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THERMODYNAMIC LIMITS, PSEUDODIFFERENTIAL OPERATORS
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\font\un = cmbx10 at 14pt \null \vskip 2cm \centerline {\un
THERMODYNAMIC LIMITS FOR HAMILTONIANS }
\medskip
\centerline {\un DEFINED AS PSEUDODIFFERENTIAL OPERATORS.} \vskip
1cm \centerline {\bf J. NOURRIGAT and Chr. ROYER } \vskip 1cm
\centerline {\bf Abstract.} If $P(h)$ is a $h-$pseudodifferential
operator in $ {\bf R} ^n$ associated to an holomorphic
semi-bounded symbol in some neighborhood of the real phase space,
with bounded derivatives, we describe the symbol of $e^{-tP(h)}$,
by inequalities where the constants depend on the bounds for the
derivatives of the symbol of $P(h)$, but not on the dimension $n$.
Some applications to thermodynamic limits (free energy) are given.
\vskip 1cm
\noindent
{\bf 1. Introduction.}
\bigskip
In [16], J. Sj\"ostrand describes the exponential of an $n-$dimensional
semiclassical Schr\"odinger operator
$$P_n(h)\ =\ -h^2 \Delta \ + \ V_n(x)\ . \leqno (1.1)$$
Since the dimension $n$
is variable, the given object is rather a sequence $(V_n)_{n\geq 1}$,
where $V_n\in C^{\infty}({\bf R}^n)$, belonging to a suitable class
of such sequences, called {\it $0-$standard}. This class is defined by
inequalities where the constants are
independent on the dimension $n$.
Instead of recalling the definition of $0-$standard sequences,
let us say that, if there are some constants $a>0$ and $M>0$,
independent on $n$, such that $V_n$ extends to an holomorphic
function $V_n$ on
$$\Omega _n (a)\ :=\ \{ x\in {\bf C} ^n, \
| Im\ x | _{\infty } 0)$, near the diagonal, where the constants in the inequalities
are independent
on $n$, and he gives some applications to thermodynamic limits when
$n\rightarrow +\infty $ (see below).
\bigskip
The aim of this paper is to replace the Schr\"odinger operator
by pseudodifferential operators $P_n(h)$ associated, by the
semiclassical Weyl calculus (see sect. 2), to symbols
$p_n(x, \xi)$ which extend to
$$\Omega _{2n} (a)\ :=\ \{ (x, \xi )\in {\bf C} ^{2n}, \
| Im\ (x, \xi) | _{\infty } 0$ and $M>0$, independent on $n$. Moreover,
we shall be interested in $e^{-tP_n(h)}$ instead of $e^{-{t\over
h}P_n(h)}$. Our aim is to give, assuming also that $Re\ p_n(x,
\xi)$ is lower bounded, a description of the {\it symbol} of the
operator $e^{-tP_n(h)}$, (and not of its {\it kernel}, like in
[16]), where the constants will be independent on the dimension
and, more precisely, to show that the symbol of this operator can
be written $e^{-q_n(x, \xi, t, h)}$, where $q_n$ satisfies
inequalities similar to (1.5). This result will rely on the
theorem of composition of pseudodifferential operators in large
dimension, (where good estimations are not satisfied by the
composed symbols, but by their logarithms), proved in a previous
paper [1] with L. Amour and Ph. Kerdelhu\'e. In [1], the results
proved here were conjectured, and proved in the formal level. In
[10], the results were proved under the assumption that the
symbols were bounded, and not only semi-bounded. To weaken the
hypothesis, we had to make important change to the technique of
proof.
\bigskip
Let us explain (in a particular case to simplify the notations),
the physical model given by J. Sj\"ostrand ([16], sect.8) as an
application of his results. At each point $j$ of the
one-dimensional lattice ${\bf Z}$, we consider a particle $A_j$
described by a Schr\"odinger operator in ${\bf R} ^k$ (where
$k\geq 1$ is fixed) $$P_1(h)\ =\ -h^2 \Delta + V(x)\ . $$ The
assumptions of [16] are satisfied if $V$ extends to an holomorphic
function in $\Omega _k(a)$ $(a>0)$, if its derivatives are bounded
in $\Omega _k(a)$, if $V$ is real for real $x$, and if $V(x)$ is
greater than a positive power of $|x|$ for large $|x|$. The
interaction between each particle and its neighbours in the
lattice is given by a potential $W(x, y)$, and the hypotheses of
[16] are satisfied if $W$ extends to a bounded holomorphic
function in $\Omega _{2k}(a)$, real for real $(x, y)$, and such
that $W(y, x)=W(x, y)$. For each $n\geq 1$, the Hamiltonian
$P_n(h)$ describing the system of particles $A_j$ $(-n\leq j\leq
n)$ in interaction is defined by (1.1), (with $n$ replaced by
$(2n+1)k$), where $$V_n(x) \ =\ \sum _{j=-n}^n V(x^{(j)})\ +\ \sum
_{j=-n} ^{n-1} W(x^{(j)}, x^{(j+1)})\ . \leqno (1.6)$$ The variable of
$ {\bf R} ^{(2n+1)k}$ is denoted by $x=(x^{(-n)}, \ldots
,x^{(n)})$, with $x^{(j)}\in {\bf R} ^k$. This sequence $(V_n)$
satisfies (1.3). Such type of Hamiltonians, given at each point of
a lattice, each of them interacting with its neighbors, is common
in the literature (see Toda [17] in the classical mechanics). In
[16], the lattice is not only {\bf Z}, but be multidimensional,
and one motivation can be the theory of anharmonic crystals. In
E. Lieb [9] and B. Simon [14], a similar model (the {\it quantum
Heisenberg model}), is studied, but with another Hamiltonian at
each point of the lattice, and another type of interaction.
\bigskip
Then, Sj\"ostrand [16] proves that, for each $h>0$ and $t>0$, the
sequence $$\Lambda _n(t, h)\ :=\ {1\over (2n+1)} \ ln\ \left [ (2 \pi h
)^{(2n+1)k} Tr\ \left ( e^{-{t\over h}P_n(h)} \right ) \right ] \leqno
(1.7)$$ has a limit $\Lambda _{\infty }(t, h)$ when $n\rightarrow
+ \infty$, that $$ | \Lambda _n(t, h)\ -\ \Lambda _{\infty }(t, h)
| \ \leq \ {C\over n}\ , \leqno (1.8)$$ and, with some more
hypotheses, that $\Lambda _{\infty }(t, h)$, (the {\it
thermodynamic limit}), has an asymptotic expansion in powers of
$h$ when $h\rightarrow 0$. (See Ruelle [13] for the notion of
thermodynamic limit, and also Helffer-Sj\"ostrand [6] for the
Lemma used to prove its existence and (1.8)). For an anharmonic
crystal, the derivative of $\Lambda _{\infty }(t, h)$ with respect
to $t$ (if it exists), is the mean energy at the temperature
$1/t$. In [16], the potentials are typically ${\cal O}( | x | )$
at infinity. For exactly quadratic potentials, see also Royer
[12], which gives a rigorous proof and a generalization of some
usual physical formulas on harmonic crystals.
\bigskip
As an application of our theorem 2.2, we shall prove similar results
when the particles $A_j$ $(j\in{\bf Z})$ are no more described by a
Schr\"odinger operator, and with $e^{-{t\over h}P_n(h)}$ replaced
by $e^{-tP_n(h)}$. Our result can be applied, be applied, for example, to
$$P_n(h)\ :=\ \sum _{j=-n}^n
\sqrt { I -h^2 \Delta _j}\ +\ V_n(x)$$
where $\Delta _j$ is the Laplacian for the variable $x^{(j)}\in {\bf R}
^k$, and $V_n(x)$ is defined by (1.6).
For such operators, we shall prove the analogous of the result
of Sj\"ostrand [16]. The similar
result for the quantum Heisenberg model seems to be still an open
problem. (In this model, $h$ is the inverse of the spin, and Lieb [9]
and Simon [14] proved only the continuity of $\Lambda _{\infty }(t, h)$
at $h=0$). In [16], the asymptotic expansion of the thermodynamic
limit in powers of $h$ relies on a study of Laplace integrals in
large dimension (cf also [2]). In our sect. 9, it relies on the study of the
greatest eigenvalue of some integral operator on ${\bf R}^p$ (for a
suitable $p$,
depending on the order of the asymptotic expansion in powers of $h$). The study
of this integral operator has some common points with the works of
B. Helffer (cf [3] and the references given in [3])
and Helffer-Ramond (cf [4]) for the Kac operator.
\bigskip
The results of Lascar [8] on pseudodifferential calculus in infinite
dimension have another motivation, related to the quantum field theory.
\bigskip
The main result is theorem 2.2, and its application to
thermodynamic limit is given in theorems 9.1 and 9.3. In sect.3,
a result on the composition of pseudodifferential operators in
large dimension may have its own interest.
\bigskip
In a preprint [10], a similar result was proved under the
additional hypothesis that $p_n$ was bounded, and an application
to thermodynamic limits for periodic symbols (with a suitable
notion of trace) was given. In this preprint some results on
inversion
and exponential of pseudodifferential operators in large dimension,
(which may have also perhaps their own interest), were used.
\bigskip
\noindent
{\bf 2. Statement of the result on the exponential.}
\bigskip
For each $(x, \xi )\in {\bf R} ^{2n}$, we set
$$\Vert (x, \xi ) \Vert _{\infty } \ =\
\sup _{j \leq n } sup ( |x_j | , | \xi _j | ). \leqno (2.1)$$
For each $a>0$, the set $\Omega _{2n}(a)$
is defined in (1.4).
\bigskip
\noindent
{\bf Definition 2.1. } {\it For each $a>0$, we denote by
$S (a)$ the set whose elements are
sequences $( f_n)_{(n\geq 1)}$,where
$f_n = f_n(h)= f_n(., h)$ is an holomorphic function in
$\Omega _{2n} (a)$, depending on parameter
$h$ in an interval $]0, h_n]$, where $h_n>0$, and satisfying the
following condition. There exists $M>0$, independent on $n$, and
a constant $C_n>0$ (which may depend on $n$) such that, if $1\leq
j\leq n$ and if $0< h\leq h_n$, $$ Re\ f_n (x, \xi , h ) \
\geq \ C_n, \hskip 1cm \forall (x , \xi) \in \Omega _{2n}(a) \ \
\ \ \forall h\in ]0, h_n], \leqno (2.2)$$ $$ | { \partial f_n
\over \partial x_j} (x, \xi , h) | + | { \partial f_n \over
\partial \xi _j} (x, \xi , h) | \ \leq \ M, \hskip 1cm \forall (x
, \xi) \in \Omega _{2n}(a) \ \ \ \ \forall h\in ]0, h_n]. \leqno
(2.3)$$ }
\bigskip
If $f$ is a bounded function in $\Omega _{2n} (a)$ (defined in (1.4)),
we set
$$\Vert f\Vert _a \ =\ \sup _{X\in \Omega _{2n} (a)}
| f(X) |. \leqno (2.4)$$
If the derivatives are also bounded, we set
$$\Vert \nabla f\Vert _a \
=\ \sup _{X\in \Omega _{2n} (a)}\ \sup _{j \leq n} \ sup \left |
{\partial f\over \partial x_j} (X)\right | , \
\left | {\partial f\over \partial \xi _j} (X)\right |.\leqno
(2.5)$$
\bigskip
By the Weyl calculus, we associate to a suitable
function $f\in C^{ \infty } ( {\bf R} ^{2n})$ the
$h-$pseudo\-differential operator $Op_h(f)$ by
$$(Op_h(f)u)(x)\ =\ (2\pi h)^{-n} \int _{ {\bf R} ^{2n}}
e^{{i\over h} (x-y).\xi } f({x+y\over 2}, \xi) u(y) \ dyd\xi\ . \leqno
(2.6)$$
\bigskip
If $(p_n)$ is in $S(a)$, we know, by the functional calculus of
Helffer-Robert [6], that, for each fixed dimension $n$, the
exponential $e^{-tOp_h(p_n)}$ is also a $h-$pseudodifferential
operator, and we want to describe its symbol uniformly with
respect to the dimension $n$. The main result of this paper is the
following.
\bigskip
\noindent
{\bf Theorem 2.2.} {\it Let $b>0$, and $p= (p_n)$ a family of
symbols in $S (b)$, independent on $h$.
Then, for each $a\in]0, b[$, and for each integer $m\geq 1$,
there exist $\varepsilon _m>0$, $C_m>0$ (independent on the
dimension $n$), and a family of functions
$q= (q_n)(x, \xi , t, h)$ in $S (a)$ $(0\leq t \leq 1)$ such that
$$e^{-tOp_h(p_n)}\ =\ Op_h(e^{-q_n(., t, h)}) \hskip 1cm if \ \ \
\ 0< nh^m \leq \varepsilon _m, \ \ \ \ \ \ \ \ n\geq 1\ . \leqno
(2.7)$$ Moreover, the symbol $q_n$ has an asymptotic expansion in
powers of
$h$
$$q_n(x, \xi , t, h)\ =\ \sum _{j= 0}^{m-1} E_n^{(j)}(x, \xi , t
)h^j + h^m R_n^{(m)} (x, \xi , t, h)\ , \leqno (2.8)$$ where the
families of functions $(E_n^{(j)})$ are in $S(a)$ and satisfy the
inequalities $(2.3)$, with the same conditions as above.
