l}-2\mathrm{i}\breve{b}-4\breve{a}+\breve{c}\right)\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2}-\rho^{2}(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+5a)(\partial_{\rho}\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2})-a\rho^{3}(\partial_{\rho}^{2}\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2})$
$\displaystyle\qquad\qquad\qquad\quad\\!\\!+\rho^{2}\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2}-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2})-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}\widetilde{\widetilde{\mathfrak{q}}}^{\mathrm{b},N,2})\right)\\!\bigg{\\}}.$

The same computations as in (2.3) thus give

$\displaystyle\quad\boldsymbol{P}_{\sigma}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2}]$
$\displaystyle=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i+1}\in\mathfrak{N}_{(i+1)(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i+1}:}(\chi\phi_{\bullet})\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{1\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+1},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}})]$
$\displaystyle\quad+\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\begin{subarray}{c}\boldsymbol{\alpha}_{i+1}\in\mathfrak{N}_{(i+1)(n+1)}^{0}\\\
\alpha_{i+2}\in\mathfrak{N}_{n+1}\end{subarray}}\partial^{:\boldsymbol{\alpha}_{i+2}:}(\chi\phi_{\bullet})\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{0\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+2},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,2,\alpha_{1},\ldots,\alpha_{i}})]$

with $B_{\alpha_{1},\ldots,\alpha_{i+2},\beta,\gamma,\delta}$ denotes the
coefficients in the remainder in front of
$\partial^{:\boldsymbol{\alpha}_{i+2}:}(\chi\phi_{\bullet})(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,2,\alpha_{1},\ldots,\alpha_{i}})$
containing a term
$\rho^{\delta}\xi_{k,l}^{\beta^{\prime}}\eta_{1}^{\beta_{1}^{\prime\prime}}\ldots\eta_{n}^{\beta_{n}^{\prime\prime}}$
with $|\beta|=\beta^{\prime}+|\beta^{\prime\prime}|$. Since
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2}$ is of order $\rho^{2}$,
the remainder above is of order $\rho^{2}$.

For all $j\in{\mathbb{N}}\setminus\\{0\\}$, we define

(9) $\displaystyle\mathfrak{p}_{\sigma,k,l,j}(\xi)$
$\displaystyle:=a_{0}\xi_{k,l}^{2}+(b_{0}-2\mathrm{i}ja_{0})\xi_{k,l}-j(\mathrm{i}b_{0}+ja_{0})+c_{0}=\mathfrak{p}_{\sigma,k+j-1,l}(\xi)$

and we assume that $0\notin\mathfrak{p}_{\sigma,k,l,j}({\mathbb{R}})$. By
induction over $N^{\prime}\in{\mathbb{N}}\setminus\\{0,1,2\\}$, we show that

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}u$
$\displaystyle=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{p}_{\sigma,k,l}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,1,\alpha_{1},\ldots,\alpha_{i}}]v_{k,l,\bullet}+\boldsymbol{K}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}u$

where

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}u$
$\displaystyle=\sum_{j=1}^{N^{\prime}}\left(\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}]\right)v_{k,l,\bullet},\qquad\boldsymbol{K}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}u=\sum_{j=1}^{N^{\prime}}\left(\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}]v_{k,l,\bullet}\right)$

with

$\displaystyle\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$
$\displaystyle:=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\boldsymbol{\alpha}_{i+j-1}\in\mathfrak{N}_{(i+j-1)(n+1)}^{0}}\partial^{:\boldsymbol{\alpha}_{i+j-1}:}(\chi\phi_{\bullet})\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,j,\alpha_{1},\ldots,\alpha_{i+j-1}},$
$\displaystyle\mathfrak{q}_{\sigma,k,l}^{\mathrm{b},N,j,\alpha_{1},\ldots,\alpha_{i+j-1}}$
$\displaystyle:=-\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{1\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+j-1},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}^{\mathrm{b},N,j-1,\alpha_{1},\ldots,\alpha_{i+j-2}})]\frac{\rho}{\mathfrak{p}_{\sigma,k,l,j}}.$

and

$\displaystyle\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$
$\displaystyle=\sum_{i=\lfloor(N+1)/2\rfloor}^{N}\sum_{\begin{subarray}{c}\boldsymbol{\alpha}_{i+j-1}\in\mathfrak{N}_{(i+j)(n+1)}^{0}\\\
\alpha_{i+j}\in\mathfrak{N}_{n+1}\end{subarray}}\partial^{:\boldsymbol{\alpha}_{i+j}:}(\chi\phi_{\bullet})\sum_{\beta\in\mathfrak{N}_{n+1}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}\sum_{0\leq\delta\leq
2}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}B_{\alpha_{1},\ldots,\alpha_{i+j},\beta,\gamma,\delta}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{b},N,j,\alpha_{1},\ldots,\alpha_{i}})].$

Besides, $\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$ and
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$ are of order $j$ in
$\rho$ and of order $-N+2(j-1)$ in $|\eta|_{\cancel{a}}$. Indeed, we can check
that

$\displaystyle\quad\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}^{\mathrm{b},N,j}$
$\displaystyle=\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\bigg{\\{}\rho^{j-1}\left(a_{0}\xi_{k,l}^{2}+(b_{0}-2\mathrm{i}ja_{0})\xi_{k,l}-j(\mathrm{i}b_{0}-ja_{0})+c_{0}\right)(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j})$
$\displaystyle\qquad\qquad\qquad\quad\\!\\!+\rho^{j}\left(\breve{a}\xi_{k,l}^{2}+(\breve{b}-2\mathrm{i}j\breve{a})\xi_{k,l}-j(\mathrm{i}\breve{b}-j\breve{a})+\breve{c}\right)(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j})$
$\displaystyle\qquad\qquad\qquad\quad\\!\\!-\rho^{j}(2\mathrm{i}a\xi_{k,l}+\mathrm{i}b+(2j+1)a)(\partial_{\rho}(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j}))-a\rho^{j+1}(\partial_{\rho}^{2}(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j}))$
$\displaystyle\qquad\qquad\qquad\quad\\!\\!+\rho^{j}\sum_{j=1}^{n}\left((\cancel{a}_{j}\eta_{j}^{2}+\cancel{b}_{j}\eta_{j})(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j})-\mathrm{i}(2\cancel{a}_{j}\eta_{j}+\cancel{b}_{j})(\partial_{\omega_{j}}(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j}))-\cancel{a}_{j}(\partial_{\omega_{j}}^{2}(\rho^{-j}\mathfrak{q}^{\mathrm{b},N,j}))\right)\\!\bigg{\\}}.$

so that defining inductively
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$ for all $1\leq j\leq
N^{\prime}+1$ according to the above formula,
$\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,N^{\prime}+1}$ is of order
$\rho^{N^{\prime}+1}$; the remainder
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,N^{\prime}+1}$ is also of
order $\rho^{N^{\prime}+1}$ as the term of order $\rho^{N^{\prime}}$ in
$\boldsymbol{P}_{\sigma}\rho^{\mathrm{i}\xi+k+l}\mathrm{e}^{\mathrm{i}\omega\cdot\eta}\mathfrak{q}_{\sigma,k,l,\bullet}^{\mathrm{b},N,N^{\prime}+1}$
precisely cancels the one in
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,N^{\prime}}$. As for the
total order in $|\eta|_{\cancel{a}}$, each iteration increase it by a factor 2
(starting from $j=2$) as $\mathfrak{p}_{\sigma,k,l,j}(\xi)^{-1}$ do not
produce any decay in the $\eta$ directions of the fiber.

The same proof as for Lem. 17 proves the following result:

######  Lemma 17.

For all $N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$,
$\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l})$
provided that $N>2(N^{\prime}-1)$ and
$0\notin\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}})$.

###  2.4. Global boundary parametrices

We prove in this section Prop. 11.

