%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% LaTeX2e %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% Preamble %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% Style section %%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt,a4paper,leqno]{amsart} \usepackage{amssymb} \usepackage{calrsfs} \usepackage[mathscr]{eucal} %%%%%%%%%%%%%%%%%%%%%% Declaration section %%%%%%%%%%%%%%%%%%% \renewcommand{\theequation}{\thesection.\arabic{equation}} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem*{thm*}{Theorem} \newtheorem*{lem*}{Lemma} \newtheorem*{prop*}{Proposition} \newtheorem*{thmA}{Theorem A} \newtheorem*{thmB}{Theorem B} \newtheorem*{thmC}{Theorem C} \newtheorem*{thmD}{Theorem D} \newtheorem*{thmE}{Theorem E} \newtheorem*{thmF}{Theorem F} \newtheorem*{thmG}{Theorem G} %%%%%%%%%%%%%%%%%%%%%%% Command section %%%%%%%%%%%%%%%%%%%%%% \newcommand{\C}{\mathcal} \newcommand{\D}{\mathbb} \newcommand{\E}{\mathscr} \newcommand{\F}{\mathfrak} \newcommand{\ad}{\operatorname{ad}} \newcommand{\dist}{\operatorname{dist}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\Lip}{\operatorname{Lip}} \renewcommand{\Re}{\operatorname{Re}} \newcommand{\slim}{\operatornamewithlimits{s-lim}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\wlim}{\operatornamewithlimits{w-lim}} \newcommand{\wslim}{\operatornamewithlimits{w*-lim}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% Document %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% Topmatter %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[]{Higher Order Estimates \\ in the Conjugate Operator Theory} \author[]{A\lowercase{nne} Boutet \lowercase{de} Monvel, V\lowercase{ladimir} Georgescu \lowercase{and} J\lowercase{aouad} Sahbani\footnotemark{$^1$}} \footnotetext[1] {Institut de Math\'ematiques de Jussieu, CNRS UMR 9994, Laboratoire de Physique math\'ematique et G\'eom\'etrie, case 7012, Universit\'e Paris 7 Denis Diderot, 2 place Jussieu, F-75251 Paris Cedex 05} %%%%%%%%%%%%%%%%%%%%%%% End Topmatter %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% Abstract %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} Let $H$ be a self-adjoint operator which admits a locally conjugate operator $A$ (i.e.\ the set $\mu^A(H)$ defined at the beginning of the introduction is not empty). Set $R(z)=(H-z)^{-1}$, let $\Pi_{\pm}$ be the spectral projection of $A$ associated to the interval $\pm \lbrack 0,\infty )$ and let $\C{H}_{s,p}$ ($s\in \D{R}, 1\leq p\leq \infty $) be the Besov scale associated to the operator $A$. We study the regularity properties of the maps $\lambda \mapsto R(\lambda \pm i0)$, $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)$ and $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)\Pi_{\pm}$ when considered with values in a space of the form $B(\C{H}_{s,p};\C{H}_{t,q})$. Our results imply optimal local decay and propagation properties of $H$ with respect to $A$, in particular estimates of the form $\Vert \langle A\rangle^t\Pi_{\mp}\exp (\mp i\tau H)\langle A\rangle^{-s}\Vert \leq c\tau^{-\alpha }$ for $\tau \geq 1$. \end{abstract} \maketitle \setcounter{section}{-1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% 0. Introduction %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{s:0} Let $H$, $A$ be two densely defined self-adjoint operators in a Hilbert space $\C{H}$ such that: (i) the intersection of their domains $D(A)\cap D(H)$ is dense in $D(H)$ (which is equipped with the graph topology) and (ii) the sesquilinear form defined on $D(A)\cap D(H)$ by $\langle Hg,Af\rangle-\langle Ag,Hf\rangle$ extends to a continuous sesquilinear form, denoted $\lbrack H,A\rbrack$ on $D(H)$. Let $E$ be the spectral measure of $H$ and let us set $E(\lambda;\varepsilon) =E((\lambda -\varepsilon ,\lambda +\varepsilon ))$ for $\lambda \in \D{R}$ and $\varepsilon >0$. Then $E(\lambda ;\varepsilon )\lbrack H,iA\rbrack E(\lambda ;\varepsilon )$ is a bounded self-adjoint operator and one may consider the set $\mu^A(H)$ of real numbers $\lambda $ such that $E(\lambda ;\varepsilon )\lbrack H,iA\rbrack E(\lambda ;\varepsilon )\geq aE(\lambda ;\varepsilon )$ for some real $a>0$, $\varepsilon >0$. The relevance of this type of inequality in the spectral theory of the operator $H$ has been understood by E.~Mourre \cite{M1,M2} and for this reason one calls it a \emph{strict Mourre estimate}. So $\mu^A(H)$ is the (open) set of real points at which a strict Mourre estimate (for $H$ relatively to $A$) holds. Mourre has shown that if $H$ is sufficiently regular (cf.\ (a),(b),(c) below) with respect to $A$ then $H$ has no singularly continuous spectrum in $\mu^A(H)$ and, moreover, for each $s>1/2$ and each $f\in \C{H}_s:=D(|A|^s)$ the limits $\lim_{\mu \rightarrow \pm 0}\langle f,R(\lambda +i\mu )f\rangle\equiv \langle f,R(\lambda \pm i0)f\rangle$ exist locally uniformly in $\lambda \in \mu^A(H)$. We have set $R(z)=(H-z)^{-1}$. The regularity conditions required by Mourre were rather restrictive: (a) the form $\lbrack H,A\rbrack$ had to be associated to a bounded operator $D(H)\rightarrow\C{H}$; (b) $D(H)$ had to be invariant under the unitary group $W_{\tau }=\exp (i\tau A)$ generated by $A$; (c) the map $\tau \rightarrow \langle W_{\tau }f,HW_{\tau }f\rangle$ had to be of class $C^2$, for each $f\in D(H)$. The condition (a) has been slightly relaxed in \cite{PSS} (with, however significant consequences in the case of $N$-body hamiltonians). In \cite{ABG1} and \cite{BGM} it has been shown that the class $C^2$ in (c) can be replaced by $C^{1+\varepsilon }$ for some $\varepsilon >0$ (and even $\varepsilon =+0$ in a certain sense); here $C^{1+\varepsilon}$ is the Besov class $B_{\infty }^{1+\varepsilon ,\infty }(\D{R})$. In \cite{BG5} the conditions (a) and (b) were completely eliminated and $C^2$ in (c) has been replaced by the Besov class $B^{1,1}_{\infty }(\D{R})$; moreover, it was shown that the regularity class $B^{1,1}_{\infty }$ is unimprovable on the Besov scale $B^{s,p}_{\infty }$ (if one wants to get the limiting absorption principle in the form $|\langle f,R(\lambda +i\mu )f\rangle|\leq c(\lambda )<\infty $ for $f\in \cap_{k\in \D{N}} D(A^k)$ and $\lambda \in \mu^A(H)$). In \cite{M1,M2} and \cite{PSS} the operator $H$ was assumed bounded from below. In \cite{CFKS} and \cite{ABG1} it has been remarked that such a condition is not needed under the hypotheses (b) and the version of (a) required in \cite{PSS}. However, the elimination of (a) and (b) in \cite{BG5} was possible only under the assumption that $H$ has a spectral gap. In \cite{JP} it has been observed that an estimate proved by Mourre in \cite{M2} allows one to replace the assumption $f\in \C{H}_s$ with $s>1/2$ by $f\in \C{H}_{1/2,1}$ (which is a certain Besov type space associated to $A$). A complete presentation of the theory with several improvements can be found in Chapter 7 of \cite{ABG2}. A natural question is the degree of regularity of the map $\lambda \mapsto \langle f,R(\lambda \pm i0)f\rangle$ assuming that $f$ and $H$ have certain regularity properties with respect to $A$. Results in this direction have been obtained in \cite{PSS}, \cite{W}, \cite{ABG1}, \cite{BGM}, \cite{JMP}. For example in \cite{PSS}, under the hypotheses (a),(b),(c), it has been shown that $\lambda \mapsto \langle f,R(\lambda +i0)f\rangle$ is locally H\H{o}lder continuous of order $\theta =2/3(s-1/2)$ if $f\in \C{H}_s$ for some $s\in (1/2,1\rbrack $, and in \cite{W} it has been proved that one may take $\theta =(s+1/2)^{-1}(s-1/2)$. The case of less regular hamiltonians $H$ is treated in \cite{ABG1} and \cite{BGM} with similar results. However, the optimal answer is $\theta =s-1/2$ and was obtained in \cite{BG2,BG3} under optimal regularity conditions on $H$. The regularity of $\lambda \mapsto \langle f,R(\lambda +i0)f\rangle$ improves if $f\in \C{H}_s$ with $s>1$ and if $C^2$ in condition (c) is replaced by $C^k$ for some integer $k\geq 3$, as it has been shown in \cite{JMP}, where results of the same nature (so not optimal) as those in \cite{PSS} and \cite{W} are obtained. Again, the optimal result is obtained in \cite{BG2,BG3}. Let us identify $\C{H}^*=\C{H}$ with the help of the Riesz isomorphism ($\C{H}^*$ is the space of antilinear continuous functionals). Then we shall have $\C{H}_s\subset \C{H}\subset\C{H}^*_s=\C{H}_{-s}$ and the results described above concern the operators $R(\lambda \pm i0):\C{H}_s\rightarrow \C{H}_{-s}$ for $s>1/2$. Let $\Pi_+$, $\Pi_-$ be the spectral projections of $A$ associated to the intervals $\lbrack 0,\infty )$ and $(-\infty ,0\rbrack $. It was observed in \cite{M2} that the operators $\Pi_{\mp}R(\lambda \pm i0)$ have much better continuity properties than $R(\lambda \pm i0)$ (and there are further improvements for $\Pi_{\mp}R(\lambda \pm i0)\Pi_{\pm}$). These operators were studied and their importance in scattering theory was pointed out in \cite{M2}, \cite{JMP}, \cite{J}. In \cite{J} it was shown, for example, that $\Pi_{\mp}R(\lambda \pm i0)$ send $\C{H}_s$ into $\C{H}_{s-1}$ if $s>1/2$ and if $H$ is sufficiently regular. Some regularity properties of the maps $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)\in B(\C{H}_s;\C{H}_t)$ are described in \cite{JMP}. However, their results are far from optimal, as it will be clear from a comparison with our results. Similar problems concerning the maps $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)\Pi_{\pm}$ are treated in \cite{JMP} and \cite{J}. We follow Mourre and we say that $A$ is {\emph{locally (strictly) conjugated to $H$ on an interval} $J$ if $J\subset \mu^A(H)$. Loosely speaking, we say that $A$ is locally conjugated to $H$ if $\mu^A(H)\not =\varnothing$. The simplest example of a pair $(H,A)$ such that $A$ is locally conjugated to $H$ is obtained as follows: $\C{H}=L^2(\D{R})$, $H$ is the operator of multiplication by a function $h:\D{R}\rightarrow \D{R}$ of class $C^1$ with $h'(x)>0$ ($\forall x$), and $A=i(d/dx)$; later on we refer to this situation as the \emph{classical case}. The developments of this paper suggest that the following proposition is, in some sense, true: when properly formulated, each assertion which holds in the classical case should hold in general. Of course, the ``proper formulation'' should involve only terms which make sense in an abstract Hilbert space setting. We hope that Section 1, which contains a quite detailed description of our results, will clarify this point of view. Our purpose in this paper is to make a systematic study of the behaviour as $\mu \rightarrow \pm 0$ of the resolvent $R(\lambda +i\mu )$ in the framework of the conjugate operator theory ($\lambda \in \mu^A(H)$). We shall obtain the best possible (on a scale that will be defined in Section 1) regularity properties of the maps $\lambda \mapsto R(\lambda \pm i0)$, $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)$ and $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)\Pi_{\pm}$ when considered with values in $B(\C{H}_{s,p};\C{H}_{t,q})$, where $\lbrace \C{H}_{s,p}\rbrace $ is the Besov scale associated to $A$ in $\C{H}$. This will be done under optimal $A$-regularity hypotheses on $H$, in a sense that will be made clear in Section 1. In our main theorems we require $H$ to have a spectral gap. The advantage is that this allows us to avoid any condition concerning the domain of $H$. Moreover, we may then treat as easily the case when $H$ ``lives'' in a Hilbert space which is smaller than that in which ``lives'' $A$. In other terms, $H$ could be a non-densely defined operator in $\C{H}$, and this is useful in several applications (see \cite{BGS} for example). Of course, the spectral gap hypothesis is annoying in some applications, e.g.\ to Stark effect hamiltonians or simply characteristic operators. For operators $H$ of $A$-regularity class $\C{C}^{\alpha }(A)$ with $1<\alpha <3/2$ this condition has been eliminated in \cite{S1,S2}. The paper is organized as follows. In Section 1 the main definitions and results are stated and discussed in detail. Moreover, we try to motivate the statements and we prove their optimality by studying the ``classical case''. In Section 2 we describe the Besov scale $\lbrace \C{H}_{s,p} \mid s\in \D{R}, 1\leq p\leq \infty \rbrace $ associated to $A$ in $\C{H}$. The main result of the section is the theorem proved in \S2.3, which will play an important role in Sections 5 and 6. We are also going to take advantage of this theorem in \S2.5 and we shall prove several interesting facts concerning the spaces $\C{H}_{s,p}$ (we could have deduced these results from more general ones obtained in Chapter 3 of \cite{ABG2}). In \S2.6 we show that the Banach space $\C{H}_{s,1}$ is weakly sequentially complete (we need this in the statement of Theorems D and E). In Section 3 we make a similar study of a Besov scale $\lbrace \C{C}^{s,p}(A) \mid s\in (0,\infty ), p\in \lbrack 1,\infty \rbrack \rbrace $ associated to the operator $\ad_A:S\mapsto \lbrack A,S\rbrack $ acting in the Banach space $B(\C{H})$. The regularity with respect to $A$ of the hamiltonian $H$ will be expressed by conditions of the form $(H-z)^{-1}\in \C{C}^{s,p}(A)$. The most important statement of the section is the theorem from \S3.7 (the proof is given in \cite{BG3}) which gives a Littlewood-Paley type description of the space $\C{C}^{s,p}(A)$: this is the main ingredient of the proof of our main results from Sections 5 and 6. Several other propositions from Section 3 could have been obtained as corollaries of the results from Ch. 5 in \cite{ABG2}, but we have made an effort in order to make the section as self-contained as possible (all the unproven facts are straightforward consequences of the theorem from \S3.7). Section 4 is devoted to the study of the resolvent (or pseudo-resolvent) families, i.e.\ resolvents of non-densely defined self-adjoint operators, in the framework of the conjugate operator theory. A similar, but less systematic, study may be found in \cite{BG5} and \cite{BGS}. In \S4.2 we give a new and very simple proof of a version of the Helffer-Sj\"ostrand formula (although we shall not use it in the rest of the paper; cf.\ \cite{D1,D2}). In \S4.5 we define the Mourre sets $\mu^A(H)$ and $\tilde{\mu }^A(H)$ and we prove our first main theorem (Theorem A from Section 1). In \S4.6 we explain how one may reduce the proof of the other theorems from the Section 1 to the case when $H$ is a bounded everywhere defined operator. It is at this point that we require the hamiltonian to have a spectral gap. Sections 5 and 6 are the technical core of the paper. Section 5 contains estimates on the ``distorted'' hamiltonian and in Section 6 are proven all the theorems stated in Section 1 (for everywhere defined $H$). Note that we do not use the method of differential ineqialities in the study of the regularity properties of the map $\lambda \mapsto R(\lambda +i0)$, neither the method of \cite{J} in order to prove (for example) that $\Pi_{-}R(\lambda +i0)\C{H}_{s,p}\subset \C{H}_{s-1,p}$. It seems to us that these methods would require significantly stronger $A$-regularity properties for $H$. Our techniques (based on Lemmas 6.1, 6.2) are natural extensions of those from \cite{BG2,BG3}. The idea of using a ``twisted'' (or ``distorted'') hamiltonian $H_{\varepsilon}$ in the study of the behaviour of $(H-\lambda -i\mu )^{-1}$ as $\mu \rightarrow +0$ goes back to \cite{AC}, \cite{BC} (the case of $A$-analytic $H$) and to \cite{M1,M2}, \cite{JMP}. The connection with the dilation analyticity (or complex scaling) techniques is particularly striking in our approach. Indeed, the twisted hamiltonian $H_{\varepsilon}$ is constructed as follows. First, in order to compensate the lack of $A$-regularity of $H$, we ``regularize'' $H$ with respect to $A$ (this is the analogue of regularizing a function $h$ by using a convolution). This gives a self-adjoint operator $H(\varepsilon )$ which is $A$-analytic and such that $\Vert H(\varepsilon )-H\Vert \leq C\varepsilon^s$ if (and only if) $H\in \C{C}^{s,\infty }(A)$. Then we take $H_{\varepsilon}=e^{-\varepsilon A}H(\varepsilon )e^{\varepsilon A}$ for $\varepsilon >0$. If $H$ itself is $A$-analytic, the first step is not necessary and the proofs become extremely natural and transparent. In fact we strongly recommend to anyone who really wants to understand the conjugate operator method to go through the proofs in the particular case of $A$-analytic hamiltonians with the distorsion $H_{\varepsilon}=e^{-\varepsilon A}H(\varepsilon )e^{\varepsilon A}$ (see \S4.12 in \cite{BG3} and also \cite{S1}). \smallskip N.B. This text first appeared as a preprint of the Institut de Math\'ematiques de Jussieu in January 1996 (some minor mistakes of the preprint are corrected here). The paper "Boundary Values of Regular Resolvent Families", submitted for publication to Journal of Functional Analysis in May 1996, contains in a rather condensed form (the next 62 pages are reduced to 33) the main results of the preprint, with one significant improvement ( Proposition 3.1, which clarifies the considerations of \S 4.5 below). This paper is available as preprint number 96-506 from the Mathematical Physics Preprint Archive of the University of Texas (http://www.ma.utexas.edu/mp\_arc/). We decided to make available to a larger audience the present detailed version because we think that the developments of the following four sections are of some independent interest. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% 1. Main results %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %------------------------------% \protect\setcounter{equation}{0} %------------------------------% \section{Description of the Main Results} \label{s:1} %------------% \subsection{} \label{ss:1.1} Generally, a family $\lbrace R(z)\mid z\in \D{C}\setminus \D{R}\rbrace $ of bounded operators in the Hilbert space $\C{H}$ is called a \emph { self-adjoint pseudo-resolvent family} if the following relations are satisfied: $R(z)^*=R(\bar{z})$ and $R(z_1)-R(z_2)=(z_1-z_2)R(z_1)R(z_2)$; in this paper we are going to abbreviate the name and just say {\em resolvent family}. We say that such a family is {\em regular with respect to a densely defined self-adjoint operator $A$ in $\C{H}$}, or simply {\em $A$-regular}, if for some $z$ the following condition is fulfilled ($W_{\tau }=\exp (i\tau A)$): \begin{equation} \label{eq:1.1} \int_0^1 \Vert W^*_{2\tau }R(z)W_{2\tau }-2W^*_{\tau }R(z)W_{\tau }+R(z)\Vert \tau^{-2}d\tau <\infty . \end{equation} Intuitively this means that the map $\tau \mapsto W^*_{\tau }R(z)W_{\tau }\in B(\C{H})$ is slightly more than of class $C^1$ (in norm). Conditions of this type will be discussed in some detail later on. The most important class of resolvent families is that associated to everywhere defined (hence bounded) self-adjoint operators $H$ through the formula $R(z)=(H-z)^{-1}$. In this case (1.1) is equivalent to \begin{equation} \label{eq:1.2} \int_0^1 \Vert W^*_{2\tau } HW_{2\tau }-2W^{\star }_{\tau }HW_{\tau }+H\Vert \tau^{-2}d\tau <\infty . \end{equation} A more general situation is that when $R(z)=(H-z)^{-1}$ with $H$ a densely defined self-adjoint operator in $\C{H}$. If the domain of $H$ is invariant under all the operators $W_{\tau}$ ($\tau \in\D{R}$) then the condition (1.1) is equivalent to $$ \int_0^1 \Vert W^*_{2\tau }HW_{2\tau }-2W^{\star }_{\tau }HW_{\tau }+H\Vert_{D(H)\rightarrow D(H)^*} \tau^{-2}d\tau <\infty.