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Semi-classical analysis, thermodynamic limit, maximum principles,  
Witten-Laplacian, Schroedinger operator
---------------0310010804683
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\title{On the spectrum of semi-classical
Witten-Laplacians and Schr\"{o}dinger operators
in large dimension}
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\author{
{\sc Oliver Matte} and {\sc Jacob Schach M{\o}ller\footnote{
Supported by a Marie Curie individual fellowship
from the EU.
}} \\
Fachbereich Mathematik,
Johannes Gutenberg-Universit\"{a}t \\
55099 Mainz, Germany \\
Email: {\tt matte@mathematik.uni-mainz.de}\\
and {\tt jacob@imf.au.dk}
}
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\date{Oktober 1, 2003}
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\begin{document}




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\begin{abstract}
\noindent We investigate the low-lying spectrum
of Witten-Laplacians on forms of arbitrary degree
in the semi-classical limit and uniformly in the 
space dimension. We show that under suitable assumptions
implying that the phase function has a unique minimum
one obtains a number of bands of discrete eigenvalues
at the bottom of the spectrum.
Moreover we are able to count the number of eigenvalues in each band.
We apply our results to certain sequences of Schr\"{o}dinger
operators having strictly convex potentials and show that 
some well-known results of semi-classical analysis hold
also uniformly in the dimension.
\vspace{2mm}

\noindent{\bf Keywords:} Semi-classical analysis, 
thermodynamic limit, maximum principles,
Witten-Laplacian, Schr\"{o}dinger operator.
\end{abstract}

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


\section{Introduction and main results}
\label{intro}


\noindent In the last years the semi-classical analysis
of Witten-Laplacians uniformly in the space dimension has
attracted some attention in connection with models of classical
statistical mechanics
\cite{BJS,BaMo,BoHe2,Co,DGI,HeBook2,HeSjmaxprin,Sj3,Sj4}.
This is mainly due to an identity by Helffer and Sj\"{o}strand
\cite{HeSjmaxprin} that represents truncated two-point correlation
functions of continuous spin lattice models as matrix elements
of the inverse of a Witten-Laplacian restricted to one-forms.
Furthermore suitable Witten-Laplacians on zero-forms are
unitarily equivalent to the generators of gradient Brownian
motions whose stationary distributions are finite dimensional
Gibbs measures.
In these models the semi-classical parameter is interpreted
as the temperature. The interest in uniform properties
with
respect to large dimensions is motivated by the fact that the physical
behaviour of these models is observed in the thermodynamic
limit.
Relevant questions connected with Witten-Laplacians on zero-forms
are for instance the existence of spectral gaps, Poincar\'{e}
and logarithmic Sobolev
inequalities. In fact they are related to the decay of 
correlations \cite{BoHe2}.
The Helffer-Sj\"{o}strand representation allows one to derive 
sharp asymptotics of truncated two-point correlation functions.
The crucial feature of the Witten-Laplacian on one-forms 
entering into the analysis 
of the correlation functions is the shape of its low-lying
spectrum. For the first time this was made explicit in
\cite{Sj3} where the leading asymptotics of correlations
functions were computed.
In this work the Hamilton function of the spin system,
which replaces the Morse function in Witten's original
paper \cite{Wi}, was assumed to be strictly convex.
In a subsequent paper \cite{BJS} this condition was relaxed
allowing in general non-convex Hamilton functions having
a unique non-degenerate minimum. The spectral information
exploited in these papers is what we call the first band of eigenvalues
and the gap above.


Here we observe that, combining the techniques of
\cite{BJS} with the Hodge theory of the Witten complex,
we obtain a number $n$ of bands at the bottom of the
spectrum
of Witten-Laplacians on forms of any degree supposing the
semi-classical
parameter $h>0$ is small enough.
The width and the separation of the bands is uniform with respect
to the dimension, as is the upper bound for $h$ which depends
only on $n$.
A similar picture of the spectrum
of the Witten-Laplacians on zero
and one forms was obtained at least
implicitely in \cite{Sj4}. In this article Sj\"{o}strand calculated
the complete low-temperature asymptotics
of two-point correlations in the thermodynamical limit.
To this end  he constructed approximate
eigenforms  of Witten-Laplacians on zero
and one forms  and used  inductively higher order Grushin problems.


We remark that because of the one-well shape of the Hamilton function
there are no phase transitions in the models considered
in \cite{BJS,Sj3,Sj4}. Rather the correlations
decrease exponentially with the distance of the lattice sites.
Using a more complicated deformation of the exterior
derivative than Witten's deformation
one can, however, also analyze correlation functions
coming from Hamilton functions that may have several
local minima but only one global minimum, for 
temperatures in the single-phase region
below some possible critical value where
phase transitions might occur \cite{BaMo}.
See \cite{DoSh} for examples of phase transitions due to local
minima of the Hamilton function.

Finally we note that in fixed dimensions there are 
of course numerous well-known results on low-lying
eigenvalues of Witten-Laplacians.
Originally Witten introduced his deformed complex
on compact Riemannian manifolds, the deformation being
induced by a Morse
function \cite{Wi}. 
Among other things he indicated that the semi-classical analysis
of the Witten-Laplacian yields a proof of the Morse
inequalities. Rigorously this was performed in the
sequel in \cite{Si1} (see also \cite{CFKS})
and also in \cite{HeSj4} using WKB methods
from \cite{HeSj1}.
An alternative presentation of Witten's proof can be found in \cite{Zh}.
Especially the techniques of \cite{HeSj4} yield
very precise asymptotics of the low-lying eigenvalues
as $h$ tends to zero.
All these results are, however, obtained without any control
with respect to the dimension. 


One crucial point in Witten's proof is that the number
of exponentially small w.r.t. $\frac{1}{h}$ 
eigenvalues of the Witten-Laplacian
on $k$-forms,
counted with multiplicity and including zero eigenvalues,
is equal to the number of critical points
of the Morse function with Morse index $k$.
On the other hand the dimension of the kernel
of the Witten-Laplacian
on $k$-forms equals $k^{\textrm{th}}$
Betti number of the manifold.
In our considerations we observe that
up to some $h$-dependent form degree $l$
the Witten-Laplacian
on $k$-forms, $0\klg k\klg l$, 
is bounded from below by a constant times
$k$. This is due to the fact that our functions $\Phi_{h,\Lambda}$
which replace the Hamilton resp. Morse functions
and which we call phase functions have only one 
non-degenerate critical point, namely their minimum.


Our original aim when starting this work was to obtain
estimates on the separation of
low-lying eigenvalues
measured relative to the ground state energy
of Schr\"{o}dinger
operators in the semi-clssical limit for large dimensions.
As is well-known a Schr\"{o}dinger
operator is apart from its simple ground state energy 
level identical
to a suitable Witten-Laplacian on zero-forms.
In order to construct this Witten-Laplacian one has to replace
the Morse function in Witten's original deformation by minus
the logarithm of the strictly positive ground state 
eigenfunction of the Schr\"{o}dinger operator.
Employing maximum principles as developped in
\cite{HeSjmaxprin,Sj1,Sj2}, 
we are able to extract enough information 
on the latter in order to apply the techniques from \cite{BJS}.
The use of the maximum principles is the reason why
we restrict our attention to a class of Schr\"{o}dinger operators
having strictly
convex potentials which are in some sense small
perturbations of harmonic oscillator potentials. 
As a special case of our results on more general
Witten-Laplacians we hence obtain a band structure
of the low-lying spectrum of  Schr\"{o}dinger
operators in the semi-clssical limit uniformly
in the space dimension. The number of eigenvalues
in the $n^{\textrm{th}}$ band equals the degeneracy
of the $n^{\textrm{th}}$ energy level of a simple
isotropic harmonic oscillator. We do not 
get any information on the first 
eigenvalue by our methods.


For some classes of Schr\"{o}dinger
operators, for instance for the Kac 
model, Helffer and Sj\"{o}strand prove that the thermodynamic
limit, i.e. the limit as the dimension tends to infinity
of the first eigenvalue divided by the dimension, exists
and converges exponentially fast with respect to the 
dimension. Moreover they obtain complete semi-classical
asymptotic expansions of the first eigenvalue and its
thermodynamic limit. 
Furthermore Helffer and Sj\"{o}strand
give uniform estimates from below 
and above of the gap between the first and second
eigenvalues for some Schr\"{o}dinger
operators with convex potentials. 
In the case of double-well potentials they show
that the splitting between the first two
eigenvalues is exponentially small with
respect to the dimension for small values of
the semi-classical parameter $h$
supposing that the dimension is $\cO(h^{-N})$,
for some $N>0$.
These results are derived in
\cite{HeDemuth,HeSjseclexp1,HeSjseclexp2,Sjpotwells1,Sjpotwells2,Sjeveq,Sj1}.
A survey and further results and references are given in 
\cite{HeDemuth}.
Finally we note that, employing classical infrared estimates
from statistical physics, one can prove that the splitting
between the first two
eigenvalues in the double-well case vanishes 
for fixed $0<h\ll1$ as the dimension
tends to infinity which indicates the existence of phase
transitions in some models of quantum statistical
mechanics \cite{He1}.

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



\subsection{The main result for the
Witten-Laplacian}\label{ssecWL}


\noindent Let $\LL$ be some arbitrary family of 
finite sets.
Furthermore
let 
$\Phi_{h,\Lambda}\in C^{\infty}(\RR^{\Lambda})$,
such that 
$\exp(-\frac{1}{h}\Phi_{h,\Lambda})\in L^2(\RR^{\Lambda})$,
for each $h\in(0,1]$ and $\Lambda\in\LL$.
In the following we define
and collect some basic properties
of an associated family of
Witten-Laplacians, $\WLL$,
$h\in(0,1],\,\Lambda\in\LL$, acting in 
$\HR\equiv\HR(\Lambda):=L^2(\RR^{\Lambda})\otimes\bigwedge\CC^{\Lambda}$.
We are interested in the behaviour of the
low-lying spectrum of $\WLL$,
as the cardinality of $\Lambda\in\LL$
increases and for small values of the semi-classical
parameter $h\in(0,1]$. Our first main result is formulated 
in Theorem~\ref{mainthmWL} and is valid under
Hypothesis~\ref{hypphi} on $\Phi_{h,\Lambda}$ given below.


Let
$\desa^{k}\equiv\desa^{k}(\Lambda):=
C_0^{\infty}(\RR^{\Lambda})
\otimes\bigwedge\!\!{}^{(k)}\CC^{\Lambda}$,
$0\klg k\klg|\Lambda|,\,\Lambda\in\LL$,
denote the space of compactly supported,
smooth, complex-valued $k$-forms.
The Witten-Laplacian $\WLL$
is just the
Hodge-Laplacian associated to the
complex
\[
0\;\longrightarrow\; \desa^{0}
\;
\stackrel{d_h^{(0)}}{\longrightarrow}
\;\desa^{1}
\;
\stackrel{d_h^{(1)}}{\longrightarrow}\;
\cdots\;
\stackrel{d_h^{(|\Lambda|-1)}}{\longrightarrow}\;
\,\desa^{|\Lambda|}
\;
\stackrel{d_h^{(|\Lambda|)}}{\longrightarrow}
\;0\;.
\]
Here $d_h$ denotes Witten's deformation of the
ordinary exterior derivative $d$,
\[
d_h\;=\;h^{1/2}\,
\exp(-\,{\textstyle\frac{1}{h}}\,\Phi_{h,\Lambda})\circ d\circ 
\exp(\,{\textstyle\frac{1}{h}}\,\Phi_{h,\Lambda})\;,
\]
and $d_h^{(k)}$ its restriction to 
$\desa^{k}$.
We introduce the short-hand notation
$\Phi'_i:=\dd{{}}{x_i}\Phi_{h,\Lambda}$, 
$\Phi''_{ij}:=\frac{\partial^2}{\partial x_i\partial
  x_j}\Phi_{h,\Lambda}$
etc. and define
\begin{equation}\label{def-ZZ*}
Z_i\;:=\;
h^{1/2}\,{\textstyle\dd{{}}{x_i}}\,+\,{\textstyle\frac{1}{h^{1/2}}}\,\Phi_i'\;,
\qquad
Z_i^*\;:=\;
-\,h^{1/2}\,{\textstyle\dd{{}}{x_i}}\,+\,{\textstyle\frac{1}{h^{1/2}}}\,\Phi_i'\;,
\end{equation}
for each $i\in\Lambda$.
Then the local expressions for $d_h$ and its formal adjoint,
$\delta_h$,
read
\[
d_h\;=\;\sum\limits_{i\in\Lambda}Z_i\otimes \ak_i\;,
\qquad
\delta_h\;=\;\sum\limits_{i\in\Lambda}Z^*_i\otimes a_i\;.
\]
We are using the fermionic calculus. That is
$\ak_i$ denotes exterior multiplication with the
$i^{\textrm{th}}$ coordinate differential and
$a_i$ interior multiplication with the $i^{\textrm{th}}$
basis vector field.
They fulfill the canonical anti-commutation relations (CAR)
\[
\{a_i\,,\,\ak_j\}\;=\;\delta_{ij}\,\id\;,
\qquad
\{a_i\,,\,a_j\}\;=\;\{\ak_i\,,\,\ak_j\}\;=\;0\;,
\]    
for $i,j\in\Lambda$. In contrast the operators
$Z_i$ and $Z_i^*$ fulfill the following
canonical commutation relations (CCR),
\[
[\,Z_i,Z_j^*\,]\;=\;2\,\Phi_{ij}''\;,
\qquad
[\,Z_i,Z_j\,]\;=\;[\,Z_i^*,Z_j^*\,]\;=\;0\;,
\]
for $i,j\in\Lambda$.
Using the CAR and CCR as well as the
complex property for $d_h$ and $\delta_h$,
i.e. $d_h^2=\delta_h^2=0$,
one easily obtains
the following representation for the Witten-Laplacian
$\WLL$, which is for the present defined on 
$\desa:=\bigoplus\limits_{k=0}^{|\Lambda|}\desa^{k}$,
\begin{equation}\label{WL-local}
\WLL\;:=\;
(\,d_h+\delta_h\,)^2
\;=\;
\WL{0}\otimes\id\,+\,2\,
\sum\limits_{i,j\in\Lambda}\Phi_{ij}''\otimes \ak_ia_j\;.
\end{equation}
Here $\WL{0}$ denotes the Witten-Laplacian restricted
to zero-forms,
\[
\WL{0}\;=\;
\sum\limits_{i\in\Lambda}Z_i^*Z_i
\;=\;-h\,\Delta\,+\,{\textstyle\frac{1}{h}}\,|\nabla \Phi_{h,\Lambda}|^2
\,-\,\Delta\Phi_{h,\Lambda}\;.
\]
Accordingly $\WL{k}$ denotes the restriction of $\WLL$
to $k$-forms, for $0\klg k\klg |\Lambda|$.
Frequently we also use the notation
\[
d\Gamma(\Phi_{h,\Lambda}''(x))\;:=\;\sum\limits_{i,j\in\Lambda}\Phi_{ij}''(x)
\otimes \ak_ia_j
\]
for the second quantization of $\Phi_{h,\Lambda}''$ at 
$x\in\RR^{\Lambda}$, and
$d\Gamma^{(k)}(\Phi_{h,\Lambda}''(x))$ for its restriction
to $k$-forms.

Since $\delta_h\subset d_h^*$ is densely defined $d_h$
is closable.
A limiting argument
shows that we again have $\bar{d}_h\circ\bar{d}_h=0$.
Therefore the sequence
\begin{equation}\label{complex-bard}
0\;\stackrel{\bar{d}_h^{(-1)}\equiv0}{\longrightarrow}\;
\mathscr{D}_0
\;\stackrel{\bar{d}_h^{(0)}}{\longrightarrow}\;
\mathscr{D}_1
\;\stackrel{\bar{d}_h^{(1)}}{\longrightarrow}\;
\cdots
\;\stackrel{\bar{d}_h^{(|\Lambda|-1)}}{\longrightarrow}\;
\mathscr{D}_{|\Lambda|}
\;\stackrel{\bar{d}_h^{(|\Lambda|)}}{\longrightarrow}\;0\;,
\end{equation}
where
$\mathscr{D}_k:=\dom({\bar{d}_h^{(k)}})\subset \HR^k$,
for $0\klg k\klg|\Lambda|$,
defines a complex of closed operators between the Hilbert spaces
\[
\HR^k\;\equiv\;\HR^k(\Lambda)\;:=\;
L^2(\RR^{\Lambda})\otimes\bigwedge\!\!{}^{(k)}\CC^{\Lambda}\;,
\qquad0\klg k\klg|\Lambda|\;.
\]
Here, as well as below, 
the symbol $\dom(T)$ denotes the
domain of an operator~$T$.
The complex
(\ref{complex-bard}) is an example of a Hilbert complex 
in the sense of \cite{BrLe}.
We also use the following abbreviations:
\[
\ran_k\;:=\;\Ran \bar{d}_h^{(k-1)}\;,\qquad
\ran_k^*\;:=\;\Ran d_h^{(k)*}\;,
\qquad0\klg k\klg|\Lambda|\;.
\]
Here we note that $d_h^{(|\Lambda|)*}\equiv0$
In the proof of our main theorem the Hodge
decomposition associated to the complex (\ref{complex-bard})
will play a crucial r\^{o}le.
We will formulate all the necessary ingredients
in the next three propositions.
For related discussions we refer to
\cite{Ar,BrLe,Jo}.


\begin{proposition}(i) Each power of $d_h+\delta_h$ is
essentially selfadjoint on $\desa$. In particular
the Witten-Laplacian $\WLL$ is essentially self-adjoint
on $\desa$ and so are its restrictions $\WL{k}$
on $\desa^{k}$, for $0\klg k\klg|\Lambda|$.
The symbols $\WLL$ resp. $\WL{k}$ will in the following always
denote the respective
unique selfadjoint extensions 
acting in $\HR$ resp.
$\HR^{k}$.

\smallskip

\noindent(ii) We have 
$\dom(\WLL)=\dom(\bar{d}_hd_h^*)\cap\dom(d_h^*\bar{d}_h)$
and
\[
\WLL\;=\;\bar{d}_hd_h^*+d_h^*\bar{d}_h\;.
\]
(iii) 
The following {\em weak Hodge decompositions}
are valid:
\begin{eqnarray}\label{weak-Hodge}
\HR^k&=&\ker\WL{k}\oplus\overline{\ran_k}
\oplus\overline{\ran_k^*}\;,\qquad1\klg k\klg|\Lambda|
\\ \label{weak-Hodge-0}
\HR^0&=&\ker\WL{0}\oplus\overline{\ran_0^*}
\end{eqnarray}
\end{proposition}

\textsc{Proof:}
Part (i) is
due to a well-known result by Chernoff \cite{Ch}.


In order to prove part (ii) we first recall that
a theorem of von Neumann implies that both
$\bar{d}_hd_h^*$ and $d_h^*\bar{d}_h$ are
selfadjoint and in particular closed. Since we have
$\SL\bar{d}_hd_h^*\phi\,,\,d_h^*\bar{d}_h\psi\SR=0$,
for all 
$\phi,\psi\in\dom(\bar{d}_hd_h^*)\cap\dom(d_h^*\bar{d}_h)$,
an easy argument shows that 
$\bar{d}_hd_h^*+d_h^*\bar{d}_h$ is closed.
Since it is also symmetric and extends $(d_h+\delta_h)^2$,
it is equal to $\WLL$ because of part (i).

