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\title{Semi-classical analysis for the
transfer operator: formal WKB constructions in large dimension}
\author{Bernard Helffer
\\
Universit\'e Paris-Sud,\\
D\'epartement de Math\'ematiques, UA 760 du CNRS,\\ Bat. 425,
F-91405 Orsay Cedex, FRANCE }
\date{November 8 , 1996}
\begin{document}
\bibliographystyle{plain}
\maketitle
%
%RESUME
%
\begin{abstract}
This paper is devoted to problems coming from statistical mechanics.
The transfer
matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical
systems to the analysis of the
spectral properties of their transfer operator. Sometimes the new problem appears to have a
semi-classical nature. Although the problem
is similar to the semiclassical study of the Kac's operator presented
in our paper with M.Brunaud \cite{BruHe1991} which was devoted to the study of
$\exp - \frac{V}{2} \cdot \exp h^2 \Delta \cdot \exp - \frac{V}{2} $
for $h$ small, new features appear for
the model $\exp - \frac{V}{2h} \cdot \exp \frac ha \Delta \cdot
\exp - \frac{V}{2h} $. Our first results concern
semi-classical analysis of the ground state for this
second operator. We then analyze the two models in
the large dimension situation. One basic technique is Sj\"ostrand's formalism
of the $0$-standard functions as introduced in \cite{Sj1993e}. The one-dimensional
case was presented in \cite{He1995b}.
\end{abstract}
% %FIN DE RESUME %
\section{Introduction} \label{intro}
We consider a model generalizing the Kac model with nearest
neighbors interaction:
\begin{equation}\label{i1}
\Phi(x)
= \sum_{j\in \zz/m\cdot\zz} V(x_j)
+ \sum_{j\in \zz/m\cdot\zz} W(x_j,x_{j+1})\;,
\end{equation}
for $x\in\rz^{mp}$, $x= (x_1,\cdots, x_m)$, $x_j=
(x_{j1},\cdots,x_{jp})\in \rz^p$,
and we assume that
\begin{equation}\label{i2}
\Phi(-x)=\Phi (x)\;.
\end{equation}
$V$ is now a potential defined on $\rz^p$ (which is usually invariant
by circular permutation of the variables)
and we shall try to follow most of the arguments with respect to $p$
in order to take the limit $p\ar\infty$ after having first taken the
limit $m\ar \infty$.
We are interested in the asymptotic properties of the measure
$$ (\pi h)^{-\frac {mp}{2}}\exp - \frac{\Phi}{h} dx^{(mp)}$$ as $m$ and
$1/h$ tend to $+\infty$.
More precisely we are mainly interested in the asymptotic behavior of the
total measure
\begin{equation}\label{i3}
\mbox{Vol}(\rz^{mp}) := \int_{\rz^{mp}} \exp - \frac{\Phi}{h} \;dx^{(mp)}
\end{equation}
as $m$ and $p$ tend to $\infty$.
As often in statistical mechanics, the order in which we take the limit
could be quite
important.\\
Following the general reduction called the "transfer matrix method",
we have seen that we have to analyze the following
transfer operator $K^{tf}=K_V$ whose kernel is given by
$$ K_V(x,y) = (\pi h)^{-\frac p2} \exp - \left (\frac {1}{2h} \cdot
(V(x) + V(y))\right)
\cdot \exp - h^{-1} W(x,y)\;.$$
We recall indeed the following relation (see for example
(12.19) in \cite{Pa1984} or \cite{He1994d}, \cite{He1994e}) between
the transfer
operator and the total volume:
\begin{equation}\label{i4}
\lim_{m\ar +\infty} \frac{\ln \mbox{Vol}(\rz^{mp}) }{m}= \ln \mu_1\;,
\end{equation}
where $\mu_1$ is the largest eigenvalue of $K^{tf}$.\\
The second step could be to analyze $\lim_{p\ar + \infty} \frac{\ln \mu_1}{p}$
but this will not be done here and we shall only speak about
uniform control with respect to $p$ of $\frac{\ln \mu_1}{p}$.\\
Although our methods are potentially more general, we shall concentrate our study
on the case when
$$ W(x,y) = \frac a4 |x-y|^2 \;,$$
with $a>0$.\\
As a typical $V$ we take
$$V^{(p)}(y) = \sum_{i=1}^p v(y_i) + b \sum_{i=1}^p |y_i-y_{i+1}|^2\;,
$$
and we assume that $b>0$ and $v$ is a single or double well one dimensional potential with the symmetry
$$
v(-t)=v(t)\;.
$$
\\
We recognize indeed a "Kac" like operator (up to a multiplicative
constant)
that we already studied
previously (\cite{BruHe1991}, \cite{He1992b})
$$ K^{tf} = \exp - \frac{V}{2h} \cdot \exp \frac ha \Delta \cdot
\exp -\frac{ V}{2h} \;,$$
with $a>0$ and $\Delta = -\sum_j \pa_{x_j}^2$,
but the semiclassical parameter is not inserted as in the standard
semiclassical analysis for which we had
$$ K^{kac} = \exp - \frac{V}{2} \cdot \exp h^2 \Delta \cdot
\exp -\frac{V}{2} \;.$$
We note that the potential $V^{(m,p)}$ can in the case $a=b$ be written in the form
$$
V^{(m,p)} = \sum_{(j,i)\in (\zz/m\zz)\times (\zz/p\zz)} v(x_{ji})
+ a \sum_{(i,j)\sim (k,\ell)} |x_{ji}-x_{k\ell}|^2\;,$$
where $(i,j)\sim (k,\ell)$ means here that $(i,j)$ and $(k,\ell)$ are
nearest neighbors in $(\zz/m\zz)\times (\zz/p\zz)$.\\
Our main study here will be to {\bf analyze
the existence of formal WKB approximations of groundstate eigenfunctions for
these operators.} Simultaneously, this will give a semiclassical expansion for the first (= largest) eigenvalue
and the other important point is that {\bf we shall analyze the behavior with respect to the dimension of the coefficients of this
expansion}. This is one step in the direction of understanding the semi-classical
analysis of the first eigenvalue with control with respect to the dimension. This step was done in the large dimension context for the
Schr\"odinger operator by J. Sj\"ostrand \cite{Sj1993a}, \cite{Sj1993b}
and \cite{Sj1993e} ( see also \cite{HeSj1992a} )
and for the transfer or Kac operator without control with respect to the
dimension in
\cite{BruHe1991}, \cite{He1992b} and \cite{He1995b}.
\section{Lower bound of the first eigenvalue}\label{Section2}
We first give a lower bound for the largest eigenvalue $\mu_1$ of $K$ which is here defined as
\begin{equation}
K^{tf}= \exp -\frac{V}{2h}\cdot \exp \frac ha\Delta\cdot\exp -\frac{V}{2h}\; .
\end{equation}
The distribution kernel of this operator is given by
\begin{equation}
K^{tf}(x,y)= (4\pi \frac{h}{a})^{-\frac p2}\;\cdot \exp
-\frac{V(x)}{2h}\cdot \exp - \frac{a}{4h} |x-y|^2\cdot\exp -\frac{V(y)}{2h}\; .
\end{equation}
We recall that the Perron-Frobenius theorem
(more precisely, its extension to compact integral operators with
strictly positive kernel) gives that the largest eigenvalue is simple
and equal to the norm of the operator with kernel $K$. Moreover, the first
eigenvector can be chosen strictly positive.\\
Let us assume that the potential $V$ satisfies
\begin{itemize}
\item
\begin{equation}
V\geq 0\;,\quad\quad\quad \quad\quad\quad
\end{equation}
\item
\begin{equation}
V(x)\ar +\infty \mbox{
as } x\ar \infty\; , \quad\quad\quad \quad
\end{equation}
\item
\begin{equation}
|D_x^\alpha V(x)|\leq C_\alpha (1+ V(x))^{(1-\rho |\alpha|)_+}\;,\quad\quad\quad
\end{equation}
for some $\rho >0$ and suitable constants $C_\alpha$,
\item
\begin{equation}
\mbox{the minima of } V \mbox{ are isolated, non-degenerate and in }
V^{-1}(0)\;.
\end{equation}
\end{itemize}
One can use Segal's Lemma (cf \cite{ReSi}, p. 333) giving a direct comparison between the
largest eigenvalue of the Kac operator and the smallest eigenvalue $\lambda_1$
of the corresponding Schr\"odinger operator
$$ - \frac ha\Delta + \frac V h\;,$$
with the following inequality
\begin{equation}\label{Segal}
\mu_1\geq \exp - \lambda_1 \;.
