# Polarimetric Information for Multi-Modal 6D Pose Estimation of
Photometrically Challenging Objects with Limited Data

Patrick Ruhkamp1,2 Daoyi Gao1 HyunJun Jung1 Nassir Navab1 Benjamin Busam1,2 1
TU Munich, 2 3Dwe.ai
p.ruhkamp {...}<EMAIL_ADDRESS>
###### Abstract

6D pose estimation pipelines that rely on RGB-only or RGB-D data show
limitations for photometrically challenging objects with e.g. textureless
surfaces, reflections or transparency. A supervised learning-based method
utilising complementary polarisation information as input modality is proposed
to overcome such limitations. This supervised approach is then extended to a
self-supervised paradigm by leveraging physical characteristics of polarised
light, thus eliminating the need for annotated real data. The methods achieve
significant advancements in pose estimation by leveraging geometric
information from polarised light and incorporating shape priors and invertible
physical constraints.

##  1 Introduction

We address limitations of 6D object pose estimation using depth sensor data
affected by inaccuracies from artifacts like multi-path interference, ambient
light, and transparency [24]. Utilising polarised light’s geometric
information for pose estimation has shown promising results, outperforming
RGB-only and RGB-D methods [9]. However, obtaining extensive annotated
datasets is challenging [27, 15]. We investigate the effectiveness of
polarimetric data in improving 6D object pose estimation through supervised
learning. Additionally, we propose a physics-induced polarimetric self-
supervision scheme to alleviate the need for large annotated real datasets
[10].

##  2 Related Work

Figure 1: Challenges in Depth Estimation for Household Objects.  Reflective
boundaries (1,3) and strong reflections (2,3) cause depth miscalculations,
while semi-transparent objects (4) become partially invisible to the depth
sensor.

Polarimetric Imaging.  Early studies on shape from polarisation (SfP) focus on
estimating surface normals and depth information by analysing the relationship
between polarisation and object surfaces. However, these works mainly operate
in controlled lab environments [3, 11, 22, 29]. Monocular polarisation images
are commonly used, but SfP can also utilise multiple views for depth
estimation [2, 6]. Polarimetric images are combined with photometric
information from stereo [1] or monocular RGB [31] to enhance depth
predictions. Additionally, polarised light improves noisy depth maps obtained
from other sensors [16]. Recent work by Ba et al. [4] employs neural networks
to predict surface normals by leveraging plausible cues extracted from
polarimetric images, demonstrating the disambiguation capability of such cues
in SfP. Inspired by these findings, our approach incorporates shape priors
from physical properties extracted from polarised light to complement pose
estimation.

6D Object Pose Estimation.  Dense correspondence-based methods have gained
popularity for 6D object pose estimation [30, 13, 17, 19, 21]. These methods
involve training neural networks to predict 2D-3D correspondences between
object pixels and their corresponding 3D locations on the object’s surface.
The correspondences are then used with PnP+RANSAC, the Umeyama algorithm, or
direct regression to compute the 6D object pose. ZebraPose [23] introduces
hierarchical feature representations, and zero-shot methods are being explored
for 6D pose estimation [20]. However, many correspondence-based methods suffer
from computationally expensive post-processing steps in RANSAC-based pose
solvers. GDR-Net [26] and its follower SO-Pose [7] address this limitation by
employing learning-based MLP networks to directly predict the target pose from
dense correspondences, thereby improving computational efficiency. In our
work, we build upon these findings to directly regress the object pose.

Geometric Depth Information.  Several methods, including FFB6D [12], Uni6D
[14], ESA6D [18], FS6D [28], and DGECN [5], incorporate depth information into
their prediction pipelines. However, these approaches heavily rely on depth
quality, which is problematic for photometrically challenging objects [9].
Geometric cues derived from polarization can potentially mitigate these
issues.

Polarimetric 6D Pose Prediction.  Recently published annotated datasets for
real-world polarimetric category-level [27] and instance-level [9] 6D pose
estimation enable the study of methods using this unexplored imaging modality.
PPP-Net [9] investigates the advantages of using polarisation for object pose
estimation and proposes a hybrid pipeline that combines physical model cues
with learning. Unlike previous methods such as Self6D [25] and Self6D++ [24],
we leverage polarimetric images and an extended differentiable renderer to
incorporate geometric representations and enable self-supervision based on our
invertible physical model.

##  3 Polarimetric Physical Model

RGB-D sensors often rely on active illumination for depth measurement.
However, they are susceptible to photometric challenges such as translucency
and reflections, leading to erroneous depth estimates. Figure 1 demonstrates
such depth artifacts. We instead leverage the surface normals obtained from
polarisation for 6D object pose estimation, where accurate 3D information is
crucial. When unpolarised light passes through a linear polariser or is
reflected at Brewster’s angle from a surface, it becomes polarised. The
combination of specular and diffuse reflection results in a partially
polarised reflected light, which carries information about the surface
properties. We propose to use surface normals obtained from polarisation to
overcome the photometric challenges faced by RGB-D sensors.

### Image Formation Model.

The polarisation image formation model [8] considers the degree of
polarization (DoP) $\rho$ and angle of polarisation (AoP) $\phi$ of the
incoming light. By analysing the oscillation state of light captured by
polarisation filters at different angles $\varphi_{pol}$, the polarised
intensities can be computed using: $I_{\varphi_{pol}}=I_{un}\cdot\ (1+\rho\
\cos(2(\phi-\varphi_{pol})))$. We find $\varphi$ and $\rho$ from the over-
determined system of linear equations using linear least squares. Depending on
the surface properties, AoP is calculated as:

$\displaystyle\left\\{\begin{array}[]{lll}\phi_{d}[\pi]&=\alpha&\text{for
diffuse reflection}\\\ \phi_{s}[\pi]&=\alpha-\frac{\pi}{2}&\text{for specular
reflection,}\end{array}\right.$ (3)

where $[\pi]$ indicates the $\pi$-ambiguity and $\alpha$ is the azimuth angle
of the surface normal n. We can further relate the viewing angle
$\theta\in[0,\pi/2]$ to the degree of polarisation by considering Fresnel
coefficients, thus DoP is similarly given by [3]

$\displaystyle\left\\{\begin{array}[]{l}\rho_{d}=\frac{(\eta-1/\eta)^{2}\sin^{2}(\theta)}{2+2\eta^{2}-(\eta+1/\eta)^{2}\sin^{2}(\theta)+4\cos(\theta)\sqrt{\eta^{2}-\sin^{2}(\theta)}}\\\
\\\
\rho_{s}=\frac{2\sin^{2}(\theta)\cos(\theta)\sqrt{\eta^{2}-\sin^{2}(\theta)}}{\eta^{2}-\sin^{2}(\theta)-\eta^{2}\sin^{2}(\theta)+2\sin^{4}(\theta),}\end{array}\right.$
(7)

with the refractive index of the observed object material $\eta$. Solving
equation 7 for $\theta$, we retrieve three solutions
$\theta_{d},\theta_{s1},\theta_{s2}$, one for the diffuse case and two for the
specular case. For each of the cases, we can now find the 3D orientation of
the surface by calculating the surface normals
$\mathbf{n}=\left(\cos{\alpha}\sin{\theta},\sin{\alpha}\sin{\theta},\cos{\theta}\right)^{\text{T}}.$
We use these plausible normals
$\mathbf{n}_{d},\mathbf{n}_{s1},\mathbf{n}_{s2}$ as physical priors per pixel
as input to the neural network.

### Invertible Physical Model.

The inverted physical model takes a rendered object surface normal map and
derives the analytical polarimetric parameters, considering different
reflection properties. The viewing angle $\theta_{v}$ is obtained from the dot
product between the rendered surface normal map and the viewing vector
$\mathbf{v}$. The analytical DoP $\hat{\rho}$ is derived for diffuse and
specular reflection cases using Equation 7. This inverted physical model
enables us to optimise the model using object shape cues, which is more robust
in challenging lighting conditions compared to active depth sensors.

##  4 Polarimetric 6D Object Pose Estimation

Figure 2: Training Scheme Overview. The upper branch, up until the pose
estimation, is equivalent to the supervised training scheme. The training
scheme utilises four polarisation images captured at different filter angles
along with polarimetric and geometric representations derived from the
physical model. The student network is optimised using pseudo labels
($L_{pseudo}$) generated by the teacher and by minimising the discrepancy
between $\rho$ from the physical model and $\hat{\rho}$ from the inverted
physical model ($L_{physics}$). During inference, the lightweight student
network predicts direct pose estimates (dark gray background).

Polarimetric images provide additional geometric information that improves the
accuracy of 6D object pose prediction [9] (cf. Inputs in Fig.2). We explore
its effectiveness in supervised training on real data and propose a self-
supervised approach leveraging the analytical physical model, thus eliminating
the need for annotated real data. Here, the supervised network is only pre-
trained on synthetic data. The polarimetric real data is leveraged as self-
supervision signal against the analytically derived polarimetric
representation of the output of a differentiable renderer based on the
predicted pose (cf. Fig.2).

Table 1: Supervised Modalities Evaluation.  Impact of input-output modalities
on supervised pose estimation (ADD).

Obj. |  Photo. Chall. | Input Modalities | Outputs | Normal Metrics | Pose
---|---|---|---|---|---
RGB | Pol RGB | Phy. N | N | NOCS |  mean$\downarrow$ |  med.$\downarrow$ |  11.25∘ $\uparrow$ |  22.5∘ $\uparrow$ |  30∘ $\uparrow$ | ADD
Cup |  | ✓ |  |  |  | ✓ | - | - | - | - | - | 91.1
| ✓ |  |  | ✓ | - | - | - | - | - | 91.3
| ✓ |  | ✓ | ✓ | 7.3 | 5.5 | 86.2 | 96.1 | 97.9 | 91.3
| ✓ | ✓ | ✓ | ✓ | 4.5 | 3.5 | 94.7 | 99.1 | 99.6 | 97.2
Fork | $\dagger\dagger$ | ✓ |  |  |  | ✓ | - | - | - | - | - | 85.4
| ✓ |  |  | ✓ | - | - | - | - | - | 86.1
| ✓ |  | ✓ | ✓ | 11.0 | 7.3 | 72.6 | 90.7 | 93.9 | 92.9
| ✓ | ✓ | ✓ | ✓ | 6.5 | 4.3 | 87.6 | 95.9 | 97.6 | 95.9

### Supervised Learning.

The overall objective for the supervised learning is composed of both
geometrical features learning and pose optimisation [9], as:
$\mathcal{L}=\mathcal{L}_{pose}+\mathcal{L}_{geo}$, with:
$\mathcal{L}_{pose}=\mathcal{L}_{R}+\mathcal{L}_{center}+\mathcal{L}_{z}$ and
$\mathcal{L}_{geo}=\mathcal{L}_{mask}+\mathcal{L}_{normals}+\mathcal{L}_{xyz}$.
Specifically, we employ separate loss terms for given ground truth rotation
$\mathbf{R}$, $(\delta_{x},\delta_{y})$ and $\delta_{z}$ as:

$\begin{cases}\mathcal{L}_{R}&=\underset{\mathbf{x}\in\mathcal{M}}{\operatorname{avg}}\|{\mathbf{R}}\mathbf{x}-\hat{\mathbf{R}}\mathbf{x}\|_{1}\\\
\mathcal{L}_{center}&=\|({\delta}_{x}-\hat{\delta}_{x},{\delta}_{y}-\hat{\delta}_{y})\|_{1}\\\
\mathcal{L}_{z}&=\|{\delta}_{z}-\hat{\delta}_{z}\|_{1}\end{cases},$ (8)

where $\hat{\bullet}$ denotes prediction. For symmetrical objects, the
rotation loss is calculated based on the smallest loss from all possible
ground-truth rotations under symmetry.

