# Object Parsing in Sequences Using CoordConv Gated Recurrent Networks

Ayush Gaud1, Y V S Harish1 and K Madhava Krishna1 1Ayush Gaud, Y V S Harish
and K Madhava Krishna are with Robotics Research Center, International
Institute of Information Technology, Hyderabad, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a monocular object parsing framework for consistent keypoint
localization by capturing temporal correlation on sequential data. In this
paper, we propose a novel recurrent network based architecture to model long-
range dependencies between intermediate features which are highly useful in
tasks like keypoint localization and tracking. We leverage the expressiveness
of the popular stacked hourglass architecture and augment it by adopting
memory units between intermediate layers of the network with weights shared
across stages for video frames. We observe that this weight sharing scheme not
only enables us to frame hourglass architecture as a recurrent network but
also prove to be highly effective in producing increasingly refined estimates
for sequential tasks. Furthermore, we propose a new memory cell, we call
CoordConvGRU which learns to selectively preserve spatio-temporal correlation
and showcase our results on the keypoint localization task. The experiments
show that our approach is able to model the motion dynamics between the frames
and significantly outperforms the baseline hourglass network. Even though our
network is trained on a synthetically rendered dataset, we observe that with
minimal fine tuning on 300 real images we are able to achieve performance at
par with various state-of-the-art methods trained with the same level of
supervisory inputs. By using a simpler architecture than other methods enables
us to run it in real time on a standard GPU which is desirable for such
applications. Finally, we make our architectures and 524 annotated sequences
of cars from KITII dataset publicly available.

##  I INTRODUCTION

Estimating semantically meaningful joint locations and hence shape and pose of
moving objects like cars in a highway driving scenario is a challenging
problem in computer vision. With the growing interest in autonomous driving,
many successful approaches use expensive sensors like LiDARs and stereo
cameras for tracking and predicting trajectory of vehicles on road. However,
there is an ongoing effort to minimize the cost of expensive sensor suite
required for such tasks. Some recent approaches have shown to recover both
shape and pose from monocular images but fail to leverage the temporal
characteristics of the data in such scenarios[1]. Our approach bases itself on
the success of the stacked hourglass architecture[2] which showcased state of
the art results for human pose prediction. While its multi-level encoder-
decoder architecture with skip layers preserves spatial information at
different scales, it fails to exploit any temporal information that is present
in sequential data and treats each frame independently which is detrimental
especially in scenarios like driving.

In this paper, we posit that leveraging the temporal information using
recurrent architectures enables us to parse objects with higher accuracy. It
is based on the notion that the corresponding part locations _(keypoints)_ of
an object is usually in the neighborhood across frames of a sequence,
especially for a rigid body. We present our findings which conforms to this
hypothesis and justifies the validity of our proposed network architecture for
performing this task. Specifically, we demonstrate that on scenarios like
driving where environment involves high dynamics, this property can be quite
useful for tasks like propagating annotations on on-road vehicles consistently
across the frames, unlike other methods[1][2][3][4] which discards the
sequential consistency of a scene by treating each frame independently to
perform predictions.

To illustrate this idea, we utilize semantically meaningful locations of an
object represented as keypoints which efficiently encapsulates the 3D
structure of an object category. These 3D keypoints are annotated on CAD
models which enables us to generate large amounts of synthetic dataset
required for training an architecture like this. The dataset is generated
using the annotated CAD models of cars from the ShapeNet[5] corpus. The cars
are rendered from different viewpoints by moving the camera in a natural
motion for generating sequences which appear temporally consistent. We observe
a consistent improvement in the accuracy of predicted keypoint locations as we
continue to improve the expressiveness of the model by making changes to the
original hourglass architecture by adding ConvGRU and then CoordConvGRU cells
to the residual layer. Our contributions are summarized as follows:

  1. 1.

We introduce an improved stacked hourglass architecture to impose spatio-
temporal consistency in consecutive video frames from a monocular camera for
keypoint prediction. Instead of employing RNNs based on vanilla LSTM, we
utilize convolutional gated recurrent networks[6] part to preserve spatial
connectivity in the image.

  2. 2.

We note further improvement using our proposed CoordConvGRU cells designed by
adding two coordinate channels, one for each axis, to the convolution layer of
ConvGRU, which learns to selectively preserve translational invariance and
effectively learn the temporal correlation between intermediate features of
the frames in a sequence.

We evaluate our hypothesis by conducting ablation experiments on different
network designs to highlight the efficacy of our approach. To further
emphasize the generalizability of our approach, we also perform benchmarks on
the images from KITTI dataset and compare the predicted keypoint locations
with various other networks. Our architecture is able to outperform several
other architectures, while at the same time is fast enough to run in real-time
on a standard GPU. In section 4 we present the qualitative and quantitative
results of our proposed architecture and demonstrate consistent improvement in
performance of the network across other architectures.

##  II RELATED WORK

Keypoint localization is a well studied topic in the literature popularized by
pose estimation for humans. DeepPose[7] shifted the attention of people from
classical approaches to deep learning based approaches. Tompson[8] further
brought a significant improvement in keypoint localization by introducing a
fully convolution based network. He proposed that generating heatmaps directly
from a hierarchy of multi-scale convolutional structure combined with a
graphical model enables learning the spatial relationships between the joints
and achieves higher accuracy. Tompson built his work[8] on the idea of cascade
refinement based on the multi-stage pose machines[9].

Newell et al.[2] introduced intermediate supervision to a cascaded network
based on the conv-deconv and encoder-decoder architectures[10][11][12][13].
The repeated bottom-up and top-down inference from stacking further allows for
reevaluation of initial estimates and features across the image. This helps in
forming higher order spatial relationships and maintaining local and global
cues.

In a few studies, temporal cues have also been
integrated[14][15][16][17][18][19] to the network for the task of pose
estimation. Using dense optical flow[20] certain approaches[18][19] have
attempted in predicting consistent position of joints preserving smooth motion
across the frames. Thin-Slicing network[19] presented improved results by
using both optical flow and a spatial-temporal model but at the cost of
computational complexity rendering it slower than other approaches.

LSTM pose machines[21] introduced using of memory augmented recurrent networks
to capture temporal consistency and was able to output perform various state
of the art approaches in 2D video pose estimation tasks. Their design was
based on another strong baseline approach Convolution Pose Machines[22] by
using essentially the same architecture but also providing a weight sharing
scheme across stages while at the same time using LSTM cells to promote
intermediate features. This helps in better utilizing historical joint
locations and achieving almost double inference speed which is critical for
real-time applications.

Siam et al.[23] investigated convolutional gated recurrent architecture
(convGRU) for video segmentation. They proposed a modification to the standard
convolutional recurrent units which are designed for processing text data, to
preserve the spatial connectivity between the pixels which would have been
otherwise lost. By replacing the dot products with convolution operation they
were able to create a much more efficient recurrent cell which can be easily
trained on images without the curse of high dimensionality of weights. They
were able to achieve significant gains on the baseline architectures by
exploiting the spatio-temporal information of the videos.

Payer et al.[24], further utilizing the effectiveness of both hourglass and
ConvGRU architectures, presented a recurrent hourglass architecture for
instance segmentation tracking. They rigorously evaluated their approach on
various cell tracking datasets and were able to achieve state-of-the-art
performance.

CoordConv layers presented by Liu et al.[25] posed a counter intuitive idea
from assuming the convolution layers to be appropriate for spatial
representations. This is justified by a trivial counterexample of learning
coordinate transforms in one-hot pixel space. They claim that by adding a few
extra coordinate channels enables the network to learn varying degrees of
translational invariance. There is a stark contrast observed in the rate of
learning and the parameters required across varying domains of applications
including image classification and object detection.

In this paper, we approach the problem of keypoint localization as a semantic
segmentation and tracking task unlike other approaches and propose a novel
architecture that can leverage the spatio-temporal features in sequences to
generate more consistent and robust detections. Using our proposed
CoordConvGRU architecture, we demonstrate consistent performance improvement
by conducting ablation studies on synthetic video sequences and compare our
performance with other state-of-the-art methods in section IV. We also make
the network architectures and annotated data publicly available. 111Project
page and annotated KITTI sequences:
https://ayushgaud.github.io/Stacked_HG_CoordConvGRU/

##  III APPROACH

###  III-A Overview

Our approach builds on the notion that in a real-world context the data
possess temporal consistency and the spatial features move gradually with
time. Hence, instead of doing processing on static images we propose a network
which can learn to model the motion dynamics in an end-to-end fashion. We
leverage the recurrent neural network architecture which jointly learns to
model spatial and temporal features from videos.

###  III-B Network Architecture

Figure 1: Two Stack Hourglass with Recurrent Cells for Skip Layers Figure 2:
Comparison of different types of GRU cells

####  III-B1 Stacked Hourglass with Intermediate Supervision

Stacked hourglass networks proposed by Newell et al.[2] shows that stacking
conserves the higher order spatial relationships. Stages of hourglass enable
inferring higher order features when intermediate supervision is applied. We
utilize a similar architecture with iterative refinement using multiple stacks
of hourglass for predicting keypoint likelihood map.

####  III-B2 Convolution Gated Recurrent Units (ConvGRU)

Conventional Recurrent Neural Networks are applied to sequence of inputs to
capture the temporal relation in the data. However, due to vanishing gradient
problem gated architectures were proposed. Long Short Term Memory (LSTM), one
of the most popular model used for RNNs has three gates namely, input, output
and forget. The latter controls the amount of information flow from the
previous states and acts as a memory for predictions. Gated Recurrent Unit
(GRU) acts just like LSTMs but with a much efficient architecture by assuming
a correlation between memorizing and forgetting. This enables it to use only a
single gate to control the two and consequently the output flow. However, for
using RNNs with images, they are vectorized into large 1D arrays which
increases the number of parameters to learn and loses the spatial connectivity
between pixels. This makes the network harder to train due to large search
space where spatial context is already lost. To mitigate this problem
convolutional recurrent units were introduced which, unlike regular GRU
replaces dot products with small convolution filters as shown in the equation
below

$z_{t}=\sigma\left(W_{hz}*h_{t-1}+W_{xz}*x_{t}+b_{z}\right)$ (1a)
$r_{t}=\sigma\left(W_{hr}*h_{t-1}+W_{xr}*x_{t}+b_{r}\right)$ (1b)
$\hat{h_{t}}=\Phi\left(W_{h}*\left(r_{t}\odot
h_{t-1}\right)+W_{x}*x_{t}+b\right)$ (1c) $h_{t}=\left(1-z_{t}\right)\odot
h_{t-1}+z\odot\hat{h}_{t}$ (1d)

Learning such small filters instead of weights for each pixel proves to be
much more efficient while at the same time preserving spatial context.

