SAN Grégory Boil

V Ũ Ngo

LONG-TIME DYNAMICS OF COHERENT STATES IN STRONG MAGNETIC FIELDS

We consider a charged particle on a plane, subject to a strong, purely magnetic external field. It is well known that quantum dynamics closely follows the classical one for short periods of time, while for times larger than ln 1 , where is Planck's constant, purely quantum phenomena are expected to happen. In this paper we follow the Schrödinger evolution of generalized coherent states for times of order 1/ . We prove that, when the initial energy is low, the initial states splits into multiple wavepackets, each one following the average dynamics of the guiding center motion but at its own speed.

1. Introduction 1.1. Motivation. The aim of this article is to study the propagation of coherent states in the 2-dimensional plane R 2 ⊂ R 3 , subject to a strong magnetic field B = Be 3 perpendicular to the plane. In this article, B will be a C ∞ smooth, non vanishing function B : R 2 → (0, +∞). From the Poincaré lemma, one can find a smooth function A : R 2 → R 2 , the magnetic potential, such that

dA := ∂ 2 A 1 -∂ 1 A 2 = B.
The classical magnetic Hamiltonian is H : R 2 × R 2 (q, p) → p -A(q) 2 ∈ R, giving rise to the Hamiltonian flow Φ t H (q 0 , p 0 ) = (q(t), p(t)) defined by Hamilton's equations (1.1)    q(t) = ∂H ∂p (q(t), p(t)) ṗ(t) = -∂H ∂q (q(t), p(t)), with initial condition q(0) = q 0 , p(0) = p 0 . The corresponding quantum operator, or magnetic Laplacian, is the differential operator given by (1.2)

L ,A := -i ∇ -A(q) 2 = (-i ∂ 1 -A 1 (q)) 2 + (-i ∂ 2 -A 2 (q)) 2 .
Date: June 10, 2018.

1

It is well known that L ,A , as an unbounded operator acting on L 2 (R 2 ), is essentially self-adjoint [START_REF] Ikebe | Uniqueness of the self-adjoint extension of singular elliptic differential operators[END_REF], and one can define the associated Schrödinger unitary group on L 2 (R 2 ) given by

(1.3)    i ∂ t ϕ t = L ,A ϕ t ϕ t=0 = ϕ 0
We shall consider the case where the initial quantum state ϕ 0 is coherent, which means strongly localized in phase space (see Section 2). The general semiclassical theory states that, for finite times (by finite, we mean a time t that does not depend on ), the quantum evolution (1.3) closely follows the classical trajectories (1.1), in the regime where is small. Early works in semiclassical physics and chemistry emphasize the importance of this idea, see [START_REF] Littlejohn | A guiding center Hamiltonian: a new approach[END_REF][START_REF] Heller | The semiclassical way to molecular spectroscopy[END_REF]. More precisely, one can define the classical limit (or semiclassical wavefront set, see Definition 2.16) of a coherent state; this is a position z = (q, p) in phase space. Then, if z 0 denotes the classical limit of the initial quantum state ϕ 0 , Egorov's theorem (Proposition 2.9) ensures that the classical limit of ϕ t coincides with the flow at time t of the classical Hamiltonian starting from z 0 . See for instance [START_REF] Hagedorn | Semiclassical quantum mechanics. i. the → 0 limit for coherent states[END_REF] for a mathematical account of this in the case of electric potentials, or [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Zworski | Semiclassical Analysis[END_REF] for a more general formulation. However, for long times, i.e. times that tend to infinity when → 0, most of the results break down. In many cases, the non-commutativity of the limits → 0 and t → ∞ is simply inextricable, at least beyond the so-called Ehrenfest time t |ln |. In this paper, we will investigate the propagation of a coherent quantum wave packet under the action of strong magnetic fields, for times of order t 1/ . How is this possible?

Classical trajectories of a charged particle under a strong magnetic field have a distinctive feature: they can be approximately described as a superposition of a fast rotation motion (cyclotron, or Larmor motion) with a slow drifting motion (often called the guiding center motion [START_REF] Littlejohn | A guiding center Hamiltonian: a new approach[END_REF]), see Figure 1. If the energy of the initial state is small enough, then this guiding center motion can be controlled, and the fast rotation is almost decoupled, giving rise to an adiabatic invariant (see the classical book [START_REF] Northrop | The adiabatic motion of charged particles[END_REF], or the recent review [START_REF] Cheverry | Mathematical perspectives in plasma turbulence[END_REF]). Then, since we work on a two-dimensional plane, the motion becomes nearly integrable. This near-integrability is a crucial element in an attempt to explain why it is possible to describe the motion for such long times. 1.2. Mathematical background. We shall assume that the magnetic field B and its derivatives have at most polynomial growth at infinity, in the sense that B belongs to a symbol class S(m) for some order function m on R 2 (see Section 2.2). For instance, m(q) = ( q 2 + 1) M for some M ∈ R. Then one can find a potential A lying in some S(m ), with an order function m on R 2 , showing that H ∈ S(m ) for an order function m on R 4 . The magnetic Laplacian defined in (1.2) is the natural Figure 1. Motion of a classical particle in a strong magnetic field. The particle oscillates around a level line of B. Here B(q 1 , q 2 ) = 2 -cos(q 1 ) + q 2 2 .

symmetric (Weyl) quantization of H, acting of the Schwartz space S (R 2 ):

L ,A := Op w (H).
Since L ,A is essentially self-adjoint on L 2 (R 2 ) we will identify L ,A with its selfadjoint extension, and we may now consider the magnetic Schrödinger evolution equation (1.3), where ϕ 0 ∈ S (R 2 ). From Stone's theorem (see e.g. [START_REF] Reed | Methods of mathematical physics[END_REF]), there is a unique continuous family of unitary operators (P t ) t∈R satisfying the propagation equation

   i ∂ t P t = L ,A P t P t=0 = id L 2 (R 2 )
and for any t ∈ R, the solution to (1.3) is given by ϕ t := P t ϕ 0 . In this work, ϕ 0 will be a coherent state in a generalized sense, as follows (see Section 2 for a precise definition). Defining for > 0 and z = (q, p) ∈ R 2 × R 2 the rescaled translation operator

T (z) • Λ : L 2 (R 2 ) f → x → 1 √ e -i 2 q•p e i x•p f x -q √
a state ϕ z will be called a coherent state if one can find a function f ∈ S (R 2 ) with normalized L 2 -norm, a family of functions (g ) ∈(0, 0 ) ∈ S (R 2 ) with seminorms that are uniformly bounded in , and a real number β > 0, such that

ϕ z := T (z)Λ • (f + β g ).
Such a coherent state is said to be centered at z in phase space. We denote by C the set of coherent states. Since we are interested in the propagation of quantum states with low energy, we shall consider coherent states centered at a vanishing point of the Hamiltonian H, i.e. z ∈ Σ with

Σ := H -1 ({0}) = (q, A(q)) | q ∈ R 2 ⊂ R 2 q × R 2 p .
On this smooth surface we define the pull-back B of the magnetic field B as follows.

Let j be the projection j : Σ (q, A(q)) → q ∈ R 2 q . Since Σ is the graph of the function A, j is invertible and then we can view the magnetic field as a function on Σ via B := B • j One can check that, when B is non-vanishing, Σ is a symplectic submanifold of the canonical phase space R 2 × R 2 = T * R 2 (in fact, the restriction to Σ of the canonical symplectic form on T * R 2 is exactly the pull-back j * (Bdq 1 ∧ dq 2 )). Hence B is now a Hamiltonian on Σ and we can consider its Hamiltonian flow Φ t B . A basic result in magnetic dynamics (see also [START_REF] Raymond | Geometry and spectrum in 2d magnetic wells[END_REF]) is that this flow gives a good approximation of the guiding center motion.

Description of the main results.

The main goal is to prove the following fact. Let ϕ 0 be a coherent state centered at a point z in the characteristic surface Σ. Classically, this state cannot move because it has zero kinetic energy. However, due to the uncertainty principle, the quantum state lives on a ball of radius 1/2 around z, and so almost all the points in this ball will actually move slowly, at various speeds of order , with a precise factor depending on the distance to Σ. But, because of energy quantization, the involved speeds will actually be quantized: only integer multiples of B 0 , for some positive constant B 0 , will play a part. Thus, the initial state will be split into a number of new coherent states evolving at theses speed scales, becoming genuinely separated from each other after a time of order -1 .

Theorem 1.1.

There exists a self-adjoint pseudodifferential operator J (the 'Quantum adiabatic invariant') on L 2 (R 2 ) such that for any α ∈ (0, 1) and K > 0, if J ∈ N is a family of integers satisfying J K -α , then the following holds. Let ϕ 0 ∈ C be a coherent state centered in z 0 ∈ Σ = H -1 (0) and consider its propagation (ϕ t ) t∈R through the Schrödinger equation (1.3):

∀t ∈ R, ϕ t := P t ϕ 0 .
Then ϕ t can be decomposed as follows:

(1.4) ϕ t = J j=0 α j ϕ t j, + r t
where (α j ) j∈N ∈ 2 (R + ), there exists T > 0 such that the remainder r t satisfies, for any N ∈ N,

(1.5) sup t∈[0,T / ] (L ,A ) N r t L 2 (R 2 ) = O( ∞ ),
and the ϕ t j, have the following properties. (1) ∀j 0, α j ϕ t j, is the orthogonal projection of ϕ t onto ker(J -

(2j + 1) ); (2) ∀ t ∈ [0, T ], (1.6) ϕ t/ j, ∈ C;
(3) we can follow the center of these coherent states : for any t ∈ [0, T ], we have

(1.7) WF ϕ t/ j, = z((2j + 1) t)
where z( t) := Φ tB z 0 , and Φ tB is the Hamiltonian flow of B on Σ.

Here, WF(f ) denotes the semiclassical wavefront set of the function f ∈ S (R 2 ), see Definition 2.16 below. The quantum adiabatic invariant J corresponds to the pure cyclotron motion that one would obtain for a homogeneous magnetic field of strength 1. Its spectrum consists of the eigenvalues {(2j + 1) ; j ∈ N}, and the corresponding eigenspaces

H j, = ker(J -(2j + 1) )
are infinite dimensional Hilbert spaces that one can call the adiabatic Landau levels of the system. Up to an error of size O( ∞ ), these spaces are preserved by the quantum dynamics. We see from (1.7) that the "adiabatic value" (2j + 1) is coupled with the original dynamics in that it defines a quantized speed of motion for a coherent state living in H j, . Since, to the best of our knownledge, the description given by Theorem 1.1 is new, it would be very interesting to observe this quantization effect in a (real or numerical) experiment; but the long time -1 clearly poses serious numerical challenges.

Notice that the remainder in (1.5) is controlled in terms of the obvious Sobolev-like norms that are preserved by the dynamics: the L 2 -norms of arbitrary powers of the magnetic Laplacian (which, in particular, control the quadratic form L ,A ψ, ψ , and hence, by the Cauchy-Schwarz inequality, the norm of the usual magnetic Sobolev space H 1 A , where derivatives are replaced by magnetic derivatives ∂ j +A j ). But recall that L ,A is not elliptic in the semiclassical sense, since the characteristic manifold Σ extends to infinity in q. Therefore, if one is interested in quantum transport or, rather, quantum localization, it is in general an important, non obvious question to check whether, and to which extent, these norms can be compared to more 'localized' norms like Schwartz seminorms or standard Sobolev norms. The answer is expected to depend on the geometry of the magnetic field. Recently, the control of Schwartz seminorms has been obtained in [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF], under the assumption that the magnetic field is uniformly non-vanishing, and satisfies the following property.

(P) : for all α ∈ N 2 , there exists C > 0 such that, for all x ∈ R 2 , ∂ α B(x) C B(x) . This assumption states that the derivatives of the magnetic field are controlled by the magnetic field itself, preventing B from 'oscillating too much' at infinity. It appears to be rather known and used in literature. In dimension d 3, this property is crucial to obtain that, under some ellipticity condition, L ,A has compact resolvent and hence no essential spectrum; and if these ellipticity conditions do not hold, there is a rather precise description of the essential spectrum (see [START_REF] Avron | Schrodinger operators with magnetic fields. i. general interactions[END_REF][START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de schrödinger avec un champ magnétique[END_REF]). The lower bound at the bottom of the essential spectrum of L ,A obtained in [START_REF] Helffer | Semiclassical analysis for the ground state energy of a schrodinger operator with magnetic wells[END_REF] is an important ingredient in the localization estimates of [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF].

In general, Property (P) is optimal to prove the compactness of the resolvent: in the paper [START_REF] Dufresnoy | Un exemple de champ magnetique dans R ν[END_REF], the author constructs a magnetic field that does not satisfy this property and such that L ,A , while enjoying some ellipticity properties, has no compact resolvent. On the other hand, in dimension 2, the diamagnetic inequality gives an immediate lower bound on the essential spectrum without needing (P). It would be interesting to investigate whether, in dimension 2, the localization estimates of [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF] could be obtained under a weaker condition than (P).

In this work, we take advantage of the results of [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF] to obtain a stronger estimate of the remainder r t in Theorem 1.1, as follows.

