Content-Type: multipart/mixed; boundary="-------------9908130711766" This is a multi-part message in MIME format. ---------------9908130711766 Content-Type: text/plain; name="99-297.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-297.keywords" Dirac operator, conjugate operator method, limiting absorption principle, resolvent, thresholds, smooth operators, absolutely continuous or singular spectrum, wave operatorsspectrum ---------------9908130711766 Content-Type: application/x-tex; name="Iftimo-Manto.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Iftimo-Manto.tex" %%%%%%%% LaTeX2e format file %%%%%%%%%%% % \documentclass[11pt,leqno]{article} %\input{mssymb.tex} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{amscd} %_____________________________________________________________ %------------------------------------------------------------- % MISE EN PAGE: % \if@twoside \oddsidemargin 21pt \evensidemargin 59pt \marginparwidth 85pt \else \oddsidemargin 5mm % avant c'etait 5mm, mais pour les labels... \evensidemargin 5mm % avant c'etait 5mm, mais pour les labels... \marginparwidth 68pt % avant c'etait 68pt mais avec 80pt c'et plus % % c'et plus pratique pour les "labels visibles" \fi \marginparsep 10pt \topmargin -1cm \headheight 12pt \headsep 25pt % \footheight 12pt \footskip 30pt \textheight 22.8cm \textwidth 16cm \columnsep 10pt \columnseprule 0pt % %*************************************************************************** % Labelling in output text: % ------------------------- %% When one is working to the paper it is profitable %% to SEE on output pages the labels of theorems, sections, equations, etc. %% Thus one has to STAMP them (i.e. give instructions to PRINT these labels). %% This can be made by using the commands: %% \stamp , \stampsec , \stampth and \stampeq %% wich have as argument the label of the environement we whant to tag. %% They always have to be placed just BEFORE the environement we want to stamp %% and never in math mode. %% When the paper is in its final form, one has to eliminate these commands %% through all the paper: this is easy to do if one obeys to the rule: %% << Each time we write these commands one have to put them at %% the end of the imput lines>> %% Then all we have to do in order to neutralise them is to give the %% edit command, e.g.: %% change all '\stamp' into '%\stamp' % %% 1) for paragraphs and all one-line references the command is: % \def\stamp#1{\reversemarginpar\marginpar{\fbox{\bf #1}$\!$\bigtriculc$\!$\bigtriculc}} % %% 2) for sections, subsections the command is: % \def\stampsec#1{\reversemarginpar\marginpar{\vspace{-8 mm}\fbox{\bf#1}$\!$\bigtriculc$\!\!$\bigtriculc$\!\!$\bigtriculc}} % %% 3) for equations the command is: % \def\stampeq#1{\reversemarginpar \marginpar{\vspace{4.5 mm}\fbox{\bf#1}\raisebox{.1ex}{$\!\rhd$}}} % %% 4) for all 'theorem environements' the command is: % 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%\protect\newlength{\thname} \settowidth{\thname}{\bf{\nameofenvir\thetheorem\ \ \,}} % \def\tabth{\par\noindent\hspace{\thname}} % \begin{\nameofenvir} }{\end{\nameofenvir}} %********************************************** %__________________________________________________________ % % CREATION of FONT FAMILIES: % % 1) lettres "creuses" : % \font\sevenrm=cmbx7 \font\tenmsb=msbm10 at 11pt \font\sevenmsb=msbm7 at 8pt \font\fivemsb=msbm5 at 6pt \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb#1{{\tenmsb\fam\msbfam#1}} % \def\gol#1{\Bbb#1} % \def\RR{\Bbb R} \def\CC{\Bbb C} \def\BB{\Bbb B} \def\NN{\Bbb N} \def\QQ{\Bbb Q} \def\ZZ{\Bbb Z} \def\PP{\Bbb P} \def\EE{\Bbb E} \def\K{\Bbb K} \def\TT{\Bbb T} % % _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ % % Lettres Bold Roman % \font\elevencmb=cmb10 at 11pt \font\eightcmb=cmb10 at 8pt \font\sixcmb=cmb10 at 6pt \newfam\cmbfam \textfont\cmbfam=\elevencmb \scriptfont\cmbfam=\eightcmb \scriptscriptfont\cmbfam=\sixcmb \def\brom#1{{\elevencmb\fam\cmbfam#1}} % % % _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ % % 2) Lettres gothiques: \font\teneuf=eufm10 at 12pt \font\seveneuf=eufm7 at 8pt \font\fiveeuf=eufm5 at 6pt \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\goth#1{{\teneuf\fam\euffam#1}} % \newfont{\secgoth}{eufm10 at 16pt} % _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ % % 3) Lettres rondes (script): \font\tenrsf=rsfs10 at 11pt \font\sevenrsf=rsfs7 at 8pt \font\fiversf=rsfs5 at 6pt \newfam\rsffam \textfont\rsffam=\tenrsf \scriptfont\rsffam=\sevenrsf \scriptscriptfont\rsffam=\fiversf \def\rond#1{{\tenrsf\fam\rsffam#1}} % \def\ical#1{\!\scriptscriptstyle \cal#1} \def\irond#1{\!\scriptscriptstyle \rond#1} % _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ % 5) "Signes etranges mais bien utiles"; voir le tableau de iftimo: % \font\tenmsa=msam10 at 11pt \font\sevenmsa=msam8 \font\fivemsa=msam6 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \def\extra#1{{\tenmsa\fam\msafam#1}} % %____________________________________________ %******************************************************* % COMMANDES COURTES pour l'usage du \boldmath % dans l'environnement mathematique: % \def\bm#1{\boldsymbol{#1}} % %************************************************************ % Abbreviations boldmath propres a l'article \def\balpha{\bm{\alpha}} \def\bP{\bm{P}} \def\bQ{\bm{Q}} \def\bS{\bm{S}} %************************************************************ % % ABBREVIATIONS generales: % % Lettres greques avec "var": % \def\veps{\varepsilon} \def\vepsilon{\varepsilon} \def\vkappa{\varkappa} \def\vphi{\varphi} \def\vrho{\varrho} \def\vtheta{\vartheta} % \def\[[{[\hspace{-.5mm}[}% def des commandes \[[ et \]] pour produire doubles \def\]]{]\hspace{-.5mm}]}% parantheses carrees pour la fermeture en norme % % d'espaces vectoriels % \def\Left[[{\left[\hspace{-1mm}\left[} % def des commandes \Left[[ et \Right]] \def\Right]]{\right]\hspace{-1mm}\right]}% pour les memes buts, mais % % de grandeur variable % _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ \def\Ast{{\textstyle \ast}} \def\iAst{_{_\Ast}} % \def\cqfd{\hfill\hbox{\vrule\vbox to 4pt{\hrule width 4pt \vfill\hrule}\vrule}} % \def\qed{\hfill \vrule width 8pt height 9pt depth-1pt \par\medskip} % \def\acolade#1{{\left\{ #1 \right\}}} \def\norm#1{\left\Vert #1\right\Vert } \def\abs#1{\left\vert\, #1\,\right\vert } \def\Norm#1{\muskip0=-2mu{\left|\mkern\muskip0\left| \mkern\muskip0\left|#1\right|\mkern\muskip0 \right|\mkern\muskip0\right|}} \def\tr#1{\hbox{\rm Tr}\,(#1)} \def\st{\,\vert\,} % such that % \renewcommand{\Re}{\goth{R\hspace{-.2 ex}e}} \renewcommand{\Im}{\goth{I\hspace{-.1 ex}m}} % \def\proof{\par\addvspace{1 ex}\noindent \hbox{\bf Proof: }} \def\remark{\par\addvspace{2 ex}\noindent \hbox{\bf Remark: }} \def\remarks{\par\addvspace{2 ex}\noindent \hbox{\bf Remarks: }} \def\example{\par\addvspace{2 ex}\noindent \hbox{\bf Example: }} \def\examples{\par\addvspace{2 ex}\noindent \hbox{\bf Examples: }} % \newcommand{\rarrow}{\rightarrow} \newcommand{\lrarrow}{\longrightarrow} \def\norarrow{\rarrow \hspace{-4.5 mm} / \hspace{3 mm}} % \newcommand{\bra}{\langle} \newcommand{\ket}{\rangle} % \newcommand{\ep}{$\spadesuit$} \def\Mid{\!\!\! \mid \!\!\!} % \protect\newcommand{\dist}{\mbox{\rm dist}\,} % \def\ess{\mbox{\rm \scriptsize ess}} \def\nin{\notin} \def\wtilde#1{\widetilde{#1\,}} \def\wcheck#1{\widecheck{#1\,}} \def\what#1{\widehat{ #1\,}} % \def\pprod{\textstyle \prod} %\def\pprod{\displaystyle \mbox{$\textstyle \prod$}_{\substack{i}}} \def\ooplus{\textstyle \bigoplus} \def\ootimes{\textstyle\bigotimes} \def\ccup{\textstyle\bigcup} \def\ccap{\textstyle\bigcap} % % Limits for ``DISPLAYED'' formulas: \protect\newcommand{\wlim}{\mbox{\rm w\,-}\!