Moreover, $E_n^{(0)}(x, \xi , t)= tp_n(x, \xi )$ and, for $j\geq
1$, we have $$\Vert E_n^{(j)}(. , t)\Vert _a \leq nC_j \hskip 1cm
\Vert R_n^{(m)}(. , t, h) \Vert _a \leq nC_m\ . $$ }
\bigskip
The explicit construction of the functions $E_n^{(j)}$ will be
made more precise in sections 6 (in general) and 9 (in some
particular cases).
\bigskip
The paper is organized as follows. In section 3 we recall a result
of [1] on the composition of pseudodifferential operators
in large dimension.
In section 4, we recall the result of [1] on the composition of
operators associated to symbols which are the exponential of
functions $p_n$ in $S (a)$. In [1], the result was stated for
the standard calculus and the function $p_n$ was assumed to be
bounded, but the proof is the same with our hypotheses. In section
5, following the thesis of the second author [11], we state and
prove a result which is like a linearized of the previous one. In
section 6, we give the formal construction of the $E_n^{(j)}$ of
(2.8), and in sections 7 and 8, the end of the proof of Theorem
2.2. The precise statement for applications to thermodynamic limit
is given and proved in section 9.
\bigskip
\noindent
{\bf 3. Oscillating integrals and composition of symbols in large
dimension (result of [1]).}
\bigskip
We denote by $\sigma $ the symplectic form in ${\bf C}^{2n}$
defined by $\sigma \big ( (x, \xi ),\ (y, \eta)\big ) = y. \xi -
x. \eta $. If $f$ and $g$ are $ C^{ \infty } $ functions on $
{\bf R} ^{2n}$, bounded with all their derivatives, we denote by
$f \sharp _hg$ the function defined by the oscillatory integral
$$(f \sharp _h g )(X) \ =\ ( \pi h)^{-2n} \int _{ {\bf R} ^{4n}}
e^{ -{2i \over h} \sigma (Y, Z)} f(X + Y ) g (X + Z) dY dZ \ . \leqno
(3.1)$$ More generally, we shall have to work with oscillating
integrals of the following form : $$(I_h(F))\ (X)\ \ =\ ( \pi
h)^{-2n} \int _{ {\bf R} ^{4n}} e^{ -{2i \over h} \sigma (Y, Z)}
F(X, Y, Z)\ dY dZ \ . \leqno (3.2)$$where $X$ is in $\R^p$ and $F$
is a suitable function in an open set of $\k^p \times
\k^{2n}\times \k^{2n}$.
\bigskip
The following theorem gives an estimation of such integrals.
\bigskip
\noindent
{\bf Theorem 3.1.} {\it Let $a>0$ and $\rho >0$, and $\Omega _{p,
2n, 2n }(a, \rho )$ be the set of $ (X, Y, Z)\in {\bf C}^{p}\times
{\bf C}^{2n}\times {\bf C}^{2n} $ such that $X$ is in $\Omega
_{p}(a)$ and $Y$ and $Z$ in $\Omega _{2n}(\rho )$. Let $F$ be a
bounded holomorphic function in $\Omega _{p, 2n, 2n}(a, \rho)$.
Let $I_h(F)$ be integral defined in (3.2). Then we have $$\Vert
I_h(F)\Vert _a \ \leq \ \sup _{(X, Y, Z)\in \Omega _{p, 2n,
2n}(a, \rho )}
| F(X, Y, Z) | \
\left ( 1 + \sqrt {{2 \over \pi }} { \sqrt {h} \over \rho }
e^{-{\rho ^2 \over h}} \right ) ^{4n}\ . \leqno (3.3)$$ }
\bigskip
This result is very similar to the Lemma 5.1 of [1]. The only
difference is that the integral was in $\R^{2n}$ and that, instead
of the quadratic form $\sigma (Y, Z)$ in $\R^{4n}$, we had the
form $y. \eta $ in $\R^{2n}$, but the idea of the proof is the
same: we diagonalize the quadratic form in order to reduce the
problem to a one dimensional integral, like in [1]. Therefore, we
shall not give the details of the proof.
\bigskip
The following corollary may have its own interest. It was not
explicitly written in [1].
\bigskip
\noindent
{\bf Theorem 3.2.} {\it Let $a$ and $b$ be such that $00)$, and let $h'_n$ and $h''_n$ be the sequences associated to
them as in Definition 2.1. Let $a\in ]0, b[$ and $m\geq 1$. Then,
there exist an element $\varphi $ of $S (a)$ and constants
$\varepsilon _m$ and $C_m>0$, depending only on $m$, $a$ and $b$,
and on the bounds of $\Vert \nabla f_n \Vert _b$ and $\Vert \nabla
g_n \Vert _b$, but not of the dimension $n$, such that
$$e^{-\varphi _n (h)} \ =\ e^{-f_n(h)} \sharp _h e^{-g_n(h)}
\hskip 1cm if \ \ \ \ 00)$, with bounded derivatives. Let $f_1$ and $f_2$
are holomorphic functions,
bounded respectively on $\Omega _{2n}(\mu_1)$ and $\Omega
_{2n}(\mu_2)$, where $0<\mu _j0$ and $\varepsilon _m>0$ depend on $R$, $R'$, $m$ and on
$\Vert \nabla p_j \Vert _R$, ($j=1,2)$, but not on $n$. }
\bigskip
Before to prove this result, we shall first introduce some
notations, recall some results of [1], and examine a similar
result in small dimension (proposition 5.7) which will next lead
us by induction to the conclusion.
\medskip
Let $\mu_1>0$ and $\mu_2>0$, we
denote by $\Omega _{4}(\mu_1,\mu_2)$ the following open set of
$\k^4$: $$\Omega _{4}(\mu_1,\mu_2)=\{(x,\xi,x',\xi') \in \k^4 \ |
\ |{\rm Im}(x,\xi)|_\infty<\mu_1,|{\rm Im}
(x',\xi')|_\infty<\mu_2\}\; ,$$ If $\mu_1=\mu_2=\mu$, we have
$\Omega _{4}(\mu_1,\mu_2)=\Omega_4(\mu)$.
\medskip
If $f$ is a bounded holomorphic
function defined on $\Omega _{4}(\mu_1,\mu_2)$, we set:
$$||f||_{\mu_1,\mu_2}=\sup_{X\in \Omega
_{4}(\mu_1,\mu_2)}|f(X)|\;.$$ If $f$ is an holomorphic function
defined on $\Omega _{4}(\mu)$, having all its derivatives bounded,
we set: $$||\nabla _{x,\xi}f||_\mu=\max(||\partial_{x}
f||_\mu,||\partial_\xi f||_\mu)\;,$$ $$||\nabla
_{x'\!,\,\xi'}f||_\mu=\max(||\partial_{x'}
f||_\mu,||\partial_{\xi'} f||_\mu)\;,$$ $$||\nabla
f||_\mu=\max(||\nabla_{x,\xi} f||_\mu,||\nabla_{x' \! , \, \xi'}
f||_\mu ) \; .$$
\medskip
If $\Omega $ is an open set in $\k^p$ with $p\in {\bf N}^*$, we
denote by $K^1(\Omega )$ the space of holomorphic functions in
$\Omega $ with bounded derivatives and lower bounded real part in
$\Omega$ .
\medskip
In this
section, we shall use the results proved in [1] with the standard
pseudodifferential calculus which is here replaced by the Weyl
one. Hence, we can prove, exactly as in [1], the following
propositions.
\bigskip
\noindent
{\bf Proposition 5.2 (Proposition 3.1 of [1]).} {\it Let $a$ and $b$ such that
$00$
depending only on $b-a$ and $||u'||_b$, independent on $||u||_b,$
and two functions $v_+$ and $v_-$, $C^\infty$ on $[0,h_1]$, valued
in $K^1(\Omega _1(a))$, verifying $$e^{-v_{\pm }(x,h)}=e^{\mp
i{\pi \over 4}}\sqrt{{2 \over \pi h}} \int_\R e^{{\pm 2iy^2 \over
h}}e^{-u(x-y)} dy\; \ \ \ \ \ if 0 < h\leq h_1\ \ \ \ v_{\pm }
(x, 0)=u(x) .$$ $$||e^{-v_{\pm}(.,h)+u(.)}-1||_a\leq {1 \over
2}\; . $$ }
\bigskip
\noindent
{\bf Proposition 5.3 (Proposition 6.1 of [1]).} {\it Let $R$ and $R'$ such that
$00$ having the following properties.
For all $\rho >0$ and $\mu >0$ such that $C_p\rho <\mu $ and for
all
holomorphic
function $f$ bounded in the closed ball $B(0,\mu)$ in $\k^4$ with
the $|.|_\infty$ norm,
we have:
$$ \sum_{0 \leq |j| \leq p}|f^{(j)}(0)| {\rho^{|j|} \over j!}
\leq \left[1+ \left({C_p\rho \over \mu -C_p\rho
}\right)^{p+1}\right]
\sup_{|z|_\infty \leq \mu}|f(z)| \;.$$}
\bigskip
After recalling these results of [1], we shall use them to
estimate quotients of the type (5.1), or more generally of the
form
$$\psi (.,h)(X, X')=
{I_h(e^{-u}f)(X, X') \over I_h(e^{-u})(X, X')}\;,\leqno (5.6)$$
where $I_h$ is the integral defined in (3.5), with $n=1$, $f$ is
bounded and holomorphic on $\Omega _4(\mu_1,\mu_2)$ and $u\in
K^1(\Omega _4(R))$ with $0<\mu_j0$), and
$f$ be a bounded holomorphic function on $\Omega_4(\mu_1,\mu_2)$,
($0<\mu _j0$,
$$\tilde{\Omega} _4(a)=\{X=(X_i)_{1 \leq i \leq 4} \in \k^4 /
|{\rm Im} X_1|+|{\rm Im} X_4|0$ depending only on $b-a$
and $||\nabla u||_b$, and functions $v_+^{(j)}$ and $v_-^{(j)}$,
$(1\leq j \leq 4)$ in $K^1(\Omega _1(a))$, depending smoothly on
$h\in [0,h_1]$, verifying if $0\leq h\leq h_1$:
$$e^{-v_{\pm
}^{(j)}(x,h)}=e^{\mp i{\pi \over 4}}\sqrt{{2 \over \pi h}} \int_\R
e^{{\pm 2iy^2 \over h}}e^{-f(x_1, ... , x_j-y, ... , x_4)} dy\;, $$
$$||e^{-v_{\pm }^{(j)}(.,h)+f(.)}-1||_a\leq {1 \over 2}\; . $$
Let us denote $v_{\pm }^{(j)}$ by $V_{\pm }^{(j)}(h)f$. Therefore,
the function $$ \tilde g_h =
V_-^{(4)}(h)V_-^{(3)}(h)V_+^{(2)}(h)V_+^{(1)}(h)\tilde u$$ is
holomorphic in $ \tilde{\Omega}_4(R')$ and satisfies
$$e^{-\tilde
g_{h}}=\tilde{I}_h \; , \hskip 1cm |||{e^{-\tilde u} \over e^{-\tilde
g_{h}}}|||_{R'} \leq 16 \; .$$
This is valid if $h \leq
\varepsilon_1$, where $\varepsilon_1$ is a function of $R-R'$ and
$||\nabla u||_R$. Therefore: $$\left\| {e^{-u(.)}\over
I(e^{-u})(.,h)} \right\|_{R'} \leq 16 \;,\quad if \quad h \leq
\varepsilon_1 (R-R',||\nabla u||_R) \; .\leqno (5.10)$$
\bigskip
\noindent
{\it Second step.} Estimation of the product
$e^{u(.)}I(e^{-u}f)(.,h)$. For each function $\Phi (X,X') \in C^{
\infty } ( {\bf R} ^{4n})$ let us set, for each $X=(x, \xi )$ and
$X'=(x', \xi' )$ in ${\bf R} ^{2n},$
$$H_{\Phi }(X, X')\ =\ \left
( - { \partial \Phi \over
\partial \xi},\ {\partial \Phi \over \partial x} \right )\ , $$ and we
define similarly $H'_{\Phi (X,X')}$ by the derivatives with
respect to $x'$ and $\xi '$. Then we set, for each $X$, $X'$, $Y$
and $Z$ in $ {\bf R} ^{2n}$ $$Q _{\Phi } (X, X', Y, Z)\ =\ \int
_0^1 H_{\Phi }(X+tY, X'+t Z)\ dt \ , $$ and we define similarly
$Q'_{\Phi}$ using $H'_{\Phi}$. With these notations, we can write,
using the symplectic form $\sigma$
$$u(X+Y, X' + Z) -u(X,X')= -
\sigma \left ( Y, Q_u(X, X', Y, Z)\right ) - \sigma \left (Z,
Q'_u(X, X', Y, Z)\right ) \ . $$
If $I_h$ is the integral defined in
(3.5), it follows that, for
$(X,X')\in \Omega_4(\lambda)$:
$$e^{u(X,X')}I_h(e^{-u}f)(X,X')=(\pi h)^{-2} \int_{\R^4}
e^{-{2i \over h} \varphi (X, X', Y,Z,h)} f(X+Y,X'+Z) dYdZ\;
,\leqno (5.11)$$ with, $$\varphi(.,h)= \sigma \left (Y + {h\over
2i}Q'_u(X, X', Y, Z) ,
Z - {h\over 2i}Q_u(X, X', Y, Z)\right ) + ... $$ $$... + \ {h^2\over 4}
\sigma \left ( Q_u(X, X', Y, Z), Q'_u(X, X', Y, Z)\right ) \ . $$
The {\it third step} of the proof will be stated as a lemma.