\end{equation}
We then get that
$\mu_1$ is bounded from below independently of $h$ for $00$,
\begin{equation}\label{3.4}
0< u_1(x;h) \leq \alpha^{-1} (\int K_V(x,y)^2 dy)^{\frac 12}\;.
\end{equation}
We then deduce the existence of a constant $C$ such that
the following estimate is satisfied, for all $h\in ]0,1]$,
\begin{equation}\label{est}
0< u_1(x;h)\leq C \cdot h^{-\frac p4} \cdot \exp - \frac{V(x)}{2h}\;.
\end{equation}
This means that the eigenfunction is strongly localized near
the minima of $V$. This will permit us to use cut-off functions in order
to construct quasimodes attached to each well. We observe here
that we are working with operators which are not local, but pseudolocal
in a weak sense. The pseudolocality is obtained through the interaction term
$ \frac ah \cdot |x-y|^2 $. This is a weaker pseudolocality than the pseudolocality
observed in the semi-classical study of the Kac operator where the
pseudolocality was obtained through an interaction in
$ \frac {1}{h^2}\cdot|x-y|^2$. This last case was analyzed
in detail in \cite{BruHe1991}
and in \cite{He1992b} (but without a control with respect to the dimension).
This will change the properties and the way of thinking for this problem.
The case when the interaction is equal to $ \frac {a}{h}\cdot|x-y|^2$ with
$a$ large will be more reminiscent of the semiclassical case as
observed in \cite{He1995b}.
\subsection{Toward improvements on the decay}
\noindent As presented in \cite{He1995b}, these estimates can be improved by
recursion.
We recall some of the arguments.
\noindent Our starting point is (\ref{3.4}). We immediately observe that we can implement
this upper bound in (\ref{3.2}) and we get, for any compact $K$,
the upper bound
\begin{equation}
0 0\;,\;\forall x\in \rz^p\;,
\end{equation}
and we assume that $V$ has a unique minimum at $0$ with
\begin{equation}
V(0)=0\;.
\end{equation}
It is clear that we have a family of convex functions $\varphi_n$ defined
by (\ref{recu}) and that there exists, for each $n$, an unique $C^\infty$ function
$x\mapsto y_n(x)$ such that
\begin{equation}\label{z1}
\varphi_n(x) = \varphi_{n-1} (y_n(x)) + \Theta (x,y_n(x))\;,
\end{equation}
with
\begin{equation}\label{z2}
\Theta (x,y) = \frac{V(x)}{2} + \frac{V(y)}{2} + \frac{a}{4} |x-y|^2\;.
\end{equation}
Fore simplicity,
we discuss only the case when the dimension is one.
We have for $\varphi_n''(x)$ the following formula
\begin{equation}\label{z3}
\varphi_n''(x) = \frac 12 [V''(x) + a] - \frac{a^2}{[2 v''(y_n(x)) +2a + 4 \varphi_{n-1}''(y_n(x))]}
\end{equation}
Given a global lower bound of
$\varphi_{n-1}''$ by $\sigma_{n-1}$,
we are looking for a lower bound for $\varphi_n''(x)$.
But we get from (\ref{z2}) and (\ref{z0}) the following lower bound
\begin{equation}\label{z4}
\varphi_n''(x)\geq \frac{1}{2} (\rho + a) - \frac{a^2}{2\rho + 2a + 4 \sigma_{n-1}}
\;.
\end{equation}
This suggests to define the following sequence $(\sigma_n)_{n\in \nz}$
by
\begin{equation} \label{z5}
\begin{array}{rl}
\sigma_0 &= \frac \rho 2\;,\\
\sigma_n& = \frac{1}{2} (\rho + a) - \frac{a^2}{2\rho + 2a + 4 \sigma_{n-1}}
\;.
\end{array}
\end{equation}
It is clear that we have
$$
0\leq \sigma_n \leq \frac 12 (\rho + a)
$$ and the only possible limit for $\sigma_n$
is
$$
\sigma_\infty = \frac{1}{2} \sqrt{(\rho+a)^2 - a^2}\;.
$$
The associated sequence
$$
w_n = \sigma_\infty - \sigma_n
$$
satisfies
$$
w_n = \frac{ [\frac 12 (\rho +a) - \sigma_\infty] w_{n-1}}
{ [\frac 12 (\rho +a) + \sigma_\infty] - w_{n-1} }
\;.
$$
Observing that
$ 0< w_0 < 2\sigma_\infty< \rho+a$,
it is easy to get the convergence of $w_n$ to $0$.
Modulo some limit argument, which we have not verified in detail
we surely obtain that the limit function $\varphi_\infty$ satisfies
\begin{equation}
\varphi_\infty''(x) \geq \frac{1}{2} \sqrt{(\rho+a)^2 - a^2}\;.
\end{equation}
Coming back to the general case, we can also obtain
\begin{equation}
\Hess\varphi_\infty(x) \geq \frac{1}{2} \sqrt{(\rho+a)^2 - a^2}\;.
\end{equation}
\noindent This gives a universal lower bound measuring the strict
convexity of\break $\Hess \varphi_\infty (x)$. The inequality is actually an equality in the quadratic case (see (\ref{quadr})).
This is of course related to the logconcavity of the first eigenfunction
of the Kac operator and we refer to Helffer \cite{He1993d} and Brascamp-Lieb
\cite{BraLi} for more details on this point.
\section {WKB constructions for $K^{tf}$}\label{s4}
\subsection{Introduction}
As a standard step for the construction of an approximate
solution, we shall look for the construction of a formal WKB solution
in the form\\
$u^{WKB} = a(x;h) \exp -\frac {\phi_0(x)}{h}$ with
$a(x;h)\sim \sum_j a_j(x) h^j$, \\
or in the form\\
$u^{WKB} = \exp -\frac {\phi(x;h)}{h}$
with $\phi(x;h)\sim \sum_j \phi_j(x) h^j$.\\
We then try to find a formal
$F(h)\sim\sum_j h^j F_j$ such that
$$ K u^{WKB} \sim \exp - F(h) \;u^{WKB}\;.$$
By formal we mean here that we just consider formal
expansions\footnote{ More precisely, we compare the expansions in
powers of $h$ of $\exp \frac{\phi_0(x)}{h} \;K\;u^{WKB}$ and of $\exp
\frac{\phi_0(x)}{h} \;\exp - F(h)\;u^{WKB}$ for $x$ in the
neighborhood of one minimum of $V$} in powers of $h$
and that the expansion of $K u^{WKB}$ is obtained by applying the
formal Laplace integral method.
\subsection{The eikonal equation}\label{ss41}
We want to control the $C^\infty$ character of the solution in the preceding construction
when we are near the minimum of the potential. We have already defined a Lipschitzian function
$\varphi_\infty$ which plays in some sense the role of the Agmon distance to the minima
of the potential $V$.\\
It was observed in \cite{HeSj1984} that this Agmon's
distance $d$ becomes $C^\infty$ in the neighborhood of the minima if they are assumed
to be non-degenerate. The distance $d$ was in this case a solution of
the eikonal equation
\begin{equation}
|\nabla d|^2 = V-\inf V\;,\; d\geq 0\;,
\end{equation}
in the neighborhood of a minimum.
\\
We shall start from the following identities that we obtain first formally by
assuming that everything
is differentiable. The starting point is
\begin{equation}
\phi(x) = \inf_y [\Theta(x,y) + \phi(y)]\;,
\end{equation}
with
\begin{equation}
\Theta(x,y) = W(x,y) + \frac{V(x)}{2} + \frac{V(y)}{2} \;.
\end{equation}
\noindent If we assume that the minimum is obtained
for a unique regular $y(x)$ (and this will be proved for $x$ small
enough), we get
\begin{equation}
\phi(x) = \Theta (x,y(x)) + \phi(y(x))\;,
\end{equation}
and
\begin{equation}\label{uti}
\pa_y \Theta (x,y(x)) + \nabla \phi(y(x))=0\;.
\end{equation}
Differentiating with respect to $x$, we first obtain
\begin{equation}
\nabla \phi(x) = \pa_x \Theta(x,y(x))\;.
\end{equation}
\noindent Following an enlightening discussion
with J.Sj\"ostrand, we are actually looking for a Lagrangian manifold
$$\Lambda_\phi = \{(x,\xi)\in \rz^{2p} | \xi = \nabla \phi(x)\}$$
which is invariant
under the canonical relation generated by $\Theta$:
\begin{equation} C_\Theta = \{ (x,\xi,y,\eta)\in \rz^{4p}| \xi = \pa_x \Theta(x,y)\,,\, \eta = -\pa_y \Theta(x,y)\}
\end{equation}
We recall that, in the case of the Schr\"odinger operator, we found a Lagrangian
manifold inside
$p(x,\xi)=:\xi ^2 - V(x)=0$ by looking for an invariant outgoing
Lagrangian
manifold for the flow $H_p$. Here we find
the same phenomenon
in a discrete version.\\
Let us express the diffeomorphism $\kappa$ whose graph is $ C_\Theta$ in the case when $W(x,y) = \frac a4 |x-y|^2$.\\
We find the equations :
\begin{equation}\label{kappa}
\begin{array}{ll}
\xi &= (\nabla V) (x) + \frac{a}{2} (x-y)\\
\eta&= - (\nabla V) (y) + \frac{a}{2} (x-y)\;.
