# A Spectral Study of the Linearized Boltzmann Equation for Diffusively
Excited Granular Media

Thomas Rey  Thomas Rey
CSCAMM, The University of Maryland
CSIC Building, Paint Branch Drive
College Park, MD 20740
USA<EMAIL_ADDRESS>
###### Abstract.

In this work, we are interested in the spectrum of the diffusively excited
granular gases equation, in a space inhomogeneous setting, linearized around
an homogeneous equilibrium.

We perform a study which generalizes to a non-hilbertian setting and to the
inelastic case the seminal work of Ellis and Pinsky [8] about the spectrum of
the linearized Boltzmann operator. We first give a precise localization of the
spectrum, which consists in an essential part lying on the left of the
imaginary axis and a discrete spectrum, which is also of nonnegative real part
for small values of the inelasticity parameter. We then give the so-called
inelastic “dispersion relations”, and compute an expansion of the branches of
eigenvalues of the linear operator, for small Fourier (in space) frequencies
and small inelasticity.

One of the main novelty in this work, apart from the study of the inelastic
case, is that we consider an exponentially weighted $L^{1}(m^{-1})$ Banach
setting instead of the classical $L^{2}(\mathcal{M}_{1,0,1}^{-1})$ Hilbertian
case, endorsed with Gaussian weights. We prove in particular that the results
of [8] holds also in this space.

###### Key words and phrases:

Inelastic Boltzmann equation, granular gases, spectrum, dispersion relations,
hydrodynamic limit, heat equation

###### 2010 Mathematics Subject Classification:

Primary: 76P05, 82C40, Secondary: 35P20, 76T25

###### Contents

  1. 1 Introduction
    1. 1.1 The Model Considered
    2. 1.2 The Linearized Operator
    3. 1.3 Functional Framework and Main Results
    4. 1.4 Method of Proof and Plan of the Paper
  2. 2 Localization of the Spectrum
    1. 2.1 Geometry of the Essential Spectrum
    2. 2.2 Behavior of the Eigenvalues for Small Inelasticity
  3. 3 Inelastic Dispersion Relations
    1. 3.1 Projection of the Eigenvalue Problem
    2. 3.2 Finite Dimensional Resolution
    3. 3.3 First Order Coefficients of the Taylor Expansion
    4. 3.4 Higher Order Expansion
  4. A Functional Toolbox on the Collision Operator

##  1\. Introduction

Let $f^{\varepsilon}:=f^{\varepsilon}(t,x,v)$ be a solution to the space
inhomogeneous collisional kinetic equation

(1) $\frac{\partial f^{\varepsilon}}{\partial
t}+v\cdot\nabla_{x}f^{\varepsilon}=\frac{1}{\varepsilon}\left(\mathcal{Q}_{\alpha}(f^{\varepsilon},f^{\varepsilon})+\varepsilon\,\Delta_{v}(f^{\varepsilon})\right),$

where $t\geq 0$, $v\in\mathbb{R}^{d}$ and $x\in\Omega$, for $\Omega$ being
either the whole space domain $\mathbb{R}^{d}$ or the torus111The case of a
square domain $[-L,L]^{d}$, for $L\geq 0$ with specular reflection on the
boundary can also be seen as a particular case of a torus made of $2^{d}$
independent copies of the initial box, using the parity of the normal
component of the velocity of $f^{\varepsilon}$ at the boundary (as noticed by
Grad in [10]). $\mathbb{T}^{d}$. The collision operator $\mathcal{Q}_{\alpha}$
is the so-called _granular gases_ operator (sometimes known as the inelastic
Boltzmann operator), describing an energy-dissipative microscopic collision
dynamics, which we will present in the following section. The parameter
$\varepsilon>0$ is the scaled _Knudsen_ number, that is the ratio between the
mean free path of particles before a collision and the length scale of
observation.

Once $\varepsilon$ goes to $0$, and then when the number of collisions per
time unit goes to infinity, the complexity of equation (1) is (formally)
greatly reduced, the solution being described almost completely by its local
hydrodynamic fields, namely its _mass_ $N\geq 0$, its _momentum_
$\bm{u}\in\mathbb{R}^{d}$ and its _temperature_ $T\geq 0$. These quantities
are obtained from a particle distribution function $f$ by computing the first
moments in velocity:

(2) $\begin{gathered}N(t,x)\,=\,\int_{\mathbb{R}^{d}}f(t,x,v)\,dv,\qquad
N(t,x)\,\bm{u}(t,x)\,=\,\int_{\mathbb{R}^{d}}f(t,x,v)\,v\,dv,\\\
T(t,x)\,=\,\frac{1}{d\,N}\int_{\mathbb{R}^{d}}f(t,x,v)\,|v-\bm{u}|^{2}\,dv.\end{gathered}$

This reduction is usually carried on using the so-called Hilbert or Chapman-
Enskog expansions of the solutions to a linearized version of the kinetic
equation (1) (see _e.g._ the book of Cercignani, Illner and Pulvirenti [5] for
a complete mathematical introduction in the elastic case).

A rigorous mathematical proof of this “contraction of the kinetic description”
(namely the hydrodynamic limit of the kinetic model towards a macroscopic one)
for the elastic case has been first given for the linear setting in the paper
of Ellis and Pinsky [8] but the inelastic case still remains to be
investigated. An important step in the proof of this elastic limit is to give
the so-called _dispersion relations_ of the collision operator, namely a
Taylor expansion of the eigenvalues of the linearization of the collision
operator, with respect to the space variable, near a global equilibrium (and
this was the main purpose of [8]). The precise knowledge of the dispersion
relations is actually of crucial interest in the study of the full nonlinear
and compressible hydrodynamic limit and it was for example used by Kawashima,
Matsumura and Nishida in [20, 13] (as a part of a rather abstract Cauchy-
Kowalevski-type argument which is also related to Niremberg [19]). The work of
Caflisch [3] also relies (but perhaps not as centrally as the previous ones)
on these dispersion relations. Let us also quote the work of Degond and Lemou
[7] where a similar analysis of the dispersion relations was conducted for the
linearized Fokker-Planck equation.

We propose to give in this paper the corresponding inelastic expansion, with
respect to both the space variable and the inelasticity parameter, allowing to
investigate in a future work first the two linearized hydrodynamic limits of
our model “à la Ellis et Pinsky” and then the nonlinear, compressible ones “à
la Nishida”. This result will allow us in particular to confirm a claim
concerning the _clustering_ behavior of granular gases made in the classical
textbook [2, p. 238] after a formal analysis, namely that

> _the smaller the inelasticity, the larger the system must be to reveal
> clusters._

###  1.1. The Model Considered

Let $\alpha\in(0,1]$ be the restitution coefficient of the microscopic
collision process, that is the ratio of kinetic energy dissipated during a
collision, in the direction of impact. Then, we can define a strong form of
the _collision operator_ $\mathcal{Q}_{\alpha}$ by

(3) $\displaystyle\mathcal{Q}_{\alpha}(f,g)(v)$
$\displaystyle=\int_{\mathbb{R}^{d}\times\mathbb{S}^{d-1}}|u|\left(\frac{\,{}^{\prime}f\,^{\prime}g_{*}}{\alpha^{2}}-f\,g_{*}\right)b(\widehat{u}\cdot\sigma)\,d\sigma\,dv_{*},$
$\displaystyle=\mathcal{Q}_{\alpha}^{+}(f,g)(v)-f(v)L(g)(v),$

where we have used the usual shorthand notation
$\,{}^{\prime}f:=f(^{\prime}v)$, $\,{}^{\prime}f_{*}:=f(^{\prime}v_{*})$,
$f:=f(v)$, $f_{*}:=f(v_{*})$ and $\widehat{u}:=u/|u|$. In (3),
$\,{}^{\prime}v$ and $\,{}^{\prime}v_{*}$ are the pre-collisional velocities
of two particles of given velocities $v$ and $v_{*}$, defined for
$\sigma\in\mathbb{S}^{d-1}$ as

$\left\\{\begin{aligned}
&{}^{\prime}v=\frac{v+v_{*}}{2}-\frac{1-\alpha}{4\,\alpha}(v-v_{*})+\frac{1+\alpha}{4\,\alpha}|v-v_{*}|\,\sigma,\\\
&{}^{\prime}v_{*}=\frac{v+v_{*}}{2}+\frac{1-\alpha}{4\,\alpha}(v-v_{*})-\frac{1+\alpha}{4\,\alpha}|v-v_{*}|\,\sigma.\end{aligned}\right.$

The unitary vector $\sigma$ is the center of the _collision sphere_ (see
Figure 1) and $u:=v-v_{*}$ is the _relative velocity_ of the pair of
particles. Finally, the function $b$ is the so-called _angular cross-section_
, describing the probability of collision between two particles. We assume
that

(4) $b\text{ is a Lipschitz, non-decreasing and convex function on }(-1,1),$

and also that it is bounded from above and below by two nonnegative constants
$b_{m}$ and $b_{M}$:

(5) $b_{m}\leq b(x)\leq b_{M},\quad\forall x\in(-1,1).$

In particular, this cross-section is integrable on the unit sphere, thus
fulfilling the so-called Grad’s cut-off assumption222Physically relevant in
the case of inelastic collisions, due to the macroscopic size of the grains
forming the gas.. The operator $\mathcal{Q}_{\alpha}^{+}(f,g)(v)$ is usually
known as the _gain_ term because it can be understood as the number of
particles of velocity $v$ created by collisions of particles of pre-
collisional velocities $\,{}^{\prime}v$ and $\,{}^{\prime}v_{*}$, whereas
$f(v)L(g)(v)$ is the _loss_ term, modeling the loss of particles of pre-
collisional velocities $\,{}^{\prime}v$.

We can also give a weak form of the collision operator. Indeed, if
$\omega\in\mathbb{S}^{d-1}$ is the direction of impact, we can parametrize the
post-collisional velocities $v^{\prime}$ and $v_{*}^{\prime}$ as

$\left\\{\begin{aligned}
v^{\prime}&=v-\frac{1+\alpha}{2}\left(u\cdot\omega\right)\omega,\\\
v_{*}^{\prime}&=v_{*}+\frac{1+\alpha}{2}\left(u\cdot\omega\right)\omega.\end{aligned}\right.$

Then we have the weak representation, for any smooth test function $\psi$,

(6)
$\int_{\mathbb{R}^{d}}Q_{\alpha}(f,g)\,\psi(v)\,dv=\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}|u|f_{*}\,g\,\left(\psi^{\prime}+\psi_{*}^{\prime}-\psi-\psi_{*}\right)b(\widehat{u}\cdot\omega)\,d\omega\,dv\,dv_{*}.$

1.9063.14160.1*cos(t)+-0.22|0.1*sin(t)+0
$v$$v_{*}$$O$$\Omega_{+}$$\Omega_{-}$$v^{\prime}$$v_{*}^{\prime}$$v^{\prime}$$v_{*}^{\prime}$$\theta$$\sigma$$\omega$
Figure 1. Geometry of inelastic collisions, $O:=(v+v_{*})/2$ and
$\Omega_{\pm}:=O\pm(v_{*}-v)\,(1-e)/2$ (dashed lines represent the elastic
case).

Thanks to this expression, we can compute the macroscopic properties of the
collision operator $\mathcal{Q}_{\alpha}$. Indeed, we have the microscopic
conservation of impulsion and dissipation of kinetic energy:

$\displaystyle v^{\prime}+v_{*}^{\prime}$ $\displaystyle=v+v_{*},$
$\displaystyle|v^{\prime}|^{2}+|v_{*}^{\prime}|^{2}-|v|^{2}-|v_{*}|^{2}$
$\displaystyle=-\frac{1-\alpha^{2}}{2}|u\cdot\omega|^{2}\leq 0.$

Then if we integrate the collision operator against
$\varphi(v)=(1,\,v\,|v|^{2})$, we obtain the preservation of mass and momentum
and the dissipation of kinetic energy:

$\int_{\mathbb{R}^{d}}\mathcal{Q}_{\alpha}(f,f)(v)\begin{pmatrix}1\\\ v\\\
|v|^{2}\end{pmatrix}dv\,=\,\begin{pmatrix}0\\\ 0\\\
-(1-\alpha^{2})D(f,f)\end{pmatrix},$

where $D(f,f)\geq 0$ is the _energy dissipation_ functional, given by

(7)
$D(f,f):=b_{1}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}f\,f_{*}\,|v-v_{*}|^{3}\,dv\,dv_{*}\geq
0,$

and $b_{1}$ is the angular momentum, depending on the cross-section $b$ and
given by

$b_{1}:=\int_{\mathbb{S}^{d-1}}(1-(\widehat{u}\cdot\omega))\,b(\widehat{u}\cdot\omega)\,d\omega<\infty.$

It is of course finite thanks to the bounds (5).

In all the following of the paper, we shall assume that the restitution
coefficient is related to the Knudsen number in the following way :

$\alpha=1-\varepsilon.$

The macroscopic properties of the collision operator, together with the
conservation of positiveness, imply that the equilibrium profiles of
$\mathcal{Q}_{\alpha}$ are trivial Dirac masses (see _e.g._ the review paper
[21] of Villani). Nevertheless, adding a thermal bath $(1-\alpha)\Delta_{v}$
will prevent this fact. Indeed, the existence of a non-trivial equilibrium
profile $F_{\alpha}$ to the space homogeneous granular gases equation with a
thermal bath is insured by the competition occurring between the dissipation
of kinetic energy occasioned by the collision operator $\mathcal{Q}_{\alpha}$
and the gain of energy given by the diffusion term $\Delta_{v}$.

More precisely, if we multiply the equation
$\mathcal{Q}_{\alpha}(f,f)+(1-\alpha)\,\Delta_{v}(f)=0$ by $|v|^{2}$,
integrate in velocity and divide by $1-\alpha$, we obtain using (7) the
balance equation

(8) $\left(1+\alpha\right)D(f,f)=2\,d.$

It has then been shown in [1, 16] that under the hypotheses (4)–(5) on the
cross-section, there exists $\alpha_{*}\in(0,1)$ such that for all
$\alpha\in[\alpha_{*},1]$, there exists an unique equilibrium profile $0\leq
F_{\alpha}\in\mathcal{S}(\mathbb{R}^{d})$ of unit mass and zero momentum:

(9) $\left\\{\begin{aligned}
&\mathcal{Q}_{\alpha}(F_{\alpha},F_{\alpha})+(1-\alpha)\,\Delta_{v}(F_{\alpha})=0,\\\
&\,\\\
&\int_{\mathbb{R}^{d}}F_{\alpha}(v)\,dv=1,\qquad\int_{\mathbb{R}^{d}}F_{\alpha}(v)\,v\,dv=0.\end{aligned}\right.$

In the last expression, $\mathcal{S}(\mathbb{R}^{d})$ denotes the Schwartz
class of $\mathcal{C}^{\infty}$ functions decreasing at infinity faster than
any polynomials. The tails of this distribution are exponentials, of order
$3/2$.

