%This document is written in LATEX . Texte principal.
\documentstyle[12pt]{article}
%
\def\nz{\ifmmode {I\hskip -3pt N} \else {\hbox {$I\hskip -3pt N$}}\fi}
\def\zz{\ifmmode {Z\hskip -4.8pt Z} \else
{\hbox {$Z\hskip -4.8pt Z$}}\fi}
\def\qz{\ifmmode {Q\hskip -5.0pt\vrule height6.0pt depth 0pt
\hskip 6pt} \'else {\hbox
{$Q\hskip -5.0pt\vrule height6.0pt depth 0pt\hskip 6pt$}}\fi}
\def\rz{\ifmmode {I\hskip -3pt R} \else {\hbox {$I\hskip -3pt R$}}\fi}
\def\cz{\ifmmode {C\hskip -4.8pt\vrule height5.8pt\hskip 6.3pt} \else
{\hbox {$C\hskip -4.8pt\vrule height5.8pt\hskip 6.3pt$}}\fi}
% divers Macros
\def\const{{\rm const.\,}} % constante
\def\div{{\rm div}\,}% divergence
\def\Hess{{\rm Hess\,}}% Hessian
\def\supp{\mathop{\rm supp} \nolimits} % Support
\def\tr{\mathop{\rm Tr \;} \nolimits} % Trace
\def\Tr{\mathop{\rm Tr \;} \nolimits} % Trace
\def\var{{\rm var}\;}% variance
\def\Cor{{\rm Cor}\,}%correlation
%
\def\qed{\hbox {\hskip 1pt \vrule width 4pt height 6pt depth 1.5pt
\hskip 1pt}\\}% cqfd
%
\def\Ag {{\cal A}} %A gothique
\def\Bg {{\cal B}} %B gothique
\def\Cg {{\cal C}} %C gothique
\def\Fg {{\cal F}}
\def\Hg {{\cal H}} %H gothique
\def\Hb {{\bf H}} %H
\def\hb {{\bf h}} %H
\def\Kb {{\bf K}} %K
\def\Ig {{\cal I}} %I gothique
\def\Jg {{\cal J}} %J gothique
\def\Kg {{\cal K}} %K gothique
\def\Lg {{\cal L}} %L gothique
\def\Ng {{\cal N}} %N gothique
\def\Og {{\cal O}} % O gothique ou grand O
\def\Pg {{\cal P}} %P gothique
\def\Qg {{\cal Q}} %Q gothique
\def\Rg {{\cal R}} %R gothique
\def\Rb {{\bf R}} %R bold
\def\Sg {{\cal S}} %S gothique
\def\Sb {{\bf S}} % S bold
\def\Vg {{\cal V}} %V gothique
\def\Wg {{\cal W}} %V gothique
%
\def\For {{\rm \;for\;}}
\def\for {{\rm \;for\;}}
\def\Or {{\rm \;or\;}}
\def\If {{\rm \;if \;}}
\def\and {{\rm \; and \;}}
\def\dist {{\rm \; dist \;}}
\def\then {{\rm \; then \;}}
\def\with {{\rm \; with \;}}
%
\newcommand {\pa}{\partial}
\newcommand {\ar}{\rightarrow}
\newcommand {\Ar}{\Rightarrow}
\newcommand {\ot}{\otimes}
\newcommand{\tnp}[1]{#1\index{#1}}
% changement de la numerotation
\makeatletter
\@addtoreset{equation}{section}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\makeatother
%
%\def\baselinestretch{2}
% Environments
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\title{Splitting in large dimension and infrared estimates II - Moment
inequalities}
\author{B. HELFFER\\
UA 760 du CNRS, D\'epartement de
math\'ematiques,\\
B\^at 425,\\
F-91405 Orsay C\'edex, FRANCE}
\date {March 18, 1997}
\begin{document}
\bibliographystyle{plain}
\maketitle
%
%RESUME
%
\begin{abstract}
This is the continuation of notes written for the NATO-ASI conference
in Il Ciocco (Sept. 96) consisting in the analysis of the links between estimating the
splitting between the two first eigenvalues for the
Schr\"odinger operator $H$ and the proof of infrared estimates for quantities attached
to Gaussian type measures. These notes were mainly reporting on the ``old''
contributions
of Dyson, Fr\"ohlich, Glimm, Jaffe, Lieb, Simon, Spencer (in the
seventies) in
connection with more recent
contributions of Pastur, Khoruzhenko, Barbulyak, Kondratev which
treat in general more sophisticated models. Here we concentrate on the simplest
model related to field theory and extend the
results of Barbulyak-Kondratev by mixing ideas coming from
Pastur-Khozurenko
related to the use of Bogolyubov's inequality with classical
inequalities due to Ginibre, Lebowitz, Sokal.... or in the case when
the temperature $T$ is zero by applying rather elementary
estimates
on Schr\"odinger operators, in order to find lower bounds for
second order moments attached to the measure $\phi \mapsto \Tr \phi
\exp - \beta H/\tr \exp - \beta H$ with $\beta=\frac 1T$. This
question
was
``left to the reader'' in lectures given by J. Fr\"ohlich in 1976
\cite{Fr}, but we think that it is worthwhile to do this ``home work''
carefully.
\end{abstract}
% %FIN DE RESUME %
\section{Introduction}\label{Sectiona}
Our aim is to understand in the large dimension limit $L \ar +\infty$
the splitting between the two lowest eigenvalues
of the following Schr\"odinger operator
\begin{equation}\label{a.1}
H_\Lambda^{per}:=- h^2 \Delta + \sum_{j\in \Lambda}v (x_j) + \frac{\Jg}{2} \sum_{j\sim k} |x_j-x_{k}|^2
\end{equation}
where
\begin{itemize}
\item $\Lambda$ is a square periodic lattice $\Lambda:= \Lambda(L) =
(\zz/L\zz)^d$ identified (up to translation)
with
$\{ 0,\cdots, L-1\}^d$ in $\zz^d$,
\item $j\sim k$ means $j$ nearest neighbor of $k$ in $\Lambda$ (seen
as a periodic lattice),
\item $v$ is a one particle even potential on $\rz^n$ defining as
typical cases either a single well or
a symmetric double well,
\item $\Jg$ is the intensity of the interaction.
\end{itemize}
As well known, the first eigenvalue of the Schr\"odinger operator is simple, so the splitting is always strictly positive.
Its behavior with respect to $L$ as $L\ar +\infty$ depends actually heavily on the nature of $v$ and on the size on $\Jg$ which is always assumed positive.
\\
The parameter $\Jg$
satisfies
the condition
\begin{equation}\label{a.2}
\Jg >0\;.
\end{equation}
The case $\Jg = 0$, which could be considered for comparison,
corresponds to the case without interaction and the analysis in this
case is immediately reduced to the semi-classical analysis of the
so-called one-particle Hamiltonian on $L^2(\rz^n)$
\begin{equation}
\label{a1.3}
H_0:= - h^2 \Delta_y + v(y)\;.
\end{equation}
This model can also be written as
\begin{equation}\label{a.4}
H_\Lambda^{per}= - h^2 \Delta + \sum_{\ell\in \Lambda} {\tilde v} (x_\ell)-\Jg
\sum_{i\sim j} x_i\cdot x_j\;,
\end{equation}
with
\begin{equation}\label{a.5}
{\tilde v}(y) = v(y) + \Jg d \,y^2\;,\;\forall y\in \rz^n\;.
\end{equation}
This sometimes leads (See \cite{BaKo}) to an other notion of one-particle operator
\begin{equation}
\label{a.6}
{\tilde H}_0:= - h^2 \Delta_y + {\tilde v}(y)\;.
\end{equation}
This article will in particularly discuss about the relevance of these
various
one-particle Hamiltonians in the problem of the phase transition.\\
\noindent For a one dimensional lattice (d=1), this model was
analyzed in \cite{He1994e}, by extension of the study of another model
analyzed in collaboration with
J. Sj\"ostrand \cite{HeSj1992b}. It was proved that
\begin{equation}\label{a.7}
\lambda_2(L;h) - \lambda_1(L;h) \leq C \exp - (\frac{\epsilon L}{h})\;,
\end{equation}
for $C$, $\epsilon$ independent of $L$ and for $h0$.
All these results were obtained
relatively easy
extensions of these contributions. Because, particularly in
the paper by \cite {BaKo}, semi-classical analysis is involved, we tried to be more precise that in the original paper, using
our more precise knowledge of the tunneling \cite{HeSj1984}. On the
other hand we got progressively the impression that some assumptions were actually
irrelevant in the problem and our main goal is to present here rather
optimal results, although limited by the bounds of this approach (See
\cite{Fr} for discussions).\\
Let us now recall the main results obtained through the infrared estimates.\\
Following \cite{BaKo}, we now consider the operator $\exp -\beta H_\Lambda^{per}$ and
the associated
\begin{equation}\label{a.10}
\langle |\frac{1}{|\Lambda|}
\sum_{k \in \Lambda} x_k |^2 \rangle_{\beta,\Lambda}:= \frac{\Tr ( |\frac{1}{|\Lambda|}
\sum_{k\in \Lambda} x_k|^2 \exp ( -\beta
H_\Lambda^{per}))}{\Tr ( \exp (- \beta
H_\Lambda^{per}))} \;.
\end{equation}
Physically the strictly positive parameter $\beta$ corresponds to the inverse of the temperature.\\
We now introduce the so called parameter of long-range
order
\begin{equation}\label{a.11}
P(\beta) = \lim_{|\Lambda|\ar \infty} P_\Lambda(\beta) \;,
\end{equation}
with
\begin{equation}\label{a.12}
P_\Lambda(\beta) =\left\langle |\frac{1}{|\Lambda|}
\sum_{k\in \Lambda} x_k|^2\right\rangle_{\beta,\Lambda}\;.