Theorem 1.2. With the same notations as in Theorem 1.1, for a magnetic field B satisfying the above property (P), assuming that there exists b 0 > 0 such that

(1.8) ∀q ∈ R 2 , B(q) b 0 ,
we get the following estimate of r t : given any T ∈ R and any Schwartz seminorm p, we have

(1.9) sup t∈[0,T / ] p(r t ) = O( ∞ ) .
We surmise that this result does not hold for general magnetic fields B. Although the control in terms of Schwartz seminorms is much more satisfying in the setting of coherent states, the price to pay for this is that the proof of Theorem 1.2 is substantially more involved than the proof of Theorem 1.1.

1.4. Organization of the article. In section 2, we define the class of coherent states that we will use, and we present some facts about their propagation. We will prove some short time propagation properties for these coherent states and some long-time estimates in the case of coherent states centered at a critical point of the classical dynamics. In section 3, we introduce the symplectic and quantum magnetic normal forms, which are here the main tools to study the long time dynamics. In section 4, we prove Theorems 1.1 and 1.2.

A general class of coherent states and related propagation results

The goal of this section is to introduce a class of 'coherent states', which is large enough to contain the usual ones (such as the Gaussian coherent states and the squeezed ones -see for instance [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF]), but also more flexible, allowing to replace exponential localization by a softer Schwartz-like 'rapid decay'. As the following discussion holds in any dimension, in this paper we will not limit ourselves to the dimension 2, but we will consider instead the general dimension n. Recall that the Schwartz space S (R n ) has the Fréchet topology induced by the seminorms

(2.1) p m,N (f ) := sup x∈R n α∈N n ,|α| m β∈N n ,|β| N x β ∂ α f (x) , m, N ∈ N.
2.1. Definition of the class. We consider first s [n] , the set of shapes of coherent states defined as follows.

Definition 2.1. We say that f ∈ s [n] if and only if one can find f ∈ S (R n ) with f L 2 (R n ) = 1, a family (g ) ∈(0, 0 ) ⊂ S (R n ) with seminorms uniformly bounded with respect to , and a real number β > 0 such that

f = f + β g .
In the definition below, we use the following unitary operators on L 2 (R n ): the rescaling operator

(Λ u)(x) = 1 n/4 u x √
and the translation operator

∀z = (q, p) ∈ R 2n , (T (z)u)(x) = e -i 2 q•p e i x•p u(x -q) .
Definition 2.2. A coherent state ϕ in the class C [n] is given by its center z in the phase space R 2 q × R 2 p , a phase δ ∈ R and a shape f ∈ s [n] by the formula ϕ = e -iδ/ T (z)Λ • f .

In other words, we have

C [n] := ϕ = e -iδ/ T (z)Λ • f δ ∈ R, z ∈ R 2n , f ∈ s [n] .
We will denote C := C [2] . In this text, unless explicitly stated, the expression 'coherent state' will always refer to an element of C. This definition generalizes the ones in [START_REF] Hagedorn | Semiclassical quantum mechanics. i. the → 0 limit for coherent states[END_REF] or [START_REF] Combescure | Coherent States and Applications in Mathematical Physics[END_REF] that only allow a independent part in the shape f . In [START_REF] Paul | On the pointwise behavior of semi-classical measures[END_REF][START_REF] Paul | Semi-classical methods with emphasis on coherent states[END_REF], the author uses a full expansion in 1/2 for the shape in S (R n ) in the Borel sense. In this paper, the shape of the coherent state admits an expansion at the first order in any β in the shape, β > 0, bounded in in S (R n ). Finally, an interesting thing in this definition is that, using results in [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF][START_REF] Combescure | Coherent States and Applications in Mathematical Physics[END_REF][START_REF] Robert | La cohérence dans tout ses états[END_REF], under short-time propagation this class must remain stable. This point will be dealt with in what follows.

Propagation results.

A first and natural question is 'How is such a coherent state propagated for finite times?'. The propagation of coherent states has been widely studied over the last few decades. A lot of results are related to approximation of the propagation of coherent states by sums of coherent states whose parameters depend on time. In [START_REF] Hagedorn | Semiclassical quantum mechanics. i. the → 0 limit for coherent states[END_REF][START_REF] Hagedorn | Semiclassical quantum mechanics. iii. the large order asymptotics and more general states[END_REF][START_REF] Hagedorn | Semiclassical quantum mechanics, iv : large order asymptotics and more general states in more than one dimension[END_REF] we find such propagation results for finite times and for Gaussian like coherent states, defined as the squeezed states in [START_REF] Combescure | Coherent States and Applications in Mathematical Physics[END_REF]. In [START_REF] Hagedorn | Semiclassical dynamics with exponentially small error estimates[END_REF][START_REF] Hagedorn | Exponentially accurate semiclassical dynamics: Propagation, localization, ehrenfest times, scattering, and more general states[END_REF], the authors investigate the propagation for longer times, reaching Ehrenfest times, i.e. times of order | log( )| and the bounds in these approximations are precised in [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF] in terms of the the linearized flow around the underlying classical dynamics, using the well-known propagation of purely Gaussian coherent states through quadratic Hamiltonians and the theory of metaplectic operators. One can find a precise review of the quadratic propagation of coherent states in [START_REF] Robert | Propagation of coherent states in quantum mechanics and applications[END_REF][START_REF] Paul | Semi-classical methods with emphasis on coherent states[END_REF]. In [START_REF] Yajima | Gevrey frequency set and semi-classical behaviour of wave packets[END_REF] the authors investigate expansion in power of instead of 1/2 of the dynamics of coherent states defined in the formalism of Lagrangian states. Nevertheless, due to the analysis of the turning points, this study holds for short times in the propagation. Finally, we can state the work done in [START_REF] Paul | A construction of quasi-modes using coherent states[END_REF][START_REF] Paul | On the pointwise behavior of semi-classical measures[END_REF] where the authors use coherent states and their propagation in order to build quasimodes for general Schrodinger operators

We now present some facts about the propagation of coherent states in the class C [n] , and some Schwartz estimates of these propagations. Our goal is not to approximate the propagation by expansions of coherent states but, because of the general definition of the shape f , to turn these results into a stability property of the class through finite time propagation. We also give estimates on these propagations. To do so, it will be convenient to introduce the semiclassical Sobolev spaces, which are more suitable for pseudodifferential computations. We recall here their definition, and refer to [START_REF] Zworski | Semiclassical Analysis[END_REF]Chapter 8] for a more detailed presentation.

An order function on R 2n is a function m : R 2n → R for which there is an integer N and a constant C > 0 such that for any

X, Y ∈ R 2n m(X) C X -Y N m(Y )
where X := (1 + X 2 ) 1/2 , • denoting the euclidian norm on R n (see for instance [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]Definition 7.4]). The symbol class associated with this function is

S(m) := a ∈ C ∞ (R 2n ; R) | ∀α ∈ N 2n , ∃C α > 0, ∀X ∈ R 2n , |∂ α a(X)| C α m(X) .
One can find an order function m such that S( m) = S(m) and m ∈ S( m); therefore we will always assume that the order function m lies in its own class, i.e.

(2.2) m ∈ S(m).

Given an 'observable' a ∈ S(m), its Weyl quantization is the operator

[Op w (a)ψ](x) := 1 (2π ) n ˆˆR 2n e i (x-y)η a x+y 2 , η ψ(y)dy dη for ψ ∈ S (R n ).
Let g := log(m) and g w := Op w (g); the exponential e ±g w is a pseudodifferential operator with symbol c ± ∈ S(m).

Definition 2.3. The generalized Sobolev space associated with m is given by

H (m) := u ∈ S (R n ) | e g w u ∈ L 2 (R n )
and the norm on H (m) is defined by

u H (m) = e g w u L 2 (R n ) .
We have the following equivalence of seminorms.

Lemma 2.4. Let p be a Schwartz seminorm. One can find two order functions m and m satisfying (2.2) and C > 0 such that for any u ∈ S (R n ) and ∈ (0, 1],

1 C u H (m) p(u) C u H ( m) .
Operators with a symbol in S(m) map the Schwartz space to itself and, more precisely, the continuity between semiclassical Sobolev spaces is given by the following proposition.

Proposition 2.5. Given two order functions

m 1 , m 2 , if a ∈ S(m 1 ), then Op w (a) : H (m 2 ) → H m 2 m 1 defines a continuous operator uniformly for ∈ (0, 1], such that as → 0 Op w (a) H (m 2 )→H (m 2 /m 1 ) = O(1).
The following well-known (and straightforward) lemma will be repeatedly used to deal with the √ -localization of coherent states.

Lemma 2.6. Let c ∈ S(m), for some order function m on R 2n . Then

T (z)Λ -1 Op w (c)T (z)Λ = Op w =1 (c 1 ), where c 1 (X) := c(z + √ X) (and hence c 1 ∈ S(m)).
We can now prove some propagator estimates.

Lemma 2.7. Let z ∈ R 2n and > 0. Then for any f, ϕ ∈ S (R n ), for any order function

ν on R 2n , T (z)Λ f H (ν) C 1 f H 1 (ν) and T (z)Λ -1 ϕ H 1 (ν) C 2 ϕ H (ν)
with C 1 , C 2 independent of , and

H 1 (ν) := H =1 (ν).
Proof. Let f , z, ν as above. Let γ := log(ν), and γ w := Op w (γ), γ w 1 := Op w =1 (γ). Then, using the unitarity of T (z)Λ ,

T (z)Λ f H (ν) = T (z)Λ -1 e γ w T (z)Λ e -γ w 1 e γ w 1 f L 2 (R n )
.

Since e ±γ w = Op w (c ± ) with c ± ∈ S ν ±1 , Lemma 2.6 gives

T (z)Λ -1 e γ w T (z)Λ = Op w =1 (c 1 ) with c 1 : X → c(z + √ X), c 1 ∈ S(ν). Therefore, T (z)Λ e γ w T (z)Λ e -γ w 1 = Op w =1 (c 1 )e -γ w 1 is bounded on L 2 (R)
because it is a pseudodifferential operator with a symbol in S [START_REF] Avron | Schrodinger operators with magnetic fields. i. general interactions[END_REF]. Finally,

T (z)Λ f H (ν) Op w =1 (c 1 )e -γ w 1 L 2 (R n ) f H 1 (ν) .
For the reverse inequality, we can write

T (z)Λ -1 ϕ H 1 (ν) = T (z)Λ e γ w 1 T (z)Λ -1 e -γ w e γ w ϕ L 2 (R n ) . But, T (z)Λ e γ w 1 T (z)Λ -1 = Op w (c -1 ) with c -1 : X → c X -z √ ,
The symbol c -1 belongs to the 'limit' class S 1/2 (ν) (see [START_REF] Zworski | Semiclassical Analysis[END_REF]Chapitre 4] for a definition and the related theory). Thus, Op w (c -1 )Op w (c -) is a pseudodifferential operator with a symbol in S 1/2 (1), and from the Calderon-Vaillancourt theorem it defines a bounded operator on L 2 (R n ). Finally,

T (z)Λ -1 ϕ H 1 (ν) Op w (c -1 )e -γ w L 2 (R n ) ϕ H (ν)
, which concludes the proof.

We prove now the following continuity property of bounded propagators on generalized Sobolev spaces.

Lemma 2.8. Let (H t ) t∈[0,1] be a real-valued time-dependent family of Hamiltonians.

We assume (H t ) t∈[0,1] to be continuous in time and to lie in S( 1) uniformly in times. Then there exists a family

(U (t, s)) t,s∈[0,1] of unitary operators on L 2 (R n ) satisfying (1) U (0, 0) = id L 2 (R n ) and for any r ∈ [0, 1], U (t, s) = U (t, r)U (r, s); (2) U (•, •) is strongly continuous on L 2 (R n ); (3) for any ψ ∈ L 2 (R n ) and any s ∈ [0, 1], the map t → U (t, s)ψ is differentiable,
and satisfies

(2.3) i ∂ t U (t, s)ψ = Op w (H t )U (t, s)ψ in L 2 (R n ) ,
and this family is unique. Moreover, for any order function ν, for any ϕ ∈ S (R n ), we have

U (t, 0)ϕ H (ν) C ϕ H (ν)
where C is -independent.

Proof. We consider the equation

(2.4) i ∂ t U (t, 0)ϕ = Op w (H t )U (t, 0)ϕ ϕ ∈ S (R n ).
Let ν be an order function on

)ϕ) ∈ C 1 ([0, 1], H (ν)), with (U (t, s)) (t,s)∈[0,1]
satisfying the above assumptions (1), ( 2) and ( 3). One should remark that U (t, 0) then defines an operator in

C 1 ([0, 1], L(H (ν))), that is unitary for ν = 1, i.e. on L 2 (R n ).
In order to prove the announced estimate, let now γ := log(ν) and γ w := Op w (γ). We set also U γ := e γ w U e -γ w , and Ĥγ

t := e γ w Op w (H t )e -γ w , so that U γ ∈ C 1 ([0, 1], L(L 2 (R n )))

and Ĥγ

t is a pseudodifferential operator with symbol in S [START_REF] Avron | Schrodinger operators with magnetic fields. i. general interactions[END_REF]

such that i ∂ t U γ (t, 0) = Ĥγ t U γ (t, 0) and -i ∂ t U γ (t, 0) * = U γ (t, 0) * Ĥ-γ t .
Then we have for ϕ

∈ L 2 (R n ), i ∂ t U γ (t, 0)ϕ 2 L 2 (R n ) = U γ (t, 0) * e -γ w [e 2γ w , Op w (H t )]e -γ w U γ (t, 0)ϕ 0 , ϕ 0 L 2 (R n ) = η w (t)U γ (t, 0)ϕ, U γ (t, 0)ϕ L 2 (R n )
where η w (t) := Op w (η(t)) = i e -γ w [e 2γ w , Op w (H t )]e -γ w whose symbol η(t) belongs to S(1) uniformly in time. Therefore η w (t)U γ (t, 0)ϕ ∈ L 2 (R n ), which justifies the above computation. Then by Proposition 2.5, we have

∂ t U γ (t, 0)ϕ 2 L 2 (R n ) η w (t) L 2 (R n )→L 2 (R n ) U γ (t, 0)ϕ 2 L 2 (R n ) which implies U γ (t, 0)ϕ L 2 (R n ) exp ˆ1 0 η w (s) L 2 (R n )→L 2 (R n ) ds ϕ L 2 (R n ) .
Considering now ψ ∈ H (ν) and ϕ = e γ w ψ proves the result.