Of course, if $\alpha=1$ (elastic, non-heated case), the distribution $F_{1}$
is nothing but the following _Maxwellian_ 333Hence, there is a bifurcation
which occur between the inelastic heated case and the elastic nonheated one.
distribution

(10) $F_{1}(v):=\mathcal{M}_{1,0,\bar{T}_{1}}(v),$

where $\mathcal{M}_{N,\,\bm{u},\,T}$ is the Maxwellian distribution of mass
$N$, velocity $\bm{u}$ and temperature $T$, only equilibria of the elastic
collision operator $\mathcal{Q}_{1}$ (see _e.g._[5] for more details), and
given by

$\mathcal{M}_{N,\,\bm{u},\,T}(v):=\frac{N}{(2\pi
T)^{d/2}}\exp\left(\frac{|v-\bm{u}|^{2}}{2T}\right)$

for $(N,\bm{u},T)\in\mathbb{R}^{d+2}$. The quantity $\bar{T}_{1}$ in (10) is
defined by passing to the limit $\alpha\to 1$ in the balance equation (8) :

$D(F_{1},F_{1})=d.$

We can then show thanks to this relation (see [16] for details) that
$\bar{T}_{1}$ is given by

(11)
$\bar{T}_{1}=\frac{1}{2}\frac{d^{2/3}}{b_{1}^{2/3}}\left(\int_{\mathbb{R}^{d}}\mathcal{M}_{1,0,1}(v)|v|^{3}\,dv\right)^{-2/3}.$

###  1.2. The Linearized Operator

As we have said in the introduction, our goal is to perform the fluid dynamic
limit $\varepsilon\to 0$ of equation (1). By rescaling the time
$\widetilde{t}=t/\varepsilon$ and introducing a new distribution
$\widetilde{f}(\widetilde{t},x,v)=f(t,x,v)$, the equation (1) now reads
(forgetting the tildas)

(12) $\frac{\partial f^{\varepsilon}}{\partial
t}+\varepsilon\,v\cdot\nabla_{x}f^{\varepsilon}=\mathcal{Q}_{\varepsilon}(f^{\varepsilon},f^{\varepsilon})+\varepsilon\,\Delta_{v}(f^{\varepsilon}).$

The hydrodynamic limit then amounts to consider the large time, small space
variations of the model (see the paper of Carlen, Chow and Grigo [4] for more
details on the scaling and on the different types of limit models it can
yields). This means as $\varepsilon=1-\alpha$ that we are studying
_fluctuations_ $g$ of $f^{\alpha}$ near the space homogeneous equilibrium
profile $F_{\alpha}$:

(13) $f^{\alpha}=F_{\alpha}+g.$

By plugging this expansion on equation (12) and using the equilibrium relation
(9), we obtain the following equation for $g$:

(14) $\frac{\partial g}{\partial
t}+(1-\alpha)\,v\cdot\nabla_{x}g=\mathcal{L}_{\alpha}\,g+(1-\alpha)\,\Gamma_{\alpha}(g,g),$

where the linearized operator $\mathcal{L}_{\alpha}$ is given for
$v\in\mathbb{R}^{d}$ by

$\mathcal{L}_{\alpha}(g)(v)\,:=\,\mathcal{Q}_{\alpha}(g,F_{\alpha})(v)+\mathcal{Q}_{\alpha}(F_{\alpha},g)(v)+(1-\alpha)\Delta_{v}(g)(v),$

and $\Gamma_{\alpha}$ is the quadratic remainder.

In order to prove rigorous results on the original model, such as nonlinear
stability, it will be crucial that the fluctuation $g$ lives in a weighted
$L^{1}$ space. Indeed, to this purpose, we shall need to connect the
properties of the linearized operator $\mathcal{L}_{\alpha}$ to the existing
$L^{1}_{3}$ _a priori_ estimates for the nonlinear operator
$\mathcal{Q}_{\alpha}$. These regularity properties were discussed extensively
by Mischler and Mouhot in the series of paper [15, 16]. As we can see in these
papers (we recalled the most important properties in the Appendix), we will
need to take $g\in L^{1}(m^{-1})$, for $m$ an _exponential weight_ function:
there exists $a>0$ and $0<s<1$ such that

(15) $m(v):=\exp(-a\,|v|^{s}).$

The expansion (13) is well defined provided that the original distribution
$f\in L^{1}\left(m^{-1}\right)$.

Let us present some basic properties of the linear operator
$\mathcal{L}_{\alpha}$. We first need to define the so-called _collision
frequency_ by

$\nu_{\alpha}(v):=L(F_{\alpha})(v)=\int_{\mathbb{R}^{d}\times\mathbb{S}^{d-1}}|v-v_{*}|\,F_{\alpha}(v_{*})\,b(\widehat{u}\cdot\sigma)\,d\sigma\,dv_{*}.$

It is known (see for example the lemma 2.3 of [15] for an elementary proof)
that for any $g\in L^{1}_{3}(\mathbb{R}^{d}$), there exists some explicit
nonnegative constants $c_{0}$, $c_{1}$ such that

$0<c_{0}\,(1+|v|)\leq L(g)(v)\leq c_{1}\,(1+|v|),\quad\forall\in
v\in\mathbb{R}^{d}.$

In particular, the collision frequency $\nu_{\alpha}$ verifies

(16)
$0<\nu_{0,\alpha}\,(1+|v|)\leq\nu_{\alpha}(v)\leq\nu_{1,\alpha}\,(1+|v|),$

for two explicit nonnegative constants $\nu_{0,\alpha}$, $\nu_{1,\alpha}$.
Then, we can rewrite the linearized collision operator as a difference of
nonlocal and local operators:

$\mathcal{L}_{\alpha}(g)\,=\,\mathcal{L}_{\alpha}^{+}(g)-\mathcal{L}^{*}(g)-\mathcal{L}^{\nu_{\alpha}}(g),$

where $\mathcal{L}_{\alpha}^{+}$ is the linearization near $F_{\alpha}$ of the
gain term, $\mathcal{L}^{*}$ a convolution operator and $\mathcal{L}^{\nu}$ is
the operator of multiplication by a function of the velocity variable $\nu$.
Classically, for $\alpha=1$, the linearized operator splits between a compact
operator on $L^{1}(m^{-1})$ (see the paper of Mouhot [17] for this particular
exponentially weighted $L^{1}$ case) and a multiplication operator:

$\displaystyle\mathcal{L}_{1}(g)$
$\displaystyle\,=\,\mathcal{L}_{1}^{c}(g)-\mathcal{L}^{\nu_{1}}(g).$

We will see in Section 2 that the same type of decomposition holds for
$\mathcal{L}_{\alpha}$.

As a first step to treat mathematically the question of the hydrodynamic limit
of equation (1), we shall forget the nonlinearity in equation (14) and study
the hydrodynamic limit of the linear equation

(17) $\frac{\partial g}{\partial
t}+(1-\alpha)\,v\cdot\nabla_{x}g=\mathcal{L}_{\alpha}\,g.$

One strategy of proof is to compare the spectrum of the linear operator

(18) $-(1-\alpha)\,v\cdot\nabla_{x}+\mathcal{L}_{\alpha},$

to the one of the linearized fluid equation associated to the limit, as done
in the seminal paper of Ellis and Pinsky [8]. As a byproduct, the study of
this spectrum will allow us to answer to the question of the stability of the
solutions to equation (17), by proving that the real part of the eigenvalues
of (18) remains nonpositive. Hence, the rest of this paper is devoted to the
computation of the spectrum (for small inelasticity and small space positions)
of (18).

In order to avoid to deal with the free transport operator in differential
form, we shall now use Fourier transform in space. More precisely, if we
define the Fourier transform in $x$ of a function
$\varphi:\mathbb{R}^{d}\to\mathbb{R}$ as

$\mathcal{F}_{x}(\varphi)(\xi):=\int_{\mathbb{R}^{d}}e^{-i\xi\,\cdot\,x}\varphi(x)\,dx,\quad\forall\,\xi\in\mathbb{R}^{d},$

it is well know that

$\mathcal{F}_{x}\left(\nabla g\right)(\xi)=i\,\xi\,\mathcal{F}_{x}(g)(\xi).$

Then using the fact that $\mathcal{L}_{\alpha}$ only acts on velocity
variables and setting

$\gamma:=\left(1-\alpha\right)\xi,$

we can write (18) in (scaled) spatial Fourier variables as

(19) $-i\,(\gamma\cdot
v)+\mathcal{L}_{\alpha}=:\mathcal{L}_{(\alpha,\,\gamma)}.$

This operator is well defined on $L^{1}\left(m^{-1}\right)$, with domain
$\operatorname{dom}(\mathcal{L}_{\alpha,\,\gamma})=W_{1}^{2,1}\left(m^{-1}\right)$.
In this particular set of variable, the equation (17) finally reads

$\frac{\partial g}{\partial t}=\mathcal{L}_{\alpha,\,\gamma}\,g,$

and we see now the need to study the spectrum of the linear operator
$\mathcal{L}_{\alpha,\,\gamma}$ for small values of the variable $\gamma$.

To finish with the definitions, let us denote by $N_{1}$ the kernel of the
elastic operator $\mathcal{L}_{1}$. It is spanned by the elastic _collisional
invariants_ , namely

$N_{1}:=\operatorname{Span}\\{F_{1},\,v_{i}\,F_{1},|v|^{2}\,F_{1}:\,1\leq
i\leq d\\},$

where $F_{1}=\mathcal{M}_{1,0,\bar{T}_{1}}$ and $T_{1}$ is the quasi-elastic
equilibrium temperature (11). For $\alpha<1$, the kernel $N_{\alpha}$ of the
inelastic operator $\mathcal{L}_{\alpha}$ is smaller, because of the lack of
energy conservation; it is given by

$N_{\alpha}:=\operatorname{Span}\\{F_{\alpha},\,v_{i}\,F_{\alpha}:\,1\leq
i\leq d\\}.$

###  1.3. Functional Framework and Main Results

Let us present some functional spaces needed in the paper. We denote by
$L_{q}^{p}$ for $p\in[1,+\infty)$ and $q\in[1,+\infty)$ the following weighted
Lebesgue spaces:

$L^{p}_{q}=\left\\{f:\mathbb{R}^{d}\rightarrow\mathbb{R}\text{ measurable;
}\|f\|_{L^{p}_{q}}:=\int_{\mathbb{R}^{d}}|f(v)|^{p}\,\langle
v\rangle^{pq}\,dv<\infty\right\\},$

where $\langle v\rangle:=\sqrt{1+|v|^{2}}$. The weighted $L^{\infty}_{q}$ is
defined thanks to the norm

$\|f\|_{L^{\infty}_{q}}:=\operatorname{supess}_{v\in\mathbb{R}^{d}}\left(|f(v)|\,\langle
v\rangle^{q}\right).$

Then, we denote for $s\in\mathbb{N}$ by $W_{q}^{s,p}$ the weighted Sobolev
space

$W_{q}^{s,p}:=\left\\{f\in L^{p}_{q};\|f\|_{W_{q}^{s,p}}^{p}:=\sum_{|k|\leq
s}\int_{\mathbb{R}^{d}}\left|\partial^{k}f(v)\right|^{p}\langle
v\rangle^{p\,q}\,dv<\infty\right\\}.$

The case $p=2$ is the Sobolev space $H^{s}_{q}:=W^{s,2}_{q}$, which can also
be defined thanks to Fourier transform by the norm

$\|f\|_{H_{q}^{s}}^{2}:=\left\|\mathcal{F}_{v}\left(f\,\langle\cdot\rangle^{s}\right)\right\|_{L^{2}_{q}}.$

We also need to define the more general weighted spaces $L^{p}(m^{-1})$ and
$W^{s,p}(m^{-1})$, where $m$ is an exponential weight function given by (15)
respectively by the norms

$\displaystyle\|f\|_{L^{p}(m^{-1})}^{p}:=\int_{\mathbb{R}^{d}}|f(v)|^{p}\,m^{-1}(v)\,dv,$
$\displaystyle\|f\|_{W^{s,p}(m^{-1})}^{p}:=\sum_{|k|\leq
s}\left\|\partial^{k}f\right\|_{L^{p}(m^{-1})}^{p}.$

For the sake of completeness, let us finally state some notions about
operators that we shall need in the following.

######  Definition 1.

A _closed_ operator $T$ defined on a Banach space $X$ is said to be a

  * •

_Fredholm_ operator of index $(\operatorname{nul}(T),\operatorname{def}(T))$
if the quantities $\operatorname{nul}(T):=\dim(\ker T)$ (the _nullity_) and
$\operatorname{def}(T):=\operatorname{codim}(\operatorname{R}(T))$ (the
_deficiency_) are finite;

  * •

_semi-Fredholm_ operator if $\operatorname{R}(T)$ is closed and at least one
of these two quantities are finite.

For such an operator, we define the

  * •

_resolvent set_ $R(T)\subset\mathbb{C}$ and the _resolvent operator_
$\mathcal{R}(T,\zeta)$ as

$R(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is invertible on $X$, of bounded
inverse }\mathcal{R}(T,\zeta)\\};$

  * •

_spectrum_ $\Sigma(T)$ of $T$ as the (closed) set

$\Sigma(T):=R(T)^{c};$

  * •

_Fredholm_ set $\mathcal{F}(T)\subset\mathbb{C}$ of $T$ as

$\mathcal{F}(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is Freholm}\\};$

  * •

_semi-Fredholm_ set $\mathcal{SF}(T)\subset\mathbb{C}$ of $T$ as

$\mathcal{SF}(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is semi-Freholm}\\};$

  * •

_essential spectrum_ $\Sigma_{ess}(T)$ of $T$ as the set

$\Sigma_{ess}(T):=\mathcal{SF}(T)^{c}\subset\Sigma(T);$

  * •

_discrete spectrum_ $\Sigma_{d}(T)$ of $T$ as the set

$\Sigma_{d}(T):=\Sigma(T)\setminus\Sigma_{ess}(T).$

The two main results of this paper are the following Theorems. We first
localize the spectrum of the operator $\mathcal{L}_{\alpha,\gamma}$ in the
space $L^{1}\left(m^{-1}\right)$, generalizing to this space the classical
$L^{2}$ result of Nicolaenko [18] (see also the chapter 7 of the book [5] of
Cercignani, Illner and Pulvirenti). Let us denote by $\Delta_{x}$ for
$x\in\mathbb{R}$ the half-plane

$\Delta_{x}:=\\{\zeta\in\mathbb{C}:\Re e\,\zeta\geq x\\}.$

We first prove the following result (which has been summarized in Figure 2).

[xAxis=true,yAxis=true,labels=none,Dx=2,Dy=2,ticksize=0pt
0]->(.8,0)(-10,-6)(7,6)
$\delta$$-\delta$$\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$$-\bar{\mu}_{\alpha}$$-\mu_{*}$$-\bar{\lambda}$$\Sigma_{d}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$
Figure 2. Localization of the eigenvalues of $\mathcal{L}_{\alpha,\,\gamma}$
for $|\gamma|\leq\gamma_{0}(\delta)$.

######  Theorem 1.1.

Let $\alpha\in(\alpha_{1},1]$, for a constructive constant $0<\alpha_{1}<1$.
There exists a constructive constant $\bar{\mu}_{\alpha}>0$ such that the
essential spectrum of the operator $\mathcal{L}_{(\alpha,\,\gamma)}$ in
$W_{1}^{2,1}\left(m^{-1}\right)$ is contained on the half-plane
$\Delta_{-\bar{\mu}_{\alpha}}^{c}$:

$\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)\subset\Delta_{-\bar{\mu}_{\alpha}}^{c}.$

The remaining part of its spectrum is composed of discrete eigenvalues. Their
behavior for small frequencies $\gamma$ is the following.

Let us fix $\delta>0$. There exist some constants
$0<\bar{\lambda}<\mu_{*}<\mu_{\alpha}$ and $\alpha_{2}\in(\alpha_{1},1]$ such
that if $\alpha\in(\alpha_{2},1]$ there exists a nonnegative number
$\gamma_{0}$ such that for all $|\gamma|\leq\gamma_{0}$, if
$\lambda\in\Sigma_{d}(\mathcal{L}_{(\alpha,\,\gamma)})$, then

$\displaystyle\lambda\in\Delta_{-\mu_{*}}\Rightarrow\left|\Im
m\,\lambda\right|\leq\delta;$
$\displaystyle\lambda\in\Delta_{-\frac{\bar{\lambda}}{2}}\Rightarrow|\lambda|\leq\delta.$

We then give a first order (in $\gamma$ and $\alpha$) Taylor expansion of the
eigenvalues of $\mathcal{L}_{\alpha,\gamma}$, which generalizes the results of
Ellis and Pinsky [8] and Mischler and Mouhot [16]. Notice that this result
also contains a part of the analysis led by Brilliantov and Pöschel in the
chapter 25 of their book [2] about the stable and unstable modes of the fluid
approximation of the granular gases equation, namely that the energy
eigenvalue is proportional to the inelasticity.

