0$ and $|\l\hat \f_n|\le
a|\l\hat \f_1|/|n|^N$ for some positive constants $a$ and $N$.
Then there exists $\e_0>0$, independent of $i$ and $\r$, such that,
if $|\l\hat\f_1|\le\e_0v_0^4/Q$, with $v_0=\sin(\p\r)$, the one-particle
Hamiltonian $\bf h$ has a gap of width $\ge |\l\hf1|/2$ around $\m$.
Moreover, $\hat\r_n(\f,\m)$ is a continuous function of $\l$, which\
converges to a continuous function of
$\l$ as $i\to\io$, and $\hat\r_0(\f,\m)=\r$.}
\*
\sub(2.3) We can write the self-consistence equation \equ(1.12) as
%
$$ \hf{n} = - \l^2 c_n(\s) \hf{n} + \l \tilde \r_n(\s,\F) \; , \quad
\s\=\l\hf1\; ,\quad \F\=\{\l\hf{n}\}_{|n|>1}\;,\Eq(2.1) $$
%
where $c_n(\s)$ depends on $\f$ only through
$\s$. We write $\hat\r_n$ as a perturbative expansion
in $\l$ (different from the power expansion in $\l$); this expansion is
described in \sec(3). If $|n|>1$, we are here defining $-\l c_n(\s) \hat\f_n$
the contribution to $\hat\r_n$ proportional to $\hf{n}$ of order $1$
in the expansion, while $-\s c_1(\s)$ is the contribution to $\hat\r_1$
proportional to $\s$ of order $\le 1$ in the expansion
(explicit expressions for $c_n(\s)$ and $c_1(\s)$ will
be given in \equ(4.9) and \equ(4.32) respectively);
$\tilde \r_n$ takes into account all the remaining terms of first
order plus all terms of order higher than 1.
The equation \equ(2.1) has of course the trivial solution $\hf{n}=0,
\forall n$,
but it is easy to see that this is not a local minimum, by using the expansion
for $\hat\r_n$ of \sec(3). Therefore we shall look for solutions such that
$\s\neq 0$, so that we can rewrite \equ(2.1) as
%
$$ \eqalignno{
& (1+\l^2 c_1(\s)) = {\l^2\tilde\r_1(\s,\F) \over \s } \; ,&\eq(2.2)\cr
& \F_n\= \l \hf{n} = {\l^2 \tilde\r_n(\s,\F) \over (1+\l^2 c_n(\s))} \; ,
\qquad |n|> 1 \; .&\eq(2.3)\cr}$$
%
Note that the equation for $n=-1$ does not appear simply because
$\r_{-1}=\r_{1}$, as a consequence of the condition
$\hat\f_n=\hat\f_{-n}\in\RRR$, see \equ(1.10).
\*
\sub(2.4) We prove Theorem \secc(1.7) in three steps as follows.\\
$\bullet$ We first study the self-consistence equation \equ(2.3),
considering $\s$ as a variable belonging to the interval
%
$$J = (\;-\exp (-\p\, v_0/\l^2)\;,\; \exp (-\p\, v_0/\l^2)\; )\;.\Eq(2.4)$$
%
We find a solution, that we denote $\F(\s)$, if $\l$ is small enough.\\
$\bullet$ We then prove that, if $L$ is large enough, the equation (in $\l$)
%
$$ 1+\l^2 c_1(\s) = {\l^2 \tilde \r_1 (\s,\F(\s))
\over \s} \Eq(2.5) $$
%
has two solutions $\s^{(\pm)}\in J$, of the form \equ(1.18).
Therefore $(\s^{(\pm)}(\l)/\l,$ $\F(\s^{(\pm)}(\l))/$ $\l)$ turn out to be
solutions of \equ(1.12), which verify, thanks to Lemma \secc(2.2),
\equ(1.13) with $L=L_i$.\\
$\bullet$ We finally prove that the Hessian matrices \equ(1.15)
corresponding to these two solutions are positive definite.
\*
\sub(2.5) {\cs Remarks.} The coefficient $\hf{1}$
has a privileged role with respect to the other coefficients.