We now turn to the propagation of coherent states for finite times. The following proposition generalizes to the class C [n] the result obtained by Robert for Gaussian states (see [ 1), the corresponding Hamiltonian vector field is uniformly bounded in times, that ensures us the global existence of the flow

: [0, 1] × R 2n → R 2n . Then, for a coherent state ϕ ∈ C [n] centered at z 0 ∈ R 2n , the state ψ := U (1, 0)ϕ is still a coherent state in C [n] , centered at z 1 := Φ t=1 H (z 0 ). Proof. Let ϕ (t) = U (t, 0)ϕ and ψ = ϕ (1). Since ϕ ∈ S (R n ), from Lemma 2.8 we know that, for all t in [0, 1], ϕ (t) ∈ S (R n ). Because ϕ ∈ C [n] , there are f 0 ∈ s [n] , δ 0 ∈ R and z 0 ∈ R 2n such that (2.5) ϕ = e -iδ 0 / T (z 0 )Λ (f + β g ), β > 0, f ∈ S (R n ) and (g ) ∈(0, 0 ) ⊂ S (R n ) (see Definition 2.2). Let (z t ) 0 t 1 denote the classical evolution from z 0 led by the time-dependent Hamiltonian (H t ) 0 t 1 , ∀t ∈ [0, 1], z t = Φ t H (z 0 ). As (H t ) t∈[0,1] lies uniformly in times in S(
(Φ t H ) t∈[0,1]
. Introducing the action integral

δ t := δ 0 + ˆt 0 (p s • qs -H s (z s ))ds - 1 2 (q t • p t -q 0 • p 0 ), we define (2.6) v (t) := e -iδt/ Λ -1 T (-z t )U (t, 0)ϕ (0).
We establish the propagation equation of v (t). Noting that ∀ψ ∈ S (R n ),

∂ t T (z t )ψ (x) = T (-z t ) - i qt p t -q t ṗt 2 + ṗt • x + qt • i ∇ x ψ (x)
and by the differentiability property (2.3),

i∂ t v (t) = 1   Λ -1 T (-z t )Op w (H t )T (z t )Λ -Λ -1 Op w (x)∂ q H t (z t ) -Op w (ξ)∂ p H t (z t ) Λ -H t (z t )   v (t) where (x, ξ) = X ∈ R 2n . Since Λ -1 T (-z t )Op w (H t )T (z t )Λ = Op w (X → H t (z t + √ X)),
we are naturally led to Taylor expanding the function

X → H t (z t + √ X) around the point z t ; let K 2 (t, X) := X T Hess(H t , z t )X
and let R (3) be the integral remaining term of order 3 in this Taylor expansion. We obtain

   i∂ t v (t) = Op w =1 K 2 (t, X) + √ R (3) (t, X) v (t)
v (t = 0) = f + β g Recall, from the definition (2.6) of v (t), that this equation admits a propagator P 1 (•, •) given by

P 1 : (t, s) ∈ [0, 1] 2 → P 1 (t, s) = T (z t )Λ -1 U (t, s)T (z t )Λ .
From this formula, P 1 inherits the group and strong continuity properties of U . Moreover, it defines a map from S (R n ) to itself that satisfies, for any v ∈ S (R n ),

i∂ t P 1 (t, 0)v = Op w =1 K 2 + √ R (3) v ,
and from Lemma 2.7 and Lemma 2.8, for any order function ν,

P 1 (t, 0)v H 1 (ν) C v H 1 (ν) ,
where C is time and independent. We consider now the following propagation equation in v (0) :

(2.7)

   i∂ t v (0) (t) = Op w =1 (K 2 (t, •))v (0) (t) v (0) (t = 0) = f .
Since it is defined by a time-dependent quadratic Hamiltonian, we may apply the result of [5, Theorem 2.8], which asserts that the propagator is well-defined as long as the classical flow z t exists. Hence for t ∈ [0, 1], Equation (2.7) admits a propagator (P q (t, s)) t,s∈[0,1] (where the subscript q stands for 'quadratic'), which is smooth in t and unitary on L 2 (R n ), satisfying P q (t, r)P q (r, s) = P q (t, s).

In order to prove that P q acts on S (R n ), we will use Gaussian coherent states. Let G 0 be the normalized Gaussian on R n ,

G 0 (X) = 1 π n/4 e -X•X 2 and define G z := T =1 (z)G 0 .
Since P q (t, 0)f ∈ L 2 (R n ), the resolution of identity property of Gaussian coherent states gives (2.8)

P q (t, 0)f = ˆR2n f, P q (t, 0) * G z G z dz.
From [6, Chapter 3, Theorem 16], there is a differentiable map

t ∈ [0, 1] → Γ t ∈ Π + n
where Π + n is the Siegel space of complex and symmetric matrices with positive-definite imaginary part, such that,

P q (t, 0)G z = G Γt zt . Here we have denoted, for Γ ∈ Π + n , G Γ 0 (X) = a Γ e -i X•ΓX 2 ,
where a Γ is the L 2 -normalization constant and zt corresponds to the Hamiltonian flow of the quadratic Hamiltonian starting from z. Since f ∈ S (R n ) we can, for any t ∈ [0, 1], integrate by parts and obtain, for any integer

N > 0 f, G zt L 2 C N (f, Γ t ) zt N .
This ensures, from usual theorems of derivation under the integral (2.8), that

[0, 1] t → P q (t, 0)f ∈ S (R n )
is continuous. We now prove that such a quadratic propagator satisfies the Schwartz estimates as in Lemma 2.8. To do so, considering ν an order function, γ := log(ν), and γ w 1 := Op w =1 (γ), we have for any Schwartz function f ,

P q (t, 0)f H 1 (ν) = e γ w 1 P q (t, 0)f L 2 (R n ) ˆˆR 2n | f, G Γt zt | e γ w 1 G z L 2 (R n ) dz.
A computation gives (see Proposition 2.12) that there is an integer l depending only on the order function ν such that

e γ w 1 G z L 2 (R n ) C z l ,
where C is a constant depending only on ν; hence one can find an order function ν such that

P q (t, 0)f H 1 (ν) C(ν) f H 1 (ν) .
So, Equation (2.7) admits a propagator family P q (t, s)

t,s∈[0,1]
that satisfies the same Schwartz continuity property as P 1 . For t ∈ [0, 1], let

v (0) (t) := P q (t, 0)f ∈ S (R n )
be the solution to (2.7); notice that v (0) (t) does not depend on and satisfies, for

any t ∈ [0, 1], v (0) (t) L 2 (R n ) = v (0) (0) L 2 (R n ) .
Using now Duhamel's principle, we get 3) )P q (s, 0)f ds.

v (t) -v (0) (t) = β P 1 (t, 0) g -i √ ˆt 0 P 1 (t, s) Op w =1 (R (
From Lemma 2.8, as the Schwartz seminorms of (g ) >0 are bounded, the same holds for P and P q (s, 0)

s∈[0,1]
with respect to (t, s) in [0, 1], that the integral 3) )P q (s, 0)f ds is uniformly bounded in S (R n ) for all t ∈ [0, 1] and ∈ (0, 1]. Finally, we have 3) )P q (s, 0)f ds and β := min(1/2, β), which proves v (1) ∈ s [n] (actually we obtain v (t) ∈ s [n] for any t ∈ [0, 1]). Finally,

ˆt 0 P 1 (t, s) Op w =1 (R (
v (1) = v (0) (1) + β g where g := β-β P 1 (t, 0) g -i 1/2- β ˆt 0 P 1 (t, s) Op w =1 (R (
ψ = e iδ 1 / T (z 1 )Λ v (1)
where δ 1 := δ t=1 and z 1 := z t=1 , which establishes the announced result.

Remark 2.10. The fact that the function f in (2.5) is normalized in L 2 is not relevant. Indeed, the proof still works for any shape function of the form f = f + β g , where f is an -independent function of L 2 (R 2 ), β > 0 and (g ) >0 is a family of functions in S (R 2 ) with bounded seminorms, uniformly with respect to ∈ (0, 1].

We now develop some very useful tools on the study of the coherent state class we introduced earlier on, that is the Wigner transform.

Wigner transform of coherent states and applications.

The main idea of the Wigner transform is to take advantage of the strong localization of coherent states (sometimes called 'peak states') to get estimates on pseudodifferential operators from the associated symbol. We refer to [START_REF] Folland | Harmonic analysis in phase space[END_REF] for the definition below of the Wigner transform. In [START_REF] Robert | Propagation of coherent states in quantum mechanics and applications[END_REF][START_REF] Combescure | Coherent States and Applications in Mathematical Physics[END_REF], authors use fast decay properties of the Wigner transform of coherent states to get strong approximation results on the propagation of coherent states depending on the regularity of the generator of the propagation. Here, we will take advantage of the Schwartz property of our class of coherent states to prove some estimates on pseudodifferential operators for symbols class S(m) for arbitrary large order functions m.

We consider a family (ϕ z ) z∈R 2n of coherent states with the same shape: for any z ∈ R 2n , ϕ z = T (z)Λ f , where f is a given shape in s [n] . Now, let a ∈ S(m) a given symbol, with m(z) z l for an integer l. We have

Op w (a)ϕ z , ϕ z L 2 (R n ) = 2 n (2π ) n × ˆRn ˆˆR 2n e 2i (x-ỹ)η a (ỹ, η) e -i 2 qp+ i p(2ỹ-x) f 2ỹ-x-q √ dỹdη e i 2 q p -i p x f x-q √ dx ,
which leads to the following definition.

Definition 2.11. We define the Wigner function W z,z associated with the family (ϕ z ) z∈R 2n of coherent states with shape f ∈ s [n] as

W z,z (y, η) = 2 n n/2 e i ω(Y -1 2 z ,z-z ) × ˆRn e 2i u η-p+p 2 f   y -q+q 2 -u √   f y -q+q 2 + u √ du
where ω is the canonical symplectic form on R 2n With this definition we have the following formula

(2.9) Op w (a)ϕ z , ϕ z L 2 (R n ) = 1 (2π ) n ˆR2n a(Y )W z,z (Y )dY ,
which highlights the Wigner function as a Kernel to compute the action of pseudodifferential operators on coherent states. Setting

Ψ := (ŷ, u) ∈ R 2n → f (ŷ -u)f (ŷ + u),
Ψ is a Schwartz function on R 2n with all Schwartz seminorm uniformly bounded with ∈ (0, 0 ) and we have the shorter expression

W z,z (y, η) = 2 n n/2 e i ω(Y -1 2 z ,z-z ) × ˆRn e 2i u η-p+p 2 Ψ   u √ , y -q+q 2 √   du.
The goal is to use the oscillating integral structure to show some fast decay with respect to Y -z+z 2 and z -z and to prove the following key property for this article. Proposition 2.12. Let a ∈ S(m) for an order function m on R 2n satisfying for

l ∈ N, C > 0, ∀Y ∈ R 2n , |m(Y )| C Y l .
Let (ϕ z ) z∈R 2n being a family of coherent states with shape f ∈ s [n] . Then we have for ε > 0, for N ,M ∈ N large enough,

Op w (a)ϕ z 2 L 2 (R n ) -3n/2 C ε (f ) f 2-ε L 2 (R n ) × ˆˆR 2n ˆˆsupp(a) Y -z+z 2 √ -N dY z -z √ -M dz with (2.10) C ε (f ) = max α∈N n ,|α| K ∂ α a(Y ) Y -l L ∞ (R n ) i∈I j∈J p i,j (f ) δ i,j
where K is an integer depending on N and M and I, J finite sets depending only on N , M , l and ε, and with p i,j being some Schwartz seminorm, δ i,j > 0 depending on ε.

Remark 2.13. The parameter ε is necessary to handle the growth of the symbol a at infinity, which is compensated by f ε . Of course, if a ∈ S(1) we don't need this analysis, and the result holds for ε = 0, directly from the Calderon-Vaillancourt theorem.

