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Gaussian Analysis, Nonlinear Holomorphic Map, Fixed Points, Critical 
Pont Convergence, Hierarchical Model of Anharmonic Quantum Oscillators
---------------9907011239731
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%\documentstyle[12pt]{article}
\documentclass[12pt]{article}
\usepackage{srcltx}
\makeatletter\@addtoreset{equation}{section}\makeatother
\def\theequation{\arabic{section}.\arabic{equation}}
\newtheorem{Tm}{Theorem}[section]
\newtheorem{Rk}{Remark}[section]
\newtheorem{Lm}{Lemma}[section]
\newtheorem{Co}{Corollary}[section]
\newtheorem{Df}{Definition}[section]
\newtheorem{Pn}{Proposition}[section]


\begin{document}

\author{Sergio Albeverio \\
        Abteilung f\"ur Stochastik, Universit\"at Bonn,\\
        D 53115 Bonn (Germany); SFB 256; \\
        BiBoS Research Center, Bielefeld (Germany);\\
        CERFIM and USI, Locarno (Switzerland) \\
        e-mail albeverio@uni-bonn.de\\
  \and
        Yuri Kondratiev \\
        Abteilung f\"ur Stochastik, Universit\"at Bonn,\\
        D 53115 Bonn (Germany); SFB 256; \\
        BiBoS Research Center, Bielefeld (Germany);\\
        Institute of Mathematics, Kiev (Ukraine)\\
        e-mail kondratiev@uni-bonn.de\\
  \and
        Yuri Kozitsky\\
        Institute of Mathematics, Maria Curie-Sklodowska University\\
        PL 20-031 Lublin (Poland);\\
        Institute for Condensed Matter Physics, Lviv (Ukraine)\\
        e-mail jkozi@golem.umcs.lublin.pl\\ }

\title{\bf Nonlinear $S$-transform and Critical Point Convergence for
 a Quantum Hierarchical Model}

\def\cit#1{\cite{[#1]}}
\def\kasten{\hfil\vrule height6pt width5pt depth-1pt\par }
\date{}
\def\oom{\omega}
\def\bs{\beta_{*}}
\def\C{{C\!\!\! C}}
\def\R{{I\!\! R}}
\def\N{{I\!\! N}}
\def\Z{{Z\!\!\! Z}}
\def\oover{\over}
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\def\t{\tau}
\def\l{\lambda}
\def\om{\omega}
\def\s{\sigma}
\def\vp{\varphi}
\def\ve{\varepsilon}
\def\vt{\vartheta}
\def\D{\Delta}
\def\G{\Gamma}
\def\L{\Lambda}
\def\Om{\Omega}
\def\ooover{\over}
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\def\EC{{\cal E}}
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\def\HH{\bf {\cal H}_{-}}
\def\HQ{{\cal H}_Q}
\def\SS{\bf S}
\def\FC{\cal F}
\def\BC{\cal B}
\def\be{\begin{equation}}
\def\th{\theta}
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\def\GG{{\bf G}^{\sigma,\theta}}
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\def\eeq{\end{eqnarray}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
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\def\QQ{{\cal Q}_{\rm fin}}
\def\FL{{\cal F}^{(e)}_{\rm Lap}}
\def\hU{\hat{U}}
\def\bu{\bar{u}}
\def\bw{\bar{w}}
\def\p{2^{\delta}}
\def\2m{2^{-\delta}}
\def\k{\kappa}
\def\FB{{\cal F}^{(e)}_{\beta,\delta}}
\def\ra{\rightarrow}
\def\oo{[\beta_{0}^{-}, \beta_{0}^{+}]}
\def\bk{\bar{\kappa}}
\def\Hos{{\cal H}^{\rm osc}}
\def\Ho{{\cal H}^{\rm osc}}
\def\Hoj{{\cal H}^{\rm osc}_j }
\def\HoL{{\cal H}^{\rm osc}_{\Lambda_{n,j}} }
\def\Lf{{\cal L}_{{\bf fin}}}
\def\Lan{{\Lambda}_{n,j}}
\def\Ln{{\cal L}_n}
\def\Lal{{\Lambda}_{n-l,s}}
\def\La{{\Lambda}_{n-1,s}}
\def\Th{{\cal T}_{{\bf hier}}}
\def\z{\zeta}
\def\sta{\stackrel{\rm def}{=}}
\def\TR{{\rm trace}}
\pagenumbering{arabic}


\maketitle
\begin{center}
%\begin{large}
{\sc AMS Classification: 28C20; 58C30; 58D30}
%\end{large}
\end{center}
\begin{abstract}


A sequence of measures $\{\nu_n \}$ on a separable Hilbert space ${\cal H}$
generated by a nonlinear map is considered. For a special choice of ${\cal H}$
and $\nu_0$, such a sequence describes the Euclidean Gibbs states of a chain of
interacting quantum anharmonic oscillators. Each ${\nu_n }$ has a Laplace
transform $F_n$, which is an entire function on ${\cal H}$. The sequence $\{F_n
\}$ can be generated by a nonlinear generalization of the $S$-transform known in
Gaussian Analysis, defined as a holomorphic map on certain spaces of entire
functions. A family of fixed points for this map is found and analyzed. In the
case where $\{F_n \}$ describes the mentioned oscillators, it is proven that
this sequence converges to both stable and unstable fixed point. The convergence
to the unstable fixed point corresponds to the appearance of the strong
dependence between the oscillators peculiar to the critical point.
\end{abstract}
\tableofcontents




\section{Introduction}


Methods of infinite dimensional analysis created during the last decades have
found important applications  in several domains of modern mathematics. Among
them is a rigorous construction of models of quantum  statistical physics
\cit{AHK}. Conversely, the problems posed by quantum statistical physics
stimulate a further  development of methods of infinite dimensional analysis. In
particular, the important task here is to generalize to the infinite dimensional
case appropriate tools of finite dimensional nonlinear analysis, especially
those used to study such essentially nonlinear phenomena as phase transitions.
This paper is intended as a step in such a direction.

The role of the $S$-transform in Gaussian Analysis is well known (see e.g.
\cit{BeKo} and \cit{HKPS}). In Section 2, we consider a sequence of
measures $\{\nu_n \}$ on a real separable Hilbert space ${\cal H}$ generated by
a nonlinear map. For a special choice of ${\cal H}$ and $\nu_0 $, such a
sequence describes the Euclidean Gibbs states of a chain of interacting quantum
anharmonic oscillators. Each measure $\nu_n $ has a Laplace transform $F_n$,
which is an entire function on ${\cal H}$. The sequence $\{ F_n \}$ may be
generated by a nonlinear generalization of the $S$-transform (NLST). Thus the
weak convergence of the sequence $\{\nu_n\}$ may be shown by proving the
convergence of $\{F_n \}$ to the NLST fixed points. We study these objects in
Section 3 where appropriate locally convex spaces of entire functions of order
of growth at most two , defined on infinite dimensional Hilbert spaces, are
introduced. Then we define the NLST, prove that it is a holomorphic map,
describe a family of fixed points for it, and study their stability by means of
the NLST Fr\'echet derivative. This information is then used to prove the
convergence of the sequences $\{ F_n \}$ to a stable and to an unstable fixed
point. The convergence to the latter occurs at a certain threshold value of the
parameter which describes the interaction between the oscillators. It
corresponds to the appearance of a strong dependence between them, which is
peculiar to the critical point. The necessary information about the model and
its description in terms of measures on infinite dimensional spaces (i.e. in
terms of Euclidean Gibbs states) is given in Section 4. Section 5 contains the
proof of the theorem stating the mentioned convergence. The proof is based upon
a number of lemmas, which in turn are proven in Section 6.

This paper may be considered as a general formulation and extension of methods
discussed in a particular case in
\cit{AKK1}--\cit{AKK4} (to which we refer for further motivations and some details).

\section{General Setup}
Consider a real separable Hilbert space ${\cal H}$ with inner product $(.,.)$
and norm $ \vert . \vert $. Let ${\cal B}({\cal H})$ stand for the set of Borel
subsets of ${\cal H}$. For $A\in {\cal B}({\cal
  H} )$ and $\om \in {\cal H}$, we write $A-\om = \{ \xi \ \vert \
\xi + \om \in A \}$ and $-A = 0-A $. Given two probability measures on
${\cal H}$, their convolution $\mu \star \nu $ is defined by
$$
(\mu \star \nu )(A) = \int_{{\cal H}} \mu(A-\om )\nu (d\om).
$$
Let ${\cal M} ({\cal H})$ be the space of all probability measures on ${\cal H}$
and let ${\cal M}_0 ({\cal H})$ be the subset of ${\cal M} ({\cal H})$
containing even probability measures (i.e. such that $\mu (-A)=
\mu (A)$) possessing the property
\be
\label{001}
\int_{{\cal H}}\exp (a\vert \om\vert^2 )\mu (d\om) < \infty , \ \
\forall a>0 .
\ee
For $M\in \N$, let $\nu^{\star M}$ stand for the convolution of $M$ copies of
the probability measure $\nu$. Define for $\nu \in {\cal M} ({\cal H})$, $\theta
\geq 0 $, and $\delta \geq 0 $:
\beq
\label{002}
{\bf N}^{\theta , \delta }_M (\nu)(d\om ) & = & \frac{1}{Z_\nu }
\exp\left( \frac{1}{2}\theta^2 M^{-1-\delta }\vert \om \vert^2 \right)
\nu^{\star M}(M^{-(1+\delta)/2}d\om), \\
\label{003}
Z_\nu & = & \int_{{\cal H}}\exp\left(\frac{\theta}{2}\vert \om \vert^2 \right)
\nu^{\star M}(d\om) . \nonumber
\eeq
Clearly, $ {\bf N}^{\theta , \delta }_M $ maps ${\cal M}_0({\cal H})$ into
itself. Let us consider the sequence $\{ \nu_n \} \subset {\cal M}_0({\cal H}) $
generated by (\ref{002}) according to
\be
\label{004}
\nu_{n+1} = {\bf N}_M^{\theta , \delta } (\nu_n ), \ \ \nu_0 = \nu \in
{\cal M}_0({\cal H}) , \ \ n\in {I\!\! N}_0 \ \sta \ \N \cup \{0\} .
\ee
For $\theta = \delta =  0 $, one may expect that $\{\nu_n \}$ converges weakly
to some Gaussian measure with the covariance operator depending on the choice of
$\nu_0$. It is also naturally to expect that this asymptotic behaviour of
$\{\nu_n \}$ is stable, i.e. the same convergence holds for small positive
values of $\theta$ and $\delta =0$. This means that for $\th$  zero or small
enough but $\delta $ positive, the corresponding sequence $\{\nu_n \}$ will
converge for $n \ra \infty$ to the $\delta$--measure concentrated at zero. Does
there exist the threshold value of $\theta$ which bounds this stability region
 and for which the corresponding sequence is no longer degenerate?
If yes, it would ensure that a strong dependence, in contrast to the weak
dependence corresponding to the case of "small" $\theta $, appears for such
$\theta$. A similar phenomenon is known in statistical physics under the name
"critical point behaviour". It occurs at a critical (threshold) value of the
temperature. In this paper, a positive answer has been found for $\delta$
restricted to the interval $\delta
\in (0,1/2)$ and for the choice of ${\cal H}$ and $\nu_0$ which we present
below. For this choice, the sequence of measures $\{\nu_n \}$ describes the
Euclidean Gibbs states of the model of hierarchically interacting quantum
anharmonic oscillators. In this case the mentioned convergence ensures the
appearance of the strong dependence between their oscillations, peculiar to the
critical point. The space ${\cal H}$ is infinite dimensional, a fact which
corresponds to the quantum character of the model. Its  hierarchical nature
appears in the fact that the measures $\nu_n $ are arranged in the sequence
described above where $n\in \N_0 $ numbers the hierarchy levels. For a detailed
information regarding such models and their description by means of measures on
infinite dimensional spaces, we refer to our previous papers
\cit{AKK1}--\cit{AKK3}.

