# The Damped Wave Equation and other Evolution Problems involving Non-selfadjoint Operators

Julien Royer

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Julien Royer. The Damped Wave Equation and other Evolution Problems involving Non-selfadjoint Operators. Mathematical Physics [math-ph]. Université Paul Sabatier (Toulouse 3), 2022. tel-03905989

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Submitted on 19 Dec 2022

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Manuscrit

présenté pour l'obtention de

l'Habilitation à Diriger des Recherches

par

Julien Royer

de l'Institut de Mathématiques de Toulouse

The Damped Wave Equation
and other Evolution Problems involving
Non-selfadjoint Operators

Soutenue le 12 décembre 2022, devant le jury suivant :

Président du jury :

Nicolas BURQ Professeur, Université Paris-Saclay

Rapporteurs :

Clotilde FERMANIAN KAMMERER Professeure, Université Paris-Est Créteil
Karel PRAVDA-STAROV Professeur, Université Rennes 1
Jared WUNSCH Professeur, Northwestern University

Examinateurs :

Dietrich HÄFNER Professeur, Université Grenoble-Alpes
Xue Ping WANG Professeur, Université de Nantes

Parrain :

Jean-Marc BOUCLET Professeur, Université Toulouse 3
<figure><img src="image_2.png" /></figure>
Remerciements

Je suis très reconnaissant envers Clotilde Fermanian Kammerer, Karel Pravda-Starov et Jared Wunsch qui me font l'immense honneur d'être rapporteurs de ce mémoire. Je les remercie vivement pour leurs lectures attentives et l'intérêt qu'ils ont témoigné pour mes travaux.

Je suis également très honoré que Nicolas Burq et Dietrich Häfner aient accepté d'être membres du jury.

Le but de ce mémoire est de faire un bilan des travaux de recherche qui ont suivi ma thèse, mais je n'oublie pas que l'histoire a commencé un peu plus tôt. Je remercie Xue Ping Wang d'avoir guidé mes premiers pas vers la recherche, le monde dissipatif et les ondes amorties, et je suis très heureux qu'il soit maintenant membre de mon jury d'habilitation.

Enfin je remercie très chaleureusement Jean-Marc Bouclet, qui m'a accueilli à l'I.M.T., qui a toujours été disponible pour répondre à toutes mes questions, mathématiques ou non, qui m'a initié entre autres à l'étude de ces basses fréquences qui occupent encore une part importante de ma recherche, et qui me fait aujourd'hui le plaisir de me parrainer pour mon H.D.R.

Je profite également de l'occasion qui m'est donnée ici de remercier tous ceux qui, d'une façon ou d'une autre, contribuent à mon épanouissement scientifique.

Je pense évidemment en premier lieu à mes collaborateurs. Sans eux, les travaux présentés ici n'existeraient tout simplement pas. Ceux qui m'ont suivi vers les ondes amorties, ceux qui m'ont amené vers de nouveaux problèmes, tous m'ont tant appris. Comme un bon repas, le plaisir de faire des mathématiques n'a véritablement de saveur que lorsqu'il est partagé, et je salue tous les amis avec qui j'ai pu discuter de maths, que cela ait déjà abouti à un théorème, ou pas encore...

Je remercie également tous les collègues qui font de l'université un cadre à la fois convivial et stimulant, où il fait bon travailler. Les amis enseignants et/ou chercheurs pour les discussions en tous genres, ainsi que les gestionnaires, bibliothécaires et informaticiens sans qui rien ne fonctionnerait. Un petit mot aussi pour les étudiants, toujours sympathiques, qui contribuent à l'équilibre enseignement/recherche que j'apprécie tant dans ce métier.

Le dernier mot sera pour mes proches, amis et famille. Pour tous ces bons moments partagés, y compris dans les périodes de doutes. À vous qui m'êtes si précieux, merci !
# Presentation of the thesis

## Avant-propos

This dissertation is about the research I have done since I arrived in Toulouse in 2011, after my Ph.D.

My Ph.D. (2007-2010) was about the high-frequency dissipative Helmholtz equation. More precisely, I proved high-frequency resolvent estimates in [2], and I studied in [3] the semiclassical measure for the solution when the source term concentrates on a bounded submanifold of the Euclidean space. The two papers [4, 5] were published later, but they were already partially contained in my Ph.D. thesis. The work [4] is about the resolvent estimates when the damping can take negative values, and [5] is a generalization of [3] to the case of an unbounded submanifold. These works will not be discussed in this report (except for their links with more recent papers). Neither will be [1], which is an undergraduate work on a completely different subject (modelization of the traffic flow).

The Helmholtz equation is the stationary version of the damped wave equation. After these works, I was motivated by the time-dependent problem. However, to state a result for high frequencies only, one needs a spectral localization. This is an example among many others of an issue completely insignificant in the selfadjoint setting which becomes a real obstacle with damping.

When I arrived in Toulouse, I discussed this question with Jean-Marc Bouclet, and our conclusion was that instead of trying to localize on high-frequencies, we should prove resolvent estimates for low frequencies and then consider the damped wave equation without spectral localization. Since Jean-Marc had just proved low frequency resolvent estimates for the Schrödinger and undamped wave equations at that time, it was a perfect question to begin a collaboration. And it turned out to be a very rich subject.

## Il était une fois l'équation des ondes amorties...

An important part of this thesis will be devoted to the damped wave equation. An introduction on the subject will be given in Chapter 1. Several settings will be considered. On the Euclidean space $\mathbb{R}^d$, the damped wave equation reads

$$ \partial_t^2 u + Pu + a(x)\partial_t u = 0, \quad \text{on } \mathbb{R}_+ \times \mathbb{R}^d, $$

where $P$ is a general Laplace operator (see (1.8) below) and $a(x) \ge 0$ is the absorption index.

I have been interested in the local energy decay for this damped wave equation. In the first results, the purpose was to recover in this setting the results known for the undamped wave equation. As explained above, the main part of the work was the analysis of the contribution of low frequencies.

To see the damped wave equation as a perturbation of the undamped case, we had to assume that the absorption index $a(x)$ decays at infinity. It turned out that when the damping is effective at infinity (for instance if $a(x)$ goes to 1 at infinity) the large time behavior of the solution is different. I studied this new phenomenon in various contexts. Since the technical issues raised by these two aspects are completely different, the discussion about
the damped wave equation is split into two chapters. The case of the wave equation with localized damping is discussed in Chapter 1 while the results about a damping effective at infinity are described in Chapter 2.

Together with the wave equation, I have also considered the local energy decay for the Schrödinger equation. These two problems share many similarities, and it is relevant to discuss them together. In some situations the Schrödinger equation can be simpler, and it has been used as an intermediate step toward the analysis of the wave equation. And finally, the ideas developed for the damped wave equation had an important application to a result about the usual (conservative) Schrödinger equation.

... et d'autres problèmes (pas toujours) linéaires impli-
quant des opérateurs (pas toujours) non-autoadjoints.

In parallel with my main research work on the damped wave equation, I have also developed various collaborations on other topics. They are often questions which arose from scientific discussions with friends and finally turned into a collaboration.

I would say that my field of research is the analysis of partial differential equations involving non-selfadjoint operators, but this is of course not restrictive. Thus in most of these problems there emerges a non-selfadjoint operator, but in some others only selfadjoint operators are involved. Most results are about a particular PDE, but some are about abstract spectral theory. And most problems are linear, but some are about non-linear equations.

All these various settings will be presented separately in Chapter 3. Some are closely related to my favorite topics, some are quite different, but they all could be the starting points of new perspectives for future research.

Finally, the last chapter will be devoted to a description of my preprints, some works in progress, and some discussions about possible future projects. As for my past research, some are questions about the damped wave equation, and some others go in different directions.

I have chosen to present in this thesis all my papers which are not included in or closely related to my Ph.D. However, since it is not possible to give a detailed description for all of them, some will be emphasized with scientific context, mains ideas of the proofs and comments, and for others I will only briefly describe the results.

A list of my publications and pre-publications, labeled from [1] to [27], is given at page 91. The other references used in the text, labeled from [Aaf21] to [Zwo12], are given at the end of the manuscript.
# Contents

<table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>1</td><td>Local Energy Decay for the damped wave equation with localized damping</td><td>1</td></tr><tr><td>1.1</td><td>Local energy decay for the wave equation</td><td>1</td></tr><tr><td>1.2</td><td>Resolvent estimates for the wave operator</td><td>7</td></tr><tr><td>1.2.1</td><td>The wave operator</td><td>7</td></tr><tr><td>1.2.2</td><td>From resolvent estimates to local energy decay</td><td>10</td></tr><tr><td>1.3</td><td>The Mourre commutators method</td><td>11</td></tr><tr><td>1.3.1</td><td>Introduction</td><td>11</td></tr><tr><td>1.3.2</td><td>The dissipative Mourre method</td><td>13</td></tr><tr><td>1.3.3</td><td>The statement</td><td>14</td></tr><tr><td>1.4</td><td>Strategy for the proofs of the resolvent estimates</td><td>16</td></tr><tr><td>1.4.1</td><td>Contribution of intermediate frequencies</td><td>17</td></tr><tr><td>1.4.2</td><td>Contribution of high frequencies</td><td>17</td></tr><tr><td>1.4.3</td><td>Contribution of low frequencies</td><td>18</td></tr><tr><td>1.5</td><td>Optimal resolvent estimate for the Schrödinger equation in an asymptotically conical setting</td><td>20</td></tr><tr><td>1.6</td><td>The damped Schrödinger equation</td><td>21</td></tr><tr><td>1.7</td><td>Asymptotic behavior for the Schrödinger equation</td><td>23</td></tr><tr><td>2</td><td>The damped wave equation with damping at infinity</td><td>27</td></tr><tr><td>2.1</td><td>Introduction</td><td>27</td></tr><tr><td>2.2</td><td>On the Euclidean space</td><td>28</td></tr><tr><td>2.2.1</td><td>The wave operator in the inhomogeneous energy space</td><td>29</td></tr><tr><td>2.2.2</td><td>Local energy decay for the heat equation</td><td>29</td></tr><tr><td>2.2.3</td><td>The contribution of high frequencies</td><td>31</td></tr><tr><td>2.2.4</td><td>A first change of contour - Separation of low and high frequencies</td><td>31</td></tr><tr><td>2.2.5</td><td>The contribution of low frequencies - Comparison with the diffusive profile</td><td>32</td></tr><tr><td>2.3</td><td>On a wave guide with damping on the boundary</td><td>35</td></tr><tr><td>2.3.1</td><td>The setting</td><td>35</td></tr><tr><td>2.3.2</td><td>Spectral properties of H<sub>α</sub></td><td>36</td></tr><tr><td>2.3.3</td><td>Contribution of low frequencies</td><td>37</td></tr><tr><td>2.3.4</td><td>Contribution of high frequencies</td><td>38</td></tr><tr><td>2.3.5</td><td>Local energy decay and asymptotic profile</td><td>39</td></tr><tr><td>2.4</td><td>On a wave guide with damping at infinity</td><td>40</td></tr><tr><td>2.5</td><td>In a periodic medium</td><td>43</td></tr><tr><td>2.5.1</td><td>The asymptotically periodic setting</td><td>43</td></tr><tr><td>2.5.2</td><td>The Floquet-Bloch decomposition and spectral properties on the torus</td><td>44</td></tr><tr><td>2.5.3</td><td>Comparison with the asymptotic profile</td><td>45</td></tr><tr><td>2.6</td><td>The wave equation with highly oscillating damping</td><td>47</td></tr><tr><td>2.7</td><td>The damped Schrödinger equation on a wave guide</td><td>49</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td><b>3 Related topics</b></td>
      <td><b>53</b></td>
    </tr>
    <tr>
      <td>3.1 Spectrum of a non-selfadjoint quantum star graph</td>
      <td>53</td>
    </tr>
    <tr>
      <td>3.1.1 Quantum graphs</td>
      <td>53</td>
    </tr>
    <tr>
      <td>3.1.2 Main spectral results</td>
      <td>54</td>
    </tr>
    <tr>
      <td>3.1.3 Ideas of proof</td>
      <td>56</td>
    </tr>
    <tr>
      <td>3.2 Agmon estimates for non-accretive Schrödinger operators</td>
      <td>58</td>
    </tr>
    <tr>
      <td>3.3 Absence of embedded eigenvalues for a magnetic Laplacian</td>
      <td>60</td>
    </tr>
    <tr>
      <td>3.4 Reduction of dimension in an abstract setting</td>
      <td>61</td>
    </tr>
    <tr>
      <td>3.5 Observability for a Kolmogorov equation</td>
      <td>63</td>
    </tr>
    <tr>
      <td>3.5.1 Setting and main result</td>
      <td>63</td>
    </tr>
    <tr>
      <td>3.5.2 Spectral properties of the Kolmogorov operator</td>
      <td>64</td>
    </tr>
    <tr>
      <td>3.5.3 Positive result: observability in large time</td>
      <td>65</td>
    </tr>
    <tr>
      <td>3.5.4 Negative result: non-observability in small time</td>
      <td>67</td>
    </tr>
    <tr>
      <td>3.6 Nonlinear evolution equations with singular potentials</td>
      <td>68</td>
    </tr>
    <tr>
      <td>3.6.1 A singularly perturbed Gross-Pitaevskii equation</td>
      <td>68</td>
    </tr>
    <tr>
      <td>3.6.2 A nonlinear Klein-Gordon equation with delta potentials</td>
      <td>70</td>
    </tr>
    <tr>
      <td><b>4 Projects and perspectives</b></td>
      <td><b>73</b></td>
    </tr>
    <tr>
      <td>4.1 New questions about local energy decay</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.1.1 Asymptotic profile for the wave equation</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.1.2 Systems on non-selfadjoint equations</td>
      <td>74</td>
    </tr>
    <tr>
      <td>4.1.3 Local energy decay in more general geometric settings</td>
      <td>77</td>
    </tr>
    <tr>
      <td>4.2 Null-controllability for a Grushin equation</td>
      <td>78</td>
    </tr>
    <tr>
      <td>4.3 Discrete spectrum at the strong magnetic field limit on a curved wave guide</td>
      <td>80</td>
    </tr>
    <tr>
      <td>4.4 Spectral theory on discrete and quantum graphs</td>
      <td>82</td>
    </tr>
    <tr>
      <td>4.4.1 Spectrum of the wave equation on a non-compact star-graph</td>
      <td>82</td>
    </tr>
    <tr>
      <td>4.4.2 Other model problems on non-compact quantum graphs</td>
      <td>83</td>
    </tr>
    <tr>
      <td>4.4.3 Resolvent estimates for the Schrödinger operator on a discrete graph</td>
      <td>84</td>
    </tr>
    <tr>
      <td>4.5 Non-linear problems</td>
      <td>85</td>
    </tr>
    <tr>
      <td>4.5.1 A non linear problem on a quantum graph</td>
      <td>85</td>
    </tr>
    <tr>
      <td>4.5.2 Nonlinear damped wave equations</td>
      <td>86</td>
    </tr>
    <tr>
      <td><b>Bibliography</b></td>
      <td><b>91</b></td>
    </tr>
    <tr>
      <td>Publications and preprints</td>
      <td>91</td>
    </tr>
    <tr>
      <td>References</td>
      <td>93</td>
    </tr>
  </tbody>
</table>
# Chapter 1

## Local Energy Decay for the damped wave equation with localized damping

In this chapter and the following, we discuss the local energy decay for the damped wave equation on unbounded domains.

There are several difficulties. Because of the damping, the corresponding operator is not selfadjoint, which raises some technical issues compared to the undamped case. In the first result (see Theorem 1.1) the purpose was to generalize to this setting some estimates already known without damping. In a second step, I considered problems specific to the dissipative case (see all the results of Chapter 2). And finally, the strategies developed for these dissipative settings gave some new ideas useful even for the selfadjoint setting (see Section 1.7).

The damped wave equation was already well studied on compact domains. In this case, there is no problem with the contribution of low frequencies (from the spectral point of view, 0 does not belong to the spectrum of the corresponding operator). In all my works, I consider the wave equation on unbounded domains. Results about the contribution of high frequencies are essentially contained in my Ph.D. [Roy10] and will not be discussed in details here. My research after my Ph.D. mostly concerns the contribution of low frequencies, and the analysis is completely different. This is the main subject of this thesis.

Finally, an important difficulty is due to the wave equation itself. In many aspects, the problems of the local energy decay for the wave and Schrödinger equations are similar. However, we will see all along this thesis that there are difficulties specific to the wave equation, especially in the dissipative case.

We begin in this chapter with the damped wave equation with localized damping and closely related problems. We will discuss the papers [6, 7, 9, 10, 11, 23]. The earlier results proved in [2, 3, 4, 5], that we do not present in this report, are also related to this setting.

### 1.1 Local energy decay for the wave equation

**The model case.** In this first chapter, we consider the wave equation in an asymptotically Euclidean setting. Given $f_0$ in $H^1(\mathbb{R}^d)$ and $g_0$ in $L^2(\mathbb{R}^d)$, the model case is the usual free wave equation on $\mathbb{R}^d$

$$ \partial_t^2 u_0 - \Delta u_0 = 0, \qquad (1.1) $$

with initial condition

$$ u_0(0) = f_0, \quad \partial_t u_0(0) = g_0. \qquad (1.2) $$
This is the simplest model for the description of waves. In many propagation phenomena, there are also important non-linear effects, but this simple model is already suitable in many concrete situations, such as the propagation of acoustic or electromagnetic waves (if the electromagnetic field does not depend on time then the Maxwell equations can be decoupled in two wave equations for the electric and magnetic fields).

A large part of this thesis deals with the long time behavior of a solution of the wave equation. A relevant way to measure the evolution of a wave is to look at the localization of its energy:

$$E_0(u_0; t) = \int_{\mathbb{R}^d} (|\nabla u_0(t, x)|^2 + |\partial_t u_0(t, x)|^2) \, dx. \quad (1.3)$$

The global energy is a constant of the motion. However, we can look at the distribution of the quantity $|\nabla u_0(t)|^2 + |\partial_t u_0(t)|^2$ to see where the wave is propagating.

An important property of the wave equation is the finite speed of propagation. With all the physical constants set to 1, a wave propagates at speed at most 1. If $f_0$ and $g_0$ are supported in the ball $B(0, R)$ of radius $R > 0$, then $u_0(t)$ is supported in $B(0, R + t)$.

In odd dimension, the wave actually propagates exactly at speed 1. This is the strong Huyghens principle. This implies that there is no diffusion of waves. If a light bulb is turned on for one second, someone looking at the bulb will see it for exactly one second, even from very far away.

At least for regular initial data, we have an explicit expression for the solution of (1.1)-(1.2) (see for instance [CH89, Eva98]). For $d = 3$, we have

$$u_0(t, x) = \frac{1}{|\partial B(x, t)|} \int_{\partial B(x, t)} (f_0(y) + \nabla f_0(y) \cdot (y - x) + tg_0(y)) \, dy.$$

If the initial condition $(f_0, g_0)$ is supported in the ball $B(0, R)$, then $u_0(t)$ is supported in $B(0, t + R) \setminus B(0, t - R)$, and in particular the energy in any compact of $\mathbb{R}^d$ vanishes after finite time. For instance,

$$\forall t \ge 2R, \quad \|\nabla u_0(t)\|_{L^2(B(0,R))}^2 + \|\partial_t u_0(t)\|_{L^2(B(0,R))}^2 = 0. \quad (1.4)$$

The same applies in any odd dimension $d \ge 3$. In dimension 1, it is still true that $\partial_x u_0(t)$ and $\partial_t u_0(t)$ are supported in $[-t - R, -t + R] \cup [t - R, t + R]$, but this is not necessarily the case for the solution itself, as can be seen from the classical d'Alembert formula

$$u_0(t, x) = \frac{f_0(x + t) + f_0(x - t)}{2} + \frac{1}{2} \int_{x-t}^{x+t} g_0(s) \, ds.$$

The behavior is slightly different in even dimension. If one drops at time $t = 0$ a pebble in a (two dimensional) pond, a circular wave will emanate outward. The pond is perfectly calm outside this circular wave, but not inside as would be the case with the strong Huyghens principle. Even if the magnitude decays rapidly, the excited state will persist indefinitely. Oral communication would be quite difficult in even dimension.

In this case, the free wave equation can be solved by the descent method (we can see a wave in dimension 2 as the trace on $\mathbb{R}^2 \times \{0\}$ of a wave in dimension 3 which does not depend on the last variable). We can see, as expected, that with $f_0$ and $g_0$ compactly supported the wave does not vanish in a fixed compact, even for large times. However, the wave mostly propagates at speed close to 1 and the energy on a compact goes to 0 as $t \to +\infty$. More precisely, we can show that for $R > 0$ there exists $C_R > 0$ such that for $(f_0, g_0)$ supported in $B(0, R)$ the solution $u_0$ of (1.1)-(1.2) satisfies

$$\|\nabla u(t)\|_{L^2(B(0,R))}^2 + \|\partial_t u(t)\|_{L^2(B(0,R))}^2 \le C_R \langle t \rangle^{-2d} (\|\nabla f_0\|_{L^2(\mathbb{R}^d)}^2 + \|g_0\|_{L^2(\mathbb{R}^d)}^2). \quad (1.5)$$
This is what we call the local energy decay. In any dimension, the energy of a wave in any compact goes to 0 or, equivalently (since the global energy is constant), the energy of the wave spreads to infinity. Notice that we have to consider localized initial data. Otherwise, a wave coming from far away could reach a fixed compact after a long time and (1.4)-(1.5) could not hold.

It is then natural to wonder if the same phenomenon occurs for waves in more general settings, when explicit descriptions of the solutions are not available. Thus, the first question is whether the energy of the wave on any compact still goes to 0 for any (localized) initial condition in perturbed settings. When this is the case, the next issues are the rate of decay and the uniformity with respect to the initial data (the time $T = 2R$ in (1.4) and the constant $C_R$ in (1.5) depend on $R$ but not on $f_0, g_0$ supported in $B(0, R)$, and in particular the right-hand side in (1.5) is proportional to the initial energy).

These questions are interesting for themselves, since they give qualitative properties for the long time behavior of the solutions. For instance, if we measure a wave on a bounded subdomain, it can be useful to know for how long there is a relevent information in this region. But the local energy decay, and the resolvent estimates that we develop for the proofs, are also crucial for the mathematical analysis of other important properties (scattering theory, non-linear problems, etc.).

Together with the wave equation, we also consider the Schrödinger equation, which plays a central role in quantum mechanics. The model problem is

$$ \begin{cases} -i\partial_t u_0 - \Delta u_0 = 0, & \text{on } \mathbb{R}_+ \times \mathbb{R}^d, \\ u_{0|t=0} = f_0, & \text{on } \mathbb{R}^d, \end{cases} \qquad (1.6) $$

for some $f_0 \in L^2(\mathbb{R}^d)$. The $L^2$-norm of the solution is constant:

$$ \|u_0(t)\|_{L^2(\mathbb{R}^d)}^2 = \|f_0\|_{L^2(\mathbb{R}^d)}^2. $$

There is no analog of the Huyghens principle for the Schrödinger equation. However, the mass also escapes to infinity: for $R > 0$ there exists $C_R > 0$ such that if $f_0$ is supported in $B(0, R)$ then the solution $u_0(t)$ of (1.6) satisfies

$$ \|u(t)\|_{L^2(B(0,R))}^2 \le C_R \langle t \rangle^{-d} \|u_0\|_{L^2(\mathbb{R}^d)}^2. \qquad (1.7) $$

This is similar to (1.5). The difference is that there is no distinction due to the parity of the dimension $d$. As for the wave equation, we are interested in this local energy decay for the Schrödinger equation in more general settings.

**The asymptotically Euclidean setting.** Our main purpose in this chapter is to discuss the local energy decay for the damped wave equation in an asymptotically Euclidean setting. We introduce simultaneously the setting for the Schrödinger equation.

We consider on $\mathbb{R}^d$ a general Laplacian of the form

$$ P = -\frac{1}{w(x)} \operatorname{div} G(x) \nabla, \qquad (1.8) $$

where the matrix $G(x)$ and the scalar $w(x)$ are positive and bounded. More precisely, we assume that there exist $G_{\min}, G_{\max}, w_{\min}, w_{\max} > 0$ such that, for all $x \in \mathbb{R}^d$ and $\xi \in \mathbb{R}^d$,

$$ G_{\min} |\xi|^2 \le \langle G(x)\xi, \xi \rangle_{\mathbb{R}^d} \le G_{\max} |\xi|^2 \quad \text{and} \quad w_{\min} \le w(x) \le w_{\max}. \qquad (1.9) $$

We also set

$$ -\Delta_G = -\operatorname{div} G(x) \nabla. $$
The definition of P includes in particular the case of the standard Laplace operator (with $G(x) = \text{Id}$ and $w(x) = 1$), a Laplacian in divergence form (with $w(x) = 1$, so that $P = -\Delta_G$) or the Laplacian associated with a metric. We recall that the Laplace-Beltrami operator associated with the metric $g(x) = (g_{j,k}(x))_{1\le j,k\le d}$ is given by

$$ \mathrm{div}_{\mathbf{g}} \nabla_{\mathbf{g}} = \frac{1}{|g(x)|^{\frac{1}{2}}} \sum_{j,k=1}^{d} \frac{\partial}{\partial x_j} |g(x)|^{\frac{1}{2}} g^{j,k}(x) \frac{\partial}{\partial x_k}, $$

where $|g(x)| = |\det(g(x))|$ and $(g^{j,k}(x))_{1\le j,k\le d} = g(x)^{-1}$. Then $P_{\mathbf{g}} = -\mathrm{div}_{\mathbf{g}} \nabla_{\mathbf{g}}$ is of the form (1.8) with $w = |g|^{\frac{1}{2}}$ and $G = |g|^{\frac{1}{2}} g^{-1}$.

We assume that P is associated with a long range perturbation of the flat metric. This means that $G(x)$ and $w(x)$ are long range perturbations of $\text{Id}$ and $1$, respectively. For some $\rho_0 \in ]0, 1]$ there exist constants $C_\alpha > 0$, $\alpha \in \mathbb{N}^d$, such that for all $x \in \mathbb{R}^d$,

$$ |\partial^\alpha (G(x) - \text{Id})| + |\partial^\alpha (w(x) - 1)| \le C_\alpha \langle x \rangle^{-\rho_0 - |\alpha|}. \quad (1.10) $$

Here and everywhere below we use the standard notation $\langle x \rangle = (1 + |x|^2)^{\frac{1}{2}}$.

For the wave equation we also introduce the absorption index $a(x)$. It is smooth, bounded, takes non-negative values, and it is of short range: choosing the constants $C_\alpha$ larger if necessary, we have

$$ |\partial^\alpha a(x)| \le C_\alpha \langle x \rangle^{-1-\rho_0-|\alpha|}. \quad (1.11) $$

The short range assumption is not just a technical issue. If $a$ decays slower at infinity, then the results are different. This will be discussed in the next chapter.

On the other hand, the case $a = 0$ is allowed in this setting, so the results about the damped wave equation presented in this chapter include in particular the case of the undamped wave equation.

We consider on $\mathbb{R}^d$ the (possibly) damped wave equation

$$ \begin{cases} \tilde{\partial}_t^2 u + Pu + a(x)\partial_t u = 0, & \text{on } \mathbb{R}_+ \times \mathbb{R}^d, \\ (u, \partial_t u)|_{t=0} = (f, g), & \text{on } \mathbb{R}^d, \end{cases} \quad (1.12) $$

where $(f, g) \in H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. For $f \in L^2(\mathbb{R}^d)$ we also consider the Schrödinger equation

$$ \begin{cases} -i\partial_t u + Pu = 0, & \text{on } \mathbb{R}_+ \times \mathbb{R}^d, \\ u|_{t=0} = f, & \text{on } \mathbb{R}^d. \end{cases} \quad (1.13) $$

In this asymptotically Euclidean setting, we define the energy of the wave by

$$ E(u; t) = \int_{\mathbb{R}^d} \left( \langle G(x) \nabla u(t, x), \nabla u(t, x) \rangle_{\mathbb{R}^d} + w(x) |\partial_t u(t, x)|^2 \right) dx. \quad (1.14) $$

This is equivalent to the usual energy (1.3), in the sense that there exists $C \ge 1$ such that $C^{-1}E_0(u; t) \le E(u; t) \le CE_0(u; t)$. This definition of the energy is adapted to the geometry of the problem, since with this choice a solution of (1.12) has a non-increasing energy. More precisely, we formally have

$$ \frac{d}{dt} E(u; t) = - \int_{\mathbb{R}^d} a(x) w(x) |\partial_t u(t, x)|^2 dx \le 0. $$

For the damped wave equation, the decay of the global energy is already an interesting issue. However, the damping can be 0, and in any case it is small at infinity, so the waves at infinity are not really damped and we cannot expect a uniform decay for the global energy.