\end{array}
\end{equation}
and this gives the expression for $(y,\eta) = \kappa (x,\xi)$
\begin{equation}\label{5.8}
\begin{array}{lll}
y &= x + \frac 2a (\nabla V) (x) - \frac 2a \xi&:=y(x,\xi)\;,\\
\eta&= - (\nabla V) (y (x,\xi)) + \xi - (\nabla V)(x)&:=\eta(x,\xi)\;.
\end{array}
\end{equation}
We recall that $\kappa$ is a canonical transformation, that is a diffeomorphism
which preserves the canonical symplectic $2$-form on $\rz^{2p}$: $\sum_j d\xi_j\wedge dx_j$.\\
One gets also a similar expression for $\kappa^{-1}$:
\begin{equation}\label{5.8b}
\begin{array}{lll}
x &= y + \frac 2a (\nabla V) (y) + \frac 2a \eta&:=x(y,\eta)\;,\\
\xi &= (\nabla V) (x (y,\eta)) + \eta + (\nabla V)(x)&:=\xi(y,\eta)\;.
\end{array}
\end{equation}
Let us look at the fixed points of this diffeomorphism. We are looking
for the points
such that $y(x,\xi)=x$, $\eta(x,\xi)=\xi$
and we get $\xi = 0$, $\nabla V (x) =0$.\\
The fixed points of the diffeomorphism $\kappa$ are the points $(x,0)$
where $x$ is a critical value of $V$.
In particular, if we assume that $0$ is a non-degenerate minimum of $V$,
$(0,0)$ is a fixed point. We now want to
apply the general theory of Smale \cite{Sm1}. We have to verify
that $(0,0)$
is an hyperbolic point for $\kappa$. This will be verified in a next
subsection by looking at the quadratic case.\\
The regularity of the "local" stable manifold
is an old result due to Smale \cite{Sm1} (See also \cite{Sm2} (p.751)
or \cite{Irw}). The property that these stable manifolds are Lagrangian
is obtained as in the continuous case
(see for example \cite{HeSj1984}) once we have observed that the number
of eigenvalues
of modulus $<1$ is exactly the dimension $p$.\\
\begin{remark}:\\
One easily deduces from the symmetry of $\Theta$
$$
\Theta(x,y)=\Theta(y,x)\;,
$$
the property that
$$(x,\xi,y, \eta)\in C_\Theta \quad \mbox{ iff}\quad (y, -\eta, x,-\xi)
\in C_\Theta\;.
$$
\end{remark}
The function $\phi$ appears simply as the generating function of the Lagrangian outgoing manifold
normalized by the condition $\phi(0)=0$ and using the preceding remark one get also
that $-\phi$ is the generating function for the ingoing manifold.\\
We have found consequently, for a given non-degenerate minimum of the
potential, a ball $B$ centered at this minimum, in which $\phi$ is defined and satisfies
$$ \phi (x) = \inf_{y\in B} [\Theta(x,y) + \phi(y)]\;,\;\forall x\in B\;.$$
\subsection {The first transport equation}
Once we have found the phase $\phi$, we arrive naturally to the study
of the
WKB problem: find $a(x;h)$ and $E(h)$ such that
\begin{equation}
(K_V a \exp - \phi(\cdot)/h)(x) \sim E(h) a(\cdot;h)
\exp - \phi(\cdot )/h\;.
\end{equation}
We want the following formal identity to be satisfied:
\begin{equation}
E(h) a(x;h) \sim h^{-\frac p2} \int a(y;h)\cdot
\exp - \frac 1h [ \Theta(x,y) + \phi(y) -\phi(x)] \;dy\;.
\end{equation}
Applying the Laplace integral method, we get
\begin{equation}\label{4.12}
E_0\, a_0(x) = C^p\cdot a_0(y(x))
(\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (y(x))\;,
\end{equation}
where $C$ is a universal constant.\\
A necessary condition, in order to solve (\ref{4.12}) is
\begin{equation}
E_0 = C^p\cdot (\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (0)\;.
\end{equation}
Then we have for $a_0$ the formula
\begin{equation}
a_0(x) = \prod_{n=1}^\infty \theta (y^{(n)}(x))\;,
\end{equation}
with by definition
\begin{equation}
y^{(n)}(x) = y (y^{(n-1)}(x))\;,
\end{equation}
and
\begin{equation}
\theta (y) = \left(\frac{\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi] (0)}
{\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi] (y)}\right)^\frac 12\;.
\end{equation}
We observe here that, for any $x$ sufficiently near the bottom $0$ of
$V$,
we have
\begin{equation}
y^{(n)}(x)\ar 0
\end{equation}
and the convergence is exponentially rapid. Let us observe also that,
for our particular function $\Theta$, the Hessian $\nabla_{yy}^2 \Theta $ is independent of $x$:
\begin{equation}
(\nabla_{yy}^2 \Theta)(y,x) = \frac a2\; I + \frac 12 \Hess V(y) \;.
\end{equation}
\noindent In order to see what is going on, let us come back to the one dimensional quadratic case.
We recall that, with $V(x)= \frac c2 x^2$, we have
\begin{equation}\label{4.19a}
\phi(y) = \frac b2 y^2
\end{equation}
with
\begin{equation}\label{4.20a}
b= \frac 12 \sqrt{c(2a+c)}
\end{equation}
and
\begin{equation}\label{4.21a}
y(x) = \frac {a}{c+a+2b}\; x\;.
\end{equation}
Of course we do not need the discussion in this case, but what will be relevant
for our discussion is the property that
\begin{equation}
||y(x)|| \leq k ||x||
\end{equation}
for $0< k <1$ for a suitable norm (as can be seen in the proof of the stable manifold theorem).
This is what will make the procedure convergent and actually exponentially
convergent.
\subsection{The analysis of the diffeomorphism in the quadratic case}
In order to verify this property of hyperbolicity, let us make
explicit the computation of the diffeomorphism $\kappa_0$,
$ (y,\eta) = \kappa_0(x,\xi) $,
attached to the quadratic form $V_0 (x)= \frac 12 \langle \Hess V (0) x,
x \rangle$ instead of $V$. After diagonalization of this potential attached to
$\Hess V$ at the minimum,
we can
reduce our considerations to the two-dimensional case.
The map $\kappa$ is in this case linear and given by
the $2\times 2$ matrix
$$
M_\kappa =
\left(
\begin{array}{cc}
\frac {(a+c)}{a} &- \frac 2a\\
-\frac {1}{2a} c (c+2a) & \frac{(a+c)}{a}
\end{array}
\right)
$$
The two eigenvalues are given by
$$
\lambda_\pm = \frac{(a+c)}{a} \pm \frac 1a \sqrt{c(c+2a)}
$$
and one verifies that
$$
0<\lambda_-<1
$$
and
$$
\lambda_-^{-1}=\lambda_+ >1\;.
$$
We have consequently an hyperbolic point (saddle point situation)
and a one dimensional subspace $\Lambda_-$ on which $\kappa$ is contracting, that
is the eigenspace attached to $\lambda_-$.
One finds the equation
$$ \xi = \frac 12 \sqrt{c(c+2a)} x $$
and, as in the case of the Schr\"odinger operator, one has, with $\phi$
introduced in (\ref{4.19a}),
$$
\xi = \phi'(x)
$$
that is $\phi$ is the generating function of $\Lambda_-$.
\subsection{Remarks in the convex case}\label{ss4.5}
We have already mentioned that in the strictly convex case a global strictly convex function
$\phi (x)$ can be defined. Moreover, we can obtain a global control of the function $x\mapsto y(x)$.
Differentiating (\ref{uti}) with respect to $x$, we obtain
\begin{equation}
\left(\Hess \phi (x) + \Hess_{yy} \Theta (x, y(x))\right) \nabla y (x)
+ \pa_x\pa_y\Theta (x,y(x)) =0\;.
\end{equation}
This gives
\begin{equation}
\nabla y (x) =
\frac a2 \left( \Hess \phi (x) + \Hess_{yy} \Theta (x, y(x))\;\right)
^{-1}\;.
\end{equation}
This gives
\begin{equation}
|| \nabla y (x)|| \leq \frac{ a}{a+\rho} <1\;,\; \forall x\;.