In what follows, as ${\cal H}$ we choose the real Hilbert space $ L^2 (I_\b )$,
$ I_\b \ \sta \ [0,\beta ]$, where $\beta >0 $ stands for the inverse
temperature $T^{-1}$. From now on we fix $M$ and $\theta$ in (\ref{002}) by
setting $M=2$ and $\theta^2 = (2^\delta
-1)2^{-1-\delta} $. Such a choice of $\th$ has a physical motivation (see
\cit{AKK1},
\cit{AKK2}). Doing so we are going to seek the threshold value mentioned above
not for $\theta$ but for the inverse temperature $\beta$ (which is a parameter
more suitable for the physical interpretation of the model). Consider the
following strictly positive trace class operator on ${\cal H} = L^2 (I_\b )$
\be
\label{005}
S = ( -m\Delta_\beta +1)^{-1} ,
\ee
where $\Delta_\beta $ stands for the Laplace operator in $L^2 (I_\beta )$ and
$m>0$ is the reduced physical mass of the oscillator. For this operator, one can
define an even Gaussian measure $\gamma_S $ on ${\cal H}$ having $S$ as a
covariance operator. This measure is uniquely determined by its Fourier
transform
\be
\label{006}
\int_{{\cal H}} \exp(i(\vp , \om ))\gamma_S (d\om ) = \exp \left(
-\frac{1}{2} (S\vp , \vp )\right), \ \ \vp \in {\cal H}.
\ee
Let
$$
\Omega_\beta = \{ \om \in C([0,\beta ]\ra \R )\ \vert \ \om(0) = \om (\beta) \}.
$$
One may show that $\gamma_S (\Omega_\beta ) =1 $ and the measure $\gamma_S$
corresponds to the oscillator bridge process of length $\beta$ \cit{Simon}. The
starting element $\nu_0 $ of the sequence $\{\nu_n \}$  is intended to describe
the anharmonic oscillator and will be obtained as a perturbation of $\gamma_S$
in a way we are going to present below. For $x\in\R$, we set
\be
\label{007}
V(x) = a_1 x^2 + a_2 x^{4}, \ \ a_1 \in \R , \ \ a_2 >0 .
\ee
Then we define  a measure $\nu_0 = \nu \in {\cal M} ({\cal H})$ in such a way
that for $\om \in \Omega_\b $
\be
\label{008}
\nu (d\om ) = \frac{1}{Y } \exp\left(\int_{I_\beta} V(\om (\tau ))d\tau
\right)\gamma_S (d\om ), \ \ \om \in \Omega_\beta .
\ee
where $Y>0$ is the normalization constant (for more details see e.g. \cit{AHK}
and \cit{GK}). Then $\nu (\Omega_\beta ) = 1$ and the integrals
\beq
\label{009}
F_{2l} (\tau_1 , \dots , \tau_{2l } ) & \sta &
\int_{{\cal H}}\om (\tau_1 ) \dots \om (\tau_{2l})\nu (d\om)  \nonumber \\
& = &
\int_{\Omega_\beta}\om (\tau_1 ) \dots  \om (\tau_{2l})\nu (d\om ),
\ l\in \N ,
\eeq
are continuous as functions of $(\tau_1 , \dots , \tau_{2l }) \in
I_\beta^{2l} $. They are invariant with respect to the shifts
\be
\label{010}
F_{2l} (\tau_1 , \dots , \tau_{2l } )=
 F_{2l} (\tau_1 +\vartheta , \dots , \tau_{2l }+\vartheta ),  \ \
\vartheta \in I_\beta ,
\ee
where addition is modulo $\beta$. Set $U_2 (\tau_1 , \tau_2 ) \ \sta
\  F_2 (\tau_1 , \tau_2 ) $ and
\beq
\label{011}
&  & U_4 (\tau_1 , \dots , \tau_4 ) \ \sta
\ F_4 (\tau_1 , \dots , \tau_4 )
 -  F_2 (\tau_1 , \tau_2 )
F_2 (\tau_3 , \tau_4 ) \nonumber \\
&  - &   F_2 (\tau_1 , \tau_3 ) F_2 (\tau_2 ,
 \tau_4 ) -  F_2 (\tau_1 , \tau_4 )  F_2 (\tau_2 , \tau_3 ).
\eeq
Then (\ref{010}) implies that the following integrals do not depend
on $\tau$
\be
\label{012}
\hat{U} (0) \ \sta \ \int_{I_\beta} U_2 (\tau , \tau' ) d\tau' ,
\ee
\be
\label{013}
X \ \sta \ \int_{I_\beta^2} U_4 (\tau, \tau , \tau' ,
\tau'' ) d\tau' d\tau'' .
\ee
For $\delta <1/2$, we introduce
\be
\label{014}
\ve = \frac{1-2\delta}{4}, \ \ u(\delta ) \ =  \frac{2^\delta - 2^{-\ve }}
{2^\delta - 1},
\ee
\be
\label{015}
\kappa (u) = \frac{2^{-\delta }}{1 - (1 - 2^{-\delta })u} , \ \  u \in (0, (1-
  2^{-\delta })^{-1}) ,
\ee
\be
\label{016}
w(\delta) = 2^{2(1-\ve )-\delta }u(\delta)\frac{1-2^{-\ve}}{2^\delta -1 }.
\ee
Then for $u<u(\delta)$
\be
\label{017}
\kappa(u) < \kappa (u(\delta )) = 2^\ve .
\ee
It is not difficult to show that the integrals (\ref{009}) depend continuously
on $\beta$, which implies via (\ref{012}) that $\hat{U} (0)$ is a continuous
function of $\beta$. Thus we define $\beta^{\pm}_0 $ as the smallest values of
$\beta$ obeying the equations
\beq
\label{018}
\hU (0) & = & 1,  \ \ \ {\rm for} \ \ \  \b = \b_0^{-}; \nonumber\\
\hU (0) & = & u(\delta),  \ \ \ {\rm for} \  \ \     \b = \b_0^{+}.
\eeq
\begin{Df}
\label{1df}
Given $\delta \in (0,1/2)$, the set ${\cal M}_\delta $ consists of measures
having the form (\ref{008}) such that:
\vskip.1cm
\begin{tabular}{ll}
{\rm (i) } & $\beta^{\pm}_{0}$ defined by (\ref{018}) exist; \\ {\rm (ii) } &
for all $\beta$ from the closed interval with the endpoints $\beta^{\pm}_0 $,
one has
\end{tabular}
\vskip.1cm
\be
\label{019}
\vert X \vert < w(\delta).
\ee
\end{Df}
Below we discuss the question whether ${\cal M}_\delta $ is actually nonempty.
Now we are in a position to state our main result. To this end we define the
following rank-one operator on $L^2 (I_\beta )$:
\be
\label{020}
(B_\b \vp)(\tau) \ =  \frac{1}{\beta}\int_{I_\beta}\vp (\t )d\t .
\ee
\begin{Tm}
\label{1tm}
For every $\nu \in {\cal M}_\delta $, there exists $\beta_* $ such that for
$\beta = \beta_* $, the sequence of measures $\{\nu_n \ \vert \ \nu_0 = \nu \}$
defined by (\ref{002}), (\ref{003}) converges weakly to the zero mean Gaussian
measure which has $B_{\b_* } $ as covariance operator. For $\beta < \beta_*$,
this  sequence converges weakly to the measure degenerate at zero.
\end{Tm}
The idea of the proof may be outlined as follows. For every measure $\nu_n $,
we define its Laplace transform
\be
\label{021}
F_n (\xi ) = \int_{{\cal H}}\exp[(\xi,\om)]\nu_n (d\om ),
\  \ \xi \in {\cal H}.
\ee
Since $\nu_n $ possesses the property (\ref{001}), $F_n $ is defined on the
whole space ${\cal H}$ and one can introduce a topological vector space
containing such functions. Then we construct the operator which maps $F_{n-1 } $
into $F_n $. It appears that this operator is a nonlinear version of the
generalized $S$--transform known in White Noise Analysis (see e.g. \cit{BeKo}
and
\cit{HKPS}), thus it is called here a nonlinear $S$--transform (NLST). The
stated weak convergence of $\{ \nu_n \}$ is then obtained by showing the
conditional compactness of this sequence and by proving the pontwise convergence
of the sequence $\{F_n \}$ to the Laplace transform of the corresponding limit
measure.

\section{Nonlinear $S$--Transform}

Let the space ${\cal H}$ introduced above be embedded densely into a real
Hilbert space ${\cal H}_{-}$ with inner product $( . \ , \ .)_{-}$ and norm
$\vert \ . \ \vert_{-}$. Assume also that the embedding operator belongs to the
Hilbert--Schmidt class. The space ${\cal H}_{-}$ is chosen in such a way that
$F_0 $, defined by (\ref{021}) with $\nu_0 \in {\cal M}_\delta $, may be
continuously extended to ${\cal H}_{-}$. This possibility follows from Theorem
1.7 of \cit{BeKo} p.109. Below we use some methods of the theory of analytic
functions of a single complex variable. To this end we introduce the
complexifications  ${\cal H}^{(c)}$, ${\cal H}^{(c)}_{-}$  of the above spaces.
For $\xi$, $\eta \in {\cal H}$, we write $\xi+i\eta = \zeta \in {\cal H}^{(c)}$,
$\vert \z \vert^2 = \vert \xi \vert^2 + \vert \eta \vert^2 $, and $A\zeta = A\xi
+ i A\eta $, $(\zeta , \zeta' ) = (\xi, \xi' ) - (\eta , \eta' ) + i (\xi, \eta'
) + i (\xi' , \eta )$, where $A: {\cal H} \ra {\cal H}$ is a linear operator.