\end{equation}
The presence
of the long range order, i.e. the strict positivity of $P(\beta)$,
will serve as a test for phase transition (cf \cite{DyLiSi} and \cite{PaKh}).\\
When the limit of $\langle\cdot\rangle_{\beta,\Lambda}$ as $|\Lambda| \ar +\infty$ exists, we shall denote it
by $\langle \cdots \rangle_\beta$. Let
\begin{equation}\label{a.13}
E(p) = \sum_{i=1}^d (1- \cos p^{(i)})\;,
\end{equation}
where $\Lambda^\star$ is the dual lattice
\begin{equation}\label{a.14}
\Lambda^\star = \{ p= (p^{(1)},\cdots,p^{(d)})\;|\; p^{(i)} = 2\pi
k^{(i)}/m\;,\; 0\leq k^{(i)} \leq m - 1\;;\; 1\leq i\leq d \}\;.
\end{equation}
The method of infrared estimates gives
\footnote{Cf \cite{DyLiSi}, (52), p. 368 (cf also
Theorem
3.2 and Theorem 5.1 in this article). }
the following lower bound
\begin{theorem}.\label{Theorema1}
If $d\geq 3$ and under the following assumption on $v$,\begin{equation}
\label{a.15}
v(x) \geq a x^2 + b \;,\;\mbox{ with } a>0\;,
\end{equation}
we have
\begin{equation}\label{a.16}
P(\beta) \geq \langle x_k^2\rangle_\beta - \frac{n}{2}\cdot\frac{1}{(2\pi)^d}
\int_{]-\pi,\pi[^d} (\frac{h^2}{\Jg E(p)})^\frac 12 \coth\left[
\left(h^2\beta^2 \Jg E(p)\right)^\frac 12\right] \;dp\;.
\end{equation}
\end{theorem}
This lower bound is deduced by a limiting argument (thermodynamic limit) from the actually more
useful
inequality (for our questions relative to the splitting), which is relative to the finite lattice case, and which
is given by the following theorem essentially due to \cite{DyLiSi}.
\begin{theorem}\label{Theorema2}.
Under the assumption (\ref{a.15}) on $v$,
we have, for any $k\in \Lambda$, the following universal estimates
\begin{equation}\label{a.17}
P_\Lambda (\beta) \geq \langle x_k^2\rangle_{\beta,\Lambda} -
\frac{n}{2}\cdot \frac{1}{|\Lambda|}
\sum_{p\in \Lambda^\star\setminus \{0\}} (\frac{h^2}{\Jg E(p)})^\frac 12 \coth\left[
\left(\beta^2 h^2\Jg E(p)\right)^\frac 12\right]\;.
\end{equation}
\end{theorem}
Pastur and Khozurenko \cite{PaKh} and more recently Barbulyak and
Kondrat'ev \cite{BaKo} try (in particular) to analyze how one can
control the quantity $\langle x_k^2\rangle_{\beta,\Lambda}$ from
below in order to deduce existence for long range order. The second
authors use semi-classical analysis for the one particle Hamiltonian
operator ${\tilde H}_0$ and standard inequalities (GKS) reducing the
problem to this case. The first authors use for a rather specific model
the Bogolyoubov inequality. Their result is no more semi-classical
and explicit conditions for long range order are given.\\
Extending preliminary results announced in \cite{He96}, we push
these results in various directions:
\begin{itemize}
\item very weak assumptions in the
limit case $\beta=+ \infty$, without using (GKS),
\item stronger but still
rather weak assumptions for the case $\beta$ large, but using
standard moment inequalities.
\end{itemize}
We get for example the following theorem
\begin{theorem}\label{Theorema3}.
Let us assume $d\geq 2$. If the potential
$ v$ satisfies (\ref{a.15}),
\begin{equation}
\label{a.18}
v(x)= v(-x)\;,
\end{equation}
\begin{equation}
\label{a.19}
v(0) > \inf v\;,
\end{equation}
then there exists $h_0$ such that we have, for $0\Jg d\;,
\end{equation}
and that:
\begin{equation}
\label{a.22}
\begin{array}{l}
\mbox{ For some } q_0>0 \mbox{ the function } {\tilde v} \mbox{
attains } \break \mbox{
its strong global minimum}\\
\quad\mbox{ at the points } \pm q_0\;.
\end{array}
\end{equation}
On the other hand the conditions given by the authors are much
stronger
(but lead to a stronger result (treatment of the case $\beta$ large) which will be analyzed later).
In any case, what seems still open is the complete analysis of the
tunneling with control of the dimension and we recall from \cite{Fr} the important
point that the Peierls approch will give additional information
particularly for $d=1$ which can not be analyzed by this method. In
reference to these enlighting lectures, what we are doing here is
essentially what was left to the reader in these notes p. 237. We
recently received a preprint of Albeverio,
Kondratiev and Rebenko \cite{AlKoRe} based on the Peierls argument.\\
This article is organized as follows. \\
\begin{itemize}
\item
In Section \ref{Sectionb}, we recall the relations between the
splitting
and some notions related to the absence of long range order. We
recall
also the general strategy of the infrared estimates.
\item In Section \ref{Sectionc}, we recall a family of estimates
extending in various directions the FKG estimates: GKS, FKG, GHS,
Lebowitz and Gaussian domination estimates.\\
These estimates will play an important role in the treatment of the
case of $\beta$ large but are not used in the treatment of
$\beta=+\infty$.
\item Section \ref{Sectiond} is devoted to the analysis of lower
bounds
for the second order moments in the case $\beta=+\infty$. This is a
combination of standard semi-classical analysis with control with
respect to the dimension (harmonic approximation) in combination with
a priori estimates. Although this would be very interesting to
analyze,
no study of the tunneling is involved and only a rather weak
localization of the first eigenfunction.
\item In Section \ref{Sectione}, we extend the results proposed by
Barbulyak-Kondratev by following the same strategy but instead of
reducing, through Ginibre type inequalities GKS, to a one
particle problem, we take advantage of a reduction to a $p$-particle
problem.
\item Section \ref{Sectionf} is mainly reporting on the partially
alternative
strategy followed by Pastur-Khozurenko and based on the use of the
Bogolyubov inequality. This approach is not semi-classical and give
more explicit estimates for specific models.
\item
Section \ref{Sectiong} proposes extensions of the model treated by
Pastur-Khozurenko by implementation of the results recalled in
Section \ref{Sectionc}. This leads to less explicit results but
permits for example to analyze the dependence on $\Jg$ of the
transition of phase.
\item The last section presents some open questions.
\end{itemize}
\section{ Connection with the Splitting.}\label{Sectionb}
We recall here rather standard relations connecting the
splitting between the two first eigenvalues of the Schr\"odinger
operator
with various quantities introduced in the preceding section. They
are quite general and true for any Schr\"odinger operator
$$
H_V:=- h^2 \Delta + V\;,
$$
where $V$ is $C^\infty$ semibounded, tending to $\infty$ as $|x|\ar
+ \infty$ and satisfying
$$
V(-x) = V(x)\;.
$$
We have the following lemma
\begin{lemma}
\label{Lemmab1}
\begin{equation}\label{b.1}
\int
|\frac {1}{|\Lambda|} \sum_i x_i|^2\phi_1^\Lambda (x^\Lambda)^2 dx^\Lambda
\leq \frac {n}{|\Lambda|}
(\lambda^\Lambda_2-\lambda^\Lambda_1)^{-1}.
\end{equation}
\end{lemma}
This is indeed just the minimax principle\footnote
{One can extend $C_0^\infty$ to slowly increasing $C^\infty$
functions in our case.} saying that \begin{equation}\label{b.2}
(\lambda^\Lambda_2 - \lambda^\Lambda _1) = \inf_{\begin{array}{c}f \in C_0^\infty\\
\int f (x^\Lambda) \phi_1^\Lambda (x^\Lambda)^2 dx^\Lambda =0 \end{array}}
\frac{\langle |\nabla f|^2\rangle}{\langle |f|^2 \rangle}\;,
\end{equation}
where $\langle \cdot \rangle$ is computed for the measure
$\phi_1^\Lambda (x^\Lambda)^2 dx^\Lambda$ : \begin{equation}\label{b.3}
\langle \cdot \rangle =
\langle\cdot\rangle_{\infty,\Lambda} \;,
\end{equation}
(remembering that it corresponds to the case $\beta=+\infty$). Here
$dx^\Lambda$ is the Lebesgue measure on $(\rz^n)^{|\Lambda|}$.\\
This is
then applied to the function \begin{equation}\label{b.4}
f^{(j)}(x^\Lambda)=
\frac{1}{|\Lambda|}\sum_{i\in \Lambda} x^{(j)}_i \;, \end{equation}
for $j=1,\cdots ,n$.
