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\begin{document}
\title[Limiting Absorption Principle for Singularly Perturbed
Operators]%
{Limiting Absorption Principle for Singularly Perturbed Operators}
\author{Walter Renger}
\address{Institut f\"ur Mathematik, TU-Clausthal,
38678 Clausthal-Zellerfeld, Germany}
\email{mawr@tu-clausthal.de}
\keywords{}
% Math Subject Classifications
\subjclass{Primary 47A40; Secondary 35J25, 47A55, 60J35, 81Q10}
\maketitle
%\newcounter{me}
\begin{abstract}
Given an operator $H_1$ for which a limiting absorption principle holds,
we study operators $H_2$ which are produced by perturbing $H_1$ in the
sense that the difference between some powers of the resolvents
is compact. We show that (except for possibly a discrete set of
eigenvalues)
a limiting absorption principle holds for $H_2$.
We apply this theory to study potential and
domain perturbations of Feller operators. While our theory mostly
reproduces
known results in the case of potential perturbations,
for domain perturbations we get results which appear to be new.
%\\[.6cm]
%{\sc R\'{e}sum\'{e}.}
%Donn\'{e} un op\'erateur $H_1$ v\'erifiant le principe d'absorption
limit\'e,
%nous \'etudions des op\'erateurs $H_2$ construits par une perturbation
de
%$H_1$ dans le sens que la diff\'erence entre certaines puissances des
%r\'esolvants est compacte. Nous montrons qu'un principe
%d'absorption limit\'{e} est v\'{e}rifi\'e pour $H_2$.
%
%Nous appliquons cette th\'eorie \'{a} l'\'etude de perturbations
%de potential et de domaine d'op\'erateurs de Feller.
%Dans le cas de perturbations de domaine - contrairement au cas de
perturbation
%de potential - nous obtenons des r\'esultats qui semblent \^etre
nouveaux.
\end{abstract}
\section{Introduction}
\setcounter{equation}{0}\label{1}
Suppose $H$ is a self-adjoint operator in a Hilbert space $\hr$,
$R(z)=(H-z)^{-1}$ its resolvent, and $\lambda$ some value in its
spectrum
$\sigma(H)$. In order to study spectral and scattering properties of
$H$ it is useful to have some control over the behavior of $R(z)$ as
$z=\lpm$ tends to $\lambda$. Since the limit $\lim_\edz R(\lpm)$
cannot exist in the operator norm on $\hr$ one tries to
establish its existence in a weaker topology.
Let $X\subset\hr$ be a dense subspace of $\hr$, $X^*$ its dual.
We say that a limiting absorption principle holds for $H$ if the
limit $\lim_\edz R(\lpm)$ exists in the norm topology of
$\BO(X,X^*)$, the space of bounded operators from $X$ to $X^*$.
Our aim in this paper is to study
the stability of such a limiting absorption principle
under perturbations, especially under singular ones.
That is, we suppose a limiting absorption principle holds for some
operator $H_1$ and try to establish that it holds
for a second operator $H_2$ which is constructed by perturbing $H_1$
in some way.
There is a huge number of publications on the limiting
absorption principle for individual operators.
Among those more closely related to our problem
we would like to mention Agmon's pioneering work
\cite{Agmon} for perturbations of the Laplacian,
the abstract theory
developed by Ben-Artzi and Devinatz \cite{BAD} mainly for
potential perturbations.
Laplacians in exterior domains were treated, e.\,g., by
Leis and Roach \cite{LR}.
More recently Mourre estimates led to many new results,
cf., e.\,g., Amrein, Boutet de Monvel, and Georgescu
\cite{ABG} for an overview and references. In particular
Pearson \cite{Pearson} treats potential
perturbations of the Laplacian with this approach,
Iwashita \cite{Iwa} differential operators in exterior domains.
We want to consider the case where a differential operator is
perturbed by introducing additional Dirichlet boundary conditions
on some set, i.\,e., domain perturbations. This includes,
but is not limited to, differential operators in exterior domains.
We present an abstract theory that includes such
perturbations as well as other types
of regular and singular perturbations.
Our approach is similar to the one presented in \cite{BAD},
but more general: The abstract theory in
\cite{BAD} is geared
towards potential perturbations; while it does cover other situations as
well
it is not applicable in our case.
In what sense do we want to perturb $H_1$?
\cite{BAD} treats the case where $H_2=H_1+V$ with $V$ some suitable
operator, the main focus being on multiplication operators.
The key assumption here is that $VR_1(z)$ is a compact
operator from $X^*$ to $X$.
In the case of domain perturbations, however,
the difference of the operators $H_2$ and $H_1$ becomes meaningless.
For instance, if $H_1$ is the Laplacian on $\MR^2$ and $H_2$ the
Dirichlet
Laplacian on $\MR^2\setminus B_1(0)$ with $B_1(0)$ the unit ball around
the origin, then $H_2-H_1$ is identically zero on the intersection
of the domains.
To get around this difficulty one can in a first step consider the
difference of the resolvents $R_2(z)-R_1(z)$ and suppose that this
resolvent difference
is compact from $X^*$ to $X$. In general
$H_1$ and $H_2$ will be
defined on different Hilbert spaces, so we have to formulate a
two-space theory; we refer to Section \ref{2} for precise definitions.
There is one further complication: If one considers, e.\,g.,
domain perturbations of the Laplacian on $\MR^d$
one easily finds sufficient conditions for the compactness of
the semigroup differences
$e^{-tH_2}-e^{-tH_1}$. From this one can directly conclude
compactness of $R_2(z)^m-R_1(z)^m$ for $m\in\MN$ large enough.
For space dimension $d\geq 4$ proving
compactness of $R_2(z)-R_1(z)$ itself, however, is more problematic.
To avoid difficulties of this type we formulate our theory
in terms of differences of powers of the resolvents
rather than in terms of the differences of the resolvents themselves.
This requires some new technical considerations which will be presented
in Section \ref{2} and at the beginning of Section \ref{3}.
In Section \ref{3} we develope the abstract theory,
leading to our main result, Theorem \ref{35}.
Our approach is classical in concept, relying strongly on
compactness arguments and in particular on the Fredholm
alternative. In fact, aside from the considerations mentioned above
and some technical simplifications, the line of argument
is quite close to \cite{BAD}.
In Section \ref{4} we specialize to the case where $X$ is a weighted
$L^2$-space. We formulate a proposition which essentially
is the 'standard bootstrap argument'.
This argument is used in many proofs of
the limiting absorption principle; we prove it on an abstract level.
In Section \ref{5}, we apply our theory
to study potential and domain perturbations of Feller operators.
A Feller operator is an operator defined as the generator of a strong
Markov
process with the Feller property, cf.\ Demuth and van Casteren
\cite{DvCBuch}, \cite{DvCFrame}, \cite{DvCHS}. We assume that a limiting
absorption principle holds for the unperturbed operator $H_1$
on $L^2(M)$ ($M$ some appropriate measure space)
and consider a perturbed operator $H_2=(H_1+V)_\Sigma$ that is produced
by restricting $H_1+V$ to a set $\Sigma\subset M$ via Dirichlet
boundary conditions. We give sufficient
conditions in terms of $V$ and the equilibrium potential of the
set $\Gamma=M\setminus\Sigma$ to ensure that a limiting absorption
principle holds for $H_2$.
If, for instance, $H_1$ is the Laplacian our theory allows to
treat the usual short range potentials, but it also allows domain
perturbations by sets $\Gamma$ which may be unbounded --
the condition we impose on $\Gamma$ is slightly stronger than requiring
that
its capacity is finite.
\section{Notation and Assumptions}
\setcounter{equation}{0}\label{2}
Let $(\hr_1,\scl{\cdot}{\cdot}_1)$ and
$(\hr_2,\scl{\cdot}{\cdot}_2)$ be Hilbert spaces,
$J_2$ a bounded map from $\hr_1$ to $\hr_2$ with
$J_2 J_2^*$ equal to the identity on $\hr_2$, so
$J_2^*J_2$ is an orthogonal projection on $\hr_1$.
To allow a more concise notation we also
introduce $J_1=J_1^*=1$ the identity in $\hr_1$.
The initial operator $H_1$ acts in $\hr_1$,
the perturbed operator $H_2$ in $\hr_2$.
As usual
$\sigma(H_k)$, $\rho(H_k)$, and $R_k(z)$ denote spectrum,
resolvent set, and resolvent, respectively,
of the operator $H_k$, $k=1,2$.
Let the Hilbert space
$(X,\scl{\cdot}{\cdot}_X)$ be densely
and continuously embedded in $\hr_1$, $(X^*,\scl{\cdot}{\cdot}_{X^*})$
its dual. Instead of identifying $X$ and $X^*$ we prefer to identify
$u\in\hr_1$ with the functional $\scl{u}{\cdot}_1$ it generates on
$X$, i.e., we consider $X^*$ as the completion of $\hr_1$
under the norm
$\norm{\cdot}_{X^*}$:
$\norm{v}_{X^*}=\sup_{u\in X}\frac{\scl{v}{u}_1}{\norm{u}_X}$. Thus
$X\subset\hr_1\subset X^*$ with dense and continuous embeddings.
Here in obvious notation $\norm{\cdot}_1$, $\norm{\cdot}_2$,
$\norm{\cdot}_X$, and $\norm{\cdot}_{X^*}$ denote the norms in $\hr_1$,
$\hr_2$, $X$, and $X^*$, respectively.
Generally, unless defined otherwise, whenever
${\mathcal K}$ is a normed space we will denote its norm by
$\norm{\cdot}_{\mathcal K}$.