Here we are interested in the local energy decay. We are going to prove this property about the time dependent problem via a spectral approach. In particular, we can separate the contributions of high and low frequencies.
**The non-trapping condition.** The contribution of high frequencies, though highly non-trivial, is now quite well understood. It is known that the propagation of a high frequency wave is well approximated by the classical rays of light (for instance, the geometric optics is a good approximation of wave optics if the wavelength is very small compared to the other lengths of the problem). This is made rigorous with semiclassical analysis (see for instance [Zwo12]). For a free wave, rays of light follow straight lines at constant speed (and necessarily escape to infinity). With an obstacle, they bounce on the obstacle following the laws of geometric optics (there are subtle behaviors for rays tangent to the boundary, which we do not discuss here). For a metric g (corresponding to an inhomogeneous refraction index), the rays of light are the geodesics. In general, for $(x, \xi) \in \mathbb{R}^{2d} \simeq T^*\mathbb{R}^d$ we set

$$p(x, \xi) = \langle w(x)^{-1}G(x)\xi, \xi \rangle_{\mathbb{R}^d}. \qquad (1.15)$$

Then rays of light are the solutions of the Hamiltonian problem associated with $p$. For $(x_0, \xi_0) \in \mathbb{R}^{2d}$ we denote by $(x(t; x_0, \xi_0), \xi(t; x_0, \xi_0))$ the solution of

$$\begin{cases} x'(t) = \partial_{\xi} p(x(t), \xi(t)), \\ \xi'(t) = -\partial_{x} p(x(t), \xi(t)), \\ x(0) = x_0, \quad \xi(0) = \xi_0. \end{cases}$$

Since high frequency waves follow these classical trajectories, one expects that they all escape to infinity if and only if all rays of light go to infinity (for the space variable). This is the non-trapping condition:

$$\forall (x_0, \xi_0) \in p^{-1}(\{1\}), \quad |x(t; x_0, \xi_0)| \xrightarrow{t \to \pm\infty} +\infty. \qquad (1.16)$$

The motion of rays of light in the phase space is only an approximation and, in fact, the local energy always goes to 0. However, the higher the frequency is, the more accurate the approximation is. So without (1.16) the high frequency wave stays trapped for a long time, and the local energy decay is very slow. Thus, without (1.16), the estimate is not uniform (proportional to $E(u; 0)$) but we have a loss of derivative (it is proportional to $\|f\|_{H^{s+1}}^2 + \|g\|_{H^s}^2$ for some $s > 0$).

The behavior of the contribution of low frequencies is completely different. Compared to high frequencies, the local energy decay for the contribution of low frequencies is always uniform with respect to the initial data. On the other hand, while the contribution of high frequencies decays very fast (at least for regular initial data) the contribution of low frequencies is responsible for the lack of time decay. For instance, the rate of decay in (1.5) is governed by low frequencies. This question will be the main issue in this and the next chapters.

**Literature about the undamped case.** There is an important literature about the local energy decay for the wave equation. An early result is [Mor61], where decay at rate $1/t$ is proved by a multiplier method for the free wave outside a star-shaped obstacle in dimension 3 (with a Dirichlet boundary condition). Then exponential decay is proved in [LMP63] via an analysis of the corresponding semigroup. In [LP62] it is proved that the local energy for the wave in a general exterior domain goes to 0. See also [Zac66] for more general hyperbolic equations. In [LP72], the Lax-Phillips method is adapted to even dimensions. We refer to the book [LP67] (or the revised version [LP89]) for a review of this theory. The non-trapping condition is already mentioned there.

It is proved in [Ral69] that this non trapping condition is necessary to have uniform local energy decay. Local energy decay outside non-trapping obstacles is considered in [Mor75, Str75, MRS77], via the existence of an escape function (an escape function is a function on the phase space which is increasing along the Hamiltonian flow). It is proved in [MRS77] that we have decay at rate $t^{-1}$ in even dimension, and exponential decay in dimension 3.
This question is then discussed in [Vai75]. Compared to the previous strategies, the properties of the time-dependent problem are now deduced from the analysis of the stationary problem. Another important step is the analysis of [Mel79], based on the propagation of singularities of [MS78]). In particular, a decay at rate $t^{-d}$ is proved in even dimensions. This has then been improved in [Kaw93]. We also refer to [Vod99] for a more general setting. Finally, it is proved in [Bur98] that outside any compact smooth obstacle we have at least logarithmic decay if we allow a loss of regularity.

The stationary problem of the wave equation (the Helmholtz equation) is closely related to the one for the Schrödinger equation. Then local energy decay has also been proved for the latter. See [Rau78] for an exponentially decaying potential, [JK79] for more precise asymptotics (of the resolvent and the propagator) and [Mur82] for more general operators. The case of an exterior domain is discussed in [Tsu84].

Finally, most of the recent papers deal simultaneously with the Schrödinger and wave equations. We refer to [Bou11a, BH12] for estimates with an $\varepsilon$-loss on an asymptotically Euclidean setting (the Laplacian is a Laplace-Beltrami operator associated with a long-range perturbation of the standard metric on $\mathbb{R}^d$). This means that the decay rate for the local energy is of size $O(t^{-d+\varepsilon})$ for the Schrödinger equation and $O(t^{-2d+\varepsilon})$ for the wave equation. The $\varepsilon$-loss has finally been removed in [BB21]. The method does not see the parity of the dimension, so this final result is optimal for Schrödinger or for the wave in even dimension, but not for the wave in odd dimension. However, it is proved [BH13] that if the metric goes faster to the flat metric at infinity then we can recover a better estimate on the local energy in odd dimensions.

In these works, the time decay is proved from resolvent estimates. And the main contributions of these papers is the analysis of low frequency resolvent estimates. High frequency resolvent estimates were already understood for the Schrödinger operator in close settings. See for instance [RT87] for the Schrödinger operator with a potential, [Rob92] for a second order perturbation of the Laplacian and [Bur02] for a general compactly supported perturbation of the Laplacian in an exterior domain (via the contradiction argument using semiclassical defect measures [Gér91], as also used in [Leb96]). Low frequency resolvent estimates were also already discussed in [Bou11b, BH10]. Earlier papers also considered the limiting absorption principle at zero energy in some particular settings (see for instance [Wan06, DS09] and references therein).

**About the damped wave equation.** Here we are mainly interested in the damped wave equation. The stabilization of the wave equation also has a long history on compact domains. In this case, we consider the global energy. The wave cannot escape to infinity, but it is dissipated and we similarly study the decay to 0. In this setting there is no difficulty with low frequencies, but the analysis of high frequencies is similar. In particular, the analog of the non-trapping condition is the so-called geometric control condition. The energy decays uniformly if and only if all the classical trajectories go through the damping region.

We refer to [RT74] for exponential decay with dissipation in the compact manifold and to [BLR92] for dissipation at the boundary (see also [BG97]). For logarithmic decay with loss of regularity without the control condition we refer to [Leb96] for internal damping and to [LR97] for damping at the boundary. Then there have been several results about intermediate situations, where the geometric control condition does not hold, but the set of undamped rays of light is small in some suitable sense. See for instance [BH07] for the problem on the stadium (giving polynomial decay of the energy), [Chr07] for a situation where the decay is exponential but not uniform, [LL17] for the so-called open book and [BG20] for a rough damping. We also refer to [Sjö00, Ana10] for more results on manifolds without boundary.

In an unbounded domain we have additional difficulties, in particular due to the contribution of low frequencies. As said above, we only consider the local energy decay in this
chapter.

In this setting, the energy on a compact has now two reasons to decay. Either because it escapes to infinity, or because it is dissipated. The expected corresponding condition on classical trajectories is that they should all escape to infinity or go through the damping region. This means that we can allow trapped trajectories if they are damped. We set

$$ \Omega_b = \{ (x_0, \xi_0) \in p^{-1}(\{1\}) : \sup_{t \in \mathbb{R}} |x(t; x_0, \xi_0)| < +\infty \}. $$

Then the condition on classical trajectories (which we can call geometric damping condition) reads

$$ \forall (x_0, \xi_0) \in \Omega_b, \exists t \in \mathbb{R}, \quad a(x(t, x_0, \xi_0)) > 0. \qquad (1.17) $$

In [AK02], L. Aloui and M. Khenissi have considered the wave equation in an exterior domain, with a compactly supported damping, via the theory of Lax-Philipps and the contradiction argument with semiclassical measures. They recover, under the assumption analogous to (1.17) in an exterior domain, the exponential decay of the local energy in odd dimension. A polynomial decay is obtained in [Khe03] in even dimension.

**The main result.** The local energy decay for the damped wave equation (1.12) (in dimension $d \ge 3$) has been proved in [6]. This is a collaboration with Jean-Marc Bouclet. It has then been slightly improved in [11].

Instead of considering compactly supported initial data and the energy on a compact, we choose initial data which decay at infinity and consider a weighted energy, which gives slightly better results. For $\delta \in \mathbb{R}$ we denote by $L^{2,\delta}(\mathbb{R}^d)$ the weighted space $L^2(\langle x \rangle^{2\delta} dx)$ and for $k \in \mathbb{N}$ we denote by $H^{k,\delta}(\mathbb{R}^d)$ the corresponding Sobolev space. The main result about the damped wave equation is the following.

**Theorem 1.1.** Assume that the geometric damping condition (1.17) holds. Let $\varepsilon > 0$ and $\delta > d + \frac{1}{2}$. There exists $C > 0$ such that for $(f,g) \in H^{1,\delta}(\mathbb{R}^d) \times L^{2,\delta}(\mathbb{R}^d)$ and $t \ge 0$ we have

$$ \begin{aligned} & \| \langle x \rangle^{-\delta} \nabla u(t) \|_{L^2(\mathbb{R}^d)} + \| \langle x \rangle^{-\delta} \partial_t u(t) \|_{L^2(\mathbb{R}^d)} \\ & \le C \langle t \rangle^{-(d-\varepsilon)} (\| \langle x \rangle^{\delta} \nabla f \|_{L^2(\mathbb{R}^d)} + \| \langle x \rangle^{\delta} g \|_{L^2(\mathbb{R}^d)}), \end{aligned} $$

where $u$ is the solution of (1.12).

Notice that in even dimension we have an $\varepsilon$-loss compared to (1.5) (where the squares of the norms are considered). When [6] and [11] were written, this was the best known result even for the undamped case. Now the $\varepsilon$-loss has been removed in the undamped case in [BB21], and it is one of the perspectives to improve this result in the general case (see Section 4.1.1).

We discuss this theorem in the next three sections (notice that the presentation is slightly different than in the original papers), and then we will turn to the Schrödinger equation (see in particular the local energy decay for (1.13) in Theorem 1.25 below).

## 1.2 Resolvent estimates for the wave operator

Theorem 1.1 is proved from the spectral point of view. More precisely, we deduce estimates on the time-dependent problem from resolvent estimates for the corresponding wave operator.

### 1.2.1 The wave operator

As usual for a wave equation, we rewrite (1.12) as a first order problem of the form

$$ \begin{cases} \partial_t U(t) - \mathcal{W}U(t) = 0, \quad \forall t \ge 0, \\ U(0) = F. \end{cases} \qquad (1.18) $$
At least formally, $u(t)$ is a solution of (1.12) if and only if $U(t) = (u(t), w\partial_t u(t))$ satisfies (1.18) with $F = (f, wg)$ and

$$ W = \begin{pmatrix} 0 & w^{-1} \\ \Delta_G & -a \end{pmatrix}. \qquad (1.19) $$

We set $\mathcal{H} = H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. It is endowed with the natural norm. We also define the energy space $\mathcal{E}$ as the Hilbert completion of $\mathcal{S} = S(\mathbb{R}^d) \times S(\mathbb{R}^d)$ for the norm defined by

$$ \|(u, v)\|_{\mathcal{E}}^2 = \int_{\mathbb{R}^d} \langle G(x) \nabla u(x), \nabla u(x) \rangle_{\mathbb{R}^d} dx + \int_{\mathbb{R}^d} \frac{|v(x)|^2}{w(x)} dx. $$

In particular, $\mathcal{H}$ is dense in $\mathcal{E}$. We set

$$ \mathrm{Dom}(W) = \{U \in \mathcal{E} : WU \in \mathcal{E}\}, \qquad (1.20) $$

where $WU$ is understood in the sense of distributions. Then we consider on $\mathcal{E}$ the operator $W$ defined by (1.19) on the domain $\mathrm{Dom}(W)$.

Notice that if $u$ is a solution of (1.12) then its energy $E(u; t)$ (see (1.14)) is equal to $\|(u(t), w\partial_t u(t))\|_{\mathcal{E}}^2$, while the usual energy $E_0$ (see (1.3)) corresponds to the square of the usual norm on $\dot{H}^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. As already observed for the energy, this choice of norm on $\mathcal{E}$ is equivalent to the usual one, but it is adapted to the operator $W$, in the sense that with this Hilbert structure the operator $W$ is skew-adjoint if $a = 0$, and $iW$ is dissipative in the general case $a \ge 0$.

There are two possible conventions for dissipativeness. We choose the convention which is usual for Schrödinger operators. We set

$$ \mathbb{C}^{\pm} = \{z \in \mathbb{C} : \pm \mathrm{Re}(z) > 0\} \quad \text{and} \quad \mathbb{C}_{\pm} = \{z \in \mathbb{C} : \pm \mathrm{Im}(z) > 0\}. $$

**Definition 1.2.** We say that the operator $T$ on the Hilbert space $\mathcal{H}$ is dissipative if

$$ \forall \varphi \in \mathrm{Dom}(T), \quad \mathrm{Im} \langle T\varphi, \varphi \rangle_{\mathcal{H}} \le 0. $$

Similarly, $T$ is accretive if

$$ \forall \varphi \in \mathrm{Dom}(T), \quad \mathrm{Re} \langle T\varphi, \varphi \rangle_{\mathcal{H}} \ge 0. $$

Then we say that the dissipative (accretive) operator $T$ is maximal dissipative (maximal accretive) if its resolvent set contains some $-\text{hence any } z \text{ in } \mathbb{C}_+ (z \in \mathbb{C}^-)$.

The other possible definition for dissipativeness, satisfied by $W$ on $\mathcal{E}$, is the property $\mathrm{Re}\langle T\varphi, \varphi \rangle_{\mathcal{H}} \le 0$ (in this case $T$ is dissipative if $(-T)$ is accretive).

With the convention of Definition 1.2, $iW$ is dissipative (or $(-W)$ is accretive) on $\mathcal{E}$ since

$$ \forall U = (u, v) \in \mathrm{Dom}(W), \quad \mathrm{Im} \langle iWU, U \rangle_{\mathcal{E}} = \mathrm{Re} \langle WU, U \rangle_{\mathcal{E}} = -\langle av, v \rangle_{L^2(\mathbb{R}^d)} \le 0. $$

Now let $\zeta \in \mathbb{C}^+$ and $F \in \mathcal{H}$. If we solve formally the equation $(W - \zeta)U = F$ for $U \in \mathrm{Dom}(W)$ we get

$$ (W - \zeta)^{-1} F = \begin{pmatrix} -R(i\zeta)(aw + \zeta w) & -R(i\zeta) \\ w - wR(i\zeta)(\zeta aw + \zeta^2 w) & -\zeta wR(i\zeta) \end{pmatrix} F, \qquad (1.21) $$

where for $z \in \mathbb{C}_+$ we have set

$$ R(z) = (-\Delta_G - iawz - wz^2)^{-1}. \qquad (1.22) $$

We also have

$$ (W - \zeta)^{-1} F = \begin{pmatrix} -\zeta^{-1}(1 + R(i\zeta)\Delta_G) & -R(i\zeta) \\ -wR(i\zeta)\Delta_G & -\zeta R(i\zeta) \end{pmatrix} F. \qquad (1.23) $$
For $R(z)$ we have again chosen a convention consistent with the Schrödinger setting ($z$ belongs to the upper half-plane and $R(z)$ looks like the resolvent of the Schrödinger equation if $a=0$). With this convention we have $R(i\zeta) = (-\Delta_G + aw\zeta + w\zeta^2)^{-1}$.

All this can be made rigorous. We first prove that $R(z)$ is well defined and then we check that the right-hand side of (1.21) or (1.23) defines a bounded inverse of $(W - \zeta)^{-1}$.

**Proposition 1.3.** The operator $(-\Delta_G - iawz - wz^2)$ with domain $H^2(\mathbb{R}^d)$ has a bounded inverse on $L^2(\mathbb{R}^d)$ for all $z \in \mathbb{C}_+$. Moreover, its inverse $R(z)$ extends to a bounded operator from $H^{-1}(\mathbb{R}^d)$ to $H^1(\mathbb{R}^d)$ and there exists $C > 0$ such that for $z \in \mathbb{C}_+$ and $\beta_1, \beta_2 \in \mathbb{N}^d$ with $|\beta_1| \le 1, |\beta_2| \le 1$, we have

$$ \| \partial_x^{\beta_1} R(z) \partial_x^{\beta_2} \|_{L^2(\mathbb{R}^d)} \le \frac{C |z|^{|\beta_1|+|\beta_2|-1}}{\mathrm{Im}(z)}. \quad (1.24) $$

**Proposition 1.4.** The operator $iW$ is maximal dissipative on $\mathcal{E}$. Moreover, for $\zeta \in \mathbb{C}^+$ the resolvent $(W - \zeta)^{-1}$ is given by (1.23), and for $F \in \mathcal{H}$ we also have (1.21).

By the Hille-Yosida Theorem, we deduce from Proposition 1.4 that $W$ generates a contractions semigroup on $\mathcal{E}$, and in particular the problem (1.18) is well posed. Then we can rewrite Theorem 1.1 with $W$ in the weighted energy spaces. For $\delta \in \mathbb{R}$ we denote by $\mathcal{E}^\delta$ the Hilbert completion of $\mathcal{S}$ for the norm given by

$$ \|(u, v)\|_{\mathcal{E}^{\delta}}^{2} = \|\langle x \rangle^{\delta} \nabla u\|_{L^{2}(\mathbb{R}^{d})}^{2} + \|\langle x \rangle^{\delta} v\|_{L^{2}(\mathbb{R}^{d})}^{2}. $$

**Theorem 1.5.** Assume that the geometric damping condition (1.17) holds. Let $\varepsilon > 0$ and $\delta > d + \frac{1}{2}$. There exists $C > 0$ such that for $F \in \mathcal{E}^\delta$ we have

$$ \|e^{tW}F\|_{\mathcal{E}^{-\delta}} \le C \langle t \rangle^{-(d-\varepsilon)} \|F\|_{\mathcal{E}^{\delta}}. $$

The main ingredients of the proof described below are given in [6]. However, in [6] there is a loss of regularity, and the estimate also depends on the $L^2$-norm of the initial condition $u_0$. Then the estimate of [6] is

$$ \| \langle x \rangle^{-\delta} \nabla u(t) \|_{L^2(\mathbb{R}^d)} + \| \langle x \rangle^{-\delta} \partial_t u(t) \|_{L^2(\mathbb{R}^d)} \le C \langle t \rangle^{-(d-\varepsilon)} ( \|u_0\|_{H^{2,\delta}(\mathbb{R}^d)} + \|u_1\|_{H^{1,\delta}(\mathbb{R}^d)} ). $$

Theorem 1.5 is the version proved in [11].

**Remark 1.6.** The fact that $\dot{H}^1$ is not included in $L^2$ raises some difficulties, and one might prefer to work in $\mathcal{H}$ instead of $\mathcal{E}$. This is in some sense more convenient, but this will not give the same results. If we estimate $U$ in (a weighed version of) $\mathcal{H}$, we are not only considering the sizes of the derivatives of the wave (the energy), but also the size of the solution itself, estimated with respect to the initial energy but also the size of the initial condition $f$ in $L^2(\mathbb{R}^d)$. This can be considered –or not– as a drawback. We are also loosing the nice structural properties of the operator $W$, but the resolvent $(W - \zeta)^{-1}$ is still defined on $\mathbb{C}^+$ and the propagator $e^{tW}$ is still defined for $t \ge 0$. We will work in $\mathcal{H}$ in the next chapter, see Section 2.2.1.

Of course, this discussion is irrelevant in situations where $\dot{H}^1$ and $H^1$ are equal as sets with equivalent norms, for instance on a bounded domain with Dirichlet boundary condition. In unbounded domains, the same applies if we work with initial data supported in some fixed compact, as is usually the case in results about the local energy decay. Similarly, if $(f, g)$ is not compactly supported but belongs to some weighted space $H^{1,\delta}(\mathbb{R}^d) \times L^{2,\delta}(\mathbb{R}^d)$, and if we are ready to work in a slightly more restrictive space, then we can use the following generalization of the Hardy inequality (see Lemma 4.1 in [11]). For $\varepsilon > 0$ there exists $C > 0$ such that for all $u_0 \in S(\mathbb{R}^d)$ we have

$$ \| \langle x \rangle^{\delta} u_0 \|_{L^2(\mathbb{R}^d)} \le C \| \langle x \rangle^{\delta+1+\varepsilon} \nabla u_0 \|_{L^2(\mathbb{R}^d)}. \quad (1.25) $$
Remark 1.7. In the literature about the undamped wave equation, it is usual to diagonalise the operator W, and the analysis reduces to a problem involving the operator $\sqrt{P}$ on $L^2(\mathbb{R}^d)$. We will not use this possibility, since with the damping it is no longer possible to diagonalise W. We could certainly deal with the non-diagonal terms, but the gain is not clear at all and we prefer to stay in the (possibly) inconvenient but (certainly) natural space $\mathcal{E}$. Moreover, it seems slightly more natural to prove estimates on $\|\langle x \rangle^{-\delta} \nabla u(t)\|_{L^2(\mathbb{R}^d)}$ rather than on $\|\langle x \rangle^{-\delta} \sqrt{P}u(t)\|_{L^2(\mathbb{R}^d)}$ for the local energy decay.

### 1.2.2 From resolvent estimates to local energy decay

To deduce properties on the wave equation from resolvent estimates for W, we write the propagator $e^{tW}$ in terms of the resolvent $(W - \zeta)^{-1}$, $\zeta \in \mathbb{C}^+$.

By density, it is enough to prove Theorem 1.5 for $F \in \mathcal{S}$. Let $\phi \in C^\infty(\mathbb{R}, [0, 1])$ be equal to 0 on $]-\infty, 1]$ and equal to 1 on $[2, +\infty[$. For $\zeta \in \mathbb{C}^+$ we set

$$F_{\zeta} = \int_{\mathbb{R}} \phi'(s) e^{s(W-\zeta)} F \, ds. \qquad (1.26)$$

We can see $F_\zeta$ as a “regularized in time” version of $F$ (we recover $F$ if we replace $\phi$ by $1_{\mathbb{R}_+}$, hence $\phi'$ by $\delta_0$). Then we have

$$(W - \zeta)^{-1} F_{\zeta} = - \int_{\mathbb{R}} \phi(s) e^{s(W - \zeta)} F \, ds.$$

We still have propagation at finite speed for a wave in our perturbed setting, so for $\delta \ge 0$ and $T > 0$ there exists $C \ge 0$ such that for $s \in [0, T]$ and $F \in \mathcal{S}$ we have (see [11, Lemma 5.1])

$$\|e^{sW}F\|_{\mathcal{E}^\delta} \le C \|F\|_{\mathcal{E}^\delta}.$$

Thus there exists $C > 0$ such that, for all $F \in \mathcal{S}$ and $\zeta \in \mathbb{C}^+$,

$$\|F_{\zeta}\|_{\mathcal{E}^{\delta}} \le C \|F\|_{\mathcal{E}^{\delta}}.$$

Let $\mu > 0$. Given $\tau \in \mathbb{R}$ we write $\zeta$ for $\mu - i\tau$. The interest of considering $F_\zeta$ instead of $F$ is that $(W - \zeta)^{-1}F_\zeta$ decays rapidly as $|\tau| \to \infty$. Then the Fourier inversion formula gives

$$\phi(t)e^{tW}F = \frac{1}{2i\pi} \int_{\mathrm{Re}(\zeta)=\mu} e^{t\zeta}(W-\zeta)^{-1}F_{\zeta} \, d\zeta, \qquad (1.27)$$

where the line $\mathrm{Re}(\zeta) = \mu$ is oriented from top to bottom (from $\mu + i\infty$ to $\mu - i\infty$). Since $\|e^{tW}\|_{\mathcal{L}(\mathcal{E})} \le 1$, the estimate of Theorem 1.5 is clear for $t$ in a compact. It is then enough to estimate (1.27), which coincides with $e^{tW}F$ for $t \ge 2$.

We deal separately with the contributions of high and low frequencies. We consider $\chi_{\mathrm{low}} \in C_0^\infty(\mathbb{R}, [0, 1])$ equal to 1 on a neighborhood of 0, and $\chi_{\mathrm{high}} = 1 - \chi_{\mathrm{low}}$. For $\ast \in \{\mathrm{low}, \mathrm{high}\}$ and $t \ge 2$ we set

$$\mathcal{U}_{\ast, \mu}(t)F = \frac{1}{2i\pi} \int_{\mathrm{Re}(\zeta)=\mu} \chi_{\ast}(\mathrm{Im}(\zeta))e^{t\zeta}(W-\zeta)^{-1}F_{\zeta} \, d\zeta. \qquad (1.28)$$

We set $\mathbb{D} = \{z \in \mathbb{C} : |z| \le 1\}$. To estimate $\mathcal{U}_{\mathrm{high}, \mu}(t)F$, we need estimates for $(W - \zeta)^{-1}$. We have the following result.

**Theorem 1.8.** Assume that the geometric damping condition (1.17) holds. Let $n \in \mathbb{N}^*$ and $\delta > n - \frac{1}{2}$. There exists $C > 0$ such that for $\zeta \in \mathbb{C}^+ \setminus \mathbb{D}$ we have

$$\|(W - \zeta)^{-n}\|_{\mathcal{L}(\mathcal{E}^\delta, \mathcal{E}^{-\delta})} \le C.$$
The uniform estimates of Theorem 1.8 are not enough to estimate directly the integral $\mathcal{U}_{\text{high},\mu}(t)F$. This is the reason why we had a loss of derivative in [6]. This point has been improved in [11], and we finally have the following result.

**Proposition 1.9.** Let $\gamma \ge 0$ and $\delta > \gamma + \frac{1}{2}$. There exists $C \ge 0$ such that for $F \in \mathcal{S}$, $\mu \in ]0, 1]$ and $t \ge 1$ we have

$$ \|\mathcal{U}_{\text{high},\mu}(t)F\|_{\mathcal{E}-\delta} \le C \langle t \rangle^{-\gamma} e^{t\mu} \|F\|_{\mathcal{E}\delta}. $$

Notice that under the geometric condition (1.17), we can estimate uniformly as many derivatives as we wish for the resolvent if the weight is strong enough. Then we get a fast decay for $\mathcal{U}_{\text{high},\mu}(t)F$. This is not the case for the contribution of $\mathcal{U}_{\text{low},\mu}(t)F$, which depends on the resolvent estimates for $\zeta \in \mathbb{C}^+$ close to 0.

**Theorem 1.10.** Let $n \in \mathbb{N}^*$, $\delta > n - \frac{1}{2}$ and $\varepsilon > 0$. There exists $C > 0$ such that for $\zeta \in \mathbb{C}^+ \cap \mathbb{D}$ we have

$$ \|(W - \zeta)^{-n}\|_{\mathcal{L}(\mathcal{E}^{\delta}, \mathcal{E}^{\delta-\delta})} \le C |\zeta|^{\min(0, d-n-\varepsilon)}. $$

For low frequencies, it is not difficult to convert the resolvent estimates into time decay. We refer to [6, 11] or [BB21, Section 5].

**Proposition 1.11.** Let $\delta > d + \frac{1}{2}$ and $\varepsilon > 0$. There exists $C > 0$ such that for all $F \in \mathcal{S}$ and $\mu > 0$ we have

$$ \|\mathcal{U}_{\text{low},\mu}(t)F\|_{\mathcal{E}-\delta} \le C \langle t \rangle^{-(d-\varepsilon)} e^{t\mu} \|F\|_{\mathcal{E}\delta}. $$

Applying Propositions 1.9 and 1.11 and letting $\mu$ go to 0, we deduce Theorem 1.5. Now we focus on the main ingredients for the proofs of the resolvent estimates. In Section 1.3 we introduce the abstract commutators method, and in Section 1.4 we explain how it is used to prove Theorems 1.8 and 1.10.