\!\!\lim} \protect\newcommand{\slim}{\mbox{\rm s\,-}\!\!\!\lim} \protect\newcommand{\nlim}{\mbox{\rm n\,-}\!\!\!\lim} \protect\newcommand{\ulim}{\mbox{\rm u\,-}\!\!\!\lim} % Limits for ``IN TEXT'' formulas: \def\Wlim#1{\mbox{\rm w\,-}\!\lim_{#1}} \def\Slim#1{\mbox{\rm s\,-}\!\lim_{#1}} \def\Nlim#1{\mbox{\rm n\,-}\!\lim_{#1}} \def\Ulim#1{\mbox{\rm u\,-}\!\lim_{#1}} % \protect\newcommand{\supp}{\mbox{\rm supp}\,} \protect\newcommand{\suppi}{\mbox{\rm \scriptsize \supp}} % \def\triculc{\raisebox{.3ex}{$\scriptscriptstyle {\extra I}$ }} \def\bigtriculc{\raisebox{.3ex}{\extra I}} % %--------------------------------Environements: \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\ds}{\displaystyle} % \def\@begintheorem#1#2{\it \trivlist \item[\hskip \labelsep{\bf #1\ #2}]} \def\@endtheorem{\endtrivlist} % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{definition}[theorem]{Definition} \newtheorem{remarknumbered}[theorem]{Remark} % \renewcommand{\thefootnote}{(\alph{footnote})} %_________________________________________________ % ABBREVIATIONS propres a cet article: % % 1) Classes de fonctions differentiables: % ------------------------------------- \def\Ck#1#2{C^{#1}({#2})} % C-ontinuous, k-times derivable: #1=k % defined on #2 (ex: #2=\RR) % \def\Cck#1#2{C_{\mbox{\rm \scriptsize c}}^{#1}({#2})} % C-ontinuous, % c-ompactly supported, % k-times derivable: #1=k % defined on #2 (ex: #2=\RR) % cas particulier: k = 0; on utilisera alors: % ---------------------- \def\Cc#1{C_{\mbox{\rm \scriptsize c}}({#1})} \def\Cbk#1#2{C_{\mbox{\rm \scriptsize b}}^{#1}({#2})} % C-ontinuous, b-ounded, % k-times derivable: #1=k % defined on #2 (ex: #2=\RR) % cas particulier: k = 0; on utilisera alors: % ---------------------- \def\Cb#1{C_{\mbox{\rm \scriptsize b}}({#1})} \def\Cbu#1{C_{\mbox{\rm \scriptsize bu}}({#1})} % % % % 2) Algebres de fonctions: % ---------------------- \def\Cast#1{C_{#1}({\Bbb R}^*)} \def\CEast#1{C^{\brom E}_{#1}({\Bbb R}^*)} \def\TTE{\TT^{\brom E}} % \def\C#1{C_{#1}({\Bbb R})} \def\CE#1{C^{\brom E}_{#1}({\Bbb R})} % \def\CEalex{C^{\brom E}({\Bbb R}_{\infty})} % \def\Csphere#1{C_{#1}(\overline{\Bbb R})} \def\CEsphere#1{C^{\brom E}_{#1}(\overline{\Bbb R})} %% \def\Csphereast#1{C_{#1}(\overline{{\Bbb R}^*})} \def\CEsphereast#1{C^{\brom E}_{#1}(\overline{{\Bbb R^*}})} % \def\MEast{{\goth M}^{\brom E}} % \def\algebra#1#2{{\goth #1}_{#2}} % % Divers: \def\Rep{\mbox{\rm Rep}\,} \def\dist{\mbox{\rm dist}\,} \def\fin{\mbox{\rm \tiny fin}} \def\rc{\mbox{\rm \tiny rc}} \def\b{\mbox{\rm \scriptsize b}} \def\c{\mbox{\rm \scriptsize c}} \def\e{\mbox{\rm e}} \def\uu{\mbox{\rm \scriptsize u}} \def\p{\mbox{\rm \scriptsize p}} \def\reg{\mbox{\rm \scriptsize reg}} \def\alg{\mbox{\rm \scriptsize alg}} \newcommand{\dd}{\,\mbox{\rm d}} \def\loc{\mbox{\rm \scriptsize loc}} \def\free#1{#1_{\mbox{\scriptsize free}}} % %\def\cerc#1{\stackrel{\textstyle\vspace{-1mm} \circ}{\vspace{1mm} #1}} % \def\limQ#1{\lim_{Q \rarrow #1 \infty}} \def\limP#1{\lim_{P \rarrow #1 \infty}} % \def\btau{\tau} \def\thres#1{{\bm \tau}_{\!\mbox{\rm \scriptsize #1}}} %threshold (in bold) \def\spec#1{{\bm \sigma}_{\!\mbox{\rm \scriptsize #1}}} %spectrum (bold) % % \hyphenation{an-i-so-tro-pic} \hyphenation{par-ti-cu-lar} %_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ % % % \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} % % %\pagebreak % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{B} %\stamp{B} \protect\setcounter{equation}{0} \protect \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 \begin{thebibliography}{AAAA0} \bibitem[Ag]{Ag} Agmon,S.: {\em Spectral properties of Schr\"odinger operators and scattering theory \/}, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) {\bf 2} (1975), 151-218. % \bibitem[ABG]{ABG} Amrein,W., Boutet de Monvel,A., Georgescu,V.: {\em $C_0$-Groups, Commutator Methods and Spectral Theory of $N$-Body Hamiltonians\/}, Progress in Mathematics, Vol. 135, Birkh\"auser, 1996. % \bibitem[B-A]{B-A} Ben-Artzi,M.: {\em Global estimates for the Schr\"odinger equation\/}, J. Funct. Anal. {\bf 107} (1992), 362-368. % \bibitem[B-AK]{B-AK} Ben-Artzi,M., Kleinerman,S.: {\em Decay and regularity for the Schr\"odinger equation\/}, J. Anal. Math. {\bf 58} (1992), 25-37. % \bibitem[BG]{BG} Boutet de Monvel,A., Georgescu,V.: {\em Spectral and scattering theory by the conjugate operator method\/}, Algebra and Analysis {\bf 4}, 3 (1992), 73-116. and St.\ Petersbourg Math.\ J.\ {\bf 4}, 3 (1993), 469-501. % \bibitem[BKM]{BKM} Boutet de Monvel,A., Kazantzeva,G., M\u{a}ntoiu,M.: {\em Anisotropic Schr\"odinger operators without singular spectrum\/}, Helv.\ Phys.\ Acta, {\bf 69} (1996), 13-25. % \bibitem[BM]{BM} Boutet de Monvel,A., M\u{a}ntoiu,M.: {\em The method of the weakly conjugate operator\/}, Lecture Notes in Phys., {\bf 488} % \bibitem[BMP]{BMP} Boutet de Monvel,A., Manda,D., Purice,R.: {\em Limiting absorption principle for the Dirac operator\/}, Annales Inst.\ H. Poincar\'e, Physique Th\'eorique {\bf 58} (1993), 413-431. % \bibitem[CS]{CS} Constantin,P., Saut,J.C.: {\em Local smoothing properties of Schr\"odinger equations\/}, Indiana Univ. Math. J. {\bf 38} (1989), 791-810. % \bibitem[GM]{GM} Georgescu,V., M\u{a}ntoiu,M.: {\em On the spectral theory of singular Dirac type hamiltonians\/}, to appear in J. of Operator Th. % \bibitem[KY]{KY} Kato,T., Yajima, K.: {\em Some examples of smooth operators and the associated smoothing effect\/}, Rev. Math. Phys. {\bf 1} (1989), 481-496. % \bibitem[KRS]{KRS} Kenig,C.E., Ruiz,A., Sogge,C.D.: {\em Uniform Sobolev inequalities and unique continuation for second-order constant coefficient differential operators\/}, Duke Math. J. {\bf 55} (1987), 329-347. % \bibitem[MP]{MP} M\u{a}ntoiu,M., Pascu,M.: {\em Global resolvent estimates for multiplication operators\/}, J. Operator Theory, {\bf 36} (1996), 283-294. % \bibitem[Mo]{Mo} Mourre,E.: {\em Absence of singular continuous spectrum for certain self-adjoint operators\/}, Comm. Math. Phys.,{\bf 78} (1981), 391--408. % \bibitem[RS1-4]{RS1-4} Reed,M., Simon,B.: {\em Methods of Modern Mathematical Physics\/}, Vol. 1-4, Academic Press, New York. % \bibitem[Si]{Si} Simon,B.: {\em Best constants in some operator smoothness estimates\/}, J. Funct. Anal. {\bf 107} (1992), 66-71. % \bibitem[Th]{Th} Thaller,B.: {\em The Dirac Equation\/}, Texts and Monographs in Physics, Springer Verlag, Berlin-Heidelberg, 1992. \end{thebibliography} \end{document} ---------------9908130711766 Content-Type: application/postscript; name="Iftimo-Manto.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Iftimo-Manto.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: Iftimo-Manto.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Iftimo-Manto.