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\begin{document}
\setcounter{section}{0}
\title{Semiclassical resolvent estimates for Schr\"odinger matrix
operators with eigenvalues crossing.}
\author{
Thierry Jecko \\
IRMAR, Universit de Rennes I, \\
Campus Beaulieu, \\
F-35042 Rennes Cdex, France\\
jecko@maths.univ-rennes1.fr}
\date{\DATUM}
\maketitle
\begin{abstract}
For semiclassical Schr\"odinger $2\times 2$-matrix operators,
the symbol of which has crossing eigenvalues, we investigate the
semiclassical Mourre theory to derive bounds $O(h^{-1})$
($h$ being the semiclassical parameter) for the boundary values
of the resolvent, viewed as bounded operator on weighted spaces.
Under the non-trapping condition on the eigenvalues of the symbol
and under a condition on its matricial structure,
we obtain the desired bounds for codimension one crossings.
For codimension two crossings, we show that a geometrical condition
at the crossing must hold to get the existence of a global escape
function, required by the usual semiclassical Mourre theory.
\vspace{2mm}
\noindent
{\bf Keywords:} Schr\"odinger matrix operators, eigenvalues crossing,
semiclassical resolvent estimates, semiclassical Mourre method,
global escape function.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction.}
\label{intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper, we consider semiclassical Schr\"odinger operators with
$2\times 2$-matrix potential. Under general assumptions, they are
self-adjoint, have continuous spectrum on the positive real axis, and,
away from the pure point spectrum, their resolvents admit boundary values
on this half-axis, as bounded operators on suitable weighted spaces.
Our purpose is to find sufficient conditions to show that these resolvents
are $O(h^{-1})$, where $h$ is the semiclassical parameter. \\
Our main motivation for this kind of estimates is the semiclassical
scattering theory for molecules and, in particular, the question
of the accuracy of the Born-Oppenheimer approximation in this context.
Matrix potentials are a convenient simplification of Born-Oppenheimer
effective potentials, which are operator-valued (see \cite{jec1}).
Semiclassical estimates of relevant objects in scattering theory may
be deduced from these resolvent estimates (see \cite{w1,rt}). \\
Using the semiclassical Mourre commutator method (see \cite{m}), these
resolvent estimates follow from non-trapping conditions on associated
classical dynamics in the scalar case (see \cite{gm,kl}), the bound
$O(h^{-1})$ being in this case a signature for non-trapping dynamics
(see \cite{w}), and in the present case when the modes, i.e. the
eigenvalues of the symbol of the operators, cross nowhere
(see \cite{jec1}). Therefore we want to study crossing modes and it
is natural to try to adapt Mourre's method. \\
As far as the modes' crossing is concerned, we do not want to cover
all cases, but our choices, inspired by \cite{ha}, are not too
restrictive.
The first one, that we call Codimension $1$ crossings, is of particular
interest for the Born-Oppenheimer approximation for diatomic molecules.
In this case, we manage to derive the expected bounds from non-trapping
assumptions on the modes (see Theorem~\ref{th-codim1}), provided some
condition on the spectral subspaces of the symbol holds true. In contrast
to \cite{jec1}, the proof is more complicated here, because,
roughly speaking, the two modes do not decouple and we had to solve
a nonlinear problem related to this fact. The condition on the spectral
subspaces, which is independent with the non-trapping condition
on the modes, enables us to construct global solutions of the nonlinear
problem. For our second type of crossings,
we have a negative result: the usual Mourre method (see
Section~\ref{mourre})
cannot apply (even under the previous non-trapping conditions), if some
geometrical condition at the crossing does not hold. Although it is only
an obstruction to Mourre method, we have some reasons to believe, the
latter being also a trapping phenomenon at the crossing (that cannot
be expressed in terms of the dynamics of the modes since they may break
down there), which excludes the desired resolvent estimates
(see Remark~\ref{resonances}). If a strengthed version of the geometrical
condition hold, we show in some weak sense that this trapping phenomenon
at the crossing does not occur (see Proposition~\ref{codim2-croisement}),
but we were not able to derive the resolvent estimates under the previous
non-trapping assumptions. \\
It is interesting to compare our present work with a part of \cite{ha}
(see Section~\ref{result}) and with \cite{fg}, although the full evolution
is not considered there. In a different way, the present paper is
complementary to \cite{ne1,ne2}. \\
Before ending this introduction, we want to mention an interesting
comment by C. Grard on the subject. He regrets that we did not use
a flow directly constructed from the symbol of the operator (via its
Liouvillean). We did try but could not overcome the problem of
non-commutation of matrix symbols. \\
In Section~\ref{result}, we precisely present the frame in which we
shall work, the assumptions we need, and the announced
results, followed by some comments. In Section~\ref{mourre},
we review the general strategy known as semiclassical Mourre method,
that we follow here. Then we focus on the problem of constructing
a global escape function, a key point in our strategy. Codimension $1$
crossings are treated in Section~\ref{codim1} while Codimension $2$ ones
are considered in Section~\ref{codim2}. Finally, some useful facts are
collected in the appendix.
\underline{Acknowledgment}: the author thanks, for fruitful
discussions and advices, V. Bach, Y. Colin de Verdire, C. Grard,
G. Hagedorn, A. Joye, N. Lerner, A. Martinez, G. Mtivier, and F. Nier.
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notation and results.}
\label{result}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First of all, let us introduce the Schr\"odinger $2\times 2$ matrix
operator we want to study. For some integer $n\geq 2$, we consider
the semiclassical operator
%
\begin{equation}\label{operateur}
\hat{P}\ := \ -h^2\Delta_x \rI _2+M(x)
\end{equation}
%
acting in $\rL ^2(\R^n;\C^2)$, where $h$ is the semiclassical parameter
($h\in ]0;h_0]$ for some $h_0>0$), $\Delta_x$ denotes the Laplacian in
$\R^n$, ${\rm I}_2$ is the $2\times 2$ identity matrix, and where $M(x)$
is the multiplication operator by a real symetric $2\times 2$ matrix
$M(x)$.
We require that $M$ is $C^\infty$ on $\R^n$ and that there
exist some $\delta>0$ and some real symetric matrix $M_\infty$ such that
%
\begin{equation}\label{decroit}
\forall \alpha \in \N^n\comma \, \forall x\in \R^n\comma \hspace{.4cm}
\bigl\|\partial _x^\alpha \bigl(M(x)-M_\infty\bigr)\bigr\|\ = \ O_\alpha
\bigl(\langle x\rangle ^{-\delta -|\alpha|}\bigr)
\end{equation}
%
where $\|\cdot \|$ denotes the operator norm on the $2\times 2$ matrices
and
$\langle x\rangle =(1+|x|^2)^{1/2}$. It is well known that, under this
assumption on $M$, the operator $\hat{P}$ is self-adjoint on the domain of
the Laplacian (see \cite{rs2} for instance). Its resolvent will be denoted
by
$R(z):=(\hat{P}-z)^{-1}$, for $z$ in the resolvent set of $\hat{P}$ (we
omit
the $h$-dependence). If, for $s\in \R$, we denote by $\rL _s^2(\R^n;\C^2)$
the weighted $\rL^2$ space of mesurable, $\C^2$-valued functions $f$ on
$\R^n$ such that $x\donne \langle x\rangle^sf(x)$ belongs to
$\rL ^2(\R^n;\C^2)$, then it follows from Mourre theory that this
resolvent
has boundary values $R(\lambda \pm i0)$ as bounded operators from
$\rL _s^2(\R^n;\C^2)$ to $\rL _{-s}^2(\R^n;\C^2)$, for any $s>1/2$ and
$\lambda$ outside the pure point spectrum of $\hat{P}$ (in
fact, we partially prove here this property again).
The operator $\hat{P}$ is a $h$-pseudodifferential operateur obtained by
Weyl quantization of the following symbol, defined on $T^\ast \R^n$ with
values in the real symetric $2\times 2$ matrices,
%
\begin{equation}\label{symbole}
P(x,\xi)\ := \ |\xi|^2\rI_2+M(x)\period
\end{equation}
%
Notice that $M(x)\ = \ u(x)\rI _2+V(x)$
where $u(x)$ is $1/2$ times the trace of $M(x)$ and
%
\[ V(x)\ := \ \left(
\begin{array}{rl}
v_1(x)&v_2(x)\\
v_2(x)&-v_1(x)
\end{array}
\right) \]
%
for smooth real functions $v_1$ and $v_2$.
The eigenvalues of $V(x)$ are $\pm \rho (x)$ with
$\rho (x)=(v_1(x)^2+v_2(x)^2)^{1/2}=(-\Det V)^{1/2}$ ($\Det V$
being the determinant of $V$) and we denote by $\Pi_\pm(x)$ the
associated eigenprojectors. While $\Pi_\pm=\rI _2$ if $\rho=0$, we have,
for $\rho\neq0$, $\Pi_\pm=(\rI _2\pm V/\rho)/2$. Similarly we introduce
the corresponding notation for $M_\infty$, namely $u_\infty$,
$V_\infty$, $v_{1,\infty}$, $v_{2,\infty}$, and $\rho_\infty$.
We also define the scalar function on the phase space
$p(x,\xi):=|\xi|^2+u(x)$, which is $1/2$ times the trace of the
symbol $P$. Then the eigenvalues of $P$ are
$p_\pm (x,\xi):=p(x,\xi)\pm \rho (x)$. Notice that
$p_+(x,\xi)\, =\, p_-(x,\xi)\ \ssi \ \rho (x)\, =\, 0$.
We denote by $\cC$ (resp. $\cC^\ast$) the zero set of $\rho$ (or $V$),
viewed in $\R^n$ (resp. $T^\ast\R^n$), that is the crossing set of the
eigenvalues of $P$. The functions $p_\pm$ and $p$ are smooth functions
at least on $T^\ast\R^n \setminus \cC^\ast$ thus generate Hamilton
flows on this set. For any Hamilton function $q$, we shall denote by
$H_q$ its Hamilton field and by $\phi^t_q$ its Hamilton flow at time $t$.
The following non-trapping condition on Hamilton flows and
connected notion of global escape function (see \cite{dg})
will play an important r\^ole in this paper.
%
\begin{definition}\label{non-captif}
A smooth, real function $q$ defined on an open subset $U^\ast$ of some
cotangent bundle ($T^\ast\R^n$ or $T^\ast \cC$ ) is said to be
non-trapping on some set $U_1^\ast\subset U^\ast$ at energy
$\lambda$ if, for any point $\alpha \in U_1^\ast\cap q^{-1}(\lambda)$, the
evolution of $\alpha$, according to the Hamilton flow $\phi^t_q$ of $q$,
in
both time directions, leaves any compact subset of
$U^\ast\cap q^{-1}(\lambda)$, that is
%
\[\forall \alpha \in U_1^\ast\cap q^{-1}(\lambda),\, \forall
K\subset\subset
U^\ast\cap q^{-1}(\lambda),\, \exists T>0; \
|t|\geq T \, \impl \,
\phi^t_q(\alpha )\not\in K\period\]
%
The largest open set $U_1^\ast$ satisfying the previous condition and
$U^\ast\setminus U_1^\ast$ are respectively the non-trapping and trapping
region of $q$ at energy $\lambda$. A trajectory $\{\phi^t_q(\alpha),
t\in \R\}$ of $q$ is trapped if one of the sets $\{\phi^t_q(\alpha),
t\in \R^+\}$ and $\{\phi^t_q(\alpha),t\in \R^-\}$ is bounded. \\
A smooth, real function $a$ defined on some cotangent bundle
($T^\ast\R^n$ or $T^\ast \cC$) is an escape function on $U_1^\ast$ for $q$
at energy $\lambda$ if there exists some $c>0$ such that
%
\[\forall \alpha \in U_1^\ast\cap q^{-1}(\lambda),\, \{q,a\}(\alpha)\ \geq
c
\comma \]
%
where $\{\cdot ,\cdot\}$ denotes the usual Poisson bracket. \\
Notice that we can replace $q^{-1}(\lambda)$ by
$q^{-1}(]\lambda -\epsilon_0;\lambda +\epsilon_0[)$ for
$\epsilon_0>0$ small enough without changing the above definitions.\\
If $U_1^\ast=U_1\times \R^n$ for some subset $U_1$ of $\R^n$, we also
say that $q$ is non-trapping on $U_1$ and that
$a$ is an escape function on $U_1$. If $U_1^\ast=U^\ast$ or $U_1=U$, we
simply say that $q$ is non-trapping at energy $\lambda$ and that $a$ is a
global escape function for $q$ at energy $\lambda$.
\end{definition}
%
We shall indeed use such global escape function for scalar Hamilton
functions like $p$ but, to avoid difficulties at $\cC^\ast$, we
need a generalized version for the matricial symbol $P$,
given in the following definition and suggested by A. Martinez.
%
\begin{definition}\label{matrice-fuite}
A smooth function $A$ on $T^\ast\R^n$, valued in the $2\times 2$
real symetric matrices, is an escape function for the
\underline{matricial} symbol $P$ at energy $\lambda$ on a subset
$U^\ast$ of $T^\ast\R^n$ (resp. on a subset $U$ of $\R^n$) if there
exist a function $\theta \in C^\infty_0(\R;\R)$, with $\theta =1$ near
$\lambda$, and some $c>0$ such that, in the matrix sense of the order
$\geq$,
the ``classical'' Mourre estimate
%
\begin{equation}\label{mourre-classique}
\theta (P)\, \{P,A\}\, \theta (P)\ \geq \ c\, \theta (P)^2
\end{equation}
%
holds true on $U^\ast$ (resp. $U\times \R^n$), where $\{\cdot ,\cdot\}$
denotes the Poisson bracket for
matrix symbols (see (\ref{poisson-matriciel})), and if the matricial
commutator $[P,A]$ vanishes on $U^\ast\cap \support\theta (P)$
(resp. $(U\times \R^n)\cap \support\theta (P)$). \\
If $U^\ast=T^\ast\R^n$ or $U=\R^n$, we say that $A$ is a global escape
function for $P$ at energy $\lambda$.
\end{definition}
%
Let us precise the energy localization in
Definition~\ref{matrice-fuite}. It corresponds to the support
of a matrix-valued function $\theta (P)$ ($\theta$ being as in
Definition~\ref{matrice-fuite}). By the functional calculus for
real symetric matrices, this support is given by
%
\[\support \, \theta (P)\ = \ \bigl\{\alpha \in T^\ast\R^n;\, \exists
\mu \in \support \theta ;\, \Det \bigl(P(\alpha)-\mu \rI _2\bigr)=0
\bigr\}\period \]
%
It is thus natural to consider the open set
%
\begin{equation}\label{loca-energie}
E(\lambda ,\epsilon_0)\ := \ \bigcup _{\atop \mu \in ]\lambda -\epsilon_0;
\lambda +\epsilon_0[}\bigl\{\alpha \in T^\ast\R^n;\, \Det
\bigl(P(\alpha)-\mu \rI _2\bigr)=0\bigr\} \comma
\end{equation}
%
for some $\epsilon_0>0$, as an energy localization near $\lambda$.