# 1.3 The Mourre commutators method

## 1.3.1 Introduction

A classical argument to prove resolvent estimates for a selfadjoint Schrödinger operator near the real axis is the commutators method of Mourre.

Let $H$ be a selfadjoint operator on a Hilbert space $\mathcal{H}$. Given another selfadjoint operator $A$ (the conjugate operator) we have formally

$$ \frac{d}{dt} \langle Ae^{-itH} \varphi, e^{-itH} \varphi \rangle_{\mathcal{H}} = \langle i[H, A]e^{-itH} \varphi, e^{-itH} \varphi \rangle_{\mathcal{H}}. $$

Thus if $i\mathcal{ad}_A(H) = i[H, A]$ is a positive operator, the observation $\langle Ae^{-itH}\varphi, e^{-itH}\varphi \rangle_{\mathcal{H}}$ is an increasing function of time. In particular, it is easy to see that if $i[H, A] \ge c_0$ for some $c_0 > 0$ then $H$ cannot have an eigenvalue. Before E. Mourre, there were already important results about the nature of the spectrum of $H$ based on a positive commutator assumption (see for instance [Put67]).

The great contribution of Mourre in [Mou81] is that the positive commutator assumption is localized in energy with respect to $H$. Given an interval $J$ of $\mathbb{R}$, the assumption is then

$$ \mathbb{1}_J(H)[H, iA]\mathbb{1}_J(H) \ge c_0 \mathbb{1}_J(H), \quad (1.29) $$

where $c_0 > 0$ and $\mathbb{1}_J(H)$ is the spectral projection of $H$ on $J$.

For the free Laplacian $H = -\Delta$ on $\mathcal{H} = L^2(\mathbb{R}^d)$, we can use the generator of dilations

$$ A = -\frac{x \cdot i\nabla + i\nabla \cdot x}{2} = -\frac{id}{2} - x \cdot i\nabla. \quad (1.30) $$
This is the quantization of the symbol $(x, \xi) \mapsto x \cdot \xi$, which is an escape function for the Laplacian (the Poisson bracket $\{\xi^2, x \cdot \xi\}$ is positive on $p^{-1}(\{1\}) = \{(x, \xi) : |\xi| = 1\}$). In this case we have $i[H, A] = 2H$. This is not greater than a positive constant, but (1.29) holds for any $J \subset \subset \mathbb{R}_+^* = ]0, +\infty[$ with $c_0 = 2 \inf(J) > 0$.

From (1.29), Mourre deduced important properties such as the limiting absorption principle and the absolute continuity of the spectrum of $H$ in $J$. For this, he proves in particular uniform resolvent estimates of the form

$$ \| \langle A \rangle^{-\delta} (H-z)^{-1} \langle A \rangle^{-\delta} \|_{\mathcal{L}(H)} \le C, \quad (1.31) $$

where $\delta \ge 1$ (then for $\delta > \frac{1}{2}$ in [Mou83]) and $z \in \mathbb{C}\backslash\mathbb{R}$ has real part in $I \subset \subset J$.

The idea is to prove estimates uniform with respect to $z$ and $\varepsilon > 0$ for

$$ F_{\varepsilon}(z) = \langle A \rangle^{-1} (H_{\varepsilon} - z)^{-1} \langle A \rangle^{-1}, \quad \text{where} \quad H_{\varepsilon} = H - i\varepsilon \mathbb{1}_J(H)[H, iA]\mathbb{1}_J(H). \quad (1.32) $$

The operator $H_\varepsilon$ is dissipative. The dissipative part is not bounded below by a positive constant, but with the positivity given by (1.29) we can prove an estimate of the form

$$ \| \mathbb{1}_J(H)(H-z)^{-1} \langle A \rangle^{-1} \| \lesssim \frac{\| F_{\varepsilon}(z) \|^{\frac{1}{2}}}{\sqrt{\varepsilon}}. $$

And for $\mathrm{Re}(z) \in I \subset \subset J$ we have a uniform estimate for $(1 - \mathbb{1}_J(H))(H - z)^{-1} \langle A \rangle^{-1}$ by the spectral theorem. Since this does not give an estimate for $F_\varepsilon(z)$ uniform in $\varepsilon$ small, we also estimate the derivative of $F_\varepsilon(z)$ with respect to $\varepsilon$. After having removed the factors $\mathbb{1}_J(H)$ we have to estimate

$$ \langle A \rangle^{-1} (H_{\varepsilon} - z)^{-1} [H, A] (H_{\varepsilon} - z)^{-1} \langle A \rangle^{-1}. $$

This is where we use the fact that the dissipative part of $H_\varepsilon$ is a commutator of $H$. The factor $H$ is absorbed by one of the resolvents, and the weight $\langle A \rangle^{-1}$ is used to absorb the factor $A$. We finally get an estimate for the derivative, from which we deduce that the limit $F_0(z)$ is bounded uniformly in $z$. We also refer to [Gér08] for an alternative approach.

The flexibility of the assumption (1.29) makes the result applicable in many situations, and the Mourre method has been extended in many directions. We refer to the book [ABG96] for a general overview on the subject.

Here we are only interested in the uniform estimates, but for the resolvent and its derivatives. The multiple resolvent estimates for a selfadjoint operator are given in [JMP84, Jen85] under additional assumptions about the multiple commutators $ad_A^n(H)$, $n \in \{1, \dots, N+1\}$, for some given $N \ge 2$. The idea is to use other resolvent estimates involving the spectral projections $\mathbb{1}_{\mathbb{R}_\pm}(A)$. For $\delta_1, \delta_2 \ge 0$ such that $\delta_1 + \delta_2 < N - 1$ and $\delta \in ]\frac{1}{2}, N[$ there exists $C > 0$ such that for $z \in \mathbb{C}_+$ with $\mathrm{Re}(z) \in I$ we have

$$ \| \langle A \rangle^{\delta_1} \mathbb{1}_{\mathbb{R}_-}(A) (H-z)^{-1} \mathbb{1}_{\mathbb{R}_+}(A) \langle A \rangle^{\delta_2} \|_{\mathcal{L}(H)} \le C, \quad (1.33) $$

$$ \| \langle A \rangle^{-\delta} (H-z)^{-1} \mathbb{1}_{\mathbb{R}_+}(A) \langle A \rangle^{\delta-1} \|_{\mathcal{L}(H)} \le C, \quad (1.34) $$

$$ \| \langle A \rangle^{\delta-1} \mathbb{1}_{\mathbb{R}_-}(A) (H-z)^{-1} \langle A \rangle^{-\delta} \|_{\mathcal{L}(H)} \le C. \quad (1.35) $$

Then we can deduce similar estimates (with different conditions for the weights) for the powers of $(H - z)^{-1}$. For instance, for $\delta > \frac{3}{2}$ we can write

$$ \begin{aligned} & \| \langle A \rangle^{-\delta} (H-z)^{-2} \langle A \rangle^{-\delta} \| \\ & \le \| \langle A \rangle^{-\delta} (H-z)^{-1} \langle A \rangle^{-(\delta-1)} \| \| \langle A \rangle^{\delta-1} \mathbb{1}_{\mathbb{R}_-}(A) (H-z)^{-1} \langle A \rangle^{-\delta} \| \\ & + \| \langle A \rangle^{-\delta} (H-z)^{-1} \mathbb{1}_{\mathbb{R}_+}(A) \langle A \rangle^{\delta-1} \| \| \langle A \rangle^{-(\delta-1)} (H-z)^{-1} \langle A \rangle^{-\delta} \|, \end{aligned} $$
and use (1.31), (1.34), (1.35) and (1.31) again. We similarly prove analogs of (1.33)-(1.35) for $(H - z)^{-2}$ and then we prove by induction estimates for higher powers of the resolvent.

### 1.3.2 The dissipative Mourre method

In [2], I generalized (1.31) to a parameter-dependent dissipative operator, of the form

$$H_{\lambda} = H_{1,\lambda} - iV_{\lambda} \qquad (1.36)$$

where $H_{1,\lambda}$ is selfadjoint and semi-bounded, and $V_{\lambda}$ is selfadjoint, non-negative and $H_{1,\lambda}$-bounded with relative bound smaller than 1. There is no convenient and general way to localize with respect to the spectrum of a non-selfadjoint operator, so the assumption of positive commutator is localized with respect to the selfadjoint part $H_{1,\lambda}$. A key observation in [2] is that the positive commutator is used to give some dissipativeness to the operator $H_e$ in (1.32). Thus, if the operator is already dissipative, we can use the dissipative part of the operator to weaken the assumption (1.29) on the commutator. More precisely, in [2] the estimate (1.31) is proved under an assumption of the form

$$1_J(H_{1,\lambda})([H_{1,\lambda}, iA_{\lambda}] + \beta_{\lambda}V_{\lambda})1_J(H_{1,\lambda}) \geq c_01_J(H_{1,\lambda}), \qquad (1.37)$$

for some $\beta_{\lambda} \in [0, 1]$. We do not give too much details here since [2] is already described in [Roy10].

For the proofs of Theorems 1.8 and 1.10 we need resolvent estimates for the powers (or, equivalently, for the derivatives) of the resolvent. Thus, we first have to generalize (1.33)-(1.35) and then the estimates for the powers of the resolvent to a dissipative setting.

But this is not enough. The commutators method cannot be applied directly to the operator $iW$, for instance because its selfadjoint part is not semibounded. Thus, we first estimate the derivatives of $R(z)$ by the Mourre method, and then we deduce estimates for the derivative of $(W - \zeta)^{-1}$ by (1.21). Notice that compared to the usual settings, the derivatives of $R(z)$ are not given by its powers. We have

$$R'(z) = R(z)(ia(x)w(x) + 2zw(x))R(z),$$

and we see by induction on $m \in \mathbb{N}$ that $R^{(m)}(z)$ can be written as a sum of terms of the form

$$T_{k; j_1, \dots, j_k}(z) = R(z)(iaw + 2zw)^{j_1} R(z) \dots (iaw + 2zw)^{j_k} R(z), \qquad (1.38)$$

where $k \in \{0, \dots, m\}$ and $j_1, \dots, j_k \in \{0, 1\}$ are such that $2k - \sum_{\ell=1}^k j_\ell = m$. Thus, we also have to take into account the factors $(iaw + 2zw)$ inserted between the factors $R(z)$ (they do not commute with $R(z)$). We proved in [6] that under some suitable behavior between these inserted factors and the conjugate operator (basically, the commutators with the conjugate operator extend to bounded operators), we can generalize to this kind of setting the multiple commutator estimates.

After [6, 11], I have been interested in the wave equation in a domain $\Omega$ with boundary, in particular in wave guides, with damping at the boundary. This will be discussed with more details in the next chapter, but this motivated the analysis of a Schrödinger operator with dissipative Robin boundary condition. Given $a \in W^{1,\infty}(\partial\Omega; \mathbb{R}_+)$, we consider on $\Omega$ the operator

$$H_a = -\Delta, \quad \mathrm{Dom}(H_a) = \{u \in H^2(\Omega), \partial_{\nu}u = iau \text{ on } \partial\Omega\}. \qquad (1.39)$$

In this case, we cannot write $H_a$ as the sum of a selfadjoint operator and a dissipative part as in (1.36), so we cannot apply the results of [2, 6]. However, we can write such a sum for the quadratic form corresponding to $H_a$ on $H^1(\Omega)$:

$$Q_{\alpha}(u) = \int_{\Omega} |\nabla u(x)|^2 \mathrm{d}x - i \int_{\partial\Omega} a(x) |u(x)|^2 \mathrm{d}\sigma(x).$$
In [9], I generalized the dissipative Mourre method to such operators (note that this includes the previous cases). This is inspired by [ABG88] about a similar question for selfadjoint operators.

This version of the dissipative Mourre method has been applied to wave guides in [9]. It also had an unexpected application to the dissipative Schrödinger equation on $\mathbb{R}^d$ (see Remark 1.24 below).

Finally, in [23] I again had to use the Mourre method in a setting which was not included in the previous versions. More details will be given in Section 1.7 below, but for this problem we have to apply the multiple resolvent estimates simultaneously for two different operators. This means that we have to estimate a product with factors given by different resolvents, and with inserted factors as above.

Moreover, in [6, 11] we have applied the Mourre method to (1.22) with $w = 1$, so it was still possible to see $R(z)$ as the resolvent of the parameter-dependant operator $-\Delta_G - iza$ with spectral parameter $z^2$. In [23] (see Section 1.7), there is no damping but $w$ is not necessarily equal to 1, so we include the spectral parameter in the operator, and just see the resolvent as the inverse of a parameter-dependent dissipative operator. Thus, we need a dissipative version of the commutators method even if the operator under study is selfadjoint.

### 1.3.3 The statement

Since it includes all the previous versions, we give here the statement of [23] for the commutators method. Notice that the proofs given in [23] are self-contained.

Let $\mathcal{H}$ and $\mathcal{K}$ be two Hilbert spaces. We assume that $\mathcal{K}$ is densely and continuously embedded in $\mathcal{H}$. We denote by $\mathcal{K}^*$ the space of continuous semilinear forms on $\mathcal{K}$ (we have $\varphi(u_1 + \lambda u_2) = \varphi(u_1) + \bar{\lambda}\varphi(u_2)$ for $\varphi \in \mathcal{K}^*$, $u_1, u_2 \in \mathcal{K}$ and $\lambda \in \mathbb{C}$). We refer to [EE87, pp. 3–4] for a discussion about this choice. We identify $\mathcal{H}$ with $\mathcal{H}^*$ (with this convention, the identification is linear). Then $\mathcal{K}$ is naturally (linearly) identified with a subset of $\mathcal{K}^*$.

We consider a selfadjoint operator $A$ on $\mathcal{H}$ with domain $\mathcal{D}_{\mathcal{H}}$. We set

$$ \mathcal{D}_{\mathcal{K}} = \{\varphi \in \mathcal{K} \cap \mathcal{D}_{\mathcal{H}} : A\varphi \in \mathcal{K}\}. \qquad (1.40) $$

By restriction, $A$ defines an operator $A_{\mathcal{K}}$ on $\mathcal{K}$ with domain $\mathcal{D}_{\mathcal{K}}$. Then $\mathcal{D}_{\mathcal{K}}$ is endowed with the graph norm of $A_{\mathcal{K}}$. We can see $A_{\mathcal{K}}$ as an operator in $\mathcal{L}(\mathcal{D}_{\mathcal{K}}, \mathcal{K})$ and $A_{\mathcal{K}}^*$ maps $\mathcal{K}^*$ to $\mathcal{D}_{\mathcal{K}}^*$.

For $S \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ we set $\mathrm{ad}_A^0(S) = S$. Then, by induction on $n \in \mathbb{N}^*$, if the commutator $\mathrm{ad}_A^{n-1}(S)A_{\mathcal{K}} - A_{\mathcal{K}}^*\mathrm{ad}_A^{n-1}(S) \in \mathcal{L}(\mathcal{D}_{\mathcal{K}}, \mathcal{D}_{\mathcal{K}}^*)$ extends to an operator in $\mathcal{L}(\mathcal{K}, \mathcal{K}^*)$, then we denote this extension by $\mathrm{ad}_A^n(S)$. We can similarly define commutators in $\mathcal{L}(\mathcal{H}, \mathcal{K})$.

For $Q \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ we set $\mathrm{Re}(Q) = (Q + Q^*)/2$ and $\mathrm{Im}(Q) = (Q - Q^*)/2i$. We say that $Q \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ is non-negative if for all $\varphi \in \mathcal{K}$ we have $\langle Q\varphi, \varphi \rangle_{\mathcal{K}^*, \mathcal{K}} \ge 0$.

All this being set, we consider $Q \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ with negative imaginary part: there exists $c_0 > 0$ such that

$$ Q_+ := -\mathrm{Im}(Q) \ge c_0 \mathcal{I}, $$

where $\mathcal{I} \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ is the natural embedding. By the Lax-Milgram Theorem, $Q$ has an inverse $Q^{-1} \in \mathcal{L}(\mathcal{K}^*, \mathcal{K}) \subset \mathcal{L}(\mathcal{H})$.

**Example 1.12.** If $H$ is a selfadjoint semibounded operator and $z \in \mathbb{C}_+$ is a spectral parameter, we recover the usual setting by choosing $Q = Q(z) = H - z$, seen as an operator from the form domain $\mathcal{K}$ of $H$ to $\mathcal{K}^*$. Then $Q_+ = \mathrm{Im}(z)$. If $H$ is a dissipative operator of the form $H = H_1 - iV$, with $H_1$ selfadjoint and $V \ge 0$, then $Q_+ = V + \mathrm{Im}(z)$.

Our purpose is to prove estimates on $Q^{-1}$. For this, we use the following notion of conjugate operator.
**Definition 1.13.** Let $N \in \mathbb{N}^*$ and $\Upsilon \ge 1$. Let $A$ be a selfadjoint operator on $\mathcal{H}$. We say that $A$ is $\Upsilon$-conjugate to $Q$ up to order $N$ if the following conditions are satisfied.

(H1) For $\varphi \in \mathcal{K}$ we have $\|\varphi\|_{\mathcal{H}} \le \Upsilon \|\varphi\|_{\mathcal{K}}$.

(H2) For all $\theta \in [-1, 1]$ the propagator $e^{-i\theta A} \in \mathcal{L}(\mathcal{H})$ defines by restriction a bounded operator on $\mathcal{K}$.

(H3) The commutator $\mathrm{ad}_A^n(Q)$ is well defined in $\mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ and satisfies $\|\mathrm{ad}_A^n(Q)\|_{\mathcal{L}(\mathcal{K}, \mathcal{K}^*)} \le \Upsilon$ for all $n \in \{1, \dots, N+1\}$.

(H4) There exist $Q_\perp \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ with $\mathrm{Im}(Q_\perp) \le 0$, $Q_\perp^+ \in \mathcal{L}(\mathcal{K}, \mathcal{K}^*)$ non-negative and $\Pi \in \mathcal{L}(\mathcal{H}, \mathcal{K})$ such that

(a) $Q = Q_\perp - iQ_\perp^+$,

(b) $\|Q_\perp^+\|_{\mathcal{L}(\mathcal{K}, \mathcal{K}^*)} \le \Upsilon$, $\|\Pi\|_{\mathcal{L}(\mathcal{H}, \mathcal{K})} \le \Upsilon$, $\|\mathrm{ad}_A(\Pi)\|_{\mathcal{L}(\mathcal{H}, \mathcal{K})} \le \Upsilon$, and for $\varphi \in \mathcal{H}$ we have $\|\Pi\varphi\|_{\mathcal{K}} \le \Upsilon \|\Pi\varphi\|_{\mathcal{H}}$,

(c) $Q_\perp$ has an inverse $R_\perp \in \mathcal{L}(\mathcal{K}^*, \mathcal{K})$ which satisfies $\|(\mathrm{Id}_\mathcal{K} - \Pi)R_\perp\|_{\mathcal{L}(\mathcal{K}^*, \mathcal{K})} \le \Upsilon$ and $\|R_\perp(\mathrm{Id}_{\mathcal{K}^*} - \Pi^*)\|_{\mathcal{L}(\mathcal{K}^*, \mathcal{K})} \le \Upsilon$.

(H5) There exists $\beta \in [0, \Upsilon]$ such that if we set

$$M = \mathrm{Re}(\mathrm{iad}_A(Q)) + \beta Q_+,$$

then $\|M\|_{\mathcal{L}(\mathcal{K}, \mathcal{K}^*)} \le \Upsilon$, $\|\mathrm{ad}_A(M)\|_{\mathcal{L}(\mathcal{K}, \mathcal{K}^*)} \le \Upsilon$, and in the sense of quadratic forms on $\mathcal{H}$ we have

$$\Pi^* M \Pi \ge \Upsilon^{-1} \Pi^* \Pi. \qquad (1.41)$$

*Example 1.14.* For the dissipative operator $H = H_1 - iV$ (as in Example 1.12) and $\mathrm{Re}(z) \in I \subset \subset J$, we recover the setting of [2] by choosing $\Pi = \chi(H_1)$ (with $\chi \in C_0^\infty(\mathbb{R}, [0, 1])$ supported in $J$ and equal to 1 on a neighborhood of $I$), $Q_\perp = H_1 - z$ and $Q_\perp^+ = V$. Then $Q_\perp^{-1}\Pi_\perp = (H_1 - z)^{-1}(1 - \chi)(H_1)$ is bounded uniformly in $z$ by the spectral theorem.

Now we can state the result for a single resolvent. In these estimates, it is important that the constants $C$ do not depend on $Q$ but only on the parameter $\Upsilon$.

**Theorem 1.15.** Let $N \in \mathbb{N}^*$ and $\Upsilon \ge 1$. Assume that $A$ is $\Upsilon$-conjugate to $Q$ up to order $N$.

(i) Let $\delta > \frac{1}{2}$. There exists $C > 0$ which only depends on $\Upsilon$ and $\delta$ such that

$$\| \langle A \rangle^{-\delta} Q^{-1} \langle A \rangle^{-\delta} \|_{\mathcal{L}(\mathcal{H})} \le C. \qquad (1.42)$$

(ii) Assume that $N \ge 2$ and let $\delta_1, \delta_2 \ge 0$ such that $\delta_1 + \delta_2 < N - 1$. There exists $C > 0$ which only depends on $N$, $\Upsilon$, $\delta_1$, $\delta_2$ and such that

$$\| \langle A \rangle^{\delta_1} \mathbb{1}_{\mathbb{R}_-}(A) Q^{-1} \mathbb{1}_{\mathbb{R}_+}(A) \langle A \rangle^{\delta_2} \|_{\mathcal{L}(\mathcal{H})} \le C. \qquad (1.43)$$

(iii) Assume that $N \ge 2$ and let $\delta \in [\frac{1}{2}, N]$. There exists $C > 0$ which only depends on $N$, $\Upsilon$ and $\delta$ such that

$$\| \langle A \rangle^{-\delta} Q^{-1} \mathbb{1}_{\mathbb{R}_+}(A) \langle A \rangle^{\delta-1} \|_{\mathcal{L}(\mathcal{H})} \le C \qquad (1.44)$$

and

$$\| \langle A \rangle^{\delta-1} \mathbb{1}_{\mathbb{R}_-}(A) Q^{-1} \langle A \rangle^{-\delta} \|_{\mathcal{L}(\mathcal{H})} \le C. \qquad (1.45)$$

Then we use the following abstract lemma to deduce multiple resolvent estimates from the estimates of Theorem 1.15.
**Lemma 1.16.** Let $\mathcal{H}$ be a Hilbert space. Let $n \in \mathbb{N}^*$ and $T_1, \dots, T_n \in \mathcal{L}(\mathcal{H})$. For $j \in \{0, \dots, n\}$ we consider on $\mathcal{H}$ a selfadjoint operator $\Theta_j \ge 1$, and $\Pi_j^-, \Pi_j^+ \in \mathcal{L}(\mathcal{H})$ such that $\Pi_j^- + \Pi_j^+ = \mathbb{1}_{\mathcal{H}}$. Let $N \in \mathbb{N}^*$. For $j \in \{1, \dots, n\}$ we assume that there exist $\nu_j \ge 0$, $\sigma_j \in [0, \nu_j]$ and a collection $C_j = \{C_j; (C_{j,\delta_1}, \delta_2); (C_{j,\delta})\}$ of constants such that for $\delta_1, \delta_2 \ge 0$ with $\delta_1+\delta_2 < N-\nu_j$ and $\delta \in [\sigma_j, N]$ we have

$$ \|\Theta_{j-1}^{-\sigma_j} T_j \Theta_j^{-\sigma_j}\|_{\mathcal{L}(\mathcal{H})} \le C_j, \qquad (1.46) $$

$$ \|\Theta_{j-1}^{\delta_1} \Pi_{j-1}^{-} T_j \Pi_j^{+} \Theta_j^{\delta_2}\|_{\mathcal{L}(\mathcal{H})} \le C_{j, \delta_1, \delta_2}, \qquad (1.47) $$

$$ \|\Theta_{j-1}^{\delta-\nu_j} \Pi_{j-1}^{-} T_j \Theta_j^{-\delta}\|_{\mathcal{L}(\mathcal{H})} \le C_{j,\delta}, \qquad (1.48) $$

$$ \|\Theta_{j-1}^{-\delta} T_j \Pi_j^{+} \Theta_j^{\delta-\nu_j}\|_{\mathcal{L}(\mathcal{H})} \le C_{j,\delta}. \qquad (1.49) $$

Let $T = T_1 \dots T_n$. We set

$$ \nu = \sum_{j=1}^{n} \nu_j, \quad \sigma_+ = \sum_{j=1}^{n-1} \nu_j + \sigma_n, \quad \sigma_- = \sigma_1 + \sum_{j=2}^{n} \nu_j. $$

Assume that $N > \nu$. We set $\Pi_- = \Pi_0^-$ and $\Pi_+ = \Pi_n^+$. There exists a collection of constants $C = \{C; (C_{\delta_-, \delta_+}); (C_\delta^-); (C_\delta^+)\}$ which only depend on the constants $C_j$, $1 \le j \le n$, and such that

$$ \|\Theta_0^{-\sigma_+} T \Theta_n^{-\sigma_-}\|_{\mathcal{L}(\mathcal{H})} \le C, \qquad (1.50) $$

for $\delta_-, \delta_+ \ge 0$ such that $\delta_- + \delta_+ < N - \nu$ we have

$$ \|\Theta_0^{\delta_-} \Pi_- T \Pi_+ \Theta_n^{\delta_+}\|_{\mathcal{L}(\mathcal{H})} \le C_{\delta_-, \delta_+}, \qquad (1.51) $$

for $\delta \in [\sigma_-, N[$ we have

$$ \|\Theta_0^{\delta-\nu} \Pi_- T \Theta_n^{-\delta}\|_{\mathcal{L}(\mathcal{H})} \le C_\delta^-, \qquad (1.52) $$

and finally, for $\delta \in [\sigma_+, N[$ we have

$$ \|\Theta_0^{-\delta} T \Pi_+ \Theta_n^{\delta-\nu}\|_{\mathcal{L}(\mathcal{H})} \le C_\delta^+. \qquad (1.53) $$

It is important that the constants in the conclusions of the lemma only depend on the constants in the assumptions. Thus, if for some operators $T_j(z)$, $1 \le j \le n$, the estimates (1.46)-(1.48) are independent of the parameter $z$, then so are the estimates (1.50)-(1.53).

Theorem 1.15 allows to apply Lemma 1.16 with $\nu_j = 1$, $\sigma_j \in ]\frac{1}{2}, 1]$, $\Theta_j = \langle A \rangle$, $\Pi_j^- = \mathbb{1}_{\mathbb{R}^*}(A)$ and $\Pi_j^+ = \mathbb{1}_{\mathbb{R}_+}(A)$, where $A$ is the conjugate operator. Notice that the assumptions of Definition 1.13 are used to prove Theorem 1.15 but no longer play a role to get the multiple resolvent estimates.

Since Lemma 1.16 is completely abstract, it can also be applied to the inserted factors. Roughly, if the commutators of $T_j$ with $A$ are bounded on $\mathcal{H}$, then the assumptions of Lemma 1.16 hold with $\nu_j = \sigma_j = 0$ (see Proposition 3.11 in [23]).

Thus, we can apply Lemma 1.16 to a product of resolvents and inserted factors as in (1.38). Moreover, the resolvents do not have to be associated with the same operator, and they do not even have to be indeed resolvents.