Figure 3: Synthesis Pipeline: The above figure depicts the synthetic dataset
generation flow. ShapeNet[5] CAD models are rendered with Blender from various
viewpoints in a smooth motion for generating sequences. Green blocks represent
the output of the pipeline which includes the sequence images with background,
its corresponding 2D keypoints and a list of visible keypoints.

####  III-B3 CoordConv Layer

CoordConv layers[25] are a simple extension to regular convolution layers.
They add two extra channels (in case of images) filled with coordinate
information which are concatenated at the end of a convolution layer. This
enables the network to flexibly learn translational invariance. It means that
zero weights for these channels would imply full translational invariance like
standard convolution while any other value would lead the network to learn
translation invariance upto a degree which becomes a training parameter
depending on the task.

####  III-B4 Proposed Network Architecture

We introduce an improved architecture based on the stacked hourglass design
for preserving both spatial and temporal context in a sequence of images.
Capitalizing on the above notions, we present a recurrent architecture by
replacing standard convolutions for skip connections with ConvGRU layers. The
RNN cells are stacked together for processing a sequence of four images, while
the weights among the stacks for each frame is shared within the graph. Since
in the original paper[2] there was only a marginal performance improvement
observed when comparing eight stack hourglass network with a two stack
network, hence we also consider a two stack architecture with intermediate
supervision. Each stack consists of five layers of convolution filters with
max pooling operation with pool size of (2,2). The output of each layer is
passed to the next convolution layer and the skip layer consisting of GRU
cell. This is followed by a series of upsampling layers and subsequent
convolutions to bring the output size to 64x64. Each convolution filter has a
channel size of 36 corresponding to each keypoint heatmap except in case of
CoordConvGRU cells where two channels are added for each $i$ and $j$
coordinates as shown in figure 2. We use four residual modules each embedded
with an RNN cell for both stacks. We train the network on 64x64 images. The
network outputs 36 heatmaps of size 64x64 at the end of every stack along with
the states of the recurrent cells. We further show that our proposed cell
which is created by modifying the ConvGRU layers, and using CoordConv layers
i.e adding two channels one for $X$ and one for $Y$ axis, leads to further
improvement in performance. The implementation of this architecture is
publicly made available and can be downloaded from here.222Tensorflow
implementation: https://github.com/ayushgaud/hourglass_CoordConvGRU

###  III-C Loss Function

Newell et al.[2] in the original stacked hourglass paper treats the keypoint
localization task as a regression problem and hence uses mean squared error
loss for optimizing over the heatmaps. We posit a different route by treating
this task as a classification problem. This notion stems from the similarity
between the semantic segmentation and keypoint localization task. Due to this,
we use sigmoid cross entropy as a loss function over heatmaps. These heatmaps
are computed by using Gaussian distribution centered at ground truth location
of keypoints with standard deviation of 1 pixel. In our case we represent the
model of vehicle using 36 keypoints annotated at semantically meaningful joint
locations. If predicted heatmaps for $i^{th}$ image in sequence of length $N$
is represented by $X_{i}$ and labeled heatmap is represented by $Z_{i}$. Then
the sigmoid cross entropy loss can be written as:

$loss=\sum_{i=1}^{N}X_{i}-X_{i}*Z_{i}+log(1+e^{-X_{i}})$ (2)

We minimize this loss between predicted heatmaps from both stacks and ground
truth heatmaps leveraging the benefits of intermediate supervision along with
standard form of supervision done at the end of the network.

###  III-D Dataset Generation and Training

####  III-D1 Synthetic Dataset Generation

Since we require large amount of data to train our network and annotated
sequences are hard to find, We use the 3D keypoint annotations from [26] and
render CAD models of cars from ShapeNet[5] using the Render for CNN [27]
pipeline. To enable sufficient variations in the data we randomly sample
initial viewpoints and their respective increments using a uniform
distribution. These viewpoint increments helps in generating desirable smooth
realistic motions. We also vary the number of light sources and their
intensity randomly for each sequence of data. An overview of the synthesis
pipeline is given in figure 3. We use 472 3D CAD models and their
corresponding annotations provided by [28]. The 36 semantically meaningful
joints on each car are projected from the viewpoints at which the images are
rendered. To identify visible set of keypoints for evaluation, we also render
depth channel data and record camera viewpoints for each frame. We then use
this information and perform ray tracing to identify the depths of each
keypoint. This depth is compared to the depth of the projected keypoints from
the rendered depth image and the mismatching keypoints are identified to be
occluded. Each synthetic image is cropped and overlayed on a realistic
background image to avoid overfitting. To maintain consistency the background
image is kept same across the frames of each sequence. A total of hundred
sequences with four frames each are generated for every CAD model.

####  III-D2 Training Details

The proposed network is built on the TensorFlow framework. We initialize the
weights using Xavier initialization scheme with a learning rate of
$2.5*10^{-4}$. We use RMSProp optimizer and decay the learning rate
exponentially at every 20000 steps by a factor of 0.96. The input to the
network are images of size is 64x64 at batches of 60 for initial training.
Once the loss plateaus, the backpropagation is performed on the unwrapped
recurrent network with shared pre-trained weights on a sequence length of 4
frames and a batch size of 16. The loss is computed on all the frames in the
sequence and used for optimizing the network. We stop the training at around
500K iterations in total which takes almost 2 days to complete on 4 Nvidia
1080Ti GPUs.

##  IV RESULTS

###  IV-A Keypoint Localization

We evaluate the performance of our network using the PCK metric. This is a
standard method used to label a keypoint detection as correct depending on
whether or not it lies within the circle of radius $\alpha*L$ centered at the
ground truth keypoint location where $L$ represents the larger dimension of
the image. Instead of using the standard $\alpha=0.1$ we use a much stricter
threshold of 0.05. Even though the network is trained on all the keypoints
irrespective of visibility, we generate quantitative results based on the
visible set of keypoints to remain consistent with the convention. It should
be noted that the wireframes are made by joining all the predicted keypoint
locations, including the occluded ones in a structural fashion. Figure 4 shows
the ground truth and the predicted heatmap from the network. We take argmax
over each of the 36 predicted heatmaps of all the joints and construct a
wireframe by joining these points.

Figure 4: Ground truth and predicted keypoint likelihood maps (heatmaps)

###  IV-B Ablation Experiments

Given that our main contributions are based on the network architecture, we
explore the effect of adding recurrent cells by comparing the 2D keypoint
localization accuracy. To make a fair comparison we train all the networks
from scratch on the same rendered synthetic dataset considering standard
hourglass network as a baseline. We train three different network
architectures including the baseline hourglass and tabulate the PCK measure at
$\alpha=0.05$ in Table I.

It is interesting to note that adding ConvGRU layers itself improves the mean
2D keypoint accuracy from 94.8% to 96.1% thereby proving to have learnt to
model the motion dynamics between the frames in sequences. We also observe
that further changing the RNN cells in residual layers from ConvGRU to
CoordConvGRU improved the result by another 0.6% at 96.7%. This improvement
could be attributed to the selective translational invariance nature of the
layer which enables it to capture the spatio-temporal consistencies in the
sequence.

Network |  PCK @ $\alpha=0.05$
---|---
Standard Hourglass (Baseline) |  94.81
Hourglass ConvGRU + Sequence |  96.10
Hourglass CoordConvGRU + Sequence |  96.71
TABLE I: PCK Comparison results between the baseline hourglass network and proposed network architectures. Network |  PCK @ $\alpha=0.1$
---|---
DDN [29] |  67.6
WN-gt-yaw [3] |  88.0
Zia et al. [28] |  73.6
DSN-2D |  27.2
DSN-3D |  76.0
plain-2D |  45.2
plain-3D |  88.4
DISCO-3D-2D |  90.1
DISCO-vis-3D-2D |  92.3
DISCO-Vgg |  83.5
DISCO [4] |  93.1
Standard Hourglass |  82.11
Hourglass CoordConvGRU |  88.27
TABLE II: PCK accuracies on KITTI 3D dataset[28] Network |  PCK @ $\alpha=0.1$
---|---
Standard Hourglass |  82.33
Hourglass CoordConvGRU |  87.25
Hourglass CoordConvGRU + Sequence |  87.81
TABLE III: PCK comparison on annotated KITTI sequences Figure 5: Wireframes
predicted by the network on KITTI image sequences

###  IV-C Results on KITTI images

We fine tune the proposed architectures on 300 real images from KITTI. The
networks are not refined on sequential data but only static images. The
qualitative results in figure 5 shows that our networks are able to handle
this data distribution transform from synthetic to real quite effectively with
minimal training. We also perform a quantitative comparison of our approach on
the annotated KITTI[30] images. Table II reports PCK accuracies for various
state-of-the-art methods including Zia et al.[28], DNN[29], WarpNet[3] and
various variants of DISCO[4]. Since the DISCO architecture is based on
supervision for intermediate tasks, we also report the comparison of PCK
values for variants depending on the supervision performed. DISCO-vis-3D-2D,
DISCO- 3D-2D, plain-3D and plain-2D are networks without pose,
pose+visibility, pose+visibility+2D and pose+visibility+3D, respectively
removed as a supervisory input to the complete deep supervision required for
DISCO. It should be noted that we only provide 2D keypoint supervision for
training and still achieve better results than most of the networks using a
much simpler architecture. The amount of supervision required for training our
network is equivalent to plan-2D and DSN-2D which perform significantly poorly
on the keypoint localization task at PCKs 45.2% and 27.2% respectively. This
shows that our proposed architecture at PCK 88.27% has far better expressive
capabilities compared to the other methods for this task. We also outperform
standard hourglass by almost 6.1% on static KITTI images from [28].

