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Dirac operator, conjugate operator method, limiting absorption principle, resolvent, thresholds, smooth operators, absolutely continuous or singular spectrum, wave operatorsspectrum
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\def\Cc#1{C_{\mbox{\rm \scriptsize c}}({#1})}

\def\Cbk#1#2{C_{\mbox{\rm \scriptsize b}}^{#1}({#2})} % C-ontinuous, b-ounded,
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\begin{document}
%
%\titlepage
%\empty
\title{Limiting Absorption Principle at Critical
Values\\ for the Dirac Operator}
%
\author{
A.\ Iftimovici
\thanks{Universit\'e de Cergy-Pontoise, France, 
e-mail: {\tt iftimo@math.pst.u-cergy.fr}}
\and
M.\ M\u{a}ntoiu
\thanks{Universit\'e de Gen\`eve, Suisse,
e-mail: {\tt Marius.Mantoiu@physics.unige.ch}, on leave from 
Institute of Mathematics of the Romanian Academy,
P.O.\ Box 1-764, RO 70700, Bucharest, Romania}}
\date{}
\maketitle
\vspace{6 mm}
%
\begin{abstract}
We prove estimates for the resolvent $(H_0 - z)^{-1}$ of
the Dirac operator
$H_0 = \balpha\cdot \bP + m\beta$, valid even for $z$
close to the critical points $\pm m$. In particular, it
is shown that the operator $(1 + |x|^{2})^{-1/2}$ is
globally $H_0$-smooth. As a by-product, the absence of the
singular spectrum as well as the existence and unitarity
of the wave operators
are obtained for a class of perturbations of
$H=H_0 +V$.
\end{abstract}
%
\vspace{6 mm}
%
%\begin{center}
%{\normalsize {\bf Contents:}} \protect\\
%\end{center}
%\begin{enumerate}
%\item \hspace{3 mm} Introduction
%\dotfill\ ~\pageref{B}
%\item \hspace{3 mm} The algebraic and geometric frameworks
%\dotfill\ ~\pageref{C}
%\item \hspace{3 mm} Propagation properties
%\dotfill\ ~\pageref{D}
%\item \hspace{3 mm} Wave operators
%\dotfill\ ~\pageref{E}
%\item \hspace{3 mm} Proof of the minimal velocity theorem
%\dotfill\ ~\pageref{F}
%\item \hspace{3 mm} Appendices
%\dotfill\ ~\pageref{A}
%\item \hspace{3 mm} \vspace{8 mm} References
%\dotfill\ ~\pageref{G}
%\end{enumerate}
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%
%\pagebreak
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}                          \label{B} %\stamp{B}
\protect\setcounter{equation}{0}
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\renewcommand{\theequation}{\thesection.\arabic{equation}}
%
\indent
We shall work in the Hilbert space
${\rond H} := L^{2}({\gol R}^{3},{\gol C}^{4})$
with scalar product
$\langle \cdot , \cdot \rangle$ and norm $\| \cdot \|$.
It will be freely identified to
${\cal H} \otimes {\gol C}^{4}$
where we set
${\cal H} := L^{2}({\gol R}^{3})$.
Then the free Dirac operator
%____________________________________________________
                                          %\stampeq{e:Baa}
\begin{equation}                            \label{e:Baa}
H_{0} = {\balpha\cdot\bP} + m\beta
\end{equation}
%_______________________________________________________
describes (in suitable units) a relativistic electron
with mass $m > 0$. Here the dot means
the usual scalar product in ${\gol R}^{3}$,
${\bP}:=-i{\bm \nabla}$ is the momentum observable and
$\beta := \alpha_{0}$, $\alpha_{1}$, $\alpha_{2}$,
$\alpha_{3}$ are the (hermitian) Dirac matrices satisfying
the anti-commutation relations: $\alpha_{j}\alpha_{k} +
\alpha_{k}\alpha_{j} = 2\delta _{jk}$.
$H_{0}$ is a self-adjoint operator in
$\rond H$ with domain ${\rond H}^{1} := {\cal H}^{1}\otimes
{\gol C}^{4}$, where ${\cal H}^{1}$ is the usual Sobolev
space of order one in ${\gol R}^{3}$. The spectrum of
$H_{0}$ is purely absolutely continuous and equal to
$(-\infty, -m] \cup [m, \infty)$. We refer to [Th] for a
comprehensive discussion of the Dirac operator.
\par
In connection with the spectral analysis of $H_{0}$ and
large classes of perturbations, extensive use was made of the
limiting absorption principle (LAP).
Let us state a rough version, only for the free operator.
For two Hilbert spaces
$\rond X$ and $\rond Y$, denote by
${\gol B}({\rond X},{\rond Y})$
the Banach space of linear, continuous
operators $:{\rond X}\rarrow {\rond Y}$;
if ${\rond X}={\rond Y} \equiv {\rond H}$ we simply set
${\gol B}({\rond H})$ and use the symbol $\|\cdot\|$ also for the norm in
${\gol B}({\rond H})$.
It is known that one cannot expect to have estimates uniform in
$|\Im z|$ for $(H_{0} - z)^{-1}$ in ${\gol B}({\rond H})$
if $\Re z$ belongs to $(-\infty, -m] \cup [m, \infty)$.
But weaker estimates are in fact available
if one avoids the points $\pm m$.
\par
Let us set $\langle x \rangle = \sqrt{1 + |x|^{2}}$ for
any $x \in {\gol R}^{3}$, $Q_{j}$ for the operator of
multiplication with $x_j$ and let
${\cal S}'$ be the space of tempered distributions in
${\gol R}^{3}$, dual of the Schwartz space ${\cal S}$.
Then 
${\cal H}_{s} :=
\{ \vphi \in {\cal S}' \mid
\|\vphi\|_{{\cal H}_{s}} :=
\|\langle Q \rangle^{s}\vphi
\|_{\cal H} < \infty \}$
is the weighted Lebesgue space of order $s \in \RR$
and we set
${\rond H}_{s} := {\cal H}_{s} \otimes {\gol C}^{4}$
(Hilbert tensor product).
\par
The LAP states that for any
$s>\frac{1}{2}$ and any compact subset
$J$ of $(-\infty,-m) \cup (m, \infty)$
there is a constant $C$ such that
for any $\lambda \in J$ and $\mu >0$:
%________________________________________________________					                                 %\stampeq{e:LAP}
\begin{equation}                  \label{e:LAP}
\|(H_{0} - \lambda \mp i\mu)^{-1}\|_{{\gol
B}({\rond H}_{s},{\rond H}_{-s})} \leq C.
\end{equation}
%________________________________________________________
{\em The fact that (\ref{e:LAP}) fails for $\lambda = \pm m$ and
$s-\frac{1}{2}$ small is one of the motivations of the
present work\/}.