## 1.4 Strategy for the proofs of the resolvent estimates

In this paragraph we explain how we use the dissipative commutators method of Section 1.3 to prove Theorems 1.8 and 1.10. This is an occasion to rewrite some arguments of [6, 11] in the spirit of [23].
### 1.4.1 Contribution of intermediate frequencies

The first step is to prove uniform estimates on $R(z)$ and then on $(W+iz)^{-1}$ for $\mathrm{Im}(z)$ positive and $\mathrm{Re}(z)$ in a compact subset of $\mathbb{R}^* = \mathbb{R}\setminus\{0\}$. For $z \in \mathbb{C}$ we set

$$P_R(z) = \mathrm{Re}(-\Delta_G - iawz - wz^2) = -\Delta_G + aw\mathrm{Im}(z) - w\mathrm{Re}(z^2).$$

With the generator of dilations $A$ defined in (1.30) we have

$$[P_R(z), iA] = 2P_R(z) + K(z) + 2\mathrm{Re}(z^2),$$

where

$$K(z) = \mathrm{div}(x \cdot \nabla G(x))\nabla - \mathrm{Im}(z)(x \cdot \nabla)(aw) + \mathrm{Re}(z^2)(x \cdot \nabla)w - 2aw\mathrm{Im}(z) + 2\mathrm{Re}(z^2)(w - 1).$$

Then we apply a compactness argument. Let $\tau \in \mathbb{R}^*$. Since 0 is not an eigenvalue of $P_R(\tau)$ (see [KT06]), the operator $1_{[-\eta, \eta]}(P_R(\tau))$ goes weakly to 0 as $\eta \to 0$. On the other hand, $1_{[-1,1]}(P_R(\tau))K(\tau)1_{[-1,1]}(P_R(\tau))$ is compact, so $1_{[-\eta, \eta]}(P_R(\tau))K(\tau)1_{[-\eta, \eta]}(P_R(\tau))$ goes to 0 as $\eta \to 0$. For $\eta > 0$ small enough we get

$$1_{[-\eta, \eta]}(P_R(\tau))[P_R(\tau), iA]1_{[-\eta, \eta]}(P_R(\tau)) \ge \frac{3\tau^2}{2} 1_{[-\eta, \eta]}(P_R(\tau)).$$

Then for $\chi \in C_0^\infty(\mathbb{R}, [0, 1])$ supported in $]-\eta, \eta]$ and equal to 1 on $[-\frac{\eta}{2}, \frac{\eta}{2}]$ we get for $z \in \mathbb{C}_+$ close to $\tau$

$$\chi(P_R(z))[P_R(z), iA]\chi(P_R(z)) \ge \tau^2 \chi(P_R(z))^2. \quad (1.54)$$

This is the main assumption in Definition 1.13 (see (1.41) with $Q = (-\Delta_G - iawz - wz^2)$, $\Pi = \chi(P_R(z))$ and $\beta = 0$). We get the estimates of Theorem 1.15 with $Q^{-1} = R(z)$.

Then we use (1.21) to deduce estimates on $(W + iz)^{-1}$. With (1.10) and (1.11) it is classical that the estimates of Theorem 1.15 also hold with $Q^{-1}$ replaced by $a$ or $w$, for any $\delta, \delta_1, \delta_2 \ge 0$, and $\delta - 1$ replaced by $\delta$ in (1.44) and (1.45)). Then the assumptions of Lemma 1.16 are satisfied with $\Theta_j = \langle A \rangle$ and $\nu_j = \sigma_j = 0$. Using also the good commutation properties of $\langle D \rangle = (1-\Delta)^{\frac{1}{2}}$ with $A$, we can finally prove with Lemma 1.16 that the estimates of Theorem 1.15 hold for $Q^{-1} = (W + iz)^{-1}$ and $A$ replaced by the operator

$$A = \begin{pmatrix} \langle D \rangle^{-1} A \langle D \rangle & 0 \\ 0 & A \end{pmatrix},$$

which is selfadjoint on $\mathcal{H}$. Applying once more Lemma 1.16, we deduce the multiple resolvent estimates for $(W + iz)^{-n}$ with weight $\langle A \rangle^{-\delta}$ in $\mathcal{H}$. Notice that $A$ does not have to be conjugate to $i\mathcal{W}$ in the sense of Definition 1.13 (and it is not).

Using the regularity given by the resolvents of $\mathcal{W}$ to remove the derivatives in the weights $\langle A \rangle^{\delta}$ (see the discussion about low frequencies below), we can finally prove the following estimates.

**Proposition 1.17.** Let $K$ be a compact subset of $\mathbb{C}^*$. Let $n \in \mathbb{N}^*$ and $\delta > n - \frac{1}{2}$. There exists $C > 0$ such that for $\zeta \in K \cap \mathbb{C}^+$ we have

$$\|(W - \zeta)^{-n}\|_{\mathcal{L}(\mathcal{E}^{\delta}, \mathcal{E}^{-\delta})} \le C.$$

### 1.4.2 Contribution of high frequencies

In this paragraph we briefly discuss the proof of Theorem 1.8. With Proposition 1.17, it is enough to consider spectral parameters $z = i\zeta$ such that $\mathrm{Im}(z) > 0$ and $|\mathrm{Re}(z)| \gg 1$.

As explained in the introduction, the behavior of the contribution of high frequencies depends on the geometries of the domain and the damping region. It was the main motivation of [2] to prove in a slightly different context the following estimate for a single resolvent.
**Theorem 1.18.** Assume that the geometric damping condition (1.17) holds. Let $\delta > \frac{1}{2}$. There exist $\tau_0 > 0$ and $c > 0$ such that for $z \in \mathbb{C}_+$ with $\mathrm{Re}(z) \ge \tau_0$ we have

$$ \left\| \langle x \rangle^{-\delta} \partial_x^{\beta_1} R(z) \partial_x^{\beta_2} \langle x \rangle^{-\delta} \right\|_{\mathcal{L}(L^2(\mathbb{R}^d))} \le \frac{c}{|z|^{1-|\beta_1|-|\beta_2|}}. $$

In [2] the estimate was given for the semiclassical Schrödinger operator with potential $-h^2\Delta + V_1(x) - ihV_2(x)$ (and without the additional derivatives), but the strategy is the same for a Laplacian associated with a metric (see [6, Section 8]). We do not give details for Theorem 1.18 since the proof is essentially the same as in [Roy10], but we can at least recall that we apply the Mourre method with a conjugate operator given by the Weyl quantization of a symbol $f$ which satisfies for some $\beta \ge 0$

$$ \{p, f\} + \beta a \ge c_0 > 0, \quad \text{on } p^{-1}(\{1\}) \qquad (1.55) $$

$(p$ is the symbol defined in (1.15)). The positivity of the Poisson bracket $\{p, f\}$ means that $f$ is increasing along the Hamiltonian flow associated with $p$. With the second term in the left-hand side of (1.55) (which corresponds to the second term in the left-hand side of (1.37) or (1.41)), the symbol does not have to be increasing along the flow in the damping region. This is why we can allow bounded classical trajectories if they go through this damping region (see (1.17)).

Then, as explained in Section 1.3, Theorem 1.18 was generalized in [6] to multiple resolvent estimates with inserted factors, to deal with terms of the form (1.38). From this we can deduce Theorem 1.8. We omit the details and refer to [6, Section 8] and [11, Sections 3.1 and 4.2].

### 1.4.3 Contribution of low frequencies

The main result in [6] is the resolvent estimates for the contribution of low frequencies. The commutators method does not directly give uniform bounds for the resolvent of a Schrödinger operator near 0. We see from (1.54) that the estimate becomes bad when $\tau \to 0$. We can apply the commutators method to the operator $|z|^{-2}(-\Delta - izaw - z^2w)$ for $z$ small, but we only get an estimate of size $O(|z|^{-2})$ for $R(z)$ in weighted spaces. This is at least uniform with respect to $\mathrm{Im}(z)$ close to 0, but we need an additional argument to get a uniform bound when $\mathrm{Re}(z) \to 0$.

A key argument to recover some smallness in the low frequency analysis is a generalization of the Hardy inequality, which roughly says that in dimension $d \ge 3$ the multiplication by $|x|^{-1}$ behaves like a derivative. And for low frequencies, a derivative can be seen as a small operator.

In this report, I present the results of [6, 11] with the point of view developed recently in [23] for the Schrödinger equation. For $r \in [0, 1]$ we set $D_r = \sqrt{-\Delta}/r$. Then for $s \in \mathbb{R}$ we denote by $H_r^s$ and $\dot{H}_r^s$ the usual Sobolev spaces $H^s$ and $\dot{H}^s$, endowed respectively with the norms defined by

$$ \|u\|_{H_r^s} = \|\langle D_r \rangle^s u\|_{L^2}, \quad \|u\|_{\dot{H}_r^s} = \|D_r^s u\|_{L^2}. \qquad (1.56) $$

In particular $\|u\|_{\dot{H}^s} = r^s \|u\|_{\dot{H}_r^s}$, and for $\alpha \in \mathbb{N}^d$ and $s \in \mathbb{R}$ the derivative $\partial_x^\alpha$ defines an operator of size $r^{|\alpha|}$ from $H_r^s$ to $H_r^{s-|\alpha|}$. Finally, we denote by $\mathcal{H}_r^s$ the Hilbert completion of $\mathcal{H}$ for the norm defined by

$$ \|U\|_{\mathcal{H}_r^s}^2 = \|ru\|_{H_r^{s+1}}^2 + \|v\|_{H_r^s}^2, \quad U = (u, v). \qquad (1.57) $$

In particular, $\mathcal{H} = \mathcal{H}_1^0$. We write $D_z, H_z^s, \dot{H}_z^s$ and $\mathcal{H}_z^s$ for $D_{|z|}, H_{|z|}^s, \dot{H}_{|z|}^s$ and $\mathcal{H}_{|z|}^s$.

Let $d_0$ be a fixed integer greater than $\frac{d}{2}$. For $\kappa \ge 0$ we denote by $\mathcal{S}^{-\kappa}$ the set of smooth functions $\phi$ such that

$$ \|\phi\|_{\mathcal{S}^{-\kappa}} = \sup_{|\alpha| \le d_0} \sup_{x \in \mathbb{R}^d} |\langle x \rangle^{\kappa+|\alpha|} \partial^\alpha \phi(x)| < +\infty. \qquad (1.58) $$
We can say that the multiplication by a decaying function behaves like a derivative, and hence is small for low frequencies, in the following sense (see Proposition 7.2 in [6] or Proposition 3.1 in [23]).

**Proposition 1.19.** Let $s \in ] -\frac{d}{2}, \frac{d}{2}[$ and $\kappa \ge 0$ such that $s - \kappa \in ] -\frac{d}{2}, \frac{d}{2}[$. Let $\eta > 0$. There exists $C \ge 0$ such that for $\phi \in S^{-\kappa-\eta}$, $u \in H^s$ and $r \in ]0, 1]$ we have

$$ \|\phi u\|_{H_r^{s-\kappa}} \le Cr^{\kappa} \|\phi\|_{S^{-\kappa-\eta}} \|u\|_{H_r^s}. $$

In particular, if $\phi \in S^{-\eta}$ for some $\eta > 0$, then for any $s \in ] -\frac{d}{2}, \frac{d}{2}[$ the multiplication by $(1+\phi)$ defines a bounded operator on $H_r^s$ uniformly in $r \in ]0, 1]$.

A first application is that for $s \in ] -\frac{d}{2}, \frac{d}{2}[$ and $\rho \in ]0, \rho_0[$ small enough the assumptions (1.10) and (1.11) give

$$ \|za(x)w(x)\|_{\mathcal{L}(H_z^{s+1}, H_z^{s-1})} + \|z^2(w(x)-1)\|_{\mathcal{L}(H_z^{s+1}, H_z^{s-1})} \lesssim |z|^{2+\rho}. \quad (1.59) $$

Thus the corresponding terms are perturbations of $-\Delta - z^2$ in $\mathcal{L}(H_z^{s+1}, H_z^{s-1})$ when $z$ is small. Notice that it is important here that $a$ is of short range.

Even if $G(x) - \mathrm{Id}$ also decays by (1.10), the same does not apply to $(\Delta_G - \Delta)$. Indeed, this term is already of order 2 and we cannot pay more regularity to get some smallness in $\mathcal{L}(H_r^{s+1}, H_r^{s-1})$ for $z$ small. The idea is then to consider first the case where $\|G(x) - \mathrm{Id}\|_{S^{-\rho}}$ is small enough, and then to add a compactly supported contribution for the metric. In [6] this is done in Section 7.3. In [23] (for the Schrödinger equation) we deal with this compactly supported perturbation in each intermediate result. This is an important technical difficulty that we omit here for simplicity. Thus we proceed as if $\|G(x) - \mathrm{Id}\|_{S^{-\rho}}$ were already small enough.

*Ideas of proof for Theorem 1.10.* We see with (1.59) that $(-\Delta_G + \mathrm{raw} + r^2w)$ is a small perturbation of $-\Delta + r^2$ in $\mathcal{L}(H_r^{s+1}, H_r^{s-1})$ for any $s \in ] -\frac{d}{2}, \frac{d}{2}[$, and in particular

$$ \|R irr)\|_{\mathcal{L}(H_r^{s-1}, H_r^{s+1})} \lesssim \frac{1}{r^2}. $$

From (1.23) we deduce, for $s \in ] -\frac{d}{2} + 1, \frac{d}{2}[$,

$$ \|(W - r)^{-1}\|_{\mathcal{L}(\mathcal{H}_r^{s-1}, \mathcal{H}_r^s)} \lesssim \frac{1}{r}. $$

Thus, for $s_1, s_2 \in [0, \frac{d}{2}]$ and $m \in \mathbb{N}^*$ such that $s_1 + s_2 \le m$ we have

$$ \|(W - r)^{-m}\|_{\mathcal{L}(\mathcal{H}_r^{s_2}, \mathcal{H}_r^{s_1})} \lesssim \frac{1}{r^m}. \quad (1.60) $$

We can similarly estimate $(W - \zeta)^{-m}$ by $|\zeta|^{-m}$ if $\mathrm{Re}(\zeta) \gtrsim |\zeta|$.

In general, we set $r = |\zeta|$ and we observe that, by the resolvent identity, we can write $(W - \zeta)^{-n}$ as a sum of terms of the form

$$ (\zeta - r)^{m-n} (W - r)^{-m} \quad (1.61) $$

for some $m \ge n$, or

$$ (\zeta - r)^{2N+\nu-n} (W - r)^{-N} (W - \zeta)^{-\nu} (W - r)^{-N}, \quad (1.62) $$

for $\nu \le n$ and $N$ as large as we wish.

The idea to estimate a term of the form (1.61) is to use the weight (remember that we need an estimate in $\mathcal{L}(\mathcal{E}^{\delta}, \mathcal{E}^{-\delta})$) to convert the elliptic regularity given by (1.60) into smallness. By Proposition 1.19 and (1.25) we can prove that for $s \in [0, \frac{d}{2}]$ and $\delta > s$ we have

$$ \|U\|_{\mathcal{E}^{-\delta}} \lesssim r^s \|U\|_{\mathcal{H}_r^s} \quad \text{and} \quad \|U\|_{\mathcal{H}_r^{-s}} \lesssim r^s \|U\|_{\mathcal{E}^{\delta}}. \quad (1.63) $$
With (1.60) and (1.63) applied with

$$s_1 = s_2 = s = \frac{1}{2} \min(d - \epsilon, m),$$

we obtain that (1.61) satisfies the estimate of Theorem 1.10.

Now we turn to the terms of the form (1.62). In such a term we still have a factor $(W - \zeta)^{-\nu}$ as at the beginning, but the interest of the decomposition (1.61)-(1.62) is that in (1.62) we now have the factors $(W - r)^{-N}$ which give elliptic regularity. The main step of the proof remains the commutators method. Notice that we have omitted here the fact that $G - \text{Id}$ is only small at infinity. In general, to take this into account for the commutators method, we can as in [BB21] replace the usual generator of dilations (1.30) by a generator of dilations at infinity

$$A_z = - \frac{(1 - \chi_z)x \cdot i\nabla + i\nabla \cdot x(1 - \chi_z)}{2}, \quad (1.64)$$

where $\chi_z(x) = \chi(|z|x)$ for some $\chi \in C_0^\infty(\mathbb{R}^d, [0, 1])$ equal to 1 on a neighborhood of 0.

Following Sections 1.3 and 1.4.1, beginning carefully with the dependence on $z$ going to 0, we prove that $A_z$ is $\Upsilon$-conjugate to $|z|^2(-\Delta - izaw - z^2w)$ for some $\Upsilon$ independent of $z \in \mathbb{C}_+$ close to 0 (and with $\text{Re}(z^2) \gtrsim |z|^2$). We get in particular an estimates of the form

$$\| \langle A_z \rangle^{-\delta} R(z) \langle A_z \rangle^{-\delta} \|_{L(L^2(\mathbb{R}^d))} \lesssim \frac{1}{|z|^2}.$$

See [23, Sec. 5.2] for the case $a = 0$. Then, as in Section 1.4.1, we can deduce with successive uses of Lemma 1.16 an estimate of the form

$$\| \langle A_z \rangle^{-\delta} (W + iz)^{-\nu} \langle A_z \rangle^{-\delta} \|_{L(\mathcal{H}_z^0)} \lesssim \frac{1}{|z|^\nu}, \quad A_z = \begin{pmatrix} \langle D_z \rangle A_z \langle D_z \rangle^{-1} & 0 \\ 0 & A_z \end{pmatrix}. \quad (1.65)$$

It remains to use the factors $(W - r)^{-N}$ to recover some smallness for (1.62). We also use this regularity to compensate the derivatives used in the weights $\langle A_z \rangle$. More precisely, for $s \in [0, \frac{d}{2}]$ and $\delta > s$ we can prove

$$\| (W - r)^{-N} \langle A_{\zeta} \rangle^{\delta} \|_{L(\mathcal{H}_z^0, \mathcal{E}^{-\delta})} \lesssim r^{s-N}, \quad \| \langle A_{\zeta} \rangle^{\delta} (W - r)^{-N} \|_{L(\mathcal{E}^{\delta}, \mathcal{H}_z^0)} \lesssim r^{s-N}. \quad (1.66)$$

Notice that we cannot directly use pseudodifferential calculus to prove these estimates, since the commutators of two operators are usually not smaller than the products. We can do the computation by hand.

Finally, with (1.65) and (1.66) we obtain that (1.62) also satisfies the estimate of Theorem 1.10. $\square$

As said above, the ideas of the proof are not presented here with the point of view of [6, 11]. For instance, in these papers we did the decomposition (1.61)-(1.62) on the expression (1.38), and not directly on the resolvent of $W$. On the other hand, in [6, 11] we rescaled the operators and not the Sobolev spaces. This is equivalent.

## 1.5 Optimal resolvent estimate for the Schrödinger equation in an asymptotically conical setting

In this paragraph, we briefly discuss the paper [7] about sharp low frequency resolvent estimates on asymptotically conical manifold. This is a collaboration with Jean-Marc Bouclet. It is related to the other works presented in this chapter since it is about low frequency resolvent estimates for a Laplace operator, but the motivations are of different nature.
The setting is a connected and asymptotically conical manifold $\mathcal{M}$ with Riemannian metric $G$ (possibly with boundary). Asymptotically conical means that there exists a compact subset $K$ of $\mathcal{M}$ such that $(\mathcal{M}\backslash K, G)$ is isometric to $]R_0, +\infty[\times\mathcal{S}$ for some $R_0 > 0$ and some closed Riemannian manifold $\mathcal{S}$. This product is endowed with a metric approaching the metric $dr^2 + r^2h_0$, where $h_0$ is a Riemannian metric on $\mathcal{S}$. More precisely, there exists a diffeomorphism

$$ \kappa: \begin{cases} \mathcal{M}\backslash K & \to \]R_0, +\infty[\times\mathcal{S} \\ m & \mapsto (r(m), \omega(m)) \end{cases} $$

through which we can write

$$ G = \kappa^*(a(r)dr^2 + 2rb(r)dr + r^2h(r)), $$

where $a(r)$ is a function on $\mathcal{S}$ going to 1, $b(r)$ is a 1-form on $\mathcal{S}$ going to 0 and $h(r)$ is a Riemannian metric on $\mathcal{S}$ going to $h_0$, in the sense that there exists $\rho > 0$ such that for $j \in \mathbb{N}$ we have

$$ \|\hat{c}_r^j(a(r) - 1)\|_{\Gamma^0(\mathcal{S})} + \|\hat{c}_r^j b(r)\|_{\Gamma^1(\mathcal{S})} + \|\hat{c}_r^j(h(r) - h_0)\|_{\Gamma^2(\mathcal{S})} \lesssim r^{-j-\rho}, $$

$\Gamma^k(\mathcal{S})$ being any seminorm on the space of smooth sections of $(T^*\mathcal{S})^{\otimes k}$.

This setting is more general than the scattering metrics of [Mel95]. Moreover, even if $\mathcal{M} = \mathbb{R}^d$, the assumption on the metric is more general than being asymptotically flat since $h_0$ does not have to be the usual metric on the sphere $\mathcal{S} = \mathcal{S}^{d-1}$.

We denote by $P$ the Laplace-Beltrami operator on $\mathcal{M}$ (with Dirichlet boundary condition if $\partial\mathcal{M} \neq \emptyset$). The main results in [7] are the following resolvent estimates.

**Theorem 1.20.** There exist a neighborhood $\mathcal{U}$ of 0 in $\mathbb{C}$ and $C > 0$ such that for $z \in \mathcal{U}\backslash\mathbb{R}$ we have

$$ \|\langle r \rangle^{-1} (P - z)^{-1} \langle r \rangle^{-1}\|_{\mathcal{L}(L^2(\mathcal{M}))} \le C. $$

**Theorem 1.21.** Let $N \in \mathbb{N}^*$ and let $J$ be a compact interval of $]0, +\infty[$. There exists $C > 0$ and $\varepsilon_0 > 0$ such that for $\varepsilon \in ]0, \varepsilon_0]$ and $z \in \mathbb{C}\backslash\mathbb{R}$ with $\mathrm{Re}(z) \in J$ we have

$$ \|\langle\varepsilon r\rangle^{-N} (\varepsilon^{-2}P - z)^{-N} \langle\varepsilon r\rangle^{-N}\|_{\mathcal{L}(L^2(\mathcal{M}))} \le C. $$

The two main interests of these results are the general geometric setting and the optimal weight $\langle r \rangle^{-1}$. The best weight in the previous papers about the low frequency resolvent estimates was of the form $\langle r \rangle^{-s}$ for $s > 1$. It was in particular motivated by the Strichartz estimates written later in [BM].

Notice that the second result corresponds to what we would get for the free Laplacian on $\mathbb{R}^d$ from an estimate on $\langle r \rangle^{-N} (-\Delta - z)^{-N} \langle r \rangle^{-N}$ by a scaling argument. Thus Theorem 1.21 says that the same estimate holds in a setting where such a scaling is meaningless.

We do not go into the details of the proofs here. The first important step is to reduce the problem to resolvent estimates for a Schrödinger operator on the pure cone $]0, +\infty[\times\mathcal{S}$. Then the main part of the proof is to develop the Mourre theory on this pure cone for such an operator.

## 1.6 The damped Schrödinger equation

In [10] we have considered with Moez Khenissi the local energy decay for a damped Schrödinger equation.

Local energy decay for a dissipative Schrödinger equation has already been studied on exterior domains, with some potential damping in the domain $(\alpha = 0$ with the notation of (1.67) below) in [AK07] or at the boundary in [AK10]. See also [BC14] for a Schrödinger equation with non-linear damping at infinity.
Nevertheless, a potential damping is not strong enough to have the smoothing effect typical for the usual Schrödinger equation. For the regularized Schrödinger equation ($\alpha = 1$ in (1.67)), we can recover the usual gain of half a derivative (see [Alo08b, Alo08a]), or a $H^{\frac{1}{2}-\epsilon}$-smoothing effect if a few classical trajectories fail to satisfy the geometric damping condition (see [AKV13], see also [Bur04] for a similar result in the selfadjoint case).

In [10], we consider this damped Schrödinger equation in the asymptotically Euclidean setting. We consider on $\mathbb{R}^d$ the problem

$$ \begin{cases} -i\partial_t u + Pu - ia(x) \langle D \rangle^\alpha a(x)u = 0, & \forall t \ge 0, \forall x \in \mathbb{R}^d, \\ u|_{t=0} = u_0, & \end{cases} \qquad (1.67) $$

where $P = -\Delta_G$ is a Laplacian in divergence form, $D = \sqrt{-\Delta}$, $\alpha \in [0, 2[$ and $a$ is of short range as in (1.11). Since $a$ appears twice, the dissipative term actually decays at least like $\langle x \rangle^{-2-2\rho_0}$ with respect to the space variable.

For the Schrödinger equation, it is the $L^2$-norm of the solution which is constant without damping and non-increasing in general. Formally,

$$ \frac{d}{dt} \|u(t)\|_{L^2(\mathbb{R}^d)}^2 = -2 \langle \langle D \rangle^\alpha a u(t), a u(t) \rangle \|_{L^2(\mathbb{R}^d)} \le 0. $$

We set

$$ P_\alpha = P - ia(x) \langle D \rangle^\alpha a(x), \quad \mathrm{Dom}(P_\alpha) = H^2(\mathbb{R}^d). $$

The strength of the damping depends on the parameter $\alpha$. This only plays a role for the contribution of high frequencies (for low frequencies we have $\langle D \rangle \simeq 1$).

Under the non-trapping condition (1.16), the damping should not play an important role. And indeed, we recover in this dissipative setting what was at that time the best result known even in the undamped case for the local energy decay (the $\epsilon$ loss has been removed later in [BB21]), and the smoothing effect.

Now assume that (1.16) does not hold but the geometric damping condition (1.17) does. If the damping is strong enough ($\alpha \ge 1$), we recover the same results as under the non-trapping condition. Of course, we cannot get a better estimate even with a stronger damping ($\alpha > 1$) since the classical trajectories at infinity do not see the damping.

If the damping is weak ($\alpha < 1$), then the high-frequency resolvent estimates are weaker in the presence of bounded trajectories. However, if we have a strong weight or if we can allow a loss of regularity, then we can recover the same decay for large times (remember that the rate of decay is actually limited by the contribution of low frequencies). However the regularizing effect (governed by high frequencies) will be weaker than usual in this case.

More precisely, we prove in [10] the following local energy decay and smoothing effect for (1.67).

**Theorem 1.22 (Local energy decay).** Let $\epsilon > 0$. Let $\delta > \frac{d+1}{2}$ if $d$ is even and $\delta > \frac{d}{2} + 1$ if $\delta$ is odd. Let $N \in \mathbb{N}$ and $\sigma \in [0, 2]$. Assume that

(i) the non-trapping condition (1.16) holds,

(ii) or the geometric damping condition (1.17) holds, $N \min(1, \alpha) + \sigma \ge 2$ and $\delta > N - \frac{1}{2}$.

Then there exists $C \ge 0$ such that for $u_0 \in H^{\sigma, \delta}(\mathbb{R}^d)$ and $t \ge 0$ we have

$$ \|e^{-itP_\alpha} u_0\|_{L^{2,-\delta}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{d}{2}+\epsilon} \|u_0\|_{H^{\sigma,\delta}(\mathbb{R}^d)}. $$

The condition $N \min(1, \alpha) + \sigma \ge 2$ means that even if $\alpha$ is small, we can apply the high-frequency resolvent estimate (see (1.68) below) with $N$ large, or we can pay a loss of derivative (typically if $\alpha = 0$).
**Theorem 1.23 (Global smoothing effect).** Assume that the geometric damping condition (1.17) holds. Then there exists $C \ge 0$ such that for all $u_0 \in L^2(\mathbb{R}^d)$ we have

$$ \int_{0}^{+\infty} \|\langle x \rangle^{-1} \langle D \rangle^{\min(1,\alpha)/2} e^{-itP_{\alpha}} u_0\|_{L^2(\mathbb{R}^d)}^2 dt \le C \|u_0\|_{L^2(\mathbb{R}^d)}^2. $$

Moreover, under the non-trapping condition (1.16), we can replace $\min(1, \alpha)$ by 1.

Notice that to have the weight $\langle x \rangle^{-1}$ in Theorem 1.23 we use the optimal resolvent estimate at low frequencies deduced from Theorem 1.20.

For high frequencies, we use the same strategy as in [2, 6] (see Section 1.4.2), except that the damping is not necessarily strong enough. For $N \in \mathbb{N}^*$ and $\delta > N - \frac{1}{2}$ we get an estimate of the form

$$ \|\langle x \rangle^{-\delta} (P_{\alpha} - z)^{-N} \langle x \rangle^{-\delta}\|_{L^2(L^2(\mathbb{R}^d))} \lesssim |z|^{-\frac{N \min(1, \alpha)}{2}}. \quad (1.68) $$

For low frequencies, we adapt the proof of [6] to a dissipative perturbation of the Laplacian which is of different nature. In particular, there is no time derivative in the dissipative term, hence no factor $z$ in the corresponding term for the resolvent. This is why we need more spatial decay for the absorption index. Thus we use Proposition 1.19 in an even more crucial way than for the wave equation, and the restriction on the Sobolev indices therein could have been a serious difficulty. We do not discuss all the details and only emphasize the fact that we unexpectedly had to use the dissipative Mourre theory in the sense of forms:

**Remark 1.24.** To apply the first version of the dissipative commutators method given in [2] to the operator $P_\alpha$ at low frequencies, the dissipative part $a(x) \langle D \rangle^\alpha a(x)$ has to be uniformly relatively bounded with respect to the selfadjoint part $P$. For this we use the decay of $a(x)$ at infinity and Proposition 1.19. $P$ defines an operator of size $O(|z|^2)$ in $\mathcal{L}(H_z^2, L^2)$. By Proposition 1.19, $a(x)$ can be seen as an operator of size $O(|z|)$ in $\mathcal{L}(H_z^{s+1}, H_z^s)$ and in $\mathcal{L}(H_z^s, H_z^{s-1})$ (we omit $\langle D \rangle^\alpha$ which does not play an important role). To have a dissipative part of size $O(|z|^2)$ in $\mathcal{L}(H_z^2, L^2)$, we have to apply this with $s = 1$. Because of the restriction of Proposition 1.19, this is only possible if $d \ge 5$.

