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

since the integrand has gained integrability. Note that it is at that very
place that we have a loss of $3$ momenta and derivatives in the estimate. We
have obtained the first inequality that allows to initiate the recursive
process of the preceding section. It remains to focus on the case
$s^{\prime}=t^{\flat}+\delta$.

We now consider the operator

$\mathfrak{I}_{\varphi}=\frac{1}{\sqrt{\varepsilon}}\left.I^{m=1}_{\varphi}\right|_{s^{\prime}=t^{\flat}+\delta}-W_{1}(t^{\flat},\zeta^{\flat})\mathcal{T}^{\flat}_{1\rightarrow
2}.$

Note first that by the construction of the function $\chi$,

$\left.I^{m=1}_{\varphi}\right|_{s^{\prime}=t^{\flat}+\delta}=\sqrt{\varepsilon}\int_{\mathbb{R}}{\rm
e}^{i\phi^{\varepsilon}(\sigma)}\chi(\sigma/\lambda){\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon}){\rm
e}^{iL\sigma}\varphi d\sigma$

Following [20] Section 5.3, we first transform the expression
$\left.I^{m=1}_{\varphi}\right|_{s^{\prime}=t^{\flat}+\delta}$ by performing
the change of variable

$z=\sigma(1+\sqrt{\varepsilon}\sigma
f(\sigma\sqrt{\varepsilon})/\mu^{\flat})^{1/2}$

and observe that
$\sigma=z(1+\sqrt{\varepsilon}zg_{1}(z\sqrt{\varepsilon}))\;\;\mbox{and}\;\;\partial_{\sigma}z=1+\sqrt{\varepsilon}zg_{2}(z\sqrt{\varepsilon})$
for some smooth bounded functions $g_{1}$ and $g_{2}$ with bounded
derivatives. Note that we have used that $\sigma\sqrt{\varepsilon}$ is of
order $\delta$, thus small, in the domain of the integral. Besides, there
exists a family of operator $\widetilde{\mathcal{Q}}^{\varepsilon}(z)$ such
that
${\mathcal{Q}}^{\varepsilon}(\sigma\sqrt{\varepsilon})=\widetilde{\mathcal{Q}}^{\varepsilon}(z\sqrt{\varepsilon})$
with $\widetilde{\mathcal{Q}}^{\varepsilon}(0)={\mathcal{Q}}^{\varepsilon}(0)$
and $\mathcal{Q}_{0}(0)=W_{1}(t^{\flat},\zeta^{\flat})$ and a compactly
supported function $\tilde{\chi}$, such that

$\displaystyle\left.I^{m=1}_{\varphi}\right|_{s^{\prime}=t^{\flat}+\delta}=\sqrt{\varepsilon}\int_{\mathbb{R}}{\rm
e}^{i\mu^{\flat}z^{2}}\tilde{\chi}(z/\lambda)\widetilde{\mathcal{Q}}^{\varepsilon}(z\sqrt{\varepsilon}){\rm
e}^{iz(1+\sqrt{\varepsilon}zg_{1}(z\sqrt{\varepsilon}))\,L}\varphi\frac{dz}{1+\sqrt{\varepsilon}zg_{2}(z\sqrt{\varepsilon})}.$

A Taylor expansion allows to write

$\displaystyle\widetilde{\mathcal{Q}}^{\varepsilon}(z\sqrt{\varepsilon}){\rm
e}^{i\sqrt{\varepsilon}z^{2}g_{1}(z\sqrt{\varepsilon}))\,L}\frac{1}{1+\sqrt{\varepsilon}zg_{2}(z\sqrt{\varepsilon})}={\mathcal{Q}}_{0}(0)+\sqrt{\varepsilon}z(\widetilde{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon})+z\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon}))$

for some smooth operator-valued maps
$z\mapsto\widetilde{\mathcal{Q}}^{\varepsilon}_{j}(z\sqrt{\varepsilon})$
mapping $\mathcal{S}({\mathbb{R}}^{d})$ into itself, such that for all
$\varphi\in{\mathcal{S}}({\mathbb{R}}^{d})$ the family
$\|{\widetilde{\mathcal{Q}}^{\varepsilon}_{j}(z\sqrt{\varepsilon})}\varphi\|_{\Sigma^{k}}\leq
c_{j}\|\varphi\|_{\Sigma^{k+2}}$ (because of the loss of regularity involved
by $L$ and $\mathcal{Q}^{\varepsilon}$) We obtain

$\mathfrak{I}_{\varphi}=\mathfrak{I}_{\varphi}^{1}+\mathfrak{I}_{\varphi}^{2}$

with

$\displaystyle\mathfrak{I}^{1}_{\varphi}=\sqrt{\varepsilon}\int_{\mathbb{R}}z\,{\rm
e}^{i\mu^{\flat}z^{2}}\tilde{\chi}(z/\lambda)(\widetilde{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon})+z\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon})){\rm
e}^{izL}\,\varphi dz,$
$\displaystyle\mathfrak{I}_{\varphi}^{2}=\mathcal{Q}_{0}(0)\int_{\mathbb{R}}{\rm
e}^{i\mu^{\flat}z^{2}}(1-\tilde{\chi})(z/\lambda)\,{\rm e}^{izL}\ \varphi dz.$

Let us study $\mathfrak{I}_{\varphi}^{1}$. Arguing by integration by parts as
previously, we obtain

$\displaystyle\mathfrak{I}^{1}_{\varphi}=$
$\displaystyle-\frac{\sqrt{\varepsilon}}{2i\mu^{\flat}}\int_{\mathbb{R}}\,{\rm
e}^{i\mu^{\flat}z^{2}}\frac{d}{dz}\left(\tilde{\chi}(z/\lambda)(\widetilde{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon})+z\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon}){\rm
e}^{izL}\right)\,dz$
$\displaystyle=-\frac{\varepsilon}{2i\mu^{\flat}}\int_{\mathbb{R}}\,{\rm
e}^{i\mu^{\flat}z^{2}}\tilde{\chi}(z/\lambda)\partial_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon}){\rm
e}^{izL}\,dz$
$\displaystyle\qquad\qquad-\frac{\sqrt{\varepsilon}}{2i\mu^{\flat}}\int_{\mathbb{R}}\tilde{\chi}(z/\lambda)(\,{\rm
e}^{i\mu^{\flat}z^{2}}\partial_{z}\left(\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon})+\sqrt{\varepsilon}z\partial_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon}){\rm
e}^{izL}\right)\,dz$
$\displaystyle\qquad\qquad-\frac{\sqrt{\varepsilon}}{2i\mu^{\flat}}\int_{\mathbb{R}}\,{\rm
e}^{i\mu^{\flat}z^{2}}\tilde{\chi}^{\prime}(z/\lambda)\left(\lambda^{-1}{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon})+\frac{z}{\lambda}\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon})\right){\rm
e}^{izL}\,dz.$

One then performs $M+2$ integration by parts in the last term of the right-
hand side that is supported in $|z|>\frac{\lambda}{2}$ and we obtain

$\displaystyle\mathfrak{I}^{1}_{\varphi}=$
$\displaystyle-\frac{\varepsilon}{2i\mu^{\flat}}\int_{\mathbb{R}}\,{\rm
e}^{i\mu^{\flat}z^{2}}\tilde{\chi}(z/\lambda)(\partial_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{1}(z\sqrt{\varepsilon})+2\partial_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon})){\rm
e}^{izL}\,dz$
$\displaystyle\qquad-\frac{\varepsilon}{2i\mu^{\flat}}\int_{\mathbb{R}}\tilde{\chi}(z/\lambda)\,{\rm
e}^{i\mu^{\flat}z^{2}}z\partial^{2}_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon}){\rm
e}^{izL}\,dz+O\left(\sqrt{\varepsilon}\left(\frac{\varepsilon}{\delta}\right)^{M+1}\|\varphi\|_{\Sigma^{k+M+3}}\right)$
$\displaystyle=$
$\displaystyle\sqrt{\varepsilon}\,\Theta^{\varepsilon,\delta}_{1}\varphi+O\left(\left(\frac{\varepsilon}{\delta}\right)^{M}\|\varphi\|_{\Sigma^{k+M+3}}\right)+\frac{\varepsilon}{(2i\mu^{\flat})^{2}}\int_{\mathbb{R}}\,{\rm
e}^{i\mu^{\flat}z^{2}}\partial_{z}\left(\tilde{\chi}(z/\lambda)\partial^{2}_{z}\widetilde{\mathcal{Q}}^{\varepsilon}_{2}(z\sqrt{\varepsilon}){\rm
e}^{izL}\right)\,dz$
$\displaystyle=\sqrt{\varepsilon}\,\Theta^{\varepsilon,\delta}_{1}\varphi+\varepsilon\,\Theta^{\varepsilon,\delta}_{2}\varphi+O\left(\left(\frac{\varepsilon}{\delta}\right)^{M}\|\varphi\|_{\Sigma^{k+M+3}}\right)$

with

$\|\Theta^{\varepsilon,\delta}_{1}\|_{\Sigma^{k}}\leq
c\,\delta\|\varphi\|_{\Sigma^{k+3}}.$

It remains to observe that the term $\mathfrak{I}_{\varphi}^{2}$ satisfies
after $M+1$ integration by parts

$\|\mathfrak{I}_{\varphi}^{2}\|_{\Sigma^{k}}\leq
C\left(\frac{\sqrt{\varepsilon}}{\delta}\right)^{M+1}\|\varphi\|_{\Sigma^{k+M+3}}.$

∎

####  5.4. Propagation of wave packets - Proof of Theorem 1.21

Let $k\in{\mathbb{N}}$ and let be $\psi^{\varepsilon}_{0}$ a polarized wave
packet as in (1.39):

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

Let $\delta>0$. By Theorem 5.9, for $t\in[t_{0},t^{\flat}-\delta]$, and in
$\Sigma^{k}_{\varepsilon}$, $\psi^{\varepsilon}(t)$ is an asymptotic sum of
wave packets and writes

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

with $\psi^{\varepsilon,N}_{\ell}(t)$ given by (5.5).

We now take the vector

$\underline{\psi}^{\varepsilon,N}(t^{\flat}-\delta)=\,^{t}\left(\underline{\psi}^{\varepsilon,N}_{1}(t^{\flat}-\delta),\underline{\psi}^{\varepsilon,N}_{2}(t^{\flat}-\delta)\right),\;\;\underline{\psi}^{\varepsilon,N}_{\ell}(t^{\flat}-\delta)=\psi^{\varepsilon,N}_{\ell}(t^{\flat}-\delta),\;\ell=1,2,$

as initial data in the system (5.7). It is a sum of $N$ wave packets. By
construction, in particular because of the linearity of the equation, we have
for all $t\in[t^{\flat}-\delta,t^{\flat}+\delta]$,

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

in $\Sigma^{k}_{\varepsilon}$. When $t=t^{\flat}+\delta$, we deduce from
Theorem 5.10 that in $\Sigma^{k}_{\varepsilon}$,

$\underline{\psi}^{\varepsilon,N}(t^{\flat}+\delta)=\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat}+\delta,t^{\flat})\;\left(\mathbb{I}+\sum_{1\leq m\leq
M}\Theta^{\varepsilon,\delta}_{m,M}\right)\;\mathcal{U}^{\varepsilon}_{\rm
diag}(t^{\flat},t^{\flat}-\delta)\underline{\psi}^{\varepsilon,N}(t^{\flat}-\delta)+O(\delta^{M}).$

By Proposition C.5 of the Appendix,
$\underline{\psi}^{\varepsilon,N}_{\ell}(t^{\flat}+\delta)$ is a sum of wave
packets

$\underline{\psi}^{\varepsilon,N}_{\ell}(t^{\flat}+\delta)=\sum_{0\leq m\leq
M}\varepsilon^{\frac{m}{2}}\underline{\psi}^{\varepsilon,m,M,N}_{\ell}(t^{\flat}+\delta)$

where each term
$\underline{\psi}^{\varepsilon,m,M,N}_{\ell}(t^{\flat}+\delta)$ involves a
term of action. When $m=0$ and $m=1$, these terms have been computed
precisely:

  * •

If $m=0$, for $\ell=1,2$,

$\displaystyle\underline{\psi}^{\varepsilon,0,M,N}_{\ell}(t^{\flat}+\delta)$
$\displaystyle={\rm
e}^{\frac{i}{\varepsilon}S_{\ell}(t^{\flat}+\delta,t^{\flat}-\delta,\zeta^{\flat})}{\rm
WP}^{\varepsilon}_{\Phi^{t^{\flat}+\delta,t_{0}}(z_{0})}\left(\mathcal{M}[F_{\ell}(t^{\flat}+\delta,t_{0},z_{0})]\pi_{\ell}(t_{0},z_{0})\vec{V}_{0}f_{0}\right)$

where we have used the property of the scalar propagation of wave packets.