As usual $\BO({\mathcal K_1},{\mathcal K_2})$ is the space of
bounded operators from a Hilbert space
$\mathcal K_1$ into another Hilbert space $\mathcal K_2$,
$\BO_\infty({\mathcal K_1},{\mathcal K_2})$ the space of compact
operators,
$\BO_2({\mathcal K_1},{\mathcal K_2})$ the space of Hilbert Schmidt
operators. The norm in $\BO({\mathcal K_1},{\mathcal K_2})$
is usually written as $\norm{\cdot}_{\BO({\mathcal K_1},{\mathcal
K_2})}$,
the norm in $\BO_2({\mathcal K_1},{\mathcal K_2})$ as
$\norm{\cdot}_{HS}$.
We will abuse notation slightly and use the same symbol for bounded
operators and their
restrictions and extensions.
For instance, $R_1(z)$ will not only be used for the resolvent
of $H_1$ (in $\BO(\hr,\hr)$) but also for its restriction to
$\BO(X,X)$ as well as its extension to $\BO(X^*,X^*)$. (That
$R_1(z)$ indeed maps $X$ to $X$ is part of Hypothesis \ref{21} below.)
Similarly we use the same symbol for operators and for their integral
kernels. For instance if $e^{-tH_1}$ is a semigroup of operators which
has an integral kernel we will denote this integral kernel
by $e^{-tH_1}(\cdot,\cdot)$.
Our basic assumptions on the operators $H_1$ and $H_2$ can then be
formulated as follows.
\begin{hyp}\label{21} Suppose that
for $k=1,2$ the operators $H_k$ act as self-adjoint semibounded
operators
in $\hr_k$ with domains $\db(H_k)$;
\begin{equation}\label{201}
\scl{H_k u}{u}_k\geq c_0\qquad\text{for all }u\in\db(H_k)
\end{equation}
with some constant $c_0\in\MR$. Moreover, suppose $H_k$ are consistent
with the space $X$ in
the sense that their resolvents $R_k(z)=(H_k-z)^{-1}$ satisfy
\begin{equation}\label{202}
J_k^* R_k(z)J_k\in\BO(X,X)\qquad\text{for all }z\in\rho(H_k)
\end{equation}
the resolvents being continuous functions of $z$ in $\BO(X,X)$.
\end{hyp}
As explained above \eqref{202} is a slight abuse of notation, because we
have
used the same symbol for the restriction to $X$ as for the operators in
$\hr_1$.
We will assume that a limiting absorption
principle holds for the operator $H_1$.
\begin{hyp}\label{22}
Let $\Delta$ be a given open subset of $\MR$: Assume that
for all $\lambda\in\Delta$
\begin{equation}\label{204}
R_1^{\pm}(\lambda):=\lim_\edz R_1(\lpm)
\end{equation}
exists and is continuous with respect to $\lambda$ in $\BO(X^*,X)$.
\end{hyp}
Finally we need our compactness assumption.
\begin{hyp}\label{22a}
Suppose there exists an $a0$ small enough $z=\lpm$ satisfies
$0<\abs{\arg(z-a)}<\frac{2\pi}{m}$ and the representation \eqref{207}
holds.
Hence
\begin{align}\label{212a}
&J_k^*F_k(z)J_k
\\ \nn
&\quad=(z-a)^m\prod_{j=0}^{m-1}\Bigl\{(-w_j)
\bigl[J_k^*J_k+w_j(z-a)J_k^*R_k\bigl(a+w_j(z-a)\bigr)J_k\bigr]\Bigr\}
\end{align}
The factors for $j\geq 1$ in this product are bounded and continuous
with respect to $z$ (even in the limit $z=\lpm\rightarrow\lambda$) in
$\BO(X,X)$ by \eqref{202}, because $a+w_j(z-a)$ stays away
from $[a,\infty)$ (even in the limit).
If $J_k^*R_k^\pm(\lambda)J_k$ exists and is continuous in $\BO(X,X^*)$
the remaining factor ($j=0$)
in \eqref{212a},
\begin{equation}\label{213}
J_k^*J_k+(z-a)J_k^*R_k(z)J_k
\end{equation}
has a well defined continuous limit in $\BO(X,X^*)$ for
$z=\lpm\rightarrow\lambda$. If, on the other hand,
$J_k^*F_k^\pm(\lambda)J_k$ exists and is continuous
in $\BO(X,X^*)$ we can invert
the factors for $j=1,\dots,m-1$ in \eqref{212a} in $\BO(X,X)$ to
conclude the existence and continuity of $J_k^*R_k^\pm(\lambda)J_k$ in
$\BO(X,X^*)$.
The representation \eqref{212} is an obvious consequence of
\eqref{207} (resp.\ \eqref{212a}),
the existence of the limits and \eqref{202}. The factors in
\eqref{207} commute; by duality we can consider $J_k^*R_k(z)J_k$ as an
operator in $\BO(X^*,X^*)$ as well as in $\BO(X,X)$, hence the factors
in
\eqref{212a} commute as well even when we restrict
$J_k^*F_k(z)J_k$ to an operator in $\BO(X,X^*)$. By passing to the limit
$z=\lpm\rightarrow\lambda\in\MR$ the same holds for \eqref{212}.
\end{proof}
With this we can finally formulate our last hypothesis.
\begin{hyp}\label{25}
Suppose that for
each compact subset $K$ of $\Delta$ there is a constant $c_K>0$
such that for all $\lambda\in K$ and for all $u\in X$ with
$\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$, $R_1^\pm(\lambda)u$ lies
in $\hr_1$ with
\begin{equation}\label{214}
\norm{R_1^\pm(\lambda)u}_1\leq c_K\norm{u}_X\,.
\end{equation}
\end{hyp}
\begin{rem}\label{27}
Hypothesis \ref{25}, as it is presented here, is rather awkward,
because it intertwines conditions on $H_1$ and conditions on $H_2$
via the operator $E$. Thus it is hard
to check directly. One way to get around this difficulty is to replace
it by the stronger condition (H).
(H) For each compact subset $K$ of $\Delta$ there is a constant $c_K>0$
such that \eqref{214} holds for all
$u\in X$ with $R_1^+(\lambda)u=R_1^-(\lambda)u$.
That condition (H) does in fact imply Hypothesis \ref{25} will follow
from Lemma \ref{31}. When we work with this stronger assumption,
however,
we do not gain optimal results.
Let us illustrate this with the standard
example, the Laplacian $-\triangle$ on $\MR^d$.
There we can set $X=\LP{}$
as defined in \eqref{401}. Hypothesis \ref{22} is satisfied for
$\varphi=(1+\abs{x})^s$, $s>\frac12$, but condition (H) only holds
for $s>1$ (cf.\ \cite{BAD}). But we would like to use a space $X$
that is as large as possible in order to keep Hypothesis \ref{22a}
as weak as possible.
In Section \ref{4} we will give a sufficient condition that guarantees
Hypothesis \ref{25} in the case of weighted $L^2$-spaces and is
significantly weaker than condition (H).
Namely, if we take $H_2=-\triangle+V$ in the above example
than if we choose $s>1$ so that condition (H) is satisfied Hypothesis
\ref{22a} only admits potentials which decay like $\abs{x}^{-2-\gamma}$,
$\gamma>0$ as $\abs{x}$ tends to infinity. The hypotheses from Section
\ref{4}, on the other hand, allow to use $s>\frac12$ and hence a
decay as $\abs{x}^{-1-\gamma}$ is sufficient.
Of course in the case of potential perturbations of the Laplacian
better results than this are known in the literature
(cf.\ \cite{Pearson} for the optimal conditions);
our main goal, however, is the treatment
of singular perturbations, not potential perturbations.
\end{rem}
We note that, aside from Hypothesis \ref{22a}, our basic
hypotheses are quite similar to those in Ben-Artzi and
Devinatz \cite{BAD}.
In particular Hypothesis \ref{22}
corresponds to Definition 2.1 in \cite{BAD}
and Hypothesis \ref{25} to Assumption 3.2.
Our conditions are slightly better insofar as we
need only continuity where \cite{BAD}
uses H\"older continuity, because we formulate the limiting
absorption principle in terms of
the resolvents rather than in terms of spectral measures.
The main change, however, is that
we use the difference of powers
of resolvents in Hypothesis \ref{22a}. Using resolvents
allows us to treat singular perturbations which could not be handled
otherwise. (This trick was also used
by Ben-Artzi, Dermenjian, and Guillot \cite{BDG}.)
By considering their powers
we can prove compactness in a wider range of cases.
(It is not clear whether or not $E\in\BO_\infty(X^*,X)$ implies
$J_2^*R_2(a)J_2-R_1(a)\in\BO_\infty(X^*,X)$; we can neither prove the
implication nor give a counterexample.)
\section{Two-Space Theory for Limiting Absorption Principle}
\setcounter{equation}{0}\label{3}
Throughout this section we will assume that the hypotheses from
Section \ref{2}, namely Hypotheses \ref{21}, \ref{22}, \ref{22a}, and
\ref{25} are satisfied.
The basic idea of our approach is to derive properties
of $F_2(z)$ from those of $F_1(z)$. They are related via the
following formula. For $z\in\MC\setminus\{a\}$ with
$0<\abs{\arg(z-a)}<\frac{2\pi}{m}$,
\begin{align}\label{301}
&J_2^*J_2 F_1(z)-J_2^*F_2(z)J_2
\\ \nn
&\quad=J_2^*F_2(z)\bigl(R_2(a)^m-(z-a)^{-m}\bigr)J_2 F_1(z)
\\ \nn
&\qquad\quad-J_2^*F_2(z)J_2\bigl(R_1(a)^m-(z-a)^{-m}\bigr)F_1(z)
\\ \nn
&\quad=J_2^*F_2(z)J_2\bigl[J_2^*R_2(a)^m J_2-R_1(a)^m\bigr]F_1(z)
\end{align}
follows from the definition \eqref{207} of $F_k(z)$.
With $E=J_2^*R_2(a)^m J_2-R_1(a)^m$ this can be rewritten as
\begin{equation}\label{302}
J_2^*J_2 F_1(z)=J_2^*F_2(z)J_2\bigl[1+EF_1(z)\bigr].
