WP}^{\varepsilon}_{\Phi(z)}(\theta(z,\cdot))\right)$ $\displaystyle={\rm
WP}^{\varepsilon}_{\Phi(z)}(\mathfrak{d}\theta(z,\cdot))+\frac{i}{\sqrt{\varepsilon}}{\rm
WP}^{\varepsilon}_{\Phi(z)}\left((\mathfrak{d}\Phi_{p}(z)x-\mathfrak{d}\Phi_{q}(z)D_{x})\theta(z,\cdot)\right)$
$\displaystyle\;\;-\frac{i}{\varepsilon}{\rm
WP}^{\varepsilon}_{\Phi(z)}\left((A(z)-iB(z))\mathfrak{d}\Phi_{p}(z)\theta(z,\cdot)\right).$

We obtain

$\displaystyle\mathfrak{d}k(z,x,y)$ $\displaystyle={\rm
e}^{\frac{i}{\varepsilon}S(z)}\Bigl{(}\mathfrak{d}u(z,x)\,\,\overline{g_{z}^{\varepsilon}(y)}\,{\rm
WP}^{\varepsilon}_{\Phi(z)}(\theta(z,\cdot))(x)+u(z,x)\,\overline{g_{z}^{\varepsilon}(y)}\,{\rm
WP}^{\varepsilon}_{\Phi(z)}(\mathfrak{d}\theta(z,\cdot))(x)$
$\displaystyle\qquad\qquad+\frac{i}{\sqrt{\varepsilon}}u(z,x)\,\overline{g_{z}^{\varepsilon}(y)}\,{\rm
WP}^{\varepsilon}_{\Phi(z)}\left((\mathfrak{d}\Phi_{p}(z)x-\mathfrak{d}\Phi_{q}(z)D_{x})\theta(z,\cdot)\right)(x)\Bigr{)}$

The result then follows from the integration in $z\in{\mathbb{R}}^{2d}$. ∎

The case of Gaussian functions $\theta$ is of particular interest. Indeed, if
$\theta(z,\cdot)=g^{\Theta(z)}$ with
$\Theta\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},\mathfrak{S}^{+}(d))$, we
have for $x\in{\mathbb{R}}^{d}$ and $z\in{\mathbb{R}}^{2d}$,

(2.17)
$(\mathfrak{d}\Phi_{p}(z)x-\mathfrak{d}\Phi_{q}(z)D_{x})g^{\Theta(z)}(x)=(\mathfrak{d}\Phi_{p}(z)-\mathfrak{d}\Phi_{q}(z)\Theta(z))x\,g^{\Theta(z)}(x).$

We set

$M_{\Theta}(z):=\mathfrak{d}\Phi_{p}(z)-\mathfrak{d}\Phi_{q}(z)\Theta(z).$

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

(2.18)
$M_{\Theta}=(C-iD)-(A-iB)\Theta=(A-iB)\left[(A-iB)^{-1}(C-iD)-\Theta\right].$

Note that this matrix $M_{\Theta}$ is invertible because
$(A+iB)^{-1}(C+iD)-\overline{\Theta}\in\mathfrak{S}^{+}(d)$ (as the sum of two
elements of $\mathfrak{S}^{+}(d)$). These observations are in the core of the
proof of the next result which is a corollary of Lemma 2.20, when applied to
Gaussian profiles.

######  Corollary 2.21.

Let $k\in{\mathbb{N}}$. Let
$\Theta\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},\mathfrak{S}^{+}(d))$ such
that $M_{\Theta}$ is bounded together with its inverse, let
$S\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{R}})$,
$u\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}})$ and $\Phi$ a smooth
diffeomorphism satisfying Assumptions 2.19. Then, in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,
we have

(2.19) $\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(xg^{\Theta}\right)\right]=O(\sqrt{\varepsilon}).$

Besides, for all
$L\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{d,d})$, in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,
we have

(2.20) $\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(Lx\cdot
xg^{\Theta}\right)\right]=\frac{1}{i}\,\mathcal{J}\left[u\,{\rm
Tr}\left(L\,M_{\Theta}^{-1}\,\mathfrak{d}\Phi_{q}\right)\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(g^{\Theta}\right)\right]+O(\varepsilon).$

with

$LM_{\Theta}^{-1}\mathfrak{d}\Phi_{q}=L\left[(A-iB)^{-1}(C-iD)-\Theta\right]^{-1}.$

######  Remark 2.22.

  1. (1)

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

$\widehat{a}={\mathcal{J}}(\widehat{a}g^{\varepsilon}_{z})={\mathcal{J}}[a(z)g^{\varepsilon}_{z}]+\sqrt{\varepsilon}\widetilde{\mathcal{J}}\left({\rm
WP}^{\varepsilon}_{z}\left(\nabla a(z)\cdot\begin{pmatrix}x\\\
D_{x}\end{pmatrix}g^{i\mathbb{I}}\right)\right)+O(\varepsilon),$

where we have used the properties of wave packets. Using $\nabla
g^{i\mathbb{I}}=xg^{i\mathbb{I}}$ allows to conclude by Corollary 2.21

$\widehat{a}={\mathcal{J}}[a(z)g^{\varepsilon}_{z}]+O(\varepsilon)$

in any space $\Sigma^{k}_{\varepsilon}$. This was already proved in [56].

  2. (2)

More can be said about the $\sqrt{\varepsilon}$-order term on the right-hand
side of (2.19). By revisiting the proof below, one sees that there exists a
real-valued smooth function $z\mapsto c(z)$ such that

$\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(xg^{\Theta}\right)\right]=-i\sqrt{\varepsilon}\mathcal{J}\left[u\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(c(z)g^{\Theta}\right)\right]+O(\varepsilon).$

######  Remark 2.23.

The latter remark allows to prove Remark 1.13 (1). We observe that in $L^{2}$,

$\psi=\widehat{a\vec{V}}(\widehat{(a^{-1})}f)+O(\varepsilon)\;\;\mbox{and}\;\;\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi=\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\left(\widehat{a\vec{V}}(\widehat{(a^{-1})}f)\right)+O(\varepsilon)$

Turning the pair $(\vec{V},f)$ into $(\widehat{a\vec{V}},\widehat{(a^{-1})}f)$
consists in replacing $\vec{V}_{\ell}(t,t_{0},z)$ by

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

The two thawed Gaussian approximation constructed in that two different manner
then differs one from the other by $O(\varepsilon)$: the analysis developed in
Section 2.3 (in particular, the arguments of Remark 2.22) shows that in
$\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$,

$\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}(\widehat{\vec{V}}f)=(2\pi\varepsilon)^{-d}\int_{{\mathbb{R}}^{2d}}{\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z)}{\langle}g^{\varepsilon}_{z},\widehat{(a^{-1})}f{\rangle}\vec{V}_{\ell,a}(t,t_{0},z)g_{\Phi^{t,t_{0}}_{\ell}(t,z)}^{\Gamma_{\ell}(t,t_{0},z),\varepsilon}dz+O(\varepsilon).$

###### Proof of Corollary 2.21.

One uses (2.17) and the first relation of Lemma 2.20 that we apply to
$\theta=g^{\Theta}$. It gives that in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,

$\displaystyle\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(xg^{\Theta}\right)\right]$
$\displaystyle=\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(M_{\Theta}^{-1}(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})g^{\Theta}\right)\right]=O(\sqrt{\varepsilon}),$

whence (2.19).

Secondly, if l
$L\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{d,d})$, we consider
the matrix $L^{\prime}$ such that $L=\,^{t}L^{\prime}M_{\Theta}$. We observe
that

$\displaystyle(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\cdot(L^{\prime}xg^{\Theta})$
$\displaystyle=\left((\,^{t}L^{\prime}(\mathfrak{d}\Phi_{p}-\mathfrak{d}\Phi_{q}\Theta))x\cdot
x-{\rm Tr}(\,^{t}L^{\prime}\mathfrak{d}\Phi_{q})\right)g^{\Theta}.$

It remains to prove that in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,
we have

(2.21) $\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left((\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\cdot(L^{\prime}xg^{\Theta})\right)\right]=O(\varepsilon).$

We first apply Lemma 2.20 to the function $\theta=L^{\prime}xg^{\Theta}$ and
we write

$\displaystyle\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left((\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\cdot(L^{\prime}xg^{\Theta})\right)\right]$
$\displaystyle\;\;=-i\sqrt{\varepsilon}\mathcal{J}\left[\mathfrak{d}u\,{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(L^{\prime}xg^{\Theta}\right)+u{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(L^{\prime}x\,\mathfrak{d}(g^{\Theta})\right)\right].$

We use the relation (2.19) and obtain in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,

(2.22) $\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left((\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\cdot(L^{\prime}xg^{\Theta})\right)\right]=-i\sqrt{\varepsilon}\mathcal{J}\left[u{\rm
e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left(L^{\prime}x\,\mathfrak{d}(g^{\Theta})\right)\right]+O(\varepsilon).$

We calculate

$\mathfrak{d}(g^{\Theta})=\frac{\mathfrak{d}c_{\Theta}}{c_{\Theta}}g^{\Theta}+(\mathfrak{d}\Theta
x\cdot x)g^{\Theta},$

with

$\displaystyle(\mathfrak{d}\Theta x\cdot x)\,x\,g^{\Theta}$
$\displaystyle=(\mathfrak{d}\Theta x\cdot
x)M_{\Theta}^{-1}(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})g^{\Theta}$
$\displaystyle=M_{\Theta}^{-1}(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\left((\mathfrak{d}\Theta
x\cdot
x)g^{\Theta}\right)-2M_{\Theta}^{-1}\mathfrak{d}\Phi_{q}\mathfrak{d}\Theta\,x\,g^{\Theta}.$

Therefore, there exists matrices $L_{1}$ and $L_{2}$ such that, setting
$\tilde{\theta}=(\mathfrak{d}\Theta x\cdot x)g^{\Theta}$, we have

$L^{\prime}x\,\mathfrak{d}(g^{\Theta})=L_{1}xg^{\Theta}+L_{2}(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\tilde{\theta}.$

We deduce

$\mathcal{J}\left[u\,{\rm e}^{\frac{i}{\varepsilon}S}\,{\rm
WP}^{\varepsilon}_{\Phi}\left((\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\cdot(L^{\prime}xg^{\Theta})\right)\right]=-i\sqrt{\varepsilon}\mathcal{J}[L_{1}xg^{\Theta}]-i\sqrt{\varepsilon}\mathcal{J}[L_{2}(\mathfrak{d}\Phi_{p}x-\mathfrak{d}\Phi_{q}D_{x})\tilde{\theta}]$

and we obtain (2.21) by Lemma 2.20 applied to the function $\tilde{\theta}$,
and by the relation (2.19), which concludes the proof ∎

#####  2.3.3. Operators built on Bargmann transform via classical quantities

We now apply the results of the preceding section to the diffeomorphism $\Phi$
given by a flow map associated to a Hamiltonian $h$. We are going to derive
the results induced by Lemma 2.20 and Corollary 2.21 for time dependent
quantities after integration in time. We will use the resulting formula for
the Hamiltonians $h_{1}$ and $h_{2}$ associated with the matrix-valued
Hamiltonian $H^{\varepsilon}$.

######  Lemma 2.24.

Let $k\in{\mathbb{N}}$. Let $h$ be a subquadratic Hamiltonian on
$I\times{\mathbb{R}}^{2d}$, $I=[t_{0},t_{0}+T]$. We consider

  1. (1)

the classical quantities associated to $h$ as in Section 1.3 on the interval
$I$:

$z\mapsto S(t,z),\Phi^{t,t_{0}}(z),F(t,t_{0},z),$

  2. (2)

a smooth function defined on $I\times{\mathbb{R}}^{2d}$, bounded and with
bounded derivatives, $(t,z)\mapsto u(t,z)$,

  3. (3)

a smooth map from $I\times{\mathbb{R}}^{2d}$ into
$\mathcal{S}({\mathbb{R}}^{d})$, $(t,z)\mapsto\theta(t,z)$,

  4. (4)

a smooth function from ${\mathbb{R}}^{2d}$ into $I$, $z\mapsto t^{\flat}(z)$.

Then, for all $\chi\in\mathcal{C}^{\infty}_{0}(I)$, we have the following
equality between operators in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,

$\displaystyle\int_{{\mathbb{R}}}\chi(t)$
$\displaystyle\mathcal{J}\left[\mathbb{I}_{t>t^{\flat}(z)}u(t)\,{\rm
e}^{\frac{i}{\varepsilon}S(t)}\,{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}}\left((\mathfrak{d}\Phi_{p}^{t}x-\mathfrak{d}\Phi^{t,t_{0}}_{q}D_{x})\theta(t)\right)\right]dt$
$\displaystyle=i\sqrt{\varepsilon}\int_{{\mathbb{R}}}\mathcal{J}\left[\mathbb{I}_{t>t^{\flat}(z)}\mathfrak{d}u(t)\,{\rm
e}^{\frac{i}{\varepsilon}S(t)}\,{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}}(\theta(t))\right]dt$
$\displaystyle+i\sqrt{\varepsilon}\int_{{\mathbb{R}}}\chi(t)\mathcal{J}\left[\mathbb{I}_{t>t^{\flat}(z)}u(t)\,{\rm
e}^{\frac{i}{\varepsilon}S(t)}\,{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}}(\mathfrak{d}\theta(t))\right]dt$
$\displaystyle-i\sqrt{\varepsilon}\mathcal{J}\left[\chi(t^{\flat})\,\mathfrak{d}t^{\flat}\,u(t^{\flat})\,\,{\rm
e}^{\frac{i}{\varepsilon}S(t^{\flat})}\,{\rm
WP}^{\varepsilon}_{\Phi^{t^{\flat},t_{0}}}(\theta(t^{\flat}))\right].$

Note that the result of this lemma is an equality. Thus, we have not
emphasized assumptions that make these operators bounded. One could for
example assume global boundedness of all the quantities involve and of their
derivatives, or, what would be enough, that $\theta$ is compactly supported in
$z$.

We also emphasize that the functions denoted by $(\chi\circ
t^{\flat})\,u(t^{\flat})\,\,{\rm e}^{\frac{i}{\varepsilon}S(t^{\flat})}\,{\rm
WP}^{\varepsilon}_{\Phi^{t^{\flat}}}(\theta(t^{\flat}))$ is the map

$z\mapsto\chi\left(t^{\flat}(z)\right)\,u\left(t^{\flat}(z),z\right)\,\,{\rm
e}^{\frac{i}{\varepsilon}S\left(t^{\flat}(z),z\right)}\,{\rm
WP}^{\varepsilon}_{\Phi^{t^{\flat}(z),t_{0}}(z)}\left(\theta\left(t^{\flat}(t),z\right)\right).$

Note also that, by construction, the flow map $\Phi^{t,t_{0}}$ and the action
$S$ satisfy Assumptions 2.19 (see [13, 53, 43]).

###### Proof.

The proof follows the lines of the one of Lemma 2.20, using the relation

(2.23)
$\mathfrak{d}\left(\mathbb{I}_{t>t^{\flat}(z)}\right)=\mathfrak{d}t^{\flat}(z)\delta(t-t^{\flat}(z))$

that produces an additional term. ∎

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

######  Corollary 2.25.

With the same assumptions as in Lemma 2.24, we additionally assume
$\theta(t)=g^{\Theta(t)}$, with
$\Theta\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{2d},\mathcal{S}^{+}(d))$.
Then, for all
$L\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d},{\mathbb{C}}^{d,d})$, we have the
following equality in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d}))$,

$\displaystyle\int\chi(t)\mathcal{J}\left[\mathbb{I}_{t>t^{\flat}(z)}u\,{\rm
e}^{\frac{i}{\varepsilon}S(t,t_{0})}\,{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}}\left(Lx\cdot
xg^{\Theta(t,t_{0})}\right)\right]dt$
$\displaystyle\;\;=\frac{1}{i}\,\int\chi(t)\mathcal{J}\left[1_{t>t^{\flat}(z)}\tilde{u}(t,t_{0})\,{\rm
e}^{\frac{i}{\varepsilon}S(t,t_{0})}\,{\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}}\left(g^{\Theta(t,t_{0})}\right)\right]dt+O(\varepsilon)$

with $\widetilde{u}(t,t_{0})={\rm
Tr}\left(L\left[(A(t,t_{0})-iB(t,t_{0}))^{-1}(C(t,t_{0})-iD(t,t_{0}))-\Theta(t,t_{0})\right]^{-1}\right)$.

###### Proof.

The proof follows the lines of the one of Corollary 2.21, using the relation
(2.23). ∎

###  Chapter 3 Convergence of the thawed and the frozen Gaussian
approximations

####  3.1. Strategy of the proofs

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

Let $k\in{\mathbb{N}}$. Let
$\psi^{\varepsilon}_{0}=\widehat{\vec{V}}\phi^{\varepsilon}_{0}$ be as in
Assumption 1.9 with $\phi^{\varepsilon}_{0}\in L^{2}$ frequency localized at
the scale $\beta\geq 0$ with $N_{\beta}>d+k+\frac{1}{2}$ (which implies
$\phi^{\varepsilon}_{0}\in\Sigma^{k}_{\varepsilon}$. Without loss of
generality, we assume $\vec{V}=\pi_{\ell}(t_{0})\vec{V}$ for some
$\ell\in\\{1,2\\}$ that is now fixed.

We start with the Gaussian frame equality (2.2)

$\psi^{\varepsilon}_{0}=(2\pi\varepsilon)^{-d}\int_{z\in{\mathbb{R}}^{2d}}{\langle}g^{\varepsilon}_{z},\widehat{\vec{V}}\phi^{\varepsilon}_{0}{\rangle}g^{\varepsilon}_{z}\,dz.$

Writing
${\langle}g^{\varepsilon}_{z},\widehat{\vec{V}}\phi^{\varepsilon}_{0}{\rangle}={\langle}\widehat{\vec{\overline{V}}}g^{\varepsilon}_{z},\phi^{\varepsilon}_{0}{\rangle}$
and using Remark 2.22, we have in $\Sigma^{k}_{\varepsilon}$,

$\psi^{\varepsilon}_{0}=\mathcal{J}[\widehat{\vec{\overline{V}}}(z)g^{\varepsilon}_{z}]^{*}(\phi^{\varepsilon}_{0})+O(\varepsilon)=\mathcal{J}[\widehat{\vec{V}}(z)g^{\varepsilon}_{z}](\phi^{\varepsilon}_{0})+O(\varepsilon\|\phi^{\varepsilon}_{0}\|_{L^{2}}).$

Corollary 2.16 yields that, in $\Sigma^{k}_{\varepsilon}$, we have

$\displaystyle\psi^{\varepsilon}_{0}$
$\displaystyle=\mathcal{J}[\mathbb{I}_{|z|<R}\widehat{\vec{V}}(z)g^{\varepsilon}_{z}](\phi^{\varepsilon}_{0})+O(\varepsilon\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}C_{\beta}R^{-n_{\beta}})$
$\displaystyle=\mathcal{J}[\widehat{\vec{V}}(z)g^{\varepsilon}_{z}]((\phi^{\varepsilon}_{0})_{R,<})+O(\varepsilon\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}C_{\beta}R^{-n_{\beta}})$

with the notations of Corollary 2.16 and setting
$n_{\beta}=N_{\beta}-k-d-\frac{1}{2}>0$.

Now that the data has been written in a convenient form, we apply the
propagator $\mathcal{U}^{\varepsilon}_{H}(t,t_{0})$ and we take advantage of
its boundedness in $\mathcal{L}(\Sigma^{k}_{\varepsilon})$ to write

$\displaystyle\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}$
$\displaystyle=(2\pi\varepsilon)^{-d}\int_{|z|\leq
R}{\langle}g^{\varepsilon}_{z},\phi^{\varepsilon}_{0}{\rangle}\left(\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\vec{V}(z)g^{\varepsilon}_{z}\right)dz+O(\varepsilon\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}})$
$\displaystyle=\mathcal{J}\left[\mathbb{I}_{|z|\leq
R}\,\mathcal{U}^{\varepsilon}_{H}(t,t_{0}){\vec{V}}(z)g^{\varepsilon}_{z}\right](\phi^{\varepsilon}_{0})+O(\varepsilon\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}).$

We then use the description of the propagation of wave packets by
$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})$, as stated in Theorem 1.21: for
$N\geq d+1$, in $\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$, we have

$\mathcal{U}^{\varepsilon}_{H}(t,t_{0}){\vec{V}}(z)g^{\varepsilon}_{z}=\psi^{\varepsilon,N}_{\ell}(t)+O(\varepsilon^{N}).$

Therefore, by (1) of Theorem 2.14 in
$\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d},{\mathbb{C}}^{m}))$,

(3.1)
$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}=\mathcal{J}\left[\mathbb{I}_{|z|\leq
R}\,\,\psi^{\varepsilon,N}_{\ell}(t)\right]+O(\varepsilon^{N-d}\,R^{d}\,\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}})$

Besides, using that $\psi^{\varepsilon,N}_{\ell}(t)$ is a linear combination
of wave packets and considering the explicit formula of Theorem 1.21, (2) of
Theorem 2.14 implies that in
$\mathcal{L}(\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d},{\mathbb{C}}^{m}))$,

$\displaystyle\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}$
$\displaystyle=\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}\left(\widehat{\vec{V}}(\phi^{\varepsilon}_{0})_{R,<}\right)+O(\sqrt{\varepsilon}\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{N-d}\,R^{d}\,\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}})$
$\displaystyle=\mathcal{J}_{\ell,{\rm
th}}^{t,t_{0}}\left(\widehat{\vec{V}}\phi^{\varepsilon}_{0}\right)+O(\sqrt{\varepsilon}\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{N-d}\,R^{d}\,\|\phi^{\varepsilon}_{0}\|_{L^{2}})+O(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}})$

where we have used again Corollary 2.16. If $\beta<\frac{1}{2}$, we perform an
appropriate choice of $R$ and $N$: we choose $R=\varepsilon^{-\gamma}$ with
$\gamma\geq\frac{1}{n_{\beta}}(\frac{1}{2}-\beta)$ and
$N\geq\frac{1}{2}+d(1+\gamma)$.