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.07.26:1900 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg 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b(as)f(w)m(ell)e(as)i(to)g(the)g(fact)g(that)g(it)f(fails) f(for)h Ft(\025)25 b Fv(=)g Fq(\006)p Ft(m)p Fv(.)259 1596 y(The)38 b(break-do)m(wn)h(of)g(the)g(resolv)m(en)m(t)g(estimates) h(near)e(some)h(sp)s(ecial)f(p)s(oin)m(ts)f(is)h(a)h(w)m(ell-kno)m(wn)f (phe-)118 1709 y(nomenon.)53 b(Let)35 b(us)f(consider)g(the)g(case)i (of)f(the)g(\(p)s(ositiv)m(e\))f(Laplace)h(op)s(erator)g(\001)d(:=)g Fq(j)p Ft(P)13 b Fq(j)3306 1676 y Fs(2)3378 1709 y Fv(=)32 b Fq(\000)3567 1641 y Fl(P)3663 1667 y Fn(n)3663 1736 y(j)t Fs(=1)3805 1709 y Ft(@)3858 1676 y Fs(2)3853 1734 y Fn(j)118 1836 y Fv(acting)f(in)e Ft(L)559 1803 y Fs(2)598 1836 y Fv(\()p Fr(R)694 1803 y Fn(n)746 1836 y Fv(\).)41 b(Then)30 b(one)h(has)f(for)g(an)m(y)h Ft(s)24 b(>)1898 1800 y Fs(1)p 1898 1815 V 1898 1867 a(2)1944 1836 y Fv(:)1223 2040 y Fq(k)p Fv(\(\001)d Fq(\000)f Ft(\025)g Fq(\007)g Ft(i\026)p Fv(\))1776 2002 y FF(\000)p Fs(1)1871 2040 y Fq(k)1916 2058 y Fg(B)s Fs(\()p FF(H)2042 2066 y Fk(s)2081 2058 y Fs(\()p Fg(R)2160 2039 y Fk(n)2199 2058 y Fs(\))p Fn(;)p FF(H)2306 2067 y Fj(\000)p Fk(s)2388 2058 y Fs(\()p Fg(R)2467 2039 y Fk(n)2505 2058 y Fs(\)\))2590 2040 y Fq(\024)25 b Ft(C)q(;)-2659 b Fv(\(1.3\))118 2244 y(uniformly)19 b(in)i Ft(\026)k(>)g Fv(0)d(and)f Ft(\025)k Fq(2)g Ft(J)35 b Fq(\032)25 b Fv(\(0)p Ft(;)15 b Fq(1)p Fv(\),)25 b Ft(J)31 b Fv(compact.)39 b(F)-8 b(or)23 b(an)e(impro)m(v)m(ed)h(v)m (ersion)f(of)h(the)g(estimate)h(\(1.3\))118 2357 y(w)m(e)32 b(refer)g(to)g([Ag].)46 b(But)32 b(one)f(cannot)i(tak)m(e)g Ft(\025)27 b Fv(=)g(0)33 b(in)d(\(1.3\);)k(the)e(op)s(erators)g(\(\001) 21 b Fq(\007)g Ft(i\026)p Fv(\))3172 2324 y FF(\000)p Fs(1)3309 2357 y Fv(:)43 b Fq(H)3464 2358 y Fd(1)p 3464 2370 31 3 v 3464 2411 a(2)3504 2385 y Fs(+)p Fn(")3596 2357 y Fv(\()p Fr(R)3691 2324 y Fn(n)3744 2357 y Fv(\))28 b Fq(7!)118 2487 y(H)195 2515 y FF(\000)260 2488 y Fd(1)p 260 2500 V 260 2541 a(2)300 2515 y FF(\000)p Fn(")392 2487 y Fv(\()p Fr(R)487 2454 y Fn(n)540 2487 y Fv(\))j(are)g(not)g(b)s (ounded)e(uniformly)e(in)j Ft(\026)25 b(>)h Fv(0)31 b(if)e Ft(")e(>)e Fv(0)31 b(is)f(small.)40 b(This)29 b(is)h(connected)h(to)h (the)f(fact)118 2626 y(that)e(\001)c(=)g Ft(h)p Fv(\()p Ft(P)13 b Fv(\))29 b(with)e Ft(h)p Fv(\()p Ft(x)p Fv(\))f(=)f Fq(j)p Ft(x)p Fq(j)1335 2593 y Fs(2)1403 2626 y Fv(and)i(0)i(is)e(a)h (critical)f(v)-5 b(alue)28 b(for)g Ft(h)p Fv(.)40 b(Actually)-8 b(,)29 b(the)f(Laplace)g(op)s(erator)h(is)118 2738 y(just)c(a)h (particular)e(case.)40 b(The)25 b(op)s(erator)h Ft(h)p Fv(\()p Ft(P)13 b Fv(\))27 b(ma)m(y)f(b)s(e)f(de\014ned)g(in)f Ft(L)2552 2705 y Fs(2)2591 2738 y Fv(\()p Fr(R)2687 2705 y Fn(n)2740 2738 y Fv(\))h(for)h(large)g(classes)f(of)h(functions)118 2851 y Ft(h)j Fv(:)g Fr(R)313 2818 y Fn(n)394 2851 y Fq(7!)g Fr(R)s Fv(.)52 b(Under)32 b(certain)g(conditions)f(on)h Ft(h)p Fv(,)i(the)e(LAP)g(can)h(also)f(b)s(e)g(pro)m(v)m(ed)h(for)f Ft(h)p Fv(\()p Ft(P)13 b Fv(\))33 b(if)e Ft(\025)i Fv(b)s(elongs)118 2964 y(to)39 b(regions)f(a)m(w)m(a)m(y)i(from)d(the)h(critical)g(set)g Fc(Cr)p Fv(\()p Ft(h)p Fv(\))h(:=)f Ft(h)2117 2891 y Fl(\002)2156 2964 y Fv(\()p Fq(r)p Ft(h)p Fv(\))2354 2931 y FF(\000)p Fs(1)2449 2964 y Fv(\()p Fq(f)p Fv(0)p Fq(g)p Fv(\))2654 2891 y Fl(\003)2694 2964 y Fv(.)63 b(The)38 b(auxiliary)e(Hilb)s(ert)g(space)118 3077 y Fq(H)195 3091 y Fn(s)232 3077 y Fv(\()p Fr(R)327 3044 y Fn(n)380 3077 y Fv(\))31 b(\(with)e Ft(s)c(>)862 3041 y Fs(1)p 862 3056 36 4 v 862 3109 a(2)907 3077 y Fv(\))31 b(is)e(pla)m(ying)g(the)i(same)g(role)f(as)g(ab)s(o)m(v)m(e.)259 3190 y(In)d(the)g(last)h(p)s(erio)s(d,)e(the)h(problem)f(of)i (obtaining)e(the)h(LAP)g(for)h(the)f(op)s(erator)h Ft(h)p Fv(\()p Ft(P)13 b Fv(\))28 b(in)e(a)i(global)f(form,)118 3303 y(i.e.)38 b(also)f(at)h(the)f(critical)g(p)s(oin)m(ts,)h(has)f(b)s (ecome)g(one)h(of)g(sp)s(ecial)d(in)m(terest)j(in)e(the)h(literature;)k (as)c(a)h(short)118 3416 y(\(and)g(sub)5 b(jectiv)m(e\))38 b(selection)f(w)m(e)i(refer)e(to)i([KY],)f([Si])f(for)g(the)h 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b(of)g(the)h(function)f Ft(h)1331 4120 y Fn(m)1425 4106 y Fv(:)c Fr(R)1537 4073 y Fs(3)1610 4106 y Fq(7!)g Fr(R)41 b Fv(with)30 b Ft(h)2086 4120 y Fn(m)2153 4106 y Fv(\()p Ft(x)p Fv(\))e(:=)2426 4028 y Fl(p)p 2517 4028 373 4 v 78 x Fq(j)p Ft(x)p Fq(j)2619 4080 y Fs(2)2679 4106 y Fv(+)20 b Ft(m)2850 4080 y Fs(2)2921 4106 y Fv(and)31 b Ft(H)3175 4120 y Fs(0)3241 4106 y Fv(=)c Fp(\013)22 b Fq(\001)f Fp(P)35 b Fv(+)21 b Ft(m\014)37 b Fv(is)118 4219 y(unitarily)23 b(equiv)-5 b(alen)m(t)25 b(to)i(the)f(op)s(erator)g ([)p Ft(h)1610 4233 y Fn(m)1677 4219 y Fv(\()p Fp(P)15 b Fv(\))c Fq(\012)g Fv(1)1966 4239 y Fg(C)2009 4220 y Fd(2)2053 4219 y Fv(])g Fq(\010)g Fv([)p Fq(\000)p Ft(h)2319 4233 y Fn(m)2387 4219 y Fv(\()p Fp(P)j Fv(\))d Fq(\012)g Fv(1)2675 4239 y Fg(C)2718 4220 y Fd(2)2763 4219 y Fv(].)39 b(Unfortunately)-8 b(,)27 b(the)f(unitary)118 4332 y(transformation)k (is)f(complicated)h(\(see)i([Th]\))e(hence)g(a)h(direct)f(treatmen)m(t) i(of)f Ft(H)2937 4346 y Fs(0)3006 4332 y Fv(is)e(preferable.)259 4445 y(Let)d(us)f(describ)s(e)e(the)j(metho)s(d)f(of)g(pro)s(of.)39 b(W)-8 b(e)26 b(recall)f(\014rst)f(that)i(one)g(of)g(the)f(strongest)h (to)s(ols)f(in)f(pro)m(ving)118 4558 y(the)34 b(LAP)g(for)f(a)h (self-adjoin)m(t)f(op)s(erator)h Ft(H)41 b Fv(is)33 b(the)h(metho)s(d)f (of)h(E.)g(Mourre)f(\(see)i([Mo],)h([ABG]\).)g(It)e(relies)118 4670 y(mainly)29 b(on)h(the)g(so-called)h(Mourre)f(estimate:)1373 4875 y Ft(E)1440 4889 y Fn(H)1507 4875 y Fv(\()p Ft(J)9 b Fv(\))p Ft(i)p Fv([)p Ft(H)r(;)15 b(A)p Fv(])p Ft(E)1970 4889 y Fn(H)2039 4875 y Fv(\()p Ft(J)9 b Fv(\))27 b Fq(\025)e Ft(aE)2406 4889 y Fn(H)2473 4875 y Fv(\()p Ft(J)9 b Fv(\))p Ft(;)-2509 b Fv(\(1.4\))118 5079 y(where)25 b Ft(E)443 5093 y Fn(H)536 5079 y Fv(is)g(the)h(sp)s(ectral)f(measure)h(of)g Ft(H)7 b Fv(,)27 b Ft(J)35 b Fv(is)25 b(a)h(suitable)e(real)i(in)m (terv)-5 b(al,)26 b Ft(a)f(>)g Fv(0)h(and)f Ft(A)h Fv(is)f(a)h (self-adjoin)m(t)118 5192 y(op)s(erator.)39 b(One)22 b(often)h(sa)m(ys)h(that)f Ft(A)g Fv(is)f Fe(c)-5 b(onjugate)g(d)24 b Fv(to)g Ft(H)30 b Fv(on)22 b(the)h(in)m(terv)-5 b(al)22 b Ft(J)9 b Fv(.)39 b(Under)22 b(some)h(extra)h(tec)m(hnical)118 5305 y(assumptions)31 b(one)h(gets)i(a)e(LAP)g(for)g Ft(H)40 b Fv(with)31 b(resp)s(ect)h(to)h(the)f(op)s(erator)h(norm)f(in) e Fr(B)14 b Fv(\()q Fu(X)30 b Ft(;)15 b Fu(X)3414 5272 y FF(0)3437 5305 y Fv(\).)47 b(A)33 b(simple)1985 5706 y(2)p eop %%Page: 3 3 3 2 bop 118 162 a Fv(but)32 b(rough)g(option)g(is)g(to)i(tak)m(e)g Fu(X)57 b Fv(as)33 b(the)g(domain)e(of)i Ft(A)g Fv(endo)m(w)m(ed)g (with)e(the)i(graph)f(norm)g(and)g Fu(X)3715 129 y FF(0)3771 162 y Fv(the)118 275 y(dual)d(of)i Fu(X)24 b Fv(.)40 b(Unfortunately)-8 b(,)31 b(if)e Ft(H)j Fv(=)25 b Ft(h)p Fv(\()p Ft(P)13 b Fv(\))32 b(the)e(b)s(est)g(c)m(hoice)h(for)f Ft(A)h Fv(is)e(essen)m(tially)1289 509 y Ft(A)c Fv(=)1488 448 y(1)p 1488 488 46 4 v 1488 572 a(2)1544 509 y Fq(f)p Fv(\()p Fq(r)p Ft(h)p Fv(\)\()p Ft(P)13 b Fv(\))22 b Fq(\001)e Ft(Q)g Fv(+)g Ft(Q)g Fq(\001)h Fv(\()p Fq(r)p Ft(h)p Fv(\)\()p Ft(P)13 b Fv(\))p Fq(g)p Ft(;)118 731 y Fv(and)31 b(\(1.4\))j(cannot)f(b)s(e)e(obtained)g(if)g(the)h(closure) g(of)g Ft(J)41 b Fv(con)m(tains)32 b(critical)f(p)s(oin)m(ts)g(of)h (the)g(function)e Ft(h)p Fv(.)46 b(The)118 844 y(strict)22 b(p)s(ositivit)m(y)e(fails.)36 b(In)21 b(order)g(to)i(o)m(v)m(ercome)h (this)c(obstruction)h(an)h(approac)m(h)g(called)f(the)h(\\)p Fe(metho)-5 b(d)27 b(of)e(the)118 957 y(we)-5 b(akly)26 b(c)-5 b(onjugate)25 b(op)-5 b(er)g(ator)10 b Fv(")26 b(w)m(as)c(dev)m(elop)s(ed)f(in)f([BKM])j(and)e([BM],)i(in)e(whic)m(h)f (the)i(LAP)g(can)g(b)s(e)f(obtained)118 1070 y(from)f(the)h(w)m(eak)m (er)i(p)s(ositivit)m(y)c(assumption)g([)p Ft(iH)r(;)c(A)p Fv(])26 b Ft(>)f Fv(0)c(and)g(some)g(extra)g(tec)m(hnical)g (conditions.)36 b(This)19 b(ma)m(y)118 1183 y(b)s(e)27 b(seen)h(as)g(an)f(extension)h(of)f(the)h(w)m(ell-kno)m(wn)f (Kato-Putnam)h(theory)g(\(see)g([RS2]\).)41 b(It)28 b(has)f(already)g (b)s(een)118 1296 y(applied)d(to)i(di\013eren)m(t)f(t)m(yp)s(es)h(of)g (partial)e(di\013eren)m(tial)h(op)s(erators)h(with)e(anisotropic)h(co)s (e\016cien)m(ts,)i(leading)e(to)118 1409 y(results)f(on)h(the)h (absence)g(of)f(the)h(singular)d(sp)s(ectrum)h(in)g(v)-5 b(arious)25 b(situations)f(in)g(whic)m(h)g(Mourre's)h(approac)m(h)118 1521 y(did)k(not)h(w)m(ork.)41 b(The)30 b(task)h(of)f(the)h(presen)m(t) f(w)m(ork)h(is)e(to)i(apply)e(this)h(metho)s(d)g(to)h(Dirac)f(op)s (erators.)118 1807 y Fw(2)135 b(Main)45 b(results)118 2010 y Fv(Let)39 b Ft(H)365 2024 y Fs(0)442 2010 y Fv(b)s(e)f(the)h (self-adjoin)m(t)e(op)s(erator)i(de\014ned)e(b)m(y)i(\(1.1\).)66 b(Recall)38 b(that)h(the)g(op)s(erator)f Ft(T)52 b Fq(2)38 b Fr(B)14 b Fv(\()p Fu(H)33 b Fv(\))39 b(is)118 2122 y(called)d Ft(H)458 2136 y Fs(0)497 2122 y Fv(-)p Fe(smo)-5 b(oth)38 b Fv(if)e(there)g(is)f(a)i(\014nite)e(constan)m(t)i Ft(C)43 b Fv(suc)m(h)36 b(that)2478 2049 y Fl(R)2521 2154 y Fg(R)2588 2122 y Fq(k)p Ft(T)13 b(e)2741 2089 y FF(\000)p Fn(itH)2903 2098 y Fd(0)2943 2122 y Ft( )s Fq(k)3050 2089 y Fs(2)3090 2122 y Ft(dt)35 b Fq(\024)g Ft(C)7 b Fq(k)p Ft( )s Fq(k)3535 2089 y Fs(2)3611 2122 y Fv(for)36 b(an)m(y)118 2235 y Ft( )29 b Fq(2)c Fu(H)h Fv(.)118 2416 y Fc(Theorem)34 b(2.1)46 b Fv(\(i\))33 b Fe(Ther)-5 b(e)33 b(exists)g Ft(C)f(>)25 b Fv(0)33 b Fe(such)f(that)i(for)f(any)h Ft(\025)25 b Fq(2)g Fr(R)41 b Fe(and)34 b Ft(\026)25 b(>)g Fv(0)p Fe(:)1342 2612 y Fq(k)p Fv(\()p Ft(H)1498 2626 y Fs(0)1558 2612 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i\026)p Fv(\))1934 2574 y FF(\000)p Fs(1)2029 2612 y Fq(k)2074 2630 y Fg(B)s Fs(\()p Ff(H)2213 2639 y Fd(1)2254 2630 y Fn(;)p Ff(H)2347 2639 y Fj(\000)p Fd(1)2429 2630 y Fs(\))2486 2612 y Fq(\024)25 b Ft(C)q(:)716 2808 y Fv(\(ii\))32 b Fe(The)g(op)-5 b(er)g(ator)36 b Fq(h)p Fp(Q)q Fq(i)1563 2775 y FF(\000)p Fs(1)1690 2808 y Fe(is)c Ft(H)1863 2822 y Fs(0)1902 2808 y Fe(-smo)-5 b(oth.)118 3012 y Fc(Corollary)35 b(2.2)46 b Fe(L)-5 b(et)42 b Ft(V)62 b Fv(:)42 b Fr(R)1170 2979 y Fs(3)1257 3012 y Fq(!)f Fr(B)14 b Fv(\()p Fr(C)1545 2979 y Fs(4)1591 3012 y Fv(\))42 b Fe(b)-5 b(e)41 b(a)h(Bor)-5 b(el,)45 b(hermitian)e(value)-5 b(d)42 b(function)g(such)g(that)h(for)f(al)5 b(l)118 3125 y Ft(x)27 b Fq(2)g Fr(R)345 3092 y Fs(3)391 3125 y Fe(,)34 b Fq(k)p Ft(V)20 b Fv(\()p Ft(x)p Fv(\))p Fq(k)738 3145 y Fg(B)t Fs(\()p Fg(C)853 3126 y Fd(4)894 3145 y Fs(\))953 3125 y Fq(\024)27 b(h)p Ft(x)p Fq(i)1173 3092 y FF(\000)p Fs(2)1268 3125 y Fe(.)45 b(Then)34 b(ther)-5 b(e)35 b(exists)f(a)g(c)-5 b(onstant)36 b Fv(\000)27 b Ft(>)g Fv(0)34 b Fe(such)g(that)h(for)f(al)5 b(l)35 b Ft(\015)d Fq(2)27 b Fv(\()p Fq(\000)p Fv(\000)p Ft(;)15 b Fv(\000\))118 3238 y Fe(the)33 b(wave)g(op)-5 b(er)g(ators)36 b(for)d(the)g(c)-5 b(ouple)34 b Ft(H)1536 3252 y Fs(0)1575 3238 y Fe(,)e Ft(H)1711 3252 y Fn(\015)1780 3238 y Fv(:=)26 b Ft(H)1978 3252 y Fs(0)2037 3238 y Fv(+)20 b Ft(\015)5 b(V)1524 3434 y Fv(\012)1590 3396 y FF(\006)1674 3434 y Fv(=)25 b(s)15 b(-)j(lim)1821 3490 y Fn(t)p FF(!\0061)2058 3434 y Fv(e)2098 3396 y Fn(itH)2205 3404 y Fk(\015)2250 3434 y Fv(e)2290 3396 y FF(\000)p Fn(itH)2452 3405 y Fd(0)118 3660 y Fe(exist)33 b(and)g(ar)-5 b(e)34 b(unitary.)43 b(In)32 b(p)-5 b(articular,)35 b Ft(H)1656 3674 y Fn(\015)1732 3660 y Fe(has)f(pur)-5 b(ely)34 b(absolutely)g(c)-5 b(ontinuous)33 b(sp)-5 b(e)g(ctrum.)118 3864 y Fv(The)31 b(corollary)g(follo)m(ws)f (easily)h(b)m(y)g(com)m(bining)f(\(i\))i(of)f(Theorem)h(2.1)g(with)e (standard)h(smo)s(othness)g(results)118 3977 y(\(see)k(e.g.)h([RS4]\).) 52 b(Notice)35 b(that)g(it)f(states)h(m)m(uc)m(h)f(more)h(than)e(just)h (the)g(absence)h(of)f(the)h(p)s(oin)m(t)e(sp)s(ectrum)118 4090 y(\(whic)m(h)26 b(migh)m(t)h(b)s(e)f(a)h(kno)m(wn)f(fact)i(for)f (p)s(erturbations)d(as)j(ab)s(o)m(v)m(e\).)41 b(As)27 b(for)g(the)g(statemen)m(t)h(\(ii\))e(in)g(Theorem)118 4203 y(2.1,)36 b(it)d(follo)m(ws)g(immediately)f(\(see)j([RS4]\))g (from)e(\(i\).)51 b(W)-8 b(e)35 b(th)m(us)e(consider)g(\(i\))h(as)g(b)s (eing)e(the)i(main)f(result)118 4316 y(of)e(the)f(presen)m(t)h(pap)s (er)e(and)h(w)m(e)h(dev)m(ote)g(the)g(Section)f(3)h(to)g(its)f(pro)s (of.)259 4429 y(As)40 b(explained)d(in)h(the)i(In)m(tro)s(duction,)g (the)f(\014rst)g(problem)f(one)h(has)g(to)h(solv)m(e)g(is)e(to)j (\014nd)c(a)j(\\w)m(eakly)118 4542 y(conjugate)32 b(op)s(erator")f Ft(A)f Fv(satisfying)f([)p Ft(iH)1577 4556 y Fs(0)1617 4542 y Ft(;)15 b(A)p Fv(])26 b Ft(>)f Fv(0.)41 b(Our)29 b(c)m(hoice)i(is)1375 4776 y Ft(A)25 b Fv(=)1574 4714 y(1)p 1574 4755 V 1574 4838 a(2)1630 4776 y(\()p Ft(H)1748 4737 y FF(\000)p Fs(1)1741 4802 y(0)1842 4776 y Fp(P)35 b Fq(\001)20 b Fp(Q)g Fv(+)g Fp(Q)g Fq(\001)h Fp(P)14 b Ft(H)2486 4737 y FF(\000)p Fs(1)2479 4802 y(0)2580 4776 y Fv(\))p Ft(:)-2522 b Fv(\(2.1\))118 5005 y(Easy)25 b(computations)f(p)s(erformed)f(on)h Fu(S)43 b Fv(sho)m(w)24 b(that)h(the)g(ab)s(o)m(v)m(e)g(comm)m(utator)h(equals)e(\001\(\001)8 b(+)g Ft(m)3508 4972 y Fs(2)3548 5005 y Fv(\))3583 4972 y FF(\000)p Fs(1)3703 5005 y Fq(\021)25 b Ft(B)5 b Fv(.)118 5118 y(Then)28 b Ft(B)34 b Fv(extends)29 b(to)i(a)e(b)s(ounded)e(op)s (erator)j(in)e Fu(H)56 b Fv(whic)m(h)28 b(satis\014es)h Fq(h)p Ft( )s(;)15 b(B)5 b( )s Fq(i)27 b Ft(>)e Fv(0)k(for)g(an)m(y)h Ft( )f Fq(2)c Fu(H)44 b Fq(n)19 b(f)p Fv(0)p Fq(g)p Fv(.)118 5231 y(This)30 b(is)h(the)h(precise)g(meaning)f(to)i(giv)m(e)f(to)h (\\w)m(eak)g(p)s(ositivit)m(y".)44 b(In)32 b([BG])h(and)e([BMP])i (another)f(op)s(erator)118 5344 y Ft(A)k Fv(w)m(as)g(used)e(in)h(order) g(to)h(get)g(the)g(Mourre)g(estimate)g(for)f(the)h(p)s(erturb)s(ed)d (hamiltonian)g Ft(H)41 b Fv(=)33 b Ft(H)3642 5358 y Fs(0)3705 5344 y Fv(+)23 b Ft(V)d Fv(,)118 5457 y(under)39 b(suitable)g (conditions)g(on)i Ft(V)20 b Fv(.)72 b(It)40 b(do)s(es)h(not)f(\014t)h (to)g(our)f(purp)s(oses.)70 b(But)41 b(one)g Fe(may)49 b Fv(use)40 b(\(2.1\))j(in)1985 5706 y(3)p eop %%Page: 4 4 4 3 bop 118 162 a Fv(order)31 b(to)i(p)s(erform)d(Mourre)h(theory)h(in) e(a)i(rather)g(simpli\014ed)c(manner.)43 b(W)-8 b(e)33 b(also)f(send)e(to)j([GM])g(in)d(whic)m(h)118 275 y(a)38 b(systematic)h(w)m(a)m(y)g(to)f(treat)h(a)f(large)g(class)g(of)g (generalizations)f(of)h(the)g(Dirac)g(op)s(erator)g(p)s(erturb)s(ed)e (b)m(y)118 388 y(highly)c(singular)g(p)s(oten)m(tials)i(is)f(giv)m(en.) 52 b(Note)35 b(ho)m(w)m(ev)m(er)h(that)e(this)f(do)s(es)h(not)h(pro)m (v)m(e)g(the)f(LAP)g(at)h(critical)118 501 y(p)s(oin)m(ts)42 b(and)h(giv)m(e)h(only)f(the)g(absence)h(of)g(the)g(singular)d(con)m (tin)m(uous)i(sp)s(ectrum)f(and)h(some)h(qualitativ)m(e)118 614 y(information)33 b(on)i(the)g(p)s(oin)m(t)f(sp)s(ectrum.)54 b(Let)35 b(us)f(men)m(tion)h(that)g(one)h(cannot)f(use)g(directly)f (the)h(abstract)118 727 y(results)k(of)i([BKM])g(and)e([BM].)j(Indeed,) g(in)d(these)i(pap)s(ers)e(some)i(tec)m(hnical)f(di\016culties)e(are)j (solv)m(ed)f(b)m(y)118 840 y(w)m(orking)33 b(in)m(tensiv)m(ely)f(with)g (the)i(ev)m(olution)f(group)g Fq(f)p Ft(W)13 b Fv(\()p Ft(t)p Fv(\))30 b(=)g Ft(e)2366 807 y Fn(itA)2504 840 y Fq(j)g Ft(t)g Fq(2)g Fr(R)s Fq(g)q Fv(.)56 b(This)31 b(is)i(not)g(con)m(v)m(enien)m(t)i(in)118 953 y(our)c(case,)i(the)f (group)f(generated)h(b)m(y)g(\(2.1\))h(b)s(eing)d(to)s(o)i(in)m(v)m (olv)m(ed.)44 b(In)31 b(exc)m(hange,)i(although)e(b)s(oth)g Ft(H)3681 967 y Fs(0)3751 953 y Fv(and)118 1066 y Ft(A)j Fv(are)f(un)m(b)s(ounded)e(op)s(erators,)j(the)g(comm)m(utators)g Ft(B)k Fv(and)33 b Ft(i)p Fv([)p Ft(B)5 b(;)15 b(A)p Fv(])34 b(are)g(b)s(ounded)d(and)h(this)h(helps)e(us)i(to)118 