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%MACROS
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\newcommand{\Z}{Z\!\!Z}
\newcommand{\Zs}{Z\!\!Z}
\newcommand{\N}{I\!\!N}
\newcommand{\Ns}{I\!\!N}
\newcommand{\R}{I\!\!R}
\newcommand{\C}{C\!\!\!\rule[.5pt]{.7pt}{6.5pt}\:\:}
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\newcommand{\cBl}{{\cal B}(l^2)}
\newcommand{\cD}{{\cal D}}
\newcommand{\cK}{{\cal K}}
\newcommand{\cH}{{\cal H}}
\newcommand{\KF}{K=-i\partial_t+H_0+V(t)}
\newcommand{\KFN}{K_N=-i\partial_t+H_0+V_N(t)}
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\begin{document}
\title{{\bf Floquet Hamiltonians with pure point spectrum}
\thanks{to appear in Commun. Math. Phys.} }
\date{}
\author{P. Duclos$^a$ and P. \v S\v tov\'\i\v cek$^b$}
\maketitle
\begin{quote} {\small
(a) Centre de Physique Th\'eorique, CNRS, 13288 Marseille--Luminy and PHYMAT,
Universit\' e de Toulon et du Var, BP 132, 83957 La Garde Cedex,
France
\\ (b) Department of Mathematics and Doppler Institute,
Faculty of Nuclear Science,
CTU, Trojanova 13, 120 00 Prague, Czech Republic} \vspace{15mm}
\noindent {\bf Abstract.} We consider Floquet Hamiltonians of the type
$K_F:=-i\partial_t+H_0+\beta V(\omega t)$ where $H_0$, a selfadjoint operator
acting in a Hilbert space ${\cal H}$, has simple discrete spectrum
$E_10$
for a given $\alpha>0$, $t\mapsto V(t)$ is $2\pi$-periodic and $r$
times strongly continuously differentiable as a bounded operator on
${\cal H}$, $\omega$ and $\beta$ are real parameters and the periodic
boundary condition is imposed in time.
We show, roughly, that provided $r$ is large enough,
$\beta$ small enough and $\omega$ non-resonant then
the spectrum of $K_F$ is pure point. The method we use
relies on a successive application of the adiabatic treatment due to Howland
and the KAM-type iteration settled by Bellissard and extended
by Combescure. Both tools are revisited,
adjusted and at some points slightly simplified.
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
Spectral analysis of Floquet Hamiltonians or, equivalently, Floquet operators
\cite{Howland,Yajima} is known to be a tool to investigate the dynamical
stability of a quantum system at least in the spirit of the RAGE theorem
(see e.g. \cite{EV}). If $K=-i\partial_t+H(t)$ is pure point then for all
initial conditions $\psi_0$ in $\cal H$, the solution $\psi_t$ of the
Schr\"odinger equation fulfills
$\lim_{a\to\infty}\sup_t \Vert {\cal E}(\vert A\vert>a)\psi_t\Vert=0$,
with $\cal E$ being the spectral measure of an arbitrary
self--adjoint operator $A$. Though it has been realized recently that such
an information is rather incomplete; in particular it seems that one cannot
predict the time behaviour of $\langle\psi_t,A\psi_t\rangle$ from the nature
of $\sigma(K)$ (see e.g. \cite{Combes}).
This paper is concerned with Floquet Hamiltonians
$K_F:=-i\partial_t+H_0+\beta V(\omega t)$ depending on two real parameters
$\beta$ and $\omega$. The unperturbed (true) Hamiltonian $H_0$ in a Hilbert
space $\cal H$ has a simple discrete spectrum
$\sigma(H_0)=\{E_1,E_2,\dots\}$ obeying the gap condition given in (2.1) below
(with $\alpha>0$). The family $V(t)$ is $2\pi$-periodic and sufficiently
many times strongly differentiable.
Except of the methods based on randomizing of some parameter \cite{AnnIHP,CHM}
and using the Kotani's trick \cite{Kotani,SW} two approaches are known to
analyze the spectrum of $K_F$. The first one is called here the KAM-type
iteration method and it was introduced and popularized by Bellissard
\cite{Bellissard}.
This method requires some kind of exponential--like decay of matrix entries
of the perturbation $V$ if expressed in the eigen--basis of the
unperturbed Floquet Hamiltonian
$K_{F,0}:=-i\partial_t\otimes 1+1\otimes H_0$. These constraints were
afterwards reduced by Combescure \cite{Combescure} to a sufficiently
fast power decay. With this hypothesis and assuming in addition that
$\beta$ is small enough and $\omega$ non-resonant one can show that $K_F$
is pure point. As observed by Howland \cite{AnnIHP}, another type of
results can be obtained using the adiabatic analysis. No restrictions are
imposed on $\beta$ and $\omega$ but the information about the spectrum is
less precise. If $V(t)$ is smooth enough then the absolutely continuous
spectrum of $K_F$ is empty.
The present paper is based on the observation that these two methods can be
applied successively. First the adiabatic algorithm is used several times
to improve the behaviour of the perturbation so that the hypothesis of the
KAM iteration method is satisfied. The main theorem is stated in
Subsection 4.1 and claims, roughly, that provided $V(t)$ is sufficiently
smooth, $\beta$ sufficiently small and $\omega$ non-resonant then $K_F$ is
pure point. As an example we recall in Subsection 4.2 the well--known quantum
Fermi accelerator. Up to now, the adiabatic analysis of this model excluded
the absolutely continuous spectrum of the Floquet Hamiltonian
\cite{AnnIHP,written notes,Seba}. Here we are able to show that the
spectrum is
even pure point provided the amplitude of oscillations is small and the
frequency non-resonant. The mentioned theorem is more or less an immediate
consequence of an extensive preparatory work contained in Sections 2 and 3
which was necessary to adjust both the KAM iteration and adiabatic tools to
our goal.
Section 2 is devoted to the Combescure's modification of the KAM iteration
method \cite{Combescure}. The basic difference is that Combescure,
wishing to treat the case $\alpha=0$, imposed some additional
decay conditions on the entries of the perturbation matrix. We restrict
ourselves to $\alpha$ strictly positive and thus we don't employ these
constraints. As a consequence we differ in some estimates, orders of decay
and constants.
Besides we modify very slightly the algorithm by not insisting on the full
diagonalization in each intermediate step when one adds a part $V^{(n)}$ of
the perturbation containing only finitely many parallels to the
diagonal. We believe that this makes the structure of the method more
transparent. For the same reason we reinterpret one key estimate due to
Bellissard \cite{Bellissard} as a bound on
the norm of an operator $\Gamma$ which inverts the commutation equation
$[D,W]=V$, i.e., $\Gamma=\mbox{ad}_D^{\ -1}$.
Section 3 is devoted to the adiabatic method.
Basically, main features of the approach we have used were contained already
in the original Howland's paper \cite{AnnIHP}. However there are
some differences. To measure the degree of compactness of the perturbation,
Howland introduced subspaces of the space of bounded operators by requiring,
in principle, a power decay independently in both indices $n$ and $m$ for
the matrix $(X_{nm})$ of a bounded operator $X$ in a suitable basis.
The subspaces introduced in our approach are characterized by a power
decay in the absolute value of differences $\vert n-m\vert$. They form even
subalgebras and seem to fit more
naturally with the adiabatic algorithm making the mechanism more transparent
and, this is the main reason, they are adjusted to the desired application
of the KAM iteration procedure.
The result due to Howland about absence of the absolutely continuous part
of spectrum was generalized by several authors who wished to treat also
multiple eigen--values \cite{Nenciu,Joye}. These authors
applied much more sophisticated machinery. As already mentioned, we restrict
ourselves to simple eigen--values but we suggest that this restriction is
not intrinsic to the approach we have chosen. We believe that owing to
the proper choice of the classes of operators we are able to stay on a more
elementary level and consequently
to treat the problem in a comparatively simple manner.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The KAM-type iteration method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection {Formal algorithm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We consider a time dependent Hamiltonian $H(t)$ in a separable Hilbert space
${\cal H}$ of the type:
$$
H(t):=H_0+V(t).
$$
\nid $H_0$ is a selfadjoint operator on ${\cal H}$ the spectrum of which
is discrete, $\sigma(H_0) = \{ E_n;\ n\in\N\}$ (remark: $0\not\in\N$),
and obeys the following fundamental gap condition:
$$ \label{spectral gaps condition}
\exists\alpha>0\ \mbox{ such that }\
\inf_{n\in\Ns}{n^{-\alpha}(E_{n+1}-E_n)}>0\, .
\eqno(2.1)
$$
Hence the spectrum $\sigma(H_0)$ is
simple. Set also
$$
\Delta E:=\min\{E_{n+1}-E_n;\ n\in\N\}.
\eqno(2.2)$$
The function $t\mapsto V(t)$ is bounded measurable and $T$-periodic with
values in the symmetric bounded operators in ${\cal H}$.
