Theorem 2.23 can be seen as the consequence of the Riesz basis property for $(\varphi_n)$ (see (2.53)), the spectral gap for the eigenvalues $(\lambda_n)$ (given by (2.52)) and the fact that all the eigenvalues have negative imaginary parts), and the estimate for the resolvent of the selfadjoint operator $L$ on $L^2(\mathbb{R}^{d-1})$.

Even if it was the original motivation, it was not possible to adapt the ideas of [8] to deal with the wave equation in [14]. First, the Riesz basis property is probably wrong if the cross section has dimension greater than 1. But even with a one dimensional cross section, the difficulty with the wave equation is that we have to deal with a Robin parameter proportional to the frequency. All the results given above are valid for any $a > 0$, but they are not uniform with respect to $a$ going to $+\infty$. In particular, the size of the strip which contains $\sigma(H_a)$ or the constant $C_a$ which appears in the Riesz basis property have no reason to be uniform with respect to $a$. This is why I finally used other ideas in [14].

All this being said, it is still interesting to understand the behavior of $\sigma(H_a)$ for large values of $a$.

The first observation is that for a fixed $n$ the map $a \in [0, +\infty[ \mapsto \mathrm{Re}(z_n(a))$ is increasing from $n\nu$ to $(n+1)\nu$, while $\mathrm{Im}(z_n(a))$ is 0 for $a = 0$, it is negative for all $a > 0$, and it goes to 0 when $a \to +\infty$. Thus, with a strong damping, we recover the behavior of an undamped problem. More precisely, when $a$ goes from 0 to $+\infty$, $\lambda_n(a)$ goes from the $n$-th Neumann eigenvalue $n^2\nu^2$ to the $n$-th Dirichlet eigenvalue $(n+1)^2\nu^2$, through the lower half-plane (it goes through the upper half-plane if $a \le 0$ goes from 0 to $-\infty$).

But all this is not uniform with respect to $n$. Roughly speaking, $z_n(a)$ is close to $n\nu$ if $a \ll n$ and close to $(n+1)\nu$ if $a \gg n$. But for the wave equation, $a$ is precisely proportional to the frequency, hence of order $n$. We can be more precise. We have

$$ \sup_{a>0} |\mathrm{Im}(z_n(a))| \underset{n \to \infty}{\sim} \ln(n), $$

so the curve $a \mapsto z_n(a)$ goes deeper and deeper in the lower half-plane for large $n$. This upper bound is obtained when $a \sim n\nu$. More precisely, for $\beta \in \mathbb{R}$ we have

$$ \mathrm{Re}(z_n(n\nu + \beta \ln(n))) - n\nu \underset{n \to +\infty}{\longrightarrow} \frac{1}{\ell} \left( \pi - \arg \left( \beta + \frac{i}{\ell} \right) \right) \in [0, \nu], $$

$$ \mathrm{Im}(z_n(n\nu + \beta \ln(n))) \underset{n \to +\infty}{\sim} -\frac{\ln(n)}{\ell}, $$

and for $\gamma \in \mathbb{R}_+^* \setminus \{1\}$,

$$ \mathrm{Im}(z_n(\gamma n \nu)) \underset{n \to +\infty}{\longrightarrow} \frac{1}{\ell} \ln \left| \frac{1+\gamma}{1-\gamma} \right| \quad \text{and} \quad \mathrm{Re} z_n(\gamma n \nu) - n\nu \underset{n \to +\infty}{\longrightarrow} \begin{cases} 0 & \text{if } \gamma < 1, \\ \nu & \text{if } \gamma > 1. \end{cases} $$

These results were finally written in an appendix of [14].

These last properties were not used in [8]. They gave an intuition for [14] but could not be used directly since the cross section was no longer of dimension 1. However, it is interesting to have a good picture of the behavior of the eigenvalues in this quite simple setting, since being able to localize (even roughly) the eigenvalues of a Robin-type non-selfadjoint Schrödinger operator for any Robin coefficient will be an important issue for the Schrödinger and wave equations on (one dimensional) quantum graphs. See the discussions in Sections 3.1 and 4.4.2. See also Section 4.1.2 about the generalization of [8] to a system of two coupled equations on the wave guide.
<figure><img src="image_4.png" /><figcaption>Figure 2.2: The eigenvalues of $H_a$ for $a$ going from 0 to $+\infty$
On the left, the eigenvalues $\lambda_1(a)$ and $\lambda_2(a)$ for various values of $a$. On the right, the graphs of $a \mapsto \operatorname{Im}(\lambda_n(a))$ for $n \in \{0, \dots, 4\}$.</figcaption></figure>
# Chapter 3

## Related topics

The analysis of the damped wave equation has been the main topic of my research, but I have also worked on slightly or completely different problems. In this chapter we will discuss quantum graphs, Agmon estimates, magnetic Laplacians, abstract spectral theory, a problem of control and some non-linear problems. These works opened new directions of research and give interesting perspectives for the future.

### 3.1 Spectrum of a non-selfadjoint quantum star graph

We begin with the analysis of the spectrum of a non selfadjoint quantum graph. With Gabriel Rivière, we were interested in the spectral properties of quantum graphs. Since we were also both interested by the damped wave equation, we chose to look at the damped wave equation on graphs. In [21] we started with a toy model, namely a non-selfadjoint Schrödinger operator on a star graph.

#### 3.1.1 Quantum graphs

A discrete graph is defined by a set $V$ of vertices and a set $E$ of edges, which can be seen as a subset of $V \times V$. Two edges $v_1$ and $v_2$ are connected if and only if the pair $(v_1, v_2)$ belongs to $E$. The graph can be oriented (the pairs $(v_1, v_2)$ and $(v_2, v_1)$ are distinct) or not. For spectral properties of discrete graphs, we refer for instance to [Chu97, CdV98] and references therein.

In our context the edges have lengths. Then an edge $e \in E$ of length $\ell_e > 0$ is identified with $]0, \ell_e[$ and a natural distance can be defined between the points of the graph (vertices or points on edges). We use the terminology quantum graph when we consider a (differential) operator on the Hilbert space given by a metric graph. More precisely, the edge $e$ is endowed with the Hilbert structure of $L^2(0, \ell_e)$. Then $\Gamma$ is endowed with the Hilbert structure defined by

$$L^2(\Gamma) = \bigoplus_{e \in E} L^2(0, \ell_e).$$

Then we can consider a differential operator on each edge, usually of Schrödinger type, with boundary conditions at the vertices. For a complete introduction about quantum graphs we refer to [BK13].

In [21], we consider the graph $\Gamma$ with the simplest non-trivial topology. Given $N \in \mathbb{N}^*$, it is defined by a set of $N + 1$ vertices $v, v_1, \dots, v_N$ and $N$ edges $e_1, \dots, e_N$. The edge $e_j$ joins the vertex $v_j$ to the central vertex $v$. Then $v$ has degree $N$ and the other vertices have degree 1.

Our graph is not oriented, but we have to choose a convention for the parametrization of the edges. The edge $e_j$ has length $\ell_j > 0$ and is parametrized by $x_j \in [0, \ell_j]$, with $x_j = 0$
corresponding to the vertex $v_j$ and $x_j = \ell_j$ corresponding to the vertex $v$.

Then, given $\alpha \in \mathbb{C}$, we define a Schrödinger operator $H_\alpha$ on $L^2(\Gamma)$ with (possibly) non-selfadjoint Robin condition at the central vertex. Its domain $\text{Dom}(H_\alpha)$ is the set of functions $u = (u_j)_{1\le j \le N}$ in $\bigoplus_{j=1}^N H^2(0, \ell_j)$ which satisfy the Dirichlet boundary condition at the exterior vertices

$$ \forall j \in \{1, \dots, N\}, \quad u_j(0) = 0, \qquad (3.1) $$

continuity at the central vertex $v$

$$ \forall j, k \in \{1, \dots, N\}, \quad u_j(\ell_j) = u_k(\ell_k), \qquad (3.2) $$

and the Robin condition

$$ \sum_{j=1}^{N} u'_{j}(\ell_{j}) + \alpha u(v) = 0, \qquad (3.3) $$

where $u(v)$ stands for the common value $u_j(\ell_j)$, $j \in \{1, \dots, N\}$. Finally, for $u = (u_j)_{1\le j \le N}$ in $\text{Dom}(H_\alpha)$ we define $H_\alpha u \in L^2(\Gamma)$ by

$$ (H_\alpha u)_j = -u_j'', \quad \forall j \in \{1, \dots, N\}. \qquad (3.4) $$

The quadratic form associated with the operator $H_\alpha$ is given by

$$ \langle H_\alpha u, u \rangle_{L^2(\Gamma)} = \sum_{j=1}^{N} \|u_j'\|_{L^2(0, \ell_j)}^2 + \alpha |u(v)|^2. \qquad (3.5) $$

The vertex condition (3.3) is the analog of the usual Robin boundary condition $\partial_\nu u + \alpha u = 0$ for Schrödinger operators on domains of $\mathbb{R}^d$ (see (2.18), with a different convention for $\alpha$). When $\alpha = 0$ we recover the usual Kirchhoff (or Neumann) condition, and we see from (3.5) that the perturbation when $\alpha \ne 0$ corresponds to a Dirac potential at the central vertex $v$.

Notice that the particular case $N = 1$ corresponds to the Schrödinger operator on a bounded interval with mixed Dirichlet and Robin conditions at the boundary, while the case $N = 2$ is the Dirichlet Schrödinger operator on a bounded interval with a singular potential in the interval.

<figure><img src="image_5.png" /><figcaption>Figure 3.1: The quantum star graph with $N = 6$ edges.</figcaption></figure>

### 3.1.2 Main spectral results

Our purpose in [21] is to discuss some spectral properties of the operator $H_\alpha$. We are mainly motivated by the non-selfadjoint case $\alpha \notin \mathbb{R}$. Contrary to the selfadjoint case (see for instance
[BK13]), little is known about the spectral properties of Schrödinger operators on graphs in the non-selfadjoint case (see however [Hus14, HKS15] for some general properties).

Here we only consider a model case, and considering more general graphs will be a natural perspective. However, this allows us to give results without any restriction on the lengths $(\ell_j)_{1 \le j \le N}$, when many results about quantum graphs are only given for rational lengths or for a generic family of lengths.

We are interested in the qualitative behavior of the large eigenvalues. The main motivation is the behavior of the imaginary parts, which are related to the decay for the corresponding time-dependent problem, but the results also give information about the real parts (and in particular we obtain new information even in the selfadjoint case). We describe the spectrum of $H_\alpha$ by comparison with the spectrum of the Kirchhoff operator $H_0$. We denote by $(\lambda_n(0))_{n \in \mathbb{N}^*}$ the non-decreasing sequence (with $\lambda_1(0) > 0$) of eigenvalues of $H_0$ repeated according to their multiplicities.

We say that an operator $T$ on the Hilbert space $\mathcal{H}$ is sectorial if there exists $\gamma \in \mathbb{R}$ and $\theta \in [0, \frac{\pi Z}{2}]$ such that the numerical range of $T$ is included in the sector $\Sigma_{\gamma, \theta} = \{\zeta \in \mathbb{C} : |\arg(\zeta - \gamma)| \le \theta\}$. Then $T$ is said to be maximal sectorial if $\sigma(T) \subset \Sigma_{\gamma, \theta}$.

We begin with a rough localization of the eigenvalues of $H_\alpha$.

**Proposition 3.1.** Let $\alpha \in \mathbb{C}$. The operator $H_\alpha$ is maximal sectorial and its spectrum is given by an infinite sequence $(\lambda_n(\alpha))_{n \in \mathbb{N}^*}$ of isolated eigenvalues whose geometric and algebraic multiplicities are finite and coincide. These eigenvalues (repeated according to multiplicities) can be labeled in such a way that for $n \in \mathbb{N}^*$ large enough we have

$$|\lambda_n(\alpha) - \lambda_n(0)| \le \mathrm{dist}(\lambda_n(0), \sigma(H_0)\setminus\{\lambda_n(0)\}).$$

Moreover, if $\tau \in [\frac{\pi Z}{\ell_j}, \frac{\pi Z}{\ell_k}]$ (non-empty if $\ell_j/\ell_k \in \mathbb{Q}$) for some $j, k \in \{1, \dots, N\}$, then $\tau^2$ is a common eigenvalue of $H_0$ and $H_\alpha$, with the same multiplicities.

This proposition says that the Robin eigenvalues are in some sense not so far from the corresponding eigenvalues of the Kirchhoff operator. This gives in particular a Weyl law, already known for the latter. For $R > 0$ we set

$$\mathcal{N}_\alpha(R) = \mathrm{Card}\{n \in \mathbb{N}^* : \mathrm{Re}(\lambda_n(\alpha)) \le R\}.$$

The Weyl law for a Laplacian on a domain $\Omega$ of $\mathbb{R}^d$ (or a manifold) depends on the dimension $d$ and the size of the domain:

$$N_{\Omega}(R) \underset{R \to +\infty}{\sim} \frac{|\Omega| \omega_d R^{\frac{d}{2}}}{(2\pi)^d},$$

where $\omega_d$ is the volume of the unit ball. Here the size of the metric graph is the sum of the lengths of the edges: $|\Gamma| = \sum_{j=1}^N \ell_j$. The following result agrees with the result already known on manifold or for selfadjoint quantum graphs.

**Theorem 3.2 (Weyl Law).** Let $\alpha \in \mathbb{C}$. We have

$$\mathcal{N}_{\alpha}(R) = \frac{|\Gamma| \sqrt{R}}{\pi} + \mathcal{O}(1).$$

For $\alpha \in \mathbb{C}$ and $n \in \mathbb{N}^*$ we set $\eta_n(\alpha) = \lambda_n(\alpha) - \lambda_n(0)$. For Theorem 3.2 we only need a uniform bound on $\eta_n(\alpha)$. The main result in [21] is a refined analysis of this difference. We set $\ell = (\ell_1, \dots, \ell_N)$.

**Theorem 3.3.** There exists a probability measure $\mu_\ell$ on $\mathbb{C}$ such that for any $\alpha \in \mathbb{C}^*$ we have in the weak sense for measures

$$\frac{1}{n} \sum_{k=1}^{n} \delta_{\frac{\eta_k(\alpha)}{\alpha}} \xrightarrow{n \to +\infty} \mu_{\ell}.$$

Moreover $\mu_\ell$ satisfies the following properties.
(i) $\mu_\ell$ is supported in $[0, 2 |\Gamma|^{-1}]$.

(ii) If $\ell_j/\ell_k \in \mathbb{Q}$ for all $j, k \in \{1, \dots, N\}$ then $\mu_\ell$ is a linear combination of Dirac distributions, including the one at 0.

(iii) If $\kappa \cdot \ell \neq 0$ for all $\kappa \in \mathbb{Z}^d \setminus \{0\}$ then $\mu_\ell$ is absolutely continuous with respect to the Lebesgue measure on $[0, 2 |\Gamma|^{-1}]$ and its support is exactly $[0, 2 |\Gamma|^{-1}]$.

If $s$ belongs to the support of $\mu_\ell$ there exists an increasing sequence $(n_k)$ in $\mathbb{N}^*$ such that $\eta_{n_k}(\alpha)$ goes to $s\alpha$ as $k \to \infty$. The last result is about the rate of convergence for this limit.

**Proposition 3.4.** Let $\alpha \in \mathbb{C}$.

(i) There exists an increasing sequence $(n_k)_{k \in \mathbb{N}^*}$ such that

$$ \eta_{n_k}(\alpha) = \mathcal{O}_{k \to \infty} \left( \frac{1}{n_k^2} \right). $$

(ii) There exists $\Omega \subset (\mathbb{R}_+^*)^N$ of Lebesgue measure 0 such that for $\ell \in (\mathbb{R}_+^*)^N \setminus \Omega$, $\alpha \in \mathbb{C}$, $s \in [0, 2 |\Gamma|^{-1}]$ and $\varepsilon > 0$ we can find an increasing sequence $(n_k)_{k \in \mathbb{N}^*}$ such that

$$ \eta_{n_k}(\alpha) = s\alpha + \mathcal{O}_{k \to \infty} \left( \frac{1}{n_k^{\frac{1}{N}-\varepsilon}} \right). $$

By the last statement of Proposition 3.1, the sequence $(\eta_n(\alpha))$ has a vanishing subsequence except in the last case of Theorem 3.3, so the first statement essentially concerns the purely irrational case. Notice in particular that 0 always belongs to the support of $\mu_\ell$.

In these results, we only describe the eigenvalues of $H_\alpha$ by comparison with those of $H_0$. However, since $\mathrm{Im}(\lambda_n(\alpha)) = \mathrm{Im}(\eta_n(\alpha))$, this gives a direct description of the imaginary parts, which was our original motivation. The spacing between the real parts of $\lambda_0(\alpha)$ and $\lambda_n(\alpha)$ is nevertheless also an important result.

### 3.1.3 Ideas of proof

In the rest of this section, we comment some ideas of the proof of Theorem 3.3.

**Comparison with the Dirichlet eigenvalues.** We denote by $H_D$ the operator defined as $H_\alpha$ with (3.3) replaced by the Dirichlet condition $u(v) = 0$ at the central vector (this corresponds to $\alpha = \infty$). Then the edges are disconnected, and it is easy to identify the non-decreasing sequence $(\lambda_{D,n})_{n \in \mathbb{N}^*}$ of eigenvalues for $H_D$ (repeated according to multiplicities). By a standard argument based on the Min-max Theorem (see for instance [BK13, Th. 1.3.8]) we have

$$ \lambda_1(0) \le \lambda_{D,1} \le \lambda_2(0) \le \lambda_{D,2} \le \dots \le \lambda_n(0) \le \lambda_{D,n} \le \dots \quad (3.6) $$

Moreover, we have a similar result with $\lambda_n(\alpha)$ instead of $\lambda_n(0)$ if $\alpha$ is real. Since the eigenvalues of the Dirichlet problem are easily identified, this gives a rough localization of $\lambda_n(\alpha)$ when $\alpha$ is real.

We cannot use this idea when $\alpha \notin \mathbb{R}$. Nevertheless, we can check that the two following facts remain valid in general. If $\lambda$ is an eigenvalue of $H_D$ of multiplicity $m \in \mathbb{N}^*$, then it is an eigenvalue of $H_\alpha$ of (geometric and algebraic) multiplicity $m-1$. On the other hand, the eigenvalues of $H_\alpha$ which do not coincide with an eigenvalue of $H_D$ are of (geometric and algebraic) multiplicity 1. The difficulty is then to localize these other eigenvalues.
**The secular equation.** Let $z \in \mathbb{C}^*$. Assume that $z^2$ is an eigenvalue of $H_\alpha$ and that $u$ is a corresponding eigenvector. By the Dirichlet condition (3.1), $u_j$ is necessarily of the form

$$u_j(x_j) = \beta_j \sin(zx_j), \quad x_j \in [0, \ell_j], \qquad (3.7)$$

for some $\beta_j \in \mathbb{C}$. With this information only, we can already prove that

$$\limsup_{\substack{\lambda \in \sigma(H_\alpha) \\ \mathrm{Re}(\lambda) \to +\infty}} |\mathrm{Im}(\lambda)| \le \frac{2}{|\Gamma|} |\mathrm{Im}(\alpha)|. \qquad (3.8)$$

This implies in particular that $\sigma(H_\alpha)$ is included in a horizontal strip of $\mathbb{C}$.

