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\centerline {\bf 1. Introduction}

\bigskip\

We consider a model of a quantum isotropic $n$-vector anharmonic crystal
\cite{Schneider}. It is a quantum variant of the classical isotropic
$n$-vector ferromagnetic model, which for $ n $ tending to infinity yields the
generalized spherical (i.e., with mean spherical constraints imposed at every
lattice site) model \cite{ABC} ,\cite{Pastur} .

Let $\Lambda $ be a finite set (typically a subset of ${\mathbf Z}^d$ ). To 
each
point of $\Lambda $ we associate the equilibrium position of a quantum 
oscillator
of mass $m$. The displacement of each oscillator from its equilibrium position
$x\in \Lambda $ is a $n$ -dimensional position operator $\vec q_x=
\left( q_x^\alpha \right) _{\alpha =1,...,n}\in {\mathbf R}^n$ on the
Hilbert space $ \otimes _{x\in \Lambda }L^2\left( {\mathbf R}^n\right) _x $. 
In
the units $ \hbar =1 $ , the model Hamiltonian is given by:

$$
H_{\Lambda ,n}=\sum\limits_{\alpha =1}^n
\begin{array}[t]{c}
\{ (1/2m)\sum\limits_{x\in \Lambda }(p_x^\alpha )^2+
(1/2)\sum\limits_{x,y\in \Lambda }\Phi _{xy}(q_x^\alpha -q_y^\alpha )^2+ \\
+\sum\limits_{x\in \Lambda }[(a_x/2)(q_x^\alpha )^2-h_x^\alpha
q_x^\alpha +w\left( \frac 1n\sum\limits_{\beta =1}^n(q_x^\beta )^2\right)
]\} ,
\end{array}
\eqno (1.1)
$$
where $p_x^\alpha =-i \partial /{\partial q_x^\alpha }$,  $\Phi
_{xy}\;(x\neq y)$ are the harmonic coupling constants between oscillators,  $
\vec h_x\in {\mathbf R}^n$ is an external (electric) field, and
$$
V_x(\vec q_x)=\frac 12(a_x+2\sum\limits_{y\in \Lambda }\Phi _{xy})\vec
q_x^2+nw(\vec q_x^2/n)
$$
is the binding potential of the oscillator with its equilibrium position;
it is assumed to be isotropic, i.e. depending only on $\left| \vec
q_x\right| $. This isotropy of the model is violated by the external field,
but choosing  $ h_x^\alpha =h_x$ for all $ \alpha =1,...,n $ and
$ x\in \Lambda $, makes the model
invariant to arbitrary permutations of the $\alpha $-components. For
simplicity, the function $w$ is chosen to be a polynomial of second degree,
with $w^{\prime \prime}>0$ to ensure confinment.
The anharmonic terms in $w$ are responsible for a nontrivial
coupling between the $\alpha$ - components of the displacement operator.

We are interested in the equilibrium states at inverse temperature $\beta $ of
the model (1.1) in the large-$n$ limit.

It is standard wisdom (and, in the case at hand, it is true) that, in the
$n=\infty $ limit, the free energy, as well as
the expectation values of observables depending only on a finite number of
$\alpha $-components, can be obtained using the Hartree-Fock (or spherical)
approximation of the Hamiltonian (1.1), i.e. (1.1) is replaced by a
selfconsistent one, describing $n$ independent systems of coupled harmonic
oscillators:

$$
\tilde H_{\Lambda ,n}({\bf c})=\sum\limits_{\alpha =1}^n\tilde H_{\Lambda
,1}({\bf c})_\alpha ,\eqno (1.2)
$$
where the $ \tilde H_{\Lambda,1}({\bf c})_\alpha $ are copies of the harmonic
Hamiltonian:

$$
\tilde H_{\Lambda ,1}({\bf c})=\frac 1{2m}\sum\limits_{x\in \Lambda
}p_x^2+\frac 12({\bf q},X({\bf c}){\bf q})-({\bf q},{\bf h}
)+\sum\limits_{x\in \Lambda }\left( w(c_x)-c_xw^{\prime }(c_x)\right) ,
\eqno (1.3)
$$
with $ {\bf q}=(q_x)_{x\in \Lambda }\in {\mathbf R}^\Lambda $; $ X({\bf c}) $
is the matrix:
$$
X({\bf c})_{xy}=\delta _{xy}\left( a_x+2w^{\prime }(c_x)+2\sum\limits_{z\in
\Lambda }\Phi _{xz}\right) -\Phi _{xy}.
\eqno (1.4)
$$
The free energies of (1.1) and (1.2) coincide in the $ n=\infty $ limit, if 
the
constants ${\bf c}=(c_x) $ are the (unique) solution of the selfconsistency
equation system given by: for all $ x\in \Lambda $,
$$
c_x=\left\langle q_x^2\right\rangle _{\tilde H_{\Lambda ,1}({\bf c})}
=\frac 1{2\sqrt{m}}\left[ X({\bf c})^{-1/2}\coth \left( \frac \beta {2
\sqrt{m}}X({\bf c})^{1/2}\right) \right] _{xx}+\left[ X({\bf c})^{-1}{\bf h}
\right] _x^2 .\eqno (1.5)
$$

A quantum model with one-component oscillators, but with anisotropy term
controlling the squared average displacement over the sample, in the form
$\left| \Lambda
\right| w\left( \frac 1{\left| \Lambda \right|}\sum\limits_{x\in \Lambda}q_x^2
\right) $, has been studied before \cite{Stamenk},\cite{vanH}, and lead to 
the same
Hartree-Fock approximation, but with one ($x$-independent) parameter $c$.

The model (1.1) as well as its spherical approximation are interesting, 
because
(already in the one-component versions \cite{Stamenk},\cite{vanH}) they show
interesting
quantum effects in the thermodynamic limit. Namely, they are prototype models
for displacement structural phase transitions, and they reveal the property
that these transitions disappear for small masses of the oscillator particles,
due to quantum fluctuations \cite{VZ1},\cite{VZ2}. Quantum fluctuations are
calculated and extensively studied \cite{VZ1} for the quantum model
\cite{Stamenk},\cite{vanH}.

In this paper, we proceed to the calculation of the limit fluctuations as
defined with the finite $n$ equilibrium states of $ H_{\Lambda ,n} $, 
Eq.(1.1).
This calculation involves all terms in a $1/n $-expansion around the limit
state. In this part (I), we control the expansion for a fixed number of
oscillators. We show that the displacement fluctuations in the $ n=\infty $
limit are quantitatively different from the calculations with the
approximation Hamiltonian (1.2). Part II will deal with the thermodynamic 
limit
problem, where qualitative differences appear concerning the clustering
properties in the phase transition region.