However, for $d \ge 3$ we can proceed similarly with $s = 0$, which means that we see the dissipative part as an operator from $H_z^1$ to $H_z^{-1}$. Thus, even if $P_\alpha$ is the sum of a selfadjoint operator and a dissipative part as in [2] (see also (1.36)), for $d = 3, 4$ we can only apply the Mourre method is the sense of forms as in [9].

Finally, the smoothing effect can be directly deduced from the resolvent estimates via the theory of relatively smooth operators, which is classical for selfadjoint operators (see [RS79, Sec. XIII.7]). For dissipative operators, this relies on the theory of selfadjoint dilations. For this we refer to Proposition 6.2 in [9].

## 1.7 Asymptotic behavior for the Schrödinger equation

We finish this chapter with the most recent result about low frequency resolvent estimates. In [23], we improve the results known for the usual selfadjoint Schrödinger equation (1.13) about the local energy decay and low frequency resolvent estimates.

The optimal low frequency resolvent estimates and then the optimal local energy decay have been proved in [BB21]. We go beyond and prove that the solution of (1.13) behaves (for low frequencies and hence for large times) like the solution of the free Schrödinger equation (1.6) for some $f_0 \in L^2(\mathbb{R}^d)$. More precisely, we prove that the local energy of the difference $u(t) - u_0(t)$ decays faster than the local energy of $u_0(t)$. This means that $u_0(t)$ is the leading term for the asymptotic expansion of $u(t)$ for large $t$. We recover in particular the same local energy decay for $u(t)$ as for $u_0(t)$. We denote by $P_0$ the usual Laplacian on $\mathbb{R}^d$. The precise result is the following.
**Theorem 1.25.** Assume that the non-trapping condition (1.16) holds. Let $\rho_1 \in [0, \rho_0[$ and $\delta \ge \frac{d}{2} + 2$. There exists $C \ge 0$ such that for $t \ge 0$ we have

$$ \left\| \langle x \rangle^{-\delta} (e^{-itP} - e^{-itP_0}w) \langle x \rangle^{-\delta} \right\|_{L(L^2(\mathbb{R}^d))} \le C \langle t \rangle^{-\frac{d}{2} - \frac{\rho_1}{2}}. $$

The factor $w$ next to $e^{-itP_0}$ means that for $f \in L^2(\mathbb{R}^d)$ we compare the solution $u$ of (1.13) with the solution $u_0$ of (1.6) with $f_0 = w f$.

Since the decay at rate $t^{-\frac{d}{2}}$ is optimal for the free Schrödinger equation, Theorem 1.25 indeed says that the difference $e^{-itP} - e^{-itP_0}w$ decays faster than $e^{-itP_0}w$ (in the sense of the local energy). Moreover, as a corollary we recover the optimal decay at rate $t^{-\frac{d}{2}}$ for the solution of (1.13).

We already know that under the non-trapping condition (1.16), the contributions of high frequencies for both problems decay fast, so Theorem 1.25 is again mostly a result about low frequencies. Thus we have to compare the resolvents of $P$ and $P_0$ near 0.

**Theorem 1.26.** Let $\rho_1 \in [0, \rho_0[$, $n \in \mathbb{N}^*$ and $\delta > n + \frac{1}{2}$. There exists $C > 0$ such that for $\zeta \in D\setminus\mathbb{R}_+$ we have

$$ \left\| \langle x \rangle^{-\delta} \left( (P - \zeta)^{-n} - (P_0 - \zeta)^{-n} w \right) \langle x \rangle^{-\delta} \right\|_{L(L^2(\mathbb{R}^d))} \le C |\zeta|^{\min(0, \frac{d+\rho_1}{2} - n)}. $$

Notice that asymptotic expansions for the resolvent of a Schrödinger operator at the low frequency limit, and then the expansion of the propagator for large times, have already been studied for perturbations by a potential. We have already mentioned [JK79]. We also refer to the recent papers [Wan20] and [Aaf21] for complex-valued potentials. The difficulty in these cases is that we can have an eigenvalue or a resonance at the bottom of the spectrum, which gives a singularity for the resolvent. This is why these results require strong decay assumptions on the potential.

We have already more or less used the setting of [23] to explain the results of [6, 11] in Sections 1.3 and 1.4. Therefore, in this section we only add some comments specific to the fact that we not only estimate resolvents but compare the resolvents of two different operators.

In particular, one of the difficulties for the proof is that we have to use the commutators method simultaneously for the two operators $P$ and $P_0$. We have already explained in Section 1.3 that this is one of the improvements of the method developed for [23].

Another problem is that the operator $P$ and $P_0$ are selfadjoint, but they are not selfadjoint on the same Hilbert space (even if $L^2(\mathbb{R}^d, dx)$ and $L^2(\mathbb{R}^d, w dx)$ are equal as sets and have equivalent norms).

We replace $(P - \zeta)^{-1}$ by $(-\Delta_G - \zeta w)^{-1}w$. The operator $(-\Delta_G - \zeta w)^{-1}$ is no longer the resolvent of a selfadjoint operator, but for $\zeta \in \mathbb{C}_+$ it is still the inverse of the dissipative operator $-\Delta_G - \zeta w$. It is dissipative on the usual space $L^2(\mathbb{R}^d)$, as is $-\Delta - \zeta$. Thus we can work in $L^2(\mathbb{R}^d)$ endowed with its usual structure. This is why in Theorem 1.15 we no longer consider resolvents but the inverse of parameter-dependent dissipative operators. In particular, even for this selfadjoint problem we have to use the dissipative version of the commutators method.

Replacing $(P - \zeta)^{-1}$ by $(-\Delta_G - \zeta w)^{-1}$ is not just a technical issue, and it is really $(-\Delta_G - \zeta w)^{-1}$ that is close to $(-\Delta - \zeta)^{-1}$ in a suitable sense. Then $(P - \zeta)^{-1}$ is close to $(-\Delta - \zeta)^{-1}w$, which explains the additional factor $w$ in Theorem 1.26 and then in Theorem 1.25.

*Ideas of proof.* We have to estimate $(P - \zeta)^{-n}w^{-1} - (P_0 - \zeta)^{-n}$ for $\zeta = z^2$ close to 0. We first multiply this difference by $|z|^{2n}$ to have a spectral parameter of order 1. For $n \in \mathbb{N}^*$ we
set

$$R^{[n]}(z) = \left( \frac{P}{|z|^2} - \frac{z^2}{|z|^2} \right)^{-n} w^{-1} = |z|^{2n} ((-\Delta_G - z^2 w)^{-1} w)^{n-1} (-\Delta_G - z^2 w)^{-1}, \quad (1.69)$$

and for consistency we also set $R_0^{[n]}(z) = |z|^{2n} (-\Delta - z^2)^{-n}$.

Since the Mourre method is designed to estimate products of resolvents, we use the resolvent identity to rewrite the difference $R^{[n]}(z) - R_0^{[n]}(z)$ as

$$\sum_{k=1}^{n-1} R^{[n-k]}(z) \theta_0(z) R_0^{[k]}(z) - \sum_{k=1}^{n} R^{[n-k+1]}(z) \theta_1(z) R_0^{[k]}(z), \quad (1.70)$$

where we have set

$$\theta_0(z) = w - 1 \quad \text{and} \quad \theta_1(z) = \frac{(-\Delta_G - z^2 w) - (-\Delta - z^2)}{|z|^2}. \quad (1.71)$$

The fact that the difference $R^{[n]}(z) - R_0^{[n]}(z)$ is smaller than $R_0^{[n]}(z)$ for $z \in \mathbb{C}_+$ close to 0 is due to the smallness of $\theta_0(z)$ and $\theta_1(z)$ in the suitable rescaled Sobolev spaces (see (1.56)), given once more by Proposition 1.19.

Notice that we have factors $w$, $\theta_0(z)$ and $\theta_1(z)$ between the resolvents, but thanks to the analysis of the damped wave equation we know that this is not a problem for the commutators method. We get for instance the following estimate (with $A_z$ being the generator of dilations at infinity defined in (1.64)):

**Proposition 1.27.** Let $\rho \in [0, \rho_0[, n_1, n_2 \in \mathbb{N}^*, \delta > n_1 + n_2 - \frac{1}{2}$ and $\sigma \in \{0, 1\}$. There exists $C > 0$ such that for $z \in \mathbb{D}_+$ we have

$$\| \langle A_z \rangle^{-\delta} R^{[n_1]}(z) \theta_{\sigma}(z) R_0^{[n_2]}(z) \langle A_z \rangle^{-\delta} \|_{L(L^2(\mathbb{R}^d))} \le C |z|^{\rho}.$$

The factor $|z|^{\rho}$ which appears in this estimate is due the small factor $\theta_{\sigma}(z)$ and gives the extra smallness in Theorem 1.26 compared to the estimate of $R_0^{[n]}(z)$ or $R^{[n]}(z)$ alone.

However, this is an estimate on the rescaled resolvents and the estimate on the resolvents would be of size $O(|z|^{\rho-2n_1-2n_2})$. As in Section 1.4.3, we recover some smallness with the weights $\langle x \rangle^{-\delta}$ by elliptic regularity and Proposition 1.19. And this regularity is also used to compensate the derivatives which appear in the powers of the conjugate operator $A_z$. $\square$

Theorems 1.25 and 1.26 are important for at least two reasons. The first is that ten years ago the motivation of [6] was to recover in a non-selfadjoint setting estimates which were already known in the selfadjoint case. Now we have reached the point where ideas developed for the dissipative case are also used to improve the analysis of the selfadjoint Schrödinger operator.

On the other hand, for the low frequency resolvent estimates, this result re-opens a topic which seemed to be closed by the optimal estimates of [BB21]. Perspectives in that direction will be discussed in Section 4.1.1.
# Chapter 2

## The damped wave equation with damping at infinity

In this chapter we continue the analysis of the damped wave equation (1.12). However, instead of assuming that the damping is localized, we now consider settings for which the damping is effective at infinity. This gives completely different properties. The results discussed in this chapter are published in [8, 14, 15, 17, 18].

### 2.1 Introduction

We have seen in Theorem 1.1 that if the damping is small enough at infinity (in the sense of (1.11)), the contribution of low frequencies for the damped wave equation (1.12) behaves as for the undamped case. We will see in this chapter that this is no longer the case if the damping is effective at infinity.

Our model case in this chapter is the free wave equation ($P = -\Delta$) with constant damping $a(x) = a_1 > 0$:

$$ \begin{cases} \partial_t^2 u - \Delta u + a_1 \partial_t u = 0, \\ (u, \partial_t u)|_{t=0} = (f, g). \end{cases} \qquad (2.1) $$

With damping everywhere, it is clear that the damping condition (1.17) is satisfied, and as expected the local energy of the contribution of high frequencies will decay rapidly. The difference is that all the classical trajectories go through the damping region, and not only the bounded ones, so for high frequencies the global energy decays uniformly exponentially. As one could have guessed, a stronger damping implies stronger decay properties for the energy.

The most interesting part is the contribution of low frequencies, whose behavior is not that simple.

We first observe that for a very slowly oscillating solution $u$, the damping term $a_1 \partial_t u$ is small, but the second order term $\partial_t^2 u$ is even smaller. Neglecting this term leads to the conjecture that $u$ should behave like a solution of the heat equation

$$ \begin{cases} a_1 \partial_t u_0 - \Delta u_0 = 0, \\ u_0|_{t=0} = f_0. \end{cases} \qquad (2.2) $$

And this is precisely what happens. This is sometimes called the diffusive phenomenon. And since the local energy decay is slower for the heat equation than for the standard wave equation (see Proposition 2.2 below), it turns out that the decay for the contribution of low frequencies is not as fast with damping everywhere as it is without any damping.
There is also a rich literature about the wave equation with damping at infinity. The local energy decay for (2.1) has been discussed in [Mat76]. Matsumura proves $L^p - L^q$ estimates by explicit computations on the Fourier transform. The corresponding semilinear problem is also discussed (see Section 4.5.2 for this question). Then the comparison with the heat equation (2.2) has been studied in many papers. More precisely, it is proved that the solution $u$ of (2.1) behaves for large times like the solution $u_0$ of (2.2) with

$$f_0 = f + \frac{g}{a_1}. \qquad (2.3)$$

Then $u_0$ is the asymptotic profile of $u$ for large times. We refer to [Nis03] for a result in dimension 3, [MN03] in dimension 1, [HO04] in dimension 2 and a more general discussion in [Nar04].

The same problem has been studied in an exterior domain. See [Ike02] for a result with constant coefficients and [AIK15] for an absorption index equal to 1 outside a compact subset.

A question that will not be discussed here is the case of a slowly decaying damping. This means that $a$ goes to 0 at infinity, but it is not of short range as in (1.11). We refer to [TY09] for the energy decay when $a(x) \approx \langle x \rangle^{-\alpha}$ for some $\alpha \in [0, 1[$, to [ITY13] for the critical case $\alpha = 1$ and to [Wak14] for the asymptotic profile when $a(x) = \langle x \rangle^{-\alpha}$, $\alpha \in [0, 1[$ (which is as expected a solution of $\langle x \rangle^{-\alpha} \partial_t u_0 - \Delta u_0 = 0$). Finally, we refer to [SW16] for the same question in an exterior domain.

There are also some abstract results in [CH04, RTY11, Nis16, RTY16]. For instance, in [RTY11] the Laplacian is replaced by a general nonnegative selfadjoint operator, while in [RTY16] another operator $C$ acts on $\partial_t^2 u$. In [Nis16], a general damping operator acts on $\partial_t u$, and an application closer to our setting is provided, since the wave equation with a Laplacian in divergence form and an absorption index which can vanish is considered. Only the decay of the full energy is discussed in this setting.

My contributions for this problem are about the wave equation on a wave guide and in an asymptotically periodic setting.

The original motivation was to consider the wave equation on a straight wave guide, with damping at the boundary. See the setting of Section 2.3. I first tried to apply the Mourre theory. This gave the paper [9], already discussed in Section 1.3, but this was not enough to get the local energy decay. Then, as an intermediate step, I considered the case of the Schrödinger equation with a one-dimensional cross-section, which is a much simpler model. This has been published in [8], which is discussed in Section 2.7 below. The problem of a wave guide with constant damping on the boundary has finally been solved in [14]. This is discussed in Section 2.3.

After this, I invited in Toulouse Mohamed Malloug, who was at that time a Ph.D. Student in Sousse with Moez Khenissi. We discussed together the case of a wave guide with internal damping at infinity. This is a simpler setting, but it was an occasion to deal with a damping which does not satisfy the geometric damping condition (1.17). See Section 2.4. It is only then that, as a byproduct of the analysis, we wrote a result about the problem on the full Euclidean space (where $a(x)$ is a long-range perturbation of 1), which is actually much easier.

After wave guides, I considered with Romain Joly the wave equation in an asymptotically periodic setting. This raised different difficulties, which will be detailed Section 2.5.

Since this report is an opportunity to rewrite history, we begin here with the Euclidean case and present in Section 2.2 some ideas on the model problem (2.1). Then, in the following sections, we will discuss the difficulties coming from wave guides and the periodic setting.

## 2.2 On the Euclidean space

In this paragraph, we discuss some general ideas on the model case (2.1). More precisely, we show from a spectral point of view why the solution of (2.1) behaves for large times like the
solution of (2.2) with $f_0$ given by (2.3). This is not written like this in any paper, since there were already other proofs for this case in the literature. However, we use this simple setting to show some ideas used in the papers [14, 15, 17], without the difficulties specific to these more sophisticated cases.

### 2.2.1 The wave operator in the inhomogeneous energy space

We are going to compare a solution of the damped wave equation with a solution of the heat equation. However, for the wave equation we usually consider the first derivatives (with respect to space and time) of the solution, while for the heat equation we estimate the solution itself. Thus, in addition to the energy space $\mathcal{E} = \dot{H}^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$, we also consider the Hilbert space $\mathcal{H} = H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$ to rewrite the wave equation (1.12) in the form (1.18).

We define the wave operator $W_\mathcal{E}$ by (1.19)-(1.20). Notice that we denote by $W_\mathcal{E}$ the operator denoted by $W$ in the previous chapter. In particular, by Proposition 1.4 the operator $iW_\mathcal{E}$ is maximal dissipative and $W_\mathcal{E}$ generates a contractions semigroup on $\mathcal{E}$. Moreover, the resolvent of $W_\mathcal{E}$ is given by (1.23).

In this chapter, we denote by $W$ the operator defined by (1.19) on $\mathcal{H}$, with domain $\text{Dom}(W) = \{U \in \mathcal{H} : WU \in \mathcal{H}\}$. This is convenient to consider the solution and not only the derivatives, but $iW$ is no longer a dissipative operator. However, we will still be able to write (1.27) in $\mathcal{H}$.

**Proposition 2.1.** The operator $W$ satisfies on $\mathcal{H}$ the following properties.

(i) $i(W - \frac{1}{2})$ is maximal dissipative on $\mathcal{H}$.

(ii) $\mathbb{C}^+ \subset \rho(W)$, and for $\zeta \in \mathbb{C}^+$ the resolvent $(W - \zeta)^{-1}$ is given by (1.21) or (1.23).

(iii) $W$ generates a $C^0$-semigroup on $\mathcal{H}$. Moreover, for $\nu \in [0, 1]$ there exists $M_\nu \ge 0$ such that $\|e^{tW}\|_{\mathcal{L}(\mathcal{H})} \le M_\nu e^{t\nu}$ for all $t \ge 0$.

### 2.2.2 Local energy decay for the heat equation

We have said that we compare the solution of (2.1) with a solution of (2.2). Before going further, we recall the local energy decay for a solution of the heat equation. For the solution itself and its first derivatives.

**Proposition 2.2.** Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $s \in [0, 1]$. Let $\kappa > 1$. There exists $C > 0$ such that for $t > 0$ we have

$$ \begin{align*} \| \langle x \rangle^{-\kappa s_1} e^{t\Delta} \langle x \rangle^{-\kappa s_2} \|_{\mathcal{L}(L^2(\mathbb{R}^d))} &\le C \langle t \rangle^{-\frac{1}{2}(s_1+s_2)}, \\ \| \langle x \rangle^{-\kappa s_1-s} \nabla e^{t\Delta} \langle x \rangle^{-\kappa s_2-s} \|_{\mathcal{L}(L^2(\mathbb{R}^d))} &\le C \langle t \rangle^{-\frac{1}{2}(1+s_1+s_2+s)}, \\ \| \langle x \rangle^{-\kappa s_1} \partial_t e^{t\Delta} \langle x \rangle^{-\kappa s_2} \|_{\mathcal{L}(L^2(\mathbb{R}^d))} &\le C \langle t \rangle^{-\frac{1}{2}(2+s_1+s_2)}. \end{align*} $$

In [15] we provided a proof based on the explicit kernel for the heat equation (see Proposition 3.1 therein). Here we describe on this model case a more general strategy which will also be used to estimate the difference between the solutions of (2.1) and (2.2) below, following [17, Prop. 4.12]. See also [14, Prop. 3.3] for a third approach.

In Proposition 2.2 we give estimates in weighted $L^2$-spaces, but in the proof they are deduced from $L^p-L^q$ estimates. This means that we can also directly state $L^p-L^q$ estimates in this context.

The first estimate can be rewritten as

$$ \| \langle x \rangle^{-\kappa s_1} e^{t\Delta} \phi \|_{L^2(\mathbb{R}^d)} \lesssim \langle t \rangle^{-\frac{1}{2}(s_1+s_2)} \| \langle x \rangle^{\kappa s_2} \phi \|_{L^2(\mathbb{R}^d)}, \quad \phi \in L^{2, \kappa s_2}(\mathbb{R}^d). $$

The parameter $s_2$ measures how localized is the initial data $\phi$. With $s_2 = 0$ we are considering a general $\phi \in L^2(\mathbb{R}^d)$, while if $\phi$ is compactly supported we get a better estimate by choosing
$s_2 = \frac{d}{2}$. Notice that we do not get a better estimate by considering a stronger weight, see Remark 2.3. Similarly, $s_1$ gives the weight for the energy. If $s_1 = 0$ we consider the $L^2$ norms of $u(t)$, $\nabla u(t)$ and $\partial_t u(t) = \Delta u(t)$, but if we are only interested in the $L^2$ norms of these quantities in a compact subset of $\mathbb{R}^d$ we get better estimates with $s_1 = \frac{d}{2}$. All the intermediate situations $s_1, s_2 \in ]0, \frac{d}{2}]$ are also possible. There is a parameter $\kappa > 1$ in the weights since if $s_j \in ]0, \frac{d}{2}]$ we need a weight $\langle x \rangle^{-\delta_j}$ with $\delta_j > s_j$, while no weight is necessary ($\delta_j = 0$) when $s_j = 0$. Finally, as we will see in the last step of the proof, we can get an extra power of $t^{-\frac{1}{2}}$ for the gradient if we can pay an extra weight $\langle x \rangle^{-1}$ on both sides of the estimate. This is the role of the parameter $s$.

*Proof.* We first notice that in the third estimate we can replace $\partial_t e^{t\Delta}$ by $\Delta e^{t\Delta}$. Let $f_0 \in S(\mathbb{R})$. For $t \ge 0$ we set $u_0(t) = e^{t\Delta}f_0$. Let

$$p_1 = \frac{2d}{d - 2s_1} \in [2, +\infty] \quad \text{and} \quad p_2 = \frac{2d}{d + 2s_2} \in [1, 2]. \qquad (2.4)$$

We denote by $p'_1 \in [1, 2]$ and $p'_2 \in [2, +\infty]$ the conjugate exponents of $p_1$ and $p_2$, respectively. Let $\alpha \in \mathbb{N}^d$. Since $L^{p_1}(\mathbb{R}^d)$ is continuously embedded in $L^{2, -\kappa s_1}(\mathbb{R}^d)$, and by continuity of the (inverse) Fourier transform (with respect to the space variable) from $L^{p'_1}(\mathbb{R}^d)$ to $L^{p_1}(\mathbb{R}^d)$, we have

$$\|D^{\alpha} u_0(t)\|_{L^{2, -\kappa s_1}(\mathbb{R}^d)} \lesssim \|D^{\alpha} u_0(t)\|_{L^{p_1}(\mathbb{R}^d)} \lesssim \|\widehat{D^{\alpha} u_0}(t)\|_{L^{p'_1}(\mathbb{R}^d)}. \qquad (2.5)$$

Similarly,

$$\|\hat{f}_0\|_{L^{p'_2}(\mathbb{R}^d)} \lesssim \|f_0\|_{L^{p_2}(\mathbb{R}^d)} \lesssim \|f_0\|_{L^{2, \kappa s_2}(\mathbb{R}^d)}. \qquad (2.6)$$

Let $p'_0 \in [1, +\infty]$ be such that

$$\frac{1}{p'_0} = \frac{1}{p'_1} - \frac{1}{p'_2} = \frac{1}{p_2} - \frac{1}{p_1} = \frac{s_1 + s_2}{d}.$$

By the Hölder inequality and the change of variable $\eta = \sqrt{t\xi}$ we get

$$\|\widehat{D^{\alpha} u_0}(t)\|_{L^{p'_1}(\mathbb{R}^d)} \lesssim \|\xi^{\alpha} e^{-t|\xi|^2}\|_{L^{p'_0}(\mathbb{R}^d)} \|\hat{f}_0\|_{L^{p'_2}(\mathbb{R}^d)} \lesssim t^{-\frac{|\alpha|}{2} - \frac{d}{2p'_0}} \|\eta^{\alpha} e^{-|\eta|^2}\|_{L^{p'_0}(\mathbb{R}^d)} \|\hat{f}_0\|_{L^{p'_2}(\mathbb{R}^d)}.$$

Finally,

$$\|D^{\alpha} u_0(t)\|_{L^{2, -\kappa s_1}(\mathbb{R}^d)} \lesssim t^{-\frac{1}{2}(|\alpha| + s_1 + s_2)} \|f_0\|_{L^{2, \kappa s_2}(\mathbb{R}^d)},$$

and the three estimates follow if $s = 0$. Now we assume that $s = 1$. Only the estimate for the gradient is concerned. Let $j \in \{1, \dots, d\}$ and $x \in \mathbb{R}^d$. We compute

$$\begin{aligned}
\partial_{x_j} u_0(t, x) &= \int_{\mathbb{R}^d} e^{ix \cdot \xi} i \xi_j e^{-t|\xi|^2} \hat{f}_0(\xi) \, d\xi \\
&= -\frac{i}{2t} \int_{\mathbb{R}^d} e^{ix \cdot \xi} \partial_{\xi_j} (e^{-t|\xi|^2}) \hat{f}_0(\xi) \, d\xi \\
&= -\frac{x_j}{2t} \int_{\mathbb{R}^d} e^{ix \cdot \xi} e^{-t|\xi|^2} \hat{f}_0(\xi) \, d\xi + \frac{i}{2t} \int_{\mathbb{R}^d} e^{ix \cdot \xi} e^{-t|\xi|^2} \partial_{\xi_j} \hat{f}_0(\xi) \, d\xi \\
&= \frac{1}{2t} (-x_j(e^{t\Delta}f_0)(x) + e^{t\Delta}(y_jf_0)(x)).
\end{aligned}$$

Then, using the first estimate of the proposition,

$$\begin{aligned}
\|\partial_{x_j} u_0(t)\|_{L^{2, -\kappa s_1-1}(\mathbb{R}^d)} &\lesssim t^{-\frac{s_1+s_2}{2}} \left( \|f_0\|_{L^{2, \kappa s_2}(\mathbb{R}^d)} + \|y_j f_0\|_{L^{2, \kappa s_2}(\mathbb{R}^d)} \right) \\
&\lesssim t^{-\frac{s_1+s_2}{2}} \|f_0\|_{L^{2, \kappa s_2+1}(\mathbb{R}^d)}.
\end{aligned}$$

This gives the estimate for the gradient when $s = 1$. The case $s \in ]0, 1[$ follows by interpolation. $\square$
Remark 2.3. The estimates of Proposition 2.2 are optimal in the sense that we do not get better decay even if we consider stronger (for instance compactly supported) weights. Indeed, for $f_0 \in C_0^\infty(\mathbb{R}^d)$, the explicit expression for $u_0$ gives uniformly for $x$ in a compact

$$u_0(t, x) = \frac{1}{(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^d} e^{-\frac{|x-y|^2}{4t}} f_0(y) \, dy \underset{t \to +\infty}{\sim} \frac{1}{(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^d} f_0(y) \, dy,$$

$$\nabla u_0(t, x) = -\frac{1}{(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^d} \frac{(x-y)}{2t} e^{-\frac{|x-y|^2}{4t}} f_0(y) \, dy \underset{t \to +\infty}{\sim} -\frac{1}{2t(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^d} (x-y) f_0(y) \, dy,$$

and

$$\partial_t u_0(t, x) = -\frac{d}{2t} u_0(t, x) + \mathcal{O}(t^{-\frac{d}{2}-2}) \underset{t \to +\infty}{\sim} -\frac{d}{2t(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^d} f_0(y) \, dy.$$

Our purpose is now to prove that the decay of the difference between the solutions of (2.1) and (2.2) is faster than the decay given by Proposition 2.2.

### 2.2.3 The contribution of high frequencies

We begin with high frequencies. As already said, since we have damping everywhere, all the classical trajectories and not only the bounded ones see the damping. From this we can prove uniform estimates for the resolvent $(W - \zeta)^{-1}$ directly in the energy space, and not in weighted versions. This implies in particular that the imaginary axis (except 0) is included in the resolvent set of $W$, with uniform estimates for the resolvent (away from a neighborhood of 0). It is classical that if the imaginary axis is included in the resolvent set with uniform estimates for the resolvent, then the propagator decays uniformly (by the Gearhart-Prüss-Greiner Theorem, see [EN00, Th. V.1.11]). We cannot apply this result directly to $W$ because of the problem at 0, but it is not surprising to recover a uniform and exponential decay for the contribution of high frequencies.

