# Quantum unique ergodicity for magnetic Laplacians on $T^2$

Léo Morin and Gabriel Rivière

November 28, 2024

## Abstract

Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schrödinger operator. Under a geometric control condition on the magnetic field, we show that every sequence of high energy eigenfunctions satisfies the quantum unique ergodicity property even if the Liouville measure is not ergodic for the underlying classical flow (the Euclidean geodesic flow on the 2-torus).

## Contents

<table><tbody><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Earlier results for vanishing magnetic fields</td><td>4</td></tr><tr><td>1.2</td><td>Quantum unique ergodicity</td><td>5</td></tr><tr><td>1.3</td><td>Strategy of proof</td><td>6</td></tr><tr><td>2</td><td>Magnetic Laplacian on the torus</td><td>7</td></tr><tr><td>2.1</td><td>Definitions</td><td>8</td></tr><tr><td>2.2</td><td>The case of a constant magnetic field</td><td>10</td></tr><tr><td>3</td><td>A magnetic quantization</td><td>10</td></tr><tr><td>3.1</td><td>Magnetic Weyl quantization</td><td>11</td></tr><tr><td>3.2</td><td>Composition formula</td><td>13</td></tr><tr><td>3.3</td><td>Calderón-Vaillancourt Theorem</td><td>17</td></tr><tr><td>4</td><td>Proof for constant magnetic fields</td><td>19</td></tr><tr><td>4.1</td><td>Averaging method</td><td>20</td></tr><tr><td>4.2</td><td>Microlocalization</td><td>22</td></tr><tr><td>5</td><td>Magnetic semiclassical measures</td><td>23</td></tr><tr><td>5.1</td><td>Lifing quantum limits</td><td>24</td></tr><tr><td>5.2</td><td>Decomposition of invariant measures</td><td>25</td></tr><tr><td>6</td><td>Periodic orbits</td><td>27</td></tr><tr><td>6.1</td><td>Analyzing the noncompact part</td><td>29</td></tr><tr><td>6.2</td><td>Analyzing the compact part</td><td>31</td></tr><tr><td>6.3</td><td>Conclusion</td><td>34</td></tr></tbody></table>
**7 Proof for variable magnetic fields** 34

**A Results for more general magnetic fields** 35

# 1 Introduction

Let $\mathbf{T}^2 = \mathbf{R}^2/\mathbf{Z}^2$ be the standard Euclidean torus of dimension 2 that we endow with a fixed smooth and real-valued 2-form $B(x_1, x_2)dx_1 \wedge dx_2$ satisfying

$$ \hat{B}_0 := \int_{\mathbf{T}^2} B(x_1, x_2)dx_1 \wedge dx_2 \in 2\pi\mathbf{Z}. $$

Under such an assumption, one can define a magnetic (or Bochner) Laplacian<fnref n="1" /> $\mathcal{L}$. We refer to Section 2 for precise definitions and let us just recall that it acts on the following type of functions

$$ \mathcal{C}^{\infty}(\mathbf{T}^2, L) := \left\{ u \in \mathcal{C}^{\infty}(\mathbf{R}^2, \mathbf{C}) : u(x_1, x_2) = e^{-\frac{i}{2}\hat{B}_0 x_1} u(x_1, x_2 - 1) = e^{-\frac{i}{2}\hat{B}_0 x_2} u(x_1 - 1, x_2) \right\}. $$

Given $V \in \mathcal{C}^{\infty}(\mathbf{T}^2, \mathbf{R})$, we are interested in solutions to the eigenvalue problem,

$$ (\mathcal{L} + V) u_{\lambda} = \lambda^2 u_{\lambda}, \quad \|u_{\lambda}\|_{L^2} = 1, \qquad (1.1) $$

as $\lambda \to +\infty$. In order to state our main result, we introduce the following function on $S^*\mathbf{T}^2 \simeq \mathbf{T}^2 \times \mathbf{S}^1$,

$$ B_{\infty}(x, \theta) := \lim_{T \to +\infty} \frac{1}{T} \int_{0}^{T} B(x + t\theta)dt, \quad (x, \theta) \in S^*\mathbf{T}^2. $$

Recall that, if $k \cdot \theta \neq 0$ for every $k \in \mathbf{Z}^2 \setminus \{0\}$, then $B_{\infty}(x, \theta) = \hat{B}_0$ while for rational values of $\theta$, the function $B_{\infty}$ may depend in a nontrivial (but smooth) way on $x \in \mathbf{T}^2$. When $B_{\infty} > 0$ everywhere, one often says that the magnetic field verifies a geometric control condition [50, 10]. Our main theorem states that solutions to (1.1) enjoy a quantum unique ergodicity property under this geometric assumption.

**Theorem 1.1.** Suppose that $B_{\infty}(x, \theta) > 0$ for every $(x, \theta) \in S^*\mathbf{T}^2$. Then, for any sequence of solutions $(u_{\lambda})_{\lambda \to +\infty}$ to (1.1) and for any $a \in S^0(T^*\mathbf{T}^2)$, one has

$$ \lim_{\lambda \to +\infty} \langle \mathrm{Op}_{\lambda-1}^{B}(a)u_{\lambda}, u_{\lambda} \rangle = \frac{1}{2\pi} \int_{S^*\mathbf{T}^2} a(x, \theta)xd\theta, $$

where $\mathrm{Op}_{\lambda-1}^B$ is the semiclassical magnetic Weyl quantization. In particular, for every $a \in \mathcal{C}^{\infty}(\mathbf{T}^2, \mathbf{C})$, one has

$$ \lim_{\lambda \to +\infty} \int_{\mathbf{T}^2} a(x)|u_{\lambda}(x)|^2dx = \int_{\mathbf{T}^2} a(x)dx. $$

Here $S^0(T^*\mathbf{T}^2)$ denotes the subspace of $\mathcal{C}^{\infty}(\mathbf{T}^2 \times \mathbf{R}^2, \mathbf{C})$ made of functions all of whose derivatives are bounded. The precise definition of the magnetic Weyl quantization [45, 33, 32, 34] is recalled in Section 3, together with its main properties. This is the analogue for the space $\mathcal{C}^{\infty}(\mathbf{T}^2, L)$

<footnote n="1">¹In fact, up to gauge choices, there is a one parameter family $\mathcal{L}_{\alpha}$ of such operators, where $\alpha$ varies in $\mathbf{R}^2/(2\pi\mathbf{Z}^2)$. Yet, our main result is the same for each parameter $\alpha$ (and for each gauge choice) and we omit it in the introduction.</footnote>
of the Weyl quantization on $\mathcal{C}^\infty(\mathbf{T}^2, \mathbf{C})$ [64]. This theorem shows that magnetic eigenfunctions become equidistributed in the high frequency limit as soon as the geometric control condition $B_\infty > 0$ is satisfied. It is for instance verified if $B > 0$ everywhere on $\mathbf{T}^2$ but this geometric condition is in fact much more general as it only involves the average of the magnetic field along the geodesic flow:

$$ \varphi^t : (x, \theta) \in S^*\mathbf{T}^2 \mapsto (x + t\theta, \theta) \in S^*\mathbf{T}^2. $$

We emphasize that this geometric control condition was already identified by Glass and Han-Kwan in [28, Prop. 5.1] as playing an important role when studying the classical dynamics induced by a magnetic field. Indeed, under this geometric assumption and given any nonempty open set $\omega \subset \mathbf{T}^2$ and any $T_0 > 0$, they proved that one can find $R_0 > 0$ such that, for every $(x_0, \xi_0) \in T^*\mathbf{T}^2$ verifying $|\xi_0| \ge R_0$, one has

$$ (x', \xi') = (\xi, B(x)\xi^{\perp}), \quad (x(0), \xi(0)) = (x_0, \xi_0) \quad \implies \quad x([0, T_0]) \cap \omega \neq \emptyset. \quad (1.2) $$

In other words, a particle with a high momentum inside a magnetic field verifying $B_\infty > 0$ explores all the configuration space. Even if we will not exploit this dynamical property explicitly as our limit measures are not invariant by the dynamics (1.2), it already illustrates how the geometric control condition on $B$ induces a kind of minimality of the magnetic dynamics on the classical side.

Coming back to the quantum side and as we shall recall it in Section 5, standard arguments of semiclassical analysis show that any accumulation point of the sequence

$$ W_\lambda : a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2, \mathbf{C}) \mapsto \langle \mathrm{Op}_{\lambda-1}^B(a)u_\lambda, u_\lambda \rangle $$

is a probability measure on $S^*\mathbf{T}^2$ that is invariant by the geodesic flow $\varphi^t$ (i.e. not by the magnetic flow appearing in (1.2)). These are the so-called semiclassical measures of $\mathcal{L} + V$. In fact, the geodesic flow has many invariant measures (e.g. invariant measures carried by closed orbit) and Theorem 1.1 shows that, due to their spectral nature, the only semiclassical measure is the Liouville measure. This is referred to as a *quantum unique ergodicity property* for the operator $\mathcal{L} + V$. As far as we know, Theorem 1.1 provides the first family of deterministic Laplace type operators on a compact manifold for which *any* sequence of high frequency eigenfunctions verifies the quantum unique ergodicity property.

It is also important to note that the underlying classical flow (meaning the geodesic flow on $S^*\mathbf{T}^2$) is completely integrable and thus far from being ergodic with respect to the Liouville measure. Hence, among other things, Theorem 1.1 also shows that quantum (unique) ergodicity does not imply classical ergodicity. Other examples of quantum ergodic systems with non-ergodic classical flow were described by Gutkin in [29] and this converse quantum ergodicity problem is discussed in more details by Zelditch in [63, §4.5]. In particular, it seems that, so far, all the known counterexamples to converse quantum ergodicity involved a $\mathbf{Z}_2$-symmetry which is not the case of Theorem 1.1.

As we shall see in the proof, our theorem remains true if we only suppose that the eigenvalue equation (1.1) is satisfied up to a $o_{L^2}(1)$ remainder. In fact, the last part of our theorem will be shown to be true if we only make the assumption that (1.1) is verified up to a $o(\lambda^{1/2})$ remainder.

Once Theorem 1.1 is proven, it is also natural to figure out if one can compute the rate of convergence. Unfortunately, our proof of this theorem is not of very quantitative nature. Yet, in the case where the magnetic field $B$ is constant and where $V$ is identically 0, we are able to provide an alternative (and more direct) proof of Theorem 1.1 using somehow a refined version of (1.2). This will allow us to prove the following quantitative statement:
**Theorem 1.2.** Suppose that $B \equiv \hat{B}_0 > 0$ and that $V \equiv 0$. Then, there exist constants $C_0, N_0 > 0$ such that, for any solution $u_\lambda$ to (1.1) and for any $a \in S^0(T^*T^2)$, one has

$$ \left| \langle \mathrm{Op}_{\lambda-1}^B (a) u_\lambda, u_\lambda \rangle - \frac{1}{2\pi} \int_{S^*T^2} a(x, \theta) \mathrm{d}x \mathrm{d}\theta \right| \le C_0 \|a\|_{\mathcal{C}^{N_0}} (1 + |\lambda|)^{-\frac{1}{2}}, $$

where $\mathrm{Op}_{\lambda-1}^B$ is the semiclassical magnetic Weyl quantization.

### 1.1 Earlier results for vanishing magnetic fields

When $B = V \equiv 0$, it follows from the works of Cooke [20] and Zygmund [65] that any accumulation point $\nu$ of a sequence $\nu_\lambda(dx) := |u_\lambda(x)|^2 dx$ is of the form $h(x)dx$ where $h \in L^2(T^2, \mathbf{R}_+)$. This result was refined by Jakobson [36] who proved that $h$ is in fact a trigonometric polynomial. This cannot be improved as it can be witnessed by considering the sequence of eigenfunctions $u_k(x_1, x_2) = \sin(2\pi x_1)e^{2i\pi kx_2}$ whose limit measure is $\sin^2(2\pi x_1)dx_1dx_2$. See also [35] for earlier results in dimension 2. Jakobson also proved that, in any dimension, any accumulation point is of the form $h(x)dx$ with $h \in L^1(T^d)$. When $V \neq 0$ but still $B \equiv 0$, Bourgain, Burq and Zworski obtained observability properties for the limit measure in dimension 2 in the case of low regularity potentials [16, 13]. Still for general (but more regular) potentials and using the notion of two-microlocal measures [26, 48, 49, 27], Anantharaman and Macià proved that the limit measures are absolutely continuous and enjoy an observability property in any dimension [40, 5]. Together with Fermanian-Kammerer [2] and Léautaud [4], they broadened the scope of applications of these two-microlocal methods and extended these results to eigenfunctions of non-degenerate completely integrable systems on $T^d$ and of Schrödinger operators on the disk. Motivated by the Aharonov-Bohm effect, Macià also proved that these results remain true in the case where the magnetic field is identically 0 but the potential vector is not [41].

Our proof of Theorem 1.1 will in fact follow this kind of semiclassical strategy through second-microlocalization methods. More precisely, it is closely related to the analysis made by Macià and the second author [43] in order to analyze eigenmodes of semiclassical operators of the form $-h^2\Delta + \varepsilon_h^2 V$ with $h \ll \varepsilon_h \ll 1$. Indeed, as we shall see in this article, looking at the high frequency limit of the operator $\mathcal{L} + V$ corresponds in some sense to a subprincipal perturbation of the operator $-h^2\Delta$ of size $h$. Following earlier results on Zoll manifolds [42, 44], the key point in [43] was to understand these perturbative results and two-microlocal procedures through the scope of the quantum averaging method appearing in Weinstein's seminal work on Schrödinger operators on $S^d$ [59]. In some sense, this will be again the mechanism at work behind our proofs and the magnetic nature of the problem will yield the extra regularity properties compared with [43].

Finally, we recall that, given an orthonormal basis of eigenfunctions of $\mathcal{L} + V$, Marklof and Rudnick proved that one can find a density one subsequence of eigenfunctions such that the limit measure is $dx$ in the configuration space $\mathbb{T}^2$ [47]. Theorem 1.1 shows that, as soon as $B$ verifies the geometric control condition $B_\infty > 0$, there is no need to extract a subsequence and equidistribution holds in the full phase space $S^*T^2$.

<footnote n="2">²Strictly speaking, the proof in that reference is given for $B = V \equiv 0$ but the proof could be adapted to deal with the general case.</footnote>
## 1.2 Quantum unique ergodicity

Marklof–Rudnick’s Theorem is in fact the toral analogue of the classical quantum ergodicity Theorem of Shnirelman [55, 56], Zelditch [60] and Colin de Verdière [19]. This theorem states that, as soon as the Liouville measure is ergodic for the geodesic flow of a compact Riemannian manifold $(M, g)$, then most of the eigenfunctions of a given orthonormal basis of Laplace eigenfunctions are equidistributed in phase space $S*M$. This result is in fact true for more general elliptic operators whose underlying Hamiltonian flow is ergodic. See e.g. the works of Helffer, Martínez and Robert [31] for semiclassical Schrödinger operators. A few years later [54], Rudnick and Sarnak made the so-called *quantum unique ergodicity conjecture* by predicting that, for *compact negatively curved manifolds, all* Laplace eigenfunctions should become equidistributed in the high frequency limit. Equivalently the Liouville measure should be the only semiclassical measure for the Laplace-Beltrami operator. Among other things, this was motivated by their result showing that, on arithmetic surfaces, there exists a specific basis of eigenfunctions, the Hecke eigenbasis, for which eigenfunctions put zero mass on closed hyperbolic geodesics in the high-frequency limit. Later on, building on earlier works with Bourgain [14], Lindenstrauss proved that these Hecke eigenmodes become equidistributed in the high frequency limit [39]. This property is often referred as *arithmetic quantum unique ergodicity*.

For more general negatively curved manifolds or more general eigenbasis, the conjecture remains completely open as far as we know. Yet, several important progress were made towards the understanding of this problem in the last twenty years. First, in the related toy-model of quantum cat-maps, it was proved by Faure, Nonnenmacher and De Bièvre that the quantum unique ergodicity conjecture does not hold true in dimension 2 [25]. On the higher dimensional analogues of this model, Kelmer even proved that arithmetic quantum unique ergodicity fails as soon as there exist invariant isotropic rational subspaces [37]. In the case of Laplace eigenfunctions on the Bunimovich stadium (for which ergodicity holds despite the weak hyperbolicity features of the billiard flow), Hassell proved that quantum unique ergodicity also fails [30]. On the positive direction, Ananthanarajan proved that, on negatively curved manifolds, any accumulation point in phase space must have positive Kolmogorov-Sinai entropy and thus discarded full concentration along closed geodesics [1]. Then, together with Koch and Nonnenmacher, they obtained more explicit lower bounds on this entropy [6, 3]. See also [52, 51] for more precise results in dimension 2. More recently, for negatively curved surfaces, Dyatlov, Jin and Nonnenmacher proved that, in the high frequency limit, Laplace eigenfunctions should put some positive mass on every nontrivial open subset of $S*M$ [22, 23]. See also [21, 9, 38] for higher dimensional results in the same vein. These recent results are again specific to chaotic (Anosov) classical dynamics and they are based on a subtle harmonic analysis property originally developed by Dyatlov and Zahl [24] to study spectral gaps in chaotic scattering and referred as a fractal uncertainty principle.

It is worth mentioning that the dynamical mechanism at work in our setting is of very different nature as in these works on quantum ergodicity. Indeed, the underlying Hamiltonian flow for the operator $\mathcal{L} + V$ is the geodesic flow on the unit tangent bundle of the torus which is a completely integrable dynamical system. Thus, it has much weaker chaotic properties: existence of Kronecker tori, but also tori of periodic orbits. Moreover, in the case of the free Laplacian on $\mathbf{T}^2$, the discussion in §1.1 shows that Laplace eigenfunctions are far from being quantum uniquely ergodic. For these reasons, the dynamical properties of the underlying classical flow cannot be the only reason for the equidistribution properties of the eigenfunctions of $\mathcal{L} + V$ and this is rather the action of the magnetic field that will produce equidistribution result in our setting.

Regarding the influence of magnetic fields towards quantum unique ergodicity, we can mention
the works of Zelditch for magnetic Laplacians on surfaces of constant negative curvature [61]. In that reference, he proved that, if $B$ is constant but strong (meaning $\hat{B}_0$ of order $\lambda/\sqrt{2}$), then the corresponding eigenfunctions become equidistributed in phase space (with no need to extract a subsequence). In that set-up, the reason for equidistribution comes from the ergodic properties of the classical flow. Indeed, in his case, the underlying classical flow is the horocycle flow (and not the geodesic flow) which has an unique invariant measure and semiclassical measures are invariant by this flow. See also [18] for recent developments by Charles and Lefeuvre in this direction and [46, 53] for related results on quantum maps by Marklof-Rudnick and Rosenzweig using unique ergodicity of the classical map. Besides the fact that the classical flow is not uniquely ergodic for the Liouville measure, a notable difference of our work with [61, 18] is that we work in the regime where $\hat{B}_0$ is fixed while these references consider the strong magnetic field regime where $|\hat{B}_0| \to \infty$. As highlighted by Cekić and Lefeuvre in [17], an interesting issue would be to understand more generally the asymptotic regularity of eigenfunctions as $\|(\lambda, \hat{B}_0)\| \to \infty$ which would encompass all the different regimes.

