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transport operator, functional model, spectral singularity
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\begin{document}
\topmargin=3.3cm
\begin{center}
{\LARGE\bf {Spectral Analysis of Transport Operator: \vskip .3cm
Functional Model Approach}}
\bigskip\medskip
{\large {Yuri A. Kuperin$^*$, Serguei N. Naboko$^{\dagger,\ddagger}$
and Roman V. Romanov$^{*,\dagger}$}}
\vskip.4cm
$^*$ Laboratory of Complex Systems Theory\\ Institute for
Physics\\ Saint Petersburg State University\\ 198904, Saint
Petersburg, Russia
\vskip.3cm
$^\dagger$Department of Mathematics, \\
University of Alabama at Birmingham,\\
CH 452, Birmingham, AL 35294-1170 USA \\
e-mail: romanov@math.uab.edu
\vskip.3cm
$^\ddagger$ Department of Mathematical Physics, Faculty of Physics\\
St.Petersburg University, 198904, St.Petersburg, Russia
\end{center}
%\Names{Kuperin, Naboko and Romanov}
%\Keywords{transport operator, functional model, spectral singularity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
This paper presents the spectral analysis of the
dissipative one-speed transport operator for the cases of a
finite slab and of a finite body, by means of the Szokefalvi-Nagy
- Foias functional model. It is shown that the essential spectrum
of the operator is absolutely continuous with one possible
spectral singularity at point $0$. The absolutely continuous
component of the operator is studied in detail in the nontrivial
case, when the spectral singularity does occur.
\end{abstract}
%\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\thispagestyle{empty}
%\Keywords{transport operator, functional model, spectral singularity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\topmargin=0cm
\leftmargin=0cm
\newsection{Introduction}
The spectral theory of nonselfadjoint operators based on the
functional model of Szokefalvi-Nagy and Foias \cite{Na} made
significant progress towards understanding the structure of
operators with discrete spectrum (see e.g. \cite{Nik}). At the
same time, the structure of operators with absolutely continuous
spectrum is less understood and the theory is still short of
applications. So far, these essentially amount to the
Schr\"odinger operator with complex potential \cite{PavEn}. In
the present paper we analyze an example of a dissipative operator
with absolutely continuous spectrum coming from physics - the
one-speed transport operator, and solve a relevant abstract
problem in the theory applying then the result to the example.
It is well-known that the nonselfadjoint one-speed transport
operator acting in the Hilbert space $L^2$ of distribution
functions has rich essential spectrum \cite{Shikhov} for various
geometrical situations. As a rule this spectrum fills a set of
positive measure in $ \C $. The general theory of functional
model is not developed far enough to be applied effectively in
this case. Another situation arises if we consider the transport
operator for a piece of multiplicative medium embedded in a pure
absorbent filling the configurational space $ \R^n $. Since the
paper \cite{JLh} it is known that in this case the essential
spectrum coincides with the real line. The analysis in that paper
was based on the direct study of the resolvent by elementary
methods. The idea of the present paper to apply the functional
model to the spectral analysis of the operator appears to be
natural in this situation. This reduces the problems in spectral
analysis of the operator to the questions of complex analysis. We
fix our attention on two geometrical situations,
{\bf 1}. A slab of multiplicative medium of finite width.
{\bf 2}. A piece of muliplicative medium of finite diameter in the
three-di\-men\-sion\-al space.
Multiplicative properties of the medium are described by the local
mean number of the secondaries per collision $c$, a nonnegative
function on the configurational space $ \Lambda $, $\Lambda = \R$
in case {\bf 1}, and $\Lambda = \R^3 $ in case {\bf 2}. We
suppose throughout that $c$ is essentially bounded.
For case {\bf 1} it was shown in \cite{JLh} that the spectrum of
the transport operator for finite slab with constant $c$ consists
of finitely many eigenvalues lying on the imaginary axis and the
essential spectrum which fills the real axis. Analysis of the
proof shows that this result holds for arbitrary compactly
supported $c\in L^\infty(\R)$ and makes it possible to give an
estimate of the Birman-Schwinger type for the dimension of the
subspace corresponding to the discrete spectrum \cite{KNR}. For
$c$ small in an appropriate sense, we also give an estimate of
the angle between the discrete and essential spectrum subspaces
of the operator in terms of $c$ in a closed form.
In case {\bf 2}, for arbitrary compactly supported $c\in L^\infty
(\R^3)$, the geometrical picture of the spectrum is shown to
remain the same as for the slab geometry, and a Birman -
Schwinger type estimate for the dimension of the discrete
spectrum subspace is also provided.
All the other results are obtained by the study of the component
corresponding to the essential spectrum by using the functional
model. For both situations we show that the essential spectrum of
the transport operator is purely absolutely continuous. It is
proven that the corresponding component of the operator is
similar to a selfadjoint operator if the function $c \notin \cE $
for a certain singular set $\cE \subset L^\infty( \Lambda )$.
Moreover, if we take the operator corresponding to the transport
in a perfect absorber for the generator of unperturbed dynamics,
then the wave operators exist and are complete.
As is well-known \cite{PavEn,Pav}, an operator with absolutely
continuous spectrum can fail to be similar to a selfadjoint one
owing to the presence of so called spectral singularities. Roughly
speaking, the spectral singularities are those points of the
absolutely continuous spectrum of the operator at which the
resolvent has growth of more than the first order. The
differential operators with spectral singularities were studied in
the papers by Naimark, Ljance and Pavlov (see
\cite{PavEn,Nai,Lja} and references therein). In the situation
under consideration for $c\in \cE$ the operator is shown to have
a unique point of spectral singularity at $0$. We prove that the
absolutely continuous component of the operator is similar to the
orthogonal sum of a selfadjoint operator and an operator with
spectrum of finite multiplicity $ \cM $, which is calculated in
terms of $c$. For the spectral component of the transport operator
corresponding to a neighborhood of the spectral singularity we
also give an estimate of the angle between the corresponding
invariant subspaces.
The proofs of these results are based on the analysis of the
behaviour of the characteristic function of the operator. The
characteristic function of a dissipative operator $ L $ is a
contractive analytic operator function in the upper half plane
which can be considered as an operator analog of the perturbation
determinant for the pair $ ( L, L^* ) $. The information about
boundary behavior of the characteristic function turns into
conclusions about structure of operators by general theorems of
functional model theory. One should mention that the questions
arising here are more difficult than those of the selfadjoint
theory where the singularities of the resolvent correspond to
zeroes of $R$-functions \cite{NabArkiv}. The study of the
characteristic function is reduced, to some extent, to that of a
scalar analytic function by the presence of a scalar multiple.
The proof of the absolute continuity of the spectrum essentially
amounts to the verification of triviality of the singular inner
factor for the scalar multiple. The existence and completeness of
the wave operators in the situation under consideration are facts
of the general theory \cite{Nab}. The splitting of the absolutely
continuous component is based on separation of some invariant
subspaces defined in the model representation of the operator
through the spectral decomposition of $ \Delta = 1 - S^* S $, $S$
being the characteristic function. These subspaces are defined in
such a way that the estimate of the angle between them is reduced
to the estimate of the function $ S^{-1} $ on the real axis. The
suggested construction of the invariant subspaces corresponding
to the absolutely continuous spectrum with the estimate of the
angle between them is the main abstract result of the paper. It
is presented in lemma \ref{An} and corollary \ref{abstrsep}.
The proofs are completely parallel for the cases {\bf 1} and {\bf
2}, with some simplification in the second case.
One should emphasize that the functional model is not involved in
the statement of results about the transport operator and is only
used as a tool for their derivation. It should also be stressed
that the appearance of the spectral singularity in our problem is
not a kind of pathology. Namely, it will be seen that for any
nonzero compactly supported $ c \in L^\infty $ the function
$\kappa c $ belongs to $ \cE $ for an infinite discrete set of
values of the constant $ \kappa $. This distinguishes the
transport operator from the other examples of operators with
spectral singularities known in mathematical physics, like the
Schr\"odinger operator with complex potential \cite{Nai,Lja}
where the spectral singularities appear {\it ad hoc}, for
specially constructed potentials.
The structure of the paper is the following. In section 2 we give
a brief description of the functional model of a dissipative
operator. We use the symmetric form of the functional model (see
\cite{Pav,N}). Then the abstract construction of invariant
subspaces is described. Sections 3 and 4 are devoted to the
spectral analysis of the transport operator for the slab and for
the finite piece respectively. In the slab case for completeness
we provide a derivation of the spectrum, further simplified as
compared to \cite{JLh,KNR,LW}, and of the estimate of the
discrete spectrum from our paper \cite{KNR} (Theorem 1, section
3.2). The spectrum of the transport operator for the finite body
is derived in Theorem 3, section 4.2. The estimate of the angle
between discrete and essential spectrum subspaces for the slab is
given in proposition \ref{hdhc}, section 3.2. The absolute
continuity of the essential spectrum is established in
propositions \ref{out}, \ref{out1}. The remaining parts of
sections 3.3 and 4.2 are concerned with the derivation of the
results on the structure of the operators in the case $ c \in \cE
$ which are resumed in Theorems 2, 4.
\newsubsection{Notations and necessary standard facts}
We use the same notation for analogous objects in sections $3,4$
wherever confusion is impossible. Throughout the paper $| \cdot
|_\infty $ stands for the $ L^\infty $-norm of functions and $ \|
\cdot \|_p $ for the norm of operators in classes ${\bf S}^p $, $
\wh{f} $ for the Fourier transform of a function $ f \in L^2 ( \R
) $, $ \chi_M $ is the indicator of a set $ M \subset \R $, $
U_\delta (z) = \left\{ z^\prime \in \C : \; | z - z^\prime | <
\delta \right\} $, $\omede (z) = U_\delta (z) \bigcap \C_+ $,
$\C_+$ being the open upper half plane; $ \Pi_\epsilon = \{ z :\;
0 < \Im z < \epsilon \} $, $[ z_1, z_2 ) = \{ z = t z_1 + (1-t)
z_2 ,\; t \in (0,1] \} $ for $ z_{1,2} \in \C $. The term
measurable refers to the Lebesgue measure on $ \R $. Parentheses
$( \cdot, \cdot ) $ stand for the angle between subspaces of a
Hilbert space. For an operator $A$ on a Hilbert space $H$ we write
$\sigma_+ ( A ) = \sigma (A) \bigcap \C_+$. We adopt the
definition of the essential spectrum of a closed operator used in
\cite{RS}, vol. 4.
%Given a bounded measurable function $ \Xi
%\colon \ \R \to {\bf B} (E) $, $ E $ being a Hilbert space, we
%often use $ \Xi $ without arguments to denote the operator of
%multiplication by this function in $ L^2 ( \R , E ) $.
In Section 2 below we introduce some spaces $ L^2 $ of vector
functions with operator weights, defined in the following way.
Let $ E $ be a Hilbert space and $ \Xi : \R \to {\bf B} ( E ) $
be a bounded measurable function such that $ \Xi \ge 0 $. By $
L^2 ( \R ; \Xi ) $, or $ L^2 ( \Xi ) $, we denote the Hilbert
space obtained by the closure of the linear set of equivalence
classes $ L^2 ( \R , E ) / \ker \Xi $ in the metric given by the
weight $ \Xi $. We do not distinguish in our notation an element
$ f \in L^2 ( \R , E ) $ and the corresponding equivalence class
in $ L^2 (\R ; \Xi ) $.
Let $L$ be a dissipative operator, $ L = L_0 +i V $, $V = \Im L $
is bounded, $ R (z) = \( L - z \)^{-1} $, $ R_0 (z) = \( L_0 - z
\)^{-1} $. Then the resolvent identity $R ( z ) - R_0 ( z ) = -i
R_0 ( z ) $ $ V R ( z ) $ holds for $ z \in \C_+ \setminus
\sigma_+ ( L ) $. Simple algebraic manipulations (see e. g.
\cite{Yaf}\footnote{In \cite{Yaf} the perturbation is assumed to
be selfadjoint, which is of no importance for the manipulations.})
show that
\begin{eqnarray} \label{Srez} & R ( z ) = R_0 ( z ) - i R_0 ( z )
\sqrt{V} \( I + i\sqrt{V} R_0 ( z ) \sqrt{V} \)^{-1}\sqrt{V} R_0 (
z ), & \\ & \( I + i\sqrt{V} R_0 ( z ) \sqrt{V} \) \( I -
i\sqrt{V} R ( z ) \sqrt{V} \) = I . & \nonumber \end{eqnarray} It
follows that $\sigma_+ ( L ) = \{ z \in \C_+ : \ -1 \in \sigma \(
Q ( z ) \) \} $ where \be\la{Qdef} Q ( z ) = i \sqrt V R_0 ( z )
\sqrt V \ee is an analytic operator function on $ \C_+ $. The
following is a version of the Weil theorem on relatively compact
perturbations convenient for our proposes.
{\bf Sublemma.} {\it If $ Q ( z ) $ is a compact operator (at
least at one point $ z \in \C_+ $ and then at all points) then $
\sigma_{ess} ( L ) = \sigma_{ess} ( L_0 ) $.}
\begin{proof} First
\be\la{ReQ} \Re Q ( z ) = \frac i2 \( \sqrt V ( R_0 ( z ) - R_0 (
\overline{z} ) ) \sqrt V \) = - \Im z \sqrt V R_0 ( z ) \cdot R_0
( \overline{z} ) \sqrt V \ee is compact along with $ Q ( z ) $. It
follows that $ R_0 ( z )\sqrt V $ is also compact for $ \Im z \ne
0 $. From the resolvent identity one now concludes that $ R ( z )
- R_0 ( z ) $ is compact for $ z \in \rho ( L ) \cap \rho ( L_0 )
$, and the result follows from the Weil theorem \cite{RS}.
\end{proof}
In the following we preserve the notation $ Q(z) $ for the
restriction of the r.h.s. of (\ref{Qdef}) to $
\overline{\mbox{Ran}\ V} $.
An analytic operator function $ A (z) $ in the upper half plane $
\C_+ $ is called an {\it $ R $ -function} if $ \Im A ( z ) \ge 0 $
for all $ z \in \C_+ $. The following remark results from a
general representation theorem for $ R $ - functions
\cite[Theorem 2.2, Remark 1]{NabArkiv}.
\begin{remark}\la{Rfun} If $ A ( z ) $ is an $ R $ - function in
$ \C_+ $ then $ \sup_{ |z| \le 1 } \Im z \len A ( z ) \rin $ is
finite.
\end{remark}
Although the following simple remark also results immediately from
the representation theorem, below we give an elementary
derivation of it.
\begin{remark}\la{Rfun1} If an $ R $ - function $ A ( z ) $ is
analytic and selfadjoint on the negative real axis, then $ A ( k
) $ is a monotone increasing function for $ k < 0 $.
\end{remark}
\begin{proof} It is enough to verify the assertion for scalar
functions. Since for any $ k_0 < 0 $ the numbers $ A ( k_0 ) $ and
$ A^\prime ( k_0 ) $ are real, it follows that $ A^\prime ( k_0 )=
\lim_{ \von \to 0+ } \von^{ -1 } \Im A ( k_0 + i \von ) $ is
nonnegative. If $ A^\prime ( k_0 ) = 0 $, then for some $ j \ge 2
$ we would have $ A ( z ) = A ( k_0 ) + \( z - k_0 \)^j A_j \( 1
+ O ( z - k_0 ) \) $ with an $ A_j \ne 0 $. Now considering $ z =
k_0 + \von e^{ i \varphi } $, $ \von \to 0 $, for an appropriate $
\varphi \in ( 0 , \pi ) $, one arrives at a contradiction with
the property of being $ R $ -function.
\end{proof}
Let $ S : \C_+ \to {\bf B} ( E ) $, $E$ being a Hilbert space, be
a bounded analytic operator - function. A scalar function $ m ( z
) \not \equiv 0 $ in $ \C_+ $ is called a scalar multiple for $S$
if there exists a bounded analytic operator - function $\Omega ( z
) $ in $ \C_+ $ such that $ m ( z ) I = S ( z ) \Omega ( z ) =
\Omega ( z ) S ( z ) $ for all $ z \in \C_+ $ \cite{N}.
The determinant $ \det( I+ A ) $ is defined for any $ A \in {\bf
S}^1 $ \cite{GK}. The function $ A \mapsto \det( I+ A ) $ is
continuous in the $ {\bf S}^1 $ - norm. Also \be \la{stand} | \det
( I + A ) | \le \exp \( \len A \rin_1 \) . \ee If $ \ker ( I + A )
\neq \{ 0 \} $ then $ F = \det ( I + A ) ( I + A )^{ -1 } $
satisfies \be \la{trace} \| F \| \le \exp \( \len A \rin_1 \) .