The energy shell of $P$ of energy $\lambda$ is defined as $E(\lambda)
:=\{\alpha \in T^\ast\R^n;\, \Det \bigl(P(\alpha)-\lambda \rI
_2\bigr)=0\}$.\\
Notice that $[P,A](\alpha)\neq 0$ implies $[P,A](\alpha)$ has a
negative eigenvalue, since its trace is $0$. Since we want to
derive some positivity of $[\hat{P},\hat{A}]$ by the sharp G\aa rding
inequality, we require $[P,A]=0$ on $\support \theta(P)$. \\
It is straighforward to verify that, under (\ref{decroit})
and for $\lambda>\|M_\infty\|$, the ``classical'' Mourre estimate
(\ref{mourre-classique}) holds true for the scalar function $A_\infty$
defined by
%
\begin{equation}\label{fuite-infini}
\forall (x,\xi)\in T^\ast\R^n,\, A_\infty (x,\xi) \ := \
a_\infty (x,\xi)\, \rI _2\ := \ x\cdot \xi \, \rI _2\ \comma
\end{equation}
%
provided $|x|$ is large enough. If $\lambda>\|M_\infty\|$ is
large enough, this function is even a global escape function
for $P$ at energy $\lambda$. \\
To use the semiclassical Mourre method, we demand that the global
escape function belongs to some class of semiclassical symbols. In
fact, it suffices to require (see \cite{gm}, \cite{jec1}) that, for
$|x|$ large enough, the global escape function coincides with $A_\infty$.
In Section~\ref{mourre}, we shall derive (almost as in the scalar case)
the following
%
\begin{theorem}\label{th-mourre}
Assume that the symbol $P$ satisfies (\ref{decroit}) and let
$\lambda$ be some real number such that there exists some
global escape function for $P$ at energy $\lambda$, which is equal
to $A_\infty$ (cf. (\ref{fuite-infini})) for $|x|$ large enough,
then, for any $s>1/2$, the boundary values of the resolvent satisfy
the estimates $R(\lambda \pm i0)=O_s(h^{-1})$ as bounded operators from
$\rL _s^2(\R^n;\C^2)$ to $\rL _{-s}^2(\R^n;\C^2)$.
\end{theorem}
%
In the scalar case (cf. \cite{gm}) (resp. the matricial case without
crossing (cf. \cite{jec1})), these resolvent estimates hold true
under a non-trapping condition at energy $\lambda$ on the symbol $P$
of the operator (resp. the eigenvalues of $P$).
Furthermore, it is also known, in the scalar case, that this
non-trapping condition is necessary (cf. \cite{w}).
What we are trying to understand in this paper is: under which
condition can we apply Theorem~\ref{th-mourre} to $P$, if its
eigenvalues cross somewhere?
We refer to \cite{ha} for the description of the different types
of crossing that may appear and focus here on two important ones.
In each case, we demand that $\cC$ is a smooth submanifold of $\R^n$,
while its codimension in $\R^n$ depends on the type. Since the
results will be different for these two types, we shall present
them together with the corresponding result concerning the
existence of a global escape function.
\underline{Codimension $1$ crossing}: We assume that $\cC$ is a
submanifold of $\R^n$ of codimension one. More precisely, we
demand that there exists some scalar $C^\infty$ function $\tau$ and
some $C^\infty$ function $\tV$, valued in the traceless, real,
symetric matrices (i.e. like $V$), such that $V=\tau \tV$ in some
vicinity of $\cC$, that $\trho :=(-\Det\tV)^{1/2}$ and the gradient
of $\tau$ does not vanish on the zero set of $\tau$, which is $\cC$.
Finally, in view of (\ref{decroit}), we require that there exist
$\epsilon,C_0>0$ and some real symetric matrix $\tV_\infty$,
with $\trho _\infty:=(-\Det\tV_\infty)^{1/2}>0$, such that
%
\begin{equation}\label{tau-decroit}
\forall \alpha \in \N^n, \forall x\in \R^n, \,
\rho(x)<\epsilon \ \impl \ |\partial _x^\alpha \tau (x)|
+ \bigl\|\partial _x^\alpha (\tV (x)-\tV_\infty )\bigr\|\ = \
O_\alpha \bigl(\langle x\rangle ^{-\delta -|\alpha|}\bigr) \period
\end{equation}
%
We point out that (\ref{tau-decroit}) is in fact an assumption at infinity
near $\cC$, since it holds automatically true, if $\cC$ is compact, under
the previous assumptions. If $\cC$ is not compact, then $V$ tend to $0$
at infinity by (\ref{decroit}), so this assumption says that, near the
crossing, the convergence of $V$ to $0$ is due to the convergence of
$\tau$ to zero, while the matrix structure of $V$ tends to some
invertible matrix $\tV_\infty$. Notice further that the difference
$\trho -\trho _\infty$ also satisfies the estimates (\ref{tau-decroit}).
Among the Codimension $1$ crossings, there is the following radial
situation,
which contains the case of Born-Oppenheimer diatomic molecules with
crossing.
\underline{Radial potential with crossing}: We assume that $M$ is
a radial function (depending only on $|x|$) and that $\cC$ is the
sphere of $\R^n$ centered at $0$ with radius $r_0>0$. We demand
further that the gradients of $v_1$ and $v_2$ do not vanish on $\cC$.
Using a Taylor expansion with rest integral near $r_0$, we see
that this is a Codimension $1$ crossing, if we choose
$\tau (x)=|x|-r_0$ near $\cC$. Notice that $\cC$ could be a finite
union of spheres centered at $0$.
Let us return to general Codimension $1$ crossings. Notice that, at
$\cC^\ast$, the eigenvalues $p_\pm=p\pm|\tau|\trho$ and their associated
eigenprojectors are not smooth. But one can easily regularize the
situation (cf. \cite{k}). Denoting by $\cC^\ast_\pm$ the regions of
$T^\ast\R^n$ where $\pm \tau>0$, we define two new functions
$\tp _\pm$ on $T^\ast\R^n$ by $\tp _\pm =p _\pm$ on
$\cC^\ast_+$, $\tp _\pm =p _\mp$ on $\cC^\ast_-$, and $\tp _\pm =p$ on
$\cC^\ast$. Similarly, we set $\tPi_\pm(x)=\Pi_\pm(x)$ on $\cC^\ast_+$ and
$\tPi_\pm(x)=\Pi_\mp(x)$ on $\cC^\ast_-$. Since $\tp _\pm =p \pm \tau
\trho$
and $\tPi_\pm=(\rI _2\pm \tV /\trho )/2$ near $\cC^\ast$, these functions
are smooth everywhere and, by (\ref{tau-decroit}),
%
\begin{equation}\label{limit-pi+-}
\forall \alpha \in \N^n, \forall x\in \R^n, \,
\rho(x)<\epsilon \ \impl \bigl\|\partial _x^\alpha (\tPi_\pm (x)-
\tPi_{\pm, \infty} )\bigr\|\ = \
O_\alpha \bigl(\langle x\rangle ^{-\delta -|\alpha|}\bigr) \comma
\end{equation}
%
where $\tPi_{\pm, \infty}=(\rI _2\pm \tV_\infty /\trho_\infty )/2$.
We just have changed the numbering of the eigenvalues
in order to get smooth ones and smooth eigenprojectors. For convenience,
we may assume that, for $\rho(x)\geq \epsilon$ and $\pm \tau (x)>0$,
$\tau(x)=\pm \rho (x)$ and $\tau (x)\tV (x)=V(x)$.
Our first main result, proved in Section~\ref{codim1}, is the following
%
\begin{theorem}\label{th-codim1}
Assume that the symbol $P$ satisfies (\ref{decroit}) and let
$\lambda >\|M_\infty \|$, the operator norm of $M_\infty$. Assume
that the Hamilton functions $\tp_\pm$ are non-trapping at energy
$\lambda$. Then there exists $\kappa >0$ such that, if
$\|\langle x\rangle^{1+\delta}(\nabla_x\tPi_+)(x)\|\leq \kappa$ for
all $x\in \R^n$, then there exists a global escape function for
$P$ at energy $\lambda$ (cf. Definition~\ref{matrice-fuite})
which equals $A_\infty$, defined in (\ref{fuite-infini}), for $|x|$ large.
In particular, Theorem~\ref{th-mourre} applies.
\end{theorem}
%
At first sight, one could argue that this result should be clear,
even without any condition on $\nabla_x\tPi_+$,
since one can smoothly diagonalize $P$ and probably decouple the
two levels. This is not true and, already in \cite{ha} where
the propagation of coherent states is studied, one can see that
the two levels do interact, however not at the leading order.
In Section~\ref{codim1}, we shall explain why we cannot simply
adapt the proof of \cite{jec1} to the present case. \\
In contrast to Codimension 2 crossings (see below), the geometrical
formulation of the problem at the crossing does not predict any local
obstruction to the existence of escape function there. We really have
the two degrees of freedom allowed by the commutation condition in
Definition~\ref{matrice-fuite}. Fixing appropriatly one of them,
attached to $\tp_+$ for instance, we demand that the other satisfies
a nonlinear, scalar p.d.e, which reduces to an ordinary differential
equation along the flow of $\tp_-$. Thanks to the condition on
$\nabla_x\tPi_+$ (see Remark~\ref{petitesse}), which does not depend on
$\tp_\pm$, the resolution of this p.d.e furnishes the second part of the
desired global escape function. The size of $\kappa$ may be estimated
in terms of the time needed by the flow of $\tp_-$ to leave some compact
region (see the proof of Theorem~\ref{th-codim1} in
Section~\ref{codim1}).\\
Notice that, if $V=\tau \tV$ everywhere with constant $\tV$, we can
simplify the proof considerably (see Remark~\ref{cas-simple-1}) since the
two
levels decouple in this case. \\
Are the non-trapping conditions necessary to get the resolvent estimates ?
If the crossing is empty, we believe that the arguments by \cite{w} may be
adapted successfully. This could be probably extended to the present case
for constant $\tV$ near $\cC$, that is for a fixed matricial
structure near the crossing. It seems impossible to remove the condition
on $\nabla_x\tPi_+$ in Theorem~\ref{th-codim1} if we follow the present
Mourre method (see Section~\ref{codim1}). Another method could perhaps
do it, but a relevant non-trapping condition on the symbol might also
be more complicated than ours.
We come now to our second type of crossing, namely
\underline{Codimension $2$ crossing}: We assume that $\cC$ is a
submanifold of $\R^n$ of codimension two. More precisely, we
demand that the gradients of $v_1$ and $v_2$ are linearly independent
on $\cC$, the intersection of their zero set.
The eigenvalues $p_\pm$ and their associated eigenprojectors are not
smooth at $\cC^\ast$, in general, and the previous regularization does
not work. It turns out that the cotangent space $T^\ast\cC$ of
$\cC$ will be important. Its fiber over some $x\in \cC$ is defined by
%
\begin{equation}\label{def-cotangent2}
T_x^\ast\cC \ := \ \bigl\{ \xi \in T_x^\ast\R^n;\, \xi \in \bigl(
\rvect (\nabla v_1(x), \nabla v_2(x))\bigr)^\perp \bigr\}\comma
\end{equation}
%
where $\cV(x):=\rvect (\nabla v_1(x), \nabla v_2(x))$ is the vector space
spanned by the vectors $\nabla v_1(x)$ and $\nabla v_2(x)$, and
where $\cV(x)^\perp $ is the space of linear forms vanishing on $\cV(x)$.
Here, we do not view it in an intrinsic way but as a submanifold of
$T^\ast \R^n$. Its importance comes from the identity (see
Appendix~\ref{geom})
%
\begin{equation}\label{champ-tangent}
T^\ast\cC \ = \ \{\alpha \in \cC^\ast,\, H_p(\alpha)\in T_\alpha
\cC^\ast\}\period
\end{equation}
%
Furthermore, we have a special,
geometrical configuration (see Appendix~\ref{geom}). For $\alpha \in
T^\ast\cC$, $T_\alpha \cC^\ast=\rvect (H_{v_1}, H_{v_2})\oplus T_\alpha
T^\ast\cC$. According to this decomposition,
%
\begin{equation}\label{decomp-champ2}
\forall \alpha \in T^\ast\cC,\, H_p(\alpha)\ = \ \bigl(\mu_1(\alpha)
H_{v_1}(\alpha)+\mu_2(\alpha)H_{v_2}(\alpha)\bigr)\, +\, H_{p'}(\alpha)
\comma
\end{equation}
%
for some real coefficients $\mu_1(\alpha), \mu_2(\alpha)$, where
$p'$ denotes the restriction of $p$ to $T^\ast\cC$.
The experience of the scalar case, the matrix case without crossing, and
the matrix case with codimension 1 crossing, says us that the Hamilton
functions $p_\pm$ should be non-trapping at the considered energy. But,
assuming this, in the sense given in Definition~\ref{non-captif},
is it sufficient? As discussed in Section~\ref{mourre}, where we compare
the present situation with the one without crossing, we have to understand
what happens at the crossing. To this end, we express the problem of
the existence of an escape function near the crossing in geometrical
terms. The previous geometrical situation reveals a local obstruction
for Codimension 2 crossings. To describe this obstruction, we need the
following
%
\begin{definition}\label{confinement}
Let $\lambda\in \R$ and $\alpha\in T^\ast\cC$. We say that the crossing
is confining at $\alpha$ for $P$ at energy $\lambda$ if
$\alpha$ belongs to the energy shell $E(\lambda)$ of $P$ and if
$\mu_1(\alpha)^2+\mu_2(\alpha)^2\leq 1$, these coefficients being
defined in (\ref{decomp-champ2}).
\end{definition}
%
This leads to the following, negative result, proved in
Section~\ref{codim2}.
%
\begin{theorem}\label{th-neg-codim2}
If the crossing is confining for $P$ at energy $\lambda$ on
some region in $T^\ast\cC$, which contains a trapped trajectory
for $p'$ (at energy $\lambda$), then there exists no global escape
function for $P$ at energy $\lambda$.
\end{theorem}
%
The assumptions of Theorem~\ref{th-neg-codim2} imply that $P$ cannot
have a scalar escape function near $T^\ast\cC$ and this also works for
Codimension $1$ crossings (see Appendix~\ref{geom}).
