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%\BOZZA
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%\vskip2.truecm
\centerline{\titolone Electrons in a lattice with an incommensurate potential}
\vskip1.truecm
\centerline{{\titolo G. Benfatto}\footnote{${}^\ast$}{\ottorm
Supported by MURST, Italy.}}
\centerline{Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata''}
\centerline{Via della Ricerca Scientifica, I-00133, Roma}
\vskip.2truecm
\centerline{{\titolo G. Gentile}\footnote{${}^\dagger$}{\ottorm
Supported by EC (TMR program).}}
\centerline{IHES, 35 Route de Chartres, F-91440 Bures sur Yvette.}
\vskip.2truecm
\centerline{\titolo V. Mastropietro${}^\ast$}
\centerline{Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata''}
\centerline{Via della Ricerca Scientifica, I-00133, Roma}
\vskip1.truecm
\line{\vtop{
\line{\hskip1.5truecm\vbox{\advance \hsize by -3.1 truecm
\0{\cs Abstract.}
{\it A system of fermions on a one-dimensional lattice,
subject to a periodic potential whose period is incommensurate
with the lattice spacing and verifies a diophantine condition, is studied.
The Schwinger functions are obtained,
and their asymptotic decay for large distances is exhibited
for values of the Fermi momentum
which are multiple of the potential period.}} \hfill} }}
\vskip1.2truecm
\centerline{\titolo 1. Introduction.}
\*\numsec=1\numfor=1
\0{\bf 1.1.} The {\sl Static Holstein model}
[P, H] describes a system of fermions (electrons) in
a linear lattice interacting with a classical phonon field.
It is obtained from a tight-binding Hamiltonian with
neglect of the vibrational kinetic energy of the lattice
(an approximation which can be justified in physical models
as the atom mass is much larger than the electron mass).
The Hamiltonian of the model, if we neglect all internal degrees of freedom
(the spin, for example), which play no role, is given by
%
$$ H=\sum_{x,y\in\L}
t_{xy}\,\psi^+_{x}\psi^-_{y} - \mu\sum_{x\in\L}
\psi^+_{x}\psi^-_{x}
-\l\sum_{x\in\L} \f_x\psi^+_{x}\psi^-_{x}
+ {1\over 2} \sum_{x\in\L} \f_x^2 \; , \Eq(1.1) $$
%
where $x,y$ are points on the one-dimensional lattice $\L$
with unit spacing, length $L$ and periodic boundary conditions; we shall
identify $\L$ with $\{x\in\ZZZ:\ -[L/2]\le x \le [(L-1)/2]\}$.
Moreover the matrix $t_{xy}$ is
defined as $t_{xy}=\d_{x,y}-(1/2)[\d_{x,y+1}+\d_{x,y-1}]$,
where $\d_{x,y}$ is the Kronecker delta.
The fields $\psi_x^{\pm}$ are creation ($+$) and annihilation ($-$)
fermionic fields, satisfying periodic boundary conditions:
$\psi_x^{\pm}=\psi_{x+L}^{\pm}$. We define also
$\psi_{\xx}^{\pm}=e^{tH}\psi_x^{\pm}e^{-Ht}$, with $\xx=(x,t)$,
$-\b/2\le t \le \b/2$ for some $\b>0$; on $t$ antiperiodic boundary
conditions are imposed.
The potential $\f_x$ is a real function
representing the classical phonon field, of a form
which will be specified below (see \S 1.3).
In \equ(1.1) $\m$ is the chemical potential,
and $\l$ is the interaction strenght.
The expectation value of an observable $\cal O$ in the Grand-canonical state
at inverse temperature $\b$ and volume $\L$ is given by $\langle {\cal O}
\rangle \= {\rm Tr} [\exp(-\b H) {\cal O}] /{\rm Tr} [\exp(-\b H)]$. If ${\cal
O} = {\bf T} [\psi^-_{\xx_1} \cdots \psi^-_{\xx_n} \psi^+_{\yy_1} \cdots
\psi^+_{\yy_n}]$, where $\bf T$ denotes the anticommuting time ordering
operator, we get the {\sl $2n$-point Schwinger functions} of the model.
The most interesting problem about the model \equ(1.1) is to find the minima
with respect to $\f_x$ of the ground state energy of the system $E(\f)$, in
the thermodynamic limit. It is easy to show that all stationary points of
$E(\f)$ satisfy the condition
%
$$\f_x= \l\r_x \; ,\qquad \r_x\= \lim_{\b\to\io} \lim_{L\to\io}
\langle {\psi^+_x \psi^-_x} \rangle \; .\Eq(1.2)$$
%
This equation has been rigorously studied, up to now, only in the case of
density $1/2$ [KL, LM]. However, if $\r= \lim_{L\to\io}L^{-1}\sum_x \r_x$ is
an irrational number, there have been recently, starting from [AAR], some
numerical studies of the model, which led,
through a strong numerical evidence, to the conjecture
that, for small coupling, the ground state energy of the system $E(\f)$ has
a minimum for a potential of the form $\f_x=\bar\f(2px)$, where $\bar\f(u)$ is
a $2\pi$-periodic real function of the real variable $u$ and $p=\p\r$.
The conjecture has a physical interest to explain the properties of
strongly anisotropic compounds which can be considered as one-dimensional
systems; in such systems one finds a charge density wave incommensurate with
the lattice, according to \equ(1.2).
In this paper, we shall not study the minimization problem of $E(\f)$, but
we shall analyze the properties of the two-point Schwinger function
$S_2(\xx;\yy)=\langle {\bf T} \psi^-_\xx \psi^+_\yy \rangle$,
for a suitable set of
values of $\mu$ and $\f_x=\bar\f(2px)$, with $p/\p$ irrational.
[Note that all the Schwinger functions can be expressed
in terms of the two-point Schwinger functions, as the
interaction is quadratic in the fermionic fields].
We shall do that by constructing a convergent expansion
for $S_2(\xx;\yy)$, that we hope
will be useful in studying the equation \equ(1.2).
In any case, this expansion allows
to prove some properties of $S_2(\xx;\yy)$, which are interesting by
themselves; these properties imply known results about the Schroedinger
equation related to the model \equ(1.1), but are not a trivial consequence of
them (see discussion in \S 1.6 below).
\*
\0{\bf 1.2.} As it is well known, the Schwinger functions can be written as
power series in $\l$, convergent for $|\l| \le \e_\b$, for some constant
$\e_\b$ (the only trivial bound of $\e_\b$ goes to zero, as
$\b\to\io$). This power expansion is constructed in the usual way in terms of
Feynman graphs (in this case only chains, since the interaction is quadratic
in the field), by using as {\sl free propagator} the function
%
$$\eqalign{
g^{L,\b}(\xx;\yy) &\= g^{L,\b}(\xx-\yy)=
{{\rm Tr} \left[e^{-\b H_0} {\bf T} (\psi^-_\xx \psi^+_\yy)\right] \over
{\rm Tr} [e^{-\b H_0}]} = \cr
&={1\over L} \sum_{k\in {\cal D}_L}
e^{-ik(x-y)} \left\{ {e^{-\t e(k)} \over 1+e^{-\b e(k)}}
\indic(\t>0) - {e^{-(\b+\t) e(k)} \over 1+e^{-\b e(k)}} \indic(\t\le 0)
\right\}\; , \cr}\Eq(1.3)$$
%
where $H_0$ is the free Hamiltonian ($\l=0$), $\xx=(x,x_0)$, $\yy=(y,y_0)$,
$\t=x_0-y_0$, $\indic(E)$ denotes the indicator function ($\indic(E)=1$, if
$E$ is true, $\indic(E)=0$ otherwise), $e(k)=1-\cos k -\mu$ and
${\cal D}_L\=\{k={2\pi n/L}, n\in \ZZZ, -[L/2]\le n \le [(L-1)/2]\}$.
It is easy to prove that, if $x_0\not= y_0$,
%
$$g^{L,\b}(\xx-\yy)= \lim_{M\to\io} {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
{e^{-i\kk\cdot(\xx-\yy)}\over -ik_0+\cos p_F-\cos k}\; , \Eq(1.4)$$
%
where $\kk=(k,k_0)$, $\kk\cdot\xx=k_0x_0+kx$, ${\cal D}_{L,\b}\={\cal D}_L
\times {\cal D}_\b$, ${\cal D}_\b\=\{k_0=2(n+1/2)\pi/\b, n\in
\ZZZ, -M\le n \le M-1\}$ and $p_F$ is the {\sl Fermi momentum},
defined so that $\cos p_F =1-\mu$ and $0< p_F < \p$.
[$M$ is an (arbitrary) integer].
Hence, if we introduce a finite set of Grassmanian
variables $\{\psi^\pm_\kk\}$, one for each of the allowed $\kk$ values, and a
linear functional $P(d\psi)$ on the generated Grassmanian algebra, such that
%
$$\int P(d\psi) \psi^-_{\kk_1}\psi^+_{\kk_2} = L\b \d_{\kk_1,\kk_2}
\hat g_{\kk_1}\;,\quad \hat g_\kk= {1\over -ik_0+\cos p_F-\cos k}
\; ,\Eq(1.5)$$
%
we have
%
$${1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}} \, e^{-i\kk\cdot(\xx-\yy)} \,
\hat g_\kk = \int P(d\psi)\psi^-_\xx \psi^+_\yy
\= g^{L,\b}(\xx;\yy) \; ,\Eq(1.6)$$
%
where the {\sl Grassmanian field} $\psi_\xx$ is defined by
%
$$\psi_\xx^{\pm}= {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}} \psi_\kk^{\pm}
e^{\pm i\kk\cdot\xx}\; .\Eq(1.7)$$
The ``Gaussian measure'' $P(d\psi)$ has a simple representation in terms of
the ``Lebesgue Grassmanian measure'' $d\psi^- d\psi^+$, defined as the linear
functional on the Grassmanian algebra, such that, given a monomial
$Q(\psi^-,\psi^+)$ in the variables $\psi_\kk^-,\psi_\kk^+$,
%
$$\int d\psi^- d\psi^+ Q(\psi^-,\psi^+) =\cases{1& if $Q(\psi^-,\psi^+)=
\prod_\kk \psi_\kk^-\psi_\kk^+ \; ,$\cr 0& otherwise $\; .$\cr} \Eq(1.8)$$
%
We have
%
$$P(d\psi) = \Big\{ \prod_\kk (L\b\hat g_\kk) \Big\}
\exp \Big\{-\sum_\kk (L\b \hat g_\kk)^{-1} \psi_\kk^+ \psi_\kk^- \Big\}
d\psi^- d\psi^+ \; .\Eq(1.9)$$
%
Note that, since $(\psi_\kk^-)^2=(\psi_\kk^+)^2=0$,
$e^{-z \psi_\kk^+ \psi_\kk^-}
=1-z \psi_\kk^+ \psi_\kk^-$, for any complex $z$.
By using standard arguments (see, for example, [NO], where a different
regularization of the propagator is used), one can show that the Schwinger
functions can be calculated as expectations of suitable functions of the
Grassmanian field with respect to the ``Gaussian measure'' $P(d\psi)$.
In particular, the two-point Schwinger function, which in our case
determines the other Schwinger functions through the Wick rule,
can be written, if $x_0\not= y_0$, as
%
$$ S^{L,\b}(\xx;\yy) = \lim_{M\to\io} {\int P(d\psi)\,
e^{\VV(\psi)}\,\psi^-_{\xx}\psi^+_{\yy}
\over\int P(d\psi)\,e^{\VV (\psi)}} \; , \Eq(1.10) $$
%
where
%
$$ \VV(\psi)=\sum_{x\in\L} \int_{-\b/2}^{\b/2} dx_0
\Big[\l \f_x\psi_\xx^+ \psi^-_\xx \Big] \; .\Eq(1.11) $$
%
If $x_0=y_0$, $S^{L,\b}(\xx;\yy)$ must be defined as the limit of \equ(1.10)
as $x_0-y_0\to 0^-$, as we shall understand always in the following.
\*
\0{\cs Remark.} The {\sl ultraviolet cutoff} $M$ on the $k_0$ variable was
introduced in order to give a precise meaning to the Grassmanian integration
(the numerator and the denominator in the r.h.s. of \equ(1.10) are indeed
finite sums),
but it does not play any essential role in this paper, since all bounds
will be uniform with respect to $M$ and they easily imply the
existence of the limit. Hence, we shall not stress anymore the dependence on
$M$ of the various quantities we shall study.
\*
\0{\bf 1.3.} We now define precisely the potential $\f_x$.
We are interested in studying potentials which, in the limit $L\to\io$,
are of the form $\f_x=\bar\f(2px)$, where $\bar\f$ is a real function on the
real line $2\pi$-periodic and $p/\p$ is an irrational number, so
that the phonon field has a period which is incommensurate
with the period of the lattice.
We also impose that $\bar\f(u)$ is of mean zero (its mean value can be
absorbed in the chemical potential), even and analytic in $u$,
so that
%
$$\bar\f(u) = \sum_{0\neq n \in \zzz}
\hat \f_n\,e^{inu} \; , \qquad |\hat \f_n| \le F_0 \, e^{-\x|n|}\; , \qquad
\hat \f_n=\hat \f_{-n}=\hat \f_n^* \; .
\Eq(1.12) $$
%
At finite volume we need a potential satisfying periodic boundary conditions;
hence, at finite $L$, we approximate $\f_x$ by
%
$$ \f^{(L)}_x = \sum_{n=-[L/2]}^{[(L-1)/2]}
\hat \f_n\,e^{2inp_Lx}\; , \Eq(1.13) $$
%
where $p_L$ tends to $p$ as $L\to\io$ and is of the form
$p_L=n_L\pi/L$, with $n_L$ an integer, relatively prime with respect to $L$.
The definition of $p_L$ implies that $2np_L$ is an allowed momentum
(modulo $2\p$), for any $n$,
and that the sum in \equ(1.13) is indeed a sum over all allowed values
of $k$, except $k=0$.
If we insert \equ(1.13) in the r.h.s. of \equ(1.11), we get
%
$$\VV(\psi) =
\sum_{n=-[L/2]}^{[(L-1)/2]} {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
\, \l \hat \f_n \, \psi^+_\kk \psi^-_{\kk+2n\pp_L} \; ,\Eq(1.14)$$
%
where $\pp_L=(p_L,0)$ and $k+2np_L$ is of course defined modulo $2\p$.
Let us now suppose that $p_F=m p_L$, for some integer $m\ge 1$, so that the
$\hat g_\kk^{-1}$ is small for $\kk\simeq \pm m\pp_L$. In this case (see
\equ(1.9)) there is no hope to
treat perturbatively the terms with $n=\pm m$ and $\kk$ near $\mp m\pp_L$,
but we can at most expect that the interacting measure is a perturbation of
the measure
%
$$\bar P_\l(d\psi) \= {1\over \NN} P(d\psi)
\exp \Big\{ \l\hat \f_m {1\over\b}\sum_{k_0\in {\cal D}_\b}
\fra1L \sum_{k\in I_-} [\psi^+_\kk \psi^-_{\kk+2m\pp_L} +
\psi^+_{\kk+2m\pp_L} \psi^-_\kk] \Big\}\; ,\Eq(1.15)$$
%
where $\NN$ is a normalization constant and $I_-$ is a small interval
centered in $-p_F$, so small that $I_-\cap I_+ =
\emptyset$, if $I_+ \= \{k=\bar k+2p_F, \bar k\in I_-\}$.
It is very easy to study the measure \equ(1.15); in fact
$\bar P_\l(d\psi)= \NN'^{-1} d\psi d\bar\psi \exp [-J(\bar\psi,\psi)]$, where
$J(\bar\psi,\psi)$ is a quadratic form, which can be simply diagonalized,
since it only couples $\psi^\pm_\kk$, $k\in I_-$,
with $\psi^\pm_{\kk+2m\pp_L}$. One can show that there is an
orthogonal transformation from the variables $\psi_\kk^\pm$ to new variables
$\c_\kk^\pm$, such that, if $I\=I_-\cup I_+$, then
%
$$J(\bar\psi,\psi) = \fra1\b \sum_{k_0\in {\cal D}_\b} \Big\{
\fra1L \sum_{k\not\in I} \c^+_\kk \c^-_\kk [-ik_0 + e(k)] +\fra1L
\sum_{k\in I} \c^+_\kk \c^-_\kk [-ik_0 + E(k)] \Big\}\; , \Eq(1.16)$$
%
where $e(k)=\cos p_F-\cos k$ is the free dispersion relation, while $E(k)$ is
the new dispersion relation near $\pm p_F$. $E(k)$ is such that
%
$$E(k) \, \hbox{sign}\, (|k|-p_F) \ge |\l\hat \f_m|\; .\Eq(1.17)$$
Note that $p_F$ is an allowed momentum, if $mn_L$ is even. In this case, there
are two eigenstates of the one-particle Hamiltonian $h_{xy}$
corresponding to \equ(1.16) with energy $\m$,
for $\l=0$ (\ie of $h_{xy} =t_{xy}$; see \equ(1.1));
the coupling removes the degeneracy and the corresponding
interacting eigenstates have energy $\m \pm \l\hat\f_m$.
