Keywords: Matrix-valued Hamiltonian, smooth crossings, codimension 1 crossings, wave-packet, Bargmann transform, Herman-Kluk propagators, thawed and frozen Gaussian approximations

Chapter 1. Introduction 1.1. First overview 1.2. The setting 1.3. Classical quantities 1.4. Thawed and frozen Gaussian approximations 1.5. Wave packets propagation at any order through generic smooth crossings 1.6. Detailed overview Part 1. Initial value representations Chapter 2. Frequency localized families 2.1. The Bargmann transform 2.2. Frequency localized families 2.3. Operators built on Bargmann transform Chapter 3. Convergence of the thawed and the frozen Gaussian approximations 3.1. Strategy of the proofs 3.2. Thawed Gaussian approximations with transfers terms 3.3. Frozen Gaussian approximations with transfers terms Part 2. Wave-packet propagation through smooth crossings Chapter 4. Symbolic calculus and diagonalization of Hamiltonians with smooth crossings 4.1. Formal asymptotic series 4.2. 'Rough' reduction 4.3. Superadiabatic projectors and diagonalization Chapter 5. Propagation of wave packets through smooth crossings 5.1. Propagation faraway from the crossing area 5.2. Propagation close to the crossing area 5.3. Propagation through the crossing set 5.4. Propagation of wave packets -Proof of Theorem 1.21 Appendix A. Matrix-valued Hamiltonians Appendix B. Elements of symbolic calculus : the Moyal product B.1. Formal expansion B.2. Symbols with derivative bounds v vi CONTENTS Appendix C. Elements of semi-classical calculus: perturbation of scalar systems C.1. Egorov Theorem C.2. Asymptotic behavior of the propagator C.3. Propagation of wave packets

Introduction

Since the early days of semi-classical analysis, operators that approximate the dynamics of a semi-classical propagator have been the object of major attention. The theory of Fourier integral operators answers to this question by proposing methods for constructing approximative propagators of a scalar semi-classical Schrödinger equation (see [START_REF] Zworski | Semiclassical analysis[END_REF]Chapter 12] or [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF]). Pioneering work about a semi-classical theory of FIOs is [START_REF] Chazarain | Spectre d'un hamiltonien quantique et mécanique classique[END_REF], extended in [START_REF] Helffer | Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques[END_REF] and [START_REF] Robert | Autour de l'approximation semi-classique volume 68 of Progress in Mathematics Birhauser[END_REF].

Few results exist for systems except for those that are called adiabatic, because the eigenvalues of the underlying Hamiltonian matrix are of constant multiplicity. The analysis of such systems can be reduced to those of scalar equations through a diagonalization process using the so-called super-adiabatic projectors. The super-adiabatic approach has been carried out by Martinez and Sordoni [START_REF] Martinez | Twisted pseudodifferential calculus and application to the quantum evolution of molecules[END_REF] as well as Spohn and Teufel [START_REF] Spohn | Adiabatic decoupling and time-dependent Born-Oppenheimer theory[END_REF], see also [START_REF] Emmrich | Geometry of the transport equation in multicomponent WKB approximation[END_REF][START_REF] Nenciu | On the adiabatic theorem of quantum mechanics[END_REF][START_REF] Nenciu | Linear adiabatic theory. Exponential estimates[END_REF][START_REF] Bily | Propagation d'états cohérents et applications[END_REF] for earlier results or [START_REF] Volker | Superadiabatic transition histories in quantum molecular dynamics[END_REF][START_REF] Panati | Space-adiabatic perturbation theory[END_REF] for more recent results in a similar direction. The present study gives the first complete construction of an integral representation of the propagator associated to a Hamiltonian generating non-adiabatic dynamics in a very general situation. It focuses on those Hamiltonian matrices that have smooth eigenprojectors, with smooth eigenvalues, though of non constant multiplicity. The framework applies to generic situations where two eigenvalues cross along a hypersurface on points where the Hamiltonian vector fields associated with these eigenvalues are transverse to the crossing hypersurface. This set-up has already been the one of the work of Hagedorn [START_REF] Hagedorn | Molecular Propagation through Electron Energy Level Crossings[END_REF]Section 5] and Jecko [START_REF] Jecko | Semiclassical resolvent estimates for Schrödinger matrix operators with eigenvalues crossings[END_REF]. The Fourier integral operators approximating the propagator associated with these non-adiabatic Hamiltonians are based on Gaussian wave-packets and the Bargmann transform, in the spirit of the Herman-Kluk propagator.

The Herman-Kluk propagator has been introduced in theoretical chemistry (see [START_REF] Heller | Time-dependent approach to semiclassical dynamics[END_REF][START_REF] Kay | Integral expressions for the semi-classical time-dependent propagator[END_REF][START_REF] Herman | A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations[END_REF][START_REF] Kay | The Herman-Kluk approximation: derivation and semiclassical corrections[END_REF]) for the analysis of molecular dynamics for scalar equations. The mathematical analysis has been performed later by Rousse and Swart [START_REF] Swart | A mathematical justification for the Herman-Kluk Propagator[END_REF] and Robert [START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF], independently. The action of the Herman-Kluk propagator consists in the continuous decomposition of the initial data into semiclassical Gaussian wave-packets and the implementation of the propagation of the wave-packets as studied in the 70s and 80s by Heller [START_REF] Heller | Time-dependent approach to semiclassical dynamics[END_REF], Combescure and Robert [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF], and Hagedorn [24]. It involves time-dependent quantities that are called classical quantities because they can be interpreted in terms of Newtonian mechanics. Such an approximative description of the propagator in terms of several Gaussian wave packets motivates numerical methods that naturally combine with probabilistic sampling techniques, see [START_REF] Kluk | Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator[END_REF] or more recently [START_REF] Lasser | Discretising the Herman-Kluk Propagator[END_REF][START_REF] Kröninger | Sampling strategies for the Herman-Kluk propagator of the wavefunction[END_REF].

We prove the convergence of two types of approximations, respectively called thawed and frozen Gaussian approximations, both built of continuous superpositions of Gaussian wave-packets, the frozen one in the spirit the original Herman-Kluk propagator. Their difference mainly consists in the way the width matrices resulting from the propagation of the individual semi-classical Gaussian wave packets are treated. The presence of crossings requires to add to the semi-classical Gaussian wave packet propagation some transitions between the crossing hypersurfaces. Therefore, these

INTRODUCTION

Fourier integral operators incorporate classical transport along the Hamiltonian trajectories associated with the eigenvalues of the Hamiltonian and a branching process along the crossing hypersurface. Some of these ideas have been introduced in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF][START_REF] Fermanian Kammerer | Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations[END_REF], in particular in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] where the propagation of wave-packets through smooth crossings has been studied. Here, we revisit and extend these results, by proving that a wave-packet propagated through a smooth generic crossing remains asymptotically a wave-packet to any order in the semi-classical parameter. We then prove uniform estimates for the associated semi-classical approximations of propagators when acting on families of initial data that are frequency localized in the sense that their L 2 -mass does not escape in phase space to ∞ when the semi-classical parameter goes to 0, neither in position, nor in momentum. This class of initial data is typically met for the numerical simulation of molecular quantum systems.

Previous results. The analysis of the propagation through smooth eigenvalue crossings has been pioneered by Hagedorn in [26, Chapter 5]. He considered Schrödinger operators with matrixvalued potentials and propagated initial data that are known as semi-classical wave packets or generalized coherent states [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]Chapter 4]. The core of the wave-packet had to be chosen such that it classically propagates to the crossing. In the same framework adjusted to the context of solid states physics, Watson and Weinstein [START_REF] Watson | Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing[END_REF] analyze the propagation of wave-packets through a smooth crossing of Bloch bands. The results developed here extend [START_REF] Hagedorn | Molecular Propagation through Electron Energy Level Crossings[END_REF]Chapter 5] and [START_REF] Watson | Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing[END_REF] in two ways. The single wave-packet is turned into an initial value representation with uniform control for frequency localized initial data. The Schrödinger and Bloch operators are generalized to Weyl quantized operators with smooth time-dependent symbol.

First overview

The remainder of the introduction specifies the mathematical setting (assumptions on the Hamiltonian operator and the initial data), discusses the classical quantities involved in the approximation, reviews the known results on the thawed and frozen initial value representations in the adiabatic setting, and then presents the main results of this paper: Theorem 1.18 on the thawed approximation with hopping trajectories, Theorem 1. [START_REF] Kammerer | An Egorov theorem for avoided crossings of eigenvalue surfaces[END_REF] and Theorem 1.20 on the frozen approximation with hopping trajectories, that are pointwise and averaged in time, respectively, and Theorem 1.21 on wave-packet propagation through smooth crossings to arbitrary order.

We prove Theorems 1.18, 1.19 and 1.20 in Chapters 2 and 3. These proofs rely on Theorem 1.21, that is proved in Chapters 4 and 5.

Chapter 2 recalls elementary facts about the Bargmann transform. Then, it introduces the new notion of frequency localization, which will be crucial for controlling the remainder estimates for both the frozen and the thawed initial value representations in Chapter 3.

The refined wave-packet analysis of Chapters 4 and 5 does not depend on the theory of initial value representations and can be read independently from Chapters 2 and 3. It propagates wavepackets through smooth crossings in two steps: using a rough diagonalisation of the Hamiltonian operator in the crossing region and super-adiabatic projectors for the outside. Both constructions rely on pseudo-differential calculus for matrix-valued symbols that is developed in Chapter 4 and complemented by additional technical points in the appendices.

Notations and conventions. All the functional sets that we shall consider in this article can have values in C (scalar-valued), C m (vector-valued) or in C m,m (matrix-valued). We denote by g, f = R d f (x)g(x)dx the inner product of L 2 (R d , C). If π is a projector, then π ⊥ denotes the projector π ⊥ = I -π. We set D x = 1 i ∂ x . In the context of Assumption 1.3, we shall say that a matrix A is diagonal if A = π 1 Aπ 1 + π 2 Aπ 2 and off-diagonal if A = π 1 Aπ 2 + π 2 Aπ 1 .

1.2. The setting 1.2.1. The Schrödinger equation. We consider the Schrödinger equation

(1.1) iε∂ t ψ ε (t) = H ε (t)ψ ε (t), ψ ε |t=t0 = ψ ε 0 . in L 2 (R d , C m ), m ≥ 2
, where H ε (t) is the semi-classical quantization of an Hermitian matrix symbol

H ε (t, z) ∈ C m,m .
Here, t ∈ R, z = (x, ξ) ∈ R d × R d and ε is the semi-classical parameter, ε 1. Moreover, for a ∈ C ∞ (R 2d ) being a smooth scalar-, vector-or matrix-valued function with adequate control on the growth of derivatives, the Weyl operator a = op w ε (a) is defined by

op w ε (a)f (x) := af (x) := (2πε) -d R 2d a x + y 2
, ξ e iξ•(x-y)/ε f (y) dy dξ for all f ∈ S(R d ).

In full generality, we could assume that the map (t, z) → H ε (t, z) is a semi-classical observable in the sense that the function H ε (t, z) is an asymptotic sum of the form j≥0 ε j H j (t, z). However, in this asymptotic sum, the important terms are the principal symbol H 0 (t, z) and the sub-principal one H 1 (t, z); the terms H j (t, z) for j ≥ 2 only affect the solution at order ε, which is the order of the approximation we are looking for. Therefore, we assume that the self-adjoint matrix H ε writes H ε (t, z) := H 0 (t, z) + εH 1 (t, z).

1.2.2. Assumptions on the Hamiltonian. We work on a time interval of the form I := [t 0 , t 0 + T ], t 0 ∈ R and T > 0 and consider subquadratic matrix-valued Hamiltonians. Definition 1.1 (Subquadratic ). The ε-dependent Hamiltonian

H ε = H 0 + εH 1 ∈ C ∞ (I × R 2d , C m,m )
is subquadratic on the time interval I if and only if one has the property:

(1.2) ∀j ∈ {0, 1}, ∀γ ∈ N d , ∃C j,γ > 0, sup (t,z)∈I×R 2d |∂ γ z H j (t, z)| ≤ C j,γ z (2-j-|γ|)+
Assuming that H ε is subquadratic on the time interval I ensures that the system (1.1) is well-posed in L 2 (R d , C m ) for t ∈ I, and, more generally (see [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]), in the functional spaces

Σ k ε (R d ) = f ∈ L 2 (R d ), ∀α, β ∈ N d , |α| + |β| ≤ k, x α (ε∂ x ) β f ∈ L 2 (R d ) , k ∈ N endowed with the norm f Σ k ε = sup |α|+|β|≤k x α (ε∂ x ) β f L 2 .

INTRODUCTION

We denote by U ε H (t, t 0 ) the unitary propagator defined by iε∂ t U ε H (t, t 0 ) = H ε (t)U ε (t, t 0 ), U ε (t 0 , t 0 ) = I C m . It is a bounded operator of the Σ k ε (R d ) spaces, uniformly in ε (see [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]): there exists C T > 0 such that sup t∈I U ε H (t, t 0 ) L(Σ k ε ) ≤ C T . We assume that the principal symbol H 0 (t, z) of H ε (t, z) has two distinct eigenvalues that present a smooth crossing in the sense of the definitions of [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]. Namely, we consider different properties of the crossings. Definition 1.2.

(1) (Smooth crossing). The matrix

H 0 ∈ C ∞ (I × R 2d , C m,m
) has a smooth crossing on the set Υ ⊆ I × R 2d if there exists h 1 , h 2 ∈ C ∞ (I × R 2d ) and two orthogonal projectors

π 1 , π 2 ∈ C ∞ (I × R 2d , C m,m ) such that H 0 = h 1 π 1 + h 2 π 2 and h 1 (t, z) = h 2 (t, z) ⇐⇒ (t, z) ∈ Υ.
(2) Set f (t, z) = 1 2 (h 1 (t, z) -h 2 (t, z)) and v(t, z) = 1 2 (h 1 (t, z) + h 2 (t, z)).

(a) (Non-degenerate crossing). The crossing is non-degenerate at (t , ζ ) ∈ Υ if

d t,z (H 0 -v I C m ) (t , ζ ) = 0
where d t,z is the one differential form in the variables (t, z). (b) (Generic crossing points). The crossing is generic at (t , ζ ) ∈ Υ if one has

(1.3) ∂ t f + {v, f }(t , ζ ) = 0.
Note that there then exists an open set Ω ⊂ I × R 2d containing (t , ζ ) such that the set Υ ∩ Ω is a manifold.

Above, we denote by {f, g} the poisson bracket of the functions f and g defined on R 2d

x,ξ :

{f, g} = ∇ ξ f • ∇ x g -∇ x f • ∇ ξ g.
With these definitions in hands, we introduce one of the main assumptions on the crossing points of the Hamiltonian H ε . Assumption 1.3 (Crossing set). The Hamiltonian H ε = H 0 + εH 1 has a smooth crossing set Υ and all the points of Υ are non degenerate and generic crossing points.

In order to consider the unitary propagators U t,t0 h1 and U t,t0 h2 and be endowed with convenient bounds on the growth of the projectors, we shall make additional assumptions on the growth of the eigenvalues and of their gap function. Our setting will be the following: Assumption 1.4 (Growth conditions for smooth crossings). Let H ε = H 0 +εH 1 ∈ C ∞ (R×R 2d ) be subquadratic on the time interval I and have a smooth crossing on the set Υ. We consider the two following assumptions :

(i) The growth of H 0 (t, x), h 1 (t, z) and h 2 (t, z) is driven by the function v(t, z), i.e. for j ∈ {1, 2}

(1.4) ∀γ ∈ N 2d , |γ| = 1, ∃C γ > 0, ∀(t, z)

∈ I × R 2d , |∂ γ z (H 0 -v I C m )(t, z)| + |f (t, z)| ≤ C γ . (ii)
The eigenvalues h 1 and h 2 are subquadratic, i.e. (1.5) ∀γ ∈ N 2d , |γ| ≥ 2, ∃C γ > 0, ∀(t, z) ∈ I × R 2d , |∂ γ z h j (t, z)| ≤ C γ .

(iii) The gap is controlled at infinity, i.e. there exist R > 0 and n 0 ∈ N such that

(1.6) ∀t ∈ I, ∀|z| > R, |f (t, z)| ≥ C z -n0 ,
and in the case n 0 = 0, the functions z → π 1 , π 2 are assumed to have bounded derivatives at infinity.

Remark 1.5.

(1) The fact that the eigenvalues h 1 (t, z) and h 2 (t, z) are of subquadratic growth guarantees the existence of the unitary propagators U ε hj (t, t 0 ) for j ∈ {1, 2} and of the classical quantities associated with the Hamiltonians h 1 and h 2 that we will introduce below.

(2) The growth conditions of Assumption 1.4 imply that the eigenprojectors π j (t), j = 1, 2, and their derivatives have at most polynomial growth. However, when n 0 = 0, they may actually grow. This is proved in Lemma A. [START_REF] Volker | Superadiabatic transition histories in quantum molecular dynamics[END_REF]. It is for this reason that we assume that the projectors have bounded derivatives when n 0 = 0 in Point (iii).

(3) If one has (1.4) and (1.6) with n 0 = 0, then (1.5) holds. However, the examples below contain interesting physical situations for which n 0 = 0.

Example 1.6.

(1) Examples of matrix-valued Hamiltonian are given in molecular dynamics (see [ Chapter 5] in [START_REF] Hagedorn | Molecular Propagation through Electron Energy Level Crossings[END_REF]) by Schrödinger operators with matrix-valued potential,

H S = - ε 2 2 ∆ x I C 2 + V (x), V ∈ C ∞ (R d , C 2×2 ).
When V presents a codimension 1 crossing (as defined in [START_REF] Hagedorn | Molecular Propagation through Electron Energy Level Crossings[END_REF], then the crossing points (x, ξ) are non degenerate and generic when ξ = 0. (2) Another class of examples appear in solid state physics in the context of Bloch band decompositions (see [START_REF] Watson | Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing[END_REF][START_REF] Chabu | Effective mass theorems with Bloch modes crossings[END_REF] for example) with Hamiltonians of the form

H A = A(-iε∇ x ) + W (x)I C 2 , A ∈ C ∞ (R d , C 2×2 ), W ∈ C ∞ (R d , C).
(3) Finally, in [START_REF] Fermanian Kammerer | Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations[END_REF], the authors have considered the operator

H k,θ = ε i d dx I C 2 + kx 0 e iθx e -iθx 0 , with d = 1, N = 2, θ ∈ R + , k ∈ R * .
1.2.3. Assumptions on the data. We consider vector-valued initial data

ψ ε 0 ∈ L 2 (R d , C m ) of the form ψ ε 0 = V φ ε 0
where z → V (z) is a smooth function, bounded together with its derivatives and φ ε 0 ∈ L 2 (R d , C) is frequency localized in the sense of the next definition. For stating it, we denote the Gaussian of expectation q, variance √ ε that oscillates along p according to (1.7)

g ε z (x) = (πε) -d/4 e -(x-q) 2 ε + i ε p•(x-q) , ∀x ∈ R d .
Definition 1.7 (Frequency localized functions). Let (φ ε ) ε>0 be a family of functions of L 2 (R d ). The family (φ ε ) ε>0 is frequency localized if the family is bounded in L 2 (R d ) and if there exist R 0 , C 0 , ε 0 > 0 and N 0 > d + 1 2 such that for all ε ∈ (0,

ε 0 ], (2πε) -d/2 | g ε z , φ ε | ≤ C 0 z -N0 for all z ∈ R d with |z| > R 0 .
One then says that (φ ε ) ε>0 is frequency localized.

INTRODUCTION

We will introduce a more precise definition in Chapter 2. For families that are frequency localized, the set of z ∈ R 2d in the identity (2.2) can be restricted to a compact set (see Lemma 2.6). The analysis of the examples given below is performed in Lemma 2.9.

Example 1.8.

(1) The Gaussian wave packets (g ε z0 ) ε>0 are frequency localized families. (2) Define (WP ε z0 (u)) ε>0 by

(1.8) WP ε z0 (u)(x) = ε -d/4 e i ε p0•(x-q0) u x -q 0 √ ε , x ∈ R d ,
for u ∈ S(R d ) and z 0 = (q 0 , p 0 ) ∈ R 2d . They are frequency localized families. (3) Lagrangian (or WKB) states

ϕ ε (x) = a(x)e i ε S(x) with a ∈ C ∞ 0 (R d , C) and S ∈ C ∞ (R d , R
), also are frequency localized families.

Our vector-valued initial data will have a scalar part consisting in a frequency localized family.

Assumption 1.9. The initial data ψ ε 0 in (1.1) satisfies (1.9)

ψ ε 0 (x) = V φ ε 0 (x), x ∈ R d where (i) The family (φ ε 0 ) ε>0 is frequency localized with constants R 0 , N 0 , C 0 , ε 0 in Definition 1.7. (ii) The function z → V (z) is a function of C ∞ (R 2d , C m )
, bounded together with its derivatives, and valued in the set of normalized vectors.

We point out that any vector-valued bounded family in L 2 (R d ) writes as a sum of data of the form V φ ε 0 (x) for (φ ε 0 ) ε>0 bounded. As a consequence, assuming the initial data ψ ε 0 satisfies 1.9 is not really restrictive. Of course, the vector valued function V can be turned into -V by changing φ ε 0 into -φ ε 0 .

Classical quantities

In this section, we introduce classical quantities associated with the Hamiltonian H ε . These quantities will be used to construct the approximations of the propagator U ε H (t, t 0 ) that are the subject of this text. They are called classical because they do not depend on the semi-classical parameter ε and are obtained by solving ε-independent equations that mainly are ODEs instead of PDEs. Thus, the numerical realization of the resulting propagator's approximations avoids the difficulties induced by the 1 ε -oscillations and is applicable in a high-dimensional setting, see [START_REF] Lasser | Computing quantum dynamics in the semiclassical regime[END_REF] for a recent review on this topic. Besides their definition, we shall also recall well-known results about their role in the description of Schrödinger propagators.

In this section, we assume that H ε = H 0 + εH 1 is subquadratic on the time interval I (as defined in Definition 1.1), with smooth eigenprojectors π 1 and π 2 , and eigenvalues h 1 and h 2 , the latter being subquadratic (as in (ii) of Assumption 1.4).

1.3.1. The flow map. Let ∈ {1, 2}, we associate with h :

I × R 2d → R, (t, z) → h (t, z)
the functions z (t) = (q (t), p (t)) which denote the classical Hamiltonian trajectory issued from a phase space point z 0 at time t 0 , that is defined by the ordinary differential equation

ż (t) = J∂ z h (t, z (t)), z (t 0 ) = z 0 1.3. CLASSICAL QUANTITIES with (1.10) J = 0 I R d -I R d 0 .
We note that J is the matrix associated with the symplectic form

σ(z, z ) = Jz, z = p • q -p • q, z = (q, p), z = (q , p ) ∈ R 2d .
The trajectory z (t) = z (t, t 0 , z 0 ) depends on the initial datum and defines the associated flow map Φ t,t0 h of the Hamiltonian function h via

z → Φ t,t0 h (z) := z (t, t 0 , z 0 ), z ∈ R 2d .
We will also use the trajectory's action integral

(1.11) S (t, t 0 , z 0 ) = t t0 (p (s) • q (s) -h (s, z (s))) ds,
and the Jacobian matrix of the flow map, also called stability matrix

(1.12) F (t, t 0 , z 0 ) = ∂ z Φ t,t0
h (z 0 ). Note that F (t, t 0 , z 0 ) is a symplectic 2d × 2d matrix, that satisfies the linearized flow equation

(1.13) ∂ t F (t, t 0 , z 0 ) = JHess z h (t, z (t)) F (t, t 0 , z 0 ), F (t 0 , t 0 , z 0 ) = I R 2d .
We denote its blocks by

(1.14) F (t, t 0 , z 0 ) = A (t, t 0 , z 0 ) B (t, t 0 , z 0 ) C (t, t 0 , z 0 ) D (t, t 0 , z 0 ) . 1.3.2.
The metaplectic transform and Gaussian states. It is standard to associate with the time-dependent symplectic map F (t, t 0 , •) a unitary evolution operator, the metaplectic transformation that acts on square integrable functions in L 2 (R d ) as a unitary transformation.

M[F (t, t 0 , z 0 )] : u 0 → u(t)
and associates with an initial datum u 0 the solution at time t of the Cauchy problem i∂ t u(t) = op w 1 Hess z h t, Φ t,t0 h (z 0 ) z • z u, u(t 0 ) = u 0 . This map is called the metaplectic transformation associated with the matrix F (t, t 0 , z 0 ) (see [START_REF] Maspero | On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms[END_REF]). It satisfies for all ε > 0 and for all symbol a compactly supported or polynomial

(1.15) M[F (t, t 0 , z 0 )] -1 op ε (a)M[F (t, t 0 , z 0 )] = op ε (F (t, t 0 , z 0 )z).
All these classical quantities are involved in the description of the propagation of Gaussian states by U ε h (t, t 0 ), that are a generalization of the Gaussian families (g ε z ) ε>0 that we have already seen. Gaussian states are wave packets WP ε z (g Γ ) with complex-valued Gaussian profiles g Γ , whose covariance matrix Γ is taken in the Siegel half-space S + (d) of d × d complex-valued symmetric matrices with positive imaginary part,

S + (d) = Γ ∈ C d×d , Γ = Γ τ , ImΓ > 0 . More precisely, g Γ depends on Γ ∈ S + (d) according to (1.16) g Γ (x) := c Γ e i 2 Γx•x , x ∈ R d , 1. INTRODUCTION where c Γ = π -d/4 det 1/4 (ImΓ) is a normalization constant in L 2 (R d ).
It is a non-zero complex number whose argument is determined by continuity according to the working environment. The propagation of Gaussian states by a metaplectic transform is well-known: for Γ 0 ∈ S + (d), we have

(1.17) M[F (t, t 0 , z 0 )]g Γ0 = g Γ (t,t0,z0) ,
where the width Γ (t, t 0 , z 0 ) ∈ S + (d) and the corresponding normalization c Γ (t,t0,z0) are determined by the initial width Γ 0 and the Jacobian F (t, t 0 , z 0 ) according to

Γ (t, t 0 , z 0 ) = (C (t, t 0 , z 0 ) + D (t, t 0 , z 0 )Γ 0 )(A (t, t 0 , z 0 ) + B (t, t 0 , z 0 )Γ 0 ) -1 (1.18) c Γ (t,t0,z0) = c Γ0 det -1/2 (A (t, t 0 , z 0 ) + B (t, t 0 , z 0 )Γ 0 ).
The branch of the square root in det -1/2 is determined by continuity in time. Besides, the action of U ε h (t, t 0 ) on Gaussian wave packets WP ε z (g Γ ) obey to

U ε h (t, t 0 )WP ε z0 (g Γ0 ) = e i ε S (t,t0,z0) WP ε Φ t,t 0 h (z0) (g Γ (t,t0,z) ) + O( √ ε)
in any space Σ k ε (R d ) (see [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]). 1.3.3. Parallel transport. For systems, the wave function is valued in L 2 (R d , C m ) and thus vector-valued. The propagation then involves transformation of the vector part of the eigenfunctions that is called parallel transport.

Denoting by π ⊥ the projector π ⊥ = I -π , we define self-adjoint matrices H adia ,1 by

π ⊥ H adia ,1 π ⊥ = 0, π H adia ,1 π = π H 1 + 1 2i {H 0 , π } π , (1.19) π ⊥ H adia ,1 π = π ⊥ (i∂ t π + i{h , π }) π . One then introduces the map R (t, t 0 , z) defined for ∈ {1, 2} by (1.20) i∂ t R (t, t 0 , z) = H adia ,1 t, Φ t,t0 h (z) R (t, t 0 , z), R (t 0 , t 0 , z) = I m . The map t → H adia ,1 t, Φ t,t0
h (z) is a locally Lipschitz map valued in the set of self adjoint matrices. Therefore, the existence of R (t, t 0 , z) comes from solving a linear time dependent ODE by the Cauchy Lipschitz Theorem.

Lemma 1.10. For all (t, z) ∈ I × R 2d and ∈ {1, 2}, the matrices R (t, t 0 , z) are unitary matrices. Besides, they satisfy

(1.21) R (t, t 0 , z)π (t 0 , z) = π t, Φ t,t0 (z) R (t, t 0 , z).
This Lemma is proved in Appendix A. The relation (1.21) implies that whenever a vector V 0 is in the eigenspace of H 0 (t 0 , z 0 ) for the eigenvalue h (t 0 , z 0 ), then the vector R (t, t 0 , z) V 0 is in the range of π (t, Φ t,t0 (z)). In other words, we have constructed a map that preserves the eigenspaces along the flow: R (t, t 0 , z) : Ran (π (t 0 , z)) → Ran π (t, Φ t,t0 (z)) . The matrices R (t, t 0 , z) are sometimes referred to as Larmor precession (see [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]).

The map V 0 → R (t, t 0 , z)π (t 0 , z) V 0 is a parallel transport in the Hermitian vector fiber bundle (t, z) → Ran(π (t, z)) over the phase space I × R 2d ⊂ R 1+2d , associated with the curve s → γ(s) = s, Φ s,t0 h (z 0 ) s∈I and the matrix H adia ,1 . Indeed, the covariant derivative along the curve (γ(s)) s∈I is given by

∇ γ(s) = ∂ t + Jdh • ∇ z
and the relation X s, Φ s,t0 h (z 0 ) = R (t, t 0 , z)π (t 0 , z) V 0 defines a smooth section along the path γ that satisfies ∇ γ(s) X(t, x) = -iH adia ,1 (t, x) X(t, x). The map R , ∈ {1, 2} plays a role on the quantum side in the adiabatic setting for the propagation of wave packets. The proof of the next statement can be found in [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]Chapter 14] and [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF], and Chapter 5 of this memoir.