The main technical result of our paper is concerning the $1/n $-expansion of
the free energy and is the following:

\medskip\
{\em The free energy per component of the model (1.1) at inverse
temperature }$\beta $:
$$
f_{\Lambda ,n}\left( {\bf h}\right) =-\frac 1{\beta n\left| \Lambda \right|
}\ln \;tr\exp \left( -\beta H_{\Lambda ,n}\right) \eqno (1.6)
$$
{\em allows a complete asymptotic series in powers of }$1/n$:
$$
f_{\Lambda ,n}\left( {\bf h}\right) \sim \sum\limits_{k=0}^\infty \left( \frac
{\beta w''}n \right) ^k \tilde f_\Lambda ^{\left( k\right) }
\left( {\bf h}\right) ,\eqno (1.7)
$$
{\em where}
$$
\tilde f_\Lambda ^{\left( 0\right) }\left( {\bf h}\right) =-\frac 1{\beta
\left| \Lambda \right| }\ln \;tr\exp \left( -\beta \tilde H_{\Lambda ,1}(
{\bf c})\right)
\eqno (1.8)
$$
{\em is the Hartree-Fock approximation, i.e. the free energy of model (1.2-5).
The explicit expressions of the coefficients }
$\tilde f_\Lambda ^{\left( k\right) }\left( {\bf h}\right) $
{\em are given below.}

\medskip\
Our technique is using the Feynman-Kac representation of the partition 
function
and realising a decoupling of the components in the anharmonic quartic term in
the Hamiltonian (1.1) using the Gaussian identity (the Bochner-Minlos Theorem,
see \cite{Hida}). Doing so, one has to introduce a regularisation of the
interaction, and to prove the necessary continuity properties in order to
remove it. This is not only our strategy allowing us to prove the asymptotic
series (1.7), but it allows also to compute the $1/n$-expansions of the
distribution functions for the $q$- and $q^2$-fluctuations.

\bigskip\

\centerline {\bf 2.Main theorem: 1/n-expansion of the finite-volume free 
energy}

\bigskip\ Following Eq. (1.6) we compute


$$
f_{\Lambda ,n}\left( {\bf h}\right) =-\frac 1{\beta n\left| \Lambda \right|
}\ln \;Z_n = -\frac 1{\beta n\left| \Lambda \right|
}\ln \frac {Z_n}{\tilde Z_n\left( {\bf c} \right)}+
\tilde f_{\Lambda} ^{\left( 0 \right)} \left( {\bf h} \right), \eqno (2.1)
$$
where

$$
Z_n=tr\exp \left( -\beta H_{\Lambda ,n}\right),\hskip .5cm
\tilde Z_n\left( {\bf c} \right)=tr\exp
\left( -\beta {\tilde H_{\Lambda ,n}\left( {\bf c} \right)} \right) =
\tilde Z_1\left( {\bf c} \right) ^n . \eqno (2.2)
$$

It is clear that our problem is reduced to the derivation of the asymptotic
expansion of $ Z_n / \tilde Z_1\left( {\bf c} \right) ^n $ which means that we
compute the corrections to the Hartree-Fock result.

To this aim, we use the Feynman-Kac formula for the kernels of
$ \exp \left( -\beta H \right) $ in calculating the traces. An element of
$ \left[ C\left( \left[ 0,\beta / m \right]\right)^{\Lambda}\right]^n = \left[
\Omega_ {\Lambda}\right]^n $ will be denoted as:

$$
\vec{\omega } \left( . \right) = \left( {\bf \omega}^{\alpha}
\left( . \right)\right)_
{\alpha =1,...,n} ; \hskip .5cm  \omega^\alpha \left( . \right)=
\left( \omega ^\alpha_x \left( . \right) \right)_{x \in \Lambda} \in
\Omega_\Lambda , \eqno (2.3)
$$
and the product of Brownian-bridge measures on $\left[
\Omega_ {\Lambda}\right]^n $ will be denoted $ d \vec{\omega} $ , i.e.

$$
d \vec{\omega} = \prod\limits_{\alpha=1}^n d  \omega ^\alpha
= \prod\limits_{\alpha
=1}^n\prod\limits_{x\in \Lambda }dq_x^\alpha P_{q_x^\alpha ,q_x^\alpha
}^{\beta /m}\left( d\omega _x^\alpha \right) , \eqno (2.4)
$$
where $ dq $ is Lebesgue measure on the line and $ P_{q,q}^t \left( d \omega
\right) $ is the conditional Wiener measure, conditioned by $\omega
\left( 0\right) =\omega \left( t\right) =q$.

One obtains

$$
\frac{Z_n}{\tilde Z_1\left( {\bf c}\right) ^n}=\left\langle \exp \left[ -
\frac{\beta w^{\prime \prime }}2 \left\| {\cal F}
_n\left( \vec{\omega }\right) \right\| ^2\right] \right\rangle _{n,{\bf c}
}
= \int d \nu _{n,{\bf c}} \exp \left[ -
\frac{\beta w^{\prime \prime }}2 \left\| {\cal F}
_n\left( \vec{\omega }\right) \right\| ^2\right],\eqno (2.5)
$$
where $ {\bf \cal F}_n\left( \vec{\omega }\right) : \left[
\Omega_ {\Lambda}\right]^n \to {\cal H}_{\Lambda}: = L^2 \left[ {0,1}\right] 
\otimes
{\mathbf R}^{\Lambda}$ is the "fluctuation functional" of paths, defined by:

$$
{\bf \cal F}_n\left( \vec{\omega }\right)_x \left( t \right)=
n^{-1/2}\sum\limits_{\alpha =1}^n
\left[ \omega _x^\alpha \left( t\beta /m\right) ^2-c_x\right], \eqno (2.6)
$$
and
$$
\|{\cal F}_n(\vec{\omega })\|^2 = \int_0^1 dt \sum_{x \in \Lambda } |{\cal
F}_n(\vec{\omega })_x(t)|^2\,.
$$
Here $ \nu _{n,{\bf c}} $ is the product
probability measure on $ \left[ \Omega_ {\Lambda}\right]^n $ of density

$$
\begin{array}[b]{l}
\left( {d\nu _{n,{\bf c}}}/{d \vec{\omega}}\right) (\vec{\omega}) = \rho
_{n,{\bf c}}(\vec \omega ) :=\\
\prod\limits_{\alpha =1}^n K\,
\exp -\beta \int\limits_0^1 \left[ \frac 12\left( {\bf  \omega}^\alpha
(t\beta /m) ,X({\bf c}) {\bf \omega}^\alpha (t\beta /m) \right)
-\left( {\bf h},\omega^\alpha (
t\beta /m) \right) \right] dt ,
\end{array}
\eqno (2.7)
$$
$K$ is a normalisation constant, and ${\bf c}$ is the solution of Eq.(1.5),
implying

$$
\left\langle  {\bf \cal F}
_n\left( \vec{\omega }\right)_x \right\rangle _{n,{\bf c}}=0 ,
\hskip .5cm \forall x\in \Lambda .
\eqno (2.8)
$$