\ee If $ A ( z ) $ is an analytic\footnote{It is enough to verify
that the function is analytic in the operator norm and locally
bounded in the ${\bf S}^1 $ - norm.} ${\bf S}^1 $ - valued
operator function on a domain $\cD \subset \C $ then $ a ( z ) =
\det \( I + A ( z )\) $ is a scalar analytic function on $\cD $.
The following lemma is essentially proved in \cite{Pav1}.
\begin{lemma} \la{abstr} If $ A ( \cdot ) $ is an $ {\bf S}^p $ -
valued ($ p \in \N $) function on a domain $\cD \subset \C $
analytic in the operator norm and satisfying $ \sup_{ z \in \cD }
\len A ( z ) \rin_p < \infty $ then the function $ s ( z ) = \det
\( I - A^p ( z )\) $ is a scalar multiple for $ S(z) = I - A ( z
) $ if $ s ( z ) \not \equiv 0 $.
\end{lemma}
\begin{proof} From (\ref{stand}) and the inequality $ \len A^p ( z )
\rin_1 \le \len A ( z ) \rin_p^p $ one sees that $ s ( z ) $ is a
bounded analytic function on $ \cD $. Assume $ s ( z ) \not \equiv
0 $. Put $ F ( z ) = s ( z ) \( I - A^p ( z ) \)^{-1}$. We have $$
s ( z ) I = F ( z ) \( I - A^p ( z ) \) = F ( z ) \( \sum_{ j = 0
}^{ p - 1 } A^j ( z ) \) \( I - A ( z ) \). $$ Put $ \Omega ( z )
= F ( z ) \sum_{ j = 0 }^{ p - 1 } A^j ( z ) $. In view of the
estimate (\ref{trace}) the function $ F (\cdot ) $ is analytic and
bounded in $\cD $, and so is $ \Omega ( \cdot ) $. Since $ \Omega
( z ) $ and $ A ( z ) $ commute, the proof is completed.
\end{proof}
\newsection{Functional Model and Angles}
Let $L$ be a closed dissipative operator in a Hilbert space $ H $
with bounded imaginary part $ V = \Im L $ such that $
\sigma_{ess} (L) \subset \R $, and let $ E = \overline{{\rm Ran
}V}$. The characteristic function $ S(z): \; E\to E $, $ z\in
\C_+$, of the operator $L$ is defined by the formula $$ S(z) = I
+ 2 i \sqrt V \( L^* - z \)^{-1} \sqrt{V}.
$$ This is a contractive analytic function in $ \C_+ $. This
function has the nontangential boundary values $ S(k) \equiv S ( k
+ i 0 ) $ on the real axis in the strong sense for a.e. $ k\in \R
$. Let $ L_{pure} $ be the completely nonselfadjoint part of $ L
$. Then the set $ \sigma ( L_{pure} ) \bigcap \R $ conicides with
the complement (in $ \R $) of the union of open intervals on which
$ S ( z ) $ is analytic and unitary. For $ z \in \C_+ \bigcap \rho
(L)$ the operator $ S ( z ) $ is boundedly invertible on $ E $.
It can easily be shown by manipulations with the resolvent
identity that $ S ( z ) $ is expressed in terms of the operator $
Q ( z ) $ given by (\ref{Qdef}) as follows \cite{Brod}:
\be\la{Char} S ( z ) = \frac{I + Q(z)}{I - Q ( z ) }, \ \ z \in
\C_+. \ee Remark that the operator $ I - Q ( z )$ has the
bounded inverse since $\Re Q(z) \le 0 $ for all $ z \in \C_+ $.
This formula shows, in particular, that $ S ( z ) - I \in {\bf
S}^p $ whenever $ Q ( z ) \in {\bf S}^p $ and that $ S ( z ) $ is
analytic at a point on the real axis if so does $ Q ( z ) $.
The matrix $\pmatrix{ I&S^*(k) \cr S(k)&I \cr} $ defines a
nonnegative operator in $ E \oplus E $ for a. e. $ k \in \R $
which allows us to define the Hilbert space $ {\cal X} = L^2
\pmatrix{ I & S^*\cr S & I\cr}$. Let $ H^2_\pm (E) $ be the Hardy
classes of $E$-valued functions $f$ analytic in $ \C_\pm $,
respectively, and satisfying $\sup_{ \von
> 0 } \int_\R \len f ( k \pm i\von ) \rin^2_E dk < \infty $.
Define for a representative $ ( \tg, g ) \in L^2 (\R, E \oplus E
) $ from the equivalence class of an element of $ {\cal X} $
\be\la{PK0} P_\cK \pmatrix{ \tg\cr g\cr} = \pmatrix{ \tg - P_+ \(
\tg + S^* g \) \cr g - P_- \( S \tg + g \) \cr} \ee where $P_\pm$
are the Riesz projections on $H^2_\pm (E)$ in $L^2( E)$. Then
\cite{Pav1}, $ P_\cK $ is a correctly defined operator in $ {\cal
X} $ and its closure in $ {\cal X} $ which will be denoted with
the same letter, is an orthogonal projection. Let $ \cK $ be the
range of $ P_\cK $ in $ {\cal X} $. Define a unitary group
$\cU_t$ of operators in ${\cal X} $ by the formula $ ( \cU_t f )
(k) = e^{ikt} f(k) $. Then the completely nonselfadjoint part of
$L$ is unitarily equivalent to the generator of the contraction
semigroup $Z_t = P_\cK \cU_t \mid_\cK$. This generator is called
the functional model of the operator $L_{pure}$.
%As any contractive analytic in $ \C_+ $ function, $ S $ admits
%canonical factorization $ S(z) = S_i (z) S_e (z) $ in the product of
%inner and outer functions. A bounded analytic function $ \Theta :
%\C_+ \to {\bf B} ( E ) $ is called {\it inner} if $ \Theta ( k ) $
%is an isometric operator for a. e. $ k \in \R $, {\it outer} if $
%\overline { \Theta H^2_+ ( E ) } = H^2_+ ( E ) $.
We use theorems on dissipative operators corresponding to theorems
stated for contractions in the book \cite{Na} without special
explanations. Also the definitions of the functional - theoretic
objects related to the upper half plane like inner and outer
functions etc. are adopted which correspond naturally to those
introduced in \cite{Na} for the unit circle. For example, a
bounded analytic function $ \Theta : \C_+ \to {\bf B} ( E ) $ is
called {\it outer} if $ \overline { \Theta H^2_+ ( E ) } = H^2_+
( E ) $. Then, the characteristic function admits the {\it
canonical factorization} in a product of two contractive analytic
$ {\bf B} ( E ) $ - valued functions of the form $ S = S_i S_e $
where $ S_e $ is an outer function, and $ S_i ( k ) $ is
isometric for a.e. $ k \in \R $.
Throughout the paper, by an {\it invariant subspace} of an
operator we mean a regular invariant subspace; that is, a subspace
$ \cH $ is called an invariant subspace of $L$ if $ \overline{( L
-\lambda )^{-1} \cH } = \cH $ for all $\lambda \in \rho (L)$.
\begin{remark} Every regular invariant subspace $ \cH $ is invariant
in the following natural sense,
$(i)$ $\cD (L) \bigcap \cH = \( L - \lambda \)^{-1} \cH $, $
\lambda \in \rho (L) $,
$(ii)$ $L \(\cD ( L ) \bigcap \cH \) \subset \cH $.
\end{remark}
\begin{proof} First, $ (ii) $ implies $ (i) $ in view of the identity
$ \( L - z \)^{-1} L f - z \( L - z \)^{-1} f = f $. Then $ i n \(
L + i n \)^{-1} \to I $ as $ n \to + \infty $ in the strong sense
by the functional calculus \cite[Theorem III.2.1]{Na}. Hence $ i n
\( L + i n \)^{-1} L f \to L f $ for any $ f \in \cD ( L ) \bigcap
\cH $. Now, the identity $ \( L + i n \)^{-1} L f = f - i n \( L
+ i n \)^{-1} f $ shows that the left hand side belongs to $ \cH $
for all $ n > 0 $ by the regular invariance of $ \cH $, and $ (ii)
$ follows.
\end{proof}
It follows that the restriction $ L_\cH = L|_\cH $ with the domain
$ \cD (L) \bigcap \cH $ is closed and densely defined.
In the following, we define some invariant subspaces of the
completely nonselfadjoint part of $ L $ in terms of its functional
model. In doing so, to keep notation at minimum, we omit in
formulas the operator that accomplishes the unitary equivalence
between $ L_{pure} $ and its functional model.
Let $ \Delta ( k ) = I - S^* ( k ) S ( k ) $. Since $ \Delta (k)
\ge 0 $ for a.e. $ k \in \R $, one can define the space $L^2 (\R;
\Delta ) $. Note that for any $ \tg\in L^2(\R; \Delta) $ the
vector $ \pmatrix{\tg \cr -S\tg \cr} $ defines an element of $
{\cal X} $ of the norm $ \len \tg \rin_{ L^2(\R; \Delta) } $.
{\it The absolutely continuous subspace $\cN_e \subset \cK $} of
the operator $L$ \cite{Pav,Sachn} is defined as the closure of
the linear set $\wt{\cN_e} $ of smooth vectors: $$ \cN_e =
\overline{\wt{\cN_e}}, \hspace{.7cm} \wt{\cN_e}\equiv\left\{ P_\cK
\pmatrix{ \tg \cr -S\tg \cr},\; \tg\in L^2(\R; \Delta)\right\}. $$
Then $\cN_e $ coincides with the invariant subspace of $L$
corresponding to the canonical factorization of the
characteristic function in the sense that the characteristic
function of $ \left. L \right|_{ \cN_e } $ coincides with the
pure part of $ S_e $ (see \cite{Pav}). We call the spectrum of
$L$ {\it purely absolutely continuous} if $ H = H_0 \oplus \cN_e $
where $H_0 $ is an invariant subspace of $L$ such that $ \left. L
\right|_{H_0} $ is a selfadjoint operator with absolutely
continuous spectrum in the sense of spectral theory of
selfadjoint operators \cite{RS}. The set of smooth vectors of a
completely nonselfadjoint dissipative operator admits the
following description in model-free terms, \cite[Theorem 4,
Corollary 1]{N} $ \wt{\cN_e} = \left\{ u \in H: \; \sqrt V \( L -
\cdot \)^{ -1 } u \in H_2^+ (E) \right\} $. This description
allows to motivate the suggested definition of the absolutely
continuous subspace from the viewpoint of the theory smooth
perturbations \cite{Nab}. Another interpretation of the
definition stems from the fact \cite{Pavli} that a suitable
normalized system of generalized eigenfunctions corresponding to
the spectrum of $ \left. L \right|_{ \cN_e } $ is complete in $
\cN_e $. These eigenfunctions are constructed in terms of
generalized limits of the resolvent $ R ( k\pm i0 )$ on the real
axis. We refer the reader to \cite{Tikhon,Ryzov} and references
therein for the further discussion of this definition.
Let us define a bounded operator $W: L^2(\R; \Delta)\to \cK$ by
setting \be\la{W2} W: \tg \mapsto P_\cK\pmatrix{\tg \cr -S\tg
\cr}. \ee The norm of a smooth vector $ W \tg $ is given
by\footnote{In this section for $E$-valued functions on the real
axis $ \| \cdot \|$ assumes the norm in $ L^2 (\R, E ) $, unless
otherwise specified.} \cite{Pavli} \be\la{PK} \len W \tg \rin^2 =
\len S \sqrt \Delta \tg \rin^2 + \len P_- \Delta \tg \rin^2 . \ee
Then, \cite[Theorem 4, Corollary 1]{N}, \be \la{inter} \( L - z
\)^{-1} W \ =\ W \( A_0 - z \)^{-1}, \hspace{ .4cm} z \in \rho (
L ) \ee where $A_0$ is the operator of multiplication by the
independent variable in $L^2 (\R; \Delta)$. Note that $ \cN_e
\subset H_{ess} $ where $ H_{ess} $ is an invariant subspace of $
L $ naturally associated with the real spectrum as follows: $
H_{ess} = \bigcap \ker \cP_d $ where $ \cP_d $ ranges over the
Riesz projections for isolated finite portions of the spectrum $
\sigma_+ ( L ) $. Indeed, it follows from (\ref{inter}) by the
integration over a closed contour in $ \C_+ $ encircling a
portion that $ \cP_d W = 0 $ for each $ \cP_d $, and the closure
gives $ \cP_d \cN_e = 0 $.
The following lemma describes a class of invariant subspaces of
the operator $L$ in $\cN_e $. Let $\{ X(k) \} $ be a measurable
family of subspaces of $ E $ defined for a. e. $k \in \R $, and
let $ X = \{ f\in L^2(\R, E):\; f(k) \in X ( k ) \mbox{ for a.e. }
k \}$. Define $ \sH $ to be the closure of $ X $ in $L^2(\R;
\Delta ) $. For an operator $ A $ in a Hilbert space $ {\cal A} $
a subspace $ {\cal J} \subset {\cal A} $ is called generating if $
{\cal A} = \bigvee_{ \lambda \in \rho ( A ) } \( A - \lambda
\)^{-1}{\cal J}$. We define the {\it multiplicity of the spectrum}
of an operator $ A $ as the number $m ( A ) = \inf \dim \cN $,
where $\cN $ ranges over the generating subspaces of $A$.
\begin{lemma}
\la{inv} $\cH = \overline{W \sH}$ is an invariant subspace of the
operator $L$, and \be \la{mul} m \( L_\cH \) \le \mbox {ess
sup}_{k\in \R } \dim X (k) .\ee
If $ S ( k ) $ is invertible a. e. on the real axis in the wide
sense, that is, $ \ker S ( k ) = \{ 0 \} $ for a. e. $ k \in \R $,
then \be\la{mul1} m \( L_\cH \) = \mbox {ess sup}_{k\in \R } \dim
\Delta ( k ) X (k) .\ee
If $\mbox {ess sup}_{k \in \R } \len \left. \( S(k) \right|_{
\Delta ( k ) X (k) } \)^{-1} \rin < \infty $ then $\left. W
\right|_\sH $ has a bounded inverse, and thus $ L_\cH $ is similar
to a selfadjoint operator. \end{lemma}
\begin{proof} Since the space $\sH$ reduces $A_0$, the invariance of
$ \cH $ follows from the intertwining relation (\ref{inter}).
Let ${\cal J}$ be a generating subspace for $\left. A_0
\right|_\sH $, thus $ \sH = \bigvee_ {\Im \lambda \neq 0}\( A_0 -
\lambda \)^{-1} {\cal J} $. Since obviously $ \bigvee_ { \lambda
\in \C_+ }\( A_0 - \lambda \)^{-1} {\cal J} = \bigvee_ { \lambda
\in \rho ( L ) \cap \C_+}\( A_0 - \lambda \)^{-1} {\cal J} $, one
obtains that $ \overline{ W {\cal J} } $ is a generating subspace
for $ L_\cH $. This implies (\ref{mul}).
By (\ref{PK}) the bounded operator $ W^{ -1 } : \cN_e \to L^2 (
\R; S^* S \Delta ) $ is defined. Let $ \wt{A}_0 $ be the operator
of multiplication by the independent variable in $ L^2 (\R; S^* S
\Delta) $. One sees from the intertwining relation \be\la{inter1}
W^{ -1 } \( L - z \)^{-1} \ = \( \wt{A}_0 - z \)^{-1} W^{ -1 },
\hspace{ .4cm} z \in \rho ( L ), \ee that for any generating
subspace $ \cN $ of $ L_\cH $ $ \overline{ W^{-1} \cN } $ is a
generating subspace for the restriction of $ \wt{A}_0 $ to $ \sH_*
$, $ \sH_* $ being the closure of $ X $ in $ L^2 ( \R; S^* S
\Delta ) $. Thus
$$ m \( L_\cH \) \ge m \( \left. \wt{A}_0 \right|_{ \sH_* } \) = m
\( \left. A \right|_{ \overline{ S^* S \Delta X } } \) = \mbox
{ess sup}_{k\in \R } \dim \Delta ( k ) X (k) $$ since $ \dim \, (
S^* S \Delta ) ( k ) X (k) = \dim \Delta ( k ) X (k) $ for a. e.
$ k \in \R $ by the assumption about $ S $. Here $ A $ is the
operator of multiplication by the independent variable in $ L^2
(\R, E ) $.