**Theorem 2.4.** There exist $\gamma > 0$ and $C > 0$ such that for all $\zeta \in \mathbb{C}$ with $\mathrm{Re}(\zeta) \ge -2\gamma$ and $|\mathrm{Im}(\zeta)| \ge 1$ we have $\zeta \in \rho(W)$ and $\|(W - \zeta)^{-1}\|_{\mathcal{L}(\mathcal{H})} \le C$.

*Ideas of proof.* Since the model case (2.1) is a problem with constant coefficients on $\mathbb{R}^d$, we can use the Fourier transform. For $F \in \mathcal{S}$ and $\xi \in \mathbb{R}^d$ we have

$$\widetilde{WF}(\xi) = M(\xi) \hat{F}(\xi), \quad \text{where} \quad M(\xi) = \begin{pmatrix} 0 & 1 \\ -\xi^2 & -a_1 \end{pmatrix}. \qquad (2.7)$$

The eigenvalues of $M(\xi)$ are

$$\lambda_{\pm}(\xi) = \frac{-a_1 \pm \sqrt{a_1^2 - 4\xi^2}}{2},$$

where we have set $\sqrt{a_1^2 - 4\xi^2} = i\sqrt{4\xi^2 - a_1^2}$ if $4\xi^2 > a_1^2$. Then $\zeta \in i(\mathbb{R}\setminus]-1, 1[)$ is not an eigenvalue of $M(\xi)$ and we see that $(M(\xi) - \zeta)^{-1}$ is bounded in $\mathcal{L}(C_\xi^2)$ uniformly in $\zeta$ and $\xi$, with $\|(\alpha, \beta)\|_{C_\xi^2} = (1 + |\xi|^2) |\alpha|^2 + |\beta|^2$. Then $(W - \zeta)^{-1}F$ is the inverse Fourier transform of $\xi \mapsto (M(\xi) - \zeta)^{-1}\hat{F}(\xi)$ and we deduce Theorem 2.4 by the Parseval identity. $\square$

### 2.2.4 A first change of contour - Separation of low and high frequencies

As with a localized damping (see Section 1.2.2), we write the propagator $e^{tW}$ in terms of the resolvent $(W - \zeta)^{-1}$. See (1.27). Since the integrand is holomorphic in $\zeta$, we can change the contour of integration $\{\mathrm{Im}(\zeta) = \mu\}$ by $\Gamma_\mu$ defined as follows (see Figure 2.1). Given $\theta \in C_0^\infty(\mathbb{R}, [0, 1])$ supported in $]-2, 2[$ and equal to 1 on $[-1, 1]$ we set

$$\Gamma_{\mu}: \tau \in \mathbb{R} \mapsto -i\tau + \mu\theta(\tau) - \gamma(1 - \theta(\tau)). \qquad (2.8)$$
<figure><img src="image_3.png" /><figcaption>Figure 2.1: Contours {$\text{Re}(z) = \mu$}, $\Gamma_\mu$ and $\Gamma_{-a/2} \cup \mathcal{C}(\lambda_+(\xi), r)$. (in grey, a region which contains the spectrum of $\mathcal{W}$ in the first two pictures, and the eigenvalues of $M(\xi)$ or $\mathcal{W}_\sigma$, for $\xi$ or $\sigma$ small, in the third)</figcaption></figure>

Then we separate the contributions of high and low frequencies. We consider $\chi_{\text{low}} \in C_0^\infty(\mathbb{R}, [0, 1])$ supported in $]-3,3[$ and equal to 1 on $[-2,2]$. Then we set $\chi_{\text{high}} = 1 - \chi_{\text{low}}$. Finally, for $* \in \{\text{low, high}\}$ we set

$$ \mathcal{U}_*(t)F = \frac{1}{2i\pi} \int_{\Gamma_\mu} e^{t\zeta} \chi_*(\operatorname{Im}(\zeta))(\mathcal{W} - \zeta)^{-1} F_\zeta d\zeta. \qquad (2.9) $$

This is analogous to (1.28). In particular, we use the notation $F_\zeta$ introduced in (1.26). Notice that it is important to write the integral on the contour $\Gamma_\mu$ before separating the contributions of low and high frequencies, since the integrands in $\mathcal{U}_*(t)F$ are no longer holomorphic in the region $\{2 \le |\operatorname{Im}(\zeta)| \le 3\}$. It is still possible to change the contour for $\mathcal{U}_{\text{low}}(t)$ in the strip $\{|\operatorname{Im}(\zeta)| \le 2\}$. In particular $\mathcal{U}_{\text{low}}(t)F$, as $\mathcal{U}_{\text{high}}(t)F$, does not depend on $\mu$.

We finish the analysis of high frequencies. Following the same idea as for Proposition 1.9, we can deduce from Theorem 2.4 that $\mathcal{U}_{\text{high}}(t)F$ decays uniformly exponentially.

**Proposition 2.5.** Let $\gamma > 0$ be given by Theorem 2.4. There exists $C > 0$ such that for $t \ge 0$ and $F \in \mathcal{H}$ we have

$$ \|\mathcal{U}_{\text{high}}(t)F\|_{\mathcal{H}} \le C e^{-\frac{t\gamma}{2}} \|F\|_{\mathcal{H}}. $$

### 2.2.5 The contribution of low frequencies - Comparison with the diffusive profile

We turn to the contribution of low frequencies. Replacing $F$ by $F_\zeta$ in (2.9) is convenient for the analysis of $\mathcal{U}_{\text{high}}(t)F$, but to obtain a precise expression of the asymptotic profile, we go back to $F$ for low frequencies. For $t \ge 1$ we set

$$ U_{\#}(t) = \frac{1}{2i\pi} \int_{\Gamma_{\mu}} e^{t\zeta} \chi_{\text{low}}(\operatorname{Im}(\zeta))(\mathcal{W} - \zeta)^{-1} F d\zeta. $$

For $\zeta \in \mathbb{C}^+$ we have

$$ (\mathcal{W} - \zeta)^{-1}(F_\zeta - F) = \int_0^2 (1 - \phi(s)) e^{s(\mathcal{W} - \zeta)} F ds, $$

so with integration by parts we can see that $\mathcal{U}_{\text{low}}(t)F - \mathcal{U}_{\#}(t)F$ is small:

**Proposition 2.6.** Let $m \in \mathbb{N}$. There exists $C \ge 0$ such that for $F \in \mathcal{H}$ and $t \ge 1$ we have

$$ \|\mathcal{U}_{\text{low}}(t)F - \mathcal{U}_{\#}(t)F\|_{\mathcal{H}} \le C \langle t \rangle^{-m} \|F\|_{\mathcal{H}}. $$

To estimate $\mathcal{U}_{\#}(t)F$ we now have several possibilities.
**First strategy.** The first option is to compare the resolvent

$$R(z) = R_{a_1}(z) = (-\Delta - iza_1 - z^2)^{-1}$$

of the wave equation (2.1) to the resolvent $R_h(z) = (-\Delta - iza_1)^{-1}$ corresponding to the heat equation (2.2). This is what led to the ideas recently developed in [23] and explained in Section 1.7.

**Proposition 2.7.** Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $\kappa > 1$. Let $\beta \in \mathbb{N}^d$ with $|\beta| \le 1$. Let $m \in \mathbb{N}$. There exists $C > 0$ such that for $z \in \mathbb{C}_+ \cap \mathbb{D}$ we have

$$\| \langle x \rangle^{-\kappa s_1} \partial_x^\beta \partial_z^m (R(z) - R_h(z)) \langle x \rangle^{-\kappa s_2} \|_{\mathcal{L}(L^2(\mathbb{R}^d))} \le C |z|^{\frac{1}{2} \min(0, s_1 + s_2 - 2m + |\beta|)}.$$

*Ideas of proof.* Here, for the model case, the resolvent identity gives

$$R(z) - R_h(z) = z^2 R(z) R_h(z). \qquad (2.10)$$

As in Section 1.4 we work in the rescaled Sobolev space, except that we now use $H^s_{|z|^{1/2}}$ instead of $H^s_{|z|}$. A derivative is now of size $O(|z|^{\frac{1}{2}})$ in $\mathcal{L}(H^s_{|z|^{1/2}}, H^s_{|z|^{1/2}})$ for any $s \in \mathbb{R}$. On the other hand, $R(z)$ and $R_h(z)$ are of size $O(|z|^{-1})$ in $\mathcal{L}(H^s_{|z|^{1/2}}, H^s_{|z|^{1/2}})$. An important difference with the setting of Chapter 1 is that for $z \in \mathbb{C}_+$ the spectral parameter is $iza_1$. Since $\mathrm{dist}(iza_1, \mathbb{R}_+) \gtrsim |z|^{-1}$, we do not need the limiting absorption principle and we will not use the commutators method of Section 1.3 in this chapter. The weights $\langle x \rangle^{-\delta}$ are only used to convert the elliptic regularity given by the resolvents into powers of $z$. By Proposition 1.19 we have

$$\| \langle x \rangle^{-\delta_1} \|_{\mathcal{L}(H^s_{|z|^{1/2}}, L^2(\mathbb{R}^d))} \lesssim |z|^{\frac{s_1}{2}} \quad \text{and} \quad \| \langle x \rangle^{-\delta_2} \|_{L^2(\mathbb{R}^d), \mathcal{L}(H^s_{|z|^{1/2}})} \lesssim |z|^{\frac{s_2}{2}}. \qquad (2.11)$$

This is the analog of (1.63) used for (1.61).

We do not go further in the details of the proof, but the rough conclusion is that we have better estimates on $R(z) - R_h(z)$ than on $R_h(z)$. Notice that we can iterate (2.10) and get an expansion of $R(z)$ in terms of $R_h(z)$ up to a rest of any order. With (1.21) we can similarly write an expansion of the resolvent $(W - \zeta)^{-1}$. $\square$

All the arguments above with the operator $W$ on $\mathcal{H}$ can be reproduced with the operator $\Delta/a_1$ on $L^2(\mathbb{R}^d)$. Then, up to rapidly decaying terms, the solution $e^{\frac{t\Delta}{a_1}} f_0$ of (2.2) is equal to

$$\frac{1}{2i\pi} \int_{\Gamma_\mu} e^{t\zeta} \chi_{\mathrm{low}}(\mathrm{Im}(\zeta)) \left(\frac{\Delta}{a_1} - \zeta\right)^{-1} f_0 \mathrm{d}\zeta = -\frac{a_1}{2i\pi} \int_{\Gamma_\mu} e^{t\zeta} \chi_{\mathrm{low}}(\mathrm{Im}(\zeta)) R_h(i\zeta) f_0 \mathrm{d}\zeta.$$

In view of (1.21) (replacing $R(z)$ by $R_h(z)$, removing the terms with extra powers of $\zeta$ and removing the constant term in the bottom left coefficient), we see that the leading term for the first component of $U_{\pm, \mu}(t)$ is the solution of (2.2) with $f_0$ as given by (2.3) (and the leading term for the second component is its time derivative).

This kind of strategy has been used in [14] and [15] for wave guides. The interest is that we can in particular consider an absorption index $a(x)$ which is a long-range perturbation of the constant $a_1$, since we can similarly estimate the difference $R_a(z) - R_{a_1}(z)$, using Proposition 1.19 again.

The drawback with this strategy is that we have to know in advance what the asymptotic profile will look like. This can lead to an impasse with a wrong guess. This is what happened for [17]. More important, in the periodic setting the perturbation does not decay at infinity and we can no longer use a result like Proposition 1.19 to recover some smallness for low frequencies. For [17], we had to proceed with a different idea. Here we explain this new strategy on the model case. This is also what gave the idea used in the proof of Proposition 2.2.
**Second strategy.** The alternative approach uses the symmetries of the problem. Here, with constant coefficients, we use again the Fourier transform. We can write

$$ \mathcal{U}_{\#}(t)F = \frac{1}{2i\pi} \int_{\xi \in \mathbb{R}^d} \int_{\zeta \in \Gamma_{\mu}} e^{ix \cdot \xi} e^{t\zeta} \chi_{\text{low}}(\text{Im}(\zeta))(M(\xi) - \zeta)^{-1} \hat{F}(\xi) \, d\zeta \, d\xi. $$

For $\rho > 0$ (to be chosen small enough) there exists $\varepsilon > 0$ such that $\text{Re}(\lambda_{\pm}(\xi)) \le -2\varepsilon$ if $|\xi| \ge \rho$. Then for $|\xi| \ge \rho$ we can replace the contour $\Gamma_{\mu}$ by $\Gamma_{-\varepsilon}$ and the corresponding contribution decays exponentially. It remains the contribution $\mathcal{U}_{\rho}(t)F$, defined as $\mathcal{U}_{\#}(t)F$ with the integral on $\xi$ over the ball $B(\rho)$ instead of $\mathbb{R}^d$.

For $\xi$ small, $M(\xi)$ has one simple eigenvalue $\lambda_{+}(\xi)$ close to 0 and the other one close to $-a$. We define $\tilde{\mathcal{U}}_{\rho}(t)F$ as $\mathcal{U}_{\rho}(t)F$ with $\Gamma_{\mu}$ replaced by $\Gamma_{-a/2}$. It decays uniformly exponentially. The difference between $\mathcal{U}_{\rho}(t)$ and $\tilde{\mathcal{U}}_{\rho}(t)$ is given by the Residue Theorem (see the third contour in Figure 2.1). We obtain that, up to exponentially decaying terms, $\mathcal{U}_{\#}(t)F$ is equal to

$$ \int_{\xi \in B(\rho)} e^{ix \cdot \xi} e^{t\lambda_{+}(\xi)} \Pi(\xi) \hat{F}(\xi) \, d\xi, \qquad (2.12) $$

where

$$ \Pi(\xi) = -\frac{1}{2i\pi} \int_{\zeta \in \mathcal{C}(\lambda_{+}(\xi), r)} (M(\xi) - \zeta)^{-1} \, d\zeta $$

is the Riesz projection on the eigenspace of $M(\rho)$ corresponding to the eigenvalue $\lambda_{+}(\xi)$ (the integral is over a small circle around $\lambda_{+}(\xi)$, oriented in the direct sense, while the circle in Figure 2.1 is oriented in the clockwise direction). For $G \in \mathbb{C}^2$ we have

$$ \Pi(\xi)G = \langle \Psi(\xi), G \rangle_{\mathbb{C}^2} \Phi(\xi), \quad \Phi(\xi) = \begin{pmatrix} 1 \\ \lambda_{+}(\xi) \end{pmatrix}, \quad \Psi(\xi) = \frac{1}{\sqrt{a_1^2 - 4\xi^2}} \begin{pmatrix} a_1 + \lambda_{+}(\xi) \\ 1 \end{pmatrix}. $$

Notice that

$$ \lambda_{+}(\xi) = -\frac{\xi^2}{a_1} + O(\xi^4), \quad \Phi(\xi) = \begin{pmatrix} 1 \\ -\frac{\xi^2}{a_1} + O(\xi^4) \end{pmatrix}, $$

and

$$ \langle \Psi(\xi), \hat{F}(\xi) \rangle_{\mathbb{C}^2} = \hat{f}(\xi) + \frac{\hat{g}(\xi)}{a_1} + O(\xi^2 \|\hat{F}(\xi)\|). $$

In particular, if in (2.12) we replace each factor by the leading term in its Taylor expansion, we expect that $\mathcal{U}_{\rho}(t)$ should be close to

$$ \mathcal{U}_{\text{heat}}(t)F = \int_{B(\rho)} e^{ix \cdot \xi} e^{-\frac{t\xi^2}{a_1}} \hat{f}_0(\xi) \begin{pmatrix} 1 \\ -\frac{\xi^2}{a_1} \end{pmatrix} \, d\xi, \qquad (2.13) $$

with $f_0$ given by (2.3). Up to the contribution of $|\xi| \ge \rho$, which is exponentially decaying, we see that the first component of (2.13) is the solution of (2.2)-(2.3) and the second component is its time derivative. To prove that (2.12) is close to (2.13) we apply the idea given in the proof of Proposition 2.2.

**Conclusion.** We finally obtain the following result, where for $\delta \in \mathbb{R}$ we have set $\mathcal{H}^{\delta} = H^{1,\delta}(\mathbb{R}^d) \times L^{2,\delta}(\mathbb{R}^d)$.

**Theorem 2.8.** Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $\kappa > 1$. Let $s \in [0, 1]$. There exists $C > 0$ such that for $(f, g) \in \mathcal{H}^{\kappa s_2}$ and $t \ge 0$ we have

$$ \begin{aligned} \|u(t) - u_0(t)\|_{L^{2, -\kappa s_1}(\mathbb{R}^d)} &\le C \langle t \rangle^{-\frac{1}{2}(2+s_1+s_2)} \|F\|_{\mathcal{H}^{\kappa s_2}}, \\ \|\nabla(u(t) - u_0(t))\|_{L^{2, -\kappa s_1-s}(\mathbb{R}^d)} &\le C \langle t \rangle^{-\frac{1}{2}(3+s_1+s_2+s)} \|F\|_{\mathcal{H}^{\kappa s_2+s}}, \\ \|\partial_t(u(t) - u_0(t))\|_{L^{2, -\kappa s_1}(\mathbb{R}^d)} &\le C \langle t \rangle^{-\frac{1}{2}(4+s_1+s_2)} \|F\|_{\mathcal{H}^{\kappa s_2}}, \end{aligned} $$

where $u(t)$ is the solution of (2.1) and $u_0(t)$ is the solution of (2.2) with $f_0$ given by (2.3).
Combining Theorem 2.8 and Proposition 2.2 we deduce in particular that the estimates of Proposition 2.2 also hold for $u(t)$.

We have already observed that the local energy of the wave equation decays slowly with a strong damping compared to the case without damping. We furthermore notice that the decay for the heat equation (2.2) is slower when $a_1$ becomes large.

## 2.3 On a wave guide with damping on the boundary

As said in the introduction of the chapter, my first work about a damped wave equation with damping at infinity was not on the Euclidean space but on a wave guide.

More precisely, after a discussion with D. Krejcirik and P. Siegl on my earlier results about the resolvent estimates by the commutator method, I was interested in the analysis of the problem on a wave guide, with damping at the boundary.

On a waveguide we have bounded classical trajectories at infinity, so the geometric damping condition is not satisfied if the damping is localized. It seemed simpler to consider the case of a constant damping. This led to the first question with damping at infinity (even if I finally first wrote [8] about the Schrödinger equation, see Section 2.7).

Notice that there are now recent results [Wanb, Wanc] about the global energy decay for the damped wave equation on (more general) wave guides with damping at the boundary.

### 2.3.1 The setting

Let $\Omega = \mathbb{R}^d \times \omega$ be a straight wave guide in $\mathbb{R}^{d+n}$. The cross section $\omega$ is a smooth, bounded, connected and open subset of $\mathbb{R}^n$. A general point of $\Omega$ will be denoted by $(x, y)$ with $x \in \mathbb{R}^d$ and $y \in \omega$.

We consider on $\Omega$ the wave equation

$$ egin{cases} 
\partial_t^2 u - \Delta u = 0, & \text{on } \mathbb{R}_+ \times \Omega, \\
\partial_\nu u + a \partial_t u = 0, & \text{on } \mathbb{R}_+ \times \partial \Omega, \\
(u, \partial_t u)|_{t=0} = (f, g), & \text{on } \Omega,
\end{cases} 
\quad (2.14) $$

where $a > 0$ is a constant absorption index on the boundary and $(f, g)$ belongs to $H^1(\Omega) \times L^2(\Omega)$. The energy is still a non-increasing function of time:

$$ \frac{d}{dt} \int_{\Omega} (|\nabla u(t)|^2 + |\partial_t u(t)|^2) = -2a \int_{\partial \Omega} |\partial_t u(t)|^2 \le 0. $$

It is still possible to rewrite the wave equation in the form (1.18). For this we now define on $\mathcal{E} = \dot{H}^1(\Omega) \times L^2(\Omega)$ the operator

$$ \mathcal{W} = \begin{pmatrix} 0 & 1 \\ \Delta & 0 \end{pmatrix}, \quad (2.15) $$

with domain

$$ \mathrm{Dom}(\mathcal{W}) = \{(u, v) \in \mathcal{E} : (v, \Delta u) \in \mathcal{E} \text{ and } \partial_{\nu} u + av = 0 \text{ on } \partial \Omega\}. \quad (2.16) $$

We can check that $i\mathcal{W}$ is a dissipative operator on $\mathcal{E}$. Then we introduce the analog of (1.22). It is given by

$$ R(z) = (H_{az} - z^2)^{-1}, \quad (2.17) $$

where for $\alpha \in \mathbb{C}$ we have defined on $L^2(\Omega)$ the operator $H_\alpha$ as in (1.39)

$$ H_{\alpha} = -\Delta, \quad \mathrm{Dom}(H_{\alpha}) = \{u \in H^2(\Omega) : \partial_{\nu} u = i\alpha u \text{ on } \partial\Omega\}. \quad (2.18) $$
One of the difficulty in [6] was that the derivatives of $R(z)$ (as defined by (1.22)) are not its powers. Here it is even worse since the operator ($H_{az} - z^2$) depends on $z$ via its domain. Thus its derivatives only make sense if it is seen as a form on $H^1(\Omega)$.

In the sense of forms it becomes clearer that $H_{az}-z^2$ is the analog of $(-\Delta-iza(x)-z^2)$ for a damping at the boundary. Indeed, the quadratic form on $H^1(\Omega)$ associated with $H_{az}-z^2$ is $-\Delta-iz\Theta_a-z^2 \in \mathcal{L}(H^1(\Omega), H^1(\Omega)^*)$, where $\Theta_a$ is defined by

$$ \forall \varphi, \psi \in H^1(\Omega), \quad \langle \Theta_a \varphi, \psi \rangle_{H^1(\Omega)*, H^1(\Omega)} = a \int_{\partial \Omega} \varphi \bar{\psi}. $$

The fact that the boundary condition is a good model for a damping concentrated at the boundary has been discussed in more details for a quantum graph in [26, Section 4].

Thus, in addition to $R(z)$ we also have to consider, at least for $z \in \mathbb{C}_+$,

$$ \tilde{R}(z) = (-\Delta - iz\Theta_a - z^2)^{-1} \in \mathcal{L}(H^1(\Omega)^*, H^1(\Omega)). $$

The link between $R(z)$ and $\tilde{R}(z)$ is that for $\varphi \in L^2(\Omega)$ and $\psi \in H^1(\Omega)$ then $u = \tilde{R}(z)(\varphi+\Theta_a\psi)$ is the unique solution in $H^2(\Omega)$ of the problem

$$ \begin{cases} (-\Delta - z^2)u = \varphi & \text{on } \Omega, \\ \partial_{\nu}u = iazu + a\psi & \text{on } \partial\Omega. \end{cases} $$

In particular, $\tilde{R}(z)\varphi = R(z)\varphi$ for all $\varphi \in L^2(\Omega)$.

Then we can check that the operator $i\mathcal{W}$ is maximal dissipative in $\mathcal{E}$, and that for $F \in H^1(\Omega) \times L^2(\Omega)$ its resolvent is given by

$$ (\mathcal{W} - \zeta)^{-1} F = \begin{pmatrix} -\tilde{R}(i\zeta)(\Theta_a + \zeta) & -\tilde{R}(i\zeta) \\ 1 - \tilde{R}(i\zeta)(\zeta\Theta_a + \zeta^2) & -\zeta\tilde{R}(i\zeta) \end{pmatrix} F. \quad (2.19) $$

We want to understand the long time behavior of the solution of (2.14), in particular the local energy decay and the asymptotic profile. Compared to (2.1) we cannot just “remove” the term $\partial_t^2 u$ in (2.14) to get a heat equation which should describe the asymptotic behavior of $u(t)$.

### 2.3.2 Spectral properties of $H_\alpha$

It is $\tilde{R}(z)$ which appears in (2.19), but since $\tilde{R}(z)$ and $R(z)$ are closely related, we can look at the properties of $R(z)$, hence at the spectral properties of the operator $H_\alpha$. For this we use the symmetries of $\Omega$. We write the Laplacian $H_\alpha$ as the sum of the second derivatives in the $x$ and $y$ directions. We denote by $L$ the usual Laplacian on $\mathbb{R}^d$ and by $T_\alpha$ the Robin Laplacian on the cross section $\omega$:

$$ T_{\alpha} = -\Delta_{\omega}, \quad \mathrm{Dom}(T_{\alpha}) = \{u \in H^2(\omega) : \partial_{\nu}u = i\alpha u \text{ on } \partial\omega\}. \quad (2.20) $$

Then, if we identify $L$ with $L \otimes \mathrm{Id}_{L^2(\omega)}$ and $T_\alpha$ with $\mathrm{Id}_{L^2(\mathbb{R}^d)} \otimes T_\alpha$, we have

$$ H_{\alpha} = L + T_{\alpha}. $$

The spectrum of $L$ is the half-line $[0, +\infty[$. On the other hand, $T_\alpha$ has a non-empty resolvent set and its domain $\mathrm{Dom}(T_\alpha)$ is compactly embedded in $L^2(\omega)$, so its spectrum is given by a sequence $(\lambda_k(\alpha))_{k \in \mathbb{N}}$ of isolated eigenvalues of finite multiplicities. Then it is a general result (see for instance [RS79, §XIII.9]) that

$$ \sigma(H_{\alpha}) = \sigma(L) + \sigma(T_{\alpha}) = \bigcup_{k \in \mathbb{N}} \{\lambda_{k}(\alpha) + r, k \in \mathbb{N}, r \in \mathbb{R}_{+}\}. \quad (2.21) $$
Thus $\sigma(H_{az})$ is a sequence of half-lines which depend on the spectral parameter $z$.

In a selfadjoint setting it is not difficult to combine the properties of the longitudinal and transverse operators to get information on the wave guide. We can proceed similarly if we have a Riesz basis of $L^2(\omega)$ made with eigenvectors of the transverse operator (see (2.54) below, see also [BK08] for another non-selfadjoint context). Here we have to proceed differently.

Let $\tau \in \mathbb{R}^*$. The eigenvalues of $T_{a\tau}$ have negative imaginary parts. Then $\tau^2$ is in the resolvent set of $H_{a\tau}$, so the resolvent $R(\tau)$ is well defined and $-i\tau \in \rho(W)$. The two important steps of the proof are then to show that the resolvent $(W + i\tau)^{-1}$ is uniformly bounded for $|\tau| \to \infty$ and to understand the behavior of the resolvent for $\tau$ close to 0.

### 2.3.3 Contribution of low frequencies

We begin with the contribution of low frequencies. Notice that $T_0$ is the Neumann Laplacian on $\omega$, so $\lambda_0(0) = 0$ is a simple eigenvalue of $T_0$ and the corresponding eigenprojection is the orthogonal projection on constant functions. Seen as an operator on $L^2(\Omega)$ it reads

$$P_{\omega}u(x, y) = \frac{1}{|\omega|} \int_{\omega} u(x, \theta) \, d\theta. \qquad (2.22)$$

The other eigenvalues of $T_0$ are positive.

Since $T_\alpha$ is analytic with respect to $\alpha$ (family of type B in the sense of Kato [Kat80]), the spectrum of $T_\alpha$ is a holomorphic function of $\alpha$. In particular, there exists a neighborhood $V$ of 0 such that for $z$ small the operator $T_{az}$ has exactly one (geometrically and algebraically) simple eigenvalue $\lambda_0(az)$ in $V$.

If $\lambda$ is another eigenvalue of $H_{az}$ and $r \in \sigma(L) = \mathbb{R}_+$, then the sum $\lambda + r$ (see (2.21)) is “far” from 0, so the intuition is that only the contribution of the “first” transverse eigenvalue $\lambda_0(az)$ of $T_{az}$ should play a role in the behavior of $(H_{az} - z^2)^{-1}$ when $z$ is small.

To make this more precise, we would like to use a kind of spectral localization, but we cannot use spectral projections given by functional calculus ($H_{az}$ is not selfadjoint) or Riesz projections (the spectrum of $H_{az}$ close to 0 is not separated from the rest of the spectrum). However, we can use simultaneously the facts that $L$ is selfadjoint and $T_{az}$ has discrete spectrum.