Our strategy is to represent the exponential in Eq.(2.5) as the characteristic
function with respect to a Gaussian process indexed by the Hilbert space
$ {\cal H}_{\Lambda} $. We introduce a convenient realisation of the latter in
terms of the covariance operator, $ R $ of the oscillator process for one 
component
$  \omega \left( t \right) $ of measure $ \nu _{1,{\bf c}} $ . The
operator $ R $ is defined on $ {\cal H}_{\Lambda} $ by the kernel:

$$
\begin{array}{c}
R_{x,y}\left( t,t^{\prime }\right) =\left\langle \left( \omega _x\left(
t\beta /m\right) -m_x\right) (\omega _y\left( t^{\prime }\beta /m\right)
-m_y)\right\rangle _{1, {\bf c}} \hfill
\\
\\ =\frac 1{2\sqrt m}\left[
\cosh \frac \beta{2\sqrt m} X\left( {\bf c}\right) ^{1/2}\left( 1-2\left|
t-t^{\prime}\right| \right)/ \left( X\left( {\bf c}\right) ^{1/2}
{\sinh \frac \beta{2\sqrt m} X\left( {\bf c}\right) ^{1/2}}\right) \right]
_{xy},
\end{array}
\eqno (2.9)
$$
where
$ m_x:=\left\langle \left( \omega _x\left( t\beta /m\right) \right)
\right\rangle _{1, {\bf c}}=\left[ X({\bf c})^{-1}{\bf h}
\right] _x $.

Obviously, $ R $ is a positive operator with finite trace, therefore $ 
R^{1/2} $
is Hilbert-Schmidt.
The nuclear space structure of the test function space
$ C^\infty_{\mathrm {per}} \left( \left[ 0,1 \right],
{\mathbf R}^{\Lambda}\right) \subset {\cal H}_\Lambda $  is defined by
means of the sequence of norms

$$
\left\| \psi \right\| _p^2 = \left\| R^{-p/2}\psi \right\| ^2;\hskip .5cm
p\in {\mathbf Z}, \psi \in C^\infty_{\mathrm {per}}\left( \left[ 0,1 \right],
{\mathbf R}^{\Lambda}\right),
\eqno (2.10)
$$
with $ \left\| \psi \right\| _0=\left\| \psi \right\| $, the Hilbert space 
norm
of $ {\cal H}_{\Lambda} $ . We denote $ {\cal S}_p $ the completion of
$ C^{\infty } $ with respect to the norm $ \left\| . \right\| _p $ .

Using the Bochner-Minlos Theorem \cite {Hida}, we realise
the probability measure $ \mu $ of the Gaussian process on the space  $ {\cal
S}_{-1} $. In this way, for all $ f\in {\cal S}_1 $ one has the representation

$$
\exp -\frac 12\left\| f \right\|^2 =\int\limits_{{\cal S}_{-1}} d\mu
({\bf v})\exp i(f,{\bf v}) , \eqno (2.11)
$$
where $ \left( .,. \right) $ is the duality defined by the scalar
product of $ {\cal H }_{\Lambda } $.

In order to apply Eq.(2.11) to the representation (2.5) one should first
regularize $ {\bf \cal F}_n\left( \vec{\omega }\right) $ such that
it belongs to $ {\cal S}_1 $ for all $ {\bf \vec{\omega}}\in \left[
\Omega_{\Lambda}
\right]^n $. We perform this by cutting its Fourier series. We shall use for
convenience the Fourier basis in the complexification
$ {\cal H}_{\Lambda}^{\mathbf C} = L^2 \left( \left[ {0,1}\right] ,{\mathbf
C} \right) \otimes {\mathbf C}^{\Lambda} $
of $ {\cal H}_{\Lambda} $ , i.e. we take as complete orthonormal system $ \{ 
e_k\otimes
\delta_x;\hskip .3cm k\in {\mathbf Z}, x\in \Lambda \} $, where $ e_k
(t)=e^{2\pi ikt}$ and $ (\delta_x)_y=\delta_{x,y} $. Let $ \pi_k $ be the
projection onto $ e_k $ in $ L^2 \left( \left[ {0,1}\right] ,{\mathbf
C} \right) $. Then the $ \Pi ^{(N)}:=\left( \sum\limits _{\left| k \right| 
\leq N} \pi_k
\right)\otimes 1 $ are projections on the real space $ {\cal H}_{\Lambda} $
and, by dominated convergence,

$$
\frac{Z_n}{\tilde Z_1\left( {\bf c}\right) ^n}=\lim\limits_{N\to \infty}
\left\langle \exp \left[ -
\frac {\beta w^{\prime \prime }}2 \left\| \Pi ^{(N)} {\cal F}
_n\left(\vec{\omega }\right) \right\| ^2\right] \right\rangle _{n,{\bf c}}.
\eqno (2.12)
$$

Since $ \Pi ^{(N)}{\cal H}_{\Lambda}\subset C_{\mathrm {per}}^\infty\left(
\left[ 0,1 \right], {\mathbf R}^{\Lambda}\right)\subset {\cal S}_1 $, one
obtains, using Eq.(2.11), the Fubini theorem and the fact that $ \nu _{n,{\bf 
c}}
$ is a product measure,

\begin{eqnarray*}
\frac{Z_n}{\tilde Z_1\left( {\bf c}\right) ^n} & = & \lim\limits_{N\to \infty}
\left\langle \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\exp
i\sqrt {\beta w^{\prime \prime }}\left( {\bf v},\Pi ^{(N)} {\cal F}
_n\left(\vec{\omega }\right)\right)\right\rangle _{n,{\bf c}}\\
& = & \lim\limits_{N\to \infty}
 \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\left\langle\exp
i\sqrt {\beta w^{\prime \prime }}\left( {\bf v},\Pi ^{(N)} {\cal F}
_n\left(\vec{\omega }\right)\right)\right\rangle _{n,{\bf c}}\\
& = & \lim\limits_{N\to \infty}
 \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\left\langle\prod\limits_{\alpha
 =1}^n \exp
i\sqrt {\frac {\beta w^{\prime \prime }}n }\left( \Pi ^{(N)}{\bf v}, {\cal F}
_1\left( \omega ^\alpha\right)\right)\right\rangle _{n,{\bf c}}\\
& = & \lim\limits_{N\to \infty}
 \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\left[\left\langle\exp
i\sqrt {\frac {\beta w^{\prime \prime }}n }\left( \Pi ^{(N)}{\bf v}, {\cal F}
_1\left( \omega \right)\right)\right\rangle _{1,{\bf c}}\right]^n ,
\end{eqnarray*}
i.e. the decoupling of the components $\omega ^\alpha $ has been achieved 
under
the Gaussian integral. We denote, for ${\bf v}\in {\cal S}_1$:
$$
f({\bf v})=\ln \left\langle\exp
i\left( {\bf v}, {\cal F}
_1\left( \omega\right)\right)\right\rangle _{1,{\bf c}}, \eqno (2.13)
$$
in terms of which
$$
\frac{Z_n}{\tilde Z_1\left( {\bf c}\right) ^n}=\lim\limits_{N\to \infty}
\int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\exp \left[ nf\left( \sqrt {\frac
 {\beta w^{\prime \prime }}n } \Pi ^{(N)}{\bf v}\right) \right]. \eqno (2.14)
$$

In order to take the limit $ N\to \infty $ in Eq. (2.14), we shall use the
dominated convergence theorem. As, obviously,
$$
\left| \left\langle\exp i\left( {\bf v}, {\cal F}
_1\left( \omega\right)\right)\right\rangle _{1,{\bf c}}\right|
\leq 1, \eqno (2.15)
$$
we need only to show pointwise convergence. This is taken care of by the
following:

\medskip\
{\bf Proposition 1.}{\it The function} $f:{\cal S}_1\to {\mathbf C}$ {\it 
defined by
Eq.(2.13) is continuous with respect to the norm} $\left\| .\right\| _{-1}$,
{\it therefore it extends by continuity to the whole} $ {\cal S}_{-1}$.