The last assertion of the lemma results from the following
estimate valid for $\tv \in \sH $ by (\ref{PK}) $$ \| W\tv \| \ge
\len S \sqrt \Delta \tv \rin \ge \len \( S|_{\sqrt \Delta X }
\)^{-1} \rin^{-1} \len \tv \rin_{L^2(\R ; \Delta)}. $$ \end{proof}
Note that a special case of the formula (\ref{mul1}) was obtained
in paper \cite{Sachn} when studying the triangle model of a
dissipative operator.
\begin{corollary} \la{spp} \cite{Pavli}
For a Borel set $\omega \subset \R $ let $\chi_\omega $ be the
operator of multiplication by the indicator of the set $\omega $
in $L^2 ( \R; \Delta )$. Define $ \cH_\omega = \overline { \Ran
\, W \chi_\omega } $. Then $ \cH_\omega $ is an invariant subspace
of $ L $. If $\mbox{ess sup}_{k \in \omega } \len S^{-1} (k) \rin
< \infty $ then $ \left. L \right|_{ \cH_\omega } $ is similar to
a selfadjoint operator, and $ \sigma \( \left. L \right|_{
\cH_\omega } \) \subset \overline \omega $. Moreover, in this
case the projection $\cP_\omega $ on the subspace $ \cH_\omega $
parallel to $ \cH_{ \R \setminus\omega } $ is bounded and
coincides with the closure of $ W \chi_\omega W^{-1} $.
\end{corollary}
The operators $\cP_\omega $ defined in this corollary, can be
considered as the spectral projections of the operator $L$ for
the sets $\omega $ if one can guarantee that the spectrum of the
restriction of $ L $ to $ \Ran ( I - \cP_\omega ) = \cH_{\R
\setminus \omega } $ is contained in $ \overline { \R \setminus
\omega } $. The operator $ I - \cP_\omega $ in this case gives the
spectral projection for the set $\R \setminus \omega $.
\begin{remark} \la{scmulloc} \cite{Na,Pavli}
If the function $ S $ has a scalar multiple then for any Borel
set $ \omega \subset \R $
$(i)$ $\sigma \(L|_{ \cH_\omega } \) \subset \overline \omega $.
$(ii)$ any invariant subspace $ \cH \subset \cN_e $ of $L$ such
that $\sigma \( \left. L \right|_\cH \) \subset \overline \omega
$ is contained in $\cH_\omega $. \end{remark}
A point $ k \in \R $ is called a {\it proper point of the operator
L} if $\sup_{ z \in \omede ( k ) } \len S_e^{-1} ( z ) \rin $ $ <
\infty $ for some $ \delta > 0 $. Following \cite{Pav} we call a
point $ k \in \R $ a {\it spectral singularity} if is not proper.
Applying lemma \ref{inv} to $ \sH = L^2 (\R, \Delta ) $ one gets
the if part of the following
\vspace{.5cm}
{\bf Nagy - Foias criterion.} {\it $ \left. L \right|_{ \cN_e } $
is similar to a selfadjoint operator if and only if $ \sup_{k \in
\R } \len S^{-1} ( k ) \rin < \infty $.}
In the usual form of the criterion \cite{Na} the supremum is taken
over the upper half plane and the assertion refers,
correspondingly, to the operator $ L $ itself, which is equivalent
but less convenient for our proposes.
Given $ X $ and defining the corresponding invariant susbspace $
\cH $ as in lemma \ref{inv}, one can always construct another
invariant subspace $ \cH^\sim $ of $L$ such that $\cN_e =
\overline{\cH \dot{+} \cH^\sim }$ by the formula $ \cH^\sim =
\overline{ W \sH^\sim } $, $ \sH^\sim $ being the closure of $
X^\perp = \{ f\in L^2(\R, E):\; f(k) \in X ( k )^\perp \mbox{ for
a.e. } k \}$ in $ L^2 (\R; \Delta ) $. In general, the angle $ (
\cH, \cH^\sim ) $ can hardly be estimated. We now choose $\{ X (
k ) \} $ in a special way so that the angle $ ( \cH, \cH^\sim ) $
admits an estimate in terms of $S$.
Define the function $D = S^* S $. Given a measurable function $ k
\mapsto \gamma_k $, $\gamma_k \in [0,1]$, $ k \in \R $, let $P_1 (
k ) $, $ P_2 ( k ) $ be the spectral projections of $ D ( k ) $
for the intervals $ [ 0, \gamma_k )$, $[\gamma_k , 1 ] $,
respectively, and let $X_{1,2} (k) = \mbox {Ran} P_{1,2} (k) $,
hence $ E = X_1 (k) \oplus X_2 (k) $ since $0 \le D \le I $.
Set\footnote{We omit the argument $ k $ after all the spaces and
operators in the following equalities which are assumed to hold
for a. e. $ k \in \R $.} $\wt{S}_1 = \sqrt D \mid_{ X_1} \oplus
I\mid_{X_2}$, $\wt{S}_2 = I\mid_{X_1} \oplus \sqrt D \mid_{X_2}$.
In what follows the contractions defined by the functions $D(
\cdot )$, $\wt{S}_{1,2}( \cdot )$ in $ L^2 (\R, E)$ will be
denoted with the same letters without arguments. Analogously we
write $ X_{1,2} = \{ f \in L^2 ( \R, E ):\; f \in X_{1,2} ( k )
\mbox { for a.e. } k \in \R \} $. By construction the pair of
orthogonal subspaces $ X_{1,2} \subset L^2 (\R, E)$ reduces
$\Delta $.
Let $ \sH_{1,2} $ be the closures of $ X_{1,2} $ in $ L^2 ( \R;
\Delta) $, hence $ L^2 ( \R; \Delta) = \sH_1\oplus \sH_2$. By
lemma \ref{inv}, $\cH_{1,2} = \overline{W \sH_{1,2}}$ are
invariant subspaces of the operator $L$.
\begin{lemma} \la{An}
\be\la{Ang} \sin \( {\cH_1, \cH_2 } \) \ge {\rm ess }\inf_{k \in
\R } \len \( S(k)\mid_{X_2(k)}\)^{-1}\rin^{-1}. \ee \end{lemma}
\begin{proof}
The assertion is nontrivial if $ \len \s2^{-1} \rin = {\rm ess }
\sup_{ k \in \R } \len \( S(k) \mid_{X_2(k)}\)^{-1} \rin $ is
finite. For $\tu \in X_1 $, $\tv\in X_2 $ we have \bequnan \len
W\tv \rin^2 \le \len \tv \rin_{L^2(\R ; \Delta)}^2 = \len \sqrt{
\Delta } \tv \rin^2 \le \len \s2^{-1} \rin^2 \len \s2 \sqrt{
\Delta } \tv \rin^2 \eequnan and \bequnan \len \s2 \sqrt{ \Delta
} \tv \rin^2 = \len \sqrt{D \Delta} \tv \rin^2 \le \len \sqrt{D
\Delta} \tu \rin^2 + \len \sqrt{D \Delta} \tv \rin^2 = \len
\sqrt{D \Delta} ( \tu - \tv )\rin^2 = \\ = \len S \sqrt \Delta (
\tu - \tv ) \rin^2 \le \len W\( \tu - \tv \) \rin^2. \eequnan
Here we used the first equality in (\ref{PK}) and the fact that $
X_{1,2} $ reduce $ \Delta $. Thus $\len \s2^{-1} \rin^{-1} $ $ \|
W \tv \| \le \| W (\tu - \tv ) \|. $ Now, using the definition of
the angle between subspaces $X,Y$ of a Hilbert space \cite{GK},
$\sin ( X, Y ) = \inf_{u\in X, v\in Y :\| v\| = 1} \| u - v \|$,
we obtain $$ \sin ( \cH_1, \cH_2 ) = \inf_{\tu \in X_1 ,\tv \in
X_2: \| W\tv\| = 1} \| W\tu - W\tv\| \ge \len \s2^{-1} \rin^{-1}
$$ since $X_{1,2}$ are dense in $\sH_{1,2}$.
\end{proof}
We shall use $ P_{ [ 0 , \delta ) } ( k ) $ for the spectral
projection of $ D ( k ) $ corresponding to the interval $ [ 0 ,
\delta ) $.
Note that it is obvious from the proof that this lemma actually
bears local character, that is, for an interval $ \omega \subset
\R $ the angle $ \( \overline{ \cH_\omega \cap \cH_1 },
\overline{ \cH_\omega \cap \cH_2 } \) $ admits the estimate
(\ref{Ang}) with the infimum taken over $ \omega $.
In the standard setup of the Nagy-Foias theory (see \cite[Theorem
VII.1.1]{Na}) invariant subspaces are related to regular
factorizations of the characteristic function in the products of
contractive analytic functions. We refer to our recent paper
\cite{NR} where the above proof of lemma \ref{An} is taken from,
for the explanation of the correspondence between the constructed
invariant subspaces $ \cH_{1,2} $ and the regular factorizations.
For any $ \beta \in ( 0, 1 ) $ one can always construct a
decomposition of $ L_e = \left. L \right|_{ \cN_e } $ into a
linear sum $ L_e = L_1^\beta \dot{+} L_2^\beta $ of operators
$L_{1,2}^\beta = \left. L \right|_{ \cH_{1,2}^\beta }$ acting in
the invariant subspaces $\cH_{1,2}^\beta $ such that $(i)$ $
L_2^\beta $ is similar to a selfadjoint operator, and $ (ii) $ $
\sin ( \cH_1^\beta, \cH_2^\beta ) \ge \beta \neq 0 $, by setting $
\gamma_k \equiv \beta $ in the construction above. If the function
$S$ is continuous up to the real axis and at $ \infty $ in the
operator norm, and the perturbation $ V $ is relatively compact
which is equivalent to saying that $ S ( z ) \in I + {\bf
S}^\infty $, then $ m \( L_1^\beta \) < \infty $ for any $\beta
\in ( 0, 1 ) $. According to the following corollary, in this case
one can choose $ \beta $ appropriately to give $ m \( L_1^\beta \)
$ its smallest value and thus obtain a decomposition of $ L_e $
with minimal possible multiplicity of the component $ L_1^\beta $
such that $ L_2^\beta $ is still similar to a selfadjoint
operator, at the penalty of loss of quantitative estimate of the
angle $ ( \cH_1, \cH_2 ) $. Let $ \cP_\delta $ be the spectral
projection of $ L $ for the $ \delta $ - neighborhood
$\sigma_0^\delta $ of the set of spectral singularities, which is
defined for $ \delta > 0 $ as a bounded operator, $ \cP_\delta =
I - \cP_\omega $, $ \omega = \R \setminus \sigma_0^\delta $.
\begin{corollary}\la{abstrsep} Suppose that $ S $ is norm continuous
in the closed upper halfplane, $ S ( k ) - I \in {\bf S}^\infty $
for all $ k \in \R $, and $ S ( k ) \to I $ in the operator norm
as $ |k| \to \infty $. Then $ \cN_e $ can be represented as a
linear sum $ \cN_e = \cH_1 \dot{+} \cH_2 $ of invariant subspaces
$ \cH_{ 1,2 } $ of the operator $ L $ such that
1. $ L_1 $ has spectrum of multiplicity $ \max_{ k \in \R } \dim
\ker S ( k ) < \infty $,
2. $L_2$ is similar to a selfadjoint operator.
3. $( \cH_1, \cH_2 ) > 0$.
4. For any $ \delta > 0 $ small enough the angle between the
subspaces $ \cP_\delta \cH_1 $ and $ \cP_\delta \cH_2 $ satisfies
$$ \( \cP_\delta \cH_1 , \cP_\delta \cH_2 \) \ge \frac p{\sqrt 2} ,
\; \; p = \min_k \, \mbox{dist} \( 0 , \sigma \( \sqrt{D ( k )} \)
\setminus \{ 0 \} \) $$ where $ k $ ranges over the set $
\sigma_0 = \{ k : \ker S ( k ) \ne 0 \} $ of spectral
singularities.
\end{corollary}
\begin{proof} Under the formulated assumptions the eigenvalues $
\xi_j ( k ) $, $ 0 \le \xi_j (k ) \le \xi_{ j + 1 } ( k ) $, $ j
\ge 1 $ of $ D ( k ) $ are continuous in $ k $. Put $ n ( k ) =
\dim \ker S ( k ) $. Then the set $ \{ k : \ n ( k ) \le m \} $ is
open for any $ m \ge 0 $. From this fact and the asymptotic of $ S
$ at infinity one gets that $ n = \max_{ k \in \R } n ( k ) <
\infty $ and $ \xi_{ n + 1 } ( k ) \ge 2 \beta > 0 $. Define the
subspaces $ \cH_{1, 2} $ according to the construction above with
$ \gamma_k \equiv \beta $. Then, since $ \mbox{ rank} \( P_{ [ 0,
\beta ) } ( k ) \) = n $ for $ k \in \omega $ for an interval $
\omega \subset \R $ and $ \ker S ( k ) = \{ 0 \} $ for a. e. $ k
\in \R $ (see e. g. \cite{Nab}), one gets assertions 1 through 3
from (\ref{mul1}) and lemma \ref{An}. Now choose $ \delta $ from
the condition $ \rho ( \delta ) < p^2/2 $ where $ \rho $ is the
modulus of continuity of the function $ D $ on $ \sigma_0 $, and
put $ \beta = p^2/2 $. Assertion 4 then follows from the remark
after lemma \ref{An}.
\end{proof}
Remark that the subspace $ \cH_1 $ in this corollary can be
chosen to belong to $ \Ran \cP_\delta $. It suffices to put $
\gamma_k = \frac {p^2}2 \chi^\delta (k) $ in the construction,
where $ \chi^\delta $ is the indicator of $ \sigma_0^\delta $.
\newsection{Transport Operator for Slab}
\newsubsection{Preliminaries}
The phase space of the one-speed transport problem for the slab is
$\Gamma = \R \times \Omega ,$ $\Omega \equiv [-1,1]$. For a
particle at a point $(x ,\mu ) \in \Gamma $, the number $x\in \R $
is the position and $ \mu \in \Omega $ is the cosine of the angle
between the particle momentum and the coordinate axis. A density
of particles is a complex-valued function on $\Gamma $ belonging
to the Hilbert space $ H = L^2(\R \times \Omega )$. The physical
parameters in the problem are the total cross-section $ \sigma
\ge 0 $, the mean number of secondaries per collision $ c: \R \to
\R_+ $ and the collision operator $K\in {\bf B} L^2(\Omega )$. We
take the operator $ K $ in the form $ K = \frac 12 \int_\Omega
\cdot \ d \mu^\prime $ which corresponds to the isotropic
scattering of the secondaries. Here $ d \mu $ stands for the
Lebesgue measure on $ \Omega $. It is assumed that $c\in L^\infty
(\R) $.
We do not distinguish the operator $ K $ and the operator $ I
\otimes K $ in $ L^2(\R\times \Omega ) = L^2(\R ) \otimes L^2(
\Omega )$ in our notation. We say that a positive function $ d\in
L^\infty (\R ) $ belongs to the class $ L_0^+ $ if there exists
an $ a > 0 $ such that $ d(x) = 0 $ for a. e. $ x \notin [-a,a]
$. Then $ {\bf 1}$ stands for the indicator of the set $\Omega$.
The evolution of a density in the space $ H $ is given by the
solution of the Cauchy problem for the Boltzmann equation
\cite{JLh}, \be\la{Evol} \cases {-i
\partial _t u = L u \cr \left. u \right|_{ t = 0 } = u_0. \cr} \ee
The generator $L$, which is called the one-speed transport
operator, is defined in $H$ by the following expression, \be\la{L}
L = i\mu \partial_x + i\sigma ( 1- c(x) K ), \ee on the domain
$\cD = \{ f \in H:\, (1) f(\cdot ,\mu )\,\mbox { is absolutely
continuous for a.e. } \mu \in \Omega ;\; (2) $ $ \mu
\partial _x f \in H \}$. The imaginary part of $L$ is bounded
while its real part $ L_0 = i\mu \partial _x$ is a selfadjoint
operator on the domain $\cD $ which corresponds to the evolution
$U_t = \exp {i L_0 t}$, $(U_t\,f)(x,\mu ) = f(x-\mu t,\mu )$, of
densities in vacuum.
Instead of $L$, it is convenient to deal with the dissipative
operator $ T = L^* + i\sigma = i\mu \partial _x + i \sigma c(x) K
$. Without loss of generality, one can set $\sigma = 1$; then $ V
= \Im T = c(x) K $. The spectral analysis of $ L $ is reduced to
that of $ T $ since $ L $ and $ -T + i $ are unitarily
equivalent, $ L = J ( - T + i ) J^* $ with $ ( J f ) ( x, \mu ) =
f ( x, - \mu ) $.