Table 2: Supervised 6D Pose Benchmark comparisons.  We compare our method
against recent RGB-D (FFB6D [12]) and RGB-only (GDR-Net [26]) methods on
challenging objects with varying photometric conditions ($\dagger$) and depth
map quality (good: $+$ to low:$-$). We report the average recall of ADD(-S)
for evaluation.

Object |  Photo. Chall. | Properties |  Depth Quality | RGB-D Split | RGB Split
---|---|---|---|---|---
Refl. | Metal | Textureless | Transp. | Symm. | FFB6D | Ours | GDR | Ours
Cup |  |  |  |  |  |  | (+) | 99.4 | 98.1 | 96.7 | 97.2
Teapot | $\dagger$ | (*) |  |  |  |  | ++ | 86.8 | 94.2 | 99.0 | 99.9
Can | $\dagger$ | * | * |  |  |  | - | 80.4 | 99.7 | 96.5 | 98.4
Fork | $\dagger\dagger$ | * | * | * |  |  | \-- | 37.0 | 72.4 | 86.6 | 95.9
Knife | $\dagger\dagger$ | * | * | * |  |  | \--- | 36.7 | 87.2 | 92.6 | 96.4
Bottle | $\dagger\dagger\dagger$ | * |  | * | * | * | None | 61.5 | 93.6 | 94.4 | 97.5
Mean |  |  |  |  |  |  |  | 67.0 | 90.9 | 94.3 | 97.6

To learn the intermediate geometrical features, we employ additional losses
with masks ${\mathbf{M}}$:

$\begin{cases}\mathcal{L}_{mask}&=\|{\mathbf{M}}-\hat{\mathbf{M}}\|_{1}\\\
\mathcal{L}_{xyz}&=\mathbf{M}\odot\|\mathbf{M}_{xyz}-\hat{\mathbf{M}}_{xyz}\|_{1}\\\
\mathcal{L}_{normal}&=1-\langle\mathbf{n},\hat{\mathbf{n}}\rangle\end{cases}$
(9)

where $\odot$ indicates the Hadamard product of element-wise multiplication,
and $\langle\bullet,\bullet\rangle$ denotes the dot product.

### Self-Supervised Learning.

The self-supervised network (cf. Fig. 2) consists of a teacher network and a
lightweight student network. Pre-trained on synthetic data, the networks use
pseudo-labels from the teacher to guide self-supervised learning of the
student on real data. Our approach enhances and modifies established student-
teacher training schemes for 6D object pose estimation [24], with detailed
explanations and justifications in the following.

The teacher network (based on PPP-Net [9] as in the supervised scenario above)
takes polarimetric intensities and shape priors as inputs. It predicts object
mask $\mathbf{\tilde{M}_{t}}$, object normal map $\mathbf{\tilde{N}t}$, and
dense correspondences map $\mathbf{\tilde{M}{{xyz}_{t}}}$. It estimates
rotation and translation vectors to represent the pose as
$\mathbf{\tilde{P}_{t}}=[\mathbf{\tilde{R}_{t}}\mid\mathbf{\tilde{t}_{t}}]$.
The teacher network uses a differentiable renderer to generate pseudo labels,
including object mask $\mathbf{\tilde{M}^{R}_{t}}$ and object normal map
$\mathbf{\tilde{N}^{R}_{t}}$.

The lightweight student network directly predicts the pose
$\mathbf{\hat{P}_{s}}$ without a geometric decoder. It is optimised using
pseudo labels from the teacher network and generates an object normal map
$\mathbf{\hat{N}_{s}}$ and mask $\mathbf{\hat{M}_{s}}$ through rendering based
on the predicted pose.

Our training scheme combines pseudo label learning and physics-induced self-
supervision using polarimetric images. The pseudo label loss
($\mathcal{L}_{\text{pseudo}}$) transfers knowledge from the teacher network
to the student network, while the physical loss term
($\mathcal{L}_{\text{physics}}$) optimizes the student’s prediction using raw
polarisation data and the inverted physical model. These terms contribute to
the overall loss ($\mathcal{L}$).

The pseudo label loss ($\mathcal{L}_{pseudo}$) is formulated as:
$\mathcal{L}_{pseudo}=\lambda_{1}\mathcal{L}_{pose}+\mathcal{L}_{geo}$, where
$\mathcal{L}_{pose}$ measures the discrepancy between the predicted pose of
the student network and the pseudo ground truth pose provided by the teacher
network. The $\mathcal{L}_{geo}$ term regularises the predicted mask and
normal map by comparing them with the rendered mask and normal map. The
weighting factor $\lambda_{1}$ depends on the alignment between the predicted
shape and pose.

The physics-induced self-supervision loss ($\mathcal{L}_{physics}$) is
formulated as:
$\mathcal{L}_{physics}=\underset{\mathbf{x}\in{\hat{\rho_{d}},\hat{\rho_{s}}}}{\min}|\rho-\mathbf{x}|1$,
where $\rho$ is the ground truth degree of polarisation (DoP) obtained from
real polarisation images, and $\hat{\rho_{d}}$ and $\hat{\rho_{s}}$ are the
analytically computed diffuse and specular DoP using the inverted physical
model. The loss term aligns the predicted DoP of the student network with the
ground truth DoP. The overall loss is given by:
$\mathcal{L}=\mathcal{L}_{pseudo}+\mathcal{L}_{physics}$.

##  5 Quantitative Results on Real Data

We conducted experiments to analyse the influence of input modality on pose
estimation accuracy, specifically focusing on the impact of polarimetric image
information. By comparing RGB-only and polar RGB inputs, we found that
polarisation modality improves accuracy slightly, for the challenging object
fork. Incorporating shape information from polarimetric images enhances pose
estimation significantly, as evidenced by improved normals prediction and
overall performance. These results highlight the network’s ability to
establish a more accurate geometrical representation when guided by
polarisation and shape cues (cf. Tab. 1).

Our experiments demonstrate the robustness of polarimetric imaging inputs in
achieving accurate 6D pose prediction for photometrically challenging objects
(cf. Tab. 2). Compared to FFB6D [12], which combines appearance and depth
information, our method leverages polarimetric information to overcome
challenges associated with photometric complexity, such as reflection or
transparency.

Table 3 presents quantitative results that demonstrate the effectiveness of
our self-supervision pipeline. We evaluate the performance of the teacher and
student networks under different conditions. The $\text{OURS}_{\text{(Lower-
Bound Teacher)}}$ and $\text{OURS}_{\text{(Lower-Bound Student)}}$ denote the
lower bound performance when trained on synthetic data and tested on real
data. The $\text{OURS}_{\text{(Upper-Bound Teacher)}}$ and
$\text{OURS}_{\text{(Upper-Bound Student)}}$ represent the upper bound
performance when trained and evaluated on real data using ground-truth labels,
where the $\text{OURS}_{\text{(Upper-Bound Teacher)}}$ is identical to PPP-Net
[9]. Our model $\text{{OURS}}_{\text{(RGB-Ps)}}$ consistently outperforms the
SOTA RGB-D method [24] and achieves comparable results to the fully supervised
upper bound baseline [9] for the cup object, which is the least complex in
terms of photometric considerations.

Table 3: Self-Supervised Quantitative Results. Average recall of ADD(-S)
metric is reported for different objects with increasing photometric
complexity. $\dagger$ as upper bound is identical to PPP-Net [9]; Self6D++
from [24]

Methods | Cup | Fork | Knife | Bottle | Mean
---|---|---|---|---|---
$\text{OURS}_{\text{(Lower-Bound Student)}}$ | 53.7 | 64.4 | 46.1 | 47.5 | 52.9
$\text{OURS}_{\text{(Lower-Bound Teacher)}}$ | 72.3 | 75.0 | 67.3 | 76.2 | 72.7
$\text{OURS}_{\text{(Upper-Bound Student)}}$ | 86.4 | 88.0 | 91.1 | 80.4 | 86.5
$\text{OURS}_{\text{(Upper-Bound Teacher)}}$ $\dagger$ | 91.4 | 91.7 | 90.0 | 89.4 | 90.6
Self6D++ | 68.4 | 14.3 | 17.8 | 33.5 | 34.0
$\text{{OURS}}_{\text{(RGB-Ps)}}$ | 93.8 | 72.4 | 78.4 | 78.2 | 80.7

##  6 Conclusion

Our proposed self-supervised framework leveraging polarimetric information has
demonstrated its effectiveness in improving 6D object pose estimation
accuracy. By integrating pseudo label learning and physics-induced self-
supervision, our approach achieves superior performance compared to existing
RGB-D methods and even approaches the results of fully supervised methods. The
utilisation of polarimetric images as additional inputs provides valuable
complementary information and helps bridging the domain gap between synthetic
and real data. These findings highlight the potential of self-supervision and
polarimetric imaging in advancing the field of object pose estimation.

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# Green’s function for the fractional KdV equation on the periodic domain via
Mittag–Leffler’s function

Uyen Le  Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>and  Dmitry E. Pelinovsky
Department of Mathematics and Statistics, McMaster University, Hamilton,
Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

The linear operator $c+(-\Delta)^{\alpha/2}$, where $c>0$ and
$(-\Delta)^{\alpha/2}$ is the fractional Laplacian on the periodic domain,
arises in the existence of periodic travelling waves in the fractional
Korteweg–de Vries equation. We establish a relation of the Green’s function of
this linear operator with the Mittag–Leffler function, which was previously
used in the context of Riemann–Liouville’s and Caputo’s fractional
derivatives. By using this relation, we prove that Green’s function is
strictly positive and single-lobe (monotonically decreasing away from the
maximum point) for every $c>0$ and every $\alpha\in(0,2]$. On the other hand,
we argue from numerical approximations that in the case of $\alpha\in(2,4]$,
the Green’s function is positive and single-lobe for small $c$ and non-
positive and non-single lobe for large $c$.