The condition of symmetry on $V$ (which is a consequence of the
symmetry of $v$ for
our specific example) gives the orthogonality of $f^{(j)} \Phi_1^\Lambda$ to
$\Phi_1^\Lambda$.\\
In the case when $x_i\in \rz^n$ with $n>1$, we have to introduce the
functions $f^{(j)} (x^\Lambda) =
\frac{1}{|\Lambda|}\sum_{i\in \Lambda} x^{(j)}_i $ (for
$j=1,\cdots,n$).\\
This means that if we can prove by other means the property that
\begin{equation} \label{b.5}
[\int
|\frac {1}{|\Lambda|} \sum_i x_i|^2\phi^\Lambda_1 (x^\Lambda)^2 dx^\Lambda]
\geq \rho >0 \;,
\end{equation}
with $\rho$ independent of $\Lambda$, then we get that
\begin{equation}\label{b.6}
\lim_{|\Lambda|\ar +\infty}
|\lambda^\Lambda_2 - \lambda^\Lambda_1
| = 0\;. \end{equation}
A lower bound for $P_\Lambda (+\infty)$ gives an upper
bound for the splitting but Theorem \ref{Theorema2} gives the
starting point for finding this lower bound and
this will then permit to prove Theorem \ref{Theorema3}. We shall
mainly follow the proof of Barbulyak-Kondrat'ev but with a small change.\\
In \cite{BaKo}, the authors take indeed first the limit $|\Lambda|\ar +
\infty$
and then the limit $\beta \ar + \infty$. We shall proceed for the
application to the splitting
in the inverse order.\\
\noindent {\bf Proof of Theorem \ref{Theorema3}, modulo a lower bound
of $\langle x_k^2\rangle_{\beta,\Lambda}$ }:\\
We assume in this part that we have proved the existence of a constant
$\delta(\beta,h) >0 $ such that for any $\Lambda$ (with sometimes a
restriction $L\geq L_0$)
\begin{equation}
\label{b.7}
\langle x_k^2\rangle_{\beta,\Lambda} \geq \delta(\beta,h)
\end{equation}
We shall sometimes get in the next sections a control of
$\delta(\beta,h)$, with $\beta$ and $h$, of the type:
\begin{itemize}
\item In the case $\beta = + \infty$,\\ (As1) there exists $\delta_0 >0$ and $h_0 >0$ such that for $h\in ]0,
h_0]$
$$
\delta(\infty,h)\geq \delta_0\;.
$$
\item
In the case $\beta$ large,\\
\begin{itemize}
\item
(As2) there exists $\delta_0 >0$, $h_0 >0$ such that for $h\in ]0,
h_0]$ there exists $\beta_0(h)$ such that, for all $\beta\geq \beta_0(h)$,
$$
\delta(\beta,h)\geq \delta_0\;,
$$
or
\item (As3) there exists $\delta_0 >0$, $h_0 >0$ and $\beta_0$ such that for $h\in ]0,
h_0]$ and $\beta\geq \beta_0$,
$$
\delta(\beta,h)\geq \delta_0\;.
$$
\end{itemize}
\end{itemize}
What is common to all these estimates is that they are independent of
$\Lambda$
with possibly the restriction $L$ large enough. This last
restriction is usually not important because
we shall always take the limit $L\ar + \infty$ at the end.\\
\noindent We start from (\ref{a.17})
\begin{equation}\label{b.8}
\begin{array}{l}
P_{\Lambda}(\beta)\\
\geq \langle x_k^2\rangle_{\beta,\Lambda}
- \frac{n\,h}{2}\cdot \frac{1}{|\Lambda|}\sum_{p\in \Lambda^\star\setminus\{0\}}
(\frac{1}{\Jg E(p)})^\frac 12 \coth\left[
\left(h^2\beta^2 \Jg E(p)\right)^\frac 12\right]\;.
\end{array}
\end{equation}
This leads, under assumption (\ref{b.7}) to
\begin{equation}
\label{b.9}
P(\beta) \geq \delta(\beta,h) - \frac {nh}{2 \Jg^\frac 12 }\cdot \frac{1}
{(2\pi)^{d}}\;
\int_{]-\pi, \pi[^d } E(p)^{-\frac 12} \coth \left [h^2\beta^2 \Jg
E(p))^\frac 12\right] dp\;.
\end{equation}
The discussion of the case $\beta$ large is then related with the analysis of
$\delta(\beta,h)$ for $\beta$ large and its comparison with the quantity \break
$\frac {nh}{2 \Jg^\frac 12}\cdot \frac{1}{(2\pi)^{d}}\;
\int_{]-\pi, \pi[^d } E(p)^{-\frac 12} \coth \left [h^2\beta^2 \Jg
E(p))^\frac 12\right] dp$. The condition $d\geq 3$ appears naturally
in this case to keep the integral appearing in the r.h.s. of (\ref{b.9}) convergent.\\
Taking the limit $\beta\ar +\infty$ in (\ref{b.8}), we first obtain
\begin{equation}\label{b.10}
\lim_{\beta\ar +\infty} P_{\Lambda}(\beta)
\geq \langle x_k^2\rangle_{+\infty,\Lambda}
- \frac{n\,h}{2}\cdot \frac{1}{|\Lambda|}\sum_{p\in \Lambda^\star\setminus\{0\}}
(\frac{1}{\Jg E(p)})^\frac 12 \;.
\end{equation}
Now, as $|\Lambda|\ar + \infty$ and if $d\geq 2$, the right hand side is estimated from
below, under the assumption (As1) by
\begin{equation} \label{b.11}
P(+\infty)\geq \delta_0
-\frac {nh}{2} \Jg^{-\frac 12} I_d\;,
\end{equation}
where we recall that
\begin{equation}\label{b.12}
I_d = \frac{1}{(2\pi)^d} \int_{]-\pi,\pi[^d}
E(p)^{-\frac 12} dp\;.
\end{equation}
We now observe that $I_d$ is finite for $d\geq 2$. Then $P(\infty)$
becomes strictly positive when $h\leq h_0$ and
\begin{equation}
\label{b.13}
\delta_0
-\frac {nh}{2} \Jg^{-\frac 12} I_d >0\;.
\end{equation}
In some cases $h_0$ will be explicitly computed (e.g in the cases
considered by \cite{PaKh}) and we can in this case discuss the
influence of $\Jg$ for fixed $h$, in particular (\ref{b.13}) is
satisfied for $\Jg$ large enough. In other cases (e.g. in the cases
considered by \cite{BaKo}) $h_0$ is not explicited by
the (semi-classical) analysis and we can only say that (\ref{b.13})
is
satisfied for $h$ small enough.
\noindent In particular we obtain the proof of Theorem
\ref{Theorema3} modulo the verification of (As1).\\
\noindent{\bf The case: $d=1$}.\\
\noindent Another interesting point is to analyze the case $d=1$.
The sum
$$ \frac{1}{|\Lambda|}\sum_{p\in \Lambda^\star\setminus\{0\}}
(\frac{1}{\Jg E(p)})^\frac 12$$ is divergent as $|\Lambda|\ar
+\infty$, but this divergence is controlled
in $ \ln(|\Lambda|)$. This leads to the statement that
the splitting remains in $\Og(\frac{1}{|\Lambda|})$ for $h\leq h_0$ and
$ |\Lambda| \leq \exp \frac Th$, where $T$ is explicitely
computable.\\
This condition is much weaker than the condition (\ref{a.8}) given in
\cite{HeSj1992b}.\\
As $h\ar 0$, the estimate still work, under assumption (As1), if
\begin{equation}\label{b.14}
\ln |\Lambda|\leq C\frac{\delta_0}{ h } [2\Jg]^\frac 12\;,
\end{equation}
where $C$ is a universal constant.
We note also that the phenomenon is effectively related to $\Jg\neq 0$.
To summarize, we prove in this case that
\begin{theorem}\label{Theoremb2}.
Let $d=1$ and $ v$ satisfying (\ref{a.15}), (\ref{a.18}) and (\ref{a.19}), then there exists a constant $C$ such that
\begin{equation}\label{b.15}
\lambda_2^\Lambda - \lambda_1^\Lambda \leq \frac{C}{|\Lambda|} \;
\frac{1}{(1- C h \log |\Lambda|)_+}
\;.
\end{equation}
\end{theorem}
\noindent As mentioned in the introduction and as was communicated to us by
J. Fr\"ohlich, this condition on
$|\Lambda|$
is due to the method (when $n=1$). An approach using the Peierls trick could be more effective in this case \cite{Fr}.\\
\section{Ginibre or Lebowitz type inequalities}\label{Sectionc}
Ginibre type inequalities\footnote
{A good reference is \cite{Si74} or \cite{Si79}. These inequalities correspond to
extensions of the so called FKG inequalities \cite{FKG}. A more recent
review can be found in \cite{La}. }
make it possible to estimate $\langle
x_k^2\rangle_{\beta,\Lambda}$
from below by the average $\langle x_k^2\rangle_{\beta,{\hat H}}$
taken over the measure corresponding to the "formal" decoupled Hamiltonian
\begin{equation}\label{c.1}
{\hat H} = - h^2 \sum_{k\in \zz^d} \frac{\pa^2}{\pa x_k^2} +
\sum_{k\in \zz^d}{\tilde v}(x_k)\equiv \sum_{k\in \zz^d}{\tilde H}_k\;,
\end{equation}
with separate variables.\\
Less formally, we introduce
\begin{equation}\label{c.2}
{\hat H}^{\Lambda} = \sum_{k\in \Lambda}{\tilde H}_k
\end{equation}
and we get (\cite{BaKo}) the
\begin{proposition}\label{Propositionc1}.
Under the assumptions (\ref{a.15}), (\ref{a.18}) and (\ref{a.21})-(\ref{a.22}) on the potential ${\tilde v}$ and $\Jg$, the following inequality is true, for any $k\in \Lambda$,
\begin{equation}\label{c.3}
\langle x_k^2\rangle_{\beta,\Lambda} \geq
\langle x_k^2\rangle_{\beta,\Lambda, {\hat H^\Lambda}} \;.
\end{equation}
\end{proposition}
Let us describe briefly the version of the Ginibre
inequalities which is
needed here.