\end{equation}
This operator identity holds in $\BO(\hr_1,\hr_1)$ and by
extension in $\BO(X,X^*)$. By Hypothesis \ref{22}
$E\in\BO_\infty(X^*,X)$ and hence $EF_1(z)\in\BO_\infty(X,X)$.
Thus we may use the Fredholm alternative: For fixed $z$ either
$\bigl(1+EF_1(z)\bigr)$ is invertible in $\BO(X,X)$ or there exists
an $u\in X$, $u\not =0$ with $\bigl(1+EF_1(z)\bigr)u=0$.
Since we want to use this formula to study the limit
$z=\lpm\rightarrow\lambda$ it will be sufficient to
study the limit operator $1+EF_1^\pm(\lambda)$ for
$\lambda\in\Delta$ and its invertibility in $\BO(X,X)$.
We note that one can in fact easily verify along the line of
proof of Lemma \ref{34} that for $0<\abs{\arg(z-a)}<\frac{2\pi}{m}$ the
operator $1+EF_1(z)$ is invertible, because otherwise $H_2$ would have
a non-real eigenvalue.
Now we need to analyze the sets where $1+EF_1^+(\lambda)$ and where
$1+EF_1^-(\lambda)$ are not invertible in more detail. First we note
that these sets in fact coincide.
\begin{lem}\label{31}
If for some $\lambda\in\Delta$ either $1+EF_1^+(\lambda)$ or
$1+EF_1^-(\lambda)$ is not invertible in $\BO(X,X)$ then there
exists an $u\in X$, $u\not=0$ such that
$u=-EF_1^+(\lambda)u=-EF_1^-(\lambda)u$ and hence neither of the
operators $1+EF_1^\pm(\lambda)$ is invertible.
\end{lem}
\begin{proof}
Without loss suppose $1+EF_1^+(\lambda)$ is not invertible. By the
Fredholm alternative there is an $u\in X$, $u\not=0$ with
$u=-EF_1^+(\lambda)u$. Since $E$ is self-adjoint we find that
\begin{equation}\label{303}
\scl{F_1^+(\lambda)u}{u}_1=-\scl{F_1^+(\lambda)u}{EF_1^+(\lambda)u}_1
\end{equation}
is real.
By inspection of \eqref{212} we see that each factor in this
formula except for the first one
$1+(\lambda-a)R_1^\pm(\lambda)$ is either self-adjoint or occurs
together with its adjoint. Since $R_1^+(\lambda)^*=R_1^-(\lambda)$
we see $F_1^+(\lambda)^*=F_1^-(\lambda)$.
Hence \eqref{303} implies that
\begin{equation}\label{304}
\Scl{\bigl(F_1^+(\lambda)-F_1^-(\lambda)\bigr)u}{u}_1=0.
\end{equation}
We claim that
\begin{equation}\label{304a}
\bigl\langle
i\bigl(F_1^+(\lambda)-F_1^-(\lambda)\bigr)v,v\bigr\rangle_1\geq 0
\qquad\text{for all }v\in X.
\end{equation}
This can be seen as follows. By \eqref{212}
\begin{align}\label{305}
&F_1^+(\lambda)-F_1^-(\lambda)
=(\lambda-a)^{m+1}\bigl[-R_1^+(\lambda)+R_1^-(\lambda)\bigr]
\\ \nn
&\qquad\qquad\times\prod_{j=1}^{m-1}\Bigl\{(-w_j)\bigl[1
+w_j(\lambda-a)R_1(a+w_j(\lambda-a)\bigr)\bigr]\Bigr\}
\\ \nn
&\quad=\lim_{\edz}(\lambda-a)^{m+1}
\bigl[-R_1(\lambda+i\eps)+R_1(\lambda-i\eps)\bigr]
\\ \nn
&\qquad\qquad\times\prod_{j=1}^{m-1}\Bigl\{(-w_j)\bigl[1
+w_j(\lambda-a)R_1(a+w_j(\lambda-a)\bigr)\bigr]\Bigr\}.
\end{align}
Since all factors on the right hand side of \eqref{305} commute we only
have to establish positivity for individual factors, \eqref{304a}
will then hold by passing to the limit.
The first factor on the right hand side of \eqref{305} is
\begin{align}\label{306}
&-R_1(\lambda+i\eps)+R_1(\lambda-i\eps)
\\ \nn
&\quad=-2i\eps R_1(\lambda+i\eps)R_1(\lambda-i\eps)
=-2i\eps\int\limits_\MR\frac{E_1(d\nu)}{(\nu-\lambda)^2+\eps^2}\ ,
\end{align}
$E_1(\cdot)$ denoting the spectral measure associated with $H_1$.
The integral on the right hand side of \eqref{306} is a positive
operator for all $\eps>0$.
If $m$ is even $w_{m/2}=-1$, the corresponding factor
in \eqref{305} is
\begin{equation}\label{307}
1-(\lambda-a)R_1(2a-\lambda).
\end{equation}
This operator is positive,
because $\norm{(\lambda-a)R_1(2a-\lambda)}_1
\leq\frac{\lambda-a}{\lambda-2a+c_0}<1$.
For each of the remaining factors containing some $w_j$ there is
a corresponding adjoint factor containing $\ol{w_j}$. The product
of an operator with its adjoint is positive, thus we indeed
find
\begin{align}\label{307a}
&\Bigl\langle i\bigl[-R_1(\lambda+i\eps)+R_1(\lambda-i\eps)\bigr]
\\ \nn
&\qquad\times\prod_{j=1}^{m-1}\Bigl\{(-w_j)\bigl[1
+w_j(\lambda-a)R_1(a+w_j(\lambda-a)\bigr)
\bigr]\Bigr\}v,v\Bigr\rangle_1
\geq 0
\end{align}
for all $v\in\hr_1$ and a fortiori for
all $v\in X$. Hence passing to the limit $\edz$ \eqref{304a} holds.
Together with \eqref{304} this implies
\begin{equation}\label{309}
F_1^+(\lambda)u=F_1^-(\lambda))u,
\end{equation}
proving the lemma.
\end{proof}
Now, as \eqref{302} suggests, we find that indeed
the limit $F_2^\pm(\lambda)$ is well defined when $1+EF_1^\pm(\lambda)$
is invertible in $\BO(X,X)$. More precisely, we obtain the following
result.
\begin{lem}\label{32}
If $1+EF_1^\pm(\lambda_0)$ is invertible in $\BO(X,X)$ for some
$\lambda_0\in\Delta$ then there exists a neighborhood $U\subset\Delta$
of $\lambda_0$ such that for all $\lambda\in U$
\begin{equation}\label{310}
J_2^*R_2^\pm(\lambda)J_2=\lim_\edz J_2^*R_2(\lpm)J_2
\end{equation}
exists and is continuous with respect to $\lambda$ in
$\BO(X,X^*)$.
\end{lem}
\begin{proof}
Let $z=\lambda_0+\delta\pm i\eps$, $\delta\in\MR$, $\eps>0$,
be close to $\lambda_0$. Then
\begin{align}\label{311}
1+EF_1(z)=\bigl(1+EF_1^\pm(\lambda_0)\bigr)\Bigl[1+
\bigl(1+EF_1^\pm(\lambda_0)\bigr)^{-1}E
\bigl(F_1(z)-F_1^\pm(\lambda_0)\bigr)\Bigr].
\end{align}
The continuity of $F_1(z)$ respectively $F_1^\pm(\lambda)$
in $\BO(X,X^*)$ allows us to pick $\eps,\delta$ small
enough so that
$\norm{\bigl(1+EF_1^\pm(\lambda_0)\bigr)^{-1}
E\bigl(F_1(z)-F_1^\pm(\lambda_0)\bigr)}_{\BO(X,X)}<1$
and hence the expression in \eqref{311} is invertible with
continuous inverse. That is, for $\lambda\in U$ where
$U$ is some small neighborhood of $\lambda_0$,
\begin{equation}\label{312}
\bigl[1+EF_1^\pm(\lambda)\bigr]^{-1}
=\lim_\edz\bigl[1+EF_1(\lpm)\bigr]^{-1}\in\BO(X,X)
\end{equation}
exists and is continuous with respect to $\lambda$.
Hence we can conclude from \eqref{302} that
\begin{align}\label{313}
&J_2^*F_2^\pm(\lambda)J_2:=\lim_\edz J_2^*F_2(\lpm)J_2
=\lim_\edz J_2^*J_2 F_1(\lpm)\bigl[1+EF_1(\lpm)\bigr]^{-1}
\end{align}
exists and is continuous with respect to $\lambda$ in
$\BO(X,X^*)$ for $\lambda\in U$. The claim
then follows from Lemma \ref{24}
\end{proof}
If $J_2^*R_2^\pm(\lambda)J_2$ exists in $\BO(X,X^*)$ in some
neighborhood
$U\subset\Delta$ of $\lambda_0$ we know (cf.\ \cite{RS4}, Thm. XIII.19
and
XIII.20) that the spectrum $\sigma(H_2)$ of $H_2$ is purely absolutely
continuous in $U$. Hence we need to analyze the set where
$1+EF_1^\pm(\lambda)$ is not invertible in more detail. We begin
with a simple observation.
\begin{lem}\label{33}
If $\lambda\in\Delta$ is an eigenvalue of $H_2$ with eigenvector
$\psi$ then $u:=-EJ_2^*\psi\in X$ satisfies
$\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$, $u\not=0$.
\end{lem}
\begin{proof}
$u\in X$ is clear from the definition, since $E\in\BO_\infty(X^*,X)$.
Next we want to establish $u\not=0$. We calculate
\begin{equation}\label{314}
u=-\bigl[J_2^*R_2(a)^mJ_2-R_1(a)^m\bigr]J_2^*\psi
=-\bigl[(\lambda-a)^{-m}-R_1(a)^m\bigr]J_2^*\psi,
\end{equation}
where we have used the fact that $J_2J_2^*\psi=\psi$
is an eigenfunction of $H_2$.