We first define a global boundary right parametrix for
$\boldsymbol{P}_{\sigma}$. Let $N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$
such that $N>2(N^{\prime}-1)$ and assume that
$0\notin\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}})$.
Let

$\displaystyle\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$
$\displaystyle:=\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{f},N}+\boldsymbol{Q}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l}).$

The mapping property $\mathcal{H}^{k,l}\to\mathcal{X}^{k+1,l}$ of
$\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$ has been established in Lem. 15
and Lem. 17. The computations carried out in Sect. 2.2 and Sect. 2.3 show that

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}u$
$\displaystyle=\chi u+\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u$

where

(10)
$\displaystyle\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\boldsymbol{k}_{\sigma,k,l,\bullet}^{N,N^{\prime}}]v_{k,l,\bullet},\qquad\boldsymbol{k}_{\sigma,k,l,\bullet}^{N,N^{\prime}}=\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N,1}+\sum_{j=1}^{N^{\prime}}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}]$

with

$\displaystyle\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N,1}$
$\displaystyle=\sum_{i=\lfloor(N+1)/2\rfloor}^{N-1}\sum_{\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{1}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|=0\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N-1,\alpha_{1},\ldots,\alpha_{i-1}})$
$\displaystyle\qquad+\sum_{i=\lceil(N+1)/2\rceil}^{N}\sum_{\boldsymbol{\alpha}_{i}\in\mathfrak{N}_{i(n+1)}^{1}}\partial^{:\boldsymbol{\alpha}_{i}:}(\chi\phi_{\bullet})\sum_{\begin{subarray}{c}\beta\in{\mathbb{N}}^{n+1}\\\
|\beta|\leq
1\end{subarray}}\sum_{\gamma\in\mathfrak{N}_{2n+3}}A_{\alpha_{i},\beta,\gamma}(\partial^{\gamma}\widetilde{\mathfrak{q}}_{\sigma,k,l}^{\mathrm{f},N,\alpha_{1},\ldots,\alpha_{i-1}}).$

Above, $\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N,1}$ is nothing but
the terms in $\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N}$ from Sect.
2.2 containing derivatives of $\chi$ (hence supported far away from
$\\{0\\}_{\rho}$) and $\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}$
are from Sect. 2.3.

It remains to prove:

######  Lemma 18.

The operator
$\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{k,l})$
is compact and if $N\geq 3N^{\prime}-2$, then
$\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{H}^{k+N-2(N^{\prime}-1),l})$.

###### Proof.

The boundedness part of the statement is clear from Lem. 15 and Lem. 17 as
$\boldsymbol{k}_{\sigma,k,l,\bullet}^{\mathrm{f},N,N^{\prime}}$ is of order
$-(N-2(N^{\prime}-1))$ in $\xi$ and $|\eta|_{\cancel{a}}$ (the
$-2(N^{\prime}-1)$ term coming from the correction in the basis) and of order
$N^{\prime}$ in $\rho$: as we gain one derivative whenever one power of $\rho$
is associated to one power of
$\frac{\rho}{a\xi^{2}+\rho|\eta|_{\cancel{a}}^{2}}$, we can increase the
regularity of the target space provided that $N-2(N^{\prime}-1)\geq
N^{\prime}$, that is $N\geq 3N^{\prime}-2$.

To show compactness, let us pick a bounded sequence
$(u_{t})_{t\in{\mathbb{N}}}\subset\mathcal{H}^{k,l}$. Using the beginning of
the proof of Lem. 15, we can check that
${\mathbb{R}}\times\mathcal{M}\times{\mathbb{R}}\times{\mathbb{R}}^{n}\ni(x:=\ln\rho,\omega,\xi,\eta)\mapsto\boldsymbol{k}_{\sigma,k,l,\bullet}^{N,N^{\prime}}(\mathrm{e}^{x},\omega,\xi,\eta)$
is a decaying Fourier symbol:

$\displaystyle(|\xi|^{N+\alpha}+|\eta_{j}|^{N+\beta})|\partial_{\omega_{j}}^{\alpha}(\rho\partial_{\rho}^{\beta})\boldsymbol{k}_{\sigma,k,l,\bullet}^{N,N^{\prime}}|$
$\displaystyle=\mathcal{O}(\rho^{N^{\prime}}).$

It follows from [Z, Thm. 4.28] (using the standard quantization
$\mathrm{Op}_{0}$ as denoted therein) that we can extract a subsequence
$(u_{t^{\prime}})_{t^{\prime}\in{\mathbb{N}}}$ from
$(u_{t})_{t\in{\mathbb{N}}}$ such that999We first make
$\rho^{-l}\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u_{t^{\prime}}$ converge
in the appropriate unweighted $L^{2}$ space thanks to [Z, Thm. 4.28], then we
can multiply by $\rho^{l}$ to get convergence in $\mathcal{H}^{0,l}$.

$\displaystyle\lim_{t^{\prime}\to+\infty}\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u_{t^{\prime}}$
$\displaystyle=u_{\infty}\in\mathcal{H}^{0,l}.$

The proof of Lem. 15 also shows that
$\partial_{\rho}^{k_{1}}\partial_{\omega}^{k_{2}}\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u=\mathrm{Op}_{\mathcal{M},\mathcal{F}}[\breve{\boldsymbol{k}}_{\sigma,k,l,\bullet}^{N,N^{\prime}}]u$
with $k_{1}+|k_{2}|=k$, where
${\mathbb{R}}\times\mathcal{M}\times{\mathbb{R}}\times{\mathbb{R}}^{n}\ni(x:=\ln\rho,\omega,\xi,\eta)\mapsto\breve{\boldsymbol{k}}_{\sigma,k,l,\bullet}^{N,N^{\prime}}(\mathrm{e}^{x},\omega,\xi,\eta)$
is also a decaying Fourier symbol; as a consequence, [Z, Thm. 4.28] entails
that we can extract a subsequence
$(u_{t^{\prime\prime}})_{t^{\prime\prime}\in{\mathbb{N}}}$ from
$(u_{t^{\prime}})_{t^{\prime}\in{\mathbb{N}}}$ such that

$\displaystyle\lim_{t^{\prime\prime}\to+\infty}\partial_{\rho}^{k_{1}}\partial_{\omega}^{k_{2}}\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u_{t^{\prime}}$
$\displaystyle=w_{\infty}\in\mathcal{H}^{0,l}.$

But then,
$\rho^{-l}\partial_{\rho}^{k_{1}}\partial_{\omega}^{k_{2}}v_{\infty}=\rho^{-l}w_{\infty}$
in the distribution sense; as $\rho^{-l}w_{\infty}\in
L^{2}(X,\frac{\mathrm{d}\rho}{\rho}\mathrm{d}\omega)$, we discover that
$\rho^{-l}\partial_{\rho}^{k_{1}}\partial_{\omega}^{k_{2}}v_{\infty}\in
L^{2}(X,\frac{\mathrm{d}\rho}{\rho}\mathrm{d}\omega)$ as well, that is
$v_{\infty}\in\mathcal{H}^{k,l}$. This completes the proof. ∎

We now construct a global boundary left parametrix for
$\boldsymbol{P}_{\sigma}$. The construction is very similar to the one for the
right parametrix above; as we will not explicitly use the left parametrix in
the sequel, we will only give the main lines of its construction. Let again
$N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$ such that $N>2(N^{\prime}-1)$
and assume that
$0\notin\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}})$.
Then define

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}$
$\displaystyle:=\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{f},N}+\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l})$

so that

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}\boldsymbol{P}_{\sigma}$
$\displaystyle=\chi\mathds{1}_{\mathcal{X}^{k+1,l}}+\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}.$

Above $\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{f},N}$ first produces
decay in the fiber,
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}$ then
creates decay in the basis and
$\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}$ is compact.