\par
The points $\pm m$ also play a special role in the
spectral theory of perturbations $H = H_{0} + V$. The main
feature is that for large classes of potentials $V$
(that roughly are sums of a short-range and a long-range
part), $\lambda = \pm m$ are the only possible accumulation points
of the point spectrum of $H$ and, moreover, any eigenvalue
that differs of $\pm m$ has finite multiplicity. We also
know that the singular continuous spectrum is absent. These
results (see e.g.\ [BMP]) are closely related to
(\ref{e:LAP}) as well as to the fact that it fails for
$\lambda =\pm m$.
\par
The break-down of the resolvent estimates near some
special points is a well-known phenomenon. Let us consider
the case of the (positive) Laplace operator
$\Delta := |P|^{2} = -\sum_{j=1}^{n}\partial_{j}^{2}$
acting in $L^{2}({\gol R}^{n})$.
Then one has for any $s > \frac{1}{2}$:
%________________________________________________________
		                                 %\stampeq{e:LAP-lap}
\begin{equation}                              \label{e:LAP-lap}
\|(\Delta - \lambda \mp i\mu)^{-1}\|_
{{\gol B}({\cal H}_{s}({\gol R}^{n}),{\cal H}_{-s}({\gol R}^{n}))}
\leq C,
\end{equation}
%________________________________________________________
uniformly in $\mu>0$ and
$\lambda\in J \subset (0, \infty)$, $J$ compact.
For an improved version of the estimate
(\ref{e:LAP-lap}) we refer to [Ag]. But one cannot take
$\lambda=0$ in (\ref{e:LAP-lap}); the operators 
$(\Delta \mp i\mu)^{-1}\,:\, 
{\cal H}_{\frac{1}{2} + \veps}({\gol R}^{n}) \mapsto  
{\cal H}_{-\frac{1}{2} - \veps}({\gol R}^{n})$ are not
bounded uniformly in $\mu > 0$ if $\veps > 0$ is small.
This is connected to the fact that $\Delta = h(P)$ with
$h(x) = |x|^{2}$ and $0$ is a critical value for $h$.
Actually, the Laplace operator is just a particular case.
The operator $h(P)$ may be defined in $L^{2}({\gol R}^{n})$
for large classes of functions $h : {\gol R}^{n}
\mapsto {\gol R}$. Under certain conditions on $h$, the
LAP can also be proved for $h(P)$ if $\lambda$ belongs to regions
away from the critical set $\mbox{\bf Cr}(h) := h
\left[(\nabla h)^{-1}(\{0\})\right]$. The auxiliary Hilbert
space ${\cal H}_{s}({\gol R}^{n})$ (with $s >
\frac{1}{2}$) is playing the same role as above.
\par
In the last period, the problem of obtaining the LAP for
the operator $h(P)$ in a global form, i.e.\ also at the
critical points, has become one of special interest in the
literature; as a short (and subjective) selection we refer
to [KY], [Si] for the laplacian and to [B-A], [B-AK],
[CS], [KRS], [MP] for the general case. The main new fact
brought by the global estimates is that one is forced to
use a smaller space in the  above estimates, namely one
has (roughly) to replace ${\cal H}_{\frac{1}{2} +
\veps}({\gol R}^{n})$ by ${\cal H}_{1}({\gol R}^{n})$.
\par
The main purpose of the present paper is to obtain similar
results,  namely a {\em global\/} LAP for the Dirac operator.
The set $\{-m, m\}$ may be viewed as a
critical set because
$m$ is the only critical value of the function
$h_{m}: {\gol R}^{3} \mapsto {\gol R}$ with
$h_{m}(x):= \sqrt{|x|^{2} + m^{2}}$
and 
$H_{0} = \balpha\cdot\bP + m\beta$ is unitarily
equivalent to the operator 
$[h_{m}(\bP) \otimes 1_{{\gol C}^{2}}] \oplus
[-h_{m}(\bP) \otimes 1_{{\gol C}^{2}}]$. Unfortunately, the
unitary transformation is complicated (see [Th]) hence a
direct treatment of $H_{0}$ is preferable.
\par
Let us describe the method of proof.
We recall first that one of the strongest
tools in proving the LAP for a self-adjoint operator $H$
is the method of E.\ Mourre (see [Mo], [ABG]).
It relies mainly on the so-called Mourre estimate:
%__________________________________________________________
		                          %\stampeq{e:Mourre}
\begin{equation}                           \label{e:Mourre}
E_{H}(J)i[H,A]E_{H}(J) \geq a E_{H}(J),
\end{equation}
%__________________________________________________________
where $E_{H}$ is the spectral measure of $H$, $J$
is a suitable real interval, $a>0$
and $A$ is a self-adjoint operator.
One often says that $A$ is {\em conjugated} to $H$
on the interval $J$.
Under some extra technical assumptions one gets a LAP for $H$ with respect
to the operator norm in ${\gol B}({\rond X},{\rond X}')$.
A simple but rough option is to take ${\rond X}$ as
the domain of $A$ endowed with the graph norm
and ${\rond X}'$ the dual of
${\rond X}$.
Unfortunately, if $H=h(P)$ the best choice for $A$ is
essentially
%--------------------------------------------------------
$$
A = \frac{1}{2} \{(\nabla h)(P) \cdot Q + Q \cdot
(\nabla h)(P) \},
$$
%--------------------------------------------------------
and (\ref{e:Mourre}) cannot be obtained if the closure of $J$
contains critical points of the function $h$. The strict
positivity fails. In order to overcome this obstruction an
approach called the ``{\em method of the weakly conjugate
operator\/}'' was developed in [BKM] and [BM],
in which the LAP can be obtained
from the weaker positivity assumption $[iH,A]>0$ and some extra
technical conditions. This may be seen as an extension
of the well-known Kato-Putnam theory
(see [RS2]). It has already been applied to different types of
partial differential operators with anisotropic
coefficients, leading to results on the absence of the
singular spectrum in various situations in which Mourre's
approach did not work. The task of the present work is to
apply this method to Dirac operators.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}                  \label{C} %\stamp{C}
\protect\setcounter{equation}{0}
\protect
\renewcommand{\theequation}{\thesection.\arabic{equation}}
%

Let $H_{0}$ be the self-adjoint operator defined by (\ref{e:Baa}).
Recall that the operator $T \in {\gol B}({\rond H})$ is called
$H_{0}$-{\em smooth} if there is a finite constant $C$
such that $\int_{\gol R} \| T e^{-itH_{0}} \psi \|^{2} dt
\leq C\|\psi\|^{2}$ for any $\psi \in {\rond H}$.