\bigskip
\noindent {\bf Lemma 5.6.} {\it If we suppose, setting now
$M=||\nabla u||_R$: $$h ||\nabla _{x' \! ,\xi'} u||_R \leq
\mu_1-\lambda \;, \quad h ||\nabla _{x,\xi} u||_R \leq
\mu_2-\lambda \quad and \quad {h M \over R-R'} \leq {1 \over 4}
\;,\leqno (5.12)$$then, for each $(X, X')$ in $\Omega _4(\lambda)$
and for each $(Y, Z)$ in $\Omega _4({ \mu _1 - \lambda \over 2},{
\mu _2 - \lambda \over 2} )$, the map $$(\widetilde Y, \widetilde
Z) \longrightarrow \left ( Y - {h\over 2i} Q'_u(X, X',\widetilde
Y, \widetilde Z),\ \ Z + {h\over 2i} Q_u(X, X',\widetilde Y,
\widetilde Z) \right ) $$ sends $\overline{\Omega_4(\mu_1-\lambda
,\mu_2-\lambda)}$ in itself and has a unique fixed point
$(\widetilde Y, \widetilde Z)$ in this set. Let us denote this
fixed point by $(\widetilde Y, \widetilde Z)= \gamma _h(X,X', Y,
Z) $. Then, under the conditions (5.12), the jacobian $J_h$ of the
map $$\Omega _4({ \mu _1 - \lambda \over 2},{ \mu _2 - \lambda
\over 2} )\ni (Y, Z)\rightarrow \gamma _h (X, X', Y, Z)\in \
\overline{\Omega_4(\mu_1-\lambda ,\mu_2-\lambda)} $$ satisfies
$$|J_h(X, X', Y, Z)|\leq 2 \hskip 1cm \forall (Y, Z)\in \Omega
_4({ \mu _1 - \lambda \over 2},{ \mu _2 - \lambda \over 2} )\hskip
1cm \forall (X, X')\in \Omega _4(\lambda).\leqno (5.13)$$
Moreover, we have, under the same conditions $$|e^{-i{h\over 2}
\sigma (Q_u(X, X',\gamma _h(X,X', Y, Z)),Q'_u(X, X',\gamma
_h(X,X', Y, Z) )) }|\leq 2\ \ {\rm if }\ \ hM^2 \leq {1\over
2}\ . \leqno (5.14)$$}
\bigskip
The proof of this lemma is elementary.
\bigskip
\noindent {\it End of the proof of Lemma 5.5.} Then the map $\R^4
\ni (Y, Z)\rightarrow \gamma _h (X,X',Y,Z): \R^4 \rightarrow \k^4
$ defines a contour of integration $\gamma _h$ in $\k^4$,
depending on $(X, X')$, and we can write, using this change of
contour, the expression (5.11) under the form
$$e^{u(X,X')}I_h(e^{-u}f)(X,X',h)=(\pi h)^{-2} \int_{\gamma _h}
e^{-{2i \over h} \varphi (X, X',Y,Z,h)} f(X+Y,X'+Z) dYdZ\; $$
$$... \ =\ (\pi h)^{-2} \int_{\R^4}e^{-{2i \over h} \varphi (X, X',
\gamma _h(X, X', Y,Z,h)} f((X, X')+ \gamma _h(X, X', Y,Z))J_h(X,
X', Y, Z)dY dZ\ . $$
By the definition of $\gamma _h$, we have
$$\varphi (X, X', \gamma _h(X, X', Y,Z) = \sigma (Y,Z) + \ldots $$
$$\ldots + \ {h^2\over
4} \sigma \left ( Q_u \left ( X, X',\gamma _h(X,X', Y, Z)\right )
\ ,\ Q'_u \left ( X, X',\gamma _h(X,X', Y, Z) \right ) \right ) \ . $$
Since the point $(X, X')+ \gamma _h(X, X', Y,Z)$ is in
$\Omega_4(\mu_1 ,\mu_2)$ if $(Y, Z)$ is in $\Omega _4({ \mu _1 -
\lambda \over 2},{ \mu _2 - \lambda \over 2} )$, we have, using
(5.13), (5.14) and Theorem 3.1, $$ \left\| e^{u(.)}
I(e^{-u}f)(.,h) \right\|_\lambda \leq 4(1+ {2\sqrt{h}\over
\mu-\lambda} e^{-(\mu-\lambda)^2/4h})^4||f||_{\mu_1,\mu_2} \;
,\leqno (5.15)$$ if the conditions (5.8) are satisfied. The
estimation (5.7) follows from (5.10) and (5.15).
\bigskip
Lemma 5.5 is not satisfactory since the constant in the inequality
is very far from $1$. Proposition 5.8, (which uses Lemma 5.7),
will give a better constant.
\bigskip
\noindent
{\bf Lemma 5.7.} {\it With the notations of Lemma 5.5,
the function $\psi(.,h)$ of Proposition 5.5 satisfies, for each
integer $m\geq 2$, for each $\lambda $ and $\lambda '$ such that
$0<\lambda < \lambda ' < \mu = \inf (\mu_1,\mu_2)$
$$||\psi(.,h)||_\lambda \leq ||f||_{\lambda
'}+ {A_mh^m \over (\lambda'- \lambda)^{2m}}
\sup_{s \in [0,h]} ||\psi(.,s)||_{\lambda '} \;
\hskip 1cm {\rm if } \ \ h\leq \varepsilon _m (R-R', \Vert \nabla u \Vert _R)
,\leqno (5.16)$$
where $A_m$ and $\varepsilon _m$ are positive functions of $m$,
$R-R'$, where $R'=\max (\mu_1,\mu_2)$ and $||\nabla u||_R$. }
\bigskip
\noindent
{\it Proof.} With the notations of Lemma 5.5,
we know, by (3.6), that the function $I_h(e^{-u})(X, X')$ defined
in (3.5) satisfies $$\displaystyle 2i{\partial I_h(e^{-u})\over
\partial h}=
({\partial^2 \over \partial {x'}\partial {\xi}}-
{\partial^2 \over \partial {x}\partial
{\xi'}})I_h(e^{-u})\ . \leqno (5.17)$$ Moreover, the function
$\psi_h(X, X')I_h(e^{-u})(X, X')=I_h(fe^{-u})(X, X')$ satisfies a
similar equation. By proposition 5.3, we know that there exists an
holomorphic function $v(X, X', h)$ satisfying (5.4)for $h$ small
enough, but perhaps in a smaller domain. It follows from (5.17),
from the similar equation satisfied by $ \psi_hI_h(e^{-u})$ and
from (5.4) that $$2i{\partial v \over
\partial h}={\partial^2 v \over \partial x'
\partial \xi}-{\partial^2 v \over \partial x \partial
\xi'}-{\partial v \over \partial x'}{\partial v \over \partial
\xi}+{\partial v \over \partial x}{\partial v \over \partial
\xi'} \;,$$ $$2i{\partial \psi \over \partial h}={\partial^2 \psi
\over
\partial x' \partial \xi}-{\partial^2 \psi \over \partial x
\partial {\xi'}}+{\partial v \over \partial x'}{\partial
\psi \over \partial \xi}-{\partial v \over \partial x}{\partial
\psi \over \partial \xi'}+ {\partial \psi \over \partial
x'}{\partial v \over \partial \xi}-{\partial \psi \over \partial
x}{\partial v \over \partial \xi'} \; .$$Therefore, setting
$z=(z_1,z_2,z_3,z_4)=(x,\xi,x',\xi')$, we can write, for all
$j\in\{1,\dots,m+1\}$: $$
\partial_h^j \psi (z,h)=\sum_{1\leq|\alpha |\leq2j}
b_{j k}(z,h)\partial_z^{\alpha} \psi (z,h) \;,
\leqno(5.18) $$ where the $b_{j \alpha}$ are linear combinations
of the products of the form $\partial_z^{\beta
_1}v...\partial_z^{\beta _k}v$, where $\beta _1, ..., \beta _k$
are multi-indices such that $|\beta _1|+ ...+ |\beta
_k|=j-|\alpha|$. Let us explain now the estimates of the functions
$b_{j \alpha}$ of (5.18), and when (5.18) is valid. Let $R''=
{R+R'\over 2}$. By Proposition 5.3, there are two constants
$\varepsilon _m$ and $M_1$, both depending only on $R-R'$ and
$||\nabla u||_R$, such that the equality (5.4)is valid if $z\in
\Omega _4(R'')$ and $h<\varepsilon _m$, and such that we have
$||\nabla v(., h)||_{R''}\leq M_1$ if $h\leq \varepsilon _m$. By
the Cauchy inequalities, and by the form of the functions $b_{j
\alpha}$ of (5.18), there exists $M_2>0$ (depending only on $m$,
$R-R'$ and $||\nabla u||_R$) such that $$||b_{j\alpha
}(.,h)||_{R'} \leq M_2 \hskip 1cm {\rm if }\ \ \ 1\leq j\leq m+1,\
\ \ \ \ 1\leq |\alpha|\leq 2j \ .\leqno (5.19) $$
\medskip
By the Taylor formula and (5.18), we can write:
$$\psi(z,h)=\sum_{0\leq|k|\leq2m} a_k(z,h) \partial_z^k \psi (z,0)
+h^{m+1}R_m(z,h) \;, \quad \forall z\in \Omega _4(\lambda )
\;,\leqno (5.20)$$ where: $$a_k(z,h)=\sum_{{|k| \over 2}\leq
j\leq m} b_{jk}(z,0){h^j \over j!}\;, \quad if \quad 1 \leq |k|
\leq2m \;,\leqno (5.21)$$ $$a_0(z,h)=1 \;,$$ and: $$|R_m(z,h)|\leq
\sup_{0\leq s\leq h} |\partial_s^{m+1}\psi (z,s)|\leq \sup_{0\leq
s\leq h} \sum_{1\leq |k| \leq 2m+2} |b_{m+1
,k}(z,s)||\partial_z^k\psi (z,s)| \;.\leqno (5.22)$$ By (5.20),
(5.21), (5.19), and by the Cauchy inequalities which, combined
with (5.22), give a bound of $R_m$, there exists a constant $B>0$,
depending on $m$, $R-R'$ and $M_2 $, such that, if $z \in \Omega
_4(\lambda )$ and $h\in[0,\varepsilon_m ]$: $$|\psi (z,h)|\leq
\sum_{0\leq |k|\leq 2m}{B^{|k|} h^{{|k| \over 2}} \over k!}
|\partial_z^k\psi (z,0)|+{Bh^{m+1}\over (\lambda'
-\lambda)^{2m+2}} \sup_{0\leq s\leq h} ||\psi (.,s)||_{\lambda '}
\;.$$ By lemma 5.4 applied to $\rho =Bh^{{1 \over 2}}$ and
$\mu=\lambda ' $, $p=2m$, we see easily that if $z\in\Omega
_4(\lambda )$ and $2C_{2m} Bh^{{1 \over 2}}<\lambda '-\lambda$,
$C_{2m}$ being the constant of lemma 5.4, we have, since $\psi (z,
0)=f(z)$ : $$ \sum_{0\leq |k|\leq2m}{B^{|k|} h^{{|k| \over 2}}
\over k!} |\partial_z^k\psi (z,0)|\leq \left[1+\left({C_{2m}Bh^{{1
\over2}} \over \lambda '-C_{2m}Bh^{{1 \over2}}
}\right)^{2m+1}\right] ||f||_{\lambda '}\;.$$ Hence, if
$2C_{2m}Bh^{{1 \over 2}}<\lambda '-\lambda$, there is $A_m$, as in
the statement of the lemma, such that (5.16) is satisfied. If $h$
verifies $2C_{2m}Bh^{{1 \over 2}}\geq \lambda '-\lambda $, the
inequality (5.16) is trivial, with another $A_m$.
\bigskip
\noindent
{\bf Proposition 5.8.} {\it Let $u$ a function in $K^1(\Omega
_4(R))$, ($R>0$), $f$ a bounded holomorphic function on
$\Omega_4(\mu_1,\mu_2)$, ($0<\mu _j0$, and we may assume that $K$ (resp. $\varepsilon $) is an
increasing (resp. decreasing) function of its last variable.
\bigskip
\noindent
{\it Proof.} By lemma 5.5, there exists $\varepsilon_1(R-R',||\nabla u||_R)>0$ such that,
setting $\lambda'=(\lambda+\mu)/2$: $$||\psi(.,h)||_{\lambda'}
\leq 64(1+ {2\sqrt{h} \over (\mu-\lambda')}
e^{-(\mu-\lambda')^2/4h})^4||f||_{\mu_1,\mu_2} \; ,\leqno (5.25)$$
if the relations (5.8) are satisfied, replacing $\lambda $ by
$\lambda '$ and $\varepsilon $ by $\varepsilon _1$. Now we shall
distinguish two cases.
\medskip
\noindent {\it First case}: $h(\mu-\lambda)^{-2} \leq 1$.
