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thermodynamic limit
pseudodifferential operator
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\vskip 2cm
\centerline {\un PSEUDODIFFERENTIAL CALCULUS }
\medskip
\centerline {\un AND THERMODYNAMIC LIMITS.}
\vskip 1cm
\centerline {\bf J. NOURRIGAT}
\vskip 1cm
\centerline {\bf Abstract.}
If $P(h)$ is a $h-$pseudidifferential operator in $ {\bf R} ^n$
associated to an holomorphic bounded symbol in some neighborhood
of the real phase space, we describe the symbol of $e^{-tP(h)}$,
by inequalities where the constants depend on the bounds for the
symbol of $P(h)$, but not on the dimension $n$. Some applications to
thermodynamic limits are given.
\vskip 1cm
\noindent
{\bf 1. Introduction.}
\bigskip
In [10], 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 [10]), 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.
\bigskip
Unfortunately, in this first paper, we have also to assume that $ |
p_n(x, \xi) | \leq Mn$ for all $(x, \xi)$ in $\Omega _{2n}(a)$. With
this hypothesis, the main result proved here is the theorem 2.2
below. In a
work in progress, Ch. Royer is trying to prove a similar result without
assuming that $p_n(x, \xi )$ is bounded,
by an improvement of the technique. Even if this work is
successful, the result will be limited essentially to first order
operator (like Klein-Gordon).
\bigskip
Let us explain (in a particular case to simplify the notations), the
physical model given by J. Sj\"ostrand ([10], 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 [10] 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 [10]
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 $kn$), 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 [11] in
the classical mechanics). In E. Lieb [5] and B. Simon [8],
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 [10] proves that,
for each $h>0$ and $t>0$, the sequence
$$\Lambda _n(t, h)\ :=\ {1\over (2n+1)} \
ln\ Tr\ \left ( e^{-{t\over h}P_n(h)} \right )\leqno (1.7)$$
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 [7] for the notion of thermodynamic limit, and also
Helffer-Sj\"ostrand [3] for the Lemma used to prove its existence and (1.8)).
In [10], the potentials are typically ${\cal O}( | x | )$ at infinity.
For exactly quadratic potentials, see also Royer [6].
\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 $e^{-tP_n(h)}$. We can hope that the work in progress of Ch.
Royer will 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 that similar results will be proved. In this paper, we
can only consider a similar model (perhaps physically non realist),
where the 1-particle Hamiltonian is periodic with respect to some
lattice in $ {\bf R} ^{2k}$, and where the interaction is also
periodic, like a system of Harper Hamiltonians in interaction.
In this case, the trace in (1.7) does not exist in the usual sense, and
has to be replaced by another notion of trace (see section 9). 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 [5]
and Simon [8] proved only the continuity of $\Lambda _{\infty }(t, h)$
at $h=0$). In [10], the asymptotic expansion of the thermodynamic
limit in powers of $h$ relies on a study of Laplace integrals in
large dimension. In our sect. 9, it relies on the study of the
greatest eigenvalue of some integral operator on the torus.
\bigskip
The results of Lascar [4] on pseudodifferential calculus in infinite
dimension have another motivation, related to the quantum field theory.
\bigskip
Besides the main result (theorem 2.2) and its application (theorems
9.1 and 9.3), some propositions may have their own interest : composition
(sect. 3), inversion (sect. 6) and exponential (sect. 7) of
pseudodifferential operators in large dimension.
\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
$\Sigma (a)$ 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 $n$, such that, if $1\leq j\leq n$ and if
$0< h\leq h_n$,
$$ | f_n (x, \xi , h ) | \ \leq \ n \ M,
\hskip 1cm \forall (x , \xi) \in \Omega _{2n}(a)
\ \ \ \ \forall h\in ]0, h_n[,
\leqno (2.2)$$
$$ | { \partial f_n \over \partial x_j} (x, \xi , h) |
+ | { \partial f_n \over \partial \xi _j} (x, \xi , h) |
\ \leq \ M,
\hskip 1cm \forall (x , \xi) \in \Omega _{2n}(a)
\ \ \ \ \forall h\in ]0, h_n[.
\leqno (2.3)$$
We set also
$$S(a)\ =\ \bigcap _{0**0$, and $p= (p_n)$ a family of
symbols in $\Sigma (b)$, independent on $h$.
Then, for each $a\in]0, b[$, and for each integer $m\geq 1$,
there exist $\varepsilon _m>0$ and a family of functions
$q= (q_n)(x, \xi , t, h)$ in $\Sigma (a)$ $(0\leq t \leq 1)$
such that
$$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 (2.6)$$
This function $q_n(., t, h)$ satisfies (2.2) and (2.3)
for some constant $M>0$, for all
$t\in [0, 1]$, $(x, \xi)\in \Omega _{2n}(a)$ and
$h>0$ and $n\geq 1$ such that $nh^m \leq \varepsilon _m$.
Moreover, the symbol $q_n$ has an asymptotic expansion in powers of
$h$
$$q_n(x, \xi , t, h)\ =\ \sum _{j= 0}^{m-1} E_n^{(j)}(x, \xi , t )h^j
+ h^m R_n (x, \xi , t, h)\leqno (2.7)$$
where the families of functions $(E_n^{(j)})$
satisfy the inequalities $(2.2)$ and $(2.3)$, and and $R_n $
satisfies (2.2) with the same conditions
as above.
}
\bigskip
The explicit construction of the functions $E_j^{(n)}$ will be
precised in sections 5 (in general) and 9 (in some particular cases).
\bigskip
The paper is organized as follows. In sections 3, 6 and 7, we give
some results on the Weyl calculus of holomorphic symbols, which have
perhaps their own interest : composition (section 3), inversion
(section 6), and exponential (section 7). In section 4, we recall
the result of [1], stated here for the Weyl calculus. In section 5, we
give the formal construction of the $E_n^{(j)}$ of (2.7), and in
section 8, the end of the proof of Theorem 2.2. The precise statement
for applications to thermodynamic limit
is given and proved in section 9.
\bigskip
\noindent
{\bf 3. Composition of symbols in large dimension.}
\bigskip
We denote by $\sigma $ the symplectic form in ${\bf C}^{2n}$ defined
by $\sigma \big ( (x, \xi ),\ (y, \eta)\big ) =
y. \xi - x. \eta $. If $f$ and $g$ are $ C^{ \infty } $ functions
on $ {\bf R} ^{2n}$, bounded with all their derivatives, we denote by
$f \sharp _hg$ the function defined by the oscillatory integral
$$(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)$$
If $f$ is a bounded function in $\Omega _{2n} (a)$ (defined in (1.4)),
we set
$$\Vert f\Vert _a \ =\ \sup _{X\in \Omega _{2n} (a)}
| f(X) | \leqno (3.2)$$If the derivatives are also bounded, we set
$$\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 |$$
\bigskip
\noindent
{\bf Theorem 3.1.} {\it Let $a$ and $b$ be such that
$00$ and $\rho >0$, and
$\Omega _{6n}(a, \rho )$ be the set of $ (X, Y, Z)\in {\bf
C}^{6n}$ such that $X$ is in $\Omega _{2n}(a)$ and $Y$ and
$Z$ in $\Omega _{2n}(\rho )$. Let $F$ be a bounded holomorphic function
in $\Omega _{6n}(a, \rho)$. Let $I_h(F)$ be the following oscillatory
integral
$$(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 (3.4)$$
Then we have
$$\Vert I_h(F)\Vert _a \ \leq \
\sup _{(X, Y, Z)\in \Omega _{6n}(a, \rho )}
| F(X, Y, Z) | \
\left ( 1 + \sqrt {{2 \over \pi }} { \sqrt {h} \over \rho }
e^{-{\rho ^2 \over h}}
\right ) ^{4n} \leqno (3.5)$$
}
\bigskip
\noindent
{\it Proof.}
We set, for each $R>0$ and $t\in {\bf R}$
$$\psi _R(t) \ =\ \left \{
\matrix {=\ -R\hfill &if &t\leq -R \cr
=t\hfill & if & -R\leq t\leq R \cr
=R\hfill & if & R\leq t \cr
}
\right.
$$
We set, for each $t\in {\bf R} ^n$
$$\varphi _R(t) \ =\ \big (
\psi _R(t_1), \ldots \psi _R(t_n))$$
We define, for each $R>0$, an integration contour
$\gamma _R$ in $\k ^{4n}$ in the following way. For each
$\theta =(s,t, \sigma , \tau )\in {\bf R} ^{4n}$, $\gamma _R(\theta )=
(Y_R(\theta ), Z _R(\theta ))$
is the point of $\k ^{4n}$ defined by
$$
\matrix {
{1 \over 2} \ \big ( y_R(\theta ) + \zeta _R(\theta ) \big ) \ =\
s + i \varphi _R(s)
\cr
{1 \over 2} \ \big ( y_R(\theta ) - \zeta _R(\theta ) \big ) \ =\
\sigma - i \varphi _R(\sigma)
\cr }
\hskip 1cm
\matrix {
{1 \over 2} \ \big ( z_R(\theta ) - \eta _R(\theta ) \big ) \ =\
t + i \varphi _R(t)
\cr
{1 \over 2} \ \big ( z_R(\theta ) + \eta _R(\theta ) \big ) \ =\
\tau - i \varphi _R(\tau)
\cr }
$$
If $2R< \rho$ and $X\in\Omega _{2n}(a)$, then $(X, Y_R(\theta ),
Z_R(\theta ))$ is in $\Omega _{6n}(a, \rho)$ for all $\theta $, and
we may replace ${\bf R}^{2n}$ by $\gamma _R$ in (3.4) and write
$$(I_h(F))\ (X)\ \ =\
( \pi h)^{-2n} \int _{ { \gamma _R} }
e^{ -{2i \over h} \sigma (Y, Z)}
F(X, Y, Z)\ dY dZ$$
in other words
$$(I_h(F))\ (X)\ \ =\
( \pi h)^{-2n} \int _{ {\bf R}^{4n} }
e^{ -{2i \over h} \sigma (Y_R(\theta ), Z_R(\theta ))}
F(X, Y_R(\theta ), Z_R(\theta ))\ \ det \ \gamma '_R( \theta )\ d\theta $$
Proposition 3.2 will follow from the next lemma (with $R = \rho /2$).