The so-called Floquet Hamiltonian
$K_F:=-i\partial_t+H(t)$ acts in $L^2(0,T)\otimes{\cal H}$.
We wish to consider the frequency $\omega=2\pi/T$ as a parameter lying in
a compact interval $\Omega:=[a,b],\ 00$ and $\sigma>0$ chosen in an appropriate way.
Let us formulate it more precisely. Assume that we are given a function
$E:\N\to\R$ fulfilling the $\alpha$--gap condition (2.1) and another
function $g:\N\times\N\times\Omega\to\C$ such that
$$
%\label{minimal conditions on g}
\begin{tabular}{ll}
$\Vert g \Vert_0$ & $:=\sup_{\omega\in\Omega;n,m\in\Ns}\vert
g(n,m;\omega)\vert<\infty$,\\
$\Vert g \Vert_1$ & $:=\sup_{\omega,\omega'\in\Omega;n,m\in\Ns}\vert
{g(n,m;\omega)-g(n,m;\omega')\over\omega-\omega'}\vert<\infty$\\
\end{tabular}
$$
and
$$
g(n,m;\omega)=-g(m,n;\omega)\, .
\eqno(2.9)$$
Notice that while $\Vert .\Vert_0$ is a norm $\Vert.\Vert_1$ is only a
seminorm. Given $\gamma$ and $\sigma$ positive we set
$$
\Omega_{\rm bad}:=
\bigcup_{\begin{array}{c}
(k,m,n)\in \Z\times\N\times\N\\
\vert k\vert+\vert n-m\vert\neq0 \end{array} }
I_{k,m,n}
$$
where
$$
I_{k,m,n}:= \{\omega\in\Omega;\ \vert
k\omega+E_n-E_m+g(n,m;\omega)\vert<\gamma(\vert k\vert + \vert
n-m\vert)^{-\sigma}\}\, .
$$
The following theorem is due to Bellissard \cite{Bellissard} who proved it
in the case $\alpha=1$.
%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Theorem 2.1.
%%%%%%%%%%%%%%%%%%%%%%%
Let $E$ and $g$ be defined as above and suppose that
\begin{eqnarray*}
\Vert g\Vert_0 & < & \min\{\Delta E,\inf\Omega\}, & (2.10) \cr
\Vert g\Vert_1 & < & 1\, . & (2.11) \cr
\end{eqnarray*}
Provided $\sigma$ obeys the inequality $\sigma>\sigma_\star(\alpha)$ where
$$
%\label{critical sigma}
\sigma_\star(\alpha):=\left\{
\begin{tabular}{ll}
${1\over\alpha}$ & \mbox{if } $0<\alpha\le 1$,\\
${2 \over 1+\alpha}$ & \mbox{if } $1<\alpha$,
\end{tabular}
\right.
\eqno(2.12)$$
then there exists $C_1\equiv C_1(E,\Omega,\sigma)\geq 0$ such that
for every $\gamma$,
$$
0<\gamma\le\min\{\Delta E,\inf\Omega\}-\Vert g\Vert_0\,
\Longrightarrow
\vert\Omega_{\rm bad}\vert\leq {C_1\over 1-\Vert g\Vert_1}\gamma\, .
$$
For the reader's convenience a sketchy proof is postponed to Appendix 1.
It is rather close to the Bellissard's original treatment since we can
use the fact that $\alpha>0$ and
consequently no weights are employed as in \cite{Combescure}.
To proceed further we shall need a subalgebra of bounded operators in
$l^2(\Z\times\N)$ distinguished by exponential decay out of the diagonal
(matrices are expressed in the standard basis).
%%%%%%%%%%%%%%%%%%%%%%%%%%
\nid{\bf Definition 2.2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%
Banach algebra ${\cal B}(\Omega,r),\ r\ge 0$, is the subspace of
$L^\infty((\Z\times\N)^2\times\Omega,\C)$
formed by matrices $A$, depending on the parameter $\omega\in\Omega$,
for which the norm $\Vert A\Vert_{\Omega,r}$ is finite ($|d|:=|d_1|+|d_2|$),
$$
\Vert A \Vert_{\Omega,r}:=\sum_{d\in\Zs\times\Zs}{\rm e}^{\vert d\vert r}
\left(
\sup_{i-j=d;\ \omega\in\Omega}\vert A(i,j;\omega)\vert+
\sup_{i-j=d;\ \omega,\omega'\in\Omega}
\left\vert
{A(i,j;\omega)-A(i,j;\omega')\over\omega-\omega'}
\right\vert\right).
$$
For a diagonal matrix $A$, $\Vert A \Vert_{\Omega,r}$ does not depend on $r$ and
we shall sometimes write it also as $\Vert A \Vert_\Omega$. We shall call
${\cal B}^{\rm off}(\Omega,r)$ the subspace of ${\cal B}(\Omega,r)$ of
elements having zero diagonals.
{\em Remark}. All the properties concerning these algebras are proven in
\cite{Combescure}.
We are now in a position to recall a theorem which gives a solution to the
small divisor problem. In addition to the mapping $E: \N\to\R$
with the same properties as above we are given a mapping
$G:\N\times\Omega\to\C$ such that, if considered as a diagonal matrix
depending on $\omega$,
$G(i,j;\omega)=G(i_2;\omega)\,\delta_{ij}$, $G$ has a finite norm
$\Vert G\Vert_\Omega$. Let finally
$\Gamma:{\cal B}^{\rm off}(\Omega,r)\to{\cal B}^{\rm off}(\Omega',r')$
be the operator defined by: $W=\Gamma V$ iff $[D_0+G,W]=V$.
Here we assume that $0\le r'\sigma_\star(\alpha)$ be as above and assume in addition that
$$
2\Vert G\Vert_\Omega<\min\{1,\Delta E,\inf \Omega\}.
$$
%
Then for every $\gamma$ obeying
$$
0<\gamma\leq \min\{\Delta E, \inf\Omega\}-2\Vert G\Vert_{\Omega},
$$
%
there exists $\Omega'\subset\Omega$ such that
$$
\vert \Omega\setminus\Omega'\vert\leq {C_1(\sigma)\over 1-2\Vert
G\Vert}\gamma,
\eqno(2.13)$$
and the norm of
$\Gamma:{\cal B}^{\rm off}(\Omega,r)\to{\cal B}^{\rm off}(\Omega',r')$
can be estimated by
$$
\Vert \Gamma \Vert\leq C_2(\sigma)\gamma^{-2}(r-r')^{-2\sigma-1}\, .
\eqno(2.14)$$
The constant $C_1(\sigma)\equiv C_1(E,\Omega,\sigma)$ was introduced in
Theorem~2.1 and
$$
C_2(\sigma)\equiv C_2(E,\Omega,\sigma):=\left({2\sigma+1\over e}
\right)^{2\sigma+1}(1+\min\{\Delta E,\inf \Omega\}).
$$
As already mentioned, the bound (2.14) means only a reinterpretation of
a known estimate \cite{Bellissard,Combescure}
and the proof can be found with some modifications in various
papers. For example, a nice presentation is given in \cite{Jauslin}.
But to make this paper self-content we recall the proof in a sketchy form
in Appendix 2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Convergence of the algorithm}%% subsubsection %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Thus it is possible to cope with the small divisor problem provided one
accepts to loose some values of $\omega$ in each step of the KAM algorithm.
In the $n^{\rm th}$ step, the estimate on small divisor will be governed
by a constant $\gamma_n$ and the lost of exponential decay is determined
by a constant $\rho_n=r_n-r_{n+1}$. We choose
$$
\gamma_n={\overline \gamma} n^{-\mu},\quad \mu>1,\quad
\rho_n={\overline \rho}n^{-\nu},\quad \nu>1\, ,
$$
for some fixed positive ${\overline \gamma}$ and ${\overline \rho}$.
We introduce also the numbers
$$
r_n:=\sum_{j=n}^\infty\rho_j,\ {\rm hence}\quad r_n\leq{\overline
r}n^{-\nu+1}\ {\rm with}\ {\overline
r}:={\nu\over \nu-1}{\overline\rho}.
$$
We set $\Omega_0=\Omega_1=\Omega$ and then in the $n^{\rm th}$ step,
$n\ge 1$, we restrict ourselves to $\Omega_{n+1}\subset\Omega_n$ so that
Theorem 2.3 is applicable. To avoid clumsy notation we shall write simply
$\Vert V_n\Vert\equiv\Vert V_n\Vert_{\Omega_n,r_n}$,
$\Vert W_n\Vert\equiv\Vert W_n\Vert_{\Omega_{n+1},r_{n+1}}$, $\dots$ .
To fulfill the hypothesis of Theorem 2.3 we shall assume now and verify
afterwards that the norms $\Vert G_n\Vert$ are bounded uniformly and
$$
%\label{bounds on Gn and gamman}
\sup_{n\geq1}\Vert G_n\Vert \leq {1\over 4}\min\{\Delta E, \inf\Omega,1\}\, .
