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
$\sum_{n~~ r} \hat{u}(x)dx\sim r^{-\alpha -2\varepsilon }$
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
$1~~**s$ such that one of the following equivalent
conditions is satisfied:
$$
\Bigl[ \int^1_0 \Vert
\varepsilon^{-s}(\C{W}_{\varepsilon
}-1)^mS\Vert^p\frac{d\varepsilon}{\varepsilon}\Bigr]^{1/p}<\infty
\ \text{ or }\
\Bigl[ \int^1_0 \lbrack\varepsilon^{-s}\omega_m(\varepsilon
)\rbrack^p\frac{d\varepsilon}{\varepsilon}
\Bigr]^{1/p}<\infty .
$$
If $p=\infty $ then these conditions must be read
$\omega_m(\varepsilon )\leq c\varepsilon^s$ for some
number
$c$ and
$\varepsilon >0$. We shall write
$\C{C}^s(A)$ for
$\C{C}^{s,\infty }(A)$.
The equivalence of the two conditions stated above is easy to
prove if
$0 s$ 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 p**

0$ 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 $0s$. (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 $00$ and $\hat{\phi}\geq 0$, then the condition is fulfilled. Another sufficient condition is $\phi \in \C{H}^{m+n/2+\varepsilon }(\D{R}^n)$ for some $\varepsilon >0$. %------------% \subsection{} \label{ss:3.9} Let $S\in B(\C{H})$ be an operator of class $\C{C}^{s,p}(A)$ for some $s>0$ and $p\in \lbrack 1,\infty \rbrack$. One can show quite easily that $S\C{H}_{s,p}\subset \C{H}_{s,p}$ (see Lemma 5.3.2 in \cite{ABG2}) and that the operator $S_0$ induced by $S$ in $\C{H}_{s,p}$ is continuous (closed graph theorem). Now assume that $1\leq p<\infty $. Since $S^*$ is of the same class as $S$, by considering the adjoint of the restriction of $S^*$ to $\C{H}_{s,p}$ we get a continuous extension $\widehat{S}$ of $S$ to $\C{H}_{-s,p'}$. If $1 0$ 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

1$). Then the final formula, although more complicated, is sometimes more useful. Let us set $R^{(k)}(z)=(d/dz)^kR(z)=k!R(z)^{k+1}$. Then the Taylor expansion of the resolvent used before can be written as follows: $$ R(\lambda +i\varepsilon )=\sum^{m-1}_{k=0} \frac{i^k}{k!} (\varepsilon -r)^kR^{(k)}(\lambda +ir)+\frac{(-i)^m}{(m-1)!}\int^r_{\varepsilon} \partial^m_{\lambda }R(\lambda +i\mu )\cdot (\mu -\varepsilon )^{m-1}d\mu . $$ This holds for an arbitrary $r>\varepsilon $. Now assume that a function $\lambda \mapsto r(\lambda )$ of class $C^m$ is given on $\D{R}$ such that $\inf_{\lambda \in \D{R}} r(\lambda )>0$ and let $0<\varepsilon <\inf r(\lambda )$. The following formula is easily verified by induction $$ \partial^m_{\lambda }\int^{r(\lambda )}_{\varepsilon} F(\lambda ,\mu )d\mu =\sum_{j+k=m-1}\partial^j_{\lambda }\lbrack r'(\lambda )F^{(k)}(\lambda ,r(\lambda ))\rbrack +\int^{r(\lambda )}_{\varepsilon} F^{(m)}(\lambda ,\mu )d\mu , $$ where $F^{(k)}(\lambda ,\mu )=\partial^k_{\lambda } F(\lambda ,\mu )$. In particular, the last term in the expansion of $R(\lambda +i\varepsilon )$ (with $r=r(\lambda )$) is equal to \begin{align*} \frac{(-i\partial_{\lambda })^m}{(m-1)!} \int^{r(\lambda )}_{\varepsilon} & R(\lambda +i\mu ) (\mu -\varepsilon )^{m-1}d\mu \\ & -\sum_{j+k=m-1}\frac{(-i)^m} {(m-1)!}\partial^j_{\lambda }\lbrack r'(\lambda )R^{(k)}(\lambda + ir(\lambda ))(r(\lambda )-\varepsilon )^{m-1}\rbrack . \end{align*} So if $\varphi \in C^m_0(\D{R})$ we obtain for $\int_{\D{R}} \varphi (\lambda )R(\lambda +i\varepsilon )d\lambda $ the following expression \begin{align*} \sum^{m-1}_{k=0}\int_{\D{R}}\Bigl[ & \frac{(-i)^k}{k!} (r(\lambda )-\varepsilon )^k\varphi (\lambda) \\ & +\frac{i^{m-k}(-i)^k}{(m-1)!}r'(\lambda )(r(\lambda ) -\varepsilon )^{m-1}\varphi^{(m-1-k)}(\lambda)\Bigr] \cdot R^{(k)}(\lambda +ir(\lambda ))d\lambda \\ & +\frac{i^m}{(m-1)!