This is related to the control of the sign of the correlations
attached to the more general measure:
\begin{equation}\label{c.4}
Z^{-1}\exp \sum_{ij} J_{ij} x_i\,x_j \prod_{j=1}^m
d\nu_j(x_j)\;. \end{equation}
We look at
the partial derivative of $\langle (x_k)^2\rangle$ with respect to $J_{ij}$.
Here $\langle\cdot \rangle$ is taken with respect to the measure (\ref{c.4}).\\
This gives $$ \pa_{J_{ij} }
\langle (x_k)^2\rangle
= \langle x_i\,x_j\, (x_k)^2\rangle
- \langle (x_k)^2\rangle\cdot \langle x_i\,x_j\, \rangle\;.
$$
The right hand side appears as the pair correlation of the two
functions
$f= x_i\;x_j$ and $g = x_k^2$.\\
We shall deduce from the Ginibre's inequalities that this
expression is
positive.
\begin{equation}\label{c.5}
\pa_{J_{ij} }
\langle (x_k)^2\rangle\geq 0\;.
\end{equation}
Let us briefly recall some elements of this theory (cf \cite{Si79}, p.119-124), for a nice exposition).
We recall the
\begin{theorem}\label{Theoremc2}.
Let $\Fg_1$ be the set of functions on $\rz$ which are nonnegative
and monotone on $[0,+\infty)$ and either even or odd. Let $\Fg_n$
be the functions on $\rz^m$ of the form $f_1(x_1)\cdots f_n(x_n)$
with $f_i\in \Fg_1$. Let $d\mu$ be a probability measure of the
form (\ref{c.4}) where $J_{ij} \geq 0$ and each $d\nu_j$ has the form
$\exp( f_j(x)) d\lambda_j(x)$ with $f_j\in \Fg_1$ and $d\lambda_j$
even. Then
\begin{equation}\label{c.6}
\begin{array}{ll}
(GKS1) & \langle f\rangle \geq 0 \\
(GKS2) & \langle f\,g \rangle \geq \langle f\rangle \cdot
\langle
g\rangle\;,
\end{array}
\end{equation}
for all $f,g \in \Fg_{m+1}$.
\end{theorem}The reader can find a proof of the theorem in \cite{Si79}.\\
\noindent The measure in consideration is attached to the restriction to the diagonal
of the distribution kernel of $\exp - \beta H_\Lambda^{per}$, where
$\Jg $ becomes a parameter in $[0, \Jg_0]$ and has not
exactly the structure which is introduced above. But using the Trotter product formula which describes the kernel
of $\exp - \beta H_\Lambda^{per}$ as the limit
in a weak sense of $(\exp - \frac{\beta V(x) }{N}\exp \frac{\beta
\Delta}{N})^N$, we observe that the measure is a limit of measures
satisfying the assumptions of the theorem for $\Jg\geq 0$.\\
We have just to control (Cf \cite{Si79} p. 120 and p.122) the limit procedure in order to get the result.\\
\noindent {\bf Application}:\\
The Hamiltonian ${\hat H}^\Lambda$
describes a system of non interacting particles. Consequently, we get
immediately, for $k\in \Lambda$,
\begin{equation}\label{c.7}
\langle x_k^2\rangle_{\beta,\Lambda;{\hat H}} =\langle
x_k^2\rangle_{\beta,{\tilde H}_k}=
\langle
x_0^2\rangle_{\beta, {\tilde H}_0}\;,
\end{equation}
for all $k\in \zz^d$.\\
Here we recall that ${\tilde H}_k$ is the one-particle Hamiltonian at $k\in
\zz^d$.\\
We consequently have obtained
\begin{proposition}\label{Propositionc3}.
Under the previous assumptions on ${\tilde v}$ and $\Jg$ (in
particular $n=1$), we have
\begin{equation}\label{c.8}
\langle x_k^2\rangle_{\beta,\Lambda} \geq
\langle x_0^2\rangle_{\beta,H_0} \;.
\end{equation}
\end{proposition}
In particular we obtain the condition (As2) mentioned in the
previous section by observing that it is sufficient for treating the
case $\beta$ large enough, to analyze the case $\beta=+\infty$ for
$\langle
x_0^2\rangle_{\beta, H_0}$ and to use weak estimates for one-particle
Hamiltonians. This will be generalized in one of the next sections.
\begin{remark}\label{Remarkc4}.
When $n>1$, we can no more apply this technique directly. The case
when we have a rotational symmetry can probably be treated by taking
polar coordinates. The results in this case is mentioned in
\cite{GlJa1987}. Pastur and Khoruzenko proceed differently in the
case of the model\break $v(x) = (1-|x|^2)^2$. We shall discuss this point later.
\end{remark}
We now present two other inequalities which may be important in order
to compare moments relative to various hamiltonians. We discover these
inequalities
in the nice paper by A. Sokal \cite{Sok} who refers to \cite{EMN} and \cite{EN}. The name of Lebowitz is in
reference
with the particular case of spin systems.
\begin{theorem}
\label{Theoremc5}{(GS and Lebowitz inequalities)}.
Let $d\mu (x)$ a probability measure on $\rz^m$ of the type
\begin{equation}\label{c.9}
d\mu(x) = Z^{-1} \exp \left[\frac 12 \sum_{i,j} J_{ij} x_i x_j +
\sum_i h_i x_i\right ]
\prod_{i=1}^m d\nu_i(x_i)\;,
\end{equation}
with
\begin{equation}\label{c.10}
J_{ij} = J_{ji} \geq 0\;,\; h_i\geq 0\;,
\end{equation}
where $d \nu_i$ is for each $i$ a measure on $\rz$ of the type
\begin{equation}\label{c.11} d\nu_i(y) = \exp - {\tilde v}_i(y) dy\;,
\end{equation}
where ${\tilde v}_i$ is an even differentiable function such that
\begin{equation}
\label{c.12}
{\tilde v}' \mbox{ is convex on } ]0,+\infty[\;.
\end{equation}
Then we have
\begin{equation}
\label{c.13}
(GS)\quad \langle x_i x_j x_k \rangle \leq \langle x_i x_j\rangle \langle
x_k \rangle +
+ \langle x_i x_k\rangle \langle x_j\rangle + \langle x_j
x_k\rangle \langle x_i\rangle - 2 \langle x_i\rangle \langle x_j\rangle \langle x_k\rangle\;,
\end{equation}
and, when $h=(h_1,\cdots, h_m) = 0$,
\begin{equation}\label{c.14}
(Leb)\quad \langle x_i x_j x_k x_\ell\rangle \leq \langle x_i x_j\rangle \langle
x_k x_\ell\rangle + \langle x_i x_\ell\rangle \langle x_j x_k\rangle
+ \langle x_i x_k\rangle \langle x_j x_\ell\rangle
\end{equation}
\end{theorem}
We shall also use the following extension mentioned by Sokal who
refers to \cite{BFS}, \cite{Ne}, \cite{Sy} and \cite{Br}.
\begin{theorem}\label{Theoremc6}{(Gaussian domination inequalities)}.
Under the assumptions of validity of Theorem \ref{Theoremc5}, we have,
with $h=(h_1,\cdots,h_m) =0$,
for any multindex $\alpha = ( \alpha_1,\cdots , \alpha_p)$ with $p$
even
and $\alpha_j\in \{ 1,\cdots, m\}$ and with $x^\alpha = \prod_{j=1}^p x_{\alpha_j}$,
\begin{equation}
\label{c.15}
\langle x^\alpha \rangle \leq \sum_{pairings} \prod \langle x_k x_\ell\rangle
\end{equation}
where the sum is over all the pairs giving a decomposition of $\alpha$.\\
\end{theorem}
\section{Semiclassical analysis of the many particle
problem, the case $\beta=+\infty$}\label{Sectiond}
We have seen in Section \ref{Sectionc} that we can estimate from below,
using the Ginibre's inequalities, the
quantity
$\langle x_j^2\rangle_{\beta,\Lambda}$ (for $j\in \Lambda$), by the corresponding
quantity $\langle y^2 \rangle_{\beta, H_{\tilde v}}$, attached to one particle potential $\tilde v$
. We can then first
analyze
the limit case $\beta = + \infty$ by semiclassical analysis and
proved that for $\beta$ large enough an analogous estimate from below
remains true.\\
This was the approach developed by Barbulyak-Kondratev \cite{BaKo}
that we further developed in \cite{He96}.\\
This result appears to be only semi-satisfactory because it does not seem
so natural to impose that $\tilde v$ creates a double well and one is
more likely waiting for
the weaker assumption that $v$ creates a double well. Of course this is not
important when $\Jg$ is sufficiently small but it appears more
important as $\Jg$ is large.\\
Our alternative idea is to treat the case $\beta= + \infty$ by
analyzing
directly the quantity $\langle x_j^2 \rangle_{\infty,\Lambda}$,
without to reduce to a one-particle Hamiltonian. If successfull, the
advantage is that we do not need any assumption related to the
application of statistical mechanics like Ginibre' inequalities.\\
This is of course not sufficient for having the result for large
$\beta$
independently of $\Lambda$ but is sufficient for the analysis
of the splitting.
The proposition is the following
\begin{proposition}\label{Propositiond1}.
Let us assume that $v$ is $C^\infty$, positive, satisfying
\begin{equation}\label{d.1}
\lim\inf_{|x|\ar +\infty} v (x) >0,
\end{equation}
such that there exists $q_0>0$, with
\begin{equation}\label{d.2}
v(x)> 0\;,\; \forall x \mbox{ s.t. } | x |< q_0 \;,
\end{equation}
and the property that there exists $y_0 \in \rz^n$ such that $|y_0| = q_0$ and
\begin{equation}\label{d.3}
v(y_0)= 0\;.