Proposition 1.11 (Vector-valued wave packets). Let k ∈ N and assume that H ε = H 0 + εH 1 satisfies Assumptions 1.3 and 1.4. Let ∈ {1, 2} and (t 0 , z 0 ) ∈ I × R d . Then, for any ϕ 0 ∈ S(R d , C) and V 0 ∈ Ran π (t 0 , z 0 ), there exists a constant C > 0 such that

sup t∈J U ε H (t, t 0 ) V 0 WP ε z0 ϕ 0 -e i ε S (t,t0,z0) V (t, t 0 ) WP ε Φ t,t 0 h (z0) ϕ ε (t) Σ k ε ≤ C √ ε,
where the profile function ϕ ε (t) is given by

ϕ ε (t) = M[F (t, t 0 , z 0 )]ϕ 0 , and V (t, t 0 , z) = R t, t 0 , Φ t,t0 h (z 0 ) V 0 .

Thawed and frozen Gaussian approximations

Thawed and frozen Gaussian approximations have been introduced in the 80's in theoretical chemistry [START_REF] Herman | A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations[END_REF][START_REF] Kay | Integral expressions for the semi-classical time-dependent propagator[END_REF][START_REF] Kay | The Herman-Kluk approximation: derivation and semiclassical corrections[END_REF]. The frozen one has become popular as the so-called Herman-Kluk approximation. They rely on the fact that the family of wave packets (g ε z ) z∈R 2d forms a continuous frame and provides for all square integrable functions f ∈ L 2 (R d ) the reconstruction formula

f (x) = (2πε) -d z∈R 2d g ε z , f g ε z (x)dz.
The leading idea is then to write the unitary propagation of general, square integrable initial data

ψ ε 0 ∈ L 2 (R d ) as U ε H (t, t 0 )ψ ε 0 = (2πε) -d z∈R 2d g ε z , ψ ε 0 U ε H (t, t 0 )g ε z dz,
and to take advantage of the specific properties of the propagation of Gaussian states to obtain an integral representation that allows in particular for an efficient numerical realization of the propagator. Such a program has been completely accomplished in the scalar case. However, the mathematical proof of the convergence of this approximation is more recent [START_REF] Swart | A mathematical justification for the Herman-Kluk Propagator[END_REF][START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF] and can be easily extended to the adiabatic setting (see [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]). We recall in the first subsection these adiabatic results and then we explain how we extend this approach to systems presenting smooth crossings via a hopping process. Surface hopping has been popularized in theoretical chemistry by the algorithm of the fewest switches (see [START_REF] Tully | Trajectory surface hopping approach to nonadiabatic molecular collisions: the reaction of H + with D 2[END_REF]) and been combined with frozen Gaussian propagation in various instances, see for example [START_REF] Wu | A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator[END_REF][START_REF] Lu | Frozen Gaussian approximation with surface hopping for mixed quantumclassical dynamics: A mathematical justification of fewest switches surface hopping algorithms[END_REF]. Here, it is the first time that the combination is achieved in a fully rigorous manner.

1.4.1. The adiabatic situation. Whenever the eigenvalues are of constant multiplicity, the classical quantities that we have introduced above are enough to construct an approximation of the propagator. For ∈ {1, 2}, we define the first order thawed Gaussian approximation for the -th mode as the operator J t,t0

,th defined on functions of the form

ψ = V f , f ∈ L 2 (R d ), (1.22) J t,t0 ,th ( V f ) = (2πε) -d R 2d e i ε S (t,t0,z) g ε z , f V (t, t 0 , z)g Γ (t,t0,z),ε Φ t,t 0 (t,z) dz, 1. INTRODUCTION with (1.23) V (t, t 0 , z) = R (t, t 0 , z)π (t 0 , z) V (z).
This family of operators is bounded in

L L 2 (R d ), Σ k ε (R d ) (see Corollary 2.17). Notice that the operator f → J t,t0
,th ( V f ) has a Schwartz distribution kernel and defines a Fourier integral operator with an explicit complex phase. Theorem 1.12 (Thawed Gaussian approximation [START_REF] Kay | Integral expressions for the semi-classical time-dependent propagator[END_REF][START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF][START_REF] Swart | A mathematical justification for the Herman-Kluk Propagator[END_REF][START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]). Assume h is an eigenvalue of constant multiplicity of a matrix H ε = H 0 + εH 1 of subquadratic growth on the time interval I. Let t 0 , T ∈ R with [t 0 , T ] ⊂ I. Then, there exists

C T > 0 such that for all φ ε 0 ∈ L 2 (R d ), V ∈ C ∞ (R 2d , C m ) bounded with bounded derivatives, for all t ∈ [t 0 , t 0 + T ] U ε H (t, t 0 ) π (t 0 ) V φ ε 0 -J t,t0 ,th ( V φ ε 0 ) L 2 ≤ C T ε φ ε 0 L 2 .
Remark 1.13.

(1) Of course, there is no unicity of the writing ψ = V f . However, changing ( V , f ) into (k V , 1 k f ) for some constant k ∈ C does not affect the result. One can also think to modifying V by multiplying it by a non-vanishing function a ∈ C ∞ (R 2d ) such that a and 1 a have bounded derivatives. Then, it is enough to turn f into a -1 f (see Remark 2.23).

(2) The approach of thawed and frozen approximations that we develop in this text allows to extend the convergence to the spaces Σ ε k provided the initial data (φ ε 0 ) ε>0 is frequency localized and k satisfies

N 0 > k + d + 1 2 (N 0 being associated to (φ ε 0 ) ε0 by Definition 1.7).
As first proposed in [START_REF] Herman | A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations[END_REF], it is also possible to get rid of the time-dependent variance matrices Γ by introducing the Herman-Kluk prefactors for the -th modes, a , defined by (1.24) a (t, t 0 , z) = 2 -d/2 det 1/2 (A (t, t 0 , z) + D (t, t 0 , z) + i(C (t, t 0 , z) -B (t, t 0 , z)) .

One then defines the first order frozen Gaussian approximation for the -th mode as the operator J t,t0 ,fr defined by

(1.25) J t,t0 ,fr ( V f ) = (2πε) -d R 2d e i ε S (t,t0,z) g ε z , f a (t, t 0 , z) V (t, t 0 , z)g ε Φ t,t 0 1 (z) dz.
Here again, this family of operators is bounded in Corollary 2.17). The next result then is a consequence of Theorem 1.12 Theorem 1.14 (Frozen Gaussians approximation [START_REF] Kay | Integral expressions for the semi-classical time-dependent propagator[END_REF][START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF][START_REF] Swart | A mathematical justification for the Herman-Kluk Propagator[END_REF][START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]). Assume h is an eigenvalue of constant multiplicity of a matrix H = H 0 + εH 1 of subquadratic growth on the time interval I.

L L 2 (R d ), Σ k ε (R d ) (see
Let t 0 , T ∈ R with [t 0 , T ] ⊂ I. Then, there exists C T > 0 such that for all φ ε 0 ∈ L 2 (R d ), V ∈ C ∞ (R 2d , C m )
bounded with bounded derivatives, and for all t ∈ [t 0 , t 0 + T ],

U ε H (t, t 0 ) π (t 0 ) V φ ε 0 -J t,t0 ,fr ( V φ ε 0 ) L 2 ≤ C T ε φ ε 0 L 2 .
The terminology thawed/frozen for these Gaussian approximations was introduced by Heller [START_REF] Heller | Time-dependent approach to semiclassical dynamics[END_REF] to put emphasis on the fact that, on the first case, the covariance of the matrix was evolving "naturally" by following the classical motion wile, on the other one, the covariance is "frozen" (constant). The possibility of freezing the covariance matrix was realized by Herman and Kluk (see [START_REF] Herman | A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations[END_REF]) by computing the kernel of the time dependent propagator.

As for Theorem 1.12, one can extend the result to an approximation in Σ k ε provided the data (φ ε 0 ) ε0 is frequency localized family and k ∈ N is chosen so that N 0 > k + d + 1 2 (N 0 being associated to (φ ε 0 ) ε0 by Definition 1.7).

In the next sections, we present our results and an extension of these statements to systems with crossings. The method we develop also allow to prove the approximations of Theorems 1.14 and 1.12 in the spaces Σ k ε (R d ), with additional assumptions on the initial data. 1.4.2. Initial value representations for codimension 1 crossing at order √ ε. Our first result consists in an extension of the range of validity of Theorems 1.12 and 1.14 to Hamiltonians presenting smooth crossing and satisfying (1.3) at the prize of a loss in the accuracy of the approximation.

Theorem 1.15 (Leading order thawed/frozen Gaussian approximation). Let k ∈ N. Assume H ε = H 0 + εH 1 satisfies Assumptions 1.3 and 1.4 on the interval I. Then, there exist constants C T,k > 0, such that for all initial data ψ ε 0 = V φ ε 0 that satisfies Assumptions 1.9 with frequency localization index N 0 > k + d + 1 2 , there exists ε 0 > 0 such that for all t ∈ I and ε ∈ (0, ε 0 ], we have

U ε H (t, t 0 ) π (t 0 ) V φ ε 0 -J t,t0 ,th/fr φ ε 0 Σ k ε ≤ C T √ ε ( φ ε 0 L 2 + C 0 ) .
The remarks below also holds for Theorems 1.18, 1.19 and 1.20.

Remark 1.16.

(1) Of course the result also holds for initial data

ψ ε 0 (x) = V φ ε 0 (x) + r ε 0 (x), x ∈ R d when the family (r ε 0 ) ε>0 satisfies r ε 0 L 2 (R d ) = O(ε) in Σ k ε for the index k considered in the statement.
(2) The fact of being frequency localized with N 0 > k + d + 1 2 implies that (φ ε 0 ) ε>0 is bounded in Σ k ε (see Section 2.2.6). Thus, (U ε H (t, t 0 )ψ ε 0 ) ε>0 also is bounded in Σ k ε and this space is the natural space where studying the approximation.

(3) The control of the approximation in terms of the initial data by φ ε 0 L 2 + C 0 instead of φ ε 0 Σ ε k is due to the method of the proof, which has to account for the presence of the crossing. The constant C 0 (and the L 2 -norm) control the Σ ε k -norm. The loss of accuracy of the approximation, in √ ε instead of ε, is also due to the presence of the crossing set Υ. It induces transitions between the modes that are exactly of order √ ε and cannot be neglected. If the initial data is frequency localized in a domain such that all the classical trajectories issued from its microlocal support at time t 0 do not reach the crossing set before the time t 0 + T , then an estimate in ε will hold. However, if these trajectories pass through the crossing, some additional terms of order √ ε have to be added to obtain an approximation at order ε. Let us now introduce the hopping trajectories that we will consider and the branching of classical quantities that we will use above the crossing set.

1.4.3. Hopping trajectories and branching process. Assume H ε = H 0 + εH 1 satisfies Assumptions 1.3 and 1.4 on the interval I. For considering initial data ψ ε 0 = V φ ε 0 that are frequency localized in a compact set K ⊂ B(0, R 0 ), we are going to make assumptions on the set K.

We consider sets K that are connected compact subsets of R 2d and that do not intersect the crossing set Υ. If one additionally assumes that the trajectories Φ t,t0 (z) issued from points z ∈ K intersect Υ on generic crossing points, then, because of their transversality to Υ, a given 1. INTRODUCTION trajectory Φ t,t0 (z) issued from z ∈ K meets Υ only a finite number of times. We then denote by (t (t 0 , z), ζ (t 0 , z)) the first crossing point in Υ: (1.26) ζ (t 0 , z) = Φ t (t0,z),t0 (z).

For the -th mode and the compact K, we define t ,max (t 0 , K) = max{t (t 0 , z), z ∈ K} and t ,min (t 0 , K) = min{t (t 0 , z), z ∈ K}.

We shall assume that K is well-prepared in the sense that all trajectories issued from K for one of the mode have passed through Υ (if they do) before the ones for the other mode start to reach Υ.

Assumption 1.17 (Well-prepared frequency domain). The set K is a connected compact subset of R 2d that does not intersect the crossing set Υ. The trajectories Φ t,t0 (z) issued from points z ∈ K intersect Υ on generic crossing points and one has t 1,max (t 0 , K) < t 2,min (t 0 , K).

A space-time crossing point (t (t 0 , z), ζ (t 0 , z)) is characterized by three parameters

µ ∈ R, (α , β ) ∈ R 2d
given by

µ (t 0 , z) = 1 2 (∂ t f + {v, f }) t (t 0 , z), ζ (t 0 , z) , (1.27) α (t 0 , z), β (t 0 , z) = J∇ z f t (t 0 , z), ζ (t 0 , z) . (1.28)
The hopping process is affected with a transition coefficient τ 1,2 (t, t 0 , z) that restrict the space time variables (t, z) to trajectories that have met the crossing set Υ (1.29) τ 1,2 (t, t 0 , z) = I t≥t 1 (t0,z) 2iπ µ (t 0 , z) .

Note that when K satisfies Assumption 1.17, then if t < t 1,min (K) and z ∈ K, one has τ 1,2 (t, t 0 , z) = 0. Moreover, if t ∈ t 1,max (K), t 2,min (K) , z → τ 1,2 (t, t 0 , z) is smooth.

One then introduces hopping trajectories by setting

(1.30) Φ t,t0 1,2 (z) = Φ t,t 1 (t0,z) 2 ζ 1 (t 0 , z) , t > t 1 (t 0 , z).
This trajectory Φ t,t0 1,2 (z) t>t (t0,z) is the branch of a generalized trajectory that has hopped from the mode = 1 to the mode = 2 at the crossing point (t 1 (t 0 , z), ζ 1 (t 0 , z)). One could define similarly trajectories hopping from the mode = 2 to = 1 by exchanging the role of the indices 1 and 2.

Along these trajectories, one defines classical quantites as follows:

(a) The function S 1,2 (t, t 0 , z) is the action accumulated along the hopping trajectories, i.e. between times t 0 and t 1 (t 0 , z) on the mode = 1 and then on the mode = 2

(1.31) S 1,2 (t, t 0 , z) = S 1 (t 1 (t 0 , z), t 0 , z) + S 2 (t, t 1 (t 0 , z), ζ 1 (t 0 , z)), (b)
The matrix Γ 1,2 (t, t 0 , z) is generated according to (1.18) for the mode = 2 along the trajectory (Φ 1,2 (t, t 0 , z)) t>t 1 (t0,z) starting at time t 1 = t 1 (t 0 , z) from the matrix

(1.32) Γ (t 0 , z) = Γ 1 (t 1 , t 0 , z) - (β -Γ 1 (t 1 , t 0 , z)α ) ⊗ (β -Γ 1 (t 1 , t 0 , z)) 2µ -α • β + α • Γ 1 (t 1 , t 0 , z)α , (c) The vector V 1,2 (t, t 0 , z
) is obtained by propagating the vector V 1 t 1 , t 0 , z for the mode = 2 along the trajectory (Φ 1,2 (t, t 0 , z)) t>t (t0,z) starting at time t 1 from the vector

π 2 (t 1 , ζ 1 ) V 1 t 1 , t 0 , z with ζ 1 = ζ 1 (t 0 , z). One has (1.33) V 1,2 (t, t 0 , z) = R 2 (t, t 1 , ζ 1 )π 2 t 1 , ζ 1 V 1 t 1 , t 0 , z .
(d) The matrices F 1,2 (t, t 0 , z) are associated with the flow maps

(1.34) F 1,2 (t, t 0 , z) = ∂ z Φ t,t0 1,2 (z) = A 1,2 (t, t 0 , z) B 1,2 (t, t 0 , z) C 1,2 (t, t 0 , z) D 1,2 (t, t 0 , z) .
(e) The transitional Herman-Kluk prefactors depend on Γ 1,2 (t, t 0 , z) and τ 1,2 (t, t 0 , z) according to

a 1,2 = τ 1,2 det 1/2 (C 1,2 -iD 1,2 -i(A 1,2 -iB 1,2 )) det 1/2 (C 1,2 -iD 1,2 -Γ 1,2 (A 1,2 -iB 1,2 )) (1.35) = τ 1,2 det 1/2 (A 1,2 + D 1,2 + i(C 1,2 -B 1,2 )) det 1/2 (D 1,2 + iC 1,2 -iΓ 1,2 (A 1,2 -iB 1,2 ))
where we have omitted to mark the dependence on (t, t 0 , z) for readability.

With these quantities in hands, we can define the correction terms of order √ ε of the thawed & frozen approximations and state our main results.

1.4.4. Thawed Gaussian approximation at order ε. With the notations of the preceding section, one defines the thawed Gaussian correction term for the mode = 1 as

(1.36) J t,t0 1,2,th ( V f ) = (2πε) -d z∈K τ 1,2 (t, t 0 , z)e i ε S1,2(t,t0,z) g ε z , f V 1,2 (t, t 0 , z)g Γ1,2(t,t0,z),ε Φ t,t 0 1,2 (z)
dz

The formula (1.36) defines a family of operators that is bounded in L(L 2 (R d ), Σ k ε (R d )) (see Corollary 2.17). The restriction t > t 1 (t 0 , z) introduces a localization of the domain of integration on one side of the hypersurface {t = t 1 (t 0 , z)}.

The thawed Gaussian correction term for the mode = 2, denoted by J t,t0

2,1,th would be defined by exchanging the roles of the indices 1 and 2. These correction terms allow to ameliorate the accuracy of the thawed gaussian approximation and to obtain an approximation at order ε.

Theorem 1.18 (Thawed Gaussian approximation with hopping trajectories). Let k ∈ N. Assume H ε = H 0 + εH 1 satisfies Assumptions 1.3 and (1.4) on the interval I. Then, there exists constants C T,k > 0, such that for all initial data ψ ε 0 = V φ ε 0 that satisfies Assumptions 1.9 in a compact K satisfying Assumption 1.17, there exists ε 0 > 0 such that for all t ∈ I we have for

1. INTRODUCTION ε ∈ (0, ε 0 ], U ε H (t, t 0 )ψ ε 0 -J t,t0 1,th π 1 (t 0 ) V φ ε 0 -J t,t0 2,th π 2 (t 0 ) V φ ε 0 - √ εJ t,t0 1,2,th π 1 (t 0 ) V φ ε 0 Σ k ε ≤ C T,k ε (C 0 + φ ε 0 L 2 ) .
This result emphasizes that for systems with smooth crossings, a term of order √ ε is generated by the crossing.

1.4.5. Frozen Gaussian approximation at order ε. In order to freeze the covariance of the Gaussians Γ 1,2 (t, t 0 , z) that appear in the formula of the thawed Gaussian correction term (1.36), we use the correction prefactors a 1,2 and a 2,1 introduceded in (1.35) and define the frozen Gaussian correction term for the mode = 1 as

(1.37) J t,t0 1,2,fr ( V f ) = (2πε) -d z∈ K a 1,2 (t, t 0 , z)e i ε S1,2(t,t0,z) g ε z , f V 1,2 (t, t 0 , z)g ε Φ t,t 0 1,2 (z) dz.
Notice that the map f → J t,t0 1,2,fr ( V f ) defines a Fourier-integral operator with a complex phase associated with the canonical transformations Φ t,t0

1,2 that define the hopping flow. We first state a point-wise approximation.

Theorem 1.19 (Point-wise time frozen Gaussian approximation with hopping trajectories). Let k ∈ N. Assume H ε = H 0 + εH 1 is of subquadratic growth and satisfies Assumptions 1.3 and 1.4 on the interval I. Then, there exists constants C T,k > 0, such that for all initial data ψ ε 0 = V φ ε 0 that satisfies Assumptions 1.9 in a compact K satisfying Assumption 1.17, there exists ε 0 > 0 such that for all t ∈ I satisfying t < t 1,min (t 0 , K) or t 1,max (t 0 , K) ≤ t < t 2,min (t 0 , K),

we have for ε ∈ (0, ε 0 ],

U ε H (t, t 0 )ψ ε 0 -J t,t0 1,fr π 1 (t 0 ) V φ ε 0 -J t,t0 2,fr π 2 (t 0 ) V φ ε 0 - √ εJ t,t0 1,2,fr π 1 (t 0 ) V φ ε 0 Σ k ε ≤ C T,K ε ( φ ε 0 L 2 + C 0 ) .
The proof of Theorem 1.19 is based on integration by parts and requires differentiability. When t < t 1,min (K), then the transfer coefficient τ 1,2 (t, t 0 , z) = 0 for all z ∈ K. When t ∈ [t 1,max (K), t 2,min (K)), then z → τ 1,2 (t, t 0 , z) is smooth. It is for that reason, that we have to restrict the time validity of the approximation.

Averaging in time allows to overcome this difficulty and to obtain an approximation result that holds almost everywhere on intervals of time such that the classical trajectories issued from K and associated with the level = 2 have not yet reached Υ.

Theorem 1.20 (Time averaged frozen Gaussians approximation with hopping trajectories). Let k ∈ N. Assume H ε = H 0 + εH 1 is of subquadratic growth and satisfies Assumptions 1.3 and 1.4 on the interval I. Then, there exists constants C T,k > 0, such that for all initial data ψ ε 0 = V φ ε 0 1.5. WAVE PACKETS PROPAGATION AT ANY ORDER THROUGH GENERIC SMOOTH CROSSINGS that satisfies Assumptions 1.9 in a compact K satisfying Assumption 1.17, there exists

ε 0 > 0 such that for all χ ∈ C ∞ 0 t 0 , t 2,min (t 0 , K) , R χ(t) U ε H (t, t 0 )ψ ε 0 -J t,t0 1,fr π 1 (t 0 ) V φ ε 0 -J t,t0 2,fr π 2 (t 0 ) V φ ε 0 - √ εJ t,t0 1,2,fr π 1 (t 0 ) V φ ε 0 dt Σ k ε ≤ C T,k ε χ L ∞ ( φ ε 0 L 2 + C 0 ) .
To go beyond the time t 2,min (t 0 , K), one has to consider new transitions that would now go from the level = 2 to the level = 1, each time a trajectory for the level = 2 hits Υ. The process can be understood as a random walk: each time a trajectory passes through Υ a new trajectory arises on the other mode with a transition rate of order √ ε.

The averaging in time can be understood as the result of a non-pointwise observation, that takes place over some time interval, that might even be a short one.

For proving Theorem 1.19, 1.20 we use an accurate analysis for the propagation of individual wave-packets. We prove that a Gaussian wave-packet stays a generalized Gaussian wave packet modulo an error term of order ε µ , for any µ ∈ N, and in any space Σ k ε , k ∈ N, as well before, or after, hitting the crossing hypersurface Υ.

The proof consists first in proving the thawed approximations and in then deriving the frozen approximation from the thawed one. The arguments developed in Section 3.3 will show that one can "freeze" the Gaussian on any state g Γ0,ε z . The choice of some Γ 0 instead of iI will imply a slight modification of the definition of the Herman-Kluk prefactors a , and the transitional ones a , , , ∈ {1, 2}.

Wave packets propagation at any order through generic smooth crossings

Our results crucially rely on the analysis of the propagation of wave-packets (including the ones with Gaussian amplitude functions) through smooth crossings. We consider a Hamiltonian H ε = H 0 + εH 1 that satisfies Assumptions 1.4 on the time interval I and presents a smooth crossing on a set Υ. We fix a point z 0 = (q 0 , p 0 ) / ∈ Υ and times t 0 , T such that (1.38) t 0 < t 1 (t 0 , z 0 ) < t 0 + T < t 2 (t 0 , z 0 ).

We assume that the point Φ

ψ ε 0 = V 0 WP ε z0 (f 0 ) with f 0 ∈ S(R d ) and V 0 ∈ C m .
Let ψ ε (t) be the solution of (1.1) with initial data ψ ε 0 . There exist κ 0 ∈ N and three families of differential operators B ,j (t) j∈N , ∈ {1, 2} and B 1→2,j (t) j∈N such that setting for δ > 0 and

1. INTRODUCTION t ∈ I δ = [t 0 , t 1 (t 0 , z 0 ) -δ] ∪ [t 1 (t 0 , z 0 ) + δ, t 0 + T ] ψ ε,N 1 (t) = e i ε S1(t,t0,z0) WP ε z1(t) (f ε 1 (t)) , ψ ε,N 2 (t) = e i ε S2(t,t0,z0) WP ε z2(t) (f ε 2 (t)) + I t>t e i ε S1,2(t,t0,z0) WP ε Φ1,2(t,t0,z0) (f ε 1→2 (t)) , with f ε (t) = R (t, t 0 ) M[F (t, t 0 )] 0≤j≤N ε j/2 B ,j (t)f 0 , ∈ {1, 2}, f ε 1→2 (t) = R 2 (t, t 1 ) M[F 2 (t, t 1 )] 1≤j≤N ε j/2 B 1→2,j (t)f 0 ,
one has the following property: for all k, N, M ∈ N, there exists

C M,N,k > 0 such for all t ∈ I δ ψ ε (t) -ψ ε,N 1 (t) + ψ ε,N 2 (t) Σ k ε ≤ C M,N,k √ ε δ N +1 δ -κ0 + δ M .
Moreover, the operators B ,j (t) are differential operators of degree ≤ 3j with time dependent smooth vector-valued coefficients and satisfy for ∈ {1, 2},

B ,0 (t) = π (t 0 , z 0 ) V 0 and B ,j (t 0 ) = 0 ∀j ≥ 1, (1.40) B ,1 (t) = |α|=3 1 α! 1 i t t0 ∂ α z h (s, z (s)) op w 1 [(F (s, t 0 , z 0 )z) α ] ds (1.41) + 1 i t t0 ∇ z H adia, 1 (s, z s ) • op 1 (F (t 0 , s)z)ds π (t 0 , z 0 ) V 0 , B 1→2 (t) = W 1 (t 1 , ζ 1 ) * T 1→2 M[F 1 (t 1 , t 0 )]π 1 (t 0 , z 0 ) V 0 (1.42)
where the scalar transfer operator T 1→2 is defined by

(1.43) T 1→2 ϕ(y) = +∞ -∞ e i(µ -α •β /2)s 2 e isβ •y ϕ(y -sα )ds, ∀ϕ ∈ S(R d )
and the transfer matrix W 1 (t , ζ ) is given by

(1.44) W 1 = π 1 H 1 π 2 + iπ 1 ∂ t π 1 + 1 2 {h 1 + h 2 , π 1 } π 2 .
In other words, Theorem 1.21 says that if ψ ε 0 a polarized wave packet, then, for t ∈ I, t = t 1 (t 0 , z 0 ), the solution ψ ε (t) of (1.1) is asymptotic at any order to an asymptotic sum of wave packets. Indeed, if n ∈ N is fixed, choosing δ = ε α and M, N large enough will give an approximation in O(ε n ).

The polarization of the wave packets ψ ε,N (t) is first described by the vectors B ,j (t 0 ) that evolves through R (t, t 0 ) M[F (t, t 0 )]. Such evolution preserves the eigenmode. Secondly, in ψ ε,N 2 (t), one sees a √ ε contribution that comes from a transfer from the mode 1 to the mode 2. The change of polarization is performed by the matrix W * 1 which maps Ran(π 1 ) to Ran(π 2 ). Indeed, one has

W * 1 = π 2 H 1 π 1 -iπ 2 ∂ t π 1 + 1 2 {h 1 + h 2 , π 1 } π 1 = π 2 H 1 π 1 + iπ 2 ∂ t π 2 + 1 2 {h 1 + h 2 , π 2 } π 1 .
The latter equation shows that the result is symmetric with respect to the modes and one can exchange their roles.

Theorem 1.21 was proved in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] up to order o(ε). The notations are compatible. However, in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF], a coefficient γ appears in the definition of the transfer operator. It corresponds to a normalization process that we avoid here by using the projector π 2 (t 1 , ζ 1 ) instead of taking the scalar product with a normalized eigenvector.

For proving the initial value representations of U ε H of Theorems 1.15 to 1.20, we shall use two consequences of Theorem 1.21:

(i) the wave packet structure up to any order in ε of U ε H V 0 WP ε z0 (g iI ), (ii) the exact value of the action of B 1,0 , B 2,0 , and of B 1→2 (t) when f 0 is the Gaussian g iI . We recall that the action of the operators R (t, s) M[F (t, s)] on focalized Gaussians preserves the Gaussian structure and the focalization: in view of (1.17) and (1.23),

R (t, s) M[F (t, s, z)]π (s, z) V 0 g iI = V (t, s, z)g Γ (t,s,z)
where V ∈ Ran(π (t, Φ t,s (z))) and the matrix Γ (t, s, z) is given by (1.18) with Γ 0 = iI. Besides, regarding the transfer term, with the notations of Corollary 3.9 of [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] and those of (1.27), (1.28), (1.32) and (1.33)

B 1→2 (t 1 )g iI = 2iπ µ (t 0 , z 0 ) V 1,2 (t 1 , t 0 , z 0 )g Γ (t 1 ,t0,z0) .
These elements may enlighten the construction of the operator J ,fr/th and J t,t0 , ,fr/th for indices , ∈ {1, 2}, = .

Detailed overview

The main results of this paper are Theorems 1.18, 1.19, 1.20, and 1.21. We prove Theorems 1.18, 1.19 and 1.20 in Chapters 2 and 3. These proofs rely on Theorem 1.21 that is proved later. Chapter 2 starts with Section 2.1 that recalls elementary facts about the Bargmann transform. Then, in Section 2.2, we analyze the notion of frequency localization that we have first introduced in this text and that is crucial in the setting of frozen and thawed initial value representations. Indeed, these approximations rely on a class of operators that is studied in Section 2.3. Endowed with these results, we are able to prove the approximations of Theorems 1.18, 1.19, 1.20 in Chapter 3. We describe the general proof strategy in Section 3.1 and develop the proof of Theorem 1.18 in Section 3.2. We explain in Section 3.3 how to pass from a thawed to a frozen approximation, and thus obtain Theorems 1.19 and 1.20.

