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correlation, semi-classical.
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\centerline{\Stor Complete asymptotics for
correlations of }
\centerline{\Stor Laplace integrals in the
semi-classical limit.}
\medskip
\centerline{{\stor Johannes
Sj\"ostrand}\footnote{*}{\noindent Centre de
Math\'ematiques, Ecole Polytechnique,
F--91128 Palaiseau cedex, and UMR 7640 of
CNRS\smallskip
Key words: correlation,
semi-classical.
1991 subject
classification: Primary 82B20, secondary:
81Q20}}
\medskip
\par\noindent \it Abstract. \rm\liten In
this paper we study the exponential
asymptotics of correlations at large
distance associated to a measure of
Laplace type. As in [S1], [BJS], we look at
a semi-classical limit. While in those
papers we got the exponential decay rates
and the prefactor only up to some factor
$(1+{\cal O}(h^{1/2}))$, where $h$ denotes
the small semi-classical parameter, we now
get full asymptotic expansions. The main
strategy is the same as in the quoted
papers, namely to use an identity ([HS])
involving the Witten Laplacian of degree 1,
and a Grushin (Feshbach) reduction for
the bottom of the spectrum of this
operator. The essential difference is
however that we now have to use higher
order Grushin problems (amounting to the
study of a larger part of the bottom of th
spectrum). In a perturbative case, the
strategy of higher order Grushin problems
was recently implemented by W.M.Wang [W] to
get a few terms in the perturbative
expansion of the decay rate.\rm
\smallskip
\par\noindent \it Resum\'e. \rm\liten Dans
cet article nous \'etudions l'asymptotique
exponentielle des
corr\'elations \`a grande distance
associ\'ees \`a une mesure de type de
Laplace. Comme dans [S1], [BJS], on se
place dans la limite semi-classique. Dans
ces travaux nous avons obtenu le
taux de d\'ecroissance exponentielle et le
pr\'efacteur \`a un facteur
$(1+{\cal O}(h^{1/2}))$ pr\`es, o\`u $h$
d\'esigne le petit param\`etre
semi-classique. Nous obtenons maintenant
des d\'eveloppements asymptotiques
compl\`ets. La strat\'egie est la m\^eme
que dans [S1,BJS]: Utiliser une
identit\'e ([HS]), qui comporte
le laplacien de Witten en degr\'e
1, ainsi qu'une r\'eduction de Grushin
(Feshbach) pour le bas du spectre de cet
op\'erateur. La nouveaut\'e est que nous
devons maintenant utiliser des probl\`emes
de Grushin d'ordre sup\'erieur, ce
qui revient \`a \'etudier une plus
large fraction du spectre. Dans un cas
perturbatif W.M.Wang [W] a r\'ecemment
utilis\'e des probl\`emes de Grushin
d'ordre sup\'erieur pour obtenir quelques
termes dans le d\'eveloppement du taux de
d\'ecroissance.\rm
\medskip
\centerline{\bf 0. Introduction.}
\medskip
\par In recent years there has been an
attempt by B. Helffer, the author and
others ([BJS], [H1--4], [HS], [J],
[S1--6], [SW], [W]) to apply direct methods
to the study of integrals and operators in
high dimension, of the type that may appear
naturally in statistical mecanics and
Euclidean field theory. In the first works,
we applied asymptotic methods and noticed
already that there is a very strong
interplay between asymptotic expansions for
integrals obtained by some variant of the
stationary phase method and asymptotic
solutions of certain Schr\"odinger type
operators obtained by the WKB method
([S3--5]). In later works ([S6], [HS],
[S4]) we noticed that a suitable version of
the maximum principle could be used in the
proof of certain asymptotic expansions and
to obtain exponential decay of
correlations. (In the the work [SW] this
was even applied to integrals in the
complex domain, and was applied to show
exponential decay of the expectation of the
Green function for discrete Schr\"odinger
operators with random potentials.)
\par In the present work we are
interested in correlations for Laplace
integrals at large distance. In
physics language we are
interested in the correlations at large
distance for continuous spin systems. Under
assumptions that imply the exponential
decay of these correlations, we want to
know the precise rate of exponential decay
and to determine the possible polynomial
prefactor. The original
inspiration came from a
talk given by R.Minlos in St Peterburg in
1993 and a corresponding joint paper by him
and E.Zhizhina [MZ], about the asymptotics
of correlations for discrete spin models at
high temperature. Even though we
never quite understood the methods used in
[MZ], it prompted us to further develop
our own methods in the continous spin case
and in [S1] we were able to get the leading
exponential decay asymptotics for
correlations associated to measures of the
type
$e^{-\phi (x)/h}dx\over \int
e^{-\phi (x)/h}dx,$
at large distance, in the semi-classical
limit ($h\to 0$). Here $\phi \in
C^\infty ({\bf R}^\Lambda ;{\bf R})$, and
$\Lambda
$ is a finite subset of ${\bf Z}^d$ or a
discrete torus of dimension $d$, and we
study the limit when $\Lambda $ is
large. Recall that the correlation of two
functions $u,v$ which do not grow too
fast at infinity is given by
\ekv{0.1}
{{\rm Cor\,}(u,v)=\langle (u-\langle
u\rangle )(v-\langle v\rangle ),}
where
$$\langle u\rangle = {\int u(x)e^{-\phi
(x)/h}dx\over \int e^{-\phi (x)/h}dx}$$
denotes the expectation. We
also observed that certain associated
Schr\"odinger operators, already found with
B.Helffer in [HS], are Witten Laplacians in
degree 0 or 1. We also managed to replace
the use of the maximum principle at many
places by
$L^2$ methods and consequently we got rid of a certain
rigidity in the conditions. The elimination
of the maximum principle was not complete
however, and the results were
flawed by a certain number of unnatural
assumptions, in particular that of global
uniform strict convexity of the function
$\phi $. Another short-coming of [S1] was
that we only determined the decay rate and
the prefactor up to a factor $(1+{\cal
O}(h^{1/2}))$. Moreover, we did not work
out the thermodynamical limit ($\Lambda
\to{\bf Z}^d$) so
oscillations within a factor $1+{\cal
O}(h^{1/2})$ could not be excluded, when
$\Lambda$ varies.
\par With V.Bach and T.Jecko [BJS] we
elimintated completely the use of the
maximum principle and were able to give
simpler and more natural
conditions. In particular we
could allow the exponent $\phi $ to be
strictly convex only near the point where
$\phi $ is minimal. The new assumptions
still imply that there is only one
critical point however. Again we obtained
the decay rate and the prefactor only up
to a factor $(1+{\cal O}(h^{1/2}))$, and we
did not treat the thermodynamical
limit.
\par The aim of the present paper is
to get full asymptotic expansions in powers
of $h$ of the decay rate and in powers of
the inverse distance and in $h$ of the
prefactor, and we shall also treat the
thermodynamical limit. To get such more
precise results, one has to get higher in
the spectrum of the associated Witten
Laplacians (or rather something close to
that), and we do so by using
higher order Grushin problems that we
explain more in detail later in this
introduction. The main idea of this strategy
was rather clear in the author's mind
since the writing of [S1], and has become
practically realizable with the
improvements of [BJS]. W.M.Wang [W] has
recently used similar ideas in order to
study the rate of exponential decay of
correlations when
$h=1$ and
$\phi $ is a small perturbation of a
non-degenerate quadratic form. For the
decay rate, she got several terms in an
expansion in powers of the perturbation
parameter. In principle the method should
give full asymptotic expansions and also the
prefactor in that case too. [W] also has an
interesting application to the exponential
decay rate of the Green function for
discrete random Schr\"odinger operators.
\par We now start to formulate the main
result of this paper, and after that we will
outline some ideas of the proof. To
understand the idea behind many of the
assumptions, it may be helpful to have in
mind the special case when $\phi =\phi
_\Lambda $ is of the form
\ekv{0.2}
{\phi (x)=\sum_{j\in\Lambda
}f(x(j))+\sum_{\vert j-k\vert _1=1}
w(x(j),x(k)),}
when
\ekv{0.3}{\Lambda =({\bf Z}/L{\bf Z})^d,} is
a discrete torus with $L\ge 2$ and $\phi $
is of the form
\ekv{0.4}
{\phi (x)=\sum_{j\in\Lambda
}f(x(j))+\sum_{{\vert j-k\vert
_1=1}\atop{j\,{\rm or}\, k\in\Lambda
}}w(\widetilde{x}(j),\widetilde{x}(k)),}
when $\Lambda$ is a finite subset of ${\bf
Z}^d$. Here $\vert \cdot \vert _p$ will
always denote the $\ell^p$ norm, and
when $x$ is the discrete torus $\Lambda $
in (0.3), we let $\vert j\vert _p$ denote
$\inf_{\widetilde{j}\in \pi _\Lambda
^{-1}(j)}\vert \widetilde{j}\vert _p$, where
$\pi _\Lambda :{\bf Z}^d\to \Lambda $ is the
natural projection. In the formula (0.4) we
let
$\widetilde{x}(j)$ be equal to $x(j)$ for
$j\in\Lambda $ and $=0$ otherwise. $f$
and $w$ are smooth real valued functions on
${\bf R}$ and ${\bf R}^2$ respectively,
with
\ekv{0.5}
{w(x,y)=w(y,x).}
In the proofs the special forms (0.2),
(0.4) will never be of any real
advantage, so we shall formulate our main
result with more general functions.
Notice that if we let $\Lambda $ grow, then
the expressions (0.2), (0.4) will in
general diverge, while the derivatives
of all orders will converge. For that
reason (and with the termodynamical limit
in mind) we will start by fixing the
limiting Hessian, and after that we will
see how to get a family of functions $\phi
(x)=\phi _\Lambda (x)$ both in the torus
case and in the subset case.
\par We say that a function $f$ on ${\bf
R}^{{\bf Z}^d}$ is smooth if it is
continuous for the $\ell^\infty $ topology,
differentiable in each of the variables
with continuous derivatives (for the
$\ell^\infty $ topology) and the derivatives
enjoy the same properties et c. Let $\Phi
_{j,k}(x)$,
$j,k\in{\bf Z}^d$ be smooth and real on
${\bf R}^{{\bf Z}^d}$ and satisfy
\ekv{{\rm A}.1}
{\Phi _{j,k}(x)=\Phi _{k,j}(x),}
\ekv{{\rm A}.2}
{\partial _{x_\ell}\Phi _{j,k}=\partial
_{x_j}\Phi _{\ell,k},}
\ekv{{\rm A}.3}
{\Phi =(\Phi _{j,k})\hbox{ is }2\hbox{
standard},}
\ekv{{\rm A}.4}
{\Phi (0)\ge {\rm Const.}>0.}
Here we use the terminology of [S2]
concerning $k$ standard
tensors. Let $a=(a_{\Lambda ;j,k}(x))$,
$x\in{\bf R}^\Lambda $, $j,k\in\Lambda $
be a family of matrices (i.e. 2 tensors)
depending on some family of finite sets
$\Lambda $. We say that $a=a_\Lambda $ is
2 standard if we have uniformly in
$x\in{\bf R}^\Lambda $, $\Lambda $, the
estimates
\ekv{0.6}
{\langle \nabla ^ka(x),t_1\otimes
...\otimes t_{k+2}\rangle ={\cal
O}_k(1)\vert t_1\vert _{p_1}..\vert
t_k\vert _{p_{k+2}},} for all $t_j\in {\bf
C}^\Lambda $ and
$p_j\in [1,+\infty ]$ with
$1={1\over p_1}+..+{1\over p_{k+2}}$. Here
$\vert \cdot \vert _p$ denotes the
standard $\ell^p$ norm on ${\bf C}^\Lambda
$. When $x$ varies in ${\bf R}^{{\bf
Z}^d}$ and $j,k\in{\bf Z}^d$, we require
the $a_{j,k}(x)$ to be smooth in the
sense mentioned earlier and say that $a$
is 2-standard if (0.6) holds with $\Lambda
={\bf Z}^d$ and $t_j\in{\bf C}^\Lambda $,
with $t_j(\lambda )\to 0$, $\Lambda \ni
\lambda \to \infty $. Notice that a
2-standard matrix is
${\cal O}(1):
\ell^p\to \ell^p$, for $1\le p\le \infty $.
\par If we think of the formal expression
(0.2) with $\Lambda ={\bf Z}^d$ and assume
that
$\nabla ^k f$ and
$\nabla ^kw$
are bounded for all $k\ge 2$, then the
matrix $\Phi (x)=(\phi ''_{j,k}(x))$
fulfills the assumptions (A.1--3). $\Phi
(0)$ is the matrix of a convolution
and (A.4) amounts to the assumption:
\ekv{0.7}
{f''(0)+4d\partial _x^2w(0,0)>4d\vert
\partial _x\partial _yw(0,0)\vert .}
\par If ${\bf Z}^d$ is replaced by a
finite set $\Lambda $, then (A.1,2) becomes
a necessary and sufficient condition for
the existence of a realvalued function
$\phi \in C^\infty ({\bf R}^\Lambda )$ with
$\phi ''_{j,k}=\Phi _{j,k}$. In the ${\bf
Z}^d$ case we shall now see how to produce
two different finite dimensional versions of
such a function.
\par Let $U\subset{\bf Z}^d$ be finite.
If $x\in{\bf R}^U$, let
$\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be
the zero extension of $x$, so that
$\widetilde{x}(j)=x(j)$ for $j\in U$,
$\widetilde{x}(j)=0$, for $j\in{\bf
Z}^d\setminus U$. Then $$\Phi
_{U;j,k}(x):=\Phi _{j,k}(\widetilde{x}),\
j,k\in U$$
is a smooth tensor on ${\bf R}^U$ which
satisfies (A.1,2) with $j,k,\ell\in U$.
Hence there exists a function $\phi
_U(x)\in C^\infty ({\bf R}^U;{\bf R})$ with
\ekv{0.8}
{\phi ''_{U;j,k}(x)=\Phi _{U;j,k}(x),\
x\in{\bf R}^U,\, j,k\in U.}
We make $\phi _U$ unique up to a constant,
by requiring that
\ekv{0.9}
{\phi '_U(0)=0.}
It is easy to check that $\phi ''_U$ is
2-standard.
\par We next do the same with $U$ replaced
by a discrete torus $\Lambda =({\bf
Z}/L{\bf Z})^d$. If $\lambda \in {\bf
Z}^d$, we define $\tau _\lambda x\in{\bf
R}^{{\bf Z}^d}$, by $(\tau _\lambda x)(\nu
)=x(\nu -\lambda )$. We will
assume translation invariance for
$\Phi $:
\ekv{{\rm A}.7}
{\Phi _{j+\lambda ,k+\lambda }(\tau
_\lambda x)=\Phi _{j,k}(x),\
j,k,\lambda \in{\bf Z}^d.}
(In section 10 we discuss a larger set of
conditions and reproduce here only the
most important ones with the same
numbering as in section 10.) Notice that if
$\Phi _{j,k}$ were the Hessian of a smooth
function
$\phi
\in C^\infty ({\bf R}^{{\bf Z}^d})$ (and the
discussion remains valid if we replace
${\bf Z}^d$ by a discrete torus $\Lambda
$) then (A.7) would be a consequence of the
simpler translation invariance property:
\ekv{0.10}
{\phi (\tau _\lambda x)=\phi (x).}
\par If $x\in{\bf R}^\Lambda $, let
$\widetilde{x}=x\circ \pi _\Lambda \in{\bf
R}^{{\bf Z}^d}$ be the corresponding $L{\bf
Z}^d$ periodic lift, where
$\pi _\Lambda :{\bf Z}^d\to\Lambda $ is the
natural projection. Replacing
$x$ by
$\widetilde{x}$ in
(A.7), we get
\ekv{0.11}
{\Phi _{j-\lambda ,k-\lambda
}(\widetilde{x})=\Phi
_{j,k}(\widetilde{x}),\ \lambda \in L{\bf
Z}^d.}
If we view $\Phi $ as a matrix, this is
equivalent to
\ekv{0.12}
{\tau _\lambda \circ \Phi
(\widetilde{x})=\Phi (\widetilde{x})\circ
\tau _\lambda ,\ \lambda \in L{\bf Z}^d,}
so $\Phi (\widetilde{x})$ maps $L{\bf
Z}^d$ periodic vectors into the same kind
of vectors. Hence there is a naturally
defined $\Lambda \times \Lambda $ matrix
$\Phi _\Lambda (x)$, defined by
\ekv{0.13}
{
\widetilde{\Phi _\Lambda (x)t}=\Phi
(\widetilde{x})\widetilde{t}, }
where again the tilde indicates that we
take the periodic lift. On the matrix
level, we get
\ekv{0.14}
{
\Phi _{\Lambda
;j,k}(x)=\sum_{\widetilde{k}\in\pi
_\Lambda ^{-1}(k)}\Phi
_{\widetilde{j},\widetilde{k}}(
\widetilde{x}), }
for any $\widetilde{j}\in \pi _\Lambda
^{-1}(j)$. Alternatively, we have
\ekv{0.15}
{\Phi _{\Lambda
;j,k}(x)=\sum_{\widetilde{j}\in\pi
_\Lambda ^{-1}(j)}\Phi
_{\widetilde{j},\widetilde{k}}(
\widetilde{x}),\ \widetilde{k}\in \pi
_\Lambda ^{-1}(k),}
and $\Phi _{\Lambda ;j,k}$ is symmetric
(cf. (A.1)).
\par
In section 10, we shall verify that $\Phi
_\Lambda $ satisfies (A.1--4), so there
exists $\phi _\Lambda \in C^\infty ({\bf
R}^\Lambda ;{\bf R})$, unique up to a
constant, such that
\ekv{0.16}
{\Phi _{\Lambda ;j,k}(x)=\partial
_{x_j}\partial _{x_k}\phi _\Lambda (x),\
\phi _\Lambda '(0)=0.}
\par In the case when $\Phi $ is the
Hessian of the formal expression (0.2)
with $\Lambda $ there replaced by ${\bf
Z}^d$, and if we assume that 0 and (0,0)
are critical points of $f$ and $w$
respectively, then it is easy to see that
$\phi _\Lambda $ is given by the
expression (0.2) when $\Lambda $ is a
discrete torus with $L\ge 3$ and by (0.4)
when
$\Lambda =U$ is a bounded subset of ${\bf
Z}^d$.
\par We assume
that $\Phi (0)$ is ferromagnetic in the
sense that
\ekv{{\rm A}.9}{\Phi _{j,k}(0)\le 0,\ j\ne
k.}
We have
\ekv{0.17}
{\Phi (0)=
1-\widetilde{v}_0*, }
where $0\le \widetilde{v}_0\in\ell^1({\bf
Z}^d)$ is even with
$\widetilde{v}_0(0)=0$ and the star
indicates that
$\widetilde{v}_0$ acts as a convolution.
Actually, the constant 1
should be replaced by a more general
constant $a>0$, but we may always reduce
ourselves to the case $a=1$, by a dilation
in $h$.
\par Assume that there exists a
finite set
$K\subset {\bf Z}^d$ such that
\ekv{{\rm A}.10}
{\widetilde{v}_0(j)>0,\, j\in K,\ {\rm
Gr\,}(K)={\bf Z}^d,}
where ${\rm Gr\,}(K)$ denotes the smallest
subgroup of ${\bf Z}^d$ which contains
$K$. We also make the following finite
range assumption:
\ekv{{\rm A.fr}}
{\exists C_0,\hbox{ such that }\Phi
_{j,k}(x)=0\hbox{ for }\vert j-k\vert
>C_0.}
\par We introduce the $2$ standard
matrix
\ekv{0.18}
{A(x)=\int_0^1 \Phi (tx) dt,}
The following assumption is a weakened
convexity assumption and will be used in
section 10 together with a maximum
principle (from [S4]) to obtain other
more explicit conditions.
\eekv{{\rm A.mp}}
{\exists \epsilon _0>0\hbox{ such that
for every $x\in {\bf R}^{{\bf Z}^d}$,
$A(x)$ satisfies (${\rm
mp\,}\epsilon _0$): If }} {\hbox{$t\in
\ell^1({\bf Z}^d)$, $s\in
\ell^\infty ({\bf Z}^d)$, and
$\langle t,s\rangle =\vert t\vert _1\vert
s\vert _\infty $, then $\langle
A(x)t,s\rangle \ge \epsilon _0 \vert
t\vert _1\vert s\vert _\infty $.}}
Notice that this assumption is fulfilled
if $A(x)=1+B(x)$ with $\Vert B(x)\Vert
_{{\cal L}(\ell^\infty ,\ell^\infty )}\le
1-\epsilon _0$. Also notice that (A.4) is
a consequence of (A.mp).
\par Let $U_j\in {\bf Z}^d$, $j=1,2,..$ be
an increasing sequence of finite sets
containing
$0$ and converging to ${\bf Z}^d$. Let
$2\le L_j\nearrow\infty $ be a sequence of
integers with
\ekv{0.19}
{U_j\subset [-{L_j\over 4},{L_j\over
4}]^d,}
and let $\Lambda =\Lambda _j=({\bf Z}/L_j{\bf
Z})^d$ be a corresponding sequence of
discrete tori,
so that we can view $U_j$ as a subset of
$\Lambda _j$ in the natural way.
\par The following is the main result of
our work:
\medskip
\par\noindent \bf Theorem 0.1. \it Let
$\Phi _{j,k}(x)\in C^\infty ({\bf R}^{{\bf
Z}^d})$ satisfy (A.1--3, 7, 9, 10, fr, mp)
and define
$\phi _U(x)\in C^\infty ({\bf R}^U;{\bf
R})$, $\phi _\Lambda \in C^\infty ({\bf
R}^\Lambda ;{\bf R})$ as above,
when $U\subset {\bf Z}^d$ is finite and
$\Lambda =({\bf Z}/L{\bf Z})^d$ is a
discrete torus. Let $U_j$, $\Lambda _j$
be as above, and put $r_j:={\rm
dist\,}(0,{\bf Z}^d\setminus U_j)$, so
that $r_j\nearrow +\infty $ when
$j\to\infty $.
\par Then there exist $C_0\ge 1$,
$j_0\in{\bf N}$, $\theta >0$, $h_0>0$, such
that for
$j\ge j_0$, $00$
uniformly in $j$. If we had assumed (as in
[S1]) that
$\phi ''(x)\ge {\rm const.}>0$ uniformly
in $x,j$, that would have been immediate
from (0.31). As in [BJS] we only assume
this at $x=0$ however, and the idea
(exploited in [BJS]) is then to make a
limited Taylor expansion,
$$\phi ''(x)=\phi ''(0)+\sum_\nu A_\nu
(x)\phi _{x_\nu }'(x),$$
to write $\phi
'_{x_\nu }(x)$ as $h^{1\over 2}(Z_\nu
+Z_\nu ^*)$, and to use a priori estimates
that give control over $\Vert Z_\nu u\Vert
$.
\par In [HS], we established a general
formula for the correlations and in
[S1] we observed that it is related to
Witten Laplacians. In this formalism and
under the normalization condition (0.25) it
reads:
\ekv{0.33}
{{\rm Cor\,}(u,v)=({\Delta _\phi
^{(1)}}^{-1}d_\phi (e^{-\phi /2h}u)\vert
d_\phi (e^{-\phi /2h}v))=h({\Delta
_\phi ^{(1)}}^{-1}(e^{-\phi /2h}du)\vert
(e^{-\phi /2h}dv)).}
In [HS], we used such a formula to
establish the exponential decay of the
correlations ${\rm Cor\,}(x_\nu ,x_\mu )$
when ${\rm dist\,}(\nu ,\mu )$ is large.
This is based on the simple idea that
since we have a uniform bound on the norm
of $(\Delta _\phi ^{(1)})^{-1}$, then we
should also have such a bound after a
conjugation of this operator by an
exponential weight $\rho (\nu
)=e^{r(\nu )}$, $\nu \in\Lambda $, provided
that $r$ does not vary too fast.
\par In [S1] we obtained the leading
behaviour of
${\rm Cor\,}(x_\nu ,x_\mu )$ for large
${\rm dist\,}(\nu ,\mu )$ by using a
Feshbach (or Grushin) approach to $\Delta
_\phi ^{(1)}$ which in many ways
amounts to study the bottom of the
spectrum of this operator. We introduced
the auxiliary operator
$R_+= R_+^{1,0}:L^2({\bf R}^\Lambda
)\to\ell^2(\Lambda )$ by
\ekv{0.34}{(R_+^{1,0}u)(j)=(u\vert
e^{-\phi /2h}dx_j)=(u_j\vert e^{-\phi
/2h}),\ j\in\Lambda ,}
where $u=\sum u_jdx_j\simeq
(u_j)_{j\in\Lambda }$, so in each
component, we project onto the kernel of
$\Delta _\phi ^{(0)}$. Let
$R_-^{1,0}=(R_+^{1,0})^*$ be the adjoint.
\par Let
$${\cal H}_1=\{ u\in L^2({\bf
R}^\Lambda );\, Z_\nu u\in L^2,\forall
\nu \in\Lambda \}$$
with the corresponding norm
$$\Vert u\Vert _{{\cal H}_1}^2=\Vert
u\Vert ^2+\sum_\nu \Vert Z_\nu \Vert ^2,$$
where $\Vert \cdot \Vert $ denotes the
$L^2$ norm. Let ${\cal H}_{-1}={\cal
H}_1^*$ denote the dual space. Then as we
shall prove below (and as was essentially
proved in [S1] and in greater generality in
[BJS]), the operator
\eekv{0.35}
{
{\cal P}^{0,1}(z)=\pmatrix{\Delta _\phi
^{(1)}-z &R_-^{0,1}\cr
R_+^{0,1}&0}:(\ell^2(\Lambda )\otimes
{\cal H}_1)\times (\ell^2(\Lambda
)\otimes {\bf C})\to}
{\hskip 6cm (\ell^2(\Lambda )\otimes {\cal
H}_{-1})\times (\ell^2(\Lambda )\otimes
{\bf C}) }
is bijective with a uniformly bounded
inverse
\ekv{0.36}
{{\cal
E}^{0,1}(z)=\pmatrix{E^{0,1}(z)&E_+^{0,1}(z)
\cr E_-^{0,1}(z) & E_{-+}^{0,1}(z) }}
for
\ekv{0.37}
{-C\le z\le 2\lambda _{{\rm min}}(\phi
''(0))-{1\over C},}
when $h$ is small enough depending on
$C$, and $C\ge 1$ may be arbitrary. \it
Here and in the following we follow the
convention that all estimates and
assumptions will be uniform w.r.t.
$\Lambda $, if nothing else is specified.
\rm By $\lambda _{{\rm min}}(\phi ''(0))$,
we denote the lowest eigenvalue of $\phi
''(0)$. (As a matter of fact, we
will need the invertibility only for $z=0$
but keeping track of the spectral
parameter will help the understanding. In
the end classes of exponential weights
will be the more appropriate objects.)
\par Further, as we shall see (and as was
established in [S1], [BJS]), we have
\eekv{0.38}
{E_+^{0,1}=R_-^{0,1}+{\cal O}(h^{1\over
2}),\
\ E_-^{0,1}=R_+^{0,1}+{\cal O}(h^{1\over
2}),} {E_{-+}^{0,1}=z-\phi ''(0)+{\cal
O}(h^{1\over 2}),}
in the respective spaces of bounded
operators. Notice that $E_{-+}^{0,1}(z)$
is invertible for $-C\le z\le \lambda
_{{\rm min}}(\phi ''(0))-{1\over C}$, i.e.
in a smaller domain than (0.37). Actually,
instead of varying the spectral parameter,
we shall take $z=0$ and conjugate ${\cal
P}^{0,1}(0)$ by an exponential
weight$\pmatrix{\rho \otimes 1&0\cr 0& \rho
\otimes 1}$, with $\rho =e^r:\Lambda \to
]0,\infty [$. We shall then see that the
conjugated operator ${\cal P}^{0,1}$ is
uniformly invertible for $\rho $ in a
large class of weights. Notice that the
inverse is simply
$$\pmatrix{\rho \otimes 1 &0\cr 0& \rho
\otimes 1}{\cal E}^{0,1}(0)\pmatrix{\rho
^{-1}\otimes 1 &0\cr 0 & \rho ^{-1}\otimes
1}.$$ Moreover we shall see that (0.38)
remains valid for the conjugated
operators. $(E_{-+}^{0,1})^{-1}$ will
cope with conjugation only with weights in a
smaller class, and starting with the case
when $\Lambda$ is a discrete torus
(implying that
$E_{-+}^{0,1}$ is a convolution), we shall
be able to analyze quite precisely
the rate of decay of this inverse, and see
that it corresponds to weights in the
larger class of weights with which ${\cal
E}^{1,0}$ accomodates conjugation. Since
$$(\Delta _\phi
^{(1)})^{-1}=E^{0,1}(0)-E_+^{0,1}(0)(E_{-+}
(0))^{-1}E_-^{0,1}(0),$$
we can apply (0.33) and get
\eekv{0.39}
{{\rm Cor\,}(x_\nu ,x_\mu
)=h(E^{0,1}(0)(e^{-\phi /2h}dx_\nu )\vert
(e^{-\phi /2h}dx_\mu ))-} {\hskip 4cm
h((E_{-+}(0))^{-1}E_-^{0,1}(0)(e^{-\phi
/2h}dx_\nu )\vert E_-^{0,1}(0)(e^{-\phi
/2h}dx_\mu )).}
Because ${\cal E}^{0,1}$ can cope with
conjugation with stronger exponential
weights than $(E_{-+}^{0,1})^{-1}$, we see
that the first term of the RHS in (0.39)
has faster decay, than the second, when
${\rm dist\,}(\nu ,\mu )\to\infty $ and
the more precise information evocated
about the inverse of $E_{-+}$ together
with (0.38) leads to a result of the type
(0.20), where a priori the $p_{1,h}$ and
$q$ will depend also on $\Lambda $ through
factors $1+{\cal O}(h^{1/2})$. So far the
ideas were already developed in
[S1] and [BJS].
\par In order to get complete expansions
as stated in the theorem, we will
introduce higher order Grushin problems.
Let ${\bf N}_j^\Lambda $ be the set of
multiindices $\alpha :\Lambda \to {\bf N}$
of length $j$: $\vert \alpha \vert =\vert
\alpha \vert _1=j$. If $J$ is a finite
subset of ${\bf N}$, we put ${\bf
N}_J=\cup_{j\in J}{\bf N}_j^\Lambda $.
Since the $Z_\nu ^*$ form a commutative
family, the operator
${1\over \alpha !}(Z^*)^\alpha $ is
well-defined. For
$u\in L^2({\bf R}^\Lambda )$, put
$$(R_+^{N,0}u)(\alpha )=(u\vert
{1\over\alpha !}(Z^*)^\alpha (e^{-\phi
/2h})),\
\vert
\alpha \vert \le N,$$
so that $R_+^{N,0}:L^2({\bf R}^\Lambda
)\to\ell ^2({\bf N}^\Lambda _{[0,N]})$.