\end{equation}
This gives that the map $\rz^p\ni x\mapsto y(x)$ is a contracting map
\begin{equation}
||y(x)|| \leq k ||x||\;,
\end{equation}
with $k= \frac{ a}{a+\rho}$.
\begin{remark}:\\
We observe also that we know (through Brascamp-Lieb \cite{BraLi}) from for example
\cite{He1993d} that the first eigenfunction is logconcave and that it is
possible to measure its logconcavity.
\end{remark}
\section{Transport equations}
\subsection{First approach}
We recall that the first equation was
\begin{equation}
E_0 a_0 (x) = C^p \cdot a_0(y(x))
(\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (y(x))
\;,
\end{equation}
where $C$ was a universal constant. We have also obtained
\begin{equation}
E_0 = C^p
(\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (0)
\;,
\end{equation}
\noindent By continuing to apply the Laplace integral method
in order to cancel the different powers of $h$, we obtain as next equation
\begin{equation}
E_1 a_0 (x) + E_0 a_1(x) = C^p \cdot a_1(y(x))
(\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (y(x))
+ c_1(x)
\;,
\end{equation}
where $c_1$ depends only on $a_0$ and on
its derivatives of order less or equal
to $2$.\\
It seems better to look for $a_1$ in the form
\begin{equation}
a_1(x) = a_0 (x) \cdot b_1(x)\;,
\end{equation}
and write
$c_1$ in the form
$$
c_1(x) = E_0\, a_0 (x) \cdot d_1(x)\;.
$$
We then obtain
$$
\begin{array}{ll}
E_1\, a_0 (x) + E_0 \,a_0 (x) \cdot b_1(x) =\quad \quad \\ \quad \quad
C^p \cdot a_0 (y(x)) \cdot b_1(y(x))
(\det [\nabla_{yy}^2 \Theta +\nabla^2 \phi])^{-\frac 12} (y(x))
\;+\; E_0\, a_0 (x) \cdot d_1(x)\;,
\end{array}
$$
or using the equation satisfied by $a_0$
$$
b_1(x) = b_1(y(x)) + (d_1(x)-\frac{E_1}{E_0})\;.
$$
We meet the necessary condition which determines actually $E_1$
\begin{equation}
d_1(0)-\frac{E_1}{E_0}=0\;,
\end{equation}
and the equation takes the structure
\begin{equation}
b_1(x) = b_1(y(x)) + \theta_1(x)\;,
\end{equation}
with
$\theta_1(x) =d_1(x) -d_1(O)\;,$
satsfying in particular
\begin{equation}
\theta_1(0) =0\;.
\end{equation}
We then obtain the following formula for $b_1$ by iterating the functional equation
\begin{equation}
b_1(x) = \sum_{n=0}^\infty \theta_1 (y^{(n)}(x) )\;.
\end{equation}
The successive transport equations are treated in the same way.
\subsection{Second approach}
As was observed by J.Sj\"ostrand in the case of the Schr\"odinger operator, it
can be easier,
particularly if our idea is to study later the thermodynamic limit,
to look for a WKB solution in the form
\begin{equation}\label{sa1}
u^{wkb}(x) = \exp -\frac{\phi(x;h)}{h}\;,
\end{equation}
with
\begin{equation}\label{sa2}
\phi(x;h)\sim \sum_{j=0}^\infty h^j\;\phi_j(x)\;.
\end{equation}
Here $\phi_0$ is the solution we have determined before in our study of the eikonal equation.
We emphasize that this approach is only adapted for the first eigenfunction which can be chosen to be
strictly positive.
\\
We shall consequently look (after a renormalization by a
multiplicative factor) for a solution satisfying formally (in the sense of formal series in powers of $h$,
the Laplace integral being considered only formally in a neighborhood
of a minimum)
\begin{equation}\label{sa3}
(\pi h)^{-\frac p2}\int_{\rz^p}\exp - \frac 1h
\left(-\phi(x;h) +\Theta (x,y) + \phi(y;h)\right) \;dy \;\sim\;
\exp - F(h)\;,\;
\end{equation}
for all $x$ in a neighborhood of the origin,
with
\begin{equation}\label{sa4}
F(h)\sim \sum_{j=0}^\infty h^j\; F_j\;.
\end{equation}
We shall follow the treatment which was used by J. Sj\"ostrand \cite{Sj1993a}
(and later by the author
\cite{He1994a}) for Laplace integrals by
reproving a Laplace integral method with parameters (the parameter will be
$x$). Following J. Sj\"ostrand, the key point is to reprove
also the Morse's Lemma which is used in the standard proof of the Laplace integral method.
The idea is actually to look for a change of
variables in the Laplace integral taking account also of the Jacobian
and in the form
\begin{equation} \label{sa5}
y\mapsto z \sim \nabla_y f (x,y;h)\;,
\end{equation}
with
\begin{equation} \label{sa6}
f(x,y;h)\sim \sum_{j=0}^\infty h^j\; f_j(x,y)\;.
\end{equation}
We find the following equation
\begin{equation}\label{sa7}
\begin{array}{l}
-\phi(x;h) +\Theta (x,y) + \phi(y;h) \\ \quad\quad \sim
|\nabla_y f (x,y;h)|^2 - h \ln\det (\nabla^2_{yy} f (x,y;h)) +
h F(h)
\;,
\end{array}
\end{equation}
that we have to solve formally in powers of $h$.\\
\noindent
Let us look first at the first equation. We recall
that we have already chosen $\phi_0$
such that the map
$
y\mapsto \Psi_x(y)= -\phi_0(x) +\Theta (x,y) + \phi_0(y)$ vanishes to
order $2$ at $y= y(x)$.\\
We find an "eikonal" equation for $f_0$,
\begin{equation} \label{sa8}
-\phi_0(x) +\Theta (x,y) + \phi_0(y) = |\nabla_y f_0 (x,y)|^2\;,
\end{equation}
and we are looking for a function $f_0$ satisfying
\begin{equation}\label{sa9}
f_0(x,y)\geq 0\;,
\end{equation}
and vanishing at the critical point $y=y(x)$ of the function
$ \Psi_x (y)$, that is,
\begin{equation}\label{sa10}
f_0(x,y(x)) = 0\;.
\end{equation}
We get also from the equation satisfied by $f_0$
\begin{equation}\label{sa10b}
(\nabla_y f_0)(x,y(x)) = 0\;.
\end{equation}
If the Hessian of $\Psi_x$ satisfies the condition
\begin{equation}\label{sa11}
(\nabla^2_{yy} \Psi_x) (y(x)) >0
\end{equation}
then $f_0$ exists with a good
control if there is a corresponding good control of $\Psi_x$.\\
In particular if
$\Hess V (0)$ is a sufficiently small perturbation of a diagonal matrix $D_0$:\\
$$\Hess V(0) = D_0 + W_0\;,$$ then the same will be true
for $\Hess \phi_0 (0)$ which will be indeed close to\break $\frac 12 \sqrt{
D_0 (D_0 + 2a I)}$.
(This remark is for the future when we shall have to control all the construction
with respect
to the dimension).\\
Let us now look at the $\Og(h)$ term in (\ref{sa7}). We find (think of $x$ as a parameter)
\begin{equation}\label{sa12}
-\phi_1(x)+\phi_1(y) = 2 \nabla_y f_0(x,y)\cdot\nabla_y f_1(x,y)
- \ln\det (\nabla^2_{yy} f_0 (x,y)) + F_0 \;.
\end{equation}
Let us explain now how we can solve this equation. We first
consider necessary conditions and note that
\begin{equation}\label{sa13}
-\phi_1(x)+\phi_1(y(x)) =
- \ln\det (\nabla^2_{yy} f_0 (x,y(x))) + F_0 \;.
\end{equation}
Here the unknowns are $\phi_1$ and $F_0$. By taking $x=0$, we
obtain the new equation
\begin{equation}\label{sa14}
F_0 = \ln \det (\nabla^2_{yy} f_0 (0,0))\;,
\end{equation}
which actually defines $F_0$.\\
Once this condition is satisfied, we shall
get the "transport" equation
\begin{equation}\label{sa15}
\phi_1 (x) = \phi_1(y(x)) + \theta_1 (x)\;,
\end{equation}
with
\begin{equation}\label{sa16}
\theta_1(x)= \ln\det \left(\;\nabla^2_{yy} f_0 \,(x,y(x))\;\right) - F_0 \;,
\end{equation}
and we recall that, by our choice of $F_0$,
\begin{equation}\label{sa17}
\theta_1(0)=0\;.