Chapters 4 and 5 are devoted to the proof of Theorem 1.21. These two chapters are independent of the preceding ones, and one can start reading them, skipping Chapters 2 and 3. In Chapter 4, we construct the different diagonalisations of the Hamiltonian H ε that we are going to use. In the crossing region, we use a rough diagonalisation (see Section 4.2), and outside this region we use super-adiabatic projectors as proposed in [START_REF] Bily | Propagation d'états cohérents et applications[END_REF][START_REF] Martinez | Twisted pseudodifferential calculus and application to the quantum evolution of molecules[END_REF][START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics Lecture Notes in Mathematics 1821[END_REF] (see Section 4.3). These constructions rely on a symbolic calculus that we present in Section 4.1. The analysis of the propagation of wave-packets is then performed in Chapter 5.

The Appendices are devoted to the proof of some technical points used in the proofs of Chapters 4 and 5.

Part 1

Initial value representations

Frequency localized families

In this chapter, we study frequency localized families as introduced in the Introduction. We shall use the more precise following definition.

Definition 2.1 (Frequency localized functions of order β). Let β ≥ 0. Let (φ ε ) ε>0 be a family of functions of L 2 (R d ). The family (φ ε ) ε>0 is frequency localized of order β if the family is bounded in L 2 (R d ) and if there exist R β , C β , ε β > 0 and N β > d + 1 2 such that for all ε ∈ (0, ε β ], (2πε) -d/2 | g ε z , φ ε | ≤ C β ε β z -N β for all z ∈ R d with |z| > R β . One then says that (φ ε ) ε>0 is frequency localized of order β on the ball B(0, R β ). Above, g, f = R d f (x)g(x)dx denotes the inner product of L 2 (R d ).
In this chapter, we first recall some facts about the Bargmann transform, then we study frequency localized families and, finally, the class of operators built by use of Bargmann transform and to which the thawed/frozen Gaussian approximations belong.

The Bargmann transform

The thawed/frozen approximations that we aim at studying are constructed thanks to the Bargmann transform. They belong to a class of operators obtained by integrating the Bargmann transform against adapted families.

Recall that the Bargman transform is the map

B B : L 2 (R d ) f → B[f ] ∈ L 2 (R 2d ), defined by (2.1) B[f ](z) = (2πε) -d 2 g ε z , f , z ∈ R 2d .
The Bargmann transform is an isometry and one has

R 2d |B[f ](z)| 2 dz = f 2 L 2 .
Indeed, the Gaussian frame identity writes

(2.2) f (x) = (2πε) -d R 2d g ε z , f g ε z (x)dz = (2πε) -d 2 R 2d B[f ]g ε z (x)dz,
where the function

g ε z is introduced in (1.7), g ε z = WP ε z (g iI ) with the notation (1.16). Equation (2.2) is equivalent to f (x) = (2πε) -d 2 R 2d B[f ](z)g ε z (x)dz, ∀f ∈ L 2 (R d ).
More generally, the Bargmann transform characterizes the Σ k ε spaces according to the next result that we prove in Section 2.2.6 below.

Lemma 2.2. Let k ∈ N, there exists a constant c k such that for all f ∈ S(R d ),

f Σ k ε ≤ c k z k B[f ] L 2 (R 2d
) . The condition of spectral localization introduced in Definition 2.1 expresses in terms of the Bargmann transform: The family (φ ε ) ε>0 is frequency localized at the scale

β ≥ 0 if there exists constants R β , C β , , ε β > 0, N β ∈ N, N β > 2d such that for |z| > R β and ε ∈ (0, ε β ], |B[φ ε ](z)| ≤ C β ε β z -N β .
In other words, the Bargmann transform of (φ ε ) ε>0 has polynomial decay at infinity and is controlled by

ε β outside a ball B(0, R β ).
The operators in which we are interested are built on the Bargmann transform. Consider a smooth family of the form

(z → θ ε z ) ∈ C ∞ (R 2d z , L 2 (R d )). We then denote by J [θ ε z ] the operator acting on φ ∈ L 2 (R d ) according to (2.3) J [θ ε z ](φ)(x) = (2πε) -d 2 R 2d B[φ](z)θ ε z (x)dz = (2πε) -d R 2d g ε z , φ θ ε z (x)dz, x ∈ R d .
The thawed/frozen operators of equations (1.22), (1.25), (1.36) and (1.37) are of that form. The Gaussian frame identity (2.2) also writes with these notations

J [g ε z ] = I L 2 (R d ) .
Note that the formal adjoint of

J [θ ε z ] is (2.4) J [θ ε z ] * : φ → (2πε) -d R 2d θ ε z , φ g ε z dz.
In the first Section 2.2, we study the properties of frequency localized families, which is the type of data we consider in our main results. Then, in Section 2.3, we analyze some properties of the operators of the form (2.3). Finally, we prove Theorems 1.15 and 1.18 in Section 3.2, and Theorems 1.19 and 1.20 in Section 3.3.

Along the next sections of this chapter, we shall use properties of wave packets that we sum-up here.

Lemma 2.3. if f, g ∈ S(R d ) and z, z ∈ R 2d , then (2.5) WP ε z (f ), WP ε z (g) = e i ε p •(q-q ) W [f, g] z -z √ ε where the function W [f, g] is the Schwartz function on R 2d defined by W [f, g](ζ) = R d f (x)g(x -q)e ip•x dx, ζ = (q, p).
Moreover, for all n ∈ N, there exists a constant C n > 0 such that

(2.6) ∀ζ ∈ R 2d , ζ n |W [f, g](ζ)| ≤ C n 0≤n ≤n f Σ n g Σ n-n .
Proof. The formula for WP ε z (f ), WP ε z (g) comes from a simple computation. Then, for α, γ ∈ N d and z = (q, p) ∈ R 2d , we observe

|q γ p α W [f, g](z)| = q γ R d D α x (f (x)g(x -q))e ix•p dx ≤ q |γ| R d D α x (f (x)g(x -q)) dx ≤ 2 |γ| 2 R d x |γ| x -q |γ| D α x (f (x)g(x -q)) dx
where we have used Peetre inequality

(2.7) ∀t ∈ R, ∀ ∈ Z, t t ≤ 2 2 t -t | | .
The conclusion then follows.

Frequency localized families

We investigate here the properties of families that are frequency localized in the sense of Definition 2.1 and we use the notation (2.1).

We point out that Definition 2.1 is enough to treat vector-valued families by saying that a vector-valued family is frequency localized at the scale β ≥ 0 if and only if all its coordinates are frequency localized at the scale β. For this reason, we focus below on scalar-valued frequency localized families.

First properties of frequency localized functions.

It is interesting to investigate the properties of this notion. The first properties are straightforward.

Proposition 2.4. The set of frequency localized function is a subspace of L 2 (R d ). Moreover, we have the following properties:

(1) If (φ ε 1 ) ε>0 and (φ ε 2 ) ε>0 are two frequency localized families at the scales β 1 and β 2 respectively, then for all a, b ∈ C, the family (aφ ε 1 + bφ ε 2 ) ε>0 is frequency localized at the scale min(β 1 , β 2 ).

(2) If (φ ε ) ε>0 is frequency localized at the scale β ≥ 0, then it is also frequency localized at the scale β for all β ∈ [0, β].

This notion is microlocal. Indeed, defining the ε-Fourier transform by

F ε f (ξ) = (2πε) -d 2 R d e i ε ξ•x φ ε (x)dx = (2πε) -d 2 f ξ ε , f ∈ S 2 (R d ).
Proposition 2.5. Let (φ ε ) ε>0 be a bounded family in L 2 (R d ). Then, (φ ε ) ε>0 is frequency localized family at the scale β ≥ 0 if and only if (F ε φ ε ) ε>0 is frequency localized at the scale β ≥ 0.

Proof. This comes from the observation that for all z ∈ R 2d ,

| g ε z , φ ε | = | g ε Jz , F ε φ ε |
where J is the matrix defined in (1.10). Thus it is equivalent to state the fact of being frequency localized for a family or for the family of its ε-Fourier transform.

Frequency localized families and Bargmann transform.

The Gaussian frame identity (2.2) allows to decompose a function of L 2 (R d ) into a (continuous) sum of Gaussians. After discretization of the integral, this sum may be turned into a finite one, which opens the may to approximation's strategies (see [START_REF] Lasser | Discretising the Herman-Kluk Propagator[END_REF] where this observation is used for numerical purposes). It is thus important to identify assumptions that allow to compactify the set of integration in z. The notion of frequency localized families plays this role according to the next result.

Lemma 2.6. Let (φ ε ) ε>0 be a frequency localized family at the scale β ≥ 0. Let R β , C β and N β be the constants associated to

(φ ε ) ε>0 according to Definition 2.1. Let k ∈ N with N β > d + k. Then, for all χ ∈ L ∞ (R) supported in [0, 2] and equal to 1 on [0, 1], there exists C > 0 such that for R > R β , φ ε -J g ε z χ |z| R (φ ε ) Σ k ε (R d ) ≤ C C β ε β |z|>R z -2(N β -k) dz 1/2 .
In the following, we will use the notation

(2.8) φ ε R,< := J g ε z χ |z| R (φ ε ) = B -1 I |z|<R B[φ ε ](z) .
Remark 2.7. Lemma 2.6 can be used in different manners.

(

) If β > 0, then J [g ε z χ(|z|/R)](φ ε ) approximates φ ε in L 2 (R d ) as ε goes to 0 in any space Σ k ε (R d ) with k ∈ N such that N β > k + d + 1 2 1
, and uniformly with respect to R > R β . (2) If β ≥ 0 (which includes β = 0), then the same approximation holds by letting R go to +∞, and it is uniform with respect to ε. In particular, when β = 0 we have lim sup

ε→0 φ ε -J g ε z χ |z| R (φ ε ) Σ k ε (R d ) ≤ CR -(N β -k-d-1 2 ) .
Proof. We set

r ε (x) = (2πε) -d |z|≥R g ε z , ϕ ε g ε z (x)dz and consider k ∈ N. For R > R β , α, γ ∈ N d with |α| + |γ| = k, we have x α (εD x ) γ r ε 2 L 2 (R d ) ≤ (2πε) -2d R d |z|>R |z |>R g ε z , ϕ ε g ε z , ϕ ε g α,γ ε,z (x) g α,γ ε,z (x) dx dz dz .
where (2.9)

g α,γ ε,z = x α (εD x ) γ g ε z = WP ε z (q + √ εy) α (p + √ εD y ) γ g iI ) , z = (q, p).
We will use that for all n ∈ N, there exists c n,k > 0 such that for all z ∈ R 2d

(2.10)

g α,γ ε,z Σ n ≤ c n,k z k . By (2.5), we obtain x α (εD x ) γ r ε 2 L 2 (R d ) ≤ C 2 β ε 2β (2πε) -d |z|>R |z |>R z -N β z -N β W [g α,γ ε,z , g α,γ ε,z ] z -z √ ε dz dz .
Besides, by (2.6), there exists a constant C n,k such that

W [g α,γ ε,z , g α,γ ε,z ](ζ) ≤ C n,k ζ -n z k z k .
We deduce the existence of c > 0 such that

x α (εD x ) γ r ε 2 L 2 (R d ) ≤ c C 2 β ε 2β ε -d |z|>R |z |>R z -N β +k z -N β +k z -z √ ε -n dzdz ≤ c C 2 β ε 2β |z|>R z -N β +k z + √ εζ -N β +k ζ -n dzdζ. Since -N β + k ≤ 0, Peetre inequality gives z + √ εζ -N β +k ≤ 2 N β -k 2 √ εζ N β -k z -N β +k ≤ 2 N β -k 2 ζ N β -k z -N β +k ,
by restricting ourselves to ε ≤ 1. Therefore, there exists a constant c > 0 such that

x α (εD x ) γ r ε 2 L 2 (R d ) ≤ c C 2 β ε 2β R 2d ζ -n+N β -k dζ |z|>R z -2(N β -k) dz
and we conclude the proof by choosing n = N β + k + 2d + 1.

Examples. A first fundamental example consists in bounded families in

L 2 (R d ) that are compactly supported. Lemma 2.8. Let (φ ε 0 ) ε>0 be a bounded family in L 2 (R d ) such that φ ε 0 = I |x|≤M φ ε 0 for some M > 0. Then, (φ ε 0 ) ε>0 is frequency localised at any scale β ≥ 0.
Proof. There exists a constant C > 0 such that for all z = (q, p) ∈ R 2d

|B[φ ε 0 ](z)| ≤ C g ε z I |x|≤M L 2 .
Besides, one can find a smooth real-valued function χ compactly supported in {|x| ≤ 2} such that

g ε z I |x|≤M 2 L 2 = (πε) -d 2 R 2d χ( x M )χ( y M )g ε z (x)g ε z (y)dxdy.
We set

L = (|p| 2 + |q -x| 2 ) -1 (-ip + q -x) • ∇ x and we observe that εLg ε z (x) = g ε z (x) for all x ∈ R d . Besides, if |z| > 8M , either |q| > 4M and if |x| ≤ 2M ≤ |q| 2 , then |x -q| > |q| 2 > 4M , or |q| ≤ 4M and |p| > 2M . In any case, (|p| 2 + |q -x| 2 ) -1 > 2|z| -1 and for all N ∈ N, there exists a constant c N,M such that (L * ) N χ( x M ) ≤ c N,M |z| -N ,
Using the vector field L, we perform N integration by parts, and we obtain the existence of a constant C > 0 such that

g ε z I |x|≤M 2 L 2 ≤ C ε N -d 2 |z| -N R 2d I |x|,|y|≤M dxdy, whence the boundedness of ε -β z N β g ε z I |x|≤M L 2 for all β ≥ 0 and N β ∈ N.
An important consequence of this result is related with the notiont notion of compacity and of ε-oscillation that are often considered in semi-classical analysis. We recall that the uniformly bounded family (φ ε ) ε>0 is said to be compact if lim sup

ε→0 |x|>R |φ ε (x)| 2 dx -→ R→+∞ 0.
It is said ε-oscillating when lim sup

ε→0 |ξ|> R ε | φ ε (ξ)| 2 dξ -→ R→+∞ 0,
or, equivalently, when its ε-Fourier transform (F ε φ ε ) ε>0 is compact. Therefore, a compact family or an ε-oscillatory family can be approached by frequency-localized families. Note however that The notion of compacity or ε-oscillation is weaker than being frequency localized. For example, the family

(2.11) u ε (x) = | ln ε| d 2 a(x| ln ε|), x ∈ R d ,
is a compact and ε-oscillating family which has no scale of frequency localization.

Let us now analyze the examples given in the Introduction.

Lemma 2.9.

(1) Let u ∈ S(R d ) and z 0 = (q 0 , p 0 ) ∈ R 2d . Then, the family (WP ε z0 (u)) ε>0 is frequency localized at the scale β for any

β ≥ 0. (2) Let a ∈ C ∞ 0 (R d ) and S ∈ C ∞ (R d ).
Then, the family (e i ε S(x) a) ε>0 is frequency localized at the scale β for any β ≥ 0.

Proof. 1-By (2.5), we have for z ∈ R 2d B[WP ε z0 (u)](z) = (2πε) -d 2 WP ε z (g iI ), WP ε z0 (u) = (2πε) -d 2 e i ε p0•(q-q0) W [g iI , u] z 0 -z √ ε .
Let N ∈ N, the estimate (2.6) implies the existence of a constant C N such that

|B[WP ε z0 (u)](z)| ≤ C N ε -d 2 z 0 -z √ ε -N . Choosing |z| > max(2|z 0 |, 1), we have 2|z 0 -z| ≥ 2(|z| -|z 0 |) ≥ |z| and we deduce z 0 -z √ ε -N = ε ε + |z -z 0 | 2 N 2 ≤ 4ε 4ε + |z| 2 N 2 ≤ (4ε) N 2 |z| -N ,
whence the existence of a constant c N > 0 such that for all z ∈ R 2d and N ∈ N,

|B[WP ε z0 (u)](z)| ≤ c N ε N -d-1 2 z -N .

2-One has

B[e i ε S(x) a](z) = (2π) -d/2 π -d/4 ε -d/4 e i ε S(q) R d a(q + √ εy) × Exp - i √ ε p • y + i ε S(q + y √ ε) e -|y| 2 2 dy.
This term has a very specific structure involving the symbol y → a(y), a rapidly decaying function

y → e -|y| 2 2
and an oscillating phase

y → Λ ε (y) := - 1 √ ε p • y + 1 ε S(q + y √ ε).
We are going to show that the terms defined for j ∈ {1, • • • , d} by

A ε j := q j R d a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy and B ε j := p j R d a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy,
have the same structure. Then, it will be enough to consider only one of these terms and to prove that they are controlled by a power of ε, this will implies the adequate control on |B[e i ε S(x) a](z)|. Let us first transform A ε j and B ε j . Indeed, we have

A ε j = R d (q j + √ εy j )a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy - √ ε R d a(q + √ εy) y j e -|y| 2 2
e iΛ ε (y) dy.

The first integral of the right hand side has the same structure with the symbol y → a(y) and the second one with the rapidly decaying function y → y j e -|y| 2 2 . Besides, observing

p j e iΛ ε (y) = -i √ ε∂ yj (e iΛ ε (y) ) -∂ yj S(q + √ εy)e iΛ ε (y) ,
we obtain with an integration by parts

B ε j = - R d ∂ yj S(q + √ εy)a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy + i √ ε R d ∂ yj a(q + √ εy)e -|y| 2 2
e iΛ ε (y) dy.

Here again the right hand side has the same structure with different symbols and rapidly decaying term.

We now focus in proving that one typical term (2.12)

L ε := R d a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy is of order ε N for all N ∈ N. The decay of y → e -|y| 2 2
allows to reduce the set of integration. Indeed, we have

|y|>ε -1 4 a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy ≤ e - √ ε 4 a L ∞ R d e -|y| 2 4 dy.
Therefore, there exists a constant c > 0 such that

|L ε | ≤ c |y|≤ε -1 4 a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy + e -1 4 √ ε
.

We now use the oscillations of the phase for treating the integral in |y| ≤ ε -1 4 . We observe that there exists

R 0 > 0 such that if |z| > R 0 , then z / ∈ {|p -∇S(q)| ≤ 1, dist(q, supp(a)) ≤ 1}.
We choose |z| > R 0 and we have the following alternative:

either dist(q, supp(a)) > 1, or (dist(q, supp(a)) ≤ 1 and |p -∇S(q)| > 1) .

If dist(q, supp(a)) > 1, there exists ε 0 > 0 such that if ε ∈ (0, ε 0 ] and |y| ≤ ε 1/4 , then q + √ εy / ∈ supp(a). The integral thus is zero and we are reduced to the case where dist(q, supp(a)) ≤ 1 and |p -∇S(q)| > 1. One can find ε 1 > 0 such that for ε ∈ (0, ε 1 ] and |y| ≤ ε 1 4 ,

∇S(q + y √ ε) -p > 1 2 .
We then consider the differential operator

L ε = √ ε ∇S(q + y √ ε) -p |∇S(q + y √ ε) -p| 2 • ∇ y
and we write

|y|≤ε -1 4 a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy = |y|≤ε -1 4 a(q + √ εy)e -|y| 2 2 (L ε ) N e iΛ ε (y) dy = |y|≤ε -1 4 (L ε ) N * a(q + √ εy)|∇S(q + y √ ε) -p| -2N e -|y| 2 2
e iΛ ε (y) dy.

There exists a constant C > 0 independent of z such that for all ε ∈ (0, ε 1 ] and |y| ≤ ε

1 4 , (L ε ) N * a(q + √ εy)|∇S(q + y √ ε) -p| -2N e -|y| 2 2 ≤ Cε N 2 e -|y| 2 2 .
We deduce

|y|≤ε -1 4 a(q + √ εy)e -|y| 2 2 e iΛ ε (y) dy ≤ Cε N 2
and (2.12) writes

|L ε | ≤ c ε N 2 + e -1 4 √ ε
for some constant c > 0. This terminates the proof.

2.2.4. Characterization of frequency localized families. The characterization of frequency localized families can be done by using other families of wave packets than Gaussian ones and the cores z can be distributed in different manners.

Proposition 2.10. The family (φ ε ) ε>0 is frequency localized at the scale β ≥ 0 if and only if for all

C 1 -diffeomorphism Φ satisfying ∃a, b > 0, ∀z ∈ R 2d , a|z| ≤ Φ(z) ≤ b|z|, for all θ ∈ S(R d ), there exists C β , N β , R β and ε β such that for all ε ∈ (0, ε β ] and for |z| > R β (2πε) -d 2 | WP ε Φ(z) (θ), φ ε | ≤ C β ε β z -N β max 1, 1 a N β θ Σ 2d+1+N β .
Moreover, for all family

(λ ε ) ε>0 bounded in L ∞ (R 2d ), ε ∈ (0, ε β ] and R > R β , J e i ε λ ε (z) WP ε Φ(z) (θ)I |z|>R (φ ε ) Σ k ε ≤ C C β ε β |z|>R z -N β dz .
Proof. We only have to prove that if (φ ε ) ε>0 is frequency localized at the scale β ≥ 0, then the property holds for some given profile θ and diffeomorhism Φ. Then, the equivalence will follow. We consider the constants C β , N β , R β and ε β given by Definition 2.1 and we take ε ∈ (0, ε β ]. We observe

(2πε) -d 2 WP ε Φ(z) (θ), φ ε = (2πε) -3d 2 R d WP ε Φ(z) (θ), g ε z g ε z , φ ε dz = I 1 + I 2 with I 1 = (2πε) -3d 2 |z |>R β WP ε Φ(z) (θ), g ε z g ε z , φ ε dz .
Let us study I 1 . Using (2.5), (2.6) and that (φ ε ) ε>0 is frequency localized at the scale β ≥ 0, we deduce the existence of c β , N β > 0 such that we have

|I 1 | ≤ c β ε β (2πε) -d |z |>R β W [θ, g iI ] z -Φ(z) √ ε z -N β dz ≤ c β θ Σ n ε β (2πε) -d R d z -Φ(z) √ ε -n z -N β dz ≤ c β θ Σ n ε β R d ζ -n Φ(z) + √ εζ -N β dζ,
where the constant c β may have changed between two successive lines. We observe that Peetre's inequality (2.7) yields

Φ(z) + √ εζ -N β ≤ 2 N β 2 Φ(z) -N β √ εζ N β ≤ 2 N β 2 Φ(z) -N β ζ N β , whence by choosing n > 2d + 1 + N β , |I 1 | ≤ c β θ Σ 2d+1+N β ε β Φ(z) -N β R d ζ -(2d+1) dζ,
for some new constant c β > 0. We conclude by observing that

Φ(z) -N β ≤ max 1, 1 a N β z -N β ,
whence, by modifying c β ,

|I 1 | ≤ c β θ Σ 2d+1+N β ε β max 1, 1 a N β z -N β .
We now study I 2 . Using (2.5), (2.6), we write for n ∈ N

|I 2 | ≤ φ ε L 2 (2πε) -3d 2 |z |≤R β z -Φ(z) √ ε -n dz . We observe that if |z| > 2aR β , then for |z | ≤ R β ≤ 1 2a |z|, we have |z -Φ(z)| ≥ |Φ(z)| -|z | ≥ 1 2a |z|. Therefore z -Φ(z) √ ε -n = ε ε + |z -Φ(z)| 2 n 2 ≤ (2a) n ε n 2 |z| -n .
Using that (φ ε ) ε>0 is a bounded family in L 2 , we obtain that there exists a constant c such that for |z| > 2aR β and any n ∈ N,

|I 2 | ≤ c ε n-3d 2 |z| -n .
The proof of the last property follows the line of the proof of Lemma 2.6 combined with adapted change of variables. This terminates the proof.

2.2.5. Frequency localized families and semi-classical pseudodifferential calculus. With these elements in hands, we can prove some properties that frequency localized families enjoy with respect to pseudodifferential calculus. Proposition 2.11. Let (φ ε ) ε>0 be a frequency localized family at the scale β ≥ 0.

(1) For all semi-classical symbol a ∈ C ∞ c (R 2d ), the family a φ ε ε>0 is frequency localized at the scale β ≥ 0.

(2) For all subquadratic Hamiltonian h ∈ C ∞ (R × R 2d ), for all t, t 0 ∈ R, the vector-valued family U ε h (t, t 0 )φ ε ε>0 is frequency localized at the scale β ≥ 0.

Proof. (1) We can assume without loss of generality that a is real-valued. We write

B[ aφ ε ] = (2πε) -d/2 ag ε z , φ ε . Since g ε
z is a wave packet, we have

ag ε z = a WP ε z (g iI ) = WP ε z (g ε a ), g ε a = a(z + √ ε•)g iI .
The function g ε a is of Schwartz class on R d and its Schwartz semi-norms are uniformly bounded in ε because a is compactly supported. We deduce from Proposition 2.10,

|B[ aφ ε ]| ≤ C β ε β z -N β g ε a 2d+1+N β , which concludes the proof. (2) We write B [U ε h (t, t 0 )φ ε ] = (2πε) -d/2 U ε h (t, t 0 )g ε z , φ ε . Since g ε z is a wave packet, we have (2.13) U ε h (t, t 0 )g ε z = e i ε S(-t,z) WP ε Φ -t,0 h (z) (g Γ(-t,z) + √ ε r ε z (t))
with the notations of the introduction. Besides, for all n ∈ N, there exists a constant C = C(n, t) such that r ε z (t) Σ n ≤ C(n, t). We deduce from Proposition 2.10,

|B [U ε h (t, t 0 )φ ε ]| ≤ C β ε β z -N β g Γ(t,z) + √ εr ε z (t) 2d+1+N β , which concludes the proof.
Remark 2.12.

(1) The proof of Proposition 2.11 (1) extends to smooth functions a with polynomial growth

∃N 0 ∈ N, ∀γ ∈ N d , ∀z ∈ R 2d , |∂ γ a(z)| ≤ z N0-|γ|
provided the integer N β associated with the frequency localisation at the scale β ≥ 0 of the family (φ ε 0 ) ε>0 verifies N β > 2d + 1 + N 0 .

(2) The proof of Proposition 2.11 (2) also extends to adiabatic smooth matrix-valued Hamiltonian H that are subquadratic according to Definition 1.1. However, it is not clear whether the same result holds for Hamiltonians with crossings, either they are smooth as in this article or conical as in the Appendix of [START_REF] Fermanian Kammerer | Propagation of Coherent States through Conical Intersections[END_REF]. Indeed, even though one knows that

U ε H (t, t 0 )(g ε z V
) is asymptotic to a wave packet, it is not clear that the remainder of the approximation has a wave-packet structure as in (2.13).

2.2.6. Frequencies localized families and Σ k ε -regularity. The size of N β in Definition 2.1 gives an information about the regularity of the family. Lemma 2.13. Let (φ ε ) ε>0 a frequency localized family at the scale β ≥ 0, let C β , N β , ε β are the constants associated by Definition 2.1. Assume such that k ∈ N is such that

N β > d + k + 1 2 , then (φ ε ) 0<ε<ε β is uniformly bounded in Σ k ε and there exists c > 0 independent of ε φ ε Σ k ε ≤ c(C β + φ ε 0 L 2 )
. This Lemma is a simple consequence of Lemma 2.2 that we are going to prove now.

Proof of Lemma 2.2. Let k ∈ N and α, γ ∈ N d such that |α| + |γ| = k. we consider the operator T α,γ = B • (x α (εD x ) γ ) • B -1 • z -k .
The kernel of this operator is the function

R 4d (X, Y ) → k ε (X, Y ) = (2πε) -d g ε X , x α (εD x ) γ g ε Y Y -k . Therefore, by (2.5), there exists a constant c k such that Y ∈R 2d sup X∈R 2d |k ε (X, Y )|dY = (2πε) -d Y ∈R 2d sup X∈R 2d W [g iI , g α,γ ε,Y ] X -Y √ ε Y -k dY.
We deduce from (2.6) and (2.5) the existence of a constant c k > 0 such that

Y ∈R 2d sup X∈R 2d |k ε (X, Y )|dY ≤ c k .
Similarly, we have

X∈R 2d sup Y ∈R 2d |k ε (X, Y )|dX = (2πε) -d X∈R 2d sup Y ∈R 2d W [g iI , g α,γ ε,Y ] X -Y √ ε Y -k dX ≤ c k .
Therefore, the Schur test yields the boundedness of T α,γ . One then deduces that for f ∈ S(R d ), one has

x α (εD x ) γ f L 2 (R d ) = B[x α (εD x ) γ f L 2 (R 2d ) = T α,γ z k B[f ] L 2 (R 2d ) ≤ c k z k B[f ] L 2 (R 2d ) ,
which concludes the proof.

Let us now prove Lemma 2.13

Proof of Lemma 2.13.

Since N β > d + k, we have for |z| > R β , z k |B[φ ε ](z)| ≤ ε β C β z -N β +k ∈ L 2 (R 2d ). Moreover, z → z 2k |B[φ ε ](z)| 2
is locally integrable and we can write

z k B[φ ε ](z) 2 L 2 (R 2d ) ≤ R β 2k |z|≤R β |B[φ ε ](z)| 2 dz + ε 2β C 2 β R 2d z -2(N β -k) dz ≤ R β 2k B[φ ε ](z) 2 L 2 (R 2d + ε 2β C 2 β R 2d z -2(N β -k) dz ≤ R β 2k φ ε 2 L 2 (R d ) + ε 2β C 2 β R 2d z -2(N β -k) dz,
whence the conclusion since the right hand side is bounded for 2(N β -k) -2d > 1.

