# Stabilization of the wave equation on larger-dimension tori with rough
dampings

M. Rouveyrol  Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay,
Bâtiment 307, 91405 Orsay Cedex<EMAIL_ADDRESS>
###### Abstract.

This paper deals with uniform stabilization of the damped wave equation. When
the manifold is compact and the damping is continuous, the geometric control
condition is known to be necessary and sufficient. In the case where the
damping is a sum of characteristic functions of polygons on a two-dimensional
torus, a result by Burq-Gérard states that stabilization occurs if and only if
every geodesic intersects the interior of the damped region or razes damped
polygons on both sides. We give a natural generalization of their result to a
sufficient condition on tori of any dimension $d\geq 3$. In some particular
cases, we show that this sufficient condition can be weakened.

###### Contents

  1. 1 Notation and main result
  2. 2 First microlocalization
  3. 3 Non-concentration estimates
  4. 4 Proof of Theorem 4
    1. 4.1 Second microlocal calculus
    2. 4.2 The model case : 4 damped prisms in a tunnel
      1. 4.2.1 Generalization to higher dimensions
    3. 4.3 Reduction of the general case and conclusion
    4. 4.4 $\zeta.\partial_{z}$ vector field over the sphere at infinity
  5. 5 Proof of Theorem 5
    1. 5.1 Directional second microlocalizations
    2. 5.2 Third microlocalization near isolated undamped normal directions

##  1\. Notation and main result

Consider the damped wave (or Klein-Gordon) equation

(1.1)
$(\partial_{t}^{2}-\Delta_{g}+a(x)\partial_{t}+m(x))u=0,(u|_{t=0},\partial_{t}u|_{t=0})=(u_{0},u_{1})\in
H^{1}(M)\times L^{2}(M),$

where $(M,g)$ denotes a smooth compact Riemannian manifold without boundary,
$g$ is the manifold’s metric, $\Delta_{g}$ is the Laplace operator on $M$, and
the damping $a$ and potential $m$ are two non-negative $L^{\infty}$ functions
over $M$. The dxenergy

(1.2)
$E_{m}(u)(t)=\frac{1}{2}\int_{M}(|\nabla_{g}u|_{g}^{2}+|\partial_{t}u|^{2}+m|u|^{2})d\operatorname{vol}_{g}$

is then decaying, as

$E_{m}(u)(t)=E_{m}(u)(0)-\int_{0}^{t}\int_{M}a(x)|\partial_{t}u(t,x)|^{2}d\operatorname{vol}_{g}(x)dt.$

We say that _uniform stabilization_ holds for the damping $a$ if one of the
following equivalent conditions is satisfied :

  1. (1)

There exists a rate $f(t)$ such that $\lim_{t\rightarrow+\infty}f(t)=0$ and
for any $(u_{0},u_{1})\in H^{1}(M)\leavevmode\nobreak\
\times\leavevmode\nobreak\ L^{2}(M)$,

$E_{m}(u)(t)\leq f(t)E_{m}(u)(0).$

  2. (2)

There exist some constants $C,c>0$ such that for any $(u_{0},u_{1})\in
H^{1}(M)\leavevmode\nobreak\ \times\leavevmode\nobreak\ L^{2}(M)$,

$E_{m}(u)(t)\leq Ce^{-ct}E_{m}(u)(0).$

  3. (3)

There exist some $T>0$ and $c>0$ such that for any $(u_{0},u_{1})\in
H^{1}(M)\leavevmode\nobreak\ \times\leavevmode\nobreak\ L^{2}(M)$, if $u$ is a
solution of the damped wave equation (1.1), then

$E_{m}(u)(0)\leq
C\int_{0}^{T}\int_{M}2a(x)|\partial_{t}u|^{2}d\operatorname{vol}_{g}.$

  4. (4)

There exist some $T>0$ and $c>0$ such that for any $(u_{0},u_{1})\in
H^{1}(M)\leavevmode\nobreak\ \times\leavevmode\nobreak\ L^{2}(M)$, if $u$ is a
solution of the _un_ damped wave equation

$(\partial_{t}^{2}-\Delta+m)u=0,\quad(u|_{t=0},\partial_{t}u|_{t=0})=(u_{0},u_{1})\in
H^{2}(M)\times L^{2}(M),$

then

$E_{m}(u)(0)\leq
C\int_{0}^{T}\int_{M}2a(x)|\partial_{t}u|^{2}d\operatorname{vol}_{g}.$

By the HUM method, uniform stabilization is equivalent to observability
estimates (see Proposition 2.1 for a precise statement) and to controllability
of the equation. We refer to [BG02, Chapters 3 and 4] for proofs concerning
the wave equation and its controllability properties.

In the case where the damping $a$ is continuous, a necessary and sufficient
condition for uniform stabilization is given by the following landmark result
:

######  Theorem 1 (Geometric control condition, [RT75, BLR92, BG97]). –

Let $m\geq 0$. Assume that the damping $a$ is continuous. For
$\rho_{0}=(x_{0},\xi_{0})\in S^{*}M$ (the unit cotangent bundle of $M$), let
$\gamma_{\rho_{0}}(s)$ denote the geodesic starting from $x_{0}$ in the
(co-)direction $\xi_{0}$. The damping $a$ then stabilizes uniformly the wave
equation if and only if :

(GCC) $\exists T,c>0\text{ such that }\inf_{\rho_{0}\in
S^{*}M}\int_{0}^{T}a(\gamma_{\rho_{0}}(s))ds\geq c,$

or equivalently if every geodesic $\gamma_{\rho_{0}}(s)$ intersects
$\\{a>0\\}$ in time $T$.

In the case where $a$ is merely $L^{\infty}$, the following classical result
is a consequence of Bardos, Lebeau and Rauch’s work :

######  Theorem 2 ([BLR92], [BG20]-Theorem 2). –

Assume that $0\leq a\in L^{\infty}(M)$. Then the _strong geometric control
condition_

(SGCC) $\begin{gathered}\exists T,c>0\text{ such that }\forall\rho_{0}\in
S^{*}\mathbb{T}^{d},\exists s\in(0,T),\delta>0\\\ \text{ such that }a\geq
c\text{ a.e. over }B(\gamma_{\rho_{0}}(s),\delta)\end{gathered}$

is sufficient for uniform stabilization, and the _weak geometric control
condition_

(WGCC) $\begin{gathered}\exists T>0\text{ such that }\forall\rho_{0}\in
S^{*}\mathbb{T}^{d},\exists s\in(0,T)\text{ such that
}\gamma_{\rho_{0}}(s)\in\operatorname{supp}(a),\end{gathered}$

where $\operatorname{supp}(a)$ is the support of $a$ in the distributional
sense, is necessary for uniform stabilization.

Finding a necessary and sufficient condition in between (SGCC) and (WGCC) in
the case of $L^{\infty}$ dampings yet remains an open problem. In [Leb92],
Gilles Lebeau proved that uniform stabilization holds on the sphere
$\mathbb{S}^{d}=\\{x\in\mathbb{R}^{d+1},|x|=1\\}$ when $a=1$ over the half-
sphere. Hui Zhu generalized this result to Zoll surfaces of revolution in
[Zhu18]. See also [AL14, Appendix B] for a case where the damped region
$\\{a=1\\}$ does not satisfy (WGCC).

A necessary and sufficient condition for uniform stabilization was given by
Nicolas Burq and Patrick Gérard in [BG20], in the case where $M$ is a
2-dimensional torus $\mathbb{T}^{2}=\mathbb{R}^{2}/A\mathbb{Z}\times
B\mathbb{Z}$ with $A,B>0$, and the damping $a$ is a sum of characteristic
functions of disjoint polygons. Their result is stated thereafter and
illustrated in Figure 1.1.

######  Theorem 3 ([BG20], Assumption 1.2 and Theorem 4). –

The damping $a$ stabilizes the wave equation if and only if the following
assumption holds : there exists some $T>0$ such that all geodesics (straight
lines) of length $T$ either encounter the interior of one of the polygons or
follow for some time one of the sides of a polygon _on the left_ and for some
time one of the sides of a polygon (possibly the same) _on the right_.

The present article is dedicated to exploring generalizations of Theorem 3 to
higher-dimensional tori. Like in [BG20], we use dampings $a$ that are
characteristic functions. This is motivated both by the fact that applied
control historically uses characteristic functions, and by a will to
understand the concentration and non-concentration properties of waves in
situations where geodesics raze the control region without entering its
interior. The case of tori is a favorable one for this study because the
absence of curvature or boundary simplifies the geometry, and because it
ensures that quasimodes of the Laplace operator satisfy good non-concentration
properties (see Proposition 3.1).

We identify a $d$-dimensional torus $\mathbb{T}^{d}$ with a cuboid
$\prod_{j=1}^{d}[0,A_{j}]$, $A_{j}>0$, with appropriate identification of the
faces. We call a _polyhedron_ of $\mathbb{T}^{d}$ the intersection of the
cuboid with a finite number of half-spaces, and we assume that $a$ is a finite
sum of characteristic functions of disjoint such polyhedrons. This choice
comes from the fact that it is always possible to concentrate solutions of the
wave equation near a geodesic that only has finite-order contacts with the
damped region (see for example [BG20, Section 5]). Thus, geodesics which
violate (SGCC) need to have infinite-order contacts with the damped zone for
stabilization to occur. By considering a damping equal to 1 over a finite
union of polyhedrons, we essentially cover all cases where the damping is a
characteristic function.

1.1.aStabilization

1.1.bNo stabilization

1.1.cNo stabilization

Figure 1.1. N. Burq and P. Gérard’s checkerboards on $\mathbb{T}^{2}$ [BG20] :
the damping $a$ is equal to 1 in the colored region and 0 elsewhere. For all
these examples, (WGCC) is satisfied but not (SGCC). The dashed lines are
geodesics which violate the condition of Theorem 3.

1.2.a

1.2.b

Figure 1.2. In these three-dimensional illustrations, the damping $a$ is equal
to 1 in the colored zone and 0 elsewhere. Figure 1.2.a illustrates the
framework of Theorem 4, as the red geodesic satisfies condition (1.3) but does
not intersect the interior of $\operatorname{supp}(a)$. In Figure 1.2, the red
geodesic does not satisfy this condition as directions $\Xi=(\pm 1,0)$ or
$(0,\pm 1)$ are not damped in the sense of Definition 1.1. In Section 5, we
prove that uniform stabilization can still occur in cases involving such
geodesics.

On $\mathbb{T}^{2}$, the conormal space to a geodesic at a point is one-
dimensional, hence the unit sphere of (co-)normal directions to the geodesic
is reduced to two points which correspond to the right and left sides of the
geodesic. The assumption of Theorem 3 thus naturally generalizes to assuming
that every geodesic is damped in each of its normal directions. Our first
result states that this condition is indeed sufficient for uniform
stabilization. It holds on any torus
$\mathbb{T}^{d}=\mathbb{R}^{d}/(A_{1}\mathbb{Z}\times\dots\times
A_{d}\mathbb{Z})$ of dimension $d\geq 3$.

######  Theorem 4. –

Assume that every geodesic of $\mathbb{T}^{d}$ either intersects the interior
of $\operatorname{supp}(a)$, or is damped in every normal direction, _ie_

(1.3) $\begin{gathered}\text{ for all }\rho_{0}=(X_{0},\Xi_{0})\in
S^{*}\mathbb{T}^{d},\Xi\text{ orthogonal to }\Xi_{0}\text{ with }|\Xi|=1,\\\
\text{ there exist some positive $\delta_{0}$ and some interval
$I\subset\mathbb{R}$ such that }\\\
\gamma_{\rho_{0}}(s)+\delta\Xi\in\operatorname{Int}(\operatorname{supp}(a)),\quad\forall
s\in I,\quad\forall\delta\in(0,\delta_{0}].\end{gathered}$

Then stabilization holds.

Note that any geodesic which encounters the interior of
$\operatorname{supp}(a)$ also satisfies assumption (1.3). A typical example of
a geodesic satisfying the assumption without entering the interior of
$\operatorname{supp}(a)$ is the red one in Figure 1.2.a. The proof of Theorem
4 consists in studying precisely the concentration of high-frequency
quasimodes of the Laplacean near such a geodesic (specifically near the red
geodesic of Figure 4.1, see section 4.2) using second microlocalization
techniques, then reducing the general case to this model case.

Condition (1.3) naturally leads to the following definition :

######  Definition 1.1. –

Consider a geodesic $\gamma_{\rho_{0}}$ of $\mathbb{T}^{d}$ with direction
$\Xi_{0}\in\mathbb{S}^{d-1}$, and some $\Xi\in\mathbb{S}^{d-1}$ orthogonal to
$\Xi_{0}$. We say that $\gamma_{\rho_{0}}$ is _damped in the normal direction_
$\Xi$ if there exists some positive $\delta_{0}$ and some interval
$I\subset\mathbb{R}$ such that

(1.4)
$\gamma_{\rho_{0}}(s)+\delta\Xi\in\operatorname{Int}(\operatorname{supp}(a)),\quad\forall
s\in I,\quad\forall\delta\in(0,\delta_{0}].$

For example, in Figure 1.2.b, the red geodesic is damped in all but four
normal directions. Note also that since the interval $I$ is independent of
$\delta$ in the definition, a geodesic can be damped in some direction only if
it razes an edge or a face of some polyhedron in that direction. Punctual
contacts between a damped polyhedron and a geodesic are not considered
sufficient for the geodesic to be damped in the normal directions that enter
the polyhedron, so as to avoid phenomena resembling that of Figure 1.1.c.

We emphasize that in dimension 2, assumption (1.3) is exactly the necessary
and sufficient condition found by N. Burq and P. Gérard in [BG20]. In
dimensions 3 and greater, the enriched geometry (specifically the fact that
the unit sphere of the conormal space to a geodesic at a point is infinite)
permits uniform stabilization under weaker conditions, as in the following
result :

######  Theorem 5. –

Identify $\mathbb{T}^{3}\simeq[-1,1]^{2}_{x}\times[-1,1]_{y}$ and assume that
$a$ is the characteristic function of the set

(1.5) $\displaystyle\\{|x_{1}|\geq\frac{1}{2}\text{ or
}|x_{2}|\geq\frac{1}{2}\\}\cup\\{-1<y<-\frac{1}{2},x_{1}>0,x_{2}<0\\}\cup\\{-\frac{1}{2}<y<0,x_{1}>0,x_{2}>0\\}$
$\displaystyle\cup\\{0<y<\frac{1}{2},x_{1}<0,x_{2}>0\\}\cup\\{\frac{1}{2}<y<1,x_{1}<0,x_{2}<0\\},$

as illustrated in Figure 5.1. Then stabilization holds.

This case is based on the shape of the damping in Figure 1.2.b, with some
geodesics damped in all but a finite number of directions, so that condition
(1.3) is not satisfied. We prove this theorem using directional second
microlocalization and third microlocalization procedures that are new to the
best of our knowledge. It leads us to conjecture that a necessary and
sufficient condition for uniform stabilization in our framework is that every
geodesic be damped on a full-Lebesgue-measure set of normal directions.

The paper is constructed as follows. In Section 2, we recall the first
microlocalization procedure for the wave equation and introduce relevant
tools. In Section 3, we prove non-concentration estimates for Laplacean
quasimodes on $\mathbb{T}^{d}$. Section 4 is dedicated to the proof of Theorem
4 and Section 5 to the refined example of Theorem 5. In the concluding
paragraphs of Section 5, we discuss possible extensions of this theorem using
our techniques.

_Acknowledgments_ The author wishes to thank Nicolas Burq for his rich and
careful advice during the writing of this article and Antoine Prouff for the
many discussions about it. This work was funded by a CDSN PhD grant from Ecole
Normale Supérieure Paris-Saclay via the Hadamard Doctoral School of
Mathematics.

##  2\. First microlocalization

The goal of this section is to introduce the first microlocal measure and the
observability estimate that will be used later on. We do so by proving that
(SGCC) implies uniform stabilization. The result of this section is valid on
any compact Riemannian manifold without boundary $M$ and for any non-negative
damping function $a\in L^{\infty}(M)$. For simplicity, we prove it on
$d$-dimensional tori. We refer to [BG20, Sections 2, 5 and Annex A] for
statements and proof of (SGCC) and (WGCC) in the general case.

Let us recall the statement of the strong geometric control condition :

######  Theorem 6. –

Assume that $0\leq a\in L^{\infty}(\mathbb{T}^{d})$. Then the _strong
geometric control condition_

(SGCC) $\begin{gathered}\exists T,c>0\text{ such that }\forall\rho_{0}\in
S^{*}\mathbb{T}^{d},\exists s\in(0,T),\delta>0\\\ \text{ such that }a\geq
c\text{ a.e. over }B(\gamma_{\rho_{0}}(s),\delta)\end{gathered}$

is sufficient for uniform stabilization.