Given a one-particle Hamiltonian $h_{xy}$ (in the model \equ(1.1)
$h_{xy} = t_{xy} -\l\f_x\d_{xy}$) with spectrum $\Si$, we define as usual
the {\sl spectral gap\ } around the level $\m$ in the following way:
%
$$\D = \inf \{E\in\Si: E>\m\} - \sup \{E\in\Si: E<\m\} \Eq(1.17a)$$
%
The bound \equ(1.17) implies that the measure $\bar P_\l(d\psi)$ is associated
with a one-particle Hamiltonian
with a spectral gap $2|\l\hat\f_m| + O(L^{-1})$ around the level $\m$.
It is also easy to prove that the zero temperature density $\r^L$,
defined as the limit as $\b\to\io$ of the finite $\b$ density, given
by
%
$$\r^{L,\b}=-\lim_{\t\to 0^-} {1\over L}\sum_x S^{L,\b}(x,\t;x,0)\; ,
\Eq(1.18)$$
%
is independent of $\l$ and $L$, for the approximated model, and is given by
$\r^L=p_F/\p$. This follows from the previous calculations and from the
remark that $\r^L$ is equal to the number of eigenvalues lower than $\m$ of the
one-particle Hamiltonian plus half the number of eigenvalues equal to $\m$,
divided by $L$; hence the two eigenstates that degenerate for $\l=0$ (if they
are present) give the same
contribution to $\r^L$ for any value of $\l$, as well as all the others,
thanks to \equ(1.17).
We shall prove that there is a diverging sequence of volumes $L_i$, such
that the measure $P(d\psi) \exp [\VV(\psi)]$ is a perturbation of
$\bar P_\l(d\psi)$, for $\l$ small enough, uniformly in $i$ and $\b$.
In particular, we shall prove that there is a spectral gap of order
$|\l\hat\f_m|$ around $\m$, independent of $i$, if $p_F=mp_{L_i}$.
We shall prove also that the density $\r^{L_i,\b}$ is a continuous function of
$\l$ near $0$, uniformly in $i$ and $\b$, as well as its limit as $\b\to\io$.
This result implies that $\lim_{\b\to\io}\r^{L_i,\b}=p_F/\p$, independently of
$\l$ and $L_i$; in fact, at finite volume and zero temperature, the density
can take only a finite set of values , hence it is constant if it is
continuous.
In order to implement this program, one must face one main difficulty,
related to the fact that $p_{L_i}$
converges to an irrational number as $L_i\to\io$,
so that there are terms in the interaction \equ(1.14),
which are almost equal to those included in the definition of
$\bar P_\l(d\psi)$, without being exactly equal. These terms can
not be simply included in the definition of $\bar P_\l(d\psi)$ and
make difficult to control the perturbation theory. This difficulty will be
cared by using the decreasing property of $\hat \f_n$, see \equ(1.12),
and a diophantine condition hypothesis on $p$.
\*
\0{\bf 1.4.} Denote by $\|\a-\b\|_{\ttt^1}$
the distance on ${\TTT}^1$ of $\a,\b\in {\TTT}^1$,
and, for $\xx=(x,x_0), \yy=(y,y_0)\in\RRR^2$, by $|\xx-\yy|$
the distance $|\xx-\yy|=\sqrt{(x-y)^2+(x_0-y_0)^2}$.
We shall prove the following theorem.
\*
\0{\cs Theorem 1.}
{\it Let us consider a sequence $L_i$, $i\in \ZZZ^+$, such that
%
$$ \lim_{i\to\io} L_i=\io \; , \qquad
\lim_{i\to\io}p_{L_i}=p \; , $$
%
and let $S^{L_i,\b}(\xx;\yy)$ be the Schwinger function \equ(1.10).
Suppose also that there is a positive integer $m$ such that
$p_F=mp_{L_i}\, (\mod 2\p)$, $\hat \f_m\not=0$ and
$p_{L_i}$ satisfies the {\sl diophantine condition}
%
$$ \|2n p_{L_i} \|_{\ttt^1} \ge C_0 |n|^{-\t} \; , \qquad
\forall \; 0\neq n \in \ZZZ \;,\qquad |n|\le {L_i\over 2} \; , \Eq(1.19) $$
%
for some positive constants $C_0$ and $\t$ independent of $i$.
Then there exists $\e_0>0$, such that, if $\l\in\RRR$ and $|\l|\le\e_0$,
the following sentences are true.
\vskip.2truecm
(i) There exists the limit
%
$$\lim_{\b\to\io \atop i\to\io} S^{L_i,\b}(\xx;\yy) = S(\xx;\yy)
=S_1(\xx;\yy)+\l S_2(\xx;\yy) \; , \Eq(1.20) $$
%
where
%
$$ S_1(\xx;\yy) = g^{(1)}(\xx;\yy) + \int {d\kk\over(2\p)^2}\,
[1-\hat f_1(\kk)]\, \phi(k,x,\sigma)\,\phi^*(k,y,\sigma) \,
{e^{-ik_0(x_0-y_0)} \over -ik_0+\e(k,\sigma)}\; ,\Eq(1.21) $$
%
with
%
$$ \eqalignno{
g^{(1)}(\xx;\yy) & = \int {d\kk\over(2\p)^2}\,
\hat f_1(\kk)\, {e^{-ik_0(x_0-y_0)} \over -ik_0+ e(k)}\; ,\cr
\s & =\l \hat \f_m\; ,\cr
\e(k,\sigma) & = [1-\cos(|k|-p_F)] \cos p_F \cr
& +\sign (|k|-p_F) \, \sqrt{ [\sin(|k|-p_F)\sin p_F]^2+\sigma^2} \; , \cr
\phi(k,x,\sigma) & =e^{-ikx} u(k,x,\sigma) \; , & \eq(1.22) \cr
u(k,x,\sigma) & = e^{i\sign(k)p_Fx}
\left[ \cos(p_Fx) \sqrt{ 1-
{\sign(|k|-p_F)\sigma \over \sqrt{(\sin(|k|-p_F)\sin p_F)^2+\sigma^2}}}
\right. \cr
& \left. - i\,\sign(k)\sin(p_Fx) \sqrt{ 1+
{\sign(|k|-p_F)\sigma\over\sqrt{(\sin(|k|-p_F)\sin p_F)^2+\sigma^2}}}
\; \right] \; . \cr} $$
%
Here $\hat f_1(\kk)$ denotes a cutoff function with support far enough from
the two singular points $\kk=(\pm p_F,0)$, see \equ(2.4) in \S2 for a precise
definition.
\vskip.2truecm
(ii) $S(\xx;\yy)$ is continuous as a function of $\l$; moreover
there are three constants $K_1$, $K_2$, $K_3$ and,
for any $N>1$, a constant $C_N$, such that, if $|\xx-\yy|\ge K_3
|\sigma|^{-1}$,
%
$$ |S_1(\xx;\yy)| , |S_2(\xx;\yy)|
\le {K_1 |\s|} {C_N \over 1 + (|\s|\,|\xx-\yy|)^N}
\; , \Eq(1.23) $$
%
while, for $|\xx-\yy|\le K_3 |\sigma|^{-1}$, one has
%
$$|S_2(\xx;\yy)| \le K_2\,(1+|\xx-\yy|)^{-1}\; .\Eq(1.24)$$
%
Finally, for any $|\xx-\yy|$, one has
%
$$|S_1(\xx;\yy) - g(\xx;\yy)| \le K_2 |\sigma| \log(|\s|^{-1}+1)\; ,
\Eq(1.24a)$$
%
where $g(\xx;\yy)\=\lim_{\b\to\io, i\to\io} g^{L_i,\b}(\xx;\yy)$
(it satisfies the bound \equ(1.24) for any $|\xx-\yy|$).
\vskip.2truecm
(iii) For any $i$, the density $\r^{L_i,\b}$, given by \equ(1.18), is a
continuous function of $\l$, uniformly in $\b$,
as well as its limit as $\b\to\io$.
\vskip.2truecm
(iv) For any $i$, there is a spectral gap $\D \ge |\s|/2$ around $\m$.
}
\*
\0{\bf 1.5.} The above theorem states that, if in the infinite volume limit
the Fermi momentum is a multiple of the period of the potential ($p_F=mp$)
and $p/\pi$ is an irrational number verifying a diophantine condition,
then the two-point Schwinger function decays as the free one $(\l=0$)
for $|\xx-\yy|\le K_3|\sigma|^{-1}$ and faster than any power if
$|\xx-\yy|\ge K_3|\sigma|^{-1}$, for some constant $K_3$.
The region in which the behaviour is
a power law enlarges taking larger and larger $m$.
As the points of the form $m p$ are dense on ${\TTT}^1$,
very small changes in the Fermi momentum
(related to changes of the density of the system)
can correspond to very different values of $m$, and so to very different
asymptotic behaviour of $S(\xx;\yy)$ (one can pass for instance
with a very small difference in $p_F$ from a situation
in which the faster than any power decay is observable to a
situation in which it occurs at so large distances to becomes unobservable).
\*
\0{\bf 1.6.} The infinite volume two-point Schwinger function is obtained as
the limit
of $S^{L_i,\b}(\xx;\yy)$, when $p_{L_i}/(2\pi)$ is a sequence of rational
numbers verifying the generalized diophantine condition \equ(1.19)
and converging to an irrational diophantine number. A
sequence with the above property is constructed in Appendix 1, for any
diophantine number.
Finally, by looking at the proof of Theorem 1, one can see that,
if there are sequences
$L_i$, $p_{F,L_i}=2\p n_{F,L_i}/L_i$, $p_{L_i}$ such that
$\lim_{i\to\io}L_i=\io$ and
%
$$ \|2np_{L_i}\|_{\ttt^1},\; \|p_{F,L_i}+np_{L_i}\|_{\ttt^1}
\ge C_0 |n|^{-\t} \; , \qquad \forall \; 0 \neq n \in \ZZZ \; ,
\qquad |n|\le {L_i\over 2}\;, \Eq(1.26) $$
%
for some positive constants $C_0$ and $\t$, then
the two-point Schwinger function is given by
%
$$ S(\xx;\yy)=g(\xx;\yy)+\l S_2(\xx;\yy) \; , \Eq(1.27) $$
%
where $g(\xx;\yy)$ is defined after \equ(1.24a) and
%
$$ |g(\xx;\yy)|, |S_2(\xx;\yy)|\le {K_4\over 1+|\xx-\yy|} \; , \Eq(1.28) $$
%
for some constant $K_4$.
However, since in this case the construction of a sequence of
$L_i, p_{F,L_i}, p_{L_i}$ verifying \equ(1.26)
seems to be much more involved, while the
renormalization group analysis, to which mainly is devoted this paper, seems
to be essentially the same, we prefer not to discuss this case here.
\*
\0{\bf 1.7.} Systems of fermions on a
lattice subject to a periodic potential {\sl incommensurate}
with the period of the lattice are widely studied,
starting from [P], in which this problem was considered relevant to
understand a system of electrons in a lattice and subject to a magnetic
field and was faced by studying
the spectrum of the finite difference Schroedinger equation
%
$$ -\psi(x+1)-\psi(x-1)+\l \f_x\psi(x)=E\psi(x) \; , \Eq(1.29) $$
%
where $\f_x$ is defined as before. This problem is
of course closely related to the
study of the spectrum of the Schroedinger equation
%
$$ -{d^2 \psi(x)\over dx^2}
+\l \f_x\psi(x)=E\psi(x)\; , \Eq(1.30) $$
%
where $\f_x=\bar \f(\oo x)$, $\oo\in\RRR^d$ is a vector
with rationally independent components and $\bar \f({\bf u})$ is
$2\p$-periodic in all its $d$ arguments.
In fact in \equ(1.29) there are two periods,
the one of the potential and the intrinsic one of the lattice,
and this makes the properties of \equ(1.29) and of \equ(1.30) (with $d=2$)
very similar to each other.
The eigenfunctions and the spectrum strongly depend on $\l$.
For large $\l$
there are eigenfunctions with an exponential decay for large distances; this
phenomenon is called {\sl Anderson localization} (for details,
see for instance [PF] and references therein).
On the other hand, for small $\l$ and for certain values of $E$,
there are eigenfunctions which are {\sl quasi-Bloch waves}
of the form $e^{ik(E)x}\,u(x)$ with
$u(x)=\bar u(px)$ for \equ(1.29) and
$u(x)=\bar u(\oo x)$ for \equ(1.30), $\bar u$ being $2\p$-periodic in its
arguments.
This was proved for \equ(1.30) in [DS], with the condition that there exist
positive constants $C_0,\t$ such that
%
$$|\oo\cdot\nn|\ge C_0 |\nn|^{-\t}\; ,\qquad
|k(E)+\oo\cdot\nn|\ge C_0 |\nn|^{-\t} \; , \qquad
\forall {\bf 0}\neq \nn\in \ZZZ^d \; , \Eq(1.31)$$
%
by using KAM techniques modulo some technical assumptions
(like the condition of large $E$) which were relaxed in [E].
An analogous statement was proved for \equ(1.29),
with the condition $\|k+np\|_{\ttt^1}\ge C_0 |n|^{-\t}$
(see Theorem 1 for notations),
in [BLT], by using essentially the same ideas as in [DS].
The existence of quasi-Bloch waves for \equ(1.30)
with $k(E)$ verifying $k(E)={1\over 2}\, \oo\cdot\nn$ and
$|\oo\cdot\nn|\ge C_0 |\nn|^{-\t}$
was proved, together with the existence of gaps in the spectrum, in [JM,MP]
with some additional assumption removed in [E].
Our results are in agreement with those contained in the papers referenced
above, but we think that they do not follow completely from them. In
particular, the properties of the Schwinger functions do not seem to us a
consequence of the known properties of the one-particle Hamiltonian spectrum.
\*
\0{\bf 1.8.} The proof of Theorem 1 is performed by
using renormalization group
techniques combined with the diophantine condition \equ(1.19).
The proof of the convergence of the perturbative series for the
two-point Schwinger function
is similar to the proof of the convergence of
the Lindstedt series for the invariant tori of a mechanical system,
[G,GM], in which a notion of resonance is introduced and it is shown
that, thanks to the diophantine condition, if one
subtracts the relevant part of the value associated to
the resonances ({\sl resonance value}, see [GM] and \S 3.3 below),
the resulting series is convergent. In the Lindstedt
series the sum of the relevant part of the resonance values is vanishing;
this is not true in this case, in which the relevant part of the resonance
value is a {\sl running coupling constant},
in the renormalization group sense. However, here a different
mechanism still ensures the convergence of the perturbative series.
\*
The paper is organized as follows. In \S 2 we introduce
the multiscale decomposition of the propagators and set up the
graph formalism, which allows us to treat all contributions
corresponding to graphs not belonging to a certain class
(graphs without resonances, see the definition in \S2.5); this will lead
to Lemma 1.
In \S 3 we show that a more refined renormalization procedure (which
consists essentially in changing suitably the ``Grassmanian
integration'' at each step of the renormalization procedure)
allows us to extend the result of \S 2 to all graphs (Lemma 2);
then the convergence of the effective potential follows.
In \S4 we study the two-point Schwinger functions, with the
same techniques of \S 3, and we prove Theorem 1.
\vskip2.truecm
\centerline{\titolo 2. Multiscale decomposition}
\*\numsec=2\numfor=1
\0{\bf 2.1.} In order to simplify the notation, in this section and in the
following one we shall not stress anymore the dependence
on $\b$ and $L\=L_i$ of the various quantities; in particular
$p_{L_i}$, $g^{L_i,\b}(\xx;\yy)$ will be written simply as
$p$, $g(\xx;\yy)$.
It is convenient to
decompose the Grassmanian integration $P(d\psi)$ into
a finite product of independent integrations:
%
$$ P(d\psi)=\prod_{h=h_\b }^1 P(d\psi^{(h)}) \; ,\Eq(2.1) $$
%
where $h_\b >-\io$ will be defined below (before \equ(2.9))
This can be done by setting
%
$$ \psi_\kk^{\pm}=\bigoplus_{h=h_\b }^1\psi_\kk^{(h)\pm} \; ,
\qquad \hat g_\kk=\sum_{h=h_\b }^1 \hat g^{(h)}_\kk \; , \Eq(2.2) $$
%
where $\psi^{(h)\pm}_\kk$ are families of Grassmanian fields
with propagators $\hat g^{(h)}_\kk$ which are defined in the
following way.
We introduce a {\sl scaling parameter} $\g>1$ and a function
$\c(\kk') \in C^{\io}(\TTT^1\times \RRR)$, $\kk'=(k',k_0)$, such that,
if $|\kk'|\=\sqrt{k_0^2+||k'||_{\ttt^1}^2}$,
%
$$ \c(\kk') = \c(-\kk') = \cases{
1 & if $|\kk'| a_0\; ,$\cr}\Eq(2.3)$$
%
where $a_0=\min \{p_F/2, (\p-p_F)/2 \}$. This definition
is such that the supports of $\c(k-p_F,k_0)$ and $\c(k+p_F,k_0)$ are
disjoint and the $C^\io$ function on $\TTT^1\times \RRR$
%
$$\hat f_1(\kk) \= 1- \c(k-p_F,k_0) - \c(k+p_F,k_0) \Eq(2.4)$$
%
is equal to $0$, if $\|\,|k|-p_F\|_{\ttt^1}^2 +k_0^2t_0 \g^{h+1}$, and $f_h(\kk')=
1$ for $|\kk'| =t_0\g^h$.
We finally define, for any $h\le 0$,
%
$$ \hat f_h(\kk) = f_h(k-p_F,k_0) + f_h(k+p_F,k_0)\; ,\Eq(2.7) $$
%
$$ \hat g^{(h)}_\kk \= { \hat f_h(\kk) \over -ik_0+\cos p_F -\cos k}
\; . \Eq(2.8) $$
%
The label $h$ is called {\sl scale} or {\sl frequency} label.