\eqno(2.15)$$
Furthermore we shall require the constant $\overline\gamma$ to satisfy
$$
{\overline \gamma}\leq {1\over 2}\min\{\Delta E,\inf\Omega,1\}
$$
and hence
${\overline \gamma}\leq \min\{\Delta E,\inf\Omega\}-
2\sup_{n\ge1}\Vert G_n\Vert$.
With these restrictions one can actually apply Theorem 2.3 and so
$|\Omega_n\setminus\Omega_{n+1}|\le C_1(1-2\| G_n\|)^{-1}\gamma_n$ and
for the operator $\Gamma_n:
{\cal B}^{\rm off}(\Omega_n,r_n)\to{\cal B}^{\rm off}(\Omega_{n+1},r_{n+1})$
we have $\Vert\Gamma_n\Vert\le C_2\gamma_n^{\, -2}\rho_n^{\, -2\sigma-1}$.
Then for
$$
\Omega':=\bigcap_{n\ge0}\Omega_n
$$
it is clearly true
\begin{eqnarray*}
& & |\Omega\setminus\Omega'|\le C_1(\sigma)
(1-2\sup_{n\ge1}\Vert G_n\Vert)^{-1}\gamma\, ,
\mbox{ where} & (2.16) \cr
& & \gamma:={\mu\over\mu-1}{\overline\gamma}\ge\sum_1^\infty\gamma_n\, .
& \cr
\end{eqnarray*}
Letting
$$
C_\Gamma:=
{C_2(\sigma) \over {\overline \gamma}^2 {\overline \rho}^{2\sigma+1}}\, ,
$$
the bound on $\Gamma_n$ can be rewritten as
$$
\Vert \Gamma_n\Vert\leq F_n:=C_\Gamma n^{2\mu+(2\sigma+1)\nu}\, .
\eqno(2.17)$$
Some other estimates follow obviously,
$$
\Vert W_n\Vert\leq \Vert\Gamma_n\Vert \Vert V_n\Vert,\quad \Vert U_n^{\pm
1}\Vert\leq\exp{\left(\sum_{m=1}^n\Vert W_m\Vert\right)}.
$$
Though the operator $\Gamma$ is defined on the space
${\cal B}^{\rm off}(\Omega,r)$
distinguished by exponential decay the matrix $V$ is required, owing to the
Combescure's trick, to exhibit only sufficiently fast power decay. We shall
characterize it by the quantity
$$
%\label{polynomial decay assumption}
C_V:=\sup_{d\in \Zs^2}(1+ |d|^\tau)\sup_{i-j=d}\vert
V(i,j)\vert<\infty\, .
\eqno(2.18)$$
The constraints on the power $\tau$ will be specified later. Notice that
the norm $\Vert V^{(n)}\Vert_{\Omega,r}$ is finite for any $r\ge0$.
In fact, since $V^{(n)}$ doesn't depend on $\omega$,
$$
\Vert V^{(n)}\Vert_{\Omega,r}\le 4n\,{\rm e}^{nr}
\sup_{|i-j|=n}|V(i,j)|\, .
$$
Setting here $r=r_n\le{\overline r}n^{-\nu+1}$ and taking into account (2.18)
we get ($n\ge1$)
$$
\nu\geq 2\Longrightarrow \Vert V^{(n)}\Vert\leq
4 C_V {\rm e}^{\overline r} n^{1-\tau}.
$$
Our goal is to show that under appropriate conditions, $\Vert
V_n\Vert\equiv\Vert V_n\Vert_{\Omega_n,r_n}$ tends to zero as $n$ tends to
infinity. It is elementary to derive the following lemma from the relations
(2.7) and (2.8) defining the algorithm provided one uses the fact that
$\Vert\cdot\Vert_{\Omega,r}$ is a norm of an algebra and the equality
$[D_0+G_n,W_n]=V_n$.
%%%%%%%%%%%%%%%%%%%%%
\proclaim Lemma 2.4.
%%%%%%%%%%%%%%%%%%%%%
For any $n\geq1$ one has
\begin{eqnarray*}
\Vert V_{n+1}\Vert &\leq & 2 e^{2\Vert W_n\Vert}\Vert W_n\Vert\Vert
V_n\Vert+\Vert U_n\Vert\Vert U_n^{-1}\Vert\Vert V^{(n+1)}\Vert & (2.19)\cr
\Vert U_n^{\pm 1}\Vert &\leq & e^{\Vert W_n\Vert}
\Vert U_{n-1}^{\pm 1}\Vert & (2.20)\cr
\Vert G_{n+1}-G_n\Vert_{\Omega_{n+1}}
&\leq & 2 e^{2\Vert W_n\Vert}\Vert W_n\Vert\Vert
V_n\Vert+\Vert U_n\Vert\Vert U_n^{-1}\Vert\Vert V^{(n+1)}\Vert\, .
& (2.21)\cr
\end{eqnarray*}
Let us now introduce an auxiliary sequence $\{ x_n\}$,
$$
x_0:=0\quad\mbox{and}\quad x_n:=F_n\Vert V_n\Vert,\ \mbox{ for }n\ge1\, ,
\eqno(2.22)$$
and constants
$$
%\label{definition of C4}
C_3:=\sup_{n\geq 1}{F_{n+1}\over F_n}\leq
2^{2\mu+(2\sigma+1)\nu},\quad C_4:=4\,{\rm e}^{\overline r}C_V C_\Gamma\, .
\eqno(2.23)$$
Multiplying both sides of the inequality (2.19) by $F_{n+1}$ we get for $n\ge0$,
$$
x_{n+1}\leq 2C_3 x_n^2{\rm e}^{2x_n}+
{C_4\over (n+1)^\beta}\exp\left(2\sum_{m=1}^nx_m\right)\, ,
\eqno(2.24)$$
where
$$
\beta:=\tau-1-2\mu-(2\sigma+1)\nu.
\eqno(2.25)$$
It is easy to solve this finite difference inequality.
%%%%%%%%%%%%%%%%%%%%%
\proclaim Lemma 2.5.
%%%%%%%%%%%%%%%%%%%%%
Assume that $\beta>1$ and a sequence $\{x_n\}_{n\ge0}$ obeys the inequality
(2.24) and $x_0=0$. Then there exists a constant
$C_4^\star\equiv C_4^\star(\beta,C_3)$ such that $C_4\le C_4^\star$
implies
$$
x_n\leq 3C_4n^{-\beta}, \mbox{ for all }n\ge1\, .
\eqno(2.26)$$
\noindent{\em Proof}. Clearly, (2.24) implies $x_1\le C_4\le 3C_4$. The proof
is then easily carried out by induction. $C_4$ should be small enough so
that the following induction step goes through,
\begin{eqnarray*}
x_n\leq 3C_4n^{-\beta}&\Longrightarrow&\left\{
\begin{tabular}{ll}
$2C_3x_n^2e^{2x_n}$ & $\leq C_4(n+1)^{-\beta}$\\
$C_4(n+1)^{-\beta}\exp(2\sum_{j=1}^n x_j)$ & $\leq 2C_4(n+1)^{-\beta}$
\end{tabular}
\right\} \cr
&\Longrightarrow & x_{n+1}\leq 3C_4(n+1)^{-\beta}\, .\qquad\QED\cr
\end{eqnarray*}
The constraints on $\tau$ follow from the requirement $\beta>1$. Since
$\mu>1,\ \nu\ge2$ and $\sigma>\sigma_\star(\alpha)$ we get
$$
\tau>2+2\mu+(2\sigma+1)\nu>4+2(2\sigma_\star+1)=4\sigma_\star+6\, .
$$
This means
$$
%\label{critical tau condition}
\tau>\tau_\star(\alpha):=\left\{
\begin{tabular}{ll}
$6+{4\over\alpha}$ &if $\alpha\leq1$,\\
$6+{8\over 1+\alpha}$ &if $\alpha>1$,
\end{tabular}
\right.
\eqno(2.27)$$
Conversely, if $\tau>\tau_\star(\alpha)$ then one can choose
$\mu>1,\ \nu\ge2$ and $\sigma>\sigma_\star(\alpha)$ so that $\beta>1$. In
that case $C_3$ and $C_4^\ast$ are fixed by (2.23) and Lemma 2.5, respectively.
According to (2.23), the inequality $C_4\le C_4^\ast$ will be satisfied
provided $C_V$ is small enough,
$$
C_V\le{1\over 4}\,\mbox{e}^{-\bar r}C_4^\star\, C_2(\sigma)^{-1}\,
\bar\rho^{2\sigma+1}\cdot\bar\gamma^2\, .