Operators built on Bargmann transform

We investigate here the properties of the operators defined in (2.3). We shall investigate two cases :

(a) The case where the family (θ ε z ) ε>0 is only uniformly bounded in L 2 (R d ), which is a light assumption, but with uniform bounds in z on adequate semi-norms or norms. (b) The case where the family (θ ε z ) ε>0 is a wave packet (up to a phase), which is a stronger assumption on the family. The thawed/frozen approximation operators belong to the type (b). We will consider operators of type (a) in the proofs of Theorems 1.15 and 1.18, when taking for the family (θ ε z ) ε>0 a term of rest appearing in the expansion of the action of the propagator on a Gaussian wave packet. The Theorems 1.19 and 1.20 are consequences of Theorem 1.18.

In the Subsection 2.3.1, we analyze the action of these operators on Σ k ε spaces. In Subsection 2.3.2, we prove special properties of the operators corresponding to families of the type (b) involving classical quantities linked with the propagation of Gaussian wave packets by Schrödinger evolution.

Action in Σ k

ε of operators built on Bargmann transform. This section is devoted to the proof of the following result.

Theorem 2.14. Let ε 0 > 0.

(1) Let R > 0. There exists c 0 > 0 such that for all measurable z-dependent family

(θ ε z ) ε>0 , for all k ∈ N, ε ∈ (0, ε 0 ], for all φ ∈ L 2 (R d ) J [θ ε z I |z|<R ](φ) Σ k ε ≤ (2πε) -d c 0 φ L 2 R 2d sup |z|≤R θ ε z Σ k ε .
(

) Assume θ ε z = λ ε (z)WP ε Φ(z) (θ) with θ ∈ S(R d ), (λ ε ) ε>0 a bounded family in L ∞ (R 2d , C) and Φ a smooth diffeomorphism of R 2d such that ∃c > 0, ∃ ∈ N, ∀z ∈ R 2d , |J Φ (z)| + |J Φ (z) -1 | ≤ c z . 2
Then, there exists

c 0 > 0 such that for all φ ∈ L 2 (R d ), k ∈ N, ε ∈ (0, ε 0 ], J [θ ε z ](φ) Σ k ε ≤ c 0 λ ε L ∞ φ L 2 θ Σ k+ +2d+1 .
The properties of the operators J [θ ε z ] extend to its adjoint (see (2.4)).

Corollary 2.15. Under the assumptions of Theorem 2.14, the family of operators J [θ ε z ] * (see (2.4)) satisfies the same kind of estimates than the family J [θ ε z ]. A straightforward consequence of Theorem 2.14 and of Lemme 2.6 is given in the next statement.

Corollary 2.16. Assume (θ ε z ) ε>0 satisfies the assumptions of Theorem 2.14 [START_REF] Arnold | Ordinary differential equations[END_REF]. Let (Φ ε ) ε>0 be a frequency localized family at the scale β ≥ 0 and C β > 0, N β ∈ N be the constants associated with Definition 2.1. Then, for all k > 0 such that N β > k + d, there exists a constant c k such that for all R > 0,

J [θ ε z ](φ ε -φ ε R,< ) Σ ε k ≤ c k C β ε β R -(N β -k-d-1 2 )
. where the family (φ ε R,< ) ε>0 is introduced in (2.8).

Proof of Theorem 2.14.

(1) The proof is similar to the first part of the proof of [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF]. By Cauchy-Schwartz inequality, for x ∈ R d , we have

J [θ ε z I |z|≤R ](φ ε ) 2 L 2 ≤ (2πε) -2d φ ε 2 L 2 |z|,|z |≤R x∈R d θ ε z (x)θ ε z (x)dx dz dz ≤ (2πε) -2d φ ε 2 L 2 |z|,|z |≤R θ ε z L 2 θ ε z L 2 dz dz ≤ c 1 R 4d (2πε) -2d φ ε 2 L 2 sup |z|≤2R θ ε z 2 L 2
where c 1 > 0 is a universal constant.

(2) Let us first prove the L 2 -estimate (k = 0). Let (x, y) → k ε (x, y) be the integral kernel of the operator J [θ ε z ]. Since the Bargmann transform is an isometry, it is equivalent to consider the operator

B • J [θ ε z ] • B -1 , the kernel of which is the function (R 2d ) 2 (X, Y ) → k ε B (X, Y ) defined by k ε B (X, Y ) = (2πε) -d R 2d g ε X (x)g ε Y (y)k ε (x, y)dxdy = (2πε) -2d z∈R 2d g ε z , g ε Y g ε X , θ ε z dz. Therefore, by (2.5), k ε B (X, Y ) satisfies |k ε B (X, Y )| ≤ (2πε) -2d z∈R 2d λ ε (z)W [g iI , g iI ] Y -z √ ε W [g iI , θ] Φ(z) -X √ ε dz.
We deduce

R 2d |k ε B (X, Y )|dX ≤ M λ ε L ∞ R 2d |W [g iI , g iI ](z)|dz R 2d |W [g iI , θ](X)|dX , R 2d |k ε B (X, Y )|dY ≤ M λ ε L ∞ R 2d |W [g iI , g iI ](Y )|dY R 2d |W [g iI , θ](z)J -1 Φ (z)|dz ,
with M = sup ε∈(0,1] λ ε L ∞ , and, by equations (2.6) and (2.10), we deduce the existence of C > 0 such that

R 2d |k ε B (X, Y )|dX + R 2d |k ε B (X, Y )|dY ≤ CM θ Σ 2d+ +1 .
We then conclude by Schur Lemma and obtain

B • J [θ ε z ] • B -1 L(L 2 (R 2d )) ≤ CM λ ε L ∞ θ Σ 2d+ +1
, and so it is for J [θ ε z ]. For concluding the proof when k = 0, we again use that for α, γ ∈ N d and φ ∈ S(R d ),

x α (ε∂ x ) γ J [θ ε z ] = J [x α (ε∂ x ) γ θ ε z ]
, and the additional observation

x α (ε∂ x ) γ WP ε z (θ) = WP ε (q + √ εx) α (p + √ εD x ) γ θ .
We then conclude by observing that, as in the estimate (2.10), we have for all n ∈ N,

(q + √ εx) α (p + √ εD x ) γ θ Σ n ≤ z k θ Σ n+k .
This finishes the proof.

Theorem 2.14 has consequences for the thawed/frozen approximation operators introduced in Chapter 1.

Corollary 2.17. Assume the Hamiltonian H ε = H 0 + εH 1 satisfies Assumptions 1.3 and 1.4. Let k ∈ N and t ∈ I.

(1) The families of operators J t,t0 ,th/fr ε>0 defined in (1.22) and (1.25)

are bounded families in L(L 2 (R d , C m ), Σ k ε (R d , C m )).
(2) Assume moreover that the compact K satisfies Assumptions 1.17. Then, the family of operators J t,t0 1,2,th/fr ε>0 defined in (1.36) and (1.37) are bounded families in the space

L(L 2 (R d , C m ), Σ k ε (R d , C m )).
Remark 2.18. If one assumes that (t, z) → ∂ t f + {v, f } is bounded from below and ∂ t f is bounded, then one can replace the compact K by R 2d in the definition of J t,t0

1,2,th and one obtains a bounded family in

L(L 2 (R d , C m ), Σ k ε (R d , C m )).
Proof. Let ∈ {1, 2}. Let us first discuss J t,t0 ,th . We write J t,t0 ,th = J [θ ε z ] with

θ ε z = λ ε (z)WP ε Φ t,t 0 (z) (g Γ (t, t 0 , z)) and λ ε (z) = e i ε S (t,t0,z) V (t, t 0 , z).
We observe that for all t ∈ I and z ∈ R 2d ,

V (t, t 0 , z) C m = V (t 0 , t 0 , z) C m = π (t 0 ) V C m ≤ V C m,m
which is independent of z. Therefore, the family (λ ε ) ε>0 is bounded in L ∞ (R 2d ). Besides, by Proposition A.4, the flow map (t, z) → Φ t,t0 (z) satisfies the assumptions of (2) of Theorem 2.14. Similarly, the map (t, z) → Γ (t, t 0 , z) is bounded on I × R 2d . Therefore, for any N ∈ N, there exists c N,t0,T > 0 such that

∀t ∈ I, x α ∂ β x g Γ (t, t 0 , •) Σ N ≤ c N,t0,T .
We then conclude by (2) of Theorem 2.14. The proof for J t,t0

,fr follows exactly the same lines. The proof for J t,t0 1,2,th/fr requires additional observations. We need to consider the transition coefficient map (t, z) → τ 1,2 (t, t 0 , z) (see (1.29)) and the matrix-valued maps z → Γ (t 0 , z) (see (1.32)), which requires the analysis of the function parametrizing the crossing (see (1.27) and (1.28)), (2.14) z → α (t 0 , z), β (t 0 , z), µ (t 0 , z) .

By the condition (1.4) of Assumption 1.3, with n 0 = 0, the derivatives of (t, z) → f (t, z) are uniformly bounded in z. Moreover, if one takes z in a compact K that satisfies Assumptions 1.17, one has the additional properties that ∂ t f and µ are bounded below. As a consequence z → α (t 0 , z), z → β (t 0 , z) and z → µ (t 0 , z) are bounded functions on R 2d for all t ∈ I, the map defined in (2.14) is smooth. One then argues as before by including the coefficient τ 1,2 (t, t 0 , z) in the definition of λ ε and the result follows from Theorem 2.14 (2).

2.3.2. Some properties of operators built on Bargmann transform via families with wave packet structure. In this section we analyze the properties of the operators J

[θ ε z ] when (θ ε z ) ε>0 is of the form (2.15) θ ε z = e i ε S(z) u(z)WP ε Φ(z) (θ(z, •)), where θ ∈ C ∞ (R 2d , S(R d )), S ∈ C ∞ (R 2d , R), u ∈ C ∞ (R 2d , C
) and Φ a smooth diffeomorphism sarisfying the assumptions of Theorem 2.14. We are interested in the case where S and Φ are linked in the same manner as when they are the flow map and the action associated with classical trajectories. Therefore, we consider the following set of Assumptions.

Assumption 2.19. Let S ∈ C ∞ (R 2d z , R), u ∈ C ∞ (R 2d z , C
) and Φ a smooth diffeomorphism. We assume the following properties:

(i) There exists c > 0 and ∈ N such that

∀z ∈ R 2d , |J Φ (z)| + |J Φ (z) -1 | ≤ c z .
(ii) Setting Φ(z) = (Φ q (z), Φ p (z)) and

∂ z Φ = A(z) B(z) C(z) D(z) ,
we have

∇ q S(z) = -p + A(z)Φ p (z) and ∇ p S(z) = B(z)Φ p (z), z = (q, p). (iii) For all k ∈ N, the z-dependent seminorms u(z) Σ k and sup |α|≤k ∂ α z S L ∞ are uniformly bounded in z.
The next technical lemma will be useful for proving our main results. It contains all the information needed to pass from the thawed approximation to the frozen one.

Lemma 2.20. C) and Φ be a smooth diffeomorphism satisfying Assumptions 2.19. Then, the following equality between operators in L(L 2 (R d ), Σ k ε ) holds for k ∈ N:

Let d = ∂ q -i∂ p . Let θ ∈ C ∞ (R 2d , S(R d )), S ∈ C ∞ (R 2d , R), u ∈ C ∞ (R 2d ,
J u e i ε S WP ε Φ ((dΦ p x -dΦ q D x )θ) = -i √ ε J du e i ε S WP ε Φ (θ) -i √ ε J u e i ε S WP ε Φ (dθ) .
Note that with the notation of Lemma 2.20, we have

(2.16) dΦ p (z) = C(z) -iD(z) and dΦ q (z) = A(z) -iB(z).
Besides, if condition (ii) of Assumption 2.19 is satisfied, then the equality of Remark 2.20 holds formally. The condition (i) ensures the boundedness of the operators involved in the estimates.

Proof. The integral kernel of the operator J u e

i ε S WP ε Φ (θ) is the function (x, y) → z∈R 2d k(z, x, y)dz defined by k(z, x, y) = u(z, x)e i ε S(z) g ε z (y)WP ε Φ(z) (θ(z, •))(x), (x, y) ∈ R d , z ∈ R 2d .
We aim at calculating dk. We observe for z = (q, p) ∈ R 2d , y ∈ R d and

dS(z) = -p + (A(z) -iB(z))dΦ p (z), d g ε z (y) = i ε [(dp + idq)(y -q) + pdq]g ε z (y) = i ε p g ε z (y), d WP ε Φ(z) (θ(z, •)) = WP ε Φ(z) (dθ(z, •)) + i √ ε WP ε Φ(z) ((dΦ p (z)x -dΦ q (z)D x )θ(z, •)) - i ε WP ε Φ(z) ((A(z) -iB(z))dΦ p (z)θ(z, •)) .
We obtain

dk(z, x, y) = e i ε S(z) du(z, x) g ε z (y) WP ε Φ(z) (θ(z, •))(x) + u(z, x) g ε z (y) WP ε Φ(z) (dθ(z, •))(x) + i √ ε u(z, x) g ε z (y) WP ε Φ(z) ((dΦ p (z)x -dΦ q (z)D x )θ(z, •)) (x)
The result then follows from the integration in z ∈ R 2d .

The case of Gaussian functions θ is of particular interest. Indeed, if θ(z,

•) = g Θ(z) with Θ ∈ C ∞ (R 2d , S + (d)), we have for x ∈ R d and z ∈ R 2d , (2.17) (dΦ p (z)x -dΦ q (z)D x )g Θ(z) (x) = (dΦ p (z) -dΦ q (z)Θ(z))x g Θ(z) (x).
We set M Θ (z) := dΦ p (z) -dΦ q (z)Θ(z).

By (2.16), we have the equality between matrix-valued functions

(2.18) M Θ = (C -iD) -(A -iB)Θ = (A -iB) (A -iB) -1 (C -iD) -Θ .
Note that this matrix M Θ is invertible because (A + iB) -1 (C + iD) -Θ ∈ S + (d) (as the sum of two elements of S + (d)). These observations are in the core of the proof of the next result which is a corollary of Lemma 2.20, when applied to Gaussian profiles.

Corollary 2.21. Let k ∈ N. Let Θ ∈ C ∞ (R 2d , S + (d)) such that M Θ is bounded together with its inverse, let S ∈ C ∞ (R 2d , R), u ∈ C ∞ (R 2d , C) and Φ a smooth diffeomorphism satisfying Assumptions 2.19. Then, in L(L 2 (R d ), Σ k ε (R d )), we have (2.19) J u e i ε S WP ε Φ xg Θ = O( √ ε). Besides, for all L ∈ C ∞ (R 2d , C d,d ), in L(L 2 (R d ), Σ k ε (R d )), we have (2.20) J u e i ε S WP ε Φ Lx • xg Θ = 1 i J u Tr L M -1 Θ dΦ q e i ε S WP ε Φ g Θ + O(ε). with LM -1 Θ dΦ q = L (A -iB) -1 (C -iD) -Θ -1 .
Remark 2.22.

(1) This result has an interesting consequence concerning the pseudodifferential calculus. Indeed, for a real-valued and with bounded derivatives, in view of (2.4)

a = J ( ag ε z ) = J [a(z)g ε z ] + √ ε J WP ε z ∇a(z) • x D x g iI + O(ε),
where we have used the properties of wave packets. Using ∇g iI = xg iI allows to conclude by Corollary 2.21 a = J [a(z)g ε z ] + O(ε) in any space Σ k ε . This was already proved in [START_REF] Swart | A mathematical justification for the Herman-Kluk Propagator[END_REF]. (2) More can be said about the √ ε-order term on the right-hand side of (2.19). By revisiting the proof below, one sees that there exists a real-valued smooth function z → c(z) such that

J u e i ε S WP ε Φ xg Θ = -i √ εJ u e i ε S WP ε Φ c(z)g Θ + O(ε).
Remark 2.23. The latter remark allows to prove Remark 1.13 [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF]. We observe that in L 2 ,

ψ = a V ( (a -1 )f ) + O(ε) and U ε H (t, t 0 )ψ = U ε H (t, t 0 ) a V ( (a -1 )f ) + O(ε)
Turning the pair ( V , f ) into ( a V , (a -1 )f ) consists in replacing V (t, t 0 , z) by

V ,a (t, t 0 , z) := R (t, t 0 , z)π (t 0 , z)(a(z) V (z)) = a(z) V (t, t 0 , z).
The two thawed Gaussian approximation constructed in that two different manner then differs one from the other by O(ε): the analysis developed in Section 2.3 (in particular, the arguments of Remark 2.22) shows that in Σ k ε (R d ),

J t,t0 ,th ( V f ) = (2πε) -d R 2d e i ε S (t,t0,z) g ε z , (a -1 )f V ,a (t, t 0 , z)g Γ (t,t0,z),ε Φ t,t 0 (t,z) dz + O(ε).
Proof of Corollary 2.21. One uses (2.17) and the first relation of Lemma 2.20 that we apply to θ = g Θ . It gives that in

L(L 2 (R d ), Σ k ε (R d )), J u e i ε S WP ε Φ xg Θ = J u e i ε S WP ε Φ M -1 Θ (dΦ p x -dΦ q D x )g Θ = O( √ ε), whence (2.19). Secondly, if l L ∈ C ∞ (R 2d , C d,d
), we consider the matrix L such that L = t L M Θ . We observe that

(dΦ p x -dΦ q D x ) • (L xg Θ ) = ( t L (dΦ p -dΦ q Θ))x • x -Tr( t L dΦ q ) g Θ . It remains to prove that in L(L 2 (R d ), Σ k ε (R d )), we have (2.21) J u e i ε S WP ε Φ (dΦ p x -dΦ q D x ) • (L xg Θ ) = O(ε).
We first apply Lemma 2.20 to the function θ = L xg Θ and we write

J u e i ε S WP ε Φ (dΦ p x -dΦ q D x ) • (L xg Θ ) = -i √ εJ du e i ε S WP ε Φ L xg Θ + ue i ε S WP ε Φ L x d(g Θ ) .
We use the relation (2.19) and obtain in

L(L 2 (R d ), Σ k ε (R d )), (2.22) J u e i ε S WP ε Φ (dΦ p x -dΦ q D x ) • (L xg Θ ) = -i √ εJ ue i ε S WP ε Φ L x d(g Θ ) + O(ε). We calculate d(g Θ ) = dc Θ c Θ g Θ + (dΘx • x)g Θ , with (dΘx • x) x g Θ = (dΘx • x)M -1 Θ (dΦ p x -dΦ q D x )g Θ = M -1 Θ (dΦ p x -dΦ q D x ) (dΘx • x)g Θ -2M -1
Θ dΦ q dΘ x g Θ . Therefore, there exists matrices L 1 and L 2 such that, setting θ = (dΘx • x)g Θ , we have

L x d(g Θ ) = L 1 xg Θ + L 2 (dΦ p x -dΦ q D x ) θ. We deduce J u e i ε S WP ε Φ (dΦ p x -dΦ q D x ) • (L xg Θ ) = -i √ εJ [L 1 xg Θ ] -i √ εJ [L 2 (dΦ p x -dΦ q D x ) θ]
and we obtain (2.21) by Lemma 2.20 applied to the function θ, and by the relation (2.19), which concludes the proof 2.3.3. Operators built on Bargmann transform via classical quantities. We now apply the results of the preceding section to the diffeomorphism Φ given by a flow map associated to a Hamiltonian h. We are going to derive the results induced by Lemma 2.20 and Corollary 2.21 for time dependent quantities after integration in time. We will use the resulting formula for the Hamiltonians h 1 and h 2 associated with the matrix-valued Hamiltonian H ε . Lemma 2.24. Let k ∈ N. Let h be a subquadratic Hamiltonian on I × R 2d , I = [t 0 , t 0 + T ]. We consider

(1) the classical quantities associated to h as in Section 1.3 on the interval I:

z → S(t, z), Φ t,t0 (z), F (t, t 0 , z), (2)
a smooth function defined on I × R 2d , bounded and with bounded derivatives, (t, z) → u(t, z), (3) a smooth map from I × R 2d into S(R d ), (t, z) → θ(t, z), (4) a smooth function from R 2d into I, z → t (z). Then, for all χ ∈ C ∞ 0 (I), we have the following equality between operators in

L(L 2 (R d ), Σ k ε (R d )), R χ(t)J I t>t (z) u(t) e i ε S(t) WP ε Φ t,t 0 (dΦ t p x -dΦ t,t0 q D x )θ(t) dt = i √ ε R J I t>t (z) du(t) e i ε S(t) WP ε Φ t,t 0 (θ(t)) dt + i √ ε R χ(t)J I t>t (z) u(t) e i ε S(t) WP ε Φ t,t 0 (dθ(t)) dt -i √ εJ χ(t ) dt u(t ) e i ε S(t ) WP ε Φ t ,t 0 (θ(t )
) . Note that the result of this lemma is an equality. Thus, we have not emphasized assumptions that make these operators bounded. One could for example assume global boundedness of all the quantities involve and of their derivatives, or, what would be enough, that θ is compactly supported in z.

We also emphasize that the functions denoted by (χ

• t ) u(t ) e i ε S(t ) WP ε Φ t (θ(t )) is the map z → χ t (z) u t (z), z e i ε S(t (z),z) WP ε Φ t (z)
,t 0 (z) θ t (t), z . Note also that, by construction, the flow map Φ t,t0 and the action S satisfy Assumptions 2.19 (see [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF][START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF][START_REF] Lasser | Computing quantum dynamics in the semiclassical regime[END_REF]).

Proof. The proof follows the lines of the one of Lemma 2.20, using the relation (2.23)

d I t>t (z) = dt (z)δ(t -t (z))
that produces an additional term.

As a Corollary, for Gaussian profiles, we have the following Corollary.

Corollary 2.25. With the same assumptions as in Lemma 2.24, we additionally assume

θ(t) = g Θ(t) , with Θ ∈ C ∞ (I × R 2d , S + (d)). Then, for all L ∈ C ∞ (R 2d , C d,d ), we have the following equality in L(L 2 (R d ), Σ k ε (R d )), χ(t)J I t>t (z) u e i ε S(t,t0) WP ε Φ t,t 0 Lx • xg Θ(t,t0) dt = 1 i χ(t)J 1 t>t (z) ũ(t, t 0 ) e i ε S(t,t0) WP ε Φ t,t 0 g Θ(t,t0) dt + O(ε) with u(t, t 0 ) = Tr L (A(t, t 0 ) -iB(t, t 0 )) -1 (C(t, t 0 ) -iD(t, t 0 )) -Θ(t, t 0 ) -1 .
Proof. The proof follows the lines of the one of Corollary 2.21, using the relation (2.23).

CHAPTER 3

Convergence of the thawed and the frozen Gaussian approximations

Strategy of the proofs

Our aim in this section is to prove the initial value representations of Theorems 1.15. We also explain the overall strategy that is also used for proving Theorems 1.18, 1.19 and 1.20.

Let k ∈ N. Let ψ ε 0 = V φ ε 0 be as in Assumption 1.9 with φ ε 0 ∈ L 2 frequency localized at the scale β ≥ 0 with

N β > d + k + 1 2 (which implies φ ε 0 ∈ Σ k ε .
Without loss of generality, we assume V = π (t 0 ) V for some ∈ {1, 2} that is now fixed.

We start with the Gaussian frame equality (2.2)

ψ ε 0 = (2πε) -d z∈R 2d g ε z , V φ ε 0 g ε z dz.
Writing g ε z , V φ ε 0 = V g ε z , φ ε 0 and using Remark 2.22, we have in Σ k ε ,

ψ ε 0 = J [ V (z)g ε z ] * (φ ε 0 ) + O(ε) = J [ V (z)g ε z ](φ ε 0 ) + O(ε φ ε 0 L 2
). Corollary 2.16 yields that, in Σ k ε , we have

ψ ε 0 = J [I |z|<R V (z)g ε z ](φ ε 0 ) + O(ε φ ε 0 L 2 ) + O(ε β C β R -n β ) = J [ V (z)g ε z ]((φ ε 0 ) R,< ) + O(ε φ ε 0 L 2 ) + O(ε β C β R -n β
) with the notations of Corollary 2.16 and setting n β = N β -k -d -1 2 > 0. Now that the data has been written in a convenient form, we apply the propagator U ε H (t, t 0 ) and we take advantage of its boundedness in L(Σ k ε ) to write

U ε H (t, t 0 )ψ ε 0 = (2πε) -d |z|≤R g ε z , φ ε 0 U ε H (t, t 0 ) V (z)g ε z dz + O(ε φ ε 0 L 2 ) + O(ε β C β R -n β ) = J I |z|≤R U ε H (t, t 0 ) V (z)g ε z (φ ε 0 ) + O(ε φ ε 0 L 2 ) + O(ε β C β R -n β ).
We then use the description of the propagation of wave packets by U ε H (t, t 0 ), as stated in Theorem 1.21: for

N ≥ d + 1, in Σ k ε (R d ), we have U ε H (t, t 0 ) V (z)g ε z = ψ ε,N (t) + O(ε N ). Therefore, by (1) of Theorem 2.14 in L(L 2 (R d ), Σ k ε (R d , C m )), (3.1) U ε H (t, t 0 )ψ ε 0 = J I |z|≤R ψ ε,N (t) + O(ε N -d R d φ ε 0 L 2 ) + O(ε β C β R -n β )
Besides, using that ψ ε,N (t) is a linear combination of wave packets and considering the explicit formula of Theorem 1.21, (2) of Theorem 2.14 implies that in

L(Σ k ε (R d , C m )), U ε H (t, t 0 )ψ ε 0 = J t,t0 ,th V (φ ε 0 ) R,< + O( √ ε φ ε 0 L 2 ) + O(ε N -d R d φ ε 0 L 2 ) + O(ε β C β R -n β ) = J t,t0 ,th V φ ε 0 + O( √ ε φ ε 0 L 2 ) + O(ε N -d R d φ ε 0 L 2 ) + O(ε β C β R -n β )
where we have used again Corollary 2.16. If β < 1 2 , we perform an appropriate choice of R and N :

we choose R = ε -γ with γ ≥ 1 n β ( 1 2 -β) and N ≥ 1 2 + d(1 + γ).
At this stage of the description, the thawed Gaussian approximation of Theorem (1.15) is proved. For obtaining the frozen one, we shall argue as in the scalar case considered in [START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF] (Lemma 3.2 and Lemma 3.4). We will detail this argument later in Section 3.3 below.

The proofs of the order ε approximations of Theorems 1.18, 1.19 and 1.20 start with the same lines. However, one includes in the approximation the two first terms of the asymptotic expansion of ψ ε,N (t): the one of order ε 0 and the one of order ε 1 2 . The terms of order √ ε are twofold:

(i) The one along the same mode as the initial data, here denoted by . This term will be proved to be of lower order because its structure allows to use the first part of Corollary 2.21. (ii) The one generated by the crossing along the other mode. This one is not negligible.

At that stage of the proofs, one will be left with the thawed approximation. The derivation of the frozen approximation from the thawed one involves the second part of Corollary 2.21. However, complications are induced in the treatment of term described in (ii) above because of the singularity in time that it contains. This difficulty is overcome by averaging in time and using Corollary 2.25. We implement this strategy in the next sections.

Thawed Gaussian approximations with transfers terms

We prove here the higher order approximation of Theorem 1.18 for initial data ψ ε 0 = V φ ε 0 with (φ ε 0 ) ε>0 frequency localized at the scale β ≥ 0 in a compact set K. As in the preceding section, we assume V = π (t 0 ) V and, without loss of generality, we suppose = 1.

We start as in the preceding section and transform equation (3.1) by taking the terms of order ε 0 and ε 1 2 in the expansion of ψ ε,N . We obtain

U ε H (t, t 0 )ψ ε 0 (x) = J I |z|<R (ψ ε,1 1 (t) + ψ ε,1 2 (t)) (φ ε 0 ) + O ε β C β R -n β + O(ε N -d R d ψ ε 0 L 2 ). The rest in O(ε N -d R d ψ ε 0 L 2 )
comes from the remainder of the approximation of U ε H (t, t 0 )g ε z while the term O(ε ψ ε 0 L 2 ) comes from the terms of order ε j for j ≥ 1 of the approximation, these terms having a wave packet structure while the rest is just known as bounded in Σ k ε . We write for ∈ {1, 2}

ψ ε,1 (t) = 1 j=0 ε j 2 ψ ε,1 ,j (t).
Because the assumptions on K induce that there is only one passage through the crossing, Theorem 1.21 implies that ψ ε,1 2,0 (t) = 0 and ψ ε,1 2,1 (t) only depends on the transfer profile f ε 1→2 (indeed, we have assumed V = π 1 (t 0 ) V ). Moreover, for the mode 1, we have for ∈ {1, 2} and j ∈ {0, 1}

ψ ε,1 ,j (t) = e i ε S (t,t0,z0) WP ε z (t) R (t, t 0 ) M[F (t, t 0 )] B ,j (t)g iI .
We recall that B ,1 is given by (1.41). We use the structure of the term

R (t, t 0 ) M[F (t, t 0 )] B ,1 (t)g iI
(see [START_REF] Robert | Propagation of coherent states in quantum mechanics and applications[END_REF] section 3 or the book [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]): it writes

R (t, t 0 ) M[F (t, t 0 )] B ,1 (t)g iI (x) = a(t)xg Γ (t,t0,z0) (x)
for some smooth and bounded vector-valued map (t, z) → a(t, z). Therefore, Corollary 2.21 yields

J I |z|<R ψ ε,1 1,1 (t) (φ ε 0 ) = O( √ ε φ ε 0 L 2 )
and we are left with

U ε H (t, t 0 )ψ ε 0 (x) = J I |z|<R (ψ ε,1 1,0 (t) + √ ε ψ ε,1 2,1 (t)) (φ ε 0 ) + O ε β C β R -n β + O(ε N -d R d ψ ε 0 L 2 ) = J ψ ε,1 1,0 (t) + √ ε ψ ε,1 2,1 (t) ((φ ε 0 ) R,< ) + O ε β C β R -n β + O(ε N -d R d ψ ε 0 L 2 ) = J ψ ε,1 1,0 (t) + √ ε ψ ε,1 2,1 (t) (φ ε 0 ) + O ε β C β R -n β + O(ε N -d R d ψ ε 0 L 2 )
by Corollary 2.16. Identifying the terms, we deduce

U ε H (t, t 0 )ψ ε 0 (x) = J t,t0 1,th V φ ε 0 + √ ε J t,t0 1,2,th V φ ε 0 + O ε β C β R -n β + O(ε N -d R d ψ ε 0 L 2 ). If β < 1, we choose R = ε -γ , N = 1 + d(γ + 1) with γ ≥ 1 n β (1 -β).
This gives Theorem 1.18. More precisely, for a general V = π 1 (t 0 ) V + π 2 (t 0 ) V , we obtain

U ε H (t, t 0 )ψ ε 0 (x) = J t,t0 1,th π 1 (t 0 ) V φ ε 0 + J t,t0 2,th π 2 (t 0 ) V φ ε 0 + √ ε J t,t0 1,2,th π 1 (t 0 ) V φ ε 0 (3.2) + O (ε(C β + ψ ε 0 L 2 )) .