%\tag{1.1} $$ Here $D(H)$ is the domain of $H$ equipped with the graph topology, $D(H)^*$ is the adjoint space, and we identify $D(H)\subset \C{H}\equiv \C{H}^*\subset D(\C{H})^*$ with the help of the Riesz's lemma. In particular $B(\C{H})\subset B(D(H);\C{H})\subset B(D(H);D(H)^*)$. If $\lbrace R(z)\rbrace $ is an arbitrary resolvent family, then the map $z\mapsto R(z)\in B(\C{H})$ is holomorphic on $\D{C}\setminus \D{R}$ and one has $\Vert R(z)\Vert \leq |\Im z|^{-1}$. Moreover, if we fix a real number $\lambda $, then there are two possibilities: either there is a constant $C=C(\lambda )<\infty $ such that $\Vert R(\lambda +i\mu )\Vert \leq C$ for all $\mu \not =0$, and this happens if and only if the map $z\mapsto R(z)\in B(\C{H})$ has a holomorphic extension to a neighbourhood of $\lambda $ in $\D{C}$, or $\Vert R(\lambda +i\mu )\Vert =|\mu |^{-1}$ for all real $\mu \not =0$. The points $\lambda $ of the second type form the {\em spectrum of the resolvent family} and it is clear that $\lim_{n\rightarrow \infty } R(\lambda +i\mu_n )$ cannot exist weakly in $B(\C{H})$ if $\lambda $ belongs to the spectrum and $\lbrace \mu_n\rbrace $ is a sequence of real numbers which tends to zero. However, the two limits $\lim_{\mu \rightarrow \pm 0} R(\lambda +i\mu )\equiv R(\lambda \pm i0)$ could exist in a space larger than $B(\C{H})$ and this fact has important consequenes in spectral and scattering theory. The purpose of our work is to study the behaviour of $R(\lambda +i\mu )$ as $\mu \rightarrow \pm 0$ and the properties of the limit operators $R(\lambda \pm i0)$ in the framework of the so-called {\em conjugate operator method}. The class of ``spaces larger than $B(\C{H})$'' usually considered in applications consists of spaces of the form $B(\C{E};\C{F})$, where $\C{E}$, $\C{F}$ are Banach spaces such that $\C{E}$ is continuously and densely embedded in $\C{H}$ and $\C{H}$ is continuously embedded in $\C{F}$; then one clearly has a canonical continuous embedding $B(\C{H})\subset B(\C{E};\C{F})$. {\it The main point of the conjugate operator theory is that it provides natural candidates for the spaces $\C{E}$ and $\C{F}$}: they can be constructed with the help of the ``conjugate operator'' $A$. But the $A$-regularity condition on $\lbrace R(z)\rbrace $, although important, is not sufficient in order to obtain interesting results. Indeed, the set of real $\lambda $ for which $R(\lambda \pm i0)$ can be given a sense in this framework depends on $A$ and could be empty. So, for the theory to be non-trivial, the resolvent family has to satisfy a certain non-degeneracy condition with respect to $A$. Such a condition has been isolated by Mourre in \cite{M1} and for this reason it is called the {\em (strict) Mourre estimate}. %------------% \subsection{} \label{ss:1.2} Before going to the general theory we think it worthwile to discuss in some detail a particular case which explains and motivates many of the later developments. More precisely we consider here the case where $H$ is the operator of multiplication by the bounded Borel function $h:\D{R}\rightarrow \D{R}$ in the Hilbert space $L^2(\D{R})$ and $A$ is the usual self-adjoint realization of $i(d/dx)$ in $L^2(\D{R})$. Later on we shall refer to this situation as the classical case. Then the $A$-regularity condition (1.2) is equivalent to the fact that $h$ belongs to the Besov space $B^{1,1}_{\infty }(\D{R})$. Let us remark that in this paper we follow the conventions of Peetre \cite{P} and denote by $B^{s,p}_q$ the Besov spaces associated to the translation group in $L^q$, but that in the cases $q=2$ and $q=\infty $ we use the following special notations: $$ B^{s,p}_2=\C{H}^{s,p} ,\ \C{H}^{s,2}=\C{H}^s;\ \ \ B^{s,p}_{\infty }=\Lambda^{s,p} ,\ \Lambda^{s,\infty }=\Lambda^s . $$ So (1.2) means that $h\in \Lambda^{1,1}$. We recall that $\Lambda^{1,1}(\D{R})\subset BC^1(\D{R})$ (space of bounded functions with bounded and continuous derivative) and that this embedding is optimal on the scale $\lbrace \Lambda^{s,p}\rbrace $. More precisely there are functions in $\cap_{p>1} \Lambda^{1,p}$ which are not Lipschitz on any interval; this explains the degree of precision of the condition (1.2). The behaviour of $\langle g,(H-\lambda -i\mu )^{-1}f\rangle$ as $\mu \rightarrow \pm 0$ is intimately related to the spectral properties of the operator $H$. The spectrum of $H$ is equal to the closure $\overline{h(\D{R})}$ of the image of $h$, so it is a compact real interval ($h$ being continuous). It is easily shown that the singular spectrum of $H$ is included in the closure of the set of critical values of $h$ (real numbers $\lambda $ such that $\lambda =h(x)$ for some $x$ with $h'(x)=0$). Now, even if $h$ is of class $C^{\infty }$, this set could be a rather arbitrary real set of Lebesgue measure zero. In order to be able to say something interesting we have to restrict ourselves to the set of non-critical values of $h$, i.e.\ the set of numbers $\lambda $ such that $h'(x)\not =0$ if $h(x)=\lambda $ (this is the non-degeneracy condition alluded to at the end of \S1.1). For simplicity we shall assume that $h'(x)>0$ for all $x\in \D{R}$. Then, for $f$, $g\in L^2(\D{R})$ and $z=\lambda +i\mu $ with $\lambda \in \D{R}$ and $\mu >0$ one has \begin{equation} \label{eq:1.3} \langle g,(H-z)^{-1}f\rangle=\int_I \frac{\overline{g(h^{-1}(x))}f(h^{-1}(x))}{x-\lambda -i\mu } \frac{dx}{h'(h^{-1}(x))} \equiv \int_I \frac{u(x)dx}{x-\lambda -i\mu } . \end{equation} Here $I=h(\D{R})$ is an open real interval whose closure is the spectrum $\sigma (H)$ of the operator $H$. We are interested in the behaviour of the resolvent $(H-z)^{-1}$ as $\mu \rightarrow +0$. Since outside $\sigma (H)$ the resolvent is a holomorphic function, we consider only values $\lambda \in I$ (the ends of the interval $I$ should be considered as critical values of $h$ and are not treated here). If we set $u(x)=0$ for $x\in \D{R}\setminus I$, so that $u\in L^1(\D{R})$, and if we denote by $\tilde{u}$ the Hilbert transform of $u$, then the limit of (1.3) as $\mu \rightarrow +0$ exists for (Lebesgue) almost every real $\lambda $ and we have \begin{equation} \label{eq:1.4} \lim_{\mu \rightarrow +0} \langle g,(H-\lambda -i\mu )^{-1}f\rangle=\pi(-\widetilde{u}(\lambda )+iu(\lambda ))=2i\int^{\infty }_{0} e^{i\lambda x}\hat{u}(x)dx . \end{equation} Here $\hat{u}(x)=(2\pi )^{-1/2}\int_{\D{R}} e^{-ixy}u(y)dy$ is the Fourier transform of $u$. By using the first equality in (1.4) and well-known results concerning the Hilbert transformation one can now easily deduce various properties of the holomorphic function $z\mapsto \langle g,(H-z)^{-1}f\rangle$ and of its boundary value function $I\ni \lambda \mapsto \langle g,(H-\lambda -i0)^{-1}f\rangle$ (which is the first member of (1.4)). For example, if $u$ is locally of class $\Lambda^{\alpha }$ on $I$ for some real $\alpha>0$ then $\tilde{u}$ has the same property, and so the limit in (1.4) exists locally uniformly in $\lambda $ on $I$ and the function $\lambda \mapsto \langle g,(H-\lambda -i0)^{-1}f\rangle$ is (locally on $I$) of the same class $\Lambda^{\alpha }$. Moreover, the holomorphic function $z\mapsto \langle g,(H-z)^{-1}f\rangle$ extends to a function of class $\Lambda^{\alpha }$ in each rectangle $a\leq \lambda \leq b$, $0\leq \mu \leq c$ with $\lbrack a,b\rbrack \subset I$ and $c>0$. From the second inequality in (1.4) one can also see that only the behaviour of the Fourier transform $\hat{u}(x)$ as $x\rightarrow +\infty $ really matters. When so formulated this type of results cannot be extended to a general and abstract framework, like that of \S1.1. Indeed, $u$ has been obtained from $f$, $g$ by a procedure which has no meaning in a purely Hilbert space setting. However, it is not difficult to find simple and natural conditions on $f$, $g$ and $h$ which make sense in the context of \S1.1 and which in the present context imply that $u$ is locally of class $\Lambda^{\alpha }$. Let us set $s=\alpha +1/2$, so that $s$ is a real number strictly larger than $1/2$. Then we can apply the Sobolev embedding theorem and get: $$ \C{H}^s\equiv B^{s,2}_2(\D{R})\subset B^{\alpha ,2}_{\infty }(\D{R})\equiv \Lambda^{\alpha ,2}\subset \Lambda^{\alpha } . $$ So if $f$, $g$ belong to $\C{H}^s$ then they belong to $\Lambda^{\alpha }$, which is an algebra, hence $\bar{f}g\in \Lambda^{\alpha }$. Moreover, $h'$ is of class $\Lambda^{\alpha }$ if and only if $h$ is of class $\Lambda^{\alpha +1}$, and then the local class $\Lambda^{\alpha }$ is stable under composition with $h^{-1}$. In conclusion, if $f$, $g\in \C{H}^s$ with $s=\alpha +1/2>1/2$ and if $h\in \Lambda^{\alpha +1}$, then $u$ is locally of class $\Lambda^{\alpha }$ on $I$, hence the assertions made above in connection with the function $z\mapsto \langle g,(H-z)^{-1}f\rangle$ remain true. The condition on $f$, $g$ now makes sense in the context of \S1.1: for example $f\in \C{H}^s$ means that $f$ belongs to the domain of the operator $|A|^s$ or, equivalently, that the vector valued function $\tau \mapsto W_{\tau }f\in \C{H}$ is of class $\Lambda^{s,2 }$. The condition $h\in\Lambda^{\alpha +1}$ can be expressed in abstract terms too: this means precisely that the operator valued function $\tau \mapsto W^*_{\tau }HW_{\tau }\in B(\C{H})$ is of class $\Lambda^{\alpha +1}$. It turns out that this point of view is convenient not only because it can be immediately considered at an abstract level but also because it allows one to study some limit cases by using the general classes $\C{H}^{s,p}$ and $\Lambda^{\alpha ,q}$. For example, if $f$, $g$ belong to $\C{H}^{1/2,1}$ and $h\in \Lambda^{1,1}$ (which is, formally, the limit case $\alpha =0$), then the limit (1.4) still exists, locally uniformly in $\lambda \in \Lambda $. Now let us consider the last member of (1.4): we see that the local regularity properties of the boundary value function $\lambda \mapsto \langle g,(H-\lambda -i0)^{-1}f\rangle$ depend only on the behaviour of $\hat{u}$ at $+\infty $. If we set $f_0(x)=f(h^{-1}(x))$ and $g_0(x)=g(h^{-1}(x))(h'(h^{-1}(x)))^{-1}$ and if we extend these functions by zero outside $I$, we get $\hat{u}(x)=(2\pi )^{1/2}\int^{\infty }_{-\infty } \hat{f_0}(x+y)\hat{g_0}^*dy$. So if $\hat{f_0}$ decays rapidly enough at $+\infty $ {\it and} $-\infty $, then $\hat{u}$ will decay as rapidly as we wish at $+\infty $ provided that $\hat{g_0}$ decays suficiently rapidly {\it at $-\infty $ only}. It is not so easy to control the decay at $-\infty $ of $\hat{g_0}$ in terms of that of $\hat{g}$ (in particular this shows that the methods that we shall use in the proof of the general theorems give better results even in the classical case). But if we take $h(x)\equiv x$ (we have assumed, until now, $h$ bounded only for the simplicity of the presentation) and if $\supp\hat{g}\subset \lbrack 0,\infty )$, then one can show quite easily that $\int^{\infty }_0 (1+|x|)^{\alpha }|\hat{u}(x)|dx<\infty $ provided that $h\in \C{H}^s$ and $g\in \C{H}^{1-s+\alpha }$ for some $s>\alpha +1/2>1/2$ (before we had to put the much stronger condition $g\in \C{H}^s$). Clearly, the preceding relation satisfied by $\hat{u}$ implies that $\lambda \mapsto \langle g,(H-\lambda -i0)^{-1}f\rangle$ is of class $\Lambda^{\alpha }$. Of course,we have proved this only for a very special function $h$, but it will turn out that the result remains true under very general hypotheses. We note that the condition $\supp\hat{g}\subset \lbrack 0,\infty )$ can be very simply stated in terms of the operator $A$: if $\Pi_-$ is the spectral projection of $A$ associated to the interval $(-\infty ,0\rbrack $ then $\supp \hat{g}\subset \lbrack 0,\infty )$ is equivalent to $\Pi_{-}g=g$. The last point that we want to discuss is the abstract analogue of the condition $h'(x)>0$, which played such an important role in the preceding analysis (we could as well assume that $h'(x)<0$ for all $x$; however, this trivial modification amounts to the replacement of the operator $A$ by $-A$ below). Note that $\lbrack H,iA\rbrack $ is just the operator of multiplication by the function $h'$ in $L^2(\D{R})$ (we recall the choice $A=i(d/dx)$), so we see that we could express the condition $h'>0$ in terms of the strict positivity of the operator $\lbrack H,iA\rbrack $. Now it turns out that such a global condition is rather difficult to check and is also quite restricitve in many applications. On the other hand, one may easily convince himself that in order to study (by the preceding methods) the behaviour of $(H-\lambda -i\mu )^{-1}$ as $\mu \rightarrow \pm 0$ for $\lambda $ in some open real set $J$ it suffices to have $h'(x)>0$ if $h(x)\in J$, i.e.\ $h'(x)>0$ on the open set $h^{-1}(J)$. The characteristic function of the set $h^{-1}(J)$, when considered as multiplication operator in $L^2(\D{R})$, is just the spectral projection $E(J)$ associated to the self-adjoint operator $H$ and to the set $J$. So the condition $h'(x)>0$ if $h(x)\in J$ can be expressed in terms of the ``strict positivity'' of the self-adjoint operator $E(J)\lbrack H,iA\rbrack E(J)$, where $E$ is the spectral measure of $H$. A rather natural ormulatio of this ``strict positivity'' condition is: there is a strictly positive real number $a$ such that $E(J)\lbrack H,iA\rbrack E(J) \geq aE(J)$. This is exactly the strict Mourre estimate. %------------% \subsection{} \label{ss:1.3} The general theory which will be developed from now on should be considered as an abstract version of the ``model'' discussed in \S1.2. Although our framework and our assumptions are simple and natural generalizations of those from \S1.2, it turns out that they cover quite large classes of hamiltonian operators that appear in applications and that our results give optimal answers to several questions important in spectral and scattering theory. We also stress the fact that several consequences of the general theory in the classical case (i.e.\ the case stated in \S1.2) are rather difficult to obtain by classical means (cf. a comment in \S1.2; see also Theorem 7.6.2 in \lbrack ABG 2\rbrack ). Our decision to treat general resolvent families (which could not be associated to densely defined self-adjoint operators in $\C{H}$) could be justified by esthetical and practical reasons as well. Our attitude towards the requirement of density in $\C{H}$ of the domain of a self-adjoint operator is the same as that of A.P.~Morse and J.P.~Randolph regarding the measurability assumptions for planar sets: it is ``usually a luxury, rarely a convenience, never a necessity'' (Trans.~Amer.~Math.~Soc.\ 55 (1944)). It was observed already in \cite{BG5} that the conjugate operator method can be extended to situations where the domain of the hamiltonian is not dense in $\C{H}$ and the usefulness of this fact was shown in \cite{BGS} by studying in detail a class of N-body hamiltonians with hard-core interactions. There are other examples in which the possibility of analysing a hamiltonian $H$ with the help of an operator $A$ which generates a unitary group in a larger Hilbert space is very convenient. %------------% \subsection{} \label{ss:1.4} In order to describe the most general resolvent family we find convenient to generalize the standard notion of self-adjoint operator. A linear operator $H:D(H)\subset \C{H}\rightarrow \C{H}$ will be called {\em self-adjoint} if $\langle Hf,g\rangle=\langle f,Hg\rangle$ for all $f$, $g\in D(H)$ and $(H\pm i)D(H)=\overline{D(H)}$ (closure of $D(H)$ in $\C{H}$). In other terms, $H$ sends $D(H)$ into its closure in $\C{H}$ and, when considered as operator in $\overline{D(H)}$, $H$ is self-adjoint in the usual sense. So a densely defined (i.e.\ $\overline{D(H)}=\C{H}$) self-adjoint operator is a self-adjoint operator in the usual sense. To a self-adjoint operator $H$ in $\C{H}$ we associate the resolvent family defined as follows: if $f\in \overline{D(H)}$ then $R(z)f=(H-z)^{-1}f$, where $(H-z)^{-1}$ is the inverse in the Hilbert space $\overline{D(H)}$ of the bijective operator $H-z:D(H)\rightarrow \overline{D(H)}$, and $R(z)f=0$ if $f$ is orthogonal to $D(H)$. Then each resolvent family is obtained by this procedure from a uniquely defined self-adjoint operator. In the context of this paper it is natural to call {\em spectral measure} a countable additive orthogonal projection valued function $E$ defined on the $\sigma $-algebra of the real Borel sets (so we do {\em not} assume $E(\D{R})=1$). Then the formulas $H=\int_{\D{R}} \lambda E(d\lambda )$ and $R(z)=\int_{\D{R}} (\lambda -z)^{-1}E(d\lambda )$ establish bijective correspondances between self-adjoint operators, spectral measures and resolvent families; note that $E(\D{R})$ is the orthogonal projection of $\C{H}$ onto $\overline{D(H)}$. The spectral measure $E$ allows one to speak about various spectral properties of the operator $H$ or of the resolvent family $\lbrace R(z)\rbrace $. In particular, the spectrum $\sigma (H)$ of $H$ (or of its resolvent family) is the support of the measure $E$. Moreover, we can define functions of $H$ by using $E$: if $\varphi :\D{R}\rightarrow \D{C}$ is continuous and convergent to zero at infinity, we set $\varphi (H)=\int_{\D{R}} \varphi (\lambda )E(d\lambda )$. >From now on we fix a self-adjoint operator $H$ in $\C{H}$ with resolvent family $\lbrace R(z)\rbrace $ and a {\em densely defined} self-adjoint operator $A$ in $\C{H}$. The commutator $\lbrack H,iA\rbrack $ is then well defined as a symmetric sesquilinear form with domain $D(A)\cap D(H)$ by the formula $\langle f,\lbrack H,iA\rbrack f\rangle=2\Re \langle Hf,iAf\rangle$. We say that $H$ {\em is of class $C^1(A)$} if there is a number $z\in \D{C}\setminus \sigma (H)$ such that the map $\tau \mapsto W^*_{\tau }R(z)W_{\tau }\in B(\C{H})$ is strongly $C^1$ (this property is independent of $z$; recall that $W_{\tau }=\exp (i\tau A)$). If this is the case then $D(A)\cap D(H)$ is a core for $H$ (i.e.\ it is dense in $D(H)$ for the graph topology), for each $\varphi \in C^{\infty }_0(\D{R})$ we have $\varphi (H)D(A)\subset D(A)\cap D(H)$, and the densely defined symmetric sesquilinear form $\varphi (H)^{\star }\lbrack H,iA\rbrack \varphi (H)$ (with domain $D(A)$) is continuous for the topology induced by $\C{H}$ on $D(A)$ (see \S4.5). We shall denote by the same symbol $\varphi (H)^*\lbrack H,iA\rbrack \varphi (H)$ the bounded (everywhere defined) symmetric operator in $\C{H}$ associated to the sesquilinear form $\varphi (H)^{\star }\lbrack H,iA\rbrack \varphi (H)$ (in \S4.5 we adopt a more pedantic notation). If $H$ is of class $C^1(A)$ we associate to it two subsets of the real line defined as follows: (i) {\em the strict Mourre set $\mu^A(H)$ of $H$ with respect to $A$} is the set of real numbers $\lambda $ with the property that there are a real function $\varphi \in C^{\infty }_0(\D{R})$ with $\varphi (\lambda )\not =0$ and a strictly positive real number $a$ such that $\varphi (H)\lbrack H,iA\rbrack \varphi (H)\geq a\varphi (H)^2$; (ii) {\em the Mourre set $\tilde{\mu}^A(H)$ of $H$ with respect to $A$} is the set of real numbers $\lambda $ such that a real function $\varphi \in C^{\infty }_0(\D{R})$ with $\varphi (\lambda )\not =0$, a strictly positive real number $a$ and a compact operator $K$ in $\C{H}$ such that $\varphi (H)\lbrack H,iA\rbrack \varphi (H)\geq a\varphi (H)^2+K$. The next result is a rather straightforward consequence of the definitions (see \S 4.5). \begin{thmA} Assume that $H$ is of class $C^1(A)$. Then $\mu^A(H)$ and $\tilde{\mu }^A(H)$ are open real sets, $\mu^A(H)\subset \tilde{\mu }^A(H)$, and the set $\tilde{\mu }^A(H)\setminus \mu^A(H)$ does not have accumulation points in $\tilde{\mu }^A(H)$. Moreover, $\tilde{\mu }^A(H)\setminus \mu^A(H)$ consists of eigenvalues of $H$ of finite multiplicity and the spectrum of $H$ in $\mu^A(H)$ is purely continuous. \end{thmA} We do not know whether the $C^1(A)$ regularity property suffices for the absence of the singularly continuous spectrum of $H$ in $\mu^A(H)$ (this is true in the classical case and also in some rather non-significative generalizations). However, we know that a very natural version of the limiting absorption principle breaks down if $H$ is only $C^1(A)$ (see Section 7.B in \cite{ABG2} for a precise statement). For such matters the regularity of $H$ expressed in (1.1), plays an important role. Before stating our next results we have to introduce several regularity classes of functions, vectors and operators. %------------% \subsection{} \label{ss:1.5} Let $E$ be a Banach space and $\phi: \D{R}\rightarrow E$ a bounded continuous function. For each integer $m\geq 1$ we define the {\em modulus of continuity} (or {\em smoothness}) of order $m$ of $\phi $ as the function $\omega_m$ given by $\omega_m(\varepsilon )=\sup_{x\in \D{R}} \Vert \sum^m_{k=0} (-1)^k\binom mk\phi (x+k\varepsilon )\Vert_E$ for $\varepsilon >0$. Let $s>0$ be a real number and $p\in \lbrack 1,\infty \rbrack $. Then $\phi $ {\em is of class} $\Lambda^{s,p}$ if there is an integer $m>s$ such that $\lbrack \int^1_0 (\varepsilon^{-s}\omega_m(\varepsilon ))^p \varepsilon^{-1}d\varepsilon \rbrack^{1/p}<\infty $ (if $p=\infty $ this means $\omega_m(\varepsilon )\leq c\varepsilon^s$ for a finite constant $c$). It is not difficult to show that this condition is independent of $m$. Moreover, one has $\Lambda^{s,p}\subset \Lambda^{t,q}$ if and only if $s>t$ or $s=t$ but $p\leq q$. The classes $\Lambda^{s,\infty }\equiv \Lambda^s$ are called {\em Lipschitz-Zygmund} (or {\em H\"older-Zygmund}) {\em classes}. If $k\geq 1$ is an integer we can also consider the classes $BC^k$ and $\Lip^{(k)}$ defined as follows: $\phi \in BC^k$ means that the derivatives of order $\leq k$ of $\phi $ exist and are bounded and norm continuous; $\phi \in \Lip^{(k)}$ means that $\omega_k(\varepsilon )\leq c\varepsilon^k$ for a constant $c$ and all $\varepsilon >0$ (this is a k-th order Lipschitz condition). We shall point out now several relations between the preceding spaces. If $k\geq 1$ is an integer then $\Lambda^{k,1}\subset BC^k\subset \Lip^{(k)}\subset \Lambda^k$, all embeddings being strict and optimal on the scale $\Lambda^{s,p}$ (the spaces $\Lambda^{k,p}$ are not comparable with $BC^k$ and $\Lip^{(k)}$ if $1
0$ is a real number. Then
$\phi \in \Lambda^{s,p}$ if and only if
$\phi \in BC^k$ and
$\phi^{(k)}\in \Lambda^{\sigma ,p}$. Note that one can always
choose
$0<\sigma \leq 1$. Let
$p=\infty $. If $0<\sigma <1$, then $\phi \in \Lambda^s$ means
$\phi \in BC^k$ and $\Vert
\phi^{(k)}(x+\varepsilon )-\phi^{(k)}(x)\Vert_E\leq
C|\varepsilon |^{\sigma }$; if
$\sigma =1$, then $\phi \in \Lambda^s$ means that $\phi \in
BC^k$ and $\Vert \phi^{(k)}(x+\varepsilon
)+\phi^{(k)}(x-\varepsilon )-2\phi^{(k)}(x)\Vert_E\leq
C|\varepsilon |$, i.e.\ $\phi^{(k)}$ has to verify a Zygmund
condition. All these results are consequences of the general
theory developed in
\cite{BB} or in Chapter 3 of \cite{ABG2}.
