# Asymptotic initial value representation of the solutions of semi-classical
systems presenting smooth codimension one crossings

Clotilde Fermanian-Kammerer  LAMA, UMR CNRS 8050, Université Paris EST, 61,
avenue du Général de Gaulle, 94010 Créteil Cedex
France<EMAIL_ADDRESS>,  Caroline Lasser  Department of
Mathematics Technische Universität München 85748 Garching bei München, Germany
<EMAIL_ADDRESS>and  Didier Robert  Laboratoire de mathématiques Jean Leray
UMR 6629 du CNRS Université de Nantes 2, rue de la Houssinière 44322 Nantes
Cedex 3, France<EMAIL_ADDRESS>This text is dedicated to the
memory of George Hagedorn (1953–2023)

###### Abstract.

This paper is devoted to the construction of approximations of the propagator
associated with a semi-classical matrix-valued Schrödinger operator with
symbol presenting smooth eigenvalues crossings. Inspired by the approach of
the theoretical chemists Herman and Kluk who propagated continuous
superpositions of Gaussian wave-packets for scalar equations, we consider
frozen and thawed Gaussian initial value representations that incorporate
classical transport and branching processes along a hopping hypersurface.
Based on the Gaussian wave-packet frame work, our result relies on an accurate
analysis of the solutions of the associated Schrödinger equation for data that
are vector-valued wave-packets. We prove that these solutions are asymptotic
to wavepackets at any order in terms of the semi-classical parameter.

###### Key words and phrases:

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

Acknowledgements. 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.

###### Contents

  1. 1 Introduction
    1. 1.1 First overview
    2. 1.2 The setting
      1. 1.2.1 The Schrödinger equation
      2. 1.2.2 Assumptions on the Hamiltonian
      3. 1.2.3 Assumptions on the data
    3. 1.3 Classical quantities
      1. 1.3.1 The flow map
      2. 1.3.2 The metaplectic transform and Gaussian states
      3. 1.3.3 Parallel transport
    4. 1.4 Thawed and frozen Gaussian approximations
      1. 1.4.1 The adiabatic situation
      2. 1.4.2 Initial value representations for codimension $1$ crossing at order $\sqrt{\varepsilon}$.
      3. 1.4.3 Hopping trajectories and branching process
      4. 1.4.4 Thawed Gaussian approximation at order $\varepsilon$
      5. 1.4.5 Frozen Gaussian approximation at order $\varepsilon$
    5. 1.5 Wave packets propagation at any order through generic smooth crossings
    6. 1.6 Detailed overview
  2. I Initial value representations
    1. 2 Frequency localized families
      1. 2.1 The Bargmann transform
      2. 2.2 Frequency localized families
        1. 2.2.1 First properties of frequency localized functions
        2. 2.2.2 Frequency localized families and Bargmann transform
        3. 2.2.3 Examples
        4. 2.2.4 Characterization of frequency localized families
        5. 2.2.5 Frequency localized families and semi-classical pseudodifferential calculus
        6. 2.2.6 Frequencies localized families and $\Sigma^{k}_{\varepsilon}$-regularity
      3. 2.3 Operators built on Bargmann transform
        1. 2.3.1 Action in $\Sigma^{k}_{\varepsilon}$ of operators built on Bargmann transform
        2. 2.3.2 Some properties of operators built on Bargmann transform via families with wave packet structure
        3. 2.3.3 Operators built on Bargmann transform via classical quantities
    2. 3 Convergence of the thawed and the frozen Gaussian approximations
      1. 3.1 Strategy of the proofs
      2. 3.2 Thawed Gaussian approximations with transfers terms
      3. 3.3 Frozen Gaussian approximations with transfers terms
  3. II Wave-packet propagation through smooth crossings
    1. 4 Symbolic calculus and diagonalization of Hamiltonians with smooth crossings
      1. 4.1 Formal asymptotic series
      2. 4.2 ‘Rough’ reduction
      3. 4.3 Superadiabatic projectors and diagonalization
        1. 4.3.1 Formal superadiabatic projectors
        2. 4.3.2 Formal adiabatic decoupling
    2. 5 Propagation of wave packets through smooth crossings
      1. 5.1 Propagation faraway from the crossing area
      2. 5.2 Propagation close to the crossing area
        1. 5.2.1 Localization up to the crossing region
        2. 5.2.2 Adiabatic decoupling close to the gap
        3. 5.2.3 Application to wave packets
      3. 5.3 Propagation through the crossing set
        1. 5.3.1 Dyson expansion
        2. 5.3.2 Analysis of the matrices $E(s,t^{\flat})$
        3. 5.3.3 Uniform estimates for the elements of the Dyson series
        4. 5.3.4 Proof of Lemma 5.12: the recursive argument
        5. 5.3.5 Proof of Lemma 5.12: the initialization of the recursion
      4. 5.4 Propagation of wave packets - Proof of Theorem 1.21
    3. A Matrix-valued Hamiltonians
    4. B Elements of symbolic calculus : the Moyal product
      1. B.1 Formal expansion
      2. B.2 Symbols with derivative bounds
    5. C Elements of semi-classical calculus: perturbation of scalar systems
      1. C.1 Egorov Theorem
      2. C.2 Asymptotic behavior of the propagator
      3. C.3 Propagation of wave packets

###  Chapter 1 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 [64, Chapter 12] or [14]). Pioneering
work about a semi-classical theory of FIOs is [7], extended in [31] and [52].

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 [46] as well as Spohn and Teufel [55], see also [17, 48,
49, 5] for earlier results or [4, 51] 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 [26, Section 5] and Jecko [35]. 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
[32, 38, 33, 39]) for the analysis of molecular dynamics for scalar equations.
The mathematical analysis has been performed later by Rousse and Swart [56]
and Robert [53], independently. The action of the Herman–Kluk propagator
consists in the continuous decomposition of the initial data into semi-
classical Gaussian wave-packets and the implementation of the propagation of
the wave-packets as studied in the 70s and 80s by Heller [32], Combescure and
Robert [13], 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 [40] or more
recently [42, 41].

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 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 [20, 21], in particular in [20] 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 $\infty$ 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 matrix-valued potentials and propagated initial
data that are known as semi-classical wave packets or generalized coherent
states [13, 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 [62] analyze the
propagation of wave-packets through a smooth crossing of Bloch bands. The
results developed here extend [26, Chapter 5] and [62] 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.

####  1.1. 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 1.19 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 wave-packets 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 ${\mathbb{C}}$ (scalar-valued),
${\mathbb{C}}^{m}$ (vector-valued) or in ${\mathbb{C}}^{m,m}$ (matrix-valued).
We denote by

$\langle g,f\rangle=\int_{{\mathbb{R}}^{d}}f(x)\overline{g}(x)dx$

the inner product of $L^{2}({\mathbb{R}}^{d},{\mathbb{C}})$. If $\pi$ is a
projector, then $\pi^{\perp}$ denotes the projector
$\pi^{\perp}=\mathbb{I}-\pi$. We set $D_{x}=\frac{1}{i}\partial_{x}$. In the
context of Assumption 1.3, we shall say that a matrix $A$ is diagonal if
$A=\pi_{1}A\pi_{1}+\pi_{2}A\pi_{2}$ and off-diagonal if
$A=\pi_{1}A\pi_{2}+\pi_{2}A\pi_{1}$.

####  1.2. The setting

#####  1.2.1. The Schrödinger equation

We consider the Schrödinger equation

(1.1)
$i\varepsilon\partial_{t}\psi^{\varepsilon}(t)=\widehat{H}^{\varepsilon}(t)\psi^{\varepsilon}(t),\;\;\psi^{\varepsilon}_{|t=t_{0}}=\psi^{\varepsilon}_{0}.$

in $L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m})$, $m\geq 2$, where
$\widehat{H}^{\varepsilon}(t)$ is the semi-classical quantization of an
Hermitian matrix symbol $H^{\varepsilon}(t,z)\in{\mathbb{C}}^{m,m}$.

Here, $t\in{\mathbb{R}}$, $z=(x,\xi)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$
and $\varepsilon$ is the semi-classical parameter, $\varepsilon\ll 1$.
Moreover, for $a\in{\mathcal{C}}^{\infty}({\mathbb{R}}^{2d})$ being a smooth
scalar-, vector- or matrix-valued function with adequate control on the growth
of derivatives, the Weyl operator $\widehat{a}={\rm op}^{w}_{\varepsilon}(a)$
is defined by

${\rm
op}^{w}_{\varepsilon}(a)f(x):=\widehat{a}f(x):=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}a\\!\left(\frac{x+y}{2},\xi\right){\rm
e}^{i\xi\cdot(x-y)/\varepsilon}f(y)\,dy\,d\xi$

for all $f\in{\mathcal{S}}({\mathbb{R}}^{d})$.

In full generality, we could assume that the map $(t,z)\mapsto
H^{\varepsilon}(t,z)$ is a semi-classical observable in the sense that the
function $H^{\varepsilon}(t,z)$ is an asymptotic sum of the form $\sum_{j\geq
0}\varepsilon^{j}H_{j}(t,z)$. 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 $j\geq 2$ only affect the solution at
order $\varepsilon$, which is the order of the approximation we are looking
for. Therefore, we assume that the self-adjoint matrix $H^{\varepsilon}$
writes

$H^{\varepsilon}(t,z):=H_{0}(t,z)+\varepsilon H_{1}(t,z).$

#####  1.2.2. Assumptions on the Hamiltonian

We work on a time interval of the form

$I:=[t_{0},t_{0}+T],\;\;t_{0}\in{\mathbb{R}}\;\;\mbox{and}\;\;T>0$

and consider subquadratic matrix-valued Hamiltonians.

######  Definition 1.1 (Subquadratic ).

The $\varepsilon$-dependent Hamiltonian

$H^{\varepsilon}=H_{0}+\varepsilon
H_{1}\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{2d},{\mathbb{C}}^{m,m})$

is subquadratic on the time interval $I$ if and only if one has the property:

(1.2) $\forall j\in\\{0,1\\},\;\;\forall\gamma\in{\mathbb{N}}^{d},\;\;\exists
C_{j,\gamma}>0,\;\;\sup_{(t,z)\in
I\times{\mathbb{R}}^{2d}}|\partial_{z}^{\gamma}H_{j}(t,z)|\leq
C_{j,\gamma}\langle z\rangle^{(2-j-|\gamma|)_{+}}$

Assuming that $H^{\varepsilon}$ is subquadratic on the time interval $I$
ensures that the system (1.1) is well-posed in
$L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m})$ for $t\in I$, and, more generally
(see [47]), in the functional spaces

$\Sigma_{\varepsilon}^{k}({\mathbb{R}}^{d})=\left\\{f\in
L^{2}({\mathbb{R}}^{d}),\;\;\forall\alpha,\beta\in{\mathbb{N}}^{d},\;\;|\alpha|+|\beta|\leq
k,\;\;x^{\alpha}(\varepsilon\partial_{x})^{\beta}f\in
L^{2}({\mathbb{R}}^{d})\right\\},\;\;k\in{\mathbb{N}}$

endowed with the norm

$\|f\|_{\Sigma^{k}_{\varepsilon}}=\sup_{|\alpha|+|\beta|\leq
k}\|x^{\alpha}(\varepsilon\partial_{x})^{\beta}f\|_{L^{2}}.$

We denote by ${\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})$ the unitary propagator
defined by

$i\varepsilon\partial_{t}\mathcal{U}^{\varepsilon}_{H}(t,t_{0})=\widehat{H}^{\varepsilon}(t)\mathcal{U}^{\varepsilon}(t,t_{0}),\;\;\mathcal{U}^{\varepsilon}(t_{0},t_{0})=\mathbb{I}_{\mathbb{C}^{m}}.$

It is a bounded operator of the $\Sigma_{\varepsilon}^{k}({\mathbb{R}}^{d})$
spaces, uniformly in $\varepsilon$ (see [47]): there exists $C_{T}>0$ such
that

$\sup_{t\in
I}\|{\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})\|_{{\mathcal{L}}(\Sigma^{k}_{\varepsilon})}\,\leq\,C_{T}.$

We assume that the principal symbol $H_{0}(t,z)$ of $H^{\varepsilon}(t,z)$ has
two distinct eigenvalues that present a smooth crossing in the sense of the
definitions of [20]. Namely, we consider different properties of the
crossings.

######  Definition 1.2.

  1. (1)

(Smooth crossing). The matrix
$H_{0}\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{2d},{\mathbb{C}}^{m,m})$
has a smooth crossing on the set $\Upsilon\subseteq I\times{\mathbb{R}}^{2d}$
if there exists $h_{1},h_{2}\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{2d})$
and two orthogonal projectors
$\pi_{1},\pi_{2}\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{2d},{\mathbb{C}}^{m,m})$
such that $H_{0}=h_{1}\pi_{1}+h_{2}\pi_{2}$ and

$h_{1}(t,z)=h_{2}(t,z)\;\Longleftrightarrow\;(t,z)\in\Upsilon.$

  2. (2)

Set
$\displaystyle{f(t,z)=\frac{1}{2}\left(h_{1}(t,z)-h_{2}(t,z)\right)\;\;\mbox{and}\;\;v(t,z)=\frac{1}{2}\left(h_{1}(t,z)+h_{2}(t,z)\right)}$.

    1. (a)

(Non-degenerate crossing). The crossing is non-degenerate at
$(t^{\flat},\zeta^{\flat})\in\Upsilon$ if

$d_{t,z}\left(H_{0}-v\,\mathbb{I}_{{\mathbb{C}}^{m}}\right)(t^{\flat},\zeta^{\flat})\not=0$

where $d_{t,z}$ is the one differential form in the variables $(t,z)$.

    2. (b)

(Generic crossing points). The crossing is generic at
$(t^{\flat},\zeta^{\flat})\in\Upsilon$ if one has

(1.3) $\partial_{t}f+\\{v,f\\}(t^{\flat},\zeta^{\flat})\neq 0.$

Note that there then exists an open set $\Omega\subset
I\times{\mathbb{R}}^{2d}$ containing $(t^{\flat},\zeta^{\flat})$ such that the
set $\Upsilon\cap\Omega$ is a manifold.

Above, we denote by $\\{f,g\\}$ the poisson bracket of the functions $f$ and
$g$ defined on ${\mathbb{R}}^{2d}_{x,\xi}$:

$\\{f,g\\}=\nabla_{\xi}f\cdot\nabla_{x}g-\nabla_{x}f\cdot\nabla_{\xi}g.$

With these definitions in hands, we introduce one of the main assumptions on
the crossing points of the Hamiltonian $H^{\varepsilon}$.

######  Assumption 1.3 (Crossing set).

The Hamiltonian $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ has a smooth
crossing set $\Upsilon$ and all the points of $\Upsilon$ are non degenerate
and generic crossing points.

In order to consider the unitary propagators ${\mathcal{U}}_{h_{1}}^{t,t_{0}}$
and $\mathcal{U}^{t,t_{0}}_{h_{2}}$ 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 $H^{\varepsilon}=H_{0}+\varepsilon
H_{1}\in\mathcal{C}^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{2d})$ be
subquadratic on the time interval $I$ and have a smooth crossing on the set
$\Upsilon$. We consider the two following assumptions :

  1. (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 $j\in\\{1,2\\}$

(1.4) $\forall\gamma\in{\mathbb{N}}^{2d},\;|\gamma|=1,\;\;\exists
C_{\gamma}>0,\;\;\forall(t,z)\in
I\times{\mathbb{R}}^{2d},\;\;|\partial_{z}^{\gamma}(H_{0}-v\,\mathbb{I}_{{\mathbb{C}}^{m}})(t,z)|+|f(t,z)|\leq
C_{\gamma}.$

  2. (ii)

The eigenvalues $h_{1}$ and $h_{2}$ are subquadratic, i.e.

(1.5) $\forall\gamma\in{\mathbb{N}}^{2d},\;|\gamma|\geq 2,\;\;\exists
C_{\gamma}>0,\;\;\forall(t,z)\in
I\times{\mathbb{R}}^{2d},\;\;\left|\partial_{z}^{\gamma}h_{j}(t,z)\right|\leq
C_{\gamma}.$

  3. (iii)

The gap is controlled at infinity, i.e. there exist $R>0$ and
$n_{0}\in{\mathbb{N}}$ such that

(1.6) $\forall t\in I,\;\;\forall|z|>R,\;\;|f(t,z)|\geq C\langle
z\rangle^{-n_{0}},$

and in the case $n_{0}\not=0$, the functions $z\mapsto\pi_{1},\pi_{2}$ are
assumed to have bounded derivatives at infinity.

######  Remark 1.5.

  1. (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
${\mathcal{U}}^{\varepsilon}_{h_{j}}(t,t_{0})$ for $j\in\\{1,2\\}$ and of the
classical quantities associated with the Hamiltonians $h_{1}$ and $h_{2}$ that
we will introduce below.

  2. (2)

The growth conditions of Assumption 1.4 imply that the eigenprojectors
$\pi_{j}(t)$, $j=1,2$, and their derivatives have at most polynomial growth.
However, when $n_{0}\not=0$, they may actually grow. This is proved in Lemma
A.4. It is for this reason that we assume that the projectors have bounded
derivatives when $n_{0}\not=0$ in Point (iii).

  3. (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
$n_{0}\not=0$.

######  Example 1.6.

  1. (1)

Examples of matrix-valued Hamiltonian are given in molecular dynamics (see
[Chapter 5] in [26]) by Schrödinger operators with matrix-valued potential,

$\widehat{H}_{S}=-\frac{\varepsilon^{2}}{2}\Delta_{x}\,\mathbb{I}_{{\mathbb{C}}^{2}}+V(x),\quad
V\in{\mathcal{C}}^{\infty}({\mathbb{R}}^{d},{\mathbb{C}}^{2\times 2}).$

When $V$ presents a codimension 1 crossing (as defined in [26], then the
crossing points $(x,\xi)$ are non degenerate and generic when $\xi\not=0$.

  2. (2)

Another class of examples appear in solid state physics in the context of
Bloch band decompositions (see [62, 12] for example) with Hamiltonians of the
form

$\widehat{H}_{A}=A(-i\varepsilon\nabla_{x})+W(x)\mathbb{I}_{{\mathbb{C}}^{2}},\quad
A\in{\mathcal{C}}^{\infty}({\mathbb{R}}^{d},{\mathbb{C}}^{2\times 2}),\quad
W\in{\mathcal{C}}^{\infty}({\mathbb{R}}^{d},{\mathbb{C}}).$

  3. (3)

Finally, in [21], the authors have considered the operator

$\widehat{H}_{k,\theta}=\frac{\varepsilon}{i}\frac{d}{dx}\mathbb{I}_{{\mathbb{C}}^{2}}+kx\begin{pmatrix}0&{\rm
e}^{i\theta x}\\\ {\rm e}^{-i\theta x}&0\end{pmatrix},$

with $d=1$, $N=2$, $\theta\in{{\mathbb{R}}}_{+}$, $k\in{\mathbb{R}}^{*}$.