Frozen Gaussian approximations with transfers terms

It remains to pass from the thawed to the frozen approximation. As we have already mentioned, we use the argument developed in Lemma 3.2 and 3.4 of [START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF]. It is based on an evolution argument which crucially uses Corollary 2.21. We now explain that step.

End of the proof of Theorem 1.15. We start from the approximation given by the first part of Theorem 1.15: in Σ k ε (R d ), we have

U ε H (t, t 0 )ψ ε 0 (x) = J t,t0 1,th π 1 (t 0 ) V φ ε 0 + O( √ ε(C β + φ ε 0 L 2 ))
and our aim is to prove that in Σ k ε (R d )

J t,t0 1,th π 1 (t 0 ) V φ ε 0 = J t,t0 1,fr π 1 (t 0 ) V φ ε 0 + O(ε).
Of course, a remainder of size O( √ ε) would be enough for proving Theorem 1.15; however, it will be usefull to have it in order to prove Theorems 1.19 and 1.20.

The strategy dates back to [START_REF] Robert | On the Herman-Kluk Semiclassical Approximation[END_REF]. We follow the presentation of [START_REF] Fermanian Kammerer | Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations[END_REF]. We set for s ∈ [0, 1] Θ(s, z) = (1 -s)Γ (t, t 0 , z) + isI where Γ is given by (1.18). We consider the partially normalised Gaussian function

g(t, s) = (π) -d/4 e i 2 Θ(s,z)x•x , x ∈ R d
and we set g

Θ(s),ε Φ t,t 0 (z) (x) = WP ε Φ t,t 0 (z) ( g(t, s)) ,
The aim is to construct a map s → a(s, z) such that for all s

∈ [0, 1] in L(L 2 (R d ), Σ k ε ), d ds J a(s, z) V (t, t 0 , z) g Θ(s),ε Φ t,t 0 (z) = O(ε).
Choosing a(0, z) = 1, we have

J t,t0
,th = J a(0, z) V (t, t 0 , z) g

Θ(0),ε Φ t,t 0 (z)
, and we will obtain that for any

f ∈ L 2 (R d ), we have in Σ k ε (R d ) J t,t0 ,th ( V f ) = J a(1, z) V (t, t 0 , z) g Θ(1),ε Φ t,t 0 (z) (f ) + O(ε) = J t,t0 ,fr ( V f ) + O(ε) provided a(1, z) = a (t, t 0 , z) as defined in (1.24).
For constructing the map s → a(s, z), we compute d ds J a(s, z) V (t, t 0 , z) g

Θ(s),ε Φ t,t 0 (z) = J ∂ s a(s, z) V (t, t 0 , z) g Θ(s),ε Φ t,t 0 (z) + i 2 J a(s, z) V (t, t 0 , z)WP ε Φ t,t 0 (z) ∂ s Θ(s)x • x g Θ(s),ε .
We use equation (2.20) of Corollary 2.21 to transform the second term of the right-hand side and obtain

J a(s, z) V (t, t 0 , z)WP ε Φ t,t 0 (z) (∂ s Θ(s)x • xg Θ(s) ) = 1 i J a(s, z) V (t, t 0 , z)WP ε Φ t,t 0 (z) (Tr(Θ 1 (s))g Θ(s) ) + O(ε) in L(L 2 (R d ), Σ k ε ) and with Θ 1 (s) = ∂ s Θ(s) (A -iB ) -1 (C -iD ) -Θ -1
where M (s, z) is associated to Θ(s, z) according to (2.18). In particular, we have

∂ s M (s) = -(A -iB )∂ s Θ(s). We deduce Θ 1 (s) = -(A -iB ) -1 ∂ s M (s)M (s) -1 (A -iB ) and Tr(Θ 1 (s)) = -Tr(∂ s M (s)M (s) -1 ) = -detM (s) -1 ∂ s (detM (s)) .
Therefore, the condition

∂ s a(s, z) - 1 2 Tr(∂ s M (s, z)M (s) -1 )a(s, z) = 0
that we have to fulfilled, is realized by a(s, z) = detM (s) detM (0) a(0, z) = a (t, t 0 , z).

Proof of Theorem 1.19. We now start from the result of Theorem 1.18, that is equation (3.2). In view of what has been done in the end proof of the proof of Theorem 1.15, we only have to prove

J t,t0 1,2,th π 1 (t 0 ) V φ ε 0 = J t,t0 1,2,fr π 1 (t 0 ) V φ ε 0 + O √ ε(C β + φ ε 0 L 2 .
As noticed in the introduction, when t < t 1,min (K), then τ 1,2 (t, t 0 , z) = 0 for all z ∈ K and when t ∈ [t 1,max (K), t 2,min (K)), z → τ 1,2 (t, t 0 , z) is smooth. Therefore, one can use the perturbative argument allowing to froze the covariances of the Gaussian terms as in the proof of Theorem 1.15 and one obtains the formula (1.35).

Proof of Theorem 1.20. One now has to cope with the discontinuity of the transfer coefficient τ 1,2 (t, t 0 , z). We use Lemma 2.24.

Part 2

Wave-packet propagation through smooth crossings CHAPTER 4

Symbolic calculus and diagonalization of Hamiltonians with smooth crossings

In this section, we revisit the diagonalization of Hamiltonians in the case of the smooth crossings in which we are interested. We settle the algebraic setting that we will use in Section 5 for the propagation of wave packets.

We will use the Moyal product about which we recall some facts: if A ε , B ε are semi-classical series, their Moyal product is the formal series

C ε := A ε B ε where C ε = j≥0 ε j C j C j (x, ξ) = 1 2 j |α+β|=j (-1) |β| α!β! (D β x ∂ α ξ A).(D α x ∂ β ξ B)(x, ξ), j ∈ N. (4.1)
We also introduce the Moyal bracket

{A ε , B ε } := A ε B ε -B ε A ε .
Let us now consider a smooth matrix-valued symbol H ε = H 0 +εH 1 , where the principal symbol H 0 = h 1 π 1 + h 2 π 2 has two smooth eigenvalues h 1 and h 2 with smooth eigenprojectors π 1 and π 2 . We allow for a non-empty crossing set Υ as in Definition 1.2. By standard symbolic calculus with smooth symbols, we have for ∈ {1, 2} the relations

(4.2) π (iε∂ t -H ε ) = (iε∂ t -h ) π = O(ε).
We are going to see two manners to replace the projector π and the Hamiltonian h by asymptotic series so that the relation above holds at a better order.

We call "rough" the first diagonalization process that we propose. It will hold everywhere, including Υ and is comparable the reduction performed in [START_REF] Volker | Superadiabatic transition histories in quantum molecular dynamics[END_REF] for avoided crossings. It is the subject of Section 4.2.

The second one, more sophisticated, will require to work in a domain that does not meet Υ. Based on the use of superadiabatic projectors, as developed in [START_REF] Bily | Propagation d'états cohérents et applications[END_REF][START_REF] Martinez | Twisted pseudodifferential calculus and application to the quantum evolution of molecules[END_REF][START_REF] Spohn | Adiabatic decoupling and time-dependent Born-Oppenheimer theory[END_REF][START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics Lecture Notes in Mathematics 1821[END_REF]. This strategy is implemented in Section 4.3. The new element comparatively to the references that we have mentioned, is that we keep a careful memory of the dependence of the constructed elements with respect to the distance of their support from Υ. For this reason, we will use a pseudodifferential setting that we precise in the next Section 4.1.

Formal asymptotic series

We consider formal semi-classical series

A ε = j≥0 ε j A j
where all the functions A j are smooth (matrix-valued) in an open set

D ⊂ R × R 2d , that is, A j ∈ C ∞ (D, C m,m ).
Notation. If A ε = j≥0 ε j A j is a formal series and N ∈ N, we denote by A ε,N the function

(4.3) A ε,N = 0≤j≤N ε j A j .
The formal series that we will consider in Section 4.3 will present two small parameters: the semi-classical parameter ε > 0 and another parameter δ > 0 that controls the growth of the symbol and of its derivatives. For our intended application, δ is related to the size of the gap between the eigenvalues of the Hamiltonian's symbol.

A ε = j≥0 ε j A j is in S µ ε,δ (D) if A j ∈ S µ-2j δ (D) for all j ∈ N. We set S ε,δ (D) := S 0 ε,δ (D). Remark 4.2. ( 1
) If A ∈ S µ δ , B ∈ S µ δ then AB ∈ S µ+µ δ while {A, B} ∈ S µ+µ -2 δ . Besides, if A ∈ S µ δ (D), then ∂ γ z A ∈ S µ-|γ| δ (D).
(2) When δ = 1, as in the next Section 4.2, then for all µ ∈ R, S µ 1 = S 0 1 coincides with the standard class of Calderón-Vaillancourt symbols, those smooth functions that are bounded together with their derivatives. Similarly, S µ ε,1 = S µ ε coincides with asymptotic series of symbols.

(3) The parameter δ can be understood as a loss that appears at each differentiation. However, in the asymptotic series, one loose δ 2 when passing from some j-th term of to the (j +1)-th. one. In Section 4.3, δ will monitor the size of the gap function f in the domain D.

The Moyal bracket satisfies the property stated in the next lemma.

Lemma 4.3. Let δ A , δ B ∈]0, 1]. If A ε and B ε are formal series of S ε,δ A (D) and S ε,δ B (D), respectively. then A ε B ε is a formal series of S ε,min(δ A ,δ B ) (D). Besides, for N ∈ N, A ε,N B ε,N = 0≤j≤N ε j C j + ε N +1 R ε,N A,B
where for all γ ∈ N 2d , there exists C N,γ independent on δ A,B and ε such that

|∂ γ z R ε,N A,B (t, z)| ≤ C N,γ [min(δ A , δ B )] -N -κ0 , ∀ε ∈]0, 1], ∀(t, z) ∈ D
, where κ 0 is a universal constant depending only on the dimension d.

Proof. The estimate is a direct consequence of [START_REF] Bouzouina | Uniform semi-classical estimates for the propagation of quantum observables[END_REF]Theorem A1]. In Appendix B, Theorem B.1 we give a detailed proof.

When δ A = 1 and δ B = δ ∈ (0, 1], min(δ A , δ B ) = δ. This shows that

S ε,1 (D) S ε,δ (D) ⊂ S ε,δ (D).
Let us conclude this Section by comments on the quantization of symbols of the classes S µ δ . The Calderón-Vaillancourt estimate for pseudodifferential operators (see [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]) states that there exists a constant C > 0 such that for all a ∈ C ∞ (R 2d ),

op ε (a) L(L 2 (R d )) ≤ C sup 0≤|γ|≤d+1 ε |γ| 2 sup z∈R d |∂ γ z a(z)|.
Actually, the article [START_REF] Calderón | On the boundedness of pseudo-differential operators[END_REF] treats the case ε = 1 and the estimate in the semi-classical case comes from the observation that

op ε (a) = Λ * ε op 1 (a( √ ε•))Λ ε
where Λ ε is the L 2 -unitary scaling operator defined on function f ∈ S(R d ) by

Λ ε f (x) = ε -d 4 f x √ ε , x ∈ R d .
One can derive an estimate in the sets Σ k ε by observing

x α (ε∂ x ) β • op ε (a) = |γ1|+|γ2|+|γ3|≤k ε |γ 1 | 2 c γ1,γ2,γ3 (ε) op ε (∂ γ1 z a) • (x γ2 (ε∂ x ) γ3 )
for some coefficients c γ1,γ2,γ3 (ε), uniformly bounded with respect to ε ∈ [0, 1]. This implies the boundedness of op ε (a) in weighted Sobolev spaces: for all k ∈ N, there exists a constant C k > 0 such that for all a ∈ S(R 2d ),

(4.4) op ε (a) L(Σ k ε ) ≤ C k 0≤|γ|≤d+k+1 ε |γ| 2 sup z∈R d |∂ γ z a(z)|. Proposition 4.4. Let A ∈ S µ-2j δ for µ ∈ R, j ∈ N. Then, for k ∈ N op ε (A) L(Σ k ε ) ≤ C k sup 0≤|γ|≤d+1 ε |γ| 2 δ µ-2j-k-|γ| Therefore, if δ ≥ √ ε (4.5) op ε (A) L(Σ k ε ) ≤ C k δ µ-k-2j
. We will use such estimates. Questions related with symbolic calculus in the classes S µ δ,ε are discussed in Appendix B.2.

'Rough' reduction

The next result gives a reduction of the Hamiltonian in a block diagonalized form. We will use this reduction on small interval of times. Theorem 4.5. Assume H ε = H 0 +εH 1 with H 0 having smooth eigenprojectors and eigenvalues. There exist matrix-valued asymptotic series

π ε 1 = π 1 + j≥1 ε j π 1,j , h ε = h I + j≥1 ε j h ,j , W ε = j≥1 ε j W j , ∈ {1, 2}
such that for all N ∈ N, π ε,N

1 and π ε,N 2 = 1 -π ε,N 1 are approximate projectors (4.6) π ε,N π ε,N = π ε,N + O(ε N +1 ), ∈ {1, 2}
and

H ε = H 0 + εH 1 reduces according to π ε,N 1 (iε∂ t -H ε ) = (iε∂ t -h ε,N 1 ) π ε,N 1 + W ε,N π ε,N 2 + O(ε N +1 ), (4.7) π ε,N 2 (iε∂ t -H ε ) = (iε∂ t -h ε,N 2 ) π ε,N 2 + (W ε,N ) * π ε,N 1 + O(ε N +1 ). (4.8)
Moreover, for all ∈ {1, 2} and j ≥ 1, the symbols π ,j and h ,j are self-adjoint, the matrices W j are the off-diagonal (see equation (1.44) for the value of W 1 ) and

h ,1 = π H 1 π + (-1) i 2 (h 1 -h 2 )π {π 1 , π 1 }π . (4.9)
If H ε also satisfies Assumption 1.4 on the time interval I, then the 4×4 matrix-valued Hamiltonian

H ε := h ε 1 W ε (W ε ) * h ε 2 is subquadratic according to Definition 1.1.
Note that in H ε , the off-diagonal blocks are of lower order than the diagonal ones since the asymptotic series W ε has no term of order 0.

Theorem 4.5 allows to put the equation (1.1) in a reduced form by setting

ψ ε = t (ψ ε 1 , ψ ε 2 ) with ψ ε = π ε ψ ε . Indeed, we then have (4.10) iε∂ t ψ ε = H ε ψ ε + O(ε ∞ ), ψ ε |t=0 = t π ε 1 ψ ε 0 , π ε 2 ψ ε 0 .
We deduce the corollary.

Corollary 4.6. Formally, we have for t ∈ I,

U ε H (t, t 0 )ψ ε 0 = ψ ε 1 + ψ ε 2
, where ψ ε solves (4.10).

Proof. The proof relies on a recursive argument. The case N = 0 is equivalent to (4.2) The case N = 1 has been proved in Lemma B.2 in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]. However, we revisit the proof in order to compute W 1 . We first compute π , by requiring π ε,( ) π ε,( ) = π ε,( ) + O(ε 2 ), which admits the solution

π 1,1 = -π 2,1 = - 1 2i π 1 {π 1 , π 1 }π 1 + 1 2i π 2 {π 1 , π 1 }π 2 .
We recall that {π 1 , π 1 } is diagonal and skew-symmetric (see Lemma B.1 in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF]). Then, we observe

π (iε∂ t -H ε ) = (iε∂ t -h ) π + εΘ + O(ε 2 ) with Θ = - 1 2i {π , H 0 } -π H 1 -i∂ t π + 1 2i {h , π } or, equivalently Θ 1 = 1 2i (h 2 -h 1 ){π 1 , π 1 } -i∂ t π 1 -π 1 H 1 + 1 i {h 1 , π 1 }π 1 + 1 2i {h 1 + h 2 , π 1 }π 2 , Θ 2 = 1 2i (h 1 -h 2 ){π 2 , π 2 } -i∂ t π 2 -π 2 H 1 + 1 i {h 2 , π 2 }π 2 + 1 2i {h 1 + h 2 , π 2 }π 1 = - 1 2i (h 2 -h 1 ){π 1 , π 1 } + i∂ t π 1 -π 2 H 1 - 1 i {h 2 , π 1 }π 2 - 1 2i {h 1 + h 2 , π 2 }π 1
We observe

Θ * 2 = - 1 2i (h 2 -h 1 ){π 1 , π 1 } -i∂ t π 1 -H 1 π 2 + 1 i π 2 {h 2 , π 1 } + 1 2i π 1 {h 1 + h 2 , π 2 } and (4.11) π 1 Θ * 2 π 2 = π 1 Θ 1 π 2 . Thus, we have to solve -π 1,1 H 0 = -h 1,1 π 1 -h 1 π 1,1 + i∂ t π 1 + Θ 1 + W 1 π 2 , -π 2,1 H 0 = -h 2,1 π 2 -h 2 π 2,1 + i∂ t π 2 + Θ 2 + W * 1 π 1 .
Multiplying on the right the first equation by π 1 and the second by π 2 , we obtain that h 1,1 and h 2,1 have to solve

h 1,1 π 1 = i∂ t π 1 π 1 + Θ 1 π 1 and h 2,1 π 2 = i∂ t π 2 π 2 + Θ 2 π 2
which is solved by taking the self-adjoint matrices

h 1,1 = i∂ t π 1 π 1 + Θ 1 π 1 -iπ 1 ∂ t π 1 π 2 + π 1 Θ * 1 π 2 , h 2,1 = i∂ t π 2 π 2 + Θ 2 π 2 -iπ 2 ∂ t π 2 π 1 + π 2 Θ * 2 π 1 .
Multiplying on the right the first equation by π 2 and the second by π 1 , we obtain that W 1 has to solve

W 1 π 2 = -(h 2 -h 1 )π 1,1 π 2 -Θ 1 π 2 and W * 1 π 1 = (h 2 -h 1 )π 2,1 π 1 -Θ 2 π 1 . Using the relations π * 1,1 = π 1,1 = -π 2,1 , π 1 ∂ t,z π 1 = ∂ t,
W 1 π 2 = π 1 W 1 = π 1 H 1 + i∂ t π 1 + 1 2 {h 1 + h 2 , π 1 } π 2 ,
whence (1.44).

One can now perform the recursive argument. Assume that we have obtained (4.6), (4.7) and (4.8) for some N ≥ 1 and let us look for π 1,N +1 , h 1,N +1 , h 2,N +1 and W N +1 such that the relations for N + 1 too.

We start with π 1,N +1 . We write

π ε,N 1 π ε,N 1 = π ε,N 1 + ε N +1 R ε ,
where R ε is an asymptotic series with first term R N . We first observe that R N is diagonal. Indeed, we have

(1 -π ε,N 1 ) π ε,N 1 π ε,N 1 = π ε,N 1 π ε,N 1 (1 -π ε,N 1 ) and (1 -π ε,N 1 ) π ε,N 1 π ε,N 1 = -ε N +1 R ε π ε,N 1 , π ε,N 1 π ε,N 1 (1 -π ε,N 1 ) = -ε N +1 π ε,N 1 R ε . This yields π ε,N 1 R ε = R ε π ε,N 1 and imply π 1 R N = R N π 1 . We now look to π 1,N +1 that must satisfy π 1,N +1 = R N + π 1,N +1 π 1 + π 1 π 1,N +1
. This relation fixes the diagonal part of π 1,N +1 according to

π 1 π 1,N +1 π 1 = -π 1 R N π 1 and π 2 π 1,N +1 π 2 = π 2 R N π 2 ,
We will see later that we do not need to prescribe off-diagonal components to π 1,N +1 .

Let us now focus on h 1,N +1 , h 2,N +1 and W N +1 . We write

π ε,N 1 (iε∂ t -H ε ) = (iε∂ t -h ε,N 1 ) π ε,N 1 + W ε,N π ε,N 2 + ε N +1 Θ ε 1
where Θ ε 1 is an asymptotic series of first term Θ 1,N . For obtaining information about Θ 1,N , we compute π ε,N (iε∂ t -H ε ) π ε,N for different choices of , ∈ {1, 2}.

• Taking = gives two relations

π ε,N 1 (iε∂ t -H ε ) π ε,N 2 = π ε,N 1 W ε,N π ε,N 2 + ε N +1 π ε,N 1 Θ ε 1 π ε,N 2 + O(ε N +2 ) π ε,N 2 (iε∂ t -H ε ) π ε,N 1 = π ε,N 2 (W ε,N ) * π ε,N 1 + ε N +1 π ε,N 2 Θ ε 2 π ε,N 1 + O(ε N +2 ), from which we deduce π 2 Θ 2,N π 1 = (π 1 Θ 1,N π 2 ) * . • Taking = gives the relations π ε,N (iε∂ t -H ε ) π ε,N = π ε,N (iε∂ t -h ε,N ) π ε,N + ε N +1 π ε,N Θ ε π ε,N + O(ε N +2 ),
whence the self-adjointness of the diagonal part of Θ ε . We now enter into the construction of h 1,N +1 , h 2,N +1 and W N +1 . We write the asymptotic series

π ε,N +1 1 (iε∂ t -H ε ) = π ε,N 1 (iε∂ t -H ε ) -ε N +1 π 1,N +1 H 0 + O(ε N +2 ), (iε∂ t -h ε,N +1 1 ) π ε,N +1 1 + W ε,N +1 π ε,N +1 2 = (iε∂ t -h ε,N 1 ) π ε,N 1 + W ε,N π ε,N 2 + ε N +1 (i∂ t π 1,N -h 1,N +1 π 1 -h 1 π 1,N +1 + W N +1 π 2 ) + O(ε N +2 ).
Therefore, we look for h 1,N +1 and W N +1 such that

-π 1,N +1 H 0 = i∂ t π 1,N -h 1,N +1 π 1 -h 1 π 1,N +1 + W N +1 π 2 + Θ 1,N or equivalently 0 = i∂ t π 1,N -h 1,N +1 π 1 + (h 2 -h 1 )π 1,N +1 π 2 + W N +1 π 2 + Θ 1,N .
By multiplying on the right by π 1 2 and π 2 , we are left with the two equations

(4.12) h 1,N +1 π 1 = Θ 1,N π 1 + i∂ t π 1,N π 1 and W N +1 π 2 = Θ 1,N π 2 + (h 2 -h 1 )π 1,N +1 π 2 + i∂ t π 1,N π 2 .
Considering similarly the conditions for the mode h 2 , we obtain that h 2,N +1 and W * N +1 have to satisfy

h 2,N +1 π 2 = Θ 2,N π 2 + i∂ t π 2,N π 2 and W * N +1 π 1 = Θ 2,N π 1 -(h 2 -h 1 )π 2,N +1 π 1 + i∂ t π 2,N π 1 .
Since π 2,N = -π 1,N for N ≥ 1, we are left with the relation

(4.13) h 2,N +1 π 2 = Θ 2,N π 2 -i∂ t π 1,N π 2 and W * N +1 π 1 = Θ 2,N π 1 + (h 2 -h 1 )π 1,N +1 π 1 -i∂ t π 1,N π 1 .
We set

h 1,N +1 = Θ 1,N π 1 + i∂ t π 1,N π 1 + π 1 Θ * 1,N π 2 -iπ 1 ∂ t π 1,N π 2 , h 2,N +1 = Θ 2,N π 2 + i∂ t π 2,N π 2 + π 2 Θ * 2,N π 1 -iπ 2 ∂ t π 2,N π 1 .
Then, h 1,N +1 and h 2,N +1 are self-adjoint and satisfy the first part of (4.12) and (4.13) respectively.

The construction of W N +1 requires to be more careful because there is a compatibility condition between (4.12) and (4.13). We look for W N +1 of the form

W N +1 = Θ 1,N π 2 + (h 2 -h 1 )π 1,N +1 π 2 + i∂ t π 1,N π 2 + U N +1 π 1 ,
which guarantees (4.12). Then, one has

W * N +1 = π 2 Θ * 1,N + (h 2 -h 1 )π 2 π 1,N +1 -iπ 2 ∂ t π 1,N + π 1 U * N +1
and

W * N +1 π 1 = π 2 Θ * 1,N π 1 -iπ 2 ∂ t π 1,N π 1 + π 1 U * N +1 π 1 = π 2 Θ 2,N π 1 -iπ 2 ∂ t π 1,N π 1 + π 1 U * N +1 π 1
where we have used the first property of the matrices Θ 1,N and Θ 2,N that we have exhibited, together with the fact that π 1,N +1 is diagonal. It is then enough to choose

U N +1 = π 1 Θ * 2,N + (h 2 -h 1 )π 1,N +1 + i∂ t π 1,N π 1 since it implies W * N +1 π 1 = π 2 Θ 2,N π 1 -iπ 2 ∂ t π 1,N π 1 + π 1 (Θ 2,N + i∂ t π 1,N + (h 2 -h 1 )π 1,N +1 )π 1 = Θ 2,N π 1 + (h 2 -h 1 )π 1,N +1 π 1 -iπ 2 ∂ t π 1,N π 1 ,
where we have used π 1,N +1 π 1 = π 1 π 1,N +1 π 1 . As a consequence, the second part of (4.13) is satisfied. This concludes the recursive argument and the proof of the Theorem 4.5 since the growth properties of the matrices that we have constructed come with the recursive equations.

Superadiabatic projectors and diagonalization

One now wants to get rid of the off-diagonal elements W ε , which is possible outside Υ. We are going to take into account how far from the crossing set we are by introducing a gap assumption.

Assumption 4.7 (Gap assumption). Let t 0 < t 1 , I an open interval of R containing [t 0 , t 1 ]
and Ω an open subset of R 2d . We say that the eigenvalue h has a gap larger than δ ∈ (0, 1] in

D := I × Ω if one has (N C δ ) d (h(t, z), Sp(H 0 (t, z))) ≥ δ, ∀(t, z) ∈ D.
The construction of superadiabatic projectors dates to [START_REF] Bily | Propagation d'états cohérents et applications[END_REF] which was inspired by the paper [START_REF] Emmrich | Geometry of the transport equation in multicomponent WKB approximation[END_REF]. It has then been carefully developed in [START_REF] Martinez | Twisted pseudodifferential calculus and application to the quantum evolution of molecules[END_REF] and [START_REF] Spohn | Adiabatic decoupling and time-dependent Born-Oppenheimer theory[END_REF] (see also the book [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics Lecture Notes in Mathematics 1821[END_REF]). We revisit here the construction of superadiabatic projectors, in order to control their norms with respect to the parameter δ.

We follow the construction of the Section 14.4 of the latest edition of [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF] (2021), that we adapt to our context. One proceeds in two steps: first by defining the formal series for the projectors and then for the Hamiltonians. In order to simplify the notations in the construction, we just consider an eigenvalue h and we will then apply the result to the eigenvalues h 1 and h 2 .

Formal superadiabatic projectors.

Theorem 4.8 (semiclassical projector evolution). Assume the eigenvalue h of the Hamiltonian H 0 satisfies Assumption 4.7 in D. Then, there exists a unique formal series j≥1 ε j-1 Π j in S -1 ε,δ (D) such that setting Π 0 (t, z) = π(t, z), the formal series

Π ε (t, z) = j≥0 ε j Π j (t, z)
is a formal projection and

(4.14) iε∂ t Π ε (t) = [H ε (t), Π ε (t)] .
Moreover the sub-principal term Π 1 (t) is an Hermitian matrix given by the following formulas:

π(t)Π 1 (t)π(t) = - 1 2i π(t){π(t), π(t)}π(t), (4.15) π(t) ⊥ Π 1 (t)π(t) ⊥ = 1 2i π(t) ⊥ {π(t), π(t)}π(t) ⊥ , π(t) ⊥ Π 1 (t)π(t) = π(t) ⊥ (H 0 (t) -h(t)) -1 π(t) ⊥ R 1 (t)π(t), where R 1 (t) = i∂ t π(t) - 1 2i ({H 0 (t), π(t)} -{π(t), H 0 (t)}) -[H 1 (t), π(t)]. Proof. With Notations 4.3, Π ε,N Π ε,N -Π ε,N = ε N +1 S N +1 + O(ε N +2 ), (4.16) iε∂ t Π ε,N -H 0 + εH 1 , Π ε,N = ε N +1 R N +1 + O(ε N +2 ). (4.