\newsubsection{Spectral Analysis: Discrete Spectrum}
We henceforth assume that the slab has a finite width, i.e., $ c
\in L_0^+ $. Depending on the context, we use $\sqrt{c} $ either
for the operator of multiplication by the function $\sqrt{c(x)} $
in $L^2 ( \Gamma ) $, or for the corresponding element of $ L^2 (
\Gamma ) $. One can naturally identify the range of $ K $ in $ H
$ with the space $L^2 (\R ) $ of functions of the variable $x$,
$K H = L^2( \R )\otimes {\bf 1} \simeq L^2( \R ) $. Thus
$\sqrt{c} $ acts in $K H = L^2 ( \R ) $ and leaves invariant $E
\equiv \overline{ \mbox {Ran}\, V} \subset L^2 ( \R ) $. In this
section we use $ \langle \cdot , \cdot \rangle $ for the inner
product in $ L^2 ( \R ) $.
Set $ R_0 ( z ) = \( L_0 - z \)^{-1} $. The kernel of the
operator $Q ( z ) = \left. i\sqrt V R_0 ( z ) \sqrt V \right|_E
$, $ z \in\C_+ $, acting in the space $ E $, has the form
\cite{JLh}\footnote{This integral kernel is actually calculated
in \cite[ \S 2]{JLh} for the case $c = q \chi_{[-a,a]} $ with
constant $ q $. Below we use an estimate on $Q$ obtained in
\cite{LW,LW2} for the case $c = q \chi_{[-a,a]} $, since the
estimate for general $ c \in L_0^+ $ follows from it.} $ -\frac
12\sqrt{c ( x ) }E ( - i z | x - y | ) \sqrt{c ( y ) }$, where $
E ( s ) = \int_1^\infty e^{-st} \frac{dt}t $ for $\Re s > 0 $.
The function $E(s)$ admits the representation \be\la{assy} E(s) =
-\ln s - \gamma +\theta (s) \ee where $\theta (s) = -
\sum_{m=1}^\infty \frac{(-s)^m}{m!m} $ is an entire function and
$\gamma $ is the Euler constant. Since $c$ is compactly
supported, it follows that $Q(z)$ is of the Hilbert - Schmidt
class $ {\bf S}^2 $ for all $z\in \C_+$. Let $ \left\{ \eta _n (
z ) \right\} _{n=1}^\infty $, $ |\eta _n|\geq |\eta_{n+1}|>0 $,
be the eigenvalues of the operator $ Q ( z ) $. Using
(\ref{assy}), one can represent $ Q(z) $ as \be\la{assy1} Q (z) =
\wt{Q}(z) +\frac 12 \Theta (z) \ee where $\tQ ( z ) $ is the
operator with kernel $\frac 12 \sqrt{c ( x ) } ( \ln ( -i z | x -
y | ) + \gamma ) \sqrt{c ( y ) }$; thus, $ \Theta (z)$ is an
entire function and $\tQ (z) $ is an analytic function in ${\cal
O } = \C \setminus \{ -it,\ t\ge 0 \} $. According to this
formula, the function $Q ( z ) $ has an analytical continuation
from $ \C_+ $ to ${\cal O } $. The Hilbert - Schmidt norm of
$\Theta (z)$ admits the following estimate, \begin{eqnarray}
\la{Theta} \len \Theta (z) \rin_2 \le \( \int c ( x ) c ( y
)\left| \sum_{m=1}^\infty \frac{ | x-y |^m |z|^m}{ m!m }
\right|^2 dx \, dy \)^{ \frac 12 } \le
|c|_1 \sum_1^\infty \frac {\( 2 a |z| \)^m} {m!} = \nonumber \\
|c|_1 \( e^{ 2a|z|} -1\),
\end{eqnarray} where $ |c|_1 = \int_\R c(x) dx $. \vspace{.5cm}
{\bf Theorem 1.} {\it Suppose that $ c \in L_0^+ $. Then the
nonreal spectrum $\sigma _+ ( T ) $ of the operator $T$ consists
of finitely many eigenvalues lying on the imaginary axis.
Moreover, $T$ has no associate vectors. The dimension $ N(c) $ of
the invariant subspace $ H_d $ corresponding to $\sigma _+ ( T )
$ satisfies the estimate \be \la{num} N (c) \le 1 + \frac 14
\int\!\!\!\int \ln^2|x - y| c(x) c(y) dx dy. \ee The essential
spectrum of $T$ coincides with the real axis: $\sigma_{ess}(T) =
\R $.}
\begin{proof} First $\sigma_{ess} ( T ) = \sigma_{ess} ( L_0 ) = \R $
by Sublemma.
We derive the inclusion $ \sigma _+ ( T ) \subset i\R $ from the
fact that $Q( z ) $ has no real eigenvalues (and, particularly the
eigenvalue $-1$) provided that $k \equiv \Re z\neq 0$. It suffices
to show that the operator $\Im Q(z) $ is strictly positive or
negative for $ k\neq 0$. Consider the operator $\Xi ( z ) = \left.
i K R_0 ( z ) \right|_{ K H}$ acting in the space $K H $. In the
Fourier representation with respect to the variable $x$, this
operator acts as the multiplication by the function \be\la{xxi}
\xi ( p , z ) = \frac i2 \int_{-1}^1 \frac{ d\mu }{ p\mu - z }
.\ee Then
\begin{eqnarray} \la{rz} r_z ( p ) \equiv \Im \xi ( p , z ) = \frac
12 \int_{-1}^1 \frac{p\mu - k}{\( p\mu - k \) ^2+\( \Im z\)
^2}d\mu = \frac 1{2|p|} \int_{-|p|-k}^{|p|-k} \frac{ \tau d \tau
}{\tau ^2+\( \Im z\) ^2 } = \nonumber \\ \frac 1{ 4 p } \ln \frac
{ \( p - k \)^2 + \( \Im z \)^2}{ \( p + k \)^2 + \( \Im z \)^2}.
\end{eqnarray}
Clearly, $ r_z ( p ) < 0 $ or $ -r_z ( p ) < 0 $ and hence $ \Im
\Xi ( z ) <0 $ or $ -\Im \Xi ( z ) < 0 $ depending on $ \mbox{sign
} k $. Since $\Im Q(z) = \left. \sqrt c \Im \Xi (z) \sqrt c
\right|_E $, it follows that for $ k \ne 0 $ either $ \Im Q ( z )
< 0 $ or $ -\Im Q ( z ) < 0 $. Note that the last equality in
(\ref{rz}) is not necessary for the conclusion that $ r_z $ has
definite sign but will be used in due time.
This property also implies immediately the absence of associate
vectors for the operator. Indeed, consider the function $ G ( w )
= \( I + Q ( \sqrt{w} ) \)^{ -1 } $. This an $ R $ - function in
the upper half plane, and applying remark \ref{Rfun} one concludes
that $ | \Re z | \len \( I + Q ( z ) \)^{ -1 } \rin $ is bounded
as $ z $ approaches a $ z_0 \in \sigma_+ ( T ) $. If follows that
$ \( I + Q ( z ) \)^{-1} $ has simple poles at the points of $
\sigma_+ (T)$, and the same is true of $ \( T - z \)^{ -1 } $ by
(\ref{Srez}).
We shall now estimate $ N ( c ) $. Let $ N_\von (c) $, $\von > 0
$, be the dimension of the spectral subspace corresponding to the
interval $[i\von, i\infty ) $. Set $ z = i \von $. Then $ r_z (p)
\equiv 0 $, and thus $ Q ( i\von ) $ is selfadjoint. Applying now
remark \ref{Rfun1} to the function $ - Q \( \sqrt w \) $, one
concludes that $ Q ( i\von ) $ is a monotone {\it increasing}
function of $\von $ and a negative operator for each $ \von > 0 $.
%In the Fourier representation,
%the operator in parentheses acts as the multiplication by the
%function $ \( p^2 + \von^2 \)^{-1} $.
It follows that $ N_\von(c) \equiv \# \{ s\ge \von: \; \ker (I +
Q (is)) \neq 0 \} = \# \{ \tau\in (0, 1]:\; \ker ( I + \tau Q
(i\von ) )\neq 0 \} $, co\-unting multiplicity. According to
(\ref{assy1}), \be\la{Q} 2 Q ( i\von ) = Q^0 + \Theta ( i\von ) +
( \gamma + \ln \von ) \ \P1 , \ee where $ \P1 = \llangle\cdot ,
\sqrt c \rrangle \sqrt c $ and $ Q^0 $ is the operator with
kernel $ \sqrt{ c(x) c(y)} \ln |x - y| $. Since $\P1 $ has unit
rank, it follows that \bequnan \# \{\tau \in (0, 1]:\; \ker (I +
\tau Q (i\von ) )\neq 0 \}\le 1 + \# \{\tau \in (0, 1]:\; \\ \ker
(I + \tau Q_1 (i\von ) ) \neq 0 \} \le 1 + \len Q_1 (i\von )
\rin_2^2 \eequnan where $ Q_1 ( i \von ) = \frac 12 ( Q^0 +
\Theta ( i \von )) $. From this using (\ref{Theta}) one obtains
\bequnan N_\von(c) \le 1 + \len Q_1 ( i\von ) \rin_2^2 = 1 +
\frac 14 \len Q^0 \rin_2^2 + O ( \von ) = 1 + \\ \frac 14
\int\!\!\!\int \ln^2 |x - y| c(x) c(y) dx dy + O ( \von ).
\eequnan Then, $ N(c) = \lim_{\von \downarrow 0} N_\von(c)$,
which gives (\ref{num}). \end{proof}
Note that $\sigma_+(T) $ is not empty for any nonzero function $ c
\in L_0^+ $, since $\| Q(it) \| \to \infty $ as $t \to 0 $ and $\|
Q(it ) \| \to 0 $ as $t\to \infty $. It is also clear that $ N (
\kappa c ) \to \infty $ as the coupling constant $ \kappa \to
\infty $.
All the assertions of this theorem, except for the estimate
(\ref{num}), has been proven in \cite{JLh,LW} for the case $ c =
q \chi _{ [ -a,a ] } $. The basic observation about
sign-definiteness properties of $ \Im Q ( z ) $, with an
implication that $ \sigma_+ ( T ) \subset i \R $, was made in
these papers. It has not been realized, however, that the
monotonicity of $ Q ( i \von ) $ and the absence of associate
vectors also follow immediately. Instead, in \cite{JLh,LW2} these
facts were established by calculating the Fourier representation
of $ Q ( i \von ) $ explicitly and observing that $ Q^\prime ( i
\von ) $ times a scalar constant is strictly positive. The latter
can be shown to imply the required first order estimate for $ \(
I + Q ( z ) \)^{-1} $ (see Appendix). Although this arguing is
also avaiable in the case of general $ c \in L_0^+ $, it is
remarkable that all these tedious calculations specific for the
problem are redundant\footnote{Below (Proposition \ref{hdhc}),
however, we employ them for another purpose.}, and the result
follows at once from the abstract theory. The finiteness of $
N(c) $ was deduced in \cite{JLh} from the monotonicity of
$Q(i\von )$ and the fact that the divergent term in the
asymptotics of $ Q ( i\von ) $ at $0$ has finite rank but it has
not been realized that these circumstances allow also to estimate
this number. A proof of the theorem more closed to the original
Lehner-Wing arguments can be found in our paper \cite{KNR}.
Define $\cP_d $ to be the Riesz projection corresponding to
$\sigma _+ ( T ) $, so $ H_d = \cP_d H $. Then $ H = H_d \dot{+}
H_{ess} $ where $ H_{ess} = ( I - \cP_d ) H $ is the invariant
subspace of $ T $ corresponding to the essential spectrum. Since
$ H_d $ is finite dimensional, the angle $ ( H_d, H_{ess} ) > 0
$. In the following Proposition we give an estimate of this angle
for small $ c $ announced in \cite{KNR} without proof.
\begin{proposition}\la{hdhc}
Let $c$ be such that the integral in (\ref{num}) $< 4$ and hence $
\dim H_d = 1 $. Then $$ \sin \, ( H_d, H_{ess} ) \ge \frac{e^{ -
\gamma }}{ 4 + \pi^2 } \frac 1{a |c|_\infty } \exp \( -\frac
2{|c|_1} \).$$
\end{proposition}
\begin{proof} We rely on the following abstract formula valid in the case
$\dim H_d = 1 $, \be\la{ea} \sin ( H_d, H_{ess} ) = \Im z_0 \left|
\llangle Q^\prime (z_0 ) e, e^* \rrangle \right| \ee where $\{ z_0
\} = \sigma_+ ( T ) $, $ e \in \ker ( I + Q ( z_0 ) ) $, $ e^* \in
\ker ( I + Q^* ( z_0 ) ) $, $ \| e \| = \len e^* \rin = 1$.
Indeed, for $ \dim H_d = 1 $ $ \cP_d = \frac 1{\llangle u, u^*
\rrangle } \llangle \cdot , u^* \rrangle u $ where $ u \in H_d $,
$ u^* \in \ker ( T^* - \overline{ z_0 } ) $. We have $ u = - i
R_0 ( z_0 ) V u $, $ u^* = i R_0 \( \overline{ z_0 } \) V u^* $.
Multiplying this by $ \sqrt V $, one obtains that $ \sqrt V u $
and $ \sqrt V u^* $ differ from $ e $ and $ e^* $, respectively,
by scalar factors. Then \bequnan \left| \llangle u, u^* \rrangle
\right| = \left| \llangle R_0 ( z_0 ) V u , R_0 ( \overline{ z_0
} ) V u^* \rrangle \right| = \left| \llangle \sqrt V R_0^2 ( z_0
) V u , \sqrt V u^* \rrangle \right| = \\ = \left| \llangle
Q^\prime ( z_0 ) \sqrt V u , \sqrt V u^* \rrangle \right|.
\eequnan By the definitions of $ u $ and $ u^* $ one has $ \Im
z_0 \len u \rin^2 = \len \sqrt V u \rin^2 $, $ \Im z_0 \len u^*
\rin^2 = \len \sqrt V u^* \rin^2 $. We now get (\ref{ea}) from the
identity $$ \sin ( H_d, H_{ess} ) \equiv \len \cP_d \rin^{-1} =
\frac{ \left| \llangle u, u^* \rrangle \right| }{ \| u \| \ \|
u^* \| }. $$
In our case $ Q(z_0 )$ is selfadjoint and thus $e = e^* $. We
shall estimate separately the factors in (\ref{ea}) from below.
One has $Q ( i\von ) = \left. \sqrt c \Xi ( i\von ) \sqrt c
\right|_E $, where the operator $\Xi ( i\von ) = \Xi^* ( i\von )$
in the Fourier representation acts as the multiplication by the
function \be\la{xi} \xi_\von ( p ) = -\frac\von 2\int_{-1}^1\frac
{d\mu}{ \left| p\mu + i\von \right|^2 } = -\frac 1{ |p| } \arctan
\frac{ |p| }\von . \ee One can now calculate the operator $
Q^\prime ( i \von ) = - i \left. \sqrt{c} \( \frac d{d\von} \Xi
(i\von) \) \sqrt{c} \right|_E $ in the Fourier representation by
differentiation. This gives \be\la{pred} \left| \llangle Q^\prime
(i \von ) e, e \rrangle \right| = \int \frac {\left|
\wh{\sqrt{c}e}( p )\right|^2}{p^2 + \von^2} \, dp . \ee
Define the operator $ G (\von) = \left. \sqrt c \Xi^2 ( i \von )
\sqrt c \right|_E $ acting in $E$. By (\ref{xi}) we have $
\llangle G(\von) e, e \rrangle = \llangle \xi_\von^2
\wh{\sqrt{c}e}, \wh{\sqrt{c} e}\rrangle $. On the other hand ($z_0
= i \von_0 $) $$ \frac 1{\left| c \right|_\infty} = \frac { \left|
\llangle Q^2 (i \von_0) e, e \rrangle \right| } { \left| c
\right|_\infty} = \frac { \len \sqrt c \Xi (i \von_0) \sqrt{c} e
\rin^2 } { \left| c \right|_\infty} \le \len \Xi (i \von_0 )
\sqrt{c} e \rin^2 = \llangle G (\von_0 ) e, e \rrangle .$$ From
this and (\ref{pred}) using the elementary inequality $$ \frac
1{p^2 + t^2} \ge \frac 1{ 1 + \frac{\pi^2} 4 }\xi_t^2 (p) $$ we
find \be\la{preda} \left| \llangle Q^\prime \( i \von_0 \) e, e
\rrangle \right| \ge \frac 1{ \(1+ \frac{\pi^2}4\) \left| c
\right|_\infty }. \ee
We now estimate $\von_0 $ from below through the variational
principle according to which $ -\eta_1 (i\von_0) = \max_{u \in E;
\| u \| = 1 } \( - \llangle Q(i\von_0) u, u \rrangle \) = 1 $. By
monotonicity of the function $E$ one has $$ 1 \ge - \frac{
\llangle Q(i\von_0) \tu, \tu \rrangle }{ \len \tu \rin^2 } =
\frac 1{ 2 |c|_1} \int\!\!\int c(x) c(y) E ( \von_0 | x - y | )
dx\, dy \ge \frac { |c|_1 }2 E (2 a \von_0 ) $$ for $\tu = \sqrt
c $. The following estimate results from the identity $$ E(s) = -
\gamma - \ln s + \int_0^s \frac{ 1 - e^{-t} }t dt $$ valid by the
definition of $ E $, $$ E(s) \ge - \gamma - \ln s,
\hspace*{.5cm} s > 0. $$ From this one finds \be\la{t0} \von_0
\ge \frac 1{2a} \exp \( - \gamma - \frac 2{|c|_1} \). \ee
Substituting this and (\ref{preda}) in (\ref{ea}), one comes to
the required estimate.