\end{equation}
Then, for any $\epsilon >0$, there exists $h_0(\epsilon) >0$ such that,
for $h\in ]0, h_0(\epsilon)]$, $ \forall j\in \Lambda$, we have,
\begin{equation}\label{d.4}
\langle x_j^2 \rangle_{\infty,\Lambda} \geq (q_0-\epsilon)^2\;.
\end{equation}
\end{proposition}
This contains the double well case. The even character of $v$ is not
assumed in this proposition but note that this
assumption is quite important in other parts of the
problem
in order for example to have $\langle x_j\rangle_{\infty,\Lambda}
=0$.\\
The argument is actually the conjonction of the existence of a
minimum of $v$ outside the origin and of the property that $v>0$ in a neighborhood of
$0$.\\
Let us explain for simplification the proof in the case $n=1$ and
when $v$ is even.\\
Let us denote by $\Phi_1$ the first normalized eigenfunction of $H_\Lambda^{per}$. We
have to estimate from below
$
\int x_j^2 \Phi_1^2\; dx\;.
$
It is trivial that, for any $\epsilon$ such that $q_0>\epsilon >0$, we have
\begin{equation}\label{d.5}
\int x_j^2 \Phi_1^2 dx \geq (q_0- \epsilon)^2 \int_{x\in
\rz^{|\Lambda|};\,|x_j| \geq q_0-\epsilon} \Phi_1^2\; dx \;.
\end{equation}
We rewrite this inequality in the form
\begin{equation}\label{d.6}
\int x_j^2 \Phi_1^2 dx \geq (q_0- \epsilon)^2 - (q_0-\epsilon)^2 \int_{x\in
\rz^{|\Lambda|};\,|x_j| \leq q_0-\epsilon} \Phi_1^2 dx\;.
\end{equation}
We are then reduced to the control of $ \int_{x\in
\rz^{|\Lambda|};\,|x_j| \leq q_0-\epsilon} \Phi_1^2 dx$. For this
estimate, we come back to the equation satisfied by $\Phi_1$ and get,
using also the invariance by translation of $H_\Lambda^{per}$,\begin{equation}
\label{d.7}
\int v(x_j) \Phi_1^2 dx \leq \lambda_1^\Lambda/|\Lambda|\;.
\end{equation}
Let $m_\epsilon = \inf_{y\in [-q_0+\epsilon, q_0 - \epsilon]}
v(y)$. This quantity is strictly positive by assumption. We get in
particular
\begin{equation}
\label{d.8}
\int_{|x_j|\leq q_0-\epsilon} \Phi_1^2 dx \leq \frac{1}{m_\epsilon}\cdot \frac{\lambda_1^\Lambda}{|\Lambda|}\;.
\end{equation}
We have to control
$\frac{\lambda_1^\Lambda}{|\Lambda|}$
independently of $|\Lambda|$. This point is the consequence of the
rather standard proof of
the existence of the thermodynamic limit (see for example
\cite{He1994e}), but we need more, that is the control with dependence on $h$.
We can add to $v$ the
potential $(y^2 - q_0^2)^2$ in order to get a potential with
two non degenerate wells at $\pm q_0$. Using the minimax principle and
the harmonic approximation, it is proved in \cite{He1994e} the
following lemma that we state here in a very weak form.
\begin{lemma}.
Let us assume that $v$ has non degenerate minima $q_i$ with $v(q_i)
=0$.
Then there exists a constant $C >$ and $h_0$ such that for
$h\in ]0,h_0]$ we have
\begin{equation}\label{d.9}
\frac{\lambda_1^\Lambda}{|\Lambda|}\leq C h\;.
\end{equation}
\end{lemma}
{\bf Proof of Theorem \ref{Theorema3}}\\
As explained in Section \ref{Sectionb}, this is an immediate consequence of the infrared estimates once we
have proven the existence of $\delta_0 >0$ such that $\langle
x_j^2\rangle_{\beta,\Lambda}\geq \delta_0$. We can first assume that
$\inf v = 0$. Under the assumption that $v(0)>0$ and (\ref{a.15}), it is
clear that there exists $y_0$ such that $v(y_0)= 0$ and such that the
conditions of Proposition \ref{Propositiond1} are satisfied.
\begin{remark}\label{Remarkd2}.
The proof gives also a ``weak'' control of the convergence of the
type, for any $\epsilon >0$, there exists $h_0(\epsilon)$ such that, for
$h\in ]0,h_0(\epsilon)]$
$$\lambda_2^\Lambda
(h)-\lambda_1^\Lambda(h)\leq n \frac{1+\epsilon}{q_0^2}\cdot
\frac{1}{|\Lambda|}\;.
$$
\end{remark}
\begin{remark}\label{Remarkd3}.
Let us emphasize that the assumptions are much weaker that the
assumptions
given by Barbulyak-Kondratev but the proof of these authors is still
valid
for $\beta$ large. This is not the case of our proof. We shall see in
the next section how we can analyze this case.\\
Let us also emphasize that we have not used any result of Section \ref{Sectionc}.
\end{remark}
\begin{remark}\label{Remarkd4}.
Similarly to the case treated in Section \ref{Sectionb}, we can have, in the case $d=1$, some estimate like
(\ref{b.15})
under the weaker assumptions of Theorem \ref{Theorema3} (\ref{a.15}),
(\ref{a.18}) and (\ref{a.19}).
\end{remark}
\section {The case $\beta$ finite, large enough} \label{Sectione}
We want to treat the case when $\beta$ is finite by improving the
ideas
of Barbulyak-Kondratev.
Our theorem will be the following
\begin{theorem}\label{Theoreme1}.
Let us assume that $v$ is a $C^\infty$ potential on $\rz$ with two
non-degenerate minima at $\pm q_0$. Then, for all $\varepsilon>0$, there
exists $h(\varepsilon)>0$ such that, for all $h\in ]0,h(\varepsilon)]$,
there exists $\beta_1(h)$ such that, for $|\Lambda|= L^d$, large
enough, for any $k\in \Lambda$,
\begin{equation}\label{e.1}
\langle x_k^2\rangle_{\beta,\Lambda} \geq (1-\varepsilon) q_0^2\;,
\end{equation}
\end{theorem}
This proof is modelled on the proof sketched by Barbulyak-Kondratev \cite{BaKo}.
These authors reduce, through the Ginibre inequalities, to the study
of a one particle problem attached to the potential $\tilde v$. The
condition $n=1$ appears only in the application of these inequalities.\\
For simplicity, we explain the case when $d=1$ . In the general case,
this will
correspond to the idea that $\Lambda$ may be decomposed in smaller
boxes.
\\
Our unique idea is to replace the reduction to a one-particle problem
by a reduction to a $p$-particle problem with $p$ large enough but
then fixed in the thermodynamic limit.\\
Let us divide $L$ by $p$. The number $p$ will be chosen large enough
later.
We start from
$$
L = p L_0 + \ell\;.
$$
This leads to a natural decomposition of $\{1,\cdots, L\}$ in $L_0 +1$
intervals $I_k$ ($k = 1,\cdots, L_0 +1$) with $I_k = \{(k-1)p,\cdots
, kp -1\}$
for $k = 1,\cdots, L_0 $.
Following \cite{BaKo}, we destroy the terms $\Jg x_j x_k$
when $j$ and $k$ do not belong to the same interval. We recall that
these interaction terms appear only for $j$ and $k$ nearest neighbors
in $\zz/L\zz$.\\
Using the Ginibre inequalities, we obtain
\begin{equation}\label{e.2}
\langle x_k^2\rangle_{\beta,\Lambda} = \frac{1}{L} \sum_{k=1}^L
\langle x_k^2\rangle_{\beta,\Lambda}\geq (1- \frac \ell L) \left(\frac 1p\;
\sum_{q=1}^p\langle x_q^2\rangle_{\beta, H^{(p)}}\right)\;.
\end{equation}
Here the last mean value is associated with the operator $\exp - \beta
H^{(p)}$
and $H^{(p)}$ is the $p$-particle operator
\begin{equation}\label{e.3}
H^{(p)} = \sum_{j=1}^p v(x_j) + \frac \Jg 2 \sum_{j=1}^{p-1}
|x_j-x_{j+1}|^2
+ \frac \Jg 2 (x_1^2 + x_p^2)\;.
\end{equation}
When $p=1$, we recover the one-particle hamiltonian $H^{(1)}={\tilde H}_0$
introduced by \cite{BaKo}. So, what we have written above is just the extension of what was written
in the case $p=1$ by these authors.\\
In the case when $p=1$, the inequalities obtained from the
semi-classical
analysis of $H^{(1)}$ were rather weak, particularly when $\Jg$ was
large, and lead to (\ref{a.21})-(\ref{a.22}).
Our aim is to eliminate this condition by playing with $p$ large
enough.\\
We now implement the ideas appearing in Section \ref{Sectiond}
but for $H^{(p)}$. The new point is that we have destroyed the
invariance by circular permutation of the model and we have to measure
the effect of this modification.\\
The first problem is that we have now for the first eigenvalue
$\lambda_1^{(p)}(h)$ of $H^{(p)}$ a weaker estimate than for the
first eigenvalue of $H^\Lambda(h)$.
\begin{lemma}
\label{Lemmae2}.
There exists $C$ and $h_0$ such that, for any $h\in ]0, h_0]$ and any
$p$,
\begin{equation}
\label{e.4}
\lambda_1^{(p)}(h)\leq C p h + C\;.