At this stage of the description, the thawed Gaussian approximation of Theorem
(1.15) is proved. For obtaining the frozen one, we shall argue as in the
scalar case considered in [53] (Lemma 3.2 and Lemma 3.4). We will detail this
argument later in Section 3.3 below.

The proofs of the order $\varepsilon$ approximations of Theorems 1.18, 1.19
and 1.20 start with the same lines. However, one includes in the approximation
the two first terms of the asymptotic expansion of
$\psi^{\varepsilon,N}_{\ell}(t)$: the one of order $\varepsilon^{0}$ and the
one of order $\varepsilon^{\frac{1}{2}}$. The terms of order
$\sqrt{\varepsilon}$ are twofold:

  1. (i)

The one along the same mode as the initial data, here denoted by $\ell$. This
term will be proved to be of lower order because its structure allows to use
the first part of Corollary 2.21.

  2. (ii)

The one generated by the crossing along the other mode. This one is not
negligible.

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

####  3.2. Thawed Gaussian approximations with transfers terms

We prove here the higher order approximation of Theorem 1.18 for initial data
$\psi^{\varepsilon}_{0}=\vec{V}\phi^{\varepsilon}_{0}$ with
$(\phi^{\varepsilon}_{0})_{\varepsilon>0}$ frequency localized at the scale
$\beta\geq 0$ in a compact set $K$. As in the preceding section, we assume
$\vec{V}=\pi_{\ell}(t_{0})\vec{V}$ and, without loss of generality, we suppose
$\ell=1$.

We start as in the preceding section and transform equation (3.1) by taking
the terms of order $\varepsilon^{0}$ and $\varepsilon^{\frac{1}{2}}$ in the
expansion of $\psi_{\ell}^{\varepsilon,N}$. We obtain

$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}(x)=\mathcal{J}\left[\mathbb{I}_{|z|<R}(\psi^{\varepsilon,1}_{1}(t)+\psi^{\varepsilon,1}_{2}(t))\right](\phi^{\varepsilon}_{0})+O\left(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}\right)+O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}}).$

The rest in $O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}})$
comes from the remainder of the approximation of
$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})g^{\varepsilon}_{z}$ while the term
$O(\varepsilon\|\psi^{\varepsilon}_{0}\|_{L^{2}})$ comes from the terms of
order $\varepsilon^{j}$ for $j\geq 1$ of the approximation, these terms having
a wave packet structure while the rest is just known as bounded in
$\Sigma^{k}_{\varepsilon}$.

We write for $\ell\in\\{1,2\\}$

$\psi^{\varepsilon,1}_{\ell}(t)=\sum_{j=0}^{1}\varepsilon^{\frac{j}{2}}\psi^{\varepsilon,1}_{\ell,j}(t).$

Because the assumptions on $K$ induce that there is only one passage through
the crossing, Theorem 1.21 implies that $\psi^{\varepsilon,1}_{2,0}(t)=0$ and
$\psi^{\varepsilon,1}_{2,1}(t)$ only depends on the transfer profile
$f^{\varepsilon}_{1\rightarrow 2}$ (indeed, we have assumed
$\vec{V}=\pi_{1}(t_{0})\vec{V}$). Moreover, for the mode $1$, we have for
$\ell\in\\{1,2\\}$ and $j\in\\{0,1\\}$

$\psi^{\varepsilon,1}_{\ell,j}(t)={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{\ell}(t)}\left({\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]\,\vec{B}_{\ell,j}(t)g^{i\mathbb{I}}\right).$

We recall that $\vec{B}_{\ell,1}$ is given by (1.41). We use the structure of
the term

${\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]\,\vec{B}_{\ell,1}(t)g^{i\mathbb{I}}$

(see [54] section 3 or the book [13]): it writes

${\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]\,\vec{B}_{\ell,1}(t)g^{i\mathbb{I}}(x)=\vec{a}(t)xg^{\Gamma_{\ell}(t,t_{0},z_{0})}(x)$

for some smooth and bounded vector-valued map $(t,z)\mapsto\vec{a}(t,z)$.
Therefore, Corollary 2.21 yields

$\mathcal{J}\left[\mathbb{I}_{|z|<R}\psi^{\varepsilon,1}_{1,1}(t)\right](\phi^{\varepsilon}_{0})=O(\sqrt{\varepsilon}\|\phi^{\varepsilon}_{0}\|_{L^{2}})$

and we are left with

$\displaystyle\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}(x)$
$\displaystyle=\mathcal{J}\left[\mathbb{I}_{|z|<R}(\psi^{\varepsilon,1}_{1,0}(t)+\sqrt{\varepsilon}\,\psi^{\varepsilon,1}_{2,1}(t))\right](\phi^{\varepsilon}_{0})+O\left(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}\right)+O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}})$
$\displaystyle=\mathcal{J}\left[\psi^{\varepsilon,1}_{1,0}(t)+\sqrt{\varepsilon}\,\psi^{\varepsilon,1}_{2,1}(t)\right]((\phi^{\varepsilon}_{0})_{R,<})+O\left(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}\right)+O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}})$
$\displaystyle=\mathcal{J}\left[\psi^{\varepsilon,1}_{1,0}(t)+\sqrt{\varepsilon}\,\psi^{\varepsilon,1}_{2,1}(t)\right](\phi^{\varepsilon}_{0})+O\left(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}\right)+O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}})$

by Corollary 2.16. Identifying the terms, we deduce

$\displaystyle\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}(x)$
$\displaystyle=\mathcal{J}_{1,{\rm
th}}^{t,t_{0}}\left(\widehat{\vec{V}}\,\phi^{\varepsilon}_{0}\right)+\sqrt{\varepsilon}\,\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}\left(\widehat{\vec{V}}\phi^{\varepsilon}_{0}\right)+O\left(\varepsilon^{\beta}\,C_{\beta}\,R^{-n_{\beta}}\right)+O(\varepsilon^{N-d}R^{d}\|\psi^{\varepsilon}_{0}\|_{L^{2}}).$

If $\beta<1$, we choose $R=\varepsilon^{-\gamma}$, $N=1+d(\gamma+1)$ with
$\gamma\geq\frac{1}{n_{\beta}}(1-\beta).$ This gives Theorem 1.18.

More precisely, for a general
$\vec{V}=\pi_{1}(t_{0})\vec{V}+\pi_{2}(t_{0})\vec{V}$, we obtain

(3.2)
$\displaystyle\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}(x)$
$\displaystyle=\mathcal{J}_{1,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)+\mathcal{J}_{2,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{2}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)+\sqrt{\varepsilon}\,\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)$
$\displaystyle\qquad+O\left(\varepsilon(C_{\beta}+\|\psi^{\varepsilon}_{0}\|_{L^{2}})\right).$

####  3.3. Frozen Gaussian approximations with transfers terms

It remains to pass from the thawed to the frozen approximation. As we have
already mentioned, we use the argument developed in Lemma 3.2 and 3.4 of [53].
It is based on an evolution argument which crucially uses Corollary 2.21. We
now explain that step.

###### End of the proof of Theorem 1.15.

We start from the approximation given by the first part of Theorem 1.15: in
$\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$, we have

$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}(x)=\mathcal{J}_{1,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)+O(\sqrt{\varepsilon}(C_{\beta}+\|\phi^{\varepsilon}_{0}\|_{L^{2}}))$

and our aim is to prove that in $\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$

$\mathcal{J}_{1,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)=\mathcal{J}_{1,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\,\phi^{\varepsilon}_{0}\right)+O(\varepsilon).$

Of course, a remainder of size $O(\sqrt{\varepsilon})$ would be enough for
proving Theorem 1.15; however, it will be usefull to have it in order to prove
Theorems 1.19 and 1.20.

The strategy dates back to [53]. We follow the presentation of [21]. We set
for $s\in[0,1]$

$\Theta(s,z)=(1-s)\Gamma_{\ell}(t,t_{0},z)+is\mathbb{I}$

where $\Gamma_{\ell}$ is given by (1.18). We consider the partially normalised
Gaussian function

$\widetilde{g}(t,s)=(\pi)^{-d/4}{\rm e}^{\frac{i}{2}\Theta(s,z)x\cdot
x},\;x\in{\mathbb{R}}^{d}$

and we set

$\widetilde{g}^{\,\Theta(s),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}(x)={\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\left(\widetilde{g}(t,s)\right),\\\
$

The aim is to construct a map $s\mapsto a(s,z)$ such that for all $s\in[0,1]$
in $\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon})$,

$\frac{d}{ds}\mathcal{J}\left[a(s,z)\vec{V}_{\ell}(t,t_{0},z)\widetilde{g}^{\,\Theta(s),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\right]=O(\varepsilon).$

Choosing $a(0,z)=1$, we have

$\mathcal{J}^{t,t_{0}}_{\ell,{\rm
th}}=\mathcal{J}\left[a(0,z)\vec{V}_{\ell}(t,t_{0},z)\widetilde{g}^{\,\Theta(0),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\right],$

and we will obtain that for any $f\in L^{2}({\mathbb{R}}^{d})$, we have in
$\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$

$\displaystyle\mathcal{J}^{t,t_{0}}_{\ell,{\rm th}}(\vec{V}f)$
$\displaystyle=\mathcal{J}\left[a(1,z)\vec{V}_{\ell}(t,t_{0},z)\widetilde{g}^{\,\Theta(1),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\right](f)+O(\varepsilon)$
$\displaystyle=\mathcal{J}^{t,t_{0}}_{\ell,{\rm fr}}(\vec{V}f)+O(\varepsilon)$

provided $a(1,z)=a_{\ell}(t,t_{0},z)$ as defined in (1.24).

For constructing the map $s\mapsto a(s,z)$, we compute

$\displaystyle\frac{d}{ds}\mathcal{J}\left[a(s,z)\vec{V}_{\ell}(t,t_{0},z)\widetilde{g}^{\,\Theta(s),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\right]$
$\displaystyle=\mathcal{J}\left[\partial_{s}a(s,z)\vec{V}_{\ell}(t,t_{0},z)\widetilde{g}^{\,\Theta(s),\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\right]$
$\displaystyle+\frac{i}{2}\mathcal{J}\left[a(s,z)\vec{V}_{\ell}(t,t_{0},z){\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}\left(\partial_{s}\Theta(s)x\cdot
x\widetilde{g}^{\,\Theta(s),\varepsilon}\right)\right].$

We use equation (2.20) of Corollary 2.21 to transform the second term of the
right-hand side and obtain

$\displaystyle\mathcal{J}\left[a(s,z)\vec{V}_{\ell}(t,t_{0},z){\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}(\partial_{s}\Theta(s)x\cdot
xg^{\,\Theta(s)})\right]$
$\displaystyle\qquad=\frac{1}{i}\mathcal{J}\left[a(s,z)\vec{V}_{\ell}(t,t_{0},z){\rm
WP}^{\varepsilon}_{\Phi^{t,t_{0}}_{\ell}(z)}({\rm
Tr}(\Theta_{1}(s))g^{\,\Theta(s)})\right]+O(\varepsilon)$

in $\mathcal{L}(L^{2}({\mathbb{R}}^{d}),\Sigma^{k}_{\varepsilon})$ and with

$\Theta_{1}(s)=\partial_{s}\Theta(s)\left[(A_{\ell}-iB_{\ell})^{-1}(C_{\ell}-iD_{\ell})-\Theta\right]^{-1}$

where $M_{\ell}(s,z)$ is associated to $\Theta(s,z)$ according to (2.18). In
particular, we have

$\partial_{s}M(s)=-(A_{\ell}-iB_{\ell})\partial_{s}\Theta(s).$

We deduce

$\Theta_{1}(s)=-(A_{\ell}-iB_{\ell})^{-1}\partial_{s}M(s)M(s)^{-1}(A_{\ell}-iB_{\ell})$

and

${\rm Tr}(\Theta_{1}(s))=-{\rm Tr}(\partial_{s}M(s)M(s)^{-1})=-{\rm
det}M(s)^{-1}\,\partial_{s}\left({\rm det}M(s)\right).$

Therefore, the condition

$\partial_{s}a(s,z)-\frac{1}{2}{\rm Tr}(\partial_{s}M(s,z)M(s)^{-1})a(s,z)=0$

that we have to fulfilled, is realized by

$a(s,z)=\frac{{\rm det}M(s)}{{\rm det}M(0)}a(0,z)=a_{\ell}(t,t_{0},z).$

∎

###### Proof of Theorem 1.19.

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

$\mathcal{J}_{1,2,{\rm
th}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)=\mathcal{J}_{1,2,{\rm
fr}}^{t,t_{0}}\left(\widehat{\pi_{1}(t_{0})\vec{V}}\phi^{\varepsilon}_{0}\right)+O\left(\sqrt{\varepsilon}(C_{\beta}+\|\phi^{\varepsilon}_{0}\|_{L^{2}}\right).$

As noticed in the introduction, when $t<t_{1,{\rm min}}(K)$, then
$\tau_{1,2}(t,t_{0},z)=0$ for all $z\in K$ and when $t\in[t_{1,{\rm
max}}(K),t_{2,{\rm min}}(K))$, $z\mapsto\tau_{1,2}(t,t_{0},z)$ is smooth.
Therefore, one can use the perturbative argument allowing to froze the
covariances of the Gaussian terms as in the proof of Theorem 1.15 and one
obtains the formula (1.35). ∎

###### Proof of Theorem 1.20.

One now has to cope with the discontinuity of the transfer coefficient
$\tau_{1,2}(t,t_{0},z)$. We use Lemma 2.24. ∎

##  Part II Wave-packet propagation through smooth crossings

###  Chapter 4 Symbolic calculus and diagonalization of Hamiltonians with
smooth crossings

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

We will use the Moyal product about which we recall some facts: if
$A^{\varepsilon},B^{\varepsilon}$ are semi-classical series, their Moyal
product is the formal series

$\displaystyle C^{\varepsilon}:=A^{\varepsilon}\circledast
B^{\varepsilon}\;\;\mbox{where}\;\;\displaystyle{C^{\varepsilon}=\sum_{j\geq
0}\varepsilon^{j}C_{j}}$ (4.1) $\displaystyle
C_{j}(x,\xi)=\frac{1}{2^{j}}\sum_{|\alpha+\beta|=j}\frac{(-1)^{|\beta|}}{\alpha!\beta!}(D^{\beta}_{x}\partial^{\alpha}_{\xi}A).(D^{\alpha}_{x}\partial^{\beta}_{\xi}B)(x,\xi),\;\;j\in{\mathbb{N}}.$

We also introduce the Moyal bracket

$\\{A^{\varepsilon},B^{\varepsilon}\\}_{\circledast}:=A^{\varepsilon}\circledast
B^{\varepsilon}-B^{\varepsilon}\circledast A^{\varepsilon}.$

Let us now consider a smooth matrix-valued symbol
$H^{\varepsilon}=H_{0}+\varepsilon H_{1}$, where the principal symbol
$H_{0}=h_{1}\pi_{1}+h_{2}\pi_{2}$ has two smooth eigenvalues $h_{1}$ and
$h_{2}$ with smooth eigenprojectors $\pi_{1}$ and $\pi_{2}$. We allow for a
non-empty crossing set $\Upsilon$ as in Definition 1.2. By standard symbolic
calculus with smooth symbols, we have for $\ell\in\\{1,2\\}$ the relations

(4.2)
$\pi_{\ell}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=(i\varepsilon\partial_{t}-h_{\ell})\circledast\pi_{\ell}=O(\varepsilon).$

We are going to see two manners to replace the projector $\pi_{\ell}$ and the
Hamiltonian $h_{\ell}$ by asymptotic series so that the relation above holds
at a better order.

We call “rough” the first diagonalization process that we propose. It will
hold everywhere, including $\Upsilon$ and is comparable the reduction
performed in [4] for avoided crossings. It is the subject of Section 4.2.

The second one, more sophisticated, will require to work in a domain that does
not meet $\Upsilon$. Based on the use of superadiabatic projectors, as
developed in [5, 46, 55, 58]. This strategy is implemented in Section 4.3. The
new element comparatively to the references that we have mentioned, is that we
keep a careful memory of the dependence of the constructed elements with
respect to the distance of their support from $\Upsilon$. For this reason, we
will use a pseudodifferential setting that we precise in the next Section 4.1.

####  4.1. Formal asymptotic series

We consider formal semi-classical series

$A^{\varepsilon}=\sum_{j\geq 0}\varepsilon^{j}A_{j}$

where all the functions $A_{j}$ are smooth (matrix-valued) in an open set
$\mathcal{D}\subset{\mathbb{R}}\times{\mathbb{R}}^{2d}$, that is,
$A_{j}\in\mathcal{C}^{\infty}(\mathcal{D},{\mathbb{C}}^{m,m})$.

Notation. If $A^{\varepsilon}=\sum_{j\geq 0}\varepsilon^{j}A_{j}$ is a formal
series and $N\in{\mathbb{N}}$, we denote by $A^{\varepsilon,N}$ the function

(4.3) $A^{\varepsilon,N}=\sum_{0\leq j\leq N}\varepsilon^{j}A_{j}.$

The formal series that we will consider in Section 4.3 will present two small
parameters: the semi-classical parameter $\varepsilon>0$ and another parameter
$\delta>0$ that controls the growth of the symbol and of its derivatives. For
our intended application, $\delta$ is related to the size of the gap between
the eigenvalues of the Hamiltonian’s symbol.

######  Definition 4.1 (Symbol spaces).

Let $\mu\in{\mathbb{R}}$ and $\delta\in(0,1]$.

  1. (i)

We denote by $\boldsymbol{S}^{\mu}_{\delta}(\mathcal{D})$ the set of smooth
(matrix-valued) functions in $\mathcal{D}$ such

$|\partial^{\gamma}_{z}A(t,z)|\leq
C_{\gamma}\delta^{\mu-|\gamma|},\;\;\forall(t,z)\in\mathcal{D}.$

Notice that the set
$\boldsymbol{S}_{\delta}(\mathcal{D}):=\boldsymbol{S}^{0}_{\delta}(\mathcal{D})$
has the algebraic structure of a ring.

  2. (ii)

We shall say that a formal series $\displaystyle{A^{\varepsilon}=\sum_{j\geq
0}\varepsilon^{j}A_{j}}$ is in
$\boldsymbol{S}_{\varepsilon,\delta}^{\mu}(\mathcal{D})$ if
$A_{j}\in\boldsymbol{S}^{\mu-2j}_{\delta}(\mathcal{D})$ for all
$j\in{\mathbb{N}}$. We set
$\boldsymbol{S}_{\varepsilon,\delta}(\mathcal{D}):=\boldsymbol{S}^{0}_{\varepsilon,\delta}(\mathcal{D})$.

######  Remark 4.2.

  1. (1)

If $A\in\mathbf{S}^{\mu}_{\delta}$, $B\in\mathbf{S}^{\mu}_{\delta^{\prime}}$
then $AB\in\mathbf{S}^{\mu+\mu^{\prime}}_{\delta}$ while
$\\{A,B\\}\in\mathbf{S}^{\mu+\mu^{\prime}-2}_{\delta}$. Besides, if
$A\in\boldsymbol{S}^{\mu}_{\delta}(\mathcal{D})$, then
$\partial_{z}^{\gamma}A\in\boldsymbol{S}_{\delta}^{\mu-|\gamma|}(\mathcal{D})$.