%========================================================
\begin{tabtheorem}{theorem}                          \label{t:our-LAP}
                                          %\stampth{t:our-LAP}
{\rm (i)}
There exists $C >0$ such that for any 
$\lambda \in {\gol R}$ and $\mu > 0$:
$$ \| (H_{0} - \lambda \mp i\mu )^{-1}
\| _{{\gol B}({\rond H}_{1},{\rond H}_{-1})}
\leq C. $$
\tabth
{\rm (ii)}
The operator $\langle {\bQ} \rangle ^{-1}$
is $H_{0}$-smooth.
\end{tabtheorem}
%========================================================
%========================================================
\begin{corollary}                            \label{c:WOp}
                                          %\stampth{c:WOp}
Let $V:\RR ^3  \rarrow \BB (\CC ^4)$ be a Borel,
hermitian valued function such that  for all $x\in \RR ^3$,
$\|V(x)\|_{\BB(\CC ^4)} \leq \langle x\rangle^{-2}$.
Then there exists a constant $\Gamma >0$
such that for all $\gamma \in (-\Gamma,\Gamma)$
the wave operators for the couple $H_0$,
$H_{\gamma}:=H_0 + \gamma V$
%
$$
\Omega^{\pm} = \slim_{t\rarrow \pm\infty}
\e^{itH_{\gamma}}\e^{-itH_0}
$$
%
exist and are unitary. In particular,
$H_{\gamma}$ has purely absolutely continuous spectrum.
\end{corollary}
%========================================================
The corollary follows easily by combining (i) of
Theorem \ref{t:our-LAP} with standard smoothness results
(see e.g.\ [RS4]). Notice that it states much more than
just the absence of the point spectrum
(which might be a known fact for perturbations as above).
As for the statement (ii) in Theorem \ref{t:our-LAP}, it follows
immediately (see [RS4]) from (i). We thus consider (i) as being the
main result of the present paper and we devote the Section \ref{D}
to its proof.
\par
As explained in the Introduction, the first problem
one has to solve is to find a ``weakly conjugate operator''
$A$ satisfying $[iH_0,A]>0$.
Our choice is
%_____________________________________________________
					%\stampeq{e:conj-op}
\begin{equation}                           \label{e:conj-op}
A = \frac{1}{2} (H_{0}^{-1}{\bP\cdot\bQ} + 
{\bQ\cdot \bP} H_{0}^{-1}).
\end{equation}
%_____________________________________________________
Easy computations performed on $\rond S$ show that
the above commutator equals
$\Delta (\Delta + m^{2})^{-1} \equiv B$.
Then $B$ extends to a bounded operator in $\rond H$ which satisfies
$\langle \psi,B\psi \rangle > 0$ for any
$\psi \in {\rond H}\setminus \{0\}$.
This is the precise meaning to give to ``weak positivity''.
In [BG] and [BMP] another operator $A$ was used in order
to get the Mourre estimate for
the perturbed hamiltonian $H=H_0 + V$,
under suitable conditions on $V$.
It does not fit to our purposes. But one {\em may\/} use 
(\ref{e:conj-op}) in order to perform Mourre theory
in a rather simplified manner. We also send to [GM]
in which a systematic way to treat a large class
of generalizations of the Dirac operator
perturbed by highly singular potentials is given.
Note however that this does not prove  the LAP
at critical points and give only the absence
of the singular continuous spectrum and some
qualitative information on the point spectrum.
Let us mention that one cannot use directly the abstract
results of [BKM] and [BM]. Indeed, in these papers
some technical difficulties
are solved by working intensively with the evolution group
$\{W(t)=e^{itA} \mid t \in {\gol R}\}$. This is not
convenient in our case, the group generated by
(\ref{e:conj-op}) being too involved.
In exchange, although both
$H_{0}$ and $A$ are unbounded operators, the commutators
$B$ and $i[B,A]$ are bounded and this helps us to
overcome the technical difficulties.
On the other hand, in an abstract setting a
supplementary space $\rond A$ appears in the LAP.
But its rigorous definition is based also on the
evolution group $\{W(t) \mid t \in {\gol R}\}$.
Moreover, in applications the space $\rond A$
is too involved and somewhat artificial.
Our direct proof puts instead in evidence
the simple and familiar space
${\rond H}_{1} = {\cal H}_{1} \otimes {\gol C}^{4}$.
%
%We did not succeed to find this simple
%statement in the literature, but see [Ku] and [Kl] for
%results concerning the point spectrum.
%In Section \ref{D} an electric potential is added to
%$H_{0}$. Since some rather complicated formulas appear in
%computing commutators, the proof is only sketched;
%relies heavily on the proof given in Section \ref{C} for
%the free operator and we stress only the new aspects.
%An interesting feature is that we allow functions $V$ which
%do not decay at infinity (see ).
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{t:our-LAP}}          \label{D} %\stamp{D}
\protect\setcounter{equation}{0}
\protect
\renewcommand{\theequation}{\thesection.\arabic{equation}}
%
Our goal is to prove the estimate
%__________________________________________________________
                                          %\stampeq{e:our-LAP}
\begin{equation}                             \label{e:our-LAP}
|\langle \psi,(H_{0} - \lambda \mp i\mu )^{-1}\psi \rangle|
\leq
C \| \psi \|_{{\rond H}_{1}}^{2},
\end{equation}
%__________________________________________________________
where the constant $C$ is independent of
$\psi \in {\rond H}_{1}$, $\lambda \in \RR$ and $\mu >0$.
As stated in the Introduction, the strategy is to find a
self-adjoint operator $A$  having positive commutator with
$H_{0}$ and fulfilling some extra technical conditions. We
define $A$ on ${\rond S}:= {\cal S} \otimes {\gol C}^{4}$
(the completion in $\| \cdot\|_{\rond H}$ of the
algebraical tensor product ${\cal S} \odot {\gol C}^{4}$)
by the formula (\ref{e:conj-op}).
Notice that 
$H_{0}^2=|{\bP}|^{2} + m^{2} =\Delta + m^{2}$, hence
$H_{0}^{-1}=H_{0}(\Delta + m^{2})^{-1}$. Thus 
%---------------------------------------------------------
$$
A = \frac{1}{2}
\left[
H_{0}(|{\bP}|^{2} + m^{2})^{-1} 
\bP\cdot{\bQ} + 
{\bQ} \cdot\bP 
(|{\bP}|^{2} + m^{2})^{-1} H_{0}
\right]
$$
%---------------------------------------------------------
leaves {\rond S} invariant. The operator $A$ may also be set in
the non-symmetric forms
%__________________________________________________________
                                          %\stampeq{e:non-sym}
\begin{equation}                             \label{e:non-sym}
A = H_{0}^{-1} {\bP\cdot \bQ} + \frac{3i}{2}H_{0}^{-1} -
\frac{i}{2}H_{0}^{-1}{\balpha\cdot\bP}H_{0}^{-1} = 
H_{0}^{-1} {\bP\cdot\bQ} + iH_{0}^{-1} +
\frac{1}{2}miH_{0}^{-1}\beta H_{0}^{-1}
\end{equation}
%__________________________________________________________
if one takes into account the commutation formulae
%
$$[Q_{j},P_{k}] = i\delta_{jk} \mbox{\  and \  }
[Q_{j},H_{0}^{-1}] = -H_{0}^{-1}[Q_{j},H_{0}]H_{0}^{-1}=
-iH_{0}^{-1}\alpha_{j}H_{0}^{-1},\ j=1,2,3.
$$
%====================================================
\begin{tabtheorem}{lemma}                 \label{l:nelson}
                                          %\stampth{l:nelson}
{\rm (i)} $A$ is an essentially self-adjoint operator.