But, for Codimension $2$ crossings, the existence of a global
escape function for $P$ implies the existence of a scalar one near
$T^\ast\cC$ (see Appendix~\ref{app-codim2}), yielding
Theorem~\ref{th-neg-codim2}. \\
The condition of Definition~\ref{confinement} roughly says that
the component of $H_p$ in $\rvect (H_{v_1}, H_{v_2})$
has a small size. Since it forbids to ``escape'' in the conormal
direction to the crossing near $T^\ast\cC$ (see the proof), it may be
seen as a kind of confining condition in that direction on the
crossing w.r.t. $p$ at the considered energy. Here we are thinking of
quantum evolution constrained to (a neighborhood of) a submanifold as
in \cite{fh}, for instance. The assumptions in Theorem~\ref{th-neg-codim2}
seem to describe a (quantum) capture phenomenon, that cannot be expressed
in terms of the classical Hamilton functions $p_+$ and $p_-$, thus to be
independent with non-trapping properties of $p_+$ and $p_-$. \\
It is not surprising that $T^\ast\cC$ plays a central r\^ole for
the existence of a global escape function if we consider the work
\cite{ha} by Hagedorn, where the evolution of coherent states
through eigenvalues crossing, with \underline{transversal} impuls at the
crossing, does not reveal any capture phenomenon. He had to avoid the
non-generic situation of a tangent impuls that we have to take into
account here, since we work on the full resolvent, and that precisely
corresponds to considering a point in $T^\ast\cC$.
%
\begin{remark}\label{resonances}
In the situation of Theorem~\ref{th-neg-codim2}, we do not know if the
resolvent estimates of Theorem~\ref{th-mourre} hold.
However we would not be surprised if they would not hold and that
resonances would run rapidly (faster than $h$) to the real axis as $h$
goes to $0$. A corresponding resonant state could be essentially a tensor
product of a microlocally confined state in the conormal direction to the
crossing and of an eigenstate on $T^\ast \cC$ (under some kind of
Bohr-Sommerfeld quantization condition). Technics from \cite{gs} and
\cite{fh} may be useful to deal with this question. At least, there is a
positive answer by \cite{ne1}. Notice that \cite{ne2} predicts
the presence of resonances near $\lambda$ in the present situation
but does not describe semiclassically their width.
\end{remark}
%
Now it is natural to ask what happens when the obstruction does not
occur. Is there a global escape function, if we assume further that
$p_+$ and $p_-$ are non-trapping at the considered energy? Unfortunately,
we did not succeed in finding a complete answer to this question.
However, our previous geometrical analysis allows us to exhibit sufficient
conditions (almost converse to the assumptions of
Theorem~\ref{th-neg-codim2}), under which there are (scalar) escape
functions near the crossing (see Proposition~\ref{codim2-croisement}
below).
Maybe these conditions, together with the non-trapping condition on $p_+$
and $p_-$, are sufficient to construct a global escape function. We could
not prove this using Proposition~\ref{codim2-croisement}, since it is
rather difficult to transform an escape function on some quite arbritary
region into a global one.
%
\begin{proposition}\label{codim2-croisement}
Assume that the symbol $P$ satisfies (\ref{decroit}) and let
$\lambda >\|M_\infty \|$, the operator norm of $M_\infty$.
If one of the following two conditions
%
\begin{enumerate}
\item the restriction $p'$ of the half-trace $p$ of $P$ to $T^\ast\cC$ is
non-trapping at energy $\lambda$,
\item there is no $\alpha\in T^\ast\cC$ at which the crossing is
confining for $P$ at energy $\lambda$,
\end{enumerate}
%
holds true then there exists a smooth, scalar function $A$, which is an
escape function for $P$ at energy $\lambda$ on $\cC^\ast$
and which equals $A_\infty$, for $|x|$ large.
\end{proposition}
%
In fact, we first show that there is an escape function near $T^\ast\cC$
and
then add appropriately some function to get an escape function near
$\cC^\ast$. \\
Assumption 1 in Proposition~\ref{codim2-croisement} allows us to construct
a global escape function for $p'$ on $T^\ast\cC$ (as in \cite{gm}),
that we extend to an escape function for $P$ near $T^\ast\cC$.
Notice that this assumption implies that $\cC$ is not
compact. \\
Under Assumption 2 in Proposition~\ref{codim2-croisement},
we can construct explicitly an escape function and really use
conormal directions to $\cC^\ast$ to ``escape''. \\
It is interesting to compare this result with the case where
$\lambda$ is very large, for which we have a global escape
function, namely $A_\infty$. If $\cC$ is given by $\{x_1=x_2=0\}$ in
$\R^3$, the kinetic energy is concentrated ``along the crossing''
so that 1 is true and 2 is false if $u$ is constant on $\cC$.
Now, if $\cC$ is a circle in $\R^3$, then 1 cannot be true but one
can verify that 2 holds true. \\
In order to exhibit clearly the relevant properties on which this result
is based, we did not really optimize the assumptions. In fact, if $\cC$
is \underline{not compact}, we can relax them as shown in
Remark~\ref{codim2-croisement-raffine} below. This comes from
the fact that we have roughly two independent escape directions, one
cotangent the crossing and one conormal to the crossing. If one is
not practicable somewhere, we can follow the other. This leads to the
following refinement of Proposition~\ref{codim2-croisement}, proved
in Section~\ref{codim2}.
%
\begin{remark}\label{codim2-croisement-raffine}
Assume that the symbol $P$ satisfies (\ref{decroit}) and let
$\lambda >\|M_\infty \|$. If the crossing is not confining for $P$ at
energy $\lambda$ on some vicinity of the (closed) trapping region of
$p'$ at energy $\lambda$ then there exists some scalar escape function
for $P$ at energy $\lambda$ on $\cC^\ast$, which equals $A_\infty$ for
$|x|$ large.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Semiclassical Mourre method.}
\label{mourre}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we describe the semiclassical Mourre method
which leads to a proof of Theorem~\ref{th-mourre}. We also
review semiclassical resolvent estimates in the scalar case
(cf. \cite{gm,w}) and in the matrix case without crossing (cf.
\cite{jec1}). A new proof of the latter case will be sketched
to enlighten the present situation.
The semiclassical Mourre method for the operator $\hat{P}$ (defined
in (\ref{operateur})) consists in seeking a so called conjugate operator
which will be a $h$-pseudodifferential operator $\hat{A}$ (the Weyl
$h$-quantization of some symbol $A$) that ``coincide'' with the generator
of dilations (i.e. $A=A_\infty$), for $|x|$ large, and satisfies
two conditions. Firstly, we should have, for the energy $\lambda$ that
we consider, the existence of some function
$\theta \in C^\infty_0(\R;\R)$, with $\theta =1$ near $\lambda$, and
some $c>0$ such that,
%
\begin{equation}\label{mourre-quantique}
\theta (\hat{P})\, i[\hat{P},\hat{A}]\, \theta (\hat{P})\ \geq \ c\cdot
h\cdot \theta (\hat{P})^2 \period
\end{equation}
%
Here $[\cdot ,\cdot ]$ denotes the commutator of (unbounded) operators.
Secondly, we need that the double commutator
$[[\hat{P},\hat{A}],\hat{A}]$ is $\hat{P}$-bounded and that
%
\begin{equation}\label{double-commutateur}
\bigl[ \theta (\hat{P})[\hat{P},\hat{A}]\theta (\hat{P}),\hat{A} \bigr]\,
(\hat{P}+i)^{-1} \ = \ O(h)\period
\end{equation}
%
A bound $O(h^2)$ is usually required (see \cite{jec1}) since it directly
holds in the scalar case. But in the matrix case, we do not expect in
general such a bound because of commutation problems. Fortunately, if
we carefully follow Mourre's arguments (cf. \cite{m}), the bound in
(\ref{double-commutateur}) suffices (see \cite{w}) to get the desired
semiclassical resolvent estimates. \\
To get (\ref{mourre-quantique}), we can use the sharp G\aa rding
inequality,
as usual, since it works for matricial symbols, as pointed out in
\cite{h}.
For convenience, we sketch a proof in Appendix~\ref{garding}, which is
the matricial version of an elegant proof indicated to us by
A. Martinez. \\
By the functional calculus of B.Helffer and J.Sj\"ostrand (see
\cite{hs,dg}),
the energy localization operator $\theta (\hat{P})$ is a
$h$-pseudodifferential operator with principal
symbol $\theta (P)$. Therefore, the principal symbol of
the r.h.s. of (\ref{mourre-quantique}) is $\theta (P)[P,A]\theta (P)$
($[\cdot ,\cdot]$ denoting here the matricial commutator), according to
the composition formula for $h$-pseudodifferential operators (see
\cite{r}).
Since $[P,A]$ is traceless, it has opposite eigenvalues, thus
$[P,A]\geq 0$ implies $[P,A]=0$. This explains the commutation condition
required in Definition~\ref{matrice-fuite} (which is trivially realized
in the scalar case), that we also use to get (\ref{double-commutateur}).
Under this condition, the principal symbol of $ih^{-1}[\hat{P},\hat{A}]$
is, using the previous composition rule,
%
\begin{eqnarray}\label{poisson-matriciel}
\{P,A\} &:=& \frac{1}{2}\bigl((\nabla_\xi P\cdot \nabla_x A-
\nabla_\xi A\cdot \nabla_x P)-(\nabla_x P\cdot \nabla_\xi A-
\nabla_x A\cdot \nabla_\xi P)\bigr)\comma \\
&=& \frac{1}{2}\bigl((\nabla_\xi P\cdot \nabla_x A-
\nabla_x P\cdot \nabla_\xi A)+(\nabla_x A\cdot \nabla_\xi P
-\nabla_\xi A\cdot \nabla_x P)\bigr)\period \nonumber
\end{eqnarray}
%
In the scalar case, we recognize the usual Poisson bracket.
{\bf Proof of Theorem~\ref{th-mourre}:} As conjugate operator
we choose the Weyl $h$-quantization $\hat{A}$ (see Appendix~\ref{garding})
of the global escape function $A$ (here we use the fact that $A=A_\infty$
for large enough $|x|$). Let $\theta, \theta'\in C_0^\infty(\R;\R)$
with $\theta'=\theta=1$ near $\lambda$ and satisfying $\theta
\theta'=\theta$, and assume the properties of
Definition~\ref{matrice-fuite}
for $\theta'$. Thanks to $[P,A]=0$ on $\support \theta'(P)$,
the principal symbol of $\theta' (\hat{P})ih^{-1}[\hat{P},\hat{A}]\theta'
(\hat{P})$ is $\theta' (P)\{P,A\}\theta' (P)$. By
(\ref{mourre-classique}),
this matricial symbol is bounded below by $c'\theta' (P)^2$, for some
$c'>0$.
Now, by the sharp G\aa rding inequality for the bounded, non-negative
symbol
$\theta' (P)\{P,A\}\theta' (P)-c'\theta' (P)^2$,
%
\[\theta' (\hat{P})\, ih^{-1}[\hat{P},\hat{A}]\, \theta' (\hat{P})
\ \geq \ c'\cdot \theta' (\hat{P})^2 \, -\, O(h)\comma \]
%
which, after left and right multiplication by $\theta (\hat{P})$, yields
the Mourre estimate (\ref{mourre-quantique}) for
$c=c'/2>0$ and $h$ small enough. The double
commutator is seen to be $\hat{P}$-bounded thanks to assumption
(\ref{decroit}) and, since $P$ and $A$ commute on $\support \theta(P)$,
(\ref{double-commutateur}) holds true. We can then use
Mourre's arguments (cf. \cite{m}), following the $h$-dependence,
to obtain the desired result (see also \cite{w}). \cqfd
As remarked in \cite{jec1}, the main problem is then to construct
a global escape function for $P$ at energy $\lambda$.
Let us review some situations where the resolvent estimates of
Theorem~\ref{th-mourre} have already been obtained. \\ In the
scalar case, the previous Mourre method was successfully
followed by C.Grard and A.Martinez in \cite{gm}, and then
by X.P. Wang (in a more general setting in \cite{w}), under the
non-trapping condition for the symbol at the considered energy. In
this case, there is a natural way, given in \cite{gm}, to construct a
global escape function $a$: let $g\in C^\infty_0(T^\ast \R^n)$ with
$0\leq g\leq 1$ and $g=1$ on a large bounded domain $D^\ast\subset
p^{-1}(\lambda)$. To construct $a$ in $D^\ast$, we set $\{p,a\}=g$ and
compose on both sides with the flow $\phi^t_p$. This leads to
$(d/dt)(a\rond \phi^t_p)=g\rond \phi^t_p$ pointwise in $T^\ast \R^n$.
By the non-trapping condition, this can be integrated as
%
\begin{equation}\label{integration-flot}
a\rond \phi^t_p \ = \ -\int_t^\infty g\rond \phi^s_p\, ds \comma
\end{equation}
%
which gives $a$ for $t=0$. To exhibit a global escape function, which
agrees with $a_\infty$ for $|x|$ large, we combine
appropriatly $a$ and $a_\infty$ (see Proposition~\ref{prol-infini} below).
\\
On a formal level, we can reproduce this in the matricial case, if we
replace the classical flow by the propagator of the Liouvillean
$\{P,\cdot\}$ of $P$, but we do not see how to ensure the commutation
condition required in Definition~\ref{matrice-fuite}. \\
In \cite{jec1}, the matrix case without
crossing was studied and we avoided global escape function for $P$
but our method gave the impression that the conjugate operator should be
scalar. This condition is irrelevant and let us
extract the real basis of the proof. \\
The initial idea was to seek a conjugate operator of the form
$F=\Pi_+\hat{a}_+\Pi_++\Pi_-\hat{a}_-\Pi_-$, where $\Pi_\pm(x)$ are the
eigenprojectors of $V(x)$ associated to the eigenvalues $\pm\rho(x)$ and
where $a_\pm(x,\xi)$ are scalar symbols, and to
reduce the commutator $[\hat{P},F]$ to scalar ones, involving $p_+$ and
$p_-$, leading to the condition that $a_\pm$ is a global escape
function for $p_\pm$ at energy $\lambda$. On this way, we met the
condition $a_+=a_-$ to cancel some uncontrolled term. Since the
energy shells $p_+^{-1}(\lambda)$ and $p_-^{-1}(\lambda)$ are
disjoint, we were able to construct such an operator by glueing
together different functions $a_+$ and $a_-$, which were global
escape functions for $p_+$ and $p_-$, respectively. \\
This construction seems artificial. From this, we learn that it
may be simplier to work on symbols rather than on operators, that
the scalarness of $F$ is irrelevant, and that the basis of the
proof is the separation of the energy shells. The latter can be expressed
by $\theta (p_+)\theta (p_-)=0$ for a function $\theta$ as in
Definition~\ref{matrice-fuite} with small enough support. Indeed we
can rewrite the proof along the following lines. Choosing $A=a_0\rI_2+
a_1V$ for scalar, smooth functions $a_1$ and $a_2$ (recall that we
have to ensure $[P,A]=0$), we have
%
\begin{equation}\label{separation}
\theta (P)\{P,A\}\theta (P) \ = \ \theta (p_+)^2\{p_+,a_+\}\Pi_+
\, + \, \theta (p_-)^2\{p_-,a_-\}\Pi_-\, + \, \theta (p_+)
\theta (p_-)B \comma
\end{equation}
%
where $B$ is some matrix (roughly the previous uncontrolled term),
$a_\pm=a_0\pm a_1\rho$. Choosing the support of $\theta$ small enough,
the last term vanishes. Choosing $a_+$ (resp. $a_-$) as a global
escape function for $p_+$ (resp. $p_-$) at energy $\lambda$, with
$a_+=a_\infty$ (resp. $a_-=a_\infty$) for $|x|$ large, we get
the ``classical'' Mourre estimate (\ref{mourre-classique}). Since
$\rho$ does not vanish, we can recover $a_0$, $a_1$ from $a_+$
and $a_-$. Theorem~\ref{th-mourre} gives now the desired resolvent
estimates. \\
In the present situation ($\cC\neq \emptyset$), the intersection of
the energy shells $p_+^{-1}(\lambda)$ and $p_-^{-1}(\lambda)$ is not
empty and is included in $\cC^\ast$. We thus have to understand the
effect of this on the construction of global escape functions. \\
However, to derive a global escape function from an escape function
on some compact region in $x$, we are able to adapt the idea in \cite{gm}.