  2. (2)

When $\delta=1$, as in the next Section 4.2, then for all
$\mu\in{\mathbb{R}}$, $S^{\mu}_{1}=S^{0}_{1}$ coincides with the standard
class of Calderón-Vaillancourt symbols, those smooth functions that are
bounded together with their derivatives. Similarly,
$S^{\mu}_{\varepsilon,1}=S^{\mu}_{\varepsilon}$ coincides with asymptotic
series of symbols.

  3. (3)

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

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

######  Lemma 4.3.

Let $\delta_{A},\,\delta_{B}\in]0,1]$. If $A^{\varepsilon}$ and
$B^{\varepsilon}$ are formal series of ${\bf
S}_{\varepsilon,\delta_{A}}(\mathcal{D})$ and ${\bf
S}_{\varepsilon,\delta_{B}}(\mathcal{D})$, respectively. then
$A^{\varepsilon}\circledast B^{\varepsilon}$ is a formal series of ${\bf
S}_{\varepsilon,\min(\delta_{A},\delta_{B})}(\mathcal{D})$. Besides, for
$N\in{\mathbb{N}}$,

$A^{\varepsilon,N}\circledast B^{\varepsilon,N}=\sum_{0\leq j\leq
N}\varepsilon^{j}C_{j}+\varepsilon^{N+1}R_{A,B}^{\varepsilon,N}$

where for all $\gamma\in{\mathbb{N}}^{2d}$, there exists $C_{N,\gamma}$
independent on $\delta_{A,B}$ and $\varepsilon$ such that

$|\partial_{z}^{\gamma}R_{A,B}^{\varepsilon,N}(t,z)|\leq
C_{N,\gamma}\,[\min(\delta_{A},\delta_{B})]^{-N-\kappa_{0}},\;\forall\varepsilon\in]0,1],\;\;\forall(t,z)\in\mathcal{D},$

where $\kappa_{0}$ is a universal constant depending only on the dimension
$d$.

###### Proof.

The estimate is a direct consequence of [6, Theorem A1]. In Appendix B,
Theorem B.1 we give a detailed proof. ∎

When $\delta_{A}=1$ and $\delta_{B}=\delta\in(0,1]$,
$\min(\delta_{A},\delta_{B})=\delta$. This shows that

${\bf S}_{\varepsilon,1}(\mathcal{D})\circledast{\bf
S}_{\varepsilon,\delta}(\mathcal{D})\subset{\bf
S}_{\varepsilon,\delta}(\mathcal{D}).$

Let us conclude this Section by comments on the quantization of symbols of the
classes ${\bf S}_{\delta}^{\mu}$. The Calderón-Vaillancourt estimate for
pseudodifferential operators (see [14, 64]) states that there exists a
constant $C>0$ such that for all
$a\in\mathcal{C}^{\infty}({\mathbb{R}}^{2d})$,

$\|{\rm op}_{\varepsilon}(a)\|_{\mathcal{L}(L^{2}({\mathbb{R}}^{d}))}\leq
C\sup_{0\leq|\gamma|\leq
d+1}\,\varepsilon^{\frac{|\gamma|}{2}}\sup_{z\in{\mathbb{R}}^{d}}|\partial_{z}^{\gamma}a(z)|.$

Actually, the article [9] treats the case $\varepsilon=1$ and the estimate in
the semi-classical case comes from the observation that

${\rm op}_{\varepsilon}(a)=\Lambda_{\varepsilon}^{*}{\rm
op}_{1}(a(\sqrt{\varepsilon}\cdot))\Lambda_{\varepsilon}$

where $\Lambda_{\varepsilon}$ is the $L^{2}$-unitary scaling operator defined
on function $f\in\mathcal{S}({\mathbb{R}}^{d})$ by

$\Lambda_{\varepsilon}f(x)=\varepsilon^{-\frac{d}{4}}f\left(\frac{x}{\sqrt{\varepsilon}}\right),\;\;x\in{\mathbb{R}}^{d}.$

One can derive an estimate in the sets $\Sigma^{k}_{\varepsilon}$ by observing

$\left(x^{\alpha}(\varepsilon\partial_{x})^{\beta}\right)\circ{\rm
op}_{\varepsilon}(a)=\sum_{|\gamma_{1}|+|\gamma_{2}|+|\gamma_{3}|\leq
k}\varepsilon^{\frac{|\gamma_{1}|}{2}}c_{\gamma_{1},\gamma_{2},\gamma_{3}}(\varepsilon)\,{\rm
op}_{\varepsilon}(\partial_{z}^{\gamma_{1}}a)\circ\left({x^{\gamma_{2}}}(\varepsilon\partial_{x})^{\gamma_{3}}\right)$

for some coefficients $c_{\gamma_{1},\gamma_{2},\gamma_{3}}(\varepsilon)$,
uniformly bounded with respect to $\varepsilon\in[0,1]$. This implies the
boundedness of ${\rm op}_{\varepsilon}(a)$ in weighted Sobolev spaces: for all
$k\in{\mathbb{N}}$, there exists a constant $C_{k}>0$ such that for all
$a\in\mathcal{S}({\mathbb{R}}^{2d})$,

(4.4) $\|{\rm
op}_{\varepsilon}(a)\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\sum_{0\leq|\gamma|\leq
d+k+1}\varepsilon^{\frac{|\gamma|}{2}}\,\sup_{z\in{\mathbb{R}}^{d}}|\partial_{z}^{\gamma}a(z)|.$

######  Proposition 4.4.

Let $A\in\mathbf{S}^{\mu-2j}_{\delta}$ for $\mu\in{\mathbb{R}}$,
$j\in{\mathbb{N}}$. Then, for $k\in{\mathbb{N}}$

$\|{\rm op}_{\varepsilon}(A)\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\sup_{0\leq|\gamma|\leq d+1}\
\varepsilon^{\frac{|\gamma|}{2}}\delta^{\mu-2j-k-|\gamma|}$

Therefore, if $\delta\geq\sqrt{\varepsilon}$

(4.5) $\|{\rm
op}_{\varepsilon}(A)\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\,\delta^{\mu-k-2j}.$

We will use such estimates. Questions related with symbolic calculus in the
classes ${\bf S}_{\delta,\varepsilon}^{\mu}$ are discussed in Appendix B.2.

####  4.2. ‘Rough’ reduction

The next result gives a reduction of the Hamiltonian in a block diagonalized
form. We will use this reduction on small interval of times.

######  Theorem 4.5.

Assume $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ with $H_{0}$ having smooth
eigenprojectors and eigenvalues. There exist matrix-valued asymptotic series

$\displaystyle\pi^{\varepsilon}_{1}$ $\displaystyle=\pi_{1}+\sum_{j\geq
1}\varepsilon^{j}\pi_{1,j},\;\;h^{\varepsilon}_{\ell}=h_{\ell}\,{\mathbb{I}}+\sum_{j\geq
1}\varepsilon^{j}h_{\ell,j},\;\;W^{\varepsilon}=\sum_{j\geq
1}\varepsilon^{j}W_{j},\;\;\ell\in\\{1,2\\}$

such that for all $N\in{\mathbb{N}}$, $\pi^{\varepsilon,N}_{1}$ and
$\pi_{2}^{\varepsilon,N}=1-\pi^{\varepsilon,N}_{1}$ are approximate projectors

(4.6)
$\pi^{\varepsilon,N}_{\ell}\circledast\pi^{\varepsilon,N}_{\ell}=\pi^{\varepsilon,N}_{\ell}+O(\varepsilon^{N+1}),\;\;\ell\in\\{1,2\\}$

and $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ reduces according to

(4.7)
$\displaystyle\pi^{\varepsilon,N}_{1}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})$
$\displaystyle=(i\varepsilon\partial_{t}-h^{\varepsilon,N}_{1})\circledast\pi_{1}^{\varepsilon,N}+W^{\varepsilon,N}\circledast\pi_{2}^{\varepsilon,N}+O(\varepsilon^{N+1}),$
(4.8)
$\displaystyle\pi^{\varepsilon,N}_{2}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})$
$\displaystyle=(i\varepsilon\partial_{t}-h^{\varepsilon,N}_{2})\circledast\pi_{2}^{\varepsilon,N}+(W^{\varepsilon,N})^{*}\circledast\pi_{1}^{\varepsilon,N}+O(\varepsilon^{N+1}).$

Moreover, for all $\ell\in\\{1,2\\}$ and $j\geq 1$, the symbols $\pi_{\ell,j}$
and $h_{\ell,j}$ are self-adjoint, the matrices $W_{j}$ are the off-diagonal
(see equation (1.44) for the value of $W_{1}$) and

(4.9) $\displaystyle h_{\ell,1}$
$\displaystyle=\pi_{\ell}H_{1}\pi_{\ell}+(-1)^{\ell}\frac{i}{2}(h_{1}-h_{2})\pi_{\ell}\\{\pi_{1},\pi_{1}\\}\pi_{\ell}.$

If $H^{\varepsilon}$ also satisfies Assumption 1.4 on the time interval $I$,
then the $4\times 4$ matrix-valued Hamiltonian

$\underline{H}^{\varepsilon}:=\begin{pmatrix}{h_{1}^{\varepsilon}}&W^{\varepsilon}\\\
(W^{\varepsilon})^{*}&{h_{2}^{\varepsilon}}\end{pmatrix}$

is subquadratic according to Definition 1.1.

Note that in $\underline{H}^{\varepsilon}$, the off-diagonal blocks are of
lower order than the diagonal ones since the asymptotic series
$W^{\varepsilon}$ has no term of order $0$.

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

$\underline{\psi}^{\varepsilon}=\,^{t}(\underline{\psi}^{\varepsilon}_{1},\underline{\psi}^{\varepsilon}_{2})\;\;\mbox{with}\;\;\underline{\psi^{\varepsilon}_{\ell}}=\widehat{\pi_{\ell}^{\varepsilon}}\psi^{\varepsilon}.$

Indeed, we then have

(4.10)
$i\varepsilon\partial_{t}\underline{\psi}^{\varepsilon}=\widehat{\underline{H}}^{\varepsilon}\underline{\psi}^{\varepsilon}+O(\varepsilon^{\infty}),\;\;\underline{\psi}^{\varepsilon}_{|t=0}=\,^{t}\left(\widehat{\pi_{1}^{\varepsilon}}\,\psi^{\varepsilon}_{0},\widehat{\pi_{2}^{\varepsilon}}\,\psi^{\varepsilon}_{0}\right).$

We deduce the corollary.

######  Corollary 4.6.

Formally, we have for $t\in I$,

$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}=\underline{\psi^{\varepsilon}_{1}}+\underline{\psi^{\varepsilon}_{2}},$

where $\underline{\psi}^{\varepsilon}$ solves (4.10).

###### Proof.

The proof relies on a recursive argument.

The case $N=0$ is equivalent to (4.2)

The case $N=1$ has been proved in Lemma B.2 in [20]. However, we revisit the
proof in order to compute $W_{1}$. We first compute $\pi_{\ell,\ell}$ by
requiring
$\pi^{\varepsilon,(\ell)}\circledast\pi^{\varepsilon,(\ell)}=\pi^{\varepsilon,(\ell)}+O(\varepsilon^{2})$,
which admits the solution

$\pi_{1,1}=-\pi_{2,1}=-\frac{1}{2i}\pi_{1}\\{\pi_{1},\pi_{1}\\}\pi_{1}+\frac{1}{2i}\pi_{2}\\{\pi_{1},\pi_{1}\\}\pi_{2}.$

We recall that $\\{\pi_{1},\pi_{1}\\}$ is diagonal and skew-symmetric (see
Lemma B.1 in [20]). Then, we observe

$\pi_{\ell}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=(i\varepsilon\partial_{t}-h_{\ell})\circledast\pi_{\ell}+\varepsilon\Theta_{\ell}+O(\varepsilon^{2})$

with

$\Theta_{\ell}=-\frac{1}{2i}\\{\pi_{\ell},H_{0}\\}-\pi_{\ell}H_{1}-i\partial_{t}\pi_{\ell}+\frac{1}{2i}\\{h_{\ell},\pi_{\ell}\\}$

or, equivalently

$\displaystyle\Theta_{1}$
$\displaystyle=\frac{1}{2i}(h_{2}-h_{1})\\{\pi_{1},\pi_{1}\\}-i\partial_{t}\pi_{1}-\pi_{1}H_{1}+\frac{1}{i}\\{h_{1},\pi_{1}\\}\pi_{1}+\frac{1}{2i}\\{h_{1}+h_{2},\pi_{1}\\}\pi_{2},$
$\displaystyle\Theta_{2}$
$\displaystyle=\frac{1}{2i}(h_{1}-h_{2})\\{\pi_{2},\pi_{2}\\}-i\partial_{t}\pi_{2}-\pi_{2}H_{1}+\frac{1}{i}\\{h_{2},\pi_{2}\\}\pi_{2}+\frac{1}{2i}\\{h_{1}+h_{2},\pi_{2}\\}\pi_{1}$
$\displaystyle=-\frac{1}{2i}(h_{2}-h_{1})\\{\pi_{1},\pi_{1}\\}+i\partial_{t}\pi_{1}-\pi_{2}H_{1}-\frac{1}{i}\\{h_{2},\pi_{1}\\}\pi_{2}-\frac{1}{2i}\\{h_{1}+h_{2},\pi_{2}\\}\pi_{1}$

We observe

$\Theta_{2}^{*}=-\frac{1}{2i}(h_{2}-h_{1})\\{\pi_{1},\pi_{1}\\}-i\partial_{t}\pi_{1}-H_{1}\pi_{2}+\frac{1}{i}\pi_{2}\\{h_{2},\pi_{1}\\}+\frac{1}{2i}\pi_{1}\\{h_{1}+h_{2},\pi_{2}\\}$

and

(4.11) $\pi_{1}\Theta_{2}^{*}\pi_{2}=\pi_{1}\Theta_{1}\pi_{2}.$

Thus, we have to solve

$\displaystyle-\pi_{1,1}H_{0}$
$\displaystyle=-h_{1,1}\pi_{1}-h_{1}\pi_{1,1}+i\partial_{t}\pi_{1}+\Theta_{1}+W_{1}\pi_{2},$
$\displaystyle-\pi_{2,1}H_{0}$
$\displaystyle=-h_{2,1}\pi_{2}-h_{2}\pi_{2,1}+i\partial_{t}\pi_{2}+\Theta_{2}+W_{1}^{*}\pi_{1}.$

Multiplying on the right the first equation by $\pi_{1}$ and the second by
$\pi_{2}$, we obtain that $h_{1,1}$ and $h_{2,1}$ have to solve

$h_{1,1}\pi_{1}=i\partial_{t}\pi_{1}\pi_{1}+\Theta_{1}\pi_{1}\;\;\mbox{and}\;\;h_{2,1}\pi_{2}=i\partial_{t}\pi_{2}\pi_{2}+\Theta_{2}\pi_{2}$

which is solved by taking the self-adjoint matrices

$\displaystyle h_{1,1}$
$\displaystyle=i\partial_{t}\pi_{1}\pi_{1}+\Theta_{1}\pi_{1}-i\pi_{1}\partial_{t}\pi_{1}\pi_{2}+\pi_{1}\Theta_{1}^{*}\pi_{2},$
$\displaystyle h_{2,1}$
$\displaystyle=i\partial_{t}\pi_{2}\pi_{2}+\Theta_{2}\pi_{2}-i\pi_{2}\partial_{t}\pi_{2}\pi_{1}+\pi_{2}\Theta_{2}^{*}\pi_{1}.$

Multiplying on the right the first equation by $\pi_{2}$ and the second by
$\pi_{1}$, we obtain that $W_{1}$ has to solve

$W_{1}\pi_{2}=-(h_{2}-h_{1})\pi_{1,1}\pi_{2}-\Theta_{1}\pi_{2}\;\;\mbox{and}\;\;W_{1}^{*}\pi_{1}=(h_{2}-h_{1})\pi_{2,1}\pi_{1}-\Theta_{2}\pi_{1}.$

Using the relations $\pi_{1,1}^{*}=\pi_{1,1}=-\pi_{2,1}$,
$\pi_{1}\partial_{t,z}\pi_{1}=\partial_{t,z}\pi_{1}\pi_{2}$ and (4.11), we
obtain

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

whence (1.44).

One can now perform the recursive argument. Assume that we have obtained
(4.6), (4.7) and (4.8) for some $N\geq 1$ and let us look for $\pi_{1,N+1}$,
$h_{1,N+1}$, $h_{2,N+1}$ and $W_{N+1}$ such that the relations for $N+1$ too.

We start with $\pi_{1,N+1}$. We write

$\pi_{1}^{\varepsilon,N}\circledast\pi_{1}^{\varepsilon,N}=\pi_{1}^{\varepsilon,N}+\varepsilon^{N+1}R^{\varepsilon},$

where $R^{\varepsilon}$ is an asymptotic series with first term $R_{N}$. We
first observe that $R_{N}$ is diagonal. Indeed, we have

$(1-\pi_{1}^{\varepsilon,N})\circledast\pi_{1}^{\varepsilon,N}\circledast\pi_{1}^{\varepsilon,N}=\pi_{1}^{\varepsilon,N}\circledast\pi_{1}^{\varepsilon,N}\circledast(1-\pi_{1}^{\varepsilon,N})$

and

$(1-\pi_{1}^{\varepsilon,N})\circledast\pi_{1}^{\varepsilon,N}\circledast\pi_{1}^{\varepsilon,N}=-\varepsilon^{N+1}R^{\varepsilon}\circledast\pi_{1}^{\varepsilon,N},\;\;\pi_{1}^{\varepsilon,N}\circledast\pi_{1}^{\varepsilon,N}\circledast(1-\pi_{1}^{\varepsilon,N})=-\varepsilon^{N+1}\pi_{1}^{\varepsilon,N}\circledast
R^{\varepsilon}.$

This yields $\pi_{1}^{\varepsilon,N}\circledast
R^{\varepsilon}=R^{\varepsilon}\circledast\pi_{1}^{\varepsilon,N}$ and imply
$\pi_{1}R_{N}=R_{N}\pi_{1}$. We now look to $\pi_{1,N+1}$ that must satisfy

$\pi_{1,N+1}=R_{N}+\pi_{1,N+1}\pi_{1}+\pi_{1}\pi_{1,N+1}.$

This relation fixes the diagonal part of $\pi_{1,N+1}$ according to

$\pi_{1}\pi_{1,N+1}\pi_{1}=-\pi_{1}R_{N}\pi_{1}\;\;\mbox{and}\;\;\pi_{2}\pi_{1,N+1}\pi_{2}=\pi_{2}R_{N}\pi_{2},$

We will see later that we do not need to prescribe off-diagonal components to
$\pi_{1,N+1}$.

Let us now focus on $h_{1,N+1}$, $h_{2,N+1}$ and $W_{N+1}$. We write

$\pi^{\varepsilon,N}_{1}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=(i\varepsilon\partial_{t}-h^{\varepsilon,N}_{1})\circledast\pi_{1}^{\varepsilon,N}+W^{\varepsilon,N}\circledast\pi_{2}^{\varepsilon,N}+\varepsilon^{N+1}\Theta^{\varepsilon}_{1}$

where $\Theta^{\varepsilon}_{1}$ is an asymptotic series of first term
$\Theta_{1,N}$. For obtaining information about $\Theta_{1,N}$, we compute

$\pi^{\varepsilon,N}_{\ell}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})\circledast\pi^{\varepsilon,N}_{\ell^{\prime}}$

for different choices of $\ell,\ell^{\prime}\in\\{1,2\\}$.

  * •

Taking $\ell\not=\ell^{\prime}$ gives two relations

$\displaystyle\pi^{\varepsilon,N}_{1}$
$\displaystyle\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})\circledast\pi^{\varepsilon,N}_{2}$
$\displaystyle=\pi^{\varepsilon,N}_{1}\circledast
W^{\varepsilon,N}\circledast\pi^{\varepsilon,N}_{2}+\varepsilon^{N+1}\pi^{\varepsilon,N}_{1}\circledast\Theta^{\varepsilon}_{1}\circledast\pi^{\varepsilon,N}_{2}+O(\varepsilon^{N+2})$
$\displaystyle\pi^{\varepsilon,N}_{2}$
$\displaystyle\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})\circledast\pi^{\varepsilon,N}_{1}$
$\displaystyle=\pi^{\varepsilon,N}_{2}\circledast(W^{\varepsilon,N})^{*}\circledast\pi^{\varepsilon,N}_{1}+\varepsilon^{N+1}\pi^{\varepsilon,N}_{2}\circledast\Theta^{\varepsilon}_{2}\circledast\pi^{\varepsilon,N}_{1}+O(\varepsilon^{N+2}),$

from which we deduce
$\pi_{2}\Theta_{2,N}\pi_{1}=\left(\pi_{1}\Theta_{1,N}\pi_{2}\right)^{*}$.