\medskip\
In proving the proposition we take advantage of the fact that ${\cal F}
_1\left( \omega\right)$ is quadratic in $\omega$, so that $f({\bf v})$
can be explicitly calculated.

\medskip\
{\bf Lemma 1}. {\it For all} ${\bf v}\in {\cal S}_1$,
$$
\begin{array}{c}
f({\bf v})=-\frac 12 tr\left[ \ln (1-2iR^{1/2}M({\bf v})R^{1/2})
+2iR^{1/2}M({\bf v})R^{1/2}\right]
\\-2\left( R^{1/2}M({\bf m}){\bf v},(1-2iR^{1/2}M({\bf v})R^{1/2})^{-1}
R^{1/2}M({\bf m}){\bf v}\right),
\end{array}
\eqno (2.16)
$$
{\it where} $M({\bf v})$ {\it denotes the operator in} $ {\cal H}_{\Lambda}$
{\it of pointwise multiplication by} ${\bf v}$, i.e. $(M({\bf v})\psi )_x(t)=
v_x(t)\psi _x(t)$. {\it Also, the scalar product in Eq.(2.16) is that
of}  $ {\cal H}_{\Lambda}$.

\medskip\
Using the continuity of ${\bf v}$ and ${\bf \omega}$, the integral in
$ \left( {\bf v}, {\cal F}_1\left( \omega\right)\right)$ is approached
by Riemann sums, hence the proof of Lemma 1 consists in performing a finite\- 
-
dimensional Gaussian integral. We omit this calculation.


The main step in proving the $ {\cal S}_{-1}$ -continuity of $ f({\bf v})$ is
the following:

\medskip\
{\bf Lemma 2}. {\it The linear map} ${\bf v}\mapsto R^{1/2}M({\bf v})R^{1/2}$
{\it is bounded from} $ {\cal S}_{-1}$ {\it to the Hilbert space of
Hilbert-Schmidt operators on}
$ {\cal H}_{\Lambda}^{\mathbf C}$.

\medskip\
{\it Proof}.  We calculate

\begin{eqnarray*}
\left\| R^{1/2}M({\bf v})R^{1/2}\right\| _{HS}^2 & = & tr\left(
RM({\bf v})\right) ^2\\
& = & \sum\limits_{x,y\in \Lambda}\int\limits_0^1 dt \int\limits_0^1 dt'\,
v_x(t)\, R_{xy}(t,t')^2\, v_y(t')\\
& = & \left( {\bf v}, R\odot R\, {\bf v}\right)\\
& = & \left( R^{1/2}{\bf v}, R^{-1/2}(R\odot R)R^{-1/2}\cdot R^{1/2}{\bf v}
\right),
\hskip 1cm (2.17)
\end{eqnarray*}
where we used the notation $ R_1\odot R_2 $ for the {\it Schur product} of
two operators $ R_1, R_2 $ defined on $ {\cal H}_{\Lambda}^{\mathbf C}$
by continuous kernels, namely, $ R_1\odot R_2 $ is the operator with kernel 
equal
to the pointwise product of the kernels of $ R_1 $ and $ R_2 $:
$$
(R_1\odot R_2 )_{xy}(t,t'):=(R_1)_{xy}(t,t')\, (R_2)_{xy}(t,t').
\eqno (2.18)
$$

 The Schur product $\odot $ is, of course, representation dependent. We use
the obvious fact that the product $\odot $
factorizes with respect to the tensor product,
i.e. if $ R_k=X_k\otimes Y_k,\; k=1,2 $, with $ X_k, Y_k $
defined by kernels on the factor spaces
$ L^2 \left[ {0,1}\right] ,\; {\mathbf C}^{\Lambda}$ respectively, then
$$
R_1\odot R_2=(X_1\odot X_2)\otimes (Y_1\odot Y_2).
\eqno (2.19)
$$
\vskip .3cm
Eq. (2.17) shows that one has to prove that
$ R^{-1/2}(R\odot R)R^{-1/2} $ is a bounded operator on
$ {\cal H}_{\Lambda}^{\mathbf C}$. To this aim, one uses the spectral
decomposition of $ R $:
$$
R=\sum\limits _{k\in {\mathbf Z}}\; \sum\limits _{\Omega ^2\in \sigma (X({\bf 
c}))}
r_{k,\Omega}\left( \pi _k\otimes P_\Omega \right) ,
\eqno (2.20)
$$
where $\pi _k$ is the projection onto $ e_k $ in $ L^2 \left[ {0,1}\right]$
and $ P_\Omega $ is the projection in $ {\mathbf C}^{\Lambda}$ onto the
corresponding eigenspace of $ X({\bf c}) $; the eigenvalues
$ r_{k,\Omega} $ are given by
$$
r_{k,\Omega}=\frac {2\beta /m}{(\beta \Omega /\sqrt m)^2+(2\pi k)^2};\;
k\in {\mathbf Z},\; \Omega ^2\in \sigma (X({\bf c})).
\eqno (2.21)
$$
Then, by Eq.(2.19) and because $ \pi _{k_1}\odot \pi _{k_2}= \pi _{k_1+k_2} $,
we obtain:
$$
R\odot R=\sum\limits _{\Omega_1^2,\Omega_2^2\in \sigma (X({\bf c}))}\; \sum
\limits _{k\in {\mathbf \mathrm Z}}\left( \sum
\limits _{k^{\prime}\in {\mathbf \mathrm Z}}r_{k-k^{\prime},\Omega_1}
r_{k^{\prime},\Omega_2}\right)\pi _k\otimes (P_{\Omega_1}\odot P_{\Omega_2}).
\eqno (2.22)
$$
Since $\pi _{k_1} \pi _{k_2}=\delta_{k_1,k_2}\pi _{k_1}$, one gets:
$$
\begin{array}{c}
R^{-1/2}(R\odot R)R^{-1/2} \hfill \\
=\sum\limits _{\Omega_1^2,\Omega_2^2,\Omega_3^2,\Omega_4^2\in \sigma (X({\bf 
c}))}\;\sum
\limits _{k\in {\mathbf \mathrm Z}}\frac 1{\sqrt {r_{k,\Omega_3}}}\left( \sum
\limits _{k^{\prime}\in {\mathbf \mathrm Z}}r_{k-k^{\prime},\Omega_1}
r_{k^{\prime},\Omega_2}\right)\frac1{\sqrt {r_{k,\Omega_4}}} \hfill \\
\\
\pi _k\otimes
\left[ P_{\Omega_3}(P_{\Omega_1}\odot P_{\Omega_2}) P_{\Omega_4}\right] .
\end{array}
\eqno (2.23)
$$
Hence, in order to prove the boundedness of this operator, taking
into account that the sums over $\Omega$ are finite and using the monotonicity
of $r_{k,\Omega}$ with respect to $\Omega$, it is sufficient to show that
$$
\sup\limits _{k\in {\mathbf \mathrm Z}}\frac 1{r_{k,\Omega_{\mathrm 
{max}}}}\sum
\limits _{k^{\prime}\in {\mathbf \mathrm Z}}r_{k-k^{\prime},\Omega_{\mathrm
{min}}}r_{k^{\prime},\Omega_{\mathrm{min}}}\,<\,\infty .
\eqno (2.24)
$$
For this, remark that $\max (\left| k'\right| , \left| k-k'\right| )\geq
\left| k\right| /2$, thus
$$
\sum\limits _{k'}r_{k-k^{\prime},\Omega}r_{k^{\prime},\Omega}\leq
r_{k/2,\Omega}\sum\limits _{k'}r_{k^{\prime},\Omega} ,
$$
and that $\sup\limits_k
r_{k/2,\Omega_{\mathrm{min}}}/r_{k,\Omega_{\mathrm{max}}}<\infty $, from which
(2.24) follows. This proves Lemma 2.  \QED