For an entire function $F: {\cal H}_{-}^{(c)}
\ra \C$ and some $\alpha >0$, we set
\beq
\label{022}
\|F\|_\a & = & \sup\left\{ \vert F(\z )\vert\exp\left(-\half \a \vert \z \vert_{-}
^2 \right) \ \vert \ \z \in {\cal H}_{-}^{(c)} \right\} ,\\
\label{023}
{\cal F}_\a & = & \{F\in{\cal F} \vert \|F\|_{\a'} <\infty, \forall
\a' > \a \} , \ \  \alpha \geq 0 .
\eeq
The latter set endowed with the topology generated by the family $\{ \| \ . \
\|_{\a' } , \a' > \a \}$ becomes a Fr{\'e}chet space.
Now let us consider the transformation (\ref{021}) with $\z = \xi + i \eta $,
 instead of real $\xi$, and $\nu_0 \in {\cal M}_\delta $. It defines $F_0 $ as a
function on ${\cal H}^{(c)}$, which can be extended to ${\cal H}_{-}^{(c)}$. One
can easily show that this extension (which is also written as $F_0 $) belongs to
${\cal F}_0$.

Let $O: {\cal H} \ra {\cal H}_{-}$ stand for the embedding operator and let $O^*
: {\cal H}_{-} \ra {\cal H}$ be its adjoint. Then
\be
\label{024}
T \ \sta \ OO^* : {\cal H}_{-} \ra {\cal H}_{-} ,
\ee
is a strictly positive trace class operator (since $O$ belongs to the
Hilbert--Schmidt class, more details can be found e.g. in \cit{BeKo} pp.3-13).
$T$ defines on ${\cal H}_{-}$ a zero mean Gaussian measure $\gamma_T$ having
such $T$ as covariance operator. Then $({\cal H}_{-} , \BC ({\cal H}_{-} ),
\gamma_T )$ is {\it the Gaussian space associated with $({\cal H}, \vert \ . \
\vert )$} ( see
\cit{HKPS}, p.3).

For positive $\s $, $\theta $ and for appropriate $F$, we define
\be
\label{0.5}
F\mapsto \GG (F) , \ \ \GG (F)(\z) = \int_{{\cal H}_{-}}
F^2 (\s \z + \theta \om  ) \gamma_T (d\om), \ \ \z \in {\cal H}_{-}^{(c)},
\ee
which can be expressed in terms of the generalized $S$--transform
${\bf S}_{\s , \theta}$ used in White
Noise Analysis
$$
\GG (F) = {\bf S}_{\s , \theta }(F^2 ).
$$
(see \cit{HKPS}, p.13, the operator ${\cal S}$ introduced in \cit{HKPS} is in
our present notations ${\bf S}_{1,1}$). To define $\GG$ as an operator on the
spaces ${\cal F}_\a $ we will need some technical background, which we are going
to provide. By means of Theorem 1.6
\cit{BeKo}, p.106 one can prove the following version of Fernique's theorem.
\begin{Pn}
\label{1.1pn}
Let $T$ and $\gamma_T$ be as above and let $A$ be a symmetric bounded linear
operator on ${\cal H}_{-}$ such that
\be
\label{0.6}
\TR (TA) <1 .
\ee
Then the expression
\be
\label{0.7}
\exp\left(\half (A\om , \om )_{-} \right)\gamma_T (d\om) =  K_D \g_D (d\om ),
\ee
defines on ${\cal H}_{-}  $ an even Gaussian measure $\g_D $ with
the covariance operator
\be
\label{0.8}
D= (1-TA)^{-1}T,
\ee
where
$$
K_D = \int_{{\cal H}_{-}}\exp\left(\half (A\om , \om )_{-} \right)\gamma_T
(d\om ) = \left[{\rm det}(1-TA)\right]^{-1/2}.
$$
\end{Pn}
Let $A$ be as above and $G$ be such that the function
\be
\label{0.9}
F(\z ) = \exp\left(\half (A\z , \z )_{-} \right) G(\z),
\ee
is in the domain of $\GG$, which is described below.
\begin{Lm}
\label{1.1lm}
Let
\be
\label{0.10}
\TR (TA) < \frac{1}{2\theta^2}.
\ee
Then for $F$ given by (\ref{0.9}):
\begin{eqnarray}
\label{0.11}
\GG (F)(\z ) & = & K_C \exp \left(\s^{2} (A\z ,\z )_{-}
+2\s^{2}\th^{2}(A\z ,CA\z )_{-}\right) \nonumber\\
       &  &{\bf G}_{C}^{\sigma , \theta} (G)((1+2\th^{2} CA)\z ),
\end{eqnarray}
where
\be
\label{0.12}
C=  (1-2\th^{2} TA)^{-1}T,
\ee
\be
\label{999}
K_C = [\det (1-2\th^{2} TA)]^{-1/2}.
\ee
\end{Lm}
{\bf Proof}. Inserting $F$ as given by (\ref{0.9}) into (\ref{0.5}) and
applying Proposition \ref{1.1pn} we obtain
\begin{eqnarray}
\label{0.13}
\GG (F)(\z ) & = & \exp \left( \s^{2} (A\z ,\z )_{-} \right) K_C \nonumber \\
 &  & \int_{\HH} G^{2}(\s \z + \th \om ) \exp \left( 2\s \th (A\z ,\om )_{-}
\right) \g_{C}(d\om ),
\end{eqnarray}
with $C$ given by (\ref{0.12}). The next step is to shift $\g_C $ (see e.g.
\cit{BeKo}, p.151) in order to include $ \exp (.) $ under the integral in
(\ref{0.13}) into the shifted measure. Let $y \in {\HH}$ be chosen such that $
C^{-1}y \in {\HH}$. Then we set
$$
\gamma_{C,y} (d\om ) =
\exp \left( - (C^{-1}y,\om )_{-} -
\half (C^{-1}y,y)_{-}\right)\gamma_C (d\om ).
$$
The shifted measure has the property
\be
\label{1}
\int \Psi (\om )\g_{C,y} (d\om ) = \int \Psi (\om - y)\g_{C} (d\om ),
\ee
that yields:
\beq
&  & \GG (F)(\z )  =  \exp \left(\s^2 (A\z ,\z )_{-}\right) K_C \nonumber\\
&  & \int_{\HH} G^2 (\s \z + \th \om ) \exp \{ 2\s \th (A\z ,\om )_{-}
+ (C^{-1} y,\om )_{-} + \half (C^{-1}y,y)_{-} \}\g_{C,y}(d\om ). \nonumber
\end{eqnarray}
Setting here
\be
\label{0.14}
y = -2\s \th CA\z ,
\ee
one arrives at
\beq
\label{0.15}
\GG (F)(\z ) & = & \exp\left(\s^2 (A\z ,\z )_{-} +
2\s^2 \th^2 (A\z ,CA\z )_{-}\right)
 \nonumber\\
&  & K_C \int_{\HH} G^{2}( \s \z + \th \om )\g_{C,y}(d\om ).
\eeq
Then by means of (\ref{1})
\beq
\GG (F)(\z ) & = & \exp\left(\s^2 (A\z ,\z )_{-} +
 2\s^2 \th^2 (A\z ,CA\z )_{-}\right)
\nonumber\\
&  & K_C \int_{\HH} G^2 (\s \z + \th ( \om + 2\s \th CA\z ))
\g_C (d\om),\nonumber
\eeq
which gives (\ref{0.11}).
\kasten
Now we may define  $\GG$ as a holomorphic map on appropriate spaces of entire
functions. Let $\|T\|$ stand for the operator norm of $T$ as an operator in
${\cal H}_{-}$ and suppose $\s < 1/\sqrt 2 $. Let also
\be
\label{0.16}
\a_{\rm max} \ \sta \ \min \left\{ \frac{1-2\s^2}{2\th^2 \|T\|} \ ; \
\frac{1}{ 2\th^2 \TR T }\right\}.
\ee
\begin{Lm}
\label{1.2lm}
For $\a < \a_{\rm max}$, the map $\GG$ (\ref{0.5}) is
holomorphic, i.e. Fr\'echet
differentiable, on ${\cal F}_{\a}$.
\end{Lm}
{\bf Proof}. For ${\a}' \in (\a, \a_{\rm max})$
and two
functions  $F_1 , F_2 \in {\FC}_\a $, we estimate
$$
 \Phi (\z ) \ \sta \
 \int_{\HH}F_1 (\s \z + \th \om )F_2 (\s \z + \th \om )\g_{T} (d\om ).
$$
The definitions (\ref{022}), (\ref{023}) yield
$$
\vert F_i (\z ) \vert \leq \|F_i \|_{{\a}'} \exp\left(\half {\a}'
 \vert \z \vert^2_{-} \right) , \ \z = \xi + i \eta .
$$
Therefore,
\beq
&  & \vert \Phi (\z ) \vert  \leq  \int_{\HH}
\vert F_1 (\s \z + \th \om ) \vert \
\vert F_2 (\s \z + \th \om ) \vert
\g_{T}(d\om )
   \leq  \|F_1 \|_{{\a}'} \|F_2 \|_{{\a}'} \nonumber \\
 &  & \int_{\HH} \exp \left\{ {\a}'(\s^2 \vert \xi \vert^2_{-} +
2 \s \th (\xi , \om )_{-}
+\s^2 \vert \eta \vert^2_{-} + \th^2 \vert \om \vert^2_{-} ) \right\}
\g_{T}(d\om ).\nonumber
\eeq
Integrating over $\HH$, one obtains
\beq
 \vert \Phi (\z ) \vert
  & \leq & \|F_1  \|_{{a}'} \|F_2 \|_{{\a}'}
K \exp ({\a}' \s^2 \vert \eta \vert^2_{-} +
{\a}' \s^2 ((1-2{\a}' \th^2 T)^{-1}\xi ,\xi )_{-})  \nonumber \\
  & \leq & K \|F_1 \|_{{\a}'} \|F_2 \|_{{\a}'} \exp \left\{
\frac{{\a}' \s^2 }{1- 2{\a}' \th^2 \|T\| } \vert \z \vert^2_{-} \right\},
\nonumber
\eeq
where
$$
K= [ \det (1-2{\a}' \th^2 T)]^{-1/2 } \leq [ 1- 2{\a}' \th^2
{\rm trace}T]^{-1/2}.
$$
This yields for ${\a}' < {\a}_{{\rm max}}$,
\be
\label{2}
\|\int_{\HH} F_1 (\s \z + \th \om ) F_2 (\s \z + \th \om ) \g_{T}(d\om )
\|_{{\a}'} \leq K \|F_1 \|_{{\a}'} \|F_2 \|_{{\a}'}.
\ee
Putting here $F_1 =F_2 = F $, one gets
\be
\label{3}
\|\GG (F)\|_{{\a}'} \leq K \|F\|^2_{{\a}'}.
\ee
For $F\in {\FC}_{\a}$, $G\in {\FC}_{\a}$, we have
\beq
\label{4}
&  & \GG (F+G) (\z ) - \GG (F) (\z ) = \\ \nonumber
 &  &2\int_{\HH} F(\s \z + \th \om )G(\s \z +\th \om ) \g_{T}(d\om ) +
\GG (G) (\z).
\eeq
The latter term can be estimated by (\ref{3}). Therefore, the Fr\'echet
derivative of $\GG$ at $F\in {\FC}_{\a}$ is the following linear operator on
${\FC}_{\a}$
\be
\label{5}
\{(\GG (F))'G \} (\z ) = 2 \int_{\HH} F(\s \z + \th \om ) G(\s \z +
\th \om ) \g_{T}(d\om ).
\ee
That $(\GG (F))'$ is bounded can be shown by taking $F_1 = F$, $F_2
= G$ in (\ref{2}).
\kasten
\begin{Rk}
\label{99rk}
We can extend $\GG$ preserving all its properties stated above to
${\FC}_{\a_{\rm max}}$  provided $ {\a}_{\rm max} $ is strictly less than
$(2\th^2 {\rm trace}T)^{-1}$, or to other spaces, e.g. the space consisting of
the functions given by (\ref{0.9}) with $G \in {\FC}_0 $ and such $A$ that the
constant (\ref{999}) exists.
\end{Rk}
\begin{Df}
\label{99df}
We denote by ${\FC}_S$  the maximal space  on which the map $\GG$ can be defined
maintaining the properties stated by Lemma \ref{1.2lm}.
\end{Df}
 The problem we
are concerned now is the existence of fixed points of the operator $\GG$ and
their stability. Among the possible fixed points one may distinguish a family,
which can be described in an evident way. It consists of Gaussian fixed points
and is defined as follows. Let $ \QQ = \{ Q \subset \N \ \vert \
\vert Q \vert < \infty \}$, where $\vert Q \vert $ means
the cardinality. Let also $\{\om_j , j\in \N \}$ be the base of $\HH$ formed by
the eigenvectors of the trace class operator $T$ (see (\ref{024})). For a
nonempty $Q\in
\QQ$, let $\HQ$ be the linear span of $\{\om_j , j\in Q \}$. Let $W_Q $ be
the orthogonal projector onto $\HQ$ and $T_Q$ be the restriction of $T$ onto
$\HQ$. For a nonempty $Q\in \QQ$, we put
\be
\label{0.18}
A_Q = \frac{1-2\s^2 }{2\th^2 } T^{-1}_{Q} W_Q,
\ee
and $A_Q = 0 $ for $Q= \emptyset$. Note that $T^{-1}_{Q}$ exists for each
nonempty $Q\in \QQ$ --  a fact, which follows from the strict positivity of $T$.
\begin{Pn}
\label{10.1pn}
The elements of the family
\be
\label{1.18}
{\AC} = \{ F_Q (\z ) = (2\s^2 )^{ \vert Q \vert \over 2}
\exp \left(\half (A_Q \z ,\z )_{-}\right)
 \ \vert \ Q\in \QQ \}
\ee
are fixed points of $\GG$.
\end{Pn}
{\bf Proof}. We have $F_{\emptyset} \equiv 1$, which is a fixed point. For a
nonempty $Q\in \QQ$, using (\ref{0.11}) with $G
\equiv 1$ and imposing corresponding conditions on $A$, we obtain (\ref{0.18}).
\kasten