  * •

Taking $\ell=\ell^{\prime}$ gives the relations

$\displaystyle\pi^{\varepsilon,N}_{\ell}$
$\displaystyle\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})\circledast\pi^{\varepsilon,N}_{\ell}$
$\displaystyle=\pi^{\varepsilon,N}_{\ell}\circledast(i\varepsilon\partial_{t}-h^{\varepsilon,N}_{\ell})\circledast\pi^{\varepsilon,N}_{\ell}+\varepsilon^{N+1}\pi^{\varepsilon,N}_{\ell}\circledast\Theta^{\varepsilon}_{\ell}\circledast\pi^{\varepsilon,N}_{\ell}+O(\varepsilon^{N+2}),$

whence the self-adjointness of the diagonal part of
$\Theta^{\varepsilon}_{\ell}$.

We now enter into the construction of $h_{1,N+1}$, $h_{2,N+1}$ and $W_{N+1}$.
We write the asymptotic series

$\displaystyle\pi^{\varepsilon,N+1}_{1}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=\pi^{\varepsilon,N}_{1}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})-\varepsilon^{N+1}\pi_{1,N+1}H_{0}+O(\varepsilon^{N+2}),$
$\displaystyle(i\varepsilon\partial_{t}-h^{\varepsilon,N+1}_{1})\circledast\pi_{1}^{\varepsilon,N+1}+W^{\varepsilon,N+1}\circledast\pi_{2}^{\varepsilon,N+1}=(i\varepsilon\partial_{t}-h^{\varepsilon,N}_{1})\circledast\pi_{1}^{\varepsilon,N}+W^{\varepsilon,N}\circledast\pi_{2}^{\varepsilon,N}$
$\displaystyle\qquad\qquad+\varepsilon^{N+1}\left(i\partial_{t}\pi_{1,N}-h_{1,N+1}\pi_{1}-h_{1}\pi_{1,N+1}+W_{N+1}\pi_{2}\right)+O(\varepsilon^{N+2}).$

Therefore, we look for $h_{1,N+1}$ and $W_{N+1}$ such that

$-\pi_{1,N+1}H_{0}=i\partial_{t}\pi_{1,N}-h_{1,N+1}\pi_{1}-h_{1}\pi_{1,N+1}+W_{N+1}\pi_{2}+\Theta_{1,N}$

or equivalently

$0=i\partial_{t}\pi_{1,N}-h_{1,N+1}\pi_{1}+(h_{2}-h_{1})\pi_{1,N+1}\pi_{2}+W_{N+1}\pi_{2}+\Theta_{1,N}.$

By multiplying on the right by $\pi_{1}2$ and $\pi_{2}$, we are left with the
two equations

(4.12)
$h_{1,N+1}\pi_{1}=\Theta_{1,N}\pi_{1}+i\partial_{t}\pi_{1,N}\pi_{1}\;\;\mbox{and}\;\;W_{N+1}\pi_{2}=\Theta_{1,N}\pi_{2}+(h_{2}-h_{1})\pi_{1,N+1}\pi_{2}+i\partial_{t}\pi_{1,N}\pi_{2}.$

Considering similarly the conditions for the mode $h_{2}$, we obtain that
$h_{2,N+1}$ and $W_{N+1}^{*}$ have to satisfy

$h_{2,N+1}\pi_{2}=\Theta_{2,N}\pi_{2}+i\partial_{t}\pi_{2,N}\pi_{2}\;\;\mbox{and}\;\;W_{N+1}^{*}\pi_{1}=\Theta_{2,N}\pi_{1}-(h_{2}-h_{1})\pi_{2,N+1}\pi_{1}+i\partial_{t}\pi_{2,N}\pi_{1}.$

Since $\pi_{2,N}=-\pi_{1,N}$ for $N\geq 1$, we are left with the relation

(4.13)
$h_{2,N+1}\pi_{2}=\Theta_{2,N}\pi_{2}-i\partial_{t}\pi_{1,N}\pi_{2}\;\;\mbox{and}\;\;W_{N+1}^{*}\pi_{1}=\Theta_{2,N}\pi_{1}+(h_{2}-h_{1})\pi_{1,N+1}\pi_{1}-i\partial_{t}\pi_{1,N}\pi_{1}.$

We set

$\displaystyle h_{1,N+1}$
$\displaystyle=\Theta_{1,N}\pi_{1}+i\partial_{t}\pi_{1,N}\pi_{1}+\pi_{1}\Theta_{1,N}^{*}\pi_{2}-i\pi_{1}\partial_{t}\pi_{1,N}\pi_{2},$
$\displaystyle h_{2,N+1}$
$\displaystyle=\Theta_{2,N}\pi_{2}+i\partial_{t}\pi_{2,N}\pi_{2}+\pi_{2}\Theta_{2,N}^{*}\pi_{1}-i\pi_{2}\partial_{t}\pi_{2,N}\pi_{1}.$

Then, $h_{1,N+1}$ and $h_{2,N+1}$ are self-adjoint and satisfy the first part
of (4.12) and (4.13) respectively.

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

$W_{N+1}=\Theta_{1,N}\pi_{2}+(h_{2}-h_{1})\pi_{1,N+1}\pi_{2}+i\partial_{t}\pi_{1,N}\pi_{2}+U_{N+1}\pi_{1},$

which guarantees (4.12). Then, one has

$W_{N+1}^{*}=\pi_{2}\Theta_{1,N}^{*}+(h_{2}-h_{1})\pi_{2}\pi_{1,N+1}-i\pi_{2}\partial_{t}\pi_{1,N}+\pi_{1}U_{N+1}^{*}$

and

$\displaystyle W_{N+1}^{*}\pi_{1}$
$\displaystyle=\pi_{2}\Theta_{1,N}^{*}\pi_{1}-i\pi_{2}\partial_{t}\pi_{1,N}\pi_{1}+\pi_{1}U_{N+1}^{*}\pi_{1}$
$\displaystyle=\pi_{2}\Theta_{2,N}\pi_{1}-i\pi_{2}\partial_{t}\pi_{1,N}\pi_{1}+\pi_{1}U_{N+1}^{*}\pi_{1}$

where we have used the first property of the matrices $\Theta_{1,N}$ and
$\Theta_{2,N}$ that we have exhibited, together with the fact that
$\pi_{1,N+1}$ is diagonal. It is then enough to choose

$U_{N+1}=\pi_{1}\left(\Theta_{2,N}^{*}+(h_{2}-h_{1})\pi_{1,N+1}+i\partial_{t}\pi_{1,N}\right)\pi_{1}$

since it implies

$\displaystyle W_{N+1}^{*}\pi_{1}$
$\displaystyle=\pi_{2}\Theta_{2,N}\pi_{1}-i\pi_{2}\partial_{t}\pi_{1,N}\pi_{1}$
$\displaystyle\qquad+\pi_{1}(\Theta_{2,N}+i\partial_{t}\pi_{1,N}+(h_{2}-h_{1})\pi_{1,N+1})\pi_{1}$
$\displaystyle=\Theta_{2,N}\pi_{1}+(h_{2}-h_{1})\pi_{1,N+1}\pi_{1}-i\pi_{2}\partial_{t}\pi_{1,N}\pi_{1},$

where we have used $\pi_{1,N+1}\pi_{1}=\pi_{1}\pi_{1,N+1}\pi_{1}$. As a
consequence, the second part of (4.13) is satisfied. This concludes the
recursive argument and the proof of the Theorem 4.5 since the growth
properties of the matrices that we have constructed come with the recursive
equations. ∎

####  4.3. Superadiabatic projectors and diagonalization

One now wants to get rid of the off-diagonal elements
$\widehat{W}^{\varepsilon}$, which is possible outside $\Upsilon$. We are
going to take into account how far from the crossing set we are by introducing
a gap assumption.

######  Assumption 4.7 (Gap assumption).

Let $t_{0}<t_{1}$, $I$ an open interval of ${\mathbb{R}}$ containing
$[t_{0},t_{1}]$ and $\Omega$ an open subset of ${\mathbb{R}}^{2d}$. We say
that the eigenvalue $h$ has a gap larger than $\delta\in(0,1]$ in
$\mathcal{D}:=I\times\Omega$ if one has

  1. $({NC}_{\delta})$

$\displaystyle{d\left(h(t,z),{\rm
Sp}(H_{0}(t,z))\right)\geq\delta,\;\;\forall(t,z)\in\mathcal{D}.}$

The construction of superadiabatic projectors dates to [5] which was inspired
by the paper [17]. It has then been carefully developed in [46] and [55] (see
also the book [58]). We revisit here the construction of superadiabatic
projectors, in order to control their norms with respect to the parameter
$\delta$.

We follow the construction of the Section 14.4 of the latest edition of [13]
(2021), that we adapt to our context. One proceeds in two steps: first by
defining the formal series for the projectors and then for the Hamiltonians.
In order to simplify the notations in the construction, we just consider an
eigenvalue $h$ and we will then apply the result to the eigenvalues $h_{1}$
and $h_{2}$.

#####  4.3.1. Formal superadiabatic projectors

######  Theorem 4.8 (semiclassical projector evolution).

Assume the eigenvalue $h$ of the Hamiltonian $H_{0}$ satisfies Assumption 4.7
in $\mathcal{D}$. Then, there exists a unique formal series $\sum_{j\geq
1}\varepsilon^{j-1}\Pi_{j}$ in
$\boldsymbol{S}_{\varepsilon,\delta}^{-1}(\mathcal{D})$ such that setting
$\Pi_{0}(t,z)=\pi(t,z)$, the formal series

$\displaystyle{\Pi^{\varepsilon}(t,z)=\sum_{j\geq
0}\varepsilon^{j}\Pi_{j}(t,z)}$

is a formal projection and

(4.14)
$i\varepsilon\partial_{t}\Pi^{\varepsilon}(t)=[H^{\varepsilon}(t),\Pi^{\varepsilon}(t)]_{\circledast}.$

Moreover the sub-principal term $\Pi_{1}(t)$ is an Hermitian matrix given by
the following formulas:

(4.15)
$\displaystyle\pi(t)\Pi_{1}(t)\pi(t)=-\frac{1}{2i}\pi(t)\\{\pi(t),\pi(t)\\}\pi(t),$
$\displaystyle\pi(t)^{\perp}\Pi_{1}(t)\pi(t)^{\perp}=\frac{1}{2i}\pi(t)^{\perp}\\{\pi(t),\pi(t)\\}\pi(t)^{\perp},$
$\displaystyle\pi(t)^{\perp}\Pi_{1}(t)\pi(t)=\pi(t)^{\perp}(H_{0}(t)-h(t))^{-1}\pi(t)^{\perp}R_{1}(t)\pi(t),$

where

$R_{1}(t)=i\partial_{t}\pi(t)-\frac{1}{2i}\left(\\{H_{0}(t),\pi(t)\\}-\\{\pi(t),H_{0}(t)\\}\right)-[H_{1}(t),\pi(t)].$

###### Proof.

With Notations 4.3,

(4.16)
$\displaystyle\Pi^{\varepsilon,N}\circledast\Pi^{\varepsilon,N}-\Pi^{\varepsilon,N}$
$\displaystyle=\varepsilon^{N+1}S_{N+1}+O(\varepsilon^{N+2}),$ (4.17)
$\displaystyle
i\varepsilon\partial_{t}\Pi^{\varepsilon,N}-\left[H_{0}+\varepsilon
H_{1},\Pi^{\varepsilon,N}\right]_{\circledast}$
$\displaystyle=\varepsilon^{N+1}R_{N+1}+O(\varepsilon^{N+2}).$

Step $N=1$. We start with $N=0$. We have
$\Pi^{(0)}=\pi\in\boldsymbol{S}^{0}_{\delta}(\mathcal{D})$. Since
$\pi^{2}=\pi$ and $[H_{0},\pi]=0$, we obtain

$S_{1}=\tfrac{1}{2i}\\{\pi,\pi\\}\;\mbox{and}\;R_{1}=i\partial_{t}\pi-\tfrac{1}{2i}(\\{H_{0},\pi\\}-\\{\pi,H_{0}\\})-[H_{1},\pi],$

and we have $R_{1},S_{1}\in\boldsymbol{S}^{0}_{\delta}(\mathcal{D})$. Two
structural observations are in order:

  1. (1)

The matrix $S_{1}$ is symmetric and satisfies $\pi
S_{1}\pi^{\perp}=\pi^{\perp}S_{1}\pi=0$.

  2. (2)

The matrix $R_{1}$ is skew-symmetric. It satisfies

$\pi
R_{1}\pi=0\quad\text{and}\quad\pi^{\perp}R_{1}\pi^{\perp}=[H_{0},\pi^{\perp}S_{1}\pi^{\perp}].$

If $H_{0}$ has only two eigenvalues, $H_{0}$ expresses only in terms of $\pi$
and the expression of $R_{1}$ given above shows that $R_{1}$ is off-diagonal.
The situation is more complicated if $H_{0}$ has strictly more than two
distinct eigenvalues. For verifying (2) in that case, one uses the Poisson
bracket rule $\\{A,BC\\}-\\{AB,C\\}=\\{A,B\\}C-A\\{B,C\\}$ two times. We
obtain

$\displaystyle\\{H_{0},\pi\\}-\\{\pi,H_{0}\\}$
$\displaystyle=\\{H_{0},\pi^{2}\\}-\\{\pi^{2},H_{0}\\}$
$\displaystyle=\\{h\pi,\pi\\}+\\{H_{0},\pi\\}\pi-
H_{0}\\{\pi,\pi\\}-\\{\pi,h\pi\\}+\\{\pi,\pi\\}H_{0}-\pi\\{\pi,H_{0}\\}$
$\displaystyle=\pi\\{h,\pi\\}-\\{\pi,h\\}\pi+[\\{\pi,\pi\\},H_{0}]+\\{H_{0},\pi\\}\pi-\pi\\{\pi,H_{0}\\},$

which implies

$\pi^{\perp}(\\{H_{0},\pi\\}-\\{\pi,H_{0}\\})\pi^{\perp}=\pi^{\perp}[\\{\pi,\pi\\},H_{0}]\pi^{\perp}.$

For determining the $\pi$-diagonal component, we choose $A=\pi$,
$B=H_{0}\pi^{\perp}$, and $C=\pi$ to obtain

$\displaystyle 0$
$\displaystyle=\\{\pi,H_{0}\pi^{\perp}\\}\pi-\pi\\{H_{0}\pi^{\perp},\pi\\}$
$\displaystyle=\\{\pi,H_{0}\\}\pi-\\{\pi,h\pi\\}\pi-\pi\\{H_{0},\pi\\}+\pi\\{h\pi,\pi\\}$
$\displaystyle=\\{\pi,H_{0}\\}\pi-\\{\pi,h\\}\pi-\pi\\{H_{0},\pi\\}+\pi\\{h,\pi\\}.$

This relation implies

$\pi(\\{H_{0},\pi\\}-\\{\pi,H_{0}\\})\pi=0.$

For constructing the matrix $\Pi_{1}$ that defines
$\Pi^{(1)}=\pi+\varepsilon\Pi_{1}$, we need to satisfy

$\pi\Pi_{1}+\Pi_{1}\pi-\Pi_{1}=-S_{1}\;\;\mbox{and}\;\;-[H_{0},\Pi_{1}]=-R_{1}.$

The first of these two equations uniquely determines the diagonal blocks of
$\Pi_{1}$, while the second equation uniquely determines the off-diagonal
blocks. We obtain

$\displaystyle\pi\Pi_{1}\pi=-\pi
S_{1}\pi\;\;\mbox{and}\;\;\pi^{\perp}\Pi_{1}\pi^{\perp}=\pi^{\perp}S_{1}\pi^{\perp},$
$\displaystyle\pi\Pi_{1}\pi^{\perp}=-\pi
R_{1}\pi^{\perp}(H_{0}-h)^{-1}\;\;\mbox{and}\;\;\pi^{\perp}\Pi_{1}\pi=(H_{0}-h)^{-1}\pi^{\perp}R_{1}\pi.$

For concluding this first step, we deduce from
$R_{1},S_{1}\in\boldsymbol{S}^{0}_{\delta}(\mathcal{D})$ that
$\Pi_{1}\in\boldsymbol{S}_{\delta}^{-1}(\mathcal{D})$.

Step $N\geq 1$. Next we proceed by induction and assume that we have
constructed the matrices
$\Pi_{j}(t)\in\boldsymbol{S}_{\delta}^{1-2j}(\mathcal{D})$ for $1\leq j\leq N$
such that (4.16) and (4.17) hold. Note that by Lemma 4.3, this implies

$R_{N+1}(t)\in\boldsymbol{S}_{\delta}^{-2N}(\mathcal{D})\;\;\mbox{and}\;\;S_{N+1}(t)\in\boldsymbol{S}_{\delta}^{-2N}(\mathcal{D}).$

Indeed, $i\varepsilon\partial_{t}\Pi^{\varepsilon,N}-\left[H_{0}+\varepsilon
H_{1},\Pi^{\varepsilon,N}\right]_{\circledast}$ is a formal series of
$\varepsilon\,\boldsymbol{S}^{-2}_{\varepsilon,\delta}(\mathcal{D})$ while
$\Pi^{N}\circledast\Pi^{\varepsilon,N}-\Pi^{\varepsilon,N}$ is a formal series
of $\varepsilon\boldsymbol{S}^{-1}_{\varepsilon,\delta}(\mathcal{D})$. In
order to go one step further, we see that $\Pi_{N+1}$ has to satisfy

$\pi\Pi_{N+1}+\Pi_{N+1}\pi-\Pi_{N+1}=-S_{N+2}\;\;\mbox{and}\;\;-[H_{0},\Pi_{N+1}]=-R_{N+2}.$

For solving these equations, and achieving the recursive process, we need to
verify that at each step

  1. (1)

The matrix $S_{N}$ is symmetric and satisfies $\pi
S_{N}\pi^{\perp}=\pi^{\perp}S_{N}\pi=0$.

  2. (2)

The matrix $R_{N}$ is skew-symmetric and off-diagonal. It satisfies

$\pi
R_{N}\pi=0\quad\text{and}\quad\pi^{\perp}R_{N}\pi^{\perp}=[H_{0},\pi^{\perp}S_{N}\pi^{\perp}].$

Then, we will be able to construct
$\Pi_{N+1}(t)\in\boldsymbol{S}_{\delta}^{-2N-1)}(\mathcal{D})$ and we will
have as a by product
$R_{N+2}(t)\in\boldsymbol{S}_{\delta}^{-2N-2}(\mathcal{D})$,
$S_{N+2}(t)\in\boldsymbol{S}_{\delta}^{-2N-2}(\mathcal{D})$ because of
equations (4.16), (4.17) and Lemma 4.3.

For proving (1), we take advantage of the fact that

$Z:=\Pi^{\varepsilon,N}\circledast\left((\Pi^{\varepsilon,N})^{2\circledast}-\Pi^{\varepsilon,N}\right)\circledast(\mathbb{I}-\Pi^{\varepsilon,N})=\varepsilon^{N+1}\pi
S_{N+1}\pi^{\perp}+O(\varepsilon^{N+2}),$

while one also has by construction

$Z=\left((\Pi^{\varepsilon,N})^{2\circledast}-\Pi^{\varepsilon,N}\right)\circledast\left(\Pi^{\varepsilon,N}-(\Pi^{\varepsilon,N})^{2\circledast}\right)={\mathcal{O}}(\varepsilon^{2N+2}).$

This implies that $\pi S_{N+1}\pi^{\perp}=0$ and, using that $S_{N+1}$ is
hermitian, we deduce that it is diagonal.

For proving (2), we argue similarly with

$Z^{\prime}:=\Pi^{\varepsilon,N}\circledast\left(i\varepsilon\partial_{t}\Pi^{\varepsilon,N}-\left[H_{0}+\varepsilon
H_{1},\Pi^{\varepsilon,N}\right]_{\circledast}\right)\circledast\Pi^{\varepsilon,N}=\varepsilon^{N+1}\pi
R_{N+1}\pi+O(\varepsilon^{N+2}),$

which also satisfies

$Z^{\prime}=\left((\Pi^{\varepsilon,N})^{2\circledast}-\Pi^{\varepsilon,N}\right)\circledast(H_{0}+\varepsilon
H_{1})\Pi^{\varepsilon,N}-\Pi^{\varepsilon,N}(H_{0}+\varepsilon
H_{1})\left((\Pi^{\varepsilon,N})^{2\circledast}-\Pi^{\varepsilon,N}\right)=O(\varepsilon^{2N+2}).$

This implies $\pi R_{N+1}\pi=0$ and one can argue similarly with
$1-\Pi^{\varepsilon,N}$ for obtaining the other relation
$\pi^{\perp}R_{N+1}\pi^{\perp}=0$. ∎

#####  4.3.2. Formal adiabatic decoupling

The second (and decisive) part of the analysis is a formal adiabatic
decoupling using the superadiabatic projectors introduced before.