There are natural classes associated to the preceding ones.
For example, if $\Omega $ is an open real set and $\phi
:\Omega \rightarrow E$ is a continuous function, then $\phi $
is locally of class
$\Lambda^{s,p}$ if
$\theta \phi \in \Lambda^{s,p}$ for each
$\theta \in C^{\infty }_0(\Omega )$.
The classes $\Lambda^{s,p}$ can be defined for functions of
several variables. We shall give just one example that we
shall need later on. Let $\D{C}_+=\lbrace z\in \D{C} \mid \Im
z>0\rbrace $ and let
$J$ be an open real set. Assume that $\phi $ is a continuous
map from $\D{C}_+\cup J$ to $E$ and let
$s$ be a strictly positive real number. We shall say that
$\phi $ is locally of class $\Lambda^s$ (on $\D{C}_+\cup J$)
if for each rectangle $K=\lbrace z\mid a\leq \Re z\leq b ,
0\leq \Im z\leq c\rbrace $, with
$\lbrack a,b\rbrack \subset J$ and
$c>0$, there are real numbers
$\delta >0$, $M>0$ and an integer $m>s$ such that $\Vert
\sum^m_{k=0} (-1)^k\binom mk\phi (z+k\varepsilon )\Vert \leq
M|\varepsilon |^s$ for all
$z\in K$ and
$\varepsilon \in \D{C}$ with
$|\varepsilon|\leq \delta $ and
$\Im \varepsilon \geq 0$ (this property is, of course,
independent of $m$).
One should notice one more fact concerning the classes
$\Lambda^s$ and
$\Lip^{(k)}$: they do not really depend on the topology of
$E$, but rather on its bornology (i.e.\ the family of bounded
sets). More precisely, if we have a new vector space topology
on $E$, and if the bounded sets associated to this topology
are the same as those of the initial $E$, then the classes
$\Lambda^s$ and $\Lip^{(k)}$ are also the same. For example the
weak
$\Lambda^s$ class (obvious definition) coincides with the norm
$\Lambda^s$ class; or if $E$ is an adjoint space, then the
weak* $\Lambda^s$ class coincides with the norm $\Lambda^s$
class. Moreover, if $E=B(\C{E};\C{F})$ (resp.
$E=B(\C{E};\C{F}^*)$) for some Banach spaces
$\C{E}$,
$\C{F}$, then the norm and the weak (resp. weak*) operator
topology on $E$ give the same classes
$\Lambda^s$ and
$\Lip^{(k)}$.
%------------%
\subsection{} \label{ss:1.6}
One can use the regularity classes $\Lambda^{s,p}$ in order to
treat in a unified way various Besov type spaces of vectors
$f\in \C{H}$ or operators $S\in B(\C{H})$ naturally associated
to the densely defined self-adjoint operator $A$ (a detailed
discussion and other equivalent characterizations of these
spaces may be found in Sections 2 and 3).
Let $s$ be a strictly positive real number and $p\in \lbrack
1,\infty \rbrack $. Then $\C{H}_{s,p}$ is the set of vectors
$f\in \C{H}$ such that the function $\tau \mapsto W_{\tau
}f\in \C{H}$ is of class
$\Lambda^{s,p}$. Similarly,
$\C{C}^{s,p}(A)$ is the set of operators
$S\in B(\C{H})$ such that the map
$\tau \mapsto W^*_{\tau }SW_{\tau }\in B(\C{H})$ is of
class $\Lambda^{s,p}$. We thus get two scales of spaces which
are totally ordered in the following sense:
$\C{H}_{s,p}\subset \C{H}_{t,q}$ and
$\C{C}^{s,p}(A)\subset \C{C}^{t,q}(A)$ if $s>t$ or if $s=t$
but $p\leq q$. There are natural topologies on
$\C{H}_{s,p}$ and
$\C{C}^{s,p}(A)$ for which these spaces become topological
vector spaces, and these topologies can be defined by
(complete) norms.
It is possible to extend the scale $\lbrace \C{H}_{s,p}\rbrace
$ to $s\leq 0$ by the following procedure (a more convenient
method is presented in Section 2; it is also possible to
define the spaces $\C{C}^{s,p}(A)$ for
$s\leq 0$, but we shall not need them). Let $\C{H}_{\infty
}=\cap_{s>0} \C{H}_{s,p}$ and let
$\C{H}^o_{s,p}$ be the closure of
$\C{H}_{\infty }$ in
$\C{H}_{s,p}$. Then
$\C{H}_{\infty }$ is dense in
$\C{H}$ and
$\C{H}^o_{s,p}=\C{H}_{s,p}$ if $1\leq p <\infty $. Now we set
$\C{H}_{-s,p'}=\lbrack \C{H}^o_{s,p}\rbrack^*$ (adjoint
space, i.e.\ the space of continuous anti-linear forms on
$\C{H}^o_{s,p}$ equipped with the strong topology). This
defines
$\C{H}_{t,q}$ for all $t\in \D{R}\setminus \lbrace 0\rbrace $
and
$1\leq q\leq \infty $. We identify $\C{H}^*=\C{H}$ with
the help of Riesz's lemma and so we get continuous embeddings
$\C{H}_{s,p}\subset \C{H}\subset \C{H}_{t,q}$ if
$s>0$ and
$t<0$. Finally, if $s=0$ we define
$\C{H}_{0,p}$ for $1\leq p\leq \infty $ by real interpolation
(see (2.11)).
A rather remarkable fact happens for $p=2$ (this is due to the
Hilbert space geometry of $\C{H}$, the unitarity of $W_{\tau
}$ plays no role; cf. Section 3.7 in \cite{ABG2}). Let $k\geq
1$ be an integer and let
$f\in \C{H}$. Then the map
$\tau \mapsto W_{\tau }f\in \C{H}$ is of class $\Lambda^{k,2}$
if and only if it is of class
$BC^k$ and also if and only if it is of class
$\Lip^{(k)}$ (see \S2.5 for an elementary proof). In
particular, $\C{H}_{k,2}=D(A^k)$. We set
$\C{H}_s=\C{H}_{s,2}$ for all
$s\in \D{R}$. Then
$\C{H}_s=D(|A|^s)$ for
$s\geq 0$, in particular $\C{H}_0=\C{H}$.
Geometrically speaking, the Banach space $B(\C{H})$ looks like
a space $L^{\infty }$. For this reason the value
$p=\infty $ plays a rather special role in this case, so we
set
$\C{C}^s(A)=\C{C}^{s,\infty }(A)$. Let
$s=k$ be an integer
$\geq 1$. Then one can also introduce the space $C^k(A)$ of
operators $S\in B(\C{H})$ such that the map
$\tau \mapsto W^*_{\tau }SW_{\tau }\in B(\C{H})$ is of
class
$\Lip^{(k)}$. One can prove that $C^k(A)\subset \C{C}^k(A)$
{\em strictly}. Moreover, $S$ belongs to
$C^k(A)$ if and only if the function
$\tau \mapsto W^*_{\tau }SW_{\tau }\in B(\C{H})$ is of
class
$BC^k$ in the strong (or weak) operator topology; if this map
is of class $BC^k$ in the norm topology, we write
$S\in C^k_u(A)$. One has the following relation between these
spaces: $\C{C}^{k,1}(A)\subset C^k_u(A)\subset C^k(A)\subset
\C{C}^k(A) $ strictly (see Section 3). We set
$C^{\infty }(A)=\cap_{k\in \D{N}} C^k(A)$.
Let $H$ be a self-adjoint operator in $\C{H}$ and $\lbrace
R(z)\rbrace $ its resolvent family. We say that
$H$, or
$\lbrace R(z)\rbrace$, {\em is of class} $\C{C}^{s,p}(A)$
($\C{C}^s(A)$ if $p=\infty $) if there is
$z\in \D{C}\setminus \sigma (H)$ such that $R(z)\in
\C{C}^{s,p}(A)$ (this property is independent of $z$). We see
that
$\lbrace R(z)\rbrace $ {\em is a regular (in the sense of {\rm
(1.1)})
resolvent family if and only if $H$ is of class
$\C{C}^{1,1}(A)$, and then
$H$ is of class $C^1(A)$}.
Assume that $H$ is of class $\C{C}^{s,p}(A)$ for some $s>0$
and let $\varepsilon \in (0,s)$ be an arbitrary small real
number. Then for each $z\in \D{C}\setminus \sigma (H)$ the
operator
$R(z):\C{H}\rightarrow \C{H}$ has a unique extension to a
continuous operator
$R(z):\C{H}_{-s+\varepsilon }\rightarrow \C{H}_{-s+\varepsilon
}$ (see \S3.9). Moreover, this extension has the property
$R(z)\C{H}_{t,q}\subset \C{H}_{t,q}$ for each real $t$ with
$|t| 0$ there is a finite set
$K\subset \D{N}$ such that
$\sum_{j\not \in K} \Vert f_n(j)\Vert_Z<\varepsilon $ for all
$n\in \D{N}$. For this we shall use Theorem 4 on page 104 of
\cite{DU} where the measure space
$(\Omega ,\mu )$ is
$\D{N}$ with the counting measure. However, in order to make
it sure that the boundedness of the measure
$\mu $ is not needed for the result of the theorem to hold, we
indicate a modification of the first few lines of the proof of
that theorem.
(iii)
Let $\C{L}$ be a bounded subset of $\ell^1(Z)$ and assume that
there is $\varepsilon >0$ such that for each finite
$K\subset \D{N}$ there is $f\in \C{L}$ with the property
$\sum_{j\not \in K} \Vert f(j)\Vert_Z>\varepsilon
$. Then for each
$n\in \D{N}$ there is
$g_n\in \C{L}$ and
$k_n>n$ such that
$\sum_{n 0$ and some $p\in \lbrack 1 ,\infty
\rbrack $, then
$\Pi_{\mp}S\Pi_{\pm}\C{H}\subset\C{H}_{\alpha,p}$. In
particular, if
$S\in \C{C}^{\alpha ,2}(A)$ then
$\Pi_{\mp}S\Pi_{\pm}\in B(\C{H}_s;\C{H}_{s+\alpha})$ for all
real
$s$ such that $-\alpha \leq s\leq 0$.
\end{thm*}
\begin{proof}
(i)
We first prove a weak-type estimate, namely we
show that
$S_0\equiv \Pi_- S\Pi_+$ sends
$\C{H}$ into
$\C{H}_{m,\infty}$ if
$S\in C^m(A)$ for some integer $m\geq 1$.
Let $\chi $ be the characteristic function of the real set
defined by
$1\leq |x|\leq 2$. Then it suffices to show that
$\Vert \chi (\varepsilon
A)S_0\Vert \leq C\varepsilon^m$ for some constant
$C$ and all
$0<\varepsilon <1$. Set
$S_{\tau }=\exp (\tau A)S_0\exp (-\tau A)$ for
$\tau \geq 0$. Then
$\tau \mapsto S_{\tau }$ is strongly of class $C^m$ on
$\lbrack 0,\infty )$ and its k-th order
derivative ($0\leq k\leq m$) is equal to
$\ad^k_AS_{\tau }=\exp (\tau A)\Pi_-(\ad^k_AS)\Pi_+\exp (-\tau
A)$.
By making a Taylor expansion up to order
$m$ we get (see
(6.1)):
$$
S_0=\sum^{m-1}_{k=0} \frac{(-1)^k}{k!}
\ad^k_AS_1+\frac{(-1)^m}{(m-1)!} \int^1_0 \ad^m_A S_{\tau
}\cdot
\tau^{m-1}d\tau .
$$
The operators $\ad^k_AS_1$ clearly send $\C{H}_{-\infty }$
into $\C{H}_{+\infty }$, so it suffices to consider the
contribution of the integral term. We have :
\begin{align*}
\int^1_0 \Vert \chi (\varepsilon A)\ad^m_AS_{\tau }\Vert
\tau^{m-1}d\tau
&\leq \Vert \ad^m_AS_0 \Vert
\int^1_0
\Vert \chi (\varepsilon A)\Pi_-e^{\tau A}\Vert
\tau^{m-1}d\tau \\
&\leq
\Vert \ad^m_AS_0\Vert \int^1_0\sup_{x>0}
\chi (\varepsilon x)e^{-\tau x}\tau ^{m-1}d\tau \\
&=\Vert
\ad^m_AS_0\Vert \int ^1_0 e^{-\tau/\varepsilon }\tau^{m-1}d\tau
\leq C\varepsilon^m ,
\end{align*}
which is the desired estimate.
(ii)
Let $\mathcal{P} :B(\C{H})\rightarrow B(\C{H})$ be the linear
continuous operator given by $\mathcal{P} S=\Pi_-S\Pi_+$. Then
$\Vert \mathcal{P} \Vert =1$ and
$\mathcal{P} C^m(A)\subset B(\C{H};\C{H}_{m, \infty })$
(by what we have shown above and the closed
graph theorem). On the space
$C^m(A)$ there is a natural Banach space structure
such that the embedding $C^m(A)\subset B(\C{H})$
be continuous. Then one can obtain the spaces
$\C{C}^{\alpha ,p}(A)$ by real interpolation:
$\C{C}^{\alpha ,p}(A)=(C^m(A),B(\C{H})_{\theta ,p}$ with
$\theta =1-\alpha /m$ if
$0<\alpha r} \hat{u}(x)dx\sim r^{-\alpha -2\varepsilon }$
as $r\rightarrow \infty $, where
$u=\overline{g}f$. On the other hand, we have (see (1.4)):
$$
\langle \Pi_{-}g,(H-\lambda -i0)^{-1}f\rangle=2i\int^{\infty
}_0 e^{i\lambda x}\hat{u}(x)dx .
$$
Since $\hat{u}$ is a positive function
one may use a wellknown result of Boas and deduce that
$\lambda \mapsto \langle \Pi_{-}g,(H-\lambda
-i0)^{-1}f\rangle$ is precisely of class $\Lambda^{\alpha
+2\varepsilon }$ (not more !). We thus see that the map
$\lambda \mapsto \Pi_{-}R(\lambda +i0)\in B(\C{H}_s;
\C{H}_{s-1-\alpha })$ cannot be of class
$\Lambda^{\beta }$ for some
$\beta >\alpha $, and this even if $H$ is of class
$C^{\infty }(A)$. The optimality of the condition
$H\in \C{C}^{s+1/2}(A)$ will be discussed later on
(cf. end of \S1.9).
%------------%
\subsection{} \label{ss:1.9}
We shall now mention two simple but useful consequences of
Theorems C and E
(see \cite{BGSh} for other result of this nature). Let $H$ be a
regular self-adjoint operator and
$\varphi
\in C^{\infty }_0(\mu^A(H))$. Then
$\varphi (H)$ leaves invariant the space $\C{H}_{1/2,1}$
(because $\varphi (H)$ is of class
$\C{C}^{1,1}(A)$) hence, if
$H$ has a spectral gap, we may use Theorem B in order
to obtain for all $f$, $g\in \C{H}_{1/2,1}$
and all real
$t>0$:
$$
\langle g,e^{-iHt}\varphi (H)f\rangle=(2\pi i)^{-1}\int_{\D{R}}
e^{-it\lambda }\langle g,R(\lambda +i0)\varphi
(H)f\rangle d\lambda .
$$
Now we use the following elementary fact: if the function
$u:\D{R}^n\rightarrow \D{C}$ belongs to
the Besov space
$B^{\alpha ,\infty }_1(\D{R}^n)$ for some real $\alpha > 0$,
then its Fourier transform $\hat{u}$ is
a continuous function such that
$|\hat{u}(t)|\leq \text{const.}\langle t\rangle^{-\alpha }$,
where
$\langle t\rangle=(1+|t|^2)^{1/2}$. So the Theorems C and E
and the uniform boundedness principle give:
\begin{thmF}
Let $H$ be a self-adjoint operator with a spectral gap and of
class $\C{C}^{s+1/2}(A)$
for some real
$s>1/2$. Let $\alpha $ be a real number such that
$0<\alpha 0$ belongs to $B(\C{H})$ if
and only if the sesquilinear form on $\C{H}_s$ associated to it
is continuous for the topology
induced by $\C{H}$ on $\C{H}_s$.
