\documentstyle[12pt]{article}
\title{Examples of discrete operators with a\\
pure point spectrum of finite multiplicity }
\author{L. Amour
\thanks{D\'epartement de Math\'ematiques, U.R.A. C.N.R.S. 1870,
Universit\'e de Reims, Moulin de la Housse, B.P. 1039,
51687 Reims Cedex 2, France.
E-mail: laurent.amour@univ-reims.fr}
\and
and
\and
J. C. Guillot
\thanks{ C.M.A.P., Ecole Polytechnique, U.R.A. C.N.R.S. 756,
91128 Palaiseau Cedex, France
and
D\'epartement de Math\'ematiques, U.R.A. C.N.R.S. 742,
Universit\'e Paris-Nord, 93430 Villetaneuse, France.
E-mail: guillot@cmapx.polytechnique.fr and
guillot@math.univ-paris13.fr}}
\date{}
\begin{document}
\maketitle
\newcommand{\fp}{\hfill $\Box$}
\newcommand{\la}{\lambda}
\newcommand{\om}{\omega}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\R}{{\bf R}}
\newcommand{\Z}{{\bf Z}}
\newcommand{\Nm}{{\bf N}^m}
\newcommand{\Nn}{{\bf N}^n}
\newcommand{\C}{{\bf C}}
\newcommand{\cM}{{\cal M}}
\newcommand{\LH}{L^2_R\times H^1_R}
\newcommand{\Ima}{\Im {\em m}}
\newcommand{\Ree}{\Re {\em e}}
\newcommand{\expo}{e^{\left(\vert \Ima \frac{k}{2}\vert+
\vert \Ima \frac{\om k}{2}\vert+\vert \Ima \frac{{\om}^2 k}{2}\vert
\right)x}}
\newcommand{\cg}{[\![}
\newcommand{\cd}{]\!]}
\newcommand{\Nd}{N^{\sharp}}
\newcommand{\Ne}{N^{\star}}
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\newcommand{\by}{{\bar{y}}}
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\newcommand{\beq}{\begin{equation}}
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\newcommand{\eeq}{\end{equation}}
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\newtheorem{th}{Theorem }
\newtheorem{lem}{Lemma }
\begin{abstract}
One constructs
operators acting on $\l^2(\Z^m)$
(or $\l^2(\Z^m)^p$), $m,p\geq 1$, with a real pure point spectrum
of finite multiplicity by perturbing diagonal matrices using
a KAM procedure. The point spectrum can be dense on an interval
or a Cantor set of measure zero. The basic fact here is to
remark that for perturbations built up with an infinite number
of block diagonals, regularly separated, it is possible to deal with
eigenvalues of multiplicity strictly greater than one. Examples of
discrete operators associated with discretization of systems of partial
differential equations are given.
\end{abstract}
\parindent=0cm
\parskip 10pt plus 1pt
\baselineskip 15pt
\section{Introduction.}
It is known that given a sequence of real numbers $(d_i)_{i\in\Z^m}$
and a sufficiently small perturbation $P$
it is possible to
construct a diagonal matrix $D$ such that the discrete operator
$D+P$ has the $d_i$'s as eigenvalues (see [C], [P]). For this construction
one use a $KAM$ procedure and it is crucial that the divisors
$d_i-d_j$ do not get too small as $\vert i-j\vert$ is large. A
typical condition is
\beq\label{1}
\vert d_i-d_j\vert\geq c \vert i-j\vert^{-\alpha},\quad
i\neq j,
\eeq
where $\alpha$ and $c$ are positive real numbers.
Each eigenvalue $d_i$ is simple.
In this paper our aim is precisely to avoid the restrictions on the
multiplicity of the eigenvalues. Therefore we replace the conditions
(\ref{1}) by weaker assumptions, balanced by a stronger hypothesis
on the perturbation $P$. This additional assumption on the
perturbation is on its block structure. Of course, to
see that the simplicity of the eigenvalues is not an essential condition
for the construction of $D$, it suffices to look at diagonal perturbations.
For $a,b\in\Z^m$
we use the notation
\[
a**0$, where $S_\sigma$ is the set of all sequences
$\sigma_0\geq\sigma_1\geq\cdots\geq 0$ with
$\sum \sigma_\nu\leq \sigma$. See [R] and the appendix of [P].
A real sequence $d=(d_i)_{i\in\Z^m}$
is a distal sequence
for $\cM$ if
\beq\label{4}
(d-T_kd)^{-1}\in\cM,\ \Vert(d-T_kd)^{-1}\Vert_\infty
\leq\Omega(\vert k\vert),
\ 0\neq k\in\Z^m
\eeq
for some approximation function $\Omega(\,\cdot\,)$.
A pair of arbitrary real sequences $d=(d_i)_{i\in\Z^m}$ and
${\tilde d}=({\tilde d}_i)_{i\in\Z^m}$ is
called a distant pair
for $\cM$ if
\beq\label{5}
(d-T_k{\tilde d})^{-1}\in\cM,
\ \Vert(d-T_k{\tilde d})^{-1}\Vert_\infty
\leq\Omega(\vert k\vert),
\ k\in\Z^m
\eeq
for some approximation function.