######  Theorem 4.9 (formal adiabatic decoupling).

Assume the eigenvalue $h$ of the Hamiltonian $H_{0}$ satisfies Assumption 4.7
in $\mathcal{D}$. There exists a formal time dependent Hermitian Hamiltonian
in $\mathcal{D}$,

$\displaystyle{H^{{\rm adia},\varepsilon}=\sum_{j\geq 0}\varepsilon^{j}H^{\rm
adia}_{j}}$

such that

(4.18)
$\Pi^{\varepsilon}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=(i\varepsilon\partial_{t}-H^{{\rm
adia},\varepsilon})\circledast\Pi^{\varepsilon}$

with the following properties:

  1. (1)

The principal symbol is $H^{\rm adia}_{0}=h\,\mathbb{I}_{{\mathbb{C}}^{m}}$.

  2. (2)

The subprincipal term $H_{1}^{\rm adia}$ is a Hermitian matrix satisfying

$\displaystyle\pi^{\perp}\,H_{1}^{\rm
adia}\pi=\pi^{\perp}\left(i\partial_{t}\pi+i\\{h,\pi\\}\right)\pi\;\;\mbox{and}\;\;\pi\,H_{1}^{\rm
adia}\pi$ $\displaystyle=\pi H_{1}\pi+\frac{1}{2i}\pi\\{H_{0},\pi\\}\pi$

(see (1.19)) and we can choose $\pi^{\perp}H^{{\rm
adia},\varepsilon}_{1}\pi^{\perp}=0.$

  3. (3)

We have

$\varepsilon^{-2}\left(H^{{\rm
adia},\varepsilon}-h\mathbb{I}_{m,m}-\varepsilon H^{\rm
adia}_{1}\right)\in{\bf S}_{\varepsilon,\delta}^{-1}(\mathcal{D}).$

  4. (4)

Finally, $\pi(t)$ satisfies a transport equation along the classical flow for
$h(t)$.

(4.19) $\partial_{t}\pi+\\{h,\pi\\}=\frac{1}{i}[H^{{\rm adia}}_{1},\pi].$

######  Remark 4.10.

Note that equations (1.19) implies that $H^{\rm adia}_{1}(t,z)$ is smooth
everywhere, including on the crossing set, if any.

The above construction applied to the Hamiltonian $H^{\varepsilon}$ with two
smooth eigenvalues $(h_{1},h_{2})$ and two smooth eigenprojectors
$(\pi_{1},\pi_{2})$ imply the construction of two pairs of formal series

(4.20) $\Pi^{\varepsilon}_{\ell}=\sum_{j\geq
0}\varepsilon^{j}\Pi_{\ell,j}\;\;{\rm and}\;\;H^{{\rm
adia},\varepsilon}_{\ell}=\sum_{j\geq 0}\varepsilon^{j}H^{\rm adia}_{\ell,j}.$

######  Corollary 4.11.

At the level of the evolution operator, the result implies

$\mathcal{U}^{\varepsilon}_{H}(t,t_{0})=\widehat{\Pi_{1}^{\varepsilon}}(t)\mathcal{U}^{\rm
adia,\varepsilon}_{1}(t,t_{0})\widehat{\Pi_{1}^{\varepsilon}}(t_{0})+\widehat{\Pi_{2}^{\varepsilon}}(t)\mathcal{U}^{\rm
adia,\varepsilon}_{2}(t,t_{0})\widehat{\Pi_{2}^{\varepsilon}}(t_{0})$

where for $\ell\in\\{1,2\\}$, $\mathcal{U}^{\rm
adia,\varepsilon}_{\ell}(t,t_{0})$ are the evolution operators associated with
the Hamiltonian $H^{{\rm adia},\varepsilon}_{\ell}$

###### Proof.

This result is Theorem 80 of Chapter 14 in [13] combined with Lemma 4.3. We
first observe that equation (4.18) reduces to proving

(4.21)
$\Pi^{\varepsilon}\circledast(i\varepsilon\partial_{t}-H^{\varepsilon})=(i\varepsilon\partial_{t}-H^{{\rm
adia},\varepsilon})\circledast\Pi^{\varepsilon}.$

For proving the latter relation, one first observes that if $H_{0}^{\rm
adia}=h$, then we have

$\displaystyle(H^{\rm adia,\varepsilon}-H_{0}-\varepsilon
H_{1})\circledast\Pi^{\varepsilon}$
$\displaystyle=\varepsilon\left((h-H_{0})\Pi_{1}+(H_{1}^{\rm
adia}-H_{1})\pi+\frac{1}{2i}\\{h-H_{0},\pi\\}\right)+i\partial_{t}\pi+\mathcal{O}(\varepsilon^{2}).$

Therefore, $H_{1}$ has to be chosen so that

$(H_{1}^{\rm
adia}-H_{1})\pi=(H_{0}-h)\Pi_{1}+\frac{1}{2i}\\{H_{0}-h,\pi\\}+i\partial_{t}\pi.$

In view of (4.15), this requires

$\pi(H_{1}^{\rm adia}-H_{1})\pi=\frac{1}{2i}\pi\\{H_{0},\pi\\}\pi,$

which is given by the second relation of (1.19), and, using again (4.15)

$\pi^{\perp}(H_{1}^{\rm
adia}-H_{1})\pi=\pi^{\perp}(R_{1}+\frac{1}{2i}\\{H_{0}-h,\pi\\}+i\partial_{t}\pi)\pi=\pi^{\perp}(i\partial_{t}\pi+\frac{1}{2i}\\{\pi,H_{0}\\}-\frac{1}{2i}\\{h,\pi\\}-H_{1})\pi$

which is also given by the first relation of (1.19) in view of the observation
that

$\pi^{\perp}\\{\pi,H_{0},\pi\\}\pi=-\pi^{\perp}\\{h,\pi\\}\pi.$

For proving this relation, one uses the Poisson bracket rule

$\\{A,BC\\}-\\{AB,C\\}=\\{A,B\\}C-A\\{B,C\\}$

several times. First, one gets

$\pi\\{\pi,\pi\\}\pi^{\perp}=0=\pi^{\perp}\\{\pi,\pi\\}\pi.$

Then, taking $A=\pi^{\perp}$, $B=\pi$, $C=H_{0}$, one gets

$\displaystyle\\{\pi^{\perp},h\pi\\}-0$
$\displaystyle=\\{\pi^{\perp},\pi\\}H_{0}-\pi^{\perp}\\{\pi,H_{0}\\},$

whence $-\pi^{\perp}\\{\pi,h\\}\pi=-\pi^{\perp}\\{\pi,H_{0}\\}\pi.$ Finally,
for concluding the construction of $H_{1}^{\rm adia}$, It remains to check
that

$\left((H_{0}-h)\Pi_{1}+\frac{1}{2i}\\{H_{0}-h,\pi\\}+i\partial_{t}\pi\right)\pi^{\perp}=0$

which comes from the latter observation about $\\{H_{0},\pi\\}$.

Now that $H_{0}^{\rm adia}$ and $H_{1}^{\rm adia}$ are constructed, one uses a
recursive argument: assume that one has constructed $H_{j}^{\rm adia}$ for
$0\leq j\leq N$ with $H_{j}^{\rm adia}\in\boldsymbol{S}^{-2j}_{\delta}$ for
$j\in\\{2,\cdots,N\\}$ and such that has (4.21) holds up to
$\mathcal{O}(\varepsilon^{N+1})$. Let us construct $H_{N+1}^{\rm adia}$.
Setting as in Notation 4.3

$H^{{\rm adia},\varepsilon,N}=\sum_{j=1}^{N}\varepsilon^{j}H_{j}^{\rm adia}$

we write

$(H^{{\rm adia},\varepsilon,N}-H_{0}-\varepsilon
H_{1})\circledast\Pi^{\varepsilon,N}=\varepsilon^{N+1}T_{N}+O(\varepsilon^{N+2})$

with $T_{N}\in\boldsymbol{S}_{\delta}^{-2N-3}$ and we look for $H^{\rm
adia}_{N+1}$ such that $\pi H^{\rm adia}_{N+1}=T_{N}$. This is doable as long
as $\pi^{\perp}T_{N}=0$, which comes form the observation that

$(H_{0}+\varepsilon H_{1}-H^{{\rm
adia},\varepsilon,N})\circledast\Pi^{\varepsilon,N}\circledast(1-\Pi^{\varepsilon,N})=\varepsilon^{N+1}\pi^{\perp}T_{N}+\mathcal{O}(\varepsilon^{N+1})=\mathcal{O}(\varepsilon^{N+1})$

by the properties of superadiabatic projectors. Besides, $H^{\rm
adia}_{N+1}\in\boldsymbol{S}_{\delta}^{-2N-3}$, which fits with (ii) of
Definition 4.1

(3) comes from Lemma 4.3.

(4) comes from (1.19). ∎

###  Chapter 5 Propagation of wave packets through smooth crossings

In this section, we prove Theorem 1.21. We consider a subquadratic Hamiltonian
$H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ satisfying Assumptions 1.2 in
$I\times\Omega$, $\Omega\subset{\mathbb{R}}^{2d}$, and we are interested in
the description of the solution to equation (1.1) for initial data that is a
wave packet as in (1.8).

The proof consists in three steps: one first propagates the wave packet from
time $t_{0}$ to some time $t^{\flat}-\delta$, $\delta>0$ in a zone that is at
a distance of $\Upsilon$ of size larger than $c\delta$ for some constant
$c>0$. In this zone, we use the superadiabatic projectors. Then, we propagate
the wave packet from time $t^{\flat}-\delta$ to $t^{\flat}+\delta$, using the
rough diagonalization in the crossing region. Finally, between times
$t^{\flat}+\delta$ and $t_{1}$, we are again at distance larger than $c\delta$
to $\Upsilon$ and the analysis with superadiabatic projectors apply. The
parameter $\delta$ will be taken afterwards as
$\delta\approx\varepsilon^{\alpha}$; the analysis of Section 5.2 will ask for
$\alpha<\frac{1}{2}$ (see Theorem 5.9).

In order to explain carefully each step of the proof, we start by proving the
propagation faraway from the crossing area in Section 5.1. That allows us to
settle the arguments, before doing it precisely close to $\Upsilon$ in Section
5.2. Then, Section 5.3 is devoted to the calculus of the transitions in the
crossing region.

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

######  Assumption 5.1 (Dynamical assumption).

We say that $\Omega_{1}$ and $t_{1}$ satisfy the dynamical assumption $(DA)$
for the mode $h_{\ell}$ if we have

  1. $(DA)$

$\Phi^{t,t_{0}}_{\ell}(\Omega_{1})\subset\Omega$ for all $t\in[t_{0},t_{1}]$.

####  5.1. Propagation faraway from the crossing area

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

$\Omega:=\\{z\in{\mathbb{R}}^{2d},\;|h_{2}(t,z)-h_{1}(t,z)|>\delta_{0},\;\;\forall
t\in[t_{0},t_{1}]\\},\;\;\mathcal{D}=I\times\Omega$

where $I$ is an open interval of ${\mathbb{R}}$ containing $[t_{0},t_{1}]$ and
where the gap condition is also satisfied. We associate with $H^{\varepsilon}$
the formal series of Theorems 4.9 and 4.8 for each of the modes:

$\Pi^{\varepsilon}_{\ell}=\sum_{j\geq 0}\varepsilon^{j}\Pi_{\ell,j}\;\;{\rm
and}\;\;H^{{\rm adia},\varepsilon}_{\ell}=\sum_{j\geq 0}\varepsilon^{j}H^{\rm
adia}_{\ell,j}$

and we will use the notation introduced in (4.3).

With $z_{0}\in\Omega$, we associate the open sets $\Omega_{0}$, $\Omega_{1}$,
$\Omega_{2}$ and $\Omega_{3}$ such that

$z_{0}\in\Omega_{0}\Subset\Omega_{2}\Subset\Omega_{1}\Subset\Omega_{3}\Subset\Omega$

where $\Omega_{0}$ and $\Omega_{2}$ are constructed so that for any initial
data $z\in\Omega_{0}$ the flows are staying in $\Omega_{2}$:

$\Phi_{h_{\ell}}^{t,t_{0}}(z)\in\Omega_{2},\;\;\forall
t\in[t_{0},t_{1}],\;\;\forall z\in\Omega_{0},\;\;\forall\ell\in\\{1,2\\}.$

We associate cut-offs to these subsets. We take $\chi_{0}\in
C_{0}^{\infty}(\Omega_{0})$ with $\chi_{0}=1$ near $z_{0}$. Then, we choose
$K_{0}$ a compact neighborhood of $z_{0}$ in $\Omega_{0}$, which implies that
$\Omega_{2}$ is a neighborhood of

$\widetilde{K}_{\ell,0}:=\\{\Phi_{h_{\ell}}^{t,t_{0}}(K_{0}),\;t_{0}\leq t\leq
t_{1}\\},\;\;\ell\in\\{1,2\\}.$

So we can choose $\chi_{2}\in C_{0}^{\infty}(\Omega)$ with $\chi_{2}=1$ on
$\widetilde{K}_{0}=\widetilde{K}_{1,0}\cup\widetilde{K}_{2,0}$. Finally, we
take $\chi_{1},\chi_{3}\in C_{0}^{\infty}(\Omega)$ with $\chi_{1}=1$ on ${\rm
supp}(\chi_{2})$ and $\chi_{3}=1$ on ${\rm supp}(\chi_{1})$.

For $\ell\in\\{1,2\\}$, we set

$\widetilde{H}^{{\rm adia},\varepsilon,N}_{\ell}(t)=\chi_{3}\,H^{{\rm
adia},\varepsilon,N}_{\ell}(t),$

which is a smooth subquadratic Hamitonian, and we consider
${\mathcal{U}}^{{\rm adia},\varepsilon,N}_{\ell}(t,s)$ the propagator
associated with the Hamiltonian $\chi_{1}\,H_{\ell}^{{\rm
adia},\varepsilon,N}(t)$.

The next result is the usual adiabatic decoupling that results from the
preceding analysis.

######  Proposition 5.2 (adiabatic decoupling - I).

Let $k\in{\mathbb{N}}$.

  1. (i)

For any $\ell\in\\{1,2\\}$, we have in
$\mathcal{L}(\Sigma^{k}_{\varepsilon})$,

(5.1)
$\displaystyle\left(i\varepsilon\partial_{t}-\widehat{H}^{\varepsilon}(t)\right){\rm
op}_{\varepsilon}\left(\chi_{1}\Pi^{N,\varepsilon}_{\ell}(t)\right)\,{\rm
op}_{\varepsilon}(\chi_{2})=$ $\displaystyle\qquad{\rm
op}_{\varepsilon}\left(\chi_{1}\Pi^{N,\varepsilon}_{\ell}(t)\right)\,\left(i\varepsilon\partial_{t}-{\rm
op}_{\varepsilon}\left(\chi_{3}H^{{\rm
adia},N,\varepsilon}_{\ell}(t)\right)\right)\,{\rm
op}_{\varepsilon}(\chi_{2})+{\mathcal{O}}(\varepsilon^{N+1}).$

  2. (ii)

Let $\psi^{\varepsilon}_{0}\in\Sigma^{k}_{\varepsilon}$ such that
$\widehat{\chi}_{0}\psi^{\varepsilon}_{0}=\psi^{\varepsilon}_{0}+{\mathcal{O}}(\varepsilon^{\infty}).$
Set

$\psi^{\varepsilon,N}_{\ell}(t)={\rm
op}_{\varepsilon}\left(\chi_{1}\Pi_{\ell}^{N,\varepsilon}(t)\right)\,{\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})\,{\rm
op}_{\varepsilon}\left(\chi_{0}\Pi_{\ell}^{N,\varepsilon}(t_{0})\right)\psi^{\varepsilon}_{0},\;\;\ell\in\\{1,2\\}.$

Then we have in $\Sigma^{k}_{\varepsilon}$,

(5.2)
${\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}=\psi^{\varepsilon,N}_{1}(t)+\psi^{\varepsilon,N}_{2}(t)+{\mathcal{O}}(\varepsilon^{N+1}),\;\forall
t\in[t_{0},t_{1}].$

######  Remark 5.3.

  1. (1)

The assumption satisfied by $(\psi^{\varepsilon}_{0})_{\varepsilon>0}$ in (ii)
of Proposition 5.2 is sometimes referred in the literature as having a compact
semi-classical wave front set.

  2. (2)

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

###### Proof.

(i) We fix $\ell\in\\{1,2\\}$. Using Theorem 4.9, we obtain

$\displaystyle(i\varepsilon\partial_{t}-H^{\varepsilon}(t))\circledast\left.(\chi_{1}\Pi^{\varepsilon,N}_{\ell}(t)\right)$
$\displaystyle=\chi_{1}(i\varepsilon\partial_{t}-H^{\varepsilon}(t))\circledast\left(\chi_{1}\Pi^{\varepsilon,N}_{\ell}(t)\right)+\varepsilon^{N+1}R^{\varepsilon}_{N}$
$\displaystyle=\chi_{1}\left(\Pi^{\varepsilon,N}_{\ell}(t)\right)\circledast\left(i\varepsilon\partial_{t}-H^{{\rm
adia},\varepsilon,N}_{\ell}(t)\right)+\varepsilon^{N+1}R^{\varepsilon}_{N}$
$\displaystyle=\left(\chi_{1}\Pi^{\varepsilon,N}_{\ell}(t)\right)\circledast\left(i\varepsilon\partial_{t}-\widetilde{H}^{{\rm
adia},\varepsilon,N}_{\ell}(t)\right)+\varepsilon^{N+1}R^{\varepsilon}_{N}$

where the $R_{N}^{\varepsilon}$s are rest terms that may change from one line
to the other one and all satisfy $\chi_{2}R_{N}^{\varepsilon}=0$ and where we
have used that $\chi_{3}\chi_{1}=\chi_{1}$. The relation on operators follows
from Theorem B.1 and Corollary B.2 (with $\delta=1$).

(ii) Let $K_{0}={\rm supp}(\chi_{0})$. We apply (5.1) and we use that
$\chi_{2}$ is identically equal to $1$ on the compact set $\widetilde{K}_{0}$.
Hence, using Egorov Theorem of Appendix C.1, we have for $\ell\in\\{1,2\\}$

$\displaystyle\widehat{\chi}_{1}\,{\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})\,\widehat{\chi}_{0}$
$\displaystyle\,=\widehat{\chi}_{1}\,{\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})\,\widehat{\chi}_{0}\,{\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})^{-1}\,{\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})$ $\displaystyle\,={\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{0})\,\widehat{\chi}_{0}+{\mathcal{O}}(\varepsilon^{N+1}).$

Hence we deduce (5.2) from (5.1) using again the Egorov Theorem and that

${\rm
op}_{\varepsilon}\left(\chi_{1}\Pi_{1}^{N,\varepsilon}+\chi_{1}\Pi_{2}^{N,\varepsilon}\right)\widehat{\chi}_{0}=\widehat{\chi}_{0}+{\mathcal{O}}(\varepsilon^{N+1}).$

∎

The adiabatic decoupling of Proposition 5.2 and Egorov Theorem (see
Proposition C.1) allows to give an explicit description at any order of the
solution of equation (1.1) for initial data that are focalized wave packets.

Indeed, by the technics of Appendix C that are classic when $\delta=1$ (see
for example the recent edition of [13]), one constructs two maps
${\mathcal{R}}_{1}(t,t_{0},z)$ and ${\mathcal{R}}_{2}(t,t_{0},z)$ introduced
in (1.20)) and one has the following result (see Proposition C.5).

######  Theorem 5.4.

Assume that $\psi^{\varepsilon}_{0}$ a polarized wave packet:

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

Let $N\geq 1$ and $k\geq 0$. Then, there exists a constant $C_{N,k}>0$ such
that the solution $\psi^{\varepsilon}(t)$ of (1.1) satisfies for all
$t\in[t_{0},t_{1}]$,

$\left\|\psi^{\varepsilon}(t)-\left(\psi^{\varepsilon,N}_{1}(t)+\psi^{\varepsilon,N}_{2}(t)\right)\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C_{N,k}\,\varepsilon^{N},\;$

with for $\ell\in\\{1,2\\}$ and for all $M\geq 0$,

$\psi_{\ell}^{\varepsilon,N}(t)={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z_{0})}\,{\rm
WP}^{\varepsilon}_{z_{\ell}(t)}\left({\mathcal{R}}_{\ell}(t,t_{0},z_{0}){\mathcal{M}}[F_{\ell}(t,t_{0})]\sum_{0\leq
j\leq
M}\varepsilon^{j/2}\vec{B}_{\ell,j}(t)f_{0}\right)+{\mathcal{O}}(\varepsilon^{M/2}),$

where $\vec{B}_{\ell,j}(t)$ are differential operators of degree $\leq 3j$
with vector-valued time-dependent coefficients satisfying (1.40) and (1.41).