###### Proof.

Let us assume that (SGCC) holds. The following result states that uniform
stabilization is equivalent to an observability estimate for solutions of the
Helmholtz equation in the high-frequency regime.

######  Proposition 2.1 ([BG20], Proposition A.5). –

Consider an $L^{\infty}$ non-negative function $a$ such that
$\int_{\mathbb{T}^{d}}a(x)dx>0$, then $a$ uniformly stabilizes the wave
equation (1.1) if and only if

(Obs) $\begin{gathered}\exists C,h_{0}>0\text{ such that }\forall
0<h<h_{0},\quad\forall(u,f)\in H^{2}(\mathbb{T}^{d})\times
L^{2}(\mathbb{T}^{d}),\\\ (h^{2}\Delta+1)u=f,\text{ there holds
}\|u\|_{L^{2}}\leq
C\left(\|a^{\frac{1}{2}}u\|_{L^{2}}+\frac{1}{h}\|f\|_{L^{2}}\right).\end{gathered}$

We then study microlocal measures associated to a sequence of quasimodes that
violate the observability estimate to prove it by contradiction, following the
idea of [Leb96, Bur02].

Assuming that (Obs) does not hold, we obtain sequences $(h_{n})\rightarrow 0$
and $(u_{n},f_{n})$ that satisfy

$(h_{n}^{2}\Delta+1)u_{n}=f_{n},\quad\|u_{n}\|_{L^{2}}=1,\quad\|a^{\frac{1}{2}}u_{n}\|_{L^{2}}=o(1)_{n\rightarrow+\infty},\quad\|f_{n}\|_{L^{2}}=o(h_{n})_{n\rightarrow+\infty}.$

Given a symbol $q\in C^{\infty}_{c}(T^{*}\mathbb{T}^{d})$, we define its
quantization as follows. Using a partition of unity, we can assume that $q$ is
supported in a local chart. In this chart, we define

(2.1)
$\operatorname{Op}_{h}(q)u(X)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}e^{i(X-Y).\Xi}q(X,h\Xi)\zeta(Y)u(Y)dYd\Xi$

where $\zeta=1$ in a neighborhood of the support of $q$. Then, up to
extracting a subsequence, $u_{n}$ admits a semiclassical measure $\nu$ over
$T^{*}\mathbb{T}^{d}$ which satisfies

$\lim_{n\rightarrow+\infty}(\operatorname{Op}_{h_{n}}(q)u_{n},u_{n})_{L^{2}}=\langle\nu,q\rangle.$

$u_{n}$ is $h_{n}$-oscillating, as

$\int_{|\Xi|\geq\frac{R}{h_{n}}}|\hat{u}_{n}(\Xi)|^{2}d\Xi\leq
R^{-4}\|h_{n}^{2}|\Xi|^{2}|\hat{u}_{n}|\|_{L^{2}}^{2}\leq\frac{C}{R^{4}}\rightarrow_{R\rightarrow+\infty}0,$

thus $\nu$ has total mass 1. The asymptotic
$\|(h_{n}^{2}\Delta+1)u_{n}\|_{L^{2}}=o(h_{n})$ gives that $\nu$ is supported
in the characteristic set

$\\{(X,\Xi)\in T^{*}\mathbb{T}^{d},|\Xi|^{2}=1\\}$

and invariant by the bicharectistic flow :

$\Xi.\nabla_{X}\nu=0.$

In particular, $\nu$ can be considered as a measure over the unit cotangent
bundle of the torus
$S^{*}\mathbb{T}^{d}=T^{*}\mathbb{T}^{d}/\mathbb{R}^{*}_{+}$. We refer to
[GL93, Bur97] and [Zwo12, Chapter 5] for the construction and properties of
semiclassical measures.

Denote then

$S=\\{X\in\mathbb{T}^{d},\text{ there exist }\delta>0,c>0\text{ such that
}a\geq c\text{ on }B(X,\delta)\\}.$

Taking some $X_{0}\in S$ and corresponding $\delta,c$, we get that for every
symbol $q(X,\Xi)$ supported in $B(X_{0},\delta)$ in the $X$ variable,

(2.2) $\displaystyle|(\operatorname{Op}_{h_{n}}(q)u_{n},u_{n})_{L^{2}}|$
$\displaystyle=|(\operatorname{Op}_{h_{n}}(q)u_{n},\mathbbm{1}_{B(X,\delta)}u_{n})_{L^{2}}|$
$\displaystyle\leq\|\operatorname{Op}_{h_{n}}(q)u_{n}\|_{L^{2}}\|\mathbbm{1}_{B(X,\delta)}u_{n}\|_{L^{2}}$
$\displaystyle\leq\frac{1}{c}\|\operatorname{Op}_{h_{n}}(q)u_{n}\|_{L^{2}}\|au_{n}\|_{L^{2}}.$

$\operatorname{Op}_{h_{n}}(q)$ is bounded uniformly with respect to $h_{n}$
over $L^{2}$, so that $(\operatorname{Op}_{h_{n}}(q)u_{n})$ is a bounded
sequence in $L^{2}$. Since $\|au_{n}\|=o(1)_{L^{2}}$, we get that
$\langle\nu,q\rangle=0$. Hence, $\nu$ vanishes in a neighborhood of every
point $\rho\in S^{*}_{X}\mathbb{T}^{d}$ for all $X\in S$. By (SGCC), every
bicharacteristic contains one such $\rho$, so that $\nu$ is identically 0.
This contradicts the fact that $\nu$ has total mass 1. ∎

Theorem 6 can be rephrased as saying that uniform stabilization occurs when
the set $\Omega=\bigcup_{\omega}\omega$ captures every geodesic in finite
time, where the union is over all open sets $\omega$ where $a\geq
c_{\omega}\mathbbm{1}_{\omega}$ a.e. for some $c_{\omega}>0$. In the case
where $a$ is the characteristic function of an open set, $\Omega$ is exactly
the interior of $\operatorname{supp}(a)$. Thus the result will later be used
in the following form, which is a direct consequence of the proof :

######  Corollary 2.2. –

Assume $a$ is the characteristic function of an open set. If a
bicharacteristic contains some $\rho=(X,\Xi)\leavevmode\nobreak\
\in\leavevmode\nobreak\ S^{*}\mathbb{T}^{d}$ with
$X\in\operatorname{Int}(\operatorname{supp}(a))$, then the first microlocal
measure $\nu$ has no support over this bicharacteristic.

##  3\. Non-concentration estimates

In this section, we prove the non-concentration estimates that will determine
the scaling of the second and third microlocalization procedures. When
performing the second microlocalization procedure, these estimates allow to
avoid dealing with the trace-operator-valued part of the 2-microlocal measure
supported at a finite distance from the origin (see [AM14, Section3], [FK00,
Theorem 1]). Our result generalizes the estimate obtained by N. Burq and P.
Gérard on the 2-dimensional torus in [BG20, Section 3A] (itself a
generalization of [BZ15, Section 3]). The proof follows theirs.

Recall that $\|(h_{n}^{2}\Delta+1)u_{n}\|_{L^{2}}=o(h_{n})$ and denote

(3.1)
$\epsilon(h_{n})=\max\left(h_{n}^{\frac{1}{6}},\left(\frac{\|(h_{n}^{2}\Delta+1)u_{n}\|}{h_{n}}\right)^{\frac{1}{6}}\right).$

$\epsilon(h_{n})$ satisfies :

(3.2) $h_{n}^{-1}\epsilon^{-6}(h_{n})\|(h_{n}^{2}\Delta+1)u_{n}\|_{L^{2}}\leq
1\text{ and }\lim_{n\rightarrow+\infty}\epsilon(h_{n})=0.$

The result we prove is the following :

######  Proposition 3.1. –

On the torus $\mathbb{T}^{d}$, assume that
$\|u_{n}\|_{L^{2}(\mathbb{T}^{d})}=\mathcal{O}(1)$ and (3.2) is satisfied,
then there exists a constant $C>0$ such that

$\forall n\in\mathbb{N},\|u_{n}\|_{L^{2}(\\{|x_{j}|\leq
h_{n}^{\frac{1}{2}}\epsilon^{-2}(h_{n})\\})}\leq
C\epsilon^{\frac{1}{2}}(h_{n}),\quad j=1,\dots,d.$

Note that $\epsilon(h_{n})\geq h_{n}^{\frac{1}{6}}$ so the width of the slice
tends to zero as $n$ goes to $+\infty$.

We prove the result for $j=1$ without loss of generality to simplify
notations. In this proof only, we denote
$X=(x_{1},y^{\prime})\in\mathbb{T}\times\mathbb{T}^{d-1}$ and
$\Xi=(\xi_{1},\eta^{\prime})\in\mathbb{R}\times\mathbb{R}^{d-1}$. Proposition
3.1 will result from the following statement :

######  Proposition 3.2. –

There exist some positive $C$ and $h_{0}$ such that for any $0<h<h_{0}$,
$1\leq\beta\leq h^{-\frac{1}{2}}$ and any $(u,f)\in H^{2}\times L^{2}$
satisfying

$(h^{2}(\partial_{x_{1}}^{2}+\Delta_{y^{\prime}})+1)u=f,$

the following estimate holds :

(3.3) $\|u\|_{L^{\infty}(\\{|x_{1}|\leq\beta h^{1/2}\\})}\leq
C\beta^{-\frac{1}{2}}h^{-\frac{1}{4}}(\|u\|_{L^{2}_{x_{1},y^{\prime}}(\\{\beta
h^{1/2}\leq|x_{1}|\leq 2\beta
h^{1/2}\\})}+h^{-1}\beta^{2}\|f\|_{L^{2}_{x_{1},y^{\prime}}(\\{|x_{1}|\leq
2\beta h^{1/2})\\})}).$

Let us show that Proposition 3.2 implies Proposition 3.1. We apply Hölder’s
inequality in the $x_{1}$ variable and Proposition 3.2 with
$\beta=\epsilon^{-3}(h_{n})\leq h^{-\frac{1}{2}}$ :

$\displaystyle\|u\|_{L^{2}(\\{|x_{1}|\leq h^{1/2}\epsilon^{-2}(h)\\}\\})}$
$\displaystyle\leq
h^{\frac{1}{4}}\epsilon^{-1}(h)\|u\|_{L^{\infty}(\\{|x_{1}|\leq
h^{\frac{1}{2}}\epsilon^{-3}(h)\\};L^{2}_{y^{\prime}}\\})}$ $\displaystyle\leq
C\epsilon^{\frac{1}{2}}(h)\left(\|u\|_{L^{2}_{x_{1},y^{\prime}}(\\{h^{1/2}\epsilon^{-3}(h)\leq|x_{1}|\leq
2h^{1/2}\epsilon^{-3}(h)\\})}\right.$
$\displaystyle\qquad\left.+h^{-1}\epsilon^{-6}(h)\|f\|_{L^{2}_{x_{1},y^{\prime}}(\\{|x_{1}|\leq
2h^{1/2}\epsilon^{-3}(h)\\})}\right)$ $\displaystyle\leq
C\epsilon^{\frac{1}{2}}(h)(\|u\|_{L^{2}}+h^{-1}\epsilon^{-6}(h)\|f\|_{L^{2}})$
$\displaystyle\leq 2C\epsilon^{\frac{1}{2}}(h).$

We now prove Proposition 3.2. Denote $v$ (resp. $g$) the partial Fourier
transform of $u$ (resp. $f$) with regards to the $y^{\prime}$ variable. For a
fixed $x_{1}$, the Plancherel equality gives

$\|v(x_{1},.)\|_{L^{2}_{\eta^{\prime}}}=K\|u(x_{1},.)\|_{L^{2}_{y^{\prime}}},$

where $K$ depends on the periods of the torus in the $y^{\prime}$ directions.
Inequality (3.3) is then equivalent to

(3.4) $\|v\|_{L^{\infty}(\\{|x_{1}|\leq\beta
h^{1/2}\\};L^{2}_{\eta^{\prime}})}\leq
C\beta^{-\frac{1}{2}}h^{-\frac{1}{4}}\left(\|v\|_{L^{2}(\\{\beta
h^{1/2}\leq|x_{1}|\leq 2\beta h^{1/2}\\};L^{2}_{\eta^{\prime}})}\right.\\\
\left.+h^{-1}\beta^{2}\|g\|_{L^{2}(\\{|x_{1}|\leq 2\beta
h^{1/2}\\};L^{2}_{\eta^{\prime}})}\right).$

Besides, by the Minkowski inequality,

$\|v\|_{L^{\infty}_{x_{1}};L^{2}_{\eta^{\prime}}}\leq\|v\|_{L^{2}_{\eta^{\prime}};L^{\infty}_{x_{1}}}$

so it is enough to prove the following one-dimensional result :

######  Proposition 3.3 ([BG20], Proposition 3.3). –

There exist some positive $C$ and $h_{0}$ such that for any $0<h<h_{0}$,
$\eta\in\mathbb{R}$, $1\leq\beta\leq h^{-\frac{1}{2}}$ and any $(v,g)$
satisfying

$\left(h^{2}\frac{d^{2}}{dx_{1}^{2}}+1-h^{2}|\eta^{\prime}|^{2}\right)v=g,$

we have

(3.5) $\|v\|_{L^{\infty}(\\{|x_{1}|\leq\beta h^{\frac{1}{2}}\\})}\leq
C\beta^{-\frac{1}{2}}h^{-\frac{1}{4}}\left(\|v\|_{L^{2}(\\{\beta
h^{\frac{1}{2}}\leq|x|\leq 2\beta
h^{\frac{1}{2}}\\})}+h^{-1}\beta^{2}\|g\|_{L^{2}(\\{|x|\leq 2\beta
h^{\frac{1}{2}}\\})}\right).$

Integrating the square of (3.5) in the $\eta^{\prime}$ variable gives (3.4).

The proof of the one-dimensional estimate (3.5) is exactly that of [BG20]. We
include it for the sake of completeness. Performing the change of variables
$x_{1}=\beta h^{\frac{1}{2}}z$, it is equivalent to prove that any solutions
of

$\left(h\beta^{-2}\partial_{z}^{2}+1-h^{2}|\eta^{\prime}|^{2}\right)v=g$

satisfy

$\|v\|_{L^{\infty}(\\{|z|\leq 1\\})}\leq C\left(\|v\|_{L^{2}(\\{1\leq|z|\leq
2\\})}+h^{-1}\beta^{2}\|g\|_{L^{2}(\\{|z|\leq 2\\})}\right).$

Setting $\tau=\beta^{2}h^{-1}(1-h^{2}|\eta^{\prime}|^{2})$, it suffices to
prove the following lemma :

######  Lemma 3.4 ([BG20], Lemma 3.4). –

There exists some positive $C$ such that for any $\tau\in\mathbb{R}$ and any
solution $(v,k)$ over $(-2,2)$ of

$(\partial_{z}^{2}+\tau)v=k,$

there holds the inequality

$\|v\|_{L^{\infty}(-1,1)}\leq C\left(\|v\|_{L^{2}(\\{1\leq|z|\leq
2\\})}+\frac{1}{\sqrt{1+|\tau|}}\|k\|_{L^{1}(-2,2)}\right).$

###### Proof of the lemma..

Let $\chi\in C^{\infty}_{c}(-2,2)$ be equal to 1 over $(-1,1)$. $w=\chi v$
satisfies

(3.6) $(\partial_{z}^{2}+\tau)w=\chi
k+2\partial_{z}(\chi^{\prime}v)-\chi^{\prime\prime}v.$

We distinguish two regimes :

Elliptic regime : $\tau\leq-1$. We multiply (3.6) by $w$ and integrate by
parts to get

$\|\partial_{z}w\|_{L^{2}(-2,2)}^{2}+|\tau|\|w\|_{L^{2}(-2,2)}^{2}=-(\chi
k-\chi^{\prime\prime}v,w)_{L^{2}}+2(\chi^{\prime}v,\partial_{z}w)_{L^{2}}.$

Thus,

$\|\partial_{z}w\|_{L^{2}(-2,2)}^{2}+|\tau|\|w\|_{L^{2}(-2,2)}^{2}\leq\\\
C\left(\|k\|_{L^{1}(-2,2)}\|w\|_{L^{\infty}}+\|v\|_{L^{2}(\\{1\leq|z|\leq
2\\})}(\|w\|_{L^{2}(\\{1\leq|z|\leq
2\\})}+\|\partial_{z}w\|_{L^{2}(-2,2)})\right).$

By the Gagliardo-Nirenberg inequality in dimension 1,

$\|w\|_{L^{\infty}}\leq
C\|\partial_{z}w\|_{L^{2}}^{\frac{1}{2}}\|w\|_{L^{2}}^{\frac{1}{2}}=C|\tau|^{-\frac{1}{4}}\|\partial_{z}w\|_{L^{2}}^{\frac{1}{2}}|\tau|^{\frac{1}{4}}\|w\|_{L^{2}}^{\frac{1}{2}}.$

Set
$A=\|\partial_{z}w\|_{L^{2}(-2,2)}+|\tau|^{\frac{1}{2}}\|w\|_{L^{2}(-2,2)}$,
then applying the Gagliardo-Nirenberg inequality again gives

$A^{2}\leq
C\left(\|k\|_{L^{1}(-2,2)}|\tau|^{-\frac{1}{4}}A+\|v\|_{L^{2}(\\{1\leq|z|\leq
2\\})}A\right).$

Applying it a third time yields

$\|v\|_{L^{\infty}(-1,1)}\leq\|w\|_{L^{\infty}}\leq
C|\tau|^{-\frac{1}{4}}A\leq C\left(\|v\|_{L^{2}(\\{1\leq|z|\leq
2\\})}+|\tau|^{-\frac{1}{2}}\|k\|_{L^{1}(-2,2)}\right).$

Since $|\tau|\geq 1$, we get the desired inequality.