Proof. Applying Plancherel's theorem along with (2.9), we first get

Op w (a)ϕ z 2 L 2 (R n ) f 2 L 2 (R n ) = 1 (2π ) 3n ˆˆR 2n ˆˆR 2n a(Y )W z,z (Y )dY 2 dz .
We now use the fact that the Wigner function is an oscillating integral. Noting that

z -z √ -2N • (1 -∆ Y ) N • e i ω(Y -z 2 ,z-z ) = e i ω(Y -z 2 ,z-z )
and, for any integer M ,

η -p+p 2 √ -2M • (1 -∆ u ) M e 2i u•(η-p+p 2 ) = 2 M e 2i u•(η-p+p 2 )
we get

Op w (a)ϕ z 2 L 2 (R n ) f 2 L 2 (R n ) 1 (2π ) 3n • 2 2n n × ˆˆR 2n ˆˆR 2n Y l z-z √ 2N Y -z+z 2 √ 2M 2N |k|=0 2 2M C k Y -l |∂ k Y a(Y )|× ˆRn y -q+q 2 √ 2M (1 -∆ u ) M Ψ [k]   u √ , y -q+q 2 √   dudY 2 dz with Ψ [k]   u √ , y -q+q 2 √   = 0 |j 1 k |,|j 2 k | 2N Ck u √ j 1 k ∂ j 2 k y Ψ   u √ , y -q+q 2 √   constants Ck not depending on , for k ∈ N 2n . As a ∈ S(m), with ∀Y ∈ R 2n |m(Y )| Y l , we have sup 0 |k| N 2 2M C k ∂ k Y a(Y ) Y -l L ∞ (dY ) C(l, M ) with C(l, M ) independent of . Moreover, Y l z-z √ 2N Y -z+z 2 √ 2M C l z l z-z 2 2N -l Y -z+z √ √ 2M -l .
Finally, setting

C l,M (Y ) = C(k, l, M )× 2N |k|=0 ˆRn y -q+q 2 √ 2M (1 -∆ u ) M Ψ [k]   u √ , y -q+q 2 √   du, from Hölder inequality in Y variable, we get Op w (a)ϕ z 2 L 2 (R n ) f 2 L 2 (R n ) C 2l z 2l • 2 2n (2π ) 3n n × ˆˆR 2n    ˆˆsupp(a)   z -z √ 2N -l Y -z+z 2 √ 2M -l   -2 dY    × ˆˆsupp(a) |C l,M (Y )| 2 dY dz . We now estimate ˆR2n |C l,M (Y )| 2 dY.
For more simplicity, we set

ŷ := y - q + q 2 η := η - p + p 2 Ŷ := Y - z + z 2 Noting that Ŷ √ -1 η √ -1
and that

2N |k|=0 ŷ √ 2M (1 -∆ u ) M Ψ [k] u √ , ŷ √ = i,j∈ Î C i,j ŷ √ i u √ j ∂ α i,j f u - ŷ √ ∂ β i,j f ŷ + u √
where the Î is a finite set and C i,j only depends on the integers k, N, M and l, but not on the shape f , we get for δ > 0 small enough

|C l,M (Y )| 2 i,j∈ Ĵ Ci,j η √ -2 M p i,j (v, w) → f v √ f w √ 2δ ×   ∂ α i,j f • √ 1-δ ∂ β i,j f • √ 1-δ   2 (2ŷ)
with again J a finite set and Ci,j > 0, these two only depending on the integers k, N, M and l, but not on the shape f , and where is the convolution product. So, integrating, we have

ˆR2n |C l,M (Y )| 2 dY i,j∈ Ĵ Ci,j ˆRn η √ -2 M dη × p i,j f • √ 2δ p i,j f • √ 2δ 2 -n × ∂ α i,j f (1-δ) • √ 2 L 2 (R n ) ∂ β i,j f (1-δ) • √ 2 L 1 (R n )
due to usual estimates on the convolution product. Noting that

∂ α i,j f (1-δ) • √ 2 L 2 (R n ) -n/2 ∂ α i,j f 2(1-2δ) L 2 (R n ) ∂ α i,j f 2δ L 1 (R n )
and that

∂ β i,j f (1-δ) • √ L 1 (R n ) -n/2 ∂ β i,j f (1-2δ) L 2 (R n ) p K (∂ β i,j f ) 2δ/(1+2δ) ˆRn x K dx
for some fixed integer K and for p K a seminorm depending on K, we use the following lemma to estimate the terms

∂ α f L 2 (R n ) and ∂ β f L 2 (R n )
. This lemma is known as the Kolmogorov inequalities.

Lemma 2.14. Let f ∈ S (R n ). Then for any L ∈ N, for any c ∈ N, c L, for any integer i such that 1 i n, we have

∂ c i f L 2 (R n ) f 1-c L L 2 (R n ) ∂ L i f c L L 2 (R n )
Then, for any f ∈ S (R n ), for c = (c 1 , . . . , c n ) ∈ N n , for L ∈ N, L max c, and L := (L, . . . , L), we have

∂ c f L 2 (R n ) f n i=1 (1-c i L ) L 2 (R n ) ∂ L f i,j (1-c i L ) c j L L 2 (R n ) . So, (2.11) ˆR2n |C l,M (Y )| 2 dY -3n/2 C L,K,N,M,l (f )× i,j∈ Ĵ f 4(1-2δ) k 1- α (k) i,j L 1- β (k) i,j L L 2
for, given i, j ∈ Ĵ, α i,j = (α

(0) i,j , . . . , α (n) i,j ), β i,j = (β (0) i,j , . . . , β (n) i,j
), and for L ∈ N, L max i,j,k (α

(k) i,j , β (k) i,j
). Let now ε > 0 : for L great enough and δ small enough, from (2.11) we get

ˆR2n |C l,M (Y )| 2 dY -3n/2 C ε (f ) f 4-ε L 2 (R n )
with C ε (f ) satisfying (2.10). We then get the expected result

Op w (a)ϕ z 2 L 2 (R n ) -3n/2 C ε (f ) f 2-ε × ˆˆR 2n ˆˆsupp(a) Y -z+z 2 √ -N dY z -z √ -M dz
We state two applications of this result. The first one is a straightforward application of the above Proposition 2.12. Proposition 2.15. Let (ϕ z ) z∈R 2n a family of coherent states with shape f ∈ s [n] and let m be any order function on R 2n . Then, given any ε > 0 we have the continuity property

(2.12) ∀a ∈ S(m), ∀z ∈ R 2n , Op w (a)ϕ z 2 L 2 (R n ) -3n/2 C(a)C ε (f ) f 2-ε L 2 (R n ) with C(a) C max 0 |α| K m -1 ∂ α a L ∞ (R 2n
) where C > 0 only depending on m and with C ε (f ) as in (2.10).

This property gives a useful boundedness property for pseudodifferential operators acting on C [n] . We will mainly use this property in the last part of the paper, where the shape will depend on a parameter j ∈ N. The second property is about the localization of elements of C [n] . We first define what mean by localization (one could also say microlocalization) of a function depending on . Definition 2.16. Let 0 > 0 and let (ϕ ) ∈(0, 0 ] be a family of Schwartz functions on R n such that ϕ L 2 (R n ) = O (1). We define its semi-classical wavefront set WF(ϕ ) as follows. Let Z 0 ∈ T * R n , then Z 0 / ∈ WF(ϕ ) if and only if there exists a ∈ S( 1) such that a(Z) γ > 0 for a fixed γ ∈ R and for Z in a small neighborhood of Z 0 with

Op w (a)ϕ L 2 (R n ) = O( ∞ ). Then a function ϕ ∈ S (R n ) is said localized at X 0 ∈ T * R n if WF(ϕ ) ⊂ {X 0 }.
Proposition 2.17. Let ϕ z be a coherent state centered at z ∈ T * R n . Let a ∈ S(m), for some order function m, and assume that one can find δ > 0 such that a vanishes on B(z, δ), then [START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de schrödinger avec un champ magnétique[END_REF] and with constants in O( ∞ ) only depending on δ.

Op w (a)ϕ z 2 L 2 (R n ) = C(a)C ε (f ) f 2-ε L 2 (R n ) O( ∞ ) with C(a) and C ε (f ) as in Proposition 2.
Proof. Let δ > 0 and a ∈ S(m) such that supp(a) ⊂ T * R n = R 2n \ B(z, δ). Then, if ε > 0, because of Proposition 2.12, we have Op w (a)ϕ z 2 L 2 (R n ) C ε (f ) f 2-ε L 2 (R n ) × ˆˆR 2n ˆˆR 2n \B(z,δ) Y -z+z 2 √ -N dY z -z √ -M
dz .

We denote

C ε := C(a)C ε (f ) f 2-ε . Then, setting Υ := Y -z and ζ := z -z, we get Op w (a)ϕ z 2 L 2 (R n ) C ε   ˆˆB(0,δ) ˆˆR 2n \B(0,δ) Υ √ -N dΥ ζ √ -M dζ (1)
+ ˆˆR 2n \B(0,δ) ˆˆR 2n \B(0,δ) Υ √ -N dΥ ζ √ -M dζ (2)
  and one can check that (1) = O( M ) and (2) = O( N ), so that for any integer M, N , we have Op w (a)ϕ z

2 L 2 (R n ) C ε O( N + M ).
From this we get the following strong localization property.

Proposition 2.18. Let z ∈ T * R n and ϕ z = T (z)Λ f ∈ C [n] . Then ϕ z is localized at z, i.e. WF(ϕ z ) = {z}. Moreover if a ∈ S(m), where m is a given order function, a vanishing near z, then for any order function µ and any ε > 0 we have

Op w (a)ϕ z H (µ) = C(a, µ)C ε (f ) f 1-ε L 2 (R n ) O( ∞ ) with C(a, µ), C ε (f ) as in Proposition 2.17.
Proof. From Proposition 2.17, we get that WF(ϕ z ) ⊂ {z}. Now, using stationary phase methods (see e.g. [START_REF] Zworski | Semiclassical Analysis[END_REF]Chapter 3]), for a ∈ S( 1) with compact support we get

Op w (a)ϕ z 2 L 2 (R n ) = |a(q, p)| 2 f 2 L 2 (R n ) + O( ) , which gives
WF(ϕ z ) = {z}. We now consider a ∈ S(m) and γ := log(µ), for some order functions m, µ. We have

Op w (a)ϕ z H (µ) = e γ w Op w (a)ϕ z L 2 (R n ) .
Since L := e γ w Op w (a) is a pseudodifferential operator with its symbol in S(mµ), we may apply Proposition 2.17 to obtain the announced result.

In order to apply the above properties to the magnetic propagation of coherent states, we now introduce the magnetic normal form background.

Geometry and propagation under magnetic field

In this section, for the reader's convenience, we recall some recent results on the geometry and analysis of magnetic fields that will be crucial for our analsis.

The first part of this section is about the geometry underlying the magnetic field, more precisely about the classical and quantum normal forms of L ,A . Classsical and quantum normal forms for the magnetic Hamiltonian were introduced by Raymond and the second author, see [START_REF] Raymond | Geometry and spectrum in 2d magnetic wells[END_REF]. We first briefly present the symplectic normal form adapted to the Hamiltonian H, and the corresponding quantum normal form. The second part of this section recalls the long time quantum propagation estimates from [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF].

3.1. Symplectic normal form. The following symplectic normal form result is about the existence of a symplectomorphism that locally around Σ ∩ Ω transform the magnetic Hamiltonian H into an almost integrable system. Theorem 3.1 ([25]). Let

H(q, p) := p -A(q) 2 , (q, p) ∈ T * R 2 = R 2 × R 2 ,
where the magnetic potential A : R 2 → R 2 is smooth. Let B := ∂ q 1 A 2 -∂ q 2 A 1 be the corresponding magnetic field. Let Ω ⊂ R 4 be a bounded open set such that B does not vanish on

Ω 0 = {q ∈ R 2 |(q, A(q)) ∈ Ω}.
Then there exists a symplectic diffeomorphism κ, defined in an open set Ω ⊂ C z 1 ×R 2 z 2 , with values in T * R 2 , wich identifies the plane {z 1 = 0}∩ Ω with the surface {H(q, p) = 0|q ∈ Ω 0 }, and such that

H • κ = |z 1 | 2 f (z 2 , |z 1 | 2 ) + O(|z 1 | ∞ ),
where f : R 2 × R → R is smooth. Moreover, the map

φ : Ω 0 q → κ -1 • j(q) ∈ ({0} × R 2 z 2 ) ∩ Ω is a local diffeomorphism and f • (φ(q), 0) = |B(q)|.
From this theorem, we get a better understanding of the dynamics near the characteristic surface Σ. Indeed, up to an O(|z 1 | ∞ ) term, that is for trajectories very close to Σ, the trajectories in the (z 1 , z 2 ) coordinates are given by the Hamiltonian

|z 1 | 2 f (z 2 , |z 1 | 2 ) that

Theorem 3.2 ([25]

). For small enough, there is a unitary Fourier integral operator

U such that U * L ,A U = N + R ,
where (1) N is a pseudodifferential operator that commutes with

I := 2 ∂ 2 x 1 + x 2 1 ;
(2) For any hermite function h j , such that I h j = (2j + 1)h j , the operator N (j) acting on L 2 (R x 2 ) by

(3.1) h j ⊗ N (j) (u) = N (h j ⊗ u)
is a classical pseudodifferential operator in S( 1) of order 1 in with its principal symbol

n (j) (x 2 , ξ 2 ) = (2j + 1)B • φ -1 (x 2 , ξ 2 )
with φ the diffeomorphism in Theorem 3.1; (3) Given any pseudodifferential operator D whose principal symbol d 0 such that

d 0 (z 1 , z 2 ) = c(z 2 )|z 1 | 2 +O(|z 1 | 3
), and any N 1, there exist pseudodifferential operators S ,N and Z N such that

R = S ,N (D ) N + Z N ,
with Z N a pseudodifferential operator whose symbol is supported away from a fixed neighborhood of

|z 1 | = 0. (4) N = H 0 +H 1 , where H 0 = Op w (H 0 ), H 0 = B •φ -1 (z 2 )|z 1 | 2 ,

and the operator

H 1 is relatively bounded with respect to H 0 with an arbitrarily small relative bound.