###### Key words and phrases:

Fractional Laplacian, Green’s function, positivity and monotonicity, periodic
domain

##  1\. Introduction

This work deals with Green’s function for the linear operator

$L_{c,\alpha}:=c+(-\Delta)^{\alpha/2},$ (1.1)

where $c>0$ is arbitrary parameter and $(-\Delta)^{\alpha/2}$, $\alpha>0$ is
the fractional Laplacian on the normalized periodic domain
$\mathbb{T}=[-\pi,\pi]$. The fractional Laplacian is defined via Fourier
series by

$f(x)=\sum_{n\in\mathbb{Z}}f_{n}e^{inx},\quad(-\Delta)^{\alpha/2}f(x)=\sum_{n\in\mathbb{Z}}|n|^{\alpha}f_{n}e^{inx}.$
(1.2)

Properties of the fractional Laplacian on the $d$-dimensional torus
$\mathbb{T}^{d}$ were studied in [33]. Recent review of boundary-value
problems for the fractional Laplacian and related applications can be found in
[23].

Green’s function denoted by $G_{\mathbb{T}}$ satisfies the periodic boundary
value problem

$\left[c+(-\Delta)^{\alpha/2}\right]G_{\mathbb{T}}(x)=\delta(x),\hskip
14.45377ptx\in\mathbb{T},$ (1.3)

where $\delta$ is the Dirac delta distribution. The solution is represented
via Fourier series by

$G_{\mathbb{T}}(x)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\frac{\cos(nx)}{c+|n|^{\alpha}}=\frac{1}{2\pi}\left(\frac{1}{c}+2\sum_{n=1}^{\infty}\frac{\cos(nx)}{c+n^{\alpha}}\right).$
(1.4)

Green’s function $G_{\mathbb{T}}$ defined by (1.3) and (1.4) arises in the
study of the stationary equation

$\left[c+(-\Delta)^{\alpha/2}\right]\psi(x)=\psi(x)^{1+p},\quad
x\in\mathbb{T},$ (1.5)

where $p\in\mathbb{N}$. The stationary equation (1.5) defines the travelling
periodic waves of the fractional Korteweg–de Vries (fKdV) equation with the
speed $c$ [6, 7, 18, 19, 26, 27] and the standing periodic waves of the
fractional nonlinear Schrödinger (fNLS) equation with the frequency $c$ [9,
17]. Periodic solutions in other nonlinear elliptic equations associated with
the fractional Laplacian were also considered, e.g., in [2, 12].

Green’s function $G_{\mathbb{T}}$ defined by (1.3) and (1.4) was used in the
proof of strict positivity of the periodic solutions of the stationary
equation (1.5) for $c>0$, $\alpha\in(0,2]$, and $p=1$ by using Krasnoselskii’s
fixed point theorem (see Theorem 2.2 in [22]). The important ingredient of the
proof is the property of strict positivity of Green’s function
$G_{\mathbb{T}}$ for every $c>0$.

The property of strict positivity of Green’s function was proven for different
boundary-value problems associated with the fractional operators in [28] for
$\alpha\in(0,1)$ and in [3] for $\alpha\in(1,2)$; however, the fractional
derivatives were considered in the Riemann–Liouville sense (see [20, 30] for
review of fractional derivatives).

Here we prove strict positivity of Green’s function $G_{\mathbb{T}}$
satisfying the boundary-value problem (1.3) on $\mathbb{T}$ for every $c>0$
and every $\alpha\in(0,2]$. Moreover, we show that $G_{\mathbb{T}}$ has the
single-lobe profile in the sense that $G_{\mathbb{T}}$ is monotonically
decreasing on $\mathbb{T}$ away from its maximum point located at $x=0$. The
following theorem presents this result.

######  Theorem 1.

For every $c>0$ and every $\alpha\in(0,2]$, Green’s function $G_{\mathbb{T}}$
defined by (1.3) and (1.4) is even, strictly positive on $\mathbb{T}$, and
monotonically decreasing on $(0,\pi)$.

The result of Theorem 1 is known in the context of Green’s function
$G_{\mathbb{R}}$ for the linear operator $\mathcal{L}_{c,\alpha}$ in (1.1)
considered on the real line $\mathbb{R}$ (see Lemma A.4 in [13]). This was
shown from similar properties of the heat kernel related to the fractional
Laplacian $(-\Delta)^{\alpha/2}$ (see Lemma A.1 in [13]). The constant $c>0$
in $\mathcal{L}_{c,\alpha}$ can be normalized to unity when
$\mathcal{L}_{c,\alpha}$ is considered on the real line $\mathbb{R}$.

The same properties hold for Green’s function $G_{\mathbb{T}}$ on the periodic
domain $\mathbb{T}$ because it can be written as the following periodic
superposition of Green’s function $G_{\mathbb{R}}$ on the real line
$\mathbb{R}$:

$G_{\mathbb{T}}(x)=\sum_{n\in\mathbb{Z}}G_{\mathbb{R}}(x-2\pi n),\quad
x\in\mathbb{T}.$ (1.6)

Hence, if $G_{\mathbb{R}}(x)>0$ for $x\in\mathbb{R}$, then
$G_{\mathbb{T}}(x)>0$ for $x\in\mathbb{T}$ and if
$G_{\mathbb{R}}^{\prime}(x)\leq 0$ for $x\geq 0$, then
$G_{\mathbb{T}}^{\prime}(x)\leq 0$ for $x\in[0,\pi]$. Here the parameter $c$
in $G_{\mathbb{T}}$ cannot be normalized to unity.

The main novelty of our work is the relation between Green’s function
$G_{\mathbb{T}}$ and the Mittag–Leffler function [25]. The Mittag–Leffler
function naturally arises in the other (Riemann–Liouville and Caputo)
formulations of fractional derivatives [20, 30] but it has not been used in
the context of the fractional Laplacian to the best of our knowledge. In
particular, we prove Theorem 1 for $\alpha\in(0,2)$ by using the integral
representations and properties of the Mittag–Leffler function and some
trigonometric series from [32]. For $\alpha=2$, the result of Theorem 1 can be
readily shown by writing $G_{\mathbb{T}}$ in the exact analytical form (see
Appendix A).

Figure 1 illustrates the statement of Theorem 1. It shows the single-lobe
positive profile of $G_{\mathbb{T}}$ for two values of $c$ in the case
$\alpha=0.5$ (left) and $\alpha=1.5$ (right). The only difference between
these two cases is that $G_{\mathbb{T}}(0)$ is bounded for $\alpha>1$ and is
unbounded for $\alpha\leq 1$.

Figure 1. Profiles of $G_{\mathbb{T}}$ for $\alpha=0.5$ (left) and
$\alpha=1.5$ (right) for specific values of $c$.

Green’s function $G_{\mathbb{R}}$ on the real line $\mathbb{R}$ is also used
to understand interactions of strongly localized waves, e.g. attractive versus
repelling interactions [16, 24] (see also [8, 29]). These interactions were
recently studied in [10, 11] in the context of the beam equation, which
corresponds to the case $\alpha=4$. The fractional cases for $\alpha\in(0,2)$
and $\alpha\in(2,4)$ are also important from applications in quantum
computing, fluid dynamics, and elasticity theory.

Theorem 1 shows that the properties of $G_{\mathbb{T}}$ for $\alpha\in(0,2)$
are similar to those for $\alpha=2$ (the same holds for $G_{\mathbb{R}}$).
However, it is an open question if the properties of $G_{\mathbb{T}}$ for
$\alpha\in(2,4)$ are similar to those for $\alpha=4$, for which
$G_{\mathbb{R}}$ has infinitely many oscillations, whereas the number of
oscillations of $G_{\mathbb{T}}$ depends on $c>0$ and becomes infinite in the
limit of $c\to\infty$ (see Appendix B). In the second part of this paper, we
present numerical results which support the following conjecture.

######  Conjecture 1.

For each $\alpha\in(2,4]$, there exists $c_{0}>0$ such that for
$c\in(0,c_{0})$, Green’s function $G_{\mathbb{T}}$ defined by (1.3) and (1.4)
is even, strictly positive on $\mathbb{T}$, and monotonically decreasing on
$(0,\pi)$. For $c\in[c_{0},\infty)$, $G_{\mathbb{T}}$ has a finite number of
zeros on $\mathbb{T}$. The number of zeros is bounded in the limit of
$c\to\infty$ if $\alpha\in(2,4)$ and unbounded as $c\to\infty$ if $\alpha=4$.

Since the limit $c\to\infty$ for Green’s function $G_{\mathbb{T}}$ can be
rescaled as Green’s funciton $G_{\mathbb{R}}$ with $c$ normalized to unity,
Conjecture 1 implies the following conjecture (relevant for interactions of
strongly localized waves in [10, 11]).

######  Conjecture 2.

For every $c>0$ and every $\alpha\in(2,4]$, Green’s function $G_{\mathbb{R}}$
is not strictly positive on $\mathbb{R}$ and is not monotonically decreasing
on $(0,\infty)$. It has a finite number of zeros on $\mathbb{R}$ if
$\alpha\in(2,4)$ and an infinite number of zeros if $\alpha=4$.

Figure 2 illustrates the statement of Conjecture 1. For $\alpha=2.5$ (top),
Green’s function $G$ has the single-lobe positive profile for $c=2$ (red
curve) but it is not positive for $c=10$ (blue curve). For $\alpha=3.5$
(bottom), it is positive for $c=1$ (red curve), has one pair of zeros for
$c=10$ (blue curve), and has two pairs of zeros for $c=60$ (black curve).

Figure 2.  Profiles of $G$ on $\mathbb{T}$ for $\alpha=2.5$ (top) and
$\alpha=3.5$ (bottom) at specific values of $c$.

The remainder of the article is structured as follow. Section 2 presents an
overview of the Mittag-Leffler function and its properties. The integral
representation of Green’s function $G_{\mathbb{T}}$ is derived in Section 3.
The proof of Theorem 1 is presented in Section 4. The validity of Conjecture 1
is discussed in Section 5. Conclusion is given in Section 6. Appendices A and
B give explicit formulas for Green’s function $G_{\mathbb{T}}$ for local cases
of $\alpha=2$ and $\alpha=4$, respectively. Appendix C contains formal
asymptotic results to support Conjecture 1 for $\alpha>2$ with small
$|\alpha-2|$.