The conditions (3.2) and (3.3) at the central vertex $v$ give $N$ linear equations with respect to the $N$ coefficients $\beta_j$, $j \in \{1, \dots, N\}$. Then $z^2$ is an eigenvalue of $H_\alpha$ if and only if the corresponding determinant is 0. In general, the determinant is some polynomial in $\sin(z\ell_j)$ and $\cos(z\ell_j)$, $j \in \{1, \dots, N\}$. Here, with a star-graph, everything can be explicit. We obtain the secular equation

$$F_0(z\ell) + \frac{\alpha}{z} F_D(z\ell) = 0, \qquad (3.9)$$

where for $y = (y_1, \dots, y_d) \in \mathbb{C}^d$ we have set

$$F_0(y) = \sum_{j=1}^{N} \cos(y_j) \prod_{k \ne j} \sin(y_k) \quad \text{and} \quad F_D(y) = \prod_{j=1}^{N} \sin(y_j).$$

The determinant $F_0(z\ell)$ corresponds to the case $\alpha = 0$. This is the determinant for the Kirchhoff operator. On the other hand, $F_D(z\ell)$ is the determinant corresponding to the Dirichlet condition at $v$. For $y \in \mathbb{C}^N$ and $z \in \mathbb{C}$ such that $F_D(y) \ne 0$ and $F_D(z\ell) \ne 0$ we set

$$\Psi(y) = -\frac{F_0(y)}{F_D(y)} = -\sum_{j=1}^{N} \cotan(y_j) \quad \text{and} \quad \psi(z) = \Psi(z\ell).$$

Then $z^2 \in \mathbb{C} \setminus \sigma(H_D)$ (remember that $\sigma(H_\alpha) \cap \sigma(H_D)$ is already understood) is an eigenvalue of $H_\alpha$ if and only if

$$\psi(z) = \frac{\alpha}{z}. \qquad (3.10)$$

**Perturbation of the Kirchhoff eigenvalues.** The family of operators $(H_\alpha)_{\alpha \in \mathbb{C}}$ is analytic with respect to $\alpha$ (family of type B in the sense of [Kat80]). At least for $\alpha$ small the eigenvalues of $H_\alpha$ are in some sense close to the eigenvalues of $H_0$. The next step is to show that the eigenvalues of $H_0$ cannot move too far as long as $\alpha$ stays in a compact set of $\mathbb{C}$.

Let $n \in \mathbb{N}^*$ such that $\lambda_n(0) \ne \lambda_{D,n}$ (in particular, $\lambda_n(0)$ is a simple eigenvalue of $H_0$). The idea is to find a bounded domain $\omega_n$ such that $\omega_n \cap \sigma(H_0) = \{\lambda_n(0)\}$ and $\partial\omega_n \cap H_{t\alpha} = \emptyset$ for $t \in [0, 1]$. By continuity of the spectrum, this will imply that $H_\alpha$ has exactly one simple eigenvalue in $\omega_n$. For the star graph, we can use the explicit expression for $\psi$. Using the Taylor expansion around $\tau_n = \sqrt{\lambda_n(0)}$, we can prove that (3.10) with $\alpha$ replaced by $t\alpha$ cannot have a solution on the circle $C(\tau_n, \frac{8|\alpha|}{\tau_n \psi'(\tau_n)})$ when $n$ is large. In particular there is an eigenvalue $\lambda_n(\alpha)$ of $H_\alpha$ which remains “close” to $\lambda_n(0)$. Then, by Taylor expansion again, we can deduce that

$$\eta_n(\alpha) = \frac{2\alpha}{\psi'(\tau_n)} + O(\tau_n^{-1}). \qquad (3.11)$$

At this stage, we have roughly localized some eigenvalues of $H_\alpha$. For $n \in \mathbb{N}^*$, if $\lambda_n(0)$ coincides with a Dirichlet eigenvalue, then it is also an eigenvalue of $H_\alpha$ and we set $\lambda_n(\alpha) = \lambda_n(0)$ (the multiplicities agree), otherwise there is an eigenvalue $\lambda_n(\alpha)$ close to $\lambda_n(0)$ (both are simple). We have used the continuity of the eigenvalues of $H_{t\alpha}$ to see how the eigenvalues of $H_0$ move to those of $H_\alpha$. It could also happen that some new eigenvalues appear at infinity. From the
analysis of $\psi$ again, we check that the eigenvalues of $H_{t\alpha}$ for $t \in [0, 1]$ all stay in a region of $\mathbb{C}$ which only has bounded connected components, so this phenomenon cannot occur here. Thus, we have identified all the eigenvalues of $H_\alpha$. For some of them we have $\eta_n(\alpha) = 0$ (this happens if some lengths are commensurable), and for the others the difference $\eta_n$ satisfies (3.11).

**The Barra-Gaspard measure and definition of $\mu_\ell$.** We observe that the leading term in (3.11) is simply linear with respect to $\alpha$. For a better understanding of the sequence $(\eta_n(\alpha))_{n \in \mathbb{N}^*}$, the next step is the analysis of the sequence $(\psi'(\tau_n))_{n \in \mathbb{N}^*}$. This is a quantity which only depends on the $\tau_n$, $n \in \mathbb{N}^*$.

For this we use the measure introduced by Barra and Gaspard in [BG00] to study the level spacing of $H_0$. See also [KMW03, BW10]. We see $F_0$ as a function on the $N$-dimensional torus $\mathbb{T}^N$. We set $\mathcal{Z} = \{y \in \mathbb{T}^N : F_0(y) = 0\}$, so that $\tau^2$ is an eigenvalue of $H_0$ if and only if $\tau\ell \in \mathcal{Z}$. For $t \ge 0$ we set $\varphi_\ell^t = [t\ell]$ in $\mathbb{T}^N$.

Here we focus on the case where $\kappa \cdot \ell \ne 0$ for all $\kappa \in \mathbb{Z}^d \setminus \{0\}$ (otherwise $\varphi_\ell^t$ lives in a subtorus of $\mathbb{T}^N$, the general case is considered in [21]). In this case $\mathcal{Z}_0 = \mathcal{Z} \setminus \{0\}$ is a submanifold of dimension $N-1$ in $\mathbb{T}^N$ to which any $\ell$ with positive components is transverse. For this last property we use again the explicit expressions available for the case of a star graph. The contribution of a small neighborhood of 0 in $\mathcal{Z}$ will be small and will not play any role. On $\mathcal{Z}_0$ we consider the Barra-Gaspard measure

$$ \mu_{\text{BG}} = |\ell \cdot \nu(y)| \mu_{\mathcal{Z}_0}, \qquad (3.12) $$

where $\mu_{\mathcal{Z}_0}$ is the Lebesgue measure on $\mathcal{Z}_0$ and $\nu(y)$ is a normal unit vector. Using in particular unique ergodicity, we see that if $g$ is a continuous and compactly supported function on $\mathcal{Z}_0$ we have

$$ \frac{1}{n} \sum_{k=1}^{n} g(\varphi_{\ell}^{\tau_k}) \xrightarrow{n \to \infty} \frac{\pi}{|\Gamma| |\mathbb{T}^N|} \int_{\mathcal{Z}_0} g \, d\mu_{\text{BG}}. $$

Then the measure $\mu_\ell$ of Theorem 3.3 is given by

$$ \mu_{\ell} = \frac{\pi}{|\Gamma| |\mathbb{T}^N|} \Phi_* \mu_{\text{BG}}, \quad \text{where} \quad \Phi(y) = \frac{2}{\nabla \Psi(y) \cdot \ell}. $$

When some lengths are commensurable, $|\mathbb{T}^N|$ has to be replaced by the size of the sub-torus in which the flow $\varphi_\ell^t$ is dense, and there is in $\mu_\ell$ a Dirac mass at 0, corresponding of the eigenvalues which do not move with $\alpha$.

From the expression of $\mu_\ell$ we can then prove the properties given in Theorem 3.3. We do not discuss this part of the proof here.

It is natural to wonder if we can recover the same results on a more general quantum graph. Some possible extensions of the results of [21] will be discussed in Section 4.4.2.

## 3.2 Agmon estimates for non-accretive Schrödinger operators

In this paragraph we briefly discuss the result of [13] about some non-accretive Schrödinger operators. This is a joint work with David Krejčířík, Nicolas Raymond and Petr Siegl.

The motivation for this paper was the Agmon estimates for a large class of non-selfadjoint Schrödinger operators. The so-called Agmon estimates measure the decay of the eigenfunctions of a Schrödinger operator in the classically forbidden region.

If we consider on $\mathbb{R}$ the case of a confining real valued potential $V$ with minimum at 0 (for instance, $V(x) = x^2$), then a classical particle of energy $E$ will necessarily stay in the neighborhood of 0 defined by $\{x \in \mathbb{R} : V(x) \le E\}$. This is not the case for a quantum particle, but an eigenfunction of the operator $-\hat{c}_x^\dagger + V(x)$ corresponding to the eigenvalue $E$
decays exponentially fast outside this region, and the rate of decay is given by the distance to the classical region as defined by the Agmon (pseudo) metric

$$ \sqrt{(V(x) - E)_+} dx^2. $$

More generally, in a non-confining setting and for more general operators, the decay of eigenfunctions was studied for eigenvalues smaller than the bottom of the essential spectrum. We refer for instance to [Agn82]. See also [HS96, Ch. 3]. The Agmon estimates have then been extended in many settings. For a review, see for instance the talk [Hel19] and references therein.

The purpose of [13] is to show that we have a similar phenomenon when the particle is confined by the imaginary part of the potential. We also consider magnetic Laplacians, and it turns out that the magnetic field actually plays exactly the same role. We can even go further and show that if the imaginary part of the potential or the magnetic field becomes large, then we can allow a (smaller) negative real part for the potential. As a typical example, we can consider on $L^2(\mathbb{R})$ the operator

$$ -\frac{d^2}{dx^2} - x^2 + ix^3, $$

whose numerical range covers the whole complex plane.

In fact, much wilder potentials are allowed. The setting of [13] is the following. We consider an open and connected subset $\Omega$ of $\mathbb{R}^d$, $V \in C^1(\bar{\Omega}, \mathbb{C})$ and $A \in C^2(\bar{\Omega}, \mathbb{R}^d)$. The magnetic matrix is $B = (\partial_j A_k - \partial_k A_j)_{1 \le j, k \le d}$. We set

$$ \langle B; V \rangle = \sqrt{1 + |B| + |V|}. $$

Conditions sufficient for all the results of the paper are

$$ (\mathrm{Re}V)_- = \underset{|x| \to +\infty}{o} \left( \langle B; V \rangle \right) \\ |\nabla V| + |\nabla B| = \underset{|x| \to +\infty}{o} \left( \langle B; V \rangle^{\frac{3}{2}} \right). $$

The second assumption prevents strong oscillations for the coefficients. The power $\frac{3}{2}$ is an improvement compared to previous papers on related subjects. The first condition says that the negative real part has to be small at infinity compared to the imaginary part or the magnetic matrix, as discussed above. The purpose is then to define and give spectral properties for the electromagnetic operator

$$ \mathcal{L} = (-i\nabla + A)^2 + V. $$

We are thus considering Schrödinger operators which are not even accretive. We first define the operator. The form domain is

$$ \mathcal{V} = \{ u \in L^2(\Omega) : (-i\nabla + A)u \in L^2(\Omega), \langle B; V \rangle^{\frac{1}{2}} u \in L^2(\Omega) \}. $$

We cannot apply a standard version of the representation theorem, so we use a generalized version given in [AH15]. This defines a closed operator $\mathcal{L}$ with domain

$$ \mathrm{Dom}(\mathcal{L}) = \{ u \in \mathcal{V} : (-i\nabla + A)^2 u + Vu \in L^2(\Omega) \}. $$

The next step is to prove that we have separation of the spectrum, namely

$$ \mathrm{Dom}(\mathcal{L}) = \{ u \in \mathcal{V} : (-i\nabla + A)^2 u \in L^2(\Omega) \text{ and } Vu \in L^2(\Omega) \}. $$

The main result is then the following. For $E \in \mathbb{R}$ we set

$$ \Sigma_E = \{ \zeta \in \mathbb{C} : \mathrm{Re}(\zeta) + |\mathrm{Im}(\zeta)| < E \}. $$
**Theorem 3.5.** (i) There exist $\gamma_1 > 0$ and $\gamma_2 \in \mathbb{R}$ such that for

$$ \mu < \liminf_{|x| \to +\infty} \langle B; V \rangle (x) $$

then any $\zeta \in \Sigma_{\gamma_1\mu-\gamma_2}$ is such that $(\mathcal{L} - \zeta)$ is Fredholm with index 0 and the spectrum of $\mathcal{L}$ in $\Sigma_{\gamma_1\mu-\gamma_2}$ consists of isolated eigenvalues with finite algebraic multiplicities.

(ii) Given $\lambda \in \Sigma_{\gamma_1\mu-\gamma_2}$ we define the corresponding Agmon distance on $\Omega$ by

$$ d_{\text{Ag},\lambda}(x, y) = \inf_{\gamma \in C^1(x,y)} \int_0^1 \sqrt{(\gamma_1 \langle B; V \rangle(\gamma(t)) - \gamma_2 - \text{Re}(\lambda) - |\text{Im}(\lambda)|)_+} |\gamma'(t)| \, dt, $$

where $C^1(x, y)$ is the set of continuous and piecewise $C^1$ path $\gamma: [0, 1] \to \Omega$ with $\gamma(0) = x$ and $\gamma(1) = y$. Let $x_0 \in \Omega$ be fixed. If $\lambda$ is an eigenvalue of $\mathcal{L}$ then for any $\psi$ in the corresponding algebraic eigenspace we have

$$ e^{\frac{1-\varepsilon}{3} d_{\text{Ag}}(\cdot, x_0)} \psi \in L^2(\Omega). $$

## 3.3 Absence of embedded eigenvalues for a magnetic Laplacian

In this paragraph we present [20]. This joint work with Nicolas Raymond is about the absence of embedded eigenvalues in the essential spectrum of a magnetic Laplacian on $\mathbb{R}^2$.

We consider a magnetic field $B$ which is invariant with respect to one variable. We set $B(x, y) = b(x)$ for some $b \in C^1(\mathbb{R}; \mathbb{R})$. A corresponding vector potential $A = (A_1, A_2)$ is given by $A_1(x, y) = 0$ and, for some $a_0 \in \mathbb{R}$,

$$ A_2(x, y) = a(x) = a_0 + \int_0^x b(s) \, ds. \qquad (3.13) $$

Then we consider the (selfadjoint) operator

$$ \mathcal{L} = (-i\nabla - A)^2 = -\partial_x^2 + (-i\partial_y - a(x))^2, $$

defined on the domain

$$ \text{Dom}(\mathcal{L}) = \{u \in L^2(\mathbb{R}^2) : (-i\nabla - A)u, (-i\nabla - A)^2 u \in L^2(\mathbb{R}^2)\}. $$

The spectrum of $\mathcal{L}$ is purely essential. The purpose is to prove, under some assumptions on $b$, that there are no eigenvalues in (some parts of) the spectrum of $\mathcal{L}$. In some cases, the spectrum of $\mathcal{L}$ can even be purely absolutely continuous. Such results cannot be true in general since for a constant and non-zero magnetic field $B$, the spectrum of $\mathcal{L}$ consists of eigenvalues of infinite multiplicities, given by the Landau levels $(2n - 1)|B|$, $n \in \mathbb{N}^*$ (so $\sigma(\mathcal{L}) = \sigma_{\text{ess}}(\mathcal{L}) = \sigma_{\text{p}}(\mathcal{L})$ in this case).

Let

$$ \phi_{\pm} = \lim_{x \to \pm\infty} a(x). $$

It is already known [Iwa85] that if $\phi_+ = \phi_- = \pm\infty$ or if $b(x)$ has two distinct finite and non-zero limits at $-\infty$ and $+\infty$, then $\mathcal{L}$ has an absolutely continuous spectrum (and in particular, no eigenvalues). Our two main results in [20] are the following.

**Theorem 3.6.** Assume that the limit $\phi_-$ exists in $\mathbb{R} \cup \{\pm\infty\}$. Assume that there exist $\alpha \in ]-1, +\infty[ \setminus \{0\}$ and $c_1, C > 0$ such that

$$ b(x) \underset{x \to +\infty}{\sim} c_1 x^{\alpha} \quad \text{and} \quad |b'(x)| \le C \langle x \rangle^{\alpha-1}, \quad x \ge 0. $$

Then $\mathcal{L}$ has no eigenvalue.
We could similarly consider the cases where $b$ is replaced by $-b$ or where the behavior of $b$ is prescribed near $-\infty$. Notice that Theorem 3.6 covers in particular situations where $b(x)$ goes to 0 or to $+\infty$ for $x \to +\infty$.

In the second theorem, we consider situations where $b$ goes faster to 0 at infinity. We assume that $\phi_-$ and $\phi_+$ are finite but with $\phi_+ - \phi_- \gg 1$ (the case $\phi_- - \phi_+ \gg 1$ is similar). For instance, we assume $\phi_- = 0$ and $\phi = \phi_+ \gg 1$.

**Theorem 3.7.** Assume that $b$ takes positive values, that it is integrable and that $b'(x) = O(x^N)$ for some $N \ge 0$ as $x \to +\infty$. We choose $a_0 = \int_{-\infty}^0 b(s) ds$ in (3.13), so that $\phi_- = 0$ and $\phi := \phi_+ = \|b\|_{L^1(\mathbb{R})}$. Then we assume that $a \in L^1(\mathbb{R}_-)$ and $a - \phi \in L^1(\mathbb{R}_+)$.

$$ (i) \sigma(\mathcal{L}) = [0, +\infty[. $$

$$ (ii) \mathcal{L} = \mathcal{L}_{\phi} \text{ has no eigenvalue in } \left[ \frac{\phi^2}{4}, +\infty[. $$

$$ (iii) \text{ Let } \eta_{\phi} \text{ be such that } \eta_{\phi} = o(\ln(\phi)^{-6}) \text{ as } \phi \to +\infty. \text{ Then there exists } \phi_0 > 0 \text{ such that} \\ \text{ for } \phi \ge \phi_0 \text{ the operator } \mathcal{L} = \mathcal{L}_{\phi} \text{ has no eigenvalue smaller than } \eta_{\phi} \phi^2. $$

The question of existence of eigenvalues in $[\eta_{\phi} \phi^2, \frac{\phi^2}{4}]$ remains open in this case.