\end{proof}
Let us explain the meaning of this proposition in terms of the
dynamics generated by $ L $. Since $ L $ and $ - T + i $ are
intertwined by $ J $, $ J H_d $, $ J H_{ess} $ are invariant
subspaces of $ L $. The further study of $ T $ by tools of the
functional model shows that under the assumption made in the
proposition the restriction of $ T $ to $ H_{ess} $ is similar to
a selfadjoint operator (see remark \ref{instab}). From this it is
easy to see that $ e^{ i L t } f = e^{ (\von_0 - 1 ) t } \wt{
\cP_d } f + O \( e^{ - t } \) $ where $ \wt{ \cP_d } = J \cP_d
J^* $. Since $ \len \wt{\cP_d} \rin^{ -1 } = \sin \, ( H_d,
H_{ess} ) $, the estimate obtained provides us with an upper bound
for the factor before exponent in the leading term of the time
asymptotics.
Of course, the estimate (\ref{t0}) for $ \von_0 $ is rather crude
and can be improved in many ways. Our only aim here is to
demonstrate the possibility of estimating the angle $ ( H_d ,
H_{ess} ) $ explicitly in terms of $ c $, and (\ref{t0}) has been
chosen for its simplicity. Let us remark, however, that
estimating $ \von_0 $ is a question of some interest of its own,
since a lower bound for $ \von_0 $ provides us with a lower bound
on the rate of the slowest decay in the corresponding physical
problem related to the operator $ L $.
\newsubsection{Spectral Analysis: Essential Spectrum}
We now proceed to study the component $T_{ess} = \left. T
\right|_{H_{ess}} $ by means of the functional model. Let $ S $ be
the characteristic function of the operator $T$. Then $ S $ is
analytic on the real axis except at the point $ 0 $.
\begin{lemma} \la{scmul} $S$ has a scalar multiple. \end{lemma}
\begin{proof} We first see that $T(z) \equiv S(z) -
I\in {\bf S}^2$ for all $z\in \C_+$. According to lemma
\ref{abstr}, to conclude that the function $m ( z ) \equiv \det \(
I - T^2 ( z ) \)$ is a scalar multiple for $ S $, it suffices to
show that $ \len T ( z ) \rin_2 $ is uniformly bounded in $\C_+$.
We first estimate it in the vicinity of $ 0 $. By (\ref{assy}),
the following representation holds:
$$T (z) = ( A + \Theta ( z ) ) \( I - Q(z) \)^{-1} + M ( z ), $$
where $\mbox{rank } M(z) = 1 $ and $ A \in {\bf S}^2 $. Since $\|
T ( z ) \| \le 2 $, we have $\len M(z) \rin_2 = \| M(z) \| \le 2 +
\| A \| + C_\delta |z| \le C_{1,\delta} $ for $|z| \le \delta $.
It follows that $ \sup_{z \in \omede (0) } \len T(z) \rin_2 \le
C_\delta $, since $ \len \Theta ( z ) \rin_2 $ is bounded for $ |
z | \le \delta $ by (\ref{Theta}). Then, Lemma 5 in \cite{LW2}
states, in our notation, that\footnote{This estimate was
essentially established in the course of the proof. The statement
of Lemma 5 in \cite{LW2} is weakened to the estimate of the
operator norm of $Q(z)$.} \be\la{estQ} \len Q(z) \rin^2_2 \le
\frac {C_\von}{ 1 +|\Re z|} \ee for $z \in \Pi_\von \bigcap \{ z:
\; | \Re z | \ge 1 \} $ for any $\von > 0$. Choosing an arbitrary
$ \von \in (0, 1] $ and taking $\delta > \sqrt 2 $, we obtain
$\len T(z) \rin_2 \le C $ for $z \in \Pi_\von $. Then, by the
obvious inequality $ |E(s)| \le E(\Re s) \le E (\Re s_0 ) $ valid
for $0 < \Re s_0 \le \Re s $, which results from the definition
of $E$, one has $\len Q(z) \rin_2 \le \len Q(i\von) \rin_2 $ for
$ \Im z \ge \von $, and thus $\sup_{z \in \C_+ \setminus \Pi_\von
} \len T(z) \rin_2 < \infty $. Combining the estimates obtained
we get $\sup_{z \in \C_+} \len T(z) \rin_2 < \infty $. \end{proof}
The estimate (\ref{estQ}) was used in \cite{LW2} for the
derivation of some pointwise asymptotics of solutions of the
Boltzmann equation. However, the derivation itself contains an
error (see below).
Let ${\cal B}_c = \{ k_n = \lim _{\von \downarrow 0 } \eta_n (
i\von ) :\ | k_n | < \infty \ \} $. We define the singular set
$\cE \subset L_0^+ $ as follows: $\cE = \{ c\ :\ -1\in {\cal B}_c
\} $. Note that for any $c\in L_0^+ $ the function $\kappa c $
belongs to $ \cE $ for the infinite discrete set $\left\{ -
k_n^{-1},\; k_n \in {\cal B}_c \right\}$ of values of the
constant $\kappa > 0 $. The following lemma motivates the
definition of the set $\cE $.
\begin{lemma}\la{sepE} If $c\notin \cE $, then $\sup_{z \in \omede
(0) } \len S^{-1} (z) \rin < \infty $ for sufficiently small
$\delta $. \end{lemma}
\begin{proof} By (\ref{Char}) we have $ S^{-1} (z) = -I + 2 \( I + Q
(z) \)^{-1} $ for $ z \in \C_+ \cap \rho ( T ) $, and thus the
desired estimate is equivalent to the uniform boundedness of $
\len \( I + Q (z) \)^{-1} \rin $ in $ z \in \omede (0) $.
Suppose, on the contrary, that $\( I + Q ( z_n ) \) \varphi_n \to
0 $ as $ z_n \to 0 $ for some sequence $ \{ \varphi_n \} ,
\|\varphi_n \| =1 $. Then $\ln |z_n| \ \P1 \varphi_n $ remains
bounded as $ n \to \infty $ (see (\ref{assy1})), and thus $ \P1
\varphi_n \to 0 $. It follows that $ \( Q(z_n) - Q(i|z_n|) \)
\varphi_n \to 0 $. This gives $ ( I + Q ( i |z_n| ) ) \varphi_n
\to 0 $ which contradicts the assumption that $c \notin \cE $.
\end{proof}
\begin{corollary}\la{sepEco}
$ 0 $ is a spectral singularity if and only if $ c \in \cE $.
\end{corollary}
\begin{proof} For $ c \in \cE $ we have $ S ( k ) f( k ) \to 0 $ as $
k \to 0 $ where $ f ( k ) $ is the normalized eigenfunction of $ Q
( i |k| ) $, $Q ( i |k| ) f ( k ) = \eta ( |k| ) f ( k ) $, such
that $\eta ( |k| ) \to -1 $ as $ k \to 0$. Indeed, \bequnan \| (
I+ Q ( k ) ) f ( k ) \| = \| ( 1 + \eta ( |k| ) ) f ( k ) + ( Q (
k ) - Q ( i |k| ) ) f ( k ) \| \le \\ | 1 + \eta ( |k| ) | + \frac
12 \frac \pi 2 \| \P1 f (k ) \| + O (|k| ) .\eequnan The first
term tends to $0$ by the choice of $ f ( k ) $ and $\P1 f ( k )
\to 0$ because $\ln |k| \ \P1 f ( k ) $ remains bounded as $ k
\to 0 $ by (\ref{Q}). Since $ \| S ( k ) u \| = \| S_e ( k ) u \|
$ for a.e. $ k \in \R $, this implies the if part of the
assertion. The only if part follows from lemma \ref{sepE} since $
S_e^{ -1 } = S^{ -1 } S_i $ is bounded whenever $ S^{ -1 } $ is
bounded. \end{proof}
Note that it follows from the proof that for $ c \in \cE $ we have
$ S ( z ) f ( \pm |z| ) \to 0 $ as $ z \to 0$ uniformly in $
\mbox{arg}\, z \in [ 0 , \pi ] $. Thus for $c\in\cE $ the function
$ S^{-1} ( \cdot ) $ is unbounded at any neighborhood of $0$. A
priori the inner factor $ S_i $ of the characteristic function can
also have a zero at the point $ 0 $ \cite{Pav}. In Proposition
\ref{out} below, we show that this does not take place in fact.
Let us first obtain another description of the set $\cE $.
\begin{lemma}
\la{S0} $ S( \cdot ) $ is continuous at $0$ in the operator norm
and \be \la{este} c \in \cE \Longleftrightarrow \ker\( I -
\tQ_0^2 - \frac 2{\vartheta_c} \llangle \cdot , \sqrt c \rrangle
\sqrt c \) \neq 0, \ee where $\tQ_0 $ is the integral operator
with kernel $\frac 12 \sqrt{c(x)} \ln \(\frac{|x - y|}{2a} \)
\sqrt{c(y)} $ and $\vartheta_c = \llangle \( I - \tQ_0 \)^{-1}
\sqrt c, \sqrt c \rrangle $.
\end{lemma}
\begin{proof} Since $ \Re Q ( z ) \le 0 $, it follows from
(\ref{assy1}) that $ \( I - \tQ (z) \)^{-1}$ exists and is
bounded for $ z\in \omede (0) $ provided that $\delta $ is
sufficiently small. Rewriting (\ref{Char}) as $ S(z) = - I + 2 ( I
- Q (z) )^{-1}$, we see from the identity \be \la{razn} ( I - Q(z)
)^{-1} - \(I - \tQ (z)\)^{-1} = - ( I- Q (z) )^{-1} \frac {\Theta
(z)}2 \( I- \tQ (z) \)^{-1} \ee that $ S ( z ) $ is continuous at
$0$ whenever so is $ \( I - \tQ (z) \)^{-1} $. Fix an arbitrary
positive $ \delta $ such that $ \tQ ( i \delta ) < 1/2 $. We then
have $\tQ (z) = \tQ ( i \delta ) + \alpha (z) \ \P1 $ where
$\alpha (z) = \frac 12 \ln \( - \delta^{ -1 } i z \) $. The
operator $ I - \tQ ( i \delta ) $ has a bounded inverse, and $
\vartheta_c ( \delta ) = \llangle \( I - \tQ ( i \delta ) \)^{-1}
\sqrt c, \sqrt c \rrangle \neq 0 $. A straightforward computation
now gives \be \la{tQ} \( I - \tQ (z) \)^{-1} = \( I - \tQ (
i\delta ) \)^{-1} + \frac { \alpha (z) } { 1 - \alpha (z)
\vartheta_c ( \delta ) } \( I - \tQ (i\delta) \)^{-1} \P1 \( I -
\tQ (i\delta) \)^{-1}. \ee The right hand side is obviously
continuous at $ 0 $. Thus, $S$ is continuous at $0$.
Set $ \tau = \frac 1{ 2a e^\gamma } $. It is easy to see that the
operator\footnote{This fact, with a somewhat lengthy derivation,
has first been observed in \cite[Lemma 7]{LW} for the case $c = q
\chi_{[-a,a]} $ when proving the finiteness of $ \sigma_+ ( T ) $,
and is actually redundant for that purpose.} $ \tQ_0 \equiv \tQ (
i \tau ) < 0 $. Indeed, one can insert the indicator of the set $
| x - y | \le 2a $ into the definition of the kernel $ \tQ_0 $,
so $ \tQ_0 = \left. \sqrt c \, \wt {\Xi}_0 \sqrt c \right|_E $
where the operator $ \wt {\Xi}_0 $ in the Fourier representation
acts as the multiplication by the function
$$ \frac 1{ 2 \sqrt { 2 \pi }} \int_{ -2a }^{ 2a } e^{ ips } \ln
\frac{|s|}{2a} \,\, ds = \frac 1{ \sqrt { 2 \pi }}\, 2a \int_0^1
\cos ( 2aps ) \ln s \, ds
$$
which is obviously negative for real $ p $. Thus, (\ref{tQ})
holds with $ \delta = \tau $, and passing to the limit $ z \to 0
$ one gets $$ S(0) = \frac{ I + \tQ_0 } { I - \tQ_0 } - \frac 2 {
\vartheta_c } \( I - \tQ_0 \)^{-1} \P1 \( I - \tQ_0 \)^{-1}. $$
Multiplying this equality on the left and right by $ I - \tQ_0 $,
one obtains (\ref{este}), since $ c \in \cE $ iff $ \ker S(0)
\neq \{ 0 \} $. \end{proof}
Let $ \cN_e $ be the absolutely continuous subspace of $T$ and $
H_0 $ be the maximal reducing subspace of $ T $ on which it
coincides with $ L_0 $.
\begin{proposition}\la{out}
The component $T_{ess}$ of the operator $T$ has purely absolutely
continuous spectrum, $ H_{ess} = \cN_e \oplus H_0 $.
\end{proposition}
\begin{proof} First, the space $ \cN_e \subset H_{ess} \ominus
H_0 $ (see the remark made after (\ref{inter})). According to the
general theory \cite{Pav}, $ H \ominus H_0 = \cN_e \dot{+} \cN_i
$ where $ \cN_i $ is the inner subspace of $T$ (see \cite{Pav}
for the definition), provided that the angle $ ( \cN_e , \cN_i ) >
0 $. Our aim will be to show that $ \cN_i = H_d $.
It is obvious from the representation (\ref{assy1}) that the
analytical continuation of the function $Q(z)$ across the real
axis to the set $ {\cal O} $ also holds in the $ {\bf S}^2 $ -
norm. This implies that the scalar multiple $ m ( z ) $ is
analytic on the real axis outside $ 0 $ since so is $ T ( z ) $
in the $ {\bf S}^2 $ - norm. This means that the singular
component $m_s $ in the canonical factorization of $m$ has the
form $$ m_s(z) = e^{i \mu_1 z - \mu_2 \frac iz }
$$ with some $\mu_1 , \mu_2 \ge 0 $.
It follows from (\ref{tQ}) that $ \(I - \tQ (z) \)^{-1} $ is a
bounded analytic function in $ U_\tau (0) \setminus [ 0, -i\tau )
$. Then, by taking into account the fact that $\| \Theta ( z ) \|
= o( 1 ) $ as $ z \to 0 $, we conclude that $ \( I - Q(z) \)^{-1}
$, and therefore $ S(z) $, extends to a bounded analytic function
on $ \wt{U}_\delta \equiv U_\delta (0) \setminus [0, -i\delta ) $
for sufficiently small $ \delta $. Arguing as in the proof of
lemma \ref{scmul}, we now see that the analytic continuation of $
T ( z ) $ into $ \wt{U}_\delta $ is also bounded in the $ {\bf
S}^2 $ - norm, and therefore, $m$ admits analytic continuation
into $ \wt{U}_\delta $ which is bounded. It follows that $ \mu_2 =
0 $. Indeed, otherwise an application of the Carlson theorem
\cite{Titch} to the function $g\( z^{ -1 } \) $, where $g (z) =
e^{\frac {\mu_2}2\frac iz } m(z) $, implies $ g \equiv 0 $. Thus,
the inner factor, $ m_i $, of $ m $ has the form $ m_i = b e^{ i
\mu_1 z } $ where $ b $ is a Blaschke product.
%\footnote{ The function $ \wt S $ is pure, otherwise there would
%exist a subspace $ E^\prime \subset E $ such that $ S ( k )
%|_{E^\prime } $ is a constant isometry which is in fact unity
%considering that $ S ( k ) \to I $ for $ |k| \to \infty $. This is
%not the case since, as it will be shown in the proof of Proposition
%\ref{simse}, $ \Im Q (k) \ne 0 $ for all $ k \ne 0 $.}
The characteristic function $ S_{ i * } $ of the restriction $
\left. T \right|_{ \cN_i } $ has the scalar multiple $ m_i $, the
inner factor of $ m $. By \cite[Theorem VI.5.1]{Na} it follows
that $ \left. T \right|_{ \cN_i } $ has the minimal function $
m_T $, which is the greatest common inner divisor of scalar
multiples for $ S_{ i * } $. Assume that $ \theta ( z ) = e^{ i d
z } $ is an inner divisor of $ m_T $ for some $ d > 0 $. Then
(\cite[Theorem III.6.3]{Na}) there exists an invariant subspace
of $ T $ such that $ \theta $ is the minimal function for the
restriction $ T^\prime $ of $ T $ to this subspace. This means
that $ \sigma ( T^\prime ) $ is empty which is impossible because
$ T $, and therefore $ T^\prime $, has bounded imaginary part.