##  2\. Properties of Mittag–Leffler function

Here we discuss properties of the Mittag–Leffler function defined by

$E_{\alpha}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(k\alpha+1)},\qquad\alpha>0,$
(2.1)

and its two-parametric generalization defined by

$E_{\alpha,\beta}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(k\alpha+\beta)},\qquad\alpha,\beta>0.$
(2.2)

Mittag–Leffler functions were introduced in the theory of analytic functions
[25]. In recent years, they became popular due to their applications in
fractional differential equations [20]. Indepth studies of the Mittag–Leffler
functions can be found in [4] and [15].

Mittag–Leffler functions are typically used to represent solutions of initial-
value problems for the fractional differential equations defined by the
Riemann-Liouville or Caputo fractional derivatives [20]. As our study involves
the boundary-value problem for the fractional Laplacian
$(-\Delta)^{\alpha/2}$, Mittag–Leffler function $E_{\alpha}(-x^{\alpha})$ is
used in the integral representation of Green’s function $G_{\mathbb{T}}$. This
integral representation is derived in Section 3.

Here we review some important properties of the Mittag–Leffler functions.

######  Lemma 2.1.

For every $\alpha>0$ and every $x\in\operatorname{\mathbb{R}}$, it is true
that

$E_{\alpha,\alpha}(x)=\alpha\frac{d}{dx}E_{\alpha}(x).$ (2.3)

###### Proof.

The result is obtained by differentiating (2.1) and using (2.2):

$\frac{d}{dx}E_{\alpha}(x)=\sum_{k=1}^{\infty}\frac{x^{k-1}}{\alpha\Gamma(\alpha
k)}=\frac{1}{\alpha}\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(\alpha
k+\alpha)}=\frac{1}{\alpha}E_{\alpha,\alpha}(x).$

The series converges absolutely for every $x\in\mathbb{R}$ since $E_{\alpha}$
and $E_{\alpha,\alpha}$ are entire functions. ∎

######  Lemma 2.2.

[31] For every $\alpha\in(0,1]$, the function $x\mapsto E_{\alpha}(-x)$ is
positive and completely monotonic for $x\geq 0$, that is

$(-1)^{m}\frac{d^{m}}{dx^{m}}E_{\alpha}(-x)\geq 0,\qquad m\in\mathbb{N},\quad
x\geq 0.$ (2.4)

Consequently, $E_{\alpha,\alpha}(-x)\geq 0$ for every $x\geq 0$.

######  Remark 2.1.

A necessary and sufficent condition for the function $x\mapsto E_{\alpha}(-x)$
to be completely monotonic for $x\geq 0$ is that $E_{\alpha}(-x)$ can be
expressed in the form

$E_{\alpha}(-x)=\int_{0}^{\infty}e^{-xt}dF_{\alpha}(t),\quad x\geq 0,$

where $F_{\alpha}$ is a nondecreasing and bounded on $(0,\infty)$. The proof
of [31] is based on the representation of $E_{\alpha}(-x)$ given by

$E_{\alpha}(-x)=\frac{1}{2i\pi\alpha}\int_{C}\frac{e^{t^{1/\alpha}}}{t+x}dt,$

with a specially selected the contour $C$ in $\mathbb{C}$.

######  Lemma 2.3.

[15] For every $\alpha\in(0,2)$, $E_{\alpha}(-x^{\alpha})$ admits the
asymptotic expansion

$E_{\alpha}(-x^{\alpha})=-\sum_{k=1}^{N}\frac{(-1)^{k}}{\Gamma(1-\alpha
k)x^{\alpha k}}+\mathcal{O}\left(\frac{1}{|x|^{\alpha
N+\alpha}}\right)\quad\mbox{\rm as}\quad x\to\infty,$ (2.5)

where $N\in\mathbb{N}$ is arbitrarily fixed. For every $\alpha\geq 2$,
$E_{\alpha}(-x^{\alpha})$ admits the asymptotic expansion

$E_{\alpha}(-x^{\alpha})=\frac{1}{\alpha}\sum_{n=-N+1}^{N}e^{a_{n}x}+\mathcal{O}\left(\frac{1}{|x|^{\alpha}}\right)\quad\mbox{\rm
as}\quad x\to\infty,$ (2.6)

where $a_{n}=e^{\frac{i\pi(2n-1)}{\alpha}}$ and $N$ is the largest integer
satisfying the bound $2N-1\leq\frac{\alpha}{2}$.

######  Remark 2.2.

Asymptotic expansions (2.5) and (2.6) can be differentiated term by term.

######  Remark 2.3.

We list the explicit cases of the Mittag–Leffler function
$E_{\alpha}(-x^{\alpha})$ for the first integers:

$\displaystyle\alpha=1,$ $\displaystyle\quad E_{1}(-x)=e^{-x},$
$\displaystyle\alpha=2,$ $\displaystyle\quad E_{2}(-x^{2})=\cos(x),$
$\displaystyle\alpha=3,$ $\displaystyle\quad
E_{3}(-x^{3})=\frac{1}{3}e^{-x}+\frac{2}{3}\displaystyle{e^{\frac{x}{2}}}\cos\left(\frac{\sqrt{3}x}{2}\right),$
$\displaystyle\alpha=4,$ $\displaystyle\quad
E_{4}(-x^{4})=\cos\left(\frac{x}{\sqrt{2}}\right)\cosh\left(\frac{x}{\sqrt{2}}\right).$

For $\alpha=1$, the asymptotic representation (2.5) admits zero leading-order
terms for every $N\in\mathbb{N}$. The asymptotic representation (2.6) is also
obvious from the exact expressions for $\alpha=2,3,4$, moreover, the remainder
term is zero for $\alpha=2$ and can be included to the summation by increasing
$N$ by one for $\alpha=3$ and $\alpha=4$.

######  Lemma 2.4.

[15] For every $\alpha\in(0,2)$ and every $x\in\operatorname{\mathbb{R}}$,
$E_{\alpha}(-x)$ satisfies the following integral representation,

$E_{\alpha}(-x^{\alpha})=\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2}\right)\int_{0}^{\infty}\frac{t^{\alpha-1}\cos(xt)}{1+2t^{\alpha}\cos\left(\frac{\pi\alpha}{2}\right)+t^{2\alpha}}dt.$
(2.7)

######  Remark 2.4.

It is claimed in [15] that the integral representation (2.7) is true for all
$\alpha>0$, however, the integral is singular for $\alpha=2$ and a discrepancy
exists at $x=0$ for $\alpha>2$. For example, when $\alpha=3$, it follows from
(2.1) that $E_{3}(0)=1$ whereas computing the integral given in (2.7) via the
change of variable $u=t^{3}$ gives

$\displaystyle E_{3}(0)$
$\displaystyle=-\frac{2}{3\pi}\int_{0}^{\infty}\frac{du}{1+u^{2}}=-\frac{1}{3}\neq
1.$

Hence, the integral representation (2.7) can only be used for
$\alpha\in(0,2)$, for which $E_{\alpha}(-x^{\alpha})$ is bounded and decaying
as $x\to+\infty$.

##  3\. Integral representation of Green’s function $G_{\mathbb{T}}$

Here, we take Green’s function $G_{\mathbb{T}}$ defined by the Fourier series
in (1.4) and rewrite it in the integral form involving the Mitag–Leffler
function $E_{\alpha,\alpha}$. The following proposition gives the result for
$\alpha\in(0,2]$.

######  Proposition 3.1.

For every $c>0$ and every $\alpha\in(0,2]$, it is true that

$G_{\mathbb{T}}(x)=\frac{1}{2\pi c}+\frac{1}{\pi
c}\int_{0}^{\infty}\left(\frac{e^{t}\cos(x)-1}{1-2e^{t}\cos(x)+e^{2t}}\right)t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})dt,\quad
x\in\mathbb{T}.$ (3.1)

###### Proof.

Assume first that $x\neq 0$ and $c\in(0,1)$. Expanding each term of the
trinometric sum in (1.4) into absolutely convergent geometric series and
interchanging the two series, we obtain

$\sum_{n=1}^{\infty}\frac{\cos(nx)}{c+n^{\alpha}}=\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^{\alpha}}\sum_{k=0}^{\infty}\left(\frac{-c}{n^{\alpha}}\right)^{k}=\sum_{k=0}^{\infty}(-c)^{k}\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^{\alpha(k+1)}}.$
(3.2)

It is known from the integral representation (1) in [32, Section 5.4.2] that
for every $x\neq 0$ and $\alpha>0$ that

$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^{\alpha(k+1)}}=\frac{1}{\Gamma(\alpha
k+\alpha)}\int_{0}^{\infty}\frac{t^{\alpha(k+1)-1}\left(e^{t}\cos(x)-1\right)}{1-2e^{t}\cos(x)+e^{2t}}dt,$
(3.3)

where $k\geq 0$. Substituting (3.3) into (3.2) and interchanging formally the
summation and the integration yields the following representation:

$\displaystyle\sum_{n=1}^{\infty}\frac{\cos(nx)}{c+n^{\alpha}}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-c)^{k}}{\Gamma(\alpha
k+\alpha)}\int_{0}^{\infty}\frac{t^{\alpha(k+1)-1}\left(e^{t}\cos(x)-1\right)}{1-2e^{t}\cos(x)+e^{2t}}dt,$
(3.4)
$\displaystyle=\int_{0}^{\infty}\left(\frac{e^{t}\cos(x)-1}{1-2e^{t}\cos(x)+e^{2t}}\right)t^{\alpha-1}\sum_{k=0}^{\infty}\frac{\left(-ct^{\alpha}\right)^{k}}{\Gamma(\alpha
k+\alpha)}dt,$ (3.5)
$\displaystyle=\int_{0}^{\infty}\left(\frac{e^{t}\cos(x)-1}{1-2e^{t}\cos(x)+e^{2t}}\right)t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})dt.$
(3.6)

This yields formally the integral formula (3.1). Let us now justify the
interchange of summation and integration in (3.4). Using the chain rule and
Lemma 2.1, we get

$t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})=-\frac{1}{c}\frac{d}{dt}E_{\alpha}(-ct^{\alpha}).$
(3.7)

It follows from (3.7) that for every $\alpha\in(0,2]$, the asymptotic
expansion (2.5) in Lemma 2.3 for $\alpha\in(0,2)$ and Remark 2.3 for
$\alpha=2$ imply that

$\sup_{t\in[0,\infty)}t^{\alpha-1}|E_{\alpha,\alpha}(-t^{\alpha})|<\infty.$
(3.8)

Hence, the integral in (3.1) converges absolutely for every $x\neq 0$ and
$\alpha\in(0,2]$. Similarly, the integral in (3.4) converges absolutely for
every $x\neq 0$ and $\alpha\in(0,2]$, whereas the numerical series converges
absolutely for every $c\in(0,1)$. Thus, the interchange of summation and
integration in (3.4) is justified by Fubini’s theorem.

