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Spectral singularities, contractive semigroups, functional model
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\begin{document}
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\title{Spectral Singularities and Asymptotics of Contractive Semigroups. I.}
\author{S. Naboko$^1$ and R. Romanov$^{2,3}$\thanks{The
work was supported in part by EPSRC Grants No. GR/ M00549 and GR/
R20885.}
\\
\\ $^1$ Department of Mathematical Physics, Faculty of Physics\\
St.Petersburg University, 198904, St.Petersburg, Russia
\\ \\ $^2$ Laboratory of Complex Systems Theory\\ Institute for
Physics\\ Saint Petersburg State University\\ 198904, Saint
Petersburg, Russia \\ \\ $^3$ Department of Computer Science\\
Cardiff University, \\
Queen's Buildings, PO Box 916\\
Newport Road, Cardiff CF24 3XF\\
UK\\
e-mail: R.V.Romanov@cs.cardiff.ac.uk}
\date{}
%\pagestyle{empty}
\maketitle
\footnotetext{AMS Subject Classification: 47B44, 47D06}
\begin{abstract}
The problem of localization of spectral singularities of
dissipative operators in terms of the asymptotic of the
corresponding exponential function is studied. We give a solution
to this problem for the singularities of higher orders in the
frame of the perturbation theory.
\end{abstract}
{\bf Introduction.} In this paper we continue the study of
relationships between spectral singularities of dissipative
operators and the asymptotic of the corresponding semigroup
started in \cite{NR}. For a dissipative operator, $ L $, with
purely absolutely continuous spectrum spectral singularities
\cite{Pav} are the points on the real axis at vicinities of which
the estimate \be\la{NF} \len \( L - z \)^{ -1 } \rin \le \frac
C{\im z}, \hspace*{7mm} \im z > 0, \ee fails to hold. The role of
spectral singularities in analysis of operators is revealed by the
Sz.-Nagy - Foias criterion \cite{Na} according to which
{\it A maximal dissipative operator $ L $ such that $ \sigma ( L )
\subset \R $ is similar to a self-adjoint one\footnote{that is,
there exists a bounded operator $ U $ with bounded inverse such
that $ L = U A U^{ -1 } $ for some self-adjoint $ A $.} if and
only if its resolvent satisfies the estimate (\ref{NF}).}
The criterion implies that if a maximal dissipative operator $ L $
with purely absolutely continuous spectrum has no spectral
singularities at finite distance and satisfies (\ref{NF}) in a
neighborhood of infinity then the operator $ Z_t = e^{ iLt } $ is
invertible for $ t \ge 0 $ and $ \len Z_t^{ -1 } \rin \le C_0 $.
In fact, according to a theorem of Sz.-Nagy, $ L $ is similar to a
self-adjoint operator iff $ L $ generates a group $ Z_t = e^{ iLt
} $ of bounded operators, and $ \sup_{ t \in \R } \len Z_t \rin <
\infty $. Our general task is to analyze in detail the impact of
spectral singularities on asymptotic behaviour of $ Z_t $ as $ t
\to - \infty $. As the first step, in \cite{NR} we suggested the
problem of localization of spectral singularities in terms of the
asymptotic in the "simplest" case of finitely many singularities
each of finite power order, that is, of calculating their orders
and locations from the asymptotic.
In the self-adjoint theory, if a self-adjoint operator, $ A $, has
an absolutely continuous spectrum and finitely many eigenvalues, $
\{ \lambda_j \} $, then the similar problem of calculating the $
\lambda_j $'s from the asymptotic of $ e^{ iAt } $ is solved
trivially because of the spectral theorem. Namely, we fix
arbitrary $ u $ and $ v $ such that $ ( u , e_j ) \ne 0 $, $ ( v ,
e_j ) \ne 0 $ for all $ e_j $, the eigenvectors of $ A $, and
consider $ ( e^{ iAt } u, v ) $. Then $$ ( e^{ iAt } u, v ) = \int
e^{ i\lambda t } d\rho (t) + \sum_{ j = 0 }^N c_j e^{ i\lambda_j t
} , $$ where $ \rho $ is a measure absolutely continuous with
respect to the Lebesgue measure and $ c_j = ( u , e_j ) ( e_j , v
) \ne 0 $ for all $ j $. The first term is $ o(1) $ as $ t \to \pm
\infty $ by the Riemann-Lebesgue lemma, and one can determine the
$ \lambda_j$'s from the finite sum in the right hand side,
provided that we know $ f ( t ) = ( e^{ iAt } u, v ) $ at large $
t $.
The nonself-adjoint problem above is considerably more complicated
since for an operator with spectral singularities we do not have a
spectral decomposition {\it converging in the topology of the
original Hilbert space} \cite{Pavli}. Also, notice that even the
self-adjoint problem gets complicated if we allow the singular
part of the spectrum to have rich structure, in particular,
because of the existence of singular measures satisfying the
Riemann-Lebesgue property \cite{Meyer}. These make it natural to
consider first the case of finitely many spectral singularities.