The proofs of these two results are based on a Fourier transform with respect to $y$ and a careful analysis of the family of one-dimensional Schrödinger operators

$$ \mathcal{L}_{\xi} = -\partial_x^2 + (\xi - a(x))^2. $$

In particular, a refined version of the harmonic approximation is provided to give a precise description of the eigenvalues of $\mathcal{L}_{\xi}$.

## 3.4 Reduction of dimension in an abstract setting

This paragraph is devoted to the paper [16], written with David Krejčířík, Nicolas Raymond and Petr Siegl.

This work was motivated by a Schrödinger operator with non-selfadjoint Robin condition in a shrinking layer. More precisely, we consider a closed, orientable, smooth hypersurface $\Sigma$ in $\mathbb{R}^d$ without boundary, and we denote by $\nu : \Sigma \to \mathbb{S}^{d-1}$ a unit normal vector field which specifies the orientation of $\Sigma$. We assume that $\Sigma$ has a tubular neighborhood: for some $\varepsilon > 0$ the map $\Theta_{\varepsilon} : (s, t) \mapsto s + \varepsilon t \nu(s)$ is injective on $\Omega = \Sigma \times [-1, 1]$ and defines a diffeomorphism from $\Omega$ to its image $\Omega_{\varepsilon} = \Theta_{\varepsilon}(\Omega)$.

We set $\Sigma_{\pm, \varepsilon} = \Theta_{\varepsilon}(\Sigma \times \{\pm 1\})$. Given a smooth function $\alpha : \Sigma \to \mathbb{C}$ we define $\alpha_{\pm, \varepsilon}$ on $\Sigma_{\pm, \varepsilon}$ by $\alpha_{\pm, \varepsilon}(s \pm \varepsilon \nu(s)) = \alpha(s)$. Then we consider on $L^2(\Omega_{\varepsilon})$ the operator $P_{\varepsilon}$ defined as the usual Laplace operator on $\Omega_{\varepsilon}$ subject to the Robin boundary condition

$$ \partial_{\nu} u + \alpha_{\pm, \varepsilon} u = 0 \quad \text{on } \Sigma_{\pm, \varepsilon}. $$

This is a very particular (PT-symmetric) choice of Robin coefficients, but a more general setting could be similarly considered. Then the purpose is to prove that when $\varepsilon \to 0$ the spectral properties of $P_{\varepsilon}$ are close to those of an effective operator $P_{\text{eff}} = -\Delta_{\Sigma} + V_{\text{eff}}$ on $L^2(\Sigma)$. Here $-\Delta_{\Sigma}$ is the usual Laplace-Beltrami operator on $\Sigma$ and $V_{\text{eff}}$ is a potential which depends on the geometry and the Robin coefficient $\alpha$:

$$ V_{\text{eff}} = |\alpha| - 2\alpha \text{Re}(\alpha) - \alpha(\kappa_1 + \cdots + \kappa_{d-1}), $$

where $\kappa_1, \dots, \kappa_{d-1}$ are the principal curvatures.

The convergence of $P_{\varepsilon}$ to $P_{\text{eff}}$ is in the sense of the norm of the resolvent. More precisely, we denote by $\Pi$ the projection of $L^2(\Omega)$ on functions which do not depend on $t$. For $u \in L^2(\Omega)$
and $(s, t) \in \Omega$ we set $(\Pi u)(s, t) = \frac{1}{2} \int_{-1}^{1} u(s, \theta) d\theta$. Moreover, for $\varepsilon > 0$ small enough, an explicit unitary operator $U_{\varepsilon} : L^2(\Omega_{\varepsilon}) \to L^2(\Omega, w_{\varepsilon} \, ds \, dt)$ is defined, for some $w_{\varepsilon} : \Omega \to \mathbb{R}_+$ bounded and bounded away from 0 uniformly in $\varepsilon$. In the following result, $P_{\text{eff}}$ is seen as an operator on $\Pi L^2(\Omega)$.

**Theorem 3.8.** Let $K$ be a compact subset of $\rho(P_{\text{eff}})$. There exist $\varepsilon_0 > 0$ and $C \ge 0$ such that for $z \in K$ and $\varepsilon \in [0, \varepsilon_0]$ we have $z \in \rho(P_{\varepsilon})$ and

$$ \|(P_{\varepsilon} - z)^{-1} - U_{\varepsilon}^{-1}(P_{\text{eff}} - z)^{-1}\Pi U_{\varepsilon}\|_{L(L^2(\Omega_{\varepsilon}))} \le C\varepsilon. $$

Notice that the convergence in the sense of the norm-resolvent is very strong and we can deduce important properties for $P_{\varepsilon}$ with $\varepsilon > 0$ small from the properties of $P_{\text{eff}}$. We say that we have reduced the dimension since a $d$-dimensional problem is reduced to a $(d-1)$-dimensional problem, which can be significantly simpler.

There were already many results of this kind in the literature for selfadjoint operators. The common aspect is that the spectral properties of a parameter-dependent operator are well described by those of an effective operator when the parameter goes to some limit. However, in the previous literature, this similarity is hidden in the various technical steps specific to each situation. In [16], we give a unified abstract result, described below. This includes various settings such as the semiclassical Born-Oppenheimer approximation, a problem on a shrinking tubular neighborhood as above, or the similar problem on a fixed domain but for a large Robin parameter $\alpha$. Reduction of dimension is already known in some of these contexts, but the abstract result also provides new results. Here we only describe the abstract result, but these examples of applications are discussed in [16].

The abstract setting is the following. Given a measure space $\Sigma$ and a family of Hilbert spaces $\mathcal{H}_s$, $s \in \Sigma$, we set $\mathcal{H} = \bigoplus_{s \in \Sigma} \mathcal{H}_s$. A vector in $\mathcal{H}$ is a family $(\phi_s)_{s \in \Sigma}$, where each $\phi_s$ belongs to $\mathcal{H}_s$, and $\int_{s \in \Sigma} \|\phi_s\|_{\mathcal{H}_s}^2 d\sigma(s) < +\infty$. We consider on $\mathcal{H}$ a selfadjoint operator of the form

$$ \mathcal{L} = S^*S + \bigoplus_{s \in \Sigma} T_s, $$

where $S$ is densely defined and $T_s$ is selfadjoint and non-negative for all $s \in \Sigma$.

Typically, if we consider on $\mathbb{R}^2$ an operator of the form $-h^2\Delta + V$ with $V \ge 0$ then we can take $\Sigma = \mathbb{R}$, $S = -ih\partial_s$ and $T_s = -\partial_t^2 + V(s, t)$ for each $s \in \mathbb{R}$.

Then we consider a function $s \in \Sigma \to \gamma_s$ such that $\gamma_s \ge \gamma$ for some $\gamma > 0$. For $s \in \Sigma$ we denote by $\Pi_s$ the spectral projection of $T_s$ on $[0, \gamma_s[$. We define $\Pi \in \mathcal{L}(\mathcal{H})$ by $\Pi\phi = (\Pi_s\phi_s)_{s \in \Sigma}$ for $\phi = (\phi_s)_{s \in \Sigma}$, and we set $\Pi^{\perp} = \text{Id} - \Pi$. We assume that $\text{Dom}(S)$ is invariant under $\Pi$.

In a typical example, the bottom of the spectrum of $T_s$ is given by a simple eigenvalue and $\gamma_s$ is the second eigenvalue of $T_s$, so that $\Pi_s$ is the orthogonal projection on the eigenspace corresponding to the first eigenvalue. After having chosen a corresponding eigenvector, $\Pi_s\phi_s$ is characterized by a scalar and $\Pi\phi$ can be identified with a function on $\Sigma$.

We set $\mathcal{L}_{\text{eff}} = \Pi\mathcal{L}\Pi$. This is an operator on $\Pi\mathcal{H}$ with domain $\Pi\mathcal{H} \cap \text{Dom}(\mathcal{L})$. The purpose is to compare $\mathcal{L}$ with the simpler operator $\mathcal{L}_{\text{eff}}$. As an intermediate step we consider the operator $\tilde{\mathcal{L}} = \mathcal{L}_{\text{eff}} + \mathcal{L}^{\perp}$, where $\mathcal{L}^{\perp} = \Pi^{\perp}\mathcal{L}\Pi^{\perp}$ on $\mathcal{H}$. Then we have $\sigma(\tilde{\mathcal{L}}) = \sigma(\mathcal{L}_{\text{eff}}) \cup \sigma(\mathcal{L}^{\perp})$ and for $z \in \rho(\tilde{\mathcal{L}})$ such that $z \notin [\gamma, +\infty[$ we have

$$ \|(\tilde{\mathcal{L}} - z)^{-1} - (\mathcal{L}_{\text{eff}} - z)^{-1}\Pi\|_{\mathcal{L}(\mathcal{H})} \le \frac{1}{\text{dist}(z, [\gamma, +\infty[)} $$

Thus the main result is about the difference between $\mathcal{L}$ and $\tilde{\mathcal{L}}$. We consider $\eta_1(z)$, $\eta_2(z)$, $\eta_3(z)$ and $\eta_4(z)$ which are explicit functions of $z \in \mathbb{C}$, $\gamma$ and $\|[S, \Pi]\|_{\mathcal{L}(\mathcal{H})}$ (see (1.5) in [16]). Then we have the following estimate.
**Theorem 3.9.** Let $z \in \rho(\widehat{\mathcal{L}})$. Assume that

$$1 - \eta_1(z) \|(\widehat{\mathcal{L}} - z)^{-1}\| - \eta_2(z) > 0.$$

Then $z \in \rho(\mathcal{L})$ and

$$\|(\mathcal{L} - z)^{-1} - (\widehat{\mathcal{L}} - z)^{-1}\| \\ \le \eta_1(z) \|(\mathcal{L} - z)^{-1}\| \|(\widehat{\mathcal{L}} - z)^{-1}\| + \eta_2(z) \|(\mathcal{L} - z)^{-1}\| + \eta_3(z) \|(\widehat{\mathcal{L}} - z)^{-1}\| + \eta_4(z).$$

Combining these two results we get information on the resolvent of $\mathcal{L}$ from properties on the resolvent of $\mathcal{L}_{\text{eff}}$.

The regret with Theorem 3.9 is that even if it is quite general, it does not include the non-selfadjoint setting of Theorem 3.8. However, the proofs follow the same ideas and use common intermediate results. It would be interesting to be able to prove an abstract theorem which also includes non-selfadjoint settings.

## 3.5 Observability for a Kolmogorov equation

In this paragraph we consider a question of control for a degenerate evolution equation. This is a joint work with Jérémi Dardé [22].

### 3.5.1 Setting and main result

We consider a two-dimensional domain $\Omega$ of the form $\mathbb{T} \times I$, where $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ and $I = ]-\ell_{-}, \ell_{+}[$ for some $\ell_{+} > 0$. A general point in $\Omega$ is denoted by $(x, y)$ with $x \in \mathbb{T}$ and $y \in I$. Let $q \in C^3(\bar{I}, \mathbb{R})$ such that $q(0) = 0$ and $\min(q') > 0$. The model case is $q(y) = y$. Let $T > 0$. We consider a Kolmogorov equation of the form

$$\left\{ \begin{array}{ll} \partial_t u + q(y)^2 \partial_x u - \partial_{yy} u = 0, & \text{on } ]0, T[ \times \Omega, \\ u = 0, & \text{on } ]0, T[ \times \partial \Omega, \\ u|_{t=0} = u_0 \in L^2(\Omega). \end{array} \right. \qquad (3.14)$$

We say that the Kolmogorov equation (3.14) is observable through $\partial\Omega$ in time $T > 0$ if there exists $C > 0$ such that for any $u_0 \in L^2(\Omega)$ the solution $u$ of (3.14) satisfies

$$\|u(T)\|_{L^2(\Omega)}^2 \le C \int_0^T \|\partial_\nu u(t)\|_{L^2(\partial\Omega)}^2 \, dt. \qquad (3.15)$$

We can replace $\partial\Omega$ by any non-empty open subset of $\partial\Omega$. We can also consider observability through a non-empty open subset $\omega$ of $\Omega$. In this case, we replace (3.15) by

$$\|u(T)\|_{L^2(\Omega)}^2 \le C \int_0^T \|u(t)\|_{L^2(\omega)}^2 \, dt. \qquad (3.16)$$

The critical time $T_c \in [0, +\infty]$ for the observability of (3.14) through $\partial\Omega$ is the infimum of the times $T > 0$ for which we have observability. In particular, (3.14) is not observable in time $T > 0$ through $\partial\Omega$ if $T < T_c$. Moreover, since (3.14) is dissipative (the norm of $u(t)$ is a non-increasing function of time), (3.14) is observable in any time $T > T_c$.

The main motivation is the null-controllability by the boundary for the adjoint problem. Given $f \in L^2(0, T; L^2(\partial\Omega))$ we consider the problem

$$\left\{ \begin{array}{ll} \partial_t u - q(y)^2 \partial_x u - \partial_{yy} u = 0, & \text{on } ]0, T[ \times \Omega, \\ u = f, & \text{on } ]0, T[ \times \partial \Omega, \\ u|_{t=0} = u_0 \in L^2(\Omega). \end{array} \right. \qquad (3.17)$$
Then we say that the Kolmogorov equation (3.17) is null-controllable by $\partial\Omega$ in time $T > 0$ if for any $u_0 \in L^2(\Omega)$ there exists $f \in L^2(0, T; L^2(\partial\Omega))$ such that the solution $u$ satisfies $u(T) = 0$. It is classical in control theory (see for instance [Cor07, Th. 2.44]) that the null-controllability for (3.17) is equivalent to the observability for (3.14).

This problem is similar to the better understood Grushin equation (see Section 4.2 below). For the Kolmogorov equation, it is proved in [Bea14] that (3.14) is observable through any open subset of $\Omega$ if $q(y)^2$ is replaced by $y$, and that there is a critical time $T_c \ge \frac{a^2}{2}$ for the observability when $q(y)^2 = y^2$ and $\omega = \mathbb{T} \times [a, b]$ for some $0 < a < b < \ell_+$. With the same $\omega$, it is also proved in [BHHR15] that (3.14) cannot be observable in any time for $q(y)^2$ replaced by $y^n$ with $n \ge 3$. There are also many results about parabolic equations generated by general quadratic operators, including the usual Kolmogorov equation (see for instance [BPS18] and references therein).

Our purpose in [22] is to prove for the Kolmogorov equation a result similar to the result of [BDE20] about the Grushin equation. The main result is the following.

**Theorem 3.10.** Let

$$T_{-} = \frac{1}{q'(0)} \int_{-\ell_{-}}^{0} |q(s)| \mathrm{d}s \quad \text{and} \quad T_{+} = \frac{1}{q'(0)} \int_{0}^{\ell_{+}} q(s) \mathrm{d}s.$$

There exists

$$T_c \in [\min(T_{-}, T_{+}), \max(T_{-}, T_{+})]$$

such that the Kolmogorov equation (3.14) is observable through $\partial\Omega$ in any time $T > T_c$, but not in time $T < T_c$.

We observe that we get exactly the critical time $T_c$ in any configuration for which $T_{-} = T_{+}$. This is in particular the case in a symmetric setting ($\ell_{-} = \ell_{+}$ and $q$ is odd).

Since the coefficients in (3.14) do not depend on $x$, we can consider the Fourier series of the solutions with respect to $x$. Setting

$$u(t, x, y) = \sum_{n \in \mathbb{Z}} u_n(t, y) e^{inx}, \quad u_n(t, y) = \frac{1}{2\pi} \int_{\mathbb{T}} e^{-inx} u(t, x, y) \mathrm{d}x,$$

the Fourier coefficients $u_n(t, y)$ are solutions of

$$\begin{cases} \partial_t u_n - \partial_{yy} u_n + inq(y)^2 u_n = 0, & \text{on } ]0, T[ \times I, \\ u_n(t, -\ell_-) = u_n(t, \ell_+) = 0, & \text{for } t \in ]0, T[, \\ u_n(0) \in L^2(I). \end{cases} \quad (3.18)$$

Then (3.15) holds if and only if we have observability for $u_n$ uniformly in $n \in \mathbb{Z}$. This means that there exists $C > 0$ such that for $n \in \mathbb{N}$ (the case $n$ negative follows by complex conjugation of (3.18)) and a solution $u_n$ of (3.18) we have

$$\|u_n(T)\|_{L^2(I)}^2 \le C \int_0^T (|\partial_y u_n(t, -\ell_-)|^2 + |\partial_y u_n(t, \ell_+)|^2) \mathrm{d}t. \quad (3.19)$$

### 3.5.2 Spectral properties of the Kolmogorov operator

The analysis of (3.18) depends on the spectral properties of the non-selfadjoint operator

$$K_n = -\partial_{yy} + inq(y)^2, \quad \text{with domain } \mathrm{Dom}(K_n) = H^2(I) \cap H_0^1(I) \subset L^2(I).$$

In particular we use the decay of the corresponding semigroup.
**Proposition 3.11.** Let

$$ \gamma < \frac{q'(0)}{\sqrt{2}}. $$

There exists $C > 0$ such that for $n \in \mathbb{N}$, a solution $u_n$ of (3.18) and $\theta_1, \theta_2 \in [0, T]$ with $\theta_1 \le \theta_2$, one has

$$ \|u_n(\theta_2)\|_{L^2(I)}^2 \le C e^{-2\gamma \sqrt{n}(\theta_2 - \theta_1)} \|u_n(\theta_1)\|_{L^2(I)}^2. $$

We notice that the semigroup $e^{-tK_n}$ is regularizing, even if it is not as small as the propagator of the heat equation for high frequencies.

We also give spectral properties for the operator $K = -\partial_y^2 + q(y)^2\partial_x$ on $\Omega$, and deduce that (3.14) is well posed, dissipative and regularizing (in particular the normal trace in (3.15) makes sense).

Near $y = 0$ (where the problem is degenerate) the coefficient $q(y)^2$ looks like $q'(0)^2y^2$. To prove Proposition 3.11, we compare $K_n$ with the Davies operator $H_n$ on $L^2(\mathbb{R})$:

$$ H_n = -\partial_y^2 + inq'(0)^2y^2, \quad \mathrm{Dom}(H_n) = \{u \in H^2(\mathbb{R}) : y^2u \in L^2(\mathbb{R})\}. $$

The basic spectral properties of $H_n$ are well known (see for instance [Hel13, §1.3]). The spectrum of $H_n$ is given by a sequence $(\lambda_{n,k})_{k \in \mathbb{N}^*}$ of (geometrically and algebraically) simple eigenvalues given by $\lambda_{n,k} = (2k-1)\sqrt{nq'(0)}e^{i\frac{\pi}{4}}$. As is often the case for a non-selfadjoint operator, the size of the resolvent of $H_n$ is far from being controled by the distance between the spectral parameter and the spectrum (see [PS06, HSV13, KSTV15]). However, given $\gamma < \frac{q'(0)}{\sqrt{2}}$ there exists $c > 0$ such that

$$ \sup_{\mathrm{Re}(z) \le \gamma \sqrt{n}} \|(H_n - z)^{-1}\|_{L(L^2(\mathbb{R}))} \le \frac{c}{\sqrt{n}}. \quad (3.20) $$

To recover similar properties for $K_n$, we compare the resolvents of $K_n$ and $H_n$. More precisely, with the natural restriction operator $1_I : L^2(\mathbb{R}) \to L^2(I)$ we have the following estimate.