\tabth
{\rm (ii)} The domain of the (self-adjoint) closure
of $A$ contains ${\rond H}_{1}$.
\end{tabtheorem}
%======================================================
\proof Obviously, $A$ is symmetric.
According to the commutator theorem of Nelson (see [RS2]
Theorem X.37, with $ N \equiv \langle {\bQ} \rangle \geq 1$)
one needs to prove for any $\psi \in {\rond S}$:
%------------------------------------------------------
$$
\| A\psi \| \leq C \, \|\langle {\bQ} \rangle \psi\| 
\mbox{\ \  and \ \ }
\left| \left\langle A\psi ,\langle {\bQ} \rangle \psi
\right\rangle - 
\left\langle \langle {\bQ} \rangle \psi ,A\psi 
\right\rangle \right| \leq
C \left\langle \psi ,\langle {\bQ} \rangle \psi
\right\rangle .
$$
%------------------------------------------------------
Since the operators $H_{0}^{-1}$ and $H_{0}^{-1}P_{j}$ are bounded
(for each $j$), the first inequality follows from (\ref{e:non-sym}).
The second one is an immediate consequence of the equality
%------------------------------------------------------
$$
i[A,\langle {\bQ}\rangle ] =
H_{0}^{-1} \frac{|{\bQ}|^{2}}{\langle {\bQ}\rangle} +
\frac{|{\bQ}|^{2}}{\langle {\bQ}\rangle} -
H_{0}^{-1}
\frac{\balpha\cdot\bQ}{\langle {\bQ}\rangle}
H_{0}^{-1}{\bP\cdot\bQ} -
{\bQ\cdot\bP}H_{0}^{-1}
\frac{\balpha\cdot\bQ}{\langle {\bQ}\rangle} H_{0}^{-1},
$$
%------------------------------------------------------
obtained by a direct commutator calculus
performed on the invariant domain $\rond S$.
\qed
%
In Section \ref{C} we introduced the notation $B$ for the bounded operator
$\Delta (\Delta + m^2 )^{-1} = 1- m^{2}(\Delta + m^2 )^{-1}$.
As a consequence of the Lemma \ref{l:nelson}, 
a standard computation performed on vectors
$\vphi_1, \ \vphi_2 \in {\rond H}_{1} \cap {\rond H}^{1}$ gives
%
%_________________________________________________________
		                          %\stampeq{e:first-com}
\begin{equation}                           \label{e:first-com}
\langle \vphi_1 , B \vphi_2 \rangle =
\langle iH_0\vphi_1 , A \vphi_2 \rangle -
\langle iA\vphi_1 , H_0 \vphi_2 \rangle .
\end{equation}
%__________________________________________________________
%
This allows us to interpret $B$ as the commutator $[iH_0 , A]$ on
${\rond H}_{1} \cap {\rond H}^{1}$.
Since it will play an important r\^ole in our proof,
we shall point out some of its properties.
If one sets
$B^{1/2} = |\bm P |(\bm{P} ^2 + m^2)^{-1/2}\in \BB (\rond H)$ we may see
$B^{1/2}$ as an operator $b(\bm P )$ defined via the functional calculus by the
Borel function $b:\RR ^3 \rarrow \RR$,
$b(\bm\kappa) = |\bm \kappa |(\bm{\kappa} ^2 + m^2)^{-1/2}$.
The left inverse $B^{-1/2}$ of $B^{1/2}$ satisfies the property
$B^{-1/2} \in \BB ({\rond H}_1 , \rond H)$, i.e. for all
$\psi \in {\rond H}_1$
%__________________________________________________________
		                          %\stampeq{e:our-Hardy}
\begin{equation}                           \label{e:our-Hardy}
\| B^{-1/2} \psi \| \leq C\|\psi\|_{{\rond H}_{1}}.
\end{equation}
%__________________________________________________________
Indeed, since
$b(\bm\kappa) \leq 1 - m| \bm{\kappa} |^{-1}$ it will be enough to show that
$|\bm P|^{-1} \in  \BB ({\rond H}_1 , \rond H)$.
But this is a direct consequence
of the Hardy inequality
$\| |\bm P |^{-1} \psi \|_{\rond H} \leq
\frac{1}{4} \|\psi\|_{{\rond H}_{1}}$,
valid for each $\psi \in {\rond H}_{1}$.
Also, remark that
$B^{-1/2} A$ is a well defined linear operator
$: {\rond S} \rarrow {\rond H}$.
We stress that it extends to an element of
$\BB ({\rond H}_{1},{\rond H})$
i.e.\ there is some $C>0$ such that
for any $\psi \in {\rond H}_{1}$
%__________________________________________________________
		                          %\stampeq{e:2.13}
\begin{equation}                           \label{e:2.13}
\| B^{-1/2} A \psi \| \leq C \|\psi\|_{{\rond H}_{1}}.
\end{equation}
%__________________________________________________________
This clearly follows from (\ref{e:our-Hardy})
and (\ref{e:non-sym}) if one takes into account that
$B^{-1/2} = |H_0|\,|\bm P |^{-1}$ on $\rond S$.
Moreover, an easy computation performed on $\rond S$ yields
%
%__________________________________________________________
\begin{eqnarray*}
[iB,A] = i[1-m^2 H_0 ^{-2}, A] & = &
m^2 H_0 ^{-1} ([iH_0 , A] H_0 ^{-1} + H_0 ^{-1} [iH_0 , A] ) H_0 ^{-1} \\
& = & m^2 H_0 ^{-1} (\Delta H_0 ^{-3} + H_0 ^{-3} \Delta ) H_0 ^{-1} =
2 m^2 \Delta H_0 ^{-5}.
\end{eqnarray*}
%__________________________________________________________
Actually, $[iB,A]$ extends to an element of $\BB (\rond H )$,
and one checks by continuity that for all
$\vphi_1, \ \vphi_2 \in {\rond H}_{1} \cap {\rond H}^{1}$,
%
%__________________________________________________________
		                          %\stampeq{e:second-com}
\begin{equation}                           \label{e:second-com}
\langle \vphi_1 , 2 m^2 \Delta H_0 ^{-5} \vphi_2 \rangle =
\langle iB\vphi_1 , A \vphi_2 \rangle -
\langle iA\vphi_1 , B \vphi_2 \rangle .