Finally, we can mention that another efficient mechanism for producing quantum uniquely ergodic basis of eigenfunctions is given by probabilistic tools. For instance, in the case of the round sphere, one has large multiplicity in the Laplace spectrum and there are thus a large choice of orthonormal basis. This set of eigenbasis can be endowed with a natural probability measure and it was proved by Zelditch that most orthonormal basis have the quantum ergodicity property [62]. This was later refined by VanderKam to show that almost all orthonormal basis on $S^d$ enjoy the quantum unique ergodicity property [58]. In a different direction but still of probabilistic nature, we refer to the works of Bourgade, Yau and Yin [11, 12] and the references therein for quantum unique ergodicity results for eigenfunctions of Wigner matrices, i.e. random matrices with matrix elements distributed by identical mean-zero random variables.

## 1.3 Strategy of proof

We start in Section 2 by giving a precise definition of the magnetic Laplacians $\mathcal{L}_\alpha$ on the torus. In particular these operators act on a space of functions with a twisted periodicity condition denoted by $L^2(\mathbf{T}^2, L)$, see (2.3). We also give an explicit description of the spectrum in the case of constant magnetic fields.

Then, in Section 3 we introduce the magnetic Weyl quantization, in the spirit of [45, 33, 32, 34]. Given a symbol $a \in \mathcal{C}^\infty(\mathbf{T}^2 \times \mathbf{R}^2)$, we show that the quantization $\mathcal{O}_{\mathbf{P}_h^B}(a)$ defines a nice operator on smooth subspaces of $L^2(\mathbf{T}^2, L)$, under some standard regularity assumptions on $a$. We also prove a product formula for this quantization, and a Calderón-Vaillancourt Theorem, following the standard techniques of pseudodifferential calculus.

These tools being introduced, we proceed in the remaining sections to the study of semiclassical measures. We begin in Section 4 with a proof of Theorem 1.2 which deals with the case of constant magnetic fields. Under this assumption, we can directly use the properties of this quantization procedure together with the so-called quantum averaging method and classical estimates on oscillatory integrals on the circle.

In the general case, up to dividing by $\lambda^2 = h^{-2}$, we can write the eigenvalue equation (1.1) in a semiclassical way,

$$h^2(\mathcal{L} + V)u_h = u_h,$$

and we consider a sequence $(u_h)_{h \to 0}$ satisfying this equation – or more generally an approximate version. We will first show that the sequence of measures $|u_h(x)|^2 dx$ has only $dx$ as an accumulation
point. To do so, we introduce in Section 5 the phase-space Wigner distribution,

$$ \langle w_h, a \rangle = \langle \mathrm{Op}_h^B(a) u_h, u_h \rangle, \quad a \in \mathcal{C}_c^\infty(\mathbf{T}^2 \times \mathbf{R}^2), $$

and study its accumulation points $d\mu$ in the limit $h \to 0$. In Lemma 5.2 we recall that a limit distribution must be a probability measure on $\mathbf{T}^2 \times \mathbf{S}^1$, which is invariant under the geodesic flow. As such, we can decompose this measure according to periodic and dense orbits. For irrational angles $\theta$, the geodesic $t \in \mathbf{R} \mapsto x + t\theta \in \mathbf{T}^2$ is dense and the measure $\mu$ restricted to these angles is the Lebesgue measure (see Lemma 5.4). Thus, we only have to describe the measure $\mu$ along periodic orbits and this is the aim of Section 6.

To do so, we use the strategy initiated by Macià in [40] through the use of 2-microlocal measures. More precisely, we proceed as in [43] by splitting the eigenfunctions between the part that is localized in a neighborhood of size $\lesssim h^{1/2}$ of a torus of periodic orbits and the one that is at distance $\gg h^{1/2}$ of such closed geodesics. We prove in Lemma 6.2 that the part at distance $\gg h^{1/2}$ gives rise to the Lebesgue measure $dx$ when $h \to 0^+$. For the part at distance $\lesssim h^{1/2}$, we show in Lemma 6.5 that the induced 2-microlocal measure is invariant by another flow involving the magnetic field $B$. When $B_\infty > 0$, this extra invariance property results into the fact that the 2-microlocal measure is 0. This will conclude the proof of the second part of Theorem 1.1. We use this property together with the properties of the magnetic Weyl quantization to proceed to the first part of Theorem 1.1 in Section 7.

Finally, in Appendix A, we discuss what happens when $B_\infty(x, \theta)$ is not positive everywhere and what partial conclusions can be obtained in that general case using the methods of the present article.

**Remark 1.3.** All along the article, we suppose that $\hat{B}_0 > 0$ for the sake exposition. Yet, the main results remain true when $\hat{B}_0 < 0$ with the assumption that $B_\infty < 0$ everywhere.

## Acknowledgements

We are most grateful to Bernard Helffer for bringing to our attention the magnetic Weyl calculus originally developed in [45] as a convenient and elegant way of quantizing observables in the presence of magnetic fields. The second author also thanks Fabricio Macià for discussions over the years regarding the study of semiclassical measures for completely integrable systems.

LM was supported by the ERC Advanced Grant MathBEC - 101095820. (This work is funded by the European Union. Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.)

GR acknowledges the support of the Institut Universitaire de France, the PRC grant ADYCT (ANR-20-CE40-0017) and the Centre Henri Lebesgue (ANR-11-LABX-0020-01).

## 2 Magnetic Laplacian on the torus

A magnetic field on a configuration space consists in a 2-form $\mathbf{B}$. The difficulty when defining a Schrödinger operator quantizing this magnetic field comes from the fact that $\mathbf{B}$ may not derive from a potential (i.e. it cannot be written $\mathbf{B} = dA$). Note also that, even if there exists such a primitive, there is not necessarily an unique choice for $\mathbf{A}$. The first difficulty is handled by dealing with functions taking their values in a certain line bundle and by considering an appropriate Laplace
type operator acting on this line bundle. In the case where the configuration space is the 2-torus, this construction is particularly simple and explicit and we will review this in this section. This will give rise to a magnetic Schrödinger operator which can be viewed as a periodized version of the magnetic Laplacian on $\mathbf{R}^2$. Finally, as we shall see, there is not a single choice of such operators and there is a whole family of allowed line bundles parametrized by some parameter $\alpha \in \mathbf{R}^2/2\pi\mathbf{Z}^2$. This is related to the Aharonov-Bohm effect in the physics literature.

### 2.1 Definitions

In the 2-dimensional case, a magnetic field $\mathbf{B}$ can be identified with a function $B$ once we have fixed the volume form $dx_1 \wedge dx_2$. Thus, we let $B \in \mathcal{C}^\infty(\mathbf{T}^2, \mathbf{R})$ be a periodic magnetic field that is fixed once and for all. It can be written as the sum of its Fourier series,

$$B(x) = \sum_{p \in \mathbf{Z}^2} \hat{B}_p e^{2i\pi p \cdot x}, \quad \hat{B}_p = \int_{\mathbf{T}^2} B(x) e^{-2i\pi p \cdot x} dx. \qquad (2.1)$$

In particular, we can find a smooth vector potential $A = A^0 + A^{\text{per}}$ such that

* $B = \partial_1 A_2 - \partial_2 A_1$,

* $A^0(x_1, x_2) = \frac{\hat{B}_0}{2}(-x_2, x_1)$,

* $A^{\text{per}}$ is periodic.

**Remark 2.1.** Note that the choice of $A^{\text{per}}$ is not unique and it can be modified by a gradient vector field $\nabla\phi$, with $\phi \in \mathcal{C}^\infty(\mathbf{T}^2, \mathbf{R})$ (the so-called gauge choice). Yet, as usual when studying magnetic Laplacians, it would result into unitarily equivalent operators and our main results are the same for each choice of periodic gauge. Recall indeed that eigenfunctions are modified by multiplication by $e^{i\phi}$ – which does not affect the spaces we will define – while eigenvalues remain the same.

We define the magnetic translations associated to $\hat{B}_0$ by

$$\mathcal{T}_m^B u(x) = e^{\frac{i}{2}\hat{B}_0 m \wedge x} u(x-m), \quad u \in L^2(\mathbf{R}^2), \quad m \in \mathbf{Z}^2, \qquad (2.2)$$

where $m \wedge x = m_1 x_2 - m_2 x_1$. The class of smooth functions invariant under these translations is denoted by

$$\mathcal{C}^{\infty}(\mathbf{T}^2, L) = \{u \in \mathcal{C}^{\infty}(\mathbf{R}^2), \mathcal{T}_m^B u = u \quad \forall m \in \mathbf{Z}^2\}. \qquad (2.3)$$

Note that, for this space to make sense, we need the commutation property $\mathcal{T}_m^B \mathcal{T}_{m'}^B = \mathcal{T}_m^B \mathcal{T}_m^B$, which translates in the flux condition $\hat{B}_0 \in 2\pi\mathbf{Z}$. From a geometrical point of view, these functions can be seen as sections of a line bundle over $\mathbf{T}^2$. The notation $L$ refers to this line bundle, even though we do not need this interpretation in the article. In fact, when considering the limit of large magnetic fields, one usually writes the line bundle as $L^{\otimes\phi}$ (where $\hat{B}_0 = 2\pi\phi$) to emphasize the dependence in $\phi$. In the present work, $\phi$ is fixed and we drop this exponent to alleviate the notations.

**Remark 2.2.** The line bundle on $\mathbf{T}^2$ is the set $L = \mathbf{R}^2 \times \mathbf{C}/\sim$ where

$$\forall m \in \mathbf{Z}^2, \quad (x, u) \sim \left(x - m, e^{\frac{i}{2}\hat{B}_0 m \wedge x} u\right).$$

The corresponding projection is given by $(x, u) \in L \mapsto x \in \mathbf{T}^2$ and $\mathcal{C}^\infty(\mathbf{T}^2, L)$ can be identified with the set of smooth sections $\mathbf{T}^2 \to L$.
Similarly we define the space of square integrable functions (resp. $H^2$ functions) invariant by the magnetic translations,

$$L^2(\mathbf{T}^2, L) = \{u \in L_{\text{loc}}^2(\mathbf{R}^2), \quad \mathcal{T}_m^B u = u \quad \forall m \in \mathbf{Z}^2\}, \qquad (2.4)$$

$$H^2(\mathbf{T}^2, L) = \{u \in H_{\text{loc}}^2(\mathbf{R}^2), \quad \mathcal{T}_m^B u = u \quad \forall m \in \mathbf{Z}^2\}. \qquad (2.5)$$

They are the closure of $\mathcal{C}^\infty(\mathbf{T}^2, L)$ for the norms

$$\|u\|_{L^2} = \left( \int_{\mathbf{T}^2} |u(x)|^2 dx \right)^{\frac{1}{2}}, \quad \|u\|_{H^2} = \left( \sum_{|\alpha| \le 2} \int_{\mathbf{T}^2} |(\partial^A)^\alpha u(x)|^2 dx \right)^{\frac{1}{2}}, \qquad (2.6)$$

where $\partial^A = \partial_x - iA^0$ and where we identify $\mathbf{T}^2$ with the fundamental domain $[0, 1]^2$. Note that, as $\partial^A$ commutes with the magnetic translations, these norms and the associated scalar products are independent of the choice of fundamental domain for $\mathbf{T}^2$.

We define a family of operators, the magnetic Laplacians on the torus.

**Theorem 2.3.** Assume $\hat{B}_0 \in 2\pi\mathbf{Z}$. Then, for all $\alpha \in \mathbf{R}^2$, the operator given by

$$\mathcal{L}_\alpha u = (-i\partial_{x_1} - \alpha_1 - A_1)^2 u + (-i\partial_{x_2} - \alpha_2 - A_2)^2 u, \quad u \in H_{\text{loc}}^2(\mathbf{R}^2),$$

commutes with the magnetic translations, i.e. $\mathcal{L}_\alpha \mathcal{T}_m^B = \mathcal{T}_m^B \mathcal{L}_\alpha$ for $m \in \mathbf{Z}^2$. Moreover, $\mathcal{L}_\alpha$ is self-adjoint on $L^2(\mathbf{T}^2, L)$ with domain

$$D(\mathcal{L}_\alpha) = H^2(\mathbf{T}^2, L),$$

and has compact resolvent. Finally, if $\alpha - \alpha' \in 2\pi\mathbf{Z}^2$, then $\mathcal{L}_\alpha$ and $\mathcal{L}_{\alpha'}$ are unitarily equivalent.

*Proof.* The operator $\mathcal{L}_\alpha$ is well defined on $H_{\text{loc}}^2(\mathbf{R}^2)$, and the commutation property with the magnetic translations can be checked by a direct calculation. In particular, $\mathcal{L}_\alpha$ maps $H^2(\mathbf{T}^2, L)$ on $L^2(\mathbf{T}^2, L)$, and it is thus a well defined unbounded operator on this Hilbert space. By convolution arguments, $\mathcal{C}^\infty(\mathbf{T}^2, L)$ (and thus $H^2(\mathbf{T}^2, L)$) is a dense subspace of $L^2(\mathbf{T}^2, L)$. Moreover, by the Rellich-Kondrachov Theorem (applied on open bounded subsets of $\mathbf{R}^2$), the injection $H^2(\mathbf{T}^2, L) \subset L^2(\mathbf{T}^2, L)$ is compact. It follows that the resolvent $(\mathcal{L}_\alpha + i)^{-1}$ is a compact operator. Using integration by parts, we know that $\mathcal{L}_\alpha$ is symmetric, since the boundary terms vanish by the translation property $\mathcal{T}_m^B u = u$. The adjoint has domain

$$D(\mathcal{L}_\alpha^*) = \{u \in L^2(\mathbf{T}^2, L), \quad \mathcal{L}_\alpha u \in L_{\text{loc}}^2(\mathbf{R}^2)\},$$

and, by elliptic regularity, this is precisely $H^2(\mathbf{T}^2, L)$. Thus $\mathcal{L}_\alpha$ is self-adjoint. To prove the last statement, note that, if $\alpha - \alpha' \in 2\pi\mathbf{Z}^2$, the function $e^{i(\alpha - \alpha') \cdot x}$ is periodic, and we have the conjugation property

$$e^{-i(\alpha - \alpha')x} \mathcal{L}_\alpha e^{i(\alpha - \alpha')x} = \mathcal{L}_{\alpha'}. \qquad \square$$

<figure><img src="image_1.png" /></figure>
### 2.2 The case of a constant magnetic field

As an example, we describe here the spectrum of $\mathcal{L}_\alpha$ in the case of a constant field $B = \hat{B}_0$ (with the gauge $A^{\text{per}} = 0$). Here we also suppose that $B \neq 0$ as, in the case $B = 0$, eigenmodes are given by the standard decomposition in Fourier series. In the case $B \neq 0$, eigenfunctions and eigenvalues are in fact also explicitly computable. To see this, we can conjugate $\mathcal{L}_\alpha$ by a phase to change the gauge,

$$ \mathcal{P}_{\alpha,B} = e^{i(\frac{B}{2}x_2 - \alpha_1)x_1} \mathcal{L}_{\alpha} e^{-i(\frac{B}{2}x_2 - \alpha_1)x_1} = -\partial_{x_1}^2 + (-i\partial_{x_2} - \alpha_2 - Bx_1)^2. $$

Of course the periodicity condition is modified. The domain of $\mathcal{P}_{\alpha,B}$ is now

$$ D(\mathcal{P}_{\alpha,B}) = \{v \in H_{\text{loc}}^2(\mathbf{R}^2), \ v(x) = e^{-i\frac{B}{2}m_1m_2}e^{-i\alpha_1m_1}e^{iBm_1x_2}v(x-m), \ m \in \mathbf{Z}^2\}. $$

In particular, if $v$ is in the domain, then $x_2 \mapsto v(x_1, x_2)$ is 1-periodic and can be written as the sum of its Fourier series,

$$ v(x) = \sum_{p \in \mathbf{Z}} e^{2i\pi p x_2} v_p(x_1). $$

The periodicity condition in $x_1$ translates in

$$ v_{p+\phi}(x_1) = e^{-i\alpha_1} v_p(x_1 - 1), \ p \in \mathbf{Z}, \ x_1 \in \mathbf{R}, $$

where $\phi = \frac{B}{2\pi} \in \mathbf{Z}$. Therefore, any function in the domain of $\mathcal{P}_{\alpha,B}$ is determined by the functions $v_0(x_1), \dots, v_{\phi-1}(x_1)$, through the formula

$$ v(x) = \sum_{n \in \mathbf{Z}} \sum_{p=0}^{\phi-1} e^{2i\pi(n\phi+p)x_2} e^{-in\alpha_1} v_p(x_1-n). $$

Now assume $v$ is an eigenfunction with eigenvalue $\lambda$. The eigenvalue equation on the $p$-mode reads

$$ -\partial_{x_1}^2 v_p + (2\pi p - \alpha_1 - Bx_1)^2 v_p = \lambda v_p, $$

which is a translated harmonic oscillator. The eigenvalues are thus $\lambda_j = (2j - 1)B$ for $j \ge 1$, with multiplicity $\phi$, and the associated eigenfunctions are given by twisted sums of translated Hermite functions.

## 3 A magnetic quantization

In this section, we introduce a magnetic quantization which is adapted to our problem. The goal is to make sense of pseudo-differential operators acting on spaces of functions with the magnetic periodicity condition, namely $\mathcal{C}^\infty(\mathbf{T}^2, L)$ introduced in (2.3). Using the complex line bundle point of view, this could be achieved by considering pseudo-differential operators with operator-valued symbols, i.e. functions lying in $\mathcal{C}^\infty(\mathbf{T}^2 \times \mathbf{R}^2, \text{Hom}(L, L))$. Here, due to the Euclidean structure of our problem, we can proceed in a slightly more explicit manner by considering twisted versions of the standard Weyl quantization [64]. Such a construction was carried out in a microlocal set-up on $\mathbf{R}^d$ by Mantoiu and Purice [45]. See also [15] for earlier related constructions, [33, 34] for further developments regarding magnetic fields and [32] for the semiclassical version. One advantage of this point of view is to ensure that all terms in the asymptotic expansions of standard theorems
of pseudo-differential calculus depend only on $B$ (and not on a choice of a local primitive of $B$ if one would have chosen to work in local charts). Here, we will need a semiclassical and *periodic* version of this magnetic pseudo-differential calculus. Hence, we review how the construction in these works can be adapted to our periodic set-up and what are the main properties of these magnetic pseudo-differential operators.

**Remark 3.1.** Taking $\hat{B}_0 = 0$ in this section, the construction below recovers the standard Weyl quantization on the torus.