\end{equation}
Imposing that
\begin{equation}\label{sa18}
\phi_1(0)=0\;,
\end{equation}
we find
\begin{equation}\label{sa19}
\phi_1 (x) = \sum_{j=0}^\infty \theta_1 (y^{(j)} (x))\;.
\end{equation}
Once we have solved (\ref{sa19}), then we can solve (\ref{sa12}) as was done for example in \cite{HeSj1984}.
The convergence is not too difficult to obtain once we have observed
the property that $||y(x)||\leq k ||x||$ with $k<1$
for a suitable norm $||\;\cdot\;||$.\\
Now the general compatibility equation is
\begin{equation}\label{sa20}
-\phi_j(x)+\phi_j(y(x)) =
- [\ln\det (\nabla^2_{yy} f (x,y(x);h))]_{j-1} + F_{j-1}\;.
\end{equation}
$[\ln\det (\nabla^2_{yy} f (x,y(x);h))]_{j-1}$ is the coefficient of
$h^{j-1} $ in the expansion of \break $\ln\det (\nabla^2_{yy} f (x,y(x);h))$ and we observe that
it depends only
on $f_\ell$ and $F_\ell$ for $\ell\in \{0, \cdots,j-1\}$. This can be solved in the same way as for $\phi_1$.
\\
Once (\ref{sa20})
is satisfied,
we can solve the equation corresponding to the vanishing
of the $j$-th coefficient of (\ref{sa7})
\begin{equation}\label{sa21}
-\phi_j(x) + \phi_j(y) =
\sum_{\ell=0}^{j}\nabla_y f_\ell (x,y)\cdot \nabla_y f_{j-\ell} (x,y) -
[ \ln\det (\nabla^2_{yy} f (x,y;h))]_{j-1} + F_{j-1}
\;.
\end{equation}
What we see here is that, once we have found that $\phi_0$ has
good properties\footnote{This means in particular that $\phi_0(x)$ is
a $C^\infty$ strictly positive function with strictly positive Hessian
at $0$ but also various controls with respect to the dimension which
will be detailed in Section \ref{sstand}.},
all the other operations appearing in the construction have essentially been considered in \cite{Sj1993a},
\cite{He1994a} and \cite{Sj1993e}. We shall indeed see later how this can be used
in order to control the construction with respect to the dimension.
\section{WKB for the Kac model}\label{s6}
Before considering the large dimension problem, we think it is worth
while to analyze in the same spirit
the other transfer matrix problem that we have first considered in
\cite{BruHe1991} and \cite {He1992b}. Trying to answer a problem posed by M. Kac,
we have considered another regime of parameters and constructed the
WKB ground state
solution for $\exp -\frac V2 \exp h^2\Delta \exp -\frac V2$ whose kernel is
\begin{equation}\label{ka1}
K_V(x,y;h)= (4\pi h^2)^{-\frac p2 }\exp - \frac{V(x)}{2}\;
\exp -\frac{|x-y|^2}{4 h^2}\;\exp - \frac{V(y)}{2}\;.
\end{equation}
We shall look, in a suitable neighborhood of a non-degenerate minimum of $V$, for a WKB solution in the form
$$ u^{wkb}(x) \sim \exp -\frac{\phi(x;h)}{h}\;,
$$
with
\begin{equation}\label{ka2}
\phi(x,h)\sim \sum_{j=0}^\infty h^j\;\phi_j(x)\;.
\end{equation}
Our previous proof was based on the observation that this operator can be considered as a $h$-pseudodifferential operator
and that we can adapt in this context the standard WKB constructions
introduced for the Schr\"odinger operator, but which were actually true for more
general Hamiltonians, if some analyticity condition in the impulsion variable
is satisfied. But it seems difficult to control this technique as
the dimension tends to $\infty$. The reason is that a theory
of $h$-pseudodifferential operators with control with respect to the dimension
is still at a primitive stage and we know only about one paper by J. Sj\"ostrand
\cite {Sj1993e} which does not seem sufficient for our purpose. We propose here a more direct approach which will use the specific structure
of the integral operator. This seems however more difficult than in the
previous case because the parameter $h$ does not appear in an homogeneous way.
We have to deal consequently with a kind of Laplace integral method
with a degenerate
critical set of minima for the phase.\\
\noindent We are consequently looking for a solution satisfying
formally
\begin{equation}\label{ka3}
\int_{\rz^p} K_V(x,y;h)\exp - \frac 1h
\left(-\phi(x;h) + \phi(y;h)\right) \;dy\; \sim \exp - F(h)
\end{equation}
with
\begin{equation}\label{ka4}
F(h)\sim \sum_{j=0}^\infty h^j\; F_j\;.
\end{equation}
As in the preceding section, we shall look for a change of
variables in the Laplace integral taking account also of the Jacobian
and in the form
\begin{equation} \label{ka5}
z \sim \nabla_y f (x,y;h)\;,
\end{equation}
with
\begin{equation} \label{ka6}
f(x,y;h)\sim \sum_{j=0}^{+\infty} h^j\; f_j(x,y)\;.
\end{equation}
The equation we find is the following
\begin{equation}\label{ka7}
\begin{array}{l}
-h^{-1}\phi(x;h) + \frac 14 h^{-2}|x-y|^2 +h^{-1} \phi(y;h) +
\frac{ V(x)}{2} +
\frac{V(y)}{2}
\\\quad \quad \quad \sim
h^{-2}|\nabla_y f (x,y;h)|^2 - \ln\det (\nabla^2_{yy} f (x,y;h)) +
F(h) -p \ln 2
\;,
\end{array}
\end{equation}
that we have to solve formally by identifying the coefficients of the formal expansion
in powers of $h$.\\
\noindent {\bf The coefficient of $h^{-2}$} is eliminated by taking
\begin{equation}\label{ka8}
f_0(x,y) = \frac 14 |x-y|^2\;.
\end{equation}
This means that the first change of variable is just the map
$y\mapsto \frac 12 (y-x) $.\\
\noindent {\bf The coefficient of $h^{-1}$}.\\
We get
\begin{equation}\label{ka9}
\begin{array}{ll}
\phi_0(y) - \phi_0 (x) &= 2( \nabla_y f_0)(x,y)\cdot(\nabla_y f_1)(x,y)\\
& = (y-x)\cdot(\nabla_y f_1)(x,y)\;,
\end{array}
\end{equation}
and we choose the initial condition
\begin{equation}\label{ka9a}
f_1(x,x)=0\;.
\end{equation}
Although, for the moment, neither $f_1$ nor $\phi_0$ are determined, we
observe as a consequence of (\ref{ka9}) that,
\begin{equation}\label{ka10}
(\nabla\phi_0) (x)= (\nabla_y f_1)(x,x)
\;.
\end{equation}
\noindent{\bf The coefficient of $h^0$}.\\
We get
\begin{equation}\label{ka11}
\begin{array}{l}
\phi_1(y) - \phi_1 (x) +\frac{V(x)}{2}+ \frac{V(y)}{2}\\ \quad
=
(y-x)\cdot(\nabla_y f_2)(x,y) +(|\nabla_y f_1|^2)(x,y) \\
\quad\quad - \ln\det
(\nabla^2_{yy} f_0 (x,y)) + F_0 + p\ln\frac 12\;,
\end{array}
\end{equation}
and we take as initial condition
\begin{equation}\label{ka12}
f_2(x,x)=0\;.
\end{equation}
According to (\ref{ka8}), we then obtain
\begin{equation}\label{ka11b}
\phi_1(y) - \phi_1 (x) +\frac{V(x)}{2}+ \frac{V(y)}{2}=
(y-x)\cdot(\nabla_y f_2)(x,y) +(|\nabla_y f_1|^2)(x,y)+F_0 \;.
\end{equation}
A necessary and sufficient condition in order to solve (\ref{ka11})-(\ref{ka12})
is to have
\begin{equation}\label{ka13}
V(x) = (|\nabla_y f_1|^2)(x,x) + F_0\;,
\end{equation}
which gives according to (\ref{ka10})
\begin{equation}\label{ka14}
V(x)= |\nabla\phi_0 (x)|^2 +F_0\;.