By lemma 5.7, there exists
$\varepsilon_2 $ and $A_m$, depending only on $m$, $R-R'$ and $
||\nabla u||_R)>0$ such that, if $h\leq \varepsilon_2$, we have
(5.16). Hence, by (5.25) and (5.16), we see that there exist
$\varepsilon _m$ and $K_m$, both depending only on $||\nabla
u||_R$ and $R-R'$, such that (5.23) is satisfied if the conditions
(5.24) are. \medskip \noindent {\it Second case}:
$h(\mu-\lambda)^{-2} \geq 1$. In this case,(5.23) is
a direct consequence of (5.7).
\bigskip
The previous proposition allows
us now to demonstrate the theorem 5.1.
\bigskip
\noindent
{\it Proof of theorem 5.1.}
\medskip
\noindent
{\it Notations.}
For all $k \leq n$, we consider the following open sets of
$\k^{4n}$:
$$ \Omega _{k,n}(\lambda ,\mu_1, \mu _2)=\{ (X,X') \in
\k^{2n}\times \k^{2n}, \ \ \ \
|{\rm Im} (X_j, X'_j)|_{\infty} <\lambda \ {\rm if} \ \ j0$.
\medskip
\noindent
{\it First step:} We
consider the sequence of function $u_k(.,h)$ on $\Omega
_{k,n}(R',R)$ with $k \in \{1,\ldots,n\}$, defined by: $$u_1
(X,X')=p_1(X)+p_2(X') \qquad (X,X') \in
\Omega_{2n}(R)\times\Omega_{2n}(R)=\Omega_{1,n}(R',R) \;,$$ and by
the following induction relation if $1\leq k\leq n-1$:
$$e^{-u_{k+1}(.,h)}=I_k(e^{-u_k})(.,h)\; .$$ With the constant
$C_m$ of proposition 5.3, we define the following sequence
$(P_k(h))_{1\leq k\leq n}$ by: $$P_k(h)=(1+h^mC_m(R-R',2||\nabla
u_1||_R))^{k-1}\;,$$ and we set $\displaystyle
c_m=\left[C_m(R-R',2||\nabla u_1||_R)^{-1} \ln 2\right]^{{1 \over
m}}$.
\medskip
With this choice for $c_m$, we remark
that: $$h\leq c_mn^{-{1 \over m}} \Rightarrow P_k(h)\leq 2\;,\quad
\forall k\in\{1,\dots,n\}\;.$$
Owing to proposition 5.3, we can choose $\varepsilon _0>0$ such that
the following property $(H_k)$ is valid for
$k\in\{1,\dots,n\}$:
\medskip
\noindent
{\it $(H_k)$: If $nh^m \leq \varepsilon _0$, then:
$$||\partial_{x_j} u_k(.,h)||_{R',R,k} \leq P_k(h)||\partial_{x_j}
u_1||_R\leq 2||\partial_{x_j} u_1||_R\;,\leqno (5.26)$$and likewise
for the derivatives in $\xi_j$, $x'_j$ and $\xi'_j$, particularly:
$$||\nabla u_k(.,h)||_{R',R,k} \leq P_k(h)||\nabla u_1||_R \leq2
||\nabla u_1||_R\;.\leqno (5.27)$$ Moreover, the real part of
$u_k(.,h)$ is lower bounded on $\Omega _{k,n}(R',R)$.}
\medskip
\noindent
{\it Second step:} We consider
another sequence of functions $\psi_k(.,h)$ on
$\Omega_{k,n}(\lambda,\mu_1,\mu_2)$ with $k \in \{1,\ldots,n
+1\}$, defined by: $$\psi _1 (X,X')=f_1(X)f_2(X') \qquad (X,X')
\in
\Omega_{2n}(\mu_1)\times\Omega_{2n}(\mu_2)=\Omega_{1,n}(\lambda,\mu_1,\mu_2)
\;,$$ and by the following induction relation if $1\leq k\leq n$:
$$\psi_{k+1}(.,h)={I_k(e^{-u_k}\psi_k)(.,h) \over
I_k(e^{-u_k})(.,h)} \;.$$ Using inequalities (5.26) and (5.27) and
proposition 5.8, we verify by induction the following property
$(I_k)$, for $k\in\{1,\dots,n+1\}$, where $\varepsilon_1$ is the
constant of proposition 5.8:
\medskip
\noindent
{\it $(I_k):$ If:
$$\left\{ \matrix {
h ||\nabla _{x'_k \! ,\xi'_k} u_1 ||_{R} \leq {\mu_1-\lambda \over
4} \cr h ||\nabla _{x_k ,\xi_k} u_1 ||_{R} \leq {\mu_2-\lambda
\over 4}\cr
} \right.
\quad and \quad nh^m \leq \min(\varepsilon _0(m,R-R',||\nabla
u_1||_R), \varepsilon_1^m (m,R-R',2||\nabla u_1||_R)) \ , $$ then:
$$||\psi_{k}(.,h)||_{\lambda,\mu_1,\mu_2,k} \leq \left(1+ {h^m
K_m(R-R',2||\nabla u_1||_{R})\over (\mu-\lambda)^{2m}}
\right)^{k-1} ||\psi _1||_{\lambda,\mu _1,\mu_2,1} \; .$$}
The definition of
$u_1$ implies that: $$||\nabla _{x'_k \! ,\xi'_k} u_1 ||_{R} \leq
||\nabla p_2||_R \; , \ \ ||\nabla _{x_k \! ,\xi_k} u_1 ||_{R}
\leq ||\nabla p_1||_R \; .$$ Hence, for $k=n+1$, setting:
$$\varepsilon _m\ =\ \varepsilon (m,R-R',||\nabla u_1||_R)\ =\
\min(\varepsilon _0(m,R-R',||\nabla u_1||_R),\varepsilon_1^m
(m,R-R',2||\nabla u_1||_R) )\;,$$ we obtain:
$$||\psi_{n+1}(.,h)||_{\lambda,\mu_1,\mu_2,n+1}\leq \left(1+ {h^m
K_m(R-R',2||\nabla u_1||_{R})\over (\mu-\lambda)^{2m}}\right)^n
||\psi_1||_{\lambda,\mu _1,\mu_2,1} \; ,$$ if $h$ satisfies (5.3).
By the definition of the sequence $(\psi _k)$, we verify easily
that: $$\psi_h(f_1,f_2)(X)=\psi_{n+1}(X,X,h) \; ,$$ and with the
above-mentioned conditions on $h$, we obtain finally (5.2), with
$K_m$ depending on $R-R'$, $m$ and $\Vert \nabla p_j \Vert _R$,
($j=1,2)$.
\bigskip
\noindent
{\bf 6. Formal construction of the symbol in theorem 2.2.}
\medskip
The formal aspect of theorem 2.2 is very similar to the Th\'eor\`eme 2.2
of [1]. The only difference is that the standard calculus is now
replaced by the Weyl one.
For each functions
$f$ and $g$, $ C^{ \infty } $ in $ {\bf R} ^{2n}$, let us set
$(f\otimes g) (y, \eta, z, \zeta) = f(y, \eta)g(z, \zeta)$ and
$$a_k(f, g)(x, \xi)\ =\ {i^k\over 2^k k!} \Bigg [ \sigma
\big ( (D_y, D_{\eta }), \ (D_z, D_{\zeta })\big ) ^k
(f\otimes g) \Bigg ] (x, \xi , x, \xi ) \ . \leqno (6.1)$$
We know that, if $f$ and $g$ are bounded with all their derivatives
$$ (f\sharp _h g )(X)\ \sim \sum _{k\geq 0} a_k(f, g) (X)h^k \ .
\leqno (6.2)$$
\bigskip
\noindent
{\bf Definition 6.1.} {\it For each integers $j$ and $k\geq 1$,
and for each functions $p$ and $\Phi _0, \ldots , \Phi _{k-1}$ in
$C^{\infty }({\bf R}^{2n})$, we denote by $c_j^{(k)}(p, \Phi _0,
\ldots ; \Phi _{k-1})$ the coefficient of $h^{j}$ in the
polynomial $e^{\Phi } a_k (p,e^{-\Phi })$, where $\Phi = \Phi _0
+ \ldots + h^{k-1} \Phi _{k-1}$. If $p = (p_n)$ is in $S(b)$
$(b>0)$, we can define a sequence of functions $E_n^{(j)}(x, \xi ,
t)$ by $E_n^{(0)}(x,\xi, t) = tp_n(x, \xi)$ and, if $k\geq 1$, by
$${\partial E_n^{(k)} \over \partial t}\ =\ \sum _{j=0}^{k-1}
c_{j}^{(k-j)} (p, E_n^{(0)}, \ldots , E_n^{(k-1}) \hskip 1cm
E_n^{(k)}(x, \xi , 0)\ =\ 0 \ . \leqno (6.3)$$ }
\bigskip
For each $m\geq 1$, let us set
$$\Phi _n^{(m)}(x, \xi , t, h) = \sum _{j=0}^{m-1}E_n^{(j)}(x, \xi ,
t)h^j\ .
\leqno (6.4)$$
\bigskip
\noindent
{\bf Proposition 6.2.} {\it With the previous notations, for each
$a\in ]0, b[$, the sequence of functions $E_n^{(j)}(x, \xi, t)$ is in
$S(a)$ and there exists a function $R_n^{(m)}(x, \xi, t, h)$
in $S(a)$ such that, we have
$${ \partial \over \partial t} e^{-\Phi _n ^{(m)}(., h, t)} \ +
\sum _{k=0}^{m-1} a_k \left ( p_n, e^{-\Phi _n^{(m)}(., h, t)} \right ) h^k
\ =\ h^m e^{-\Phi _n^{(m)}(., h, t)} R_n^{(m)}(., h, t)\ .
\leqno (6.5)$$
Moreover, $E_n^{(0)}(x, \xi , t)= tp_n(x, \xi)$ and, for $j\geq 1$ and
$m\geq 1$, and we have
$$ \Vert E_n^{(j)}(. , t) \Vert _a \leq nC_j \quad {\rm and} \quad
\Vert R_n^{(m)}(. , t, h) \Vert _a \leq nC_m,\leqno (6.6)$$
where $C_j$ is independent on $n$, and the function defined by (6.4)
satisfies
$$||\nabla \Phi _n^{(m)}(., t ,h)||_R\leq Mt \;,\quad
\forall t \in[0,1] \;,
\quad \forall h\in]0,1]\;.\leqno (6.7)$$
}
\bigskip
\noindent {\it Proof.} Here, and many times in the sequel, we
omit the subscript $n$. By the Lemma 9.1 of [1], if $00$, depending
only on $k$, and on the norms of $\Vert \nabla p \Vert _b$ and
$\Vert \nabla \Phi _j \Vert _b$ $(0\leq j \leq k-1)$
such that the coefficients $c_j^{(k)}$ of this polynomial satisfy
$$ \Vert \nabla c_j^{(k)}(p, \Phi _0, \ldots ; \Phi _{k-1}) \Vert
_a \ \leq C_m \ \ \ \ and, \ \ if \ \ \ \ k\geq 1, \ \ \ \ \
\Vert c_j^{(k)}(p, \Phi _0, \ldots ; \Phi _{k-1}) \Vert _a
\ \leq \ n C_m \ . \leqno (6.8)$$ Therefore, the left hand side of
(6.5), multiplied by $e^{\Phi ^{(m)}(., h, t)}$, is a polynomial
in $h$, of degree depending only on $m$, with coefficients
satisfying inequalities like (6.8). The induction relation
defining the $E^{(j)}$ shows that the coefficient of $h^j$ in this
polynomial vanishes if $j\leq m-1$. Therefore, we can write (6.5),
(6.6) and (6.7).
\bigskip
We can write the first coefficients of the formal series :
$E^{(0)}(x , \xi , t)= t p(x, \xi)$, $E^{(1)}= 0$ and $$E^{(2)} \
= \ {t^2\over 8} \sum _{jk}\Bigg [ { \partial ^2p\over \partial
x_j
\partial x_k} \ {\partial ^2p \over \partial \xi _j \partial \xi
_k}\ - \ {\partial ^2p \over \partial x_j \partial \xi _k}\
{\partial ^2p\over \partial \xi _j \partial x_k}\Bigg ] + \ldots
\leqno (6.9)$$ $$\ldots \ +\ {t^3 \over 24}\ \sum _{jk} \Bigg [ {
\partial ^2p\over \partial x_j \partial x_k}\ {\partial p\over
\partial \xi _j}\ {\partial p\over \partial \xi _k}\ +\
{\partial ^2p\over \partial \xi _j \partial \xi _k}\ {\partial
p\over
\partial x_j}\ {\partial p\over \partial x_k}\
-\ 2\ {\partial ^2p\over \partial x_j \partial \xi _k }\ {\partial
p\over \partial \xi _j}\ {\partial p\over \partial x_k} \Bigg ]
\ . $$
\bigskip
Now we have a formal approximation. Since we hope that $e^{-\Phi
_n^{(m)}(., h, t)}$ will be an approximation of the symbol of
$e^{-tOp_h(p_n)}$, it will be useful to give a majorization of the
function $r_n^{(m)}(., t, h)$ defined by $${ \partial \over
\partial t} e^{-\Phi _n^{(m)}(., t, h)} \ = -p_n \ \sharp _h \
e^{-\Phi _n ^{(m)}(., t, h)} \ + \ e^{-\Phi _n ^{(m)}(., t,
h)}r_n^{(m)}(., t, h)\ . \leqno (6.10)$$
\bigskip
The following proposition of Sj\"ostrand will be used in the proof
of the bounds of $r_n^{(m)}$ in (6.10), and
several times later. Let us recall it now.