Finally, we note that $\WL{k}$ is identical to
the unique selfadjoint operator associated
to the closed form 
${\sf q}_k(\phi,\psi):=
\SL\bar{d}_h^{(k)}\phi\,,\,\bar{d}_h^{(k)}\psi\SR
+\SL d_h^{(k-1)*}\phi\,,\,d_h^{(k-1)*}\psi\SR$,
for $\phi,\psi\in\dom(\bar{d}_h)\cap\dom(d_h^*)$.
Here $d_h^{(-1)*}\equiv0$.
Part (iii)  now follows from \cite[Proposition~A.1]{Ar}.
\hfill$\blacksquare$

\smallskip

\begin{proposition}\label{prop-Hodge}
Let $l\in\{1,\dots,|\Lambda|\}$ and
suppose that $\WL{k}$ is strictly positive,
for $1\klg k\klg l$. Then we have:

\smallskip


\noindent(i) For $1\klg k\klg l$, the subspaces
$\ran_k$ and $\ran_k^*$ are closed and the
following {\em Hodge decompositions} are valid:
\begin{eqnarray}\label{Hodge-0}
\HR^0&=&\CC\,\exp(-\,{\textstyle\frac{1}{h}}\,\Phi_{h,\Lambda})
\oplus R_0^*\;,
\\ \label{Hodge-k}
\HR^k&=&R_{k}\oplus R_k^*\;,
\qquad1\klg k\klg l\;.
\end{eqnarray}
(ii) For $0\klg k\klg l$, there exist
unitary operators 
$U_k:\ran_k^*\rightarrow\ran_{k+1}$
having the property
\[
\WL{k}\!\!\upharpoonright_{\ran_k^*}\;=\;
U_k^*\,\WL{k+1}\!\!\upharpoonright_{\ran_{k+1}}U_k\;.
\]
Consequently
\[
\sigma(\WL{k}\!\!\upharpoonright_{\ran_k^*})
\;=\;
\sigma(\WL{k+1}\!\!\upharpoonright_{\ran_{k+1}})\;,
\qquad0\klg k\klg l\;.
\]
\end{proposition}

\textsc{Proof:} We note that
$\ker\WL{0}=\ker d_h^{(0)*}\bar{d}_h^{(0)}
=\ker\bar{d}_h^{(0)}
=\CC\,\exp(-\,{\textstyle\frac{1}{h}}\,\Phi_{h,\Lambda})$.
The remaining
assertions follow directly from
\cite[Theorem 1.3]{Jo}
together with the weak Hodge decompositions
(\ref{weak-Hodge}) and (\ref{weak-Hodge-0}).
\hfill$\blacksquare$

\smallskip


 From now on we assume that 
$\Phi_{h,\Lambda}$ fulfills Hypothesis~\ref{hypphi}
below.

\smallskip

\noindent\textbf{Convention:}{\em We call a 
constant uniform, if it neither depends on $\Lambda\in\LL$
nor $h\in(0,h_0]$, 
for some $\Lambda$-independent $h_0\in(0,1]$.
}

\smallskip






\begin{hypothesis}\label{hypphi}
(i) For each $h\in(0,1]$ and $\Lambda\in\LL$,
the phase function 
$\Phi_{h,\Lambda}\in C^{\infty}(\RR^{\Lambda})$ 
has some critical point $x_0=x_0(h,\Lambda)\in\RR^{\Lambda}$
such that 
\begin{equation}\label{est-bijk}
|\,\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\,|
\;\klg\;\sum\limits_{k\in\Lambda}
b_{ijk}(h,\Lambda)\,|\Phi_k'(x)|\;,
\end{equation}
for all $i,j\in\Lambda$ and 
$x\in\RR^{\Lambda}$. 
Here, for $\Lambda\in\LL$
and $i,j,k\in\Lambda$, the $b_{ijk}(h,\Lambda)$ 
are non-negative
numbers such that
\begin{equation}\label{BB'}
\max\limits_{i\in\Lambda}\sum\limits_{k\in\Lambda}
b_{iik}(h,\Lambda)\;\klg\;B\;,
\qquad
\max\limits_{i\in\Lambda}\sum\limits_{j,k\in\Lambda}
b_{ijk}(h,\Lambda)\;\klg\;B'\;,
\end{equation}
where $0<B,B'<\infty$ are uniform constants.


\noindent (ii) Moreover we assume that there exist 
uniform constants
$0<\muu\klg\moo<\infty$ such that
\begin{equation}\label{muu-moo}
\muu\,\id\;\klg\;2\,\Phi_{h,\Lambda}''(x_0)\;\klg\;\moo\,\id\;,
\end{equation} 
for all $h\in(0,1]$ and $\Lambda\in\LL$.
\end{hypothesis}


\noindent 
This hypothesis is taken over from
\cite{BJS} in order to apply the estimation 
techniques developed there 
in our Section~\ref{sec-form-bound}. 
It implies
that $\Phi_{h,\Lambda}$ has only one critical point,
namely its non-degenerate minimum
at $x_0$, but is 
in general not convex~\cite{BJS}. 





\begin{proposition}\label{shmuel} 
Assume that Hypothesis~\ref{hypphi}
holds. 
Then, for each $l\in\NN$, there is some
$h_0\in(0,1]$ such that $\WL{k}$ is strictly
positive, for $1\klg k\klg l$ and $h\in(0,h_0]$.
\end{proposition}

\textsc{Proof:}
This is an immediate consequence of Lemma~\ref{asymp-WL1}
below.
\hfill$\blacksquare$

\smallskip



\noindent 
If the spaces $\ran_{k}$ resp. $\ran_k^*$ 
are closed, for $0\klg k\klg l$,
we will often refer to them
as the subspaces
of closed resp. non-closed $k$-forms.
This is justified by the fact that 
in this situation
the first $l$ Betti
numbers of our Hilbert complex vanish except
for the zeroth which equals one. 
Since
$\bar{d}_h$ and $d_h^*$ both commute with $\WLL$ the
subspaces
of closed and non-closed $k$-forms
are in fact spectral subspaces of $\WL{k}$.



Before we formulate the main result
of this subsection we introduce some 
further abbreviating notation:
If $T$ is a selfadjoint operator in some Hilbert space
and if $E_J(T)$ denotes its spectral projection 
corresponding to some interval $J$, we write
\begin{equation}\label{def-DJ}
D_J(T)\;:=\;\dim\Ran E_J(T)\;.
\end{equation}

\begin{theorem}\label{mainthmWL}
Assume that Hypothesis \ref{hypphi} holds.
Let $n\in\NN$ and suppose that
\[
(n+1)\,\muu
\,-\,n\,\moo\;=:\;
\triangle_n\;>\;0\;.
\]
Then, for each $0<\eta<\,\frac{2}{5}\,\triangle_n$,
there exists some $h_0=h_0(n,\eta)>0$ 
such that, for all $h\in(0,h_0]$,
$\Lambda\in\LL$, and $0\klg k\klg \min\{n,|\Lambda|\}$,
the spectrum of $\WL{k}$ below $(n+1)\,\muu\,-\,\eta$
is purely discrete and has a band structure:
We set 
\[
I_l^{\eta}\;:=\;[\,l\,\muu-\,\eta,l\,\moo+\,\eta\,]\;.
\]
Then we have, for $k\klg l\klg n$, 
\begin{equation}\label{number-of-ev0}
D_{I_l^{\eta}}(\WL{k})
\;=\;
{|\Lambda|+l-k-1\choose l-k}{|\Lambda|\choose k}\;.
\end{equation}
All eigenforms of $\WL{k}$ corresponding to the
eigenvalues lying in $I_k^{\eta}$ are closed,
for $0\klg k\klg n$.
For $k+1\klg l\klg\min\{|\Lambda|-1,n\}$,
the number of eigenvalues among those counted in 
(\ref{number-of-ev0})
corresponding to non-closed
eigenforms
equals
\begin{equation}
\label{number-of-ev1}
D_{I_l^{\eta}}(\WL{k}\!\!\upharpoonright_{\ran_k^*})
\;=\;
\sum\limits_{\alpha=0}^{l-1-k}(-1)^{l-k-1-\alpha}
{|\Lambda|+\alpha-1 \choose \alpha}
{|\Lambda|\choose l-\alpha}\;.
\end{equation}
If $n\grg|\Lambda|$ and $|\Lambda|\klg l\klg n$, the 
last formula has to be replaced by
\begin{equation}
D_{I_l^{\eta}}(\WL{k}\!\!\upharpoonright_{\ran_k^*})
\;=\;
\label{number-of-ev2}
\sum\limits_{\alpha=0}^{|\Lambda|-1-k}(-1)^{|\Lambda|-k-1-\alpha}
{|\Lambda|+\alpha+l-1 \choose \alpha+l-|\Lambda|}
{|\Lambda|\choose |\Lambda|-\alpha}
\;.
\end{equation}
For $k=0$ the formulas
(\ref{number-of-ev1}) and (\ref{number-of-ev2})
both simplify to
\[ 
D_{I_l^{\eta}}(\WL{0}\!\!\upharpoonright_{\ran_0^*})
\;=\;{|\Lambda|+l-1\choose l}\;.
\]
Moreover, for $0\klg k klg\min\{n,|\Lambda|\}$ 
there is no spectrum of $\WL{k}$
contained in the set ($\delta_{0k}$ is Kronecker's symbol)
\[
\big(-\infty\,,\,k\,\muu\,-(1-\delta_{0k})\,\eta\,\big)\cup
\bigcup_{l=k}^n\big(\,l\,\moo\,+\,\eta\,,\,
(l+1)\,\muu\,-\,\eta\,\big)
\;.
\]
\end{theorem}

\noindent We emphasize that the width and the separation
of the bands of eigenvalues described in Theorem \ref{mainthmWL}
is uniform in the dimension $|\Lambda|$. Note also that
the number ${|\Lambda|+l-1\choose l}$ is just the degeneracy
of the $l^{\mathrm{th}}$ energy level
of an isotropic $|\Lambda|$-dimensional harmonic oscillator.


The first step in the proof of Theorem~\ref{mainthmWL}
is to show that, for small
$h\in(0,1]$, the Witten-Laplacian $\WLL$ is a small perturbation of
some simpler comparison operator $\COO$ in the 
quadratic form sense uniformly in $\Lambda\in\LL$. 
For $\WL{1}$, this is already done
in \cite{BJS}. The generalization
to the full Witten-Laplacian is straight forward.
The numbers $b_{ijk}(h,\Lambda)$ occuring in Hypothesis~\ref{hypphi}
are used to compensate for summations over $\Lambda$
in the derivation of this relative bound.
We present all this in detail in Section~\ref{sec-form-bound}
below.
In Section~\ref{proof-mainthmWL} we then prove Theorem~\ref{mainthmWL}
by combining the spectral information given by $\COO$ with
the Hodge decomposition.



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\subsection{Application to Schr\"{o}dinger operators with
strictly convex potentials}\label{ssecappl}


\noindent Now we describe an application of Theorem~\ref{mainthmWL}
to the semi-classical analysis of Schr\"{o}dinger operators
$\PP:=-h^2\Delta+V_{\Lambda}$, $h\in(0,1]$, acting in 
$\HR^0$ uniformly in $\Lambda\in\LL$.
The potentials $V_{\Lambda}\in C^{\infty}(\RR^{\Lambda})$ 
are small perturbations
of harmonic oscillator potentials in a sense made explicite
in Hypothesis \ref{hyp} below. In particular they are strictly 
convex.
More precisely there are probably $\Lambda$-dependent
constants $0<c<c'<\infty$, and $E\in\RR$, such that
$c\,x^2\klg V_{\Lambda}-E\klg c'\,x^2$. So
$\PP$ can be
realized as a selfadjoint operator on
$\dom(\PP)
=\dom(\Delta)\cap\dom(x^2)$.
The space of test functions $C_0^{\infty}(\RR^{\Lambda})$
is a domain
of essential selfadjointness \cite[Theorem X.28]{RS2}.
Furthermore, since $V_{\Lambda}(x)\rightarrow\infty$,
as $|x|\rightarrow\infty$,
we know that each
$\PP$ 
has purely discrete spetrum. This is for example
a consequence of a well-known theorem of
Persson \cite{Pe} (see also \cite[Theorem XIII.67]{RS4}).
The connection between this family of
Schr\"{o}dinger operators and the Witten-Laplacian considered
in the last subsection is based on the following
well-known observation. Under our hypothesis on $V_{\Lambda}$
the usual Perron-Frobenius argument 
\cite[Theorem XIII.47]{RS4}
implies
that the lowest eigenvalue, $\mu_0(h,\Lambda)$,
of $\PP$ is simple
and that the corresponding eigenfunction, $\phi_0$, 
can be chosen
strictly
positive. 
Hence it can be expressed as 
$\phi_0=\exp(-\frac{1}{h}\,\Phi_{h,\Lambda})$.
As suggested by our notation we use the phase $\Phi_{h,\Lambda}$
of the ground state eigenfunction 
to construct a Witten deformation as in the preceeding
subsection. 
The relation of the resulting Witten-Laplacian
to the Schr\"{o}dinger operator under
investigation is obtained through the eigenvalue equation
for 
$\exp(-\frac{1}{h}\,\Phi_{h,\Lambda})$. Since the latter function is strictly
positive we get
\begin{eqnarray}\nonumber
V_{\Lambda}\,-\,\mu_0(h,\Lambda)&=&|\nabla\Phi_{h,\Lambda}|^2\,-
\,h\,\Delta\Phi_{h,\Lambda}
\\ \label{conn-PP-WL}
\Longrightarrow
\qquad
\PP\,-\,\mu_0(h,\Lambda)&=&
h\,\WL{0}\;.
\end{eqnarray}
Consequently apart from the factor $h$
the spectrum of $\WL{0}$ is identical to the 
spectrum of the ``renormalized''
Schr\"{o}dinger operator 
$\PP\,-\,\mu_0(h,\Lambda)$.
Therefore we would also obtain a band structure
of the  low-lying spectrum of $\PP$, 
for small values of $h\in(0,1]$, uniformly in $\Lambda\in\LL$,
if we were
able to verify Hypothesis~\ref{hypphi}, for the
phase $\Phi_{h,\Lambda}$ of the ground state eigenfunction
of $\PP$. This is done in Section~\ref{sec-est-on-the-phase}
below. The numbers $b_{ijk}(h,\Lambda)$ occuring in Hypothesis~\ref{hypphi}
will essentially be given as products of exponentials
of minus the distance between the points
$i,j,k\in\Lambda$ in some metric we now introduce.
For each $\Lambda\in\LL$,
we assume that a metric 
$d_{\Lambda}:\Lambda\times\Lambda\rightarrow[0,\infty)$
is given 
such that
\begin{equation}\label{def-Cd}
\max\limits_{k\in\Lambda}\sum\limits_{l\in\Lambda}
e^{-d_{\Lambda}(k,l)}\;\klg\;C_d\;<\;\infty\;,
\end{equation}
for some uniform constant $C_d$.
This is the only condition on $d_{\Lambda}$
which is otherwise arbitrary. In fact we only
use the properties of a semi-metric
when dealing with $d_{\Lambda}$ in this article
so that the distance between distinct points may
also be zero.


The main example we have in mind
is when the sets $\Lambda$ are given as finite subsets
of $\ZZ^{\nu}$ for some $\nu\in\NN$ and the metrics 
are given as, e.g., the $\ell^{1}$-distance on 
$\ZZ^{\nu}$ supplemented by some boundary
conditions. For example we could chose periodic
boundary conditions, i.e. succesively
increasing cubes contained in
$\ZZ^{\nu}$ could be interpreted as disrete 
tori. In this case, for different $\Lambda\Subset\ZZ^{\nu}$, 
the metrics $d_{\Lambda}$ would actually
give different distances for corresponding
pairs of points.



For all $o\in\Lambda$, we denote
the $\Lambda\times\Lambda$-diagonal matrix with entries 
$e^{-\vk\,d_{\Lambda}(o,j)}\,\delta_{ij}$, 
$i,j\in\Lambda,\,\vk\in\RR$, 
by $\Theta_{\Lambda,o}^{\vk}$.
Furthermore, for $p\in[1,\infty]$, let
$\el{p}{o}{\vk}\equiv\el{p}{o}{\vk}(\Lambda)$ be
the vector space $\RR^{\Lambda}$ equipped with the
norm
\[
\|\,v\,\|_{\el{p}{o}{\vk}} \;:=\;
\|\,\Theta_{\Lambda,o}^{\vk}v\,\|_{\ell^p(\Lambda)}\;,
\qquad v\in\RR^{\Lambda}\;.
\]
We note that we preserve the notation $|\cdot|$ for the standard
Euklidean norm on~$\RR^{\Lambda}$.


Using the notation introduced above we assume the following

\begin{hypothesis}\label{hyp} For $\Lambda\in\LL$,
the potential
$V_{\Lambda}$ can be expressed as
\[
V_{\Lambda}(x)\;=\;(D_{\Lambda}x)^2\,+\,W_{\Lambda}(x)\;,
\qquad 
x\in\RR^{\Lambda}\;, 
\]
where $D_{\Lambda}$ is some constant real
diagonal $\Lambda\times\Lambda$-matrix and
$W_{\Lambda}\in C^{\infty}(\RR^{\Lambda})$.
There exist two uniform real numbers, 
$\lambda_{\min},\lambda_{\max}$, such that
\[
0\;<\;\lambda_{\min}\;\klg\; D_{\Lambda}
\;\klg\;\lambda_{\max}\;<\;\infty\;,\qquad \Lambda\in\LL\;.
\]
Moreover, there is some uniform constant
$\gamma$ such that
\begin{equation}\label{hyp1}
\|\,W_{\Lambda}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}
\;=\;
\|\,\Theta_{\Lambda,o}^{\vk}\,W_{\Lambda}''(x)\,\Theta^{-\vk}_{\Lambda,o}\,\|_{\LO(\ell^{\infty}(\Lambda))}
\;\klg\;
2\,\gamma\;\klg\;2\,\lambda_{\min}^2\;,
\end{equation}
for $\vk=0,1,2$ and
for all $\Lambda\in\LL$, $o\in\Lambda$,
$x\in\RR^{\Lambda}$.
Additionally $\gamma$ fulfills
\begin{equation}\label{def-Cgamma}
C_{\gamma}\;:=\;\frac{\gamma}{\lambda_{\min}+\sqrt{\lambda_{\min}^2-\gamma}}
\;<\;\frac{\lambda_{\min}}{C_d}\;.
\end{equation}
Furthermore the third derivative of $W_{\Lambda}$
obeys the following bound,
\begin{equation}\label{hyp2}
\|\,W_{\Lambda}^{(3)}(x)\,\|
_{\LO(\el{\infty}{o}{1},\,\LO(\el{\infty}{o}{1},\el{\infty}{o}{2}))}\;\klg\;\Gamma\;<\;\infty\;,
\end{equation}
for each $\Lambda\in\LL$, $o\in\Lambda$,
$x\in\RR^{\Lambda}$,
where $\Gamma$ is some uniform constant.
\end{hypothesis}

\noindent We remark that (\ref{hyp1}) and (\ref{hyp2}) are
implied by
\[
\max_{k\in
  \Lambda}\sum\limits_{l\in\Lambda}e^{2\,d_{\Lambda}(k,l)}\,|W_{kl}''(x)|
\;\klg\;2\,\gamma\;,\qquad
\max_{j\in\Lambda}\sum\limits_{k,l\in\Lambda}
e^{d_{\Lambda}(j,k)+d_{\Lambda}(j,l)}\,|W_{jkl}^{(3)}(x)|
\;\klg\;\Gamma\;,
\]
for each $\Lambda\in\LL$, $x\in\RR^{\Lambda}$,
which is easily verified using the triangle inequality for~$d_{\Lambda}$.
Under Hypothesis~\ref{hyp} the potentials $V_{\Lambda}$ model
a system of weakly coupled
harmonic oscillators. 
The conditions (\ref{hyp1}) and (\ref{hyp2})
allow for an unbounded w.r.t. $\Lambda$ interaction range 
but impose an exponential decay
on the interaction strength 
relative to $d_{\Lambda}(i,j)$, $i,j\in\Lambda$.
Hypothesis~\ref{hyp} implies the following
two theorems. Theorem~\ref{thm-sj1} is due to \cite{Sj2}, and
Theorem~\ref{thm-sj2} is obtained in \cite{Sj1} under the
assumption $D_{\Lambda}=\id$. 




\begin{theorem}\label{thm-sj1}
Under Hypothesis~\ref{hyp} we have 
$\Phi_{h,\Lambda}(x)=\frac{1}{2}\,x\cdot 
D_{\Lambda}x +\Psi_{h,\Lambda}(x)$
where $\Psi_{h,\Lambda}$ obeys the bound
\[
\sup\limits_{x\in\RR^{\Lambda}}
\|\,\Psi_{h,\Lambda}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}
\;\klg\; C_{\gamma}\;,
\]
for $\vk=0,1,2$,
and for all $h\in(0,1]$, $\Lambda\in\LL$,
$o\in\Lambda$.
\end{theorem}



\begin{theorem}\label{thm-sj2}
Under Hypothesis~\ref{hyp} there exists some uniform
constant $C^{(3)}>0$ such that
\[
\sup\limits_{x\in\RR^{\Lambda}}\|\Psi_{h,\Lambda}^{(3)}(x)\|
_{\LO(\el{\infty}{o}{1},
\LO(\el{\infty}{o}{1},\el{\infty}{o}{2}))}
\;\klg\;
C^{(3)}\;,
\]
for all $h\in(0,1]$, $\Lambda\in\LL$, and $o\in\Lambda$.
\end{theorem}


Theorems~\ref{thm-sj1} and~\ref{thm-sj2} 
are both derived by applying so-called
maximum principles developped
in \cite{HeSjmaxprin,Sj1,Sj2}.
To make this article self-contained we give
detailed proofs of both theorems in Section~\ref{max-prin}.
These proofs require an asymptotic analysis of the logarithm
of the first eigenfunction of the operator
$-\Delta+x^2+t\,U$, where $t\in[0,1]$, and $U$ is some
compactly supported smooth function.
This analysis has to be uniform in $t\in[0,1]$.
It is carried out in \cite{Sj1} without taking
the parameter $t$
explicitly into account.
Since one has to fill in
some more details in that paper we felt that it 
might be worthwhile to work out everything
carefully. This is done in Section~\ref{sec-harm-osc}.