(We will see in section 4 that this operator
is uniformly bounded.) Notice that
${1\over\alpha !}(Z^*)^\alpha (e^{-\phi
/2h})$ are Hermite functions when $\phi $
is a quadratic form. When
$u\in
\ell^2(\Lambda )\otimes L^2({\bf
R}^\Lambda )$, we put
$(R_+^{N,1}u)(j,\alpha
)=(R_+^{N,0}u_j)(\alpha )$, $(j,\alpha
)\in \Lambda \times {\bf
N}_{[0,N]}^\Lambda $. Let
$R_-^{N,k}=(R_+^{N,k})^*$,
$k=0,1$, and introduce the auxiliary
(Grushin) operators for $k=0,1$:
\eekv{0.40}
{{\cal P}^{N,k}(z)=\pmatrix{ \Delta _\phi
^{(k)} &R_+^{N,k}\cr R_-^{N,k}
&0}:}
{\cases{ {\cal H}_1\times \ell^2({\bf
N}_{[0,N]}^\Lambda )\to{\cal H}_{-1}\times
\ell^2({\bf N}_{[0,N]}^\Lambda ),\ k=0,
\cr (\ell^2(\Lambda )\otimes {\cal
H}_1)\times (\ell^2(\Lambda )\otimes
\ell^2({\bf N}_{[0,N]}^\Lambda
))\to(\ell^2(\Lambda )\otimes {\cal
H}_{-1})\times
(\ell^2(\Lambda )\otimes \ell^2({\bf
N}_{[0,N]}^\Lambda )),\ k=1.}}
We will see in section 5 that ${\cal
P}^{N,0}(z)$ is uniformly invertible
for\hfill\break
$-C\le z\le (N+1)\lambda _{{\rm min}}(\phi
''(0))-{1\over C}$ and that the same is
true for ${\cal P}^{N,1}(z)$ in the range
$-C\le z\le (N+2)\lambda _{{\rm min}}(\phi
''(0))-{1\over C}$. This will be proved
following the inductice scheme
$$(N,1)\to (N+1,0)\to (N+1,1),$$
starting with the case $(-1,1)$, where by
definition ${\cal P}^{-1,1}(z)=\Delta
_\phi ^{(1)}-z$.
\par If
$${\cal E}^{N,k}(z)=\pmatrix{E^{N,k}(z)
&E_+^{N,k}(z)\cr E_-^{N,k}(z)
&E_{-+}^{N,k}(z)}$$
denotes the inverse of ${\cal P}^{N,k}(z)$
and $E^{N,k}_{-+;\nu ,\mu }(z)$ denotes
the operator matrix element of
$E_{-+}^{N,0}$ corresponding to the
decomposition $\ell^2({\bf
N}_{[0,N]}^\Lambda )=\oplus_{\nu
=0}^N\ell^2({\bf N}_\nu ^\Lambda )$, we
will further see that
\ekv{0.41}
{E_{-+;\nu ,\mu }^{N,0}(z)=h^{{1\over
2}\vert \nu -\mu\vert }B^N_{\nu ,\mu
}(z;h)+{\cal O}(h^{{1\over 2}(\vert N+1-\nu
\vert +\vert N+1-\mu \vert )})\hbox{ in
}{\cal L}(\ell^2,\ell^2),}
where $B^N_{\nu ,\mu }$ has a complete
asymptotic expansion in powers $h^\ell$,
$\ell\in{\bf N}$. Essentially the same
result holds for
$E_{-+}^{N,1}$ and similar results hold
for $E_{\pm}^{N,k}$. The idea behind this
result is to consider the matrix of
$\Delta _\phi ^{(0)}$ (and similarly for
$\Delta _\phi ^{(1)}$) with respect to the
decomposition
$$L^2({\bf R}^\Lambda )={\cal
L}_0\oplus..\oplus{\cal L}_N\oplus{\cal
L}_{[0,N]}^\perp ,$$
where ${\cal L}_j=R_-^{N,0}(\ell^2({\bf
N}_j^\Lambda ))$ and ${\cal
L}_{[0,N]}^\perp$ is the orthogonal of
${\cal L}_0\oplus ..\oplus{\cal L}_N={\cal
L}_{[0,N]}$, for which the corresponding
matrix elements of $(\Delta _\phi
^{(0)})_{\nu ,\mu }$ should behave as in
(0.41).
Notice that (0.41) gives increasing
precision in the asymptotics for a fixed
$(\mu ,\nu )$, when $N$ increases. It is
possible to describe ${\cal E}^{M,k}$ in
terms of ${\cal E}^{N,k}$, for $M\le N$,
and using this with $M=0$ and $N\to \infty
$, we arrive at a complete asymptotic
expansion of $E_{-+}^{0,1}(z;h)$ and at
similar almost complete descriptions of
$E_{\pm}^{0,1}(z;h)$. In other words, by
using higher order Grushin problems it is
possible to improve (0.38) and to get
full asymptotics. This improvement also
survives the conjugation by exponential
weights in a sufficiently large class, and
leads to a complete asymptotic
description of $(E_{-+}^{0,1}(z))^{-1}$,
including the decay rate at
large distances. Finally we use this
improved information in (0.39) to get
complete asymptotics of the correlations.
The handling of the thermodynamical limit
requires some additional arguments that we
do not discuss here.
\par A major motivation for this paper was
the hope (yet to be fulfilled) that the
use of higher order Grushin problems may
be useful in the study of correlations in
cases when $\phi $ is only weakly
convex at its critical point. In such cases
we do not always expect the correlations to
decay exponentially any more and one may
expect phenomena like phonons in
crystals. Though we are still far from
such a result, we may point out that the
parameter $N$ can be interpreted as a
maximum number of particles under
consideration, and that the $k$ particle
space
$\ell^2({\bf N}_k^\Lambda )$ can be
identified with the $k$ fold symmetric
tensor product of $\ell^2(\Lambda )$ with
itself. In other words, our particles are
bosons.
\par In sections 1--9, we do all the
essential work, adding successively the
assumptions that we need. At the end of
section 9, we arrive at the main result.
In section 10, we consider a slightly less
general framework and extract a main
result which is more easily formulated.
\par\noindent In section 1 we review some
standard facts about Witten Laplacians.
\par\noindent In section 2 we introduce
some special Sobolev spaces, which are the
natural ones for our variational
point of view.
\par\noindent In section 3 we discuss how
to reshuffle creation and annihilation
operators. The reason for doing so
will appear very naturally, and we are
aware of the fact that such reorderings also
appear in quantum field theory.
\par\noindent In section 4 we apply the
result of the preceding section to study
certain scalar producs.
\par\noindent Section 5 is devoted to the
well-posedness of higher order Grushin
problems.
\par\noindent In section 6 we get
asymptotics for the solutions of these
problems and in section 7, we show that
these asymptotics for ${\cal
P}^{N,1}$ remain after introducing certain
exponential weights on the $\ell^2(\Lambda
)$ component of
${\cal P }^{N,1}$.
\par\noindent In section 8, we study the
effect of parameter dependence in order to
treat the thermodynamical limit.
\par\noindent In section 9 we arrive at
the main result on the asymptotics of the
correlations also in the thermodynamical
limit.
\par\noindent In section 10 we extract the
main result as it is formulated in Theorem
0.1 above.
\par\noindent The two appendices can be
read when referred to in the main text.
\medskip
\par\noindent \bf Acknowledgements. \rm
This work was supported by the
TMR--network FMRX-CT 96-0001 "PDE and QM".
We have benefitted from interesting
discussions with W.M.Wang.
%\vfill\eject
\bigskip
\centerline{\bf 1. Assumptions on $\phi $.}
\medskip
\par Let $\phi \in C^\infty ({\bf
R}^\Lambda ;{\bf R})$, where $\Lambda $ is
a finite set. We shall let $\Lambda $ and
consequently $\phi $ vary with some
parameter, but all assumptions are uniform
w.r.t. $\Lambda $, if nothing else is
specified. Our first assumption is
\eeekv{{\rm H}1}
{\phi ^{(2)}=\phi ''\hbox{ is 2 standard in
the sense that for every $k\ge 2$, we have
uniformly} }
{\hbox{in $\Lambda $ and in $x\in{\bf
R}^\Lambda $}: \langle \phi
^{(k)}(x),t_1\otimes\dots\otimes
t_k\rangle ={\cal O}(1)\vert t_1\vert
_{p_1}\dots \vert t_k\vert _{p_k},\
t_j\in{\bf C}^\Lambda ,} {\hbox{whenever
}1\le p_j\le
\infty ,\hbox{ and }1={1\over p_1}+\dots
+{1\over p_k}.}
Here $\phi ^{(k)}=\nabla ^k\phi $ is the
symmetric tensor of $k$th order
derivatives. See [S2] for definitions and
basic properties concerning standard
tensors. Recall that by complex
interpolation it suffices to have the
estimate in (H1) in the extreme cases
$$p_\nu =\cases{1,\ \nu =j,\cr \infty ,\
\nu \ne j,}$$
for $j=1,..,k$. Notice that (H1) implies
that
$\phi ''(x):\ell^p\to\ell^p$ is uniformly
bounded for $x\in{\bf R}^\Lambda $, $1\le
p\le \infty $.
\par The next three assumptions imply that
$x=0$ is a non-degenerate critical point of
$\phi $ and the only critical point:
\ekv{{\rm H}2}{\phi '(0)=0,}
\ekv{{\rm H}3}{\phi ''(0)\ge {\rm
const.}>0,}
\eekv{{\rm H}4}
{\phi '(x)=A(x)x\hbox{ where $A(x)$ is 2
standard and has an}}
{\hbox{inverse }B(x)\hbox{ which is }{\cal
O}(1):\ell^p\to\ell^p,\ 1\le p\le \infty .}
We observe that $B$ will also be 2
standard. Also notice that (H1), (H2)
imply that
\ekv{1.1}
{\vert \phi '(x)\vert _p\le {\cal
O}(1)\vert x\vert _p,\ 1\le p\le \infty ,}
while (H4) implies the reverse estimate
\ekv{1.2}{\vert x\vert _p\le{\cal O}(1)\vert
\phi '(x)\vert _p,\ 1\le p\le \infty .}
It would be of interest to know if
conversely (1.2) and (H1--3) imply (H4).
Also notice that (H4) (or (1.2)) implies
that
$\phi ''(0)^{-1}$ exists and is ${\cal
O}(1):\ell^p\to\ell^p$. When checking
(H4), a natural candidate for $A(x)$ is
$\int_0^1\phi ''(tx)dt$, which is 2
standard by (H1).
\par We end this section by introducing
Witten Laplacians and related objects (cf.
[S]). For that purpose we shall work on
${\bf R}^\Lambda $, where $\Lambda $ is some
finite set. Let $d=\sum_{\ell\in \Lambda
}dx_\ell^\wedge\otimes \partial _{x_\ell}$
denote the DeRahm exterior differentiation
which takes differential $k$ forms on ${\bf
R}^\Lambda $ to differential $k+1$ forms.
Here $dx_\ell^\wedge$ denotes the operator
of left exterior multiplication by
$dx_\ell$ and we let $dx_\ell^\rfloor$
denote the adjoint operator of
contraction, which is well defined if we
view ${\bf R}^\Lambda $ as a Riemannian
manifold with the standard metric. Recall
that
$d$ is a complex in the sense that $d\circ
d=0$. Using the standard scalar product on
the space of smooth $k$ forms, we can define
the adjoint
$d^*=\sum_{\ell\in\Lambda
}dx_\ell^\rfloor\otimes (-\partial
_{x_\ell})$, taking $k+1$ forms into
$k$ forms. The corresponding Hodge
Laplacian is then $d^*d+dd^*$. It
conserves $k$ forms and commutes with $d$
and $d^*$.
\par The Witten exterior differentiation
is obtained from $d$ by conjugation by
$e^{\phi /h}$ and multiplication by a
cosmetic factor:
\ekv{1.3}
{d_\phi :=h^{1/2}e^{-\phi /2h}\circ d\circ
e^{\phi /2h}=\sum_{\ell\in\Lambda
}dx_\ell^\wedge\otimes Z_\ell,} where
\ekv{1.4}{Z_\ell=e^{-\phi /2h}\circ
h^{1/2}\partial _{x_\ell}\circ e^{\phi
/2h}=h^{1/2}\partial
_{x_\ell}+h^{-1/2}\partial _{x_\ell}\phi
/2.} We view $Z_\ell$ as annihilation
operators. The corresponding creation
operators are
\ekv{1.5}{Z_\ell^*=-h^{1/2}\partial
_{x_\ell}+h^{-1/2}\partial _{x_\ell}\phi/2
.}
We have the commutation relations:
\ekv{1.6}{[Z_j,Z_k]=0,\ [Z_j,Z_k^*]=\phi
_{j,k}''(x),\ j,k\in \Lambda .}
\par $d_\phi $ is a complex and the
corresponding Hodge Laplacian is called
the Witten Laplacian and is given by:
\ekv{1.7}{\Delta _\phi =d_\phi ^*d_\phi
+d_\phi d_\phi ^*.}
It conserves the degree of forms and we
denote by $\Delta _\phi ^{(k)}$ the
restriction to $k$ forms. Only the cases
$k=0,1$ will be of importance to us and
maybe the explanation of this fact is
that by working with differential forms,
we impose a fermionic structure, while
the problems in this paper have a bosonic
structure with the degree $k$ viewed as
the number of particles. It would be
interesting to know if there are some other
operators better adapted to the bosonic
structure. A general formula for
$\Delta _\phi $ is
\ekv{1.8}{\Delta _\phi =I\otimes\Delta _\phi
^0+\sum_{j,k}\phi
_{j,k}''(x)dx_j^\wedge\,dx_k^\rfloor ,}
where we from now adopt the convention
of letting the form component be the
first factors and the function
components to be the last factors when
we represent differential forms and
corresponding operators as tensor
products. $\Delta _\phi ^{(0)}$ acts on
scalar functions and is given by:
\ekv{1.9}
{\Delta _\phi ^{(0)}=\sum_j Z_j^*Z_j.}
When $k=1$, the formula (1.8) simplifies
to
\ekv{1.10}
{\Delta _\phi ^{(1)}=I\otimes \Delta _\phi
^{(0)}+\phi ''(x),}
provided that we view $1$ forms as
functions with values in ${\bf C}^\Lambda
$.
Again $d_\phi $ and $d_\phi ^*$ commute
with $\Delta _\phi $ and in particular,
\ekv{1.11}{d_\phi \Delta _\phi
^{(0)}=\Delta _\phi ^{(1)}d _\phi ,\
d_\phi^* \Delta _\phi ^{(1)}=\Delta _\phi
^{(0)}d _\phi^*.}
\par Under the assumptions (H1-4) we know
(see for instance [BJS] or [Jo]) that
$\Delta _\phi ^{(k)}$ can be realized as a
selfadjoint operator by means of the
Friedrichs extension. We will use the same
symbol to denote this selfadjoint
operator. Moreover, the spectrum is
discrete and contained in $[0,+\infty [$.
When $k=0$ the lowest eigenvalue is simple
and equal to $0$. The corresponding
eigenspace is spanned by $e^{-\phi /2h}$.
When $k=1$, the lowest eigenvalue is $>0$
(see for instance [S]).
%\vfill\eject
\bigskip
\centerline{\bf 2. The spaces ${\cal
H}_{\pm 1}$.}
\medskip
\par There will be two versions of these
spaces, one for scalar (${\bf C}$ valued)
functions and one for functions with
values in ${\bf C}^\Lambda $. We start
with the scalar case. We assume
(H1--4) throughout this section.
\par If $u\in C_0^\infty ({\bf R}^\Lambda
)$, we put
\ekv{2.1}{\Vert u\Vert _{{\cal
H}_1}^2=\Vert u\Vert _1^2=\Vert u\Vert
^2+\sum_{\ell\in\Lambda }\Vert Z_\ell
u\Vert ^2,}
and let ${\cal H}_1$ be the closure of
$C_0^\infty $ for this norm. ${\cal H}_1$
is the form domain of $\Delta _\phi
^{(0)}$, and by a standard regularization
argument we know that
\ekv{2.2}{{\cal H}_1=\{ u\in L^2({\bf
R}^\Lambda );\, Z_\ell u\in L^2,\,\forall
\ell\in\Lambda \}.}
Let ${\cal H}_{-1}$ be the dual space.
Using the standard $L^2$ inner product, we
view ${\cal H}_{-1}$ as a space of
temperate distributions and have the
natural inclusions:
\ekv{2.3}
{{\cal S}({\bf R}^\Lambda )\subset{\cal
H}_1\subset{\cal H}_0\subset{\cal
H}_{-1}\subset{\cal S}'({\bf R}^\Lambda ).}
Here the two inclusions in the middle
correspond to inclusion operators of norm
$\le 1$ and
${\cal H}_0$ denotes
$L^2({\bf R}^\Lambda )$. ${\cal H}_1$ is a
Hilbert space with scalar product
\ekv{2.4}{[u\vert v]_1=(u\vert
v)+\sum_{\ell\in
\Lambda }(Z_\ell u\vert Z_\ell
v)=((1+\Delta _\phi ^{(0)})u\vert v),} where
$(\cdot
\vert
\cdot
\cdot )$ is the usual innerproduct in
$L^2$. From this it follows that $1+\Delta
_\phi ^{(0)}$ is unitary from ${\cal
H}_1$ to ${\cal H}_{-1}$. We also remark
that ${\cal H}_{-1}$ is the space of all
\ekv{2.5}{u=u^0+\sum Z_\ell^*u_\ell ,}
with $u^0,u_\ell\in L^2$. Moreover $\Vert
u\Vert _{-1}^2$ is the infimum of $\Vert
u^0\Vert ^2+\sum \Vert u_\ell\Vert ^2$
over all decompositions as in (2.5).
\par We now pass to spaces of 1 forms,
whenever there is a possibility of
confusion we indicate the degree of the
forms by a superscript $(k)$, so that
the spaces just defined are ${\cal
H}_{\pm 1}^{(0)}$. Put
\ekv{2.6}
{{\cal H}_{\pm 1}^{(1)}=\ell^2(\Lambda
)\otimes {\cal H}_{\pm 1}^{(0)}.}
The corresponding scalar product of two 1
forms $u=\sum u_jdx_j$, $v=\sum v_jdx_j$
is then
\ekv{2.7}
{[u\vert v]_1=\sum [u_j\vert v_j]_1=\sum_j
(u_j\vert v_j)+\sum_{j,k}(Z_ju_k\vert
Z_jv_k)=(u\vert v)+\sum_{j,k}(Z_ju_k\vert
Z_jv_k).}
It can also be written $((1+1\otimes\Delta
_\phi ^{(0)})u\vert v)$. Again ${\cal
H}_{-1}^{(1)}$ is the dual space of
${\cal
H}_{1}^{(1)}$, and
\ekv{2.8}{(1+1\otimes\Delta
_\phi ^{(0)}):{\cal H}_1^{(1)}\to{\cal
H}_{-1}^{(1)}\hbox{ is unitary.}}
\par Later we will need to approximate
$\phi ''(x)$ by $\phi ''(0)$ in these
spaces, and for that we shall use the
following lemma.
\medskip
\par\noindent \bf Lemma 2.1. \it The
operator $u(x)\mapsto (\phi ''(x)-\phi
''(0))u(x)$ is bounded ${\cal
H}_1^{(1)}\to {\cal H}_{-1}^{(1)}$ and of
norm ${\cal O}(h^{1/2})$.
\medskip
\par\noindent \bf Proof. \rm Using
Proposition A.1, we see that
\ekv{2.9}
{\phi_{j,k} ''(x)-\phi_{j,k} ''(0)=h\phi
^{(1)}_{j,k}(x)+h^{1/2}\sum_{\ell}Z_\ell^*\circ
\phi ^{(0)}_{j,k,\ell}(x)+h^{1/2}
\sum_{\ell}
\phi ^{(0)}_{j,k,\ell}(x)\circ Z_\ell,}
where $\phi ^{(\nu )}$ are standard
tensors. Let $u,v\in C_0^\infty ({\bf
R}^\Lambda )$ and use (2.9) to get
\eekv{2.10}
{((\phi ''(x)-\phi ''(0))u\vert
v)}{\hskip 1cm =h^{1/2}\sum_{j,k,\ell}(
\phi _{j,k,\ell}^{(0)}u_k\vert
Z_\ell v_j)+h^{1/2}\sum_{j,k,\ell}(\phi
_{j,k,\ell}^{(0)}Z_\ell u_k\vert
v_j)+h\sum_{j,k}(\phi ^{(1)}_{j,k}u_k\vert
v_j).}
Since $\phi ^{(1)}$ is 2 standard, the
last sum is ${\cal O}(h)\Vert u\Vert
\Vert v\Vert $. For the two other sums, we
use Lemma B.2 and get
for the first sum:
\ekv{2.11}
{\vert \sum_{\ell ,j,k}
\phi _{j,k,\ell}^{(0)}u_k
\overline{Z_\ell v_j}\vert \le {\cal
O}(1)(\sum_k
\vert u_k(x)\vert ^2)^{1/2}(\sum
_{j,\ell}\vert Z_\ell v_j\vert ^2)^{1/2}.}
This implies that the first sum in (2.10)
is
${\cal O}(1)\Vert u\Vert \Vert v\Vert _1$.
Similarly the second sum is ${\cal
O}(1)\Vert u\Vert _1\Vert v\Vert $. We
then get
\ekv{2.12}
{((\phi ''(x)-\phi ''(0))u\vert v)={\cal
O}(h^{1/2})\Vert u\Vert _1\Vert v\Vert _1,}
which implies the lemma.\hfill{$\#$}
%\vfill\eject
\bigskip
\centerline{\bf 3. Reshuffling of $Z$ and
$Z^*$.}
\medskip
\par Let $J=\{ 1,..,N\}$, $K=\{ 1,..,M\}$,
$f\in C^\infty ({\bf R}^\Lambda )$. Then
for $j\in \Lambda ^J$, $k\in \Lambda ^K$,
we want to rewrite
\ekv{3.1}{(\prod_{\nu \in J}Z_{j(\nu
)})\circ f\circ (\prod_{\mu \in K}Z_{k(\mu
)}^*)}
as a sum of similar terms with the $Z^*$
to the left and the $Z$ to the right. We
first move each factor $Z_{j(\nu )}$ as
far as possible to the right, taking into
account the appearance of commutator
terms, due to the relations
\ekv{3.2}
{[Z_j,f]=h^{1\over 2}\partial
_{x_j}f(x)=[f,Z_j^*],\ [Z_j,Z_k^*]=\phi
_{j,k}''(x).}
After that, we move the surviving factors
$Z_k^*$ as a far as possible to the left,
generating new commutator terms. The
expression (3.1) becomes
\eekv{3.3}
{\hskip -1cm\sum_{P\ge
0}{1\over P!}\sum_{{{J=J_0\cup
..\cup J_{P+1}}\atop{K=K_0\cup
..\cup
K_{P+1}}}\atop{{{\rm partitions\
with}}\atop {J_p\ne\emptyset\ne K_p},\ {\rm
for}\ 1\le p\le P}}(\prod_{\mu \in
K_0}Z_{k(\mu )}^*)\circ h^{{1\over 2}(\#
J_{P+1}+\# K_{P+1})}((\prod_{\mu \in
K_{P+1},\atop \nu \in J_{P+1}}\partial
_{x_{k(\mu )}}\partial _{x_{j(\nu
)}})f)\times }
{\prod_{p=1}^P(h^{-1+{1\over 2}(\# J_p+\#
K_p)}(\prod_{\mu \in K_p\atop \nu \in
J_p}(\partial _{x_{k(\mu )}}\partial
_{x_{j(\nu )}}))\phi )\prod_{\nu \in
J_0}Z_{j(\nu )}. }
Here and in the following we use the term
partition for a union of pairwise disjoint
sets. The factor ${1\over P!}$ can be
eliminated if we let the second summation
be over all similtaneous partitions of $J$
and $K$
which are non-ordered in the indices $1\le
p\le P$.
\par Define a map $m: \Lambda ^N\to {\bf
N}^\Lambda $, by
\ekv{3.4}
{m(j)(\lambda )=\# \{k;\, j(k)=\lambda
\},\ \lambda \in\Lambda . }
If $\alpha\in {\bf N}^\Lambda $, we put
$\vert \alpha \vert =\vert \alpha \vert
_1=\sum_{\lambda \in\Lambda }\alpha
(\lambda )$. Then $\vert m(j)\vert =N$. We
write
\ekv{3.5}
{
{\bf N}_N^\Lambda =\{ \alpha \in{\bf
N}^\Lambda ; \vert \alpha \vert =N\}, }
and more generally
\ekv{3.6}
{
{\bf N}_A^\Lambda =\{ \alpha \in{\bf
N}^\Lambda ;\, \vert \alpha \vert \in A\}, }
if $A\subset{\bf N}$.
\par If $j\in \Lambda ^J$, $k\in \Lambda
^K$ as above, then
\ekv{3.7}
{
\prod_{\nu \in J}Z_{j(\nu )}=Z^\alpha ,\
\prod_{\mu \in K}Z^*_{k(\mu
)}=(Z^*)^\beta , }
where $\alpha =m(j)$, $\beta =m(k)$, and
where we use standard multiindex notation,
$Z^\alpha =\prod_{\lambda \in\Lambda
}Z_\lambda ^{\alpha (\lambda )}$. Conversely
for a given
$\alpha
\in {\bf N}_N^\Lambda $, the number of $j\in
\Lambda ^J$ with $m(j)=\alpha $ is equal
to ${N!\over\alpha !}={\vert \alpha \vert
!\over\alpha !}$. For a typical term in
(3.3), write
\ekv{3.8}
{
\prod_{\mu \in K_p}\partial _{x_{k(\mu
)}}=\partial _x^{\beta _p},\ \prod_{\nu
\in J_p}\partial _{x_{j(\nu )}}=\partial
_x^{\alpha _p}, }
and similarly with $\partial _x$ replaced
by $Z$ or $Z^*$. Then
\ekv{3.9}
{\alpha =\alpha _0+..+\alpha _{P+1},\ \beta
=\beta _0+..+\beta_{P+1},\hbox{ with
}\alpha _p\ne 0\ne\beta _p\hbox{ when
}1\le p\le P. }
Conversely, for such a decomposition of
$\alpha =m(j)$, we consider the
decomposition $\alpha
(\lambda )=\alpha _0(\lambda )+..+\alpha
_{P+1}(\lambda )$ for every $\lambda
\in\Lambda $, and see that there are
$\displaystyle {\alpha !\over
\alpha _0!..\alpha _{P+1}!}$ corresponding
partitions of $J$ into $J_0\cup ..\cup
J_{P+1}$. The equality of the expressions
in (3.1) and in (3.3) becomes
\eeekv{3.10}
{
{Z^\alpha \over \alpha !}\circ f\circ
{(Z^*)^\beta \over \beta !}= }
{
\sum_{P\ge 0}{1\over P!}\sum_{{{\alpha
=\alpha _0+..+\alpha
_{P+1},}\atop{\beta =\beta
_0+..+\beta _{P+1},}}\atop{\alpha _j,\beta
_j\ne 0{\rm \, for\, }1\le j\le P.}}
{(Z^*)^{\alpha _0}\over \alpha
_0!}h^{{1\over 2}(\vert \alpha _{P+1}\vert
+\vert \beta _{P+1}\vert )}{\partial
_x^{\alpha _{P+1}+\beta _{P+1}}f\over
\alpha _{P+1}!\beta _{P+1}!}\times
}{\hskip 5cm \prod_{p=1}^P (h^{-1+{1\over
2}(\vert
\alpha _p\vert +\vert \beta _p\vert
)}{\partial _x^{\alpha _p+\beta _p}\phi
\over \alpha _p!\beta _p!}){Z^{\beta
_0}\over \beta _0!}.}
\par We shall transform our expression
further by using non-commutative
expansions of the tensors appearing in
(3.3), (3.10). For this, it seems easier to
work with (3.3), and we assume that $f$ is
0 standard, or possibly $M$ standard,
depending on $M$ additional indices. For
simplicity, we write
$$\prod_{\mu \in K_0}Z^*_{k(\mu
)}=Z^*_{k\vert K_0},\ \ (\prod_{\mu \in
K_p}\partial _{x_{k(\mu )}}\prod_{\nu
\in J_p}\partial _{x_{j(\nu )}})\phi =\phi
_{k\vert K_p,j\vert J_p}.$$
Now apply Proposition A.1 to one of the
tensors:
\eekv{3.11}
{
\phi _{k\vert K_p,j\vert J_p}(x)=\phi
_{k\vert K_p,j\vert J_p}(0)+h^{1\over
2}\sum_{\ell\in\Lambda }Z_\ell^*\circ
\phi ^{(0)}_{k\vert K_p,j\vert
J_p,\ell}(x)+}{\hskip 4cm
h^{1\over
2}\sum_{\ell\in\Lambda }\phi
^{(0)}_{k\vert K_p,j\vert
J_p,\ell}(x)\circ Z_\ell + h\phi
^{(1)}_{k\vert K_p,j\vert J_p}(x).} When
substituting this into (3.3), the effect of
the first term of the RHS will be to freeze
the corresponding factor to
$x=0$. For the contributions of the second
term of the RHS of (3.11) in (3.3), we
move the $Z_\ell^*$ to
the left, until either it joins the
factors $Z^*_{k\vert K_0}$ or until it
forms a commutator with a $\phi _{k\vert
K_q,j\vert J_q}$ or with $f_{k\vert
K_{P+1},j\vert J_{P+1}}$, that we denote
by $\phi _{k\vert K_q,j\vert J_q,\ell}$
(also for $q=P+1$). In the second case the
$\ell$ summation amounts to the
contraction of two standard tensors, which
produces a standard tensor and an
additional power of $h$. For the
contribution of the last sum in (3.11), we
move the factors $Z_\ell$ as far as
possible to the right and repeat the
same discussion. The contribution from the
last term in (3.11) in (3.3) is simply to
introduce an extra power of $h$. The
procedure can be iterated a finite number
of times, and we see that the general term
term in (3.3) becomes a finite sum of terms
of the type
$$\eqalignno{
\sum_{\ell\in \Lambda ^{L_1\cup ..\cup
L_{Q+1}}\atop r\in \Lambda ^{R_1\cup
..\cup R_{Q+1}}}&h^X Z_{k\vert
K_0}^*Z^*_{\ell\vert L_1\cup ..\cup
L_{Q+1}}\circ \Phi ^{(1)}_{k\vert
K_1,j\vert J_1,\ell\vert L_1,r\vert
R_1}(x)..&(3.12)\cr&\Phi ^{(Q+1)}_{k\vert
K_{Q+1},j\vert J_{Q+1},\ell\vert
L_{Q+1},r\vert R_{Q+1}}(x)\circ Z_{j\vert
J_0}Z_{r\vert R_1\cup ..\cup R_Q}.}
$$
Here $K=K_0\cup ..\cup K_{Q+1}$,
$J=J_0\cup ..\cup J_{Q+1}$ are partitions
and $K_q\ne\emptyset\ne J_q$ for $1\le
q\le Q$. $\Phi ^{(q)}$ are standard,
$L_q$, $R_q$ are finite disjoint sets,
possibly empty, and
$$X={1\over 2}\# (L_1\cup ..\cup
L_Q)+{1\over 2}\# (R_1\cup ..\cup
R_Q)+N+\sum_1^Q({1\over 2}(\# K_q+\#
J_q)-1)+{1\over 2}(\# K_{Q+1}+\#
J_{Q+1}),$$
where $N\in {\bf N}$, and we have arranged
that $\Phi ^{(Q+1)}$ is the factor which
contains the contribution from $f$ under
the contraction procedure.
\par The point with the further Taylor
expansions of the term (0.3) is to arrive at
terms with constant factors $\Phi
^{(\nu )}$. More precisely, we can
introduce a stopping rule, so that we only
get terms of the form (3.12) with
\ekv{3.13}
{\# (K_0\cup L_1\cup ..\cup L_{Q+1})\le A,}
\ekv{3.14}
{\# (J_0\cup R_1\cup ..\cup R_{Q+1})\le B,}
\ekv{3.15}
{N\le N_0,}
where $A,B,N_0$ are given integers $\ge 0$
with $A\ge \# K$, $B\ge \#J$, and
so that the factors $\Phi ^{(\nu
)}$ are constant for all terms for which we
have strict inequality in all the three
equations (3.13--15).