Note that, if $\kk\in {\cal D}_{L,\b}$, then $|k_0|\ge \p/\b$, implying that
$\hat f_h(\kk)=0$ for any $h< h_\b = \min \{h:t_0\g^{h+1} > \p/\b \}$.
Hence, if $\kk\in {\cal D}_{L,\b}$, the definitions \equ(2.4) and \equ(2.7),
together with the identity \equ(2.6), imply that
%
$$1=\sum_{h=h_\b }^1 \hat f_h(\kk) \; .\Eq(2.9)$$
The definition \equ(2.7) implies also that, if $h\le 0$, the support of
$\hat f_h(\kk)$ is the union of two disjoint sets, $A_h^+$ and $A_h^-$. In
$A_h^+$, $k$ is strictly positive and $||k-p_F||_{\ttt^1}\le a_0\g^h \le a_0$,
while, in $A_h^-$, $k$ is strictly negative and
$||k+p_F||_{\ttt^1}\le a_0\g^h$.
Therefore, if $h\le 0$, we can write $\psi^{(h)\pm}_{\kk}$ as the sum of two
independent Grassmanian variables $\psi_{\kk,\o}^{(h)\pm}$ with propagator
%
$$ \int P(d\psi^{(h)})\,\psi^{(h)-}_{\kk_1,\o_1}
\psi^{(h)+}_{\kk_2,\o_2} = L\b \d_{\kk_1,\kk_2}\,\d_{\o_1,\o_2}\,
\hat g^{(h)}_{\o_1}(\kk_1) \; , \Eq(2.10)$$
%
so that
%
$$ \psi^{(h)\pm}_{\kk}=\bigoplus_{\o=\pm 1}\psi^{(h)\pm}_{\kk,\o}
\; , \qquad \hat g^{(h)}_\kk=\sum_{\o=\pm 1} \hat g^{(h)}_\o(\kk) \; ,
\Eq(2.11) $$
%
$$ \hat g^{(h)}_\o(\kk)={\theta(\o k) \, \hat f_h(\kk) \over -ik_0
+ \cos p_F - \cos k }\; , \Eq(2.12) $$
%
where $\theta(k)$ is the (periodic) step function.
If $\o k> 0$, we will write in the following $k=k'+\o p_F$,
where $k'$ is the {\sl momentum measured from the Fermi surface} and we shall
define, if $h\le 0$,
%
$$ \tilde g^{(h)}_{\o}(\kk') \= \hat g^{(h)}_\o(\kk)=
{f_h(\kk') \over -i k_0+v_0 \o\sin k'+(1-\cos k')\cos p_F } \; , \Eq(2.13) $$
%
where $v_0=\sin p_F$.
In order to simplify the notation, it will be useful in the following to
denote $\hat g^{(1)}_\kk$ also as $\tilde g^{(1)}_{1}(\kk')$, with
$k=k'+p_F$.
It is easy to prove that, for any $h\le 1$ and any $\o$,
%
$$ |\tilde g^{(h)}_\o(\kk')|\le G_0 \g^{-h} \; , \Eq(2.14) $$
%
for a suitable positive constant $G_0$, depending on $p_F$ and diverging as
$a_0\to 0$.
In the following we shall use also the definitions
$$\eqalign{
\psi^{(\le h)\pm}_{\kk,\o} &= \bigoplus_{j=h_\b}^h
\psi^{(j)\pm}_{\kk,\o}\;,\qquad \tilde g^{(\le h)}_\o(\kk')
=\sum_{j=h_\b}^h \tilde g^{(j)}_\o(\kk') \; ,\cr
\psi^{(\le h)\pm}_{\kk} &= \bigoplus_{j=h_\b}^h
\psi^{(j)\pm}_{\kk}\; ,\qquad \hat g^{(\le h)}_\kk
=\sum_{j=h_\b}^h \hat g^{(j)}_\kk\; .\cr} \Eq(2.15)$$
%
Of course $\psi^{\pm}_{\kk} \= \psi^{(\le 1)\pm}_{\kk}$,
and $\hat g_\kk \= \hat g^{(\le 1)}(\kk)$.
\*
\0{\bf 2.2.} The most naive definition for the
{\sl effective potential} ``at scale'' $h$ is the following:
%
$$e^{\VV^{(h)}(\psi^{(\le h)}) + E_h}
=\int P(d\psi^{(h+1)})\ldots \int P(d\psi^{(1)}) e^{\VV(\psi^{(\le 1)})}
\; , \Eq(2.16) $$
%
where $E_h$ is defined so that $\VV^{(h)}(0)=0$.
If we define $\pp=(p,0)$ and $\pp_F=(p_F,0)$, $\VV(\psi^{(\le
1)})$ can be written as
%
$$ \VV(\psi^{(\le 1)}) =
\sum_{n=-[L/2]}^{[(L-1)/2]}{1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
\, \, \l \hat \f_n \,
\psi^{(\le 1)+}_\kk \psi^{(\le 1)-}_{\kk+2n\pp} \; ,\Eq(2.17) $$
%
with $\hat \f_0=0$, see \equ(1.12).
Hence the effective potential on scale $h\le 0$ can be represented as
%
$$\VV^{(h)}(\psi^{(\le h)}) =
\sum_{n=-[L/2]}^{[(L-1)/2]} {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
\; \WW_n^{(h)}(\kk) \,
\psi^{(\le h)+}_{\kk,\o} \psi^{(\le h)-}_{\kk+2n\pp,\o'} \; .
\Eq(2.18) $$
Note that here, as always in the following, the momentum $k$ is defined modulo
$2\p$.
The kernel $\WW_n^{(h)}(\kk)$ admits the diagrammatic
representation in terms of chain graphs
described in \S 2.3 below. Note that a sum over the
labels $\o,\o'$ could be introduced in
\equ(2.18), but it is useless as the labels
$\o$ and $\o'$ are uniquely determined by the signs of $k$
and $k+2np$ respectively: $\o=\sign(k)$ and $\o'=\sign(k+2np)$
(see comments after \equ(2.12)).
We shall study the convergence of the effective potential in terms of the norm
%
$$\|\VV^{(h)}\| \= \sum_{n=-[L/2]}^{[(L-1)/2]} \sup_{\kk\in {\cal D}_h} \left|
\WW_n^{(h)}(\kk) \right|\; ,\Eq(2.19)$$
%
where ${\cal D}_h \= \{\kk\in {\cal D}_{L,\b}:
\sum_{h'=h_\b}^h \hat f_{h'}(\kk)\not=0\}$.
\*
\0{\bf 2.3.} A graph $\th$ of order $q$
(see Fig.2.1 below) is a chain of $q+1$ lines
$\ell_1,\ldots,\ell_{q+1}$ connecting
a set of $q$ ordered points ({\sl vertices}) $v_1,\ldots,v_q$, so that
$\ell_i$ enters $v_i$ and $\ell_{i+1}$ exits from $v_i$; $\ell_1$ and
$\ell_{q+1}$ are the {\sl external lines} of the graph and both have
a free extreme, while the others are the {\sl internal lines}; we shall denote
${\rm int}(\th)$ the set of all internal lines.
We say that $v_i2^\t$,
%
$$ \sup_{\kk\in{\cal D}_h} \Big|\tilde \WW^{(h)}_{n,q}(\kk) \Big| \le
(|\l| B_1)^q \, e^{-{\x\over2}|n|} \; , \Eq(2.25) $$
%
for some constant $B_1$.}
\*
The proof of Lemma 1 is in Appendix 2, \S A2.2.
\*
To obtain a bound like \equ(2.25) also for graphs with resonances
(and {\it so} for all graphs), a more refined procedure is required,
which next section will be devoted to.
\vskip2.truecm
\centerline{\titolo 3. Renormalization}
\*\numsec=3\numfor=1
\0{\bf 3.1.}
We introduce a {\sl localization operator}
$\LL$, which acts on the effective potential in the following way:
%
$$ \eqalign{
&\LL \left\{ {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
\psi^{(\le h)+}_{\kk'+\o_1 \pp_F,\o_1}
\psi^{(\le h)-}_{\kk'+\o_2 \pp_F+2n\pp,\o_2}\,
\WW_n^{(h)}(\kk'+\o_1 \pp_F) \right\} \cr
& = \d_{(\o_1-\o_2)p_F+2np,0} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
\psi^{(\le h)+}_{\kk'+\o_1 \pp_F,\o_1}
\psi^{(\le h)-}_{\kk'+\o_2 \pp_F,\o_2}\,
\WW_n^{(h)}(\o_1 \pp_F) \; . \cr} \Eq(3.1) $$
%
Note that $\kk'=0$ is not an allowed value, but
$\WW_n^{(h)}(\kk'+\o_1 \pp_F)$ is a well defined expression
for any real values of $\kk'$, so that
$\WW_n^{(h)}(\o_1 \pp_F)$ is well defined.
The effect of this operator is to ``isolate'' the problem connected to the
resonances, in order to treat it separately in a way that we shall discuss
below.
We say that $\LL \VV^{(h)}(\psi^{(\le h)})$ is the
{\sl relevant part} (or {\sl localized part})
of the effective potential $\VV^{(h)}(\psi^{(\le h)})$.
We perform the integration $P(d\psi)$ in the following way.
First we integrate the field with frequency $h=1$
(ultraviolet integration), which can be written, up to a constant,
%
$$ \eqalign{
P(d\psi^{(1)}) & =
\prod_{\kk} d\psi^{(1)+}_{\kk} d\psi^{(1)-}_{\kk} \cr
& \exp \Big\{ - {1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
\, \hat f_1^{-1}(\kk)
\Big[ \big( -ik_0+\cos p_F - \cos k \big)
\psi^{(1)+}_{\kk} \psi^{(1)-}_{\kk}
\Big] \Big\} \; , \cr} \Eq(3.2) $$
%
and we obtain $\VV^{(0)}(\psi^{(\le 0)})$ as a power series in $\l$,
convergent in the norm \equ(2.19), for $|\l|$ small enough,
say $|\l| \le \bar\e_0$, by \equ(2.25).
Then we decompose $\psi^{(\le0)}$ as in \equ(2.11), and we write,
using also the evenness of the potential,
%
$$ \LL \VV^{(0)}=\n_0 F_\n^{(0)} + s_0 F_\sigma^{(0)} \; , \Eq(3.3) $$
%
where $F_\n^{(0)}$ and $F_\s^{(0)}$ are given by
%
$$ \eqalign{
F_\n^{(h)} & = \sum_{\o=\pm1}{1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
\psi^{(\le h)+}_{\kk'+\o \pp_F,\o}
\psi^{(\le h)-}_{\kk'+\o \pp_F,\o} \; , \cr
%
F_\s^{(h)} & = \sum_{\o=\pm1}{1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
\psi^{(\le h)+}_{\kk'+\o\pp_F,\o}
\psi^{(\le h)-}_{\kk'-\o\pp_F,-\o} \; , \cr}
\Eq(3.4) $$
%
with $h=0$. Note that, if $|\l|$ is small enough, by \equ(2.25) there exists
$A_0$ such that
%
$$|s_0-\l\hat \f_m| \le A_0|\l|^2 e^{-m\x/2}\; ,\qquad |\n_0| \le A_0|\l|^2
\; . \Eq(3.5)$$
%
We have to study
%
$$\int P(d\psi^{\le 0}) \,
e^{\VV^{(0)}(\psi^{\le 0})} \; , \Eq(3.6) $$
%
where $P(d\psi^{(\le 0)})$ is the Grassmanian integration
with propagator
%
$$ \eqalign{
& g^{(\le 0)}(\xx;\yy) =
\sum_{\o,\o'=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
e^{-i\kk'\cdot(\xx-\yy)}\,e^{-i(\o x - \o' y)p_F}
\, \tilde g^{(\le 0)}_{\o,\o'}(\kk') \; , \cr
& \tilde g^{(\le 0)}_{\o,\o'}(\kk') = \d_{\o,\o'} \,
\tilde g^{(\le 0)}_{\o}(\kk') \; , \cr} \Eq(3.7) $$
with
%
$$ \tilde g^{(\le 0)}_\o(\kk') = {C_0^{-1}(\kk') \over
-ik_0 + (1-\cos k') \cos p_F + v_0\o \sin k' } \; ,
\qquad C_h^{-1}(\kk')=\sum_{j=h_\b}^h f_{j}(\kk') \; , \Eq(3.8) $$
%
see \equ(2.13), \equ(2.15).
We write
%
$$\int P(d\psi^{(\le 0)}) \, e^{\VV^0(\psi^{(\le 0)})} =
{1 \over \NN_0}\int \tilde P(d\psi^{(\le 0)})
\, e^{\tilde \VV^{(0)}(\psi^{\le 0})} \; , \Eq(3.9) $$
%
where $\NN_0$ is a suitable constant and, again up to a constant,
%
$$ \eqalign{
\tilde P(d\psi^{(\le 0)}) = &
\prod_{\kk}\prod_{\o=\pm1} d\psi^{(\le 0)+}_{\kk'+\o\pp_F,\o}
d\psi^{(\le 0)-}_{\kk'+\o\pp_F,\o} \cr
\exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
C_0(\kk') \Big[\Big( -ik_0-(\cos k'-1)\cos p_F +\o v_0\sin k' \Big) \cr
& \psi^{(\le0)+}_{\kk'+\o\pp_F,\o} \psi^{(\le0)-}_{\kk'+\o\pp_F,\o}
- \sigma_{0}(\kk') \, \psi^{(\le0)+}_{\kk'+\o\pp_F,\o}
\psi^{(\le0)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.10) $$
%
with $\sigma_{0}(\kk')=C_0^{-1}(\kk')\,s_0$ and
$\tilde \VV^{(0)}=\LL \tilde \VV^{(0)}+(1-\LL) \VV^{(0)}$, if
%
$$ \LL\tilde \VV^{(0)}=\n_0 F_\nu^{(0)} \; . \Eq(3.11) $$
The r.h.s of \equ(3.9) can be written as
%
$$ {1 \over \NN_0}\int P(d\psi^{(\le -1)}) \int \tilde
P(d\psi^{(0)}) \, e^{\tilde \VV^{(0)}(\psi^{(\le 0)})} \; , \Eq(3.12) $$
%
where $ P(d\psi^{(\le -1)})$ and $\tilde P(d\psi^{(0)})$ are given
by \equ(3.10) with $C_0(\kk')$ replaced with
$C_{-1}(\kk')$ and $f_0^{-1}(\kk')$ respectively,
and $\psi^{(\le 0)}$ replaced with
$\psi^{(\le -1)}$ and $\psi^{(0)}$ respectively.