For $R>0$, we set
%
\begin{equation}\label{br}
B_R\ := \ \{x\in \R^n;\, |x|< R\}\hspace{.6cm}\mbox{and}\hspace{.6cm}
B_R^\ast\ := \ B_R\times \R^n \period
\end{equation}
%
\begin{proposition}\label{prol-infini}
Assume that we have a smooth function $A$, which is bounded on
$E(\lambda ;\epsilon_0)$ (cf. (\ref{loca-energie})) for $\epsilon_0>0$
small enough, such that, for $R>0$ large enough, it is an escape function
for $P$ at energy $\lambda$ on $B_R$. Then, we can find a smooth function
$\tA$, which coincides with $A_\infty$ for $|x|$ large enough, and which
is a global escape function for $P$ at energy $\lambda$.
\end{proposition}
%
\Pf Let $R_1>0$ be large enough to have the assumption for $R_1$ and
large enough so that there exists some $c_0>0$ such that $\{P,A_\infty\}
\geq c_0\rI _2$ on $E(\lambda ;\epsilon_0)\setminus B_{R_1}^\ast$.
Let $\tA:=A_\infty+d\chi A$ with $\chi \in C^\infty_0(\R^n;\R)$,
$0\leq \chi \leq 1$, $\chi =1$ on $B_{R_1}$, and with $d>0$ large enough
to ensure
%
\[\sup _{B_{R_1}^\ast\cap E(\lambda ;\epsilon_0)}\bigl\| \{P,A_\infty\}
\bigr\| \ < \ d\, c_1 \comma \]
%
where $c_1$ is given by the assumption for $R_1$ (i.e.
$\{P,A\}\geq c_1\rI _2$ on $B_{R_1}^\ast\cap E(\lambda ;\epsilon_0)$).
Since $\{P,\chi \}$ is scalar, we have
%
\begin{equation}\label{dev-crochet}
\{P,\tA\} \ = \ \{P,A_\infty\}\, + \, d\{P,\chi \}A\, +\, d\chi
\{P,A\} \period
\end{equation}
%
Then, on $E(\lambda ;\epsilon_0)\cap B_{R_1}^\ast$,
$\{P,\tA\}\geq c'_1\rI _2$, for some $c'_1>0$. Now, we choose
the variation of $\chi$ such that, on $E(\lambda ;\epsilon_0)$,
$\|d\{P,\chi \}A\|\leq c_0/2$. Let $R>R_1$ be large enough such that
$\support\chi \subset B_R$. On $E(\lambda ;\epsilon_0)\cap B_R^\ast$,
for $c>0$ given by the assumption for $R$, $d\chi \{P,A\}\geq
d\chi c\rI _2\geq 0$, thus $\{P,\tA\} \geq \min (c'_1,c_0/2)\rI _2$.
Finally $\{P,\tA\} \geq c_0\rI _2$ on $E(\lambda ;\epsilon_0)\setminus
B_R^\ast$ and $\tA$ coincides with $A_\infty$
for $|x|\geq R$. \cqfd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Codimension 1 crossings.}
\label{codim1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section is devoted to the proof of Theorem~\ref{th-codim1}. In other
words, we are going to construct a global escape function for $P$ for
\underline{Codimension 1 crossings}.
According to the discussion in Section~\ref{mourre}, we should look
at the situation at the crossing. In fact, one can make the same
geometrical analysis near $\cC^\ast$ or rather $T^\ast\cC$ as for
Codimension 2 crossing (see Appendix~\ref{geom}). Since, locally on
$T^\ast\cC$, the existence of an escape function does not imply the
existence of a scalar one, as for Codimension 2 crossings, we do not
expect that the geometrical situation at $T^\ast\cC$ produces a local
obtruction as in Section~\ref{codim2}. However, this situation
does produce such an obstruction to the existence of a scalar escape
function at $T^\ast\cC$ (as shown in Appendix~\ref{geom}). So we
really have to exploit the two degrees of freedom given by the
commutation condition, namely look for a function $A=a_0\rI _2+a_1\tV
=a_+\tPi _++a_-\tPi _-$ (so that $[A,V]=0$ everywhere) with
eventually non-zero $a_1$. Furthermore, like in Section~\ref{mourre},
we have
%
\begin{eqnarray}\label{crochet-codim1}
\{P,A\}&=&\{\tp _+,a_+\}\tPi _+
\, + \, \{\tp _-,a_-\}\tPi _-\\
&&+\, 2\trho \bigl(2a_1\xi -\tau (\nabla_\xi a_0)\bigr) \cdot
\bigl(\tPi _+(\nabla_x \tPi _+)\tPi _-\,
+\, \tPi _-(\nabla_x \tPi _+)\tPi _+\bigr) \comma \nonumber
\end{eqnarray}
%
but this time we cannot eliminate the last term by energy localization.
Unless this term is zero (this is the case if $\nabla \tPi _+=0$
everywhere, that is for a fixed matricial structure of $V$, see
Remark~\ref{cas-simple-1} below), we need to control it. \\
By energy localization, if we ensure the positivity of
$\{\tp _\pm,a_\pm\}$ on $\tp _\pm^{-1}(\lambda)$, we need the positivity
of
(\ref{crochet-codim1}) on $\tp _+^{-1}(\lambda)\cap
\tp _-^{-1}(\lambda)$. Recall that $\tau$ is small on this region and,
since
we are looking for a function $A$ that coincides with $A_\infty$ for $|x|$
large, the term containing $\tau$ in (\ref{crochet-codim1}) should be
uniformly a $O(|\tau |)$. So it is reasonable to neglect it.
For all $(x,\xi)\in T^\ast\R^n$, let $f_\pm(x)\in \C^2$ be a normalized
vector generating the range of $\tPi _\pm(x)$ and let $\psi (x,\xi)=
\langle f_-(x),(2\xi \cdot \nabla_x \tPi _+(x))f_+(x)\rangle \in \C$ where
$\langle \cdot ,\cdot \rangle$ denotes the usual scalar product in $\C^2$.
The positivity of the matrix
%
\begin{eqnarray}\label{crochet-principal}
\{P,A\}'&:=& \{P,A\}\, +\, 2\trho \tau (\nabla_\xi a_0)\cdot
\bigl(\tPi _+(\nabla_x \tPi _+)\tPi _-\,
+\, \tPi _-(\nabla_x \tPi _+)\tPi _+\bigr)\\
&=& \{\tp _+,a_+\}\tPi _+ \, + \, \{\tp _-,a_-\}\tPi _-
\, +\, 4a_1\trho \xi \cdot
\bigl(\tPi _+(\nabla_x \tPi _+)\tPi _-\,
+\, \tPi _-(\nabla_x \tPi _+)\tPi _+\bigr) \nonumber
\end{eqnarray}
%
on $\tp _+^{-1}(\lambda)\cap \tp _-^{-1}(\lambda)$, is garanteed by
$\{\tp _+,a_+\}>0$ on $\tp _+^{-1}(\lambda)$ and
%
\begin{equation}\label{positivite}
\{\tp _+,a_+\}\, \{\tp _-,a_-\}
\ > \ |\psi |^2\, (a_+-a_-)^2
\end{equation}
%
on $\tp _+^{-1}(\lambda)\cap \tp _-^{-1}(\lambda)$, since
$a_1\trho =(a_+-a_-)/2$. So, if we require $\{\tp _+,a_+\}>0$ on
$\tp _+^{-1}(\lambda)$ and
%
\begin{equation}\label{positivite'}
r_0\, \{\tp _-,a_-\}
\ > \ |\psi |^2\, (a_+-a_-)^2
\end{equation}
%
on $\tp _-^{-1}(\lambda)$, for some positive function $r_0$ which
coincides with $\{\tp _+,a_+\}$ on $\tp _+^{-1}(\lambda)\cap
\tp _-^{-1}(\lambda)$, we get the ``classical'' Mourre estimate
(\ref{mourre-classique}), locally in $x$, if the neglected term is
really small enough. \\
We see that we have to deal with a nonlinear problem. Given a global
escape
function $a_+$ for $\tp _+$, we try to solve for $a_-$ the following
nonlinear p.d.e.
%
\begin{equation}\label{edp-non-lineaire}
r_0\, \{\tp _-,a_-\}\ = \ |\psi |^2\, (a_+-a_-)^2\, +\, rr_0
\comma
\end{equation}
%
on $\tp _-^{-1}(\lambda)$ for positive functions $r,r_0$, with
$r_0=\{\tp _+,a_+\}$ near the crossing. If we compose
with the flow $\phi^t_{\tp _-}$, we need in fact to solve a family of
nonlinear, differential equations of Ricatti's type. In particular, we
need to adjust $a_+$, $r$, and $r_0$ in order to avoid explosion in
finite time, to ensure the boundness of $a_--a_\infty$ (needed in
Proposition~\ref{prol-infini}), and to guarantee a suitable smallness of
the
neglected term. In fact, we do not
solve these Ricatti's equations but just show the global (time-)existence
of some solutions that ensure the required properties on $a_-$.
To this end, we need the following (known?, partially known?) result on
special Ricatti's differential equations.
%
\begin{proposition}\label{eq-diff}
Let $a,b$ be non-negative, integrable functions on $\R^+$ such that
$I>0$ and $4IJ<1$, where
%
\begin{equation}\label{integrales}
I\ = \ \int_0^{+\infty}a(t)\, dt\hspace{.3cm}\mbox{and}\hspace{.3cm}
J\ = \ \int_0^{+\infty}b(t)\, dt\period
\end{equation}
%
Then the solution of the following Cauchy problem
%
\begin{equation}\label{eq-z}
z'\ := \ \frac{dz}{dt}\ = \ z^2a\, +\, b\comma \ z(0)=z_0<0
\end{equation}
%
is defined and bounded on $\R^+$, provided $-2Iz_0\in ]1-\sqrt{1-4IJ};
1+\sqrt{1-4IJ}[$. In this case, the solution satisfies
%
\begin{equation}\label{bornes-z}
\forall t\in \R^+, \ z_0\leq z(t)<0\period
\end{equation}
%
\end{proposition}
%
{\bf Proof:} see Appendix~\ref{app-codim1}. \cqfd
%
\begin{remark}\label{petitesse}
Let us point out that, if there exist constants $a_0,b_0>0$ such that
$a\geq a_0$ and $b\geq b_0$ on some $[t_0,t_0+T]$ with $t_0\geq 0$ and
$\pi \leq T\sqrt{a_0b_0}$, then the solution blows up at some $T^\ast\leq
t_0+T$. Indeed, it suffices to integrate the inequality $z'\geq
a_0z^2+b_0$.
Therefore, we do need a smallness condition in Proposition~\ref{eq-diff}.
This explains the requirement that $4IJ<1$. To ensure this condition
in the frame of Theorem~\ref{th-codim1}, we cannot simply choose $a_+$
small enough since $|\psi|^2/r_0$ would be large. So, we use another
degree of freedom, which does not depend on the Hamilton flows, namely
the variation of $\tV /\trho$.
\end{remark}
%
{\bf Proof of Theorem~\ref{th-codim1}: } Let us choose some $\epsilon_0>0$
such that the functions $\tp _\pm$ are non-trapping on
$\tp _\pm^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$, respectively.
Recall that we can find $R_1>0$ such that $A_\infty=a_\infty\rI _2$
(defined in (\ref{fuite-infini})) is an escape function for $P$ at
energy $\lambda$ outside $B_{R_1}^\ast$ (defined in (\ref{br})).