### 3.1 Magnetic Weyl quantization

The symbols $a(x, \zeta)$ of such pseudo-differential operators are periodic with respect to $x$, i.e. $a \in C^\infty(\mathbf{T}^2 \times \mathbf{R}^2, \mathbf{C})$. The construction relies on the magnetic Weyl operators $W_h^B(\eta, \zeta)$, defined for $\eta \in 2\pi\mathbf{Z}^2$ and $\zeta \in \mathbf{R}^2$ by

$$ W_h^B(\eta, \zeta)u(x) = e^{\frac{i\eta\hat{B}_0}{2}}\zeta \wedge x e^{i\eta\cdot(x+\frac{h\zeta}{2})}u(x+h\zeta), \quad u \in L^2(\mathbf{T}^2, L). \quad (3.1) $$

It should be noted that $W_h^B(\eta, \zeta)$ commutes with the magnetic translations,

$$ W_h^B(\eta, \zeta) \circ \mathcal{T}_m^B = \mathcal{T}_m^B \circ W_h^B(\eta, \zeta), $$

which makes it a well defined isometry $W_h^B(\eta, \zeta) : L^2(\mathbf{T}^2, L) \to L^2(\mathbf{T}^2, L)$. The adjoint is also explicitly given by

$$ W_h^B(\eta, \zeta)^* = W_h^B(-\eta, -\zeta). \quad (3.2) $$

Finally we have the following composition formula for Weyl operators.

**Proposition 3.2.** For all $\eta, \eta' \in 2\pi\mathbf{Z}^2$, $\zeta, \zeta' \in \mathbf{R}^2$, we have the composition rule

$$ W_h^B(\eta, \zeta) \circ W_h^B(\eta', \zeta') = e^{i\frac{h}{2}\Omega_{hB}(\eta, \zeta; \eta', \zeta')} W_h^B(\eta + \eta', \zeta + \zeta'), $$

where

$$ \Omega_{hB}(\eta, \zeta; \eta', \zeta') := \eta' \cdot \zeta - \eta \cdot \zeta' + h\hat{B}_0\zeta' \wedge \zeta. $$

*Proof.* For $u \in L^2(\mathbf{T}^2, L)$, we have

$$ \begin{align*} W_h^B(\eta, \zeta) \circ W_h^B(\eta', \zeta') u(x) &= e^{i\eta \cdot (x + \frac{h\zeta}{2})} + \frac{i\eta\hat{B}_0}{2}\zeta \wedge x W_h^B(\eta', \zeta') u(x + h\zeta) \\ &= e^{i\eta \cdot (x + \frac{h\zeta}{2})} + \frac{i\eta\hat{B}_0}{2}\zeta \wedge x e^{i\eta' \cdot (x + h\zeta + \frac{h\zeta'}{2})} + \frac{i\eta\hat{B}_0}{2}\zeta' \wedge (x + h\zeta) u(x + h\zeta + h\zeta') \\ &= e^{i\frac{h}{2}(\eta' \cdot \zeta - \eta \cdot \zeta' + h\hat{B}_0\zeta' \wedge \zeta)} e^{i(\eta + \eta') \cdot (x + \frac{h\zeta}{2} + \frac{h\zeta'}{2})} + \frac{i\eta\hat{B}_0}{2}(\zeta + \zeta') \wedge x u(x + h\zeta + h\zeta') \\ &= e^{i\frac{h}{2}(\eta' \cdot \zeta - \eta \cdot \zeta' + h\hat{B}_0\zeta' \wedge \zeta)} W_h^B(\eta + \eta', \zeta + \zeta') u(x). \end{align*} $$

□

Given a symbol $a : \mathbf{T}^2 \times \mathbf{R}^2 \to \mathbf{C}$ which is periodic with respect to the first variable, we formally define the magnetic quantization using the Weyl operators $W_h^B$ by the following formula

$$ \mathrm{Op}_h^B(a) = \frac{1}{(2\pi)^2} \sum_{\eta \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^2} \mathcal{F}(a)(\eta, \zeta) W_h^B(\eta, \zeta) d\zeta, \quad (3.3) $$
where we denoted by $\mathcal{F}$ the Fourier transform of the symbol $a(x, \xi)$, namely

$$ \mathcal{F}(a)(\eta, \zeta) = \int_{\mathbf{T}^2 \times \mathbf{R}^2} a(x, \xi) e^{-i\eta \cdot x} e^{-i\zeta \cdot \xi} dx d\xi. \qquad (3.4) $$

Note that the inverse formula for $\mathcal{F}$ is

$$ \mathcal{F}^{-1}(a)(x, \xi) = \frac{1}{(2\pi)^2} \sum_{\eta \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^2} a(\eta, \zeta) e^{ix \cdot \eta} e^{i\xi \cdot \zeta} d\zeta. $$

This definition (3.3) is analog to the standard Weyl quantization, which correspond to the case $\hat{B}_0 = 0$. In that case, one can write (3.3) in terms of the Fourier coefficients of $u$ – see e.g. [43, Appendix]. However when $\hat{B}_0 \neq 0$, $u$ is not periodic and these techniques no longer make sense. That is a reason why we work instead with the representation (3.3). This quantization is the analogue in a periodic setting of the one introduced by Mantoiu and Purice (see [45, 33, 32, 34] for instance), and we hope that this article opens further potential applications of their formalism to spectral geometry.

Before studying this quantization for general symbols, let us give a couple of important examples. First of all, if $a(x, \xi) = \xi_j$ (for $j = 1, 2$) then we obtain the magnetic derivative

$$ \mathrm{Op}_h^B(\xi_j) = -ih\partial_{x_j} - hA_j^0, \qquad (3.5) $$

where we recall that $A^0(x_1, x_2) = \frac{\hat{B}_0}{2}(-x_2, x_1)$ is our choice of gauge generating the uniform field of strength $\hat{B}_0$. When $a$ is independent of $\xi$, we obtain the multiplication operator

$$ \mathrm{Op}_h^B(a(x)) = a(x), \qquad (3.6) $$

as in the non-magnetic case. More generally, when $a$ belongs to a suitable class of symbol, $\mathrm{Op}_h^B(a)$ defines a continuous operator acting on the space $\mathcal{C}^\infty(\mathbf{T}^2, L)$ of smooth magnetic-periodic functions, introduced in (2.3).

**Proposition 3.3.** Let $m \in \mathbf{R}$, and assume $a \in S^m$, where

$$ S^m = \{a \in \mathcal{C}^\infty(\mathbf{T}^2 \times \mathbf{R}^2, \mathbf{C}), \forall \alpha, \beta \in \mathbf{N}_0^2, |\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| \le C_{\alpha\beta} \langle\xi\rangle^m\}. $$

Then $\mathrm{Op}_h^B(a) : \mathcal{C}^\infty(\mathbf{T}^2, L) \to \mathcal{C}^\infty(\mathbf{T}^2, L)$ defines a continuous operator. Moreover, the formal adjoint is given by

$$ \mathrm{Op}_h^B(a)^* = \mathrm{Op}_h^B(\bar{a}). \qquad (3.7) $$

*Proof.* Here the magnetic field adds no significant problems, so we can use the usual strategy through oscillatory integral arguments [64, §3.6]. For $u \in \mathcal{C}^\infty(\mathbf{T}^2, L)$, we formally integrate by parts in the definition of $\mathrm{Op}_h^B(a)$ until we get a convergent integral. These formal calculations correspond to interpreting (3.3) in distributional sense as in [64, Th. 3.18]. Using the Fourier multiplier $\langle D_x \rangle = (1 - \partial_{x_1}^2 - \partial_{x_2}^2)^{\frac{1}{2}}$ and properties of the Fourier transform we have, for $\beta, \gamma, N \in 2\mathbf{N}$,

$$ \mathrm{Op}_h^B(a)u(x) = \frac{1}{(2\pi)^2} \sum_{\eta \in 2\pi\mathbf{Z}^2} \langle\eta\rangle^{-\gamma} \int_{\mathbf{R}^2} \langle D_\zeta \rangle^N \left[ \langle\zeta\rangle^{-\beta} \mathcal{F}(\langle\xi\rangle^{-N} \langle D_y \rangle^\gamma \langle D_\xi \rangle^\beta a) \right] W_h^B(\eta, \zeta)u(x)d\zeta. $$
When we integrate by part, $D_ζ$ now acts on $W_h^B(η, ζ)u(x)$,

$$ \mathrm{Op}_h^B(a)u(x) = \frac{1}{(2\pi)^2} \sum_{\eta \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{-\gamma} \int_{\mathbb{R}^2} \langle\zeta\rangle^{-\beta} \mathcal{F}(\langle\xi\rangle^{-N} \langle D_y\rangle^{\gamma} \langle D_\xi\rangle^{\beta} a) \langle D_\zeta\rangle^N [W_h^B(\eta, \zeta)u(x)] d\zeta. $$

The derivatives with respect to $ζ$ of the Weyl operator are bounded by seminorms of $u$ and powers of $η$,

$$ |\mathrm{Op}_h^B(a)u(x)| \le C \sum_{\eta \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{-\gamma} \int_{\mathbb{R}^2} \langle\zeta\rangle^{-\beta} |\mathcal{F}(\langle\xi\rangle^{-N} \langle D_y\rangle^{\gamma} \langle D_\xi\rangle^{\beta} a)| \langle\eta\rangle^N \|u\|_{\mathcal{C}^N} d\zeta. $$

If $a \in S^m$ and $N$ large enough, then the Fourier transform $\mathcal{F}$ appearing in the previous integral is a bounded function. Therefore,

$$ |\mathrm{Op}_h^B(a)u(x)| \le C \sum_{\eta \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{N-\gamma} \int_{\mathbb{R}^2} \langle\zeta\rangle^{-\beta} d\zeta \|u\|_{\mathcal{C}^N}, $$

and the above sum and integral are convergent for $γ$ and $β$ large enough. Similarly, we obtain bounds on the derivatives of $\mathrm{Op}_h^B(a)u$, and deduce that $\mathrm{Op}_h^B(a)$ defines a continuous operator on $\mathcal{C}^\infty(\mathbb{T}^2, L)$. The formula for the adjoint follows from the definition (3.3) and property (3.2). □

## 3.2 Composition formula

We can now turn to the first main property of this quantization procedure.

**Theorem 3.4.** Let $m_1, m_2 \in \mathbb{R}$ and assume $a \in S^{m_1}, b \in S^{m_2}$. Then,

1. There exists a $h$-dependent symbol $a \star b \in S^{m_1+m_2}$ such that

$$ \mathrm{Op}_h^B(a) \circ \mathrm{Op}_h^B(b) = \mathrm{Op}_h^B(a \star b). $$

2. The symbol $a \star b$ is explicitly given by the oscillatory integral

$$ a \star b(x, \xi) = \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{T}^4} e^{i(\eta y + \eta' y' + 2\hat{B}_0 y \wedge y')} \\ \times a\left(x - y, \xi + \frac{h\eta'}{2} + h\hat{B}_0 y^{\perp}\right) b\left(x - y', \xi - \frac{h\eta}{2} + h\hat{B}_0 y'^{\perp}\right) dy dy', $$

where $y^{\perp} = (-y_2, y_1)$.

3. For all $N \ge 1$ we have the asymptotic expansion

$$ a \star b(x, \xi) = \sum_{n=0}^{N-1} \left(\frac{ih}{2}\right)^n \Omega_{hB}(D_x, D_\xi, D_{x'}, D_{\xi'})^n a(x, \xi) b(x', \xi') \delta_{(x, \xi)=(x', \xi')} + \mathcal{O}_{S^{m_1+m_2}}(h^N). $$

4. In particular,

$$ a \star b = ab + \mathcal{O}_{S^{m_1+m_2}}(h), $$

and

$$ a \star b - b \star a = \frac{h}{i} \{a, b\} + \frac{\hat{B}_0 h^2}{i} (\partial_{\xi_2} a \partial_{\xi_1} b - \partial_{\xi_1} a \partial_{\xi_2} b) + \mathcal{O}_{S^{m_1+m_2}}(h^3). \quad (3.8) $$
*Proof.* Using the composition rule for Weyl operators (Proposition 3.2), we find

$$ \mathbf{Op}_h^B(a) \circ \mathbf{Op}_h^B(b) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} e^{i\frac{h}{2}(\eta' \cdot \zeta - \eta \cdot \zeta')} e^{\frac{i\hbar^2}{2}\hat{B}_0\zeta' \wedge \zeta} \\ \times \mathcal{F}(a)(\eta, \zeta)\mathcal{F}(b)(\eta', \zeta')W_h^B(\eta + \eta', \zeta + \zeta')d\zeta d\zeta' $$

We translate the variables $\eta$ and $\zeta$ to get

$$ \mathbf{Op}_h^B(a) \circ \mathbf{Op}_h^B(b) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} e^{i\frac{h}{2}(\eta' \cdot (\zeta - \zeta') - (\eta - \eta') \cdot \zeta')} e^{\frac{i\hbar^2}{2}\hat{B}_0\zeta' \wedge \zeta} \\ \times \mathcal{F}(a)(\eta - \eta', \zeta - \zeta')\mathcal{F}(b)(\eta', \zeta')W_h^B(\eta, \zeta)d\zeta d\zeta' $$

Thus $\mathbf{Op}_h^B(a) \circ \mathbf{Op}_h^B(b) = \mathbf{Op}_h^B(a \star b)$ with

$$ \mathcal{F}(a \star b)(\eta, \zeta) = \frac{1}{(2\pi)^2} \sum_{\eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^2} e^{i\frac{h}{2}(\eta' \cdot (\zeta - \zeta') - (\eta - \eta') \cdot \zeta')} e^{\frac{i\hbar^2}{2}\hat{B}_0\zeta' \wedge \zeta} \quad (3.9) \\ \times \mathcal{F}(a)(\eta - \eta', \zeta - \zeta')\mathcal{F}(b)(\eta', \zeta')d\zeta d\zeta' $$

Inverting the Fourier transform, we find

$$ a \star b(x, \xi) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} e^{i\eta \cdot x} e^{i\zeta \cdot \xi} e^{i\frac{h}{2}(\eta' \cdot (\zeta - \zeta') - (\eta - \eta') \cdot \zeta')} e^{\frac{i\hbar^2}{2}\hat{B}_0\zeta' \wedge \zeta} \\ \times \mathcal{F}(a)(\eta - \eta', \zeta - \zeta')\mathcal{F}(b)(\eta', \zeta')d\zeta d\zeta' $$

We translate $\eta$ and $\zeta$ back to obtain

$$ a \star b(x, \xi) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} e^{i(\eta+\eta') \cdot x} e^{i(\zeta+\zeta') \cdot \xi} e^{i\frac{h}{2}(\eta' \cdot \zeta - \eta \cdot \zeta')} e^{\frac{i\hbar^2}{2}\hat{B}_0\zeta' \wedge \zeta} \\ \times \mathcal{F}(a)(\eta, \zeta)\mathcal{F}(b)(\eta', \zeta')d\zeta d\zeta' $$

We recall the notation $\Omega_{hB}(\eta, \zeta; \eta', \zeta') = \eta'\zeta - \eta\zeta' + h\hat{B}_0\zeta' \wedge \zeta$, and insert the definition of the Fourier transform for $a$ and $b$,

$$ a \star b(x, \xi) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} \int_{(\mathbf{T}^2 \times \mathbf{R}^2)^2} e^{i(x, \xi) \cdot (\eta, \zeta)} e^{i(x, \xi) \cdot (\eta', \zeta')} e^{i\frac{h}{2}\Omega_{hB}(\eta, \zeta, \eta', \zeta')} \\ \times a(w)b(w')e^{-iw \cdot (\eta, \zeta)}e^{-iw \cdot (\eta', \zeta')}dwdw'd\zeta d\zeta' $$

Now we change $w$ and $w'$ by $z - w$ and $z - w'$ with $z = (x, \xi)$ to obtain

$$ a \star b(z) = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^4} \int_{(\mathbf{T}^2 \times \mathbf{R}^2)^2} e^{i\frac{h}{2}\Omega_{hB}(\eta, \zeta; \eta', \zeta')} e^{iw \cdot (\eta, \zeta)} e^{iw \cdot (\eta', \zeta')} \\ \times a(z - w)b(z - w')dwdw'd\zeta d\zeta' \\ = \int_{(\mathbf{T}^2 \times \mathbf{R}^2)^2} G_h(w, w')a(z - w)b(z - w')dwdw', \quad (3.10) $$
where

$$G_h(w, w') = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^4} e^{i\frac{\eta}{2}\Omega_{h_B}(\eta, \zeta; \eta', \zeta')} e^{iw \cdot (\eta, \zeta)} e^{iw' \cdot (\eta', \zeta')} d\zeta d\zeta,$$

is the inverse Fourier transform of $\exp(i\frac{\eta}{2}\Omega_{h_B})$. We can compute $G_h$ explicitly as a sum of $\delta$ measures. Indeed, using the notations $w = (y, \theta)$ and $w' = (y', \theta')$, we have

$$G_h(w, w') = \frac{1}{(2\pi)^4} \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^4} e^{\frac{i\eta}{2}\hat{B}_0\zeta'\wedge\zeta} e^{\frac{i\eta'}{2}(\eta'\cdot\zeta - \eta\cdot\zeta')} e^{i(y\cdot\eta + \theta\cdot\zeta + y'\cdot\eta' + \theta'\cdot\zeta')} d\zeta d\zeta'$$

and, using the Poisson formula twice,

$$G_h(w, w') = \frac{h^4}{\pi^4} \sum_{m, m' \in \mathbb{Z}^2} e^{2i\hat{B}_0(y'+m')\wedge(y+m)} e^{\frac{2i}{h}\theta'(y+m)} e^{-\frac{2i}{h}\theta(y'+m')}.$$

Since $\hat{B}_0 m' \wedge m \in 2\pi\mathbb{Z}$, we deduce

$$G_h(w, w') = \frac{h^4}{\pi^4} e^{2i\hat{B}_0 y' \wedge y} e^{\frac{2i}{h}(\theta' y - \theta y')} \frac{1}{(2\pi)^4} \sum_{m \in \mathbb{Z}^2} e^{im\left(\frac{2}{h}\theta' + 2\hat{B}_0 y'^{\perp}\right)} \sum_{m' \in \mathbb{Z}^2} e^{im'\left(-\frac{2}{h}\theta - 2\hat{B}_0 y'^{\perp}\right)},$$

where $y^{\perp} = (-y_2, y_1)$ is such that $y \wedge m = m \cdot y^{\perp}$. Using the Poisson formula again we find

$$G_h(w, w') = e^{-2i\hat{B}_0 y' \wedge y} \sum_{\eta \in 2\pi\mathbb{Z}^2} e^{i\eta y} \delta_{\{\theta' = -h\hat{B}_0 y'^{\perp} + \frac{h\eta}{2}\}} \sum_{\eta' \in 2\pi\mathbb{Z}^2} e^{i\eta' y'} \delta_{\{\theta = -h\hat{B}_0 y'^{\perp} - \frac{h\eta'}{2}\}} \quad (3.11)$$