The stability of the above fixed points can be studied by means of the
eigenvalues of the Fr\'echet derivative of $\GG$ at the points $F_Q$. Thus let
us consider the eigenvalue problem
\be
\label{1.20}
(\GG (F_Q))'G = \l G.
\ee
Let $G(\z ) = F_Q (\z ) P(\z )$ with $P$ being an even polynomial -- we restrict
ourselves to this case since only even measures, and hence even functions $F_n$,
are to be considered here. Inserting such $G$ into (\ref{5}) we obtain
\beq
2(2\s^2 )^{\vert Q \vert } \int_{\HH}
\exp \left[ (A_Q (\s \z + \th \om ), \s \z + \th \om )_{-} \right] P(\s
\z + \th \om ) \g_{T}(d\om)  \nonumber\\
=\l (2\s )^{\vert Q \vert
\oover 2 } \exp \left(\half (A_Q \z , \z )_{-} \right) P(\z ).  \nonumber
\eeq
By means of Lemma \ref{1.1lm} the latter equality can be transformed into the
equation:
\be
\label{1.21}
2\int_{\HH} P(\s L_Q \z + \th \om )\g_{C_Q }(d\om ) = \l P(\z ),
\ee
where
\be
\label{1.22} \s L_Q = \s ( 1 - W_Q ) + \frac{1}{2 \s } W_Q,
\ee
\be
\label{1.23}
C_Q = (1- W_Q ) T  + \frac{1}{2\s^2 } W_Q T.
\ee
Let $\deg P = 2k$, then one may put
\be
\label{1.230} P=P_k +  \delta P , \hskip0.5cm \deg P_k = 2k, \ P_k (\a
\z ) = \a^{2 k} P_k (\z), \ \deg \delta P < 2k,
\ee
and obtain from (\ref{1.21})
\be
\label{35.35} 2P_k (\s L_Q \z ) + {P}' (\z) = \l P_k (\z) + \l \delta
P(\z),
\ee
with $P'$ such that $\deg {P}' <2 k $.
Let ${\cal H}^{(1)} =
\{ \z = \xi + i \eta \ \vert \ \xi \in \HQ \ {\rm or} \ \eta \in \HQ
 \}$ and ${\cal H}^{(2)} = \{ \z = \xi + i \eta \ \vert \ \xi \in {\HH}
\setminus \HQ \ {\rm or} \ \eta \in {\HH} \setminus \HQ \}$.
We set
\beq
\label{1.231}
{\PC}& =& \{P \ \vert \ P_k (\z) = 0, \ \forall \z \in {\cal H}^{(1)} \},\\
\nonumber {\PP}& = &
\{ P \ \vert \ P_k (\z) = 0, \ \forall \z \in {\cal H}^{(2)} \},
\eeq
and obtain
\beq
\label{1.232}
P_k (\s L_Q \z ) &=& P_k (\s \z ) , \hskip0.2cm \ {\rm for}  \ \ \
 P \in \PC, \\
\nonumber P_k (\s L_Q \z  ) &=& P_k ( {1 \over 2 \s } \z ), \ {\rm for} \
 \ P\in \PP.
\eeq
Inserting such $P$ into  (\ref{35.35}) and taking into account the
homogeneity of $P_k$ (\ref{1.230}), as well as (\ref{1.232}), we
arrive at
\be
\label{1.24}
\l_k = 2 \s^{2k }; \ \ P\in \PC, \ 2 k = \deg P, \ k\in
\Z_{+},
\ee
\be
\label{1.25}
\tilde{\l}_k = \frac{2}{(2\s )^{2k }}; \ \ P\in \PP, \ 2 k= \deg P, \ k\in \N .
\ee
We recall that a fixed point $F_Q$ is stable if all the eigenvalues of $(\GG
(F_Q))'$ do not exceed one. $F_Q$ is $l$-unstable  $(l\in \N)$ if there exist
$l$ eigenvalues which are greater than one. For every unstable fixed point,
there exist "directions" such that arbitrary small perturbations $\d F$ of $F$
in these directions produce the effect that the sequence $\{ F_n , \ n\in \Z_{+}
\}$, defined recursively
\be
\label{1.26}
F_n = \GG (F_{n-1}) , \ \ \  F_0 = F_Q + \d F,
\ee
 does not approach $F_Q$ as $n \ra \infty$.
Such directions correspond to the eigenvalues $\l$ of $(\GG (F_Q))'$ which are
greater than one. In our case, $\l_0 = 2 >1$, thus the sequence $\{ F_n , \ n\in
\Z_{+} \}$ with $F_0 (\z )  = K_0 \exp (\half (A_Q \z ,\z )_{-} )$, $ K_0 \neq
(2\s^2 )^{\vert Q
\vert / 2}$ does not converge to the fixed point $F_Q$. To exclude this trivial
instability and having in mind that each $F_n $ ought to be the Laplace
transform of a probability measure, we redefine the NLST by putting instead of
(\ref{0.5})
\be
\label{1.27}
\GH (F)(\z ) = \frac{ \int_{\HH} F^2 (\s \z +\th \om )\g_T (d\om )}
{\int_{\HH} F^2 (\th \om )\g_T (d\om )}.
\ee
Accordingly, the family $\AC$ (\ref{1.18}) transforms into
\be
\label{1.28}
\hat{\AC} = \{ F_Q (\z )= \exp \left(\half (A_Q \z , \z )_{-}\right)
\ \vert \ Q\in \QQ \}.
\ee
Let us set
\be
\label{1.30}
\hat{\FC}  = \{ F\in {\FC}_S \ \vert \ F(\z ) = F(- \z ),
  \ F(0) = 1\}.
\ee
Then $\hat{\AC} \subset \hat{\FC}$, and $\GH$ maps $\hat{\FC}$ into itself. The
stability of each $F_Q \in \hat{\AC}$ under the perturbations within $\hat{\FC}$
is defined by the eigenvalues (\ref{1.24}), (\ref{1.25}) but now with $k\in
\N$, i.e. $k\neq 0$. Since $\s < 1/\sqrt 2$, all $\l_k <1$, $k\in \N$. At the same time
 we have $\tilde{\l}_1
>1 $, which means that $F_Q$ is unstable whenever ${\PP}$ is nonempty. The only
possibility for ${\PP}$ to be empty is $A_Q =0$, which occurs for $Q =
\emptyset$. Thus the only stable fixed point in $\hat{\AC}$ is $F_{\emptyset}
\equiv 1$. In what follows, the sequence $\{F_n \in \hat{\FC} , \ n \in \Z_{+}
\}$ defined recursively
\be
\label{1.33}
F_n = \GH (F_{n-1}), \ n \in \N ,
\ee
with $\s \in (0, 1/\sqrt 2)$, may converge as $n \ra \infty$ to the stable fixed
point $ F_{\emptyset} \equiv 1$, or to some unstable fixed point $F_Q \in
\hat{\AC}$.
 An example can be constructed by choosing
$F_0 (z) = \exp (\half (A_0 \z,\z)_{-})$ with a suitable operator $A_0$. In this
case we have $F_n (\z ) = \exp (\half (A_n \z ,\z )_{-})$, and the sequence
$\{A_n
\}$ may be described explicitly. We do not consider this simple example (which
describes the harmonic oscillators) and will be concerned with the case of a
 non-Gaussian $F_0$, where all $F_n $ are also non--Gaussian and describe the
anharmonic oscillators. For every $\nu_n$, the Laplace transform (\ref{021})
belongs to $\hat{\cal F}$ and one can easily prove the following statement.
\begin{Pn}
\label{N1pn}
Let the sequence of measures $\{\nu_n \} $ be defined by (\ref{002}),
(\ref{003}) with the operator ${\bf N}^{\theta, \delta }_2 $. Then the sequence
of the Laplace transforms \{$F_n \}$ defined by (\ref{021}) obeys the recursion
(\ref{1.33}) with $\s = 2^{ - (1+\delta)/2}$.
\end{Pn}
It should be stressed  here that the explicit form of the operator $T$, as well
as of the space $\HH$, has no relevance -- for any pair $({\cal H}_{-}, T)$
obeying the above conditions, Proposition \ref{N1pn} holds. An example, which
corresponds to our choice ${\cal H} = L^2 (I_\beta )$, is ${\cal H}_{-} =
W^{-1}_2 (I_\beta ) $, where the latter is the Sobolev space of square
integrable generalized functions of index $-1$.