Apply the factorization \eqref{206}
and use the fact that all its factors commute. The expression in
\eqref{314} cannot vanish unless
\begin{equation}\label{315}
\bigl[R_1(a)-w_j(\lambda-a)^{-1}\bigr]J_2^*\psi=
(\lambda-a)^{-1}\bigl[\lambda-a-w_j(H_1-a)\bigr]R_1(a)J_2^*\psi=0
\end{equation}
for some $j\in\{0,\dots,m-1\}$. But \eqref{315} would imply
that $\zeta=a+\frac{1}{w_j}(\lambda-a)$ is an eigenvalue of $H_1$.
For $j\not=0$ this is impossible because then
$\zeta\in\MC\setminus\MR$, for $j=0$ $\zeta=\lambda\in\Delta$
cannot be an eigenvalue of $H_1$ by \eqref{204}.
It remains to verify $\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$. We
calculate
\begin{align}\label{316}
&EF_1(\lpm)u=-EF_1(\lpm)\bigl[(\lambda-a)^{-m}-R_1(a)^m\bigr]J_2^*\psi
\\ \nn
&\quad=E\bigl[R_1(a)^m-(\lpm-a)^{-m}\bigr]^{-1}
\bigl[R_1(a)^m-(\lambda-a)^{-m}\bigr]J_2^*\psi
\\ \nn
&\quad=EJ_2^*\psi+EF_1(\lpm)\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]
J_2^*\psi\,.
\end{align}
In order to prove our claim we thus have to verify
$EF_1(\lpm)\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]J_2^*\psi
\xrightarrow[\edz]{} 0$. By Lemma \ref{24}
\begin{equation}\label{317}
F_1(\lpm)\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]
\xrightarrow[\edz]{} 0\text{ in }\BO(X,X^*)
\end{equation}
since $(\lpm-a)^{-m}-(\lambda-a)^{-m}\xrightarrow[\edz]{} 0$.
With the factorization \eqref{207} and a similar factorization
for $\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]$ we calculate
\begin{align}\label{318}
&F_1(\lpm)\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]
\\ \nn
&\quad=\prod_{j=0}^{m-1}\Bigl\{(-w_j)\bigl[
1+w_j(\lpm-a)R_1\bigl(a+w_j(\lpm-a)\bigr)\bigr]
\\ \nn
&\qquad\qquad\times
\bigl(\lambda-a-w_j(\lpm-a)\bigr)(\lambda-a)^{-1}\Bigr\}.
\end{align}
This operator remains bounded in $\BO(\hr_1,\hr_1)$ as $\eps$ tends
to $0$, because the only factor that could possibly be unbounded in
this limit, namely the one for $j=0$, is bounded because of
\begin{equation}\label{319}
\Norm{R_1(\lpm)(\mp i\eps)}_{\BO(\hr_1,\hr_1)}\leq 1.
\end{equation}
Boundedness in $\BO(\hr_1,\hr_1)$ and convergence on the dense
subset $X\subset\hr_1$ imply
\begin{equation}\label{320}
F_1(\lpm)\bigl[(\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr]
\xrightarrow[\edz]{w} 0\text{ in }\BO(\hr_1,\hr_1)
\end{equation}
($\xrightarrow{w}$ denoting weak convergence). Thus \eqref{316}
yields
\begin{equation}\label{321}
EF_1^\pm(\lambda)u=\lim_\edz EF_1(\lpm)u=EJ_2^*\psi=-u.
\end{equation}
\end{proof}
Using Hypothesis \ref{25} we can establish the converse of the
previous lemma.
\begin{lem}\label{34}
Suppose $\bigl(1+EF_1^\pm(\lambda)\bigr)$ is {\em not} invertible
in $\BO(X,X)$ for some $\lambda\in\Delta$; let $u\in X$, $u\not=0$
satisfy $u=-EF_1^\pm(\lambda)u$. Then $\lambda$ is an
eigenvalue of $H_2$ with eigenvector $\psi=R_2(a)J_2F_1^\pm(\lambda)u
\in\db(H_2)$, $\psi\not=0$.
\end{lem}
\begin{proof}
By Hypothesis \ref{25} $R_1^\pm(\lambda)u\in\hr_1$ and hence by
\eqref{212} $v=F_1^\pm(\lambda)u\in\hr_1$. Hence in the space $X^*$
\begin{align}\label{322}
&\bigl(R_1(a)^{m}-(\lambda-a)^{-m}\bigr)v
=\lim_\edz\bigl(R_1(a)^m-(\lambda-a)^{-m}\bigr)F_1(\lpm)u
\\ \nn
&\quad=u+\lim_\edz\bigl((\lpm-a)^{-m}-(\lambda-a)^{-m}\bigr)F_1(\lpm)u
=u.
\end{align}
Equation \eqref{322} implies $v\not=0$ and also allows us to rewrite
$\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$ as
\begin{equation}\label{323}
\bigl[\bigl(R_1(a)^m-(\lambda-a)^{-m}\bigr)+E\bigr]v
=\bigl[J_2^*R_2(a)^m J_2-(\lambda-a)^{-m}\bigr]v=0.
\end{equation}
Multiplying \eqref{323} by the projection $1-J_2^*J_2$ we see
$(1-J_2^*J_2)v=0$, hence $J_2^*J_2 v\not=0$ and with
\eqref{206} equation \eqref{323} becomes
\begin{align}\label{324}
&J_2^*\bigl[R_2(a)^m-(\lambda-a)^{-m}\bigr]J_2v
\\ \nn
&\quad=J_2^*\prod_{j=0}^{m-1}\Bigl\{(\lambda-a)^{-1}
\bigl[\lambda-a-w_j(H_2-a)\bigr]R_2(a)\Bigr\}J_2 v=0.
\end{align}
Since $a+\frac{1}{w_j}(\lambda-a)\in\MC\setminus
[a,\infty)\subset\rho(H_2)$
for $j=1,\dots,m-1$ this implies that the factor for $j=0$ satisfies
\begin{equation}\label{325}
(\lambda-H_2)R_2(a)J_2 v=0.
\end{equation}
\end{proof}
Lemmas \ref{33} and \ref{34} yield our main abstract theorem.
\begin{thm}\label{35}
Suppose $H_1$ and $H_2$ satisfy Hypotheses \ref{21}, \ref{22},
\ref{22a},
and \ref{25}. Then $\sigma(H_2)\cap\Delta$ is, except for possibly
a discrete set $\sigma_p(H_2)\cap\Delta$
of eigenvalues of finite multiplicity,
purely absolutely continuous.
For all $\lambda\in\Delta\setminus\sigma_p(H_2)$
\begin{equation}\label{325a}
J_2^*R_2^\pm(\lambda)J_2:=\lim J_2^*R_2(\lpm)J_2
\end{equation}
exists and is continuous in $\BO(X,X^*)$.
\end{thm}
\begin{proof}
(i) We begin by establishing that the set of eigenvalues of $H_2$,
$\sigma_p(H_2)$ is
discrete in $\Delta$. Suppose not. Then there exists a
sequence $(\lambda_k)_\kin\subset\bigl(\Delta\cap\sigma_p(H_2)\bigr)$
with $\lambda_k\askinf\lambda\in\Delta$; the limit value $\lambda$
may or may not be an eigenvalue. By Lemma \ref{33} for each $\lambda_k$
there
exists a vector $u_k\in X$ with
\begin{equation}\label{326}
\bigl(1+EF_1^\pm(\lambda_k)\bigr)u_k=0,\qquad \kin;
\end{equation}
without loss suppose $\norm{u_k}_X=1$. Since the sequence
$(u_k)_\kin$ is bounded there exists a subsequence, without loss
the sequence itself, such that
$u_k-u_l\xrightarrow[k,l\rightarrow\infty]{w}0$ in $X$
($\xrightarrow{w}$ denoting weak convergence). From \eqref{326} we infer
\begin{align}\label{327}
u_k-u_l=&-EF_1^\pm(\lambda)(u_k-u_l)
\\ \nn
&-E\bigl(F_1^\pm(\lambda_k)-F_1^\pm(\lambda)\bigr)u_k
+E\bigl(F_1^\pm(\lambda_l)-F_1^\pm(\lambda)\bigr)u_l\,.
\end{align}
the first term on the right hand side of \eqref{327} converges strongly
to $0$ in $X$, because $EF_1^\pm(\lambda)\in\BO_\infty(X,X)$. The
remaining
two terms converge because $F_1^\pm(\lambda)$ is continuous in
$\lambda$ and $(u_k)_\kin$ is bounded.
Hence we have in fact strong convergence,
$u_k-u_l\xrightarrow[k,l\rightarrow\infty]{}0$ in $X$.
Set $u=\lim_\kin u_k$. From $\norm{u_k}_X=1$ we conclude
$\norm{u}_X=1$; from \eqref{326} and the continuity of
$F_1^\pm(\cdot)$ that
\begin{equation}\label{328}
\bigl(1+EF_1^\pm(\lambda)\bigr)u=0.
\end{equation}
By Lemma \ref{34} for $\kin$ there exist vectors
$\psi_k,\psi\in\db(H_2)$ with $(H_2-\lambda_k)\psi_k=0$,
$(H_2-\lambda)\psi=0$. Together with Hypothesis \ref{25}
this lemma also yields
\begin{equation}\label{329}
\norm{\psi_k}_2\leq\norm{R_2(a)}_{\BO(\hr_2,\hr_2)}
\norm{J_2}_{\BO(\hr_1,\hr_2)}
\norm{F_1^\pm(\lambda_k)u_k}_1
\leq c
\end{equation}
with $c>0$.