#####  1.2.3. Assumptions on the data

We consider vector-valued initial data $\psi^{\varepsilon}_{0}\in
L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m})$ of the form

$\psi^{\varepsilon}_{0}=\widehat{\vec{V}}\phi^{\varepsilon}_{0}$

where $z\mapsto\vec{V}(z)$ is a smooth function, bounded together with its
derivatives and $\phi^{\varepsilon}_{0}\in
L^{2}({\mathbb{R}}^{d},{\mathbb{C}})$ is frequency localized in the sense of
the next definition. For stating it, we denote the Gaussian of expectation
$q$, variance $\sqrt{\varepsilon}$ that oscillates along $p$ according to

(1.7) $g^{\varepsilon}_{z}(x)=(\pi\varepsilon)^{-d/4}{\rm
e}^{-\frac{(x-q)^{2}}{\varepsilon}+\frac{i}{\varepsilon}p\cdot(x-q)},\;\;\forall
x\in{\mathbb{R}}^{d}.$

######  Definition 1.7 (Frequency localized functions).

Let $(\phi^{\varepsilon})_{\varepsilon>0}$ be a family of functions of
$L^{2}({\mathbb{R}}^{d})$. The family $(\phi^{\varepsilon})_{\varepsilon>0}$
is frequency localized if the family is bounded in $L^{2}({\mathbb{R}}^{d})$
and if there exist $R_{0},C_{0},\varepsilon_{0}>0$ and $N_{0}>d+\frac{1}{2}$
such that for all $\varepsilon\in(0,\varepsilon_{0}]$,

$(2\pi\varepsilon)^{-d/2}\left|\langle
g^{\varepsilon}_{z},\phi^{\varepsilon}\rangle\right|\leq C_{0}\,\langle
z\rangle^{-N_{0}}\;\;\ \text{for all}\ z\in{\mathbb{R}}^{d}\ \text{with}\
|z|>R_{0}.$

One then says that $(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency
localized.

We will introduce a more precise definition in Chapter 2. For families that
are frequency localized, the set of $z\in{\mathbb{R}}^{2d}$ 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. (1)

The Gaussian wave packets $(g^{\varepsilon}_{z_{0}})_{\varepsilon>0}$ are
frequency localized families.

  2. (2)

Define $({\rm WP}^{\varepsilon}_{z_{0}}(u))_{\varepsilon>0}$ by

(1.8) ${\rm WP}^{\varepsilon}_{z_{0}}(u)(x)=\varepsilon^{-d/4}{\rm
e}^{\frac{i}{\varepsilon}p_{0}\cdot(x-q_{0})}u\left(\frac{x-q_{0}}{\sqrt{\varepsilon}}\right),\;\;x\in{\mathbb{R}}^{d},$

for $u\in\mathcal{S}({\mathbb{R}}^{d})$ and
$z_{0}=(q_{0},p_{0})\in{\mathbb{R}}^{2d}$. They are frequency localized
families.

  3. (3)

Lagrangian (or WKB) states $\varphi^{\varepsilon}(x)=a(x){\rm
e}^{\frac{i}{\varepsilon}S(x)}$ with
$a\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{d},{\mathbb{C}})$ and
$S\in\mathcal{C}^{\infty}({\mathbb{R}}^{d},{\mathbb{R}})$, 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 $\psi^{\varepsilon}_{0}$ in (1.1) satisfies

(1.9)
$\psi^{\varepsilon}_{0}(x)=\widehat{\vec{V}}\phi^{\varepsilon}_{0}(x),\;\;x\in{\mathbb{R}}^{d}$

where

  1. (i)

The family $(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ is frequency localized
with constants $R_{0},N_{0},C_{0},\varepsilon_{0}$ in Definition 1.7.

  2. (ii)

The function $z\mapsto\vec{V}(z)$ is a function of
$\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{m})$, bounded together
with its derivatives, and valued in the set of normalized vectors.

We point out that any vector-valued bounded family in
$L^{2}({\mathbb{R}}^{d})$ writes as a sum of data of the form
$\widehat{\vec{V}}\phi^{\varepsilon}_{0}(x)$ for
$(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ bounded. As a consequence, assuming
the initial data $\psi^{\varepsilon}_{0}$ satisfies 1.9 is not really
restrictive. Of course, the vector valued function $\vec{V}$ can be turned
into $-\vec{V}$ by changing $\phi^{\varepsilon}_{0}$ into
$-\phi^{\varepsilon}_{0}$.

####  1.3. Classical quantities

In this section, we introduce classical quantities associated with the
Hamiltonian $H^{\varepsilon}$. These quantities will be used to construct the
approximations of the propagator $\mathcal{U}^{\varepsilon}_{H}(t,t_{0})$ that
are the subject of this text. They are called classical because they do not
depend on the semi-classical parameter $\varepsilon$ and are obtained by
solving $\varepsilon$-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
$\frac{1}{\varepsilon}$-oscillations and is applicable in a high-dimensional
setting, see [43] 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 $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ is
subquadratic on the time interval $I$ (as defined in Definition 1.1), with
smooth eigenprojectors $\pi_{1}$ and $\pi_{2}$, 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 $\ell\in\\{1,2\\}$, we associate with
$h_{\ell}:I\times{\mathbb{R}}^{2d}\to{\mathbb{R}}$, $(t,z)\mapsto
h_{\ell}(t,z)$ the functions

$z_{\ell}(t)=(q_{\ell}(t),p_{\ell}(t))$

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

$\dot{z}_{\ell}(t)=J\partial_{z}h_{\ell}(t,z_{\ell}(t)),\;\;z_{\ell}(t_{0})=z_{0}$

with

(1.10) $J=\begin{pmatrix}0&\mathbb{I}_{{\mathbb{R}}^{d}}\\\
-\mathbb{I}_{{\mathbb{R}}^{d}}&0\end{pmatrix}.$

We note that $J$ is the matrix associated with the symplectic form

$\sigma(z,z^{\prime})=\langle Jz,z^{\prime}\rangle=p\cdot
q^{\prime}-p^{\prime}\cdot q,\qquad
z=(q,p),\,z^{\prime}=(q^{\prime},p^{\prime})\in{\mathbb{R}}^{2d}.$

The trajectory $z_{\ell}(t)=z_{\ell}(t,t_{0},z_{0})$ depends on the initial
datum and defines the associated flow map $\Phi_{h_{\ell}}^{t,t_{0}}$ of the
Hamiltonian function $h_{\ell}$ via

$z\mapsto\Phi_{h_{\ell}}^{t,t_{0}}(z):=z_{\ell}(t,t_{0},z_{0}),\qquad
z\in{\mathbb{R}}^{2d}.$

We will also use the trajectory’s action integral

(1.11)
$S_{\ell}(t,t_{0},z_{0})=\int_{t_{0}}^{t}\left(p_{\ell}(s)\cdot\dot{q}_{\ell}(s)-h_{\ell}(s,z_{\ell}(s))\right)ds,$

and the Jacobian matrix of the flow map, also called stability matrix

(1.12) $F_{\ell}(t,t_{0},z_{0})=\partial_{z}\Phi_{h_{\ell}}^{t,t_{0}}(z_{0}).$

Note that $F_{\ell}(t,t_{0},z_{0})$ is a symplectic $2d\times 2d$ matrix, that
satisfies the linearized flow equation

(1.13) $\partial_{t}F_{\ell}(t,t_{0},z_{0})=J{\rm
Hess}_{z}h_{\ell}(t,z_{\ell}(t))\,F_{\ell}(t,t_{0},z_{0}),\;\;\;\;F_{\ell}(t_{0},t_{0},z_{0})=\mathbb{I}_{{\mathbb{R}}^{2d}}.$

We denote its blocks by

(1.14)
$F_{\ell}(t,t_{0},z_{0})=\begin{pmatrix}A_{\ell}(t,t_{0},z_{0})&B_{\ell}(t,t_{0},z_{0})\\\
C_{\ell}(t,t_{0},z_{0})&D_{\ell}(t,t_{0},z_{0})\end{pmatrix}.$

#####  1.3.2. The metaplectic transform and Gaussian states

It is standard to associate with the time-dependent symplectic map
$F_{\ell}(t,t_{0},\cdot)$ a unitary evolution operator, the metaplectic
transformation that acts on square integrable functions in
$L^{2}({\mathbb{R}}^{d})$ as a unitary transformation.

${\mathcal{M}}[F_{\ell}(t,t_{0},z_{0})]:\;u_{0}\mapsto u(t)$

and associates with an initial datum $u_{0}$ the solution at time $t$ of the
Cauchy problem

$i\partial_{t}u(t)={\rm op}^{w}_{1}\left({\rm
Hess}_{z}h_{\ell}\left(t,\Phi^{t,t_{0}}_{h_{\ell}}(z_{0})\right)z\cdot
z\right)u,\;\;u(t_{0})=u_{0}.$

This map is called the metaplectic transformation associated with the matrix
$F_{\ell}(t,t_{0},z_{0})$ (see [47]). It satisfies for all $\varepsilon>0$ and
for all symbol $a$ compactly supported or polynomial

(1.15) ${\mathcal{M}}[F_{\ell}(t,t_{0},z_{0})]^{-1}{\rm
op}_{\varepsilon}(a){\mathcal{M}}[F_{\ell}(t,t_{0},z_{0})]={\rm
op}_{\varepsilon}(F_{\ell}(t,t_{0},z_{0})z).$

All these classical quantities are involved in the description of the
propagation of Gaussian states by
${\mathcal{U}}^{\varepsilon}_{h_{\ell}}(t,t_{0})$, that are a generalization
of the Gaussian families $(g^{\varepsilon}_{z})_{\varepsilon>0}$ that we have
already seen. Gaussian states are wave packets ${\rm
WP}^{\varepsilon}_{z}(g^{\Gamma})$ with complex-valued Gaussian profiles
$g^{\Gamma}$, whose covariance matrix $\Gamma$ is taken in the Siegel half-
space ${\mathfrak{S}}^{+}(d)$ of $d\times d$ complex-valued symmetric matrices
with positive imaginary part,

${\mathfrak{S}}^{+}(d)=\left\\{\Gamma\in{\mathbb{C}}^{d\times d},\
\Gamma=\Gamma^{\tau},\ \text{\rm Im}\Gamma>0\right\\}.$

More precisely, $g^{\Gamma}$ depends on $\Gamma\in{\mathfrak{S}}^{+}(d)$
according to

(1.16) $g^{\Gamma}(x):=c_{\Gamma}\,{\rm e}^{\frac{i}{2}\Gamma x\cdot x},\quad
x\in{\mathbb{R}}^{d},$

where $c_{\Gamma}=\pi^{-d/4}{\rm det}^{1/4}(\text{\rm Im}\Gamma)$ is a
normalization constant in $L^{2}({\mathbb{R}}^{d})$. 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 $\Gamma_{0}\in{\mathfrak{S}}^{+}(d)$, we have

(1.17)
${\mathcal{M}}[F_{\ell}(t,t_{0},z_{0})]g^{\Gamma_{0}}=g^{\Gamma_{\ell}(t,t_{0},z_{0})},$

where the width $\Gamma_{\ell}(t,t_{0},z_{0})\in\mathfrak{S}^{+}(d)$ and the
corresponding normalization $c_{\Gamma_{\ell}(t,t_{0},z_{0})}$ are determined
by the initial width $\Gamma_{0}$ and the Jacobian $F_{\ell}(t,t_{0},z_{0})$
according to

(1.18) $\displaystyle\Gamma_{\ell}(t,t_{0},z_{0})$ $\displaystyle=$
$\displaystyle(C_{\ell}(t,t_{0},z_{0})+D_{\ell}(t,t_{0},z_{0})\Gamma_{0})(A_{\ell}(t,t_{0},z_{0})+B_{\ell}(t,t_{0},z_{0})\Gamma_{0})^{-1}$
$\displaystyle c_{\Gamma_{\ell}(t,t_{0},z_{0})}$ $\displaystyle=$
$\displaystyle c_{\Gamma_{0}}\,{\rm
det}^{-1/2}(A_{\ell}(t,t_{0},z_{0})+B_{\ell}(t,t_{0},z_{0})\Gamma_{0}).$

The branch of the square root in ${\rm det}^{-1/2}$ is determined by
continuity in time. Besides, the action of
$\mathcal{U}^{\varepsilon}_{h_{\ell}}(t,t_{0})$ on Gaussian wave packets ${\rm
WP}^{\varepsilon}_{z}(g^{\Gamma})$ obey to

$\mathcal{U}^{\varepsilon}_{h_{\ell}}(t,t_{0}){\rm
WP}^{\varepsilon}_{z_{0}}(g^{\Gamma_{0}})={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}_{h_{\ell}}(z_{0})}(g^{\Gamma_{\ell}(t,t_{0},z)})+O(\sqrt{\varepsilon})$

in any space $\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$ (see [13]).

#####  1.3.3. Parallel transport

For systems, the wave function is valued in
$L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m})$ and thus vector-valued. The
propagation then involves transformation of the vector part of the
eigenfunctions that is called parallel transport.

Denoting by $\pi_{\ell}^{\perp}$ the projector
$\pi_{\ell}^{\perp}=\mathbb{I}-\pi_{\ell}$, we define self-adjoint matrices
$H^{{\rm adia}}_{\ell,1}$ by

(1.19) $\displaystyle\pi_{\ell}^{\perp}\,H_{\ell,1}^{\rm
adia}\pi_{\ell}^{\perp}=0,\;\;\;\;\;\pi\,H_{\ell,1}^{\rm
adia}\pi=\pi_{\ell}\left(H_{1}+\frac{1}{2i}\\{H_{0},\pi_{\ell}\\}\right)\pi_{\ell},$
$\displaystyle\pi_{\ell}^{\perp}\,H_{\ell,1}^{\rm
adia}\pi_{\ell}=\pi_{\ell}^{\perp}\left(i\partial_{t}\pi_{\ell}+i\\{h_{\ell},\pi_{\ell}\\}\right)\pi_{\ell}.$

One then introduces the map ${\mathcal{R}}_{\ell}(t,t_{0},z)$ defined for
$\ell\in\\{1,2\\}$ by

(1.20) $i\partial_{t}{\mathcal{R}}_{\ell}(t,t_{0},z)=H^{{\rm
adia}}_{\ell,1}\left(t,\Phi_{h_{\ell}}^{t,t_{0}}(z)\right){\mathcal{R}}_{\ell}(t,t_{0},z),\;\;{\mathcal{R}}_{\ell}(t_{0},t_{0},z)=\mathbb{I}_{m}.$

The map $t\mapsto H^{{\rm
adia}}_{\ell,1}\left(t,\Phi_{h_{\ell}}^{t,t_{0}}(z)\right)$ is a locally
Lipschitz map valued in the set of self adjoint matrices. Therefore, the
existence of ${\mathcal{R}}_{\ell}(t,t_{0},z)$ comes from solving a linear
time dependent ODE by the Cauchy Lipschitz Theorem.

######  Lemma 1.10.

For all $(t,z)\in I\times{\mathbb{R}}^{2d}$ and $\ell\in\\{1,2\\}$, the
matrices ${\mathcal{R}}_{\ell}(t,t_{0},z)$ are unitary matrices. Besides, they
satisfy

(1.21)
${\mathcal{R}}_{\ell}(t,t_{0},z)\pi_{\ell}(t_{0},z)=\pi_{\ell}\left(t,\Phi^{t,t_{0}}_{\ell}(z)\right){\mathcal{R}}_{\ell}(t,t_{0},z).$

This Lemma is proved in Appendix A. The relation (1.21) implies that whenever
a vector $\vec{V}_{0}$ is in the eigenspace of $H_{0}(t_{0},z_{0})$ for the
eigenvalue $h_{\ell}(t_{0},z_{0})$, then the vector
${\mathcal{R}}_{\ell}(t,t_{0},z)\vec{V}_{0}$ is in the range of
$\pi_{\ell}(t,\Phi^{t,t_{0}}_{\ell}(z))$. In other words, we have constructed
a map that preserves the eigenspaces along the flow:

${\mathcal{R}}_{\ell}(t,t_{0},z):{\rm
Ran}\left(\pi_{\ell}(t_{0},z)\right)\mapsto{\rm
Ran}\left(\pi_{\ell}(t,\Phi^{t,t_{0}}_{\ell}(z))\right).$

The matrices ${\mathcal{R}}_{\ell}(t,t_{0},z)$ are sometimes referred to as
Larmor precession (see [13]).

The map
$\vec{V}_{0}\mapsto{\mathcal{R}}_{\ell}(t,t_{0},z)\pi_{\ell}(t_{0},z)\vec{V}_{0}$
is a parallel transport in the Hermitian vector fiber bundle $(t,z)\mapsto{\rm
Ran}(\pi_{\ell}(t,z))$ over the phase space
$I\times{\mathbb{R}}^{2d}\subset{\mathbb{R}}^{1+2d}$, associated with the
curve $s\mapsto\gamma(s)=\left(s,\Phi^{s,t_{0}}_{h_{\ell}}(z_{0})\right)_{s\in
I}$ and the matrix $H^{\rm adia}_{\ell,1}$. Indeed, the covariant derivative
along the curve $(\gamma(s))_{s\in I}$ is given by

$\nabla_{\dot{\gamma}(s)}=\partial_{t}+Jdh_{\ell}\cdot\nabla_{z}$

and the relation
$\vec{X}\left(s,\Phi^{s,t_{0}}_{h_{\ell}}(z_{0})\right)={\mathcal{R}}_{\ell}(t,t_{0},z)\pi_{\ell}(t_{0},z)\vec{V}_{0}$
defines a smooth section along the path $\gamma$ that satisfies
$\nabla_{\dot{\gamma}(s)}\vec{X}(t,x)=-iH_{\ell,1}^{\rm
adia}(t,x)\vec{X}(t,x)$.

The map $\mathcal{R}_{\ell}$, $\ell\in\\{1,2\\}$ 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 [13, Chapter 14] and [20], and Chapter 5
of this memoir.