\section{Quantum Hierarchical Model}
As it was pointed out above, the non--Gaussian sequence of functions generated
by the map (\ref{1.27}) with the starting element $F_0$ being the Laplace
transform of the measure (\ref{008}) describes the Euclidean Gibbs states of the
model of hierarchically interacting quantum anharmonic oscillators. It is
characterized by a number of parameters. A part of them describes the individual
properties of the oscillators, they define the starting element $F_0$ (see
(\ref{007}), (\ref{008})). The other parameters describe the interaction between
the oscillators and define in turn the properties of the map $\GH$. This
interaction grows proportionally to the inverse temperature $\b$. When the
inverse temperature reaches its critical value $ \b_{*}>0 $, the oscillations
become strongly dependent, which is known as the critical point behaviour of the
model. Having in mind Proposition \ref{N1pn} we fix $\s$ and $\theta$
\be
\label{3.6}
\th^2 = (2^\delta -1)2^{-1 -\delta} =   \half - \s^2 , \ \
\s = 2^{ - (1+\d)/2}, \  \    \d \in (0,1/2),
\ee
and set for this choice, $\GB \ \sta \ \GH$. Then the sequence $\{ F_n \}$ is
given  as follows
\beq
\label{3.8}
F_n (\z ) & = & \GB (F_{n-1})(\z) \\ \nonumber
& = & \frac{\int_{\Hm} F_{n-1}^2 (2^{-(1+\d)/2}\z + 2^{-(1+\d)/2}
\sqrt{2^{\d} -1} \om ) \g_{T}(d\om )}{\int_{\Hm}F^2 (2^{-(1+\d)/2}
\sqrt{ 2^{\d} -1 } \om ) \g_{T}(d\om )},  \ \z \in {\cal H}_{-}^{(c)} , \\
  F_0 (\vp) & = & \int_{{\cal H}}\exp((\vp,\oom))\nu_0 (d\oom),
 \ \vp \in {\cal H}^{(c)} ,  \ \ \nu_0 \in {\cal M}_\delta . \nonumber
\eeq
As an even entire function $F_n$ can be written in the form of
the expansion
\be
\label{100.1}
F_n = \sum_{l=0}^{\infty} \frac{1}{(2l)!}F^{(n)}_{2l} ,
\ee
which is absolutely convergent on the whole ${\cal H}^{(c)}_{-}$.
Since $F_n (0) =1$, the function
\be
\label{47}
U^{(n)}  \ \sta \  \log F_n ,
\ee
is holomorphic in a neighborhood of zero in ${\cal H}^{(c)}_{-} $,
where the expansion
\be
\label{48}
U^{(n)}  = \sum_{l=1}^{\infty}{1 \oover (2l)!} U^{(n)}_{2l}
\ee
converges absolutely. Both $F^{(n)}_{2l}$ and  $U^{(n)}_{2l}$  are homogeneous
$2l$-linear continuous functions on the corresponding tensor products of ${\cal
H}^{(c)}_{-} $. The embedding operator $O: {\cal H} \rightarrow {\Hm}$ belongs
to the  Hilbert-Schmidt class, thus the kernel theorem \cit{BeKo} implies that
for $\vp
\in {\cal H}^{(c)}$, these functions possess
 the integral representations:
\beq
\label{49}
F^{(n)}_{2l}  =  \int_{I_\b^{2l }} F^{(n)}_{2l}
(\t_1 , \t_2 , \dots , \t_{2l})
\vp (\t_1 ) \dots \vp( \t_{2l})d\t_1 d\t_2 \dots d\t_{2l},
\eeq
\be
\label{49.1}
U^{(n)}_{2l}
 =  \int_{I_\b^{2l}}
U^{(n)}_{2l} (\t_1 , \t_2 , \dots, \t_{2l}) \vp (\t_1 )
\dots \vp (\t_{2l})
d\t_1 d\t_2 \dots d\t_{2l}.
\ee
A particular case of (\ref{49}) has appeared as (\ref{009}). Just as in that
particular case the above kernels belong to $C( I_{\beta}^{2l})$ and are
invariant with respect to the shifts (\ref{010}). Therefore, as in (\ref{012}),
(\ref{013}) the following integrals do not depend on $\t $:
\be
\label{5.9}
\hat{U}_{n}(0) \ \sta \ \int_{I_\b}U^{(n)}_2 (\t ,\t' )d\t',
\ee
\be
\label{5.10}
X_n \ \sta \ \int_{I_\b}U^{(n)}_4 (\t , \t , \t' , \t'' )d\t' d\t''.
\ee
Now let us describe the fixed points of $\GB$ which are the Laplace transforms
of the limiting Gaussian measures mentioned in Theorem \ref{1tm}. For the choice
of $\th$ and $\s$ (\ref{3.6}), the operator $A_Q$ is $T_Q^{-1}W_Q $ (see
(\ref{0.18})). The stable fixed point $F_\emptyset $ corresponds to $A_Q = 0$.
It is the Laplace transform of the $\delta$-measure concentrated at zero, which
is the limiting Gaussian measure for $\b < \b_{*}$. In the case $\b
= \b_*$ one should have
$$
(A_Q \vp , \vp)_{-} = (B_{\b_*} \vp , \vp),
$$
where the rank--one operator $B_\b $ was introduced in (\ref{020}). This means
that $Q = \{1\}$ and $W_Q $ is the projector on the one--dimensional subspace
${\cal H}_Q \subset L^2 (I_\b ) \subset W_2^{-1} (I_\b )$ consisting of constant
functions and spanned by $\om_1 \equiv 1/\sqrt{\b} $. The subspace ${\cal H}_Q$
corresponds to the eigenvalue $s_1 = 1$, thus $ \bb{\om_1}_{-} = \bb{\om_1} =
1$, $\BQ{T} = s_1
= 1$, and $\a_{\rm max} = 1$ (see Remark \ref{99rk}). Then
\beq
\label{5.101}
\exp\left({1\over 2}(A_{Q} \vp , \vp )_{-}\right) & = &
\exp\left({1\over 2} (\vp , \om_1 )^2 \right) \\
& = & \exp \left( {1\over 2 }
 (B_\b \vp , \vp )\right) \in \hat{\AC}. \nonumber
\eeq
Since $A_Q$ has rank one, the fixed point (\ref{5.101}) is one-dimensional. This
means in turn that the measure which has it as a Laplace transform is supported
on the one-dimensional subspace of $L^2 (I_\b )$ consisting of constants just
like in the classical (nonquantum) case (see
\cit{AKK4}). The classical
character of the critical point in the hierarchical quantum spin model
was shown in \cit{MoSch}.

In the sequel we use also the operator
description of the model under consideration. The starting measure
(\ref{008}) is connected with a
one-particle Hamiltonian. It is
the following operator
\be
\label{2.6}
H= -{1\over 2m} \left({d\over dx}\right)^2 + \frac{1}{2}x^2 +
V(x), \ \ x\in \R ,
\ee
densely defined in the Hilbert space $\Ho = L^2 (\R , dx)$, which describes a
quantum particle performing one-dimensional oscillations around the equilibrium
position $x=0$. Here $m$ is the same as in (\ref{005}) and the function $V$ is
given by (\ref{007}). Let $q$ denote the position operator which describes the
oscillator displacements from the equilibrium and acts in $\Ho$ as follows:
$(q\psi)(x) = x\psi (x)$. For all $\b>0$, the operators $ q^l \exp (-\b H), \
l\in \Z_{+}$ belong to the trace class. Let us consider a countable chain of
copies of the described oscillator, numbered by $j\in \N$. Introduce
\be
\label{10.1}
\Lan = \{ s \in \N \vert \  2^n (j-1) +1 \leq s \leq 2^n j \}, \ j \in \N ,
\ n \in \Z_{+} .
\ee
 Then
\be
\label{10.2}
\Lan = \bigcup_{s\in \Lambda_{l,j}} \Lal, \ l = 1,2, \dots , n.
\ee
For each $\Lan$, a Hamiltonian $H_{\Lan }$, which describes the particles
located in $\Lan$, is defined recursively on the base of (\ref{10.2}):
\be
\label{10.4}
H_{\Lan} = - \half (2^{\delta} -1) 2^{-n(1+\delta)} (\sum_{s\in \Lan} q_s )^2
+ \sum_{s\in \Lambda_{1,j}} H_{\La},
\ee
where $\delta$ was introduced in (\ref{002}), and the starting element
$H_{\Lambda_{0,j}}$ is the one--particle Hamiltonian given by (\ref{2.6}).
 Each $H_{\Lan}$ is an essentially self--adjoint operator in the
space $\HoL$, which is the tensor product of the ${\Hos}$ copies indexed by
$s\in
\Lan $. The Gibbs state of the particles located in $\Lan$ may be described by
means of temperature Green functions, which are defined as follows. For $0\leq
\tau_1 \leq \tau_2 \leq \dots \leq \tau_l $ , $l \in \N$, we set
\beq
\label{10.5}
&  & \Gamma^{\b , \Lan}_{A_1 , \dots , A_l } (\tau_1 , \dots , \tau_l ) =
 \frac{1}{Z_{\Lan}} \\
&  & \TR \left\{A_1 \exp(-(\tau_2 - \tau_1 )H_{\Lan}) \dots
A_l \exp( -(\b -\tau_l + \tau_1)
H_{\Lan}) \right\},\nonumber
\eeq
where
$$
Z_{\Lan} = \TR \left\{\exp( -\b H_{\Lan}) \right\}.
$$
$A_i , \  i=1, \dots , l$ are densely defined operators in $\HoL$, and such that
the trace in (\ref{10.5}) exists. Put
\be
\label{10.6}
A_i = A_{\Lan} = 2^{- \half n(1+\delta)}\sum_{s\in \Lan }q_s .
\ee
Note that $\Lan$ consists of $2^n $ points, which means
that the normalization of the above sum, due to the presence of the
factor $2^{- n \delta /2}$ with $\delta >0$, is abnormal
(comparing to that used in the case of the weak dependence described by
the central limit theorem). Therefore, if the dependence
between the displacements $q_s$ is weak, the sequence
$\{ A_{\Lan}, n\in \N \}$  ought to degenerate at zero.
The latter means that the
sequences of Green functions defined by (\ref{10.5}) with such $A_i$ converge
to zero for all values of their variables and for all $l\in \N$.
To simplify notations we set
\be
\label{10.7}
\Gamma^{(n)}_{2l} (\tau_1 , \dots , \tau_{2l})  =
\Gamma^{\b , \Lan }_{A_1 , \dots , A_{2l}} (\tau_1 , \dots , \tau_{2l}),
\ee
where the $A_i$ are given by (\ref{10.6}). The next assertion establishes the
relationship between the operator and functional integral approaches in quantum
statistical mechanics. It directly follows from \cit{AHK} and was used in our
works
\cit{AKK1}, \cit{AKK2} devoted to this model, to which
we refer for a more detailed description of the model and technical background.
\begin{Pn}
\label{0.0pn}
For all $\vp \in L^2 (I_\b )$, one has
$$
F_n (\vp) = \sum_{l=1}^{\infty} \frac{1}{(2l)!}
\int_{I_\b^{2l}} \Gamma_{2l}^{(n)}
(\tau_1 , \dots , \tau_{2l} ) \vp (\t_1 )  \dots
\vp (\t_{2l}) d\tau_1 \dots d\tau_{2l}.
$$
In other words, the functions introduced in (\ref{49}) coincide with those
defined by (\ref{10.5}) -- (\ref{10.7}).
\end{Pn}