The Grassmanian integration $\tilde P(d\psi^{(\le 0)})$
has propagator
%
$$ g^{(0)}(\xx;\yy) = \sum_{\o,\o'=\pm1}
e^{-i(\o x - \o' y)p_F}\,
g^{(0)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.13) $$
%
if
%
$$ g^{(0)}_{\o,\o'}(\xx;\yy)\=\int \tilde P(d\psi^{(0)})\,
\psi^{(0)-}_{\xx,\o}\psi^{(0)+}_{\yy,\o'} \Eq(3.14) $$
%
is given by
%
$$ g^{(0)}_{\o,\o'}(\xx;\yy)={1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
e^{-i\kk'\cdot(\xx-\yy)}f_0(\kk')[T_{0}^{-1}(\kk')]_{\o,\o'}
\; , \Eq(3.15) $$
%
where the $2\times2$ matrix $T_{0}(\kk')$ has elements
%
$$ \cases{
[T_{0}(\kk')]_{1,1} =
\left(-ik_0-(\cos k'-1)\cos p_F+ v_0\sin k' \right) \; , & \cr
[T_{0}(\kk')]_{1,2} = [T_{0}(\kk')]_{2,1} = - \sigma_{0}(\kk') \; , & \cr
[T_{0}(\kk')]_{2,2} =
\left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.16) $$
%
which is well defined on the support of $f_0(\kk')$, so that,
if we set
%
$$ A_{0}(\kk') = \det T_0(\kk') =
[ -ik_0-(\cos k'-1)\cos p_F ]^2 - (v_0\sin k')^2
- [\sigma_{0}(\kk')]^2 \; ,\Eq(3.17) $$
%
then
%
$$ T_{0}^{-1}(\kk')= {1\over A_{0}(\kk') }
\left( \matrix{
[\t_{0}(\kk')]_{1,1} & [\t_{0}(\kk')]_{1,2} \cr
[\t_{0}(\kk')]_{2,1} & [\t_{0}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.18) $$
%
with
%
$$ \cases{
[\t_{0}(\kk')]_{1,1} = \left[-ik_0-(\cos k'-1)
\cos p_F-v_0\sin k'\right] \; , & \cr
[\t_{0}(\kk')]_{1,2} = [\t_{0}(\kk')]_{2,1} =
\sigma_{0}(\kk') \; , & \cr
[\t_{0}(\kk')]_{2,2} = \left[-ik_0-(\cos k'-1)\cos p_F
+ v_0 \sin k' \right] \; . & \cr} \Eq(3.19) $$
%
We perform the integration
%
$$ \int \tilde P(d\psi^{(0)}) \, e^{\tilde \VV^{(0)}(\psi^{(\le 0)})}
\= e^{\VV^{-1}(\psi^{(\le -1)}) + \tilde E_0} \; , \Eq(3.20) $$
%
where $\tilde E_0 = \log \int \tilde P(d\psi^{(0)}) \, \exp
\{\tilde \VV^{(0)}(\psi^{(0)}) \}$. We can write
%
$$ \LL \VV^{(-1)}= \g^{-1}\n_{-1} F_\nu^{(-1)}+s_{-1} F_\sigma^{(-1)}
\; , \Eq(3.21) $$
%
with suitable constants $\n_{-1}$ and $s_{-1}$, and,
by following the same procedure which led from
\equ(3.9) to \equ(3.12), we have
%
$$ \eqalign{
\int P(d\psi^{(\le -1)}) \, e^{\VV^{(-1)}(\psi^{(\le -1)})}
& ={1\over \NN_1} \int \tilde P (d\psi^{(\le -1)}) \, e^{\tilde
\VV^{(-1)}(\psi^{(\le -1)})} \cr & =
{1\over \NN_1}\int P(d\psi^{(\le -2)}) \int \tilde P(d\psi^{(-1)}) \,
e^{\tilde \VV^{(-1)} (\psi^{(\le -1)})} \; , \cr} \Eq(3.22) $$
%
where, up to a constant,
%
$$ \eqalign{
\tilde P(d\psi^{(\le -1)}) &=
\prod_{\kk'}\prod_{\o=\pm1} d\psi^{(\le -1)+}_{\kk'+\o\pp_F,\o}
d\psi^{(\le -1)-}_{\kk'+\o\pp_F,\o} \cr
\exp \Big\{ &-\sum_{\o=\pm1} {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}}\,
C_{-1}(\kk') \, \Big[ \Big(-ik_0-(\cos k'-1)\cos p_F
+\o v_0\sin k' \Big) \cr
& \qquad\psi^{(\le-1)+}_{\kk'+\o\pp_F,\o} \psi^{(\le-1)-}_{\kk'+\o\pp_F,\o}
- \sigma_{-1}(\kk') \, \psi^{(\le-1)+}_{\kk'+\o\pp_F,\o}
\psi^{(\le-1)-}_{\kk'-\o\pp_F,-\o} \Big] \Big\} \; , \cr} \Eq(3.23) $$
%
with $\sigma_{-1}(\kk')$ $=$
$\sigma_0(\kk')+C_{-1}^{-1}(\kk') s_{-1}$
and
%
$$ \LL\tilde \VV^{(-1)}=
\g^{-1} \n_{-1} F_\nu^{(-1)} \; . \Eq(3.24) $$
The above procedure can be iterated, and at each step one has
to perform the integration
%
$$ \eqalignno{
&\quad\int P(d\psi^{(\le h)}) \, e^{\VV^{(h)}(\psi^{(\le h)})} ={1\over \NN_h}
\int \tilde P(d\psi^{(\le h)})\,e^{\tilde \VV^{(h)}(\psi^{(\le h)})} =
&\eq(3.25)\cr & ={1\over \NN_h}
\int P(d\psi^{(\le h-1)})\int \tilde P(d\psi^{(h)}) \,
e^{\tilde \VV^{(h)}(\psi^{(\le h)})} =
{1\over \NN_h} \int P(d\psi^{(\le h-1)}) e^{\VV^{(h-1)}(\psi^{(\le h-1)})
+\tilde E_h}\; , \cr}$$
%
which gives $\sigma_{h-1}(\kk')$ $=$
$\sigma_h(\kk')+C_{h}^{-1}(\kk') s_{h}$, and defines the propagator
%
$$ g^{(h)}(\xx;\yy) = \sum_{\o,\o'=\pm1}
e^{-i(\o x - \o' y)p_F}\,
g^{(h)}_{\o,\o'}(\xx;\yy) \; , \Eq(3.26) $$
%
with
%
$$ \eqalign{
g^{(h)}_{\o,\o'}(\xx;\yy) & \=
\int \tilde P(d\psi^{(h)})\,
\psi^{(h)-}_{\xx,\o}\psi^{(h)+}_{\yy,\o'} \cr
& = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
e^{-i\kk'\cdot(\xx-\yy)}f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'}
\; , \cr} \Eq(3.27) $$
%
and
%
$$ \cases{
[T_{h}(\kk')]_{1,1} = \left(-ik_0-(\cos k'-1)\cos p_F+
v_0\sin k' \right) \; , & \cr
[T_{h}(\kk')]_{1,2} = [T_{h}(\kk')]_{2,1} = - \sigma_{h}(\kk') \; , & \cr
[T_{h}(\kk')]_{2,2} =
\left(-ik_0-(\cos k'-1)\cos p_F-v_0\sin k'\right) \; , \cr} \Eq(3.28) $$
%
so that
%
$$ T_{h}^{-1}(\kk')= {1\over A_{h}(\kk') }
\left( \matrix{
[\t_{h}(\kk')]_{1,1} & [\t_{h}(\kk')]_{1,2} \cr
[\t_{h}(\kk')]_{2,1} & [\t_{h}(\kk')]_{2,2} \cr} \right) \; , \Eq(3.29) $$
%
where
%
$$ \cases{
[\t_{h}(\kk')]_{1,1} = [-ik_0-(\cos k'-1) \cos p_F-v_0\sin k'] \; , & \cr
[\t_{h}(\kk')]_{1,2} = [\t_{h}(\kk')]_{2,1} = \sigma_{h}(\kk') \; , & \cr
[\t_{h}(\kk')]_{2,2} = [-ik_0-(\cos k'-1)\cos p_F
+ v_0 \sin k' ] \; , & \cr} \Eq(3.30) $$
%
and
%
$$A_{h}(\kk') =
\left[-ik_0-(\cos k'-1)\cos p_F\right]^2
- \left(v_0\sin k'\right)^2 - [\s_{h}(\kk')]^2 \; .\Eq(3.31) $$
%
We can define also
%
$$ \eqalign{
g^{(\le h)}_{\o,\o'}(\xx;\yy) & \=
\int \tilde P(d\psi^{(\le h)})\,
\psi^{(\le h)-}_{\xx,\o}\psi^{(\le h)+}_{\yy,\o'} \cr
& = {1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \,
e^{-i\kk'\cdot(\xx-\yy)}C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'}
\; , \cr} \Eq(3.32) $$
%
where the last identity follows from \equ(3.23) (with $h$ in place of
$-1$) and \equ(3.25). Set
%
$$ \tilde g^{(h)}_{\o,\o'}(\kk')
= f_h(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \qquad
\tilde g^{(\le h)}_{\o,\o'}(\kk')
= C_h^{-1}(\kk')[T_{h}^{-1}(\kk')]_{\o,\o'} \; , \Eq(3.33) $$
%
so that
%
$$ \eqalign{
g^{(h)}_{\o,\o'}(\xx;\yy) & =
{1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}
\, \tilde g^{(h)}_{\o,\o'}(\kk')\; , \cr
%
g^{(\le h)}_{\o,\o'}(\xx;\yy) & =
{1\over L\b} \sum_{\kk'\in {\cal D}_{L,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}
\, \tilde g^{(\le h)}_{\o,\o'}(\kk')\; .
\cr} \Eq(3.34) $$
The localized part of the effective potential will be written as
%
$$ \LL \tilde \VV^{(h)}=\g^h \n_h F_\n^{(h)} \; , \Eq(3.35) $$
%
which defines the {\sl running coupling constants} $\n_h$. Moreover
%
$$\s_h(\kk')=\sum_{j=h}^{0} C_{j}^{-1}(\kk') \, s_j \; . \Eq(3.36) $$
%
Note that, thanks to the definition of $\c(\kk')$, see \equ(2.3),
if $f_h(\kk') \not= 0$, we have
%
$$ \sigma_h(\kk')= C_h^{-1}(\kk')
\, s_h + \sum_{j=h+1}^{0} s_j \; . \Eq(3.37) $$
%
Hence, by \equ(2.5) and the second equation in \equ(3.8), $\s_h(\kk')$ is a
smooth function on $\TTT^1\times \RRR$, such that
$\s_h(\kk')= \sum_{j=h}^0 s_j$ for $0\le |\kk'|\le t_0\g^h$ and
$\s_h(\kk')= \sum_{j=h+1}^0 s_j$ for $|\kk'|= a_0\g^h$;
we define $\sigma_h\equiv \sum_{j=h}^0 s_j$.
Note that
%
$$ \Re {[A_h(\kk')+\s_h(\kk')^2]} = -k_0^2 -4 \sin^2 {k'\over 2}
\sin \Big(p_F+ {k'\over 2}\Big) \sin \Big(p_F- {k'\over 2}\Big)\; ,\Eq(3.38)$$
%
and $p_F \pm k'/2 >0$ on the support of $f_h(\kk')$.
Hence there is a constant $G_3$, such that, on the support of $f_h(\kk')$
%
$$|A_h(\kk') + \s_h(\kk')^2 | \ge G_3^2 \g^{2h}\; .\Eq(3.39)$$
%
Let us now define, for any complex $\l$ with $|\l| \le \bar\e_0$,
%
$$h^* \= \inf \{h\ge h_\b:G_3 \g^h \ge 2\bar\s \} \; ,\qquad
\bar\s = |\l \hat \f_m| \not= 0 \; ,\Eq(3.40)$$
%
and let us suppose that there exists $\e_0 \le \bar\e_0$, such that,
for $|\l|\le \e_0$ and $h\ge h^*$,
%
$${1\over 2}\bar\s \le |\s_h(\kk')| \le {3\over 2}\bar\s\; ;\Eq(3.41)$$
%
such an assumption will be justified by Lemma 2, in \S 3.4.
It follows that there exists a constant $G_1$, such that, for any $h\ge h^*$,
%
$$|\tilde g^{(h)}_{\o,\o'}(\kk')| \leq G_1 \g^{-h} \; ,\Eq(3.42)$$
%
and, if $\l$ is real,
%
$$|\tilde g^{(\le h^*)}_{\o,\o'}(\kk')| \leq G_1 \g^{-h^*} \; .\Eq(3.43)$$
%
Note that, if $\l$ is real, the factor $2$ in front of $\bar\s$ in the
definition of $h^*$ could be substituted with any constant, without loosing
the bounds \equ(3.42) and \equ(3.43)
Finally, since $|s_h|\le |\s_h|+|\s_{h+1}| \le3\bar\s$,
it easy to prove that, for $|\l|\le \e_0$, $h\ge h^*$, $0\le t \le 1$
and any $\qq$,
%
$$\left| {d\over dt} \tilde g_{\o,\o'}^{(h)} (t\kk'+\qq)\right|
\le G_2 |\kk'|\g^{-2h}\; ,\Eq(3.44)$$
%
for a suitable constant $G_2$, a bound which will play an important role in
the following.
\*
\0{\bf 3.2.} The new effective potential $\tilde \VV^{(h)}$ can be written
as in \equ(2.18), by substituting the kernel $\WW_n^{(h)}(\kk)$ with
a new kernel $\WW_{\RR,n}^{(h)}(\kk)$, which admits
a graph representation in terms of new labeled graphs $\th_\RR$, the
{\sl renormalized graphs}, which differ from
the ones described in \S 2.3 in the following points:\\
%
$\bullet$ there is no resonant vertex with $n_v=\pm m$;\\
$\bullet$ there are new resonant vertices with $n_v=0$, produced by the
renormalization procedure, to which we associate a label $h_v\le 0$
and a factor $\g^{h_v} \n_{h_v}$;\\
$\bullet$ at least one of the lines emerging from the resonant vertices with
label $h_v$ has frequency label $h_v$,
while the other has frequency label $h_v$ or $h_v-1$ (it is an immediate
consequence of the renormalization procedure and momentum conservation);\\
$\bullet$ the internal lines $\ell$'s carry
two labels $\o_{\ell}^i$, $i=1,2$;\\
$\bullet$ given a line $\ell$, the corresponding propagator is
$\tilde g^{(h_\ell)}_{\o_{\ell}^1,\o_{\ell}^2}(\kk_{\ell}')$;\\
$\bullet$ on each resonant cluster $V$ (including $\th_\RR$ itself, if it
is a resonance) the $\RR\=\openone-\LL$ operator acts;\\
$\bullet$ the conservation of the momentum measured from the Fermi surface in
each vertex gives the constraint
$$ k'_{\ell_v} = k' +\sum_{\tilde v \le v} [ 2n_{\tilde v}p +
( \o^1_{\ell_{\tilde v'}} - \o^2_{\ell_{\tilde v}} )p_F] \; . \Eq(3.45)$$
Then the second equation in \equ(2.21) is replaced with
%
$$\WW^{(h)}_{\RR,n,q}(\kk) =
\sum_{\th_{\RR}\in\TT_{\RR,n,q}^{h}} {\rm Val}(\th_{\RR}) \; , \Eq(3.46) $$
%
where $\TT_{\RR,n,q}^{h}$ denotes the family of renormalized
graphs of order $q$ and scale $h$, such that $\sum_{v\in\th_\RR} 2n_v p +
\sum_{\ell\in {\rm int}(\th)} ( \o^1_\ell-\o^2_\ell )p_F = 2np \;$ and
$-[L/2] \le n \le [(L-1)/2]$, and ${\rm Val}(\th_\RR)$ is computed
following the rules listed above.
\*
\0{\bf 3.3.}
The renormalization procedure allows us to improve the bound
of the graph values, and to extend Lemma 1 to cover also
the case of graphs with resonances. As an example,
let us consider a resonant cluster $V$,
which does not contain other resonant clusters;
we can associate to it a {\sl resonance value}
%
$$ \X_V^h(\kk') = \prod_{\ell\in V}
\tilde g^{(h_\ell)}_{\o_{\ell}^1,\o_{\ell}^2}(\kk'_{\ell}) \; ; \Eq(3.47) $$
%
the lines $\ell_V^i$ and $\ell_V^o$ have a momentum measured from the
Fermi surface (by the Definition in \S 2.5) $k'\=k_{\ell_V^i}'=
k_{\ell_V^o}'$, and $h=\min_{\ell\in V}\{h_{\ell}\}$ is the scale of $V$.
The effect of the localization operator is to replace $\X_V^h(\kk')$ with
$\LL \X_V^h(\kk') \= \X_V^h(\V 0)$, so that
the effect of the $\RR$ operator is to replace $\X_V^h(\kk')$ with
%
$$ \RR \X_V^h(\kk') \= \X_V^h(\kk') - \X_V^h(\V 0)=
\int_0^1 dt\Big[ {d\over dt}
\X_V^h(t\kk') \Big] \; . \Eq(3.48) $$
%
Hence, by using \equ(3.42) and \equ(3.44), we can bound $\RR\X_V^h(\kk')$
in the following way:
%
$$ |\RR \X_V^h(\kk') | \le \sum_{\ell' \in V}
a_0 G_2 G_1^{L_V-1} \g^{h_V^e -h_{\ell'}} \prod_{\ell\in V} \g^{-h_\ell}\; ,
\Eq(3.49) $$
%
where $h_V^e$ is the external scale of $V$ (see \S 2.4).
Hence, with respect to the unrenormalized bound,
$\RR$ produces an extra factor of the form $\g^{h_V^e -h_{\ell'}}$, which
can be used to compensate the lack of the small factor associated to non
resonant clusters, as a consequence of the condition \equ(1.26).
Concerning the resonant vertices, the renormalization procedure eliminated
those with $n_v=\pm m$, but introduced new resonant vertices with $n_v=0$.
The new vertices carry a factor $\g^h$, which is a real gain
in the power counting, if one can prove that $\n_h$ is uniformly bounded.
In fact the discussion can be generalized to graphs containing an
arbitrary number of re\-so\-nan\-ces: all these improvements will be used
in Appendix 2, \S A2.3, to prove the following
extension of Lemma 1.
\*
\0{\bf 3.4.} {\cs Lemma 2.} {\it If $\hat \f_m\not=0$
and $\g>2^{\t}$, there exists $\e_0 \le \bar\e_0$, such that, for
$|\l|\le\e_0$ and $h^*\le h\le 0$, we have
%
$$\eqalignno{\sup_{\kk\in {\cal D}_h}
\Big| \WW^{(h)}_{\RR,n,q}(\kk) \Big| &\le (|\l|B_2)^q \,
e^{-{\x\over2}|n|}\; , &\eq(3.50)\cr
|\s_h-\l\hat \f_m| &\le A |\l|^2 e^{-{\x\over2}m}\; ,\qquad |\n_h|
\le B_3|\l| \; , &\eq(3.51)\cr} $$
%
for some constants $B_2$, $B_3$, $A$.}
\*
\0{\cs Remark.} Lemma 2 implies that the series
defining the kernel of the effective potential is convergent in the norm
\equ(2.19), uniformly in $L=L_i$ and $\b$.