Since each $\psi_k$ is eigenvector to a different eigenvalue $\lambda_k$
it is orthogonal to all other eigenvectors $\psi_l$,
$l\not=k$.
Bessel's inequality then implies that
$\DS\widetilde{\psi}_k=\frac{\psi_k}{\norm{\psi_k}_2}
\xrightarrow[k\rightarrow\infty]{w}0$ in $\hr_2$ and hence by
\eqref{329} $\psi_k\xrightarrow[k\rightarrow\infty]{w}0$ in
$\hr_2$ and a fortiori
\begin{align}\label{330}
J_2^*\psi_k\xrightarrow[k\rightarrow\infty]{w}0
\end{align}
in $X^*$. On the other hand $u_k\rightarrow u$ and the continuity
of $F_1^\pm(\cdot)$ imply
\begin{equation}\label{331}
J_2^*\psi_k=J_2^*R_2(a)J_2F_1^\pm(\lambda_k)u_k\askinf J_2^*\psi
\quad\text{in }X^*.
\end{equation}
Since $\norm{u}_X=1$ we know $\psi\not=0$ by Lemma \ref{34}
in contradiction to \eqref{330}.
(ii) A similar argument may be used to show that each eigenvalue
can have only finite multiplicity. Suppose not. Then there exists
a sequence $(\psi_k)_\kin$ of orthogonal eigenvectors. We can
define $u_k$ by Lemma \ref{33}, norm by $\norm{u_k}_X=1$
and proceed as in (i).
(iii) If $\lambda_0\in\Delta$ is not an eigenvalue of $H_2$
then by Lemma \ref{34} $\bigl(1+EF_1^\pm(\lambda_0)\bigr)$
is invertible in $\BO(X,X)$ and by Lemma \ref{32}
the limit $J_2^*R_2^\pm(\lambda)J_2$ exists for all $\lambda$ in some
neighborhood $U\subset\Delta$ of $\lambda_0$. By the general theory
(cf., e.\,g., \cite{RS4}, Thm. XIII.19, XIII.20) this establishes
the claim.
\end{proof}
\begin{rem}\label{37}
We have established that the point spectrum $\sigma_p(H_2)$
of $H_2$ does not accumulate in $\Delta$. Of course it might
still accumulate at the boundary of $\Delta$. We can, however, exclude
this behavior, if Hypothesis \ref{22} holds for some $\Delta$ with
$\sigma(H_2)\subset\Delta$.
\end{rem}
\section{A sufficient condition for Hypothesis \ref{25} for
weighted $L^2$-spaces}
\setcounter{equation}{0}\label{4}
In most applications of the limiting absorption principle
some weighted $L^2$-space is used for $X$. In this setup we can replace
Hypothesis \ref{25} by a different Hypothesis \ref{41} if we
strengthen the assumption on $E$, i.\,e., Hypothesis \ref{22a} slightly.
The main advantage of this approach is that,
while Hypothesis \ref{25} intertwines
conditions on $H_1$ and conditions on $H_2$ in a very implicit way and
hence is hard to check, this new hypothesis will impose separate
conditions
which can be verified much more easily.
We work in the following setting.
Let $M$ be a measure space with Borel field $\EM$, $\Sigma\in\EM$ a
measurable set. We define $\hr_1=L^2:=L^2(M)$,
$\hr_2=L^2(\Sigma)$ and
\begin{align}\label{401}
X=L^2_\varphi:=\bigl\{u\in L^2(E):\ \norm{u}_\varphi:=
\norm{\varphi u}<\infty\bigr\}\,.
\end{align}
Here $\varphi:\ M\rightarrow[1,\infty)$ is a measurable function.
(In this and the following section norm and scalar product
without a subscript denote the usual $L^2$-norm and
scalar product.)
$\LP{s}$, $s\geq 0$ is defined by a formula analogous to
\eqref{401}, $\LP{-s}$ as its dual $(L^2_{\varphi^s})^*$;
in particular $\LP{-1}=(L^2_{\varphi})^*$.
In this setup we can replace Hypotheses \ref{22}, \ref{22a}, and
\ref{25} by
\begin{hyp}\label{41}
Suppose there is an open set $\Delta\subset\MR$ such that for all
$\lambda\in\Delta$,
\begin{equation}\label{402}
R_1^\pm(\lambda):=\lim_\edz R_1(\lpm)
\end{equation}
exists and is continuous in $\BO(\LP{},\LP{-1})$.
Furthermore, assume that for
each compact subset $K$ of $\Delta$ there exists a constant
$c_K>0$ such that for all $\lambda\in K$ and all $u\in L^2_{\varphi^2}$
with \mbox{$R_1^+(\lambda)u=R_1^-(\lambda)u$}
\begin{equation}\label{403}
\norm{R_1^\pm(\lambda)u}\leq c_K\norm{u}_{\varphi^2}\,.
\end{equation}
\end{hyp}
and
\begin{hyp}\label{41a}
Suppose there exists an $a0$.
\end{hyp}
Hypothesis \ref{41a} is, as we will see in the proof of Proposition
\ref{43} a slight strengthening of Hypothesis \ref{22a} formulated
in the setting of weighted $L^2$-spaces.
Equation \eqref{402} is a reformulation of Hypothesis \ref{22}.
But what is the difference between \eqref{403} and
Hypothesis \ref{25}?
On one hand in Hypothesis \ref{25} some condition
is imposed on those $u$ for which $\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$,
while in \eqref{403} we impose a condition on the larger class of
all those $u$ for which $R_1^+(\lambda)u=R_1^-(\lambda)u$. This larger
class has the advantage of being far easier to describe.
On the other hand the condition that is imposed in Hypothesis \ref{25}
is stronger: There we require that $\norm{R_1^\pm(\lambda)u}$
can be estimated by the $L^2_\varphi$-norm of $u$,
in \eqref{403} we merely require that it can be estimated by the
(larger) $L^2_{\varphi^2}$-norm.
In a sense \eqref{403} involves the same power of
$\varphi$ as \eqref{402} while Hypothesis \ref{25} involves a smaller
power: \eqref{402} implies $\scl{R_1^\pm(\lambda)u}{v}\leq c_K
\norm{\varphi u}\,\norm{\varphi v}$; \eqref{403} is
of the form $\scl{R_1^\pm(\lambda)u}{v}\leq c_K\norm{\varphi^2
u}\,\norm{v}$.
Hypothesis \ref{25}, on the other hand, is of the form
$\scl{R_1^\pm(\lambda)u}{v}\leq c_K\norm{\varphi u}\,\norm{v}$.
In Section \ref{5} we will discuss Hypothesis \ref{41a} in
more detail for the case of potential and domain perturbations.
We will give a sufficient condition for this Hypothesis in terms of
the potential and the set on which we introduce additional
boundary conditions. Since we will always consider Hypothesis \ref{41}
as a given let us mention some operators for which it holds.
\begin{rem}\label{42b}
(i) If $H_1=-\triangle$ is the Laplacian on $\MR^d$ then Hypothesis
\ref{41} holds with $\Delta=(0,\infty)$ and $\varphi=(1+\abs{x})^s$,
$s>\frac12$, see \cite{BAD}.
(ii) Case (i) can be extended to functions of the Laplacian.
If $f\in C^2(\MR_+)$, $f(t)>C t^{\gamma/2}$ with
some $C,\gamma>0$, then
$H_1=f(-\triangle)$ satisfies the hypothesis with
$\Delta=(f(0),\infty)$,
$\varphi=\bigl(1+\abs{x}\bigr)^s$, $s>\frac12$, see \cite{BAN}.
In particular this includes the relativistic Hamiltonian
$\sqrt{-\triangle+1}-1$.
(iii) If $H_1$ is an elliptic operators with constant coefficients the
hypothesis holds as well, see \cite{BAD}.
\end{rem}
\begin{prop}\label{43}
Hypotheses \ref{41} and \ref{41a} imply Hypotheses \ref{22},
\ref{22a}, and \ref{25}.
\end{prop}
\begin{proof}
We can rewrite \eqref{404} as
$\varphi^{2+\gamma}E\in\BO_\infty(L^2,L^2)$. Then by Schauder's theorem
(cf., e.\,g., Kato \cite{Kato} Thm. III.4.10)
$E \varphi^{2+\gamma}=(\varphi^{2+\gamma}E)^*\in\BO_\infty(L^2,L^2)$.
Since $\varphi^{2+\gamma}E$ is a compact operator in a Hilbert space
we can approximate it by operators of finite rank (cf., e.\,g., Pietsch
\cite{Pietsch} Section 10.2), say $\varphi^{2+\gamma}E_k$. Now we use
the
interpolation theory as presented by Lions and Magenes \cite{LiMag}.
By the definition of the space $\LP{}$ (cf.\ \eqref{401})
the interpolation space according to this approach
(cf.\ \cite{LiMag} Chapter 1, Sections 3--5)
is $[\LP{},L^2]_\theta=
\bigl\{u\in L^2:\ \norm{\varphi^{1-\theta}
u}<\infty\bigr\}=\LP{1-\theta}$.
Then by the interpolation
theorem (cf.\ \cite{LiMag} Chapter I, Thm.\ 5.1)
$\norm{E-E_k}_{\BO(L^2_{\varphi^{-\theta}},L^2_{\varphi^{2+\gamma-\theta}})}
\askinf 0$ for all $\theta\in[0,2+\gamma]$. Hence
\begin{equation}\label{405}
E\in\BO_\infty(L^2_{\varphi^{-\theta}},L^2_{\varphi^{2+\gamma-\theta}})
\quad\text{for all }\theta\in[0,2+\gamma].
\end{equation}
A fortiori this also means
$E\in\BO_\infty(L^2_{\varphi^{-1}},L^2_{\varphi})$, i.\,e.,
Hypothesis \ref{22a} is satisfied. Hypothesis \ref{22} is
equation \eqref{402}, hence it only remains to verify Hypothesis
\ref{25}.