  * •

If $m=1$, only the term with $\ell=2$ contributes and

$\displaystyle\underline{\psi}^{\varepsilon,1,M,N}_{2}(t^{\flat}+\delta)$
$\displaystyle={\rm
e}^{\frac{i}{\varepsilon}S_{2}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})+\frac{i}{\varepsilon}S_{1}(t^{\flat},t_{0},z_{0})}{\rm
WP}^{\varepsilon}_{\Phi^{t^{\flat}+\delta,t^{\flat}}_{2}(\zeta^{\flat})}(\varphi_{2})$

with

$\varphi_{2}=\mathcal{M}[F_{2}(t^{\flat}+\delta,t^{\flat},\zeta^{\flat})]W_{1}(t^{\flat},\zeta^{\flat})^{*}\mathcal{T}^{\flat}_{1\rightarrow
2}\mathcal{M}[F_{1}(t^{\flat},t_{0},z_{0})]\pi_{1}(t_{0},z_{0})\vec{V}_{0}f_{0},$

where $W_{1}$ is the off-diagonal matrix computed in (1.44).

At that stage of the proof, we have obtained that
$\psi^{\varepsilon}(t^{\flat}+\delta)$ is a sum of wave packet in
$\Sigma^{k}_{\varepsilon}$ up to
$\mathcal{O}\left(\left(\frac{\varepsilon}{\delta^{2}}\right)^{N}\delta^{-\kappa_{0}}+\delta^{M+1}\right)$
and we know precisely the terms of order $\varepsilon^{0}$ and
$\varepsilon^{\frac{1}{2}}$.

For concluding, we take the vector

$\psi^{\varepsilon}_{\rm
app}(t^{\flat}+\delta):=\underline{\psi}^{\varepsilon,N}_{1}(t^{\flat}+\delta)+\underline{\psi}^{\varepsilon,N}_{2}(t^{\flat}+\delta)$

as initial data at time $t=t^{\flat}+\delta$ in the equation (1.1). The
function $\psi^{\varepsilon}_{\rm app}(t^{\flat}+\delta)$ is an approximation
of $\psi^{\varepsilon}(t^{\flat}+\delta)$ at order
$\mathcal{O}\left(\left(\frac{\varepsilon}{\delta^{2}}\right)^{N}\delta^{-\kappa_{0}}+\delta^{M+1}\right)$
in $\Sigma^{k}_{\varepsilon}$. By construction and because of the linearity of
the equation, for all times $t\in[t^{\flat}+\delta,t_{0}+T]$,

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

We then applies Theorem 5.9 between times $t^{\flat}+\delta$ and $t$. Indeed,
the classical trajectories involved in the construction do not meet $\Upsilon$
again and we are in an adiabatic regime, as in theorem 5.9. This concludes the
proof of Theorem 1.21.

###  Appendix A Matrix-valued Hamiltonians

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

In this section, we work with $m\times m$ ($m\in{\mathbb{N}}$) matrix-valued
Hamiltonians $H$ that are subquadratic:

(A.1) $\forall\beta\in{\mathbb{N}}^{d},\;\;\exists
C_{\beta}>0,\;\;\forall(t,z)\in
I\times{\mathbb{R}}^{d},\;\;|\partial^{\beta}H(t,z)|_{{\mathbb{C}}^{m,m}}\leq
C_{\beta}\langle z\rangle^{(2-|\beta|)+}.$

######  Lemma A.1.

Assume that the matrix-valued function
$H\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{d},{\mathbb{C}}^{m,m})$
satisfies (A.1). Assume that for all $(t,z)\in I\times{\mathbb{R}}^{2d}$,
$H(t,z)$ has a smooth eigenvalue $h$ with a smooth eigenprojector $\pi(t,z)$
of constant rank for $|z|>m$, $>0$. Then, there exists a constant $C>0$ such
that for all $(t,z)\in I\times{\mathbb{R}}^{2d}$ with $|z|>m$,

$|h(t,z)|\leq C\langle z\rangle^{2},\;\;|\nabla h(t,z)|\leq C\langle
z\rangle.$

###### Proof.

The relation $H\pi=h\pi$ implies $|h(t,z)|\leq C\langle z\rangle^{2}$.
Moreover, writing

$H=h\pi+H(1-\pi)=h\pi+(1-\pi)H$

and differentiating these two relations, we obtain for all $j\in\\{1,\cdots
2d\\}$, denoting $\partial_{z_{j}}$ by $\partial_{j}$

(A.2)
$\partial_{j}H=\partial_{j}h\,\pi+h\partial_{j}\pi+\partial_{j}H(1-\pi)-H\partial_{j}\pi=\partial_{j}h\,\pi+h\partial_{j}\pi+(1-\pi)\partial_{j}H-\partial_{j}\pi
H.$

Multiplying from the left and the right with $\pi$ and using that
$\partial\pi$ is off-diagonal, we obtain the relation
$\pi\partial_{j}H\pi=\partial_{j}h\pi$, whence with $c={\rm Rank}(\pi)=cte$,

$\partial_{j}h=c\,{\rm Tr}(\pi\partial_{j}H\pi).$

This implies $|\partial_{j}h(t,z)|\leq C\langle z\rangle$. ∎

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

######  Example A.2.

Assume $d=1$. Let $\rho,\theta\in\mathcal{C}^{\infty}({\mathbb{R}})$ such that
$\rho$ has with bounded derivatives and with

$\rho(x)=\frac{1}{x^{3}}\;\;\mbox{and}\;\;\theta(x)=x\ln(x)-x\;\;\mbox{for}\;\;|x|>1.$

Define

$H(x)=\rho(x)\begin{pmatrix}\cos(\theta(x))&\sin(\theta(x))\\\
\sin(\theta(x))&-\cos(\theta(x))\end{pmatrix}.$

The eigenvalues of $H$ are $\pm\rho$ and the eigenprojector associated with
the eigenvalue $\rho$ is

$\pi(x)=\frac{1}{2}\left(1+\begin{pmatrix}\cos(\theta(x))&\sin(\theta(x))\\\
\sin(\theta(x))&-\cos(\theta(x))\end{pmatrix}\right).$

Its derivative

$\pi^{\prime}(x)=\frac{1}{2}\theta^{\prime}(x)\begin{pmatrix}-\sin(\theta(x))&\cos(\theta(x))\\\
\cos(\theta(x))&\sin(\theta(x))\end{pmatrix}$

is not bounded. On the other side, $\rho$ and $H$ are sub-quadratic. Indeed,
for $|x|>1$, the derivatives of the coefficient of $H$ are of the form

$\frac{1}{x^{3}}\left(p_{1}\left(\frac{1}{x},\ln
x\right)\cos\theta(x)+p_{2}\left(\frac{1}{x},\ln x\right)\sin\theta(x)\right)$

for $p_{1}$ and $p_{2}$ two polynomial functions of two variables and thus
bounded.

A manner of controlling the growth of the potential consists in requiring a
lower bound on the gap function $f$ at infinity.

######  Lemma A.3.

Let $\ell\in\\{1,2\\}$. Assume that the matrix-valued function
$H\in\mathcal{C}^{\infty}(I\times{\mathbb{R}}^{d},{\mathbb{C}}^{m,m})$
satisfies (A.1). Assume that for all $(t,z)\in I\times{\mathbb{R}}^{2d}$,
$H(t,z)$ has a smooth eigenvalue $h$ with smooth associated eigenprojectors
$\pi(t,z)$ of constant rank for $|z|>m$. Assume there exists $C,n_{0}>0$ such
that for $(t,z)\in I\times{\mathbb{R}}^{d}$ with $|z|>m$,

${\rm dist}\left(h(t,z),{\rm Sp}(H(t,z))\setminus\\{h(t,z)\\}\right)\geq
C\langle z\rangle^{-n_{0}}.$

Then, for all $\gamma\in{\mathbb{N}}^{2d}$ with $|\gamma|\geq 2$, there exists
a constant $C_{\gamma}>0$ such that

$\forall(t,z)\in
I\times{\mathbb{R}}^{2d},\;\;|\partial_{z}^{\gamma}\pi(t,z)|\leq
C_{\gamma}\langle
z\rangle^{|\gamma|n_{0}+\ell-1}\;\mbox{and}\;|\partial_{z}^{\gamma}h(t,z)|\leq
C_{\gamma}\langle z\rangle^{(|\gamma|-1)n_{0}+2(\ell-1)}.$

###### Proof.

We work for $|z|>m$ and fix $j\in\\{1,2\\}$. The relation (A.2) also implies

$\partial_{j}\pi(H-h)=(H-h)\partial_{j}\pi=\partial_{j}(H-h)$

where we keep the notation $\partial_{j}:=\partial_{z_{j}}$ for
$j\in\\{1,\cdots,2d\\}$. Using that $\partial_{j}\pi$ is off diagonal and
$(H-h)$ invertible on ${\rm Range}(1-\pi)$, we deduce

$\partial_{j}\pi=(1-\pi)\partial_{j}\pi\,\pi+\pi\,\partial_{j}\pi(1-\pi)$

with

$(1-\pi)\partial_{j}\pi\,\pi=(H-h)^{-1}(1-\pi)\partial_{j}(H-h)$

and

$\pi\,\partial_{j}\pi(1-\pi)=\pi\,\partial_{j}(H-h)(1-\pi)(H-h)^{-1}=\pi\partial_{j}(H-h)(H-h)^{-1}(1-\pi).$

On the range of $\pi$, the resolvent $(H-h)^{-1}$ is invertible, more
precisely, there exists $c>0$ such that

$\|(H-h)^{-1}(1-\pi)\|_{\mathcal{L}({\rm Ran}(1-\pi))}\leq c\,{\rm
dist}\left(h,{\rm Sp}(H)\setminus\\{h\\}\right)^{-1}\leq c\,C\langle
z\rangle^{n_{0}+\ell-1},$

whence $|\partial_{j}\pi(t,z)|\leq C\langle z\rangle^{n_{0}}$.

For analyzing the derivatives of $\pi$, one observes that the relation
$\pi=\pi^{2}$ implies

$\partial^{\gamma}\pi=\pi\partial^{\gamma}\pi+\partial^{\gamma}\pi\,\pi+\sum_{1<|\alpha|,|\beta|<|\gamma|}c_{\alpha,\beta}\partial^{\alpha}\pi\partial^{\beta}\pi$

for some coefficients $c_{\alpha,\beta}$. A recursive argument then gives the
estimate on the growth of the eigenprojectors.

Let us now consider the eigenvalue $h$. The relation $H\pi=\pi H=h\pi$ gives

$\partial h\pi=\partial H\pi-\partial\pi(H-h)=\pi\partial H-(H-h)\partial\pi.$

Multiplying on both side by $\pi$, using $(H-h)\pi=0$ and taking the matricial
trace, we obtain

$\partial h\,{\rm Tr}_{{\mathbb{C}}^{m,m}}(\pi)={\rm
Tr}_{{\mathbb{C}}^{m,m}}(\pi\partial H).$

This implies $|\partial h|\leq C\langle z\rangle^{\ell-1}$.