**Proposition 3.12.** Let $\gamma < \frac{q'(0)}{2}$. Then for $n$ large enough the eigenvalues of $K_n$ have real parts greater than $\gamma\sqrt{n}$. Moreover,

$$ \sup_{\mathrm{Re}(z) \le \gamma \sqrt{n}} \|(K_n - z)^{-1} - 1_I(H_n - z)^{-1} 1_I^*\|_{L(L^2(I))} = o\left(\frac{1}{\sqrt{n}}\right). $$

We deduce Proposition 3.11 from Proposition 3.12 and the standard semigroup theory (see for instance [EN00, Th. V.1.11]).

Another consequence of Proposition 3.12 is that the spectrum of $K_n$ is in some sense “close” to the spectrum of $H_n$. More precisely, we are interested in the “smallest” eigenvalue. By perturbation theory, we get the following result.

**Proposition 3.13.** There exists a sequence $(\lambda_n)_{n \in \mathbb{N}}$ in $\mathbb{C}$ such that $\lambda_n$ is an eigenvalue of $K_n$ for all $n$ and

$$ \lambda_n = \sqrt{nq'(0)}e^{\frac{i\pi}{4}} + o\left(\sqrt{n}\right). \quad (3.21) $$

### 3.5.3 Positive result: observability in large time

To prove that we have observability in time $T > \max(T_-, T_+)$ we proceed in two steps. The first step consists in proving that (3.19) actually holds in arbitrarily small time. But not uniformly in $n$. For large $n$ we have the following estimate.
**Proposition 3.14.** Let $\tau_1, \tau_2 \in ]0, T]$ with $\tau_1 < \tau_2$ and

$$ \kappa > \max \left( \frac{1}{\sqrt{2}} \int_{0}^{\ell_{+}} q(s) \, ds, \frac{1}{\sqrt{2}} \int_{-\ell_{-}}^{0} |q(s)| \, ds \right) = \frac{q'(0)}{\sqrt{2}} \max(T_{-}, T_{+}). $$

There exist $n_0 \in \mathbb{N}$ and $C > 0$ such that for $n \ge n_0$ and a solution $u_n$ of (3.18) we have

$$ \|u_n(\tau_2)\|_{L^2(I)}^2 \le C e^{2\kappa\sqrt{n}} \int_{\tau_1}^{\tau_2} (|\partial_y u_n(t, -\ell_-)|^2 + |\partial_y u_n(t, \ell_+)|^2) \, dt. $$

The estimate is not uniform for high frequencies, but we control the dependence in $n$. Then, the second step consists in... waiting. We see from Proposition 3.11 that for large times the contribution of high frequencies is actually small. The time $\max(T_-, T_+)$ is precisely how long we have to wait to ensure that the smallness in Proposition 3.11 (applied with $\theta_2 = T$ and $\theta_1 = \tau_2$) compensates the bad estimate of Proposition 3.14 (applied with $\tau_2$ small enough).

Proposition 3.14 can be seen as a quantified version of a unique continuation result (in particular, if $u$ is a solution of (3.18) with $\partial_\nu u = 0$ on $[0, T] \times \partial I$ then $u(t) = 0$ for all $t \in [0, T]$). It is usual to use a Carleman estimate for this kind of result.

*Ideas of proof.* Given $n \in \mathbb{N}$ and a solution $u$ of (3.14) (we omit the index $n$) we set $w = e^{-\phi}u$ for some $\phi \ge 0$. We prove estimates on $w$ and then deduce estimates on $u$ (then $\phi$ should be chosen as small as possible).

If $\phi$ is large near $t = \tau_1$ and $t = \tau_2$, then $w$ satisfies the assumptions of the following proposition with $[a, b] = [-\ell_-, \ell_+]$ and $g = 0$.

**Proposition 3.15.** Let $n \in \mathbb{N}$, $\tau_1, \tau_2 > 0$ with $\tau_1 < \tau_2$, $a, b \in \mathbb{R}$ with $a < b$, and $g \in L^2([\tau_1, \tau_2] \times [a, b])$. Let $\phi \in C^4([\tau_1, \tau_2] \times [a, b], \mathbb{R}_+)$. We consider $w \in C^0([\tau_1, \tau_2], H^2(a, b)) \cap C^1([\tau_1, \tau_2], L^2(a, b))$ such that

$$ \partial_t w - \partial_y^2 w + \operatorname{inq}(y)^2 w + \partial_t \phi w - 2\partial_t \phi \partial_t w - (\partial_y \phi)^2 w - \partial_y^2 \phi w = g. \quad (3.22) $$

We assume that $w$ also satisfies the Dirichlet boundary condition $w(t, a) = w(t, b) = 0$ for $t \in [\tau_1, \tau_2[$, and the initial and final conditions $w(\tau_1, y) = w(\tau_2, y) = 0$ and $\partial_y w(\tau_1, y) = \partial_y w(\tau_2, y) = 0$ for $y \in ]a, b[$. Then we have

$$ \int_{\tau_1}^{\tau_2} \int_a^b (\Phi_0 |w|^2 + \Phi_1 |\partial_y w|^2) \, dy \, dt \le - \int_{\tau_1}^{\tau_2} [\partial_y \phi |\partial_y w|^2]_a^b \, dt + \frac{1}{2} \int_{\tau_1}^{\tau_2} \int_a^b |g|^2 \, dy \, dt, \quad (3.23) $$

where

$$ \Phi_0 = -2(\partial_y \phi)^2 \partial_y^2 \phi - \frac{\partial_t^2 \phi}{2} + \frac{\partial_y^4 \phi}{2} + 2\partial_t \partial_y \phi \partial_y \phi - \frac{n^{\frac{3}{2}} q^2 q'}{\sqrt{2}}, \quad \Phi_1 = -2\partial_y^2 \phi - \sqrt{2nq'} \quad (3.24) $$

With $g = 0$ we can deduce an observability estimate for $w$ if $\Phi_0$ and $\Phi_1$ are positive. Thus the purpose is to construct $\phi \ge 0$ as small as possible and such that $\Phi_0$ and $\Phi_1$ are bounded away from 0. We construct $\phi$ in such a way that the first terms in $\Phi_0$ and $\Phi_1$ are positive. Because of the last terms in the expressions of $\Phi_0$ and $\Phi_1$, $\phi$ should be at least of size $\sqrt{n}$. We construct $\phi$ of the form $\phi(t, y) = \sqrt{n\theta(t)}\psi(y)$. The second to fourth terms in $\Phi_0$ will be smaller for large $n$.

The function $\theta$ is chosen in such a way that $\theta \ge 1$ (otherwise the first term in each expression would be too small), $\theta$ goes to $+\infty$ near $\tau_1$ and $\tau_2$ (for the boundary conditions in Proposition 3.15) and $\theta = 1$ on $[\frac{2\tau_1+\tau_2}{3}, \frac{\tau_1+2\tau_2}{3}]$ (see (3.25) below). Then it remains to define $\psi$ to have, for some $\varepsilon > 0$,

$$ \psi \ge \varepsilon, \quad -2(\psi')^2 \psi'' - \frac{q^2 q'}{\sqrt{2}} \ge \varepsilon, \quad -2\psi'' - \sqrt{2}q' \ge \varepsilon. $$
We see that $\psi''$ has to be negative. Moreover, $|\psi'|$ has to be bounded away from 0, but the sign of $\psi'$ is not important. We also see from (3.23) that if $\psi'$ is positive it is enough to observe on the left, and that it is enough to observe from the right if $\psi'$ is negative. We could indeed observe from one side only, but it would be expansive to go through 0. Since we observe from both sides, we proceed as follows. Given $\varepsilon > 0$ and then $\delta > 0$ small enough, we apply Proposition 3.15 with $\psi' > 0$ on $[-\ell_-, \delta]$ and with $\psi' < 0$ on $[-\delta, \ell_+]$. More precisely, for $y \in [-\delta, \ell_+]$ we set

$$ \psi_+(y) = \left(\frac{1}{\sqrt{2}} + \varepsilon\right) \int_y^{\ell_+} (q(s) + \varepsilon) \, ds + c_+, $$

and for $y \in [-\ell_-, \delta]$

$$ \psi_-(y) = \left(\frac{1}{\sqrt{2}} + \varepsilon\right) \int_{\ell_-}^y (-q(s) + \varepsilon) \, ds + c_-. $$

The constants $\varepsilon > 0$ and $c_\pm \ge 0$ are chosen in such a way that $\psi_-(0) = \psi_+(0) \le \kappa$. Since $w$ does not vanish at $\pm\delta$, we have to use cut-off functions to apply Proposition 3.15 on $[-\ell_-, \delta]$ and $[-\delta, \ell_+]$. Then $g$ is no longer 0 in (3.23), but the corresponding term on the right can be absorbed by the left-hand side. Gluing the two estimates, we get for $u$

$$ \int_{\tau_1}^{\tau_2} \int_I (n^{\frac{3}{2}} |u|^2 + \sqrt{n} |u_y|^2) e^{-2\sqrt{n}\varphi} \, dy \, dt \le \sqrt{n} \int_{\tau_1}^{\tau_2} (|u_y(t, -\ell_-)|^2 + |u_y(t, \ell_+)|^2) \, dt. $$

What we need is an estimate on $u(\tau_2)$. However, since $u(\tau_2)$ is smaller than $u(s)$ for any $s \in [\tau_1, \tau_2]$, we have

$$ \|u(\tau_2)\|_{L^2(I)}^2 \le \frac{3e^{\kappa\sqrt{n}}}{\tau_2 - \tau_1} \int_{\frac{2\tau_1 + \tau_2}{3}}^{\frac{\tau_1 + 2\tau_2}{3}} \|e^{-\sqrt{n}\varphi}u(s)\|_{L^2(I)}^2 \, ds. \quad (3.25) $$

We only use the estimate for $s \in [\frac{2\tau_1 + \tau_2}{3}, \frac{\tau_1 + 2\tau_2}{3}]$ since otherwise $\theta$ (and hence $\phi$) can be large. Proposition 3.14 follows. $\square$

### 3.5.4 Negative result: non-observability in small time

In this paragraph, we discuss the non-observability for (3.14) when $T < \min(T_-, T_+)$. For this, we construct a sequence of solutions which contradicts (3.15). We need a solution of (3.14) which is small at the boundary. Since the (imaginary) potential $nq(y)^2$ is large away from 0 when $n$ is large, we use the Agmon estimates described in Section 3.2. Following [13], we prove the following estimate.

**Proposition 3.16.** Let $E > 0$ and $\varepsilon \in ]0, 1[$. For $n \in \mathbb{N}$ and $y \in \bar{I}$ we set

$$ W_{n,\varepsilon}(y) = \frac{1-\varepsilon}{\sqrt{2}} \left| \int_{0}^{y} \sqrt{(nq(s)^2 - \sqrt{n}(E+\varepsilon))}_{+} \, ds \right|. \quad (3.26) $$

There exists $C > 0$ such that for $n \in \mathbb{N}$, $u \in \text{Dom}(K_n)$ and $\lambda \in \mathbb{C}$ with $|\text{Re}(\lambda)| + |\text{Im}(\lambda)| \le E\sqrt{n}$, we have

$$ \|e^{W_{n,\varepsilon}u'}\|_{L^2(I)}^2 + \sqrt{n} \|e^{W_{n,\varepsilon}u}\|_{L^2(I)}^2 \le C\sqrt{n} \|u\|_{L^2(I)}^2 + \frac{C}{\sqrt{n}} \|e^{W_{n,\varepsilon}(K_n - \lambda)}u\|_{L^2(I)}^2. $$

For large $n$ we see that $W_{n,\varepsilon}(y)$ is close to $\sqrt{n}\kappa_\varepsilon(y)$, where

$$ \kappa_\varepsilon(y) = \frac{(1-\varepsilon)}{\sqrt{2}} \int_0^y q(s) \, ds. $$

We consider the eigenvalue $\lambda_n$ of $K_n$ given by Proposition 3.13 and a corresponding normalized eigenfunction $\psi_n$. Since $\psi_n'' = -\lambda_n\psi_n + inq(y)^2$, we can deduce the following estimate.
**Proposition 3.17.** There exists $C > 0$ such that for $n \in \mathbb{N}^*$ we have

$$ \|e^{\sqrt{n\kappa_{\varepsilon}}\psi'_{n}}\|_{L^{\infty}(I)}^{2} \leq Cn \|\psi_{n}\|_{L^{2}(I)}^{2}. $$

Finally, we set $u_m(t, x, y) = e^{-\lambda_m t}e^{imx}\psi_m(y)$, and we get a sequence of solutions of (3.14) for which (3.15) cannot hold if $T < \min(T_-, T_+)$.

## 3.6 Nonlinear evolution equations with singular potentials

### 3.6.1 A singularly perturbed Gross-Pitaevskii equation

Most of my works concern spectral theory and linear PDEs. I have also developed with Stefan Le Coz a collaboration around some non-linear problems. The starting point was a question of Stefan about the spectral properties of the linearized operator which appeared in one of his problems. We discussed more and more about it and, one thing leading to another, I was aware of the whole problem and I finally joined the project.

The purpose of [12] (with Isabella Ianni and Stefan Le Coz) is the analysis of the black solitons of the Gross-Pitaevskii equation with Dirac potential

$$ -i\partial_t u - \partial_x^2 u + \gamma\delta u = (1 - |u|^2)u, \qquad (3.27) $$

subject to the boundary condition

$$ |u(t, x)| \underset{|x| \to \infty}{\longrightarrow} 1, \quad \forall t \in \mathbb{R}. \qquad (3.28) $$

The difficulty is then twofold. Technical problems arise from the presence of the singular potential and the non standard condition at infinity. For instance, one of the difficulties due to (3.28) is that we have to work in a functional space which is not even a vector space:

$$ \mathcal{E} = \{u \in H_{\text{loc}}^{1}(\mathbb{R}) : \partial_{x}u \in L^{2}(\mathbb{R}), (1 - |u|^{2}) \in L^{2}(\mathbb{R})\}. $$

It can however be endowed with a complete metric structure with the distance

$$ d_0(u, v) = \|\partial_x u - \partial_x v\|_{L^2(\mathbb{R})} + \| |u|^2 - |v|^2 \|_{L^2(\mathbb{R})} + |u(0) - v(0)|. $$

This work has been an occasion to get familiar with questions and technics of non-linear analysis. The question here is the stability of the stationary solutions (or black solitons), solutions of

$$ -\partial_x^2 u + \gamma \delta u = (1 - |u|^2)u. $$

The solutions can be explicit. Up to phase shift, they are given by

$$ \kappa(x) = \tanh\left(\frac{x}{\sqrt{2}}\right), \quad b_{\gamma}(x) = \tanh\left(\frac{|x| - c_{\gamma}}{\sqrt{2}}\right), \quad \tilde{b}_{\gamma}(x) = \coth\left(\frac{|x| + c_{\gamma}}{\sqrt{2}}\right), $$

where $c_\gamma$ is an explicit parameter ($\tilde{b}_\gamma$ is only defined for $\gamma < 0$). We get a first result of stability by minimization of the energy

$$ E(u) = \frac{1}{2} \int_{\mathbb{R}} |\partial_x u|^2 dx + \frac{1}{4} \int_{\mathbb{R}} (1 - |u|^2)^2 dx + \frac{\gamma}{2} |u(0)|^2. \qquad (3.29) $$

**Theorem 3.18.** Let $\gamma \in \mathbb{R}^*$.

(i) The energy $E$ reaches its (finite) minimum on

$$ \mathcal{G}_{\gamma} = \begin{cases} \{e^{i\theta} b_{\gamma}, \theta \in \mathbb{R}\} & \text{if } \gamma > 0, \\ \{e^{i\theta} \tilde{b}_{\gamma}, \theta \in \mathbb{R}\} & \text{if } \gamma < 0. \end{cases} $$
(ii) The set $\mathcal{G}_\gamma$ is stable for the flow of (3.27):

$$ \forall \varepsilon > 0, \exists \delta > 0, \forall u_0 \in \mathcal{E}, \quad d_0(u_0, \mathcal{G}_\gamma) < \delta \implies \sup_{t \in \mathbb{R}} d_0(u(t), \mathcal{G}_\gamma) < \varepsilon. $$

The other stationary solutions are expected to be unstable. In [12] we prove the linear instability of $\kappa$. If we write $u = \kappa + \eta$, then the perturbation $\eta$ satisfies an equation of the form $\partial_t\eta + L\eta + N(\eta) = 0$ where $N(\eta)$ is non-linear in $\eta$ and

$$ L\eta = -i(\partial_x^2\eta - \gamma\delta\eta + (1 - \kappa^2)\eta - 2\kappa^2\mathrm{Re}(\eta)). $$

Then $\kappa$ is said to be linearly unstable if 0 is an unstable solution of the linear equation $\partial_t\eta + L\eta = 0$.

**Theorem 3.19.** If $\gamma > 0$ then $\kappa$ is linearly unstable.

The proof relies on the spectral analysis of the $\mathbb{R}$-linear operator $L$. We can rewrite $L$ as a $\mathbb{R}$-linear matrix operator acting on $(\mathrm{Re}(\eta), \mathrm{Im}(\eta))$:

$$ \mathcal{L} = \begin{pmatrix} 0 & -L_{-}^{\gamma} \\ L_{+}^{\gamma} & 0 \end{pmatrix}, \quad L_{-}^{\gamma} = -\partial_{x}^{2} - (1 - \kappa^{2}), \quad L_{+}^{\gamma} = -\partial_{x}^{2} + 2 - 3(1 - \kappa^{2}). $$

The operators $L_-$ and $L_+$ are defined on the same domain

$$ \mathcal{D}_{\gamma} = \{u \in H^2(\mathbb{R}^*) \cap H^1(\mathbb{R}) : u'(0^+) - u'(0^-) = \gamma u(0)\}. \quad (3.30) $$

Then a careful spectral analysis of $L_-$ and $L_+$ shows that $L$ has a negative eigenvalue whenever $\gamma > 0$.

Unexpectedly, the main difficulty and hence the main part in [12] turned out to be the analysis of the Cauchy problem (see already [Gér06] without singular potential). Before all the discussion above, we check that the problem (3.27) is globally well posed and that the energy $E$ is conserved. The first step is the analysis of the propagator of the linear part, generated by the operator associated with the quadratic form $u \mapsto \|u'\|_{L^2(\mathbb{R})}^2 + \gamma|u(0)|^2$. It is given by

$$ H_{\gamma} = -\partial_{x}^{2}, \quad \mathrm{Dom}(H_{\gamma}) = \mathcal{D}_{\gamma}. $$

The difficulty is to deal with this not so usual operator in the completely unusual space $\mathcal{E}$. By Fourier transform we can see that $e^{-itH_0} - \mathrm{Id}$ maps $\mathcal{E}$ to $H^1(\mathbb{R})$. Since $\mathcal{E} + H^1(\mathbb{R}) = \mathcal{E}$, we deduce that $e^{-itH_0}$ maps $\mathcal{E}$ into itself.