We define the spaces
\begin{equation}
\widetilde{L}_s(\lambda):=\bigl\{u\in L^2_{\varphi^s}:\
R_1^+(\lambda)u=R_1^-(\lambda)u\bigr\},\quad s\geq 1.
\end{equation}
Since $R_1^\pm(\lambda)\in\BO(L^2_{\varphi},L^2_{\varphi^{-1}})$
by \eqref{402} we know that for $s\geq 1$ the space $\LT{s}$ is a closed
subspace of $\LP{s}$. Further we note $\LT{s}=\LT{1}\cap\LP{s}$,
i.\,e., we can consider $\LT{s}$, $s\geq 1$ as a subspace of
$\LT{1}$. Again consider the interpolation spaces
$[\LT{1},\LT{2}]_\theta$ defined according to \cite{LiMag}.
Here $\LT{2}$ can be considered as
$\LT{2}=\bigl\{u\in\LT{1}:\ \norm{\varphi u}_{\LT{1}}<\infty\bigr\}$.
So, for $0\leq\theta\leq 1$,
\begin{equation}\label{407}
[\LT{1},\LT{2}]_\theta
=\bigl\{u\in\LT{1}:\ \norm{\varphi^\theta u}_{\LT{1}}<\infty\bigr\}
=\LT{1+\theta}.
\end{equation}
We know from \eqref{402} and \eqref{403} that
$R_1^\pm(\lambda)\in\BO(\LT{1},\LP{-1})\cap\BO(\LT{2},L^2)$ for all
$\lambda\in\Delta$ and that for each compact subset $K$ of
$\Delta$ we can find a constant $c_K>0$ such that for all $\lambda\in K$
\begin{equation}\label{408}
\norm{\Repm}_{\BO(\LT{1},\LP{-1})}\leq c_K, \qquad
\norm{\Repm}_{\BO(\LT{2},L^2)}\leq c_K.
\end{equation}
Hence by the interpolation theorem
(\cite{LiMag} Thm.\ 5.1) $\Repm\in\BO(\LT{1+\theta},\LP{-1+\theta})$
for all $\theta\in [0,1]$. Upon closer inspection
of the proof of this theorem we see that we know more.
The space $[\LT{1},\LT{2}]_\theta$ is the space of all traces
of functions $w:\MR\rightarrow \LT{2}$ in the space
\begin{equation}\label{408a}
W_1=\bigl\{w\in L^2(\MR;\LT{2}):\
\abs{\tau}^{1/2\theta}\widehat{w}\in L^2(\MR; \LT{1}\bigr\}.
\end{equation}
Here $\widehat{w}$ denotes the Fourier transform of $w$. (This is a
function
of $\tau$, $\tau\in\MR$.) The norm on $W_1$ is defined
as the sum of the norm of $w$ in $L^2(\MR;\LT{2})$ and the
norm of $\abs{\tau}^{1/2\theta}\widehat{w}$ in $L^2(\MR; \LT{1})$; for
details we refer to \cite{LiMag}, Chapter 1, Section 4.
Then (\cite{LiMag} Theorem 4.2) if
$u\in[\LT{1},\LT{2}]_\theta=\LT{1+\theta}$
then $u=w(0)$ with some $w\in W_1$ and
\begin{equation}\label{408c}
\frac{1}{c}\norm{w}_{W_1}\leq\norm{u}_{\LT{1+\theta}}\leq
c\norm{w}_{W_1}
\end{equation}
with some $c>0$. Moreover, one verifies that
this constant $c$ depends only on $\theta$ and is
independent of the spaces between which one interpolates.
Similarly $[\LP{-1},L^2]_\theta=\LP{-1+\theta}$ is the space of traces
of functions in a space $W_2$ and an estimate analogous to \eqref{408c}
holds.
Define the action of $R_1^\pm(\lambda)$
on $w\in W_1$ by
$\bigl(R_1^\pm(\lambda)w\bigr)(t)=R_1^\pm(\lambda)(w(t))$. Then the
interpolation theorem together with \eqref{408} and
\eqref{408c} yields
\begin{align}\label{408b}
&\norm{R_1^\pm(\lambda)u}_{\LP{-1+\theta}}\leq
c\norm{R_1^\pm(\lambda)w}_{W_2}
\\ \nn
&\quad\leq c\max\bigl\{\norm{\Repm}_{\BO(\LT{1},\LP{-1})},
\norm{\Repm}_{\BO(\LT{2},L^2)}\bigr\}\norm{w}_{W_1}
\\ \nn
&\quad\leq c^2 c_K\norm{u}_{\LT{1+\theta}}.
\end{align}
That is,
\begin{equation}\label{409}
\norm{\Repm}_{\BO(\LT{1+\theta},\LP{-1+\theta})}\leq c(K,\theta)
\end{equation}
with $c(K,\theta)$ depending on $K$ and $\theta$ only.
Now suppose
$u\in\LP{}$ with $\bigl(1+EF_1^\pm(\lambda)\bigr)u=0$.
By Lemma \ref{31} $u\in\LT{1}$, hence
$R_1^\pm(\lambda)u\in\LP{-1}$ and by Lemma \ref{24} also
$F_1^\pm(\lambda)u\in\LP{-1}$. Hence from \eqref{405}
with $\theta=1$ we get $u=-EF_1^\pm(\lambda)u\in\LP{1+\gamma}\,$.
Moreover, one sees that
$\norm{u}_{\varphi^{1+\gamma}}\leq c_K\norm{u}_\varphi$ with
$c_K$ some constant independent of $\lambda$ for
$\lambda\in K$.
We iterate this procedure: By Lemma \ref{31} we now know
$u\in\LT{1+\gamma}$, hence $\Repm u\in\LP{-1+\gamma}\,$. While
Lemma \ref{24} does not apply directly any more we still have the
factorization \eqref{212}. By Hypothesis \ref{21} and duality
$R_1(z)\in\BO(L^2,L^2)\cap \BO(\LP{-1},\LP{-1})$ for
$z\in\rho(H_1)$.
Thus we can interpolate to conclude
$R_1(z)\in\BO(\LP{-\theta},\LP{-\theta})$
for $\theta\in[0,1]$.
Again the norm can be bounded by a constant for $z$ in a compact
subset of $\rho(H_1)$.
Hence $F_1^\pm(\lambda)u\in\LP{-1+\gamma}\,$, by \eqref{405}
$u=-EF_1^\pm(\lambda)u\in\LP{1+2\gamma}\,$. After $n$ steps we find
$u\in\LP{1+n\gamma}$ with $\norm{u}_{\varphi^{1+n\gamma}}\leq c_{n,K}
\norm{u}_\varphi$ with some $c_{n,K}>0$. Pick $n$ with $1\leq
n\gamma<1+\gamma$ and we have
$\norm{u}_{\varphi^2}\leq\norm{u}_{\varphi^{1+n\gamma}}
\leq c_{n,K}\norm{u}_\varphi$.
Since $u\in\LT{2}$ by Lemma \ref{31}
\eqref{214} follows from \eqref{403}.
\end{proof}
\section{Application to Feller operators}
\setcounter{equation}{0}\label{5}
In this section we want to give an outline how the abstract theory we
have developed can be applied to singular as well as regular
perturbations
of Feller operators, i.\,e. operators defined as generators of
strong Markov processes with the Feller property. For
a detailed treatment of such operators we refer to
Demuth and van Casteren \cite{DvCBuch}, \cite{DvCFrame},
\cite{DvCHS},
\cite{DvCResults}, and
van Casteren \cite{vC}.
See also Fukushima, Oshima, and Takeda \cite{FOT},
Ma and R\"ockner \cite{MaRoe}.
Let $M$ be a locally compact second countable Hausdorff space with
Borel field $\EM$, $dx$ a non-negative Radon measure on $M$. We define
the operators $H_1$ and $H_2$ through their semigroups which are in turn
defined via strong Markov processes.
Let $p:\ \MR^+\times M\times M\rightarrow\MR^+$ be a continuous function
which satisfies the following assumptions (BASSA, cf.\ \cite{DvCBuch},
\cite{DvCFrame}, \cite{DvCHS}).
(i) For all $s,t>0$, $x,y\in M$,
\begin{align}\label{501}
&\int_M p(t,x,y)dy\leq 1,\quad\int_M p(s,x,z)p(t,z,y) dz=p(s+t,x,y),
\\ \nn
& p(t,x,y)=p(t,y,x).
\end{align}
(ii) If $f\in C_\infty(M)$, the space of continuous functions which
vanish at infinity, then
\begin{equation}\label{502}
\lim_{t\downarrow 0}\int_M p(t,x,y) f(y) dy=f(x),
\quad\int(p(t,\cdot,y)f(y)dy\in C_\infty(M).
\end{equation}
$\bigl(p(t,\cdot,\cdot)\bigr)_{t>0}$ is a Markovian transition function,
see \cite{FOT} Section 1.4 or \cite{MaRoe}, Section II.4.
By the continuity condition \eqref{502} the corresponding operator
semigroup is strongly continuous (cf. \cite{FOT} Lemma 1.4.3)
and we can define $H_1$ as the generator of
the semigroup
\begin{equation}\label{503}
\bigl(e^{-tH_1}f\bigr)(x)=\int_M p(t,x,y)f(y)dy.
\end{equation}
$H_1$ is self-adjoint and positive by Hille-Yosida.
We need one additional assumption on $p$, namely, that
this semigroup has finite spectral dimension, i.\,e.,
that there exist constants $c_S,\alpha_S,d_S>0$ such that
\begin{equation}\label{504}
\sup_{x,y\in M} p(t,x,y)=\Norm{e^{-tH_1}}_{\BO(L^1,L^\infty)}
\leq c_S e^{\alpha_s t}t^{-d_s/2}\,.