We also use our labeled 524 sequences of images from KITTI for evaluation of
performance of the network on sequences. In Table III we have tabulated the
PCK values on this data. It is clear that again our proposed network surpasses
our baseline by a significant margin of around 6% and is 0.5% higher than
without sequence. It should also be noted that the disparity between
performance is significant when compared to synthetic data. This can be
because of the better convergence behavior of the CoordConv[25] unit of the
RNN cells. Since we fine-tune the network on a limited set of images this
behavior is not contrary to the expectations.

###  IV-D Run-Time Analysis

While using RNNs on sequential data gives significant performance improvements
in terms of accuracy, they do come at a price. Run time of ConvGRU and
CoordConvGRU is higher as they perform more operations than a regular
convolution layer. At the same time since the CoordConv layers add two more
channels to each convolution layer inside the RNN cell, this further increases
the forward pass time of the network. Our network is much simpler in
architecture than the state-of-the-art DISCO[4] as it based on the very deep
VGG network architecture with fully connected layers for intermediate task
supervision. This enables us to run this network for applications requiring
real-time time predictions. We evaluated the forward pass times of all the
three networks and the results are presented in Table IV. Since while doing
forward pass, the network only requires images and hidden states of the
previous RRN cells hence the run time is computed for a single frame and not
the sequence.

Network | Forward Pass Time
---|---
Standard Hourglass | 11ms
Hourglass ConvGRU | 19ms
Hourglass CoordConvGRU | 21ms
TABLE IV: Run time of different networks on an Nvidia 1080Ti GPU

##  V CONCLUSIONS

We present a framework for keypoint localization on an object category and
leverage the spatio-temporal correlation between frames in a sequence. The
experimental results prove our hypothesis and demonstrate that our network
performs at par with other networks which are trained at same level of
supervision, using a relatively simpler architecture. We also demonstrate that
our network is fast enough for real time applications like autonomous driving.
We showcase a new type of RNN cell and conduct ablation experiments to
highlight the performance benefits of using it. The results from the proposed
architecture can further be improved by adding multiple levels of supervision
and deserves further examination. We make our proposed network architecture
and annotated data for KITII sequences publicly available for critical
analysis and extension of this work.

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$r\in[r_{-},r_{+}]$, and a similar argument applies. $\mu_{\pm}>0$ on
$\overline{\mathrm{Supp}(\chi_{\pm})}$.

Let also
$\phi_{\bullet},\widetilde{\phi}_{\bullet}\in\mathcal{C}^{\infty}(\mathbb{S}^{2},[0,1])$
with $\bullet\in\\{N,E,E^{\prime},S\\}$ ($N$ for north pole, $E,E^{\prime}$
for equator – we need two open sets to cover this region of the sphere – and
$S$ for south pole) such that $\phi_{\bullet}\equiv 1$ near the north
pole/equator/south pole of $\mathbb{S}^{2}$,
$\phi_{N}+\phi_{E}+\phi_{E^{\prime}}+\phi_{S}=1$ and
$\widetilde{\phi}_{\bullet}\phi_{\bullet}=\phi_{\bullet}$ (thus the compact
smooth manifold $\mathcal{M}$ in Sect. 2 is $\mathbb{S}^{2}$). We then
identify $\mathbb{S}^{2}\cap\mathrm{Supp}(\widetilde{\phi}_{\bullet})$ with an
open set of ${\mathbb{R}}^{2}$. While it needs no particular care to construct
the radial cut-offs, the angular cut-offs must respect some additional
properties in order to ensure ellipticity of $\cancel{P}_{\sigma}$. For
$\bullet\in\\{N,S\\}$, we require:

$\displaystyle\phi_{\bullet}(x_{*,\bullet},y_{*,\bullet})$
$\displaystyle=0\qquad\text{if }x_{*,\bullet}^{2}+y_{*,\bullet}^{2}\geq 1.$

On $\mathrm{Supp}(\chi_{\pm})\times\mathrm{Supp}(\phi_{\bullet})$ with
$\bullet\in\\{E,E^{\prime}\\}$, we have
$\boldsymbol{P}_{\sigma}=\boldsymbol{\rho}_{\pm}^{-1}P_{\sigma}+\cancel{P}_{\\!\sigma}$
where

$\displaystyle P_{\sigma}$
$\displaystyle=a(\boldsymbol{\rho}_{\pm})(\boldsymbol{\rho}_{\pm}D_{\boldsymbol{\rho}_{\pm}})^{2}+b(\boldsymbol{\rho}_{\pm},\theta,\sigma)\boldsymbol{\rho}_{\pm}D_{\boldsymbol{\rho}_{\pm}}+c(\boldsymbol{\rho}_{\pm},\theta,\sigma),\qquad\cancel{P}_{\\!\sigma}=\cancel{a}_{\theta}(\boldsymbol{\rho}_{\pm},\theta)D_{\theta}^{2}+\cancel{b}_{\theta}(\boldsymbol{\rho}_{\pm},\theta)D_{\theta},$

and (the expressions of $c$ and $c_{\bullet}$ below have to be slightly
modified, _cf._ the end of the section)

$\displaystyle a(\boldsymbol{\rho}_{\pm})$
$\displaystyle=\mu_{\pm}(\boldsymbol{\rho}_{\pm}),$ $\displaystyle
b(\boldsymbol{\rho}_{\pm},\theta,\sigma)$ $\displaystyle=\mp
2\frac{\rho^{2}}{\Omega^{2}}\left(g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}})\right)\pm\mathrm{i}\boldsymbol{\rho}_{\pm}\Omega^{2}\partial_{r}\\!\left(\frac{\mu_{\pm}}{\Omega^{2}}\right),$
$\displaystyle c(\boldsymbol{\rho}_{\pm},\theta,\sigma)$
$\displaystyle=\boldsymbol{\rho}_{\pm}\bigg{(}-\mathrm{i}\Omega^{2}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\left(g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}})\right)\right)+m^{2}\frac{\rho^{2}}{\Omega^{2}}$
$\displaystyle\qquad\quad\;\;-\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}t_{*}}(\sigma+qA_{t_{*}})^{2}-2\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}\varphi_{*}}(\sigma+qA_{t_{*}})(\ell+qA_{\varphi_{*}})-\frac{\rho^{2}}{\Omega^{2}}g^{\varphi_{*}\varphi_{*}}(\ell+qA_{\varphi_{*}})^{2}\bigg{)},$
$\displaystyle\cancel{a}_{\theta}(\boldsymbol{\rho}_{\pm},\theta)$
$\displaystyle=\kappa,$
$\displaystyle\cancel{b}_{\theta}(\boldsymbol{\rho}_{\pm},\theta)$
$\displaystyle=-\mathrm{i}\frac{\Omega^{2}}{\sin\theta}\partial_{\theta}\\!\left(\frac{\sin\theta}{\Omega^{2}}\right).$

Notice that $a(0)$, $b(0,\theta,\sigma)$ and $c(0,\theta,\sigma)=0$ are purely
radial for all $(\theta,\sigma)\in(0,\pi)\times{\mathbb{C}}$.

On $\mathrm{Supp}(\chi_{\pm})\times\mathrm{Supp}(\phi_{\bullet})$ with
$\bullet\in\\{N,S\\}$, we have
$\boldsymbol{P}_{\sigma}=\boldsymbol{\rho}_{\pm}^{-1}P_{\sigma}+\cancel{P}_{\\!\sigma}$
where

$\displaystyle P_{\sigma}$
$\displaystyle=a(\boldsymbol{\rho}_{\pm})(\boldsymbol{\rho}_{\pm}D_{\boldsymbol{\rho}_{\pm}})^{2}+b(\boldsymbol{\rho}_{\pm},\omega,\sigma)\boldsymbol{\rho}_{\pm}D_{\boldsymbol{\rho}_{\pm}}+c_{\bullet}(\boldsymbol{\rho}_{\pm},\omega,\sigma),$
$\displaystyle\cancel{P}_{\\!\sigma}$
$\displaystyle=\cancel{a}_{x_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})D_{x_{*,\bullet}}^{2}+\cancel{b}_{x_{*,\bullet}}(\boldsymbol{\rho}_{\pm},x_{*,\bullet},y_{*,\bullet})D_{x_{*,\bullet}}+\cancel{a}_{y_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})D_{y_{*,\bullet}}^{2}+\cancel{b}_{y_{*,\bullet}}(\boldsymbol{\rho}_{\pm},x_{*,\bullet},y_{*,\bullet})D_{y_{*,\bullet}}$