\bigskip
\noindent
{\bf Lemma 3.3.} {\it With the previous notations, we have, for all $R>0$
and $h>0$
$$ (\pi h)^{-2n} \ \int _{{\bf R}^{4n}}
\Bigg | e^{ -{2i\over h} \sigma (Y_R(\theta), Z _R(\theta))}
\ {\rm det} \ \gamma ' _R(\theta) \Bigg | \ d\theta
\ \leq \
\left ( 1 + { \sqrt {h} \over R \sqrt {2\pi} }e^{-{4R ^2 \over h}}
\right ) ^{4n}
$$
}
\bigskip
\noindent
{\it Proof of Lemma 3.3.}
It suffices to prove this lemma when $n=1$. Let us set in this case
$$F _R(\theta )\ =\ \Bigg |
e^{ -{2i\over h} \sigma (Y_R(\theta ) , Z_R(\theta ))}
\ {\rm det} \ \gamma ' _R(\theta ) \Bigg | $$
We see easily that, for each $\theta \in {\bf R}^4$, (denoting its
components by $(\theta _1, \ldots , \theta _4)$ instead of
$(s, \sigma , t, \tau)$)
$$ F _R(\theta )\ \leq \ 4 \
\prod _{j=1}^4 \Big [ 1 + \varphi ' _R(\theta _j)^2 \Big ] ^{1/2}
\
e^{ -{4 \over h}\theta \ .\ \varphi _R(\theta ) }
$$
For each $\theta \in {\bf R} ^4$, let us set $N(\theta )
= \sharp \{ j\leq 4,\ | \theta _j | \geq R \}$, and let
$A_j$ $(0\leq j \leq 4)$ be the set of $\theta \in {\bf R} ^4 $
such that $N(\theta )= j$.
We can see that, if $\theta \in A_0$
$$F _R(\theta) \ \leq \ 16 \ e^{-{4\over h} | \theta | ^2 } $$
If $\theta \in A_1$ and, for example, ${\rm sup } | \theta _j | =
|\theta _1|$, we have
$$F _R(\theta ) \ \leq \ 8 \sqrt {2} \ e^{-{4\over h}
( R |\theta _1|+ \theta _2^2 + \theta _3^2 + \theta _4^2 )} $$
If $\theta \in A_2$ and, for example, $ | \theta _1 | $ and
$ | \theta _2 | \geq R$, and $ | \theta _3 | $ and
$ | \theta _4 | \leq R$, we have
$$F _R(\theta ) \ \leq \ 8 \ e^{-{4\over h}
(R|\theta _1| + R | \theta _2 | + \theta _3^2 + \theta _4^2 )} $$
If $\theta \in A_3$ and, for example, $ | \theta _1 | $,
$ | \theta _2 | $ and $ | \theta _3 | \geq R$, and
$ | \theta _4 |\leq R $, we have
$$F _R(\theta ) \ \leq \ 4 \sqrt {2} \ e^{-{4\over h}
(R|\theta _1| + R | \theta _2 | + R | \theta _3 | + \theta _4^2 )} $$
If $\theta \in A_4$, we have
$$F _R(\theta ) \ \leq \ 4
e^{-{4\over h} (R|\theta _1| + R | \theta _2 | + R | \theta _3 |
+R | \theta _4 |)} $$
Therefore, if $0\leq j\leq 4$
$$(\pi h )^{-2} \int _{A_j} F _R(\theta )\ d\theta
\ \leq \
(\pi h)^{-2} 4 (\sqrt {2})^{4-j}\ \ C_4^j \left [
2 \int _R^{\infty } e^{-{4R\theta \over h}}\ d\theta \right ]^j
\ \left [ \int _{- \infty }^{+ \infty } e^{-{4\theta ^2 \over h}}
d\theta \right ]^{4-j}$$
$$\ldots \ \leq \ (\pi h)^{-2} 4 (\sqrt {2})^{4-j}\ C_4^j\
\left [ {h\over 2R}e^{-{4R^2 \over h}} \right ]^j\
\left [ { \sqrt { \pi h} \over 2 } \right ]^{4-j}$$
Therefore
$$(\pi h)^{-2} \int _{ {\bf R} ^{4}} F _R(\theta )\ d\theta \
\leq \ 4 (\pi h)^{-2}
\left [ \sqrt { {\pi h \over 2} }\ +\
{h\over 2R}e^{-{4R^2 \over h}} \right ]^4$$
The lemma is proved.
\bigskip
\noindent
{\bf 4. Composition of exponentials of symbols (result of [1]).}
\bigskip
We shall use the theorems 2.4 and 2.6 of the paper [1] with L. Amour
and Ph. Kerdelhu\'e. Unfortunately, these results are written in [1]
with the standard calculus, and here we use the Weyl calculus. It is
useful to write here the statement for the Weyl calculus, and to
recall the formal construction, but the
proof, which is the same as in [1], will be omitted.
\medskip
Let us explain first the formal aspects.
Let $ b>0$, and $f= (f_n)$ and $g = (g_n)$ be functions in
$S(b)$, which are first supposed to be independent on $h$. We want to
find first a formal series (in powers of $h$) $\varphi _n(. , h)$ such that
$e^{\varphi _n(. , h)}\sim e^{f _n} \sharp _h
e^{g _n}$. For that, we can
find a unique formal serie
$$u(y, \eta, z, \zeta , h) \ \sim \
\sum _{k\geq 0} u_k(y, \eta, z, \zeta )h^k$$
which is a formal solution of the Cauchy problem (the substrict $n$
being omitted)
$$2i\ {\partial e^u\over \partial h}\ =\
\sigma \big ( (\partial _y, \partial _{\eta }),\ (\partial _z, \partial
_{\zeta })\big )\ e^u\leqno (4.1)$$
$$u(y, \eta, z, \zeta , 0)\ =\ f(y, \eta ) + g(z, \zeta )\leqno (4.2)$$
Then the formal serie $\varphi (., h)$ will be given by
$\varphi (x, \xi, h)= u(x, \xi, x, \xi, h)$.
Let us prove now that the coefficients of this formal series are in
$S(b)$.
The equation (4.1) can be written
$$2 i { \partial u \over \partial h}\ \sim \ \sum _{j=1}^n
{ \partial ^2u \over \partial z_j \partial \eta _j} \
-\ {\partial ^2u \over \partial y_j \partial \zeta _j} \
+\ {\partial u\over \partial z_j}{\partial u\over \partial \eta _j} \
-\ {\partial u\over \partial y_j}{\partial u\over \partial \zeta _j}$$
Therefore the coefficients $u_k$ are defined by
$u_0(y, \eta, z, \zeta)= f(y, \eta ) + g(z, \zeta )$ and, if $k\geq
1$, by
$$2ik u_k\ =\ \sum _{j=1}^n\Bigg [
{\partial^2u_{k-1}\over \partial z_j\partial \eta _j} \ -\
{\partial^2u_{k-1}\over \partial y_j\partial \zeta _j}\
+\ \sum _{p+q=k-1} \Big (
{\partial u_p\over \partial z_j}{\partial u_q\over \partial
\eta _j} \ -\ {\partial u_p\over \partial y_j}{\partial u_q
\over \partial \zeta _j} \Big ) \Bigg ]
\leqno (4.3)$$
We set $c_k(f, g)(x, \xi) := u_k(x, \xi , x, \xi)$.
Then the definition of
$c_k(f, g)$ can be extended to functionf $f$ and $g$ depending on
$h$ (using the explicit expression (4.3), not the differential
equation (4.1)). It is easy to
see, using Cauchy inequalities and the
Proposition 1.1 of [9], recalled below for reader's convenience,
that the $c_k(f, g)$ are in $S(b)$. More precisely, if
$00$, depending only on $k$, $b-a$, $\Vert \nabla f \Vert _b$ and
$\Vert \nabla g \Vert _b$. We remark also that, if $f$ and $g$ are
polynomials in $h$ of degree $\leq m$, then $c_k(f, g)$ is a
polynomial of degree $\leq (k+1)m$.