######  Proposition 1.11 (Vector-valued wave packets).

Let $k\in{\mathbb{N}}$ and assume that $H^{\varepsilon}=H_{0}+\varepsilon
H_{1}$ satisfies Assumptions 1.3 and 1.4. Let $\ell\in\\{1,2\\}$ and
$(t_{0},z_{0})\in I\times{\mathbb{R}}^{d}$. Then, for any
$\varphi_{0}\in{\mathcal{S}}({\mathbb{R}}^{d},{\mathbb{C}})$ and
$\vec{V}_{0}\in{\rm Ran}\,\pi_{\ell}(t_{0},z_{0})$, there exists a constant
$C>0$ such that

$\sup_{t\in
J}\left\|{\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})\,{\vec{V}_{0}}\,{{\rm
WP}}^{\varepsilon}_{z_{0}}\varphi_{0}-{\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z_{0})}\,\vec{V}_{\ell}(t,t_{0})\,{{\rm
WP}}^{\varepsilon}_{\Phi_{h_{\ell}}^{t,t_{0}}(z_{0})}\varphi^{\varepsilon}(t)\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C\sqrt{\varepsilon},$

where the profile function $\varphi^{\varepsilon}(t)$ is given by

$\varphi^{\varepsilon}(t)={\mathcal{M}}[F_{\ell}(t,t_{0},z_{0})]\varphi_{0},\;\;\mbox{and}\;\;\vec{V}_{\ell}(t,t_{0},z)={\mathcal{R}}_{\ell}\left(t,t_{0},\Phi^{t,t_{0}}_{h_{\ell}}(z_{0})\right)\vec{V}_{0}.$

####  1.4. Thawed and frozen Gaussian approximations

Thawed and frozen Gaussian approximations have been introduced in the 80’s in
theoretical chemistry [33, 38, 39]. The frozen one has become popular as the
so-called Herman–Kluk approximation. They rely on the fact that the family of
wave packets $(g^{\varepsilon}_{z})_{z\in{\mathbb{R}}^{2d}}$ forms a
continuous frame and provides for all square integrable functions $f\in
L^{2}({\mathbb{R}}^{d})$ the reconstruction formula

$f(x)=(2\pi\varepsilon)^{-d}\int_{z\in{\mathbb{R}}^{2d}}\langle
g^{\varepsilon}_{z},f\rangle g^{\varepsilon}_{z}(x)dz.$

The leading idea is then to write the unitary propagation of general, square
integrable initial data $\psi^{\varepsilon}_{0}\in L^{2}({\mathbb{R}}^{d})$ as

${\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}=(2\pi\varepsilon)^{-d}\int_{z\in{\mathbb{R}}^{2d}}\langle
g^{\varepsilon}_{z},\psi^{\varepsilon}_{0}\rangle\,{\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})g^{\varepsilon}_{z}\,dz,$

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
[56, 53] and can be easily extended to the adiabatic setting (see [20]). 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 [60]) and been combined with frozen
Gaussian propagation in various instances, see for example [63, 44]. 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 $\ell\in\\{1,2\\}$, we define the first
order thawed Gaussian approximation for the $\ell$-th mode as the operator
$\mathcal{J}_{\ell,{\rm th}}^{t,t_{0}}$ defined on functions of the form
$\psi=\widehat{\vec{V}}f$, $f\in L^{2}({\mathbb{R}}^{d})$,

(1.22) $\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}(\widehat{\vec{V}}f)=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}{\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z)}{\langle}g^{\varepsilon}_{z},f{\rangle}\vec{V}_{\ell}(t,t_{0},z)g_{\Phi^{t,t_{0}}_{\ell}(t,z)}^{\Gamma_{\ell}(t,t_{0},z),\varepsilon}dz,$

with

(1.23)
$\vec{V}_{\ell}(t,t_{0},z)=\mathcal{R}_{\ell}(t,t_{0},z)\pi_{\ell}(t_{0},z)\vec{V}(z).$

This family of operators is bounded in
$\mathcal{L}\left(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})\right)$
(see Corollary 2.17).
Notice that the operator $f\mapsto\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}(\widehat{\vec{V}}f)$ has a Schwartz distribution kernel and
defines a Fourier integral operator with an explicit complex phase.

######  Theorem 1.12 (Thawed Gaussian approximation [38, 53, 56, 20]).

Assume $h_{\ell}$ is an eigenvalue of constant multiplicity of a matrix
$H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ of subquadratic growth on the time
interval $I$. Let $t_{0},T\in{\mathbb{R}}$ with $[t_{0},T]\subset I$. Then,
there exists $C_{T}>0$ such that for all $\phi^{\varepsilon}_{0}\in
L^{2}({\mathbb{R}}^{d})$,
$\vec{V}\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{m})$ bounded
with bounded derivatives, for all $t\in[t_{0},t_{0}+T]$

$\left\|\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\left(\widehat{\pi}_{\ell}(t_{0})\widehat{\vec{V}}\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}(\widehat{\vec{V}}\phi^{\varepsilon}_{0})\right\|_{L^{2}}\leq
C_{T}\,\varepsilon\,\|\phi^{\varepsilon}_{0}\|_{L^{2}}.$

######  Remark 1.13.

  1. (1)

Of course, there is no unicity of the writing $\psi=\widehat{\vec{V}}f$.
However, changing $(\vec{V},f)$ into $(k\vec{V},\frac{1}{k}f)$ for some
constant $k\in{\mathbb{C}}$ does not affect the result. One can also think to
modifying $\vec{V}$ by multiplying it by a non-vanishing function
$a\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d})$ such that $a$ and $\frac{1}{a}$
have bounded derivatives. Then, it is enough to turn $f$ into $a^{-1}f$ (see
Remark 2.23).

  2. (2)

The approach of thawed and frozen approximations that we develop in this text
allows to extend the convergence to the spaces $\Sigma^{\varepsilon}_{k}$
provided the initial data $(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ is
frequency localized and $k$ satisfies $N_{0}>k+d+\frac{1}{2}$ ($N_{0}$ being
associated to $(\phi^{\varepsilon}_{0})_{\varepsilon_{0}}$ by Definition 1.7).

As first proposed in [33], it is also possible to get rid of the time-
dependent variance matrices $\Gamma_{\ell}$ by introducing the Herman-Kluk
prefactors for the $\ell$-th modes, $a_{\ell}$, defined by

(1.24) $a_{\ell}(t,t_{0},z)=2^{-d/2}{\rm
det}^{1/2}\left(A_{\ell}(t,t_{0},z)+D_{\ell}(t,t_{0},z)+i(C_{\ell}(t,t_{0},z)-B_{\ell}(t,t_{0},z)\right).$

One then defines the first order frozen Gaussian approximation for the
$\ell$-th mode as the operator $\mathcal{J}_{\ell,{\rm fr}}^{t,t_{0}}$ defined
by

(1.25) $\mathcal{J}_{\ell,{\rm
fr}}^{t,t_{0}}(\vec{V}f)=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}{\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z)}{\langle}g^{\varepsilon}_{z},f{\rangle}a_{\ell}(t,t_{0},z)\vec{V}_{\ell}(t,t_{0},z)g_{\Phi_{1}^{t,t_{0}}(z)}^{\varepsilon}dz.$

Here again, this family of operators is bounded in
$\mathcal{L}\left(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})\right)$
(see Corollary 2.17). The next result then is a consequence of Theorem 1.12

######  Theorem 1.14 (Frozen Gaussians approximation [38, 53, 56, 20]).

Assume $h_{\ell}$ is an eigenvalue of constant multiplicity of a matrix
$H=H_{0}+\varepsilon H_{1}$ of subquadratic growth on the time interval $I$.
Let $t_{0},T\in{\mathbb{R}}$ with $[t_{0},T]\subset I$. Then, there exists
$C_{T}>0$ such that for all $\phi^{\varepsilon}_{0}\in
L^{2}({\mathbb{R}}^{d})$,
$\vec{V}\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{m})$ bounded
with bounded derivatives, and for all $t\in[t_{0},t_{0}+T]$,

$\left\|\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\left(\widehat{\pi}_{\ell}(t_{0})\widehat{\vec{V}}\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{\ell,{\rm
fr}}^{t,t_{0}}(\widehat{\vec{V}}\phi^{\varepsilon}_{0})\right\|_{L^{2}}\leq
C_{T}\,\varepsilon\,\|\phi^{\varepsilon}_{0}\|_{L^{2}}.$

The terminology thawed/frozen for these Gaussian approximations was introduced
by Heller [32] 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 [33]) by computing the kernel of the time dependent propagator.

As for Theorem 1.12, one can extend the result to an approximation in
$\Sigma^{k}_{\varepsilon}$ provided the data
$(\phi^{\varepsilon}_{0})_{\varepsilon_{0}}$ is frequency localized family and
$k\in{\mathbb{N}}$ is chosen so that $N_{0}>k+d+\frac{1}{2}$ ($N_{0}$ being
associated to $(\phi^{\varepsilon}_{0})_{\varepsilon_{0}}$ 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
$\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$, with additional assumptions on
the initial data.

#####  1.4.2. Initial value representations for codimension $1$ crossing at
order $\sqrt{\varepsilon}$.

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 $k\in{\mathbb{N}}$. Assume $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$
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
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ that satisfies
Assumptions 1.9 with frequency localization index $N_{0}>k+d+\frac{1}{2}$,
there exists $\varepsilon_{0}>0$ such that for all $t\in I$ and
$\varepsilon\in(0,\varepsilon_{0}]$, we have

$\left\|\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\left(\widehat{\pi_{\ell}(t_{0}){\vec{V}}}\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{\ell,{\rm
th}/{\rm
fr}}^{t,t_{0}}\phi^{\varepsilon}_{0}\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C_{T}\,\sqrt{\varepsilon}\,\left(\|\phi^{\varepsilon}_{0}\|_{L^{2}}+C_{0}\right).$

The remarks below also holds for Theorems 1.18, 1.19 and 1.20.

######  Remark 1.16.

  1. (1)

Of course the result also holds for initial data

$\psi^{\varepsilon}_{0}(x)=\widehat{\vec{V}}\phi^{\varepsilon}_{0}(x)+r^{\varepsilon}_{0}(x),\;\;x\in{\mathbb{R}}^{d}$

when the family $(r^{\varepsilon}_{0})_{\varepsilon>0}$ satisfies
$\|r^{\varepsilon}_{0}\|_{L^{2}({\mathbb{R}}^{d})}=O(\varepsilon)$ in
$\Sigma^{k}_{\varepsilon}$ for the index $k$ considered in the statement.

  2. (2)

The fact of being frequency localized with $N_{0}>k+d+\frac{1}{2}$ implies
that $(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ is bounded in
$\Sigma^{k}_{\varepsilon}$ (see Section 2.2.6). Thus,
$(\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0})_{\varepsilon>0}$
also is bounded in $\Sigma^{k}_{\varepsilon}$ and this space is the natural
space where studying the approximation.

  3. (3)

The control of the approximation in terms of the initial data by
$\|\phi^{\varepsilon}_{0}\|_{L^{2}}+C_{0}$ instead of
$\|\phi^{\varepsilon}_{0}\|_{\Sigma^{\varepsilon}_{k}}$ 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
$\Sigma^{\varepsilon}_{k}$-norm.

The loss of accuracy of the approximation, in $\sqrt{\varepsilon}$ instead of
$\varepsilon$, is also due to the presence of the crossing set $\Upsilon$. It
induces transitions between the modes that are exactly of order
$\sqrt{\varepsilon}$ 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 $\varepsilon$ will hold. However, if these
trajectories pass through the crossing, some additional terms of order
$\sqrt{\varepsilon}$ have to be added to obtain an approximation at order
$\varepsilon$. 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 $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ satisfies Assumptions 1.3 and
1.4 on the interval $I$. For considering initial data
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ that are frequency
localized in a compact set $K\subset B(0,R_{0})$, we are going to make
assumptions on the set $K$.

We consider sets $K$ that are connected compact subsets of ${\mathbb{R}}^{2d}$
and that do not intersect the crossing set $\Upsilon$. If one additionally
assumes that the trajectories $\Phi^{t,t_{0}}_{\ell}(z)$ issued from points
$z\in K$ intersect $\Upsilon$ on generic crossing points, then, because of
their transversality to $\Upsilon$, a given trajectory
$\Phi_{\ell}^{t,t_{0}}(z)$ issued from $z\in K$ meets $\Upsilon$ only a finite
number of times. We then denote by
$(t^{\flat}_{\ell}(t_{0},z),\zeta_{\ell}(t_{0},z))$ the first crossing point
in $\Upsilon$:

(1.26)
$\zeta_{\ell}^{\flat}(t_{0},z)=\Phi^{t^{\flat}_{\ell}(t_{0},z),t_{0}}_{\ell}(z).$

For the $\ell$-th mode and the compact $K$, we define

$t_{\ell,{\rm max}}^{\flat}(t_{0},K)=\max\\{t_{\ell}^{\flat}(t_{0},z),\;z\in
K\\}\;\;\mbox{and}\;\;t_{\ell,{\rm
min}}^{\flat}(t_{0},K)=\min\\{t_{\ell}^{\flat}(t_{0},z),\;z\in K\\}.$

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 $\Upsilon$ (if they
do) before the ones for the other mode start to reach $\Upsilon$.

######  Assumption 1.17 (Well-prepared frequency domain).

The set $K$ is a connected compact subset of ${\mathbb{R}}^{2d}$ that does not
intersect the crossing set $\Upsilon$. The trajectories
$\Phi^{t,t_{0}}_{\ell}(z)$ issued from points $z\in K$ intersect $\Upsilon$ on
generic crossing points and one has

$t_{1,{\rm max}}^{\flat}(t_{0},K)<t_{2,{\rm min}}^{\flat}(t_{0},K).$

A space-time crossing point
$(t^{\flat}_{\ell}(t_{0},z),\zeta^{\flat}_{\ell}(t_{0},z))$ is characterized
by three parameters

$\mu^{\flat}\in{\mathbb{R}},\qquad(\alpha^{\flat},\beta^{\flat})\in{\mathbb{R}}^{2d}$

given by

(1.27)
$\displaystyle\mu^{\flat}(t_{0},z)=\frac{1}{2}\left(\partial_{t}f+\\{v,f\\}\right)\left(t^{\flat}_{\ell}(t_{0},z),\zeta^{\flat}_{\ell}(t_{0},z)\right),$
(1.28)
$\displaystyle\left(\alpha^{\flat}(t_{0},z),\beta^{\flat}(t_{0},z)\right)=J\nabla_{z}f\left(t^{\flat}_{\ell}(t_{0},z),\zeta^{\flat}_{\ell}(t_{0},z)\right).$

The hopping process is affected with a transition coefficient
$\tau_{1,2}(t,t_{0},z)$ that restrict the space time variables $(t,z)$ to
trajectories that have met the crossing set $\Upsilon$

(1.29) $\tau_{1,2}(t,t_{0},z)=\mathbb{I}_{t\geq
t^{\flat}_{1}(t_{0},z)}\sqrt{\frac{2i\pi}{\mu^{\flat}(t_{0},z)}}.$

Note that when $K$ satisfies Assumption 1.17, then if $t<t^{\flat}_{1,{\rm
min}}(K)$ and $z\in K$, one has $\tau_{1,2}(t,t_{0},z)=0$. Moreover, if
$t\in\left(t^{\flat}_{1,{\rm max}}(K),t^{\flat}_{2,{\rm min}}(K)\right)$,
$z\mapsto\tau_{1,2}(t,t_{0},z)$ is smooth.

One then introduces hopping trajectories by setting

(1.30)
$\Phi_{1,2}^{t,t_{0}}(z)=\Phi_{2}^{t,t^{\flat}_{1}(t_{0},z)}\left(\zeta_{1}^{\flat}(t_{0},z)\right),\;\;t>t^{\flat}_{1}(t_{0},z).$

This trajectory $\left(\Phi_{1,2}^{t,t_{0}}(z)\right)_{t>t^{\flat}(t_{0},z)}$
is the branch of a generalized trajectory that has hopped from the mode
$\ell=1$ to the mode $\ell=2$ at the crossing point
$(t^{\flat}_{1}(t_{0},z),\zeta^{\flat}_{1}(t_{0},z))$. One could define
similarly trajectories hopping from the mode $\ell=2$ to $\ell=1$ by
exchanging the role of the indices $1$ and $2$.

Along these trajectories, one defines classical quantites as follows:

  1. (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 $t^{\flat}_{1}(t_{0},z)$ on the
mode $\ell=1$ and then on the mode $\ell=2$

(1.31)
$S_{1,2}(t,t_{0},z)=S_{1}(t^{\flat}_{1}(t_{0},z),t_{0},z)+S_{2}(t,t^{\flat}_{1}(t_{0},z),\zeta^{\flat}_{1}(t_{0},z)),$

  2. (b)

The matrix $\Gamma_{1,2}(t,t_{0},z)$ is generated according to (1.18) for the
mode $\ell=2$ along the trajectory
$\left(\Phi_{1,2}(t,t_{0},z)\right)_{t>t^{\flat}_{1}(t_{0},z)}$ starting at
time $t_{1}^{\flat}=t^{\flat}_{1}(t_{0},z)$ from the matrix

(1.32)
$\Gamma^{\flat}(t_{0},z)=\Gamma_{1}(t^{\flat}_{1},t_{0},z)-\frac{(\beta^{\flat}-\Gamma_{1}(t^{\flat}_{1},t_{0},z)\alpha^{\flat})\otimes(\beta^{\flat}-\Gamma_{1}(t^{\flat}_{1},t_{0},z))}{2\mu^{\flat}-\alpha^{\flat}\cdot\beta^{\flat}+\alpha^{\flat}\cdot\Gamma_{1}(t^{\flat}_{1},t_{0},z)\alpha^{\flat}},$

  3. (c)

The vector $\vec{V}_{1,2}(t,t_{0},z)$ is obtained by propagating the vector
$\vec{V}_{1}\left(t_{1}^{\flat},t_{0},z\right)$ for the mode $\ell=2$ along
the trajectory $\left(\Phi_{1,2}(t,t_{0},z)\right)_{t>t^{\flat}(t_{0},z)}$
starting at time $t_{1}^{\flat}$ from the vector
$\pi_{2}(t_{1}^{\flat},\zeta_{1}^{\flat})\vec{V}_{1}\left(t_{1}^{\flat},t_{0},z\right)$
with $\zeta_{1}^{\flat}=\zeta_{1}^{\flat}(t_{0},z)$. One has

(1.33)
$\vec{V}_{1,2}(t,t_{0},z)=\mathcal{R}_{2}(t,t_{1}^{\flat},\zeta_{1}^{\flat})\pi_{2}\left(t_{1}^{\flat},\zeta_{1}^{\flat}\right)\vec{V}_{1}\left(t_{1}^{\flat},t_{0},z\right).$

  4. (d)

The matrices $F_{1,2}(t,t_{0},z)$ are associated with the flow maps

(1.34)
$F_{1,2}(t,t_{0},z)=\partial_{z}\Phi^{t,t_{0}}_{1,2}(z)=\begin{pmatrix}A_{1,2}(t,t_{0},z)&B_{1,2}(t,t_{0},z)\\\
C_{1,2}(t,t_{0},z)&D_{1,2}(t,t_{0},z)\end{pmatrix}.$

  5. (e)

The transitional Herman–Kluk prefactors depend on $\Gamma_{1,2}(t,t_{0},z)$
and $\tau_{1,2}(t,t_{0},z)$ according to

(1.35) $\displaystyle a_{1,2}$ $\displaystyle=\tau_{1,2}\frac{{\rm
det}^{1/2}(C_{1,2}-iD_{1,2}-i(A_{1,2}-iB_{1,2}))}{{\rm
det}^{1/2}(C_{1,2}-iD_{1,2}-\Gamma_{1,2}(A_{1,2}-iB_{1,2}))}$
$\displaystyle=\tau_{1,2}\frac{{\rm
det}^{1/2}(A_{1,2}+D_{1,2}+i(C_{1,2}-B_{1,2}))}{{\rm
det}^{1/2}(D_{1,2}+iC_{1,2}-i\Gamma_{1,2}(A_{1,2}-iB_{1,2}))}$

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
$\sqrt{\varepsilon}$ of the thawed & frozen approximations and state our main
results.