\medskip\
{\it Proof of Proposition 1}.  The first term in Eq.(2.16) is the logarithm
of the regularized determinant of the Hilbert-Schmidt operator
$2iR^{1/2}M({\bf v})R^{1/2}$. As the regularized determinant is continuous on
the Hilbert space of Hilbert-Schmidt operators (cf. \cite{Dunford},
Thm.XI.6.26), and as $2iR^{1/2}M({\bf v})R^{1/2}$ has imaginary eigenvalues, 
the
continuity of this term follows from Lemma 2.

As far as the continuity of the second term is concerned, we use Lemma 2,
and the boundedness of the map
${\bf v}\mapsto R^{1/2}M({\bf m}){\bf v}$ from ${\cal S}_{-1}$ to
${\cal H}_{\Lambda}$, which itself amounts to prove that the operator
$R^{1/2}M({\bf m})R^{-1/2}$ acting on ${\cal H}_{\Lambda}$ is bounded. This is
proved again by using the spectral representation Eq.(2.20) for $R$
and the fact that ${\bf m}$ is independent
of $t$, yielding the formula
$$
R^{1/2}M({\bf m})R^{-1/2}=\sum\limits_{\Omega _1,\Omega_2}\left( \sum\limits 
_k
\left( r_{k,\Omega_1}/r_{k,\Omega_2}\right)^{1/2}\pi _k \right)\otimes
\left( P_{\Omega _1}M({\bf m})P_{\Omega _1}\right) ,
$$
which is a finite sum of bounded operators, and therefore bounded. \QED

\bigskip\
The final point of the above discussion is that Eq.(2.1) has the following
representation:

$$
\begin{array}[b]{ll}
{\displaystyle  {n}\left( f_{\Lambda ,n}\left( {\bf h}\right)
-\tilde f_{\Lambda} ^{\left( 0 \right)} \left( {\bf h} \right) \right)
= -\frac 1{\beta \left| \Lambda \right|
}\ln \frac {Z_n}{\tilde Z_1\left( {\bf c} \right)^n} }\\
\\
{\displaystyle =-\frac 1{\beta \left| \Lambda \right|
}\ln \; \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})\exp \left[ nf\left( \sqrt 
{\frac
 {\beta w^{\prime \prime }}n } {\bf v}\right) \right] ,}
\end{array}
\eqno (2.25)
$$
where $f$ is the (continuous extension to ${\cal S }_{-1}$ of the) function
(2.16) and where
$$
\begin{array}[b]{ll}
\left| \Lambda \right| {\tilde f}_\Lambda ^{\left( 0\right) }\left( {\bf 
h}\right)
=-\frac 1\beta \ln \tilde Z_1\left( {\bf c} \right) \\
\\
=\frac 1\beta tr\,\ln \left[ 2\sinh \left( \frac \beta {2 \sqrt{m}}
X({\bf c})^{1/2}\right) \right] - ({\bf h},X({\bf c})^{-1}{\bf h}
)+ \sum\limits_{x\in \Lambda }\left( w(c_x)-c_xw^{\prime }(c_x)\right) .
\end{array}
\eqno (2.26)
$$

The limit of Eq.(2.25), providing ${\tilde f}_\Lambda ^{\left( 1 \right) 
}({\bf
h})$ of Eq.(1.7), is easily obtained by dominated convergence, using the
uniform bound provided by Eq.(2.15) and the explicit form of $f$, Eq.(2.16),
yielding the pointwise limit: for all ${\bf v}\in {\cal S }_{-1}$,
$$
\begin{array}[b]{ll}
\lim\limits _{n\to \infty} nf\left( \sqrt {\frac {\beta w''}n}{\bf v} \right) 
=
-\beta w'' \left[ tr(R^{1/2}M({\bf v})R^{1/2}) ^2 + 2
\left\| R^{1/2}M({\bf m}){\bf v}\right\| ^2 \right] \\
= -\frac 12 \left( {\bf v}, 2\beta w'' \left[ R\odot R + 2M({\bf m})RM({\bf
m}) \right] \;{\bf v} \right) .
\end{array}
\eqno (2.27)
$$

The operator appearing in Eq. (2.27):
$$
A:= 2\beta w'' \left[ R\odot R + 2M({\bf m})RM({\bf m})
\right]
\eqno (2.28)
$$
plays a central role in the following. In fact, $A$ is the Hartree-Fock
covariance of ${\cal F}_1 (\omega )$ (compare with Eq. (2.13):
$$
A_{xy} (t,t')=\beta w''\left\langle (\omega _x(t)^2-c_x)(\omega _y(t)^2-c_y)
\right\rangle _{1,{\bf c}}, \eqno (2.29)
$$
We remark that $A$ is positive and of
finite trace, therefore one can define on ${\cal S }_{-1}$ a Gaussian
probability measure $ \mu_ A $ with perturbed covariance, i.e. absolutely
continuous with respect to $\mu $ of density
$$
\frac {d\mu _A}{d\mu }({\bf v}) = N_A^{-1}\exp -\frac 12 ({\bf v}, A{\bf v})
\eqno (2.30)
$$
where
$$
N_A = \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})
\exp -\frac 12 ({\bf v}, A{\bf v})= \det (1+A)^{-1/2}.
\eqno (2.31)
$$

With this notation, one has simply:
$$
{\tilde f}_\Lambda ^{\left( 1 \right) }({\bf h})=
-\frac 1{\beta \left| \Lambda \right| }\frac 1{\beta w''}\ln N_A
\eqno (2.32)
$$
and Eq. (2.25) can be written in terms of the measure $ \mu _A $ :
$$
\begin{array}[b]{ll}
 {n}\left( f_{\Lambda ,n}\left( {\bf h}\right)
-\tilde f_{\Lambda} ^{\left( 0 \right)} \left( {\bf h} \right) \right) \\
= \beta w'' {\tilde f}_\Lambda ^{\left( 1 \right) }({\bf h})
-\frac 1{\beta \left| \Lambda \right| }\ln
\int\limits_{{\cal S}_{-1}}d\mu _A({\bf v})
\exp \left[ nf\left( \sqrt {\frac
 {\beta w^{\prime \prime }}n } {\bf v}\right) +\frac 12 ({\bf v}, A{\bf v})
\right] .
\end{array}
\eqno (2.33)
$$

\medskip\
The asymptotic series is formally obtained from Eq. (2.33) by expanding the
exponential in power series in $ n ^{-1/2} $ . The following two lemmas 
justify
this procedure.