In such a case we say that $S$ extends to a bounded operator in
$\C{H}$.
We set $\C{H}_{\infty }=\cap_{s\in \D{R}}\C{H}_s$ and
$\C{H}_{-\infty }=\cup_{s\in
\D{R}}\C{H}_s$ and we extend
$\Vert \cdot \Vert_s$ to all of $\C{H}_{-\infty }$ by setting
$\Vert \cdot \Vert_s=\infty $ if
$f\not \in \C{H}_s$. The space
$\C{H}_{\infty }$ has a natural Fr\'echet space topology and
there is a canonical identification of its adjoint space with
$\C{H}_{-\infty }$; we shall equip $\C{H}_{-\infty }$ with the
strong adjoint topology. Notice that if
$\varphi:\D{R}\rightarrow \D{C}$ is a Borel function such that
$|\varphi (x)|\leq c\langle x\rangle^{\sigma }$ for some $c>0$
and
$\sigma \in \D{R}$ then $\varphi (A)$ has a unique extension
to a continuous operator
$\varphi (A):\C{H}_{-\infty }\rightarrow \C{H}_{-\infty}$ and
this operator sends each
$\C{H}_s$ continuously into
$\C{H}_{s-\sigma}$.
%------------%
\subsection{} \label{ss:2.2}
If $S\in B(\C{H}_s;\C{H}_{-s})$ and $k\geq 0$
is an integer then we define
$\ad^k_AS\in B(\C{H}_{s+k};\C{H}_{-s-k})$ by induction:
$\ad^0_AS=S$,
$\ad_AS\equiv \ad^1_AS\equiv \lbrack A,S\rbrack =AS-SA$, and
$\ad^{k+1}_AS=\ad_A(\ad^k_AS)$.
The following formula holds
\begin{equation} \label{eq:2.4}
\ad^k_AS=\sum_{i+j=k} \frac{k!}{i!j!} (-1)^jA^iSA^j .
\end{equation}
If $S\in B(\C{H})$
then it is more convenient to interpret this formula in terms
of sesquilinear forms, namely
$\ad^k_AS$ is the continuous sesquilinear form on
$\C{H}_k(\equiv D(A^k)$ domain of $A^k$ in
$\C{H}$) given for
$f$,
$g\in D(A^k)$ by
$$
\langle f,(\ad^k_AS)g\rangle=\sum_{i+j=k} \frac
{k!(-1)^j}{i!j!}
\langle A^if,SA^jg\rangle .
$$
The following observation is useful. \emph{Let $S\in B(\C{H})$
such
that for some integer $m\geq 2$ the sesquilinear form
$\ad^m_AS$ on $D(A^m)$ is continuous for the topology induced
by $\C{H}$ on $D(A^m)$. Then for each integer $k\in \lbrack
0,m\rbrack $ the sesquilinear form $\ad^k_AS$ is continuous for
the topology induced by
$\C{H}$ on
$D(A^k)$}. For the proof note that if
$f$, $g\in D(A^m)$ and $W_x=e^{iAx}$ then $x\mapsto \langle
W_xf,SW_xg\rangle$ is a function of class $C^m$ on
$\D{R}$. Let
$\psi \in C^{\infty }_0(\D{R})$ with
$\supp \psi \subset \lbrack 0,1\rbrack $,
$\int x^k\psi (x)dx=i^kk!$ and
$\int x^j\psi (x)dx=0$ if
$0\leq j(t,q)$. In this case the embedding is continuous; the
embedding is dense if and only if
$q\not =\infty $. Moreover,
$\C{H}_{\infty }$ is a dense subspace of $\C{H}_{t,q}$ if and
only if $q\not =\infty $. We denote by
$\C{H}^o_{t,\infty }$ the closure of
$\C{H}_{\infty }$ in
$\C{H}_{t,\infty }$.
Recall that $\C{H}^*_{\infty }=\C{H}_{-\infty }$. If
$p\not =\infty $ then we have a continuous dense embedding
$\C{H}_{\infty }\subset \C{H}_{s,p}$, hence we get a canonical
embedding
$(\C{H}_{s,p})^*\subset \C{H}_{-\infty }$. One can show
that
\begin{equation} \label{eq:2.9}
\lbrack \C{H}_{s,p}\rbrack^*=\C{H}_{-s,p'}
\text{ if } 1\leq p<\infty \text{ and } p^{-1}+{p'}^{-1}=1 .
\end{equation}
The adjoint of the space $\C{H}_{s,\infty }$
is not a Besov space (in fact it can not be realized
as a subspace of
$\C{H}_{-\infty }$). However we have
$\C{H}_{\infty }\subset \C{H}^o_{s,\infty }$
continuously and densely, so the space adjoint to
$\C{H}^o_{s,\infty }$ can be realized as a subspace of
$\C{H}_{-\infty }$, and in fact one can show that
\begin{equation} \label{eq:2.10}
\lbrack \C{H}^o_{s,\infty }\rbrack^*
=\C{H}_{-s,1 } .
\end{equation}
The Besov scale is stable under real interpolation. More
precisely, let $s$, $t\in \D{R}$ and $p$,
$q\in \lbrack 1,\infty \rbrack $.
Then for each $\theta \in (0,1)$ and $r\in \lbrack 1,\infty
\rbrack $ we have
\begin{equation} \label{eq:2.11}
(\C{H}_{s,p }, \C{H}_{t,q})_{\theta ,r}
=\C{H}_{(1-\theta )s+\theta t,r}
\end{equation}
as topological vector spaces.
%------------%
\subsection{} \label{ss:2.5}
The theorem proved in \S2.3 allows us to give other descriptions
of the Besov spaces $\C{H}_{s,p}$. Let $\psi :\D{R}\rightarrow
\D{C}$ be a locally bounded Borel function and assume that
there are numbers
$b>a>0$ and $c>0$ such that $|\psi (x)|\geq c^{-1}$ on
$\lbrack a,b\rbrack $. Let $n$ be the first
integer such that
$(b/a)^{n+1}\geq 2$. Then for
$1\leq x\leq 2$ we have
$\sum_{0\leq k\leq n} c|\psi (a^{k+1}b^{-k}x)|\geq 1$
hence,
if we denote by $\chi_{12}$ the
characteristic function of the interval
$\lbrack 1,2\rbrack $, then we have
$$
\Bigl[ \int^{\infty }_1 \Vert \tau^s\chi_{12}(A/\tau )f
\Vert^p\frac{d\tau}{\tau} \Bigr]^{1/p}\leq
\sum^n_{k=0}\frac{ca^{(k+1)s}}{b^{ks}} \Bigl[
\int^{a^{k+1}b^{-k}}_0 \Vert \varepsilon^{-s}\psi
(\varepsilon A)f\Vert^p\frac{d\varepsilon}{\varepsilon}
\Bigr]^{1/p} .
$$
If
$|\psi |$ is also bounded from below by a strictly
positive constant on an interval
$\subset (-\infty ,0)$ then we shall have a similar
estimate but with $\chi_{12}(A/\tau )$ replaced by
$\chi_{12}(-A/\tau )$. This will give us an upper bound
for $\Vert f\Vert_{s,p}$ in terms of the
function
$\psi $. In order to get a lower bound we use (2.7) with
$s=0$ and take into account that
$\Vert g\Vert \leq c_1\Vert g\Vert_{0,2}\leq c_2\Vert
g\Vert_{0,1}$. We finally obtain the following
result.\emph{ Let
$\psi :\D{R}\rightarrow \D{C}$ be a locally bounded Borel
function such that
$|\psi (x)|\geq \text{const.}>0$ for $x\in J$, where $J$ is an
open set with $J\cap (-\infty ,0)\not =\emptyset $ and
$J\cap (0,\infty )\not =\emptyset$; let
$s\in \D{R}$ and
$p\in \lbrack 1,\infty \rbrack $ and assume that
$|\psi (x)|\leq c|x|^s\cdot \min (|x|^{\nu},|x|^{-\nu })$
for some constants
$c,\nu >0$. Then there is a constant $C>0$
such that for all $f\in \C{H}_{-\infty }$:}
\begin{equation} \label{eq:2.12}
C^{-1}\Vert f\Vert_{s,p}\leq
\Vert E_A(\lbrack -2,2\rbrack )f\Vert
+\lbrack \int^1_0\Vert \varepsilon^{-s}\psi
(\varepsilon A)f\Vert^p\varepsilon^{-1}d\varepsilon
\rbrack^{1/p}\leq C\Vert f\Vert_{s,p} .
\end{equation}
We mention the following possible choices:
(i) if $s<0$ then one may take $\psi $ equal to the
characteristic function of the interval
$\lbrack -1,1\rbrack $;
(ii) if $s>0$ then $\psi $ can be chosen as the
charcteristic function of the set
$(-\infty ,-1\rbrack \cup \lbrack 1,\infty )$;
(iii) if $s>0$ and if $m$ is an integer strictly larger
than $s$, then one
may take $\psi (x)=\lbrack x(x+i)^{-1}\rbrack^m$.
A more interesting example in the case $s>0$ is obtained by
choosing $\psi (x)=(e^{ix}-1)^m$ with
$m>s$ integer. Let us set
$W_{\sigma }=e^{iA\sigma }$ for $\sigma \in \D{R}$.
Now let $s$ be a strictly positive real number,
$m>s$ an integer, and
$p\in \lbrack 1,\infty \rbrack $. Then there
is a constant $C>0$ such that
\begin{equation} \label{eq:2.13}
C^{-1}\Vert f\Vert_{s,p}\leq
\Vert f\Vert +\lbrack \int^1_0 \Vert
\varepsilon^{-s}(W_{\varepsilon}-1)^mf\Vert^p
\varepsilon^{-1}d\varepsilon \rbrack^{1/p}\leq C\Vert
f\Vert_{s,p} .
\end{equation}
Note the following difference between the description of
$\C{H}_{s,p}$ given by the gauge (2.6) and that associated
to the gauge which appears in the middle term of (2.13) (for
$s>0$): the first one gives a characterization of the property
$f\in
\C{H}_{s,p}$ in terms of the behaviour of $f$ at infinity in a
spectral representation of $A$, while the second one describes
the property $f\in \C{H}_{s,p}$ in terms of local regularity
conditions on the function $\D{R}\ni \sigma \mapsto W_{\sigma
}f\in \C{H}$.
Finally, we explain how one may obtain the description of
$\C{H}_{s,p}$ in terms of moduli of continuity of the function
$\sigma \mapsto W_{\sigma }f$ (see Section 1). If $f\in \C{H}$
and
$m\geq 1$ is an integer we set $\omega_m(\varepsilon
)=\sup_{|\sigma |\leq \varepsilon } \Vert (W_{\sigma
}-1)^mf\Vert$ and
$\omega (\varepsilon )\equiv \omega_1(\varepsilon )$. Let
$s\in (0,m)$ and
$p\in \lbrack 1,\infty \rbrack $. Then the function
$\sigma \mapsto W_{\sigma }f\in \C{H}$ is of class
$\Lambda^{s,p}$ if and only if $\bigl[ \int^1_0 \lbrack
\varepsilon^{-s}\omega_m(\varepsilon
)\rbrack^p\varepsilon^{-1}d\varepsilon \bigr]^{1/p}<\infty $.
It is clear that this implies
$f\in \C{H}_{s,p}$ (see (2.13)). The reciprocal assertion is
not so obvious: we shall prove it here in the case
$m=1$ (for the general case see the remarks after the proof of
Theorem 3.4.6 in \cite{ABG2}). Observe that if
$\sigma ,\varepsilon $ are non-zero real numbers then
\begin{align*}
e^{iA\sigma }-1&=(e^{iA\sigma }-1)\frac{e^{iA\varepsilon
}-1}{iA\varepsilon }+(e^{iA\sigma }-1)\lbrack
1-\frac{e^{iA\varepsilon }-1}{iA\varepsilon }\rbrack
\\
&=\frac{\sigma }{\varepsilon}\frac{e^{iA\sigma }}{iA\sigma
}(e^{iA\varepsilon }-1)+(e^{iA\sigma }-1)\int^1_0\lbrack
1-e^{iA\varepsilon \tau }\rbrack d\tau .
\end{align*}
This clearly gives for $f\in \C{H}$:
$$
\Vert(e^{iA\sigma }-1)\Vert \leq |\sigma/\varepsilon |\cdot
\Vert (e^{iA\varepsilon }-1)f\Vert +2\int^1_0\Vert
(e^{iA\varepsilon \tau }-1)f\Vert d\tau .
$$
In particular we have
$$
\omega (\varepsilon )\leq \Vert (W_{\varepsilon}-1)f\Vert
+2\int^1_0 \Vert (W_{\varepsilon \tau }-1)f\Vert d\tau .
$$
It is now straightforward to show that $\bigl[ \int^1_0
\lbrack\varepsilon^{-s}\omega (\varepsilon)
\rbrack^p\varepsilon^{-1}d\varepsilon \bigr]^{1/p}<\infty $ if
$f\in \C{H}_{s,p}$ (with $0
k$ is an integer. A new description of $\C{H}_k$
can be obtained in terms of the modulus of
continuity
$\omega_k$ of order
$k$ by taking into account the identity
$$
\Vert \varepsilon^{-k}(W_{\varepsilon}-1)^kf\Vert^2=
\int_{\D{R}} \biggl\vert\frac{e^{i\varepsilon
\lambda }-1}{\varepsilon}\biggr\vert^{2k}\Vert E_A(d\lambda )
f\Vert^2 .
$$
By using Fatou lemma we see that $f\in \C{H}_k$ if and only if
$f\in \C{H}$ and
$\liminf_{\varepsilon \rightarrow 0}\Vert
\varepsilon^{-k}(W_{\varepsilon}-1)^kf\Vert <\infty $,
and the second
condition is in fact equivalent to
$\omega_k(\varepsilon )\leq c\varepsilon^k$ and also to the
aparently much stronger condition that the
function
$\sigma \mapsto W_{\sigma }f\in \C{H}$ be strongly of class
$C^k$.
%------------%
\subsection{} \label{ss:2.6}
Besides the norm topology it will be convenient to consider on
$\C{H}_{s,p}$ the topology defined by the family of seminorms
$f\mapsto |\langle f,g\rangle|$ where $g\in \C{H}_{-s,p'}$; we
shall call it w-{\it topology}. If
$1s$ is an integer, then
$(3.9)$ holds for each
$\theta $ such that
$\theta^{(k)}\in \C{M}(\D{R})$ for
$0\leq k\leq m$ and $\theta^{(k)}(0)=0$ for $0\leq k\leq m-1$.
\end{thm*}
The proof of this result can be found in \cite{BG3}.
One may take $\theta (x)=(e^{ix}-1)^m$, but this choice is not
useful for our purposes in Section 5.
As a first application of the preceding theorem we shall give
a simple proof of the embedding
$\C{C}^{s,p}(A)\subset \C{C}^{t,q}(A)$ for $(s,p)>(t,q)$. The
fact that $\C{C}^{s,\infty }(A)\subset \C{C}^{t,1}(A)$ if
$s>t>0$ is an immediate consequence of the definition, so it
suffices to consider the case $s=t$ and
$1\leq p0$ if and only if
$1<|x|<2$ and
$\varphi (x)=1$ if
$1<|x|<4$. Then
$\theta (x)=\theta (x)\varphi (\nu x)$ for all $x\in \D{R}$
and $1\leq \nu \leq 2$, hence $\theta (\varepsilon x)=\theta
(\varepsilon x)\varphi (\tau x)$ for all
$x\in \D{R}$ and
$0<\varepsilon \leq \tau \leq 2\varepsilon $, in particular
$\theta (\varepsilon \C{A})=\theta (\varepsilon \C{A})\varphi
(\tau \C{A})$ for such
$\varepsilon $,
$\tau $. This implies
$\Vert \theta (\varepsilon \C{A})S\Vert \leq \Vert \theta
\Vert_{\C{M}}\Vert \varphi (\tau
\C{A})S\Vert
$ for $S\in B(\C{H})$, so there is a constant
$C$ (depending only on
$s$ and
$p$) such that for all $\varepsilon >0$
$$
\Vert \varepsilon^{-s}\theta (\varepsilon \C{A})S\Vert \leq
C\left\lbrack \int^{2\varepsilon }_{\varepsilon} \Vert
\tau^{-s}\varphi (\tau \C{A})S\Vert^p\tau^{-1}d\tau
\right\rbrack^{1/p} .
$$
If $\C{C}^{s,p}(A)$ then the r.h.s.\ above is uniformly
bounded in
$\varepsilon >0$ (by the preceding theorem) hence $S\in
\C{C}^{s,\infty }(A)$ (by a new application of the theorem).
Finally, let us prove that $\C{C}^{k,1}(A)\subset C^k_u(A)$
if $k\geq 1$ is an integer. Let
$\varphi \in C^{\infty }_0(\D{R})$ with
$\varphi (x)=1$ on a neighbourhood of the origin, and let
us set $S_{\varepsilon}=\varphi (\varepsilon
\C{A})S$ for some $S\in \C{C}^{k,1}(A)$. Then
$S_{\varepsilon}\in C^{\infty }(A)$ for
$\varepsilon \not =0$ and
$\lim_{\varepsilon \rightarrow 0} S_{\varepsilon}=S$ in the
strong operator topology. Now for
$j=0,\dots,k$ and $\varepsilon \not =0$ let us set
$S_{j\varepsilon }=\C{A}^jS_{\varepsilon}$. Then we have
$$
S'_{j\varepsilon }\equiv \frac{d}{d\varepsilon }
S_{j\varepsilon }=\C{A}^{j+1}\varphi' (\varepsilon
\C{A})S=\varepsilon^{-j-1}\varphi_j(\varepsilon \C{A})S ,
$$
where $\varphi_j(x)=x^{j+1}\varphi'(x)$. An application of
the theorem stated before in this paragraph
gives:
$S\in \C{C}^{j,1}(A)$ if and only if $\int^1_0 \Vert
S'_{j\varepsilon }\Vert
d\varepsilon <\infty $.
Since
$S$ is of class
$\C{C}^{k,1}(A)$ we shall have this property for each
$j=0,1,\cdot \cdot \cdot ,k$. In particular
$\lim_{\varepsilon \rightarrow 0} \C{A}^jS_{\varepsilon}$
exists in norm for
$j=0,1,\dots,k$. But clearly $C^k_u(A)$ is a
Banach space for the norm $\sum^k_{j=0}
\Vert \C{A}^jT\Vert $. Hence
$S\in C^k_u(A)$.
%------------%
\subsection{} \label{ss:3.8}
We have mentioned before that the spaces $C^k(A)$,
$C^k_u(A)$ and $\C{C}^{s,p}(A)$ are full
involutive subalgebras of
$B(\C{H})$. We shall now prove that they are stable
under a much larger class of operations. The
following result will be needed.
\begin{lem*} \label{eq:3.10}
For each integer $m\geq 1$ there is a number $C_m$
such that for any bounded operator $B$ and any bounded
self-adjoint operator $S$ the next estimate holds
\begin{equation}
\Vert \ad^m_B(e^{iS})\Vert \leq C_m \sum \Vert
\ad^{m_1}_B(S)\Vert \dots \Vert
\ad^{m_k}_B(S)\Vert .
\end{equation}
The sums runs over all decompositions $m=m_1+\dots +m_k$
of $m$ into a sum of integers
$m_1,\dots ,m_k \geq 1$.
\end{lem*}
\begin{proof}
For $m=1$ we use
\begin{align*}
\ad_B(e^{iS})&=(B-e^{iS}Be^{-iS})e^{iS}=-\int^1_0
\frac{d}{dt}e^{itS}Be^{i(1-t)S}dt\\
&=\int^1_0 e^{itS}\ad_B(iS)e^{i(1-t)S}dt ,
\end{align*}
which gives $\Vert \ad_Be^{iS}\Vert \leq \Vert \ad_BS\Vert $.