Actually, the above theorem is a slightly different formulation of the one stated in [START_REF] Raymond | Geometry and spectrum in 2d magnetic wells[END_REF]. Indeed, the authors introduced a Fourier integral operator Ũ that is unitary only microlocally in a fixed neighborhood of Σ. It will be useful for us to have a genuinely unitary operator, which microlocally satisfies the same assumptions. Since the canonical transformation associated to U is obtained by the flow of a compactly supported, time-dependent Hamiltonian, one can in fact obtain U as the quantum flow of a time-dependent, uniformly bounded self-adjoint propagator, as in Lemma 2.8; see [START_REF] Zworski | Semiclassical Analysis[END_REF]Chapter 11].

3.3. Quantum propagation under magnetic field. As is well known, for a general Schrödinger operator, and for times bigger than the Erhenfest time |ln |, propagated coherent states may not remain coherent. Thus, in order to obtain a rough control on the localization of the propagated state, it is natural to estimate the growth of its Schwartz seminorms. In the case of a purely magnetic propagation, this is given by the following theorem, proved in [START_REF] Boil | On the stability of the schwartz class under the magnetic schrödinger flow[END_REF]. For the purpose of this article, we only state it in the 2-dimensional case.

Theorem 3.3 ( [2]

). Let P t be the propagator of L ,A , given by Stone's theorem,

P : t ∈ R → P t = e -it L ,A .
Under property (P), and assuming B b 0 > 0,we have

∀t ∈ R, P t S (R d ) ⊂ S (R d ) .
More precisely, for all M ∈ N * , for any Schwartz seminorm p, there exist 0 > 0, C > 0, N ∈ N * and a seminorm p, such that, for all ∈ (0, 0 ), and for all ψ 0 ∈ S (R d ), and all t ∈ [0, -M ], p(P t ψ 0 ) C -N p(ψ 0 ) .

Propagation of C-class states

Our goal is to propagate a state ϕ 0 ∈ C through the magnetic Schrödinger equation, (4.1)

   i ∂ t ϕ t = L ,A ϕ t ϕ t=0 = ϕ 0 ,
where the initial state ϕ 0 is such that

WF(ϕ 0 ) ⊂ {X 0 } for X 0 ∈ Σ ∩ Ω.
We consider the Fourier integral operator U from Theorem 3.2. Since U is the time-1 flow of a time-dependent quantum Hamiltonian, the same holds for the adjoint (or inverse) U * , and we may apply Proposition 2.9, which shows that ψ 0 := U * ϕ 0 is still a coherent state of the class C. Moreover, because the canonical transformation associated with U * is the symplectomorphism κ -1 , we obtain that ψ 0 is centered at a point z 0 := κ -1 (X 0 ) ∈ κ -1 (Σ) = {z 1 = 0}. Our strategy is to first propagate ψ 0 through the quantum normal form N and then to relate it to the propagation of ϕ 0 through the initial Schrödinger equation (4.1). 4.1. Propagation through the normal form. The normal form N of Theorem 3.2 defines an essentially self-adjoint operator on S (R 2 ). Indeed, as

N = H 0 + H 1
with H 1 relatively bounded with respect to H 0 , it suffices to prove that H 0 is essentially self-adjoint. This follows from the fact that H 0 is a tensor product of two essentially self-adjoint operators on S (R). Thus, we can study the following Cauchy problem:

(4.2)    i ∂ t ψ t = N ψ t ψ t=0 = ψ 0 .
From Stone's theorem, we have a unique family of propagators (Q t ) t∈R , unitary on L 2 (R 2 ) and satisfying

∀ψ ∈ S (R 2 ) i ∂ t Q t ψ = N Q t ψ.
Then, we denote ψ t := Q t ψ 0 the solution to (4.2). As explained above, ψ 0 ∈ C is localized on κ -1 (X 0 ) = (0, z 0 2 ). Considering the harmonic oscillator in the variable z 1 ,

I = -2 ∂ 2 x 1 + x 2 1
, and the associated Hermite functions (h j ) j∈N ⊂ S (R x 1 ), we get the following Lemma.

Lemma 4.1.

There is a family of states

(f 0 j ) j∈N ⊂ S (R x 2 ) and a sequence (α j ) j∈N ⊂ 2 (R + ) such that ψ 0 = ∞ j=0 h j ⊗ f 0 j
with for any integer j, if α j = 0 then α -1 j f 0 j ∈ C [1] , and WF x 2 (f 0 j ) = {z 0 2 }. In the case of α j = 0, we have

f j = O L 2 ( β ) and WF x 2 (f 0 j ) ⊂ {z 0 2 }. Proof. From Definition 2.2, one can find a function f in S (R 2 ), with f L 2 (R 2 ) = 1,
and a family (g ) ∈(0, 0 ) ⊂ S (R 2 ), with seminorms in S (R 2 ) bounded uniformily in , such that

ψ 0 = T (0, z 0 2 )Λ (f + β g ). Then we let (4.3) f 0 j := ψ 0 , h j dx 1 = T (z 2 ) (z 0 2 )Λ (z 2 ) (f + β g ), h ( =1) j dx 1 ∈ S (R x 2 ). Since (h j ) j∈N is a Hilbert basis of L 2 (R x 1 ), after defining ∀j ∈ N, α j := f, h ( =1) j L 2 (Rx 1 ) L 2 (Rx 2 ) , we get α 2 j = f 2 L 2 (R 2 )
= 1 and, for j ∈ N, α j = 0, α -1 j f 0 j ∈ C [1] , centered at z 0 2 . In the case of α j = 0, we get f 0 j = O S ( β ) and applying Proposition 2.17 we have WF x 2 (f 0 j ) ⊂ {z 0 2 }, with equality if g is not O S ( ∞ ) (in that case, we have WF x 2 (f 0 j ) = ∅). Lemma 4.1 expresses ψ 0 as a sum of functions, each one corresponding to a 'Landau level' of the harmonic oscillator: f 0 j corresponds to the occupation of the j th energy level. The following lemma gives the energy distribution of these levels, in the semiclassical limit → 0. Lemma 4.2. Taking the -independent sequence (α j ) j∈N ⊂ 2 (R + ) introduced above, there is an -dependent sequence (ε( )) j ⊂ R such that for any j ∈ N f 0 j 2 = α 2 j + β ε j ( ) with j 0 α 2 j = 1 and where j 0 |ε j ( )| is bounded independently of .

Proof. From the proof of Lemma 4.1 we have j 0 α 2 j = 1 and we can compute, using (4.3):

f 0 j 2 L 2 (Rx 2 ) = f, h ( =1) j dx 1 2 dx 2 + β 2Re f, h ( =1) j dx 1 , g , h ( =1) j dx 1 dx 2 + β g , h ( =1) j dx 1 2 dx 2 = α 2 j + β ε j ( ), where |ε j ( )| 1 2 ( f, h ( =1) j dx 1 2 dx 2 + g , h ( =1) j dx 1 2 dx 2 ) + β g , h ( =1) j dx 1 2 dx 2 and j 0 |ε j ( )| g 2 L 2 (R 2 ) ( 1 2 + β ) + 1,
wihch is uniformly bounded for ∈ (0, 0 ).

Remark 4.3. The result of Lemma 4.2 can be extended to the functions x γ ∂ δ x ψ 0 , for any γ, δ in N 2 . Indeed, we have

x γ ∂ δ x T (z)Λ = -|δ| T (z)Λ [q + √ x] γ [ip + √ ∂ x ] δ where we denote z = (z 1 , z 2 ) ∈ C × R 2 , z 1 = q 1 + ip 1 , z 2 = (q 2 , p 2
) and finally q = (q 1 , q 2 ), p = (p 1 , p 2 ). Then, for ψ 0 = T (z)Λ • (f + β g ), with f and the family (g ) as in Definition 2.2, we get

x γ ∂ δ x ψ 0 = -|δ| T (z)Λ (f γ,δ + β g γ,δ
), with β = min(1/2, β), where the function f γ,δ ∈ S (R 2 ) is independent of , and the family (g γ,δ ) belongs to the Schwartz class with bounded seminorms, uniformly in , satisfying the definition of C. Then, letting

x γ ∂ δ x ψ 0 = -|δ| j∈N h j ⊗ f γ,δ j with f γ,δ j = |δ| x γ ∂ δ x ψ 0 , h j dx 1 , mimicking the above proof, we get f γ,δ j 2 L 2 (dx 2 ) = |α γ,δ j | 2 + β ε γ,δ j ( )
where (α γ,δ j ) j and (ε γ,δ j ) are sequences as in Lemma 4.2 except that j |α γ,δ j | 2 is not necessarily equal to 1.

Applying Lemma 4.1, we now study the propagation of

ψ 0 = ∞ j=0 h j ⊗ f 0 j .
From (3.1), the solution to (4.2) can be written as

ψ t = ∞ j=0 h j ⊗ f t j
where for all j ∈ N, f t j satisfies the evolution equation

(4.4)    i∂ t f t j = 1 N (j) f t j f t=0 j = f 0 j
where 1 N (j) is a semi classical pseudodifferential operator whose principal symbol is (2j + 1)B • φ -1 . In order to apply usual propagation results to (4.4) we rescale time as follows. Given j ∈ N, we set τ j := t × (2j + 1) so that τ j represents the reduced time corresponding to the slow drift of the j th landau level. So for a global time propagation t ∈ [0, T -1 ], we have a reduced time τ j ∈ [0, (2j + 1)T ]. We let now f τ j j := f t j and get the following equivalent propagation equation

   i ∂ τ j f τ j j = F (j) f τ j j f τ j =0 j = f 0 j , where F (j) := 1 (2j + 1) N (j)
is a pseudodifferential operator acting on L 2 (R x 2 ) whose principal symbol is B • φ -1 . At this point, we may highlight that even if the total symbol of F (j) depends on j, its principal symbol does not. Since N (j) is the restriction of an essentially selfadjoint operator to a stable subspace, it is essentially selfadjoint on L 2 (R x 2 ). Therefore, from Stone's theorem we get a family of propagators (Q τ j j ) τ j ∈R for this equation; coming back to the initial propagation time t, we obtain, for t ∈ [0, T / ],

f t j = f τ j j = Q (2j+1) t j
f 0 j . Furthermore, from Lemma 4.1 and Proposition 2.9, we know that α -1 j f τ j j is a coherent state in C [1] when τ j is fixed, and the evolution of its wavefront set is governed by the Hamiltonian flow of K := B • φ -1 , which is the pull back of the magnetic field B from Ω 0 ⊂ R 2 q on the zero energy surface {z 1 = 0} by φ = κ • (j |Σ ) -1 . More precisely, let Φ t K denote the Hamiltonian flow of K, and let T > 0 be such that Φ

(2j+1) t K (z 0 2 ) stays in the open set Ω for t ∈ [0, T ]. Then, since WF x 2 (f t=0 j ) ⊂ {z 0 2 }, we get WF x 2 ( f τ j j ) ⊂ {Φ τ j K (z 0
2 )} for τ j ∈ [0, (2j + 1)T ]. In other words, for any t ∈ [0, T ], the wavefront set of f t/ j is given by (4.5)

WF x 2 (f t/ j ) ⊂ {Φ (2j+1) t K (z 0
2 )}, while the solution to (4.2) is given by

∀t ∈ [0, T / ], ψ t = j∈N h j ⊗ f t j .
We can then at this stage of the proof give an informal explanation about the quantum phenomenon occurring here. Actually, our coherent state splits into a weighted sum of coherent states, each one with a wavefront set with its own dynamics. For times t of order -1 , the localization of ψ t is no more a point in phase space but a sequence of points lying on the integral curve of the Hamiltonian provided by the magnetic field in restriction to the characteristic set, or the zero set. The point labelled by 'j' corresponds to the occupation of the j th Landau levels of I . Furthermore, since the Schrödinger evolution preserves the L 2 -norm, that is

(4.6) f t j L 2 (Rx 2 ) = f 0 j L 2 (Rx 2 )
, the occupied energy level of the harmonic oscillator remains occupied with the same amount of energy, that is α 2 j . In other words, we do not have any shift of energy to higher or lower levels during the evolution. 4.2. Back to the initial magnetic laplacian. After the study of the propagation in the normal form setting, we need to return to the initial laplacian L ,A . The first step is the following lemma. Recall that from Lemma 4.1 that we write (4.7)

ψ 0 = ∞ j=0 h j ⊗ f 0 j .
Lemma 4.4. Let α ∈ (0, 1) and for K > 0 consider an dependent integer J such that J K -α . Then, for