Hyperbolic regime : $\tau\geq-1$. Set $\sigma=\sqrt{\tau}\in\mathbb{R}^{+}\cup
i[0,1]$. Integrating (3.6) gives

$w(x)=\int_{z=-2}^{x}g(z)\int_{y=z}^{x}e^{i\sigma(2y-x-z)}dydz,$

where $g=\chi
k-\chi^{\prime\prime}v+2\partial_{z}(\chi^{\prime}v)=g_{1}+\partial_{z}g_{2}.$
Since for any $x,z\in[-2,2]$

$\left|\int_{y=z}^{x}e^{i\sigma(2y-x-z)}dy\right|\leq\frac{C}{1+|\sigma|}=\frac{C}{1+|\tau|^{\frac{1}{2}}},$

the contribution of $g_{1}$ is bounded uniformly by

$\frac{C}{1+|\tau|^{\frac{1}{2}}}\left(\|\chi
k\|_{L^{1}(-2,2)}+\|v\|_{L^{1}(\\{1\leq|z|\leq 2\\})}\right).$

Integrating by parts, we get that the contribution of $\partial_{z}g_{2}$ is
bounded by

$C\|\chi^{\prime}v\|_{L^{1}(-2,2)}\leq C^{\prime}\|v\|_{L^{1}(\\{1\leq|z|\leq
2\\})}.$

Summing both contributions gives the result. This concludes the proof of
Proposition 3.1. ∎

##  4\. Proof of Theorem 4

This section is dedicated to the proof of Theorem 4, namely that stabilization
occurs when every geodesic either intersects the interior of the damped zone
or is damped in every normal direction. The first subsection is dedicated to
the proof of $L^{2}$-boundedness of pseudodifferential operators and a Gårding
inequality for our 2-microlocal calculus. Well-known semiclassical analysis
techniques are used to give an appropriate and precise framework to define
2-microlocal measures. In the second subsection, we study the 2-microlocal
measure and prove uniform stabilization for a model damping. In the third, we
reduce the general case to this model. The last subsection contains the proof
of a geometric lemma used in the second one.

###  4.1. Second microlocal calculus

We start by constructing the 2-microlocal pseudodifferential calculus. We deal
with more general symbols than in [BG20], as we only assume decay in the
$\zeta$ variable rather than polyhomogeneity. This weakened assumption will be
used in the third microlocalization in Section 5.

We introduce symbol classes and the second quantization for these symbols,
then show that this quantization provides a pseudodifferential calculus in
$\epsilon^{2}(h)$ with sufficient tools to construct 2-microlocal measures.
Our symbols take variables in
$\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{q}\times\mathbb{R}^{q}$.
Case $q=d-1$ will be used in the present section, and $d=3$, $q\in\\{1,2\\}$
in Section 5.

######  Definition 4.1 (Symbol classes). –

  1. (1)

We define $S^{0}$ to be the class of smooth functions $b$ of
$(X,\Xi,z,\zeta)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{q}\times\mathbb{R}^{q}$
which are compactly supported in the $(X,\Xi)$ variables and satisfy the
following decay estimate in the $\zeta$ variable :

$\forall\alpha,\beta\in\mathbb{N}^{q},\quad\exists C_{\alpha,\beta}\quad\text{
s. t.
}\quad\sup_{(X,\Xi,z,\zeta)\in\mathbb{R}^{2d}\times\mathbb{R}^{2q}}|\partial^{\alpha}_{z}\partial^{\beta}_{\zeta}b(X,\Xi,z,\zeta)|\leq
C_{\alpha,\beta}\langle\zeta\rangle^{-|\beta|}$

where $\langle.\rangle$ denotes the usual
$\langle\zeta\rangle=(1+|\zeta|^{2})^{\frac{1}{2}}$.

  2. (2)

For any integer $m\geq 0$, we define $S^{m}_{H}$ as the set of smooth
functions which are compactly supported in the $(X,\Xi)$ variables and
polyhomogeneous of degree $m$ with respect to the $(z,\zeta)$ variables with
limits in the radial direction :

$\lim_{r\rightarrow+\infty}\frac{1}{r^{m}}a\left(X,\Xi,\frac{(rz,r\zeta)}{\|(z,\zeta)\|}\right)=\tilde{a}\left(X,\Xi,\frac{(z,\zeta)}{\|(z,\zeta)\|}\right).$

We have $S^{0}_{H}\subset S^{0}$. Besides, when $m=0$ functions in $S^{0}_{H}$
are identified with smooth compactly supported functions on
$\mathbb{R}^{2d}\times\overline{B(0,1)}_{\tilde{z},\tilde{\zeta}}$ via the
change of variables

$(z,\zeta)\mapsto(\tilde{z},\tilde{\zeta})=\frac{(z,\zeta)}{\sqrt{1+|z|^{2}+|\zeta|^{2}}}.$

Consider some function $\epsilon(h)$ satisfying

(4.1) $\lim_{h\rightarrow 0}\epsilon(h)=0,\quad\epsilon(h)\geq
h^{\frac{1}{2}},\quad
h^{\frac{1}{2}}\epsilon^{-2}(h)\rightarrow_{h\rightarrow{0}}0.$

Given a symbol $b$ belonging to $S^{m}_{H}$ or $S^{0}$, its second
quantization is defined by :

(4.2)
$\operatorname{Op}_{h}(b)u(X)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}_{Y}\times\mathbb{R}^{d}_{\Xi}}e^{i(X-Y).\Xi}b\left(X,h\Xi,\frac{\epsilon(h)}{h^{\frac{1}{2}}}x,\epsilon(h)h^{\frac{1}{2}}\xi\right)u(Y)dYd\Xi$

where $X=(x,x^{\prime}),\Xi=(\xi,\xi^{\prime})$ belong to
$\mathbb{R}^{q}\times\mathbb{R}^{d-q}$. In other words,

(4.3)
$\operatorname{Op}_{h}(b)=b\left(x,x^{\prime},hD_{x},hD_{x^{\prime}},\frac{\epsilon(h)}{h^{\frac{1}{2}}}x,\epsilon(h)h^{\frac{1}{2}}D_{x}\right).$

We first prove uniform boundedness over $L^{2}(\mathbb{R}^{d})$ and over the
space $L^{2}_{ul}$ defined in (4.4). Boundedness over $L^{2}_{ul}$ will allow
to define 2-microlocal measures for periodic functions over $\mathbb{R}^{d}$.

######  Proposition 4.2. –

Consider some symbol $b\in S^{0}$, then there exists some positive $h_{0}$
such that the following statements hold :

  1. (1)

$\operatorname{Op}_{h}(b)$ is a bounded operator over $L^{2}(\mathbb{R}^{d})$
uniformly wrt $h\in(0,h_{0}]$.

  2. (2)

We introduce a partition of unity $\chi\in C^{\infty}(\mathbb{R}^{d})$ such
that $\operatorname{supp}(\chi)\subset[-1,1]^{d}$, $0\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \chi\leavevmode\nobreak\ \leq\leavevmode\nobreak\ 1$
and $\sum_{p\in\mathbb{Z}^{d}}\chi_{p}=1$ where $\chi_{p}=\chi(\cdot-p)$. We
define the set of uniformly locally $L^{2}$ functions over $\mathbb{R}^{d}$ by

(4.4) $L^{2}_{ul}=\left\\{u\in
L^{2}_{loc},\quad\sup_{p\in\mathbb{Z}^{d}}\|\chi_{p}u\|_{L^{2}(\mathbb{R}^{d}}<+\infty\right\\}$

endowed with the norm
$\|u\|_{L^{2}_{ul}}=\sup_{p\in\mathbb{Z}^{d}}\|\chi_{p}u\|_{L^{2}(\mathbb{R}^{d})}$.
Then $\operatorname{Op}_{h}(b)$ is a bounded operator over
$L^{2}_{ul}(\mathbb{R}^{d})$ uniformly wrt $h\in(0,h_{0}]$.

###### Proof.

We first prove uniform boundedness over $L^{2}$. Setting
$\tilde{x}=\frac{\epsilon(h)}{h^{\frac{1}{2}}}x$,
$\tilde{y}=\frac{\epsilon(h)}{h^{\frac{1}{2}}}y$, and
$\tilde{\xi}=\epsilon(h)h^{\frac{1}{2}}\xi$ in (4.2), we obtain that

(4.5)
$\operatorname{Op}_{h}(b)=T_{h}^{*}b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\epsilon(h)h^{\frac{1}{2}}D_{x},D_{x^{\prime}},x,\epsilon^{2}(h)D_{x}\right)T_{h}$

where $T_{h}$ is the unitary operator defined by
$T_{h}u(x,x^{\prime})=\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}\right)^{\frac{q}{2}}u\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime}\right)$.
Since $\epsilon(h)\geq h^{\frac{1}{2}}$, the $h$-dependent symbol

$\check{b}(X,\Xi)=b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\epsilon(h)h^{\frac{1}{2}}\xi,h\xi^{\prime},x,\epsilon^{2}(h)\xi\right)$

has bounded derivatives uniformly wrt $h$ small enough. By the Calderón-
Vaillancourt theorem [Zwo12, Theorem 5.1], $\operatorname{Op}_{1}(\check{b})$
is bounded over $L^{2}$, which proves uniform $L^{2}$-boundedness for
$\operatorname{Op}_{h}(b)$ by unitary conjugation.

To generalize the result to $L^{2}_{ul}$, consider the symbol $\check{b}$
defined above and write, up to unitary conjugation :

$\chi_{r}\operatorname{Op}_{h}(b)u(X)=\chi_{r}(X)\sum_{p,q\in\mathbb{Z}^{d}}\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}\times\mathbb{R}^{d}}e^{i(X-Y).\Xi}\chi_{p}(X)\check{b}(X,\Xi)\chi_{q}(Y)u(Y)dYd\Xi.$

Our goal is to obtain some bound on
$\|\chi_{r}\operatorname{Op}_{h}(b)u\|_{L^{2}}$ that is uniform wrt
$h\in(0,h_{0}]$ and $r\in\mathbb{Z}^{d}$. If $|p-r|\geq 2$ then
$\operatorname{supp}(\chi_{p})\cap\operatorname{supp}(\chi_{r})$ has measure
zero, hence the the sum over $p$ is finite. Then, whenever $|p-q|\geq 2$, we
integrate by parts in the $\Xi$ variable using that $\check{b}$ is compactly
supported and that for any integer $N$

$\left(\frac{1-\Delta_{\Xi}}{1+|X-Y|^{2}}\right)^{N}\left(e^{i(X-Y).\Xi}\right)=e^{i(X-Y).\Xi}.$

Since $X$ is localized near $p$ and $Y$ is localized near $q$, when $|p-q|\geq
2$ there exists some $c>0$ such that

$\frac{1}{c}\frac{1}{1+|p-q|}\leq\frac{1}{1+|X-Y|}\leq\frac{c}{1+|p-q|}.$

The sum over $q$ is thus convergent for every fixed $p$ and $N$ large enough.
The Calderón-Vaillancourt theorem then allows to conclude, as

$\displaystyle\|\chi_{r}\operatorname{Op}_{h}(b)u\|$ $\displaystyle\leq
C\sum_{|p-r|\leq
1}\sum_{q\in\mathbb{Z}^{d}}\left\|\frac{\chi_{p}}{(1+|p-q|)^{N}}\operatorname{Op}_{h}((1-\Delta_{\Xi})^{N}b)\chi_{q}u\right\|_{L^{2}}$
$\displaystyle\leq C\sum_{|p-r|\leq 1}\left(\sum_{|p-q|\geq
2}\frac{1}{(1+|p-q|)^{N}}\|\chi_{q}u\|_{L^{2}}+\sum_{|p-q|<2}\|\chi_{q}u\|_{L^{2}}\right)$
$\displaystyle\leq
C\left(\sum_{q^{\prime}\in\mathbb{Z}^{d}}\frac{1}{(1+|q^{\prime}|)^{N}}\right)\|u\|_{L^{2}_{ul}}.$

∎

We now prove Gårding’s weak inequality with small parameter $\epsilon^{2}(h)$.
The result is the following :

######  Proposition 4.3. –

Let $b\in S^{0}$ be non-negative. Then, for any $\alpha>0$, there exist some
$h_{0}(\alpha)>0$ and $C(\alpha)>0$ such that for any $h\in(0,h_{0}]$, $u\in
L^{2}(\mathbb{R}^{d})$ :

(4.6)
$\begin{gathered}\operatorname{Re}(b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\epsilon(h)h^{\frac{1}{2}}D_{x},hD_{x^{\prime}},x,\epsilon^{2}(h)D_{x}\right)u,u)_{L^{2}}\geq-(\alpha+C\epsilon^{2}(h))\|u\|_{L^{2}}^{2}.\\\
\left|\operatorname{Im}(b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\epsilon(h)h^{\frac{1}{2}}D_{x},hD_{x^{\prime}},x,\epsilon^{2}(h)D_{x}\right)u,u)_{L^{2}}\right|\leq
C\epsilon^{2}(h)\|u\|_{L^{2}}^{2}.\end{gathered}$

By unitary conjugation, we conclude that the same inequality holds for the
operator

$b\left(x,x^{\prime},hD_{x},hD_{x^{\prime}},\frac{\epsilon(h)}{h^{\frac{1}{2}}}x,\epsilon(h)h^{\frac{1}{2}}D_{x}\right).$

###### Proof.

We first note that
$b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\epsilon(h)h^{\frac{1}{2}}D_{x},hD_{x^{\prime}},x,\epsilon^{2}(h)D_{x}\right)$
is the $\epsilon^{2}(h)$-quantization of the non-negative $h$-dependent symbol

(4.7)
$\tilde{b}_{h}(X,\Xi)=b\left(\frac{h^{\frac{1}{2}}}{\epsilon(h)}x,x^{\prime},\frac{h^{\frac{1}{2}}}{\epsilon(h)}\xi,\frac{h}{\epsilon^{2}(h)}\xi^{\prime},x,\xi\right).$

Fix some $\alpha>0$ and consider the symbol
$c_{h}=\sqrt{\tilde{b}_{h}+\alpha}$. Since
$h^{\frac{1}{2}}\epsilon^{-1}(h)\leq 1$, all the derivatives of
$\tilde{b}_{h}$ are bounded over $\mathbb{R}^{d}\times\mathbb{R}^{d}$
uniformly wrt $h$ small enough and derivatives of order $m$ in the $\Xi$
variable decay at rate $\langle\Xi\rangle^{-m}$. Thus, $c_{h}$ also has
bounded derivatives and a similar rate of decay. Since $c_{h}$ is real-valued,
we get that

$\operatorname{Op}_{\epsilon^{2}(h)}(c_{h})^{*}=\operatorname{Op}_{\epsilon^{2}(h)}(c_{h})+\mathcal{O}_{\mathcal{L}(L^{2})}\left(\epsilon^{2}(h)\right).$

We then apply the usual composition formulas (see for example [Zwo12, Theorem
4.14] which generalizes to our framework) to get

$(\operatorname{Op}_{\epsilon^{2}(h)}(\tilde{b}_{h})u,u)_{L^{2}}+\alpha\|u\|_{L^{2}}^{2}=\|\operatorname{Op}_{\epsilon^{2}(h)}(c_{h})u\|_{L^{2}}^{2}+\mathcal{O}(\epsilon^{2}(h))\|u\|_{L^{2}}^{2}.$

Taking the real and imaginary parts then gives the desired inequalities (4.6)
for $h_{0}(\alpha)$ small enough and some constant $C(h_{0},\alpha)$. ∎

###  4.2. The model case : 4 damped prisms in a tunnel

Using the 2-microlocal calculus for symbols in $S^{0}_{H}$, we turn to the
study of the concentration properties of a sequence $(u_{n})$ of quasimodes
over the three-dimensional torus $\mathbb{T}^{3}$, under a specific choice of
damping function $a$. We will then generalize this model case to higher
dimensions and reduce the general case to it.