\par We will also need a slight variation
of the arguments above. Assume for
simplicity that $f=1$. Then from (3.3), we
see that (3.1) takes the the form
\ekv{3.16}
{
\sum_{P\ge 0}{1\over P!}\sum_{{J=J_0\cup
..\cup J_P\atop K=K_0\cup ..\cup
K_P}\atop{{\rm partitions\, with}\atop
J_p\ne\emptyset\ne K_p{\rm \, for\,}1\le
p\le P}} Z^*_{k\vert
K_0}\prod_{p=1}^P(h^{-1+{1\over 2}(\#
J_p+\# K_p)}\partial _{x_{k\vert
K_p}}\partial _{x_{j\vert J_p}}\phi
(x))Z_{j\vert J_0}. }
We now want the coefficients to the left, so
we move all the factors $\partial _{x_{k\vert
K_p}}\partial _{x_{j\vert J_p}}\phi(x)$ to
the left, taking into account the
commutator terms. Then the expression (3.1)
becomes:
\ekv{3.17}
{\sum_{P\ge 0}\sum_{{J=J_0\cup ..\cup
J_P\atop K=K_0\cup ..\cup K_P}\atop{{\rm
partitions\, with}\atop
J_p\ne\emptyset\ne K_p{\rm \, for\,}1\le
p\le
P}}C^{J_0,..,J_P}_{K_0,..,K_P}\prod_{p=1}^P(h^{-1+{1\over
2}(\# J_p+\# K_p)}\partial _{x_{k\vert
K_p}}\partial _{x_{j\vert J_p}}\phi (x))
Z^*_{k\vert K_0}Z_{j\vert J_0}. }
Here the combinatorial coefficients
$C^{\cdots }_{\cdots }$ are independent of
$\Lambda $.
%\vfill\eject
\bigskip
\centerline{\bf 4. Study of $\displaystyle
( {1\over
\alpha !}(Z^*)^\alpha (e^{-\phi /h})\vert
{1\over \beta !}(Z^*)^\beta (e^{-\phi /h}))
$.}
\medskip
\par After adding a $h$-dependent constant
to $\phi $, we assume that
\ekv{4.1}
{\int e^{-\phi (x)/h}dx=1.}
We want to study the matrix formed by the
scalarproducts in the title of this
section, for $\vert \alpha \vert , \vert
\beta \vert \le N_0$, for some $N_0\in{\bf
N}$. Equivalently, we want to study,
\ekv{4.2}
{
(Z^*_{k\vert K}(e^{-\phi /2h})\vert
Z^*_{j\vert J}(e^{-\phi /2h})), }
for $J=\{ 1,..,N\}$, $K=\{ 1,..,M\}$,
$0\le N,M\le N_0$, $k\in\Lambda ^K$,
$j\in\Lambda ^J$. Here we use the notation
of section 3. We write this as
\ekv{4.3}
{
(Z_{j\vert J}Z^*_{k\vert K}(e^{-\phi
/2h})\vert e^{-\phi /2h}), } and apply (
3.12), with $f=1$, in which case the factor
$\Phi ^{(Q+1)}$ drops out. Since
$Z(e^{-\phi /2h})=0$, we can further
restrict our attention to the terms with
$K_0,L_q,J_0,R_q$ all empty, and it follows
that (4.2) is a finite sum of terms of the
type
\ekv{4.4}
{
h^X(\Phi ^{(1)}_{k\vert K_1,j\vert
J_1}..\Phi ^{(Q)}_{k\vert K_Q,j\vert
J_Q}e^{-\phi /2h}\vert e^{-\phi /2h}). }
Here $K=K_1\cup ..\cup K_Q$, $J=J_1\cup
..\cup J_Q$ are partitions with $K_q\ne
\emptyset\ne J_q$ for all $q$. Further,
\ekv{4.5}
{X=N+\sum_1^Q ({1\over 2}(\# K_q+\#
J_q)-1),\ N\in [0,N_1]\cap {\bf N}, }
where $N_1$ is any fixed sufficiently large
integer
$\ge 0$, and as we saw in section 3, we
may arrange that $\Phi ^{(\nu )}$ are
constant for the terms with $N0$ such that the
following holds for $h>0$ small enough:
\smallskip
\par\noindent (A) If $-C\le z\le
(N+1)\lambda _{\min }(\phi ''(0))-1/C$,
then (Gr($N$,0)) has a unique solution
$(u,u_-)\in{\cal H}_1\times \ell^2$ for
every $(v,v_+)\in{\cal
H}_{-1}\times\ell^2$, and
\ekv{5.10}
{
\Vert u\Vert _{{\cal H}_1}+\vert u_-\vert
_2\le \widetilde{C}(\Vert v\Vert _{{\cal
H}_{-1}}+\vert v_+\vert _2). }
\smallskip
\par\noindent (B) If $-C\le z\le
(N+2)\lambda _{\min }(\phi ''(0))-1/C$,
then (Gr($N$,1)) has a unique solution
$(u,u_-)\in{\cal H}_1\times \ell^2$ for
every $(v,v_+)\in{\cal H}_{-1}\times
\ell^2$ and (5.10) holds.
\smallskip
\par Here $\lambda _{\min}(\phi ''(0))>0$
denotes the smallest eigenvalue of $\phi
''(0)$ and ${\cal H}_{\pm 1}={\cal
H}_{\pm 1}^{(\nu )}$ in case $\nu $,
$\ell^2=\ell^2({\bf N}^\Lambda _{[0,N]})$
in the case $\nu =0$,
$\ell^2=\ell^2(\Lambda )\otimes\ell^2({\bf
N}^\Lambda _{[0,N]})$, when $\nu
=1$.\rm\medskip
\par We shall prove the proposition
following the inductive scheme
$${\rm Gr\,}(k,1)\to{\rm
Gr\,}(k+1,0)\to{\rm Gr\,}(k+1,1),$$
where we start by considering ${\rm
Gr}(-1,1)$, which by definition is the
problem
\ekv{5.11}
{
(\Delta _\phi ^{(1)}-z)u=v,\ u\in{\cal
H}_1,\, v\in{\cal H}_{-1}. }
\medskip
\par\noindent \bf Lemma 5.2. \it For all
$C\ge 1$, $-C\le z\le \lambda
_{\min}(\phi ''(0))-1/C$, and $h$
sufficiently small, depending on $C$, the
problem (5.11) has a unique solution
$u\in{\cal H}_{1}$, for every $v\in{\cal
H}_{-1}$. Moreover,
\ekv{5.12}
{\Vert u\Vert _{{\cal
H}_1}\le\widetilde{C}\Vert v\Vert _{{\cal
H}_{-1}},}
where $\widetilde{C}>0$ depends on $C$
but not on $z,h,\Lambda $.\rm\medskip
\par\noindent \bf Proof. \rm Recall that
$\Delta _\phi ^{(1)}=1\otimes\Delta _\phi
^{(0)}+\phi ''(x)$ and consider first the
simplified problem,
\ekv{5.13}
{(1\otimes\Delta _\phi ^{(0)}+\phi
''(0)-z)u=v,\ u\in{\cal H}_1,\, v\in{\cal
H}_{-1}.} If $u$ solves (5.13), take the
scalar product of this equation with $u$
and get
$$\eqalign{& \Vert v\Vert _{{\cal
H}_{-1}}\Vert u\Vert _{{\cal H}_1}\ge
(v\vert u)=((1\otimes\Delta _\phi
^{(0)}+\phi ''(0)-z)u\vert u)\cr&\ge
\epsilon ((1\otimes \Delta _\phi
^{(0)}+1)u\vert u)+((\phi
''(0)-z-\epsilon )u\vert u)\ge \epsilon
\Vert u\Vert _{{\cal H}_1}^2 ,}$$ for
$\epsilon >0$ small enough, so
\ekv{5.14}
{
\Vert u\Vert _{{\cal H}_1}\le C\Vert
v\Vert _{{\cal H}_{-1}}. }
This gives injectivity and the analogue of
(5.12) for the problem (5.13). Since
$(1\otimes\Delta _\phi ^{(0)}+\phi
''(0)-z)$ is a bounded selfadjoint operator
${\cal H}_1\to{\cal H}_{-1}$, it is also
surjective, so (5.13) is uniquely solvable
and satisfies (5.14). To get the lemma it
suffices to use that $\phi ''(x)-\phi
''(0)={\cal O}(h^{1/2}):{\cal
H}_{1}\to{\cal H}_{-1}$.\hfill{$\#$}
\medskip
\par The preceding lemma gives
well-posedness for ${\rm Gr\,}(-1,1)$ in
the appropriate range. Let us now perform
the step ${\rm Gr\,}(-1,1)\to{\rm
Gr\,}(0,0)$, so consider
\ekv{5.15}
{
\cases{(\Delta _\phi
^{(0)}-z)u+R_-^{0,0}u_-=v\cr
R_+^{0,0}u=v_+,} }
with $v\in{\cal H}_{-1}$, $u\in{\cal
H}_1$, $u_-,v_+\in{\bf C}$,
$R_+^{0,0}u=(u\vert e^{-\phi /2h})$. We let
$z$ be in the range of the lemma above,
and we first prove uniqueness in (5.15).
Let $v=0$, $v_+=0$ in (5.15). Since $d_\phi
R_-^{0,0}=0$, $d_\phi \Delta _\phi
^{(0)}=\Delta _\phi ^{(1)}d_\phi $, we get
by applying $d_\phi $ to the first
equation in (5.15):
\ekv{5.16}
{(\Delta _\phi ^{(1)}-z)d_\phi u=0.}
Here we only know apriori that $d_\phi
u\in L^2$, so we cannot apply Lemma 5.2
directly. However, it is easy and standard
to show that every $L^2$ solution $w$
of $(\Delta ^{(1)}_\phi -z)w=0$, has to
belong to ${\cal S}$ and in particular to
${\cal H}_1$. Consequently, we can apply
Lemma 5.2 and conclude that $d_\phi u=0$.
Since $d_\phi =h^{1/2}e^{-\phi /2h}\circ
d\circ e^{\phi /2h}$, it follows that
$u=\lambda e^{-\phi /2h}$ for some constant
$\lambda \in{\bf R}$. Using also that
$v_+=0$ in (5.15), we see that $\lambda
=0$, so $u=0$ and then $u_-=0$, and we
have proved uniqueness for solutions of
(5.15). Define $z_0\in [0,\infty [$ by
\ekv{5.17}
{
z_0=\inf_{u\in{\cal H}_1\cap (e^{-\phi
/2h})^\perp\atop \Vert u\Vert =1}(\Delta
_\phi ^{(0)}u\vert u). }
Since the inclusion map ${\cal H}_1\to L^2$
is compact, there exists $u_0\in{\cal
H}_1\cap(e^{-\phi /2h})^\perp$ with $\Vert
u_0\Vert =1$, such that $z_0=(\Delta _\phi
^{(0)}u_0\vert u_0)$, i.e. $((\Delta _\phi
^{(0)}-z_0)u_0\vert u_0)=0$, while
$((\Delta _\phi ^{0)}-z_0)u\vert u)\ge 0$
for general $u\in{\cal H}_1\cap (e^{-\phi
/2h})^\perp$. It follows that $(\Delta
_\phi ^{(0)}-z_0)u_0=\mu e^{-\phi /2h}$ for
some $\mu \in{\bf R}$ and since $u_0\perp
e^{-\phi /2h}$ and $e^{-\phi /2h}\in{\rm
Ker\,}\Delta _\phi ^{(0)}$, we see that
$\mu =0$. Hence $(\Delta _\phi
^{(0)}-z_0)u_0=0$, so $u=u_0$, $u_-=0$ is
a solution of (5.15) with $v=0$, $v_+=0$
and $z=z_0$. Since we know that (5.15) is
injective for
$0\le z\le \lambda _{\rm \min}(\phi
''(0))-1/2C$, for $h$ small enough
depending on $C$, we conclude that
\ekv{5.18}
{
z_0\ge \lambda _{\rm min}(\phi
''(0))-{1\over 2C}. }
\par Let us now restrict the attention to
$-C\le z\le \lambda _{\rm min}(\phi
''(0))-1/C$ and derive an apriori estimate
for solutions to (5.15). Let first $v_+=0$
in (5.15), and take the scalar product of
the first equation there with $u$, and use
that $(R_-^{0,0}u_-\vert u)=(u_-\vert
R_+^{0,0}u)=0$. We get
\ekv{5.19}
{
((\Delta _\phi ^{(0)}-z)u\vert u)=(v\vert
u). }
With $\delta >0$ small enough, write
$$\Delta _\phi ^{(0)}-z=\delta \Delta
_\phi ^{(0)}+(1-\delta )(\Delta _\phi
^{(0)}-z_0)+(1-\delta )(z_0-{z\over
1-\delta }),$$
and get
$$((\Delta _\phi ^{(0)}-z)u\vert u)\ge
\delta (\Delta _\phi ^{(0)}u\vert
u)+(1-\delta )(z_0-{z\over 1-\delta
})\Vert u\Vert ^2\ge \delta \Vert u\Vert
_{{\cal H}_1}^2.$$
Hence from (5.19), we get
$$\Vert u\Vert _{{\cal H}_1}^2\le
\widetilde{C}\Vert v\Vert _{{\cal
H}_{-1}}\Vert u\Vert _{{\cal H}_1},$$
\ekv{5.20}
{
\Vert u\Vert _{{\cal H}_1}\le
\widetilde{C}\Vert v\Vert _{{\cal
H}_{-1}}, }
for solutions of (5.15) with $v_+=0$.
\par Now take the scalar product of the
first equation in (5.15) with
$R_-^{0,0}u_-$ and get
$$-z(u\vert R_-^{0,0}u_-)+\vert u_-\vert
^2=\overline{u}_-(v\vert e^{-\phi /h}).$$
With (5.20), this gives
$\vert u_-\vert ^2\le\widehat{C}\Vert v\Vert
_{{\cal H}_{-1}}\vert u_-\vert $ and hence
\ekv{5.21}
{\vert u_-\vert \le \widetilde{C}\Vert
v\Vert _{{\cal H}_{-1}},}
where we let $\widetilde{C}$ denote a new
constant in every new formula.
\par If $v_+\ne 0$, consider
$\widetilde{u}:=u-v_+e^{-\phi /2h}$, which
solves
\ekv{5.22}
{
\cases{(\Delta _\phi
^{(0)}-z)\widetilde{u}+R_-^{0,0}u_-=
v+zv_+e^{-\phi /2h}\cr
R_+^{0,0}\widetilde{u}=0. }
}
Applying (5.20), (5.21) to this system, we
get
$$\Vert \widetilde{u}\Vert _{{\cal
H}_1}+\vert u_-\vert
\le\widetilde{C}(\Vert v\Vert _{{\cal
H}_{-1}}+\vert v_+\vert ),$$
leading to
\ekv{5.23}
{
\Vert u\Vert _{{\cal H}_1}+\vert u_-\vert
\le\widetilde{C}(\Vert v\Vert _{{\cal
H}_{-1}}+\vert v_+\vert )
,}
for solutions of (5.15). Since this
problem is selfadjoint, we also have
existence and we have proved the
proposition for (Gr(0,0)).
\par Les us now prove that if for some
$N\in{\bf N}$ the proposition is valid for
(Gr($N$,0)) then it is valid for
(Gr($N$,1)). So we assume for a fixed $N$
that (A) holds for all
$C$ with $h>0$ small enough depending on
$C$, and we want to prove (B) with the same
$N$. Using again that $\phi ''(x)-\phi
''(0)={\cal O}(h^{1/2}):{\cal H}_1\to{\cal
H}_{-1}$, we see that it suffices to treat
the simplified problem
\ekv{5.24}
{
\cases{(1\otimes\Delta _\phi ^{(0)}+\phi
''(0)-z)u+R_-^{N,1}u_-=v\cr
R_+^{N,1}u=v_+.} }
Since we have (A), for the chosen value
of $N$, we know from the preceding
discussion that
\ekv{5.25}
{
\inf_{w\in{\cal H}_1,\, \Vert w\Vert
=1\atop R_+^{N,0}w=0}(\Delta _\phi
^{(0)}w\vert w)\ge (N+1)\lambda
_{\min}(\phi ''(0))-{1\over 2C}, }
for $h>0$ small enough depending on $C$.
Consider (5.24) in the case $v_+=0$. Then
$R_+^{N,0}u_j=0$ for each component $u_j$
of $u$ and consequently
$$((1\otimes\Delta _\phi ^{(0)})u\vert
u)\ge ((N+1)\lambda _{{\rm
min}}-{1\over 2C})\Vert u\Vert ^2.$$
Since $(\phi '(0)u\vert u)\ge \lambda
_{\rm min}\Vert u\Vert ^2$, we get
\ekv{5.26}
{
((1\otimes\Delta _\phi ^{(0)}+\phi
''(0)-z)u\vert u)\ge {1\over 2C}\Vert
u\Vert ^2, }
for $z$ in the range of values of (B). As
before, this leads to
\ekv{5.27}
{
((1\otimes\Delta _\phi ^{(0)}+\phi
''(0)-z)u\vert u)\ge \delta \Vert u\Vert
_{{\cal H}_1}^2, }
for some $\delta >0$. Take the scalar
product of the first equation in (5.24)
with $u$, and use that $(R_-^{N,1}u_-\vert
u)=(u_-\vert R_+^{N,1}u)=0$. Then
$$\delta \Vert u\Vert _{{\cal H}_1}^2\le
\Vert v\Vert _{{\cal H}_{-1}}\Vert u\Vert
_{{\cal H}_1},$$
which gives,
\ekv{5.28}
{\Vert u\Vert _{{\cal
H}_1}\le\widetilde{C}\Vert v\Vert
_{{\cal H}_{-1}}}
for solutions of (5.24) with $v_+=0$,
when $z$ is in the range of (B).
\par We next want to take the
scalarproduct with $R_-^{N,1}u_-$, and as
a preparation we need to
establish two results about $R_\pm$.
\medskip
\par\noindent \bf Lemma 5.3. \it
$R_+^{N,0}R_-^{N,0}$ is ${\cal
O}(1):\ell^2({\bf N}^\Lambda
_{[0,N]})\to\ell^2({\bf N}^\Lambda
_{[0,N]})$ and has a uniformly bounded
inverse. Moreover, if $r_P:\ell^2({\bf
N}^\Lambda _{[0,N]})\to\ell^2({\bf
N}^\Lambda _P)$ is the natural
restriction operator, then for $0\le
P,Q\le N$.
\ekv{5.29}
{
r_PR_+^{N,0}R_-^{N,0}r_Q^*\sim\sum_{\nu
=0}^\infty h^{\nu +{1\over 2}\vert
P-Q\vert }M_{\nu ;P,Q}, }
in ${\cal L}(\ell^2({\bf N}^\Lambda
_Q),\ell^2({\bf N}_P^\Lambda ))$
uniformly with repect to $\Lambda $.
Here
\ekv{5.30}
{
M_{0;P,P}^{(N)}=\phi ''(0)\odot ..\odot\phi
''(0). }
\rm\medskip
\par\noindent \bf Proof. \rm For
simplicity, we work with the equivalent
operators $\widetilde{R}_{\pm}^{N,0}$
between $L^2({\bf R}^\Lambda )$ and
$\ell^2_b(\Lambda ^0\cup\Lambda ^1\cup
..\cup\Lambda ^N)$, where the subscript
$b$ indicates that we take
the "Bosonic" subspace of permutation
invariant elements of $\ell^2$. Then the
matrix of
$r_P\widetilde{R}_+^{N,0}
\widetilde{R}_-^{N,0}r_Q^*$ is given by
\ekv{5.31}
{{1\over\sqrt{P!Q!}}(Z^*_{p\vert
{\cal P}}(e^{-\phi /2h})\vert Z^*_{q\vert
{\cal Q}}(e^{-\phi /2h})),}
with ${\cal P}=\{ 1,..,P\}$, ${\cal Q}=\{
1,..,Q\}$, $p\in\Lambda ^{\cal Q}$,
$p\in\Lambda ^{\cal P}$. The uniform
asymptotic expansion (5.29) then follows
from Proposition 4.1. Moreover, the matrix
$\widetilde{M}_{0,P,P}^{(N)}$
(corresponding to $M^{(N)}_{0,P,P}$) has
the elements
\ekv{5.32}
{
{1\over P!}\sum_{\pi \in{\rm
Perm\,}(K)}\prod_{\nu \in{\cal P}}\phi
_{p(\nu ),q(\pi (\nu ))}''(0),}
which has the same action on
$\ell^2_b(\Lambda ^P)$ as the matrix
\ekv{5.33}
{
\prod_{\nu \in{\cal P}}\phi _{p(\nu
),q(\nu )}''(0), }
which is simply the matrix $\phi
''(0)\otimes ..\otimes \phi
''(0)$.\hfill{$\#$}
\medskip
\par By tensoring all the spaces with
$\ell^2(\Lambda )$, we get the obvious
analogue of Lemma 5.3
for $R_+^{N,1}R_-^{N,1}$. It also follows
from Lemma 5.3, that $R_-^{N,0}$ is
uniformly ${\cal O}(1):\ell^2\to L^2$.
Consequently $R_+^{N,0}={\cal
O}(1):L^2\to\ell^2$, and we have
the corresponding facts for $R_\pm^{N,1}$.
This can be strengthened:
\medskip
\par\noindent \bf Lemma 5.4. \it
$R_-^{N,0}$ is uniformly bounded:
$\ell^2({\bf N}^\Lambda _{[0,N]})\to{\cal
H}_1$. Consequently $R_+^{N,0}$ is
uniformly bounded ${\cal
H}_{-1}\to\ell^2({\bf N}^\Lambda
_{[0,N]})$.
\medskip
\par\noindent \bf Proof. \rm Again we
think it is more convenient to work with
the equivalent operator
$\widetilde{R}_-^{N,0}$. Let $1\le M\le
N$, $u\in\ell_b^2(\Lambda ^M)$ and consider
\ekv{5.34}
{
Z_j\widetilde{R}_-^{N,0}u=\sum_{m\in\Lambda
^{{\cal
M}}}{1\over\sqrt{M!}}Z_jZ^*_{m\vert {\cal
M}}(e^{-\phi /2h})u(m), }
where ${\cal M}=\{ 1,..,M\}$. We apply the
expression (3.17) to obtain
\ekv{5.35}
{
Z_j\widetilde{R}_-^{N,0}u=\sum_{m\in\Lambda
^{{\cal M}}}\sum_{{{{\cal
M}=M_0\cup M_1}\atop{{\rm
partition\, with}}}\atop {M_1\ne
\emptyset}}C_{M_0,M_1}h^{-{1\over
2}+{1\over 2}\#M_1}\partial _{x_j}\partial
_{x_{m\vert M_1}}\phi (x)Z^*_{m\vert
M_0}(e^{-\phi /2h})u(m), } where
$C_{M_0,M_1}$ is independent of
$\Lambda $, and equal to $1/\sqrt{M!}$ when
$\#M_1=1$. For $u\in \ell_b^2(\Lambda ^M)$,
$v\in
\ell_b^2(\Lambda ^P)$, $1\le P,M\le N$, we
get
\ekv{5.36}
{
\sum_j(Z_j\widetilde{R}_-^{N,0}u\vert
Z_j\widetilde{R}_-^{N,0}v)=(B^{P,M}u\vert
v), }
where $B^{P,M}$ is given by a matrix
$B_{p,m}^{P,M}$, $p\in\Lambda ^P$,
$m\in\Lambda ^M$, which is a finite linear
combination of terms
\ekv{5.37}
{
h^{-1+{1\over 2}\#M_1+{1\over
2}\# P_1}(Z_{p\vert P_0}\sum_j(\partial
_{x_{p\vert P_1}}\partial _{x_j} \phi
)(\partial _{x_j}\partial _{x_{m\vert
M_1}}\phi )Z^*_{m\vert M_0}(e^{-\phi
/2h})\vert e^{-\phi /2h}),}
where ${\cal M}=M_0\cup M_1$, ${\cal
P}=P_0\cup P_1$ are partitions with
$M_1\ne\emptyset\ne P_1$. Here
$$\Phi _{p\vert P_1,m\vert
M_1}:=\sum_j(\partial _{x_{p\vert
P_1}}\partial _{x_j} \phi )(\partial
_{x_j}\partial _{x_{m\vert M_1}}\phi )$$
is a standard tensor, being the
contraction of two standard tensors of
size $1+\#P_1$ and $1+\#M_1$, with at
least one of the sizes $\ge 2$ (cf. Lemma
8.2).
\par As in sections 3,4, in particular the
discussion leading to (3.12-15), we see that
(5.37) is a finite sum of terms
\ekv{5.38}
{
h^X(\Phi ^{(1)}_{p\vert
\widetilde{P}_1,m\vert
\widetilde{M}_1}..\Phi _{p\vert
\widetilde{P}_Q,m\vert \widetilde{M}_Q
}^{(Q)}e^{-\phi /2h}\vert
e^{-\phi /2h}),} where ${\cal
P}=\widetilde{P}_1\cup
..\cup\widetilde{P}_Q$, ${\cal
M}=\widetilde{M}_1\cup
..\cup\widetilde{M}_Q$ are partitions with
$\widetilde{P}_q,\widetilde{M}_q\ne
\emptyset$, $\widetilde{P}_1\supset P_1$,
$\widetilde{M}_1\supset M_1$. $\Phi
^{(q)}$ are standard tensors and
$$X=\widetilde{N}+\sum_1^Q({1\over
2}(\#\widetilde{M}_q+\#\widetilde{P}_q)-1),\
\widetilde{N}\in[0,N_1]\cap{\bf N}.
$$
Here we can fix any $N_1\in{\bf
N}$, and the
$\Phi ^{(q)}$ are independent of $x$, when
$\widetilde{N}0$ as large as we like).
\par As before we conclude that
$$\inf_{u\in{\cal H}_1,\,\Vert u\Vert
=1\atop R_+^{N+1,0}u=0}(\Delta _\phi
^{(0)}u\vert u)\ge (N+2)\lambda
_{\min}(\phi ''(0))-{1\over 2C},$$
for every $C>0$ when $h>0$ is small
enough depending on $C$. By repeating
earlier arguments, we obtain the apriori
estimate (5.10) for solutions to
Gr($N$+1,0), as well as existence of such
solutions for arbitrary $v\in{\cal
H}_{-1}$ and $v_+\in\ell^2$. In other
words, we get part (A) of the proposition
with $N$ replaced by $N+1$ and this
completes the inductive proof of
Proposition 5.1.
\medskip\it Remark 5.5. \rm Let us compute
$(\Delta _\phi
^{(0)}\widetilde{R}_-^{N,0}u\vert
\widetilde{R}_-^{N,0}v)$ to leading order
for $u,v\in\ell_b^2(\Lambda ^0\cup ..\cup
\Lambda ^N)$, i.e. modulo ${\cal
O}(1)h^{1/2}\vert u\vert _2\vert v\vert
_2$. The proof of Lemma 5.4 shows that the
searched expression involves a block
diagonal matrix, so we may assume that
$u,v\in\ell_b^2(\Lambda ^{{\cal P}})$,
${\cal P}=\{ 1,..,P\}$, for $1\le P\le N$.
(The case $P=0$ will give 0.) Then if
$\equiv$ indicates equality modulo ${\cal
O}(1)h^{1/2}\vert u\vert _2\vert v\vert
_2$, we get
$$\eqalign{
&(\Delta _\phi
^{(0)}\widetilde{R}_-^{N,0}u\vert
\widetilde{R}_-^{N,0}v)=\sum_{j\in\Lambda
}(Z_j\widetilde{R}_-^{N,0}u\vert Z_j
\widetilde{R}_-^{N,0}v)\equiv\cr
&
{1\over P!}\sum_{j\in\Lambda
}\sum_{\widehat{p},\widehat{q}=
1}^P\sum_{p,q\in\Lambda ^{{\cal P}}}\phi
''_{q(\widehat{q}),j}(0)\phi
''_{j,p(\widehat{p})}(0)(Z^*_{p\vert
{\cal
P}\setminus\{\widehat{p}\}}(e^{-\phi /2h})
\vert
Z^*_{q\vert {\cal P}\setminus\{
\widehat{q}\}}(e^{-\phi
/2h}))u(p)\overline{v(q)}. }$$
Using that $u,v\in\ell_b^2$, we can reduce
the sum to the case
$\widehat{p}=\widehat{q}=1$, and get
$$
\eqalign{
&{P^2\over P!}\sum_{j\in\Lambda
} \sum_{p,q\in\Lambda ^{{\cal P}}}\phi
''_{q(1),j}(0)\phi
''_{j,p(1)}(0)(Z^*_{p\vert {\cal
P}\setminus\{ 1\} }(e^{-\phi
/2h})\vert Z^*_{q\vert {\cal
P}\setminus\{ 1\}}(e^{-\phi
/2h}))u(p)\overline{v(q)}\cr
&
={P^2\over P!}\sum_{p,q\in\Lambda ^{{\cal
P}}}(\phi ''(0)^2)_{q(1),p(1)}(Z^*_{p\vert
{\cal P}\setminus\{ 1\}}(e^{-\phi
/2h})\vert Z^*_{q\vert {\cal P}\setminus\{
1\}}(e^{-\phi /2h}))u(p)\overline{v(q)}\cr
&\equiv {P^2\over P!}\sum_{p,q\in\Lambda
^{{\cal P}}}(\phi
''(0)^2)_{q(1),p(1)}\sum_{\pi \in{\rm
Perm}(\{ 2,..,P\} )}(\prod_{\nu =2}^P \phi
''_{p(\nu ),q(\pi (\nu
))}(0))u(p)\overline{v(q)}=
\cr &
P\sum_{p,q\in\Lambda ^{{\cal P}}}(\phi
''(0)^2)_{q(1),p(1)}\prod_{\nu =2}^P (\phi
''_{p(\nu ), q(\nu
)}(0))u(p)\overline{v(q)}=(P\phi
''(0)^2\otimes \phi ''(0)\otimes ..\otimes
\phi ''(0)u\vert v)_{\ell^2}
\cr&
\hskip 2cm =(P(\phi ''(0)\otimes 1\otimes
..\otimes 1)(\phi ''(0)^{1\over 2}\otimes
..\otimes
\phi ''(0)^{1\over 2})u\vert (\phi
''(0)^{1\over 2}\otimes ..\otimes \phi
''(0)^{1\over 2})v),
}$$
where we again used that $u,v\in\ell_b^2$.