\vskip2.truecm
\centerline{\titolo 4. The two-point Schwinger function}
\*\numsec=4\numfor=1
\0{\bf 4.1.}
In this section we define a perturbative expansion, similar to the one
discussed for the effective potential in \S 3,
for the two-point Schwinger function, defined by \equ(1.10),
which can be rewritten
%
$$S^{L,\b}(\xx;\yy) = \lim_{M\to\io}
{\partial^2\over\dpr\phi_\xx^+\partial\phi^-_\yy}
{1\over \NN_1} \int P(d\psi) \, e^{\VV(\psi)+ \int d\xx
\big( \phi^+_\xx \psi^-_\xx + \psi^+_\xx \phi^-_\xx \big) }
\; \Big|_{\phi^+=\phi^-=0} \; , \Eq(4.1) $$
%
where $\int d\xx$ is a shortcut for $\sum_{x\in\L} \int_{-\b/2}^{\b/2} dx_0$,
$\NN_1 = \int P(d\psi)\,e^{\VV(\psi)}$ and $\{\phi_\xx^\pm\}$ are
Grassmanian variables (the {\sl external field}), anticommuting with
$\{\psi_\xx^\pm\}$. Note that all objects appearing in the r.h.s. of
\equ(4.1), as well as the other defined below, depend on $L$ and $\b$, but
we shall not indicate explicitly this dependence, in order to simplify the
notation.
Setting $\psi=\psi^{(\le 0)}+\psi^{(1)}$
and performing the integration over the
field $\psi^{(1)}$ (ultraviolet integration), we find
%
$$ \eqalignno{
S^{L,\b}(\xx;\yy) & = \lim_{M\to\io}
{\partial^2 \over \dpr \phi_\xx^+\partial\phi^-_\yy}
e^{\int d\xx d\yy\, \phi_\xx^+ \, V^{(0)}_{\phi,\phi}(\xx;\yy) \, \phi_\yy^- }
& \eq(4.2) \cr
& {1\over\NN_0}\,\int P(d\psi^{(\le 0)})\, e^{\int d\xx
\big( \phi^+_\xx\psi^{(\le 0)-}_\xx +
\psi^{(\le 0)+}_\xx \phi^-_\xx \big)} e^{\VV^{(0)}(\psi^{(\le 0)})
+ W^{(0)}(\psi^{(\le 0)},\phi) } \; \Big|_{\phi^+=\phi^-=0} \; ,\cr} $$
%
where $\NN_0 = \int P(d\psi^{(\le 0)})\,e^{\VV^{(0)}(\psi^{(\le 0)})}$,
%
$$ W^{(0)}(\psi^{(\le 0)},\phi) = \int d\xx d\yy
\Big( \phi_\xx^+ \, K^{(0)}_{\phi,\psi}(\xx;\yy) \, \psi_\yy^{(\le 0)-} +
\psi_\xx^{(\le 0)+} \, K^{(0)}_{\psi,\phi}(\xx;\yy) \, \phi_\yy^- \Big)
\; ,\Eq(4.3) $$
%
$$V^{(0)}_{\phi,\phi}(\xx;\yy)= g^{(1)}(\xx;\yy)+
K^{(0)}_{\phi,\phi}(\xx;\yy)\; , \Eq(4.4)$$
%
and
$$ \eqalign{
\int d\xx d\yy \, \c_\xx^{(1)+} & K^{(0)}_{\c^{(1)},\c^{(2)}}(\xx;\yy) \,
\c_\yy^{(2)-} \cr & = \sum_{n=L/2}^{[(L-1)/2]}
{1\over L\b} \sum_{\kk\in {\cal D}_{L,\b}}
\, \hat K^{(0)}_{\c^{(1)},\c^{(2)},n}
(\kk) \c_{\kk}^{(1)+} \c_{\kk+2n\pp}^{(2)-} \; . \cr} \Eq(4.5) $$
%
The kernels $\hat K^{(0)}_{\c^{(1)},\c^{(2)},n}(\kk)$ can be represented
as sums of graphs of the same type of those appearing in the graph expansion
of the effective potential $\VV^{(0)}$; the new graphs differ only in the
following respects:\\
$\bullet$ if $\c^{(2)}=\phi$, the right external line is associated to the
$\phi^-$ field and the graph ends with a vertex carrying no $\l\hat \f_n$
factor;\\
$\bullet$ if $\c^{(1)}=\phi$, the left external line is associated to the
$\phi^+$ field and the graph begins with a vertex carrying no $\l\hat \f_n$
factor;\\
Note that there is at least one vertex carrying a $\l\hat \f_n$ factor.
The propagators of the internal lines emerging from vertices without a
$\l\hat \f_n$ factor will be called the {\sl external propagators}.
We have, in particular,
%
$$ K^{{(0)}}_{\phi,\phi}(\xx;\yy) = \sum_{q=3}^{\io}
\sum_{n=-[L/2]}^{[(L-1)/2]}
\sum_{\th \in \TT^{\phi\phi,0}_{n,q}} {1\over L\b}
\sum_{\kk\in {\cal D}_{L,\b}}
\, e^{-i\kk\cdot(\xx-\yy) + 2 i n p y} \,
{\rm Val(\th)} \; , \Eq(4.6) $$
%
where $\TT^{\phi\phi,0}_{n,q}$ is the set of all labeled graphs of order
$q$ with two external propagators, such that $\sum_{v\in\th}n_v=n$ and
$h_{\ell}=1$ $\forall\ell$; moreover, ${\rm Val(\th)}$ is obtained from
\equ(2.20) by adding the two external propagators.
It is very easy to take the limit $M\to\io$ in \equ(4.2). In fact, for $M$
large enough, the measure $P(d\psi^{(\le 0)})$ is independent of $M$; hence,
the limit is obtained by taking the limit as $M\to\io$ of the r.h.s. of
\equ(4.4). The limit of $K^{(0)}_{\phi,\phi}(\xx;\yy)$ is trivial; in fact
each graph contributing to it behaves as $1/|k_0|^2$, as $k_0\to\io$, so that
the sum over $k_0$ is absolutely convergent. On the contrary, the limit of
$g^{(1)}$ has to be done carefully, since it involves a sum over
$k_0$, which is not absolutely convergent; however, by using standard
techniques, one can show easily that the limit does exist and, uniformly in
$L$ and $\b$, if $|x_0-y_0| \le \b/2$ and $|x-y| \le L/2$,
%
$$ \Big| g^{(1)}(\xx;\yy)\Big|
\le {C_N\over 1+ |\xx-\yy|^N} \; , \Eq(4.6a) $$
%
for any $N\ge 0$ and suitable constants $C_N$. Hence, from now on, we shall
suppose that the limit ${M\to\io}$ has been performed, but we shall still
use the same notation for $g^{(1)}$ and $K^{(0)}_{\phi,\phi}(\xx;\yy)$.
Equation \equ(4.2) can be written
%
$$S^{L,\b}(\xx;\yy) = V^{(0)}_{\phi,\phi}(\xx;\yy) + S^{(0)}(\xx;\yy)
\; , \Eq(4.7)$$
where
$$\eqalign{
S^{(0)}(\xx;\yy) &= {\partial^2 \over \dpr \phi_\xx^+\partial\phi^-_\yy}
{1\over\NN_0}\,\int P(d\psi^{(\le 0)})\cr
& e^{\int d\xx \big( \phi^+_\xx\psi^{(\le 0)-}_\xx +
\psi^{(\le 0)+}_\xx \phi^-_\xx \big)} e^{\VV^{(0)}(\psi^{(\le 0)})
+ W^{(0)}(\psi^{(\le 0)},\phi) } \; \Big|_{\phi^+=\phi^-=0}
\; .\cr}\Eq(4.8) $$
%
\*
\0{\bf 4.2.}
We proceed now as in \S 3, using the same notations. We write
%
$$ \eqalign{
S^{(0)}(\xx;\yy) & = {\dpr^2 \over \dpr\phi_\xx^+\dpr\phi^-_\yy}
{1\over \tilde\NN_0} \, \int \tilde P(d\psi^{(\le 0)}) \cr
& e^{ \int d\xx
\big(\phi^+_\xx \psi^{(\le 0)-}_\xx + \psi^{(\le 0)+}_\xx\phi^-_\xx
\big)} e^{\tilde \VV^{(0)}(\psi^{(\le 0)}) + W^{(0)}(\psi^{(\le 0)},\phi) } \;
\Big|_{\phi^+=\phi^-=0}\; , \cr} \Eq(4.9) $$
%
and decompose $\tilde \VV^{(0)} = \LL \tilde \VV^{(0)} + \RR \VV^{(0)}$.
On the contrary, we do not split $W^{(0)}$ into a relevant and an
irrelevant part.
The integration over $\psi^{(0)}$ gives
%
$$ \eqalign{
S^{(0)}(\xx;\yy) & = {\dpr^2\over\dpr\phi_\xx^+\dpr\phi^-_\yy} \;
e^{\int d\xx d\yy \phi_\xx^+ V^{(-1)}_{\phi,\phi}(\xx;\yy) \phi_\yy^- }
{1\over \tilde \NN_1} \,\int \tilde P(d\psi^{(\le -1)}) \cr
& e^{\int d\xx \big( \phi^+_\xx\psi^{(\le -1)-}_\xx+\psi^{(\le-1)+}_\xx
\phi^-_\xx \big)}
e^{\VV^{(-1)}(\psi^{(\le -1)}) + W^{(-1)}(\psi^{(\le-1)},\phi) }
\;\Big|_{\phi^+=\phi^-=0} \; , \cr} \Eq(4.10) $$
%
with
%
$$W^{(-1)}(\psi^{(\le -1)},\phi) = \int d\xx d\yy
\Big( \phi_\xx^+ \, K^{(-1)}_{\phi,\psi}(\xx,\yy) \, \psi_\yy^{(\le -1)-}
+ \psi_\yy^{(\le -1)+} \, K^{(-1)}_{\psi,\phi}(\xx;\yy)\, \phi_\xx^-
\Big) \; ,\Eq(4.11) $$
%
$$V^{(-1)}_{\phi,\phi}(\xx;\yy)= g^{(0)}(\xx;\yy)+
K^{(-1)}_{\phi,\phi}(\xx;\yy) \; . \Eq(4.12) $$
%
The kernels $\hat K^{(-1)}_{\c^{(1)},\c^{(2)},n}(\kk)$
can be represented as sums of graphs of the same type of those appearing
in the graph expansion of the effective potential $\VV^{(-1)}$;
the new graphs differ only in the following respects:\\
$\bullet$ if $\c^{(2)}=\phi$, the right external line is associated to the
$\phi^-$ field and the graph ends with a vertex carrying no $\l\hat \f_n$
factor;\\
$\bullet$ if $\c^{(1)}=\phi$, the left external line is associated to the
$\phi^+$ field and the graph begins with a vertex carrying no $\l\hat \f_n$
factor;\\
$\bullet$ $\RR\=\openone$ on resonances containing an external propagator
(defined as before).\\
$\bullet$ $h_\th=0$ for all graphs, if $\c^{(1)}=\c^{(2)}=\phi$.
\*
\0{\bf 4.3.} The above construction can be iterated and we find, for any
$h^* \le h\le 0$,
%
$$S^{L,\b}(\xx;\yy) = \sum_{h'=h}^0 V^{(h')}_{\phi,\phi}(\xx;\yy) +
S^{(h)}(\xx;\yy) \; , \Eq(4.13)$$
%
where
%
$$\eqalign{
S^{(h)}(\xx;\yy) &= {\partial^2 \over \dpr \phi_\xx^+\partial\phi^-_\yy}
{1\over\NN_h}\,\int P(d\psi^{(\le h)}) \cr
& e^{\int d\xx \big( \phi^+_\xx\psi^{(\le h)-}_\xx +
\psi^{(\le h)+}_\xx \phi^-_\xx \big)} e^{\VV^{(h)}(\psi^{(\le h)})
+ W^{(h)}(\psi^{(\le h)},\phi) } \; \Big|_{\phi^+=\phi^-=0} \; ,\cr}
\Eq(4.14) $$
%
$$W^{(h)}(\psi^{(\le h)},\phi) = \int d\xx d\yy
\Big( \phi_\xx^+ \, K^{(h)}_{\phi,\psi}(\xx,\yy) \, \psi_\yy^{(\le h)-}
+ \psi_\yy^{(\le h)+} \, K^{(h)}_{\psi,\phi}(\xx;\yy)\, \phi_\xx^-
\Big) \; ,\Eq(4.15) $$
%
$$V^{(h)}_{\phi,\phi}(\xx;\yy)= g^{(h+1)}(\xx;\yy)+K^{(h)}_{\phi,\phi}
(\xx;\yy) \; . \Eq(4.16)$$
The kernels $\hat K^{(h)}_{\c^{(1)},\c^{(2)},n}(\kk)$ can be represented
as sums of graphs of the same type of those appearing in the graph expansion
of the effective potential $\VV^{(h)}$; the new graphs differ only in the
following respects:
\item{$\bullet$} if $\c^{(2)}=\phi$, the right external line is associated
to the $\phi^-$ field and the graph ends with a vertex carrying no
$\l\hat \f_n$ factor;
\item{$\bullet$} if $\c^{(1)}=\phi$, the left external line is associated
to the $\phi^+$ field and the graph begins with a vertex carrying no
$\l\hat \f_n$ factor;
\item{$\bullet$} $\RR\=\openone$ on resonances containing an external
propagator (defined as before);
\item{$\bullet$} $h_\th=h+1$ for all graphs, if $\c^{(1)}=\c^{(2)}=\phi$.
Let us now suppose that $L=L_i$, $i\in\ZZZ^+$, so that the condition
\equ(1.19) is satisfied.
The integration over the field $\psi^{(\le h^*)}$ can be performed in a single
step, since the covariance $g^{(\le h^*)}$ satisfies the same bound as
$g^{(h^*)}$, see \equ(3.42) and \equ(3.43).
Then the functional derivatives in \equ(4.1) give
%
$$S^{L_i,\b}(\xx;\yy) =
\sum_{h=h^*}^{0} \Big( g^{(h+1)}(\xx;\yy)+K_{\phi,\phi}^{(h)}(\xx;\yy)\Big)
+ g^{(\le h^*)}(\xx;\yy) + K_{\phi,\phi}^{(1$,
%
$$ \Big| S_2(\xx;\yy) \Big|
\le C_N \sum_{h=h^*}^{h_x-1} \g^h +
\sum_{h=h_x}^0 {C_N\g^h\over \g^{Nh} |\xx-\yy|^N}
\le \g^{h_x} C_N \le {C_N\over 1+ |\xx-\yy|} \; .\Eq(4.35)$$
%
On the other hand,
if $|\xx-\yy|\ge \g G_3(2|\s|)^{-1}$, \equ(4.33) implies that
%
$$ \Big| S_2(\xx;\yy) \Big|
\le {C_N\over |\xx-\yy|^N}\sum_{h=h^*}^0 \g^{-(N-1)h}
\le {C_N\over |\xx-\yy|^N} |\s|^{-N+1}
\le {C_N|\s|\over 1+ |\s|^N|\xx-\yy|^N} \; , \Eq(4.34)$$
%
provided that $N>1$.
The proof of \equ(1.24a) is an easy consequence of the definition \equ(3.26).
In fact, by proceeding as in Appendix 3, one can prove that
%
$$\left|g^{(h)}(\xx;\yy) - \sum_{\o=\pm 1} \int{d\kk'\over (2\p)^2}
\tilde g_\o^{(h)}(\kk')\right| \le \fra{C_0 \g^{2h}}{\g^{2h} + \bar\s^2}
\left[ \fra{\bar\s^2 \g^h}{\g^{2h} + \bar\s^2} + \bar\s\right] \le C_0
\bar\s\; .\Eq(4.36)$$
%
A similar bound is valid for $g^{(\le h^*)}(\xx;\yy)$; hence we have
%
$$|S_1(\xx;\yy) - g(\xx-\yy)| \le \sum_{h=h^*}^0 C_0 \bar\s \le
C_0 \bar\s \log (\bar\s^{-1})\; .\Eq(4.37)$$
The continuity of $S(\xx;\yy)$ as a function of $\l\in\RRR$, $|\l|\le \e_0$,
is completely trivial for $\l\not= 0$ and, in $\l=0$, immediately follows from
\equ(1.20), \equ(1.23) and \equ(1.24a).
\*
\0{\bf 4.5.}
The proof of item (iii) of Theorem 1 is based on similar arguments, applied
to the finite $L=L_i$ and $\b$ case, but one has to consider more carefully
the contribution of the scales $h a_0\g^h\}$.
In fact, for $h\o$. We define, for $i\ge 0$,
$$\eqalign{
h_+(x) &= p_{2i} + \fra{p_{2i+2}-p_{2i}}{q_{2i+2}-q_{2i}} (x-q_{2i})
\quad \hbox{if}\; q_{2i}\le x \le q_{2i+2} \cr
h_-(x) &= p_{2i+1} + \fra{p_{2i+3}-p_{2i+1}}{q_{2i+3}-q_{2i+1}} (x-q_{2i+1})
\quad \hbox{if}\; q_{2i+1}\le x \le q_{2i+3} \cr} \Eqa(A3.1)$$
Note that the graph of $h_+(x)$ ($h_-(x)$) is made by a sequence of segments
joining the points $(q_{2i},p_{2i})$ and $(q_{2i+2},p_{2i+2})$
($(q_{2i+1},p_{2i+1})$ and $(q_{2i+3},p_{2i+3})$).
The well known properties of the convergents (see, for example, [D]) imply
that\\
(a) $h_+(x) > \o x > h_-(x)$, $\forall x\ge q_1$.\\
(b) $\d_+(x)\equiv h_+(x)-\o x$ and $\d_-(x)\equiv \o x-h_-(x)$ are
strictly decreasing functions and $\lim_{x\to\io} \d_{\pm}(x)=0$.\\
(c) If $k,n\in\NNN$, $n\ge q_0$ and $\o n-k<0$, then $k-\o n\ge \d_+(n)$,
the equality being satisfied iff $k=p_{2i}$, $n=q_{2i}$, $i\ge 0$; vice versa,
if $k,n\in\NNN$, $n\ge q_1$ and $\o n-k>0$, then $\o n-k\ge \d_-(n)$, the
equality being satisfied iff $k=p_{2i+1}$, $n=q_{2i+1}$, $i\ge 0$.