We insert this in (3.10) to find the product formula,

$$a \star b(x, \xi) = \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{T}^4} e^{i(\eta y + \eta' y' + 2\hat{B}_0 y \wedge y')} \\ \times a\left(x - y, \xi + \frac{h\eta'}{2} + h\hat{B}_0 y^{\perp}\right) b\left(x - y', \xi - \frac{h\eta}{2} + h\hat{B}_0 y'^{\perp}\right) dy dy'$$

Let us now argue that $a \star b$ belongs to the class $S^{m_1+m_2}$. We use the Fourier multiplier $\langle D_y \rangle$, integrate by part, and use that $a \in S^{m_1}$ and $b \in S^{m_2}$,

$$\begin{aligned}
|a \star b(x, \xi)| &= \left| \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{T}^4} \langle\eta\rangle^{-N} \langle\eta'\rangle^{-N} \langle D_y \rangle^N \langle D_{y'} \rangle^N (e^{i(\eta y + \eta' y')}) \right. \\
&\qquad \left. \times e^{2i\hat{B}_0 y \wedge y'} a\left(x - y, \xi + \frac{h\eta'}{2} + h\hat{B}_0 y^{\perp}\right) b\left(x - y', \xi - \frac{h\eta}{2} + h\hat{B}_0 y'^{\perp}\right) dy dy' \right| \\
&= \left| \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{-N} \langle\eta'\rangle^{-N} \int_{\mathbb{T}^4} e^{i(\eta y + \eta' y')} \langle D_y \rangle^N \langle D_{y'} \rangle^N (e^{2i\hat{B}_0 y \wedge y'} \right. \\
&\qquad \left. \times a\left(x - y, \xi + \frac{h\eta'}{2} + h\hat{B}_0 y^{\perp}\right) b\left(x - y', \xi - \frac{h\eta}{2} + h\hat{B}_0 y'^{\perp}\right) dy dy' \right| \\
&\le C \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{-N} \langle\eta'\rangle^{-N} \int_{\mathbb{T}^4} \left(\xi + \frac{h\eta'}{2} + h\hat{B}_0 y^{\perp}\right)^{m_1} \left(\xi - \frac{h\eta}{2} + h\hat{B}_0 y'^{\perp}\right)^{m_2} dy dy' \\
&\le C \langle\xi\rangle^{m_1+m_2},
\end{aligned}$$
for $N$ is large enough. We can use the same argument to bound derivatives of $a \star b$ and we deduce that $a \star b \in S^{m_1+m_2}$. To obtain the expansion of $a \star b$, we start from (3.9), and use a Taylor expansion at order $N$ of the exponential.

$$ 
\begin{aligned}
\mathcal{F}(a \star b)(\eta, \zeta) &= \frac{1}{(2\pi)^2} \sum_{\eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^2} e^{i\frac{\eta}{2}\Omega_{hB}(\eta-\eta', \zeta-\zeta', \eta', \zeta')} \mathcal{F}(a)(\eta-\eta', \zeta-\zeta')\mathcal{F}(b)(\eta', \zeta')d\zeta' \\
&= \frac{1}{(2\pi)^2} \sum_{n=0}^{N-1} \left(\frac{ih}{2}\right)^n \sum_{\eta' \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^2} \Omega_{hB}(\eta-\eta', \zeta-\zeta', \eta', \zeta')^n \\
&\qquad \times \mathcal{F}(a)(\eta-\eta', \zeta-\zeta')\mathcal{F}(b)(\eta', \zeta')d\zeta' + h^N \mathcal{F}(\mathcal{R}_N).
\end{aligned}
$$

We recognize the Fourier multiplier

$$ \mathcal{F}(a \star b)(\eta, \zeta) = \sum_{n=0}^{N-1} \left(\frac{ih}{2}\right)^n \mathcal{F}\left(\Omega_{hB}(D_x, D_\xi, D_{x'}, D_{\xi'})^n a(x, \xi)b(x', \xi')\delta_{(x,\xi)=(x',\xi')}\right) + h^N \mathcal{F}(\mathcal{R}_N). \quad (3.12) $$

We can use the inverse Fourier transform and control the remainder term $\mathcal{R}_N$ in the same way as above to deduce

$$ a \star b(x, \xi) = \sum_{n=0}^{N-1} \left(\frac{ih}{2}\right)^n \Omega_{hB}(D_x, D_\xi, D_{x'}, D_{\xi'})^n a(x, \xi)b(x', \xi')\delta_{(x,\xi)=(x',\xi')} + \mathcal{O}_{S^{m_1+m_2}}(h^N). $$

Observe that the terms $n=1$ and $n=2$ are of the form

$$ 
\begin{aligned}
(n=1) \qquad & \frac{ih}{2}(\partial_x a \partial_\xi b - \partial_\xi a \partial_x b) + \frac{ih^2}{2}\hat{B}_0(\partial_{\xi_1} a \partial_{\xi_2} b - \partial_{\xi_2} a \partial_{\xi_1} b), \\
(n=2) \qquad & -\frac{h^2}{4}(\partial_\xi^2 a \partial_x^2 b + \partial_x^2 a \partial_\xi^2 b - 2\partial_x \partial_\xi a \partial_x \partial_\xi b) + \mathcal{O}_{S^{m_1+m_2}}(h^3),
\end{aligned}
$$

and the theorem follows. □

Using this composition formula, we can already observe that

$$ h^2 \mathcal{L}_\alpha = \sum_{j=1}^{2} \mathcal{O}_{h}^{B} (\xi_j - h\alpha_j - hA_j^{\text{per}})^2 = \mathcal{O}_{h}^{B} (|\xi|^2 + hc_1(x) \cdot \xi + h^2 c_0(x)), \quad (3.13) $$

where $c_0$ and $c_1$ belong to $\mathcal{C}^\infty(\mathbb{T}^2, \mathbb{R})$ and $\mathcal{C}^\infty(\mathbb{T}^2, \mathbb{C}^2)$ respectively. Here, we used that the symbols are linear in the $\xi$-variable so that all the high order terms appearing in the Taylor formula used to prove the composition formula cancels out. In view of proving Theorem 1.2, it is also worth collecting the following consequence of the composition formula:

**Corollary 3.5.** Let $a \in S^m$ and $\alpha \in \mathbb{R}^2$. Then, one has

$$ [\mathcal{O}_{h}^{B}(|\xi - h\alpha|^2), \mathcal{O}_{h}^{B}(a)] = \mathcal{O}_{h}^{B} \left( \frac{2h}{i}(\xi - h\alpha) \cdot \partial_x a - \frac{2h^2 \hat{B}_0}{i}(\xi - h\alpha)^\perp \cdot \partial_\xi a \right). $$
The important observation here is that this is an exact formula. In particular, there is no remainder term (even $\mathcal{O}(h^\infty)$). This is due to the fact that $|\xi-h\alpha|^2$ is quadratic in $\xi$ and independent of $x$, which implies that all the higher order terms in the Taylor formula used to prove Theorem 3.4 cancel. For later purpose, it is useful to compute the flow induced by the vector field appearing in this formula, namely

$$ \mathbf{X}_h(x, \xi) := 2(\xi - h\alpha) \cdot \partial_x - 2h\hat{B}_0(\xi - h\alpha)^\perp \cdot \partial_\xi. $$

An explicit computation shows that the induced flow is given by

$$ \Phi_h^t(x, \xi) = \left( x + \frac{1}{h\hat{B}_0} \left( e^{2th\hat{B}_0 J} - \mathrm{Id} \right) (\xi - h\alpha)^\perp, h\alpha + e^{2th\hat{B}_0 J} (\xi - h\alpha) \right), \quad (3.14) $$

where $J := \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. From this expression, one can remark on the one hand that

$$ \lim_{h \to 0} \Phi_h^t(x, \xi) = (x + 2t\xi, \xi) = \varphi^{2t}(x, \xi) $$

is the standard geodesic flow (up to reparametrization of time by a factor 2). On the other hand, the flow $\Phi_h^t$ is $\pi/(h\hat{B}_0)$-periodic which is somehow a manifestation of the cyclotron motion at work in magnetic fields.

## 3.3 Calderón-Vaillancourt Theorem

In view of applying this composition formula in concrete situations, one needs to ensure the boundedness of pseudo-differential operators on $L^2$ spaces. This is the content of the Calderón-Vaillancourt Theorem which can be stated as follows.

**Theorem 3.6.** If $a \in S^0$ then $\mathbf{Op}_h^B(a)$ is a bounded operator on $L^2(\mathbf{T}^2, L)$ and

1. If $h = 1$ then

$$ \| \mathbf{Op}_1^B(a) \|_{L^2 \to L^2} \le C_0 \sum_{|\gamma|+|\beta| \le N_0} \| \partial_x^\gamma \partial_\xi^\beta a \|_{\infty} $$

for some $C_0 > 0$ and $N_0 > 0$ independent of $a$.

2. If $h \in (0, 1]$ then

$$ \| \mathbf{Op}_h^B(a) \|_{L^2 \to L^2} \le C_0 \sum_{|\gamma|+|\beta| \le N_0} h^{|\beta|} \| \partial_x^\gamma \partial_\xi^\beta a \|_{\infty}. $$

*Proof.* We follow the strategy from the proof of the Calderón-Vaillancourt Theorem given in [64, Theorem 4.23]. Let $\chi$ be a smooth compactly supported cutoff function such that $0 \le \chi \le 1$ and

$$ \sum_{k \in \mathbb{Z}^2} \chi_k(\xi) = 1, \quad \chi_k(\xi) = \chi(\xi - k), \quad \forall \xi \in \mathbb{R}^2. $$

Then we decompose $a$ as a sum of compactly supported symbols,

$$ a = \sum_{k \in \mathbb{Z}^2} a_k, \quad a_k(x, \xi) = a(x, \xi)\chi_k(\xi). $$
For $k, \ell \in \mathbb{Z}^2$ we introduce the symbol $b_{k\ell} = \bar{a}_k \star a_\ell$. Let us first explain why the operators $\mathbf{Op}_1^B(b_{k\ell})$ are bounded in $L^2$, with uniform bounds of the form

$$ \| \mathbf{Op}_1^B(b_{k\ell}) \|_{L^2 \to L^2} \le C_N \langle k - \ell \rangle^{-N}, \quad (3.15) $$

for all $N > 0$. Using the composition formula in Theorem 3.4, and using the Fourier multipliers $\langle D_y \rangle, \langle D_{y'} \rangle$ we have

$$ b_{k\ell}(x, \xi) = \int_{\mathbb{T}^4} \sum_{\eta, \eta' \in 2\pi\mathbb{Z}^2} \langle\eta\rangle^{-M} \langle\eta'\rangle^{-M} e^{i(\eta y + \eta' y')} \langle D_y \rangle^M \langle D_{y'} \rangle^M \left( e^{2i\hat{B}_0 y \wedge y'} \times \bar{a}_k(x - y, \xi + \frac{\eta'}{2} + \hat{B}_0 y^\perp) a_\ell(x - y', \xi - \frac{\eta}{2} + \hat{B}_0 y'^\perp) \right) dy dy'. $$

We choose the fundamental domain $(0, 1)^2$ to integrate, and bound the symbol as

$$ |b_{k\ell}(x, \xi)| \le \sum_{\eta, \eta'} \langle\eta\rangle^{-M} \langle\eta'\rangle^{-M} \int_{(0,1)^4} |\langle D_y \rangle^M \langle D_{y'} \rangle^M \left( e^{2i\hat{B}_0 y \wedge y'} \times \bar{a}_k(x - y, \xi + \frac{\eta'}{2} + \hat{B}_0 y^\perp) a_\ell(x - y', \xi - \frac{\eta}{2} + \hat{B}_0 y'^\perp) \right)| dy dy'. $$

Due to the support of $a_k$ and $a_\ell$, the integrand is vanishing unless

$$ |\xi + \frac{\eta'}{2} + \hat{B}_0 y^{\perp} - k| \le C, \quad \text{and} \quad |\xi - \frac{\eta}{2} + \hat{B}_0 y'^{\perp} - \ell| \le C, $$

which implies

$$ \langle k - \ell \rangle \le C \langle \eta \rangle + C \langle \eta' \rangle, \quad \text{and} \quad \langle \xi - \frac{k+\ell}{2} \rangle \le C \langle \eta \rangle + C \langle \eta' \rangle, $$

where the constant $C$ may depend on $\hat{B}_0$. It also depends on at most $2M$ derivatives of $a$. Therefore,

$$ |b_{k\ell}(x, \xi)| \le C \sum_{\eta, \eta'} \langle\eta\rangle^{2N-M} \langle\eta'\rangle^{2N-M} \langle k - \ell \rangle^{-N} \langle\xi - \frac{k+\ell}{2}\rangle^{-N} $$

Taking $M$ large enough the sum converges, and arguing similarly for derivatives of $b$ we obtain

$$ |\partial_x^\alpha \partial_\xi^\beta b(x, \xi)| \le C_N \langle k - \ell \rangle^{-N} \langle\xi - \frac{k+\ell}{2}\rangle^{-N}. \quad (3.16) $$

We now use the definition (3.3) of the quantization to find

$$ \begin{aligned} \| \mathbf{Op}_1^B(b_{k\ell}) \|_{L^2 \to L^2} &\le \frac{1}{(2\pi)^2} \sum_{\eta \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^2} |\mathcal{F}(b_{k\ell})(\eta, \zeta)| d\zeta \\ &\le C \sum_{\eta \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^2} \langle\eta\rangle^{-M} \langle\zeta\rangle^{-M} |\mathcal{F}(\langle D_x \rangle^M \langle D_\xi \rangle^M b_{k\ell})| d\zeta \\ &\le C \sum_{\eta \in 2\pi\mathbb{Z}^2} \int_{\mathbb{R}^2} \langle\eta\rangle^{-M} \langle\zeta\rangle^{-M} \|\langle\eta\rangle^3 \langle D_x \rangle^M \langle D_\xi \rangle^M b_{k\ell}\|_{L^1} d\zeta \\ &\le C \langle k - \ell \rangle^{-N}, \end{aligned} $$
and note that $C$ only depends on a finite number of derivatives of $a$. Since

$$ \mathrm{Op}_1^B(b_{k\ell}) = \mathrm{Op}_1^B(a_k)^* \mathrm{Op}_1^B(a_\ell), $$

we deduce with $N$ large enough that

$$ \sup_k \sum_\ell \| \mathrm{Op}_1^B(a_k)^* \mathrm{Op}_1^B(a_\ell) \|^{1/2} \le C(a), \qquad (3.17) $$

for some $C(a)$ independent of $k, l$, but depending on finitely many derivatives of $a$. Similarly,

$$ \sup_k \sum_\ell \| \mathrm{Op}_1^B(a_k) \mathrm{Op}_1^B(a_\ell)^* \|^{1/2} \le C(a), \qquad (3.18) $$

and it follows from the Cotlar-Stein Theorem [64, C.5] that $\mathrm{Op}_1^B(a) = \sum_k \mathrm{Op}_1^B(a_k)$ is a bounded operator on $L^2$, with norm smaller than $C(a)$. The statement for $h \in (0, 1]$ follows from the Remark 3.7 below. $\square$

**Remark 3.7.** It is useful to observe that

$$ \mathrm{Op}_h^B(a(x, \delta\xi)) = \mathrm{Op}_{h\delta}^B(a(x, \xi)). \qquad (3.19) $$

In particular, we can deduce from the upper bound

$$ \| \mathrm{Op}_1^B(a) \|_{L^2 \to L^2} \le C_0 \sum_{|\gamma|+|\beta| \le N_0} \| \partial_x^\gamma \partial_\xi^\beta a \|_{\infty} $$

that

$$ \| \mathrm{Op}_h^B(a) \|_{L^2 \to L^2} \le C_0 \sum_{|\gamma|+|\beta| \le N_0} h^{|\beta|} \| \partial_x^\gamma \partial_\xi^\beta a \|_{\infty}, $$

which will be useful in view of picking symbols oscillating along the $\xi$-variable.

We finally deduce from Theorems 3.4 and 3.6 the following useful properties of our quantization procedure.

**Corollary 3.8.** Let $a, b \in S^0$. Then one has the following properties.

$$ 1. \quad \| \mathrm{Op}_h^B(a) \mathrm{Op}_h^B(b) - \mathrm{Op}_h^B(ab) \|_{L^2 \to L^2} = \mathcal{O}(h), $$

$$ 2. \quad \left\| \left[ \mathrm{Op}_h^B(a), \mathrm{Op}_h^B(b) \right] - \frac{h}{i} \mathrm{Op}_h^B(\{a, b\}) - \frac{\hat{B}_0 h^2}{i} \mathrm{Op}_h^B(\partial_{\xi_2} a \partial_{\xi_1} b - \partial_{\xi_1} a \partial_{\xi_2} b) \right\|_{L^2 \to L^2} = \mathcal{O}(h^3), $$

$$ 3. \quad \text{If } \mathrm{supp} a \cap \mathrm{supp} b = \emptyset \text{ then } \| \mathrm{Op}_h^B(a) \mathrm{Op}_h^B(b) \|_{L^2 \to L^2} = \mathcal{O}(h^\infty). $$

# 4 Proof for constant magnetic fields

In this Section, we will give a proof of Theorem 1.2, meaning that we will deal with the simplified case where $B$ is constant equal to $\hat{B}_0$ and where $V$ is identically 0. In that case, we are able to give a more direct proof of our quantum unique ergodicity property based on the exact formula given in Corollary 3.5 together with appropriate stationary phase estimates. This allows to illustrate the
dynamical mechanisms at work behind our proof even if it will not appear as explicitly when dealing with more general magnetic fields. Note that our proof does not make use of the exact expression of the eigenfunctions derived in Section 2.

All along this section, $a$ will denote an element in $S^0$ and we will decompose it in Fourier series along the $x$ variable as follows:

$$a(x, \xi) := \sum_{k \in \mathbb{Z}^2} \hat{a}_k(\xi) e^{2i\pi k \cdot x}.$$

We also fix a function $\chi \in \mathscr{C}^\infty(\mathbf{R}, \mathbf{R}_+)$ which is identically equal to 1 (resp. 0) on $[-1/4, 1/4]$ (resp. outside $[-1/2, 1/2]$). From $\chi$ and given any $0 < \delta < 1$, we define the smooth function

$$\chi_{\delta}(\xi) = \chi\left(\frac{|\xi - h\alpha| - 1}{\delta}\right).$$

## 4.1 Averaging method

The following averaging lemma is key step in the proof of Theorem 1.2.