1178 y(o)m(v)m(ercome)i(the)e(tec)m(hnical)f(di\016culties.)45 b(On)32 b(the)h(other)g(hand,)f(in)f(an)i(abstract)g(setting)g(a)g (supplemen)m(tary)118 1291 y(space)j Fu(A)57 b Fv(app)s(ears)35 b(in)g(the)h(LAP)-8 b(.)36 b(But)g(its)f(rigorous)g(de\014nition)e(is)i (based)h(also)f(on)h(the)g(ev)m(olution)f(group)118 1404 y Fq(f)p Ft(W)13 b Fv(\()p Ft(t)p Fv(\))27 b Fq(j)f Ft(t)h Fq(2)e Fr(R)s Fq(g)q Fv(.)48 b(Moreo)m(v)m(er,)34 b(in)c(applications)f (the)i(space)g Fu(A)53 b Fv(is)30 b(to)s(o)h(in)m(v)m(olv)m(ed)g(and)f (somewhat)i(arti\014cial.)118 1517 y(Our)d(direct)h(pro)s(of)g(puts)f (instead)h(in)f(evidence)h(the)h(simple)d(and)i(familiar)e(space)j Fu(H)3074 1531 y Fs(1)3138 1517 y Fv(=)25 b Fq(H)3311 1531 y Fs(1)3370 1517 y Fq(\012)20 b Fr(C)3521 1484 y Fs(4)3567 1517 y Fv(.)118 1801 y Fw(3)135 b(Pro)t(of)45 b(of)g(Theorem)g(2.1)118 2004 y Fv(Our)29 b(goal)i(is)f(to)h(pro)m(v)m (e)g(the)f(estimate)1295 2195 y Fq(jh)p Ft( )s(;)15 b Fv(\()p Ft(H)1568 2209 y Fs(0)1629 2195 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i\026)p Fv(\))2005 2157 y FF(\000)p Fs(1)2100 2195 y Ft( )s Fq(ij)26 b(\024)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)2568 2157 y Fs(2)2568 2218 y Ff(H)2641 2227 y Fd(1)2681 2195 y Ft(;)-2588 b Fv(\(3.1\))118 2386 y(where)23 b(the)h(constan)m(t)h Ft(C)31 b Fv(is)23 b(indep)s(enden)m(t)e(of)j Ft( )29 b Fq(2)c Fu(H)1933 2400 y Fs(1)1972 2386 y Fv(,)g Ft(\025)g Fq(2)g Fr(R)33 b Fv(and)23 b Ft(\026)i(>)g Fv(0.)39 b(As)24 b(stated)g(in)f(the)h(In)m(tro)s(duction,)118 2498 y(the)f(strategy)h(is)d(to)i(\014nd)e(a)i(self-adjoin)m(t)f(op)s (erator)h Ft(A)f Fv(ha)m(ving)g(p)s(ositiv)m(e)g(comm)m(utator)h(with)f Ft(H)3342 2512 y Fs(0)3403 2498 y Fv(and)g(ful\014lling)118 2611 y(some)31 b(extra)f(tec)m(hnical)g(conditions.)39 b(W)-8 b(e)32 b(de\014ne)d Ft(A)h Fv(on)g Fu(S)43 b Fv(:=)25 b Fq(S)i(\012)19 b Fr(C)2549 2578 y Fs(4)2624 2611 y Fv(\(the)31 b(completion)e(in)g Fq(k)20 b(\001)g(k)3541 2627 y Ff(H)3668 2611 y Fv(of)30 b(the)118 2724 y(algebraical)c(tensor) h(pro)s(duct)e Fq(S)19 b(\014)13 b Fr(C)1394 2691 y Fs(4)1439 2724 y Fv(\))27 b(b)m(y)f(the)h(form)m(ula)f(\(2.1\).)41 b(Notice)27 b(that)h Ft(H)2915 2691 y Fs(2)2908 2749 y(0)2979 2724 y Fv(=)d Fq(j)p Fp(P)15 b Fq(j)3206 2691 y Fs(2)3258 2724 y Fv(+)e Ft(m)3422 2691 y Fs(2)3486 2724 y Fv(=)25 b(\001)13 b(+)g Ft(m)3835 2691 y Fs(2)3872 2724 y Fv(,)118 2837 y(hence)31 b Ft(H)454 2799 y FF(\000)p Fs(1)447 2863 y(0)573 2837 y Fv(=)25 b Ft(H)745 2851 y Fs(0)784 2837 y Fv(\(\001)20 b(+)g Ft(m)1086 2804 y Fs(2)1126 2837 y Fv(\))1161 2804 y FF(\000)p Fs(1)1255 2837 y Fv(.)41 b(Th)m(us)874 3069 y Ft(A)25 b Fv(=)1073 3008 y(1)p 1073 3048 46 4 v 1073 3132 a(2)1144 2996 y Fl(\002)1182 3069 y Ft(H)1258 3083 y Fs(0)1297 3069 y Fv(\()p Fq(j)p Fp(P)15 b Fq(j)1463 3032 y Fs(2)1523 3069 y Fv(+)k Ft(m)1693 3032 y Fs(2)1733 3069 y Fv(\))1768 3032 y FF(\000)p Fs(1)1862 3069 y Fp(P)35 b Fq(\001)20 b Fp(Q)g Fv(+)g Fp(Q)h Fq(\001)f Fp(P)14 b Fv(\()p Fq(j)p Fp(P)h Fq(j)2589 3032 y Fs(2)2649 3069 y Fv(+)20 b Ft(m)2820 3032 y Fs(2)2859 3069 y Fv(\))2894 3032 y FF(\000)p Fs(1)2989 3069 y Ft(H)3065 3083 y Fs(0)3104 2996 y Fl(\003)118 3281 y Fv(lea)m(v)m(es)32 b Fu(S)e Fv(in)m(v)-5 b(arian)m(t.)40 b(The)30 b(op)s(erator)h Ft(A)f Fv(ma)m(y)h(also)f(b)s(e)g(set)h(in)e (the)i(non-symmetric)e(forms)469 3507 y Ft(A)d Fv(=)f Ft(H)742 3468 y FF(\000)p Fs(1)735 3533 y(0)836 3507 y Fp(P)34 b Fq(\001)21 b Fp(Q)f Fv(+)1182 3445 y(3)p Ft(i)p 1182 3486 77 4 v 1197 3569 a Fv(2)1269 3507 y Ft(H)1352 3468 y FF(\000)p Fs(1)1345 3533 y(0)1466 3507 y Fq(\000)1574 3445 y Ft(i)p 1567 3486 46 4 v 1567 3569 a Fv(2)1622 3507 y Ft(H)1705 3468 y FF(\000)p Fs(1)1698 3533 y(0)1800 3507 y Fp(\013)g Fq(\001)g Fp(P)15 b Ft(H)2098 3468 y FF(\000)p Fs(1)2091 3533 y(0)2217 3507 y Fv(=)25 b Ft(H)2396 3468 y FF(\000)p Fs(1)2389 3533 y(0)2490 3507 y Fp(P)35 b Fq(\001)20 b Fp(Q)h Fv(+)e Ft(iH)2940 3468 y FF(\000)p Fs(1)2933 3533 y(0)3055 3507 y Fv(+)3156 3445 y(1)p 3156 3486 V 3156 3569 a(2)3212 3507 y Ft(miH)3406 3468 y FF(\000)p Fs(1)3399 3533 y(0)3500 3507 y Ft(\014)5 b(H)3639 3468 y FF(\000)p Fs(1)3632 3533 y(0)118 3507 y Fv(\(3.2\))118 3718 y(if)29 b(one)i(tak)m(es)h(in)m(to)e(accoun)m(t)i (the)f(comm)m(utation)g(form)m(ulae)412 3909 y([)p Ft(Q)509 3923 y Fn(j)545 3909 y Ft(;)15 b(P)643 3924 y Fn(k)687 3909 y Fv(])25 b(=)g Ft(i\016)904 3924 y Fn(j)t(k)1010 3909 y Fv(and)60 b([)p Ft(Q)1314 3923 y Fn(j)1351 3909 y Ft(;)15 b(H)1474 3870 y FF(\000)p Fs(1)1467 3935 y(0)1569 3909 y Fv(])25 b(=)g Fq(\000)p Ft(H)1869 3870 y FF(\000)p Fs(1)1862 3935 y(0)1963 3909 y Fv([)p Ft(Q)2060 3923 y Fn(j)2097 3909 y Ft(;)15 b(H)2213 3923 y Fs(0)2252 3909 y Fv(])p Ft(H)2360 3870 y FF(\000)p Fs(1)2353 3935 y(0)2480 3909 y Fv(=)25 b Fq(\000)p Ft(iH)2761 3870 y FF(\000)p Fs(1)2754 3935 y(0)2855 3909 y Ft(\013)2913 3923 y Fn(j)2950 3909 y Ft(H)3033 3870 y FF(\000)p Fs(1)3026 3935 y(0)3127 3909 y Ft(;)46 b(j)31 b Fv(=)25 b(1)p Ft(;)15 b Fv(2)p Ft(;)g Fv(3)p Ft(:)118 4275 y Fc(Lemma)33 b(3.1)46 b Fv(\(i\))32 b Ft(A)h Fe(is)g(an)g(essential)5 b(ly)33 b(self-adjoint)h(op)-5 b(er)g(ator.)641 4388 y Fv(\(ii\))32 b Fe(The)g(domain)j(of)e(the)g(\(self-adjoint\))h(closur)-5 b(e)33 b(of)g Ft(A)g Fe(c)-5 b(ontains)34 b Fu(H)3072 4402 y Fs(1)3111 4388 y Fe(.)118 4564 y Fc(Pro)s(of:)48 b Fv(Ob)m(viously)-8 b(,)27 b Ft(A)h Fv(is)f(symmetric.)40 b(According)27 b(to)i(the)g(comm)m(utator)g(theorem)g(of)f(Nelson)g (\(see)h([RS2])118 4677 y(Theorem)h(X.37,)i(with)d Ft(N)35 b Fq(\021)25 b(h)p Fp(Q)q Fq(i)g(\025)g Fv(1\))31 b(one)g(needs)f(to)h (pro)m(v)m(e)g(for)f(an)m(y)h Ft( )e Fq(2)c Fu(S)18 b Fv(:)633 4868 y Fq(k)p Ft(A )s Fq(k)27 b(\024)e Ft(C)d Fq(kh)p Fp(Q)q Fq(i)p Ft( )s Fq(k)61 b Fv(and)106 b Fq(jh)q Ft(A )s(;)15 b Fq(h)p Fp(Q)q Fq(i)p Ft( )s Fq(i)21 b(\000)f(h)q(h)p Fp(Q)p Fq(i)p Ft( )s(;)15 b(A )s Fq(i)r(j)26 b(\024)f Ft(C)c Fq(h)q Ft( )s(;)15 b Fq(h)p Fp(Q)q Fq(i)p Ft( )s Fq(i)h Ft(:)118 5058 y Fv(Since)30 b(the)i(op)s(erators)f Ft(H)1000 5020 y FF(\000)p Fs(1)993 5085 y(0)1125 5058 y Fv(and)g Ft(H)1386 5020 y FF(\000)p Fs(1)1379 5085 y(0)1480 5058 y Ft(P)1538 5072 y Fn(j)1606 5058 y Fv(are)h(b)s(ounded)c (\(for)k(eac)m(h)g Ft(j)5 b Fv(\),)33 b(the)e(\014rst)f(inequalit)m(y)g (follo)m(ws)g(from)118 5171 y(\(3.2\).)43 b(The)30 b(second)g(one)h(is) e(an)h(immediate)g(consequence)h(of)f(the)h(equalit)m(y)507 5417 y Ft(i)p Fv([)p Ft(A;)15 b Fq(h)p Fp(Q)r Fq(i)p Fv(])26 b(=)f Ft(H)1052 5379 y FF(\000)p Fs(1)1045 5443 y(0)1156 5356 y Fq(j)p Fp(Q)p Fq(j)1285 5323 y Fs(2)p 1156 5396 169 4 v 1166 5480 a Fq(h)p Fp(Q)p Fq(i)1355 5417 y Fv(+)1456 5356 y Fq(j)p Fp(Q)p Fq(j)1585 5323 y Fs(2)p 1456 5396 V 1466 5480 a Fq(h)p Fp(Q)p Fq(i)1655 5417 y(\000)20 b Ft(H)1829 5379 y FF(\000)p Fs(1)1822 5443 y(0)1933 5356 y Fp(\013)g Fq(\001)h Fp(Q)p 1933 5396 214 4 v 1965 5480 a Fq(h)p Fp(Q)p Fq(i)2157 5417 y Ft(H)2240 5379 y FF(\000)p Fs(1)2233 5443 y(0)2334 5417 y Fp(P)35 b Fq(\001)20 b Fp(Q)g Fq(\000)g Fp(Q)g Fq(\001)h Fp(P)14 b Ft(H)2978 5379 y FF(\000)p Fs(1)2971 5443 y(0)3082 5356 y Fp(\013)20 b Fq(\001)h Fp(Q)p 3082 5396 V 3114 5480 a Fq(h)p Fp(Q)q Fq(i)3306 5417 y Ft(H)3389 5379 y FF(\000)p Fs(1)3382 5443 y(0)3483 5417 y Ft(;)1985 5706 y Fv(4)p eop %%Page: 5 5 5 4 bop 118 162 a Fv(obtained)30 b(b)m(y)g(a)h(direct)f(comm)m(utator)i (calculus)d(p)s(erformed)f(on)j(the)f(in)m(v)-5 b(arian)m(t)30 b(domain)f Fu(S)18 b Fv(.)p 3831 154 67 67 v 259 319 a(In)35 b(Section)f(2)i(w)m(e)f(in)m(tro)s(duced)f(the)h(notation)g Ft(B)k Fv(for)c(the)g(b)s(ounded)e(op)s(erator)j(\001\(\001)23 