\eqno(2.28)$$
We are still due to verify the condition (2.15) on $\Vert G_n\Vert$. From
(2.21) and the proof of Lemma 2.5 one obtains immediately that
\begin{eqnarray*}
\Vert G_{n+1}-G_n\Vert_{\Omega_{n+1}} & \le & F_{n+1}^{\ -1}
\big(2C_3x_n^{\, 2}{\rm e}^{2x_n}+C_4(n+1)^{-\beta}
\exp\left(2\sum_{1\le m\le n}x_m\right)\big) \cr
& \le & 3C_4 F_{n+1}^{\ -1}(n+1)^{-\beta}
=12\,{\rm e}^{\overline r}C_V(n+1)^{1-\tau}\, . \cr
\end{eqnarray*}
Consequently, recalling that $G_1=V^{(0)}$,
\begin{eqnarray*}
\Vert G_n\Vert & \leq &\Vert G_1\Vert+\sum_{m=2}^\infty
12 {\rm e}^{\overline r}C_Vn^{1-\tau}\leq \Vert V^{(0)}\Vert +
{12\over \tau-2}{\rm e}^{\overline r}C_V \cr
& \leq & C_V\left(1+{12\over \tau-2}{\rm e}^{\overline r}\right)
\leq C_V(1+3\,{\rm e}^{\overline r})\, , \cr
\end{eqnarray*}
where in the last line we used the fact that $\tau$ is always greater
than 6. So again, one can always satisfy the bound (2.15) by taking $C_V$
small enough.
We have arrived at the conclusion that $\Vert V_n\Vert_{\Omega',0}$ tends
to $0$ as $n\to\infty$ (c.f. (2.17), (2.22) and (2.26)).
By the Cauchy criterion, $G_n$ tends to some matrix
$G_\infty$ in the norm $\Vert\cdot\Vert_{\Omega',0}$ and necessarily
$G_\infty$ is diagonal. Similarly, since
$$
\sum_{n=1}^\infty\Vert W_n\Vert_{\Omega',0}\le\sum_{n=1}^\infty
F_n\Vert V_n\Vert<\infty\, ,
$$
$U_n^{\,\pm1}$ tends to $U_\infty^{\,\pm1}$ in the same norm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stability of the pure point spectrum}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One can show by induction that the relations (2.7) and (2.8) mean ($n\ge1$)
$$
D_0+G_n+V_n=U_{n-1}(D_0+\sum_{m=0}^n V^{(m)})U_{n-1}^{\ -1}\, .
\eqno(2.29)$$
It is possible to perform the limit $n\to\infty$ since the convergence in
the norm $\Vert\cdot\Vert_{\Omega',0}$ implies the convergence in the
standard operator norm in $\cB(l^2)$ for any fixed $\omega\in\Omega'$
\cite{Combescure}. In addition to the final conclusion of the preceding
subsection we note that obviously
$$
\sum_{m=0}^n V^{(n)}\to V\, ,\ \mbox{as}\quad n\to\infty\, ,
$$
in the norm $\Vert\cdot\Vert_{\Omega',0}$ and hence in the standard operator
norm as well. However, respecting the fact that $D_0$ is unbounded though
self--adjoint with its natural domain, let us make a short comment about the
convergence procedure since this point is usually carelessly skipped.
First note that $W_n$ preserves the domain $D_0$ and $[D_0,W_n]$ is bounded.
To this end it is enough to observe that $D_0$, being diagonal, can be
applied formally
to any sequence (without any restrictions on summability) and it holds
formally
$$
D_0W_n=W_n(D_0+G_n)-G_nW_n+V_n\, .
$$
Consequently we find that $\exp(\pm W_n)$ preserves the domain of $D_0$ and
$$
\exp(\pm W_n)D_0\exp(\mp W_n)-D_0
$$
is bounded. This follows from the formal equality
$$
D_0\exp(W_n)=\exp(W_n)\, D_0+\sum_{k=1}^\infty{1\over k!}\sum_{j=0}^{k-1}
W_n^{\, j}[D_0,W_n]\, W_n^{\, k-1-j}\, .
$$
Hence $U_n^{\, -1}$ preserves the domain of $D_0$ as well and the equality
(2.29) can be rewritten
$$
D_0U_{n-1}^{\ -1}=U_{n-1}^{\ -1}(D_0+G_n+V_n)-(\sum_{m=0}^n V^{(m)})\,
U_{n-1}^{\ -1}\, .
$$
Now to perform the limit procedure it is enough to notice that $D_0$ is
closed. We conclude that $U_\infty^{\, -1}$ preserves the domain of
$D_0$, too, and
$$
U_\infty(D_0+V)U_\infty^{\, -1}=D_0+G_\infty\, .
$$
We note also, as it is quite obvious from their constuction (cf. (2.4), (2.8))
that all the matrices $W_n(j,k)$ are anti-Hermitian and the diagonal elements
$G_n(i_2)$ are real provided the matrix $V(j,k)$ is Hermitian. Consequently,
in that case $U_\infty$ is unitary.
Thus after an obvious rescaling we get
%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Theorem 2.6.
%%%%%%%%%%%%%%%%%%%%%%%
Let $\Omega=[a,b]\subset ]0,\infty[$ and $\Omega\ni\omega\mapsto
K_\omega:=D_0+V$ be a family of selfadjoint operators acting in
$l^2(\Z\times\N)$. $D_0$ is supposed to be diagonal in the standard basis
of $l^2(\Z\times\N)$ with entries on the diagonal given by
$$
\forall i\in \Z\times\N,\quad D_0(i):=\omega i_1+ E_{i_2}\, ,
$$
where $E:\N\to\R$ fulfills the gap condition
$$
\inf\{n^{-\alpha}(E_{n+1}-E_n);\ n\in\N\}>0
$$
for some strictly positive $\alpha$. We assume also that $V$ obeys the
following power law decay condition
$$
\sup_{d\in\Zs^2}(1+\vert d\vert^\tau)\sup_{i-j=d}\vert V(i,j)\vert =: C_V<\infty
$$
for some $\tau$ strictly greater than $\tau_\star(\alpha)$ given in (2.27).\\
\indent Then
there exist two constants $\gamma^\star>0$, $C^\star>0$ such that the
inequalities $0<\gamma<\gamma^\star$ and $0\le C_V\le C^\star\gamma^2$
imply that one can find
a subset $\Omega'\subset\Omega$ fulfilling
$\vert \Omega\setminus\Omega'\vert<\gamma$ and for every $\omega\in\Omega'$,
$K_\omega$ is pure point.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Adiabatic method}%%%% subsection %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Classes of operators}% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The assumptions on the unperturbed Hamiltonian $H_0$ and its spectrum
$\sigma(H_0)=\{ E_1,E_2,\dots\}$, including the $\alpha$-gap condition (2.1)
with some $\alpha>0$, are the same as in Section 2.
Using the eigen--basis of $H_0$ we identify the Hilbert space $\cal H$
with $l^2=l^2(\N)$.
This means that the eigen--vectors of $H_0$ coincide with the vectors of
the standard basis $\{ e_n\}$ in $l^2$. The aim is to modify the Howland's
adiabatic treatment in order to show that provided $V(t)$ is smooth enough
then the Floquet operator $K=-i\partial_t+H_0+V(t)$ ( with periodic
boundary conditions in $t$) is unitarily equivalent to another one,
$\tilde K=-i\partial_t+H_0+\tilde V(t)$, and the new
perturbation $\tilde V(t)$ enables application of the KAM algorithm.
In this situation the value of the period is inessential.
We start from definition of the classes of bounded operators as announced
in Introduction.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\nid{\bf Definition 3.1.}
%%%%%%%%%%%%%%%%%%%%%%%%%%
For $k\in\Z_+$ ($0\in\Z_+$) we say that a bounded operator $X$ belongs to
the class ${\cal A}_k\subset{\cal B}(l^2)$ if and only if
(ad$_{H_0})^k\cdot X\in{\cal B}(l^2)$.
Thus by definition ${\cal A}_0={\cal B}(l^2)$ and we set
also ${\cal A}_\infty=\bigcap_{k\ge0}{\cal A}_k$.
Notice that a matrix $(X_{nm})$, if expressed in the standard basis,
corresponds to an operator $X\in{\cal A}_k$ if and only if the diagonal
sequence $\{X_{nn}\}$ belongs to $l^\infty$ and the matrix with entries
$(E_n-E_m)^kX_{nm}$ corresponds to a bounded operator. It is so because the
last two conditions imply that the matrix $(X_{nm})$ itself
corresponds to a bounded operator. This assertion can be easily verified using
the fact that the $\alpha$-gap condition (2.1) implies
$$
\inf_{n\not=m} \vert n^{1+\alpha}-m^{1+\alpha}\vert^{-1}
\vert E_n-E_m\vert >0\, .
\eqno(3.1)
$$
Thus it is enough to show the boundedness of the operator $Y$ corresponding
to the matrix $(Y_{nm})$, $Y_{nn}=0$ and
$Y_{nm}=\vert n^{1+\alpha}-m^{1+\alpha}\vert^{-k}$
for $n\not=m$ ($k\ge 1$). But since
$1\ge z^\beta+(1-z)^\beta$ for any $ z\in [0,1]$ and $\beta\ge1$, we have
$\vert x^\beta-y^\beta\vert\ge\vert x-y\vert^\beta$ for any
$x>0,\ y>0$. Then the Schur--Holmgren criterion gives the result,
$$
\sup_n\sum_m Y_{nm}\le\sup_n\sum_{m\not=n}\vert n-m\vert^{-(1+\alpha)k}\le
2\sum_{j=1}^\infty j^{-(1+\alpha)k} \, .