}\int_{\D{R}}\int^{r(\lambda)}_{\varepsilon} \varphi^{(m)}(\lambda )R(\lambda +i\mu )(\mu -\varepsilon )^{m-1}d\mu d\lambda . \end{align*} Now let us assume $m\geq 2$ (this assures the convergence in norm of the integrals below; in the case $m=1$ the next formulas are to be interpreted in the weak topology, cf.\ Section 6.1 in \cite{ABG2}). Then the next limit clearly exists in norm and we have \begin{equation} \label{eq:4.3} \lim_{\varepsilon \rightarrow +0}\int_{\D{R}} \varphi (\lambda )R(\lambda +i\varepsilon )d\lambda = \end{equation} \begin{align*} \sum^{m-1}_{k=0} \int_{\D{R}} \Bigl[ \frac{1}{k!} \varphi (\lambda ) & +\frac{ir'(\lambda )}{(m-1)!} (ir(\lambda ))^{m-1-k}\varphi^{(m-1-k)}(\lambda )\Bigr] \cdot \\ & (-ir(\lambda ))^kR^{(k)}(\lambda +ir(\lambda ))d\lambda \\ & +\frac{i^m}{m!}\int_{\D{R}}\int^{r(\lambda )}_0 \varphi^{(m)}(\lambda )R(\lambda +i\mu )d\mu^{m}d\lambda . \end{align*} By using (4.1) as before one gets a new expression for $\varphi (H)$ which has the advantage that it will hold for a larger class of functions $\varphi $. Indeed, we assumed until now that $\varphi $ has compact support, but the final formula will clearly remain valid for all $\varphi $ such that the integrals from the r.h.s.\ of the formula are norm convergent. Assume, for example, that $\supp \varphi \subset \lbrack 1,\infty )$ and take $r(\lambda )=\lambda $ for $\lambda >1$. Then $$ \lim_{\varepsilon \rightarrow +0} \int_{\D{R}} \varphi (\lambda )R(\lambda +i\varepsilon )d\lambda = $$ \begin{align*} \sum^{m-1}_{k=0} \int_{\D{R}} \Bigl[ \frac{1}{k!} \varphi (\lambda ) & +\frac{i(i\lambda )^{m-1-k}} {(m-1)!}\varphi^{(m-1-k)}(\lambda)\Bigr]\cdot (-i\lambda )^kR^{(k)}(\lambda +i\lambda )d\lambda \\ & +\frac{i^m}{m!} \int_{\D{R}} \int^{\lambda }_0 \varphi^{(m)}(\lambda )R(\lambda +i\mu )d\mu^md\lambda . \end{align*} We have $\Vert \lambda^kR^{(k)}(\lambda +i\lambda )\Vert \leq \lambda^{-1}$ so the integrals from the sum above are convergent provided that $\int^{\infty }_1 |\lambda^j\varphi^{(j)} (\lambda )|\lambda^{-1}d\lambda <\infty $ for $0\leq j\leq m-1$. Since $\int^{\lambda }_{0} \Vert R(\lambda +i\mu )\Vert d\mu^m\leq m\int^{\lambda }_0 \mu^{m-2}d\mu =m(m-1)^{-1}\lambda^{m-1}$ if $m\geq 2$, we see that the convergence of the last integral is assured by the condition $\int^{\infty }_1 |\lambda^m\varphi^{(m)}(\lambda ) |\lambda^{-1}d\lambda <\infty $. If these conditions are satisfied then by taking into account that $\partial^k_{\lambda }R(\lambda +i\lambda ) =(1+i)^kR^{(k)}(\lambda +i\lambda )$ we also obtain \begin{equation} \label{eq:4.4} \lim_{\varepsilon \rightarrow +0} \int_{\D{R}} \varphi (\lambda )R(\lambda +i\varepsilon )d\lambda = \int_{\D{R}} \sum^{m-1}_{k=0} (1+i)^{-k}\partial^k_{\lambda } \Bigl[ \frac{(i\lambda )^k}{k!} \varphi (\lambda )+ \end{equation} $$ \frac{i(i\lambda )^{m-1}}{(m-1)!}\varphi^{(m-1-k)} (\lambda )\Bigr] \cdot R(\lambda +i\lambda )d\lambda + \frac{i^m}{m!} \int_{\D{R}} \int^{\lambda }_0 \varphi^{(m)}(\lambda )R(\lambda +i\mu )d\mu^md\lambda . $$ \bigskip %------------% \subsection{} \label{ss:4.3} Let us denote by $\C{S}(\C{H})$ the set of all self-adjoint resolvent families on $ \C{H}$ (or, equivalently, the set of all self-adjoint operators in $\C{H}$) and by $\C{U}(\C{H})$ the set of all unitary operators in $\C{H}$. The \emph{Cayley transform} of a resolvent family $\lbrace R(z)\rbrace $ (or of a self-adjoint operator $H$) is the unitary operator $U=1-2iR(-i)$ (or $U=(H-i)(H+i)^{-1}$ with the convention $Uf=f$ if $f$ is orthogonal to $\overline{D(H)}$). The unitarity of $U$ follows from $U^*=1+2iR(i)$, which gives $U^*U=UU^*=1$. We obtain in this way a map $\C{S}(\C{H})\rightarrow\C{U}(\C{H})$ which is bijective: the injectivity follows from the fact that $R(-i)$ determines $R(\cdot )$ (because $R(\cdot )$ is holomorphic and all its derivatives at $z=-i$ are known once $R(-i)$ is known), while the surjectivity follows from the formula $Hf=i(1+U)(1-U)^{-1}$ for $f\in D(H)\equiv$ range of $(U-1)$, where $(1-U)^{-1}$ is the inverse of the restriction of $1-U$ to the orthogonal complement of its kernel. Observe also that by taking $z_0=-i$ in (4.5) one gets the following expression for $R(z)$ in terms of $U$: if $\zeta =(z-i)(z+i)^{-1}$ then $(z+i)R(z)=(1-U)(U-\zeta )^{-1}$. $\C{U}(\C{H})$ is a complete metric space for the metric induced by the norm. So if we equip $\C{S}(\C{H})$ with the metric $$ \delta (H_1,H_2)=\Vert (H_1+i)^{-1}-(H_2+i)^{-1}\Vert =\frac{1}{2} \Vert U_1-U_2\Vert , $$ where $U_j$ is the Cayley transform of the self-adjoint operator $H_j$, \emph{ we provide $\C{S}(\C{H})$ with a complete metric space structure}. We have $\lim_{n\rightarrow \infty } H_n=H$ in $\C{S}(\C{H})$ if and only if $H_n$ converges to $H$ in norm-resolvent sense. \emph{The set of densely defined self-adjoint operators is dense in $\C{S}(\C{H})$}. Indeed, $H\in \C{S}(\C{H})$ is densely defined if and only if its Cayley transform $U$ has not $1$ as eigenvalue, so it suffices to show that a unitary operator $U$ which has $1$ as eigenvalue is a norm limit of unitary operators $U_{\lambda }$ which do not have $1$ as eigenvalue. Let $P$ be the orthogonal projection of $\C{H}$ onto $ker(U-1)$ and let $V$ be the unitary operator in $(P\C{H})^{\perp }$ such that $U=V\oplus P$ relatively to the decomposition $\C{H}=(P\C{H})^{\perp }\oplus P\C{H}$. Then we set $U_{\lambda }=V\oplus \lambda P$ with $|\lambda |=1$,$\lambda \not =1$. Since $V-1$ is injective, $U_{\lambda }-1=(V-1)\oplus (\lambda -1)P$ is injective too and $\Vert U_{\lambda }-U\Vert =|\lambda -1|\rightarrow 0$ as $\lambda \rightarrow 1$. The operator $H\in \C{S}(\C{H})$ is everywhere defined (hence bounded) if and only if the number $1$ does not belong to the spectrum of its Cayley transform $U$. One can easily prove as above that \emph{the set of (bounded) everywhere defined self-adjoint operators is dense in $\C{S}(\C{H})$}. \emph{For each function $\varphi :\D{R}\rightarrow \D{C}$ continuous and convergent to zero at infinity the map $\C{S}(\C{H})\ni H\mapsto \varphi (H)\in B(\C{H})$ is norm continuous}. Indeed, the set of functions $\varphi $ that have this property is stable for addition, multiplication and conjugation, and contains the function $\varphi (x)=(x+i)^{-1}$; then we apply the Stone-Weierstrass theorem. Note that $\C{S}(\C{H})$ has the weakest topology for which this property holds. %------------% \subsection{} \label{ss:4.4} Assume now that a densely defined self-adjoint operator $A$ is given in $\C{H}$ and let $\lbrace R(z)\rbrace $ be the resolvent family associated to a self-adjoint operator $H$. We shall say that $\lbrace R(z)\rbrace $ (or $H$) \emph{is of class $C^k(A)$, $C^k_u(A)$, or $\C{C}^{s,p}(A)$} if there is a complex number $z_0$ outside the spectrum of $H$ such that the bounded operator $R(z_0)$ is of class $C^k(A)$, $C^k_u(A)$ or $\C{C}^{s,p}(A)$ respectively. Note that if this property holds for some $z_0$ then it holds for all complex numbers $z$ outside the spectrum of $H$. Indeed, the operator $1-(z-z_0)R(z_0)$ will then be invertible in $\C{H}$ with inverse equal to $1+(z-z_0)R(z)$, and so \begin{equation} \label{eq:4.5} R(z)=R(z_0) \lbrack 1-(z-z_0)R(z_0)\rbrack^{-1}. \end{equation} Hence the assertion follows from the fact that $C^k(A)$, $C^k_u(A)$ and $\C{C}^{s,p}(A)$ are full subalgebras of $B(\C{H})$. In particular, $H$ belongs to one of the preceding regularity classes with respect to $A$ if and only if its Cayley transform $U$ belongs to the same class. The map $\lambda \mapsto (\lambda -i)(\lambda +i)^{-1}$ extends to a homeomorphism of the one point compactification $\D{R}\cup \lbrace \infty \rbrace $ of the real line $\D{R}$ onto the unit circle $\Sigma =\lbrace \zeta \in \D{C}\mid |\zeta |=1\rbrace $ which sends $\infty $ into $1$. So the rule $\varphi (\lambda )=\varphi^{\#}((\lambda -i)(\lambda +i)^{-1})$ will give us a bijective correspondence between complex functions $\varphi $ on $\D{R}$ which are continuous and tend to zero at infinity and functions $\varphi^{\#}:\Sigma \rightarrow \D{C}$ continuous and such that $\varphi^{\# }(1)=0$. Then clearly we have $\varphi (H)=\varphi^{\#}(U)$ if $U$ is the Cayley transform of the self-adjoint operator $H$. This fact allows us to use the Proposition from \S3.8 in order to show that the bounded operator $\varphi (H)$ is of the same regularity class with respect to $A$ as $H$ if the function $\varphi $ is sufficiently smooth. For example, by taking into account the remark made after the proof of the quoted proposition we see that \emph{if the function $\varphi :\D{R}\rightarrow \D{C}$ has compact support and is of Sobolev class $\C{H}^{m+1+\varepsilon }(\D{R})$ for some integer $m\geq 1$ and some real $\varepsilon >0$ then $\varphi (H) $ is of class $C^m(A)$, or $C^m_u(A)$, or $\C{C}^{s,p}(A)$ if $H$ is of class $C^m(A)$, or $C^m_u(A)$, or $\C{C}^{s,p}(A)$ respectively and if $0-1$, $\varphi_0$ 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 $0<\sigma 0$ such that $\sum_{l+k=m}|\partial^l_{\lambda}\partial^k_{\varepsilon}F( \lambda ,\varepsilon )|\leq M\varepsilon^{\sigma -m}$ on $\widetilde{J}$. Then the limit $\lim_{\varepsilon \rightarrow 0}F(\lambda ,\varepsilon )\equiv F_0(\lambda )$ exists uniformly in $\lambda \in J$ and the function $F_0:J\rightarrow \D{C}$ is locally of class $\Lambda^{\sigma }$. Moreover, there is a constant $C_m$ (depending only on $m$) such that \begin{equation} \label{eq:6.2} |\lbrack (T_{\nu }-1)^mF_0\rbrack (\lambda )| \leq C_mM\sigma^{-1}|\nu |^{\sigma } \end{equation} if $\lambda \in J$ and $\nu \in \D{R}$ have the properties $|\nu |<\varepsilon_0$ and $\lambda +t\nu \in J$ for all $t\in \lbrack 0,m\rbrack $. In $(6.2)$ the translation operator $T_{\nu }$ acts according to $(T_{\nu }g)(\lambda )=g(\lambda +\nu )$. \end{lem} We keep the notations and assumptions of the preceding section. In particular $H$ is a bounded everywhere defined self-adjoint operator and $J$ is an open real set such that the conditions stated at the beginning of Section 5 are fulfilled. Note that the regularity hypotheses that we make below imply $H\in \C{C}^{1,1}(A)$, which in turn implies $H\in C^1_u(A)$. We recall that $\D{C}_+=\lbrace z\in \D{C}\mid \Im z>0\rbrace $ and we set $R(z)=(H-z)^{-1}$. \begin{thm} \label{thm:6.3} Assume that $H\in \C{C}^{1+l,1}(A)$ for some integer $l\geq 0$ and set $s=l+1/2$. Then for each $f\in \C{H}_{s,1}$ the holomorphic map $\D{C}_{+}\ni z\mapsto \langle f,R(z)f\rangle$ extends to a function of class $C^l$ on $\D{C}_+\cup J$, i.e.\ for each integer $0\leq k\leq l$ the holomorphic function on $\D{C}_+$ given by $(d/dz)^k\langle f,R(z)f\rangle=\langle f,k!R(z)^{k+1}f\rangle$ has a continuous extension to $\D{C}_+\cup J$. The limit $\lim_{\mu \rightarrow 0} \langle f,R(\lambda +i\mu )f\rangle\equiv \langle f,R(\lambda +i0)f\rangle$ exists uniformly in $\lambda \in J$, the boundary value function $\lambda \mapsto \langle f,R(\lambda +i0)f\rangle$ is of class $C^l$ on $J$, and for $0\leq k\leq l$ integer one has \begin{equation} \label{eq:6.3} \frac{d^k}{d\lambda^k}\langle f,R(\lambda +i0)f\rangle=\lim_{\mu \rightarrow +0}\langle f,k!R(\lambda +i\mu )^{k+1}f\rangle \end{equation} uniformly in $\lambda \in J$. \end{thm} \begin{proof} Let $L_{\varepsilon}=L_{\varepsilon}(z)=\varphi (\varepsilon A)G_{\varepsilon}(z)\varphi (\varepsilon A)$ where $\varphi $ is a function in $\C{S}(\D{R})$ with $\varphi (0)=1$ and $0\leq \varepsilon \leq \varepsilon_0$, $z=\lambda +i\mu $ with $\lambda \in J$, $\mu >0$. Clearly \begin{equation} \label{eq:6.4} L^{(l,0)}_{\varepsilon}= \partial^l_{\lambda }L_{\varepsilon}=(\frac{d}{dz})^l\varphi (\varepsilon A)G_{\varepsilon}(z)\varphi (\varepsilon A)= \varphi (\varepsilon A)l!G_{\varepsilon}(z)^{l+1}\varphi (\varepsilon A). \end{equation} Note that by Proposition 5.2 (b) the map $\varepsilon\mapsto L^{(l,0) }_{\varepsilon}\in B(\C{H})$ is strongly $C^1$ on the closed interval $\lbrack 0,\varepsilon_0\rbrack $ and $L^{(l,0)}_0=\partial^l_zR(z)=l!R(z)^{l+1}$. Now let us fix $f\in \C{H}_{s,1}$ and define $h(\varepsilon )=\langle f,L^{(l,0)}_{\varepsilon }f\rangle$ for $0\leq \varepsilon\leq \varepsilon_0$. Then for $\varepsilon >0$ and $m\geq 0$ integer we have $h^{(m)}(\varepsilon )=\langle f,L^{(l,m)}_{\varepsilon}f\rangle$ which can be estimated as in (5.23). So there is $C<\infty $, independent of $\varepsilon $, $\lambda $, $\mu $ and $f$, such that \begin{align} \label{eq:6.5} |\varepsilon^mh^{(m)}(\varepsilon )| & \leq C\sum_{{a+b=m,}\atop {i\leq a,j\leq b}}\varepsilon^{-l}|\Vert \varphi_{i,a}(\varepsilon A)f|\Vert \cdot |\Vert \varphi_{j,b}(\varepsilon A)f|\Vert \\ & +C|\Vert f|\Vert^2\sum_{0\leq j\leq m-1} \varepsilon^{-l}\Vert \varepsilon^jK^{(j)}_{\varepsilon}\Vert .\notag \end{align} By Proposition 5.2 (d) the condition $H\in \C{C}^{1+l,1}(A)$ is equivalent to the integrability with respect to the measure $\varepsilon^{-1}d\varepsilon$ on $(0,\varepsilon_0)$ of the second term on the r.h.s.\ of (6.5). We claim that if $m>2l$ then each term of the sum from (6.5) is also integrable (with respect to the same measure). Indeed, if $a+b=m$ then either $a>l$ or $b>l$. In the first case we have $$ \int^1_0 \varepsilon^{-l}|\Vert \varphi_{i,a} (\varepsilon A)f|\Vert \cdot |\Vert \varphi_{j,b}(\varepsilon A)f|\Vert \varepsilon^{-1}d\varepsilon \leq $$ $$ C'|\Vert f|\Vert \int^1_0 \Vert \varepsilon^{-l}\varphi_{i,a}(\varepsilon A) f\Vert_{1/2,1}\varepsilon^{-1}d\varepsilon\leq C''|\Vert f|\Vert \cdot \Vert f\Vert_{s,1} $$ due to the theorem from \S2.3 (observe that $\varphi_{i,a}$ has a zero of order $\geq a>l$ at the origin). Let us fix an integer $m>2l$. We have seen that there is a function $\chi :(0,\varepsilon_0)\rightarrow \D{R}$, independent of $\lambda $ and $\mu $, such that $|\varepsilon^mh^{(m)}(\varepsilon )|\leq \chi (\varepsilon )$ and $\int^{\varepsilon_0}_0\chi (\varepsilon)\varepsilon^{-1}d\varepsilon <\infty $. So we can apply Lemma 6.1 and thus obtain \begin{equation} \label{eq:6.6} \langle f,\partial^l_zR(z)f\rangle =\sum^{m-1}_{k=0}\frac{(-\varepsilon_0)^k}{k!}\langle f,L^{(l,k)}_{\varepsilon_0}f\rangle \end{equation} $$ +\frac{(-1)^m}{(m-1)!}\int^{\varepsilon_0}_0\langle f,L^{(l,m)}_{\varepsilon}f\rangle\varepsilon^{m-1}d\varepsilon. $$ According to Proposition 5.1, for each $\varepsilon \in \lbrack 0,\varepsilon_0\rbrack $ the function $z\mapsto G_{\varepsilon}=(H_{\varepsilon}-z)^{-1}$ is holomorphic in the region $\lambda \in J$, $\mu >-a\varepsilon $, where $a>0$. So each term in the sum from (6.6) extends to a holomorphic function of $z$ below the real axis if $\Re z\in J$ (see (5.22) for example). For the integral in (6.6) we can use the dominated convergence theorem in order to deduce that its limit as $\mu \rightarrow +0 $ exists uniformly in $\lambda \in J$. We have shown that $\lim_{\mu \rightarrow 0}\langle f,\partial^l_zR(z)f\rangle$ exists uniformly in $\lambda \in J$. Clearly the arguments still work if $l$ is replaced by a small integer. \end{proof} It is convenient to reformulate Theorem 6.3 in slightly different terms. For an arbitrary self-adjoint operator $H$ the map $z\mapsto R(z)\in B(\C{H})$ is holomorphic on $\D{C}_+$. Recall that we have continuous embeddings \begin{equation} \label{eq:6.7} B(\C{H})\subset B(\C{K};\C{K}^*) \subset B(\C{H}_{s,1};\C{H}_{-s,\infty }) \end{equation} if $s\geq 1/2$. So, for example, $z\mapsto R(z)\in B(\C{K};\C{K}^*)$ is a holomorphic map on $\D{C}_+$. Now assume that $H\in \C{C}^{1,1}(A)$, i.e.