Let $M$ be the space of all matrices
$P=\left(P(i,j)\right)_{(i,j)\in\Z^{2m}}$ satisfying
\[ P_k=\left(P(i,i+k)\right)_{i\in\Z^m}\in\cM.
\]
for every $k\in\Z^m$.
As in [P] we define a scale of Banach spaces
\[
M^s=\{ P\in M; \Vert P\Vert_s <\infty\}
\]
where
\[
\Vert P\Vert_{M^s}=\sup_{k\in\Z^m}\Vert P_k\Vert_\infty
e^{\vert k\vert s},\
0\leq s\leq\infty.
\]
In what follows we consider a subset of diagonal matrices $D$
and we find a diagonal matrix $\hat{D}$ such that $\hat{D}+P$
is similar to $D$ when $P$ is a sufficiently small perturbation.
Thus we construct a subset of matrices each one being isospectral
to $D$. The spectrum of $D$ is pure point and can be dense in an
interval or a Cantor set of measure zero.
Our starting point is a set of real sequences $d=(d_i)_{i\in\Z^m}$
in $\cM$ such that some subsequences satisfy diophantine conditions.
$N,\Nd$ and $\Ne=N/\Nd$ are chosen as above. Consider a real
sequence $d=(d_i)_{i\in\Z^m}$ in $\cM$. For every $\alpha\in\Nm$
with $0\leq \al0$ depends only on
$m$ and $N$.
Then there
exists a diagonal matrix $\hat{D}$ and an invertible matrix $V$
such that
\beq\label{8}
V^{-1}(P+\hat{D})V=D.
\eeq
\end{th}
{\it Remark 1.}
In theorem 1 it possible to consider the given
sequence $d=(d_{i})_{i\in\Z^m}$ with each $d_i$ at most repeated
$\pi_{\Ne}$ times.
Define
\[
I_k=\left\{ \al\in\Nm,\ 0\leq \al-k\Nd<\Nd\right\}
\]
where $0\leq k<\Ne$.
If $k\not =k'$, $I_k\cap I_{k'}=\o$.
Thus the pair $(\al,\be)\in I_k\times I_{k'}$ with $k\not =k'$ is
not concerned with hypothesis $(ii)$ that needs not to be checked.
Therefore, it is possible to choose $d_{\al}=d_{\be}$ and more
generally it possible to choose
\[
d_{\al^1}=d_{\al^2}=\ \cdots\ =d_{\al^{\pi_{\Ne}}},
\]
for any $(\al^1,\al^2,\cdots,\al^{\pi_{\Ne}})\in
I_{k^1}\times I_{k^2}\times\cdots\times I_{k^{\pi_{\Ne}}}$
with $k^i\not =k^j$ for $i\not =j$.
{\it Remark 2.}
When $N=(1,\cdots,1)$ no pair $(\al,\be)$ is concerned with
the hypothesis $(ii)$ that needs not to be checked.
$\al=(0,\cdots,0)$ is the only $m$-integer concerned with the
hypothesis $(i)$. Thus $d_\al=d$ and theorem 1 is nothing else
than the theorems of Craig and P\"oschel(cf [C] and [P]).
{\it Remark 3.}
Given $N$ it is with
$\Nd=(1,\cdots,1)$ that the chosen eigenvalues $d_i$ can have
the highest multiplicity which is always $\pi_N$ by remark 2.
In that case the perturbations are matrices with an infinite
number of diagonals regularly separated by $N-(1,\cdots,1)$
zero coefficients.
{\it Remark 4.}
In the situation described in remark 1 we can choose $d_{\al^i}$
dense in an interval or a Cantor set of measure zero as in [P].
Consequently the sequence $d$ will be dense in the same interval
or Cantor set.
\section{Results for discretized systems.}
In this section we use the Hilbert space ${l^2(\Z^m)}^N$
where $N$ is a $n$-integer.
$u$ is in ${l^2(\Z^m)}^N$ if and only if $u=(u_\al)_{0\leq \al< N}$
where $u_\al$ is in $l^2(\Z^m)$ for every $n-$integer $\al$.
A similar result to theorem 1 can be stated for discrete operators
on ${l^2(\Z^m)}^N$.
Such an operator is a matrix
$Q=(Q_{\alpha\beta})_{0\leq\al,\be0$ depends only on $N$. This proves (\ref{v0}).
\fp
{\it Remark 5.}
It can be proved similarly that the embedding
\[
L_{N,\Nd}^s\hookrightarrow
{\cal A}
\cap M^{s/\left(1+\inf_{1\leq i\leq m}(N_i+\Nd_i)\right)}
\] is continuous.
\begin{lem}\label{ba}
(i) $\mu(P_1P_2)=\mu(P_1)\mu(P_2)$ for $P_1, P_2\in M$.
\newline
(ii) $ \mu^{-1}(Q_1Q_2)=\mu^{-1}(Q_1)\mu^{-1}(Q_2)$
for $Q_1, Q_2\in \Lambda_N$.
\end{lem}
{\it Proof of lemma \ref{ba}: }We have $\left(\mu(P)\right)_{\al\be}(i,j)=
P(\al+iN,\be+jN)$. Then
$
\mu(P_1)\mu(P_2)_{\al\be}=
\sum_{0\leq\gamma**