To get the relation on higher derivatives, we differentiate this relation,
which gives for $i,j\in{\mathbb{N}}^{2d}$,

$\displaystyle\partial^{2}_{ij}h\,{\rm Tr}_{{\mathbb{C}}^{m,m}}(\pi)$
$\displaystyle=\partial_{i}\left({\rm
Tr}_{{\mathbb{C}}^{m,m}}(\pi\partial_{j}H)\right)-\partial_{j}h{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial_{i}\pi).$

Since ${\rm Tr}_{{\mathbb{C}}^{m,m}}(\partial_{i}\pi)=0$, we are left with

$\displaystyle\partial^{2}_{ij}h\,{\rm Tr}_{{\mathbb{C}}^{m,m}}(\pi)$
$\displaystyle={\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial_{i}\pi\partial_{j}H)+{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\pi\partial_{ij}H).$

We obtain a control of the form $\langle z\rangle^{n_{0}+2(\ell-1)}$. Besides,
it allows to perform a recursive argument by writing for
$\gamma\in{\mathbb{N}}^{2d}$,

$\displaystyle\partial^{\gamma}\partial^{2}_{ij}h\,{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\pi)$ $\displaystyle=\partial^{\gamma}\left({\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial_{i}\pi\partial_{j}H)+{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\pi\partial_{ij}H)\right)$
$\displaystyle\qquad-\sum_{2\leq|\alpha|,\alpha\leq\gamma}c_{\alpha}\partial^{\gamma-\alpha}\partial^{2}_{ij}h\,{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial^{\alpha}\pi)$
$\displaystyle=\sum_{\alpha\leq\gamma}c_{\alpha}\left({\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial^{\gamma-\alpha}\partial_{i}\pi\,\partial^{\alpha}\partial_{j}H)+{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial^{\gamma-\alpha}\pi\partial^{\alpha}\partial_{ij}H)\right)$
$\displaystyle\qquad-\sum_{2\leq|\alpha|,\alpha\leq\gamma}c_{\alpha}\partial^{\gamma-\alpha}\partial^{2}_{ij}h\,{\rm
Tr}_{{\mathbb{C}}^{m,m}}(\partial^{\alpha}\pi)$

for some coefficients $c_{\alpha}$. One can then conclude recursively to
$|\partial^{\gamma}\partial^{2}_{ij}h|\leq c^{\prime}_{\gamma}\langle
z\rangle^{(|\gamma|+1)n_{0}+2(\ell-1)}$. ∎

These two Lemmata allow to derive the consequences of Assumptions 1.4 for a
Hamiltonian $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$. We now fix $m=2$.

######  Proposition A.4.

Assume that $H^{\varepsilon}=H_{0}+\varepsilon H_{1}$ satisfies the
Assumptions 1.4. Then, for $j\in\\{1,2\\}$ we have the following properties:

  1. (1)

For all $\gamma\in{\mathbb{N}}^{2d}$ with $|\gamma|\geq 2$, there exists a
constant $C_{\gamma}>0$ such that

$\forall(t,z)\in
I\times{\mathbb{R}}^{2d},\;\;|\partial_{z}^{\gamma}\pi_{j}(t,z)|\leq
C_{\gamma}\langle
z\rangle^{|\gamma|n_{0}}\;\mbox{and}\;|\partial_{z}^{\gamma}h_{j}(t,z)|\leq
C_{\gamma}\langle z\rangle^{(|\gamma|-1)n_{0}}.$

  2. (2)

If moreover $n_{0}=0$ in (1.6), then the maps
$z\mapsto\partial^{\gamma}\pi_{j}(t,z)$ and
$z\mapsto\partial_{z}^{\gamma}h_{j}(t,z)$ for $|\gamma|\geq 2$ are bounded. As
a consequence, the Hamiltonian trajectories $\Phi^{t_{0},t}_{h}(z)$ are
globally defined for all $z\in{\mathbb{R}}$. Besides, there exists
$C^{\prime}>0$ such that

$|\Phi^{t_{0},t}_{h_{j}}(z)|\leq C|z|{\rm e}^{C|t-t_{0}|}$

and the Jacobian matrices $F_{j}(t,z)=\partial_{z}\Phi^{t,t_{0}}_{h_{j}}(z)$
(see (1.12)) satisfy

$\|F_{j}(t,z)\|_{{\mathbb{C}}^{2d,2d}}\leq C{\rm e}^{C|t-t_{0}|}.$

######  Remark A.5.

Note that under the assumptions of Proposition A.4, for all
$j\in\\{1,\cdots,2d\\}$, the matrices

$f\partial_{z_{j}}\pi_{1}=-f\partial_{z_{j}}\pi_{2}=\frac{1}{2}\left(\partial_{z_{j}}(H-v)-\partial_{z_{j}}f(\pi_{2}-\pi_{1})\right)$

are bounded. This comes from the differentiation of the relation
$H=v\mathbb{I}+f(\pi_{2}-\pi_{1})$.

###### Proof.

Note first that Point 2 is a consequence of Point 1. We thus focus on Point 1.
We use that $\pi_{2}$ is the projector of the matrix $H_{0}-v\mathbb{I}$ for
the eigenvalue $f$. And the matrix $H_{0}-v\mathbb{I}$ satisfies the
assumptions of Lemma A.3 with $\ell=1$. Therefore, we have for $(t,z)\in
I\times{\mathbb{R}}^{d}$ with $|z|\geq 1$ and for $\gamma\in{\mathbb{N}}^{d}$.

$|\partial_{z}^{\gamma}\pi_{j}(t,z)|\leq C_{\gamma}\langle
z\rangle^{|\gamma|n_{0}}\;\mbox{and}\;|\partial_{z}^{\gamma}f(t,z)|\leq
C_{\gamma}\langle z\rangle^{(|\gamma|-1)n_{0}}.$

One concludes by observing that the function $v=2{\rm Tr}(H)$ is subquadratic,
whence the property of $h_{2}=v+f$. One argues similarly for $h_{1}$. ∎

We close this Section with the proof of Lemma 1.10

###### Proof of Lemma 1.10.

The map $t\mapsto\mathcal{R}_{\ell}(t,t_{0},z)$ is valued in the set of
unitary maps because the matrix $H^{{\rm adia}}_{\ell,1}$ is self adjoint.
Besides, the map

$(t,z)\mapsto
Z_{\ell}(t,z)=\pi_{\ell}(\Phi^{t,t_{0}}_{\ell}(z){\mathcal{R}}_{\ell}(t,t_{0},z)\pi_{\ell}^{\perp}(z)$

satisfies the ODE

$i\partial_{t}Z_{\ell}=-i\left(\pi_{\ell}(\partial_{t}\pi_{\ell}+\\{h,\pi_{\ell}\\})\right)\circ\Phi^{t,t_{0}}_{\ell}Z_{\ell},\;\;\;\;Z_{\ell}(t_{0},z)=0,$

and thus coincides with the solution $Z_{\ell}(t)=0$. ∎

###  Appendix B Elements of symbolic calculus : the Moyal product

In this section, we revisit results about the remainder estimate for the Moyal
product, aiming at their extension to the setting of the sets
$\boldsymbol{S}_{\delta}^{\mu}(\mathcal{D})$ that we have introduced in
Definition 4.1.

####  B.1. Formal expansion

We first recall the formal product rule for quantum observables with Weyl
quantization. Let $A,B\in{\mathcal{S}}({\mathbb{R}}^{2d},{\mathbb{C}}^{m,m})$.
The Moyal product $C:=A\circledast B$ is the semi-classical observable $C$
such that $\widehat{A}\circ\widehat{B}=\widehat{C}$. Some computations with
the Fourier transform give the following well known formula [34, Theorems
18.1.8]

(B.1)
$C(x,\xi)=\exp\left(\frac{i\varepsilon}{2}\sigma(D_{q},D_{p};D_{q^{\prime}},D_{p^{\prime}})\right)A(q,p)B(q^{\prime},p^{\prime})|_{(q,p)=(q^{\prime},p^{\prime})=(x,\xi)},$

where $\sigma$ is the symplectic bilinear form
$\sigma((q,p),(q^{\prime},p^{\prime}))=p\cdot q^{\prime}-p^{\prime}\cdot q$
and $D=i^{-1}\nabla$. By expanding the exponential term, we obtain

(B.2) $C(x,\xi)=\sum_{j\geq
0}\frac{\varepsilon^{j}}{j!}\left(\frac{i}{2}\sigma(D_{q},D_{p};D_{q^{\prime}},D_{p^{\prime}})\right)^{j}A(q,p)B(q^{\prime},p^{\prime})|_{(q,p)=(q^{\prime},p^{\prime})=(x,\xi)}.$

So that $C=\sum_{j\geq 0}\varepsilon^{j}C_{j}$ is a formal power series in
$\varepsilon$ with coefficients given by (4.1).

####  B.2. Symbols with derivative bounds

For $\mu\geq 0$ denote by ${\bf P}(\mu)$ the linear space of matrix-valued
$\mathcal{C}^{\infty}$ symbols
$A:{\mathbb{R}}^{2d}\rightarrow{\mathbb{C}}^{m,m}$ such that for any
$\gamma\in{\mathbb{N}}^{2d}$ with $|\gamma|\geq\mu$, there exists
$C_{\gamma}>0$ such that

$|\partial_{z}^{\gamma}A(z)|\leq C_{\gamma}\;\;\forall z\in{\mathbb{R}}^{2d}.$

Assuming that $A\in{\bf P}(\mu_{A})$, $B\in{\bf P}(\mu_{B})$ it is known that,
for $\varepsilon$ fixed, $A\circledast B=C$ where $C\in{\bf P}(\mu_{C})$ with
$\mu_{C}\geq\max\\{\mu_{A},\mu_{B}\\}$ (see e.g. the proof [34, Theorem
18.1.8])

We aim at having better estimates for small $\varepsilon>0$ and a control of
the derivatives of $C$ in terms of those of $A$ and $B$. The following
estimate and its proof are a particular case of [6, Theorem A.1].

######  Theorem B.1.

For every $N\in{\mathbb{N}}$ and $\gamma\in{\mathbb{N}}^{2d}$, there exists a
constant $K_{N,\gamma}$ such that for any $A\in{\bf P}(\mu_{A})$, $B\in{\bf
P}(\mu_{B})$ the Moyal remainder

(B.3) $R_{N}(A,B;z;\varepsilon):=(A\circledast B)(z)-\sum_{0\leq j\leq
N}\varepsilon^{j}C_{j}(z)$

satisfies for every $z\in{\mathbb{R}}^{2d}$ and $\varepsilon\in(0,1]$,

(B.4)
$\left|\partial_{z}^{\gamma}R_{N}(A,B;z;\varepsilon)\right|\leq\varepsilon^{N+1}K_{N,\gamma}\sum_{N+1\leq|\alpha|,|\beta|\leq
N+\kappa_{0}+|\gamma|}\|\partial_{z}^{\alpha}A\|_{L^{\infty}}\|\partial_{z}^{\beta}B\|_{L^{\infty}},$

with $\kappa_{0}=4d+2$.

The estimate 4.4 allows to evaluate the norms of the operators involved in
Theorem B.1.

######  Corollary B.2.

If $A\in\mathbf{S}_{\varepsilon,\delta}^{\mu}$ and
$B\in\mathbf{S}_{\varepsilon,\delta}^{\mu^{\prime}}$, then $A\circledast
B\in\mathbf{S}^{\mu+\mu^{\prime}}_{\varepsilon,\delta}$ (see Remark 4.2) and
for all $N,k\in{\mathbb{N}}$, there exists a constant $C_{N,k}>0$ such that

$\|{\rm
op}_{\varepsilon}(R_{N}(A,B;z;\varepsilon))\|_{\Sigma^{k}_{\varepsilon}}\leq
C_{N,k}\,\varepsilon^{N+1}\,\delta^{\mu+\mu^{\prime}-2(N+1+k+\kappa_{0})}.$

###### Proof.

By Fourier transform computations and application of the Taylor formula, we
get the following formula for the remainder,

(B.5)
$R_{N}(A,B;z;\varepsilon)=\frac{1}{N!}\left(\frac{i\varepsilon}{2}\right)^{N+1}\int_{0}^{1}(1-t)^{N}R_{N,t}(z;\varepsilon)dt,$

where

$\displaystyle R_{N,t}(z;\varepsilon)=$ $\displaystyle(2\pi\varepsilon
t)^{-2d}\int\int_{{\mathbb{R}}^{2d}\times{\mathbb{R}}^{2d}}\exp\left(-\frac{i}{2t\varepsilon}\sigma(u,v)\right)\sigma^{N+1}(D_{u},D_{v})A(u+z)B(v+z)dudv.$

Notice that the integral is an oscillating integral as we shall see below. We
now use Lemma B.3 for $A,B\in{\mathcal{S}}({\mathbb{R}}^{2d})$ with the
integrand