\eqno(3.2)$$
Let us summarize basic properties of the classes ${\cal A}_k$.
%%%%%%%%%%%%%%%%%%%
\proclaim Lemma 3.2. The classes ${\cal A}_k$ are nested,
%%%%%%%%%%%%%%%%%%%
$$
{\cal A}_{k+1}\subset{\cal A}_k,\ \hbox{ for } k=0,1,\dots\ .
\eqno(3.3)$$
For every $k$, ${\cal A}_k$ is a $\ast$-subalgebra in ${\cal B}(l^2)$.
\nid{\em Proof.} Concerning the property (3.3), for $k=0$ it is true by definition
and for $k\ge1$, $X\in{\cal A}_{k+1}$ implies that the operator
(ad$_{H_0})^k\cdot X$ with vanishing diagonal
belongs to ${\cal A}_1\subset{\cal B}(l^2)$.
${\cal A}_k$ is closed with respect to multiplication because of (3.3) and
the relation
$$
(\hbox{ad}_{H_0})^k\cdot XY=\sum_{j=0}^k
\left(\begin{array}{c} k\\ j \end{array}\right)\,
\left( (\ad_{H_0})^j\cdot X\right)\, \left( (\ad_{H_0})^{k-j}\cdot Y\right)
\, . $$
$\cAk$ is closed with respect to Hermitian conjugation because of
$$
(\ad_{H_0})^k\cdot X^\ast =(-1)^k\,((\ad_{H_0})^k\cdot X)^\ast \, .\quad \QED
$$
The following definition takes into account the time dependence.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\nid{\bf Definition 3.3.}
%%%%%%%%%%%%%%%%%%%%%%%%%%
We shall say that a family $X(t)$ of bounded operators depending periodically on
$t$ (with a given period) belongs to $C^r(\cAk)$ if and only if $X(t)$ is
strongly $C^r$ and $(\ad_{H_0})^k\cdot(d/dt)^sX(t)$ is bounded and strongly
continuous for $s=0,1,\dots,r$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{One step of the adiabatic method}% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose we are given $V(t)\in C^{r+1}(\cAk)$ and set $H(t):=H_0+V(t)$. We wish to
diagonalize partially $H(t)$ at every moment $t$ while taking care
about differentiability. Partially means starting from some sufficiently high
eigen--value and this turns out to be possible owing to the gap condition.
Necessarily this
subsection is mostly technical but the techniques involved are very standard.
This concerns the adiabatic method applied to the lower part of the spectrum and the
regular perturbation theory \cite{Kato} applicable to higher eigen--values.
We acquire some basic steps from \cite{AnnIHP}.
We are looking for a family of unitary operators $J(t)$ such that the domain
$\cD(H_D(t))=J(t)^\ast\big(\cD(H(t))\big)$ contains the standard basis and
the corresponding matrix $H_D(t)=J(t)^\ast H(t) J(t)$ fulfills
\begin{eqnarray*}
H_D(t)_{nm} & = & 0\, ,\qquad\qquad\ \,
\mbox{ for } n\le N,\ m>N\mbox{ or } n>N,\ m\le N, & \cr
& = & E_n(t)\,\delta_{nm}\, ,\quad
\mbox{ for } n>N,\ m>N. & (3.4)\cr
\end{eqnarray*}
Here $N\in\N$ is chosen sufficiently large so that the regular perturbation theory
is applicable to $H(t)$ starting from the $N^{th}$ eigen--value. The
eigen--values $\{ E_n(t)\}_{n>N}$ form the spectrum of $H(t)$ on the interval
$](E_N+E_{N+1})/2,+\infty[$. Let us recall briefly the construction.
Set $r_n=4^{-1}\min\{E_n-E_{n-1},\, E_{n+1}-E_n\}$ for $n>1$ and
$r_1=4^{-1}(E_2-E_1)$. $N$ is chosen so that
$r_n\ge 3\,\sup\Vert V(t)\Vert$,
for all $n\ge N$. Let us denote by $\Gamma_n$ the positively oriented circle with
radius $r_n$ and centred at $E_n$ (local notation; $\Gamma_n$ shouldn't be
confused with the $\Gamma$ operator of the KAM method).
Then the projectors ($n>N,\ R(z,t)\equiv (H(t)-z)^{-1}$)
$$
P_n(t):=-{1\over 2\pi i}\int_{\Gamma_n}R(z,t)\, dz
$$
are rank one and
$$
\varphi_n(t)=\|P_n(t)e_n\|^{-1}P_n(t)e_n\,
$$
are eigen--vectors corresponding to $E_n(t)$. They are strongly $C^{r+1}$
and periodic in $t$. Denote by $Q_0$ the orthogonal projector onto
span$\{ e_1,\dots,e_N\}$ and by $Q(t)$ the spectral projector of $H(t)$
onto the interval $]-\infty,(E_N+E_{N+1})/2]$. Then rank $Q(t)=N$.
Moreover, using the Kato's proof of the adiabatic theorem one can show that
there exists a partial isometry $U(t)$ mapping Ran$(Q_0)$ onto Ran$(Q(t))$
and vanishing on Ker$(Q_0)=\mbox{Ran}(Q_0)^\perp$. Furthermore, $U(t)$ and
$U(t)^\ast$ are periodic in $t$ and strongly $C^{r+1}$. Now we are able to
define $J(t)$,
\begin{eqnarray*}
J(t) Q_0 & = & U(t), & (3.5) \cr
J(t) e_n & = & \varphi_n(t),\ \mbox{for }n>N\, . & \cr
\end{eqnarray*}
To be able to cope with differentiability of $J(t)$ we shall need
the following lemma which can be extracted from \cite{AnnIHP}, Part I, Sec. 5.
%%%%%%%%%%%%%%%%%%%%%
\proclaim Lemma 3.4.
%%%%%%%%%%%%%%%%%%%%%
The following estimates hold
\begin{eqnarray*}
|\langle\varphi_n(t),(d/dt)^s\varphi_n(t)\rangle | & \le &
c_1(s)\, n^{-\alpha}, & (3.6) \cr
|\langle\varphi_n(t),(d/dt)^s\varphi_m(t)\rangle | & \le &
c_2(s)\, |E_n-E_m|^{-1}, & (3.7) \cr
\end{eqnarray*}
for $n,m>N$, $n\not= m$, and $s=1,\dots,r+1$.
Now we can state the basic observation of this subsection.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Proposition 3.5.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If $V(t)$ is strongly $C^{r+1}$ then the same is
true for $J(t)$. Furthermore,
$$
H_D(t)-H_0\in C^{r+1}(\cA_\infty).
\eqno(3.8)
$$
\noindent{\em Proof}. It is clear from the defining relation (3.5) that
$J(t)$ is strongly $C^{r+1}$ on Ran$\, Q_0$. Thus we have to show the same
also for the subspace Ran$(\bI-Q_0)$. Let ${\cal L}$ be the linear hull of
$\{ e_n\}_{n>N}$. Then $\overline{\cal L}={\rm Ran}(\bI-Q_0)$ and
$J(t)x$ is $C^{r+1}$ for every $x\in {\cal L}$. Denote by
$J^{(s)}(t)$ the operator on ${\cal L}$ defined by $J^{(s)}(t)x:=(d/dt)^sJ(t)x$.
It suffices to show that the norms $\| J^{(s)}(t)\|$ are bounded uniformly in
$t$ for $s=1,\dots,r+1$. Instead of $J^{(s)}(t)$ we shall consider two
operators $Q_0J(t)^\ast J^{(s)}(t)$ and $(\bI-Q_0)J(t)^\ast J^{(s)}(t)$.
Using repeatedly the identity
$$
Q_0J(t)^\ast J^{(j+1)}(t)=-(d/dt)U(t)^\ast\cdot J^{(j)}(t)+
(d/dt)(Q_0J(t)^\ast J^{(j)}(t))
$$
one finds that
$$
Q_0J(t)^\ast J^{(s)}(t)=\sum_{j=0}^{s-1}\alpha_{sj}\,
{d^{s-j}\over dt^{s-j}}U(t)^\ast\cdot J^{(j)}(t)
$$
with some constants $\alpha_{sj}$. Thus we deduce that provided $J(t)$
is strongly $C^{s-1}$ then the norm $\| Q_0J(t)^\ast J^{(s)}(t)\|$ is
uniformly bounded in $t$. Consequently it is sufficient to show that the
norms $\|(\bI- Q_0)J(t)^\ast J^{(s)}(t)\|$ are uniformly bounded in $t$ for
$s=1,\dots,r+1$. But the matrix $(Y^{(s)}(t)_{nm})_{n>N,m>N}$ corresponding
to $(\bI- Q_0)J(t)^\ast J^{(s)}(t)$ restricted to $\mbox{Ran}(\bI-Q_0)$
has entries
$Y^{(s)}(t)_{nm}=\langle\varphi_n(t),(d/dt)^s\varphi_m(t)\rangle$. Thus
we can use the estimate (3.7) combined with (3.1) and recall once more
the Schur--Holmgren criterion which again leads to the inequalities (3.2),
now with $k=1$.