**Lemma 4.1.** There exist constants $C_0, N_0 > 0$ such that, for any $0 < h < 1$, any $a \in S^0$ and any $u_h$ solution to

$$h^2 \mathscr{L}_{\alpha} u_h = u_h, \quad \|u_h\|_{L^2} = 1,$$

one has

$$\left| \langle \mathbf{O}_{\mathbf{p}_h^B}(a)u_h, u_h \rangle - \left\langle \mathbf{O}_{\mathbf{p}_h^B} \left( \chi_1(\xi) \frac{1}{2\pi} \int_{S^*\mathbf{T}^2} a(x, |\xi - h\alpha|\theta + h\alpha) \mathrm{d}x \mathrm{d}\theta \right) u_h, u_h \right\rangle \right| \le C_0 \|a\|_{\mathscr{C}^{N_0}} h^{\frac{1}{2}}.$$

The idea behind the proof of this lemma is that the “quantum” flow (3.14) induced by $h^2 \mathscr{L}_{\alpha}$ is periodic with period $\pi/(h\hat{B}_0)$. This allows to apply an averaging argument similar to the one used by Weinstein in his seminal work on eigenvalues of Schrödinger operators on the sphere [59]. See also [42, 44, 8, 7, 18] for the use of this averaging argument in the context of semiclassical measures. The extra difficulty compared with these references where the “quantum” flow is $2\pi$-periodic is that we average over very long semiclassical times of order $h^{-1}$. Yet, the nice properties of our quantization procedure (namely Corollary 3.5) ensure that this averaging procedure makes sense. In summary, this periodicity allows to replace $a$ by its average along a circle of radius of order $h^{-1}$. Then, a standard stationary phase argument allows to replace this first average of $a$ by its average over the $x$-variable.

*Proof.* We start by writing

$$\langle \mathbf{O}_{\mathbf{p}_h^B}(a)u_h, u_h \rangle = \langle \mathbf{O}_{\mathbf{p}_h^B}(a\chi_1)u_h, u_h \rangle + \langle \mathbf{O}_{\mathbf{p}_h^B}(a(1-\chi_1))u_h, u_h \rangle.$$

From the composition formula and (3.13), we can write

$$\langle \mathbf{O}_{\mathbf{p}_h^B}(a(1-\chi_1))u_h, u_h \rangle = \langle \mathbf{O}_{\mathbf{p}_h^B}(a(1-\chi_1)(|\xi - h\alpha|^2 - 1)^{-1})(h^2 \mathscr{L}_{\alpha} - 1)u_h, u_h \rangle + \mathcal{O}_a(h),$$

where the constant in the remainder depends on a finite number of derivatives of $a$. Then, the eigenvalue equation $h^2 \mathscr{L}_{\alpha} u_h = u_h$ allows us to deduce that

$$\langle \mathbf{O}_{\mathbf{p}_h^B}(a)u_h, u_h \rangle = \langle \mathbf{O}_{\mathbf{p}_h^B}(a\chi_1)u_h, u_h \rangle + \mathcal{O}_a(h). \quad (4.1)$$
Using the eigenvalue equation one more time together with Corollary 3.5, we also get that, for every $a$ in $S^0$,

$$ \langle \mathbf{Op}_h^B (\chi_1 \mathbf{X}_h(a)) u_h, u_h \rangle = 0, $$

where we have used that $\mathbf{X}_h(\chi_1) = 0$ and the exact commutation formula from Corollary 3.5. We now apply this formula to $a \circ \Phi_h^t$, where $\Phi_h^t$ is the flow induced by $\mathbf{X}_h$ and defined by (3.14). In particular, we find that, for all $t \in \mathbf{R}$,

$$ \langle \mathbf{Op}_h^B (\chi_1 \mathbf{X}_h (a \circ \Phi_h^t)) u_h, u_h \rangle = 0. $$

Integrating this expression, we get that

$$ \langle \mathbf{Op}_h^B (\chi_1 a \circ \Phi_h^t) u_h, u_h \rangle = \langle \mathbf{Op}_h^B (\chi_1 a) u_h, u_h \rangle. $$

Then, we can integrate this invariance relation over a period of the flow $\Phi_h^t$ and we obtain

$$ \langle \mathbf{Op}_h^B (a) u_h, u_h \rangle = \left\langle \mathbf{Op}_h^B \left( \chi_1 \frac{h \hat{B}_0}{\pi} \int_0^{\frac{\pi}{h \hat{B}_0}} a \circ \Phi_h^t dt \right) u_h, u_h \right\rangle + \mathcal{O}_a(h). \quad (4.2) $$

We can now recall the exact expression (3.14) for the flow $\Phi_h^t$ to infer

$$ \langle \mathbf{Op}_h^B (a) u_h, u_h \rangle = \langle \mathbf{Op}_h^B (\chi_1(\xi) \langle a \rangle_h(x, \xi)) u_h, u_h \rangle + \mathcal{O}_a(h), $$

where

$$ \langle a \rangle_h(x, \xi) := \frac{1}{2\pi} \int_0^{2\pi} a \left( x + \frac{(\xi - h\alpha)^\perp}{h\hat{B}_0} - J \frac{e^{tJ}}{h\hat{B}_0} (\xi - h\alpha), e^{tJ}(\xi - h\alpha) + h\alpha \right) dt. $$

For $\xi - h\alpha \neq 0$, this can rewritten as an integral over the circle:

$$ \langle a \rangle_h(x, \xi) = \frac{1}{2\pi} \int_{S^1} a \left( x + \frac{(\xi - h\alpha)^\perp}{h\hat{B}_0} - J \frac{|\xi - h\alpha|\theta}{h\hat{B}_0}, |\xi - h\alpha|\theta + h\alpha \right) d\theta, $$

which can be also decomposed using the Fourier decomposition of $a$ in the $x$-variable,

$$ \langle a \rangle_h(x, \xi) = \frac{1}{2\pi} \int_{S^1} \hat{a}_0 (|\xi - h\alpha|\theta + h\alpha) d\theta \\ + \frac{1}{2\pi} \sum_{k \neq 0} e^{2i\pi k \cdot \left(x + \frac{(\xi - h\alpha)^\perp}{h\hat{B}_0}\right)} \int_{S^1} e^{i \frac{2\pi |\xi - h\alpha| Jk}{h\hat{B}_0} \cdot \theta} \hat{a}_k (|\xi - h\alpha|\theta + h\alpha) d\theta. $$

According to the Calderón-Vaillancourt Theorem 3.6, each $k \neq 0$ yields the following contribution to the value of $\langle \mathbf{Op}_h^B (a) u_h, u_h \rangle$:

$$ \sum_{|\gamma|+|\beta| \le N_0} h^{|\beta|} \langle k \rangle^{|\gamma|} \left\| \partial_{\xi}^{\beta} \left( \chi_1(\xi) e^{\frac{2i\pi k \cdot \left(\frac{\xi-h\alpha}{h\hat{B}_0}\right)^\perp}{h\hat{B}_0}} \int_{S^1} e^{\frac{i2\pi|\xi-h\alpha|Jk}{h\hat{B}_0} \cdot \theta} \hat{a}_k (|\xi-h\alpha|\theta+h\alpha) d\theta \right) \right\|_{\infty}. $$

Recall now that the function $\chi_1(\xi)$ ensures that $|\xi - h\alpha|$ lies in the interval $[1/2, 3/2]$ as it depends on $\xi$ through the variable $|\xi - h\alpha|$. Hence, the above sum can be bounded by

$$ C_1 \langle k \rangle^{N_0} \sum_{|\beta|, \ell \le N_0} \sup_{\frac{1}{2} \le |\xi+h\alpha| \le \frac{3}{2}} \left| \int_{S^1} e^{\frac{i2\pi|\xi+h\alpha|Jk}{h\hat{B}_0} \cdot \theta} \left( \frac{k}{|k|} \cdot \theta \right)^{\ell} (\partial^{\beta} \hat{a}_k) (|\xi-h\alpha|\theta+h\alpha) d\theta \right|, $$
where $C_1 > 0$ depends only on $\chi$ and on $\hat{B}_0$. We are now in position to apply a standard stationary phase asymptotics [57, Th. 1, §VIII.3.1] to deduce that the oscillatory integral in this sum is bounded by

$$ \left| \int_{S^1} e^{i \frac{2\pi |\xi - h\alpha| Jk}{h \hat{B}_0} \cdot \theta} \left( \frac{k}{|k|} \cdot \theta \right)^\ell (\partial^\beta \hat{a}_k) (|\xi - h\alpha|\theta + h\alpha) \, d\theta \right| \le C_2 h^{\frac{1}{2}} \|\hat{a}_k\|_{\mathscr{C}^{N_2}} \quad (4.3) $$

where $C_2, N_2 > 0$ do not depend on $a$ and $\alpha$ and where $C_2$ (but not $N_2$) depends also on $\hat{B}_0$. Thanks to (4.2), we finally get that

$$ \langle \mathbf{Op}_h^B(a) u_h, u_h \rangle = \left\langle \mathbf{Op}_h^B \left( \chi_1(\xi) \frac{1}{2\pi} \int_{S^1} \hat{a}_0 (|\xi - h\alpha|\theta + h\alpha) \, d\theta \right) u_h, u_h \right\rangle + \mathcal{O}_a(h^{\frac{1}{2}}), $$

where the constant in the remainder depends on a finite number of derivatives of $a$ and on $\hat{B}_0$. This concludes the proof of the Lemma. □

## 4.2 Microlocalization near the unit cotangent bundle

We are now in position to prove Theorem 1.2. To do so and according to Lemma 4.1, it is sufficient to analyze the following quantity:

$$ \left\langle \mathbf{Op}_h^B \left( \chi_1(\xi) \int_{S^*T^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dx d\theta \right) u_h, u_h \right\rangle. $$

It can be split in two parts that we will analyze independently,

$$ A_1(h) := \left\langle \mathbf{Op}_h^B \left( \chi_1(\xi) \chi_h^{\frac{1}{2}}(\xi) \int_{S^*T^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dx d\theta \right) u_h, u_h \right\rangle, $$

and

$$ A_2(h) := \left\langle \mathbf{Op}_h^B \left( \chi_1(\xi) \left(1 - \chi_h^{\frac{1}{2}}(\xi)\right) \int_{S^*T^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dx d\theta \right) u_h, u_h \right\rangle. $$

We begin with the case of $A_2(h)$. To that aim, we first observe that the symbol

$$ \frac{h^{\frac{1}{2}}(1 - \chi_h^{\frac{1}{2}}(\xi))}{|\xi - h\alpha|^2 - 1} \chi_1(\xi) \int_{S^*T^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dx d\theta $$

involves the function $\tilde{\chi}(t) := t^{-1}(1 - \chi(t))$ which is a smooth function that is identically equal to 0 on $[-1/4, 1/4]$. Hence, even if the symbol has all its derivatives with respect to $\xi$ bounded, it is not strictly speaking an element in $S^0$ as the derivatives are not bounded uniformly with respect to $h$. In fact, one has

$$ \partial_{\xi}^{\beta} \left( \tilde{\chi} \left( \frac{|\xi - h\alpha| - 1}{h^{\frac{1}{2}}} \right) \right) = \mathcal{O}(h^{-\frac{|\beta|}{2}}). $$

In particular, observing that all the involved symbols depend only on the $\xi$-variable and applying the composition formula from Theorem 3.4 together with the Calderón-Vaillancourt Theorem 3.6, we find that, up to $\mathcal{O}_a(h)$ remainder, $A_2(h)$ is equal to

$$ \left\langle \mathbf{Op}_h^B \left( \frac{\chi_1(\xi)}{|\xi - h\alpha| + 1} \tilde{\chi} \left( \frac{|\xi - h\alpha| - 1}{h^{\frac{1}{2}}} \right) \int_{S^*T^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dx d\theta \right) \frac{(h^2 \mathcal{L}_{\alpha} - 1)}{h^{\frac{1}{2}}} u_h, u_h \right\rangle. $$
Using the eigenvalue equation for $u_h$, this is in fact equal to 0 and we get that the proof of Theorem 1.2 reduces to estimating $A_1(h)$. To deal with this term, we apply the Taylor formula to write down

$$ \int_{S^*\mathbf{T}^2} a(x, |\xi - h\alpha|\theta + h\alpha) \, dxd\theta = \int_{S^*\mathbf{T}^2} a(x, \theta) \, dxd\theta \\ + \int_0^1 \int_{S^*\mathbf{T}^2} (\partial_\xi a)(x, (1-t)\theta + t(|\xi - h\alpha|\theta + h\alpha)) \cdot ((|\xi - h\alpha| - 1)\theta + h\alpha) \, dtdxd\theta. $$

Again, the symbol

$$ \chi_1(\xi)\chi_{h^{\frac{1}{2}}}^B(\xi) \int_0^1 \int_{S^*\mathbf{T}^2} (\partial_\xi a)(x, (1-t)\theta + t(|\xi - h\alpha|\theta + h\alpha)) \cdot ((|\xi - h\alpha| - 1)\theta + h\alpha) \, dtdxd\theta $$

does not have its derivative uniformly bounded with respect to $h$. It only verifies that the derivatives of order $\beta$ are of size $\mathcal{O}(h^{\frac{1-|\beta|}{2}})$. Using the Calderón-Vaillancourt Theorem 3.6, this implies that

$$ A_1(h) = \int_{S^*\mathbf{T}^2} a(x, \theta) \, dxd\theta \left\langle \mathcal{O}_{\mathbf{p}_h^B} \left( \chi_1(\xi)\chi_{h^{\frac{1}{2}}}(\xi) \right) u_h, u_h \right\rangle + \mathcal{O}_a(h^{\frac{1}{2}}). $$

Applying our proof with $a \equiv 1$ implies that

$$ \left\langle \mathcal{O}_{\mathbf{p}_h^B} \left( \chi_1(\xi) \chi_{h^{\frac{1}{2}}}(\xi) \right) u_h, u_h \right\rangle = 1 + \mathcal{O}(h^{\frac{1}{2}}), $$

which concludes the proof of Theorem 1.2.

# 5 Magnetic semiclassical measures

In the following, we will say that a probability measure $\nu$ on $\mathbf{T}^2$ is a quantum limit if there exists a sequence $(u_{h_n})_{n \ge 1}$ in $H^2(\mathbf{T}^2, L)$ solving the approximate eigenvalue equation

$$ h_n^2 \mathcal{L}_\alpha u_{h_n} = u_{h_n} + o_{L^2} \left( h_n^{\frac{3}{2}} \right), \quad \|u_{h_n}\|_{L^2(\mathbf{T}^2)} = 1, \quad h_n \to 0^+, \qquad (5.1) $$

and such that

$$ \forall a \in \mathcal{C}^0(\mathbf{T}^2, \mathbf{C}), \quad \lim_{n \to \infty} \int_{\mathbf{T}^2} a(x) |u_{h_n}(x)|^2 dx = \int_{\mathbf{T}^2} a(x) \nu(dx). \qquad (5.2) $$

Note that, as $u_{h_n}$ belongs to $L^2(\mathbf{T}^2, L)$, the function $|u_{h_n}(x)|^2$ belongs to $L^1(\mathbf{T}^2, \mathbf{R}_+)$. Moreover, as $\mathbf{T}^2$ is a compact metric space, the set of probability measures on $\mathbf{T}^2$ is compact (for the weak-⋆ topology) and we can always extract converging subsequences from a given sequence of probability measures. The set of quantum limits for $\mathcal{L}_\alpha$ will be denoted by $\mathcal{N}(\mathcal{L}_\alpha)$.

The goal of this section is to recall how these quantum limits can be lifted to the phase space $T^*\mathbf{T}^2$ using the magnetic Weyl quantization and the notion of semiclassical measures. We also describe some elementary properties of these lifted measures.

**Remark 5.1.** As usual with semiclassical methods, we will drop the index $n$ in the following and rather write $(u_h)_{h \to 0^+}$. We will also refer to such families as sequences though the notation may be misleading.
## 5.1 Lifting quantum limits to the cotangent bundle

In order to study quantum limits, it is standard to lift these measures to distributions on $T^*\mathbf{T}^2 = \mathbf{T}^2 \times \mathbf{R}^2$ through the following definition

$$w_h : a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2, \mathbf{C}) \mapsto \langle \mathbf{Op}_h^B(a)u_h, u_h \rangle_{L^2(\mathbf{T}^2, \mathbf{L})},$$

where we recall that the scalar product $\langle u, v \rangle = \int_{\mathbf{T}^2} u(x)\bar{v}(x)dx$ for $u, v \in L^2(\mathbf{T}^2, \mathbf{L})$ is independent of the choice of the fundamental domain for $\mathbf{T}^2$. We list a few properties of these distributions in the following Lemma:

**Lemma 5.2.** Let $(u_h)_{h \to 0^+}$ be a sequence verifying<fnref n="3" /> (5.1). Then the following holds:

1. The corresponding sequence $(w_h)_{h \to 0^+}$ is bounded in $\mathcal{D}'(T^*\mathbf{T}^2)$;

2. Any accumulation point $\mu$ of $(w_h)_{h \to 0^+}$ is a probability measure on

$$S^*\mathbf{T}^2 := \{(x, \xi) \in T^*\mathbf{T}^2 : |\xi| = 1\},$$

which is invariant by the geodesic flow

$$\varphi^t : (x, \xi) \in T^*\mathbf{T}^2 \mapsto (x + t\xi, \xi) \in T^*\mathbf{T}^2;$$

3. One has $\pi_*\mu = \nu$ where $\pi : (x, \xi) \in T^*\mathbf{T}^2 \mapsto x \in \mathbf{T}^2$ and where $\nu$ is a quantum limit for $(u_h)_{h \to 0^+}$, i.e. it satisfies (5.2).

In the following, these probability measures will be refered as (magnetic) semiclassical measures and we will denote their set by $M(\mathcal{L}_\alpha)$.

*Proof.* The proof of these facts is now standard in the case of Laplace eigenfunctions [64, Ch. 5] and we briefly explain how it can be directly translated in our magnetic setting. For the first point, this follows directly from the fact that

$$\| \mathbf{Op}_h^B(a) \|_{L^2(\mathbf{T}^2, \mathbf{L}) \to L^2(\mathbf{T}^2, \mathbf{L})} \le \frac{1}{(2\pi)^4} \sum_{\xi \in 2\pi\mathbf{Z}^2} \int_{\mathbf{R}^2} |\mathcal{F}(a)(\eta, \zeta)| d\zeta \le C \|a\|_{\mathcal{C}^8},$$

where the constant depends only on the support of $a$. Here we used (3.3) but we could as well have used the Calderón-Vaillancourt Theorem.

For the second point, we fix some smooth function $a \ge 0$ compactly supported in $T^*\mathbf{T}^2$ and some $\varepsilon > 0$. We set $a_\varepsilon := \sqrt{a + \varepsilon} \in S^0$. Then, one can use the composition rule in Theorem 3.4 together with the formula (3.7) for the adjoint and the Calderón-Vaillancourt Theorem 3.6 to write that

$$\| \mathbf{Op}_h^B(a_\varepsilon)u_h \|_{L^2}^2 = \langle \mathbf{Op}_h^B(a_\varepsilon)^* \mathbf{Op}_h^B(a_\varepsilon)u_h, u_h \rangle = \langle \mathbf{Op}_h^B((a + \varepsilon))u_h, u_h \rangle + \mathcal{O}_\varepsilon(h).$$

Hence, for every $a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2, \mathbf{R}_+)$ and for every $\varepsilon > 0$, $\langle \mu, a \rangle \ge -\varepsilon$ from which we can infer that $\mu$ is a positive distribution, hence a measure. From the fact that $u_h$ is normalized in $L^2$, we also know that it has total mass $\le 1$.