\medskip\
{\bf Lemma 3.} {\it For any}  $ \rho > 0 $,
$$
\int\limits_{\left\| {\bf v}\right\| _{-1} > \rho \sqrt n}d \mu ({\bf v})
\exp nf\left( \sqrt {\frac
 {\beta w ^{\prime \prime }}n } {\bf v}\right) \sim 0\; (n\to \infty )
  \eqno (2.34)
$$
{\it (i.e. it tends to zero faster than any power of $ n ^ {-1} $ as
$ n \to \infty $ )}

\medskip\
{\it Proof.}  By Eqs. (2.13) and (2.15), it is sufficient to show that
$ \int\limits_{\left\| {\bf v}\right\| _{-1} > \rho \sqrt n}d \mu ({\bf v})$
has the property. As $ \left\| {\bf v}\right\| _{-1}^2=({\bf v}, R{\bf v})
>\rho ^2 n $ in the integration domain, we have for small $ \epsilon >0$,
$$
\int\limits_{\left\| {\bf v}\right\| _{-1} > \rho \sqrt n}d \mu ({\bf v})
\leq \int\limits_{{\cal S}_{-1}}d\mu ({\bf v})
\exp \frac {\epsilon }2 \left[({\bf v}, R{\bf v})-n\rho ^2\right]=
\det(1-\epsilon R)^{-1/2} \exp -\epsilon \frac {n\rho ^2}2 ,
$$
proving the assertion. \QED

\medskip\
The following lemma shows the analyticity of $f$ around zero in $ {\cal
S}_{-1}$. We formulate it in a way convenient for our purpose.

\medskip\
{\bf Lemma 4.} {\it There exists} $ \rho >0 $, {\it such that for every}
${\bf v}\in {\cal S}_{-1}$ {\it with}
$ \| {\bf v} \| _{-1} \leq \rho $,
{\it the function of a complex variable} $ z $ {\it defined by}:
$$
\phi _{\bf v}(z)=\frac {\beta w''}{z^2}f(z{\bf v})+
\frac 12 ({\bf v}, A{\bf v}) , \; \phi _{\bf v}(0)=0 ,
\eqno (2.35)
$$
{\it is defined and analytic on the closed unit disk} $ \{ |z| \leq 1 \}$.
{\it Moreover, the family} $\{ \phi _{\bf v};\; {\bf v}\in {\cal
S}_{-1},\, \left\| {\bf v}\right\| _{-1} \leq \rho \} $ {\it is uniformly
bounded on} $\{ |z| \leq 1 \} $,
{\it hence, for every} $k\geq 1$, {\it the family of} $k${\it -th derivatives}
$\{ \phi _{\bf v}^{(k)};\; {\bf v}\in {\cal
S}_{-1},\, \| {\bf v}\| _{-1} \leq \rho \} $
{\it is likewise bounded}.

\medskip\
{\it Proof.} The analyticity follows from the analyticity of $\ln (1+z)$ and 
of
$(1+z)^{-1}$ in $ \{ |z| < 1 \}$, providing a HS-norm-convergent power series
of the corresponding operator functions $\ln (1+X)$ and
$(1+X)^{-1}$, and from
$$
|tr(X^k)| \leq \| X \| _{HS}^k \; (k\geq 2)
$$
applied for $X=2izR^{1/2}M({\bf v})R^{1/2} $ combined with Lemma 2. The
rest is due to the Montel compact principle for analytic functions, see
e.g.\ \cite{Markushevich}. \QED

\medskip\
Lemma 3 implies that the function $f_{\Lambda ,n}({\bf h})$ is asymptotically
equivalent to
$$
f_{\Lambda ,n}\left( {\bf h}\right) \sim
\tilde f_{\Lambda} ^{\left( 0 \right)} \left( {\bf h} \right) +
\frac {\beta w''}n \tilde f_{\Lambda} ^{\left( 1 \right)} \left( {\bf h} 
\right) +
\frac {\beta w''}n \Phi _n(1) ,
\eqno (2.36)
$$
where
$$
\Phi _n(z)=-\frac 1{\beta \left| \Lambda \right| }\frac 1{\beta w''}\ln
\int\limits_{\left\| {\bf v}\right\| _{-1} \leq \rho \sqrt n}d\mu _A({\bf v})
\exp \phi _{{\bf v}\sqrt {\frac {\beta w''}n}}(z)
\eqno (2.37)
$$
is defined on the unit disk of ${\mathbf C}$. All derivatives of $\Phi _n(z)$
exist at $z=0$. The
$$
k!\; \tilde f_{\Lambda} ^{\left( \frac k2 +1 \right)} \left( {\bf h}
\right) :=
\lim\limits _{n\to \infty} \left( \frac n{\beta w''} \right) ^{k/2}
\Phi _n^{(k)}(0),\;(k\geq 1)
\eqno (2.38)
$$
are linear combinations of Gaussian integrals of homogeneous polynomials of
degree $k$ in the variable ${\bf v}$. Therefore,
$\tilde f_{\Lambda} ^{\left( \frac k2 +1 \right)} \left( {\bf h} \right)=0$ 
for
$k$ odd. Furthermore, Lemma 4 implies that for all $N$ :
$$
\sup\limits _{|z|\leq 1} \left| n^{N-1}\Phi _n^{(2N-1)}(z)\right|
=O(n^{-1/2}) $$
finally proving:

\medskip\
{\bf Theorem 1.} {\it With the above notations (2.26), (2.32) and (2.38), for
all N, one has}
$$
\lim\limits _{n\to \infty} n^N\left( f_{\Lambda ,n}\left( {\bf h}\right) -
\sum\limits _{k=0}^N  \left( \frac
{\beta w''}n \right) ^k \tilde f_\Lambda ^{\left( k\right) }
\left( {\bf h}\right) \right) = 0,
\eqno (2.39)
$$
{\it hence the series (1.7) is a complete asymptotic series.}

\bigskip\

\centerline {\bf 3. Displacement fluctuations}

\bigskip\

As mentioned in the introduction, expectation values of observables
depending on a finite number of components do converge in the $n
\rightarrow \infty $ limit to expectation values for the Hartree-Fock
approximation.  We illustrate this fact as a simple corrollary of
Theorem~1.  We derive the complete asymptotic series for these expectations
from which one reads off the mentioned property.