\*
\0{\bf A1.2.} {\cs Lemma 3.}
{\it If $k,n\in\NNN$, $i\ge 2$ and $q_1\le n\le q_i/2$, then
$$|n \fra{p_i}{q_i} -k| \ge \d_n\equiv \fra12 \min \{ \d_+(n),\d_-(n) \}
\; . $$
}
\*
\0{\bf A1.3.} {\it Proof of Lemma 3.}
Suppose that $i\ge 2$ is even; the property (a) implies that
$n \fra{p_i}{q_i} -k > n\o -k$, so that, by (c),
$\forall k,n\in\NNN$, $n\ge q_1$,
$$n\o-k>0 \;\Rightarrow\; n \fra{p_i}{q_i} -k>\d_-(n) \; . $$
%
If $n\o-k<0$ and $q_1\le n\le q_i/2$, we have, by (a), (b) and (c),
%
$$\eqalign{
&-n \fra{p_i}{q_i} +k = -n\o+k -n(\fra{p_i}{q_i} -\o) \ge
\d_+(n)- \fra{n}{q_i}(p_i-\o q_i) =\cr
&=\d_+(n)- \fra{n}{q_i} \d_+(q_i) \ge \d_+(n)- \fra12 \d_+(q_i)
\ge \fra12 \d_+(n) \; . \cr}$$
%
Hence, if $i\ge 2$ is even and $q_1\le n\le q_i/2$,
$|n \fra{p_i}{q_i} -k|\ge \min \{\fra12 \d_+(n),\d_-(n)\}$.
Analogously, if $i$ is odd, one can show that,
if $q_1\le n\le q_i/2$, $|n \fra{p_i}{q_i} -k|\ge \min \{\fra12 \d_-(n),
\d_+(n)\}$.
The claim of Lemma 3 immediately follows from the previous remarks.
\qed
\*
\0{\bf A1.4.} {\cs Lemma 4.}
{\it If there exist $c>0$ and $\t\ge 1$, such that $|n\o-k|\ge cn^{-\t}$,
$\forall k,n\in\NNN$, $n>0$, then
%
$$\d_n \ge \fra12 \fra{c}{n^\t} \; ,
\qquad \forall n\ge q_1 \; . $$
%
}
\*
\0{\bf A1.5.} {\it Proof of Lemma 4.}
The function $\d_+(x)$ is a convex function, linear between $q_{2i}$ and
$q_{2i+2}$; moreover
$$\d_+(q_{2i}) = p_{2i}-\o q_{2i} \ge \fra{c}{q_{2i}^\t} \; . $$
Since $c x^{-\t}$ is a convex function too, we have
%
$$\d_+(x) \ge \fra{c}{x^\t} \; , \qquad \forall x\ge q_0 \; . $$
%
We can show analogously that $\d_-(x)\ge c x^{-\t}\;
\forall x\ge q_1$ and
Lemma 4 immediately follows from the definition of $\d_n$. \qed
\*
Lemma 3 and Lemma 4 immediately imply the following result.
\*
\0{\bf A1.6.} {\cs Proposition 1.}
{\it If there exist $c>0$ and $\t\ge 1$, such that $|n\o-k|\ge cn^{-\t}$,
$\forall k,n\in\NN$, $n>0$, then, for any $i\ge 2$,
%
$$|n \fra{p_i}{q_i} -k| \ge \fra12 \fra{c}{n^\t} \quad , \qquad \hbox{if}\;
q_1 \le n \le \fra12 q_i \; . $$
%
}
\*
By using Proposition 1, one can define the sequence of $p^{L_i}$ verifying
\equ(1.19) with $C_0=\p c$, by setting $L_i=q_i$, $p^{L_i}=
{\pi p_i\over q_i}$.
\vskip2.truecm
\centerline{\titolo Appendix 2. Proof of Lemmata 1 and 2}
\*\numsec=2\numfor=1
\0{\bf A2.1.} Let us consider the quantity
$\WW_{n,q}^{(h)}(\kk)$ introduced in \equ(2.21):
%
$$ \WW^{(h)}_{n,q}(\kk) =
\sum_{\th\in\TT_n^{q,h}} {\rm Val}(\th) \; , \Eqa(A1.1) $$
%
where, if we denote by ${\V T}$ the set of clusters contained in $\th$
(including $\th$),
%
$$\sum_{\th\in\TT_{n,q}^{h}} {\rm Val}(\th) =
\sum_{n_{v_1},\ldots,n_{v_q} \atop n_{v_1}+\ldots+ n_{v_q} = n\;\mod L}
\l^q \, \hat\f_{n_{v_1}}\ldots\hat\f_{n_{v_q}} \sum_{ \{h_{\ell}\}}
\Big( \prod_{\ell\in{\rm int}(\th)}
\tilde g^{(h_{\ell})}_{\o_{\ell}}(\kk_{\ell}') \Big) \Eqa(A1.2) $$
%
can be rewritten as
%
$$\sum_{\th\in\TT_{n,q}^{h}} {\rm Val}(\th) =
\sum_{n_{v_1},\ldots,n_{v_q} \atop n_{v_1}+\ldots+ n_{v_q} = n\;\mod L}
\l^q \, \hat\f_{n_{v_1}}\ldots\hat\f_{n_{v_q}} \sum_{ \{h_{\ell}\} }
\prod_{T\in\V T} \Big( \prod_{\ell\in T_0}
\tilde g^{(h_{T})}_{\o_{\ell}}(\kk_{\ell}') \Big) \; ,\Eqa(A1.3) $$
%
where the first product is over all the clusters contained
in $\th$ (which are uniquely determined by the frequency labels
assignment), $h_T$ is the scale of the cluster $T$ and
$T_0$ is the collection of lines inside $T$ which are
outside the clusters internal to $T$ (so that
the last product is over the lines on scale $h_T$
contained in $T$, see \S 2.4).
Finally, we shall suppose that $\kk\in {\cal D}_h$.
\*
\0{\bf A2.2.} {\it Proof of Lemma 1.}
The case $q=1$ is trivial. Let us suppose that $q\ge 2$ and
let us consider one of the graphs contributing to the sum in the r.h.s. of
\equ(A1.3) and suppose that it satisfies the
non resonance condition assumed in the statement of
Lemma 1; this means that there are neither clusters nor vertices for
which the resonance conditions \equ(2.24) can occur.
We start by considering a cluster $T\in {\cal T}_1(\th)$, that is a minimal
cluster (see \S 2.4). By \equ(2.14) and \equ(1.12), we have
%
$$\Big| \Big( \prod_{v\in T} \hat\f_{n_v} \Big)
\Big( \prod_{\ell\in T} \tilde
g^{(h_{\ell})}_{\o_{\ell}}(\kk_{\ell}') \Big) \Big|
\le F_0^{M^{(2)}_T}
\Big(\prod_{v\in T} e^{-\x|n_v|} \Big) (G_0 \g^{-h_T})^{L_T}
\; , \Eqa(A1.4)$$
%
where $M_T^{(2)}=M_T$ as we are considering clusters with depth $D_T=1$,
(see \S 2.4 for notations).
If $h_T=1$, the fact that the vertices are not resonant gives no constraint
on the values of $n_v$. However, if $h_T\le 0$, by the support properties of
$f_h(\kk')$ and \equ(1.19), we have, for any $v\in T$,
%
$$\eqalign{
2 a_0 \g^{h_T} & \ge |k_{\ell_v}'-k_{\ell_{v'}}'| =
|2n_v p+(\o_{\ell_v}-\o_{\ell_{v'}})p_F| =\cr
&= |2n_v p+(\o_{\ell_v}-\o_{\ell_{v'}})m p| \ge
C_0 \left( |n_v|+m \right)^{-\t} \; , \cr} \Eqa(A1.5)$$
%
since $2n_v + (\o_{\ell_v}-\o_{\ell_{v'}})m \not=0$ by hypothesis. Hence, if
we define $C_1 = (C_0/2 a_0)^{1/\t}$, we have
%
$$|n_v| \ge C_1\g^{-h_T/\t} - m \; .\Eqa(A1.6)$$
The inequalities \equ(2.14), \equ(A1.4) and \equ(A1.6) easily imply that,
for any $T\in {\cal T}_1(\th)$,
%
$$ \eqalign{
\Big| \Big( \prod_{v\in T} & \hat\f_{n_v} \Big)
\Big( \prod_{\ell\in T} \tilde
g^{(h_{\ell})}_{\o_{\ell}}(\kk_{\ell}') \Big) \Big|
\le G_0^{L_T} (F_0 \bar C_2)^{M^{(2)}_T}
\Big( \prod_{v\in T} e^{-{3\x\over 4}|n_{v}|} \Big)
\, e^{- 2^{-3} \x |n_T|} \cr
& \Big[ \g^{-h_T L_T} \,
e^{-2^{-3}\x M^{(2)}_T C_1\g^{-h_T/\t} } \Big] \; , \cr}
\Eqa(A1.7) $$
%
where $\bar C_2= \max \{ e^{m\x/8}, e^{C_1\g^{-1/\t}\x/8} \}$ and we used the
trivial bound $\sum_{v\in T}|n_v| \ge |n_T|$.
Next we consider a cluster $T\in {\cal T}_2(\th)$, that is a cluster of depth
$D_T=2$. Since $h_T<1$, we can use again \equ(A1.5) for any $v\in
T_0$; moreover, given a cluster $\tilde T \subset T$, since $2n_T +
(\o_{\ell_T^o} -\o_{\ell_T^i})m \not=0$, we have an analogous bound:
%
$$2 a_0 \g^{h_T} \ge |k_{\ell_{\tilde T}^i}'-
k_{\ell_{\tilde T}^o}'| \ge
C_0 \left( |n_T|+m \right)^{-\t} \;, \Eqa(A1.8) $$
%
where $k_{\ell_T^i}$ are $k_{\ell_T^o}$ are defined as in \S 2.4.
By using \equ(A1.5) and \equ(A1.8), it is easy to see that
%
$$\eqalign{
\Big| \Big( \prod_{{\tilde T} \subset T} &
e^{-2^{-3} \x |n_{\tilde T}|} \Big)
\Big( \prod_{v \in T_0} \hat\f_{n_v} \Big)
\Big( \prod_{\ell \in T_0 } \tilde g^{(h_l)}_{\o_{\ell}}
(\kk_{\ell}') \Big) \Big| \cr
& \le G_0^{L_T} F_0^{M^{(2)}_T} \bar C_2^{M_T} \,
\Big( \prod_{v\in T_0} e^{-{3\x\over 4}|n_v|} \Big)
\, e^{-2^{-4} \x |n_{T}|} \cr &
\Big[ \g^{-h_T L_T}\,
e^{-2^{-4} \x M_T C_1 \g^{-h_T/\t} } \Big] \; ,
\cr} \Eqa(A1.9) $$
%
where we used the bound
$\sum_{\tilde T\subset T} |n_{\tilde T}| + \sum_{v\in T_0} |n_v| \ge |n_T|$.
By iterating the previous procedure, and noting that\\
$\bullet$ $\sum_{T\in {\V T}} \sum_{v\in T_0} |n_v| =
\sum_{v\in\th} |n_v|\ge \Big| \sum_{v\in\th}
n_v \Big| = |n|$,\\
$\bullet$ $\sum_{T\subset\V T} M_T^{(2)} =
\sum_{T\subset\V T} L_T +1 = q$,\\
$\bullet$ $M_T^{(1)}+M_T^{(2)}\=M_T=L_T+1$,\\
we obtain in the end
%
$$ \eqalign{
\sup_{\kk\in{\cal D}_h} \Big| &\tilde \WW^{(h)}_{n,q}(\kk) \Big| \le
|\l|^q e^{-{\x\over 2}|n|} G_0^{q-1} F_1^q \bar C_2^{M_{\V T}} \, \cr
& \sum_{\{h_{\ell}\} } \prod_{T\in {\V T} }
\g^{-h_TL_T} e^{- 2^{-(D_T+2)} \x C_1 M_T\g^{-h_T/\t} }
\; , \cr} \Eqa(A1.10) $$
%
where $F_1 = F_0 \sum_{n\in\zzz}e^{-\x|n|/4}$ and $M_{\V T} = \sum_{T\in
{\V T}} M_T$.
The r.h.s. of \equ(A1.10) can be further bounded by\\
(1) neglecting the ordering relation
between the frequency labels, and\\
(2) taking into account
only the fact that, if a cluster $T$ has depth $D_T$,
then $h_T\le -D_T+2$.\\
We get
%
$$ \eqalign{
\sup_{\kk\in{\cal D}_h} \Big| &\tilde \WW^{(h)}_{n,q} (\kk) \Big| \le
e^{-{\x\over 2}|n|}
G_0^{q-1} F_1^q \, |\l|^q\cr
& {\sum_{\V T}}^* \bar C_2^{M_{\V T}} \prod_{T\in{\V T}} \Big[
\sum_{h_T\le -D_T+2} \g^{h_T}\Big(
\g^{-h_T} e^{- 2^{-(D_T+2)} \x C_1 \g^{-h_T/\t} } \Big)^{M_T}
\Big] \; , \cr} \Eqa(A1.11) $$
%
where ${\sum_{\V T}}^*$ is the sum over all the possible choices
of arrangements of the clusters over
a chain of $q$ vertices, which is bounded by $4^q$. Hence we have
%
$$ \eqalign{
\sup_{\kk\in{\cal D}_h} \Big| &\tilde \WW^{(h)}_q(\kk;\kk+ 2n\pp) \Big| \le
e^{-{\x\over 2}|n|}
G_0^{q-1} (4 F_1)^q \, |\l|^q\cr
& \max_{\V T} \left\{ \bar C_2^{M_{\V T}}
\prod_{T\in{\V T}} \left[\;
\sum_{r=D_T-2}^{\io} \g^{-r} \Big(
\g^{r} e^{-2^{-4} \x C_1\left( {\g^{1/\t} /2 }
\right)^r } \Big)^{M_T} \right] \right\} \; . \cr} \Eqa(A1.12) $$
%
Suppose now $\g$ so large that
$\tilde \g \= \g^{1/\t}/2>1$, and note that, $\forall N>0$,
$\exists$ $C_N>0$ such that
%
$$ e^{-2^{-4}C_1\x\tilde\g^r} \le
{C_N \over 1 + (2^{-4}C_1\x\tilde\g^r)^N } \; , \Eqa(A1.13) $$
%
(one can take $C_N=1+N!$).
Choose $N$ so that $\tilde \g^N\ge 2\g$; then, since $M_T\ge 1$,
%
$$ \eqalign{
\sum_{r=D_T-2}^{\io} & \g^{-r} \Big(
\g^{r} e^{-2^{-4}\x C_1 \tilde\g^r } \Big)^{M_T} \le \cr
& \le
\g \Big( \sum_{r=D_T-2}^{\io }
{ C_N \g^r \over 1 + (2^{-4}C_1\x)^N (2\g)^r }\Big)^{M_T}
\le \g (C_4 2^{-D_T})^{M_T} \; , \cr} \Eqa(A1.14) $$
%
where $C_4= 8C_N/(2^{-4}C_1\x)^N$. Since $\sum_{T\in {\V T}}M_T\le 2q$,
\equ(A1.12) and \equ(A1.14) yield \equ(2.25) for some constant $B_1$.
\qed
\*
\0{\bf A2.3.} {\it Proof of Lemma 2.}
If there are resonances, the proof in \S A2.2
does not apply, as \equ(A1.6) and \equ(A1.8) do not hold for resonances, and
we have to carefully analyze the effect of the renormalization procedure
described in \S 3. In particular we shall need the bound \equ(3.44), which
depend on the hypothesis \equ(3.41). However, to check the validity of
\equ(3.41), we need a bound on the effective potential; hence the proof will
be inductive. We shall suppose that $\hat \f_m\not=0$, $h\le -1$ and that,
if $h+1\le h'\le 0$,
%
$$|\s_{h'}-\l\hat \f_m| \le A |\l|^2 e^{-m{\x\over 2}}\; ,\qquad
|\n_{h'}| \le B_3|\l| \; , \Eqa(A1.15)$$
%
and we shall prove that it is possible to choose $A$ and $B_3$ so that
\equ(A1.15) is true also for $h'=h$, together with
the bound \equ(3.50) on the effective potential. The proof of Lemma 2 will
follow from the remark that \equ(A1.15) is verified for $h=-1$, by
\equ(3.5), if $A\ge A_0$.
Let $\th_\RR\in \TT^{h}_{\RR,n,q}$ and $q>1$ (the case $q=1$ is trivial,
except for the $\n_h$ vertex).