#####  1.4.4. Thawed Gaussian approximation at order $\varepsilon$

With the notations of the preceding section, one defines the thawed Gaussian
correction term for the mode $\ell=1$ as

(1.36) $\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}(\vec{V}f)=(2\pi\varepsilon)^{-d}\int_{z\in
K}\tau_{1,2}(t,t_{0},z){\rm
e}^{\frac{i}{\varepsilon}S_{1,2}(t,t_{0},z)}\langle
g_{z}^{\varepsilon},f\rangle\vec{V}_{1,2}(t,t_{0},z)g_{\Phi_{1,2}^{t,t_{0}}(z)}^{\Gamma_{1,2}(t,t_{0},z),\varepsilon}dz$

The formula (1.36) defines a family of operators that is bounded in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$
(see Corollary 2.17). The restriction $t>t_{1}^{\flat}(t_{0},z)$ introduces a
localization of the domain of integration on one side of the hypersurface
$\\{t=t_{1}^{\flat}(t_{0},z)\\}$.

The thawed Gaussian correction term for the mode $\ell=2$, denoted by
$\mathcal{J}_{2,1,{\rm th}}^{t,t_{0}}$ 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 $\varepsilon$.

######  Theorem 1.18 (Thawed Gaussian approximation with hopping
trajectories).

Let $k\in{\mathbb{N}}$. Assume $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$
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
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ that satisfies
Assumptions 1.9 in a compact $K$ satisfying Assumption 1.17, there exists
$\varepsilon_{0}>0$ such that for all $t\in I$ we have for
$\varepsilon\in(0,\varepsilon_{0}]$,

$\displaystyle\Bigl{\|}\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}$
$\displaystyle-\mathcal{J}_{1,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{2,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{2}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)$
$\displaystyle-\sqrt{\varepsilon}\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)\Bigr{\|}_{\Sigma^{k}_{\varepsilon}}\leq
C_{T,k}\,\varepsilon\left(C_{0}+\|\phi^{\varepsilon}_{0}\|_{L^{2}}\right).$

This result emphasizes that for systems with smooth crossings, a term of order
$\sqrt{\varepsilon}$ is generated by the crossing.

#####  1.4.5. Frozen Gaussian approximation at order $\varepsilon$

In order to freeze the covariance of the Gaussians $\Gamma_{1,2}(t,t_{0},z)$
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 $\ell=1$ as

(1.37) $\mathcal{J}_{1,2,{\rm
fr}}^{t,t_{0}}(\vec{V}f)=(2\pi\varepsilon)^{-d}\int_{z\in
K}a_{1,2}(t,t_{0},z){\rm e}^{\frac{i}{\varepsilon}S_{1,2}(t,t_{0},z)}\langle
g_{z}^{\varepsilon},f\rangle\vec{V}_{1,2}(t,t_{0},z)g^{\varepsilon}_{\Phi_{1,2}^{t,t_{0}}(z)}dz.$

Notice that the map $f\mapsto\mathcal{J}_{1,2,{\rm fr}}^{t,t_{0}}(\vec{V}f)$
defines a Fourier-integral operator with a complex phase associated with the
canonical transformations $\Phi_{1,2}^{t,t_{0}}$ 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 $k\in{\mathbb{N}}$. Assume $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ 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
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ that satisfies
Assumptions 1.9 in a compact $K$ satisfying Assumption 1.17, there exists
$\varepsilon_{0}>0$ such that for all $t\in I$ satisfying

$t<t_{1,\rm min}^{\flat}(t_{0},K)\;\;\mbox{or}\;\;t^{\flat}_{1,{\rm
max}}(t_{0},K)\leq t<t^{\flat}_{2,{\rm min}}(t_{0},K),$

we have for $\varepsilon\in(0,\varepsilon_{0}]$,

$\displaystyle\biggl{\|}\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}-\mathcal{J}_{1,{\rm
fr}}^{t,t_{0}}$
$\displaystyle\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{2,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{2}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)$
$\displaystyle-\sqrt{\varepsilon}\mathcal{J}_{1,2,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)\Bigr{\|}_{\Sigma^{k}_{\varepsilon}}\leq
C_{T,K}\,\varepsilon\,\left(\|\phi^{\varepsilon}_{0}\|_{L^{2}}+C_{0}\right).$

The proof of Theorem 1.19 is based on integration by parts and requires
differentiability. When $t<t^{\flat}_{1,{\rm min}}(K)$, then the transfer
coefficient $\tau_{1,2}(t,t_{0},z)=0$ for all $z\in K$. When
$t\in[t^{\flat}_{1,{\rm max}}(K),t^{\flat}_{2,{\rm min}}(K))$, then
$z\mapsto\tau_{1,2}(t,t_{0},z)$ 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
$\ell=2$ have not yet reached $\Upsilon$.

######  Theorem 1.20 (Time averaged frozen Gaussians approximation with
hopping trajectories).

Let $k\in{\mathbb{N}}$. Assume $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ 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
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ that satisfies
Assumptions 1.9 in a compact $K$ satisfying Assumption 1.17, there exists
$\varepsilon_{0}>0$ such that for all
$\chi\in\mathcal{C}^{\infty}_{0}\left(\left(t_{0},t^{\flat}_{2,{\rm
min}}(t_{0},K)\right)\right)$,

$\displaystyle\biggl{\|}\int_{\mathbb{R}}\chi(t)\biggl{(}\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}-\mathcal{J}_{1,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)-\mathcal{J}_{2,{\rm
fr}}^{t,t_{0}}\,\left(\widehat{\pi_{2}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)$
$\displaystyle\qquad\qquad-\sqrt{\varepsilon}\mathcal{J}_{1,2,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)\biggr{)}dt\Bigr{\|}_{\Sigma^{k}_{\varepsilon}}$
$\displaystyle\qquad\qquad\qquad\qquad\leq
C_{T,k}\,\varepsilon\,\|\chi\|_{L^{\infty}}\,\left(\|\phi^{\varepsilon}_{0}\|_{L^{2}}+C_{0}\right).$

To go beyond the time $t^{\flat}_{2,{\rm min}}(t_{0},K)$, one has to consider
new transitions that would now go from the level $\ell=2$ to the level
$\ell=1$, each time a trajectory for the level $\ell=2$ hits $\Upsilon$. The
process can be understood as a random walk: each time a trajectory passes
through $\Upsilon$ a new trajectory arises on the other mode with a transition
rate of order $\sqrt{\varepsilon}$.

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
$\varepsilon^{\mu}$, for any $\mu\in{\mathbb{N}}$, and in any space
$\Sigma^{k}_{\varepsilon}$, $k\in{\mathbb{N}}$, as well before, or after,
hitting the crossing hypersurface $\Upsilon$.

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
$g^{\Gamma_{0},\varepsilon}_{z}$. The choice of some $\Gamma_{0}$ instead of
$i\mathbb{I}$ will imply a slight modification of the definition of the
Herman–Kluk prefactors $a_{\ell}$, and the transitional ones
$a_{\ell,\ell^{\prime}}$, $\ell,\ell^{\prime}\in\\{1,2\\}$.

####  1.5. 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 $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$
that satisfies Assumptions 1.4 on the time interval $I$ and presents a smooth
crossing on a set $\Upsilon$. We fix a point
$z_{0}=(q_{0},p_{0})\notin\Upsilon$ and times $t_{0},T$ such that

(1.38) $t_{0}<t^{\flat}_{1}(t_{0},z_{0})<t_{0}+T<t_{2}^{\flat}(t_{0},z_{0}).$

We assume that the point
$\Phi^{t_{1}^{\flat}(t_{0},z_{0}),t_{0}}_{h_{1}}(z_{0})$ is a non degenerate
generic crossing point (see Definition 1.2) that we denote by
$\Phi^{t_{1}^{\flat},t_{0}}_{h_{1}}(z_{0})$ for simplicity. We use the
notations introduced in Section 1.4.4 (namely equations (1.26), (1.27),
(1.28), (1.30) and (1.31)). For $\ell\in\\{1,2\\}$, we set

$z_{\ell}(t)=\Phi^{t,t_{0}}_{\ell}(z_{0}).$

######  Theorem 1.21.

Let $(t_{0},z_{0})$ satisfy (1.38), let $\psi^{\varepsilon}_{0}$ be a
polarized wave packet

(1.39) $\psi^{\varepsilon}_{0}=\vec{V}_{0}{\rm
WP}^{\varepsilon}_{z_{0}}(f_{0})\;\;\mbox{with}\;\;f_{0}\in{\mathcal{S}}({\mathbb{R}}^{d})\;\;\mbox{and}\;\;\vec{V}_{0}\in{\mathbb{C}}^{m}.$

Let $\psi^{\varepsilon}(t)$ be the solution of (1.1) with initial data
$\psi^{\varepsilon}_{0}$. There exist $\kappa_{0}\in{\mathbb{N}}$ and three
families of differential operators
$\left(\vec{B}_{\ell,j}(t)\right)_{j\in{\mathbb{N}}}$, $\ell\in\\{1,2\\}$ and
$\left(\vec{B}_{1\rightarrow 2,j}(t)\right)_{j\in{\mathbb{N}}}$ such that
setting for $\delta>0$ and $t\in
I_{\delta}=[t_{0},t_{1}^{\flat}(t_{0},z_{0})-\delta]\cup[t_{1}^{\flat}(t_{0},z_{0})+\delta,t_{0}+T]$

$\displaystyle\psi_{1}^{\varepsilon,N}(t)={\rm
e}^{\frac{i}{\varepsilon}S_{1}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{1}(t)}\left(f^{\varepsilon}_{1}(t)\right),$
$\displaystyle\psi_{2}^{\varepsilon,N}(t)={\rm
e}^{\frac{i}{\varepsilon}S_{2}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{2}(t)}\left(f^{\varepsilon}_{2}(t)\right)+{\mathbb{I}}_{t>t^{\flat}}{\rm
e}^{\frac{i}{\varepsilon}S_{1,2}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{\Phi_{1,2}(t,t_{0},z_{0})}\left(f^{\varepsilon}_{1\rightarrow
2}(t)\right),$

with

$\displaystyle f^{\varepsilon}_{\ell}(t)$
$\displaystyle={\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]\sum_{0\leq
j\leq N}\varepsilon^{j/2}\,\vec{B}_{\ell,j}(t)f_{0},\;\;\ell\in\\{1,2\\},$
$\displaystyle f^{\varepsilon}_{1\rightarrow 2}(t)$
$\displaystyle={\mathcal{R}}_{2}(t,t^{\flat}_{1})\,{\mathcal{M}}[F_{2}(t,t^{\flat}_{1})]\sum_{1\leq
j\leq N}\varepsilon^{j/2}\,\vec{B}_{1\rightarrow 2,j}(t)f_{0},$

one has the following property: for all $k,N,M\in{\mathbb{N}}$, there exists
$C_{M,N,k}>0$ such for all $t\in{\color[rgb]{0,1,1}I_{\delta}}$

$\left\|\psi^{\varepsilon}(t)-\left(\psi^{\varepsilon,N}_{1}(t)+\psi^{\varepsilon,N}_{2}(t)\right)\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C_{M,N,k}\left(\left(\frac{\sqrt{\varepsilon}}{\delta}\right)^{N+1}\delta^{-\kappa_{0}}+\delta^{M}\right).$

Moreover, the operators $\vec{B}_{\ell,j}(t)$ are differential operators of
degree $\leq 3j$ with time dependent smooth vector-valued coefficients and
satisfy for $\ell\in\\{1,2\\}$,

(1.40)
$\displaystyle\vec{B}_{\ell,0}(t)={\pi_{\ell}(t_{0},z_{0})}\vec{V}_{0}\;\;\mbox{
and}\;\;\vec{B}_{\ell,j}(t_{0})=0\;\;\forall j\geq 1,$ (1.41)
$\displaystyle\vec{B}_{\ell,1}(t)=\biggl{(}\sum_{|\alpha|=3}\frac{1}{\alpha!}\frac{1}{i}\int_{t_{0}}^{t}\partial_{z}^{\alpha}h_{\ell}(s,z_{\ell}(s))\,{\rm
op}_{1}^{w}[(F_{\ell}(s,t_{0},z_{0})z)^{\alpha}]\,ds$
$\displaystyle\qquad\qquad+\frac{1}{i}\int_{t_{0}}^{t}\nabla_{z}H^{{\rm
adia},\ell}_{1}(s,z_{s})\cdot{\rm
op}_{1}(F_{\ell}(t_{0},s)z)ds\biggr{)}{\pi_{\ell}(t_{0},z_{0})}\vec{V}_{0},$
(1.42) $\displaystyle\vec{B}_{1\rightarrow
2}(t)=W_{1}(t^{\flat}_{1},\zeta_{1}^{\flat})^{*}\,\mathcal{T}^{\flat}_{1\rightarrow
2}\,\mathcal{M}[F_{1}(t^{\flat}_{1},t_{0})]\pi_{1}(t_{0},z_{0})\vec{V}_{0}$

where the scalar transfer operator $\mathcal{T}^{\flat}_{1\rightarrow 2}$ is
defined by

(1.43) ${\mathcal{T}}^{\flat}_{1\rightarrow
2}\varphi(y)=\int_{-\infty}^{+\infty}{\rm
e}^{i(\mu^{\flat}-\alpha^{\flat}\cdot\beta^{\flat}/2)s^{2}}{\rm
e}^{is\beta^{\flat}\cdot
y}\varphi(y-s\alpha^{\flat})ds,\;\;\forall\varphi\in\mathcal{S}({\mathbb{R}}^{d})$

and the transfer matrix $W_{1}(t^{\flat},\zeta^{\flat})$ is given by

(1.44)
$W_{1}=\pi_{1}H_{1}\pi_{2}+i\pi_{1}\left(\partial_{t}\pi_{1}+\frac{1}{2}\\{h_{1}+h_{2},\pi_{1}\\}\right)\pi_{2}.$

In other words, Theorem 1.21 says that if $\psi^{\varepsilon}_{0}$ a polarized
wave packet, then, for $t\in I$, $t\not=t^{\flat}_{1}(t_{0},z_{0})$, the
solution $\psi^{\varepsilon}(t)$ of (1.1) is asymptotic at any order to an
asymptotic sum of wave packets. Indeed, if $n\in{\mathbb{N}}$ is fixed,
choosing $\delta=\varepsilon^{\alpha}$ and $M,N$ large enough will give an
approximation in $\mathcal{O}(\varepsilon^{n})$.

The polarization of the wave packets $\psi_{\ell}^{\varepsilon,N}(t)$ is first
described by the vectors $\vec{B}_{\ell,j}(t_{0})$ that evolves through
${\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]$. Such
evolution preserves the eigenmode. Secondly, in $\psi_{2}^{\varepsilon,N}(t)$,
one sees a $\sqrt{\varepsilon}$ 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 ${\rm Ran}(\pi_{1})$ to ${\rm Ran}(\pi_{2})$.
Indeed, one has

$\displaystyle W_{1}^{*}$
$\displaystyle=\pi_{2}H_{1}\pi_{1}-i\pi_{2}\left(\partial_{t}\pi_{1}+\frac{1}{2}\\{h_{1}+h_{2},\pi_{1}\\}\right)\pi_{1}$
$\displaystyle=\pi_{2}H_{1}\pi_{1}+i\pi_{2}\left(\partial_{t}\pi_{2}+\frac{1}{2}\\{h_{1}+h_{2},\pi_{2}\\}\right)\pi_{1}.$

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 [20] up to order $o(\varepsilon)$. The notations
are compatible. However, in [20], a coefficient $\gamma^{\flat}$ appears in
the definition of the transfer operator. It corresponds to a normalization
process that we avoid here by using the projector
$\pi_{2}(t^{\flat}_{1},\zeta_{1}^{\flat})$ instead of taking the scalar
product with a normalized eigenvector.

For proving the initial value representations of
$\mathcal{U}^{\varepsilon}_{H}$ 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 $\varepsilon$ of
$\mathcal{U}^{\varepsilon}_{H}\vec{V}_{0}{\rm
WP}^{\varepsilon}_{z_{0}}(g^{i\mathbb{I}})$,

  * (ii)

the exact value of the action of $\vec{B}_{1,0}$, $\vec{B}_{2,0}$, and of
$\vec{B}_{1\rightarrow 2}(t)$ when $f_{0}$ is the Gaussian $g^{i\mathbb{I}}$.