####  5.2. Propagation close to the crossing area

Our goal in this section is to extend the result of Theorem 5.4 up a the time
$t^{\flat}-c\delta$ for some $c>0$ and $\delta\ll 1$. We follow the same
strategy as in the preceding section and checks carefully the dependence in
$\delta$ of the estimates.

More precisely, the situation is the following: we consider a wave packet at
initial time $t_{0}$ that is focalized along the mode $h_{1}$ at some point
$z_{0}$. We let it evolve along that mode according to Theorem 5.4, up to
$(t_{1},z_{1})$ conveniently chosen and we consider
$\psi^{\varepsilon}(t_{1})$ as a new initial data, knowing that it is a wave-
packet, modulo ${\mathcal{O}}(\varepsilon^{\infty})$. The point
$(t_{1},z_{1})$ is chosen close enough to $(t^{\flat},\zeta^{\flat})$ such
that $|t_{1}-t^{\flat}|+|z_{1}-\zeta^{\flat}|\leq\eta_{0}$ where $\eta_{0}$ is
defined in the next Lemma.

######  Lemma 5.5.

Assume $(t^{\flat},\zeta^{\flat})$ is a generic smooth crossing point as in
Definition 1.2 and consider $\delta\in(0,1]$.

  1. (1)

There exist $\eta_{0}>0$ and $c_{0}>0$ such that we have

$|f(t,\Phi_{h_{1}}^{t,t_{1}}(z))|\geq c_{0}|t-t^{\flat}|\;\mbox{
if}\;\;|t-t^{\flat}|+|z-\zeta^{\flat}|\leq\eta_{0}.$

  2. (2)

There exists $c,M>0$ such that for all $(t,z)$ satisfying
$\left|z-\Phi_{h_{1}}^{t,t_{1}}(z_{1})\right|\leq c\delta$, we have

$|f(t,z)|\geq c_{0}\delta-Mc\delta\geq\frac{c_{0}}{2}\delta.$

###### Proof.

The result comes readily from the transversality of the curve
$t\mapsto\Phi_{h_{1}}^{t,t_{1}}(z)$ to the set $\Upsilon=\\{f=0\\}$. Recall
that this transversality is due to Point (b) of Assumption 1.2. ∎

Our goal is to prove accurate estimates for the evolution of the solution
$\psi^{\varepsilon}(t)$ of the Schrödinger equation with the initial data
$\psi^{\varepsilon}(t_{1})$, for $t\in[t_{1},t^{\flat}-c\delta]$. We thus have
to improve in this precise setting the accuracy of the estimates obtained
before for fixed $\delta=\delta_{0}$.

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

In several place we need to replace $\delta$ by $c\delta$, for a finite number
of $0<c=c_{0},c_{1},\cdots,c_{L}$ ($L\in{\mathbb{N}}$). We will not mention
that point each time.

#####  5.2.1. Localization up to the crossing region

We construct the cut-off functions by using thin tubes along the classical
trajectories. We use a straightening theorem for non singular vector fields.
We set $D(z_{1},\rho_{1})=\\{|z-z_{1}|\leq\rho_{1}\\}$ and consider a branch
of trajectory

${\mathcal{T}}_{1}:=\\{\Phi_{h}^{t,t_{1}}(z_{1}),\;t\in[t_{1},t_{1}^{+}]\\},\;\;t_{1}^{+}>t^{\flat}.$

######  Lemma 5.6.

[2] Let be $P_{1}$ a transverse hyperplane to the curve ${\mathcal{T}}_{1}$ in
$z_{1}$. There exist $\rho_{1}>0$ and $t_{1}^{-}<t_{1}<t^{\flat}<t_{1}^{+}$
such that the map

$(t,z)\mapsto\Phi^{t,t_{1}}_{h}(z)$

is a diffeomorphism from $]t_{1}^{-},t_{1}^{+}[\times D(z_{1},\rho_{1})$ onto
a neighborhood ${\mathcal{W}}_{1}$ of ${\mathcal{T}}_{1}$ in $P_{1}$.

Hence for any $z$ in the tube ${\mathcal{W}}_{1}$, we have

$z=\Phi_{h}^{\tau(z),t_{1}}(Y(z))$

where $\tau$ and $Y$ are smooth functions of $z$,
$\tau(z)\in[t_{1},t_{1}^{+}]$, $Y(z)\in D(z_{1},\rho_{1})$.

We then define the cut-off functions as follows: consider

  * •

$\zeta\in C_{0}^{\infty}(]-2,2[)$ equal to 1 in $[-1,1]$,

  * •

$\theta\in C^{\infty}({\mathbb{R}})$ with $\theta(t)=0$ if $t\leq-1$ and
$\theta(t)=1$ if $t\geq 1$,

we set for $\delta>0$,

$\chi^{\delta}(z)=\theta\left(\frac{\tau(z)-t_{1}^{-}}{\eta}\right)\left(1-\zeta\right)\left(\frac{\tau(z)-t^{\flat}}{c\delta}\right)\zeta\left(\frac{|z-\Phi_{h}^{\tau(z),t_{1}}(z_{1})|^{2}}{(C\delta)^{2}}\right),$

where $c>0,C>0$ and $\eta>0$ is a small enough constant.

By adapting the constants $c$ and $C$ conveniently, we construct some
functions

$\chi^{\delta}_{j}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{2d},[0,1]),\;\;j\in\\{1,2,3\\},$

such that

  1. (1)

$\chi^{\delta}_{j}=1$ on $\displaystyle{\bigcup_{t_{0}\leq t\leq
t_{1}}B\left(\Phi_{h}^{t_{0},t}(z_{0}),c_{j}\delta\right)}$ and
$\chi^{\delta}_{j}$ is supported in $\displaystyle{\bigcup_{t_{0}\leq t\leq
t_{1}}B\left(\Phi_{h}^{t_{0},t}(z_{0}),2c_{j}\delta\right)}$,

  2. (2)

for all $\gamma\in{\mathbb{N}}^{2d}$, there exists $C_{\gamma}$ such that for
all $z\in{\mathbb{R}}^{2d}$

$|\partial_{z}^{\gamma}\chi_{j}^{\delta}(z)|\leq
C_{\gamma}\,\delta^{-|\gamma|},$

  3. (3)

$\chi_{3}^{\delta}=1$ on supp$\chi_{1}^{\delta}$ and $\chi_{1}^{\delta}=1$ on
supp$\chi_{2}^{\delta}$.

Finally, with $\chi_{0}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}}^{2d},[0,1])$
satisfying $\chi_{0}=1$ on $B(0,1)$ and $\chi_{0}(z)=0$ for $|z|\geq 2$, we
associate

$\chi^{\delta}_{0}(z)=\chi_{0}\left(\frac{z_{1}-z}{\delta}\right).$

And we consider $\chi_{4}$ a smooth fixed cut-off ($\delta$-independent).

#####  5.2.2. Adiabatic decoupling close to the gap

Omitting the mode index, we set

(5.3) $\widetilde{H}^{{\rm
adia},N,\varepsilon}(t)=\chi_{4}\left(h(t)+\varepsilon H_{1}^{\rm
adia}(t)\right)+\chi_{3}^{\delta}\left(\sum_{2\leq j\leq
N}\varepsilon^{j}H^{\rm adia}_{j}(t)\right).$

Notice that, because the crossing is smooth, the eigenavalues $h_{\ell}$,
$\pi_{\ell}$ and the first adiabatic correctors $H_{\ell,1}^{{\rm adia}}$ are
smooth, even in a neighborhood of $(t^{\flat},\zeta^{\flat}$).

Let ${\mathcal{U}}^{{\rm adia},N,\varepsilon}(t,s)$ be the quantum propagator
associated with the Hamiltonian $\widetilde{H}^{{\rm adia},N,\varepsilon}(t)$
(omitting once again the index $\ell=1$). The following result is a
consequence of the sharp estimates given in [6] concerning propagation of
quantum observables (see (ii) of the Egorov Theorem C.1).

######  Proposition 5.7.

Consider the cut-off functions $\chi_{0}^{\delta}$ and $\chi_{2}^{\delta}$
defined above and set for $t\in[t_{1},t^{\flat}-c\delta]$

${\rm op}_{\varepsilon}(\chi_{0}^{\delta}(t,t_{1})):={\mathcal{U}}^{{\rm
adia},\varepsilon,N}(t,t_{1})\,{\rm
op}_{\varepsilon}(\chi^{\delta}_{0})\;{\mathcal{U}}^{{\rm
adia},\varepsilon,N}(t_{1},t).$

Then, for any $M\geq 1$, $z\in{\mathbb{R}}^{2d}$ and
$t\in[t_{1},t^{\flat}-c\delta]$, we have:

$(1-\chi_{2}^{\delta})\circledast\chi_{0}^{\delta}(t,t_{0},z)=\left(\frac{\varepsilon}{\delta^{2}}\right)^{M}\zeta_{M}(t)\;\;\mbox{with}\;\;\zeta_{M}(t,z)\in\boldsymbol{S}_{\delta^{2}}(\mathcal{D}).$

Revisiting the proof of Proposition 5.2, using Lemma 4.3 for the formal series
$\Pi^{\varepsilon}_{\ell}\in\boldsymbol{S}_{\delta^{2}}^{0}(\mathcal{D})$ and
using (3) of Theorem 4.9 about $H^{\varepsilon,{\rm adia}}_{\ell}$, we obtain
the following result.

######  Proposition 5.8 (adiabatic decoupling - II).

With the previous notations, we have the following properties.

(ii) For $t_{1}\leq t\leq t^{\flat}-\delta$, we have

(5.4)
$\displaystyle\left(i\varepsilon\partial_{t}-\widehat{H}^{\varepsilon}(t)\right)\,{\rm
op}_{\varepsilon}\left(\chi_{1}^{\delta}\,\Pi^{N,\varepsilon}(t)\right)\,{\rm
op}_{\varepsilon}(\chi^{\delta}_{2})$ $\displaystyle\qquad={\rm
op}_{\varepsilon}\left(\chi_{1}^{\delta}\,\Pi^{N,\varepsilon}(t)\right)\left(i\varepsilon\partial_{t}-{\rm
op}_{\varepsilon}(\widetilde{H}^{{\rm adia},N,\varepsilon}(t))\right)\,{\rm
op}_{\varepsilon}(\chi_{2}^{\delta})+{\mathcal{O}}\left(\left(\frac{\varepsilon}{\delta^{2}}\right)^{N+1}\delta^{-\kappa_{0}}\right)$

where $\kappa_{0}\in{\mathbb{N}}$ is $N$-independent.

(ii) Set for $\ell=1$,

$\psi^{\varepsilon,N}_{\ell}(t)={\rm
op}_{\varepsilon}\left(\chi_{1}^{\delta}\Pi_{\ell}^{N,\varepsilon}(t)\right){\mathcal{U}}^{{\rm
adia},N,\varepsilon}_{\ell}(t,t_{1})\,{\rm
op}_{\varepsilon}\left(\chi_{0}^{\delta}\Pi_{\ell}^{N,\varepsilon}(t_{1})\right)\psi^{\varepsilon}(t_{1})$

where
$\psi^{\varepsilon}(t_{1})={\mathcal{U}}^{\varepsilon}_{H}(t_{1},t_{0})\psi^{\varepsilon}(t_{0})$.
Then we have, for $N\geq 2$ and for all $t\in[t_{1},t^{\flat}-\delta]$,

${\mathcal{U}}^{\varepsilon}_{H}(t,t_{0})\psi^{\varepsilon}_{0}=\psi^{\varepsilon,N}_{1}(t)+\psi^{\varepsilon,N}_{2}(t)+{\mathcal{O}}\left(\left(\frac{\varepsilon}{\delta^{2}}\right)^{N+1}\delta^{-\kappa_{0}}\right).$

where ${\mathcal{U}}^{{\rm adia},N,\varepsilon}_{\ell}$ is the propagator
associated with the Hamiltonian $\widetilde{H}^{{\rm adia},N,\varepsilon}(t)$.

The Remark 5.3 is valid also for this Proposition, furthermore here the
coefficients of the expansion of $\psi^{\varepsilon,N}_{j}(t)$ in
$\sqrt{\varepsilon}$ are $\delta$-dependent for the order $\varepsilon^{k/2}$
for $k\geq 2$.

Note that the integer $\kappa_{0}$ stems from the symbolic calculus estimates
of Theorem B.1.

#####  5.2.3. Application to wave packets

As in Theorem 5.4, the previous results have consequences for wave packets
propagation and give an asymptotic expansion mod
${\mathcal{O}}(\varepsilon^{\infty})$ for any $\alpha<1/2$ if
$\delta\approx\varepsilon^{\alpha}$. In other words the super-adiabatic
approximation is valid for times $t$ such that
$|t-t^{\flat}|\geq\varepsilon^{1/2-\eta}$, for any $\eta>0$. The results of
Appendix C give the following result.

######  Theorem 5.9.

Consider

$\psi^{\varepsilon}_{1}:=\psi^{\varepsilon}(t_{1})={\rm
WP}^{\varepsilon}_{z_{1}}(\varphi_{1}^{\varepsilon}),\;\;\varphi_{1}^{\varepsilon}\in{\mathcal{S}}({\mathbb{R}}^{d})\;\mbox{
modulo}\;{\mathcal{O}}(\varepsilon^{\infty}).$

There exist $N_{0}\in{\mathbb{N}}$ and two families of differential operators
$\left(\vec{B}_{\ell,j}(t)\right)_{j\in{\mathbb{N}}}$, $\ell\in\\{1,2\\}$ such
that setting for $t\in[t_{1},t^{\flat}-\delta]$

(5.5) $\psi_{\ell}^{\varepsilon,N}(t)={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t,t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{\ell}(t)}\left({\mathcal{R}}_{\ell}(t,t_{0})\,{\mathcal{M}}[F_{\ell}(t,t_{0})]\,\sum_{0\leq
j\leq 2N}\varepsilon^{j/2}\vec{B}_{\ell,j}(t)\varphi_{0}\right),$

one has the following property: for all $k\in{\mathbb{N}}$,
$N\in{\mathbb{N}}$, there exists $C_{N,k}>0$ such that the solution
$\psi^{\varepsilon}(t)$ of (1.1) satisfies for all
$t\in[t_{1},t^{\flat}-\delta]$.

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

Moreover the operators $B_{\ell,j}(t)$ are differential operators of degree
$\leq 3j$ with time dependent smooth vector-valued coefficients and satisfy
(1.40) and (1.41).

####  5.3. Propagation through the crossing set

We now use the rough reduction of section 4.2 to treat the zone around the
crossing. We fix the point $(t^{\flat},\zeta^{\flat})\in\Upsilon$ and consider
trajectories $z_{1}(t)$ and $z_{2}(t)$ arriving simultaneously at time
$t^{\flat}$ in the point $\zeta^{\flat}$. We consider $N\in{\mathbb{N}}$ and
we set

$\underline{\psi}_{\ell}^{\varepsilon,N}(t)=\widehat{\pi_{\ell}^{\varepsilon,N}(t)}\psi^{\varepsilon}(t),\;\;\ell\in\\{1,2\\}.$

By Theorem 4.5, if $k\in{\mathbb{N}}$, the solution $\psi^{\varepsilon}(t)$ of
the Schrödinger equation (1.1) satisfies in $\Sigma^{k}_{\varepsilon}$,

(5.6)
$\psi^{\varepsilon}(t)=\underline{\psi}^{\varepsilon,N}_{1}(t)+\underline{\psi}^{\varepsilon,N}_{2}(t)+O(\varepsilon^{N+1}).$

Our aim in this section is to determine $\psi^{\varepsilon}(t^{\flat}+\delta)$
in terms of $\psi^{\varepsilon}(t^{\flat}-\delta)$ by using the description
(5.6) of $\psi^{\varepsilon}(t)$.

The family
$\underline{\psi}^{\varepsilon,N}=\,^{t}(\underline{\psi}^{\varepsilon,N}_{1},\underline{\psi}^{\varepsilon,N}_{2})$
satisfies

(5.7)
$i\varepsilon\partial_{t}\underline{\psi}^{\varepsilon,N}=\widehat{\underline{H}^{\varepsilon,N}}(t)\underline{\psi}^{\varepsilon,N}$

with

$\underline{H}^{\varepsilon,N}(t):=\begin{pmatrix}h_{1}^{\varepsilon,N}(t,z)&0\\\
0&h_{2}^{\varepsilon,N}(t,z)\end{pmatrix}+\begin{pmatrix}0&W^{\varepsilon,N}(t)\\\
W^{\varepsilon,N}(t)^{*}&0\end{pmatrix}.$

According to Theorem 4.5, the Hamiltonian $\underline{H}^{\varepsilon}$ is
subquadratic (see Definition 1.1), thus for $k,N\in{\mathbb{N}}$ there exists
$C_{k,N}>0$ such that for all $\varepsilon>0$ and $t\in I$,

(5.8)
$\|\widehat{W}^{\varepsilon,N}(t)\|_{\mathcal{L}(\Sigma^{k+1}_{\varepsilon},\Sigma^{k}_{\varepsilon})}\leq
C_{k,N}\varepsilon.$

We have used here the fact that the asymptotic series $W^{\varepsilon,}$
starts with the term of order $\varepsilon$ and we recall that the value of
$W_{1}$ is given in (1.44).

Let us summarize the information about the data that comes from the preceding
section. Let $\delta>0$, for all
$s\in(t^{\flat}-\delta,t^{\flat}-\frac{\delta}{2})$,

(5.9) $\underline{\psi}^{\varepsilon,N}(s)=\,^{t}\left({\rm
WP}^{\varepsilon}_{z_{1}(s)}(\varphi^{\varepsilon,N}_{1}(s)),{\rm
WP}^{\varepsilon}_{z_{2}(s)}(\varphi^{\varepsilon,N}_{2}(s))\right),$

with for $\ell=1,2$,

$\varphi^{\varepsilon,N}_{\ell}=\sum_{j=0}^{N}\varepsilon^{\frac{j}{2}}\varphi_{j,\ell},\;\;\varphi_{j,\ell}\in\mathcal{S}({\mathbb{R}}^{d}).$

Our aim is to prove that the description of
$\underline{\psi}^{\varepsilon,N}(s)$ given in Equation (5.9) extends to
$s=t^{\flat}+\delta$ and to derive precise formula for
$\varphi_{j,\ell}(t^{\flat}+\delta)$ when $j=\\{0,1\\}$ and
$\ell\in\\{1,2\\}$.

We consider the Hamiltonians

$H_{\rm
diag}^{\varepsilon,N}(t,z)=\begin{pmatrix}h_{1}^{\varepsilon,N}(t,z)&0\\\
0&h_{2}^{\varepsilon,N}(t,z)\end{pmatrix}$

and

$\varepsilon H_{\rm
adiag}^{\varepsilon,N}(t,z)=\begin{pmatrix}0&W^{\varepsilon,N}(t)\\\
W^{\varepsilon,N}(t)^{*}&0\end{pmatrix}.$

so that $\underline{H}^{\varepsilon,N}=H_{\rm
diag}^{\varepsilon,N}+\varepsilon H_{\rm adiag}^{\varepsilon,N}$. We fix $N$
large enough and, for simplifying the notations, we drop the mentions of $N$
in the following. We use the notations

$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,s)\;\;{\rm
and}\;\;\mathcal{U}^{\varepsilon}_{H_{\rm diag}}(t,s)$

for the propagators associated to the truncated Hamiltonians
$H^{\varepsilon,\delta,N}(t)$ and $H^{\varepsilon,N}_{\rm diag}(t)$
respectively, omitting the mention of $\delta$ in
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,s)$. The action of
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,s)$ on wave packets is described
by the next Theorem on which we focus now. It gives a precise description of
the action of
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t^{\flat}+\delta,t^{\flat}-\delta)$
on a wave packet and describes the propagation of a wave packet through the
crossing set, in particular the exchange of modes at the crossing points.