$F_{N,\gamma}(z;u,v)=\pi^{-2d}\
\partial_{z}^{\gamma}\left(\sigma^{N+1}(D_{u},D_{v})A(u+z)B(v+z)\right)$

and the parameter $\lambda=1/(2t\varepsilon)$. We then have

$|\partial_{z}^{\gamma}R_{N,t}(z;\varepsilon)|\leq
C_{d}\sup_{u,v\in{\mathbb{R}}^{2d}\atop|\alpha|+|\beta|\leq
4d+1}|\partial^{\alpha}_{u}\partial^{\beta}_{v}F_{N,\gamma}(z;u,v)|.$

Moreover, there holds the elementary estimate

$|\sigma^{N+1}(D_{u},D_{v})A(u)B(v)|\leq(2d)^{N+1}\sup_{|\alpha|+|\beta|=N+1}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}A(x,\xi)\partial_{y}^{\beta}\partial_{\eta}^{\alpha}B(y,\eta)|.$

Together with the Leibniz formula, we then get the claimed results with
universal constants. For symbols $A\in{\bf P}(\mu_{A})$ and $B\in{\bf
P}(\mu_{B})$ we argue by localisation. We use $A_{\eta}(u)={\rm e}^{-\eta
u^{2}}A(u)$ and $B_{\eta}(v)={\rm e}^{-\eta v^{2}}B(v)$ for $\eta>0$ and pass
to the limit $\eta\rightarrow 0$. ∎

######  Lemma B.3.

There exists a constant $C_{d}>0$ such that for any
$F\in{\mathcal{S}}({\mathbb{R}}^{2d}\times{\mathbb{R}}^{2d},{\mathbb{C}}^{m,m})$
the integral

(B.6)
$I(\lambda)=\lambda^{2d}\int_{{\mathbb{R}}^{2d}\times{\mathbb{R}}^{2d}}\exp[-i\lambda\sigma(u,v)]F(u,v)dudv.$

satisfies

(B.7) $|I(\lambda)|\leq
C_{d}\sup_{u,v\in{\mathbb{R}}^{2d}\atop|\alpha|+|\beta|\leq
4d+1}|\partial^{\alpha}_{u}\partial^{\beta}_{v}F(u,v)|.$

###### Proof of Lemma B.3.

The lemma is proved in a standard way, using integration by parts and
stationary phase argument. For the sake of completeness, we give here a proof.
We introduce a cut-off $\chi_{0}\in\mathcal{C}_{0}^{\infty}({\mathbb{R}})$
such that

$\chi_{0}(x)=1\;\;\mbox{ for}\;\;|x|\leq 1/2\;\;\mbox{
and}\;\;\chi_{0}(x)=0\;\;\mbox{for }\;\;|x|\geq 1.$

We split $I(\lambda)$ into too pieces and write
$I(\lambda)=I_{0}(\lambda)+I_{1}(\lambda)$ with

$\displaystyle I_{0}(\lambda)$
$\displaystyle=\lambda^{2d}\int\\!\\!\int_{{\mathbb{R}}^{2d}\times{\mathbb{R}}^{2d}}\exp[-i\lambda\sigma(u,v)]\chi_{0}(u^{2}+v^{2})F(u,v)dudv,$
$\displaystyle I_{1}(\lambda)$
$\displaystyle=\lambda^{2d}\int\\!\\!\int_{{\mathbb{R}}^{2d}\times{\mathbb{R}}^{2d}}\exp[-i\lambda\sigma(u,v)](1-\chi_{0})(u^{2}+v^{2}))F(u,v)dudv.$

We notice that $(u,v)\mapsto\sigma(u,u)$ is a quadratic non-degenerate real
form on ${\mathbb{R}}^{4d}$.
Let us estimate $I_{1}(\lambda)$. We can integrate by parts with the
differential operator

$L=\frac{i}{|u|^{2}+|v|^{2}}\left(Ju\cdot\frac{\partial}{\partial
v}-Jv\cdot\frac{\partial}{\partial u}\right),$

using that $L{\rm e}^{-i\lambda\sigma(u,v)}=L{\rm e}^{-i\lambda Ju\cdot
v}=\lambda{\rm e}^{-i\lambda\sigma(u,v)}$. For $I_{1}(\lambda)$, the integrand
is supported outside the ball of radius $1/\sqrt{2}$ in ${\mathbb{R}}^{4d}$.
Performing $4d+1$ integrations by parts for gaining enough decay to ensure
integrability in $(u,v)\in{\mathbb{R}}^{4d}$, we get a constant $c_{d}$ such
that

$|I_{1}(\lambda)|\leq c_{d}\sup_{u,v\in{\mathbb{R}}^{2d}\atop|\mu|+|\nu|\leq
4d+1}|\partial^{\mu}_{u}\partial^{\nu}_{v}F(u,v)|.$

To estimate $I_{0}(\lambda)$ we apply the stationary phase. The symmetric
matrix of the quadratic form $\sigma(u,v)$ is

$A_{\sigma}=\begin{pmatrix}0&-J\\\ J&0\end{pmatrix}.$

So the stationary phase Theorem ([34], Vol.I, section 7.7), we obtain the
existence of two constants $c_{1},c_{2}>0$ such that

(B.8) $|I_{0}(\lambda)-\lambda^{-2d}c_{1}|\leq
c_{2}\sup_{u,v\in{\mathbb{R}}^{2d}\atop|\alpha|\leq
2}|\partial^{\alpha}(\chi_{0}(u^{2}+v^{2})F(u,v)|.$

∎

###  Appendix C Elements of semi-classical calculus: perturbation of scalar
systems

In this Appendix, we revisit several well-known results concerning a
Hamiltonian $K(t)$ valued in the set ${\mathbb{C}}^{m,m}$ of $m\times m$
matrices ($m\in{\mathbb{N}}$), and which is a perturbation of a scalar
function $k(t)$. We consider an interval $I_{\delta}\subset{\mathbb{R}}$ that
may depend on $\delta>0$ and assume that $K(t)$ is defined on $I_{\delta}$ and
of the form

(C.1) $K(t)=k(t){\mathbb{I}}_{m}+\varepsilon
K_{1}(t)+\cdots\varepsilon^{N}K_{N}(t)$

with $k$ scalar-valued and $k(t){\mathbb{I}}_{m}+\varepsilon K_{1}(t)$ is
subquadratic on the time interval $I_{\delta}$ according to Definition 1.1.

The difference with the classical setting is that we assume

$\forall t\in
I_{\delta},\;\;\frac{1}{\varepsilon^{2}}\left(K(t)-k(t)\mathbb{I}_{m}-\varepsilon
K_{1}(t)\right)\in\boldsymbol{S}_{\varepsilon,\delta}^{-1}(\mathcal{D}).$

Therefore, we have to revisit the results to take care of the loss in $\delta$
and control all the classical estimates with respect to this parameter. We
denote by $\mathcal{U}_{K}^{\varepsilon}(t_{0},t)$ the unitary propagator
associated with $K(t)$.

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

####  C.1. Egorov Theorem

The Egorov Theorem describes the evolution of an observable when it is
conjugated by the propagator $\mathcal{U}_{K}^{\varepsilon}(t,s)$ associated
with the operator $K(t)$. It is important to notice that this propagator maps
$\Sigma^{k}_{\varepsilon}$ in itself for all $k\in{\mathbb{N}}$. Indeed, for
$1\leq j\leq d$,

$\displaystyle
i\varepsilon\partial_{t}(x_{j}\psi^{\varepsilon}(t))-\widehat{K}(t)(x_{j}\psi^{\varepsilon}(t))=f^{\varepsilon}_{j}(t)\;\;\mbox{and}\;\;i\varepsilon\partial_{t}(\varepsilon\partial_{x_{j}}\psi^{\varepsilon}(t))-\widehat{K}(t)(\varepsilon\partial_{x_{j}}\psi^{\varepsilon}(t))=g^{\varepsilon}_{j}(t),$

with $f^{\varepsilon}_{j}(t)=[x_{j},\widehat{K}(t)]\psi^{\varepsilon}(t)$ and
$g^{\varepsilon}_{j}(t)=[\varepsilon
D_{x_{j}},\widehat{K}(t)]\psi^{\varepsilon}(t)$ uniformly bounded in $L^{2}$.
Therefore, one deduces the $L^{2}$-boundedness of the families
$(x_{j}\psi^{\varepsilon}(t))$ and $(\varepsilon
D_{x_{j}}\psi^{\varepsilon}(t))$ for all $1\leq j\leq d$, whence the
boundedness of $\psi^{\varepsilon}(t)$ in $\Sigma^{1}_{\varepsilon}$ for all
$t\in{\mathbb{R}}$. The reader will have understood that a recursive process
will give the boundedness of $\psi^{\varepsilon}(t)$ in any
$\Sigma^{k}_{\varepsilon}$, for $t\in{\mathbb{R}}$ and $k\in{\mathbb{N}}$. In
this setting, our aim is to revisit the evolution of
$\mathcal{U}_{K}^{\varepsilon}(t_{\rm
in},t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,t_{\rm in})$ for matrix-
valued observables $A\in\boldsymbol{S}_{\delta}$ and in spaces
$\Sigma^{k}_{\varepsilon}$, with a precise estimate of the remainders.

######  Proposition C.1.

With the above notations, for any matrix-valued symbol
$A\in\boldsymbol{S}_{\delta}$, there exists a formal series $(t,t_{\rm
in})\mapsto\sum_{j\geq 0\varepsilon^{j}}A_{j}(t,t_{\rm in})$ defined on
$I_{\delta}\times I_{\delta}$ and such that for any $J\geq 1$, we have for all
$t,t_{\rm in}\in I^{\delta}$,

$\mathcal{U}_{K}^{\varepsilon}(t_{\rm
in},t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,t_{\rm in})=\sum_{0\leq
j\leq J}\varepsilon^{j}\widehat{A}_{j}(t,t_{\rm
in})+\varepsilon^{J+1}\widehat{R}^{\varepsilon}_{J}(t,t_{\rm in}),$

with $A_{j}(t,t_{\rm in})\in\boldsymbol{S}^{-2j}_{\delta}$ uniformly in
$\varepsilon\in]0,1]$. Besides, for all $k\in{\mathbb{N}}$, there exists a
constant $C_{k,J}$ such that

$\|\widehat{R^{\varepsilon}_{J}}(t,t_{\rm
in})\|_{\Sigma^{k}_{\varepsilon}}\leq c_{k}\delta^{-2(J+1+\kappa_{0})},$

and the matrix $A_{0}(t,t_{\rm in})$ is given by

(C.2) $A_{0}(t,t_{\rm in},z)=\left({\mathcal{R}}^{*}(t,t_{\rm
in})A{\mathcal{R}}(t,t_{\rm in})\right)\circ(\Phi_{k}^{t,t_{\rm in}})(z),$

where the unitary matrices ${\mathcal{R}}(t,t_{\rm in},z)$ solve the transport
equation

(C.3) $i\partial_{t}{\mathcal{R}}(t,t_{\rm
in},z)=K_{1}\left(t,\Phi_{k}^{t,t_{\rm in}}(z)\right){\mathcal{R}}(t,t_{\rm
in},z),\;{\mathcal{R}}(t_{\rm in},t_{\rm in},z)=\mathbb{I}_{m}.$

######  Remark C.2.

In particular we have the propagation law of the supports:

${\rm supp}(A_{j}(t,t_{\rm in}))=\Phi_{k}^{t_{\rm in},t}({\rm
supp}(A))\;\;\mbox{ for any}\;\;j\geq 0.$

In the scalar time independent case case ($k=k(z)$ and $K_{j}=0$ for $j\geq
1$), the Egorov theorem

$\mathcal{U}_{K}^{\varepsilon}(t_{\rm
in},t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,t_{\rm
in})=\widehat{A\circ\Phi_{k}^{t,t_{in}}}+O(\varepsilon)$

is well-known (see [13, 14, 64] for example). In the time-dependent matrix-
valued case considered here, the dynamics on the observable is driven by the
classical flow twisted by the precession ${\mathcal{R}}$ (see also Section
1.3.3 where such terms appear). The proof also requires a careful treatment of
the time.