At the end of this section let us discuss the question whether the set of
measures ${\cal M}_\delta $ introduced by Definition \ref{1df} is actually
nonempty. First we remark that the statement analogous to Theorem \ref{1tm}
which describes the classical hierarchical models was proved in \cit{Koz}, where
it was shown that the corresponding set of measures is nonempty if the
parameters of $V$ (\ref{007}) satisfy certain conditions. Second, each measure
of the type of (\ref{008}) converges weakly, when $m \ra +\infty $, to the
measure describing the classical oscillator -- an element of the mentioned
classical hierarchical model. This was proved in \cit{AKK4}. Therefore, the
family ${\cal M}_\delta $ is nonempty for an appropriate choice of $V$ and large
values of the reduced physical mass $m$. On the other hand, as it was proved in
\cit{AKK2}, there exists $m_* > 0$ such that, for $m<m_*$, one has $\hat{U}
(0)<1$ for all $\beta$. This means that no threshold $\b_* $ exists, and hence
no strong dependence like that described by Theorem \ref{1tm} may occur, in this
case. In physical terms this means that small values of $m$ make critical
phenomena impossible in such models.

\section{Proof of the Theorem}


Introduce
\be
\label{51}
{\KC} = \{ k \ \vert \ k={2\pi \over \b} \k , \ \k\in \Z \},
\ee
and
\be
\label{52}
\hU_n (k) = \int_{I_\b} U^{(n)}_2 (0,\t) \cos (k\t)d\t, \ \ k\in \KC .
\ee
Then the value of $\hU_n (k)$ for $k=0$ coincides with the one given by
(\ref{5.9}) and
\be
\label{529}
U_2^{(n)}(\t_1 , \t_2 ) = \frac{1}{\b} \sum_{k\in {\cal K}}\hU_n (k)
\cos [k(\t_1 - \t_2 )] .
\ee
We also set
\beq
\label{5.31}
Y_n & = & \int_{I_{_\b }^3} U^{(n)}_4 (\t_1 , \t_2 , \t_3 , \t_4 )d\t_2
d\t_3 d\t_4 \nonumber\\
& = & {1 \over \b } \int_{I_{\b}^4} U^{(n)}_4 (\t_1 , \t_2 , \t_3 , \t_4 )
d\t_1 d\t_2 d\t_3 d\t_4.
\eeq
The proof of our theorem relies upon the lemmas stated just below, which hold
for any choice of the initial element $\nu_0 \in {\cal M}_\delta $.
\begin{Lm}
\label{5.7lm}
For all $n\in \Z_{+}$ and $l\in \N$, the
following estimates hold true everywhere
\beq
\label{5.32}
&  & \int_{I_\b^2} U^{(n)}_4 (\t , \t , \t_1 , \t_2 )d\t_1 d\t_2 \nonumber\\
& \leq & \int_{I_\b^2} U^{(n)}_4 (\t,\t',\t_1 ,\t_2 )d\t_1 d\t_2 ;
\eeq
\be
\label{5.33}
(-1)^{l-1} U^{(n)}_{2l} (\t_1 , \t_2 , \dots , \t_{2l} ) \geq 0;
\ee
\be
\label{20.33}
F^{(n)}_{2l} (\t_1 , \t_2 , \dots , \t_{2l}) \geq 0  .
\ee
\end{Lm}
\begin{Lm}
\label{5.5lm}
There exists $\b_{*}$ such that for $\b = \b_{*}$, the sequences $\{ X_n , n\in
\Z_{+} \}$ , $\{ Y_n , n\in \Z_{+} \}$ converge to zero, and the sequence $\{
\hU_{n}(0) , n\in \Z_{+} \}$ converges to one, as $n\ra \infty$. For $\b < \b_{*}$, all these
sequences converge to zero.
\end{Lm}
\begin{Lm}
\label{5.8lm}
For every $\b>0$, $n\in \N$, and arbitrary nonzero $k\in{\KC}$,
\be
\label{20.331}
0\leq \hat{U}_n (k) \leq \frac{1}{mk^2 }\vert {\Lambda}_n \vert^{-\delta}
= \frac{2^{-n\delta}}{mk^2 }.
\ee
\end{Lm}
The latter statement has two evident and important corollaries. Set
\be
\label{20.332}
{\cal K}_N = \{ k \in {\cal K} \ \vert \ k=\frac{2\pi}{\b}\kappa , \
\vert \kappa \vert \leq N , \ N\in \N\}, \ \ {\cal K}_N^c =
{\cal K} \setminus {\cal K}_N .
\ee
\begin{Co}
\label{N1co}
For every $\b>0$,
$$
\sup_{n\in \Z_+ }\left\{ \sum_{k\in {\cal K}_N^c } \hat{U}_n (k) \right\}
 \longrightarrow 0 ,
\ \ \ N\ra +\infty ,
$$
hence the sequence $\{\nu_n \}$ is conditionally
compact ( see \cit{Pa}, p.154).
\end{Co}
\begin{Co}
\label{N2co}
For every $\b$,
$$
\sum_{k\in {\cal K}\setminus \{0\}} \hat{U}_n (k)
 \longrightarrow  0 , \ \ \ n\ra +\infty .
$$
\end{Co}
This fact follows from the positivity of $\delta$, which defines the abnormal
normalization corresponding to the critical point convergence. Therefore, the
general conclusion following from the above statement is that for $\b$ finite,
the one-dimensional (i.e. classical)
 fixed points characterize the critical point convergence in such models.

Let $\xi_1 \in L^2 (I_\b )$ be such that $\xi_1 (\t)  \equiv 1$.
\begin{Lm}
\label{5.9lm}
For every $n\in \N $, the following representation holds
\be
\label{5.34}
F_n (z\xi_1 )  = \prod_{j=1}^{\infty} (1+ c_j^{(n)} z^2 ), \ \ z \in \C ,
\ee
where
$$
c_j^{(n)} >0, \; \; \sum_j c_j^{(n)} < \infty , \; \; c_1^{(n)} \geq c_j^{(n)}.
$$
For $\beta \leq \beta_*$, there exists $c>0$ such that, for all $n\in \N$,
\be
\label{5.341}
c_1^{(n)} \leq c .
\ee
\end{Lm}
\begin{Co}
\label{5.10lm}
Let $c$ be as above. Then for every $n\in \N$, the following estimates hold
\beq
\label{5.35}
\bb{Y_n} &< & 6c \hU_n (0), \\
\label{5.351}
 \int_{I_\b^{2l}}\lv U^{(n)}_{2l} (\t_1 , \dots , \t_{2l} ) \rv d\t_1 \dots
d\t_{2l}
& < & { (2l-1)! \over 6} c^{l-2} \b \bb{Y_n},  \ l> 2 .
\eeq
\end{Co}
{\bf Proof}. From the representations (\ref{47})--(\ref{49.1}) and
 (\ref{5.34}), one has ( $l\geq 1$):
\be
\label{5.36}
\int_{I_\b^{2l}}U^{(n)}_{2l} (\t_1 , \dots , \t_{2l} ) d\t_1 \dots d\t_{2l} =
2(2l-1)!(-1)^{l-1}\sum_{j=1}^{\infty}\left(c_j^{(n)}\right)^l .
\ee
Taking into account (\ref{5.9}), (\ref{5.31}), the estimates (\ref{5.341}), and
$c_1^{(n)} \geq c_j^{(n)}$, one obtains
\beq
\label{5.3610}
\b \hat{U}_n (0) & = & 2\sum_j c_j^{(n)} > 2c_1^{(n)} , \\
\b \lv Y_n \rv & = & 12 \sum_j \left(c_j^{(n)}\right)^2 < 6c_1^{(n)}\b
\hat{U}_n (0) \leq 6c \b \hat{U}_n (0). \nonumber
\eeq
The latter yields (\ref{5.35}), the estimate (\ref{5.351}) follows from
(\ref{5.36}) and (\ref{5.33}).
\kasten
For a continuous complex valued function $\vp : I_{\b} \rightarrow \C$,
i.e. for $\vp \in C(I_{\b}) $, we set
$$
\|\vp\|_{C(I_{\b})} = \sup \{\vert \vp (\tau ) \vert \
\vert \ \tau \in I_{\b} \}.
$$
\begin{Co}
\label{5.12lm}
For every $\vp \in C(I_{\b})$ and $n \in \Z_{+}$, $F_{n}(z \vp )$ is
an entire function of $z \in \C$, which obeys the estimate
\be
\label{87}
\vert F_n (z \vp) \vert \leq \exp \left(\half \b \hat{U}_n (0) \vert z \vert^2
 \| \vp \|^{2}_{C(I_{\b})} \right) .
\ee
\end{Co}
{\bf Proof}. To estimate $F_n $ on $C(I_{\b})$  we exploit its representation
(\ref{100.1}). Having in mind that $C(I_{\b}) \subset {\cal H}^{(c)}$ we may use
the representation (\ref{49}), the positivity property (\ref{20.33}),
(\ref{5.34}), (\ref{5.3610}), and obtain
\beq
\label{899.1}
\vert F_n (z \vp) \vert & \leq & \sum_{l=0}^{\infty} \frac{1}{(2l)!}
 \lv z \rv^{2l} \|\vp\|^{2l}_{C(I_{\b})}
\int_{I^{2l}_{\b}} \vert F_{2l}^{(n)} (\tau_1 ,
\tau_2 , \dots , \tau_{2l} ) \vert d\tau_1 d\tau_2 \dots d\tau_{2l} \nonumber\\
& = & F_n (\lv z \rv \|\vp\|_{C(I_{\b})} \xi_1 ) =
\prod_{j=1}^{\infty} (1+ c_j^{(n)} \lv z \rv^2 \|\vp\|_{C(I_{\b})}^2 ) \nonumber \\
& \leq & \prod_{j=1}^{\infty} \exp (c_j^{(n)} \vert z \vert^2
\|\vp \|^{2}_{C(I_{\b})})
= \exp \left(\frac{1}{2} \b \hat{U}_n (0) \vert z \vert^2
\|\vp \|^{2}_{C(I_{\b})} \right). \nonumber
\eeq
For every $z \in \C$, $F_n (z \vp )$ is an entire function of $z\vp \in
{\cal H}^{(c)}_{-} $, thus it is an entire function of $z$
for every fixed $\vp \in
C(I_{\b}) \subset {\cal H}_{-}^{(c)} $.
\kasten
{\bf Proof of Theorem \ref{1tm}. } If we prove the pointwise convergence of the
sequences $\{F_n \}$ on the dense subset $C(I_\b )\setminus {\cal H}^{(c)}$ to
the Laplace transforms of the corresponding measures (for $\b < \b_*$ and for
$\b =
\b_*$), then the weak convergence of $\{\nu_n \}$ will follow from Corollary
\ref{N1co} by Lemma 2.1 \cit{Pa}, p.153. For $\b
\leq \b_* $, let $b ( \b ) = 0$ if $ \b < \b_* $, and $b(\b_* ) = 1 $. Then we
should show that, for all $\vp \in C(I_\b )$,
\be
\label{a}
F_n (\vp ) \ra \exp\left\{ \frac{b(\b )}{2\b }\left( \int_{I_\b }\vp (\t)d\t
\right)^2 \right\},  \ \ n\ra \infty, \ \ \b \leq \b_* .
\ee
First we prove that, for $\|\vp\|_{C(I_\b )} < 1/\sqrt{c}$ ($c$ was
introduced in (\ref{5.341})),
$$
\log F_n (\vp ) = U^{(n)} (\vp ) \ra
\frac{b(\b )}{2\b }\left( \int_{I_\b }\vp (\t)d\t \right)^2 .
$$
Applying (\ref{48}), (\ref{49.1}) one has
\beq
\label{501}
&  & \lv U^{(n)} (\vp ) - \frac{b(\b )}{2\b }\left(\int_{I_{\b_{*}}}
\vp (\tau )d\tau \right)^2 \rv  \nonumber  \\
& \leq   & \half \| \vp\|^{2}_{C(I_{\b)}}
 \int_{I^{2}_{\b}} \left\vert U^{(n)}_2 (\tau_1 , \tau_2 )
- \frac{b(\b )}{\b}
\right\vert d\tau_1 d\tau_2   \\ \
& +  & \sum_{l=2}^{\infty} \frac{1}{(2l)!} \| \vp \|^{2l}_{C(I_{\b})}
 \int_{I^{2l}_{\b}}
\vert U^{(n)}_{2l} (\tau_1 , \tau_2 , \dots , \tau_{2l}) \vert
d\tau_1 d\tau_2 \dots \tau_{2l}. \nonumber
\eeq
The second term can be estimated by means of (\ref{5.351}), which gives its
upper bound
$$
- \frac{\b }{12c^2 }\vert Y_n \vert \log \left(1-
c \| \vp \|_{C(I_\b )}^2 \right) \longrightarrow 0 , \ \ n\ra \infty ,
$$
since $Y_n \ra 0$ for $\b \leq \b_*$ by Lemma \ref{5.5lm}. To estimate the first
term on the right-hand side of (\ref{501}) we use (\ref{529}), which gives
\begin{eqnarray*}
&  & \int_{I^{2}_{\b}} \left\vert U^{(n)}_2 (\tau_1 , \tau_2 )
- \frac{b(\b )}{\b}
\right\vert d\tau_1 d\tau_2 \\
&  = & \frac{1}{\b}\int_{I_\b^2 }
\lv \hU_n (0) -b(\b) + \sum_{k\in {\cal K}\setminus \{0\}}
\hU_n (k) \cos[k(\t_1 - \t_2 )] \rv d\t_1 d\t_2 \\
& \leq & \b \lv \hU_n (0) - b(\b )\rv + \b \sum_{k\in {\cal K}\setminus \{0\}}
\hU_n (k) .
\end{eqnarray*}
Here the first term tends to zero by Lemma \ref{5.5lm} whereas the second one
tends to zero by Corollary \ref{N2co}. Thus we have proved the pointwise
convergence in the ball $ \|\vp \|_{C(I_\b )} < 1/\sqrt{c} $. This convergence
implies that, for such $\vp$, the sequence $\{ F_n (z\vp ) \}$ of entire
functions of a single complex variable  converges for $\lv z \rv <1 $. Since the
sequence $\{
\hat{U}_n (0) \}$  is bounded for $\b \leq \b_* $ (see Lemma \ref{5.5lm}), the
sequence $\{F_n (z \vp ) \}$ is bounded in view of (\ref{87}). Therefore, by
Vitali's theorem it converges uniformly on compact subsets of $\ \C$, and hence
pointwise on $\ \C$. For every $\tilde{\vp} \in C(I_\b )$, one finds $z\in \C $
and $\vp $ in the mentioned ball such that $\tilde{\vp} = z \vp $, which
completes the proof.
\kasten