Further we note that from the regular perturbation theory one can deduce
that for $n>N$, absolute values of the differences $E_n(t)-E_n$ are
bounded uniformly in $n$ and $t$ and the same is true for the derivatives
$(d/dt)^sE_n(t),\ s=1,\dots,r+1$. Since by construction $H_D(t)$ fulfills
(3.4) the property (3.8) follows immediately. \QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Improving decay of the perturbation matrix} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The decay property we are interested in is characterized by the class of
bounded operators $\cA_k$ as introduced above. Let us proceed to the
formulation of the basic result concerning the adiabatic algorithm.
%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Theorem 3.6.
%%%%%%%%%%%%%%%%%%%%%%%%
Assume that $H_0$ acting in a separable Hilbert
space $\cal H$ has a discrete spectrum with simple multiplicity and that
the eigen--values fulfill the $\alpha$--gap condition (2.1) with some
$\alpha>0$. Assume further that we are given a $T$-periodic family of
potentials $V(t)\in C^{r+1}(\cA_k)$, with $k,r\in\Z_+$. Then the Floquet
Hamiltonian $K=-i\partial_t+H_0+V(t)$ acting in $\cK=L^2(]0,T[,\cH,dt)$
(with periodic boundary condition)
is unitarily equivalent to $\tilde K=-i\partial_t+H_0+\tilde V(t)$, with
$\tilde V(t)\in C^r(\cA_{k+1})$.
\noindent{\em Proof}. First we shall show that $J(t)\in C^{r+1}(\cA_{k+1})$.
We know from Lemma 3.5 that $J(t)$ is strongly $C^{r+1}$. The equality
$J(t)^\ast H(t)J(t)=H_D(t)$ can be rewritten as
$$
\mbox{ad}(H_0)J(t)=J(t)(H_D(t)-H_0)-V(t)J(t)\, .
\eqno(3.9)
$$
Since the RHS of (3.9) is bounded we have $J(t)\in C^0(\cA_1)$. Now by the
fact that $\cA_s$ is an algebra, by the assumption on $V(t)$ and because of
the property (3.8) one can proceed by induction in $s$ to show that
$J(t)\in C^0(\cA_s)$ for $s=1,\dots,k+1$. Differentiating step by step
$(r+1)$-times the relation (3.9) and using the same reasoning in each step
one obtains $J(t)\in C^{r+1}(\cA_{k+1})$.
The family $J(t)$ acting by multiplication in $L^2(]0,T[,\cH,dt)$ determines
a unitary operator $\bf J$ in this Hilbert space. One gets
${\bf J}^\ast K{\bf J}=:\tilde K=-i\partial_t+H_0+\tilde V(t)$ where
($J'(t)\equiv(d/dt)J(t)$)
$$
\tilde V(t)=H_D(t)-H_0-iJ(t)^\ast J'(t)\, .
$$
If $J(t)$ is strongly $C^{r+1}$ then the same is true for $J(t)^\ast$.
This fact follows immediately from unitarity (for the zero order) and from
the equality $(d/dt)J(t)^\ast =-J(t)^\ast J'(t)J(t)^\ast$. Hence
$J(t)^\ast\in C^{r+1}(\cA_{k+1})$ and $J(t)^\ast J'(t)\in C^r(\cA_{k+1})$.
Thus in view of (3.8) we have $\tilde V(t)\in C^r(\cA_{k+1})$. \QED
Applying repeatedly Theorem 3.6 we obtain
%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Corollary 3.7.
%%%%%%%%%%%%%%%%%%%%%%%%
With the same assumptions on $H_0$, if $V(t)$ is
bounded and strongly $C^{r+k}$, with $r,k\in\Z_+$, then $K$ is unitarily
equivalent to $\tilde K$ with $\tilde V(t)\in C^r(\cA_k)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bf Remark 3.8.}
%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us now make an observation important in the
sequel about dependance of $V$ on an additional parameter $\beta$.
Let us assume in Theorem 3.6 that $V(t;\beta)$ belongs to $C^{r+1}(\cA_k)$
in $t$ for all values of $\beta$ lying in some neighbourhood of $0$. We shall
require even more, namely that
$\beta^{-1}\mbox{ad}_{H_0}^j\cdot(d/dt)^sV(t,\beta)$ is bounded uniformly
for all $t$ and sufficiently small $\beta$ whenever $j=0,1,\dots,k$, and
$s=0,1,\dots,r+1$. This means that $V(t;\beta)$ and all its derivatives up
to the order $r+1$ depend in the norm on $\beta$ as $O(\beta)$. Then the
unitary mapping $J(t;\beta)$ constructed with the help of adiabatic and
perturbation methods will possess similar property:
$\beta^{-1}(J(t,\beta)-\bI)$ and $\beta^{-1}(d/dt)^jJ(t,\beta)$, for
$j=1,\dots,r+1$, are uniformly bounded in $t$ and $\beta$. The same is true
for the operators $\beta^{-1}(H_D(t;\beta)-H_0)$ and
$\beta^{-1}(d/dt)^jH_D(t;\beta)$, for $j=1,\dots,r+1$. Now reexamining
the proof of Theorem 3.6 one can claim that $\tilde V(t;\beta)$ will depend on
$\beta$ in the same manner though with obvious changes in orders related
to the new class $C^r(\cA_{k+1})$. Particularly, if we set in Corollary 3.7
$V(t;\beta)=\beta V_0(t)$, with $V_0(t)$ being strongly $C^{r+k}$,
then $\beta^{-1}\mbox{ad}_{H_0}^j\cdot(d/dt)^s\tilde V(t;\beta)$ will be
uniformly bounded in $t$ and $\beta$ for $j=0,1,\dots,k$, and
$s=0,1,\dots,r$.
Let us conclude this section with some comments concerning absence of
the absolutely continuous spectrum and the multiplicity of eigen--values.
As observed by Howland in \cite{AnnIHP}, if the perturbation $V(t)$
is sufficiently smooth then the absolutely continuous spectrum of the
Floquet operator $K$ is empty. Here this result is obtained as a byproduct
as demonstrated by
%%%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Proposition 3.9.
%%%%%%%%%%%%%%%%%%%%%%%%%%
If $X\in\cA_k$ with $k>\alpha^{-1}$ and
{\rm diag}$(X)=0$ then $X$ is trace class.
\noindent{\em Proof}. The proof follows from the estimate
$\| X\|_{trace}\le\sum_{n,m}|X_{nm}|$. The RHS can be shown to be finite with
the help of the integral criterion. \QED
We recall too that the condition on zero diagonal means no principal
restriction on $\tilde V(t)$ since diag$\, \tilde V(t)$ can be written as a sum of a
bounded constant part plus a time dependent part but with the time average
equal to zero. The constant part can be combined with $H_0$ and the time
dependent part can be removed by a gauge transformation \cite{AnnIHP}.
{\em Remark}. Note that the assumption about simplicity of the spectrum
of $H_0$ was needed in fact only in some neighbourhood of infinity.
But we even suggest that this version of treating the adiabatic process
can be extended
to the more general case considered by Nenciu \cite{Nenciu}
and Joye \cite{Joye}. Namely,
rather than imposing the $\alpha$-gap condition on eigen--values of $H_0$
one assumes that the spectrum of $H_0$ can be written as a union of groups
of eigen--values, $\sigma(H_0)=\bigcup\sigma_n$, with multiplicities uniformly
bounded and the $\alpha$-gap condition is then imposed on these groups of
eigen--values, dist$(\sigma_n,\sigma_{n+1})\ge c\, n^\alpha$. Apparently
the main complication caused by this generalization is that infinite matrices
should be split into finite--dimensional blocks which may lead to a more
complicated notation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pure point spectrum}%%%% subsection %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Main Theorem}% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we can formulate the main result of this paper. The Floquet Hamiltonian
will be assumed to depend also on a real coupling constant $\beta$,
$K_{\omega,\beta}=-i\omega\partial_t+H_0+\beta V(t)$. The conditions on
$H_0$ are the same as in the previous two sections, $V(t)$ is
$2\pi$-periodic and strongly $C^{r+k}$, with $r,k\in\Z_+$ to be determined.
According to Corollary 3.7 and Remark 3.8, after $k$ steps of the adiabatic
method we get another but unitarily equivalent Floquet Hamiltonian
$\tilde K_{\omega,\beta}=-i\omega\partial_t+H_0+\tilde V(t;\beta)$, with
$\tilde V(t;\beta)\in C^r(\cA_k)$ and, in the sense specified in Remark 3.8,
$\tilde V(t;\beta)$ is of order $O(\beta)$. We shall suppress the tilde.