###### Proof.

We perform a recursive argument. The starting point comes from the analysis of
the auxiliary map defined for $\tau,t\in I_{\delta}$ and valued in
$\mathbf{S}_{\delta}$ by

$A\mapsto
A(t,\tau):=\left(\mathcal{R}(\tau,t)A\,\mathcal{R}(\tau,t)^{*}\right)\circ\Phi^{t,\tau}_{k}.$

Because we are going to differentiate in $\tau$, we use the relation

$i\partial_{\tau}\Phi^{t,\tau}_{k}=-J\nabla k(\tau,\Phi^{t,\tau}_{k}),$

(where $J$ is the matrix defined in (1.10)), which implies for all
$z\in{\mathbb{R}}^{2d}$, using also that the flow map is symplectic and
preserves the Poisson bracket,

$\partial_{\tau}A(t,\tau,z)+\\{k(\tau),A(t,\tau)\\}(z)=\frac{1}{i}[K_{1}(\tau,z),A(t,\tau,z)],\;\;A(t,t)=A.$

Let us know starts with the proof of the result for $J=0$. We choose
$s,\tau,t\in I_{\delta}$ and consider the quantity

$Q^{\varepsilon}_{A}(t,s,\tau)=\mathcal{U}^{\varepsilon}_{K}(s,\tau)\widehat{A}(t,\tau)\mathcal{U}^{\varepsilon}_{K}(\tau,s).$

The times $s,\tau,t$ can be understood as $s\leq\tau\leq t$ with $s$ an
initial time (that will be taken as $s=t_{\rm in}$) and $t$ the time at which
we want to prove the property. We then have the boundary properties

$Q^{\varepsilon}_{A}(t,s,t)=\mathcal{U}^{\varepsilon}_{K}(s,t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,s)\;\;\mbox{and}\;\;Q^{\varepsilon}_{A}(t,s,s)=\widehat{A}(t,s).$

Differentiating in $\tau$, we have

$\displaystyle\frac{d}{d\tau}Q^{\varepsilon}_{A}(t,s,\tau)$
$\displaystyle=\mathcal{U}^{\varepsilon}_{K}(s,\tau)\left(\left[-\frac{1}{i\varepsilon}\widehat{K}(\tau),\widehat{A}(t,\tau)\right]+\partial_{\tau}\widehat{A}(t,\tau)\right)\mathcal{U}^{\varepsilon}_{K}(\tau,s)$
$\displaystyle=\mathcal{U}^{\varepsilon}_{K}(s,\tau)\left(\left[-\frac{1}{i\varepsilon}\widehat{K}(\tau),\widehat{A}(t,\tau)\right]-\widehat{\\{k,A(t,\tau)\\}}+\frac{1}{i}\widehat{[K_{1}(\tau),A(t,\tau)]}\right)\mathcal{U}^{\varepsilon}_{K}(\tau,s)$
$\displaystyle=\varepsilon\,\mathcal{U}^{\varepsilon}_{K}(s,\tau)\,\widehat{B_{1}^{\varepsilon}}(t,\tau)\,\mathcal{U}^{\varepsilon}_{K}(\tau,s)$

where the matrix $B^{\varepsilon}_{1}\in\mathbf{S}^{-2}_{\varepsilon,\delta}$
stems from the Moyal product (see Corollary B.2). We deduce by integration
between the times $s$ and $t$

$\mathcal{U}^{\varepsilon}_{K}(s,t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,s)=\widehat{A}(t,s)+\varepsilon\int_{s}^{t}\mathcal{U}^{\varepsilon}_{K}(s,\tau)\,\widehat{B_{1}^{\varepsilon}}(t,\tau)\,\mathcal{U}^{\varepsilon}_{K}(\tau,s)d\tau,$

which gives the first step of the recursive argument.

We now assume that we have obtained for $J\geq 0$

$\mathcal{U}^{\varepsilon}_{K}(s,t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,s)=\sum_{j=0}^{J}\varepsilon^{j}\widehat{A}_{j}(t,s)+\varepsilon^{J+1}\int_{s}^{t}\mathcal{U}^{\varepsilon}_{K}(s,\tau)\,\widehat{B_{J+1}^{\varepsilon}}(t,\tau)\,\mathcal{U}^{\varepsilon}_{K}(\tau,s)d\tau$

with $B^{\varepsilon}_{J+1}\in\mathbf{S}^{-2(J+1)}_{\delta}$. We write

$B^{\varepsilon}_{J+1}=B_{J+1}+B^{1,\varepsilon}_{J+2}$

with $B_{J+1}\in\mathbf{S}_{\delta}^{-2(J+1)}$ and
$B^{\varepsilon}_{J+2}\in\mathbf{S}^{-2(J+2)}_{\varepsilon,\delta}$. Then, the
preceding equation writes

(C.4)
$\displaystyle\mathcal{U}^{\varepsilon}_{K}(s,t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,s)$
$\displaystyle=\sum_{j=0}^{J}\varepsilon^{j}\widehat{A}_{j}(t,s)$
$\displaystyle+\varepsilon^{J+1}\int_{s}^{t}Q^{\varepsilon}_{B_{J+1}}(t,s,\tau)d\tau+\varepsilon^{J+2}\int_{s}^{t}\mathcal{U}^{\varepsilon}_{K}(s,\tau)\,\widehat{B_{J+2}^{1,\varepsilon}}(t,\tau)\,\mathcal{U}^{\varepsilon}_{K}(\tau,s)d\tau.$

We focus on the term involving $Q^{\varepsilon}_{B_{J+1}}(t,s,\tau)$ that we
treat as in the preceding step. We obtain

$Q^{\varepsilon}_{B_{J+1}}(t,s,\tau)=\mathcal{R}(s,\tau)B_{J+1}(t,\tau)\mathcal{R}(\tau,s)^{*}+\varepsilon\int_{s}^{\tau}\mathcal{U}^{\varepsilon}_{K}(s,\tau^{\prime})\widehat{B}_{J+2}^{2,\varepsilon}(\tau,\tau^{\prime})\mathcal{U}^{\varepsilon}_{K}(\tau^{\prime},s)d\tau^{\prime}$

with $B_{J+2}^{2,\varepsilon}\in\mathbf{S}^{-2(J+2)}_{\varepsilon,\delta}$. We
set

$A_{J+1}(s,t)=\int_{s}^{t}\mathcal{R}(s,\tau)B_{J+1}(t,\tau)\mathcal{R}(\tau,s)^{*}\in\mathbf{S}^{-2(J+1)}_{\delta}$

and

$B^{\varepsilon}_{J+2}(t,\tau)=B^{1,\varepsilon}_{J+2}(t,\tau)+\int_{s}^{\tau}B_{J+2}^{2,\varepsilon}(\tau^{\prime},\tau)d\tau^{\prime}\in\mathbf{S}^{-2(J+1)}_{\varepsilon,\delta}.$

The equation (C.4) then becomes

$\mathcal{U}^{\varepsilon}_{K}(s,t)\widehat{A}\,\mathcal{U}^{\varepsilon}_{K}(t,s)=\sum_{j=0}^{J+1}\varepsilon^{j}\widehat{A}_{j}(t,s)\\\
+\varepsilon^{J+2}\int_{s}^{t}\mathcal{U}^{\varepsilon}_{K}(s,\tau)\,\widehat{B_{J+2}^{\varepsilon}}(t,\tau)\,\mathcal{U}^{\varepsilon}_{K}(\tau,s)d\tau$

and this concludes the proof. ∎

####  C.2. Asymptotic behavior of the propagator

In this section, we analyze the propagator
$\mathcal{U}^{\varepsilon}_{K}(t,t_{\rm in})$ and compare it with
$\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm in})$ the propagator for

$KS(t)=k(t)\mathbb{I}_{m}+\varepsilon K_{1}(t).$

######  Lemma C.3.

For all $J\in{\mathbb{N}}$, there exists

$W^{\varepsilon}(t,t_{\rm in})=\sum_{0\leq j\leq
J}\varepsilon^{j}\,W_{j}(t,t_{\rm
in})+\varepsilon^{J+1}R_{J}^{\varepsilon}(t,t_{\rm in})$

with $W_{j}(t,t_{\rm in})\in\boldsymbol{S}_{\delta}^{-2j}$ such that for all
$t\in{\mathbb{R}}$,

$\mathcal{U}_{K}^{\varepsilon}(t,t_{\rm
in})=\,\widehat{W}^{\varepsilon}(t,t_{\rm
in})\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm in}).$

Besides, $W_{0}(t,t_{\rm in})=\mathbb{I}_{m}$ and for all
$\gamma\in{\mathbb{N}}^{2d}$, there exists $C_{J,\gamma}>0$ such that

$\,\sup_{z\in{\mathbb{R}}^{d}}|\partial_{z}^{\gamma}R_{J}^{\varepsilon}(t,t_{\rm
in},z)|\leq C_{J,\gamma}\,\delta^{-2(J+1+|\gamma|+\kappa_{0})}$

where $\kappa_{0}$ is the universal constant of Theorem B.1.

######  Remark C.4.

Using the estimate (4.4), we deduce that for all $k\in{\mathbb{N}}$, there
exists $C_{k}>0$ such that

$\|{\rm op}_{\varepsilon}(R_{J}^{\varepsilon}(t,t_{\rm
in}))\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\left(\frac{\varepsilon}{\delta^{2}}\right)^{\frac{|\gamma|}{2}}\,\delta^{-2(J+1+k+\kappa_{0})}.$

As a consequence, for $\delta=\varepsilon^{\alpha}$ with
$\alpha\in(0,\frac{1}{2}]$, we have

(C.5) $\|{\rm op}_{\varepsilon}(R_{J}^{\varepsilon}(t,t_{\rm
in}))\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\,\delta^{-2(J+1+k+\kappa_{0})}.$

Similarly, by (4.5), for such $\delta$,

(C.6) $\|{\rm op}_{\varepsilon}(W_{j}(t,t_{\rm
in}))\|_{\mathcal{L}(\Sigma^{k}_{\varepsilon})}\leq
C_{k}\,\delta^{-2j-k},\;\;j\in{\mathbb{N}}.$

###### Proof.

If such a $W^{\varepsilon}(t,t_{\rm in})$ exists, it must satisfy the
following equation

$i\varepsilon\partial_{t}\widehat{W}^{\varepsilon}(t,t_{\rm
in})\widehat{W}^{\varepsilon}(t,t_{\rm
in})\mathcal{U}_{KS}^{\varepsilon}(t_{\rm
in},t)\left(\widehat{K}(t)-\widehat{KS}(t)\right)\,\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm
in})^{*},\;\;\widehat{W}^{\varepsilon}(t_{\rm in},t_{\rm in})=\mathbb{I}_{m}$

Applying Egorov Theorem of Propostion C.1, we know that

$\mathcal{U}_{KS}^{\varepsilon}(t_{\rm
in},t)\left(\widehat{K}(t)-\widehat{KS}(t)\right)\,\mathcal{U}_{KS}^{\varepsilon}(t_{\rm
in},t)^{*}=\varepsilon^{2}\widehat{L}^{\varepsilon}(t,t_{\rm in})$

where

$\displaystyle{L^{\varepsilon}(t,t_{\rm in})=\sum_{j\geq
0}\varepsilon^{j}L_{j}(t,t_{\rm in})}\in{\bf S}^{-1}_{\varepsilon,\delta}$
with estimates on $L_{j}(t,t_{\rm in})$ and on the remainder term in the
asymptotic expansion. Using the estimates proved in Appendices A and B, it is
enough to solve as formal series in $\varepsilon$ the equation

(C.7) $i\partial_{t}\left(\sum_{j\geq 0}\varepsilon^{j}W_{j}(t,t_{\rm
in})\right)=\varepsilon\left(\sum_{j\geq 0}\varepsilon^{j}W_{j}(t,t_{\rm
in})\right)\circledast\left(\sum_{k\geq 0}\varepsilon^{j}L_{j}(t,t_{\rm
in})\right).$

∎

####  C.3. Propagation of wave packets

When $\psi^{\varepsilon}_{0}$ is a wave packet, the action of
$\mathcal{U}^{\varepsilon}_{KS}(t,t_{1})\psi^{\varepsilon}_{0}$ on
$\psi^{\varepsilon}_{0}$ can be described precisely. Following Section 14.2 of
[13], Theorem 77, we have the following result.