We cannot use the Fourier transform for $e^{-itH_\gamma}$ when $\gamma \neq 0$. Instead, we compute explicitly the kernel of $\Gamma(t) = e^{-itH_\gamma} - e^{-itH_0}$. For instance, for $\gamma < 0$ it is given by

$$ \Gamma(t; x, y) = -\frac{|\gamma|}{2} \int_{0}^{+\infty} e^{-\frac{|\gamma|s}{2}} K_{0}(t, s - |x| - |y|) ds + \frac{|\gamma|}{2} e^{\frac{i\gamma^2 t}{4}} e^{-\frac{|\gamma|(|x|+|y|)}{2}}, $$

where $K_0(t, \zeta)$ is the kernel of the free Schrödinger equation. By tedious computation, we can show in particular that $\Gamma(t)$ maps $\mathcal{E}$ to $H^1(\mathbb{R})$ (and $\|\Gamma(t)u\|_{H^1(\mathbb{R})}$ is controled by $\|\partial_x u\|_{L^2(\mathbb{R})} + |u(0)|$).

Conservation of the energy is also a non-trivial issue. Finally, the analysis of the Cauchy problem follows the standard strategy but, again, this standard strategy had to be adapted to our non standard setting. We finally have a well-posedness result.

**Theorem 3.20.** For $u_0 \in \mathcal{E}$ there exists a unique solution $u \in C^0(\mathbb{R}, \mathcal{E})$ of (3.27) such that $u(0) = u_0$. Moreover $E(u(t)) = E(u_0)$ for all $t \in \mathbb{R}$.

Here, a solution is in the sense given by the Duhamel formula. For a more regular $u_0$ we have a solution in a stronger sense (in particular with $\partial_t u(t) \in C^0(\mathbb{R}, L^2(\mathbb{R}))$). We also have a result of continuity with respect to the initial condition, but for a slightly different distance
on $\mathcal{E}$.

Notice that a problem on $\mathbb{R}$ with singular potential can be seen has a problem on a graph with two edges of infinite length. The jump condition which appears in (3.30) is then the analog of the Robin condition at the unique vertex (see (3.3)).

### 3.6.2 A nonlinear Klein-Gordon equation with delta potentials

After [12], we started to discuss with François Genoud and Elek Csobo (who was at that time Ph.D. student under the supervision of F. Genoud and S. Le Coz) another problem of stability for standing waves, suggested by a question of Masahito Ohta.

We now consider a non-linear Klein-Gordon equation, again on the real line with a singular potential. We also consider a singular term involving $\partial_t u$. This looks like a singular damping, but with a purely imaginary (conservative) coefficient. More precisely, given $m > 0$ and $\gamma, \alpha \in \mathbb{R}$, the problem considered in [19] is

$$ egin{cases} 
\partial_t^2 u - \partial_x^2 u + m^2 u + \gamma \delta u + i\alpha \delta \partial_t u - |u|^{p-1} u = 0, & \text{on } \mathbb{R}_+ \times \mathbb{R}, \\ 
u(t, x) \underset{|x| \to \infty}{\longrightarrow} 0, & \forall t > 0, \\ 
(u, \partial_t u)|_{t=0} = (f, g) \in H^1(\mathbb{R}) \times L^2(\mathbb{R}). 
\end{cases} 
\quad (3.31) $$

The first step in the analysis of (3.31) is again the Cauchy problem. Here there is a subtil- ity due to the term $\gamma \delta u$, which is not usual for a wave equation. The operator corresponding to the linear part is

$$ W = \begin{pmatrix} 0 & 1 \\ \partial_x^2 - m^2 & 0 \end{pmatrix}, $$

with domain

$$ \mathrm{Dom}(W) = \{ (u, v) \in (H^2(\mathbb{R}^*) \cap H^1(\mathbb{R})) \times H^1(\mathbb{R}) : u'(0^+) - u'(0^-) = \gamma u(0) + i\alpha v(0) \}. $$

This defines a skew-adjoint operator on the space $\mathcal{H} = H^1(\mathbb{R}) \times L^2(\mathbb{R})$, endowed with the norm defined by

$$ \|(u, v)\|_{\mathcal{H}, m, \gamma}^2 = \|u'\|_{L^2(\mathbb{R})}^2 + m^2 \|u\|_{L^2(\mathbb{R})}^2 + \gamma |u(0)|^2 + \|v\|_{L^2(\mathbb{R})}^2. $$

The problem is that if $\gamma$ is too negative, this norm is not a norm. In this case, we replace $m$ by some $\mu \ge m$ large enough. We lose the skew-adjointness, but $W$ still generates on $(\mathcal{H}, \|\cdot\|_{\mathcal{H}, \mu, \gamma})$ a strongly continuous semigroup. Then we can proceed with the local well-posedness for (3.31). Moreover, the following energy is a constant of the motion for $(u, v) = (u, \partial_t u)$:

$$ E(u, v) = \frac{1}{2} \|\partial_x u\|_{L^2(\mathbb{R})}^2 + \frac{m^2}{2} \|u\|_{L^2(\mathbb{R})}^2 + \frac{1}{2} \|v\|_{L^2(\mathbb{R})}^2 + \frac{\gamma}{2} |u(0)|^2 - \frac{1}{p+1} \int_{\mathbb{R}} |u|^{p+1} \, dx. $$

The charge is another constant of the motion:

$$ Q(u, v) = \mathrm{Im} \int_{\mathbb{R}} u \bar{v} \, dx - \frac{\alpha}{2} |u(0)|^2. $$

The main issue of [19] is then the question of orbital stability for the standing waves. For this we follow the general theory of Grillakis, Shatah and Strauss [GSS87].

A standing wave is a solution of (3.31) of the form $e^{i\omega t}\Phi_\omega(x)$, where $\Phi_\omega = (\varphi_\omega, i\omega\varphi_\omega)$ is a solution of the stationary equation

$$ E'(\Phi_\omega) + \omega Q'(\Phi_\omega) = 0. $$
Non-trivial solutions exist if and only if $m^2 - \omega^2 > (\gamma - \alpha\omega)^2/4$. A standing wave $e^{i\omega t}\Phi_\omega(t)$ is said to be orbitally stable if for any $\varepsilon > 0$ there exists $\delta > 0$ such that the following property holds for any solution $U(t)$ of (3.31):

$$ \|U(0) - \Phi_{\omega}\|_{\mathcal{H}} \le \delta \implies \sup_{t} \inf_{\theta \in \mathbb{R}} \|U(t) - e^{i\theta}\Phi_{\omega}\|_{\mathcal{H}} \le \varepsilon. $$

This property implies in particular that a solution starting close to $\Phi_\omega$ is defined for all $t \in \mathbb{R}$.

The method is based on the symplectic structure of the problem (see [DBGRN15]). We can rewrite (3.31) in the form

$$ \mathcal{J}U'(t) = E'(U(t)), \quad U(t) = (u(t), \partial_t u(t)), $$

where $\mathcal{J}(u, v) \in \mathcal{H} \mapsto (-i\alpha\delta v, u) \in \mathcal{H}^*$ is skew-symmetric.

Then the criterion for orbital stability of the standing wave $e^{it\omega_0}\Phi_{\omega_0}$ depends on two quantities. The first is the sign of

$$ \left. \frac{dQ(\Phi_{\omega})}{d\omega} \right|_{\omega=\omega_0}. $$

Since $\Phi_\omega$ is explicitly known, this can be analysed by tedious but straightforward computations. To conclude to orbital stability or instability (we also discuss linear stability and instability), we also need some spectral properties for the operator $\tilde{R}^{-1}L_\omega''(\Phi_\omega)$, where $\tilde{R}: \mathcal{H} \to \mathcal{H}^*$ is defined by $(-\partial_x^2 + 1, \mathrm{Id}_{L^2})$ and $L_\omega = E + \omega Q$. I do not detail the assumptions here, but it concerns the positiveness of essential spectrum and the number of negative eigenvalues.

This is the main part of the paper [19]. However, all these properties depend on the parameters $\gamma$, $\alpha$ and $\omega$ (the strength $p$ of the non-linearity also plays a role), so the results of stability and instability have to be split into many different cases and I choose not to give the precise statements here. See Section 4 in [19].
# Chapter 4

# Projects and perspectives

After having discussed my past research in the previous chapters, the remaining part of this thesis is devoted to the present and the future. In this last chapter I present unpublished (submitted) results, works in progress and some longer term perspectives.

Some perspectives discussed here are parts of ongoing projects and can be continuations of results already mentioned above, but this thesis is also an occasion to propose new directions of research.

## 4.1 New questions about local energy decay

We begin with the projects which are close to the main subject of this thesis, namely the local energy decay for wave or Schrödinger equations.

### 4.1.1 Asymptotic profile for the wave equation

In [23] (see the discussion in Section 1.7), it is proved that the solution of the Schrödinger equation (1.13) behaves for large times like a solution of the free problem (1.6), in the sense of Theorem 1.25. There are two natural continuations for this work.

These two questions, though quite different, are expected to be related to the same difficulty. This difficulty is in fact already present in [23]. The time decay in Theorem 1.25 depends on the parameter $ρ_0$ which measures the spatial decay of the coefficients in (1.10). This parameter is assumed to be not greater than 1. It is not the purpose of this report to go into technical details, but the reason is that the smallness given by the decay of the coefficients of the operators $θ_σ(z)$ in (1.71) is limited (as always because of the restriction of the Sobolev indices in Proposition 1.19), and our proof would not give a better result with $ρ_0 > 1$. In [23] this is not a problem since we are mostly interested in the case $ρ_0 > 0$ small, but this becomes a more serious problem for the following two perspectives.

The main perspective in that direction is to prove an analog of Theorem 1.25 for the wave equation (1.12). The proof of Theorem 1.25 is written in such a way that it is robust with respect to dissipative perturbations, and adding a damping should not be a major problem. However, even for the undamped case, the asymptotic profile is not completely clear, and the problem comes from the gradient of the solution.

The method of [23] should work for the solution itself and its time derivative. We expect that they should look like a solution of the free wave equation (1.1). From (1.21) we can guess what should be the initial condition ($f_0, g_0$) for the asymptotic profile. However, there is an additional difficulty for the gradient of the solution. We have said above that we cannot gain as much as we wish with the inserted factors. Here, because of the gradient, the method does not give anything at all. There is something more to understand. In particular, we could be
in the same situation as for Theorem 2.19, where the asymptotic profile for the gradient is
not the gradient of the asymptotic profile.

The motivation for having such a result for the wave equation is twofold. In even dimension, we already have the optimal decay in [BB21] for the undamped case (for the solution itself but not for its derivatives), so a result like Theorem 1.25 would give the leading term of the asymptotic expansion (which would be new in any case) and in particular the optimal decay (which is not known in the dissipative case). This would already be an important improvement.

In odd dimension, the situation is different. The proofs of resolvent estimates and local energy decay based on the Mourre method do not see the parity of the dimension, so the results in odd dimensions are actually limited. Comparing the solution of the perturbed problem to the solution of the model case (which decays much faster in odd dimensions) is a way to reintroduce the difference of parity in the method, which is necessary to improve the results in odd dimensions. A generalization of [23] to the wave equation would improve the estimates of [BB21] in odd dimensions, even for the undamped equation.

Another natural continuation of [23] is to give a more precise asymptotic expansion for the solution of the Schrödinger equation (1.13). In [23], we only give the first term of the expansion. It could be interesting to go further.

The leading term for the resolvent is simply given by the resolvent identity (1.70). We can apply the resolvent identity once more to get rests with two factors $\theta_{\sigma}(z)$. The same strategy should work to estimate these rests, and we can iterate as many times as we wish. In fact, the restriction in the proof of [23] is not about the number of terms in the asymptotic expansion, but about the size of the rest. As above, we could probably consider $N$ factors $\theta_{\sigma}(z)$ in the rests, but we would not get an optimal result if $N\rho_0 \ge 1$. Going further would require a new improvement in the proof.

Finally, a related question is to consider the settings of Chapter 2. With damping at infinity we have already worked by comparison with a model problem, this is even what motivated [23] and the perspectives above. But I have never considered a general metric perturbation of the Laplacian. We have proved a result for a perturbed setting in [17] (see Proposition 2.20), but the assumptions on the perturbation are too strong. This is something I have not tried yet, but considering a long range perturbation of (2.1) would probably raise the same kind of difficulties as the long range perturbation of (1.12).

We could also consider all the intermediate situations (see the introduction of Chapter 2). In particular, not much is known and I am still curious about the critical case $a(x) \simeq C \langle x \rangle^{-1}$.

We can also consider problems with a damping going to infinity (see [FST18] for some basic properties of the wave operator in this case).

### 4.1.2 Systems on non-selfadjoint equations

I have recently started to consider questions of energy decay for a system of two equations. So far I have only considered model problems, but this kind of systems naturally appear in physics. For instance, there are already results about the energy decay for the Lamé system in [BL01, DDK10].

**System of Schrödinger equations on a wave guide** In 2019, I invited Radhia Ayechi and Ilhem Boukhris for two months in Toulouse. They were at that time Ph.D. students in Sousse with Moez Khenissi. I suggested to look together at a generalization of [8] (see Section 2.7). Then 2020 was a complicated time, but Ilhem came one more month. We finally finished the work at the end of 2021, which gave the preprint [24].

The setting in [24] is as in [8] a straight wave guide with one-dimensional cross section: $\Omega = \mathbb{R}^{d-1} \times ]0, \ell[$ $\subset \mathbb{R}^d$ for some $\ell > 0$. We consider on $\Omega$ a system of two Schrödinger equations with damping and coupling at the boundary. Here they are only effective on $\mathbb{R}^{d-1} \times \{0\}$, and
we consider a Neumann boundary condition on the other side (as in [8], various situations could be similarly considered). Given $a > 0$ (the absorption coefficient) and $b \in \mathbb{R}^*$ (the coupling coefficient), we consider the problem

$$ \begin{cases} -i\partial_t u - \Delta u = 0, & \text{on } \mathbb{R}_+ \times \Omega, \\ -i\partial_t v - \Delta v = 0, & \text{on } \mathbb{R}_- \times \Omega, \end{cases} \qquad (4.1) $$

with boundary conditions

$$ \left\{ \begin{aligned} \partial_{\nu} u(t; x, 0) &= iau(t; x, 0) + ibv(t; x, 0), \\ \partial_{\nu} v(t; x, 0) &= -ibu(t; x, 0), \\ \partial_{\nu} u(t; x, \ell) &= \partial_{\nu} v(t; x, \ell) = 0, \end{aligned} \quad \forall t > 0, \forall x \in \mathbb{R}^{d-1} \right. \qquad (4.2) $$

and with initial data

$$ (u, v)|_{t=0} = (f, g) \in L^2(\Omega) \times L^2(\Omega). \qquad (4.3) $$

Notice that only the first component $u$ is dissipated at the boundary. However, the second component $v$ is indirectly dissipated through the coupling terms. The masses of $u$ and $v$ separately have no reasons to be non-increasing, but it is the case for their sum:

$$ \frac{d}{dt} ( \|u(t)\|_{L^2(\Omega)}^2 + \|v(t)\|_{L^2(\Omega)}^2 ) = -2a \int_{\mathbb{R}^{d-1}} |u(t; x, 0)|^2 dx \le 0. $$

The question is the decay of this total mass. The answer is that even if neither the damping nor the coupling satisfies the geometric condition (the rays of light parallel to the boundary do not see the boundary), and even if the second component $v$ is not dissipated, we have exponential decay for both components.

**Theorem 4.1.** Let $a > 0$ and $b \in \mathbb{R}^*$. There exist $\gamma > 0$ and $C > 0$ such that for $(f, g) \in L^2(\Omega) \times L^2(\Omega)$ and $t > 0$ we have

$$ \|u(t)\|_{L^2(\Omega)}^2 + \|v(t)\|_{L^2(\Omega)}^2 \le C e^{-\gamma t} (\|f\|_{L^2(\Omega)}^2 + \|g\|_{L^2(\Omega)}^2), $$

where $(u, v)$ is the solution of (4.1)-(4.3).

As in [8], the analysis on the wave guide is related to the properties on $L^2(0, \ell) \times L^2(0, \ell) \simeq L^2(0, \ell; \mathbb{C}^2)$ of the transverse operator

$$ \mathcal{T}_{a,b} = \begin{pmatrix} -\partial_t^2 & 0 \\ 0 & -\partial_t^2 \end{pmatrix}, $$

defined on the domain

$$ \{U \in H^2(0, \ell; \mathbb{C}^2) : U'(0) + iM_{a,b}U(0) = 0, \quad U'(\ell) = 0\}, \quad M_{a,b} = \begin{pmatrix} a & b \\ -b & 0 \end{pmatrix}. $$

We prove that we have a spectral gap for the eigenvalues of $\mathcal{T}_{a,b}$, and that we have a Riesz basis of $L^2(0, \ell; \mathbb{C}^2)$ made with corresponding (generalized) eigenfunctions.

In [8], the eigenvalues $(\lambda_n(0))$ of $T_0$ are simple, and a square root $z_n(a)$ of $\lambda_n(a)$ cannot cross the lines $\mathrm{Re}(z) \in \mathbb{N}\nu$ (remember that $\nu = \pi/\ell$), so by continuity it is possible to conclude that the eigenvalues of $T_a$ remain simple with a rough localization.

Here it is no longer that simple. The model operator $\mathcal{T}_{0,0}$ has double eigenvalues $(n\nu)^2$, $n \in \mathbb{N}$. We can check that for $(a, b)$ small the eigenvalue $(n\nu)^2$ splits into two simple eigenvalues if $a^2 \ne 4b^2$ and it stays a double eigenvalue if $a^2 = 4b^2$ (in this case we have a Jordan block if $a \ne 0$). Moreover, the square roots of the eigenvalues can cross the lines $\mathrm{Re}(z) \in \nu\mathbb{N}$ and even for large $n$ the (generalized) eigenfunctions are not close to being orthogonal. However, despite of these additional difficulties, we can prove that we indeed have a spectral gap
and the Riesz basis property.