17)

Step N = 1. We start with N = 0. We have Π (0) = π ∈ S 0 δ (D). Since π 2 = π and [H 0 , π] = 0, we obtain

S 1 = 1 2i {π, π} and R 1 = i∂ t π -1 2i ({H 0 , π} -{π, H 0 }) -[H 1 , π],
and we have R 1 , S 1 ∈ S 0 δ (D). Two structural observations are in order:

(1) The matrix S 1 is symmetric and satisfies πS 1 π ⊥ = π ⊥ S 1 π = 0.

(2) The matrix R 1 is skew-symmetric. It satisfies

πR 1 π = 0 and π ⊥ R 1 π ⊥ = [H 0 , π ⊥ S 1 π ⊥ ].
If H 0 has only two eigenvalues, H 0 expresses only in terms of π and the expression of R 1 given above shows that R 1 is off-diagonal. The situation is more complicated if H 0 has strictly more than two distinct eigenvalues. For verifying [START_REF] Arnold | Ordinary differential equations[END_REF] in that case, one uses the Poisson bracket rule {A, BC} -{AB, C} = {A, B}C -A{B, C} two times. We obtain

{H 0 , π} -{π, H 0 } = {H 0 , π 2 } -{π 2 , H 0 } = {hπ, π} + {H 0 , π}π -H 0 {π, π} -{π, hπ} + {π, π}H 0 -π{π, H 0 } = π{h, π} -{π, h}π + [{π, π}, H 0 ] + {H 0 , π}π -π{π, H 0 }, which implies π ⊥ ({H 0 , π} -{π, H 0 })π ⊥ = π ⊥ [{π, π}, H 0 ]π ⊥ .
For determining the π-diagonal component, we choose A = π, B = H 0 π ⊥ , and C = π to obtain

0 = {π, H 0 π ⊥ }π -π{H 0 π ⊥ , π} = {π, H 0 }π -{π, hπ}π -π{H 0 , π} + π{hπ, π} = {π, H 0 }π -{π, h}π -π{H 0 , π} + π{h, π}.
This relation implies π({H 0 , π} -{π, H 0 })π = 0.

For constructing the matrix Π 1 that defines Π (1) = π + εΠ 1 , we need to satisfy

πΠ 1 + Π 1 π -Π 1 = -S 1 and -[H 0 , Π 1 ] = -R 1 .
The first of these two equations uniquely determines the diagonal blocks of Π 1 , while the second equation uniquely determines the off-diagonal blocks. We obtain

πΠ 1 π = -πS 1 π and π ⊥ Π 1 π ⊥ = π ⊥ S 1 π ⊥ , πΠ 1 π ⊥ = -πR 1 π ⊥ (H 0 -h) -1 and π ⊥ Π 1 π = (H 0 -h) -1 π ⊥ R 1 π.
For concluding this first step, we deduce from R 1 , S 1 ∈ S 0 δ (D) that Π 1 ∈ S -1 δ (D).

Step N ≥ 1. Next we proceed by induction and assume that we have constructed the matrices Π j (t) ∈ S 1-2j δ (D) for 1 ≤ j ≤ N such that (4.16) and (4.17) hold. Note that by Lemma 4.3, this implies

R N +1 (t) ∈ S -2N δ (D) and S N +1 (t) ∈ S -2N δ (D). Indeed, iε∂ t Π ε,N -H 0 + εH 1 , Π ε,N is a formal series of ε S -2 ε,δ (D) while Π N Π ε,N -Π ε,N is a formal series of εS -1 ε,δ (D).
In order to go one step further, we see that Π N +1 has to satisfy

πΠ N +1 + Π N +1 π -Π N +1 = -S N +2 and -[H 0 , Π N +1 ] = -R N +2 .
For solving these equations, and achieving the recursive process, we need to verify that at each step [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF] The matrix S N is symmetric and satisfies πS N π ⊥ = π ⊥ S N π = 0.

(2) The matrix R N is skew-symmetric and off-diagonal. It satisfies

πR N π = 0 and π ⊥ R N π ⊥ = [H 0 , π ⊥ S N π ⊥ ].
Then, we will be able to construct Π N +1 (t) ∈ S -2N -1) δ

(D) and we will have as a by product

R N +2 (t) ∈ S -2N -2 δ (D), S N +2 (t) ∈ S -2N -2 δ
(D) because of equations (4.16), (4.17) and Lemma 4.3. For proving (1), we take advantage of the fact that

Z := Π ε,N (Π ε,N ) 2 -Π ε,N (I -Π ε,N ) = ε N +1 πS N +1 π ⊥ + O(ε N +2 ),
while one also has by construction

Z = (Π ε,N ) 2 -Π ε,N Π ε,N -(Π ε,N ) 2 = O(ε 2N +2 ).
This implies that πS N +1 π ⊥ = 0 and, using that S N +1 is hermitian, we deduce that it is diagonal. For proving (2), we argue similarly with

Z := Π ε,N iε∂ t Π ε,N -H 0 + εH 1 , Π ε,N Π ε,N = ε N +1 πR N +1 π + O(ε N +2 ),
which also satisfies

Z = (Π ε,N ) 2 -Π ε,N (H 0 + εH 1 )Π ε,N -Π ε,N (H 0 + εH 1 ) (Π ε,N ) 2 -Π ε,N = O(ε 2N +2 ).
This implies πR N +1 π = 0 and one can argue similarly with 1-Π ε,N for obtaining the other relation

π ⊥ R N +1 π ⊥ = 0.
4.3.2. Formal adiabatic decoupling. The second (and decisive) part of the analysis is a formal adiabatic decoupling using the superadiabatic projectors introduced before. Theorem 4.9 (formal adiabatic decoupling). Assume the eigenvalue h of the Hamiltonian H 0 satisfies Assumption 4.7 in D. There exists a formal time dependent Hermitian Hamiltonian in D,

H adia,ε = j≥0 ε j H adia j such that (4.18) Π ε (iε∂ t -H ε ) = (iε∂ t -H adia,ε ) Π ε
with the following properties:

(1) The principal symbol is

H adia 0 = h I C m . (2)
The subprincipal term H adia 1 is a Hermitian matrix satisfying

π ⊥ H adia 1 π = π ⊥ (i∂ t π + i{h, π}) π and π H adia 1 π = πH 1 π + 1 2i π{H 0 , π}π (see (1.19
)) and we can choose π ⊥ H adia,ε

1 π ⊥ = 0. (3) We have ε -2 H adia,ε -hI m,m -εH adia 1 ∈ S -1 ε,δ (D). ( 4
) Finally, π(t) satisfies a transport equation along the classical flow for h(t).

(4.19) ∂ t π + {h, π} = 1 i [H adia 1 , π].
Remark 4.10. Note that equations (1.19) implies that H adia 1 (t, z) is smooth everywhere, including on the crossing set, if any.

The above construction applied to the Hamiltonian H ε with two smooth eigenvalues (h 1 , h 2 ) and two smooth eigenprojectors (π 1 , π 2 ) imply the construction of two pairs of formal series

(4.20) Π ε = j≥0 ε j Π ,j and H adia,ε = j≥0 ε j H adia ,j .
Corollary 4.11. At the level of the evolution operator, the result implies

U ε H (t, t 0 ) = Π ε 1 (t)U adia,ε 1 (t, t 0 ) Π ε 1 (t 0 ) + Π ε 2 (t)U adia,ε 2 (t, t 0 ) Π ε 2 (t 0 )
where for ∈ {1, 2}, U adia,ε (t, t 0 ) are the evolution operators associated with the Hamiltonian H adia,ε Proof. This result is Theorem 80 of Chapter 14 in [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF] combined with Lemma 4.3. We first observe that equation (4.18) reduces to proving

(4.21) Π ε (iε∂ t -H ε ) = (iε∂ t -H adia,ε ) Π ε .
For proving the latter relation, one first observes that if H adia 0 = h, then we have

(H adia,ε -H 0 -εH 1 ) Π ε = ε (h -H 0 )Π 1 + (H adia 1 -H 1 )π + 1 2i {h -H 0 , π} + i∂ t π + O(ε 2 ).
Therefore, H 1 has to be chosen so that

(H adia 1 -H 1 )π = (H 0 -h)Π 1 + 1 2i {H 0 -h, π} + i∂ t π.
In view of (4.15), this requires

π(H adia 1 -H 1 )π = 1 2i π{H 0 , π}π,
which is given by the second relation of (

π ⊥ (H adia 1 -H 1 )π = π ⊥ (R 1 + 1 2i {H 0 -h, π} + i∂ t π)π = π ⊥ (i∂ t π + 1 2i {π, H 0 } - 1 2i {h, π} -H 1 )π 1.19), and, using again (4.15)
which is also given by the first relation of (1.19) in view of the observation that π ⊥ {π, H 0 , π}π = -π ⊥ {h, π}π.

For proving this relation, one uses the Poisson bracket rule

{A, BC} -{AB, C} = {A, B}C -A{B, C} several times. First, one gets π{π, π}π

⊥ = 0 = π ⊥ {π, π}π. Then, taking A = π ⊥ , B = π, C = H 0 , one gets {π ⊥ , hπ} -0 = {π ⊥ , π}H 0 -π ⊥ {π, H 0 }, whence -π ⊥ {π, h}π = -π ⊥ {π, H 0 }π. Finally, for concluding the construction of H adia 1 , It remains to check that (H 0 -h)Π 1 + 1 2i {H 0 -h, π} + i∂ t π π ⊥ = 0
which comes from the latter observation about {H 0 , π}. where Ω 0 and Ω 2 are constructed so that for any initial data z ∈ Ω 0 the flows are staying in Ω 2 :

Now that H adia

Φ t,t0 h (z) ∈ Ω 2 , ∀t ∈ [t 0 , t 1 ], ∀z ∈ Ω 0 , ∀ ∈ {1, 2}.
We associate cut-offs to these subsets. We take χ 0 ∈ C ∞ 0 (Ω 0 ) with χ 0 = 1 near z 0 . Then, we choose K 0 a compact neighborhood of z 0 in Ω 0 , which implies that Ω 2 is a neighborhood of

K ,0 := {Φ t,t0 h (K 0 ), t 0 ≤ t ≤ t 1 }, ∈ {1, 2}. So we can choose χ 2 ∈ C ∞ 0 (Ω) with χ 2 = 1 on K 0 = K 1,0 ∪ K 2,0
. Finally, we take χ 1 , χ 3 ∈ C ∞ 0 (Ω) with χ 1 = 1 on supp(χ 2 ) and χ 3 = 1 on supp(χ 1 ).

For ∈ {1, 2}, we set

H adia,ε,N (t) = χ 3 H adia,ε,N (t),
which is a smooth subquadratic Hamitonian, and we consider U adia,ε,N (t, s) the propagator associated with the Hamiltonian χ 1 H adia,ε,N (t).

The next result is the usual adiabatic decoupling that results from the preceding analysis. (i) For any ∈ {1, 2}, we have in

L(Σ k ε ), iε∂ t -H ε (t) op ε χ 1 Π N,ε (t) op ε (χ 2 ) = (5.1) op ε χ 1 Π N,ε (t) iε∂ t -op ε χ 3 H adia,N,ε (t) op ε (χ 2 ) + O(ε N +1 ). (ii) Let ψ ε 0 ∈ Σ k ε such that χ 0 ψ ε 0 = ψ ε 0 + O(ε ∞ ). Set ψ ε,N (t) = op ε χ 1 Π N,ε (t) U adia,N,ε (t, t 0 ) op ε χ 0 Π N,ε (t 0 ) ψ ε 0 , ∈ {1, 2}. Then we have in Σ k ε , (5.2) U ε H (t, t 0 )ψ ε 0 = ψ ε,N 1 (t) + ψ ε,N 2 (t) + O(ε N +1 ), ∀t ∈ [t 0 , t 1 ]. Remark 5.3. (1)
The assumption satisfied by (ψ ε 0 ) ε>0 in (ii) of Proposition 5.2 is sometimes referred in the literature as having a compact semi-classical wave front set.

(2) In the proof below, the reader will notice that we do not need to assume that H ε (t) is sub-quadratic, we only need to know that H ε (t) defines a unitary Schrödinger propagator in L 2 (R d , C m ). However, we use the boundedness of the derivatives of the projectors.

Proof. (i) We fix ∈ {1, 2}. Using Theorem 4.9, we obtain

(iε∂ t -H ε (t)) (χ 1 Π ε,N (t) = χ 1 (iε∂ t -H ε (t)) χ 1 Π ε,N (t) + ε N +1 R ε N = χ 1 Π ε,N (t) iε∂ t -H adia,ε,N (t) + ε N +1 R ε N = χ 1 Π ε,N (t) iε∂ t -H adia,ε,N (t) + ε N +1 R ε N
where the R ε N s are rest terms that may change from one line to the other one and all satisfy χ 2 R ε N = 0 and where we have used that χ 3 χ 1 = χ 1 . The relation on operators follows from Theorem B.1 and Corollary B.2 (with δ = 1).

(ii) Let K 0 = supp(χ 0 ). We apply (5.1) and we use that χ 2 is identically equal to 1 on the compact set K 0 . Hence, using Egorov Theorem of Appendix C.1, we have for ∈ {1, 2}

χ 1 U adia,N,ε (t, t 0 ) χ 0 = χ 1 U adia,N,ε (t, t 0 ) χ 0 U adia,N,ε (t, t 0 ) -1 U adia,N,ε (t, t 0 ) = U adia,N,ε (t, t 0 ) χ 0 + O(ε N +1 ).
Hence we deduce (5.2) from (5.1) using again the Egorov Theorem and that

op ε χ 1 Π N,ε 1 + χ 1 Π N,ε 2 χ 0 = χ 0 + O(ε N +1 ).
The adiabatic decoupling of Proposition 5.2 and Egorov Theorem (see Proposition C.1) allows to give an explicit description at any order of the solution of equation (1.1) for initial data that are focalized wave packets.

Indeed, by the technics of Appendix C that are classic when δ = 1 (see for example the recent edition of [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]), one constructs two maps R 1 (t, t 0 , z) and R 2 (t, t 0 , z) introduced in (1.20)) and one has the following result (see Proposition C.5).

Theorem 5.4. Assume that ψ ε 0 a polarized wave packet:

ψ ε 0 = V 0 WP ε z0 (f 0 ), with f 0 ∈ S(R d ) and V 0 ∈ C m . Let N ≥ 1 and k ≥ 0. Then, there exists a constant C N,k > 0 such that the solution ψ ε (t) of (1.1) satisfies for all t ∈ [t 0 , t 1 ], ψ ε (t) -ψ ε,N 1 (t) + ψ ε,N 2 (t) Σ k ε ≤ C N,k ε N ,
with for ∈ {1, 2} and for all M ≥ 0,

ψ ε,N (t) = e i ε S (t,t0,z0) WP ε z (t)   R (t, t 0 , z 0 )M[F (t, t 0 )] 0≤j≤M ε j/2 B ,j (t)f 0   + O(ε M/2 ),
where B ,j (t) are differential operators of degree ≤ 3j with vector-valued time-dependent coefficients satisfying (1.40) and (1.41).

Propagation close to the crossing area

Our goal in this section is to extend the result of Theorem 5.4 up a the time t -cδ for some c > 0 and δ 1. We follow the same strategy as in the preceding section and checks carefully the dependence in δ of the estimates.

More precisely, the situation is the following: we consider a wave packet at initial time t 0 that is focalized along the mode h 1 at some point z 0 . We let it evolve along that mode according to Theorem 5.4, up to (t 1 , z 1 ) conveniently chosen and we consider ψ ε (t 1 ) as a new initial data, knowing that it is a wave-packet, modulo (1) There exist η 0 > 0 and c 0 > 0 such that we have

O(ε ∞ ). The point (t 1 , z 1 ) is chosen close enough to (t , ζ ) such that |t 1 -t | + |z 1 -ζ | ≤ η 0 where η 0 is defined in the next Lemma.
|f (t, Φ t,t1 h1 (z))| ≥ c 0 |t -t | if |t -t | + |z -ζ | ≤ η 0 .
(2) There exists c, M > 0 such that for all (t, z) satisfying z -Φ t,t1 h1 (z 1 ) ≤ cδ, we have

|f (t, z)| ≥ c 0 δ -M cδ ≥ c 0 2 δ.
Proof. The result comes readily from the transversality of the curve t → Φ t,t1 h1 (z) to the set Υ = {f = 0}. Recall that this transversality is due to Point (b) of Assumption 1.2.

Our goal is to prove accurate estimates for the evolution of the solution ψ ε (t) of the Schrödinger equation with the initial data ψ ε (t 1 ), for t ∈ [t 1 , t -cδ]. We thus have to improve in this precise setting the accuracy of the estimates obtained before for fixed δ = δ 0 .

We use the control in the small parameter δ of the Moyal product rule for ε-Weyl quantization as stated in Lemma 4.3 and the estimates in the Egorov Theorem for symbols in the classes S ε,δ . Finally, the construction of the cut-off functions relies on the fact that due to Point (b) of Assumption 1.2, we can apply a straightening theorem for vector fields.

In several place we need to replace δ by cδ, for a finite number of 0

< c = c 0 , c 1 , • • • , c L (L ∈ N).
We will not mention that point each time.

5.2.1.

Localization up to the crossing region. We construct the cut-off functions by using thin tubes along the classical trajectories. We use a straightening theorem for non singular vector fields. We set D(z 1 , ρ 1 ) = {|z -z 1 | ≤ ρ 1 } and consider a branch of trajectory

T 1 := {Φ t,t1 h (z 1 ), t ∈ [t 1 , t + 1 ]}, t + 1 > t .
Lemma 5.6. [START_REF] Arnold | Ordinary differential equations[END_REF] Let be P 1 a transverse hyperplane to the curve T 1 in z 1 . There exist ρ 1 > 0 and t -

1 < t 1 < t < t + 1 such that the map (t, z) → Φ t,t1 h (z) is a diffeomorphism from ]t - 1 , t + 1 [×D(z 1 , ρ 1 ) onto a neighborhood W 1 of T 1 in P 1 .
Hence for any z in the tube W 1 , we have

z = Φ τ (z),t1 h (Y (z))
where τ and Y are smooth functions of z, τ (z) ∈ [t 1 , t + 1 ], Y (z) ∈ D(z 1 , ρ 1 ). We then define the cut-off functions as follows: consider 2 , where c > 0, C > 0 and η > 0 is a small enough constant. By adapting the constants c and C conveniently, we construct some functions

• ζ ∈ C ∞ 0 (] -2, 2[) equal to 1 in [-1, 1], • θ ∈ C ∞ (R) with θ(t) = 0 if t ≤ -1 and θ(t) = 1 if t ≥ 1, we set for δ > 0, χ δ (z) = θ τ (z) -t - 1 η (1 -ζ) τ (z) -t cδ ζ |z -Φ τ (z),t1 h (z 1 )| 2 (Cδ)
χ δ j ∈ C ∞ 0 (R 2d , [0, 1]), j ∈ {1, 2, 3}, such that (1) χ δ j = 1 on t0≤t≤t1 B Φ t0,t h (z 0 ), c j δ and χ δ j is supported in t0≤t≤t1 B Φ t0,t h (z 0 ), 2c j δ , (2)
for all γ ∈ N 2d , there exists C γ such that for all z ∈ R 2d

|∂ γ z χ δ j (z)| ≤ C γ δ -|γ| , (3) χ δ 3 = 1 on suppχ δ 1 and χ δ 1 = 1 on suppχ δ 2 . Finally, with χ 0 ∈ C ∞ 0 (R 2d , [0, 1]
) satisfying χ 0 = 1 on B(0, 1) and χ 0 (z) = 0 for |z| ≥ 2, we associate

χ δ 0 (z) = χ 0 z 1 -z δ .
And we consider χ 4 a smooth fixed cut-off (δ-independent).

5.2.2.

Adiabatic decoupling close to the gap. Omitting the mode index, we set

(5.3) H adia,N,ε (t) = χ 4 h(t) + εH adia 1 (t) + χ δ 3   2≤j≤N ε j H adia j (t)   .
Notice that, because the crossing is smooth, the eigenavalues h , π and the first adiabatic correctors H adia ,1 are smooth, even in a neighborhood of (t , ζ ). Let U adia,N,ε (t, s) be the quantum propagator associated with the Hamiltonian H adia,N,ε (t) (omitting once again the index = 1). The following result is a consequence of the sharp estimates given in [START_REF] Bouzouina | Uniform semi-classical estimates for the propagation of quantum observables[END_REF] concerning propagation of quantum observables (see (ii) of the Egorov Theorem C.1).

Proposition 5.7. Consider the cut-off functions χ δ 0 and χ δ 2 defined above and set for t ∈ [t 1 , t -cδ] op ε (χ δ 0 (t, t 1 )) := U adia,ε,N (t, t 1 ) op ε (χ δ 0 ) U adia,ε,N (t 1 , t). Then, for any M ≥ 1, z ∈ R 2d and t ∈ [t 1 , t -cδ], we have:

(1 -χ δ 2 ) χ δ 0 (t, t 0 , z) = ε δ 2 M ζ M (t) with ζ M (t, z) ∈ S δ 2 (D).
Revisiting the proof of Proposition 5.2, using Lemma 4.3 for the formal series Π ε ∈ S 0 δ 2 (D) and using (3) of Theorem 4.9 about H ε,adia , we obtain the following result.

Proposition 5.8 (adiabatic decoupling -II). With the previous notations, we have the following properties.

(ii) For t 1 ≤ t ≤ t -δ, we have

iε∂ t -H ε (t) op ε χ δ 1 Π N,ε (t) op ε (χ δ 2 ) (5.4) = op ε χ δ 1 Π N,ε (t) iε∂ t -op ε ( H adia,N,ε (t)) op ε (χ δ 2 ) + O ε δ 2 N +1 δ -κ0
where κ 0 ∈ N is N -independent.

(ii) Set for = 1,

ψ ε,N (t) = op ε χ δ 1 Π N,ε (t) U adia,N,ε (t, t 1 ) op ε χ δ 0 Π N,ε (t 1 ) ψ ε (t 1 )
where ψ ε (t 1 ) = U ε H (t 1 , t 0 )ψ ε (t 0 ). Then we have, for N ≥ 2 and for all t ∈ [t 1 , t -δ],

U ε H (t, t 0 )ψ ε 0 = ψ ε,N 1 (t) + ψ ε,N 2 (t) + O ε δ 2 N +1 δ -κ0 .
where U adia,N,ε is the propagator associated with the Hamiltonian H adia,N,ε (t).

The Remark 5.3 is valid also for this Proposition, furthermore here the coefficients of the expansion of ψ ε,N j (t) in √ ε are δ-dependent for the order ε k/2 for k ≥ 2. Note that the integer κ 0 stems from the symbolic calculus estimates of Theorem B.1. 5.2.3. Application to wave packets. As in Theorem 5.4, the previous results have consequences for wave packets propagation and give an asymptotic expansion mod O(ε ∞ ) for any α < 1/2 if δ ≈ ε α . In other words the super-adiabatic approximation is valid for times t such that |t -t | ≥ ε 1/2-η , for any η > 0. The results of Appendix C give the following result.

Theorem 5.9. Consider

ψ ε 1 := ψ ε (t 1 ) = WP ε z1 (ϕ ε 1 ), ϕ ε 1 ∈ S(R d ) modulo O(ε ∞ ).
There exist N 0 ∈ N and two families of differential operators B ,j (t) j∈N , ∈ {1, 2} such that setting for t ∈ [t 1 , t -δ]

(5.5)

ψ ε,N (t) = e i ε S (t,t0,z0) WP ε z (t)   R (t, t 0 ) M[F (t, t 0 )] 0≤j≤2N ε j/2 B ,j (t)ϕ 0   ,
one has the following property: for all k ∈ N, N ∈ N, there exists C N,k > 0 such that the solution ψ ε (t) of (1.1) satisfies for all t ∈ [t 1 , t -δ].

ψ ε (t) -ψ ε,N 1 (t) + ψ ε,N 2 (t) Σ k ε ≤ C N,k ε δ 2 N +1 δ -κ0 .
Moreover the operators B ,j (t) are differential operators of degree ≤ 3j with time dependent smooth vector-valued coefficients and satisfy (1.40) and (1.41).

Propagation through the crossing set

We now use the rough reduction of section 4.2 to treat the zone around the crossing. We fix the point (t , ζ ) ∈ Υ and consider trajectories z 1 (t) and z 2 (t) arriving simultaneously at time t in the point ζ . We consider N ∈ N and we set ψ ε,N (t) = π ε,N (t)ψ ε (t), ∈ {1, 2}. By Theorem 4.5, if k ∈ N, the solution ψ ε (t) of the Schrödinger equation (1.1) satisfies in Σ k ε , (5.6)

ψ ε (t) = ψ ε,N 1 (t) + ψ ε,N 2 (t) + O(ε N +1
). Our aim in this section is to determine ψ ε (t +δ) in terms of ψ ε (t -δ) by using the description (5.6) of ψ ε (t).

The family

ψ ε,N = t (ψ ε,N 1 , ψ ε,N 2 ) satisfies (5.7) iε∂ t ψ ε,N = H ε,N (t)ψ ε,N with H ε,N (t) := h ε,N 1 (t, z) 0 0 h ε,N 2 (t, z) + 0 W ε,N (t) W ε,N (t) * 0 .
According to Theorem 4.5, the Hamiltonian H ε is subquadratic (see Definition 1.1), thus for k, N ∈ N there exists C k,N > 0 such that for all ε > 0 and t ∈ I,

(5.8) W ε,N (t) L(Σ k+1 ε ,Σ k ε ) ≤ C k,N ε.
We have used here the fact that the asymptotic series W ε, starts with the term of order ε and we recall that the value of W 1 is given in (1.44).

Let us summarize the information about the data that comes from the preceding section. Let δ > 0, for all s ∈ (t -δ, t -δ 2 ),

(5.9)

ψ ε,N (s) = t WP ε z1(s) (ϕ ε,N 1 (s)), WP ε z2(s) (ϕ ε,N 2 (s)) ,
with for = 1, 2,

ϕ ε,N = N j=0 ε j 2 ϕ j, , ϕ j, ∈ S(R d ).
Our aim is to prove that the description of ψ ε,N (s) given in Equation (5.9) extends to s = t + δ and to derive precise formula for ϕ j, (t + δ) when j = {0, 1} and ∈ {1, 2}. We consider the Hamiltonians

H ε,N diag (t, z) = h ε,N 1 (t, z) 0 0 h ε,N 2 (t, z) and εH ε,N adiag (t, z) = 0 W ε,N (t) W ε,N (t) * 0 .
so that H ε,N = H ε,N diag + εH ε,N adiag . We fix N large enough and, for simplifying the notations, we drop the mentions of N in the following. We use the notations

U ε H (t, s) and U ε H diag (t, s)
for the propagators associated to the truncated Hamiltonians H ε,δ,N (t) and H ε,N diag (t) respectively, omitting the mention of δ in U ε H (t, s). The action of U ε H (t, s) on wave packets is described by the next Theorem on which we focus now. It gives a precise description of the action of U ε H (t + δ, t -δ) on a wave packet and describes the propagation of a wave packet through the crossing set, in particular the exchange of modes at the crossing points.

Theorem 5.10. Let k, N, M ∈ N with M ≤ N . Let δ > 0 such that δ 2 ≥ √ ε. Then, there exists C > 0 and an operator Θ ε,δ M such that for all ε ∈ (0, 1), (5.10)

U ε H (t + δ, t -δ) = U ε diag (t + δ, t ) Θ ε,δ M U ε diag (t , t -δ) + R ε,δ M with R ε,δ M L(Σ k+M +1 ε ,Σ k ) ≤ Cδ M +1 and Θ ε,δ M (t ) = I + 1≤m≤M Θ ε,δ m,M (t ).
Moreover, there exists ε 0 > 0, and a family of operators

(T ε,δ m,M ) m≥1 such that for all ϕ ∈ S(R d , C 2 ), m ≥ 1, ε ∈ (0, ε 0 ), (5.11) Θ ε,δ m,M WP ζ (ϕ) = WP ε ζ (T ε,δ m,M ϕ) + O( √ ε ϕ Σ k+2m+2 ) with (5.12) T ε,δ m,M ϕ Σ k ≤ c k,m,M ε m 2 | log ε| max(0,m-1) ϕ Σ k+2m+1
for some constants c k,m > 0. Besides, with the notation (1.43)

(5.13)

T ε,δ 1,M = 0 W 1 (t , ζ )T 2→1 W 1 (t , ζ ) * T 1→2 0 .
We point out that some additional action effects will appear when applying U ε H (t + δ, t -δ) to a wave packet via the operators U ε diag (t + δ, t ) and U ε diag (t , t -δ). When applied to a Gaussian wave packets, i.e. when ϕ ,1 = g Γ in (5.9), the leading order correction term at time t + δ due to the crossing is

√ ε e i ε S1(t +δ,t ,ζ )+ i ε S2(t ,t0,z0) WP ε z1(t +δ) (ϕ 1 ) e i ε S2(t +δ,t ,ζ )+ i ε S1(t ,t0,z0) WP ε z2(t +δ) (ϕ 2 )
with

ϕ 1 = M[F 1 (t + δ, t , ζ )]W 1 (t , ζ )T 2→1 M[F 2 (t , t 0 , z 0 )]g Γ 2 , ϕ 2 = M[F 2 (t + δ, t , ζ )]W 1 (t , ζ ) * T 1→2 M[F 1 (t , t 0 , z 0 )]g Γ 1 .
Recall that W 1 is the off-diagonal matrix described in (1.44).