To construct $\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{f},N}$, we
first let

$\displaystyle(\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{f},1}u)(\rho)$
$\displaystyle\quad:=\frac{1}{(2\pi)^{n+1}}\int_{{\mathbb{R}}_{\xi}}\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}F_{k,l}(\rho,\xi)\frac{\chi(\rho)\phi_{\bullet}(\omega)y}{a(y,\zeta,\sigma)\xi_{k,l}^{2}+y|\eta|_{\cancel{a}}^{2}}(y^{-l}\partial_{y}^{k}v_{k,l,\bullet})(y,\zeta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}\mathrm{d}\eta\mathrm{d}\xi$

where
$v_{k,l,\bullet}(\rho,\omega):=\widetilde{\phi}_{\bullet}(\omega)(\partial_{\rho}^{k}u)(\rho,\omega)$
(once again, $\mathcal{M}\cap\mathrm{Supp}(\widetilde{\phi}_{\bullet})$ is
identified as an open subset of ${\mathbb{R}}^{n}$ so that the Fourier
transform with respect to
$\zeta\in\mathcal{M}\cap\mathrm{Supp}(\widetilde{\phi}_{\bullet})$ is well
defined). Then:

$\displaystyle\quad\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}\frac{y^{-l}F_{k,l}(\rho,\xi)y}{a(y)\xi_{k,l}^{2}+y|\eta|_{\cancel{a}}^{2}}\left(\left(-\frac{a(y\partial_{y})^{2}}{y}-\sum_{j=1}^{n}\cancel{a}_{j}\partial_{\zeta_{j}}^{2}\right)\partial_{y}^{k}v_{k,l,\bullet}\right)\\!(y,\eta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}$
$\displaystyle=\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}\frac{y^{-l}F_{k,l}(\rho,\xi)}{a(y)\xi_{k,l}^{2}+y|\eta|_{\cancel{a}}^{2}}\left(\left(-a\partial_{y}y\partial_{y}-\sum_{j=1}^{n}\cancel{a}_{j}\partial_{\zeta_{j}}^{2}\right)\partial_{y}^{k}v_{k,l,\bullet}\right)\\!(y,\eta)\mathrm{d}\zeta\mathrm{d}y$
$\displaystyle=\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}\frac{y^{-l}F_{k,l}(\rho,\xi)}{a(y)\xi_{k,l}^{2}+y|\eta|_{\cancel{a}}^{2}}\left(\left(\frac{a\xi_{k,l}^{2}}{y}+|\eta|_{\cancel{a}}^{2}+\mathrm{subprincipal\
terms}\right)\partial_{y}^{k}v_{k,l,\bullet}\right)\\!(y,\eta)\mathrm{d}\zeta\mathrm{d}y$
$\displaystyle=\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}y^{-l}F_{k,l}(\rho,\xi)(\partial_{y}^{k}v_{k,l,\bullet})(y,\eta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}+\mathrm{remainder}.$

Above, the subprincipal terms come from commutators (we write
$-a\partial_{y}y\partial_{y}-\sum_{j=1}^{n}\cancel{a}_{j}\partial_{\zeta_{j}}^{2}=-\partial_{y}y\partial_{y}a-[a,\partial_{y}]y\partial_{y}-\partial_{y}y[a,\partial_{y}]-\sum_{j=1}^{n}(\cancel{a}_{j}\partial_{\zeta_{j}}^{2}+[\cancel{a}_{j},\partial_{\zeta_{j}}^{2}])$
and so on) and the remainder term is the integral of terms of order $1$ in $y$
and $-1$ in $\xi$ and $|\eta|_{\cancel{a}}$. Using Lem. 14, we obtain

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{f},1}\boldsymbol{P}_{\sigma}$
$\displaystyle=\chi\mathds{1}_{\mathcal{X}^{k+1,l}}+\widetilde{\boldsymbol{K}}_{\sigma,k,l,\bullet}^{\mathrm{f},1}$

where $\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{\mathrm{f},1}$ has the same
properties as the remainder term in the last line above. That
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{f},1}\in\mathcal{B}(\mathcal{H}^{k,l},\mathcal{X}^{k+1,l})$
then follows using the same arguments as Lem. 15. We then inductively
construct
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{f},2},\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{f},3},\ldots$
to decrease the total order in $\xi$ and $|\eta|_{\cancel{a}}$ of the
remainder, then we define

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{f},N}$
$\displaystyle:=\sum_{j=1}^{N}\sum_{\bullet}\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{f},j}.$

To construct $\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{b},N}$, we
first let

$\displaystyle(\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{b},1}u)(\rho)$
$\displaystyle\quad:=\frac{1}{(2\pi)^{n+1}}\int_{{\mathbb{R}}_{\xi}}\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}F_{k,l}(\rho,\xi)\frac{\chi(\rho)\phi_{\bullet}(\omega)y\widetilde{\boldsymbol{k}}_{\sigma,k,l,\bullet}(y,\zeta)}{\mathfrak{p}_{\sigma,k,l}^{1}(\xi)}(y^{-l}\partial_{y}^{k}v_{k,l,\bullet})(y,\zeta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}\mathrm{d}\eta\mathrm{d}\xi$

where is defined in (7) and
$\widetilde{\boldsymbol{k}}^{\mathrm{f},N,1}_{\sigma,k,l,\bullet}(y,\zeta)$ is
such that

$\displaystyle\sum_{\bullet}\chi\phi_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[F_{k,l}](\widetilde{\boldsymbol{k}}^{\mathrm{f},N,1}_{\sigma,k,l,\bullet}v_{k,l,\bullet})$
$\displaystyle=-\widetilde{\boldsymbol{K}}^{\mathrm{f},N}_{\sigma,k,l}u$

As we can write

(11) $\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle=\frac{-a_{0}(\rho\partial_{\rho})^{2}+(b_{0}-2\mathrm{i}a_{0})(\rho\partial_{\rho})-\mathrm{i}b_{0}-a_{0}+c_{0}}{\rho}+\mathcal{O}(\partial_{\rho},\partial_{\rho}^{2},\rho\partial_{\omega_{1}},\rho\partial_{\omega_{1}}^{2},\ldots,\rho\partial_{\omega_{n}},\rho\partial_{\omega_{n}}^{2}),$

integrations by parts yield

$\displaystyle\sum_{\bullet}\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{b},1}\boldsymbol{P}_{\sigma}u$
$\displaystyle=-\widetilde{\boldsymbol{K}}^{\mathrm{f},N}_{\sigma,k,l}u+\mathrm{remainder}$

with a remainder having as decay in $\xi$ and $|\eta|_{\cancel{a}}$ as the
$\widetilde{\boldsymbol{k}}_{\sigma,k,l,\bullet}^{\mathrm{f},N,1}$ but one
more power of $y$. We then inductively construct
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{b},N,2},\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{b},N,3},\ldots$
to increase the total order in $y$ of the remainder using the polynomials
$\mathfrak{p}_{\sigma,k,l,j}(\xi)$ defined in (9); note that each iteration
now increases the total order of $|\eta|_{\cancel{a}}$ by 2 since we have to
cancel the remainder in (11) that does not simplify with the terms
$\mathfrak{p}_{\sigma,k,l,j}(\xi)$. Then we define

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{\mathrm{b},N,N^{\prime}}$
$\displaystyle:=\sum_{j=1}^{N^{\prime}}\sum_{\bullet}\widetilde{\boldsymbol{Q}}_{\sigma,k,l,\bullet}^{\mathrm{b},N,j}.$

We have then:

$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}\boldsymbol{P}_{\sigma}$
$\displaystyle=\chi\mathds{1}_{\mathcal{X}^{k+1,l}}+\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}.$

Compactness of
$\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{X}^{k+1,l})$
then follows again from [Z, Thm. 4.28] (using the quantization
$\mathrm{Op}_{1}$ therein) with the arguments of Lem. 18 above. Furthermore,
using the well known property of the Mellin transform $\mathcal{M}[y\mapsto
y^{N^{\prime}}U(y)](\xi)=\mathcal{M}[U](\xi+\mathrm{i}N^{\prime})$, the extra
term $y^{N^{\prime}}$ from the basis correction can be transformed into
$\rho^{N^{\prime}}$ upon shifting the contour intgral in $\xi$ from
${\mathbb{R}}$ to ${\mathbb{R}}-\mathrm{i}N^{\prime}$, ensuring the Sobolev
regularity $\mathcal{H}^{k+1+N-2(N^{\prime}-1),l}$; this amounts to changing
$\xi_{k,l}$ to $\xi_{k+N^{\prime},l}$, whence the extra assumption
$0\notin\bigcup_{0\leq\alpha\leq
N^{\prime}}\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k+j-1,l}({\mathbb{R}}-\mathrm{i}\alpha)$.
As a result,
$\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k,l}\in\mathcal{B}(\mathcal{X}^{k+1+N-2(N^{\prime}-1),l})$.