Now assume that (3.10) has been proved for all integers
$\leq m$ and all
$S$. From the preceding identity we obtain
\begin{align*}
\ad_B^{m+1}(e^{iS})&=\int^1_0 \ad_B^{m}\lbrack
e^{itS}\ad_B(iS)e^{i(1-t)S}\rbrack dt \\
&=\sum_{{a+b+c=m,}\atop {a,b,c\geq 0}}
\int^1_0
\frac{m!}{a!b!c!}
\ad_B^{a}(e^{itS})\ad_B^{b+1}(iS)\ad_B^{c}(e^{i(1-t)S})dt .
\end{align*}
Then the induction hypothesis gives
\begin{align*}
\Vert \ad_B^{m+1}(e^{})\Vert \leq
C(m)\int^1_0 dt
\sum
&\Vert \ad_B^{a_1}(tS)\Vert \dots \Vert
\ad_B^{a_n}(tS)\Vert \cdot
\Vert \ad_B^{b+1}(iS)\Vert \cdot \\
&\Vert \ad_B^{c_1}((1-t)S\Vert\dots\Vert
\ad_B^{c_l}((1-t)S\Vert
\end{align*}
where the sum runs over all decompositions of
$a$ and $c$ into sum of integers $a_1,
\dots ,a_n$ and $c_1,\dots ,c_l$
respectively, with
$a_j\geq 1$,
$c_j\geq 1$. This clearly implies (3.10) with
$m$ replaced by $m+1$.
\end{proof}
Let us fix now an integer $n\geq 1$ and a family
$S_1,\dots,S_n$ of bounded self-adjoint operators on
$\C{H}$. We set
$\D{S}=(S_1,\dots,S_n)$ and for $x=(x_1,\dots,x_n)\in \D{R}^n$ we
denote
$x\D{S}=x_1S_1+\dots+x_nS_n$. We shall define a
bounded operator
$\phi (\D{S}) $ for each bounded continuous function
$\phi :\D{R}^n\rightarrow \D{C}$ such that the
Fourier transform
$\hat{\phi }$ is a bounded measure by setting
$\phi (\D{S})=\int_{\D{R}^n} \exp (ix\D{S})\hat{\phi }(x)dx$.
Here
$\phi (y) =\int\exp (ixy)\hat{\phi }(dx)$, with a slightly
formal notation. We clearly have
$\Vert \phi (\D{S})\Vert \leq \int |\hat{\phi }(x)|dx$ (=total
variation of the measure
$\hat{\phi }$). If the operators
$S_1,\cdot \cdot \cdot ,S_n$ are pairwise commuting, then
there is a unique spectral measure $F$ on
$\D{R}^n$ such that
$\int_{\D{R}^n} y_jF(dy)=S_j$ for each
$j$ ($F$ is the joint spectral measure of the family $\D{S}$).
In
this case one clearly has $\phi (\D{S})=\int_{\D{R}^n} \phi
(y)F(dy)$; note that this depends only on the restriction of
$\phi $ to the joint spectrum of
$\D{S}$. In particular, if
$S=S_1+iS_2$ is a normal (e.g. unitary) operator, then the
operator $\phi (\D{S})$ defined by the preceding procedure
coincides with that defined by standard functional calculus.
\begin{prop*}
Assume that $\phi $ satisfies $\int _{\D{R}^n}
\langle x\rangle^m|\hat{\phi }(x)|dx<\infty $ for some integer
$m\geq 1$. If the bounded self-adjoint operators
$S_1,\dots ,S_n$ are of class $C^m(A)$,
or $C^m_u(A)$, or $\C{C}^{s,p}(A)$ for some
$0
s$.
(ii)
An application of the lemma proved above in this paragraph
with the choices $B=A_{\varepsilon}$ and
$S=x\D{S}$ will clearly give us
\begin{align*}
\Vert \ad_{A_{\varepsilon}}^{m}(e^{ix\D{S}})\Vert
&\leq C_m
\sum_{{m_1+\dots+m_k=m,}\atop{m_1,\dots ,m_k\geq 1}}
\sum_{1\leq j_1,\dots,j_k\leq n} \Vert
x_{j_1}\ad_{A_{\varepsilon }}^{m_1}S_{j_1}\Vert\dots
\Vert x_{j_k}\ad_{A_{\varepsilon }}^{m_k}S_{j_k}\Vert \\
&\leq C\langle x\rangle^m\sum \Vert \ad_{A_{\varepsilon
}}^{m_1}S_{j_1}\Vert\dots\Vert
\ad_{A_{\varepsilon}}^{m_k}S_{j_k}\Vert .
\end{align*}
(iii)
Assume first that $S_1,\dots,S_n$ are of class
$C^m(A)$. Then, for example,
$\Vert \ad_{A_{\varepsilon}}^{m_1}S_{j_1}\Vert
\leq \text{const.}<\infty $ if
$m_1\leq m$ and
$0<\varepsilon <1$. Hence we shall have
$$
\Vert \ad_{A_{\varepsilon}}^{m}\phi (\D{S})\Vert \leq
\int_{\D{R}^n} \Vert \ad_{A_{\varepsilon }}^{m}e^{ix\D{S}}\Vert
\cdot |\hat{\phi }(x)|\leq C'\int_{\D{R}^n}
\langle x\rangle^m|\hat{\phi }(x)|dx ,
$$
from which we get $\phi (\D{S})\in C^m(A)$. If the
operators
$S_j$ are of class
$C^m_u(A)$, then the map
$\varepsilon \mapsto W^*_{\varepsilon}
\exp (ix\D{S})W_{\varepsilon}\equiv \exp \lbrack
ixW^*_{\varepsilon}\D{S}W_{\varepsilon}\rbrack $ is
clearly of class
$C^m$ in norm, for each
$x\in \D{R}^n$. Moreover, the estimate proved at
the step (ii) implies the following bound for its
derivative of order $k=0,1,\dots ,m$
with respect to $\varepsilon $ at $\varepsilon =0$:
$\Vert \ad^k_A\lbrack \exp (ix\D{S})\rbrack \Vert \leq c_k
\langle
x\rangle^k$. So by using the dominated convergence theorem we
see that
$\varepsilon \mapsto W^*_{\varepsilon}\phi
(\D{S})W_{\varepsilon}$
is norm
$C^m$, i.e.\ $\phi (\D{S})\in C^m_u(A)$.
(iv)
Finally, let us assume that $S_1, \dots ,S_n$ are of
class $\C{C}^{s,p}(A)$. By the estimate obtained at (ii) we
have
$$
\Vert \varepsilon^{m-s}\ad^m_{A_{\varepsilon}}e^{ix\D{S}}\Vert
\leq C\langle x\rangle^m\sum \Vert
\varepsilon^{m_1-s_1}\ad^{m_1}_{A_{\varepsilon}}S_{j_1}\Vert
\dots \Vert
\varepsilon^{m_k-s_k}\ad^{m_k}_{A_{\varepsilon}}S_{j_k}\Vert .
$$
Here, for each decomposition
$m=m_1+\cdot \cdot \cdot +m_k$ with
$m_j\geq 1$ integer we have chosen a decomposition
$s=s_1+\cdot \cdot \cdot +s_k$ with
$0-1$,
$\varphi_0$ of compact support, and
$\varphi_+(x)=0$ if
$x<1$).
%------------%
\subsection{} \label{ss:4.5}
The commutator $\lbrack H,iA\rbrack $ is defined as
the symmetric sesquilinear form on the domain
$D(A)\cap D(H)$ given by
$\langle f,\lbrack H,iA\rbrack f\rangle=2\Re \langle
Hf,iAf\rangle$. Now assume that $H$ is of class $C^1(A)$; then
$D(A)\cap D(H)$ is a dense subspace of
$D(H)$ (for the graph topology). Indeed, we have
$R(z)D(A)\subset D(A)$ (see \S3.9) and $R(z)$ is a
continuous surjective map
$\C{H}$ onto
$D(H)$ ($z$ does not belong to the spectrum of $H$);
since $D(A)$ is dense in $\C{H}$ we see that
$R(z)D(A)$ is a dense subspace of $D(H)$ and
$R(z)D(A)\subset D(A)\cap D(H)$.
Let $\varphi $, $\psi \in C^{\infty }_0(\D{R})$ real and such
that $x\varphi (x)=\psi (x)\varphi (x)$. Then
$\varphi (H)\in C^1(A)$, hence for
$f\in D(A)$ we have
$\varphi (H)f\in D(A)\cap D(H)$ (see \S3.9) and
$$
\langle \varphi (H)f,\lbrack H,iA\rbrack \varphi
(H)f\rangle=2\Re
\langle H\varphi (H)f,iA\varphi (H)\rangle
$$
$$
=2\Re \langle \psi (H)\varphi (H)f,iA\varphi
(H)f\rangle=\langle
\varphi (H)f,\lbrack
\psi (H),iA\rbrack \varphi (H)f\rangle .
$$
In other terms we have $\varphi (H)D(A)\subset D(A)\cap D(H)$
and
\begin{equation} \label{eq:4.6}
\varphi (H)\lbrack H,iA\rbrack \varphi (H)=\varphi
(H)\lbrack \psi (H),iA\rbrack \varphi (H)
\end{equation}
as sesquilinear forms on $D(A)$. But $\psi (H)\in C^1(A)$, so
the r.h.s.\ of (4.6) extends to the bounded operator
$\varphi (H)i\C{A}\lbrack \psi (H)\rbrack \varphi (H)$ on
$\C{H}$, hence the sesquilinear form $\varphi (H)\lbrack
H,iA\rbrack \varphi (H)$ with domain
$D(A)$ (dense in
$\C{ H}$) extends to a bounded operator, denoted
$\varphi (H)i\C{A}\lbrack H\rbrack \varphi
(H)$, on
$\C{H}$ and we have
$\varphi (H)\C{A}\lbrack H\rbrack \varphi (H)=\varphi
(H)\C{A}\lbrack \psi (H)\rbrack \varphi
(H)$.
We can now define the \emph{strict Mourre set $\mu^A(H)$ of $H$
with
respect to $A$} as the set of real numbers
$\lambda $ such that there are a real function
$\varphi\in C^{\infty }_0(\D{R})$ with $\varphi (\lambda )\not
=0$ and a strictly positive real number
$a$ such that
$\varphi (H)i\C{A}\lbrack H\rbrack \varphi
(H)\geq a\varphi (H)^2$. This is clearly an open subset
of
$\D{R}$.
In non-trivial practical situations it is impossible to find
explicitly the set $\mu^A(H)$. For this reason it is useful to
introduce the \emph{ Mourre set $\tilde{\mu}^A(H)$ of $H$ with
respect to $A$}, defined as the set of real numbers $\lambda $
for which there are a real function $\varphi \in C^{\infty
}_0(\D{R})$ with
$\varphi (\lambda )\not =0$, a strictly positive real number
$a$ and a compact operator $K$ on
$\C{H}$ such that
$\varphi (H)i\C{A}\lbrack H\rbrack \varphi (H)\geq a\varphi
(H)^2+K$. It turns out that in many
interesting cases one can describe
$\tilde {\mu}^A(H)$ rather explicitly. For this reason
the next result is important. Note that
$\tilde {\mu}^A(H)$ is an open set and $\mu^A(H)\subset
\tilde{\mu}^A(H)$.
\begin{thm*}
The set $\tilde{\mu}^A(H)\setminus \mu^A(H)$ does
not have accumulation points inside
$\tilde{\mu}^A(H)$ and it consists of eigenvalues of
$H$ of finite multiplicity. The spectrum of $H$ in $\mu^A(H)$
is purely continuous.
\end{thm*}
\begin{proof}
(i) We first show that the \emph{Virial Theorem} holds
true, namely that if $f\in D(H)$ is an eigenvector of
$H$ then $\langle f,\varphi (H)\C{A}\lbrack H\rbrack \varphi
(H)f\rangle=0$ for all $\varphi \in C^{\infty }_0(\D{R})$
real. Let
$\psi \in C^{\infty }_0(\D{R})$ real such that
$x\varphi (x)=\psi (x)\varphi (x)$. Then
\begin{align*}
\langle f,\varphi (H)\C{A}\lbrack H\rbrack \varphi (H)f\rangle
&=\langle f,\varphi (H)\C{A}\lbrack \psi (H)\rbrack
\varphi (H)f\rangle \\
& =\varphi (\lambda )^2\lim_{\varepsilon
\rightarrow 0}
\langle f,\lbrack \psi (H),A_{\varepsilon
}\rbrack f\rangle
\end{align*}
where $\lambda $ is the
eigenvalue of
$H$ associated to
$f$ and
$A_{\varepsilon}=(i\varepsilon )^{-1}(W_{\varepsilon}-1)$. But
\begin{align*}
\langle f,\lbrack \psi (H),A_{\varepsilon}\rbrack
f\rangle & =\langle
\psi (H)f,A_{\varepsilon}f\rangle-\langle f,A_{\varepsilon}\psi
(H)f\rangle \\
& =\psi (\lambda )\langle f,A_{\varepsilon}f\rangle-\psi (\lambda
)\langle f,A_{\varepsilon}f\rangle=0 ,
\end{align*}
so the virial theorem is proved.
(ii)
Now assume that $\varphi $ is a real function of class
$C^{\infty }_0(\D{R})$, $a>0$ is a real number and
$K$ is compact operator such that $\varphi (H)i\C{A}
\lbrack H\rbrack \varphi (H)\geq a\varphi
(H)^2+K$. If
$f\in D(H)$ is an eigenvector of
$H$ associated to the eigenvalue
$\lambda $ then
$0\geq a\varphi (\lambda )^2\Vert f\Vert^2+\langle
f,Kf\rangle$. It follows that for each $\varepsilon >0$ there
is at most a finite number of eigenvalues
$\lambda $ of
$H$ with
$|\varphi (\lambda )|\geq \varepsilon $ and each has
finite multiplicity. Otherwise there is an infinite
orthonormal sequence
$\lbrace f_n\rbrace $ consisting of eigenvectors with
eigenvalues $\lambda_n$ such that $|\varphi
(\lambda_n)|\geq \varepsilon $, hence
$\langle f_n,Kf_n\rangle\leq -a\varepsilon <0$; but
$\lim \langle f_n,Kf_n\rangle=0$ due to the compacity of
$K$, so we have a contradiction. If $K=0$ then clearly
there are no eigenvalues $\lambda $ of $H$ with
$\varphi (\lambda )\not =0$.
(iii)
At this stage we have shown that there are no eigenvalues in
$\mu^A(H)$ and that the eigenvalues in
$\tilde{\mu }^A(H)$ are of finite multiplicity and do not
have accumulation points inside
$\tilde{\mu }^A(H)$. It remains to be shown that the points
from $\tilde{\mu }^A(H)\setminus
\mu^A(H)$ are eigenvalues of $H$. For this it suffices to
prove the following assertion: if $\lambda
$ is not an eigenvalue of $H$ and if there are $\varphi_0\in
C^{\infty }_0(\D{R})$ real with
$\varphi_0(\lambda )\not =0$ a real number
$a_0>0$ and a compact operator $K$ such that
$\varphi_0(H)i\C{A}\lbrack H\rbrack
\varphi_0(H)\geq a_0\varphi_0(H)^2+K$, then for each
$a< a_0$ there is
$\varphi \in C^{\infty }_0(\D{R})$ real with
$\varphi (\lambda )\not =0$ and such that
$\varphi (H)i\C{A}\lbrack H\rbrack \varphi (H)\geq
a\varphi (H)^2$.
(iv)
Let us choose a real function $\psi \in C^{\infty }_0(\D{R})$
such that $\psi (x)=x$ on
$\supp \varphi_0$, and let us set
$B=i\C{A}\lbrack \psi (H)\rbrack $, so that
$B$ is a bounded self-adjoint operator. Then we have
$\varphi_0(H)i\C{A}\lbrack H\rbrack
\varphi_0(H)=\varphi_0(H)B\varphi_0(H)$ hence
$\varphi_0(H)B\varphi_0(H)\geq a\varphi_0(H)^2+K$. We can
assume that
$\varphi_0(x)=1$ on a neighbourhood of
$\lambda $ (otherwise we left and right multiply the
preceding inequality by $\eta (H)$, where $\eta
\in C^{\infty }_0(\D{R})$ is real and such that
$\eta (x)=\varphi_0(x)^{-1}$ on a neighbourhood of $\lambda $;
hence it suffices to replace
$\varphi_0$ by
$\varphi_0\eta $ and
$K$ by
$\eta (H)K\eta (H)$). Now let
$\varphi_n\in C^{\infty }_0(\D{R})$ such that
$0\leq \varphi_n\leq 1$, $\varphi_n(x)=1$ if
$|x-\lambda |\leq 2^{-n}$ and
$\varphi_n(x)=0$ if
$|x-\lambda |\geq 2^{-n+1}$. Then for
$n\in \D{N}$ large enough we have
$\varphi_n(H)B\varphi_n(H)\geq
a_0\varphi_n(H)^2+\varphi_n(H)K\varphi_n(H)$. Since
$\lambda $ is not an eigenvalue of $H$, we have
$\lim_{n\rightarrow \infty } \varphi_n(H)=0$ strongly, hence
$\Vert \varphi_n(H)K\varphi_n(H)\Vert \rightarrow 0$ as
$n\rightarrow \infty $. So there is
$n\in \D{N}$ such that
$\varphi_n(H)K\varphi_n(H)\geq a-a_0$, in particular
$\varphi_nB\varphi_n(H)\geq a_0\varphi_n(H)^2+a-a_0$. Upon
pre-and post multiplication of this inequality by
$\varphi_{n+1}(H)$ we get
$\varphi_{n+1}(H)B\varphi_{n+1}(H)\geq
a_0\varphi_{n+1}(H)^2$, and so it suffices to take
$\varphi =\varphi_{n+1}$.
\end{proof}
%------------%
\subsection{} \label{ss:4.6}
We shall say that the self-adjoint operator $H$ (or the
resolvent family $\lbrace R(z)\rbrace $ associated to it)
\emph{ has
a spectral gap} if its spectrum is not equal to $\D{R}$. This
class of operators is convenient because the study of its
resolvent can easily be reduced to the study of the resolvent
of a bounded, everywhere defined self-adjoint operator.
Indeed, let
$\lambda_0$ be a real number outside the spectrum of
$H$ and let $R=-R(\lambda_0)=(\lambda_0-H)^{-1}$. Then $R$ is
a bounded self-adjoint operator
$R:\C{H}\rightarrow \C{H}$ and for
$\Im z\not =0$
\begin{equation} \label{eq:4.7}
R(z)=(\lambda_0-z)^{-1}R\lbrack
R-(\lambda_0-z)^{-1}\rbrack^{-1}.
\end{equation}
In the rest of this paragraph we shall keep the notations
introduced above and we shall explain how
one may reduce the proof of the theorems stated in Section
1 to the proof of the corresponding
results with $H$ replaced by $R$. First we have to relate
the (strict) Mourre set of $H$ to that of
$R$.
\begin{prop*}
$H$ is of class $C^1(A)$ if and only if $R$ is of class
$C^1(A)$. A real number
$\lambda \not =\lambda_0$ belongs to
$\mu^A(H)$ (resp. $\tilde{\mu }^A(H)$) if and only if
$(\lambda_0-\lambda )^{-1}$ belongs to
$\mu^A(R)$ (resp.
$\tilde{\mu }^A(R)$).
\end{prop*}
The proof of this result is straightforward and will be not
given; see Proposition 8.3.4 in \cite{ABG2} and
note that in our context one can replace the class $C^1_u$ by
the class $C^1$ (cf.\ Propositions 7.2.5 and 7.2.7 of
\cite{ABG2} for the case of densely defined
operators).
Let us set $\zeta =(\lambda_0-z)^{-1}$. Then $z\mapsto \zeta $ is
a holomorphic diffeomorphism of $\D{C}\setminus \lbrace
\lambda_0\rbrace $ onto
$\D{C}\setminus \lbrace 0\rbrace $ which leaves the upper and
the lower half-planes invariant and restricts to a
$C^{\infty }$ diffeomorphism of
$\mu^A(H)\setminus \lbrace \lambda_0\rbrace $ onto
$\mu^A(R)\setminus \lbrace 0\rbrace $. For
$f$,
$g\in \C{H}$ and $\Im z\not =0$ (so $\Im \zeta \not =0$) we
have as a consequence of (4.7):
\begin{equation} \label{eq:4.8}
\langle g,R(z)f\rangle=\zeta \langle g,(R-\zeta
)^{-1}Rf\rangle.