For $x=0$, we note that $G_{\mathbb{T}}(0)<\infty$ if $\alpha>1$ and
$G_{\mathbb{T}}(0)=\infty$ if $\alpha\in(0,1]$. Since
$E_{\alpha,\alpha}(-x^{\alpha})=1+\mathcal{O}(x^{\alpha})$ as $x\to 0$, the
integral in (3.1) converges absolutely for $x=0$ and $\alpha\in(1,2]$ and
diverges for $x=0$ and $\alpha\in(0,1]$. Hence, the integral representation
(3.1) holds again for $x=0$, $c\in(0,1)$, and $\alpha\in(0,2]$.

In order to extend the integral representation (3.1) from $c\in(0,1)$ to every
$c>0$, we use real analyticity of Green’s function $G_{\mathbb{T}}$ and the
integral in (3.1) in $c$ for $c>0$. Due to uniqueness of the analytical
continuation of both $G_{\mathbb{T}}$ and the integral in (3.1) in $c$, the
equality in (3.1) is uniquely continued from $c\in(0,1)$ to $c>0$. ∎

The integral representation (3.1) of Green’s function $G_{\mathbb{T}}$ can be
justified for $\alpha>2$ provided that $c$ is sufficiently small. This result
is described by the following proposition.

######  Proposition 3.2.

For every $\alpha>2$, there exists $c_{\alpha}>0$ given by

$c_{\alpha}:=\left[\cos\left(\frac{\pi}{\alpha}\right)\right]^{-\alpha},$
(3.9)

such that for every $c\in(0,c_{\alpha})$, the integral representation (3.1) is
true for every $x\in\mathbb{T}$.

###### Proof.

The asymptotic expansion (2.6) in Lemma 2.3 implies for every $c>0$ and
$\alpha>2$ that

$\sup_{t\in[0,\infty)}e^{-t\cos\left(\frac{\pi}{\alpha}\right)}t^{\alpha-1}|E_{\alpha,\alpha}(-t^{\alpha})|<\infty,$
(3.10)

where we have used again the connection formula (3.7). In addition,
$E_{\alpha,\alpha}(-x^{\alpha})=1+\mathcal{O}(x^{\alpha})$ as $x\to 0$. Due to
the above properties, the integral in (3.1) converges absolutely for every
$x\in\mathbb{T}$ if $c\in(0,c_{\alpha})$, where $c_{\alpha}$ is given by
(3.9). This justifies the formal computations in the proof of Proposition 3.1.
∎

######  Remark 3.1.

For $c\geq c_{\alpha}$ and $\alpha>2$, the Fourier series representation (1.4)
suggests that $|G_{\mathbb{T}}(x)|<\infty$ for every $x\in\mathbb{T}$.
However, the integral in (3.1) does not converge absolutely, hence it is not
clear if the integral representation (3.1) can be used in this case. Our
numerical results in Section 5 show that the integral representation (3.1)
cannot be used for $c>c_{\alpha}$.

##  4\. Green’s function $G_{\mathbb{T}}$ for $\alpha\in(0,2)$

Here, we prove Theorem 1 by using the integral representation (3.1) in terms
of the Mittag–Leffler function $E_{\alpha,\alpha}$. It follows from (1.4) that
$G_{\mathbb{T}}$ is even for every $c>0$ and $\alpha>0$. Furthermore, if
$\alpha\in(0,1]$, then $\lim\limits_{x\to 0}G_{\mathbb{T}}(x)=+\infty$, and if
$\alpha>1$, then

$G_{\mathbb{T}}(0)=\frac{1}{2\pi}\left(\frac{1}{c}+2\sum_{n=1}^{\infty}\frac{1}{c+n^{\alpha}}\right)>0.$

We shall prove that $G_{\mathbb{T}}^{\prime}(x)\leq 0$ for $x\in(0,\pi)$ and
$G_{\mathbb{T}}(\pi)>0$ for every $c>0$ and $\alpha\in(0,2]$. For $\alpha=2$,
this result follows from the exact analytical representation of
$G_{\mathbb{T}}$ in Appendix A. Therefore, we focus on the case
$\alpha\in(0,2)$ here. The following proposition gives an integral
representation for $G_{\mathbb{T}}(\pi)$ which implies its strict positivity
for every $c>0$ and $\alpha\in(0,2)$

######  Proposition 4.1.

For every $c>0$ and every $\alpha\in(0,2)$, it is true that

$G_{\mathbb{T}}(\pi)=\frac{\sin(\frac{\alpha\pi}{2})}{\pi
c^{1-\frac{1}{\alpha}}}\int_{0}^{\infty}\frac{s^{\alpha}\operatorname{csch}(\pi
c^{\frac{1}{\alpha}}s)}{1+2s^{\alpha}\cos(\frac{\alpha\pi}{2})+s^{2\alpha}}ds,$
(4.1)

which implies $G_{\mathbb{T}}(\pi)>0$.

###### Proof.

Evaluating the integral representation (3.1) at $x=\pi$, we obtain

$G_{\mathbb{T}}(\pi)=\frac{1}{2\pi c}-\frac{1}{\pi
c}\int_{0}^{\infty}\frac{1}{1+e^{t}}t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})dt.$
(4.2)

Substituting (3.7) into (4.2), integrating by parts, and using the asymptotic
representation (2.5) to get zero contribution in the limit of $t\to\infty$, we
obtain

$G_{\mathbb{T}}(\pi)=\frac{1}{\pi
c}\int_{0}^{\infty}\frac{e^{t}}{(1+e^{t})^{2}}E_{\alpha}(-ct^{\alpha})dt,$
(4.3)

where the integral converges absolutely for every $c>0$ and $\alpha\in(0,2)$.
Substituting the integral representation (2.7) for $E_{\alpha}(-ct^{\alpha})$
from Lemma 2.4 into (4.3), we obtain

$G_{\mathbb{T}}(\pi)=\frac{2}{\pi^{2}c}\sin\left(\frac{\alpha\pi}{2}\right)\int_{0}^{\infty}\frac{e^{t}}{(1+e^{t})^{2}}\int_{0}^{\infty}\frac{s^{\alpha-1}\cos(c^{\frac{1}{\alpha}}ts)}{1+2s^{\alpha}\cos(\frac{\pi\alpha}{2})+s^{2\alpha}}dsdt.$
(4.4)

Since both integrands belong to $L^{1}(0,\infty)$, the order of integration in
(4.4) can be interchanged to get

$G_{\mathbb{T}}(\pi)=\frac{2}{\pi^{2}c}\sin\left(\frac{\alpha\pi}{2}\right)\int_{0}^{\infty}\frac{s^{\alpha-1}}{1+2s^{\alpha}\cos(\frac{\pi\alpha}{2})+s^{2\alpha}}\int_{0}^{\infty}\frac{e^{t}\cos(c^{\frac{1}{\alpha}}st)}{(1+e^{t})^{2}}dtds.$
(4.5)

The inner integral is evaluated exactly with the help of integral (7) in [32,
Section 2.5.46]:

$\displaystyle\int_{0}^{\infty}\frac{e^{t}\cos(c^{\frac{1}{\alpha}}st)}{(1+e^{t})^{2}}dt=\frac{\pi}{2}c^{\frac{1}{\alpha}}s\operatorname{csch}(\pi
c^{\frac{1}{\alpha}}s),$

When it is substituted into (4.5), it yields the integral representation
(4.1). The integrand is positive and absolutely integrable for every $c>0$ and
$\alpha\in(0,2)$, which implies that $G_{\mathbb{T}}(\pi)>0$. ∎

######  Remark 4.1.

Positivity of $G_{\mathbb{T}}(\pi)$ for $c>0$ and $\alpha\in(0,1]$ also
follows from the representation (4.3) due to positivity of
$E_{\alpha}(-ct^{\alpha})$ for every $t>0$ in Lemma 2.2. However,
$E_{\alpha}(-ct^{\alpha})$ is not positive for all $t>0$ when $\alpha>1$,
hence, the representation (4.3) is not sufficient for the proof of positivity
of $G_{\mathbb{T}}(\pi)$ if $\alpha\in(1,2)$.

It remains to prove that $G_{\mathbb{T}}^{\prime}(x)\leq 0$ for every
$x\in(0,\pi)$. The proof is carried differently for $\alpha\in(0,1]$ and for
$\alpha\in(1,2)$. In the former case, we obtain the integral representation
for $G_{\mathbb{T}}^{\prime}(x)$, which is strictly negative for
$x\in(0,\pi)$. In the latter case, we employ the variational method to verify
that the unique solution $G_{\mathbb{T}}$ of the boundary-value problem (1.3)
admits the single lobe profile, with the only maximum located at the point of
symmetry at $x=0$. The following two propositions give these two results.

######  Proposition 4.2.

For every $c>0$ and every $\alpha\in(0,1]$, $G_{\mathbb{T}}^{\prime}(x)<0$ for
every $x\in(0,\pi)$.

###### Proof.

Differentiating the integral representation (3.1) in $x$ yields

$\displaystyle G_{\mathbb{T}}^{\prime}(x)$ $\displaystyle=\frac{1}{\pi
c}\int_{0}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})\frac{d}{dx}\left(\frac{e^{t}\cos(x)-1}{1-2e^{t}\cos(x)+e^{2t}}\right)dt,$
$\displaystyle=-\frac{\sin(x)}{\pi
c}\int_{0}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})\frac{e^{t}(e^{2t}-1)}{\left(1-2e^{t}\cos(x)+e^{2t}\right)^{2}}dt,$
(4.6)

where the integrand is absolutely integrable. It follows by Lemma 2.2 that
$E_{\alpha,\alpha}(-ct^{\alpha})\geq 0$ for $t>0$. Since $\sin(x)>0$ for
$x\in(0,\pi)$, and the integrand is positive, it follows from the integral
representation (4.6) that $G_{\mathbb{T}}^{\prime}(x)<0$ for $x\in(0,\pi)$. ∎

######  Proposition 4.3.

For every $c>0$ and every $\alpha\in(1,2)$, $G_{\mathbb{T}}^{\prime}(x)\leq 0$
for every $x\in(0,\pi)$.

###### Proof.

The proof consists of the following two steps. First, we obtain a variational
solution to the boundary-value problem (1.3). Second, we use the fractional
Polya–Szegö inequality to show that the solution $G_{\mathbb{T}}$ has a
single-lobe profile on $\mathbb{T}$ with the only maximum located at the point
of symmetry at $x=0$.