In fact, as we shall see in \sec(5),
the properties of the system when only $\hf{1}$ is different from $0$
are very close to the properties of the case in which
all the coefficients are non vanishing.
This suggests that the ``important" equation is \equ(2.2),
so explaining the strategy outlined above.
The previous remark also implies that $1+\l^2 c_1(\s) \simeq 0$. It follows
that $1+\l^2 c_n(\s) \simeq 0$, for all $n$ such that $2\p\r n \simeq 2\p\r$
($\mod 2\p$). Since $\min_{|n|>1} |2\p\r n - 2\p\r|=2\p/Q$, we can expect
that our bounds will not be uniform in $Q$. This is the reason why Theorem
\secc(1.7) can not be extended to irrational density; at most one can hope
that a Diophantine condition on $\r$ is needed, but we have only been able
to prove that the $Q$ dependence can be substituted with a dependence on the
Diophantine constants in some of the bounds described below.
Note also that, if $Q=2$, the only equation to discuss
is just the equation \equ(2.2) with $\F=0$ and the r.h.s. equal to zero;
its solution is well known in this case, see [KL,LM] for example.
If $Q=3$, again \equ(2.2) is the only equation to discuss, but the r.h.s.
is different from zero; however it is easy to prove that the solution has
essentially the same properties as in the case $Q=2$.
Hence, in the following we shall consider only the case $Q\ge 4$.
The following lemma, furnishing a bound on the
constants $c_n(\s)$ and their derivatives, is proven in \sec(4.8).
\*
\sub(2.6) {\cs Lemma.} {\it There exists a constant $C$, independent of
$i$ and $\r$, such that, if $|n|\ge 2$,
%
$$|c_n(\s)| \le {C\over v_0}
\left(1+\log{1\over v_0}\right) \log Q\; ,\Eq(2.6)$$
%
$$\left|{\dpr c_n(\s)\over \dpr\s}\right|
\le {C\over v_0 |\s|}\; ,\Eq(2.6a)$$
}
\*
\sub(2.8)
Fixed $L=L_i$, $\F$ is a finite sequence of $Q-3$ elements, which can be
thought as a vector in $\RRR^{Q-3}$, which is a function of $\s$.
In order to study the equation \equ(2.2) for $\s$, we shall need a bound on
$\F$ and on the derivative of $\F$ with respect to $\s$. Hence we consider
the space $\FF=\CC^1(\,J\;,\RRR^{Q-3})$ of $C^1$-functions of $\s\in J$
with values
in $\RRR^{Q-3}$; the solutions of \equ(2.3) can be seen
as fixed points of the operator ${\bf T}_{\l}: \FF\to \FF$,
defined by the equation:
%
$$[{\bf T}_{\l}(\F)]_n(\s) =
{\l^2 \tilde\r_n(\s,\F(\s)) \over (1+\l^2 c_n(\s))} \; ,\Eq(2.7)$$
We shall define, for each positive integer $N$, a norm in $\FF$ in
the following way:
%
$$\|\F\|_\FF\= \sup_{|n|>1,\s\in J} \left\{
|n|^N \left[ |\s|^{-1} |\F_n(\s)|
+\left|{\dpr\F_n\over \dpr\s}(\s)\right|\right] \right\} \;.\Eq(2.8)$$
We shall also define
%
$$\BB = \{\F\in\FF : \|\F\|_\FF\le 1\}\; ;\Eq(2.9)$$
%
$$R(\F)_n(\s) = \tilde\r_n(\s,\F(\s))\;,\quad |n|\ge 2\;.\Eq(2.9a)$$
The following two lemmata, to be proved in \sec(5.5) and \sec(5.6),
respectively,
resume the main properties of $R(\F)$.