\bigskip
\noindent
{\bf Proposition 6.3. (Proposition 1.1 of [15]).} {\it Let $a$ and
$b$ such that $00$ (independent on
$n$) with the following property. Let $f(y, \zeta )$ be an
holomorphic function in $\Omega _{2n}(b)$ such that $ | \nabla f |
_{\infty }$ is bounded in $\Omega _{2n}(b)$. Then we have $$\Vert
\sum _{j=1}^n { \partial^2f \over \partial y_j \partial \zeta _j}
\Vert _a \leq Cn \Vert \nabla f \Vert _b \hskip 1cm \Vert
\nabla \left ( \sum _{j=1}^n { \partial^2f \over \partial y_j
\partial \zeta _j} \right ) \Vert _a \ \leq \ C \Vert \nabla f
\Vert _b \ . \leqno (6.11)$$ Moreover, if $u= (u_1, \ldots u_n)\in
{\bf C}^n$, and $\rho >0$ such that $a + \rho | u | _{\infty }
\leq b$, we have $$\Vert \sum _{j=1}^n u_j {\partial f\over
\partial y_j} \Vert _a \ \leq \ {1\over \rho }\ \Vert f\Vert
_{a + \rho | u | _{\infty }} \ . \leqno (6.12)$$ }
For the majorization of $r_n$ defined in (6.10), we shall need the
following Proposition, which gives a bound of the error term in
the composition of symbols, in our unusual situation.
\bigskip
\noindent
{\bf Proposition 6.4.} {\it If $00$, depending only on $a$,
$b$, $m$, and on $\Vert \nabla \Phi \Vert _b$ and $\Vert \nabla p
\Vert _b$, but not on $n$ and $h$ such that, with the previous
relations $$\Vert T_n^{(m)}(., p_n, \Phi _n, h)\Vert _a \ \leq \
nC_m \ . \leqno (6.14) $$ }
\bigskip
\noindent
{\it Proof. } For each function $\Phi \in C^{ \infty } ( {\bf
R} ^{2n})$ let us set, for each $(x, \xi ) \in {\bf R} ^{2n}$
$$H_{\Phi }(x, \xi )\ =\ \left ( - { \partial \Phi \over \partial
\xi},\ {\partial \Phi \over \partial x} \right ) \ , $$ and, for each
$X$ and $Y$ in $ {\bf R} ^{2n}$
$$Q _{\Phi } (X)\ =\ \int _0^1
H_{\Phi }(X+tY)\ dt \ . $$ Thus we have
$$\Phi (X+Y) \ - \ \Phi (X)\ =\
\sigma ( Q_{\Phi }(X, Y),\ Y) \ . $$
With these notations, we can
write, for each $\Phi \in S(b)$
$$e^{\Phi } \Big [ p\ \sharp _h
e^{- \Phi } \Big ] (X)\ =\ (\pi h)^{-2n} \int _{ {\bf R} ^{4n}}
e^{-{2i\over h} \sigma ( Y - {ih\over 2} Q_{\Phi } (X, Z) , Z )}
p(X+Y) \ dYdZ\leqno (6.15)$$
$$\ldots \ =\ (\pi h)^{-2n} \int _{
{\bf R} ^{4n}} e^{-{2i\over h} \sigma ( Y , Z )} F_{h}(X, Y, Z)\
dYdZ \ , $$ where
$$F_h(X, Y, Z)\ =\ p(X+Y + {ih\over 2} Q_{\Phi } (X,
Z)) \ . \leqno (6.16)$$
Of course, this is not a change of variables,
but a change of contour. Let $a$ , $a_1$ and $a_2$ such that
$a0$,
independent on $n$, such that, if $X$ and $X+Z$ are in $\Omega
_{2n }(a_2)$, then $ | Q_{\Phi } (X, Z) | _{\infty } \leq C$.
Therefore, the change of contour is allowed and, by Proposition
6.3, there is some other constant $C>0$, independent on $h$, such
that, for $h$ small enough,
$$ | \nabla F_{h}(X, Y, Z) | _{\infty
}\ \leq \ C\leqno (6.17)$$
if $X$, $X+Y$ and $X+Z$ are in $\Omega
_{2n}(a_2)$. Let $L$ be the following operator in $ {\bf R} ^{4n}$
$$L\ =\ {1\over 2i}\ \sigma (\partial _Y, \ \partial _Z).$$
For the integral defined in (6.15), we have, by Taylor formula
$$e^{\Phi _n} \Big [ p_n\ \sharp _h e^{-\Phi _n } \Big ] (X)\ =\
\sum _{k=0}^{m-1} {1\over k!} h^k (L^k F_h)(X, 0, 0)\ + h^m
A_n^{(m)}(X, h) \ , \leqno (6.18) $$
where
$$A_n^{(m)}(X, h) =\ \int
_0^1 {(1-\theta )^{m-1}\over (m-1)!} (\pi h)^{-2n} \int _{ {\bf R}
^{4n}} e^{-{2i\over h} \sigma ( Y , Z )} (L^mF_h)(X,\sqrt {
\theta }Y, \sqrt {\theta } Z) dYdZd\theta .$$
The coefficients in the asymptotic expansion (6.18) depend on $h$,
but we can write,
for each $k\leq m-1$
$$(L^k F_h)(X, 0, 0)\ =\ \sum _{j=0}^{m-k-1}
h^j g_{(j,k)}(X)\ +\ h^{m-k} R^{(m,k)}(X, h) \ , \leqno (6.19)$$
where, by Proposition 6.3, $g_{(j,k)}$ are in $S(a)$ and $R^{(m,k)}$ satisfy
$$\Vert R_n^{(m, k)} \Vert _a \leq C_m \ . \leqno (6.20) $$
Thus, we obtain an asymptotic
expansion of the left hand side of (6.18). We have also the expansion
(6.2), which is very classical, but, in our situation, does not give
a good estimation of the error term. By an identification of these
two expansions, we obtain
$$\sum _{k=0}^{m-1} {1\over k!} h^k (L^k F_h)(X, 0, 0) \ -\
e^{\Phi}\
\sum _{k=0}^{m-1} a_k \left ( p, e^{-\Phi(.)} \right ) h^k
\ = \
h^m \sum _{k=0}^{m-1}{1\over k!} R_{n}^{(m, k)}(X, h)\ . \leqno (6.21) $$
It remains to give a bound of $A_n^{(m)}(X, h)$.
By (6.17) and Proposition 6.3, there exists $C>0$, independent on
$n$, such that the function $F_h$ defined in (6.16) satisfies $$
| \sigma (D_Y, \ D_Z)^mF_h(X, Y, Z) | \leq Cn \hskip 1cm |\nabla
\Big ( \sigma (D_Y, \ D_Z)^m F_h(X, Y, Z) \Big ) |_{\infty } \
\leq \ C$$ if $X$, $X+Y$ and $X+Z$ are in $\Omega _{2n}(a_1)$.
Therefore, it follows from Theorem 3.1 that $$\Vert A_n^{(m)}\Vert
_a \ \leq \ Cn \left ( 1 + { \sqrt {h} \over a_1-a }e^{-{(a_1-a)^2
\over h}} \right ) ^{4n}\ \leq \ Cne^4 \leqno (6.22)$$ if $00$ such that
$\Phi ^{(m)}(t,h)$ defined in (6.4) is in $S(R)$,
and $R'\in ]0, R[$. Then there exist $\varepsilon_m$ and $C_m$,
depending only on $R$, $R'$, $m$ and $\sup _{n} \Vert \nabla
\Phi _n^{(m)} \Vert _R,$ but not of $n$, such that, if $t_0>0$
and $t_1>0$ and $t_0+t_1 \leq1$,
$$ \left\| {e^{-\Phi
^{(m)}(t_1,h)} \sharp_h e^{-\Phi ^{(m)}(t_0,h)} \over e^{-\Phi
^{(m)}(t_1+t_0,h)}} \right\|_{R'} \leq 1+C_m nh^m \; ,\leqno
(7.1)$$ if $nh^m\leq \varepsilon _m$. }
\bigskip
\noindent
{\it Proof.}
We know by Theorem 4.1 that, if $t_0>0$ and $t>0$, there exists a function
$\Psi^{(m)}(t,t_0,h)\in S(R')$ and constants $\varepsilon _m>0$
and $C_m>0$ such that:
$$e^{-\Psi^{(m)}(t,t_0,h)}=e^{-\Phi^{(m)}(t,h)}
\sharp_h e^{-\Phi^{(m)}(t_0,h)} \; ,\leqno (7.2) $$
if $nh^m\leq\varepsilon _m$, and that $\Psi^{(m)}(t,t_0,h)$
has an asymptotic expansion in powers of $h$:
$$\Psi^{(m)}(.,t,t_0,h) = \sum_{j=0}^{m-1}\psi^{(j)}(.,t,t_0)h^j \ +\
g^{(m)}(., t, t_0, h) \; , \leqno (7.3)$$
where $g^{(m)}(., t, t_0, h)$ satisfies, if $nh^m\leq \varepsilon _m$,
and $00$ such that the
function $\Phi^{(m)}(t,h)$ of Proposition 6.1 is in $S(R)$, and
$M>0$ a constant such that (6.7) is satisfied. Let $\alpha $ such that
$0<\alpha 0$ et $K_{m}>0$, both
depending only on $m$, $R$, $\alpha $ and $M$, but not on $n$,
with the following
properties.
If $b_1$, $b_2$ satisfy $0<\alpha < \min (b_1, b_2) \leq \max (b_1,
b_2) < R$, if $f_1$ and $f_2$ are holomorphic functions,
bounded respectively on $\Omega _{2n}(b_1)$ and $\Omega
_{2n}(b_2)$ and if $t_1>0$ and $t_2>0$ satisfy $t_1+t_2\leq1$,
if we set $b= \min (b_1, b_2) $, then we have $$\left\|
{(e^{-\Phi^{(m)}(t_1,h)}f_1) \sharp _h (e^{-\Phi^{(m)}(t_2,h)}f_2)
\over e^{-\Phi^{(m)}(t_1+t_2,h)}} \right\| _{\alpha } \leq
2||f_1||_{b_1}||f_2||_{b_2} \left( 1+{K_{m} h^{m+1} \over
(b-\alpha)^{2m+2}} \right)^n \; ,\leqno (7.6)$$ if $nh^m\leq \varepsilon
_m$ and $h$ satisfies the
following conditions: $$ \left\{ \matrix {
h ||\nabla \Phi^{(m)}(t_2,h)||_R \leq {b_1-a \over 4} \cr
h ||\nabla \Phi^{(m)}(t_1,h)||_R \leq {b_2-a \over 4} \cr
} \right.
.\leqno (7.7) $$
}
\bigskip
We may assume that $\varepsilon _m$ (resp. $K_m$) is a
decreasing (resp. increasing ) function of $\alpha $.
\bigskip
\noindent
{\it Proof.}
By Theorem 5.1, with $m$ replaced by $m+1$, there exist
$\varepsilon '_m>0$ and $K'_{m}$ depending on $m$,
$R$, $\alpha $ and $M$ such that:
$$\left\| {(e^{-\Phi^{(m)}(t_1,h)}f_1) \sharp _h (e^{-\Phi^{(m)}(t_2,h)}f_2)
\over
e^{-\Phi^{(m)}(t_1,h)} \sharp _h e^{-\Phi^{(m)}(t_2,h)} } \right\|
_{\alpha }
\leq ||f_1||_{b_1}||f_2||_{b_2} \left( 1+{K'_{m} h^{m+1} \over
(b-\alpha )^{2m+2}}
\right)^n \; ,\leqno (7.8)$$
if $nh^{m+1}\leq \varepsilon '_m$ and if the conditions of (7.7) are
satisfied.
By Proposition 7.1, there exist
$\varepsilon''_m>0$ and $K''_m>0$, such that
$$ \left\| {e^{-\Phi
^{(m)}(t_1,h)} \sharp_h e^{-\Phi ^{(m)}(t_2,h)} \over e^{-\Phi
^{(m)}(t_1+t_2,h)}} \right\|_{\alpha } \leq 1+K''_m nh^m \; \leqno (7.9)
$$
if $nh^m\leq\varepsilon ''_m$.
If $nh^m$ is small enough, (7.6) follows from (7.8) and (7.9).
\bigskip
Now, we shall compose several factors instead of two like in the
previous proposition, but the number of factors will be of the same
order than the dimension.
\bigskip
\noindent
{\bf Proposition 7.3.} {\it
Let $m\geq 2$ and $R>0$ such that
the function $\Phi^{(m)}(.,t,h)$ defined in (6.4) is in $S(R)$, and
$M>0$ a constant such that (6.7) is valid.
Let $A\geq1 $ be an integer, and $a$, $b$ such that $00$,
depending only on $m$, $M$, $A$, $a$, $b$ and $R$,
with the following properties.