Step 1: Let us consider the following minimization problem,

$\mathcal{B}_{c}:=\min_{u\in H^{\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})}\\{B_{c}(u)-u(0)\\},$ (4.7)

where the quadratic functional $B_{c}(u)$ is given by

$B_{c}(u)=\frac{1}{2}\int_{\mathbb{T}}\left[\left(D^{\frac{\alpha}{2}}u\right)^{2}+cu^{2}\right]dx.$
(4.8)

Since $c>0$, we have

$\displaystyle\frac{1}{2}\min(1,c)\|u\|_{H^{\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})}$ $\displaystyle\leq
B_{c}(u)\leq\frac{1}{2}\max(1,c)\|u\|_{H^{\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})},$

hence, $B_{c}(u)$ is equivalent to the squared $H^{\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})$ norm. Moreover, for $\alpha\in(1,2)$, $\delta\in
H^{-\frac{\alpha}{2}}_{\rm per}(\mathbb{T})$, the dual of
$H^{\frac{\alpha}{2}}_{\rm per}(\mathbb{T})$ since

$\|\delta\|_{H^{-\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})}=\sum_{\xi\in\mathbb{Z}}\frac{1}{\left(1+|\xi|^{2}\right)^{\frac{\alpha}{2}}}<\infty.$

Thus, by Lax–Milgram theorem (see Corollary 5.8 in [5]), there exists a unique
$G_{\mathbb{T}}\in H^{\frac{\alpha}{2}}_{\rm per}(\mathbb{T})$ such that
$G_{\mathbb{T}}$ is the global minimizer of the variational problem (4.7), for
which the Euler–Lagrange equation is equivalent to the boundary-value problem
(1.3). By uniqueness of solutions of the two problems, $G_{\mathbb{T}}$ is
equivalently written as the Fourier series (1.4), from which it follows that
$G_{\mathbb{T}}(\pi)<G_{\mathbb{T}}(0)$. Hence, $G_{\mathbb{T}}$ is different
from a constant function on $\mathbb{T}$.

######  Remark 4.2.

The variational method and in particular the Lax–Milgram theorem cannot be
applied to the case $\alpha\in(0,1]$ since the Dirac delta distibution
$\delta$ does not belong to the dual space of $H^{\frac{\alpha}{2}}_{\rm
per}(\mathbb{T})$ when $\alpha\in(0,1]$.

Step 2: We utilize the fractional Polya–Szegö inequality, proved in the
appendix of [9], to show that a symmetric decreasing rearrangement of the
minimizer $G_{\mathbb{T}}$ on $\mathbb{T}$ does not increase $B_{c}(u)$. For
completeness, we state the following definition and lemma.

######  Definition 4.1.

Let $m$ be the Lebesgue measure on $\mathbb{T}$ and
$f(x):\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$ be a $2\pi$
periodic function. The symmetric and decreasing rearrangement $\tilde{f}$ of
$f$ on $\mathbb{T}$ is given by

$\tilde{f}(x)=\inf\\{t:\;\;m(\\{z\in\mathbb{T}:\;\;f(z)>t\\})\leq
2|x|\\},\hskip 14.45377ptx\in\mathbb{T}.$ (4.9)

The rearrangement $\tilde{f}$ satisfies the following properties:

  * i)

$\tilde{f}(-x)=\tilde{f}(x)$ and $f^{\prime}(x)\leq 0$ for $x\in(0,\pi)$.

  * ii)

$\tilde{f}(0)=\max_{x\in\mathbb{T}}f(x)$.

  * iii)

$\|\tilde{f}\|_{L^{2}(\mathbb{T})}=\|f\|_{L^{2}(\mathbb{T})}$.

######  Lemma 4.1.

[9] For every $\alpha>1$ and every $f\in
H^{\frac{\alpha}{2}}_{per}(\mathbb{T})$, it is true that

$\int_{-\pi}^{\pi}|D^{\frac{\alpha}{2}}\tilde{f}|^{2}dx\leq\int_{-\pi}^{\pi}|D^{\frac{\alpha}{2}}f|^{2}dx.$
(4.10)

The argument of the proof in the second step goes as follows. Suppose
$\widetilde{G}_{\mathbb{T}}$ is the symmetric and decreasing rearrangement of
$G_{\mathbb{T}}$, then by Lemma 4.1 and by property (iii) of Definition 4.9 we
have $B_{c}(\widetilde{G}_{\mathbb{T}})\leq B_{c}(G_{\mathbb{T}})$. Since the
global minimizer of the variational problem (4.7) is uniquely given by
$G_{\mathbb{T}}$, $\widetilde{G}_{\mathbb{T}}$ coincides with $G_{\mathbb{T}}$
up to a translation on $\mathbb{T}$. However, it follows from (1.4) that
$G_{\mathbb{T}}(-x)=G_{\mathbb{T}}(x)$ and
$G_{\mathbb{T}}(\pi)<G_{\mathbb{T}}(0)$, hence an internal maximum at
$x_{0}\in(0,\pi)$ would contradicts to the single-lobe profile of
$G_{\mathbb{T}}$ and the only maximum of $G_{\mathbb{T}}$ is located at $0$,
so that $G_{\mathbb{T}}(x)=\widetilde{G}_{\mathbb{T}}(x)$ for every
$x\in\mathbb{T}$. It follows from property (i) of Definition 4.9 that
$G_{\mathbb{T}}^{\prime}(x)\leq 0$ for $x\in(0,\pi)$. ∎

##  5\. Green’s function $G_{\mathbb{T}}$ for $\alpha>2$

Here we provide numerical approximations of the Green’s function
$G_{\mathbb{T}}$ for $\alpha>2$, which support Conjecture 1. The profiles of
$G_{\mathbb{T}}$ are depicted on Figure 2. We only give details on how zeros
of $G_{\mathbb{T}}(\pi)$ depend on parameters $(c,\alpha)$.

It follows from the Fourier series (1.4) that $G_{\mathbb{T}}(\pi)$ can be
computed by the numerical series

$G_{\mathbb{T}}(\pi)=\frac{1}{2\pi}\left(\frac{1}{c}+2\sum_{n=1}^{\infty}\frac{(-1)^{n}}{c+n^{\alpha}}\right),$
(5.1)

where the series converges absolutely if $\alpha>1$. On the other hand,
$G_{\mathbb{T}}(\pi)$ can also be computed from the integral representation
(4.3), that is,

$G_{\mathbb{T}}(\pi)=\frac{1}{\pi
c}\int_{0}^{\infty}\frac{e^{t}}{(1+e^{t})^{2}}E_{\alpha}(-ct^{\alpha})dt,$
(5.2)

which converges absolutely for $c\in(0,c_{\alpha})$, see Proposition 3.2,
where $c_{\alpha}$ is given by (3.9).

Figure 3. Difference between computations of $G_{\mathbb{T}}(\pi)$ in (5.1)
and (5.2) for $\alpha=2.5$ (left) and $\alpha=3.5$ (right) versus parameter
$c$.

Figure 3 shows the difference of $G_{\mathbb{T}}(\pi)$ computed from (5.1) and
(5.2) for $\alpha=2.5$ (left) and $\alpha=3.5$ (right) in logarithmic scale
versus parameter $c$. The Fourier series (5.1) is truncated such that the
remainder is of the size $\mathcal{O}(10^{-10})$. For the integral
representation of $G_{\mathbb{T}}(\pi)$ in (5.2), we numerically compute the
Mittag–Leffler function $E_{\alpha}(-ct^{\alpha})$ on the half line; this task
is accomplished by using the Matlab code provided in [14], where the Mittag-
Leffler functions are approximated with relative errors of the size
$\mathcal{O}(10^{-15})$. As follows from Fig. 3, the difference between the
two computations is constantly small if $c<c_{\alpha}$, when the integral
representation (5.2) converges absolutely, where $c_{\alpha=2.5}\approx 18.8$
and $c_{\alpha=3.5}\approx 5.2$. However, the accuracy of numerical
computations based on the integral representation (5.2) deteriorates for $c$
approaching $c_{\alpha}$ and as a result, the difference between two
computations quickly grows for $c>c_{\alpha}$.

Figure 4. Top: Location of the first five roots of $G_{\mathbb{T}}(\pi)$ on
the $(c,\alpha)$ plane. Bottom: The first root of $G_{\mathbb{T}}(\pi)$
relative to the boundary $c_{\alpha}$ (left). Coalescence of the 2nd and 3rd
roots (upper right) and the 4th and 5th roots (lower right).

Roots of $G_{\mathbb{T}}(\pi)$ in $c$ for each fixed $\alpha>2$ are computed
from the Fourier series representation (5.1) using the bisection method.
Figure 4 (top) shows the first five zeros of $G_{\mathbb{T}}(\pi)$ on the
$(c,\alpha)$ plane, where the dots show the roots of $G_{\mathbb{T}}(\pi)$
computed from the exact solutions in Appendix B for $\alpha=4$. The first root
exists for every $\alpha>2$ and is located inside $(0,c_{\alpha})$, see the
bottom left panel. The other roots are located outside $(0,c_{\alpha})$ and
disappear via pairwise coalescence as $\alpha$ is reduced towards $\alpha=2$,
see the bottom right panels. The 2nd and 3rd roots coalesce at $\alpha\approx
3.325$ and the 4th and 5th roots coalesce at $\alpha\approx 3.89$. The number
of terms in the Fourier series of $G_{\mathbb{T}}(\pi)$ is increased to
compute the 4th and 5th roots such that the remainder is of the size of
$\mathcal{O}(10^{-14})$ because $G_{\mathbb{T}}(\pi)$ becomes very small near
the location of these roots.

Table 1 compares the error between the numerically detected roots at
$\alpha=4$ and the roots of $G_{\mathbb{T}}(\pi)$ obtained from solving the
transcendental equation (B.7) in Appendix B.

Root | Error
---|---
1st | 1.9915 e-11
2nd | 7.1495 e-08
3rd | 3.3182 e-06
4th | 0.0031
5th | 0.0156
Table 1. Difference between locations of the first five roots of
$G_{\mathbb{T}}(\pi)$ for $\alpha=4$ computed from (5.1) and (B.7).

Green’s function $G_{\mathbb{T}}$ was computed versus $x$ using the Fourier
series representation (1.4) for fixed values of $(c,\alpha)$. The plots of
$G_{\mathbb{T}}$ are shown in Figures 1 and 2. Since the first root of
$G_{\mathbb{T}}(\pi)$ occurs at $c\approx 2.507$ for $\alpha=2.5$ and at
$c\approx 1.446$ for $\alpha=3.5$, see Fig. 4 (bottom left panel), the
threshold $c_{0}$ in Conjecture 1 reduces with the larger value of $\alpha$.
Appendix C gives a formal asymptotic approximation of the threshold $c_{0}$ as
$\alpha\to 2$.