\end{equation}
\end{lemma}
This gives in particular that $\lambda_1^{(p)}(h)/p$ will be small for
large $p$ and small $h$.\\
Following our previous analysis, we now observe that
\begin{equation}
\label{e.5}
\sum_{j=1}^p \int_{\rz^p} v(x_j) \left(\varphi_1^{(p)}(x)\right)^2\; dx^{(p)} \leq \lambda_1^{(p)}(h)\;.
\end{equation}
The difference with the previous argument is that we can no more study
independently
the different terms of the sum in the left hand side of (\ref{d.6}).
This does not appear to be a big problem.\\
As for (\ref{d.6}), we can write
\begin{equation}\label{e.6}
\sum_j \int x_j^2\left(\varphi_1^{(p)}(x)\right)^2\; dx^{(p)} \geq p (q_0 -\epsilon)^2
- q_0^2 \sum_j \int_{|x_j|\leq q_0-\epsilon}\left(\varphi_1^{(p)}(x)\right)^2\; dx^{(p)}\;.
\end{equation}
We then obtain
\begin{equation}\label{e.7}
\begin{array}{l}
\sum_j \int x_j^2\left(\varphi_1^{(p)}(x)\right)^2\; dx^{(p)}
\\
\quad\quad \geq p (q_0 -\epsilon)^2
-\frac{q_0^2}{m_\epsilon}\cdot \sum_j \int_{|x_j|\leq
q_0-\epsilon}
v(x_j)\left(\varphi_1^{(p)}(x)\right)^2\; dx^{(p)}\;,
\end{array}
\end{equation}
hence
\begin{equation}\label{e.8}
\begin{array}{ll}
\sum_j \int x_j^2\;|\varphi_1^{(p)}(x)|^2\; dx^{(p)}&\geq p (q_0 -\epsilon)^2
-\frac{q_0^2 }{m_\epsilon}\cdot \sum_j \int
v(x_j)\;|\varphi_1^{(p)}(x)|^2\; dx^{(p)}\;,\\&\geq
p (q_0 -\epsilon)^2
-\frac{q_0^2 }{m_\epsilon}\cdot \lambda_1^{(p)} (h)\;.
\end{array}
\end{equation}
Finally, using Lemma \ref{Lemmae2}, we prove,
\begin{equation}\label{e.9}
\sum_j \int x_j^2\;|\varphi_1^{(p)}(x)|^2\; dx^{(p)}\geq p (q_0 -\epsilon)^2
- { \tilde C} (ph +1)\;.
\end{equation}
This is equivalent, dividing by $p$, to
\begin{equation}\label{e.10}
\frac 1p \left(\sum_j \int x_j^2\;|\varphi_1^{(p)}(x)|^2\; dx^{(p)}\right)\geq (q_0 -\epsilon)^2
- { \tilde C} h -{\tilde C} \frac 1p\;.
\end{equation}
So we have proved the following lemma.
\begin{lemma}\label{Lemmae3}.
For any $\epsilon>0$, there exists $h_\epsilon$ and $p_\epsilon$ such
that, for $h\in ]0,h_\epsilon]$ and $p\geq p_\epsilon$,
\begin{equation}\label{e.11}
\frac 1p\left (\sum_{j=1}^p \int x_j^2 \;|\varphi_1^{(p)}|^2\; dx^{(p)}\right)\geq (q_0
-\epsilon)^2\;.
\end{equation}
\end{lemma}
Using (\ref{e.2}), this gives Theorem \ref{Theoreme1}. The argument
for comparing the situation $\beta = +\infty$ and the situation
$\beta$ large
is now possible, once $p$ is fixed.
\section {\bf About the Bogolyubov's inequality}\label{Sectionf}
Let us first recall some of the results obtained by Pastur and
Khozurenko. We shall then analyze the technique which is involved in
order to prepare the improvements of the next section.
\noindent These authors discussed the case when
\begin{equation}
\label{f.1} v(x) = - \frac a2 |x|^2 + \frac b4 |x|^4
\end{equation} with $a>0$, $b>0$. \\
They prove the following theorem,
\begin{theorem}\label{Theoremf1}.
If $d\geq 3$ and if
\begin{equation}\label{f.2}
\Jg > h^2(n+2)^2 b^2 I_d^2/4 a^2\;,
\end{equation}
then
there exists a temperature $\beta_0^{-1}$ such that, for $\beta
>\beta_0$,
the corresponding $P(\beta)$ is strictly positive.
\end{theorem}
The control of $\langle x_k^2\rangle_{\beta,\Lambda}$ is
obtained by using the Bogolyubov's inequality. We recall that this
inequality (See \cite{Ru1969}, Lemma 5.5.1) says the following
\begin{lemma}\label{Lemmaf2}.
Under suitable assumptions\footnote{This is discussed in detail in \cite{BoMa}} on the various domains of the possibly not
bounded operator $C$ and of the selfadjoint operator $H$,
\begin{equation}\label{f.3}
\langle [\;[C^\star, H], C]\rangle_{\beta,\Lambda} \geq 0\;.
\end{equation}
\end{lemma}
In our context, this
inequality is applied with
\begin{equation}\label{f.4}
C = \frac 1i \sum_{j\in \Lambda}\pa_{x_j}\;.
\end{equation}
This leads, by
considering
the limit $\beta\ar + \infty$ before to take the thermodynamic limit,
to the following result for the splitting between the two first eigenvalues as $|\Lambda|$ tends to $\infty$.
\begin{theorem}\label{Theoremf3}.
If $d\geq 2$, then, for $ v$ defined by (\ref{f.1}) and $h_0 =
\frac {2a \Jg^\frac 12}{(n+2)b I_d}$, we have, for $h \in ]0,h_0[$,
\begin{equation}\label{f.5}
\lim_{|\Lambda|\ar + \infty} |\lambda_2^\Lambda - \lambda_1^\Lambda| =
0\;.
\end{equation}
\end{theorem}
For this theorem, we need only the Bogolyubov inequality in the
limiting case $\beta= +\infty$. In this case and when $H = - \Delta +
V$ is a Schr\"odinger operator, the inequality (\ref{f.3}) becomes
simply
\begin{equation}\label{f.6}
\int ([\;[C^\star, H], C] \Phi)(x) \Phi(x) dx \geq 0\;,
\end{equation}
where $\Phi$ is the first normalized positive eigenfunction of $H$.
This is easily and directly obtained by the minimax principle.
In particular, if $C$ is given by (\ref{f.4}), we get
\begin{lemma}\label{Lemmaf6}.
\begin{equation}\label{f.7}
\sum_{jk} \int ( \pa^2 V/\pa{x_j}\pa{x_k})(x) \Phi(x)^2 dx\geq 0\;.
\end{equation}
\end{lemma}
When $V(x)$ has the form (we take for simplification $n=1$)
$$
V(x) =\sum_j v(x_j) + J \sum_{i\sim j} |x_i - x_j|^2\;,
$$
we simply get
\begin{equation}\label{f.8}
\sum_{j} \int v''(x_j) \Phi(x)^2 dx\geq 0\;.
\end{equation}
By invariance of $\Phi$ in this case, we get,
\begin{proposition}\label{Propositionf5}. For any $j$,
\begin{equation}\label{f.9}
\int v''(x_j) \Phi(x)^2 dx\geq 0\;.
\end{equation}
More generally, for any $\beta$, we have the property
\begin{equation}
\label{f.10}
\langle v''(x_j)\rangle_{\beta,\Lambda}\geq 0\;.
\end{equation}
\end{proposition}
In the more specific case when $v(x) = \frac b4 x^4 - \frac a2 x^2$
we obtain
\begin{equation}\label{f.11}
\int x_j^2 \Phi(x)^2 dx\geq \frac{a}{3b}\;.
\end{equation}
The proof through the infrared estimates is then easy using the
arguments presented in Section \ref{Sectionb}.
\section{Extensions of Pastur-Khozurenko: universal lower bounds}\label{Sectiong}
\subsection{Introduction}
If we compare (\ref{f.11}) with the semi-classical lower bound
obtained by Barbulyak-Kondratev, we note that when
$n=1$ and $ a > 2\Jg d$, the minimum of $\tilde v$ is for $q= q_0$
with $ q_0^2 = \frac{a-2\Jg d}{b}$. This leads asymptotically (as
$h\ar 0$)
to (\ref{f.11}) with $\frac {a}{3b}$ replaced by $ \frac{a-2\Jg d}{b}
- \varepsilon$ with $\varepsilon>0$. The improvement we gave in
Section \ref{Sectiond} leads asymptotically (as
$h\ar 0$) to a lower bound by $ \frac{a}{b}
- \varepsilon$ with $\varepsilon>0$, which seems optimal in the
semi-classical regime.\\
The main goal of this section is to extend the results proposed by Pastur-Khozurenko.
\begin{theorem}\label{Theoremg1}.
Let us assume that $n=1$ and that $v$ satisfies (\ref{a.15}),
(\ref{a.18}) and the condition
\begin{equation}\label{g.1}
\begin{array}{lr}
\quad& v'' (x) \leq - \gamma_0 + \gamma_1 x^2\;,
\end{array}
\end{equation}
for some real $\gamma_0$, $\gamma_1$ with $\gamma_0 >0$.
Then, for any $h>0$,
\begin{equation}\label{g.2}
\frac{\gamma_0}{\gamma_1}\leq \langle x_j^2 \rangle_{\beta,\Lambda}\;.