If $f_1,\dots,f_k$ ( $1\leq k\leq An$), are bounded
holomorphic
functions in $\Omega _{2n}(b)$, if $\theta _i\geq 0$
( $1\leq i\leq k$), if $h\in]0, 1]$, and if the following
condition is satisfied :
$$\sup _{\alpha \leq k} ||f_\alpha||_b \leq {1\over 4e^2}
\hskip 1cm nh^{m}\leq \varepsilon _m
\hskip 1cm \theta _1+\dots+\theta _k\leq1, \leqno (7.10)$$
then we have
$$\left\| {(e^{-\Phi^{(m)}(\theta _1,h)}f_1)\sharp
_h\dots\sharp_h
(e^{-\Phi^{(m)}(\theta _k,h)}f_k)
\over
e^{-\Phi^{(m)}(\theta _1+\dots+\theta_k,h)}}\right\|_a \leq
\left [ \sup _{\alpha \leq k} ||f_\alpha||_b \right ]^{{ k+1 \over 2}}
.\leqno (7.11) $$
}
\bigskip
\noindent
{\it Proof.}
For $k \in \{1,\dots,An\}$, we set:
$$a_ k=b-(b-a){\ln k \over \ln (An)} \;.$$
Therefore $a=a_{An}<\dots0$, depending only on $m$,
$M$, $a$, $b$, $R$ and $A$
such that, if $nh^{m}\leq \varepsilon''_m$, the conditions (7.13) are
satisfied and therefore (7.12) is valid.
By (7.14), there exists $\varepsilon''' _m>0$, depending only on the
same parameters such that, if $nh^{m}\leq \varepsilon'''_m$, we have
$$ \left(1+{K_m h^{m+1}\over (a_i-a_k)^{2m+2}}\right)^n
\leq \left (1+{1\over n} \right )^n\leq e \;. \leqno (7.15)$$
With our induction hypothesis, we have:
$$||F||_{a_i}\leq \lambda ^{{i+1 \over 2 }}
\hskip 1cm
||G||_{a_{k-i}}\leq \lambda ^{{k-i+1 \over 2 }}
\hskip 1cm if \ \ \ \ \
\lambda = \sup _{\alpha \leq k} ||f_{\alpha }||_b \leq {1\over 4e^2}
\ \ \ \ nh^{m} \leq \varepsilon _m .
\leqno (7.16) $$
If we choose now $\varepsilon _m=
\min(\varepsilon'' _m, \varepsilon''' _m)$, the inequalities (7.12), (7.15)
and (7.16)
imply:
$$||H||_{a_k}\leq 2 \lambda ^{{i+1 \over 2 }}\lambda
^{{k-i+1\over 2 }} e
\leq \lambda ^{{k+1 \over 2 }} \hskip 1cm
{\rm if } \ \ \ \lambda \leq {1\over 4e^2}\ \ \ \ \
{\rm and }\ \ \ \ nh^m \leq \varepsilon _m \ .$$
\medskip
Therefore, with this choice of $\varepsilon _m$, the property
$(X_k)$
is proved for all $k\leq An$, and the proposition itself follows
since $a_k\geq a$.
\bigskip
\noindent
{\bf 8. End of the proof of theorem 2.2.}
\bigskip
Theorem 2.2 will follow from the next Proposition (where
$f\sharp _h g$ is the composition law defined in (3.1)).
\bigskip
\bigskip
\noindent
{\bf Proposition 8.1.} {\it Let $p=(p_n)_{n\geq1}$ an element of
$S(b)$ ($b>0$), $a\in ]0, b[$, and $m$ an integer $\geq 2$. Then
there exists $\varepsilon_m >0$ and $C_m>0$, depending only on
the family $p$ and on $a$, $b$ and $m$, and a
sequence of holomorphic bounded functions $u=(u_n)(x,\xi,t,h)$ on
$\Omega_{2n}(a)$ depending in a $C^\infty$ way on $t\in[0,1]$ such
that, if $nh^m\leq\varepsilon_m$:
$$ \left \{ \matrix {
\displaystyle {\partial u \over \partial t} (.,t,h)=-p\,\sharp_h
u(.,t,h) \;, \cr u(.,0,h)=1\;.\cr } \right . \leqno (8.1) $$
Moreover $u$ can be written:
$$u(.,t,h)=e^{-\Phi^{(m)}(.,t,h)+R^{(m)}(.,t,h)} \;, \quad if
\quad nh^m\leq \varepsilon _m \;, \leqno (8.2)$$ where
$\Phi^{(m)}(t,h)\in S(a)$ is the family of polynomials in
$h$ defined in (6.4), and $R^{(m)}(.,t,h)$ a family of holomorphic
bounded functions on $\Omega _{2n}(a)$ such that:
$$||R^{(m)}(.,t,h)||_a \leq C_mnh^m \;,\quad \forall t\in[0,1]
\quad {\rm if } \ \ \ nh^m\leq \varepsilon _m \;.
\leqno (8.3)$$ }
\bigskip
\noindent
{\bf Proof.}
If $\Phi ^{(m)}(.,t,h)$ is the function (polynomial in $h$)
defined in (6.4),
we shall find a solution of problem (8.1) of the following form:
$$u(.,t,h)=e^{-\Phi^{(m)}(t,h)}+\int_0^te^{-\Phi^{(m)}(t-s,h)}
\sharp _h[e^{-\Phi^{(m)}(s,h)}v_h(.,s)]ds \; , \leqno (8.4)$$
where $v_h(., t)$ will be a function in a suitable space. If we
replace $u$ by the previous expression in (8.1), using (6.10), we
see that $u$ is a solution if and only if $v_h$ satisfies:
$$v_h-T_h(r_h)v_h=r_h \; ,\leqno (8.5)$$ where, for each functions
$u(., t)$ and $v(., t)$ in suitable spaces, we set :
$$\left [T_h(u)v)\right ](.,t)=e^{\Phi^{(m)}(t,h)}\int_0^t[e^{-\Phi^{(m)}(t-s,h)}u(t-s)]
\sharp _h[e^{-\Phi^{(m)}(s,h)}v(s)]ds \;,\leqno (8.6)$$
and where
$r_h(.,t)$ is defined in (6.10). It remains to find a space $E$ in
which some power of $T_h(r_h)$ will be a contractive map. For each
$a_1$ and $a_2$ such that $00$. Let $u(., t)$
be a function in
$E(R', R')$ ($R'$), $[a_1, a_2]$ be a compact interval in
$]0, R'[$ and $m\geq 2$ be an integer. Then :
\smallskip
\noindent i)There exists $\varepsilon _m>0$, depending only on
$m$, $R$, $R'$, $a_1$, $a_2$ and $M$,
such that, if $$ nh^m \leq \varepsilon _m
\hskip 1cm \Vert u \Vert _{E(R',R')} \leq {1\over 4e^2} ,\leqno
(8.8)$$
the map $\left [ T_h(u)\right ] ^{(2m+2)n+1}$ is contractive in
$E(a_1, a_2)$, with a norm $\leq {1\over 2}$.
\smallskip
\noindent ii) If $00$, depending only on the same parameters
and on $a_3$ and $a_4$, such that, if $u\in E(R', R')$, $v\in E(a_3,
a_4)$, if (8.8) is satisfied and if $1\leq k\leq (2m+2)n+1$,
$$\Vert \left [ T_h(u)\right ] ^{k}v\Vert _{E(a_1, a_2)} \leq 2e\
\Vert u \Vert _{E(R', R')}^{(k+1)/2} \ \Vert v \Vert _{E(a_3,
a_4)}\ . \leqno (8.9)$$ }
\bigskip
\noindent
{\it Proof of the Lemma. First step (points i and ii).}
If $v\in E(a_1, a_2)$, we have, for all integer $k$ and $t>0$
$$\left [T_h(u)^{k}v\right ](.,t)=
\int_{\Delta ^{k}(t)} G_k(t_1, \ldots , t_k , t;
u, v)dt_1\ldots dt_k \ , \leqno (8.10)$$
where $\Delta^{k}(t)=
\{(t_1,\dots,t_{k}), \, 0\leq t_1\leq \dots \leq t_{k}\leq t \}$
and, for each $(t_1, \ldots , t_k)\in \Delta ^k (t)$, and each
function $u\in E(R', R')$ and $v\in E(a_1, a_2)$, we set
$$G_k(t_1, \ldots , t_k
, t; u, v)\ =\
e^{\Phi^{(m)}(t,h)}[e^{-\Phi^{(m)}(t-t_{k},h)}
u(t-t_{k})]\sharp _h \dots \ \ \ \ \ \ \ \ \ \ \ \leqno
(8.11)$$
$$\ \ \ \ \ \ \ \ \ \ldots \sharp
_h[e^{-\Phi^{(m)}(t_2-t_{1},h)}u(t_2-t_{1})] \sharp
_h[e^{-\Phi^{(m)}(t_{1},h)}v(t_1)]\ .$$
Let $\mu $ in $]0, R[$. By using Proposition 7.3 with $f_1 =
u(t-t_k)$, $\ldots$, $f_k = u(t_2-t_1)$, $A = 2m +3$, we know
that there exists $\varepsilon' _m >0$, depending only on $R$, $R'$, $M$
(the constant such that the function $\Phi^{(m)}_n(t,h)$ defined
in (6.4) satisfies (6.7) in $\Omega _{2n}(R)$), $m$, and $\mu $,
such that, if $$\Vert u\Vert _{E(R',R')} \leq ( 4e^2)^{-1} \hskip
1cm nh^m \leq \varepsilon '_m \hskip 1cm k\leq (2m+3)n \
,\leqno (8.12)$$ there exists a bounded holomorphic function $F_h
(t_1, \ldots , t_k, u)$ in $\Omega _{2n}(\mu)$ satisfying
$$[e^{-\Phi^{(m)}(t-t_{k},h)}u(t-t_{k})]\sharp _h \dots \sharp
_h[e^{-\Phi^{(m)}(t_2-t_{1},h)}u(t_2-t_{1})]=e^{-\Phi^{(m)}(t-t_1,h)}
F_h(t_1,\dots,t_{k},t, u) \; ,$$ $$||F_h(t_1,\dots,t_{k},t, u)
||_{\mu}\leq \Vert u\Vert _{E(R',R')} ^{{k+1\over 2}} \; .$$
If $u\in E(R', R')$
and $v\in E(\lambda , \mu )$ ($0< \lambda < \mu < R'$), if $0\leq
t_1 ... \leq t_k \leq t \leq 1$, we have $$G_k(t_1, ... , t_k, t,
u, v) \ = \ e^{ \Phi ^{(m)} (t, h)} \left [ \left ( e^{- \Phi
^{(m)} (t-t_1, h)}F_h (t_1, \ldots , t_k,t, u)\right ) \ \sharp _h
\left ( e^{- \Phi ^{(m)} (t_1, h)}v(t_1) \right ) \right ] \ , $$
where $F_h$ is the function defined above. Let $\alpha \in
]0, \lambda [$. By Proposition 7.2, there exist $\varepsilon'' _m>0$
and $K''_m>0$, depending only on $M$, $\alpha $, $\lambda $, $R$ and $m$,
such that, if $\alpha \leq \sigma < A(t_1, \lambda , \mu)$,
$$\Vert G_k(t_1, ... , t_k, t, u, v) \Vert _{\sigma } \ \leq \ 2
\Vert F_h (t_1, \ldots , t_k, t, u)\Vert _{\mu } \ \Vert v(t_1)
\Vert _{A(t_1,\lambda, \mu )} \left ( 1 + {K''_m h^{m+1} \over
(A(t_1, \lambda , \mu) - \sigma )^{2m+2}}\right )^n$$
if
$$hM (t- t_1) \leq {A(t_1, \lambda , \mu) - \sigma \over 4} \hskip
1cm hMt_1 \leq {\mu - \sigma \over 4} \hskip 1cm nh^m \leq
\varepsilon '' _m \ , \leqno (8.13) $$
and therefore, by the definition of $E(\lambda , \mu)$, if (8.12)
and (8.13) are satisfied $$\Vert G_k(t_1, ... , t_k, t, u, v)
\Vert _{\sigma } \ \leq \ 2 \Vert u\Vert _{E(R',R')} ^{{k+1\over 2}}
\ \Vert v \Vert _{E(\lambda , \mu)} \left ( 1 + {K_m h^{m+1} \over
(A(t_1, \lambda , \mu ) - \sigma )^{2m+2}}\right )^n \ . $$
\bigskip
\noindent {\it Proof of the Lemma. Second step. Point i)} If $v\in
E(a_1, a_2)$, we apply the first step with $\lambda =
a_1$, $\mu = a_2$, $k= (2m+2)n+1$, $\sigma = A(t, a_1, a_2)$. By
the definition (8.7) of $A(t, \lambda , \mu)$, the two first
conditions of (8.13) are satisfied if $hM\leq {a_2 - a_1 \over
4}$. Therefore, we can find $\varepsilon _m $, as in the
statement of i) such that, if $nh^m \leq \varepsilon _m$ $$\Vert
G_{(2m+2)n+1}(t_1, ... , t_{(2m+2)n+1}, t, u, v) \Vert _{A(t, a_1,
a_2) } \ \leq \ 2 \Vert u\Vert _{E(R',R')} ^{(m+1)n+1} \ \Vert v
\Vert _{E(a_1 , a_2)} \left ( {2 \over (t-t_1
)^{2m+2}}\right )^n\ . $$
We remark that
$$\int _{ \{ (t_2, \ldots ,
t_{(2m+2)n+1}),\ t_1 0$).