Using this property once more, we can
replace $P(\phi ''(0)\otimes 1\otimes
..\otimes 1)$ by the more suggestive
expression
\ekv{5.53}{\Phi _P:=\phi ''(0)\otimes
1\otimes ..\otimes 1+1\otimes \phi
''(0)\otimes 1 ..\otimes 1+..+1\otimes
..\otimes 1\otimes
\phi ''(0).} If $\lambda _1,..,\lambda
_{\#\Lambda }$ denote the eigenvalues of
$\phi ''(0)$, then the eigenvalues of
(5.53) are of the form
\ekv{5.54}
{\sum_{\nu =1}^P\lambda _{p(\nu )},\
p\in\{ 1,..,\#\Lambda \}^{\cal P}.}
\par Summing up the discussion, for $u\in
\ell_b^2(\Lambda ^{{\cal P}})$,
$v\in\ell_b^{{\cal Q}}$, ${\cal P}=\{
1,..,P\}$, ${\cal Q}=\{ 1,..,Q\}$, we have
\eekv{5.55}
{(\Delta _\phi
^{(0)}\widetilde{R}_-^{N,0}u\vert
\widetilde{R}_-^{N,0}v)}{={\cal
O}(h^{1\over 2})\vert u\vert _2\vert v\vert
_2+\cases{ 0,\hbox{ if }P\ne Q\cr
(\Phi _P(\phi
''(0)^{1\over 2}\otimes ..\otimes \phi
''(0)^{1\over 2})u\vert (\phi
''(0)^{1\over 2}\otimes ..\otimes \phi
''(0)^{1\over 2})v),\ P=Q. }}
This should be compared with the following
consequence of section 4:
\eekv{5.56}
{(\widetilde{R}_-^{N,0}u\vert
\widetilde{R}_-^{N,0}v)}{={\cal
O}(h^{1\over 2})\vert u\vert _2\vert v\vert
_2+\cases{ 0,\hbox{ if }P\ne Q\cr
((\phi
''(0)^{1\over 2}\otimes ..\otimes \phi
''(0)^{1\over 2})u\vert (\phi
''(0)^{1\over 2}\otimes ..\otimes \phi
''(0)^{1\over 2})v),\ P=Q. }}
%\vfill\eject
\bigskip
\centerline{\bf 6. Asymptotics of the
solutions of the Grushin problems.}
\medskip
\par We first work with the scalar case
and denote by ${\cal L}_j$ the span of all
${(Z^*)^\alpha \over\alpha !}(e^{-\phi
/2h})$, $\vert \alpha \vert =j$.
Equivalently,
${\cal L}_j$ is equal to
$R_-^{N,0}(\ell^2({\bf N}_j^\Lambda ))$, if
$j\le N$. If $A$ is a finite subset of ${\bf
N}$, we write ${\cal L}_A=\oplus_{j\in
A}{\cal L}_j\subset L^2.$
Notice that the orthogonal projection onto
${\cal L}_{[0,N]}$ is given by
\ekv{6.1}
{
R_-^{N,0}(R_+^{N,0}R_-^{N,0})^{-1}R_+^{N,0}.
} By section 4 we know that
\ekv{6.2}
{
\Vert R_-^{N,0}v_+\Vert \sim\vert v_+\vert
,\ u\in\ell^2({\bf N}_{[0,N]}^\Lambda ). }
We can identify ${\cal L}_j$ with
$\ell^2({\bf N}_j^\Lambda )$ by means of
$r_jR_+^{N,0}$, where $r_j:\ell^2({\bf
N}_{[0,N]}^\Lambda )\to\ell^2({\bf
N}_j^\Lambda )$ is the natural restriction
map, and again by section 4 we know that
\ekv{6.3}
{
\vert R_+^{N,0}u\vert _2\sim \Vert u\Vert
,\ u\in{\cal L}_{[0,N]}. }
\par
We have the decomposition
\ekv{6.4}
{
L^2({\bf R}^\Lambda )={\cal L}_0\oplus
..\oplus{\cal L}_N\oplus {\cal
L}_{[0,N]}^\perp,\
u=u_0+..+u_N+u_{N+1}\in L^2, }
and correspondingly
\ekv{6.5}
{\Vert u\Vert ^2\sim\sum_0^{N+1}\Vert
u_j\Vert^2. }
For $j\le N$, the projection onto ${\cal
L}_j$ is given by
$$\Pi
_j=R_-^{N,0}r_j^*r_j{(R_+^{N,0}R_-^{N,0})}^
{-1} R_+^{N,0}.$$ Lemma 5.4 and (6.2) imply
that
\ekv{6.6}
{
\Vert u\Vert _{{\cal H}_1}\le{\cal
O}(1)\Vert u\Vert ,\ u\in{\cal L}_{[0,N]}, }
and the same lemma with (6.3) implies that
\ekv{6.7}
{
\Vert u\Vert \le{\cal O}(1)\Vert u\Vert
_{{\cal H}_{-1}},\ u\in{\cal L}_{[0,N]}. }
In other words, the norms of ${\cal H}_1$,
${\cal H}_{-1}$ and $L^2$ are (uniformly)
equivalent on ${\cal L}_{[0,N]}$, and we
also know that the projections (6.1) and
$\Pi _j$ are bounded in these spaces.
\par We are interested in the block matrix
of $\Delta _\phi ^{(0)}$, viewed as an
operator
$$\eqalignno{
\Delta _\phi ^{(0)}:&{\cal L}_0\oplus{\cal
L}_1\oplus ..\oplus{\cal L}_N\oplus ({\cal
H}_1\cap{\cal L}_{[0,N]}^\perp)\to&(6.8)
\cr&{\cal L}_0\oplus{\cal
L}_1\oplus ..\oplus{\cal L}_N\oplus ({\cal
H}_{-1}\cap{\cal L}_{[0,N]}^\perp)
.}$$
(3.12) shows that for $j\in\Lambda $,
${\cal M}=\{ 1,..,M\}$, $0\le M\le N$,
$m\in\Lambda ^{{\cal M}}$:
\eekv{6.9}
{Z_jZ_{m\vert {\cal M}}^*(e^{-\phi
/2h})=\hbox{ a finite sum of terms of the
type}} {
\sum_{\ell\in\Lambda ^L}h^XZ_{m\vert
M_0}^*Z_{\ell\vert L}^*\circ \Phi
_{j,\ell\vert L, m\vert M_1}(x)(e^{-\phi
/2h}),\ \# M_0+\# L\le N, }
where ${\cal M}=M_0\cup M_1$ is a partition
with
$M_1\ne\emptyset$, $L$ is finite,
and $\Phi $ is standard. Moreover,
\ekv{6.10}
{
X={1\over 2}\#L+\widetilde{N}+{1\over
2}(\# M_1-1),\ 0\le\widetilde{N}\le
N_1\in{\bf N}. }
Here $N_1$ is any sufficiently large
integer and
$\Phi _{j,\ell\vert L,m\vert M_1}$ is
independent of $x$, when $\# M_0+\# L0$ be constants
with $d_{N+1}=1$, such that
\ekv{6.27}
{
d_{j+1}/d_j\in [h^{{1\over
2}},h^{-{1\over 2}}],\ 0\le j\le N, }
or satifying the sharper assumption
\ekv{6.28}
{d_{j+1}/d_j\in [\delta ,1/\delta ], \
0\le j\le N,}
for some $h^{1\over 2}\le \delta \le 1$.
Let $\widetilde{d}:\ell^2({\bf
N}^\Lambda _{[0,N]})\to\ell^2({\bf
N}^\Lambda _{[0,N]})$ be given by the
block diagonal matrix ${\rm
diag\,}(d_j)_{0\le j\le N}$, with respect
to the orthogonal decomposition
$\ell^2({\bf N}^\Lambda
_{[0,N]})=\oplus_{j=1}^N \ell^2({\bf
N}_j^\Lambda )$. Put
$$K_+=(R_+R_-)^{-{1\over 2}}R_+,\
K_-=R_-(R_+R_-)^{-{1\over 2}}$$
so that
$$K_+^*=K_-,\ K_+K_-=1,\ K_-K_+=\Pi =\Pi
_{[0,N]}.$$
Put $d=K_-\widetilde{d}K_++(1-\Pi )$,
where we notice that the first term
commutes with $\Pi $; $\Pi
K_-\widetilde{d}K_+=K_-\widetilde{d}K_+\Pi
=K_-\widetilde{d}K_+$. We observe that $d$
and $\widetilde{d}$ are selfadjoint and
that $d^{-1}$ corresponds to
$\widetilde{d}^{-1}$:
$d^{-1}=K_-\widetilde{d}^{-1}K_+ +(1-\Pi
)$.
\par Consider
$\widetilde{d}^{-1}R_+d=\widetilde{d}^{-1}
(R_+R_-)^{1\over
2}\widetilde{d}(R_+R_-)^{-{1\over
2}}R_+$. Here we know from Proposition 4.1
that the block matrix elements
$((R_+R_-)^{1\over 2})_{j,k}$ are ${\cal
O}(h^{\vert j-k\vert /2})$ and it follows
that
$$\widetilde{d}^{-1}(R_+R_-)^{1\over
2}\widetilde{d}={\cal
O}(1):\ell^2\to\ell^2$$ under the
assumption (6.27) and that
$$\widetilde{d}^{-1}(R_+R_-)^{1\over
2}\widetilde{d}-(R_+R_-)^{1\over 2}={\cal
O}(1) {h^{1\over 2}\over \delta
}:\ell^2\to\ell^2$$ under the assumption
(6.28). We conclude that under the latter
assumption
\eekv{6.29}
{\widetilde{d}^{-1}R_+d-R_+={\cal
O}(1){h^{1\over 2}\over \delta }:{\cal
H}_{-1}\to\ell^2} {d^{-1}R_-
\widetilde{d}-R_-={\cal O}(1){h^{1\over
2}\over \delta }:\ell^2\to{\cal
H}_{1}.}
Here the second relation follows from the
first by duality and in both relations, we
are allowed to replace $(\widetilde{d},d)$
be $(\widetilde{d}^{-1}, d^{-1})$.
\par Now recall that the ${\cal H}_{\pm 1}$
norms and the $L^2$ norm are all equivalent
on
${\cal L}_{[0,N]}$, and consider
$$\eqalign{&d^{-1}\Delta _\phi
^{(0)}d-\Delta _\phi ^{(0)}=\cr
&K_-(\widetilde{d}^{-1}(R_+R_-)^{-{1\over
2}}R_+\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over
2}}\widetilde{d}-(R_+R_-)^{-{1\over
2}}R_+\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over 2}})K_+\cr
&+K_-(\widetilde{d}^{-1}-1)(R_+R_-)^{-{1
\over 2}}R_+\Delta _\phi ^{(0)}(1-\Pi
)+(1-\Pi )\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over
2}}(\widetilde{d}-1)K_+.}$$
Here the block matrix element of
$(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over 2}}$ at
$(j,k)$ is ${\cal O}(h^{{1\over 2}\vert
j-k\vert })$, so
$$\eqalign{\widetilde{d}^{-1}(R_+R_-)^{-{1\over
2}}R_+\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over
2}}\widetilde{d}-(R_+R_-)^{-{1\over
2}}R_+\Delta
_\phi^{(0)}R_-(R_+R_-)^{-{1\over
2}}\cr={\cal O}(1){h^{1\over 2}\over \delta
}:\ell^2\to
\ell^2.}
$$
Similarly
$$\eqalign{(\widetilde{d}^{-1}-1)
(R_+R_-)^{-{1\over 2}}R_+\Delta
_\phi ^{(0)}(1-\Pi )&={\cal
O}(1){h^{{1\over 2}}\over \delta }:L^2\to
\ell^2\cr
(1-\Pi )\Delta _\phi
^{(0)}R_-(R_+R_-)^{-{1\over
2}}(\widetilde{d}-1)&={\cal
O}(1){h^{{1\over 2}}\over \delta
}:\ell^2\to L^2,
}$$
and we conclude that
\ekv{6.30}
{
d^{-1}\Delta _\phi ^{(0)}d-\Delta _\phi
^{(0)}={\cal O}(1){h^{1\over 2}\over
\delta }:{\cal H}_1\to{\cal H}_{-1}. }
Define
\ekv{6.31}
{D=\pmatrix{d &0\cr 0
&\widetilde{d}}={\cal H}_{\pm 1}\times
\ell^2({\bf N}^\Lambda _{[0,N]})\to {\cal
H}_{\pm 1}\times \ell^2({\bf N}^\Lambda
_{[0,N]}).}
If
\ekv{6.32}
{
{\cal P}^{N,0}=\pmatrix{ \Delta _\phi
^{(0)}-z&R_-^{N,0}\cr R_+^{N,0}&0}, }
then under the assumption (6.27)
\ekv{6.33}
{
D^{-1}{\cal P}^{N,0}D={\cal O}(1):{\cal
H}_1\times \ell^2\to{\cal H}_{-1}\times
\ell^2, }
and if (6.28) holds, then
\ekv{6.34}
{
D^{-1}{\cal P}^{N,0}D-{\cal P}^{N,0}={\cal
O}(1)h^{1\over 2}/\delta :{\cal
H}_1\times \ell^2\to{\cal H}_{-1}\times
\ell^2. }
Under the assumptions of Proposition
5.1(A), we introduce
\ekv{6.35}
{
{\cal E}^{N,0}=({\cal P}^{N,0})^{-1}:{\cal
H}_{-1}\times \ell^2\to{\cal H}_1\times
\ell^2. }
Under the assumption (6.27), we have
\ekv{6.36}
{
D^{-1}{\cal E}^{N,0}D={\cal O}(1):{\cal
H}_{-1}\times \ell^2\to{\cal H}_{1}\times
\ell^2, }
(noticing that we have
(6.28) with $\delta =Ch^{1/2}$ and $C$
large enough) and if we assume (6.30),
then
\ekv{6.37}
{
D^{-1}{\cal E} ^{N,0}D -{\cal E}^{N,0}={\cal
O}(1)h^{1\over 2}/\delta . }
\par Write
\ekv{6.38}
{
{\cal
E}^{N,0}=\pmatrix{E^{N,0}&E_+^{N,0}\cr
E_-^{N,0}&E_{-+}^{N,0}}. }
We shall derive approximations of
$E_{\pm}$, $E_{-+}$, where we sometimes
drop the superscript $N,0$ and for that we
look for an approximate solution of the
system
\ekv{6.39}
{\cases{(\Delta_\phi^{(0)}
-z)u+R_-u_-=0\cr R_+u=v_+.}}
Try
\ekv{6.40}
{
u_0=R_-(R_+R_-)^{-1}v_+=:E_+^0v_+,
}
so that $R_+u_0=v_+$. We will choose
$u_-=u_-^0$ in order to satisfy the ${\cal
L}_{[0,N]}$ component of the first
equation of (6.39). Since the orthogonal
projection onto that component is given by
$R_-(R_+R_-)^{-1}R_+$, this means that we
look for $u_-^0\in\ell^2$, such that
$$R_+(\Delta _\phi
^{(0)}-z)u_0+R_+R_-u_-^0=0,$$
i.e. we take
\ekv{6.41}
{
u_-^0=(R_+R_-)^{-1}R_+(z-\Delta _\phi
^{(0)})R_-(R_+R_-)^{-1}v_+=:E_{-+}^0v_+. }
If $v_+\in\ell^2({\bf N}_M^\Lambda )$,
$0\le M\le N$, then
\ekv{6.42}
{\cases{
(\Delta _\phi
^{(0)}-z)E_+^0v_++R_-E_{-+}^0v_+=
\sum_{0\le \widetilde{M}\le N}h^{{1\over
2}\vert N+1-\widetilde{M}\vert
}D_{N+1,\widetilde{M}
;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}}
(R_+R_-)^{-1}v_+\cr R_+E_+^0v_+=v_+, }}
where $D_{P,M;N+1}$ is defined as in
(6.22), with
$\widetilde{R}_-^{N,0}$ replaced by the
equivalent operator
$R_-^{N,0}$.
Then (dropping the superscripts in (6.38))
we get
\ekv{6.43}
{
\cases{E_+v_+=E_+^0v_+-\sum_{0\le \widetilde{M}\le N}h^{{1\over
2}\vert N+1-\widetilde{M}\vert
}ED_{N+1,\widetilde{M}
;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}}
(R_+R_-)^{-1}v_+\cr
E_{-+}v_+=E_{-+}^0v_+-\sum_{0\le \widetilde
{M}\le N}h^{{1\over 2}\vert
N+1-\widetilde{M}\vert
}E_-D_{N+1,\widetilde{M}
;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}}
(R_+R_-)^{-1}v_+. }} Recall that
$\Pi _j$, $j=0,..,N+1$ are the projections
associated to the decomposition (6.4)
and that $\Pi
_j=R_-r_j^*r_j(R_+R_-)^{-1}R_+$, $0\le
j\le N$, $\Pi _{N+1}=1-\Pi _{[0,N]}$. Let
$A:{\cal H}_{-1}\to {\cal H}_1$. We claim
that the following two statements are
equivalent:
\smallskip
\par\noindent (a) $d^{-1}Ad={\cal
O}(1):{\cal H}_{-1}\to{\cal H}_1$ for all
$(d_j)$ satisfying (6.27).
\smallskip
\par\noindent (b) $\Pi _jA\Pi _k={\cal
O}(h^{{1\over 2}\vert j-k\vert }):{\cal
H}_{-1}\to{\cal H}_1$ for all $j,k\in\{
0,..,N+1\}$.
\smallskip
\par To see that, we introduce the
orthogonal projections $\widehat{\Pi }_j$,
$0\le j\le N+1$, with $\widehat{\Pi
}_j\widehat{\Pi }_k=0$ for $j\ne k$,
$1=\widehat{\Pi }_0+..+\widehat{\Pi
}_{N+1}$, by
$$\widehat{\Pi }_j=R_-(R_+R_-)^{-{1\over
2}}r_j^*r_j(R_+R_-)^{-{1\over
2}}R_+\hbox{ for }0\le j\le N,\
\widehat{\Pi } _{N+1}=\Pi _{N+1}.$$
Then $d=\sum_0^{N+1}d_j\widehat{\Pi }_j$
and
\ekv{*}
{d^{-1}Ad=\sum_{j,k=0}^{N+1}{d_k\over
d_j}\widehat{\Pi }_jA\widehat{\Pi }_k.}
We also notice that $\widehat{\Pi
}_j={\cal O}(1)={\cal H}_{\pm 1}\to{\cal
H}_{\pm 1}$. We shall show that (a) and
(b) are both equivalent to the statement
\smallskip
\par\noindent (c) $\widehat{\Pi
}_jA\widehat{\Pi }_k={\cal O}(h^{{1\over
2}\vert j-k\vert }):{\cal H}_{-1}\to{\cal
H}_1$, for all $0\le j,k\le N+1$.
\smallskip
\par That (c) implies (a) is obvious if we
use ($*$), and to get from (a) to (c), it
suffices to write
$${\cal O}(1)=\widehat{\Pi
}_jd^{-1}Ad\widehat{\Pi }_k={d_k\over
d_j}\widehat{\Pi }_jA\widehat{\Pi }_k,$$
and choose $d_\nu $ satisfying (6.27) such
that $d_k/d_j=h^{-{1\over 2}\vert j-k\vert
}$.
\par The equivalence between (b) and (c)
is an easy consequence of the following
estimates
$$\Pi _j\widehat{\Pi }_k,\, \widehat{\Pi
}_j\Pi _k={\cal O}(h^{{1\over 2}\vert
j-k\vert }):{\cal H}_{\pm 1}\to{\cal
H}_{\pm 1},$$
that we shall verify:
\par When $j=k=N+1$, we have $\Pi
_{N+1}\widehat{\Pi }_{N+1}=\widehat{\Pi
}_{N+1}\Pi _{N+1}=\Pi _{N+1}=\widehat{\Pi
}_{N+1}.$
\par When $j\ne k$ and $N+1\in\{ j,k\}$,
then $\widehat{\Pi }_j\Pi _k=\Pi
_j\widehat{\Pi }_k=0$.
\par For $0\le j,k\le N$, we get
$$\Pi _j\widehat{\Pi
}_k=R_-r_j^*r_j(R_+R_-)^{-{1\over
2}}r_k^*r_k(R_+R_-)^{-{1\over 2}}R_+$$
and the block matrix element
$r_j(R_+R_-)^{-{1\over 2}}r_k^*$ is ${\cal
O}(h^{{1\over 2}\vert j-k\vert
}):\ell^2\to\ell^2$. Consequently $\Pi
_j\widehat{\Pi }_k={\cal O}(h^{{1\over
2}\vert j-k\vert }):{\cal H}_{-1}\to {\cal
H}_{1}$. Similarly,
$$\widehat{\Pi }_j\Pi
_k=R_-(R_+R_-)^{-{1\over
2}}r_j^*r_j(R_+R_-)^{1\over
2}r_k^*r_k(R_+R_-)^{-1}R_+={\cal
O}(h^{{1\over 2}\vert j-k\vert }):{\cal
H}_{-1}\to{\cal H}_1.$$
Combining (6.24), (6.36) and
(6.43), we get:
\medskip
\par\noindent \bf Proposition 6.2. \it
With $E_+^0$, $E_{-+}^0$ given by (6.40),
(6.41) and under the assumptions of
Proposition 5.1(A), we have
\ekv{6.44}
{
\cases{
\Pi _P(E_+^{N,0}-E_+^0)r_Q^*={\cal O}(1)
h^{{1\over 2}(\vert N+1-Q\vert +\vert
N+1-P\vert )}:\ell^2\to{\cal H}_1
\cr
r_Q(E_-^{N,0}-E_-^{0})\Pi _P={\cal O}(1)
h^{{1\over 2}(\vert N+1-Q\vert +\vert
N+1-P\vert )}:{\cal H}_{-1}\to \ell^2
\cr r_{\widetilde{P}}(E_{-+}^{N,0}-E_{-+}^0)
r_Q^* ={\cal O}(1)h^{{1\over 2}(\vert
N+1-Q\vert +\vert
N+1-\widetilde{P}\vert )}:\ell^2\to\ell^2,
}
}
for $0\le P\le N+1$, $0\le
\widetilde{P},Q\le N$, where
$E_-^0:=(E_+^0)^*$.\rm\medskip
\par Here the second equation in (6.44) is
obtained by duality, using that $\Pi
_{N+1}$ is the orthogonal projection
(6.1) and that $\Pi _P$ is given after
(6.5).
\par Now let $M>N$ and let us compare
$${\cal P}^{N,0}, {\cal P}^{M,0}$$ and
their inverses for $z$ in the domain of
wellposedness of the "smaller" problem
${\cal P}^{N,0}$. Let $r=r_{[0,N]}$ denote
the natural restriction operator:
$\ell^2({\bf N}^\Lambda
_{[0,M]})\to\ell^2({\bf N}^\Lambda
_{[0,N]})$, and notice that
\ekv{6.45}
{
R_+^{N,0}=r_{[0,N]}R_+^{M,0},\
R_-^{N,0}=R_-^{M,0}r_{[0,N]}^*, }
where $r_{[0,N]}^*$ is the adjoint:
$\ell^2({\bf N}^\Lambda _{[0,N]})\to
\ell^2({\bf N}^\Lambda _{[0,N]})$. To
shorten the notations, we write ${\cal
P}={\cal P}^{N,0}$, $\widetilde{{\cal
P}}={\cal P}^{M,0}$, and similarly for the
associated quantities. In order to solve
Gr($N$,0):
\ekv{6.46}
{
\cases{
(\Delta _\phi ^{(0)}-z)u+R_-u_-=v\cr
R_+u=v_+,
}
}
we consider the bigger problem Gr($M$,0)
\ekv{6.47}
{\cases{
(\Delta _\phi
^{(0)}-z)u+\widetilde{R}_-\widetilde{u}_-
=v\cr
\widetilde{R}_+u=\widetilde{v}_+, }}
and write the solution as
\ekv{6.48}
{\cases{
u=\widetilde{E}v+\widetilde{E}_
+\widetilde{v}_+\cr
\widetilde{u}_-=\widetilde{E}_-v+
\widetilde{E}_{-+}\widetilde{v}_+. }}
We want (6.46) to be fulfilled, so we get
the condition
\ekv{6.49}
{
R_-u_-=\widetilde{R}_-\widetilde{u}_-.
}
The necessary and sufficient condition on
$\widetilde{u}_-$ for (6.49) to have a
solution $u_-$ is
\ekv{6.50}
{
r_{[N+1,M]}\widetilde{u}_-=0,}
where
$r_{[N+1,M]}:\ell^2({\bf N}^\Lambda
_{[0,M]})\to\ell^2({\bf N}^\Lambda
_{[N+1,M]})$ is the restriction operator,
and the corresponding $u_-$ is then
\ekv{6.51}
{
u_-=r_{[0,N]}\widetilde{u}_-.
}
We then get a solution of (6.46) iff
\ekv{6.52}
{
r_{[N+1,M]}\widetilde{E}_-v+r_{[N+1,M]}
\widetilde{E}_{-+}\widetilde{v}_+=0, }
\ekv{6.53}
{
r_{[0,N]}\widetilde{v}_+=v_+.
}
(6.52) is equivalent to
\ekv{6.54}
{
\widetilde{E}_{-+}
\widetilde{v}_++r^*_{[0,N]}w_-=-\widetilde{E}
_-v,\hbox{ for some }w_-\in\ell^2({\bf
N}^\Lambda _{[0,N]}). }
(6.54), (6.53) lead to a new
Grushin problem, namely to invert the
matrix
\ekv{6.55}
{
\widetilde{{\cal
E}}_{-+}:=\pmatrix{\widetilde{E}_{-+}
&r^*_{[0,N]}\cr
r_{[0,N]}&0}, }
which is wellposed in the range of
wellposedness of ${\cal P}$ given in
Proposition 5.1(A).
This follows from Remark 5.5 and
the fact that
$$\widetilde{E}_{-+}=(\widetilde{R}_+
\widetilde{R}_-)^{-1} \widetilde{R}_+
(z-\Delta _\phi
^{(0)})\widetilde{R}_-(
\widetilde{R}_+\widetilde{R}_-)^{-1}+
{\cal O}(h^{1\over 2}),$$
by Proposition 6.2. Modulo ${\cal
O}(h^{1\over 2})$ we obtain a block
diagonal matrix and the diagonal block at
$(j,j)$ with $0\le j\le M$ is given by
$$(\phi ''(0)^{-{1\over 2}}\otimes
..\otimes \phi ''(0)^{-{1\over
2}})(z-(\phi ''(0)\otimes 1\otimes
..\otimes 1+...1\otimes ..\otimes 1\otimes
\phi ''(0)))(\phi ''(0)^{-{1\over 2}}\otimes
..\otimes \phi ''(0)^{-{1\over
2}}).$$ Here the tensor products are of
length $j$ and for $j=0$ the expression
above should be replaced by $z$.
\par Let
\ekv{6.56}
{\pmatrix{F&F_+\cr F_-&F_{-+}}}
be the inverse of (6.55), so that
\ekv{6.57}
{
\pmatrix{\widetilde{v}_+\cr
w_-}=\pmatrix{F &F_+\cr
F_-&F_{-+}}\pmatrix{-\widetilde{E}_-v\cr
v_+}, }
i.e.
\ekv{6.58}
{\widetilde{v}_+=-F\widetilde{E}_-v+F_+v_+,\
w_-=-F_-\widetilde{E}_-v+F_{-+}v_+.}
The solution of (6.46) is then given from
(6.48), (6.51), (6.58):
$$\eqalign{
u&=\widetilde{E}v+\widetilde{E}_+(-F
\widetilde{E}_-v+F_+v_+)\cr
u_-&=r_{[0,N]}(\widetilde{E}_-v+\widetilde{E}_{-+}(-F
\widetilde{E}_-v+F_+v_+)), }$$
i.e.
\ekv{6.59}
{\cases{
u=(\widetilde{E}- \widetilde{E}_+ F
\widetilde{E}_-) v +\widetilde{E}_+ F_+v_+
\cr
u_-=r_{[0,N]}(\widetilde{E}_--\widetilde{E}_
{-+} F \widetilde{E}_- )v+ r_{[0,N]}
\widetilde{E}_{-+} F_+v_+. }}
This can be further simplified, if we use
the identity
$\widetilde{E}_{-+}F+r^*F_-=1$, with
$r=r_{[0,N]}$ for short. Then
$$(\widetilde{E}_--\widetilde{E}_{-+}F
\widetilde{E}_-)= (1-\widetilde{E}_{-+}
F)\widetilde{E}_-= r^*F_-\widetilde{E}_-,$$
and since $rr^*=1$, we get
\ekv{6.60}
{\cases{
u=(\widetilde{E}-\widetilde{E}_+
F\widetilde{E}_-)v+ \widetilde{E}_+ F_+ v_+
\cr
u_-=F_-\widetilde{E}_-v+r_{[0,N]}
\widetilde{E}_{-+}F_+v_+. }}
We next use the relation
$\widetilde{E}_{-+} F_++r^*F_{-+}=0$, to
rewrite the last equation as
$$u_-=F_-\widetilde{E}_-v-rr^*F_{-+}v_+,$$
and using again that $rr^*=1$, we get the
solution of the small problem ${\cal
P}={\cal P}^{N,0}$ as
\ekv{6.61}
{\cases{
u=Ev+E_+v_+\cr u_-=E_-v+E_{-+}v_+
}
\hbox{ with }
\cases{
E=\widetilde{E}-\widetilde{E}_+F
\widetilde{E}_-,\ E_+=\widetilde{E}_+F_+\cr
E_-=F_-\widetilde{E}_-,\ E_{-+}=-F_{-+} }.
}
\par The next goal is to get
asymptotics for $F_{-+}$, $F_-$ similar to
(6.44) with $N$ replaced by $M$. Define
$\widetilde{E}_+^0$
and $\widetilde{E}_{-+}^0$ as in (6.40)
(6.41):
\ekv{6.62}
{\cases{\widetilde{E}_+^0=\widetilde{R}_-
(\widetilde{R}_+ \widetilde{R}_-)^{-1}
\cr\widetilde{E}_{-+}^0=(
\widetilde{R}_+\widetilde{R}_-)^{-1}
\widetilde{R}_+(z-\Delta _\phi
^{(0)})\widetilde{R}_-(\widetilde{R}_
+\widetilde{R}_-)^{-1},}}
so that analogously to (6.44)
\ekv{6.63}
{\cases{
\Pi
_P(\widetilde{E}_+-\widetilde{E}_+^0)
r_Q^*={\cal O}(1) h^{{1\over 2}(\vert
M+1-Q\vert +\vert M+1-P\vert
)}:\ell^2\to{\cal H}_1
\cr
r_Q(\widetilde{E}_--\widetilde{E}_-^0)\Pi
_P={\cal O}(1) h^{{1\over 2}(\vert
M+1-Q\vert +\vert M+1-P\vert )}:{\cal
H}_{-1}\to \ell^2
\cr
r_{\widetilde{P}}(\widetilde{E}_{-+}
-\widetilde{E}_{-+}^0) r_Q^* ={\cal
O}(1)h^{{1\over 2}(\vert M+1-Q\vert +\vert
M+1-\widetilde{P}\vert
)}:\ell^2\to\ell^2,} }
for $0\le P\le M+1$, $0\le
Q,\widetilde{P}\le M$, where
$\widetilde{E}_-^0$ is defined to be the
adjoint of $\widetilde{E}_+^0$.