and

$\displaystyle c_{\bullet}(\boldsymbol{\rho}_{\pm},\theta,\sigma)$
$\displaystyle=\boldsymbol{\rho}_{\pm}\bigg{(}-\mathrm{i}\Omega^{2}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\left(g^{t_{*}r}(\sigma+qA_{t_{*}})+g^{r\varphi_{*}}(\ell+qA_{\varphi_{*}})\right)\right)+m^{2}\frac{\rho^{2}}{\Omega^{2}}$
$\displaystyle\qquad\quad\;\;-\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}t_{*}}(\sigma+qA_{t_{*}})^{2}-2\frac{\rho^{2}}{\Omega^{2}}g^{t_{*}\varphi_{*}}(\sigma+qA_{t_{*}})(\ell+qA_{\varphi_{*}})-\frac{\rho^{2}}{\Omega^{2}}\breve{g}^{x_{*,\bullet}y_{*,\bullet}}\ell^{2}$
$\displaystyle\qquad\quad\;\;-\mathrm{i}q\Omega^{2}\cos\theta\partial_{x_{*,\bullet}}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}(g^{x_{*,\bullet}x_{*,\bullet}}A_{x_{*,\bullet}}+g^{x_{*,\bullet}y_{*,\bullet}}A_{y_{*,\bullet}})\right)$
$\displaystyle\qquad\quad\;\;-\mathrm{i}q\Omega^{2}\cos\theta\partial_{y_{*,\bullet}}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}(g^{x_{*,\bullet}y_{*,\bullet}}A_{x_{*,\bullet}}+g^{y_{*,\bullet}y_{*,\bullet}}A_{y_{*,\bullet}})\right)$
$\displaystyle\qquad\quad\;\;-q^{2}\frac{\rho^{2}}{\Omega^{2}}\left(g^{x_{*,\bullet}x_{*,\bullet}}A_{x_{*,\bullet}}^{2}+2g^{x_{*,\bullet}y_{*,\bullet}}A_{x_{*,\bullet}}A_{y_{*,\bullet}}+g^{y_{*,\bullet}y_{*,\bullet}}A_{y_{*,\bullet}}^{2}\right)\bigg{)},$
$\displaystyle\cancel{a}_{x_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})$
$\displaystyle=\cancel{a}_{y_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})=\kappa\cos^{2}\theta,$
$\displaystyle\cancel{b}_{x_{*,\bullet}}(\boldsymbol{\rho}_{\pm},x_{*,\bullet},y_{*,\bullet})$
$\displaystyle=\mathrm{i}\Omega^{2}\cos\theta\left(\partial_{x_{*,\bullet}}\\!\left(\frac{\rho^{2}g^{x_{*,\bullet}x_{*,\bullet}}}{\Omega^{4}\cos\theta}\right)+\partial_{y_{*,\bullet}}\\!\left(\frac{\rho^{2}g^{x_{*,\bullet}y_{*,\bullet}}}{\Omega^{4}\cos\theta}\right)\right)-\mathrm{i}\frac{\rho^{2}}{\Omega^{2}}\breve{g}^{x_{*,\bullet}y_{*,\bullet}}x_{*,\bullet}$
$\displaystyle\quad+2q\frac{\rho^{2}}{\Omega^{2}}(g^{x_{*,\bullet}x_{*,\bullet}}A_{x_{*,\bullet}}+g^{x_{*,\bullet}y_{*,\bullet}}A_{y_{*,\bullet}}),$
$\displaystyle\cancel{b}_{y_{*,\bullet}}(\boldsymbol{\rho}_{\pm},x_{*,\bullet},y_{*,\bullet})$
$\displaystyle=\mathrm{i}\Omega^{2}\cos\theta\left(\partial_{y_{*,\bullet}}\\!\left(\frac{\rho^{2}g^{y_{*,\bullet}y_{*,\bullet}}}{\Omega^{4}\cos\theta}\right)+\partial_{x_{*,\bullet}}\\!\left(\frac{\rho^{2}g^{x_{*,\bullet}y_{*,\bullet}}}{\Omega^{4}\cos\theta}\right)\right)-\mathrm{i}\frac{\rho^{2}}{\Omega^{2}}\breve{g}^{x_{*,\bullet}y_{*,\bullet}}y_{*,\bullet}$
$\displaystyle\quad+2q\frac{\rho^{2}}{\Omega^{2}}(g^{x_{*,\bullet}y_{*,\bullet}}A_{x_{*,\bullet}}+g^{y_{*,\bullet}y_{*,\bullet}}A_{y_{*,\bullet}}).$

Above,

$\displaystyle\breve{g}^{x_{*}y_{*}}$
$\displaystyle:=\frac{g^{x_{*}y_{*}}}{x_{*}y_{*}}=\frac{\Omega^{2}}{\rho^{2}}\left(\frac{\kappa^{2}-(\kappa+\kappa(\bullet))\widetilde{\kappa}_{\bullet}}{\kappa}-\frac{\kappa(\bullet)^{2}(\mu\widetilde{c}_{\pm}^{2}+a(1+\lambda)\widetilde{c}_{\pm})}{(1+\lambda)^{2}}\right)$

with $\kappa(N):=\kappa(0)$ and $\kappa(S):=\kappa(\pi)$. We then used that

$\displaystyle-\frac{\rho^{2}}{\Omega^{2}}\left(g^{x_{*,\bullet}x_{*,\bullet}}+\breve{g}^{x_{*,\bullet}y_{*,\bullet}}y_{*,\bullet}^{2}\right)$
$\displaystyle=-\frac{\rho^{2}}{\Omega^{2}}\left(g^{y_{*,\bullet}y_{*,\bullet}}+\breve{g}^{x_{*,\bullet}y_{*,\bullet}}x_{*,\bullet}^{2}\right)=\frac{\kappa(\bullet)^{2}}{\kappa}-\frac{\kappa^{2}-(\kappa+\kappa(\bullet))\widetilde{\kappa}_{\bullet}}{\kappa}\sin^{2}\theta=\kappa\cos^{2}\theta.$

We also used the identity

$\displaystyle-2x_{*,\bullet}y_{*,\bullet}D_{x_{*,\bullet}}D_{y_{*,\bullet}}$
$\displaystyle=D_{\bar{\varphi}_{*}}^{2}-y_{*,\bullet}^{2}D_{x_{*,\bullet}}^{2}-x_{*,\bullet}^{2}D_{y_{*,\bullet}}^{2}-\mathrm{i}(x_{*,\bullet}D_{x_{*,\bullet}}+y_{*,\bullet}D_{y_{*,\bullet}})$

in order to remove any mixed term $D_{x_{*,\bullet}}D_{y_{*,\bullet}}$ from
$\cancel{P}_{\sigma}$, entering thereby the abstract setting of Sect. 2.

We now check that Assumptions 9 are fulfilled. To do this, we need to slighlty
modify the symbol $\mathfrak{q}_{\sigma,k,-1/2}^{\mathrm{f},1}$ in Sect. 2.2
and replace its denominator by
$a(\boldsymbol{\rho}_{\pm})(\xi_{k,-1/2}^{2}+(k-1/2+\epsilon)^{2})+\boldsymbol{\rho}_{\pm}|\eta|_{\cancel{a}}^{2}$
for any $\epsilon>0$, and then respectively replace $c$, $c_{\bullet}$ above
by $c-a(k-1/2)^{2}$, $c_{\bullet}-a(k-1/2)^{2}$. As we have

$\displaystyle a(\boldsymbol{\rho}_{\pm})>0\text{ for all
}\boldsymbol{\rho}_{\pm}\in\overline{\mathrm{Supp}(\chi_{\pm})}\subset[0,r_{+}-r_{-}),$
$\displaystyle\cancel{a}_{x_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})=\cancel{a}_{y_{*,\bullet}}(x_{*,\bullet},y_{*,\bullet})=\kappa(1-x_{*,\bullet}^{2}-y_{*,\bullet}^{2})>0\text{
for all
}(x_{*,\bullet},y_{*,\bullet})\in\overline{\mathrm{Supp}(\phi_{\bullet})}\subset\\{x_{*,\bullet}^{2}+y_{*,\bullet}^{2}<1\\},$

the new denominator of $\mathfrak{q}_{\sigma,k,-1/2}^{\mathrm{f},1}$ cancels
for no $(\xi,\eta)\in{\mathbb{R}}^{3}$, and the last two assumptions on the
principal symbol follow from the positivity of $a$,
$\cancel{a}_{x_{*,\bullet}}$ and $\cancel{a}_{y_{*,\bullet}}$ as noticed in
Rmk. 10.

###  3.5. Proof of Thm. 1

We use the abstract setting developped in Sect. 2 to prove Thm. 1. In Sect. 2,
we replace $\rho$ by $\boldsymbol{\rho}_{\pm}$, $\chi$ by $\chi_{\pm}$, we
define $\mathcal{X}_{\pm}^{k,-1/2}$ and $\mathcal{H}_{\pm}^{k,-1/2}$
accordingly to the boundary considered (near
$\\{\boldsymbol{\rho}_{\pm}=0\\}$), then we replace
$u\in\mathcal{H}_{\pm}^{k,-1/2}$ by $\widetilde{\chi}_{\pm}u$ with $u\in
H^{k}$ – we take $l=-1/2$ because
$\|\widetilde{\chi}_{\pm}u\|_{L^{2}((r_{-},r_{+})\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)}=\|\boldsymbol{\rho}_{\pm}^{1/2}\widetilde{\chi}_{\pm}u\|_{L^{2}((0,+\infty),\frac{\mathrm{d}\boldsymbol{\rho}_{\pm}}{\boldsymbol{\rho}_{\pm}})}$.