We recall that the action of the operators
${\mathcal{R}}_{\ell}(t,s)\,{\mathcal{M}}[F_{\ell}(t,s)]$ on focalized
Gaussians preserves the Gaussian structure and the focalization: in view of
(1.17) and (1.23),

${\mathcal{R}}_{\ell}(t,s)\,{\mathcal{M}}[F_{\ell}(t,s,z)]\pi_{\ell}(s,z)\vec{V}_{0}g^{i\mathbb{I}}=\vec{V}_{\ell}(t,s,z)g^{\Gamma_{\ell}(t,s,z)}$

where $\vec{V}_{\ell}\in{\rm Ran}(\pi_{\ell}(t,\Phi^{t,s}_{\ell}(z)))$ and the
matrix $\Gamma_{\ell}(t,s,z)$ is given by (1.18) with
$\Gamma_{0}=i\mathbb{I}$. Besides, regarding the transfer term, with the
notations of Corollary 3.9 of [20] and those of (1.27), (1.28), (1.32) and
(1.33)

$\displaystyle\vec{B}_{1\rightarrow
2}(t^{\flat}_{1})g^{i\mathbb{I}}=\sqrt{\frac{2i\pi}{\mu^{\flat}(t_{0},z_{0})}}\,\vec{V}_{1,2}(t_{1}^{\flat},t_{0},z_{0})g^{\Gamma^{\flat}(t_{1}^{\flat},t_{0},z_{0})}.$

These elements may enlighten the construction of the operator
$\mathcal{J}_{\ell,{\rm fr/th}}^{\ell}$ and
$\mathcal{J}_{\ell,\ell^{\prime},{\rm fr/th}}^{t,t_{0}}$ for indices
$\ell,\ell^{\prime}\in\\{1,2\\}$, $\ell\not=\ell^{\prime}$.

####  1.6. 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 $H^{\varepsilon}$ 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 [5,
46, 58] (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 I Initial value representations

###  Chapter 2 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 $\beta$).

Let $\beta\geq 0$. Let $(\phi^{\varepsilon})_{\varepsilon>0}$ be a family of
functions of $L^{2}({\mathbb{R}}^{d})$. The family
$(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency localized of order $\beta$
if the family is bounded in $L^{2}({\mathbb{R}}^{d})$ and if there exist
$R_{\beta},C_{\beta},\varepsilon_{\beta}>0$ and $N_{\beta}>d+\frac{1}{2}$ such
that for all $\varepsilon\in(0,\varepsilon_{\beta}]$,

$(2\pi\varepsilon)^{-d/2}\left|\langle
g^{\varepsilon}_{z},\phi^{\varepsilon}\rangle\right|\leq
C_{\beta}\,\varepsilon^{\beta}\,\langle z\rangle^{-N_{\beta}}\;\;\ \text{for
all}\ z\in{\mathbb{R}}^{d}\ \text{with}\ |z|>R_{\beta}.$

One then says that $(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency
localized of order $\beta$ on the ball $B(0,R_{\beta})$.

Above, $\langle g,f\rangle=\int_{{\mathbb{R}}^{d}}f(x)\overline{g}(x)dx$
denotes the inner product of $L^{2}({\mathbb{R}}^{d})$. 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.

####  2.1. 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 $\mathcal{B}$

$\mathcal{B}:\;\;L^{2}({\mathbb{R}}^{d})\ni f\mapsto\mathcal{B}[f]\in
L^{2}({\mathbb{R}}^{2d}),$

defined by

(2.1) $\mathcal{B}[f](z)=(2\pi\varepsilon)^{-\frac{d}{2}}\langle
g^{\varepsilon}_{z},f\rangle,\;\;z\in{\mathbb{R}}^{2d}.$

The Bargmann transform is an isometry and one has

$\int_{{\mathbb{R}}^{2d}}|\mathcal{B}[f](z)|^{2}dz=\|f\|^{2}_{L^{2}}.$

Indeed, the Gaussian frame identity writes

(2.2) $f(x)=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}\langle
g^{\varepsilon}_{z},f\rangle
g^{\varepsilon}_{z}(x)dz=(2\pi\varepsilon)^{-\frac{d}{2}}\int_{{\mathbb{R}}^{2d}}\mathcal{B}[f]g^{\varepsilon}_{z}(x)dz,$

where the function $g^{\varepsilon}_{z}$ is introduced in (1.7),
$g^{\varepsilon}_{z}={\rm WP}^{\varepsilon}_{z}(g^{i\mathbb{I}})$ with the
notation (1.16). Equation (2.2) is equivalent to

$f(x)=(2\pi\varepsilon)^{-\frac{d}{2}}\int_{{\mathbb{R}}^{2d}}\mathcal{B}[f](z)g^{\varepsilon}_{z}(x)dz,\;\;\forall
f\in L^{2}({\mathbb{R}}^{d}).$

More generally, the Bargmann transform characterizes the
$\Sigma^{k}_{\varepsilon}$ spaces according to the next result that we prove
in Section 2.2.6 below.

######  Lemma 2.2.

Let $k\in{\mathbb{N}}$, there exists a constant $c_{k}$ such that for all
$f\in\mathcal{S}({\mathbb{R}}^{d})$,

$\|f\|_{\Sigma^{k}_{\varepsilon}}\leq c_{k}\|\langle
z\rangle^{k}\mathcal{B}[f]\|_{L^{2}({\mathbb{R}}^{2d})}.$

The condition of spectral localization introduced in Definition 2.1 expresses
in terms of the Bargmann transform: The family
$(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency localized at the scale
$\beta\geq 0$ if there exists constants
$R_{\beta},C_{\beta},,\varepsilon_{\beta}>0$, $N_{\beta}\in{\mathbb{N}}$,
$N_{\beta}>2d$ such that for $|z|>R_{\beta}$ and
$\varepsilon\in(0,\varepsilon_{\beta}]$,

$\left|B[\phi^{\varepsilon}](z)\right|\leq
C_{\beta}\,\varepsilon^{\beta}\,\langle z\rangle^{-N_{\beta}}.$

In other words, the Bargmann transform of
$(\phi^{\varepsilon})_{\varepsilon>0}$ has polynomial decay at infinity and is
controlled by $\varepsilon^{\beta}$ outside a ball $B(0,R_{\beta})$.

The operators in which we are interested are built on the Bargmann transform.
Consider a smooth family of the form

$\left(z\mapsto\theta^{\varepsilon}_{z}\right)\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d}_{z},L^{2}({\mathbb{R}}^{d})).$

We then denote by $\mathcal{J}[\theta^{\varepsilon}_{z}]$ the operator acting
on $\phi\in L^{2}({\mathbb{R}}^{d})$ according to

(2.3)
$\mathcal{J}[\theta^{\varepsilon}_{z}](\phi)(x)=(2\pi\varepsilon)^{-\frac{d}{2}}\int_{{\mathbb{R}}^{2d}}\mathcal{B}[\phi](z)\theta^{\varepsilon}_{z}(x)dz=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}\langle
g^{\varepsilon}_{z},\phi\rangle\theta^{\varepsilon}_{z}(x)dz,\;\;x\in{\mathbb{R}}^{d}.$

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

$\mathcal{J}[g^{\varepsilon}_{z}]=\mathbb{I}_{L^{2}({\mathbb{R}}^{d})}.$

Note that the formal adjoint of ${\mathcal{J}}[\theta^{\varepsilon}_{z}]$ is

(2.4)
${\mathcal{J}}[\theta^{\varepsilon}_{z}]^{*}:\phi\mapsto(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}\langle\theta^{\varepsilon}_{z},\phi\rangle
g^{\varepsilon}_{z}dz.$

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 $f,g\in\mathcal{S}({\mathbb{R}}^{d})$ and
$z,z^{\prime}\in{\mathbb{R}}^{2d}$, then

(2.5) $\langle{\rm WP}_{z}^{\varepsilon}(f),{\rm
WP}^{\varepsilon}_{z^{\prime}}(g)\rangle={\rm
e}^{\frac{i}{\varepsilon}p^{\prime}\cdot(q-q^{\prime})}W[f,g]\left(\frac{z^{\prime}-z}{\sqrt{\varepsilon}}\right)$

where the function $W[f,g]$ is the Schwartz function on ${\mathbb{R}}^{2d}$
defined by

$W[f,g](\zeta)=\int_{{\mathbb{R}}^{d}}\overline{f}(x)g(x-q){\rm e}^{ip\cdot
x}dx,\;\;\zeta=(q,p).$

Moreover, for all $n\in{\mathbb{N}}$, there exists a constant $C_{n}>0$ such
that

(2.6)
$\forall\zeta\in{\mathbb{R}}^{2d},\;\;\langle\zeta\rangle^{n}\left|W[f,g](\zeta)\right|\leq
C_{n}\sum_{0\leq n^{\prime}\leq
n}\|f\|_{\Sigma^{n^{\prime}}}\|g\|_{\Sigma^{n-n^{\prime}}}.$

###### Proof.

The formula for $\langle{\rm WP}_{z}^{\varepsilon}(f),{\rm
WP}^{\varepsilon}_{z^{\prime}}(g)\rangle$ comes from a simple computation.
Then, for $\alpha,\gamma\in{\mathbb{N}}^{d}$ and
$z=(q,p)\in{\mathbb{R}}^{2d}$, we observe

$\displaystyle\left|q^{\gamma}p^{\alpha}W[f,g](z)\right|$
$\displaystyle=\left|q^{\gamma}\int_{{\mathbb{R}}^{d}}D_{x}^{\alpha}(\overline{f}(x)g(x-q)){\rm
e}^{ix\cdot p}dx\right|$ $\displaystyle\leq\langle
q\rangle^{|\gamma|}\int_{{\mathbb{R}}^{d}}\left|D_{x}^{\alpha}(\overline{f}(x)g(x-q))\right|dx$
$\displaystyle\leq 2^{\frac{|\gamma|}{2}}\int_{{\mathbb{R}}^{d}}\langle
x\rangle^{|\gamma|}\langle
x-q\rangle^{|\gamma|}\left|D_{x}^{\alpha}(\overline{f}(x)g(x-q))\right|dx$

where we have used Peetre inequality

(2.7) $\forall
t\in{\mathbb{R}},\;\;\forall\ell\in{\mathbb{Z}},\;\;\frac{\langle
t\rangle^{\ell}}{\langle t^{\prime}\rangle^{\ell}}\leq
2^{\frac{\ell}{2}}\langle t-t^{\prime}\rangle^{|\ell|}.$

The conclusion then follows. ∎

####  2.2. 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
$\beta\geq 0$ if and only if all its coordinates are frequency localized at
the scale $\beta$. For this reason, we focus below on scalar-valued frequency
localized families.

#####  2.2.1. 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
$L^{2}({\mathbb{R}}^{d})$. Moreover, we have the following properties:

  1. (1)

If $(\phi^{\varepsilon}_{1})_{\varepsilon>0}$ and
$(\phi^{\varepsilon}_{2})_{\varepsilon>0}$ are two frequency localized
families at the scales $\beta_{1}$ and $\beta_{2}$ respectively, then for all
$a,b\in{\mathbb{C}}$, the family
$(a\phi^{\varepsilon}_{1}+b\phi^{\varepsilon}_{2})_{\varepsilon>0}$ is
frequency localized at the scale ${\rm min}(\beta_{1},\beta_{2})$.

  2. (2)

If $(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency localized at the scale
$\beta\geq 0$, then it is also frequency localized at the scale
$\beta^{\prime}$ for all $\beta^{\prime}\in[0,\beta]$.

This notion is microlocal. Indeed, defining the $\varepsilon$-Fourier
transform by

$\mathcal{F}^{\varepsilon}f(\xi)=(2\pi\varepsilon)^{-\frac{d}{2}}\int_{{\mathbb{R}}^{d}}{\rm
e}^{\frac{i}{\varepsilon}\xi\cdot
x}\phi^{\varepsilon}(x)dx=(2\pi\varepsilon)^{-\frac{d}{2}}\widehat{f}\left(\frac{\xi}{\varepsilon}\right),\;\;f\in\mathcal{S}^{2}({\mathbb{R}}^{d}).$

######  Proposition 2.5.

Let $(\phi^{\varepsilon})_{\varepsilon>0}$ be a bounded family in
$L^{2}({\mathbb{R}}^{d})$. Then, $(\phi^{\varepsilon})_{\varepsilon>0}$ is
frequency localized family at the scale $\beta\geq 0$ if and only if
$(\mathcal{F}^{\varepsilon}\phi^{\varepsilon})_{\varepsilon>0}$ is frequency
localized at the scale $\beta\geq 0$.

###### Proof.

This comes from the observation that for all $z\in{\mathbb{R}}^{2d}$,

$\left|\langle
g^{\varepsilon}_{z},\phi^{\varepsilon}\rangle\right|=\left|\langle
g^{\varepsilon}_{Jz},\mathcal{F}^{\varepsilon}\phi^{\varepsilon}\rangle\right|$

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
$\varepsilon$-Fourier transform. ∎

#####  2.2.2. Frequency localized families and Bargmann transform

The Gaussian frame identity (2.2) allows to decompose a function of
$L^{2}({\mathbb{R}}^{d})$ 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 [42] 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 $(\phi^{\varepsilon})_{\varepsilon>0}$ be a frequency localized family at
the scale $\beta\geq 0$. Let $R_{\beta},C_{\beta}$ and $N_{\beta}$ be the
constants associated to $(\phi^{\varepsilon})_{\varepsilon>0}$ according to
Definition 2.1. Let $k\in{\mathbb{N}}$ with $N_{\beta}>d+k$. Then, for all
$\chi\in L^{\infty}({\mathbb{R}})$ supported in $[0,2]$ and equal to $1$ on
$[0,1]$, there exists $C>0$ such that for $R>R_{\beta}$,

$\left\|\phi^{\varepsilon}-\mathcal{J}\left[g^{\varepsilon}_{z}\chi\left(\frac{|z|}{R}\right)\right](\phi^{\varepsilon})\right\|_{\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})}\leq
C\,C_{\beta}\,\varepsilon^{\beta}\left(\int_{|z|>R}\langle
z\rangle^{-2(N_{\beta}-k)}{dz}\right)^{1/2}.$

In the following, we will use the notation

(2.8)
$\phi^{\varepsilon}_{R,<}:=\mathcal{J}\left[g^{\varepsilon}_{z}\chi\left(\frac{|z|}{R}\right)\right](\phi^{\varepsilon})=\mathcal{B}^{-1}\left(\mathbb{I}_{|z|<R}\mathcal{B}[\phi^{\varepsilon}](z)\right).$

######  Remark 2.7.

Lemma 2.6 can be used in different manners.

  1. (1)

If $\beta>0$, then
$\mathcal{J}[g^{\varepsilon}_{z}\chi({|z|/R})](\phi^{\varepsilon})$
approximates $\phi^{\varepsilon}$ in $L^{2}({\mathbb{R}}^{d})$ as
$\varepsilon$ goes to $0$ in any space
$\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$ with $k\in{\mathbb{N}}$ such that
$N_{\beta}>k+d+\frac{1}{2}$, and uniformly with respect to $R>R_{\beta}$.

  2. (2)

If $\beta\geq 0$ (which includes $\beta=0$), then the same approximation holds
by letting $R$ go to $+\infty$, and it is uniform with respect to
$\varepsilon$. In particular, when $\beta=0$ we have

$\limsup_{\varepsilon\rightarrow
0}\left\|\phi^{\varepsilon}-\mathcal{J}\left[g^{\varepsilon}_{z}\chi\left(\frac{|z|}{R}\right)\right](\phi^{\varepsilon})\right\|_{\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})}\leq
CR^{-(N_{\beta}-k-d-\frac{1}{2})}.$

###### Proof.

We set

$r^{\varepsilon}(x)=(2\pi\varepsilon)^{-d}\int_{|z|\geq R}\langle
g^{\varepsilon}_{z},\varphi^{\varepsilon}\rangle g^{\varepsilon}_{z}(x)dz$

and consider $k\in{\mathbb{N}}$. For $R>R_{\beta}$,
$\alpha,\gamma\in{\mathbb{N}}^{d}$ with $|\alpha|+|\gamma|=k$, we have

$\displaystyle\|x^{\alpha}(\varepsilon
D_{x})^{\gamma}r^{\varepsilon}\|^{2}_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle\;\leq(2\pi\varepsilon)^{-2d}\int_{{\mathbb{R}}^{d}}\int_{|z|>R}\int_{|z^{\prime}|>R}\langle
g^{\varepsilon}_{z},\varphi^{\varepsilon}\rangle\,\overline{\langle
g^{\varepsilon}_{z^{\prime}},\varphi^{\varepsilon}\rangle}g_{\varepsilon,z}^{\alpha,\gamma}(x)\,\overline{g_{\varepsilon,z^{\prime}}^{\alpha,\gamma}(x)}\,dx\,dz\,dz^{\prime}.$

where

(2.9) $g_{\varepsilon,z}^{\alpha,\gamma}=x^{\alpha}(\varepsilon
D_{x})^{\gamma}g^{\varepsilon}_{z}={\rm
WP}^{\varepsilon}_{z}\left((q+\sqrt{\varepsilon}y)^{\alpha}(p+\sqrt{\varepsilon}D_{y})^{\gamma}g^{i\mathbb{I}})\right),\;\;z=(q,p).$

We will use that for all $n\in{\mathbb{N}}$, there exists $c_{n,k}>0$ such
that for all $z\in{\mathbb{R}}^{2d}$

(2.10) $\|g_{\varepsilon,z}^{\alpha,\gamma}\|_{\Sigma^{n}}\leq c_{n,k}\langle
z\rangle^{k}.$

By (2.5), we obtain

$\displaystyle\|x^{\alpha}(\varepsilon
D_{x})^{\gamma}r^{\varepsilon}\|^{2}_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle\qquad\leq
C_{\beta}^{2}\,\varepsilon^{2\beta}\,(2\pi\varepsilon)^{-d}\int_{|z|>R}\int_{|z^{\prime}|>R}\langle
z\rangle^{-N_{\beta}}\langle
z^{\prime}\rangle^{-N_{\beta}}\,W[g_{\varepsilon,z}^{\alpha,\gamma},g_{\varepsilon,z^{\prime}}^{\alpha,\gamma}]\left(\frac{z-z^{\prime}}{\sqrt{\varepsilon}}\right)dz\,dz^{\prime}.$

Besides, by (2.6), there exists a constant $C^{\prime}_{n,k}$ such that

$\left|W[g_{\varepsilon,z}^{\alpha,\gamma},g_{\varepsilon,z^{\prime}}^{\alpha,\gamma}](\zeta)\right|\leq
C^{\prime}_{n,k}\,\langle\zeta\rangle^{-n}\langle z\rangle^{k}\langle
z^{\prime}\rangle^{k}.$

We deduce the existence of $c>0$ such that

$\displaystyle\|x^{\alpha}(\varepsilon
D_{x})^{\gamma}r^{\varepsilon}\|^{2}_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle\leq
c\,C_{\beta}^{2}\,\,\varepsilon^{2\beta}\,\varepsilon^{-d}\int_{|z|>R}\int_{|z^{\prime}|>R}\langle
z\rangle^{-N_{\beta}+k}\langle
z^{\prime}\rangle^{-N_{\beta}+k}\left\langle\frac{z-z^{\prime}}{\sqrt{\varepsilon}}\right\rangle^{-n}dzdz^{\prime}$
$\displaystyle\leq
c\,C_{\beta}^{2}\,\varepsilon^{2\beta}\,\int_{|z|>R}\int\langle
z\rangle^{-N_{\beta}+k}\langle
z+\sqrt{\varepsilon}\zeta\rangle^{-N_{\beta}+k}\langle\zeta\rangle^{-n}dzd\zeta.$

Since $-N_{\beta}+k\leq 0$, Peetre inequality gives

$\langle z+\sqrt{\varepsilon}\zeta\rangle^{-N_{\beta}+k}\leq
2^{\frac{N_{\beta}-k}{2}}\langle\sqrt{\varepsilon}\zeta\rangle^{N_{\beta}-k}\langle
z\rangle^{-N_{\beta}+k}\leq
2^{\frac{N_{\beta}-k}{2}}\langle\zeta\rangle^{N_{\beta}-k}\langle
z\rangle^{-N_{\beta}+k},$

by restricting ourselves to $\varepsilon\leq 1$. Therefore, there exists a
constant $c>0$ such that

$\displaystyle\|x^{\alpha}(\varepsilon
D_{x})^{\gamma}r^{\varepsilon}\|^{2}_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle\leq
c\,C_{\beta}^{2}\,\varepsilon^{2\beta}\left(\int_{{\mathbb{R}}^{2d}}\langle\zeta\rangle^{-n+N_{\beta}-k}d\zeta\right)\left(\int_{|z|>R}\langle
z\rangle^{-2(N_{\beta}-k)}dz\right)$

and we conclude the proof by choosing $n=N_{\beta}+k+2d+1$. ∎

#####  2.2.3. Examples

A first fundamental example consists in bounded families in
$L^{2}({\mathbb{R}}^{d})$ that are compactly supported.