Thus, there exists $c_1>0$ such that $\{P,A_\infty\}\geq c_1\rI _2$
on $E(\lambda;\epsilon_0)\setminus B_{R_1}^\ast$. By
(\ref{crochet-codim1}), this implies that
$\{\tp _\pm,a_\infty\}\geq c_1$ on $\tp _\pm^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\setminus B_{R_1}^\ast$. \\
For some $R>R_1$ large enough, we construct, as in \cite{gm}, a global
escape function $a_+$ for $\tp _+$ at energy $\lambda$, which coincides
with $a_\infty$ on $T^\ast\R^n\setminus B_R^\ast$ and satisfies, for
some $c>0$, $\{\tp _+,a_+\}\geq c$ on $\tp _+^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$. \\
Now, we choose $T>0$ bigger than the
supremum of $|a_\infty|$ on $E(\lambda;\epsilon_0)
\cap B_R^\ast$. Therefore $E(\lambda;\epsilon_0)
\cap B_R^\ast \subset \{\beta \in T^\ast\R^n; -T\leq a_\infty (\beta)
\leq T\}\cap E(\lambda;\epsilon_0)$. Denoting $\phi_{\tp _-}^t$ by
$\phi^t$ for simplicity, we point out that we have a smooth diffeomorphism
$\Phi$ from $\R\times [\{\beta \in T^\ast\R^n; a_\infty (\beta)=-T\}
\cap \tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)]$ onto
$\tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$, given by
$\Phi (t,\beta)=\phi^t(\beta)$, since $\tp _-$ is non-trapping on
$\tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$ and $a_\infty$
is an escape function for $\tp _-$ on $\tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\setminus B_R^\ast$. The compact set $\Phi^{-1}
(\tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)\cap B_R^\ast)$
is contained in some $[0;T_0]\times K^\ast$, where $T_0>0$ and where
$K^\ast$ is a compact subset of $\{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}
\cap \tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$. This way
to isolate the region $\tp _-^{-1}(]\lambda-\epsilon_0; \lambda+
\epsilon_0[)\cap B_R^\ast$ is due to \cite{gs}. \\
Let $\chi_1\in C_0^\infty (\R;\R)$ with $\chi_1(t)\geq \rsup (t;1/2)$,
$0\leq \chi_1'\leq 1$, $\chi_1(t)=t$ on $[1;+\infty[$, and
$\chi_1=1/2$ on $]-\infty;0]$. In view of
(\ref{edp-non-lineaire}), we choose
%
\begin{eqnarray}
\rsup \bigl(\{\tp _+,a_+\};c/2\bigr)\, \leq \, r_0&:=& c\,
\chi_1\bigl(\{\tp _+,a_+\}/c\bigr)\comma \label{r0}\\
\rsup \bigl(\{\tp _-,a_+\};c_1/2\bigr)\, \leq \, r&:=& c_1\,
\chi_1\bigl(\{\tp _-,a_+\}/c_1\bigr)\period \label{r}
\end{eqnarray}
%
For $\alpha \in \{\beta \in T^\ast\R^n; a_\infty (\beta)=-T\}
\cap \tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$, let
$k(t;\alpha):=(|\psi|^2/r_0)\rond \phi^t(\alpha) \geq \linebreak0$,
$h(t;\alpha):=r\rond \phi^t(\alpha)$, and $g(t;\alpha)=a_+\rond
\phi^t(\alpha)$. Notice that the function $(h-g')(\cdot;\alpha)$ is
nonnegative and has compact support included in $\R^+$, since
$r\geq \{\tp _-,a_+\}$ with equality on $\tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\setminus B_R^\ast$. We define
%
\begin{equation}\label{integrales-alpha}
I^{\pm}(\alpha)\ := \ \pm \int_0^{\pm \infty}k(t;\alpha)\, dt\ \geq
0\comma \
J(\alpha)\ := \ \int_0^{+\infty}(h-g')(t;\alpha)\, dt\ \geq 0\period
\end{equation}
%
Due to the non-trapping assumption (on $\tp _-$), there exists some finite
$T_R>0$ such that, for all $\alpha \in \tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$, the Lebesgue measure of the set $\{t\in \R;
\phi^t(\alpha)\in B_R^\ast\}$ is $\leq T_R$. The properties of $\chi_1$
ensure that
%
\begin{equation}\label{borne-j}
J(\alpha)\ \leq \ T_R \max \Bigl(1/2\, ;\, \sup _{B_R^\ast}|\{\tp
_-,a_+\}|
\Bigr) \ =: \ T_1\comma
\end{equation}
%
for all $\alpha \in \{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$,
and this bound still holds if we increase $T$. By (\ref{r0}) and
(\ref{limit-pi+-}), we see that, for all
$\alpha \in \{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)$,
%
\begin{equation}\label{borne-i}
I^\pm(\alpha)\ \leq \ \pm(2/c)\, \kappa\, 2\sqrt{\lambda}\,
\int_0^{\pm\infty}
\langle q(t;\alpha)\rangle ^{-1-\delta}\, dt \comma
\end{equation}
%
where we wrote $\phi^t_{\tp _-}(\alpha)=\phi^t(\alpha)=
(q(t;\alpha);p(t;\alpha))$. We need some more information on this flow,
given in the following lemma. We set $\phi_0^t(\alpha)=(q_0(t;\alpha);
p_0(t;\alpha)):=(x+2t\xi,\xi)$, for $\alpha =(x,\xi)$.
%
\begin{lemma}\label{borne-flot}
There exist some $C>0$ and $\sigma \in [0;1[$ such that,
for $R'\geq R$ large enough and $T$ bigger than the supremum of
$|a_\infty|$ on $E(\lambda;\epsilon_0)\cap B_{R'}^\ast$, $\langle
q(t;\alpha)\rangle \geq C\langle t\rangle$, the derivative with
respect to $\alpha$ of $\phi^t(\alpha)$ satisfy
$|D_\alpha (\phi^t-\phi_0^t)(\alpha)|\leq 1-\sigma$ on
$\R\times [\{\beta \in T^\ast\R^n; a_\infty (\beta)=-T\}\cap
\tp _-^{-1}(]\lambda-\epsilon_0; \lambda+\epsilon_0[)\setminus K^\ast]$,
$\R^-\times K^\ast$, and $\R^+\times \phi^{T_0}(K^\ast)$. In particular,
there exists some $D>0$ such that, on these three regions,
%
\[|D_\xi q(t;\alpha)|\ \leq \ D\langle t\rangle \hspace{.6cm} \mbox{and}
\hspace{.6cm} |D_x q(t;\alpha)|+|D_x p(t;\alpha)|+|D_\xi p(t;\alpha)|
\ \leq \ D\period \]
%
\end{lemma}
%
\Pf This lemma follows essentially from results in \cite{dg}, Chapters
$1$ and $2$. However, for sake of completeness, we sketch a proof in
Appendix~\ref{app-codim1}.
\cqfd
By (the first result in) Lemma~\ref{borne-flot}, we derive from
(\ref{borne-i}) the finitness of the integrals $I^\pm(\alpha)$. \\
Actually, we even obtain that $I^-(\alpha)\leq D_1(R')^{-\delta/2}$,
uniformly for $\alpha \in \{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0;\lambda+\epsilon_0[)$.
We choose $R'$ large enough such that $D_1(R')^{-\delta/2}\leq
1/(8T_1)$. Let $T$ be bigger than the supremum of
$|a_\infty|$ on $E(\lambda;\epsilon_0)\cap B_{R'}^\ast$. Now,
considerind (\ref{borne-i}) again, we choose $\kappa$ small enough such
that, for all $\alpha \in \{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0;\lambda+\epsilon_0[)$,
$I^+(\alpha)\leq 1/(8T_1)$. In view of (\ref{edp-non-lineaire}),
we consider the maximal solution $t\donne z(t;\alpha)$ of the Ricatti's
differential equation, for $\alpha \in \{\beta \in T^\ast\R^n; a_\infty
(\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0;\lambda+\epsilon_0[)$,
%
\begin{eqnarray}\label{ricatti}
z'(t;\alpha)&:=& \frac{dz}{dt}(t;\alpha)\ = \ z^2(t;\alpha)k(t;\alpha)\,
+\ h(t;\alpha)\, -\, g'(t;\alpha)\comma \\
z(0;\alpha)&=& -4T_1\comma \nonumber
\end{eqnarray}
%
keeping in mind that this solution will be $(a_--a_+)\rond
\phi^t(\alpha)$. Thanks to (\ref{borne-j}), we can apply
Proposition~\ref{eq-diff} (if $I^+(\alpha)=0$, the corresponding solution
is defined on $\R^+$ and bounded above by $J(\alpha)$, for any
$z_0\in \R^-$). Thus, the maximal solution is defined on $\R^+$ and
satisfies $z_0\leq z\leq T_1$. On $\R^-$, (\ref{ricatti}) reduces to
$z'=z^2k$. Since $I^-(\alpha)\leq 1/(8T_1)$, the solution is defined on
$\R^-$ and satisfies $-4T_1/(1-4T_1I^-)\leq z\leq -4T_1$. \\
Now, we define $a_-$ to be a smooth function on
$T^\ast\R^n$ such that, $a_-\rond \phi^t(\alpha)=a_+\rond
\phi^t(\alpha)+z(t;\alpha)$, for all $t\in \R$ and $\alpha \in \{\beta
\in T^\ast\R^n; a_\infty (\beta)=-T\}\cap \tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$. We can choose this function such that
$a_--a_\infty$ is bounded on $E(\lambda; \epsilon_0)$. Since it satisfies
(\ref{edp-non-lineaire}) thus (\ref{positivite'}) on
$\tp _-^{-1}(]\lambda-\epsilon_0;\lambda+\epsilon_0[)$, we obtain
(\ref{positivite}) on $\tp _+^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\cap \tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$,
yielding the positivity of (\ref{crochet-principal}) on
$\tp _+^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\cap \tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$. More precisely, we have, for the
matrix (\ref{crochet-principal}), the lower bound $(cc_1/4)\rI _2$ on
$\tp _+^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\cap \tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)$. \\
Assume, for a while, that the function $(\nabla_\xi (a_-+a_+))\cdot
\nabla_x\tPi _+$ is bounded on $E(\lambda; \epsilon_0)$, then we choose
$\epsilon_0'\in ]0;\epsilon_0]$ small enough to ensure that the term
containing $\tau$ in (\ref{crochet-principal}) is $\leq (cc_1/5)\rI _2$
on $\tp _+^{-1}(]\lambda-\epsilon_0';\lambda+\epsilon_0'[)\cap
\tp _-^{-1}(]\lambda-\epsilon_0';\lambda+\epsilon_0'[)$. On
$E(\lambda; \epsilon_0')\setminus [\tp _+^{-1}(]\lambda-\epsilon_0';
\lambda+\epsilon_0'[)\cap \tp _-^{-1}(]\lambda-\epsilon_0';
\lambda+\epsilon_0'[)]$, the ``classical'' Mourre
estimate (\ref{mourre-classique}) holds true since we ensured
$\{\tp _\pm,a_\pm\}\geq c'>0$ on $\tp _\pm^{-1}(]\lambda-\epsilon_0';
\lambda+\epsilon_0'[)$. But $a_-$ does not coincide with $a_\infty$ for
$|x|$ large. Let $\chi \in C_0^\infty(\R^n;\R)$ with $0\leq \chi \leq 1$
and $\chi =1$ on $B_R$. Since $a_--a_\infty$ is bounded on
$E(\lambda; \epsilon_0')$, we can choose
the variation of $\chi$ small enough such that $a_+\tPi_++(\chi a_-
+(1-\chi )a_\infty)\tPi_-$ is a global escape function for $P$ at energy
$\lambda$. \\
So, to end the proof, we show that $(\nabla_\xi (a_-+a_+))\cdot
\nabla_x\tPi _+$ is bounded on $E(\lambda; \epsilon_0)$. By
(\ref{limit-pi+-}), the function $(\nabla_\xi a_\infty)\cdot
\nabla_x\tPi _+$ is bounded there, so it suffices to show that
$(\nabla_\xi (a_--a_+))$ is bounded, since $a_+=a_\infty$ for $|x|$
large enough. We just have to show the boundness of
$\nabla_\xi (a_--a_+)$ on $\tp _-^{-1}(]\lambda-\epsilon_0;
\lambda+\epsilon_0[)\setminus \Phi ([0;T_0]\times K^\ast)$,
since we remove a compact set. By (the second result in)
Lemma~\ref{borne-flot}, it suffices to bound $D_\alpha [(a_--a_+)
\rond \phi^t(\alpha)]$ on the three regions given in
Lemma~\ref{borne-flot}. In the two first regions, the function
$(a_--a_+)\rond \phi^t(\alpha)$ coincides with \- the function $z$,
while, on the third region, it equals the function $\tz (t;\alpha)
:=z(t+T_0;\phi^{-T_0}(\alpha))$, which is the solution of the equation
(\ref{ricatti}) with initial condition $\tz (0;\alpha)=
z(T_0;\phi^{-T_0}(\alpha))<0$. But, on the relevant regions, we have
the following explicit formula
%
\begin{equation}\label{z-explicite}
z(t;\alpha)\ = \ \frac{-(1+\epsilon)T_1}{-(1+\epsilon)T_1+\int_0^t
k(s;\alpha)\, ds}\
\end{equation}
%
for $z$ and the same for $\tz$ with $-(1+\epsilon)T_1$ replaced by
$z(T_0;\phi^{-T_0}(\alpha))$, since the function $(h-g')(t;\alpha)$
vanishes. Notice that $D_\alpha [z(T_0;\phi^{-T_0}(\alpha))]$ is bounded
on $\phi^{T_0}(K^\ast)$. By Lemma~\ref{borne-flot},
%
\begin{eqnarray*}
\Bigl|D_\xi\int_0^tk(s;\alpha)\, ds\Bigr|&\leq &\int_0^{+\infty}\Bigl\{
\bigl|D_x[|\psi|^2/r_0)]\rond \phi^t(\alpha)\bigr|\cdot
\bigl|D_\xi q(t;\alpha)\bigr|\\
&&\hspace{1.3cm}+ \, \bigl|D_\xi[|\psi|^2/r_0]\rond
\phi^t(\alpha)\bigr|\cdot \bigl|D_\xi p(t;\alpha)\bigr|\Bigr\}\, dt\\
&\leq &C'\int_0^{+\infty}\bigl\{\langle q(t;\alpha)\rangle ^{-3-2\delta}
\langle t\rangle \, + \, \langle q(t;\alpha)\rangle ^{-2-2\delta}\bigr\}
\, dt\, \leq \, C''\comma
\end{eqnarray*}
%
uniformly on the regions given in Lemma~\ref{borne-flot}. On the same
regions, we find in the same way a bound for $|D_x\int_0^tk(s;\alpha)\,
ds|$.
This yields the boundness of $D_\alpha z(t;\alpha)$ and $D_\alpha
\tz(t;\alpha)$ on the relevant regions. \cqfd
%
\begin{remark}\label{cas-simple-1}
In some particular case, we do not need to consider
a nonlinear problem and can directly derive Theorem~\ref{th-codim1}.
Assume indeed that $V=\tau \tV$ everywhere and that $\tV$ is a constant
matrix near $\cC$. Then, in (\ref{crochet-codim1}), the last term equals
zero. Therefore, it suffices to use the non-trapping condition on
$\tp _\pm$ to construct independent escape functions $a_\pm$ for $\tp
_\pm$,
which equal $a_\infty$ for large $|x|$, as in \cite{gm}. Then the function
$A=a_+\tPi _++a_-\tPi _-$ is a global escape function for $P$,
which equals $A_\infty$ for large $|x|$.
\end{remark}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Codimension 2 crossings.}
\label{codim2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we still consider the problem of the existence of
global escape function for $P$, but in the case of
\underline{Codimension 2 crossings}. First of all, we exhibit the
obstruction announced in Section~\ref{result} and prove
Theorem~\ref{th-neg-codim2}. Then, we show
Proposition~\ref{codim2-croisement} and its refinement in
Remark~\ref{codim2-croisement-raffine}.
As already pointed out, if there is some escape function for $P$ near
$T^\ast\cC$, then there is a scalar one, for instance its half-trace (see
Appendix~\ref{app-codim2}). Therefore, we restrict the question to scalar
functions. So, we are looking for a smooth function $a:T^\ast \R^n\dans
\R$
such that, for some $\epsilon_0>0$,
%
\begin{equation}\label{positivite2}
\{p,a\}\ > \ \sqrt{\{v_1,a\}^2+\{v_2,a\}^2}
\end{equation}
%
on $T^\ast\cC \cap E(\lambda ,\epsilon_0)$, where the energy localization
$E(\lambda ,\epsilon_0)$ is defined in (\ref{loca-energie}).