Let us consider the collection ${\V V}_1$ of
{\sl maximal resonances}, \ie resonances which are
not strictly contained in any other resonance. If $V$ is such a resonance,
$\ell_V^i$ and $\ell_V^o$ are its external lines,
and $k_{\ell_V^i}'=k_{\ell_V^o}'$. Then
(recall the definition in item (5) of \S 2.4)
%
$${\rm Val(\th_\RR)} = \Big( \prod_{v\in\th_\RR}F_v \Big)
\Big( \prod_{\ell\cap \V V_1 = \emptyset}
\tilde g_\ell \Big) \prod_{V \in \V V_1}
\Big[ \tilde g_{\ell_V^i} \, \tilde g_{\ell_V^o}\,
\RR \X^{h_V}_{V}(\kk_{\ell_V^o}') \Big] \; ,\Eqa(A1.16) $$
%
where $\tilde g_\ell$ is a shorthand for
$\tilde g^{(h_{\ell})}_{\o^1_{\ell},\o^2_{\ell}}(\kk_{\ell}')$, if $\ell$ is
an internal line of $\th_\RR$, $\tilde g_\ell = 1$ otherwise;
$\prod_{\ell\cap \V V_1 = \emptyset} \tilde g_\ell =1$, if $\th_\RR$
itself is a resonance (so that all lines intersect $\V V_1$);
$F_v = \g^{h_v}\n_{h_v}$ if $n_v=0$, $F_v=\l\hat\f_{n_v}$ otherwise.
Finally the {\sl resonance value} $\X^{h_V}_{V}(\kk_{\ell_V^o}')$
is given by
%
$$\X^{h_V}_{V}(\kk_{\ell_V^o}') =
\Big( \prod_{\ell\in V \, : \, \ell\cap \V V_2 = \emptyset}
\tilde g_\ell \Big) \prod_{V' \in \V V_2 \cap V} \Big[
\tilde g_{\ell_{V'}^i} \, \tilde g_{\ell_{V'}^o} \,
\RR \X^{h_{V'}}_{V'}(\kk_{\ell_{V'}^o}') \Big] \; , \Eqa(A1.17) $$
%
where ${\V V_2}$ is the collection of resonances which
are strictly contained inside some resonance in ${\V V_1}$,
and which are maximal, and $\V V_2 \cap V$ is the subset of
resonances in $\V V_2$ which are contained in $V$.
Note that \equ(A1.17) extends \equ(3.47) to the case in
which $V$ contains other resonances.
We can write $\RR \X^{h_V}_{V}(\kk_{\ell_V^o}')$ as in \equ(3.48), that is
%
$$ \RR \X_V^{h_V}(\kk'_{\ell_V^o}) \= \X_V^{h_V}(\kk'_{\ell_V^o}) -
\X_V^h(\V 0)= \int_0^1 dt\Big[ {d\over dt}
\X_V^h(t\kk'_{\ell_V^o}) \Big] \; . \Eqa(A1.18) $$
Note that $\X_V^h(t\kk'_{\ell_V^o})$ can be written as in \equ(A1.17), by
substituting the momentum $\kk'_\ell$ of any line with $t \kk'_{\ell_V^o}+
\qq_\ell$, for suitable values of $\qq_\ell$. Therefore
the r.h.s. of \equ(A1.18) can be written as a sum of terms of the form
\equ(A1.17) with a derivative $d/dt$ acting either\\
(1) on one of the propagators corresponding to a line outside $\V V_2$, or\\
(2) on one of the $\RR \X^{h_{V'}}_{V'}$.
In case (2), we write
%
$$ {d\over dt} \RR
\X^{h_{V'}}_{V'}(t\kk'_{\ell_{V}^o} + \qq_{\ell_{V'}^o}) =
{d\over dt} \Big[
\X^{h_{V'}}_{V'}(t\kk'_{\ell_{V}^o} + \qq_{\ell_{V'}^o}) -
\X^{h_{V'}}_{V'}(\V0) \Big]
= {d\over dt}
\X^{h_{V'}}_{V'}(t\kk'_{\ell_{V}^o} + \qq_{\ell_{V'}^o}) \; ,
\Eqa(A1.19) $$
%
so that, if the derivative corresponding to a resonance $V$
acts on the value of some resonance $V'\subset V$, one can
replace with $\openone$ the $\RR$ operator corresponding to $V'$.
We can now iterate this procedure, by applying to
$\X^{h_{V'}}_{V'}(t\kk'_{\ell_{V}^o} + \qq_{\ell_{V'}^o})$ the equation
\equ(A1.17), with $\V V_3$ (the family of resonances which are strictly
contained
inside some resonance belonging to $\V V_2$ in place of $\V V_2$), and so on.
At the end the r.h.s. of \equ(A1.18) can be written as a sum of $q_V-1$
terms, if $q_V$ denotes the number of vertices contained in $V$, which can be
described in the following way.\\
(1) There is one term for each line $\bar\ell \in V$;\\
(2) if $\bar\ell \in T_0$, where $T$ is a cluster contained in $V$ (see
item (1) in \S 2.4 and note that $T$ can be equal to $V$),
and $T=T^{(r)}\subset T^{(r-1)} \ldots \subset T^{(1)}=V$ is the chain of $r$
clusters containing $T$ and contained in $V$, then
the graph value can be computed by replacing with $\openone$
the $\RR$ operator
acting on $T^{(i)}$, $i=1,\ldots,r$, even if $T^{(i)}$ is a resonance,
because of the comments after \equ(A1.19);\\
(3) the $\RR$ operation acts on all other resonances contained in $V$;\\
(4) the derivative $d/dt$ acts on the propagator of $\bar\ell$, whose momentum
is of the form $t \kk'_{\ell_V^o}+\qq_\ell$.
A similar decomposition of the resonance value is now applied, for each
term of the previous sum, to all resonant clusters, which are still affected
by the $\RR$ operation. This procedure is iterated, until no $\RR$ operation
is explicitly present; it is easy to see that we end with an expression of the
form
%
$${\rm Val(\th_\RR)} = \sum \int dt_1 \ldots dt_s
\Big( \prod_{v\in\th_\RR} F_v \Big)
\prod_{T\in\V T} \Big[ \prod_{\ell\in T_0}
\Big({d\over dt_{i(\ell)}}\Big)^{d_\ell} \tilde g_\ell \Big] \; , \Eqa(A1.20)$$
%
where the sum is over all possible choices of $s$, $\{d_\ell\}$ and
$\{i(\ell)\}$,
which satisfy the following conditions:\\
(1) $d_\ell$ is equal to $0$ or $1$;\\
(2) if $d_\ell=0$, $i(\ell)$ is arbitrarily defined,
otherwise $i(\ell)\in \{1,\ldots,s\}$ and $i(\ell)\not=i(\ell')$, if
$\ell\not=\ell'$;\\
(3) the number of lines for which $d_\ell=1$ is equal to the number of
interpolating parameters $s$;\\
(4) for each derived line $\ell$ there is a chain of $r$ clusters
$T=T^{(r)}\subset T^{(r-1)} \ldots \subset T^{(1)}=V$, such that $\ell\in T_0$
and $V$ is a resonance;\\
(5) no cluster can belong to more than one chain of clusters;\\
(6) each resonance belongs to one of the chains of clusters;\\
(7) the momentum of the derived line is of the form $t_{i(\ell)}\kk'+\qq_l$,
with $|\kk'| \le a_o \g^{-h_V^e}$,
(in general $\kk'$ is not $\kk_{\ell_V^0}'$, but it
can depend also on the interpolation parameters
corresponding to resonances containing $V$, if any),
where $h_V^e$ is the external scale of $V$,
that is the scale of the smaller cluster containing it.
The item (7) above implies that, for each derived line, by \equ(3.44),
%
$$\Big|{d\over dt_{i(\ell)}} \tilde g_\ell\Big| \le a_0 G_2 \g^{h_V^e-h_\ell}
\g^{-h_\ell} \; . \Eqa(A1.21)$$
%
Note that
%
$$h_V^e-h_\ell = \sum_{i=1}^r [h^e_{T^{(i)}} - h_{T^{(i)}}]\; ; \Eqa(A1.22) $$
%
hence the ``gain'' $\g^{h_V^e-h_\ell}$ in the bound \equ(A1.21),
with respect to the bound of a non derived propagator,
can be divided between the clusters of
the chain associated to the derived line $\ell$, so that each cluster has a
factor $\g^{h^e_{T^{(i)}} - h_{T^{(i)}}}\le 1$ associated with it;
in particular we
have a factor of this type associated with each resonance, for each term in
the sum of \equ(A1.20). Since the number of terms in this sum is bounded
by $2^{q-1}$, we can write, if we denote by $\V V$ the family of resonant
clusters,
%
$$\Big|{\rm Val(\th_\RR)}\Big| \le 2^{q-1}
\prod_{v\in\th_\RR} |F_v| \prod_{T\in\V T} (G_3 \g^{-h_T})^{L_T}
\prod_{T\in {\V V}} \g^{h_T^e-h_T} \; ,\Eqa(A1.23)$$
%
where $G_3=\max \{G_1,a_0 G_2\}$.
We shall now consider, as in the proof of Lemma 1, a minimal cluster
$T\in\TT_1(\th_\RR)$ with $h_T\le 0$ and note that
%
$$\prod_{v\in T} |F_v| \le |\n_{h_T} \g^{h_T}|^{M_T^{(\n)}}
(|\l|F_0)^{\tilde M_T^{(2)}} \prod_{v\in T} e^{-\x|n_v|} \; ,\Eqa(A1.24)$$
%
where $M_T^{(\n)}$ is the number of resonant vertices contained in $T$ and
$\tilde M_T^{(2)}$ is the number of non resonant vertices, so that
$M_T=M_T^{(2)}=\tilde M_T^{(2)} + M_T^{(\n)}=L_T+1$. If we now
recall that $n_T=\sum_{v\in T}n_v + \sum_{\ell\in T}
(\o^1_\ell-\o^2_\ell)m/2$ and
we use \equ(A1.6) for non resonant vertices, we get
%
$$\prod_{v\in T} e^{-\x|n_v|} \le \bar C_2^{\tilde M_T^{(2)}+L_T}
\Big( \prod_{v\in T} e^{-{3\x\over 4}|n_{v}|} \Big)
\, e^{- 2^{-3} \x |n_T|}
e^{-2^{-3}\x \tilde M^{(2)}_T C_1\g^{-h_T/\t} } \; . \Eqa(A1.25)$$
Hence, if $|\n_h|\le B_3|\l|$, by \equ(A1.24) and \equ(A1.25) we have
%
$$\eqalign{
\left(\prod_{v\in\th} |F_v| \right) &
(G_3 \g^{-h_T})^{L_T} \le (|\l| C_2^4)^{M_T}
\Big( \prod_{v\in T} e^{-{3\x\over 4}|n_{v}|} \Big)\cr
&\, e^{- 2^{-3} \x |n_T|} \g^{h_T}
\left[ \g^{-h_T} e^{-2^{-3}\x C_1\g^{-h_T/\t} }\right]^{\tilde M^{(2)}_T}
\; ,\cr}\Eqa(A1.26)$$
%
where $C_2 = \max \{B_3,G_3,\bar C_2, F_0\}$.
Next we consider a cluster $T\in\TT_2(\th_\RR)$;
by using \equ(A1.6) for non resonant
vertices and \equ(A1.8) for non resonant clusters, whose number will be called
$\tilde M_T^{(1)}$, we get
%
$$\eqalign{
\Big( \prod_{{\tilde T} \subset T} &
e^{-2^{-3} \x |n_{\tilde T}|} \Big) \Big( \prod_{v \in T_0} |F_v| \Big)
\left(G_3 \g^{-h_T}\right)^{L_T} \le |\l|^{M^{(2)}_T} C_2^{4 M_T} \cr
& \Big( \prod_{v\in T_0} e^{-{3\x\over 4}|n_v|} \Big)
\, e^{-2^{-4} \x |n_{T}|} \g^{-h_T M_T^{(r)}} \g^{h_T}
\Big[ \g^{-h_T}\, e^{-2^{-4} \x C_1 \g^{-h_T/\t} } \Big]^{\tilde M_T} \; ,
\cr} \Eqa(A1.27) $$
%
where $M_T^{(r)}=M_T^{(1)}-\tilde M_T^{(1)}$ is the number of resonant
clusters strictly contained in $T$ and $\tilde M_T = \tilde M_T^{(1)} +\tilde
M_T^{(2)}$.
We iterate the previous procedure, as in the proof of Lemma 1, and we get
%
$$\eqalign{
|{\rm Val(\th_\RR)}| &\le (2|\l|)^q C_2^{4 M_{\V T}} e^{-{\x\over 2}|n|}
\Big( \prod_{v\in \th_\RR} e^{-{\x\over 4}|n_v|} \Big) \; \cdot \cr
& \cdot \; \left\{ \prod_{T\in{\V T}} \g^{-h_T M_T^{(r)}} \g^{h_T}
\Big[ \g^{-h_T}\, e^{-2^{-4} \x C_1 \g^{-h_T/\t} } \Big]^{\tilde M_T}
\right\} \prod_{T\in {\V V}} \g^{h_T^e-h_T} \; ,\cr}\Eqa(A1.28)$$
%
where $\V V$ is the family of all resonant clusters strictly contained
in $\th_\RR$.
Note that
%
$$\left[ \prod_{T\in{\V T}} \g^{-h_T M_T^{(r)}} \right]
\left[ \prod_{T\in {\V V}, T\not=\th_\RR} \g^{h_T^e-h_T} \right] =
\prod_{T\in {\V {\V V}}, T\not=\th_\RR} \g^{-h_T} \; ;\Eqa(A1.29)$$
%
hence \equ(A1.28) can be written also as
%
$$ \eqalign{
\Big|{\rm Val(\th_\RR )}\Big| & \le \g^{h_{\th_\RR}} (2|\l|)^q C_2^{4 M_{\V T}}
e^{-{\x\over 2}|n|} \Big( \prod_{v\in \th_\RR} e^{-{\x\over 4}|n_v|} \Big)\
\cr & \qquad\qquad
\left\{\prod_{T\in {\V T}} \Big[ \g^{-h_T}\, e^{-2^{-4} \x C_1
\g^{-h_T/\t} } \Big]^{\tilde M_T} \right\} \; . \cr} \Eqa(A1.30)$$
In order to bound $W^{(h)}_{\RR,n,q} (\kk)$ we have to perform the sum
of \equ(A1.30) over the $n_v$, $\o_{\ell}^i$ and $h_T$ labels.
The sum over $n_v$ is trivial,
as well as the sum over $h_T$, for the clusters with $\tilde M_T \not= 0$.
The sum over $h_T$ would give some bad factor, when $\tilde M_T = 0$, but it
turns out that there is indeed no sum in this case. In fact, if all the
clusters and vertices strictly contained in $T$ are resonant, then $T$ itself
must be a
resonance and all its internal lines have the same $\kk'$ as the external
ones, implying, by support properties of the $f_h$ functions, that the
frequency label of the external lines is equal to $h_T-1$. Hence we can
proceed as in the proof of Lemma 1 and we get
%
$$\Big| \WW^{(h)}_{\RR,n,q} (\kk) \Big| \le e^{-{\x\over 2}|n|} B_2^q
\Eqa(A1.31) $$
%
for a suitable constant $B_2>B_1$.
We still have to check that the bound \equ(A1.15) is satisfied also by
$\s_h$ and $\n_h$. Note that, $\forall h<0$,
%
$$\eqalign{
s_h &=\s_h-\s_{h+1}=\sum_{q=2}^\io \bar\WW^{(h)}_{\RR,m,q} (-m\pp)\;,\cr
\n_h &= \g \n_{h+1} + \g^{-h} \sum_{q=2}^\io \bar\WW^{(h)}_{\RR,0,q}(-m\pp)\;,
\cr}\Eqa(A1.32)$$
%
where $\bar\WW^{(h)}_{\RR,0,q}(-m\pp)$ and $\bar\WW^{(h)}_{\RR,m,q}(-m\pp)$
admit an expansion in terms of graphs $\th_\RR$, differing from the
corresponding expansion of
$\WW^{(h)}_{\RR,0,q}(-m\pp)$ and $\WW^{(h)}_{\RR,m,q}(-m\pp)$ in the
following respects:\\
(1) the $\RR$ operation on the whole graph, which
is necessarily a resonance, is substituted with the localization operation,
hence in the previous analysis $\th_\RR$ must not be included in the set
${\V V}$;\\
(2) the internal scale of $\th_\RR$ is equal to $h+1$, that is there is in the
graph at least one line of frequency $h+1$;\\
(3) if all maximal clusters strictly contained in $\th_\RR$ are resonant,
as well as the vertices belonging to $\th_{\RR 0}$ (see item (3) in \S 2.4),
that is if $\tilde M_{\th_\RR}=0$, then ${\rm Val}(\th_\RR)=0$.
Item (3) follows from the support properties of the propagators, the
definition of resonance in \S 2.5 and
from the observation that all lines $\ell\in\th_{\RR 0}$
would have $\kk_\ell'=0$, if $\tilde M_{\th_\RR}=0$,
since $\kk_\ell'=0$ for the external lines.
Item (2) easily implies that the bound \equ(A1.30) is valid also for the
new graphs, possibly with a different value of $C_2$, even if there is no
$\RR$ operation on $\th_\RR$. Even more, items (2) and (3) together imply that
bound \equ(A1.31) can be improved and we can write, for any $N$ and
a suitable constant $C_N$, for $n=0$ or $n=m$,
%
$$|\bar\WW^{(h)}_{\RR,n,q}(-m\pp)| \le \g^{Nh} (|\l| C_N)^q \; . \Eqa(A1.33)$$
%
Note that item (2) alone implies the bound \equ(A1.33) with $N=1$,
by \equ(A1.30), which is sufficient for iterating the bound
on $\n_h$.