\section{\bf Proof of Lemmas}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We start with the following assertion.
\begin{Pn}
\label{6.2lm}
For arbitrary $n\in \N$, the following inequalities hold:
\be
\label{6.8}
\hU_{n}(0) \leq \k (\hU_{n-1}(0) ) \hU_{n-1}(0);
\ee
\be
\label{6.10}
\hU_n (0) \geq \k(\hU_{n-1}(0)) \hU_{n-1}(0) + \half (1 - \2m )
\left[\k(\hU_{n-1}(0))\right]^3 2^{2\d -1} X_{n-1};
\ee
\be
\label{6.11}
2^{2\d -1} \left[\k (\hU_{n-1}(0))\right]^4  X_{n-1} \leq X_n < 0;
\ee
\be
\label{6.120}
0\leq \hU_n (k) \leq \hU_n (0),
\ee
where $\k(u)$ and $X_n$ are defined by (\ref{015}) and (\ref{5.10})
respectively.
\end{Pn}
The proof of this Proposition, as well as of Lemma \ref{5.7lm}, may be given in
the framework of a lattice approximation technique similar to that developed in
constructive quantum field theory (see  \cit{Si}, \cit{Simon}). This technique
was applied in
\cit{AKK2} where some of the mentioned estimates were proved (e.g. (\ref{6.8})
in Lemma 2.2). The full exposition of it will be done in our forthcoming paper
\cit{AKK5}.

The following assertion is the main tool in proving Lemma \ref{5.5lm}.
\begin{Lm}
\label{6.3lm}
Let ${\IC}_n $,
$n \in {\Z_{+}}$, be the
triple of statements $(i_n^1 , i_n^2 , i_n^3 )$ where
\beq
i_n^1 & = & \{ \exists \b_n^{+} \in \oo  : \ \hU_n (0) = u(\delta),   \
\b = \b_n^{+} ; \
\hU_n (0) < u (\delta ),  \ \forall \b < \b_n^{+} \}; \nonumber\\
i_n^2 & = & \{ \exists \b_n^{-} \in \oo  : \ \hU_n (0) = 1 , \
\b = \b_n^{-}; \
\hU_n (0) < 1 , \ \forall \b < \b_n^{-} \}; \nonumber\\
i_n^3 & = & \{ \forall \b \in (0, \b_n^{+} ) : \ \bb{X_n} < w (\delta) \}. \nonumber
\eeq
Then
{\rm (i)}   ${\IC}_0 $ is true; {\ }
{\rm (ii)}  ${\IC}_{n-1}$ implies ${\IC}_n$.
\end{Lm}
{\bf Proof}. ${\IC}_0$ is true in view of Definition \ref{1df}.
Let us prove the implication. For $\b = \b_{n}^{+}$,
$\k (\hU_n (0) )  = 2^\ve $
and  $\k (\hU_n (0) ) < 2^\ve $ for $ \b < \b_n^{+}$ (see (\ref{014}) -- (\ref{017})). Let $ \b =
\b_{n-1}^{+}$, then (\ref{016}), (\ref{6.10}), and $i_{n-1}^3$
yield
\beq
\label{6.15}
\hU_n (0) & \geq & 2^\ve u (\delta)+ \half (1-\2m) 2^{3 \ve}
 2^{2\d -1} X_{n-1}\nonumber\\
& > & 2^\ve  u (\delta ) \left[ 1 - 2^{2(\ve - 1)+\delta}
(2^\delta -1)\frac{w(\delta)}{u(\delta)}\right] = u(\delta) .
\eeq
For $\b = \b_{n-1}^{-}$, the estimate (\ref{6.8}) gives
\be
\label{6.16}
\hU_{n}(0) \leq 1.
\ee
Taking into account the $\b$-continuity of $\hU_n (0) $ (which one may prove
recursively having this property for $n=0$) and the estimates (\ref{6.15}) and
(\ref{6.16}), one concludes that there exists at least one value
$\tilde{\b}_n^{+} \in (\b_{n-1}^{-}, \b_{n-1}^{+})$ such that $\hU_n (0) = u
(\delta) $. Then we put $\b_n^{+} \ \sta \  \min \tilde{\b_n^{+}}$. The
mentioned continuity of $\hU_n (0)$ yields also $\hU_n (0) < u(\delta) $ for $
\b < \b_n^{+}$. Thus $i_n^1 $ is true. The existence of $ \b_n^{-} \in
[\b_{n-1}^{-}, \b_{n-1}^{+})$ can be proven similarly with  $\b_n^{-} =
\min \{ \b \in [\b_{n-1}^{-} , \b_{n-1}^{+}) \ \vert \ \hU_n (0) = 1 \}$. Thus
$i_n^2 $ is also true. For $\b < \b_n^{+} < \b_{n-1}^{+}$, we have
$\k (\hU_{n -1}(0)) < 2^\ve $, which yields
\be
\label{6.17}
\lv{X_n }\rv < 2^{2\d -1} 2^{4 \ve }\lv {X_{n-1}} \rv
 = \lv{X_{n-1}}\rv < w (\delta),
\ee
hence $i_n^3 $ is true as well. The proof is concluded by remarking that
\be
\label{6.18}
[\b_n^{-}, \b_n^{+}] \subset [\b_{n-1}^{-}, \b_{n-1}^{+}) \subset \oo.
\ee
\kasten