If we denote by $\sigma$ the second fundamental form on $T^\ast\R^n$,
we can rewrite this condition as
%
\begin{equation}\label{positivite2-bis}
\sigma (H_p,H_a)\ > \ \sqrt{\sigma (H_{v_1},H_a)^2+\sigma
(H_{v_2},H_a)^2}
\end{equation}
%
on $T^\ast\cC \cap E(\lambda ,\epsilon_0)$. Since $T^\ast\cC$ is
symplectic, we have, for all $\alpha \in T^\ast\cC$,
%
\begin{equation}\label{somme-ortho}
T_\alpha T^\ast \R^n \ = \ T_\alpha T^\ast \cC \, \oplus \bigl(T_\alpha
T^\ast \cC \bigr)^\sigma
\end{equation}
%
where the second term is the orthogonal space of $T_\alpha T^\ast \cC$
with respect to $\sigma$ evaluated at $\alpha$. If we define $p'$
(resp. $a'$) as the restriction of $p$ (resp. $a$) to $T^\ast \cC$,
then, according to the decomposition (\ref{somme-ortho}), we
have $H_p=H_{p'}+(H_p-H_{p'})$ (resp. $H_a=H_{a'}+(H_a-
H_{a'})$), as shown in Appendix~\ref{geom}. Therefore,
$\sigma (H_p,H_a)=\sigma (H_{p'},H_{a'})+
\sigma (H_p-H_{p'},H_a-H_{a'})$. Since $T^\ast \cC\subset \cC^\ast$
and $(T_\alpha \cC^\ast)^\sigma =\rvect (H_{v_1},H_{v_2})$
(see Appendix~\ref{geom}), $\rvect (H_{v_1},H_{v_2})\subset
(T_\alpha T^\ast\cC)^\sigma$ and
$\sigma (H_{v_j},H_a)$ reduces to $\sigma (H_{v_j},H_a-H_{a'})$,
for $j=1,2$. So, (\ref{positivite2-bis}) may be equivalently rewritten as
%
\begin{equation}\label{positivite2-ter}
\sigma (H_{p'},H_{a'})+\sigma (H_p-H_{p'},H_a-H_{a'})
\ > \ \sqrt{\sigma (H_{v_1},H_a-H_{a'})^2+\sigma
(H_{v_2},H_a-H_{a'})^2}
\end{equation}
%
on $T^\ast\cC \cap E(\lambda ,\epsilon_0)$. Furthermore, since $H_p$
belongs to $T_\alpha \cC^\ast$, $H_p-H_{p'}$ is in fact in $\rvect
(H_{v_1},H_{v_2})$, as expressed in (\ref{decomp-champ2}) and shown in
Appendix~\ref{geom}. Now, we can
guess what kind of obstruction may appear. Since $T^\ast\cC$ is
symplectic,
$\sigma (H_{p'},H_{a'})$ can be viewed as a Poisson bracket of functions
on
$T^\ast\cC$ and if $\cC$ is compact, for instance, then so is $T^\ast\cC
\cap E(\lambda ,\epsilon_0)$ and $p'$ cannot have a global escape
function.
Therefore, the term $\sigma (H_{p'},H_{a'})$ cannot be positive everywhere
on $T^\ast\cC \cap E(\lambda ,\epsilon_0)$. Thus, at some point $\alpha
\in T^\ast\cC \cap E(\lambda ,\epsilon_0)$,
%
\begin{equation}\label{positivite-normale}
\sigma (H_p-H_{p'},H_a-H_{a'})
\ > \ \sqrt{\sigma (H_{v_1},H_a-H_{a'})^2+\sigma
(H_{v_2},H_a-H_{a'})^2}\comma
\end{equation}
%
which implies (see Appendix~\ref{app-codim2}) that $\alpha$ is not
confining for $P$ at the considered energy, according to
Definition~\ref{confinement}. Here, we have introduced the main
ingredients of the proofs of Theorem~\ref{th-neg-codim2} and
Proposition~\ref{codim2-croisement}.
{\bf Proof of Theorem~\ref{th-neg-codim2}:} Assume that there
is an escape function for $P$ on $T^\ast\cC$. Then, by
Appendix~\ref{app-codim2}, there is a scalar one: $a$. By
the previous discussion, there is some $\epsilon_0>0$ such that
(\ref{positivite2-ter}) holds true on $T^\ast\cC \cap E(\lambda ,
\epsilon_0)$. By assumption, there is a trapped trajectory for $p'$,
contained in the confining region for $P$. On this trajectory,
there is a point $\alpha \in T^\ast\cC \cap
E(\lambda ,\epsilon_0)$ where $\sigma (H_{p'},H_{a'})$ vanishes.
So, (\ref{positivite-normale}) holds true at $\alpha$, which is
a contradiction, since the crossing is confining at $\alpha$. \cqfd
Now, we want to prove Proposition~\ref{codim2-croisement},
constructing first an escape function near $T^\ast\cC$ and then
near $\cC^\ast$. To this end, we need some notation
and notions. Let us define
%
\begin{equation}\label{u}
\cU\ := \ \bigl\{x\in \R^n;\, |\nabla v_1(x)|\, |\nabla v_2(x)|\ > \
|\nabla v_1(x)\cdot \nabla v_2(x)|\bigr\}\hspace{.4cm}
\mbox{and}\hspace{.4cm}\cU^\ast\ := \ \cU\times \R^n
\period
\end{equation}
%
By assumption, the open set $\cU$ contains $\cC$. We recall (\ref{br}),
since the escape function should coincide with $A_\infty$ outside
some $B_R^\ast$. We have already described $T^\ast\cC$, namely in
(\ref{def-cotangent2}). For $x\in \cC$, we have the direct sum
%
\begin{equation}\label{conormal-cotangent}
T^\ast_x\R^n \ = \ T^\ast_x\cC \, \oplus \, N^\ast_x\cC \comma
\end{equation}
%
where the conormal space $N^\ast_x\cC$ to $\cC$ over $x$ is
defined by
%
\begin{equation}\label{def-conormal}
N^\ast_x\cC \ := \ \bigl\{\xi \in T^\ast_x\R^n ; \, \xi \in
(T_x\cC)^\perp \bigr\}\period
\end{equation}
%
We denote by $P_x^N$ (resp. $P_x^T$) the projection onto $N^\ast_x\cC$
(resp. $T^\ast_x\cC$) associated to the decomposition
(\ref{conormal-cotangent}).
The assumptions 1 and 2 in Proposition~\ref{codim2-croisement} are
independent, as the previous discussion shows. Thus, we consider them
separately to prove this proposition. Some more work will give the
refinement given in Remark~\ref{codim2-croisement-raffine}.
Assume that assumption 1 of Proposition~\ref{codim2-croisement} holds
true.
In view of (\ref{positivite2-ter}), we look for a function $a$
such that $H_a-H_{a'}=0$ and such that $a'$ is an escape function
for $p'$. So, we construct such an escape function, which is defined
on $T^\ast \cC$ only, and extend it to some neighborhood of $T^\ast \cC$.
To this end, avoiding local coordinates, we use
a vector field that is conormal to $\cC^\ast $. Precisely, we consider,
the differential equation
%
\begin{equation}\label{equa-diff-2}
\frac{dy_t}{dt}\ = \ -\nabla \rho^2(y_t)\comma \ y_0\ = \ x\period
\end{equation}
%
There exists some open set $\cU_1$ with $\cC\subset\cU_1\subset\cU$
such that, for $x\in \cU_1$, the maximal solution of (\ref{equa-diff-2})
is well defined for all $t\geq 0$, the limit $y(x)$ of $y(t;x)$, as
$t\tend +\infty$, exists and defines a smooth function of $x$ on $\cU_1$
with values in $\cC$, which coincides with the identity on $\cC$.
Furthermore, for $(x,\xi)\in T^\ast\cU_1$, we can construct
$\eta (x,\xi)\in T^\ast_{y(x)}\cU_1$ such that $\eta$
is smooth on $T^\ast\cU_1$ with values in $T^\ast\cC$, and such that
$\eta (x,\xi)=\xi$ and $\nabla v_j\cdot \nabla _\xi \eta (x,\xi)=0$ on
$T^\ast\cC$, for $j=1,2$. All these properties are proved in
Appendix~\ref{app-codim2}.
%
\begin{proposition}\label{fuite-le-long-cotangent}
Under the assumptions of Proposition~\ref{codim2-croisement}, with
condition 1, there exist a smooth, scalar function $A=a\rI _2$, which
equals $A_\infty$ for $|x|$ large enough, and some $R>0$ large
enough such that $A$ is an escape function for $P$ at energy $\lambda$
on $[T^\ast\cC \cap B_R^\ast]\cup T^\ast\R^n\setminus B_R^\ast$.
\end{proposition}
%
{\bf Proof:} Let $R_1>0$ be large enough such that, for some $c_1>0$,
$\{P,A_\infty\}\geq c_1\rI _2$ on $E(\lambda ,\epsilon_0)\setminus
B_{R_1}^\ast$. Let $g'\in C_0^\infty(T^\ast \cC;\R)$ with
$0\leq g'\leq 1$ and $g'=1$ on $T^\ast \cC\cap B_{R_1}^\ast$.
Since $p'$ is non-trapping at energy $\lambda$, we can find a smooth,
bounded function $a'$ on $T^\ast\cC$ which satisfies
(\ref{integration-flot})
(where $g'$ and $p'$ replace $g$ and $p$, respectively) on $(p')^{-1}
(]\lambda -\epsilon_0;\lambda +\epsilon_0[)=T^\ast \cC\cap
E(\lambda ,\epsilon_0)$ (decreasing eventually $\epsilon_0$).
Now, we choose a smooth function $a_1$ on $T^\ast\R^n$ such that
$a_1(x,\xi)=a'(y(x),\eta(x,\xi))$ on $\cU_1^\ast$ and it is also bounded,
and we set $A_1=a_1\rI _2$. By the choice of $\eta$, its restriction to
$T^\ast \cC$ is $a'$ and $H_{a_1}-H_{a_1'}$ vanishes on
$T^\ast \cC$. This implies, by (\ref{positivite2-ter}), that
$\{P,A_1\}\geq \rI _2$ (resp. $\{P,A_1\}\geq 0$) on $E(\lambda
,\epsilon_0)
\cap T^\ast\cC\cap B_{R_1}^\ast$ (resp. $E(\lambda ,\epsilon_0)\cap
T^\ast\cC$). We set $A=A_\infty+d\chi A_1$ for some constant $d>0$ and
some cut-off $\chi$, as in Propostion~\ref{prol-infini}. Following the
arguments of the proof of Propostion~\ref{prol-infini}, we can choose $d$
and $\chi$ such that $A$ is a scalar escape function for $P$ at energy
$\lambda$ on $[B_R^\ast\cap T^\ast \cC]\cup T^\ast\R^n
\setminus B_R^\ast$, for $R\geq R_1$ large enough. Moreover, $A$ equals
$A_\infty$ for $|x|$ large. \cqfd
Next, we consider the other condition in
Proposition~\ref{codim2-croisement}, condition 2. Under this condition,
we want to obtain (\ref{positivite2-ter}) through
(\ref{positivite-normale}). Therefore, it is natural to seek
a function $a_1$ of the form $a_1(x,\xi ):=\beta_1(x,\xi )\xi
\cdot \nabla v_1(x)+\beta_2(x,\xi )\xi \cdot \nabla v_2(x)$,
which satisfies $a_1=0$ and $\nabla_\xi a_1=\beta_1\nabla v_1(x)+\beta_2
\nabla v_2$ on $T^\ast\cC$. In view of (\ref{positivite-normale}) and
its meaning (see Appendix~\ref{app-codim2}), in view of
(\ref{decomp-champ2}) rewritten near $T^\ast\cC$ with smooth functions
$\mu_1$ and $\mu_2$ (see Appendix~\ref{geom}), we choose the bounded,
smooth functions on $\cU^\ast$
%
\begin{eqnarray}
\beta_1&=&\bigl(\mu_1|\nabla v_2|^2-\mu_2(\nabla v_1\cdot \nabla v_2)
\bigr)/\mu ^2 \comma \label{beta_1}\\
\beta_2&=&\bigl(\mu_2|\nabla v_1|^2-\mu_1(\nabla v_1\cdot \nabla v_2)
\bigr)/\mu ^2 \comma \label{beta_2}
\end{eqnarray}
%
where $\mu =(1+\mu_1^2+\mu_2^2)^{1/2}\geq 1$. This defines a
bounded, smooth function $a_1$ on $\cU^\ast$.
%
\begin{proposition}\label{fuite-normale-cotangent}
Under the assumptions of Proposition~\ref{codim2-croisement}, with
condition 2, there exist a smooth, scalar function $A=a\rI _2$, which
equals $A_\infty$ for $|x|$ large enough, and $R>0$ large enough such
that $A$ is an escape function for $P$ at energy $\lambda$ on $[T^\ast
\cC\cap B_R^\ast]\cup T^\ast\R^n\setminus B_R^\ast$.
\end{proposition}
%
{\bf Proof:} Let $A_1=a_1\rI _2$, for $a_1:=\beta_1(\xi
\cdot \nabla v_1)+\beta_2(\xi \cdot \nabla v_2)$, where the functions
$\beta_1,\beta_2$ are defined in (\ref{beta_1}) and (\ref{beta_2}).
Since $a_1=0$ on $T^\ast\cC$, (\ref{positivite2})
reduces to (\ref{positivite-normale}). Notice that
(\ref{positivite-normale}) only depends on $\nabla_\xi a_1=
\beta_1\nabla v_1+\beta_2\nabla v_2$. Thus (\ref{positivite-normale})
holds true on $T^\ast \cC$, by the choice of $\beta_1,\beta_2$
(see Appendix~\ref{geom}). We set $A=A_\infty+d\chi A_1$ for some constant
$d>0$ and some cut-off $\chi$, as in Propostion~\ref{prol-infini}.
As in the proof of Propostion~\ref{prol-infini},
we can find such $A$ satisfying the requirement of
Proposition~\ref{fuite-normale-cotangent}. \cqfd
Now, we try to construct an escape function near $\cC^\ast$.
Our idea is to add to the previous escape function on $T^\ast\cC$ a
scalar function $\xi \cdot \nabla w^2(x)$, where $w(x)$ is small near
$\cC$, in order to get rid of the matricial structure of $P$. Indeed,
the off-diagonal terms of $\{P,\xi \cdot \nabla w^2\}$ are small, while
the diagonal, up to a small term, is $4|\xi \cdot \nabla w|^2\rI _2$.
It is quite natural to choose $w=\rho$, since, for $x\in \cC$ and
$\xi\not\in
T^\ast_x\cC$, if $|\xi|^2$ is positive so is also $|P_x^N\xi|^2$,
yielding the positivity we want to use. We define the smooth, scalar
function $A_N=a_N\rI _2$ with
%
\begin{equation}\label{aN}
\forall (x,\xi)\in T^\ast\R^n\comma \hspace{.7cm} a_N(x,\xi)\ = \ \xi
\cdot
\nabla \rho ^2(x) \period
\end{equation}
%
Notice that it is bounded on $E(\lambda ,\epsilon_0)$.