As a byproduct of this analysis, we also deduce the Weyl Law for the eigenvalues of $\mathcal{T}_{a,b}$. We can see $\mathcal{T}_{a,b}$ as a Schrödinger operator on a graph with two edges of same length $\ell$, with a non-standard vertex condition between them. We denote by $\mathcal{N}_{a,b}(R)$ the number of eigenvalues of $\mathcal{T}_{a,b}$ (counted with multiplicities) with real part smaller than $R$. Then the following result is analogous to Theorem 3.2 with $|\Gamma| = 2\ell$:

$$ \mathcal{N}_{a,b}(R) = \frac{2\ell\sqrt{R}}{\pi} + \mathcal{O}(1). \quad (4.4) $$

Even if it was not necessary for the proof of Theorem 4.1 or for (4.4), we have continued the analysis of the localization of the eigenvalues of $\mathcal{T}_{a,b}$, and in particular the question of the number of square roots of eigenvalues in the region $\mathbb{C}_n = \{\mathrm{Re}(z) \in \mathbb{C} : |\mathrm{Re}(z)| \le n\nu, (n+1)\nu \le \mathrm{Im}(z)\}$. We prove in particular the following observations.

* $(n\nu)^2$ is a double eigenvalue of $\mathcal{T}_{0,0}$. Its square root $n\nu$ splits into $z_+(a, b)$ and $z_-(a, b)$ such that $z_+(a, b)^2$ and $z_-(a, b)^2$ are eigenvalues of $\mathcal{T}_{a,b}$.

* One (say $z_+(a, b)$) always belongs to $\mathbb{C}_n$.

* If $a^2 > 4b^2 > 0$, then it is also the case for $z_-(a, b)$ and $z_-(a, b) \ne z_+(a, b)$. The eigenvalues of $\mathcal{T}_{a,b}$ are simple.

* If $a^2 = 4b^2$, then $z_+(a, b) = z_-(a, b)$ and the eigenvalues of $\mathcal{T}_{a,b}$ have algebraic multiplicities 2 (but geometric multiplicities 1).

* If $a^2 < 4b^2$, then $z_-(a, b)$ can go to $\mathbb{C}_n$ or $\mathbb{C}_{n-1}$ depending on the value of $\theta \in ]\frac{a}{2\nu}, +\infty[$ such that

$$ 4b^2 = a^2 + \frac{a^2}{4\theta^2\pi^2} \ln \left( \frac{2\theta\nu + a}{2\theta\nu - a} \right)^2. $$

Then $z_-(a, b)$ is in $\mathbb{C}_{n-1}$ if $\theta < n$, in $\mathbb{C}_n$ if $\theta > n$ and $\mathrm{Re}(z_-(a, b)) = n$ if $\theta = n$.

This kind of additional remarks is motivated by curiosity (this is already a good reason), but not only. For Theorem 4.1 it is enough to understand the eigenvalues $\lambda_n(a, b)$ for $n \gg a + |b|$, but this is not the case for the wave equation. Thus such a localization of all the eigenvalues is an interesting step toward the analysis of the wave equation.

A natural continuation of this work would be to consider a system of wave equations on a wave guide. We will discuss this issue in the Euclidean space in the next paragraph. But, as will be discussed in Section 4.4.2, we are also interested in the wave equation in general quantum graphs.

It would be interesting to prove rough localization of the eigenvalues on a general compact quantum graphs. To get a better intuition, we could begin by computing what happens for the same problem with $N$ equations, or considering only 2 edges but with a general non-selfadjoint vertex. We will continue this discussion in Section 4.4.2.

**System of wave equations** In Chapter 2, we have discussed several problems about the wave equation with damping at infinity. Each situation (Euclidean space, wave guide, periodic setting) raised different and challenging difficulties. A new question, which turns out to be again different and challenging, is the case of a system of several equations with coupling.

With Lassaad Aloui and Moez Khenissi, we have started to discuss the simplest case, namely a system of two wave equations with constant coupling and damping on $\mathbb{R}^d$. More precisely, given $a > 0$ and $b \in \mathbb{R}^*$ as above, we consider on $\mathbb{R}^d$ the system

$$ \begin{cases} \partial_t^2 u - \Delta u + m_1 u + b \partial_t v + a \partial_t u = 0, \\ \partial_t^2 v - \Delta v + m_2 v - b \partial_t u = 0, \end{cases} \quad (4.5) $$
with initial conditions $(u, \partial_t u)|_{t=0} = (f_0, f_1)$ and $(v, \partial_t v)|_{t=0} = (g_0, g_1)$ in $H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. The coefficients $m_1$ and $m_2$ are non-negative. When $m_j = 0$ the corresponding equation is a wave equation, while $m_j > 0$ gives a Klein-Gordon equation.

Compared to the previous case, we have damping and coupling everywhere. However, as above, only the first component is dissipated and $v$ can only be dissipated via the coupling.

We look at the decay of the energy

$$E(u, v; t) = \|\partial_t u(t)\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla u(t)\|_{L^2(\mathbb{R}^d)}^2 + m_1 \|u(t)\|_{L^2(\mathbb{R}^d)}^2 \\ + \|\partial_t v(t)\|_{L^2(\mathbb{R}^d)}^2 + \|\nabla v(t)\|_{L^2(\mathbb{R}^d)}^2 + m_2 \|v(t)\|_{L^2(\mathbb{R}^d)}^2.$$

The first result is that the contributions of high frequencies for the global energy decay uniformly and exponentially, so both components are dissipated.

For low frequencies, the wave and Klein-Gordon equations have different properties, and the four situations ($m_1 = 0/m_1 \neq 0$ and $m_2 = 0/m_2 \neq 0$) will give four completely different behaviors.

When $m_1 > 0$ and $m_2 > 0$, there is no difficulty with low frequencies and the global energy of $u$ and $v$ decays exponentially. On the other hand, if $m_1 = 0$ or $m_2 = 0$ then 0 belongs to the spectrum of the corresponding operator

$$W = \begin{pmatrix} 0 & 1 & 0 & 0 \\ \Delta - m_1 & -a & 0 & -b \\ 0 & 0 & 0 & 1 \\ 0 & b & \Delta - m_2 & 0 \end{pmatrix},$$

defined on the Hilbert completion $\mathcal{E}$ of $S(\mathbb{R}^d)^4$ for the norm

$$\|(u_0, u_1, v_0, v_1)\|_{\mathcal{E}}^2 = \|\nabla u_0\|_{L^2}^2 + m_1 \|u_0\|_{L^2}^2 + \|u_1\|_{L^2}^2 + \|\nabla v_0\|_{L^2}^2 + m_2 \|v_0\|_{L^2}^2 + \|v_1\|_{L^2}^2.$$

We have to understand the contribution of low frequencies.

As in Section 2.2 we can use the Fourier transform. Let $M(\xi)$, $\xi \in \mathbb{R}^d$, be defined as $W$ with $-\Delta$ replaced by $|\xi|^2$ (as in (2.7)). For a system Wave-Klein-Gordon ($m_1 = 0, m_2 > 0$), 0 is a simple eigenvalue of $M(0)$ and we recover a behavior similar to the case of a single equation. In particular, we can observe that $u$ behaves like a solution of the heat equation. If $m_2 = 0$ then 0 is a double eigenvalue of $M(0)$. The behavior of the contribution of low frequencies is governed by the behavior of these two eigenvalues (counted with multiplicities) for $\xi \neq 0$ small (as well as the corresponding eigenprojections). And this strongly depends on the parameters $m_1, m_1, a$ and $b$. We should get the asymptotic profile in each case, but with a much wilder behavior than with a single equation.

We have already four parameters in all the computations, but we could (should) consider another one. Everywhere, we have considered the speed of propagation $c$ normalized to 1. This is harmless for a single equation, but for (4.5) this means that we consider the same speed for the two components $u$ and $v$. It would be interesting to introduce the quotient of these two speeds and see how the results would depend on this new parameter.

The problem (4.5) is the simplest possible system of wave equations on an unbounded domain. It will then be natural to consider more evolved settings, as is done for a single equation, and systems of equations which appear in concrete problems.

### 4.1.3 Local energy decay in more general geometric settings

Most of my works about the damped wave equation are set on asymptotically Euclidean settings. I also considered the problem on a wave guide and on an asymptotically periodic setting, and this highlighted new interesting phenomena.

In [7] (see Section 1.5) we have proved resolvent estimates for the Schrödinger equation on an asymptotically conical manifold. It would be interesting to generalize the results about the wave equation to this setting.
Recently, V. Grasselli has proved in this setting all the results of [BB21]. See [Gra]. Thus, she proved in this context low frequency resolvent estimates and the corresponding local energy decay for the Schrödinger and wave equations. In a similar context, R. Wang [Wana] has also generalized the results of [BJ16] about the exponential decay for the Klein-Gordon equation.

Concerning the Schrödinger equation, the obvious question is the adaptation of the result of [23] to this setting. Is it true that solution of the Schrödinger equation on an asymptotically conical manifold $M$ looks like (in the sense of Theorem 1.25) the solution of the free Schrödinger equation on the corresponding conical model $M_0$ for large times ? And how does the geometry affect the initial condition for the asymptotic profile ?

Similarly, a result analogous to those of Chapter 2 could be proved on asymptotically conical manifolds with damping at infinity, or on more general wave guides. And, again, it would be interesting to see what would be the asymptotic profiles in these settings.

The results on the asymptotically conical manifold are expected to be proved with the techniques developed in this setting on the one hand, and the strategy developed for the Euclidean setting on the other hand.

A less marked path would be to consider more general space-time geometries, in the spirit of the recent papers [MST20] and [Kof] (the latter being in some sense a generalization of [6] on a Lorentzian manifold).

## 4.2 Null-controllability for a Grushin equation

In 2021, a few months after the publication of [22] about observability for a Kolmogorov equation (see Section 3.5), Armand Koenig arrived as a post-doc in Toulouse. Armand had already several results on similar problems (see [Koe17, DK20]), so it was natural to continue this study together with Jérémi Dardé. In this paragraph, we discuss the preprint [25] and further perspectives on the subject.

Let $T > 0$, $\ell_\pm > 0$, $I = ] -\ell_-, \ell_+[$ and $\Omega = I \times \mathbb{T}$. Given an open subset $\omega$ of $\Omega$, we are interested in the null-controllability for the following Baouendi-Grushin equation

$$ \left\{ \begin{array}{ll} \partial_t u - \partial_x^2 u - q(x)^2 \partial_y^2 u = \mathbb{1}_\omega f, & \text{on } ]0, T[ \times \Omega, \\ u(t, \cdot) = 0, & \text{on } \partial\Omega, \text{ for all } t \in ]0, T[, \\ u|_{t=0} \in L^2(\Omega). & \end{array} \right. \quad (4.6) $$

This is analogous to (3.17), except that the degenerate term is now a diffusive term. Less important, the control is in the domain and not at the boundary (and not important at all, we have also switched the roles of $x$ and $y$). The coefficient $q$ is again regular and vanishes at and only at 0. Then (4.6) is a heat equation which is degenerate on the line $x = 0$.

In the first results about the Grushin equation, the control (or observability) region $\omega$ is usually rectangular (of the form $[a, b] \times \mathbb{T}$, for $[a, b] \subset I$). In [BCG14], for $q(x)^2 = |x|^\gamma$ and $0 < a < b < \ell_+$, it is proved that there is always observability if $\gamma < 2$ (for any open $\omega$ in this case), never if $\gamma > 2$, and that there is a finite critical time $T_c \ge \frac{a^2}{2}$ if $\gamma = 2$. In [BMM15], it is proved that $T_c = 0$ if $a = 0$ and that the critical time is exactly $\frac{a^2}{2}$ if $\omega = \mathbb{T} \times ([-b, -a] \cup [a, b])$. Finally, [BDE20] deals with the case where $q$ is general and the observation is on one side of the boundary, which is essentially equivalent to observing through a rectangular neighborhood of the boundary (this is the result that we have proved for the Kolmogorov equation in [22]).

In [Koe17, DK20], results are given for more general control domains. However, these results are only proved for the model case $q(x) = x$. Our purpose in [25] is to extend this analysis for more general coefficients (see the precise assumptions for each result).
<figure><img src="image_6.png" /><figcaption>Figure 4.1: A domain of control and the corresponding critical time.</figcaption></figure>

In this picture, a domain of control $\omega$ as in Theorem 4.4, a path $\gamma$ as in Theorem 4.2 (dotted line, with $\delta(\gamma_x^-) = \max(\delta(\gamma_x(s)))$) and a line segment $[a, b] \times \{y_0\}$ as in Theorem 4.3. We can choose $a$ and $\gamma_x^-$ arbitrarily close to $\gamma_2^-$, which gives the critical time $T_c = \delta(\gamma_2^-)/q'(0)$.

Theorem 4.2 is a positive result of null-controllability. It gives a sufficient condition on $\omega$ to prove observability for large times, which gives an upper bound for the critical time of null-controllability.

On the contrary, Theorem 4.3 provides a condition on $\omega$ for which we have a positive lower bound for the critical time, which means that we cannot have null-controllability in small times.

Finally, Theorem 4.4 gives a family of domains $\omega$ which satisfy the assumptions of both results and for which the lower and upper bounds for the critical time coincide. In this case, we have the precise critical time for the null-controllability of the Baouendi-Grushin equation (4.6).

For these statements we set

$$ \delta(x) = \int_{0}^{x} q(s) \, ds, \quad \forall x \in I, \quad \text{and} \quad \delta(\pm \ell_{\pm}) = +\infty. $$

**Theorem 4.2.** Assume that $q \in C^3(\bar{I})$ is such that $q(0) = 0$ and $\inf_I q' > 0$. Assume that there exists a closed path $\gamma = (\gamma_x, \gamma_y) \in C^0(\mathbb{T}; \omega)$ such that $\{-\ell_-\} \times \mathbb{T}$ and $\{\ell_+\} \times \mathbb{T}$ are included in different connected components of $(\bar{I} \times \mathbb{T}) \setminus \gamma(\mathbb{T})$. Then the Baouendi-Grushin equation (4.6) is null-controllable on $\omega$ in any time $T$ such that

$$ T > \frac{1}{q'(0)} \max_{s \in \mathbb{T}} \delta(\gamma_x(s)). $$

The main result in [25] is the following.

**Theorem 4.3.** Assume that $q \in C^2(\bar{I})$ is such that $q(0) = 0$, $q'(0) > 0$ and $q(x) \neq 0$ whenever $x \neq 0$. Let $\omega$ be an open subset of $I \times \mathbb{T}$. Assume that there exist $a \in [-\ell_-, 0[, b \in ]0, \ell_+]$ and $y_0 \in \mathbb{T}$ such that, in $\bar{I} \times \mathbb{T}$,

$$ ([a, b] \times \{y_0\}) \cap \bar{\omega} = \emptyset. $$

Then the generalized Baouendi-Grushin equation (4.6) is not null-controllable on $\omega$ in time $T$ such that

$$ T < \frac{1}{q'(0)} \min(\delta(a), \delta(b)). $$

In particular, we never have null-controllability if $] -\ell_-, \ell_+ [ \times \{y_0\} \cap \bar{\omega} = \emptyset$ for some $y_0 \in \mathbb{T}$.

**Theorem 4.4.** Assume that $q \in C^3(\bar{I})$ is such that $q(0) = 0$ and $\inf_I q' > 0$. Let $\gamma_1, \gamma_2 \in C^0(\mathbb{T}; I)$ such that $\gamma_1(y) < \gamma_2(y)$ for all $y \in \mathbb{T}$. We set

$$ \gamma_1^+ = \max(0, \max_{\mathbb{T}} \gamma_1) \quad \text{and} \quad \gamma_2^- = \min(0, \min_{\mathbb{T}} \gamma_2). $$
Then the critical time for the null-controllability of (4.6) on $\omega = \{(x, y) \in I \times \mathbb{T} : \gamma_1(y) < x < \gamma_2(y)\}$ is

$$ \mathcal{T}_c = \frac{1}{q'(0)} \max(\delta(\gamma_2^{-}), \delta(\gamma_1^{+})). $$

We do not discuss the general strategy of the proof but only emphasize one argument of the proof of Theorem 4.3. Instead of the (non-selfadjoint) operator $K_n$ which appeared in the analysis of the Kolmogorov equation in Section 3.5, we get after Fourier transform the (selfadjoint) operator

$$ P_n = -\hat{\partial}_x^2 + n^2 q(x)^2 \quad (\text{with } \text{Dom}(P_n) = H_0^1(I) \cap H^2(I)). $$

To prove Theorem 4.3 we contradict the corresponding observability estimate (see (3.16)) if $T$ is too small.

A key argument depends on the first eigenvalue $\lambda_n$ of $P_n$ (which is again close to the eigenvalue $q'(0)n$ of the harmonic oscillator), and a corresponding eigenfunction $\varphi_n$. Setting, for some $\varepsilon > 0$,

$$ \gamma_{t,x}(n) = e^{-t(\lambda_{n+1}-q'(0)(n+1))} \varphi_{n+1}(x) e^{(n+1)\delta(x)(1-\varepsilon)} $$

we have to prove the following estimate on polynomials

$$ \left\| \sum \gamma_{t,x}(n) a_n z^n \right\|_{L^\infty(X)} \lesssim \left\| \sum a_n z^n \right\|_{L^\infty(V)}, $$

uniformly in $t \in ]0, T[$ and $x \in I$, where $X$ is a compact subset of $\mathbb{C}$ and $V$ is an open and star-shaped (with respect to 0) neighborhood of $X$. For this, we use a result (see [Koe17, Th. 18]) which requires estimates on a holomorphic extension of $\gamma_{t,x}$ on some domain of $\mathbb{C}$. In particular, we need properties on the analytic extensions of $\lambda_\nu$ and $\varphi_\nu$ for $\text{Re}(\nu) > 0$. Thus we have to study $P_\nu$ for some non-real parameters $\nu$ and, even in this setting, spectral properties of non-selfadjoint operators play an important role.

The results about null-controllability for the Grushin (discussed here) or the Kolmogorov (see Section 3.5) equations still only concern particular situations. Even if more and more general control domains $\omega$ and coefficients $q(x)$ are considered, much remains to be understood.

In [25], we have provided examples of control domains $\omega$ which are not covered by our results. Some geometries are not so complicated and it is frustrating not to be able to conclude in these cases.

Similarly, another assumption that we would like to relax is the increasingness of $q$ in Theorem 4.2. We could also consider the case where $q$ vanishes twice (then $q^2$ would form a double-well potential). The difficulty does not come from the arguments based on spectral theory, since these more general settings are more or less already understood. The main obstacle is probably the Carleman estimate used for the positive result (Proposition 3.14 in the context of [22]).

Of course, we would like to go further. For me, this first means a better understanding of the classical techniques in control theory (Carleman estimates, the method of moments, how we can use the classical tools of microlocal analysis in this kind of context, etc.). This is a very interesting possible perspective.

## 4.3 Discrete spectrum at the strong magnetic field limit on a curved wave guide

In this section we discuss the preprint [27], written with Engerran Bon-Lavigne, Loïc Le Treust and Nicolas Raymond. This started with a discussion at the CIRM in June 2021.