\par Let
\ekv{6.64}
{
\pmatrix{F^0&F_+^0\cr
F_-^0&F_{-+}^0}={\pmatrix{
\widetilde{E}_{-+}^0&r_{[0,N]}^*\cr
r_{[0,N]}&0}}^{-1}=:(\widetilde{{\cal
E}}_{-+}^0)^{-1}. }
Let $\lambda _j>0$, $0\le j\le M+1$ and
let $\lambda _{[0,M]}={\rm diag\,}(\lambda
_j)_{0\le j\le M}$, $\lambda _{[0,N]}={\rm
diag\,}(\lambda _j)_{0\le j\le N}$. Let
\ekv{6.65}{\Lambda =\pmatrix{\lambda
_{[0,M]}&0\cr 0&\lambda _{[0,N]}},}
viewed as an operator on
$$(\oplus_0^M\ell^2({\bf N}_j^\Lambda
))\times (\oplus_0^N\ell^2({\bf
N}_j^\Lambda )).$$
As before we define the action of $\lambda
$ on ${\cal H}_{\pm 1}$ and on $L^2({\bf
R}^\Lambda )$. We also need a second
system of weights $\mu _j$ and define $\mu
_{[0,N]}$, $\mu _{[0,M]}$ and
${\cal M} =\pmatrix{\mu _{[0,M]}&0\cr 0&\mu
_{[0,N]}}$ analogously. (6.63) can
be reformulated as
\ekv{6.66}
{\cases{\lambda
(\widetilde{E}_+-\widetilde{E}_+^0)\mu
_{[0,M]}^{-1}={\cal O}(1):\ell^2\to{\cal
H}_1,
\cr
\mu
_{[0,M]}(\widetilde{E}_--
\widetilde{E}_-^0) \lambda ^{-1}={\cal
O}(1): {\cal H}_{-1}\to\ell^2,
\cr\lambda
_{[0,M]}(\widetilde{E}_{-+}-
\widetilde{E}_{-+}^0)\mu _{[0,M]}^{-1}={\cal
O}(1):\ell^2\to\ell^2,}}
for all $\lambda $, $\mu $ as above with
\ekv{6.67}
{
{\lambda _{j+1}\over \lambda _j}, {\mu
_{j+1}\over\mu _j}\in [h^{1\over
2},h^{-{1\over 2}}],\ 0\le j\le M, }
\ekv{6.68}
{\lambda _{M+1}=\mu _{M+1}.}
It follows from (6.66) that
\ekv{6.69}
{\Lambda (\widetilde{{\cal
E}}_{-+}-\widetilde{{\cal E}}^0_{-+}){\cal
M}^{-1}={\cal O}(1),}
for all $\mu $, $\lambda $ which satisfy
(6.67,68). We also have
\ekv{6.70}
{\Lambda \widetilde{{\cal E}}_{-+}\Lambda
^{-1}={\cal O}(1)}
and similarly with $\Lambda $ replaced by
one of $\Lambda ^{-1}$, ${\cal M}^{\pm 1}$
and/or $\widetilde{{\cal E}}_{-+}$ replaced
by $(\widetilde{{\cal E}}^0_{-+})^{\pm
1}$, $(\widetilde{{\cal E}}_{-+})^{-1}$.
Then
\ekv{6.71}
{\Lambda (\widetilde{{\cal
E}}_{-+}^{-1}-(\widetilde{{\cal
E}}^0_{-+})^{-1}){\cal M}^{-1}=\Lambda
\widetilde{{\cal E}}_{-+}^{-1}\Lambda
^{-1}\Lambda (\widetilde{{\cal
E}}_{-+}^0-\widetilde{{\cal E}}_{-+}){\cal
M}^{-1}{\cal M}(\widetilde{{\cal E
}}_{-+}^0)^{-1}{\cal M}^{-1}={\cal O}(1),}
for all $\mu $, $\lambda $ satisfying
(6.67), (6.68). Equivalently, if we
introduce the block matrix notation
$A_{j,k}=r_jAr_k^*$, then
$$\eqalignno{
(F-F^0)_{j,k}&={\cal O}(h^{{1\over
2}(\vert M+1-j\vert +\vert M+1-k\vert
)}),\ 0\le j,k\le M,&(6.72)\cr
(F_+-F_+^0)_{j,k}&= {\cal O}(h^{{1\over
2}(\vert M+1-j\vert +\vert M+1-k\vert
)}),\ 0\le j\le M,\ 0\le k\le N,\cr
(F_--F_-^0)_{j,k}&={\cal O}(h^{{1\over
2}(\vert M+1-j\vert +\vert M+1-k\vert
)}),\ 0\le j\le N,\ 0\le k\le M,\cr
(F_{-+}-F_{-+}^0)_{j,k}&={\cal O}(h^{{1\over
2}(\vert M+1-j\vert +\vert M+1-k\vert
)}),\ 0\le j,k\le N.
}$$
\par For $\widetilde{E}_{-+}^0$ we have a
complete asymptotic expansion
\ekv{6.73}{\widetilde{E}^0_{-+;j,k}\sim
h^{{1\over 2}\vert j-k\vert }\sum_{\nu
=0}^\infty h^\nu A_{j,k}^\nu \hbox{ in
}{\cal L}(\ell^2,\ell^2),}
and it follows that the inverse (cf.
(6.64)) of the corresponding Grushin
problem
$\widetilde{{\cal E}}_{-+}^0$ has the same
structure.
$$\eqalignno{F_{j,k}^0&\sim h^{{1\over
2}\vert j-k\vert }\sum_{\nu =0}^\infty
h^\nu B_{j,k}^{0,\nu },\ 0\le j,k\le
M,&(6.74)\cr
F_{+;j,k}^0&\sim h^{{1\over
2}\vert j-k\vert }\sum_{\nu =0}^\infty
h^\nu B_{+;j,k}^{0,\nu },\ 0\le j\le M,\
0\le k\le N,
\cr
F_{-;j,k}^0&\sim h^{{1\over
2}\vert j-k\vert }\sum_{\nu =0}^\infty
h^\nu B_{-;j,k}^{0,\nu },\ 0\le j\le N,\
0\le k\le M,
\cr
F_{-+;j,k}^0&\sim h^{{1\over
2}\vert j-k\vert }\sum_{\nu =0}^\infty
h^\nu B_{-+;j,k}^{0,\nu },\ 0\le j,k\le N .
}$$
Combining this with (6.72) and letting
$M\to \infty $ we obtain a complete
asymptotic expansion for
$F_{-+}=-E_{-+}^{N,0}$:
\ekv{6.75}
{
-F_{-+;j,k}\sim h^{{1\over 2}\vert
j-k\vert }\sum_{\nu =0}^\infty h^\nu
B^\nu _{-+;j,k},\ 0\le j,k\le N. }
For $F_{\pm}$ we only have a partial
asymptotics with the limitations of (6.72)
but again it will be advantageous to let
$M\to\infty $.
\par We now look at $E_+$. The first
equation in (6.63) tells us that
$$(\widetilde{E}_+-\widetilde{E}_+^0)r_Q^*=
{\cal O}(1)h^{{1\over 2}\vert M+1-Q\vert
}:\ell^2\to {\cal H}_1.$$
Write
$$E_+=\widetilde{E}_+F_+=(\widetilde{E}_+-
\widetilde{E}_+^0)F_++\widetilde{E}_+^0(
F_+-F_+^0)+ \widetilde{E}_+^0 F_+^0.$$
Here
$$(\widetilde{E}_+-\widetilde{E}_+^0
)F_+= {\cal O}(1)
\sum_{Q=0}^Mh^{{1\over 2} (\vert
M+1-Q\vert +(Q-N)_+) }:\ell^2\to{\cal
H}_1,$$
so that
$(\widetilde{E}_+-\widetilde{E}_+^0
)F_+={\cal O}(1)h^{{1\over 2}\vert
M+1-N\vert }:\ell^2\to{\cal H}_1$. We also
have
$$\widetilde{E}_+^0(F_+-F_+^0)={\cal
O}(1)\sum_{Q=0}^M h^{{1\over 2}(\vert
M+1-Q\vert +\vert M+1-N\vert )} ={\cal
O}(1)h^{{1\over 2}\vert M+1-N\vert
}:\ell^2\to{\cal H}_1.$$
Consequently,
$$E_+^{N,0}=\widetilde{E}_+^0F_+^0+{\cal
O} (1)h^{{1\over 2}(M+1-N)}:\ell^2 \to
{\cal H}_1.$$
Here $F_+^0$ has a complete asymptotic
expansion given by (6.74) and
$\widetilde{E}_+^0$ is given by (6.62):
$$E_+^{N,0}=R_-^{M,0}(R_+^{M,0}R_-^{M,0}
)^{-1} F_+^0+{\cal O}(1)h^{{1\over
2}(M+1-N)}:\ell^2\to{\cal H}_1,$$
where $(R_+^{M,0}R_-^{M,0})^{-1}$ has a
complete asymptotic expansion of the same
type as $R_+^{M,0}R_-^{M,0}$ (c.f.
(5.29)), so we conclude that for every
$M\ge N$:
\ekv{6.76}
{E_+^{N,0}=R_-^{M,0}C^M+{\cal
O}(1)h^{{1\over 2}(M+1-N)}:\ell^2\to {\cal
H}_1,}
where
\ekv{6.77}
{
C_{j,k}^M\sim \sum_{\nu =0}^\infty
h^{{1\over 2}\vert j-k\vert +\nu
}D_{j,k}^{M,\nu },\hbox{ in }{\cal
L}(\ell^2,\ell^2), }
for $0\le j\le M$, $0\le k\le N$.
Summing up, we have
\medskip
\par\noindent \bf Proposition 6.3. \it
$E_{-+}^{N,0}$ has a complete asymptotic
expansion in ${\cal L}(\ell^2,\ell^2)$,
that can be written at the level of block
matrix elements:
\ekv{6.78}
{
E_{-+;j,k}^{N,0}\sim h^{{1\over 2}\vert
j-k\vert }\sum_{\nu =0}^\infty h^\nu
B^\nu _{-+;j,k},\ 0\le j,k\le N. }
For every $M\ge N$, we have (6.76,77) for
$E_+^{N,0}$.\rm\medskip
\par Using Proposition 6.2, (6.40),
(6.41), we also get the leading terms in
the asymptotic expansions (6.78), (6.77):
\eekv{6.79}
{B^0_{-+;j,j}=(\phi ''(0)\otimes ..\otimes
\phi ''(0))^{-{1\over 2}}}{\hskip
2cm (z-(\phi ''(0)\otimes 1\otimes ..\otimes
1+..+1\otimes ..\otimes 1\otimes \phi
''(0)))(\phi ''(0)\otimes ..\otimes \phi
''(0))^{-{1\over 2}},}
\ekv{6.80}
{D_{j,j}^{M,0}=(\phi ''(0)\otimes ..\otimes
\phi ''(0))^{-1},\ 0\le j\le M.}
\par We now want to do the same job with
Gr($N$,1) as we did with Gr($N$,0), and
the only slightly new thing is to analyze
the block matrix of $\phi ''(x)$, viewed
as an operator
\eekv{6.81}
{\ell^2(\Lambda )\otimes ({\cal L}_0\oplus
..\oplus{\cal L}_N\oplus({\cal
H}_1\cap{\cal L}_{[0,N]}^\perp))\to}
{\ell^2(\Lambda )\otimes ({\cal L}_0\oplus
..\oplus{\cal L}_N\oplus({\cal
H}_{-1}\cap{\cal L}_{[0,N]}^\perp)).}
According to (3.12) (or by reviewing more
directly the arguments leading to that
result), if ${\cal M}=\{ 1,..,M\}$, for
$1\le M\le N$, or ${\cal M}=\emptyset $,
then for
$\nu ,\mu \in\Lambda $, $m\in\Lambda
^{{\cal M}}$,
$\phi ''_{\nu ,\mu}(x)Z^*_{m\vert {\cal
M}}(e^{-\phi /2h})$ is a finite sum of terms
of the type
\ekv{6.82}
{\sum_{\ell\in\Lambda ^L}h^XZ^*_{m\vert
M_0}Z^*_{\ell\vert L}(\Phi _{\nu ,\mu
,m\vert M_1,\ell\vert L}(x)e^{-\phi /2h}),}
where ${\cal M}=M_0\cup M_1$ is a
partition and $L$ is a finite set,
possibly with $M_1$ or $L$ empty. $\Phi $
is a standard tensor and
\ekv{6.83}
{X={1\over 2}(\# M_1+\# L)+\widetilde{N},\
{\bf N}\ni\widetilde{N}\le N_1\in{\bf N},}
where $N_1\in {\bf N}$ is any
fixed number. Moreover
$\Phi $ is independent of $x$, when
$$\cases{\# M_0+\# L0$ sufficiently small
and uniformly for all $\rho \in W_a$, we
have the asymptotic expansion in ${\cal
L}(\ell^2\otimes \ell^2,\ell^2\otimes
\ell^2)$, that can be written for the
block diagonal elements:
\ekv{7.23}
{
(\rho \otimes 1)^{-1}E_{-+;j,k}^{N,1}(\rho
\otimes 1)\sim h^{{1\over 2}\vert j-k\vert}
\sum_{\nu =0}^\infty h^\nu (\rho
\otimes 1)^{-1}B_{-+;j,k}^\nu (\rho
\otimes 1),\ 0\le j,k\le N. }
Here $B_{-+;j,k}^\nu $ are the same as in
(6.99).
\par For $M\ge N$, we have:
\ekv{7.24}
{
(\rho \otimes 1)^{-1}E_+^{N,1}(\rho
\otimes 1)=R_-^{M,1}(\rho \otimes
1)^{-1}C^M(\rho \otimes 1)+{\cal
O}(1)h^{{1\over 2}(M+1-N)}:\ell^2\otimes
\ell^2\to \ell^2\otimes {\cal H}_1. }
Here $C^M$ is the same as in Proposition
6.5, and we have the asymptotic expansion
for the block matrix elements, valid
uniformly with respect to $\rho\in W_a $:
\ekv{7.25}
{
(\rho \otimes 1)^{-1}C_{j,k}^M(\rho
\otimes 1)\sim
\sum_{\nu =0}^\infty h^{{1\over 2}\vert
j-k\vert +\nu }(\rho \otimes
1)^{-1}D_{j,k}^{M,\nu }(\rho \otimes 1), }
for $0\le j\le M$, $0\le k\le
N$.\rm\medskip
%\vfill\eject
\bigskip
\centerline{\bf 8. Parameter dependent
exponents.}
\medskip
\par In this section we carry out an
essential preparation for controling the
thermodynamical limit of the correlations.
For that, we need to estimate the
variation of the correlations, when the
exponent $\phi =\phi _t(x)$ depends on a
parameter
$t\in [0,1]$:
\ekv{8.1{\rm H}}
{\phi _t(x)=\phi _{t,0}(x)+C(t;h),}
with $C(t;h)$ independent of $x$ and with
$\phi _{t,0}$ independent of $h$. We
assume that $\phi _t(x)$ is of class $C^1$
in $t$ and smooth in $x\in{\bf R}^\Lambda
$. Further assumptions will be given later
on. We assume that $C(t;h)$ is chosen so
that
\ekv{8.2{\rm H}}{\int e^{-\phi
_t(x)/h}dx=1.}
\par We start the section by
making some formal computations. After
that we will introduce some precise
assumptions on
$\phi _t$ that justify the formal
computations. Finally we will estimate the
various terms that we get. The estimates
will be summed up in Proposition 8.4.
\par We are interested in
\ekv{8.3}
{{\rm Cor}_t(u,v)=\int e^{-\phi
_t(x)/h}(u-\langle u\rangle _t)(v-\langle
v\rangle _t)dx,}
where $\langle u\rangle _t$ denotes the
expectation of $u$ with respect to
$e^{-\phi _t/h}dx$, and where $u$ and $v$
are supposed to be independent of $t$. Since
$u-\langle u\rangle _t$ and $v-\langle
v_t\rangle $ have expectation 0, we get
\ekv{8.4}
{-\partial _t{\rm Cor}_t(u,v)=\int
e^{-\phi _t(x)/h}{\partial _t\phi
_t(x)\over h}(u-\langle u\rangle
_t)(v-\langle v\rangle _t)dx.}
\par Assume that
\ekv{8.5{\rm H}}
{\partial _t\phi _{t,0}(0)=0,\ \partial
_x\partial _t\phi _{t,0}(0)=0.}
Then
\ekv{8.6}
{\partial _t\phi
_{t,0}(x)=\sum_{j,k}\widetilde{\Phi
}_{j,k}(x)x_jx_k=\langle \widetilde{\Phi
}(x)x,x\rangle ,}
with $\widetilde{\Phi }=\widetilde{\Phi
}_t$ given by
\ekv{8.7}
{\widetilde{\Phi
}_{j,k}(x)=\int_0^1(1-s)\partial
_{x_j}\partial _{x_k}\partial _t\phi
_{t,0}(sx)ds.} Now assume that
\ekv{8.8{\rm H}}
{\partial _x\phi _t(x)=A_t(x)x,}
where the matrix $A_t(x)$ is $C^1$ in $t$,
$C^\infty $ in $x$ and invertible.
Combining this with (8.6), we get
\ekv{8.9}
{\partial _t\phi
_{t,0}(x)=\sum_{j,k}\Phi
_{j,k}(x)\partial _{x_j}\phi \partial
_{x_k}\phi =\langle \Phi (x)\partial
_x\phi (x),\partial _x\phi (x)\rangle ,}
with
\ekv{8.10}
{\Phi
(x)={^t\hskip -1pt
A}(x)^{-1}\widetilde{\Phi }(x)A(x)^{-1}.}
Here and in the following, we often drop
the subscript $t$. Later on it will be
useful to keep in mind that $\Phi
_{j,k}(x)$ is symmetric.
\par With $\phi =\phi _t$, we define
$Z_j,Z_j^*$ as in section 1. A straight
forward computation shows that
\ekv{8.11}
{
e^{-\phi /2h}{\partial _t\phi \over
h}=\sum_{j,k}Z_j^*Z_k^*(e^{-\phi /2h}\Phi
_{j,k})+h^{1\over 2}\sum_j Z_j^*(e^{-\phi
/2h}\Psi _j)+D(x;h)e^{-\phi /2h}, }
where
\ekv{8.12}
{
\Psi _j=2\sum_k\partial _{x_k}\Phi _{j,k},
}
\ekv{8.13}
{
D=\sum_{j,k}(\partial
_{x_j}\partial_{x_k}\phi )\Phi
_{j,k}+h\sum_{j,k}\partial _{x_j}\partial
_{x_k}\Phi _{j,k}+{\partial
_tC(t;h)\over h}. }
Using this in (8.4), we get
\ekv{8.14}
{
-\partial _t{\rm Cor}_t(u,v)={\rm I}+{\rm
II}+{\rm III}, }
where
\ekv{8.15}
{
\cases{{\rm
I}=\int\sum_{j,k}Z_j^*Z_k^*(e^{-\phi
/2h}\Phi _{j,k})e^{-\phi /2h}(u-\langle
u\rangle )(v-\langle v\rangle )dx,
\cr {\rm II}=\int h^{1\over 2}\sum_j
Z_j^*(e^{-\phi /2h}\Psi _j)e^{-\phi
/2h}(u-\langle u\rangle )(v-\langle
v\rangle )dx,\cr
{\rm III}=\int e^{-\phi /2h}De^{-\phi
/2h}(u-\langle u\rangle )(v-\langle
v\rangle )dx.
} }
Since $Z_j\circ e^{-\phi /2h}=e^{-\phi
/2h}h^{1/2}\partial _{x_j}$, we get after
an integration by parts,
\eeekv{8.16}
{
{\rm I}=\int\sum_{j,k}e^{-\phi /2h}\Phi
_{j,k}Z_jZ_k(e^{-\phi /2h}(u-\langle
u\rangle )(v-\langle v\rangle ))dx }
{
=h\int \sum_{j,k}e^{-\phi /2h}\Phi
_{j,k}e^{-\phi /2h}\partial _{x_j}\partial
_{x_k}((u-\langle u\rangle )(v-\langle
v\rangle ))dx }
{
={\rm I}_1+{\rm I}_2+{\rm I}_3,
}
where
\ekv{8.17}
{
{\rm I}_1=2h\int \sum_{j,k}\Phi
_{j,k}(e^{-\phi /2h}\partial
_{x_j}u)(e^{-\phi /2h}\partial _{x_k}v) dx,
} and
$$\cases{
{\rm I}_2=h\int\sum_{j,k}e^{-\phi /2h}\Phi
_{j,k}(\partial _{x_j}\partial
_{x_k}u)e^{-\phi /2h}(v-\langle v\rangle
)dx,\cr
{\rm I}_3=h\int\sum_{j,k}e^{-\phi /2h}\Phi
_{j,k}(\partial _{x_j}\partial
_{x_k}v)e^{-\phi /2h}(u-\langle u\rangle
)dx.}
$$
Here we used the symmetry of $(\Phi
_{j,k})$ to get ${\rm I}_1$. We need
to transform ${\rm I}_2$,
${\rm I}_3$ further.
\par Later in this section we shall solve
\ekv{8.18}
{\cases{
e^{-\phi /2h}(u-\langle u\rangle )=\sum_\nu
Z_\nu ^*(\widetilde{u}_\nu )\,\,(=d_\phi
^*(\sum \widetilde{u}_\nu dx_\nu )),\cr
e^{-\phi /2h}(v-\langle v\rangle )=\sum_\nu
Z_\nu ^*(\widetilde{v}_\nu ),\cr
e^{-\phi /2h}D=\sum Z_\nu
^*(\widetilde{D}_\nu ).
}}
As for the last equation, we see formally
that
\ekv{8.19}
{
\langle D\rangle =0.
}
This follows from (8.11), since
$$\langle {\partial _t\phi \over
h}\rangle =-\partial _t\int e^{-\phi
_t/h}dx=-\partial _t(1)=0,$$
and
$$\int e^{-\phi /2h}Z_j^*wdx=\int
Z_j(e^{-\phi /2h})w dx=0,$$
unders suitable assumptions on $w$ which
will be verified.
\par Substitution of the second equation
of (8.18) into the expression for ${\rm
I}_2$ and integration by parts gives
\ekv{8.20}
{
{\rm I}_2=h^{3\over 2}\int\sum_{j,k,\nu
}e^{-\phi /2h}\partial _{x_\nu }(\Phi
_{j,k}\partial _{x_j}\partial
_{x_k}u)\widetilde{v}_\nu
dx={\rm I}_{2,1}+{\rm I}_{2,2}, }
where
\ekv{8.21}
{\cases{
{\rm I}_{2,1}=h^{3\over 2}\int\sum_{j,k,\nu
}e^{-\phi
/2h}(\partial _{x_\nu }\Phi
_{j,k})(\partial _{x_j}\partial
_{x_k}u)\widetilde{v}_\nu dx,\cr {\rm
I}_{2,2}=h^{3\over 2}\int\sum_{j,k,\nu
}e^{-\phi /2h}\Phi _{j,k}(\partial _{x_\nu
}\partial _{x_j}\partial
_{x_k}u)\widetilde{v}_\nu dx.
}}
Since ${\rm I}_3$ is obtained from ${\rm
I}_2$ by exchanging $u$ and $v$, we get
${\rm I}_3={\rm I}_{3,1}+{\rm I}_{3,2}$,
with
${\rm I}_{3,1}$,
${\rm I}_{3,2}$ as in (8.21), with $u$
replaced by $v$ and $\widetilde{v}_\nu $ by
$\widetilde{u}_\nu $.
\par After an integration by parts and
application of (8.18), we get
\ekv{8.22}
{
{\rm II}={\rm II}_1+{\rm II}_2,
}
\ekv{8.23}
{\cases{
{\rm II}_1=h\int\sum_{j,\nu }\Psi
_j(e^{-\phi /2h}\partial _{x_j}u)Z_\nu
^*\widetilde{v}_\nu dx,\cr
{\rm II}_2=h\int\sum_{j,\nu }\Psi
_j(e^{-\phi /2h}\partial _{x_j}v)Z_\nu
^*\widetilde{u}_\nu dx. }}
We observe that ${\rm II}_1$ and ${\rm
II}_2$ differ only by a permutation of $u$
and $v$ and their related quantities. By
integration by parts, we get
\ekv{8.24}
{
{\rm II}_1={\rm II}_{1,1}+{\rm II}_{1,2},
}
\ekv{8.25}
{\cases{
{\rm II}_{1,1}=h^{3\over 2}\int\sum_{j,\nu
}(\partial _{x_\nu }\Psi _j)(e^{-\phi
/2h}\partial _{x_j}u)\widetilde{v}_\nu
dx,\cr
{\rm II}_{1,2}=h^{3\over 2}\int\sum_{j,\nu
}\Psi _j(e^{-\phi
/2h}\partial _{x_\nu }\partial
_{x_j}u)\widetilde{v}_\nu dx.
}}
Similarly, we have ${\rm II}_2={\rm
II}_{2,1}+{\rm II}_{2,2}$, where ${\rm
II}_{2,i}$ is obtained from ${\rm
II}_{1,i}$, by replacing $u$ by $v$ and
$\widetilde{v}_\nu $ by $\widetilde{u}_\nu
$.
\par Next consider ${\rm III}$ in (8.15).
Using (8.18) and an integration by parts,
we get
\ekv{8.26}
{
{\rm III}={\rm III}_1+{\rm III}_2,
}
\ekv{8.27}
{\cases{
{\rm III}_1=h^{1\over 2}\int\sum_\nu
\widetilde{D}_\nu (\partial _{x_\nu
}u)e^{-\phi /2h}(v-\langle v\rangle )dx,\cr
{\rm III}_2=h^{1\over 2}\int\sum_\nu
\widetilde{D}_\nu (\partial _{x_\nu
}v)e^{-\phi /2h}(u-\langle u\rangle )dx.
}}
Again the two terms are analogous.
Applying (8.18) and integrating by parts,
we get
\ekv{8.28}
{
{\rm III}_1={\rm III}_{1,1}+{\rm
III}_{1,2}, }
\ekv{8.29}
{\cases{
{\rm III}_{1,1}=h^{1\over 2}\int\sum_{\nu
,\mu }\widetilde{v}_\mu (Z_\mu
\widetilde{D}_\nu )(\partial _{x_\nu
}u)dx,\cr
{\rm III}_{1,2}=h\int\sum_{\nu ,\mu
}\widetilde{v}_\mu \widetilde{D}_\nu
(\partial _{x_\mu }\partial _{x_\nu }u)dx.
}}
Clearly ${\rm III}_2={\rm III}_{2,1}+{\rm
III}_{2,2}$, where ${\rm III}_{2,i}$ is
obtained from ${\rm III}_{1,i}$ by
replacing $u$ by $v$ and
$\widetilde{v}_\mu $ by $\widetilde{u}_\mu
$. This completes our formal calculations:
\eeekv{8.30}
{-\partial _t{\rm Cor\,}(u,v)={\rm
I}_{1}+{\rm I}_{2,1}+{\rm I}_{2,2}+{\rm
I}_{3,1}+{\rm I}_{3,2}+}
{\hskip 3cm {\rm II}_{1,1}+{\rm
II}_{1,2}+{\rm II}_{2,1}+{\rm
II}_{2,2}+}
{\hskip 4cm {\rm III}_{1,1}+{\rm
III}_{1,2}+{\rm III}_{2,1}+{\rm
III}_{2,2}.}
\par Let $W=W_\Lambda $ be a set of
positive weight functions $\rho :\Lambda
\to ]0,\infty [$, with $1\in W$ and such
that $\rho \in W\Rightarrow 1/\rho \in W$.
First of all we assume that $\phi =\phi
_t$ satisfies the assumptions of the
earlier sections uniformly w.r.t. $t$.
(Actually in the next sections we shall see
that the sets of weights we use in this
section are smaller than the
corresponding sets of weights in sections
7.) More precisely we assume that
($\widetilde{{\rm H}1}$) (section 7) holds
uniformly in $t$. We assume (H2) and we
assume that (H3) holds uniformly in $t$.
We strengthen (H4) to:
\eeeekv{\widetilde{{\rm H}4}}
{\phi _t'(x)=A_t(x)x\hbox{, where }\rho
^{-1}A_t(x)\rho \hbox{ is 2 standard
and }\rho (\ell )(\partial
_{x_\ell}A_t)\circ \rho ^{-1}, } {\rho
^{-1}\circ \rho (\ell )\partial
_{x_\ell} A_t\hbox{ are 3 standard uniformly
for }\rho
\in W,\ t\in [0,1].\hbox{ Moreover,
}A_t(x)\hbox{ has} } {\hbox{an inverse
}B_t(x)\hbox{ such that }\rho
^{-1}B_t(x)\rho ={\cal
O}(1):\ell^p\to\ell^p,\ 1\le p\le \infty
,}{\hbox{uniformly for }0\le t\le 1,\ \rho
\in W.}
It follows that $\rho ^{-1}A_t^{-1}(x)\rho
$ is 2 standard and that $\rho (\ell )(\partial
_{x_\ell}A_t^{-1})\circ \rho ^{-1}$, $\rho
^{-1}\circ \rho (\ell )\partial
_{x_\ell} A_t^{-1}$ are 3 standard,
uniformly for
$0\le t\le 1$, $\rho \in W$. The most
natural choice of $A_t$ seems to be
$A_t=\int_0^1 \phi _t''(sx)ds$. With that
choice we only have to check the statement
about the inverse of $A_t$, since the
other properties follow from the previous
assumptions on $\phi $. Let
$W_a\subset W$ be a set of weights as with
$\rho
\in W_a\Rightarrow 1/\rho \in W_a$ such that
(7.2) holds, with $\phi =\phi _t$. We let
$a$ be fixed with
\ekv{8.32}
{
00$. Then for the solution in
$\ell^2\otimes {\cal H}_1$ of the last
equation in (8.18), we have
\ekv{8.41}
{\sum_{\nu }\Vert \rho _1(\nu
)\widetilde{D}_\nu \Vert ^2+\sum_\nu
\sum_\mu \Vert \rho _1(\nu
)Z_\mu \widetilde{D}_\nu \Vert ^2\le {\cal
O}(1)h.}
\par The justification of our derivation
of (8.30) is now immediate, and next we
shall estimate the various terms that
appear in that equation.
\smallskip
\par\noindent \it Estimate of $I_1$
(see (8.17)). \rm We rewrite the integrand
in (8.17) as
\ekv{8.42}
{
e^{-\phi /h}\sum_{j,k}(\rho _0(j)\rho
_0(k)\Phi _{j,k}(x)){1\over \rho _u(j)\rho
_0(j)}(\rho _u(j)\partial _{x_j}u){1\over
\rho _v(k)\rho _0(k)}(\rho _v(k)\partial
_{x_k}v). }
In the proof of Lemma 8.1 we have seen
that $\rho _0(j)\rho _0(k)\Phi _{j,k}(x)$
is 2 standard, so the double sum in (8.42)
is
$${\cal O}(1){1\over\inf \rho _u\rho
_0}{1\over\inf \rho _v\rho _0}\vert \rho
_u(j)\partial _{x_j}u\vert _2\vert \rho
_v(k)\partial _{x_k}v\vert _2.$$
Combining this with (8.37) and the
analogous estimate for $v$, we get
\ekv{8.43}
{
{\rm I}_1={\cal O}(1)h{1\over \inf \rho
_u\rho _0}{1\over \inf \rho _v\rho _0}. }
\smallskip
\par\noindent \it Estimate of
$I_{2,1}$ (see (8.21)). \rm We write
the integrand in (8.21) as
\ekv{8.44}
{
\sum_{j,k,\nu }(\rho _0(j)\rho
_0(\nu )\partial _{x_\nu }\Phi
_{j,k}){1\over\rho _u(j)\rho
_0(j)}(e^{-\phi /2h}\rho _u(j)\partial
_{x_j}\partial _{x_k}u){1\over \rho_v(\nu
)\rho _0(\nu ) }(\rho _v(\nu
)\widetilde{v}_\nu ). }
Here we observe that $\rho _0(j)\rho
_0(\nu )\partial _{x_\nu }\Phi _{j,k}$ is
(3,$\infty $) standard. If
$a$ is (2,2) standard, then
$\vert a\vert _2={\cal O}(1)$ by Lemma B.1.