##  6\. Conclusion

The main contribution of this work is the novel relation between Green’s
function for the linear operator $c+(-\Delta)^{\alpha/2}$ on the periodic
domain $\mathbb{T}$ and the Mittag–Leffler function. With the help of this
relation, we have proved that Green’s funciton is strictly positive on
$\mathbb{T}$ and single-lobe (monotonically decreasing away from the maximum
point) for every $c>0$ and $\alpha\in(0,2]$. The same property is also true
for sufficiently small $c$ and $\alpha\in(2,4]$ but we give numerical and
asymptotic results that Green’s function has a finite number of zeros on
$\mathbb{T}$ for sufficiently large $c$, the number of zeros is bounded in the
limit $c\to\infty$ for $\alpha\in(2,4)$ but is unbounded for $\alpha=4$.
Rigorous proof of properties of Green’s function for $\alpha\in(2,4)$ is an
open problem left for further studies.

##  Appendix A Green’s function $G_{\mathbb{T}}$ for $\alpha=2$

Here we derive the exact analytic form of Green’s function $G_{\mathbb{T}}$
for $\alpha=2$. The following proposition reproduces Theorem 1 for $\alpha=2$.

######  Proposition A.1.

For every $c>0$, Green’s function $G_{\mathbb{T}}$ at $\alpha=2$ is even,
strictly positive on $\mathbb{T}$, and strictly monotonically decreasing on
$(0,\pi)$.

###### Proof.

For $\alpha=2$, Green’s function $G_{\mathbb{T}}$ satisfies the second-order
differential equation

$-G_{\mathbb{T}}^{\prime\prime}(x)+cG_{\mathbb{T}}(x)=\delta(x),\quad
x\in\mathbb{T},$ (A.1)

where $c>0$. It follows from the theory of Dirac delta distributions that
$G_{\mathbb{T}}$ is continuous, even, periodic on $\mathbb{T}$, and have a
jump discontinuity of the first derivative at $x=0$.

To see the jump condition of $G_{\mathbb{T}}^{\prime}(x)$ across $x=0$, we
integrate (A.1) on $(-\varepsilon,\varepsilon)$ and then take the limit as
$\varepsilon\to 0$.

$\lim\limits_{\varepsilon\to
0}\int_{-\varepsilon}^{\varepsilon}\left(-G_{\mathbb{T}}^{\prime\prime}(x)+cG_{\mathbb{T}}(x)\right)dx=\lim\limits_{\varepsilon\to
0}\int_{-\varepsilon}^{\varepsilon}\delta(x)dx=1,$ (A.2)

where the last equality follows from properties of $\delta$. Since
$G_{\mathbb{T}}\in C^{0}(\mathbb{R})$, the second term on the left hand side
vanishes as $\varepsilon\to 0$, which yields
$-G_{\mathbb{T}}^{\prime}(0^{+})+G_{\mathbb{T}}^{\prime}(0^{-})=1$. Since
$G_{\mathbb{T}}$ is even on $\mathbb{R}$, we obtain

$G_{\mathbb{T}}^{\prime}(0^{+})=-\frac{1}{2}.$ (A.3)

Additionally, it follows from the Fourier series representation (1.4) with
$\alpha=2$ that

$G_{\mathbb{T}}(0)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\frac{1}{c+n^{2}}=\frac{\coth(\sqrt{c}\pi)}{2\sqrt{c}},$
(A.4)

where we have used numerical series (4) in [32, Section 5.1.25].

The differential equation (A.1) is solved for even $G_{\mathbb{T}}$ as
follows:

$\displaystyle
G_{\mathbb{T}}(x)=G_{\mathbb{T}}(0)\cosh(\sqrt{c}x)+G_{\mathbb{T}}^{\prime}(0^{+})\frac{\sinh(\sqrt{c}|x|)}{\sqrt{c}},\quad
x\in\mathbb{T}.$

Due to (A.3) and (A.4), this can be rewritten in the closed form as

$\displaystyle
G_{\mathbb{T}}(x)=\frac{\cosh(\sqrt{c}(\pi-|x|))}{2\sqrt{c}\sinh(\sqrt{c}\pi)},\quad
x\in\mathbb{T}.$ (A.5)

It follows from (A.5) that

$\displaystyle
G_{\mathbb{T}}^{\prime}(x)=-\frac{\sinh(\sqrt{c}(\pi-x))}{2\sinh(\sqrt{c}\pi)}<0,\quad
x\in(0,\pi),$ (A.6)

and hence $G_{\mathbb{T}}$ is strictly monotonically decreasing on $(0,\pi)$.
On the other hand,

$\displaystyle G_{\mathbb{T}}(\pi)=\frac{1}{2\sinh(\sqrt{c}\pi)}>0,\quad c>0,$
(A.7)

and hence $G_{\mathbb{T}}$ is strictly positive on $\mathbb{T}$. Note that the
exact expression for $G_{\mathbb{T}}(\pi)$ in (A.7) also follows from
numerical series (6) in [32, Section 5.1.25]. ∎

######  Remark A.1.

It follows from (A.5) that $G_{\mathbb{T}}^{\prime}(\pi)=0$, due to smoothness
and periodicity of even $G_{\mathbb{T}}(x)$ across $x=\pm\pi$. Therefore, the
exact expression in (A.5) and the relation for $G_{\mathbb{T}}(0)$ in (A.4)
can be alternatively found by solving the differential equation (A.1) for even
$G_{\mathbb{T}}$ subject to the boundary conditions
$G_{\mathbb{T}}^{\prime}(0^{\pm})=\mp\frac{1}{2}$ and
$G_{\mathbb{T}}^{\prime}(\pm\pi)=0$.

##  Appendix B Green’s function $G_{\mathbb{T}}$ for $\alpha=4$

Here we derive the exact analytic form of Green’s function $G_{\mathbb{T}}$
for $\alpha=4$. The following proposition proves Conjecture 1 for $\alpha=4$.

######  Proposition B.1.

There exists $c_{0}>0$ such that for $c\in(0,c_{0})$, Green’s function
$G_{\mathbb{T}}$ at $\alpha=4$ is even, strictly positive on $\mathbb{T}$, and
strictly monotonically decreasing on $(0,\pi)$. For $c\in[c_{0},\infty)$,
$G_{\mathbb{T}}$ has a finite number of zeros on $\mathbb{T}$, which becomes
unbounded as $c\to\infty$.

###### Proof.

For $\alpha=4$, Green’s function $G_{\mathbb{T}}$ satisfies the fourth-order
differential equation

$G_{\mathbb{T}}^{\prime\prime\prime\prime}(x)+cG_{\mathbb{T}}(x)=\delta(x),\quad
x\in\mathbb{T},$ (B.1)

where $c>0$. It follows from the theory of Dirac delta distributions that
$G_{\mathbb{T}}$ is continuous, even, periodic on $\mathbb{T}$, and have a
jump discontinuity of the third derivative at $x=0$. Similarly to the
computation in (A.2), it follows that Green’s function solves the boundary-
value problem with the boundary conditions

$G_{\mathbb{T}}^{\prime}(0)=G_{\mathbb{T}}^{\prime}(\pm\pi)=G_{\mathbb{T}}^{\prime\prime\prime}(\pm\pi)=0,\quad
G_{\mathbb{T}}^{\prime\prime\prime}(0^{\pm})=\pm\frac{1}{2}.$ (B.2)

Due to the boundary conditions (B.2), it is easier to solve the differential
equation (B.1) for $G_{\mathbb{T}}^{\prime}$ on $[0,\pi]$. By using the
parametrization $c=4a^{4}$, we obtain

$\displaystyle G_{\mathbb{T}}^{\prime}(x)$ $\displaystyle=$ $\displaystyle
c_{1}\cosh(ax)\cos(ax)+c_{2}\cosh(ax)\sin(ax)$
$\displaystyle+c_{3}\sinh(ax)\cos(ax)+c_{4}\sinh(ax)\sin(ax),\quad
x\in[0,\pi],$

where $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ are some coefficients. We can
find $c_{1}=0$ and $c_{4}=\frac{1}{4a^{2}}$ from the two boundary conditions
(B.2) at $x=0^{+}$. The other two boundary conditions (B.2) at $x=\pi$ gives
the linear system for $c_{2}$ and $c_{3}$:

$\displaystyle\begin{bmatrix}\cosh(\pi a)\sin(\pi a)&\sinh(\pi a)\cos(\pi
a)\\\ \sinh(\pi a)\cos(\pi a)&-\cosh(\pi a)\sin(\pi
a)\end{bmatrix}\begin{bmatrix}c_{2}\\\
c_{3}\end{bmatrix}=-c_{4}\begin{bmatrix}\sinh(\pi a)\sin(\pi a)\\\ \cosh(\pi
a)\cos(\pi a)\end{bmatrix}.$

By Cramer’s rule, we find the unique solution

$\displaystyle c_{2}=-c_{4}\frac{\sinh(2\pi a)}{\cosh(2\pi a)-\cos(2\pi
a)},\quad c_{3}=c_{4}\frac{\sin(2\pi a)}{\cosh(2\pi a)-\cos(2\pi a)},$

which results in the exact analytical expression

$G_{\mathbb{T}}^{\prime}(x)=\frac{1}{4a^{2}}\frac{\sinh(ax)\sin
a(2\pi-x)-\sin(ax)\sinh a(2\pi-x)}{\cosh(2\pi a)-\cos(2\pi a)},\quad
x\in[0,\pi].$ (B.3)

Integrating (B.3) in $x$ yields the exact analytical expression for
$G_{\mathbb{T}}$:

$G_{\mathbb{T}}(x)=\frac{1}{8a^{3}}\frac{g(x)}{\cosh(2\pi a)-\cos(2\pi
a)},\quad x\in[0,\pi],$ (B.4)

where

$\displaystyle g(x)$ $\displaystyle:=$ $\displaystyle\sinh(ax)\cos
a(2\pi-x)+\cosh(ax)\sin a(2\pi-x)$ $\displaystyle+\sin(ax)\cosh
a(2\pi-x)+\cos(ax)\sinh a(2\pi-x)$

and the constant of integration is set to zero due to the differential
equation (B.1).

We verify the validity of the exact solution (B.4) by comparing
$G_{\mathbb{T}}(0)$ and $G_{\mathbb{T}}(\pi)$ with the Fourier series
representation (1.4) for $\alpha=4$:

$G_{\mathbb{T}}(0)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\frac{1}{4a^{4}+n^{2}}=\frac{1}{8a^{3}}\frac{\sinh(2\pi
a)+\sin(2\pi a)}{\cosh(2\pi a)-\cos(2\pi a)}$ (B.5)

and

$\displaystyle
G_{\mathbb{T}}(\pi)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\frac{(-1)^{n}}{4a^{4}+n^{2}}=\frac{1}{4a^{3}}\frac{\sinh(\pi
a)\cos(\pi a)+\sin(\pi a)\cosh(\pi a)}{\cosh(2\pi a)-\cos(2\pi a)}.$ (B.6)

Indeed, the exact expressions coincide with those found from the numerical
series (1) and (2) in [32, Section 5.1.27].