\bigskip
\noindent
{\bf Proposition 4.1 (Proposition 1.1 of [9]).}
{\it Let $a$ and $b$ such that $00$
(independent on $n$) with
the following property. Let $f(y, \zeta )$ be an holomorphic
function in $\Omega _{2n}(b)$ such that $ | \nabla f | _{\infty }$ is
bounded in $\Omega _{4n}(b)$. Then we have
$$\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
$$
Moreover, if $u= (u_1, \ldots u_n)$ a vector, and $\rho >0$
such that $a + \rho | u | _{\infty } < b$, we have
$$\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 }}$$
}
Now, instead of a formal series, we want to find a function
$\varphi _n$ satisfying
$e^{f _n(. , h)} \sharp _h e^{g _n(. , h)}$
The answer is given by theorems 2.4 and 2.6
of [1] (or, more precisely, their analogues for the Weyl calculus),
which can be written as follows.
\bigskip
\noindent
{\bf Theorem 4.2.} {\it Let $f$ and $g$
be functions in $\Sigma (b)$ $(b>0)$, and let $h'_n$ and $h''_n$ be
the sequences associated to them as in Definition 1.1.
Let $a\in ]0, b[$ and $m\geq 1$. Then, there exist an element $\varphi $
of $\Sigma (a)$ and a constant $\varepsilon _0$ such that, if we
set
$$h_n \ = \ inf \ (h'_n, h''_n, \varepsilon _0 n^{- 1/m}),\leqno
(4.4)$$
we have
$$e^{\varphi _n (h)} \ =\
e^{f_n(h)} \sharp _h e^{g_n(h)}
\hskip 1cm
if \ \ \ \ 00$, depending only on $m$, such that,
if $0< h \leq h_n$
$${ 1 \over n} \Vert c_k(f, g)\Vert _a \
+ \ { 1 \over n} \Vert R(., h) \Vert _a \
+ \ \Vert \nabla c_k(f, g)\Vert _a \
+\ \Vert \nabla R(., h) \Vert _a \
\leq \ C_m
$$
}
It will be useful later to know how $c_k(f, g)$ is changed by a small
perturbation of $f$ or $g$ (of course, the dependence is not linear).
\bigskip
\noindent
{\bf Proposition 4.3.} {\it Let $f= (f_n)$, $g= (g_n)$
and $\widetilde g= (\widetilde g_n)$
be functions in $\Sigma (b)$ $(b>0)$. Then, for
each $a\in ]0, b[$ and $k\geq 0$, there exists $C_k>0$, independent on
$n$, such that
$${1\over n} \Vert c_k(f, g + h^m \widetilde g)\ -\ c_k(f, g)\Vert _a
\ +\ \Vert \nabla \left ( c_k(f, g + h^m \widetilde g)\ -\ c_k(f, g)
\right ) \Vert _a
\ \leq \ C_kh^m \leqno (4.7)$$
}
\bigskip
\noindent
{\it Proof.} Let $u_k$ and $w_k$
$(k\geq 0)$ be the sequences of functions defined by (4.3) and by
$$u_0(y, \eta, z, \zeta)= f(y, \eta )+ g(z, \zeta)
\hskip 1cm
w_0(y, \eta, z, \zeta)= f(y, \eta )+ g(z, \zeta)
+ h^m \widetilde g(z, \zeta)$$
Let $v_k= w_k -u_k$.
We have $v_0(y, \eta, z, \zeta)= h^m \widetilde g(z, \zeta)$ and, if
$k\geq 1$
$$2ikv_k\ =\ \sum _{j=1}^n \Bigg [
{ \partial ^2v_{k-1} \over \partial z_j \partial \eta _j} \
-\ {\partial ^2v_{k-1} \over \partial y_j \partial \zeta _j} \
+\ \sum _{p+q=k-1} \Big (
{\partial u_p\over \partial z_j}{\partial v_q\over \partial \eta _j} \
-\ {\partial u_p\over \partial y_j}{\partial v_q\over \partial \zeta _j}
\ +\ \ldots \leqno (4.8)$$
$$\ldots \ + \ {\partial v_p\over \partial z_j}{\partial u_q\over \partial \eta _j} \
-\ {\partial v_p\over \partial y_j}{\partial u_q\over \partial \zeta _j}
+ {\partial v_p\over \partial z_j}{\partial v_q\over \partial \eta _j} \
-\ {\partial v_p\over \partial y_j}{\partial v_q\over \partial \zeta _j}
\Big ) \Bigg ] $$
Now let $p$ be an integer $\geq 1$, and let $a_0, a_1, \ldots, a_p$ be real
numbers such that $a= a_p < a_{p-1}< \ldots < a_10)$ (independent on $h$) and $m$ be an integer $\geq 0$.
Then there exist
functions $\Phi ^{(m)}(., h, t)$ and $R^{(m)}(., h, t)$ in $S(b)$
$(t\in {\bf R} )$ such that
$\Phi ^{(m)}$ is a polynomial in $h$ of degree $\leq m-1$ and, (omitting
the subscript $n$)
$${ \partial \over \partial t} e^{\Phi ^{(m)}(., h, t)} \ =
\sum _{k=0}^{m-1} a_k \left ( p, e^{\Phi^{(m)}(., h, t)} \right ) h^k
+ h^m e^{\Phi ^{(m)}(., h, t)} R^{(m)}(., h, t)
\leqno (5.3)$$
}
\bigskip
\noindent
{\it Proof.} We take $\Phi ^{(0)}(x, \xi , h, t) = tp(x, \xi )$. Once
the polynomial
$\Phi ^{(m)}$ of degree $\leq m-1$ in $h$ is constructed, we denote
by $c_k^{(m)}(p, \Phi ^{(m)})$ the coefficient of $h^{m}$ in the
polynomial $e^{-\Phi ^{(m)}} a_k (p, e^{\Phi ^{(m)}})$. Then we denote
by
$E^{(m)}(x, \xi, t)$ the function satisfying
$${\partial E^{(m)} \over \partial t}\ =\ \sum _{k=0}^m
\ c_k^{(m)} (p, \Phi ^{(m)})
\hskip 1cm E^{(m)}(x, \xi , 0)= 0$$
and we set $\Phi ^{(m+1)} = \Phi ^{(m)} + h^m E^{(m)}$. We see easily that
the sequence $\Phi^{(m)}$ defined by this way satisfies (5.3) and,
using the argument of the section 9 of [1], we see that the $E^{(j)}$ are
in
$S(b)$.
\bigskip
We can write the first coefficients of the formal series :
$E^{(0)}(x , \xi , t)= tp(x, \xi)$, $E^{(1)}= 0$ and
$$E^{(2)} \ = \ {t^2\over 8}
\sum _{jk}\Bigg [ { \partial ^2p\over \partial x_j \partial x_k} \
{\partial ^2p \over \partial \xi _j \partial \xi _k}\ - \
{\partial ^2p \over \partial x_j \partial \xi _k}\
{\partial ^2p\over \partial \xi _j \partial x_k}\Bigg ] + \ldots
\leqno (5.4)$$
$$\ldots \ +\ {t^3 \over 24}\ \sum _{jk} \Bigg [
{ \partial ^2p\over \partial x_j \partial x_k}\ {\partial p\over
\partial \xi _j}\ {\partial p\over \partial \xi _k}\ +\
{\partial ^2p\over \partial \xi _j \partial \xi _k}\ {\partial p\over
\partial x_j}\ {\partial p\over \partial x_k}\
-\ 2\
{\partial ^2p\over \partial x_j \partial \xi _k }\
{\partial p\over \partial \xi _j}\ {\partial p\over \partial x_k}
\Bigg ] $$
\bigskip
We know that there exists another
function $\widetilde R^{(m)}(., h, t)$ in $S(b)$ such that
$${ \partial \over \partial t} e^{\Phi ^{(m)}(., h, t)} \ =
\sum _{k=0}^{m-1} a_k \left ( e^{\Phi^{(m)}(., h, t)}, p \right )h^k
+ h^m e^{\Phi ^{(m)}(., h, t)} \widetilde R^{(m)}(., h, t)
\leqno (5.5)$$
\bigskip
Now we shall obtain the following
consequence of this proposition.
\bigskip
\noindent
{\bf Proposition 5.2.} {\it With the notations of proposition 5.1,
there exist a family $S^{(m)}(., h, t)$ in $S(b)$ such that
$${ \partial \over \partial t} e^{\Phi ^{(m)}(., h, t)} \ =
p \ \sharp _h \ e^{\Phi ^{(m)}(., h, t)}
\ +\ h^m e^{\Phi ^{(m)}(., h, t)}S^{(m)}(., h, t)
\leqno (5.6)$$
More precisely, for each $a\in ]0, b[$, there exists $\varepsilon
_0>0$ and $M>0$, independent on $n$, such that $S^{(m)}$ satisfies
$(H_1)$ and $(H_2)$ in $\Omega _{2n}(a)$ if
$00$, 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 there is some other
constant $C>0$, independent on $h$, such that, for $h$ small enough,
$$ | F_{h}(X, Y, Z) | \ \leq \ Cn
\hskip 1cm
| \nabla F_{h}(X, Y, Z) | _{\infty }\ \leq \ C\leqno (5.11)$$
if $X$, $X+Y$ and $X+Z$ are in $\Omega _{2n}(a_2)$.