\*
\sub(2.9) {\cs Lemma.} {\it There are two constants $C_1>1$ and $C_2$,
independent of $i$, $\r$ and $N$, such that, if $\F,\F'\in\BB$ and
%
$$C_1 Q v_0^{-4} |\s|[1+\log (v_0^2/|\s|)]\le 1\;,\Eq(2.9b)$$
%
then
%
$$\|R(\F)-R(\F')\|_\FF \le {C_2 3^N N! \over v_0}
\left(1+\log{1\over v_0}\right)
\|\F-\F'\|_\FF\; .\Eq(2.10)$$}
\*
\sub(2.10) {\cs Lemma.} {\it There is $C>1$, such that, if
%
$$C Q v_0^{-3} |\s|^{1/2} [1+\log (v_0^2/|\s|)] \le 1\;,\Eq(2.9c)$$
%
then
%
$$\|R(0)\|_\FF \le {C\over v_0} \left(1+\log{1\over v_0}\right)
\sup_{|n|>1} \left\{ |n|^N \Big({|\s|\over v_0^2}\Big)^{|n|\over 10}
\right\} \; .\Eq(2.11)$$
}
\*
\sub(2.11) {\cs Lemma.} {\it There are $\e,c,K$, independent of $i$, $\r$
and $N$, such that, if $\s\in J$ and
%
$$\l^2 \le \e\,{v_0^2 (1+\log v_0^{-1})^{-1}
\over K^N N! \log(c\,Q/v_0^4)}\;,\Eq(2.11a)$$
%
there exists a unique solution $\F\in\BB$ of \equ(2.3);
moreover the solution satisfies the bound
%
$$\|\F\|_\FF \le \left({\l^2\over v_0}\right)^N\; .\Eq(2.12)$$
}
\*
\sub(2.12) {\it Proof of Lemma {\secc(2.11)}.}
It is easy to see that, if $\s\in J$, the
conditions on $\s$ of Lemma \secc(2.9) and Lemma \secc(2.10) are
satisfied, if
%
$$\l^2 \le \e_0/\log(c Q/v_0^4)\; ,\Eq(2.13)$$
%
with suitable values of $\e_0$ and $c$. Moreover, if $\e_0\le \e_1 v_0
(1+\log v_0^{-1})^{-1}$ and $\e_1$ is chosen small
enough, \equ(2.6) and \equ(2.13) imply that $\l^2|c_n(\s)|\le 1/2$,
so that, by using \equ(2.6a), \equ(2.7) and Lemma \secc(2.9), we have that,
if $\F\in\BB$,
%
$$ \|{\bf T}_{\l} (\F)\|_\FF \le 4 \l^{2} \left(1+\l^2{C\over v_0}\right)
\left[ \|R(0)\|_\FF + {C_2\over v_0}\left(1+\log{1\over v_0}\right)
3^N N! \|\F\|_\FF\right]\; . \Eq(2.14)$$
Therefore, by \equ(2.11) and \equ(2.4), there exist constants $C_3$ and $C_4$,
such that, if $\e_1\le \e v_0 (C_4^N N!)^{-1}$ and $\e$ is small enough,
%
$$ \|{\bf T}_{\l} (\F)\|_\FF
\le {C_3\l^2\over v_0^2} \left(1+\log{1\over v_0}\right)
\left[3^N N! + \sup_{|n|>1}
|n|^N \exp \Big(-{\p v_0 |n|\over 10 \l^2 }\Big)
\right] \le 1\; . \Eq(2.15)$$
Moreover, by \equ(2.10), if $\F,\F'\in\BB$ and
similar conditions on $\l$ are satisfied, we have
%
$$\|{\bf T}_{\l}(\F)-{\bf T}_{\l}(\F')\|_\FF \le {C_5^N N!\l^2\over v_0^2}
\left(1+\log{1\over v_0}\right) \|\F-\F'\|_\FF \le
\fra12 \|\F-\F'\|_\FF \; .\Eq(2.16)$$
The bounds \equ(2.15) and \equ(2.16) imply that $\BB$ is invariant
under the action of ${\bf T}_{\l}$ and that ${\bf T}_{\l}$ is a
contraction on $\BB$. Hence, by the contraction mapping principle, there is
a unique fixed point $\bar\F$ of ${\bf T}_{\l}$ in $\BB$, which can
be obtained as the limit of the sequence $\F^{(k)}$ defined through the
recurrence equation $\F^{(k+1)}={\bf T}_{\l}(\F^{(k)})$, with any initial