######  Theorem 5.10.

Let $k,N,M\in{\mathbb{N}}$ with $M\leq N$. Let $\delta>0$ such that
$\delta^{2}\geq\sqrt{\varepsilon}$. Then, there exists $C>0$ and an operator
$\Theta^{\varepsilon,\delta}_{M}$ such that for all $\varepsilon\in(0,1)$,

(5.10)
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t^{\flat}+\delta,t^{\flat}-\delta)=\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat}+\delta,t^{\flat})\;\Theta^{\varepsilon,\delta}_{M}\;\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat},t^{\flat}-\delta)+R^{\varepsilon,\delta}_{M}$

with
$\|R^{\varepsilon,\delta}_{M}\|_{\mathcal{L}(\Sigma_{\varepsilon}^{k+M+1},\Sigma^{k})}\leq
C\delta^{M+1}$ and

$\Theta^{\varepsilon,\delta}_{M}(t^{\flat})=\mathbb{I}+\sum_{1\leq m\leq
M}\,\Theta^{\varepsilon,\delta}_{m,M}(t^{\flat}).$

Moreover, there exists $\varepsilon_{0}>0$, and a family of operators
$(\mathcal{T}_{m,M}^{\varepsilon,\delta})_{m\geq 1}$ such that for all
$\vec{\varphi}\in\mathcal{S}({\mathbb{R}}^{d},{\mathbb{C}}^{2})$, $m\geq 1$,
$\varepsilon\in(0,\varepsilon_{0})$,

(5.11) $\Theta^{\varepsilon,\delta}_{m,M}{\rm
WP}_{\zeta^{\flat}}(\varphi)={\rm
WP}^{\varepsilon}_{\zeta^{\flat}}(\mathcal{T}^{\varepsilon,\delta}_{m,M}\vec{\varphi})+O(\sqrt{\varepsilon}\|\varphi\|_{\Sigma^{k+2m+2}})$

with

(5.12)
$\|\mathcal{T}^{\varepsilon,\delta}_{m,M}\vec{\varphi}\|_{\Sigma^{k}}\leq
c_{k,m,M}\,\varepsilon^{\frac{m}{2}}|\log\varepsilon|^{\max(0,m-1)}\,\|\vec{\varphi}\|_{\Sigma^{k+2m+1}}$

for some constants $c_{k,m}>0$. Besides, with the notation (1.43)

(5.13)
$\mathcal{T}^{\varepsilon,\delta}_{1,M}=\begin{pmatrix}0&W_{1}(t^{\flat},\zeta^{\flat})\mathcal{T}^{\flat}_{2\rightarrow
1}\\\ W_{1}(t^{\flat},\zeta^{\flat})^{*}\mathcal{T}^{\flat}_{1\rightarrow
2}&0\end{pmatrix}.$

We point out that some additional action effects will appear when applying
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t^{\flat}+\delta,t^{\flat}-\delta)$
to a wave packet via the operators $\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat}+\delta,t^{\flat})$ and $\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat},t^{\flat}-\delta)$. When applied to a Gaussian wave packets,
i.e. when $\varphi_{\ell,1}=g^{\Gamma}_{\ell}$ in (5.9), the leading order
correction term at time $t^{\flat}+\delta$ due to the crossing is

$\displaystyle\sqrt{\varepsilon}\begin{pmatrix}{\rm
e}^{\frac{i}{\varepsilon}S_{1}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})+\frac{i}{\varepsilon}S_{2}(t^{\flat},t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{1}(t^{\flat}+\delta)}(\varphi_{1})\\\ {\rm
e}^{\frac{i}{\varepsilon}S_{2}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})+\frac{i}{\varepsilon}S_{1}(t^{\flat},t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{z_{2}(t^{\flat}+\delta)}(\varphi_{2})\end{pmatrix}$

with

$\displaystyle\varphi_{1}$
$\displaystyle=\mathcal{M}[F_{1}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})]W_{1}(t^{\flat},\zeta^{\flat})\mathcal{T}^{\flat}_{2\rightarrow
1}\mathcal{M}[F_{2}(t^{\flat},t_{0},z_{0})]g^{\Gamma}_{2},$
$\displaystyle\varphi_{2}$
$\displaystyle=\mathcal{M}[F_{2}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})]W_{1}(t^{\flat},\zeta^{\flat})^{*}\mathcal{T}^{\flat}_{1\rightarrow
2}\mathcal{M}[F_{1}(t^{\flat},t_{0},z_{0})]g^{\Gamma}_{1}.$

Recall that $W_{1}$ is the off-diagonal matrix described in (1.44).

The remainder of this section is devoted to the proof of Theorem 5.10. The use
of Dyson series allows to obtain the decomposition (5.10) (see Section 5.3.1).
Then, the analysis of each terms of the series is made in Sections 5.3.2 and
5.3.3. Finally, we recall how to compute explicitly the quantities
$\mathcal{T}^{\varepsilon,\delta}_{m,M}$ and $S_{1,m}^{\flat}$ in Section
5.3.5, which was already done in [20].

Before starting the proof, we introduce a cut-off
$\chi_{\delta}(t)=\chi\left(\frac{t-t^{\flat}}{\delta}\right)$, $\chi_{0}\in
C_{0}^{\infty}]-1,1[$, $\chi_{0}(t)=1$ if $|t|\leq 1/2$. We set

$\underline{H}^{\varepsilon,\delta,N}(t)=H^{\varepsilon,N}_{\rm
diag}(t)+\varepsilon\chi_{\delta}(t-t^{\flat})H^{\varepsilon,N}_{\rm
adiag}(t)$

and we consider the propagator
$\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t,s)$ associated with
$\underline{H}^{\varepsilon,\delta,N}$. We claim that if Theorem 5.10 holds
for $\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t,s)$, then it also
holds for $\mathcal{U}^{\varepsilon}_{\underline{H}}(t,s)$.

Indeed, we have for $t\in[t^{\flat}+\delta,t^{\flat}+\frac{\delta}{2}]$,

$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,t^{\flat}-\delta)=\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t^{\flat}+\delta,t^{\flat}-\delta)\underline{\psi}^{\varepsilon,N}(t^{\flat}-\delta)+i\int_{t^{\flat}-\delta}^{t}\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t,s)(1-\chi_{\delta})(s-t^{\flat})\widehat{\underline{H}_{\rm
adiag}^{\varepsilon,N}}(s)\mathcal{U}^{\varepsilon}_{\underline{H}}(s,t^{\flat}-\delta)ds.$

This formula allows to obtain (5.10) for
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,t^{\flat}-\delta)$. It remains to
consider the action of
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t,t^{\flat}-\delta)$ on asymptotic
sum of wave packets We observe that in the support of the integral,
$|t^{\flat}-s|>\frac{\delta}{2}$ and $-\delta\leq
s-t^{\flat}\leq\frac{\delta}{2}$. Therefore,
$s\in[t^{\flat}-\delta,t^{\flat}-\frac{\delta}{2}]$ on the support of the
integral. where we know that
$\mathcal{U}^{\varepsilon}_{\underline{H}}(s,t^{\flat}-\delta)$ propagates
wave packets, whence the expansion in wave packets To conclude, we observe
that since $|t^{\flat}-s|>\frac{\delta}{2}$ on the support of the integral
term, we have

$\mathcal{U}^{\varepsilon}_{\underline{H}}(t^{\flat}+\delta,t^{\flat}-\delta)=\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t^{\flat}+\delta,t^{\flat}-\delta)+O(\delta^{2})=\mathcal{U}^{\varepsilon}_{\underline{H}_{\delta}}(t^{\flat}+\delta,t^{\flat}-\delta)+o(\sqrt{\varepsilon})$

so the formula for the first two terms of the asymptotic assumptions are the
same.

In view of these considerations, we focus in the next sections in proving
Theorem 5.10 for the Hamiltonian $\underline{H}^{\varepsilon,\delta,N}(t)$.

#####  5.3.1. Dyson expansion

We perform a Dyson expansion via the Duhamel formula. A first use of Duhamel
formula gives for $t_{1},t_{2}\in{\mathbb{R}}$,

(5.14)
$\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},t_{1})=\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},t_{1})+i^{-1}\int_{t_{1}}^{t_{2}}\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\,\mathcal{U}^{\varepsilon}_{H_{\rm diag}}(s_{1},t_{1})ds_{1}.$

With one iteration of the Duhamel formula, we obtain

$\displaystyle\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},t_{1})$
$\displaystyle=\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},t_{1})+i^{-1}\int_{t_{1}}^{t_{2}}\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\mathcal{U}^{\varepsilon}_{H_{\rm diag}}(s_{1},t_{1})ds_{1}$
$\displaystyle\qquad-\int_{t_{1}}^{t_{2}}\int_{s_{1}}^{t_{2}}\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},s_{2})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{2})\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{1},t_{1})ds_{1}ds_{2}.$

With two iterations, we have

$\displaystyle\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},t_{1})=\;$
$\displaystyle\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},t_{1})+i^{-1}\int_{t_{1}}^{t_{2}}\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\mathcal{U}^{\varepsilon}_{H_{\rm diag}}(s_{1},t_{1})ds_{1}$
$\displaystyle\;-\int_{t_{1}}^{t_{2}}\int_{s_{1}}^{t_{2}}\mathcal{U}^{\varepsilon}_{\rm
diag}(t_{2},s_{2})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{2})\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{1},t_{1})ds_{1}ds_{2}$
$\displaystyle\;-\frac{1}{i}\int_{t_{1}}^{t_{2}}\int_{s_{2}}^{t_{2}}\int_{s_{1}}^{t_{2}}\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},s_{3})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{3})\mathcal{U}^{\varepsilon}_{\rm
diag}(s_{3},s_{2})\widehat{H^{\varepsilon}_{\rm adiag}}(s_{2})$
$\displaystyle\qquad\qquad\qquad\times\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{2},s_{1})\widehat{H^{\varepsilon}_{\rm
adiag}}(s_{1})\,\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(s_{1},t_{1})ds_{1}ds_{2}$

After $M$ iterations, $M\in{\mathbb{N}}$, we have the Dyson formula

$\displaystyle\mathcal{U}^{\varepsilon}_{\underline{H}}(t_{2},t_{1})=\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{2},t_{1})\bigg{(}\mathbb{I}\,+$ $\displaystyle\sum_{1\leq m\leq
M}(i)^{-m}\int_{\mathcal{P}(t_{2},t_{1})}F^{\varepsilon}(s_{1},\cdots,s_{m},t_{1})ds_{m}\cdots
ds_{1}\biggr{)}+R_{M}^{\varepsilon}(t_{2},t_{1})$

with

(5.15)
$F^{\varepsilon}(s_{1},\cdots,s_{m},t_{1})=E(s_{m},t_{1})E(s_{m-1},t_{1})\cdots
E(s_{2},t_{1})E(s_{1},t_{1}),$

where the operators $E(s,t_{1})$ are given by

(5.16) $E(s,t_{1})=\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t_{1},s)\widehat{H^{\varepsilon}_{\rm
adiag}}(s)\mathcal{U}^{\varepsilon}_{H_{\rm diag}}(s,t_{1}),$

and the set of integration $\mathcal{P}(t_{2},t_{1})\subset{\mathbb{R}}^{M}$
satisfies

$\mathcal{P}(t_{2},t_{1})=\\{t_{1}\leq s_{M}\leq\dots\leq s_{2}\leq s_{1}\leq
t_{2}\\}.$

Besides, by (5.8) there exists a constant $C>0$ such that

$\|R_{M}^{\varepsilon}(t_{2},t_{1})\|_{\mathcal{L}(\Sigma^{k+M+1}_{\varepsilon},\Sigma^{k}_{\varepsilon})}\leq
C|t_{2}-t_{1}|^{M+1}.$

We apply this formula to $t_{1}=t^{\flat}$, $t_{2}=t^{\flat}+\delta$,

$\displaystyle\mathcal{U}^{\varepsilon}_{\underline{H}}(t^{\flat}+\delta,t^{\flat}-\delta)=\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t^{\flat}+\delta,t^{\flat})\bigg{(}\mathbb{I}\,+$
$\displaystyle\sum_{1\leq m\leq
M}(i)^{-m}\int_{s_{M}\in{\mathbb{R}}}\int_{-\infty}^{s_{M}}\cdots\int_{-\infty}^{s_{2}}F^{\varepsilon}(s_{1},\cdots,s_{M},t^{\flat})ds_{M}\cdots
ds_{1}\biggr{)}\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t^{\flat},t^{\flat}-\delta)$
$\displaystyle+R_{M}^{\varepsilon}(t^{\flat}+\delta,t^{\flat})\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t^{\flat},t^{\flat}-\delta)$

which gives equation (5.10) with

$\Theta^{\varepsilon,\delta}_{m,M}=(i)^{-m}\int_{s_{M}\in{\mathbb{R}}}\int_{-\infty}^{s_{M}}\cdots\int_{-\infty}^{s_{2}}F^{\varepsilon}(s_{1},\cdots,s_{M},t^{\flat})ds_{M}\cdots
ds_{1}$

and
$R^{\varepsilon,\delta}_{M}=R_{M}^{\varepsilon}(t^{\flat}+\delta,t^{\flat}-\delta)\mathcal{U}^{\varepsilon}_{H_{\rm
diag}}(t^{\flat},t^{\flat}-\delta)$ satisfies

$\|R^{\varepsilon,\delta}_{M}\|_{\mathcal{L}(\Sigma_{\varepsilon}^{k+M+1},\Sigma^{k})}\leq
C\delta^{M+1}.$

The operators $\Theta^{\varepsilon,\delta}_{m,M}$ contain all the information
about the interactions between the modes $h_{1}$ and $h_{2}$ modulo
$O(\delta^{\infty})$ when $M$ goes to $+\infty$.

In the next sections, we focus in understanding the action of
$\Theta^{\varepsilon,\delta}_{m,M}$ on wave packets of the form

${\rm WP}^{\varepsilon}_{\zeta^{\flat}}(\vec{\varphi})=\begin{pmatrix}{\rm
WP}^{\varepsilon}_{\zeta^{\flat}}(\varphi_{1})\\\ {\rm
WP}^{\varepsilon}_{\zeta^{\flat}}(\varphi_{2})\end{pmatrix}$

and in proving equations (5.11), (5.12) and (5.13).

#####  5.3.2. Analysis of the matrices $E(s,t^{\flat})$

We have

$E(s,t^{\flat})=\begin{pmatrix}0&{\mathcal{I}}(s,t^{\flat})\\\
{\mathcal{I}}^{*}(s,t^{\flat})&0\end{pmatrix},\;\;s\in[t^{\flat}-\delta,t^{\flat}+\delta]$

with

(5.17)
${\mathcal{I}}(s,t^{\flat})=\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)\chi_{\delta}(s-t^{\flat})W^{\varepsilon,N}(s)\,\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat}).$

This operator combines conjugation of the pseudodifferential operator
$\chi_{\delta}(s-t^{\flat})W^{\varepsilon,N}(s)$ by the propagator
$\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})$ and
composition by two different propagators
$\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)$ and
$\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(t^{\flat},s)$. Indeed, we
can write

${\mathcal{I}}(s,t^{\flat})=\left(\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)W^{\varepsilon,N}(s)\,\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(s,t^{\flat})\right)\circ\left(\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})\right).$

The conjugation of a pseudo by a propagator is perfectly understood and is
described in our setting by the Egorov Theorem of Appendix C (with
$\delta=1$). The operator
$\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(t^{\flat},s)W^{\varepsilon,N}(s)\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})$
has an asymptotic expansion.

$\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(t^{\flat},s)W^{\varepsilon,N}(s)\,\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})=\sum_{j\geq
1}\varepsilon^{j}\underline{W}_{j}(s).$

Similarly, for ${\mathcal{I}}^{*}(s,t^{\flat})$, one writes

${\mathcal{I}}^{*}(s,t^{\flat})=\left(\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(t^{\flat},s)W^{\varepsilon,N}(s)^{*}\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})\right)\circ\left(\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(t^{\flat},s)\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(s,t^{\flat})\right).$

Note that the actions of these two operators are perfectly adapted to the
geometric context: ${\mathcal{I}}(s,t^{\flat})$ picks the contribution of the
second component (the lower one), which lives on the mode $2$, transforms it
into something related with the mode $1$ (the upper one), via the operator
$\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})$,
and then, an operator related to the first mode acts on what is now a
component living on this precise mode. And conversely for
${\mathcal{I}}^{*}(s,t^{\flat})$.

The action on wave packets of two different propagators acting one backwards
and the other one forwards has been studied in [20] (see Section 5.2). Using
Egorov theorem, the action of scalar propagators on wave packets, and the
precise computation of the operator
$\mathcal{U}^{\varepsilon}_{h_{1}^{\varepsilon,N}}(t^{\flat},s)\,\mathcal{U}^{\varepsilon}_{h_{2}^{\varepsilon,N}}(s,t^{\flat})$
performed therein (which involves the canonical transformation of the phase
space $z\mapsto\Phi_{1}^{t^{\flat},s}\circ\Phi_{2}^{s,t^{\flat}}(z)$), one
obtains the analogue of Lemma 5.3 of [20], which writes in our context as
follows.

######  Lemma 5.11.

Let $k\in{\mathbb{N}}$. There exist

  * -

A smooth real-valued map $s\mapsto\Lambda(s)$ with $\Lambda(0)=0$,
$\dot{\Lambda}(0)=0$,
$\ddot{\Lambda}(0)=2\mu^{\flat}+\alpha^{\flat}\cdot\beta^{\flat}$,

  * -

A smooth vector-valued map $s\mapsto z(s)=(q(s),p(s))$ with $z(0)=0$,
$\dot{z}(0)=(\alpha^{\flat},\beta^{\flat})$,

  * -

A smooth map $\sigma\mapsto{\mathcal{Q}}^{\varepsilon}(s)$ of
pseudodifferential operators, that maps Schwartz functions to Schwartz
functions, with
${\mathcal{Q}}^{\varepsilon}(s)=\sum_{j=0}^{M}\varepsilon^{j}{\mathcal{Q}}_{j}(s)+\mathcal{Q}^{\varepsilon}_{M+1}$,

$\mathcal{Q}_{0}(0)=W_{1}(t^{\flat},\zeta^{\flat})^{*}$

such that for all $\varphi\in\mathcal{S}({\mathbb{R}}^{d})$,

$I(s,t^{\flat})^{*}{\rm WP}^{\varepsilon}(\varphi)(y)={\rm
WP}^{\varepsilon}\left({\rm
e}^{\frac{i}{\varepsilon}\Lambda(s-t^{\flat})}{\mathcal{Q}}^{\varepsilon}(s-t^{\flat}){\rm
e}^{ip_{\varepsilon}(s-t^{\flat})\cdot(y-q_{\varepsilon}(s-t^{\flat}))}\varphi(y-q_{\varepsilon}(s-t^{\flat}))+R^{\varepsilon}\varphi\right)$

with, for some $c_{M}>0$

$\displaystyle\|\mathcal{Q}_{j}(s)\varphi\|_{\Sigma^{k}}\leq
c_{M}\|\varphi\|_{\Sigma^{k+1}},\;\;\forall j\in\\{1,\cdots,M\\},$
$\displaystyle\|\mathcal{Q}_{M+1}(s)\varphi\|_{\Sigma^{k}}\leq
c_{M}\|\varphi\|_{\Sigma^{k+1+\kappa_{0}}},$
$\displaystyle\|R^{\varepsilon}\varphi\|_{\Sigma^{k}}\leq
c_{M}\,\|\varphi\|_{\Sigma^{k+1+\kappa_{0}}},$

where we have used the scaling notation
$z_{\varepsilon}(s)=z(s)/\sqrt{\varepsilon}$ and where $\kappa_{0}$ is the
universal constant of Theorem B.1.

A similar result holds for $I(s,t^{\flat})$ by replacing
$W_{1}(t^{\flat},\zeta^{\flat})^{*}$ by $W_{1}(t^{\flat},\zeta^{\flat})$ and
exchanging the roles of the modes $h_{1}$ and $h_{2}$.

#####  5.3.3. Uniform estimates for the elements of the Dyson series

We now focus on the operators $\Theta^{\varepsilon,\delta}_{m,M}$. For
$s\in[t^{\flat}-\delta,t^{\flat}+\delta]$, we define recursively the
quantities $J_{m}(s)$ for $m\in\\{1,\cdots M\\}$ by

$J_{1}(s)=-i\int_{-\infty}^{s}E(s_{1},t^{\flat})ds_{1}$

and for $m\geq 2$,

$J_{m}(s)=-i\int_{-\infty}^{s}E(s_{m},t^{\flat})J_{m-1}(s_{m})ds_{m}.\;\;$

We recall that $E_{1}(s,t^{\flat})$ is supported on $|t^{\flat}-s|\leq\delta$
due to the cut-off function $\chi_{\delta}(s-t^{\flat})$ that appears in
(5.17). With these notations,

$\Theta^{\varepsilon,\delta}_{m,M}=J_{m}(t^{\flat}+\delta).$

We are reduced to proving the existence of operators
$s\mapsto\mathcal{T}_{m}^{\varepsilon,\delta}(s)$ such that for all
$s\in[t^{\flat}-\delta,t^{\flat}+\delta]$ and
$\vec{\varphi}\in\mathcal{S}({\mathbb{R}}^{d},{\mathbb{C}}^{2})$,

(5.18) $J_{m}(s){\rm WP}^{\varepsilon}_{\zeta^{\flat}}(\vec{\varphi})={\rm
WP}^{\varepsilon}_{\zeta^{\flat}}(\mathcal{T}_{m}^{\varepsilon,\delta}(s)\vec{\varphi}),$

with for all $k\in{\mathbb{N}}$, the estimate 5.12. Note that we are omitting
the index $M$ of the notations of Theorem 5.10.