\end{equation}
This is actually the standard eikonal equation that we can
solve locally as in \cite{HeSj1984} under the condition
\begin{equation}\label{ka14a}
F_0 =0\;,
\end{equation}
adding the conditions
$$
\phi_0 (x)\geq 0\;,\; \phi_0 (0) =0\;.
$$
There is no surprise in the
sense that we have already proved in \cite{BruHe1991}, \cite{He1992b}
that the WKB solution
exists and has this form. But the study which is presented here
will have the specific advantage of permitting an easier approach for the large dimension problem. This study of the phase $\phi_0$ was already done
in \cite{Sj1993a}.\\
Once $\phi_0$ is determined, we can determine $f_1$
as a solution of (\ref{ka9})-(\ref{ka9a}). \\
If we solve (\ref{ka11}), we get as a consequence
the following relation between $\phi_1$ and $f_2$
by differentiating with respect to $x$ and taking $x=y$
\begin{equation}\label{ka15}
-(\nabla \phi_1 (x)) + \frac 12 (\nabla V) (x) = -
(\nabla_y f_2)(x,x)
+ (\nabla_x(|\nabla_y f_1|^2))(x,x)
\end{equation}
This gives us a relation between $\nabla \phi_1$ and $\nabla_y f_2$ restricted
to the diagonal $y=x$ of the same nature as in (\ref{ka10}).\\
\noindent {\bf The coefficient of $h$}.\\
The new equation is
\begin{equation}\label{ka16}
\begin{array}{l}
\phi_2(y) - \phi_2 (x) \\=
(y-x)\cdot(\nabla_y f_3)(x,y)
+2 (\nabla_y f_1\cdot \nabla_y f_2)(x,y)
\\- \left(\ln\det (\nabla^2_{yy} f (x,y;h)\right)_1 +F_1 \;,
\end{array}
\end{equation}
with the initial condition
\begin{equation}\label{ka16a}
f_3(x,x)=0\;.
\end{equation}
This can be rewritten as
\begin{equation}\label{ka16b}
\begin{array}{l}
\phi_2(y) - \phi_2 (x) \\=
(y-x)\cdot(\nabla_y f_3)(x,y)
+2 (\nabla_y f_1\cdot \nabla_y f_2)(x,y)
\\- (\Tr \nabla^2_{yy} f_1)(x,y) +F_1\quad \;.
\end{array}
\end{equation}
The necessary and sufficient condition in order to solve is
\begin{equation}\label{ka17}
2 (\nabla_y f_1)\cdot (\nabla_y f_2)(x,x)
- (\Tr \nabla^2_{yy} f_1)(x,x) +F_1\;=\;0\;.
\end{equation}
This can be written in the form
\begin{equation}\label{ka18}
2 (\nabla \phi_0)(x)\cdot (\nabla_y f_2)(x,x)
- (\Tr \nabla^2_{yy} f_1)(x,x) +F_1=0\;.
\end{equation}
But now $(\nabla_y f_2)(x,x)$ can be expressed, according to (\ref{ka15}), in terms of $\nabla\phi_1$
and this gives the transport equation for $\phi_1$, which takes the form
\begin{equation}\label{ka19}
\begin{array}{l}
2 (\nabla \phi_0)(x)\cdot (\nabla \phi_1)(x)
- (\Tr \nabla^2_{yy} f_1)(x,x) +F_1=\\
(\nabla \phi_0)(x)\cdot(\nabla V) (x)
- 2(\nabla \phi_0)(x)\cdot (\nabla_x(|\nabla_y f_1|^2))(x,x) \;.
\end{array}
\end{equation}
We meet here, in order to solve with the initial condition $\phi_1(0)=0$,
a condition determining $F_1$ (take $x=0$)
\begin{equation}\label{ka19a}
F_1= (\Tr \nabla^2_{yy} f_1)(0,0) \;,
\end{equation} (which is not astonishing if we compare with the Schr\"odinger operator)
and we can then determine
$\phi_1$. Coming back to (\ref{ka11}), we have also determined $f_2$.\\
But we now observe from (\ref{ka16}) the following relation between $\nabla \phi_2$
and $(\nabla_y f_3)(x,x)$,
\begin{equation}\label{ka20}
- \nabla_x\phi_2 (x) =
-(\nabla_y f_3)(x,x)
+2 \nabla_x((\nabla_y f_1\cdot \nabla_y f_2))(x,x)-
(\Tr \nabla_x\nabla^2_{yy} f_1)(x,x)
\;.
\end{equation}
To summing up at this stage, we have determined $\phi_0$, $\phi_1$, $f_0$,
$f_1$, $f_2$ and $F_0$, $F_1$. At this stage we can find for any $\phi_2$,
$f_3$ and $f_3$ satisfies in particular (\ref{ka20}).\\
\noindent {\bf The coefficient of $h^2$}.\\
After this step, the general way for solving these equations becomes clear. The next equation is
\begin{equation}\label{ka21}
\begin{array}{ll}
\phi_3(y) - \phi_3 (x) &=
(y-x)\cdot(\nabla_y f_4)(x,y)
+2 (\nabla_y f_1\cdot \nabla_y f_3)(x,y) \\
&\quad + |\nabla_y f_2 (x,y)|^2
- [\ln\det (\nabla^2_{yy} f (x,y;h))]_2 + F_2\;,
\end{array}
\end{equation}
with the initial condition
\begin{equation}\label{ka22}
f_4(x,x)=0\;.
\end{equation}
The necessary condition at $y=x$ takes the form
\begin{equation}\label{ka23}
2 (\nabla_y f_1\cdot \nabla_y f_3)(x,x) + |\nabla_y f_2 (x,x)|^2
- [\ln\det ((\nabla^2_{yy} f) (x,x;h))]_2 + F_2\;,
\end{equation}
and using (\ref{ka20}) we find a transport equation for $\phi_2$ which can be
solved under a condition determining $F_2$. This permits to find $f_3$. (\ref{ka21}) gives us again
a relation between $\nabla \phi_3$ and $(\nabla_y f_4)(x,x)$ which will
be used in the next step.\\
\noindent It is now time to establish the general transport equation corresponding to the coefficient of $h^k$
with $k\geq 2$. We assume that we have already found $f_k, F_{k-1},
\phi_{k-2}$
and we have seen that we can determine $f_{k+1}$ once we found some
$\phi_{k}$
satisfying some condition relating
$(\nabla_y f_{k+1})(x,x)$ and $\nabla \phi_{k}(x)$.\\
The equation corresponding to the cancellation of the coefficients of
$h^k$ in (\ref{ka7}) is
\begin{equation}\label{trkack}
\begin{array}{l}
-\phi_{k+1}(x) + \phi_{k+1}(y)
\\\quad \quad \quad=
\nabla_y f_0 (x,y)\cdot\nabla_y f_{k+2} (x,y)
+ \nabla_y f_1 (x,y)\cdot\nabla_y f_{k+1} (x,y)\\
\quad \quad \quad \quad + \sum_{j=2}^{k+2}
\nabla_y f_j (x,y)\cdot\nabla_y f_{k+2-j}(x,y)
- (\ln\det (\nabla^2_{yy} f (x,y;h)))_k
+ F_{k}
\;,
\end{array}
\end{equation}
We have only here to know that $(\ln\det (\nabla^2_{yy} f (x,y;h)))_k $ depends only
on the $f_\ell$ for $\ell\leq k$.
\\
In order to solve the equation, we meet the following compatibility condition for $x=y$,
\begin{equation}
\begin{array}{l}
\nabla_y f_1 (x,x)\cdot (\nabla_y f_{k+1}(x,x)) + \\
+\sum_{j=2}^{k+2} \nabla_y f_j (x,x)\cdot\nabla_y f_{k+2-j}(x,x)
- (\ln\det (\nabla^2_{yy} f (x,x;h)))_k
+ F_{k} =0\;.
\end{array}
\end{equation}
But the relation obtained by recursion and relating
$(\nabla_y f_{k+1})(x,x)$ and $\nabla \phi_{k}(x)$ gives the existence
of $x\mapsto v_k(x)\in \rz^p$ such that
\begin{equation}
\begin{array}{l}
\nabla \phi_0(x) \cdot \nabla \phi_{k}(x) \\
+ \sum_{j=2}^{k+2} \nabla_y f_j (x,x)\cdot\nabla_y f_{k+2-j}(x,x)\\
- (\ln\det (\nabla^2_{yy} f (x,x;h)))_k
+ F_{k} + \nabla \phi_0 (x)\cdot v_{k}(x)=0\;.
\end{array}
\end{equation}
This gives a transport equation in the form
\begin{equation}
\nabla \phi_0 (x) \cdot \nabla \phi_{k}(x) = e_{k}(x) - F_{k}\;,
\end{equation}
where $x\mapsto e_{k}(x)$ is a known function
and this equation can be solved under the condition that
\begin{equation}
F_{k}= e_{k}(0)
\;,
\end{equation}
which determines $F_k$.\\
Once we have determined $F_{k}$, we solve the equation for $\phi_{k}$
by imposing the initial condition $\phi_{k}(0)=0$. From the
preceding step (that is (\ref{trkack}) for $k-1$), we can now
get $f_{k+1}$. Coming back to (\ref{trkack}) we see that we can find
$f_{k+2}$ if we know $\phi_{k+1}$ and, by differentiating with respect
to $x$ (and taking $y=x$),
we get the following necessary condition relating $\nabla \phi_{k+1}$
and $\nabla_y f_{k+2}(x,x)$
\begin{equation}\label{trkack1}
\begin{array}{l}
\nabla \phi_{k+1}(x)
\\\quad \quad \quad =
(\nabla_y f_{k+2})(x,x)
- \nabla_x (\nabla_y f_1 \cdot\nabla_y f_{k+1})(x,x) \\
\quad \quad \quad \quad -
\nabla_x(\sum_{j=2}^{k+2} \nabla_y f_j \cdot\nabla_y f_{k+2-j})(x,x)
+ \nabla_x (\ln\det (\nabla^2_{yy} f (\cdot,\cdot;h)))_k)(x,x)
\;.