b(+)g Ft(m)3453 286 y Fs(2)3492 319 y Fv(\))3527 286 y FF(\000)p Fs(1)3655 319 y Fv(=)32 b(1)24 b Fq(\000)118 431 y Ft(m)198 398 y Fs(2)237 431 y Fv(\(\001)i(+)f Ft(m)550 398 y Fs(2)590 431 y Fv(\))625 398 y FF(\000)p Fs(1)719 431 y Fv(.)66 b(As)38 b(a)h(consequence)g(of)g(the)g(Lemma)g(3.1,)i(a)e (standard)f(computation)h(p)s(erformed)e(on)118 544 y(v)m(ectors)32 b Ft(')486 558 y Fs(1)526 544 y Ft(;)46 b(')656 558 y Fs(2)721 544 y Fq(2)25 b Fu(H)896 558 y Fs(1)955 544 y Fq(\\)20 b Fu(H)1151 511 y Fs(1)1221 544 y Fv(giv)m(es)1164 729 y Fq(h)p Ft(')1258 743 y Fs(1)1299 729 y Ft(;)15 b(B)5 b(')1472 743 y Fs(2)1512 729 y Fq(i)25 b Fv(=)g Fq(h)p Ft(iH)1810 743 y Fs(0)1850 729 y Ft(')1909 743 y Fs(1)1949 729 y Ft(;)15 b(A')2116 743 y Fs(2)2156 729 y Fq(i)21 b(\000)f(h)p Ft(iA')2496 743 y Fs(1)2536 729 y Ft(;)15 b(H)2652 743 y Fs(0)2692 729 y Ft(')2751 743 y Fs(2)2791 729 y Fq(i)p Ft(:)-2733 b Fv(\(3.3\))118 914 y(This)41 b(allo)m(ws)i(us)f(to)i(in)m(terpret)f Ft(B)k Fv(as)d(the)f(comm)m(utator)h([)p Ft(iH)2332 928 y Fs(0)2372 914 y Ft(;)15 b(A)p Fv(])44 b(on)f Fu(H)2777 928 y Fs(1)2845 914 y Fq(\\)28 b Fu(H)3049 881 y Fs(1)3089 914 y Fv(.)79 b(Since)42 b(it)h(will)d(pla)m(y)118 1027 y(an)d(imp)s(ortan)m(t)f(r^)-45 b(ole)36 b(in)g(our)g(pro)s(of,)i(w)m (e)f(shall)e(p)s(oin)m(t)h(out)h(some)g(of)g(its)f(prop)s(erties.)59 b(If)36 b(one)h(sets)g Ft(B)3681 994 y Fs(1)p Fn(=)p Fs(2)3827 1027 y Fv(=)118 1140 y Fq(j)p Fp(P)15 b Fq(j)p Fv(\()p Fp(P)364 1104 y Fs(2)430 1140 y Fv(+)25 b Ft(m)606 1107 y Fs(2)646 1140 y Fv(\))681 1107 y FF(\000)p Fs(1)p Fn(=)p Fs(2)886 1140 y Fq(2)39 b Fr(B)14 b Fv(\()p Fu(H)33 b Fv(\))39 b(w)m(e)g(ma)m(y)h(see)f Ft(B)1849 1107 y Fs(1)p Fn(=)p Fs(2)1998 1140 y Fv(as)g(an)g(op)s(erator)g Ft(b)p Fv(\()p Fp(P)15 b Fv(\))39 b(de\014ned)e(via)i(the)g(functional) 118 1253 y(calculus)31 b(b)m(y)h(the)h(Borel)f(function)f Ft(b)e Fv(:)g Fr(R)1534 1220 y Fs(3)1607 1253 y Fq(!)g Fr(R)s Fv(,)39 b Ft(b)p Fv(\()p Fp(\024)p Fv(\))29 b(=)f Fq(j)p Fp(\024)p Fq(j)p Fv(\()p Fp(\024)2356 1220 y Fs(2)2417 1253 y Fv(+)22 b Ft(m)2590 1220 y Fs(2)2629 1253 y Fv(\))2664 1220 y FF(\000)p Fs(1)p Fn(=)p Fs(2)2829 1253 y Fv(.)47 b(The)31 b(left)h(in)m(v)m(erse)h Ft(B)3628 1220 y FF(\000)p Fs(1)p Fn(=)p Fs(2)3824 1253 y Fv(of)118 1365 y Ft(B)192 1332 y Fs(1)p Fn(=)p Fs(2)332 1365 y Fv(satis\014es)d(the)g(prop)s(ert) m(y)g Ft(B)1269 1332 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1459 1365 y Fq(2)25 b Fr(B)13 b Fv(\()q Fu(H)1724 1379 y Fs(1)1769 1365 y Ft(;)i Fu(H)27 b Fv(\),)k(i.e.)40 b(for)30 b(all)g Ft( )e Fq(2)d Fu(H)2700 1379 y Fs(1)1570 1550 y Fq(k)p Ft(B)1689 1513 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1854 1550 y Ft( )s Fq(k)h(\024)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)2307 1565 y Ff(H)2380 1574 y Fd(1)2420 1550 y Ft(:)-2327 b Fv(\(3.4\))118 1735 y(Indeed,)35 b(since)f Ft(b)p Fv(\()p Fp(\024)p Fv(\))f Fq(\024)f Fv(1)23 b Fq(\000)g Ft(m)p Fq(j)p Fp(\024)p Fq(j)1330 1702 y FF(\000)p Fs(1)1459 1735 y Fv(it)34 b(will)e(b)s(e)i(enough)g(to)i(sho)m(w)e(that)i Fq(j)p Fp(P)14 b Fq(j)2852 1702 y FF(\000)p Fs(1)2979 1735 y Fq(2)32 b Fr(B)14 b Fv(\()p Fu(H)3251 1749 y Fs(1)3296 1735 y Ft(;)h Fu(H)27 b Fv(\).)54 b(But)35 b(this)118 1848 y(is)c(a)g(direct)g(consequence)h(of)g(the)f(Hardy)g(inequalit)m (y)f Fq(kj)p Fp(P)15 b Fq(j)2200 1815 y FF(\000)p Fs(1)2294 1848 y Ft( )s Fq(k)2401 1863 y Ff(H)2526 1848 y Fq(\024)2634 1812 y Fs(1)p 2634 1827 36 4 v 2634 1879 a(4)2679 1848 y Fq(k)p Ft( )s Fq(k)2831 1863 y Ff(H)2904 1872 y Fd(1)2944 1848 y Fv(,)32 b(v)-5 b(alid)30 b(for)h(eac)m(h)i Ft( )d Fq(2)c Fu(H)3833 1862 y Fs(1)3872 1848 y Fv(.)118 1970 y(Also,)i(remark)f(that)h Ft(B)922 1937 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1086 1970 y Ft(A)f Fv(is)g(a)g(w)m(ell)f(de\014ned)g (linear)g(op)s(erator)i(:)d Fu(S)43 b Fq(!)25 b Fu(H)i Fv(.)40 b(W)-8 b(e)28 b(stress)f(that)h(it)f(extends)118 2083 y(to)k(an)f(elemen)m(t)h(of)g Fr(B)14 b Fv(\()p Fu(H)974 2097 y Fs(1)1019 2083 y Ft(;)h Fu(H)27 b Fv(\))k(i.e.)f(there) h(is)e(some)i Ft(C)h(>)25 b Fv(0)30 b(suc)m(h)g(that)h(for)g(an)m(y)f Ft( )f Fq(2)c Fu(H)3183 2097 y Fs(1)1536 2268 y Fq(k)p Ft(B)1655 2230 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1820 2268 y Ft(A )s Fq(k)h(\024)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)2341 2283 y Ff(H)2414 2292 y Fd(1)2454 2268 y Ft(:)-2361 b Fv(\(3.5\))118 2465 y(This)32 b(clearly)i(follo)m(ws)f(from)h(\(3.4\))i (and)d(\(3.2\))j(if)d(one)i(tak)m(es)g(in)m(to)f(accoun)m(t)i(that)f Ft(B)3064 2432 y FF(\000)p Fs(1)p Fn(=)p Fs(2)3260 2465 y Fv(=)c Fq(j)p Ft(H)3463 2479 y Fs(0)3502 2465 y Fq(j)15 b(j)p Fp(P)g Fq(j)3673 2432 y FF(\000)p Fs(1)3802 2465 y Fv(on)118 2578 y Fu(S)j Fv(.)41 b(Moreo)m(v)m(er,)32 b(an)f(easy)g(computation)f(p)s(erformed)f(on)h Fu(S)48 b Fv(yields)512 2762 y([)p Ft(iB)5 b(;)15 b(A)p Fv(])26 b(=)f Ft(i)p Fv([1)d Fq(\000)d Ft(m)1190 2725 y Fs(2)1230 2762 y Ft(H)1313 2724 y FF(\000)p Fs(2)1306 2788 y(0)1407 2762 y Ft(;)c(A)p Fv(])84 b(=)f Ft(m)1858 2725 y Fs(2)1897 2762 y Ft(H)1980 2724 y FF(\000)p Fs(1)1973 2788 y(0)2074 2762 y Fv(\([)p Ft(iH)2241 2776 y Fs(0)2281 2762 y Ft(;)15 b(A)p Fv(])p Ft(H)2497 2724 y FF(\000)p Fs(1)2490 2788 y(0)2612 2762 y Fv(+)20 b Ft(H)2786 2724 y FF(\000)p Fs(1)2779 2788 y(0)2881 2762 y Fv([)p Ft(iH)3013 2776 y Fs(0)3052 2762 y Ft(;)15 b(A)p Fv(]\))p Ft(H)3303 2724 y FF(\000)p Fs(1)3296 2788 y(0)1624 2900 y Fv(=)83 b Ft(m)1858 2863 y Fs(2)1897 2900 y Ft(H)1980 2862 y FF(\000)p Fs(1)1973 2926 y(0)2074 2900 y Fv(\(\001)p Ft(H)2268 2862 y FF(\000)p Fs(3)2261 2926 y(0)2383 2900 y Fv(+)20 b Ft(H)2557 2862 y FF(\000)p Fs(3)2550 2926 y(0)2651 2900 y Fv(\001\))p Ft(H)2845 2862 y FF(\000)p Fs(1)2838 2926 y(0)2965 2900 y Fv(=)25 b(2)p Ft(m)3186 2863 y Fs(2)3225 2900 y Fv(\001)p Ft(H)3384 2862 y FF(\000)p Fs(5)3377 2926 y(0)3478 2900 y Ft(:)118 3085 y Fv(Actually)-8 b(,)45 b([)p Ft(iB)5 b(;)15 b(A)p Fv(])42 b(extends)g(to)g(an)g(elemen)m(t)g (of)g Fr(B)14 b Fv(\()p Fu(H)33 b Fv(\),)45 b(and)c(one)h(c)m(hec)m(ks) i(b)m(y)d(con)m(tin)m(uit)m(y)h(that)g(for)g(all)118 3198 y Ft(')177 3212 y Fs(1)217 3198 y Ft(;)k(')347 3212 y Fs(2)412 3198 y Fq(2)25 b Fu(H)587 3212 y Fs(1)646 3198 y Fq(\\)20 b Fu(H)842 3165 y Fs(1)882 3198 y Fv(,)1034 3382 y Fq(h)p Ft(')1128 3396 y Fs(1)1168 3382 y Ft(;)15 b Fv(2)p Ft(m)1333 3345 y Fs(2)1373 3382 y Fv(\001)p Ft(H)1532 3344 y FF(\000)p Fs(5)1525 3408 y(0)1626 3382 y Ft(')1685 3396 y Fs(2)1725 3382 y Fq(i)26 b Fv(=)f Fq(h)p Ft(iB)5 b(')2081 3396 y Fs(1)2121 3382 y Ft(;)15 b(A')2288 3396 y Fs(2)2328 3382 y Fq(i)21 b(\000)f(h)p Ft(iA')2668 3396 y Fs(1)2709 3382 y Ft(;)15 b(B)5 b(')2882 3396 y Fs(2)2921 3382 y Fq(i)p Ft(:)-2863 b Fv(\(3.6\))118 3579 y(It)31 b(is)e(also)h(clear)h(that)g Ft(B)979 3546 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1143 3579 y Fv(\001)p Ft(H)1302 3541 y FF(\000)p Fs(5)1295 3605 y(0)1396 3579 y Ft(B)1470 3546 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1665 3579 y Fv(extends)f(to)h(a)g(b)s(ounded)d(op)s(erator)j(in)e Fu(H)e Fv(.)259 3692 y(W)-8 b(e)31 b(start)f(no)m(w)g(the)f(main)g (part)g(of)h(the)g(pro)s(of)e(of)i(the)g(Theorem)f(2.1)i(b)m(y)e (de\014ning)f(for)h(eac)m(h)i Ft(\025)25 b Fq(2)g Fr(R)38 b Fv(and)118 3805 y Ft(\026)33 b(>)g Fv(0)j(the)f(family)f(of)h(op)s (erators)g Ft(T)1416 3772 y FF(\006)1403 3828 y Fn(")1475 3805 y Fv(\()p Ft(\025;)15 b(\026)p Fv(\))34 b(:=)g Ft(H)1933 3819 y Fs(0)1995 3805 y Fq(\000)23 b Ft(\025)h Fq(\007)f Ft(i\026)g Fq(\007)g Ft(i"B)5 b Fv(,)37 b(indexed)d(b)m(y)h Ft(")f