If the functions $\mathcal{T}_{m}^{\varepsilon,\delta}$ exist, they satisfy
for $s^{\prime}\in[t^{\flat}-\delta,t^{\flat}+\delta]$ the recursive equations

${\rm
WP}^{\varepsilon}_{\zeta^{\flat}}\left(T_{m+1}^{\varepsilon,\delta}(s)\vec{\varphi}\right)_{2}=-i\int_{-\infty}^{s^{\prime}}I(s,t^{\flat})^{*}{\rm
WP}^{\varepsilon}_{\zeta^{\flat}}\left(T_{m}^{\varepsilon,\delta}(s)\vec{\varphi}\right)_{1}ds$

and

${\rm
WP}^{\varepsilon}_{\zeta^{\flat}}\left(T_{m+1}^{\varepsilon,\delta}(s)\vec{\varphi}\right)_{1}=-i\int_{-\infty}^{s^{\prime}}I(s,t^{\flat}){\rm
WP}^{\varepsilon}_{\zeta^{\flat}}\left(T_{m}^{\varepsilon,\delta}(s)\vec{\varphi}\right)_{2}ds.$

Therefore, by Lemma 5.11, if the functions
$\mathcal{T}_{m}^{\varepsilon,\delta}$ exist, they satisfy for
$s^{\prime}\in[t^{\flat}-\delta,t^{\flat}+\delta]$ the recursive equations

$\left(\mathcal{T}_{m+1}^{\varepsilon,\delta}(s)\vec{\varphi}\right)_{2}=\int_{-\infty}^{s^{\prime}}{\rm
e}^{\frac{i}{\varepsilon}\Lambda(s-t^{\flat})}{\mathcal{Q}}^{\varepsilon}(s-t^{\flat}){\rm
e}^{ip_{\varepsilon}(s-t^{\flat})\cdot(y-q_{\varepsilon}(s-t^{\flat}))}(\mathcal{T}_{m}^{\varepsilon,\delta}(s)\vec{\varphi})_{1}(y-q_{\varepsilon}(s-t^{\flat}))ds$

and some analogue equation for the other components.

At the stage of the proof, the operators
$\mathcal{T}_{m}^{\varepsilon,\delta}$ are defined by a recursive process that
we are going to study for proving (5.12). Therefore, we focus on the analysis
for scalar valued functions $\varphi$ of

$I_{\varphi}^{m}:=\int_{-\infty}^{s^{\prime}}{\rm
e}^{\frac{i}{\varepsilon}\Lambda(s-t^{\flat})}{\mathcal{Q}}^{\varepsilon}(s-t^{\flat}){\rm
e}^{ip_{\varepsilon}(s-t^{\flat})\cdot(y-q_{\varepsilon}(s-t^{\flat}))}(T_{m}^{\varepsilon,\delta}(s)\varphi)(y-q_{\varepsilon}(s-t^{\flat}))ds.$

In the following sections, we prove (5.18) recursively:

  1. (1)

In Section 5.3.4, we prove that if $m>1$ and if (5.18) holds for
$m^{\prime}=1,2,\cdots,m-1$, then it also holds for $m^{\prime}=m$.

  2. (2)

In Section 5.3.5, we prove (5.18) when $m=1$.

#####  5.3.4. Proof of Lemma 5.12: the recursive argument

We use Lemma 5.11 to perform a recursive argument on the structure of
$J_{m}(s)$.

######  Lemma 5.12.

Assume there exists $m\in\\{1,2,\cdots,M-1\\}$ such that for all
$k\in{\mathbb{N}}$, we have (5.18) with the inequality (5.12) for $m$ and for
the integers between $1$ and $m$. Then, for all $k\in{\mathbb{N}}$, there
exists a constant $C_{k,m+1}$ such that for all
$\varphi\in\mathcal{S}({\mathbb{R}}^{d})$, we have

$\|I_{\varphi}\|_{\Sigma^{k}}\leq
C_{k,m+1}\varepsilon^{\frac{m+1}{2}}|\log\varepsilon|^{\max(0,m)}\|\varphi\|_{\Sigma^{k+2(m+1)+1}}.$

This lemma, together with the precise computation of $J_{1}(t^{\flat}+\delta)$
(see [20] and Section 5.3.5 below) concludes the proof of Theorem 5.10.

Note that at each step of the recursion one loose 2 degrees of regularity. One
degree is lost because the operator $\mathcal{Q}^{\varepsilon}$ that may has
linear growth, and another loss is due to an integration by parts that will
involve the term $y\cdot p_{\varepsilon}$ in the phase and the argument
$q_{\varepsilon}$ inside the function $\varphi$. The initial loss of
regularity when $m=1$ comes from similar reasons: one step is due to the
presence of the operator $\mathcal{Q}^{\varepsilon}$ and the two other ones by
integration by parts.

Let us start by exhibiting basic properties of $I_{\varphi}$. After the change
of variables $s=t^{\flat}+\sigma\sqrt{\varepsilon}$ and letting appear a cut-
off $\widetilde{\chi}$ such that $\chi\tilde{\chi}=\chi$, we write

$\displaystyle I_{\varphi}=$
$\displaystyle\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{\frac{i}{\varepsilon}\Lambda(\sigma\sqrt{\varepsilon})}\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon}){\rm
e}^{ip_{\varepsilon}(\sigma\sqrt{\varepsilon})\cdot(y-q_{\varepsilon}(\sigma\sqrt{\varepsilon}))}$
$\displaystyle\qquad\times(T_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi)(y-q_{\varepsilon}(\sigma\sqrt{\varepsilon}))d\sigma$

with

$\lambda=\frac{\delta}{\sqrt{\varepsilon}}.$

One sees that the change of variable has exhibited a power
$\sqrt{\varepsilon}$, which is exactly what one wants to earn for the
recursive process. However, even though the integrand is bounded, the size of
the support of the integral is large: it is of size
$\lambda=\frac{\delta}{\sqrt{\varepsilon}}$, which spoils that gain of
$\sqrt{\varepsilon}$. This integral will turn out to be smaller than what
gives this rough estimate because of the oscillations of the phase. The proof
then consists in integration by parts. For this reason, we are interested in
derivatives and we observe that the recursive assumption yields

(5.19)
$\|\partial_{s}T_{m}^{\varepsilon,\delta}(s)\vec{\varphi}\|_{\Sigma^{k}}\leq
C_{k,m}\varepsilon^{\frac{m-1}{2}}|\log\varepsilon|^{\max(0,m-2)}\|\vec{\varphi}\|_{\Sigma^{k+2(m-1)+1}}$

and similarly for the integers between $2$ and $m$.

Analysis of the phase. We know analyze the phase of the integral
$I_{\varphi}$. We set

$\phi^{\varepsilon}(\sigma)=\Lambda(\sigma\sqrt{\varepsilon})-\frac{1}{2}p_{\varepsilon}(\sigma\sqrt{\varepsilon})\cdot(y-q_{\varepsilon}(\sigma\sqrt{\varepsilon}))\;\;\mbox{and}\;\;L=\beta^{\flat}\cdot
y-\alpha^{\flat}\cdot D_{y}.$

We will use that ${\rm e}^{isL}$ maps $\Sigma^{k}$ into itself continuously
for all $s\in{\mathbb{R}}$ and $k\in{\mathbb{N}}$. Besides, for
$\delta\leq\delta_{0}$, $\delta_{0}>0$ small enough, we have

$\left|\frac{d}{ds}\phi^{\varepsilon}(s)\right|\geq
c_{0}|s|,\;\forall|s|\leq\lambda$

and

${\rm
e}^{i\phi^{\varepsilon}(s)}=\frac{1}{s}b^{\varepsilon}(s)\partial_{s}{\rm
e}^{i\phi^{\varepsilon}(s)}\;\;\mbox{with}\;\;b^{\varepsilon}(s):=\frac{s}{i\partial_{s}\phi^{\varepsilon}(s)}.$

The function $b^{\varepsilon}(s)$ is uniformly bounded, as well as its
derivatives, for $|s|\geq 1/2$. Then, following [20], we have

$I_{\varphi}^{m}=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}{\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon})\chi(\sigma/\lambda){\rm
e}^{iL\sigma}T_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi
d\sigma$

and there exists a smooth function $f$ such that

$\phi^{\varepsilon}(\sigma)=\mu^{\flat}\sigma^{2}+\sqrt{\varepsilon}\,\sigma^{3}\,f(\sigma\sqrt{\varepsilon}).$

At the stage of the proof, all the elements have been collected to perform the
recursive argument.

Because of the considerations we have made on the support of the integral and
because the phase $\phi^{\varepsilon}(s)$ is oscillating for $s$ far away from
$0$, we use the cut-off function $\chi$ to write

$I^{m}_{\varphi}=I_{\varphi}^{m,1}+I_{\varphi}^{m,2}$

with

$I_{\varphi}^{m,1}=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\chi(\sigma)\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon}){\rm
e}^{iL\sigma}\mathcal{T}_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi
d\sigma$

The compactly supported term. The term $I_{\varphi}^{m,1}$ is the easiest to
deal with since it has compact support, independently of $\varepsilon$ and
$\delta$. Therefore,

$\displaystyle\|I_{\varphi}^{m,1}\|_{\Sigma^{k}}\ $ $\displaystyle\leq
C\,\sqrt{\varepsilon}\sup_{s\in[t^{\flat}-\delta,t^{\flat}+\delta]}\|\mathcal{T}_{m}^{\varepsilon,\delta}(s)\varphi\|_{\Sigma^{k+1}}$
$\displaystyle\leq
C\varepsilon^{\frac{m+1}{2}}|\log\varepsilon|^{\max(0,m-1)}\,\|\varphi\|_{\Sigma^{k+2m+2}}.$

Note that the presence of $\mathcal{Q}^{\varepsilon}$ induces a loss of
regularity of $1$.

The oscillating term. For dealing with the term $I_{\varphi}^{m,2}$, we take
advantage of the oscillating phase for compensating the fact that the support
is large and we perform integration by parts. We write

$\displaystyle I_{\varphi}^{m,2}$
$\displaystyle=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}\partial_{\sigma}\left({\rm
e}^{i\phi^{\varepsilon}(\sigma)}\right)\frac{1}{\sigma}b^{\varepsilon}(\sigma)(1-\chi)(\sigma)\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon})$
$\displaystyle\qquad\times{\rm
e}^{iL\sigma}(\mathcal{T}_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi)d\sigma$
$\displaystyle=-\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\partial_{\sigma}\left(\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}\mathcal{T}_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi\right)d\sigma$
$\displaystyle\qquad\qquad+\sqrt{\varepsilon}\left.\left({\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}\right)\right|_{\sigma=\frac{s^{\prime}-t^{\flat}}{\sqrt{\varepsilon}}}\mathcal{T}_{m}^{\varepsilon,\delta}(s^{\prime})\varphi$

with

$b^{\varepsilon}_{1}(\sigma)=(1-\chi)(\sigma)\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon}).$

Note that the operator-valued functions $\sigma\mapsto
b^{\varepsilon}_{1}(\sigma)$ and
$\sigma\mapsto\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma)$ are bounded form
$\Sigma^{k+1}_{\varepsilon}$ to $\Sigma^{k}_{\varepsilon}$ (with a loss due to
the presence of the operator $\mathcal{Q}^{\varepsilon}$), and similarly for
its derivatives. We write

$\displaystyle I_{\varphi}^{m,2}$
$\displaystyle=-\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\partial_{\sigma}\left(\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma)\right){\rm
e}^{iL\sigma}(\mathcal{T}_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi)d\sigma$
$\displaystyle-\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}(iL\mathcal{T}_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi+\sqrt{\varepsilon}\partial_{s}T_{m}^{\varepsilon,\delta}(t^{\flat}+\sigma\sqrt{\varepsilon})\varphi)d\sigma$
$\displaystyle+\sqrt{\varepsilon}\left.\left({\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}\right)\right|_{\sigma=\frac{s^{\prime}-t^{\flat}}{\sqrt{\varepsilon}}}\mathcal{T}_{m}^{\varepsilon,\delta}(s^{\prime})\varphi$

and we check

$\displaystyle\left\|I_{\varphi}^{m,2}\right\|_{\Sigma^{k}}\leq
C\,\sqrt{\varepsilon}\,$
$\displaystyle\;\times\,\sup_{s\in[t^{\flat}-\delta,t^{\flat}+\delta]}\biggl{(}\,\|\mathcal{T}_{m}^{\varepsilon,\delta}(s)\varphi\|_{\Sigma^{k+1}}+(\|L\mathcal{T}_{m}^{\varepsilon,\delta}(s)\varphi\|_{\Sigma^{k+1}}+\sqrt{\varepsilon}\|\partial_{s}\mathcal{T}_{m}^{\varepsilon,\delta}(s)\varphi\|_{\Sigma^{k+1}})\int_{\frac{1}{2}\leq|s|\leq\frac{\delta}{\sqrt{\varepsilon}}}\frac{d\sigma}{\sigma}\biggr{)}$
$\displaystyle\qquad\qquad\leq
C\left(\sqrt{\varepsilon}\,\varepsilon^{\frac{m}{2}}|\log\varepsilon|^{\max(0,m-1)}\,\|\varphi\|_{\Sigma^{k+2m+3}}+\varepsilon|\log\varepsilon|\varepsilon^{\frac{m-1}{2}}|\log\varepsilon|^{\max(0,m-1)}\|\varphi\|_{\Sigma^{k+2m}}\right)$
$\displaystyle\qquad\qquad\leq
C\varepsilon^{\frac{m+1}{2}}|\log\varepsilon|^{\max(0,m)}\,\|\varphi\|_{\Sigma^{k+2(m+1)+1}}.$

We point out that it is at that very last stage that we loose some
$\log\varepsilon$ coefficient. Note also that the loss of regularity in the
recursive process is covered by the $m\rightarrow m+1$ process. It will appear
in the initialization process ($m=1$) in which we will get rid of the
logarithmic loss $|\log\varepsilon|$ by trading it as a loss of regularity.

#####  5.3.5. Proof of Lemma 5.12: the initialization of the recursion

For initializing the recursive process, we have to study $J_{1}(s)$, i.e. the
integral $I_{\varphi}^{m}$ replacing the transfer operator of the integrand by
$\mathbb{I}$, the identity operator. We obtain

$I_{\varphi}^{m=1}=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}{\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon})\chi(\sigma/\lambda){\rm
e}^{iL\sigma}\varphi\,d\sigma.$

Besides the estimate of the norm of $I_{\varphi}^{m=1}$, we also want to
calculate the leading order term when $s=t^{\flat}+\delta$. The main
difference with the preceding analysis is that we can push the integration by
parts at any order because the integrand is simpler.

######  Lemma 5.13.

Let $k\in{\mathbb{N}}$. Then, there exists a constant $C_{k,1}$ such that for
all $\varphi\in\mathcal{S}({\mathbb{R}}^{d})$

$\|I_{\varphi}^{m=1}\|_{\Sigma^{k}}\leq
C_{k,1}\sqrt{\varepsilon}\,\|\varphi\|_{k+3}.$

Moreover, there exists an operator
$\Theta^{\varepsilon,\delta}=\Theta^{\varepsilon,\delta}_{1}+\sqrt{\varepsilon}\Theta^{\varepsilon,\delta}_{2}$
and a constant $c_{k,M}$ such that for all $M\in{\mathbb{N}}$,

$\|\left.I_{\varphi}^{m=1}\right|_{s^{\prime}=t^{\flat}+\delta}-\sqrt{\varepsilon}W_{1}(t^{\flat},\zeta^{\flat})^{*}\mathcal{T}^{\flat}_{1\rightarrow
2}\varphi-\varepsilon\,\Theta^{\varepsilon,\delta}\varphi\|_{\Sigma^{k}}\leq
C_{k,M}\sqrt{\varepsilon}\left(\frac{\sqrt{\varepsilon}}{\delta}\right)^{M+1}\|\varphi\|_{\Sigma^{k+M+3}}$

with for $\ell=1,2$,
$\|\Theta^{\varepsilon,\delta}_{\ell}\varphi\|_{\Sigma_{k}}\leq
c_{k,M}\|\varphi\|_{\Sigma^{k+3}}$.

We recall that $\mathcal{T}^{\flat}_{1\rightarrow 2}$ is the transfer operator
defined in (1.43). In [20], this Lemma has already be proved with $M=1$. We
ameliorate here this result. This terminates the proof of Theorem 5.10 with
$(\mathcal{T}_{1,M}^{\varepsilon,\delta})_{2}=W_{1}(t^{\flat},\zeta^{\flat})^{*}\mathcal{T}^{\flat}_{1\rightarrow
2}$.

###### Proof.

Following the same lines of proofs than above, we write
$I_{\varphi}^{m=1}=I_{\varphi}^{1,1}+I_{\varphi}^{1,2}$ with

$I_{\varphi}^{1,1}=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\chi(\sigma)\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon}){\rm
e}^{iL\sigma}\varphi d\sigma.$

We point out that we use here the same notation than in the preceding section
for a simpler integrand. Like before, this term satisfies the estimate

$\|I_{\varphi}^{1}\|_{\Sigma^{k}}\ \leq
C\,\sqrt{\varepsilon}\,\|\varphi\|_{\Sigma^{k+1}}.$

We proceed to integration by parts in $I_{\varphi}^{2}$. We obtain

$\displaystyle I_{\varphi}^{1,2}$
$\displaystyle=-\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\partial_{\sigma}\left(\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma)\right){\rm
e}^{iL\sigma}\varphi d\sigma$
$\displaystyle-i\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}\,L\ \varphi d\sigma+\sqrt{\varepsilon}\left.\left({\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma){\rm
e}^{iL\sigma}\right)\right|_{\sigma=\frac{s^{\prime}-t^{\flat}}{\sqrt{\varepsilon}}}\varphi.$

We write

$\partial_{\sigma}\left(\frac{1}{\sigma}b^{\varepsilon}_{1}(\sigma)\right)=b^{\varepsilon}_{2}(\sigma)+\frac{1}{\sigma}b^{\varepsilon}_{3}(\sigma)\;\mbox{with}\;b^{\varepsilon}_{3}(\sigma)=\partial_{\sigma}b^{\varepsilon}_{1}(\sigma)+\sqrt{\varepsilon}(1-\chi)(\sigma)\chi(\sigma/\lambda)\partial_{s}\mathcal{Q}^{\varepsilon}(\sqrt{\varepsilon}\sigma).$

where the maps $\sigma\mapsto b^{\varepsilon}_{2}(\sigma)$, $\sigma\mapsto
b^{\varepsilon}_{3}(\sigma)$ and
$\sigma\mapsto\frac{1}{\sigma}b^{\varepsilon}_{3}(\sigma)$ are bounded from
$\Sigma^{k+1}_{\varepsilon}$ to $\Sigma^{k}_{\varepsilon}$ (with a loss of
regularity because of the presence of $\mathcal{Q}^{\varepsilon}(s)$) with the
additional property

$\int_{\mathbb{R}}\|b^{\varepsilon}_{2}(\sigma)\|_{\mathcal{L}(\Sigma^{k+1}_{\varepsilon},\
\Sigma^{k}_{\varepsilon})}d\sigma<c_{0}<+\infty$

for some constant $c_{0}$ independent of $\varepsilon$ and $\delta$.
Therefore, in $\Sigma^{k}$,

$\displaystyle I_{\varphi}^{1,2}$
$\displaystyle=-\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\frac{1}{\sigma}(b^{\varepsilon}_{3}(\sigma)+ib^{\varepsilon}_{1}(\sigma)L){\rm
e}^{iL\sigma}\varphi d\sigma+O(\sqrt{\varepsilon}\|\varphi\|_{\Sigma^{k+1}}).$

Another integration by parts gives

$\displaystyle I_{\varphi}^{1,2}$
$\displaystyle=\sqrt{\varepsilon}\int_{-\infty}^{(s^{\prime}-t^{\flat})/\sqrt{\varepsilon}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\partial_{\sigma}\left(\frac{1}{\sigma^{2}}b^{\varepsilon}(\sigma)(b^{\varepsilon}_{3}(\sigma)+ib^{\varepsilon}_{1}(\sigma)L)\right){\rm
e}^{iL\sigma}\varphi d\sigma$
$\displaystyle-\sqrt{\varepsilon}\left.\left({\rm