Before stating the result, let us define the eigenvalue problem we want to
deal with: finding a triple $(\lambda,\gamma,h)$ such that

(20) $\left(-i(\gamma\cdot v)+\mathcal{L}_{\alpha}\right)h=\lambda\,h,$

for $\gamma\in\mathbb{R}^{d}$, $\lambda\in\mathbb{C}$ and $h\in
L^{1}\left(m^{-1}\right)$.

######  Theorem 1.2.

There exist $\alpha_{*}\in(\alpha_{2},1]$, some open sets $U_{1}\times
U_{2}\subset\mathbb{R}\times\mathbb{C}$, neighborhood of $(0,0)$, and
functions

$\left\\{\begin{aligned} &\lambda^{(j)}:U_{1}\times(\alpha_{*},1]\to
U_{2}&\forall\,j\in\\{-1,\ldots,d\\},\\\
&h^{(j)}:U_{1}\times\mathbb{S}^{d-1}\times(\alpha_{*},1]\to
L^{1}\left(m^{-1}\right)&\forall\,j\in\\{-1,\ldots,d\\},\end{aligned}\right.$

such that

  1. (1)

The triple
$\left(\rho\,\omega,\,\lambda^{(j)}(\rho,\alpha),\,h^{(j)}(\rho,\omega,\alpha\right)$
is solution to the eigenvalue problem (20), for all $\alpha\in(\alpha_{*},1]$,
$\rho\in U_{1}$, $\omega\in\mathbb{S}^{d-1}$, $j\in\\{-1,\ldots,d\\}$;

  2. (2)

The eigenvalue $\lambda^{(j)}$ is analytic on $U_{1}\times(\alpha_{*},1]$ and
verifies

$\left\\{\begin{aligned}
&\lambda^{(j)}(0,1)=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\
&\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},&&\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=0,&&\forall\,j\in\\{2,\ldots,d\\},\\\
&\frac{\partial^{2}\lambda^{(j)}}{\partial\rho^{2}}(0,1)<0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\
&\frac{\partial\lambda^{(0)}}{\partial\alpha}(0,1)=-\frac{3}{\bar{T}_{1}},&&&&\frac{\partial\lambda^{(j)}}{\partial\alpha}(0,1)=0,&&\forall\,j\in\\{-1,1,\ldots,d\\};\end{aligned}\right.$

  3. (3)

For $\alpha\in(\alpha_{*},1]$, if a triple $(\rho\omega,\lambda,h)$ is
solution to the problem (20) for $(\rho,\lambda)\in U_{1}\times U_{2}$, then
necessarily $\lambda=\lambda^{(j)}$ for some $j\in\\{-1,\ldots,d\\}$;

  4. (4)

For $j\in\\{-1,0,1\\}$, $\alpha\in(\alpha_{*},1]$ and
$\omega\in\mathbb{S}^{d-1}$, the function $v\mapsto
h^{(j)}(\rho\,\omega,\alpha)(v)$ depends only on $|v|$ and $v\cdot\omega$.

######  Remark 1.

We notice in this result that the eigenvalues depend only on $|\gamma|$ and
not on $\gamma$ itself. This is due to the rotational invariance of the
linearized collision operator. However, this is not the case of the
eigenvectors, which can also depend on the angular coordinates.

######  Remark 2.

As a consequence of this result, we can write for $\left(\rho,\alpha\right)\in
U_{1}\times(\alpha_{*},1]$

$\lambda^{(j)}(\rho,\alpha)=i\lambda_{1}^{(j)}\rho-\lambda_{2}^{(j)}\rho^{2}-e_{1}^{(j)}\left(1-\alpha\right)+\mathcal{O}\left(\rho^{2}+(1-\alpha)^{2}\right),$

for explicit (see Section 3.4) constants $\lambda_{1}^{(j)}\in\mathbb{R}$,
$\lambda_{2}^{(j)}\in\mathbb{R}_{+}$ and $e_{1}^{(j)}\in\mathbb{R}_{+}$. In
particular, we obtain that for small space frequencies and small values of the
inelasticity, the spectrum of the linear operator remains at the left of the
imaginary axis in the complex plane. This phenomenon has at least two
important consequences on the behavior of the solution to the granular gases
equation:

  * •

The solutions to the linear collision equation (17) are
$L^{1}\left(m^{-1}\right)$ stable;

  * •

The clustering phenomenon (see _e.g._[2]) is not possible for quasi-elastic
collisions $\alpha\sim 1$, or for systems with a small typical length scale.

###  1.4. Method of Proof and Plan of the Paper

The proof of Theorem 1.1, concerning the rough444By rough, we mean more
precisely that this result do not establish whether or not the eigenvalues can
cross the vertical axis of the complex plane. localization of the spectrum of
the linear operator $\mathcal{L}_{(\alpha,\,\gamma)}$ is given in Section 2,
and can be summarized as follows:

  * •

We first decompose this operator as a sum of compact and Schrödinger-like
operators:

$\displaystyle\mathcal{L}_{(\alpha,\,\gamma)}\,h$
$\displaystyle=-\left[i\,(\gamma\cdot
v)+\nu_{\alpha}(v)\right]h+(1-\alpha)\Delta_{v}h+2\,\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},h\right)-F_{\alpha}\,L(h)$
$\displaystyle=D_{(\alpha,\,\gamma)}h+\mathcal{L}_{\alpha}^{c}h.$

  * •

We then compute the spectrum in $L^{1}$ of the Schrödinger-like part
$D_{(\alpha,\,\gamma)}$ and apply a Banach variant of Weyl’s Theorem (found
_e.g._ in [12]) about the stability of the essential spectrum under relatively
compact perturbation

$\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)=\Sigma_{ess}\left(D_{(\alpha,\,\gamma)}\right).$

  * •

Once the essential spectrum has been localized, we take advantage of some
space homogeneous coercivity and spectral gap estimates (proven in [15]) to
establish the existence of the eigenvalues.

  * •

We finally combine some information about the asymptotic ($\alpha\to 1$)
behavior of the space homogeneous eigenvalues (also taken from [15]) and the
decay properties of the semi-group of the operator $D_{(\alpha,\,\gamma)}$ to
finally give a rough localization of the spectrum.

The proof of Theorem 1.2 concerning the Taylor expansion of the eigenvalues of
$\mathcal{L}_{(\alpha,\,\gamma)}$ is given in Section 3. Up to a certain
extent. this proof is a generalization of Ellis & Pinsky’s arguments for [8],
namely:

  * •

We start by reformulating the eigenvalue problem (20) as a functional
equation, using bounded operators:

$\eqref{pbEigenValueLambdaGammaRho}\Longleftrightarrow\text{Finding
}(\lambda,\gamma,h)\text{ s.t.
}h=\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}h,$

where $\Pi$ is a projection operator, $\Phi$ a “multiplication” operator, and
$\Psi$ a “small” perturbation of the identity.

  * •

We then project this new problem onto the space of _elastic_ collisional
invariants, allowing to rewrite completely (20) as a finite dimensional system
of linear equations of the form

$\left(A_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)X_{(\lambda,\gamma,\alpha)}=0,$

for a non-invertible square matrix $A$.

  * •

We finally solve this system of equation taking advantage of some elastic and
space homogeneous techniques, from both [8] and [15].

##  2\. Localization of the Spectrum

In this section, we shall give a rough localization of the spectrum of the
linearized collision operator $\mathcal{L}_{\alpha}$, proving Theorem (1.1).

###  2.1. Geometry of the Essential Spectrum

We start by describing the “easy part”, namely the essential spectrum. As we
have to deal with the Banach space $L^{1}\left(m^{-1}\right)$, we cannot apply
directly the classical Weyl’s Theorem about the stability of the spectrum
under relatively compact perturbations, because of the lack of Hilbertian
structure. We shall rather apply the more general version stating only the
stability of the semi-Fredholm set, which is well suited for our definition of
the essential spectrum.

######  Proposition 2.1.

Let $\alpha\in(\alpha_{0},1]$, where $\alpha_{0}$ is defined in Lemma A.1.
There exists a constructive constant $\bar{\mu}_{\alpha}>0$ such that the
essential spectrum of the operator $\mathcal{L}_{(\alpha,\,\gamma)}$ in
$W_{1}^{2,1}\left(m^{-1}\right)$ is contained on the half-plane
$\Delta_{-\bar{\mu}_{\alpha}}^{c}$:

$\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)\subset\Delta_{-\bar{\mu}_{\alpha}}^{c}.$

The remaining part of its spectrum is composed of discrete eigenvalues.

###### Proof.

Let us use the expression (19) of the collision operator in spatial Fourier
variables, and decompose it for $h\in L^{1}\left(m^{-1}\right)$ as a local and
a non local part:

$\displaystyle\mathcal{L}_{(\alpha,\,\gamma)}\,h$
$\displaystyle=-\left[i\,(\gamma\cdot
v)+\nu_{\alpha}(v)\right]h+(1-\alpha)\Delta_{v}h+2\,\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},h\right)-F_{\alpha}\,L(h)$
(21) $\displaystyle=D_{(\alpha,\,\gamma)}h+\mathcal{L}_{\alpha}^{c}h,$

where

(22) $\left\\{\begin{aligned} &D_{(\alpha,\,\gamma)}:=-\left[i\,(\gamma\cdot
v)+\nu_{\alpha}(v)\right]\operatorname{Id}+(1-\alpha)\Delta_{v},\\\
&\mathcal{L}_{\alpha}^{c}:=2\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},\cdot\right)-F_{\alpha}\,L(\cdot).\end{aligned}\right.$

We start by the spectrum of $D_{(\alpha,\,\gamma)}$ in
$L^{1}\left(m^{-1}\right)$. This operator is the difference of a Laplace
operator with the operator of multiplication by
$C_{(\alpha,\,\gamma)}(v):=i\,(\gamma\cdot v)+L(F_{\alpha})$. This quantity
verifies according to the lower bound of the collision frequency (16)

$\Re e\,C_{\alpha,\,\gamma}(v)>\nu_{0,\alpha},$

where $\nu_{0,\alpha}$ is the lower bound of the loss term $v\to
L(F_{\alpha})(v)$ (thanks to the smoothness of the profile $F_{\alpha}$ stated
in Proposition A.2).

It is known from _e.g._[11, 14] that the spectrum of the Schrödinger-like
operator $D_{(\alpha,\,\gamma)}$ is independent of the weighted $L^{p}$ space
(for $p\in[1,+\infty)$) where we study it. Let us compute this spectrum in
$L^{2}$. To this end, we shall look at the stability properties of the semi-
group generated by $D_{(\alpha,\,\gamma)}$ on $L^{2}\left(m^{-1}\right)$: let
$h=h(t,v)\in\mathcal{C}\left(0,+\infty;W_{1}^{2}\right)$ be a weak solution to

(23) $\frac{\partial h}{\partial t}=D_{(\alpha,\,\gamma)}h.$

If we multiply this equation by $\bar{h}$ and integrate in the velocity space,
we have thanks to Stokes Theorem and for $\alpha$ close to $1$

(24) $\displaystyle\frac{\partial}{\partial
t}\|h(t)\|_{L^{2}\left(m^{-1}\right)}^{2}\leq-\|C_{(\alpha,\,\gamma)}\,h(t)\|_{L^{2}\left(m^{-1}\right)}\leq-\nu_{0,\alpha}\|h(t)\|_{L^{2}\left(m^{-1}\right)}^{2}.$

We then obtain that

$\|h(t)\|_{L^{2}\left(m^{-1}\right)}\leq e^{-\nu_{0,\alpha}\,t/2}.$

Hence, there exists a constant $0<\bar{\mu}_{\alpha}<\nu_{0,\alpha}$ such that
the spectrum of the operator $D_{(\alpha,\,\gamma)}$ is included in the set
$\Delta_{-\bar{\mu}_{\alpha}}^{c}$.

Moreover, thanks to the Hölder continuity of the inelastic gain term in
operator norm (with loss of weight) stated in Proposition A.2, the operator
$\mathcal{L}_{\alpha}^{c}$ is $D_{(\alpha,\,\gamma)}$–compact (and this is
here that the weak inelasticity assumption $\alpha\in(\alpha_{0},1]$ is used).
Notice that we have chosen to define the essential spectrum of an operator $S$
“à la Kato”, namely as the complement of the semi-Fredholm set of $S$ in
$\mathbb{C}$. Then we can apply the Banach version of Weyl’s Theorem (see
_e.g._[12, Theorem IV.5.26 and IV.5.35]), stating that the semi-Fredholm set
is stable under relatively compact perturbation. Hence, the essential spectrum
of $\mathcal{L}_{(\alpha,\,\gamma)}$ is included in
$\Delta_{-\bar{\mu}_{\alpha}}^{c}$.

The set $\Delta_{-\bar{\mu}_{\alpha}}$ is then equal to the Fredholm set
$\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$. It remains to show
that this set only contains the eigenvalues and the resolvent set. We know
from the discussion in [12, Chapter IV, Section 6, and Theorem 5.33] that
$\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$ is an open set,
composed of the union of a countable number of components $\mathcal{F}_{n}$,
characterized by the value of the index: for any $n\in\mathbb{N}$, the
functions

$\operatorname{nul}:\zeta\to\operatorname{nul}\left(\mathcal{L}_{(\alpha,\,\gamma)}-\zeta\right),\qquad\operatorname{def}:\zeta\to\operatorname{def}\left(\mathcal{L}_{(\alpha,\,\gamma)}-\zeta\right)$

are constant on $\mathcal{F}_{n}$, except for a countable set of isolated
values of $\zeta$. In our case, we have
$\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)=\Delta_{-\bar{\mu}_{\alpha}}$
which is connected; it has only one component, which means that
$\operatorname{nul}(\zeta)$ and $\operatorname{def}(\zeta)$ are constant on
$\Delta_{-\bar{\mu}_{\alpha}}$, except for a countable set of isolated values
of $\zeta$.

We will prove that these constant values are
$\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0$, meaning that $\zeta$
belongs to the resolvent set of $\mathcal{L}_{(\alpha,\,\gamma)}$. The
remaining isolated values $\zeta$, being in the Fredholm set, then verify
$0<\operatorname{nul}(\zeta)<+\infty$ and
$0<\operatorname{def}(\zeta)<+\infty$, which exactly characterizes the
eigenvalues. We shall follow closely the proof of [17, Proposition 3.4], and
exhibit an uncountable set $I\subset\Delta_{-\bar{\mu}_{\alpha}}$ such that
$\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0$ for all $\zeta\in I$.

Let us use the decomposition (72) introduced initially in [15]

$\mathcal{L}_{\alpha,\,\gamma}=A_{\delta}-B_{\alpha,\,\delta}\left(i(\gamma\cdot
v)\right),$

for $\delta>0$ (see Section A for more details). We know from Lemma A.1 that
$A_{\delta}$ is compact on $L^{1}\left(m^{-1}\right)$, and that
$B_{\alpha,\delta}$ satisfies the coercivity estimate

(25)
$\left\|B_{\alpha,\,\delta}(\zeta)\,g\right\|_{L^{1}\left(m^{-1}\right)}\geq\left\|\left(\nu_{1}+\Re
e\,\zeta\right)g\right\|_{L^{1}\left(m^{-1}\right)}-\varepsilon(\delta)\left\|\left(\nu_{1}+\Re
e\,\zeta\right)g\right\|_{L^{1}\left(m^{-1}\right)},$

where $\varepsilon(\delta)\to 0$ where $\delta\to 0$. If we fix $r_{0}>0$
sufficiently big and $\delta>0$ small enough, then we have according to (25)
for all $r\geq r_{0}$

$\left\|B_{\alpha,\,\delta}\left(r+i(\gamma\cdot
v)\right)g\right\|_{L^{1}\left(m^{-1}\right)}\geq\frac{\nu_{0,1}+r_{0}}{2}\|g\|_{L^{1}\left(m^{-1}\right)}.$

Thus, the operator $B_{\alpha,\,\delta}\left(r+i(\gamma\cdot v)\right)$ is
invertible on $L^{1}\left(m^{-1}\right)$, for all $r\geq r_{0}$, and then it
is the same for
$\mathcal{L}_{\alpha,\,\gamma}-r=A_{\delta}-B_{\alpha,\,\delta}\left(r+i(\gamma\cdot
v)\right)$ by compacity of $A_{\delta}$. It finally means that the interval
$I:=[r_{0},+\infty)$ is included on the resolvent set of
$\mathcal{L}_{\alpha,\,\gamma}$, and then that

$\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0,\quad\forall\,\zeta\in[r_{0},+\infty),$

which concludes the proof. ∎

###  2.2. Behavior of the Eigenvalues for Small Inelasticity

We shall now focus on the discrete spectrum of this operator, namely its
eigenvalues. A major difference with the elastic case in the classical
Hilbertian $L^{2}$ setting is that the operator we deal with is not a
nonpositive operator, and we cannot conclude thanks to the last proposition
that this operator has a _spectral gap_ (namely a negative bound for its
eigenvalues). Nevertheless, we know from [17] for the elastic case and [16]
for the weak inelasticity case $\alpha\in(\alpha_{0},1)$ that
$\mathcal{L}_{\alpha}$ has a spectral gap $-\bar{\lambda}$ in
$L^{1}\left(m^{-1}\right)$, verifying for $\alpha$ sufficiently small (say
$\alpha\in(\alpha_{1},1]$ for $1>\alpha_{1}>\alpha_{0}$)

$0<\bar{\lambda}<\mu_{*}<\bar{\mu}_{\alpha},$

for a nonnegative constant $\mu_{*}$ depending on $\alpha$.