\smallskip
\noindent
{\it Second step.}
Let $L$ be the following operator in $ {\bf R} ^{4n} $
$$L\ =\ {i\over 2}\ \sigma (D_Y, \ D_Z)$$
For the integral defined in (5.10), we have, by Taylor formula
$$e^{-\Phi } \Big [ p\ \sharp _h e^{\Phi } \Big ] (X)\ =\
\sum _{k=0}^{m-1} {1\over k!} h^k (L^k F_h)(X, 0, 0)\ +
h^m T^{(m)}(X)$$
where
$$T^{(m)}(X)\ =\ \int _0^1 {(1-\theta )^{m-1}\over (m-1)!}
(\pi h)^{-2n} \int _{ {\bf R} ^{4n}} e^{-{2i\over h}
\sigma ( Y , Z )} (L^mF_h)(X, \theta Y, \theta Z) dYdZd\theta $$
We can write, for each $k\leq m-1$
$$(L^k F_h)(X, 0, 0)\ =\ \sum _{j=0}^{m-k-1} h^j g_j(X)\ +\
h^{m-k} R_{mk}(X, h)$$
where $g_j$ and $R_{mk}$ are in $S(a)$.
Therefore, we have
$$e^{-\Phi}\
\sum _{k=0}^{m-1} a_k \left ( p, e^{\Phi(.)} \right ) h^k
\ -\
\sum _{k=0}^{m-1} {1\over k!} h^k (L^k F_h)(X, 0, 0)\ = \
h^m R'^{(m)}(., h)$$
where $R'^{(m)}$ is in $S(a)$.
If we apply that to the function $\Phi ^{(m)}$ of Proposition 5.1,
we obtain the equality (5.6) with $S^{(m)}= R^{(m)}+ R'^{(m)} - T^{(m)}$. It remains
to prove that $T^{(m)}$ is in $S(a)$.
By (5.12) and Proposition 4.1,
there exists $C>0$, independent
on $n$, such that the function $F_h$ defined in (5.11) with $\Phi =
\Phi^{(m)}$ satisfies
$$ | \sigma (D_Y, \ D_Z)^mF(X, Y, Z) | \leq Cn
\hskip 1cm
|\nabla \Big ( \sigma (D_Y, \ D_Z)^m F(X, Y, Z) \Big ) |_{\infty }
\ \leq \ C$$
if $X$, $X+Y$ and $X+Z$ are in $\Omega _{2n}(a_1)$. Therefore, it
follows from Proposition 3.2 that
$$\Vert T^{(m)}\Vert _a \ \leq \ Cn
\left ( 1 + { \sqrt {h} \over a_1-a }e^{-{(a_1-a)^2 \over h}}
\right ) ^{4n} $$
If $00$ (independent on $n$), with the following properties.
If $f$ is a bounded holomorphic function in $\Omega _{2n}(b)$ $(n\geq
1)$ such that $\Vert f\Vert _b\leq \varepsilon _0$, and if $h>0$
satisfies $h\leq \varepsilon _0 (1 + Log\ n)^{-3}(b-a)^2$, then there exists
a bounded holomorphic function $g_h$ in $\Omega _{2n}(a)$ such that
$$ (1+f)\ \sharp _h \ (1+g_h)\ =\ 1 \leqno (6.1)$$
$$\Vert g_h \Vert _a \ \leq 5 \Vert f \Vert _b\leqno (6.2)$$
}
\bigskip
The proof will rely on the following lemma, which is stated in a more
general form since it will be used also in Section 7. The role of the
integer $4n+1$ will appear there.
\bigskip
\noindent
{\bf Lemma 6.2.} {\it There
exists $\varepsilon _0>0$ (independent on $n$), with the following properties.
If $00$ satisfies
$h\leq \varepsilon _0 (1 + Log\ n)^{-3}(b-a)^2$, then
$$\Vert f_1 \ \sharp _h \ \ldots \ \sharp _h \ f_k\Vert _a
\ \leq \ \varepsilon ^{(k+1)/2}$$
}
\bigskip
\noindent
{\it Proof of Lemma 6.2.} Let us set
$$\varepsilon _0 \ = \ inf \ \Bigg ( \Big ( Log\ {10\over
9} \Big )^2, \ e^{-8} \Bigg )
\hskip 1cm
a_j \ =\ b\ -\ (b-a)\ {Log \ j \over Log\ (4n+1)} $$
Therefore $a= a_{4n+1} < \ldots < a_1 = b$. We shall prove, by induction
on $j$ $(1 \leq j \leq k)$, the following property.
\smallskip
\noindent
$(X_j)$ If $F_j$ is a product (for the composition law $\sharp _h$) of
$k$ functions chosen among $f_1,\ \ldots \ , f_k$ (which satisfy (6.3)),
and if $0< h\leq \varepsilon _0 (1 + Log\ (4n+1))^{-3}(b-a)^2$, then we have
$$\Vert F_j \Vert _{a_j} \ \leq \ \varepsilon ^{ (j+1)/2}$$
\smallskip
For $j=1$, $(X_1)$ is our hypothesis. Suppose that $j\geq 2$, and that
$(X_i)$ is proved for all $ia_1$ we have $\Vert R^{(k)} \Vert _{a_1} \leq \varepsilon
^{(n+1)/2} k^{-4n}$. Therefore $G_1 = \sum _{k= 1}^{\infty } R^{(k)}$
satisfies
$$(1-R)\ \sharp _h\ (1+ G_1) \ =\ 1
\hskip 1cm
\Vert G_1 \Vert _{a_1} \ \leq \ \varepsilon ^{(n+1)/2} \sum
_1^{\infty } {1\over k^{4n} } \ \leq \ 2 \varepsilon ^{(n+1)/2}$$
Therefore the function $g = G + G_1 + G \sharp _h G_1$ satisfies
$(1+f) \sharp _h (1+g)= 1$ and
$$\Vert g \Vert _a
\ \leq \ \Vert G \Vert _{a_1}
\ + \ \Vert G_1 \Vert _{a_1} \ +\
\Vert G \Vert _{a_1} \Vert G_1 \Vert _{a_1}
\left ( 1 + { \sqrt {h} \over a_1-a }e^{-{(a_1-a)^2 \over h}}
\right ) ^{4n} $$
If $h\leq (a_1-a)^2/(1+Log\ n)$ and $\varepsilon < (200)^{-8}$, it follows
that $\Vert g\Vert _a \leq 5 \varepsilon = 5 \Vert f\Vert _b$.
The proposition is proved.
\bigskip
\noindent
{\bf 7. Exponential of a small operator.}
\bigskip
\noindent
{\bf Proposition 7.1.} {\it There exists $\varepsilon _0>0$ and $C>0$
(independent on the dimension $n$) with the following
properties. If $00$ and $\varepsilon _0>0$
(independent on $n$), and a function $U_m$ in $\Sigma (a)$ such that
$$e^{\Phi ^{(m)}(., h, t)}\ \sharp _h \
e^{\Phi ^{(m)}(., h, -t)}\ =\ 1 + h^m U_m(., h, t)\leqno (8.1)$$
satisfies
$${1\over n}\ \Vert U_m (., h, t)\Vert _a
\ +\ \Vert \nabla U_m (., h, t)\Vert _a
\ \leq \ C \leqno (8.2)$$
is $nh^m \leq \varepsilon _0$ and $ | t | \leq 1$.
}
\bigskip
\noindent
{\it Proof.} By theorem 4.2, there exists $\Psi ^{(m)}(., h, t)$ in
$\Sigma (a)$ such that, if $nh^m$ is small enough,
$$e^{\Phi ^{(m)}(., h, t)}\ \sharp _h \
e^{\Phi ^{(m)}(., h, -t)}\ =\
e^{\Psi ^{(m)}(., h, t)}\leqno (8.3)$$
Moreover, $\Psi ^{(m)}$ has an asymptotic expansion
$$\Psi ^{(m)}(., h, t) \ =\ \sum _{k=0}^{m-1}
\psi _k(x, \xi, t) h^k\ +\ h^m \widetilde \Psi _m(x, \xi, h, t)$$
where $\Vert \widetilde \Psi _m \Vert _a \leq Cn$.
In order to find the $\psi _k$, we differentiate (8.3) with respect
to $t$, using (5.5) and (5.13)(5.13) (with $t$ replaced by $-t$). We
obtain, for each $(x, \xi , t)$
$$e^{\Psi ^{(m)}(x, \xi, h, t)} {\partial \Psi ^{(m)}
\over \partial t} (x, \xi, h, t)\ =\ {\cal O}(h^m)$$
In other words, $\psi _k=0$ for $k\leq m-1$, and
$\Psi^{(m)}=h^m \widetilde \Psi ^{(m)}$. Therefore, if $nh^m$ is small
enough, there exists a function $U_m(., h, t)$ in
$\Sigma (a)$ such that $e^{h^m \widetilde \Psi ^{(m)}} =
1 + h^m U_m(., h, t)$, and (8.1) is satisfied. The lemma is proved.