Finally, the regularity in $\sigma$ statement for both the right and left
parametrices is a mere consequence of the parametrix construction which uses
rational functions of the coefficients of $\boldsymbol{P}_{\sigma}$ (the
denominators never cancel far away from the critical strips).

###  2.5. Local index 0 Fredholm property

We finally prove Prop. 13.

For the sake of definiteness, we assume that $a(\rho,\omega,\sigma)>0$ and
$\cancel{a}_{j}(\rho,\omega,\sigma)>0$ on
$\overline{X^{\prime}}\times\mathcal{M}$. Let $C>0$ and
$N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$ such that $N>2(N^{\prime}-1)$ as
in Prop 11. We define $\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$ and
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}$ with
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$ replaced by
$\frac{\rho}{a\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}+\rho C}$. We claim
that
$\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{\prime
k,l},\mathcal{X}^{\prime k+1,l})$ and
$\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{\prime
k,l},\mathcal{X}^{\prime k+1,l})$ are invertible for $C$ sufficiently large.
To see this, we use the above computations to obtain

(12) $\displaystyle(\boldsymbol{P}_{\sigma}+\rho
C)\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$
$\displaystyle=\mathds{1}_{\mathcal{H}^{\prime
k,l}}+\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}},$ (13)
$\displaystyle\widetilde{\boldsymbol{Q}}_{\sigma,k,l}^{N,N^{\prime}}(\boldsymbol{P}_{\sigma}+\rho
C)$ $\displaystyle=\mathds{1}_{\mathcal{X}^{\prime
k+1,l}}+\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}$

for some
$\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{H}^{\prime
k,l})$ and
$\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}\in\mathcal{B}(\mathcal{X}^{\prime
k+1,l})$ with operator norms of order $o_{C\to+\infty}(1)$. Indeed, let us we
write
$\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}u=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[\boldsymbol{k}_{\sigma,k,l,\bullet}]v_{k,l,\bullet}$
with
$v_{k,l,\bullet}(\rho,\omega):=\chi(y)\widetilde{\phi}_{\bullet}(\omega)(\rho^{-l}\partial_{\rho}^{k}u)(\rho,\omega)$
(recall that $\chi_{|X^{\prime}}=\mathds{1}_{X^{\prime}}$; here we implicitly
use a Sobolev extension of $u$ to $\mathrm{Supp}(\chi)$ so that we can
consider $u$ as a function defined on $(0,+\infty)$ vanishing on a
neighborhood of $+\infty$).

On the one hand,

$\displaystyle\quad\bigg{|}\int_{{\mathbb{R}}_{\xi}}\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\mathds{1}_{\rho/y\leq
1}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)v_{k,l,\bullet}(y,\zeta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}\mathrm{d}\eta\mathrm{d}\xi\bigg{|}$
$\displaystyle\leq\int_{{\mathbb{R}}_{\xi}}\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\mathds{1}_{\rho/y\leq
1}\left|\frac{\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)}{\rho}\right||v_{k,l,\bullet}(y,\zeta)|\mathrm{d}\zeta\frac{\rho}{y}\mathrm{d}y\mathrm{d}\eta\mathrm{d}\xi$
$\displaystyle\leq\int_{{\mathbb{R}}_{\xi}}\int_{{\mathbb{R}}^{n}_{\eta}}\sup_{\rho/y\leq
1/2}\left|\frac{\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)}{\rho}\right|\mathrm{d}\eta\mathrm{d}\xi\sqrt{\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}|v_{k,l,\bullet}(y,\zeta)|^{2}\mathrm{d}\zeta\frac{\mathrm{d}y}{y}}\sqrt{\int_{\mathrm{Supp}(\chi)}y\mathrm{d}y}$

by Cauchy-Schwartz inequality; since
$\boldsymbol{k}_{\sigma,k,l,\bullet}/\rho$ is uniformly bounded in
$(\rho,\omega)\in[0,+\infty)\times\mathcal{M}$ and analytic in $\xi$ and in
$\eta$, we can use the residue theorem (choosing $N$ large enough for
integrability) to see that the integral over $X^{\prime}\times\mathcal{M}$ of
the square of the left hand side above is bounded multiple of
$\|u\|_{\mathcal{H}^{\prime k,l}}$ of order $o_{C\to+\infty}(1)$ (integrals
with respect to $\rho$ and $\omega$ are harmless for the estimate as we
integrate over bounded domains).

On the other hand, when $\rho/y>1$, we can use the residue theorem for the
integral over ${\mathbb{R}}_{\xi}$ and sum up over all the residues of
$\xi\mapsto\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)$
in the upper half plane ${\mathbb{C}}^{+}$ (since
$\left|\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\right|$ decays
exponentially fast as a function of $\xi$ with $\mathrm{Im}(\xi)>0$ when
$\rho/y>1$); the residues comes from the falling factorial term
$(\mathrm{i}\xi+k+l)_{(k)}$ in the denominator of $F_{k,l}$ (whose the nearest
pole to ${\mathbb{R}}$ is $\xi=\mathrm{i}/2$), from the denominator
$\frac{\rho}{a\xi_{k,l}^{2}+\rho|\eta|_{\cancel{a}}^{2}+\rho C}$ and from the
indicial polynomials (both yielding poles with imaginary part greater or equal
to $\mathrm{i}(k+l)$). Since $k+l>0$ by assumption, we can factor out of the
sum of the residues a term $y^{\epsilon}$ with $\epsilon>0$:

$\displaystyle\quad\bigg{|}\int_{{\mathbb{R}}_{\xi}}\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\mathds{1}_{\rho/y>1}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\mathrm{e}^{\mathrm{i}(\omega-\zeta)\cdot\eta}\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)v_{k,l,\bullet}(y,\zeta)\mathrm{d}\zeta\frac{\mathrm{d}y}{y}\mathrm{d}\eta\mathrm{d}\xi\bigg{|}$
$\displaystyle\leq\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\eta}}\int_{{\mathbb{R}}^{n}_{\zeta}}\mathds{1}_{\rho/y>1}\sum\left|\underset{\,{\mathbb{C}}^{+}}{\mathrm{Res}}\left(\xi\mapsto\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\frac{\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)}{y^{\epsilon}}\right)\right||v_{k,l,\bullet}(y,\zeta)|\mathrm{d}\zeta\frac{y^{\epsilon}}{y}\mathrm{d}y\mathrm{d}\eta$
$\displaystyle\leq\int_{{\mathbb{R}}^{n}_{\eta}}\sup_{\rho/y>1}\sum\left|\underset{\,{\mathbb{C}}^{+}}{\mathrm{Res}}\left(\xi\mapsto\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi}\frac{\boldsymbol{k}_{\sigma,k,l,\bullet}(\rho,y,\omega,\zeta,\xi,\eta)}{y^{\epsilon}}\right)\right|\mathrm{d}\eta\sqrt{\int_{(0,+\infty)_{y}}\int_{{\mathbb{R}}^{n}_{\zeta}}|v_{k,l,\bullet}(y,\zeta)|^{2}\mathrm{d}\zeta\frac{\mathrm{d}y}{y}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\sqrt{\int_{0}^{\rho}\frac{y^{2\epsilon}}{y}\mathrm{d}y}.$

It remains to integrate over $X^{\prime}\times\mathcal{M}$ (putting
$y^{\epsilon}$ into the denominator of $\boldsymbol{k}_{\sigma,k,l,\bullet}$
takes away the term $\rho^{-\epsilon}$ from
$\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi_{*}}=\rho^{-\epsilon}\left(\frac{\rho}{y}\right)^{\mathrm{i}\xi_{*}-\epsilon}y^{\epsilon}$
where $\xi_{*}\in{\mathbb{C}}^{+}$ is a pole; since the integral over
$(0,\rho)$ above is equal to $\rho^{2\epsilon}/(2\epsilon)$, integrability
near 0 is ensured). As above, the denominator of
$\boldsymbol{k}_{\sigma,k,l,\bullet}$ makes that we get a multiple of
$\|u\|_{\mathcal{H}^{\prime k,l}}$ of order $o_{C\to+\infty}(1)$.