\end{equation}
We can now prove Theorems B and C from \S1.7 assuming that
they are known in the case of the bounded everywhere defined
self-adjoint operator $R$. Note that $H$ is of the same class as
$R$ and that it suffices to assume that $\lambda $ is outside
a neighbourhood of
$\lambda_0$ (because $\lambda_0$ is in the resolvent set of
$H$, so $R(\cdot )$ is holomorphic on a neighbourhood of
$\lambda_0$). If $H$ is of class $\C{C}^{1,1}(A)$, then
$R\in \C{C}^{1,1}(A)$, hence it leaves invariant
$\C{H}_{1/2,1}$ (by \S3.9). So if, $f$, $g\in
\C{H}_{1/2,1}$ then
$Rf$ belongs to
$\C{H}_{1/2,1}$ and (4.8) clearly shows that Theorem B holds
for $H$ if it holds for $R$. If $H$ is of class
$\C{C}^{s+1/2}(A)$ for some
$s>1/2$, then
$R\in \C{C}^{s+1/2}(A)$, hence $Rf\in \C{H}_{s,p}$ if $f\in
\C{H}_{s,p}$ ($p =\infty $ and
$p=1$ are of interest). So we obtain Theorem C by using (4.8)
again (for the class $\Lambda^{s-1/2}$ we may use the uniform
boundedness principle, cf.\ the end of \S1.5).
Now assume that we are in the conditions of Theorem D of
\S1.8. By (4.7) and by what we have seen before we clearly
have for
$\lambda \in \mu^A(H)\setminus \lbrace
\lambda_0\rbrace $:
$$
\Pi_-R(\lambda +i0)=(\lambda_0-\lambda )^{-1}\Pi_-\lbrack
R-(\lambda_0-\lambda )^{-1}-i0)\rbrack^{-1}R.
$$
Since $R\in B(\C{H}_{s,p})$ we clarly obtain the result stated
in Theorem D (assuming that it holds for
$R$), and similarly for Theorem E of \S1.8.
The argument is slightly more involved in the case of Theorem
G of \S1.10 (or, more generally, Theorems 6.5 and 6.8). We
write (here $\Pi_{\pm}$ are as in \S3.10):
$$
\langle \Pi_-g,R(z)\Pi_+f\rangle=\zeta \langle \Pi_-g,(R-\zeta
)^{-1}R\Pi_+f\rangle
$$
$$
=\zeta \langle \Pi_-g,(R-\zeta
)^{-1}\Pi_+R\Pi_+f\rangle+\zeta \langle \Pi_-g,(R-\zeta
)^{-1}\Pi_-R\Pi_+f\rangle .
$$
The first term in the last member here is treated exactly as
before (i.e.\ Theorem G applies directly, because
$R\Pi_+f\in\C{H}_{s,\infty }$ if
$f\in \C{H}_{s,\infty }$). For the last term we first use the
theorem from \S3.10. The operator $R$ belongs to
$\C{C}^{2+t-s}(A)$ with
$s<1/2$, $t>-1/2$ (cf. Theorem G). Since $-(2+t-s)0$ such that the following condition is satisfied: there is
an open set
$J_0$ with $\dist(J,\D{R}\setminus J_0)\equiv \inf \lbrace
|x-y| \mid x\in J, y\not \in J_0\rbrace =\delta >0$ and there
is a number
$a_0>a$ such that
$E(J_0)i\C{A}\lbrack H\rbrack E(J_0)\geq a_0E(J_0)$.
Our first result contains a version of the so-called
quadratic estimate of Mourre; see \cite{M1},
\cite{ABG1,ABG2},
\cite{BG3}.
\begin{prop} \label{prop:5.1}
Let $\lbrace
H_{\varepsilon}\rbrace_{\varepsilon \geq 0}$ be a family of
bounded operators in
$\C{H}$ such that
$H_0=H$,
$\Vert H_{\varepsilon}-H\Vert \rightarrow 0$ and
$\Vert \varepsilon^{-1}\Im H_{\varepsilon}+i\C{A}\lbrack
H_{\varepsilon}\rbrack \Vert \rightarrow 0$
as
$\varepsilon \rightarrow 0$. Then there are strictly
positive numbers $\varepsilon_0, b$ such that, for
each
$\varepsilon \in \lbrack 0,\varepsilon_0\rbrack $ and each
$z\in \D{C}$ with
$\Re z\in J$ and
$\Im z>-a\varepsilon $, the operator
$H_{\varepsilon}-z:\C{H}\rightarrow \C{H}$ is bijective
and its inverse
$G_{\varepsilon}=G_{\varepsilon}(z)=(H_{\varepsilon}-z)^{-1}\in
B(\C{H})$ satisfies the estimates
\begin{equation} \label{eq:5.1}
\Vert G^{(\pm )}_{\varepsilon}f\Vert^2
\leq \pm \frac{1}{a\varepsilon +\Im z} \Im
\langle f,G_{\varepsilon}f\rangle+\frac{b\varepsilon
}{(a\varepsilon +\Im z)\lbrack \delta^2+(\Im z)^2\rbrack }
\Vert f\Vert^2
\end{equation}
for all $f\in \C{H}$.
We have set
$G^{(+)}_{\varepsilon}=G_{\varepsilon}$, $G^{(-)}_{\varepsilon
}=G^*_{\varepsilon}$.
In particular, one has
\begin{equation} \label{eq:5.2}
\Vert G_{\varepsilon}(z)\Vert
\leq \frac{1}{a\varepsilon +\Im
z}+\left\lbrack
\frac{b\varepsilon }{(a\varepsilon +\Im
z)\lbrack\delta^2+(\Im z)^2\rbrack }\right\rbrack^{1/2}.
\end{equation}
\end{prop}
The following consequences of the inequalities (5.1) and (5.2)
will be especially useful later on:
if $\Im z\geq 0$ then for
$0<\varepsilon \leq \varepsilon_0$ one has
\begin{equation} \label{eq:5.3}
\Vert G^{(\pm )}_{\varepsilon}f\Vert^2
\leq \pm \frac{1}{a\varepsilon } \Im
\langle f,G_{\varepsilon}f\rangle+\frac{b}{a\delta^2}\Vert
f\Vert^2 ,
\end{equation}
\begin{equation} \label{eq:5.4}
\Vert G_{\varepsilon}\Vert
\leq \frac{1}{a\varepsilon } +
\left(\frac{b}{a\delta^2}\right)^{1/2} .
\end{equation}
\begin{proof}[Proof of Proposition 5.1]
(i)
We first establish a preliminary
estimate involving the bounded everywhere defined self-adjoint
operator $S=i\C{A}\lbrack H\rbrack $. Let $\nu $ be a strictly
positive real number and let us set $P=1-E(J_0)$. Since $\pm
2\Re C\leq \nu +\nu^{-1}C^*C$ holds for all bounded
operators
$C$, we have
$$
a_0E(J_0)\leq (1-P)S(1-P)=S-2\Re (SP)+PSP\leq S+\nu
+P(S+\nu^{-1}S^2)P .
$$
Hence
$$
a_0-\nu =a_0E(J_0)-\nu +a_0P\leq S+P(a_0+S+\nu^{-1}S^2)P\leq
S+\Vert a_0+S+\nu^{-1}S^2\Vert P .
$$
If $\Re z\in J $ then $P|H-z|^{-2}$ is a bounded operator with
norm smaller than $d(z)^{-2}$, where
$d(z)={\rm dist} (z,\D{R}\setminus J_0)\geq \lbrack \delta^2+(\Im
z)^2\rbrack^{1/2}$. So, by writing
$P=P|H-z|^{-2}|H-z|^2$, we get for all
$f\in \C{H}$:
\begin{equation} \label{eq:5.5}
(a_0-\nu )\Vert f\Vert^2\leq \langle
f,Sf\rangle+\frac{\Vert a_0+S+\nu^{-1}S^2\Vert }{\delta^2+(\Im
z)^2}\Vert (H-z)f\Vert^2
\end{equation}
(ii)
Now let us set $z=\lambda +i\mu $ with $\lambda \in J$ and
$\mu \in \D{R}$ and let $C=C(\nu ,\mu )=\Vert
a_0+S+\nu^{-1}S^2\Vert (\delta^2+\mu^2)^{-1}$. Then for an
arbitrary operator $K\in B(\C{H})$ and an arbitrary
self-adjoint operator
$T\in B(\C{H})$ we have as a consequence of (5.5):
$$
(a_0-\nu )\Vert f\Vert^2\leq \langle f,Tf\rangle+\langle
f,(S-T)f\rangle+2C\Vert (K-z)f\Vert^2+2C\Vert (H-K)f\Vert^2 .
$$
Since $C(\nu ,\mu )\leq C(\nu ,0)$, we get
$$
\bigl[ a_0-\nu -\Vert S-T\Vert -2\delta^{-2}
\Vert a_0+S+\nu^{-1}S^2\Vert \cdot \Vert H-K\Vert
\bigr]\cdot \Vert f\Vert^2\leq \langle f,Tf\rangle+2C\Vert
(K-z)f\Vert^2 .
$$
We define $H_{\varepsilon}=H^*_{-\varepsilon }$ if
$\varepsilon <0$ and we take $K=H_{\varepsilon}$
and
$T=\varepsilon^{-1}\Im H^*_{\varepsilon}$ with
$\varepsilon \not =0$. Since
$H_{\varepsilon}\rightarrow H$ and
$\varepsilon^{-1}\Im H^*_{\varepsilon}\rightarrow S$
in norm as
$\varepsilon \rightarrow 0$, by choosing first a
small enough number $\nu >0$ and then $\varepsilon_0>0$, we
obtain for
$\varepsilon \in \D{R}$,
$0<|\varepsilon |\leq \varepsilon_0$:
\begin{equation} \label{eq:5.6}
a\Vert f\Vert^2\leq \varepsilon^{-1}
\Im \langle
H_{\varepsilon}f,f\rangle+2C\Vert (H_{\varepsilon}-z)f\Vert^2 .
\end{equation}
For $0\leq \varepsilon \leq \varepsilon_0$
we then get
\begin{equation} \label{eq:5.7}
(a\varepsilon +\mu )\Vert f\Vert^2
\leq \Im \langle
(H_{\varepsilon}-z)f,f\rangle+2C\varepsilon \Vert
(H_{\varepsilon }-z)f\Vert^2 .
\end{equation}
Now let us consider (5.6)
with $\varepsilon $ replaced by $-\varepsilon $ and $\mu $ by
$-\mu $. Then, again for
$0\leq \varepsilon \leq \varepsilon_0 $,
we obtain:
\begin{equation} \label{eq:5.8}
(a\varepsilon +\mu )\Vert f\Vert^2\leq -\Im \langle
(H_{\varepsilon}-z)^*f,f\rangle+2C\varepsilon \Vert
(H_{\varepsilon}-z)^*f\Vert^2 .
\end{equation}
Until now $\mu $ was arbitrary. If $a\varepsilon +\mu >0$,
then (5.7) implies that $\Vert
(H_{\varepsilon}-z)f\Vert \geq \text{const.}\Vert f\Vert $ for
some strictly positive constant and all $f\in
\C{H}$, so
$H_{\varepsilon}-z$ is injective with closed range.
Since by (5.8), its adjoint operator is also
injective, we get that
$H_{\varepsilon}-z:\C{H}\rightarrow \C{H}$ is bijective
and bounded, so its inverse
$G_{\varepsilon}$ is also bounded.
We obtain (5.1) with
$b=2\Vert a_0+S+\nu^{-1}S^2\Vert $ if we replace
$f$ in (5.7) and (5.8) by $G_{\varepsilon}f$ and
$G^*_{\varepsilon}f$ respectively. Finally, (5.1)
implies (5.2) because from (5.1) we get
$$
\Vert G_{\varepsilon}\Vert^2\leq \frac{1}{a\varepsilon +\mu
}\Vert G_{\varepsilon}\Vert +\frac{b\varepsilon
}{(a\varepsilon +\mu )(\delta^2+\mu^2)} .
$$
\end{proof}
Now let us assume that the family $\lbrace
H_{\varepsilon}\rbrace $ from Proposition 5.1 has two more
properties:
(1)
$H_{\varepsilon}$ is of class $C^1(A)$ if $0<\varepsilon
<\varepsilon_0 $;
(2)
the map $\varepsilon \mapsto H_{\varepsilon}\in B(\C{H})$ is
strongly $C^1$ on
$(0,\varepsilon_0)$.
\\
Let $z$ be a complex number with
$\Re z\in J$ and
$\Im z\geq 0$ and let $0<\varepsilon <\varepsilon_0$. Then
$G_{\varepsilon}\in C^1(A)$ and $\C{A}\lbrack
G_{\varepsilon}\rbrack =-G_{\varepsilon}\C{A}\lbrack
H_{\varepsilon}\rbrack G_{\varepsilon}$. Indeed, if for
$\tau \not =0$ we set
$A_{\tau }=(i\tau )^{-1}(e^{iA\tau }-1)$ then we clearly have
$\lbrack A_{\tau },G_{\varepsilon}\rbrack
=G_{\varepsilon}\lbrack H_{\varepsilon},A_{\tau }\rbrack
G_{\varepsilon}$ and the result follows by taking the limit as
$\tau \rightarrow 0$ and by using, for example, the fact that
$\lbrack H_{\varepsilon},A_{\tau }\rbrack \rightarrow
\C{A}\lbrack H_{\varepsilon}\rbrack $ strongly as
$\tau \rightarrow 0$. Furthermore, the map $\varepsilon
\mapsto G_{\varepsilon}\in B(\C{H})$ is strongly
$C^1$ on
$(0,\varepsilon_0 )$ and its derivative is given by
$G'_{\varepsilon}\equiv \frac{d}{d\varepsilon
}G_{\varepsilon}=-G_{\varepsilon}H'_{\varepsilon}G_{\varepsilon}$
(this is an easy consequence of (5.4)).
In particular we get
\begin{equation} \label{eq:5.9}
G'_{\varepsilon}=
\C{A}\lbrack G_{\varepsilon}\rbrack
+G_{\varepsilon}(\C{A}\lbrack H_{\varepsilon}\rbrack
-H'_{\varepsilon})G_{\varepsilon} .
\end{equation}
This equation plays a fundamental role in the theory.
In this paper we shall choose
$H_{\varepsilon}$ of the form
$H=\xi (\varepsilon \C{A})H$ where $\xi :\D{R}\rightarrow
\D{C}$ is a function such that the preceding expression makes
sense, at least for small enough
$\varepsilon $ (notice that only the behaviour of
$H_{\varepsilon}$ as $\varepsilon \rightarrow 0$ matters).
Other choices for $H_{\varepsilon}$ are sometimes convenient
but will not be considered here (see
\cite{ABG1},
\cite{BGM}). Let us see what conditions should $\xi $ satisfy
for $H_{\varepsilon}$ to have the properties required in
Proposition 5.1. For $H_0=H$ we demand that $\xi (0)=1$. Then,
at least formally, we have
$H^*_{\varepsilon}=\xi^+(\varepsilon \C{A})H$, hence
$2i\Im H^*_{\varepsilon}=\lbrack
\overline{\xi }(-\epsilon \C{A})-\xi (\varepsilon \C{A})\rbrack H$.
So the condition
$\lim_{\varepsilon \rightarrow 0}\varepsilon^{-1}\Im H^{\star
}_{\varepsilon}=i\C{A}H$ is formally satisfied if
$\xi $ is of class
$C^1$ and
$\Re \xi' (0)=1$. For simplicity we would also like to have
$H^*_{\varepsilon
}=H_{-\varepsilon }$ (see the Proposition 5.1), which
formally follows from
$\xi^+(x)=\xi (-x)$, i.e.\ $\xi $ should be real.
In conclusion, if we take $H_{\varepsilon}=\xi (\varepsilon
\C{A})H$, then the function $\xi $ on
$\D{R}$ has to be real and to satisfy
$\xi (0)=\xi'(0)=1$. Then, again formally, we have
$$
\C{A}H_{\varepsilon}-H'_{\varepsilon}=\C{A}\xi
(\varepsilon \C{A})H-\frac{d}{d\varepsilon }\xi
(\varepsilon \C{A})H=\C{A}\xi (\varepsilon
\C{A})H-\C{A}\xi'(\varepsilon\C{A})H=\frac{1}{\varepsilon
}\eta (\varepsilon \C{A})H ,
$$
where $\eta (x)\equiv x\xi (x)-x\xi' (x)=O(x^2)$
as
$x\rightarrow 0$. So (5.9) becomes
\begin{equation} \label{eq:5.10}
G'_{\varepsilon}=\C{A}\lbrack G_{\varepsilon}\rbrack
+\varepsilon^{-1}G_{\varepsilon}\eta (\varepsilon
\C{A})\lbrack H\rbrack G_{\varepsilon} .
\end{equation}
We shall now give three examples, which are not relevant
for our approach, but explain the
constructions from \cite{M1,M2} and \cite{JMP}.
(i)
Assume $H\in C^2(A)$. Then one may take $\xi (x)=1+x$, which
gives $\eta (x)= x^2$, hence
\begin{equation} \label{eq:5.11}
G'_{\varepsilon}=
\lbrack G_{\varepsilon},A\rbrack +\varepsilon
G_{\varepsilon}\lbrack A,\lbrack A,H\rbrack
\rbrack G_{\varepsilon}.
\end{equation}
(ii) Let $H\in C^k(A)$ for some integer $k\geq 2$.
Then one can take $\xi
(x)=\sum^{k-1}_{j=0}x^j/j!$, hence
$\eta (x)=x^k/(k-1)!$, so
\begin{equation} \label{eq:5.12}
G'_{\varepsilon}
=\lbrack G_{\varepsilon},A\rbrack +\frac{\varepsilon
^{k-1}}{(k-1)!}G_{\varepsilon }\C{A}^k\lbrack H\rbrack
G_{\varepsilon}.
\end{equation}
Notice that for $k=3$ the second term in the r.h.s.\ of
the preceding identity is bounded as
$\varepsilon \rightarrow 0$, while for $k>3$ it is
an $O(\varepsilon^{k-3})$, so it vanishes as $\varepsilon
\rightarrow 0$. Moreover, and this is the main fact,
these estimates are independent of $z$ (with
$\Re z\in J$ and
$\Im z\geq 0$) as follows from (5.4).
(iii)
The best choice can be made if $H$ is $A$-analytic: then we
take $\xi (x)=e^x$, so that $\eta =0$. Note that this time the
expession $H_{\varepsilon}=e^{\varepsilon
\C{A}}H=e^{-\varepsilon
\C{A}}He^{\varepsilon \C{A}}$ has a meaning only if
$|\varepsilon |$ is small enough (unless
$H$ is
$A$-entire). Then we have
$G'_{\varepsilon}=\C{A}G_{\varepsilon}$. This situation
appears in the theory of dilation-analytic hamiltonians
\cite{AC},
\cite{BC}.
Our choice for $H_{\varepsilon}$ is related to (iii): the
point is that $H$ being non-analytic in general, we shall have
to regularize it first according to the general procedure
described in \S 3.6. Let $\theta \in C^{\infty }_0(\D{R})$ be
a real even function with $\theta (x)=1$ on a neighbourhood of
zero. From now on in this section we take $\xi (x)= e^x\theta
(x)$ and $H_{\varepsilon }=\xi (\varepsilon \C{A})H$ for all
$\varepsilon \in \D{R}$. Note that the operator
$H_{\varepsilon}$ is not self-adjoint in general, but we have
$H^*_{\varepsilon}=H_{-\varepsilon }$. The function
$\eta $ which appears in (5.10) is now given by
$\eta (x)=x(\xi (x)-\xi '(x))=-e^xx\theta '(x)$, so that
$\eta \in C^{\infty }_0(\D{R}\setminus \lbrace 0\rbrace )$.
The fact that
$0 $ does not belong to the support of $\eta $ is quite
important for what follows.
It is convenient to have in mind a slightly different
expression for $H_{\varepsilon}$. Set
$H^{\varepsilon}=\theta (\varepsilon \C{A})H$ for
$\varepsilon \in \D{R}$. Then
$H^{\varepsilon}$ is a self-adjoint operator and for
$\varepsilon \not =0$ the operators $H^{\varepsilon}$
and
$H_{\varepsilon}$ are
$A$-entire and are related by
$H_{\varepsilon}=e^{\varepsilon \C{A}}H^{\varepsilon}$
(see the end of \S 3.5).