\end{equation}
\end{theorem}
In many cases, we can take $\gamma_0 = - v''(0)$. The point $0$
corresponds for example
to the top between the two wells. \\
We note also that this approach does not make use of the Ginibre
inequalities. This leaves possible extensions for the case $n >1$
in the case when $\Hess v (0)$ has a strictly negative eigenvalue
which seems generic for a double well problem and also true in the
radial case considered by \cite{PaKh}.\\
We observe also that the left hand side in (\ref{g.2}) is independent of $h$.\\
\subsection{Proof of the theorem}
The Bogolyubov inequality was giving
\begin{equation}\label{g.3}
\langle v''(x_j)\rangle_{\beta,\Lambda} \geq 0\;.
\end{equation}
The theorem follows immediately from (\ref{g.1}).\\
Because the left hand side of (\ref{g.2}) is independent of $\beta$
and $\Lambda$, the arguments developed in Section \ref{Sectionb}
work in the same way: long range order for $h$ small enough or $\Jg$
large enough. Moreover, the control of ``$h$ small enough'' is
explicit.\\
We recall that in the semi-classical approach developed in the
spirit of Barbulyak-Kondratev ``$h$ small enough'' is implicit.\\
As already mentioned, the condition (\ref{g.2}) can be verified when
\begin{equation}\label{g.4}
v''(0) <0\;,
\end{equation}
which is a condition of concavity at $0$,
and when the condition at $\infty$
\begin{equation}
v''(x) \leq C x^2\;, \; \forall x > C\;,
\end{equation}
is satisfied.
The first one seems natural in the context. The second one is
apparently
more technical, at least in the semi-classical context, as we shall see in the next subsection for an
example.
\subsection{The case $n=1$, $v(x) =b x^{2k} - a x^2$}
In this case we have
\begin{equation}\label{g.5}
v''(x) = 2k (2k-1)b x^{2k-2} - 2a \;,
\end{equation}
with $a>0$.
We then obtain the inequality, for any $j\in \Lambda$,
\begin{equation}\label{g.6}
\frac{a}{k(2k-1)b}\leq \langle x_j^{(2k-2)}\rangle_{\beta,\Lambda}\;,
\end{equation}
as a consequence of the Bogolyubov inequality.\\
For $k=2$, this was enough for having a universal lower bound for
$\langle x_j^2\rangle_{\beta,\Lambda}$ which was decisive for the use
of the infrared estimate.\\
The problem is then to treat the case $k>2$. The Gaussian domination
inequalities give a rather satisfactory answer. They give indeed the
inequality
\begin{equation}\label{g.7}
\langle x_j^{(2k-2)}\rangle_{\beta,\Lambda} \leq C(k) \langle
x_j^2\rangle ^{k-1}\;,
\end{equation}
where $C(k)$ is a universal constant counting the number of pairings.
Newman \cite{Ne} gives,
\begin{equation}\label{g.8}
C(k) =\frac{ (2k -2)!}{2^{k-1} (k-1)!}\;.
\end{equation}
Consequently, a lower bound for $\langle x_j^2\rangle_{\beta,\Lambda}$ (which has to be
uniform
with respect to $\Lambda$, $\beta$ and $j$) will be a consequence
of an upper bound for $\langle x_j^{2k}\rangle_{\beta,\Lambda}$. We
note here that everything is actually independent of $j$.\\
The first idea could be to analyze first the case when $\beta = +
\infty$. This leads to some results which are not basically better
than the results obtained in Section \ref{Sectiond}. We just mention
that these results are obtained by combination of (\ref{f.9}) and of the inequality
\begin{equation}\label{g.9}
\langle v(x_j) \rangle_{\infty, \Lambda} \leq \frac{
\lambda_1^\Lambda}{|\Lambda|}\;.
\end{equation}
In the case $\beta$ large,
the arguments of the preceding example work if we have a control of $\langle x_j^{2k}
\rangle_{\beta, \Lambda}$ by a constant (using in this case an
H\"older inequality) or if we can more directly have a control of $\langle x_j^{2k-2}
\rangle_{\beta, \Lambda}$ by a power of $\langle x_j^{2}
\rangle_{\beta, \Lambda}$. In the next section, we shall exploit this
idea in combination with the Ginibre's inequalities.
\subsection{ Using Lebowitz inequality or gaussian domination inequalities.}
As already said, the general princip of the Lebowitz inequalities is to control any
correlation by the pair correlations
under rather weak assumptions. Our strategy will be to modify the
initial potential $v$ in order to get a new potential $v_1$ satisfying
the assumptions of Theorem \ref{Theoremc6}.
This will lead to the following improvement of Theorem \ref{Theoremg1}
\begin{theorem}
\label{Theoremg2}.
Let us assume that $n=1$, that $v$ satisfies (\ref{a.15}),
(\ref{a.18}) and the condition
\begin{equation}\label{g.10}
v''(0) <0\;.
\end{equation}
Let us also assume the existence of $\ell$ and $C$ such that
\begin{equation}\label{g.11}
|v''(x)| \leq C x^\ell, \forall x> C\;.
\end{equation}
Then, there exists $ \rho >0$ such that, for any $h>0$, $\beta>0$ and
$\Lambda$,
we have
\begin{equation}\label{g.12}
\rho \leq \langle x_j^2 \rangle_{\beta,\Lambda}\;.
\end{equation}
\end{theorem}
Outside this unfortunate condition $n=1$ due to the fact that we use
Ginibre's
inequality, this theorem is rather satisfactory. The computation of
$\rho$ is not very explicit but can be done on examples.\\
{\bf Proof of Theorem \ref{Theoremg2}}
\noindent The main new step is the following lemma
\begin{lemma}
\label{Lemmag3}.
Under the assumption of Theorem \ref{Theoremg2}, there exists $k\in \nz$,
$\alpha_0>0$ and
$\alpha_1 >0$
such that
\begin{equation}
\label{g.13}
v_1(x) = v(x) + \alpha_0 x^4 + \alpha_1 x^{2k}
\end{equation}
satisfies:
\begin{equation}
\label{g.14}
v'_1(x)\mbox{ is convex on } \rz^+\;.
\end{equation}
\end{lemma}
We observe that $v_1$ satisfies (\ref{a.15}), (\ref{a.18}) and for
suitable $\gamma_0 >0$ and $\gamma_1>0$ the estimate
\begin{equation}
\label{g.15}
v_1''(x) \leq - \gamma_0 + \gamma_1 x^{2k}
\end{equation}
The key remark is that the Ginibre's inequalities give the estimate
\begin{equation}
\label{g.16}
\langle x_j^2\rangle_{\beta,\Lambda, v}
\geq \langle x_j^2\rangle_{\beta,\Lambda, v_1}\;.
\end{equation}
Here $\langle x_j^2\rangle_{\beta,\Lambda, v}$(resp. $\langle
x_j^2\rangle_{\beta,\Lambda, v_1}$) means that we consider
$H_\Lambda^{per}$ with the one particle potential $v$ (resp $v_1$).
We are now reduced to the problem of finding a lower bound for
$\langle x_j^2\rangle_{\beta,\Lambda, v_1}$. With the properties of
convexity
of $v_1'$ and the estimate (\ref{g.16}), this case is easily treated
by combination of the Bogolyubov inequality and of the gaussian
domination inequality.
\section{Conclusion}\label{Sectionh}
In this article, we have given explicit conditions on the potential
$v$
under which a good lower bound can be found for the second order
moment. \\
When $\beta=+\infty$, the results are not far from the optimal one
and no restriction in the dimension $n$ is involved. Nethertheless,
restrictions on $d$ appear when applying these inequalities in the
infrared estimate strategy for proving transition of phase.\\
When we want to extend this study to the case $\beta$ large, we meet
immediately the condition $n=1$ (which could probably be extended to
the case $n>1$ but with invariance by $SO(n)$) and this is due to the
fact that we use these monotonicity arguments related to various
extensions of FKG inequalities.\\
The importance of the two restrictions on $d$ and $n$ is sometimes
technical, sometimes perhaps deeper and has surely to be investigated more
carefully.\\
The analysis of the splitting given here is rather rough due to our
weak
control of the localization of the first eigenfunction in the
thermodynamic limit. This has also to be analyzed in the future in order to see the effects
of the tunneling between the wells in the spirit of our previous
works with J. Sj\"ostrand \cite{HeSj1992a}, \cite{HeSj1992b}.\\
\noindent {\bf Acknowledgements :} \\
\noindent Preliminary results were announced at the conference in
Ascona (June 1996) and the NATO conference in
Castel Vecchio (September 1996). We thank J. Fr\"ohlich for a
motivating discussion in Ascona.\\
We thank also the European
Union which partially supported this research through the TMR Programme of the
European Commission - Network Postdoctoral training programme in
partial differential equations and application in quantum
mechanics-.\\
\begin{thebibliography}{99}
\bibitem [{AlKoRe}]{AlKoRe} Albeverio, S., Kondratiev, A.Yu and Rebenko,
A.L. (1996)
\newblock Peierls argument and long-range order behaviour of quantum
lattice systems with unbounded spins,
\newblock Preprint.
\bibitem[{BaKo1}]{BaKo} Barbulyak, V.S. and Kondrat'ev, Y. (1992)
\newblock The quasiclassical limit for the Schr\"odinger operator
and phase transitions in Quantum statistical physics,
\newblock {\it Functional Analysis and applications}, {\bf 26}(2),
61-64.
\bibitem[{BaKo2}]{BaKo90} Barbulyak, V.S. and Kondrat'ev, Y. (1990)
\newblock Methods of functional analysis in Problems of
mathematical Physics (in Russian), Institut Mat. AN USSR, Kiev,
30-41.