######  Proposition C.5.

Assume we have $\psi^{\varepsilon}={\rm
WP}_{z_{0}}^{\varepsilon}(f^{\varepsilon})\vec{V}$ with
$\vec{V}\in{\mathbb{C}}^{m}$,

$f^{\varepsilon}=\sum_{0\leq j\leq
J}\varepsilon^{j/2}f_{j},\;\;f_{j}\in{\mathcal{S}}({\mathbb{R}}^{d}).$

There exists a family $(\vec{U}_{j}(t))_{j\geq 0}$ defined on the interval
$I_{\delta}$ such that

  1. (i)

For all $j\in{\mathbb{N}}$ and $t\in I_{\delta}$,
$\vec{U}_{j}(t)\in\mathcal{S}({\mathbb{R}}^{d})$,

  2. (ii)

For all $k,j\in{\mathbb{N}}$, there exists a constant $C_{k,j}$ such that

$\sup_{t\in
I_{\delta}}\,\sup_{|\alpha|+|\beta|=k}\|x^{\alpha}\partial_{x}^{\beta}U_{j}(t)\|_{L^{\infty}}\leq
C_{k,j}\delta^{-j}.$

  3. (iii)

For all $k\in{\mathbb{N}}$ and $N\in{\mathbb{N}}$, there exists $C_{k,N}$ and
$N_{k}$ such that for all $t\in{\mathbb{R}}$, we have

$\left\|\mathcal{U}^{\varepsilon}_{K}(t,t_{\rm in})\psi^{\varepsilon}-{\rm
e}^{\frac{i}{\varepsilon}S(t,t_{\rm in},z_{0})}{\rm
WP}^{\varepsilon}_{\Phi_{k}^{t,t_{\rm
in}}(z_{0})}\left(\sum_{j=0}^{2N}\varepsilon^{\frac{j}{2}}\vec{U}_{j}(t)\right)\right\|_{\Sigma^{k}_{\varepsilon}}\leq
C_{k,N}\left(\frac{\varepsilon}{\delta^{2}}\right)^{N}\delta^{-N-N_{k}}.$

Besides

(C.8) $\vec{U}_{0}(t)={\mathcal{R}}(t,t_{\rm in})\,{\mathcal{M}}[F(t,t_{\rm
in})]f_{0}\vec{V}\;\;\mbox{and}\;\;\vec{U}_{1}(t)={\mathcal{R}}(t,t_{\rm
in})\,{\mathcal{M}}[F(t,t_{\rm in})]{\bf b}_{1}(t,t_{\rm in})f_{0}\vec{V}$

where

$\displaystyle{\bf b}_{1}(t,t_{\rm in})$
$\displaystyle=\frac{1}{i}\sum_{|\alpha|=3}\frac{1}{\alpha!}\int_{t_{\rm
in}}^{t}\partial_{z}^{\alpha}h(s,z_{s}){\rm op}_{1}^{w}((F(t_{\rm
in},s)z)^{\alpha})ds$ (C.9) $\displaystyle\;+\frac{1}{i}\int_{t_{\rm
in}}^{t}\partial_{z}H_{1}(s,z_{s}){\rm op}_{1}(F(t_{\rm in},s)z)ds,$

$F(t,t_{\rm in})$ is the stability matrix for the flow
$z_{t}:=\Phi_{k}^{t,t_{\rm in}}(z_{0})$ (see (1.12)) and
${\mathcal{R}}(t,t_{\rm in})$ satisfies the equation (C.3).

###### Proof.

Let $k\in{\mathbb{N}}$. Using Lemma C.3 and the estimate (C.5), we obtain

$\displaystyle\mathcal{U}_{K}^{\varepsilon}(t,t_{\rm in})\psi^{\varepsilon}$
$\displaystyle=\,\widehat{W}^{\varepsilon}(t,t_{\rm
in})\,\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm in})\psi^{\varepsilon}$
$\displaystyle=\sum_{0\leq j\leq J}\varepsilon^{j}\,\widehat{W}_{j}(t,t_{\rm
in})\,\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm
in})\psi^{\varepsilon}+\varepsilon^{J+1}\widehat{R}_{J}^{\varepsilon}(t,t_{\rm
in})\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm in})\psi^{\varepsilon}$
$\displaystyle=\sum_{0\leq j\leq J}\varepsilon^{j}\,\widehat{W}_{j}(t,t_{\rm
in})\,\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm
in})\psi^{\varepsilon}+O\left(\varepsilon^{J+1}\delta^{-2(J+1+\kappa_{0})}\right)$

in $\Sigma^{k}_{\varepsilon}({\mathbb{R}}^{d})$. We then use the standard
result of propagation of wave packets for $KS(t)$ (see [13])

$\mathcal{U}_{KS}^{\varepsilon}(t,t_{\rm in})\psi^{\varepsilon}={\rm
e}^{\frac{i}{\varepsilon}S(t,t_{\rm in},z_{0})}{\rm
WP}^{\varepsilon}_{z_{t}}\left(\sum_{j=1}^{2J}\varepsilon^{\frac{j}{2}}\vec{B}_{j}(t)\right)+O(\varepsilon^{J+1})$

with $\vec{B}_{j}(t)=\vec{U}_{j}(t)$ as in (C.8) for $j=0,1$, and
$\vec{B}_{j}(t)$ is determined by a recursive equation in terms of
$\vec{B}_{0}(t),\cdots,\vec{B}_{j-1}(t)$. This description relies on the
observation that setting

$\varphi^{\varepsilon}_{j}(t)={\rm e}^{\frac{i}{\varepsilon}S(t,t_{\rm
in},z_{0})}{\rm WP}^{\varepsilon}_{z_{t}}\left(\vec{B}_{j}(t)\right)$

we have the two relations

$\displaystyle i\varepsilon\partial_{t}\varphi^{\varepsilon}_{j}(t)={\rm
e}^{\frac{i}{\varepsilon}S}{\rm
WP}^{\varepsilon}_{z_{t}}\left(-\partial_{t}(S+\xi_{t}\cdot
x_{t})\vec{B}_{j}+\sqrt{\varepsilon}\dot{z}_{t}\cdot\widehat{z}\vec{B}_{j}+i\varepsilon\partial_{t}\vec{B}_{j}\right)$
$\displaystyle\widehat{KS(t)}\varphi^{\varepsilon}_{j}(t)={\rm
e}^{\frac{i}{\varepsilon}S}{\rm
WP}^{\varepsilon}_{z_{t}}\left(\widehat{KS(t,z_{t}+\sqrt{\varepsilon}z)}\vec{B}_{j}(t)\right)$
$\displaystyle\qquad={\rm e}^{\frac{i}{\varepsilon}S}{\rm
WP}^{\varepsilon}_{z_{t}}\left({\rm
op}_{1}\left(k(t,z_{t})+\sqrt{\varepsilon}\nabla k(t,z_{t})\cdot
z+\frac{\varepsilon}{2}{\rm Hess}\,k(t,z_{t})z\cdot z+\varepsilon
K_{1}(t,z_{t})+O(\varepsilon^{\frac{3}{2}}\right)\vec{B}_{j}\right)$

where we have set $z_{t}=(x_{t},\xi_{t})$. These relations also prove (C.8)
with the additional remark that $W_{0}(t,t_{\rm in})=\mathbb{I}$ and, for
$j\in{\mathbb{N}}$,

$\widehat{W}_{j}(t,t_{\rm in}){\rm
WP}^{\varepsilon}_{z_{t}}\left(\vec{B}_{j}(t)\right)={\rm
WP}^{\varepsilon}_{z_{t}}\left({\rm op}_{1}\left(W_{j}(t,t_{\rm
in},z_{t}+\sqrt{\varepsilon}z)\right)\vec{B}_{j}(t)\right)$

and using the estimate (C.6) with $\varepsilon=1$. ∎

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### Index


Verifying liquidity of recursive Bitcoin contracts

M. Bartoletti]Massimo Bartoletti
University of Cagliari, Italy

S. Lande]Stefano Lande
University of Cagliari, Italy

M. Murgia]Maurizio Murgia
University of Trento, Italy

R. Zunino]Roberto Zunino
University of Trento, Italy

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# A $G$-covering subgroup system of a finite group
for some classes of $\sigma$-soluble groups ††thanks: Research was supported
by the NNSF of China (No. 11771409.)

A-Ming Liu, W. Guo,
School of Science, Hainan University, Haikou, 570228, P.R. China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Inna N. Safonova
Department of Applied Mathematics and Computer Science,
Belarusian State University, Minsk 220030, Belarus
E-mail<EMAIL_ADDRESS>

Alexander N. Skiba
Department of Mathematics and Technologies of Programming,
Francisk Skorina Gomel State University,
Gomel 246019, Belarus
E-mail<EMAIL_ADDRESS>
###### Abstract

Let ${\mathfrak{F}}$ be a class of group and $G$ a finite group. Then a set
$\Sigma$ of subgroups of $G$ is called a _$G$ -covering subgroup system_ for
the class ${\mathfrak{F}}$ if $G\in{\mathfrak{F}}$ whenever
$\Sigma\subseteq{\mathfrak{F}}$.

We prove that: If a set of subgroups $\Sigma$ of $G$ contains at least one
supplement to each maximal subgroup of every Sylow subgroup of $G$, then
$\Sigma$ is a $G$-covering subgroup system for the classes of all
$\sigma$-soluble and all $\sigma$-nilpotent groups, and for the class of all
$\sigma$-soluble $P\sigma T$-groups.

This result gives positive answers to questions 19.87 and 19.88 from the
Kourovka notebook.

00footnotetext: Keywords: finite group, $G$-covering subgroup system,
$\sigma$-soluble group, $\sigma$-nilpotent group, $P\sigma
T$-group.00footnotetext: Mathematics Subject Classification (2010): 20D10

##  1 Introduction

Throughout this paper, all groups are finite and $G$ always denotes a finite
group. Moreover, $\mathbb{P}$ is the set of all primes and $\sigma$ is some
partition of $\mathbb{P}$, that is, $\sigma=\\{\sigma_{i}\mid i\in I\\}$,
where $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and
$\sigma_{i}\cap\sigma_{j}=\emptyset$ for all $i\neq j$.

Before continuing, recall some concepts of the papers [1, 2, 3, 4] which play
fundamental role in the theory of $\sigma$-properties of groups.

The group $G$ is said to be: _$\sigma$ -primary_ if $G$ is a
$\sigma_{i}$-group for some $i=i(G)$; _$\sigma$ -decomposable_ or _$\sigma$
-nilpotent_ if $G=G_{1}\times\dots\times G_{n}$ for some $\sigma$-primary
groups $G_{1},\ldots,G_{n}$; _$\sigma$ -soluble_ if every chief factor of $G$
is $\sigma$-primary.

A set ${\cal H}$ of subgroups of $G$ is a _complete Hall $\sigma$-set_ of $G$
if every member $\neq 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$
for some $\sigma_{i}\in\sigma$ and ${\cal H}$ contains exactly one Hall
$\sigma_{i}$-subgroup of $G$ for every $i$.

Recall that a subgroup $A$ of $G$ is said to be _$\sigma$ -permutable_ in $G$
if $G$ possesses a complete Hall $\sigma$-set ${\cal H}$ such that
$AH^{x}=H^{x}A$ for all $H\in{\cal H}$ and all $x\in G$ .

We say that $G$ is a $P\sigma T$-group if ${\sigma}$-permutability is a
transitive relation in $G$, that is, if $K$ is a ${\sigma}$-permutable
subgroup of $H$ and $H$ is a ${\sigma}$-permutable subgroup of $G$, then $K$
is a ${\sigma}$-permutable subgroup of $G$.

Let ${\mathfrak{F}}$ be a class of group and $G$ a finite group. Then a set
$\Sigma$ of subgroups of $G$ is called a _$G$ -covering subgroup system_ [5]
for the class ${\mathfrak{F}}$ if $G\in{\mathfrak{F}}$ whenever
$\Sigma\subseteq{\mathfrak{F}}$.

In this paper, we prove the following

Theorem A. Suppose that a set of subgroups $\Sigma$ of $G$ contains at least
one supplement to each maximal subgroup of every Sylow subgroup of $G$. Then
$\Sigma$ is a $G$-covering subgroup system for any class $\mathfrak{F}$ in the
following list:

(i) $\mathfrak{F}$ is the class of all $\sigma$-soluble groups.

(ii) $\mathfrak{F}$ is the class of all $\sigma$-nilpotent groups.

(iii) $\mathfrak{F}$ is the class of all $\sigma$-soluble $P\sigma T$-groups.