\bigskip
\noindent
{\it End of the proof of Theorem 2.2.}
Let $a\in ]0, b[$. Let $a_1$, $a_2$ and $a_3$ such that
$a0$, $C>0$, and a
function $V_m(., h, t)$ in $\Sigma (a_3)$ such that,
if $nh^m \leq \varepsilon _0$
$$ \big ( 1 + h^m U_m(., h, t)\big )\ \sharp _h \
\big (1 + h^m V_m(., h, t) \big )\ =\ 1
\hskip 1cm
\Vert V_m(., h, t)\Vert _{a_3}\ \leq Cn \leqno (8.4)$$
Let us set
$$f_m(., h, t)\ =\ e^{\Phi ^{(m)}(., h, -t)}\ \sharp _h\
\left (1 + h^m V_m(., h, t) \right ) \ \sharp _h\
\left ( h^m e^{\Phi^{(m)}(., h, t)}
S^{(m)}(., h, t) \right )$$
where $S^{(m)}$ is the function of Proposition 5.2. If $nh^m$ is small
enough, we can write
$$1 + h^m V_m(., h, t) \ =\ e^{h^m \widetilde V_m(., h, t)}
\hskip 1cm
1 + h^m S_m(., h, t) \ =\ e^{h^m \widetilde S_m(., h, t)}$$
where $\widetilde V_m$ and $\widetilde S_m$ are in $\Sigma (a_3)$.
Therefore
$$1+f_m(., h, t)\ =\
e^{\Phi ^{(m)}(., h, -t)}\ \sharp _h\
e^{h^m \widetilde V_m(., h, t)}\ \sharp _h \
e^{\Phi^{(m)}(., h, t)+h^m \widetilde S_m(., h, t) }\leqno (8.5)$$
By Theorem 4.2 and Proposition 4.4, if $nh^m$ is small enough, there
exists
$F^{(m)}(., h, t)$ in $\Sigma (a_2)$ such that
$1+f_m(., h, t)= e^{F^{(m)}(., h, t)}$. Moreover, $F^{(m)}$ has
an asymptotic expansion
$$F^{(m)}(x, \xi , h, t)\ = \ \sum _{k=0}^{m-1}
a_k(x, \xi , t)h^k\ +\ h^m \widetilde F^{(m)}(., h, t)$$
where $\Vert \widetilde F^{(m)}(., t, h)\Vert _{a_2} \leq Cn$. By
Proposition 4.3, the coefficients $a_k$ are the same as if
$\widetilde V_m $ and $\widetilde S_m$ were replaced by $0$ in (8.5).
Hence, by Lemma 8.1, we have $a_k=0$ for $k\leq m-1$. Therefore
$$\Vert f_m(., h, t)\Vert _{a_2}\ = \
\Vert e^{h^m \widetilde F^{(m)}(., h, t)}-1\Vert _{a_2}
\ \leq \ Cnh^m$$
Therefore, by Proposition 7.1, if $nh^m$ is small enough, there exists
an holomorphic function $v_m(., h, t)$ in $\Omega _{2n}(a_1)$ such that
$${\partial v_m\over \partial t }(., h, t) \ =\
- f_m(., h, t)\ \sharp _h\ v_m(., h, t)
\hskip 1cm
v_m(., h, 0)\ =\ 1\leqno (8.6)$$
$$\Vert v_m(., h, t)\ -\ 1\Vert _{a_1} \ \leq \ Cnh^m
\leqno (8.7)$$
where $C>0$ is independent on $n$. By (5.6), (8.6), (8.5), (8.1)
and (8.4), the function
$u_m(., h, t)= e^{\Phi^{(m)}(., h, t) }\sharp _hv_m(., h, t)$
satisfies
$${\partial u_m\over \partial t}(., h, t)\ =\
p \ \sharp _h\ u_m(., h, t)
\hskip 1cm
u_m(., h, 0)\ =\ 1$$
Hence, $u_m(., h, t)$ is the symbol of the operator $e^{tP_n(h)}$.
Now, let us study the symbol
$$w_m(., h, t)\ =\ e^{-\Phi ^{(m)}(., h, t)} \ u_m(., h, t)\ =\
e^{-\Phi ^{(m)}(., h, t)}\
\Big [ e^{\Phi^{(m)}(., h, t) }\sharp _hv_m(., h, t) \Big ] $$
For that, we use the argument of the first step in the proof of
Proposition 5.2, and the function $Q_{\Phi ^{(m)}}$ satisfying (5.9),
(with $\Phi = \Phi ^{(m)}$). With these notations, we can write
$$w_m(X, h, t)\ =\ (\pi h)^{-2n} \ \int _{ {\bf R} ^{4n}}
e^{-{2i\over h} \sigma (Y, Z)}\ e^{\sigma (Q_{\Phi^{(m)}}(X, Y), Y)}
\ v_m(X+Z, h, t)\ dYdZ$$
and, by a change of contour which is allowed here
$$w_m(X, h, t)\ -\ 1\ = \ (\pi h)^{-2n} \ \int _{ {\bf R} ^{4n}}
e^{-{2i\over h} \sigma (Y, Z)}\
\Big [ v_m(X+Z+ {ih\over 2} Q_{\Phi^{(m)}}(X, Y), h, t)\ -\ 1
\Big ] \ dYdZ$$
By (8.7), there exist $C>0$ and $\rho >0$ such that
$$ | v_m(X+Z+ {ih\over 2} Q_{\Phi^{(m)}}(X, Y), h, t)\ -\ 1 |
\ \leq \ C nh^m$$
if $X$ is in $\Omega _{2n}(a)$ and $Y$ and $Z$ in
$\Omega _{2n}(\rho )$. By Proposition 3.2, it follows that
$$\Vert w_m(., h, t) \ -\ 1 \Vert _a \ \leq \
Cnh^m\ \left ( 1 \ +\ {\sqrt {h}\over
\rho } e^{-{\rho ^2\over h }}\right )^{4n}$$
If $nh^m$ is small enough, it follows that
$$\Vert w_m(., h, t)\ -\ 1\Vert _a
\ \leq \ Ce^4nh^m \ \leq \ {1\over 2}$$
Therefore we can write $w_m= e^{ h^m \widetilde w_m}$, where
$\widetilde w_m$ is in $\Sigma (a)$. The symbol of
$e^{tP_n(h)}$ can be written $e^{ \Phi ^{(m)}(x, \xi, h, t)
+ h^m \widetilde w_m(x, \xi, h, t)}$. The Theorem 2.2 is proved.
\bigskip
\noindent
{\bf 9. Periodic symbols and 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 a bounded holomorphic
function in $\Omega _{2k}(a)$, (resp. $\Omega _{4k}(a)$), defined as
in (1.4). We assume also that there exist a lattices $\Gamma $
in $ {\bf R} ^{2k}$, such that
$ {\bf R} ^{2k}/\Gamma $ is compact,
ans such that $A$ (resp. B) is periodic with respect to
$\Gamma $ (resp. $\Gamma \times \Gamma $).
We assume also that $A$ and $B$ are real for real $(x, \xi)$, and that
$B(y, \eta, x, \xi)= B(x, \xi , y, \eta)$.
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
$X^{(n+1)}=X^{(-n)}$,
$$p_n(X )\ =\ \sum _{j=-n}^n
\Big ( A(X^{(j)}) \ + B(X^{(j)}, X^{(j+1)}) \Big ) \leqno (9.1)$$
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).
We see easily that the sequences
$(p_n)$ and $(\widetilde p_n)$ are in $\Sigma (a)$.
\bigskip
We want to associate, to the sequence of operators $P_n(h)$, a
notion of thermodynamic limit. A classical possible definition of a
thermodynamic limit is the limit of (1.7), if it exists.
Unfortunately, $e^{-tP_n(h)}$ is not here of trace class. 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 $e^{-q_n(x, \xi , t, h)}$.
$$e^{-tP_n(h)} \ = \ Op_h \left ( e^{-q_n(. , t, h)}\right )
\leqno (9.3)$$
and we have also a symbol $\widetilde q_n(x, \xi , t, h)$
with similar properties for $\widetilde P_n(h)$.
We see easily that
$q_n$ and $\widetilde q_n$ are periodic with the lattice
$\Gamma _n= (\Gamma )^{2n+1}$
of $ {\bf R} ^{2k(2n+1)}$.
\bigskip
It is natural to associate to a
$h-$pseudodifferential
operator $A_n(h)$ associated, by the Weyl calculus, to a symbol
$a_n(x, \xi)$ which is periodic with respect to $\Gamma _n$,
not the usual trace, but the following one
$$ T \big ( A(h) \big ) \ =\
\ {1 \over | E_n | }\ \int _{E_n} a_n(x, \xi ) \ dx d\xi \leqno (9.4)$$
where $E_n$ is a fundamental domain of $\Gamma _n$.
If $p$ and $q$ are $ C^{ \infty } $, periodic with respect to $\Gamma
_n$, and if $P(h)$ and $Q(h)$ are the corresponding
$h-$pseudodifferential operators, we have
$$T(P(h)\circ Q(h)) \ =\ {1 \over | E_n | }\
\int _{E_n} p(x, \xi)q(x, \xi)\ dxd\xi\ =\
T(Q(h)\circ P(h))\leqno (9.5)$$
If, moreover, $P(h)$ is a positive self-adjoint operator,
we have
$$\vert T(P(h)\circ Q(h)) \vert \ \leq \
T(P(h))\ \Vert Q(h) \Vert \leqno (9.6)$$
where $ \Vert Q(h) \Vert$ is the norm of $ Q(h) $
as an operator in $L^2({\bf R}^n)$.