By Lemma B.2 in the same section, we know
that if $b$ is (3,$\infty$) standard, $a$ a
2-tensor and
$c$ a 1 tensor, then
$$\langle b,a\otimes c\rangle ={\cal
O}(1)\vert a\vert _2\vert c\vert _2.$$
Hence the expression (8.44) is
\ekv{8.45}
{
{\cal O}(1){1\over \inf (\rho _u\rho
_0)}e^{-\phi /2h}{\cal O}(1){1\over\inf
(\rho _v\rho _0) }\vert \rho _v(\nu
)\widetilde{v}_\nu (x)\vert _2. }
We use this and (8.40) in (8.21), and get
\ekv{8.46}
{
{\rm I}_{2,1}={\cal O}(1){h^2\over\inf (\rho
_u\rho _0)\inf (\rho _v\rho _0)}. }
\smallskip
\par\noindent \it Estimate of $I_{2,2}$
(see (8.21)). \rm Rewrite the integrand in
(8.21) as
\ekv{8.47}
{\sum_{j,k,\nu }\rho _0(j)\Phi
_{j,k}e^{-\phi /2h}{1\over \rho _0(j)\rho
_u(j)}(\rho _u(j)\partial _{x_\nu
}\partial _{x_j}\partial
_{x_k}u)\widetilde{v}_\nu =e^{-\phi
/2h}\sum_{j,k}\rho _0(j)\Phi _{j,k}B_{j,k},}
with
\ekv{8.48}
{B_{j,k}={1\over \rho _0(j)\rho
_u(j)}\sum_{\nu }(\rho _u(j)\partial
_{x_\nu }\partial _{x_j}\partial
_{x_k}u)\widetilde{v}_\nu .}
Here we recall that $\rho _u(j)\partial
_{x_\nu }\partial _{x_j}\partial
_{x_k}u$ is (3,2) standard, so if we view
$B=(B_{j,k})$ as a matrix,
\ekv{8.49}
{
\Vert B\Vert _{{\cal L}(\ell^\infty
,\ell^1)}={\cal O}(1){1\over\inf (\rho
_0\rho _u)}\vert \widetilde{v}_\nu \vert
_2. }
On the other hand $\rho _0(j)\Phi _{j,k}$
is 2 standard and hence ${\cal
O}(1):\ell^1\to \ell^1$. If we view the
last sum in (8.47) as ${\rm tr\,}((\rho
_0(j)\Phi _{j,k})\circ {^t\hskip -1pt B})$,
we conclude by the trace lemma that it is
${\cal O}(1){1\over\inf (\rho _0\rho
_u)}\vert \widetilde{v}_\nu \vert _2$.
Using this in (8.47) and then in (8.21),
we get
\ekv{8.50}
{
{\rm I}_{2,2}={\cal O}(h^{3\over 2}){1\over
\inf (\rho _0\rho _u)}\Vert \widetilde{v}\Vert
_{\ell^2\otimes L^2}={\cal
O}(h^2){1\over\inf (\rho _0\rho _u)}. }
Here we used (8.40) in the last step, with
$\rho _v$ replaced by 1. By playing with
$\rho _v$ also we could reach the estimate
$${\rm I}_{2,2}={\cal O}(h^2){1\over\inf
(\rho _0\rho _u)}{1\over\inf (\rho
_0\rho _v)},$$
provided that we add to (H6), the
assumption that
$$\sum_{j,k,\nu }\rho _u(j){\rho
_v(k)\over\rho _v(\nu )}(\partial _{x_\nu
}\partial _{x_j}\partial _{x_k}u)
t_js_kr_\nu ={\cal O}(1)\vert t\vert
_\infty \vert s\vert _\infty \vert r\vert
_2.$$
\smallskip
\par\noindent \it Estimate of $II_{1,1}$
(see (8.25)). \rm Write the integrand in
(8.25) as
$$\sum_{j,\nu }{1\over\rho _0(j)\rho
_u(j)}(\rho _0(j)\partial _{x_\nu }\Psi
_j)(e^{-\phi /2h}\rho _u(j)\partial
_{x_j}u)\widetilde{v}_\nu ={{\cal
O}(1)\over\inf (\rho _0\rho _u)}e^{-\phi
/2h}\vert \widetilde{v}_\nu \vert _2.$$
Here we used that $\rho _0(j)\partial
_{x_\nu }\Psi _j$ is 2 standard by Lemma
8.1, and that $\rho _u(j)\partial _{x_j}u$
is (1,2) standard by (H6). Inserting this
into (8.25) and using (8.40) with $\rho
_v$ replaced by 1, we get
\ekv{8.51}
{
{\rm II}_{1,1}={\cal O}(1){h^2\over\inf
(\rho _0\rho _u)}. }
\smallskip
\par\noindent \it Estimate of $II_{1,2}$
(see (8.25)). \rm Write the integrand in
(8.25) as
\ekv{8.52}
{
e^{-\phi /2h}\sum_{j,\nu }(\rho _0(j)\Psi
_j){1\over \rho _0(j)\rho _u(j)}(\rho
_u(j)\partial _{x_\nu} \partial
_{x_j}u)\widetilde{v}_\nu ={\cal
O}(1)e^{-\phi /2h}{1\over\inf (\rho _0\rho
_u)}\vert \widetilde{v}_\nu \vert _2, }
where we used Lemma 8.1 and (H6). Using
this with (8.40) in (8.25), we get
\ekv{8.53}
{
{\rm II}_{1,2}={\cal O}(1){h^2\over\inf
(\rho _0\rho _u)}. }
\smallskip
\par\noindent \it Estimate of $III_{1,1}$
(see (8.29)). \rm We write the integrand
in (8.29) as
\ekv{8.54}
{
\sum_{\nu ,\mu }\widetilde{v}_\mu (\rho
_1(\nu )Z_\mu \widetilde{D}_\nu ){1\over
\rho _1(\nu )\rho _u(\nu )}(\rho _u(\nu
)\partial _{x_\nu }u)={{\cal
O}(1)\over\inf (\rho _1\rho _u)}\vert
\widetilde{v}\vert _2\vert \rho _1(\nu
)Z_\mu \widetilde{D}_\nu \vert
_{\ell^2\otimes \ell^2}, }
where we used (H6). Using this in the
expression for ${\rm III}_{1,1}$ together
with (8.40), (8.41), we get
\ekv{8.55}
{
{\rm III}_{1,1}={\cal O}(1){h^{3\over
2}\over\inf (\rho _1\rho _u)}. }
\smallskip
\par\noindent \it Estimate of $III_{1,2}$
(see (8.29)). \rm Write the integrand as
\ekv{8.56}
{
\sum_{\nu ,\mu }\widetilde{v}_\mu (\rho
_1(\nu )\widetilde{D}_\nu ){1\over \rho
_1(\nu )\rho _u(\nu )}(\rho _u(\nu
)\partial _{x_\mu }\partial _{x_\nu
}u)={\cal O}(1){1\over\inf (\rho _1\rho
_u)}\vert \widetilde{v}_\mu \vert _2\vert
\rho _1(\nu )\widetilde{D}_\nu \vert _2, }
since $\vert \rho _u(\nu )\partial _{x_\mu
}\partial _{x_\nu }u\vert _{\ell^2\otimes
\ell^2}={\cal O}(1)$ by (H6) and Lemma B.1.
Using this in (8.29) with (8.40), (8.41), we
get
\ekv{8.57}
{{\rm III}_{1,2}={\cal
O}(1){h^2\over\inf (\rho _1\rho _u)}.}
\par Recall that for $X={\rm I},{\rm
II},{\rm III}$ and $i=1,2$, we get
$X_{2,i}$ from $X_{1,i}$ by exchanging
$u$ and $v$ as well as their associated
quantities. This means that we get the
estimates for $X_{2,i}$ from those for
$X_{1,i}$, by exchanging $u$ and $v$ to the
right, and we therefore obtain estimates
for all terms in (8.30). Summing up, we
have
\medskip
\par\noindent \bf Proposition 8.4. \it Let
$\phi _t(x)=\phi _t(x;h)$, $0\le t\le 1$,
$x\in{\bf R}^\Lambda $ be
$C^1$ in
$t$ and smooth in $x$, of the form
(8.1H), satisfying (8.2H). Let $W$ be a
set of weights $\rho :\Lambda \to ]0,\infty
[$, with $\rho \in W\Rightarrow 1/\rho \in
W$, $1\in W$. Assume that $\phi =\phi _t$
satisfies ($\widetilde{H1}$) (section 7),
(H2), (H3) (section 1),
($\widetilde{H4}$) and (H5) of this
section uniformly with respect to $t\in
[0,1]$. Here $W_a$ is defined prior to
(H5) with some fixed $a$ as in (8.32). Let
$u,v\in C^\infty ({\bf R}^\Lambda ;{\bf
R})$ be independent of $t$ and satisfy
(H6). Finally choose $\rho _0\ge \rho
_1\in W_a$ such that (H7) holds. (Cf Lemma
8.1.) Then
\ekv{8.58}
{\partial _t{\rm Cor}_{\phi _t}(u,v)={\cal
O}(1)({h\over \inf (\rho _u\rho _0)\inf
(\rho _v\rho _0)}+{h^{3\over 2}\over\inf
(\rho _1\rho _u)}+{h^{3\over 2}\over \inf
(\rho _1\rho _v)}).}\rm\medskip
\par The estimate (8.58) could certainly
be improved to the price of some further
assumptions. Also notice that the
assumptions of standardness could be
weakened, since we only use derivatives up
to some fixed finite order.
%\vfill\eject
\bigskip
\centerline{\bf 9. Asymptotics of the
correlations.}
\medskip
\par This section is divided into three
parts. In part A we make only the
assumptions of section 1 and consider the
correlation of two functions with
(1,2)-standard gradients, which are
independent of $h$. We show that it has an
asymptotic expansion in powers of $h$, and
that this expansion is valid uniformly with
respect to
$\Lambda $. In part B we let $\Lambda
=({\bf Z}/L{\bf Z})^d$ with $L\in \{
2,3,..\}$. Adding assumptions on $\phi
''(0)$; an assumption of translation
invariance, as well as the assumption
($\widetilde{{\rm H}1}$) of section 7 for a
suitable family of weights, we study the
asymptotics of ${\rm Cor\,}(x_\nu ,x_\mu
)$ for $\nu ,\mu \in \Lambda $, when $1\ll
{\rm dist\,}(\nu ,\mu )\ll L$ and obtain
the product of an exponentially decaying
factor and a factor with power behaviour,
in the limit $\nu -\mu \to \infty $. The
exponent in the exponential factor is
positively homogeneous of degree $1$ in
$\nu -\mu $ and we show that it has an
asymptotic expansion in powers of $h$. We
obtain a similar result for the power
factor. In this result all terms in the
asymptotic expansions, may depend on
$\Lambda $ but they remain bounded and the
asymptotic expansions are valid uniformly in
$\Lambda $. In part C we make some
additional assumptions that allow us to
pass to the thermodynamical limit. This
part contains the final result of the
paper, and the results here remain valid
also with
$\Lambda $ equal to a finite subset
of ${\bf Z}^d$ which contains a large ball
centered at $0$. (In section 10 we derive
simplified sets of assumptions in order to
reach the formulation of the main result
in the introduction.) Throughout the whole
section we make the assumptions (H1--4) of
section 1 and let the functions $\phi $ be
normalized as in (8.2H).
\smallskip
\par\noindent $\underline{{\rm A.}}$ In this
part we only assume that $u$, $v$ are
functions on
${\bf R}^\Lambda $ independent of $h$,
such that $\nabla u$, $\nabla v$ are (1,2)
standard (as defined in section 8). We are
interested in the asymptotics of
\ekv{9.1}
{
{\rm Cor\,}(u,v)=(e^{-\phi /2h}(u-\langle
u\rangle )\vert e^{-\phi /2h}(v-\langle
v\rangle ))=h({\Delta _\phi
^{(1)}}^{-1}e^{-\phi /2h}du\vert e^{-\phi
/2h}dv), }
as $h\to 0$. The second equality was
established in this explicit form in [S1],
but already effectively used in earlier
work by Helffer and the author [HS]. Under
the present assumptions the derivation is
very simple: Let $\widetilde{u}\in{\cal S}$
solve (8.18). After an integration by
parts, we get ${\rm Cor\,}(u,v)=(d_\phi
\widetilde{u}\vert d_\phi (e^{-\phi
/2h}v))$. In view of (8.38), we have
$d_\phi \widetilde{u}=(\Delta _\phi
^{(1)})^{-1}d_\phi u$, which gives the last
expression in (9.1).
\par We apply Proposition A.1, which
extends to ($P$,2) standard tensors, and
write
\ekv{9.2}
{
du\,e^{-\phi /2h}=\sum_{n=0}^{N+1}\sum_{\nu
=0}^M\sum_{j\in\Lambda ^n}h^{{1\over
2}n+\nu }Z^*_j(\widetilde{u}^n_{j,\nu
}e^{-\phi /2h}), }
where $\widetilde{u}^n_{j,\nu }$ is
($1+j$,2) standard, and for $\nu 0\hbox{
for } j\in K,\hbox{ and }{\rm Gr}(K)={\bf
Z}^d, }
where ${\rm Gr}(K)$ denotes the smallest
subgroup of ${\bf Z}^d$ which contains
$K$. Put
\ekv{9.14}
{
F_{\widetilde{v}_0}(\eta )=\sum_{k\in {\bf
Z}^d}e^{k\cdot \eta }\widetilde{v}_0(k),\
\eta \in{\bf R}^d, }
where we know ([S1]) that
\ekv{9.15}
{\{ \eta \in {\bf R}^d;\,
F_{\widetilde{v}_0}(\eta )<\infty \}}
is convex and that $F_{\widetilde{v}_0}$ is
a convex function, which is smooth on the
interior of the set (9.15). Assume
\eekv{{\rm H}11}
{
\exists \hbox{ an open convex even set
}\widetilde{\Omega }\subset{\bf
R}^d,\hbox{ independent of }\Lambda
,\hbox{ such that }}
{F_{\widetilde{v}_0}(\eta
) \hbox{ is uniformly bounded on every
compact subset of }\widetilde{\Omega }. }
Here we define even sets to be the ones
which are symmetric around 0. Then from [S1]
(based on the fact that $K$ is contained in
no hyperplane of
${\bf R}^d$) we know that $F_{\widetilde{v}_0}$ is
strictly convex:
\ekv{9.16}
{
\nabla ^2F_{\widetilde{v}_0}(\eta )\ge
{1\over{\cal O}(1)},\ \eta
\in\widetilde{\Omega }. }
(We even have that $\log
F_{\widetilde{v}_0}$ is strictly convex.)
\par Let $\Omega
\subset\subset\widetilde{\Omega }$ be
an open even convex set, which is
independent of $\Lambda $ and assume that
\ekv{9.17{\rm H}}
{\liminf_{\Omega \ni \eta \to\partial
\Omega }F_{\widetilde{v}_0}(\eta )\ge
1+3\epsilon _0,}
where $\epsilon _0>0$ is independent of
$\Lambda $. $F_{\widetilde{v}_0}(\eta )$ is
then uniformly bounded in $\Omega $
and its derivatives are uniformly bounded
on every fixed relatively compact subset
of $\Omega $. Using also (9.16), it is
clear that the sets
\ekv{9.18}
{
\Omega _b:=\{ \eta \in\Omega ;
F_{\widetilde{v}_0}**0$
arbitrarily large, provided that we choose
$\epsilon _1$ small enough. Combining this
with (9.20), we see as in [BJS] (and [S1],
[SW])
\ekv{9.21}
{\sum_{x\in{\bf
Z}^d}e^{\widetilde{r}(x)}\widetilde{v}_0(x)\le
1+\epsilon _0.}
\par Let
$r\in C^{1,1}(({\bf R}/L{\bf Z})^d)$ be
real and assume that $\nabla r(x)\in\Omega
_{1+\epsilon _0/2}$, $\vert \nabla
^2r\vert \le \epsilon _1$ everywhere,
where $\epsilon _1$ is small enough. Then
\ekv{9.22}
{
\sum_{x\in\Lambda
}e^{r(x)}v_0(x)=\sum_{x\in{\bf
Z}^d}e^{r\circ \pi _\Lambda
(x)}\widetilde{v}_0(x)\le 1+\epsilon _0, }
since $\widetilde{r}:=r\circ \pi _\Lambda
$ satisfies the earlier assumptions.
\par If $r'(x)$ is merely Lipschitz on
$({\bf R}/L{\bf Z})^d$ with $r'(0)=0$, and
$\nabla r'(x)\in\Omega _{1+\epsilon _0/2}$
a.e., then by regularization,we can find
$r$ with the above properties, such that
$r-r'={\cal O}_{\epsilon _0,\epsilon
_1}(1)$.
\par Let $a=-\epsilon _0$, and let $W=W_a$
consist of all weights $\rho
(x)=e^{r(x)}$, $x\in ({\bf R}/L{\bf
Z})^d$, for which $\nabla r\in\Omega
_{1+\epsilon _0/2}$, and $\vert \nabla
^2r\vert
\le \epsilon _1$, with $\epsilon _1>0$
sufficiently small. Using Shur's lemma and
(9.22) (with $r$ there replaced by
$r(x)-r(0)$) we see that for all $\rho \in
W$:
\ekv{9.23}
{
\Vert \rho (v_0*)\rho ^{-1}\Vert _{{\cal
L}(\ell^2,\ell^2)}, \Vert \rho^{-1} (v_0*)
\rho \Vert _{{\cal L}(\ell^2,\ell^2)}\,\le
1+\epsilon _0,}
so that
\ekv{9.24}
{
(\rho ^{-1}\phi ''(0)\rho u\vert u)\ge
-\epsilon _0\vert u\vert ^2,\
u\in\ell^2(\Lambda ). }
We fix $\epsilon _0$ with
\ekv{9.25}
{
0<\epsilon _0<1-\vert v_0\vert
_1-{1\over{\cal O}(1)}, }
so that
\ekv{9.26}
{-\epsilon _0>-\lambda _{\min}(\phi
''(0))+{1\over{\cal O}(1)}=-1+\vert
v_0\vert _1+{1\over{\cal O}(1)}.}
\par As for the higher derivates of $\phi
$, we will assume ($\widetilde{{\rm H}1})$
(section 7), with $W$ equal to the set of
weights defined above.
\par We apply Proposition 7.1 with $N=0$,
$z=0$, and we shall drop the superscript
(0,1) for simplicity. We recall the formula
\ekv{9.27}
{
{\Delta _\phi
^{(1)}}^{-1}=E-E_+E_{-+}^{-1}E_-, }
that we shall use in (9.1), with $u=x_\mu
$, $v=x_\nu $, $\mu ,\nu \in\Lambda $.
\par Let
us first consider the contribution from
$E$. According to Proposition 7.1, we
know that $(\rho ^{-1}\otimes 1)E(\rho
\otimes 1)={\cal O}(1):\ell^2\otimes
L^2\to \ell^2\otimes L^2$, for all $\rho
\in W$, and in view of the observation
after (9.22), we know that this remains
true for $\rho =e^r$, with $r\in{\rm
Lip\,}(({\bf R}/L{\bf Z})^d)$, $\nabla
r\in\Omega _{1+\epsilon _0/2}$ a.e. Now
recall that we have introduced the norm
$p_b$ on ${\bf R}^d$ in (9.19). Let
$d_b=d_b^\Lambda $ be the corresponding
distance on $\Lambda $, given
by
$$d_b(\nu ,\mu )=\inf_{\widetilde{\nu
}\in \pi _\Lambda ^{-1}(\nu )\atop
\widetilde{\mu }\in \pi _\Lambda ^{-1}(\mu
)}p_b(\widetilde{\nu }-\widetilde{\mu
}).$$
Then $(\rho _\mu \otimes 1)e^{-\phi
/2h}dx_\mu ={\cal O}(1)$ in $\ell^2\otimes
L^2$ with
$\rho _\mu =e^{d_{1+\epsilon _0/2}(\mu
,\cdot )}$. By the weighted boundedness
result for $E$, that we have established
above, we have $(\rho _\mu \otimes
1)E(e^{-\phi /2h}dx_\mu )={\cal O}(1)$ in
$\ell^2\otimes L^2$. It follows that (cf.
(9.1))
\ekv{9.28}
{
h(Ee^{-\phi /2h}dx_\mu \vert e^{-\phi
/2h}dx_\nu )={\cal O}(1)he^{-d_{1+\epsilon
_0/2}(\nu ,\mu )}. }
As a matter of fact, since $e^{-\phi
/2h}dx_\mu $, $e^{-\phi /2h}dx_\nu $
belong to the image of $R_-$ and $ER_-=0$,
the expression (9.28) vanishes, However
the weaker formulation in (9.28) may be of
interest for more general correlations. The
main contribution to (9.1) will come from
the second term in (9.27) and is equal to
\ekv{9.29}
{
-h(E_{-+}^{-1}E_-(e^{-\phi /2h}dx_\mu
)\vert E_-(e^{-\phi /2h}dx_\nu
))=-h(E_{-+}^{-1}\delta _\mu ,\delta _\nu
), }
since $e^{-\phi /2h}dx_\mu =R_-\delta _\mu
$ and $E_-R_-=1$. Here we are in the
presence of convolution matrices. Indeed,
from (H8) we deduce (cf. [S1]) a certain
translation invariance for
${\cal P}^{0,1}$ and its inverse, which
implies that
$E_{-+}$ and its inverse are convolutions
and
\ekv{9.30}
{
E_-(e^{-\phi /2h}dx_\mu )=\tau _\mu
E_-(e^{-\phi /2h}dx_0). }
Proposition 7.4 can be applied together
with the remark after (9.22) to show that
\ekv{9.31}
{
E_{-+}= -(1-v*),\ v\sim\sum_{\nu
=0}^{\infty }h^\nu v_\nu \hbox{ in }{\cal
L}(\rho \ell^2,\rho \ell^2), }
uniformly when $\rho =e^r$, with $r$
Lipschitz, such that $\nabla r\in\Omega
_{1+\epsilon _0/2}$ a.e., with $v_0$ as
before. Here we equip $\rho \ell^2$ with
the natural norm $\Vert \rho u\Vert _{\rho
\ell^2}=\Vert u\Vert _{\ell^2}$. This
implies that
\ekv{9.32}
{
\vert v(\ell )\vert \le {\cal
O}(1)e^{-d_{1+\epsilon _0/2}(\ell )},\
\vert v_\nu (\ell )\vert \le {\cal O}_\nu
(1)e^{-d_{1+\epsilon _0/2}(\ell )},\
\ell\in\Lambda . }
\par We have already assumed that $v_0$ has
a lift $\widetilde{v}_0$ to ${\bf Z}^d$
with certain properties including (9.13)
and we know that $\widetilde{v}_0(\ell
)={\cal O}(1)e^{-p_{1+\epsilon
_0/2}(\ell)}$. For $v_{\nu}$, $\nu \ge 1$
and $v-v_0$ we use the following lift: If
$\ell\in\Lambda $, let $A(\ell
)\subset{\bf Z}^d$ be the set of
$\widetilde{\ell }$ in $\pi _\Lambda
^{-1}(\ell )$ for which $p_{1+\epsilon
_0/2}(\widetilde{\ell})$ is minimal
($=d_{1+\epsilon _0/2}(0,\ell )$). Let
$\widetilde{\Lambda }$ be the union of all
such $A(\ell )$, and define
$\widetilde{v}_\nu (\ell )$ to be the
unique function on ${\bf Z}^d$ with
support in $\widetilde{\Lambda }$, such
that
\ekv{9.33}
{
v_\nu (\ell )=\sum_{\widetilde{\ell}\in\pi
_\Lambda ^{-1}(\ell )}\widetilde{v}_\nu
(\widetilde{\ell}), }
and such that $\widetilde{v}_\nu $ is
constant on each $A(\ell )$. Define
$\widetilde{v}-\widetilde{v}_0$ (where
$\widetilde{v}_0$ is already known) by the
same construction. This means that we have
defined $\widetilde{v}$. Then
\ekv{9.34}
{
v(\ell )=\sum_{\widetilde{\ell}\in\pi
_\Lambda ^{-1}(\ell )}\widetilde{v}(\ell
), }
\ekv{9.35}
{
\widetilde{v}\sim \sum_0^\infty h^\nu
\widetilde{v}_\nu }
in $\ell^\infty $ and even in
$e^{-p_{1+\epsilon _0/2}}\ell ^\infty $.
Let
\ekv{9.36}
{
\widehat{\widetilde{v}}(\xi
)=\sum_{\ell\in{\bf
Z}^d}\widetilde{v}(\ell )e^{-i\ell \xi },\
\xi \in ({\bf R}/2\pi {\bf Z})^d=:{\bf
T}^d, }
denote the Fourier transform of
$\widetilde{v}$. From the above asymptotic
expansions, it follows that
$\widehat{\widetilde{v}}$ extends to a
holomorphic function in ${\bf T}^d+i\Omega
_{1+\epsilon _0/2}$ which is uniformly
bounded and has a uniform asymptotic
expansion
\ekv{9.37}
{
\widehat{\widetilde{v}}(\zeta
)\sim\sum_{\nu =0}^\infty h^\nu
\widehat{\widetilde{v}}_\nu (\zeta ), }
in ${\bf T}^d+i\Omega _b$, for every fixed
$b<1+\epsilon _0/2$.
\par As in [S1], we can study the
asymptotic behaviour of
$(1-\widetilde{v}*)^{-1}$ by means of
Fourier inversion. In that paper (as well
as in [BJS]) we only knew that
$\widehat{\widetilde{v}}(\zeta
)=\widehat{\widetilde{v}}_0(\zeta )+{\cal
O}(h^{1/2})$ in ${\bf T}^d+i\Omega _b$,
while we now have the full asymptotic
expansion (9.37), but the discussion in the
above mentioned papers goes through
without any essential changes and will
give the full $h$ asymptotics. We only
recall some steps. If
\ekv{9.38}
{
\widetilde{F}*=(1-\widetilde{v}*)^{-1},
}
then
\ekv{9.39}
{
\widetilde{F}(k)={1\over (2\pi
)^d}\int_{{\bf T}^d}{e^{ik\cdot \xi
}\over 1-\widehat{\widetilde{v}}(\xi
)}d\xi . }
In section 4 of [S1], we discussed the
corresponding inverse $F_0*$ of
$(1-\widetilde{v}_0*)$,
\ekv{9.40}
{\widetilde{F}_0(k)={1\over (2\pi
)^d}\int_{{\bf T}^d}{e^{ik\cdot \xi
}\over 1-\widehat{\widetilde{v}}_0}d\xi ,}
denoted by $E(k)$ there. We observed that
$1-\widehat{\widetilde{v}}_0(\zeta )\ne 0$
for $\zeta =\xi +i\eta $, with $\eta \in
\Omega _1$, and that
$1-\widehat{\widetilde{v}}_0(\zeta )$
vanishes for $\eta \in\partial \Omega _1$
precisely for $\xi =0$. (Here is where the
full power of (H10) is used.) We do not
repeat the proof here, but simply recall
that $\widehat{\widetilde{v}}_0(i\eta
)=F_{\widetilde{v}_0}(\eta )$.
For $k\in{\bf R}^d\setminus\{ 0\}$, let
$\eta _0(k)=\eta _0(k/\vert k\vert
)\in\partial \Omega _1$ be the unique
point where the exterior normal of $\Omega
_1$ is equal to a positive multiple of
$k$. We can write
\ekv{9.41}
{
\eta _0(k)=\eta _0'(k)+p_1({k\over\vert
k\vert }){k\over \vert k\vert},\ \eta
_0'(k)\in (k)^\perp }
and because of the strict convexity, we
can represent the boundary of $\partial
\Omega _1$ in a neighborhood of $\eta
_0(k)$ as
\ekv{9.42}
{
\partial \Omega _1=\{ \eta _0'(k)+\eta
'+(p_1({k\over \vert k\vert })-g_{{k\over
\vert k\vert },0}(\eta ')){k\over \vert
k\vert };\, \eta '\in (k)^\perp\cap{\rm
neigh\,}(0)\}, }
where $g$ is a real and analytic function
vanishing to second order at $0$ and with
\ekv{9.43}
{
g''_{{k\over \vert k\vert },0}(0)>0.
}
Near $i\eta (k)$, we can view the complex
hypersurface
$1=\widehat{\widetilde{v}}_0(\zeta )$ as
the complexification of the real-analytic
hypersurface $i\partial \Omega _1$, so we
get
\ekv{9.43}
{
(1-\widehat{\widetilde{v}}_0)^{-1}(0)=\{
i\eta '_0(k)+\zeta '+i(p_1({k\over \vert
k\vert })-g_{{k\over \vert k\vert
},0}({\zeta '\over i})){k\over \vert k\vert
};\,
\zeta '\in{\rm neigh\,}(0),\, k\cdot
\zeta '=0\} . }
By contour deformation and residues, we
got in [S1]:
\eeekv{9.44}
{
\widetilde{F}_0(k)=}{{ie^{-p_1(k)}\over
(2\pi )^{d-1}}\int_{\xi '\in V\cap
(k)^\perp}{e^{\vert k\vert g_{k/\vert
k\vert ,0}(-i\xi ')}
\over -({k\over \vert k\vert }\cdot
\partial )(\widehat{\widetilde{v}}_0)(\xi
'+i\eta _0'(k)+i(p_1({k\over \vert k\vert
})-g_{{k\over \vert k\vert },0}({\xi\over i
}')){k\over \vert k\vert }) }d\xi
'}{\hskip 8cm +{\cal O}(1)e^{-p_1(k)-\delta
_0\vert k\vert }, }
where $V$ is a small real neighborhood of
$0$ in ${\bf R}^d$ and $\delta _0>0$ some
fixed constant.
\par When passing from $\widetilde{v}_0$ to
$\widetilde{v}$ very little changes. Let
$\Omega _1(h)$ be the set of $\eta
\in\Omega _{1+\epsilon _0/2}$ such that
$F_{\widetilde{v}}(\eta )\le 1$. This set
is very close to $\Omega _1$, and is
strictly convex. Again, we have
$\widehat{\widetilde{v}}(i\eta
)=F_{\widetilde{v}}(\eta )$. For $k\in
{\bf R}^d\setminus\{ 0\}$, let $\eta
(k)\sim\sum \eta _\nu ({k\over \vert
k\vert })h^\nu $ be the unique point in
$\partial \Omega _1(h)$, where the
exterior unit normal of $\Omega _1(h)$ is
equal to a positive multiple of $k$.
Write
\ekv{9.45}
{
\eta (k)=\eta '(k)+p_{1,h}({k\over \vert
k\vert }){k\over \vert k\vert },\ \eta
'(k)\in (k)^\perp, }
where $p_{1,h}(k)=\sup_{\eta \in\Omega
_1(h)}k\cdot \eta $ is the support
function of $\Omega _1(h)$. Then
\ekv{9.46}
{
p_{1,h}\sim p_1+\sum_{\nu =1}^\infty
p_1^{(\nu )}(k), }
with $p_1^{(\nu )}$ positively
homogeneous of degree 1. In a
neighborhood of $\eta (k)$, we can
represent $\partial \Omega _1(h)$ as
\ekv{9.47}
{
\partial \Omega _1(h)=\{ \eta '(k)+\eta
'+(p_{1,h}({k\over \vert k\vert
})-g_{{k\over \vert k\vert }}(\eta
')){k\over \vert k\vert };\, \eta '\in
(k)^\perp \cap{\rm neigh\,}(0)\}, }
where $g$ is real and analytic and has
the uniform asymptotic expansion
\ekv{9.48}
{
g_{k\over \vert k\vert }(\eta
')\sim\sum_{\nu =0}^\infty g_{{k\over
\vert k\vert },\nu }(\eta ')h^\nu , }
for $\eta '\in
(k)^\perp\cap\widetilde{V}$, where
$\widetilde{V}$ is a fixed complex
neighborhood of $0$.