\ the hypothesis of Theorem 6.3 holds with $l=0$. Then the theorem says that the preceding function extends to a weak* continuous function on $\D{C}_+\cup J$, in fact $\lim_{\mu \rightarrow +0}R(\lambda +i\mu )\equiv R(\lambda +i0)\in B(\C{K};\C{K}^*)$ exists in the weak* topology of $B(\C{K};\C{K}^*)$, uniformly in $\lambda \in J$. So the boundary value function $\lambda \mapsto R(\lambda +i0)\in B(\C{K};\C{K}^*)$ is well defined and weak* continuous on $J$. According to (6.7), we may consider the map $\lambda \mapsto R(\lambda +i0)\in B(\C{H}_{s,1};\C{H}_{-s,\infty })$ for each $s\geq 1/2$; clearly it is a weak* continuous function (recall that $\C{H}_{-s,\infty }=\C{H}^*_{s,1}$, which defines the weak* topology of the preceding space). Now assume that $H\in \C{C}^{1+l,1}(A)$ for some integer $l\geq 1$. Then the Theorem 6.3 says that the map $\lambda \mapsto R(\lambda +i0)\in B(\C{H}_{s,1}; \C{H}_{-s,\infty })$ is of class $C^l$ on $J$ in the weak* topology if $s=l+1/2$. Moreover its weak* derivatives are given by \begin{equation} \label{eq:6.8} \frac{d^k}{d\lambda^k}R(\lambda +i0)=\lim_{\mu \rightarrow +0}k!R(\lambda +i\mu )^{k+1}\equiv k!R^{k+1}(\lambda +i0) \end{equation} where the limit exists in the weak* topology of $B(\C{H}_{s,1};\C{H}_{-s,\infty})$, uniformly in $\lambda \in J$. $\C{K}^*=\C{H}_{-1/2,\infty }$ is the smallest space in the Besov scale associated to $A$ which contains the set $R(\lambda +i0)\C{H}_{\infty }$ (if $\lambda \in J$ is a spectral value of $H$). We show now that the operator $\Pi_-R(\lambda +i0)\C{H}_{\infty }$ behaves much better. Here $\Pi_-=E_A((-\infty ,0\rbrack )$ extends to a continuous operator in $\C{H}_{-\infty }$ which leaves invariant each $\C{H}_{s,p}$; hence the product $\Pi_-R(\lambda +i0)$ is well defined and belongs to $B(\C{K};\C{K}^*)$. Observe that in the next theorems we implicitly use the facts established in \S3.9. For example, under the conditions of Theorem 6.4 we have $R(z)\C{H}_{s,p}\subset \C{H}_{s,p}$, hence the r.h.s.\ of (6.9) makes sense. \begin{thm} \label{thm:6.4} Let $H\in \C{C}^{s+1/2,p}(A)$ for some real number $s>1/2$ and some $p\in \lbrack 1,\infty \rbrack $. Then for all $\lambda \in J$ one has $\Pi_-R(\lambda +i0)\C{H}_{s,p}\subset \C{H}_{s-1,p}$. Let $l\geq 0$ be an integer such that $l \alpha \equiv s-1/2$. Then the integral over the interval $(0,1)$ with respect to the measure $\varepsilon^{-1}d\varepsilon$ of the first term on the r.h.s.\ of (6.10) is bounded by \begin{align*} C\Bigl[ \int^1_0|\Vert \varepsilon^{\alpha -l} \Pi_-e^{\varepsilon A}g|\Vert^{p'} & \varepsilon^{-1}d\varepsilon \Bigr]^{1/p'} \cdot \Bigl[ \int^1_0 |\Vert \varepsilon^{-\alpha }\zeta (\varepsilon A)f|\Vert^p\varepsilon^{-1}d\varepsilon \Bigr]^{1/p} \\ & \leq C'\Vert g\Vert_{1/2-\alpha +l,p'}\Vert f\Vert_{1/2+\alpha ,p} \end{align*} We have used the theorem from \S2.3 which is allowed by the fact that $\alpha -l>0$, $0<\alpha 0$ real and $r\in \lbrack 1,\infty \rbrack $. Let $l\in \D{N}$ with $l<\alpha $, let $s$ be a real number such that $1/2-(\alpha -l)\leq s\leq 1/2$, and let us denote $t=s-1+(\alpha -l)$, so that $-1/2\leq t\leq -1/2+(\alpha -l)$. Finally, let $f\in \C{H}_{s,p}$ and $g\in \C{H}_{-t,q'}$ where $p$, $q\in \lbrack 1,\infty \rbrack $ are such that \textup{(i)} if $s=1/2-(\alpha -l)$ then $p=r'$ and $q=\infty $; \textup{(ii)} if $s=1/2$ then $p=1$ and $q=r$; \textup{(iii)} if $1/2-(\alpha -l)1/2$, 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 s-1/2\equiv \beta $ there is a number $C(l,m)$, independent of $\varepsilon ,\lambda ,\mu $, such that \begin{equation} \label{eq:6.20} \Vert L^{(l,m)}_{\varepsilon}\Vert_{\C{H}_{s,\infty } \rightarrow \C{H}_{s-1-\alpha ,1} }\leq C(l,m)\varepsilon^{\alpha -l-m}. \end{equation} We use Proposition 5.6. Then (6.15) with $\chi =\zeta $ (which vanishes of order $m>\beta $ at the origin, see (5.24)) implies $|\Vert \zeta (\varepsilon A)f|\Vert \leq C'\varepsilon^{\beta }\Vert f\Vert_{s,\infty }$. On the other hand the Theorem from \S 2.3 implies for $\beta -\alpha >0$ \begin{equation} \label{eq:6.21} \varepsilon^{\beta -\alpha }|\Vert \Pi_-e^{\varepsilon A}g|\Vert \leq C''\Vert g\Vert_{1/2-\beta +\alpha ,\infty }=C''\Vert g\Vert_{-s+\alpha ,\infty } . \end{equation} Hence the first term on the r.h.s.\ of (5.25) is bounded by a constant times $\varepsilon^{\alpha}\Vert g\Vert_{1-s+\alpha , \infty }\Vert f\Vert_{s,\infty }$. Now we bound the terms of the sum from (5.25) by using $|\Vert (\varepsilon A)^b\psi^{(j)}(\varepsilon A)f|\Vert \leq C'|\Vert f|\Vert \leq C''\Vert f\Vert_{s,\infty }$ and Proposition 5.2(d). We shall get terms of the form $C'''\varepsilon^{\beta}|\Vert \Pi_-(\varepsilon A)^ae^{}g|\Vert \cdot \Vert f\Vert_{s,\infty }$. By an estimate similar to (6.21) (use the Theorem from \S2.3 again) we finally obtain $$ |\langle g,\varepsilon^{l+m}L^{(l,m)}_{\varepsilon}f\rangle|\leq C\varepsilon^{\alpha }\Vert g\Vert_{1-s+\alpha ,\infty }\Vert f\Vert_{s,\infty } . $$ This implies (6.20) because $\C{H}_{1-s+\alpha ,\infty } =(\C{H}_{s-1-\alpha ,1})^*$. (ii) Let $F(\lambda ,\varepsilon )=\langle g,L_{\varepsilon }(\lambda +i\mu )f\rangle$ with $f\in \C{H}_{s,\infty }$ and $g\in \C{H}_{1+\alpha -s,\infty }$. If $l$, $m\geq 0$ are integers and $m>\beta $ then (6.20) gives $$ |\partial^l_{\lambda }\partial^m_{\varepsilon}F(\lambda ,\varepsilon )|\leq C(l,m)\Vert f\Vert_{s,\infty }\Vert g\Vert_{1+\alpha -s,\infty } \varepsilon^{\alpha -l-m} . $$ Now the proof can be finished as in the case of Theorem 6.6. \end{proof} \begin{thm} \label{thm:6.8} Assume that $H\in \C{C}^{1+\alpha }(A)$ for some $\alpha >0$. Let $\beta $, $s$, $t$ be real numbers such that $0<\beta <\alpha ,1/2-(\alpha -\beta )\leq s\leq 1/2$ and $t=s-1+(\alpha -\beta )$, so that $-1/2\leq t\leq -1/2+(\alpha -\beta )$. Finally, let $p$, $q\in \lbrack 1,\infty \rbrack $ be such that \textup{(i)} if $s=1/2-(\alpha -\beta )$ then $p=q=\infty $ ; \textup{(ii)} if $s=1/2$ then $p=q=1$ ; \textup{(iii)} if $1/2-(\alpha -\beta )0$ 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{MMMI} \bibitem[AC]{AC} J.~Aguilar and J.-M.~Combes, A Class of Analytic Perturbations of One-Body Schr\H{o}dinger Hamiltonians, \emph{Comm.\ Math.\ Phys.} \textbf{22} (1971) 269--279. \bibitem[ABG1]{ABG1} W.~Amrein, A.~Boutet de Monvel, and V.~Georgescu, On Mourre's approach to spectral theory, \emph{Helv.\ Phys.\ Acta} \textbf{62} (1989) 1--20. \bibitem[ABG2]{ABG2} W.~Amrein, A.~Boutet de Monvel, and V.~Georgescu, \emph{$C_0$-Groups, Commutator Methods and Spectral Theory of $N$-Body Hamiltonians}, Progress in Math.\ Ser., \textbf{135}, Birkh\"auser, B\^ale, 1996. \bibitem[ABG3]{ABG3} W.~Amrein, A.~Boutet de Monvel, and V.~Georgescu, Notes on the $N$-Body Problem, II, Preprint Universit\'e de Gen\`eve, UGVA-DPT, 1991/04-178 (1991) 160--423. \bibitem[BB]{BB} P.L.~Butzer and H.~Berens, \emph{Semi-Groups of Operators and Approximations}, Springer, Berlin, 1967. \bibitem[BC]{BC} E.~Balslev and J.-M.~Combes, Spectral Properties of Many-Body Schr\H{o}dinger Operators with Dilation-Analytic Interactions, \emph{Comm.\ Math.\ Phys.} \textbf{22} (1971) 280--294. \bibitem[BG1]{BG1} A.~Boutet de Monvel and V.