Before stating the next theorem we remark further
that in the present situation we know \`{a} priori that
the associated Witten-Laplacian has 
a compact resolvent:
As a consequence of Theorem~\ref{thm-sj1} and 
(\ref{hyp1}) with $\vk=0$, we obtain
the estimate
\begin{equation}\label{susi}
0\;<\;\lambda_{\min}\,-\,C_{\gamma}
\;\klg\;
\sup\limits_{x\in\RR^{\Lambda}}\|\,\Phi_{h,\Lambda}''(x)\,\|
_{\LO(\ell^{\infty}(\Lambda))}\;\klg\;\lambda_{\max}\,+\,C_{\gamma}\;.
\end{equation}
Since $\Phi_{h,\Lambda}''(x)$ is symmetric we have the same bounds
also
in the norm on $\LO(\ell^{1}(\Lambda))$, and, by the Riesz-Thorin
interpolation theorem, also in the norm on $\LO(\ell^{2}(\Lambda))$.
For $1\klg k\klg|\Lambda|$, the second
quantization $d\Gamma^{(k)}(\Phi_{h,\Lambda}'')$ thus
defines a bounded operator on
$\HR^{k}$,
satisfying
$\|d\Gamma^{(k)}(\Phi_{h,\Lambda}'')\|\,\klg
\,k\,(\lambda_{\max}\,+\,C_{\gamma})$.
The representation (\ref{WL-local}) and the fact that
$\WL{0}\otimes\id$ has a compact resolvent for trivial
reasons imply that $\WLL$ is self-adjoint on
$\dom(\PP)\otimes\bigwedge\CC^{\Lambda}$ and that
\[
(\WLL+1)^{-1}\,=\,(\WL{0}\otimes\id+1)^{-1}
-(\WL{0}\otimes\id+1)^{-1}\,2d\Gamma(\Phi_{h,\Lambda}'')\,
(\WLL+1)^{-1}
\]
is compact. We also note that the lower bound in (\ref{susi})
with the $\ell^{\infty}$-norm replaced by the $\ell^2$-norm
shows that $\Phi_{h,\Lambda}$ is strictly convex. 
Actually, this is already
a direct consequence of Hypothesis \ref{hyp} by a classical
result of Brascamp and Lieb \cite{BrLi}.
Consequently $\WL{k}$ is strictly positive, for
$1\klg k\klg|\Lambda|$, which is evident from (\ref{WL-local}).
Therefore the Hodge decompositons (\ref{Hodge-0})
and (\ref{Hodge-k}) are valid without any restrictions
on $h$.


Our main result on the family of
Schr\"{o}dinger operators $\PP$ is
 
 

\begin{theorem}\label{main-thm}
Assume that Hypothesis \ref{hyp} holds and set
\[
\muu\;:=\;\lambda_{\min}-C_{\gamma}\;,\qquad
\moo\;:=\;\lambda_{\max}+C_{\gamma}\;.
\]
Let $n\in\NN$
and suppose that
\[
(n+1)\,\mueu
\,-\,n\,\mueo\;=:\;
\triangle_n\;>\;0\;.
\]
Then, for each $0<\eta<\,\frac{2}{5}\,\triangle_n$,
there exists some $h_0=h_0(n,\eta)>0$ such that, for all $h\in(0,h_0]$,
$\Lambda\in\LL$, and $0\klg k\klg n$, 
we have
\[ 
D_{\mu_0(h,\Lambda)+h\,[\,k\,\mueu\,-\,\eta\,,\,
k\,\mueo\,+\,\eta\,]}(\PP)
\;=\;
{|\Lambda|+k-1\choose k}
\;.
\]
Furthermore there is no spectrum of 
$\PP$
contained in the set
\[
\mu_0(h,\Lambda)\,+\,h\,\Big\{\,
\big(\,0\,,\,\mueu\,-\,\eta\,\big)
\cup
\bigcup_{k=1}^n\big(\,k\,\mueo\,+\,\eta\,,\,
(k+1)\,\mueu\,-\,\eta\,\big)\;\Big\}
\;.
\]
\end{theorem}

\noindent Again we point out that the crucial
fact in Theorem~\ref{main-thm}
is the uniformity in $\Lambda$, since, for fixed dimension,
everything is well-known.

As already indicated above, the proof of Theorem~\ref{main-thm}
given in Section~\ref{sec-est-on-the-phase} consists
of verifying the conditions of Hypothesis~\ref{hypphi}
and then applying Theorem~\ref{mainthmWL}
in view of (\ref{conn-PP-WL}).
As ingredients we need Theorem~\ref{thm-sj1}
and~\ref{thm-sj2}.



\bigskip



{\em Remark on perturbation theory.}

We note that our methods also give 
$\Lambda$-independent results in the
perturbative case. By this we mean the case where
$h=1$ is fixed and $\gamma$ tends to zero in
Hypothesis~\ref{hyp}. 
Here it is essential
that the unperturbed Schr\"{o}dinger operator, 
$-\Delta+(D_{\Lambda}x)^2$,
is a sum of non-interacting harmonic oscillators.
 

\begin{theorem}
Let $n\in\NN$
and suppose that
\[
(n+1)\,\lambda_{\max}
\,-\,n\,\lambda_{\min}\;=:\;
\triangle_n\;>\;0\;,
\qquad\eta\in(0\,,\,{\textstyle\frac{2}{5}}\,\triangle_n)\;.
\]
Then there exists some $\gamma_0=\gamma_0(n,\eta)>0$
such that, for all $W_{\Lambda}$ such that
Hypothesis~\ref{hyp}
is fulfilled, for some $\gamma\in(0,\gamma_0]$,
\begin{eqnarray*}
D_{\mu_0(1,\Lambda)+[\,k\,\lambda_{\min}\,-\,\eta\,,\,
k\,\lambda_{\max}\,+\,\eta\,]}(P_{1,\Lambda})
\;=\;
{|\Lambda|+k-1\choose k}
\;,
& &
\\
\sigma(P_{1,\Lambda})\cap
\Big\{
\mu_0(1,\Lambda)\,+\,
\big(\,k\,\lambda_{\max}+(1-\delta_{0k})\,\eta\;,\,
(k+1)\,\lambda_{\min}-\eta\,\big)\Big\}
&=&\varnothing\;,
\end{eqnarray*}
holds true, for $0\klg k\klg n$.
\end{theorem}

\textsc{Proof:} We construct the Witten-Laplacian
$\Delta_{1,\Lambda}$ associated to $P_{1,\Lambda}$
and define a selfadjoint comparison operator
$\mathscr{A}_{\Lambda}$ on 
$\dom(P_{1,\Lambda})\otimes\bigwedge\CC^{\Lambda}$ by
\[
\mathscr{A}_{\Lambda}\;:=\;
\Delta^{(0)}_{1,\Lambda}\otimes\id
\,+\,2\,d\Gamma(D_{\Lambda})\;.
\]
From Theorem~\ref{thm-sj1} we get 
$\Psi_{1,\Lambda}''\,\klg\, C_{\gamma}\id\,
\klg\,\frac{C_{\gamma}}{\lambda_{\min}}\,D_{\Lambda}$.
We recall that $C_{\gamma}=\cO(\gamma)$, as $\gamma\rightarrow0$.
With this we readily derive the following
quadratic form bound,
\[
\pm\big(
\Delta_{1,\Lambda}\,-\,\mathscr{A}_{\Lambda}
\big)
\;=\;
\pm 2\,d\Gamma(\Psi_{1,\Lambda}'')
\;\klg\;
{\textstyle\frac{C_{\gamma}}{\lambda_{\min}}}
\:\mathscr{A}_{\Lambda}\;.
\]
Using this form bound we can proceed as in the proofs
of Corollary~\ref{proj-cor} resp. Theorem~\ref{mainthmWL}
given in Section~\ref{sec-form-bound} 
resp. Section~\ref{proof-mainthmWL} to get the assertion.
\hfill$\blacksquare$

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\section{Uniform semi-classical asymptotics of~$\WLL$}
\label{sec-form-bound}


\noindent First we show that, 
in the quadratic form sense and
for sufficiently small $h\in(0,1]$, the Witten-Laplacian
$\WLL$ is a small perturbation of 
\[
\COO\;:=\;\WL{0}\otimes\id\,+\,2\,d\Gamma(\Phi_{h,\Lambda}''(x_0))\;.
\]
Since $d\Gamma(\Phi_{h,\Lambda}''(x_0))$ is bounded,
the operator $\COO$ is apparently selfadjoint
on $\dom(\WL{0})\otimes\bigwedge\CC^{\Lambda}$
and essentially selfadjoint on $\desa$.
The form bound derived in Lemma~\ref{asymp-WL1}
enables us to compare the dimensions of
spectral subspaces corresponding to $\WL{k}$ and $\CO{k}$,
the restriction of $\COO$ to $\HR^k$,
for small $h\in(0,1]$. This is done in a succeeding
corollary.


In the following  
$N:=d\Gamma(\id)=\sum\limits_{i\in\Lambda}\ak_ia_i$
denotes the number operator on the exterior algebra 
(or the fermionic Fock space) $\bigwedge\CC^{\Lambda}$.
The form domain of $\WLL$ is denoted by 
$\fdom(\WLL)=\dom(\bar{d}_h)\cap\dom(d_h^*)$.

\begin{lemma}[\cite{BJS}]\label{asymp-WL1}
Assume that Hypothesis \ref{hypphi} holds
and let $h'\in(0,1]$ be strictly smaller than
the uniform constant $B$
appearing in (\ref{BB'}).
Then there exists some uniform constant $C=C(h')>0$
such that, for all $h\in(0,h']$ and $\Lambda\in\LL$,
we have
\begin{eqnarray}
\pm\big(\,\WLL-\COO\,\big)
&=&\pm 2\,
d\Gamma(\Phi_{h,\Lambda}''(\cdot)-\Phi_{h,\Lambda}''(x_0))\nonumber
\\
&\klg&\label{Donald}
C\,h^{1/2}\,\big(\,
\WL{0}\otimes N\,+\,2\,d\Gamma(\Phi_{h,\Lambda}''(x_0))\,\big)\;,
\end{eqnarray}
on $\fdom(\WLL)$ in the sense of quadratic forms.
In particular we have
\begin{equation}
\pm\big(\,\WL{k}-\CO{k}\,\big)\;\klg\;
C\,k\,h^{1/2}\,\CO{k}\;,\qquad\textrm{for}\;\;1\klg k\klg|\Lambda|\;.
\end{equation}
\end{lemma}

\textsc{Proof:}
We note that, as a domain of essential selfadjointness,
$\desa$ is also a form core for both $\WLL$ and $\COO$.
To begin with we have, for 
$v\in \desa$,
\begin{eqnarray}
\lefteqn{  \nonumber
\Big|\Big\langle
v\,,\sum\limits_{i,j\in\Lambda}
\big(\Phi_{ij}''-\Phi_{ij}''(x_0)\big)
\otimes \ak_ia_j\,v \Big\rangle  \Big|
}
\\ \nonumber
&\klg&
\int\limits_{\RR^{\Lambda}}
\sum\limits_{i,j\in\Lambda}\big|\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\big|
\,\|a_iv(x)\|\,\|a_jv(x)\|\,dx
\\ \label{WL1-est1}
&\klg&
\int\limits_{\RR^{\Lambda}}
\sum\limits_{i,j\in\Lambda}\big|\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\big|
\,\|a_iv(x)\|^2\,dx\;,
\end{eqnarray}
by Cauchy-Schwarz. Now (\ref{est-bijk}) and (\ref{BB'}) imply,
for each $i\in\Lambda$
and $x\in\RR^{\Lambda}$,
\begin{eqnarray}
\nonumber
\sum\limits_{j\in\Lambda}\big|\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\big|
&\klg&
\sum\limits_{j,k\in\Lambda}
b_{ijk}(h,\Lambda)\,|\Phi_k'(x)|
\\ \label{WL1-est2}
&\klg&
\frac{1}{2}\:
\Big(
B'\,h^{1/2}\,+\,h^{-1/2}\sum\limits_{j,k\in\Lambda}
b_{ijk}(h,\Lambda)\,|\Phi_k'(x)|^2\,
\Big)
\end{eqnarray}
Furthermore we have, for each $k\in\Lambda$,
in the sense of quadratic forms in $L^2(\RR^{\Lambda})$,
\begin{eqnarray}
\nonumber
|\Phi_k'(x)|^2
&\klg&
|\Phi_k'(x)|^2\,-\,h\,\Phi_{kk}''(x)
\,+\,\,h\,\Phi_{kk}''(x_0)
\,+\,
h\,|\Phi_{kk}''(x)-\Phi_{kk}''(x_0)\big|
\\ \nonumber%\label{WL1-est3}
&\klg&
h\,Z_k^*Z_k\,+\,\,h\,\Phi_{kk}''(x_0)
\,+\,h\,\sum\limits_{l\in\Lambda}b_{kkl}(h,\Lambda)\,|\Phi_l'(x)|
\\ \nonumber
&\klg&
h\,Z_k^*Z_k\,+\,\,\frac{h\,\moo}{2}
\:+\,\frac{h}{4}\,B\,+\,
h\,\sum\limits_{l\in\Lambda}b_{kkl}(h,\Lambda)\,|\Phi_l'(x)|^2\;,
\end{eqnarray}
where we used the notation introduced in (\ref{def-ZZ*}).
This is equivalent to
\begin{equation}\label{WL1-est4}
\sum\limits_{l\in\Lambda}\big(\,\delta_{kl}\,-\,
h\,b_{kkl}(h,\Lambda)\,\big)\,|\Phi_l'(x)|^2
\;\klg\;
h\,Z_k^*Z_k\,+\,h\,C'\;,
\end{equation}
with $C':=\moo/2+B/4$.
Since the $\Lambda\times\Lambda$-matrix $\Xi$
with entries $\Xi_{kl}=b_{kkl}(h,\Lambda)$ is bounded by $B$ in the
norm of $\LO(\ell^{\infty}(\Lambda))$ by (\ref{BB'}),
for $h\klg h'<B$, the inverse
of the $\Lambda\times\Lambda$-matrix
$\id-h\,\Xi$ is given by a 
$\LO(\ell^{\infty}(\Lambda))$-norm-convergent
Neumann series.
Furthermore each entry of $(\id-h\,\Xi)^{-1}$
is strictly positive, so that we can apply
$(\id-h\,\Xi)^{-1}$ formally to (\ref{WL1-est4}),
giving
\begin{eqnarray}\nonumber
|\Phi_k'(x)|^2
&\klg&
\sum\limits_{l\in\Lambda}(\id-h\,\Xi)^{-1}_{kl}
\big(\,h\,Z_l^*Z_l\,+\,h\,C'\,\big)
\\ \nonumber
&\klg&
\|\,(\id-h\,\Xi)^{-1}\,\|_{\LO(\ell^{\infty}(\Lambda))}
\Big(\,\sum\limits_{l\in\Lambda}h\,Z_l^*Z_l\,+\,h\,C'\,\Big)
\\ \label{WL1-est5}
&\klg&
\frac{1}{1-h\,B}\:\Big(\,h\,\WL{0}\,+\,h\,C'\,\Big)\;,
\end{eqnarray}
for $h\in(0,h']$.
Combining (\ref{WL1-est1}), (\ref{WL1-est2})
and (\ref{WL1-est5}), we arrive at
\begin{eqnarray}
\lefteqn{  \nonumber
\Big|\Big\langle
v\,,\sum\limits_{i,j\in\Lambda}
\big(\Phi_{ij}''-\Phi_{ij}''(x_0)\big)
\otimes \ak_ia_j\,v \Big\rangle  \Big|
}
\\
&\klg&\nonumber
\frac{B'}{2}\: \Big\langle v\,,\,
\sum\limits_{i\in\Lambda}
\Big(\,h^{1/2}\,\id\,+\,
\frac{h^{-1/2}}{1-h\,B}\:\Big(\,h\,\WL{0}\,+\,h\,C'\,\Big)
\otimes \ak_ia_i\,v\,\Big\rangle
\\  \nonumber%\label{WL1-est6}
&\klg&
C''\,\Big\langle
v\,,\,
\Big(\,h^{1/2}\,\WL{0}\otimes N
\,+\,h^{1/2}\,\id\otimes N\,\Big)
\,v\,\Big\rangle\;,
\end{eqnarray}
for $v\in \desa$, where $C''>0$
is uniform in $h\in(0,h']$ and $\Lambda\in\LL$.
Now the claim follows since $\Phi_{h,\Lambda}''(x_0)\grg\muu/2$, thus
$\frac{2}{\muu}\,d\Gamma(\Phi_{h,\Lambda}''(x_0))\grg N$.
\hfill$\blacksquare$


\bigskip

\noindent In the next corollary we again use the
notation introduced in (\ref{def-DJ}) and the symbol
$\vr(T)$ for
the resolvent set of an operator $T$.


\begin{corollary}\label{proj-cor}
Assume that Hypothesis~\ref{hypphi} holds and
let  
$l\in\{1,\dots,|\Lambda|\}$.
Let 
$a,b\grg0$ and $0<\ve<1$ be such that $a+2\,\ve<b$,
\begin{equation}\label{res-sets}
(a-\ve,a+\ve)\subset\vr(\CO{l}),\quad \textrm{and}\quad 
(b-\ve,b+\ve)\subset\vr(\CO{l})\;.
\end{equation}
Furthermore let $J$ denote one of the intervals
$(a,b),\,(a,b],\,[a,b),$ or $[a,b]$.
Then there exists some $h_0\in(0,1]$
such that
\begin{equation}\label{dim-eq}
D_{J}(\WL{l})\,=\,D_{J}(\CO{l})\;,
\end{equation}
for all $h\in(0,h_0]$ and $\Lambda\in\LL$.
Moreover, if $k\in\{1,\dots,|\Lambda|\}$
and if (\ref{res-sets}) holds, for all
$1\klg l\klg k$, 
then 
(\ref{dim-eq}) holds simultaneously
for all $1\klg l\klg k$, $h\in(0,h_0]$ and $\Lambda\in\LL$.
\end{corollary}


\textsc{Proof:}
Let $k\in\{1,\dots,|\Lambda|\}$ and
(\ref{res-sets}) be fulfilled,
for all $1\klg l\klg k$.
We show that, 
for $1\klg l\klg k$, and sufficiently small $h\in(0,1]$,
we get
$\|E_{J}(\WL{l})-E_{J}(\CO{l})\|<1$.
Clearly this implies (\ref{dim-eq}).