It follows from (B.6) that $G_{\mathbb{T}}(\pi)$ vanishes for $c=4a^{4}>0$ if
and only if $a>0$ is a solution of the transcendental equation

$\tanh(\pi a)+\tan(\pi a)=0.$ (B.7)

Elementary graphical analysis on Figure 5 shows that there exist a countable
sequence of zeros $\\{a_{n}\\}_{n\in\mathbb{N}}$ such that
$a_{n}\in\left(n-\frac{1}{4},n\right)$, $n\in\mathbb{N}$. Hence,
$G_{\mathbb{T}}$ is not positive for $a\in(a_{1},\infty)$.

Figure 5. Countable sequence of zeros $\\{a_{n}\\}_{n\in\mathbb{N}}$ of B.7

Let us now show that the profile of $G_{\mathbb{T}}$ is strictly,
monotonically decreasing on $(0,\pi)$ for small $a$. It follows from (B.3)
that $G_{\mathbb{T}}^{\prime}(x)<0$ for $x\in(0,\pi)$ if and only if

$\frac{\sin(ax)}{\sinh(ax)}>\frac{\sin a(2\pi-x)}{\sinh a(2\pi-x)},\qquad
x\in(0,\pi).$ (B.8)

The function

$x\mapsto\frac{\sin(ax)}{\sinh(ax)}$

is monotonically decreasing on $[0,2\pi]$ as long as

$\cos(ax)\sinh(ax)-\sin(ax)\cosh(ax)\leq 0,\qquad x\in[0,2\pi],$ (B.9)

which is true at least for $a\in(0,\frac{1}{2})$. Hence, $G_{\mathbb{T}}$ is
strictly motonically decreasing on $(0,\pi)$ with $G_{\mathbb{T}}(\pi)>0$ for
$a\in(0,a_{0})$, where $a_{0}\in(\frac{1}{2},1)$. On the other hand, it is
obvious that there exists $a_{*}\in(1,\frac{3}{2})$ such that the inequality
(B.9) [and hence the inequality (B.8)] is violated at $x=\pi$ for
$a\in(a_{*},2)$, for which $G_{\mathbb{T}}^{\prime}(x)>0$ at least near
$x=\pi$.

The first part of the proposition is proven due to the relation $c=4a^{4}$. It
remains to prove that $G_{\mathbb{T}}$ has a finite number of zeros on
$\mathbb{T}$ for fixed $a\in[a_{0},\infty)$ which becomes unbounded as
$a\to\infty$. To do so, we simplify the expression (B.4) for $G_{\mathbb{T}}$
in the asymptotic limit of large $a$ for every fixed $x\in(0,\pi)$:

$G_{\mathbb{T}}(x)=\frac{1}{8a^{3}}\left[e^{-ax}\cos(ax)+e^{-ax}\sin(ax)+\mathcal{O}(e^{-a(2\pi-x)})\right]\quad\mbox{\rm
as}\quad a\to\infty.$ (B.10)

Thus, as $a$ gets large, there are finitely many zeros of $G_{\mathbb{T}}$ on
$(0,\pi)$ but the number of zeros of $G_{\mathbb{T}}$ grows unbounded as
$a\to\infty$. ∎

######  Remark B.1.

The leading-order term in the asymptotic expansion (B.10) represents Green’s
function $G_{\mathbb{R}}$. The proof of Conjecture 2 for $\alpha=4$ follows
from this explicit expression.

######  Remark B.2.

Figure 6 shows boundaries on the $(a,x)$ plane between positive (yellow) and
negative (blue) values of $G_{\mathbb{T}}$ (left) and
$G_{\mathbb{T}}^{\prime}$ (right). It follows from the figure that the zeros
of $G_{\mathbb{T}}$ and $G_{\mathbb{T}}^{\prime}$ are monotonically decreasing
with respect to parameter $a$ and the number of zeros only grows as $a$
increases. In other words, zeros of $G_{\mathbb{T}}$ cannot coalesce and
disappear. We were not able to prove these properties for every $a>0$ inside
$(0,\pi)$ (however, the proof can be given at $x=\pi$).

Figure 6. Left: areas on $(a,x)$ plane where $G_{\mathbb{T}}$ is positive
(yellow) and negative (blue). Right: the same but for
$G_{\mathbb{T}}^{\prime}$.

##  Appendix C Asymptotic approximation of the first zero of
$G_{\mathbb{T}}(\pi)$

Here, we obtain the formal asymptotic dependence of the first zero of
$G_{\mathbb{T}}(\pi)$ as $(c,\alpha)\to(\infty,2)$. We use the integral
representation (5.2). Replacing the Mittag–Leffler function $E_{\alpha}$ by
its leading-order asymptotic expression (2.6) for $2\leq\alpha<6$, we obtain
formally

$G_{\mathbb{T}}(\pi)=\frac{1}{2\pi\alpha c}\left[I(a,b)+\mbox{\rm error
terms}\right],$ (C.1)

where

$I(a,b):=\int_{0}^{\infty}{\rm
sech}^{2}\left(\frac{t}{2}\right)e^{at}\cos(bt)dt$ (C.2)

with

$a:=c^{\frac{1}{\alpha}}\cos\left(\frac{\pi}{\alpha}\right),\quad
b:=c^{\frac{1}{\alpha}}\sin\left(\frac{\pi}{\alpha}\right).$ (C.3)

The limit $(c,\alpha)\to(\infty,2)$ such that $c<c_{\alpha}$ corresponds to
the limit $b\to\infty$ with $a<1$.

The integral $I(a,b)$ is the rapidly oscillating integral in the limit
$b\to\infty$. We split it into two parts:

$\displaystyle I(a,b)$ $\displaystyle=$ $\displaystyle
I_{1}(a,b)+I_{2}(a,b)\qquad\qquad$ (C.4) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm
sech}^{2}\left(\frac{t}{2}\right)\cosh(at)\cos(bt)dt+\int_{0}^{\infty}{\rm
sech}^{2}\left(\frac{t}{2}\right)\sinh(at)\cos(bt)dt,$

where $I_{1}(a,b)$ is exponentially small in $b$ and $I_{2}(a,b)$ is
algebraically small in $b$. Indeed, by Darboux principle [1], we evaluate the
first integral for $a<1$ with the residue theorem:

$\displaystyle I_{1}(a,b)$ $\displaystyle=\frac{1}{2}{\rm
Re}\int_{-\infty}^{\infty}{\rm
sech^{2}}\left(\frac{t}{2}\right)\cosh(at)e^{ibt}dt\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle={\rm Re}4\pi i{\rm Res}_{z=\pi
i}\left[\frac{e^{z}\cosh(az)e^{ibz}}{(1+e^{z})^{2}}\right]+\mathcal{O}(e^{-3\pi
b})$ $\displaystyle=4\pi\left[b\cos(\pi a)+a\sin(\pi a)\right]e^{-\pi
b}+\mathcal{O}(e^{-3\pi b}).$

Integrating the second integral by parts several times for $a<1$ (see Section
5.2 in [21]), we obtain

$\displaystyle I_{2}(a,b)$ $\displaystyle=\frac{1}{b}\sin(bt)\sinh(at){\rm
sech}^{2}\left(\frac{t}{2}\right)\biggr{|}_{t=0}^{t\to\infty}-\frac{1}{b}\int_{0}^{\infty}\sin(bt)\frac{d}{dt}\left[{\rm
sech}^{2}\left(\frac{t}{2}\right)\sinh(at)\right]dt\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=\frac{\cos(bt)}{b^{2}}\frac{d}{dt}\left[{\rm
sech}^{2}\left(\frac{t}{2}\right)\sinh(at)\right]\biggr{|}_{t=0}^{t\to\infty}-\frac{1}{b^{2}}\int_{0}^{\infty}\cos(bt)\frac{d^{2}}{dt^{2}}\left[{\rm
sech}^{2}\left(\frac{t}{2}\right)\sinh(at)\right]dt$
$\displaystyle=-\frac{a}{b^{2}}+\mathcal{O}\left(\frac{a}{b^{4}}\right).$

Finding zero of $I(a,b)$ in $a$ as $b\to\infty$ yields the approximation

$a=4\pi b^{3}e^{-\pi
b}\left[1+\mathcal{O}\left(\frac{1}{b^{2}}\right)\right].$ (C.5)

Substituting (C.3) into (C.5) then taking the logarithm of both sides yields
the following transcendental equation for $c^{\frac{1}{\alpha}}$

$2\ln\left(c^{\frac{1}{\alpha}}\right)-\pi\sin\left(\frac{\pi}{\alpha}\right)c^{\frac{1}{\alpha}}+\mathcal{O}(c^{-\frac{2}{\alpha}})=\ln\left(\frac{\cos(\frac{\pi}{\alpha})}{4\pi\sin^{3}\left(\frac{\pi}{\alpha}\right)}\right).$
(C.6)

In order to determine the dependence of $c$ in terms of $\alpha$ we first
factor $c^{\frac{1}{\alpha}}$ on the left hand side, then expand around
$\alpha=2$ to obtain

$-\pi
c^{\frac{1}{\alpha}}\left[1+\mathcal{O}\left((\alpha-2)^{2}\right)+\mathcal{O}\left(\frac{\ln\left(c^{\frac{1}{\alpha}}\right)}{c^{\frac{1}{\alpha}}}\right)\right]=\ln\left[\frac{\alpha-2}{16}+\mathcal{O}\left((\alpha-2)^{2}\right)\right].$

Since $c^{\frac{1}{\alpha}}$ is of order $\mathcal{O}(\ln(\alpha-2))$, the
above equation becomes

$-\pi
c^{\frac{1}{\alpha}}=\ln\left(\frac{\alpha-2}{16}\right)\left[1+\mathcal{O}\left(\frac{\ln|\ln(\alpha-2)|}{|\ln(\alpha-2)|}\right)\right],$

which, as $\alpha\to 2$, implies

$c=\frac{1}{\pi^{2}}\ln^{2}\left(\frac{\alpha-2}{16}\right)\left[1+\mathcal{O}\left(\frac{\ln|\ln(\alpha-2)|}{|\ln(\alpha-2)|}\right)\right].$
(C.7)

The asymptotic approximation (C.7) suggests that the first zero of
$G_{\mathbb{T}}(\pi)$ at $c=c_{0}(\alpha)$ satisfies $c_{0}(\alpha)\to\infty$
as $\alpha\to 2$. However, we note that the asymptotic approximation of the
root of $I(a,b)$ given by (C.7) is derived without analysis of the error terms
in (C.1).

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