(4.8) ψ0 := j J h j ⊗ f 0 j , we have ψ 0 -ψ0 = O S ( ∞ ).
Proof. Let us first recall a useful fact about Hermite coefficients of Schwartz functions. Consider some function ψ in S (R). We have ψ = j∈N c j h j with c j := ψ, h j L 2 (R) . Since ψ ∈ S (R), for any N ∈ N, for any > 0, I N ψ ∈ S (R). Furthermore, we have

I N ψ = j∈N N (2j + 1) N c j h j .
Hence, taking the L 2 -norm of I N ψ, and = 1, we get (4.9)

j∈N (2j + 1) 2N |c j | 2 < ∞
and so for any non negative integer N the sequence (j 2N |c j | 2 ) j∈N is bounded. Consider now the above function ψ 0 ∈ C ⊂ S (R 2 ). We have

ψ 0 = T (0, z 2 )Λ • (f + β g )
with f and (g ) as in Definition 2.2. In (4.7) we have f 0 j = ψ 0 , h j L 2 (dx 1 ) and f 0 j 2 L 2 (dx 2 ) = α 2 j + β ε( ) according to Lemma 4.2. Then, from (4.9), and by definition of the sequence (α j ) j , for any non negative integer N , (α 2 j j 2N ) j is bounded (and independent of ). Moreover, from the -boundedness of (g ) in S (R 2 ), we get that for any N , I N g L 2 (R 2 ) is bounded uniformly with ∈ (0, 0 ). From this, we have that for any non negative integer N , (j 2N ε j ( )) j is bounded uniformly with ∈ (0, 0 ). We now turn to (4.8); let

d := ψ 0 -ψ0 = j>J h j ⊗ f 0 j ,
and we wish to prove that, for any γ, δ ∈ N 2

x γ ∂ δ x d ∞ = O( ∞ ).
To do so, we note that

x γ ∂ δ x d = j>J x γ 1 1 ∂ δ 1 x 1 h j ⊗ x γ 2 2 ∂ δ 2 x 2 f j . First, if we set f γ 2 δ 2 j := δ 2 (x γ 2 2 ∂ δ 2
x 2 f j ), then because of the definition of the sequence (f j ) j and applying Remark 4.3, we get that there is a sequence of real numbers (α γ 2 δ 2 j ) j independent of and a family of sequences (ε γ 2 δ 2 j ( )) j uniformly bounded with respect to ∈ (0, 0 ) such that

f γ 2 δ 2 j 2 L 2 (dx 2 ) = α γ 2 δ 2 j 2 + β ε γ 2 δ 2 j ( ).
We now study the x γ 1 1 ∂ δ 1 x 1 h j term. One can note that from the 1-D recursion relations

h j = j 2 h j-1 - j + 1 2 h j+1
and

xh j = j 2 h j-1 + j + 1 2 h j+1 we get x γ 1 1 ∂ δ 1 x 1 h j = j+(γ 1 +δ 1 ) k=j-(γ 1 +δ 1 ) C γ 1 δ 1 j k h k where (C γ 1 δ 1 j k
) k,j is a sequence of real numbers such that for any integers γ 1 , δ 1 , there exists C γ 1 ,δ 1 > 0 such that for any k, j in N,

|C γ 1 δ 1 j k | C γ 1 δ 1 j γ 1 +δ 1 2
.

We finally get

x γ ∂ δ x d = -δ 2 j>J   j+γ 1 +δ 1 k=j-(γ 1 +δ 1 ) C γ 1 δ 1 j k h k   ⊗ f γ 2 δ 2 j
that we rewrite as

x γ ∂ δ x d = -δ 2 k>J -(γ 1 +δ 1 ) h k ⊗   k+γ 1 +δ 1 j=k-(γ 1 +δ 1 ) C γ 1 δ 1 j k f γ 2 δ 2 j   . Denoting f γδ k := k+γ 1 +δ 1 j=k-(γ 1 +δ 1 ) C γ 1 δ 1 j k f γ 2 δ 2 j ,
which is a Schwartz function as a finite sum of Schwartz functions on R, we get that

x γ ∂ δ x d 2 L 2 (R 2 ) -2δ 2 k>J γ 1 δ 1 f γδ k 2 L 2 (dx 2 ) with J γ 1 δ 1 := J -(γ 1 + δ 1 ). Note that there is a constant K > 0 such that J γ 1 δ 1 K -α . The sequence (2k +1) N f γδ k L 2 (dx 2 ) k
, is bounded uniformly with ∈ (0, 0 ). Indeed, from above computations,

(2k + 1) N f γδ k L 2 (dx 2 ) C γ 1 δ 1 k+(γ 1 +δ 1 ) j=k-(γ 1 +δ 1 ) j γ 1 +δ 1 2 +N 2k + 1 j N (α γ 2 δ 2 j ) 2 + β 0 |ε γ 2 δ 2 j ( )| C N γ 1 δ 1 k+(γ 1 +δ 1 ) j=k-(γ 1 +δ 1 ) j γ 1 +δ 1 2 +N (α γ 2 δ 2 j ) 2 + j γ 1 +δ 1 2 +N β 0 |ε γ 2 δ 2 j ( )| 2(γ 1 + δ 1 ) + 1 C N γ 1 δ 1 j γ 1 +δ 1 2 +N (α γ 2 δ 2 j ) 2 ∞ + β 0 j γ 1 +δ 1 2 +N ε γ 2 δ 2 j ( ) ∞
and from what precedes, for any non negative integer N , the sequences (α γ 2 δ 2 j ) 2 j 2N j and ε γ 2 δ 2 j ( )j 2N are bounded uniformly in ∈ (0, 0 ). Finally, from the boundedness of (2k + 1) N f γδ k L 2 , we get

x γ ∂ δ x d 2 L 2 (R 2 ) = -2δ 2 k>J γ 1 δ 1 f γδ k 2 L 2 (dx 2 ) = -2δ 2 k>J γ 1 δ 1 f γδ k 2 L 2 (dx 2 ) k 2N (k α ) -2N 2N α C γδN 2(N α-δ 2 ) k>J γ 1 δ 1 (k α ) -2N
for any non negative integer N . Then, by sum-integral comparison methods, we have

x γ ∂ δ x d 2 L 2 (R 2 ) = O( (2N -1)α-2δ 2 ), from what we obtain that x γ ∂ δ x d = O L 2 ( ∞ )
p(f 0 j ) δ < ∞
We will use this property later on.

Thus, this lemma teaches us that only the first Landau levels of our initial state ψ 0 matter in the propagation, but we need to consider a large number of levels in order to get a nice estimation of the remainder. We now relate the propagation of the state ψ0 := j J ψ 0 j, through N to the propagation of ϕ 0 through the initial laplacian.

Propagation under general magnetic fields.

In what follows, we assume that the magnetic field B does not vanish on Ω 0 , but we don't require Property (P). Recall also from the introduction that B belongs to a symbol class S(m). Theorem 3.2 gives the relation between L ,A and N ,

L ,A = U N U * + U R U * .
For j ∈ N and t ∈ R, we let ψ t j, := h j ⊗ f t j . From (4.4) we get the following propagation equation:

i ∂ t U ψ t j, = L ,A U ψ t j, -U R ψ t j, .
Considering α ∈ (0, 1) and J ∈ N as in Lemma 4.4 and summing for j J , we get the following propagation equation for ψt := j J h j ⊗ f t j :

(4.10)

   i ∂ t U ψt = L ,A U ψt + e t ψt=0 = ψ0
with e t = U R ψt . We compare the solution to (4.10) to the solution ϕ t to (4.1). For any > 0 and for any t ∈ R, setting D t = ϕ -U ψt , the Duhamel principle gives (4.11) D t = P t D 0 -i ˆt 0 P t-s e s ds where P t is the propagator associated with (4.1), given by Stone's theorem. We aim at proving that for a given time T and a given integer N , uniformly for times

t ∈ [0, T / ], L ,A N D t L 2 (R 2 )
= O( ∞ ).

Since P t is unitary and commutes with L ,A , we have

L ,A N D t L 2 (R 2 ) L ,A N D 0 L 2 (R 2 ) + T 2 sup s∈[0,T / ] L ,A N e s L 2 (R 2 )
.

Since L ,A N is a pseudo-differential operator with symbol in S(m N ) and U is unitary, we have due to Lemma 4.4

L ,A N D 0 L 2 (R 2 ) j>J ψ 0 j, H (m -N ) = O( ∞ ). Noting that L ,A N e s L 2 (R 2 )
R ψs H (m -N ) , it remains to prove (4.12) sup

s∈[0,T / ] R ψs H (m -N ) = O( ∞ ).
Actually, we prove that for any order function µ, R ψs

H (µ) = O( ∞ ),
which implies (4.12). To do so, we begin with giving an explicit expression of R .

Using D = H 0 in Item 3 of Theorem 3.2, we get that for any N ∈ N, there are pseudodifferential operators S ,N , Z N such that R = S ,N H N 0 + Z N . Then, for any order function µ, as j K -α

(4.14) R j J ψ t j, H (µ) C N (1-α)N j J S ,N Op w B • φ -1 (h j ⊗ f t j ) H (µ) + j J Z N ψ t j, H (µ)
.

From Proposition 2.15, we get that for t ∈ [0, T / ], for a given ε ∈ (0, 1),

(4.15) S ,N Op w B • φ -1 (h j ⊗ f t j ) H (µ) -3/2 C(µ, N )C(ψ t j, ) f 0 j 1-ε L 2 (Rx 2 )
because of (4.6), with C(ψ t j, ) as in (2.12). To estimate this constant, we note that ψ t j, = id L 2 (Rx 1 ) ⊗ Q t j • [h j ⊗ f 0 j ] where t = t ∈ [0, T ] so from Lemma 2.8 and because of n (j) ∈ S(1), we have for any order function µ We now estimate the Schwartz seminorms of Z N ψ t . Using Proposition 2.18, as Z N is a pseudodifferential operator with symbol N supported away from a fixed neignborhood of {z 1 = 0}, where the states ψ t j, are centered at any time t ∈ [0, T / ], and because of (4.16), we get for any order function µ and for any ε > 0, (4.18) sup

t∈[0,T / ] Z N ψ t j, H (µ) C( N , µ)C ε (f 0 j ) f 0 j 1-ε L 2 (R) O( ∞ ).
Finally, using (4.17

M C N C(µ, N ) (1-α)N -3/2 j J f 0 j 1-ε L 2 (R 2 ) + C( N , µ)O( ∞ ) j J C ε (f 0 j ) f 0 j 1-ε L 2 (R) .
Now, from Remark 4.5, we get that

j J (1 + C ε (f 0 j )) f 0 j 1-ε L 2 (R)
is a convergent series whose sum is bounded uniformly with , so that

L ,A N ϕ t -U ψt L 2 (R 2 )
= O( ∞ ).

Propagation under magnetic fields satisfying the property (P). We now establish the propagation result for magnetic fields satisfying (P). We assume henceforth that (P) holds for B. We assume also there exists b 0 > 0 such that ∀q ∈ R 2 , B(q) b 0 . Considering again D t = ϕ -U ψt , from (4.11) we have D t = P t D 0 -i ˆt 0 P t-s e s ds and we now aim at proving that for a given T > 0, uniformly for times t ∈ [0, T / ],

D t = O S ( ∞ ).
We will reach this goal in two steps. We first prove that P t D 0 = O S ( ∞ ). We have

U * D 0 = j>J ψ 0 j,
then from Lemma 4.4, we get that (4.21)

U * D 0 = O S ( ∞ ).
Applying Lemma 2.8 to U , which is a time-1 quantum flow, we get

D 0 = U U * D 0 = O S ( ∞ ) .
Finally, applying Theorem 3. In other words, letting ϕ t j, = U ψ t j, , we have proved where (ϕ t ) t∈R is the solution to (4.1), 4.3. End of the proof of Theorems 1.1 and 1.2. We consider for any j the subspaces

H j, = U • span(h j ) ⊗ L 2 (R x 2 )
and the operator J = U I U * , which is pseudodifferential by the Egorov theorem. Since (h j ) j∈N is a Hilbert basis, the spaces H j, are in direct sum, and ∀j ∈ N, ∀f ∈ H j, , J f = (2j + 1) f. For all t ∈ R, ϕ t j, := U ψ t j, , and ψ t j, is the projection of ψ t on the space span(h j ) ⊗ L 2 (R x 2 ). Therefore, ϕ j, is the projection of ϕ t on H j, , so ϕ t j, ∈ H j, , and

ϕ t = j∈N ϕ t j, .
This with (4.20) proves (1.4) and (1.5). Now, for each j, using (4.5) and Proposition 2.9 applied to the Fourier integral operator U , we see that the wavefront set of ϕ t -1 j, is located at the point

κ(0, Φ (2j+1)t K z 0 2 ) = Φ (2j+1)t B X 0
which proves (1.7). Finally, (1.6) is proved by Lemma 4.1. This ends the proof of Theorem 1.1. In order to prove Theorem 1.2, we still need to estimate the remainder r in Schwartz seminorms. These estimates are given by (4.23), which gives (1.9).