condition $\F^{(0)}\in\BB$. If we choose $\F^{(0)}=0$, we get, by
\equ(2.16),
%
$$\|\bar\F\|_\FF \le \sum_{i=1}^\io \|\F^{(i)}-\F^{(i-1)}\|_\FF
\le \sum_{i=1}^\io {1\over 2^{i-1}} \|\F^{(1)}\|_\FF \le
\|\F^{(1)}\|_\FF \; .\Eq(2.17)$$
%
On the other hand, by \equ(2.11),
%
$$\|\F^{(1)}\|_\FF = \|{\bf T}_{\l}(0)\|_\FF \le
{C_6^N N! \l^2\over v_0^2} \left(1+\log{1\over v_0}\right)
\left({\l^2\over v_0}\right)^N\; ,\Eq(2.18)$$
%
which immediately implies the bound \equ(2.12), if $\e_1\le \e
v_0(C_6^N N!)^{-1}$, with $\e$ small enough. \qed
\*
\sub(2.13) Let us now consider the equation \equ(2.5). We want to prove that
it has two solutions of the form \equ(1.18), if $\s\in J$ and $L_i$ is large
enough.
In order to achieve this result, we need some detailed properties of the
function $c_1(\s)$, which are described in the following Lemma \secc(2.13a),
to be proved in \sec(4.9). We need also the bounds on $\tilde\r_1(\s,\F(\s))$
and its derivative with respect to $\s$, contained in Lemma \secc(2.13b), to
be proved in \sec(5.7).
\*
\sub(2.13a) {\cs Lemma.}
{\it There is a constant $C$, such that, if
%
$${v_0\over L_i |\s|} \le \tilde\e \le {1\over 8\p}
\; ,\qquad {|\s|\over v_0^2}\le 1\;,\Eq(2.18a)$$
%
then
%
$$-c_1(\s) = {1\over 2\p v_0} \left[ \log {v_0^2\over |\s|} + r_1(\s)
\right]\;,\Eq(2.19)$$
%
with
%
$$\eqalign{
|r_1(\s)| &\le C \left(1+\log{1\over v_0}\right)\;,\cr
\left|{\dpr r_1(\s)\over \dpr\s}\right| &\le C \left( {1\over v_0^2}
+{\tilde\e\over |\s|} \right)\;.\cr}\Eq(2.19a)$$
}
\*
\sub(2.13b) {\cs Lemma.}
{\it If $\s\in J$, $\l$ satisfies the inequality \equ(2.11a), with
$\e$ small enough, $\F(\s)$ is
the solution of the equation \equ(2.3) described in Lemma \secc(2.11) and
%
$$r_2(\s)\= {2\p v_0 \tilde \r_1 (\s,\F(\s))\over \s}\;,\Eq(2.19b)$$
%
then there is a constant $C$, such that
%
$$\eqalign{
|r_2(\s)| &\le C %\left(1+\log{1\over v_0}\right)
\left[\left({|\s|\over
v_0^2}\right)^{1/4} + \left({\l^2\over v_0}\right)^N \right]
\;,\cr
\left|{\dpr r_2(\s)\over \dpr\s}\right| &\le {C
\over |\s|}
\left[\left({|\s|\over
v_0^2}\right)^{1/4} + \left({\l^2\over v_0}\right)^N \right]
\;.\cr}\Eq(2.19c)$$
}
\*
\sub(2.14) {\cs Lemma.}
{\it There exist positive constants $\e$, $\tilde\e$, $c$ and $K$,
independent of $i$, $\r$ and $N$, such that,
if $\l$ satisfies the inequalities \equ(1.16),
there are two solutions $\s^{(\pm)}(\l)\in J$ of equation \equ(2.5)
of the form \equ(1.18).}
\*
\sub(2.15) {\it Proof of Lemma {\secc(2.14)}.}
By using the definitions of $r_1(\s)$ and $r_2(\s)$ given in \equ(2.19) and
\equ(2.19b), we can write the equation \equ(2.5) in the form
%
$$F(\s)\=\log {v_0^2\over |\s|} -{2\p v_0\over \l^2}+r(\s)
=0\;,\Eq(2.20)$$
%
where $r(\s)=r_1(\s)+r_2(\s)$.