In the same spirit, we can establish the same estimate for the $\mathcal{H}$
norm of derivatives of $\boldsymbol{K}_{\sigma,k,l}u$, concluding that
$\|\boldsymbol{K}_{\sigma,k,l}\|_{\mathcal{B}(\mathcal{H}^{\prime
k,l})}=o_{C\to+\infty}(1)$. Moreover, if
$\widetilde{\boldsymbol{K}}_{\sigma,k,l}^{N,N^{\prime}}u=\sum_{\bullet}\mathrm{Op}_{\mathcal{M},\mathcal{F}}[\widetilde{\boldsymbol{k}}_{\sigma,k,l,\bullet}]v_{k,l,\bullet}$
then $\widetilde{\boldsymbol{k}}_{\sigma,k,l,\bullet}/y$ is uniformly bounded
so that we can multiply by $y$ in the integral with respect to $y$, removing
once again the troublesome $\frac{1}{y}$ term before using Cauchy-Schwartz
inequality (this time we do not need to integrate by parts using
$\partial_{\xi}$). The conclusion is
$\|\widetilde{\boldsymbol{K}}_{\sigma,k,l}\|_{\mathcal{B}(\mathcal{X}^{\prime
k+1,l})}=o_{C\to+\infty}(1)$ as well.

Now we choose $C$ lare enough (depending on $|\sigma|$, $k$ and $|l|$) so that
the right hand sides in (12) and (13) are invertible: this proves that
$\boldsymbol{P}_{\sigma}+C\in\mathcal{B}(\mathcal{X}^{\prime
k+1,l},\mathcal{H}^{\prime k,l})$ is invertible. Back into (12), we get the
invertibility of
$\boldsymbol{Q}_{\sigma,k,l}\in\mathcal{B}(\mathcal{H}^{\prime
k,l},\mathcal{X}^{\prime k+1,l})$.

Finally, Sect. 2.4 tells us that

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$
$\displaystyle=\mathds{1}_{\mathcal{H}^{\prime
k,l}}+\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}$

with
$\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}\equiv\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}(C)$
still defined with the aforementioned modification of
$\mathfrak{q}_{\sigma,k,l}^{\mathrm{f},1}$ and
$\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}\equiv\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}(C)$
compact on $\mathcal{H}^{\prime k,l}$ (but not necessarily invertible as
$C\to+\infty$). Since both $\boldsymbol{Q}_{\sigma,k,l}^{N,N^{\prime}}$ and
$\mathds{1}_{\mathcal{H}^{\prime
k,l}}+\boldsymbol{K}_{\sigma,k,l}^{N,N^{\prime}}$ are index 0 Fredholm
operators, so is $\boldsymbol{P}_{\sigma}$. The proof is complete.

##  3\. The charged KG operator near cosmological accelerating and rotating
charged black holes

We apply in this section the results obtained in Sect. 2 to charged KG scalar
fields near cosmological accelerating and rotating charged black holes.

###  3.1. The metric in Boyer-Lindquist coordinates

We follow _cf._ [PG] and [H1, Sect. 3.2 & 3.3]. Let $M,\Lambda>0$,
$a,Q,\alpha\in{\mathbb{R}}$ and define the horizon function:

$\displaystyle\mu(r)$ $\displaystyle:=(r^{2}+a^{2})\left(1-\frac{\Lambda
r^{2}}{3}\right)+\left(-2Mr+Q^{2}\left(1+\frac{\Lambda
a^{2}}{3}\right)^{2}\right)(1-\alpha^{2}r^{2}).$

On
$\mathbb{R}_{t}\times(r_{-},r_{+})_{r}\times\mathbb{S}^{2}_{\theta,\varphi}$
with $r_{\pm}>0$ to be fixed below, we define the DSKN metric in Boyer-
Lindquist coordinates

(14) $\displaystyle g$
$\displaystyle=\frac{1}{\Omega^{2}}\left(\frac{\mu}{(1+\lambda)^{2}\rho^{2}}\big{(}\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\varphi\big{)}^{2}-\frac{\kappa\sin^{2}\theta}{(1+\lambda)^{2}\rho^{2}}\big{(}a\mathrm{d}t-(r^{2}+a^{2})\mathrm{d}\varphi\big{)}^{2}-\rho^{2}\left(\frac{\mathrm{d}r^{2}}{\mu}+\frac{\mathrm{d}\theta^{2}}{\kappa}\right)\right)$

where

$\displaystyle\Omega$ $\displaystyle=1-\alpha
r\cos\theta,\qquad\quad\lambda=\frac{\Lambda
a^{2}}{3},\qquad\quad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle\kappa$ $\displaystyle=1-2\alpha
M\cos\theta+\left(\alpha^{2}(a^{2}+Q^{2}(1+\lambda)^{2})+\lambda\right)\cos^{2}\theta.$

(Notice that we use the coordinates $\bar{t}$ and $\bar{\varphi}$ in [PG, eq.
(4)] while $e=Q(1+\lambda)$ and $g=0$.) If

$\displaystyle A$
$\displaystyle:=-\frac{Qr}{\rho^{2}}(\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\varphi),$

then $(g,A)$ solves the coupled Einstein-Maxwell equation with electromagnetic
potential $\mathrm{d}A$. The volume form $\mathrm{dVol}_{g}$ associated to $g$
is given by:

$\displaystyle\mathrm{dVol}_{g}$
$\displaystyle=\frac{1}{\Omega^{4}}\frac{\rho^{2}\sin\theta}{(1+\lambda)^{2}}\mathrm{d}t\mathrm{d}r\mathrm{d}\theta\mathrm{d}\varphi.$

The coefficients of the inverse metric $g^{-1}$ are:

$\displaystyle g^{tt}$ $\displaystyle=-\Omega^{2}\frac{(1+\lambda)^{2}(\mu
a^{2}\sin^{2}\theta-(r^{2}+a^{2})^{2}\kappa)}{\mu\kappa\rho^{2}},\qquad
g^{t\varphi}=-\Omega^{2}\frac{a(1+\lambda)^{2}(\mu-(r^{2}+a^{2})\kappa)}{\mu\kappa\rho^{2}},$
$\displaystyle g^{\varphi\varphi}$
$\displaystyle=-\Omega^{2}\frac{(1+\lambda)^{2}(\mu-a^{2}\kappa\sin^{2}\theta)}{\mu\kappa\rho^{2}\sin^{2}\theta},\qquad
g^{rr}=-\Omega^{2}\frac{\mu}{\rho^{2}},\qquad
g^{\theta\theta}=-\Omega^{2}\frac{\kappa}{\rho^{2}}.$

This metric describes the exterior of a cosmological, accelerating and
rotating charged black hole. $M$ is the mass of the black hole, $Q$ is its
electric charge, $a$ is its angular velocity, $\alpha$ its acceleration and
$\Lambda$ is the cosmological constant. The polar coordinates
$(\theta,\varphi)$ are chosen such that the rotation of the black hole is
azimuthal. When $\alpha=0$, $g$ boils down to the De Sitter-Kerr-Newman
metric.

######  Assumptions 19.

  1. 1.

We assume the there exist real numbers $0<r_{-}<r_{+}$ such that $\mu$ is
positive in $(r_{-},r_{+})$ and $\mu(r_{\pm})=0$.

  2. 2.

We assume that $\Omega,\kappa>0$ on
$[r_{-},r_{+}]_{r}\times\mathbb{S}^{2}_{\theta,\varphi}$.

######  Remark 20.

When $a=\alpha=0$, [Mo, eq. (5)] or equivalently [H1, eq. (3.7)] gives
necessary and sufficient conditions on $M,\Lambda$ and $|Q|$ for $\mu$ to have
four simple real roots $r_{n}<0<r_{c}<r_{-}<r_{+}$ and to be positive in
$(r_{-},r_{+})$; the implicit function theorem ensures that $\mu$ still has
four simple real roots, still denoted by $r_{n}<0<r_{c}<r_{-}<r_{+}$, when
$a,\alpha$ are small enough, and therefore Assumption 1 is verified in this
setting.