It is not yet clear whether the so-called family $\lbrace
H_{\varepsilon}\rbrace $ satisfies or not the hypotheses of
Proposition 5.1. In fact it does not if $H$ is only of class
$C^1(A)$, as we explain in the next proposition:
\begin{prop} \label{prop:5.2}
The family
$\lbrace H_{\varepsilon}\rbrace_{\varepsilon \in \D{R}}$
defined above satisfies the hypotheses of Proposition $5.1$ if
and only if the operator $H$ is of class $C^1_u(A)$. Assume
that $H\in C^1_u(A)$ and let
$z\in \D{C}$ with $\Re z\in J$ and
$\Im z>0$.
\textup{(a)}
For $0\leq \varepsilon \leq \varepsilon_0$ one has
$G_{\varepsilon}\in C^1_u(A)$ and $\C{A}\lbrack
G_{\varepsilon}\rbrack =-G_{\varepsilon}\C{A}\lbrack
H_{\varepsilon}\rbrack G_{\varepsilon}$; if
$0<\varepsilon <\varepsilon_0$ then
$G_{\varepsilon}\in C^{\infty }(A)$.
\textup{(b)}
The map $\varepsilon \mapsto H_{\varepsilon}$
is of class $C^1$ in norm on $\D{R}$ and is of class
$C^{\infty }$ on
$\D{R}\setminus \lbrace 0\rbrace $. The map
$\varepsilon \mapsto G_{\varepsilon}$ is of class
$C^1$ in norm on the closed interval
$\lbrack 0,\varepsilon_0\rbrack $,
where its derivative is given by
$G'_{\varepsilon}=
-G_{\varepsilon}H'_{\varepsilon}G_{\varepsilon}$,
and is of class
$C^{\infty }$ on
$(0,\varepsilon_0\rbrack $.
\textup{(c)}
Set $K_{\varepsilon}=\varepsilon^{-1}\eta
(\varepsilon\C{A})H$ for $\varepsilon \not =0$,
where $\eta \in
C^{\infty }_0(\D{R}\setminus \lbrace 0\rbrace )$
is given by
$\eta (x)=-e^xx\theta '(x)$. Then $K_{\varepsilon}
\in C^{\infty }(A)$,
$\varepsilon \mapsto K_{\varepsilon}$ is of class
$C^{\infty }$ on $\D{R} \setminus \lbrace 0\rbrace $,
and for each $0< \varepsilon \leq \varepsilon_0 $
one has
\begin{equation} \label{eq:5.13}
G'_{\varepsilon}=\C{A}\lbrack G_{\varepsilon}\rbrack
+G_{\varepsilon}K_{\varepsilon}G_{\varepsilon} .
\end{equation}
\textup{(d)}
Set $K^{(j)}_{\varepsilon}=(d/d\varepsilon )^jK_{\varepsilon}$
and let $\alpha >-1$ real and $p\in
\lbrack 1,\infty \rbrack $. Then
$H$ is of class $\C{C}^{1+\alpha ,p}(A)$
if and only if the condition
\begin{equation} \label{eq:5.14}
\left\lbrack \int^1_0 \Vert
\varepsilon^{-\alpha
+j}K^{(j)}_{\varepsilon}\Vert^p\varepsilon^{-1}d\varepsilon
\right\rbrack^{1/p}<\infty
\end{equation}
holds for $j=0$.
If this is the case then $(5.14)$ holds for each integer $j\geq
0$.
\end{prop}
\begin{proof}
We define a real even function $\rho \in C^{\infty
}_0(\D{R})$ by $\rho (0)=1$ and $\rho (x) =x^{-1}\sinh x\cdot
\theta (x)$ if $x\not =0$. Then for an arbitrary bounded
self-adjoint operator
$H$ we have
$\varepsilon^{-1}\Im H^*_{\varepsilon}=i\C{A}\rho
(\varepsilon \C{A})H\equiv S_{\varepsilon}$ (see
the Proposition from \S 3.5).
Assume first that $\lim_{\varepsilon \rightarrow 0}
S_{\varepsilon}$ exists in norm in $B(\C{H})$ and denote by
$S$ the limit. Since
$C^{\infty }(A)$ is a subspace of the norm-closed space
$C^0_u(A)$ and $S_{\varepsilon}\in
C^{\infty }(A)$ if $\varepsilon \not =0$, we get
$S\in C^0_u(A)$. For
$f\in D(A)$ we have
$$
\langle f,S_{\varepsilon}f\rangle=\langle f,\lbrack \rho
(\varepsilon
\C{A})H,iA\rbrack f\rangle =2\Re \langle (\rho
(\varepsilon\C{A})H)f,iAf\rangle
$$
which converges to $2\Re \langle Hf,iAf\rangle$ as $\varepsilon
\rightarrow 0$. So we have $2\Re \langle Hf,iAf\rangle=\langle
f,Sf\rangle$ for
$f\in D(A)$, i.e.\
$i\C{A}H=S\in C^0_u(A)$. This clearly means
$H\in C^1_u(A)$ (see \S 3.3).
Reciprocally, if $H\in C^1_u(A)$ then $H$ is of class
$C^0_u(A)$ hence $\Vert H_{\varepsilon}-H\Vert
\rightarrow 0$ as
$\varepsilon \rightarrow 0$. Moreover, we shall also have
$S_{\varepsilon}=i\rho (\varepsilon \C{A})\C{A}H$
(see \S 3.5) and $\C{A}H\in C^0_u(A)$, so
$\Vert S_{\varepsilon}-i\C{A}H\Vert \rightarrow 0$ as
$\varepsilon\rightarrow 0$. Hence the family
$\lbrace H_{\varepsilon}\rbrace_{\varepsilon \geq 0}$
satisfies the hypotheses of Proposition 5.1.
The proof of the assertions (a), (b) and (c) is easy, see the
arguments which follow the proof of Proposition 5.1. For part
(d) we use the Theorem from \S 3.7. Observe that $\eta \in
C^{\infty }_0(\D{R}\setminus \lbrace 0\rbrace )$ is not
identically zero on $(-\infty ,0)$ and on
$(0,\infty )$, so if (5.14) holds with
$j=0$ then
$H\in \C{C}^{1+\alpha ,p}(A)$. Reciprocally,
if $H$ has this property then we have (5.14) for all
$j$ because
$\varepsilon^jK^{(j)}_{\varepsilon}=\eta_j(\varepsilon \C{A})H$
for some
$\eta_j\in C^{\infty }_0(\D{R}\setminus \lbrace 0\rbrace )$.
\end{proof}
The next proposition contains one of the basic estimates of
the theory. In the proof we use Mourre's method of
differential inequalities (see \cite{M1}, \cite{M2})
together with a version of the Gronwall
lemma obtained in \cite{BGM}.
We first introduce some notations and make some conventions
for the rest of this section. We denote by $\Vert |\cdot \Vert
|$ either the norm in the Banach space $\C{K}=\C{H}_{1/2,1}$
or the norm associated to it in
$B(\C{K};\C{K}^*)$, and we recall that we have
continuous embeddins $\C{K}\subset
\C{H}\subset \C{K}^*$ and
$B(\C{H})\subset B(\C{K};\C{K}^*)$.
\emph{From now on we assume that
$H$ is (at least) of class $\C{C}^{1,1}(A)$.}
We write $z=\lambda +i\mu $ and the numbers
$\varepsilon $,
$\lambda $,
$\mu $ are supposed to verify
$0<\varepsilon <\varepsilon_0$, $\lambda \in J$,
$\mu >0$. One shoukd think of $\mu $ rather as a parameter,
but it is important that the various
constants that appear below are independent of
$\mu $. If $F$ is a function of $(\lambda ,\varepsilon )
\in J\times (0, \varepsilon_0)$ we denote by
$F^{(k,m)}\equiv \partial^k_{\lambda }
\partial^m_{\varepsilon}F$ its derivative of order
$k$ with respect to
$\lambda $ and of order $m$ with respect
to $\varepsilon $. We also set $F^{(m)}=F^{(0,m)}$. The operator
$G_{\varepsilon}=G_{\varepsilon}(z)=
G_{\varepsilon}(\lambda +i\mu )$ will be considered as a function
of
$(\lambda ,\varepsilon )\in J\times (0, \varepsilon_0)$;
we clearly have for $k\in \D{N}$:
\begin{equation} \label{eq:5.15}
G^{(k,0)}_{\varepsilon}=\partial^k_{\lambda }
G_{\varepsilon}(\lambda +i\mu
)=k!G^{k+1}_{\varepsilon} .
\end{equation}
\begin{prop} \label{prop:5.3}
If $H$ is of class $\C{C}^{1,1}(A)$
then for each $k$, $m\in \D{N}$ there is a
number
$C<\infty $, independent of
$\varepsilon \in (0,\varepsilon_0)$,
$\lambda \in J$ and $\mu >0$, such that
\begin{equation} \label{eq:5.16}
|\Vert G^{(k,m)}_{\varepsilon}|\Vert \leq C\varepsilon^{-k-m} ,
\end{equation}
\begin{equation} \label{eq:5.17}
\Vert G^{(k,m)}_{\varepsilon}\Vert_{\C{K}
\rightarrow \C{H}}+\Vert
G^{(k,m)}_{\varepsilon}\Vert_{\C{H}
\rightarrow \C{K}^*}\leq C\varepsilon^{-k-m-1/2} .
\end{equation}
\end{prop}
\begin{proof}
(i) We first prove (5.16), (5.17) in the case
$k=m=0$. Fix a number $\varepsilon_1\in \lbrack
0,\varepsilon_0)$ and a family
$\lbrace F_{\varepsilon}\rbrace_{\varepsilon_1<\varepsilon
\leq \varepsilon_0}$ of vectors in $D(A)$ such that
the function
$\varepsilon \mapsto f_{\varepsilon}\in \C{H}$ is of class
$C^1$. We set
$F_{\varepsilon}=\langle f_{\varepsilon},G_{\varepsilon
}f_{\varepsilon}\rangle$ for
$\varepsilon_1<\varepsilon \leq \varepsilon_0$ and we
get by using (5.13):
$$
F'_{\varepsilon
}=\langle f'_{\varepsilon}-Af_{\varepsilon
},G_{\varepsilon}f_{\varepsilon}\rangle+\langle G^{*
}_{\varepsilon}f_{\varepsilon},f'_{\varepsilon
}+Af_{\varepsilon}\rangle+\langle G^{*
}_{\varepsilon}f_{\varepsilon},K_{\varepsilon}G_{\varepsilon
}f_{\varepsilon}\rangle .
$$
Denote $l_{\varepsilon}=\Vert f'_{\varepsilon}\Vert +\Vert
Af_{\varepsilon}\Vert $. Then (5.3) implies
$$
|F'_{\varepsilon}|\leq l_{\varepsilon}(\Vert G_{\varepsilon}
f_{\varepsilon}\Vert +\Vert
G^*_{\varepsilon}f_{\varepsilon}\Vert )+\Vert K_{\varepsilon}
\Vert \cdot
\Vert G_{\varepsilon}f_{\varepsilon}\Vert \cdot \Vert
G^*_{\varepsilon}f_{\varepsilon
}\Vert
$$
$$
\leq 2l_{\varepsilon}a^{-1/2}(\varepsilon^{-1/2}
|F_{\varepsilon}|^{1/2}+b^{1/2}\delta^{-1}\Vert
f_{\varepsilon}\Vert )+\Vert K_{\varepsilon}\Vert
a^{-1}(\varepsilon^{-1}|F_{\varepsilon}|+b\delta^{-2}\Vert
f_{\varepsilon}\Vert^2).
$$
So there is a constant
$c>0$, depending only on $a$,
$b$ and
$\delta $, such that for
$\varepsilon_1<\varepsilon \leq \varepsilon_0$:
$$
c^{-1}|F'_{\varepsilon}|\leq l_{\varepsilon}
\Vert f_{\varepsilon}\Vert +\Vert K_{\varepsilon}
\Vert \cdot \Vert
f_{\varepsilon}\Vert^2+l_{\varepsilon}
\varepsilon^{-1/2}|F_{\varepsilon}|^{1/2}+
\Vert K_{\varepsilon}\Vert
\varepsilon^{-1}|F_{\varepsilon}| .
$$
According to Proposition
3.1 from
\cite{BGM} the preceding estimate implies
\begin{align} \label{eq:5.18}
|F_{\varepsilon_1}| & \leq
2\Bigl\{
|F_{\varepsilon_0}|+c\int^{\varepsilon_0}_{\varepsilon_1}
\lbrack l_{\tau }\Vert f_{\tau }\Vert +\Vert K_{\tau }\Vert
\cdot
\Vert f_{\tau }\Vert^2\rbrack d\tau
\\ &
+c^2\Bigl[ \int^{\varepsilon_0}_{\varepsilon_1}
l_{\tau }\tau^{-1/2}d\tau
\Bigr]^2\Bigr\} \exp
\int^{\varepsilon_0}_{\varepsilon_1} c\Vert
K_{\tau }\Vert \tau^{-1}d\tau .\notag
\end{align}
By Proposition 5.2 (d) we have
$\int^{\varepsilon_0}_0 \Vert K_{\tau }\Vert
\tau^{-1}d\tau $ if and only if
$H\in \C{C}^{1,1}(A)$. Now let
$f_{\varepsilon}\in \C{H}_{1/2,1}$ and
$f_{}=\theta ((\varepsilon -\varepsilon_1)A)f$,
with the same function
$\theta $ as in the definition of $H_{\varepsilon}$.
If we set $\tilde{\theta }(x)=x\theta' (x)$ and
$\theta_{(1)}(x)=x\theta (x)$, then
$$
\int^{\varepsilon_0}_{\varepsilon_1} l_{\tau }
\tau^{-1/2}d\tau =\int^{\varepsilon_0-\varepsilon_1}_{0} (\Vert
\tilde{\theta }(\sigma A)f\Vert +\Vert
\theta_{(1)}(\sigma A)f\Vert )\frac{d\sigma }
{\sigma (\sigma +\varepsilon_1)^{1/2}}
$$
$$
\leq c'\Vert
f\Vert_{\C{H}_{1/2,1}}=c'|\Vert f|\Vert
$$
where $c'$ is a finite constant depending
only on
$\varepsilon_0$ and
$\theta $. Now by using (5.18) we easily see that
there is a constant
$c''<\infty $ such that
$|\langle f,G_{\varepsilon}f\rangle|\leq c''|\Vert f|\Vert^2$
for
$0<\varepsilon \leq \varepsilon_0$,
$\lambda \in J$,
$\mu >0$ and
$f\in \C{K}$. The polarization identity will then give
$|\Vert G|\Vert \leq \text{const}$. Finally the estimate
(5.17) with $k=m=0$ is an immediate consequence of the
preceding one and of (5.3).
(ii)
Now we treat the case where one of the numbers $k$, $m$ is not
zero. If $m=0$ then the estimates follow easily from those
with $k=m=0$ by taking into account (5.4) and (5.15), so we
can assume
$m\geq 1$. Then by Proposition 5.2 (b) the operator
$G^{(m)}_{\varepsilon}$ is a linear combination of terms of
the form
$$
G_{\varepsilon}H^{(m_1)}_{\varepsilon}G_{\varepsilon}
H^{(m_2)}_{\varepsilon} \dots G_{\varepsilon
}H^{(m_n)}_{\varepsilon}
$$
with $m_1,\dots ,m_n\geq 1$ integers and
$m_1+ \dots +m_n=m$. So from (5.15) it follows that
$G^{(k,m)}_{\varepsilon}$ is a linear combination of terms
of the form
$$
G^{k_0+1}_{\varepsilon}H^{(m_1)}_{\varepsilon}
G^{k_1+1}_{\varepsilon}H^{(m_2)}_{\varepsilon}
G^{k_2+1}_{\varepsilon}\dots
H^{(m_n)}_{\varepsilon}G^{k_n+1}_{\varepsilon}
$$
with $m_1,\dots ,m_n$ as above and
$k_0,k_1,\dots ,k_n\in \D{N}$ such that
$k_0+k_1+\dots +k_n=k$. The norm in
$B(\C{K};\C{K}^*)$ of such a term is bounded by
$$
\Vert G_{\varepsilon}\Vert_{\C{H}\rightarrow
\C{K}^*}\Vert G_{\varepsilon}\Vert^{k_0}\Vert
H^{(m_1)}_{\varepsilon}\Vert
\cdot
\Vert G_{\varepsilon}\Vert^{k_1+1}\dots
\Vert H^{(m_n)}_{\varepsilon}\Vert \cdot \Vert G_{\varepsilon
}\Vert^{k_n}
\Vert G_{\varepsilon}\Vert_{\C{K}\rightarrow \C{H}}
$$
$$
\leq \text{const.} \varepsilon^{-1/2}\cdot
\varepsilon^{-k_0}\Vert H^{(m_1)}_{\varepsilon}\Vert \cdot
\varepsilon^{-k_1-1}
\dots
\Vert H^{(m_n)}_{\varepsilon}\Vert \varepsilon^{-k_n}
\cdot \varepsilon^{-1/2}
$$
where we have used (5.17) with
$k=m=0$ and (5.4). Similarly, the norm in
$B(\C{K};\C{H})$ is bounded by
$$
\Vert G_{\varepsilon}\Vert^{k_0+1}\Vert
H^{(m_1)}_{\varepsilon}\Vert \cdot \Vert
G_{\varepsilon}\Vert^{k_1+1}\dots
\Vert H^{(m_n)}_{\varepsilon}\Vert\cdot \Vert
G_{\varepsilon}\Vert^{k_n}\Vert G_{\varepsilon
}\Vert_{\C{K}\rightarrow \C{H}}
$$
$$
\leq \text{\text{const.}}
\varepsilon^{-k_0-1}\Vert H^{(m_1)}_{\varepsilon}\Vert\cdot
\varepsilon^{-k_1-1}\dots\Vert
H^{(m_n)}_{\varepsilon}\Vert
\cdot
\varepsilon^{-k_n}\cdot \varepsilon^{-1/2} .
$$
We see that the assertions of the proposition are a
consequence of the estimate
$\Vert H^{(m)}_{\varepsilon}\Vert
\leq c_m\varepsilon^{1-m}$ for
$m\geq 1$ integer and $\varepsilon >0$. But we have
(see (3.7), the end of \S 3.6 and the Proposition from
\S3.5):
$$
H^{(m)}_{\varepsilon}=\partial^m_{\varepsilon}\xi
(\varepsilon\C{A})H=\C{A}^m\xi^{(m)}
(\varepsilon\C{A})H
=\varepsilon^{1-m}(\varepsilon\C{A})^{m-1}\xi^{(m)}
(\varepsilon\C{A})\C{A}H
$$
$$
=\varepsilon^{1-m}\varphi
(\varepsilon\C{A})\C{A}H .
$$
where $\varphi (x)=x^{m-1}\xi^{(m)}(x)$
is a function of class
$C^{\infty }_0(\D{R})$. Hence
$$
\Vert H^{(m)}_{\varepsilon}\Vert
\leq \varepsilon^{1-m}\Vert \varphi \Vert_{\C{M}}\Vert
\C{A}H\Vert .
$$
\end{proof}
\begin{lem} \label{lem:5.4}
Set
$\widetilde{G}_{\varepsilon}=G_{\varepsilon}K_{\varepsilon}
G_{\varepsilon}$,
where $K_{\varepsilon}$ is as in Proposition
\textup{5.2 (c)}. Then for
each $k$, $m\in \D{N}$ there is a finite constant $C$,
independent of
$\varepsilon ,
\lambda , \mu $, such that
\begin{equation} \label{eq:5.19}
|\Vert \widetilde {G}^{(k,m)}_{\varepsilon}|\Vert
\leq C\varepsilon^{-k-m-1}\sum^m_{j=0}\Vert
\varepsilon^jK^{(j)}_{\varepsilon}\Vert .
\end{equation}
In particular, if $H\in \C{C}^{1+\alpha }(A)$ for
some $\alpha >0$, then we have $|\Vert
\widetilde{G}^{(k,m)}_{\varepsilon}|\Vert \leq
c\varepsilon^{\alpha -k-m-1}$.