First we pick some smooth function 
$\chi\in C^{\infty}_0([a-\ve,b+\ve],[0,1])$, such that
$\chi\equiv1$ on $[a+\ve,b-\ve]$, 
and 
\begin{equation}\label{Kleiner1/2}
\big|\,\chi-\id_{J}\,\big|_{\infty}\;\klg\;\frac{1}{2}\;.
\end{equation}
Furthermore we may assume that there is some 
$J$- and $\ve$-independent $K>0$ such that
\begin{equation}\label{chinuest}
\int\limits_{\RR}|\partial^{\nu}\chi|\;\klg\;
K\,\ve^{-(\nu-1)}\;,
\qquad\nu=1,2,3\;.
\end{equation}
Assumption (\ref{res-sets}) implies
\begin{equation}\label{E=chi}
E_{J}(\CO{l})\;=\;\chi(\CO{l})\;,\qquad1\klg l\klg k\;.
\end{equation}
Due to (\ref{Kleiner1/2}) and (\ref{E=chi}) 
we have the estimate
\begin{equation}\label{gisela}
\|\,E_{J}(\WL{l})-E_{J}(\CO{l})\,\|
\,\klg\;
\frac{1}{2}
\;+\,
\|\,\chi(\WL{l})-\chi(\CO{l})\,\|\;,
\end{equation}
for $1\klg l\klg k$. So it suffices to show that
the norm on the right side of the last equation
is less than $\frac{1}{2}$, for small $h$.
To this end we employ the following
formula due to Amrein et al. \cite[Theorem 6.1.4(b)]{ABdMBG} valid for
any selfadjoint operator $T$ in some Hilbert space and $\nu\in\NN$:
\begin{eqnarray}\nonumber
\chi(T)&=&
\sum\limits_{\vk=0}^{\nu-1}\frac{1}{\pi\vk!}\:
\int\limits_{\RR}\chi^{(\vk)}(\lambda)
\,\Im\big[
i^{\vk}(T-\lambda-i)^{-1}\big]\,d\lambda
\\
& &\label{Amrein}
\;+\;\int\limits_0^1\frac{t^{\nu-1}}{\pi(\nu-1)!}\:
\int\limits_{\RR}\chi^{(\nu)}(\lambda)
\,\Im\big[
i^{\nu}(T-\lambda-it)^{-1}\big]\,d\lambda\,dt\;.
\end{eqnarray}
To make use of this formula we need the estimate
\begin{eqnarray*}\lefteqn{
\big\|\,(\WL{l}-\lambda\pm it)^{-1}
(\WL{l}-\CO{l})(\CO{l}-\lambda\pm it)^{-1}\,\big\|
}
\\
&\klg&
\big\|\,(\WL{l}-\lambda\pm it)^{-1}(\WL{l}+1)^{1/2}\,\big\|
\\
& &\;\cdot\big\|\,(\WL{l}+1)^{-1/2}(\CO{l}+1)^{1/2}\,\big\|
\\
& &\;\cdot\big\|\,(\CO{l}+1)^{-1/2}
(\WL{l}-\CO{l})(\CO{l}+1)^{-1/2}\,\big\|
\\
& &\;\cdot\big\|\,(\CO{l}+1)^{1/2}(\CO{l}-\lambda\pm it)^{-1}\,\big\|\;,
\end{eqnarray*}
for $t>0$.
The norms in the second and fifth line of 
the last equation are less or equal
to some uniform constant times
$\sqrt{b+1}/t$, for $\lambda\in\supp\,\chi$. 
The norm in the fourth line
is less or equal to $Ch^{1/2}l\klg Ch^{1/2}k $,
for sufficiently small $h$,
because of Lemma~\ref{asymp-WL1}.
In order to bound the norm in the third line we
consider the polar decomposition
$(\WL{l}+1)^{-1/2}(\CO{l}+1)^{1/2}=UG^{1/2}$, where
$U$ is a partial isometry and
\begin{eqnarray*}
\lefteqn{
0\;\klg\;G\;:=\;
(\CO{l}+1)^{1/2}\,(\WL{l}+1)^{-1}\,(\CO{l}+1)^{1/2}
}
\\
&=&
\Big(\,\id\,-\,(\CO{l}+1)^{-1/2}\,(\CO{l}-\WL{l})\,(\CO{l}+1)^{-1/2}\,
\Big)^{-1}\;.
\end{eqnarray*}
The norm of $G$ is less or equal to
$(1-C\,k\,h^{1/2})^{-1}$, for $1\klg l\klg k$, and
for sufficiently small $h$ satisfying
$C\,k\,h^{1/2}<1$.
Collecting all estimates and using formula (\ref{Amrein})
with $\nu=3$, the second resolvent equation and (\ref{chinuest}) 
we see that there is some $h_0\in(0,1]$ such that
\begin{equation}\label{agartha}
\|\,\chi(\WL{l})-\chi(\CO{l})\,\|
\;\klg\;\const\;\frac{h^{1/2}\,k\,(b+1)(b-a+1)}{\ve^2}\;,
\end{equation}
for $h\in(0,h_0]$ and $0<\ve<1$.
The constant appearing in (\ref{agartha}) is uniform.
In view of (\ref{gisela}) this implies the assertion.
\hfill$\blacksquare$


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\section{Proof of Theorem~\ref{mainthmWL}}
\label{proof-mainthmWL}

\noindent We prove the theorem by induction on $n\in\NN$.
To this end we fix $\Lambda\in\LL$.
We observe, however, that the restrictions on the semi-classical
parameter $h\in(0,1]$ below are uniform in $\Lambda$.
We start by proving the theorem for $n=1$, 
then for $1\klg n<|\Lambda|$, and finally
for $n\grg|\Lambda|$.
In all steps we use the following simple facts:
Suppose that the spectrum of $\WL{0}$ below 
or equal to some $E>0$ 
is purely discrete and consists of
the eigenvalues
$\mu_0\klg\dots\klg\mu_L$, for some $L\in\NN_0$.
Moreover suppose that $\WL{0}$ has no spectrum
contained in $(E,E')$, where $E+\moo<E'+\muu$.
Let $\phi_j$  be
the eigenfunction corresponding to $\mu_j$, for
$0\klg j\klg L$. 
We recall that $\mu_0=0$ is simple with corresponding
eigenfunction $\phi_0=\exp(-\frac{1}{h}\Phi_{h,\Lambda})$.
Accordingly, let
$m_j$ 
denote the $j^{\textrm{th}}$ eigenvalue 
(again counted with multiplicity)
of $2\,\Phi_{h,\Lambda}''(x_0)$,
and let $v_j$ denote
the corresponding eigenvector, for
$1\klg j\klg|\Lambda|$.
By Hypothesis~\ref{hypphi}(ii)
we know that $\muu\klg m_j\klg \moo$, for $1\klg j\klg|\Lambda|$,
uniformly in $h\in(0,1]$ and $\Lambda\in\LL$.
Then the spectrum of
\[
\CO{k}\;=\;\WL{0}\otimes\id\;+\;2\,d\Gamma^{(k)}(\Phi_{h,\Lambda}''(x_0))
\]
strictly below $E'+\muu$ is given by
\begin{eqnarray*}
\lefteqn{
\sigma(\CO{k})\cap(-\infty,E'+\muu)
}
\\
&=&
\Big\{\,
\mu_j+\,\sum\limits_{\alpha=1}^km_{i_\alpha}:
0\klg j\klg L\,,\;1\klg i_1<\dots<i_k\klg|\Lambda|\,
\Big\}\;.
\end{eqnarray*}
The eigenvalue $\mu_j\,+\,\sum\limits_{\alpha=1}^km_{i_\alpha}$
corresponds to the eigenfunction
$\phi_j\otimes v_{i_1}\wedge\dots\wedge v_{i_k}$.
In particular 
\begin{equation}\label{low-bound-WLk}
\CO{k}
\;\grg\;k\,\muu\;,\qquad1\klg k\klg|\Lambda|\;.
\end{equation}



\framebox[1.6\width]{\textbf{(1) $n=1$}}
Let $0<\eta<\frac{2}{5}\,\big(2\,\muu-\moo\big)$.
We set $\ve:=\eta/2$.
Due to
Propositions~\ref{prop-Hodge}
and~\ref{shmuel} there is some $h_0\in(0,1]$
such that
the Hodge decompositions
\begin{equation}\label{heidi1}
\HR^0\;=\;\CC\,\exp(-\textstyle{\frac{1}{h}}\Phi_{h,\Lambda})
\oplus \ran_0^*\;,\qquad
\HR^k\;=\;\ran_{k-1}\oplus \ran_k^*\;,
\qquad k=1,2\;,
\end{equation}
and unitary equivalences
\begin{equation}\label{heidi2}
\WL{k}\!\!\upharpoonright_{\ran_k^*}
\;=\;
U_k^*\,\WL{k+1}\!\!\upharpoonright_{\ran_{k+1}}\,U_k\;,
\qquad k=0,1\;,
\end{equation}
are valid, for all $h\in(0,h_0]$.
First we observe that (\ref{low-bound-WLk}) together with
Corollary~\ref{proj-cor}
implies that there is a $h_1\in(0,h_0]$
such that 
\[
\sigma(\WL{1})\cap(-\infty,\muu-\ve\,]\;=\;\varnothing\;,
\]
for all $h\in(0,h_1]$.
Since (\ref{heidi1}) and (\ref{heidi2}) imply
$\sigma(\WL{0})\setminus\{0\}\subset\sigma(\WL{1})$,
it follows that 
\[
\sigma(\WL{0})\cap(-\infty,\muu-\ve\,]\;=\;\{0\}\;,
\]
for all $h\in(0,h_1]$.
By assumption we have $\moo+5\,\ve<2\,\muu$.
Hence $\CO{1}=\WL{0}+2\Phi_{h,\Lambda}''(x_0)$ has exactly
$|\Lambda|$ eigenvalues in $[\,\muu,\moo\,]$,
namely those of $2\,\Phi_{h,\Lambda}''(x_0)$.
Besides, $\CO{1}$ has empty spectrum in
$(\moo,2\,\muu-\ve)$.
Now Corollary~\ref{proj-cor}
implies that there is a $0<h_2\klg h_1$, such that
\begin{eqnarray}\label{ernest}
D_{[\,\muu-\ve,\moo+\ve\,]}(\WL{1})
&=&
D_{[\,\muu-\ve,\moo+\ve\,]}(\CO{1})\;=\;|\Lambda|\;,
\\ \nonumber
D_{(\,\moo+\ve,2\,\muu-2\,\ve\,)}(\WL{1})
&=&
D_{(\,\moo+\ve,2\,\muu-2\,\ve\,)}(\CO{1})\;=\;0\;,
\end{eqnarray}
for $h\in(0,h_2]$.
It follows from $\CO{2}\grg2\,\muu$
and Corollary~ \ref{proj-cor}
that 
\[
\sigma(\WL{2})\cap (-\infty,2\,\muu-\ve)\;=\;\varnothing\;,
\]
for $h\in(0,h_3]$, where $0<h_3\klg h_2$.
Since
$\WL{1}\!\!\upharpoonright_{\ran_1^*}$
is unitarily equivalent to
$\WL{2}\!\!\upharpoonright_{\ran_2}$
all
the eigenforms of $\WL{1}$ corresponding 
to its eigenvalues 
in the interval
$[\,\muu-\ve,\moo+\ve\,]$ are closed, for $h\in(0,h_3]$. 
Thus the unitary equivalence of
$\WL{0}\!\!\upharpoonright_{\ran_0^*}$ and
$\WL{1}\!\!\upharpoonright_{\ran_1}$
implies
\[
\sigma(\WL{0})\cap
(-\infty,2\,\muu\,-2\,\ve\,)\;=\;
\{0\}\cup\sigma(\WL{1})\cap(-\infty,2\,\muu\,-2\,\ve\,)
\]
with equal multiplicities of all eigenvalues,
for $h\in(0,h_3]$. 
This together with (\ref{ernest})
proves the theorem for $n=1$.


\bigskip



\framebox[1.4\width]{\textbf{(2) $1\klg n<|\Lambda|$}}
Suppose $n\in\{2,\dots,|\Lambda|-1\}$, and assume that
the theorem is true for all $1\klg k\klg n-1$.


Let $0<\eta'<\frac{2}{5}\,\big((n+1)\muu-n\,\moo\big)$ and
set $\ve:=\eta'/2$. We pick some $h_0\in(0,1]$ 
such that the
assertions of the theorem for $1\klg k\klg n-1$
hold with $\eta=\ve$, 
and such that
the decompositions
\begin{equation}\label{willi1}
\HR^0\;=\;\CC\,\exp(-\textstyle{\frac{1}{h}}\Phi_{h,\Lambda})
\oplus \ran_0^*\;,\qquad
\HR^k\;=\;\ran_{k}\oplus \ran_k^*\;,
\qquad1\klg k\klg(n+1)\;,
\end{equation}
and unitary equivalences
\begin{equation}\label{willi2}
\WL{k}\!\!\upharpoonright_{\ran_k^*}
\;=\;
U_k^*\,\WL{k+1}\!\!\upharpoonright_{\ran_{k+1}}\,U_k\;,
\qquad0\klg k\klg n\;,
\end{equation}
are valid,
for all $h\in(0,h_0]$.
Then we know that, for $1\klg k\klg n$ and
$h\in(0,h_0]$,
$\CO{k}$ has exactly 
${|\Lambda|+n-k-1\choose n-k}{|\Lambda|\choose k}$
eigenvalues in the interval $[\,n\,\muu-\ve,n\,\moo+\ve\,]$,
namely those corresponding to the eigenvectors
\[
\phi_j\otimes v_{i_1}\wedge\dots\wedge v_{i_k}\;,
\]
where $\phi_j$ is some eigenvector of $\WL{0}$
with corresponding eigenvalue
\[
\mu_j\in[\,(n-k)\,\muu-\ve\,,\,(n-k)\,\moo+\ve\,]\;.
\]
Moreover, 
\begin{eqnarray*}
\sigma(\CO{k})\cap
\big(\,(n-1)\,\moo+\ve,n\,\muu-\ve\big)
&=&\varnothing\;,
\\
\sigma(\CO{k})\cap
\big(\,n\,\moo+\ve,(n+1)\,\muu-\ve\big)
&=&
\varnothing\;.
\end{eqnarray*}
Since $(n+1)\muu-n\,\moo>5\,\ve$ we can
apply Corollary~\ref{proj-cor} and find
some $h_1\in(0,h_0]$,
such that, for all $h\in(0,h_1]$
and $1\klg k\klg n$,
\begin{eqnarray}\label{graf-zahl}
D_{[\,n\,\muu-2\,\ve\,,\,n\,\moo+2\,\ve\,]}(\WL{k})
&=&
{|\Lambda|+n-k-1\choose n-k}{|\Lambda|\choose k}\;,
\\ \nonumber
D_{(\,(n-1)\,\moo+2\,\ve\,,\,n\,\muu-2\,\ve\,)}
(\WL{k})
&=&0\;,
\\ \nonumber
D_{(\,n\,\moo+2\,\ve\,,\,(n+1)\,\muu-2\,\ve\,)}
(\WL{k})
&=&0\;.
\end{eqnarray}
Moreover we know that
$\CO{n+1}\grg (n+1)\,\muu>n\,\moo+5\,\ve$, i.e.
by Corollary~\ref{proj-cor} we find some
$h_2\in(0,h_1]$ such that
\[
\sigma(\WL{n+1})\cap(-\infty,(n+1)\,\muu-\ve\,]
\;=\;\varnothing\;,
\] 
for $h\in(0,h_2]$.
This together with (\ref{willi2})
shows that all ${|\Lambda|\choose n}$ eigenvalues of
$\WL{n}$ lying in $[\,n\,\muu-2\,\ve,n\,\moo+2\,\ve\,]$
correspond to closed eigenforms.
The decompositions~(\ref{willi1})
and unitary equivalences~(\ref{willi2})
together with (\ref{graf-zahl})
now lead recursively to the formulas
\[
D_{[\,n\,\muu-2\,\ve\,,\,n\,\moo+2\,\ve\,]}
(\WL{k}\!\!\upharpoonright_{\ran_k^*})
\;=\;
\sum\limits_{\alpha=0}^{n-1-k}(-1)^{n-k-1-\alpha}
{|\Lambda|+\alpha-1 \choose \alpha}
{|\Lambda|\choose n-\alpha}
\;,
\]
for $0\klg k< n$.
These remarks together with (\ref{comb-le1})
prove the theorem for 
$1\klg n<|\Lambda|$.


\bigskip


\framebox[1.6\width]{\textbf{(3) $n\grg|\Lambda|$}}
Suppose $n\grg|\Lambda|$ and assume that Theorem~\ref{main-thm}
holds for all $1\klg k\klg n-1$.
Let $0<\eta'<\frac{2}{5}\,\big((n+1)\,\muu-n\,\moo\big)$.
We set $\ve:=\eta'/2$
and pick some $h_0\in(0,1]$ such that the
assertions of the theorem for $1\klg k\klg n-1$
hold with $\eta=\ve$, 
and such that
the decompositions
\[
\HR^0\;=\;\CC\,\exp(-\textstyle{\frac{1}{h}}\Phi_{h,\Lambda})
\oplus \ran_0^*\;,\qquad
\HR^k\;=\;\ran_{k}\oplus \ran_k^*\;,
\qquad1\klg k\klg|\Lambda|\;,
\]
and unitary equivalences
\[
\WL{k}\!\!\upharpoonright_{\ran_k^*}
\;=\;
U_k^*\,\WL{k+1}\!\!\upharpoonright_{\ran_{k+1}}\,U_k\;,
\qquad1\klg k\klg|\Lambda|\;,
\]
are valid,
for all $h\in(0,h_0]$.
Exactly as in the preceeding step we see
that we can
find
some $h_1\in(0,h_0]$,
such that, for all $h\in(0,h_1]$
and $1\klg k\klg n$,
there are precisely
${|\Lambda|+n-k-1\choose n-k}{|\Lambda|\choose k}$
eigenvalues of $\WL{k}$
in the interval $[\,n\,\muu-2\,\ve,n\,\moo+2\,\ve\,]$
and no eigenvalues in 
$\big(\,(n-1)\,\moo+2\,\ve,n\,\muu-2\,\ve\big)
\cup\big(\,n\,\moo+2\,\ve,(n+1)\,\muu-2\,\ve\big)$.
Trivially all eigenforms of $\WL{|\Lambda|}$
are closed so that we now obtain the formulas
\begin{eqnarray*}
\lefteqn{
D_{[\,n\,\muu-2\,\ve\,,\,n\,\moo+2\,\ve\,]}
(\WL{k}\!\!\upharpoonright_{\ran_k^*})
}
\\
&=&
\sum\limits_{\alpha=0}^{|\Lambda|-1-k}(-1)^{|\Lambda|-k-1-\alpha}
{|\Lambda|+\alpha+(n-|\Lambda|)-1 \choose \alpha+(n-|\Lambda|)}
{|\Lambda|\choose|\Lambda| -\alpha}
\;,
\end{eqnarray*}
for $0\klg k\klg|\Lambda|$.
Together with (\ref{comb-le2}) this finishes
the proof of Theorem~\ref{mainthmWL}.
\hfill$\blacksquare$



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\section{Harmonic oscillators perturbed by\\
compactly supported potentials}
\label{sec-harm-osc}

In this section we work out 
\cite[Section 1]{Sj1}. In particular we verify
that everything works uniformly in the parameter
$t$ introduced below and give a proof of
Lemma~\ref{le-Wasow} which was left to the
reader in \cite{Sj1}.
Our goal is to obtain the following
proposition. Its proof is carried out throughout the
whole section.

\begin{proposition}\label{main-le-sec4}
Let $n\in\NN$ and $U\in C_0^{\infty}(\RR^n,\RR)$.
Let
$\mu_t$ denote the first
eigenvalue and
$\exp(-\Phi_t)$ the first normalized strictly positive
eigenfunction
of the unique selfadjoint extension of
$-\Delta+x^2+t\,U$ starting from $C_0^{\infty}(\RR^n)$, 
for $t\in[0,1]$.
Then it follows that 
$\Phi_t(x)=x^2/2+\Psi_t(x)$,
where, for each multi-index $\beta\in\NN_0^{n}$, we have
\[
\partial^{\beta}\Psi_t(x)
\;\longrightarrow\;0\;,\qquad\textrm{as}\;|x|\rightarrow\infty\;,
\]
uniformly in $t\in[0,1]$.
\end{proposition}

\smallskip

\noindent We start with a well-known result, which we state 
in a slightly more general form for later reference:

 Let $\{V_s\}_{s\grg0}$
be a family of smooth potential defined on $\RR^n$
such that
\begin{eqnarray}\label{Cinfty-conv-Vs}\label{john}
\lim\limits_{|x|\rightarrow\infty}V_s(x)
&=&\infty\;,\qquad\textrm{for}\;\;s\grg0\;,
\\
\textrm{and}\qquad
\sup\limits_{x\in K}|\,\partial^{\beta}(V_s-V)\,|
&\longrightarrow&0\;,\qquad
\textrm{as}\;\;s\rightarrow s_0\;,
\end{eqnarray}
for each compact set $K\Subset\RR^n$ and each multi-index
$\beta\in\NN_0^n$.
We consider the family of Schr\"{o}dinger operators
$P_s$ obtained as the closures
of $-\Delta+V_s$, which are defined and
essentially selfadjoint on
$C_0^{\infty}(\RR^n)$, for $s\grg 0$.
Due to (\ref{john}) each $P_s$ has purely discrete spectrum.
We assume furthermore that $P_s$ converges to $P_0$
in the norm resolvent sense, as $s\rightarrow 0$.
In the following lemma we study uniform convergence
properties of the unique normalized strictly positive
ground state eigenfunction of $P_s$, which we denote
by $u_s$, for $s\grg0$. The corresponding lowest
eigenvalue of $P_s$ is denoted by $e_s$, for $s\grg0$.
Finally,
$E_s$ stands for the
orthogonal projection onto the space spanned by $u_s$, for $s\grg0$.

\begin{lemma}\label{conv-le-sec5}
We have
\begin{eqnarray}
\sup\limits_{x\in K}|\,\partial^{\beta}(u_s-u_0)\,|
&\longrightarrow&0\;,\qquad\textrm{as}\;\;
s\rightarrow 0\;,\label{conv-us}
\\
\sup\limits_{x\in K}|\,\partial^{\beta}(\ln u_s-\ln u_0)\,|
&\longrightarrow&0\;,\qquad\textrm{as}\;\;
s\rightarrow 0\;,\label{conv-lnus}
\end{eqnarray}
for each compact set $K\Subset\RR^n$ and each multi-index
$\beta\in\NN_0^n$.
\end{lemma}

{\sc Proof:}
By norm resolvent convergence we get
$e_s\rightarrow e_0$,
and $\|\,E_0-E_s\,\|\rightarrow0$, 
as $s\rightarrow 0$ 
\cite[Theorem VIII.23]{RS1}.
Since $u_s$ is strictly positive
and normalized, for $s\grg 0$,
this implies
\[
1\;\grg\;\SL u_s\,,\,u_0 \SR\;=\;\|\,\SL u_s\,,\,u_0 \SR\,u_0\,\|
\;=\;\|\,E_0u_s\,\|
\;\grg\;
1\,-\,\|\,E_0-E_s\,\|
\;\rightarrow\;1\;,
\]
as $s\rightarrow 0$.
Hence
$\|\,u_s-u_0\,\|^2\,=\,2\,-\,2\,\SL u_s\,,\,u_0\SR\rightarrow0$,
as $s\rightarrow 0$.
Assertion (\ref{conv-us}) follows from these remarks
and (\ref{Cinfty-conv-Vs}) by considering the identity
\[
\Delta(u_s-u_0)\;=\;
(e_0\,-\,V_0)\,(u_0-u_s)\,+\,(e_0-e_s)\,u_s
\,+\,(V_s-V_0)\,u_s\;,
\]  
using recursively the regularity theorem for the Laplacian
and applying the Sobolev imbedding theorem
as in \cite[\textsection3]{Hebook}.
Furthermore, we know that $u_0^{-1}$ is bounded  
on each compact set $K\Subset\RR^n$.
So the family 
$(u_s^{-1})_{s_1\grg s\grg 0}$
is uniformly bounded on $K$, for some sufficiently small
$s_1>0$, because of (\ref{conv-us}) with $\beta=0$.
Therefore
assertion (\ref{conv-lnus}) follows easily from
(\ref{conv-us}). 
\hfill$\blacksquare$


\smallskip

\noindent Now we come back to the proof of
Proposition~\ref{main-le-sec4}.

First we note that $\mu_t$ is continuous
in $t$ on $[0,1]$.
Hence it is possible to
pick some $\rho_0>\sup_{t\in[0,1]}\sqrt{\mu_t}$
such that $\supp\,U\Subset\{|x|<\rho_0\}$.
We define $S:=\{|x|=\rho_0\}$
and $\Omega:=\{|x|\grg\rho_0\}$.
Then $\exp(-\Phi_t)$
is a solution of the Dirichlet problem
\begin{eqnarray}\nonumber
(\,-\Delta\,+\,x^2\,-\,\mu_t\,)\,u
&=&0
\quad\textrm{on}\;\stackrel{\circ}{\Omega}\;,\\ \label{Dirichletexp}
u&\equiv&\exp(-\Phi_t)\!\!
\upharpoonright_S
\quad\textrm{on}\;S\;.
\end{eqnarray}
Therefore we will analyze the homogenous Dirichlet problem
with inhomogenous boundary conditions
\begin{eqnarray}\nonumber
(\,-\Delta\,+\,x^2\,-\,\mu_t\,)\,u
&=&0
\quad\textrm{on}\;\stackrel{\circ}{\Omega}\;,\\ \label{Dirichlet1}
u&\equiv&\tilde{v}_t\quad\textrm{on}\;\;S\;,
\end{eqnarray}
where $\tilde{v}_t\in C^{\infty}(S)$.
We are only interested in square integrable solutions
of (\ref{Dirichlet1}).
First we transform (\ref{Dirichlet1}) into an inhomogenous 
Dirichlet problem with homogenous boundary conditions.
To this end we extend $\tilde{v}_t$ to a function
$\hat{v}_t\in C^{\infty}(\Omega)$ such that
$\hat{v}_t\equiv0$ on $\{|x|\grg r_0\}$, for some $r_0>\rho_0$.
For example, we may set $\hat{v}_t(x)=\tilde{v}_t(x/|x|)\,\chi(|x|)$,
for some appropriate smooth cut-off function $\chi$.
Then we define $v_t:=-(-\Delta+x^2-\mu_t)\hat{v}_t$,
so that $v_t\in C^{\infty}(\Omega)\cap L^2(\Omega)$.
We denote by $P$ the Dirichlet realization of
$-\Delta+x^2$ on $\Omega$. This is identical to the 
Friedrichs extension of $-\Delta+x^2$ starting
from $C_0^{\infty}(\stackrel{\circ}{\Omega})$ and its 
domain $\dom(P)$ is contained in
$\Sob$,
the closure of $C_0^{\infty}(\stackrel{\circ}{\Omega})$
with respect to the norm on $H^{1}(\RR^n)$ 
(see, e.g., \cite{Pe}).
Its spectrum is purely discrete and 
contained in $[\rho_0,\infty)$ \cite{Pe}.
Therefore $P-\mu_t$
is continuously invertible and
$w_t:=(P-\mu_t)^{-1}v\in\Sob$.
We readily verify that the standard elliptic
regularity theorems (e.g. in the form 
presented in \cite[\textsection  5.9]{Ra})
apply and show that $w_t\in C^{\infty}(\Omega)$,
$w_t\equiv 0$ on $S$.
In summary we see that $w_t$ solves the problem
\begin{eqnarray*}
(\,-\Delta\,+\,x^2\,-\,\mu_t\,)\,w_t
&=&v_t
\quad\textrm{on}\;\stackrel{\circ}{\Omega}\;,\\
w_t&\equiv&0\quad\textrm{on}\;S\;.
\end{eqnarray*}
By construction $u_t:=w_t+\hat{v}_t\in C^{\infty}(\Omega)$
is then a solution of the original
problem~(\ref{Dirichlet1}). 
Using weighted $L^2$-estimates, regularity theorems
for the Laplace operator, and the Sobolev imbedding 
theorem as in \cite[\textsection6]{DiSj} and
\cite[\textsection3]{Hebook} we verify that
$u_t$ decays exponentially together with all its derivatives:

\begin{lemma}\label{exp-decay-wt}
Let $\alpha:=\min\limits_{t\in[0,1]}\sqrt{(\rho_0^2-\mu_t)/2}$
and suppose that 
$\sup\limits_{t\in[0,1]}\max\limits_{y\in S}
|\tilde{v}_{t}(y)|<\infty$.
Then, for each multi-index $\beta\in\NN_0^n$, and $\ve>0$,
there exists some $t$-independent
constant $c_{\beta,\ve}>0$ such that
\[
\sup\limits_{t\in[0,1]}|\partial^{\beta}_xu_t(x)|\;\klg\;c_{\beta,\ve}\,
e^{-(\alpha-\ve)\,|x|}
\;,\qquad x\in\Omega\;.
\]
\end{lemma}

This implies
  

\begin{lemma}\label{cor-max-prin}
(i) Any square-integrable solution $\tilde{u}_t$ 
of (\ref{Dirichlet1}) is identical to $u_t:=w_t+\hat{v}_t$.
In particular
the solution $u_t$ of (\ref{Dirichlet1})
does not depend
on the choice of~$\hat{v}_t$.

\smallskip

\noindent (ii) We have 
$\min \{\,0\,,\,\min\limits_{S}\tilde{v}_t\,\}\,\klg\,u_t\,
\klg\,
\max\{\,0\,,\,\max\limits_{S}\tilde{v}_t\,\}$. In particular
$\tilde{v}_t\grg0$ implies $u_t\grg0$.
\end{lemma}

\textsc{Proof:}
(i) We have $(-\Delta+x^2-\mu_t)(u_t-\tilde{u}_t)=0$ on
$\stackrel{\circ}{\Omega}$
and $u_t-\tilde{u}_t\equiv0$ on $S$.
Furthermore we find, for each $\ve>0$, some $R>\rho_0$
such that $|u_t|,|\tilde{u}_t|\klg\ve$
on $\{|x|\grg R\}$.
The classical maximum principle 
(see, e.g. \cite[\textsection 5.11. Corollary 7]{Ra})
implies that $|u_t-\tilde{u}_t|\klg2\ve$
on $\{\rho_0\klg|x|\klg R\}$.
Since $\ve>0$ is arbitrarily small and 
$R$ arbitrarily large this gives the assertion
\smallskip

\noindent (ii) For each $\ve>0$, we find some $R>0$
such that $|u_t|\klg\ve$ on $\{|x|\grg R\}$.
The maximum principle again implies
$\min\{-\ve,\min\limits_{S}\tilde{v}_t\}
\,\klg\,u_t\,\klg\,
\max\{\ve,\max\limits_{S}\tilde{v}_t\}$
on $\{|x|\grg R\}$.
\hfill $\blacksquare$

\bigskip

\noindent Direct calculation shows that if
$K(\tilde{v}_t):=u_t$ is a solution of (\ref{Dirichlet1})
with boundary condition $\tilde{v}_t$ then
$u_t\circ R=K(\tilde{v}_t\circ R)$,
for any rotation matrix $R\in\textrm{SO}(n,\RR)$.
Denoting the expressions for $u_t$ and $\tilde{v}_t$
in polar coordinates $x=(r,\theta)$ again
by the same symbols this implies 
$XK(\tilde{v}_t)=K(X\tilde{v}_t)$,
for each vector field $X$ on $S^{n-1}$
that is the infinitesimal generator of a rotation.
In particular 
\begin{equation}\label{partial-theta-commute}
\partial_{\theta}^{\alpha}K(\tilde{v}_t)
\,=\,K(\partial_{\theta}^{\alpha}\tilde{v}_t)\;,
\quad\textrm{for each multi-index}\;\alpha\;.
\end{equation} 




Now we focus our attention on radial symmetric
solutions. That is we suppose that $\tilde{v}_t\equiv1$
in (\ref{Dirichlet1}). Due to (\ref{partial-theta-commute}) the corresponding
solution $u_t^{\circ}$ depends only on the absolut value
of $x\in\Omega$ and can therefore be written
in the form $u_t^{\circ}(x)=u_t^{\circ}(|x|)$, for $x\in\Omega$.
After passing to polar coordinates the 
radial Dirichlet problem
reads
\begin{equation}\label{Dir-u0}
\Big(-\partial_r^2-\,\frac{n-1}{r}\,\partial_r\,+\,r^2\,-\,\mu_t\Big)
\,u^{\circ}\,=\,0\quad\textrm{on}\;[\rho_0,\infty)\;,
\qquad
u^{\circ}(\rho_0)\,=\,1\;,
\end{equation}
and we consider solutions which are 
in $L^2([\rho_0,\infty),r^{n-1}dr)$. A further substitution
$f(r):=r^{\frac{n-1}{2}}u^{\circ}(r)$, for $r\in[\rho_0,\infty)$,
turns the Dirichlet problem above into
\begin{equation}\label{Dir-f}
\Big(-\partial_r^2-\,\frac{(n-1)(n-3)}{4r^2}\,+\,r^2\,-\,\mu_t\Big)
\,f\;=\;0\quad\textrm{on}\;[\rho_0,\infty)\;,
\quad
f(\rho_0)\,=\,\rho_0^{\frac{n-1}{2}}\,,
\end{equation}
where $f\in L^2([\rho_0,\infty),dr)$.
  In the following we will construct an asymptotic
solution of (\ref{Dir-f}).
More precisely we have
\begin{lemma}\label{le-asymp-sol}
For $t\in[0,1]$, there is a smooth function
$\vp_t:[\rho_0,\infty)\rightarrow\RR$, 
such that
\begin{equation}
\label{asymp1}
\Big(-\partial_r^2-\,\frac{(n-1)(n-3)}{4r^2}\,+\,r^2\,-\,\mu_t\Big)
\,\exp(-\varphi_t)
\;=\;\widetilde{R}_t\,\exp(-\varphi_t)
\;,
\end{equation}
where, for each $\beta\in\NN_0$,
\begin{equation}\label{asymp-tildeR} 
\partial_r^{\beta}\widetilde{R}_t\;\sim\;0\qquad
\textrm{as}\qquad
r\rightarrow\infty\;,\;
\textrm{uniformly in}\;t\;(\textrm{u.i.}\,t)\;.
\end{equation}
There is a sequence of continuous
functions $(a_{\nu})_{\nu\in\NN_0}$,
$a_{\nu}\in C([0,1],\RR)$, for $\nu\in\NN_0$,
such that, for each $\beta\in\NN_0$,
\begin{equation}\label{asymp2}
\partial_r^{\beta+1}\varphi_t(r)
\;\sim\;
r\,\delta_{\beta,0}\,+\,
\sum\limits_{\nu=0}^{\infty}
a_{\nu}(t)\,(-1)^{\beta}\,
\frac{(\beta+2\nu)!}{(2\nu)!}\,r^{-1-2\nu-\beta}
\;,\quad(r\rightarrow\infty,\,\textrm{u.i.}\,t)\;.
\end{equation}
\end{lemma}

\smallskip


\noindent Equation (\ref{asymp-tildeR}) means that,
for all $\beta\in\NN_0$ and $N\in\NN$, there exist
two $t$-independent constants 
$C_{\beta,N},R_{\beta,N}>0$, such that
\[
\big|\,\partial_r^{\beta}\widetilde{R}_t(r)\,\big|
\;\klg\;
\frac{C_{\beta,N}}{r^{N}}\;,\qquad
r\grg R_{\beta,N},\,t\in[0,1]\;.
\]
In particular,
\begin{equation}\label{hugo2}
\forall\,\beta\in\NN_0:\;\;
\partial_r^{\beta}\widetilde{R}_t(r)
\;\longrightarrow\;0\;,
\qquad\textrm{as}\;\;
r\rightarrow\infty,\,\textrm{u.i.}\,t\;.
\end{equation}
Correspondingly (\ref{asymp2}) says that,
for all $\beta\in\NN_0$ and $N\in\NN$,
there are two $t$-independent constants 
$D_{\beta,N},S_{\beta,N}>0$, such that
\[
\Big|\:
\partial_r^{\beta+1}\varphi_t(r)\,-\,
r\,\delta_{\beta,0}\,-\,
\sum\limits_{\nu=0}^{N}
a_{\nu}(t)\,(-1)^{\beta}\,
\frac{(\beta+2\nu)!}{(2\nu)!}\,r^{-1-2\nu-\beta}
\:\Big|
\;\klg\;
\frac{D_{\beta,N}}{r^{2N+3+\beta}}\;,
\]
for all $r\grg S_{\beta,N} $ and $t\in[0,1]$.
In particular
\begin{eqnarray}\nonumber
\varphi_t'(r)-r&=&\cO(\textstyle{\frac{1}{r}})\;,\qquad\qquad
\varphi_t''(r)-1\;=\;\cO(\textstyle{\frac{1}{r^2}})\;,
\\ \label{hugo}
\partial^{\beta}\varphi_t(r)&=&\cO(\textstyle{\frac{1}{r^{\beta+1}}})\;,
\qquad\beta\grg3\;,
\end{eqnarray}
where the $\cO$-symbols are all unifom in $t\in[0,1]$.



\smallskip

{\sc Proof of Lemma~\ref{le-asymp-sol} (Sketch):}
First we plug the formal power series
$\rho_t(r)=r+\sum\limits_{\nu=0}^{\infty}a_{\nu}(t)\,r^{-1-2\nu}$
into the Riccati equation
\[
\rho_t'\,-\,\rho_t^2\,
-\,\frac{(n-1)(n-3)}{4r^2}\,+\,r^2\,-\,\mu_t\;=\;0
\]
and convince ourselfs that the coefficients
$a_{\nu}(t)$ are determined by recursive formulas.
Moreover each coefficient $a_{\nu}(t)$ is a polynomial
in $\mu_t$ and therefore continuous in $t\in[0,1]$.
Now we carry through a Borel procedure:
We pick some $\chi\in C^{\infty}(\RR,[0,1])$
such that $\chi\equiv0$ on $(-\infty,1]$
and $\chi\equiv1$ on $[2,\infty)$ and define
\[
\varphi_t'(r)\;=\;
r\,+\,\sum\limits_{\nu=0}^{\infty}
a_{\nu}(t)\,
\chi_{\nu,t}(r)
\,r^{-1-2\nu}\;,
\quad
\varphi_t(r)\;=\;\int\limits_{\rho_0}^r\varphi_t'(s)\,ds\;,
\]
for $r\in[\rho_0,\infty)$,
where $\chi_{\nu,t}(r):=\chi(r\cdot2^{-A_{\nu}(t)})$
and $A_{\nu}(t):=1+2\nu+\max\limits_{0\klg\vk\klg\nu}|a_{\vk}(t)|$. 
By construction we have
\[
\big|\,r^N\,a_{\nu}(t)\,
\chi_{\nu,t}(r)
\,r^{-1-2\nu}\,\big|
\;\klg\;\frac{|a_{\nu}(t)|}{2^{(1+2\nu-N)\,A_{\nu}(t)}}
\qquad(\nu\in\NN_0)\;.
\]
Similar estimates show that all involved series
converge geometrically.
Besides, the supports of $\partial^{\beta}\chi_{\nu,t}$ are disjoint
for different $\nu$ and $\beta\in\NN$.
Using these remarks
and the continuity
of $t\mapsto A_{\nu}(t)$
on $[0,1]$, we verify (\ref{asymp1}) with
\[
\widetilde{R}_t\;:=\;
\varphi_t''\,-\,\varphi_t'{}^2\,
-\,\frac{(n-1)(n-3)}{4r^2}\,+\,r^2\,-\,\mu_t
\]
and (\ref{asymp2}) by direct calcuation.
\hfill$\blacksquare$


\smallskip 

Next we claim that the exact solution of (\ref{Dir-f})
can be written in the form $f_t=e^{-\varphi_t-R_t}$,
where $\partial^{\alpha}R_t\sim 0$
uniformly in $t\in[0,1]$, for each $\alpha\in\NN$. 
Plugging this expression for $f_t$
into the differential equation in (\ref{Dir-f})
we see that $R_t'$ is necessarily
a solution of 
\[
R_t''\,-\,R_t'{}^2\,-\,2\,\varphi_t'\,R_t'\,+\,\widetilde{R}_t\;=\;0
\qquad\textrm{on}\;[\rho_0,\infty)\;.
\]



\begin{lemma}\label{le-Wasow}
For $\xi>\rho_0$ big enough, there is a 
smooth solution $m_t$
of
\begin{equation}\label{ODE-le-Wasow}
m_t'\;=\;2\,\varphi_t'\,m_t\,+\,m_t^2
\,-\,\tilde{R}_t
\qquad\textrm{on}\;\;[\,\xi,\infty)\;,
\end{equation}
that is asymptotically zero together with all its
derivatives, $\partial^{\alpha}m_t\sim0$, 
uniformly in $t\in[0,1]$, for $\alpha\in\NN_0$.
In particular, for each $\alpha\in\NN_0$,
$\partial^{\alpha}m_t(r)$ converges to zero as 
$r\rightarrow\infty$,
uniformly in $t\in[0,1]$.
\end{lemma}

\textsc{Proof:} We proceed along the lines of 
\cite[Section 14]{Wa}.
For $\eta\grg\rho_0$, 
let 
$\mathscr{X}_{\eta}=\{f\in C([\eta,\infty)):
\|\,f\,\|_{\infty}\klg\frac{1}{2}\}$. 
This is a closed subset of
the Banach space
$\SL C_b^{\infty}([\,\eta,\infty)),\|\cdot\|_{\infty}\SR$.
We define a mapping 
$\mathscr{P}_{\eta}^{(t)}$ on 
$\mathscr{X}_{\eta}$
by
\begin{equation}\label{def-P}
\mathscr{P}^{(t)}_{\eta}f(r)\;=\;
-\int\limits_r^{\infty}e^{-(s^2-r^2)}\,
\Big(-\widetilde{R}_t(s)\,+\,\big(2\,\varphi_t'(s)-2\,s\big)\,f(s)
\,+\,f^2(s)
\Big)\,ds\,,
\end{equation}
for $r\in[\,\eta,\infty)$, $f\in\mathscr{X}_{\eta}$.
We want to show that $\mathscr{P}_{\xi}^{(t)}$ possesses a fixpoint
$m_t\in\mathscr{X}_{\xi}$ which is asymptotically zero 
uniformly in $t\in[0,1]$,
if we choose the $t$-independent parameter
$\xi$ large enough. Since 
$\mathscr{P}^{(t)}_{\xi}f\in C^{k+1}([\,\xi,\infty))$
if $f\in C^{k}([\,\xi,\infty)) $,
for all $k\in\NN_0$, any fixpoint of $\mathscr{P}^{(t)}_{\xi}$
is automatically smooth.
Moreover, differentiating the fixpoint equation shows that
any fixpoint of $\mathscr{P}^{(t)}_{\xi}$ solves (\ref{ODE-le-Wasow}).