If $A_mnh^m \leq {1 \over
4e^2}$, the second condition of (8.8) is satisfied. Therefore, if
$nh^m \leq \varepsilon _m$ (for some $\varepsilon _m>0$ independent of
the dimension), the map $\left [ T_h(r_h) \right ]^{(2m+2)n+1}$ is
contractive in $E(a_1, a_2)$, with a norm $\leq {1\over 2}$. Hence,
there exists $v_h\in E(a_1, a_2)$ such that
$$\left ( I \ -\ T_h(r_h)^{(2m+2)n+1} \right ) v_h \ =\
\sum _{k=0}^{(2m+2)n} T_h(r_h)^k r_h \ . $$
By Lemma 8.2 ii), applied with $a_3=a_4= {a_2 +R'\over 2}$, we have,
under the same type of conditions, if $k \leq (2m+2)n$
$$\Vert T_h(r_h) ^k r_h\Vert _{E(a_1, a_2)} \ \leq \ 2e \left [ \Vert
r_h\Vert _{E(R', R')} \right ]^{{k+1\over 2 }+1}\ . $$
Therefore the solution $v_h$ satisfies
$$\Vert v_h \Vert _{E(a_1, a_2)} \ \leq \ 2 \sum _{k=0}^{\infty } 2e
(A_m nh ^m)^{{k+1\over 2}+1}\ . \leqno (8.14)$$
In order to prove that $v_h$ satisfies (8.5), we set $w_h = \left (I -
T_h(r_h) \right ) v_h \ -\ r_h$. Let $[a'_1 , a'_2]$ be a compact
interval in $]a, a_1[$. By Lemma 8.2 ii), under the same type of
conditions, $w_h$ is in $E(a'_1, a'_2)$ and satisfies
$\left ( I \ -\ T_h(r_h)^{(2m+2)n+1} \right ) w_h =0$. Since
$T_h(r_h)^{(2m+2)n+1}$ is also contractive in $E(a'_1, a'_2)$ under
the same type of conditions, it follows that $w_h=0$, and therefore
that $v_h$ satisfies (8.5), and that $u_h$ defined by (8.4) is a
solution of the Cauchy problem (8.1). Equation (8.4) can be written
$$u(.\ , \ h)\ =\ e^{-\Phi ^{(m)}(., h)} \left [ 1 + T_h(1)
v_h\right ] \ .$$
By (8.14) and Lemma 8.2 ii), we have $\Vert T_h(1)v_h\Vert _{E(a'_1,
a'_2)} \leq A_mnh^m \leq {1\over 2}$ if $nh^m$ is small enough, and
therefore there exists an holomorphic function $R^{(m)} (.,t, h)$ in
$E(a'_1, a'_2)$ such that $1 + T_h(1) v_h = e^{R^{(m)}(., h)}$ and
$$\Vert R^{(m)}(., h) \Vert _{E(a'_1, a'_2)} \ \leq 2 \Vert
T_h(1)v_h \Vert _{E(a'_1, a'_2)} \ \leq \ 2 A_m nh^m\ . $$
Proposition is proved, and Theorem 2.2 is another equivalent formulation.
\bigskip
\noindent
{\bf 9. Application to thermodynamic limits.}
\bigskip
At each point $j$ of the lattice $ {\bf Z}$, we consider a particle
$A_j$ described, when there is no interaction, by an hamiltonian
$A(x, \xi )\in C^{ \infty } ( {\bf R} ^{2k})$,
where $k\geq 1$ is a fixed integer. The interaction between $A_j$ and
$A_{j+1}$ will be described by a function $B(x, \xi , y, \eta)\in
C^{ \infty } ( {\bf R} ^{4k})$, ($A$, $B$ and $k$ are independent on
$j$).
\bigskip
We assume that $A$ (resp. $B$) extends to an holomorphic
function
in $\Omega _{2k}(a)$, (resp. $\Omega _{4k}(a)$), defined as
in (1.4), that $B$ is bounded in $\Omega _{4k}(a)$,
and that $A$ has bounded derivatives in $\Omega _{2k}(a)$, and that
its real part is lower semi-bounded in this domain.
We assume also that $A$ and $B$ are real for real $(x, \xi)$, and that
$B(y, \eta, x, \xi)= B(x, \xi , y, \eta)$, and that
there exist $c>0$, $R>0$ and $\delta >0$ such that
$$(x,\xi ) \in \R ^{4k}, \quad
| (x, \xi ) |\geq R \;
\Rightarrow A(x, \xi) \ \geq c | (x, \xi ) |^{\delta }.$$
We shall write now the hamiltonian describing the system of particles
$A_j$ $( | j | \leq n)$ with interaction. If there is no
interaction between $A_n$ and $A_{-n}$, this hamiltonian can be
written, setting $X=(x, \xi)$ and denoting by
$(X^{(-n)}, \ldots , X^{(n)})$ the variable of $ {\bf R} ^{2k(2n+1)}$
(with $X^{(j)}= (x^{(j)}, \xi ^{(j)})\in {\bf R} ^{2k}$)
$$\widetilde p_n(X )\ =\ \sum _{j=-n}^n
A(X^{(j)}) \ + \sum _{j=-n}^{n-1} B(X^{(j)}, X^{(j+1)}) \ . \leqno (9.1)$$
If there is an interaction between $A_{n}$ and $A_{-n}$ (case some
formulas will be simpler), the hamiltonian becomes, setting
$X^{(n+1)}=X^{(-n)}$,
$$p_n(X )\ =\ \sum _{j=-n}^n
\Big ( A(X^{(j)}) \ + B(X^{(j)}, X^{(j+1)}) \Big ) \ . \leqno (9.2)$$
We see easily that the sequences
$(p_n)$ and $(\widetilde p_n)$ are in $S (a)$.
\bigskip
We denote by
$P_n(h)$ and $\widetilde P_n(h)$ the $h-$pseudodif\-ferential operator in
$L^2( {\bf R} ^{k(2n+1)})$, associated to the symbols
$p_n$ and $\widetilde p_n$ by the Weyl calculus (2.5).
Given the sequence of operators $P_n(h)$, we define the thermodynamic limit
as the limit, if it exists, of the sequence
$$\Lambda _n(t, h)\ :=\ {1\over (2n+1)} \ ln\ \left [ (2 \pi h
)^{(2n+1)k} Tr\ \left ( e^{-tP_n(h)} \right ) \right ] \leqno
(9.3)$$
(the same as in (1.7), but with $t/h$ replaced by $t$), has a limit
$\Lambda _{\infty }(t, h)$ when $n\rightarrow
+ \infty$. We define also a similar
sequence
$\widetilde \Lambda _n(t, h)$ for the operator $\widetilde P_n(h)$.
\bigskip
By theorem 2.2, we know that $e^{-tP_n(h)}$ is an
$h-$pseudodifferential operator associated, by the Weyl calculus,
to a symbol of the form $e^{-q_n(x, \xi , t, h)}$.
$$e^{-tP_n(h)} \ = \ Op_h \left ( e^{-q_n(. , t, h)}\right )\ ,
\leqno (9.4)$$
and we have also a symbol $\widetilde q_n(x, \xi , t, h)$
with similar properties for $\widetilde P_n(h)$.
\bigskip
The main results of this section are the theorems 9.1 (existence of
the thermodynamic limit) and 9.3 (asymptotic expansion in powers of
$h$).
\bigskip
\noindent
{\bf Theorem 9.1.} {\it With the previous notations, for each $h>0$
and $t>0$, the sequence $\Lambda _n(t, h)$ defined in (9.3) and its
analogue $\widetilde \Lambda _n(t, h)$ for the operator
$\widetilde P_n(h)$ have a common limit $\Lambda (t, h)$
when $n\rightarrow \infty $. There is a constant $C>0$ such that
$$ | \Lambda _n (t, h)\ - \ \Lambda (t, h) | \ +\
| \widetilde \Lambda _n(t, h) \ -\ \Lambda (t, h) | \
\leq \ {C \over n}
\hskip 1cm
\forall h\in ]0, 1], \ \ \ \ \ \
\forall t\in [0, 1]\ . \leqno (9.5)$$
}
\bigskip
The proof relies on the following Lemma.
\bigskip
\noindent
{\bf Lemma 9.2.} {\it There exists $C>0$ such that,
for each $m$ and
$n\geq 1$, for each $h$ and $t$ in $]0, 1]$
$$ \Delta _{mn}(t, h)\ :=\ | (2m+2n+1) \widetilde \Lambda _{m+n}(t, h) \ -\
(2m+1)\widetilde \Lambda _m (t, h)\ -\ (2n+1)\widetilde \Lambda _n(t, h) | \
\leq \ C \ . \leqno (9.6)$$
}
\bigskip
\noindent
{\it Proof of the Lemma.} For $m\geq 1$ and $n\geq 1$,
let us denote by $(X, Y) = (X^{(-m-n)}, \ldots , X^{(m+n)}, Y)$ the
variable of $ {\bf R} ^{2k(2m+2n+2)}$. We define two
functions in $ {\bf R} ^{2k(2m+2n+2)}$ by
$$a_{mn}(X, Y)\ =\ \widetilde p_{m+n}(X)\ +\ A(Y), $$
$$b_{mn}(X, Y)\ =\ B(X^{(-m-1)}, X^{(-m)})\ +\ B (X^{(m)}, X^{(m+1)})
-B(X^{(-m-1)}, Y)\ -\ B(Y, X^{(m+1)}). $$
We denote by $A_{mn}(h)$ and
$B_{mn}(h)$ the $h-$operators in
$L^2( {\bf R} ^{k(2m+2n+2)})$ associated to the symbols
$a_{mn}(x, \xi)$ and $b_{mn}(x, \xi)$, and by $P_0(h)$ the
$h-$operator associated to $A(y, \eta )$.
We see easily that
$$(2m+2n+1) \widetilde \Lambda _{m+n}(t, h)\ =\
Log \ \left [(2 \pi h)^{2k(m+n+1)} Tr\big ( e^{-tA_{mn}(h) } \big ) \right ]
\ -\ Log \left [(2\pi h)^{k} Tr\big ( e^{-tP_0(h) } \big ) \right ],$$
$$(2m+1)\widetilde \Lambda _m (t, h) + (2n+1) \widetilde \Lambda _n (t, h) =
\ Log \left [ (2 \pi h)^{2k(m+n+1)}
Tr\big ( e^{-tA_{mn}(h) +tB_{mn}(h)} \big ) \right ]. $$
If we set $F(\theta )=
Log \Big ( Tr \big ( e^{-tA_{mn}(h) + \theta t B_{mn}(h)}\big ) \Big )$,
we can write, for the left hand side $\Delta _{mn}(t, h)$ of (9.6)
$$\Delta _{mn}(t, h) \ \leq \
| F(1)-F(0) | \ +\ \left | Log \ \left [ (2\pi h)^{k} Tr \left (e^{-tP_0(h)}
\right ] \right )
\right | . $$
We obtain
$$ | F(1)-F(0) | \leq \ \sup _{\theta \in [0, 1]} | F'(\theta) |
\ =\ \sup _{\theta \in [0, 1] }
{ | Tr\big (t B_{mn}(h) e^{-tA_{mn}(h) +t\theta B_{mn}(h) } \big ) | \over
| Tr\big ( e^{-tA_{mn}(h) +t\theta B_{mn}(h) } \big ) | } . $$
Since $e^{-tA_{mn}(h) +t\theta B_{mn}(h) }$
is a positive self-adjoint operator, we have
$$ | Tr\big ( B_{mn}(h) e^{-tA_{mn}(h) + t\theta B_{mn}(h)} \big ) | \ \leq \
\Vert B_{mn}(h)\Vert \
| Tr\big ( e^{-tA_{mn}(h)+ t\theta B_{mn}(h) } \big ) |.$$
Since the norm of $B_{mn}(h)$ is bounded independently on $m$, $n$
and $h\in ]0, 1]$, the Lemma is proved.
\bigskip
\noindent
{\it Proof of Theorem 9.1.} Since the sequence $\widetilde \Lambda
_n(t, h)$
satisfies (9.6), it follows classically (cf. Helffer-Sj\"ostrand
[6], lemma 2.5)
that $\widetilde \Lambda _n(t, h)$ has a limit $\Lambda (t, h)$ when
$n \rightarrow \infty$, and that
$$ | \widetilde \Lambda _n (t, h)\ -\ \Lambda (t, h) | \ \leq \ {C\over n}
\ \ \ \ \ \ \ \ \forall n\geq 1
\ \ \ \ \forall h\in]0, 1]. $$
The proof of Lemma 9.2 shows also that,
for some constant $C>0$
$$ | \Lambda _n (t, h)\ -\ \widetilde \Lambda _n(t, h) | \
\leq \ {C \over n}
\hskip 1cm
\forall n\geq 1\ . $$
Therefore (9.5) is proved.