\bigskip
This notion of trace is
often used in the literature of $C^*-$algebras.
Therefore, it is natural to define, for our sequence $P_n(h)$, the
thermodynamic limit as the limit, if it exists, of
$$\Lambda _n(t, h)\ =\ {1\over 2n+1} Log \Big [ T\big ( e^{-tP_n(h)}
\big ) \Big ] \leqno (9.7)$$
where $T$ is the 'trace' defined in (9.4). We define also a similar
sequence
$\widetilde \Lambda _n(t, h)$ for the operator $\widetilde P_n(h)$.
\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 (9.7) 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
$$ | \Lambda _n (t, h)\ - \ \Lambda (t, h) | \ +\
| \widetilde \Lambda _n(t, h) \ -\ \Lambda (t, h) | \
\leq \ {C \over n}
\hskip 1cm
\forall h\in ]0, 1], \ \ \ \ \ \
\forall t\in [0, 1] \leqno (9.8)$$
}
\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
$n\geq 2$, for each $h$ and $t$ in $]0, 1]$
$$ \Delta _{mn}(t, h)\ :=\ | (2m+2n+1) \widetilde \Lambda _{m+n}(t, h) \ -\
(2m+1)\widetilde \Lambda _m (t, h)\ -\ (2n+1)\widetilde \Lambda _n(t, h) | \
\leq \ C\leqno (9.9)$$
}
\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
$$(2m+2n+1) \widetilde \Lambda _{m+n}(t, h)\ =\
Log \left [ T\big ( e^{-tA_{mn}(h) } \big ) \right ]
\ -\ Log \left [ T\big ( e^{-tP_0(h) } \big ) \right ]$$
$$
(2m+1)\widetilde \Lambda _m (t, h) + (2n+1) \widetilde \Lambda _n (t, h) =\ Log \left [
T\big ( e^{-tA_{mn}(h) +tB_{mn}(h)} \big ) \right ] $$
If we set $F(\theta )= Log \Big ( T \big ( e^{-tA_{mn}(h) + \theta t B_{mn}(h)}\big ) \Big )$,
we can write, for the left hand side $\Delta _{mn}(t, h)$ of (9.9)
$$\Delta _{mn}(t, h) \ \leq \
| F(1)-F(0) | \ +\ \left | Log \ T \left (e^{-tP_0(h)} \right )
\right | $$
Using (9.5), we obtain
$$ | F(1)-F(0) | \leq \ \sup _{\theta \in [0, 1]} | F'(\theta) |
\ =\ \sup _{\theta \in [0, 1] }
{ | T\big (t B_{mn}(h) e^{-tA_{mn}(h) +t\theta B_{mn}(h) } \big ) | \over
| T\big ( e^{-tA_{mn}(h) +t\theta B_{mn}(h) } \big ) | } $$
By (9.6), since $e^{-tA_{mn}(h) +t\theta B_{mn}(h) }$
is a positive self-adjoint operator, we have
$$ | T\big ( C_{mn}(h) e^{-tA_{mn}(h) + t\theta B_{mn}(h)} \big ) | \ \leq \
\Vert B_{mn}(h)\Vert \
| T\big ( e^{-tA_{mn}(h)+ t\theta B_{mn}(h) } \big ) |$$
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 (9.10), it follows classically (cf. Helffer-Sj\"ostrand
[3], 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 (9.8) 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
$L^2(E)$ (where $E$ is a fundamental domain of
the lattice $\Gamma $), defined by
$$(S_0(t)u)(X)\ =\ {1 \over | E | } \
\int _{E} e^{ -{t\over 2}( A(X) + 2 B(X, Y) + A(Y))}
\ u(Y)\ dY \hskip 1cm \forall u\in L^2(E)\leqno (9.10)$$
This operator 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 $h\rightarrow 0$
$$\Lambda (t, h)\ \sim \ \sum _{j\geq 0} \gamma _j (t) h^j \leqno (9.11)$$
where $\gamma _j(t)$ are real numbers, and
$\gamma _0(t) =Log ( \Vert S_0(t) \Vert )$, where $S_0(t)$ defined in (9.10). }
\bigskip
If $B=0$, (when there is no interaction), the `classical'
thermodynamic limit $\gamma _0(t)$ is given by
$$\gamma _0(t)\ =\ Log\ \left ( {1\over | E_0 | } \int _{E_0}
e^{-tA(X)}dX \right )$$
\bigskip
\noindent
{\it Formal construction of the $\gamma _j(t)$.}
By theorem 2.2, we know that $ q_n(x, \xi , t, h)$, (the symbol
satisfying (9.3)), has the
asymptotic expansion (2.7) with coefficients $E_n^{(j)}(x, \xi , t)$.
By the construction of section 5, we know that
$E_n^{(0)}= t\widetilde p_n$, $E_n^{(1)}= 0$, and that
$E_n^{(2)}$ is given like in (5.4), where $p$ is replaced by
$ p_n$ defined in (9.2). By (5.4) 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 in $S(a)$ (holomorphic
with bounded derivatives in
$\{ X\in {\bf C} ^{8k}, | Im\ X | _{\infty } < b\}$ for all
$b\in ]0, a[$). More generally, if we follow the induction in the
proof of Proposition 5.1, 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
$$q_n ^{(m)}(x, \xi , t, h)\ :=\
\sum _{j=0}^{m}E_j(x, \xi , t)h^j\ =\
\sum _{j=-n}^n F^{(m)}(X^{(j)}, \ldots , X^{(j+ d_m)}, t, h)\leqno (9.12)$$
For example, we have $d_0= d_1 = 1$, $d_2=3$. The approximation
$q_n ^{(m)}$ can be written in the form (9.12), but not in the unique
way. For $m=0$, $F^{(0)}$ (which is independent on $h$) can be
written
$$F^{(0)} (X, t)\ =\ t \left ( {A(X^{(1)})\over 2}\ +\ B(X^{(1)},
X^{(2)})\ +\ {A(X^{(2)})\over 2}\right )$$
For $m$ arbitrary and $h=0$, we can take, for all $X$ in
${\bf R} ^{2k(1+d_m)} $
$$F^{(m)}(X^{(j)}, \ldots , X^{(j+ d_m)}, t, 0)\ =\
{t\over d_m+1}\ \sum _{j=1}^{1+d_m} A(X^{(j)})\ +\ {t\over d_m}\
\sum _{j=1}^{d_m} B(X^{(j)}, X^{(j+1)})
\leqno (9.13)$$
Using this form of the coefficients $E_j(X, t)$, we shall define, for
each $m$, an integral operator whose greatest eigenvalue will have, up
to the order $m$, the same asymptotic expansion in $h$ that the
exponential of the thermodynamic limit.
For that, we introduce the operator
$S_m(t, h)$ in $L^2(E^{d_m})$ (where $E$ is a fondamental domain of
$\Gamma $) defined, for all $u\in L^2(E^{d_m})$ by
$$(S_m(t, h) u) (Y)\ =\
{1\over | E | ^{d_m}} \
\int _{E^{d_m}} e^{-\Phi _m(X, Y, t, h)} u(X)\ dX\leqno (9.14)$$
where we set $X= (X^{(1)}, \ldots , X^{(d_m)})$,
$Y= (Y^{(1)}, \ldots, Y^{(d_m)})$, $X^{(j+d_m)}= Y^{(j)}$ and:
$$\Phi _m(X, Y, t, h)\ =\ \sum _{j=1}^{d_m}
F^{(m)} (X^{(j)}, \ldots , X^{(j+d_m)}, t, h)\leqno (9.15)$$
We shall prove that $S_m(t, 0)$ is almost isospectral to
the power $S_0(t)^{d_m}$ of $S_0(t)$ defined in (9.10).
We remark that, for each integer $n$, $S_0(t)$ satisfies
$$Tr (S_0(t)^n)\ =\ {1\over | E | ^n}\ \int _{E^n} e^{-t\Psi _n(X)} dX$$
where $Tr$ is the usual trace of operators, (here in $L^2(E^n)$), and
$$\Psi _n(X)\ =\ B(X^{(n)}, X^{(1)})\ +\
\sum _{j=1}^n A(X^{(j)})\ +\ \sum _{j=1} ^{(n-1)} B(X^{(j)},
X^{(j+1)})\hskip 1cm \forall X\in {\bf R} ^{2nk}$$
We remark also that $S_m(t,h)$ defined in (9.14) is of trace class
and that, for each integer $q$, we have
$$Tr( S_m(h)^q)\ =\ {1\over | E | ^{qd_m}}\ \int _{E^{qd_m}}
e^{-\theta _q(X,t, h)}dX\leqno (9.16)$$
where, setting now $X ^{(qd_m+j)}=X^{(j)}$ for $j\geq 1$
$$\theta _q(X, t, h)\ =\
\sum _{j=1}^{qd_m} F^{(m)} (X^{(j)}, \ldots , X^{(j+d_m)}, t,
h)\leqno (9.17)$$
It follows from (9.13) that
$\theta _q(X, t, 0)= t \Psi _{qd_m} (X)$ for all $X$ in
$ {\bf R} ^{2kqd_m}$, and therefore
$S_m(t, 0)$ is related to the operator $S_0(t)$ of (9.10),
for each integer $q$,
by
$$Tr (S_m(t, 0)^q)\ =\ Tr (S_0(t)^{qd_m})$$
Therefore, $S_m(t, 0)$
has $\Vert S_0(t)\Vert ^{d_m}$ as a simple eigenvalue,
and all others eigenvalues have a modulus strictly
smaller than $\Vert S_0(t)\Vert ^{d_m}$. 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 $\lambda _m(t, 0)= \Vert S_0(t)\Vert ^{d_m}$.