\end{equation}
%__________________________________________________________
%
It is also clear that $B^{-1/2}\Delta H_0 ^{-5}B^{-1/2}$
extends to a bounded operator in $\rond H$.
\par
We start now the main part of the proof of the Theorem \ref{t:our-LAP} by
defining for each $\lambda \in \RR$ and $\mu > 0$ the family of operators
$T_{\veps}^{\pm}(\lambda, \mu) := H_0 -\lambda \mp i\mu \mp i\veps B$,
indexed by $\veps \in [0, 1)$. Remark first that they are (linear)
homeomorphisms $: {\rond H}^1 \rarrow {\rond H}$.
Indeed, for $\mu > 0$ and $\vphi \in {\rond H}^1$,
the estimate
%__________________________________________________________
                                           %\stampeq{e:mimi}
\begin{equation}                           \label{e:mimi}
\|T_{\veps}^{\pm}\vphi\|^{2} =
\| (H_0 -\lambda \mp i\veps B)\vphi\|^2 + \mu^2 \|\vphi\|^2 +
2\Re \langle (H_0 -\lambda \mp i\veps B)\vphi , \mp i\mu\vphi \rangle 
\geq \mu ^2\|\vphi\|^2
\end{equation}
%__________________________________________________________
based on the weak positivity of $B$, shows that
$T_{\veps}^{\pm} \equiv T_{\veps}^{\pm}(\lambda, \mu)$ are injective. Since
$T_{\veps}^{\pm} = (T_{\veps}^{\mp})^*$ their ranges are dense.
For each $\veps < 1$ they are closed operators and their surjectivity follows
easily. We may thus conclude by the open mapping theorem.
\par
Set now $G_{\veps}^{\pm} \equiv G_{\veps}^{\pm}(\lambda, \mu) :=
(T_{\veps}^{\pm}(\lambda, \mu))^{-1}$.
They are operators of $\BB ({\rond H}, {\rond H}^1 ) \subset \BB (\rond H)$
which verify $G_{\veps}^{\pm} = (G_{\veps}^{\mp})^*$. For a vector
$\psi$ arbitrarily fixed in $\rond H$ we shall systematically use the notation
$F_{\veps}^{\pm} = \langle \psi , G_{\veps}^{\pm}(\lambda, \mu)\psi\rangle$.
The function $\veps \mapsto F_{\veps}^{\pm}$ is continuous on $[0,1)$ and in
particular in $\veps = 0$ as it is shown by
%__________________________________________________________
\begin{eqnarray*}
|F_{\veps}^{\pm} -F_{0}^{\pm} | \hspace{-2mm} & = & \hspace{-2mm}
| \langle \psi , G_{\veps}^{\pm}(\lambda, \mu)\psi\rangle -
\langle \psi ,( H_0 -\lambda \mp i\mu)^{-1} \psi\rangle | \leq
| \langle \psi ,
G_{\veps}^{\pm}(\lambda, \mu)i\veps B( H_0 -\lambda \mp i\mu)^{-1}
\psi\rangle | \\
\hspace{-2mm}& \leq & \hspace{-2mm}\veps \|B\| \, \|G_{\veps}^{\pm}(\lambda,
\mu)\| \,
\|( H_0 -\lambda \mp i\mu)^{-1}\|\, \|\psi\|^2 \leq 
\frac{\veps}{\mu ^2} \|\psi\|^2,
\end{eqnarray*}
%__________________________________________________________
which is an easy consequence of the second resolvent
identity and of (\ref{e:mimi}).
Hence $\lim_{\veps \rarrow 0} F_\veps = F_0$ exists,
and the formula (\ref{e:our-LAP}) to be proved may be rewritten as
%__________________________________________________________
		                          %\stampeq{e:tito}
\begin{equation}                           \label{e:tito}
|F_{0}^{\pm}| \leq C\|\psi\|_{{\rond H}_1}^2 .
\end{equation}
%__________________________________________________________
In order to show (\ref{e:tito}) we need detailed information about the
family of resolvents $\{G_{\veps}^{\pm}\}_\veps$. An important feature is that
it leaves ${\rond H}_1$ invariant:
%__________________________________________________________
		                          %\stampeq{e:invar}
\begin{equation}                           \label{e:invar}
G_{\veps}^{\pm}{\rond H}_1 \subset  {\rond H}_1.
\end{equation}
%__________________________________________________________
For, let us assume that $\psi \in {\rond H}_1$
i.e.\  the application
$\RR ^3 \ni \bm{x} \mapsto \e ^{i\bm{x\cdot Q}}\psi \in {\rond H}$ is $C^1$.
Then
%__________________________________________________________
\begin{eqnarray*}
\e ^{i\bm{x\cdot Q}}G_{\veps}^{\pm}\psi
\hspace{-2mm} & = & \hspace{-2mm}
(\e ^{i\bm{x\cdot Q}}H_0\e ^{-i\bm{x\cdot Q}}
-\lambda \mp i\mu  \mp i\veps\e ^{i\bm{x\cdot Q}}B\e ^{-i\bm{x\cdot Q}} )^{-1}
\e ^{i\bm{x\cdot Q}}\psi\\
\hspace{-2mm} & = & \hspace{-2mm}
\left( H_0 -\bm{\alpha \cdot x} -\lambda \mp i\mu  \mp i\veps
\frac{(\bm{P}-\bm{x})^2}{(\bm{P}-\bm{x})^2 + m^2}\right)^{-1}
\e ^{i\bm{x\cdot Q}}\psi
\end{eqnarray*}
%__________________________________________________________
is a $C^1$ function of $\bm x$ in the topology of $\rond H$, because
%__________________________________________________________
\begin{eqnarray*}
\RR ^3 \ni \bm{x} \mapsto
\left( H_0 -\bm{\alpha \cdot x} -\lambda \mp i\mu  \mp i\veps
(\bm{P}-\bm{x})^{2}\left[ (\bm{P}-\bm{x})^2 + m^2\right]^{-1}\right)^{-1}
\in \BB(\rond H)
\end{eqnarray*}
%__________________________________________________________
is $C^1$ in the operator norm.
\par
Technical estimates on $G_{\veps}^{\pm}$ are gathered below.
%
%======================================================
\begin{tabtheorem}{lemma}                             \label{l:G-epsilon}
                                          %\stampth{l:G-epsilon}
{\rm (i)}
$\|B^{1/2}G_{\veps}^{\pm}\psi\|^2 \leq {\veps}^{-1}|F_{\veps}^{\pm}|$
for each  $\psi \in \rond H$.
\tabth
{\rm (ii)}
$\|B^{1/2}G_{\veps}^{\pm}\psi\| \leq
C{\veps}^{-1}\|\psi\|_{{\rond H}_1}$ for each $\psi \in {\rond H}_1$
($C$ is some positive constant).