\bigskip
Let us prove now that $\Lambda (t, h)$ has an asymptotic expansion
in powers of $h$. In order to define the first term of this
expansion, we need the following integral operator $S_0(t)$ in
$L^2({\bf R}^{2k})$ defined by
$$(S_0(t)u)(X)\ =\
\int _{{\bf R}^{2k}} e^{ -{t\over 2}( A(X) + 2 B(X, Y) + A(Y))}
\ u(Y)\ dY \hskip 1cm \forall u\in L^2({\bf R}^{2k})\ . \leqno (9.7)$$
This operator, which is somewhat similar to the Kac operator
studied by Helffer [3] and Helffer-Ramond [4], is self-adjoint, of trace class. The norm
$ \Vert S_0(t) \Vert $ is a simple eigenvalue of $S_0(t)$
(by Krein-Rutman theorem), and all other eigenvalues
have a modulus strictly smaller than $ \Vert S_0(t) \Vert $.
\bigskip
\noindent
{\bf Theorem 9.3.} {\it The thermodynamic limit $\Lambda (t, h)$
of Theorem 9.1 has an asymptotic
expansion when $h\rightarrow 0$
$$\Lambda (t, h)\ \sim \ \sum _{j\geq 0} \gamma _j (t) h^j\ , \leqno (9.8)$$
where $\gamma _j(t)$ are real numbers, and
$\gamma _0(t) =Log ( \Vert S_0(t) \Vert )$, where $S_0(t)$ defined in
(9.7). }
\bigskip
If $B=0$, (when there is no interaction), the `classical'
thermodynamic limit $\gamma _0(t)$ is given by
$$\gamma _0(t)\ =\ Log\ \left ( \int _{{\bf R}^{2k}}
e^{-tA(X)}dX \right ). $$
\bigskip
We shall not define explicitely the others coefficients
$\gamma _j(t)$ but, for each integer $m$, we shall define an analytic
function of $h$ (denoted below by ${1\over d_m} ln\ \lambda _m(t, h)$,
and we shall prove later (lemma 9.5) that the difference between
this function and $\Lambda (t, h)$ is ${\cal O}(h^m)$. The function
$\lambda _m(t, h)$ will be an eigenvalue of an operator
$S_m(t, h)$, and the construction of this operator will be our first
step.
\bigskip
\noindent
{\it Construction of the operator $S_m(t, h)$.}
\medskip
By theorem 2.2, we know that $ q_n(x, \xi , t, h)$, (the symbol
satisfying (9.4)), has the
asymptotic expansion (2.8) with coefficients $E_n^{(j)}(x, \xi , t)$.
By the construction of section 6, we know that
$E_n^{(0)}= t p_n$, $E_n^{(1)}= 0$, and that
$E_n^{(2)}$ is given like in (6.9)), where $p$ is replaced by
$ p_n$ defined in (9.2). By (6.9) and (9.2),
we see that there is a function
$f^{(2)}\in C^{ \infty } ( {\bf R} ^{8k}\times {\bf R} )$ such that
$$ E_n^{(2)}(x, \xi , t)\ =\ \sum _{j=1}^n
f^{(2)}(X^{(j)}, X^{(j+1)}, X^{(j+2)}, X^{(j+3)}, t). $$
We have set $X^{(n+j)} = X^{(-n+ j -1)}$ for $j\geq 1$.
As a function of $t$,
$f^{(2)}$
is a polynomial. As a function of $X$, it is holomorphic
with bounded derivatives in
$\{ X\in {\bf C} ^{8k}, | Im\ X | _{\infty } < a\}$.
More generally, if we follow the induction in the
proof of Proposition 6.1, we see that, for each integer $m$, there is
some integer $d_m\geq 1$, and a function $F^{(m)}(.,t, h)\in C^{ \infty }
( {\bf R} ^{2k(1+d_m )} ) $ such that the function $\Phi _n^{(m)}$
defined in (6.4) can be written
$$\Phi _n ^{(m)}(x, \xi , t, h)\ =\
\sum _{j=-n}^n F^{(m)}(X^{(j)}, \ldots , X^{(j+ d_m)}, t, h)\ . \leqno
(9.9)$$
For example, we have $d_0= d_1 = 1$, $d_2=3$. The approximation
$\Phi _n ^{(m)}$ can be written in the form (9.9), but not in the unique
way. For $m=1$, $F^{(1)}$ (which is independent on $h$) can be
written
$$F^{(1)} (X, t)\ =\ t \left ( {A(X^{(1)})\over 2}\ +\ B(X^{(1)},
X^{(2)})\ +\ {A(X^{(2)})\over 2}\right ). $$
For $m$ arbitrary and $h=0$, we can take, for all $X$ in
${\bf R} ^{2k(1+d_m)} $,
$$F^{(m)}(X, t, 0)\ =\
{t\over d_m+1}\ \sum _{j=1}^{1+d_m} A(X^{(j)})\ +\ {t\over d_m}\
\sum _{j=1}^{d_m} B(X^{(j)}, X^{(j+1)}).
\leqno (9.10)$$
Now, we introduce the operator
$S_m(t, h)$ in $L^2({\bf R}^{2kd_m})$ defined by
$$(S_m(t, h) u) (Y)\ =\
\int _{{\bf R}^{2kd_m}} e^{-\Psi _m(X, Y, t, h)} u(X)\ dX
\hskip 1cm \forall u\in L^2({\bf R}^{2kd_m}), \leqno (9.11)$$
where we set here $X= (X^{(1)}, \ldots , X^{(d_m)})$,
$Y= (Y^{(1)}, \ldots, Y^{(d_m)})$, $X^{(j+d_m)}= Y^{(j)}$ and:
$$\Psi _m(X, Y, t, h)\ =\ \sum _{j=1}^{d_m}
F^{(m)} (X^{(j)}, \ldots , X^{(j+d_m)}, t, h). \leqno (9.12)$$
\bigskip
\noindent
{\bf Lemma 9.4.} {\it For each $t>0$, the operator $S_m(t, h)$
is of trace class, with a trace norm bounded independently on $h$
small enough, and it
has, for $h$ small enough, a simple
eigenvalue $\lambda _m(t, h)$ such that $\lambda _m(t, 0)= \Vert
S_0(t)\Vert ^{d_m}$, and all other eigenvalues
are strictly smaller in modulus.In other words, there exist
$\rho _1$ and $\rho _2$ such that $0<\rho _1 < \rho _2$,
$ | \lambda _m(t, h)| \geq \rho _2$ and
$\sigma (S_m(t, h)) \setminus \{ \lambda _m(t, h)\}
\subset B(0, \rho _1)$ for $h$ small enough.
}
\bigskip
\noindent
{\it Proof.}
We shall prove that $S_m(t, 0)$ is isospectral to
the power $S_0(t)^{d_m}$ of $S_0(t)$ defined in (9.7).
We remark that, for each integer $n$, $S_0(t)$ satisfies
$$Tr (S_0(t)^n)\ =\ \int _{{\bf R}^{2kn}} e^{-tG _n(X)} dX, $$
where
$$G _n(X)\ =\ B(X^{(n)}, X^{(1)})\ +\
\sum _{j=1}^n A(X^{(j)})\ +\ \sum _{j=1} ^{n-1} B(X^{(j)},
X^{(j+1)})\hskip 1cm \forall X\in {\bf R} ^{2nk}. $$
Similarly, $S_m(t,h)$ defined in (9.11) is of trace class
and, for each integer $q$, we have
$$Tr( S_m (t, h)^q)\ =\ \int _{{\bf R}^{2kqd_m}}
e^{-\theta _q(X,t, h)}dX\ , \leqno (9.13)$$
where, setting now $X ^{(qd_m+j)}=X^{(j)}$ for $j\geq 1$
$$\theta _q(X, t, h)\ =\
\sum _{j=1}^{qd_m} F^{(m)} (X^{(j)}, \ldots , X^{(j+d_m)}, t,
h)\ . \leqno (9.14)$$
It follows from (9.10) that
$\theta _q(X, t, 0)= t G _{qd_m} (X)$ for all $X$ in
$ {\bf R} ^{2kqd_m}$, and therefore
$S_m(t, 0)$ is related to the operator $S_0(t)$ of (9.7),
for each integer $q$,
by
$$Tr (S_m(t, 0)^q)\ =\ Tr (S_0(t)^{qd_m}).$$
Therefore, $S_m(t, 0)$ is isospectral to $S_0(t)^{d_m}$,
hence has $\Vert S_0(t)\Vert ^{d_m}$ as a simple eigenvalue,
and all others eigenvalues have a modulus strictly
smaller. Since
$S_m(t, h)$ depends analytically on $h$, by the classical
theory of perturbation of a single eigenvalue,
$S_m(t, h)$ has an eigenvalue $\lambda _m(t, h)$ which depends analytically
on $h$, such that $\lambda _m(t, 0)= \Vert S_0(t)\Vert ^{d_m}$.
\bigskip
For $h$ small enough, $Log \ \lambda _m(t, h)$ is well defined.
Since $m$ is arbitrary, the theorem 9.3 will follow from
\bigskip
\noindent
{\bf Lemma 9.5.} {\it With the previous notations, we have, for each
integer $m$, and for $h$ small enough
$$\Lambda (t, h)\ =\ {1\over d_m} Log\ (\lambda _m(t, h)) \
+\ {\cal O} (h^m)\ . \leqno (9.15)$$}
\bigskip
\noindent
{\it Proof. First step. }
It is natural to approximate $\Lambda _n(t, h)$
(defined in (9.3)), by
$$\Lambda _n^{(m)}(t, h)\ =\ {1\over 2n+1}\ Log\ \left [
\ \int _{{\bf R}^{2k(2n+1)}}
e^{- \Phi _n^{(m)}(x, \xi , t, h)}\ dxd\xi \right ] ,
\leqno (9.16)$$
where $\Phi _n^{(m)}$ is the approximation of (6.4).
We shall prove that
$$\left | \Lambda _n^{(m)} (t, h)\ - {1\over d_m} Log\ (\lambda _m
(t, h))
\right | \ \leq \ {C\over k}
\hskip 1cm if\ \ \ 2n+1 = kd_m\leqno (9.17)$$
for some constant $C$.
When $2n+1= kd_m$, we have
$$\Phi _n^{(m)}(X^{(-n)}, \ldots X^{(n)}, t, h) =
\theta _k(X^{(-n)}, \ldots X^{(n)}, t, h), $$
where $\theta _k$ is the function defined in (9.14), and therefore
$$
\Lambda _{n}^{(m)}(t, h) \ =\ {1\over kd_m}\ Log\ \left [
\ Tr \ ( S_m(h)^k ) \right ]\hskip 1cm if\ \ \ \ \
2n+1= kd_m.$$
By Lemma 9.4, we have, for some $C>0$
$$ \left | {1\over k} Log \left ( Tr \ (S_m(h)^k) \right ) \ -\
Log\ ( \lambda_m(t, h)) \right |\ \leq \ {C \over k}$$
and (9.17) follows.
\medskip
\noindent
{\it Second step.}
Theorem 2.2 shows that there is $K_m>0$ and $\varepsilon _m>0$,
independent on $n$,
such that, for each $h$ small enough, $n$ and $t$ satisfying
$0\leq t \leq 1$ and $nh^m \leq \varepsilon _m$,
and for real $(x, \xi )$, we have
$$ | q_n(x, \xi , t, h) \ - \ \Phi _n^{(m)}(x, \xi , t, h) | \ \leq \
K_m nh^m\ , \leqno (9.18)$$
where $q_n$ is the symbol satisfying (9.4) and
$\Phi _n^{(m)}$ is its approximation, defined in (6.4).
Since we used the Weyl calculus and since $e^{-t P_n(h)}$
is self-adjoint, the function $ q_n(x, \xi , t, h) $
is real-valued. Therefore, it follows from (9.18) that, for some constants
$\varepsilon _m>0$ and $K_m>0$
$$nh^m \ \leq \ \varepsilon _m\ \Longrightarrow
| \Lambda _n(t, h)\ -\ \Lambda _n^{(m)}(t, h) | \ \leq \
K_m h^m \ . \leqno (9.19)$$
For $h$ small enough, let
$n= n(h)$ be an integer such that $2n+1$ is a multiple
of $d_m$, and ${\varepsilon _m \over 2} \leq n h^m
\leq \varepsilon _m $, where
$\varepsilon _m$ is a constant such that (9.19) is valid.
Then we have
$$ | \Lambda (t, h)\ -\ {1\over d_m} Log\ \lambda _m(t, h)) | \
\leq \
| \Lambda (t, h)\ -\ \Lambda _n(t, h) | \ +\
| \Lambda _n(t, h)\ -\ \Lambda _n^{(m)}(t, h) | \ +\ \ldots $$
$$\ldots \ +\
| \Lambda _n^{(m)}(t, h)\ -\ {1\over d_m} Log\ \lambda _m(t, h)) | $$
Using (9.5), (9.19) and (9.17), it follows that
$$ | \Lambda (t, h)\ -\ {1\over d_m} Log\ \lambda _m(t, h))| \
\leq \ K_m h^m \ +\ {2C\over n} \ \leq \ \left ( K_m + {4C\over
\varepsilon _m}\right ) h^m\ . $$
The lemma is proved, and Theorem 9.3 follows directly.
\bigskip
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\smallskip
\noindent
[17] M. TODA, {\it Theory of nonlinear lattices.} Springer, 1981.
\bigskip
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D\'epartement de Math\'ematiques (UMR CNRS 6056)
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Universit\'e de Reims
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Moulin de la Housse
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B.P. 1039
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51687 Reims Cedex, France
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jean.nourrigat@univ-reims.fr
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christophe.royer@univ-reims.fr
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