Fq(2)e Fv([0)p Ft(;)15 b Fv(1\).)58 b(Remark)118 3918 y(\014rst)33 b(that)g(they)h(are)f(\(linear\))f(homeomorphisms)g (:)e Fu(H)2088 3885 y Fs(1)2158 3918 y Fq(!)f Fu(H)e Fv(.)49 b(Indeed,)33 b(for)g Ft(\026)d(>)f Fv(0)34 b(and)e Ft(')f Fq(2)e Fu(H)3673 3885 y Fs(1)3712 3918 y Fv(,)34 b(the)118 4031 y(estimate)452 4216 y Fq(k)p Ft(T)563 4178 y FF(\006)550 4238 y Fn(")622 4216 y Ft(')p Fq(k)726 4178 y Fs(2)792 4216 y Fv(=)25 b Fq(k)p Fv(\()p Ft(H)1044 4230 y Fs(0)1104 4216 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i"B)5 b Fv(\))p Ft(')p Fq(k)1645 4178 y Fs(2)1706 4216 y Fv(+)20 b Ft(\026)1852 4178 y Fs(2)1891 4216 y Fq(k)p Ft(')p Fq(k)2040 4178 y Fs(2)2101 4216 y Fv(+)g(2)p Fm(R)-9 b(e)p Fq(h)p Fv(\()p Ft(H)2496 4230 y Fs(0)2556 4216 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i"B)5 b Fv(\))p Ft(';)15 b Fq(\007)p Ft(i\026')p Fq(i)28 b(\025)d Ft(\026)3522 4178 y Fs(2)3561 4216 y Fq(k)p Ft(')p Fq(k)3710 4178 y Fs(2)118 4216 y Fv(\(3.7\))118 4400 y(based)32 b(on)g(the)g(w)m(eak)h (p)s(ositivit)m(y)d(of)i Ft(B)5 b Fv(,)32 b(sho)m(ws)g(that)g Ft(T)2057 4367 y FF(\006)2044 4423 y Fn(")2144 4400 y Fq(\021)27 b Ft(T)2308 4367 y FF(\006)2295 4423 y Fn(")2367 4400 y Fv(\()p Ft(\025;)15 b(\026)p Fv(\))33 b(are)f(injectiv)m(e.)45 b(Since)31 b Ft(T)3477 4367 y FF(\006)3464 4423 y Fn(")3564 4400 y Fv(=)d(\()p Ft(T)3764 4367 y FF(\007)3751 4423 y Fn(")3823 4400 y Fv(\))3858 4367 y FF(\003)118 4513 y Fv(their)33 b(ranges)i(are)f(dense.)51 b(F)-8 b(or)35 b(eac)m(h)g Ft(")d(<)f Fv(1)k(they)f(are)h(closed)e(op)s(erators)i(and) e(their)g(surjectivit)m(y)g(follo)m(ws)118 4626 y(easily)-8 b(.)40 b(W)-8 b(e)32 b(ma)m(y)f(th)m(us)f(conclude)g(b)m(y)g(the)g(op)s (en)g(mapping)f(theorem.)259 4739 y(Set)i(no)m(w)g Ft(G)678 4706 y FF(\006)678 4762 y Fn(")762 4739 y Fq(\021)26 b Ft(G)931 4706 y FF(\006)931 4762 y Fn(")989 4739 y Fv(\()p Ft(\025;)15 b(\026)p Fv(\))27 b(:=)f(\()p Ft(T)1457 4706 y FF(\006)1444 4762 y Fn(")1516 4739 y Fv(\()p Ft(\025;)15 b(\026)p Fv(\)\))1769 4706 y FF(\000)p Fs(1)1864 4739 y Fv(.)42 b(They)30 b(are)h(op)s(erators)g(of)g Fr(B)14 b Fv(\()p Fu(H)33 b Ft(;)15 b Fu(H)3193 4706 y Fs(1)3232 4739 y Fv(\))26 b Fq(\032)g Fr(B)14 b Fv(\()p Fu(H)33 b Fv(\))e(whic)m(h)118 4852 y(v)m(erify)42 b Ft(G)455 4819 y FF(\006)455 4874 y Fn(")559 4852 y Fv(=)j(\()p Ft(G)782 4819 y FF(\007)782 4874 y Fn(")841 4852 y Fv(\))876 4819 y FF(\003)916 4852 y Fv(.)76 b(F)-8 b(or)44 b(a)e(v)m(ector)i Ft( )i Fv(arbitrarily)40 b(\014xed)i(in)f Fu(H)69 b Fv(w)m(e)43 b(shall)d(systematically)i(use)g(the)118 4965 y(notation)34 b Ft(F)556 4932 y FF(\006)543 4987 y Fn(")645 4965 y Fv(=)c Fq(h)p Ft( )s(;)15 b(G)955 4932 y FF(\006)955 4987 y Fn(")1015 4965 y Fv(\()p Ft(\025;)g(\026)p Fv(\))p Ft( )s Fq(i)p Fv(.)51 b(The)33 b(function)f Ft(")f Fq(7!)f Ft(F)2220 4932 y FF(\006)2207 4987 y Fn(")2312 4965 y Fv(is)j(con)m(tin)m(uous)g(on)g([0)p Ft(;)15 b Fv(1\))35 b(and)e(in)f(particular)118 5078 y(in)d Ft(")d Fv(=)f(0)31 b(as)f(it)g(is)f(sho)m(wn)h(b)m(y)141 5263 y Fq(j)p Ft(F)237 5225 y FF(\006)224 5285 y Fn(")316 5263 y Fq(\000)20 b Ft(F)478 5224 y FF(\006)465 5289 y Fs(0)537 5263 y Fq(j)36 b Fv(=)g Fq(jh)p Ft( )s(;)15 b(G)939 5225 y FF(\006)939 5285 y Fn(")999 5263 y Fv(\()p Ft(\025;)g(\026)p Fv(\))p Ft( )s Fq(i)22 b(\000)e(h)p Ft( )s(;)15 b Fv(\()p Ft(H)1675 5277 y Fs(0)1735 5263 y Fq(\000)20 b Ft(\025)h Fq(\007)e Ft(i\026)p Fv(\))2111 5225 y FF(\000)p Fs(1)2206 5263 y Ft( )s Fq(ij)27 b(\024)e(jh)p Ft( )s(;)15 b(G)2685 5225 y FF(\006)2685 5285 y Fn(")2745 5263 y Fv(\()p Ft(\025;)g(\026)p Fv(\))p Ft(i"B)5 b Fv(\()p Ft(H)3221 5277 y Fs(0)3281 5263 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i\026)p Fv(\))3657 5225 y FF(\000)p Fs(1)3752 5263 y Ft( )s Fq(ij)598 5422 y(\024)36 b Ft(")p Fq(k)p Ft(B)5 b Fq(k)15 b(k)p Ft(G)1043 5385 y FF(\006)1043 5445 y Fn(")1103 5422 y Fv(\()p Ft(\025;)g(\026)p Fv(\))p Fq(k)g(k)p Fv(\()p Ft(H)1537 5436 y Fs(0)1598 5422 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i\026)p Fv(\))1974 5385 y FF(\000)p Fs(1)2069 5422 y Fq(k)15 b(k)p Ft( )s Fq(k)2281 5385 y Fs(2)2348 5422 y Fq(\024)2480 5361 y Ft(")p 2454 5402 95 4 v 2454 5485 a(\026)2509 5459 y Fs(2)2558 5422 y Fq(k)p Ft( )s Fq(k)2710 5385 y Fs(2)2751 5422 y Ft(;)1985 5706 y Fv(5)p eop %%Page: 6 6 6 5 bop 118 162 a Fv(whic)m(h)23 b(is)g(an)h(easy)h(consequence)f(of)h (the)f(second)g(resolv)m(en)m(t)h(iden)m(tit)m(y)e(and)h(of)g(\(3.7\).) 40 b(Hence)25 b(lim)3426 176 y Fn(")p FF(!)p Fs(0)3583 162 y Ft(F)3641 176 y Fn(")3704 162 y Fv(=)g Ft(F)3858 176 y Fs(0)118 275 y Fv(exists,)30 b(and)g(the)h(form)m(ula)e(\(3.1\))j (to)g(b)s(e)d(pro)m(v)m(ed)i(ma)m(y)g(b)s(e)f(rewritten)f(as)1676 479 y Fq(j)p Ft(F)1772 441 y FF(\006)1759 505 y Fs(0)1831 479 y Fq(j)d(\024)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)2202 442 y Fs(2)2202 503 y Ff(H)2275 512 y Fd(1)2315 479 y Ft(:)-2222 b Fv(\(3.8\))118 684 y(In)32 b(order)g(to)h(sho)m(w)f (\(3.8\))i(w)m(e)f(need)f(detailed)g(information)e(ab)s(out)i(the)h (family)e(of)h(resolv)m(en)m(ts)h Fq(f)p Ft(G)3566 651 y FF(\006)3566 706 y Fn(")3625 684 y Fq(g)3670 698 y Fn(")3707 684 y Fv(.)47 b(An)118 797 y(imp)s(ortan)m(t)30 b(feature)g(is)g(that)h(it)f(lea)m(v)m(es)h Fu(H)1582 811 y Fs(1)1652 797 y Fv(in)m(v)-5 b(arian)m(t:)1741 1001 y Ft(G)1813 963 y FF(\006)1813 1023 y Fn(")1872 1001 y Fu(H)1961 1015 y Fs(1)2025 1001 y Fq(\032)25 b Fu(H)2210 1015 y Fs(1)2249 1001 y Ft(:)-2156 b Fv(\(3.9\))118 1205 y(F)-8 b(or,)31 b(let)g(us)e(assume)i(that)g Ft( )d Fq(2)d Fu(H)1328 1219 y Fs(1)1397 1205 y Fv(i.e.)31 b(the)f (application)f Fr(R)2230 1172 y Fs(3)2301 1205 y Fq(3)c Fp(x)g Fq(7!)g Fv(e)2628 1172 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)2787 1205 y Ft( )j Fq(2)d Fu(H)57 b Fv(is)30 b Ft(C)3270 1172 y Fs(1)3309 1205 y Fv(.)40 b(Then)694 1409 y(e)735 1372 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)893 1409 y Ft(G)965 1372 y FF(\006)965 1432 y Fn(")1024 1409 y Ft( )f Fv(=)c(\(e)1304 1372 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)1462 1409 y Ft(H)1538 1423 y Fs(0)1577 1409 y Fv(e)1618 1372 y FF(\000)p Fn(i)p Fb(x)p Fa(\001)p Fb(Q)1851 1409 y Fq(\000)20 b Ft(\025)g Fq(\007)g Ft(i\026)h Fq(\007)f Ft(i")p Fv(e)2418 1372 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)2576 1409 y Ft(B)5 b Fv(e)2690 1372 y FF(\000)p Fn(i)p Fb(x)p Fa(\001)p Fb(Q)2903 1409 y Fv(\))2938 1372 y FF(\000)p Fs(1)3033 1409 y Fv(e)3073 1372 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)3232 1409 y Ft( )1122 1619 y Fv(=)1228 1491 y Fl(\022)1295 1619 y Ft(H)1371 1633 y Fs(0)1430 1619 y Fq(\000)20 b Fp(\013)k Fo(\001)f Fp(x)d Fq(\000)g Ft(\025)g Fq(\007)g Ft(i\026)g Fq(\007)g Ft(i")2397 1558 y Fv(\()p Fp(P)35 b Fq(\000)20 b Fp(x)p Fv(\))2719 1525 y Fs(2)p 2281 1598 592 4 v 2281 1681 a Fv(\()p Fp(P)36 b Fq(\000)20 b Fp(x)p Fv(\))2604 1655 y Fs(2)2664 1681 y Fv(+)g Ft(m)2835 1655 y Fs(2)2884 1491 y Fl(\023)2951 1508 y FF(\000)p Fs(1)3060 1619 y Fv(e)3101 1581 y Fn(i)p Fb(x)p Fa(\001)p Fb(Q)3259 1619 y Ft( )118 1881 y Fv(is)30 b(a)g Ft(C)357 1848 y Fs(1)426 1881 y Fv(function)g(of)g Fp(x)g Fv(in)f(the)i(top)s(ology)g(of)f Fu(H)d Fv(,)j(b)s(ecause)378 2119 y Fr(R)438 2082 y Fs(3)508 2119 y Fq(3)25 b Fp(x)g Fq(7!)796 2018 y Fl(\020)850 2119 y Ft(H)926 2133 y Fs(0)985 2119 y Fq(\000)20 b Fp(\013)j Fo(\001)h Fp(x)c Fq(\000)g Ft(\025)g Fq(\007)g Ft(i\026)g Fq(\007)g Ft(i")p Fv(\()p Fp(P)35 b Fq(\000)20 b Fp(x)p Fv(\))2148 2082 y Fs(2)2203 2046 y Fl(\002)2241 2119 y Fv(\()p Fp(P)35 b Fq(\000)20 b Fp(x)p Fv(\))2563 2082 y Fs(2)2623 2119 y Fv(+)g