\end{equation}
By Kolmogorov's theorem $p$ is the transition function of a strong
Markov process. Let $E_x[\cdot]$ be the expectation
for this process (starting point $x$), $X_s$ the
position at time $s$. Then we can perturb $H_1$ by a
Kato Feller potential $V$, i.\,e., a potential satisfying
\begin{equation}\label{505}
\lim_{t\downarrow 0}\sup_{x\in M}\int_0^t
E_x\bigl[V_-(X_s)+1_B V_+(X_s)\bigr]ds=0
\end{equation}
for all $B\in\EM$ compact, where $1_B$ is the characteristic function of
$B$.
We define an
operator $H_V$ via the Feynman-Kac formula as the generator of
\begin{align}\label{506}
&\bigl(e^{-tH_V}f\bigr)(x)=E_x\bigl[e^{-\int_0^t V(X_s)ds} f(X_t)\bigr].
\end{align}
(Cf. \cite{DvCBuch} Thm.\ 2.5, \cite{vC} Thm.\ 2.8,
\cite{DvCFrame}, \cite{DvCHS}.)
If $\db(H_1)\cap\db(V)$ is a form core for $H_1+V$ (defined as
the operator sum on $\db(H_1)\cap\db(V)$) our operator $H_V$ coincides
with the Friedrich's extension of $H_1+V$;
confer \cite{DvCBuch} Chapter II or \cite{DvCHS} for
further details. In the following when
we write an operator sum of the type $H_1+V$ we will understand this
to be defined through an equation analogous to \eqref{506}.
Then we perturb this operator $H_V$ by Dirichlet boundary
conditions. Let $\Sigma\in\EM$ be an open subset of $M$,
$S$ the first penetration time of the set
$\Gamma=M\setminus\Sigma$,
\begin{equation}\label{508}
S=\inf\bigl\{t>0:\ \int_0^t 1_\Gamma(X_s)ds>0\bigr\}.
\end{equation}
We define $H_2$ by
\begin{align}\label{507}
&\bigl(e^{-tH_2}f\bigr)(x)=E_x\bigl[e^{-\int_0^t V(X_s)ds} f(X_t):\
S>t\bigr].
\end{align}
Intuitively $H_2$ is a domain perturbation of $H_V$, namely
$H_2=(H_1+V)_\Sigma$, where $(\cdot)_\Sigma$ indicates restriction
of the wavefunctions to the set $\Sigma$ via Dirichlet boundary
conditions.
In fact one verifies that $e^{-tH_2}$ is the strong limit of
$e^{-t(H_V+\beta 1_\Gamma)}$ as $\beta$ tends to infinity
(cf.\ \cite{DvCBuch} Lemma 7.1). Now suppose $H_V$ is local in
the sense that $H_Vf\in L^2(\Sigma)$
(i.e., vanishes on $\Gamma=M\setminus\Sigma$)
for all $f\in L^2(\Sigma)\cap\db(H_V)$.
Then one can show (cf.\ Baumg\"artel and Demuth \cite{BG})
that $H_2$ coincides with the Friedrich's extension of the operator
$H_{min}=H_V\lvert_{L^2(\Sigma)\cap\db(H_V)}$\,.
($H_{min}$ is the operator which acts like $H_V$ but has domain
$L^2(\Sigma)\cap\db(H_V)$.)
As suggested in Section \ref{4}, $X$ will be a weighted $L^2$-space,
$X=\LP{}$ with $\varphi:\ M\rightarrow [1,\infty)$ a measurable function
which also satisfies
\begin{hyp}\label{51}
Suppose there
exist constants $\gamma_\varphi>0$ and $c_\varphi,\alpha_\varphi>0$
such that for all $x\in M$
\begin{equation}\label{509}
E_x\bigl[\varphi(X_t)^{4+\gamma_\varphi}\bigr]
\leq c_\varphi e^{\alpha_\varphi t}\varphi(x)^{4+\gamma_\varphi}\,.
\end{equation}
\end{hyp}
\begin{rem}\label{52}
(i) By H\"older's inequality \eqref{509} implies similar estimates
for all exponents smaller than $4+\gamma_\varphi$.
(ii) While Hypothesis \ref{51} may seem a bit technical it is
satisfied in a wide range of examples. In particular one can easily
verify that it holds for all polynominally bounded functions $\varphi$
if $p$ can be estimated by a Gaussian kernel in $\MR^d$,
$p(t,x,y)\leq const. e^{-\abs{x-y}^2/(2t)}$ or in the case where
$H_1=\sqrt{-\triangle+1}-1$ is the relativistic Hamiltonian.
We refer to \cite{DvCBuch}, Remark 1 after Proposition 8.21 for
details.
(iii) If $V\equiv 0$ we can replace \eqref{509} by the weaker condition
$E_x[\varphi(X_t)]\leq c_\varphi e^{\alpha_\varphi t}\varphi(x)$.
\end{rem}
\begin{lem}\label{53}
If $H_1$ and $H_2$ defined by \eqref{503} respectively \eqref{507}
satisfy Hypothesis \ref{51} they satisfy Hypothesis \ref{21}.
\end{lem}
\begin{proof}
As noted above the Markovian transition function $p$ defines a
strongly continuous operator semigroup, by Hille-Yosida its
generator $H_1$ is a positive self-adjoint
operator. By Khas'minskii's lemma (\cite{DvCBuch}, Prop.\ 3.5,
see also Simon \cite{Simon})
$H_V=H_1+V$ and hence
also $H_2$ is bounded from below by a constant $c_0$, without loss
$c_0\leq\inf(\sigma(H_1))=0$.
It remains to verify \eqref{202}. By Hypothesis \ref{51} $H_1$ satisfies
\begin{equation}\label{510}
\bigl(e^{-tH_1}\varphi\bigr)(x)=E_x [\varphi(X_t)]\leq
c_\varphi e^{\alpha_\varphi t}\varphi(x).
\end{equation}
Hence $\norm{e^{-tH_1}}_{\BO(\LP{},\LP{})}=
\norm{\varphi^{-1} e^{-tH_1}\varphi}_{\BO(L^2,L^2)}
\leq c_\varphi e^{\alpha_\varphi t}$.
Now consider $e^{-tH_1}$ as an operator in $\BO(\LP{},\LP{})$.
We can define its resolvent in this space via Laplace transform.
Clearly this resolvent coincides with $R_1(z)$ on $\LP{}$, proving
that $R_1(z)$ may be restricted to a bounded operator in
$\BO(\LP{},\LP{})$ for all $z\in\rho(H_1)$.
The argument for $H_2$ is similar. Here we replace \eqref{510} by
\begin{align}\label{511}
&\bigl(e^{-tH_2}\varphi\bigr)(x)\leq\bigl(e^{-tH_V}\varphi\bigr)(x)
=E_x \bigl[e^{-\int_0^tV(X_s)ds}\varphi(X_t)\bigr]
\\ \nn
&\quad\leq \Bigl(E_x \bigl[\varphi(X_t)^2\bigr]\Bigr)^{1/2}
\Bigl(E_x \bigl[e^{-2\int_0^tV(X_s)ds}\bigr]\Bigr)^{1/2}
\leq c e^{\alpha t}\varphi(x)
\end{align}
with some $c,\alpha>0$. The last estimate in this equation
uses the fact that
\begin{equation}\label{511a}
\sup_{x\in M} E_x \bigl[e^{-2\int_0^tV(X_s)ds}\bigr]
=\Norm{e^{-t(H_1+2V)}}_{\BO(L^\infty,L^\infty)}
\end{equation}
is exponentially bounded by Khas'minskii's lemma. Proceed as above.
\end{proof}
Now suppose the operator $H_1$ satisfies Hypothesis \ref{41},
see Remark \ref{42b} for a few examples.
Then we want to give a sufficient condition to obtain a limiting
absorption principle for $H_2$, i.\,e.,
a sufficient condition for Hypothesis \ref{41a},
in terms of $V$ and $\Sigma$.
\begin{hyp}\label{54}
Assume there exists a $\gamma_v>0$ such that for all $\delta>0$
we can pick a potential $V_\delta\in L^\infty$ with
\begin{equation}\label{512}
\norm{\varphi^{2+\gamma_v}V_\delta}_\infty<\delta
\end{equation}
($\norm{\cdot}_\infty$ denoting $L^\infty$-norm) and
\begin{equation}\label{513}
\varphi^{4+2\gamma_v}(V-V_\delta)\in L^1(M).
\end{equation}
Furthermore, suppose $\Sigma$ satisfies
\begin{equation}\label{514}
\varphi^{4+2\gamma_v} v_\Gamma\in L^1(M),
\end{equation}
where $\Gamma=M\setminus\Sigma$ and $v_\Gamma$ is the
modified equilibrium potential
\begin{equation}\label{515}
v_\Gamma(x)=E_x[e^{-S},\ S<\infty].
\end{equation}
As before $S=\inf\{t>0:\ \int_0^t 1_\gamma(X_s)ds>0\}$ is the
penetration
time of $\Gamma$.
\end{hyp}
We call $v_\Gamma$ the modified equilibrium potential, because the
equilibrium potential $\widetilde{v}_\Gamma$ is normally defined as
\begin{equation}\label{515a}
\widetilde{v}_\Gamma(x)=E_x[e^{-T},\ T<\infty],
\end{equation}
where $T$ is the first hitting time of $\Gamma$,
\begin{equation}\label{515b}
T=\inf\{t>0:\ X_t\in\Gamma\}.
\end{equation}
These two definitions coincide if $\Gamma^r=(\oin\Gamma)^r$,
where $\Gamma^r=\{x\in M:\ P_x[T=0]=1\}$, $\oin\Gamma$
denotes the open interior of $\Gamma$, and $(\oin\Gamma)^r$ is
defined analogously to $\Gamma^r$.
To better understand \eqref{514}
recall that the capacity of $\Gamma$ is equal to the $L^1$-norm of the
equilibrium potential $\widetilde{v}_\Gamma$ (cf.\
\cite{FOT}, \cite{MaRoe}) so \eqref{514} is slightly stronger
than assuming that the capacity of $\Gamma$ is finite.