By Prop. 11, for all $N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$ such that
$N>2(N^{\prime}-1)$, there exist a right boundary parametrix
$\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}\in\mathcal{B}(\mathcal{H}_{\pm}^{k-1,-1/2},\mathcal{X}_{\pm}^{k,-1/2})$
as well as a left boundary parametrix
$\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}\in\mathcal{B}(\mathcal{H}_{\pm}^{k-1,-1/2},\mathcal{X}_{\pm}^{k,-1/2})$
such that:

$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}$
$\displaystyle=\chi_{\pm}\mathds{1}_{\mathcal{H}_{\pm}^{k-1,-1/2}}+\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm},$
$\displaystyle\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}\widetilde{\boldsymbol{P}}_{\sigma}$
$\displaystyle=\chi_{\pm}\mathds{1}_{\mathcal{X}_{\pm}^{k,-1/2}}+\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}.$

The remainders
$\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}\in\mathcal{B}(\mathcal{H}_{\pm}^{k-1,-1/2})$
and
$\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,\pm}\in\mathcal{B}(\mathcal{X}_{\pm}^{k,-1/2})$
are compact. We then glue together the boundaries parametrices and compact
remainders:

$\displaystyle\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1}$
$\displaystyle:=\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1,-1/2,-}+\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1,-1/2,+},\qquad\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1}:=\boldsymbol{K}_{\sigma,k,-1/2,-}+\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1,-1/2,+},$
$\displaystyle\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1}$
$\displaystyle:=\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,-}+\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,+},\qquad\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1}:=\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,-}+\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1,-1/2,+}.$

Using the facts that $\chi_{-}+\chi_{+}=1$,
$\chi_{\pm}\mathcal{H}_{\pm}^{k-1,-1/2}\hookrightarrow H^{k-1}$ and
$\chi_{\pm}\mathcal{X}_{\pm}^{k,-1/2}\hookrightarrow\mathcal{X}^{k}$ (as
$\boldsymbol{P}_{\sigma}\chi_{\pm}=[\boldsymbol{P}_{\sigma},\chi_{\pm}]+\chi_{\pm}\boldsymbol{P}_{\sigma}$),
we finally obtain:

(15)
$\displaystyle\boldsymbol{P}_{\sigma}\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1}$
$\displaystyle=\mathds{1}_{H^{k}}+\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1},$
$\displaystyle\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1}\boldsymbol{P}_{\sigma}$
$\displaystyle=\mathds{1}_{\mathcal{X}^{k}}+\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1}.$

It is well known that operators of the form ”identity $+$ compact” are index 0
Fredholm operators (_cf._ [Hö, Cor. 19.1.8]); as a result,
$\boldsymbol{P}_{\sigma}$ as well as
$\boldsymbol{Q}^{N,N^{\prime}}_{\sigma,k-1}$ and
$\widetilde{\boldsymbol{Q}}^{N,N^{\prime}}_{\sigma,k-1}$ are Fredholm
operators (_cf._ [Hö, Cor. 19.1.9]). We show that the index of
$\boldsymbol{P}_{\sigma}$ is 0 as in the proof of Prop. 13 (we use there that
$a,\cancel{a}_{\theta},\cancel{a}_{x_{*},\bullet},\cancel{a}_{y_{*},\bullet}>0$).
Since $\boldsymbol{P}_{\sigma}$ depends analytically on
$\sigma\in{\mathbb{C}}$, Prop. 11 entails that
$\boldsymbol{K}^{N,N^{\prime}}_{\sigma,k-1}$ and
$\widetilde{\boldsymbol{K}}^{N,N^{\prime}}_{\sigma,k-1}$ are analytic in
$\sigma$ provided that
$0\notin\bigcup_{j=1}^{N^{\prime}}\mathfrak{p}_{\sigma,k-1+j-1,-1/2,\pm}({\mathbb{R}})$;
by Rmk. 12,

$\displaystyle\mathfrak{p}_{\sigma,k-1+j-1,-1/2,\pm}(\xi)$
$\displaystyle=|(\partial_{r}\mu)(r_{\pm})|(\xi-\mathrm{i}(k-1/2+j-1))\left(\xi-\mathrm{i}(k-1/2+j-1)+\frac{b(0,\theta,\sigma)}{a(0)}\right)$

where

$\displaystyle\frac{b(0,\theta,\sigma)}{a(0)}$
$\displaystyle=-\frac{2}{|(\partial_{r}\mu)(r_{\pm})|}\left((1+\lambda)(r_{\pm}^{2}+a^{2})\left(\sigma-\frac{qQr_{\pm}}{r_{\pm}^{2}+a^{2}\cos^{2}\theta}\right)+a(1+\lambda)\left(\ell+\frac{qQr_{\pm}a\sin^{2}\theta}{r_{\pm}^{2}+a^{2}\cos^{2}\theta}\right)\right).$

The indicial polynomial $\mathfrak{p}_{\sigma,k-1+j-1,-1/2,\pm}$ can thus
cancel at some $\xi\in{\mathbb{R}}$ if and only if $\sigma$ is such that
$-(k-1/2+j-1)-\frac{2(1+\lambda)(r_{\pm}^{2}+a^{2})\mathrm{Im}(\sigma)}{|(\partial_{r}\mu)(r_{\pm})|}=0$,
whence the restriction of $\sigma$ to ${\mathbb{C}}_{k}$ for $N^{\prime}=1$.

Notice finally that, when $a=0$, $\boldsymbol{P}_{\sigma}$ is elliptic inside
$(r_{-},r_{+})\times\mathbb{S}^{2}$ (_cf._ [BeHa, Sect. 2]) thus fits into the
abstract setting of 2 without any harmonical restriction. This completes the
proof.

######  Remark 22.

Observe that multiplication by $\rho^{2}\Omega^{-2}$ in the definition of
$\boldsymbol{P}_{\sigma}$ does not change the critical strip as only the ratio
$\frac{b(r_{\pm},\theta,\sigma)}{a(r_{\pm})}$ appears in
$\mathfrak{p}_{\sigma,k-1+j-1,-1/2,\pm}(\xi)$.

######  Remark 23.

The exact structure of the characteristic set of the charged KG operator in
the cotangent bundle, especially the existence of radial sets at the cospheres
at infinity above the horizons $\\{r=r_{\pm}\\}$, is not relevant for our
purpose. We refer to [V, Sect. 2.2] for the definition of the radial sets in
the De Sitter-Kerr setting; see also [V, Sect. 2.4] for propagation estimates
therein (as it turns out, one obtains the same estimates at the characteristic
set whether one microlocalises the analysis near the radial sets or not).

##  4\. Numerical scheme

In this last section, we present a method to numerically compute resonances
using the right parametrix previously constructed for
$\boldsymbol{P}_{\sigma}$ (the angular momentum $\ell\in{\mathbb{Z}}$ is still
fixed here, _cf._ Section 3.3). We will use the notations of Sect. 3.5. We
place ourselves in the context of Thm. 5, that is, we assume that
$\boldsymbol{P}_{\sigma_{0}}^{-1}$ exists for some
$\sigma_{0}\in{\mathbb{C}}_{k}$ with $k\in{\mathbb{N}}\setminus\\{0\\}$ (so
that Thm. 1 combined with analytic Fredholm theory ensure that
${\mathbb{C}}_{k}\ni\sigma\mapsto(\boldsymbol{P}_{\sigma}^{-1},\mathcal{B}(H^{k-1},\mathcal{X}^{k}))$
is meromorphic).

###  4.1. Trace class property

In this section, we prove the following result:

######  Lemma 24.

Let $k,N,N^{\prime}\in{\mathbb{N}}\setminus\\{0\\}$ be such that $N\geq
3N^{\prime}-2$ (so that Prop. 11 applies) and $N-2(N^{\prime}-1)>3$. Then for
all $\sigma\in{\mathbb{C}}_{k,N^{\prime}}$ (where
${\mathbb{C}}_{k,N^{\prime}}$ is defined in (3)), the realization
$(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(H^{k-1}))$ is of
trace class.

Recall that the _singular values_ $s_{0}\geq s_{1}\geq\ldots\geq
s_{j}\geq\ldots$ of a compact operator $K\in\mathcal{B}(H_{1},H_{2})$ acting
between two Hilbert spaces $H_{1}$ and $H_{2}$ are the (positive) square roots
of the eigenvalues of the compact, non-negative self-adjoint operator $K^{*}K$
(_cf._ [DZ, Prop. B.13]), and then $K$ is of trace class if and only if

$\displaystyle\|K\|_{1}$ $\displaystyle:=\sum_{j=0}^{+\infty}s_{j}<+\infty.$

As for finite rank operators, the trace is then defined by

$\displaystyle K$
$\displaystyle\longmapsto\sum_{j=0}^{+\infty}\sum_{k=0}^{+\infty}\langle
f_{k},Ke_{j}\rangle_{H_{2}}$

for any orthonormal Hilbert bases $(e_{j})_{j\in{\mathbb{N}}}\subset H_{1}$
and $(f_{k})_{k\in{\mathbb{N}}}\subset H_{2}$ (the trace being a linear thus
bounded functional, this invariance property extends to the space of all trace
class operators $H_{1}\to H_{2}$).

###### Proof of Lem. 24.

The assumption $\sigma\in{\mathbb{C}}_{k,N^{\prime}}$ ensures that
$(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(H^{k-1}))$ is well-
defined. Put
$\bar{H}^{k-1}:=H^{k-1}([r_{-},r_{+}]\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)$.
Let us define
$\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}\in\mathcal{B}(\bar{H}^{k-1})$
by

$\displaystyle(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}u)(r,\omega)$
$\displaystyle:=\begin{cases}(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}u)(r,\omega)&\text{if
$(r,\omega)\in(r_{-},r_{+})\times\mathbb{S}^{2}$},\\\ 0&\text{if
$r=r_{\pm}$}.\end{cases}$

The boundary values (defined as limits as $r\to r_{\pm}$) are consistent with
the fact that the total order in $|r-r_{\pm}|$ in
$\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}$ is $N^{\prime}>0$. We show that
the realization
$(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(\bar{H}^{k-1}))$
is trace class; as we suppose that $N\geq 3N^{\prime}-2$, we have
$\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}\in\mathcal{B}(\bar{H}^{k-1},\bar{H}^{k-1+N-2(N^{\prime}-1)})$.
We will then deduce from Lem. 27 that
$(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(H^{k-1}))$ is trace
class as well.