The question is about the existence of discrete spectrum under the essential spectrum for a magnetic Laplacian on a two dimensional curved wave guide, at the strong magnetic field limit.
We consider a smooth and injective curve $\gamma : \mathbb{R} \to \mathbb{R}^2$ with $|\gamma'| = 1$. We assume that the algebraic curvature $\kappa$ of $\gamma$ is compactly supported. Then our curved wave guide is defined by

$$ \Omega = \Omega_{\gamma, \delta} = \Theta(\Omega_{0, \delta}), \quad \Omega_{0, \delta} = \mathbb{R} \times ]-\delta, \delta[, \quad \Theta(s, t) = \gamma(s) + tN(s), $$

where $\delta > 0$ is small enough and $N(s) = \gamma'(s)^\perp$ (with $(a, b)^\perp = (-b, a)$).

We consider on $\Omega_{\gamma, \delta}$ a uniform magnetic field of intensity $h^{-1}$, with $0 < h \ll 1$. If $A$ is a well chosen vector potential corresponding to a field equal to 1, we consider on $L^2(\Omega)$ the operator

$$ P_{h,\gamma,\delta} = (-ih\nabla - A)^2, \quad \mathrm{Dom}(P_{h,\gamma,\delta}) = H_0^1(\Omega) \cap H^2(\Omega). $$

Without magnetic field, the Dirichlet Laplacian always has discrete spectrum if the wave guide is not straight [DE95]. However, it is also known that the magnetic field plays against the existence of discrete spectrum. It has even been conjectured by P. Duclos and P. Exner in the mid-nineties that the discrete spectrum of $P_{h,\gamma,\delta}$ should be empty when $h$ is small enough. In that direction, it is proved in [KR14] that if the magnetic field is compactly supported then there is no discrete spectrum in the strong field limit.

However, we give in [27] a sufficient condition to have discrete spectrum for a strong uniform magnetic field.

**Theorem 4.5.** Assume that $\kappa^2$ has a unique maximum, which is non-degenerate. There exist $\delta_0 > 0$ and $h_0 > 0$ such that for $\delta \in ]0, \delta_0]$ and $h \in ]0, h_0]$ we have

$$ \inf \sigma(P_{h,\gamma,\delta}) < \inf \sigma_{\text{ess}}(P_{h,\gamma,\delta}). $$

In particular, $P_{h,\gamma,\delta}$ has non-empty discrete spectrum.

For the essential spectrum, we prove that it is equal to the essential spectrum of the straight wave guide $\Omega_{0,\delta}$, for which we can compute a lower bound. On the other hand, we use the Min-max Theorem to get an upper bound for the minimum of the spectrum. The argument is based on the equality

$$ \begin{aligned} \inf \sigma(P_{h,\gamma,\delta}) - h &= \inf_{\psi \in H_0^1(\Omega) \setminus \{0\}} \frac{\|(-ih\nabla - A)\psi\|_{L^2(\Omega)}^2 - h \|\psi\|_{L^2(\Omega)}^2}{\|u\|^2} \\ &= \inf_{u \in H_0^1(\Omega) \setminus \{0\}} \frac{4h^2 \int_{\Omega} e^{-\frac{2\phi}{h}} |\partial_{\bar{z}} u|^2 dx}{\int_{\Omega} e^{-\frac{2\phi}{h}} |u|^2 dx}. \end{aligned} $$

The main parts of the proof are the construction of a good function $\phi$ and then the definition of a suitable function $u$ which is in particular holomorphic on $\Omega$ except near the boundary (to satisfy the Dirichlet condition). This function $\phi$ already plays an important role in the choice of the vector field $A$.

In addition to its own interest for the Duclos-Exner conjecture, this result is a step toward the understanding of a similar property for the magnetic Dirac operator

$$ P_{h,\gamma,\delta} = ((-ih\partial_{x_1} - A_1)\sigma_1 + (-ih\partial_{x_2} - A_2)\sigma_2)^2 = \begin{pmatrix} |-ih\nabla - A|^2 - h & 0 \\ 0 & |-ih\nabla - A|^2 - h \end{pmatrix}. $$

This is the model Hamiltonian for a non-relativistic spin-$\frac{1}{2}$ particle, constrained to move in $\Omega$, interacting with a magnetic field that is perpendicular to the plane. Spectral properties of this operator are studied in [BLTRS21] when $\Omega$ is bounded. Our project is to consider as above the operator on a wave guide.

A first step could be as in [27] the existence of discrete spectrum, but the perspective is to prove as in [BLTRS21] more precise asymptotics for the first eigenvalues. For the Duclos-Exner conjecture an upper bound for the bottom of the spectrum was enough, but we would like to prove asymptotic properties for the $k$-th eigenvalue at the strong field limit, both for the magnetic Laplacian and the Dirac operator. This is much more precise and would require in particular a better understanding on our wave guides of the functional spaces used in [BLTRS21] and wave-guide versions of classical results such as the Paley-Wiener theorem.
## 4.4 Spectral theory on discrete and quantum graphs

### 4.4.1 Spectrum of the wave equation on a non-compact star-graph

In Section 3.1, we have discussed the spectrum of a non-selfadjoint compact star graph. Here we discuss the result of the preprint [26] about the damped wave equation on a non-compact star-graph. This is a joint work with David Krejčířík.

We have already seen that the energy of the wave does not necessarily decrease faster with a stronger damping. A natural question is then the optimal damping which will give the best decay.

This question has been addressed in [BRT82] in the case of a wave on a finite interval with singular damping in the middle. This was motivated by the analysis of stringed instruments. They argue that there is indeed an optimal strength for the damping. The problem of the wave on a finite string with Dirac damping has also been analysed in [ATH00, AHT01, CH08] (see also [AN15, §4.1.1]). The damped wave equation has also been discussed in star-shaped graphs in [AJ04] (damping at the central vertex), [AJK05] (damping at a boundary vertex) or [AJK16] (local energy decay with one infinite edge).

In [26] we consider a star-graph with $N$ infinite edges and damping at the only vertex. The graph can be identified with $(\mathbb{R}_+^*)^N$ (the edges are parametrized from the vertex to infinity). Given any $\alpha \in \mathbb{C}$ we consider the wave equation

$$ \partial_t^2 u_j(t, x) - \partial_x^2 u_j(t, x) = 0, \quad 1 \le j \le N, t \ge 0, x > 0, \qquad (4.7) $$

with continuity at the vertex

$$ u_j(t, 0) = u_k(t, 0), \quad 1 \le j, k \le N, t \ge 0, \qquad (4.8) $$

(we denote by $u(t, 0)$ this common value) and the damping vertex condition

$$ \sum_{j=1}^{N} \partial_x u_j(t, 0) + \alpha \partial_t u(t, 0) = 0. \qquad (4.9) $$

We really have damping when $\alpha < 0$ (the case $\alpha > 0$ would correspond to a supply of energy, while the case $\alpha \in i\mathbb{R}$ is conservative and has an interpretation in quantum mechanics, see Section 7 in [26]). Notice that the case $N = 1$ corresponds to a semi-infinite string with damping at the end, while the case $N = 2$ is an infinite string with singular damping.

The corresponding operator is defined as follows. We denote by $\mathcal{E}$ the set of pairs $(u, v)$ such that $u'_j, v_j \in L^2(0, \infty)$ for all $j$ and $u_j(0) = u_k(0)$ for all $j, k$ (we denote by $u(0)$ this common value). This is a Hilbert space if the first component is quotiented by the space of constant functions on the graph. For $u \in \dot{H}^2(\mathbb{R}_+^*)^N$ we set $u'' = (u_j'')$. Then we define the operator $\mathcal{W}$ by

$$ \mathcal{W}\begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ u'' \end{pmatrix}, \quad \mathrm{Dom}(\mathcal{W}) = \left\{ U = (u, v) \in \mathcal{E} : \mathcal{W}U \in \mathcal{E} \text{ and } \sum_{j=1}^{N} u'_j(0) + \alpha v(0) = 0 \right\}. $$

We are interested in the spectral properties of this operator $\mathcal{W}$, and the corresponding counterparts for the time dependent problem (4.7)-(4.9). We can check that $\pm \mathcal{W}$ is maximal accretive if $\pm \mathrm{Re}(\alpha) \ge 0$ (it is skew-adjoint if $\mathrm{Re}(\alpha) = 0$). In particular, if $\mathrm{Re}(\zeta) \ne 0$ and $\mathrm{Re}(\zeta)\mathrm{Re}(\alpha) \le 0$ then $\zeta \in \rho(\mathcal{W})$ and

$$ \|(\mathcal{W} - \zeta)^{-1}\|_{\mathcal{L}(\mathcal{E})} \le \frac{1}{|\mathrm{Re}(\zeta)|}. $$

We are interested in the spectrum in the half-plane where $\mathrm{Re}(\zeta)$ and $\mathrm{Re}(\alpha)$ have the same sign. This is a difficult question in general and little is known in that direction, but for this model setting we can proceed with explicit computation.

As already observed for a finite string, there is an abrupt change of properties for particular values of $\alpha$. More precisely, we have the following result about the spectrum of $\mathcal{W}$.
**Theorem 4.6.** The spectrum of $W_\alpha$ is

$$ \begin{cases} \frac{i\mathbb{R}}{\mathbb{C}_+} & \text{if } \alpha \in \mathbb{C}\setminus\{\pm N\}, \\ \frac{\mathbb{C}_-}{\mathbb{C}_-} & \text{if } \alpha = N, \\ \mathbb{C}_- & \text{if } \alpha = -N. \end{cases} $$

Moreover, $i\mathbb{R}$ contains no eigenvalue nor residual spectrum of $W_\alpha$, and if $\alpha = \pm N$ then any $\zeta \in \mathbb{C}_\pm$ is an eigenvalue of $W_\alpha$ of geometric multiplicity 1 and infinite algebraic multiplicity.

To understand this behavior, we compute the resolvent of $W$. As expected, for $\zeta \in \mathbb{C}_\pm$ the norm of $(W - \zeta)^{-1}$ in $\mathcal{L}(\mathcal{E})$ goes to $+\infty$ when $\alpha \to \pm N$.

**Proposition 4.7.** There exist $c_0, C > 0$ such that for $\alpha \in \mathbb{C}_\pm \setminus \{\pm N\}$ and $\zeta \in \mathbb{C}_\pm$ we have

$$ \max \left( \frac{1}{|\mathrm{Re}(\zeta)|}, \frac{c_0}{|\zeta| |\alpha \mp N|} \right) \le \|(W - \zeta)^{-1}\|_{\mathcal{L}(\mathcal{E})} \le \frac{C}{|\mathrm{Re}(\zeta)|} \left( 1 + \frac{1}{|\alpha \mp N|} \right). $$

The values $\alpha = \pm N$ also have a particular role for the wave equation (4.7)-(4.9). Notice that on each edge, the wave has an outgoing part which never sees the damping and behaves as the free wave equation on $\mathbb{R}$. We are interested in the incoming part of the wave, which hits the vertex. The vertex conditions describe how the waves coming from all edges are reflected. In particular, the sum of the energies of all the reflected waves is equal (smaller, larger, respectively) than the energies of the incident waves if $\mathrm{Re}(\alpha) = 0$ ($\mathrm{Re}(\alpha) < 0$, $\mathrm{Re}(\alpha) > 0$, respectively). It turns out that in the particular case $\alpha = -N$ the incident waves are completely absorbed. And the problem is ill posed in the case $\alpha = N$ (we have no solution or an infinite number of solutions for positive times, depending on the initial conditions). Of course, these properties are reversed if we go backward in time. We refer to [26, Th. 2.4] for a more quantitative statement.

### 4.4.2 Other model problems on non-compact quantum graphs

The paper [21] and the preprint [26] were about particular cases of non-selfadjoint quantum graphs, and it would be interesting to go further in various directions.

The first obvious question is the generalization of [21] to more general graphs. The difficulties will come from the steps of the proof where we have used the explicit expression for the secular equation (3.9). For instance, it has been used to prove that the Neumann eigenvalues cannot escape too far (see the discussion before (3.11)) or to see that $\ell$ is always transverse to the submanifold $\mathcal{Z}_0$ of $\mathbb{T}^N$ (see before (3.12)). We have already thought about this while writing [21]. Considering general compact quantum graphs will not be trivial, but we can certainly go beyond the star-graph.

Another natural continuation of [21] is to consider the spectrum of the wave operator on a compact star graph (and then in a more general situation).

The main difference between the Schrödinger and wave equations is that instead of the operator $H_\alpha$ defined by (3.1)-(3.4), we have to consider the operator $H_{az}$ where $a$ is the absorption coefficient and $z$ is the spectral parameter (this is completely similar to the wave with damping at the boundary, see (2.18)). Thus we have to work with the parameter $\alpha = az$ which goes to infinity when the spectral parameter $z$ itself goes to infinity. Then the key argument that the eigenvalues of $H_\alpha$ cannot go too far from those of $H_0$ is no longer valid.

We have seen as a byproduct of studies on wave guides that for one edge (see Section 1.6) or for two edges with particular vertex conditions (see Section 4.1.2) we can prove that eigenvalues stay in some bounded region of $\mathbb{C}$ for any $\alpha$, but these results were again based on explicit computation (and we already had more difficulties with two edges than with one), so we have to find a more general argument to deal with more general graphs.
Notice that when all the edges have the same length, it is proved in [FL17] that there is a finite number of so-called spectral abscissas. With the terminology of Theorem 3.3, this means that the corresponding measure $\mu_\ell$ is a linear combination of Dirac masses (this is the analog of the second case therein). We expect that a similar result will hold with damping at the vertices. The difficulty, as in [21], is to give a precise statement for any set of lengths (or at least a generic set of lengths, but including the case of irrationally dependent lengths). For a review of some known results about the stabilization of the wave equation on quantum graphs, we also refer to [Zua13].

All the questions around [21] concern the contribution of high frequencies. If we consider non-compact quantum graphs, there will also be interesting phenomena for the contribution of low frequencies, as is the case for the wave on non-compact manifolds.

A non compact graph can be a graph with a finite number of infinite edges, as in [26], or a graph with an infinite number of edges. We give several examples (all with finite edges of equal length) in Figure 4.2.

<figure><img src="image_7.png" /><figcaption>Figure 4.2: Examples of non-compact graphs</figcaption></figure>

Let us briefly discuss for instance the case of trees. This is a particular example of quantum graph which has many applications.

Again, there is less literature on the spectral properties of quantum graphs than on the discrete ones, but this is now an active field. We refer for instance to [Car97, Sol04] (and references therein) for early works about the spectrum of the Laplacian on quantum trees, and to [AISW21] for an example of recent development.

It would be interesting to see what happens if we add some damping on the edges or at the vertices. Since I have not started the analysis, I have to remain vague about what can be expected, but this is a setting that I find exciting and that I would like to explore.

Another question which could be tested on a simple graph is the wave equation with random damping. Consider for instance the case of a wave equation on the line $\mathbb{R}$ with random singular damping at each integer. The damping coefficient $a_n$ at $n \in \mathbb{Z}$ could be independent random variables (for instance they could all follow a Bernoulli distribution, so that there is damping with probability $p$ and nothing with probability $(1-p)$). As usual, we could start with the Schrödinger operator, in which case we are closer to the classical Anderson model (see for instance [Kir08] for an introduction to random Schrödinger operators).

### 4.4.3 Resolvent estimates for the Schrödinger operator on a discrete graph

When I mention graphs, I mainly think about quantum graphs. However, even if discrete graphs have already been extensively studied, there are still exciting questions to be discussed.

As for quantum graphs, we can consider many different interesting settings, but I will only consider one problem in this paragraph. This is a project that I have started with Olivier Bourget and Diomba Sambou.

With Amal Taarabt, they have studied in [BST20] various spectral properties (limiting absorption principle, resonances, etc.) for a non-selfadjoint Schrödinger operator on $\ell^2(\mathbb{Z})$.
We recall that the one-dimensional discrete Laplacian on $ℓ^2(\mathbb{Z})$ is defined by

$$ (H_0 u)_n = 2u_n - u_{n-1} - u_{n+1}, \quad n \in \mathbb{Z}, $$

for $u \in ℓ^2(\mathbb{Z})$ (we could also consider the analog on $ℓ^2(\mathbb{Z}^d)$ for any dimension $d \in \mathbb{N}$*). Its spectrum is

$$ \sigma(H_0) = \sigma_{ac}(H_0) = [0, 4]. $$

In [BST20] they add a complex potential $V$ (a potential simply acts as a diagonal matrix on $ℓ^2(\mathbb{Z})$: $(Vu)_n = V_n u_n$).

In particular, they prove the limiting absorption principle with the Mourre commutators method.

We recall that this method requires a suitable conjugate operator. In the papers discussed in Chapter 1, the operator was always a perturbation of the usual Laplacian on $\mathbb{R}^d$, and the conjugate operator was always a perturbation of the generator of dilations. Dilations have no obvious analog in the discrete setting. However, if we set $A_0 = \text{Im}(S)X+X\text{Im}(S)$, where $X$ is the position operator (defined by $(Xu)_n = nu_n$) and $S$ is the shift operator $((Su)_n = u_{n+1})$, then we have

$$ \mathrm{ad}_{iA_0}(H_0) = 4H_0 - H_0^2. $$

With this conjugate operator, they are able to apply the Mourre theory away from the thresholds 0 and 4 (notice that there are more thresholds in higher dimensions).

This naturally raises the question of the properties (such as resolvent estimates) near the thresholds. This is precisely the analog of studying the resolvent of usual Schrödinger operators near 0.

The discussion is only at its early stage, but it is natural to try to adapt the strategy described in Chapter 1. As for the use of the generator of dilations, many arguments used in the Euclidean setting are meaningless in the discrete case, but there is a reasonable hope that we can adapt the ideas to get resolvent estimates for a discrete Schrödinger operator. Of course it is also possible to consider the wave equation in this setting.

# 4.5 Non-linear problems

## 4.5.1 A non linear problem on a quantum graph

After [12] and [19] (see Section 3.6), I have started a new project with François Genoud and Stefan Le Coz. It is again about a non-linear problem in dimension 1. Another common feature between our two previous works was the singular potential. As observed above, a problem on the real line with a singular potential can be seen as a problem on a graph with two infinite edges, the singular potential being interpreted as a “boundary condition” at the vertex.

Here we really consider a graph. Since our analysis should be essentially local, we consider the model case: a star graph $\Gamma$, with $N$ infinite edges (as in Section 4.4.1). We recall that $\Gamma$ can be seen as $N$ copies of $\mathbb{R}_+^*$, and we define $L^2(\Gamma)$ accordingly.