Since $ m_T $ is a divisor of $ m_i $, in our case this implies
that $ b $ is a scalar multiple for $ S_{ i
* } $. Hence $ \cN_i $ is generated by the root vectors of $ T $,
and so $ \cN_i = H_d $. Comparing the decompositions $ H \ominus
H_0 = \( H_{ess} \ominus H_0 \) \dot{+} H_d $ and $ H \ominus H_0
= \cN_e \dot{+} \cN_i $, we now obtain the result.
\end{proof}
Let us mention that the inner factor $ m_i $ is in fact the finite
Blaschke product corresponding to the set $ \sigma_+ ( T ) $, that
is, $ \mu_1 = 0 $ as well, since $ \len Q ( i \von ) \rin_2 \to 0
$ as $ \von \to + \infty $ and therefore $ m ( i \von ) \to 1 $.
This arguing which avoids references to abstract facts about the
minimal function, was used in the proof of this proposition in our
earlier paper \cite{KNR}.
\begin{proposition} \la{simse}
$T_{ess} $ is similar to a selfadjoint operator if and only if
$c\notin \cE $.
\end{proposition}
\begin{proof} According to the Nagy-Foias criterion, one needs to
show that $\sup_{ k \in \R } $ $ \len S^{-1} (k) \rin < \infty $
if $c\notin \cE $ (the only if part is already proven). From
(\ref{rz}) we have $ r_z (p) \to \frac 1{ 2 p } \ln \left| \frac {
p - k }{ p + k }\right| $ as $\Im z \to 0 $, $\Re z = k \neq 0$.
Since the Fourier transform of $ \sqrt{c} u $, $ u \in L^2 ( \R )
$, is continuous owing to the compactness of the support of $ c $,
and the singularity of $ r_k $ at $ p = \pm k $ is weak, one can
pass to the limit $\Im z \to 0 $ in the quadratic form of $ \Im Q
( z ) $, which gives $ \llangle \Im Q ( k ) u, u \rrangle =
\llangle r_k \wh{ \sqrt{c} u }, \wh{ \sqrt{c} u } \rrangle $.
This implies that $( \mbox {sign } k ) \Im Q (k) < 0 $ and so
$\ker S ( k ) = \{ 0 \} $ for real $k \neq 0 $. Then the estimate
(\ref{estQ}) shows that $ S ( k ) \to I $ as $ |k| \to \infty $
in the operator norm, and it follows that \be \la{estme} \sup_{k
\in \R \setminus [ -\delta, \delta ] } \len S^{-1} (k) \rin <
\infty .\ee Then, $\sup_{k \in [ -\delta, \delta ] } \len S^{-1}
(k) \rin < \infty $ for $c\notin \cE $ by lemma \ref{sepE}, and we
are done.
\end{proof}
\begin{corollary} \la{wvo} If $ c \notin \cE $ then the wave
operators $$ W_\pm ( L, L_0 + i\sigma ) \equiv W_\pm ( T^*, L_0 )
= {\mbox {s - lim}}_{ t \to \pm \infty } e^{ i T^* t } e^{ - i L_0
t } $$ exist on $ H $ and are complete, $ \mbox{ Ran } W_\pm = J
H_{ess} \equiv H_d^\perp $. \end{corollary}
\begin{proof} According to theorem 6 in \cite{Nab}, the fact that $
S ( z ) - I $ is compact for $ \Im z \ge 0 $ yields that the wave
operators $ W_\pm ( T, L_0 ) $ exist on a dense set, $ \cR $, in $
H \ominus H_0 $ and are complete, $ \overline { W_\pm ( T, L_0 )
\cR } = \cN_e $, for all $ c \in L_0^+ $. For $ c \notin \cE $
they are bounded and, therefore, exist on the whole of $ H $, and
admit inverse. It remains to note that obviously $ J W_\pm ( T^*,
L_0 ) J = W_\mp ( T, L_0 ) $. \end{proof}
We thus have the following picture of the behaviour of the
operator in coupling constant implied by monotonicity of the
function $ Q ( i \von ) $. In the remark below $ c = \kappa \wt c
$ with a fixed nonzero $ \wt c \in L_0^+ $.
\begin{remark} \la{instab}
For $ \kappa > 0 $ small enough the operator $ T $ has one
eigenvalue, and the component $ T_{ess} $ is similar to a
selfadjoint operator. In particular, $ c \notin \cE $ if the
integral in (\ref{num}) is $ < 4 $. Then, at $ \kappa = \kappa_1
= - 1/\eta_2 ( + i0 ) $, a spectral singularity occurs at the
point $ 0 $, and a bunch of eigenvalues monotone increasing in $
\kappa $ along the imaginary axis emerges\footnote{point $0$
itself is never an eigenvalue} from $ 0 $ for $ \kappa > \kappa_1
$: if $ c \in \cE $ then $ N ( ( 1 + \von ) c ) > N ( c ) $ for
any $ \von > 0 $. The spectral singularity disappears for $ \kappa
> \kappa_1 $, and $ T_{ess} $ is similar to a selfadjoint operator
as long as $ \kappa \in (\kappa_1 , \kappa_2 )$, $\kappa_2 = -
1/\eta_3 ( + i0 ) $. At $ \kappa = \kappa_2 $ the singularity
occurs again, and a new bunch of eigenvalues emerges, and so on.
\end{remark}
We now restrict our consideration to the case $ c \in \cE $. Then,
in view of (\ref{estme}), the point $ 0 $ is the only spectral
singularity of $T$. As we have seen, the function $ S $ satisfies
the assumptions of corollary \ref{abstrsep}. Thus, one can
decompose $T_{ess}$ into a linear sum, one component of which is
similar to a selfadjoint operator while the other has spectrum of
multiplicity $ \cM = \dim \ker S ( 0 ) $. Moreover, we have a
local estimate of the angle between the corresponding invariant
subspaces in the sense given by item 4 of corollary
\ref{abstrsep} with $ p = \mbox{ dist } ( 0, \sigma( S ( 0
))\setminus \{ 0 \} )$. According to the construction suggested,
it suffices to choose $ \delta \neq 0 $ small enough so that the
norm of $ D ( k ) - D ( 0 ) $ be lesser than $ {p^2}/2 $ for all
$k \in (-\delta, \delta ) $. Specific for our problem is the fact
that one can estimate the modulus of continuity of $ D $ at $ 0 $
and, thus, find a sufficient $ \delta $ explicitly in terms of $
c $ and $ p $.
\begin{lemma}
\la{fuct1} Let $ |c|_2 = \len c \rin_{L^2 (\R)} $ and \be\la{de2}
\delta_* = \frac 1 { 2 a } \exp \( - \frac { 64 a \Upsilon^2
|c|_2^2 }{ p^2 |c|_1} - \gamma \) \ee where $ \Upsilon = \(
\int_{-1/2}^{1/2} \int_{-1/2}^{1/2} \ln^4 | x - y | dx dy \)^{
1/4} $. Then $ \| D ( k ) - D ( 0 ) \| < p^2/2 $ for all $k \in
(-\delta_*, \delta_* ) $.
\end{lemma}
\begin{proof} Since $\tQ_0 < 0 $, we have $\Re \tQ (z) < 0 $ for $ z
\in U^\circ \equiv \overline{\omega_\tau ( 0 ) }$, $ z \neq 0 $.
This allows to define for $ z \in U^\circ $ the contractive
function $$ \Sigma (z) = \frac {I + \tQ (z)}{I - \tQ (z)} = - I +
2\(I - \tQ (z)\)^{-1} $$ which is continuous in $z$ and satisfies
\bequnan \len \Sigma(z) - \Sigma (0) \rin = 2 \len \left[ \frac {
\alpha (z) } { 1 - \alpha (z) \vartheta_c } + \frac 1 \vartheta_c
\right] \( I - \tQ_0 \)^{-1} \P1 \( I - \tQ_0 \)^{-1} \rin \le \\
2 \len \( I - \tQ_0 \)^{-1} \rin^2 \len \P1 \rin \left| \frac 1 {
\vartheta_c \( 1 - \alpha (z) \vartheta_c \) } \right| \le \frac
{2 |c|_1} {\vartheta_c \left| 1 - \alpha (z) \vartheta_c \right|
} \le \frac {2 |c|_1} {\vartheta_c^2 | \alpha (i |z| ) |}.
\eequnan Thus \be\la{SS} \len (\Sigma^*\Sigma) (z) - \Sigma^2 (0)
\rin \le \frac {4 |c|_1} {\vartheta_c^2 | \alpha (i |z| ) |} = -
\frac {8 |c|_1} {\vartheta_c^2 \ln ( 2 a e^\gamma |z| ) }. \ee
Then, $$ S(z) - \Sigma(z) = 2\( \( I - Q(z) \)^{-1} - \( I - \tQ
(z) \)^{-1} \). $$ By (\ref{razn}) this implies $ \len S(z) -
\Sigma (z) \rin \le \| \Theta (z) \| $ as $\len \( I - \tQ (z)
\)^{-1} \rin \le 1$, and thus $$ \| (S^* S)(z) - (\Sigma^* \Sigma
)(z) \| \le 2 \| \Theta (z) \| $$ for $ z \in U^\circ $.
Combining this with (\ref{SS}), one gets $$ (S^* S) ( z ) - S^2
(0) = ( S^* S ) ( z ) - (\Sigma^* \Sigma) (z) + (\Sigma^* \Sigma)
(z) - S^2 (0) \equiv \Omega_1 (z) + \Omega_2 (z) $$ where $ \|
\Omega_1 (z) \| \le 2 |c|_1 \( e^{2a|z|} - 1 \) $ by
(\ref{Theta}) and $ \| \Omega_2 (z) \| $ is estimated from above
by the expression in the right hand side of (\ref{SS}). It
follows that
$$ \len (S^* S) ( z ) - S^2 (0) \rin < \frac { p^2}2 $$ if $ |z| <
\delta $ with a $ \delta $ satisfying the following system of
elementary inequalities, $$ 2 |c|_1 \( e^{ 2 a \delta } - 1 \)
\le \frac { p^2 }4 ; \hspace{8mm} - \frac {8 |c|_1}
{\vartheta_c^2 \ln \( 2 a e^\gamma \delta \) } \le \frac {p^2}4
.$$ To make the estimate of $\delta $ more effective we take into
account that $$ \vartheta_c \ge \frac {|c|_1} { 1 + \len \tQ_0
\rin } \ge \frac {|c|_1} { 2 \len \tQ_0 \rin } $$ because $ c \in
\cE $ and thus $ \len \tQ_0 \rin \ge 1 $. Then, $ \len \tQ_0 \rin
\le \len \tQ_0 \rin_2 \le \frac \Upsilon 2 \sqrt{ 2 a } |c|_2 $
by the Schwartz inequality. From this we get that the required
estimate holds if $$ |z| < \delta \le \frac 1 { 2 a } \min
\left\{ \ln \( 1 + \frac { p^2 }{ 8 |c|_1 }\), e^{ - \frac { 32
\Upsilon^2 |c|_1 }{ p^2 } \( \frac { \sqrt{ 2 a} |c|_2 }{ |c|_1}
\)^2 - \gamma } \right\}. $$ Using the inequality $ e^x - 1 > x $
for $ x > 0 $ with $ x = \frac { 8 |c|_1 }{p^2} $ one can see
that the second entry is not greater than the first one, since $
\frac { \sqrt{ 2 a } |c|_2 }{ |c|_1} \ge 1 $ by the Schwartz
inequality and $ \Upsilon > 1 $.
\end{proof}
Choose a positive $ \delta \le \delta_* $ and define the
invariant subspaces $ \cH_{1,2} $ according to the construction
of Section 2 with $ \gamma_k = \frac{p^2}2 \chi_{ [ - \delta,
\delta ] }( k ) $. Let $ \cP_\delta $ be the spectral projection
of $T$ for the interval $ [-\delta, \delta ]$ which is defined to
be $\cP_\delta = I - \cP_\omega $ with $ \omega = \R \setminus
[-\delta, \delta ] $. Applying corollary \ref{abstrsep}, we thus
get the following \vspace{.5cm}
{\bf Theorem 2.} {\it If $c\in \cE $, then for any positive $
\delta \le \delta_* $, where $ \delta_* $ is given by
(\ref{de2}), the operator $T_{ess}$ can be represented as the
linear sum $T_{ess} = T_1 \dot{+} T_2 $ of operators $T_{1,2}$
acting in invariant subspaces $\cH_{1,2}$ such that
1. $T_1$ has the spectrum $\sigma (T_1) = [- \delta ,\delta ]$ of
multiplicity $\cM = \dim \ker S (0) $,
2. $T_2$ is similar to a selfadjoint operator and $\sigma (T_2) =
\R $,
3. The angle $ ( \cH_1, \cH_2 ) > 0$. The angle $( \cH_1,
\cP_\delta \cH_2 ) $ admits the estimate \be\la{sinabc} \sin (
\cH_1, \cP _\delta \cH_2 ) \ge \frac p{\sqrt{2}} \ee where $p$ is
defined before lemma \ref{fuct1}.}
\begin{proof}
It remains to verify the assertions about the spectrum. Both
follow from the intertwining relation (\ref{inter}). Indeed, it
is obvious from (\ref{Char}) that $ \| S ( k ) f \| < \| f \| $
for $ f = ( I - Q ( k ) ) \varphi $ with $ \Re Q ( k ) \varphi
\ne 0 $. Since the operator $ \Re Q ( k ) $ has a logarithmic
singularity at zero, it cannot have finite rank. Therefore, $
\mbox{rank} \Delta ( k ) = \infty $ for all $ k \in \R $, and the
assertion $ \sigma ( T_2 ) = \R $ is immediate. That $\sigma
(T_1) \subset [- \delta ,\delta ]$ is implied by the existence of
a scalar multiple, according to remark \ref{scmulloc}; the
equality now results from the intertwining relation in the form
(\ref{inter1}).
\end{proof}
There is a partial case when the spectrum of $ T_1 $ is simple
(that is, $ \cM = 1 $), and the number $ p $ admits an explicit
lower bound. For $ \xi > 0 $ let $ p ( \xi ) = \mbox{dist} \( -1,
\sigma( \tQ ( i\xi ))\)$.
\begin{remark}
Assume that $ p( \xi ) \neq 0 $ for some $ \xi > 0 $. Then $ \cM
= 1 $ and \be\la{del} p \ge \frac { p ( \xi ) }3 \ee for any $\xi
\in ( 0, \tau ) $. \end{remark}
\begin{proof} If $ \cM \neq 1 $ then there exists a nonzero
$ \varphi \in \ker ( I + \tQ (i\von)) $ such that $ \llangle
\varphi, \sqrt{c} \rrangle = 0 $ which contradicts the assumption
$ p ( \xi ) \not\equiv 0 $.
Put $ \wt{{\cal B}_c } = \{ \wt{\eta}_n ( i 0 + ) \} $, $ \wt{
\eta }_n ( z ) $ being the set of eigenvalues of $ \tQ ( z ) $,
hence $\wt{{\cal B}_c } = {\cal B}_c $. By the functional calculus
we have for some $ \wt{\eta_*} \in {\cal B}_c $,
$$ p = \frac { | 1 + \wt{\eta_*} |} { 1 - \wt{\eta_*} } \ge \min
\left\{ \frac 13 , \frac {p (\xi)}3 \right\} , $$ where the cases
having been considered separately are $ \wt{\eta_*} \le -2 $ and $
\wt{\eta_*} \ge -2 $. It remains to note that $p (\xi) < 1 $ since
$\tQ (i\xi )$ is a compact operator of infinite rank.
\end{proof}
Under the assumption of this remark, substituting (\ref{del}) in
(\ref{sinabc}) and (\ref{de2}), one gets weaker estimates which
are, however, more explicit with respect to $ c $ since they do
not require calculating a matrix element of the inverse operator
$ \( I - \tQ_0 \)^{-1} $, which is incorporated in $ p $ through $
\vartheta_c $.