\par Again, as in [S1], we get
\eeekv{9.49}
{
\widetilde{F}(k)=}{{ie^{-p_{1,h}(k)}\over
(2\pi )^{d-1}}\int_{\xi '\in V\cap
(k)^\perp}{e^{\vert k\vert g_{k/\vert
k\vert }({\xi '\over i};h)}
\over -({k\over \vert k\vert }\cdot
\partial )(\widehat{\widetilde{v}})(\xi
'+i\eta '(k)+i(p_{1,h}({k\over \vert
k\vert })-g_{{k\over \vert k\vert }}({\xi
'\over i})){k\over \vert k\vert }) }d\xi
'}{\hskip 8cm +{\cal
O}(1)e^{-p_{1,h}(k)-\delta _0\vert k\vert },
}
with $V$
and $\delta _0$ as in (9.44).
$g_{k\over \vert k\vert }({\xi '\over
i};h)$ vanishes to second order at $\xi
'=0$ and\break
${\rm Re\,}{\rm Hess\,}g_{{k\over \vert
k\vert }}(0;h)\le -{\rm Const.}<0$.
The method stationary phase gives the
large $k$ asymptotics of $F(k)$ uniformly
in $\Lambda $ and in $h$ (for $h\le
h_0>0$ sufficiently small), where all the
involved functions have uniform
asymptotic expansions in powers of $h$:
\ekv{9.50}
{
\widetilde{F}(k)={\cal O}(1)
e^{-(p_{1,h}(k)+\delta _0\vert k\vert
)}+e^{-p_{1,h}(k)}q(k;h), }
\ekv{9.51}
{
q(k;h)\sim\sum_{-\infty }^0 q_{-{d-1\over
2}-\nu }(k;h),\ k\to\infty , }
where $q_j(k;h)$ is smooth and positively
homogeneous of degree $j$ in $k$ and has
an asymptotic expansion,
\ekv{9.52}
{q_j(k;h)\sim\sum_{\nu =0}^\infty h^\nu
q_{j,\nu }(k),\ h\to 0 ,}
where $q_{j,\nu }$ is also positively
homogeneous of degree $j$. In [S1], the
leading term was computed:
\ekv{9.53}
{
q_{-{d-1\over 2},0}(k)={1\over (2\pi \vert
k\vert )^{{d-1\over 2}}}\,{(({k\over \vert
k\vert }\cdot \partial _\eta
)F_{\widetilde{v}_0}(\eta (k)))^{{d-3\over
2}}
\over\sqrt{
\det (\partial _\eta
'^2F_{\widetilde{v}_0})(\eta (k)) } }, }
where $\eta '$ indicates some orthonormal
coordinates in $(k)^{\perp}$.
\par The convolution operator $1-v*$ on
$\Lambda $ has the inverse $F*$, where
\ekv{9.54}
{
F(k)=\sum_{\widetilde{k}\in\pi _\Lambda
^{-1}(k)}\widetilde{F}(\widetilde{k}). }
For $\delta >0$, let $\Lambda _\delta $,
be the set of $k\in\Lambda $, such that
\smallskip
\par\noindent $\rm 1^o$ there is a unique
$\widetilde{k}(k)\in\pi _\Lambda ^{-1}(k)$,
such that
$d_1(k,0)=p_1(\widetilde{k}(k))$,
\smallskip
\par\noindent $\rm 2^o$ $p_1(\ell )\ge
(1+\delta )d_1(k,0)$, whenever
$\widetilde{k}(k)\ne \ell\in\pi _\Lambda
^{-1}(k)$.\smallskip
Fix $\delta >0$. Then from (9.50), (9.54)
we get the uniform asymptotics for
$k\in\Lambda _\delta $:
\ekv{9.55}
{
F(k)={\cal O}(1)e^{-(d_{1,h}(0,k)+\delta
_0\vert \widetilde{k}(k)\vert
)}+e^{-d_{1,h}(0,k)}q(\widetilde{k}(k);h),
}
where $d_{1,h}$ is the distance on
$\Lambda $, induced by $p_{1,h}$ and
$\delta _0>0$ a constant. We also have the
bound
\ekv{9.56}
{
F(k)={\cal O}(1) \vert k\vert ^{-{d-1\over
2}}e^{-d_{1,h}(0,k)},\ k\in\Lambda . }
Since $E_{-+}^{-1}=-F*$, (9.55), (9.56)
give us an asymptotic expansion for
$E_{-+}^{-1}(\mu ,\nu )$ for $\mu -\nu
\in\Lambda _\delta $ and a precise upper
bound for all $\mu ,\nu \in\Lambda $.
\par Before using this, it may be
instructive to study (cf. (9.30))
\ekv{9.57}
{E_-(e^{-\phi /2h}dx_0),}
having in mind also more general
correlations. Since $E_-=E_+^*$, we can
apply Proposition 7.4 and conclude that
\eekv{9.58}
{
E_-(e^{-\phi
/2h}dx_0)\equiv (C^M)^*R_+^{M,1}(e^{-\phi
/2h}dx_0)\equiv (C^M)^*(\delta _0)} {\hskip
4cm \equiv\sum_{\nu =0}^{\widetilde{M}}h^\nu
D_{0,0}^{M,\nu }(\delta _0\otimes e_0)+{\cal
O}(1)h^{M/2},}
modulo ${\cal O}(h^{M/2})$
in $e^{-d_{1+\epsilon _0/2}}\ell^2$, where
$e_0\in \ell_b^2(\Lambda ^0\cup\Lambda
^1\cup ..\cup\Lambda ^N)$ is the element
given by $1\in{\bf C}\simeq \ell^2(\Lambda
^0)$. Here
$M$ can be chosen arbitrarily large, so we
get a full asymptotic expansion
\ekv{9.59}
{
E_-(e^{-\phi /2h}dx_0)\sim\sum_{\nu
=0}^\infty h^\nu f_\nu \hbox{ in
}e^{-d_{1+\epsilon _0/2}}\ell^2, }
where we also know that $f_0=\delta _0$.
\par Now we combine this with (9.31),
(9.27), (9.28), (9.29), (9.55), (9.56) and
get
\medskip
\par\noindent \bf Proposition 9.2. \it
Assume (H1--4) of section 1, (H8--11) of
this section, (9.17H), and
($\widetilde{{\rm H}1}$) of section 7 for
the set of weights $\rho =e^{r(x)}$ with
$\nabla r(x)\in\Omega _{1+\epsilon _0/2}$
a.e. (discussed after (9.22)). Then
\ekv{9.60}
{{\rm Cor\,}(x_\nu ,x_\mu )={\cal
O}(h){\rm dist\,}(\nu ,\mu )^{-{d-1\over
2}}e^{-d_{1,h}(\nu ,\mu )},\ \nu ,\mu
\in\Lambda ,}
\ekv{9.61}
{{\rm Cor\,}(x_\nu ,x_\mu )={\cal O}(h)
e^{-(d_{1,h}(\nu ,\mu )+\delta _0{\rm
dist\,}(\nu ,\mu ))}+he^{-d_{1,h}(\nu
,\mu )}q(\widetilde{k}(\nu -\mu );h),\ \nu
,\mu \in\Lambda _\delta
, }
where $q$, $\widetilde{k}$, $d_{1,h}$
have been defined above (cf. (9.50),
(9.55)).\rm
\medskip
\par\noindent $\underline{\rm C.}$ Let
$U_j$, $V_j$ be increasing sequences of
bounded subsets of ${\bf Z}^d$ with
$U_j\subset V_j$, $U_j\nearrow {\bf Z}^d$,
$j\to \infty $. Let $\rho _0=\rho
_{0,j}:{\bf Z}^d\to ]0,\infty [$ be the
weight
\ekv{9.62}
{
\rho _{0,j}(\nu )=\exp (\theta \,{\rm
dist\,}(\nu ,{\bf Z}^d\setminus U_j)), }
for some fixed (possibly small) $\theta
>0$, and where ${\rm dist}$ denotes the
standard Euclidean distance on ${\bf Z}^d$.
Let
$\phi =\phi _j\in C^\infty ({\bf
R}^{V_j};{\bf R})$ satisfy the assumptions
of section 1 (with $V_j=\Lambda $) and
assume
\eekv{{\rm H}12}
{
\hbox{If }k>j,\hbox{ then }(\rho _0\otimes
\rho _0)(\phi _j\oplus\psi _{k,j}-\phi
_k)''\hbox{ is 2 standard,} }
{
\hbox{if }\psi _{k,j}\hbox{ is defined on
}{\bf R}^{V_k\setminus V_j}\hbox{ with
}\psi _{k,j}''\hbox{ 2 standard.} }
Notice that this condition is independent
of the choice $\psi _{k,j}$, and we could
for instance just take zero.
\par We also assume that $\phi =\phi _j$
satisfies ($\widetilde{{\rm H}1}$)
(section 7), ($\widetilde{{\rm H}4}$)
(section 8) with
\ekv{9.63}
{W=\{ \rho =e^r;\, \vert r(\nu )-r(\mu
)\vert \le \theta \vert \nu -\mu \vert \}
,\ \vert \cdot \vert =\vert \cdot \vert
_{\ell^2},} so that $\rho _0\in W$. Let
\ekv{9.64}
{
\rho _1(\nu )=\rho _0(\nu ) e^{-\theta
\vert \nu \vert /4},}
\ekv{9.65}
{S=S_j=\{ \nu \in U_j;\, \vert \nu \vert
\le {\rm dist\,}(\nu ,{\bf Z}^d\setminus
U_j)\},\ r_j={\rm dist\,}(0,{\bf
Z}^d\setminus U_j),}
so that
\ekv{9.66}
{
\rho _1(\nu )\ge e^{\theta r_j/4},\ \nu
\in S_j. }
Put
$$\phi _{j,k,t}=t\phi _k+(1-t)(\phi
_j\oplus \psi _{k,j}),\ 0\le t\le 1,$$
with $\psi _{k,j}(x)=\sum_{\nu \in
V_k\setminus V_j}x_\nu ^2$, where we drop
the normalization constant (cf. (8.1H))
for simplicity. Assume that $\phi
_{j,k,t}$ satisfies ($\widetilde{{\rm
H}4}$) with $W$ given in (9.63). Then we can
apply Proposition 8.4 and obtain for
$\nu ,\mu \in S_j$:
\ekv{9.67}
{{\rm Cor\,}_{\phi _k}(x_\nu ,x_\mu )-{\rm
Cor\,}_{\phi _j}(x_\nu ,x_\mu )={\cal
O}(1)h e^{-\theta r_j/4}.}
Here we also used that
$${\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu
)={\rm Cor\,}_{\phi _j\oplus\psi
_{k,j}}(x_\nu ,x_\mu ).$$
\par Let $\Lambda =\Lambda _j=({\bf
Z}/L_j{\bf Z})^d$ be a sequence of
discrete tori with $L_j\nearrow \infty $
large enough so that there exists a natural
embedding
\ekv{9.68}
{
V_j\subset\Lambda _j.
}
We can view $\rho _0=\rho _{0,j}$ as a
function on $\Lambda _j$, with $\rho _0=1$
outside $V_j$. Assume that
$\widetilde{\phi }_j\in C^\infty ({\bf
R}^{\Lambda _j};{\bf R})$ is a family
which satisfies the assumptions of
subsection B with a new set of weights $W$
that contains $\rho _0$, such that
\eekv{9.69}
{
(\rho _0\otimes \rho _0)(\phi _j\oplus\psi
_j-\widetilde{\phi }_j)''\hbox{ is 2
standard, if } }
{
\psi _j\in C^\infty ({\bf R}^{\Lambda
_j\setminus V_j};{\bf R})\hbox{ and }\psi
_j''\hbox{ is 2 standard.} }
Here $\rho _0$ is defined as in (9.62)
with ${\bf Z}^d$ replaced by $\Lambda _j$.
We also assume that
$t\widetilde{\phi }_j+(1-t)(\phi _j\oplus \psi _j)$, $0\le
t\le 1$, satisfies ($\widetilde{{\rm H}4}$)
of section 8. Similarly to (9.67), we get
\ekv{9.70}
{
{\rm Cor\,}_{\widetilde{\phi }_j}(x_\nu
,x_\mu ) -{\rm Cor\,}_{\phi _j}(x_\nu
,x_\mu )={\cal O}(1) h e^{-\theta r_j/4},\
\nu ,\mu \in S_j. }
On the other hand we can apply Proposition
9.2 to ${\rm Cor\,}_{\widetilde{\phi
}_j}(x_\nu ,x_\mu )$ and get
\ekv{9.71}
{{\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu
)={\cal O}(h)e^{-\theta r_j/4}+h
e^{-p_{1,h}^j(\nu -\mu )}q^j(\nu -\mu
;h),\ \nu ,\mu \in S_j, }
with $p_{1,h}^j$, $q_j$ as in subsection B.
\par (9.67) gives a thermodynamical limit
of the correlations, while (9.71)
describes their asymptotic behaviour. We
now combine the two results, in order to
show that $p_{1,h}^j$ has a limit when
$j\to \infty $ and that the same thing
holds for the terms in the large $\nu $
asymptotic expansion of $q^j$, as well as
for the terms in the $h$ asymptotic
expansions of these quantities. Let $k\ge
j\gg 1$ and take $\mu =0$. For every
sufficiently large
$C_0\ge 1$, there is a $C_1>0$, such that
$$\vert e^{-p_{1,h}^j(\nu )}q^j(\nu
;h)-e^{-p_{1,h}^k(\nu )}q^k(\nu ;h)\vert
\le {\cal O}(1) e^{-(p_{1,h}^j(\nu
)+r_j/C_1)},$$
for $r_j/C_0^2\le \vert \nu \vert \le
r_j/C_0$. This implies that with a new
constant $C_1>0$:
\ekv{9.72}
{\vert 1-e^{p_{1,h}^j(\nu )-p_{1,h}^k(\nu
)}
{q^k(\nu ;h)\over q^j(\nu ;h)}\vert \le
{\cal O}(1)e^{-r_j/C_1},\ {r_j\over
C_0^2}\le \vert \nu \vert \le {r_j\over
C_0}. }
\par Here it will be convenient to write
\ekv{9.73}
{q^j(\nu ;h)=\vert \nu \vert ^{-{d-1\over
2}}e^{-s^j(\nu ;h)},\ \vert \nu \vert \ge
C_0,} where
\ekv{9.74}
{
s^j(\nu ;h)\sim\sum_{-\infty }^0
s^j_\alpha (\nu ;h),\ \vert \nu \vert
\to\infty , }
uniformly with respect to $h,j$,
with $s_\alpha ^j$ positively homogeneous
of degree $-\alpha $ in $\nu $, and
\ekv{9.75}
{
s_\alpha ^j\sim\sum_{\beta =0}^\infty
s_{\alpha ,\beta }^j(\nu )h^\beta ,\ h\to
0, } with $s_{\alpha ,\beta }^j$ also
positively homogeneous of degree $-\alpha
$ in $\nu $. All these functions are
smooth and uniformly bounded in the
appropriate spaces when $j$ varies. From
(9.72), we deduce that
$$p_{1,h}^j(\nu )-p_{1,h}^k(\nu )+s^j(\nu
;h)-s^k(\nu ;h)={\cal O}(1) e^{-r_j/C_1},\
{r_j\over C_0^2}\le \vert \nu \vert
\le{r_j\over C_0}.$$
Using (9.74), we get
\ekv{9.76}
{p_{1,h}^j(\nu )-p_{1,h}^k(\nu
)+\sum_{-N}^0 (s_\alpha ^j(\nu
;h)-s_\alpha (\nu ;h))={\cal
O}(r_j^{-(N+1)}),\
{r_j\over C_0^2}\le \vert \nu \vert
\le{r_j\over C_0},}
where we recall that $C_0$ can be chosen
arbitrarily large. For $m\in {\bf R}$,
$a:{\bf Z}^d\setminus\{ 0\}\to{\bf R}$,
put
\ekv{9.77}
{
(D_ma)(\nu )=a(\nu )-2^{-m}a(2\nu ).
}
If $a$ is the restriction of a function on
${\bf R}^d\setminus\{0\}$ which is
positively homogeneous of degree $n\in
{\bf R}$, then $D_ma=(1-2^{n-m})a$, and we
observe that the prefactor $1-2^{n-m}$
vanishes precisely for $n=m$. If $-N\le
n\le 1$, we apply
$$\prod_{m\in\{ -N,..,1\}\setminus\{
n\}}D_m$$
to (9.76) and conclude that
\ekv{9.78}
{p^j_{1,h}(\nu )-p_{1,h}^k(\nu )={\cal
O}(r_j^{-(N+1)})\ \ (n=1)}
\ekv{9.79}
{
s_\alpha ^j(\nu ;h)-s_\alpha ^k(\nu
;h)={\cal O}(r_j^{-(N+1)}),\ \ (n=\alpha
) }
for $-N\le \alpha \le 0$, ${r_j\over
2C_N}\le \vert \nu \vert \le {r_j\over
C_N}$, $ N\in{\bf Z}^d$, for some $C_N\in
[C_0,C_0^2/2]$.
\medskip
\par\noindent \bf Lemma 9.3. \it Let
$\widetilde{\Omega }\subset\subset\Omega
\subset{\bf R}^d$ be open, $N_0\in\{
1,2,..\} $, ${\cal E}\subset ]0,1]$,
$0\in\overline{{\cal E}}$. Let
$u=u_\epsilon (x)\in C^\infty (\Omega )$,
$\epsilon \in{\cal E}$, and assume that
\ekv{9.80}
{\vert \partial _x^\alpha u_\epsilon
(x)\vert \le C_\alpha ,\ x\in\Omega
,\,\epsilon \in{\cal E},}
\ekv{9.81}
{\vert u_\epsilon (x)\vert \le \epsilon
^{N_0},\ x\in\epsilon {\bf Z}^d\cap\Omega
,}
where $C_\alpha $ is independent of
$\epsilon $. Then,
\ekv{9.82}
{\vert \partial _x^\alpha u_\epsilon
(x)\vert \le \widetilde{C}_\alpha \epsilon
^{N_0-\vert \alpha \vert },\
x\in\widetilde{\Omega },\, \vert \alpha
\vert \le N_0.}\rm\medskip
\par\noindent \bf Proof. \rm For $1\le
j\le d$, let $e_j$ be the $j$th unit
vector in ${\bf R}^d$ and put
$$D_{j,\epsilon }u(x)={u(x+\epsilon
e_j)-u(x)\over \epsilon }=\int_0^1
(\partial _{x_j}u)(x+t\epsilon e_j)dt.$$
Then
\eekv{9.83}
{
D_{j_1,\epsilon }..D_{j_k,\epsilon
}u(x)=\int_0^1..\int_0^1(\partial
_{x_{j_1}}..\partial
_{x_{j_k}}u)(x+\epsilon
t_1e_1+..+\epsilon t_ke_{j_k})dt_1..dt_k }
{
=\partial
_{x_{j_1}}..\partial
_{x_{j_k}}u(x)+{\cal O}_k(\epsilon
)\sup_{\vert y-x\vert _\infty \le
k\epsilon }\max_{\vert \alpha \vert
=k+1}\vert \partial ^\alpha u(y)\vert . }
Let $\widetilde{\Omega
}=:\Omega _{N_0}\subset\subset\Omega
_{N_0-1}\subset\subset
...\subset\subset\Omega _1\subset\subset
\Omega $ and choose $\epsilon >0$ small
enough. We first see that
\ekv{9.84}
{D_{j_1,\epsilon }..D_{j_k,\epsilon
}u(x)={\cal O}(\epsilon ^{N_0-k}),\
x\in\Omega _1\cap\epsilon {\bf Z}^d,\ k\le
N_0-1,} then using also (9.83), that
$\partial ^\alpha u={\cal O}(\epsilon )$,
$\vert \alpha \vert \le N_0-1$. Using
(9.83) again, we see that
$$\partial ^\alpha u={\cal O}(\epsilon
^2),\ \vert \alpha \vert \le
N_0-2,\, x\in\Omega _2.$$
Iterating this argument, we get (9.82).
\hfill{$\#$}
\medskip
\par We apply the lemma to (9.78), (9.79)
with
$\epsilon ={1\over r_j}$, after the
change of variables $\nu =r_j\mu $. Using
also the homogeneity, and that $N$ can be
chosen arbitrarily large, we conclude that
\ekv{9.85}
{
\partial ^\beta
(p^j_{1,h}-p^k_{1,h})={\cal O}_{N,\beta
}(r_j^{-N})\vert \nu \vert ^{1-\vert \beta
\vert },\ \nu
\in{\bf R}^d\setminus\{ 0\} , }
\ekv{9.86}
{
\partial ^\beta (s_\alpha ^j-s_\alpha
^k)={\cal O}_{N,\beta ,\alpha
}(r_j^{-N})\vert \nu \vert ^{\alpha -\vert
\beta \vert },\ \nu \in{\bf
R}^d\setminus\{ 0\} ,}
for all multiindices $\beta $, when $k\ge
1$.
\par From (9.85) we conclude that there
exists $p_{1,h }^\infty (\nu
)\in C^\infty ({\bf R}^d\setminus\{ 0\} )$
positively homogeneous of degree 1 in $\nu
$, and uniformly bounded in $C^\infty $,
when $h$ varies, such that
\ekv{9.87}
{
\partial ^\beta (p_{1,h}^\infty
-p_{1,h}^j)={\cal O}_{N,\beta
}(r_j^{-N})\vert \nu \vert ^{1-\vert \beta
\vert },\ \nu \in{\bf R}^d, }
for every multiindex $\beta $.
\par Consider the truncated asymptotic
expansion of $p_{1,h}^j$:
\ekv{9.88}
{
p_{1,h}^j(\nu )=\sum_{\ell
=0}^{M-1}p_{1,\ell}^j(\nu ) h^\ell
+R_M^j(\nu ;h), }
with
\ekv{9.89}
{
\partial ^\beta R_M^j={\cal O}_{M,\beta
}(h^M)\vert \nu \vert ^{1-\vert \beta \vert
,} }
uniformly in $j$. Since all functions
will be homogeneous for a while, we
restrict the attention to a spherical
shell in $\nu $, and drop the obvious
powers of $\vert \nu \vert $. Using (9.85),
we see that
$$p_{1,0}^j-p^k_{1,0}={\cal
O}(r_j^{-N}+h),$$
for all $N$, implying
\ekv{9.90}
{
p_{1,0}^j-p_{1,0}^k={\cal O}(r_j^{-N}).
}
Now we can use (9.85,88) once more, to see
that
\ekv{9.91}
{
p_{1,1}^j-p_{1,1}^k={\cal
O}({r_j^{-2N}\over h}+h), }
for every $N$. Choose $h=r_j^{-N}$, to get
\ekv{9.92}
{
p_{1,1}^j-p_{1,1}^k={\cal O}(r_j^{-N}).
}
Continuing this way, we get
\ekv{9.93}
{
p_{1,\ell}^j-p_{1,\ell}^k={\cal O}_\ell
(r_j^{-N}), }
for every $N$, and the same estimates hold
for $\partial ^\beta
(p_{1,\ell}^j-p_{1,\ell}^k)$. Using this
and (9.85) in (9.89), we get
\ekv{9.94}
{
R_M^j-R_M^k={\cal O}(r_j^{-N}),\ \forall N.
}
On the other hand, $R_M^j-R_M^k={\cal
O}(h^M)$, by (9.89), so interpolation with
(9.94) gives $R_M^j-R_M^k={\cal
O}(h^{M-1}r_j^{-N})$ for every $N\ge 0$.
Use this estimate with $M$ replaced by
$M+1$, as well as (9.93) in the identity
\ekv{9.95}
{
R_M^j-R_M^k=(p_{1,M}^j-p_{1,M}^k)h^M+
(R_{M+1}^j-R_{M+1}^k). }
We get
\ekv{9.96}
{R_M^j-R_M^k={\cal O}(h^M r_j^{-N})\vert
\nu \vert ,\ \nu \in{\bf R}^d\setminus\{
0\},}
and the analogous estimate holds for the
$\beta $th derivative, with $\vert \nu
\vert $ replaced by $\vert \nu \vert
^{1-\vert \beta \vert }$. From (9.93), we
get the existence of $p_{1,\ell}^\infty
\in C^\infty ({\bf R}^d\setminus\{ 0\} )$,
such that
\ekv{9.97}
{
\vert p_{1,\ell}^j-p_{1,\ell}^\infty \vert
={\cal O}_{\ell ,N}(r_j^{-N})\vert \nu
\vert , }
and similarly for the derivatives.
Similarly
\ekv{9.98}
{
\vert R_M^j-R_M^\infty \vert ={\cal
O}_{M,N}(h^M r_j^{-N})\vert \nu \vert . }
We conclude that
\ekv{9.99}
{
p_{1,h}^\infty (\nu )\sim\sum_{\ell
=0}^\infty p_{1,\ell}^\infty (\nu )h^\ell .
}
\par The same arguments apply to
\ekv{9.100}
{
s_\alpha ^j\sim\sum_{\beta =0}^\infty
s_{\alpha ,\beta }^j(\nu ) h^\beta , }
and we get
\ekv{9.101}
{
s_\alpha ^j-s_\alpha ^\infty ={\cal
O}_{N,\alpha }(r_j^{-N})\vert \nu \vert
^{-\alpha }, }
\ekv{9.102}
{
s_{\alpha ,\beta }^j-s_{\alpha ,\beta
}^\infty ={\cal O}_{N,\beta ,\alpha
}(r_j^{-N})\vert \nu \vert ^{-\alpha }, }
\ekv{9.103}
{s_\alpha ^\infty \sim\sum_{\beta
=0}^\infty s_{\alpha ,\beta }^\infty (\nu
)h^\beta .}
\par We combine (9.67), (9.71) to get for
$j\le k$:
\ekv{9.104}
{\vert e^{-p_{1,h}^j(\nu )}q^j(\nu
;h)-e^{-p_{1,h}^k(\nu )}q^k(\nu ;h)\vert
\le {\cal O}(1) e^{-\theta r_j/4},\ \vert
\nu \vert \le {r_j\over C_0}.}
We know that $p_{1,h}^j(\nu )$ is
uniformly of the order of $\vert \nu \vert
$, so after multiplying with
$e^{p_{1,h}^j(\nu )}$, we get (possibly
with a new $C_0$):
\ekv{9.105}
{
\vert q^j(\nu ;h)-q^k(\nu
;h)+(1-e^{p_{1,h}^j(\nu )-p_{1,h}^k(\nu
)})q^k(\nu ;h)\vert \le {\cal O}(1)
e^{-r_j/C_0}, \vert \nu \vert \le
{r_j\over C_0}. }
Here $q^k={\cal O}((\vert \nu \vert
+1)^{-{d-1\over 2}})$ uniformly with
respect to $k$ and we have (9.85), so
$$(1-e^{p_{1,h}^j-p_{1,h}^k})q^k={\cal
O}(r_j^{-N}),\ \vert \nu \vert \le
{r_j\over C_0}.$$
Using this in (9.105), we get
\ekv{9.106}
{
\vert q^j(\nu ;h)-q^k(\nu ;h)\vert \le
{\cal O}(1) r_j^{-N},\ \vert \nu \vert \le
{r_j\over C_0}, }
for every $N$. We conclude (cf. (9.87))
that there exists a function $q^\infty
(\nu ;h)$, $\nu \in{\bf Z}^d$, $00.}
If we had been working on a finite
dimensional space, then (A.1,2) would have
been a necessary and sufficient condition
for the existence of a smooth realvalued
function $\phi $ with $\phi ''_{j,k}=\Phi
_{j,k}$. Such a function does not
in general exist in the infinite
dimensional case, but we shall now see how
to produce two different finite dimensional
versions of such a function.
\par Let $U\subset{\bf Z}^d$ be finite.
If $x\in{\bf R}^U$, let
$\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be
the zero extension of $x$, so that
$\widetilde{x}(j)=x(j)$ for $j\in U$,
$\widetilde{x}(j)=0$, for $j\in{\bf
Z}^d\setminus U$. Then $$\Phi
_{U;j,k}(x):=\Phi _{j,k}(\widetilde{x}),\
j,k\in U$$
is a smooth tensor on ${\bf R}^U$ which
satisfies (A.1,2) with $j,k,\ell\in U$.
Hence there exists a function $\phi
_U(x)\in C^\infty ({\bf R}^U;{\bf R})$ with
\ekv{10.1}
{\phi ''_{U;j,k}(x)=\Phi _{U;j,k}(x),\
x\in{\bf R}^U,\, j,k\in U.}
We make $\phi _U$ unique up to a constant,
by requiring that
\ekv{10.2}
{\phi '_U(0)=0.}
It is obvious that $\phi ''_U$ is
2-standard, so we have (H1) (section 1)
with $\Lambda $ replaced by $U$, and
(H2,3) hold. In order to have (H4) of
section 1, we introduce the $2$ standard
matrix
\ekv{10.3}
{A(x)=\int_0^1 \Phi (tx) dt,}
and assume
\eekv{{\rm A}.5}
{A(x):\ell^p\to \ell^p\hbox{ has an
inverse
}B(x):\ell^p\to\ell^p,}{\hbox{which is
uniformly bounded for }x\in{\bf R}^{{\bf
Z}^d},\, 1\le p\le \infty .}
Since $A$ is $2$ standard, we see that
$B(x)$ is 2 standard.
\par With $U$ as above, we take $x\in{\bf
R}^U$ and let as before $\widetilde{x}$
denote the $0$ extension of $x$ to ${\bf
R}^{{\bf Z}^d}$. Then we can introduce the
2 standard matrix
\ekv{10.4}
{A_U(x)=\int_0^1 \phi ''_U(tx)dt=r_U
A(\widetilde{x})r_U^*,}
where $r_U:{\bf R}^{{\bf Z}^d}\to {\bf
R}^U$ is the restriction map. We assume in
addition to (A.5), that
\eekv{{\rm A}.6}
{A_U(x)\hbox{ has an inverse }B_U(x)
\hbox{ which}}{\hbox{is uniformly bounded
for }x\in{\bf R}^U,\, 1\le p\le \infty
, } uniformly for all $U$ in some class of
finite $U$ under consideration. With these
assumptions we have obtained smooth
functions $\phi _U\in C^\infty ({\bf
R}^U)$ which satisfy (H.1--4) for $U$ in
some class of finite subsets of ${\bf
Z}^d$.
\par We next do the same with $U$ replaced
by a discrete torus $\Lambda =({\bf
Z}/L{\bf Z})^d$. If $\lambda \in {\bf
Z}^d$, we define $\tau _\lambda x\in{\bf
R}^{{\bf Z}^d}$, by $(\tau _\lambda x)(\nu
)=x(\nu -\lambda )$. Eventually, we will
assume complete translation invariance for
$\Phi $:
\ekv{{\rm A}.7}
{\Phi _{j+\lambda ,k+\lambda }(\tau
_\lambda x)=\Phi _{j,k}(x),\
j,k,\lambda \in{\bf Z}^d.}
Notice that if $\Phi _{j,k}$ were the
Hessian of a smooth function on $\phi \in
C^\infty ({\bf R}^{{\bf Z}^d})$ (and the
discussion remains valid if we replace
${\bf Z}^d$ by a discrete torus $\Lambda
$) then (A.7) would be a consequence of the
simpler translation invariance property:
\ekv{10.4}
{\phi (\tau _\lambda x)=\phi (x).}
However, to begin with, we only assume
the weaker assumption
$$\Phi _{j+\lambda ,k+\lambda
}(\tau _\lambda x)=\Phi _{j,k}(x),\
j,k\in{\bf Z}^d,\, \lambda \in L{\bf
Z}^d,\eqno ({\rm A}.7)_L$$
for some given $L\in\{ 1,2,..\}$.