\end{array}
\end{equation}
The recursion is satisfied in the general case.\\
\noindent Everything is consequently determined for fixed dimension
and it remains only to
follow all the construction with respect to the dimension in order
to achieve what was done already for the Schr\"odinger operator and the Laplace integral.
This will be rather long but we have now a general scheme to follow
which is quite analogous
to what was done already in the previous papers mainly
of J. Sj\"ostrand. This will be sketched in Section \ref{skacld}.\\
{\bf Applications}:\\
Once we have constructed the formal WKB construction, it is easy to
construct an approximate solution by introducing\footnote
{or
$$ u_M^{app}(x;h) =
h^{-\frac p4}\;\chi_0(x)
\;\exp - \frac {\sum_{j=0}^M \phi_j(x)\;h^j}{h}$$
}
for example the function
$$ u_M^{app}(x;h) =
h^{-\frac p4}\;\chi_0(x)(\sum_{j=0}^M a_j(x)\;h^j)
\exp - \frac{\phi_0(x)}{h}$$
where $\chi_0$ is a cutoff function with compact support in the neighborhood of the minimum where
we were able to perform the construction, and which is equal to $1$ in another
smaller neighborhood of the minimum.\\
The pseudo-local character of these transfer operators permits us to verify that
the cutoff function introduces only exponentially small errors.\\
The cutoff of the sum at $M$ leads to an error of order $\Og (h^{M+1})$.\\
\section{ Brief presentation of Sj\"ostrand's formalism of
the $0$-standard functions}\label{sstand}
We have until now described, in two situations presented respectively
in Sections \ref{s4} and \ref{s6}, a WKB construction for the largest eigenvalue of the transfer operator. This gives
the existence of a formal expansion in powers of $h$ of this WKB-eigenvalue. The next step
is then to control the behavior with respect to the dimension $p$. Our
main goal is here to obtain that
for
$ F(h;p)\sim \sum_{j=0}^\infty h^j F_j(p)$ and for all $j \in \nz$,
there exists $C_j$ such that
\begin{equation}\label{stand0}
|F_j(p)|\leq C_j \cdot (p+1)\;.
\end{equation}
This is the type of results which was obtained by J. Sj\"ostrand in \cite{Sj1993a} for the Schr\"odinger operator.
We also hope that it is a first step in order to get, under additional conditions on the potential
(see \cite{HeSj1992a}), semi-classical expansions
on the thermodynamic limit
$$
\varphi_j=\lim_{p\ar + \infty} \frac 1p F_j(p)\;.
$$
\noindent We just mention that this is just the beginning of the program and
that the next
step would be to prove that the largest eigenvalue $E(h;p)$ has effectively
this behavior, that is to have a semiclassical expansion of
$$
-\lim_{p\ar + \infty} \frac 1p \ln (E(h;p))\sim \sum_j \varphi_j h^j\;.
$$
>From now on, in order to facilitate the reference to the papers by J. Sj\"ostrand, {\bf we change
$p$ into $m$ for the dimension and the character $p$ is used for the norms $\ell^p$ on $\rz^m$.}
We refer the interested reader to what has been done in the case of
the Schr\"odinger operator \cite{HeSj1992a}
and to the case of the Laplace integrals \cite{He1994a} for the description of what are the
next questions and statements.
\noindent We recall here the formalism introduced
by J.Sj\"ostrand \cite{Sj1993e} and some of the basic results of this theory.\\
\begin{definition}\label{defstand1}:\\
A scalar function\footnote
{We shall all the time take as $\Omega$ a suitable $\ell^\infty$-
ball with $m$-independent radius. } $a(x)$
on $\Omega\subset \rz^m$ is $0$-standard if, for $k=1, 2,\cdots,$
\begin{equation}\label{stand1}
|\langle \nabla^k a(x),t_1\otimes \cdots \otimes t_k \rangle|
\leq C_k \;|t_1|_{p_1}\cdots |t_k|_{p_k}\;,
\end{equation}
for all $p_j\in [1,\infty]$ with
$1= \frac{1}{p_1}+\cdots + \frac{1}{p_k}$ and all $t_j$ in $\rz^m$.
\end{definition}
We recall that, for $u\in \rz^m$, $ |u|_p$ is the norm in $\ell^p$ of
$u$,
$|u|_p :=\left(\sum_{j=1}^m |u_j|^p\right)^\frac 1p$.\\
{\bf Although it is not clear for the moment, it would be better to say
that we are actually looking for a family of scalar functions which are indexed by the dimension
$m$ and that the constant $C_k$ are $m$-independent. }
\noindent We shall also meet different extensions of this definition to
\begin{itemize}
\item the case of a function depending also on a finite\footnote
{Here we mean by finite that $\ell$ is fixed and independent of our play with
the dimension.}
number $\ell$ of variables\footnote
{
In this case we assume that
\begin{equation}\label{stand1par}
|\pa_s^\alpha\langle \nabla_x^k a(x,s),t_1\otimes \cdots \otimes t_k \rangle|
\leq C_{k,\alpha} \;|t_1|_{p_1}\cdots |t_k|_{p_k}\;.
\end{equation}
}, denoted by $s=(s_1,s_2,\cdots , s_\ell)$,
\item the case when we have to consider a function $(x,y)\mapsto a(x,y)$
depending on two variables $(x,y)\in \Omega\times\Omega'\subset \rz^m\times\rz^m$,
\item the case when we are considering a sequence $(a^{(n)})_{n\in \nz}$ of $0$-standard
functions.
\end{itemize}
J.Sj\"ostrand considers also the case with weights which is also quite
important in different contexts (\cite{HeSj1993})
but we will not consider this case in a first step.
\\
We shall meet also naturally the notion of $k_0$-standard function that we explain
now.\\ For a given integer $k_0\geq 1$, we say that $u(x)$ is $k_0$-standard for $k_0\geq 1$ if $u$ is a smooth function
of $x\in \Omega\subset \rz^m$ with values in the dual of
$\rz^{J_1}\otimes\cdots\otimes\rz^{J_{k_0}}$, where $J_j$ are some subsets of $\{1,\cdots,m\}$, such that for every $k=0, 1, 2, \cdots$
\begin{equation}
|\langle \langle \nabla^k u, t_1\otimes\cdots\otimes t_k\rangle, r_1\otimes\cdots
r_{k_0}\rangle|\leq C_k \left(\prod_1^k |t_j|_{p_j}\right)\cdot
\left(\prod_1^{k_0} |r_\ell|_{{\tilde p}_\ell}\right)\;,
\end{equation}
with
$$ 1= \frac{1}{p_1}+\cdots +\frac{1}{p_k} + \frac{1}{{\tilde p}_1}
+\cdots +\frac{1}{{\tilde p}_{k_0}}\;.
$$
Of course, in order to be completely rigorous, we would have to
be precise in
the order of the operations, but we shall work most of the time with
symmetric tensors and so a notation like
$\langle \nabla^ku,r_1\otimes\cdots\otimes r_{k_0}\otimes t_1\otimes
\cdots\otimes t_k\rangle$ will sometimes be used instead of
$\langle \langle \nabla^k u, t_1\otimes\cdots\otimes t_k\rangle,
r_1\otimes\cdots\otimes
r_{k_0}\rangle$.\\
The introduction of these $k_0$-standard functions appears more natural if we observe the
\begin{lemma}\label{Lemmastand2}:\\
Let $u(x)$ be $k_0$-standard (possibly depending on parameters in the sense of (\ref{stand1par})).
Then $\nabla_x^ku$ viewed as a ($k_0+k$)-dual tensor is $(k_0+k)$-standard.
\end{lemma}
The other important lemma in our study of the $WKB$ construction is
\begin{lemma}\label{Lemmastand3}:\\
Let $u(x)$ be $k_0$-standard (possibly in the parameter dependent sense).
Then $\Delta_x u$ is $k_0$-standard.