<footnote n="3">³In fact, for this lemma to be true, only a $o_{L^2}(h)$ is required.</footnote>
To see that it has total mass equal to 1, we fix a smooth function $a$ on $T^*\mathbf{T}^2$ which is identically equal to 1 for $|\xi| \le 2$ and identically 0 for $|\xi| \ge 4$. We also fix $R > 1$ and we set $a_R(x, \xi) = a(x, \xi/R)$. From the composition rule in Theorem 3.4 and the Calderón-Vaillancourt Theorem 3.6, one finds

$$ 
\begin{aligned}
1 &= \langle \mathbf{Op}_h^B(a_R)u_h, u_h \rangle + \langle \langle \mathbf{Op}_h^B(1-a_R)u_h, u_h \rangle \\
  &= \langle \mathbf{Op}_h^B(a_R)u_h, u_h \rangle + \langle \mathbf{Op}_h^B(b_R) (\mathbf{Op}_h^B(|\xi|^2) - \mathrm{Id}) u_h, u_h \rangle + \mathcal{O}(h),
\end{aligned}
$$

where $b_R(x, \xi) = (1 - a_R(x, \xi))/(|\xi|^2 - 1)$ belongs to $S^{-2}(T^*\mathbf{T}^2)$. Recall now from (3.13) that

$$ h^2 \mathcal{L}_\alpha = \mathbf{Op}_h^B ((\xi_1 - h\alpha_1 - hA_1^{\text{per}})^2 + (\xi_2 - h\alpha_2 - hA_2^{\text{per}})^2) = \mathbf{Op}_h^B (|\xi|^2 + hc_1(x)\cdot\xi + h^2c_0(x)), \quad (5.3) $$

where $c_0$ and $c_1$ belong to $\mathcal{C}^\infty(\mathbf{T}^2, \mathbf{R})$ and to $\mathcal{C}^\infty(\mathbf{T}^2, \mathbf{R}^2)$ respectively. Hence, applying the composition rule one more time together with the Calderón-Vaillancourt Theorem, one finds that

$$ 1 = \langle \mathbf{Op}_h^B(a_R)u_h, u_h \rangle + \langle \mathbf{Op}_h^B(b_R) (h^2 \mathcal{L}_\alpha - \mathrm{Id}) u_h, u_h \rangle + \mathcal{O}(h). $$

Using the eigenmode equation (5.1), we find (after letting $h \to 0^+$) that $\langle \mu, a_R \rangle = 1$. Applying the dominated convergence Theorem, we find that $\mu$ is a probability measure. Along the way, we can verify with the same argument that $\mu$ is supported on $S^*\mathbf{T}^2$. Indeed, taking a function $a$ that is supported outside $\{|\xi| = 1\}$, we find with the exact same argument that $\langle \mu, a \rangle = 0$. Hence, $\mathrm{supp}(\mu) \subset \{|\xi| = 1\}$.

Recall that so far we only used that the error in the quasimode equation (5.1) is $o(1)$ as $h \to 0^+$. We are left with proving the invariance by the geodesic flow that will require the remainder to be $o(h)$. To see this, we use the quasimode equation to write

$$ \langle [\mathbf{Op}_h^B(a), h^2 \mathcal{L}_\alpha] u_h, u_h \rangle = o(h). $$

Using the composition rule for pseudodifferential operators one last time, we find that

$$ \langle \mathbf{Op}_h^B (\{|\xi|^2, a\}) u_h, u_h \rangle = o(1), $$

from which we deduce that $\int_{S^*\mathbf{T}^2} \xi \cdot \partial_x a(x, \xi)\mu(dx, d\xi) = 0$. This implies the invariance by the geodesic flow. Finally, the last item is obtained by using symbols $a(x)$ independent of $\xi$, by considering test functions of the form $a_R(x, \xi) = a(x)\chi(\|\xi\|^2/R)$ (with $\chi$ compactly supported and equal to 1 near 0) and by letting $R \to \infty$ (after letting $h \to 0^+$). $\square$

## 5.2 Decomposition of invariant measures

The quantum limits $\nu$ we are interested in are the pushforward of invariant measures $\mu$ carried on $S^*\mathbf{T}^2$. In order to analyze the regularity of $\nu$, we can study the regularity of $\mu$. To that aim, we follow the strategy of [40, 5] and decompose $\mu$ according to the periodic orbits of the geodesic flow. Namely, we denote by $\mathcal{L}_1$ the set of primitive sublattices $\Lambda$ of $\mathbf{Z}^2$ that are of rank 1. This means that $\dim\langle\Lambda\rangle = 1$ and that $\langle\Lambda\rangle \cap \mathbf{Z}^2 = \Lambda$ where $\langle\Lambda\rangle$ is the $\mathbf{R}$-vector space generated by $\Lambda$. For every $\Lambda \in \mathcal{L}_1$, we fix $\mathfrak{e}_\Lambda$ such that $\mathbf{Z}\mathfrak{e}_\Lambda = \Lambda$ and we denote by $\mathfrak{e}_\Lambda^\perp$ the vector that is directly orthogonal to it, with same length. We set $L_\Lambda := |\mathfrak{e}_\Lambda|$ and $\Lambda^\perp := \mathbf{R}\mathfrak{e}_\Lambda^\perp$.
On the one hand, the geodesic flow on $T^*\mathbf{T}^2$ is periodic in each direction $\Lambda^{\perp}$, i.e. for every $a \in \mathcal{C}_c^{\infty}(T^*\mathbf{T}^2)$, one has

$$ \forall (x, \xi) \in \mathbf{T}^2 \times (\Lambda^{\perp} \setminus \{0\}), \quad \lim_{T \to +\infty} \frac{1}{T} \int_0^T a(x + t\xi, \xi)dt = \mathcal{I}_{\Lambda}(a)(x, \xi), $$

where

$$ \mathcal{I}_{\Lambda}(a)(x, \xi) := \sum_{k \in \Lambda} \hat{a}_k(\xi) e^{2i\pi k \cdot x}. $$

**Remark 5.3.** When $a$ is independent of $\xi$, we observe that the function $\mathcal{I}_{\Lambda}(a)$ is also independent of $\xi$. Note that, for $a(x, \xi) = B(x)$ and for $\theta$ belonging to $\Lambda^{\perp} \cap \mathbf{S}^1$, one recovers the function $B_{\infty}$ from the introduction as $\mathcal{I}_{\Lambda}(B)(x, \theta) = B_{\infty}(x, \theta)$ in that case.

On the other hand, for every $\xi$ in

$$ \Omega_2 := \mathbf{R}^2 \setminus \left( \bigcup_{\Lambda \in \mathcal{L}_1} \Lambda^{\perp} \right), $$

one has, for every $a \in \mathcal{C}_c^{\infty}(T^*\mathbf{T}^2)$,

$$ \forall x \in \mathbf{T}^2, \quad \lim_{T \to +\infty} \frac{1}{T} \int_0^T a(x + t\xi, \xi)dt = \int_{\mathbf{T}^2} a(y, \xi)dy. $$

Thanks to these observations, we can now decompose each invariant probability measure $\mu$ on $S^*\mathbf{T}^2$ as follows:

$$ \mu = \mu|_{\mathbf{T}^2 \times (\Omega_2 \cap \mathbf{S}^1)} + \sum_{\Lambda \in \mathcal{L}_1} \mu|_{\mathbf{T}^2 \times (\Lambda^{\perp} \cap \mathbf{S}^1)}, $$

where each term in this sum is a nonnegative and finite measure that is invariant by the geodesic flow. We can also write the Fourier decomposition of $\mu$, i.e.

$$ \mu(x, \xi) := \sum_{k \in \mathbf{Z}^2} \hat{\mu}_k(\xi) e^{2i\pi x \cdot k}, \quad \text{and} \quad \forall \Lambda \in \mathcal{L}_1, \mathcal{I}_{\Lambda}(\mu) = \sum_{k \in \Lambda} \hat{\mu}_k(\xi) e^{2i\pi x \cdot k}. $$

We have the following result [5, §2]:

**Lemma 5.4.** Let $\mu$ be a probability measure on $S^*\mathbf{T}^2$ that is invariant by the geodesic flow. Then, the following holds:

1. For any $\Lambda$ in $\mathcal{L}_1$, $\mathcal{I}_{\Lambda}(\mu)$ is a finite nonnegative Radon measure on $S^*\mathbf{T}^2$.

2. For any $\Lambda$ in $\mathcal{L}_1$, we have $\mu|_{\mathbf{T}^2 \times (\Lambda^{\perp} \cap \mathbf{S}^1)} = \mathcal{I}_{\Lambda}(\mu)|_{\mathbf{T}^2 \times (\Lambda^{\perp} \cap \mathbf{S}^1)}$.

3. $\hat{\mu}_0$ is a finite nonnegative Radon measure on $S^*\mathbf{T}^2$ and $\mu|_{\mathbf{T}^2 \times (\Omega_2 \cap \mathbf{S}^1)} = \hat{\mu}_0|_{\mathbf{T}^2 \times (\Omega_2 \cap \mathbf{S}^1)}$.

Note that the proof of this Lemma is purely dynamical: it does not require $\mu$ to be a semiclassical measure. The fact that our semiclassical measures $\mu$ are indeed invariant by the geodesic flow allows us to decompose them accordingly

$$ \mu = \hat{\mu}_0|_{\mathbf{T}^2 \times (\Omega_2 \cap \mathbf{S}^1)} + \sum_{\Lambda \in \mathcal{L}_1} \mathcal{I}_{\Lambda}(\mu)|_{\mathbf{T}^2 \times (\Lambda^{\perp} \cap \mathbf{S}^1)}. $$

Hence, in order to understand the regularity of $\mu$ along the $x$-variable (and thus of its pushforward $\nu$ on $\mathbf{T}^2$), we need to understand the regularity of each individual measure $\mathcal{I}_{\Lambda}(\mu)|_{\mathbf{T}^2 \times (\Lambda^{\perp} \cap \mathbf{S}^1)}$. This is the content of the next section.
## 6 Analyzing the semiclassical measure along periodic orbits

Given any accumulation point $\mu$ of $(w_h)_{h \to 0}$, we study in this section the measure $\mathcal{I}_\Lambda(\mu)$, for any $\Lambda \in \mathcal{L}_1$. We will split the analysis into two, depending on whether the direction $\xi$ is close to $\Lambda^\perp$ or not. We fix a sequence $R_m \to +\infty$ and we let $\chi$ be a smooth function on $\mathbb{R}$ which is identically equal to 1 (resp. 0) on $[-1, 1]$ (resp. outside $[-2, 2]$). We define

$$ \chi_{R_m h^{\frac{1}{2}, \Lambda}}(\xi) := \chi \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{R_m h^{\frac{1}{2}} L_\Lambda} \right), $$

where we recall that $L_\Lambda = \|\mathfrak{e}_\Lambda\|$, and we split $w_h$ accordingly:

$$ \forall a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2), \quad \langle w_h, a \rangle = \langle w_h, a\chi_{R_m h^{\frac{1}{2}, \Lambda}} \rangle + \langle w_h, a(1 - \chi_{R_m h^{\frac{1}{2}, \Lambda}}) \rangle. $$

Again, we will drop the index for the sequence $(R_m)_{m \ge 1}$ and just write $R \to \infty$. This leads to the following definitions:

$$ w_{h,R,\Lambda} : a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2) \mapsto \langle w_h, a\chi_{Rh^{\frac{1}{2},\Lambda}} \rangle, $$

and

$$ w_{h,R}^{\Lambda} : a \in \mathcal{C}_c^{\infty}(T^*\mathbf{T}^2) \mapsto \langle w_h, a(1 - \chi_{Rh^{\frac{1}{2}},\Lambda}) \rangle. $$

Then, we have the following analogue of Lemma 5.2. Here when we use the notation

$$ (w_{h,R,\Lambda})_{h \to 0^+, R \to +\infty}, $$

we implicitly mean that we take the limit $h \to 0^+$ and then $R \to \infty$, in this precise order.

**Lemma 6.1.** Let $(u_h)_{h \to 0^+}$ be a sequence verifying (5.1) with a unique<fnref n="4" /> semiclassical measure $\mu$. Then the following holds:

1. The corresponding sequences $(w_{h,R,\Lambda})_{h \to 0^+, R \to +\infty}$ and $(w_{h,R}^\Lambda)_{h \to 0^+, R \to +\infty}$ are bounded in $\mathcal{S}'(T^*\mathbf{T}^2)$;

2. Any accumulation point $\mu_\Lambda$ and $\mu^\Lambda$ of $(w_{h,R,\Lambda})_{h \to 0^+, R \to +\infty}$ and $(w_{h,R}^\Lambda)_{h \to 0^+, R \to +\infty}$ are finite measure on $S^*\mathbf{T}^2$ which are invariant by the geodesic flow;

3. $\mu = \mu^\Lambda + \mu_\Lambda$.

Let us draw a few consequences of this Lemma. First, one has

$$ \mu|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} = \mu^\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} + \mu_\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)}. \quad (6.1) $$

Also, since $\chi_{Rh^{\frac{1}{2},\Lambda}}$ is supported on an $h^{\frac{1}{2}}$-neighborhood of $\Lambda^\perp$, it follows by construction that $\mu_\Lambda$ is supported on $\Lambda^\perp$ i.e. $\mu_\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} = \mu_\Lambda$. Thanks to the invariance by the geodesic flow, we can apply Lemma 5.4 to the measures $\mu$, $\mu^\Lambda$ and $\mu_\Lambda$ and we deduce

$$ \mu|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} = \mathcal{I}_\Lambda(\mu)|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} = \mathcal{I}_\Lambda(\mu^\Lambda)|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} + \mathcal{I}_\Lambda(\mu_\Lambda). \quad (6.2) $$

In particular, in view of dealing with the regularity of $\mu|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)}$, we only need to deal with the regularity of $\mu_\Lambda$ and $\mu^\Lambda$ and to work with test functions having all their Fourier coefficients $k$ in $\Lambda$.

<footnote n="4">⁴Up to extraction, this is always the case.</footnote>
Proof of Lemma 6.1. Using (3.19), we can rescale the quantization and write

$$ \langle w_{h,R,\Lambda}, a \rangle = \langle \mathbf{Op}_{h^{\frac{1}{2}}}^{B} \left( a(x, h^{\frac{1}{2}}\xi) \chi_{R,\Lambda}(\xi) \right) u_h, u_h \rangle, $$

and

$$ \langle w_{h,R,\Lambda}^{\Lambda}, a \rangle = \langle \mathbf{Op}_{h^{\frac{1}{2}}}^{B} \left( a(x, h^{\frac{1}{2}}\xi)(1 - \chi_{R,\Lambda}(\xi)) \right) u_h, u_h \rangle. $$

Without loss of generality, we can assume $0 < h < 1$ and $R > 1$ so that all the derivatives of the symbol $a(x, h^{\frac{1}{2}}\xi)\chi_{R,\Lambda}(\xi)$ are uniformly bounded in terms of $h$ and $R$, and the involved symbols belong to the class $S^0$. Hence, arguing as in the proof of item 1 of Lemma 6.1, we get the first and the third items of the present Lemma and the fact that the accumulation points are nonnegative measures. As $\mu$ is supported on $S^*\mathbf{T}^2$, we deduce that both $\mu_\Lambda$ and $\mu^\Lambda$ are also supported on $S^*\mathbf{T}^2$. Hence, the only thing to check is the invariance by the geodesic flow. To see this, we repeat the same argument as in the proof of Lemma 5.2 with the semiclassical parameter $h^{\frac{1}{2}}$ instead of $h$. Using the eigenvalue equation (5.1) and dividing by $h$ we have

$$ \langle [\mathbf{Op}_{h^{\frac{1}{2}}}^{B} (a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)), h\mathcal{L}_{\alpha}] u_h, u_h \rangle = o(h^{\frac{1}{2}}), \quad (6.3) $$

for both $\tilde{\chi}_{R,\Lambda} = \chi_{R,\Lambda}$ or $1 - \chi_{R,\Lambda}$. Recall that $h\mathcal{L}_{\alpha} = \mathbf{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2 + h^{\frac{1}{2}}c_1(x) \cdot \xi + hc_0(x))$, where $c_0$ and $c_1$ belong to $\mathcal{C}^\infty(\mathbf{T}^2, \mathbf{C})$ and to $\mathcal{C}^\infty(\mathbf{T}^2, \mathbf{C}^2)$ respectively. Applying the composition rule from Theorem 3.4 together with the fact that $|\xi|^2$ is quadratic of order 2 (and independent of $x$) and the support properties of $a$, $\tilde{\chi}_{R,\Lambda}$ and $\nabla\tilde{\chi}_{R,\Lambda}$, we find that

$$ \langle [\mathbf{Op}_{h^{\frac{1}{2}}}^{B} (a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)), \mathbf{Op}_{h^{\frac{1}{2}}}^{B} (|\xi|^2)] u_h, u_h \rangle = 2ih^{\frac{1}{2}} \langle \mathbf{Op}_{h^{\frac{1}{2}}}^{B} (\xi \cdot \partial_x a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)) u_h, u_h \rangle + o(h^{\frac{1}{2}}), \quad (6.4) $$

where the remainder term may depend on $R$. Similarly, we find that the remaining term satisfies

$$ \langle [\mathbf{Op}_{h^{\frac{1}{2}}}^{B} (a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)), \mathbf{Op}_{h^{\frac{1}{2}}}^{B} (h^{\frac{1}{2}}c_1(x)\xi + hc_0(x))] u_h, u_h \rangle = o(h^{\frac{1}{2}}). \quad (6.5) $$

Note that, since the symbols involved are not all in $S^0$, equations (6.4) and (6.5) are not straightforward. We provide some details in the case of (6.5), the other one being similar. First note that the $h^{1/2}a(x, h^{\frac{1}{2}}\xi)\tilde{\chi}_{R,\Lambda}(\xi)$ is uniformly in $S^{-1}$. Therefore, since $c_1(x)\xi \in S^1$, the composition Theorem 3.4 gives

$$ \langle [\mathbf{Op}_{h^{\frac{1}{2}}}^{B} (a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)), \mathbf{Op}_{h^{\frac{1}{2}}}^{B} (h^{\frac{1}{2}}c_1(x)\xi)] u_h, u_h \rangle = \frac{h^{\frac{1}{2}}}{i} \langle \mathbf{Op}_{h^{\frac{1}{2}}}^{B} \{h^{\frac{1}{2}}a\tilde{\chi}_{R,\Lambda}, c_1(x)\xi\} u_h, u_h \rangle + o(h^{\frac{1}{2}}). $$

Using the support properties of $a$ and $\tilde{\chi}_{R,\Lambda}$, we find that

$$ \{a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi), c_1(x)\xi\} \in S^0, $$

uniformly. Thus,

$$ \langle [\mathbf{Op}_{h^{\frac{1}{2}}}^{B} (a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi)), \mathbf{Op}_{h^{\frac{1}{2}}}^{B} (h^{\frac{1}{2}}c_1(x)\xi)] u_h, u_h \rangle = o(h^{\frac{1}{2}}). $$
The term $c_0(x)$ in (6.5) is easily estimated since it is in $S^0$.