\par If $x\in{\bf R}^\Lambda $, let
$\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be
the corresponding $L{\bf Z}^d$ periodic
lift. Replacing $x$ by $\widetilde{x}$ in
$({\rm A}.7)_L$, we get
\ekv{10.5}
{\Phi _{j-\lambda ,k-\lambda
}(\widetilde{x})=\Phi
_{j,k}(\widetilde{x}),\ \lambda \in L{\bf
Z}^d.}
If we view $\Phi $ as a matrix, this is
equivalent to
\ekv{10.6}
{\tau _\lambda \circ \Phi
(\widetilde{x})=\Phi (\widetilde{x})\circ
\tau _\lambda ,\ \lambda \in L{\bf Z}^d,}
so $\Phi (\widetilde{x})$ maps $L{\bf
Z}^d$ periodic vectors into the same kind
of vectors. Hence there is a $\Lambda
\times \Lambda $ matrix
$\Phi _\Lambda (x)$, defined by
\ekv{10.7}
{
\widetilde{\Phi _\Lambda (x)t}=\Phi
(\widetilde{x})\widetilde{t}, }
where again the tilde indicates that we
take the periodic lift. On the matrix
level, we get
\ekv{10.8}
{
\Phi _{\Lambda
;j,k}(x)=\sum_{\widetilde{k}\in\pi
_\Lambda ^{-1}(k)}\Phi
_{\widetilde{j},\widetilde{k}}(
\widetilde{x}), }
for any $\widetilde{j}\in \pi _\Lambda
^{-1}(j)$, where $\pi _\Lambda :{\bf
Z}^d\to\Lambda $ is the natural
projection. Alternatively, we have
\ekv{10.9}
{\Phi _{\Lambda
;j,k}(x)=\sum_{\widetilde{j}\in\pi
_\Lambda ^{-1}(j)}\Phi
_{\widetilde{j},\widetilde{k}}(
\widetilde{x}),\ \widetilde{k}\in \pi
_\Lambda ^{-1}(k),}
and $\Phi _{\Lambda ;j,k}$ is symmetric
(cf. (A.1)).
\par Let us verify the analogue
of (A.2). For $j,k,\ell \in\Lambda $ we
have
\ekv{10.10}
{
\partial _{x_\ell}\Phi _{\Lambda
;j,k}(x)=\partial
_{x_\ell}\sum_{\widetilde{k}\in\pi
_\Lambda ^{-1}(k)}\Phi
_{\widetilde{j},\widetilde{k}}
(\widetilde{x})=\sum_{\widetilde{k}\in\pi
_\Lambda
^{-1}(k)}\sum_{\widetilde{\ell}\in\pi
_\Lambda ^{-1}(\ell )}\partial
_{x_{\widetilde{\ell}}}\Phi
_{\widetilde{j},\widetilde{k}}(\widetilde{x}
). }
From (A.1,2) we know that $\partial
_{x_{\widetilde{\ell}}}\Phi
_{\widetilde{j},\widetilde{k}}
(\widetilde{x})=\partial
_{x_{\widetilde{k}}}\Phi
_{\widetilde{j},\widetilde{\ell}}
(\widetilde{x})$, so the last expression
in (10.10) is symmetric in $\ell,k$ and
we get
\ekv{10.11}
{\partial _{x_{\ell}}\Phi _{\Lambda
;j,k}(x)=\partial _{x_k}\Phi _{\Lambda
;j,\ell}(x).}
Using also the symmetry of $\Phi _{\Lambda
;j,k}$, we get the analogue of (A.2). It
is now clear that there exists $\phi
_\Lambda \in C^\infty ({\bf R}^\Lambda
;{\bf R})$, unique up to a constant,
such that
\ekv{10.12}
{\Phi _{\Lambda ;j,k}(x)=\partial
_{x_j}\partial _{x_k}\phi _\Lambda (x),\
\phi _\Lambda '(0)=0.}
\par Let us verify that $\phi
''_\Lambda $ is 2 standard. If $k\ge 2$,
$t_1,..,t_k\in {\bf C}^\Lambda $,
$x\in{\bf R}^\Lambda $, we have
\ekv{10.13}
{\langle \nabla ^k\phi _\Lambda
(x),t_1\otimes ..\otimes t_k\rangle
=\langle \nabla ^{k-2}\Phi _\Lambda
,t_1\otimes ..\otimes t_k\rangle =\langle
\nabla ^{k_2}\Phi
(\widetilde{x}),1_E\widetilde{t}_1\otimes
\widetilde{t}_2\otimes ..\otimes
\widetilde{t}_k\rangle ,}
if $E\subset{\bf Z}^d$ is a fundamental
domain for $L{\bf Z}^d$ and
$\widetilde{x}$,
$\widetilde{t}_j$ denote the periodic
lifts. Using that $\Phi $ is 2 standard,
we deduce that
\ekv{10.14}
{\langle \nabla ^k\phi _\Lambda
(x),t_1\otimes ..\otimes t_k\rangle ={\cal
O}_k(1)\vert t_1\vert _1\vert t_2\vert
_\infty ..\vert t_k\vert _\infty .}
Here the index 1, can be replaced by any
other index in $\{ 1,..,k\}$, and the RHS in
(10.14) can therefore be replaced by
\ekv{10.15}
{{\cal O}_k(1)\vert t_j\vert
_1\prod_{{1\le \nu \le k,}\atop {\nu
\ne j}}\vert t_\nu \vert _\infty .}
By complex interpolation, we get the
desired relation
\ekv{10.16}
{\langle \nabla ^k\phi _\Lambda (x),
t_1\otimes ..\otimes t_k\rangle ={\cal
O}_k(1) \vert t_1\vert _{p_1}..\vert
t_k\vert _{p_k},}
uniformly in $x$, $t_j$ and $p_j$, when
$1\le p_j\le \infty $, $1={1\over
p_1}+..+{1\over p_k}$.
\par We next check that $\phi _\Lambda $
satisfies (H.3) (section 1), so we put
$x=0$, $\widetilde{x}=0$ and omit these
quantities in the formulas. Choose a
fundamental domain $E$ and let $(\Psi
_{j,k})$ be the block matrix of $\Phi $
with respect to the decomposition
$$\ell^2({\bf Z}^d)=\oplus_{k\in{\bf
Z}^d}\ell^2(E_k),$$
where $E_k:=kL+E$. Then $\Psi _{j,k}=\Psi
_{j-k}$ by slight abuse of notation. Since
$\Phi ={\cal O}(1):\ell^p\to\ell^p$, $1\le
p\le \infty $, we know that $\Phi _{j,k}$
satisfies the (equivalent) Shur condition
$$\sup_j\sum_k \vert \Phi _{j,k}\vert ,\,
\sup_k\sum_j\vert \Phi _{j,k}\vert <\infty
,$$
and this implies that $\sum_{k}\Vert \Psi
_k\Vert <\infty $, where $\Vert \cdot
\Vert $ denotes the norm in ${\cal
L}(\ell^2(E),\ell^2(E))$. Now we can write
\ekv{10.17}
{
\langle \Phi _\Lambda t,t\rangle
=\sum_k\langle \Psi _kt,t\rangle , }
identifying $t\simeq 1_E\widetilde{t}$. We
compare this with
\eeekv{10.18}{{1\over \# B(0,R)}\sum_{\vert
j\vert ,\vert k\vert \le R}\langle \Psi
_{j-k}t,t\rangle =}
{{1\over \#
B(0,R)}\sum_{{\vert j\vert \le (1-\epsilon
)R}\atop {k\in {\bf Z}^d}}.. - {1\over
\# B(0,R)}\sum_{{\vert j\vert \le
(1-\epsilon )R}\atop{\vert k\vert >R}}..
+{1\over
\# B(0,R)}\sum_{{(1-\epsilon )R<\vert
j\vert \le R}\atop {\vert k\vert \le R}}..
}
{={\rm I}+{\rm II}+{\rm III},}
where $B(0,R):=\{ j\in{\bf Z}^d;\vert
j\vert \le R\}$. Here
\ekv{10.19}
{{\rm I}={\# B(0,(1-\epsilon )R)\over \#
B(0,R)}\sum \langle \Psi _{-k}t,t\rangle =
{\# B(0,(1-\epsilon )R)\over \#
B(0,R)}\langle \Phi _\Lambda t,t\rangle ,}
\ekv{10.20}
{
\vert {\rm II}\vert \le \sum_{\vert
\ell\vert >\epsilon R}\Vert \Psi _\ell
\Vert \vert t\vert ^2=o_{\epsilon ,t}(1),\
R\to \infty , }
\ekv{10.21}
{
\vert {\rm III}\vert \le {\#
(B(0,R)\setminus B(0,(1-\epsilon )R))\over
\# B(0,R)}\sum_k \Vert \Psi _k\Vert \vert
t\vert ^2=o_t(1),\ \epsilon \to 0. }
It follows that
\ekv{10.22}
{
\langle \Phi _\Lambda t,t\rangle
=\lim_{R\to \infty }{1\over \#
B(0,R)}\sum_{\vert j\vert ,\vert k\vert
\le R}\langle \Psi _{j-k}t,t\rangle
=\lim_{R\to\infty }\langle \Phi
\widetilde{t}_R,\widetilde{t}_R\rangle , }
where
$$\widetilde{t}_R={1\over \sqrt{\#
B(0,R)}}\sum_{\vert k\vert \le
R}1_{E_k}\widetilde{t}.$$
The sum is orthogonal in $\ell^2$, so
\ekv{10.23}
{\vert \widetilde{t}_R\vert ^2=\vert
1_E\widetilde{t}\vert ^2=\vert t\vert ^2. }
On the other hand,
by (A.4) we have
$\langle \Phi
\widetilde{t}_R,\widetilde{t}_R\rangle \ge
{1\over {\cal O}(1)}\vert
\widetilde{t}_R\vert ^2$, so from this and
(10.23,22) we get (H.3) for
$\Phi _\Lambda $ with the same constant as
in (A.4).
\par Next we verify (H.4) for $\phi
_\Lambda $. For that we notice that we can
define a gradient $\phi '(x)$ at
$x\in{\bf R}^{{\bf Z}^d}$ if $\vert x\vert
_\infty <\infty $, by
\ekv{10.24}
{\phi '(x)=\int_0^1\Phi (tx)xdt=A(x)x,}
or more explicitly by
\ekv{10.25}
{\phi _j'(x)=\sum_k \int_0^1\Phi
_{j,k}(tx)x_k dt,}
and we verify that
$$\partial _{x_\ell}\phi _j'(x)=\Phi
_{j,\ell } (x)\ \ \ (=\partial _{x_j}\phi
'_\ell (x)\, )$$
by a straight forward computation:
$$\eqalign{&\partial _{x_\ell}\phi
_j'(x)=\sum_k\int_0^1 (\partial
_{x_\ell}\Phi _{j,k})(tx)tx_k dt+\int_0^1
\Phi _{j,\ell }(tx) dt=\cr & \int_0^1\sum_k
(\partial _{x_k}\Phi _{j,\ell })(tx) tx_k
dt +\int \Phi _{j,\ell}(tx) dt=\int_0^1
(t\partial _t+1)(\Phi _{j,\ell}(tx))dt=\Phi
_{j,\ell}(x).}$$
Put $A_\Lambda (x)=\int_0^1 \Phi _\Lambda
(tx) dt$ so that
\ekv{10.26}
{\phi '_\Lambda (x)=A_\Lambda (x)x.}
The relation between $A_\Lambda (x)$ and
$A(\widetilde{x})$ is the same as between
$\Phi _\Lambda $ and $\Phi $:
\ekv{10.27}
{\widetilde{A_\Lambda
(x)t}=A(\widetilde{x})\widetilde{t}.}
Since $A(\widetilde{x})$ is uniformly
invertible in $\ell^\infty ({\bf Z}^d)$ it
has the same property on the invariant
subspace of $L{\bf Z}^d$ periodic vectors.
This means that $A_\Lambda (x):\ell^\infty
(\Lambda )\to\ell^\infty (\Lambda )$ has a
uniformly bounded inverse. Since
$A_\Lambda $ is symmetric, we have the
same property on $\ell^1(\Lambda )$ and by
interpolation on $\ell^p (\Lambda )$.
\par Assume that for every $C_0>0$
\eekv{{\rm A}.8}
{\rho ^{-1}\Phi (x)\rho \hbox{ is 2
standard, uniformly for every $\rho :{\bf
Z}^d\to ]0,\infty [$ of the }}
{\hbox{form $\rho (j)=e^{r(j)}$ with
$r:{\bf R}^d\to {\bf R}$ of Lipschitz
class with $\vert \nabla r\vert \le C_0$
a.e.}}
Let us then verify ($\widetilde{{\rm
H}.1}$) (section 7) for $\phi _U$ and $\phi
_\Lambda
$ with a suitable class of weights $W$. In
the first case we let $W=W_U$ be the set
of weights of the form ${\rho _\vert }_U$
with $\rho $
as in (A.8) for an arbitrary but fixed
$C_0>0$. Then ($\widetilde{{\rm H}.1})$
holds for $\phi _U$. In the second case,
we let $W_\Lambda $ be the set of $\rho
(j)=e^{r(j)}$ with $r:({\bf R}/L{\bf
Z})^d\to{\bf R}$ of Lipschitz class with
$\vert \nabla r\vert \le C_0$ a.e., and
again $C_0>0$ is arbitrary but fixed.
Let $\widetilde{\rho
}=e^{\widetilde{r}}={\bf Z}^d\to ]0,\infty
[$ be the corresponding periodic lift.
Then we have the analogue of (10.13):
\ekv{10.28}{\langle \nabla ^{k-2}\rho
^{-1}\Phi _\Lambda (x)\rho ,t_1\otimes
..\otimes t_k\rangle =\langle \nabla
^{k-2}\widetilde{\rho }^{-1}\Phi
(\widetilde{x})\widetilde{\rho },
1_E\widetilde{t}_1\otimes
\widetilde{t}_2\otimes ..\otimes
\widetilde{t}_k\rangle ,\ k\ge 2,}
where again the index 1 can be replaced by
any other index in $\{ 1,..,k\}$. As
before, we see that $\rho ^{-1}\Phi
_\Lambda (x)\rho $ is 2 standard uniformly
for $\rho \in W_\Lambda $, so $\phi
_\Lambda $ satisfies ($\widetilde{{\rm
H}.1}$) with $W_\Lambda $ as above, for
every fixed $C_0>0$.
\par We want to apply the discussion
of section 9, and we assume (A.1--8) from
now on. Since we now have adopted (A.7),
completely, we see that $\phi _\Lambda
(0)$ is a convolution matrix. We assume
that $\Phi (0)$ is ferromagnetic in the
sense that
\ekv{{\rm A}.9}{\Phi _{j,k}(0)\le 0,\ j\ne
k.}
Then (H.9) holds for $\phi _\Lambda $ and
we have
\ekv{10.29}
{\Phi (0)=
I-\widetilde{v}_0*,\ \ \phi_\Lambda
''(0)=(1+o(1))I _\Lambda -v_0*,\
L\to\infty }
and $v_0$ and
$\widetilde{v}_0$ are related by (9.12) and
$v_0(\nu )$ is even
$\ge 0$ and vanishes for $\nu =0$. Again
we chose the constant 1 for simplicity, by
a dilation in $h$ we can always reduce
ourselves to that case.
\par Assume that there exists a finite set
$K\subset {\bf Z}^d$ such that
\ekv{{\rm A}.10}
{\widetilde{v}_0(j)>0,\, j\in K,\ {\rm
Gr\,}(K)={\bf Z}^d,}
where ${\rm Gr\,}(K)$ denotes the smallest
subgroup of ${\bf Z}^d$ which contains
$K$. This is precisely the assumption
(H.10) for the functions $\phi _\Lambda $.
Since $\widetilde{v}_0(k)={\cal
O}_{C_0}(1)e^{-C_0\vert k\vert }$
for every $C_0>0$, the function
$F_{\widetilde{v}_0}(\eta )$ in (9.14) is
well defined on ${\bf R}^d$ and from
(A.10) and the fact that
$\widetilde{v}_0>0$, we see that
$\lim_{\vert \eta \vert \to \infty
}F_{\widetilde{v}_0}(\eta )=+\infty $. We
then have(H.11) and (9.17H) for suitable
sets
$\Omega $, $\widetilde{\Omega }$. Then
the whole discussion of part B of section
9 applies and we have Proposition 9.2 for
$\phi _\Lambda $ (when $L$ is large
enough).
\par We next look at part C of section 9,
where we shall take $V_j=U_j\nearrow {\bf
Z}^d$, $j\to \infty $ with $U_j$ bounded.
For $U\subset {\bf Z}^d$ finite, define
$\rho _0=\rho _{0,U}:{\bf Z}^d\to
]0,\infty [$ as in (9.62):
\ekv{10.30}
{
\rho _0(\nu )=\exp \theta {\rm dist\,}(\nu
,{\bf Z}^d\setminus U), }
for some fixed $\theta >0$. In order to
have (H12) (which implies (H5) in the
discussion in section 9) we need an
assumption:
\ekv{{\rm A}.11}
{\big( \rho _{0,U}(j)\rho _{0,U}(k)(\Phi
_{j,k}(1_Ux)-\Phi _{j,k}(x))\big)_{j,k\in
U}\hbox{ is 2 standard.}}
Here $1_U$ is the characteristic function
of $U$, so $(1_Ux)(j)$ is equal to $x(j)$
when $j\in U$ and equal to $0$ when
$j\in{\bf Z}^d\setminus U$. We observe
that (A.11) and (A.8) follow from (A.1--3)
and the following finite range condition:
\ekv{{\rm A.fr}}
{\exists C_0,\hbox{ such that }\Phi
_{j,k}(x)=0\hbox{ for }\vert j-k\vert
>C_0.}
In fact, if (A.fr) holds, then $\Phi
_{j_1,j_2,..,j_m}:=\partial
_{x_{j_3}}..\partial _{x_{j_m}}\Phi $
vanishes if $\vert j_\nu -j_\mu \vert
>C_0$ for some $\nu ,\mu $, and (A.3) is
equivalent to the statement that $\Phi
_{j_1,..,j_m}(x)={\cal O}_m(1)$ for
$m=2,3,..$. Similarly (A.8) is equivalent
to $${\rho (j_2)\over\rho (j_1)}\Phi
_{j_1,..,j_m}(x)={\cal O}_m(j),$$
for $\rho $ as in (A.8). Since $(\rho
(j_2)/\rho (j_1))\Phi _{j_1,..,j_m}={\cal
O}(1)\Phi _{j_1,..,j_m}$, we see that
(A.8) follows from (A.3) and (A.fr).
Moreover,
$\partial
_{x_\ell}\Phi _{j,k}=\partial _{x_j}\Phi
_{\ell ,k}=..$ vanishes if $\vert
j-\ell\vert $ or $\vert k-\ell\vert $ is
$>C_0$ and consequently the expression
(A.11) vanishes as soon as ${\rm
dist\,}(j,{\bf Z}^d\setminus U)$ or ${\rm
dist\,}(k,{\bf Z}^d\setminus U)$ is larger
than $C_0$. This means that we can replace
$\rho _{0,U}$ by some uniformly bounded
functions $\widetilde{\rho }_{0,U}$
without changing the expression in (A.11),
and (A.11) then follows from the 2
standardness of $\Phi _{j,k}(1_Ux)$ and of
$\Phi _{j,k}(x)$. More generally (A.11)
holds if we assume that there is a $C_0>0$
such that $\Phi _{j,k}(x)=\Phi _{j,k}(0)$
whenever $\vert j-k\vert >C_0$. Indeed, we
again obtain that $\partial _{\ell}\Phi
_{j,k}=0$ if $\vert k-\ell\vert >C_0$ or
$\vert j-\ell\vert >C_0$.
We add (A.11) to our assumptions from now
on, and verify (H.12). If
$U\subset\widetilde{U} \subset {\bf Z}^d$
are finite, then $(\rho _{0,U}\otimes \rho
_{0,U})(\phi _U\oplus 0-\phi
_{\widetilde{U}})''$ is equal to the tensor
\ekv{10.31}
{
\rho _0(j)\rho _0(k)(\Phi
_{j,k}(1_Ux)(1_Uj)(1_Uk)-\Phi
_{j,k}(1_{\widetilde{U}}x)),\ j,k\in
\widetilde{U}, }
and we split this tensor into four,
according to the cases $j\in U$ or not,
$k\in U$ or not. In the three cases where
at least one of $j,k$ is in
$\widetilde{U}\setminus U$, we know from
(A.8) that both $\rho _0(j)\rho _0(k)\Phi
_{j,k}(1_Ux)$ and $\rho _0(j)\rho
_0(k)\Phi _{j,k}(1_{\widetilde{U}}x)$ are
2 standard, and in the remaining case when
both $j$ and $k$ belong to $U$, the tensor
(10.31) is 2 standard by (A.11). This means
that we have verified (H.12) for $\phi
_j=\phi _{U_j}$.
\par We have already checked
($\widetilde{\rm H.1}$) (section 7) with the
set of weights $W$ above and we next look
at ($\widetilde{\rm H.4}$) (section 8),
that we need to check for the new and
smaller set of weights in (9.63), of the
form $\rho =e^r$, $\vert \nabla r\vert \le
2\theta $ a.e. Using the observation after
the statement of ($\widetilde{\rm H.4}$),
we only need to check that the inverse
$B_U(x)$ of $A_U(x)$ (which exists and is
uniformly bounded by (A.6)) remains
uniformly bounded after conjugation with a
weight $\rho $ as in (9.63), if $\theta
>0$ is sufficiently small. However, using
the Shur class remark, we see that $\Vert
\rho ^{-1}A_U\rho -A_U\Vert _{{\cal
L}(\ell^p,\ell^p)}$, $1\le p\le \infty $
is as small as we like if $\theta $ is
small enough (but independent of $U$), and
consequently $(\rho ^{-1}A_U\rho )^{-1}$
is uniformly bounded, so we have checked
($\widetilde{{\rm H}.4}$) for $\phi _U$.
\par We also need $(\widetilde{\rm H.4})$
for $\phi _{j,k,t}$ given after (9.66), so
that
\eekv{10.32}
{\hskip -1cm (\phi ''_{j,k,t}(x)_{\nu ,\mu
}=t\Phi _{\nu ,\mu
}(1_{U_k}x)+(1-t)(1_{U_j}(\nu )1_{U_j}(\mu
)\Phi _{\nu ,\mu}
(1_{U_j}x)+1_{U_k\setminus U_j}(\mu )\delta
_{\nu ,\mu }),}{\hskip 7cm\ \nu ,\mu
\in U_k,\, x\in{\bf R}^{U_k}.}
The corresponding matrix $A$ becomes
$$
A_{j,k,t}(x)=tA_{U_k}(x)+(1-t)(1_{U_j}
A_{U_j}(x)1_{U_j} +1_{U_k\setminus
U_j}),$$
and we assume in analogy with (A.6) that
\eekv{{\rm A}.12}
{A_{j,k,t}(x)\hbox{ has an inverse
}B_{j,k,t}(x):\ell^p\to \ell^p,\hbox{
which}} {\hbox{is uniformly bounded for
}x\in{\bf R}^{U_k},\, 1\le p\le \infty .}
Using the Shur class point of view, we see
as before that $(\widetilde{\rm H4})$ is
fulfilled with $W$ as in (9.63).
\medskip
\par\noindent \bf Remark 10.1. \rm Using
the maximum principle of [S4] we can get a
simple condition which implies (A.5),
(A.6), (A.12) and the similar condition
(A.13) below. Assume for $A(x)=\int_0^1 \Phi
(sx)ds$:
\eekv{{\rm A.mp}}
{\exists \epsilon _0>0\hbox{ such that
for every $x\in {\bf R}^{{\bf Z}^d}$,
$A(x)$ satisfies (${\rm
mp\,}\epsilon _0$): If }} {\hbox{$t\in
\ell^1({\bf Z}^d)$, $s\in
\ell^\infty ({\bf Z}^d)$, and
$\langle t,s\rangle =\vert t\vert _1\vert
s\vert _\infty $, then $\langle
A(x)t,s\rangle \ge \epsilon _0 \vert
t\vert _1\vert s\vert _\infty $.}}
It is easy to check (first for $p=1,\infty
$ and then by interpolation for
intermdiate values of $p$) that
$A(x):\ell^p\to\ell^p$ has a uniformly
bounded inverse $B(x)$, so (A.mp) implies
(A.5). Moreover, if $A(x)$ satifies
(mp$\epsilon _0$), so does $1_UA(x)1_U^*$
(as a $U\times U$ matrix), so we get
(A.6). Finally, the set of matrices which
satisfy (mp$\epsilon _0$) is convex, so
$A_{j,k,t}(x)$ will also satisfy
(mp$\epsilon _0$) and consequently we will
have (A.12).
\medskip
\par So if we add the assumption (A.12),
or replace (A.5,6) by (A.mp), then
($\widetilde{\rm H}4)$ holds for $\phi
_{j,k,t}$, and we get (9.67).
\par Now let $\Lambda =\Lambda _j=({\bf
Z}/L_j{\bf Z})^d$ be a sequence of
discrete tori with
\ekv{10.33}
{U_j\subset [-{L_j\over 4},{L_j\over
4}]^d,}
so that we can view $U_j$ as a subset of
$\Lambda _j$ in the natural way. Let
$\widetilde{\phi }_j=\phi _{\Lambda _j}$.
We need to check (9.69) with $\rho _0(\nu
)=\rho _{0,j}(\nu )=\exp \theta {\rm
dist\,}(\nu ,\Lambda _j\setminus U_j)$,
$\nu \in\Lambda _j$. As before, we see
that it suffices to check the 2
standardness of
\ekv{10.34}
{\rho _0(\nu )\rho _0(\mu )(\Phi _{\nu
,\mu }(1_{U_j}x)-\Phi _{\Lambda _j,\nu
,\mu }(x)),\ \nu ,\mu \in U_j.}
Recall that $U_j$ is viewed as a subset of
$\Lambda _j$ and let $\widetilde{x}$
denote the $L_j{\bf Z}^d$ periodic lift of
$x$. Write (10.34) as the difference of
the following two expressions:
\ekv{10.35}
{\rho _0(\nu )\rho _0(\mu )(\Phi _{\nu
,\mu }(1_{U_j}\widetilde{x})-\Phi _{\nu
,\mu }(\widetilde{x})),}
and
\ekv{10.36}
{\rho _0(\nu )\rho _0(\mu )\sum_{0\ne
\alpha
\in{\bf Z}^d}\Phi _{\nu ,\mu +L_j\alpha
}(\widetilde{x}).}
Thanks to (A.8), the last expression is 2
standard if we replace $\widetilde{x}$ by
a general $x\in{\bf R}^{{\bf Z}^d}$. The
same holds for (10.35) by (A.11). As with
$\Phi _\Lambda $, we then see that
(10.35), (10.36) are 2 standard, and that
completes the verification of (9.69).
\par We finally need to check that $t\phi
_{\Lambda _j}+(1-t)(\phi _{U_j}\oplus \psi
_j)$ satisfies ($\widetilde{\rm H.4}$) of
section 8, with $\psi _j(x)=\sum_{\nu \in
\Lambda _j\setminus U_j}{1\over 2}x_\nu
^2$, and as before, we see that we only
need the uniform invertibility in
${\cal L}(\ell^p,\ell^p)$ of
\ekv{10.37}
{\widetilde{A}_{j,k}=tA_{\Lambda
_j}(x)+(1-t)(1_{U_j}A_{U_j}(1_{U_j}x)
1_{U_j}+1_{\Lambda _j\setminus U_j}),\
x\in{\bf R}^{\Lambda _j}.}
Assume
\ekv{{\rm A}.13}
{\widetilde{A}_{j,t}(x):\ell^p\to\ell^p
\hbox{ is uniformly invertible as in
(A.12).}}
Fortunately (A.13) is also a consequence
of (A.mp). To see that, it suffices to
verify that with $\Lambda =\Lambda _j$,
$A_{\Lambda }(x)$ has the property
(mp$\epsilon _0$) as we shall now do: Let
$t\in\ell^1(\Lambda )$, $s\in\ell^\infty
(\Lambda )$ satisfy $\langle t,s\rangle
=\vert t\vert _1\vert s\vert _\infty $. We
have the obvious analogue of (10.22):
\ekv{10.38}
{\langle A_\Lambda t,s\rangle
=\lim_{R\to\infty }\langle
A\widetilde{t}_R,\widetilde{s}_R\rangle ,}
with $\widetilde{t}_R$, $\widetilde{s}_R$
defined as after (10.22). Here $\langle
\widetilde{t}_R,\widetilde{s}_R\rangle
_{\ell^2({\bf Z}^d)}=\langle t,s\rangle
_{\ell^2(\Lambda )}$, while
$$\vert \widetilde{t}_R\vert _1\vert
\widetilde{s}_R\vert _\infty =\vert t\vert
_1\vert s\vert _\infty =\langle t,s\rangle
=\langle
\widetilde{t}_R,\widetilde{s}_R\rangle ,$$
so (A.mp) implies that $\langle
A\widetilde{t}_R,\widetilde{s}_R\rangle
\ge \epsilon _0\vert \widetilde{t}_R\vert
_1\vert \widetilde{s}_R\vert _\infty
=\epsilon _0\vert t\vert _1\vert s\vert
_\infty .$ Hence by (10.38), $\langle
A_\Lambda t,s\rangle \ge \epsilon _0\vert
t\vert _1\vert s\vert _\infty $, and we
have checked that $A_\Lambda $ satisfies
(mp$\epsilon _0$) and hence that we have
(A.13) when (A.mp) holds.
Summing up, we have verified that the
assumptions (A.1--13) imply the
results of part C in section 9, as will
be restated in the main theorem below. We
have also seen that the more
explicit conditions (A.fr) and (A.mp)
permit to reduce the number of conditions
and to simplify them in the sense that
they only concern $\Phi _{j,k}$ and not
the particular choice of sequences $U_j$
and $\Lambda _j$. Indeed we have verified
the implications:
$$\hbox{(A.1--3), (A.fr)$\Rightarrow$
(A.8), (A.11),}$$
$$\hbox{(A.1--3), (A.mp)$\Rightarrow$
(A.5,6,12),}$$
$$\hbox{(A.1--3), (A.7), (A.mp)$\Rightarrow$
(A.13).}$$
Also notice that (A.4) follows
from (A.mp). Especially
(A.1--3,7,9,10,fr,mp) imply (A.1--13).
\medskip
\par\noindent \bf Theorem 10.2. \it Let
$\Phi _{j,k}(x)\in C^\infty ({\bf R}^{{\bf
Z}^d})$ satisfy (A.1--5,7--10) and
define \break
$\phi _U\in C^\infty ({\bf R}^U;{\bf
R})$, $\phi _\Lambda \in C^\infty ({\bf
R}^\Lambda ;{\bf R})$ as above,
when $U\subset {\bf Z}^d$ is finite and
$\Lambda =({\bf Z}/L{\bf Z})^d$ is a
discrete torus. Let $U_j\subset {\bf
Z}^d$, $j=1,2,..$ be an increasing
sequence of finite subsets with $0\in
U_1$, and assume that $r_j:={\rm
dist\,}(0,{\bf Z}^d\setminus U_j)\to
\infty $, $j\to \infty $. Choose $\Lambda
_j=({\bf Z}/L_j{\bf Z})^d$ with
$U_j=[-{L_j\over 4},{L_j\over 4}]$. Assume
also that (A.6, 11--13) hold and recall
that the assumptions (A.1--3,7,9,10),
(A.fr), (A.mp) imply (A.1--13).
Then there exist $C_0\ge 1$, $j_0\in{\bf
N}$, $\theta >0$, $h_0>0$, such that for
$j\ge j_0$, $0**