Hence, we have, for $\l$ small enough and $h<0$,
%
$$\eqalignno{
|s_h| &\le B_4 \g^h |\l|^2 e^{-\fra{m}2 \x} \; ,&\eqa(A1.34)\cr
|\n_h| & \le \g|\n_{h+1}| + B_4 |\l|^2\; ,&\eqa(A1.35)\cr}$$
%
where $B_4$ is a suitable constant.
The constant $B_4$ depends in principle on the constants
$A$ and $B_3$, appearing in the
inductive hypothesis \equ(A1.15), through the constant $C_3$ in
\equ(A1.30), defined after \equ(A1.26).
However, it is easy to prove that in fact it can be
chosen independently of $A$ and $B_3$, if $\l$ is small enough. The
independence of $A$ follows from the remark that the
constant $C_2$ of \equ(A1.30) can be made independent of the constant $A$,
if $\l$ is chosen so small that \equ(3.41) is satisfied.
The independence of $B_3$ is a bit more involved. One has to observe that
there is no graph contributing to $\bar\WW^{(h)}_{\RR,n,2}(-m\pp)$, $n=0,m$,
(that is no second order contribution to $\n_h$ and $s_h$),
containing resonant vertices.
In fact one can construct graphs of this type, but their value is zero, since
they contain necessarily a line with $\kk_\ell'=0$, whose propagator vanishes
by its support properties, (see item (2) after \equ(A1.32)).
It easily follows that it is possible to choose
$B_4$ independent of $B_3$, if $\l$ is small enough.
By iterating the bound \equ(A1.35) and using the bound \equ(3.5) on
$\n_0$, we get, for $h\ge h^*$,
%
$$|\n_h| \le \g^{-h} \left[|\n_0| + B_4 |\l|^2 \sum_{j=h}^{-1} \g^j\right] \le
\g^{-h^*} |\l|^2 \left(A_0 + B_4 \sum_{r=1}^\io \g^{-r}\right)\;.\Eqa(A1.36)$$
%
Moreover, the definition \equ(3.40) of $h^*$ implies that
$\g^{-h^*} |\l| \le G_3/(2|\hat\f_m|)$.
Hence \equ(A1.15) is satisfied also for $h'=h$, if we choose $A=A_0+B_4
\sum_{r=0}^\io \g^{-r}$, $B_3=A G_3/(2|\hat\f_m|)$. \qed
\vskip2.truecm
\centerline{\titolo Appendix 3.
Proof of the bounds \equ(4.17) and \equ(4.20)}
\*\numsec=3\numfor=1
\0{\bf A3.1.} {\it Proof of \equ(4.17).}
By using \equ(3.27)$\div$\equ(3.31), it is easy to prove
that, for any $N\ge 0$, there is a constant $G_N$
such that, if $0\ge h\ge h^*$, $N_0,N_1\ge 0$ and $N_0+N_1=N$,
%
$$ \Big| D_0^{N_0} D_1^{N_1} \tilde g_{\o,\o'}^{(h)}(\kk') \Big|
\le G_N \g^{-hN}\,
{ \max\{\g^{h},|\s_{h}| \} \over \g^{2h} + \s_{h}^2 } \; , \Eqa(A2.1) $$
%
where $D_0$ and $D_1$ denote the discrete derivative with respect to $k_0$
and $k'$, respectively.
Hence, if $|x_0-y_0| \le \b/2$ and $|x-y|\le L_i/2$, we have
%
$$ \eqalign{
& \left( {\sqrt{2}\over \p} \right)^N |x_0-y_0|^{N_0} |x-y|^{N_1}
\Big| g^{(h)}(\xx;\yy) \Big| \le \cr
&\le \left| {\b\over 2\p} [e^{-i{2\p\over \b}(x_0-y_0)}-1] \right|^{N_0}
\left| {L_i\over 2\p} [e^{-i{2\p\over L_i}(x-y)}-1] \right|^{N_1}
\Big| g^{(h)}(\xx;\yy) \Big| = \cr
& = \Big| \sum_{\o,\o'=\pm1} e^{-i(\o x-\o' y)p_F}
{1\over L_i\b} \sum_{\kk\in {\cal D}_{L_i,\b}} \, e^{-i\kk'\cdot(\xx-\yy)}
\, D_0^{N_0} D_1^{N_1} \tilde g^{(h)}_{\o,\o'}(\kk') \Big| \;\le\cr
& \le C_N \g^h (\max \{\g^h,L^{-1}\}) \,\g^{-hN} {\max\{\g^h,|\s_h|\}
\over \g^{2h}+\s_h^2} \le C_N \g^{-hN}(\max \{\g^h,L^{-1}\})\; , \cr}
\Eqa(A2.2) $$
%
where $C_N$ denotes a varying constant, depending only on $N$, and
the factor $(\max \{\g^h,L^{-1}\})$ arises from the sum over $\kk'$ (note that
the sum over $k_0$ always gives a factor $\g^h$, since $h_\b\le h^*\le h$).
Therefore we have
%
$$ \Big| g^{(h)}(\xx;\yy) \Big| \le
{C_N \max \{\g^h,L^{-1}\} \over 1 + \g^{hN}|\xx-\yy|^N} \; .\Eqa(A2.3) $$
%
and a similar bound is verified for $g^{(\le h^*)}(\xx;\yy)$, if $\l$ is real.
\qed
\*
\0{\bf A3.2.} {\it Proof of \equ(4.20).}
Let $\th_\RR$ be one of the graphs contributing to the kernel
$K_{\phi,\phi}^{(h)}(\xx;\yy)$, see \equ(4.19) and let us consider the two
vertices, $v_1$ and $v_q$, connected to the external lines (which are
associated with the external field).
Suppose first that neither $v_1$ nor $v_q$, the {\sl external vertices},
are contained in any cluster,
different from $\th_\RR$ itself.
In this case, we can bound ${\rm Val}(\th_\RR)$ as in
\S A2.3, by taking into account that\\
(1) there is no factor associated to the external vertices;\\
(2) $h_{\th_\RR}=h+1$;\\
(3) there are at least two lines of scale $h+1$, the external propagators.\\
Hence we get a bound differing from \equ(A1.30) only because the power of
$|\l|$ is $q-2$ instead of $q$ and each external propagator gives a
contribution proportional to $\g^{-h_{\th_\RR}}$:
%
$$ \eqalign{
|{\rm Val(\th_\RR)}| \le & \g^{-h_{\th_\RR}} (2|\l|)^{q-2} C_2^{4 M_{\V T}}
e^{-{\x\over 2}|n|} \Big( \prod_{v\in \th} e^{-{\x\over 4}|n_v|} \Big)
\cr & \qquad\qquad
\left\{\prod_{T\in {\V T}} \Big[ \g^{-h_T}\, e^{-2^{-4} \x C_1
\g^{-h_T/\t} } \Big]^{\tilde M_T} \right\} \; , \cr} \Eqa(A2.6)$$
%
where the same notation of \S A2.3 is used, except for the definition of
$\tilde M_T$, which differs from the previous one, since we do not consider
the external vertices in the calculation of $\tilde M_T^{(2)}$; moreover we
assigned a label $n_v=0$ to the external vertices.
Suppose now that $v_1$ is contained in some cluster strictly contained in
$\th_\RR$ and that the scale of the external propagator emerging from $v_1$ is
$h_1$. In this case, there is a chain of clusters $T^{(1)} \subset T^{(2)}
\ldots \subset T^{(r)} = \th_\RR$, such that $v_1\in T^{(i)}$ and $h_{T^{(1)}}
=h_1$; moreover $\RR=\openone$ on $T^{(i)}$, $i=1,\ldots,r$,
even if $T^{(i)}$ is a resonance.
We proceed again as in \S A2.3, by we have to take into account the
lack of the factor $\g^{h_{T^{(i)}}^e-h_{T^{(i)}}}$, which was present before,
when $T^{(i)}$ is a resonance. Since $\th_\RR$ is not a resonance
(by definition)
and $h_{T^{(i)} }^e=h_{T^{(i+1)}}$, we loose at most a factor
$\g^{h_{\th_\RR}-h_{T^{(1)}}}= \g^{h+1-h_1}$.
If we also consider the bound of the
external propagator emerging from $v_1$, we see that the overall effect of
the vertex $v_1$ in the bound of ${\rm Val(\th_\RR)}$ is to add a factor
$\g^{-h-1}$ to the expression in the r.h.s. of \equ(A1.30), that is the same
effect that we should get, if the only cluster containing $v_1$ was $\th_\RR$.
A similar argument can be used for studying the effect of the vertex $v_q$.
Hence we get the bound \equ(A2.6) for all graphs contributing to
$K_{\phi,\phi}^{(h)}(\xx;\yy)$.
We can now bound as in \S A2.3 the sum over $\th_\RR$ in the r.h.s.
of \equ(4.19). Since the sum over $\kk'$ gives a factor
$\g^{h}\max \{\g^h,L^{-1}\}$, we get, for $\l$ sufficiently small,
%
$$ \Big| K_{\phi,\phi}^{(h)}(\xx;\yy) \Big| \le B_5
\sum_{n=1}^{\io} \sum_{q=3}^{\io} B_2^q |\l|^{q-2}
e^{-{\x\over 2}|n|}\max \{\g^h,L^{-1}\} \le |\l| B_6 \max \{\g^h,L^{-1}\}
\; , \Eqa(A2.7) $$
%
for suitable constants $B_5$ and $B_6$.
In the same way, for any $N_1,N_2\ge 0$, $N=N_1+N_2>0$, we have:
%
$$ \eqalign{
& \left( {\sqrt{2}\over \p} \right)^N |x_0-y_0|^{N_0} |x-y|^{N_1}
\Big| K^{(h)}_{\phi,\phi}(\xx;\yy) \Big| \le \cr
& \le \left| \sum_{n=-[L/2]}^{[(L-1)/2]}
{1\over L_i\b} \sum_{\kk\in {\cal D}_{L_i,\b}} \,
e^{-i\kk\cdot(\xx-\yy) + 2 i n p y}
\left[ D_{N_0} D_{N_1} \hat K_{\phi,\phi,n}^{(h)}(\kk)\right]
\right| \; . \cr} \Eqa(A2.8)$$
We can now proceed as in \S A3.1, by using \equ(A2.1); we get
%
$$ |\xx-\yy|^N \Big| K_{\phi,\phi}^{(h)}(\xx;\yy) \Big| \le
C_N \g^{-hN} \, |\l| \max \{\g^h,L^{-1}\} \; ; \Eqa(A2.9) $$
%
a similar bound can be obtained for $K_{\phi,\phi}^{(h^*$, the difference between
the two quantities can be bounded by a constant times $(1/L_i+1/\b)$.
In fact one gets essentially the same bounds as in the proof of
\equ(4.17) and \equ(4.20), up to a factor $\g^{-h}(1/L_i+1/\b)$,
coming from the comparison of the integral and the corresponding
Riemann sum; we shall not give the details, which are completely
straightforward. It follows that
%
$$|S(\xx;\yy)-S^{L_i,\b}(\xx;\yy)| \le C |h^*| (1/L_i+1/\b)
\tende{i,\b\to\io} 0\; .\Eqa(A2.10)$$
\vskip2.truecm
\centerline{\titolo Appendix 4.}
\*\numsec=4\numfor=1
\0{\bf A4.1.}
In this section we shall discuss how the analysis of \S A2.3 has to be
modified, if the support functions $\hat f_h$ do not depend on $k_0$.
As we have discussed in \S4.6, we must use now the bounds \equ(4.45),
instead of \equ(2.14), \equ(3.42) and \equ(3.43).
The main difference (see \equ(4.47)) is that the bound \equ(A1.21) has to be
replaced by
%
$$ \Big|{d\over dt_{i(\ell)}} \tilde g_\ell\Big| \le a_1 {G_2\over2}
\left[ { |-i\t_Vk_0+\g^{h_V^e}| \over
|-it_{i(\ell)}\t_V k_0 + \g^{h_{\ell}}|^2 } +
{\g^{h_V^e}\over \g^{h_{\ell}}
|-it_{i(\ell)}\t_V k_0 + \g^{h_{\ell}}| } \right] \; , \Eqa(A4.5) $$
%
with $a_1=\max\{a_0,1\}$ and $\t_V=\prod_i t_i$,
where the product is over all resonances strictly containing $V$.
In \S4.6 we remarked that this bound is not good for $k_0$ large; however
this problem can be cared by using, for each resonance, the decay of the
one of the propagators external to it, if the set of such propagators is not
empty. Of course, it is not possible to use one fixed propagator for two
different resonances, hence we decide to use,
for this ``balance'' of large $k_0$ behaviour, only the propagator on
the right of each resonance. It follows that we can define the
set of ``bad'' resonances as the the set $\tilde \V V$ of the
resonances $V$ with $\ell_V^o=\ell_{q+1}$, that it the set of resonances,
which have no internal line of the graph at the right. In the evaluation of
${\rm Val(\th_\RR)}$ we shall not use for such resonances the interpolation
formula \equ(A1.18), but we shall simply bound
$|\RR \X_V^{h_V}(\kk'_{\ell_V^o})|$ by
$2\sup_{\kk'_{\ell_V^o}}\{ |\X_V^{h_V}(\kk'_{\ell_V^o})|\}$.
Note that this implies
that, in \equ(A4.5), $\t_V$ must be substituted with the product over all
interpolating parameters associated with the resonances strictly containing
$V$, but not belonging to $\tilde \V V$.
It is important to note that $\tilde \V V$ can contain at most two resonances
and that, if $|\tilde \V V|=2$, one of them must coincide with the whole
graph. This claim easily follows from the remark that, if a graph $\th_\RR$
with external scale $h$ contains a resonance $V$ with $\ell_V^o=\ell_{q+1}$,
then the internal scale of $\th_\RR$ must be equal to $h+1$; hence no other
cluster, except $\th_\RR$ itself, can contain $V$.
\*
\0{\bf A4.2.} In \S A2.3 the interpolating formula \equ(A1.18) was used
to produce a ``gain factor'', that allowed to control the sum over the
scale index of the cluster containing the resonance.
The remark above implies that the bounds \equ(A1.31), \equ(A1.34), \equ(A1.35)
would not have been modified, if we did not exploit the gain factor associated
with the resonances belonging to $\tilde \V V$. We shall now prove that they
survive also to the use of \equ(A4.5) instead of \equ(3.44).
Note that, instead of \equ(A1.23), one has
%
$$ \eqalign{
\Big|{\rm Val(\th_\RR)}\Big| & \le 2^{q-1}
\Big( \prod_{v\in\th_\RR} |F_v| \Big)
\Big( \prod_{T\in\V T} {G_3^{L_T} \over
\left|-it_T\t_Tk_0+\g^{h_T} \right|^{L_T} } \Big) \cr
& \Big( \prod_{T\in {\V V} \setminus {\tilde \V V}}
{ |-i\t_Tk_0+\g^{h_T^e}| \over
|-it_T\t_Tk_0+\g^{h_T}| } \Big) \; , \cr} \Eqa(A4.7)$$
%
where $t_T$ is the interpolation parameter corresponding
to the resonance $T$, if $T\in{\V V}$, while $t_T=1$, if $T$ is not a
resonance. Note that, for $T\in{ \tilde \V V}$, $t_T=\t_T=1$.
If we recall that $L_T=\tilde M_T+M_T^{(r)} + M_T^{(\n)}-1$
(see \S 2.4 and \S A2.3 for notations), we can write
%
$$\eqalign{
&\Big( \prod_{T\in\V T} {1\over \left| -it_T\t_Tk_0+\g^{h_T}
\right|^{L_T}} \Big)
\Big( \prod_{T\in \V T} \g^{h_T M_T^{(\n)}} \Big) \le\cr
&\qquad \le \Big( \prod_{T\in\V T} {1\over \left| -it_T\t_Tk_0+\g^{h_T}
\right|^{\tilde M_T + M_T^{(r)}-1} } \Big)\; ,\cr}\Eqa(A4.7a)$$
%
and we can proceed as in the proof of \equ(A1.30); the role of
\equ(A1.29) is taken by
%
$$\eqalign{
&\Big( \prod_{T\in\V T} {1\over \left|-it_T\t_Tk_0+\g^{h_T}
\right|^{M_T^{(r)}}} \Big)
\Big( \prod_{T\in {\V V} \setminus {\tilde \V V}}
{|-i\t_Tk_0+\g^{h_T^e}| \over |-it_T\t_Tk_0+\g^{h_T}|}\Big) =\cr
&\quad = \Big(\prod_{V\in{\tilde \V V}}
{1\over |-ik_0 + \g^{h_{V}^e}|} \Big)
\Big( \prod_{T\in {\V V} \setminus {\tilde \V V}}
{1\over |-it_T\t_Tk_0+\g^{h_T}|}\Big)\; .}\Eqa(A4.8)$$
By the remarks above about $\tilde \V V$, the r.h.s. of \equ(A4.8) can be
bounded by a constant times the r.h.s. of \equ(A1.29). Hence we get again the
bound \equ(A1.30), with different values of the constants.
\vskip1.truecm
\0{\bf Acknowledgments.} We are indebted to G. Gallavotti for many
discussions and suggestions. We want also to thank the Erwin Schroedinger
Institute, where part of this work was done, during the Workshops on ``Field
Theoretical Methods for Fermion Systems'' and on ``Discrete
Geometry and Condensed Matter Physics'', January 21 -- February 17, 1996.
\pagina
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\*
\ciao
ENDBODY