Let us now study the behavior of the discrete spectrum of
$\mathcal{L}_{(\alpha,\,\gamma)}$ for small values of the frequency $\gamma$.
We shall show that if $\gamma\to 0$, then the eigenvalues of this operator
converge first towards the real axis and then towards $0$.

######  Proposition 2.2.

Let $\delta>0$. There exists $\alpha_{2}\in(\alpha_{1},1]$ such that if
$\alpha\in(\alpha_{2},1]$ there exists a nonnegative number $\gamma_{0}$ such
that for all $|\gamma|\leq\gamma_{0}$, if
$\lambda\in\Sigma_{d}(\mathcal{L}_{(\alpha,\,\gamma)})$, then

(26) $\displaystyle\lambda\in\Delta_{-\mu_{*}}\Rightarrow\left|\Im
m\,\lambda\right|\leq\delta;$ (27)
$\displaystyle\lambda\in\Delta_{-\frac{\bar{\lambda}}{2}}\Rightarrow|\lambda|\leq\delta.$

###### Proof.

Let us first notice that if $\lambda$ is an eigenvalue of
$\mathcal{L}_{(\alpha,\,\gamma)}$ and $h$ an associated eigenvector, then
using the decomposition (21) introduced in the proof of Proposition 2.1, we
can write

(28) $\mathcal{L}_{\alpha}^{c}h=\left(\lambda-D_{(\alpha,\,\gamma)}\right)h,$

where $\mathcal{L}_{\alpha}^{c}$ is compact on $L^{1}\left(m^{-1}\right)$
(thanks to the sharp estimates of Lemma A.1) and

$D_{(\alpha,\,\gamma)}=-\left[i\,(\gamma\cdot
v)+\nu_{\alpha}(v)\right]\operatorname{Id}+(1-\alpha)\Delta_{v}.$

We will proceed by contradiction using the representation (28). Concerning the
first implication, if, for $\delta>0$, there exist a sequence
$(\gamma_{n})_{n}\subset\mathbb{R}^{d}$ converging towards $0$, a sequence of
functions $(h_{n})_{n}\in L^{1}\left(m^{-1}\right)$ of unit norm, and a
sequence of complex numbers
$\lambda_{n}\in\Sigma_{d}\left(\mathcal{L}_{(\alpha,\,\gamma_{n})}\right)$
verifying

(29) $\left\\{\begin{aligned}
&\mathcal{L}_{\alpha}^{c}h_{n}=\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)h_{n},\\\
&\left|\Im m\,\lambda_{n}\right|>\delta,\quad\Re
e\,\lambda_{n}\geq-\mu_{*},\end{aligned}\right.$

then we must have $\limsup\left|\Im m\,\lambda_{n}\right|<\infty$. Indeed, the
operator $\mathcal{L}_{\alpha}^{c}$ is compact on $L^{1}\left(m^{-1}\right)$,
and then the sequence $(\mathcal{L}_{\alpha}^{c}h_{n})_{n}$ converges (up to
an extraction) towards $g\in L^{1}\left(m^{-1}\right)$. Thus we can write
using (29)

(30)
$g=\lim_{n\to\infty}\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)h_{n}.$

We have seen in the proof of Proposition 2.1 that the semi-group
$S_{t}^{(\alpha,\,\gamma)}$ associated to the operator $D_{(\alpha,\,\gamma)}$
in $L^{1}$ is exponentially decaying in time, uniformly in $\alpha$ and
$\gamma$. But, we know (see _e.g._[9], chap. II) that if
$\mathcal{R}(D_{(\alpha,\,\gamma)},\cdot)$ is the resolvent operator of
$D_{\alpha,\,\gamma}$, we have the integral representation for all $\zeta\in
R(D_{(\alpha,\,\gamma)})$

$\mathcal{R}\left(D_{(\alpha,\,\gamma)},\zeta\right)=\lim_{t\to+\infty}\int_{0}^{t}e^{-t\,\zeta}S_{t}^{(\alpha,\,\gamma)}\,dt.$

Thus, using the decay of $S_{t}^{(\alpha,\,\gamma)}$, we have
$\mathcal{R}\left(D_{(\alpha,\,\gamma)},0\right)=:D_{(\alpha,\,\gamma)}^{-1}$
bounded in $L^{1}$, uniformly in $\gamma$, and then according to (30)

(31)
$\lim_{n\to\infty}h_{n}=\left(\lim_{n\to\infty}\lambda_{n}-D_{(\alpha,\,0)}\right)^{-1}g.$

But, we also have for $v\in\mathbb{R}^{d}$

$\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)^{-1}g(v)=\frac{1}{\lambda_{n}+i(\gamma_{n}\cdot
v)+\nu_{\alpha}(v)}\left(\operatorname{Id}-\frac{1-\alpha}{\lambda_{n}+i(\gamma_{n}\cdot
v)+\nu_{\alpha}(v)}\Delta_{v}\right)^{-1}g(v),$

and then by considering again the behavior of the solutions to equation (23),
which gives inequality (24), we obtain a constant $C$ independent on
$(\alpha,\lambda_{n},\gamma_{n})$ such that

(32)
$\left\|\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)^{-1}g\right\|_{L^{\infty}}\leq\frac{C}{|\lambda_{n}|-\nu_{0,\alpha}}\|g\|_{L^{\infty}}.$

Finally, if $\lim\left|\Im m\,\lambda_{n}\right|=\infty$ we would have
according to the limit (31) and the estimation (32)

$\lim_{n\to\infty}\|h_{n}\|_{L^{\infty}}=0$

with $\|h_{n}\|_{L^{1}\left(m^{-1}\right)}=1$, which is not possible. Hence,
$\left|\Im m\,\lambda_{n}\right|\leq C$ for an infinite number of indices $n$
and $C>0$.

But, we also have $-\mu_{*}\leq\Re e\,\lambda_{n}<r_{0}$ (where $r_{0}>0$ is
defined in the proof of Proposition 2.1), and then we can extract another
subsequence $\left(\lambda_{n_{k}}\right)_{k}$ converging towards
$\lambda\in\mathbb{C}$ such that $\Im m\,\lambda\geq\delta>0$. Using the fact
that $\gamma_{n}\to 0$ and the smoothness of the map
$\lambda\mapsto\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}$, we obtain in (31)

$\lim_{k\to\infty}h_{n_{k}}=\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}g=:h\in
L^{1}\left(m^{-1}\right),$

with $\|h\|_{L^{1}\left(m^{-1}\right)}=1$. Hence we conclude by inversion of
$\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}$ and by the smoothness of the
nonlocal part of $\mathcal{L}_{\alpha}$ that

$\displaystyle\left(\lambda-
D_{(\alpha,\,0)}\right)\,h=g=\lim_{n\to\infty}\mathcal{L}_{\alpha}^{c}h_{n}=\mathcal{L}_{\alpha}^{c}h$

which means according to the definition of $\mathcal{L}_{\alpha}$ that

$\lambda h=\mathcal{L}_{\alpha}h.$

This is absurd because $\left|\Im m\,\lambda\right|\geq\delta>0$, $\Re
e\,\lambda\geq-\mu_{*}$ for $\mu_{*}$ close to the spectral gap of
$\mathcal{L}_{\alpha}$, and yet we know from [15] that the eigenvalues of
$\mathcal{L}_{\alpha}$ can be made arbitrarily close (with respect to
$1-\alpha$) to the ones of $\mathcal{L}_{1}$, which are real according to
[17].

We shall now give the proof of the implication (27), also by contradiction. If
for $\delta>0$ there exist a sequence $(\gamma_{n})_{n}$ converging towards
$0$, a sequence $(h_{n})_{n}\in L^{1}\left(m^{-1}\right)$ of unit norm, and
some complex numbers
$\lambda_{n}\in\Sigma_{d}\left(\mathcal{L}_{\varepsilon,\gamma_{n}}\right)$
such that

$-\bar{\lambda}/2\leq\Re e\,\lambda_{n}\leq-\delta,$

then $\lambda_{n}\in\Delta_{-\mu_{*}}$ and according to the relation (26) we
have $\left|\Im m\,\lambda_{n}\right|\leq\delta$. We can then extract a
subsequence $\left(\lambda_{n_{k}}\right)_{k}$ which converges towards a
complex number $\lambda$ also verifying $-\bar{\lambda}/2\leq\Re
e\,\lambda\leq-\delta$. When $k\to\infty$, the same argument than before gives
$\mathcal{L}_{\alpha}h=\lambda h$ with $\lambda\neq 0$. By using again the
spectral properties of $\mathcal{L}_{\alpha}$, we then have $\Re
e\,\lambda\leq-\bar{\lambda}$, which is absurd. ∎

This concludes the proof of Theorem 1.1. We also summarized the results of
this proposition in Figure 2. Moreover, it gives us some rough information on
the behavior of the resolvent operator of $\mathcal{L}_{(\alpha,\gamma)}$.

######  Corollary 2.1.

If $\alpha\in(\alpha_{1},1]$, the resolvent operator
$\mathcal{R}(\mathcal{L}_{(\alpha,\,\gamma}),\zeta)$ is well defined for
$\zeta\in\Delta_{-\mu_{*}}$ such that $\Im m\,\zeta>\delta$.

##  3\. Inelastic Dispersion Relations

Our goal in this section is to precise the localization results of the
previous section, by proving Theorem 1.2, that is to give a Taylor expansion
of the eigenvalues of $\mathcal{L}_{(\alpha,\gamma)}$ in $\alpha$ and
$\gamma$. The purpose of this expansion is twofold: on the one hand, we want
to establish that, at least for small values of $\alpha$ and $\gamma$, the
eigenvalues of the linear operator $\mathcal{L}_{(\alpha,\gamma)}$ stay at the
left of the imaginary axis. This could be useful if _e.g._ one wants to prove
nonlinear stability of the solutions to (1). On the other hand, obtaining this
decomposition up to the second order in the spatial frequency $\gamma$ and to
first order in $\alpha$ is necessary to establish the validity of the
linearized quasi-elastic hydrodynamic limit of our model, in the same way than
[8]. To obtain this expansion, we shall refine the method of proof of this
paper, together with the use of some ideas introduced in [16] in order to deal
with the quasi-elastic setting.

We recall for the reader’s convenience that we are interested in the following
eigenvalue problem: finding $\lambda\in\mathbb{C}$, $\gamma\in\mathbb{R}^{d}$
and $h\in L^{1}(m^{-1})$ such that

$\left(-i(\gamma\cdot v)+\mathcal{L}_{\alpha}\right)h=\lambda\,h,$

which can be reformulated thanks to the decomposition (21) as finding
$\lambda\in\mathbb{C}$, $\gamma\in\mathbb{R}^{d}$ and $h\in L^{1}(m^{-1})$
such that

(33) $\mathcal{L}_{\alpha}^{c}h=\left(\lambda+\nu_{\alpha}(v)+i(\gamma\cdot
v)-(1-\alpha)\Delta_{v}\right)h.$

###  3.1. Projection of the Eigenvalue Problem

Let us now define the scalar product we will use in the following. If
$\phi,\psi$ are such that the following expression has a meaning, we will set

$\langle\phi,\psi\rangle:=\int_{\mathbb{R}^{d}}\phi(v)\,\overline{\psi}(v)\,F_{1}^{-1}(v)\,dv,$

where $F_{1}=\mathcal{M}_{1,0,\bar{T}_{1}}$ is the quasi-elastic equilibrium
and $\bar{T}_{1}$ is given by (11). Indeed, our goal is to introduce a
_spectral decomposition_ of $L^{1}\left(m^{-1}\right)$ as a direct sum of
$\mathcal{L}_{\alpha}$-invariant spaces. The inner product we use for this
purpose is the one of $L^{2}\left(F_{1}^{-1}\right)$ because $f=m+h\in
L^{1}\left(m^{-1}\right)$ if and only if $h\in L^{1}\left(m^{-1}\right)$ and
$f\in L^{2}\left(F_{1}^{-1}\right)$ if and only if $h\in
L^{2}\left(F_{1}^{-1}\right)$. This hilbertian structure allows us to define
the spectral projections.

Let us start by decomposing the operator $\mathcal{L}_{\alpha}^{c}$ as

(34)
$\mathcal{L}_{\alpha}^{c}=\nu_{\alpha}^{1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2},$

where $\Pi$ is the projection on the space

$\mathcal{N}_{\alpha}:=\nu_{\alpha}^{1/2}N_{1}=\operatorname{Span}\\{\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}v_{i}\,F_{1},\,\nu_{\alpha}^{1/2}|v|^{2}\,F_{1}:\,1\leq
i\leq d\\},$

and $\mathcal{S}_{\alpha}$ is given by

$\mathcal{S}_{\alpha}:=\nu_{\alpha}^{-1}\mathcal{L}_{\alpha}^{c}-\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}.$

Actually, $\mathbb{P}:=\nu_{\alpha}^{-1/2}\Pi$ is the spectral projection on
the null space of $\mathcal{L}_{1}$, and can be defined using the resolvent
operator of $\mathcal{L}_{1}$ as

$\displaystyle\mathbb{P}$
$\displaystyle=\frac{1}{2i\pi}\int_{\zeta\in\mathbb{C}:|\zeta|=r}\mathcal{R}(\mathcal{L}_{1},\zeta)\,d\zeta$

where $r<\delta$ for $\delta$ sufficiently small (see the discussion in
Section $5$ of [15]). In particular, $\mathbb{P}$ commutes with
$\mathcal{L}_{1}$. Moreover, the operator $\mathcal{L}_{\alpha}^{c}$ being
compact on $L^{1}(m^{-1})$, $\Pi+\mathcal{S}_{\alpha}$ is compact on the same
space. Given that the rank of $\Pi$ is finite, $\mathcal{S}_{\alpha}$ is then
a compact operator on $L^{1}(m^{-1})$.

We first need to prove a result concerning the eigenvalues of the operator
$\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$.

######  Lemma 3.1.

On the space $L^{1}(m^{-1})$, $\lambda=1$ is not an eigenvalue of
$\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$.