We identify $\mathbb{T}^{3}$ with $[-1,1]^{3}$ where opposite faces are
identified. The damping zone $\\{a\leavevmode\nobreak\ =\leavevmode\nobreak\
1\\}$ is represented in Figure 4.1. More explicitly, denoting
$X\leavevmode\nobreak\ =\leavevmode\nobreak\ (x_{1},x_{2},y)$ and
$\Xi\leavevmode\nobreak\ =\leavevmode\nobreak\ (\xi_{1},\xi_{2},\eta)$, $a$ is
the characteristic function of the set

(4.8) $\displaystyle\\{|x_{1}|\geq\frac{1}{2}\text{ or
}|x_{2}|\geq\frac{1}{2}\\}$
$\displaystyle\cup\\{-1<y<-\frac{1}{2},x_{1}>0,x_{2}<\alpha_{R}x_{1}\\}\cup\\{-\frac{1}{2}<y<0,x_{2}>0,x_{1}>-\alpha_{T}x_{2}\\}$
$\displaystyle\cup\\{0<y<\frac{1}{2},x_{1}<0,x_{2}>\alpha_{L}x_{1}\\}\cup\\{\frac{1}{2}<y<1,x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}.$

The first set denotes a neighborhood of the lateral faces of the cube as
represented in the central picture of Figure 4.1 and the four other sets
correspond to the prisms represented on the right, from the back to the front.
$\alpha_{L},\alpha_{R},\alpha_{T},\alpha_{B}>0$ are four fixed positive
coefficients which should be thought of as small. The case
$\alpha_{L}=\alpha_{R}=\alpha_{T}=\alpha_{B}=0$ (when the triangle-shaped
damped sections vanish and the four prisms are reduced to cubes) is dealt with
in section 5.

In this situation, the only two bicharacteristics that do not intersect the
interior of $\operatorname{supp}(a)$ are

$\\{x_{1}=x_{2}=0,\xi_{1}=\xi_{2}=0,\eta=\pm 1\\},$

which are represented in red in the figure (as one geodesic, traveled in two
directions). Thus, the first microlocal measure $\nu$ is supported in the
union of these two bicharacteristics. The corresponding geodesic satisfies
assumption (1.3) if all $\alpha_{L,R,T,B}$ coefficients are positive but
violates it if any of the coefficients vanishes.

$x_{3}=y$$x_{2}$$x_{1}$

Figure 4.1. Illustration of set (4.8). The coordinate system is represented on
the left. The dual coordinates are denoted $(\xi_{1},\xi_{2},\xi_{3}=\eta)$.
We also denote $x=(x_{1},x_{2})$, $\xi=(\xi_{1},\xi_{2})$. In the center is
the torus $\mathbb{T}^{3}$ identified with the unit cube $[-1,1]^{3}$. We
color in blue the zone
$\\{|x_{1}|\geq\frac{1}{2}\\}\cup\\{|x_{2}|\geq\frac{1}{2}\\}$ where $a=1$,
thus damping the lateral faces. This leaves an undamped tunnel at the center
of the cube. The picture on the right fits inside this tunnel but is
represented separately for better legibility. On the right, we represent in
blue the set where $a=1$ inside
$\\{|x_{1}|\leq\frac{1}{2},|x_{2}|\leq\frac{1}{2}\\}$. The red geodesic
$\\{x=0\\}$ is the only one which does not encounter the interior of
$\operatorname{supp}(a)$.

We now perform the second microlocalization around this geodesic. Consider a
sequence $(u_{n})$ of functions over $\mathbb{T}^{3}$ satisfying

(4.9) $(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}(1)_{L^{2}}.$

We identify $u_{n}\in L^{2}(\mathbb{T}^{3})$ with its periodic extension over
$\mathbb{R}^{3}$. In particular, the periodic extension of $u_{n}$ belongs to
$L^{2}_{ul}$ as defined in (4.4). Using the $L^{2}_{ul}$-boundedness of
operators defined by (4.2) and Gårding’s inequality, we can extract a
subsequence (also denoted $(u_{n})$) such that there exists a positive measure
$\tilde{\mu}$ on $T^{*}\mathbb{T}^{2}\times\overline{N}$ satisfying for any
polyhomogeneous symbol $b\in S^{0}_{H},$

$\lim_{n\rightarrow+\infty}(\operatorname{Op}_{h_{n}}(b)u_{n},u_{n})_{L^{2}}=\langle\tilde{\mu},\tilde{b}\rangle.$

Here, $\overline{N}$ denotes the sphere compactification of
$\mathbb{R}^{4}_{z,\zeta}$ and $\tilde{b}$ is a continuous function on
$T^{*}\mathbb{R}^{3}\times\overline{N}$ which is defined by the value of $b$
in the interior and by

$\tilde{b}(x,y,\xi,\eta,\tilde{z},\tilde{\zeta})=\lim_{r\rightarrow+\infty}b(x,y,\xi,\eta,r\tilde{z},r\tilde{\zeta})$

on the sphere at infinity. Since the $u_{n}$ are periodic, the measure
$\tilde{\mu}$ is also periodic i, the $X$ variable, hence it naturally defines
a measure $\mu$ on $T^{*}\mathbb{T}^{3}\times\overline{N}$.

The rest of the subsection is dedicated to studying such a 2-microlocal
measure $\mu$ for the sequence $(u_{n})$ introduced in Section 2. Recall that
$(u_{n})$ satisfies $\|u_{n}\|_{L^{2}(\mathbb{T}^{d})}=1$ and the asymptotics
:

$(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}(h_{n}\epsilon(h_{n}))_{L^{2}}\quad;\quad
au_{n}=o(1)_{L^{2}}.$

The properties of the second microlocal measure $\mu$ are gathered in the
following statement :

######  Proposition 4.4. –

  1. (1)

Assume only that

$(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}(1)_{L^{2}},$

then $\mu(T^{*}\mathbb{T}^{3}\times\overline{N})=1$.

  2. (2)

The first microlocal measure $\nu$ is the projection of $\mu$ onto the
$(x,y,\xi,\eta)$ variables. Assume besides that

$(h_{n}^{2}\Delta+1)u_{n}=o(h_{n})_{L^{2}},\quad au_{n}=o(1)_{L^{2}},$

then $\mu$ is supported 1-microlocally in the red bicharacteristics of Figure
4.1,

$\\{x=0,\quad\xi=0,\quad\eta=\pm 1\\}.$

  3. (3)

Assume now that

$(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}(h_{n}\epsilon(h_{n}))_{L^{2}},$

then $\mu$ is supported in the sphere at infinity in the $(z,\zeta)$
variables.

  4. (4)

Let $R$ be one of the four prisms

$\\{-1<y<-\frac{1}{2},x_{1}>0,x_{2}<\alpha_{R}x_{1}\\},\\{-\frac{1}{2}<y<0,x_{2}>0,x_{1}>-\alpha_{T}x_{2}\\},\\\
\\{0<y<\frac{1}{2},x_{1}<0,x_{2}>\alpha_{L}x_{1}\\},\\{\frac{1}{2}<y<1,x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}$

(see Figure 4.1), then at every point of $\partial R$ $\mu$ vanishes
2-microlocally in the direction of the polyhedron $R$. Precisely (from back to
front along the red geodesic of Figure 4.1) :

(4.10)
$\begin{gathered}\mu(\\{x=0,\quad\xi=0,\quad-1<y<-\frac{1}{2},\quad\eta=\pm
1,\quad z_{1}>0,\quad z_{2}<\alpha_{R}z_{1}\\})=0,\\\
\mu(\\{x=0,\quad\xi=0,\quad-\frac{1}{2}<y<0,\quad\eta=\pm 1,\quad
z_{2}>0,\quad z_{1}>-\alpha_{T}z_{2}\\})=0,\\\ \mu(\\{x=0,\quad\xi=0,\quad
0<y<\frac{1}{2},\quad\eta=\pm 1,\quad z_{1}<0,\quad
z_{2}>\alpha_{L}z_{1}\\})=0,\\\
\mu(\\{x=0,\quad\xi=0,\quad\frac{1}{2}<y<1,\quad\eta=\pm 1,\quad z_{2}<0,\quad
z_{1}<-\alpha_{B}z_{2}\\})=0.\end{gathered}$

  5. (5)

The measure $\mu$ satisfies the conservation law :

(4.11) $(\eta\partial_{y}+\zeta.\partial_{z})\mu=0$

where $\zeta.\partial_{z}$ is a vector field on the sphere at infinity of
$\mathbb{R}^{4}_{z,\zeta}$.

###### Proof.

The first point is a direct consequence of the $h_{n}$-oscillation property
proved in section 2.

To prove the second statement, consider a symbol $b(X,\Xi)$ which does not
depend on the 2-microlocal variables $z,\zeta$, then
$\langle\nu,b\rangle=\langle\mu,b\circ\pi\rangle$ by definition of both
measures. Said otherwise, $\nu\leavevmode\nobreak\ =\leavevmode\nobreak\
\pi_{*}\mu$ where

$\pi:(X,\Xi,z,\zeta)\mapsto(X,\Xi).$

The rest of the second point comes from Corollary 2.2.

To prove the third point, we take some function $\chi\in
C^{\infty}_{c}(T^{*}\mathbb{T}^{3}\times\mathbb{R}^{4}_{z,\zeta})$. $\chi$
belongs to $S^{0}_{H}$ and there exists some positive $A$ such that
$\operatorname{supp}(\chi)\subset\\{|z|\leq A\\}$. Thus,

(4.12) $\displaystyle|(\operatorname{Op}_{h_{n}}(\chi)u_{n},u_{n})_{L^{2}}|$
$\displaystyle=|(\operatorname{Op}_{h_{n}}(\chi)u_{n},\mathbbm{1}_{\\{|x|\leq
Ah_{n}^{\frac{1}{2}}\epsilon^{-1}(h_{n})\\}}u_{n})_{L^{2}}|$
$\displaystyle\leq C\|u_{n}\|_{L^{2}}\|u_{n}\|_{L^{2}(\\{|x|\leq
h_{n}^{\frac{1}{2}}\epsilon^{-2}(h_{n})\\})}$ $\displaystyle\leq
C\|u_{n}\|_{L^{2}(\\{|x_{1}|\leq h_{n}^{\frac{1}{2}}\epsilon^{-2}(h_{n})\\})}$
$\displaystyle\leq
C\epsilon^{\frac{1}{2}}(h_{n})\rightarrow_{n\rightarrow+\infty}0.$

where we have applied the $L^{2}$-boundedness of
$\operatorname{Op}_{h_{n}}(\chi)$ and Proposition 3.1. Thus,

$\langle\mu,\chi\rangle=0,$

which proves that $\mu$ is supported in the sphere at infinity in the
$(z,\zeta)$ variables. Note that it is enough for the non-concentration
estimate to hold in the cylinder $\\{|x|\leq
h_{n}^{\frac{1}{2}}\epsilon^{-2}(h_{n})\\}$ centered around the geodesic
rather than the slice $\\{|x_{1}|\leq
h_{n}^{\frac{1}{2}}\epsilon^{-2}(h_{n})\\}$. This fact will be used in the
general case at the end of the next subsection.

We now come the microlocal vanishings. We show that $\mu$ vanishes over

$\\{x=0,\xi=0,\frac{1}{2}<y<1,\eta=+1,z_{2}<0,z_{1}<-\alpha_{B}z_{2}\\}.$

Fix some small enough $\delta_{0}>0$ and define the following cutoffs :

  * –

$\psi\in C^{\infty}(\mathbb{R})$, equal to 0 over $(-\infty,1]$ and 1 over
$[2,+\infty)$,

  * –

$\chi\in C^{\infty}_{c}(-1,1)$ equal to 1 over $(-\frac{1}{2},\frac{1}{2})$,
and

  * –

$\tilde{\chi}\in C^{\infty}_{c}(\frac{1}{2},1)$ equal to 1 over
$(\frac{1}{2}+\delta_{0},1-\delta_{0})$.

Consider the symbol

(4.13) $b(x,y,\xi,\eta,z,\zeta)=\\\
\chi\left(\frac{2|x|}{\delta_{0}}\right)\tilde{\chi}(y)\chi(|\xi|)\chi(\eta-1)\psi\left(-\frac{z_{2}}{\delta_{0}|\zeta|}\right)\psi\left(\frac{-z_{1}-\alpha_{B}z_{2}}{\delta_{0}|\zeta|}\right)\psi\left(\frac{|z|^{2}+|\zeta|^{2}}{\delta_{0}}\right).$

Cutoff $\chi\left(\frac{2|x|}{\delta_{0}}\right)\tilde{\chi}(y)$ is supported
in $B(0,\frac{\delta_{0}}{2})_{x}\times(\frac{1}{2},1)_{y}$ and cutoff
$\psi\left(-\frac{z_{2}}{\delta_{0}|\zeta|}\right)\psi\left(\frac{-z_{1}-\alpha_{B}z_{2}}{\delta_{0}|\zeta|}\right)$
localizes in the $\\{x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}$ zone, thus

$\mathbbm{1}_{\\{x\in
B(0,\frac{\delta_{0}}{2}),x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}}\mathbbm{1}_{\\{y\in(\frac{1}{2},1)\\}}\operatorname{Op}_{h_{n}}(b)=\operatorname{Op}_{h_{n}}(b),$

wherefrom

(4.14)
$\displaystyle|\langle\operatorname{Op}_{h_{n}}(b)u_{n},u_{n}\rangle_{L^{2}}|$
$\displaystyle=|\langle\operatorname{Op}_{h_{n}}(b)u_{n},\mathbbm{1}_{\\{x\in
B(0,\frac{\delta_{0}}{2}),x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}}\mathbbm{1}_{\\{y\in(\frac{1}{2},1)\\}}u_{n}\rangle_{L^{2}}|$
$\displaystyle=|\langle\operatorname{Op}_{h_{n}}(b)u_{n},\mathbbm{1}_{\\{x\in
B(0,\frac{\delta_{0}}{2}),x_{2}<0,x_{1}<-\alpha_{B}x_{2}\\}}\mathbbm{1}_{\\{y\in(\frac{1}{2},1)\\}}au_{n}\rangle_{L^{2}}|$
$\displaystyle\leq
C\|u_{n}\|_{L^{2}}\|au_{n}\|_{L^{2}}\rightarrow_{n\rightarrow+\infty}0.$

Thus $\langle\mu,b\rangle=0$. $\mu$ being a positive measure, this implies
$\mu(\\{b=1\\})=0$ and

$\begin{gathered}\mu\left(\left\\{x=0,\xi=0,y\in\left[\frac{1}{2}+\delta_{0},1-\delta_{0}\right],\eta=1,z_{2}<-2\delta_{0}|\zeta|,-(z_{1}+\alpha_{B}z_{2})>2\delta_{0}|\zeta|\right\\}\right)=0.\end{gathered}$

Taking the limit $\delta_{0}\rightarrow 0$ then yields the result.

The other 2-microlocal vanishings are obtained by changing the cutoffs in the
$y$, $\eta$ and $\frac{z}{|\zeta|}$ variables in (4.13).