\end{tabtheorem}
%======================================================
\proof Let $\psi \in \rond H$. Then (i) follows from
%__________________________________________________________
$$
|F_{\veps}^{\pm}| \geq
\mp \Im \langle \psi, G_{\veps}^{\pm}\psi \rangle =
\mp \Im \langle (H_0 -\lambda \mp i\mu \mp i\veps B) G_{\veps}^{\pm}\psi,
G_{\veps}^{\pm}\psi \rangle =
\mu \|G_{\veps}^{\pm}\psi\|^2 + \veps\|B^{1/2}G_{\veps}^{\pm}\psi\|^2.
$$
%__________________________________________________________
For (ii), let $\psi \in {\rond H}_1$. As a consequence of
(\ref{e:our-Hardy}),
%__________________________________________________________
$$
|F_{\veps}^{\pm}| =
|\langle \psi, G_{\veps}^{\pm}\psi \rangle | \leq
\|B^{-1/2}\psi\| \, \|B^{1/2}G_{\veps}^{\pm}\psi\| \leq
C \|\psi\|_{{\rond H}_1} \, \|B^{1/2}G_{\veps}^{\pm}\psi\|.
$$
%__________________________________________________________
Thus one may use (i) in order to find the estimate of (ii).
%but also the inequality
%__________________________________________________________
%		                          %\stampeq{e:F-eps}
%\begin{equation}                           \label{e:F-eps}
%|F_{\veps}^{\pm}| =
%C \|\psi\|_{{\rond H}_1}\veps ^{-1/2}|F_{\veps}^{\pm}|^{1/2}
%\end{equation}
%__________________________________________________________
\qed
\par
We are now ready for the last step of the proof. It relies mainly on a
differential inequality in terms of $F_\veps$.
We fix a $\psi \in {\rond H}_{1}$ and denote for simplicity
$G_{\veps} \equiv G_{\veps}^{+}$ and $F_{\veps} \equiv F_{\veps}^{+}$.
The proof for the case $G_{\veps} \equiv G_{\veps}^{-}$ follows analogously.
Notice first that  the resolvent identity gives
$F'_{\veps}\equiv \frac{d}{d \veps} F_{\veps} = 
i\langle G_{\veps}^* \psi,B G_{\veps}\psi \rangle $.
If one puts
$\vphi_1 = G_{\veps}^* \psi$ and $\vphi_2 = G_{\veps} \psi$,
(\ref{e:invar}), (\ref{e:first-com})
and (ii) of Lemma \ref{l:nelson} yield
%__________________________________________________________
\begin{eqnarray*}
F_{\veps}'
\hspace{-2mm} & = & \hspace{-2mm}
\langle A G_{\veps}^* \psi,H_0 G_{\veps}\psi \rangle -
\langle H_0 G_{\veps}^* \psi,A G_{\veps}\psi \rangle \\
\hspace{-2mm} & = & \hspace{-2mm}
\langle G_{\veps}^* \psi,A \psi \rangle -
\langle A \psi, G_{\veps}\psi \rangle +
i\veps ( \langle B G_{\veps}^* \psi,A G_{\veps}\psi \rangle -
\langle A G_{\veps}^* \psi,B G_{\veps}\psi \rangle ) \\
\hspace{-2mm} & = & \hspace{-2mm}
\langle G_{\veps}^* \psi,A \psi \rangle -
\langle A \psi, G_{\veps}\psi \rangle -
2m^2 \veps \langle G_{\veps}^* \psi,\Delta H_{0}^{-5} G_{\veps}\psi \rangle .
\end{eqnarray*}
%__________________________________________________________
For the last equality we used (\ref{e:second-com}). Further, (\ref{e:2.13})
and Lemma \ref{l:G-epsilon} furnish the estimate
%__________________________________________________________
\begin{eqnarray*}
|F_{\veps}'|
\hspace{-2mm} & \leq & \hspace{-2mm}
\|B^{-1/2} A \psi\| \,
(\|B^{1/2}G_{\veps}^* \psi\| + \|B^{1/2}G_{\veps} \psi\| ) +
C\veps \|B^{1/2}G_{\veps}^* \psi\| \, \|B^{1/2}G_{\veps} \psi\| \\
\hspace{-2mm} & \leq & \hspace{-2mm}
\tilde{C} \veps^{-1/2}|F_\veps|^{1/2} \|\psi\|_{{\rond H}_1}.
\end{eqnarray*}
%__________________________________________________________
After integration on $[\rho,\rho_0] \subset (0,1)$ one gets
%__________________________________________________________
$$
|F_{\rho}| \leq |F_{\rho_0}| + 
C\|\psi\|_{{\rond H}_1}\int^{\rho_0}_{\rho}
\veps^{-1/2} |F_\veps|^{1/2}\dd \veps
$$
%__________________________________________________________
which may be treated with the help of a Gronwall type lemma.
Indeed, if one
takes $\theta = 1/2$ in Lemma 7.A.1 of [ABG] we may find a
positive constant $C$ such that
%__________________________________________________________
$$
|F_{\rho}| \leq 
C\left( |F_{\rho_0}|^{1/2} +
\|\psi\|_{{\rond H}_1}(\sqrt{\rho_0}-\sqrt{\rho})\right) ^2 .
$$
%__________________________________________________________
On the other hand, (\ref{e:our-Hardy}) and (ii) of
Lemma \ref{l:G-epsilon} ensure the existence
of a constant $C>0$ such that
%__________________________________________________________
$$
|F_{\rho_0}|^{1/2} \equiv |\langle \psi,G_{\rho_0}\psi \rangle |^{1/2}
\leq 
\|B^{-1/2} \psi\|^{1/2} \, \|B^{1/2}G_{\rho_0} \psi\|^{1/2} \leq
C\|\psi\|_{{\rond H}_1}.
$$
%__________________________________________________________
If one replaces the above estimate in the previous one and let $\rho\rarrow 0$
we obtain (\ref{e:tito}), which finishes the proof of the
Theorem \ref{t:our-LAP}.
\medskip\\
\noindent{\bf Acknowledgements.} The second author is indebted to the
Swiss National Science Foundation for financial support.