\begin{Lm}
\label{Newlm}
There exists $\b_* \in [\b_0^{-}, \b_0^{+}]$ such that for $\b = \b_*$,
the following inequalities hold for all $n\in \Z_{+}$:
\be
\label{new}
1 \leq \hat{U}_n (0) < u(\delta).
\ee
For $\b < \b_* $, the above upper estimate also holds, moreover, there
exists $K(\beta )>0$ such that, for all $n\in \Z_+$
\be
\label{neww}
\hU_n (0) \leq K(\b )2^{-n\delta }    .
\ee
\end{Lm}
{\bf Proof.}
 Consider the set $\Delta _{n}\ \stackrel{\rm def}{=}%
\{\beta \in (0,{\beta }_{n}^{+})\vert 1\leq\hat{U}_{n} (0)< u(\delta)\}$.
Just above we have shown that it is nonempty and
$\Delta _{n}\subseteq [{\beta }_{n}^{-},{\beta }_{n}^{+} )$.
Let us prove that $\Delta _{n}\subseteq \Delta
_{n-1}$. Suppose there exists some $\beta \in \Delta _{n}$, which does not
belong to $\Delta _{n-1}$. For this $\beta $, either ${\hat U}_{n-1} (0) < 1$ or
${\hat U}_{n-1} (0)\geq u(\delta )$. Hence either ${\hat U}_{n} (0)<1$ or ${\hat
U}_{n} (0)> u(\delta)$ (it can be proved as above). This is in conflict with the
assumption $\beta
\in \Delta _{n}$. Now let
$D_n$ be the closure of $\Delta_n$, then
\be
\label{newv}
D_n = \{ \b \in [\b_n^- , \b_n^+ ]\vert 1\leq \hU_n (0) \leq u(\delta) \},
\ee
is nonempty, closed, and $D_n \subseteq D_{n-1} \subseteq \dots
\subset [\b_0^- , \b_0^+ ]$.
Set $D_{\ast }=\bigcap_{n}D_{n}$, then $D_{\ast }\subset [\b_0^{-}, \b_0^{+}]$
is also nonempty and closed. Now let us show that, for every $\b \in D_{\ast}$,
the sharp upper bound in (\ref{new}) holds for all $n\in \N$. Suppose $\hU_n (0)
= u(\delta )$ for some $n\in \N$. Then (\ref{6.15}) yields $\hU_m (0) > u(\delta
)$ for all $m>n$, which means that this $\b$ does not belong to all $D_m $, and
hence to $D_{\ast}$. Set $\beta_{\ast }=\min D_{\ast }$. We have just proved
that, for $\beta
=\beta _{\ast }$, (\ref{new}) holds, thus it remains to prove the second
part of the Lemma. Take $\beta <\beta _{*}.$ If $\hat{U}_{n} (0)\geq 1$ for all
$ n\in {I\!\! N},$ then either (\ref{new}) holds or there exists such $n_{0}$
that $\hat{U}_{n_{0}} (0)\geq  u(\delta)$. This means either $\beta \in D_{*}$
or $\beta >\inf {\beta }_{n}^{+}.$ Both these cases contradict the assumption
$\b <
\beta _{*}$. Hence there exists $ n_{0}$ such that $\hat{U }%
_{n} (0)<1$ for all $n\geq n_{0}.$ In what follows, the definition (\ref
{015}) and the estimate (\ref{6.8}) imply that the sequences $\{\hat{U}%
_{n} (0), \ n\geq n_{0}\}$ and $\{\kappa(\hat{U}_{n} (0)),
\ n\geq n_{0}\} $ are
strictly decreasing. Then for all $n>n_{0}$, one has (see (\ref{6.8}))
\beq
\hat{U}_{n} (0) & \leq & \kappa (\hat{U}_{n-1} (0)) \hat{U}_{n-1} (0)
\leq \dots \nonumber \\
& \leq & \kappa(\hU_{n-1} (0)
\kappa (\hU_{n-2} (0) ) \dots \kappa (\hU_{n_0} (0) )\hU_{n_0} (0)
 < (\kappa (\hU_{n_0}(0))^{n-n_{0}}. \nonumber
\eeq
Since $\kappa (\hU_{n_{0}}(0))<1,$ one gets $\sum_{n=0}^{\infty }%
\hat{U}_{n} (0)<\infty.$ Thus
\begin{equation}
\prod_{n=1}^{\infty } \left[1-(1-2^{-\delta})\hU_{n-1} (0)\right]^{-1}
 \stackrel{\rm def}{=}K_{0} < \infty .  \label{C0}
\end{equation}
Finally, we apply (\ref{6.8}) once again and obtain
\begin{eqnarray*}
\hat{U}_{n} (0) & \leq & 2 ^{-n\delta}
\left[1-(1-2^{-\delta})\hU_{n-1} (0)\right]^{-1} \dots
\left[1-(1-2^{-\delta})\hU_{0} (0)\right]^{-1} \hU_0 (0) \\
& < & 2^{-n \delta}K_0 u(\delta )
\stackrel{\rm def}{=}  K (\b )2^{-n \delta} .
\end{eqnarray*}
\kasten
{\bf Proof of Lemma \ref{5.5lm}.} Consider the case $\b = \b_* $ where
the estimate (\ref{new}) hold. First we show that $X_n \ra 0$. Making use
of (\ref{6.11}) we obtain
$$
0 > X_n \geq 2^{2\delta - 1}\left[\k (\hU_{n-1}(0))\right]^4 X_{n-1}
> X_{n-1} > X_{n-2} > \dots > - w(\delta).
$$
Therefore, $\{X_n \}$ is strictly increasing and bounded, hence it converges and
its limit, say $X_* $, obeys the condition $X_* > X_0 >
 -w(\delta)$. Assume that
$X_* <0$. Then (\ref{6.11}) yields $\k (\hU_n (0)) \ra 2^\ve $ hence
$\hU_n (0) \ra u(\delta)$. Passing to the limit $n\ra \infty$ in
(\ref{6.10}) one obtains $X_* = - w(\delta)$ which contradicts the above
condition. Thus $X_* = 0$. To show $\hU_n (0) \ra 1$ we set
\be
\label{newx}
\Xi_n = \half (1-2^{-\delta})\left[\k(\hU_{n-1} (0))\right]^3 2^{2\delta -1}
X_{n-1}.
\ee
Combining (\ref{6.8}) and (\ref{6.10}) one obtains
\be
\label{newy}
0\geq \hU_n (0) - \k (\hU_{n-1} (0) ) \hU_{n-1} (0) \geq \Xi_n \ra 0 .
\ee
For $\b = \b_*$, the sequence $\{\hU_n (0) \}\subset [1, u(\delta))$ in view of
Lemma \ref{Newlm}. By (\ref{newy})  all its accumulation points
in the interval $\in [1, u(\delta)]$  ought to solve the equation
$$
U- \k (U) U =0
$$
There is only one such $U_* =1$, which is thus the limit of the whole sequence
$\{\hat{U}_n (0) \}$. For $\b <\b_* $, $\hU_n (0) \ra 0 $ in view of
(\ref{neww}), which yields in turn $\k (\hU_n (0) ) \ra 2^{-\delta}$. The latter
and (\ref{6.11}) imply $X_n \ra 0$. To prove the convergence of $\{Y_n \}$ we
use (\ref{5.10}), (\ref{5.31}), and Lemma \ref{5.7lm}, which yields $\vert Y_n
\vert \leq \vert X_n \vert $ and hence $Y_n \ra 0$ for $\b \leq \b_*$.
\kasten
{\bf Proof of Lemma \ref{5.8lm}.} The lower bound in (\ref{20.331})
follows from (\ref{6.120}). By the definitions (\ref{47}) -- (\ref{49.1})
and (\ref{10.5}) -- (\ref{10.7}), and by means of
Proposition \ref{0.0pn} one obtains
\beq
\label{old}
U_2^{(n)} (0, \tau ) & = &  \Gamma_2^{(n)} (0, \tau)  \\
& = & \frac{1}{Z_{\Lan}}\TR\left\{A_{\Lan} \exp\left[-\tau H_{\Lan}\right]
A_{\Lan} \exp\left[-(\beta -\tau) H_{\Lan}\right]\right\}. \nonumber
\eeq
It may be shown that every $H_{\Lan} $ is invertible on its domain and its
inverse is a compact positive operator on $\HoL$. Let $\{ E_p^{(n)}, p\in \N
\}$ be the set of the eigenvalues of $H_{\Lan}$. We denote the corresponding
eigenfunctions by $\Psi_p^ {(n)}$ and set
$$
A_{pp'}^{(n)} \ \sta \ (A_{\Lan}\Psi_p^{(n)} , \Psi_{p'}^{(n)})_{\HoL} .
$$
Then one gets from (\ref{old})
$$
U_2^{(n)} (0,\tau) = \frac{1}{Z_{\Lan}} \sum_{p, p'}\left( A_{pp'}^{(n)}
\right)^2 \exp\left[ -\b E_{p'}^{(n)} + \tau (E_{p}^{(n)} - E_{p'}^{(n)})
\right] ,
$$
which can be used together with the definition (\ref{52}) to obtain
\begin{eqnarray}
\label{oldx}
\hU_n (k)& = &\frac{1}{Z_{\Lan}} \sum_{p,p'}\left(A_{pp'}^{(n)}\right)^2
\frac{E_p^{(n)} - E_{p'}^{(n)}}{k^2 + (E_p^{(n)} - E_{p'}^{(n)})^2 }
\left(\exp[-\b E_{p'}^{(n)}] - \exp[-\b  E_{p}^{(n) }]\right)
\nonumber \\
&\leq & \frac{1}{k^2} \frac{1}{Z_{\Lan}} \sum_{p,p'}
\left(A_{pp'}^{(n)}\right)^2
(E_p^{(n)} - E_{p'}^{(n)})
\left(\exp[-\b E_{p'}^{(n)}] - \exp[-\b  E_{p}^{(n)} ]\right) \nonumber \\
&=&  \frac{1}{k^2} \frac{1}{Z_{\Lan}} \TR\left\{\left[A_{\Lan},
\left[H_{\Lan}, A_{\Lan}\right]\right]\exp\left(-\b H_{\Lan}\right)\right\},
\end{eqnarray}
where $[\ . \ , \ . \ ]$ stands for the commutator. The operator $H_{\Lan}$ can
by written in the following form (see (\ref{10.4}) and (\ref{2.6}))
$$
H_{\Lan} = -\frac{1}{2m}\sum_{s\in \Lan}\left(\frac{\partial}{\partial x_s }
\right)^2 + V_{\Lan} (q),
$$
where $V_{\Lan}$ is a polynomial in $q_s , s\in \Lan$. Thus the double
commutator in (\ref{oldx}) may be computed explicitly. It equals $\lv \Lan
\rv^{-\delta }/m $, which yields (\ref{20.331}).
\kasten
{\bf Proof of Lemma \ref{5.9lm} }. Within the lattice approximation
 approach on the base of the Lieb - Sokal theorem \cit{LS}, one can show
 that for all $n\in \Z_{+}$, the function $f_n (z) = F_n (z\xi_1 ) $ is
 different from zero on $ \{ z\in \C \ \vert {\rm Re}z \neq
 0 \}$. This function is entire and  even, its growth is of order
 less then two. Then it should possess purely imaginary zeros, hence
 its Hadamard's representation ought to be (\ref{5.36}).
The bound $c$ used in (\ref{5.341}) one may compute on the base (\ref{5.3610}),
that  yields $\b_* u(\delta)/2$, where (\ref{new}) was used.
 \kasten

\section{\bf Acknowledgments}

The authors are extremely grateful to Leonard Gross for his many stimulating
remarks and constructive criticism on previous versions of this paper. The
financial support by the DFG (Research Project AL 214 / 91-2) and the
Volkswagenstiftung (GUS - project) are also gratefully acknowledged.  Yuri
Kozitsky is grateful for the kind hospitality extended to him at the Research
Center BiBoS (Bielefeld) where the main part of his work on this paper was
performed. He also thanks the financial support under the Grant KBN No 2 P03A
02915 of the Polish Scientific Research Committee.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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