To prove the last statement, we compute the bracket
$[h_{n}^{2}\Delta+1,\operatorname{Op}_{h_{n}}(q)]$ for some $q\in S^{0}_{H}$
using the second quantization formula (4.2). We get

$[h\Delta,\operatorname{Op}_{h}(q)]=\operatorname{Op}_{h}\left((2i\Xi.\partial_{X}+2i\zeta.\partial_{z}+h\Delta_{X}+2h\epsilon(h)h^{-\frac{1}{2}}(\partial_{x}.\partial_{z})+h(\epsilon(h)h^{-\frac{1}{2}})\Delta_{z})q\right)$

so that

(4.15)
$\frac{1}{2ih_{n}}[h_{n}^{2}\Delta+1,\operatorname{Op}_{h_{n}}(q)]=\frac{1}{2i}[h_{n}\Delta,\operatorname{Op}_{h_{n}}(q)]\\\
=\operatorname{Op}_{h_{n}}((\xi.\partial_{x}+\eta\partial_{y}+\zeta.\partial_{z})q)-i\frac{h_{n}}{2}\operatorname{Op}_{h_{n}}(\Delta_{X}q)-i\frac{h_{n}}{2}(\epsilon(h_{n})h_{n}^{-\frac{1}{2}})\operatorname{Op}_{h_{n}}((\partial_{x}.\partial_{z})q)\\\
-i\frac{h_{n}}{2}(\epsilon(h_{n})h_{n}^{-\frac{1}{2}})^{2}\operatorname{Op}_{h_{n}}(\partial_{z}^{2}q).$

Unfolding the bracket and using that $(h_{n}^{2}\Delta+1)u_{n}=o(h_{n})$, we
obtain

$\frac{1}{2ih_{n}}([h_{n}^{2}\Delta+1,\operatorname{Op}_{h_{n}}(q)]u_{n},u_{n})_{L^{2}}\rightarrow_{n\rightarrow+\infty}0,$

so

(4.16)
$(\operatorname{Op}_{h_{n}}((\xi.\partial_{x}+\eta\partial_{y}+\zeta.\partial_{z})q)u_{n},u_{n})_{L^{2}}=o(1).$

Since the measure $\mu$ is supported in $\\{\xi=0\\}$, conservation law (4.11)
follows. ∎

We can then conclude the contradiction argument by studying the dynamics of
the vector field $\zeta.\partial_{z}$ over the sphere at infinity. The
corresponding result is illustrated in Figure 4.3 of subsection 4.4 and summed
up in the following lemma. We postpone the proof of the lemma to that
subsection.

######  Lemma 4.5. –

Consider an initial point $(z_{0},\zeta_{0})$,
$|z_{0}|^{2}+|\zeta_{0}|^{2}=1$, in the sphere at infinity
$\mathbb{S}^{2d-3}_{\infty}$ of $\mathbb{R}^{2d-2}$, and denote
$\phi^{s}(z_{0},\zeta_{0})$ the flow of $\zeta.\partial_{z}$ at time $s$
starting from this initial point, then

(4.17)
$\phi^{s}(z_{0},\zeta_{0})=\frac{1}{\left(|z_{0}+s\zeta_{0}|^{2}+|\zeta_{0}|^{2}\right)^{\frac{1}{2}}}\left(z_{0}+s\zeta_{0},\zeta_{0}\right).$

In other words, the flow of of $\zeta.\partial_{z}$ over the sphere at
infinity is the projection of the flow at finite distance from the origin, as
depicted in Figure 4.3.

In particular, if $\zeta_{0}\neq 0$ then

(4.18)
$\phi^{s}(z_{0},\zeta_{0})\rightarrow_{s\rightarrow+\infty}\left(\frac{\zeta_{0}}{|\zeta_{0}|},0\right),$

and $(z_{0},\zeta_{0})$ is a fixed point of the flow iff $\zeta_{0}=0$.

Now, by point (1) of Proposition 4.4, $\mu$ has non-empty support. Consider a
point

$(x=0,y,\xi=0,\eta=\pm 1,z,\zeta)\in\operatorname{supp}(\mu),$

with $(z,\zeta)$ belonging to the sphere at infinity of $\mathbb{R}^{4}$. By
the conservation law (4.11), the point

$(x=0,y+s\eta,\xi=0,\eta=\pm 1,\phi^{s}(z,\zeta))$

also belongs to $\operatorname{supp}(\mu)$. The flow $\phi_{s}(z,\zeta)$
converges to some $(z_{\infty},0)$ with $z_{\infty}\neq 0$, and by the fourth
point $(z_{\infty},0)$ belongs to at least one open set of the sphere at
infinity where $\mu$ vanishes 2-microlocally along the geodesic. Thus, $\mu$
has support in an open set where it vanishes, which gives a contradiction.
This concludes the proof of stabilization for the model damping in Figure 4.1.

####  4.2.1. Generalization to higher dimensions

Since both the non-concentration estimate of Proposition 3.1 and Lemma 4.5
hold for any $d\geq 3$, this model case can be generalized to higher
dimensions. Assume that $d\geq 3$. We consider a damping $a$ over the
$d-$dimensional torus $\mathbb{T}^{d}\simeq[-1,1]^{d}$ such that :

  * –

$a=1$ outside of a tunnel, namely in the zone
$\cup_{i=1}^{d-1}\\{|x_{i}|\geq\frac{1}{2}\\}$. By Corollary 2.2, the first
microlocal measure is then supported in the one-directional set of
bicharacteristics $\\{\xi_{i}=0,1\leq i\leq d-1,\xi_{d}=\pm 1\\}.$

  * –

Inside that tunnel, $a=1$ over a finite union of polyhedrons such that the
geodesic $\\{x_{1}=\dots=x_{d-1}=0\\}$ satisfies Assumption (1.3) and is the
only geodesic not entering $\operatorname{Int}(\operatorname{supp}(a))$. For
example, one can choose $a$ to be equal to 1 over the set

$\bigcup_{i=1}^{d-1}\left\\{x_{i}<0,\quad-1+\frac{i-1}{d-1}<y<-1+\frac{i}{d-1}\right\\}\\\
\cup\bigcup_{i=1}^{d-1}\left\\{x_{i}>0,\quad\frac{i-1}{d-1}<y<\frac{i}{d-1}\right\\}.$

We then perform the same second quantization, as defined per formula (4.2),
with $z=\frac{\epsilon(h)}{h^{\frac{1}{2}}}(x_{1},\dots,x_{d-1})$ and
$\zeta=\epsilon(h)h^{\frac{1}{2}}(\xi_{1},\dots,\xi_{d-1})$. By the
2-microlocal calculus on
$\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d-1}\times\mathbb{R}^{d-1}$,
the sequence $(u_{n})$ still admits a 2-microlocal measure $\mu$. The first
two points of Proposition 4.4 are then proved exactly as previously. The proof
of the third point is identical since the non-concentration estimates of
Proposition 3.1 hold for $d-$dimensional tori. The proof of the microlocal
vanishings follow the same lines, and equations (4.15) and (4.16) still hold
so that the fifth point is unchanged.

We then conclude in the same manner since Lemma 4.5 is satisfied on any sphere
of dimension $2d-3$.

###  4.3. Reduction of the general case and conclusion

We now reduce the general case to the preceding model case. Consider a damping
$a$ on the $d$-dimensional torus $\mathbb{T}^{d}=\mathbb{R}^{d}/\Gamma$ with
$\Gamma$ an orthogonal lattice $A_{1}\mathbb{Z}\times\dots\times
A_{d}\mathbb{Z}^{d}$, $A_{1},\dots,A_{d}>0$. Assume that $a$ is a finite sum
of characteristic functions of disjoint polyhedrons in $\mathbb{T}^{d}$, and
that all geodesics of the torus which do not enter the interior of the damped
region satisfy assumption (1.3).

First we deal with non-closed geodesics. Then, we use an argument from [BZ12,
Section 3] to microlocalize near a single geodesic, the direction of which is
isolated in $\operatorname{supp}(\nu)$. We conclude by using orthonormal
changes of coordinates to reduce this geodesic to the one of the model case.

######  Proposition 4.6. –

Non-closed geodesics which are damped in every normal direction (_ie_
satisfying assumption (1.3)) intersect the interior of a polyhedron where the
damping equals 1.

F$\gamma_{\rho_{0}}$$\Xi_{0}^{\perp}$polyhedron where $a=1$ Figure 4.2. An
illustration of Proposition 4.6. The non-closed geodesic $\gamma_{\rho_{0}}$
is damped by some polyhedron in the direction $\Xi_{0}^{\perp}\in F$. By
density in $F$, it enters that polyhedron. The dashed violet edges and paler-
shade part of the polyhedron are below the plane $F$. The dashed blue and red
lines are inside the polyhedron.

###### Proof.

Consider a non-closed geodesic $\gamma_{\rho_{0}}$ of $\mathbb{T}^{d}$,
$\rho_{0}=(X_{0},\Xi_{0})$. $\gamma_{\rho_{0}}$ is then dense in some
$(X_{0}+F)/\Gamma$ where $\rho_{0}=(X_{0},\Xi_{0})$ and $F$ is a subspace of
$\mathbb{R}^{d}$ of dimension strictly greater than one.

The rest of the proof is illustrated in Figure 4.2. Denote $\Xi_{0}\in F$ the
direction of the geodesic and $\Xi_{0}^{\perp}$ some unitary vector in $F$,
orthogonal to $\Xi_{0}$. By (1.3), there exists an interval
$I\subset\mathbb{R}$ and some positive $\delta_{0}$ such that $a=1$ in an open
neighborhood of $\left\\{\gamma(s)+\delta\Xi_{0}^{\perp},s\in
I,\delta\in\left[\frac{\delta_{0}}{2},\delta_{0}\right]\right\\}\subset
X_{0}+F.$ Denote by $V$ such an open neighborhood, then
$V\subset\operatorname{Int}(\operatorname{supp}(a))$ and by density of the
geodesic, $\gamma_{\rho_{0}}$ enters $V$. ∎

The support of $\nu$ thus contains only points $(X,\Xi)$ belonging to closed
geodesics, with rational $\Xi\in\mathbb{S}^{d-1}$.

Defining $\pi_{\Xi}:(X,\Xi)\mapsto\Xi$, we get that
$\pi_{\Xi}(\operatorname{supp}(\nu))$ is countable and closed (using
compactness of the torus for the latter). By Baire’s lemma, any closed subset
with no isolated points (_aka_ a perfect set) in a complete metric space is
uncountable. Thus, $\pi_{\Xi}(\operatorname{supp}(\nu))$ contains an isolated
direction, which we denote $\Xi_{0}$.

Let us microlocalize around this isolated direction. Consider a closed
geodesic $\gamma$ in the support of $\nu$, with direction $\Xi_{0}$. Consider
a neighborhood of $\Xi_{0}$ which contains no other directions in
$\pi(\operatorname{supp}(\nu))$, and $\chi(hD_{X})$ a Fourier multiplier with
a symbol supported in this neighborhood in the $\Xi$ variable. Define

$v_{n}=\chi(hD_{X})u_{n}$

and denote $\nu_{v}$ the corresponding 1-microlocal measure. $\nu_{v}$
satisfies

(4.19)
$\nu_{v}=\chi(\xi)^{2}\nu,\quad\nu_{v}(T^{*}\mathbb{T}^{d})\geq\nu(\gamma)>0,\quad(h_{n}^{2}\Delta+1)v_{n}=o(h_{n}),$

since $h_{n}^{2}\Delta$ and $\chi(hD_{x})$ commute. We have thus
microlocalized around a single geodesic of $\operatorname{supp}(\nu)$ with an
isolated direction.

Since the geodesic is closed, there exist some integers
$n_{1},\dots,n_{d}\in\mathbb{Z}$ such that

$\Xi_{0}=\frac{(n_{1}A_{1},\dots,n_{d}A_{d})}{\left(\sum_{j=1}^{d}n_{j}^{2}A_{j}^{2}\right)^{\frac{1}{2}}}.$

The next lemma explains the induction step which allows to reduce the geodesic
to one living in a $(d-1)$-dimensional torus. It is an adaptation of [BZ12,
Lemma 2.2], to the $d$-dimensional setting (see also [BG20, Lemma 4.1]).

######  Lemma 4.7. –

Assume that $n_{d}$ is non-zero. Consider $u_{n}$ a $\Gamma$-periodic function
and denote $\Xi_{j}$ the $j$-th vector of the canonical basis of
$\mathbb{R}^{d}$ for every $j\in\\{1,\dots,d-2\\}$. Denote

(4.20)
$\Xi_{d-1}=\frac{(0,\dots,0,n_{d-1}A_{d-1},n_{d}A_{d})}{S_{d-1}},\Xi_{d}=\frac{(0,\dots,0,-n_{d}A_{d},n_{d-1}A_{d-1})}{S_{d-1}},\\\
S_{d-1}=\left(n_{d-1}^{2}A_{d-1}^{2}+n_{d}^{2}A_{d}^{2}\right)^{\frac{1}{2}},$

so that $(\Xi_{j})_{j=1,\dots,d}$ is a direct orthonormal basis and
$\Xi_{0}=\sum_{j=1}^{d-2}n_{j}A_{j}\Xi_{j}+S_{d-1}\Xi_{d-1}$. Set

$F_{d-1}(x)=\sum_{j=1}^{d}x_{j}\Xi_{j}.$

Then

(4.21) $u\circ
F_{d-1}(x_{1}+k_{1}A_{1},\dots,x_{d-2}+k_{d-2}A_{d-2},x_{d-1}+k_{d-1}S_{d-1},x_{d}+k_{d}\alpha)=\\\
u\circ F_{d-1}(x_{1},\dots,x_{d-2},x_{d-1}-k_{d}\beta,x_{d}),\forall
k\in\mathbb{Z}^{d},x\in\mathbb{R}^{d},$

where, for any fixed $p,q\in\mathbb{Z}$,

$\alpha=\frac{qn_{d-1}A_{d-1}A_{d}-pn_{d}A_{d}A_{d-1}}{S_{d-1}},\quad\beta=\frac{pn_{d-1}A_{d-1}^{2}+qn_{d}A_{d}^{2}}{S_{d-1}},\quad\alpha\neq
0.$

###### Proof.

Since $F$ preserves the first $d-2$ vectors of the canonical basis and $u$ is
$\Gamma$-periodic, it is sufficient to find some $\alpha,\beta$ such that for
any $k_{d-1},k_{d}\in\mathbb{Z}$ there exist $p,q\in\mathbb{Z}$ satisfying

(4.22)
$k_{d-1}S_{d-1}\Xi_{d-1}+k_{d}\alpha\Xi_{d}+k_{d}\beta\Xi_{d-1}=\begin{pmatrix}0\\\
\vdots\\\ 0\\\ pA_{d-1}\\\ qA_{d}\end{pmatrix}$

The $S_{d-1}$ coefficient in the first term is chosen such that it is
sufficient to deal with the case $k_{d-1}=0$, $k_{d}=1$. The last two lines of
(4.22) then give an invertible $2\times 2$ linear system which we solve by
taking the scalar product with $\Xi_{d-1}$ and $\Xi_{d}$. ∎

We thereby get an orthonormal basis $(\Xi_{1},\dots,\Xi_{d})$ and a
corresponding coordinate system $(x_{1},\dots,x_{d})$ such that $u$ is
$\left(A_{1}\mathbb{Z}\times\dots\times A_{d-2}\mathbb{Z}\times
S_{d-1}\mathbb{Z}\right)$-periodic wrt to the $d-1$ first variables, and the
geodesic is contained in a $(d-1)$-dimensional torus. Note that if the
coefficient $n_{d}$ is zero, the latter already occurs, so the assumption in
the lemma is not restrictive.

Since the non-concentration estimate requires periodicity of $u_{n}$ in $d-1$
orthogonal directions, we can apply it to $u\circ F_{d-1}$. We get

$\|u_{n}\circ F_{d-1}\|_{L^{2}(\\{|x_{d}|\leq
h_{n}\epsilon(h_{n})^{-2}\\})}\leq C\epsilon(h_{n})^{\frac{1}{2}}.$

Thus

(4.23) $\|u_{n}\circ
F_{d-1}\|_{L^{2}(F_{d-1}^{-1}(\gamma+B(0,h_{n}\epsilon(h_{n})^{-2})))}\leq
C\epsilon(h_{n})^{\frac{1}{2}},$

as a cylinder surrounding the closed geodesic is contained in any
$(d-1)$-dimensional slice of identical width.

We now iterate the construction of Lemma 4.7 and show that it preserves the
non-concentration estimate. We use the periodicity of $u_{n}$ with respect to
the $(d-1)$ first coordinates to apply the lemma in successive steps : we get
$(d-2)$ orthonormal changes of coordinates $F_{d-2},\dots,F_{1}$ (possibly
equal to identity if no change of coordinates is necessary) such that for
every $j\in\\{1,\dots,d-1\\}$ :

  * –

each $F_{j}$ preserves the first $(j-1)$ vectors of the canonical basis,

  * –

$u_{n}\circ F_{d-1}\circ\ldots\circ F_{j}$ is periodic wrt its $j$ first
variables,

  * –

and the image $F_{j}^{-1}\circ\ldots\circ F_{d-1}^{-1}(\gamma)$ of the
geodesic is contained in a torus of dimension $j$.