 
To begin with we observe that, since 
$\frac{1}{r}\,(2\,\varphi_t'(r)-2r)\rightarrow0$,
as $r\rightarrow\infty$, u.i.$t$,
there is some $\tilde{\xi}\in[\max\{2,\rho_0\}\,,\,\infty)$
such that
\begin{equation}\label{hubert}
\Big|\,\frac{1}{r}\,\big(2\,\varphi_t'(r)\,-\,2\,r\big)\,\Big|
\;\klg\;
\frac{1}{2}\;,
\end{equation}
for all $r\grg\tilde{\xi}$ and $t\in[0,1]$. 
In particular, we then have
\begin{equation}\label{norbert}
\Big|\,\frac{1}{r}\,\big(2\,\varphi_t'(r)\,-\,2\,r\big)\,(z_1-z_2)\,+\,
\frac{1}{r}\,\big(z_1^2-z_2^2\big)
\,\Big|
\;\klg\;
|z_1-z_2|\;,
\end{equation}
for all $r\grg\tilde{\xi}\grg2$, $t\in[0,1]$,
and $z_1,z_2\in\RR$
with $|z_1|,|z_2|<1/2$.
Furthermore, by (\ref{hugo2}), 
for each $\omega\in\NN$,
there exist $\xi_{\omega}\in[\,\tilde{\xi},\infty)$
and $0<c_{\omega}<\infty$
such that
\begin{equation}\label{est-tildeR-div-r}
\Big|\,\frac{\widetilde{R}_t(r)}{r}\,\Big|
\;\klg\;
c_{\omega}\,r^{-\omega}
\;\klg\;
\frac{1}{2}\;,
\end{equation}
for all $r\grg\xi_{\omega}$ 
and $t\in[0,1]$.
We may suppose that the sequence $(\xi_{\omega})_{\omega\in\NN}$
is increasing. We stress that due to (\ref{hugo}) and (\ref{hugo2})
we can choose each $\xi_{\omega}$
and $c_{\omega}$ independend from $t$.
In our argumentation below we will also need
the following observation:
For some $\chi\in C_b([1,\infty)$
such that $|\chi(r)|\klg c\,r^{-\omega}$ with some
fixed $\omega\in\NN$, we have
\begin{equation}\label{est-chi}
\Big|\,\int\limits_r^{\infty}e^{-(s^2-r^2)}\,s\,\chi(s)\,ds
\,\Big|
\;\klg\;
\frac{c}{2}\,r^{-\omega}\;,\qquad r\in[1,\infty)\;.
\end{equation}
First we note that it follows from (\ref{hubert}),
(\ref{est-tildeR-div-r}) and (\ref{est-chi})
that $\mathscr{P}_{\xi_{\omega}}$ actually maps 
$\mathscr{X}_{\xi_{\omega}}$
into itself, for each $\omega\in\NN$.
Furthermore, using (\ref{norbert}) and (\ref{est-chi})
it is easily seen that each $\mathscr{P}_{\xi_{\omega}}$
is a contraction with Lipschitz constant $\frac{1}{2}$
on $\mathscr{X}_{\xi_{\omega}}$.






Now we consider, for $\omega\in\NN$,
the iteration sequence (in the following we drop
the parameter $t$ in the notation)
\[
m_0^{(\omega)}\;\equiv\,0\;,\qquad m^{(\omega)}_{\nu+1}
\;=\;\mathscr{P}_{\xi_{\omega}}m^{(\omega)}_{\nu}\;,\qquad\nu\in\NN_0\;,
\]
which converges uniformly to the unique fixpoint $m^{(\omega)}$
of $\mathscr{P}_{\xi_{\omega}}$ in 
$\mathscr{X}_{\xi_{\omega}}$.
We will show the following two inequalities
by induction on $\nu\in\NN_0$:
\begin{equation}\label{est-iteration}
|\,m^{(\omega)}_{\nu+1}\,-\,m^{(\omega)}_{\nu}\,|
\;\klg\;
\frac{1}{2^{\nu+1}}\,c_{\omega}\,r^{-\omega}\;,
\qquad
|\,m^{(\omega)}_{\nu+1}\,|
\;\klg\;
c_{\omega}\,r^{-\omega}\;\klg\;\frac{1}{2}\;,
\end{equation}
for all $\nu\in\NN_0$ and $r\in[\,\xi_{\omega},\infty)$.
Clearly this implies
\begin{equation}\label{hugo3}
|\,m^{(\omega)}(r)\,|\;\klg\;c_{\omega}\,r^{-\omega}\;,
\qquad r\in[\,\xi_{\omega},\infty)\;,
\end{equation}
for $\omega\in\NN$. Evidently, 
$m^{(\tilde{\omega})}\!\!\upharpoonright_{[\,\xi_{\omega},\infty)}
=m^{(\omega)}$, for all $\omega,\tilde{\omega}\in\NN$,
$\omega\grg\tilde{\omega}$.
So, setting $\xi:=\xi_1$, the lemma is proved
if (\ref{est-iteration}) is true.
The case $\nu=0$
follows immediatly by definition of $m^{(\omega)}_0$
and by (\ref{def-P}), (\ref{est-tildeR-div-r})
and (\ref{est-chi}). So let $\nu\in\NN$
and suppose
that (\ref{est-iteration}) holds true, for
all $0\klg\vk\klg\nu-1$. In particular, we then
have 
$|m^{(\omega)}_{\nu-1}|,|m^{(\omega)}_{\nu}|\klg1/2$.
Using the estimate (\ref{norbert}) 
we get
\[
\Big|\,\mathscr{P}_{\xi_{\omega}}m^{(\omega)}_{\nu}(r)
\,-\,\mathscr{P}_{\xi_{\omega}}m^{(\omega)}_{\nu-1}(r)\Big|
\;\klg\;
\int\limits_r^{\infty}e^{-(s^2-r^2)}\,s\,
\big|\,m^{(\omega)}_{\nu}(s)
\,-\,m^{(\omega)}_{\nu-1}(s)\,\big|\,ds\;,
\]
which is by the induction 
hypothesis 
less than or equal to
\[
\int\limits_r^{\infty}e^{-(s^2-r^2)}\,s\,
\frac{1}{2^{\nu}}\,c_{\omega}\,s^{-\omega}\,ds
\;\klg\;
\frac{1}{2^{\nu+1}}\,c_{\omega}\,r^{-\omega}\;,
\]
for all $r\in[\,\xi_{\omega},\infty)$.
Here we again used (\ref{est-chi}).
This gives the first inequality in (\ref{est-iteration}).
The second 
follows from
\[
|\,m^{(\omega)}_{\nu+1}(r)\,|
\;\klg\;
\sum\limits_{\vk=0}^{\nu}
|m^{(\omega)}_{\vk+1}-m^{(\omega)}_{\vk}|
\;\klg\;
\frac{c_{\omega}}{2}\,r^{-\omega}
\,\sum\limits_{\vk=0}^{\nu}\frac{1}{2^{\vk}}
\;\klg\;
c_{\omega}\,r^{-\omega}\;,
\]
for all $r\in[\,\xi_{\omega},\infty)$.

So far we have $m_t\sim0$,
uniformly in $t\in[0,1]$. 
Differentiating equation (\ref{ODE-le-Wasow})
succesively we also get 
the asymptotics
$\partial_r^{\alpha}m_t\sim0$, for
$\alpha\in\NN_0$, 
uniformly in $t\in[0,1]$ by induction.
\hfill$\blacksquare$


\bigskip


\noindent Now we return to our discussion
of the exact solution $f_t$ of (\ref{Dir-f}).
Having derived the previous lemma we define
$R_t(r)\;:=\;\int_{\xi}^rm_t(s)\,ds\,-\,\varphi_t(\xi)
\,-\,\ln(f_t(\xi))$, for $r\in[\xi,\infty)$,
so that $f_t(\xi)=e^{-\varphi_t(\xi)-R_t(\xi)}$.
We conclude that
$f_t\equiv e^{-\varphi_t-R_t}$ on $[\,\xi,\infty)$
using maximum principles as in 
Corollary~\ref{cor-max-prin}.
Writing $f_t=e^{-\psi_t}$ we thus have
\begin{equation}\label{asymp-psit}
\partial_r^{\beta+1}\psi_t(r)
\;\sim\;
r\,\delta_{\beta,0}\,+\,
\sum\limits_{\nu=0}^{\infty}
a_{\nu}(t)\,(-1)^{\beta}\,
\frac{(\beta+2\nu)!}{(2\nu)!}\,r^{-1-2\nu-\beta}
\;,\quad r\rightarrow\infty,\,\textrm{u.i.}\,t\;,
\end{equation}
for all $\beta\in\NN_0$.
The expression for $u_t^{\circ}=u_t^{\circ}(t)$ in Cartesian coordinates 
is then given by $u_t^{\circ}(x)=e^{-g_t(|x|)}$ where
$g_t(r):=\psi_t(r)+\frac{n-1}{2}\,\ln(r)$, for $r\in[\xi,\infty)$.
We have
$\partial_rg_t(r)=r+\cO(\frac{1}{r})$ and
\begin{eqnarray}\nonumber
\partial_{x_i}g_t(|x|)&=&x_i\,+\,\cO(\textstyle{\frac{1}{|x|}})\;,
\qquad
\partial_{x_i}\partial_{x_j}g_t(|x|)\;=\;
\delta_{ij}\,+\,\cO(\textstyle{\frac{1}{|x|^2}})\;,
\\ \label{hansi}
\partial^{\alpha}_xg_t(|x|)&=&\cO(\textstyle{\frac{1}{|x|^{|\alpha|}}})\;,
\end{eqnarray}
for $i,j\in\Lambda$ and each multiindex $\alpha$ with $|\alpha|\grg3$.
Due to (\ref{hugo}) and Lemma~\ref{le-Wasow} these $\cO$-symbols
are all uniform in $t$.

We come back to our original problem
(\ref{Dirichletexp}) and abbreviate
$\tilde{v}_t:=\exp(-\Phi_t)\!\!
\upharpoonright_S$. Of coures $\tilde{v}_t$ is
in general not radially symmetric. 
By Lemma~\ref{conv-le-sec5} we know, however,
that the functions
$t\mapsto \min_S\tilde{v}_t$ resp.
$t\mapsto \max_S\tilde{v}_t$ are continuous on $[0,1]$
and strictly positive. Hence 
$\vu:=\inf\limits_{t\in[0,1]}\min\limits_S\tilde{v}_t>0$
and $\vo:=\sup\limits_{t\in[0,1]}\max\limits_S\tilde{v}_t<\infty$ 
are attained
and we can use them to compare solutions of (\ref{Dirichletexp})
with radially symmetric solutions.
Recall that the $L^2$-solution of (\ref{Dirichletexp}) is
given by $u_t=\exp(-\Phi_t)$.
Corollary~\ref{cor-max-prin} implies
\[
\vu\,e^{-g_t}\;\klg\;u_t\;\klg\;\vo\,e^{-g_t}\qquad
\textrm{on}\;\;\Omega\;,
\]
for each $t\in[0,1]$.
Hence there is a smooth function $k_t$ on $\Omega$
such that 
\begin{equation}\label{karl}
\ln(\vu)\;\klg\; k_t\;\klg\;\ln(\vo)\qquad
\textrm{and}\qquad
u_t\;=\;e^{-g_t+k_t}\;,\qquad
t\in[0,1]\;.
\end{equation}
Moreover, we have 
$K(\partial_{\theta_i}\tilde{v}_t)
=\partial_{\theta_i}u_t=e^{-g_t+k_t}\,\partial_{\theta_i}k_t$.
This together with Corollary~\ref{cor-max-prin} gives
\[
-\infty\,<\,e^{-g_t}\,
\inf\limits_{\tau\in[0,1]}\min\limits_{S}\partial_{\theta_i}\tilde{v}_{\tau}
\,\klg\,
e^{-g_t+k_t}\,\partial_{\theta_i}k_t\,\klg\,e^{-g_t}
\sup\limits_{\tau\in[0,1]}\max\limits_{S}\partial_{\theta_i}\tilde{v}_{\tau}
\,<\,\infty\,,
\]
and therefore
\begin{equation}\label{fuzzi}
-\infty\;<\;\vo^{-1}\:\inf\limits_{\tau\in[0,1]}\min\limits_{S}
\partial_{\theta_i}\tilde{v}_{\tau}
\;\klg\;
\partial_{\theta_i}k_t\;\klg\;
\vu^{-1}\:\sup\limits_{\tau\in[0,1]}\max\limits_{S}
\partial_{\theta_i}\tilde{v}_{\tau}\;<\;\infty\;.
\end{equation}
Proceeding on in this way we get by induction
\begin{equation}\label{dalphak}
\sup_{\Omega}|\partial_{\theta}^{\alpha}k_t|\;\klg\;C(\alpha)
\;<\;\infty
\end{equation}
uniformly in $t\in[0,1]$,
for each multi-index $\alpha$. 



To obtain Proposition~\ref{main-le-sec4} some estimates on 
the radial derivatives of $k_t$ are still missing.
The differential equation (\ref{Dirichletexp})
for $u_t=u_t^{\circ}\,e^{k_t}$ written in polar coordinates
implies, for
each $\alpha\in\NN_0^{n-1}$,
\[
\Big(
-\partial_r^2\,-\,\frac{n-1}{r}\,\partial_r\,+\,r^2\,-\,\mu_t\,
\Big)\,u_t^{\circ}(r)\,\partial_{\theta}^{\alpha}e^{k_t}
\;=\;
\frac{u_t^{\circ}(r)}{r^2}\,\partial_{\theta}^{\alpha}\Delta_{\theta}e^{k_t}\;.
\]
Since $u_t^{\circ}$ solves (\ref{Dir-u0}) we obtain
\[
\Big(
\partial_r\,+\,\frac{2\,\partial_ru_t^{\circ}}{u_t^{\circ}}\,
+\,\frac{n-1}{r}\,
\Big)\,\partial_r\partial_{\theta}^{\alpha}e^{k_t}
\;=\;
\frac{1}{r^2}\,\partial_{\theta}^{\alpha}\Delta_{\theta}e^{k_t}\;.
\]
Due to (\ref{dalphak}) the right side of the last equation
is $\cO(\frac{1}{r^2})$ uniformly in $t\in[0,1]$.
Since $\frac{2\,\partial_ru_t^{\circ}}{u_t^{\circ}}\,
+\,\frac{n-1}{r}=-2\psi_t'$
we thus get
\begin{equation}\label{rudi}
\partial_r\big(\,e^{-2\,\psi_t}\,\partial_r
\,\partial_{\theta}^{\alpha}e^{k_t}\,\big)
\;=\;e^{-2\,\psi_t}\cO(\textstyle{\frac{1}{r^2}})\;.
\end{equation}
Using Lemma~\ref{exp-decay-wt} we easily see that
$e^{-2\,\psi_t}\,\partial_r
\,\partial_{\theta}^{\alpha}e^{k_t}\longrightarrow0$,
as $r\rightarrow\infty$.
Integrating (\ref{rudi}) we hence obtain
\[
\partial_r\partial_{\theta}^{\alpha}e^{k_t}\;=\;
\int\limits_{r}^{\infty}e^{2\,(\psi_t(r)\,-\,\psi_t(s)\,)}
\cO(\textstyle{\frac{1}{s^2}})
\,ds\;.
\]
Since we have the asymptotic expansion (\ref{asymp-psit})
for $\psi_t$, and since
$r^2-s^2\klg2r(r-s)$, for $r\klg s$,
we have
$2\,(\psi_t(r)\,-\,\psi_t(s)\,)\klg(2r-c)(r-s)$, for some
$t$-independent constant $c>0$.
This yields
$\partial_r\partial_{\theta}^{\alpha}e^{k_t}\;=
\;\cO(\textstyle{\frac{1}{r^3}})$
u.i.$t$.
Differentiating (\ref{rudi}) succesively with respect to
$r$ and using (\ref{asymp-psit}) we can repeat the previous
argument and get by induction
\[
\partial^{\beta}_r\partial^{\alpha}_{\theta}\,e^{k_t}
\;=\;
\cO(\textstyle{\frac{1}{r^{2+\beta}}})\;,\qquad
\textrm{for}\;\beta\in\NN\;,
\qquad\textrm{u.i.}\,t\,.
\]
Differentiating $k_t=\ln(e^{k_t})$ we derive from this
again by induction
\[
\partial^{\beta}_r\partial^{\alpha}_{\theta}\,k_t
\;=\;
\cO(\textstyle{\frac{1}{r^{2+\beta}}})\;,\qquad
\textrm{for}\;\beta\in\NN\;,
\qquad\textrm{u.i.}\,t\,.
\]
In Cartesian coordinates this reads
\begin{equation}\label{hansi2}
\partial^{\alpha}_xk_t\;=\;
\cO(\textstyle{\frac{1}{|x|^{|\alpha|}}})\;,\qquad
\textrm{for}
\;\alpha\in\NN_0^{n},\,\alpha\not=0\,,\;
\qquad\textrm{u.i.}\,t\,.
\end{equation}
Since 
$\Phi_t=g_t-k_t$,
Proposition~\ref{main-le-sec4} finally follows from 
(\ref{hansi}) and (\ref{hansi2}).




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






\section{Maximum principles and the phase of the ground state
eigenfunction of $\PP$}
\label{max-prin}

In this section we prove Theorems~\ref{thm-sj1} and~\ref{thm-sj2}.

\bigskip


\textsc{Proof of Theorem~\ref{thm-sj1}:}
We fix $h\in(0,1]$ and $\Lambda\in\LL$. To begin with
we pick some $\rho\in C^{\infty}(\RR,[0,1])$ such that
$\rho\equiv1$ on $(-\infty,1]$ and 
$\rho\equiv0$ on $[2,\infty)$. We set 
$\chi_{\ve}(x)=\rho(\ve\ln|x|)$, for  
$\ve>0,\,x\in\RR^{\Lambda}\setminus\{0\}$, and
$\chi_{\ve}(0)=1$, for $\ve>0$,
such that $\chi_{\ve}\equiv1$ on $\{|x|\klg e^{1/\ve}\}$
and $\chi_{\ve}\equiv0$ on $\{|x|\grg e^{2/\ve}\}$.
Then $\partial^{\alpha}_x\chi_{\ve}=\cO(\ve/|x|^{|\alpha|})$,
for each multi-index $\alpha$.
 