Let us now suppose that $\l$ satisfies the
inequalities \equ(2.11a) and that $\s$ belongs to the interval
%
$$\tilde J= \left( v_0^2 e^{-4\p v_0/\l^2}\;,\; v_0^2 e^{-\p
v_0/\l^2} \right)\subset J\;.\Eq(2.21)$$
%
If $L_i$ is large enough and the constant $\e$ in \equ(2.11a) is chosen small
enough, the conditions \equ(2.18a) of Lemma \secc(2.13a) are satisfied, for
$\s\in \tilde J$, and
%
$${4\p v_0\over \l^2} \le \log(\tilde\e v_0 L_i)\;.\Eq(2.21a)$$
%
Moreover, if $\tilde \e$ and $\e$ (hence $|\s|v_0^{-2}$) are small enough,
%
$$\left| {\dpr r(\s)\over \dpr\s} \right| \le {1\over 2}
\left| {\dpr \over \dpr\s} \log {v_0^2\over\s}\right|\;;\Eq(2.22)$$
%
hence $F(\s)$ is a monotone decreasing function of $\s$ in $\tilde J$.
If we define
%
$$\s^* = v_0^2 e^{-2\p v_0/\l^2}\;,\qquad M=\sup_{\s\in\tilde J} |r(\s)|\;,
\Eq(2.23)$$
%
we have that $F(\s^*\exp (-2M))>0$ and $F(\s^*\exp (2M))<0$. Moreover, the
interval $(\s^*\exp (-2M)),\s^*\exp (2M)))$ is contained in $\tilde J$, if
$\e$ is small enough, since the bounds \equ(2.19a) and \equ(2.19c)
imply that $M\le C (1+\log v_0^{-1})$.
Hence there is a unique solution $\s^{(+)}(\l)$ of
\equ(2.20) in $\tilde J$, which can be written as
%
$$\s^{(+)}(\l)= v_0^2 e^{-{2\p v_0+\b^{(+)}(\l)\over \l^2} }\;,\Eq(2.24)$$
%
with $|\b^{(+)}(\l)| \le C\l^2(1+\log v_0^{-1})$.
In the same manner, we can show that there is solution $\s^{(-)}(\l)$
in the interval
%
$$ (-v_0^2 e^{-{\p v_0 \over \l^2}}\;,\;
-v_0^2 e^{-{4\p v_0\over \l^2}}) \subset J \; , \Eq(2.25) $$
%
with the same properties. \qed
\*
\sub(2.16) {\cs Lemma.} {\it The constants $\e$, $\tilde\e$, $c$ and $K$,
appearing in \equ(1.16), can be chosen so that the Hessian matrix \equ(1.15)
is positive definite.}
\*
\sub(2.17) The proof of Lemma \secc(2.16) is in \sec(5.8).
This completes the proof of Theorem \secc(1.7).
%\vskip1.truecm
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section(3,Graph formalism)
\sub(3.1) In this section we shall describe the expansion of $\r_x(\f,\m)$,
used to get the results of this paper.
Let us consider the operators
$\psi_{\xx}^{\pm}=e^{tH}\psi_x^{\pm}e^{-Ht}$, with $\xx=(x,t)$,
$-\b/2\le t \le \b/2$ for some $\b>0$; on $t$ antiperiodic boundary
conditions are imposed. As explained, for example, in [BGM], there is a simple
(well known) relation between $\r_x(\f,\m)$ and the {\sl two-point Schwinger
function}, defined by
%
$$S^{L,\b}(\xx;\yy) = {{\rm Tr} \left[\exp(-\b H)
\left(\theta(x_0>y_0)\psi^-_{\xx} \psi^+_{\yy}-
\theta(x_0