Denote $F=F_{d-1}\circ\ldots\circ F_{1}$. Since every change of variables is
isometric, (4.23) becomes

(4.24) $\|u_{n}\circ
F\|_{L^{2}(F^{-1}(\gamma+B(0,h_{n}\epsilon^{-2}(h_{n}))))}\leq
C\epsilon(h_{n})^{\frac{1}{2}}.$

Since
$F^{-1}(\gamma+B(0,h_{n}\epsilon^{-2}(h_{n})))=F^{-1}(\gamma)+B(0,h_{n}\epsilon^{-2}(h_{n}))$,
the estimate (4.24) is exactly the one that allows to show that the
2-microlocal measure is supported at infinity (see the proof of Proposition
4.4, point 3.). Besides, $F^{-1}(\gamma)$ is a closed geodesic depending on
only one cartesian coordinate and satisfying assumption (1.3), so that we are
microlocally reduced to the case of subsection 4.2. The rest of the
contradiction argument follows the same lines as previously, which completes
the proof of Theorem 4.

###  4.4. $\zeta.\partial_{z}$ vector field over the sphere at infinity

There remains to study the dynamics of vector field $\zeta.\partial_{z}$ over
the sphere at infinity $\mathbb{S}^{2d-3}_{\infty}$ of
$\mathbb{R}^{d-1}_{z}\times\mathbb{R}^{d-1}_{\zeta}$, $d\geq 3$, in order to
prove Lemma 4.5. The proof relies on symplectic transformations which simplify
the expression of the dynamics, and on passing to spherical coordinates (as is
done on $\mathbb{S}^{1}$ in [BG20, Proposition 3.5, 5)]).

First, we reduce the dynamics over $\mathbb{S}^{2d-3}_{\infty}$ to dynamics
over $\mathbb{S}^{3}_{\infty}$ where only one angle coordinate is non-
constant. We then show that the dynamics over $\mathbb{S}^{2d-3}_{\infty}$ can
be obtained by projecting onto the sphere at infinity the dynamics of
$\zeta.\partial_{z}$ at finite distance from the origin (a straight line
traveled at constant velocity $|\zeta(0)|$).

Denote $(z^{(0)},\zeta^{(0)})$ an initial point belonging to
$\mathbb{S}^{2d-3}_{\infty}$. Since $\mathbb{S}^{2d-3}$ is a compact manifold,
the flow of $\zeta.\partial_{z}$ is defined for all times. We denote
$(z(s),\zeta(s))$ the image of $(z^{(0)},\zeta^{(0)})$ by the flow of
$\zeta.\partial_{z}$ at time $s$. The first result is the following :

######  Proposition 4.8. –

  1. (1)

There exists an angle coordinate system $(\theta_{1},\dots,\theta_{2d-3})$ on
the sphere at infinity in which the dynamics of $\zeta.\partial_{z}$ satisfy :

(4.25)
$\begin{gathered}\dot{\theta_{1}}=-\cos(\theta_{2})\sin^{2}(\theta_{1})\quad;\quad\theta_{2}=\theta_{2}^{(0)}\quad;\quad\theta_{3}=\text{
cst }=\pm 1\quad;\quad\theta_{i}=0,\quad\forall i\geq 4.\end{gathered}$

In other words, in this coordinate system we have
$\zeta.\partial_{z}=-\cos(\theta_{2})\sin^{2}(\theta_{1})\partial_{\theta_{1}}$
over the sphere at infinity.

  2. (2)

$\theta_{1}(s)\rightarrow 0\mod\pi$ as $s$ goes to infinity, so that
$(z(s),\zeta(s))\rightarrow\left(\pm\frac{\zeta^{(0)}}{|\zeta^{(0)}|}\right)$
if $\zeta^{(0)}\neq 0$. The $(z^{(0)},\zeta^{(0)})$ such that $\zeta^{(0)}=0$
are exactly the fixed points of $\zeta.\partial_{z}$.

###### Proof.

If $\zeta^{(0)}=0$, $(z^{(0)},\zeta^{(0)})$ is a fixed point of the dynamic
since we are solving a first order initial value problem with zero initial
derivative. Thus, we assume $\zeta^{(0)}\neq 0$. We introduce a cartesian
coordinate system $(\tilde{z},\tilde{\zeta})$ such that
$\zeta.\partial_{z}=\tilde{\zeta}.\partial_{\tilde{z}}$, $\tilde{\zeta}$ has
zero coordinates except for the first one and $\tilde{z}$ has zero coordinates
except for the first two. Choosing a rotation $R\in SO(d-1)$ such that

$\tilde{\zeta}^{(0)}:=R^{T}\zeta^{(0)}=\begin{pmatrix}\tilde{\zeta}_{1}^{(0)}\\\
0\\\ \vdots\\\ 0\end{pmatrix},$

we set

(4.26) $\zeta=R\tilde{\zeta}\quad;\quad z=R^{T}\tilde{z}.$

This transformation leaves the vector field unchanged :
$\zeta.\partial_{z}=\tilde{\zeta}.\partial_{\tilde{z}}$. Then along the
trajectory of $\zeta.\partial_{z}$ starting from $(z^{(0)},\zeta^{(0)})$,
$\tilde{\zeta}_{i}(s)$ vanishes for every $i\geq 2$ and all $s\in\mathbb{R}$.
Thus,
$\zeta.\partial_{z}=\tilde{\zeta}.\partial_{\tilde{z}}=\tilde{\zeta}_{1}\partial_{\tilde{z}_{1}}$
along this trajectory.

Proceeding similarly with the $d-1$ last coordinates of $\tilde{z}^{(0)}$, we
can guarantee that $\tilde{z}_{3}^{(0)}\leavevmode\nobreak\
=\leavevmode\nobreak\ \ldots=\leavevmode\nobreak\
\tilde{z}_{d}^{(0)}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0$ without
changing the coordinates of $\tilde{\zeta}^{(0)}$. Thus, $\tilde{z}_{i}(s)=0$
for all $s\in\mathbb{R}$ and every $i\geq 3$.

We now change to spherical coordinates $(r,(\theta_{j})_{1\leq j\leq
2d-3})\in\mathbb{R}^{*}_{+}\times[0,\pi]^{2d-4}\times\mathbb{S}^{1}$, allowing
$r$ to go to $+\infty$. We set $\tilde{z}_{1}\leavevmode\nobreak\
=\leavevmode\nobreak\ r\cos\theta,\tilde{\zeta}_{1}\leavevmode\nobreak\
=\leavevmode\nobreak\
r\sin\theta_{1}\cos\theta_{2},\tilde{z}_{2}\leavevmode\nobreak\
=\leavevmode\nobreak\ r\sin\theta_{1}\sin\theta_{2}\cos\theta_{3}$, and more
generally :

(4.27) $\forall
k\in\\{1,\dots,d-1\\},\quad\tilde{z}_{k}=r\left(\prod_{i=1}^{2k-2}\sin\theta_{i}\right)\cos\theta_{2k-1}\quad;\quad\\\
\tilde{\zeta}_{k}=r\left(\prod_{i=1}^{2k-1}\sin\theta_{i}\right)\cos\theta_{2k}\text{
if }k\leq
d-2\quad;\quad\tilde{\zeta}_{d-1}=r\left(\prod_{i=1}^{2d-3}\sin\theta_{i}\right).$

Since all $(\tilde{\zeta}_{j})_{2\leq j\leq d-1}$ and $(\tilde{z}_{j})_{3\leq
j\leq d-1}$ vanish, we are reduced to a situation where
$\sin\theta_{3}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0$ and
$\cos\theta_{3}\leavevmode\nobreak\ =\leavevmode\nobreak\ \pm 1$. We now
compute $\tilde{\zeta}_{1}\partial_{\tilde{z}_{1}}$ in the $(r,\theta)$
variables. Since none of the angles $\theta_{i}$ depend on $\tilde{z}_{1}$
except for $\theta_{1}$, we get

$\tilde{\zeta}_{1}\partial_{\tilde{z}_{1}}=r\sin\theta_{1}\cos\theta_{2}\left(\frac{\partial
r}{\partial\tilde{z}_{1}}\partial_{r}+\frac{\partial\theta_{1}}{\partial\tilde{z}_{1}}\right).$

We compute both partial derivatives :
$\cos\theta_{1}=\frac{\tilde{z}_{1}}{(|\tilde{z}|^{2}+|\tilde{\zeta}|^{2})^{\frac{1}{2}}}$
gives
$\frac{\partial\theta_{1}}{\partial\tilde{z}_{1}}=-\frac{1}{r}\sin\theta_{1}$.
$\frac{\partial r}{\partial\tilde{z}_{1}}=\frac{\tilde{z}_{1}}{r}$ is
homogeneous of order zero wrt $r$. Given a symbol $q$ that is polyhomogeneous
of order zero, $r\partial_{r}q$ is polyhomogeneous of order -1 so that the
$\partial_{r}$ term above vanishes as $r\rightarrow+\infty$. Thus, over the
sphere at infinity, we have

$\forall q\in
S^{0},\quad\zeta.\partial_{z}q|_{\mathbb{S}^{2d-3}}=-\cos\theta_{2}\sin^{2}\theta_{1}\partial_{\theta_{1}}\tilde{q}(X,\Xi,r\rightarrow+\infty,\theta)$

where $\tilde{q}$ is a notation for $q$ in $(X,\Xi,r,\theta)$ coordinates. ∎

We now give an interpretation of the dynamics of this vector field as the
projection onto the sphere at infinity of the flow map of $\zeta.\partial_{z}$
starting from a point $(z^{(0)},\zeta^{(0)})\in\mathbb{S}^{1}$. This result is
illustrated in Figure 4.3.

$\mathbb{S}^{2d-3}_{\infty}$$\mathbb{S}^{2d-3}_{1}$Transport over
$\mathbb{S}^{2d-3}_{\infty}$initial point
$(z_{0},\zeta_{0})$projectionTransport over $\mathbb{R}^{2d-2}$ Figure 4.3.
Illustration of Proposition 4.9 about the transport by vector field
$\zeta.\partial_{z}$ over the sphere at infinity.

In the following statement, we denote indifferently points belonging to the
unit sphere and the sphere at infinity as they are identified by projection.

######  Proposition 4.9. –

Denote for all $s\in\mathbb{R}$

(4.28)
$\left(\check{z}(s),\check{\zeta}(s)\right):=\frac{1}{{\left(|\tilde{z}^{(0)}+s\tilde{\zeta}^{(0)}|^{2}+|\tilde{\zeta}^{(0)}|^{2}\right)^{\frac{1}{2}}}}\left(\tilde{z}^{(0)}+s\tilde{\zeta}^{(0)},\tilde{\zeta}^{(0)}\right)\in\mathbb{S}^{2d-3},$

then

(4.29) $(\tilde{z}(s),\tilde{\zeta}(s))=(\check{z}(s),\check{\zeta}(s))$

where $(\tilde{z}(s),\tilde{\zeta}(s))\in\mathbb{S}^{2d-3}$ were introduced in
(4.26).

###### Proof.

Recall that $(\tilde{z}^{(0)},\tilde{\zeta}^{(0)})$ has zero-coordinates
except maybe for
$\tilde{z}^{(0)}_{1},\tilde{\zeta}^{(0)}_{1},\tilde{z}^{(0)}_{2}$. Then by
(4.28), every coordinate of $(\check{z},\check{\zeta})$ vanishes except maybe
$\check{z}_{1},\check{\zeta}_{1},\check{z}_{2}$.

The result then follows from the same passage to angle coordinates as in
(4.27). Like before, the vanishing coordinates impose constraints on
$\check{\theta}_{3}$, so that we set

$\check{z}_{1}=r\cos\check{\theta}_{1}\quad;\quad\check{\zeta_{1}}=r\sin\check{\theta}_{1}\cos\check{\theta}_{2}\quad;\quad\check{z}_{2}=r\sin\check{\theta}_{1}\sin\check{\theta}_{2}\cos\check{\theta}_{3}=\pm
r\sin\check{\theta}_{1}\sin\check{\theta}_{2}.$

where $\check{\theta}_{1},\check{\theta}_{2}\in[0,\pi]$ and
$\check{\theta}_{3}\in[0,\pi]$ if $d>3$ and $\mathbb{S}^{1}$ if $d=3$.
Conversely, the $\check{\theta}_{j}$s satisfy :

(4.30)
$\check{\theta}_{1}=\arccos\left(\frac{\check{z}_{1}}{\sqrt{\sum_{j=1}^{p}\check{z}_{j}^{2}+\sum_{j=1}^{p}\check{\zeta}_{j}^{2}}}\right),\quad\check{\theta}_{2}=\arccos\left(\frac{\check{\zeta}_{1}}{\sqrt{\sum_{j=2}^{p}\check{z}_{j}^{2}+\sum_{j=1}^{p}\check{\zeta}_{j}^{2}}}\right).$
(4.31) $\displaystyle\text{If }p>2:$ $\displaystyle\quad\check{\theta}_{3}=$
$\displaystyle\arccos\left(\frac{\check{z}_{2}}{\sqrt{\sum_{j=2}^{p}z_{j}^{2}+\sum_{j=2}^{p}\zeta_{j}^{2}}}\right)=$
$\displaystyle\quad 0\mod\pi.$ $\displaystyle\text{If }p=2:$
$\displaystyle\quad\check{\theta}_{3}=$
$\displaystyle\arccos\left(\frac{\check{z}_{2}}{\sqrt{\sum_{j=2}^{p}z_{j}^{2}+\sum_{j=2}^{p}\zeta_{j}^{2}}}\right)$
$\displaystyle\text{ if }\check{\zeta}_{2}\geq 0,$ $\displaystyle\quad
2\pi-\arccos\left(\frac{\check{z}_{2}}{\sqrt{\sum_{j=2}^{p}z_{j}^{2}+\sum_{j=2}^{p}\zeta_{j}^{2}}}\right)$
otherwise.

Only the first coordinate of $\tilde{\zeta}^{(0)}$ is non-zero so by (4.28),
$\check{\theta}_{2}$ and $\check{\theta}_{3}$ are time-independent. Since the
$\theta_{j}$s and $\check{\theta}_{j}$s have identical initial values, we need
to show that

$\frac{d}{dt}\check{\theta}_{1}(t)=-\cos\check{\theta}_{2}\sin^{2}\check{\theta}_{1}(t)$

to conclude. We differentiate the expression of $\cos(\check{\theta}_{1}(t))$
to get

$-\sin(\check{\theta}_{1}(t))\frac{d\check{\theta}_{1}(t)}{dt}(t)=\frac{(|\tilde{z}^{(0)}+t\tilde{\zeta}^{(0)}|^{2}+|\tilde{\zeta}^{(0)}|^{2})\tilde{\zeta}_{1}^{(0)}-(\tilde{z}_{1}^{(0)}+t\tilde{\zeta}_{1}^{(0)})(\tilde{\zeta}^{(0)}.(\tilde{z}^{(0)}+t\tilde{\zeta}^{(0)}))}{(|\tilde{z}^{(0)}+t\tilde{\zeta}^{(0)}|^{2}+|\tilde{\zeta}^{(0)}|^{2})^{\frac{3}{2}}}.$

Simplifying this expression by removing vanishing coordinates and using the
expression of $\cos(\check{\theta}_{2})$, we get

$\displaystyle-\sin(\check{\theta}_{1}(t))\frac{d\check{\theta}_{1}(t)}{dt}(t)$
$\displaystyle=\cos\check{\theta}_{2}(t)\left(\frac{(\tilde{\zeta}^{(0)}_{1})^{2}+\sum_{j=2}^{p}(\tilde{z}^{(0)}_{j})^{2}}{|\tilde{z}^{(0)}+t\tilde{\zeta}^{(0)}|^{2}+|\tilde{\zeta}^{(0)}|^{2}}\right)^{\frac{3}{2}}$
$\displaystyle=\cos\check{\theta}_{2}(t)(1-\cos^{2}\check{\theta}_{1}(t))^{\frac{3}{2}}=\cos\check{\theta}_{2}(t)\sin^{3}\check{\theta}_{1}(t),$

hence the result. ∎

##  5\. Proof of Theorem 5

$x_{3}=y$$x_{2}$$x_{1}$

Figure 5.1. The damping $a$ is equal to 1 on the blue zone. The flat torus
$\mathbb{T}^{3}$ is identified to the unit cube :
$\mathbb{T}^{3}\simeq[-1,1]^{3}/(X\sim X+2e_{i}).$ The four cubes on the right
fit into the hollow part of the bigger cube. The coordinate system is still
$X=(x_{1},x_{2},y)=(x,y),\Xi=(\xi_{1},\xi_{2},\eta)=(\xi,\eta)$.