The analysis of an operator with isolated power singularities
still constitutes a highly nontrivial problem, which has important
applications. For instance, the Schr\"odinger operator on the real
axis with complex potential decreasing like $ O \( \exp ( - C
\sqrt x )\) $ studied in \cite{PavSch1}, is of the class under
consideration.
Results in the present paper are aimed towards the solution of the
above problem of localization. In \cite{NR} we have shown that the
power estimate of the resolvent of arbitrary real positive order
when approaching the real axis implies the corresponding power
estimate for $ \len Z_t^{ -1 } \rin $ for $ t > 0 $. We now show
that in the situation under consideration this upper estimate is
sharp in the power scale. The precise definition of what we call a
spectral singularity of the real order $ p $ is given in terms of
the boundary behaviour of the characteristic function \cite{Na} of
the operator. In terms of the localization, one can say that this
result answers the question of calculating the maximal order of
spectral singularities. A similar result is obtained for
logarithmic singularities. As an application we consider the three
- dimensional Boltzmann operator studied in \cite{KNR}.
The proof of this result basically amounts to an explicit
construction of the Cauchy data realizing the estimate with
arbitrarily close norm growth exponent in terms of the functional
model of Sz.-Nagy and Foias. We then derive an asymptotic for the
solution of the Cauchy problem for this data and show that the
improper eigenfunction of the spectral singularity appears
naturally as a leading term coefficient in the asymptotic
expansion outside arbitrary small vicinity of the singularity in
the spectral representation of the real part of the operator. This
matches the expectation coming from consideration of the discrete
spectrum case. This observation also suggests that the improper
eigenfunctions corresponding to spectral singularities, first
introduced by Ljance \cite{Lja} when attempting to construct a
spectral decomposition for the Schr\"{o}dinger operator with
exponentially decreasing complex potential, make a meaningful
object, despite the abstract fact \cite{Pavli} that a system of
generalized eigenfunctions of the absolutely continuous spectrum
is complete in the absolutely continuous subspace. Then we study
the asymptotic behaviour of the local norms for the evolution
operator and obtain an upper estimate for the norm of $ P_\omega
Z_{ -t } $ where $ P_\omega $ is the spectral projection of the
real part of the operator $ L $ corresponding to the interval $
\omega \subset \mathbb{R} $. These results allow us to conclude
that the spectral singularities of higher orders can indeed be
localized ``algorithmically'' in terms of the asymptotic of $ Z_t
$ as $ t \to - \infty $. More precisely, one can localize all the
singularities of the orders greater than $ p - 1/2 $ where $ p $
is the maximal order.
The main new results are formulated in theorems 3 (exactness in
the power case), 6 and 6a (upper estimate for local norms), 8
(asymptotic of the nonsmooth vector) and corollary 9 (algorithm
for localization). The structure of the paper is as follows. In
\S1 we give a brief description of the functional model and
present some necessary preliminaries. We use the symmetric form of
the model suggested by Pavlov \cite{Pav}. \S2 contains the
abstract results, and the application to the Boltzmann operator is
given in \S3.
Throughout the paper the notion almost everywhere (a.e.) refers to
the Lebesgue measure on the real axis. For a set $ \omega \subset
\R $ we denote by $ \chi_\omega $ the indicator function of this
set. By the essential spectrum of a closed operator we mean the
subset of its spectrum obtained by removing the isolated points
such that the corresponding invariant subspaces are finite
dimensional.
\bigskip
{\bf \S1}. 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 $. Denote by $ E \subset H $ the
subspace $E = \overline{\Ran V}$. By the {\it characteristic
function} of $L$ we call the contractive analytic
function $S(z)\colon E\to E$, $ z \in \C_+ $, defined by the
formula $$ S(z) = I + 2 i \sqrt V \( L^* - z \)^{-1} \sqrt{V}, \;
\; \; z\in \C_+. $$ This function has boundary values on the real
axis, $S(k) \equiv S ( k + i 0 ) $, in the strong sense for a.e. $
k\in \R $. Let $ L_{pure} $ be the completely nonself-adjoint part
of $ L $. Then the set $ \sigma ( L_{pure} ) \bigcap \R $
coincides with the complement (in $ \R $) of the union of open
intervals on which the function $ S $ is analytic and unitary. The
set $ \C_+ \bigcap \rho (L) $ coincides with the set of such $ z
\in \C_+ $ that the operator $ S (z) $ has bounded inverse defined
on the whole of $ E $. Denote by ${\cal X} = L^2 \pmatrix{ I &
S^*\cr S & I\cr}$ the Hilbert space obtained by the closure of the
linear set $ L^2 (\R, E \oplus E ) $ in the metric given by the
weight $\pmatrix{ I&S^* \cr S&I \cr} $, after factoring modulo the
set of elements of zero norm. 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{cPK} 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{Pav,Pavli}, $ 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 nonself-adjoint part of $L$ is unitarily equivalent to
the generator of the contraction semigroup $ B_t = P_\cK \cU_t
\mid_\cK $, $ t \ge 0 $. This generator is called the functional
model of the operator $ L_{pure} $.