%\pagebreak
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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%
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\end{thebibliography}
\end{document}
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3602 y Fs(1)2206 3635 y Fv(:=)e Fq(H)2405 3602 y Fs(1)2460
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(comprehensiv)m(e)g(discussion)e(of)i(the)h(Dirac)f(op)s(erator.)259
3974 y(In)i(connection)g(with)f(the)i(sp)s(ectral)f(analysis)e(of)j
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y Fs(1)p Fn(=)p Fs(2)2750 396 y Fv(d)o Ft(")118 645 y
Fv(whic)m(h)30 b(ma)m(y)h(b)s(e)f(treated)i(with)e(the)h(help)e(of)i(a)
g(Gron)m(w)m(all)g(t)m(yp)s(e)g(lemma.)41 b(Indeed,)31
b(if)e(one)j(tak)m(es)g Ft(\022)c Fv(=)e(1)p Ft(=)p Fv(2)32
b(in)118 758 y(Lemma)f(7.A.1)h(of)e([ABG])i(w)m(e)f(ma)m(y)g(\014nd)d
(a)j(p)s(ositiv)m(e)f(constan)m(t)h Ft(C)37 b Fv(suc)m(h)30
b(that)1167 987 y Fq(j)p Ft(F)1250 1001 y Fn(\032)1291
987 y Fq(j)25 b(\024)g Ft(C)1524 886 y Fl(\020)1578 987
y Fq(j)p Ft(F)1661 1001 y Fn(\032)1697 1010 y Fd(0)1737
987 y Fq(j)1762 950 y Fs(1)p Fn(=)p Fs(2)1892 987 y Fv(+)20
b Fq(k)p Ft( )s Fq(k)2135 1002 y Ff(H)2208 1011 y Fd(1)2248
987 y Fv(\()2283 926 y Fq(p)p 2359 926 87 4 v 61 x Ft(\032)2406
1001 y Fs(0)2466 987 y Fq(\000)2557 926 y(p)p 2633 926
48 4 v 61 x Ft(\032)p Fv(\))2715 886 y Fl(\021)2769 909
y Fs(2)2824 987 y Ft(:)118 1200 y Fv(On)32 b(the)h(other)f(hand,)h
(\(3.4\))h(and)e(\(ii\))g(of)g(Lemma)h(3.2)h(ensure)e(the)g(existence)h
(of)g(a)g(constan)m(t)h Ft(C)h(>)29 b Fv(0)k(suc)m(h)118
1313 y(that)672 1426 y Fq(j)p Ft(F)755 1440 y Fn(\032)791
1449 y Fd(0)830 1426 y Fq(j)855 1388 y Fs(1)p Fn(=)p
Fs(2)991 1426 y Fq(\021)25 b(jh)p Ft( )s(;)15 b(G)1321
1440 y Fn(\032)1357 1449 y Fd(0)1396 1426 y Ft( )s Fq(ij)1518
1388 y Fs(1)p Fn(=)p Fs(2)1655 1426 y Fq(\024)25 b(k)p
Ft(B)1870 1388 y FF(\000)p Fs(1)p Fn(=)p Fs(2)2035 1426
y Ft( )s Fq(k)2142 1388 y Fs(1)p Fn(=)p Fs(2)2268 1426
y Fq(k)p Ft(B)2387 1388 y Fs(1)p Fn(=)p Fs(2)2497 1426
y Ft(G)2569 1440 y Fn(\032)2605 1449 y Fd(0)2643 1426
y Ft( )s Fq(k)2750 1388 y Fs(1)p Fn(=)p Fs(2)2886 1426
y Fq(\024)g Ft(C)7 b Fq(k)p Ft( )s Fq(k)3206 1441 y Ff(H)3279
1450 y Fd(1)3319 1426 y Ft(:)118 1585 y Fv(If)38 b(one)g(replaces)f
(the)i(ab)s(o)m(v)m(e)g(estimate)f(in)f(the)h(previous)e(one)i(and)g
(let)g Ft(\032)f Fq(!)h Fv(0)g(w)m(e)h(obtain)e(\(3.8\),)42
b(whic)m(h)118 1698 y(\014nishes)28 b(the)j(pro)s(of)e(of)i(the)g
(Theorem)f(2.1.)118 1856 y Fc(Ac)m(kno)m(wledgemen)m(ts.)39
b Fv(The)25 b(second)h(author)g(is)e(indebted)h(to)h(the)g(Swiss)e
(National)h(Science)h(F)-8 b(oundation)118 1969 y(for)30
b(\014nancial)f(supp)s(ort.)118 2253 y Fw(References)118
2456 y Fv([Ag])251 b(Agmon,S.:)48 b Fe(Sp)-5 b(e)g(ctr)g(al)38
b(pr)-5 b(op)g(erties)38 b(of)e(Schr\177)-46 b(odinger)37
b(op)-5 b(er)g(ators)39 b(and)e(sc)-5 b(attering)36 b(the)-5
b(ory)38 b Fv(,)c(Ann.)532 2569 y(Scuola)c(Norm.)h(Sup.)e(Pisa,)h(Cl.)f
(Sci.)h(\(4\))h Fc(2)g Fv(\(1975\),)i(151-218.)118 2752
y([ABG])161 b(Amrein,W.,)39 b(Boutet)f(de)f(Mon)m(v)m(el,A.,)j
(Georgescu,V.:)56 b Ft(C)2610 2766 y Fs(0)2650 2752 y
Fe(-Gr)-5 b(oups,)41 b(Commutator)g(Metho)-5 b(ds)532
2865 y(and)44 b(Sp)-5 b(e)g(ctr)g(al)44 b(The)-5 b(ory)44
b(of)f Ft(N)10 b Fe(-Bo)-5 b(dy)43 b(Hamiltonians)7 b
Fv(,)46 b(Progress)41 b(in)f(Mathematics,)45 b(V)-8 b(ol.)41
b(135,)532 2978 y(Birkh\177)-45 b(auser,)29 b(1996.)118
3161 y([B-A])202 b(Ben-Artzi,M.:)63 b Fe(Glob)-5 b(al)45
b(estimates)e(for)g(the)g(Schr\177)-46 b(odinger)44 b(e)-5
b(quation)7 b Fv(,)45 b(J.)40 b(F)-8 b(unct.)42 b(Anal.)f
Fc(107)532 3274 y Fv(\(1992\),)33 b(362-368.)118 3456
y([B-AK])131 b(Ben-Artzi,M.,)38 b(Kleinerman,S.:)48 b
Fe(De)-5 b(c)g(ay)37 b(and)h(r)-5 b(e)g(gularity)39 b(for)e(the)h
(Schr\177)-46 b(odinger)38 b(e)-5 b(quation)7 b Fv(,)38
b(J.)532 3569 y(Anal.)30 b(Math.)h Fc(58)g Fv(\(1992\),)i(25-37.)118