\medskip\
{\bf Theorem 2.} {\it Let $\langle \cdot \rangle_n$ be the canonical Gibbs
state for $H_{\Lambda ,n}(1.1)$, then for all $\boldsymbol{\lambda} ^\alpha
,\boldsymbol{\mu}
^\alpha \in \mathbf C^\Lambda $, $\alpha  = 1,\ldots,N$ the characteristic
function}
$$
E(\boldsymbol\lambda ,\boldsymbol\mu )_n \equiv \left\langle e^{i\
\sum_{\alpha = 1}^N
(\lambda ^\alpha ,q^\alpha )}e^{i\ \sum_{\beta = 1}^N
(\mu ^\beta ,p^\beta )}\right\rangle_n
\eqno (3.1)
$$
{\it has a complete asymptotic series in $\frac{1}{n}$ and}
$$
\lim_{n \rightarrow \infty } E(\boldsymbol\lambda ,\boldsymbol\mu )_n =
\prod_{\alpha = 1}^N
\left\langle e^{i(\lambda ^\alpha ,q^\alpha )} e^{i(\mu ^\alpha
,p^\alpha )}\right\rangle_{1,c}
\eqno (3.2)
$$

\medskip\
{\it Proof.}  We use the Feynman-Kac representation where we denote the
first $N$ components $\boldsymbol\eta ^\alpha $, $\alpha = 1,\ldots N$, the
remaining
$n - N$ are again denoted $\omega ^\beta $, $\beta = 1,\ldots,n-N$.  Due to
the presence of $\exp i(\boldsymbol\mu ^\alpha ,{\bf p}^\alpha )$
operators, the reference measure for the $\eta$ variables is modified to
$$
\sigma (\boldsymbol\lambda ,\boldsymbol\mu ;d\vec\eta ) = \prod_{\alpha =
1}^N \prod_{x \in \Lambda
} e^{i\ \lambda _x^\alpha q_x^\alpha } \ dq_x^\alpha \ P_{q_x^\alpha
,q_x^\alpha + \vec\mu _x^\alpha }^{\beta /m}(d\eta _x^\alpha )\,.
$$
Therefore, in the notation of Eq~(2.7):
$$
E(\boldsymbol\lambda ,\boldsymbol\mu )_n = \frac{\tilde Z_1({\bf
c})^n}{Z_n} \int \sigma (\boldsymbol\lambda ,\boldsymbol\mu
;d\vec\eta ) \rho _{N,{\bf c}}(\vec\eta ) \int d\nu _
{n-N,{\bf c}}(\vec\omega
)e^{-\frac{\beta \vec\omega ''}{2}} \|{\cal F}_n(\vec \eta,\vec\omega
)\|^2\,. $$
Applying again Eq~(2.11), one gets
$$
\begin{array}[b]{c}
{\displaystyle E(\boldsymbol\lambda ,\boldsymbol\mu )_n = \frac{\tilde
Z_1({\bf c})^n}{Z_n} \int d\mu ({\bf v}) \left(\int \sigma
(\lambda ,\mu ;d\vec\eta )\rho _{N,{\bf c}}(\vec\eta
)e^{i\sqrt{\frac{\beta
\vec\omega ''}{n}}({\bf v},{\cal F}_N(\vec\eta ))}\right)}\\
\exp(n-N) f\left(\sqrt{\frac{\beta \vec\omega '' }{n}}{\bf v}\right)
\end{array}
\eqno (3.3)
$$
where $f$ is the function (2.13).  As $N$ is fixed, the analysis of $n
\rightarrow E(\boldsymbol\lambda ,\boldsymbol\mu )_n$ is reduced to the one
of section~2, and the
proof of this theorem follows from Theorem~1.  In particular, remembering
formula (2.25) and the fact that for any ${\bf v}$:
$$
\lim_{n \rightarrow \infty } \int \sigma (\boldsymbol\lambda
,\boldsymbol\mu ;d\vec\eta
)\rho _{N,c}(\vec\eta )e^{i\sqrt{\frac{\beta \vec\omega ''}{n}}({\bf
v},{\cal F}_N(\vec \eta))}
$$
is the Hartree-Fock expectation value of the right hand side of (3.2), the
limit stated in (3.2) holds. \QED

\medskip\

The situation changes drastically if one considers expectation values of
global observables e.g.\ if one computes the moments or the characteristic
function of fluctuations.  The basic fact is that fluctuations are very
sensitive to different boundary conditions.  Loosely speaking the
fluctuations behave like $\sqrt n$ for large $n$, and also the difference
$H_{\Lambda ,n} - \tilde H_{\Lambda ,n}({\bf c}) \simeq O(\sqrt n)$.  Hence
considering expectation values with respect to the full Hamiltonian
$H_{\Lambda ,n}$, or in Hartree-Fock approximation is already making a
mistake of the same size as a fluctuation.  Here we are considering
fluctuations with respect to the number of components.  The phenomenon
described above, is well known in computations of fluctuations with respect
to the thermodynamic limit.  For classical systems this programme is very
well worked out for the Curie-Weiss model
\cite{Ellis1},\cite{Ellis2},\cite{VZ3} with the thermodynamic
limit.  For the limit of infinitely many components in the classical
$n$-vector model we refer to \cite{ABC}.  The
situation for
quantum systems is more complicated because of the noncommutativity.  For
the strongly coupled BCS-model, fluctuations have been computed recently
\cite{Fannes} of course again for the thermodynamic limit.\newline
In the following we compute the $q$- and $q^2$-fluctuations for the
anharmonic crystal model, and we get explicitly the deviations from the
Hartree-Fock results.

First remark that the expectation values $\langle q^\alpha \rangle_n$ and
$\langle(q^\alpha )^2\rangle_n$ differ from ${\bf m} = \langle q^\alpha
\rangle_{1,{\bf c}}$ and ${\bf c} = \langle(q^\alpha )^2\rangle_{1,{\bf
c}}$,
the Hartree-Fock expectations.  But from Theorem~2, it follows that these
differences can be controlled up to all orders of $1/n$, e.g.
\begin{eqnarray*}
\langle q_x^\alpha \rangle_n &=& m_x + O\left(\frac{1}{n}\right)\\
\langle(q_x^\alpha )^2\rangle_n &=& c_x +
O\left(\frac{1}{n}\right)\,.
\end{eqnarray*}
Therefore, we define the fluctuations of $q$ and $q^2$ here as:

$$
F_n(q)_x = \frac{1}{\sqrt n} \sum_{\alpha = 1}^n (q_x^\alpha - m_x)
\eqno (3.4)
$$
$$
F_n(q^2)_x = \frac{1}{\sqrt n} \sum_{\alpha = 1}^n ((q_x^\alpha )^2 -
c_x) \eqno (3.5)
$$
Then we have:

\medskip\
{\bf Theorem 3.} {\it The characteristic functions: for $\boldsymbol\lambda
\in \mathbf R^\Lambda $,}
$$
\langle e^{i(\boldsymbol\lambda,{\bf F}_n(q^2))}\rangle_n\ ,\
\langle e^{i(\boldsymbol\lambda ,{\bf F}_n(q))}\rangle_n
$$
{\it have a complete asymptotic series in $1/n$, as analytic functions of
$\boldsymbol\lambda $ in $\mathbf C^\Lambda $.  In particular, if $n
\rightarrow \infty
$, the fluctuations tend in distribution to Gaussian random variables in
$\mathbf R^\Lambda $ with covariance matrices respectively:}

$$
\langle F(q_x^2)\,F(q_y^2)\rangle = \frac{1}{\beta \omega ''}
\left[A(1 + A)^{-1}\right]_{xy} (0,0)
\eqno (3.6)
$$

$$
\langle F(q_x)\,F(q_y)\rangle = \left[R - 4\beta \omega ''RM({\bf m})(1
+ A)^{-1}M({\bf m})R\right]_{xy} (0,0)
\eqno (3.7)
$$
{\it where $A$ is given by (2.29) and $R$ by (2.9) and which are the
Hartree-Fock covariance matrices.}

\medskip\
{\it Proof}.  The computation of the characteristic function $\langle
e^{i(\boldsymbol\lambda ,{\bf F}_n(q^2))}\rangle_n$ is technically the most
straightforward
one, in view of the Feynman-Kac representation and the Bochner-Minlos
trick.  One gets immediately
$$
\langle e^{i(\boldsymbol\lambda ,{\bf F}_n(q^2))}\rangle_n = \frac{\int
d\mu ({\bf
v}) \exp nf \left(\sqrt{\frac{\beta \omega ''}{n}} {\bf v} +
\frac{\boldsymbol\lambda }{\sqrt n}\delta _0\right)}{\int d\mu ({\bf v})
\exp nf \left(\sqrt{\frac{\beta \omega ''}{n}}{\bf v}\right)}
$$
where $\boldsymbol\lambda \delta _0 \in S_{-1}$ (and stands for $\lambda
\otimes \delta _0$, with $\delta _0$ the Dirac measure at zero).  Use
Lemma~3 to restrict the domain of integration over $S_{-1}$ to a sphere
$\|{\bf v}\|_{-1} \leq \rho \sqrt n$, $\rho  > 0$.  Then applying Lemma~4,
we
obtain the existence of the limits corresponding to (2.39) (see Theorem~1)
uniformly for $\boldsymbol\lambda $ in compacts.

Now we compute the limit $n \rightarrow \infty $, and obtain (see formula
(2.27)):
$$
\frac{\int d\mu ({\bf v})\ e^{-\frac{1}{2}\left({\bf v} +
\frac{\boldsymbol\lambda\delta
_0}{\sqrt{\beta \omega ''}},A\left({\bf v} + \frac{\boldsymbol\lambda
\delta _0}{\sqrt{\beta \omega ''}}\right)\right)}}{\int d\mu ({\bf v})\
e^{-\frac{1}{2}({\bf v},A{\bf v})}} = \exp - \frac{1}{2}\,\frac{1}{\beta
\omega ''} (\boldsymbol\lambda \delta _0,A(1 + A)^{-1}\boldsymbol\lambda
\delta _0) $$
where we use the fact that $A\delta _0 \in S_1$.

Finally we deal with the characterstic function of the displacement
fluctuation.  Now
$$
\left\langle e^{i(\boldsymbol\lambda ,{\bf F}(q))}\right\rangle_n =
\frac{\int d\mu ({\bf v}) \exp n\ f \left(\sqrt{\frac{\beta \omega
''}{n}}{\bf v},\frac{\boldsymbol\lambda}{\sqrt
n}\right)}{\int d\mu ({\bf v}) \exp n\ f\left(\sqrt{\frac{\beta \omega
''}{n}}{\bf v}\right)}
$$
where, as $R^{1/2}\boldsymbol\lambda \delta _0 \in S_0$, the function
\begin{eqnarray*}
f({\bf v},\lambda ) &\equiv& \ln \left\langle e^{i(\boldsymbol\lambda
(\omega (0) - m))} e^{i({\bf F}_1(\omega ),{\bf v})}\right\rangle_{1,c}\\
&=& f({\bf v}) - 2\left(R^{1/2}\boldsymbol\lambda \delta _0,\left(1 +
2iR^{1/2}M({\bf v})R^{1/2}\right)^{-1} R^{1/2} M(m){\bf v}\right)\\
&&\hspace*{1cm}-\frac{1}{2}\left(R^{1/2}\boldsymbol\lambda \delta
_0,\left(1
+ 2iR^{1/2}M({\bf v})R^{1/2}\right)^{-1} R^{1/2} \boldsymbol\lambda \delta
_0\right) \end{eqnarray*}
is well defined and integrable with respect to $\mu $.  The proof is along
the same lines as above for the $q^2$-case, but now using Lemma~4 with the
analytic function
$$
z \rightarrow \varphi _{\bf v}(\boldsymbol\lambda ,z) = \frac{\beta \omega
''}{z^2} f\left(z{\bf v},\frac{z\boldsymbol\lambda }{\sqrt{\beta \omega
''}}\right) + \frac{1}{2}({\bf v},A{\bf v})
$$
satisfying $\varphi _{\bf v}(\boldsymbol\lambda ,0) = -\frac{1}{2}
\|R^{1/2}\boldsymbol\lambda \delta _0\|^2$. \QED

\medskip\
Remark that the difference of the covariance matrices for the system
$H_{\Lambda ,n}$ and the Hartree-Fock system $\tilde H_{\Lambda ,n}({\bf
c})$ are the following:\newline
(i)~for the $q$-fluctuation, it is given by
$$
-4\beta \omega ''RM({\bf m})(1 + A)^{-1}M({\bf m})R
$$
i.e.\ a quantity vanishing in the limit of the external field ${\bf h}
\rightarrow 0$, because then ${\bf m} = 0$ (see (2.9)).  This is a result
to be expected.\newline
(ii)~On the other hand for the $q^2$-fluctuation one gets for the deviation
$$
A - A(1 + A)^{-1} = A^2(1 + A)^{-1}\,.
$$
If ${\bf h} = 0$, then $A = 4\beta \omega ''R \odot R \neq 0$ and one
obtains a nontrivial difference.

This settles our results about the $q$- and $q^2$-fluctuations for a large
number $n$ of components.  Here we did not consider momentum fluctuations.
This comes over as a nontrivial extension of what we did above, revealing
the typical quantum character of the model.  Another interesting question
related to the $1/n$-expansion is the study of the dynamics of these
fluctuations.

In a following contribution we concentrate our attention to the
thermodynamic limit ($\Lambda \rightarrow  {\mathbf Z}^d$) of our
results presented here.  We make explicit the corrections on the
cluster properties as compared to the Hartree-Fock approximation.\newline
\vfill

\noindent{\bf Acknowledgements}\\*[3mm]
N.A. and V.A.Z. would like to thank the Instituut voor Theoretische Fysica,
K.U.Leuven for hospitality and financial support.


\newpage\

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