Ft(m)2794 2082 y Fs(2)2833 2046 y Fl(\003)2871 2065 y FF(\000)p Fs(1)2965 2018 y Fl(\021)3019 2041 y FF(\000)p Fs(1)3139 2119 y Fq(2)25 b Fr(B)14 b Fv(\()p Fu(H)33 b Fv(\))118 2354 y(is)d Ft(C)282 2321 y Fs(1)351 2354 y Fv(in)f(the)h(op)s(erator)h(norm.)259 2467 y(T)-8 b(ec)m(hnical)30 b(estimates)h(on)g Ft(G)1262 2434 y FF(\006)1262 2489 y Fn(")1350 2467 y Fv(are)g(gathered)g(b)s (elo)m(w.)118 2654 y Fc(Lemma)i(3.2)46 b Fv(\(i\))32 b Fq(k)p Ft(B)915 2621 y Fs(1)p Fn(=)p Fs(2)1025 2654 y Ft(G)1097 2621 y FF(\006)1097 2677 y Fn(")1156 2654 y Ft( )s Fq(k)1263 2621 y Fs(2)1328 2654 y Fq(\024)25 b Ft(")1466 2621 y FF(\000)p Fs(1)1561 2654 y Fq(j)p Ft(F)1657 2621 y FF(\006)1644 2677 y Fn(")1717 2654 y Fq(j)32 b Fe(for)h(e)-5 b(ach)34 b Ft( )29 b Fq(2)c Fu(H)h Fe(.)641 2767 y Fv(\(ii\))32 b Fq(k)p Ft(B)914 2734 y Fs(1)p Fn(=)p Fs(2)1024 2767 y Ft(G)1096 2734 y FF(\006)1096 2790 y Fn(")1154 2767 y Ft( )s Fq(k)26 b(\024)f Ft(C)7 b(")1497 2734 y FF(\000)p Fs(1)1591 2767 y Fq(k)p Ft( )s Fq(k)1743 2782 y Ff(H)1816 2791 y Fd(1)1889 2767 y Fe(for)33 b(e)-5 b(ach)34 b Ft( )29 b Fq(2)24 b Fu(H)2501 2781 y Fs(1)2573 2767 y Fe(\()p Ft(C)39 b Fe(is)33 b(some)g(p)-5 b(ositive)34 b(c)-5 b(onstant\).)118 2955 y Fc(Pro)s(of:)48 b Fv(Let)30 b Ft( )f Fq(2)c Fu(H)i Fv(.)40 b(Then)30 b(\(i\))g(follo)m(ws)f(from)172 3159 y Fq(j)p Ft(F)268 3121 y FF(\006)255 3181 y Fn(")327 3159 y Fq(j)d(\025)f(\007)p Fm(I)-5 b(m)p Fq(h)p Ft( )s(;)15 b(G)880 3121 y FF(\006)880 3181 y Fn(")940 3159 y Ft( )s Fq(i)26 b Fv(=)f Fq(\007)p Fm(I)-5 b(m)q Fq(h)p Fv(\()p Ft(H)1503 3173 y Fs(0)1562 3159 y Fq(\000)20 b Ft(\025)h Fq(\007)f Ft(i\026)g Fq(\007)g Ft(i"B)5 b Fv(\))p Ft(G)2269 3121 y FF(\006)2269 3181 y Fn(")2328 3159 y Ft( )s(;)15 b(G)2502 3121 y FF(\006)2502 3181 y Fn(")2561 3159 y Ft( )s Fq(i)26 b Fv(=)f Ft(\026)p Fq(k)p Ft(G)2952 3121 y FF(\006)2952 3181 y Fn(")3011 3159 y Ft( )s Fq(k)3118 3121 y Fs(2)3179 3159 y Fv(+)20 b Ft(")p Fq(k)p Ft(B)3431 3121 y Fs(1)p Fn(=)p Fs(2)3541 3159 y Ft(G)3613 3121 y FF(\006)3613 3181 y Fn(")3671 3159 y Ft( )s Fq(k)3778 3121 y Fs(2)3819 3159 y Ft(:)118 3363 y Fv(F)-8 b(or)31 b(\(ii\),)f(let)g Ft( )f Fq(2)c Fu(H)852 3377 y Fs(1)891 3363 y Fv(.)41 b(As)30 b(a)h(consequence)g(of) f(\(3.4\),)668 3567 y Fq(j)p Ft(F)764 3530 y FF(\006)751 3590 y Fn(")823 3567 y Fq(j)c Fv(=)f Fq(jh)p Ft( )s(;)15 b(G)1204 3530 y FF(\006)1204 3590 y Fn(")1264 3567 y Ft( )s Fq(ij)26 b(\024)f(k)p Ft(B)1627 3530 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1792 3567 y Ft( )s Fq(k)15 b(k)p Ft(B)2033 3530 y Fs(1)p Fn(=)p Fs(2)2144 3567 y Ft(G)2216 3530 y FF(\006)2216 3590 y Fn(")2274 3567 y Ft( )s Fq(k)26 b(\024)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)2727 3583 y Ff(H)2800 3592 y Fd(1)2855 3567 y Fq(k)p Ft(B)2974 3530 y Fs(1)p Fn(=)p Fs(2)3084 3567 y Ft(G)3156 3530 y FF(\006)3156 3590 y Fn(")3215 3567 y Ft( )s Fq(k)p Ft(:)118 3772 y Fv(Th)m(us)29 b(one)i(ma)m(y)g(use)f(\(i\))g(in)f(order)h(to)h(\014nd)e (the)i(estimate)g(of)f(\(ii\).)p 3831 3764 67 67 v 259 3934 a(W)-8 b(e)33 b(are)e(no)m(w)h(ready)f(for)g(the)g(last)g(step)h (of)f(the)g(pro)s(of.)43 b(It)31 b(relies)f(mainly)g(on)h(a)h (di\013eren)m(tial)d(inequalit)m(y)118 4047 y(in)40 b(terms)g(of)h Ft(F)670 4061 y Fn(")708 4047 y Fv(.)72 b(W)-8 b(e)42 b(\014x)e(a)h Ft( )46 b Fq(2)d Fu(H)1495 4061 y Fs(1)1574 4047 y Fv(and)e(denote)g(for)g(simplicit)m(y)d Ft(G)2712 4061 y Fn(")2791 4047 y Fq(\021)k Ft(G)2976 4014 y Fs(+)2976 4070 y Fn(")3075 4047 y Fv(and)f Ft(F)3321 4061 y Fn(")3400 4047 y Fq(\021)i Ft(F)3585 4014 y Fs(+)3572 4070 y Fn(")3644 4047 y Fv(.)72 b(The)118 4160 y(pro)s(of)36 b(for)g(the)g(case)h Ft(G)939 4174 y Fn(")1011 4160 y Fq(\021)d Ft(G)1188 4127 y FF(\000)1188 4183 y Fn(")1283 4160 y Fv(follo)m(ws)h (analogously)-8 b(.)59 b(Notice)37 b(\014rst)e(that)i(the)g(resolv)m (en)m(t)f(iden)m(tit)m(y)g(giv)m(es)118 4273 y Ft(F)189 4240 y FF(0)176 4296 y Fn(")239 4273 y Fq(\021)361 4237 y Fn(d)p 344 4252 69 4 v 344 4304 a(d")423 4273 y Ft(F)481 4287 y Fn(")544 4273 y Fv(=)25 b Ft(i)p Fq(h)p Ft(G)778 4240 y FF(\003)778 4296 y Fn(")817 4273 y Ft( )s(;)15 b(B)5 b(G)1065 4287 y Fn(")1102 4273 y Ft( )s Fq(i)p Fv(.)40 b(If)24 b(one)h(puts)f Ft(')1766 4287 y Fs(1)1831 4273 y Fv(=)h Ft(G)1999 4240 y FF(\003)1999 4296 y Fn(")2038 4273 y Ft( )j Fv(and)d Ft(')2356 4287 y Fs(2)2421 4273 y Fv(=)g Ft(G)2589 4287 y Fn(")2625 4273 y Ft( )s Fv(,)i(\(3.9\),)h (\(3.3\))e(and)e(\(ii\))g(of)h(Lemma)118 4386 y(3.1)32 b(yield)640 4590 y Ft(F)711 4553 y FF(0)698 4613 y Fn(")771 4590 y Fv(=)j Fq(h)p Ft(AG)1052 4553 y FF(\003)1052 4613 y Fn(")1091 4590 y Ft( )s(;)15 b(H)1269 4604 y Fs(0)1309 4590 y Ft(G)1381 4604 y Fn(")1418 4590 y Ft( )s Fq(i)21 b(\000)f(h)p Ft(H)1738 4604 y Fs(0)1777 4590 y Ft(G)1849 4553 y FF(\003)1849 4613 y Fn(")1888 4590 y Ft( )s(;)15 b(AG)2130 4604 y Fn(")2167 4590 y Ft( )s Fq(i)771 4728 y Fv(=)35 b Fq(h)p Ft(G)984 4691 y FF(\003)984 4751 y Fn(")1023 4728 y Ft( )s(;)15 b(A )s Fq(i)22 b(\000)e(h)p Ft(A )s(;)15 b(G)1680 4742 y Fn(")1718 4728 y Ft( )s Fq(i)21 b Fv(+)f Ft(i")p Fv(\()p Fq(h)p Ft(B)5 b(G)2216 4691 y FF(\003)2216 4751 y Fn(")2256 4728 y Ft( )s(;)15 b(AG)2498 4742 y Fn(")2535 4728 y Ft( )s Fq(i)21 b(\000)f(h)p Ft(AG)2919 4691 y FF(\003)2919 4751 y Fn(")2958 4728 y Ft( )s(;)15 b(B)5 b(G)3206 4742 y Fn(")3243 4728 y Ft( )s Fq(i)p Fv(\))771 4866 y(=)35 b Fq(h)p Ft(G)984 4828 y FF(\003)984 4888 y Fn(")1023 4866 y Ft( )s(;)15 b(A )s Fq(i)22 b(\000)e(h)p Ft(A )s(;)15 b(G)1680 4880 y Fn(")1718 4866 y Ft( )s Fq(i)21 b(\000)f Fv(2)p Ft(m)2052 4828 y Fs(2)2092 4866 y Ft(")p Fq(h)p Ft(G)2241 4828 y FF(\003)2241 4888 y Fn(")2280 4866 y Ft( )s(;)15 b Fv(\001)p Ft(H)2541 4828 y FF(\000)p Fs(5)2534 4892 y(0)2636 4866 y Ft(G)2708 4880 y Fn(")2744 4866 y Ft( )s Fq(i)p Ft(:)118 5070 y Fv(F)-8 b(or)31 b(the)g(last)f(equalit)m(y)g(w)m(e)h (used)e(\(3.6\).)43 b(F)-8 b(urther,)30 b(\(3.5\))i(and)e(Lemma)h(3.2)g (furnish)c(the)k(estimate)522 5274 y Fq(j)p Ft(F)618 5237 y FF(0)605 5297 y Fn(")643 5274 y Fq(j)36 b(\024)f(k)p Ft(B)929 5237 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1094 5274 y Ft(A )s Fq(k)15 b Fv(\()p Fq(k)p Ft(B)1438 5237 y Fs(1)p Fn(=)p Fs(2)1550 5274 y Ft(G)1622 5237 y FF(\003)1622 5297 y Fn(")1661 5274 y Ft( )s Fq(k)21 b Fv(+)f Fq(k)p Ft(B)1999 5237 y Fs(1)p Fn(=)p Fs(2)2109 5274 y Ft(G)2181 5288 y Fn(")2217 5274 y Ft( )s Fq(k)p Fv(\))h(+)f Ft(C)7 b(")p Fq(k)p Ft(B)2704 5237 y Fs(1)p Fn(=)p Fs(2)2814 5274 y Ft(G)2886 5237 y FF(\003)2886 5297 y Fn(")2925 5274 y Ft( )s Fq(k)15 b(k)p Ft(B)3166 5237 y Fs(1)p Fn(=)p Fs(2)3277 5274 y Ft(G)3349 5288 y Fn(")3385 5274 y Ft( )s Fq(k)704 5412 y(\024)831 5389 y Fv(~)810 5412 y Ft(C)7 b(")924 5375 y FF(\000)p Fs(1)p Fn(=)p Fs(2)1089 5412 y Fq(j)p Ft(F)1172 5426 y Fn(")1210 5412 y Fq(j)1235 5375 y Fs(1)p Fn(=)p Fs(2)1345 5412 y Fq(k)p Ft( )s Fq(k)1497 5427 y Ff(H)1570 5436 y Fd(1)1610 5412 y Ft(:)1985 5706 y Fv(6)p eop %%Page: 7 7 7 6 bop 118 162 a Fv(After)31 b(in)m(tegration)f(on)g([)p Ft(\032;)15 b(\032)1103 176 y Fs(0)1143 162 y Fv(])26 b Fq(\032)f Fv(\(0)p Ft(;)15 b Fv(1\))32 b(one)f(gets)1173 396 y Fq(j)p Ft(F)1256 410 y Fn(\032)1297 396 y Fq(j)26 b(\024)f(j)p Ft(F)1527 410 y Fn(\032)1563 419 y Fd(0)1602 396 y Fq(j)c Fv(+)f Ft(C)7 b Fq(k)p Ft( )s Fq(k)1963 412 y Ff(H)2036 421 y Fd(1)2091 273 y Fl(Z)2181 299 y Fn(\032)2217 308 y Fd(0)2141 479 y Fn(\032)2272 396 y Ft(")2314 359 y FF(\000)p Fs(1)p Fn(=)p Fs(2)2479 396 y Fq(j)p Ft(F)2562 410 y Fn(")2599 396 y Fq(j)2624 359 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 ---------------9908130711766--