The remainder of this section is devoted to the proof of Theorem 5.10. The use of Dyson series allows to obtain the decomposition (5.10) (see Section 5.3.1). Then, the analysis of each terms of the series is made in Sections 5.3.2 and 5.3.3. Finally, we recall how to compute explicitly the quantities T ε,δ m,M and S 1,m in Section 5.3.5, which was already done in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF].

Before starting the proof, we introduce a cut-off

χ δ (t) = χ t-t δ , χ 0 ∈ C ∞ 0 ] -1, 1[, χ 0 (t) = 1 if |t| ≤ 1/2. We set H ε,δ,N (t) = H ε,N diag (t) + εχ δ (t -t )H ε,N adiag (t)
and we consider the propagator U ε H δ (t, s) associated with H ε,δ,N . We claim that if Theorem 5.10 holds for U ε H δ (t, s), then it also holds for U ε H (t, s). Indeed, we have for t ∈ [t + δ, t

+ δ 2 ], U ε H (t, t -δ) = U ε H δ (t +δ, t -δ)ψ ε,N (t -δ)+i t t -δ U ε H δ (t, s)(1-χ δ )(s-t ) H ε,N adiag (s)U ε H (s, t -δ)ds.
This formula allows to obtain (5.10) for U ε H (t, t -δ). It remains to consider the action of U ε H (t, t -δ) on asymptotic sum of wave packets We observe that in the support of the integral, |t -s| > δ 2 and -δ ≤ s -t ≤ δ 2 . Therefore, s ∈ [t -δ, t -δ 2 ] on the support of the integral. where we know that U ε H (s, t -δ) propagates wave packets, whence the expansion in wave packets To conclude, we observe that since |t -s| > δ 2 on the support of the integral term, we have

U ε H (t + δ, t -δ) = U ε H δ (t + δ, t -δ) + O(δ 2 ) = U ε H δ (t + δ, t -δ) + o( √ ε)
so the formula for the first two terms of the asymptotic assumptions are the same.

In view of these considerations, we focus in the next sections in proving Theorem 5.10 for the Hamiltonian H ε,δ,N (t).

Dyson expansion.

We perform a Dyson expansion via the Duhamel formula. A first use of Duhamel formula gives for t 1 , t 2 ∈ R,

(5.14) U ε H (t 2 , t 1 ) = U ε H diag (t 2 , t 1 ) + i -1 t2 t1 U ε H (t 2 , s 1 ) H ε adiag (s 1 ) U ε H diag (s 1 , t 1 )ds 1 .
With one iteration of the Duhamel formula, we obtain

U ε H (t 2 , t 1 ) = U ε H diag (t 2 , t 1 ) + i -1 t2 t1 U ε H diag (t 2 , s 1 ) H ε adiag (s 1 )U ε H diag (s 1 , t 1 )ds 1 - t2 t1 t2 s1 U ε H (t 2 , s 2 ) H ε adiag (s 2 ) U ε H diag (s 2 , s 1 ) H ε adiag (s 1 ) U ε H diag (s 1 , t 1 )ds 1 ds 2 .
With two iterations, we have

U ε H (t 2 , t 1 ) = U ε H diag (t 2 , t 1 ) + i -1 t2 t1 U ε H diag (t 2 , s 1 ) H ε adiag (s 1 )U ε H diag (s 1 , t 1 )ds 1 - t2 t1 t2 s1 U ε diag (t 2 , s 2 ) H ε adiag (s 2 ) U ε H diag (s 2 , s 1 ) H ε adiag (s 1 ) U ε H diag (s 1 , t 1 )ds 1 ds 2 - 1 i t2 t1 t2 s2 t2 s1 U ε H diag (t 2 , s 3 ) H ε adiag (s 3 )U ε diag (s 3 , s 2 ) H ε adiag (s 2 ) × U ε H diag (s 2 , s 1 ) H ε adiag (s 1 ) U ε H diag (s 1 , t 1 )ds 1 ds 2
After M iterations, M ∈ N, we have the Dyson formula

U ε H (t 2 , t 1 ) = U ε H diag (t 2 , t 1 ) I + 1≤m≤M (i) -m P(t2,t1) F ε (s 1 , • • • , s m , t 1 )ds m • • • ds 1 + R ε M (t 2 , t 1 ) with (5.15) F ε (s 1 , • • • , s m , t 1 ) = E(s m , t 1 )E(s m-1 , t 1 ) • • • E(s 2 , t 1 )E(s 1 , t 1 ),
where the operators E(s, t 1 ) are given by (5.16)

E(s, t 1 ) = U ε H diag (t 1 , s) H ε adiag (s)U ε H diag (s, t 1 ),
and the set of integration P(t 2 , t 1 ) ⊂ R M satisfies

P(t 2 , t 1 ) = {t 1 ≤ s M ≤ • • • ≤ s 2 ≤ s 1 ≤ t 2 }.
Besides, by (5.8) there exists a constant C > 0 such that

R ε M (t 2 , t 1 ) L(Σ k+M +1 ε ,Σ k ε ) ≤ C|t 2 -t 1 | M +1 .
We apply this formula to t 1 = t , t 2 = t + δ,

U ε H (t + δ, t -δ) = U ε H diag (t + δ, t ) I + 1≤m≤M (i) -m s M ∈R s M -∞ • • • s2 -∞ F ε (s 1 , • • • , s M , t )ds M • • • ds 1 U ε H diag (t , t -δ) + R ε M (t + δ, t )U ε H diag (t , t -δ) which gives equation (5.10) with Θ ε,δ m,M = (i) -m s M ∈R s M -∞ • • • s2 -∞ F ε (s 1 , • • • , s M , t )ds M • • • ds 1 and R ε,δ M = R ε M (t + δ, t -δ)U ε H diag (t , t -δ) satisfies R ε,δ M L(Σ k+M +1 ε ,Σ k ) ≤ Cδ M +1 .
The operators Θ ε,δ m,M contain all the information about the interactions between the modes h 1 and h 2 modulo O(δ ∞ ) when M goes to +∞.

In the next sections, we focus in understanding the action of Θ ε,δ m,M on wave packets of the form

WP ε ζ ( ϕ) = WP ε ζ (ϕ 1 ) WP ε ζ (ϕ 2
) and in proving equations (5.11), (5.12) and (5.13).

Analysis of the matrices E(s, t ).

We have

E(s, t ) = 0 I(s, t ) I * (s, t ) 0 , s ∈ [t -δ, t + δ]
with (5.17)

I(s, t ) = U ε h ε,N 1 (t , s)χ δ (s -t )W ε,N (s) U ε h ε,N 2 (s, t ).
This operator combines conjugation of the pseudodifferential operator χ δ (s -t )W

I(s, t ) = U ε h ε,N 1 (t , s)W ε,N (s) U ε h ε,N 1 (s, t ) • U ε h ε,N 1 (t , s)U ε h ε,N 2 (s, t ) .
The conjugation of a pseudo by a propagator is perfectly understood and is described in our setting by the Egorov Theorem of Appendix C (with δ = 1). The operator

U ε h ε,N 2 (t , s)W ε,N (s)U ε h ε,N 2 (s, t ) has an asymptotic expansion. U ε h ε,N 2 (t , s)W ε,N (s) U ε h ε,N 2 (s, t ) = j≥1 ε j W j (s).
Similarly, for I * (s, t ), one writes

I * (s, t ) = U ε h ε,N 2 (t , s)W ε,N (s) * U ε h ε,N 2 (s, t ) • U ε h ε,N 2 (t , s)U ε h ε,N 1 (s, t ) .
Note that the actions of these two operators are perfectly adapted to the geometric context: I(s, t ) picks the contribution of the second component (the lower one), which lives on the mode 2, transforms it into something related with the mode 1 (the upper one), via the operator

U ε h ε,N 1 (t , s)U ε h ε,N

2

(s, t ), and then, an operator related to the first mode acts on what is now a component living on this precise mode. And conversely for I * (s, t ).

The action on wave packets of two different propagators acting one backwards and the other one forwards has been studied in [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] (see Section 5.2). Using Egorov theorem, the action of scalar propagators on wave packets, and the precise computation of the operator

U ε h ε,N 1 (t , s) U ε h ε,N 2 (s, t )
performed therein (which involves the canonical transformation of the phase space z → Φ t ,s

1 • Φ s,t
2 (z)), one obtains the analogue of Lemma 5.3 of [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF], which writes in our context as follows.

Lemma 5.11. Let k ∈ N. There exist -A smooth real-valued map s → Λ(s) with Λ(0) = 0, Λ(0) = 0, Λ(0) = 2µ + α • β , -A smooth vector-valued map s → z(s) = (q(s), p(s)) with z(0) = 0, ż(0) = (α , β ), -A smooth map σ → Q ε (s) of pseudodifferential operators, that maps Schwartz functions to Schwartz functions, with

Q ε (s) = M j=0 ε j Q j (s) + Q ε M +1 , Q 0 (0) = W 1 (t , ζ ) *
such that for all ϕ ∈ S(R d ),

I(s, t ) * WP ε (ϕ)(y) = WP ε e i ε Λ(s-t ) Q ε (s -t )e ipε(s-t )•(y-qε(s-t )) ϕ(y -q ε (s -t )) + R ε ϕ with, for some c M > 0 Q j (s)ϕ Σ k ≤ c M ϕ Σ k+1 , ∀j ∈ {1, • • • , M }, Q M +1 (s)ϕ Σ k ≤ c M ϕ Σ k+1+κ 0 , R ε ϕ Σ k ≤ c M ϕ Σ k+1+κ 0 ,
where we have used the scaling notation z ε (s) = z(s)/ √ ε and where κ 0 is the universal constant of Theorem B.1.

A similar result holds for I(s, t ) by replacing W 1 (t , ζ ) * by W 1 (t , ζ ) and exchanging the roles of the modes h 1 and h 2 . 5.3.3. Uniform estimates for the elements of the Dyson series. We now focus on the operators Θ ε,δ m,M . For s ∈ [t -δ, t +δ], we define recursively the quantities J

m (s) for m ∈ {1, • • • M } by J 1 (s) = -i s -∞ E(s 1 , t )ds 1 and for m ≥ 2, J m (s) = -i s -∞ E(s m , t )J m-1 (s m )ds m .
We recall that E 1 (s, t ) is supported on |t -s| ≤ δ due to the cut-off function χ δ (s -t ) that appears in (5.17). With these notations, Θ ε,δ m,M = J m (t + δ).

We are reduced to proving the existence of operators s → T ε,δ m (s) such that for all s ∈ [t -δ, t + δ] and ϕ ∈ S(R d , C 2 ), (5.18)

J m (s)WP ε ζ ( ϕ) = WP ε ζ (T ε,δ m (s) ϕ)
, with for all k ∈ N, the estimate 5.12. Note that we are omitting the index M of the notations of Theorem 5.10.

If the functions T ε,δ m exist, they satisfy for s ∈ [t -δ, t + δ] the recursive equations

WP ε ζ T ε,δ m+1 (s) ϕ 2 = -i s -∞ I(s, t ) * WP ε ζ T ε,δ m (s) ϕ 1 ds and WP ε ζ T ε,δ m+1 (s) ϕ 1 = -i s -∞ I(s, t )WP ε ζ T ε,δ m (s) ϕ 2 ds.
Therefore, by Lemma 5.11, if the functions T ε,δ m exist, they satisfy for s ∈ [t -δ, t +δ] the recursive equations

T ε,δ m+1 (s) ϕ 2 = s -∞ e i ε Λ(s-t ) Q ε (s -t )e ipε(s-t )•(y-qε(s-t )) (T ε,δ m (s) ϕ) 1 (y -q ε (s -t ))ds
and some analogue equation for the other components.

At the stage of the proof, the operators T ε,δ m are defined by a recursive process that we are going to study for proving (5.12). Therefore, we focus on the analysis for scalar valued functions ϕ of

I m ϕ := s -∞ e i ε Λ(s-t ) Q ε (s -t )e ipε(s-t )•(y-qε(s-t )) (T ε,δ m (s)ϕ)(y -q ε (s -t ))ds.
In the following sections, we prove (5.18) recursively:

(1) In Section 5.3.4, we prove that if m > 1 and if (5.18) holds for m = 1, 2, • • • , m -1, then it also holds for m = m. (2) In Section 5.3.5, we prove (5.18) when m = 1. 5.3.4. Proof of Lemma 5.12: the recursive argument. We use Lemma 5.11 to perform a recursive argument on the structure of J m (s). Lemma 5.12. Assume there exists m ∈ {1, 2, • • • , M -1} such that for all k ∈ N, we have (5.18) with the inequality (5.12) for m and for the integers between 1 and m. Then, for all k ∈ N, there exists a constant C k,m+1 such that for all ϕ ∈ S(R d ), we have

I ϕ Σ k ≤ C k,m+1 ε m+1 2 | log ε| max(0,m) ϕ Σ k+2(m+1)+1 .
This lemma, together with the precise computation of J 1 (t + δ) (see [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] and Section 5.3.5 below) concludes the proof of Theorem 5.10.

Note that at each step of the recursion one loose 2 degrees of regularity. One degree is lost because the operator Q ε that may has linear growth, and another loss is due to an integration by parts that will involve the term y • p ε in the phase and the argument q ε inside the function ϕ. The initial loss of regularity when m = 1 comes from similar reasons: one step is due to the presence of the operator Q ε and the two other ones by integration by parts.

Let us start by exhibiting basic properties of I ϕ . After the change of variables s = t + σ √ ε and letting appear a cut-off χ such that χ χ = χ, we write

I ϕ = √ ε (s -t )/ √ ε -∞ e i ε Λ(σ √ ε) χ(σ/λ)Q ε (σ √ ε)e ipε(σ √ ε)•(y-qε(σ √ ε)) × (T ε,δ m (t + σ √ ε)ϕ)(y -q ε (σ √ ε))dσ with λ = δ √ ε .
One sees that the change of variable has exhibited a power √ ε, which is exactly what one wants to earn for the recursive process. However, even though the integrand is bounded, the size of the support of the integral is large: it is of size λ = δ √ ε , which spoils that gain of √ ε. This integral will turn out to be smaller than what gives this rough estimate because of the oscillations of the phase. The proof then consists in integration by parts. For this reason, we are interested in derivatives and we observe that the recursive assumption yields (5. [START_REF] Kammerer | An Egorov theorem for avoided crossings of eigenvalue surfaces[END_REF])

∂ s T ε,δ m (s) ϕ Σ k ≤ C k,m ε m-1 2 | log ε| max(0,m-2) ϕ Σ k+2(m-1)+1
and similarly for the integers between 2 and m. Analysis of the phase. We know analyze the phase of the integral I ϕ . We set

φ ε (σ) = Λ(σ √ ε) - 1 2 p ε (σ √ ε) • (y -q ε (σ √ ε)) and L = β • y -α • D y .
We will use that e isL maps Σ k into itself continuously for all s ∈ R and k ∈ N. Besides, for δ ≤ δ 0 , δ 0 > 0 small enough, we have The function b ε (s) is uniformly bounded, as well as its derivatives, for |s| ≥ 1/2. Then, following [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF], we have

I m ϕ = √ ε (s -t )/ √ ε -∞ e iφ ε (σ) Q ε (σ √ ε)χ(σ/λ)e iLσ T ε,δ m (t + σ √ ε)ϕdσ
and there exists a smooth function f such that

φ ε (σ) = µ σ 2 + √ ε σ 3 f (σ √ ε).
At the stage of the proof, all the elements have been collected to perform the recursive argument.

Because of the considerations we have made on the support of the integral and because the phase φ ε (s) is oscillating for s far away from 0, we use the cut-off function χ to write

I m ϕ = I m,1 ϕ + I m,2 ϕ with I m,1 ϕ = √ ε (s -t )/ √ ε -∞ e iφ ε (σ) χ(σ)χ(σ/λ)Q ε (σ √ ε)e iLσ T ε,δ m (t + σ √ ε)ϕdσ
The compactly supported term. The term I m,1 ϕ is the easiest to deal with since it has compact support, independently of ε and δ. Therefore,

I m,1 ϕ Σ k ≤ C √ ε sup s∈[t -δ,t +δ] T ε,δ m (s)ϕ Σ k+1 ≤ Cε m+1 2 | log ε| max(0,m-1) ϕ Σ k+2m+2 .
Note that the presence of Q ε induces a loss of regularity of 1.

The oscillating term. For dealing with the term I m,2 ϕ , we take advantage of the oscillating phase for compensating the fact that the support is large and we perform integration by parts. We write

I m,2 ϕ = √ ε (s -t )/ √ ε -∞ ∂ σ e iφ ε (σ) 1 σ b ε (σ)(1 -χ)(σ)χ(σ/λ)Q ε (σ √ ε) × e iLσ (T ε,δ m (t + σ √ ε)ϕ)dσ = - √ ε (s -t )/ √ ε -∞ e iφ ε (σ) ∂ σ 1 σ b ε 1 (σ)e iLσ T ε,δ m (t + σ √ ε)ϕ dσ + √ ε e iφ ε (σ) 1 σ b ε 1 (σ)e iLσ σ= s -t √ ε T ε,δ m (s )ϕ with b ε 1 (σ) = (1 -χ)(σ)χ(σ/λ)Q ε (σ √ ε). Note that the operator-valued functions σ → b ε 1 (σ) and σ → 1 σ b ε 1 (σ) are bounded form Σ k+1 ε to Σ k ε
(with a loss due to the presence of the operator Q ε ), and similarly for its derivatives. We write

I m,2 ϕ = - √ ε (s -t )/ √ ε -∞ e iφ ε (σ) ∂ σ 1 σ b ε 1 (σ) e iLσ (T ε,δ m (t + σ √ ε)ϕ)dσ - √ ε (s -t )/ √ ε -∞ e iφ ε (σ) 1 σ b ε 1 (σ)e iLσ (iLT ε,δ m (t + σ √ ε)ϕ + √ ε∂ s T ε,δ m (t + σ √ ε)ϕ)dσ + √ ε e iφ ε (σ) 1 σ b ε 1 (σ)e iLσ σ= s -t √ ε T ε,δ m (s )ϕ
and we check

I m,2 ϕ Σ k ≤ C √ ε × sup s∈[t -δ,t +δ] T ε,δ m (s)ϕ Σ k+1 + ( LT ε,δ m (s)ϕ Σ k+1 + √ ε ∂ s T ε,δ m (s)ϕ Σ k+1 ) 1 2 ≤|s|≤ δ √ ε dσ σ ≤ C √ ε ε m 2 | log ε| max(0,m-1) ϕ Σ k+2m+3 + ε| log ε|ε m-1 2 | log ε| max(0,m-1) ϕ Σ k+2m ≤ Cε m+1 2 | log ε| max(0,m) ϕ Σ k+2(m+1)+1 .
We point out that it is at that very last stage that we loose some log ε coefficient. Note also that the loss of regularity in the recursive process is covered by the m → m + 1 process. It will appear in the initialization process (m = 1) in which we will get rid of the logarithmic loss | log ε| by trading it as a loss of regularity.

5.3.5. Proof of Lemma 5.12: the initialization of the recursion. For initializing the recursive process, we have to study J 1 (s), i.e. the integral I m ϕ replacing the transfer operator of the integrand by I, the identity operator. We obtain

I m=1 ϕ = √ ε (s -t )/ √ ε -∞ e iφ ε (σ) Q ε (σ √ ε)χ(σ/λ)e iLσ ϕ dσ.
Besides the estimate of the norm of I m=1 ϕ , we also want to calculate the leading order term when s = t + δ. The main difference with the preceding analysis is that we can push the integration by parts at any order because the integrand is simpler. Lemma 5.13. Let k ∈ N. Then, there exists a constant C k,1 such that for all ϕ ∈ S(R d )

I m=1 ϕ Σ k ≤ C k,1 √ ε ϕ k+3 .
Moreover, there exists an operator

Θ ε,δ = Θ ε,δ 1 + √ εΘ ε,δ 2
and a constant c k,M such that for all M ∈ N,

I m=1 ϕ s =t +δ - √ εW 1 (t , ζ ) * T 1→2 ϕ -ε Θ ε,δ ϕ Σ k ≤ C k,M √ ε √ ε δ M +1 ϕ Σ k+M +3 with for = 1, 2, Θ ε,δ ϕ Σ k ≤ c k,M ϕ Σ k+3 .
We recall that T 1→2 is the transfer operator defined in (1.43). In [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF], this Lemma has already be proved with M = 1. We ameliorate here this result. This terminates the proof of Theorem 5.10 with (T

ε,δ 1,M ) 2 = W 1 (t , ζ ) * T 1→2 .
Proof. Following the same lines of proofs than above, we write

I m=1 ϕ = I 1,1 ϕ + I 1,2 ϕ with I 1,1 ϕ = √ ε (s -t )/ √ ε -∞ e iφ ε (σ) χ(σ)χ(σ/λ)Q ε (σ √ ε)e iLσ ϕdσ.
We point out that we use here the same notation than in the preceding section for a simpler integrand. Like before, this term satisfies the estimate

I 1 ϕ Σ k ≤ C √ ε ϕ Σ k+1 .
We proceed to integration by parts in I 2 ϕ . We obtain

I 1,2 ϕ = - √ ε (s -t )/ √ ε -∞ e iφ ε (σ) ∂ σ 1 σ b ε 1 (σ) e iLσ ϕdσ -i √ ε (s -t )/ √ ε -∞ e iφ ε (σ) 1 σ b ε 1 (σ)e iLσ L ϕdσ + √ ε e iφ ε (σ) 1 σ b ε 1 (σ)e iLσ σ= s -t √ ε ϕ.
We write

∂ σ 1 σ b ε 1 (σ) = b ε 2 (σ) + 1 σ b ε 3 (σ) with b ε 3 (σ) = ∂ σ b ε 1 (σ) + √ ε(1 -χ)(σ)χ(σ/λ)∂ s Q ε ( √ εσ).
where the maps σ

→ b ε 2 (σ), σ → b ε 3 (σ) and σ → 1 σ b ε 3 (σ) are bounded from Σ k+1 ε to Σ k ε (with a loss of regularity because of the presence of Q ε (s)) with the additional property R b ε 2 (σ) L(Σ k+1 ε , Σ k ε ) dσ < c 0 < +∞
for some constant c 0 independent of ε and δ. Therefore, in Σ k ,

I 1,2 ϕ = - √ ε (s -t )/ √ ε -∞ e iφ ε (σ) 1 σ (b ε 3 (σ) + ib ε 1 (σ)L)e iLσ ϕdσ + O( √ ε ϕ Σ k+1 ).
Another integration by parts gives

I 1,2 ϕ = √ ε (s -t )/ √ ε -∞ e iφ ε (σ) ∂ σ 1 σ 2 b ε (σ)(b ε 3 (σ) + ib ε 1 (σ)L) e iLσ ϕdσ - √ ε e iφ ε (σ) 1 σ 2 b ε (σ)(b ε 3 (σ) + ib ε 1 (σ)L)e iLσ ϕ σ= s -t √ ε + O( √ ε ϕ Σ k+1 ) = O( √ ε ϕ Σ k+3 ),
since the integrand has gained integrability. Note that it is at that very place that we have a loss of 3 momenta and derivatives in the estimate. We have obtained the first inequality that allows to initiate the recursive process of the preceding section. It remains to focus on the case s = t + δ.

We now consider the operator

I ϕ = 1 √ ε I m=1 ϕ s =t +δ -W 1 (t , ζ )T 1→2 .
Note first that by the construction of the function χ,

I m=1 ϕ s =t +δ = √ ε R e iφ ε (σ) χ(σ/λ)Q ε (σ √ ε)e iLσ ϕdσ
Following [START_REF] Fermanian Kammerer | Propagation of wave packets for systems presenting codimension one crossings[END_REF] Section 5.3, we first transform the expression I m=1 ϕ s =t +δ by performing the change of variable

z = σ(1 + √ εσf (σ √ ε)/µ ) 1/2
and observe that σ = z(1

+ √ εzg 1 (z √ ε)) and ∂ σ z = 1 + √ εzg 2 (z √ ε)
for some smooth bounded functions g 1 and g 2 with bounded derivatives. Note that we have used that σ √ ε is of order δ, thus small, in the domain of the integral. Besides, there exists a family of operator Q

ε (z) such that Q ε (σ √ ε) = Q ε (z √ ε) with Q ε (0) = Q ε (0) and Q 0 (0) = W 1 (t , ζ
) and a compactly supported function χ, such that

I m=1 ϕ s =t +δ = √ ε R e iµ z 2 χ(z/λ) Q ε (z √ ε)e iz(1+ √ εzg1(z √ ε)) L ϕ dz 1 + √ εzg 2 (z √ ε) .
A Taylor expansion allows to write

Q ε (z √ ε)e i √ εz 2 g1(z √ ε)) L 1 1 + √ εzg 2 (z √ ε) = Q 0 (0) + √ εz( Q ε 1 (z √ ε) + z Q ε 2 (z √ ε))
for some smooth operator-valued maps z → Q

ε j (z √ ε) mapping S(R d ) into itself, such that for all ϕ ∈ S(R d ) the family Q ε j (z √ ε)ϕ Σ k ≤ c j ϕ Σ k+2
(because of the loss of regularity involved by L and Q ε ) We obtain

I ϕ = I 1 ϕ + I 2 ϕ with I 1 ϕ = √ ε R z e iµ z 2 χ(z/λ)( Q ε 1 (z √ ε) + z Q ε 2 (z √ ε))e izL ϕdz, I 2 ϕ = Q 0 (0) R e iµ z 2 (1 -χ)(z/λ) e izL ϕdz.
Let us study I 1 ϕ . Arguing by integration by parts as previously, we obtain

I 1 ϕ = - √ ε 2iµ R e iµ z 2 d dz χ(z/λ)( Q ε 1 (z √ ε) + z Q ε 2 (z √ ε)e izL dz = - ε 2iµ R e iµ z 2 χ(z/λ)∂ z Q ε 1 (z √ ε)e izL dz - √ ε 2iµ R χ(z/λ)( e iµ z 2 ∂ z Q ε 2 (z √ ε) + √ εz∂ z Q ε 2 (z √ ε)e izL dz - √ ε 2iµ R e iµ z 2 χ (z/λ) λ -1 Q ε 1 (z √ ε) + z λ Q ε 2 (z √ ε) e izL dz.
One then performs M + 2 integration by parts in the last term of the right-hand side that is supported in |z| > λ 2 and we obtain

I 1 ϕ = - ε 2iµ R e iµ z 2 χ(z/λ)(∂ z Q ε 1 (z √ ε) + 2∂ z Q ε 2 (z √ ε))e izL dz - ε 2iµ R χ(z/λ) e iµ z 2 z∂ 2 z Q ε 2 (z √ ε)e izL dz + O √ ε ε δ M +1 ϕ Σ k+M +3 = √ ε Θ ε,δ 1 ϕ + O ε δ M ϕ Σ k+M +3 + ε (2iµ ) 2 R e iµ z 2 ∂ z χ(z/λ)∂ 2 z Q ε 2 (z √ ε)e izL dz = √ ε Θ ε,δ 1 ϕ + ε Θ ε,δ 2 ϕ + O ε δ M ϕ Σ k+M +3 with Θ ε,δ 1 Σ k ≤ c δ ϕ Σ k+3 .
It remains to observe that the term I 2 ϕ satisfies after M + 1 integration by parts

I 2 ϕ Σ k ≤ C √ ε δ M +1 ϕ Σ k+M +3 .

Propagation of wave packets -Proof of Theorem 1.21

Let k ∈ N and let be ψ ε 0 a polarized wave packet as in (1.39):

ψ ε 0 = V 0 WP ε z0 (f 0 ) with f 0 ∈ S(R d ) and V 0 ∈ C m .
Let δ > 0. By Theorem 5.9, for t ∈ [t 0 , t -δ], and in Σ k ε , ψ ε (t) is an asymptotic sum of wave packets and writes

ψ ε (t) = ψ ε,N 1 (t) + ψ ε,N 2 (t) + O ε δ 2 N δ -κ0 .
with ψ ε,N (t) given by (5.5).

We now take the vector

ψ ε,N (t -δ) = t ψ ε,N 1 (t -δ), ψ ε,N 2 (t -δ) , ψ ε,N (t -δ) = ψ ε,N (t -δ), = 1
, 2, as initial data in the system (5.7). It is a sum of N wave packets. By construction, in particular because of the linearity of the equation, we have for all t ∈ [t -δ, t + δ],

ψ ε (t) = ψ ε,N 1 (t) + ψ ε,N 2 (t) + O ε δ 2 N δ -κ0
in Σ k ε . When t = t + δ, we deduce from Theorem 5.10 that in Σ k ε ,

ψ ε,N (t + δ) = U ε diag (t + δ, t )   I + 1≤m≤M Θ ε,δ m,M   U ε diag (t , t -δ)ψ ε,N (t -δ) + O(δ M ).
By Proposition C.5 of the Appendix, ψ ε,N (t + δ) is a sum of wave packets

ψ ε,N (t + δ) = 0≤m≤M ε m 2 ψ ε,m,M,N (t + δ)
where each term ψ ε,m,M,N (t + δ) involves a term of action. When m = 0 and m = 1, these terms have been computed precisely:

• If m = 0, for = 1, 2, ψ ε,0,M,N (t + δ) = e i ε S (t +δ,t -δ,ζ ) WP ε Φ t +δ,t 0 (z0) M[F (t + δ, t 0 , z 0 )]π (t 0 , z 0 ) V 0 f 0 where we have used the property of the scalar propagation of wave packets. where W 1 is the off-diagonal matrix computed in (1.44). At that stage of the proof, we have obtained that ψ ε (t + δ) is a sum of wave packet in Σ k ε up to O ε δ 2 N δ -κ0 + δ M +1 and we know precisely the terms of order ε 0 and ε 1 2 . For concluding, we take the vector ψ ε app (t + δ) := ψ ε,N 1 (t + δ) + ψ ε,N 2 (t + δ) as initial data at time t = t + δ in the equation (1.1). The function ψ ε app (t + δ) is an approximation of ψ ε (t + δ) at order O ε δ 2 N δ -κ0 + δ M +1 in Σ k ε . By construction and because of the linearity of the equation, for all times t ∈ [t + δ, t 0 + T ],

ψ ε (t) = U H (t, t + δ)ψ ε app (t + δ) + O ε δ 2 N δ -κ0 + δ M +1 .
We then applies Theorem 5.9 between times t + δ and t. Indeed, the classical trajectories involved in the construction do not meet Υ again and we are in an adiabatic regime, as in theorem 5.9. This concludes the proof of Theorem 1.21.