Let us mention that given $c$ it may happens that $ p(\xi ) = 0 $
for all $\xi > 0 $ which means that $ \llangle \varphi,
\sqrt{c}\rrangle = 0 $ for a nonzero $\varphi \in \ker \( I +
\tQ_0 \)$. This situation arises, for instance, if $ c \in \cE $
is of the form $ c = \kappa \wt{c} $ where $\wt{c} $ is an even
function from $ L_0^+ $, under the appropriate choice of the
constant $ \kappa >0 $, owing to the commutation of $ \tQ $ with
the reflection operator $ R $, $ ( R f ) ( x ) = R ( - x ) $. In
this case the logarithmic estimate for the norm of $ \( I + Q (z)
\)^{ -1 } $ at the vicinity of $ 0 $ wrongly asserted by Lemma 6
in \cite{LW2} obviously fails to hold.
The main difficulty in estimating $ ( \cH_1, \cH_2 ) $, that is,
in getting a "global" estimate of the angle, is that we do not
know the way to estimate $\len S^{-1} ( k ) \rin $ in the
intermediate range i. e. for those $ k $ outside the vicinity of $
0 $ given by lemma \ref{fuct1} that the estimate (\ref{estQ}) is
not yet essential.
Finally, in view of the untary equivalence between $ T $ and $
-T^* $, all the assertions of Theorem 2 hold for the restriction
of $ L - i \sigma $ to its invariant subspace $ J H_{ess} $.
Strictly speaking, we have defined the absolutely continuous
subspace for dissipative operators only while $ L - i \sigma $ is
"minus dissipative". However if we adopt the natural
generalization of the notion of the absolutely continuous subspace
to the nondissipative case suggested in \cite{N}
% according to which
%$$ \cN_e = \overline{\wt{\cN_e}}, \hspace{.7cm}
%\wt{\cN_e}\equiv\left\{ P_\cK \pmatrix{ \tg \cr g \cr}:\; \chi_- (
%\tg + S^* g ) + \chi_+ (S \tg + g ) = 0 \right\} $$
%
%\bequnan & \cN_e = \overline{\wt{\cN_e}}, & \\ &
%\wt{\cN_e}\equiv\left\{ u \in H: \; \chi_+ \sqrt{|V|} \( A -
%\lambda \)^{ -1 } u \in H_2^+ ( E ) ; \; \chi_- \sqrt{|V|} \( A -
%\lambda \)^{ -1 } u \in H_2^- ( E ) \right\} & \eequnan
%where $ \chi_\pm $ are the projections on positive and negative
%parts of spectrum of $ V = \Im A $,
then the spectrum of $ \left. ( L - i \sigma ) \right|_ { J
H_{ess} } $ automatically becomes absolutely continuous, that is,
$ J H_{ess} = \cN_e ( T^* ) \oplus H_0 $.
\newsection{Three Dimensional Transport Operator}
\newsubsection{Preliminaries}
The phase space of the one-speed transport problem in the three
dimensional case is $ \Gamma = \R^3 \times \bS^2 $. For a
particle at a point $( q ,\wh{p} ) \in \Gamma $, the vectors $
q\in \R^3 $ and $\wh{p} \in \bS^2 $ are the position and the
momentum direction, respectively. The physical parameters in the
problem are the number $ \sigma \ge 0 $ and the function $ c: \R^3
\to \R_+ $ having the same physical meaning as before. We suppose
that the distribution of the secondaries is isotropic i.e. the
collision operator $ K \in {\bf B} L^2\( \bS^2 \)$ is given by $
K = \frac 1{4 \pi}\int_{\bS^2} \cdot \ d \Omega $, $ d \Omega $
being the Lebesgue measure on the unit sphere. It is assumed
throughout that the function $ c $ is bounded ($ c \in L^\infty
(\R^3) $) and compactly supported.
Define the operator $ T $ in the Hilbert space $H = L^2 ( \Gamma
) $ by the following expression, $$ T = i\wh{p}\cdot \nabla _q +
i \sigma c ( q ) K, $$ on the domain $\cD = \{f\in H:\, (1)\ F(t)
= f\(q + t\wh{p} ,\wh{p} \) \mbox { is absolutely continu-}$
$\mbox{ous for } $ $ \mbox{a. e. } (q, \wh{p} )\in\Gamma ;\; (2)
\left. \frac {dF}{dt} \right|_{t = 0} \in H\}$ of its real part $
L_0 = i\wh{p}\cdot \nabla _q $. Then $ T $ is a dissipative
operator with a bounded imaginary part. The evolution of a
density $ u_0 \in H $ is given by solution of the Cauchy problem
of the form (\ref{Evol}) where the one-speed transport operator
$L$ is linked with the operator $ T $ by the relation $ T = L^* +
i\sigma $. The operator $L_0$ corresponds to the evolution $U_t =
\exp {i L_0 t}$, $(U_t\,f) (q,\hp )= f\( q -\wh{p} t,\wh{p} \)$,
of densities in vacuum. The spectral analysis of $ L $ is reduced
to that of $ T $ since $ L $ and $ -T + i $ are unitarily
equivalent, $ L = J ( - T + i ) J^* $ with $ ( J f ) ( x, \wh{p}
) = f ( x, - \wh{p} ) $. Without loss of generality, we set
$\sigma = 1$; then $ V = \Im T = c ( q ) K $. One can naturally
identify the range of $ K $ in $ H $ with the space $L^2 ( \R^3 )
$ of functions of the variable $ q $, $K H = L^2 ( \R^3) \otimes
{\bf 1}\simeq L^2(\R^3) $ where $ {\bf 1} \in L^2\( \bS^2 \)$ is
the indicator of the unit sphere.
Remark that the transport operator studied in Section 3 is
obtained from $ T $ when separating variables in the case of
function $ c $ depending on one space coordinate only.
\newsubsection{Spectral Analysis}
We use the representation of the resolvent $ R_0 (z) = \( L_0 - z
\)^{-1} $ for $ z \in \C_+ $ given by the Laplace transform of
$U_t$, $$ ( R_0 (z) f ) (q, \hp ) = i \int_0^\infty e^{izt} f (q +
\hp t, \hp ) dt. $$ Let us calculate the kernel of the operator $Q
( z ) = \left. i\sqrt{V} R_0 ( z ) \sqrt{V} \right|_E $, $ z \in
\C_+ $, acting in the space $ E \equiv \overline{\mbox {Ran}\, V}
\subset L^2 ( \R^3 ) $. We have \bequnan ( Q(z) f ) ( q ) = -
\frac 1{4\pi} \sqrt{ c(q) } \int_{ \bS^2} d\Omega_p \int_0^\infty
e^{izt}f (q + \hp t ) \sqrt{ c (q + \hp t ) } dt =
\\ - \frac 1{4\pi } \sqrt{c(q)} \int_{ \R^3 } e^{iz | q^\prime | }
f(q + q^\prime ) \sqrt{c(q + q^\prime) } \frac {d^3 q^\prime} {
\left| q^\prime \right|^2} = \\ - \frac 1{ 4\pi } \sqrt{c(q)}
\int_{ \R^3 } \frac{ e^{ iz \left| q^\prime -q \right|}} {\left|
q^\prime - q \right|^2 } \sqrt{ c( q^\prime) } f( q^\prime ) d^3
q^\prime . \eequnan Thus, \be \la{Q1} ( Q(z)f )(q) = - \frac
1{4\pi } \sqrt{c(q)} \int_{\R^3} \frac{e^{iz\left| q^\prime
-q\right|}} {\left| q^\prime -q\right|^2 } \sqrt{c( q^\prime)} f(
q^\prime) d^3 q^\prime .\ee The actual integration in this formula
is over a ball of finite radius. The kernel of $Q(z)$ has a weak
($2<3$) singularity for all $z \in \C $. Initially defined in $
\C_+ $, the function $Q(z)$ thus extends to an entire function.
By the lemma on iteration of kernels with weak singularity
\cite{IE} $ Q^2 ( z ) $ is of the Hilbert-Schmidt class for any $
z \in \C $, and hence $ Q ( z ) \in {\bf S}^4 $ for all $ z \in
\C $. \vspace{.5cm}
{\bf Theorem 3.} {\it The nonreal spectrum $\sigma_+ ( T ) $ of
the operator $T$ consists of finitely many eigenvalues lying on
the imaginary axis. Moreover, $T$ has no associate vectors. The
dimension $ N(c) $ of the invariant subspace $ H_d $
corresponding to $\sigma _+ ( T ) $ satisfies the estimate
\be\la{EST} N(c) \le \frac 1{256\pi^4} \int\!\!\!\int c(q)
c(q^\prime )\(\int \frac{c(\tau ) d^3 \tau}{ |q -\tau |^2
|q^\prime -\tau|^2}\)^2 d^3q d^3q^\prime. \ee The essential
spectrum of $T$ coincides with the real axis: $\sigma_{ess}( T ) =
\R $.}
\begin{proof} First, $\sigma_{ess} ( T ) = \sigma_{ess} ( L_0 ) = \R
$ by Sublemma. Remark that the finiteness of $ N ( c ) $ follows
immediately from the analyticity of $ Q $ on the real axis by the
analytical Fredholm theorem.
Consider the operator $\wt{ \Xi } ( z ) = \left. i K R_0 ( z
)\right|_{ K H}$ acting in the space $ K H = L^2\( \R^3\) $. In
the Fourier representation with respect to the variable $q$ this
operator acts as the multiplication by the function $$ \wt{\xi} (
s , z ) = \frac i{ 4 \pi } \int_{\bS^2 } \frac { d\Omega_p }{\hp
s - z} = $$ By the rotation symmetry the integral does not depend
on direction of $\hat{s} = \frac s{|s|}$. We continue the
equality taking $ \hat{s} $ for the axis of a spherical
coordinate system, $$ = \frac i{4\pi } \int_0^{2\pi} \int_0^\pi
\frac { \sin \theta d \theta d\varphi } { |s|\cos \theta - z } =
\frac i2 \int_{-1}^1 \frac{ d \mu }{|s|\mu - z }. $$ Comparing
this with (\ref{xxi}) one sees that $ \wt{\xi} ( s, z ) = \xi ( |
s | , z ) $. In particular, $ \Im \wt{\xi} ( s, z ) = r_z ( | s |
) $. Therefore $ \Im Q ( z ) = \left. \sqrt{c} \Im \wt{ \Xi } ( z
) \sqrt{c}\right|_E $ is sign-definite for $ \Re z \neq 0 $, and
$ \Im Q ( z ) = 0 $ for $ \Re z = 0 $. Now reproducing verbatim
the arguing following (\ref{rz}), one concludes that $ \sigma_+ (
T ) \subset i\R $, $T$ has no associate vectors, and that $Q (
i\von ) $ is a negative operator and a monotone increasing
function of $\von $. Let $ N_\von (c) $ be the dimension of the
spectral subspace corresponding to the interval $ [i\von,i\infty
) $. We then have
\begin{eqnarray}\la{Nc1} N_\von(c) \equiv \# \{ s\ge \von: \;
\ker (I + Q ( i s) )\ne 0 \} = \# \{ \tau \in (0, 1]:\; \nonumber
\\ \ker (I + \tau Q (i\von ) )\ne 0 \}\le \len Q (i\von )\rin_4^4
= \len Q^2 (i\von) \rin_2^2.
\end{eqnarray} This gives $ N(c) = \lim_{ \von \downarrow 0}
N_\von (c) \le \len Q^2 ( 0) \rin_2^2$, which is (\ref{EST}).
\end{proof}
Contrary to the slab geometry case, one sees that $\sigma_+(T)=
\emptyset $ for small coupling. Comparing (\ref{EST}) and
(\ref{num}), one should mention the following drawback of
(\ref{num}). Consider $ c ( x ) = c_1 ( x - b ) + c_1 ( x + b ) $
where $ c_1 $ is a compactly supported function and $ b $ is a
large parameter. Then, obviously, the right hand side of
(\ref{num}) diverges as $ b \to \infty $. This is obscure from
the viewpoint of physics, and, indeed, one can show that $ N(c) $
is actually uniformly bounded in $b$. At the same time, the right
hand side of the estimate (\ref{EST}) is clearly uniformly bounded
in $ b $ as $ b \to \infty $ in the analogous problem for the
three-dimensional operator.
A different statement of the transport problem was used in the
paper \cite{Uk}. Under the assumption of convexity of the set $\{
x: \; c(x) \neq 0 \} $, it led again to the study of the equation
$ ( I + Q (z) ) \varphi = 0 $ for determination of the discrete
modes. Note that this equation was studied in \cite{Uk} for all $
z \in \C $. In fact, the properties of monotonicity and continuity
of $Q(i \von )$ in $ \von $ which are sufficient for the
conclusion that $ N ( c ) $ is finite, have already been
mentioned in \cite{Uk}. The estimate (\ref{EST}) for $ N ( c ) $
is apparently new.
Let $ S ( z ) $ be the characteristic function of the operator
$T$. This function is analytic on the real axis and satisfies $
S( z ) - I \in {\bf S}^\infty $. Define $\cE = \{ c \ : \ \ker (
I + Q (0) )\neq 0 \} $. One immediately concludes from the
continuity of $ S $ on the real axis that $ 0 $ is a spectral
singularity point for $c \in \cE $. The following lemma provides
us with an estimate of the function $ S^{-1} $ in a neighbourhood
of zero.
\begin{lemma} \la{l7}
1. If $c\notin \cE $, then $ \sup_{ z \in \Pi_\epsilon } \len
S^{-1} ( z ) \rin < \infty $ for $\epsilon $ sufficiently small.
2. If $c\in \cE$, then $ C_1 |z|^{-1} \le \len S^{-1} ( z ) \rin
\leq C |z|^{-1} $ as $ z \in \omede ( 0 ) $ for sufficiently small
$\delta $. \end{lemma}
\begin{proof} As well as in the proof of lemma \ref{sepE}, it
suffices to obtain the desired upper estimates for $ \( I + Q ( z
) \)^{-1} $.
1. Arguing as in the proof of Proposition \ref{simse}, one
concludes that $ \ker S ( k ) = \{ 0 \} $ for $ k \ne 0 $. Hence
we are done if we show that $ \len \Re Q (z) \rin \to 0 $ as $ |
\Re z | \to \infty $ uniformly in $ \Im z $ for $ 0 \le \Im z \le
\epsilon $. We have $ \Re Q ( z ) = \left. \sqrt{c} \Re \wt{ \Xi
} ( z ) \sqrt{c} \right|_E $ where $ \Re \wt{ \Xi }( z ) $ in the
Fourier representation acts as the multiplication by the function
$ \Re \xi ( |s| , z ) = - \Im \left[ \frac 1{ |s| } \int_{ -|s|
}^{ |s| } \frac { d \tau } { \tau - z } \right] $. Then for $ x
\in [ 0, C ] $, $ C < \Re z $, we have $ | \Re \xi ( x , z ) | \le
\frac 2{ | C - \Re z | } $. Since $ \left| \Im \( \ln \frac { z -
x } { z + x } \) \right| \le 4 \pi $ one estimates $ | \Re \xi ( x
, z ) | \le \frac { 4 \pi}x \le \frac { 4 \pi }C $ for $ x \in [
C, + \infty ) $. Taking $ C \to \infty $, $ C = o ( \Re z ) $ for
$ | \Re z | \to \infty $, one gets $ \len \Re \wt{ \Xi } (z) \rin
\to 0 $.
2. For $z \in U_\delta (0)$ we have $Q ( z ) = Q(0) - i z A + O \(
\left| z \right|^2 \) $ in the operator norm. Then $ Q^\prime ( i
\von ) = - i \left. \sqrt{c} \( \frac d{d\von} \wt\Xi (i\von) \)
\sqrt{c} \right|_E $, $\von > 0 $. In the Fourier representation,
the operator in parentheses acts as the multiplication by the
function $ \( |p|^2 + \von^2 \)^{ -1 } $. Thus, $ i Q^\prime ( i
\von ) $ is a positive operator and a monotone decreasing function
of $\von $. Hence $ A > 0 $. The upper bound now follows from
lemma \ref{deriv}. The lower bound is obvious since $\| ( I + Q(z)
) \varphi \| = O(|z|) $ for $\varphi \in \ker (I + Q(0)) $.
\end{proof}
Note that had we tried to obtain the upper bound in part 2 by
applying abstract remark \ref{Rfun}, say, to the function $ \( I
+ Q ( \sqrt{w} ) \)^{ -1 } $, we would have got an estimate of a
worse order due to the singularity of the conformal
transformation.
Define $\cP_d $ to be the Riesz projection corresponding to
$\sigma _+ ( T ) $, then $ H_{ess} = ( I - \cP_d ) H $ is the
invariant subspace of $T$ corresponding to the essential
spectrum. Part 2 of the above lemma implies the following
conclusion about the group $ Z_t^e = \exp ( i T_{ess} t ) $, $
T_{ess} = \left. T \right|_{ H_{ess} } $.
\begin{corollary} If $ c \in \cE $ then $$ \len Z_{-t}^e \rin
\le C_1 + C_2 t, \ \ \ t \ge 0, $$ and this estimate is exact in
the power scale, in the sense that for any $ \von \in (0, 1 ) $
there exists $ f \in H_{ess} $ such that $ \len Z_{-t}^e f \rin =
C_\von t^{ 1 - \von } ( 1 + o ( 1)) $, $ C \ne 0 $, for $ t \to +
\infty $.