We can choose $\eta > 0$ in such a way that if we set

$$\mathcal{G} = \{\zeta \in \mathbb{C} : \mathrm{Re}(\zeta) < \eta, |\mathrm{Im}(\zeta)| < \eta\},$$

then for $\alpha$ small enough we have $\mathcal{G} \cap \sigma(T_\alpha) = \{\lambda_0(\alpha)\}$ and $\partial\mathcal{G} \cap \sigma(T_\alpha) = \emptyset$. Let $\zeta \in \mathcal{G} \cap \rho(H_\alpha)$ and $\sigma \in \partial\mathcal{G}$. We have $\zeta - \sigma \in \mathbb{C}\backslash\mathbb{R}_+ = \rho(L)$, so we can write the resolvent identity

$$(H_{\alpha} - \zeta)^{-1} (T_{\alpha} - \sigma)^{-1} = (T_{\alpha} - \sigma)^{-1} (L - (\zeta - \sigma))^{-1} - (H_{\alpha} - \zeta)^{-1} (L - (\zeta - \sigma))^{-1}. \quad (2.23)$$

We set

$$P_{\alpha} := -\frac{1}{2i\pi} \int_{\partial\mathcal{G}} (T_{\alpha} - \sigma)^{-1} \, d\sigma.$$

This is the Riesz projection of $T_\alpha$ on its one-dimensional eigenspace $\ker(T_\alpha - \lambda_0(\alpha))$. On the other hand, since $L$ is selfadjoint we have for $\zeta \in \mathcal{G}$

$$-\frac{1}{2i\pi} \int_{\partial\mathcal{G}} (L - (\zeta - \sigma))^{-1} \, d\sigma = \mathrm{Id}.$$

Finally we can check (see [14, Prop.4.5]) that the operator

$$B_{\alpha}(\zeta) = \frac{1}{2i\pi} \int_{\partial\mathcal{G}} (T_{\alpha} - \sigma)^{-1} (L - (\zeta - \sigma))^{-1} \, d\sigma$$
is a holomorphic function of $\zeta \in \mathcal{G}$. Thus, after integration of (2.23) with respect to $\sigma \in \partial \mathcal{G}$ we have

$$(H_{\alpha} - \zeta)^{-1} = (H_{\alpha} - \zeta)^{-1} P_{\alpha} + B_{\alpha}(\zeta) = (L + \lambda_{0}(\alpha) - \zeta)^{-1} P_{\alpha} + B_{\alpha}(\zeta),$$

and for $z$ small we get

$$R(z) = (H_{az} - z^2)^{-1} = (L + \lambda_0(az) - z^2)^{-1} P_{az} + B_{az}(z^2).$$

Then we compute

$$\lambda'_{0}(0) = -i\Upsilon, \quad \Upsilon = \frac{|\partial\omega|}{|\omega|}.$$

At this point, we expect that the wave resolvent $R(z)$ should behave like $(L - ia\Upsilon z)^{-1} P_{\omega}$ for $z$ small. This is the resolvent corresponding to a heat equation in the $x$-direction, and the projection on constant functions with respect to $y$. We can indeed prove the following result.

**Theorem 2.9.** For $z \in \mathbb{D}_+$ we have

$$R(z) = (L - ia\Upsilon z)^{-1} P_{\omega} + \tilde{R}(z) = -\frac{1}{a\Upsilon} \left(-\frac{L}{a\Upsilon} + iz\right)^{-1} P_{\omega} + R_{1}(z),$$

where $R_1(z)$ satisfies the following property. For $m \in \mathbb{N}$, $s \in [0, \frac{d}{2}]$, $\delta > s$ and $\beta_x \in \mathbb{N}^d$, $\beta_y \in \mathbb{N}^n$ such that $|\beta_x| + |\beta_y| \le 1$ there exists $C > 0$ such that for $z \in \mathbb{D} \cap \mathbb{C}_+$ we have

$$\| \langle x \rangle^{-\delta} \partial_x^{\beta_x} \partial_y^{\beta_y} R_1^{(m)}(z) \langle x \rangle^{-\delta} \|_{L^2(\Omega)} \le C |z|^{\min(0, -m+s + \frac{|\beta_x|}{2})}.$$

We can also write the Taylor expansions of $\lambda_0(az)$ and $P_{az}$ up to any order to get a more precise expansion of $R(z)$. Then from (2.19) we deduce the asymptotics for the resolvent of $W$ near 0.

### 2.3.4 Contribution of high frequencies

Since we are not on the Euclidean space as in the previous chapter or in Section 2.2, we also have to discuss the contribution of high frequencies.

The classical trajectories follow straight lines in $\Omega$, bouncing on the boundary according to the laws of geometrical optics. The rays of light parallel to the boundary never see the damping, so we do not expect uniform exponential decay for the global energy. However, all the bounded rays reach the boundary, so we should have uniform decay of the local energy (notice that the set of undamped classical trajectories is unstable, so we should have a good estimate on the global energy decay with loss of regularity, but I did not go in that direction in [14]).

We already know that the resolvent $(H_{a\tau} - \tau^2)^{-1}$ is well defined for any $\tau \ne 0$. To prove uniform estimates for $|\tau| \gg 1$ we again have to combine the properties of $L$ and $T_{a\tau}$. We begin with the transverse operator. We have the following resolvent estimates.

**Proposition 2.10.** There exist $\tau_0 > 0$, $\gamma > 0$ and $c > 0$ such that for $|\tau| \ge \tau_0$ and $\zeta \in \mathbb{C}$ with $|\mathrm{Re}(\zeta) - \tau^2| \le \gamma\tau^2$ and $\mathrm{Im}(\zeta) \ge -\gamma|\tau|$ the resolvent $(T_{a\tau} - \zeta)^{-1}$ is well defined and we have

$$\|(T_{a\tau} - \zeta)^{-1}\|_{\mathcal{L}(L^2(\omega))} \le \frac{c}{|\tau|}.$$

The proof, inspired by [Mil00], is based on the usual contradiction argument with semi-classical measures on the compact domain $\omega$. Compared to my previous results about high frequencies, the damping is at the boundary (as in [BLR92]).

To prove estimates for $R(z)$, we separate the contributions of high frequencies close to longitudinal directions and high frequencies close to transverse directions.
Let $\chi \in C_0^\infty(\mathbb{R}, [0, 1])$ be supported in $]-\gamma, \gamma[$ and equal to 1 on $]-\frac{\gamma}{2}, \frac{\gamma}{2}[$. We use the functional calculus, available for $L$, and we estimate separately $\chi(L/\tau^2)R(\tau)$ and $(1-\chi)(L/\tau^2)R(\tau)$.

For the first term, the idea is that with this localization with respect to $L$, $R(\tau)$ is in some sense not too far from $(T_{a\tau} - \tau^2)^{-1}$. And, indeed, from Proposition 2.10 we can prove that

$$ \|\chi(L/\tau^2)R(\tau)\|_{L(L^2(\Omega))} \lesssim \frac{1}{\tau}. \qquad (2.24) $$

This is Proposition 7.3 in [14]. We gave a proof based on (2.23), but we can also use the Fourier transform with respect to the $x$ variable to conclude. Notice that we are dealing with the contribution of frequencies which are not too large in the direction $x$. In terms of classical trajectories, this corresponds to rays of light going in a direction transverse to the wave guide, so they see the damping. It is no surprising that there is no weight in (2.24).

This is of course different for $(1-\chi)(L/\tau^2)R(\tau)$, which corresponds to the contribution of rays of light not far from being parallel to the wave guide.

Assume that we can replace $T_{a\tau}$ by one of its eigenvalues $\lambda_k(a\tau)$. Then we have to estimate $(1-\chi)(L/\tau^2)(L+\lambda_k(a\tau)-\tau^2)^{-1}$. If $\tau^2 - \text{Re}(\lambda_k(a\tau)) \le \gamma\tau^2/4$, then we can apply the spectral theorem for $L$. If $\tau^2 - \text{Re}(\lambda_k(a\tau)) \ge \gamma\tau^2/4$ then, since $\text{Im}(\lambda_k(a\tau)) < 0$, we can apply the (high-frequency) limiting absorption principle with the spectral parameter $\tau^2 - \lambda_k(a\tau)$ for the usual Laplacian $L$ on $\mathbb{R}^d$.

The rigorous proof does not work this way, since we cannot just sum the contributions of each $\lambda_k(a\tau)$. Nevertheless, we can adapt the strategy known for the uniform resolvent estimates for $L$ in the setting of the wave guide to prove directly that for $\delta > \frac{1}{2}$ we have

$$ \|\langle x \rangle^{-\delta} (1-\chi)(L/\tau^2)R(\tau) \langle x \rangle^{-\delta}\|_{L(L^2(\Omega))} \lesssim \frac{1}{|\tau|}. \qquad (2.25) $$

Summing (2.24) and (2.25) gives the following result.

**Proposition 2.11.** Let $\delta > \frac{1}{2}$. There exist $\tau_0 > 0$ and $c > 0$ such that for $|\tau| \ge \tau_0$ we have

$$ \|\langle x \rangle^{-\delta} R(\tau) \langle x \rangle^{-\delta}\|_{L(L^2(\Omega))} \le \frac{c}{|\tau|}. $$

With (2.19) we can get a uniform estimate for $(W + i\tau)^{-1}$ in weighted space. Because of the weights, we cannot apply the same strategy as in Section 2.2. We apply the same strategy as in Chapter 1 instead. In particular, we have to prove uniform estimates for the derivatives of $R(\tau)$. As explained above, this is not obvious, and this can only be done in the sense of forms. To apply a result like Lemma 1.16, we need for instance estimates of $R(\tau)$ in incoming and outgoing regions. For this, we have to adapt all the proofs known in the Euclidean setting (we omit this part of the proof, see [14, Sec. 7.3]). We finally get the following result.

**Proposition 2.12.** Let $n \in \mathbb{N}$ and $\delta > n - \frac{1}{2}$. There exists $C > 0$ such that for $|\tau| \ge 1$ we have

$$ \|(W + i\tau)^{-n}\|_{L(\mathcal{E}^\delta, \mathcal{E}^{-\delta})} \le C. $$

### 2.3.5 Local energy decay and asymptotic profile

Now that we have estimated the resolvent, we can deduce the large time behavior for the solution of (2.14). Since the contribution of high frequencies decays fast, the result depends as usual on low frequencies. By Theorem 2.9 we expect that the asymptotic profile should be a solution of

$$ a\Upsilon\partial_t u_0 + Lu_0 = 0, \qquad (2.26) $$

seen as a problem on $\Omega$. By (2.19) and Theorem 2.9, the initial data for $u_0$ should be given by

$$ f_0 = \frac{1}{a\Upsilon} P_\omega(\Theta_a f + g). \qquad (2.27) $$
We can compute

$$P_{\omega} \Theta_a f = a \Upsilon P_{\partial \omega} f, \quad \text{where} \quad P_{\partial \omega} f(x, y) = \frac{1}{|\partial \omega|} \int_{\partial \omega} f(x, \theta) \, d\sigma(\theta).$$

This defines a function $P_{\partial\omega}f \in L^2(\Omega)$ which only depends on $x \in \mathbb{R}^d$. Then we set

$$f_0 = P_{\partial\omega}f + \frac{1}{a\Upsilon}P_{\omega}g. \qquad (2.28)$$

This is analogous to (2.3). Since $f$ is “multiplied” by the damping in (2.27), only its values at the boundary play a role in the expression of $f_0$, while $f_0$ depends on the values of $g$ on the whole $\Omega$ (notice that $g$ is only in $L^2(\Omega)$ and its trace on $\partial\Omega$ does not make sense).

On the other hand, the strength of the damping is given by the product $a\Upsilon$. The coefficient $a$ is the strength of the damping at each point of the boundary, and $\Upsilon$ depends on the shape of the cross-section. If we normalize the section $\omega$ by $|\omega| = 1$, then $\Upsilon$ is the size of $\partial\omega$. It is not surprising that for a fixed value of $a$, the effect of the damping in $\Omega$ is stronger when $\Upsilon$ is large.

Compared to Theorem 2.8 we cannot give estimates without weights (case $s_1 = s_2 = s = 0$) since we also need weights for the contribution of high frequencies. Of course, even for high frequencies we can get weaker decay with weaker weights, but for simplicity we do not detail this possibility and only write the statement which gives the best local energy decay in this case.

We observe in particular that the power of $t$ only depends on the number $d$ of longitudinal dimensions.

**Theorem 2.13.** Let $\delta > \frac{d}{2} + 2$. There exists $C > 0$ such that for $F = (f,g) \in \mathcal{H}^\delta$ and $t \ge 0$ we have

$$\| \langle x \rangle^{-\delta} \nabla (u(t) - u_0(t)) \|_{L^2(\Omega)} + \| \langle x \rangle^{-\delta} \partial_t (u(t) - u_0(t)) \|_{L^2(\Omega)} \le C \langle t \rangle^{-\frac{d}{2}-2} \|F\|_{\mathcal{H}^\delta},$$

where $u(t)$ is the solution of (2.14) and $u_0(t)$ is the solution of (2.26)-(2.28).

## 2.4 On a wave guide with damping at infinity

After [14], we discussed with Mohamed Malloug the wave equation with damping inside the waveguide $\Omega$. Since there are less technical problems than with damping at the boundary, it was an occasion to consider other difficulties. Thus, in [15] we consider a situation where the damping is not constant (and in particular breaks the symmetry of the wave guide) and the geometric damping condition is not satisfied.

**The setting.** Let $\Omega = \mathbb{R}^d \times \omega \subset \mathbb{R}^{d+n}$ be a straight wave guide as in the previous section. We consider on $\Omega$ an absorption index $a(x, y) \ge 0$ which is a long range perturbation of a positive constant $a_1$: for some $\rho \in [0, 1]$ we have, for $\beta \in \mathbb{N}^d$ and $(x, y) \in \Omega$,

$$|\partial_x^\beta (a(x, y) - a_1)| \le C_\beta \langle x \rangle^{-\rho - |\beta|}. \qquad (2.29)$$

Then we consider on $\Omega$ the damped wave equation with Neumann boundary conditions

$$\left\{
\begin{array}{ll}
\partial_t^2 u - \Delta u + a \partial_t u = 0, & \text{on } \mathbb{R}_+ \times \Omega, \\
\partial_\nu u = 0, & \text{on } \mathbb{R}_+ \times \partial\Omega, \\
(u, \partial_t u)|_{t=0} = (f, g). &
\end{array}
\right.
\qquad (2.30)$$

Notice that we can also consider Dirichlet boundary conditions. This is actually much easier since in this case $0$ is not in the spectrum of the corresponding operator, so there is no difficulty with low frequencies (see Theorem 4.2 in [15]).
In this problem, a is a perturbation of a constant damping, but it can vanish in a bounded subset of $\Omega$. In particular, we may have undamped bounded classical trajectories. This is only possible for rays of light going in a transverse direction, so the set of undamped rays is unstable. For the contribution of high frequencies, this problem is actually similar to the problem on the stadium with damping on the half-disks (see [BH07]).

In this setting the resolvent of the wave equation is

$$R(z) = (-\Delta_N - ia(x, y)z - z^2)^{-1},$$

where $-\Delta_N$ the Neumann Laplacian in $\Omega$.

**Resolvent for a damping independent of the transverse variable.** Even if $a$ depends on $y$, we use in the proof the case where it does not. Let $\alpha$ be a bounded function on $\Omega$ which only depends on $x \in \mathbb{R}^d$. We can write

$$-\Delta_N - i\alpha(x) = (L - i\alpha(x)) + T_N,$$

where $L$ is as above the standard Laplacian on $\mathbb{R}^d$ and the transverse operator $T_N$ is the usual non-negative and selfadjoint Neumann Laplacian on $L^2(\Omega)$. Compared to the setting of the previous paragraph, we consider here the damping as part of the longitudinal operator.

We denote by $(\lambda_k)_{k \in \mathbb{N}}$ the non-decreasing sequence of eigenvalues (repeated according to multiplicities) of $T_N$, and we consider a corresponding orthonormal basis of eigenvectors $(\varphi_k)_{k \in \mathbb{N}}$. Compared to the previous section, the fact that we have an orthonormal basis for the transverse operator simplifies the analysis. We have $\lambda_0 = 0$, $\lambda_1 > 0$ and $\varphi_0$ is constant.

For $f \in L^2(\Omega)$ we can write $f = \sum_{k \in \mathbb{N}} f_k \otimes \varphi_k$ with $f_k \in L^2(\mathbb{R}^d)$, and for $\zeta$ in the resolvent set of $-\Delta_N - i\alpha(x)$ we have

$$(-\Delta_N - i\alpha(x) - \zeta)^{-1} f = \sum_{k \in \mathbb{N}} (L - i\alpha(x) - (\zeta - \lambda_k))^{-1} f_k \otimes \varphi_k. \quad (2.31)$$

Thus we can deduce estimates for $(-\Delta_N - i\alpha(x) - \zeta)^{-1}$ in $\mathcal{L}(L^2(\Omega))$ from similar estimates for $(L - i\alpha(x) - (\zeta - \lambda_k))^{-1}$, $k \in \mathbb{N}$, in $\mathcal{L}(L^2(\mathbb{R}^d))$.

**Contribution of low frequencies.** Now we go back to the analysis of $R(z)$. We begin with the contribution of low frequencies. If $a$ only depends on $x$, then we see that in (2.31) (with $\alpha = az$ and $\zeta = z^2$) only the first term should play a role for $z$ small (the other terms are holomorphic around $z = 0$).

We have not assumed that $a(x, y)$ only depends on $x$, but for low frequencies only the values at infinity play an important role. Thus it is expected that, up to a rest, we can replace $a(x, y)$ by $a_1$ (and remove the term $z^2$) in $R(z)$ when $z$ is small. In other words, for $z$ small the resolvent $R(z)$ should be close to the resolvent of the heat equation

$$R_h(z) = (-\Delta_N - iza_1)^{-1}.$$

On the other hand, by (2.31), $R_h(z)$ should be close to $R_h(z)P_\omega$, where $P_\omega$ (defined by (2.22)) is the orthogonal projection on $\text{span}(\varphi_0)$. All this holds in the following precise sense.

**Proposition 2.14.** Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $\kappa > 1$. Let $\rho_1 \in ]0, \rho[$. Let $\beta \in \mathbb{N}^{d+n}$ with $|\beta| \le 1$. Let $m \in \mathbb{N}^*$. There exists $C > 0$ such that for $z \in \mathbb{D} \cap \mathbb{C}_+$ we have

$$\left\| \langle x \rangle^{-\kappa s_1} \partial_{x,y}^{\beta} \partial_z^m (R(z) - R_h(z)P_\omega) \langle x \rangle^{-\kappa s_2} \right\|_{\mathcal{L}(L^2(\Omega))} \le C |z|^{\min(0, \frac{1}{2}(s_1+s_2+\rho_1+|\beta|)-m-1)}.$$

*Ideas of proof.* To estimate the difference between $R(z)$ and $R_h(z)$ we write the resolvent identity

$$R(z) - R_h(z) = R_h(z)(iz(a - a_1) + z^2)R(z).$$
Then we proceed as in the previous chapter (see Section 1.7) or for Proposition 2.7. Here the rescaled Sobolev spaces (recall (1.56)) are only in the $x$ direction. More precisely, for $s \in \mathbb{R}$ and $z \in \mathbb{C}^*$ we set

$$ \|u\|_{H_z^s L^2(\Omega)} = \|\langle \sqrt{L/|z|} \rangle^s u\|_{L^2(\Omega)}. $$

For any $s \in \mathbb{R}$ and $z \in \mathbb{C}_+$, the resolvent $R_h(z)$ is of size $O(|z|^{-1})$ in $\mathcal{L}(H_z^{s-1}L^2(\Omega), H_z^{s+1}L^2(\Omega))$. On the other hand, by an analog of Proposition 1.19, we see that $(iz(a - a_1) + z^2)$ is of size $O(|z|^{1+\frac{\rho_1}{2}})$ in $\mathcal{L}(H_z^{s_2}L^2(\Omega), H_z^{s_1}L^2(\Omega))$ if $s_1, s_2 \in [-\frac{d}{2}, \frac{d}{2}]$ are such that $s_2 - s_1 \ge \rho_1$. In particular $R(z) - R_h(z)$ is smaller than $R_h(z)$ for $z$ small, and $R(z)$ is of size $O(|z|^{-1})$ in $\mathcal{L}(H_z^{s-1}L^2(\Omega), H_z^{s+1}L^2(\Omega))$ for any $s \in [-\frac{d}{2}, \frac{d}{2}]$. Notice also that $\nabla_x$ is of size $O(|z|^{\frac{1}{2}})$ in $\mathcal{L}(H_z^{s_2}L^2(\Omega), H_z^{s_1}L^2(\Omega))$ and we can use (2.31) to see that $\nabla_y R_h(z)$ is bounded in a neighborhood of $z = 0$. Finally, we use the weights $\langle x \rangle^{-\kappa s_j}$ to convert regularity into powers of $|z|$ as in (2.11). $\square$

**Contribution of high frequencies.** We turn to the contribution of high frequencies. Since we have damping everywhere at infinity, we can consider the global energy, and hence resolvent estimates without weights. On the other hand, since the geometric damping condition (the analog of (1.17) in $\Omega$) does not necessarily hold, we cannot expect an estimate as good as in the previous settings.

**Proposition 2.15.** Let $\beta_1, \beta_2 \in \mathbb{N}^{d+n}$ with $|\beta_1| \le 1$ and $|\beta_2| \le 1$. Let $\tau_0 > 0$. There exists $C \ge 0$ such that for $\tau \in \mathbb{R}\setminus[-\tau_0, \tau_0]$ we have

$$ \|\partial^{\beta_1} R(\tau) \partial^{\beta_2}\|_{\mathcal{L}(L^2(\Omega))} \le C |\tau|^{|\beta_1|+|\beta_2|+1} $$

and

$$ \|(W + i\tau)^{-1}\|_{\mathcal{L}(\mathcal{F})} \le C\tau^2. \qquad (2.32) $$

*Ideas of proof.* As is usual for high frequencies, we can replace $a(x, y)$ by a smaller damping. We choose $\tilde{a}(x)$ which only depends on $x$ and such that $\tilde{a}(x) \le a(x, y)$ for all $(x, y) \in \Omega$. Then we can use (2.31). Since we can take $\tilde{a}$ going to 1 at infinity, it satisfies the damping assumption on $\mathbb{R}^d$, so $(L - i\tau\tilde{a} - \tau^2)^{-1}$ if of size $O(\tau^{-1})$ in $\mathcal{L}(L^2(\mathbb{R}^d))$. We get a similar estimate for $(L - i\tau\tilde{a} - (\tau^2 - \lambda_k))^{-1}$ if $\tau^2 - \lambda_k$ is large. The loss is due to the contributions of $k$ such that $\tau^2 - \lambda_k$ is small. This corresponds to rays of light orthogonal to the boundary, possibly undamped, and explains why we do not have a uniform bound in (2.32). $\square$

**Local energy decay.** Finally, we have to convert resolvent estimates to large times estimates. For high frequencies we adapt the method of [Leb96, BH07]. Notice that there are abstract results in the case where (2.32) holds for all $\tau \in \mathbb{R}$ (with $\tau^2$ replaced by $\langle \tau \rangle^2$, see for instance [BEP506, BD08, BT10]), but we cannot apply them because of the problem at 0. For the contribution of low frequencies, the method is similar to the previous cases. From Proposition 2.14 we can deduce that the asymptotic profile $u_0(t)$ for the solution of (2.30) is the function on $\Omega$ which does not depend on $y \in \omega$ and is solution with respect to $x$ to the heat equation $a_1\partial_t u_0(t) + Lu_0(t) = 0$, with initial condition $f_0 = P_\omega(f + g/a_1)$:

$$ u_0(t) = e^{-\frac{iL}{a_1}} P_\omega \left( f + \frac{g}{a_1} \right). \qquad (2.33) $$

Finally, we get the following result.

**Theorem 2.16.** Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $\kappa > 1$. Let $\rho_1 \in [0, \rho]$. Let $\alpha \in \mathbb{N}$ and $\beta \in \mathbb{N}^{d+n}$ with $\alpha + |\beta| \le 1$. Let $k \in \mathbb{N}^*$. There exists $C > 0$ such that for $F = (f, g) \in \mathcal{H}^{\kappa s_2}(\Omega) \cap \text{Dom}(W^k)$
we have

$$ \begin{aligned} & \left\| \langle x \rangle^{-\kappa s_1} \partial_t^\alpha \partial_{x,y}^\beta (u(t) - u_0(t)) \right\|_{L^2(\Omega)} \\ & \le C \left( \langle t \rangle^{-\frac{1}{2}(s_1+s_2+2\alpha+|\beta|+\rho_1)} \|F\|_{\mathcal{H}^{\kappa s_2}} + \frac{\ln(t)^{\frac{k}{2}+1}}{t^{\frac{k}{2}}} \|(W-1)^k F\|_{\mathcal{E}} \right), \end{aligned} $$

where $u(t)$ is the solution of (2.30) and $u_0(t)$ is given by (2.33).

## 2.5 In a periodic medium

The motivation for [14] was the analysis of the damped wave equation on a wave guide. It is only while proving this result that I realized that the damped wave equation with damping at infinity was a really exciting problem in itself. When Romain Joly wrote a paper about the damped Klein-Gordon equation with periodic damping (then improved to a more general setting with Nicolas Burq in [BJ16]), giving the behavior of the contribution of high frequencies for the corresponding damped wave equation, it was a natural question to study the contribution of low frequencies for the damped wave equation with periodic damping. This is what we did together in [17].

### 2.5.1 The asymptotically periodic setting

We consider the damped wave equation (1.12) on the Euclidean space, with a general Laplace operator $P$ as in (1.8), but the coefficients (including the absorption index) are now asymptotically periodic. This means that we can write

$$ G(x) = G_{\mathbf{p}}(x) + G_0(x), \quad w(x) = w_{\mathbf{p}}(x) + w_0(x) \quad \text{and} \quad a(x) = a_{\mathbf{p}}(x) + a_0(x), $$

where $G_{\mathbf{p}}$, $w_{\mathbf{p}}$ and $a_{\mathbf{p}}$ are $\mathbb{Z}^d$-periodic and $G_0$, $w_0$ and $a_0$ are of long range: there exist $\rho_G, \rho_a > 0$ (we use two parameters since $\rho_G$ and $\rho_a$ will not play the same role in (2.46) below) and $C_G, C_a \ge 0$ such that

$$ |G_0(x)| \le C_G \langle x \rangle^{-\rho_G} \quad \text{and} \quad |w_0(x)| + |a_0(x)| \le C_a \langle x \rangle^{-\rho_a}. \qquad (2.34) $$

We first consider the purely periodic case, $G_0 = 0$, $w_0 = 0$ and $a_0 = 0$. We denote by $W_{\mathbf{p}}$ the wave operator (1.19) with coefficients $G_{\mathbf{p}}$, $w_{\mathbf{p}}$ and $a_{\mathbf{p}}$.

The damped wave equation with periodic metric had already been studied in [OZP01]. Compared to our setting, the initial condition is localized (at least in $L^1$) and the absorption index is constant (in particular it does not vanish and we can use some tools of the selfadjoint problem).

Our absorption index $a_{\mathbf{p}}$ cannot be identically zero, but it is allowed to vanish. This means that at infinity it takes positive values but it can also be 0. This is an intermediate situation between the setting of Chapter 1, where $a$ is small at infinity, and the settings of Sections 2.2 and 2.4, where $a$ is asymptotically close to a positive constant.

We will see that we recover in this case the diffusive phenomenon typical for the damped wave equation with damping effective at infinity.

Under the geometric damping condition for high frequencies (see (2.45) below), the large time behavior of the wave is as usual given by the contribution of low frequencies. And if the wave length of the solution is very large compared to the period of the medium, this solution only sees, in some sense, the mean value of $a_{\mathbf{p}}$. This mean value is positive.

Thus, as above the solution of the periodic damped wave equation should look like the solution of some heat equation. It is natural to expect that the corresponding heat equation is obtained by removing the second derivative in time in the wave equation. After multiplication by $w$ this gives

$$ -\Delta_{G_{\mathbf{p}}} u + a_{\mathbf{p}}(x) w_{\mathbf{p}}(x) \partial_t u = 0 \quad \text{on } \mathbb{R}_+ \times \mathbb{R}^d. \qquad (2.35) $$
This is indeed the case, but we want to go further. As said above, the low frequency wave only sees the mean value of $a_p w_p$. We expect that in (2.35) we can replace $a_p w_p$ by its mean value. Similarly, we would like to replace $G_p$ by a simpler (constant) matrix. The reader familiar with homogenization (which was not our case when starting this work) knows that the mean value of $G_p$ is not the good guess.

## 2.5.2 The Floquet-Bloch decomposition and spectral properties on the torus

All the coefficients of the equation are $\mathbb{Z}^d$-periodic, but this is not the case for the initial condition $(f, g) \in H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. The solution itself will not be periodic, so the problem does not reduce to a problem on the torus.