The symbols $V_{mn}(t;\beta)$ stand for matrix elements in the eigen--basis
of $H_0$. Then for $i,j\in\Z\times\N$,
$$
V(i,j,;\beta)={1\over 2\pi}\int_0^{2\pi} V_{i_2j_2}(t;\beta)\,
{\rm e}^{-{\rm i}(i_1-j_1)t}\, dt\, .
$$
Integrating $r$-times by parts and using $V\in C^r(\cA_k)$ we get
$$
|E_{i_2}-E_{j_2}|^k|V(i,j;\beta)|\le c'|\beta|(1+|i_1-j_1|)^{-r}\, .
$$
Since, in virtue of (3.1),
$$
|E_n-E_m|\ge c''|n^{1+\alpha}-m^{1+\alpha}|\ge c''|n-m|^{1+\alpha}\, ,
$$
we arrive finally at the estimate
$$
|V(i,j;\beta)|\le c|\beta|(1+|i_1-j_1|)^{-r}(1+|i_2-j_2|)^{-(1+\alpha)k}\, .
\eqno(4.1)$$
Now to apply Theorem 2.6 we need $C_V$ to be finite (c.f. (2.18)).
A sufficient condition is that both $r$ and $(1+\alpha)k$ are strictly
greater than $\tau_\star(\alpha)$. In the final formulation of the result
we shall rescale back the time.
%%%%%%%%%%%%%%%%%%%%%%
\proclaim Theorem 4.1.
%%%%%%%%%%%%%%%%%%%%%%
Assume that $H_0$ acting in a separable Hilbert space $\cal H$ has a simple
discrete spectrum and its eigen--values $\{ E_n\}$ fulfill the
$\alpha$-gap condition (2.1) with some $\alpha>0$. Assume further that there
is given a $2\pi$-periodic strongly continuous family $V(t)$ of
self--adjoint bounded operators in $\cal H$. The Floquet Hamiltonian
$K_{\omega,\beta}:=-i\partial_t+H_0+\beta\, V(\omega t)$ is supposed to act
in ${\cal K}=L^2(]0,T[,{\cal H}, dt),\ T=2\pi/\omega$, with periodic
boundary condition in $t$ and with the frequency $\omega$ lying in an
interval $\Omega=[a,b],\ 00$, $\beta^\star>0$, such that the inequalities
$$
0<\gamma<\gamma^\star,\ 0\le\beta\le\beta^\star\gamma^2\, ,
$$
imply that one can find $\Omega'\equiv\Omega'(\gamma,\beta)\subset\Omega$ with
properties
$$
|\Omega\setminus\Omega'|<\gamma\ \mbox{ and }\ K_{\omega,\beta}\
\mbox{ is pure point}
$$
for every $\omega\in\Omega'$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: Fermi accelerator}% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To give at least one illustration to Theorem 4.1 let us mention the well
known Fermi accelerator. In this case the Hilbert space itself depends on
$t$, $\cH_t=L^2(]0,y(t)[, dx)$, with $y(t)$ being a $T$-periodic strictly
positive function. The time--dependent Hamiltonian is
$H(t)=-d^2/dx^2$ with Dirichlet boundary conditions. The Floquet Hamiltonian
acts in $\cK=L^2(M,dt\, dx),\ M=\{(t,x);\ 0From the above transformation it follows (c.f. (4.3)) that
$\tilde V(t;\beta)$ is of order $O(\beta)$ and
strongly $C^r$ provided $z\in C^{r+2}$. In this case $H_0$ fulfills the
$\alpha$-gap condition with $\alpha=1$. Recall that $N(1)=17$ (c.f. (4.2)).
Now we can apply Theorem 4.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim Proposition 4.2. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If $z\in C^{19}$ then there exist constants $\gamma^\star>0$ and
$\beta^\star>0$ such that the inequalities
$$
0<\gamma<\gamma^\star,\ 0\le\beta\le\beta^\star\gamma^2\, ,
$$
imply that one can find $\Omega'\equiv\Omega'(\gamma,\beta)\subset\Omega$
with properties: $|\Omega\setminus\Omega'|<\gamma$ and for each
$\omega\in\Omega'$, the Floquet Hamiltonian $K$ related to the Fermi
accelerator is pure point.
\noindent{\em Remark.} We claim that the condition $z\in C^{19}$ in this
proposition can be weakened to $z\in C^{18}$. The point is that the
regularized potential $\tilde V(t)$ itself exhibits some decay. Assuming
that $z\in C^{18}$ we have $\tilde V(t)=v(t)\,x^2$, with $v\in C^{16}$.
One can calculate explicitly the matrix elements of the potential
$x^2$ in the eigen-basis of $H_0$. The eigen-vectors are
$\varphi_n(x)=\sqrt{2}\sin(n\pi x)$, the eigen-values $E_n=n^2\pi^2$,
$n\in\N$, and the resulting matrix elements equal
$(-1)^{m+n}8mn/\pi^2(m^2-n^2)^2$, for $m\not=n$. Consequently,
$$
\sup_{m,n}\vert n-m\vert^\beta\vert(d/dt)^s
\tilde V_{n,m}(t)\vert<\infty\, ,
$$
with $\beta=2$ and $0\le s\le16$. After five steps of the adiabatic
algorithm we obtain a transformed potential, $\tilde{\tilde V}(t)$,
which is strongly $C^{11}$ and obeys a similar condition but now with
$\beta=12$ and $0\le s\le11$. This property is already sufficient
for the application of the KAM-type iteration as claimed in Theorem 2.6.
We note that it is possible to pass from this observation to a
general assertion but we restrict ourselves to just a sketchy
remark. One can introduce another type of classes of operators
${\cal C}_\beta\subset{\cal B}(l^2)$, in addition to those having
been introduced in Definition 3.1. An operator $X\in{\cal B}(l^2)$
belongs to ${\cal C}_\beta$ if and only if its matrix elements
$(X_{nm})$ in the standard basis obey the condition
$$
\sup_{n,m}\vert n-m\vert^\beta\vert X_{n,m}\vert<\infty\, .
$$
It is not difficult to verify that ${\cal C}_\beta$ is a
$\ast$-subalgebra of ${\cal B}(l^2)$ provided $\beta>1$ (or $\beta=0$).
Clearly, ${\cal C}_0={\cal B}(l^2)$ and
${\cal A}_k\subset{\cal C}_{k(1+\alpha)}$ (and is not equal), for
$k=1,2\dots$. Let us also define the symbol
$C^r({\cal C}_\beta)\equiv{\cal C}^r_\beta$ in a way quite analogous
to Definition 3.3. Adapting the proof of Theorem 3.6 one can show
that the final assertion of this theorem can be reformulated as\\
\noindent{\em Assume further that we are given a $T$-periodic family of
potentials $V(t)\in C^{r+1}_\beta$, with $r\in\Z_+$, $\beta=0$
or $\beta>1$. Then the Floquet
Hamiltonian $K=-i\partial_t+H_0+V(t)$ acting in $\cK=L^2(]0,T[,\cH,dt)$
is unitarily equivalent to $\tilde K=-i\partial_t+H_0+\tilde V(t)$, with
$\tilde V(t)\in C^r_{\beta+\alpha+1}$.}
%\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 24pt
\noindent{\Large\bf Appendix 1. Proof of Theorem 2.1}%%%%%%%% SECTION %%%%%%%%%%
\vskip 12pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since obviously $I_{k,m,n}=I_{-k,n,m}$ it is sufficient to consider only
non-negative $k$. Where convenient we shall reparameterize $(k,m,n)$ as
$$
p:=m-n\ \mbox{ and }\ d:=k+m-n=k+p\, .
$$
Proof will be done in several steps.
{\em Step 1: excluding the cases $k=0$ and $n=m$}.
One has obviously for all $n\neq m$,
\begin{eqnarray*}
& & \Delta E>\Vert g\Vert_0\ {\rm and}\
\gamma\leq \Delta E-\Vert g\Vert_0\Longrightarrow \cr
& &
\forall\omega\in\Omega,\ \vert E_n-E_m+g(n,m;\omega)\vert\geq\gamma\ge
\gamma\vert n-m\vert^{-\sigma}\Longrightarrow \cr
& & I_{0,n,m}=\emptyset\, . \cr
\end{eqnarray*}
Similarly for all $k\neq 0$,
$$
\gamma\le\inf\Omega\Longrightarrow \forall\omega\in\Omega,\
\vert k\omega\vert\geq\inf\Omega\ge \gamma\vert k\vert^{-\sigma}
\Longrightarrow I_{k,n,n}=\emptyset\, .
$$
Moreover, notice that $I_{k,m,n}=\emptyset$ as soon as $n\geq m$.