\bibitem[{BoMa}] {BoMa} Bouziane, M. and Martin, Ph. A. (1976)
\newblock Bogolyubov inequality for unbounded operators and the Bose
gas,
\newblock {\it J. Math. Phys. } {\bf 17}(10), 1848-1851.
\bibitem[{BraLi}]{BraLi1976} Brascamp, H.J. and Lieb, E. (1976)
\newblock On extensions of the Brunn-Minkovski
and Pr{\'e}kopa-Leindler Theorems, including inequalities
for Logconcave functions, and with an application to diffusion
equation, \newblock{\it Journal of Functional Analysis} {\bf 22},
366-389.
\bibitem[{Br}]{Br} Bricmont, J. (1977)
\newblock The gaussian inequalities for multicomponent rotators,
\newblock {\it J. Stat. Phys} {\bf 17}, 289-300.
\bibitem [{BFS}]{BFS} Brydges, D., Fr\"ohlich, J., and Spencer,
T. (1978)
\newblock The random walk representation of classical spin systems and
correlation inequalities,
\newblock {\it Commun. Math. Phys.}{\bf 83}, 123-150 .
\bibitem [{DyLiSi}]{DyLiSi} Dyson, F.J., Lieb,E.H. and Simon, B. (1978)
\newblock Phase transitions in quantum spins systems with isotropic and nonisotropic interactions,
\newblock {\it J. Statistic. Phys.} {\bf 18}(4), 335-383.
\bibitem[{EN}]{EN}Ellis, R.S. and Newman, C.M.(1978)
\newblock Necessary and sufficient conditions for the GHS inequality
with applications to analysis and probability
\newblock {\it Transactions of the AMS} {\bf 237}, 83-99.
\bibitem[{EMN}]{EMN}Ellis, R.S., Monroe, J.L. and Newman, C.M.(1976)
\newblock The GHS and other correlations ineqqualities for a class of
even ferromagnets,
\newblock {\it Commun. math. Phys.} {\bf 46}, 167-182.
\bibitem[{FKG}]{FKG} Fortuin, C., Kastelyn, P. and Ginibre, J. (1971)
\newblock Correlation inequalities on some partially ordered sets,
\newblock {\it Comm. in Math. Physics} {\bf 22}, 89-103.
\bibitem[{Fr}]{Fr} Fr\"ohlich, J. (1976)
\newblock Phase transitions, Goldstone bosons and topological
superselection rules,
\newblock {\it Acta Phys. Austriaca} Suppl. XV, 133-269.
\bibitem[{FILS}]{FILS} Fr\"ohlich, J., Israel, R., Lieb, E.H. and
Simon, B. (1978)
\newblock Phase transition and reflection positivity, I General theory
and long range lattice models,
\newblock {\it Comm. in Math. Physics} {\bf 62} (1), 1-34.
\bibitem[{FSS}]{FSS} Fr\"ohlich, J., Simon, B. and Spencer, T. (1976)
\newblock Infrared bounds, phase transitions, and continuous symmetry breaking,
\newblock {\it Comm. Math. Phys.} {\bf 50}, 79-85.
\bibitem[{GlJa}]{GlJa1987} Glimm, J. and Jaffe, A. (1987)
\newblock {\it Quantum physics (a functional integral point of view),}
\newblock Springer Verlag, Second edition.
\bibitem[{GJS}] {GJS} Glimm, J., Jaffe, A. and Spencer, T. (1975)
\newblock Phase transitions for $\phi_2^4$ quantum fields,
\newblock {\it Comm. Math. Phys.} {\bf 45}, 203-216.
\bibitem [{GlKo}]{GlKo} Globa, S.A. and Kondrat'ev, Yu. G. (1987)
\newblock Application of functional analysis in Problems of
mathematical Physics (in Russian), Institut Mat. AN USSR, Kiev,
4-16.
\bibitem[{He1}]{He1987} Helffer, B. (1988)
\newblock {\it Introduction to the semiclassical analysis
for the Schr\"odinger operator and applications, }
\newblock Springer lecture Notes in Math., n$^0$ 1336.
\bibitem[{He2}]{He1994e} Helffer, B. (1996)
\newblock Recent results and open problems on Schr\"odinger operators,
Laplace integrals and transfer operators in large dimension,
\newblock Advances in PDE, in Schr\"odinger operators, Markov
semigroups, Wavelet analysis, operator algebras,
\newblock edited by M. Demuth, E. Schrohe, B.W. Schultze and J. Sj\"ostrand,
Akademie Verlag, 11-162.
\bibitem[{He3}]{He1995a} Helffer, B. (1995)
\newblock {\it Semiclassical analysis for Schr\"odinger operators,
Laplace integrals and transfer operators
in large dimension: an introduction,}
\newblock Paris 11- Edition.
\bibitem[{He4}]{He1995b} Helffer, B. (1996)
\newblock On Laplace integrals and transfer operators
in large dimension: examples in the non convex case,
\newblock
{\it Letters in Math. Physics} 38, 297-312 (1996).
\bibitem [{He5}]{He96} Helffer, B. (1996)
\newblock Splitting in large dimension and infrared estimates,
\newblock To appear 1997, in Proceedings of the NATO-ASI conference, Kluwer.
\bibitem[{HeSj1}]{HeSj1984} Helffer, B. and Sj{\"o}strand, J. (1984)
\newblock Multiple wells in the semiclassical limit,
\newblock {\it Comm. in PDE} {\bf 9} (4), 337-408.
\bibitem[{HeSj2}]{HeSj1992a} Helffer, B. and Sj{\"o}strand, J. (1992)
\newblock Semiclassical expansions of the
thermodynamic limit for a Schr{\"o}dinger equation,
\newblock {\it Ast{\'e}risque} {\bf 210}, 135-181.
\bibitem[{HeSj3}]{HeSj1992b} Helffer, B. and Sj{\"o}strand, J. (1992)
\newblock Semiclassical expansions of the thermodynamic limit
for a Schr{\"o}dinger equation II,
\newblock {\it Helvetica Physica Acta} {\bf 65}, 748-765
and Erratum {\bf 67} (1994), 1-3.
\bibitem[{HeSj4}]{HeSj1993} Helffer, B. and Sj{\"o}strand, J. (1994)
\newblock On the correlation for Kac like models in the convex case,
\newblock
{\it J. Statist. Physics}, {\bf 74} (1-2), 349-369.
\bibitem[{La}]{La} Laroche, E. (1993)
\newblock In\'egalit\'es de corr\'elation sur $\{-1,+1\}^n$ et dans
$\rz^n$,
\newblock {\it Ann. Inst. Henri Poincar\'e} {\bf 29}(4), 531-567.
\bibitem[{Leb1}] {Leb72} Lebowitz, J. (1972)
\newblock Bounds on the correlations and analyticity properties of
ferromagnetic Ising spin systems,
\newblock{\it Commun. Math. Phys.} {\bf 28}, 313-322.
\bibitem[{Leb2}] {Leb74} Lebowitz, J. (1974)
\newblock GHS and other inequalities,
\newblock{\it Commun. Math. Phys.} {\bf 35}, 87-92.
\bibitem[{LebPre}]{LebPre} Lebowitz, J., Pressutti, E. (1976)
\newblock Statistical Mechanics of systems of unbounded spins,
\newblock Comm. in Math. Phys. {\bf 50}, 195-218.
\bibitem[{Ne}]{Ne} Newman, C.M. (1975)
\newblock Gaussian correlation inequalities for ferromagnets,
\newblock {\it Z. Wahrscheinlichkeitstheorie verw. Gebiete} {\bf 33},
75-93.
\bibitem[{PaKh}]{PaKh} Pastur, L.A. and Khoruzhenko, B. A. (1987)
\newblock Phase transitions in quantum models of rotators and ferroelectrics,
\newblock {\it Teor. Mat.
Fiz.}, {\bf 73} (1), 111-124 .
\bibitem[{Re}]{Re} Rebenko, A.L. (1996)
\newblock Peierls argument and long range behavior of quantum lattice
systems
with unbounded spins.
\newblock Lecture in Ascona (June 1996).
\bibitem[{Ru}]{Ru1969} Ruelle, D. (1969)
\newblock {\it Statistical mechanics,}
\newblock Math. Physics monograph series,
W.A.Benjamin, Inc..
\bibitem[{Si1}]{Si74} Simon, B. (1974)
\newblock {\it The $P(\phi)_{2}$ Euclidean quantum field theory,}
\newblock Princeton Series in Physics.
\bibitem[{Si2}]{Si79} Simon, B. (1979)
\newblock {\it Functional integration and quantum physics.}
\newblock Pure and Applied mathematics, $n^{0}$ 86, Academic press, New
york.
\bibitem[{Sj1}]{Sj1993b} Sj{\"o}strand, J. (1993)
\newblock Potential wells in high dimension II, more about the one
well case,
\newblock{\it Ann. Inst. H. Poincar\'e}, Section Physique th\'eorique
{\bf 58}(1), 1-41.
\bibitem[{Sj2}]{Sj1993e} Sj{\"o}strand, J. (1994)
\newblock Evolution equations in a large number of variables,
\newblock Mathematische Nachrichten {\bf 166}, 17-53.
\bibitem[{Sok}]{Sok} Sokal, A.D. (1982)
\newblock Mean-field bounds and correlation inequalities,
\newblock {\it Journal of statistical Physics}, {\bf 28} (3), 431-439.
\bibitem[{Sy}]{Sy} Sylvester, G.S. (1975)
\newblock Representations and inequalities for Ising model Ursell functions,
\newblock {\it Comm. Math. Phys.} {\bf 42}, 209-220.
\end{thebibliography}
\end{document}