Now, combining (6.3), (6.4) and (6.5), and dividing by $h^{\frac{1}{2}}$ we find

$$ \lim_{h \to 0} \langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( \xi \cdot \partial_x a(x, h^{\frac{1}{2}}\xi) \tilde{\chi}_{R,\Lambda}(\xi) \right) u_h, u_h \rangle = 0. $$

We deduce that the measures $\mu_\Lambda$ and $\mu^\Lambda$ are invariant by the geodesic flow.

<figure><img src="image_2.png" /></figure>

## 6.1 Analyzing the noncompact part

We start by analyzing the “noncompact” part of the measure, meaning $\mu^\Lambda$.

**Lemma 6.2.** Let $k \in \Lambda \setminus \{0\}$ and let $a \in \mathcal{C}_c^\infty(\mathbf{R}^2)$. Then, one has

$$ \int_{S^*\mathbf{T}^2} e^{2i\pi k \cdot x} a(\xi) \mu^\Lambda(dx, d\xi) = 0. $$

In particular, the nonnegative Radon measure $\mathcal{I}_\Lambda(\mu^\Lambda)(x, \xi) = \int_{\mathbf{T}^2} \mu^\Lambda(dx, \xi)$ is independent of $x$.

As a direct corollary of this Lemma and of (6.1) and (6.2), one has

$$ \mu|_{\mathbf{T}^2 \times (\Lambda^{\perp} \setminus \{0\})} = \mathcal{I}_{\Lambda}(\mu_{\Lambda}) + \int_{\mathbf{T}^2} \mu^{\Lambda}(dx, \xi)|_{\mathbf{T}^2 \times (\Lambda^{\perp} \setminus \{0\})}. \quad (6.6) $$

In particular, in view of proving that $\mu|_{\mathbf{T}^2 \times (\Lambda^{\perp} \setminus \{0\})}$ is independent of $x$, it only remains to analyze the properties of $\mathcal{I}_\Lambda(\mu_\Lambda)$ which will be the content of Lemmas 6.4 and 6.5 below.

*Proof.* Let $k \in \Lambda \setminus \{0\}$ and let $a \in \mathcal{C}_c^\infty(\mathbf{R}^2)$. We want to analyze the limit of $\langle w_{h,R}^\Lambda, a(\xi)e^{ik \cdot x} \rangle$ as $h \to 0^+$ and $R \to \infty$ (in this order). To do this, we introduce the following symbol

$$ s_{h,R}(x, \xi) = e^{2i\pi k \cdot x} \frac{a(h^{\frac{1}{2}}\xi)}{\xi \cdot \mathcal{e}_\Lambda} \left( 1 - \chi \left( \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R} \right) \right), \quad (6.7) $$

which belongs to the class $S^0$ uniformly with respect to $h$. From the eigenvalue equation (5.1) and the Calderón-Vaillancourt Theorem 3.6, we know that

$$ \langle [\mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_{h,R}), h^2 \mathcal{L}_{\alpha}] u_h, u_h \rangle = o(h^{\frac{3}{2}}). \quad (6.8) $$

We write $\mathcal{L}_\alpha$ in $h^{1/2}$-quantization,

$$ \begin{aligned} h^2 \mathcal{L}_{\alpha} &= \mathrm{Op}_{h}^{B}(|\xi|^2 + hc_1(x) \cdot \xi + h^2 c_0(x)) \\ &= h \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2 + h^{\frac{1}{2}} c_1(x) \cdot \xi + hc_0(x)), \end{aligned} $$

and divide by $h$ to find

$$ \langle [\mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_{h,R}), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2 + h^{\frac{1}{2}}c_1(x) \cdot \xi + hc_0(x))] u_h, u_h \rangle = o(h^{\frac{1}{2}}). \quad (6.9) $$
We study the symbol of the commutator using (3.8) with semiclassical parameter $h^{\frac{1}{2}}$, and we deal with each term one by one. First of all, $s_{h,R}$ and $c_0$ belong to the symbol class $S^0$ uniformly with respect to $h$. Thus,

$$ \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_{h,R}), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(hc_0(x)) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_{h,R}^{1}), \quad \text{with} \quad \sigma_{h,R}^{1} \in h^{\frac{3}{2}}S^{0}. \qquad (6.10) $$

For the second term, note that $s_{h,R}$ is not uniformly in $S^{-1}$, but $h^{\frac{1}{2}}s_{h,R}$ is. Since $c_1(x) \cdot \xi \in S^1$ we deduce

$$ \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_{h,R}), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(h^{\frac{1}{2}}c_1(x) \cdot \xi) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_{h,R}^{2}), \quad \text{with} \quad \sigma_{h,R}^{2} = \frac{h^{\frac{1}{2}}}{i} \{s_{h,R}, h^{\frac{1}{2}}c_1(x) \cdot \xi\} + hS^0. \qquad (6.11) $$

Similarly, $hs_{h,R}$ is in the class $S^{-2}$ and $|\xi|^2 \in S^2$ so that

$$ \begin{gathered} \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_{h,R}), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_{h,R}^{3}), \\ \text{with} \quad \sigma_{h,R}^{3} = \frac{h^{\frac{1}{2}}}{i} \{s_{h,R}, |\xi|^2\} + \frac{2\hat{B}_{0}h}{i} (\xi_{1}\partial_{\xi_{2}}s_{h,R} - \xi_{2}\partial_{\xi_{1}}s_{h,R}), \end{gathered} \qquad (6.12) $$

where the remainder is 0 as in Corollary 3.5 due to the fact that $||\xi||^2$ is quadratic and independent of $x$. We insert the estimates (6.10), (6.11) and (6.12) in (6.9) together with the Calderón-Vaillancourt Theorem to deduce

$$ \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( \frac{h^{\frac{1}{2}}}{i} \{s_{h,R}, |\xi|^2\} + \frac{2\hat{B}_0 h}{i} (\xi_1 \partial_{\xi_2} s_{h,R} - \xi_2 \partial_{\xi_1} s_{h,R}) + \frac{h}{i} \{s_{h,R}, c_1(x) \cdot \xi\} \right) u_h, u_h \right\rangle = o(h^{\frac{1}{2}}). \qquad (6.13) $$

We calculate the remaining symbols and find that

$$ \frac{2\hat{B}_0 h^{\frac{1}{2}}}{i} (\xi_1 \partial_{\xi_2} (h^{\frac{1}{2}} s_{h,R}) - \xi_2 \partial_{\xi_1} (h^{\frac{1}{2}} s_{h,R})) + \frac{h^{\frac{1}{2}}}{i} \{h^{\frac{1}{2}} s_{h,R}, c_1(x) \cdot \xi\} = \mathcal{O}(h^{\frac{1}{2}} R^{-1}) \quad \text{in } S^0. $$

Therefore when we divide (6.13) by $h^{\frac{1}{2}}$ we find

$$ \lim_{R \to \infty} \lim_{h \to 0} \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( \frac{1}{i} \{s_{h,R}, |\xi|^2\} \right) u_h, u_h \right\rangle = 0. \qquad (6.14) $$

The symbol $s_{h,R}$ was chosen such that

$$ \{s_{h,R}, |\xi|^2\} = -\frac{4i\pi|k|}{L_{\Lambda}} e^{2i\pi k \cdot x} a(h^{\frac{1}{2}}\xi) \left(1 - \chi\left(\frac{\xi \cdot \mathfrak{e}_{\Lambda}}{L_{\Lambda}R}\right)\right), $$

and thus

$$ \frac{L_{\Lambda}}{4i\pi|k|} \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\{s_{h,R}, |\xi|^2\}) = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(e^{ikx}a(h^{\frac{1}{2}}\xi)(1 - \chi(\frac{\xi \cdot \mathfrak{e}_{\Lambda}}{L_{\Lambda}R})) = \mathrm{Op}_{h}^{B}(e^{ikx}a(\xi)(1 - \chi_{Rh^{\frac{1}{2}},\Lambda}(\xi))). $$

Therefore equation (6.14) becomes

$$ \langle \mu^{\Lambda}, e^{ikx} a(\xi) \rangle = \lim_{R \to \infty} \lim_{h \to 0^{+}} \langle w_{h,R}^{\Lambda}, e^{ikx} a(\xi) \rangle = 0, $$

which is the expected result.

<figure><img src="image_3.png" /></figure>
## 6.2 Analyzing the compact part

In this section we prove that $\mu_\Lambda$ vanishes, under our geometric control assumption on the magnetic field.

**Proposition 6.3.** Let $\mu$ be any accumulation point of $(w_h)_{h \to 0}$, and consider the associated measure $\mu_\Lambda$ given by Lemma 6.1. If $\mathcal{I}_\Lambda(B) > 0$ everywhere, then $\mu_\Lambda = 0$.

Recall from Remark 5.3 that $\mathcal{I}_\Lambda(B)$ is independent of $\xi$ and that $\mathcal{I}_\Lambda(B)(x) = B_\infty(x, \theta)$ when $\theta \in \Lambda^\perp \cap \mathbf{S}^1$. We introduce the two-microlocal distribution:

$$ \mu_{h,R,\Lambda} : a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2 \times \mathbf{R}) \mapsto \left\langle w_h, a \left(x, \xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda h^{\frac{1}{2}}}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R h^{\frac{1}{2}}}\right) \right\rangle. $$

In the following we denote by $\eta$ the last variable of $a$ so that $a(x, \xi, \eta)$. Mimicking the proof of Lemma 6.1, we can verify that

**Lemma 6.4.** Let $(u_h)_{h \to 0^+}$ be a sequence verifying (5.1). Then the following holds:

1. The corresponding sequence $(\mu_{h,R,\Lambda})_{h \to 0^+, R \to +\infty}$ is bounded in $\mathcal{D}'(T^*\mathbf{T}^2 \times \mathbf{R})$;

2. Any accumulation point $\tilde{\mu}_\Lambda$ (as $h \to 0^+$ and $R \to +\infty$ in this order) of $(\mu_{h,R,\Lambda})_{h \to 0^+, R \to +\infty}$ is a finite measure on $\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1) \times \mathbf{R}$, which is invariant by the lifted geodesic flow

$$ \tilde{\varphi}^t(x, \xi, \eta) = (x + t\xi, \xi, \eta); $$

3. $\mu_\Lambda(x, \xi) = \int_{\mathbf{R}} \tilde{\mu}_\Lambda(x, \xi, d\eta)$.

Observe from the invariance by the geodesic flow that $\langle \tilde{\mu}_\Lambda, e^{ik \cdot x} a(\xi, \eta) \rangle = 0$ for every $k \notin \Lambda$ and for every $a \in \mathcal{C}_c^\infty(\mathbf{R}^2 \times \mathbf{R})$. Hence, one has

$$ \tilde{\mu}_{\Lambda} = \mathcal{I}_{\Lambda}(\tilde{\mu}_{\Lambda}). \qquad (6.15) $$

Also, from the support properties of $\tilde{\mu}_\Lambda$, we can split this measure as

$$ \tilde{\mu}_{\Lambda}^{\pm} := \tilde{\mu}_{\Lambda}|_{\mathbf{T}^2 \times \{\pm \epsilon_{\Lambda}^{\pm} / L_{\Lambda}\} \times \mathbf{R}}. $$

Our last step consists in determining an extra-regularity property for the measure $\tilde{\mu}_\Lambda$.

**Lemma 6.5.** Let $\tilde{\mu}_\Lambda$ be an accumulation point as in Lemma (6.4). Then, one has, for every $a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2 \times \mathbf{R})$,

$$ \left\langle \tilde{\mu}_{\Lambda}^{\pm}, \eta \frac{\mathcal{e}_{\Lambda}}{L_{\Lambda}} \cdot \partial_x \mathcal{I}_{\Lambda}(a) \pm \mathcal{I}_{\Lambda}(B) \partial_{\eta} \mathcal{I}_{\Lambda}(a) \right\rangle = 0. $$

Before proving Lemma 6.5, let us explain how to deduce Proposition 6.3.

*Proof of Proposition 6.3.* Lemma 6.4 together with (6.15) ensure that $\tilde{\mu}_\Lambda$ is a measure on $\mathbf{T}^2 \times \{\pm \epsilon_\Lambda^\pm / L_\Lambda\} \times \mathbf{R}$ with only Fourier coefficients along $\Lambda$. Hence, it can be identified with the two measures $\tilde{\mu}_\Lambda^\pm$. Thanks to Lemma 6.5, these measures are invariant by one of the flows $\phi_{\Lambda, \pm}^t$ generated by

$$ X_{\Lambda, \pm} := \eta \frac{\mathcal{e}_{\Lambda}}{L_{\Lambda}} \cdot \partial_x \pm \mathcal{I}_{\Lambda}(B) \partial_{\eta}. $$

All the trajectories of this flow escape to infinity, as soon as $\mathcal{I}_\Lambda(B) > 0$ everywhere. As $\tilde{\mu}_\Lambda^\pm$ is an invariant finite measure, it is necessarily 0. □
Remains to prove Lemma 6.5. Let us first be slightly more precise on the symbol of the operator $h^2\mathcal{L}_\alpha$. Recall that

$$ \mathrm{Op}_h^B(\xi_j - hA_j^{\mathrm{per}} - h\alpha_j) = -ih\partial_{x_j} - hA_j^0 - hA_j^{\mathrm{per}} - h\alpha_j, $$

so that, from the composition rule for the magnetic Weyl quantization, one finds

$$ h^2 \mathcal{L}_\alpha = \mathrm{Op}_h^B (|\xi|^2 - 2h(A^{\mathrm{per}} + \alpha) \cdot \xi + h^2|A^{\mathrm{per}} + \alpha|^2). $$

As before, we shall write for short

$$ c_1(x) := -2(A^{\mathrm{per}}(x) + \alpha), \quad \text{and} \quad c_0(x) := |A^{\mathrm{per}}(x) + \alpha|^2 = \frac{|c_1(x)|^2}{4}. \qquad (6.16) $$

*Proof of Lemma 6.5.* Let $a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2 \times \mathbf{R})$ be such that $\mathcal{I}_\Lambda(a) = a$, i.e. $a$ has only Fourier coefficients along $\Lambda$. We introduce the symbol $s_h$, which belongs to the symbol class $S^0$ uniformly with respect to $h$,

$$ s_h(x, \xi) = a \left( x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda} \right) \chi \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda R} \right), $$

and which satisfies

$$ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h) = \mathrm{Op}_{h}^{B}\left(a\left(x, \xi, \frac{\xi \cdot \mathfrak{e}_{\Lambda}}{L_{\Lambda}h^{\frac{1}{2}}}\right) \chi\left(\frac{\xi \cdot \mathfrak{e}_{\Lambda}}{L_{\Lambda}Rh^{\frac{1}{2}}}\right)\right). $$

Again, we write the quasimode equation (5.1),

$$ \left\langle \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h), h^2 \mathcal{L}_{\alpha} \right] u_h, u_h \right\rangle = o(h^{\frac{3}{2}}). $$

Recall the symbol of $\mathcal{L}_\alpha$,

$$ h^2 \mathcal{L}_\alpha = h \mathrm{Op}_{h^{\frac{1}{2}}}^B (|\xi|^2 + h^{\frac{1}{2}} c_1(x) \cdot \xi + hc_0(x)), $$

from which we deduce

$$ \left\langle \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2 + h^{\frac{1}{2}}c_1(x) \cdot \xi + hc_0(x)) \right] u_h, u_h \right\rangle = o(h^{\frac{1}{2}}). \qquad (6.17) $$

As in the proof of Lemma 6.2, we check the contribution of each term one by one, using the composition rule Theorem 3.4. First, since $c_0(x)$ and $s_h$ belong to $S^0$ uniformly with respect to $h$ we have

$$ \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(hc_0(x)) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_h^1), \quad \text{with} \quad \sigma_h^1 \in h^{\frac{3}{2}}S^0. \qquad (6.18) $$

For the second term, note that $c_1(x) \cdot \xi \in S^1$ and $h^{\frac{1}{2}}s_h$ is in $S^{-1}$ uniformly with respect to $h$ so that

$$ \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(h^{\frac{1}{2}}c_1(x) \cdot \xi) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_h^2), \quad \text{with} \quad \sigma_h^2 = \frac{h^{\frac{1}{2}}}{i}\{s_h, h^{\frac{1}{2}}c_1(x) \cdot \xi\} + hS^0. \qquad (6.19) $$

Finally, the exact formula from Corollary 3.5 gives

$$ \begin{gathered} \left[ \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(s_h), \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(|\xi|^2) \right] = \mathrm{Op}_{h^{\frac{1}{2}}}^{B}(\sigma_h^3), \\ \text{with } \sigma_h^3 = \frac{h^{\frac{1}{2}}}{i}\{s_h, |\xi|^2\} + \frac{2\hat{B}_0h}{i}(\xi_1\partial_{\xi_2}s_h - \xi_2\partial_{\xi_1}s_h). \end{gathered} \qquad (6.20) $$
We insert the estimates (6.18), (6.19) and (6.20) in (6.17) together with the Calderón-Vaillancourt Theorem to deduce

$$ \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( \frac{h^{\frac{1}{2}}}{i} \{s_h, |\xi|^2\} + \frac{2\hat{B}_0 h}{i} (\xi_1 \partial_{\xi_2} s_h - \xi_2 \partial_{\xi_1} s_h) + \frac{h}{i} \{s_h, c_1(x) \cdot \xi\} \right) u_h, u_h \right\rangle = o(h^{\frac{1}{2}}). \quad (6.21) $$

Recalling that $a = \mathcal{I}_\Lambda(a)$, one has that

$$ \begin{aligned} \{|\xi|^2, s_h\} &= 2\xi \cdot \partial_x a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right) \\ &= 2 \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot \partial_x a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right). \end{aligned} $$

Regarding the term involving the subsprincipal symbol $hc_1(x) \cdot \xi$, one finds

$$ \begin{aligned} h \{c_1(x) \cdot \xi, s_h\} &= hc_{1,\Lambda}(x) \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot \partial_x a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right) \\ &\quad + h \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda} \left\{ c_{1,\Lambda}(x), a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right) \right\} \\ &\quad + h \frac{\xi \cdot \mathcal{e}_\Lambda^\perp}{L_\Lambda} \left\{ c_{1,\Lambda}^\perp(x), a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right) \right\}. \\ &= -h \frac{\xi \cdot \mathcal{e}_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda}\right) \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot \partial_x c_{1,\Lambda}^\perp(x) \chi \left(\frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R}\right) \\ &\quad + \mathcal{O}_R(h) + \mathcal{O}(h^{\frac{1}{2}}R^{-1}) \quad \text{in } S^0, \end{aligned} $$

where we have set

$$ c_{1,\Lambda}(x) := \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot c_1(x), \quad \text{and} \quad c_{1,\Lambda}^\perp(x) := \frac{\mathcal{e}_\Lambda^\perp}{L_\Lambda} \cdot c_1(x). $$