3752 y([BG])229 b(Boutet)36 b(de)f(Mon)m(v)m(el,A.,)i(Georgescu,V.:)51
b Fe(Sp)-5 b(e)g(ctr)g(al)39 b(and)e(sc)-5 b(attering)38
b(the)-5 b(ory)38 b(by)e(the)h(c)-5 b(onjugate)532 3865
y(op)g(er)g(ator)29 b(metho)-5 b(d)9 b Fv(,)26 b(Algebra)c(and)g
(Analysis)f Fc(4)p Fv(,)j(3)f(\(1992\),)j(73-116.)f(and)d(St.)g(P)m
(etersb)s(ourg)h(Math.)532 3978 y(J.)31 b Fc(4)p Fv(,)f(3)h(\(1993\),)i
(469-501.)118 4161 y([BKM])146 b(Boutet)26 b(de)d(Mon)m(v)m(el,A.,)k
(Kazan)m(tzev)-5 b(a,G.,)29 b(M\025)-45 b(an)m(toiu,M.:)39
b Fe(A)n(nisotr)-5 b(opic)28 b(Schr\177)-46 b(odinger)28
b(op)-5 b(er)g(ators)532 4274 y(without)34 b(singular)f(sp)-5
b(e)g(ctrum)7 b Fv(,)33 b(Helv.)d(Ph)m(ys.)g(Acta,)i
Fc(69)f Fv(\(1996\),)i(13-25.)118 4457 y([BM])217 b(Boutet)45
b(de)e(Mon)m(v)m(el,A.,)49 b(M\025)-45 b(an)m(toiu,M.:)67
b Fe(The)45 b(metho)-5 b(d)47 b(of)e(the)f(we)-5 b(akly)46
b(c)-5 b(onjugate)45 b(op)-5 b(er)g(ator)10 b Fv(,)532
4569 y(Lecture)31 b(Notes)h(in)d(Ph)m(ys.,)h Fc(488)118
4752 y Fv([BMP])155 b(Boutet)45 b(de)f(Mon)m(v)m(el,A.,)49
b(Manda,D.,)f(Purice,R.:)67 b Fe(Limiting)45 b(absorption)i(principle)f
(for)f(the)532 4865 y(Dir)-5 b(ac)33 b(op)-5 b(er)g(ator)10
b Fv(,)34 b(Annales)29 b(Inst.)h(H.)h(P)m(oincar)m(\023)-43
b(e,)32 b(Ph)m(ysique)d(Th)m(\023)-43 b(eorique)30 b
Fc(58)h Fv(\(1993\),)i(413-431.)118 5048 y([CS])247 b(Constan)m(tin,P)
-8 b(.,)34 b(Saut,J.C.:)45 b Fe(L)-5 b(o)g(c)g(al)36
b(smo)-5 b(othing)37 b(pr)-5 b(op)g(erties)37 b(of)e(Schr\177)-46
b(odinger)36 b(e)-5 b(quations)7 b Fv(,)35 b(Indi-)532
5161 y(ana)c(Univ.)e(Math.)j(J.)e Fc(38)h Fv(\(1989\),)i(791-810.)118
5344 y([GM])210 b(Georgescu,V.,)38 b(M\025)-45 b(an)m(toiu,M.:)51
b Fe(On)37 b(the)g(sp)-5 b(e)g(ctr)g(al)39 b(the)-5 b(ory)38
b(of)f(singular)g(Dir)-5 b(ac)38 b(typ)-5 b(e)37 b(hamiltoni-)532
5457 y(ans)7 b Fv(,)32 b(to)f(app)s(ear)f(in)f(J.)h(of)h(Op)s(erator)f
(Th.)1985 5706 y(7)p eop
%%Page: 8 8
8 7 bop 118 162 a Fv([KY])225 b(Kato,T.,)31 b(Y)-8 b(a)5
b(jima,)30 b(K.:)40 b Fe(Some)32 b(examples)h(of)f(smo)-5
b(oth)34 b(op)-5 b(er)g(ators)35 b(and)e(the)f(asso)-5
b(ciate)g(d)34 b(smo)-5 b(oth-)532 275 y(ing)33 b(e\013e)-5
b(ct)9 b Fv(,)30 b(Rev.)g(Math.)i(Ph)m(ys.)e Fc(1)g Fv(\(1989\),)j
(481-496.)118 463 y([KRS])175 b(Kenig,C.E.,)38 b(Ruiz,A.,)h
(Sogge,C.D.:)55 b Fe(Uniform)39 b(Sob)-5 b(olev)40 b(ine)-5
b(qualities)38 b(and)i(unique)d(c)-5 b(ontinu-)532 576
y(ation)41 b(for)f(se)-5 b(c)g(ond-or)g(der)42 b(c)-5
b(onstant)41 b(c)-5 b(o)g(e\016cient)40 b(di\013er)-5
b(ential)41 b(op)-5 b(er)g(ators)7 b Fv(,)43 b(Duk)m(e)c(Math.)f(J.)g
Fc(55)532 689 y Fv(\(1987\),)33 b(329-347.)118 876 y([MP])219
b(M\025)-45 b(an)m(toiu,M.,)51 b(P)m(ascu,M.:)72 b Fe(Glob)-5
b(al)48 b(r)-5 b(esolvent)48 b(estimates)f(for)g(multiplic)-5
b(ation)49 b(op)-5 b(er)g(ators)7 b Fv(,)53 b(J.)532
989 y(Op)s(erator)30 b(Theory)-8 b(,)31 b Fc(36)g Fv(\(1996\),)i
(283-294.)118 1177 y([Mo])236 b(Mourre,E.:)37 b Fe(A)n(bsenc)-5
b(e)24 b(of)i(singular)g(c)-5 b(ontinuous)26 b(sp)-5
b(e)g(ctrum)27 b(for)e(c)-5 b(ertain)26 b(self-adjoint)h(op)-5
b(er)g(ators)7 b Fv(,)532 1290 y(Comm.)30 b(Math.)i(Ph)m(ys.,)p
Fc(78)f Fv(\(1981\),)i(391{408.)118 1477 y([RS1-4])126
b(Reed,M.,)41 b(Simon,B.:)55 b Fe(Metho)-5 b(ds)41 b(of)e(Mo)-5
b(dern)41 b(Mathematic)-5 b(al)41 b(Physics)7 b Fv(,)41
b(V)-8 b(ol.)38 b(1-4,)j(Academic)532 1590 y(Press,)30
b(New)h(Y)-8 b(ork.)118 1778 y([Si])287 b(Simon,B.:)37
b Fe(Best)27 b(c)-5 b(onstants)28 b(in)f(some)g(op)-5
b(er)g(ator)30 b(smo)-5 b(othness)29 b(estimates)7 b
Fv(,)27 b(J.)d(F)-8 b(unct.)24 b(Anal.)g Fc(107)532 1891
y Fv(\(1992\),)33 b(66-71.)118 2078 y([Th])247 b(Thaller,B.:)40
b Fe(The)33 b(Dir)-5 b(ac)33 b(Equation)7 b Fv(,)31 b(T)-8
b(exts)31 b(and)f(Monographs)g(in)g(Ph)m(ysics,)f(Springer)g(V)-8
b(erlag,)532 2191 y(Berlin-Heidelb)s(erg,)28 b(1992.)1985
5706 y(8)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
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