###### Proof.

If there is an $h\in L^{1}(m^{-1})$ such that
$\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}h=h$, then according to (34),

$\displaystyle\mathcal{L}_{1}\,h$
$\displaystyle=-\nu_{1}h+\nu_{1}^{1/2}(\Pi+\mathcal{S}_{1})\,\nu_{1}^{1/2}h$
(35) $\displaystyle=\nu_{1}^{1/2}\,\Pi\,\nu_{1}^{1/2}h.$

Projecting this equation upon the null space of $\mathcal{L}_{1}$, using (35),
we obtain for all $\varphi\in N_{1}$ that

$\displaystyle 0$ $\displaystyle=\langle\mathcal{L}_{1}h,\varphi\rangle$
$\displaystyle=\langle\Pi\,\nu_{1}^{1/2}h,\nu_{1}^{1/2}\,\varphi\rangle,$

and then $\Pi\,\nu_{1}^{1/2}h$ is orthogonal to $\nu_{1}^{1/2}N_{1}$, which is
the whole range of $\Pi$. Then, necessarily, as $\Pi$ is a projection, we have
$\Pi\,\nu_{1}^{1/2}h=0$. It means according to (35) that
$\mathcal{L}_{1}\,h=0$, and then that $h\in N_{1}$, which is absurd because
$\Pi$ is a projection onto $\nu_{1}^{1/2}N_{1}$ thus

$\nu_{1}^{1/2}h=\Pi\,\nu_{1}^{1/2}h=0.$

∎

Let us now denote by $\Phi_{(\lambda,\gamma,\alpha)}$ the operator

$\Phi_{(\lambda,\gamma,\alpha)}=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot
v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v).$

If the triple $(\gamma,\lambda,h)$ is solution to the eigenvalue problem (33),
then we can write using the decomposition (34)

$\displaystyle h$ $\displaystyle=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot
v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)\,(\nu_{\alpha}(v))^{-1}\mathcal{L}_{\alpha}^{c}h$
(36)
$\displaystyle=\Phi_{(\lambda,\gamma,\alpha)}\nu_{\alpha}^{-1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}h.$

This will allow us to rewrite the eigenvalues problem with bounded operators.
To this purpose, we state a technical lemma about the asymptotic behavior of
the operator $\Phi_{(\lambda,\gamma,\alpha)}$.

######  Lemma 3.2.

For all $g\in W^{2,1}(m^{-1})$, we have

$\left\|\left(\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)g\right\|_{L^{1}(m^{-1})}\leq\varepsilon(\lambda,\gamma,\alpha)\|g\|_{W^{2,1}(m^{-1})},$

where
$\lim_{(\lambda,\gamma,\alpha)\to(0,0,1)}\varepsilon(\lambda,\gamma,\alpha)=0$.

###### Proof.

If we set $C_{(\lambda,\,\alpha,\,\gamma)}=\lambda+i\,(\gamma\cdot
v)+\nu_{\alpha}$, we can write for all $v\in\mathbb{R}^{d}$

$\displaystyle\Phi_{(\lambda,\gamma,\alpha)}g(v)-g(v)$
$\displaystyle=\left(C_{(\lambda,\,\alpha,\,\gamma)}(v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)g(v)-g(v)$
$\displaystyle=\frac{1}{C_{(\lambda,\,\alpha,\,\gamma)}}\left(\operatorname{Id}-\frac{1-\alpha}{C_{(\lambda,\,\alpha,\,\gamma)}(v)}\Delta_{v}\right)^{-1}C_{(0,0,1)}g(v)-g(v).$

Then, to prove the Lemma, we need to prove that the norm of the operator

$\mathcal{T}_{\varepsilon}:=\left(\operatorname{Id}-\varepsilon\,\Delta_{v}\right)^{-1}-\operatorname{Id},$

defined from $W^{2,1}$ onto $L^{1}$, can be made arbitrarily small for small
$\varepsilon$.

Let us first reformulate this operator using the resolvent of the Laplace
operator $\Delta_{v}$. Since we have

$\left(\operatorname{Id}-\varepsilon\Delta_{v}\right)\mathcal{T}_{\varepsilon}=\varepsilon\Delta_{v},$

we can rewrite $\mathcal{T}_{\varepsilon}$ as555If it has been possible to
conduct this study on a unweighted $L^{2}$ space, it would have been enough to
notice the following Fourier representation:
$\mathcal{F}_{v}\left(\mathcal{T}_{\varepsilon}\,f\right)(\xi)=-\frac{\varepsilon\,|\xi|^{2}}{1+\varepsilon\,|\xi|^{2}}\mathcal{F}_{v}(f)(\xi).$

(37) $\displaystyle\mathcal{T}_{\varepsilon}$
$\displaystyle=\left(\frac{1}{\varepsilon}\operatorname{Id}-\Delta_{v}\right)^{-1}\Delta_{v}=:\mathcal{R}\left(\frac{1}{\varepsilon},\Delta_{v}\right)\Delta_{v}.$

It is known from [6] (and classical for unweighted $L^{p}$ spaces) that the
resolvent of the Laplace operator on exponentially weighted $L^{1}$ spaces
verifies for an explicit real constant $a$ and a nonnegative constant $C$

(38)
$\left\|\mathcal{R}\left(\lambda,\Delta_{v}\right)h\right\|_{L^{1}(m^{-1})}\leq\frac{C}{\lambda-a}\left\|h\right\|_{L^{1}(m^{-1})},$

for any $\lambda$ in an unbounded angular sector of the complex plane which
does not contains $a$. In particular, using this inequality in the identity
(37), we have that for any $g\in W^{2,1}(m^{-1})$

$\left\|\mathcal{T}_{\varepsilon}\,g\right\|_{L^{1}(m^{-1})}\leq\frac{\varepsilon\,C}{1-a\,\varepsilon}\left\|g\right\|_{W^{2,1}(m^{-1})}\to_{\varepsilon\to
0}0,$

which concludes the proof of the Lemma.

∎

In all the following of this section, we shall use the polar decomposition
$\gamma=\rho\,\omega$ for $\rho\geq 0$ and $\omega\in\mathbb{S}^{d-1}$ of the
frequency $\gamma$. We prove an invertibility result, which is needed to
rewrite the eigenvalue problem (33) using bounded operators:

######  Lemma 3.3.

There exist $\alpha_{3}\in(\alpha_{2},1]$ and some open sets $U_{1}\times
U_{2}\subset\mathbb{R}\times\mathbb{C}$, neighborhood of $(0,0)$ such that if
$(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{3},1]$, then for all
$\omega\in\mathbb{S}^{d-1}$, the operator

$\Psi_{(\lambda,\,\rho\,\omega,\,\alpha)}:=\operatorname{Id}\,-\,\Phi_{(\lambda,\,\rho\,\omega,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}$

has a bounded inverse on $L^{1}(m^{-1})$.

###### Proof.

We have seen that $\nu_{\alpha}^{1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}$
is a compact operator on $L^{1}(m^{-1})$. Moreover, we have according to the
representation (28) (with $D_{(\alpha,\,\gamma)}$ given by (22))

$\displaystyle\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}$
$\displaystyle=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot
v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)$
$\displaystyle=\left(\lambda-D_{(\alpha,\,\gamma)}\right)^{-1}\nu_{\alpha}(v)$

which is a bounded operator (at least for small frequencies $\gamma$), thanks
to the analysis we led on Section 2, and particularly from the localization of
the discrete spectrum of $\mathcal{L}_{(\alpha,\,\gamma)}$ of Proposition 2.2.
Hence, the operator

$\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}=\left(\Phi_{(\lambda,\,\rho\,\omega,\alpha)}\nu_{\alpha}^{-1}\right)\left(\nu_{\alpha}^{1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\right)$

is compact on $L^{1}(m^{-1})$.

By Fredholm alternative, it just remains to show that for $(\rho,\lambda)$
small enough and $\alpha$ close to $1$, there is no non-trivial solutions
$h\in L^{1}(m^{-1})$ of

(39)
$h=\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}h.$

Let us do this by contradiction. Assume that there are sequences
$(\lambda_{n},\,\gamma_{n},\,\alpha_{n})_{n}\to(0,0,1)$ and
$(h_{n})_{n}\subset L^{1}(m^{-1})$, $\|h_{n}\|=1$ solutions to (39). First, we
notice thanks to the continuity of the equilibrium profiles $F_{\alpha}$ with
respect to $\alpha$ (recalled in Proposition A.4) and to the smoothness
properties of these profiles (recalled in Proposition A.3) that we have

$\lim_{\alpha\to 1}\|\nu_{\alpha}-\nu_{1}\|_{L^{\infty}}=0.$

Moreover, according to Lemma A.1, the operator $\mathcal{L}_{\alpha}$
converges towards $\mathcal{L}_{1}$ in the norm of graph in $L^{1}(m^{-1})$.
Then, the operator
$\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}$,
which is compact on $L^{1}(m^{-1})$, converges towards the operator
$\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$, and we have up to a
subsequence,

$\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}\to_{n\to\infty}g\in
L^{1}(m^{-1}).$

Thus, if we write $\Phi_{n}:=\Phi_{(\lambda_{n},\,\gamma_{n},\,\alpha_{n})}$,
given that $h_{n}$ is solution to (39), we have

(40)
$h_{n}=\Phi_{n}\left[\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}-g\right]+\Phi_{n}\,g.$

But, according to Lemma 3.2 we have

(41) $\lim_{n\to\infty}\|\Phi_{n}g-g\|_{L^{1}(m^{-1})}=0.$

Hence, by Lebesgue dominated convergence Theorem, this implies with identity
(40) that the sequence $(h_{n})_{n}$ strongly converges towards $g$ (using
also the fact that the sequence $(\Phi_{n}g)_{n}$ is bounded in
$L^{1}(m^{-1})$ for large $n$ thanks to (41)). Therefore, we have by
continuity

$g=\lim_{n\to\infty}\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}=\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}g,$

namely $g$ is an eigenvector666This justifies the use of Lemma 3.2 in equation
(41). Indeed, the solutions $h$ to this eigenvalue problem are such that
$\|h\|_{W^{2,1}(m^{-1})}\leq C\,\|h\|_{L^{1}(m^{-1})},$ for $C\geq 0$, as was
shown for instance in [15] using the decomposition (72):
$0=(\mathcal{L}_{\alpha,\gamma}-\lambda)h=\left(A_{\delta}-B_{\alpha,\,\delta}\left(\lambda+i\left(\gamma\cdot
v\right)\right)\right)h.$ for $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$
associated to the eigenvalue $1$, which is absurd according to Lemma 3.1.
Finally, the operator
$\operatorname{Id}\,-\,\Phi_{(\lambda,\,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}S\,\nu_{\alpha}^{1/2}$
is invertible on $L^{1}(m^{-1})$ for small $(\lambda,\,\gamma)$ and $\alpha$
close to $1$.

∎

Let us now rewrite the relation (36) as

$\displaystyle\Psi_{(\lambda,\gamma,\,\alpha)}h$
$\displaystyle=\left[\operatorname{Id}-\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\right]\Phi_{(\lambda,\gamma)}\nu_{\alpha}^{-1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}h$
$\displaystyle=\Phi_{(\lambda,\gamma,\,\alpha)}\left\\{\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}+\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\left[\operatorname{Id}-\Phi_{(\lambda,\gamma)}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}\right]\right\\}h.$

According to Lemma 3.3, $\Psi_{(\lambda,\gamma,\,\alpha)}$ is invertible for
small $(\lambda,\gamma)$ and $\alpha\in(\alpha_{3},1]$. Then, provided that
$h$ is solution to (36), we have

(42)
$h=\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}h.$

Let us introduce the “conjugated operator”
$\mathcal{P}:=\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}=\mathbb{P}\,\nu_{\alpha}^{1/2}$,
where $\mathbb{P}$ is the projection onto the space of elastic collisional
invariants $N_{1}$. We can use it to rewrite (42) (and then the eigenvalue
problem (33)) as the following finite dimensional system of equations

(43)
$\mathcal{P}h=\mathcal{P}\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\mathcal{P}h,$

which can be understood as:

> _Finding
> $X_{(\lambda,\gamma,\,\alpha)}=(x_{0},\ldots,x_{d+1})\in\mathbb{R}^{d+2}$
> such that_
> $\left(A_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)X_{(\lambda,\gamma,\,\alpha)}=0,$
> _where $A=\mathcal{P}\Psi^{-1}\Phi\in\mathcal{M}_{d+2,\,d+2}(\mathbb{R})$._

We shall find in the following section some conditions on $\lambda$ in order
to have

$\det\left(A_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)=0.$

Under such conditions, the abstract problem will admit a non-trivial solution
$X$. Coming back to the original problem, given that
$X=\mathcal{P}h=\mathbb{P}\nu_{\alpha}^{1/2}\,h$, we will obtain thanks to
equation (42) a solution $h$ to the original eigenvalue problem (33).

###  3.2. Finite Dimensional Resolution

We shall study the vector $X$ component-wise, using the normalization

(44) $X=\mathbb{P}\,\nu_{\alpha}^{1/2}\,h(v)=x_{0}\,F_{1}(v)+(x\cdot
v)\,F_{1}(v)+x_{d+1}\left(|v|^{2}-c_{\nu}\right)F_{1}(v),$

where $c_{\nu}$ is such that
$\langle\nu_{\alpha},|v|^{2}\rangle=\langle\nu_{\alpha},c_{\nu}\rangle$. If we
compute the product of (44) with elements of

$\mathcal{N}_{\alpha}=\operatorname{Span}\\{\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}v_{i}\,F_{1},\,\nu_{\alpha}^{1/2}\left(|v|^{2}-c_{\nu}\right)F_{1}:\,1\leq
i\leq d\\},$

we find using the definition of $\mathbb{P}$ and the orthogonality of elements
of $N_{\alpha}$ for $\langle\cdot,\cdot\rangle$ that

(45) $\left\\{\begin{aligned}
&\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}F_{1}\rangle=x_{0}\left\langle\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}F_{1}\right\rangle,\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}\,v_{i}\,F_{1}\rangle=x_{i}\left\langle\nu_{\alpha}^{1/2}\,v_{i}\,F_{1},\nu_{\alpha}^{1/2}\,v_{i}\,F_{1}\right\rangle,\quad\forall
1\leq i\leq d,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}\left(|v|^{2}-c_{\nu}\right)F_{1}\rangle=x_{d+1}\left\langle\nu_{\alpha}^{1/2}\left(|v|^{2}-c_{\nu}\right)F_{1},\nu_{\alpha}^{1/2}\left(|v|^{2}-c_{\nu}\right)F_{1}\right\rangle.\end{aligned}\right.$

For clarity sake, let us set in the following

$\langle\phi,\psi\rangle_{F_{1}}:=\int_{\mathbb{R}^{d}}\phi(v)\,\overline{\psi}(v)\,F_{1}(v)\,dv,$

in order to have

$\left\langle\nu_{\alpha}^{1/2}\,h_{1}\,F_{1},\nu_{\alpha}^{1/2}\,h_{2}\,F_{1}\right\rangle=\langle\nu_{\alpha}h_{1},h_{2}\rangle_{F_{1}}.$

Using (42) together with the relations (45), we obtain for all $1\leq i\leq d$

(46) $\left\\{\begin{aligned}
&x_{0}\left\langle\nu_{\alpha},1\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha},T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha},T_{\gamma}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&x_{i}\left\langle\nu_{\alpha}\,v_{i},v_{i}\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&x_{d+1}\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),|v|^{2}-c_{\nu}\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \
+x_{d+1}\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\gamma}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\end{aligned}\right.$

where we have set for fixed $(\lambda,\alpha)$

(47)
$T_{\gamma}:=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}.$

The system (46) is the componentwise version of the projected problem (43). We
are now going to decompose this system of $d+2$ equations in
$X=(x_{0},\ldots,x_{d+1})$ in a closed system of $3$ equations in $x_{0}$,
$x\cdot\omega$ and $x_{d+1}$ for a fixed $\omega\in\mathbb{S}^{d-1}$
(corresponding to the _longitudinal_ sound waves of the Boltzmann equation,
see also the work of Nicolaenko [18]) together with a scalar relation in
$x_{i}$ for all $1\leq i\leq d$ (corresponding to the _transverse_ sound
waves). For this, we need the following technical lemma:

######  Lemma 3.4.

Let $x,y\in\mathbb{R}^{d}$, $\bm{e}:=(1,0,\ldots,0)^{\mathsf{T}}$ and
$\gamma=\rho\,\omega$ for $\rho\in\mathbb{R}$ and $\omega\in\mathbb{S}^{d-1}$.
Then, we have