Similarly, the magnetic term is

$$ 2\hat{B}_0 h (\xi_1 \partial_{\xi_2} s_h - \xi_2 \partial_{\xi_1} s_h) = 2\hat{B}_0 h \frac{\xi \cdot e_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left(x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot e_\Lambda}{L_\Lambda}\right) \chi \left(\frac{\xi \cdot e_\Lambda}{L_\Lambda R}\right) \\ + \mathcal{O}_R(h) + \mathcal{O}(h^{\frac{1}{2}}R^{-1}) \quad \text{in } S^0. $$

Therefore, equation (6.21) gives, after dividing by $h^{\frac{1}{2}}$ and using the Calderón-Vaillancourt Theorem,

$$ \begin{aligned} & -2 \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( \left( \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda} \right) \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot \partial_x a \left( x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda} \right) \chi \left( \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R} \right) \right) u_h, u_h \right\rangle \\ & + \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( h^{\frac{1}{2}} \frac{\xi \cdot \mathcal{e}_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left( x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda} \right) \frac{\mathcal{e}_\Lambda}{L_\Lambda} \cdot \partial_x c_{1,\Lambda}^\perp(x) \chi \left( \frac{\xi \cdot \mathcal{e}_\Lambda}{L_\Lambda R} \right) \right) u_h, u_h \right\rangle \\ & - 2\hat{B}_0 \left\langle \mathrm{Op}_{h^{\frac{1}{2}}}^{B} \left( h^{\frac{1}{2}} \frac{\xi \cdot e_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left( x, h^{\frac{1}{2}}\xi, \frac{\xi \cdot e_\Lambda}{L_\Lambda} \right) \chi \left( \frac{\xi \cdot e_\Lambda}{L_\Lambda R} \right) \right) u_h, u_h \right\rangle = o_R(1) + \mathcal{O}(R^{-1}). \end{aligned} $$
We then rescale back to the h-quantization,

$$ -2 \left\langle \mathrm{Op}_h^B \left( \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda h^{\frac{1}{2}}} \right) \frac{\mathfrak{e}_\Lambda}{L_\Lambda} \cdot \partial_x a \left( x, \xi, \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda h^{\frac{1}{2}}} \right) \chi \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda R h^{\frac{1}{2}}} \right) \right) u_h, u_h \right\rangle \\ + \left\langle \mathrm{Op}_h^B \left( \frac{\xi \cdot \mathfrak{e}_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left( x, \xi, \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda h^{\frac{1}{2}}} \right) \frac{\mathfrak{e}_\Lambda}{L_\Lambda} \cdot \partial_x c_{1,\Lambda}^\perp(x) \chi \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda R h^{\frac{1}{2}}} \right) \right) u_h, u_h \right\rangle \\ - 2 \hat{B}_0 \left\langle \mathrm{Op}_h^B \left( \frac{\xi \cdot \mathfrak{e}_\Lambda^\perp}{L_\Lambda} \partial_\eta a \left( x, \xi, \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda h^{\frac{1}{2}}} \right) \chi \left( \frac{\xi \cdot \mathfrak{e}_\Lambda}{L_\Lambda R h^{\frac{1}{2}}} \right) \right) u_h, u_h \right\rangle = o_R(1) + \mathcal{O}(R^{-1}). $$

Hence, letting $h \to 0^+$ and $R \to +\infty$ (in this order), one finds that, for every $a \in \mathcal{C}_c^\infty(T^*\mathbf{T}^2 \times \mathbf{R})$ verifying $\mathcal{I}_\Lambda(a) = a$,

$$ \left\langle \tilde{\mu}_{\Lambda}, -2\eta \frac{\mathfrak{e}_{\Lambda}}{L_{\Lambda}} \cdot \partial_x a + \frac{\xi \cdot \mathfrak{e}_{\Lambda}^{\perp}}{L_{\Lambda}} \left( \frac{\mathfrak{e}_{\Lambda}}{L_{\Lambda}} \cdot \partial_x c_{1,\Lambda}^{\perp} - 2\hat{B}_0 \right) \partial_{\eta} a \right\rangle = 0, $$

where we recall that $\xi \cdot \mathfrak{e}_\Lambda^\perp = \pm L_\Lambda$ on the support of $\tilde{\mu}_\Lambda$. Using (6.16) together with the invariance by the geodesic flow and the fact that $\tilde{\mu}_\Lambda$ is carried by $\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1) \times \mathbf{R}$, one finds the expected result

$$ \left\langle \tilde{\mu}_{\Lambda}, \eta \frac{\mathfrak{e}_{\Lambda}}{L_{\Lambda}} \cdot \partial_x a + \frac{\xi \cdot \mathfrak{e}_{\Lambda}^{\perp}}{L_{\Lambda}} \mathcal{I}_{\Lambda}(B) \partial_{\eta} a \right\rangle = 0. $$

<figure><img src="image_4.png" /></figure>

## 6.3 Conclusion

Gathering all these results, we are in position to prove the following Theorem.

**Theorem 6.6.** Suppose that, for every $\Lambda$ in $\mathcal{L}_1$, $\mathcal{I}_\Lambda(B) > 0$ everywhere. Then, one has

$$ \mathcal{N}(\mathcal{L}_\alpha) = \{\text{Leb}\}. $$

More precisely, any $\mu(dx, d\xi) \in \mathcal{M}(\mathcal{L}_\alpha)$ is of the form $dx \nu_0(d\xi)$, where $\nu_0$ is a probability measure on $\mathbf{S}^1$.

*Proof.* Let $\nu$ be an element in $\mathcal{N}(\mathcal{L}_\alpha)$. Denote by $\mu$ a semiclassical measure associated to the sequence of eigenfunctions $(u_h)_{h \to 0^+}$ used to generate $\nu$. According to Lemma 5.4, it is sufficient to analyze $\mu|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)}$ in order to conclude. Thanks to (6.1), one has that

$$ \mu|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} = \mu^\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)} + \mu_\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)}. $$

According to Proposition 6.3, our assumption on $B$ ensures that $\mu_\Lambda \equiv 0$. Finally, (6.6) allows to conclude that $\mu^\Lambda|_{\mathbf{T}^2 \times (\Lambda^\perp \cap \mathbf{S}^1)}$ has no nonzero Fourier coefficients which ends the proof of the theorem. $\square$

<figure><img src="image_5.png" /></figure>

# 7 Proof for variable magnetic fields

In this final section, we consider slightly better quasimodes, namely we suppose that $(u_{h_n})_{n \ge 1}$ is a sequence in $H^2(\mathbf{T}^2, L)$ such that

$$ h_n^2(\mathcal{L}_\alpha + V)u_{h_n} = u_{h_n} + o_{L^2}(h_n^2), \quad \|u_{h_n}\|_{L^2(\mathbf{T}^2)} = 1, \quad h_n \to 0^+. \quad (7.1) $$
In particular, $(u_{h_n})_{n \ge 1}$ verifies (5.1) and, up to extraction, we denote by $\mu$ a semiclassical measure of this sequence of quasimodes. Thanks to Lemma 5.2 and to Theorem 6.6, one knows that $\mu(dx, d\xi)$ is a probability measure on $S^*T^2$ that is of the form $dx\nu_0(d\xi)$ with $\nu_0$ a probability measure on $S^1$. Our final step to prove Theorem 1.1 consists in showing that $\nu_0$ is the Lebesgue measure along the circle $S^1$. To see this, it is sufficient to pick a smooth test function $a(\xi) \in \mathcal{C}_c^\infty(\mathbb{R}^2)$ and to write the quasimode equation (7.1) one last time:

$$ \langle [\mathbf{Op}_h^B(a), h^2(\mathcal{L}_\alpha + V)] u_h, u_h \rangle = o(h^2). $$

Using the composition Theorem 3.4 together with the Calderón-Vaillancourt Theorem 3.6, one finds that

$$ \langle \mathbf{Op}_h^B (\{a(\xi), c_1(x) \cdot \xi\} - 2\hat{B}_0(\xi_1 \partial_{\xi_2} - \xi_2 \partial_{\xi_1})a) u_h, u_h \rangle = o(1). $$

Letting $h$ to 0 and using (6.16), one finds that

$$ \langle \mu, \xi_1 \{a(\xi), A_1^{\text{per}}(x)\} + \xi_2 \{a(\xi), A_2^{\text{per}}(x)\} + \hat{B}_0(\xi_1 \partial_{\xi_2} - \xi_2 \partial_{\xi_1})a \rangle = 0. $$

From this together with the fact that $\mu$ is of the form $dx\nu_0(d\xi)$, we get that

$$ \langle \nu_0, \hat{B}_0(\xi_1 \partial_{\xi_2} - \xi_2 \partial_{\xi_1})a \rangle = 0. $$

As $\hat{B}_0 \ne 0$, we can conclude that $\nu_0$ is the Lebesgue measure along the circle. This concludes the proof of Theorem 1.1.

## A Results for more general magnetic fields

Recall that Theorem 1.1 depends on the fact that

$$ B_{\infty}(x, \theta) = \lim_{T \to +\infty} \frac{1}{T} \int_{0}^{T} B(x + t\theta)dt, \quad (x, \theta) \in S^*T^2, $$

is a positive function. When this geometric control condition is not satisfied, we can still conclude some regularity properties of the elements in $\mathcal{N}(\mathcal{L}_\alpha)$ following the lines of [43]. More precisely, we can define a set of critical geodesics along $\Lambda$ (with respect to $B$) as

$$ \mathcal{Z}_{\Lambda}(B) := \{x_0 \in T^2 : \exists \theta \in S^1 \cap \Lambda^{\perp} \text{ such that } B_{\infty}(x_0, \theta) = 0\}. $$

This is a set of closed geodesics in the direction $\Lambda^{\perp}$ that could be either finite or infinite. We denote then by $\mathcal{N}_{\Lambda}(B)$ the convex closure of the measures $c\delta_\gamma$, where $c \in [0, 1]$ and where $\delta_\gamma$ is the normalized Lebesgue measure along a closed geodesic $\gamma \in \mathcal{Z}_{\Lambda}(B)$. As a result of our analysis, we can prove the following theorem which is an analogue for magnetic fields of [43, Th. 1.1].

**Theorem A.1.** Suppose that $\hat{B}_0 > 0$. Then, one has:

1. If $B_\infty \ge 0$, then any $\nu \in \mathcal{N}(\mathcal{L}_\alpha)$ can be decomposed as

$$ \nu = c_0 + \sum_{\Lambda \in \mathcal{L}_1} \nu_{\text{sing}}^{\Lambda}, $$

where $c_0 \in [0, 1]$ and $\nu_{\text{sing}}^{\Lambda} \in \mathcal{N}_{\Lambda}(B)$.
2. Without any assumption on $B_\infty$, one has, for any $\nu \in \mathcal{N}(\mathcal{L}_\alpha)$ and for any closed geodesic $\gamma$,

$$ \nu(\gamma) > 0 \implies \exists \Lambda \in \mathcal{L}_1 \text{ such that } \gamma \subset \mathcal{Z}_\Lambda(B) $$

3. Without any assumption on $B_\infty$ and if $k \neq 0$, any $\nu \in \mathcal{N}(\mathcal{L}_\alpha)$ verifies

$$ \tilde{\nu}_k \neq 0 \implies \exists \Lambda \in \mathcal{L}_1 \text{ such that } k \in \Lambda \text{ and } \mathcal{Z}_\Lambda(B) \neq \emptyset. $$

As an illustration of the first item, if $B \ge 0$ and if there exists only a single geodesic (issued from the point $(x_0, \pm\theta_0)$) such that $B_\infty(x_0, \pm\theta_0) = 0$, then any $\nu \in \mathcal{N}(\mathcal{L}_\alpha)$ is of the form $c_0dx+(1-c_0)\delta_{\gamma_0}$, where $\gamma_0$ is the closed geodesic issued from $(x_0, \theta_0) \in S^*\mathbf{T}^2$.

In view of understanding the scope of this theorem, it is worth noting that

$$ \hat{B}_0 > 0 \implies \#\{\Lambda \in \mathcal{L}_1 : \mathcal{Z}_\Lambda(B) \neq \emptyset\} < \infty. $$

Indeed, recalling Remark 5.3, one can observe that, for every $x \in \mathbf{T}^2$ and for every $\varepsilon > 0$,

$$ \mathcal{I}_\Lambda(B)(x) \ge \hat{B}_0 - \sum_{k \in \mathbb{Z} \setminus \{0\}} |\hat{B}_{k\varepsilon_\Lambda}| \ge \hat{B}_0 - \frac{1}{L_\Lambda^{\frac{1}{2}+2\varepsilon}} \left( \sum_{k \ne 0} \frac{1}{k^{1+2\varepsilon}} \right)^{\frac{1}{2}} \|B - \hat{B}_0\|_{\dot{H}^{\frac{1}{2}+\varepsilon}(\mathbf{T}^2)}, $$

from which we can deduce the above property as $\{\Lambda : L_\Lambda < R\} < \infty$ for every $R > 0$. In particular, as soon as $\hat{B}_0 > 0$, any $\nu \in \mathcal{N}(\mathcal{L}_\alpha)$ has its Fourier coefficients along a finite number of primitive rank 1 sublattices of $\mathbb{Z}^2$ (that depend only on $B$).

*Proof.* Let $\nu$ be an element in $\mathcal{N}(\mathcal{L}_\alpha)$. Up to another extraction, we can suppose that the sequence of quasimodes used to generate $\nu$ gives rise to an unique semiclassical measure $\mu$ as in Lemma 5.2. According to this Lemma and to Lemma 5.4, one has $\nu = \pi_*(\mu)$ and

$$ \mu(x, \xi) = \hat{\mu}_0(\xi)|_{\mathbf{T}^2 \times (\Omega_2 \cap \mathbf{S}^1)} + \sum_{\Lambda \in \mathcal{L}_1} \nu_\Lambda^{\pm}(x) \otimes \delta\left(\xi \mp \frac{\mathfrak{e}_\Lambda^{\pm}}{L_\Lambda}\right), $$

where $\nu_\Lambda^{\pm}(x)$ has only Fourier coefficients along $\Lambda$. Thanks to Lemma 6.1, $\mu$ can be further decomposed according to $\mu_\Lambda$ and $\mu^\Lambda$. Then, Lemma 6.2 (together with another application of Lemma 5.4) ensures that the contribution of $\mu^\Lambda|_{\mathbf{T}^2 \times \Lambda^\perp}$ has no nonzero Fourier coefficients. Hence, if we denote by $\tilde{\nu}_\Lambda^\pm$ the contribution of $\mu_\Lambda$, one finds that

$$ \nu(dx) = c_0dx + \sum_{\Lambda \in \mathcal{L}_1} (\tilde{\nu}_\Lambda^+(dx) + \tilde{\nu}_\Lambda^-(dx)), $$

where $c_0 \in [0, 1]$ is a constant. Moreover, $\tilde{\nu}_\Lambda^\pm$ has only Fourier coefficients along $\Lambda$ and, thanks to Lemma 6.5, it is the pushforward of a finite measure$^5$ $\tilde{\mu}_\Lambda^\pm(x, \eta)$ on $\mathbf{T}^2 \times \mathbf{R}$ which has only Fourier coefficients along $\Lambda$ and which is invariant by the vector field

$$ X_{\Lambda, \pm} = \eta \frac{\mathfrak{e}_\Lambda}{L_\Lambda} \cdot \partial_x \pm \mathcal{I}_\Lambda(B) \partial_\eta. $$

<footnote>$^5$Compared with Lemma 6.5, we make the small abuse of notations to remove the $\xi$-variable as the measure is carried along $\pm\mathfrak{e}_\Lambda/L_\Lambda$ along this variable.</footnote>
Equivalently, $\tilde{\mu}_{\Lambda}^{\pm}(x, \eta)$ can be identified with a measure $\rho_{\Lambda}^{\pm}(y, \eta)$ on $T^*\mathbf{T}_{\Lambda}$, with $\mathbf{T}_{\Lambda} = \langle\Lambda\rangle/\Lambda$, which is invariant by the vector field $Y_{\Lambda}^{\pm}(y, \eta) = \eta\partial_y \pm I_{\Lambda}(B)(y)\partial_{\eta}$.

In the case where $I_{\Lambda}(B) \ge 0$, the function $I_{\Lambda}(B)$ is not identically 0 as $\hat{B}_0 > 0$. In particular, all the trajectories of the flow generated by $Y_{\Lambda}^{\pm}$ escape to infinity or converge to a point in $\{(y, 0) : I_{\Lambda}(B) = 0\}$. Hence, any invariant probability measure by this vector field is in the (closed) convex hull of the Dirac masses carried by this critical set. This concludes the first part of the proof of the theorem.

Let us now discuss the general case where $I_{\Lambda}(B)$ may change sign. In that case, the set of invariant probability measures for the flow generated by $Y_{\Lambda}^{\pm}(y, \eta)$ is much larger and the support of the measure $\rho_{\Lambda}^{\pm}(y, \eta)$ may not be contained in $\{\eta = 0\}$ (for instance it may contain nontrivial closed orbits of this flow). In order to prove the second item, suppose that there exists a closed geodesic $\gamma_0$ such that $\nu(\gamma_0) > 0$ and denote by $\Lambda_0$ the rank 1 sublattice such that $\Lambda_0^{\perp}$ is parallel to $\gamma_0$. One can pick a smooth function $\chi_{\gamma_0} \in C^{\infty}(\mathbf{T}^2, [0, 1])$ having all its Fourier coefficients in $\Lambda_0$, equal to 1 in an $\epsilon$-neighborhood of $\gamma_0$ and 0 outside a $2\epsilon$-neighborhood. One has $\nu(\chi_{\gamma_0}) \ge \nu(\gamma_0)$. Taking $\epsilon$ small enough, one also has that $\hat{\nu}_0 \int_{\mathbb{T}^2} \chi_{\gamma_0} dx \le \nu(\gamma_0)/2$. Hence, there exists some $k \in \Lambda_0 \setminus \{0\}$ such that $\hat{\nu}_k \ne 0$. In particular, the second item is a consequence of the third one which itself follows from Proposition 6.3. $\square$

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Léo Morin

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COPENHAGEN, UNIVERSITETSPARKEN 5,
DK-2100 COPENHAGEN Ø, DENMARK

*E-mail address: lpdm@math.ku.dk*

Gabriel Rivière

LABORATOIRE DE MATHÉMATIQUES JEAN LERAY, NANTES UNIVERSITÉ, UMR CNRS 6629,
2 RUE DE LA HOUSSINIÈRE, 44322 NANTES CEDEX 03, FRANCE

INSTITUT UNIVERSITAIRE DE FRANCE, PARIS, FRANCE

*E-mail address: gabriel.riviere@univ-nantes.fr*