######  Lemma 2.8.

Let $(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ be a bounded family in
$L^{2}({\mathbb{R}}^{d})$ such that
$\phi^{\varepsilon}_{0}=\mathbb{I}_{|x|\leq M}\phi^{\varepsilon}_{0}$ for some
$M>0$. Then, $(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ is frequency localised
at any scale $\beta\geq 0$.

###### Proof.

There exists a constant $C>0$ such that for all $z=(q,p)\in{\mathbb{R}}^{2d}$

$|\mathcal{B}[\phi^{\varepsilon}_{0}](z)|\leq
C\|g^{\varepsilon}_{z}\mathbb{I}_{|x|\leq M}\|_{L^{2}}.$

Besides, one can find a smooth real-valued function $\chi$ compactly supported
in $\\{|x|\leq 2\\}$ such that

$\|g^{\varepsilon}_{z}\mathbb{I}_{|x|\leq
M}\|_{L^{2}}^{2}=(\pi\varepsilon)^{-\frac{d}{2}}\int_{{\mathbb{R}}^{2d}}\chi(\frac{x}{M})\chi(\frac{y}{M})g^{\varepsilon}_{z}(x)\overline{g^{\varepsilon}_{z}}(y)dxdy.$

We set

$L=(|p|^{2}+|q-x|^{2})^{-1}(-ip+q-x)\cdot\nabla_{x}$

and we observe that $\varepsilon
Lg^{\varepsilon}_{z}(x)=g^{\varepsilon}_{z}(x)$ for all
$x\in{\mathbb{R}}^{d}.$

Besides, if $|z|>8M$, either $|q|>4M$ and if $|x|\leq 2M\leq\frac{|q|}{2}$,
then $|x-q|>\frac{|q|}{2}>4M$, or $|q|\leq 4M$ and $|p|>2M$. In any case,
$(|p|^{2}+|q-x|^{2})^{-1}>2|z|^{-1}$ and for all $N\in{\mathbb{N}}$, there
exists a constant $c_{N,M}$ such that

$\left|\left(L^{*}\right)^{N}\chi(\frac{x}{M})\right|\leq c_{N,M}|z|^{-N},$

Using the vector field $L$, we perform $N$ integration by parts, and we obtain
the existence of a constant $C>0$ such that

$\|g^{\varepsilon}_{z}\mathbb{I}_{|x|\leq M}\|_{L^{2}}^{2}\leq
C\,\varepsilon^{N-\frac{d}{2}}|z|^{-N}\int_{{\mathbb{R}}^{2d}}\mathbb{I}_{|x|,|y|\leq
M}dxdy,$

whence the boundedness of $\varepsilon^{-\beta}\langle
z\rangle^{N_{\beta}}\|g^{\varepsilon}_{z}\mathbb{I}_{|x|\leq M}\|_{L^{2}}$ for
all $\beta\geq 0$ and $N_{\beta}\in{\mathbb{N}}$. ∎

An important consequence of this result is related with the notiont notion of
compacity and of $\varepsilon$-oscillation that are often considered in semi-
classical analysis. We recall that the uniformly bounded family
$(\phi^{\varepsilon})_{\varepsilon>0}$ is said to be compact if

$\limsup_{\varepsilon\rightarrow
0}\int_{|x|>R}|\phi^{\varepsilon}(x)|^{2}dx\mathop{\longrightarrow}\limits_{R\rightarrow+\infty}0.$

It is said $\varepsilon$-oscillating when

$\limsup_{\varepsilon\rightarrow
0}\int_{|\xi|>\frac{R}{\varepsilon}}|\widehat{\phi}^{\varepsilon}(\xi)|^{2}d\xi\mathop{\longrightarrow}\limits_{R\rightarrow+\infty}0,$

or, equivalently, when its $\varepsilon$-Fourier transform
$(\mathcal{F}^{\varepsilon}\phi^{\varepsilon})_{\varepsilon>0}$ is compact.
Therefore, a compact family or an $\varepsilon$-oscillatory family can be
approached by frequency-localized families. Note however that The notion of
compacity or $\varepsilon$-oscillation is weaker than being frequency
localized. For example, the family

(2.11)
$u^{\varepsilon}(x)=|\ln\varepsilon|^{\frac{d}{2}}a(x|\ln\varepsilon|),\;\;x\in{\mathbb{R}}^{d},$

is a compact and $\varepsilon$-oscillating family which has no scale of
frequency localization.

Let us now analyze the examples given in the Introduction.

######  Lemma 2.9.

  1. (1)

Let $u\in\mathcal{S}({\mathbb{R}}^{d})$ and
$z_{0}=(q_{0},p_{0})\in{\mathbb{R}}^{2d}$. Then, the family $({\rm
WP}^{\varepsilon}_{z_{0}}(u))_{\varepsilon>0}$ is frequency localized at the
scale $\beta$ for any $\beta\geq 0$.

  2. (2)

Let $a\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{d})$ and
$S\in\mathcal{C}^{\infty}({\mathbb{R}}^{d})$. Then, the family $({\rm
e}^{\frac{i}{\varepsilon}S(x)}a)_{\varepsilon>0}$ is frequency localized at
the scale $\beta$ for any $\beta\geq 0$.

###### Proof.

1- By (2.5), we have for $z\in{\mathbb{R}}^{2d}$

$\mathcal{B}[{\rm
WP}^{\varepsilon}_{z_{0}}(u)](z)=(2\pi\varepsilon)^{-\frac{d}{2}}\langle{\rm
WP}_{z}^{\varepsilon}(g^{i\mathbb{I}}),{\rm
WP}^{\varepsilon}_{z_{0}}(u)\rangle=(2\pi\varepsilon)^{-\frac{d}{2}}\,{\rm
e}^{\frac{i}{\varepsilon}p_{0}\cdot(q-q_{0})}\,W[g^{i\mathbb{I}},u]\left(\frac{z_{0}-z}{\sqrt{\varepsilon}}\right).$

Let $N\in{\mathbb{N}}$, the estimate (2.6) implies the existence of a constant
$C_{N}^{\prime}$ such that

$|\mathcal{B}[{\rm WP}^{\varepsilon}_{z_{0}}(u)](z)|\leq
C_{N}^{\prime}\,\varepsilon^{-\frac{d}{2}}\,\left\langle\frac{z_{0}-z}{\sqrt{\varepsilon}}\right\rangle^{-N}.$

Choosing $|z|>\max(2|z_{0}|,1)$, we have $2|z_{0}-z|\geq
2(|z|-|z_{0}|)\geq|z|$ and we deduce

$\left\langle\frac{z_{0}-z}{\sqrt{\varepsilon}}\right\rangle^{-N}=\left(\frac{\varepsilon}{\varepsilon+|z-z_{0}|^{2}}\right)^{\frac{N}{2}}\leq\left(\frac{4\varepsilon}{4\varepsilon+|z|^{2}}\right)^{\frac{N}{2}}\leq{(4\varepsilon)^{\frac{N}{2}}}\,{|z|^{-N}},$

whence the existence of a constant $c_{N}>0$ such that for all
$z\in{\mathbb{R}}^{2d}$ and $N\in{\mathbb{N}}$,

$|\mathcal{B}[{\rm WP}^{\varepsilon}_{z_{0}}(u)](z)|\leq
c_{N}\,\varepsilon^{\frac{N-d-1}{2}}\,\langle z\rangle^{-N}.$

2- One has

$\displaystyle\mathcal{B}[{\rm e}^{\frac{i}{\varepsilon}S(x)}a](z)$
$\displaystyle=(2\pi)^{-d/2}\pi^{-d/4}\varepsilon^{-d/4}{\rm
e}^{\frac{i}{\varepsilon}S(q)}\int_{{\mathbb{R}}^{d}}a(q+\sqrt{\varepsilon}y)$
$\displaystyle\qquad\qquad\times{\rm
Exp}\left(-\frac{i}{\sqrt{\varepsilon}}p\cdot
y+\frac{i}{\varepsilon}S(q+y\sqrt{\varepsilon})\right){\rm
e}^{-\frac{|y|^{2}}{2}}dy.$

This term has a very specific structure involving the symbol $y\mapsto a(y)$,
a rapidly decaying function $y\mapsto{\rm e}^{-\frac{|y|^{2}}{2}}$ and an
oscillating phase

$y\mapsto\Lambda^{\varepsilon}(y):=-\frac{1}{\sqrt{\varepsilon}}p\cdot
y+\frac{1}{\varepsilon}S(q+y\sqrt{\varepsilon}).$

We are going to show that the terms defined for $j\in\\{1,\cdots,d\\}$ by

$A_{j}^{\varepsilon}:=q_{j}\int_{{\mathbb{R}}^{d}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm
e}^{i\Lambda^{\varepsilon}(y)}dy\;\;\;\mbox{and}\;\;\;B_{j}^{\varepsilon}:=p_{j}\int_{{\mathbb{R}}^{d}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy,$

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 $\varepsilon$, this
will implies the adequate control on $|\mathcal{B}[{\rm
e}^{\frac{i}{\varepsilon}S(x)}a](z)|$.

Let us first transform $A_{j}^{\varepsilon}$ and $B_{j}^{\varepsilon}$.
Indeed, we have

$\displaystyle A_{j}^{\varepsilon}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}(q_{j}+\sqrt{\varepsilon}y_{j})a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm
e}^{i\Lambda^{\varepsilon}(y)}dy-\sqrt{\varepsilon}\int_{{\mathbb{R}}^{d}}a(q+\sqrt{\varepsilon}y)\left(y_{j}{\rm
e}^{-\frac{|y|^{2}}{2}}\right){\rm e}^{i\Lambda^{\varepsilon}(y)}dy.$

The first integral of the right hand side has the same structure with the
symbol $y\mapsto a(y)$ and the second one with the rapidly decaying function
$y\mapsto y_{j}{\rm e}^{-\frac{|y|^{2}}{2}}$. Besides, observing

$p_{j}{\rm
e}^{i\Lambda^{\varepsilon}(y)}=-i\sqrt{\varepsilon}\partial_{y_{j}}({\rm
e}^{i\Lambda^{\varepsilon}(y)})-\partial_{y_{j}}S(q+\sqrt{\varepsilon}y){\rm
e}^{i\Lambda^{\varepsilon}(y)},$

we obtain with an integration by parts

$\displaystyle B_{j}^{\varepsilon}$
$\displaystyle=-\int_{{\mathbb{R}}^{d}}\partial_{y_{j}}S(q+\sqrt{\varepsilon}y)a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy$
$\displaystyle\;\;\;\;+i\sqrt{\varepsilon}\int_{{\mathbb{R}}^{d}}\partial_{y_{j}}\left(a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}\right){\rm e}^{i\Lambda^{\varepsilon}(y)}dy.$

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) $L^{\varepsilon}:=\int_{{\mathbb{R}}^{d}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy$

is of order $\varepsilon^{N}$ for all $N\in{\mathbb{N}}$. The decay of
$y\mapsto{\rm e}^{-\frac{|y|^{2}}{2}}$ allows to reduce the set of
integration. Indeed, we have

$\left|\int_{|y|>\varepsilon^{-\frac{1}{4}}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy\right|\leq{\rm
e}^{-\frac{\sqrt{\varepsilon}}{4}}\|a\|_{L^{\infty}}\int_{{\mathbb{R}}^{d}}{\rm
e}^{-\frac{|y|^{2}}{4}}dy.$

Therefore, there exists a constant $c>0$ such that

$|L^{\varepsilon}|\leq
c\left(\left|\int_{|y|\leq\varepsilon^{-\frac{1}{4}}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy\right|+{\rm
e}^{-\frac{1}{4\sqrt{\varepsilon}}}\right).$

We now use the oscillations of the phase for treating the integral in
$|y|\leq\varepsilon^{-\frac{1}{4}}$. We observe that there exists $R_{0}>0$
such that if $|z|>R_{0}$, then

$z\notin\\{|p-\nabla S(q)|\leq 1,\;{\rm dist}(q,{\rm supp}(a))\leq 1\\}.$

We choose $|z|>R_{0}$ and we have the following alternative:

$\mbox{either}\;{\rm dist}(q,{\rm supp}(a))>1,\;\mbox{ or}\;\left({\rm
dist}(q,{\rm supp}(a))\leq 1\;\mbox{ and}\;|p-\nabla S(q)|>1\right).$

If ${\rm dist}(q,{\rm supp}(a))>1$, there exists $\varepsilon_{0}>0$ such that
if $\varepsilon\in(0,\varepsilon_{0}]$ and $|y|\leq\varepsilon^{1/4}$, then
$q+\sqrt{\varepsilon}y\notin{\rm supp}(a)$. The integral thus is zero and we
are reduced to the case where ${\rm dist}(q,{\rm supp}(a))\leq 1$ and
$|p-\nabla S(q)|>1$. One can find $\varepsilon_{1}>0$ such that for
$\varepsilon\in(0,\varepsilon_{1}]$ and $|y|\leq\varepsilon^{\frac{1}{4}}$,

$\left|\nabla S(q+y\sqrt{\varepsilon})-p\right|>\frac{1}{2}.$

We then consider the differential operator

$L^{\varepsilon}=\sqrt{\varepsilon}\frac{\nabla
S(q+y\sqrt{\varepsilon})-p}{|\nabla
S(q+y\sqrt{\varepsilon})-p|^{2}}\cdot\nabla_{y}$

and we write

$\displaystyle\int_{|y|\leq\varepsilon^{-\frac{1}{4}}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy$
$\displaystyle=\int_{|y|\leq\varepsilon^{-\frac{1}{4}}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}(L^{\varepsilon})^{N}\left({\rm
e}^{i\Lambda^{\varepsilon}(y)}\right)dy$
$\displaystyle=\int_{|y|\leq\varepsilon^{-\frac{1}{4}}}{(L^{\varepsilon})^{N}}^{*}\left(a(q+\sqrt{\varepsilon}y)|\nabla
S(q+y\sqrt{\varepsilon})-p|^{-2N}{\rm e}^{-\frac{|y|^{2}}{2}}\right){\rm
e}^{i\Lambda^{\varepsilon}(y)}dy.$

There exists a constant $C>0$ independent of $z$ such that for all
$\varepsilon\in(0,\varepsilon_{1}]$ and $|y|\leq\varepsilon^{\frac{1}{4}}$,

$\left|{(L^{\varepsilon})^{N}}^{*}\left(a(q+\sqrt{\varepsilon}y)|\nabla
S(q+y\sqrt{\varepsilon})-p|^{-2N}{\rm
e}^{-\frac{|y|^{2}}{2}}\right)\right|\leq C\varepsilon^{\frac{N}{2}}{\rm
e}^{-\frac{|y|^{2}}{2}}.$

We deduce

$\displaystyle\left|\int_{|y|\leq\varepsilon^{-\frac{1}{4}}}a(q+\sqrt{\varepsilon}y){\rm
e}^{-\frac{|y|^{2}}{2}}{\rm e}^{i\Lambda^{\varepsilon}(y)}dy\right|\leq
C\varepsilon^{\frac{N}{2}}$

and (2.12) writes

$|L^{\varepsilon}|\leq c\left(\varepsilon^{\frac{N}{2}}+{\rm
e}^{-\frac{1}{4\sqrt{\varepsilon}}}\right)$

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 $(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency localized at
the scale $\beta\geq 0$ if and only if for all
$\mathcal{C}^{1}$-diffeomorphism $\Phi$ satisfying

$\exists a,b>0,\;\;\forall z\in{\mathbb{R}}^{2d},\;\;a|z|\leq\Phi(z)\leq
b|z|,$

for all $\theta\in\mathcal{S}({\mathbb{R}}^{d})$, there exists $C_{\beta}$,
$N_{\beta}$, $R_{\beta}$ and $\varepsilon_{\beta}$ such that for all
$\varepsilon\in(0,\varepsilon_{\beta}]$ and for $|z|>R_{\beta}$

$(2\pi\varepsilon)^{-\frac{d}{2}}|\langle{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta),\phi^{\varepsilon}\rangle|\leq
C_{\beta}\,\varepsilon^{\beta}\,\langle
z\rangle^{-N_{\beta}}\,\max\left(1,\frac{1}{a}\right)^{N_{\beta}}\,\|\theta\|_{\Sigma^{2d+1+N_{\beta}}}.$

Moreover, for all family $(\lambda^{\varepsilon})_{\varepsilon>0}$ bounded in
$L^{\infty}({\mathbb{R}}^{2d})$, $\varepsilon\in(0,\varepsilon_{\beta}]$ and
$R>R_{\beta}$,

$\left\|\mathcal{J}\left[{\rm
e}^{\frac{i}{\varepsilon}\lambda^{\varepsilon}(z)}{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta)\mathbb{I}_{|z|>R}\right](\phi^{\varepsilon})\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C\,C_{\beta}\,\varepsilon^{\beta}\left(\int_{|z|>R}\langle
z\rangle^{-N_{\beta}}dz\right).$