Furthermore we define, for $\ve>0$ and $t\in[0,1]$,
\begin{eqnarray}\label{def-Vve}
V_{\ve}&=&
\lambda_{\min}^2\:(1-\chi_{\ve})\,x^2\,+\,\chi_{\ve}\,V_{\Lambda}
\;,\nonumber
\\ \label{def-Vvet}
V_{\ve,t}&=&\lambda_{\min}^2\,(1-t)\,x^2+\,
t\, V_{\ve}\;
=\;
\lambda_{\min}^2\,x^2+\,t\,\chi_{\ve}
\big(\,V_{\Lambda}-
\lambda_{\min}^2\,x^2\,\big)\;.
\end{eqnarray}
Then we observe that
\begin{equation}\label{nadja1}
{\textstyle\frac{1}{2}}\:
V_{\ve,t}''\;=\;D_{\ve,t}^2\,+\,Y_{\ve,t}
\end{equation}
holds with
\begin{equation}\label{achim}
D_{\ve,t}^2\;:=\;t\,\big(
\chi_{\ve}D_{\Lambda}^2\,+\,\lambda_{\min}^2(1-\chi_{\ve})\,\id\big)
\,+\,(1-t)\,\lambda_{\min}^2\,\id
\;\longrightarrow\;\lambda_{\min}^2\,\id\;,
\end{equation}
as $|x|\rightarrow\infty$.
Due to (\ref{hyp1}) and the properties
of $\chi_{\ve}$,
the smooth
matrix-valued function $Y_{\ve,t}$ satisfies
\begin{equation}\label{Yest}
\sup\limits_{x\in\RR^{\lambda}}
\|\,Y_{\ve,t}(x)\,\|_{\LO(\el{\infty}{o}{\vk})}
\;\klg\;t\,\gamma\,+\,t\,\cO_{\Lambda}(\ve)\;.
\end{equation}
Here the $\Lambda$-index on the $\cO$-symbol
indicates that the latter might depend on~$\Lambda$.
This causes no problem since our results
in the limits $t\rightarrow1$ and $\ve\rightarrow0$ below will again
be uniform in $\Lambda$.



For later use  we note that
\begin{equation}
\label{norm-res-conv-t}
\big\|\,(-h^2\Delta+V_{\ve,t}+i)^{-1}-(-h^2\Delta+V_{\ve,t_0}+i)^{-1}\,\big\|
\;\longrightarrow\;0\quad\textrm{as}\;\;
t\rightarrow t_0\;,
\end{equation}
for each $t_0\in[0,1]$, and   
\begin{equation}\label{norm-res-conv-ve}
\big\|\,(-h^2\Delta+V_{\ve}+i)^{-1}-(-h^2\Delta+V_{\Lambda}+i)^{-1}\,\big\|
\;\longrightarrow\;0\quad\textrm{as}\;\;
\ve\rightarrow0\;.
\end{equation}
Here (\ref{norm-res-conv-ve}) is true since we can write
\begin{eqnarray*}
\lefteqn{
(-h^2\Delta+V_{\ve}+i)^{-1}-(-h^2\Delta+V_{\Lambda}+i)^{-1}
}
\\
&=&
(-h^2\Delta+V_{\Lambda}+i)^{-1}\,(1-\chi_{\ve})\,\big\{
(V_{\Lambda}-\lambda_{\min}^2\,x^2)(-h^2\Delta+V_{\ve}+i)^{-1}
\big\}\;,
\end{eqnarray*}
where the operator in the curly brackets is bounded uniformly in $\ve>0$,
$(1-\chi_{\ve})$ converges weakly to zero, as $\ve\rightarrow0$,
and $(-h^2\Delta+V+i)^{-1}$ is compact.

Let
$\exp(-\,\frac{1}{h}\,\Phi_{\ve,t})$
denote the normalized strictly positive eigenfunction
corresponding to the lowest eigenvalue $\mu_{\ve,t}$
of $-\,h^2\,\Delta+V_{\ve,t}$, for $\ve>0,\,t\in[0,1]$.
Of course both $\Phi_{\ve,t}$ and $\mu_{\ve,t}$ also
depend on $h$ and $\Lambda$.
Then, after a scaling, Proposition~\ref{main-le-sec4} 
and (\ref{achim}) imply
\begin{eqnarray}\nonumber
\Phi_{\ve,t}''(x)\,-
\,\lambda_{\min}\,\id
&\rightarrow&0\qquad\textrm{as}\;\;|x|\rightarrow\infty\;,
\\ \label{nadja2}
\Rightarrow\quad
\Psi_{\ve,t}''(x)\;:=\;\Phi_{\ve,t}''(x)\,-\,D_{\ve,t}(x)
&\rightarrow&0\qquad\textrm{as}\;\;|x|\rightarrow\infty\;,
\end{eqnarray}
for each $\ve>0$, $t\in[0,1]$.
So the supremum in the next definition is 
actually attained,
\begin{equation}\label{sup-Psivet}
m_{\ve}(t)\;:=\;\sup\limits_{x\in\RR^{\Lambda}}
\|\,\Psi_{\ve,t}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}\;,
\qquad\ve>0\;,\,t\in[0,1]\;.
\end{equation}


We indicate briefly why
$m_{\ve}$ is uniformly continuous on $[0,1]$
as a function of~$t$. 
In order to show this it is of course sufficient to prove that
$m_{\ve}$ is continuous at $t_0\in[0,1]$.
Due to (\ref{def-Vvet}) and (\ref{norm-res-conv-t}) the assumptions of 
Proposition~\ref{main-le-sec4} and Lemma~\ref{conv-le-sec5}
are fulfilled by $V_{\ve,t}$ after a scaling, 
for each fixed $\ve>0$.
So let $\eta>0$. By Proposition~\ref{main-le-sec4} there is some
$R=R(\ve,\eta,\Lambda)>0$ such that
\[
\sup\limits_{t\in[0,1]}
\sup\limits_{|x|>R}\|\,\Psi_{\ve,t}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}
\;<\;\frac{\eta}{2}\;.
\]
Having chosen such an $R>0$ it suffices to observe that due to
Lemma~\ref{conv-le-sec5} there is some
$\delta=\delta(R)>0$ such that
\[
\sup\limits_{t\in[0,1]\cap(t_0-\delta\,,\,t_0+\delta)}
\sup\limits_{|x|\klg R}\|\,\Psi_{\ve,t}''(x)\,-\,\Psi_{\ve,t_0}''(x)\,
\|_{\LO(\el{\infty}{o}{\vk})}
\;<\;\eta\;.
\]
This shows that $m_{\ve}$ is uniformly continuous on $[0,1]$.



Proceeding on in the proof of the lemma
we note that the eigenvalue equation for 
$\exp(-\,\frac{1}{h}\,\Phi_{\ve,t})$ yields
\[
V_{\ve,t}\,-\,\mu_{\ve,t}
\;=\;\,(\nabla \Phi_{\ve,t})^2\,-\,h\,
\,\Delta \Phi_{\ve,t}\;.
\]
Differentiating this identity twice we get
\begin{eqnarray}
\lefteqn{
\SL v\,,V_{\ve,t}''\,w\SR\,
-\,2\,\nabla\Phi_{\ve,t}\cdot\nabla\SL v\,,D_{\ve,t}\,w\SR
\,+\,h\,\Delta\SL v\,,D_{\ve,t}\,w\SR
}  \nonumber  
\\  \label{2.Ableitung}
&=&2\,\nabla\Phi_{\ve,t}\cdot\nabla\SL v\,,\Psi_{\ve,t}''\,w\SR
\,-\,\,h\,\Delta\SL v\,,\Psi_{\ve,t}''\,w\SR
\,+\,2\,\SL v\,,(\Phi_{\ve,t}'')^2\,w\SR\;,
\end{eqnarray}
for $w\in\el{\infty}{o}{\vk}$, $v\in\el{1}{o}{-\vk}$.
In view of (\ref{nadja1}) and (\ref{nadja2}) 
and since $\Psi_{\ve,t}''$ is symmetric
(\ref{2.Ableitung}) is
equivalent to
\begin{eqnarray}
\lefteqn{
\SL v\,,Y_{\ve,t}\,w\SR\,
-\,\nabla\Phi_{\ve,t}\cdot\nabla\SL v\,,D_{\ve,t}\,w\SR
\,+\,{\textstyle\frac{h}{2}}\:\Delta\SL v\,,D_{\ve,t}\,w\SR
}  \nonumber
\\ \nonumber
&=&\nabla\Phi_{\ve,t}\cdot\nabla\SL v\,,\Psi_{\ve,t}''\,w\SR
\,-\,\,{\textstyle\frac{h}{2}}\:\Delta\SL v\,,\Psi_{\ve,t}''\,w\SR
\\
& &  \label{2.Ableitung'}
+\,
\SL D_{\ve,t} v\,,\Psi_{\ve,t}''\,w\SR
\,+\,\SL \Psi_{\ve,t}''\,v\,,D_{\ve,t}\,w\SR
\,+\,\SL v\,,(\Psi_{\ve,t}'')^2\,w\SR\,,
\end{eqnarray}
for $w\in\el{\infty}{o}{\vk}$, $v\in\el{1}{o}{-\vk}$.
We observe that,
by our choice of $\chi_{\ve}$,
we have
\begin{equation}\label{oben1}
\nabla D_{\ve,t}=t\,\cO_{\Lambda}(\ve/|x|)\;,\qquad
\Delta D_{\ve,t}=t\,\cO_{\Lambda}(\ve/|x|^2)\;.
\end{equation}
Writing
\[
\nabla\Phi_{\ve,t}(x)\;=\;\nabla\Phi_{\ve,t}(0)
\,+\,\int\limits_0^1\Phi_{\ve,t}''(\tau\,x)\,x\,d\tau
\]
and observing that $\|x\|_{\el{\infty}{o}{\vk}}\klg|x|$ and
\begin{equation}\label{bounds-Dvet}
\lambda_{\min}\,\id\;\klg\;D_{\ve,t}(x)\;\klg\;
\lambda_{\max}\,\id\;,
\qquad
x\in\RR^{\Lambda}\;,\,\ve>0\;,\,t\in[0,1]\;,
\end{equation}
we get
\begin{equation}\label{oben2}
|\nabla\Phi_{\ve,t}(x)|\;\klg\;
|\nabla\Phi_{\ve,t}(0)|\,+\,\lambda_{\max}\,|x|
\,+\,m_{\ve}(t)\,|x|\;.
\end{equation}
Here $|\nabla\Phi_{\ve,t}(0)|\klg c$, for some $c<\infty$,
uniformly in $t\in[0,1]$ and $\ve>0$ because of
(\ref{norm-res-conv-t}), (\ref{norm-res-conv-ve}) and
Lemma~\ref{conv-le-sec5}.
Due to (\ref{Yest}), (\ref{oben1}), and (\ref{oben2}),
we have the following upper bound on the left hand side of
(\ref{2.Ableitung'}),
for 
$\|v\|_{\el{1}{o}{-\vk}}=\|w\|_{\el{\infty}{o}{\vk}}=1$, 
\begin{eqnarray}
\lefteqn{\label{ulf}
\SL v\,,Y_{\ve,t}\,w\SR\,
-\,\nabla\Phi_{\ve,t}\cdot\nabla\SL v\,,D_{\ve,t}\,w\SR
\,+\,{\textstyle\frac{h}{2}}\:\Delta\SL v\,,D_{\ve,t}\,w\SR
}
\\
&\klg&\nonumber
\|\,Y_{\ve,t}\,+\,\nabla\Phi_{\ve,t}\cdot 
\nabla D_{\ve,t}\,+\,
\,{\textstyle\frac{h}{2}}\:\Delta D_{\ve,t}
\,\|_{\LO(\el{\infty}{o}{\vk})}
\\
&\klg&\nonumber
t\,\big(\gamma+\cO_{\Lambda}(\ve)\big)\,+\,t\,\big(\,c\,+\,\lambda_{\max}|x|
\,+\,m_{\ve}(t)\,|x|\,\big)\,\cO_{\Lambda}
({\textstyle\frac{\ve}{|x|}})
\,+\,t\,h\,\cO_{\Lambda}({\textstyle\frac{\ve}{|x|^2}})\;.
\end{eqnarray}
Let
$x_{\ve,t}\in\RR^{\Lambda}$ denote the point
where the supremum (\ref{sup-Psivet}) it attained.
We find $r\in\el{\infty}{o}{\vk}$, $s\in\el{1}{o}{-\vk}$ 
with $\|r\|_{\el{\infty}{o}{\vk}}=\|s\|_{\el{1}{o}{-\vk}}=1$,
such that $\SL s\,,\Psi_{\ve}''(x_{\ve,t})\,r\SR=m_{\ve}(t)$.
In particular
$\SL s\,,\Psi_{\ve,t}''(\cdot)\,r\SR\in 
C^{\infty}(\RR^{\Lambda},\RR)$
has a local maximum at $x_{\ve,t}$, that is
\begin{equation}\label{ulf2}
\nabla\Phi_{\ve,t}(x_{\ve,t})\cdot\nabla\SL s\,,\Psi_{\ve,t}''(x_{\ve,t})\,r\SR
\;=\;0\;,\qquad
-\,{\textstyle\frac{h}{2}}\:
\Delta\SL s\,,\Psi_{\ve,t}''(x_{\ve,t})\,r\SR\;\grg\;0\;.
\end{equation}
From (\ref{2.Ableitung'}), (\ref{ulf}) and (\ref{ulf2}) we obtain
\begin{eqnarray}
\lefteqn{\nonumber
t\,\big(\gamma+\cO_{\Lambda}(\ve)\big) 
\,+\,t\,m_{\ve}(t)\,\cO_{\Lambda}(\ve)
}
\\ \nonumber
&\grg&\SL D_{\ve,t}(x_{\ve,t}) s\,,\Psi_{\ve,t}''(x_{\ve,t})\,r\SR
\,+\,
\SL \Psi_{\ve,t}''(x_{\ve,t})\,s\,,D_{\ve,t}(x_{\ve,t})\,r\SR
\\
& &\label{kalle}
+\SL s\,,(\Psi_{\ve,t}''(x_{\ve,t}))^2\,r\SR\;.
\end{eqnarray}
Since
\begin{eqnarray*}
\lefteqn{
\SL  s\,,\Psi_{\ve,t}''(x_{\ve,t})\,r\SR
\;=\;
\,\SL \Psi_{\ve,t}''(x_{\ve,t})\,s\,,r\SR
\;=\;
\|s\|_{\el{1}{o}{-\vk}}\,\|\Psi_{\ve,t}''(x_{\ve,t})\,r\|_{\el{\infty}{o}{\vk}}
}
\\
&=&
\|\Psi_{\ve,t}''(x_{\ve,t})\,s\|_{\el{1}{o}{-\vk}}\,\|r\|_{\el{\infty}{o}{\vk}}
\;=\;
\|\Psi_{\ve,t}''(x_{\ve,t})\|_{\LO(\el{\infty}{o}{\vk})}
\;=\;m_{\ve}(t)\;,
\end{eqnarray*}
we see that
$s_j(\Psi_{\ve,t}''(x_{\ve,t})\,r)_j\grg0$ and
$(\Psi_{\ve,t}''(x_{\ve,t})\,s)_jr_j\grg0$, for each $j\in\Lambda$.
Since $D_{\ve,t}$ is diagonal, this 
together with (\ref{bounds-Dvet}) implies
\[
\SL D_{\ve,t}(x_{\ve,t}) s\,,\Psi_{\ve,t}''(x_{\ve,t})\,r\SR
\;\grg\;
\lambda_{\min}\,m_{\ve}(t)\;.
\]
Altogether the previous estimate, its analogue
for the second term in the second line
of (\ref{kalle}), and (\ref{kalle}) itself, lead to
\[
t\,(\gamma\,+\,\cO_{\Lambda}(\ve))
\;\grg\;\big(\,2\,\lambda_{\min}\,-\,t\,\cO_{\Lambda}(\ve)\,\big)
\,m_{\ve}(t)\,-\,m_{\ve}(t)^2\;.
\]
Now we assume that $m_{\ve}(t)\klg\lambda_{\min}$.
Solving the last inequality for $m_{\ve}(t)$ 
shows that, for sufficiently small $\ve>0$,
the only possibility 
for this to be fulfilled is
\begin{equation}\label{bound-mvet}
m_{\ve}(t)\;\klg\;
\lambda_{\min}\,-\,(\lambda_{\min}^2-t\,\gamma)^{1/2}\,
+\,t\,\cO_{\Lambda}(\ve)
\,+\,t^{1/2}\,\cO_{\Lambda}(\ve^{1/2})\;.
\end{equation}


From this we obtain the theorem as follows.
Let $0<\eta<(\lambda_{\min}^2-\gamma)^{1/2}$.
By (\ref{bound-mvet}) we find some $\ve_0>0$
such that, for all $0<\ve\klg\ve_0$ and $t\in[0,1]$,
we have
\[
m_{\ve}(t)\klg\lambda_{\min}\quad
\Longrightarrow\quad
m_{\ve}(t)\klg
\lambda_{\min}-(\lambda_{\min}^2-\gamma)^{1/2}+\eta
\,=\,C_{\gamma}+\eta
\,<\,\lambda_{\min}\;.
\]
The uniform continuity of $m_{\ve}$ and $m_{\ve}(0)=0$ show that,
for all $0<\ve\klg\ve_0$, there
is some $t_0\in(0,1]$
such that
$m_{\ve}(t_0)\klg\lambda_{\min}$.
Then we get after a simple bootstrap argument
\begin{equation}\label{bound-mve}
\sup\limits_{x\in\RR^{\Lambda}}
\|\,\Psi_{\ve,1}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}
\;=\;
m_{\ve}(1)\;\klg\;
C_{\gamma}+\eta\;,
\end{equation}
for all $0<\ve\klg\ve_0$.
Due to (\ref{norm-res-conv-ve})
we are again allowed to apply
Lemma~\ref{conv-le-sec5}.
Thus we see that, for all $R>0$, there is
some $0<\ve_1\klg\ve_0$ such that, for all $0<\ve\klg \ve_1$,
we have
\[
\sup\limits_{|x|\klg R}\|\Psi''(x)\|_{\LO(\el{\infty}{o}{\vk})}
\;\klg\;
\sup\limits_{|x|\klg R}
\|\,\Psi_{\ve,1}''(x)\,\|_{\LO(\el{\infty}{o}{\vk})}\,+\,\eta
\;\klg\;
C_{\gamma}\,+\,2\,\eta\;.
\] 
Letting $R$ tend to infinity
gives the result since $\eta>0$ was arbitrarily small.
\hfill$\blacksquare$ 