The theory of $P\sigma T$-groups was built in the works [1, 2, 3]. Theorem A
gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook
[6] and, also, allows us to give the following new characterization of
$\sigma$-soluble $P\sigma T$-groups.

Corollary 1.1. $G$ is a $\sigma$-soluble $P\sigma T$-group if and only if each
maximal subgroup of every Sylow subgroup of $G$ has a supplement $T$ in $G$
such that $T$ is a $\sigma$-soluble $P\sigma T$-group.

In the classical case when $\sigma=\sigma^{1}=\\{\\{2\\},\\{3\\},\ldots\\}$:
$G$ is $\sigma^{1}$-soluble (respectively $\sigma^{1}$-nilpotent) if and only
if $G$ is soluble (respectively nilpotent); ${\sigma}^{1}$-permutable
subgroups are also called _$S$ -permutable_ [7]; in this case a $P\sigma
T$-group is also called a _$PST$ -group_ [7].

A significant place to the theory of $PST$-groups is given in the book [6].
From Theorem A we get also the following result in this line researches.

Corollary 1.2. $G$ is a soluble $PST$-group if and only if each maximal
subgroup of every Sylow subgroup of $G$ has a supplement $T$ in $G$ such that
$T$ is a soluble $PST$-group.

##  2 Basic lemmas

If $n$ is an integer, the symbol $\pi(n)$ denotes the set of all primes
dividing $n$; as usual, $\pi(G)=\pi(|G|)$, the set of all primes dividing the
order of $G$. $G$ is said to be a $D_{\pi}$-group if $G$ possesses a Hall
$\pi$-subgroup $E$ and every $\pi$-subgroup of $G$ is contained in some
conjugate of $E$.

By the analogy with the notation $\pi(n)$, we write $\sigma(n)$ to denote the
set $\\{\sigma_{i}|\sigma_{i}\cap\pi(n)\neq\emptyset\\}$;
$\sigma(G)=\sigma(|G|)$. $G$ is said to be: a _$\sigma$ -full group of Sylow
type_ [1] if every subgroup $E$ of $G$ is a $D_{\sigma_{i}}$-group for every
$\sigma_{i}\in\sigma(E)$.

Lemma 2.1 (See Theorem A [8]). Every $\sigma$-soluble group is a $\sigma$-full
group of Sylow type.

Lemma 2.2 (Theorem 1 in [9]). $G$ is $\pi$-separable if and only if

(i) $G$ has a Hall $\pi$-subgroup and a Hall $\pi^{\prime}$-subgroup;

(ii) $G$ has a Hall $\pi\cup\\{p\\}$-subgroup and a Hall
$\pi^{\prime}\cup\\{q\\}$-subgroup for all $p\in\pi^{\prime}$ and $q\in\pi$.

Lemma 2.3  (See Corollary 2.4 and Lemma 2.5 in [1]). The class of all
$\sigma$-nilpotent groups ${\mathfrak{N}}_{\sigma}$ is closed under taking
products of normal subgroups, homomorphic images and subgroups. Moreover, if
$E$ is a normal subgroup of $G$ and $E/(E\cap\Phi(G))$ is $\sigma$-nilpotent,
then $E$ is $\sigma$-nilpotent.

In view of Lemma 2.3, the class ${\mathfrak{N}}_{\sigma}$, of all
$\sigma$-nilpotent groups, is a hereditary saturated formation and so from
Proposition 2.2.8 in [10] we get the following

Lemma 2.4 (See Proposition 2.2.8 in [10]). If $N$ is a normal subgroup of $G$,
then $(G/N)^{{\mathfrak{N}}_{\sigma}}=G^{{\mathfrak{N}}_{\sigma}}N/N.$

In this lemma, $G^{\mathfrak{N_{\sigma}}}$ denotes the _$\sigma$ -nilpotent
residual_ of $G$, that is, the intersection of all normal subgroups $N$ of $G$
with $\sigma$-nilpotent quotient $G/N$.

Lemma 2.5 (See Theorem A in [2]). If $G$ is a $\sigma$-soluble $P\sigma
T$-group and $D=G^{\mathfrak{N_{\sigma}}}$, then the following conditions
hold:

(i) $G=D\rtimes M$, where $D$ is an abelian Hall subgroup of $G$ of odd order,
$M$ is $\sigma$-nilpotent and every element of $G$ induces a power
automorphism in $D$;

(ii) $O_{\sigma_{i}}(D)$ has a normal complement in a Hall
$\sigma_{i}$-subgroup of $G$ for all $i$.

Conversely, if Conditions (i) and (ii) hold for some subgroups $D$ and $M$ of
$G$, then $G$ is a $P\sigma T$-group.

##  3 Proof of Theorem A

Proof of Theorem A.  Assume that this theorem is false. We can assume without
loss of generality that
$\sigma(G)=\\{\sigma_{1},\sigma_{2},\ldots,\sigma_{t}\\}$.

(I) $G$ is not $\sigma$-nilpotent. Hence $t>1$ and
$D:=G^{{\mathfrak{N}}_{\sigma}}\neq 1$.

Indeed, assume that $G$ is $\sigma$-nilpotent. Then $G$ is $\sigma$-soluble.
Hence Statements (i) and (ii) hold for $G$. Moreover, in this case for every
$i$ the product $H_{i}$, of all normal $\sigma_{i}$-subgroups of $G$, is the
unique normal Hall $\sigma_{i}$-subgroup of $G$ and $G=H_{1}\times
H_{2}\times\cdots\times H_{t}$. Hence every subgroup of $G$ is
$\sigma$-permutable in $G$. Thus Statement (iii) also holds for $G$, contrary
to our assumption on $G$. Hence (I) hods.

(i) Assume that this assertion is false and let $G$ be a counterexample of
minimal order.

(*) $G$ has no non-identity normal $\sigma$-primary subgroups.

Assume that $G$ has a minimal normal $\sigma$-primary subgroup, $R$ say.

Let $P/R$ be any non-identity Sylow subgroup of $G/R$. Then for some prime $p$
and for a Sylow $p$-subgroup $G_{p}$ of $G$ we have $G_{p}R/R=P/R$, so $G_{p}$
is non-identity.

Now let $V/R$ be any maximal subgroup of $P/R$, that is, $|P:V|=|P/R:V/R|=p$.
Then $V=R(G_{p}\cap V)$, so

$p=|G_{p}R:R(G_{p}\cap V)|=(|G_{p}||R|:|G_{p}\cap R|):(|R||(G_{p}\cap
V|:|R\cap G_{p}\cap V|)=|G_{p}:G_{p}\cap V|,$

so $G_{p}\cap V$ is a maximal subgroup of $G_{P}$. Hence $G$ has a
$\sigma$-soluble subgroup $T$ such that $(G_{p}\cap V)T=G$.

But then $RT/R\simeq T/(T\cap R)$ is a $\sigma$-soluble subgroup of $G/R$ such
that

$(V/R)(RT/R)=(R(G_{p}\cap V)/R)(TR/R)=G/R.$

Therefore the hypothesis holds for $G/R$, so $G/R$ is $\sigma$-soluble by the
choice of $G$. But then $G$ is$\sigma$-soluble, a contradiction. Hence we have
(*).

(**) $t=2$, that is, $\sigma(G)=\\{\sigma_{1},\sigma_{2}\\}$.

Assume that $t>2$ and let $P_{i}$ be a Sylow $p_{i}$-subgroup of $G$ for some
$p_{1}\in\sigma_{1}\cap\pi(G)$, $p_{2}\in\sigma_{2}\cap\pi(G)$ and
$p_{3}\in\sigma_{3}\cap\pi(G)$. Let $V_{i}$ be a maximal subgroup of $P_{i}$.
Then, by hypothesis, $G$ has $\sigma$-soluble subgroups $T_{1}$, $T_{2}$ and
$T_{3}$ such that $G=V_{i}T_{i}$ for $i=1,2,3.$

Let $R$ be a minimal normal subgroup of $T_{1}$. Then $R$ is $\sigma$-primary,
$R$ is a $\sigma_{k}$-group say. Since
$|G:T_{2}|=|T_{1}T_{2}:T_{2}|=|T_{1}:T_{1}\cap T_{2}|$ is a $p_{2}$-number and
$|T_{1}:T_{1}\cap T_{3}|$ is a $p_{3}$-number, where $p_{2}\in\sigma_{2}$ and
$p_{3}\in\sigma_{3}$, we have either $R\leq T_{1}\cap T_{2}$ or $R\leq
T_{1}\cap T_{3}$, $R\leq T_{1}\cap T_{2}$ say. Hence
$R^{G}=R^{T_{1}T_{2}}=R^{T_{2}}\leq T_{2}$, so $G$ has a non-identity normal
$\sigma$-primary subgroup, contrary to Claim (*). Thus (**) holds.

The final contradiction for (i).  Let $\pi=\sigma_{1}\cap\pi(G)$. Since
$T_{i}$ is $\sigma$-soluble, $T_{i}$ has a Hall $\sigma_{k}$-subgroup for all
$k$ by Lemma 2.1. Then a Hall $\sigma_{2}$-subgroup of $T_{1}$ is a Hall
$\pi^{\prime}$subgroup of $G$ and a Hall $\sigma_{1}$-subgroup of $T_{2}$ is a
Hall $\pi-$subgroup of $G$.

Now we show that $G$ has a Hall $\pi\cup\\{p\\}$-subgroup for every
$p\in\sigma_{2}\cap\pi(G)$. If $|\sigma_{2}\cap\pi(G)=1$ it is evident. Now
assume that $|\sigma_{2}\cap\pi(G)|>1$ and let
$q\in(\sigma_{2}\cap\pi(G))\setminus\\{p\\}.$ Let $V$ be a maximal subgroup of
a Sylow $q$-subgroup $Q$ of $G$. And let $T$ be a $\sigma$-soluble supplement
to $V$ in $G$. Then $T$ is $\pi$-separable by Claim (**). Hence $T$ has a Hall
$\pi\cup\\{p\\}$-subgroup $H$ by Lemma 2.2. But $|G:T|$ is a $\\{q\\}$-number,
where $p\neq q\not\in\sigma_{1}$, so $H$ is a Hall $\pi\cup\\{p\\}$-subgroup
of $G$.

Similarly it can be proved that $G$ has a Hall
$\pi^{\prime}\cup\\{p\\}$-subgroup for all $p\in\pi$. Therefore $G$ is
$\pi$-separable by Lemma 2.2 and so $G$ is $\sigma$-soluble by Claim (**),
contrary to the choice of $G$. Hence Statement (i) holds.

(iii) Assume that this assertion is false and let $G$ be a counterexample of
minimal order. Then $G$ is $\sigma$-soluble by Part (i).

(1) If $R$ is a non-identity normal subgroup of $G$, then the hypothesis holds
for $G/R$. Hence $G/R$ is a $\sigma$-soluble $P\sigma T$-group (See the proof
of Claim (*)).

(2) If $R$ is an abelian minimal normal subgroup of $G$, then $R$ is not a
Sylow subgroup of $G$.

Indeed, assume that $R$ is Sylow subgroup of $G$ and let $V$ be an y maximal
subgroup of $R$. Then for every supplement $T$ to $V$ in $G$ we have that
$T\cap R$ is normal in $G$, the minimality of $R$ implies that $T=G$. Hence
$G$ is a $\sigma$-soluble $P\sigma T$-group, a contradiction. Hence (2) holds.

(3) $D$ is $\sigma$-nilpotent.

Assume that this is falls. Then $D$ is not $\sigma$-primary. Let $R$ be a
minimal normal subgroup of $G$, so $R$ is a $\sigma_{i}$-group for some $i$
since $G$ is $\sigma$-soluble. Moreover, from Lemmas 2.3 and 2.4 we get that

$(G/R)^{{\mathfrak{N}}_{\sigma}}=G^{{\mathfrak{N}}_{\sigma}}R/R=DR/R\simeq
D/(D\cap R)$

is a Hall $\sigma$-nilpotent subgroup of $G/R$ by Claim (1). Hence $R$ is the
unique minimal normal subgroup of $G$, $R<D$ and $R\nleq\Phi(G)$ since $D$ is
not $\sigma$-nilpotent. Therefore $C_{G}(R)\leq R$ and $D/R$ is a Hall
subgroup of $G/R$. Moreover, $D/R$ is not a $\sigma_{i}$-group since $D$ is
not $\sigma$-primary. Let $p$ be a prime dividing $|D/R|$ such that
$p\not\in\sigma_{i}$. And let $P$ be a Sylow $p$-subgroup of $D$. Then $P\cap
R=1$ and $P$ is a Sylow $p$-subgroup of $G$ since $D/R$ is a Hall subgroup of
$G/R$.