Let us denote by
$\widetilde{\bar{\boldsymbol{K}}}_{\sigma,k-1}^{N,N^{\prime}}$ the realization
$(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(\bar{H}^{k-1},\bar{H}^{k-1+N-2(N^{\prime}-1)}))$.
Let
$(x_{1},x_{2},x_{3}):=(r,\theta,\varphi)\in[r_{-},r_{+}]\times[0,\pi]\times[0,2\pi]$,
$(\ell_{1},\ell_{2},\ell_{3}):=(r_{+}-r_{-},\pi,2\pi)$,
$\omega:=(\omega_{1},\omega_{2},\omega_{3}):=(\frac{2\pi}{\ell_{1}},\frac{2\pi}{\ell_{2}},\frac{2\pi}{\ell_{3}})$
then define

$\displaystyle\mathrm{e}_{\mathfrak{n},k}(x)$
$\displaystyle:=\frac{1}{\langle(\mathfrak{n}\omega)^{k}\rangle}\prod_{j=1}^{3}\frac{\mathrm{e}^{\mathrm{i}\mathfrak{n}_{j}\omega_{j}x_{j}}}{\sqrt{\ell_{j}}}\qquad\qquad\forall\mathfrak{n}:=(\mathfrak{n}_{1},\mathfrak{n}_{2},\mathfrak{n}_{3})\in{\mathbb{Z}}^{3}.$

Above,
$\langle(\mathfrak{n}\omega)^{k}\rangle^{2}:=1+(\mathfrak{n}_{1}\omega_{1})^{2k}+(\mathfrak{n}_{2}\omega_{2})^{2k}+(\mathfrak{n}_{3}\omega_{3})^{2k}$.
Following the proof of [DZ, Prop. B.21], we show that
$\iota:\bar{H}^{k-1+N-2(N^{\prime}-1)}\hookrightarrow\bar{H}^{k-1}$ is trace
class with131313For all $(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}$ and
$a,b\in{\mathbb{N}}$ such that $b-a\geq 1$, we have
$\displaystyle(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{b-a}$ $\displaystyle\leq
3^{b-a-1}(x_{1}^{2(b-a)}+x_{2}^{2(b-a)}+x_{3}^{2(b-a)})$ so that
$\displaystyle(x_{1}^{2a}+x_{2}^{2a}+x_{3}^{2a})(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{b-a}$
$\displaystyle\leq
3^{b-a-1}(x_{1}^{2a}+x_{2}^{2a}+x_{3}^{2a})(x_{1}^{2(b-a)}+x_{2}^{2(b-a)}+x_{3}^{2(b-a)})\leq
3^{b-a}(x_{1}^{2b}+x_{2}^{2b}+x_{3}^{2b})$ by Young’s inequality for products.
Consequently:  $\displaystyle\|\iota\|_{1}$ $\displaystyle\leq
1+\sqrt{2}\sum_{\mathfrak{n}\in{\mathbb{Z}}^{3}\setminus\\{0_{{\mathbb{Z}}^{3}}\\}}\sqrt{\frac{(\mathfrak{n}_{1}\omega_{1})^{2(k-1)}+(\mathfrak{n}_{2}\omega_{2})^{2(k-1)}+(\mathfrak{n}_{3}\omega_{3})^{2(k-1)}}{(\mathfrak{n}_{1}\omega_{1})^{k-1+N-2(N^{\prime}-1)}+(\mathfrak{n}_{2}\omega_{2})^{k-1+N-2(N^{\prime}-1)}+(\mathfrak{n}_{3}\omega_{3})^{k-1+N-2(N^{\prime}-1)}}}$
$\displaystyle\leq 1+\sqrt{2\cdot
3^{N-2(N^{\prime}-1)}}\sum_{\mathfrak{n}\in{\mathbb{Z}}^{3}\setminus\\{0_{{\mathbb{Z}}^{3}}\\}}\frac{1}{((\mathfrak{n}_{1}\omega_{1})^{2}+(\mathfrak{n}_{2}\omega_{2})^{2}+(\mathfrak{n}_{3}\omega_{3})^{2})^{(N-2(N^{\prime}-1))/2}}.$
:

$\displaystyle\|\iota\|_{1}$
$\displaystyle=\sum_{\mathfrak{n}\in{\mathbb{Z}}^{3}}\frac{\langle(\mathfrak{n}\omega)^{k-1}\rangle}{\langle(\mathfrak{n}\omega)^{k-1+N-2(N^{\prime}-1)}\rangle}<+\infty.$

This comes from the decomposition (_cf._ [DZ, eq. (B.3.3)])

$\displaystyle\iota$
$\displaystyle=\sum_{\mathfrak{n}\in{\mathbb{Z}}^{3}}s_{\mathfrak{n}}\left\langle\mathrm{e}_{\mathfrak{n},k-1+N-2(N^{\prime}-1)},\bullet\right\rangle_{\bar{H}^{k-1+N-2(N^{\prime}-1)}}\mathrm{e}_{\mathfrak{n},k-1}$

from which we get:

$\displaystyle s_{\mathfrak{n}}$
$\displaystyle=\left\|\iota\mathrm{e}_{\mathfrak{n},k-1+N-2(N^{\prime}-1)}\right\|_{\bar{H}^{k-1}}=\left\|\mathrm{e}_{\mathfrak{n},k-1+N-2(N^{\prime}-1)}\right\|_{\bar{H}^{k-1}}=\frac{\langle(\mathfrak{n}\omega)^{k-1}\rangle}{\langle(\mathfrak{n}\omega)^{k-1+N-2(N^{\prime}-1)}\rangle}.$

That the series converges follows from the assumption $N-2(N^{\prime}-1)>3$.
It remains to use [DZ, eq. (B.4.7)] to obtain:

(16) $\displaystyle\|\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}$
$\displaystyle=\|\iota\widetilde{\bar{\boldsymbol{K}}}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}\leq\|\iota\|_{1}\|\widetilde{\bar{\boldsymbol{K}}}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(\bar{H}^{k-1},\bar{H}^{k-1+N-2(N^{\prime}-1)})}<+\infty.$

This completes the proof. ∎

######  Remark 25.

Observe that assuming $N-2(N^{\prime}-1)>3$ is necessary because of the
coordinate singularities at the poles of $\mathbb{S}^{2}$; if we could use
only the Boyer-Lindquist coordinates then $N-2(N^{\prime}-1)>2$ would have
been sufficient for the remainder to be trace class.

Observe also that increasing $N^{\prime}$ creates new critical strips in
${\mathbb{C}}$; however each increment of $N^{\prime}$ adds such a strip below
all the others. As a consequence, when one deforms contour integration from
the upper half plane up to the upper critical strip below ${\mathbb{R}}$ to
obtain resonances expansion (1) (as _e.g._ in [BoHa, Sect. 3.2]), the exact
value of $N^{\prime}>0$ does not matter.

######  Remark 26.

Let $\varepsilon>0$ and denote by $N_{\varepsilon}$ the number of eigenvalues
$\lambda$ of $\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}$ satisfying
$|\lambda|\geq\varepsilon$. Then

$\displaystyle\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}$
$\displaystyle\geq\sum_{\lambda\text{
eigenvalue}}|\lambda|\geq\sum_{\begin{subarray}{c}\lambda\text{ eigenvalue}\\\
|\lambda|\geq\varepsilon\end{subarray}}|\lambda|\geq\varepsilon
N_{\varepsilon},$

hence

$\displaystyle N_{\varepsilon}$
$\displaystyle\leq\frac{1}{\varepsilon}\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}.$

######  Lemma 27.

We use the notations of the proof of Lem. 24. The realization
$(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(\bar{H}^{k-1}))$
is trace class if and only if
$(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(H^{k-1}))$ is.
Moreover, both the operators have the same singular values.

###### Proof.

Let $(\bar{e}_{j})_{j\in{\mathbb{N}}}$ be an orthonormal Hilbert basis of
$\bar{H}^{k-1}$ and set
$e_{j}:=(\bar{e}_{j})_{|(r_{-},r_{+})\times\mathbb{S}^{2}}$; then
$(e_{j})_{j\in{\mathbb{N}}}$ is clearly an orthonormal Hilbert basis of
$H^{k-1}$. The converse is also true: given $e_{j}\in H^{k}$ an element of a
basis of $H^{k-1}$, we can form a basis for $\bar{H}^{k-1}$ by extending
$e_{j}$ to a neighborhood of $[r_{-},r_{+}]\times\mathbb{S}^{2}$.

Let us denote by $\langle\bullet,\bullet\rangle$ the standard inner product of
$L^{2}((r_{-},r_{+})\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)$; as
$\\{r_{\pm}\\}\times\mathbb{S}^{2}$ are measure zero sets,
$\langle\bullet,\bullet\rangle$ coincides with the standard inner product of
$L^{2}([r_{-},r_{+}]\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)$.

If $\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$ is trace class, then we
can write

$\displaystyle\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$
$\displaystyle=\sum_{j\in{\mathbb{N}}}s_{j}\langle\bar{e}_{j},\bullet\rangle\bar{f}_{j}$

where

$\displaystyle\sum_{j\in{\mathbb{N}}}s_{j}$ $\displaystyle<+\infty,$

$(\bar{e}_{j})_{j\in{\mathbb{N}}}$ is an orthonormal Hilbert basis of
$\bar{H}^{k-1}$ composed of eigenvectors of
$(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}})^{*}\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$
and
$\bar{f}_{j}=s_{j}^{-1}\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}\bar{e}_{j}$
when $s_{j}\neq 0$, and $\bar{f}_{j}=0$ otherwise (_cf._ the proof of [DZ,
Prop. B.13]). Then for all $u\in H^{k-1}$ and all
$(r,\omega)\in(r_{-},r_{+})\times\mathbb{S}^{2}$,

$\displaystyle(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}u)(r,\omega)$
$\displaystyle=(\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}u)(r,\omega)=\sum_{j\in{\mathbb{N}}}s_{j}\langle\bar{e}_{j},u\rangle\bar{f}_{j}(r,\omega)=\sum_{j\in{\mathbb{N}}}s_{j}\langle
e_{j},u\rangle f_{j}(r,\omega)$

and $\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}$ is trace class with singular
values $(s_{j})_{j\in{\mathbb{N}}}$. The converse is true for
$f_{j}(r_{\pm},\omega)=0$ for all $\omega\in\mathbb{S}^{2}$ by construction of
$\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$. ∎

######  Remark 28.

It is a matter of fact that Lem. 24 is not necessary for the numerical scheme
below: indeed, it is enough to work with the extension
$\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$ on
$H^{k-1}([r_{-},r_{+}]\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)$ and
observe that $-1$ is an eigenvalue of
$\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}}$ if and only if it is for
$\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}$; it then follows that the zeros
(above the suitable critical strip) of
$\det(\mathds{1}_{H^{k-1}([r_{-},r_{+}]\times\mathbb{S}^{2},\mathrm{d}r\mathrm{d}\omega)}+\bar{\boldsymbol{K}}_{\sigma,k-1}^{N,N^{\prime}})$
are exactly the resonances of $\boldsymbol{P}_{\sigma}$. Note that either
approach yields the same computations in practice.