As pointed out in the introduction, the sufficient condition of Theorem 4 is
also necessary in dimension 2 [BG20]. The purpose of the present section is to
investigate how this condition can be weakened on higher-dimensional tori by
proving Theorem 5. We assume that the damping $a$ equals 1 on the blue zone
pictured in Figure 5.1. Compared to the model case of section 4.2, the four
prisms in the tunnel are reduced to cubes. Said otherwise, we take
$\alpha_{L}=\alpha_{R}=\alpha_{T}=\alpha_{B}=0$ in (4.8) so that $a$ is the
characteristic of the set given in (1.5).

Two main differences emerge in this example compared to the model case of
Figure 4.1. First, the bicharacteristics which do not intersect
$\operatorname{Int}(\operatorname{supp}(a))$ are the red bicharacteristic
$\\{x=0\\}$, and parallel ones which raze the faces of the cubes and are
contained in the same horizontal and vertical planes as the red one, or more
explicitly

(5.1) $\\{(x_{1}=0,|x_{2}|\leq 1/2)\text{ or }(|x_{1}|\leq
1/2,x_{2}=0),\quad\xi=0,\quad\eta=\pm 1\\}.$

Second, none of these geodesics (in particular not the red one) satisfy
Assumption (1.3). The geodesic $\\{x=0\\}$ has four undamped directions while
the others have two.

In section 5.1 we tackle the first issue. By dropping the $(z_{1},\zeta_{1})$
or $(z_{2},\zeta_{2})$ variable in the second quantization of section 4.2, we
construct 2-microlocal measures $\mu_{1},\mu_{2}$ in each direction
$x_{1},x_{2}$, such that the second microlocal measure $\mu$ is absolutely
continuous wrt each. We then show that $\mu$ has no support on any geodesic
except the red one. In section 5.2, we perform a third microlocalization
around the red geodesic to show that the sequence $u_{n}$ of quasimodes cannot
concentrate around it either. We conclude by discussing the scope of this
example and some obstacles to generalizing it.

###  5.1. Directional second microlocalizations

For a given $j\in\\{1,2\\}$, the second quantization in the direction $x_{j}$
of a symbol $b_{j}(X,\Xi,z_{j},\zeta_{j})$ is defined by

(5.2)
$\operatorname{Op}_{h}^{(j)}(b_{j})u(X)=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}_{Y}\times\mathbb{R}^{3}_{\Xi}}e^{i(X-Y).\Xi}b_{j}\left(X,h\Xi,\frac{\epsilon(h)}{h^{\frac{1}{2}}}x_{j},\epsilon(h)h^{\frac{1}{2}}\xi_{j}\right)u(Y)dYd\Xi.$

This quantization is a specific case of (4.2) with $d=3$, $q=1$, so $u_{n}$
admits two microlocal measures $\mu_{1}$, $\mu_{2}$ associated to each
quantization $\operatorname{Op}_{h}^{(1)}$, $\operatorname{Op}_{h}^{(2)}$.
$\mu_{1}$ (resp. $\mu_{2}$) measures concentration in the horizontal (resp.
vertical) direction, at the 2-microlocal scale
$h^{\frac{1}{2}}\epsilon(h)^{-1}$. We also define the second microlocal
measure $\mu$ as formerly. The properties of the $\mu_{j}$s and of $\mu$ are
summarized in the following statement :

######  Proposition 5.1 (Properties of $\mu_{1}$ and $\mu_{2}$). –

The directional 2–microlocal measures have the following properties :

  1. (1)

Assume that $(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}_{L^{2}}(1)$, then $\mu_{1}$,
$\mu_{2}$ and $\mu$ are probability measures.

  2. (2)

Assume that $(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}_{L^{2}}(h_{n})$ and
$\|au_{n}\|_{L^{2}}\leavevmode\nobreak\ =\leavevmode\nobreak\ o(1)$, and
define the projection $\pi_{j}(X,\Xi,z,\zeta)=(X,\Xi,z_{j},\zeta_{j})$ for
$j=1,2$. Then, $\mu_{j}$ is the pushforward measure of $\mu$ by the projection
$\pi_{j}$. Besides, the pushforward measure of each measure $\mu$, $\mu_{1}$,
$\mu_{2}$ by the projection onto the $(X,\Xi)$ variables is the first
microlocal measure $\nu$.

Thus,

(5.3) $\operatorname{supp}(\mu)\subset\\\
\left\\{(X,\Xi,z,\zeta),(x_{1}=0,|x_{2}|\leq 1/2)\text{ or }(|x_{1}|\leq
1/2,x_{2}=0),\xi=0,\eta=\pm 1\right\\}$

and the same inclusions hold for $\operatorname{supp}(\mu_{j}),j=1,2$,
replacing $(z,\zeta)$ with $(z_{j},\zeta_{j})$.

  3. (3)

Assume now that
$(h_{n}^{2}\Delta+1)u_{n}=\mathcal{O}(h_{n}\epsilon(h_{n}))_{L^{2}}$, then
each measure $\mu_{j}$ is supported in the sphere at infinity
$\mathbb{S}^{1}_{\infty}$ in the $(z_{j},\zeta_{j})$ variables.

  4. (4)

Along the geodesics that raze one of the faces of the four damped cubes in
Figure 5.1, directional 2-microlocal measures vanish 2-microlocally in the
direction of the cube. More precisely, $\mu_{1}$ satisfies :

(5.4)
$\displaystyle\mu_{1}\left\\{x_{1}=0,x_{2}\in\left[-\frac{1}{2},0\right),y\in\left(-1,-\frac{1}{2}\right),\xi=0,\eta=\pm
1,z_{1}>0\right\\}$ $\displaystyle=0,$
$\displaystyle\mu_{1}\left\\{x_{1}=0,x_{2}\in\left(0,\frac{1}{2}\right],y\in\left(-\frac{1}{2},0\right),\xi=0,\eta=\pm
1,z_{1}>0\right\\}$ $\displaystyle=0,$
$\displaystyle\mu_{1}\left\\{x_{1}=0,x_{2}\in\left(0,\frac{1}{2}\right],y\in\left(0,\frac{1}{2}\right),\xi=0,\eta=\pm
1,z_{1}<0\right\\}$ $\displaystyle=0,$
$\displaystyle\mu_{1}\left\\{x_{1}=0,x_{2}\in\left[-\frac{1}{2},0\right),y\in\left(\frac{1}{2},1\right),\xi=0,\eta=\pm
1,z_{1}<0\right\\}$ $\displaystyle=0,$

and similar vanishings hold for $\mu_{2}$, which we do not state for the sake
of conciseness.

  5. (5)

Consider $j\in\\{1,2\\}$, then the microlocal measure $\mu_{j}$ satisfies the
conservation law

(5.5) $(\eta\partial_{y}+\zeta_{j}\partial_{z_{j}})\mu_{j}=0.$

Since $\mu_{j}$ is supported in the sphere at infinity in the
$(z_{j},\zeta_{j})$ variables, we can use polar coordinates
$(z_{j},\zeta_{j})=(r\cos\theta,r\sin\theta)$, $r\rightarrow+\infty$ with
$\theta\in\mathbb{R}/2\pi\mathbb{Z}$. In these coordinates, equation (5.5)
becomes

(5.6) $(\eta\partial_{y}-\sin^{2}(\theta)\partial_{\theta})\mu_{j}=0.$

  6. (6)

The two preceding items imply that

(5.7)
$\displaystyle\operatorname{supp}(\mu_{1})\subset\\{(X,\Xi,z,\zeta):|x_{1}|\leq\frac{1}{2},x_{2}=0,\xi=0,\eta=\pm
1,(z_{1},\zeta_{1})\in\mathbb{S}^{1}_{\infty}\\}$
$\displaystyle\operatorname{supp}(\mu_{2})\subset\\{(X,\Xi,z,\zeta):x_{1}=0,|x_{2}|\leq\frac{1}{2},\xi=0,\eta=\pm
1,(z_{2},\zeta_{2})\in\mathbb{S}^{1}_{\infty}\\}$

Since the support of $\mu$ is a subset of those of $\mu_{1}$ and $\mu_{2}$, we
get that $\mu$ is supported in the undamped great circles of the sphere at
infinity $\mathbb{S}^{3}_{\infty}$ along the red bicharacteristics :

(5.8) $\operatorname{supp}(\mu)\subset\\{x=0,\xi=0,\eta=\pm
1,(z_{1},\zeta_{1})=(0,0)\text{ or }(z_{2},\zeta_{2})=(0,0)\\}.$

###### Proof.

The first point derives from $h_{n}$-oscillation.

As in the proof of the second point of Proposition 4.4, we have that

$\langle\mu_{j},b\rangle=\langle\mu,b\circ\pi_{j}\rangle$

for any symbol $b(X,\Xi,z_{j},\zeta_{j})$. The rest of the proof of the second
point is identical.

Concerning (3), we consider a function $\chi\in
C^{\infty}_{o}(T^{*}\mathbb{T}^{3}\times\mathbb{R}^{2}_{z_{j},\zeta_{j}})$.
There exists some $A>0$ such that $\chi$ is supported in the set
$\\{|x_{j}|\leq A\\}$. Then by Proposition 3.1,

(5.9)
$\displaystyle|\langle\operatorname{Op}_{h_{n}}(\chi)u_{n},u_{n}\rangle_{L^{2}}|$
$\displaystyle=|\langle\operatorname{Op}_{h_{n}}(\chi)u_{n},\mathbbm{1}_{\\{|x_{j}|\leq
Ah_{n}^{\frac{1}{2}}\epsilon^{-1}(h_{n})\\}}u_{n}\rangle_{L^{2}}|$
$\displaystyle\leq\|\operatorname{Op}_{h_{n}}(\chi)u_{n}\|_{L^{2}}\|u_{n}\|_{L^{2}(\\{|x_{j}|\leq
Ah_{n}^{\frac{1}{2}}\epsilon^{-1}(h_{n})\\})}$ $\displaystyle\leq
C\epsilon^{\frac{1}{2}}(h_{n})\rightarrow_{n\rightarrow+\infty}0.$

We now prove the first microlocal vanishing of $\mu_{1}$. All other vanishings
are showed similarly. Hence we are set to show that

$\mu_{1}\left\\{x_{1}=0,x_{2}\in\left[-\frac{1}{2},0\right),y\in\left(-1,-\frac{1}{2}\right),z_{1}>0\right\\}=0.$

Consider some positive $\delta$ and $\delta_{0}$ and the symbol

(5.10)
$b_{1}(X,\Xi,z_{1},\zeta_{1})=\chi_{1}\left(\frac{x_{1}}{\delta}\right)\chi_{2}(x_{2})\chi_{3}(y)\chi_{1}(\xi)\chi_{1}(\eta-1)\psi\left(\frac{z_{1}}{\delta_{0}|\zeta_{1}|}\right)\psi(|z_{1}|^{2}+|\zeta_{1}|^{2})$

where $\chi_{i}$, $i=1,2,3$ and $\psi$ are $[0,1]$-valued cutoff functions in
$C^{\infty}(\mathbb{R})$ such that :

  * –

$\chi_{1}$ equals 1 over $[-\frac{1}{2},\frac{1}{2}]$ and is supported in
$[-1,1]$,

  * –

$\chi_{2}$ equals 1 over $[-\frac{1}{2},\delta_{0}]$ and is supported in
$[-\frac{1}{2}-\delta_{0},0]$,

  * –

$\chi_{3}$ equals 1 over $[-1+\delta_{0},-\frac{1}{2}-\delta_{0}]$ and is
supported in $[-1,-\frac{1}{2}]$.

  * –

$\psi$ equals 1 over $[1,+\infty)$ and is supported inside
$[\frac{1}{2},+\infty)$.

We then proceed as in Proposition 4.4 to prove that
$\langle\mu_{1},b_{1}\rangle=0$ and deduce the 2-microlocal vanishing from the
limit $\delta_{0}\rightarrow 0$, giving the fourth point.

In the fifth point, (5.5) follows from computing the bracket
$[h_{n}^{2}\Delta+1,\operatorname{Op}_{h_{n}}^{(j)}(q)]$ and passing to the
limit $h\rightarrow 0$ to prove . Indeed, (4.15) becomes

(5.11)
$\frac{1}{2ih_{n}}[h_{n}^{2}\Delta+1,\operatorname{Op}_{h_{n}}^{(j)}(q)]=\frac{1}{2i}[h_{n}\Delta,\operatorname{Op}_{h_{n}}(q)]\\\
=\operatorname{Op}_{h_{n}}((\xi.\partial_{x}+\eta\partial_{y}+\zeta_{j}\partial_{z_{j}})q)-i\frac{h_{n}}{2}\operatorname{Op}_{h_{n}}(\Delta_{X}q)-i\frac{h_{n}}{2}(\epsilon(h_{n})h_{n}^{-\frac{1}{2}})\operatorname{Op}_{h_{n}}((\partial_{x_{j}}.\partial_{z_{j}})q)\\\
-i\frac{h_{n}}{2}(\epsilon(h_{n})h_{n}^{-\frac{1}{2}})^{2}\operatorname{Op}_{h_{n}}(\partial_{z_{j}}^{2}q).$

(5.5) then derives from the same argument as in the proof of (4.11). Thus for
any polyhomogeneous symbol $b_{j}$ of degree 0,

(5.12) $\displaystyle\zeta_{j}\partial_{z_{j}}q_{j}$
$\displaystyle=\lim_{r\rightarrow+\infty}(-\sin^{2}(\theta)\partial_{\theta}+r\cos\theta\sin\theta\partial_{r})\tilde{q}(X,\Xi,r,\theta)$
$\displaystyle=-\sin^{2}(\theta)\partial_{\theta}\lim_{r\rightarrow+\infty}\tilde{q}(X,\Xi,r,\theta).$

Note that this computation from [BG20, p. 640] is the equivalent on
$\mathbb{S}^{1}$ of Proposition 4.8 and Lemma 4.5.

Lastly, we prove the first inclusion of (5.7) by contradiction, as was done in
subsection 4.2. Assume that $\operatorname{supp}(\mu_{1})$ contains a point
$(x,y,\xi=0,\eta=\pm 1,\theta)$ with $x_{1}=0$,
$x_{2}\in[-\frac{1}{2},\frac{1}{2}]\setminus\\{0\\}$ and $\theta$ defined by
$(z_{1},\zeta_{1})=r(\cos\theta,\sin\theta)$, $r\rightarrow+\infty$. We denote
$\phi_{s}(\theta_{0})$ the flow of $\dot{\theta}=-\sin^{2}\theta$ at time $s$
starting from a point $\theta_{0}$. The conservation law for $\mu_{1}$ gives
that

(5.13) $(x,y+s\eta,\xi=0,\eta,\phi_{s}(\theta))$

also belongs to $\operatorname{supp}(\mu_{1})$ for any $s\in\mathbb{R}$. A
quick study of the flow of equation $\dot{\theta}=-\sin^{2}\theta$ (see [BG20,
p. 641]) shows that $\phi_{s}(\theta_{0})$ converges to $0(\text{mod }\pi)$ so
that the corresponding $(z_{1}(s),\zeta_{1}(s))$ converges to $(\pm 1,0)$ on
the sphere at infinity as $s$ goes to $+\infty$. For $s$ large enough, this
means that a point of the form (5.13) belongs to one of the zones where $\mu$
vanishes two-microlocally, which gives a contradiction.

The proof for $\operatorname{supp}(\mu_{2})$ is identical. We come to the
localization of $\operatorname{supp}(\mu)$. Like in Section 4.2, $\mu$
satisfies the conservation law (4.11) and the 2-microlocal vanishings (4.10)
with $\alpha_{R}=\alpha_{T}=\alpha_{L}=\alpha_{B}=0$. By Lemma 4.5 and the
contradiction argument of section 4.2, $\mu$ is supported in great circles of
the sphere at infinity $\mathbb{S}^{3}_{z,\zeta}$ that do not intersect any of
the four quadrants $\\{\varepsilon_{1}z_{1}>0,\varepsilon_{2}z_{2}>0\\}$,
$\varepsilon_{1},\varepsilon_{2}\in\\{-1,1\\}$. This implies inclusion (5.8).

∎

###  5.2. Third microlocalization near isolated undamped normal directions

To conclude the contradiction argument and show that the second microlocal
measure $\mu$ vanishes everywhere, we need to show that it has no weight over
the undamped directions $(z_{1},\zeta_{1})=(0,0)$ and
$(z_{2},\zeta_{2})=(0,0)$ of the sphere at infinity. To do this, we perform a
third microlocalization which allows to add inhomogeneity in the symbols
considered and split $\mu$ into two 2-microlocal measures $\mu_{+},\mu_{-}$
supported in each hemisphere of the sphere at infinity.