As for $\kappa$, we can check that $\kappa=0$ if and only if

$\displaystyle\cos\theta$ $\displaystyle=\frac{\alpha
M}{\alpha^{2}(a^{2}+Q^{2}(1+\lambda)^{2})+\lambda}\left(1\pm\sqrt{1-\frac{\alpha^{2}(a^{2}+Q^{2}(1+\lambda)^{2})+\lambda}{\alpha^{2}M^{2}}}\right)$

when $\alpha\neq 0$. Consequently, Assumption 2 is verified when $\alpha$ is
small enough with respect to $r_{-}$ (in particular, it is sufficient to
assume that $|\alpha|\ll 1/2M$ in the limits $M\to+\infty$ and $\Lambda,a,Q\to
0$, as then $r_{-}\to 2M$).

######  Remark 21.

It will be useful to notice that
$\kappa\equiv\kappa(\cos\theta)=\kappa(0)+\widetilde{\kappa}_{N}\sin^{2}\theta$
where

$\displaystyle\widetilde{\kappa}_{N}(\theta)$
$\displaystyle:=\frac{\kappa(\theta)-\kappa(0)}{\sin^{2}\theta}=\frac{2\alpha
M(1-\cos\theta)}{\sin^{2}\theta}-\left(\alpha^{2}(a^{2}+Q^{2}(1+\lambda)^{2})+\lambda\right)$

is smooth on a neighborhood of the north pole $\\{\theta=0\\}$. We similarly
define

$\displaystyle\widetilde{\kappa}_{S}(\theta)$
$\displaystyle:=\frac{\kappa(\theta)-\kappa(\pi)}{\sin^{2}\theta}=-\frac{2\alpha
M(1+\cos\theta)}{\sin^{2}\theta}-\left(\alpha^{2}(a^{2}+Q^{2}(1+\lambda)^{2})+\lambda\right)$

near the south pole $\\{\theta=\pi\\}$.

The set
${\mathbb{R}}_{t}\times\\{r_{-}\\}_{r}\times\mathbb{S}^{2}_{\theta,\varphi}$
is the event (or black hole) horizon,
${\mathbb{R}}_{t}\times\\{r_{+}\\}_{r}\times\mathbb{S}^{2}_{\theta,\varphi}$
is the cosmological horizon ($r_{+}$ thus being the radius of the observable
universe – this model describes no expansion of the universe); in the case
where $r_{n,a}$ and $r_{c,a}$ also exist,
${\mathbb{R}}_{t}\times\\{r_{c,a}\\}_{r}\times\mathbb{S}^{2}_{\theta,\varphi}$
is the Cauchy (or inner) horizon while the negative root $r_{n,a}$ is
associated to no physical set.

###  3.2. The metric in $*$-coordinates

The expression (14) degenerates at $\\{r_{-}\\}$, $\\{r_{+}\\}$ as well as at
the poles of $\mathbb{S}^{2}$ due to coordinate singularities there.

We first smoothen the metric coefficients near the horizons $r_{\pm}$
following [HV, Sect. 3.2]. We introduce the new coordinates $t_{*}:=t-T(r)$
and $\varphi_{*}:=\varphi-\Phi(r)$ where $T,\Phi$ satisfy near $r_{\pm}$:

$\displaystyle T^{\prime}(r)$
$\displaystyle:=\pm\left(\frac{(1+\lambda)(r^{2}+a^{2})}{\mu(r)}+c_{\pm}(r)\right),\qquad\Phi^{\prime}(r):=\pm\left(\frac{a(1+\lambda)}{\mu(r)}+\widetilde{c}_{\pm}(r)\right).$

To explicit the functions $c_{\pm}$ and $\widetilde{c}_{\pm}$, we first
introduce

$\displaystyle\mathfrak{c}^{-2}$ $\displaystyle:=\max\\{\mu_{0}(r)\ |\
r\in(r_{-,0},r_{+,0})\\}=:\mu_{0}(\mathfrak{r}),\qquad\nu(r):=\begin{cases}\sqrt{1-\mu_{0}(r)\mathfrak{c}^{2}}&\text{if
$r\leq\mathfrak{r}$,}\\\ -\sqrt{1-\mu_{0}(r)\mathfrak{c}^{2}}&\text{if
$r\geq\mathfrak{r}$}.\end{cases}$

Observe that $\nu$ is smooth in a neighborhood of $[r_{-,0},r_{+,0}]$. Then
taking $\varpi\in\mathcal{C}^{\infty}({\mathbb{R}},[0,1])$ such that
$\varpi(r)=1$ if $r\leq r_{1}$ and $\varpi(r)=0$ if $r\geq r_{2}$ for some
$r_{-}<r_{1}<r_{2}<r_{+}$, we let:

$\displaystyle c_{-}(r)$ $\displaystyle:=\frac{-1\mp\nu(r)}{\mu(r)},\qquad
c_{+}(r):=-\left(\frac{2(1+\lambda)(r^{2}+a^{2})}{\mu(r)}+\frac{-1\mp\nu(r)}{\mu(r)}\right)\varpi(r)+\nu(r)(1-\varpi(r)),$
$\displaystyle\widetilde{c}_{-}(r)$
$\displaystyle:=0,\qquad\widetilde{c}_{+}(r):=-\frac{2a(1+\lambda)}{\mu(r)}\varpi(r).$

This ensures that $T,\Phi\in\mathcal{C}^{\infty}((r_{-},r_{+}),{\mathbb{R}})$
are well-defined (up to an additive constant) and the expressions of
$T^{\prime}$ and $\Phi^{\prime}$ coincide in $[r_{1},r_{2}]$. In the
$(t_{*},r,\theta,\varphi_{*})$ coordinates with $\theta\in(0,\pi)$, we have
$\mathrm{dVol}_{g}=\frac{\rho^{2}\sin\theta}{\Omega^{4}(1+\lambda)^{2}}\mathrm{d}t_{*}\mathrm{d}r\mathrm{d}\theta\mathrm{d}\varphi_{*}$
and the new coefficients of the inverse metric read (’$+$’ sign for $r>r_{1}$,
’$-$’ sign for $r<r_{2}$, _cf._ [HV, eq. (3.17)]):

$\displaystyle\Omega^{-2}\rho^{2}g^{t_{*}t_{*}}$ $\displaystyle=-\mu
c_{\pm}^{2}-2(1+\lambda)(r^{2}+a^{2})c_{\pm}-\frac{(1+\lambda)^{2}a^{2}\sin^{2}\theta}{\kappa},$
$\displaystyle\Omega^{-2}\rho^{2}g^{t_{*}r}$ $\displaystyle=\pm\mu
c_{\pm}\pm(1+\lambda)(r^{2}+a^{2}),$
$\displaystyle\Omega^{-2}\rho^{2}g^{t_{*}\varphi_{*}}$ $\displaystyle=-\mu
c_{\pm}\widetilde{c}_{\pm}-a(1+\lambda)c_{\pm}-(1+\lambda)(r^{2}+a^{2})\widetilde{c}_{\pm}-\frac{a(1+\lambda)^{2}}{\kappa},$
$\displaystyle\Omega^{-2}\rho^{2}g^{r\varphi_{*}}$
$\displaystyle=\pm\mu\widetilde{c}_{\pm}\pm a(1+\lambda),$
$\displaystyle\Omega^{-2}\rho^{2}g^{\varphi_{*}\varphi_{*}}$
$\displaystyle=-\mu\widetilde{c}_{\pm}^{2}-2a(1+\lambda)\widetilde{c}_{\pm}-\frac{(1+\lambda)^{2}}{\kappa\sin^{2}\theta}.$

In order to remove the singularities at the poles of $\mathbb{S}^{2}$, we
set101010When $\alpha=0$, no change of variable is needed as it is explained
below eq. (3.17) in [HV].