###  4.2. Numerical approximation of resonances

In this second section, we gather all the previous results to approximate
resonances and estimate the associated numerical error (thus proving Thm. 5).

Let $k\in{\mathbb{N}}\setminus\\{0\\}$. Under the assumptions of Lem. 24
above, $(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}},\mathcal{B}(H^{k-1}))$ is
of trace class therefore the Fredholm determinant of
$\mathds{1}_{H^{k-1}}+\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}$ is well
defined and we have (_cf._ [DZ, Prop. B.28]):

$\displaystyle\mathds{1}_{H^{k-1}}+\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\text{
is invertible}\qquad$ $\displaystyle\Longleftrightarrow\qquad
D(\sigma):=\det(\mathds{1}_{H^{k-1}}+\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}})\neq
0.$

In virtue of (15) and the index 0 property of
$(\boldsymbol{P}_{\sigma},\mathcal{X}^{k})$, we deduce that141414One usually
calls resonances the poles of $\boldsymbol{P}_{\sigma}^{-1}$ lying above the
upper critical strip; in the present paper, we indifferently use the term
resonances for all these poles lying outside the critical strips.

$\sigma\in{\mathbb{C}}_{k,N^{\prime}}$ is a resonance
$\displaystyle\Longleftrightarrow\qquad D(\sigma)=0$

which proves the first part of Thm. 5. The right hand side above is an
analytic complex valued function whose zeros can be approximated or counted
inside a fixed domain in ${\mathbb{C}}$.

To see how, consider a family $(\Pi_{R})_{R\in{\mathbb{N}}\setminus\\{0\\}}$
of projectors onto some $R$ dimensional linear subspaces of $H^{k-1}$,
identified to ${\mathbb{C}}^{R}$. As $R\to+\infty$,
$\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(H^{k-1})}$
by compactness; moreover, the proof of [DZ, Prop. B.20] shows that (recall
that $s_{j}(A)$ are the singular values of $A$)

(17) $\displaystyle\mathfrak{T}_{R}(\sigma)$
$\displaystyle:=\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}\leq\sum_{j\in{\mathbb{N}}}\min\big{\\{}s_{j}(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}),\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(H^{k-1})}\big{\\}}\longrightarrow
0$

and if we set

$\displaystyle D_{R}(\sigma)$
$\displaystyle:=\det(\mathds{1}_{{\mathbb{C}}^{R}}+\Pi_{R}\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}),$

then (see _e.g._ the remark below [S, Thm. 5.1]):

(18) $\displaystyle\mathfrak{D}_{R}(\sigma)$
$\displaystyle:=|D(\sigma)-D_{R}(\sigma)|\leq\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}\mathrm{e}^{1+\max\\{\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1},\,\|\Pi_{R}\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}\\}}.$

Combining (17) and (18) together, we come with the following error estimate:

(19) $\displaystyle\mathfrak{D}_{R}(\sigma)$
$\displaystyle\leq\mathfrak{T}_{R}(\sigma)\mathrm{e}^{1+\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}+\mathfrak{T}_{R}(\sigma)}.$

We can now work with $D_{R}(\sigma)$ which is nothing but the determinant of a
matrix (so that we can evaluate it with a computer). We finally take advantage
of the analiticity in $\sigma\in{\mathbb{C}}_{k,N^{\prime}}$: for any
positively oriented contour $\Gamma\subset{\mathbb{C}}_{k,N^{\prime}}$ which
does not intersect any resonances, let $N(\Gamma)$ and $S(\Gamma,n)$
($n\in{\mathbb{N}}\setminus\\{0\\}$) be respectively the number of resonances
lying inside $\Gamma$ and the sum of the $n$-th power of these resonances
(counted with their multiplicity):

$\displaystyle N(\Gamma)$
$\displaystyle:=\frac{1}{2\pi\mathrm{i}}\oint_{\Gamma}\frac{\partial_{\sigma}D(\sigma)}{D(\sigma)}\mathrm{d}\sigma,\qquad
S(\Gamma,n):=\frac{1}{2\pi\mathrm{i}}\oint_{\Gamma}\frac{\partial_{\sigma}D(\sigma)}{D(\sigma)}\sigma^{n}\mathrm{d}\sigma.$

If $N_{R}(\Gamma)$ and $S_{R}(\Gamma,n)$ are defined as above but with
$D_{R}(\sigma)$ insted of $D(\sigma)$, then:

$\displaystyle|N(\Gamma)-N_{R}(\Gamma)|$
$\displaystyle\leq\frac{1}{2\pi}\oint_{\Gamma}\frac{1}{|D_{R}(\sigma)|}\left(\frac{|\partial_{\sigma}D(\sigma)||D(\sigma)-D_{R}(\sigma)|}{|D(\sigma)|}+|\partial_{\sigma}D_{R}(\sigma)-\partial_{\sigma}D(\sigma)|\right)|\mathrm{d}\sigma|,$
$\displaystyle|S(\Gamma,n)-S_{R}(\Gamma,n)|$
$\displaystyle\leq\frac{1}{2\pi}\oint_{\Gamma}\frac{1}{|D_{R}(\sigma)|}\left(\frac{|\partial_{\sigma}D(\sigma)||D(\sigma)-D_{R}(\sigma)|}{|D(\sigma)|}+|\partial_{\sigma}D_{R}(\sigma)-\partial_{\sigma}D(\sigma)|\right)|\sigma|^{n}|\mathrm{d}\sigma|.$

Using Cauchy formula for derivatives, we see that

$\displaystyle|\partial_{\sigma}D(\sigma)|$
$\displaystyle\leq\frac{1}{\delta(\sigma)}\max_{|\sigma^{\prime}-\sigma|\leq\delta(\sigma)}\\{|D(\sigma^{\prime})|\\},\qquad|\partial_{\sigma}D_{R}(\sigma)-\partial_{\sigma}D(\sigma)|\leq\frac{1}{\delta(\sigma)}\max_{|\sigma^{\prime}-\sigma|\leq\delta(\sigma)}\\{\mathfrak{D}_{R}(\sigma^{\prime})\\}$

where $\delta(\sigma)>0$ is such that
$\\{\sigma+\delta\mathrm{e}^{\mathrm{i}t}\ |\ 0\leq t\leq
2\pi\\}\subset{\mathbb{C}}_{k,N^{\prime}}$ for all $\sigma\in\Gamma$; on the
other hand, [DZ, eq. (B.5.11)] shows that:

$\displaystyle|D(\sigma)|$
$\displaystyle\leq\mathrm{e}^{\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{1}}.$

Consequently, we eventually get:

$\displaystyle|N(\Gamma)-N_{R}(\Gamma)|$
$\displaystyle\leq\frac{1}{2\pi}\oint_{\Gamma}\frac{1}{|D_{R}(\sigma)|}\frac{1}{\delta(\sigma)}\max_{|\sigma^{\prime}-\sigma|\leq\delta(\sigma)}\left\\{\frac{\mathrm{e}^{\|\boldsymbol{K}_{\sigma^{\prime},k-1}^{N,N^{\prime}}\|_{1}}}{|D(\sigma)|}\mathfrak{D}_{R}(\sigma)+\mathfrak{D}_{R}(\sigma^{\prime})\right\\}|\mathrm{d}\sigma|,$
$\displaystyle|S(\Gamma,n)-S_{R}(\Gamma,n)|$
$\displaystyle\leq\frac{1}{2\pi}\oint_{\Gamma}\frac{1}{|D_{R}(\sigma)|}\frac{1}{\delta(\sigma)}\max_{|\sigma^{\prime}-\sigma|\leq\delta(\sigma)}\left\\{\frac{\mathrm{e}^{\|\boldsymbol{K}_{\sigma^{\prime},k-1}^{N,N^{\prime}}\|_{1}}}{|D(\sigma)|}\mathfrak{D}_{R}(\sigma)+\mathfrak{D}_{R}(\sigma^{\prime})\right\\}|\sigma|^{n}|\mathrm{d}\sigma|.$

The rougher estimates of Thm. 5 are obtained as follows: we pick any contour
$\Gamma^{\prime}\subset{\mathbb{C}}_{k,N^{\prime}}$ containing $\Gamma$,
denote by $\delta^{\prime}>0$ the distance between $\Gamma$ and
$\Gamma^{\prime}$, then we take the maximum and minimum respectively of the
numerator and denominator of the integrand over $\Gamma$ (for the $\sigma$
variable) or $\Gamma^{\prime}$ (for the $\sigma^{\prime}$ variable). The
constants of the latter estimate are thus:

(20) $\displaystyle C_{R}(\Gamma)$
$\displaystyle=\frac{\mathrm{lg}(\Gamma)}{2\pi\delta^{\prime}\min_{\sigma\in\Gamma}\\{|D_{R}(\sigma)|\\}}\left(\frac{\max_{|\sigma^{\prime}-\sigma|\leq\delta^{\prime}}\\{\mathrm{e}^{\|\boldsymbol{K}_{\sigma^{\prime},k-1}^{N,N^{\prime}}\|_{1}}\\}}{\min_{\sigma\in\Gamma}\\{|D_{R}(\sigma)|\\}-\max_{\sigma\in\Gamma}\\{\mathfrak{D}_{R}(\sigma)\\}}\max_{\sigma\in\Gamma}\\{\mathfrak{D}_{R}(\sigma)\\}+\max_{|\sigma^{\prime}-\sigma|\leq\delta^{\prime}}\\{\mathfrak{D}_{R}(\sigma^{\prime})\\}\right),$
$\displaystyle\widetilde{C}_{R}(\Gamma,n)$
$\displaystyle=C_{R}(\Gamma)|\Gamma|^{n}.$

Above $\mathrm{lg}(\Gamma)$ denotes the length of $\Gamma$ and
$|\Gamma|:=\max\\{|\sigma|\ |\ \sigma\in\Gamma\\}$. Notice that
$C_{R}(\Gamma),\widetilde{C}_{R}(\Gamma,n)=\mathcal{O}(\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(H^{k-1})})$
as we can deduce from (17) and (19).