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 9.4.} {\it With the previous notations, we have, for each
integer $m$, and for $h$ small enough
$$\Lambda (t, h)\ =\ {1\over d_m} Log\ (\lambda _m(t, h)) \
+\ {\cal O} (h^m)\leqno (9.18)$$}
\bigskip
\noindent
{\it Proof. First step. }
It is natural to approximate $\Lambda _n(t, h)$
(defined in (9.7)), by
$$\Lambda _n^{(m)}(t, h)\ =\ {1\over 2n+1}\ Log\ \left [ {1 \over |
E_n | }\
\ \int _{E_n}
e^{- q_n^{(m)}(x, \xi , t, h)}\ dxd\xi \right ] .
\leqno (9.19)$$
where $q_n^{(m)}$ is defined in (9.12). We shall prove that
$$\left | \Lambda _n^{(m)} (t, h)\ - {1\over d_m} Log\ (\lambda _m
(t, h))
\right | \ \leq \ {C\over n}\leqno (9.20)$$
for some constant $C$.
Following the proof of Lemma 9.2, we see that
$\Lambda _n^{(m)}(t, h)$ satisfies an inequality
analogous to (9.9). By the Lemma 2.5 of [3], $\Lambda _n^{(m)}(t, h)$ has a limit
$\mu (t, h)$ when $n\rightarrow +\infty $ and, for some constant
$C>0$, we have $ | \Lambda _n^{(m)}(t, h)- \mu (t, h) | \leq C/n$.
Let us show that $\mu (t, h)= (1/d_m)Log\ \lambda _m(t,h))$.
By (9.18), (9.12), (9.17) and (9.16), we can write, when $2n+1= kd_m$
$$
\Lambda _{n}^{(m)}(t, h) \ =\ {1\over kd_m}\ Log\ \left [
\ Tr \ ( S_m(h)^k ) \right ]\hskip 1cm
2n+1= kd_m.$$
Since $S_m(h)$ is of trace class, since $\lambda _m(t, h)$ is a single
eigenvalue and since all the other eigenvalues have,
for $h$ small enough, a modulus strictly smaller than
$ | \lambda _m(t, h) | $, it follows that
$$\lim _{k\rightarrow +\infty } {1\over k} Log \left (
Tr \ (S_m(h)^k) \right ) \ =\
Log\ ( \lambda_m(t, h))$$
and, therefore, $\mu (t, h)= (1/d_m)Log\ \lambda _m(t, h))$,
and (9.20) is proved.
\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 $h$, $n$ and $t$ satisfying
$0\leq t \leq 1$, $00$ and $K_m>0$
$$nh^m \ \leq \ \varepsilon _m\ \Longrightarrow
| \Lambda _n(h)\ -\ \Lambda _n^{(m)}(h) | \ \leq \
K_mh^m\leqno (9.22)$$
For each $h\in ]0, 1]$, let
$n= n(h)$ be the integer part of $\varepsilon _m / h^m$, where
$\varepsilon _m$ is a constant such that (9.22) is valid. Then we have
$$ | \Lambda (t, h)\ -\ {1\over d_m} Log\ \lambda _m(t, h)) | \
\leq \
| \Lambda (t, h)\ -\ \Lambda _n(h) | \ +\
| \Lambda _n(h)\ -\ \Lambda _n^{(m)}(h) | \ +\ \ldots $$
$$\ldots \ +\
| \Lambda _n^{(m)}(h)\ -\ {1\over d_m} Log\ \lambda _m(t, h)) | $$
Using (9.8), (9.22) and (9.20), 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
\centerline {\bf References.}
\bigskip
\noindent
[1]L. AMOUR, Ph. KERDELHUE, J. NOURRIGAT, Calcul pseudodiff\'erentiel
en grande dimension. {\it Asymptotic Analysis,} {\bf 26} (2001), p.
135-161.
\smallskip
\noindent
[2] B. HELFFER, D. ROBERT, Calcul fonctionnel par la transformation de
Mellin et op\'erateurs admissibles,
{\it Journal of Functional Analysis}, {\bf 53}, 3), (1983), 246-268.
\smallskip
\noindent
[3] B. HELFFER, J. SJ\"OSTRAND, Semiclassical expansions of the
thermodynamic limit for a Schr\"odinger equation. I. The one
well case. {\it M\'ethodes semi-classiques, Volume 2}, Ast\'erique
210, S.M.F. (Paris), 1992.
\smallskip
\noindent
[4] B. LASCAR, Op\'erateurs pseudo-diff\'erentiels en dimension
infinie. {\it Journal d'Analyse Math.} {\bf 33}, (1978), 39-104.
\smallskip
\noindent
[5] E. LIEB, The classical limit of quantum spin systems. {Comm.
Math. Phys,} {\bf 31} (1973), 327-340.
\smallskip
\noindent
[6] Ch. ROYER, Formes quadratiques et calcul pseudodiff\'erentiel
en grande dimension. {\it Pr\'e\-publication 00.05.} Reims, 2000.
\smallskip
\noindent
[7] D. RUELLE, {\it Statistical Mechanics: rigorous results.}
Addison-Wesley, 1969.
\smallskip
\noindent
[8] B. SIMON, The classical limit of quantum partition functions.
{\it Comm. Math. Phys,} {\bf 71} (1980), 247-276.
\smallskip
\noindent
[9] J. SJ\"OSTRAND, Potential wells in high dimension I,
{\it Ann. I. H. P. Phys. Th.} {\bf 58}, (1)
(1993), 1-43.
\smallskip
\noindent
[10] J. SJ\"OSTRAND, Evolution equations in a large number
of variables, {\it Math. Nachr.} {\bf 166} (1994), 17-53.
\smallskip
\noindent
[11] M. TODA, {\it Theory of nonlinear lattices.} Springer, 1981.
\bigskip
\hskip 6cm
Jean Nourrigat
\par
\hskip 6cm
D\'epartement de Math\'ematiques (UMR CNRS 6056)
\hskip 6cm
Universit\'e de Reims
\hskip 6cm
Moulin de la Housse
\hskip 6cm
B.P. 1039
\hskip 6cm
51687 Reims Cedex, France
\hskip 6cm
jean.nourrigat@univ-reims.fr
\end
\noindent
[11] J. SJ\"OSTRAND, W.M. WANG, Supersymmetric measures and maximum
principles in the complex domain. Exponential decay of Green's
functions. {\it Ann. Sc. Ec. Norm. Sup}, {\bf 32}, (1999), 347-414.
\smallskip
\noindent
[12]E. BEDOS, An introduction to 3D discrete magnetic Laplacians
and non commutative 3-tori. {\it J. Geom. Phys,} 30 (1999),
3, 204-232.
\smallskip
\noindent
[13] B. HELFFER, J. SJ\"OSTRAND, Equation de Schr\"odinger avec champ
magn\'etique et \'equation de Harper. {\it Schr\"odinger Operators,
Proceedings, S\o nderborg 1988}, H. Holden et A. Jensen, ed. Lecture
Notes in Physics 345, Springer.
\smallskip
\noindent
[14]M. A. SHUBIN, Discrete magnetic Laplacian. {\it Comm. Math.
Phys,} 164 (1994), 2, 259-275.
\smallskip
\noindent
[15]Y. COLIN DE VERDIERE, {\it Spectres de graphes}. Cours
sp\'ecialis\'es. S.M.F, 1998.
\end
\noindent
[3]J. SJOSTRAND, {\it Singularit\'es analytiques microlocales.}
Ast\'erisque 95, S.M.F. (Paris), 1982.
\smallskip
\noindent
[1] D. FUJIWARA, The stationary phase method with an estimate of the
remainder term on a space of large dimension. {\it Nagoya Math. J.}
{\bf 124}, (1991), 61-97.
\smallskip
\noindent
[2] B. HELFFER, Around a stationary phase theorem in large dimension.
{\it Journal of Functional Analysis}, {\bf 119}, (1) (1994), 217-252.