Usually $\varphi$ will be locally bounded; then the conditions on the
potential
$V$ in \eqref{512} and \eqref{513} mean that $V$ must be locally
$L^1$ and must decay faster than $\varphi^{-2}$
for $\abs{x}\rightarrow\infty$.
As desired Hypothesis \ref{54} together with the assumptions at the
beginning
of this section implies Hypothesis \ref{41a}.
\begin{lem}\label{56}
Let $H_1$ and $H_2$ be Feller operators defined by \eqref{503}
and \eqref{507}, respectively, which
satisfy Hypotheses \ref{51} and \ref{54}. Then there exists an $m\in\MN$
and a
$\gamma>0$ such that for all $a\in\MR$ small enough
($a<\inf\sigma(H_k)$, $k=1,2$)
\begin{equation}\label{516}
E=J_2^* R_2(a)^m J_2-R_1(a)^m\in\BO_\infty(L^2,\LP{2+\gamma}).
\end{equation}
\end{lem}
\begin{proof}
For each $\delta>0$ we can decompose $V$ into
$V=V_\delta+V_+ +V_-$ with $V_\delta$ as in Hypothesis \ref{54} and
$\pm V_\pm\geq 0$. Via the Feynman-Kac formula we can define the
following
operators:
\begin{align}\label{517}
&D_V:=J_2^*e^{-tH_2}J_2-e^{-tH_V}
\\ \nn
&D_+:=e^{-tH_V}-e^{-t(H_1+V_\delta+V_-)}
\\ \nn
&D_-:=e^{-t(H_1+V_\delta+V_-)}-e^{-t(H_1+V_\delta)}
\\ \nn
&R_\delta(z)=(H_1+V_\delta-z)^{-1},\qquad z\in\MC\setminus[c_0,\infty).
\end{align}
As a first step we want to establish
$D_V,D_+,D_-\in\BO_2(L^2,\LP{2+\gamma})$,
the space of Hilbert Schmidt operators from $L^2$ to $\LP{2+\gamma}$.
(Here $\gamma>0$ may be chosen as small as necessary.) Actually we
prefer
to establish the equivalent statement
$\varphi^{2+\gamma}D_V$, $\varphi^{2+\gamma}D_+$,
$\varphi^{2+\gamma}D_-\in\BO_2(L^2,L^2)$. Let $\norm{\cdot}_{HS}$ denote
the
Hilbert Schmidt norm, i.\,e., the norm in $\BO_2(L^2,L^2)$. We use
\begin{align}\label{518}
&\norm{\varphi^{2+\gamma} D_V}_{HS}^2
\\ \nn
&\quad=\int_M\int_M dxdy\, \varphi(x)^{4+2\gamma}
\Bigl(E_x\bigl[e^{-\int_0^S V(X_u) du}
e^{-(t-S)H_V}(X_s,y):\ S0$. Thus \eqref{518} becomes
\begin{align}\label{520}
&\norm{\varphi^{2+\gamma} D_V}_{HS}^2
\leq ce^{\alpha t} t^{-d_s/2}
\int_M\int_M dxdy\,\varphi(x)^{4+2\gamma}\int_{S1$
small enough we can obtain $q(4+2\gamma)\leq 4+\gamma_\varphi$ so
that with Hypothesis \ref{51} and \cite{DvCBuch} Theorem 2.9 we find
\begin{align}\label{523a}
&\int_M dx\,\varphi(x)^{4+2\gamma}e^{-sH_V}(x,z)
\\ \nn
&\quad\leq (c_\varphi e^{\alpha_\varphi s})^{1/q}\varphi(z)^{4+2\gamma}
\Norm{e^{-s(H_1+pV)}}_{\BO(L^\infty,L^\infty)}^{1/p}
\\ \nn
&\quad\leq c e^{\alpha s}\varphi(z)^{4+2\gamma}
\end{align}
with some $c,\alpha>0$ provided $\gamma>0$ is small enough.
Thus, using \cite{DvCBuch} Theorem 2.9 again, \eqref{522} becomes
\begin{align}\label{524}
&\norm{\varphi^{2+\gamma} D_+}_{HS}^2
\\ \nn
&\quad\leq\int_0^t ds\, ce^{\alpha s}\int_M dz\,\varphi(z)^{4+2\gamma}
V_+(z)\Norm{e^{-(2t-s)(H_1+V_\delta+V_-)}}_{\BO(L^1,L^\infty)}
\\ \nn
&\quad\leq c' e^{\alpha' t} t^{-d_S/2} t\int_M dz\,
\varphi(z)^{4+2\gamma} V_+(z)
\end{align}
with constants $c',\alpha'>0$. The estimate for $D_-$ is completely
analogous, it yields
\begin{align}\label{525}
\norm{\varphi^{2+\gamma} D_-}_{HS}^2
\leq c e^{\alpha t} t^{1-d_S/2}\int_M dz\,
\varphi(z)^{4+2\gamma} \abs{V_-(z)}
\end{align}
with $c,\alpha>0$.
Combining \eqref{520}, \eqref{524}, and \eqref{525} we find
\begin{align}\label{526}
&\Norm{\varphi^{2+\gamma}\bigl(J_2^*e^{-tH_2}J_2
-e^{-t(H_1+V_\delta)}\bigr)}_{HS}
\\ \nn
&\quad\leq
c e^{\alpha t} t^{-d_S/4}\Bigl(
\norm{\varphi^{4+2\gamma}(V-V_\delta)}_{L^1}^{1/2}
+\norm{\varphi^{4+2\gamma}v_\Gamma}_{L^1}^{1/2}\Bigr)\,
\end{align}
with some $c,\alpha>0$. If we pick $\gamma>0$ small enough the right
hand side of \eqref{526} is finite by Hypothesis \ref{54}.
Now pick $m\in\MN$ such that $m-\frac{d_S}{4}>-1$ and $a\in\MR$
such that $a+\alpha<0$ where $\alpha$ is the constant from \eqref{526}.
Then
\begin{align}\label{527}
&\Norm{\varphi^{2+\gamma}\bigl(J_2^*R_2(a)^mJ_2
-R_\delta(a)^m\bigr)}_{HS}
\\ \nn
&\quad\leq\frac{1}{\Gamma(m)}\int_0^\infty dt\,t^{m-1} e^{a t}
\Norm{\varphi^{2+\gamma}\bigl(J_2^*e^{-tH_2}J_2
-e^{-t(H_1+V_\delta)}\bigr)}_{HS}
<\infty\,
\end{align}
(here $\Gamma(m)$ denotes the $\Gamma$-function). Finally we have
\begin{align}\label{528}
&\varphi^{2+\gamma}E
=\varphi^{2+\gamma}\bigl(J_2^*R_2(a)^mJ_2-R_1(a)^m\bigr)
\\ \nn
&\quad=\varphi^{2+\gamma}\bigl(J_2^*R_2(a)^mJ_2-R_\delta(a)^m\bigr)
+\varphi^{2+\gamma}\bigl(R_\delta(a)^m-R_1(a)^m\bigr).
\end{align}
For every $\delta>0$ the first operator in \eqref{528} is Hilbert
Schmidt,
hence compact. By \eqref{512} the
second operator tends to $0$ in Operator norm as $\delta$
tends to $0$, thus $\varphi^{2+\gamma}E$ is compact
(i.\,e., element of $\BO_\infty(L^2,L^2)$) as the norm limit of a
sequence of
compact operators.
Hence $E\in\BO_\infty(L^2,\LP{2+\gamma})$.
\end{proof}
So we have proved our main application theorem.
\begin{thm}\label{57}
Suppose $H_1$ and $H_2$ are Feller operators satisfying Hypotheses
\ref{51} and \ref{54}, and $H_1$ also satisfies Hypothesis \ref{41}.
Then a limiting absorption principle holds for $H_2$:
Its spectrum inside the set $\Delta$, $\sigma(H_2)\cap\Delta$ is,
except for possibly a discrete set $\sigma_p(H_2)\cap\Delta$
of eigenvalues of finite multiplicity,
purely absolutely continuous. For all
$\lambda\in\Delta\setminus\sigma_p(H_2)$
\begin{equation}\label{529}
J_2^*R_2^\pm(\lambda)J_2:=\lim J_2^*R_2(\lpm)J_2
\end{equation}
exists and is continuous in $\BO(\LP{},\LP{-1})$.
\end{thm}
Let us illustrate the strength of our approach with the simplest
example. Suppose $H_1=-\triangle$ is the Laplacian on $\MR^d$.
This operator satisfies Hypothesis \ref{41} with $\Delta=(0,\infty)$
and $\varphi=(1+\abs{x})^s$, $s>\frac12$ (cf.\ \cite{BAD}).
Then Hypothesis \ref{54} allows all potentials $V$ which are at least
locally $L^1$ and decay as $\abs{x}^{-1-\gamma}$ as $\abs{x}$ tends
to infinity, i.\,e., the usual short range potentials.
The condition \eqref{514} on the set $\Sigma$ respectively
$\Gamma=M\setminus\Sigma$ is a bit more implicit.
For comparison the capacity of $\Gamma$ is the $L^1$-norm of
the equilibrium potential. For the Laplacian \eqref{514}
becomes the condition $(1+\abs{x})^{2+\gamma}v_\gamma\in L^1$,
a slightly stronger restriction than the finiteness of the capacity.
That is, $\Gamma$ may be an arbitrary compact set or an unbounded
set provided it is small enough in the sense of \eqref{514}.
\section*{Acknowledgments}
The author wants to thank Michael Demuth for stimulating discussions
and especially for making him aware of
the reference \cite{DMcG} which provided some very
useful ideas. He is also indebted to Matania Ben-Artzi
for valuable comments.
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\end{document}