~Georgescu, Locally Conjugate Operators, Boundary Values of the Resolvent and Wave Operators, \emph{C.~R.~Acad.~Sci.~Paris S\'er.\ I Math.} \textbf{313} (1991) 13--18. \bibitem[BG2]{BG2} A.~Boutet de Monvel and V.~Georgescu, On the Regularity of the Boundary Values of a Resolvent Family, \emph{C.~R.~Acad.~Sci.~Paris S\'er.\ I Math.} \textbf{321} (1995) 1299--1304. \bibitem[BG3]{BG3} A.~Boutet de Monvel and V.~Georgescu, Boundary Values of the Resolvent of a Self-Adjoint Operator: Higher Order Estimates, in \emph{Algebraic and Geometric Methods in Mathematical Physics}, Kaciveli 1993, Math.\ Physics Studies \textbf{19}, Kluwer Academic Publ.\ (1996), 9--52. \bibitem[BG4]{BG4} A.~Boutet de Monvel and V.~Georgescu, Spectral and Scattering Theory by the Conjugate Operator Method, \emph{Algebra and Analysis} \textbf{4} (3) (1992) 79--116 (=\emph{St Petersburg Math.\ J.} \textbf{4} (3) (1993) 469--501). \bibitem[BG5]{BG5} A.~Boutet de Monvel and V.~Georgescu, Some Developments and Applications of the Abstract Mourre Theory, \emph{Ast\'erisque} \textbf{210} (1992) 27--48. \bibitem[BGM]{BGM} A.~Boutet de Monvel, V.~Georgescu, and M.~Mantoiu, Locally Smooth Operators and the Limiting Absorption Principle for $N$-Body Hamiltonians, \emph{Rev.\ Math.\ Phys.} \textbf{5} (1) (1993) 105--189. \bibitem[BGS]{BGS} A.~Boutet de Monvel, V.~Georgescu, and A.~Soffer, $N$-Body Hamiltonians with Hard Core Interactions, \emph{Rev.\ Math. Phys.} \textbf{6} (4) (1994) 515--596. \bibitem[BGSh]{BGSh} A.~Boutet de Monvel, V.~Georgescu, J.~Sahbani, Boundary Values of Resolvent Families and Propagation Properties, \emph{C.~R.~Acad.~Sci.~Paris S\'er.\ I Math.} \textbf{322} (1996) 289--294. \bibitem[CFKS]{CFKS} H.L.~Cycon, R.G.~Froese, W.~Kirsch, B.~Simon, \emph{Schr\H{o}dinger Operators}, Springer-Verlag, Berlin, 1986. \bibitem[D1]{D1} E.B.~Davies, \emph{Spectral Theory and Differential Operators}, Cambridge Studies in Advanced Math.\ \textbf{42}, Cambridge, 1995. \bibitem[D2]{D2} E.B.~Davies, The Functional Calculus, \emph{J.~London Math.~Soc.} (1995). \bibitem[DU]{DU} J.~Diestel and J.J.~Uhl, Jr, \emph{Vector Measures}, Math.~Surveys \textbf{15}, Amer. Math. Soc., Providence, Rhode Island, 1977. \bibitem[J]{J} A.~Jensen, Propagation Estimates for Schrodinger-Type Operators, \emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{291} (1985) 129--144. \bibitem[JMP]{JMP} A.~Jensen, E.~Mourre, and P.~Perry, Multiple Commutator Estimates and Resolvent Smoothness in Quantum Scattering Theory, \emph{Ann.\ Inst.\ H.~Poincar\'e} \textbf{41} (1984) 207--225. \bibitem[JP]{JP} A.~Jensen and P.~Perry, Commutator Methods and Besov Space Estimates for Schrodinger Operators, \emph{J.~Op.~Theory} \textbf{14} (1985) 181--189. \bibitem[M1]{M1} E.~Mourre, Absence of Singular Continuous Spectrum for Certain Self-Adjoint Operators, \emph{Comm.~Math.~Phys.} \textbf{78} (1981) 391--408. \bibitem[M2]{M2} E.~Mourre, Op\'erateurs conjugu\'es et propri\'et\'es de propagation, \emph{Comm.~Math.~Phys.} \textbf{91} (1983) 279--300. \bibitem[P]{P} J.~Peetre, \emph{New Thoughts on Besov Spaces}, Duke Univ.\ Math.\ Series I, Durham, 1976. \bibitem[PSS]{PSS} P.~Perry, I.M.~Sigal, and B.~Simon, Spectral Analysis of $N$-Body Schr\H{o}dinger Operators, \emph{Ann.~Math.} \textbf{114} (1981) 519--567. \bibitem[S1]{S1} J.~Sahbani, Th\'eor\`emes de propagation, hamiltoniens localement r\'eguliers et applications, \emph{Th\`ese Universit\'e Paris 7}, juillet 1996. \bibitem[S2]{S2} J.~Sahbani, The Conjugate Operator Method for Locally Regular Hamiltonians, \emph{J.\ Operator Theory}, to appear. \bibitem[W]{W} R. Weder, ``Spectral Analysis of Strongly Propagative Systems'', \emph{J. f\H{u}r Reine Angew. Math.}, 354, (1984), 95-122. \end{thebibliography} \end{document}