Conclusion

In this paper, we have shown that localized quantum particles can be split into distinct pieces by long time magnetic propagation, a property that, in similar contexts, was sometimes called quantum ubiquity. On a mathematical level, this splitting always holds modulo a small O( ∞ ) term in L 2 norm (Theorem 1.1). In order to get the stronger Schwartz estimates (1.9), in our study it was necessary for the magnetic field to satisfy property (P) and the ellipticity condition (1.8). These properties allow the use of Theorem 3.3 which is crucial in handling the remainder terms. It would be interesting to investigate whether, under magnetic confinement leading to discrete spectrum, as in [START_REF] Raymond | Geometry and spectrum in 2d magnetic wells[END_REF], we could get rid of property (P). In this case, thanks to the control on the Hamiltonian dynamics (see [START_REF] Raymond | Geometry and spectrum in 2d magnetic wells[END_REF]Section 3]), one expects to get good estimates for quantum propagators up to times of order -M for any M > 0. In this regime, the split wave-packets should self-interfere, giving rise to magnetic revivals (see [START_REF] Robinett | Quantum wave packet revivals[END_REF][START_REF] Lablée | Autour de la dynamique semi-classique de certains systèmes complètement intégrables[END_REF]). We hope to return to this question in the future.

  ω((a, b), (c, d)) = b • c -a • d and where Y := (y, η).

3 . 2 .

 32 gives the center-guide dynamics, whose center z 2 (t) follows the Hamiltonian flow led by f (•, |z 1 | 2 ). Moreover, as f (z 2 , |z 1 | 2 ) = B(z 2 ) + O(|z 1 | 2 ), still near the surface Σ, the dynamics of the center z 2 (t) is given by the magnetic field pulled back on Σ, B := B • φ -1 . Quantum normal form. The following theorem is the quantum version of Theorem 3.1. It introduces the normal form of L ,A .

id L 2 (L 2 (

 22 Rx 1 ) ⊗ Q t j • [h j ⊗ f 0 j ] H (µ) C(µ) f 0 j H (µ) .Then using the link between the Schwartz seminorms and the weighted Sobolev norms in Lemma 2.4, we get(4.16) sup t∈[0,T / ] C(ψ t j, ) C(f 0 j )and because of the definition of f 0 j , f 0 j H (ν) j∈N is a bounded sequence for any order function ν, and so is C(f 0 j ) Op w B • φ -1 (h j ⊗ f t j ) Rx 2 ) .

  ) and (4.18) with (4.14) we get, for any t ∈ [0, T / ],

  (µ) = O( ∞ )for any order function µ. This proves (4.13), and then we get(4.20) supt∈[0,T / ]

  T / ]p (e s ) = O( ∞ ). Now, applying Theorem 3.3 again, the same estimate holds for p(P t-s e s ), and by the semi-norm property we may integrate and obtain(4.22). This gives sup t∈[T / ] p D t = O( ∞ ).

  for any multi-indices γ, δ in N 2 . So, giving any order function m, we get d H (m) = O( ∞ ), which is the expected result. It appears in the above proof that for any Schwartz seminorm p and for any δ > 0, we have

	Remark 4.5. j∈N

  3 to P t , t ∈ [0, T / ], we have the long-time estimate sup t∈[0,T / ]p P t D 0 = O( ∞ ). P t-s e s ds = O S ( ∞ ).We first prove that uniformly for times s ∈ [0, T / ], e s = O S ( ∞ ). Recall that e t = U R j J ψ t j, and the estimate (4.19), holding for any order function µ. Using Lemma 2.4 and Lemma 2.8, we get for any Schwartz seminorm p,

	It remains now to be shown that (4.22) ˆt 0
	sup
	t∈[0,T / ]

Acknowledgement :

The authors are grateful to Nicolas Raymond for his important contribution to the article by bringing to their attention the property (P), leading to the proof of Theorem 3.3.


Ronan-Alexandre Cherrueau

Adrien Lebre

Dimitri Pertin

Fetahi Wuhib

João Monteiro Soares

Edge Computing Resource Management System: a Critical Building Block! Initiating the debate via OpenStack

While it is clear that edge infrastructures are required for emerging use-cases related to IoT, VR or NFV, there is currently no resource management system able to deliver all features for the edge that made cloud computing successful (e.g., an OpenStack for the edge). Since building a system from scratch is seen by many as impractical, this paper provides reflections regarding how existing solutions can be leveraged. To that end, we provide a list of the features required to operate and use edge computing resources, and investigate how an existing IaaS manager (i.e., OpenStack) satisfies these requirements. Finally, we identify from this study two approaches to design an edge infrastructure manager that fulfils our requirements, and discuss their pros and cons. This paper aims at initiating the discussion in our community.

Introduction

While several academic studies have highlighted the advantages of the edge computing paradigm in various domains [START_REF] Bonomi | Fog Computing and its role in the Internet of Things[END_REF][START_REF] Satyanarayanan | The emergence of Edge Computing[END_REF][START_REF] Shi | Edge Computing: Vision and challenges[END_REF][START_REF] Yi | Fog Computing: Platform and applications[END_REF][START_REF] Zhang | The Cloud is not enough: Saving IoT from the Cloud[END_REF], progress on how to operate and use infrastructures that serve it are marginal. Solutions like Akamai Cloudlets [START_REF]Akamai Cloudlets[END_REF] or AWS Lambda [START_REF]Amazon Lambda@Edge[END_REF] are close to the initial fog proposal that allows to run domainspecific applications on NFV-enabled infrastructures (at the edge) and centralized clouds [START_REF] Bonomi | Fog Computing and its role in the Internet of Things[END_REF]. Since these solutions do not allow to run stateful workloads in isolated environments (e.g. containers, virtual machines (VM)), they do not fully cover the requirements from developers and operators (DevOps) who expect to find most features that made current cloud solutions successful also at the edge. Therefore, such solutions are not considered in this (paper's) initial debate.

The ETSI Mobile Edge Computing Industry Specification Group defined in 2016 an architecture to orchestrate distinct independent cloud systems, a.k.a. Virtual Infrastructure Managers (VIM) [START_REF] Sabella | Mobile-Edge Computing architecture: The role of MEC in the Internet of Things[END_REF]. The idea consists in federating VIMs of the different Data Centers (DCs) that compose the edge infrastructure. By reusing VIMs, ETSI targets edge computing resource managers that behave in the same fashion as traditional ones, while mitigating development requirements. Although there is no implementation available, the idea of federating VIMs seems promising as several projects have been built on similar concepts. ONAP [START_REF]Open Network Automation Platform[END_REF], an industry-driven solution, enables the orchestration and automation of virtual network functions across distinct VIMs. From the academic side, FogBow [START_REF] Brasileiro | Fogbow: A middleware for the federation of IaaS Clouds[END_REF] aims to support federations of Infrastructure-as-a-Service (IaaS) providers. More recently, NIST initiated a collaborative effort with IEEE to advance Federated cloud through the development of a conceptual architecture and a vocabulary 1 .

Although all these projects provide valuable contributions, they all have been designed by only considering the DevOps' perspective. They provide abstractions to manage the life cycle of geo-distributed applications, but do not address administrative requirements. However, edge computing infrastructures differ from federated cloud systems in various aspects [START_REF] The | Cloud Edge Computing: Beyond the Data Center (White Paper)[END_REF], for instance: edge sites are potentially unmanned and therefore must be administered remotely; management systems should be designed to cope with intermittent network access to sites; distinct operators might be interested in interconnecting their infrastructures (like network peering).

To capitalize on the advent of edge computing, our community should take part in current discussions and actions in order to deliver a well-suited resource management system. A system that will (i) let an operator aggregate, supervise and expose the massively distributed resources of the infrastructure, and (ii) let DevOps implement new kinds of services on top of an infrastructure that may be deployed and managed on-demand.

In this paper, we present reflections to initiate discussions through our community.

• First, we introduce a classification of the features expected by both administrators and DevOps. This classification is valuable to identify missing mecha- nisms in resource management systems. • Second, based on the identified requirements, we discuss how an edge resource management system should be designed. In particular, we study pros and cons of top-down and bottom-up approaches. The former consists in interacting with each site only through exposed VIM APIs, such as in federated approaches. The latter aims at revising internal mechanisms of VIMs to enable native collaborations. Since there are many possible edge infrastructure designs, we highlight that the infrastructure considered in this study is composed of several individually-managed and geo-distributed micro DCs (up to thousands) composed of up to one hundred servers (nearly two racks). Figure 1 depicts such an infrastructure. The expected latency and bandwidth between elements may fluctuate, in particular because networks can be wired or wireless. Moreover, disconnections between sites may occur, leading to network split-brain situations [START_REF] Markopoulou | Characterization of failures in an operational ip backbone network[END_REF]. Finally, it is possible to consider additional DCs at the extreme edge, within private institutions or public transports. From the software viewpoint, since we investigate how existing cloud managers can be extended to the edge, we consider as our default VIM OpenStack [START_REF]OpenStack: Open source software for creating private and public Clouds[END_REF]: the de-facto opensource solution for cloud computing infrastructures. 2 The remainder of the paper is organized as follows. Section 2 provides a list of features expected by administrators and DevOps. Section 3 studies how OpenStack satisfies these features, highlighting the need for collaboration across the entire edge infrastructure. Section 4 discusses two approaches to design a resource management system for the edge. Section 5 concludes the paper.

Admin/DevOps' Requirements

Herein we classify features administrators and DevOps expect to find in the context of edge computing infras-tructures. The classification is based on 5 levels, starting from the easiest aspects, i.e., interacting with a single site (considered in level 1, L1), to more complex aspects like managing multiple sites (L2), up to taking into account that sites can be owned by different operators (L5). Table 1 summarizes the classification we detail below.

As previously mentioned, a large part of these features are common to the ones offered by current IaaS resource management systems. They are implemented by various services, each of which is in charge of the management of a particular aspect of the infrastructure [START_REF] Moreno-Vozmediano | IaaS Cloud architecture: From virtualized datacenters to federated Cloud infrastructures[END_REF].

L1. Operate/use any site This level considers the actions both administrators and DevOps expect to perform on a single, reachable site. Most operations are elementary from the edge viewpoint because they correspond to the ones already provided by a VIM such as OpenStack. In other words, each edge site can be considered as an independent cloud at this level. The unmanned aspect only impacts this level by requiring to perform all operations remotely if needed. Furthermore, the resource management system should provide means to ensure the integrity of the hardware resources taking part to the edge infrastructure. Strategies such as enabling/disabling physical interactions with the equipment should be considered.

L2. Operate/use several sites In L2, L1 features are considered but over several sites. This includes operations such as provisioning/managing multiple resources or gathering information from various sites simultaneously. Operations can be either intra-service (same service from different sites) or inter-service (different services from different sites). Examples of such operations include configuring users' access on a per-site basis; listing available VM images or pushing new ones on multiple sites. From the DevOps viewpoint, a user should be able to boot a VM on Site 1 using an image defined in Site 2 (inter-service operation). Similarly to L1, DevOps also expect metrics to be collected from several sites and collaboration mechanisms regarding the security (e.g., secret key sharing, network encryption).

Because collaboration between sites can be either explicit (i.e., the targeted sites are explicitly specified in the operation), or implicit (i.e., the resource management system is in charge of selecting resources), we have defined two sub-levels: L2.1 and L2.2, as depicted in Table 1. The implicit manner suggests that policies (e.g., performance objectives, energy consumption) and constraints (e.g., affinity rules, hardware requirements) are provided by admins/DevOps so that the resource management system takes the right decisions regarding the defined desiderata and the state of the infrastructure (e.g., auto-scaling, relocating workloads between sites, re-scheduling faulty resources across sites). Since split-brains might impact differently alreadyprovisioned resources and management services, we refine L3 into two sub-features. L3.1 is related to features that allow already-deployed applications to continue to serve local requests without being impacted. For instance, an apache server or a storage backend should be able to satisfy requests coming from the same geographical area, even if management services cannot be reached. L3.2 features are related to the provisioning of new resources and other management operations as described in the previous paragraph. Note that guaranteeing L2 features will not be always possible at the L3.2 level because information cannot be gathered in case of a disconnection. For instance, guaranteeing quotas across the infrastructure might be a challenge without first relaxing the consistency requirement of the information. New approaches will have to be proposed for such operations.

Finally, the possibly intermittent network connectivity between edge resources requires the ability for sites to join and leave the infrastructure.

L4. Multiple cloud environments

Delivering a resource management system at large scale in a unified manner poses a great challenge from the software viewpoint. Since different versions and types of infrastructure managers might co-exist at the same time across the whole infrastructure, L4 gathers the related requirements. More precisely, L4.1 considers L3 features when different versions of the same software stack co-exist across the infrastructure. L4.2 increases the complexity by considering the collaboration of mutiple systems, including possibly different concepts (e.g., OpenStack for VMs and Kubernetes for Containers). Such a requirement implies the ability to get sites' capabilities in order to only allow meaningful collaborations.

L5. Multiple operators L5 corresponds to the holy grail in terms of expected features. In addition to L4, it includes the possibility to use sites owned by different operators. We do not specify any requirement for administrators at this level as an operator is unlikely to let other operators administer its own site. However, operators should be able to collaborate to offer their sites' resources to any DevOps like it has been done for a while for cellular networks. The requirements here are more related to the collection and sharing of relevant metrics enabling each operator to perform charging/billing. Summary As previously mentioned, each level of our classification increases the complexity in terms of design and development constraints. Note however that since L4 and L5 both extend L3, they can be considered at the same level and can be swapped as a consequence.