\end{corollary}
This corollary is a partial case of an abstract result related to
the operators with spectral singularities of finite power order.
Its proof, which also uses the functional model, will be published
elsewhere.
\begin{lemma} \la{scmul1} $S$ has a scalar multiple. \end{lemma}
\begin{proof} Examining the proof of lemma \ref{l7}, part 1, one
concludes that the norm of $ S^{ -1 } ( z ) $ is bounded at any
strip $ \Pi_\epsilon $, $ \epsilon > 0 $, if $ z \notin U^\circ
(\delta ) $ where $ U^\circ (\delta) $ is the union of an
arbitrary open neighborhood of $ \sigma_+ (T) $ and $ \omede (0)
$, regardless to the condition $c\notin \cE $. Then $ S (z ) \to I
$ as $ \Im z \to + \infty $ uniformly in $ \Re z $ since $ \| Q (
z ) \| \le \frac { \| V \| }{ \Im z } $. Taking this into account,
one obtains for all $ \delta > 0 $ \be\la{estme1} \sup_{ z \in
\C_+ \setminus U^\circ (\delta ) } \len S^{-1} (z) \rin < \infty .
\ee It now follows from lemma \ref{l7}, part 2, that any function
of the form $ m ( z ) = b ( z ) z \wt{m} ( z) $ where $ \wt{m} \in
H^\infty $ and satisfies $ |\wt{m} (z) | \le C \( 1 + |z| \)^{ -1
} $, and $ b $ is the Blaschke product with simple zeroes at the
points of $ \sigma_+ (T) $, is a scalar multiple for $ S $, since
the poles of $ \( L - z \)^{ -1 } $ at the points of $ \sigma_+
(T) $ are simple. \end{proof}
Taking $ \wt{m} $ to be outer one gets the canonical factorization
$ m = b m_e $, $ m_e (z) = z \wt{m} (z) $, of the scalar multiple.
We thus arrive at the following
\begin{proposition} \la{out1}
The component $T_{ess}$ of the operator $T$ has purely absolutely
continuous spectrum, $ H_{ess} = \cN_e \oplus H_0 $.
\end{proposition}
The proof of this assertion for the slab case required more
analysis due to the fact that we were not able to estimate $ S^{
-1 } $ at $ 0 $. In fact, to know that $ S $ is analytic at $ 0 $
would be enough for the above proof of Proposition \ref{out1}, and
the exact estimate of lemma \ref{l7} is redundant for that
purpose.
%\begin{corollary}\la{spsin} For $c\in \cE $ the operator $T$ has the
%unique spectral singularity point at $0$. \end{corollary}
\begin{proposition} \la{simse1}
$T_{ess} $ is similar to a selfadjoint operator if and only if
$c\notin \cE $. If $ c \notin \cE $ then the wave operators $$
W_\pm ( L, L_0 + i\sigma ) \equiv W_\pm ( T^*, L_0 ) = {\mbox {s -
lim}}_{ t \to \pm \infty } e^{ i T^* t } e^{ - i L_0 t } $$ exist
on $ H $ and are complete, $ \mbox{ Ran } W_\pm = J H_{ess} \equiv
H_d^\perp $.
\end{proposition}
\begin{proof} The first assertion follows immediately from item 1 of
lemma \ref{l7} by the Nagy-Foias criterion. The statement about
wave operators follows from theorem 6 in \cite{Nab}, as it does in
the slab case. \end{proof}
The picture of the behaviour of the operator $ T $ in coupling
constant remains the same as in the slab case: remark
\ref{instab} can be taken verbatim except, of course, for the
fact that in the situation under consideration $ T $ has no
eigenvalues for small $ \kappa $.
The following lemma plays a role quite analogous to that of lemma
\ref{fuct1}. Let $\left| c \right|_R^2 = \int \frac{ c(q)
c(q^\prime )} {\left| q - q^\prime \right|^2} d^3 q d^3 q^\prime
$ be the Rolnick norm of $c$ \cite{RS}.
\begin{lemma} \la{fact2}
Set $ \tp = \mbox{ dist }\( -1,\,\sigma(Q(0))\setminus \{-1\}\)$,
$ \cM = \dim \ker (I + Q ( 0 ) ) $ and \be\la{del2} \delta_* =
\frac \pi{18} \frac {\tp^2}{\left| c\right|_R }.\ee Then $ \| D (
k ) - D ( 0 ) \| < p^2/2 $ for all $k \in (-\delta_*, \delta_* )
$.
\end{lemma}
\begin{proof}
Let $\Theta (z) = Q(z) - Q(0)$. We then have $$\| S(z) - S ( 0 )
\| = 2\len \( I- Q (z) \)^{-1} \Theta (z) \( I- Q (0) \)^{-1} \rin
\le 2 \| \Theta (z) \| $$ and thus \be \la{razn1} \| (S^* S)(z) -
S^2 ( 0 )\| \le 4 \| \Theta (z) \| . \ee The kernel of $\Theta
(z)$ has the form $ - \frac 1{4\pi } \sqrt{c(q)} \frac{e^{ i z |
q^\prime - q | } - 1 } {\left| q^\prime -q\right|^2 } $ $
\sqrt{c( q^\prime)} $. Using the inequality $\left| e^{ix} -1
\right|\le |x| $, one estimates for $ k \in \R $, \bequnan \len
\Theta (k) \rin^2 \le \len \Theta (k) \rin_2^2 = \frac 1{16\pi^2
} \int \!\!\! \int c(q) \frac{\left| e^{ ik | q^\prime - q | } -
1 \right|^2} {\left| q^\prime - q\right|^4 } c( q^\prime) d^3 q
d^3 q^\prime \le \\ \le \frac {|k|^2}{16\pi^2 } \int\!\!\!\int
\frac{c(q) c(q^\prime )} {\left| q^\prime -q\right|^2 } d^3 q d^3
q^\prime \equiv \frac {|k|^2}{16\pi^2 } \left| c \right|_R^2 .
\eequnan Substituting this in (\ref{razn1}), we obtain for $k \in
(-\delta_*, \delta_* ) $ $$ \| D ( k ) - D ( 0 )\| \le \frac {
\left| c \right|_R }\pi |k| < \frac {\tp^2}{18}. $$ It remains to
note that by the functional calculus
$$ p^2 \equiv \mbox{dist}\( 0, \,\sigma \(S^2 ( 0 ) \) \setminus
\{0\}\) \ge \( \min \left\{ \frac 13 , \frac {\tp}3 \right\} \)^2
\ge \frac {\tp^2}9 $$ since $ Q (0) \le 0 $, and thus $\tp\le 1$.
\end{proof}
Reproducing the arguing followed by lemma \ref{fuct1} and theorem
2, we now get the following\vspace{.5cm}
{\bf Theorem 4.} {\it If $c\in \cE $ then for any positive $
\delta \le \delta_* $ (see (\ref{del2})), the operator $T_{ess}$
can be represented as the linear sum $T_{ess} = T_1 \dot{+} T_2 $
of operators $T_{1,2}$ acting in invariant subspaces $\cH_{1,2}$
such that
1. $T_1$ has the spectrum $\sigma (T_1) = [- \delta ,\delta ]$ of
multiplicity $\dim \ker ( I + Q(0) )$,
2. $T_2$ is similar to a selfadjoint operator and $\sigma (T_2) =
\R $,
3. The angle $( \cH_1, \cH_2 ) > 0$. The angle $ ( \cH_1,
\cP_\delta \cH_2 )$ admits the estimate $$ \sin (\cH_1, \cP_\delta
\cH_2 ) \ge \frac \tp{3\sqrt{2}} $$ where $\tp$ is defined in
lemma \ref{fact2} and $\cP_\delta \equiv I - \cP_\omega $, $
\omega = \R \setminus [ - \delta , \delta ]$, is the spectral
projection of $T$ for the interval $[-\delta , \delta ]$ defined
after corollary \ref{spp}.}
The assertions of this theorem are extended to the operator $ L -
i \sigma $ in the same way as it takes place in the slab case,
see the remark made at the end of section 3.
\newsection{Concluding remarks}
Let us discuss an exactly solvable limit case, when $c(x)\equiv
\kappa $ with a $ \kappa >0 $ for all $x \in \R $, of the
transport operator for slab geometry. After Fourier
transformation with respect to the variable $x$ the operator $T$
is fibred. Its fibres $T_p = p\mu + i\frac \kappa 2 \langle\cdot
,{\bf 1}\rangle {\bf 1}$, $p \in \R $, act in $L^2 (\Omega)$ and
are rank $ 1 $ perturbations of the selfadjoint multiplication
operators $L_{o,p} = p\mu \cdot \, $. Thus, the essential spectrum
of $ T_p $ is $ [-p,p] $. The spectrum of $ T_p $ in $ \C_+ $ is
determined from the equation $ 1 + \xi ( p , z ) = 0 $ to give
that $ T_p $ has one eigenvalue $\lambda _p = i p\cot \frac
p\kappa $ for $|p| < \frac \pi 2\kappa $ and has none for $|p|\ge
\frac \pi 2\kappa $. From this it follows that the spectrum
$\sigma (T) = \bigcup_p \sigma ( T_p ) = \R \bigcup [ i \kappa, 0
) $ (that $\sigma (T)$ cannot get larger is obvious since $ \xi (
p , z ) \to 0 $ for fixed $ z \in \C_+ $ as $ |p| \to \infty $
(see (\ref{xxi}))). In this way, let us remark that each $ \lambda
\in [ i \kappa, 0 ) $ can be shown to be an accumulation point of
eigenvalues of the operators $ T $ corresponding to $ c = \kappa
\chi_{ [-a,a] } $ when $ a \to \infty $.
It should be noted that in this limit case $T$ is not a spectral
operator (see \cite{Dun}) {\it for all $ \kappa > 0 $}. Indeed the
spectral projection $\cP_p $ with respect to the eigenvalue $
\lambda_p $ of the operator $ T_p $ looks like $$\cP_p = -\frac
{p^2}{2\sin^2 \frac p\kappa } \llangle \cdot , \overline{ \Psi_p }
\rrangle_{ L^2 ( \Omega ) } \Psi_p $$ where $ \Psi_p ( \mu ) =
\frac 1{p\mu - i\rho (p) } $, $ \rho (p) = p \,\cot \frac p\kappa
$. Obviously, the operator $\cP_\alpha = \int_{-\frac\pi 2
\alpha}^{\frac\pi 2 \alpha} \oplus \cP_p $ is bounded whenever
$\alpha < \kappa $ and gives the spectral projection of $T$ for
the interval $\left[ i \rho \( \frac\pi 2 \alpha \), i\kappa
\right] $. However, $\| \cP_\alpha \| \to \infty $ as $\alpha \to
\kappa $ which contradicts $(B)$ property of spectral operators.
A special interest from the part of physics is paid to the
solutions of the problem (\ref{Evol}) exponentially increasing in
time. Obviously, such solutions may only exist when $\left| c
\right|_\infty >1 $ (the supercritical transport problem),
otherwise the operator $L$ is dissipative. Since $ L $ and $ - T
+ i $ are unitarily equivalent, one observes that $ \sup_{ \von >
0 } N_{1+\von} ( c ) $ is precisely the number of exponentially
increasing modes for the supercritical transport problem. Thus,
for $\left| c \right|_\infty > 1 $ one can use (\ref{num}) and the
estimate obtained by taking $ \von = 1 $ in (\ref{Nc1}) as
estimates on this number. In this respect, the estimate
(\ref{num}) appears to be rather crude due to the fact that in the
slab case the estimate of $ N_\von ( c ) $, that is, before going
to the limit $ \von \to 0 $, is worse than that of $ N ( c ) $.
Notice that the perturbations treated in this paper are not of
relative trace class: the condition $ \( H - z \)^{-1} - \( H_0 -
z \)^{-1} \in {\bf S}^1 $ obviously fails for the pair $ ( L, L_0
+ i \sigma ) $ because $ \Re Q ( z ) \notin {\bf S}^1 $, and
therefore $ R_0 (z) V R_0 (z) \notin {\bf S}^1 $ (see
(\ref{ReQ})).
It should also be noted that, once we have established the
similarity of $ T_{ ess } $ to a selfadjoint operator for $ c
\notin \cE $, the existence and completeness of the wave
operators follow directly from the theory of smooth perturbations
\cite{RS,Kato}. For the three - dimensional problem this is
straightforward, since obviously $ \| Q ( z ) \| \le \| Q (0) \| $
for $ \Im z \ge 0 $, and thus the operator $ \sqrt V $ is $ L_0 $
- smooth \cite[Theorem XIII.25]{RS}. That the operator $ \sqrt V $
is $ T_{ess} $ - smooth whenever $ c \notin \cE $ is an abstract
fact: the operator $ \sqrt { \Im D } $ is $ D $ - smooth for any
dissipative operator $D$ similar to a selfadjoint one. For the
slab problem the operator $ \sqrt V $ is not $ L_0 $ - smooth
since $ Q ( i \von ) = \Re Q ( i \von ) $ is unbounded as $ \von
\to 0 $ (see \cite[Theorem XIII.25, it. 4]{RS}); however, it is $
L_0 $ - smooth locally with respect to any closed interval not
containing zero, and this suffices.
Finally, it should be mentioned that the main difficulty in
physical interpretation of the results obtained is that the
natural statement of the transport problem assumes $ L^1 ( \Gamma
) $ for the space of distribution functions rather than $ L^2 $.
It can be shown, however, that for the slab case the sets of $ L^2
$ and $ L^1 $ - eigenfunctions of the operator $ L $ coincide,
hence the estimate (\ref{num}) remains valid for the operator $ L
$ in $ L^1 $. It is also believed that if we will manage to prove
the existence and completeness of the wave operators in $ L^1 $
for $ c \notin \cE $ then the stationary approach for calculating
the scattering operator $ {\cal S} = W_+^{-1} W_- $ could be
applied. In particular, it is easy to see that if the wave
operators exist for the problem in $ L^1 $ with $ c \notin \cE $
then they are consistent with the ones defined in Proposition
\ref{simse1}, that is, coincide on $ L^2 \cap L^1 $.
\begin{acknowledgements}
We are indebted to V.I.Vasyunin for helpful remarks and an
important correction.
\end{acknowledgements}
\bigskip
\bigskip
\centerline{\large\bf {Appendix}}
\renewcommand{\thetheorem}{A.1}
\begin{lemma} \la{deriv} Let $ A (z) $ be
an analytic operator function on a domain $ \cD \subset \C $ whose
values are compact operators in a Hilbert space $ K $ and $ \ker (
I + A ( z_0 ) ) \ne \{ 0 \} $ for some $ z_0 \in \cD $. Assume $ c
A^\prime ( z_0 ) > 0 $ with some $ c \in \C $ and $ A ( z_0 ) =
A^* ( z_0 ) $. Then
$$ \len \( I + A ( z ) \)^{ -1 } \rin \le C\left| z - z_0
\right|^{ -1 } $$ for $ z \in U_\delta ( z_0 ) $, $ \delta $
small enough.
\end{lemma}
\addvspace{0.3cm}
\Proof Put $ \cN = \ker ( I + A ( z_0 ) ) $. Representing
operators by $ 2 \times 2 $ matrices with respect to the
decomposition $ K = \cN \oplus \cN^\perp $ one has \bequnan I + A
( z ) = \pmatrix{ 0 & 0\cr 0 & A_0\cr} + ( z - z_0 ) \pmatrix{
B_{00} & B_{01}\cr B_{10} & B_{11}\cr} + O \( \( z - z_0 \)^2 \) =
\\ \pmatrix{ (z - z_0) B_{00} & 0\cr 0 & A_0\cr} \left[ \pmatrix{
I & B_{00}^{-1} B_{01} \cr 0 & I \cr} + O ( z - z_0 ) \right].
\eequnan Here we took into account that the assumption about $
A^\prime ( z_0 ) $ guarantees the invertibility of $ B_{00} $
since $ \dim \cN < \infty $ by compactness of $ A ( z ) $. Then
\bequnan \( I + A ( z ) \)^{-1} = \left[ \pmatrix{ I &
-B_{00}^{-1} B_{01} \cr 0 & I \cr} + O ( z - z_0 ) \right]
\pmatrix{ \( z - z_0 \)^{-1} B_{00}^{-1} & 0\cr 0 & A_0^{-1}\cr}.
\eequnan \hfill \rule{2.5mm}{2.5mm} \vspace{0.25cm}
\bigskip
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\end{document}
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