Furthermore, for any $R>0$, there exists some $c_R>0$ such that, on
$E(\lambda ,\epsilon_0)\cap B_R^\ast$,
%
\begin{equation}\label{crochet-P-aN}
\{P,A_N\}(x,\xi)\ \geq \ c_R|P_x^N\xi|^2\rI _2\, +\, O_R(|\rho
(x)|)\period
\end{equation}
%
\begin{proposition}\label{prol-hors-cotangent}
Assume that we have a smooth function $A$, which coincides with $A_\infty$
for $|x|$ large enough and is an escape function for $P$ at energy
$\lambda$ on $[T^\ast\cC\cap B_{R'}^\ast]\cup
T^\ast\R^n\setminus B_{R'}^\ast$, for some
$R'>0$. Then, we can find a smooth, scalar function $\tA$, which coincides
with $A_\infty$ for $|x|$ large enough, and $R>0$ large enough, such that
$\tA$ is an escape function for $P$ at energy $\lambda$ on $[\cC
\cap B_R]\cup \R^n\setminus B_R$.
\end{proposition}
%
\Pf Let $R_1\geq R'$ be large enough such that $A=A_\infty$ outside
$B_{R_1}^\ast$ and such that there exists $c_1>0$ for which
$|\xi|^2\geq c_1$ and $\{P,A_\infty\}\geq c_1/2$ on
$E(\lambda ;\epsilon_0)\cap T^\ast \R^n\setminus B_{R_1}^\ast$.
There exists some $c'_1>0$ such that $\{P,A\}\geq c'_1\rI _2$ on
$E(\lambda ;\epsilon_0)\cap T^\ast\cC\cap B_{R_1}^\ast$.
By a continuity argument, we can find $\sigma >0$ such that
$(x,\xi)\in E(\lambda ;\epsilon_0)\cap
\cC ^\ast\cap B_{R_1}^\ast$ and $|P_x^N\xi|< \sigma |\xi|$ imply
that $\{P,A\}(x,\xi)\geq c'_1/2\rI _2$. Now, by (\ref{crochet-P-aN}),
$(x,\xi)\in E(\lambda ;\epsilon_0)\cap \cC ^\ast\cap B_{R_1}^\ast$ and
$|P_x^N\xi|\geq \sigma |\xi|$ imply that $\{P,A_N\}(x,\xi)\geq
c_{R_1}\sigma ^2c_1\rI _2$, where $c_{R_1}>0$ is given by
(\ref{crochet-P-aN}) for $R=R_1$. Let $d>0$ be a constant large enough
such that
%
\[\sup _{B_{R_1}^\ast\cap E(\lambda ;\epsilon_0)}\bigl\| \{P,A\}
\bigr\| \ < \ d\, c_{R_1}\, \sigma ^2 c_1\period \]
%
As in the proof of Proposition~\ref{prol-infini}, we set
$\tA=A+d\chi A_N$ where $\chi \in C^\infty_0(\R^n;\R)$,
$0\leq \chi \leq 1$, $\chi =1$ on $B_{R_1}$, and write
(\ref{dev-crochet}).
On $E(\lambda ;\epsilon_0)\cap \cC ^\ast\cap B_{R_1}^\ast$, we have
$\{P,\tA\}\geq c'\rI _2$, for some $c'>0$, since $\chi =1$ on $B_{R_1}$.
Now we can follow the arguments in the proof of
Proposition~\ref{prol-infini} to ensure the desired result. \cqfd
{\bf Justification of Remark~\ref{codim2-croisement-raffine}:}
Under the assumptions of this remark, it suffices to construct an escape
function on $T^\ast\cC$, since the expected result will then follow from
Proposition~\ref{prol-hors-cotangent}.\\
Let $\cB\subset (p')^{-1}(\lambda)$ be the closure of the trapping region
of
$p'$. We assume that there are open sets $U_1^\ast, U_2^\ast$ of
$T^\ast \R^n$ such that $\cB\subset (p')^{-1}(\lambda)\cap U_1^\ast
\subset (p')^{-1}(\lambda)\cap \overline{U_1^\ast}\subset
(p')^{-1}(\lambda)\cap U_2^\ast$, and that the last region is not
confining for $P$ at energy $\lambda$. Notice that $(p')^{-1}(\lambda)=
E(\lambda)\cap T^\ast \cC $. \\
Let $R_1>0$ be large enough such that, for some $c_1>0$,
$\{P,A_\infty\}\geq c_1\rI _2$ on $E(\lambda )\setminus
B_{R_1}^\ast$. Let $g'\in C_0^\infty(T^\ast \cC;\R)$ with
$0\leq g'\leq 1$, $g'=1$ on $B_{R_1}^\ast \cap T^\ast \cC \cap
(T^\ast \R^n \setminus U_2^\ast)$ and $\support g'\subset T^\ast \R^n
\setminus U_1^\ast$. As in \cite{gm} (see (\ref{integration-flot})),
we can construct a smooth, bounded
function $a'$ on $T^\ast \cC$ such that $\{p',a'\}'=g'\geq 0$ on
$(p')^{-1}(\lambda)$, where $\{\cdot ,\cdot \}'$ denotes the Poisson
bracket on $T^\ast \cC$. Let $\chi '\in C_0^\infty(T^\ast \cC;\R)$
with $0\leq \chi '\leq 1$, $\chi'=1$ on $B_{R_1}^\ast \cap T^\ast \cC
\cap U_1^\ast$, $\support \chi '\subset T^\ast \cC\cap U_2^\ast$, and
$\chi '+g'\geq 1/2$ on $B_{R_1}^\ast \cap T^\ast \cC$.
Now we construct a smooth, bounded function
$a_1$ as in the proof of Proposition~\ref{fuite-le-long-cotangent}.
Similarly, we extend $\chi'$ on $\cU_1^\ast$ to $\chi$.
Let $a_2=\beta_1(x,\xi )\xi \cdot \nabla v_1(x)+\beta_2(x,\xi )
\xi \cdot \nabla v_2(x)$ on $\cU^\ast$, where the functions $\beta_1$ and
$\beta_2$ are given in (\ref{beta_1}) and (\ref{beta_2}).
In particular, there exists $c_2>0$ such that, on $E(\lambda )
\cap B_{R_1}^\ast\cap T^\ast \cC$, $\{P,\chi a_2\rI _2\}=\chi'
\{P,a_2\rI _2\}\geq c_2\chi' \rI _2$, since $a_2=0$ on $T^\ast \cC$.
Let $a=a_1+a_2$ and $A=a\rI _2$. \\
On $E(\lambda )\cap B_{R_1}^\ast\cap T^\ast \cC$,
we thus have $\{P,A\}\geq c\rI _2$, for some $c>0$. Notice that
$\{P,A\}\geq 0$ on $E(\lambda )\cap T^\ast \cC$. Now,
we define $\tA =a_\infty+d\chi_0A$ like in the proof of
Proposition~\ref{prol-infini}, that we can follow to get
Remark~\ref{codim2-croisement-raffine}. \cqfd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{.5cm}
{\bf \Large Appendix.}
\begin{appendix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Sharp G\aa rding inequality.}\label{garding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we sketch a proof of the sharp G\aa rding inequality for
matricial symbols. We just adapt a known proof for scalar symbols to
matricial ones.
For bounded symbols $A$, valued in the non-negative, real symetric
matrices, we want to show that there exists a constant $C$ such that,
in the sense of bounded self-adjoint operators on $L^2$, $\hat{A}\
\geq C\, h$, where $\hat{A}$ is the Weyl $h$-quantization of $A$ :
%
\[C_0^\infty(\R^n;\C^2)\ni u\ \donne \ (\hat{A}u)(x)\, = \,
(2\pi h)^{-n}\int_{\R^n} e^{i\xi\cdot (x-y)/h}A\bigl((x+y)/2,\xi\bigr)
u(y)\, dyd\xi \period \]
%
For any bounded, matrix-valued symbol $A$, we define $A_1$ by
%
\[A_1(x,\xi)\ := \ (\pi h)^{-n}\int_{T^\ast\R^n}
e^{-[(x-y)^2+(\xi-\eta)^2]/h}A(y,\eta )\, dyd\eta \period \]
%
We observe that $A_1$ is also a bounded symbol and that the difference
$A-A_1$ is $O(h)$ in this class of symbols. Let $\tilde{A}$ be
the Anti-Wick $h$-quantization of $A$, that is $\hat{A}_1$, the Weyl
$h$-quantization of $A_1$. By the Calderon-Vaillancourt theorem
(which works for matricial pseudodifferential operators), the bounded
operator $\hat{A}-\tilde{A}$ on $L^2$ is $O(h)$ in the corresponding
norm. So it suffices to prove $\tilde{A}\geq 0$, provided $A\geq 0$, to
get
the result. To this end, we can write, for all $u\in C_0^\infty$,
%
\[\langle u,\tilde{A}u\rangle \ = \ \int_{T^\ast\R^n} e^{-y^2/h}\,
|U(y,\eta)|^2\, A(y,\eta)\, dyd\eta \comma \]
%
where $U$ may be expressed in terms of $u$. Thus, if $A$
is nonnegative so is $\tilde{A}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometrical properties.}\label{geom}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this part, we collect some geometrical facts for both types
of crossings. We use the notation introduced in
Section~\ref{codim2}. Since we want to consider functions defined
near $\cC^\ast$, we need to extend the geometrical objects of
Section~\ref{codim2} to some neighborhood of $\cC ^\ast$. \\
We set, for $\epsilon=(\epsilon_1,\epsilon_2)\in\R^2\setminus \{(0,0)\}$,
$\cC(\epsilon):=v_1^{-1}(\epsilon_1)\cap v_2^{-1}(\epsilon_2)$, the
intersection of the $\epsilon_1$-level set of $v_1$
and the $\epsilon_2$-level set of $v_2$. The set
$\cC(\epsilon)\cap \cU$ is a codimension 2 submanifold of $\cU$.
We define the sets $\cC^\ast(\epsilon):=
\cC(\epsilon)\times \R^n$. For $x\in \cC(\epsilon)\cap \cU$,
we have, as in (\ref{conormal-cotangent}), $T^\ast_x\R^n = T^\ast_x
\cC(\epsilon) \oplus N^\ast_x\cC(\epsilon)$ where the cotangent
space $T^\ast_x\cC(\epsilon)$ (resp. the conormal space $N^\ast_x
\cC(\epsilon)$) of $\cC(\epsilon)$ at $x$ is defined as in
(\ref{def-cotangent2}) (resp. (\ref{def-conormal})). We still denote by
$P_x^T$ (resp. $P_x^N$) the natural projection onto
$T^\ast_x\cC(\epsilon)$ (resp. $N^\ast_x\cC(\epsilon)$)
associated to this direct sum. Notice that, for $\epsilon =(0,0)$, we
find objects, defined in Section~\ref{codim2}. So, in this case, we
simply pull out the symbol ``$(0,0)$'' at the end
keeping this way a coherent notation. \\
Recall first that the Hamilton field $H_a(\beta)$ of some smooth function
$a$ on $T^\ast \R^n$ at $\beta \in T^\ast \R^n$ is the unique vector $w
\in T_\beta T^\ast \R^n$ such that the differential $da(\beta)$ of $a$
at $\beta$ is given by $\sigma _\beta (w,\cdot)$, where $\sigma _\beta$
is the value of the fundamental 2-form at $\beta$. Furthermore,
if $\cD^\ast$ is some level set of $a$, then, for any $\beta\in\cD^\ast$,
the orthogonal set to $T_\beta \cD^\ast$ w.r.t. the form $\sigma _\beta$
is $(T_\beta \cD^\ast)^\sigma=\rvect (H_a(\beta))$.\\
For $\alpha\in T^\ast \cC(\epsilon)\cap \cU^\ast$, the space
$T_\alpha T^\ast \cC(\epsilon)$ is symplectic and we have
%
\begin{equation}\label{somme-ortho-eps}
T_\alpha T^\ast \R^n \ = \ T_\alpha T^\ast \cC(\epsilon) \, \oplus
\bigl(T_\alpha T^\ast \cC(\epsilon) \bigr)^\sigma \period
\end{equation}
%
For any smooth function $a$ defined on $\cU^\ast$,
we denote by $a'$ its restriction to $T^\ast \cC(\epsilon)\cap \cU^\ast$.
Since the restriction of $da$ of $a$ to
$T_\alpha T^\ast \cC(\epsilon)$ equals $da'$, we have, by definition of
the Hamilton fields, $H_a(\alpha)-
H_{a'}(\alpha)\in (T_\alpha T^\ast \cC(\epsilon))^\sigma$. Therefore,
according to (\ref{somme-ortho-eps}), $H_a(\alpha)$ splits into
$H_{a'}(\alpha)+(H_a(\alpha)-H_{a'}(\alpha))$. For Codimension 2
crossings,
we have, since $H_{v_1}(\alpha)$ and
$H_{v_2}(\alpha)$ are independent, $(T_\alpha \cC^\ast(\epsilon))^\sigma=
\rvect (H_{v_1}(\alpha),H_{v_2}(\alpha))$. Furthermore, since
$v_1$ and $v_2$ only depend on $x$, $\sigma (H_{v_1},H_{v_2})$ is zero
at $\alpha$. Thus $(T_\alpha \cC^\ast(\epsilon))^\sigma \subset
T_\alpha \cC^\ast(\epsilon)$. Obviously, $T_\alpha T^\ast \cC(\epsilon)
\subset T_\alpha \cC^\ast(\epsilon)$, so $\rvect (H_{v_1}(\alpha),
H_{v_2}(\alpha))\subset (T_\alpha T^\ast \cC(\epsilon))^\sigma$.
By an argument of dimension, we even have $T_\alpha \cC^\ast(\epsilon)=
\rvect (H_{v_1}(\alpha),H_{v_2}(\alpha))\oplus T_\alpha T^\ast
\cC(\epsilon)$. Since $p=|\xi|^2+u(x)$, we see, thanks to
(\ref{def-cotangent2}), that $\sigma_\alpha (H_{v_1}(\alpha),H_p(\alpha))
=\sigma_\alpha (H_{v_2}(\alpha),H_p(\alpha))=0$, that is
$H_p(\alpha)\in T_\alpha \cC^\ast(\epsilon)$. Therefore, according
to the previous decomposition,
%
\begin{equation}\label{decomp-champ2-eps}
\forall \alpha \in T^\ast\cC(\epsilon)\cap \cU^\ast,\hspace{.6cm}
H_p(\alpha)\ = \ \bigl(\mu_1(\alpha)H_{v_1}(\alpha)+\mu_2(\alpha)
H_{v_2}(\alpha)\bigr)\, +\, H_{p'}(\alpha)\comma
\end{equation}
%
for some smooth, real functions $\mu_1,\mu_2$. Here, $p'$
is the restriction of $p$ to $T^\ast \cC(\epsilon)$.