In this kind of context it is usual to use the Bloch waves (see for instance [BLP78, Sec. 4.3]). Compared to the plane waves $e^{ix \cdot \xi}$ used for the Fourier transform (suitable for linear problems with constant coefficients), the Bloch waves are of the form $\psi(x) = e^{ix \cdot \sigma} \phi(x)$ where $\phi$ is $\mathbb{Z}^d$-periodic. They can be chosen in such a way that $\psi$ is an eigenfunction of $-\Delta_{G_p}$. Then $\phi$ is an eigenfunction for the operator

$$-\Delta_{G_p, \sigma} := -(\mathrm{div} + i\sigma^T)G_p(x)(\nabla + i\sigma).$$

Compared to the undamped case (or with constant damping), we cannot use a basis of Bloch waves which are eigenfunctions of our operator, but we can still use the Floquet-Bloch decomposition. Let $\mathbb{T}^d = [-\frac{1}{2}, \frac{1}{2}]^d$. For $u \in S(\mathbb{R}^d)$ and $x \in \mathbb{R}^d$ we write

$$u(x) = \int_{\sigma \in 2\pi\mathbb{T}^d} e^{ix \cdot \sigma} u_{\sigma}(x) d\sigma, \quad \text{where} \quad u_{\sigma}(x) = \frac{1}{(2\pi)^d} \sum_{n \in \mathbb{Z}^d} u(x+n)e^{-i(x+n) \cdot \sigma}. \quad (2.36)$$

Notice that $u_\sigma$ is periodic for all $\sigma \in \mathbb{R}^d$. This decomposition naturally shares some useful properties with the Fourier transform. In particular, we have the Parseval identity for the Floquet-Bloch decomposition: for $u, v \in S(\mathbb{R}^d)$ we have

$$\langle u, v \rangle_{L^2(\mathbb{R}^d)} = (2\pi)^d \int_{\sigma \in 2\pi\mathbb{T}^d} \langle u_\sigma, v_\sigma \rangle_{L_p^2} d\sigma, \quad (2.37)$$

where the space $L_p^2$ of $\mathbb{Z}^d$-periodic and $L_{\mathrm{loc}}^2$ functions is endowed with the natural Hilbert structure. We also define the periodic Sobolev spaces $H_p^k$ for $k \in \mathbb{N}$.

Now for $U = (u, v) \in \mathcal{S}$ and $x \in \mathbb{R}^d$ we have

$$(W_p U)(x) = \int_{\sigma \in 2\pi\mathbb{T}^d} e^{ix \cdot \sigma} W_\sigma U_\sigma(x) d\sigma,$$

where for $\sigma \in \mathbb{R}^d$ we have set $U_\sigma = (u_\sigma, v_\sigma)$ and

$$W_\sigma = \begin{pmatrix} 0 & w_p^{-1} \\ \Delta_{G_p, \sigma} & -a_p \end{pmatrix}.$$

This defines a family $(W_\sigma)_{\sigma \in 2\pi\mathbb{T}^d}$ of operators on $H_p^1 \times L_p^2$ with domain $H_p^2 \times H_p^1$. The interest of this decomposition is that each operator $W_\sigma$ has a compact resolvent, so its spectrum is given by a sequence of isolated eigenvalues. For $U = (u, v) \in H_p^2 \times H_p^1$ and $\lambda \in \mathbb{C}$ we have

$$W_\sigma U = \lambda U \iff \begin{cases} (-\Delta_{G_p, \sigma} + \lambda a_p w_p + \lambda^2 w_p)u = 0, \\ v = \lambda w_p u. \end{cases} \quad (2.38)$$

Then we proceed as explained on the model case in Section 2.2. We first observe that 0 is the only possible purely imaginary eigenvalue of $W_\sigma$, and this happens if and only if $\sigma = 0$. Moreover, 0 is a (geometrically and algebraically) simple eigenvalue of $W_0$ and a
corresponding eigenvector is given by $\Phi_0 = (1, 0)^T$ (notice that we would have difficulties to work with $\Phi_0$ in the periodic energy space $\dot{H}_p^1 \times L_p^2$).

By continuity of the spectrum with respect to $\sigma$, we deduce that only one simple eigenvalue of $W_\sigma$ for $\sigma$ small will play a role for the contribution of low frequencies.

**Proposition 2.17.** There exist $r > 0$ and $\gamma_0, \gamma_1, \gamma_2 > 0$ such that

(i) for $\sigma \in 2\pi\mathbb{T}^d \setminus B(0, r)$ the eigenvalues of $W_\sigma$ with imaginary parts in $[-3, 3]$ have real parts smaller than $-\gamma_0$;

(ii) for $\sigma \in B(0, r)$ the operator $W_\sigma$ has a unique simple eigenvalue $\lambda_\sigma$ such that $|\lambda_\sigma| \le \gamma_1$, and the other eigenvalues of $W_\sigma$ with imaginary parts in $[-3, 3]$ have real parts smaller than $-\gamma_2$ ($\gamma_1 < \gamma_2$).

We denote by $\Pi_\sigma$ the Riesz projection of $W_\sigma$ corresponding to $\lambda_\sigma$ and $\Phi_\sigma = \Pi_\sigma \Phi_0 / \|\Pi_\sigma \Phi_\sigma\|$. Then for some $\varphi_\sigma \in H_p^2$ (smooth with respect to $\sigma$) we have $\Phi_\sigma = (\varphi_\sigma, \lambda_\sigma w_p \varphi_\sigma)^T$.

Then, up to exponentially decaying terms, we see that the important contribution is the analog $\mathcal{U}_{r,p}(t)F$ of (2.12), defined for $t \ge 0$ and $F \in \mathcal{S}$ by

$$ \mathcal{U}_{r,p}(t)F = \int_{\sigma \in B(0,r)} e^{ix \cdot \sigma} e^{t \lambda_{\sigma}} \Pi_{\sigma} F_{\sigma} \, d\sigma. \qquad (2.39) $$

The next step is to replace $\lambda_\sigma$, $\Phi_\sigma$ and $\Pi_\sigma$ by their Taylor expansions for $\sigma$ small. From (2.38) we can prove the following properties.

**Proposition 2.18.** (i) Let $i\psi_\sigma$ be the linear term in the Taylor expansion of $\varphi_\sigma$ at $\sigma = 0$. Then we have

$$ -\operatorname{div} G_p(x)(\sigma + \nabla \psi_\sigma) = 0. \qquad (2.40) $$

We denote by $W(x)$ the $\mathbb{Z}^d$-periodic matrix such that, for all $\xi \in \mathbb{R}^d$,

$$ W(x)\xi = \xi + \nabla\psi_{\xi}(x). \qquad (2.41) $$

(ii) We have

$$ \lambda_{\sigma} = -\frac{1}{a_{h}w_{h}} \langle G_{h}\sigma, \sigma \rangle + O(|\sigma|^{3}), $$

where $G_h$ is the mean value of $W(x)^T G_p(x) W(x)$:

$$ \langle G_h \sigma, \sigma \rangle = \int_{\mathbb{T}^d} \langle G_p(x)(\sigma + \nabla \psi_\sigma(x)), \sigma + \nabla \psi_\sigma(x) \rangle \, dx. \qquad (2.42) $$

(iii) For $\sigma \in B(0, r)$ we denote by $\Psi_\sigma \in L_p^2 \times L_p^2$ the vector which satisfies, for all $F$,

$$ \Pi_{\sigma} F = \langle \Psi_{\sigma}, F \rangle_{L_{p}^{2} \times L_{p}^{2}} \Phi_{\sigma}. $$

Then $\Psi_\sigma$ is a smooth function of $\sigma$ and we have

$$ \Psi_0 = \frac{1}{a_h w_h} \begin{pmatrix} a_p w_p \\ 1 \end{pmatrix}. $$

The definition of $G_h$ is classical in homogenization theory (the matrix $W(x)$ is the so-called corrector matrix). It is interesting to notice how it appears as the Hessian matrix of the eigenvalue $\lambda_\sigma$ at $\sigma = 0$.

### 2.5.3 Comparison with the asymptotic profile

From the eigenelements of $W_\sigma$ for $\sigma$ small we can define the asymptotic profile for the contribution of low frequencies. Since the contribution of high frequencies decays fast under the suitable geometric condition, we finally get a result of local energy decay for the difference between the solution of the damped wave equation and the asymptotic profile.
**The asymptotic profile** We compare (2.39) with

$$ \mathcal{U}_{r,h}(t)F = \int_{\sigma \in B(0,r)} e^{ix \cdot \sigma} e^{-\frac{t \langle G_h \sigma, \sigma \rangle}{a_h w_h}} \langle F_\sigma, \Psi_0 \rangle \Phi_0 \, d\sigma. \quad (2.43) $$

For this we use the strategy of Proposition 2.2, actually first developed for this point in [17] (see Proposition 4.12 therein). We replace the continuity of the Fourier transform (remember (2.5)-(2.6) with notation (2.4)) by the following property of the Floquet-Bloch transform (see also [17, Cor. 4.2])

$$ \|u\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \lesssim \|u_\sigma\|_{L^{p'_1}(2\pi\mathbb{T}^d; L_p^2)}, \quad \|u_\sigma\|_{L^{p'_2}(2\pi\mathbb{T}^d; L_p^2)} \lesssim \|u\|_{L^{2,\kappa s_2}(\mathbb{R}^d)}. $$

This can be proved with (2.37) and the following equality for $u \in S(\mathbb{R}^d)$ and $\psi \in L_p^2$:

$$ \langle u_\sigma, \psi \rangle_{L_p^2} = \int_{x \in \mathbb{R}^d} e^{ix \cdot \sigma} u(x) \overline{\psi(x)} \, dx. $$

The first component of (2.43) gives the asymptotic profile (the second component gives its time derivative, multiplied by $w_p$). For $t \ge 0$ we set

$$ u_h(t) = e^{-\frac{t}{a_h w_h} P_h} u_h(0), \quad \text{where} \quad u_h(0) = \langle F, \Psi_0 \rangle = \frac{a_p w_p f + w_p g}{a_h w_h}. \quad (2.44) $$

To get the asymptotic profile for $\nabla u(t)$, we cannot just take the gradient of $u_h(t)$. Indeed, the leading term for $\varphi_\sigma$ is just $\varphi_0 = 1$, but the leading term of $\nabla \varphi_\sigma$ is $i \nabla \psi_\sigma$ (see Proposition 2.18). With the factor $i\sigma$ coming from the factor $e^{ix \cdot \sigma}$ we see that $\nabla u_p(t)$ should be compared with

$$ \int_{\sigma \in B(0,r)} e^{ix \cdot \sigma} e^{-\frac{t \langle G_h \sigma, \sigma \rangle}{a_h w_h}} \langle F_\sigma, \Psi_0 \rangle i(\sigma + \psi_\sigma) \, d\sigma. $$

This is why we compare $\nabla u_p(t)$ with $W(x)\nabla u_p(t)$ in Theorem 2.19 below. Notice also that we cannot apply the trick of Proposition 2.2 to get an extra power of $t^{-\frac{1}{2}}$ with stronger weight (case $s = 1$). Then, with a strong weight, the estimate of $\nabla u_p(t) - W\nabla u_h(t)$ in Theorem 2.19 is the same as the estimate of $\nabla u_h(t)$ in Proposition 2.2. Thus, it is not clear that $W\nabla u_h(t)$ is the asymptotic profile for $\nabla u_p(t)$, but this still gives at least the optimal decay for $\nabla u_p(t)$.

**Contribution of high frequencies.** For the contribution of high frequencies, we can deduce Proposition 2.5 from the high frequency resolvent estimates given in [BJ16]. Notice that we cannot use the assumption (1.17) in this context. In (1.17), it is only assumed that each bounded classical trajectory goes once through the damping region. However, since bounded trajectories live in a compact subset in the context of Chapter 1, the time needed to reach the damping region in uniform, and each ray of light actually goes regularly through the damping region. This is no longer automatic if we consider classical trajectories at infinity, so we have to be more precise about the damping condition. With the same notation as for (1.17), we assume

$$ \exists T > 0, \exists \alpha > 0, \quad \forall (x_0, \xi_0) \in p^{-1}(\{1\}), \quad \int_0^T a(x(t); x_0, \xi_0) \, dt \ge \alpha. \quad (2.45) $$

It is clear that (2.45) (for all trajectories, of the bounded ones) implies (1.17), and on the other hand we can check that if (1.17) holds in the context of Chapter 1 then bounded trajectories satisfy (2.45).
**Local energy decay.** Now that the contributions of low and high frequencies are understood, we can state the main result of [17].

**Theorem 2.19.** Assume that the geometric damping condition (2.45) holds. Let $s_1, s_2 \in [0, \frac{d}{2}]$ and $\kappa > 1$. There exists $C > 0$ such that for $t \ge 0$ and $F = (f,g) \in \mathcal{H}^{\kappa s_2}$ we have

$$ \|u_p(t) - u_h(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{1}{2}-\frac{s_1+s_2}{2}} \|F\|_{\mathcal{H}^{\kappa s_2}}, $$

$$ \|\nabla u_p(t) - W\nabla u_h(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{s_1+s_2}{2}} \|F\|_{\mathcal{H}^{\kappa s_2}}, $$

$$ \|\partial_t u_p(t) - \partial_t u_h(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{3}{2}-\frac{s_1+s_2}{2}} \|F\|_{\mathcal{H}^{\kappa s_2}}, $$

where $u_p(t)$ is the solution of (1.12) with coefficients $(G_p, w_p, a_p)$ and $u_h(t)$ is given by (2.44).

As for Theorem 2.8, we could expand $\lambda_\sigma$ and $\Pi_\sigma$ further in (2.39) to get a more precise asymptotics for $u_p(t)$ and its derivatives.

By a perturbative argument, we get the following result for the general case with asymptotically periodic coefficients.

**Proposition 2.20.** Assume that the damping condition (2.45) holds. Let $\kappa > 1$ and $s_1, s_2, \eta \ge 0$ be such that

$$ \max(s_1, s_2) + \eta < \min\left(\frac{d}{2}, \rho_G, \rho_a + 1\right). \qquad (2.46) $$

Then there exists $C \ge 0$ such that for $U_0 = (u_0, u_1) \in \mathcal{H}^{\kappa s_2}$ and $t \ge 0$ we have

$$ \|u(t) - u_p(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{s_1+s_2}{2}-\frac{\eta}{2}} \|U_0\|_{\mathcal{H}^{\kappa s_2}}, $$

$$ \|\nabla u(t) - \nabla u_p(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-\frac{1}{2}-\frac{s_1+s_2}{2}-\frac{\eta}{2}} \|U_0\|_{\mathcal{H}^{\kappa s_2}}, $$

$$ \|\partial_t u(t) - \partial_t u_p(t)\|_{L^{2,-\kappa s_1}(\mathbb{R}^d)} \le C \langle t \rangle^{-1-\frac{s_1+s_2}{2}-\frac{\eta}{2}} \|U_0\|_{\mathcal{H}^{\kappa s_2}}, $$

where $u(t)$ and $u_p(t)$ are the solutions of (1.12) with coefficients $(G, w, a)$ and $(G_p, w_p, a_p)$, respectively.

Because of the restriction (2.46), this proposition is probably far from being optimal. The purpose was to show that even if the proof of Theorem 2.19 strongly depends on the symmetry of the setting, the result is actually robust with respect to perturbations.

Nonetheless, Proposition 2.20 gives in any case a result for the global energy and for a general initial data ($s_1 = s_2 = 0$). It also provide interesting estimates for a rapidly decaying perturbation ($\rho_G$ and $\rho_a$ not too small).

## 2.6 The wave equation with highly oscillating damping

The discussions above mostly concern the contribution of low frequencies for the wave equation. However, while preparing [17], a technical detail about the high frequencies estimates in [BJ16] drew my attention.

The analysis of high frequencies is based on semiclassical analysis, which requires quite a lot of regularity for all the coefficients. However, contrary to the case of low frequencies for which the overdamping phenomenon occurs, for high frequencies we have better decay properties with a stronger damping, so if the absorption index $a$ is not smooth, it is enough to prove the estimates with $a$ replaced by a smooth $\tilde{a}$ such that $\tilde{a} \le a$.

However, some regularity is still required on $a$. In [BJ16] it is assumed that $a$ is at least uniformly continuous. Similarly, in [2] we had to apply the Mourre theory with the original absorption index $a$, so we assumed existence and boundedness of its first two radial derivatives.
In [18], I have tried to show on a very simple model case that a highly oscillating absorption index should not be a problem, and that the only important quantity is the average damping seen by the classical rays of light, as measured by the integral in (2.45). Note that a more refined analysis of the damped wave equation with rough damping on the torus has then been published in [BG20].

We consider the Klein-Gordon equation with a periodic damping whose period goes to 0. Let $a \in C^\infty(\mathbb{R}^d, \mathbb{R}_+)$ be periodic. For $\eta \ge 1$ and $x \in \mathbb{R}^d$ we set $a_\eta(x) = a(\eta x)$. For some $m > 0$ and $(f,g) \in \mathcal{H}$ we consider the problem

$$ \begin{cases} \partial_t^2 u - \Delta u + mu + a_\eta(x)\partial_t u = 0, & \text{on } \mathbb{R}_+ \times \mathbb{R}^d, \\ (u, \partial_t u)|_{t=0} = (f, g), & \text{on } \mathbb{R}^d. \end{cases} \qquad (2.47) $$

The interest of considering the Klein-Gordon equation is that it behaves exactly as the wave equation for high frequencies (the additional term $mu$ is much smaller than the others) but there is no problem with low frequencies, since 0 is not in the spectrum of the operator

$$ \begin{pmatrix} 0 & 1 \\ \Delta - m & -a_\eta \end{pmatrix}, $$

defined on $\mathcal{H}$ with natural domain. Since we consider the free Laplacian in (2.47), we have $p(x, \xi) = |\xi|^2$ and classical trajectories are given by $\phi^t(x_0, \xi_0) = (x_0 + 2t\xi_0, \xi_0)$.

We have said that under the geometric damping condition on all classical trajectories we have uniform exponential decay for the Klein-Gordon equation (see [BJ16]). The first result in [18] is that if (2.45) is uniform in $\eta \ge 1$, then the solution $u_\eta$ of (2.47) decays exponentially, uniformly in the initial condition and uniformly in $\eta \ge 1$. Even if the sizes of the derivatives of the absorption index $a_\eta$ blow up with $\eta$. Thus we assume

$$ \exists T > 0, \exists \alpha > 0, \forall \eta \ge 1, \forall (x_0, \xi_0) \in \mathbb{R}^d \times \mathbb{S}^{d-1}, \quad \int_0^T a_\eta(x + 2t\xi) \, dt \ge \alpha. \qquad (2.48) $$

In our particular case, we observe that if the damping condition (2.45) holds for $\eta = 1$, then it holds uniformly in $\eta \ge 1$.

**Theorem 2.21.** Assume that the uniform geometric damping condition (2.48) holds. Then there exist $\gamma > 0$ and $C > 0$ such that for $\eta \ge 1$, $(f,g) \in \mathcal{H}$ and $t \ge 0$ we have

$$ \|u(t)\|_{H^1(\mathbb{R}^d)} + \|\partial_t u(t)\|_{L^2(\mathbb{R}^d)} \le C e^{-\gamma t} (\|f\|_{H^1(\mathbb{R}^d)} + \|g\|_{L^2(\mathbb{R}^d)}), $$

where $u(t)$ is the solution of (2.47).

The second part of the paper is about the case where (2.48) is not satisfied. We know that we cannot have a uniform decay, but we can still have energy decay if we allow a loss of regularity.

Since the damping is regularly distributed for any $\eta \ge 1$, we can apply [BJ16, Th. 1.3] to see that we have at least logarithmic decay. The particular case of a periodic damping has been analysed in [Wun17], and in this case we actually have polynomial decay, with the loss of one derivative. This is consistent with the fact that in the periodic setting the set of rays of light which do not see the damping is unstable. We look at the dependence in $\eta$ for this polynomial decay.

The second result of [18] is again that despite of the sizes of the derivatives of $a_\eta$, we recover the same result as in [Wun17], uniformly in $\eta \ge 1$. Moreover, the estimate is even better for large $\eta$. This is in fact natural, since for large $\eta$ the classical rays of light see in some sense the same quantity of damping, but this damping is more uniformly distributed in $\mathbb{R}^d$.
**Theorem 2.22.** There exists $c > 0$ such that for $\eta \ge 1$, $(f,g) \in \mathcal{H}$ and $t \ge 0$ we have

$$ \|u(t)\|_{H^1(\mathbb{R}^d)} + \|\partial_t u(t)\|_{L^2(\mathbb{R}^d)} \\ \le \frac{c}{\sqrt{1+t}} \left( \|f\|_{H^1(\mathbb{R}^d)} + \|g\|_{L^2(\mathbb{R}^d)} + \frac{\|\Delta f\|_{L^2(\mathbb{R}^d)} + \|\nabla g\|_{L^2(\mathbb{R}^d)}}{\eta^2} \right), $$

where $u(t)$ is the solution of (2.47).

## 2.7 The damped Schrödinger equation on a wave guide

In this paragraph we present the results of [8], which was actually written before all the papers discussed in this chapter.

We consider the Schrödinger equation on a straight wave guide with one dimensional cross section. Given $d \ge 2$ and $\ell > 0$, we set $\Omega = \mathbb{R}^{d-1} \times ]0, \ell[$. Then for $a > 0$ and $f \in L^2(\Omega)$ we consider on $\Omega$ the Schrödinger equation

$$ \left\{ \begin{array}{ll} -i\partial_t u - \Delta u = 0, & \text{on } \mathbb{R}_+ \times \Omega, \\ \partial_\nu u = iau, & \text{on } \mathbb{R}_+ \times \partial\Omega, \\ u|_{t=0} = f. & \end{array} \right. \qquad (2.49) $$

The $L^2$-norm of the solution is a non-increasing function of time. We have in fact uniform exponential decay:

**Theorem 2.23.** There exist $\gamma > 0$ and $C \ge 0$ such that for any $f \in L^2(\Omega)$ the solution $u$ of (2.49) satisfies

$$ \forall t \ge 0, \quad \|u(t)\|_{L^2(\Omega)} \le C e^{-\gamma t} \|f\|_{L^2(\Omega)}. $$

Instead of a constant damping, we can consider $a$ such that $a_0 \le a \le a_1$ on $\partial\Omega$, for some positive constants $a_0$ and $a_1$. The same result also holds if $a$ satisfies this property on one side of the boundary and vanishes on the other side. We can even consider the case where the damping is a positive constant on one side and a negative one on the other side (with a positive sum, so that the damping is positive on average, in the spirit of [4]).

It is remarkable that even if the classical trajectories parallel to the boundary never see the damping, we can have a uniform decay for the global energy of the Schrödinger equation.

The operator corresponding to (2.49) is the Laplacian with Robin boundary condition $H_a$, as defined by (1.39) or (2.18). To prove Theorem 2.23, we show that we have a spectral gap for $H_a$.

**Theorem 2.24.** There exist $\gamma_1 > 0$ and $C_1 \ge 0$ such that any $z \in \mathbb{C}$ with $\mathrm{Im}(z) \ge -\gamma_1$ belongs to the resolvent set of $H_a$ and

$$ \|(H_a - z)^{-1}\|_{L(L^2(\Omega))} \le C_1. $$

As in Section 2.3, we can deduce the spectral properties of the Laplacian on the wave guide $\Omega$ from the properties of the transverse operator $T_a$ (see (2.20)) on $L^2(0, \ell)$. In dimension 1 it reads

$$ T_a = -\partial_y^2, \quad \mathrm{Dom}(T_a) = \{u \in H^2(0, \ell) : u'(0) = -iau(0), u'(\ell) = iau(\ell)\}. $$

Then [8] is mainly about the spectral properties of $T_a$.

Since $T_a$ is a Laplacian in dimension 1, we can get many spectral properties by explicit computations. In particular, for $z \in \mathbb{C}^*$ we can check that $z^2$ is an eigenvalue of $T_a$ if and only if

$$ e^{2iz\ell} = \left( \frac{z-a}{z+a} \right)^2. \qquad (2.50) $$
From this, we deduce some qualitative properties for the sequence of eigenvalues and in particular its asymptotic behavior.

We recall that $T_0$ is the Neumann Laplacian on $]0, \ell[$. The spectrum of $T_0$ is given by the sequence of simple eigenvalues $(n^2\nu^2)_{n\in\mathbb{N}}$, where

$$ \nu = \frac{\pi}{\ell}. $$

Let $n \in \mathbb{N}^*$. By the Implicit Function Theorem there exists an analytic function $a \mapsto z_n(a)$ defined on a neighborhood $V_n$ of 0, such that $z_n(0) = n\nu$ and $z_n(a)$ is a solution of (2.50) for $a \in V_n$. Moreover,

$$ z_n(a) = n\nu - \frac{2ia}{\pi n} + \gamma a^2 + O(a^3), \quad \text{with} \quad \mathrm{Re}(\gamma) = \frac{4\ell}{n^3\pi^3}. $$

In particular, $\mathrm{Re}(z_n(a)) \in]n\nu, (n+1)\nu[$ for $a > 0$ small. On the other hand, we can check that for $a$ in a fixed compact the solutions of (2.50) are in a horizontal strip of $\mathbb{C}$ and cannot be on the vertical lines $\mathrm{Re}(\zeta) \in \nu\mathbb{N}$. Then, by continuity of the solutions of (2.50) (using for instance the Rouché Theorem), we can deduce that for any $a > 0$ there is exactly one solution of (2.50) with real part in $]n\nu, (n+1)\nu[$. It is denoted by $z_n(a)$. Finally, we check that for $a > 0$ fixed and $n$ large, we have the same asymptotic expansion as above:

$$ z_n(a) = n\nu - \frac{2ia}{\pi n} + O(n^{-2}). \qquad (2.51) $$

Thus if we set $\lambda_n(a) = z_n(a)^2$, the spectrum of $T_a$ is given by the sequence $(\lambda_n(a))_{n\in\mathbb{N}}$ and we have

$$ \lambda_n(a) = n^2\nu^2 - \frac{4ia}{\ell} + O(n^{-1}). \qquad (2.52) $$

Since the operator $T_a$ depends analytically on $a$ (in the sense of Kato [Kat80]), since the eigenvalues of $T_0$ are simple, and since the eigenvalues of $T_s$ for $s \in [0, a]$ never meet, we deduce that the eigenvalues of $T_a$ are simple. In particular, there is no Jordan block.

Then we discuss the properties of the eigenfunctions of $T_a$. An eigenvector corresponding to the eigenvalue $\lambda_n(a)$ is given by

$$ \varphi_n(y) = \varphi_n(a; y) = A_n(a) \left( e^{iz_n(a)y} + \frac{z_n(a) + a}{z_n(a) - a} e^{-iz_n(a)y} \right), \quad y \in ]0, \ell[. $$

where $A_n(a)$ is for instance chosen positive and such that $\|\varphi_n(a)\|_{L^2(0,\ell)} = 1$. With (2.51) we see that $\varphi_n(a)$ is close to $\varphi_n(0)$ for large $n$. Since the family $(\varphi_n(0))$ is orthonormal, we deduce from a perturbation argument (see [Kat80, Th. V.2.20]) that the family $(\varphi_n(a))$ is a Riesz basis of $L^2(0, \ell)$.

We recall that a sequence $(\beta_k)$ in a Hilbert space $\mathcal{H}$ is a Riesz basis if there exist an orthonormal basis $(e_k)$ and a bounded operator $\Theta \in \mathcal{L}(\mathcal{H})$ with bounded inverse such that $\beta_k = \Theta e_k$ for all $k \in \mathbb{N}$. In particular, for $\psi \in \mathcal{H}$ there exists a unique sequence $(\psi_k)_{k \in \mathbb{N}} \in \ell^2(\mathbb{N})$ such that $\psi = \sum_{k=0}^{\infty} \psi_k \beta_k$ and we have, for some $C \ge 1$ independent of $\psi$,

$$ C^{-1} \sum_{k=0}^{\infty} |\psi_k|^2 \le \|\psi\|^2_{\mathcal{H}} \le C \sum_{k=0}^{\infty} |\psi_k|^2. \qquad (2.53) $$

With this Riesz basis property, we can use the symmetry of the domain $\Omega$ almost the same way as with a selfadjoint Laplacian (or as in (2.31)). In particular, we can write the resolvent of $H_a$ in terms of the resolvent of $L$. For $u = u(x, y)$ in $L^2(\Omega)$ we write $u(x, y) = \sum_{n \in \mathbb{N}} u_n(x) \varphi_n(y)$ and for $z \in \rho(H_a)$ we have

$$ (H_a - z)^{-1} u = \sum_{n \in \mathbb{N}} ((L - (z - \lambda_n(a)))^{-1} u_n) \otimes \varphi_n. \qquad (2.54) $$