###### Proof.

We only have to prove that if $(\phi^{\varepsilon})_{\varepsilon>0}$ is
frequency localized at the scale $\beta\geq 0$, then the property holds for
some given profile $\theta$ and diffeomorhism $\Phi$. Then, the equivalence
will follow. We consider the constants $C_{\beta}$, $N_{\beta}$, $R_{\beta}$
and $\varepsilon_{\beta}$ given by Definition 2.1 and we take
$\varepsilon\in(0,\varepsilon_{\beta}]$. We observe

$\displaystyle(2\pi\varepsilon)^{-\frac{d}{2}}\langle{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta),\phi^{\varepsilon}\rangle$
$\displaystyle=(2\pi\varepsilon)^{-\frac{3d}{2}}\int_{{\mathbb{R}}^{d}}\langle{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta),g^{\varepsilon}_{z^{\prime}}\rangle\langle
g^{\varepsilon}_{z^{\prime}},\phi^{\varepsilon}\rangle
dz^{\prime}=I_{1}+I_{2}$

with

$I_{1}=(2\pi\varepsilon)^{-\frac{3d}{2}}\int_{|z^{\prime}|>R_{\beta}}\langle{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta),g^{\varepsilon}_{z^{\prime}}\rangle\langle
g^{\varepsilon}_{z^{\prime}},\phi^{\varepsilon}\rangle dz^{\prime}.$

Let us study $I_{1}$. Using (2.5), (2.6) and that
$(\phi^{\varepsilon})_{\varepsilon>0}$ is frequency localized at the scale
$\beta\geq 0$, we deduce the existence of $c_{\beta},N_{\beta}>0$ such that we
have

$\displaystyle|I_{1}|$ $\displaystyle\leq
c_{\beta}\,\varepsilon^{\beta}\,(2\pi\varepsilon)^{-d}\int_{|z^{\prime}|>R_{\beta}}\left|W[\theta,g^{i\mathbb{I}}]\left(\frac{z^{\prime}-\Phi(z)}{\sqrt{\varepsilon}}\right)\right|\langle
z^{\prime}\rangle^{-N_{\beta}}dz^{\prime}$ $\displaystyle\leq
c_{\beta}\,\|\theta\|_{\Sigma^{n}}\,\varepsilon^{\beta}\,(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{d}}\left\langle\frac{z^{\prime}-\Phi(z)}{\sqrt{\varepsilon}}\right\rangle^{-n}\langle
z^{\prime}\rangle^{-N_{\beta}}dz^{\prime}$ $\displaystyle\leq
c_{\beta}\,\|\theta\|_{\Sigma^{n}}\,\varepsilon^{\beta}\,\int_{{\mathbb{R}}^{d}}\left\langle\zeta\right\rangle^{-n}\langle\Phi(z)+\sqrt{\varepsilon}\zeta\rangle^{-N_{\beta}}d\zeta,$

where the constant $c_{\beta}$ may have changed between two successive lines.
We observe that Peetre’s inequality (2.7) yields

$\langle\Phi(z)+\sqrt{\varepsilon}\zeta\rangle^{-N_{\beta}}\leq
2^{\frac{N_{\beta}}{2}}\langle\Phi(z)\rangle^{-N_{\beta}}\langle\sqrt{\varepsilon}\zeta\rangle^{N_{\beta}}\leq
2^{\frac{N_{\beta}}{2}}\langle\Phi(z)\rangle^{-N_{\beta}}\langle\zeta\rangle^{N_{\beta}},$

whence by choosing $n>2d+1+N_{\beta}$,

$\displaystyle|I_{1}|$ $\displaystyle\leq
c_{\beta}\,\|\theta\|_{\Sigma^{2d+1+N_{\beta}}}\,\varepsilon^{\beta}\,\langle\Phi(z)\rangle^{-N_{\beta}}\int_{{\mathbb{R}}^{d}}\left\langle\zeta\right\rangle^{-(2d+1)}d\zeta,$

for some new constant $c_{\beta}>0$. We conclude by observing that

$\langle\Phi(z)\rangle^{-N_{\beta}}\leq\max\left(1,\frac{1}{a}\right)^{N_{\beta}}\langle
z\rangle^{-N_{\beta}},$

whence, by modifying $c_{\beta}$,

$|I_{1}|\leq
c_{\beta}\,\|\theta\|_{\Sigma^{2d+1+N_{\beta}}}\,\varepsilon^{\beta}\,\max\left(1,\frac{1}{a}\right)^{N_{\beta}}\langle
z\rangle^{-N_{\beta}}.$

We now study $I_{2}$. Using (2.5), (2.6), we write for $n\in{\mathbb{N}}$

$|I_{2}|\leq\|\phi^{\varepsilon}\|_{L^{2}}(2\pi\varepsilon)^{-\frac{3d}{2}}\int_{|z^{\prime}|\leq
R_{\beta}}\left\langle\frac{z^{\prime}-\Phi(z)}{\sqrt{\varepsilon}}\right\rangle^{-n}dz^{\prime}.$

We observe that if $|z|>2aR_{\beta}$, then for $|z^{\prime}|\leq
R_{\beta}\leq\frac{1}{2a}|z|$, we have

$|z^{\prime}-\Phi(z)|\geq|\Phi(z)|-|z^{\prime}|\geq\frac{1}{2a}|z|.$

Therefore

$\left\langle\frac{z^{\prime}-\Phi(z)}{\sqrt{\varepsilon}}\right\rangle^{-n}=\left(\frac{\varepsilon}{\varepsilon+|z^{\prime}-\Phi(z)|^{2}}\right)^{\frac{n}{2}}\leq{(2a)^{n}}{\varepsilon^{\frac{n}{2}}}|z|^{-n}.$

Using that $(\phi^{\varepsilon})_{\varepsilon>0}$ is a bounded family in
$L^{2}$, we obtain that there exists a constant $c^{\prime}$ such that for
$|z|>2aR_{\beta}$ and any $n\in{\mathbb{N}}$,

$\displaystyle|I_{2}|$ $\displaystyle\leq
c^{\prime}\varepsilon^{\frac{n-3d}{2}}|z|^{-n}.$

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 $(\phi^{\varepsilon})_{\varepsilon>0}$ be a frequency localized family at
the scale $\beta\geq 0$.

  1. (1)

For all semi-classical symbol
$a\in\mathcal{C}^{\infty}_{c}({\mathbb{R}}^{2d})$, the family
$\bigl{(}\widehat{a}\,\phi^{\varepsilon}\bigr{)}_{\varepsilon>0}$ is frequency
localized at the scale $\beta\geq 0$.

  2. (2)

For all subquadratic Hamiltonian
$h\in\mathcal{C}^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{2d})$, for all
$t,t_{0}\in{\mathbb{R}}$, the vector-valued family
$\bigl{(}\mathcal{U}^{\varepsilon}_{h}(t,t_{0})\phi^{\varepsilon}\bigr{)}_{\varepsilon>0}$
is frequency localized at the scale $\beta\geq 0$.

###### Proof.

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

$\mathcal{B}[\widehat{a}\phi^{\varepsilon}]=(2\pi\varepsilon)^{-d/2}\langle\widehat{a}g^{\varepsilon}_{z},\phi^{\varepsilon}\rangle.$

Since $g^{\varepsilon}_{z}$ is a wave packet, we have

$\widehat{a}g^{\varepsilon}_{z}=\widehat{a}\,{\rm
WP}^{\varepsilon}_{z}(g^{i\mathbb{I}})={\rm
WP}^{\varepsilon}_{z}(\mathfrak{g}^{\varepsilon}_{a}),\;\;\mathfrak{g}^{\varepsilon}_{a}=\widehat{a(z+\sqrt{\varepsilon}\cdot)}g^{i\mathbb{I}}.$

The function $\mathfrak{g}^{\varepsilon}_{a}$ is of Schwartz class on
${\mathbb{R}}^{d}$ and its Schwartz semi-norms are uniformly bounded in
$\varepsilon$ because $a$ is compactly supported. We deduce from Proposition
2.10,

$\left|\mathcal{B}[\widehat{a}\phi^{\varepsilon}]\right|\leq
C_{\beta}\,\varepsilon^{\beta}\,\langle
z\rangle^{-N_{\beta}}\|\mathfrak{g}^{\varepsilon}_{a}\|_{2d+1+N_{\beta}},$

which concludes the proof.

(2) We write

$\mathcal{B}\left[\mathcal{U}^{\varepsilon}_{h}(t,t_{0})\phi^{\varepsilon}\right]=(2\pi\varepsilon)^{-d/2}\langle\mathcal{U}^{\varepsilon}_{h}(t,t_{0})g^{\varepsilon}_{z},\phi^{\varepsilon}\rangle.$

Since $g^{\varepsilon}_{z}$ is a wave packet, we have

(2.13) $\mathcal{U}^{\varepsilon}_{h}(t,t_{0})g^{\varepsilon}_{z}={\rm
e}^{\frac{i}{\varepsilon}S(-t,z)}{\rm
WP}^{\varepsilon}_{\Phi^{-t,0}_{h}(z)}(g^{\Gamma(-t,z)}+\sqrt{\varepsilon}\,r^{\varepsilon}_{z}(t))$

with the notations of the introduction. Besides, for all $n\in{\mathbb{N}}$,
there exists a constant $C=C(n,t)$ such that
$\|r^{\varepsilon}_{z}(t)\|_{\Sigma^{n}}\leq C(n,t)$. We deduce from
Proposition 2.10,

$\left|\mathcal{B}\left[\mathcal{U}^{\varepsilon}_{h}(t,t_{0})\phi^{\varepsilon}\right]\right|\leq
C_{\beta}\,\varepsilon^{\beta}\,\langle
z\rangle^{-N_{\beta}}\|g^{\Gamma(t,z)}+\sqrt{\varepsilon}r^{\varepsilon}_{z}(t)\|_{2d+1+N_{\beta}},$

which concludes the proof. ∎

######  Remark 2.12.

  1. (1)

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

$\exists N_{0}\in{\mathbb{N}},\;\;\forall\gamma\in{\mathbb{N}}^{d},\;\;\forall
z\in{\mathbb{R}}^{2d},\;\;\left|\partial^{\gamma}a(z)\right|\leq\langle
z\rangle^{N_{0}-|\gamma|}$

provided the integer $N_{\beta}$ associated with the frequency localisation at
the scale $\beta\geq 0$ of the family
$(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ verifies $N_{\beta}>2d+1+N_{0}$.

  2. (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 [18]. Indeed, even though one knows that
$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})(g^{\varepsilon}_{z}\vec{V})$ 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
$\Sigma^{k}_{\varepsilon}$-regularity

The size of $N_{\beta}$ in Definition 2.1 gives an information about the
regularity of the family.

######  Lemma 2.13.

Let $(\phi^{\varepsilon})_{\varepsilon>0}$ a frequency localized family at the
scale $\beta\geq 0$, let $C_{\beta},N_{\beta},\varepsilon_{\beta}$ are the
constants associated by Definition 2.1. Assume such that $k\in{\mathbb{N}}$ is
such that $N_{\beta}>d+k+\frac{1}{2}$, then
$(\phi^{\varepsilon})_{0<\varepsilon<\varepsilon_{\beta}}$ is uniformly
bounded in $\Sigma^{k}_{\varepsilon}$ and there exists $c>0$ independent of
$\varepsilon$

$\|\phi^{\varepsilon}\|_{\Sigma^{k}_{\varepsilon}}\leq
c(C_{\beta}+\|\phi^{\varepsilon}_{0}\|_{L^{2}}).$

This Lemma is a simple consequence of Lemma 2.2 that we are going to prove
now.

###### Proof of Lemma 2.2.

Let $k\in{\mathbb{N}}$ and $\alpha,\gamma\in{\mathbb{N}}^{d}$ such that
$|\alpha|+|\gamma|=k$. we consider the operator

$T_{\alpha,\gamma}=\mathcal{B}\circ(x^{\alpha}(\varepsilon
D_{x})^{\gamma})\circ\mathcal{B}^{-1}\circ\langle z\rangle^{-k}.$

The kernel of this operator is the function

${\mathbb{R}}^{4d}\ni(X,Y)\mapsto
k^{\varepsilon}(X,Y)=(2\pi\varepsilon)^{-d}\langle
g^{\varepsilon}_{X},x^{\alpha}(\varepsilon
D_{x})^{\gamma}g^{\varepsilon}_{Y}\rangle\langle Y\rangle^{-k}.$

Therefore, by (2.5), there exists a constant $c_{k}$ such that

$\int_{Y\in{\mathbb{R}}^{2d}}\sup_{X\in{\mathbb{R}}^{2d}}|k^{\varepsilon}(X,Y)|dY=(2\pi\varepsilon)^{-d}\int_{Y\in{\mathbb{R}}^{2d}}\sup_{X\in{\mathbb{R}}^{2d}}\left|W[g_{i{\mathbb{I}}},g^{\alpha,\gamma}_{\varepsilon,Y}]\left(\frac{X-Y}{\sqrt{\varepsilon}}\right)\right|\langle
Y\rangle^{-k}dY.$

We deduce from (2.6) and (2.5) the existence of a constant $c_{k}>0$ such that

$\int_{Y\in{\mathbb{R}}^{2d}}\sup_{X\in{\mathbb{R}}^{2d}}|k^{\varepsilon}(X,Y)|dY\leq
c_{k}.$

Similarly, we have

$\displaystyle\int_{X\in{\mathbb{R}}^{2d}}\sup_{Y\in{\mathbb{R}}^{2d}}|k^{\varepsilon}(X,Y)|dX$
$\displaystyle=(2\pi\varepsilon)^{-d}\int_{X\in{\mathbb{R}}^{2d}}\sup_{Y\in{\mathbb{R}}^{2d}}\left|W[g_{i{\mathbb{I}}},g^{\alpha,\gamma}_{\varepsilon,Y}]\left(\frac{X-Y}{\sqrt{\varepsilon}}\right)\right|\langle
Y\rangle^{-k}dX\leq c_{k}.$

Therefore, the Schur test yields the boundedness of $T_{\alpha,\gamma}$. One
then deduces that for $f\in\mathcal{S}({\mathbb{R}}^{d})$, one has

$\displaystyle\|x^{\alpha}(\varepsilon
D_{x})^{\gamma}f\|_{L^{2}({\mathbb{R}}^{d})}$
$\displaystyle=\|\mathcal{B}[x^{\alpha}(\varepsilon
D_{x})^{\gamma}f\|_{L^{2}({\mathbb{R}}^{2d})}$
$\displaystyle=\left\|T^{\alpha,\gamma}\left(\langle
z\rangle^{k}\mathcal{B}[f]\right)\right\|_{L^{2}({\mathbb{R}}^{2d})}\leq
c_{k}\|\langle z\rangle^{k}\mathcal{B}[f]\|_{L^{2}({\mathbb{R}}^{2d})},$

which concludes the proof. ∎

Let us now prove Lemma 2.13

###### Proof of Lemma 2.13.

Since $N_{\beta}>d+k$, we have for $|z|>R_{\beta}$,

$\langle
z\rangle^{k}|\mathcal{B}[\phi^{\varepsilon}](z)|\leq\varepsilon^{\beta}C_{\beta}\langle
z\rangle^{-N_{\beta}+k}\in L^{2}({\mathbb{R}}^{2d}).$

Moreover, $z\mapsto\langle
z\rangle^{2k}|\mathcal{B}[\phi^{\varepsilon}](z)|^{2}$ is locally integrable
and we can write

$\displaystyle\|\langle
z\rangle^{k}\mathcal{B}[\phi^{\varepsilon}](z)\|_{L^{2}({\mathbb{R}}^{2d})}^{2}$
$\displaystyle\leq\langle R_{\beta}\rangle^{2k}\int_{|z|\leq
R_{\beta}}|\mathcal{B}[\phi^{\varepsilon}](z)|^{2}dz+\varepsilon^{2\beta}C_{\beta}^{2}\int_{{\mathbb{R}}^{2d}}\langle
z\rangle^{-2(N_{\beta}-k)}dz$ $\displaystyle\leq\langle
R_{\beta}\rangle^{2k}\|\mathcal{B}[\phi^{\varepsilon}](z)\|^{2}_{L^{2}({\mathbb{R}}^{2d}}+\varepsilon^{2\beta}C_{\beta}^{2}\int_{{\mathbb{R}}^{2d}}\langle
z\rangle^{-2(N_{\beta}-k)}dz$ $\displaystyle\leq\langle
R_{\beta}\rangle^{2k}\|\phi^{\varepsilon}\|^{2}_{L^{2}({\mathbb{R}}^{d})}+\varepsilon^{2\beta}C_{\beta}^{2}\int_{{\mathbb{R}}^{2d}}\langle
z\rangle^{-2(N_{\beta}-k)}dz,$

whence the conclusion since the right hand side is bounded for
$2(N_{\beta}-k)-2d>1$. ∎

####  2.3. Operators built on Bargmann transform

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

  1. (a)

The case where the family $(\theta^{\varepsilon}_{z})_{\varepsilon>0}$ is only
uniformly bounded in $L^{2}({\mathbb{R}}^{d})$, which is a light assumption,
but with uniform bounds in $z$ on adequate semi-norms or norms.

  2. (b)

The case where the family $(\theta^{\varepsilon}_{z})_{\varepsilon>0}$ 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 $(\theta^{\varepsilon}_{z})_{\varepsilon>0}$ 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
$\Sigma^{k}_{\varepsilon}$ 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.

#####  2.3.1. Action in $\Sigma^{k}_{\varepsilon}$ of operators built on
Bargmann transform

This section is devoted to the proof of the following result.

######  Theorem 2.14.

Let $\varepsilon_{0}>0$.