Although we tried to be exhaustive, this list of features could probably be extended, e.g., by considering different edge infrastructure scenarios (e.g., including smaller and limited devices). However, we believe it is already valuable as it delivers significant insights on the design and implementation of an edge resource management system. In the next section, we study whether OpenStack can fulfill the L1, L2, and L3 levels. The discussion of L4 and L5 is left as future work.

OpenStack at the Edge

This section studies the use of OpenStack to control an edge infrastructure. Such an analysis is meaningful as the OpenStack code base has been evolving to deal with large scale and multi-site objectives for the two last years. Particularly, the section evaluates OpenStack, including its latest improvements, with respect to the requirements defined in the previous section.

At a high level, OpenStack has two types of nodes: data nodes delivering XaaS capabilities (compute/storage/network, i.e., data plane); control nodes executing OpenStack services (i.e., control plane). Whenever users issue a request to OpenStack, the control plane processes the request which may potentially also affect the data plane in some manner. Key control plane services include keystone, nova, glance, and neutron, respectively responsible for authentication/authorization, VM life cycle management, VM image management and associated network management.

Because OpenStack comes with several deployment alternatives and because the edge sites considered in our study can host data and control nodes, we elected to discuss two scenarios: a centralized management scenario and a multiple regions scenario described next.

Centralized (Remote) Management

In this scenario, OpenStack operates an edge infrastructure as a traditional single DC environment, the key difference being the wide-area networking found between the control and compute nodes [START_REF]Toward Fog, Edge, and NFV Deployments: Evaluating Open-Stack WANwide, Boston OpenStack Summit[END_REF]. The distinction between the different edge sites can be realized by leveraging the concept of host aggregates within OpenStack.

From the requirements' viewpoint, L1 and most L2 requirements can be fulfilled in a straightforward way (because the infrastructure can be spread over several network domains, some L2 operations including specific network actions cannot be satisfied). For L3, only L3.1 requirements can be satisfied while L3.2 cannot be met due to the centralized control plane. For instance, most OpenStack services collaborate through a message system and through the manipulation of logical objects that are persisted in shared databases (DBs). While this enables services to easily collaborate, it imposes a requirement of permanent connectivity between the services located at the compute nodes and the control services like the databases and message buses located in the DC. In other words, while this scenario provides a "single pane of glass" for administrators and DevOps, it has the drawback of being a "single point of failure" preventing Dev-Ops to use edge resources in case of network split-brains.

Multiple Regions

In this scenario, each edge site corresponds to a region in the OpenStack terminology, which is a complete deployment of OpenStack with all control services and a "shared" Keystone. The main advantage of this deployment is related to the independency of each site in case of network disconnections. The downside relates to the fact that the current codebase does not provide any mechanism to allow the collaboration between several regions and thus L2 requirements cannot be met. 3

Effective Collaboration is Needed

Despite the fallibility of the network, and frequent isolation risks of an edge site from the rest of the infras-tructure, an edge infrastructure may be able to meet L3 requirements. This can be achieved by supposing a collaboration a la peer-to-peer, that is, an edge site always serves local resources and collaborates with other edge sites if need be. To develop such a resource management system, two fundamental design options exist: top-down or bottom-up. Both designs differently impact the way this required collaboration can be handled.

A top-down collaboration design implements the collaboration by federating VIMs' API. As a consequence the existing VIM codebases are used without introducing modifications/extensions. Examples of approaches following this design are: ONAP [START_REF]Open Network Automation Platform[END_REF], Kingbird [START_REF]OpenStack Kingbird[END_REF], Fog-Bow [START_REF] Brasileiro | Fogbow: A middleware for the federation of IaaS Clouds[END_REF] and p2p-OpenStack [START_REF] Soares | Re-designing Cloud platforms for massive scale using a P2P architecture[END_REF].

A bottom-up collaboration design lays on making VIMs mechanisms/services directly collaborative [START_REF] Lebre | Revising OpenStack to operate Fog/Edge Computing infrastructures[END_REF]. For example, having two OpenStack Nova services able to cooperate and communicate directly would be a realization of a bottom-up design. Such design implies either the modification/extension of existing VIMs or the creation of a completely new system.

Design Discussion

In this section, we discuss the pros and cons of the top-down and bottom-up designs targeted at the end of the previous section. More precisely, our discussion is driven by the following questions. (i) On the one hand, while the top-down design is the most common approach, can it fulfill all the expected requirements listed in our classification without requiring changes in the VIM codebase? (ii) On the other hand, while the bottom-up design seems to disrupt the design principles by requiring a la peer-to-peer strategies in VIM's internal mechanisms, should it be discarded? Top-Down Design The top-down approach consists in designing a set of overlay components that interact with existing VIM APIs to avoid modifying VIM codebases. Avoiding codebase modifications is of particular importance in fast-growing software stacks. For example, a new version of OpenStack is released every six months with a lot of changes in the codebase, whereas changes rarely impact the APIs. Therefore, a top-down design makes VIM development and the overlay system components independent. If designed to do so, the system can easily allow L4, i.e., the support of different versions and types of VIMs -as demonstrated by FogBow [START_REF] Brasileiro | Fogbow: A middleware for the federation of IaaS Clouds[END_REF].

Unfortunately, a top-down design cannot satisfy all L2 requirements without extending or revising the existing VIM codebase. For example, OpenStack Tricircle [START_REF]OpenStack Tricircle[END_REF], a top-down project to allow virtual networking across different sites, ended up "breaking" the core of OpenStack by introducing specific L2 mechanisms. Such intrusive modifications negate the aforementioned independence. Moreover, L2 features in general require reimplementing many low-level VIM functionalities at the overlay level. For instance, the OpenStack "boot a VM" process looks as follows from a bird's-eye view: (1) Get the URL of the image by looking up in the database, (2) Schedule and boot the VM. Thus, booting a VM on Site 1 using an image defined in Site 2 would require implementing a dedicated workflow at the overlay level in order to interact with both sites and copy the image from the image manager (i.e., Glance) of Site 2 to the one of Site 1 before booting the VM. This is valid for most L2 features such as a fine-grained authorization management with different rights in different edge sites (L2.1) or a cross-site scheduling functionality when sites are specified implicitly (L2.2). Such a mechanism will be similar to the placement workflow already available in Nova.

Bottom-Up Design In a bottom-up design, the system is not limited by what is available through VIM APIs. Specifically, there are two possibilities: i) rearchitecting the services/components of existing cloud platforms, in our case OpenStack; ii) through a clean-slate approach.

By rearchitecting OpenStack to allow native collaboration, several 'local' features can be supported across the entire edge infrastructure for free. For instance, the aforementioned OpenStack "boot a VM on Site 1 with image on Site 2" process would be feasible without modifying the VIM codebase if Site 1 can either directly contact the image manager of Site 2, or share the database backend with Site 2. However, since OpenStack has not been designed to be collaborative, most mechanisms must be revised to consider side-effects related to collaboration operations. For instance, a VM boot process initiated on Site 1 can be completed on Site 2. The question is then to define where the states related to this VM should be stored, keeping in mind the split-brain challenge. It is noteworthy that identifying all these aspects requires the understanding of existing code, which is a daunting task in respect to the OpenStack ecosystem size.

Therefore, the complexity of rearchitecting existing services/components might make a clean-slate approach as the most likely approach for a bottom-up design.

Finally, L4 requirements cannot be intrinsically satisfied by the bottom-up approach while L5 implies strong limitations regarding how collaborations should be implemented (for instance sharing internal states of different edge sites between operators looks unlikely).

Summary There are pros and cons for both approaches, and none individually seems to meet all requirements. From the technical perspective, the bottomup design seems to be the most appropriate to cope with L1, L2 and L3 requirements, while the top-down is the only one to satisfy L4 and L5. From the business perspective, a top-down approach is likely to fulfill the short and medium-term Time-To-Market (TTM) requirements while a bottom-up approach is required to fulfill all requirements. From the codebase perspective, top-down approach has limited impact on existing VIM codebase while the bottom-up, despite the impact on existing codebase, allows implementing functionalities efficiently, without code duplication. All in all, our community should investigate a long-term solution that builds on a bottom-up approach. This will require first diving into OpenStack and understanding its intricate internal mechanisms leveraging tools like EnOS [START_REF] Cherrueau | Toward a holistic framework for conducting scientific evaluations of OpenStack[END_REF] to mitigate the efforts, and second, to address scientific challenges such as the definition of a reference architecture for a resource management system for edge infrastructures, how to share internal system states in an edge context etc.

Conclusion

The emergence of Network Function Virtualization (NFV) technologies as well as Internet of Things (IoT) and Augmented/Virtual Reality (AR/VR) applications herald a new era of cloud computing where infrastructures need to leverage resources at the edge of the network in order to cope with latency requirements. Despite this growing need, there is no cloud management system that is designed with such an infrastructure in mind.

In this paper, we presented reflections to initiate discussions on this hot topic. We outlined a systematic grouping of the features that administrators/DevOps expect and the requirements an edge infrastructure puts on a cloud management system. We briefly studied the use of existing IaaS managers (i.e., OpenStack) to control an edge infrastructure highlighting the need for an effective collaboration between the edge sites. We then discussed pros and cons of two possible strategies to follow when designing a solution: (1) a top-down approach which designs overlay components that interact with underlying IaaS managers without modifying their codebase; (2) a bottom-up approach that extensively improves existing software to fulfill the requirements. We concluded that in the short-term, a top-down approach is required to have a timely working solution available. In the long-term, both approaches should be considered simultaneously: bottom-up to realize a native and efficient system for the edge; top-down to enable collaboration between different cloud stacks (e.g., OpenStack, Kubernetes).

Finally, it is noteworthy that our study considers one kind of edge infrastructure. Other variants can be envisioned where, for example, third-party hardware resources can dynamically join and leave the infrastructure. For such cases, this study needs to be extended to integrate additional features not discussed in this paper.

Figure 1 :

 1 Figure 1: Edge Computing Infrastructure [11]. The red dashed lines depict a split-brain situation that isolates Site 1 from other sites.

Edge Frontier Edge Frontier Extreme Edge Frontier

		Cloud Computing
		Cloud latency > 100ms
	Enterprise network
	Site 1	Inter sites latency [50-100ms]	Site 2
			Intra sites latency
			< 10ms
		Micro/Nano DC
	Site 3	Edge Computing	Site 4
		Sites
			Wired link
			Wireless link

Table 1 :

 1 Classification of the requirements to operate and use edge computing infrastructures in 5 levels.L3. Robustness w.r.t. network split-brains L3 includes L2 operations but with the possibility to face situations where the infrastructure is partitioned due to network disconnections. Figure1depicts such a case where Site 1 is isolated from the other sites. In this scenario, administrators/DevOps that can reach Site 1 (i.e., located in the same geographical area) should be able to perform L1 operations on Site 1 despite the split-brain. Such a requirement makes sense as L1 is limited to one site and does not impact other ones. For the remaining sites, L1 and L2 must be guaranteed.

	Levels	Administrators	DevOps	Both
	L1: Operate/use any site	Manage any site: install, upgrade site's	Provision compute, storage, network	Collect metrics and ensure security,
		services; manage users, flavors, quotas	resources on-demand on any site	integrity and resiliency for any site
	L2: Operate/use several sites	Manage multiple sites simultaneously	Cross-site collaborative resources	L1 but over a set of sites
	-L2.1 Explicit manner:	Manage a specific set of sites	Provision on a specific set of sites	Aggregated metrics from multiple sites
	-L2.2 Implicit manner:	Cross-site autonomous management	Cross-site autonomous provisioning	and collaborative security mechanisms
	L3: Robustness w.r.t. split brains			L1 for an isolated site; L1 and L2 for
	-L3.1 Application robustness:	--	Access reachable applications	isolated sets of sites
	-L3.2 Management service robustness: Manage reachable site(s)	Provision on reacheable site(s)	Support intermittent connectivity
	L4: Multiple Cloud environments			L3 with different IaaS environments
	-L4.1 Different IaaS versions:	Manage different IaaS versions	Provision on different IaaS versions	Discover sites' capabilities and
	-L4.2 Different IaaS technologies:	Manage different IaaS technos	Provision on different IaaS technos	compatibility
	L5: Multiple operators	--	Provision on one or many sites	L4 with multiple operators

Acknowledgement

A large part of the content presented in this document results from numerous discussions with the Open-Stack community, notably through FEMDC SiG and Edge Working Sessions. Further information available at: https://wiki.openstack.org/wiki/Fog _ Edge _ Massively _ Distributed _ Clouds.

Notes

1 https://collaborate.nist.gov/twiki-cloud-computing/ bin/view/CloudComputing/FederatedCloudPWGFC (March 2018). 2 Note that our study applies to any other resource manager built on the similar building blocks as OpenStack (e.g., Kubernetes). Such building blocks are conceptually identified in [START_REF] Moreno-Vozmediano | IaaS Cloud architecture: From virtualized datacenters to federated Cloud infrastructures[END_REF]. 3 Due to space limitation, we did not discuss a third scenario leveraging the OpenStack cells concept. However, we underline that such an approach has the drawback of the two discussed scenarios.