The eigenvalues of H are ±ρ and the eigenprojector associated with the eigenvalue ρ is π(x) = 1 2 1 + cos(θ(x)) sin(θ(x)) sin(θ(x)) -cos(θ(x))

.

Its derivative π (x) = 1 2 θ (x) -sin(θ(x)) cos(θ(x)) cos(θ(x)) sin(θ(x)) is not bounded. On the other side, ρ and H are sub-quadratic. Indeed, for |x| > 1, the derivatives of the coefficient of H are of the form 1 x 3 p 1 1 x , ln x cos θ(x) + p 2 1 x , ln x sin θ(x)

for p 1 and p 2 two polynomial functions of two variables and thus bounded.

A manner of controlling the growth of the potential consists in requiring a lower bound on the gap function f at infinity. We obtain a control of the form z n0+2( -1) . Besides, it allows to perform a recursive argument by writing for γ ∈ N 2d ,

∂ γ ∂ 2 ij h Tr C m,m (π) = ∂ γ (Tr C m,m (∂ i π∂ j H) + Tr C m,m (π∂ ij H)) - 2≤|α|,α≤γ c α ∂ γ-α ∂ 2 ij h Tr C m,m (∂ α π) = α≤γ c α Tr C m,m (∂ γ-α ∂ i π ∂ α ∂ j H) + Tr C m,m (∂ γ-α π∂ α ∂ ij H) - 2≤|α|,α≤γ c α ∂ γ-α ∂ 2 ij h Tr C m,m (∂ α π)
for some coefficients c α . One can then conclude recursively to |∂ γ ∂ 2 ij h| ≤ c γ z (|γ|+1)n0+2( -1) .

These two Lemmata allow to derive the consequences of Assumptions 1.4 for a Hamiltonian H ε = H 0 + εH 1 . We now fix m = 2.

Proposition A.4. Assume that H ε = H 0 + εH 1 satisfies the Assumptions 1.4. Then, for j ∈ {1, 2} we have the following properties:

(1) For all γ ∈ N 2d with |γ| ≥ 2, there exists a constant C γ > 0 such that ∀(t, z) ∈ I × R 2d , |∂ γ z π j (t, z)| ≤ C γ z |γ|n0 and |∂ γ z h j (t, z)| ≤ C γ z (|γ|-1)n0 . (2) If moreover n 0 = 0 in (1.6), then the maps z → ∂ γ π j (t, z) and z → ∂ γ z h j (t, z) for |γ| ≥ 2 are bounded. As a consequence, the Hamiltonian trajectories Φ t0,t h (z) are globally defined for all z ∈ R. Besides, there exists C > 0 such that |Φ t0,t

hj (z)| ≤ C|z|e C|t-t0| and the Jacobian matrices F j (t, z) = ∂ z Φ t,t0 hj (z) (see (1.12)) satisfy F j (t, z) C 2d,2d ≤ Ce C|t-t0| .

Remark A.5. Note that under the assumptions of Proposition A.4, for all j ∈ {1, • • • , 2d}, the matrices

f ∂ zj π 1 = -f ∂ zj π 2 = 1 2 ∂ zj (H -v) -∂ zj f (π 2 -π 1 )
are bounded. This comes from the differentiation of the relation H = vI + f (π 2 -π 1 ).

Proof. Note first that Point 2 is a consequence of Point 1. We thus focus on Point 1. We use that π 2 is the projector of the matrix H 0 -vI for the eigenvalue f . And the matrix H 0 -vI satisfies the assumptions of Lemma A.3 with = 1. Therefore, we have for (t, z) ∈ I × R d with |z| ≥ 1 and for γ ∈ N d .

|∂ γ z π j (t, z)| ≤ C γ z |γ|n0 and |∂ γ z f (t, z)| ≤ C γ z (|γ|-1)n0 . One concludes by observing that the function v = 2Tr(H) is subquadratic, whence the property of h 2 = v + f . One argues similarly for h 1 .

We close this Section with the proof of Lemma 1.10

Proof of Lemma 1.10. The map t → R (t, t 0 , z) is valued in the set of unitary maps because the matrix H adia ,1 is self adjoint. Besides, the map (t, z) → Z (t, z) = π (Φ t,t0 (z)R (t, t 0 , z)π ⊥ (z) satisfies the ODE Proof of Lemma B.3. The lemma is proved in a standard way, using integration by parts and stationary phase argument. For the sake of completeness, we give here a proof. We introduce a cut-off χ 0 ∈ C ∞ 0 (R) such that χ 0 (x) = 1 for |x| ≤ 1/2 and χ 0 (x) = 0 for |x| ≥ 1.

i∂ t Z = -i (π (∂ t π + {h, π })) • Φ t,
We split I(λ) into too pieces and write I(λ) = I 0 (λ) + I 1 (λ) with

I 0 (λ) = λ 2d R 2d ×R 2d
exp[-iλσ(u, v)]χ 0 (u 2 + v 2 )F (u, v)dudv,

I 1 (λ) = λ 2d R 2d ×R 2d
exp[-iλσ(u, v)](1 -χ 0 )(u 2 + v 2 ))F (u, v)dudv.

We notice that (u, v) → σ(u, u) is a quadratic non-degenerate real form on R 4d . Let us estimate I 1 (λ). We can integrate by parts with the differential operator

L = i |u| 2 + |v| 2 Ju • ∂ ∂v -Jv • ∂ ∂u ,
using that Le -iλσ(u,v) = Le -iλJu•v = λe -iλσ(u,v) . For I 1 (λ), the integrand is supported outside the ball of radius 1/ √ 2 in R 4d . Performing 4d + 1 integrations by parts for gaining enough decay to ensure integrability in (u, v) ∈ R 4d , we get a constant c d such that

|I 1 (λ)| ≤ c d sup u,v∈R 2d |µ|+|ν|≤4d+1 |∂ µ u ∂ ν v F (u, v)|.
To estimate I 0 (λ) we apply the stationary phase. The symmetric matrix of the quadratic form σ(u, v) is

A σ = 0 -J J 0 .
So the stationary phase Theorem ( [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I-III[END_REF], Vol.I, section 7.7), we obtain the existence of two constants

C.2. Asymptotic behavior of the propagator

In this section, we analyze the propagator U ε K (t, t in ) and compare it with U ε KS (t, t in ) the propagator for KS(t) = k(t)I m + εK 1 (t).

Lemma C.3. For all J ∈ N, there exists

W ε (t, t in ) = 0≤j≤J ε j W j (t, t in ) + ε J+1 R ε J (t, t in )
with W j (t, t in ) ∈ S -2j δ such that for all t ∈ R, U ε K (t, t in ) = W ε (t, t in )U ε KS (t, t in ). Besides, W 0 (t, t in ) = I m and for all γ ∈ N 2d , there exists C J,γ > 0 such that

sup z∈R d |∂ γ z R ε J (t, t in , z)| ≤ C J,γ δ -2(J+1+|γ|+κ0)
where κ 0 is the universal constant of Theorem B.1.

Remark C.4. Using the estimate (4.4), we deduce that for all k ∈ N, there exists

C k > 0 such that op ε (R ε J (t, t in )) L(Σ k ε ) ≤ C k ε δ 2 |γ| 2 δ -2(J+1+k+κ0) .
As a consequence, for δ = ε α with α ∈ (0,

C.3. Propagation of wave packets

When ψ ε 0 is a wave packet, the action of U ε KS (t, t 1 )ψ ε 0 on ψ ε 0 can be described precisely. Following Section 14.2 of [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF], Theorem 77, we have the following result.

Proposition C.5. Assume we have

ψ ε = WP ε z0 (f ε ) V with V ∈ C m , f ε = 0≤j≤J ε j/2 f j , f j ∈ S(R d ).
There exists a family ( U j (t)) j≥0 defined on the interval I δ such that (i) For all j ∈ N and t ∈ I δ , U j (t) ∈ S(R d ), (ii) For all k, j ∈ N, there exists a constant C k,j such that sup

t∈I δ sup |α|+|β|=k x α ∂ β x U j (t) L ∞ ≤ C k,j δ -j .
(iii) For all k ∈ N and N ∈ N, there exists C k,N and N k such that for all t ∈ R, we have ∂ z H 1 (s, z s )op 1 (F (t in , s)z)ds, (C.9) F (t, t in ) is the stability matrix for the flow z t := Φ t,tin k (z 0 ) (see (1.12)) and R(t, t in ) satisfies the equation (C.3).

U ε K (t,
Proof. Let k ∈ N. Using Lemma C.3 and the estimate (C.5), we obtain

U ε K (t, t in )ψ ε = W ε (t, t in ) U ε KS (t, t in )ψ ε = 0≤j≤J ε j W j (t, t in ) U ε KS (t, t in )ψ ε + ε J+1 R ε J (t, t in )U ε KS (t, t in )ψ ε = 0≤j≤J ε j W j (t, t in ) U ε KS (t, t in )ψ ε + O ε J+1 δ -2(J+1+κ0)
in Σ k ε (R d ). We then use the standard result of propagation of wave packets for KS(t) (see [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF])

U ε KS (t,

Definition 4 . 1 (

 41 Symbol spaces). Let µ ∈ R and δ ∈ (0, 1]. (i) We denote by S µ δ (D) the set of smooth (matrix-valued) functions in D such |∂ γ z A(t, z)| ≤ C γ δ µ-|γ| , ∀(t, z) ∈ D. Notice that the set S δ (D) := S 0 δ (D) has the algebraic structure of a ring. (ii) We shall say that a formal series

0 and H adia 1 are 3 H

 13 constructed, one uses a recursive argument: assume that one has constructed H adia j for 0 ≤ j ≤ N with H adia j ∈ S -2j δ for j ∈ {2, • • • , N } and such that has (4.21) holds up to O(ε N +1 ). Let us construct H adia N +1 . Setting as in Notation 4.adia,ε,N = N j=1 ε j H adia j

Proposition 5 . 2 (

 52 adiabatic decoupling -I). Let k ∈ N.

Lemma 5 . 5 .

 55 Assume (t , ζ ) is a generic smooth crossing point as in Definition 1.2 and consider δ ∈ (0, 1].

  d ds φ ε (s) ≥ c 0 |s|, ∀|s| ≤ λ and e iφ ε (s) = 1 s b ε (s)∂ s e iφ ε (s) with b ε (s) := s i∂ s φ ε (s).

• 2 =

 2 If m = 1, only the term with = 2 contributes andψ ε,1,M,N 2 (t + δ) = e i ε S2(t +δ,t ,ζ )+ i ε S1(t ,t0,z0) WP ε Φ t +δ,t 2 M[F 2 (t + δ, t , ζ )]W 1 (t , ζ ) * T 1→2 M[F 1 (t , t 0 , z 0 )]π 1 (t 0 , z 0 ) V 0 f 0 ,

Lemma A. 3 .

 3 Let ∈ {1, 2}. Assume that the matrix-valued function H ∈ C ∞ (I × R d , C m,m ) satisfies (A.1). Assume that for all (t, z) ∈ I × R 2d , H(t, z) has a smooth eigenvalue h with smooth associated eigenprojectors π(t, z) of constant rank for |z| > m. Assume there exists C, n 0 > 0 such that for (t, z)∈ I × R d with |z| > m, dist (h(t, z), Sp(H(t, z)) \ {h(t, z)}) ≥ C z -n0 .Then, for all γ ∈ N 2d with |γ| ≥ 2, there exists a constant C γ > 0 such that∀(t, z) ∈ I × R 2d , |∂ γ z π(t, z)| ≤ C γ z |γ|n0+ -1 and |∂ γ z h(t, z)| ≤ C γ z (|γ|-1)n0+2( -1) . Proof. We work for |z| > m and fix j ∈ {1, 2}. The relation (A.2) also implies∂ j π(H -h) = (H -h)∂ j π = ∂ j (H -h)where we keep the notation ∂ j := ∂ zj for j ∈ {1, • • • , 2d}. Using that ∂ j π is off diagonal and (H -h) invertible on Range(1 -π), we deduce∂ j π = (1 -π)∂ j π π + π ∂ j π(1 -π) with (1 -π)∂ j π π = (H -h) -1 (1 -π)∂ j (H -h) and π ∂ j π(1 -π) = π ∂ j (H -h)(1 -π)(H -h) -1 = π∂ j (H -h)(H -h) -1 (1 -π).On the range of π, the resolvent (H -h) -1 is invertible, more precisely, there exists c > 0 such that(H -h) -1 (1 -π) L(Ran(1-π)) ≤ c dist (h, Sp(H) \ {h}) -1 ≤ c C z n0+ -1 , whence |∂ j π(t, z)| ≤ C z n0 .For analyzing the derivatives of π, one observes that the relation π = π 2 implies∂ γ π = π∂ γ π + ∂ γ π π + 1<|α|,|β|<|γ| c α,β ∂ α π∂ β πfor some coefficients c α,β . A recursive argument then gives the estimate on the growth of the eigenprojectors.Let us now consider the eigenvalue h. The relation Hπ = πH = hπ gives∂hπ = ∂Hπ -∂π(H -h) = π∂H -(H -h)∂π.Multiplying on both side by π, using (H -h)π = 0 and taking the matricial trace, we obtain∂h Tr C m,m (π) = Tr C m,m (π∂H).This implies |∂h| ≤ C z -1 . To get the relation on higher derivatives, we differentiate this relation, which gives for i, j ∈ N 2d ,∂ 2 ij h Tr C m,m(π) = ∂ i (Tr C m,m (π∂ j H)) -∂ j hTr C m,m (∂ i π). Since Tr C m,m (∂ i π) = 0, we are left with ∂ 2 ij h Tr C m,m (π) = Tr C m,m (∂ i π∂ j H) + Tr C m,m (π∂ ij H).

( 1 -

 1 t0 Z , Z (t 0 , z) = 0, and thus coincides with the solution Z (t) = 0.with κ 0 = 4d + 2.The estimate 4.4 allows to evaluate the norms of the operators involved in Theorem B.1.Corollary B.2. If A ∈ S µ ε,δ and B ∈ S µ ε,δ , then A B ∈ S µ+µ ε,δ(see Remark 4.2) and for all N, k ∈ N, there exists a constant C N,k > 0 such thatop ε (R N (A, B; z; ε)) Σ k ε ≤ C N,k ε N +1 δ µ+µ -2(N +1+k+κ0) . Proof. By Fourier transform computations and application of the Taylor formula, we get the following formula for the remainder,(B.5) R N (A, B; z; ε) t) N R N,t (z; ε)dt,whereR N,t (z; ε) = (2πεt) -2d R 2d ×R 2d exp -i 2tε σ(u, v) σ N +1 (D u , D v )A(u + z)B(v + z)dudv.Notice that the integral is an oscillating integral as we shall see below. We now use Lemma B.3 for A, B ∈ S(R 2d ) with the integrandF N,γ (z; u, v) = π -2d ∂ γ z σ N +1 (D u , D v )A(u + z)B(v + z)and the parameter λ = 1/(2tε). We then have|∂ γ z R N,t (z; ε)| ≤ C d sup u,v∈R 2d |α|+|β|≤4d+1 |∂ α u ∂ β v F N,γ (z; u, v)|.Moreover, there holds the elementary estimate|σ N +1 (D u , D v )A(u)B(v)| ≤ (2d) N +1 sup |α|+|β|=N +1 |∂ α x ∂ β ξ A(x, ξ)∂ β y ∂ α η B(y, η)|.Together with the Leibniz formula, we then get the claimed results with universal constants. For symbols A ∈ P(µ A ) and B ∈ P(µ B ) we argue by localisation. We use A η (u) = e -ηu 2 A(u) and B η (v) = e -ηv 2 B(v) for η > 0 and pass to the limit η → 0.Lemma B.3. There exists a constant C d > 0 such that for anyF ∈ S(R 2d × R 2d , C m,m ) the integral (B.6) I(λ) = λ 2d R 2d ×R 2dexp[-iλσ(u, v)]F (u, v)dudv. satisfies (B.7) |I(λ)| ≤ C d sup u,v∈R 2d |α|+|β|≤4d+1 |∂ α u ∂ β v F (u, v)|.

c 1 , c 2 >

 12 0 such that (B.8) |I 0 (λ) -λ -2d c 1 | ≤ c 2 sup u,v∈R 2d |α|≤2 |∂ α (χ 0 (u 2 + v 2 )F (u, v)|.

  1 2 ], we have(C.5) op ε (R ε J (t, t in )) L(Σ k ε ) ≤ C k δ -2(J+1+k+κ0) . Similarly, by (4.5), for such δ,(C.6) op ε (W j (t, t in )) L(Σ k ε ) ≤ C k δ -2j-k , j ∈ N.Proof. If such a W ε (t, t in ) exists, it must satisfy the following equationiε∂ t W ε (t, t in ) W ε (t, t in )U ε KS (t in , t) K(t) -KS(t) U ε KS (t, t in ) * , W ε (t in , t in ) = I mApplying Egorov Theorem of Propostion C.1, we know thatU ε KS (t in , t) K(t) -KS(t) U ε KS (t in , t) * = ε 2 L ε (t, t in ) where L ε (t, t in ) = j≥0 ε j L j (t, t in ) ∈ S -1ε,δ with estimates on L j (t, t in ) and on the remainder term in the asymptotic expansion. Using the estimates proved in Appendices A and B, it is enough to solve as formal series in ε the equation

									
	(C.7)	i∂ t		ε j W j (t, t in )  = ε		ε j W j (t, t in ) 		ε j L j (t, t in )	 .
			j≥0		j≥0		k≥0

  t in )ψ ε -e

		i ε S(t,tin,z0) WP ε Φ t,t in k	(z0)	 	2N j=0	ε	 2 U j (t) j 	ε Σ k	≤ C k,N	ε δ 2	N	δ -N -N k .
	Besides
	(C.8)	U |α|=3	1 α!	t tin	∂ α z h(s, z s )op w 1 ((F (t in , s)z) α )ds
		+	1 i	t tin

0 (t) = R(t, t in ) M[F (t, t in )]f 0 V and U 1 (t) = R(t, t in ) M[F (t, t in )]b 1 (t, t in )f 0 V where b 1 (t, t in ) = 1 i

  t in )ψ ε = e J+1 ) with B j (t) = U j (t) as in (C.8) for j = 0, 1, and B j (t) is determined by a recursive equation in terms of B 0 (t), • • • , B j-1 (t). This description relies on the observation that setting WP ε zt -∂t (S + ξ t • x t ) B j + √ ε żt • z B j + iε∂ t B j WP ε zt op 1 k(t, z t ) + √ ε∇k(t, z t ) • z + ε 2 Hess k(t, z t )z • z + εK 1 (t, z t ) + O(εwhere we have set z t = (x t , ξ t ). These relations also prove (C.8) with the additional remark that W 0 (t, t in ) = I and, for j ∈ N,W j (t, t in )WP ε zt B j (t) = WP ε zt op 1 W j (t, t in , z t + √ εz) B j (t)and using the estimate (C.6) with ε = 1.

	i ε S(t,tin,z0) WP ε zt j (t) = e i ε S(t,tin,z0) WP ε   2J ε j 2 B j (t)  zt B j (t) zt KS(t, z t + we have the two relations iε∂ t ϕ ε j (t) = e i ε S KS(t)ϕ ε j (t) = e i ε S WP ε √ εz) B j (t)  + O(ε ϕ ε = e i ε S 3 2	B j
	j=1

FREQUENCY LOCALIZED FAMILIES

Clotilde Fermanian Kammerer acknowledges the support of the Region Pays de la Loire, Connect Talent Project High Frequency Analysis of Schrödinger equations (HiFrAn). Caroline Lasser was supported by the Centre for Advanced Study in Oslo, Norway, research project Attosecond Quantum Dynamics Beyond the Born-Oppenheimer Approximation.

we write (H adia,ε,N -H 0 -εH 1 ) Π ε,N = ε N +1 T N + O(ε N +2 ) with T N ∈ S -2N -3 δ and we look for H adia N +1 such that πH adia N +1 = T N . This is doable as long as π ⊥ T N = 0, which comes form the observation that

by the properties of superadiabatic projectors. Besides, H adia N +1 ∈ S -2N -3 δ , which fits with (ii) of Definition 4.1

(3) comes from Lemma 4.3.

(4) comes from (1.19). CHAPTER 5

Propagation of wave packets through smooth crossings

In this section, we prove Theorem 1.21. We consider a subquadratic Hamiltonian H ε = H 0 +εH 1 satisfying Assumptions 1.2 in I ×Ω, Ω ⊂ R 2d , and we are interested in the description of the solution to equation (1.1) for initial data that is a wave packet as in (1.8).

The proof consists in three steps: one first propagates the wave packet from time t 0 to some time t -δ, δ > 0 in a zone that is at a distance of Υ of size larger than cδ for some constant c > 0. In this zone, we use the superadiabatic projectors. Then, we propagate the wave packet from time t -δ to t + δ, using the rough diagonalization in the crossing region. Finally, between times t + δ and t 1 , we are again at distance larger than cδ to Υ and the analysis with superadiabatic projectors apply. The parameter δ will be taken afterwards as δ ≈ ε α ; the analysis of Section 5.2 will ask for α < 1 2 (see Theorem 5.9). In order to explain carefully each step of the proof, we start by proving the propagation faraway from the crossing area in Section 5.1. That allows us to settle the arguments, before doing it precisely close to Υ in Section 5.2. Then, Section 5.3 is devoted to the calculus of the transitions in the crossing region.

All along Section 5, we will use Assumption 4.7 and the following dynamical Assumption 5.1.

Assumption 5.1 (Dynamical assumption). We say that Ω 1 and t 1 satisfy the dynamical assumption (DA) for the mode h if we have (DA) Φ t,t0 (Ω 1 ) ⊂ Ω for all t ∈ [t 0 , t 1 ].

Propagation faraway from the crossing area

In this section, we analyze the propagation of wave packets in a region where the gap is bounded from below. It gives the opportunity to introduce the method that we shall use in the next section for a small gap region. So, we fix δ = δ 0 , δ 0 > 0 small but independent on ε and we work in the open set

where I is an open interval of R containing [t 0 , t 1 ] and where the gap condition is also satisfied. We associate with H ε the formal series of Theorems 4.9 and 4.8 for each of the modes:

and we will use the notation introduced in (4.3). With z 0 ∈ Ω, we associate the open sets Ω 0 , Ω 1 , Ω 2 and Ω 3 such that

Matrix-valued Hamiltonians

We explain here the set-up and the technical assumptions that we make on the Hamiltonian H ε . It is the occasion of motivating the set of Assumptions 1.4 and deriving their consequences. The objectives of these assumptions are first to ensure the existence of the propagators associated with the full matrix-valued Hamiltonian and with its eigenvalues, and secondly to guarantee adequate properties of growth at infinity which are used in our analysis.

In this section, we work with m × m (m ∈ N) matrix-valued Hamiltonians H that are subquadratic:

and differentiating these two relations, we obtain for all j ∈ {1,

Multiplying from the left and the right with π and using that ∂π is off-diagonal, we obtain the relation π∂ j Hπ = ∂ j hπ, whence with c = Rank(π) = cte, ∂ j h = c Tr(π∂ j Hπ).

This implies |∂

This proof shows that the study of higher derivatives of the eigenvalues requires a control on the derivatives of the eigenprojectors. The following example shows that the situation may become very intricate and one can have smooth subquadratic eigenvalues while the derivatives of the projectors are unbounded.

.

Elements of symbolic calculus : the Moyal product

In this section, we revisit results about the remainder estimate for the Moyal product, aiming at their extension to the setting of the sets S µ δ (D) that we have introduced in Definition 4.1.

B.1. Formal expansion

We first recall the formal product rule for quantum observables with Weyl quantization. Let A, B ∈ S(R 2d , C m,m ). The Moyal product C := A B is the semi-classical observable C such that A • B = C. Some computations with the Fourier transform give the following well known formula [34, Theorems 18.1.8]

where σ is the symplectic bilinear form σ((q, p), (q , p )) = p • q -p • q and D = i -1 ∇. By expanding the exponential term, we obtain

So that C = j≥0 ε j C j is a formal power series in ε with coefficients given by (4.1).

B.2. Symbols with derivative bounds

For µ ≥ 0 denote by P(µ) the linear space of matrix-valued C ∞ symbols A : R 2d → C m,m such that for any γ ∈ N 2d with |γ| ≥ µ, there exists C γ > 0 such that Theorem B.1. For every N ∈ N and γ ∈ N 2d , there exists a constant K N,γ such that for any A ∈ P(µ A ), B ∈ P(µ B ) the Moyal remainder

satisfies for every z ∈ R 2d and ε ∈ (0, 1],

Elements of semi-classical calculus: perturbation of scalar systems

In this Appendix, we revisit several well-known results concerning a Hamiltonian K(t) valued in the set C m,m of m × m matrices (m ∈ N), and which is a perturbation of a scalar function k(t). We consider an interval I δ ⊂ R that may depend on δ > 0 and assume that K(t) is defined on I δ and of the form

with k scalar-valued and k(t)I m + εK 1 (t) is subquadratic on the time interval I δ according to Definition 1.1.

The difference with the classical setting is that we assume

Therefore, we have to revisit the results to take care of the loss in δ and control all the classical estimates with respect to this parameter. We denote by U ε K (t 0 , t) the unitary propagator associated with K(t).

These assumptions are those satisfied by the Hamiltonian that we consider in the adiabatic region (see Section 5.2): by (3) of Theorem 4.9, the Hamiltonians H adia,N,ε (t), defined for ∈ {1, 2} in (5.3), satisfy the assumptions made on the Hamiltonian K(t) on the interval [t 0 , t -δ] and on the interval [t + δ, t 0 + T ], for adequate domains D given by the cut-offs. Therefore, the analysis below allows to deduce Theorem 5.9 from Proposition 5.8. In the gap region, we also use these results in the simpler case δ = 1 (see Sections 5.3 and 5.4).

C.1. Egorov Theorem

The Egorov Theorem describes the evolution of an observable when it is conjugated by the propagator U ε K (t, s) associated with the operator K(t). It is important to notice that this propagator maps Σ k ε in itself for all k ∈ N. Indeed, for 1

Therefore, one deduces the L 2 -boundedness of the families (x j ψ ε (t)) and (εD xj ψ ε (t)) for all 1 ≤ j ≤ d, whence the boundedness of ψ ε (t) in Σ 1 ε for all t ∈ R. The reader will have understood that a recursive process will give the boundedness of ψ ε (t) in any Σ k ε , for t ∈ R and k ∈ N. In this setting, our aim is to revisit the evolution of U ε K (t in , t) A U ε K (t, t in ) for matrix-valued observables A ∈ S δ and in spaces Σ k ε , with a precise estimate of the remainders.

Proposition C.1. With the above notations, for any matrix-valued symbol A ∈ S δ , there exists a formal series (t, t in ) → j≥0ε j A j (t, t in ) defined on I δ × I δ and such that for any J ≥ 1, we have for all t, t in ∈ I δ , J+1+κ0) , and the matrix A 0 (t, t in ) is given by

where the unitary matrices R(t, t in , z) solve the transport equation

Remark C.2. In particular we have the propagation law of the supports:

supp(A j (t, t in )) = Φ tin,t k (supp(A)) for any j ≥ 0.

In the scalar time independent case case (k = k(z) and K j = 0 for j ≥ 1), the Egorov theorem

is well-known (see [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF][START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] for example). In the time-dependent matrix-valued case considered here, the dynamics on the observable is driven by the classical flow twisted by the precession R (see also Section 1.3.3 where such terms appear). The proof also requires a careful treatment of the time.

Proof. We perform a recursive argument. The starting point comes from the analysis of the auxiliary map defined for τ, t ∈ I δ and valued in S δ by A → A(t, τ ) := (R(τ, t)A R(τ, t) * ) • Φ t,τ k . Because we are going to differentiate in τ , we use the relation i∂ τ Φ t,τ k = -J∇k(τ, Φ t,τ k ), (where J is the matrix defined in (1.10)), which implies for all z ∈ R 2d , using also that the flow map is symplectic and preserves the Poisson bracket,

Let us know starts with the proof of the result for J = 0. We choose s, τ, t ∈ I δ and consider the quantity s). The times s, τ, t can be understood as s ≤ τ ≤ t with s an initial time (that will be taken as s = t in ) and t the time at which we want to prove the property. We then have the boundary properties

Differentiating in τ , we have

where the matrix B ε 1 ∈ S -2 ε,δ stems from the Moyal product (see Corollary B.2). We deduce by integration between the times s and t

which gives the first step of the recursive argument.

We now assume that we have obtained for J ≥ 0

. We write

. Then, the preceding equation writes

We focus on the term involving Q ε B J+1 (t, s, τ ) that we treat as in the preceding step. We obtain