Let $V$ be a maximal subgroup of $P$ and $T$ a supplement to $V$ in $G$ such
that $T$ is a $P\sigma T$-group. Then $T^{{\mathfrak{N}}_{\sigma}}\leq D$ and
$T^{{\mathfrak{N}}_{\sigma}}$ is a Hall abelian subgroup of $T$ such that
every subgroup of $T^{{\mathfrak{N}}_{\sigma}}$ is normal in $T$ by Lemma 2.5.
Moreover, $R\leq T$ since $|G:T|$ is a $\\{p\\}$-number. Hence
$T^{{\mathfrak{N}}_{\sigma}}\cap R$ is a normal abelian Hall subgroup of $R$.
Hence either $T^{{\mathfrak{N}}_{\sigma}}\cap R=1$ or
$T^{{\mathfrak{N}}_{\sigma}}\cap R=R$ and so $R\leq
T^{{\mathfrak{N}}_{\sigma}}$.

First assume that $T^{{\mathfrak{N}}_{\sigma}}\cap R=1$. Then
$T^{{\mathfrak{N}}_{\sigma}}\leq C_{G}(R)$, so $T^{{\mathfrak{N}}_{\sigma}}=1$
and hence $T$ is $\sigma$-nilpotent. From $P=P\cap VT=V(P\cap T)$ it follow
that $T$ is not a $\sigma_{i}$-group. Hence for a Hall
$\sigma_{i}^{\prime}$-subgroup $E$ of $T$ we have $E\neq 1$ and $E\leq
C_{G}(R)\leq R$, a contradiction. Therefore $R\leq
T^{{\mathfrak{N}}_{\sigma}}$, so $R=T^{{\mathfrak{N}}_{\sigma}}$ is a
$q$-group for some prime $q\neq p$ since $C_{G}(R)\leq R$ and
$T^{{\mathfrak{N}}_{\sigma}}$ is abelian. Let $Q$ be a Sylow $q$-subgroup of
$T$. Then $R=Q$ since $R=T^{{\mathfrak{N}}_{\sigma}}$ is a Hall subgroup of
$T$. Moreover, $R$ is a Sylow $q$-subgroup of $G$ since $p\neq q$ and $|G:T|$
is a $\\{p\\}$-number, contrary to Claim (2). Hence we have (3).

(4) $D$ is nilpotent.

Assume that this false and let $R$ be a minimal normal subgroup of $G$. Then
$R\leq D$ and $C_{G}(R)\leq R$ and $D/R$ is a Hall subgroup of $G/R$ (see the
proof of Claim (3)). Hence $D\leq O_{\sigma_{i}}(G)$ for some $i$ by Claim
(3). Let $P$ be a Sylow $p$-subgroup of $G$, where
$p\in\pi(G)\setminus\sigma_{i}$. Let $V$ be a maximal subgroup of $P$ and $T$
a supplement to $V$ in $G$ such that $T$ is a $P\sigma T$-group. Then $R\leq
D\leq T$, so $T^{{\mathfrak{N}}_{\sigma}}\cap R$ is a normal abelian Hall
subgroup of $R$. Hence $R\leq T^{{\mathfrak{N}}_{\sigma}}$ (see the proof of
Claim (*)). On the other hand, $T^{{\mathfrak{N}}_{\sigma}}\leq D$. Therefore
$R=T^{{\mathfrak{N}}_{\sigma}}$ is a Sylow $q$-subgroup of $G$ for some $q\neq
p$, contrary to Claim (2).

(5) $D$ is a Hall subgroup of $G$.

Suppose that this is false and let $P$ be a Sylow $p$-subgroup of $D$ such
that $1<P<G_{p}$, where $G_{p}\in\text{Syl}_{p}(G)$. We can assume without
loss of generality that $G_{p}\leq H_{1}$.

(a) $D=P$ is a minimal normal subgroup of $G$.

Let $R$ be a minimal normal subgroup of $G$ contained in $D$. Since $D$ is
nilpotent by Claim (4), $R$ is a $q$-group for some prime $q$. Moreover,
$D/R=(G/R)^{\mathfrak{N}_{\sigma}}$ is a Hall subgroup of $G/R$ by Claim (1)
and Lemma 2.3. Suppose that $PR/R\neq 1$. Then $PR/R\in\text{Syl}_{p}(G/R)$.
If $q\neq p$, then $P\in\text{Syl}_{p}(G)$. This contradicts the fact that
$P<G_{p}$. Hence $q=p$, so $R\leq P$ and therefore $P/R\in\text{Syl}_{p}(G/R)$
and we again get that $P\in\text{Syl}_{p}(G)$. This contradiction shows that
$PR/R=1$, which implies that $R=P$ is the unique minimal normal subgroup of
$G$ contained in $D$. Since $D$ is nilpotent by Claim (4), a
$p^{\prime}$-complement $E$ of $D$ is characteristic in $D$ and so it is
normal in $G$. Hence $E=1$, which implies that $R=D=P$.

(b) $D\nleq\Phi(G)$. Hence for some maximal subgroup $M$ of $G$ we have
$G=D\rtimes M$.

(c) If $G$ has a minimal normal subgroup $L\neq D$, then $G_{p}=D\times(L\cap
G_{p})$. Hence $O_{p^{\prime}}(G)=1$.

Indeed, $DL/L\simeq D$ is a Hall subgroup of $G/L$ by Claim (1). Hence
$G_{p}L/L=RL/L$, so $G_{p}=D\times(L\cap G_{p})$. Thus $O_{p^{\prime}}(G)=1$
since $D<G_{p}$ by Claim (a).

(d) $V=C_{G}(D)\cap M$ is a normal subgroup of $G$ and $C_{G}(D)=D\times V\leq
H_{1}$.

In view of Claim (b), $C_{G}(D)=D\times V$, where $V=C_{G}(D)\cap M$ is a
normal subgroup of $G$. By Claim (a), $V\cap D=1$ and hence $V\simeq DV/D$ is
$\sigma$-nilpotent by Lemma 2.2. Let $W$ be a $\sigma_{1}$-complement of $V$.
Then $W$ is characteristic in $V$ and so it is normal in $G$. Therefore we
have (d) by Claim (c).

The final contradiction for (5). Let $Q$ be a Sylow $q$-subgroup of $G$, where
$q\in\pi(G)\setminus\pi(H_{1})$. Let $V$ be a maximal subgroup of $P$ and $T$
a supplement to $V$ in $G$ such that $T$ is a $P\sigma T$-group. Then
$T^{{\mathfrak{N}}_{\sigma}}\leq D$ and $T^{{\mathfrak{N}}_{\sigma}}$ is a
Hall abelian subgroup of $T$. Then $D$ is not a Sylow $q$-subgroup of $T$ and
so $T^{{\mathfrak{N}}_{\sigma}}=1$, which implies that $T$ is
$\sigma$-nilpotent. But then for a Sylow $q$-subgroup $T_{q}$ of $T$ we have
$1<T_{q}\leq C_{G}(D)\leq H_{1}$, a contradiction.

(6) Every subgroup $H$ of $D$ is normal in $G$. Hence every element of $G$
induces a power automorphism in $D$.

Since $D$ is nilpotent by Claim (4), it is enough to consider the case when
$H\leq O_{p}(D)$ for some $p\pi(D)$.

Let $R$ be any Sylow $r$-subgroup of $G$, where $r\not\in\pi(D)$. Let
$V_{1},V_{2},\ldots,V_{t}$ be the set of all maximal subgroups of $R$. Let
$T_{i}$ be a supplement to $V_{i}$ in $G$ such that $T_{i}$ is a $P\sigma
T$-group with $D_{i}=T^{\mathfrak{N}}$.

Since $G=V_{i}T_{i}$, $R=V_{i}(T_{i}\cap R)$. Hence for some $a_{i}\in
T_{i}\cap R$ we have we have $a_{i}\not\in V_{i}$. We show that $a_{i}\in
N_{G}(H)$.

First observe that $D\leq T_{i}$ since $|G:T|$ is a $q$-number, where
$r\not\in\pi(D)$. Moreover, $D_{i}\leq D$. But $D_{i}$ is a Hall subgroup of
$T_{i}$ and every subgroup of $D_{i}$ is normal in $T_{i}$, so $D_{i}$ is a
Hall subgroup of $D$. So either $H\leq O_{p}(D)\leq D_{i}$ or $O_{p}(D)\cap
D_{i}=1$. In the former case we have $a_{i}\in N_{G}(H)$ since every subgroup
of $D_{i}$ is normal in $T_{I}$. Now assume that $O_{p}(D)\cap D_{i}=1$, so
$D_{i}\cap O_{p}(D)\langle a_{i}\rangle=1$ since $D_{i}\leq D$ and
$r\not\in\pi(D)$, so $O_{p}(D)\langle a_{i}\rangle\simeq D_{i}O_{p}(D)\langle
a_{i}\rangle/D_{i}$ is $\sigma$-nilpotent. Hence $[O_{p}(D),a_{i}]=1$, so
$a_{i}\in N_{G}(H)$.

Let $V=\langle a_{1},a_{2},\ldots,a_{i}\rangle$. Then $V\leq N_{G}(H)$.
Moreover, if $V<R$, then for some ${i}$ we have $V\leq V_{i}$. But then
$a_{i}\not\in V_{i}$ and $a_{i}\in V\leq V_{i}\leq V_{i}$, a contradiction.
Therefore $V=R\leq N_{G}(H)$. Hence $R^{G}\leq N_{G}(H)$. Therefore $E^{G}\leq
N_{G}(H)$, where $E$ is a Hall $\pi(D)^{\prime}$-subgroup of $G$. But then
$E^{G}D/E^{G}\simeq D/(D\cap E^{G})$ is nilpotent, so $D\leq E^{G}$ and hence
$G=GE=E^{G}$. Hence we have (6).

(7) If $p$ is a prime such that $(p-1,|G|)=1$, then $p$ does not divide $|D|$.
In particular, $|D|$ is odd.

Assume that this is false. Then, by Claim (4), $D$ has a maximal subgroup $E$
such that $|D:E|=p$ and $E$ is normal in $G$. It follows that $C_{G}(D/E)=G$
since $(p-1,|G|)=1$. Since $D$ is a Hall subgroup of $G$, it has a complement
$M$ in $G$. Hence $G/E=(D/E)\times(ME/E)$, where $ME/E\simeq M\simeq G/D$ is
$\sigma$-nilpotent. Therefore $G/E$ is $\sigma$-nilpotent. But then $D\leq E$,
a contradiction. Hence $p$ does not divide $|D|$. In particular, $|D|$ is odd.

(8) $D$ is abelian.

In view of Claim (5), $D$ is a Dedekind group. Hence $D$ is abelian since
$|D|$ is odd by Claim (7).

From Claims (5)–(8) we get that $G$ is $\sigma$-soluble $P\sigma T$-group,
contrary to the choice of $G$. Hence Statement (iii) holds.

(ii) Assume that this assertion is false and let $G$ be a counterexample of
minimal order. Then $G$ is a $\sigma$-soluble $P\sigma T$-group by Part (iii)
since every $\sigma$-nilpotent group is a $\sigma$-soluble $P\sigma T$-group.
Then $G^{{\mathfrak{N}}_{\sigma}}$ is a Hall subgroup of $G$ of odd order and
every subgroup of $G^{{\mathfrak{N}}_{\sigma}}$ is normal in $G$ by lemma 2.5.
Moreover, the hypothesis holds on $G/R$ for every minimal normal subgroup $R$
of $G$ and hence $G/R$ is $\sigma$-nilpotent by the choice of $G$, so
$R=G^{{\mathfrak{N}}_{\sigma}}$ is a group of prime order $p$ for some prime
$p$ and $R$ is a Sylow $p$-subgroup of $G$. But then the maximal $V$ subgroup
of $R$ is identity and so $G$ is the unique supplement to $V$ in $G$, so $H$
is $\sigma$-nilpotent, a contradiction. Therefore Statement (ii) holds.

The theorem is proved.

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