\bigskip



\textsc{Proof of Theorem~\ref{thm-sj2}:}
The parameter $t$ is no longer needed. That is we set
$t=1$ in the definition of the objects introduced
in the preceeding proof and drop this from the notation.
Differentiating (\ref{2.Ableitung}) once more gives, for $\ve>0$,
$u\in\el{\infty}{o}{1}$, $v\in\el{1}{o}{-2}$,
$w\in\el{\infty}{o}{1}$
\begin{eqnarray*}
\lefteqn{
\Big\langle
v\,,\big(V^{(3)}_{\ve}u-2\,\nabla\Phi_{\ve}\cdot\nabla
D_{\ve}'u\big)w\Big\rangle
}
\\
& & -2\,\nabla(\Phi_{\ve}'u)\cdot\nabla\langle v\,,D_{\ve}w\rangle
\,+\,h\,\Delta\langle v\,,(D_{\ve}'u)w\rangle
\\
&=&
2\,\nabla\Phi_{\ve,t}\cdot\nabla\langle v\,,(\Psi_{\ve}^{(3)}u)w\rangle
\,+\,
2\,\nabla(\Phi_{\ve}'u)\cdot\nabla\langle v\,,\Psi_{\ve}''w\rangle
\\
& &  \,-\,
h\,\Delta\langle v\,,(\Psi_{\ve}^{(3)}u)w\rangle
\,+\,2\,\langle v\,,(\Phi_{\ve}^{(3)}u)\Phi_{\ve}''w\rangle
\,+\,2\,\langle v\,,\Phi_{\ve}''(\Phi_{\ve}^{(3)}u)w\rangle\;.
\end{eqnarray*}
Since $\lambda_{\max}$ is an upper bound
for the eigenvalues of $D_{\ve}(x)$,
uniformly in $x\in\RR^{\Lambda}$ and $\ve>0$, 
Recalling $\Phi_{\ve}''=D_{\ve}+\Psi_{\ve}''$
we see that this is equivalent to
\begin{eqnarray}
\lefteqn{\nonumber
\nabla\Phi_{\ve}\cdot\nabla\langle v\,,(\Psi_{\ve}^{(3)}u)w\rangle
\,+\,3\lambda_{\max}\,\langle v\,,(\Psi_{\ve}^{(3)}u)w\rangle
}
\\
& &\nonumber
\,+\,\langle
v\,,\big(D_{\ve}-\lambda_{\max}\,\id+\Psi_{\ve}''\big)
\,(\Psi_{\ve}^{(3)}u)w\rangle
\\
& &\nonumber
\,+\,\langle
v\,,(\Psi_{\ve}^{(3)}u)\,
\big(D_{\ve}-\lambda_{\max}\,\id+\Psi_{\ve}''\big)w\rangle
\\
& &\nonumber
\,+\,\langle
v\,,\big(\Psi_{\ve}^{(3)}
\big(D_{\ve}-\lambda_{\max}\,\id+\Psi_{\ve}''\big)u\big)w\rangle
\\
& &\nonumber
\,-\,
{\textstyle\frac{h}{2}}\,\Delta\langle v\,,(\Psi_{\ve}^{(3)}u)w\rangle
\\
&= &\nonumber
\Big\langle
v\,,\big({\textstyle\frac{1}{2}}\,
V^{(3)}_{\ve}u-\nabla\Phi_{\ve}\cdot\nabla
D_{\ve}'u\big)w\Big\rangle
\,+\,
{\textstyle\frac{h}{2}}\,\Delta\langle v\,,(D_{\ve}'u)w\rangle
\\
& &\label{karin}
\,-\,
\langle v\,,\Phi_{\ve}''(D_{\ve}'u)w\rangle
\,-\,
\langle v\,,(D_{\ve}'u)\Phi_{\ve}''w\rangle
\,-\,
\langle v\,,(D_{\ve}'\Phi_{\ve}''u)w\rangle\;.
\end{eqnarray}
First we estimate the right hand side of (\ref{karin})
from above assuming that $u$, $v$, and $w$ are
normalized,
\begin{equation}\label{uvw}
\|u\|_{\el{\infty}{o}{1}}\;=\;\|v\|_{\el{1}{o}{-2}}
\;=\;\|w\|_{\el{\infty}{o}{1}}\;=\;1\;.
\end{equation} 
The proof of the previous theorem,
in particular (\ref{bounds-Dvet}), (\ref{oben2}) and the remark following
it, as well as (\ref{bound-mve}),
shows that, for all $x\in\RR^{\Lambda}$,
\begin{eqnarray}\label{hasso1}
|\nabla\Phi_{\ve}(x)|&\klg&
c\,+\,\big(\,
\lambda_{\max}\,+\,C_{\gamma}\,+\,\cO_{\Lambda}(\ve)\,
\big)\,|x|\;,
\\ \label{hasso2a}
\|\Psi_{\ve}''(x)\|_{\LO(\el{\infty}{o}{1})}
&\klg&
C_{\gamma}\,+\,\cO_{\Lambda}(\ve)
\;=\; 
\lambda_{\min}-(\lambda_{\min}^2-\gamma)^{1/2}
\,+\,\cO_{\Lambda}(\ve)\;,
\\ \label{hasso2}
\|\Phi_{\ve}''(x)\|_{\LO(\el{\infty}{o}{1})}
&\klg&
\lambda_{\max}\,+\,C_{\gamma}\,+\,\cO_{\Lambda}(\ve)
\;.
\end{eqnarray}
We recall that 
\begin{equation}\label{hasso}
\partial_x^{\beta}D_{\ve}\;=\;\cO_{\Lambda}(\ve/|x|^{|\beta|})\;,
\end{equation}
for each multi-index $\beta\in\NN_0^{\Lambda}$.
A straight forward calculation 
using (\ref{hasso}) and (\ref{hyp2}) 
yields
\begin{equation}\label{hasso99}
\|\,V^{(3)}_{\ve}\,\|_{\LO(\el{\infty}{o}{1},
\LO(\el{\infty}{o}{1},\el{\infty}{o}{2}))}
\;\klg\;
\Gamma\,+\,\cO_{\Lambda}(\ve)\;.
\end{equation}
Combining (\ref{hasso1}), (\ref{hasso2}), (\ref{hasso}),
and (\ref{hasso99}),
we find that, under the assumption (\ref{uvw}),
the right hand side of (\ref{karin})
is bounded from above by $\Gamma/2+\cO_{\Lambda}(\ve)$.







Now we consider the left hand side of (\ref{karin}).
Proposition~\ref{main-le-sec4} again implies that
$\Psi_{\ve}^{(3)}(x)\rightarrow0$, as 
$|x|\rightarrow\infty$. Proceeding as in the last theorem
we pick some $x_{\ve}\in\RR^{\Lambda}$ where
$\|\Psi_{\ve}^{(3)}(\cdot)\|
_{\LO(\el{\infty}{o}{1},\LO(\el{\infty}{o}{1},\el{\infty}{o}{2}))}$
attains its maximum. 
We fix $u$, $v$, and $w$, such that 
\begin{equation}\label{hasso3}
\SL v\,,(\Psi_{\ve}^{(3)}(x_{\ve})u)w\SR\;=\;
\|\Psi_{\ve}^{(3)}(x_{\ve})\|
_{\LO(\el{\infty}{o}{1},
\LO(\el{\infty}{o}{1},\el{\infty}{o}{2}))}\;=:\;M_{\ve}\;,
\end{equation}
in addition to (\ref{uvw}).
So we have
\begin{equation}\label{hasso4}
\nabla\Phi_{\ve}(x_{\ve})\cdot\nabla\langle v\,,
(\Psi_{\ve}^{(3)}(x_{\ve})u)w\rangle
\;=\;0\;,
\qquad
-\,
{\textstyle\frac{h}{2}}
\,\Delta\langle v\,,(\Psi_{\ve}^{(3)}(x_{\ve})u)w\rangle
\;\grg\;0\;.
\end{equation}
Furthermore, (\ref{bounds-Dvet}) implies
\begin{equation}\label{hasso5}
\|D_{\ve}(x_{\ve})-\lambda_{\max}\,\id\|_{\LO(\el{\infty}{o}{1})}
\;\klg\;\lambda_{\max}-\lambda_{\min}\;.
\end{equation}
If we combine 
(\ref{uvw}), (\ref{hasso2a}), (\ref{hasso3}), (\ref{hasso4}),  
and (\ref{hasso5}), with our upper bound on the
right hand side of (\ref{karin})
we arrive at
\[
3\lambda_{\max}\,M_{\ve}
-
3\big(\lambda_{\min}-(\lambda_{\min}^2-\gamma)^{1/2}
+\cO_{\Lambda}(\ve)+\lambda_{\max}-\lambda_{\min}\big)M_{\ve}
\,\klg\,
{\textstyle\frac{\Gamma}{2}}+\cO_{\Lambda}(\ve)
\,.
\]
This gives
\[
M_{\ve}
\;\klg\;
\frac{1}{6}\,\frac{\Gamma\,+\,\cO_{\Lambda}(\ve)}
{(\lambda_{\min}^2-\gamma)^{1/2}-\cO_{\Lambda}(\ve)}\;,
\]
for $\ve>0$ small enough. 
Now the theorem follows from arguments similar to those
at the end of the proof of the preceeding theorem.
\hfill$\blacksquare$







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






\section{Weighted estimates on the phase
of the\\ ground state eigenfunction of $\PP$}
\label{sec-est-on-the-phase}

\noindent In the following
proposition we apply the estimates on the second
and third derivatives of $\Phi_{h,\Lambda}$ obtained from 
Theorem~\ref{thm-sj1} and~\ref{thm-sj2}
to show that $\Phi_{h,\Lambda}$ fulfills Hypothesis~\ref{hypphi}. 


\begin{proposition}\label{prop-est-on-the-phase}
The phase $\Phi_{h,\Lambda}$ of the ground state 
eigenfunction of $\PP$
is strictly convex. More precisely we have
\begin{equation}\label{low-up-bounds-Phi}
0\;<\;\lambda_{\min}\,-\,C_{\gamma}\;\klg\;\Phi_{h,\Lambda}''(x)
\;\klg\;\lambda_{\max}\,+\,C_{\gamma}\;,
\end{equation}
uniformly in $h>0$, $\Lambda\in\LL$, $x\in\RR^{\Lambda}$.
Let $x_0=x_0(h)\in\RR^{\Lambda}$ denote the point
where
$\Phi_{h,\Lambda}$ attains its unique minimum.
Then there is some uniform constant 
$K>0$
such that
\begin{equation}\label{est-sec4}
|\,\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\,|
\;\klg\;K\;\sum\limits_{k\in\Lambda}
e^{-d_{\Lambda}(i,j)-d_{\Lambda}(j,k)}\,|\Phi_k'(x)|
\end{equation}
for all $h\in(0,1]$, $\Lambda\in\LL$, $i,j\in\Lambda$ and 
$x\in\RR^{\Lambda}$. 
\end{proposition}



\textsc{Proof of Theorem~\ref{main-thm}:}
The inequalities (\ref{low-up-bounds-Phi})
show
in particular that
condition (\ref{muu-moo}) holds with 
$\muu:=2(\lambda_{\min}\,-\,C_{\gamma})$ and
$\moo:=2(\lambda_{\max}\,+\,C_{\gamma})$.
Furthermore we readily verify that (\ref{est-bijk}) and (\ref{BB'})
hold by (\ref{est-sec4}) if we set
$b_{ijk}(h,\Lambda):=K\,e^{-d_{\Lambda}(i,j)-d_{\Lambda}(j,k)}$, 
for $h\in(0,1]$, $\Lambda\in\LL$, $i,j,k\in\Lambda$,
and $B:=K\,C_d$, $B':=K^2\,C_d^2$.
Theorem~\ref{main-thm} is then a direct consequence
of Theorem~\ref{mainthmWL} because of (\ref{conn-PP-WL}).
\hfill $\blacksquare$


\bigskip


\textsc{Proof of Proposition~\ref{prop-est-on-the-phase}:} 
The validity of (\ref{low-up-bounds-Phi})
has been
observed earlier. See for the remarks below (\ref{susi}).


In order to prove (\ref{est-sec4})
we we first observe that,
for $h\in(0,1]$, $\Lambda\in\LL$, 
and $x\in\RR^{\Lambda}$,
we have the following estimate on the second partial
derivatives of $\Psi_{h,\Lambda}=\Phi_{h,\Lambda}
-\,\frac{1}{2}\,x\cdot D_{\Lambda}x$ 
with respect to $k,l\in\Lambda$,
\begin{eqnarray}
|\,\Psi_{kl}''(x)\,|
&=&
|\SL e_k\,,\,\Psi_{h,\Lambda}''(x)\,e_l\SR|\nonumber
\\  \nonumber
&\klg&
\|\,\Psi_{h,\Lambda}''(x)\,\|_{\LO(\el{\infty}{k}{2})}
\,\|e_k\|_{\el{1}{k}{-2}} 
\,\|e_l\|_{\el{\infty}{k}{2}}
\\  \label{est-Psikl}
&=&
e^{-2\,d_{\Lambda}(k,l)}\,
\|\,\Psi_{h,\Lambda}''(x)\,\|_{\LO(\el{\infty}{k}{2})}
\;\klg\;
C_{\gamma}\,e^{-2\,d_{\Lambda}(k,l)}\;.
\end{eqnarray}
In the second and third step we have observed that
the dual space of $\el{\infty}{k}{2}$
is $\el{1}{k}{-2}$ and 
$\|e_k\|_{\el{1}{k}{-2}}=1$.
The last inequality is a direct consequece
of (\ref{hyp1}) with $\vk=2$
together with Theorem \ref{thm-sj1}.

In a similar fashion (\ref{hyp1}) with $\vk=1$ and
(\ref{hyp2}) together with 
Theorem \ref{thm-sj2}
show that 
\begin{eqnarray}  
|\,\Psi^{(3)}_{ijk}(x)\,|
&=&
|\SL e_i\,,\,(\Psi_{h,\Lambda}^{(3)}(x)e_j)\,e_k\SR| \nonumber
\\ \nonumber
&\klg&
\|\,\Psi_{h,\Lambda}^{(3)}(x)\,\|_{\LO(\el{\infty}{i}{1},\,\LO(\el{\infty}{i}{1},\el{\infty}{i}{2}))}\,
\|e_i\|_{\el{1}{i}{-2}}
\,\|e_j\|_{\el{\infty}{i}{1}}
\,\|e_k\|_{\el{\infty}{i}{1}}
\\ \nonumber
&\klg&
C^{(3)}\,
\|e_i\|_{\el{1}{i}{-2}}
\,\|e_j\|_{\el{\infty}{i}{1}}
\,\|e_k\|_{\el{\infty}{i}{1}}
\\ \label{est-Psiijk}
&=&
C^{(3)}\,e^{-d_{\Lambda}(i,j)-d_{\Lambda}(i,k)}\;.
\end{eqnarray}
Since $x_0$ is the unique global minimum of
$\Phi_{h,\Lambda}$, we can write
\begin{eqnarray*}
\Phi'_k(x)\;=\;\Phi'_k(x)-\Phi'_k(x_0)&=&
\int\limits_0^1\nabla\Phi'_k(t\hat{x}+x_0)\cdot\hat{x}\,dt
\\
&=&
\lambda_k\,\hat{x}_k\,+\,\int\limits_0^1\sum\limits_{l\in\Lambda}
\Psi_{kl}''(t\hat{x}+x_0)\,\hat{x}_l\,dt\;,
\end{eqnarray*}
where $\hat{x}:=x-x_0$ and
$\lambda_k$ denotes the $k^{\textrm{th}}$
diagonal entry of $D_{\Lambda}$, $k\in\Lambda$.
Therefore the estimate (\ref{est-Psikl}) yields
\[
|\,\Phi_k'(x)\,|\;\grg\;\lambda_k\,|\hat{x}_k|\,-\,
C_{\gamma}\,\sum\limits_{l\in\Lambda}
e^{-2\,d_{\Lambda}(k,l)}\,|\hat{x}_l|\;.
\]
This implies, for each $o\in\Lambda$, $x\in\RR^{\Lambda}$,
\begin{eqnarray*}
\lefteqn{
\sum\limits_{k\in\Lambda} e^{-d_{\Lambda}(o,k)}\,|
\,\Phi_k'(x)\,|
}
\\
&\grg&
\lambda_{\min}
\sum\limits_{k\in\Lambda} e^{-d_{\Lambda}(o,k)}\,|\hat{x}_k|
\,-\,
C_{\gamma}\,\sum\limits_{k,l\in\Lambda}
e^{-(\,d_{\Lambda}(o,k)\,+\,d_{\Lambda}(k,l)\,)\,
-\,d_{\Lambda}(k,l)}\,|\hat{x}_l|
\\
&\grg&
\lambda_{\min}
\sum\limits_{k\in\Lambda} e^{-d_{\Lambda}(o,k)}\,|\hat{x}_k|
\,-\,
C_{\gamma}\,C_d\,
\sum\limits_{l\in\Lambda}e^{-d_{\Lambda}(o,l)}\,|\hat{x}_l|\;.
\end{eqnarray*}
Thus we get
\[
\sum\limits_{k\in\Lambda} e^{-d_{\Lambda}(o,k)}\,|\hat{x}_k|
\;\klg\;
\frac{1}{\lambda_{\min}-C_{\gamma}\,C_d}\:
\sum\limits_{k\in\Lambda} e^{-d_{\Lambda}(o,k)}
\,|\,\Phi_k'(x)\,|\;,
\]
for all $o\in\Lambda$ and $x\in\RR^{\Lambda}$.
To derive this inequality we used 
(\ref{def-Cgamma}) for the first time.
Finally this together with
(\ref{est-Psiijk}) implies, for fixed $i,j\in\Lambda$,
$x\in\RR^{\Lambda}$, and suitable
$\xi_{x,k}=\xi_{x,k}(h)\in\RR^{\Lambda}$, $k\in\Lambda$, the desired estimate,
\begin{eqnarray*}
|\,\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\,|
&\klg&
\sum\limits_{k\in\Lambda}|\Psi_{ijk}^{(3)}(\xi_{x,k})|\,|\hat{x}_k|
\\
&\klg&
C^{(3)}\sum\limits_{k\in\Lambda}
e^{-d_{\Lambda}(i,j)-d_{\Lambda}(j,k)}\,|\hat{x}_k|
\\
&\klg&
\frac{C^{(3)}}{\lambda_{\min}-C_{\gamma}\,C_d}\:
\sum\limits_{k\in\Lambda}e^{-d_{\Lambda}(i,j)-d_{\Lambda}(j,k)}
\,|\,\Phi_k'(x)\,|\;.
\end{eqnarray*}
\hfill$\blacksquare$


\bigskip


\noindent {\em Remark.}
Using the symmetry of $\Psi^{(3)}_{ijk}$ in $i,j$ and $k$ we
obtain
\[
|\,\Psi^{(3)}_{ijk}(x)\,|\;\klg\;
C^{(3)}\,
e^{-\frac{2}{3}\,(\,d_{\Lambda}(i,j)+d_{\Lambda}(j,k)+d_{\Lambda}(k,i)\,)}
\] 
from (\ref{est-Psiijk}). Exactly as in the preceeding proof
we derive from this the more symmetric estimate
\[
|\,\Phi_{ij}''(x)-\Phi_{ij}''(x_0)\,|
\;\klg\;\const\;\sum\limits_{k\in\Lambda}
e^{-\frac{2}{3}\,(\,d_{\Lambda}(i,j)
+d_{\Lambda}(j,k)+d_{\Lambda}(k,i)\,)}\,|\Phi_k'(x)|
\]
under the assumption $\max\limits_{j\in\Lambda}\sum\limits_{k\in\Lambda}
e^{-\frac{2}{3}\,d_{\Lambda}(k,j)}\klg C_d$. This, however, would
not give any substantial improvement.


\bigskip



\bigskip


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




\noindent\textbf{Acknowledgments}

\bigskip


\noindent We have the pleasure to thank V.~Bach, H.-P.~Heinz
and J.~Sj\"{o}strand
for helpful discussions and suggestions.
Moreover we thank F.~Herau, T.~Jecko and F.~Nier, 
who organized a nice conference in Rennes,
where we had 
the opportunity to present the content
of this paper.
The first author gratefully acknowledges the support
by the TMR grant HPRN-CT-2002-00277 from the EU.
The second author would like to thank
Dokuz Eyl\"{u}l University for hospitality.



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



\begin{appendix}

\section{Some elementary combinatorial identities}

\begin{lemma}\label{comb-lemma}
For $k,l\in\NN,\,r\in\NN_0$, the following formulas are true:
\begin{eqnarray}\label{comb-le1} 
\sum\limits_{\alpha=0}^k(-1)^{k-\alpha}{l+\alpha-1 \choose \alpha}
{l\choose k-\alpha+1}
&=&
{l+k\choose k+1}
\\ \label{comb-le2}
\sum\limits_{\alpha=0}^{l-1}(-1)^{l-1-\alpha}{l+\alpha+r-1 \choose \alpha+r}
{l\choose l-\alpha}
&=&
{2\,l+r-1\choose l+r}
\end{eqnarray}
\end{lemma}

\textsc{Proof:} 
We recall the elementary identity 
$(1-x)^{-l}=\sum_{s=0}^{\infty}{l+s-1\choose s}\,x^s$,
for $l\in\NN$. Then we observe that the left side of 
(\ref{comb-le1})
is just the coefficient of $x^k$ in the Cauchy product
of 
$(1-x)^{-l}$ with $\frac{1}{x}\,-\,\frac{(1-x)^l}{x}$.
On the other hand the right side of (\ref{comb-le1})
is the coefficient of $x^k$ in the power series
$\frac{(1-x)^{-l}}{x}\,-\,\frac{1}{x}$.
The second identity (\ref{comb-le2}) follows
in the same manner by replacing $(1-x)^{-l}$
in the above argument by the expression
$\frac{(1-x)^{-l}}{x^r}\,-\,\sum_{s=0}^{r-1}
{l+s-1\choose s}\,\frac{x^s}{x^r}$.
\hfill$\blacksquare$


\end{appendix}


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






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