$\displaystyle\bar{\varphi}_{*,N}$
$\displaystyle:=\frac{\kappa(0)}{1+\lambda}\varphi_{*}\quad\text{near the
north
pole},\qquad\bar{\varphi}_{*,S}:=\frac{\kappa(\pi)}{1+\lambda}\varphi_{*}\quad\text{near
the south pole},$

then

$\displaystyle x_{*,\bullet}$
$\displaystyle:=\sin\theta\cos\bar{\varphi}_{*,\bullet},\qquad
y_{*,\bullet}:=\sin\theta\sin\bar{\varphi}_{*,\bullet},\qquad\bullet\in\\{N,S\\}$

near the appropriate pole. We henceforth work near the north pole as
computations are analogous near the south pole (we essentially replace
$\kappa(0)$ by $\kappa(\pi)$); we omit the subscript $N$ to lighten notations.
We have

$\displaystyle\begin{pmatrix}\mathrm{d}x_{*}\\\\[2.84526pt]
\mathrm{d}y_{*}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\theta\cos\bar{\varphi}_{*}&-y_{*}\\\\[2.84526pt]
\cos\theta\sin\bar{\varphi}_{*}&x_{*}\end{pmatrix}\begin{pmatrix}\mathrm{d}\theta\\\\[2.84526pt]
\mathrm{d}\bar{\varphi}_{*}\end{pmatrix},\qquad\begin{pmatrix}\mathrm{d}\theta\\\\[2.84526pt]
\mathrm{d}\bar{\varphi}_{*}\end{pmatrix}=\begin{pmatrix}\frac{\cos\bar{\varphi}_{*}}{\cos\theta}&\frac{\sin\bar{\varphi}_{*}}{\cos\theta}\\\\[2.84526pt]
-\frac{\sin\bar{\varphi}_{*}}{\sin\theta}&\frac{\cos\bar{\varphi}_{*}}{\sin\theta}\end{pmatrix}\begin{pmatrix}\mathrm{d}x_{*}\\\\[2.84526pt]
\mathrm{d}y_{*}\end{pmatrix}$

and $\mathrm{dVol}_{g}=\frac{\rho
2}{\Omega^{4}\kappa(0)(1+\lambda)\cos\theta}\mathrm{d}t_{*}\mathrm{d}r\mathrm{d}x_{*}\mathrm{d}y_{*}$.
We compute (using Rmk. 21) the new inverse metric coefficients:

$\displaystyle\Omega^{-2}\rho^{2}g^{t_{*}x_{*}}$ $\displaystyle=\left(\mu
c_{\pm}\widetilde{c}_{\pm}+a(1+\lambda)c_{\pm}+(1+\lambda)(r^{2}+a^{2})\widetilde{c}_{\pm}+\frac{a(1+\lambda)^{2}}{\kappa}\right)\frac{\kappa(0)}{1+\lambda}y_{*},$
$\displaystyle\Omega^{-2}\rho^{2}g^{t_{*}y_{*}}$ $\displaystyle=-\left(\mu
c_{\pm}\widetilde{c}_{\pm}+a(1+\lambda)c_{\pm}+(1+\lambda)(r^{2}+a^{2})\widetilde{c}_{\pm}+\frac{a(1+\lambda)^{2}}{\kappa}\right)\frac{\kappa(0)}{1+\lambda}x_{*},$
$\displaystyle\Omega^{-2}\rho^{2}g^{rx_{*}}$
$\displaystyle=\mp\left(\mu\widetilde{c}_{\pm}+a(1+\lambda)\right)\frac{\kappa(0)}{1+\lambda}y_{*},$
$\displaystyle\Omega^{-2}\rho^{2}g^{ry_{*}}$
$\displaystyle=\pm\left(\mu\widetilde{c}_{\pm}+a(1+\lambda)\right)\frac{\kappa(0)}{1+\lambda}x_{*},$
$\displaystyle\Omega^{-2}\rho^{2}g^{x_{*}x_{*}}$
$\displaystyle=-\frac{\kappa(0)^{2}}{\kappa}+\frac{\kappa^{2}-(\kappa+\kappa(0))\widetilde{\kappa}_{N}}{\kappa}x_{*}^{2}-\frac{\kappa(0)^{2}(\mu\widetilde{c}_{\pm}^{2}+a(1+\lambda)\widetilde{c}_{\pm})}{(1+\lambda)^{2}}y_{*}^{2},$
$\displaystyle\Omega^{-2}\rho^{2}g^{y_{*}y_{*}}$
$\displaystyle=-\frac{\kappa(0)^{2}}{\kappa}+\frac{\kappa^{2}-(\kappa+\kappa(0))\widetilde{\kappa}_{N}}{\kappa}y_{*}^{2}-\frac{\kappa(0)^{2}(\mu\widetilde{c}_{\pm}^{2}+a(1+\lambda)\widetilde{c}_{\pm})}{(1+\lambda)^{2}}x_{*}^{2},$
$\displaystyle\Omega^{-2}\rho^{2}g^{x_{*}y_{*}}$
$\displaystyle=\frac{\kappa^{2}-(\kappa+\kappa(0))\widetilde{\kappa}_{N}}{\kappa}x_{*}y_{*}+\frac{\kappa(0)^{2}(\mu\widetilde{c}_{\pm}^{2}+a(1+\lambda)\widetilde{c}_{\pm})}{(1+\lambda)^{2}}x_{*}y_{*}.$

Since $\cos\theta=\sqrt{1-x_{*}^{2}-y_{*}^{2}}$ is smooth near
$\\{x_{*}=y_{*}=0\\}$, $g$ is indeed smooth and non degenerate on
${\mathbb{R}}\times(r_{-},r_{+})\times\mathbb{S}^{2}$.

###  3.3. Spectral family

Fix $\ell\in{\mathbb{Z}}$ and $k\in{\mathbb{N}}\setminus\\{0\\}$. Let
$L^{2}:=L^{2}((r_{-},r_{+})\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)$,
$H^{k}\equiv
H^{k}(\ell):=H^{k}((r_{-},r_{+})\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)\cap\mathop{\rm
Ker}\nolimits(D_{\varphi}+\ell)$ with norm111111Far away from the poles of
$\mathbb{S}^{2}$, we can take $(\omega_{1},\omega_{2})=(\theta,\varphi)$ while
near the north/south pole, we can take
$(\omega_{1},\omega_{2})=(x_{*,N/S},y_{*,N/S})$.

$\displaystyle\|u\|_{H^{k}}^{2}$
$\displaystyle:=\|u\|_{L^{2}}^{2}+\|\partial_{r}^{k}u\|_{L^{2}}^{2}+\sum_{j=1}^{2}\|\partial_{\omega_{j}}^{k}u\|_{L^{2}}^{2}.$

We define the (harmonic) spectral family associated to the charged KG operator

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle:=\Omega^{-2}\rho^{2}\mathrm{e}^{\mathrm{i}(\sigma
t_{*}+\ell\varphi+qR(r))}(\cancel{\\!\\!\\!\\!\\!\\!\qed}_{g}+m^{2})\mathrm{e}^{-\mathrm{i}(\sigma
t_{*}+\ell\varphi+qR(r))}$

where

$\displaystyle\cancel{\\!\\!\\!\\!\\!\\!\qed}_{g}$
$\displaystyle=(\nabla_{\gamma}-\mathrm{i}qA_{\gamma})(\nabla^{\gamma}-\mathrm{i}qA^{\gamma}).$

Above $\sigma\in{\mathbb{C}}$ and
$R\in\mathcal{C}^{\infty}((r_{-},r_{+}),{\mathbb{R}})$ is defined (up to an
additive constant) by

$\displaystyle R^{\prime}(r)$
$\displaystyle=\frac{Qr}{\rho^{2}}(T^{\prime}(r)-a\sin^{2}\theta\Phi^{\prime}(r)),$

removing thereby the singular term $A_{r}\equiv A_{r}(r)$ from the potential
$A$ written in the $(t_{*},r,\theta,\varphi_{*})$ coordinates (thus
$\partial_{r}-\mathrm{i}qA_{r}$ has become $\partial_{r}$); notice that the
term
$-\mathrm{i}\Omega^{2}(g^{t_{*}r}\sigma+g^{r\varphi_{*}}\ell)\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)$
is missing in [BeHa], where $\alpha=0$ and thus $\Omega=1$). We realize
$\boldsymbol{P}_{\sigma}$ on the following domain:

$\displaystyle\mathcal{X}^{k}$ $\displaystyle:=\\{u\in H^{k}\ |\
\boldsymbol{P}_{\sigma}u\in
H^{k-1}\\},\qquad\|u\|_{\mathcal{X}^{k}}^{2}:=\|u\|_{\mathcal{H}^{k}}^{2}+\|\boldsymbol{P}_{\sigma}u\|_{\mathcal{H}^{k-1}}^{2}.$

In the $(t_{*},r,\theta,\varphi_{*})$ coordinates with $\theta\in(0,\pi)$, we
have

$\displaystyle A$
$\displaystyle=A_{t}\mathrm{d}t_{*}+(A_{t}T^{\prime}(r)+A_{\varphi}\Phi^{\prime}(r))\mathrm{d}r+A_{\varphi}\mathrm{d}\varphi=:A_{t_{*}}\mathrm{d}t_{*}+A_{r}\mathrm{d}r+A_{\varphi_{*}}\mathrm{d}\varphi_{*}$

and (using $\partial_{\varphi_{*}}=\partial_{\varphi}$)

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle=\Omega^{2}D_{r}\frac{\mu}{\Omega^{2}}D_{r}+\frac{\Omega^{2}}{\sin\theta}D_{\theta}\frac{\kappa\sin\theta}{\Omega^{2}}D_{\theta}+\frac{\rho^{2}}{\Omega^{2}}\left[g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}}),D_{r}\right]_{+}$
$\displaystyle\quad-\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}t_{*}}(\sigma+qA_{t_{*}})^{2}-\frac{\rho^{2}}{\Omega^{2}}g^{\varphi_{*}\varphi_{*}}(\ell+qA_{\varphi_{*}})^{2}-2\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}\varphi_{*}}(\sigma+qA_{t_{*}})(\ell+qA_{\varphi_{*}})$
$\displaystyle\quad-\mathrm{i}\Omega^{2}(g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}}))\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)+m^{2}\frac{\rho^{2}}{\Omega^{2}}.$

In the $(t_{*},r,x_{*},y_{*})$ coordinates with $x_{*}^{2}+y_{*}^{2}<1$, the
potential vector $A$ becomes:

$\displaystyle A$
$\displaystyle=A_{t_{*}}\mathrm{d}t_{*}+A_{r}\mathrm{d}r-A_{\varphi_{*}}\frac{1+\lambda}{\kappa(0)}\frac{\sin\bar{\varphi}_{*}}{\sin\theta}\mathrm{d}x_{*}+A_{\varphi_{*}}\frac{1+\lambda}{\kappa(0)}\frac{\cos\bar{\varphi}_{*}}{\sin\theta}\mathrm{d}y_{*}$
$\displaystyle=:A_{t_{*}}\mathrm{d}t_{*}+A_{r}\mathrm{d}r+A_{x_{*}}\mathrm{d}x_{*}+A_{y_{*}}\mathrm{d}y_{*}.$

Note that $A_{x_{*}}$ and $A_{y_{*}}$ are smooth near the poles thanks to a
term $\sin^{2}\theta$ in $A_{\varphi_{*}}=A_{\varphi}$. The spectral family
reads (_cf._ App. 5 for details):

$\displaystyle\boldsymbol{P}_{\sigma}$
$\displaystyle=-\frac{\rho^{2}}{\Omega^{2}}(D_{x_{*,\bullet}}-qA_{x_{*,\bullet}})(g^{{x_{*,\bullet}}{x_{*,\bullet}}}(D_{x_{*,\bullet}}-qA_{x_{*,\bullet}})+g^{{x_{*,\bullet}}{y_{*,\bullet}}}(D_{y_{*,\bullet}}-qA_{y_{*,\bullet}}))$
$\displaystyle\quad-\frac{\rho^{2}}{\Omega^{2}}(D_{y_{*,\bullet}}-qA_{y_{*,\bullet}})(g^{{x_{*,\bullet}}{y_{*,\bullet}}}(D_{x_{*,\bullet}}-qA_{x_{*,\bullet}})+g^{{y_{*,\bullet}}{y_{*,\bullet}}}(D_{y_{*,\bullet}}-qA_{y_{*,\bullet}}))$
$\displaystyle\quad+\Omega^{2}D_{r}\frac{\mu}{\Omega^{2}}D_{r}+\frac{\rho^{2}}{\Omega^{2}}[g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{t_{*}\varphi_{*}}(\ell+qA_{\varphi_{*}}),D_{r}]_{+}$
$\displaystyle\quad+\mathrm{i}\Omega^{2}\cos\theta\partial_{x_{*,\bullet}}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{{x_{*,\bullet}}{x_{*,\bullet}}}(D_{x_{*,\bullet}}-qA_{x_{*,\bullet}})+g^{{x_{*,\bullet}}{y_{*,\bullet}}}(D_{y_{*,\bullet}}-qA_{y_{*,\bullet}}))$
$\displaystyle\quad+\mathrm{i}\Omega^{2}\cos\theta\partial_{y_{*,\bullet}}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{{x_{*,\bullet}}{y_{*,\bullet}}}(D_{x_{*,\bullet}}-qA_{x_{*,\bullet}})+g^{{y_{*,\bullet}}{y_{*,\bullet}}}(D_{y_{*,\bullet}}-qA_{y_{*,\bullet}}))$
$\displaystyle\quad-\mathrm{i}\Omega^{2}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}}))$
$\displaystyle\quad-\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}t_{*}}(\sigma+qA_{t_{*}})^{2}-2\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}\varphi_{*}}(\sigma+qA_{t_{*}})(\ell+qA_{\varphi_{*}})+m^{2}\frac{\rho^{2}}{\Omega^{2}}.$

###  3.4. Verification of Assumptions 9

We now put the operator $\boldsymbol{P}_{\sigma}$ into the framework of Sect.
2.

Let
$\chi_{\pm},\widetilde{\chi}_{\pm}\in\mathcal{C}^{\infty}([r_{-},r_{+}],[0,1])$
such that $\chi_{\pm}(r_{\pm})=1$, $\chi_{-}+\chi_{+}=1$ and
$\widetilde{\chi}_{\pm}\chi_{\pm}=\chi_{\pm}$. We then identify
$(r_{-},r_{+})\cap\mathrm{Supp}(\widetilde{\chi}_{\pm})$ with a neighborhood
of $0$ on the half axis $(0,+\infty)$ using the boundary defining functions
$\boldsymbol{\rho}_{\pm}:=|r-r_{\pm}|$ (we will use bold symbols for these in
order to emphasize the difference with the function $\rho$ introduced in Sect.
3.1). We will abusively write $\chi_{\pm}(\boldsymbol{\rho}_{\pm})$ instead of
$\chi_{\pm}(r_{\pm}\mp\boldsymbol{\rho}_{\pm})$ then consider $\chi_{\pm}$ as
defined on all $(0,+\infty)$ by taking $\chi_{\pm}(\boldsymbol{\rho}_{\pm})=0$
whenever $\boldsymbol{\rho}_{\pm}\geq r_{+}-r_{-}$ (and similarly for
$\widetilde{\chi}_{\pm}$). Notice that

$\displaystyle D_{r}$ $\displaystyle=\mp
D_{\boldsymbol{\rho}_{\pm}},\qquad\mu(r)=\mu_{\pm}(\boldsymbol{\rho}_{\pm})\boldsymbol{\rho}_{\pm}$

where121212To see this, we write
$\mu(r)=-\frac{\Lambda}{3}(r-r_{n,a})(r-r_{c,a})(r-r_{-})(r-r_{+})$ to
discover that $\frac{\mu(r)}{r-r_{-}}>0$ for all $r\in(r_{c,a},r_{+})$ and
$\frac{\mu(r)}{r_{+}-r}>0$ for all $r\in(r_{-},+\infty)$. In the case where
$\mu$ only has two real roots $0<r_{-}<r_{+}$, then we can write
$\mu(r)=-\frac{\Lambda}{3}f(r)(r-r_{-})(r-r_{+})$ with $f(r)>0$ for