>From now on we shall only consider $k\geq1$ and $1\leq n
\vert k(\omega-\omega')+g-g'\vert=\vert\omega-\omega'\vert
\left\vert k+ {g-g'\over\omega-\omega'}\right\vert
\geq \vert\omega-\omega'\vert(k-\Vert g\Vert_1).
$$
Since $\Vert g\Vert_1<1$, we find
$$
\vert I_{k,m,n}\vert \leq {2\gamma d^{-\sigma} \over d-p-\Vert g\Vert_1}.
\eqno(A1.1)$$
{\em Step 3: counting the $I_{k,m,n}$}.
If $I_{k,m,n}$ is not empty, this means that there exits $\omega$ such that,
on the one hand, ($b=\sup\Omega$)
\begin{eqnarray*}
E_m-E_n & = &
-(k\omega+E_n-E_m+g)+k\omega+g< \gamma d^{-\sigma}+k\omega+\Vert g\Vert_0
& \cr
& \le & \gamma +(d-1)\omega+\Vert g\Vert_0\le d\omega\le db\, . & (A1.2)\cr
\end{eqnarray*}
On the other hand, owing to the gap condition
%(\ref{spectral gaps condition})
(2.1) with $\alpha>0$ (it is crucial now), there exists $C_E>0$ such that
$$
C_E(n^\alpha+m^\alpha)\leq {E_m-E_n\over m-n}
\eqno(A1.3)$$
Combining (A1.2) and (A1.3) we get for fixed $p$ and $d$,
$$
1+(n+p)^\alpha\le n^\alpha+(n+p)^\alpha\leq {b\over C_E}{d \over p}.
$$
Set
$$
r:=b/C_E
$$
(only local notation, for purpose of this proof). Then we deduce that
$p\le(rd)^{1/(1+\alpha)}$ and for given $d$ and $p$ there are at most
$((rd/p)-1)^{1/\alpha}-p$ values of $n$. Finally one can see easily that
$d>(2^\alpha+1)/r$.
{\em Step 4: upper bound on the measure of $\Omega_{\rm bad}$}.
According to the above discussion (c.f. (A1.1) and the estimate of number
of $n$'s) the measure of $\Omega_{\rm bad}$ is bounded by
$$
\sum_{d=d_{\min}}^\infty\sum_{p=1}^{p_{\max}(d)}
\big(\left({rd\over p}\right)^{1/\alpha}-p\big)
{2\gamma d^{-\sigma}\over d-p-\Vert g\Vert_1}\, ,
\eqno(A1.4)$$
where $p_{\max}(d):=\min\{[(r d)^{1/(1+\alpha)}],d-1\}$ and
$d_{\min}:=[(2^\alpha+1)/r]$.
Further we shall use the estimate
$$
(d-p-\Vert g\Vert_1)^{-1}\leq {2\over 1-\Vert g\Vert_1}
\max\{1,(2r)^{1\over\alpha}\}{1\over d}.
\eqno(A1.5)$$
To see (A1.5) it is enough to notice that
$d-p-\Vert g\Vert_1\geq(d-p)(1-\Vert g\Vert_1)$
and afterwards to consider separately the cases $d\le 2(2r)^{1/\alpha}$ and
$d\ge 2(2r)^{1/\alpha}$.
Next, replacing $p_{\max}(d)$ by $[(rd)^{1/(1+\alpha)}]$ and using
$\sum_1^{[y]}p^{-1/\alpha}\le1+\int_1^y p^{-1/\alpha}\, dp$,
we can estimate
$$
\sum_{p=1}^{p_{\max}(d)} \left({rd\over p}\right)^{1\over\alpha}-p\leq
\varphi_\alpha(rd)\, ,
\eqno(A1.6)$$
where
$$
\varphi_\alpha(x):={x^{1/\alpha}\over \alpha-1}
\left({\alpha+1\over 2}
x^{(\alpha-1)/\alpha(\alpha+1)}-1\right).
$$
In the case $\alpha=1$ a simple limit shows that
$\varphi_1(x)=(x/2)\,\ln(x)$.
Finally it remains to perform the summation over $d$. Taking into account
the asymptotic of $\varphi_\alpha(x)$, $x\to+\infty$, one finds that
$\sigma$ must be chosen larger than $\sigma_\star(\alpha)$
(see %(\ref{critical sigma})
(2.12) for its definition) in order
to insure the convergence of the sum. The summation of $\varphi_\alpha(rd)
d^{-1-\sigma}$ will give a constant which depends on $r$, $\alpha$ and
$\sigma$ and since $r$ depends on $E$ and $\Omega$ (in fact, only on
$\sup\Omega$) we conclude that
$$
\vert \Omega_{\rm bad}\vert\leq {C_1(E,\Omega,\sigma)\over 1-\Vert
g\Vert_1} \gamma.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 24pt
\noindent{\Large\bf Appendix 2. Proof of Theorem 2.3}%%%%%%%% SECTION %%%%%%%%%%
\vskip 12pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We define $g(n,m;\omega):=G(n;\omega)-G(m;\omega)$. Clearly, both
$\Vert g\Vert_i$, $i=0,1$, are estimated by $2\Vert G\Vert_\Omega$
and the condition (2.9) is fulfilled.
Moreover, since $2\Vert G\Vert_{\Omega}<\min\{1,\Delta E, \inf\Omega\}$,
the conditions (2.10) and (2.11) are satisfied as well.
Let $W=\Gamma V$.
The estimate of $\Vert W\Vert_{\Omega',r'}$ is rather technical; to avoid
cumbersome formulas we introduce temporarily the shorter notations
$$
d:=\vert i-j\vert,\quad \delta:=D_0(i;\omega)-D_0(j;\omega)+g(i_2,j_2;\omega),
\quad V:=V(i,j;\omega),
$$
with the same convention for $W$ and $g$. Furthermore, $\delta'$,
$V',\ \dots$, designate $\delta$, $V,\ \dots$, where $\omega$ has been replaced
by $\omega'$. Since all conditions to apply Theorem 2.1 were verified there
exists a subset $\Omega'$ of $\Omega$ so that for all
$(i,j,\omega)\in(\Z\times\N)^2\times\Omega'$ it holds
$|\delta|\ge\gamma d^{-\sigma}$ (and similarly for $\delta'$).
Then we have $W=V/\delta,\ W'=V'/\delta'$,
$$
\vert W\vert + \left\vert{W-W'\over\omega-\omega'}\right\vert\leq
\left\vert{V\over\delta}\right\vert
\big(1+\left\vert{1\over\delta'}
{\delta-\delta'\over\omega-\omega'}\right\vert\big)+
\left\vert{1\over\delta'}{V-V'\over\omega-\omega'}\right\vert\, ,
$$
and
$$
\left\vert{\delta-\delta' \over \omega-\omega'}\right\vert=
\left\vert i_1-j_1 +{g-g'\over\omega-\omega'}\right\vert
\leq d+2\Vert G \Vert_\Omega.
$$
Because $d\geq1$ and
$\gamma\le\gamma_0:=\min\{\Delta E,\inf \Omega\}-2\Vert G\Vert_\Omega$
we get
$$
1+\left\vert {1\over\delta'}{\delta-\delta'\over\omega-\omega'}\right\vert
\leq 1+d^{\sigma}\gamma^{-1}(d+2\Vert
G\Vert_\Omega)\leq\gamma^{-1}d^{\sigma+1}(\gamma_0+1+2\Vert G\Vert_\Omega).
$$
Consequently,
$$
\vert W\vert + \left\vert{W-W'\over\omega-\omega'}\right\vert\leq
\gamma^{-2}d^{2\sigma+1}(1+\gamma_0+2\Vert G\Vert_\Omega)
\big(\vert V\vert+\left\vert
{V-V'\over \omega-\omega'}\right\vert\big)
$$
and
$$
\sup_{\vert i-j\vert=d;\omega,\omega'\in\Omega'}
\big(\vert W\vert+\left\vert {W-W'\over
\omega-\omega'}\right\vert\big)\leq c_2 \gamma^{-2}d^{2\sigma+1}
\sup_{\vert i-j\vert=d;\omega,\omega'\in\Omega'}
\big(\vert V\vert+\left\vert {V-V'\over
\omega-\omega'}\right\vert\big)\, ,
$$
with $c_2:=1+\min\{\Delta E,\inf \Omega\}$ .
It remains to sum over $d$. Since for positive $\rho$:
$$
\sup_{d\geq 0}\mbox{e}^{-\rho d}d^{2\sigma+1}=
\left({2\sigma+1\over \mbox{e}\rho}\right)^{2\sigma+1}\, ,
$$
the theorem is proved. The bound on $\vert \Omega\setminus\Omega'\vert$
is just that one given by Theorem 2.1.
\vskip 5mm
{\bf Acknowledgements} P.S. wishes to express his gratitude to his hosts in
Centre de Physique Th\'eorique of Marseille Luminy and Universit\'e de Toulon et
du Var where this work was initiated and finished. A partial support from the
grant No. 201/94/0708 of Czech GA is also gratefully acknowledged. P.D.
thanks also the Faculty of Nuclear Sciences of CTU for its hospitality during
the process of this work.
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