We briefly explain how to evaluate $C_{R}(\Gamma)$ and
$\widetilde{C}_{R}(\Gamma,n)$: the minimum over $\Gamma$ of $|D_{R}|$ which
can be numerically evaluated, as well as maxima over $\Gamma$ and
$\Gamma^{\prime}$ of $\mathfrak{D}_{R}$; by (19), the latter can be bounded if
we have an estimate on the singular values
$s_{j}(\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}})$ and
$\|(\mathds{1}_{H^{k-1}}-\Pi_{R})\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(H^{k-1})}$.
As the singular values can themselves be estimated using (16) (replacing
$\|\iota\|_{1}$ by $s_{j}(\iota)$) and a uniform bound of151515That both these
norms coincide follows from the fact that after $k-1$ derivaitves, we consider
the quantization of a symbol containing at least one power of $|r-r_{\pm}|$
near $\\{r=r_{\pm}\\}$.
$\|\widetilde{\bar{\boldsymbol{K}}}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(\bar{H}^{k-1},\bar{H}^{k-1+N-2(N^{\prime}-1)})}=\|\boldsymbol{K}_{\sigma,k-1}^{N,N^{\prime}}\|_{\mathcal{B}(H^{k-1},H^{k-1+N-2(N^{\prime}-1)})}$
over $\Gamma$ and $\Gamma^{\prime}$, we are done.

##  5\. Appendix: Explicit computation of $\boldsymbol{P}_{\sigma}$ in the
$(t_{*},r,x_{*,\bullet},y_{*,\bullet})$ coordinates

We give a detailed derivation of the spectral family near the poles of
$\mathbb{S}^{2}$ (_cf._ Sect. 3.3). We drop the index $*,\bullet$ to lighten
notations.

Using
$\partial_{\bar{\varphi}}=\frac{1+\lambda}{\kappa(\bullet)}\partial_{\varphi}$,
$x\partial_{y}-y\partial_{x}=\partial_{\bar{\varphi}}$ as well as
$g^{tx}=-yg^{t\varphi}\frac{\kappa(\bullet)}{1+\lambda}$,
$g^{ty}=xg^{t\varphi}\frac{\kappa(\bullet)}{1+\lambda}$,
$g^{rx}=-yg^{r\varphi}\frac{\kappa(\bullet)}{1+\lambda}$ and
$g^{ry}=xg^{r\varphi}\frac{\kappa(\bullet)}{1+\lambda}$, we compute:

$\displaystyle\nabla_{\gamma}\nabla^{\gamma}$
$\displaystyle=g^{tt}\partial_{t}^{2}+g^{tr}\partial_{t}\partial_{r}+g^{tx}\partial_{t}\partial_{x}+g^{ty}\partial_{t}\partial_{y}$
$\displaystyle\quad+\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{tr}\partial_{t}+g^{rx}\partial_{x}+g^{ry}\partial_{y})-\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\frac{\mu}{\Omega^{2}}\partial_{r}+\partial_{r}(g^{tr}\partial_{t}+g^{rx}\partial_{x}+g^{ry}\partial_{y})$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{x}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{tx}\partial_{t}+g^{rx}\partial_{r}+g^{xx}\partial_{x}+g^{xy}\partial_{y})+\partial_{x}(g^{tx}\partial_{t}+g^{rx}\partial_{r}+g^{xx}\partial_{x}+g^{xy}\partial_{y})$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{y}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{ty}\partial_{t}+g^{ry}\partial_{r}+g^{xy}\partial_{x}+g^{yy}\partial_{y})+\partial_{y}(g^{ty}\partial_{t}+g^{ry}\partial_{r}+g^{xy}\partial_{x}+g^{yy}\partial_{y})$
$\displaystyle=g^{tt}\partial_{t}^{2}+2g^{t\varphi}\partial_{t}\partial_{\varphi}-\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\frac{\mu}{\Omega^{2}}\partial_{r}+\partial_{x}(g^{xx}\partial_{x}+g^{xy}\partial_{y})+\partial_{y}(g^{xy}\partial_{x}+g^{yy}\partial_{y})$
$\displaystyle\quad+[g^{tr}\partial_{t}+g^{t\varphi}\partial_{\varphi},\partial_{r}]_{+}+\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{tr}\partial_{t}+g^{r\varphi}\partial_{\varphi})$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{x}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xx}\partial_{x}+g^{xy}\partial_{y})+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{y}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xy}\partial_{x}+g^{yy}\partial_{y}).$

Using next $xA_{y}-yA_{x}=\frac{1+\lambda}{\kappa(\bullet)}A_{\varphi}$,
$g^{tx}A_{x}+g^{ty}A_{y}=g^{t\varphi}A_{\varphi}$,
$g^{rx}A_{x}+g^{ry}A_{y}=g^{r\varphi}A_{\varphi}$, we compute:

$\displaystyle\nabla_{\gamma}A^{\gamma}+A_{\gamma}\nabla^{\gamma}$
$\displaystyle=[\partial_{\gamma},g^{\gamma\delta}A_{\delta}]_{+}+\frac{1}{\sqrt{\det
g}}\partial_{\gamma}\\!\left(\sqrt{\det g}\right)g^{\gamma\delta}A_{\delta}$
$\displaystyle=[\partial_{t},g^{tt}A_{t}+g^{tx}A_{x}+g^{ty}A_{y}]_{+}+[\partial_{r},g^{tr}A_{t}+g^{rx}A_{x}+g^{ry}A_{y}]_{+}$
$\displaystyle\quad+[\partial_{x},g^{tx}A_{t}+g^{xx}A_{x}+g^{xy}A_{y}]_{+}+[\partial_{y},g^{ty}A_{t}+g^{xy}A_{x}+g^{yy}A_{y}]_{+}$
$\displaystyle\quad+\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{tr}A_{t}+g^{rx}A_{x}+g^{ry}A_{y})+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{x}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{tx}A_{t}+g^{xx}A_{x}+g^{xy}A_{y})$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{y}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{ty}A_{t}+g^{xy}A_{x}+g^{yy}A_{y})$
$\displaystyle=2g^{tt}A_{t}\partial_{t}+2g^{t\varphi}(A_{t}\partial_{\varphi}+A_{\varphi}\partial_{t})+[g^{tr}A_{t}+g^{r\varphi}A_{\varphi},\partial_{r}]_{+}$
$\displaystyle\quad+[g^{xx}A_{x}+g^{xy}A_{y},\partial_{x}]_{+}+[g^{xy}A_{x}+g^{yy}A_{y},\partial_{y}]_{+}$
$\displaystyle\quad+\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{tr}A_{t}+g^{r\varphi}A_{\varphi})+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{x}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xx}A_{x}+g^{xy}A_{y})$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{y}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xy}A_{x}+g^{yy}A_{y}).$

Finally:

$\displaystyle A_{\gamma}A^{\gamma}$
$\displaystyle=(g^{tt}A_{t}+g^{tx}A_{x}+g^{ty}A_{y})A_{t}+(g^{tx}A_{t}+g^{xx}A_{x}+g^{xy}A_{y})A_{x}+(g^{ty}A_{t}+g^{xy}A_{x}+g^{yy}A_{y})A_{y}$
$\displaystyle=g^{tt}A_{t}^{2}+2g^{t\varphi}A_{t}A_{\varphi}+(g^{xx}A_{x}+g^{xy}A_{y})A_{x}+(g^{xy}A_{x}+g^{yy}A_{y})A_{y}.$

Gathering everything together, we obtain:

$\displaystyle(\nabla_{\gamma}-\mathrm{i}qA_{\gamma})(\nabla^{\gamma}-\mathrm{i}qA^{\gamma})$
$\displaystyle=(\partial_{x}-\mathrm{i}qA_{x})(g^{xx}(\partial_{x}-\mathrm{i}qA_{x})+g^{xy}(\partial_{y}-\mathrm{i}qA_{y}))$
$\displaystyle\quad+(\partial_{y}-\mathrm{i}qA_{y})(g^{xy}(\partial_{x}-\mathrm{i}qA_{x})+g^{yy}(\partial_{y}-\mathrm{i}qA_{y}))$
$\displaystyle\quad-\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\frac{\mu}{\Omega^{2}}\partial_{r}+[g^{tr}(\partial_{t}-\mathrm{i}qA_{t})+g^{t\varphi}(\partial_{\varphi}-\mathrm{i}A_{\varphi}),\partial_{r}]_{+}$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{x}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xx}(\partial_{x}-\mathrm{i}qA_{x})+g^{xy}(\partial_{y}-\mathrm{i}qA_{y}))$
$\displaystyle\quad+\frac{\Omega^{4}\cos\theta}{\rho^{2}}\partial_{y}\\!\left(\frac{\rho^{2}}{\Omega^{4}\cos\theta}\right)(g^{xy}(\partial_{x}-\mathrm{i}qA_{x})+g^{yy}(\partial_{y}-\mathrm{i}qA_{y}))$
$\displaystyle\quad+\frac{\Omega^{4}}{\rho^{2}}\partial_{r}\\!\left(\frac{\rho^{2}}{\Omega^{4}}\right)(g^{tr}(\partial_{t}-\mathrm{i}qA_{t})+g^{r\varphi}(\partial_{\varphi}-\mathrm{i}qA_{\varphi}))$
$\displaystyle\quad+g^{tt}(\partial_{t}-\mathrm{i}qA_{t})^{2}+2g^{t\varphi}(\partial_{t}-\mathrm{i}qA_{t})(\partial_{\varphi}-\mathrm{i}qA_{\varphi}).$

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