Of course, we have the same situation for Codimension 1 crossings,
that is, for $\cU=\{x\in \R^n;\nabla\tau (x)\neq 0\}$ and $\alpha \in
\cC^\ast(\epsilon)\cap \cU^\ast$, $T_\alpha \cC^\ast(\epsilon)=
\rvect (H_\tau(\alpha))\oplus T_\alpha T^\ast \cC(\epsilon)$,
$H_p(\alpha)\in T_\alpha \cC^\ast(\epsilon)$, and the corresponding
decomposition
%
\begin{equation}\label{decomp-champ1-eps}
\forall \alpha \in T^\ast\cC(\epsilon)\cap \cU^\ast,\, H_p(\alpha)\ = \
\mu (\alpha)H_\tau(\alpha)\, +\, H_{p'}(\alpha)\comma
\end{equation}
%
for some smooth, real function $\mu$. The phenomenon described in
Section~\ref{codim2} appears also here. If we consider a
\underline{scalar} function $a$, then the positivity of $\{P,a\rI _2\}$
on $E(\lambda)$ implies the inequality
%
\[\sigma (H_{p'},H_{a'})\, +\, \sigma (H_p-H_{p'},H_a-H_{a'})
\ > \ \trho |\sigma (H_\tau,H_a-H_{a'})|\]
%
on $T^\ast \cC\cap E(\lambda)$. If $\sigma (H_{p'},H_{a'})$ is not
everywhere positive on $T^\ast \cC\cap E(\lambda)$ (if
$\cC$ is compact, for instance), then
we must have $\mu >\trho$ everywhere on $T^\ast \cC\cap E(\lambda)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Codimension 1.}\label{app-codim1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the construction of escape functions, we need some result
on a special case of Ricatti's differential equations, which
might be not completely included in the litterature, and
some properties of Hamilton flows, essentially contained in
\cite{dg}, that we sketch here.
{\bf Proof of Proposition~\ref{eq-diff} :} By the Cauchy-Lipschitz
theorem, we have local existence and uniqueness of solution. We
consider the maximal solution, defined on $[0;T^\ast[$, for some
$T^\ast>0$. Assume that there exists some $t_1\in [0;T^\ast[$
such that $z(t_1)=0$. Then, since $|z(t)|\leq |z_0|$ on $[0;t_1]$,
$z'\leq az_0z+b$ on $[0;t_1]$. Therefore, we obtain on $[0;t_1]$,
%
\begin{eqnarray*}
z(t)&\leq & z_0\rexp \bigl(z_0\int_0^ta(s)\, ds\bigr)\, +\,
\int_0^tb(s)\rexp \bigl(z_0\int_s^ta(v)\, dv\bigr)\, ds\\
&\leq & z_0\rexp \bigl(z_0\int_0^ta(s)\, ds\bigr)\, +\,
\int_0^tb(s)\, ds\comma
\end{eqnarray*}
%
since $z_0<0$ and $a\geq 0$. Using the convexity of the exponential
function, we see that
%
\[0\ = \ z(t_1)\ \leq \ z_0\, +\,
z_0^2\int_0^ta(s)\, ds\, +\, \int_0^tb(s)\, ds\ \leq \ z_0\, +\, z_0^2I\,
+\, J\period \]
%
Since $1-4IJ>0$, we arrive at a contradiction for $2Iz_0\in ]-1-
\sqrt{1-4IJ};-1+\sqrt{1-4IJ}[\subset ]-\infty ;0]$. Thus, for such
$z_0$, the solution is defined on $R^+$ and negative. Since $z$
is nondecreasing, it satisfies (\ref{bornes-z}). \cqfd
{\bf Proof of Lemma~\ref{borne-flot} :} Let $a$ be a global escape
function for $\tp _-$ at energy $\lambda$, such that $a=a_\infty$
outside $B_{R'}^\ast$. Thus, there exists $c_1>0$ such that
$\{\tp _-,a\}\geq c_1$ on $\tp _-^{-1}(]\lambda -\epsilon_0;
\lambda +\epsilon_0[)$, for small enough $\epsilon_0>0$. On this
region, if we take $\beta$ with $a(\beta)=0$ then,
for all $t$, $c_2\langle q(t;\beta)\rangle \geq |a\rond \phi^t(\beta)|
\geq c_1|t|$, for $c_2>0$ independent of $t$ and $\beta$ (see Theorem
2.4.3 in \cite{dg}). Thus, there exists $c>0$ such that, uniformly,
$\langle q(t;\beta)\rangle \geq c\langle t\rangle$, yielding the first
result of Lemma~\ref{borne-flot} in the last two regions. For the
first one, $\langle q(t;\alpha)\rangle =\langle q(t-t_1;\beta)\rangle$,
for some $\beta$ satisfying $a(\beta)=0$ and some $t_1\geq 0$. Using
the previous estimate for $q(t;\beta)$ and the fact that $t_1\leq T/c_1$,
we get the first result for the first region. \\
In the spirit of \cite{dg}, we introduce a time-dependent effective force.
Let $\chi \in C^\infty (\R;\R)$ such that $0\leq \chi \leq 1$, $\chi =0$
on
$]-\infty;1/2]$, and $\chi =1$ on $[1;+\infty[$. We set, for $t\in\R$ and
$x\in \R^n$, $F(t;x)=\chi (\langle x\rangle /(C\langle t\rangle))
\chi (x/R')(-\nabla_x (u-\tau \trho))(x)$.
For the three regions we consider, $(q(t;\alpha);p(t;\alpha))$ is
the solution of the Hamilton system
%
\[\frac{dq}{dt}(t;\alpha)\ = \ 2p(t;\alpha)\comma \hspace{.8cm}
\frac{dp}{dt}(t;\alpha)\ = \ F\bigl(t,q(t;\alpha)\bigr)\comma \]
%
starting at $\alpha =(x,\xi)$ at $t=0$. As usual, this is equivalent
to
%
\[\frac{dq}{dt}(t;\alpha)\ = \ 2p(t;\alpha)\comma \hspace{.8cm}
q(t;\alpha)\ = \ q_0(t;\alpha)\, +\, 2\int_0^t(t-s)
F\bigl(s,q(s;\alpha)\bigr)\, ds\period \]
%
The function $t\donne z(t;\alpha):=q(t;\alpha)-q_0(t;\alpha)$ is solution
of $\cP (z)=z$ where
%
\begin{equation}\label{transf-p}
\cP (v)(t)\ := \ 2\int_0^t(t-s)F\bigl(s,v(s)+q_0(t;\alpha)\bigr)\,
ds\period
\end{equation}
%
Let $\cB$ be the Banach space of the functions $t\donne v(t)$, such that
$\|v\|_{\cB}:=\sup _{t\in \R}|v(t)/t|<\infty$, equiped with the norm
$\|\cdot \|_{\cB}$. As in Theorem 1.5.1 in \cite{dg}, we want to use
the fixed point Theorem. By definition of $F$, we see that
$\cP (v)$ belongs to $\cB$. Derivating w.r.t. $v$, we see that
$\|D_v\cP (v) \|_{\cB}\leq c(R')^{-\delta /2}$ by (\ref{decroit}),
showing that $\cP$ is a contraction on $\cB$, for $R'$ large enough.
Similarly we get $\|D_\xi \cP (v)\|\leq c(R')^{-\delta /2}$, since
$|D_\xi q_0(s;\alpha)|\leq C\langle s\rangle$. Since $z$ is a fixed
point of $\cP$, $D_\xi \cP (z)=(1-D_v \cP (z))D_\xi z$. Therefore
$D_\xi z$ belongs to $\cB$ and $\|D_\xi z\|_{\cB}\leq c(R')^{-\delta /2}$.
Along the same lines, we show that $D_x z$ belongs to $\cB$ and that
$\|D_x z\|_{\cB}\leq c(R')^{-\delta /2}$. Choosing $R'$ large enough,
we can ensure the second result of Lemma~\ref{borne-flot}. \cqfd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Codimension 2.}\label{app-codim2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this part, we prove that the existence of an escape function
for $P$ near $T^\ast\cC$, for a Codimension 2 crossing, implies the
existence of a scalar one, we relate the confining condition
to some positivity, and we consider some tools used in
Proposition~\ref{fuite-le-long-cotangent}.
{\bf Scalar escape function for Codimension 2 crossings:}
We want to show that the existence of an escape function $A$ for
$P$ at energy $\lambda$ on $T^\ast\cC$ implies the existence
of a scalar one, namely its half-trace $a_0$. \\
Notice that $a_0$ is a smooth function on $T^\ast\R^n$. Since $[P,A]=0$,
we can write $A=a_0\rI _2+a_1V$ outside $\cC^\ast$, for a smooth function
$a_1$ on $T^\ast\R^n\setminus \cC^\ast$. Since $A$ is smooth everywhere,
so is the function $a_1V$ and so are the functions $v_1a_1$ and $v_2a_1$.
Since $v_1$ and $v_2$ are independent coordinates near $\cC$, we
see, using Taylor expansion with rest integral for the
functions $v_1a_1$ and $v_2a_1$, that $a_1$ extends
to a smooth function on $T^\ast\R^n$. Next we compute $\{P,A\}$, defined
in (\ref{poisson-matriciel}), and obtain
%
\[\{P,A\}\ = \ \bigl(\{p,a_0\}-(1/2)\nabla_\xi a_1\cdot \nabla \rho^2
\bigr)\, \rI _2\, +\, \{p,a_1\}\, V\, +\, \{V,a_0\}\, +\, 2a_1\xi
\cdot \nabla V\period \]
%
For $x\in \cC$, $(\nabla \rho^2)(x)=0$ and $V(x)=0$, and for
$(x,\xi)\in T^\ast\cC$, $\xi \cdot \nabla V(x)=0$. Therefore, on
$T^\ast\cC$, $\{P,A\}=\{P,a_0\}$. This proves the expected result.
\cqfd
{\bf Confining condition:} In the main text, we use the following
result for the independent vectors $\nabla v_1(x)$ and $\nabla v_2(x)$
when $x$ belongs to $\cC$. \\
Let $e_1,e_2$ be two independent vectors
in $\R^m$ ($m\geq 2$). We shall prove that, given $\mu_1,\mu_2\in \R$,
there exist $\lambda_1,\lambda_2 \in \R$ such that
%
\begin{equation}\label{contrainte}
(\lambda_1e_1+\lambda_2e_2)\, \cdot \, (\mu_1e_1+\mu_2e_2)
\ > \ \sqrt{(\lambda_1|e_1|^2+\lambda_2e_1\cdot e_2)^2+
(\lambda_1e_1\cdot e_2+\lambda_2|e_2|^2)^2}\comma
\end{equation}
%
if and only if $\mu_1^2+\mu_2^2>1$. And, in this case,
we can exhibit $\lambda_1,\lambda_2$ satisfying (\ref{contrainte}).\\
To this end, we set $x=\lambda_1|e_1|^2+\lambda_2e_1\cdot e_2$ and
$y=\lambda_1e_1\cdot e_2+\lambda_2|e_2|^2$. Notice that $(\lambda_1e_1+
\lambda_2e_2)\cdot (\mu_1e_1+\mu_2e_2)=\mu_1x+\mu_2y$.
So, the condition (\ref{contrainte}) means that the scalar product
of the vectors $(\mu_1,\mu_2)$ and $(x,y)$ in $\R^2$ must
be greater than the norm of $(x,y)$. This is only possible if
the norm of $(\mu_1,\mu_2)$ is greater than $1$. In this
case, we have the solution $(x,y)=(\mu_1,\mu_2)$. Since
$e_1$ and $e_2$ are independent, $\Delta :=|e_1|^2|e_2|^2-
(e_1\cdot e_2)^2>0$ so we can recover $(\lambda_1,\lambda_2)$ from
$(x,y)$.
In particular, (\ref{contrainte}) is satisfied for
$\lambda_1=(\mu_1|e_2|^2-\mu_2(e_1\cdot e_2))/\Delta$
and $\lambda_2=(\mu_2|e_1|^2-\mu_1(e_1\cdot e_2))/\Delta$. \cqfd
{\bf Properties of the differential equation (\ref{equa-diff-2}):}
Let $z\in \cC$. Since $z\in \cU$, we can find a bounded open set $\cU_z$,
with $z\in \cU_z\subset \cU$, and a smooth diffeomorphism
%
\[\begin{array}{rcl}
\phi_z\, :\, \cU_z&\dans &\cW_z \\
x&\donne &\bigl(v_1(x), v_2(x), v_3(x), \cdots , v_n(x)\bigr)\comma
\end{array}\]
%
onto some open set $\cW_z$ in $\R^n$, such that $x\in \cU_z$ and
$a_1^2+a_2^2\leq \rho^2(x)$ imply $(a_1, a_2, v_3(x),
\linebreak \cdots , v_n(x))\in \cW_z$. For the maximal solution of
(\ref{equa-diff-2}), defined on $[0;T^\ast[$, starting at $x\in \cU_z$,
$(d/dt)\rho^2(y_t)=
-|\nabla \rho^2|^2(y_t)$, so the function $t\donne \rho^2(y_t)$ is
nonincreasing. Thus, $y_t$ stays in $\cU_z$, for all $t\in [0;T^\ast[$.
Therefore $T^\ast=\infty$. Furthermore, we can find some $c>0$ such
that $|\nabla \rho^2|^2\geq c\rho^2$ on $\cU_z$. This implies
that $(d/dt)\rho^2(y_t)\leq -c\rho^2(y_t)$ and further that
$\rho^2(y_t)\leq \exp(-ct)\rho^2(x)$. Since $\phi_z(y_t)=(v_1(y_t),
v_2(y_t), v_3(x), \cdots , v_n(x))$, we conclude that $y(x):=
\lim_{t\tend +\infty}y_t$ exists, belongs to $\cC$, and that $y$ equals
the smooth function $x\donne \phi_z^{-1}(0,0, v_3(x), \cdots , v_n(x))$
on $\cU_z$. So it is well defined and smooth on
$\cU_1:=\cup_{z\in \cC}\cU_z$. Since $\phi_z\rond y\rond \phi_z^{-1}$
is the restriction to $\cW_z$ of a projection, we see that $y'(x)$ is
bijective from $T_x\cC(\epsilon)$, for $\epsilon=(v_1(x),v_2(x))$, onto
$T_{y(x)}\cC$ and equals the identity for $x\in \cC$. Thus, so is its
transposed map $^t(y'(x))$ from $T^\ast_{y(x)}\cC$ onto
$T^\ast_x\cC(\epsilon)$. For any $\xi\in T^\ast_x\R^n$, there
exists an unique $\eta(x,\xi)\in T^\ast_{y(x)}\cC$ such that
$^t(y'(x))\eta(x,\xi)=P_x^T\xi$. This defines a smooth function $\eta$,
which satisfies, on $T^\ast\cC$, $\eta (x,\xi)=\xi$ and
$\nabla v_j\cdot \nabla _\xi \eta (x,\xi)=0$ for $j=1,2$.
\end{appendix}
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\end{document}