(48) $\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}},$
(49) $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot
v),T_{\gamma}\,1\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,1\right\rangle_{F_{1}},$
(50) $\displaystyle\begin{split}\left\langle\nu_{\alpha}\,(x\cdot
v),T_{\gamma}\,(y\cdot
v)\right\rangle_{F_{1}}&=(x\cdot\omega)(y\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}\\\
&+\left[x\cdot\omega-(x\cdot\omega)(y\cdot\omega)\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}\,v_{2}\right\rangle_{F_{1}},\end{split}$
(51) $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot
v),T_{\gamma}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},$
(52)
$\displaystyle\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}.$

###### Proof.

According to the definitions of $T_{\gamma}$ and
$\Psi_{(\lambda,\gamma,\,\alpha)}$, the $\gamma$–dependency of $T_{\gamma}$ is
only happening through the operator $\Phi_{(\lambda,\gamma,\,\alpha)}$. But,
for $M\in\mathcal{O}(d)$ the orthogonal group of $\mathbb{R}^{d}$ (namely,
$MM^{*}=\bf I_{d}$) one has for $v\in\mathbb{R}^{d}$ and
$g\in\operatorname{dom}\left(\Phi_{(\lambda,\gamma,\,\alpha)}\right)$, using
the fact that $\nu_{\alpha}$ is a radial function, for all
$v\in\mathbb{R}^{d}$,

$\displaystyle\left(\Phi_{(\lambda,\gamma,\,\alpha)}\,g\right)(Mv)$
$\displaystyle=\left(\nu_{\alpha}(Mv)+\lambda+i\,(\gamma\cdot
Mv)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(Mv)\,g(Mv),$
$\displaystyle=\left(\Phi_{(\lambda,M^{-1}\,\gamma,\,\alpha)}Mg\right)(v),$

where we have set $Mg\,(v):=g(Mv)$. Then

$\left(T_{\gamma}\,g\right)(Mv)=\left(T_{M^{-1}\gamma}\,Mg\right)(v),\quad\forall\,v\in\mathbb{R}^{d}.$

Especially, if $g$ is a radial function, there exists a function $\Gamma_{g}$
such that

$\left(T_{\gamma}\,g\right)(v)=\Gamma_{g}(\gamma\cdot v,|v|).$

One has thanks to this result

$\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}$ $\displaystyle=\left\langle
T_{\gamma}^{*}\,\nu_{\alpha},(x\cdot v)\right\rangle_{F_{1}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\gamma\cdot
v,|v|)\,(x\cdot v)\,{F_{1}}(v)\,dv.$

Let $M\in\mathcal{O}(d)$ such that $M^{-1}\,\omega=\bm{e}$. Thanks to the
change of variables $v=M\xi$ and using the polar coordinates
$\gamma=\rho\,\omega$ one has $\gamma\cdot
v=\rho\,M^{-1}\,\omega\cdot\xi=\rho\,\xi_{1}$ and then

$\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\rho\,\xi_{1},|\xi|)\,(M^{-1}x\cdot\xi)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}^{odd}(\rho\,\xi_{1},|\xi|)\,(M^{-1}x\cdot\xi)\,{F_{1}}(\xi)\,d\xi,$

where $g^{odd}(a,\cdot)=\left(g(a,\cdot)-g(-a,\cdot)\right)/2$. Given that
${F_{1}}$ is a radial function of $v$, $\langle
g^{odd},h\rangle_{F_{1}}=\langle g,h^{odd}\rangle_{F_{1}}$ and
$(M^{-1}x)_{1}=(M^{-1}x)\cdot M^{-1}\omega$, one has

$\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot
v)\right\rangle_{F_{1}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}^{odd}(\rho\,\xi_{1},|\xi|)\,(M^{-1}x)_{1}\,\xi_{1}\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=(M^{-1}x)_{1}\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\rho\,\xi_{1},|\xi|)\,\xi_{1}\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=x\cdot\omega\left\langle
T_{\rho\bm{e}}^{*}\,\nu_{\alpha},v_{1}\right\rangle_{F_{1}},$

which proves (48).

Thanks to the same arguments

$\displaystyle\left\langle\nu_{\alpha}\,(x\cdot
v),T_{\gamma}\,1\right\rangle_{F_{1}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(v)\,(x\cdot
v)\,\Gamma_{1}(\gamma\cdot v,|v|)\,{F_{1}}(v)\,dv$
$\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M^{-1}x\cdot\xi)\,\Gamma_{1}(\rho\,\xi_{1},|\xi|)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M^{-1}x\cdot\xi)\,\Gamma_{1}^{odd}(\rho\,\xi_{1},|\xi|)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}},$

which proves (49). Concerning the next identity, one has

$\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,(y\cdot
v)\right\rangle_{F_{1}}$ $\displaystyle=\sum_{1\leq i,j\leq
d}x_{i}\,y_{j}\int_{\mathbb{R}^{d}}\nu_{\alpha}(v)\,v_{i}\,(T_{\gamma}\,v_{j})(v)\,{F_{1}}(v)\,dv$
$\displaystyle=\sum_{1\leq i,j\leq
d}x_{i}\,y_{j}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M\xi)_{i}\left(T_{\rho\bm{e}}(M\xi)_{j}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=\sum_{1\leq i,j,k,l\leq
d}x_{i}\,y_{j}\,M_{il}\,M_{jk}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{l}\left(T_{\rho\bm{e}}\,\xi_{k}\right)(\xi)\,{F_{1}}(\xi)\,d\xi.$

If $k\neq l$, this integral is zero (it is clear by doing the transformation
$\xi\to-\xi$). In the other case, one has

$\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,(y\cdot
v)\right\rangle_{F_{1}}$ $\displaystyle=\sum_{1\leq i,j\leq
d}x_{i}\,y_{j}\,M_{i1}\,M_{j1}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{1}\left(T_{\rho\bm{e}}\,\xi_{1}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle+\sum_{\begin{subarray}{c}1\leq i,j\leq d\\\ 2\leq l\leq
d\end{subarray}}x_{i}\,y_{j}\,M_{il}\,M_{jl}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{2}\left(T_{\rho\bm{e}}\,\xi_{2}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$
$\displaystyle=(x\cdot\omega)(y\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}$
$\displaystyle+\sum_{1\leq i,j\leq
d}x_{i}\,y_{j}\left[(MM^{*})_{ij}-M_{i1}\,M_{j_{1}}\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}\,v_{2}\right\rangle_{F_{1}},$

which is (50) because $MM^{*}=I_{d}$. The inequalities (51) and (52) are
finally obtained using the same methods of proof. ∎

Applying this lemma to system (46), we find for all $1\leq i\leq d$

(53) $\displaystyle
x_{0}\left\langle\nu_{\alpha},T_{\rho\bm{e}}1-1\right\rangle_{F_{1}}+x\cdot\omega\left\langle\nu_{\alpha},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0,$
(54)
$\displaystyle\begin{split}x_{i}\langle\nu_{\alpha}\,v_{i},v_{i}\rangle_{F_{1}}&=\omega_{i}\,x_{0}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}+\left[x_{i}-\omega_{i}\,(x\cdot\omega)\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}v_{2}\right\rangle_{F_{1}}\\\
&+\omega_{i}\,(x\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}+\omega_{i}\,x_{d+1}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\end{split}$
(55)
$\displaystyle\begin{split}x_{0}\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\,1\right\rangle_{F_{1}}&+x\cdot\omega\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}\\\
&+x_{d+1}\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)-\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0.\end{split}$

Now, on the one hand, if we multiply (54) by $\omega_{i}$ and sum over all
$i$, using the fact that $|\omega|=1$, we find that

(56)
$x_{0}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}+x\cdot\omega\,\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}v_{1}-v_{1}\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0.$

The system (53)–(56)–(55) is closed in $(x_{0},x\cdot\omega,x_{d+1})$ for a
fixed $\omega\in\mathbb{S}^{d-1}$. Coming back to a more abstract form, there
exists solutions to this system if and only if

(57) $D(\lambda,\rho,\alpha)=0,$

where we have defined $D$ as the following Gram-like matrix (remember that
$T_{\gamma}$ is given by (47) and depends on $(\lambda,\gamma,\alpha)$)

(58) $D(\lambda,\rho,\alpha):=\\\
\begin{vmatrix}\left\langle\nu_{\alpha},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\left(|v|^{2}-c_{\nu}\right),\left(T_{\rho\bm{e}}-\operatorname{Id}\right)\left(|v|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\end{vmatrix}.$

On the other hand, if one multiplies (56) by $\omega_{i}$ and subtract this
expression to (54), one finds

(59)
$\left[x_{i}-\omega_{i}\,(x\cdot\omega)\right]D_{\omega}(\lambda,\rho,\alpha)=0,$

where we have set

(60)
$D_{\omega}(\lambda,\rho,\alpha):=\left\langle\nu_{\alpha}\,v_{1},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}.$

Then, if one solves (56) in $C\cdot\omega$, the relation (59) will give the
expression of $x_{i}$, provided that the equation
$D_{\omega}(\lambda,\rho,\alpha)=0$ admits an unique solution $\lambda$.

We will simplify these expressions thanks to the following Lemma.

######  Lemma 3.5.

Let $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{3},1]$. If $g,h$
are elastic collisional invariants, namely if

$g,h\in
N_{1}=\operatorname{Span}\\{F_{1},\,v_{i}\,F_{1},|v|^{2}\,F_{1}:\,1\leq i\leq
d\\},$

then we can write for all $\omega\in\mathbb{S}^{d-1}$ and
$\gamma=\rho\,\omega$

$\displaystyle\left(\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h,$
$\displaystyle\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}h\right\rangle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h\right\rangle.$

###### Proof.

By definition of $\mathcal{N}_{\alpha}$, we have
$\nu_{\alpha}^{1/2}h\in\mathcal{N}_{\alpha}$, and then
$\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}h=0$. But, we know that

$\Psi_{(\lambda,\gamma,\alpha)}=\operatorname{Id}-\Phi_{(\lambda,\gamma,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}.$

Thus, we have $\Psi_{(\lambda,\gamma,\alpha)}h=h$, and given that
$\Psi_{(\lambda,\gamma,\,\alpha)}$ is invertible for $(\rho,\lambda,\alpha)\in
U_{1}\times U_{2}\times(\alpha_{3},1]$ and $\gamma=\rho\,\omega$, we have

(61) $\Psi^{-1}_{(\lambda,\gamma,\alpha)}h=h,$

which proves the first relation. Using the orthogonality of the collisional
invariants and (61), we obtain the second equality:

$\displaystyle\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}h\right\rangle$
$\displaystyle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}h\right\rangle-\left\langle\nu_{\alpha}g,h\right\rangle$
$\displaystyle=\left\langle\nu_{\alpha}g,\left(\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)h\right\rangle$
$\displaystyle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)h\right\rangle.$

∎

Let us set
$\Upsilon_{(\lambda,\gamma,\,\alpha)}:=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)$.
Thanks to this lemma, to the definition of $c_{\nu}$ and by the nullity of the
odd moments of the centered Gaussian $F_{1}$, we can write (58) in a “simpler”
form, namely

(62)
$D(\lambda,\rho,\alpha)=\begin{vmatrix}\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\
\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\
\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\end{vmatrix},$

where we have set $g(v):=|v|^{2}-c_{\nu}$. We can also write (60) the same way

(63)
$D_{\omega}(\lambda,\rho,\alpha)=\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}.$

Before solving these equations, we need a last lemma.

######  Lemma 3.6.

Let $h$ be an elastic collisional invariant (namely $h\in N_{1}$). If
$\Psi^{*}$ denotes the adjoint operator of $\Psi$, then we have

$\displaystyle\left(\Psi^{*}_{(0,0,1)}\right)^{-1}\,\nu_{1}h=\nu_{1}h.$

###### Proof.

Let $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{2},1]$ and
$\omega\in\mathbb{S}^{d-1}$ and set $\gamma=\rho\,\omega$. If $T$ is an
invertible operator on a Banach space, it is known that
$(T^{*})^{-1}=(T^{-1})^{*}$. Moreover, provided that $\nu_{1}\in\mathbb{R}$,
the adjoint operator $\Phi^{*}_{(\lambda,\gamma,\,1)}$ is the operator of
multiplication by

$\frac{\nu_{1}(v)}{\nu_{1}(v)+\bar{\lambda}-i\,(\gamma\cdot v)}$

Then, if $(\lambda,\rho)\to(0,0)$ we have
$\Phi_{(\lambda,\rho\,\omega,\,1)}^{*}\to\operatorname{Id}$ strongly. But, we
can also compute

$\displaystyle\Psi_{(\lambda,\,\rho\,\omega,\,\alpha)}^{*}$
$\displaystyle=\operatorname{Id}\,-\,\left(\Phi_{(\lambda,\rho\,\omega)}\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}\right)^{*}$
$\displaystyle=\operatorname{Id}\,-\,\nu_{1}^{1/2}\mathcal{S}_{1}^{*}\,\nu_{1}^{-1/2}\,\Phi_{(\lambda,\rho\,\omega)}^{*},$

and as $h\in N_{1}$, we have

$\nu_{1}^{1/2}\mathcal{S}_{1}^{*}\,\nu_{1}^{-1/2}\,\nu h=0.$

Finally, we can write

$\displaystyle\Psi_{(0,0,1)}^{*}\nu_{1}h$
$\displaystyle=\left(\operatorname{Id}\,-\,\nu_{1}^{1/2}\mathcal{S}_{1}^{*}\,\nu_{1}^{-1/2}\right)\nu_{1}h$
$\displaystyle=\nu_{1}h,$

which concludes the proof after inversion. ∎

######  Remark 3.

The eigenvalues and eigenvectors of $\mathcal{L}_{\alpha,\,\rho\omega}$ are
analytic function or $\rho$. Indeed, thanks to the hard spheres kernel and
estimates (16), there exists a nonnegative constant $M$ such that

$\left\|(\omega\cdot v)\,h\right\|_{L^{1}(m^{-1})}\leq
M\left(\left\|h\right\|_{L^{1}(m^{-1})}+\left\|\mathcal{L}_{\alpha}h\right\|_{L^{1}(m^{-1})}\right).$

We can then apply [12, Thm. VII.2.6 and Rem. VII.2.7] about the analyticity of
the spectrum of a closed operator on a Banach space.

###  3.3. First Order Coefficients of the Taylor Expansion

We can now study in details for what values of the parameters $\lambda$ and
$\alpha$ one can solve the projected eigenvalue problem (59). We start by
considering the behavior of the transverse sound waves.