As an example, we focus on the undamped direction
$(z_{1},z_{2},\zeta_{1},\zeta_{2})=(z_{1},0,\zeta_{1},0)$ with $z_{1}>0$.
Under the flow of $\zeta.\partial_{z}$, $\zeta_{1}$ becomes arbitrarily small
for long times so that we want to localize on the region of the sphere where
$z_{1}\gg|\zeta|$ and $z_{1}\gg|z_{2}|$. We denote $b_{-}$ an operator
localizing in this region :

(5.14)
$b_{-}(X,\Xi,z,\zeta)=\chi_{\delta}(|x|)\chi_{\delta}(|\xi|)\tilde{\chi}(y)\chi_{\delta}(\eta-1)\psi\left(\frac{z_{1}}{\alpha|\zeta|}\right)\chi_{\alpha}\left(\frac{z_{2}}{z_{1}}\right)\psi(|z|^{2}+|\zeta|^{2}),$

where all cut-offs are smooth functions and satisfy :

  * –

$\chi_{\delta}$ equals 1 over $[-\frac{\delta}{2},\frac{\delta}{2}]$ and 0 out
of $[-\delta,\delta]$. $\chi_{\alpha}$ is the same with a different small
parameter.

  * –

$\tilde{\chi}$ equals 1 over $[-1+\delta_{0},-\frac{1}{2}-\delta_{0}]$ and 0
out of $[-1,-\frac{1}{2}]$.

  * –

$\psi$ equals 1 over $[1,+\infty)$ and 0 over $(-\infty,\frac{1}{2}]$.

$b_{-}$ belongs to the smooth polyhomegeneous symbol class $S^{0}_{H}$ of the
2-microlocal pseudodifferential calculus. The third microlocalization consists
in multiplying $\operatorname{Op}_{h_{n}}(b_{-})$ by
$\psi(\pm\epsilon^{\frac{1}{2}}(h)z_{2})$. This allows to localize both near
the undamped direction and in the upper and lower damped zones. The procedure
is explained in the following statement and illustrated in Figure 5.2 :

$z_{1}$$z_{2}$$\mathbb{S}^{3}_{\infty}$$\operatorname{supp}(\chi_{\alpha}(\frac{z_{2}}{z_{1}}))$$\alpha$

$z_{1}$$z_{2}$$\operatorname{supp}(\psi_{-})$$\operatorname{supp}(\psi_{+})$$(+1,0)$$(-1,0)$

Figure 5.2. On the left-hand side, we show the $|z_{2}|\ll|z_{1}|$ zone of the
sphere at infinity where the $\chi_{\alpha}(\frac{z_{2}}{z_{1}})$ cutoff
localizes. On the right-hand-side, we display the supports of the $\psi_{+}$
and $\psi_{-}$ cutoffs over the sphere at infinity (in the $(z_{1},z_{2})$
variables). The violet circle in the horizontal plane represents the
$\\{z_{2}=0,\zeta_{2}=0\\}$ great circle of $\mathbb{S}^{3}$ (the third
coordinate is then $\zeta_{1}$). The goal of the third microlocalization is to
deal with points of $\operatorname{supp}(\mu)$ belonging to this great circle.
The flow of $\zeta.\partial_{z}$ transports any point in this great circle
close to one of the two violet disks $(z_{1},z_{2})\leavevmode\nobreak\
=\leavevmode\nobreak\ (\pm 1,0)$ of $\mathbb{S}^{3}_{\infty}$ as time goes to
infinity. The goal of the third microlocalization is to distinguish points
$(z_{1},z_{2})=(\pm 1,0^{+})$ and $(\pm 1,0^{-})$.

######  Proposition 5.2. –

  1. (1)

Denote
$\psi_{\pm}(x_{2})=\psi\left(\pm\frac{\epsilon^{\frac{3}{2}}(h)}{h^{\frac{1}{2}}}x_{2}\right)$.
Denote also $\psi_{\pm}$ the associated multiplication operators. Then there
exist two positive Radon measures $\mu_{\pm}$ such that for every $b\in
S^{0}_{H}$,

$\langle\psi_{\pm}\operatorname{Op}_{h_{n}}(b)u_{n},u_{n}\rangle_{L^{2}}\rightarrow_{n\rightarrow+\infty}\langle\mu_{\pm},b\rangle.$

  2. (2)

For any 2-microlocal symbol $b\in S^{0}_{H}$, we have

$\langle(\psi_{+}+\psi_{-})\operatorname{Op}_{h_{n}}(b)u_{n},u_{n}\rangle_{L^{2}}\rightarrow_{n\rightarrow+\infty}\langle\mu,b\rangle,$

so that $\mu=\mu_{+}+\mu_{-}$. In particular,
$\operatorname{supp}(\mu_{\pm})\subset\operatorname{supp}(\mu)$.

  3. (3)

$\mu_{+}$ and $\mu_{-}$ are supported in the sphere at infinity in the
$(z,\zeta)$ variables. Besides, $\mu_{+}$ is supported in the $\\{z_{2}\geq
0\\}$ closed hemisphere of the sphere at infinity and $\mu_{-}$ is supported
in the $\\{z_{2}\leq 0\\}$ hemisphere.

  4. (4)

The $\mu_{\pm}$ measures satisfy the conservation laws

(5.15) $\mathbbm{1}_{\pm z_{2}\geq
0}\left(\eta\partial_{y}+\zeta.\partial_{z}\right)\mu_{\pm}=0.$

Since $\operatorname{supp}(\mu_{\pm})\subset\\{\pm z_{2}\geq 0\\}$, the
characteristic functions can be omitted so the $\mu_{\pm}$ satisfy the same
conservation law (4.11) as $\mu$.

  5. (5)

Consider $b_{-}$ the symbol given by (5.14), then
$\langle\mu_{-},b_{-}\rangle=0$. Similarly, replacing $\tilde{\chi}$ by a
function equal to 1 over $[-\frac{1}{2}+\delta_{0},-\delta_{0}]$ and supported
in $[-\frac{1}{2},0]$ in (5.14) defines a symbol $b_{+}$ such that
$\langle\mu_{+},b_{+}\rangle=0$. Thus for some small enough positive $\alpha$,

(5.16)
$\displaystyle\mu_{-}\left\\{x=0,y\in\left(-1,-\frac{1}{2}\right),\xi=0,\eta=\pm
1,z_{1}>\frac{2}{\alpha}|z_{2}|,z_{1}>\alpha|\zeta|\right\\}$
$\displaystyle=0,$
$\displaystyle\mu_{+}\left\\{x=0,y\in\left(0,-\frac{1}{2}\right),\xi=0,\eta=\pm
1,z_{1}>\frac{2}{\alpha}|z_{2}|,z_{1}>\alpha|\zeta|\right\\}$
$\displaystyle=0.$

###### Proof.

The $\psi_{\pm}$ are multiplication operators by smooth, uniformly bounded
functions so they are $h$-uniformly bounded operators over $L^{2}_{ul}$.
$\operatorname{Op}_{h_{n}}(b)$ is also a uniformly bounded operator.

Besides,
$\psi_{\pm}\operatorname{Op}_{h_{n}}(b)=\operatorname{Op}_{h_{n}}(\psi(\epsilon^{\frac{1}{2}}z_{2})b)$
is the 2-microlocal quantization of a function in the generalized symbol class
$S^{0}$ of our 2-microlocal calculus. Since

$b\geq 0\Rightarrow\psi(\epsilon(h)^{\frac{1}{2}}z_{2})b\geq 0,$

the Gårding inequality (4.6) still holds. The classical proof of existence of
semiclassical measures then gives existence and positivity of the Radon
measures $\mu_{\pm}$, hence the first point.

The second point derives from the non-concentration estimate of Proposition
3.1. The function
$1-\left(\psi\left(\frac{\epsilon^{\frac{3}{2}}(h)x_{2}}{h^{\frac{1}{2}}}\right)+\psi\left(-\frac{\epsilon^{\frac{3}{2}}(h)x_{2}}{h^{\frac{1}{2}}}\right)\right)$
is indeed supported in the $\\{|x_{2}|\leavevmode\nobreak\
\leq\leavevmode\nobreak\ h^{\frac{1}{2}}\epsilon(h)^{-\frac{3}{2}}\\}$ slice.
As in the previous proof, this yields

$\langle(1-(\psi_{+}+\psi_{-}))\operatorname{Op}_{h_{n}}(b)u_{n},u_{n}\rangle\rightarrow_{n\rightarrow+\infty}0,$

hence the result.

Thus, $\mu_{\pm}\ll\mu$ and the 3-microlocal measures are supported in the
sphere at infinity in the $(z,\zeta)$ variables.

To prove the second part of statement (3), consider a symbol $b$ in
$S^{0}_{H}$ supported in the $\\{z_{2}>0\\}$ (resp. $\\{z_{2}<0\\}$) part of
the sphere at infinity. Then $\psi_{-}b$ (resp. $\psi_{+}b$) vanishes in a
neighborhood of the sphere at infinity, so that $\langle\mu_{-},b\rangle=0$
(resp. $\langle\mu_{+},b\rangle=0$).

We derive point (4) from computing the bracket
$\frac{1}{2ih}[h^{2}\Delta+1,\psi_{\pm}\operatorname{Op}_{h}(b)]$ :

(5.17)
$\displaystyle\frac{1}{2ih}[h^{2}\Delta+1,\psi\left(\pm\frac{\epsilon^{\frac{3}{2}}(h)}{h^{\frac{1}{2}}}x_{2}\right)\operatorname{Op}_{h}(b)]u\quad=\frac{h}{2i}\Delta(\psi_{\pm}\operatorname{Op}_{h}(b)u)-\psi_{\pm}\operatorname{Op}_{h}(b)\left(\frac{h}{2i}\Delta
u\right)$
$\displaystyle=\frac{h}{2i}\left(\frac{\epsilon^{3}(h)}{h}\psi_{\pm}^{{}^{\prime\prime}}\operatorname{Op}_{h}(b)u\pm
2\frac{\epsilon(h)}{h^{\frac{1}{2}}}\psi_{\pm}^{{}^{\prime}}\partial_{x_{2}}(\operatorname{Op}_{h}(b)u)+\psi_{\pm}[\Delta,\operatorname{Op}_{h}(b)]u\right)$
$\displaystyle=\frac{1}{2i}\left(\psi_{\pm}^{{}^{\prime\prime}}\epsilon^{2}(h)\operatorname{Op}_{h}(b)u\pm
2\psi_{\pm}^{\prime}\epsilon^{\frac{3}{2}}(h)h^{\frac{1}{2}}\operatorname{Op}_{h}\left(i\xi_{2}b+\partial_{x_{2}}b+\frac{\epsilon(h)}{h^{\frac{1}{2}}}\partial_{z_{2}}b\right)+\psi_{\pm}[h\Delta,\operatorname{Op}_{h}(b)]u\right)$
$\displaystyle=\epsilon^{\frac{1}{2}}(h)\psi_{\pm}^{\prime}\operatorname{Op}_{h}(\zeta_{2}b)+\psi\left(\pm\frac{\epsilon^{\frac{3}{2}}(h)}{h^{\frac{1}{2}}}x_{2}\right)\operatorname{Op}_{h}((\eta\partial_{y}+\xi.\partial_{x}+\zeta.\partial_{z})b)+l.o.t.$

The lower order terms are the second derivatives of $\psi_{\pm}$ and the
remaining terms of the bracket (4.15). They vanish as $h_{n}$ goes to zero.
The first term also does. Over the sphere at infinity,
$\psi\left(\pm\frac{\epsilon^{\frac{3}{2}}(h)}{h^{\frac{1}{2}}}x_{2}\right)$
is equal to $\mathbbm{1}_{\pm z_{2}\geq 0}$, giving the result.

To prove the last point, fix some small $\alpha>0$ independent of $h$ and
consider the symbol $b_{-}$ given by (5.14), then for any $n$,
$\operatorname{supp}(\psi_{-}\operatorname{Op}_{h_{n}}(b_{-})u_{n})\subset\left\\{x_{1}>0,x_{2}<0,y\in(-1,-\frac{1}{2})\right\\}$
so that

$\displaystyle\langle\psi_{-}\operatorname{Op}_{h_{n}}(b_{-})u_{n},u_{n}\rangle$
$\displaystyle=\langle\mathbbm{1}_{\left\\{x_{1}>0,x_{2}<0,y\in(-1,-\frac{1}{2})\right\\}}\psi_{-}\operatorname{Op}_{h_{n}}(b_{-})u_{n},u_{n}\rangle$
$\displaystyle=\langle\psi_{-}\operatorname{Op}_{h_{n}}(b_{-})u_{n},\mathbbm{1}_{\left\\{x_{1}>0,x_{2}<0,y\in(-1,-\frac{1}{2})\right\\}}au_{n}\rangle$

Since $\|au_{n}\|_{L^{2}}\rightarrow 0$ and
$\psi_{-}\operatorname{Op}_{h_{n}}(b_{-})u_{n}$ is bounded uniformly wrt small
$h_{n}$, we get that $\langle\mu_{-},b_{-}\rangle=0$. The result then derives
from arguments similar to the proof of Proposition 5.1, point 4. ∎

As we have seen in the previous section, the flow of $\zeta.\partial_{z}$
converges to some point $(z_{1},z_{2},\zeta=0)$ of the sphere at infinity,
with $(z_{1},z_{2})\in\\{(\pm 1,0),(0,\pm 1)\\}$. Point 5 above shows that
both $\mu_{+}$ and $\mu_{-}$ vanish near the point $(z=(+1,0),\zeta=0)$ of the
sphere at infinity over some portion of the geodesic. Replacing $z_{1}$ by
$-z_{1}$ in cutoff (5.14) gives that $\mu_{+}$ and $\mu_{-}$ also vanish near
the point $(z=(-1,0),\zeta=0)$.

Consider a point $(x=0,y,\xi=0,\eta=\pm 1,z_{0},\zeta_{0})$ which we assume to
belong to $\operatorname{supp}(\mu)$. Since $\mu_{+}+\mu_{-}=\mu$, this point
belongs either to $\operatorname{supp}(\mu_{+})$ or to
$\operatorname{supp}(\mu_{-})$. By Lemma 4.5, $\phi_{s}(z_{0},\zeta_{0})$
converges to some point in $\\{(\pm 1,0),(0,\pm 1)\\}$ as
$s\rightarrow+\infty$. If the limit is $(\pm 1,0)$ then the flow of
$\zeta.\partial_{z}$ transports $(z_{0},\zeta_{0})$ to some point near which
either $\mu_{+}$ or $\mu_{-}$ vanishes along the geodesic. If it is $(0,\pm
1)$, we perform the same third microlocalization in the $|z_{2}|\gg|z_{1}|$
zone of the sphere at infinity and replace operators $\psi_{\pm}$ with
multipliers in the $x_{1}$ variable. Thus, we obtain a contradiction and $\mu$
is identically zero, which proves stabilization for the damping of Figure 5.1.

We finish with some concluding remarks concerning possible generalizations of
this counter-example (and some obstacles to them) :

  * –

Based on the result and techniques of the present section, we conjecture that
a necessary and sufficient condition for uniform stabilization of the wave
equation on 3-dimensional tori using damped polyhedrons is for every geodesic
to be damped in every normal direction but a finite number. Proof of the
necessary condition would likely resemble that of Nicolas Burq and Patrick
Gérard in [BG20, Section 5], using the necessity of their generalized
geometric control condition. The sufficient condition also seems difficult to
prove with the techniques of the present article. The main obstacle is that
both the directional second microlocalization of section 5.1 and the third
microlocalization of section 5.2 can be performed only in planes where the
non-concentration estimates hold (this is crucial to Proposition 5.1, point
(3) and Proposition 5.2, point (2)).

  * –

For the proof to hold, such planes need to be rational (that is, they cannot
be dense in the torus). This is needed due to the non-concentration estimates,
which require periodicity of quasimodes in $(d-1)$ directions. Hence, our
techniques allow to deal with one-directional sheets of geodesics contained in
a rational hyperplane of the torus (as in subsection 5.1) and intersections of
such sheets (using the third microlocalization), but not with cases where such
sheets of geodesics are contained in irrational planes. Thus, they allow to
prove stabilization on $\mathbb{T}^{3}$ in the case where every geodesic is
damped in all normal directions but a finite number of _rational_ ones.

  * –

Using as many additional microlocalizations as the dimension, it may even be
possible to obtain generalizations of this sufficient condition to higher-
dimensional tori as long as the damping is a characteristic function of
polyhedrons with rational faces. However, the generalized geometric control
condition conjectured by N. Burq and P. Gérard in [BG20, Section 5] indicates
that rationality or irrationality of the undamped normal directions to the
geodesics should not be relevant to uniform stabilization. For that reason, we
have chosen not to confront the technicalities of such generalizations until
additional work allows to bypass the non-concentration estimates.

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