  1. (1)

Let $R>0$. There exists $c_{0}>0$ such that for all measurable $z$-dependent
family $(\theta^{\varepsilon}_{z})_{\varepsilon>0}$, for all
$k\in{\mathbb{N}}$, $\varepsilon\in(0,\varepsilon_{0}]$, for all $\phi\in
L^{2}({\mathbb{R}}^{d})$

$\|\mathcal{J}[\theta^{\varepsilon}_{z}\mathbb{I}_{|z|<R}](\phi)\|_{\Sigma^{k}_{\varepsilon}}\leq(2\pi\varepsilon)^{-d}\,c_{0}\|\phi\|_{L^{2}}\,R^{2d}\sup_{|z|\leq
R}\|\theta^{\varepsilon}_{z}\|_{\Sigma^{k}_{\varepsilon}}.$

  2. (2)

Assume $\theta^{\varepsilon}_{z}=\lambda^{\varepsilon}(z){\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta)$ with
$\theta\in\mathcal{S}({\mathbb{R}}^{d})$,
$(\lambda^{\varepsilon})_{\varepsilon>0}$ a bounded family in
$L^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}})$ and $\Phi$ a smooth
diffeomorphism of ${\mathbb{R}}^{2d}$ such that

$\exists c>0,\;\exists\ell\in{\mathbb{N}},\;\;\forall
z\in{\mathbb{R}}^{2d},\;\;|J_{\Phi}(z)|+|J_{\Phi}(z)^{-1}|\leq c\langle
z\rangle^{\ell}.$

Then, there exists $c^{\prime}_{0}>0$ such that for all $\phi\in
L^{2}({\mathbb{R}}^{d})$, $k\in{\mathbb{N}}$,
$\varepsilon\in(0,\varepsilon_{0}]$,

$\|\mathcal{J}[\theta^{\varepsilon}_{z}](\phi)\|_{\Sigma^{k}_{\varepsilon}}\leq
c^{\prime}_{0}\,\|\lambda_{\varepsilon}\|_{L^{\infty}}\|\phi\|_{L^{2}}\|\theta\|_{\Sigma^{k+\ell+2d+1}}.$

The properties of the operators ${\mathcal{J}}[\theta^{\varepsilon}_{z}]$
extend to its adjoint (see (2.4)).

######  Corollary 2.15.

Under the assumptions of Theorem 2.14, the family of operators
${\mathcal{J}}[\theta^{\varepsilon}_{z}]^{*}$ (see (2.4)) satisfies the same
kind of estimates than the family ${\mathcal{J}}[\theta^{\varepsilon}_{z}]$.

A straightforward consequence of Theorem 2.14 and of Lemme 2.6 is given in the
next statement.

######  Corollary 2.16.

Assume $(\theta^{\varepsilon}_{z})_{\varepsilon>0}$ satisfies the assumptions
of Theorem 2.14 (2). Let $(\Phi^{\varepsilon})_{\varepsilon>0}$ be a frequency
localized family at the scale $\beta\geq 0$ and $C_{\beta}>0$,
$N_{\beta}\in{\mathbb{N}}$ be the constants associated with Definition 2.1.
Then, for all $k>0$ such that $N_{\beta}>k+d$, there exists a constant $c_{k}$
such that for all $R>0$,

$\|\mathcal{J}[\theta^{\varepsilon}_{z}](\phi^{\varepsilon}-\phi^{\varepsilon}_{R,<})\|_{\Sigma^{\varepsilon}_{k}}\leq
c_{k}\,C_{\beta}\,\varepsilon^{\beta}R^{-(N_{\beta}-k-d-\frac{1}{2})}.$

where the family $(\phi^{\varepsilon}_{R,<})_{\varepsilon>0}$ is introduced in
(2.8).

###### Proof of Theorem 2.14.

(1) The proof is similar to the first part of the proof of (1). By Cauchy-
Schwartz inequality, for $x\in{\mathbb{R}}^{d}$, we have

$\displaystyle\|\mathcal{J}[\theta^{\varepsilon}_{z}\mathbb{I}_{|z|\leq
R}](\phi^{\varepsilon})\|_{L^{2}}^{2}$
$\displaystyle\leq(2\pi\varepsilon)^{-2d}\|\phi^{\varepsilon}\|_{L^{2}}^{2}\int_{|z|,|z^{\prime}|\leq
R}\left|\int_{x\in{\mathbb{R}}^{d}}\theta^{\varepsilon}_{z}(x)\overline{\theta^{\varepsilon}_{z^{\prime}}}(x)dx\right|\,dz\,dz^{\prime}$
$\displaystyle\leq(2\pi\varepsilon)^{-2d}\|\phi^{\varepsilon}\|_{L^{2}}^{2}\int_{|z|,|z^{\prime}|\leq
R}\|\theta^{\varepsilon}_{z}\|_{L^{2}}\|\theta^{\varepsilon}_{z^{\prime}}\|_{L^{2}}\,dz\,dz^{\prime}$
$\displaystyle\leq
c_{1}R^{4d}(2\pi\varepsilon)^{-2d}\|\phi^{\varepsilon}\|_{L^{2}}^{2}\sup_{|z|\leq
2R}\|\theta^{\varepsilon}_{z}\|_{L^{2}}^{2}$

where $c_{1}>0$ is a universal constant.

(2) Let us first prove the $L^{2}$-estimate ($k=0$). Let $(x,y)\mapsto
k^{\varepsilon}(x,y)$ be the integral kernel of the operator
$\mathcal{J}[\theta^{\varepsilon}_{z}]$. Since the Bargmann transform is an
isometry, it is equivalent to consider the operator
$\mathcal{B}\circ\mathcal{J}[\theta^{\varepsilon}_{z}]\circ\mathcal{B}^{-1}$,
the kernel of which is the function $({\mathbb{R}}^{2d})^{2}\ni(X,Y)\mapsto
k^{\varepsilon}_{\mathcal{B}}(X,Y)$ defined by

$\displaystyle k^{\varepsilon}_{\mathcal{B}}(X,Y)$
$\displaystyle=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}\overline{g}^{\varepsilon}_{X}(x)g^{\varepsilon}_{Y}(y)k^{\varepsilon}(x,y)dxdy=(2\pi\varepsilon)^{-2d}\int_{z\in{\mathbb{R}}^{2d}}\langle
g^{\varepsilon}_{z},g^{\varepsilon}_{Y}\rangle\langle
g^{\varepsilon}_{X},\theta^{\varepsilon}_{z}\rangle dz.$

Therefore, by (2.5), $k^{\varepsilon}_{\mathcal{B}}(X,Y)$ satisfies

$|k^{\varepsilon}_{\mathcal{B}}(X,Y)|\leq(2\pi\varepsilon)^{-2d}\int_{z\in{\mathbb{R}}^{2d}}\left|\lambda^{\varepsilon}(z)W[g^{i\mathbb{I}},g^{i\mathbb{I}}]\left(\frac{Y-z}{\sqrt{\varepsilon}}\right)W[g^{i\mathbb{I}},\theta]\left(\frac{\Phi(z)-X}{\sqrt{\varepsilon}}\right)\right|dz.$

We deduce

$\displaystyle\int_{{\mathbb{R}}^{2d}}|k^{\varepsilon}_{\mathcal{B}}(X,Y)|dX$
$\displaystyle\leq
M\|\lambda^{\varepsilon}\|_{L^{\infty}}\left(\int_{{\mathbb{R}}^{2d}}|W[g^{i\mathbb{I}},g^{i\mathbb{I}}](z)|dz\right)\left(\int_{{\mathbb{R}}^{2d}}|W[g^{i\mathbb{I}},\theta](X)|dX\right),$
$\displaystyle\int_{{\mathbb{R}}^{2d}}|k^{\varepsilon}_{\mathcal{B}}(X,Y)|dY$
$\displaystyle\leq
M\|\lambda^{\varepsilon}\|_{L^{\infty}}\left(\int_{{\mathbb{R}}^{2d}}|W[g^{i\mathbb{I}},g^{i\mathbb{I}}](Y)|dY\right)\left(\int_{{\mathbb{R}}^{2d}}|W[g^{i\mathbb{I}},\theta](z)J_{\Phi}^{-1}(z)|dz\right),$

with $M=\sup_{\varepsilon\in(0,1]}\|\lambda^{\varepsilon}\|_{L^{\infty}}$,
and, by equations (2.6) and (2.10), we deduce the existence of $C>0$ such that

$\displaystyle\int_{{\mathbb{R}}^{2d}}|k^{\varepsilon}_{\mathcal{B}}(X,Y)|dX+\int_{{\mathbb{R}}^{2d}}|k^{\varepsilon}_{\mathcal{B}}(X,Y)|dY$
$\displaystyle\leq CM\|\theta\|_{\Sigma^{2d+\ell+1}}.$

We then conclude by Schur Lemma and obtain

$\|\mathcal{B}\circ\mathcal{J}[\theta^{\varepsilon}_{z}]\circ\mathcal{B}^{-1}\|_{\mathcal{L}(L^{2}({\mathbb{R}}^{2d}))}\leq
CM\ \|\lambda^{\varepsilon}\|_{L^{\infty}}\,\|\theta\|_{\Sigma^{2d+\ell+1}},$

and so it is for $\mathcal{J}[\theta^{\varepsilon}_{z}]$.

For concluding the proof when $k\not=0$, we again use that for
$\alpha,\gamma\in{\mathbb{N}}^{d}$ and $\phi\in\mathcal{S}({\mathbb{R}}^{d})$,

$x^{\alpha}(\varepsilon\partial_{x})^{\gamma}\mathcal{J}[\theta^{\varepsilon}_{z}]=\mathcal{J}[x^{\alpha}(\varepsilon\partial_{x})^{\gamma}\theta^{\varepsilon}_{z}],$

and the additional observation

$x^{\alpha}(\varepsilon\partial_{x})^{\gamma}{\rm
WP}^{\varepsilon}_{z}(\theta)={\rm
WP}^{\varepsilon}\left((q+\sqrt{\varepsilon}x)^{\alpha}(p+\sqrt{\varepsilon}D_{x})^{\gamma}\theta\right).$

We then conclude by observing that, as in the estimate (2.10), we have for all
$n\in{\mathbb{N}}$,

$\|(q+\sqrt{\varepsilon}x)^{\alpha}(p+\sqrt{\varepsilon}D_{x})^{\gamma}\theta\|_{\Sigma^{n}}\leq\langle
z\rangle^{k}\|\theta\|_{\Sigma^{n+k}}.$

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 $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ satisfies
Assumptions 1.3 and 1.4. Let $k\in{\mathbb{N}}$ and $t\in I$.

  1. (1)

The families of operators $\left(\mathcal{J}_{\ell,{\rm
th/fr}}^{t,t_{0}}\right)_{\varepsilon>0}$ defined in (1.22) and (1.25) are
bounded families in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d},{\mathbb{C}}^{m}))$.

  2. (2)

Assume moreover that the compact $K$ satisfies Assumptions 1.17. Then, the
family of operators $\left(\mathcal{J}_{1,2,{\rm
th/fr}}^{t,t_{0}}\right)_{\varepsilon>0}$ defined in (1.36) and (1.37) are
bounded families in the space
$\mathcal{L}(L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d},{\mathbb{C}}^{m}))$.

######  Remark 2.18.

If one assumes that $(t,z)\mapsto\partial_{t}f+\\{v,f\\}$ is bounded from
below and $\partial_{t}f$ is bounded, then one can replace the compact $K$ by
${\mathbb{R}}^{2d}$ in the definition of $\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}$ and one obtains a bounded family in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d},{\mathbb{C}}^{m}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d},{\mathbb{C}}^{m}))$.

###### Proof.

Let $\ell\in\\{1,2\\}$. Let us first discuss $\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}$. We write $\mathcal{J}^{t,t_{0}}_{\ell,{\rm
th}}=\mathcal{J}[\theta^{\varepsilon}_{z}]$ with

$\displaystyle\theta^{\varepsilon}_{z}$
$\displaystyle=\lambda^{\varepsilon}(z){\rm
WP}^{\varepsilon}_{\Phi_{\ell}^{t,t_{0}}(z)}(g^{\Gamma}_{\ell}(t,t_{0},z))\;\;\mbox{and}\;\;\lambda^{\varepsilon}(z)={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z)}\vec{V}_{\ell}(t,t_{0},z).$

We observe that for all $t\in I$ and $z\in{\mathbb{R}}^{2d}$,

$\|\vec{V}_{\ell}(t,t_{0},z)\|_{{\mathbb{C}}^{m}}=\|\vec{V}_{\ell}(t_{0},t_{0},z)\|_{{\mathbb{C}}^{m}}=\|\pi_{\ell}(t_{0})\vec{V}\|_{{\mathbb{C}}^{m}}\leq\|\vec{V}\|_{{\mathbb{C}}^{m,m}}$

which is independent of $z$. Therefore, the family
$(\lambda^{\varepsilon})_{\varepsilon>0}$ is bounded in
$L^{\infty}({\mathbb{R}}^{2d})$.

Besides, by Proposition A.4, the flow map
$(t,z)\mapsto\Phi^{t,t_{0}}_{\ell}(z)$ satisfies the assumptions of (2) of
Theorem 2.14. Similarly, the map $(t,z)\mapsto\Gamma_{\ell}(t,t_{0},z)$ is
bounded on $I\times{\mathbb{R}}^{2d}$. Therefore, for any $N\in{\mathbb{N}}$,
there exists $c_{N,t_{0},T}>0$ such that

$\forall t\in
I,\;\;\|x^{\alpha}\partial_{x}^{\beta}g^{\Gamma}_{\ell}(t,t_{0},\cdot)\|_{\Sigma^{N}}\leq
c_{N,t_{0},T}.$

We then conclude by (2) of Theorem 2.14. The proof for $\mathcal{J}_{\ell,{\rm
fr}}^{t,t_{0}}$ follows exactly the same lines.

The proof for $\mathcal{J}_{1,2,{\rm th/fr}}^{t,t_{0}}$ requires additional
observations. We need to consider the transition coefficient map
$(t,z)\mapsto\tau_{1,2}(t,t_{0},z)$ (see (1.29)) and the matrix-valued maps
$z\mapsto\Gamma^{\flat}(t_{0},z)$ (see (1.32)), which requires the analysis of
the function parametrizing the crossing (see (1.27) and (1.28)),

(2.14)
$z\mapsto\left(\alpha^{\flat}(t_{0},z),\beta^{\flat}(t_{0},z),\mu^{\flat}(t_{0},z)\right).$

By the condition (1.4) of Assumption 1.3, with $n_{0}=0$, the derivatives of
$(t,z)\mapsto f(t,z)$ 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 $\partial_{t}f$ and $\mu^{\flat}$ are bounded below. As a
consequence $z\mapsto\alpha^{\flat}(t_{0},z)$,
$z\mapsto\beta^{\flat}(t_{0},z)$ and $z\mapsto\mu^{\flat}(t_{0},z)$ are
bounded functions on ${\mathbb{R}}^{2d}$ for all $t\in I$, the map defined in
(2.14) is smooth. One then argues as before by including the coefficient
$\tau_{1,2}(t,t_{0},z)$ in the definition of $\lambda^{\varepsilon}$ 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
$\mathcal{J}[\theta^{\varepsilon}_{z}]$ when
$(\theta^{\varepsilon}_{z})_{\varepsilon>0}$ is of the form

(2.15) $\theta^{\varepsilon}_{z}={\rm e}^{\frac{i}{\varepsilon}S(z)}u(z){\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta(z,\cdot)),$

where
$\theta\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},\mathcal{S}({\mathbb{R}}^{d}))$,
$S\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{R}})$,
$u\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}})$ and $\Phi$ a smooth
diffeomorphism sarisfying the assumptions of Theorem 2.14. We are interested
in the case where $S$ and $\Phi$ 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 $S\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d}_{z},{\mathbb{R}})$,
$u\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d}_{z},{\mathbb{C}})$ and $\Phi$ a
smooth diffeomorphism. We assume the following properties:

  1. (i)

There exists $c>0$ and $\ell\in{\mathbb{N}}$ such that

$\forall z\in{\mathbb{R}}^{2d},\;\;|J_{\Phi}(z)|+|J_{\Phi}(z)^{-1}|\leq
c\langle z\rangle^{\ell}.$

  2. (ii)

Setting $\Phi(z)=(\Phi_{q}(z),\Phi_{p}(z))$ and

$\partial_{z}\Phi=\begin{pmatrix}A(z)&B(z)\\\ C(z)&D(z)\end{pmatrix},$

we have

$\nabla_{q}S(z)=-p+A(z)\Phi_{p}(z)\;\;\mbox{and}\;\;\nabla_{p}S(z)=B(z)\Phi_{p}(z),\;\;z=(q,p).$

  3. (iii)

For all $k\in{\mathbb{N}}$, the $z$-dependent seminorms
$\|u(z)\|_{\Sigma^{k}}$ and $\sup_{|\alpha|\leq
k}\|\partial_{z}^{\alpha}S\|_{L^{\infty}}$ 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.

Let $\mathfrak{d}=\partial_{q}-i\partial_{p}$. Let
$\theta\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},\mathcal{S}({\mathbb{R}}^{d}))$,
$S\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{R}})$,
$u\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}})$ and $\Phi$ be a
smooth diffeomorphism satisfying Assumptions 2.19. Then, the following
equality between operators in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon})$ holds for
$k\in{\mathbb{N}}$:

$\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left((\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\theta\right)\right]=-i\sqrt{\varepsilon}\,\mathcal{J}\left[\mathfrak{d}u\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}(\theta)\right]-i\sqrt{\varepsilon}\,\mathcal{J}\left[u\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}(\mathfrak{d}\theta)\right].$

Note that with the notation of Lemma 2.20, we have

(2.16)
$\mathfrak{d}\Phi_{p}(z)=C(z)-iD(z)\;\;\mbox{and}\;\;\mathfrak{d}\Phi_{q}(z)=A(z)-iB(z).$

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 $\mathcal{J}\left[u\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm WP}^{\varepsilon}_{\Phi}(\theta)\right]$ is
the function

$(x,y)\mapsto\int_{z\in{\mathbb{R}}^{2d}}k(z,x,y)dz$

defined by

$k(z,x,y)=u(z,x){\rm
e}^{\frac{i}{\varepsilon}S(z)}\overline{g_{z}^{\varepsilon}(y)}{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta(z,\cdot))(x),\;\;(x,y)\in{\mathbb{R}}^{d},\;\;z\in{\mathbb{R}}^{2d}.$

We aim at calculating $\mathfrak{d}k$. We observe for
$z=(q,p)\in{\mathbb{R}}^{2d}$, $y\in{\mathbb{R}}^{d}$ and

$\displaystyle\mathfrak{d}S(z)$
$\displaystyle=-p+(A(z)-iB(z))\mathfrak{d}\Phi_{p}(z),$
$\displaystyle\mathfrak{d}\left(\overline{g^{\varepsilon}_{z}(y)}\right)$
$\displaystyle=\frac{i}{\varepsilon}[(\mathfrak{d}p+i\mathfrak{d}q)(y-q)+p\mathfrak{d}q]g^{\varepsilon}_{z}(y)=\frac{i}{\varepsilon}p\,g^{\varepsilon}_{z}(y),$
$\displaystyle\mathfrak{d}\left({\rm