######  Proposition 3.1.

Let $\omega\in\mathbb{S}^{d-1}$. There exist $\rho_{0}>0$ and
$\alpha_{4}\in(\alpha_{3},1]$ such that the problem of solving the equation

$D_{\omega}(\lambda,\rho,\alpha)=0$

has a unique solution
$\lambda_{\omega}=\lambda_{\omega}(\rho,\alpha)\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{0},\bar{\rho}_{0})\times(\alpha_{4},1]\right)$,
verifying

$\lambda_{\omega}(0,1)=\frac{\partial\lambda_{\omega}}{\partial\rho}(0,1)=\frac{\partial\lambda_{\omega}}{\partial\alpha}(0,1)=0.$

###### Proof.

Let us write thanks to the compact expression (63) of $D_{\omega}$

$\displaystyle 0$ $\displaystyle=-D_{\omega}(\lambda,\rho,\alpha)$
$\displaystyle=-\left\langle\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{*}\right)^{-1}(\nu_{\alpha}v_{1}),\left(\Phi_{(\lambda,\,\rho\bm{e},\,\alpha)}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}$
$\displaystyle=-\int_{\mathbb{R}^{d}}\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{*}\right)^{-1}(\nu_{\alpha}v_{1})\left[\left(\nu_{\alpha}(v)+\lambda+i\,(\rho\bm{e}\cdot
v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)-\operatorname{Id}\right](v_{1})\,{F_{1}}(v)\,dv$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{*}\right)^{-1}(\nu_{\alpha}v_{1})\left(\nu_{\alpha}(v)+\lambda+i\,\rho\,v_{1}-(1-\alpha)\Delta_{v}\right)^{-1}\left(\lambda+i\,\rho\,v_{1}-(1-\alpha)\Delta_{v}\right)(v_{1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{F_{1}}(v)\,dv.$

Let us now set $z=\lambda/\rho$ and $s=1-\alpha$. We shall take the limit
$(\rho,s)\to(0,0)$ in $D_{\omega}$. For this, we define a new function
$G_{\omega}$ as

$G_{\omega}(z,\rho,s):=\frac{1}{\rho}D_{\omega}(\rho z,\rho,1-s).$

Then, as $\Delta_{v}(v_{1})=0$, we will have
$D_{\omega}(\lambda,\rho,\alpha)=0$ if and only if

$\displaystyle 0$ $\displaystyle=-G_{\omega}(z,\rho,s)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(\rho
z,\rho\bm{e},1-s)}^{*}\right)^{-1}(\nu_{1-s}v_{1})\left(\nu_{1-s}(v)+\rho
z+i\,\rho\,v_{1}-s\Delta_{v}\right)^{-1}\left((z+iv_{1})\,v_{1}\right){F_{1}}(v)\,dv.$

Moreover, if $\alpha\to 1$, thanks to the continuity of the equilibrium
profiles $F_{\alpha}$ with respect to $\alpha$ (recalled in Proposition A.4)
and to the smoothness properties of these profiles (recalled in Proposition
A.3), we have $\nu_{\alpha}(v)\to\nu_{1}(v)$, uniformly in $v$. Hence, if we
take the limit $(\rho,s)\to(0,0)$, we find thanks to Lemma 3.6 that

$\displaystyle 0$ $\displaystyle=-G_{\omega}(z,0,1)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(0,\,0,\,1)}^{*}\right)^{-1}(\nu_{1}v_{1})\frac{z+iv_{1}}{\nu_{1}}(v_{1})\,{F_{1}}(v)\,dv$
$\displaystyle=z\int_{\mathbb{R}^{d}}v_{1}^{2}\,{F_{1}}(v)\,dv=z\,\bar{T}_{1}.$

Provided that $\bar{T}_{1}$ is nonzero, we have $z=0$.

It just remains to apply the implicit function theorem to the map
$(z,\rho,s)\mapsto G_{\omega}(z,\rho,s)$ in $(0,0,0)$. Provided that we have

$\left\\{\begin{aligned} &G_{\omega}(0,0,0)=0,\\\ &\frac{\partial
G_{\omega}}{\partial z}(0,0,0)={\bar{T}_{1}},\end{aligned}\right.$

there exist two real constants $\bar{\rho}_{0}>0$,
$\alpha_{4}\in(\alpha_{3},1]$ and a mapping
$z_{\omega}\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{0},\bar{\rho}_{0})\times[0,1-\alpha_{4})\right)$
such that if $|\rho|\leq\bar{\rho}_{0}$ and $s\in[0,1-\alpha_{4})$, then

$\frac{1}{\rho}D_{\omega}\left(\rho
z_{\omega}(\rho,s),\rho,1-s\right)=G_{\omega}\left(z_{\omega}(\rho,s),\rho,s\right)=0.$

To conclude the proof, we set $\lambda_{\omega}(\rho,\alpha):=\rho
z_{\omega}\left(\rho,1-\alpha\right)$ and this function has the properties we
were looking from. ∎

Let us now turn to the dispersion relations (57), corresponding to the
longitudinal sound waves. We recall the simplified expression of $D$ for the
reader convenience:

$D(\lambda,\rho,\alpha)=\begin{vmatrix}\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\
\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\
\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\end{vmatrix}.$

We prove the following result concerning the behavior of the eigenvalues for
small frequency and inelasticity.

######  Proposition 3.2.

For $\lambda\in U_{2}$ (see Lemma 3.3), there exists $\bar{\rho}>0$ and
$\alpha_{5}\in(\alpha_{4},1]$ such that for $\alpha\in(\alpha_{5},1]$ the
elastic dispersion relation $D(\lambda,\rho,\alpha)=0$ has exactly three
branches of solutions $\lambda^{(j)}(\rho,\alpha)$ for all $j\in\\{-1,0,1\\}$
and $\rho\in(-\bar{\rho}_{1},\bar{\rho}_{1})$. These solutions are of class
$\mathcal{C}^{\infty}(-\bar{\rho}_{1},\bar{\rho}_{1})$ and verify

$\left\\{\begin{aligned} &\lambda^{(j)}(0,1)=0,&&\forall\,j\in\\{-1,0,1\\},\\\
&\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},\\\
&\frac{\partial\lambda^{(0)}}{\partial\alpha}(0,1)=-\frac{3}{\bar{T}_{1}},\end{aligned}\right.$

where, $\lambda^{(0)}$ is the so-called _energy eigenvalue_ and $\bar{T}_{1}$
is given by (11). Finally, we also have the symmetry properties

(64)
$\lambda^{(j)}(-\rho,\alpha)=\overline{\lambda^{(j)}}(\rho,\alpha)=\lambda^{(-j)}(\rho,\alpha).$

###### Proof.

We shall use the ideas introduced in the proof of Proposition 3.1: instead of
solving directly the equation (57), we want to solve an equivalent one
depending on $z=\lambda/\rho$ and we set to simplify $s=1-\alpha$. We then
introduce a function $G=G(z,\rho,s)$ by setting

$G(z,\rho,s):=\frac{1}{\rho^{3}}D(\rho z,\rho,1-s).$

According to the simplified expression (62) of $D$, all the components of the
matrix found in $G(z,\rho,s)$ can be written for $h_{1},\ h_{2}\in N_{1}$

$\frac{1}{\rho}\left\langle\nu_{1-s}\,h_{1},\Upsilon_{(\rho
z,\,\rho\bm{e},\,1-s)}\,h_{2}\right\rangle_{F_{1}}=\int_{\mathbb{R}^{d}}\left(\Psi_{(\rho
z,\,\rho\bm{e},\,1-s)}^{*}\right)^{-1}(\nu_{1-s}h_{1})(v)\\\
\left(\nu_{1-s}(v)+\rho
z+i\,\rho\,v_{1}-s\Delta_{v}\right)^{-1}\left(z+iv_{1}-\frac{s}{\rho}\Delta_{v}\right)(h_{2})(v)F_{1}(v)\,dv.$

By doing the same computations than in the proof of Proposition 3.1, this
quantity becomes for $\rho=s=0$

$\int_{\mathbb{R}^{d}}h_{1}(v)({z+iv_{1}})h_{2}(v)\,F_{1}(v)\,dv.$

Moreover, according to the definition of the Maxwellian distribution $F_{1}$,
we have

$\int_{\mathbb{R}^{d}}\begin{pmatrix}1\\\ |v|^{2}\\\
v_{1}^{2}\,|v|^{2}\end{pmatrix}F_{1}(v)\,dv=\begin{pmatrix}1\\\ \bar{T}_{1}\\\
\left(d+2\right)\bar{T}_{1}^{2}\end{pmatrix}.$

Thus, we can write $D$ as

$\displaystyle G(z,0,1)$ $\displaystyle=\begin{vmatrix}\langle
1,z+iv_{1}\rangle_{F_{1}}&\langle 1,(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle
1,(z+iv_{1})g\rangle_{F_{1}}\\\ \langle v_{1},z+iv_{1}\rangle_{F_{1}}&\langle
v_{1},(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle
v_{1},(z+iv_{1})g\rangle_{F_{1}}\\\ \langle g,z+iv_{1}\rangle_{F_{1}}&\langle
g,(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle g,(z+iv_{1})g\rangle_{F_{1}}\\\
\end{vmatrix}$
$\displaystyle=\begin{vmatrix}z&i\,\bar{T}_{1}&z\left(d\,\bar{T}_{1}-c_{\nu}\right)\\\
i\,\bar{T}_{1}&z\,\bar{T}_{1}&i\,\bar{T}_{1}\left(\left(d+2\right)\bar{T}_{1}-c_{\nu}\right)\\\
z\left(d\,\bar{T}_{1}-c_{\nu}\right)&i\,\left(\left(d+2\right)\bar{T}_{1}-c_{\nu}\right)&z\left(\left(d\,\bar{T}_{1}-c_{\nu}\right)^{2}+2d\,\bar{T}_{1}^{2}\right)\end{vmatrix}$
$\displaystyle=2\,\bar{T}_{1}^{2}\,z\left(dz^{2}+d\,\bar{T}_{1}+2\,\bar{T}_{1}^{2}\right)$
$\displaystyle=2d\,\bar{T}_{1}^{2}\,(z-z_{-1})(z-z_{0})\,(z-z_{+1}),$

where we have set for any $j\in\\{-1,0,+1\\}$

$z_{j}:=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}}.$

Hence, provided that $G(z,0,1)$ has no multiple root, we have shown that

$\left\\{\begin{aligned} &G(z_{j},0,0)=0,\\\ &\frac{\partial G}{\partial
z}(z_{j},0,0)\neq 0,\end{aligned}\right.$

and we can apply again the implicit function theorem to show that in a
neighborhood
$\mathcal{B}\times(-\bar{\rho}_{1},\bar{\rho}_{1})\times[0,1-\alpha_{5})$ of
$(z_{j},0)$, there exists an unique function
$\widetilde{z_{j}}\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{1},\bar{\rho}_{1})\times[0,1-\alpha_{5})\right)$
such that if $|\rho|\leq\bar{\rho}_{1}$ and $s\in[0,1-\alpha_{5})$, then
$G(\widetilde{z_{j}}(\rho,s),\rho,s)=0$ (and of course
$\widetilde{z_{j}}(0,0)=z_{j}$). We finally set
$\lambda^{(j)}(\rho,s):=\rho\widetilde{z_{j}}(\rho,s)$, which give a solution
to (57) (with $\alpha=1-s$) verifying

$\left\\{\begin{aligned} &\lambda^{(j)}(0,0)=0,\\\
&\frac{\partial\lambda^{(j)}}{\partial\rho}(0,0)=z_{j}.\\\
\end{aligned}\right.$

We now have to prove that these three branches are the only solutions to
$D(\lambda,\rho,\alpha)=0$ for small $\rho$ and $1-\alpha$.

For this, following once more [8], we shall use tools from complex analysis.
Let us fix $|\rho|\leq\bar{\rho}_{1}$ and $\alpha\in(\alpha_{5},1]$; according
to the definition of $D$, the map $\lambda\mapsto D(\lambda,\rho,\alpha)$ is
holomorphic on the set $U_{2}$ (defined Lemma 3.3). Moreover, following the
previous computations, we also have $D(\lambda,0,1)=\lambda^{3}H(\lambda)$ for
a function $H$ holomorphic on $U_{2}$ such that $H(0)=1$. Hence, if $\lambda$
is defined along a circle $\mathcal{C}$ around $0$, then $D(\lambda,0,1)$ will
encircle the origin exactly three times. Using the strong convergence of the
multiplication operator $\Phi_{(\lambda,\,\rho\bm{e},\,\alpha)}$ towards
$\operatorname{Id}$ when $(\rho,\,\alpha)\to(0,1)$, we can write

$\lim_{(\rho,\,\alpha)\to(0,1)}\sup_{\lambda\in
U_{2}}|D(\lambda,\rho,\alpha)-D(\lambda,0,1)|=0.$

Hence, for small $(\rho,1-\alpha)$, the function $\lambda\mapsto
D(\lambda,\rho,\alpha)$ encircles the origin also only three times when
$\lambda$ traverses $\mathcal{C}$. This function then only has three roots for
fixed $\rho$ and $\alpha$.

Next, we compute the partial derivative with respect to $\alpha$ of the energy
eigenvalue. This eigenvalue is given by the solution of the dispersion
relation that depends on $\rho$ only at second order, namely $\lambda^{(0)}$.
Let $h^{(0)}_{(\rho\,\omega,\,\alpha)}$ be the associated eigenvector. We then
have for all $\omega\in\mathbb{S}^{d-1}$ and $\rho\geq 0$

(65)
$\mathcal{L}_{(\alpha,\gamma)}\,h^{(0)}_{(\rho\,\omega,\,\alpha)}(v)\,=\,\lambda^{(0)}(\rho,\alpha)\,h^{(0)}_{(\rho\,\omega,\,\alpha)}(v),\quad\forall\,v\in\mathbb{R}^{d}.$

In particular, the “elastic, space homogeneous” energy eigenvector
$h^{(0)}_{(0,1)}$ is defined thanks to the Maxwellian profile $F_{1}$ (given
in (11)) as

(66) $h^{(0)}_{(0,1)}=c_{0}\left(|v|^{2}-d\,\bar{T}_{1}\right)F_{1},$

where $c_{0}$ is a normalizing constant. We have by construction, using some
elementary properties of Gaussian functions

$\left\|h^{(0)}_{(0,1)}\right\|_{L^{1}(m^{-1})}=1,\quad
N\left(h^{(0)}_{(0,1)}\right)=0,\quad\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=2\,c_{0}\,d\,\bar{T}_{1}^{2},$

where we have defined the _mass_ $N(f)$ and the _kinetic energy_
$\mathcal{E}(f)$ of a given distribution $f$ as

$N(f):=\int_{\mathbb{R}^{d}}f(v)\,dv,\quad\mathcal{E}(f):=\int_{\mathbb{R}^{d}}f(v)\,|v|^{2}\,dv.$

By integrating the eigenvalue equation (65) against $|v|^{2}$ we obtain
according to the expression of the energy dissipation functional (7)

$\lambda^{(0)}(\rho,\alpha)\mathcal{E}\left(h^{(0)}_{(\rho\,\omega,\,\alpha)}\right)=\\\
-2(1-\alpha^{2})\,D\left(F_{\alpha},h^{(0)}_{(\rho\,\omega,\alpha)}\right)+2dN(1-\alpha)\left(h^{(0)}_{(\rho\,\omega,\,\alpha)}\right)+i{\rho}\omega\cdot\int_{\mathbb{R}^{d}}h^{(0)}_{(\rho\,\omega,\,\alpha)}(v)\,v\,|v|^{2}\,dv.$

As $\rho$ tends to $0$, dividing by $1-\alpha$ yields

$\frac{\lambda^{(0)}(0,\alpha)}{1-\alpha}\mathcal{E}\left(h^{(0)}_{(0,\alpha)}\right)=-2(1+\alpha)\,D\left(F_{\alpha},h^{(0)}_{(0,\alpha)}\right)+2dN\left(h^{(0)}_{(0,\alpha)}\right).$

Now, we use the rate of convergence of the inelastic profile $F_{\alpha}$
towards the elastic one $F_{1}$ recalled in Proposition A.4 and the smoothness
of $h^{(0)}_{(0,\alpha)}$ with respect to $\alpha$ obtained thanks to the use
of the implicit functions theorem. We then obtain thanks to the nullity of the
mass of $h^{(0)}_{(0,1)}$

(67)
$\frac{\lambda^{(0)}(0,\alpha)}{1-\alpha}\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=-2(1+\alpha)\,D\left(F_{1},h^{(0)}_{(0,1)}\right)+\mathcal{O}(1-\alpha).$

Finally, we compute thanks to the expression of the elastic energy eigenvector
(66) and to the definition (11) of the equilibrium temperature the quantities

$\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=2\,d\,c_{0}\,\bar{T}_{1}^{2},\quad
D\left(F_{1},h^{(0)}_{(0,1)}\right)=\frac{3}{2}d\,c_{0}\,\bar{T}_{1}.$

Gathering these relations and passing to the limit $\alpha\to 1$ in (67) gives
the result.

Concerning the last assertion of the proposition, we notice thanks to the
invariance of the eigenvalue problem (33) under the composition of the convex
conjugation and the reflection $\gamma\to-\gamma$ that
$\overline{D}(\lambda,\rho,\alpha)=D(\overline{\lambda},-\rho,\alpha)=D(\overline{\lambda},\rho,\alpha)$.
∎

######  Remark 4.

As a consequence of the symmetry relation (64)

$\lambda^{(j)}(-\rho,\alpha)=\overline{\lambda^{(j)}}(\rho,\alpha)=\lambda^{(-j)}(\rho,\alpha),$

we have $\lambda^{(0)}(\rho,\alpha)\in\mathbb{R}$.