Given $\gamma \in \mathbb{R}$, we consider on $\Gamma$ the non-linear Schrödinger equation

$$ -i\partial_t u_j - \partial_{x_j}^2 u_j + \gamma\delta u - |u_j|^4 u_j = 0, \quad 1 \le j \le N, \qquad (4.10) $$

where $\delta$ is the Dirac distribution at the central vertex. A solution on the interval $I$ of $\mathbb{R}$ is a function $u = (u_j)_{1 \le j \le N} \in C^0(I, H^1(\mathbb{R}_+^*)^N) \cap C^1(I, (H^1(\mathbb{R}_+^*)^*)^N)$ such that $u_j$ is a solution on each edge, $u(t)$ is continuous at the vertex (as in (4.8)) for all $t \in I$, and it satisfies the Robin condition

$$ \sum_{j=1}^{N} u'_{j}(t, 0) = \gamma u(t, 0). $$
The Cauchy problem is completed by an initial condition $u_0$ at time $t = 0$. Notice that since $\gamma$ is real, the generator of the linear part (the Laplacian on each edge, with these vertex conditions in the domain) is selfadjoint. Two constants of the motion are given by the mass and the energy:

$$M(u) = \frac{\|u\|_{L^2(\Gamma)}^2}{2}, \quad E(u) = \frac{\|\partial_x u\|_{L^2(\Gamma)}^2}{2} + \frac{\gamma |u(0)|^2}{2} - \frac{\|u\|_{L^6(\Gamma)}^6}{6}.$$

Our question on this model is the existence of blowing-up solutions. It is known that for the equation on the line,

$$-i\partial_t u - \partial_x^2 u - |u|^4 u = 0, \quad u|_{t=0} = u_0, \qquad (4.11)$$

the threshold between global existence of all the solutions and existence of blowing-up solutions is the mass of the ground state $Q: x \mapsto 3^{\frac{1}{4}} \text{sech}(2x)^{\frac{3}{2}}$ (solution of $Q'' - Q + Q^5 = 0$). If $M(u_0) < M(Q)$ then the solution of (4.11) is well defined for all times, and there exists $u_0$ with $\|u_0\|_{L^2(\mathbb{R})} = \|Q\|_{L^2(\mathbb{R})}$ such that the corresponding solution of (4.11) blows up in finite time. Indeed, given $T > 0$ we can consider the solution given by pseudo-conformal transform of the standing wave $e^{it}Q(x)$, given by

$$u(t, x) = \frac{1}{\sqrt{T-t}} e^{\frac{i}{T-t}} e^{-\frac{i \pi^2}{4(T-t)}} Q \left( \frac{x}{T-t} \right).$$

Then we have

$$\|u(t)\|_{L^2(\mathbb{R})} = \|Q\|_{L^2(\mathbb{R})} \quad \text{and} \quad \|\partial_x u(t)\|_{L^2(\mathbb{R})} \underset{t \to T^{-}}{\sim} \frac{1}{T-t}.$$

On the graph, it is expected that the blowing-up solution with minimal mass has a similar behavior with blow-up at some point of one of the edges. If we only consider radial solutions $(u_j(t, x)$ does not depend on $j$), then the threshold for global existence is increased. With $\gamma = 0$, we can construct a similar solution by replacing $Q$ by $Q_\Gamma$, defined by considering a copy of $Q_{\mathbb{R}_+}$ on each edge. This is no longer a solution if $\gamma \neq 0$. Our purpose is to construct a blowing-up solution at the minimal mass $M(Q_\Gamma)$. The idea is still to construct a blowing-up solution based on the profile $Q_\Gamma$. More precisely, we apply the method of [RS11, LMR16], where the existence of minimal mass blowing-up solutions is proved in other contexts. We also refer to the recent preprint [TX], dealing with the problem on the line.

The expected result is that given $E^* \in \mathbb{R}$ there exists $T > 0$ and a radial solution $u$ on $[0, T[$ such that $M(u) = M(Q_\Gamma)$, $E(u) = E^*$, and

$$\|\partial_x u(t)\|_{L^2(\Gamma)} \underset{t \to T^{-}}{\sim} \frac{1}{(T-t)^{\frac{2}{3}}}.$$

### 4.5.2 Nonlinear damped wave equations

All the works about the damped wave equation presented in Chapters 1 and 2 concern in various settings the linear equation (1.12). I have also worked on some non-linear problems (see Sections 3.6 and 4.5.1), but it was not about the damped wave equation.

As said in introduction, in many physical situations, propagation of waves is modeled by an equation involving nonlinear terms. A possible perspective for the future is to consider some questions about non-linear damped wave equations. I have not started anything in that direction yet, so I am essentially ignorant, but one can at least discuss some natural questions on the subject.

A first perspective is to continue the analysis of one-dimensional problems with singular potential. In particular, the paper [19] discussed in Section 3.6 is about the non-linear
Klein-Gordon equation. It would be quite natural to consider the same kind of setting, for Klein-Gordon or the wave equation, but with dissipation. This corresponds to replacing $i\alpha$ by $a > 0$ in (3.31). Beginning with one-dimensional settings would be quite convenient, and the difficulties due to the singular potentials are, as already discussed, related to quantum graphs which is another aspect that I would like to develop. The natural first questions are the analysis of the local and global existence of solutions, and the decay of the energy. One remembers from Section 4.4.1 that there are already strange behaviors for the linear problem for some particular values of the damping, and it would be interesting to know what happens if we add a non-linear perturbation.

Nevertheless, the long term perspective is to start to add non-linear terms to the general damped wave equation (1.12). The typical example is to consider a source term of the form

$$f(u) = bu |u|^{p-1}, \qquad (4.12)$$

for some $p > 1$. Another possibility is to consider a non-linear dissipative term. The model example with both non-linearities is then an equation of the form

$$\partial_t^2 u - \Delta u + a \partial_t u |\partial_t u|^{\rho-1} = f(u), \qquad (4.13)$$

for some $\rho \ge 1$.

The first question is the local well-posedness for the Cauchy problem. Then comes the question of global well-posedness or existence of blowing-up solutions. When the solution is globally defined, we can look at the long time behavior of the energy

$$E(u; t) = \frac{1}{2} \|\partial_t u\|_{L^2(\Omega)}^2 + \frac{1}{2} \|\nabla u\|_{L^2(\Omega)}^2 - \frac{b}{p+1} \|u\|_{L^{p+1}(\Omega)}^{p+1}.$$

As for the linear problem, it is natural to wonder if it goes to 0 and, in this case, what is the rate of decay.

It is too early to state a conjecture here, but this report is an occasion to look at the literature. It is of course impossible to be exhaustive about the non-linear wave equation (or the related Schrödinger equation). We refer to the books [Tao06, Caz03, Sog08] for introductions to nonlinear dispersive equations and many references, and in this paragraph we only mention some papers dealing with the damped wave equation.

As for the linear setting, the problem is much better understood on compact domains (typically with Dirichlet boundary conditions), or for the Klein-Gordon equation (with an additional term $mu$ on the left in (4.13)). We refer to [HZ88] for the inhomogeneous equation. In [GT94] it is proved that for

$$1 < p < \begin{cases} \frac{d}{d-2} & \text{if } d \ge 3, \\ +\infty & \text{if } d \le 2, \end{cases} \qquad (4.14)$$

then (4.13)-(4.12) is globally well-posed if $p \le \rho$ (in the sense that $u \in C^0(\mathbb{R}_+, H_0^1(\Omega))$ and $\partial_t u \in C^0(\mathbb{R}_+, L^2(\Omega)) \cap L_{\text{loc}}^{\rho+1}(\Omega)$), while there exists a solution which blows up in $L^\infty(\Omega)$ if $p > \rho$.

Concerning the decay of the energy, we refer for instance to [Har85, HZ88, Zua88, Zua90, JL13, JL20] for results on a bounded domain and [Zua91, Deh01, DLZ03, AIN11] for the Klein-Gordon equation. Typically, these works use multipliers method or the method based on semiclassical defect measures. Notice that in addition to the geometric condition, the proofs rely on an assumption of unique continuation. Moreover, an important observation used in many of these articles is that once the solution is small, the non-linear term no longer play any important role and the solution of the non-linear problem has the same decay properties as a solution of the linear problem (see [Gér96]).
As for the linear problem, it is a different issue to consider the wave equation in an unbounded domain. Moreover, if the damping is effective at infinity, it is natural to wonder if we have a diffusive phenomenon as described in Chapter 2 for the linear equation. As for the linear equation, there are many results about the problem with constant coefficients ($a(x) = 1$ in (4.13)). The results are different in the defocusing (typically, (4.12) with $b = -1$) of focusing ($b = 1$) cases.

We begin with the case where $f(u)$ is equal to or of the form $-|u|^{p-1}u$. The paper [Mat76], which is one of the firsts about the linear problem (see the discussion in Chapter 2), also dealt with the semilinear equation. The results have been refined in [KNO95] (in particular, general initial data are allowed). In [Kar00] (see Theorem 2.3 therein), the diffusion phenomenon has been proved for initial data in $(H^1 \cap L^1) \times (L^2 \cap L^1)$, when $d \le 3$ and for $p > 1 + \frac{4}{d}$ such that (4.14) holds. Then the asymptotic profile is given by a solution of the heat equation as in the linear case. In dimension 1, a better result is given in [HKN07a]. In [INZ06], the large time asymptotics for the supercritical case $p > 1 + \frac{2}{d}$ is proved in dimension $d \le 3$. The solution is actually close to a solution of the semilinear heat equation

$$ \partial_t v - \Delta v + v |v|^{p-1} = 0, $$

which behaves itself as a solution of the linear heat equation in the supercritical case (the critical exponent $p = 1 + \frac{2}{d}$ is known as the Fujita's critical exponent [Fuj66]). The result has then been improved for $d = 3$ and extended to $d = 4$ in [Nis06]. Less is known in the critical and subcritical cases. For results about the critical exponent we refer to [HKN07b, HKN06, HN17]. For the subcritical case we refer to [Ham10]. Notice that in these cases the problem is not necessarily globally well posed.

Now we turn to the focusing case, where $f(u)$ is typically of the form $|u|^p$ or $u|u|^{p-1}$. In general, the solution blows up in finite times for large initial data (see [Lev74, Section IV]). Results about the existence of global solutions or blow-up (for small initial data) can be found in [NO93, LZ95, TY01, Zha01]. It is proved that for $p > 1 + \frac{2}{d}$ the solution with “small” initial data is defined for all time, while there are solutions which blow up when $p \le 1 + \frac{2}{d}$. These results were proved under strong decay assumption in the initial data (typically, compactly supported). We refer to [Nak01] for the analysis on an exterior domain with a damping which can vanish.

Then there have been several papers with results about local well-posedness for large initial data and global well-posedness for small initial data in $(H^1 \cap L^1) \cap (L^2 \cap L^1)$ under some conditions on $r$ and $p$. We do not mention them all and refer to the introduction of the recent paper [IIW17], which study the large data local well-posedness and the small data global well-posedness with slowly decaying initial data.

The asymptotic behavior in terms of solutions of the heat equation is also proved in [IIW17]. There are many references dealing with the large time asymptotics in more and more general settings. We mention for instance an early result in dimension 1 [GR98], [KU13] ($d \le 3$), [HKN04] (any dimension, small solutions in particular in $L^1$) or [NN08] (slowly decaying data). The recent paper [IIW19] improves the results for the linear and the non-linear problems. Higher order expansions are also available [KT16].

All the papers mentioned above concern the semilinear wave equation with linear damping ($\rho = 1$ in (4.13)). If $\rho > 1$, we notice that if the energy of the wave becomes small, the damping term becomes smaller than in the linear case. Here re-emerges the question whether the energy goes to 0 as $t \to +\infty$ (even with $f = 0$). Depending on the precise setting and on the initial conditions, there are results of decay (the energy goes to 0) and results of non-decay (it does not) for this kind of problem, on the Euclidean space or an exterior domain. It is expected that in the nondecay case, that is when the damping term becomes very small, the solution of (4.13) behaves like a solution of the undamped wave equation. This is indeed the case and it is proved under suitable assumptions in the literature. For
more precise statements we refer to [MM95, MM96, Mat03, TY07, TUY09, Dao18]. See also
[Nak13, TY15] for results with non-linear damping and a non-linear potential.

There are actually many other variants of the non-linear damped wave equation in the literature. For instance, various systems of wave equations [GR14, BD13, BR20], friction and viscoelastic damping [MMR20], wave equation with non-linear boundary damping [CDCL07], higher order problems [CX20], time-dependent damping [IW20], delayed damping [Kaf21], variable exponents for the non-linear terms [MTAS17], etc.

We also refer to results about more or less general damped hyperbolic equations: see
[LS97, LPS98, CTR21] (global existence of solutions), [GGH16c] (decay estimates), [GGH16a]
(asymptotic behavior), [GGH16b] (smoothing effect of solutions with strong damping). See
also the (older) lecture notes [Har81] and the book [Har87].

Finally, to make the link with the quantum graphs, we mention the paper [ABM21] about
the non-linear damped wave equation on a star-graph, with damping at infinity on one edge.

Even if this bibliography is far from being complete, we already see that there is a very
wide variety of problems about the non-linear damped wave equation. Of course, for future
works, it would be reasonable to begin with questions closely related to my centers of interest.

The most natural perspective about the non-linear damped wave equation is to continue
with the settings introduced in Chapters 1 and 2. As for the linear problem, the difficulty
compared to the previous literature is to deal with the contributions of low frequencies in
quite general geometric settings. The papers mentioned above about the wave equation on
an unbounded domain essentially consider the equation with constant coefficients, sometimes
in an exterior domain. It would be interesting to generalized these results, as is done for the
linear problem.

An important tool for the analysis of non-linear dispersive equations are the Strichartz estimates, which are space-time integrability properties. They are estimates on the solution of the linear problem, but they are used to estimate the integral term given by the Duhamel formula, which can be crucial for the well-posedness of the Cauchy problem for the non-linear equation. See again [Tao06, Caz03, Sog08] for an introduction to the Strichartz estimates. There are again many papers on this question, for the Schrödinger or the (undamped) wave equations. We can mention for instance the original work [Str77] and the proof of the important endpoint cases in [KT98]. See also [BGT04] in an exterior domain and [BT08] in the asymptotically Euclidean setting. We finally refer to the recent work [BM].

It seems that not much is known about the Strichartz estimates for the damped wave
or Schrödinger equations. Some estimates about the regularized Schrödinger equation (1.67)
are given in [Kar15]. There are also recent results about the damped wave equation (2.1),
see [Wat17, Inu19, IW].

These Strichartz estimates are dispersive estimates, in the spirit of the local energy decay presented in Chapters 1 and 2. Before considering general non-linear problems, it would be completely natural to begin with Strichartz estimates for the damped wave equation (1.12).

It is already a problem which is far from simple, since the standard arguments for the
Strichartz estimates as in [KT98] do not apply to a non-selfadjoint setting.
# Publications and preprints

## Pre-publications

[27] E. Bon-Lavigne, L. Le Treust, N. Raymond, and J. Royer. On Duclos-Exner's conjecture about waveguides in strong uniform magnetic fields. Preprint HAL-03678608.

[26] D. Krejčířík and J. Royer. Spectrum of the wave equation with Dirac damping on a non-compact star graph. Preprint HAL-03652993.

[25] J. Dardé, A. Koenig, and J. Royer. Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets. Preprint HAL-03618415.

[24] R. Ayechi, I. Boukhris, and J. Royer. Energy decay for a system of Schrödinger equations in a wave guide. Preprint HAL-03420199.

## Publications

[23] J. Royer. Low frequency asymptotics and local energy decay for the Schrödinger equation. *Journal of the European Mathematical Society*. In press.

[22] J. Dardé and J. Royer. Critical time for the observability of Kolmogorov-type equations. *Journal de l'École Polytechnique, Mathématiques*, 8:859-894, 2021.

[21] G. Rivière and J. Royer. Spectrum of a non-selfadjoint quantum star graph. *Journal of Physics A. Mathematical and Theoretical*, 53(49), 2020.

[20] N. Raymond and J. Royer. Absence of embedded eigenvalues for translationally invariant magnetic Laplacians. *Journal of Mathematical Physics* 7, 2019.

[19] E. Csobo, F. Genoud, M. Ohta and J. Royer. Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials. *Journal of Differential Equations* 268: 353-358, 2018.

[18] J. Royer. Energy decay for the Klein-Gordon equation with highly oscillating damping. *Annales Henri Lebesgue* 1: 297-312, 2018.

[17] R. Joly and J. Royer. Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation. *Journal of the Mathematical Society of Japan* 70 (4): 1375-1418, 2018.

[16] D. Krejčířík, N. Raymond, J. Royer, and P. Siegl. Reduction of dimension as a consequence of norm-resolvent convergence and applications. *Mathematika*, 64(2): 406-429, 2018.

[15] M. Malloug and J. Royer. Energy decay in a wave guide with damping at infinity. *ESAIM: Control, Optimisation and Calculus of Variations* 24: 519-549, 2018.
[14] J. Royer. Local energy decay and diffusive phenomenon in a dissipative wave guide. *Journal of Spectral Theory*, 8(3): 769-841, 2018

[13] D. Krejčířík, N. Raymond, J. Royer, and P. Siegl. Non-accretive Schrödinger operators and exponential decay of their eigenfunctions. *Israel Journal of Mathematics*, 221 (2): 779-802, 2017.

[12] I. Ianni, S. Le Coz, and J. Royer. On the cauchy problem and the black solitons of a singularly perturbed gross-pitaevskii equation. *SIAM Journal on Mathematical Analysis*, 49(2):1060–1099, 2017.

[11] J. Royer. Local decay for the damped wave equation in the energy space. *Journal of the Institute of Mathematics of Jussieu*, 17(3): 509-540, 2016.

[10] M. Khenissi and J. Royer. Local energy decay and smoothing effect for the damped Schrödinger equation. *Analysis & PDE*, 10(6):1285–1315, 2017.

[9] J. Royer. Mourre's commutators method for a dissipative form perturbation. *Journal of Operator Theory*, 76(1):351–385, 2016.

[8] J. Royer. Exponential decay for the Schrödinger equation on a dissipative wave guide. *Annales Henri Poincaré*, 16(8):1807–1836, 2015.

[7] J.-M. Bouclet and J. Royer. Low frequency resolvent estimates on asymptotically conical manifolds. *Communication in Mathematical Physics*, 335(2):809–850, 2015.

[6] J.-M. Bouclet and J. Royer. Local energy decay for the damped wave equation. *Journal of Functional Analysis*, 266(2):4538–4615, 2014.

[5] J. Royer. Semiclassical measure for the solution of the Helmholtz equation with an unbounded source. *Asymptotic Analysis*, 91:11–32, 2015.

[4] J. Royer. Uniform resolvent estimates for a non-dissipative Helmholtz equation. *Bulletin de la Société Mathématiques de France*, 142(4):591–633, 2014.

[3] J. Royer. Semiclassical measure for the solution of the dissipative Helmholtz equation. *Journal of Differential Equations*, 249:2703–2756, 2010.

[2] J. Royer. Limiting absorption principle for the dissipative Helmholtz equation. *Com- munication in Partial Differential Equations*, 35(8):1458–1489, 2010.

[1] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle, and J. Royer. A traffic-flow model with constraints for the modeling of traffic jams. *Mathematical Models and Methods in Applied Sciences*, 18:1269–1298, 2008.
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