\end{lemma}
This lemma is based on the following result on matrices
\begin{equation}\label{trace}
|\Tr A\,|\leq ||A||_{\Lg(\ell^\infty,\ell^1)}\;.
\end{equation}
\begin{lemma}\label{Lemmastand4}:\\
Let $u$ be $(k_0+\ell_0)$-standard and let $u_1, \cdots, u_{\ell_0}$ be
$1$-standard with values in appropriate spaces, so that
$\langle u(x),u_1\otimes\cdots\otimes u_{\ell_0}\rangle$ is a well defined $k_0$-tensor by
$$\langle \langle u(x),u_1\otimes\cdots\otimes u_{\ell_0}\rangle, r_1\otimes\cdots\otimes r_{k_0}\rangle\;.$$
Then $\langle u(x),u_1\otimes\cdots\otimes u_{\ell_0}\rangle$ is $k_0$-standard.
\end{lemma}
As particular case we shall meet the property that
\begin{itemize}
\item
If $u$ and $v$ are $1$-standard
then $\langle u, v\rangle$ is $0$- standard.
\item
If $a$ and $b$ are $0$-standard then $\langle \nabla a,\nabla b\rangle$
is $0$-standard.
\end{itemize}
\begin{lemma}\label{Lemmastand5}:\\
If $x\mapsto y=v(x)$ is $1$-standard and $ y\mapsto z=u(y)$ is $k_0$-standard,
then $x\mapsto z=u(v(x))$ is $k_0$-standard.
\end{lemma}
\noindent We now consider the implicit function theorem.
\begin{lemma}\label{Lemmastand6}:\\
Let $x\mapsto y=f(x)$ be a $1$-standard map from $\Omega \subset \rz^m$
into $\rz^m$. Let us assume that $(\nabla f) (x)$ is uniformly invertible
in $\Lg(\ell^p)$ for $p\in [1,+\infty]$, then the local\footnote
{The mention ``local'' means here that if for some $x_0 (m)$ such
that $B_\infty( x_0 (m);\delta_0)\subset \Omega(m)$
($\delta_0$ independent of
the dimension) and
$(\nabla f) (x_0(m))$
is invertible and satisfies $||(\nabla f)
(x_0(m))^{-1} )||_{\Lg(\ell^p)}\leq C$, then there exists $\delta_1$
(independent of the dimension)
such that, for $y\in B_\infty( f(x_0(m)); \delta_1)$, there exists a unique solution $x \in B_\infty ( x_0(m); \delta_0)$ of $f(x)=y$.}
inverse $g=f^{-1}$
is $1$-standard.
\end{lemma}
This lemma admits the following useful extension. Let us consider the case
when we have three groups of variables $(x,y,z)$ where $y$ and $z$
contain the same number of variables and
we assume that the map $ (x,y)\mapsto z=f(x,y)$ is $1$-standard
with uniformly invertible differential $\pa_y f$. Then the local
solution $y=g(x,z)$ of $f(x,y)=z$ is $1$-standard. In particular, the solution
$y=g(x,0)$ of $f(x,y)=0$ is $1$-standard.\\
\noindent We now recall what was obtained by J.Sj\"ostrand concerning the flow associated to
$1$-standard vector fields $v(x,\pa_x)$.\\
The main result is the following lemma
\begin{lemma}\label{Lemmastand7}:\\
Let us denote by $t \mapsto \langle u(x),u_1\otimes \cdots
\otimes u_{\ell_0}\rangle$ the solution of
\begin{equation}
\pa_t \Phi(t,x) = v(\Phi(t,x))\;,\; \Phi(0,x) = x\;.
\end{equation}
Then the function $x\mapsto \Phi(t,x)$ is $1$-standard (in the $t$-dependent
sense for $|t|\leq T$).
\end{lemma}
\noindent The last results we shall recall are more deep in nature and only true under
stronger assumptions.
\begin{proposition}\label{Propositionstand8}:\\
Let $\phi\in C^\infty(B_\infty (0,b);\rz)$ be a $0$-standard function
with $\phi(0)=\nabla\phi(0)=0$,
$\Hess \phi (0) = D + A$, where $D$ is diagonal $D\geq r_0$,
$||A||_{\Lg(\ell^p)}\leq r_1$,
where $00$.
\end{proposition}
\noindent Finally we recall that the existence of the solution
of the eikonal equation was also proven.
\begin{proposition}\label{eiksch}:\\
Let $V\in C^\infty(B_\infty (0,b);\rz)$ be a $0$-standard function
such that $V(0)=\nabla V(0)=0$,
$\Hess V (0) = D + A$, where $D$ is diagonal $D\geq r_0$,
$||A||_{\Lg(\ell^p)}\leq r_1$,
where $00$.
\end{proposition}
The proof is by recursion. For $n=1$, this is simply the fact that $x\mapsto y(x)$
is $1$-standard. We now consider
$$ y^{(n)}(x) = y^{(n-1)}(y(x))\;,$$
and differentiate this identity. In order to see the main problems we just
analyze the two first derivatives and refer to \cite{Sj1994} for the general
technique. It consists simply in analyzing more precisely the constants in the
proof of Lemma \ref{Lemmastand5}.\\
We first analyze the first derivative and write for any $t\in \rz^m $
\begin{equation} \label{rec1}
\langle \nabla y^{(n)} (x), t\rangle = \langle \nabla y^{(n-1)}(y(x)),
\langle \nabla y (x), t\rangle\rangle\;.
\end{equation}
If we assume that we have found a constant $C_1$ up to order $n-1$,
we get from
(\ref{zmaj}) and (\ref{tr5}) with $\exp -\frac{1}{\lambda_0}=k$
$$
|\langle \nabla_x y^{(n)}(x), t\rangle|_{p}\leq C_1 k^{n-1}
|\langle \nabla y (x), t\rangle|_{p_1}\leq C_1 k^n |t|_{p_1}\;.
$$
Here we have observe that the matrix $\nabla y$ is symmetric and this permits
to obtain first from (\ref{tr5}) that
$$
||\nabla y(x)||_{\Lg(\ell^1)}\leq k\;,
$$ and by interpolation
\begin{equation}
||\nabla y(x)||_{\Lg(\ell^p)}\leq k\;, \forall p\in [1,+\infty]\;.
\end{equation}
By choosing $C_1$ in order to have the result for $n=1$, we have consequently treated the case $k=1$ (all $n$).\\
The case $k=2$ shows the general new difficulty. When differentiating (\ref{rec1}),
we get the sum of two terms
\begin{equation} \label{rec2}
\begin{array}{ll}
\langle \nabla^2 y^{(n)} (x), t_1\otimes t_2\rangle
&= \langle \nabla^2 y^{(n-1)}(y(x)),
\langle \nabla y (x), t_1\rangle\otimes \langle \nabla y (x),
t_2\rangle\rangle\\
&\quad + \langle \nabla y^{(n-1)}(y(x)), \langle \nabla^2 y,t_1\otimes t_2\rangle\rangle
\;.
\end{array}
\end{equation}
If we assume the result for $k=1$ and the case $k=2$ till $n-1$ for some constant
$C_2$, we obtain that it will be verified to step $n$ if $C_2$ satisfies the condition
$$ C_2 k^{n}\geq C_2 k^{n+1} + C C_1 k^{n-1}\;,$$
where $C$ is an upperbound for the norm of $\nabla^2 y$.\\
We consequently get that $C_2$ satisfies
$$
C_2\geq \frac{ C C_1}{k(1-k)}\;,
$$
and we see that the recursion is satisfied if we take
$$ C_2 \geq \max (\frac{ C C_1}{k(1-k)},C).$$
The general case can be done in the same way.\\
Once we have this property, we obtain through (\ref{sa19})
that $\phi_1$ is $0$-standard and we can then
solve the transport equation (\ref{sa12}) giving $f_1$ in the class
of the $0$ standard functions with respect to $(x,y)$. This leads to
the equation
(\ref{sa21}) with $j=2$. We first get the compatibilty condition (\ref{sa20}) permitting to determine
$\phi_2$ as a $0$-standard function and the estimate on $F_2$ and
we then obtain $f_2$ as a $0$-standard function.
The other equations do not lead to new problems
if one compares with what was done for Laplace integrals in \cite{He1994a}.\\
\noindent We have finally obtained {\bf (coming back to our previous notation for the dimension) } the following theorem
\begin{theorem}\label{thfin}:\\
Let $V\in C^\infty(B_\infty (0,b);\rz)$ be a $0$-standard function
such that $ V(0)=\nabla V(0)=0$,
$\Hess V (0) = D + A$, where $D$ is diagonal $D\geq r_0$,
$||A||_{\Lg(\ell^q)}\leq r_1$,
and $01$,
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\end{thebibliography}
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