Thanks to this proposition, we can construct the $d+2$ normalized hydrodynamic
eigenvectors

$\left(h^{(j)}_{(\rho\,\omega,\,\alpha)}\right)_{j\in\\{-1,\ldots,d\\}}$

of the inelastic linearized collision operator, for small $\rho$, $\alpha$
close to $1$ and a given $\omega\in\mathbb{S}^{d-1}$. Indeed, on the one hand,
for $j\in\\{2,\ldots,d\\}$, we take $\lambda=\lambda_{\omega}(\rho,\alpha)$
for $|\rho|\leq\bar{\rho}_{0}$ and $\alpha\in(\alpha_{4},1]$ given by
Proposition 3.1 and choose in (44) $x_{0}=x_{d+1}=0$ and any vector
$x\in\omega^{\perp}$. The relation (42) then allows us to construct the
eigenvectors $h^{(j)}_{(\rho,\,\omega,\,\alpha)}$ associated to the
conservation of momentum.

On the other hand, for $j\in\\{-1,0,1\\}$, we pick a solution
$\lambda=\lambda^{(j)}(\rho,\,\alpha)$ for $|\rho|\leq\bar{\rho}_{1}$ and
$\alpha\in(\alpha_{5},1]$ to the dispersion relation
$D(\lambda,\rho,\alpha)=0$ given by Proposition 3.2 and choose the vector
$\left(x_{0},x\cdot\omega,x_{d+1}\right)$ to be a solution to the system
(53)–(56)–(55) corresponding to this eigenvalue. We then set
$x=(x\cdot\omega)\,\omega$ and recover through (44)

$\mathcal{P}h^{(j)}_{(\rho\,\omega,\,\alpha)}(v)=x_{0}(\rho,\,\alpha)+(x\cdot
v)(\rho,\omega\cdot
v,\,\alpha)+x_{d+1}(\rho,\,\alpha)\left(|v|^{2}-c_{\nu}\right).$

Inserting this expression in (42) finally gives us the eigenvalue, depending
on $\rho$, $\alpha$ (as a $\mathcal{C}^{\infty}$ function), $|v|$ and
$v\cdot\omega$. With this procedure, we have constructed three independent
solutions (corresponding to the acoustic waves and the kinetic energy)
$h^{(j)}=h^{(j)}_{(\rho\,\omega,\,\alpha)}\in L^{1}(m^{-1})$ to the eigenvalue
problem

$\left(-i\rho(\omega\cdot
v)+\mathcal{L}_{\alpha}\right)h^{(j)}=\lambda^{(j)}\,h^{(j)},\quad\forall\,j\in\\{-1,0,1\\}.$

###  3.4. Higher Order Expansion

We are interested in this section to give an expression for the expansion of
the eigenvalues with respect to the spatial coordinate $\gamma=\rho\,\omega$.
We have seen in Remark 3 that for a fixed $\omega\in\mathbb{S}^{d-1}$, the
eigenvalues $\lambda^{(j)}(\rho,\,\alpha)$ and eigenvectors
$h^{(j)}_{(\rho\,\omega,\,\alpha)}$ are analytic functions of the radial
coordinate $\rho$ and the inelasticity $1-\alpha$. Hence, we have for any
$v\in\mathbb{R}^{d}$

(68) $\displaystyle\lambda^{(j)}(\rho,\,\alpha)=\sum_{n\geq
0}\lambda^{(j)}_{n}\rho^{n}+(1-\alpha)e^{(j)}_{1}+\mathcal{O}\left((1-\alpha)^{2}+(1-\alpha)\rho\right),$
(69) $\displaystyle h^{(j)}_{(\rho\,\omega,\,\alpha)}(v)=\sum_{n\geq
0}h^{(j)}_{n}(\omega)(v)\rho^{n}+(1-\alpha)f^{(j)}_{1}(v)+\mathcal{O}\left((1-\alpha)^{2}+(1-\alpha)\rho\right).$

According to the computations of the previous subsection, the first order
components of this expansion are given by

$\left\\{\begin{aligned}
&\lambda^{(j)}_{0}=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\
&\lambda^{(j)}_{1}=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},&&\lambda^{(j)}_{1}=0,&&\forall\,j\in\\{2,\ldots,d\\},\\\
&e_{1}^{(0)}=-\frac{3}{\bar{T}_{1}},&&&&e_{1}^{(j)}=0,&&\forall\,j\in\\{-1,1,\ldots,d\\}.\end{aligned}\right.$

We also have for any $v\in\mathbb{R}^{d}$

$h^{(0)}_{0}(v)=c_{0}\left(|v|^{2}-d\,\bar{T}_{1}\right)F_{1},$

for a nonnegative normalizing constant $c_{0}$. Since the triple
$\left(\rho\,\omega,\lambda^{(j)}(\rho,\,\alpha),h^{(j)}_{(\rho\,\omega,\,\alpha)}\right)$
is solution to the eigenvalue problem (20), we can equate the power of $\rho$
and $1-\alpha$ in (68)–(69) to obtain

(70) $\left\\{\begin{aligned}
&\mathcal{L}_{0}\,h^{(j)}_{0}(\omega)=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\
&\mathcal{L}_{0}\,h^{(j)}_{1}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot
v)\right)h^{(j)}_{0}(\omega),&&\forall\,j\in\\{-1,\ldots,d\\},\\\
&\mathcal{L}_{0}\,h^{(j)}_{n}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot
v)\right)h^{(j)}_{n-1}(\omega)+\sum_{k=2}^{n}\lambda^{(j)}_{k}h^{(j)}_{n-k}(\omega),&&\forall\,j\in\\{-1,\ldots,d\\},\
n\geq 2,\end{aligned}\right.$

where we also used the smoothness of $\mathcal{L}_{\alpha}$ with respect to
$1-\alpha$ (Proposition A.2).

Hence, the coefficients of the expansion can be computed by induction. For
example, to compute $\lambda^{(j)}_{2}$, we can integrate the eigenvalue
problem (20) with respect to $|v|^{2}$ and use the equations (70) with $n=2$
to obtain

(71)
$\lambda^{(j)}_{2}=-\frac{i}{2\,d\,c_{0}\,\bar{T}_{1}^{2}}\,\omega\cdot{q\left(h^{(j)}_{1}(\omega)\right)},$

where we have set

$q(h):=\int_{\mathbb{R}^{d}}h(v)\,v\,|v|^{2}\,dv.$

Now, using again (70) for $n=1$, we know that

$\mathcal{L}_{0}\,h^{(j)}_{1}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot
v)\right)h^{(j)}_{0}(\omega).$

Since $\lambda^{(j)}_{1}$ is an imaginary number and $h^{(j)}_{0}$ a real
number (it is the elastic, space homogeneous eigenvector), we have that
$h^{(j)}_{1}(\omega)(v)$ is also imaginary for all $v\in\mathbb{R}^{d}$.
Gathering this information with the explicit representation (71), we obtain
that for any $j\in\\{-1,\ldots,d\\}$, the second order expansion in $\rho$ of
$\lambda^{(j)}$, denoted by $\lambda^{(j)}_{2}$ is nonpositive777Some explicit
computations are given in the $L^{2}$ case in [8, Section 4].. The higher
order expansions can be computed by the same induction process.

This concludes the proof of Theorem 1.2.

## Acknowledgment

The research of the author was granted by the ERC Starting Grant 2009 #239983
(NuSiKiMo), NSF Grants #1008397 and #1107444 (KI-Net) and ONR grant
#000141210318. The author would like to thanks F. Filbet and C. Mouhot for
their careful reading and fruitful comments on the manuscript.

##  Appendix A Functional Toolbox on the Collision Operator

Let us present some important properties concerning the granular gases
operator we heavily used on this paper.

To be consistent with [15], we shall define for $\delta>0$ the regularized
operator

$\mathcal{L}_{1,\delta}\,=\,\mathcal{L}_{1,\delta}^{+}-\mathcal{L}^{*}-\mathcal{L}^{\nu},$

where $\mathcal{L}_{1,\delta}^{+}$ is the regularization of the truncated gain
term introduced in [17]. One of the key properties of the regularized operator
is that it converges towards $\mathcal{L}_{1}$ when $\delta\to 0$ in the norm
of graph of $L^{1}(m^{-1})$ (and also in the weighted Sobolev spaces
$W^{k,1}_{q}(m^{-1})$) but with a loss of integration weights:

######  Proposition A.1 (Proposition 5.5 of [15]).

For any $k,q\in\mathbb{N}$, we have

$\left\|\left(\mathcal{L}_{1,\delta}-\mathcal{L}_{1}\right)g\right\|_{W^{k,1}_{q}(m^{-1})}\leq\varepsilon(\delta)\,\|g\|_{W^{k,1}_{q+1}(m^{-1})},$

where $\varepsilon(\delta)$ is an explicit constant, going to $0$ as
$\delta\to 0$.

We then state a result about the Hölder continuity (in the norm of the graph)
of the gain term of the granular gases operator with respect to the
restitution coefficient $\alpha$.

######  Proposition A.2 (Proposition 3.2 of [15]).

For any $\alpha,\alpha^{\prime}\in(0,1]$, and any $g\in L^{1}_{1}(m^{-1})$,
$f\in W^{1,1}_{1}(m^{-1})$, there holds

$\left\\{\begin{aligned}
&\left\|\mathcal{Q}_{\alpha}^{+}(g,f)-\mathcal{Q}_{\alpha^{\prime}}^{+}(g,f)\right\|_{L^{1}(m^{-1})}\leq\varepsilon\left(\alpha-\alpha^{\prime}\right)\|f\|_{W^{1,1}_{1}(m^{-1})}\|g\|_{L^{1}_{1}(m^{-1})},\\\
&\left\|\mathcal{Q}_{\alpha}^{+}(f,g)-\mathcal{Q}_{\alpha^{\prime}}^{+}(f,g)\right\|_{L^{1}(m^{-1})}\leq\varepsilon\left(\alpha-\alpha^{\prime}\right)\|f\|_{W^{1,1}_{1}(m^{-1})}\|g\|_{L^{1}_{1}(m^{-1})},\end{aligned}\right.$

where we have set

$\varepsilon(r)=C\,r^{\frac{1}{3+4s}}$

for a constant $s$ given by the weight function
$m(v)=\exp\left(-a\,|v|^{s}\right)$.

We also need to estimate the smoothness, the tail behavior and the pointwise
lower bound (uniformly with respect to the restitution coefficient $\alpha$)
of the equilibrium profiles $F_{\alpha}$ solutions to (9). We have the
following result.

######  Proposition A.3 (Propositions 2.1 and 2.3 of [16]).

Let us fix $\alpha_{0}\in(0,1)$. There exist some positive constants
$a_{1},a_{2},a_{3},a_{4}$ (independent of $\alpha$) and, for any
$k\in\mathbb{N}$ a positive constant $C_{k}$ such that for all
$\alpha\in[\alpha_{0},1)$

$\displaystyle\|F_{\alpha}\|_{L^{1}\left(e^{a_{1}}|v|\right)}\leq
a_{2},\quad\|F_{\alpha}\|_{H^{k}(\mathbb{R}^{d})}\leq C_{k},$ $\displaystyle
F_{\alpha}(v)\geq a_{3}\,e^{-a_{4}|v|^{8}},\quad\forall\,v\in\mathbb{R}^{d}.$

Moreover, these profiles converge in $L^{1}_{2}$ towards the elastic
Maxwellian $F_{1}$, with an explicit rate:

######  Proposition A.4 (Proposition 3.1 of [16]).

For any $\varepsilon>0$, there exists $C_{\varepsilon}$ such that

$\|F_{\alpha}-F_{1}\|_{L^{1}_{2}}\leq
C_{\varepsilon}(1-\alpha)^{\frac{1}{2+\varepsilon}}.$

We now define for $\zeta\in\mathbb{C}$ and $\delta>0$ the operators

(72) $A_{\delta}:=\mathcal{L}_{1,\delta}^{+}-\mathcal{L}^{*}\quad\text{ and
}\quad
B_{\alpha,\,\delta}(\zeta):=\mathcal{L}^{\nu_{1}}+\mathcal{I}_{\alpha}+\zeta-\left(\mathcal{L}_{1}^{+}-\mathcal{L}_{1,\delta}^{+}\right),$

where $\mathcal{I}_{\alpha}:=\mathcal{L}_{1}-\mathcal{L}_{\alpha}$ is the
difference between the elastic and inelastic linearized operators. We can then
write the problem of computing the inverse of resolvent operator of
$\mathcal{L}_{\alpha}$ as the perturbation equation

$\mathcal{L}_{\alpha}-\zeta=A_{\delta}-B_{\alpha,\,\delta}\left(\zeta\right).$

We state a result of convergence of the linearized granular gases operator
towards the linearized elastic operator (which is a consequence of Proposition
A.2), as well as estimates on the operator $B_{\alpha,\,\delta}$.

######  Lemma A.1 (Lemmas 5.9 of [15] and 5.2 of [16]).

For any $k,q\in\mathbb{N}$ and any exponential weight function $m$, the
following properties hold:

  1. (1)

There exist a constructive $\alpha_{0}\in(0,1]$ and some nonnegative constant
$C=C(k,q,m)$ such that for any $\alpha\in(\alpha_{0},1]$,

$\displaystyle\|\mathcal{L}_{\alpha}\|_{W^{k+2,1}_{q+1}(m^{-1})\to
W^{k,1}_{q}(m^{-1})}\leq C,$
$\displaystyle\left\|\mathcal{L}_{\alpha}-\mathcal{L}_{1}\right\|_{W^{3,1}_{3}(m^{-1})\to
L^{1}(m^{-1})}\leq C\,(1-\alpha).$

  2. (2)

For any $\delta>0$, the operator $A_{\delta}:L^{1}\to
W^{\infty,1}_{\infty}\left(m^{-1}\right)$ is a bounded linear operator (more
precisely, it maps function $L^{1}$ into $\mathcal{C}^{\infty}$ functions with
compact support).

  3. (3)

There exists some constants $\delta^{*}>0$ and $\alpha_{1}\in(\alpha_{0},1)$
such that for any $\zeta\in\Delta_{-\nu_{0}}$, $\delta<\delta_{*}$ and
$\alpha\in[\alpha_{1},1]$ the operator

$B_{\alpha,\,\delta}(\zeta):W^{k+2,1}_{q+1}(m^{-1})\to W^{k,1}_{q}(m^{-1})$

is invertible. Moreover, its inverse operator satisfies

$\displaystyle\left\|B_{\alpha,\,\delta}(\zeta)^{-1}\right\|_{W^{k,1}_{q}(m^{-1})\to
W^{k,1}_{q}(m^{-1})}\leq\frac{C_{1}}{|\nu_{0}-\Re e\,\zeta|},$
$\displaystyle\left\|B_{\alpha,\,\delta}(\zeta)^{-1}\right\|_{W^{k,1}_{q}(m^{-1})\to
W^{k+2,1}_{q+1}(m^{-1})}\leq\frac{C_{2}}{|\nu_{0}-\zeta|}$

for some explicit constants $C_{1},C_{2}$ depending on
$k,q,\delta^{*},\alpha_{1}$.

As a consequence of these results, we also have the following proposition.

######  Proposition A.5 (Proposition 3.8 of [15]).

For any $k,q\in\mathbb{N}$, any exponential weight function $m$, and any
$\alpha\in(\alpha_{0},1]$,

$\left\|\mathcal{L}_{\alpha}^{+}-\mathcal{L}_{1}^{+}\right\|_{W^{k,1}_{q}(m^{-1})\to
W^{k,1}_{q+1}(m^{-1})}\leq\varepsilon\,(1-\alpha)$

where $\varepsilon$ has been defined in Proposition A.2.

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