\end{lem}
\begin{proof}
By Leibnitz formula, and since $K_{\varepsilon}$
does not depend on $\lambda $,
$\widetilde{G}^{(k,m)}_{\varepsilon} $ is a linear
combination of terms of the form
$\widetilde{G}^{(a,u)}_{\varepsilon}\tilde{K}^{(w)}_{\varepsilon}
\widetilde{G}^{(b,v)}_{\varepsilon}$, with $a$,
$b$, $u$, $v$,
$w\in \D{N}$ and $a+b=k$, $u+v+w=n$. Then Proposition 5.3 implies
\begin{align*}
|\Vert & G^{(a,u)}_{\varepsilon}K^{(w)}_{\varepsilon}
G^{(b,v)}_{\varepsilon}|\Vert
\leq \Vert G^{(a,u)}_{\varepsilon}\Vert_{\C{H}
\rightarrow \C{K}^*} \Vert
K^{(w)}_{\varepsilon}\Vert \cdot \Vert
G^{(b,v)}_{\varepsilon}\Vert_{\C{K}\rightarrow
\C{H}}
\\ &
\leq \text{const.}\varepsilon^{-a-u-1/2}\Vert
K^{(w)}_{\varepsilon}\Vert \cdot
\varepsilon^{-b-v-1/2}=\text{const.}
\varepsilon^{-k-m-1}\Vert
\varepsilon^{w}K^{(w)}_{\varepsilon}\Vert .
\end{align*}
For the proof of the next estimates we need
a generalization of the identity (5.13). Assume that
we are under the hypotheses of Proposition 5.2 and let
$\widetilde{G}_{\varepsilon}=G_{\varepsilon
}K_{\varepsilon}G_{\varepsilon}$. Then for all
$l$, $k\in \D{N}$ with $k\geq 1$ and all
$\varepsilon \in (0,\varepsilon_0)$,
$z=\lambda +i\mu $, $\lambda \in J $, $\mu >0$ we have
\begin{equation} \label{eq:5.20}
G^{(l,k)}_{\varepsilon}=l!\C{A}^k\lbrack
G^{l+1}_{\varepsilon}\rbrack +\sum^{k-1}_{r=0}
\C{A}^{k-r-1}\lbrack {\widetilde G}^{(l,r)}_{\varepsilon}\rbrack .
\end{equation}
If $l=0$, $k=1$ this is just (5.13). (5.20) follows from this
special case by taking successively derivatives with respect
to $\varepsilon $ and $\lambda $ and by using the Lemma from
\S3.2.
\end{proof}
Now let us fix two functions $\varphi $, $\psi \in
\C{S}(\D{R})$ and let us define the operator
$L_{\varepsilon}\equiv L_{\varepsilon}(z):
\C{H}_{-\infty }\rightarrow \C{H}_{+\infty }$
by:
\begin{equation} \label{eq:5.21}
L_{\varepsilon}(z)=
\varphi (\varepsilon A)G_{\varepsilon}(z)\psi (\varepsilon A)
\end{equation}
for $0<\varepsilon <\varepsilon_0$ and $z=\lambda
+i\mu $ with $\lambda \in J$ and $\mu >0$. Let $l$,
$m\in \D{N}$. By using Leibnitz formula and by taking
into account the relation
$\partial^i_{\varepsilon}\varphi (\varepsilon A)
=A^i\varphi^{(i)}(\varepsilon
A)=\varepsilon^{-i}\varphi_i(\varepsilon A)$ with
$\varphi_i(x)=x^i\varphi^{(i)}(x)$ we obtain
$$
L^{(l,m)}_{\varepsilon}=\sum_{i+j+k=m}
\frac{m!}{i!j!k!}\varepsilon^{k-m}\varphi_i(\varepsilon
A)G^{(l,k)}_{\varepsilon}\psi_j(\varepsilon A) ,
$$
where the indices $i$, $j$, $k$ run over $\D{N}$.
If we use (5.20) the expression in the
r.h.s.\ above becomes
\begin{align*}
L^{(l,m)}_{\varepsilon} & =\sum_{i+j+k=m}\frac{l!m!}{i!j!k!}
\varepsilon^{k-m}\varphi_i(\varepsilon A)
\C{A}^k\lbrack G^{l+1}_{\varepsilon}\rbrack
\psi_j(\varepsilon A)
\\ &
+\sum_{{i+j+k=m,k\geq
1,}\atop{n+r=k-1}}\frac{m!}{i!j!k!}
\varepsilon^{k-m}\varphi_i(\varepsilon A)\C{A}^n\lbrack
\widetilde{G}^{(l,r)}_{\varepsilon}\rbrack \psi_j
(\varepsilon A) .
\end{align*}
Then by taking into
account the identity (3.1) we get
\begin{align} \label{eq:5.22}
& \varepsilon^mL^{(l,m)}_{\varepsilon}
=\sum_{i+j+p+q=m}\frac{l!m!}{i!j!p!q!}(-\varepsilon
A)^p(\varepsilon A)^i\varphi^{(i)}(\varepsilon
A)G^{l+1}_{\varepsilon}(\varepsilon
A)^{j+q}\psi^{(j)}(\varepsilon A)
\\
& +\sum_{{i+j+p+q+r}\atop{=m-1}}\frac{m!(p+q)!(-1)^p
\varepsilon^{r+1}}{i!j!p!q!(m-i-j)!}(\varepsilon
A)^{i+p}\varphi^{(i)}(\varepsilon A)\widetilde{G}^{(l,r)}_
{\varepsilon}
(\varepsilon A)^{j+q}\psi^{(j)}(\varepsilon A). \notag
\end{align}
\medskip
\begin{prop} \label{prop:5.5}
Let
$\varphi $,
$\psi \in \C{S}(\D{R})$ and let
$L_{\varepsilon}=L_{\varepsilon}(z)$ be defined
by $L_{\varepsilon}=\varphi (\varepsilon A)G_{\varepsilon}\psi
(\varepsilon A)$. Then for each
$l,m\in \D{N}$ there is a constant
$C$, independent of $\varepsilon , \lambda ,
\mu $, such that for all $f, g\in \C{H}_{-\infty
}$:
\begin{align} \label{eq:5.23}
|\langle g,\varepsilon^{l+m}L^{(l,m)}_{\varepsilon
}f\rangle|
& \leq
C\sum_{{{a+b=m,}\atop{0\leq i\leq a,}}\atop{0\leq j\leq
b}}|\Vert
\varphi_{i,a}(\varepsilon A)g|\Vert
\cdot |\Vert\psi_{j,b}(\varepsilon A)f|\Vert
\\ &
+C\sum_{{{a+b+c\leq m-1,}\atop{0\leq i\leq a,}}
\atop{ 0\leq j\leq b}}|\Vert
\varphi_{i,a}(\varepsilon A)g|\Vert
\cdot |\Vert
\psi_{j,b}(\varepsilon A)f|\Vert \cdot \Vert
\varepsilon^cK^{(c)}_{\varepsilon}\Vert .\notag
\end{align}
Here the functions
$\varphi_{i,a}$ and
$\psi_{j,b}$ are defined by
$\varphi_{i,a}(x)=x^a\varphi^{(i)}(x)$ and
$\psi_{j,b}(x)=x^b\psi^{(j)}(x)$.
\end{prop}
\begin{proof}
We use (5.22) and the estimates $|\Vert
\varepsilon^lG^{l+1}_{\varepsilon}|\Vert \leq C(l)$ and
$|\Vert
\varepsilon^{l+r+1}\widetilde{G}^{(l,r)}_{\varepsilon}|\Vert$
$ \leq
C(l,r)\sum_{0\leq c\leq r}\Vert
\varepsilon^cK^{(c)}_{\varepsilon}\Vert $ which have been
obtained in Proposition 5.3 and Lemma 5.4.
\end{proof}
It is clear that the first sum from (5.22) becomes much more
simpler if $\varphi $ is a function such that
$\varphi^{(i)}(x)=\varphi (x)$ for all $x$. But the only
function which has this property is
$\varphi (x)=e^x$ and it does not belong to
$\C{S}(\D{R})$. However, one can circumvent this difficulty if
in place of $L_{\varepsilon}$ one considers the operator
$\Pi_-L_{\varepsilon}$ where
$\Pi_-=E_A((-\infty ,0\rbrack )$ is the spectral projection of
$A$ associated with the interval
$(-\infty ,0\rbrack $. Then we take a function
$\varphi \in \C{S}(\D{R})$ such that $\varphi (x)=e^x$ if
$x\leq 0$. Observe that for $j$, $q$ fixed with
$n=m-j-q>0$ one has
$\sum_{i+p=n} (i!p!)^{-1}(-x)^px^j=0$. Hence, after left
multiplication by
$\Pi_-$ of (5.22), in the first sum on the r.h.s.\ will remain
only terms with $j+q=m$, so $i=p=0$.
On the other hand:
\begin{equation} \label{eq:5.24}
\sum_{j+q=m}\frac{m!}{j!q!}x^{j+q}\psi^{(j)}(x)
=x^m(1+\frac{d}{dx})^m\psi (x)\equiv \zeta (x) .
\end{equation}
Hence we obtain :
$$
\varepsilon^m\Pi_-L^{(l,m)}_{\varepsilon}=l!
\Pi_-e^{\varepsilon A}G^{l+1}_{\varepsilon}\zeta (\varepsilon
A)+
$$
$$
\sum_{{i+j+p+q+r}\atop{=m-1}}\frac{m!(p+q)!(-1)^p
\varepsilon^{r+1}}
{i!j!p!q!(m-i-j)!}\Pi_-(\varepsilon
A)^{i+p}e^{\varepsilon A}\widetilde{G}^{(l,r)}_{\varepsilon}
(\varepsilon A)^{j+q}\psi^{(j)}(\varepsilon A) .
$$
By the same argument as in the proof of
Proposition 5.5 we get, with a slight change of notation:
\begin{prop} \label{prop:5.6}
Let $\psi \in \C{S}(\D{R})$,
define $\zeta $ by $(5.24)$, and let us set
$L_{\varepsilon}=\Pi_-e^{\varepsilon A}G_{\varepsilon}\psi
(\varepsilon A)$. Then for each
$l$,
$m\in \D{N}$ there is a constant
$C$, independent of $\varepsilon , \lambda , \mu $,
such that for all $f,g\in \C{H}_{-\infty}$:
\begin{equation} \label{eq:5.25}
|\langle g,\varepsilon^{l+m}L^{(l,m)}_{\varepsilon
}f\rangle|\leq C|\Vert \Pi_-e^{\varepsilon A}g|\Vert \cdot
|\Vert
\zeta (\varepsilon A)f|\Vert +
\end{equation}
$$
C\sum_{{a+b+c\leq m-1,}\atop{0\leq j\leq b}}|\Vert
\Pi_-(\varepsilon
A)^ae^{\varepsilon A}g|\Vert
\cdot
\Vert (\varepsilon A)^b\psi^{(j)}(\varepsilon A)f|\Vert
\cdot \Vert \varepsilon^cK^{(c)}_{\varepsilon}\Vert .
$$
\end{prop}
This estimate can be further simplified by a special choice of
$\psi $. Note that if $\psi (x)=e^{-x}$ then
$\zeta =0$. Of course this choice is not allowed by the
condition $\psi \in \C{S}(\D{R})$.
However, if we take
$\psi $ of class
$\C{S}(\D{R})$ and such that $\psi (x)=e^{-x}$
if $x\geq 0$, then $\Pi_+\zeta (\varepsilon A)f=0$
for each
$f\in \C{H}_{-\infty }$.
Hence Proposition 5.6 immediately implies the next one. Here
$\Pi_+=E_A(\lbrack 0,\infty ))$.
\begin{prop} \label{prop:5.7}
Let $L_{\varepsilon}=\Pi_-e^{\varepsilon
A}G_{\varepsilon}e^{-\varepsilon A}\Pi_+$. Then for each $l$,
$m\in \D{N}$ with
$m\geq 1$ there is
$C<\infty $, independent of
$\varepsilon ,
\lambda,
\mu $, such that for all $f,
g\in \C{H}_{-\infty }$:
\begin{equation} \label{eq:5.26}
|\langle
g,\varepsilon^{l+m}L^{(l,m)}_{\varepsilon}f\rangle|
\end{equation}
$$
\leq C\sum_{a+b+c\leq m-1} |\Vert \Pi_-(\varepsilon
A)^ae^{\varepsilon A}g|\Vert
\cdot |\Vert
\Pi_+(\varepsilon A)^be^{-\varepsilon A}f|\Vert
\cdot \Vert \varepsilon^cK^{(c)}_{\varepsilon}\Vert .
$$
\end{prop}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% %%%%%%%%%%%
%%%%%%%% 6. Resolvent of Bounded Regular Operators %%%%%%%%%%%
%%%%%%%% %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%------------------------------%
\protect\setcounter{equation}{0}
%------------------------------%
\section{Resolvents of Bounded Regular Operators} \label{s:6}
This section contains the main results of the paper. For their
proof we shall use the estimates obtained in Section 5 and the
following elementary lemmas. The proof of Lemma 6.1 is quite
easy and will not given; that of Lemma 6.2 can be found in
\cite{BG3}.
\begin{lem} \label{lem:6.1}
Let $h:(0,\varepsilon_0\rbrack\rightarrow\D{C}$ be
a function of class $C^m$ for some integer
$m\geq 1$ and some real
$\varepsilon_0>0$. Assume that
$
\int^{\varepsilon_0}_0|
\varepsilon^{m-1}h^{(m)}(\varepsilon)|d\varepsilon<\infty
$.
Then
$\lim_{\varepsilon \rightarrow 0}h(\varepsilon )\equiv h(0)$
exists and
\begin{equation} \label{eq:6.1}
h(0)=\sum^{m-1}_{k=0}
\frac{(-\varepsilon_0)^k}{k!}h^{(k)}(\varepsilon_0)+
\frac{(-1)^m}{(m-1)!}\int^{\varepsilon_0}_0
h^{(m)}(\varepsilon )\varepsilon^{m-1}d\varepsilon .
\end{equation}
\end{lem}
\begin{lem} \label{lem:6.2}
Let $J\subset \D{R}$ be an open set,
$\varepsilon_0>0$ a real number and
$\widetilde{J}=\lbrace (\lambda ,\varepsilon )\in
\D{R}^2\mid \lambda \in J,
0<\varepsilon<\varepsilon_0\rbrace
$. Let
$F:\widetilde{J}\rightarrow \D{C}$ be a function of class
$C^m$ for some integer $m\geq 1$ and assume that there are real
numbers $\sigma , M$ with
$0<\sigma 1/2$, and if
$\chi $ has a zero of order $>\alpha $ at the origin
(i.e.\ $|\chi (x)|\leq c|x|^{\beta }$ for some
$\beta >\alpha $), then there is a constant
$C<\infty $ such that for all
$\varepsilon >0$:}
\begin{equation} \label{eq:6.15}
\Vert \chi (\varepsilon A)\Vert_{\C{H}_{s,\infty }
\rightarrow \C{H}_{1/2,1}}+\Vert \chi
(\varepsilon A)\Vert_{\C{H}_{-1/2,\infty }\rightarrow
\C{H}_{-s,1}}\leq C\varepsilon^{\alpha }.
\end{equation}
\bigskip
\begin{thm} \label{thm:6.6}
Let $H\in \C{C}^{1+\alpha }(A)$ for some real
$\alpha >0$ and let us set $s=\alpha
+1/2$. Then the function
\begin{equation} \label{eq:6.16}
J\ni \lambda \mapsto R(\lambda +i0)\in
B(\C{H}_{s,\infty };\C{H}_{s,1})
\end{equation}
is locally of class $\Lambda^{\alpha }$.
\end{thm}
\begin{proof}
(i)
Let $L_{\varepsilon}$ be as in the proof of
Theorem 6.3. We first prove that for each $l$,
$m\in \D{N}$ with
$m>2\alpha $ we have
\begin{equation} \label{eq:6.17}
\Vert L^{(l,m)}_{\varepsilon}\Vert_{\C{H}_{s,\infty }
\rightarrow \C{H}_{-s,1}}\leq
C(l,m)\varepsilon^{\alpha -l-m}
\end{equation}
for a number $C(l,m)<\infty $ independent of $\varepsilon
\in (0,\varepsilon_0)$, $\lambda \in J$ and
$\mu >0$. For this purpose we use the Proposition 5.5. Note
that for each term of the first sum on the
r.h.s.\ of (5.23) we have either
$a>\alpha $ or $b>\alpha $. If, for example $a>\alpha $,
we use the estimate (6.15) with $\chi
=\varphi_{i,a}$ and get that the corresponding term is
bounded by a constant times
$\varepsilon^{\alpha }\Vert g\Vert_{s,\infty }|\Vert f|\Vert $,
and this is better than needed (because
$s>1/2$). A typical term of the second sum on the r.h.s.\ of
(5.23) is dominated by
$\text{const.}|\Vert g|\Vert \cdot |\Vert f|\Vert \cdot \Vert
\varepsilon^cK^{(c)}_{\varepsilon}\Vert $ and now we may use
Proposition 5.2 (d).
(ii)
Now let
$f\in \C{H}_{s,\infty }$ and
$F(\lambda ,\varepsilon )=\langle f,L_{\varepsilon}(\lambda
+i\mu )f\rangle$. Then (6.17) gives
\begin{equation} \label{eq:6.18}
|\partial^l_{\lambda }\partial^m_{\varepsilon
}F(\lambda ,\varepsilon )|\leq C(l,m)\Vert f\Vert^2_{s,\infty
}\varepsilon^{\alpha -l-m} .
\end{equation}
This implies the hypothesis of Lemma 6.2, namely
$|\partial^l_{\lambda
}\partial^k_{\varepsilon}F(\lambda,\varepsilon )|\leq
M\varepsilon^{\alpha -m}$ if
$l+k=m$, with
$M=\text{const.}\Vert f\Vert^2_{s,\infty }$. Indeed, if
$l=0$ this is a particular case of (6.18). If $l\geq 1$ we
integrate (6.18) $l$ times with respect to
$\varepsilon $ over an interval of the form $(\tau
,\varepsilon_0)$ with $0<\tau <\varepsilon_0$; since
$\alpha -m<0$ we shall get
$|\partial^l_{\lambda }\partial^{m-l}_{\tau }F(\lambda ,\tau
)|\leq M\tau^{\alpha -m}$, which is the estimate we were
looking for. Now we use Lemma 6.2. Since
$F_0=\langle f,R(z)f\rangle$ and
$\C{H}_{s,\infty }=(\C{H}_{-s,1})^*$, the estimate
(6.2) implies the assertion of the theorem.
\end{proof}
We remark that the proof gives more than stated in Theorem
6.6: the function $z\mapsto R(z)\in B(\C{H}_{s,\infty
};\C{H}_{-s,1})$ is in fact of class
$\Lambda^{\alpha }$ (and not only locally) on the set $\lbrace
z\in \D{C}\mid \Re z\in J,
\Im z\geq 0\rbrace $.
\begin{thm} \label{thm:6.7}
Let $s$, $\alpha $ be real numbers such that
$0<\alpha 0$ such that
\begin{equation} \label{eq:6.22}
\Vert L^{(l,m)}_{\varepsilon}\Vert_{\C{H}_{s,p}
\rightarrow \C{H}_{t,q}}\leq
C(l,m)\varepsilon^{\beta -l-m} .
\end{equation}
In order to prove this we use the inequality established in
Proposition 5.7. Each term in the r.h.s.\
of (5.26) is of the form $|\Vert \varphi (\varepsilon A)g|\Vert
\cdot |\Vert \psi (\varepsilon A)f|\Vert
\cdot \Vert \varepsilon^cK^{(c)}_{\varepsilon}\Vert$ where
$\varphi ,\psi \in \C{S}(\D{R})$
but do not vanish at zero in general.
By Proposition 5.2(d) such a term is bounded by a
constant times
\begin{equation} \label{eq:6.23}
\varepsilon^{\alpha }|\Vert
\varphi (\varepsilon A)g|\Vert
\cdot |\Vert \psi (\varepsilon A)f|\Vert
=\varepsilon^{\beta }|\Vert \epsilon^{\alpha -\beta -\sigma }
\varphi
(\varepsilon A)g|\Vert \cdot |\Vert
\varepsilon^{\sigma }
\psi (\varepsilon A)f|\Vert
\end{equation}
where
$\sigma $ could be an arbitrary real number. If
$0<\sigma <\alpha -\beta $ then the r.h.s.\ of (6.23) can be
estimated with the help of the Theorem from \S 2.3. We clearly
get a bound of the form
$c\varepsilon^{\beta }\Vert g\Vert_{1/2-\alpha +\beta +\sigma
,\infty }\Vert f\Vert_{1/2-\sigma ,\infty }$. We set
$s=1/2-\sigma $ and we obtain (6.22) by a simple argument. The
limit cases
$\sigma =0$ and
$\sigma =\alpha -\beta $ are treated similarly.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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