\smallskip
\bigskip
\noindent
{\bf 8. Magnetic matrices and thermodynamic limits.}
\bigskip
Let $\Gamma $ be a lattice on $ {\bf R} ^n$ such that $ {\bf R} ^n
/\Gamma $ is compact. Let $\Gamma ^*$ be the dual lattice
(the set of $\xi \in ( {\bf R} ^n)^*$ such that $\gamma . \xi $
is in $2\pi {\bf Z}$ for all $\gamma \in \Gamma$), and let
$E^* $ be a fundamental domain of $\Gamma ^*$. For each symbol
$p(x, \xi )$ in $S(a)$ $(a>0)$ such that $p(x, \xi + \gamma ^*)
=p(x, \xi)$ for all $\gamma ^*\in \Gamma ^*$, we define an operator
$R(a)$ in $\ell ^2 (\gamma )$ by setting, for each $u\in \ell ^2
(\Gamma)$
$$\Big ( R(p) \ u \Big ) (\alpha)\ =\
\sum _{\beta \in \Gamma } p_{\alpha , \beta } u(\beta )
\hskip 1cm
p_{\alpha , \beta }\ =\ {1 \over | E^* | }
\int _{ E^*} e^{i(\alpha - \beta ).\xi }
p\left ( {\alpha + \beta \over 2},\ \xi \right )\ d\xi $$
If $Op (p)$ denotes the operator (1.2) in $L^2( {\bf R} ^n)$,
(with $h=1$), we see that, for each $u\in {\cal S}( {\bf R} ^n)$,
denoting by $\rho ( u)$ the restriction of $u$ to $\Gamma $,
we have $\rho \big (Op (p)u \big )= R(p) \big ( \rho (u) \big )$.
This is an easy consequence of Poisson sommation formula.
Therefore, if $\sharp $ denotes the composition law of (3.1) (with
$h=1$), it follows that $R(p\ \sharp \ q) = R(p)\circ R(q)$ for all
$p$ and $q$ in $S(a)$.
Let $B $ a bilinear antisymmetric form on
$ {\bf R} ^n \times {\bf R} ^n$. For each $\alpha \in \Gamma $, we
denote by $T_{\alpha }^B$ the {\it magnetic translation} defined by
$$(T_{\alpha }^B u)(x) \ =\ e^{{i\over 2} **** }
u(x-\alpha )\leqno (9.1)$$
A {\it B-magnetic matrix} is an operator $A$ in $\ell ^2(\Gamma )$
which commutes with all the magnetic translations
$T_{\alpha }^B$ $(\alpha \in \Gamma)$. Such an operator can be
written on the following form, for some function $f_A$ on $\Gamma $
$$(Au)(\alpha ) \ =\ \sum _{\beta \in \Gamma }
e^{{i\over 2} **** }
f_A(\alpha - \beta)\ u(\beta)\leqno (9.2)$$
We are interested in the case where $f$ satisfies, for some
constant $C>0$
$$ | f_A(\alpha ) | \ \leq \ C e^{-{ | \alpha | \over C}}
\hskip 1cm
\forall \alpha \in \Gamma \leqno (9.3)$$
For each $B-$magnetic matrix $A$ such that $f_A$ satisfies
(4.3), we define a function $\sigma (A)$ in $ {\bf R} ^{2n}$ by
$$\sigma (A) (x, \xi)\ =\ \sum _{\alpha \in \Gamma }
f_A(\alpha )\ e^{i\ell _{\alpha }(x, \xi)}
\hskip 1cm
\ell _{\alpha } (x, \xi)\ =\ <\alpha , \xi > \ -\ {1\over 2}
****$$
\end
If $p_n$ is defined in (1.1), we set $P_n(h)= Op_h(p_n)$ and we
should like to know if, under suitable hypotheses, for $h$ small
enough, the sequence
has a limit, called {\it thermodynamic limit}, $\Lambda (h)$
(cf. Ruelle [8]).
Another problem is to know if $\Lambda (h)$ has, when $h\rightarrow 0$,
an asymptotic expansion in powers of $h$.
Another kind of thermodynamic limit is studied in [4].
\bigskip
We suppose here that $\Phi$ is a bounded holomorphic function on
$$\{ (x, y, \xi , \eta)\in \k ^4,
\ | Im\ (x, y, \xi , \eta ) | 0$.
In this case, the symbol $p_n$ defined in (1.1) belongs to the class
$S(a)$, already introduced in the paper [1] with L. Amour et
Ph. Kerdelhu\'e.
\bigskip
If $p_n$ is in the class $S(b)$ $(b>0)$, the trace of
$exp\ (tP_n(h))$ does not exist if we consider $P_n(h)$ as a bounded
operator in $L^2( {\bf R} ^n)$. However, if $p_n$ is of the
form (1.1), and if $\Phi $ is periodic with respect to some lattice
of $ {\bf R} ^4$, we may also consider $P_n(h)$
as an operator in $L^2({\bf Z}^n)$ and, in this case, the trace
of $exp\ (tP_n(h))$ exists. We shall see in section 8 that, in this
case, the thermodynamic
limit $\Lambda (h)$ exists and, applying
theorem 1.2, we shall see that it has an
asymptotic expansion in powers of $h$.
\vfill
\eject
such that the sequence $(V
This paper is devoted to the proof of a conjecture stated in a
previous work with L. Amour and Ph. Kerdelhu\'e [1], and to some
applications.
\bigskip
The first aim of this paper is the following. We shall define a set
$S(a)$ $(a>0)$, the elements of which are sequences $(p_n)_{n\geq 1}$,
where $p_n$ is some function in $ C^{ \infty } ( {\bf R} ^{2n})$,
satisfying some inequalities where the constants are independent on
$n$. The precise definition of $S(a)$ will be given later, with some
examples playing perhaps some role in Statistical Mechanics.
To each sequence $(p_n)$ in $S(a)$ we associate, as above, a sequence
$Op_h(p_n)$, where $Op_h(p_n)$ is a bounded operator in
$L^2( {\bf R} ^n)$, depending on the parameter $h$.
We shall prove
that there is another sequence of functions $(q_n(., t, h)$ in the
same class $S(a)$ such that
$$e^{tOp_h(p_n)}\ =\ Op_h\left ( e^{q_n(., t, h)}\right ) \leqno (1.2)$$
Unfortunately, this will be valid only if $n$ and $h$ satisfy some
inequalities (cf. Theorem 1.2 below).
\bigskip
See the paper [] of Sj\"ostrand for a study of the heat equation
in large dimension, for another class of symbols.
\bigskip
This result may be applied to systems of $n$ identical particles $A_1,
\ldots A_n$ in quantum
mechanics, each of them moving in $ {\bf R} ^k$, $(k\geq 1)$,
and interacting with
his two neighbours ($A_n$ may interact or not with $A_1$).
The
hamiltonian $p_n$ (function in $ {\bf R} ^{2kn}$) describing such a system
is of the following form (if there is an interaction between $A_n$
and $A_1$)
$$p_n(x, \xi )\ =\ \sum _{j=1}^{n} \Phi \big (x^{(j)}, x^{(j+1)},
\xi ^{(j)}, \xi ^{(j+1)} \big )\leqno (1.3)$$
where $x= (x^{(1)}, \ldots , x^{(n)})$ denotes the variable
of $ {\bf R} ^{kn}$ (with $x^{(j)}\in {\bf R} ^k$ and
$x^{(n+1)}= x^{(1)}$), and
$\Phi (x, y, \xi , \eta)$ is a $ C^{ \infty } $ function on
$ {\bf R} ^{4k}$.
\bigskip
The class of symbols $S(a)$, (already introduced in the paper [] with
L. Amour and Ph. Kerdelhu\'e), is defined in such a way that it
contains sequences of functions $(p_n)_{n\geq 1}$ of the previous form
if $\Phi (x, y, \xi , \eta)$ extends to a bounded holomorphic function
in $\Omega _{4k}(a)= \{ (x, y, \xi , \eta) \in {\bf C}^{4k}, \
| Im (x, y, \xi , \eta) | _{\infty } < a \}$, where $ | \ |
_{\infty }$ denotes the $\ell _{\infty }$ norm.
\bigskip
In Quantum Statistical Mechanics, a notion of thermodynamic limit
can be defined for some sequences $(P_n(h))_{n\geq 1}$ where
$P_n(h)$ is some operator in $L^2 ( {\bf R} ^n)$. It is the following limit
$\Lambda (h)$, if it exists
$$\Lambda (h)\ =\ \lim _{ n \rightarrow +\infty }
{1\over n} Log \Big [ (2\pi h)^n \
Tr\ e^{-tP^{(n)}(h)} \Big ] \leqno (1.4)$$
(where $t>0$ is fixed). The two main problems are the existence of the
thermodynamical limit $\Lambda (h)$, and the existence of its
asymptotic expansion in powers of $h$ when $h\rightarrow 0$.
\bigskip
The second aim of this paper is to prove, for the sequence of
operators $P_n(h)= Op_h(p_n)$, where $p_n$ is of the form (1.3), where
$\Phi $ is a bounded holomorphic function in $\Omega _{4k}(a)$, which
is periodic with repect to some lattice of $ {\bf R} ^{4k}$, the
existence of a (suitably modified) thermodynamic limit
$\Lambda (h)$, and of its asymptotic expansion in powers of $h$, up to any
order, when $h\rightarrow 0$. We cannot use the notion of
thermodynamic limit defined as above, since our operator $P_n(h)$ is
bounded, and the trace in (1.4) does not exist. We have only to replace,
in (1.4), the usual notion of trace by another one, commonly used
for pseudodifferential operators with periodic symbols (cf. section 8).
\bigskip
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**