By the general theory (see \cite{Na}), there exists a contractive
outer function in the upper half plane, $ S_e $, such that for
a.e. $ k \in \R $ we have $ S (k ) = \Xi ( k ) S_e ( k ) $ where $
\Xi ( k ) $ is an isometric operator. The function $ S_e $ is
called the {\it outer factor of $ S $}. Recall that a bounded
analytic function $ \Theta : \C_+ \to {\bf B} ( E ) $ is called
{\it outer} if $ \overline { \Theta H^2_+ ( E ) } = H^2_+ ( E ) $.
We now define the absolutely continuous subspace of the completely
nonself-adjoint part of $ L $. Let $ H^\prime $ be the minimal
reducing subspace of $L$ containing $ E $. Then {\it the
absolutely continuous subspace} $\cN_e \subset H $ of the operator
$L_{pure} $ \cite{Pav,N} is defined as the closure of the set
$\wt{\cN_e} $ of smooth vectors $$ \cN_e = \overline{\wt{\cN_e}},
\hspace{.7cm} \wt{\cN_e}\equiv \left\{ u \in H^\prime: \sqrt V \(
L - z \)^{ -1 } u \in H^2_+ ( E ) \right\} . $$ The subspace
$\cN_e $ is a regular invariant subspace of the operator $ L $,
that is, for all $\lambda \in \rho (L)$ we have $ \overline{( L
-\lambda )^{-1} \cN_e } = \cN_e $. It is easy to show that $ e^{
iLt } \cN_e \subset \cN_e $ for all $ t \in \R $, and hence the
restriction $ L|_{\cN_e } $ with the domain $ \cD (L) \bigcap
\cN_e $ is defined as a closed operator. The characteristic
function of $ \left. L \right|_{ \cN_e } $ coincides (see
\cite{Pav}) with the pure part \cite{Na} of the outer factor $ S_e
$. We say that the completely nonself-adjoint part of $ L $ has
{\it absolutely continuous spectrum} if $ H_0 = H \ominus \cN_e $
is a reducing subspace of $ L $ such that $ L|_{H_0} $ is a
self-adjoint operator.
The set of smooth vectors admits the following description in
terms of the functional model \cite{Pav}. In what follows, to keep
notation at minimum, we omit in formulas the operator that
accomplishes the unitary equivalence between $ L_{pure} $ and its
functional model. Define an a.e. nonnegative function $ \Delta $
by $ \Delta (k) = I - S^*(k) S(k) $ and the space $L^2 (\R; \Delta
) $ to be the closure of $ L^2 ( \R; E ) / \{ f \colon \ \Delta f
= 0 \} $ in the metric given by the weight $ \Delta $. Then, 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) } $. Let us define a bounded operator $ W:
L^2(\R; \Delta)\to \cK $ by setting $ W: \tg \mapsto
P_\cK\pmatrix{\tg \cr -S\tg \cr}$. Then $ \wt{\cN_e} = \Ran W $.
We shall not distinguish in our notation an element $ f \in L^2 (
\R , E ) $ and the corresponding equivalence class in $ L^2 (\R ;
\Delta ) $. Note that $ \sqrt \Delta u $ for $ u \in L^2 (\R;
\Delta) $ is a well - defined element of $ L^2 ( \R, E ) $. Then,
the norm of a smooth vector $ W \tg $ is given
by\footnote{Throughout the paper $ \| \cdot \|$ for $E$-valued
functions on the real axis means the norm in $ L^2 (\R, E ) $.}
\cite{Pavli} \be\la{PK} \len W \tg \rin^2 = \len S \sqrt \Delta
\tg \rin^2 + \len P_- \Delta \tg \rin^2 , \ee and the following
intertwining relations hold \cite{N,Nab} \begin{eqnarray}
&\la{inter} \( L - z \)^{-1} W\ =\ W \( A_0 - z \)^{-1}, \hspace{
.4cm} z \in \rho ( L ) & \\& \la{interex} e^{ itL } W\ =\ W e^{
itA_0} , \hspace{ .4cm} t \in \mathbb{R}, & \end{eqnarray} where
$A_0$ is the operator of multiplication by the independent
variable in $ L^2 (\R; \Delta)$.
Remark that it is obvious from (\ref{PK}) that $ W $ is invertible
if $ S $, or, equivalently, $ S_e $, is boundedly invertible on
the real axis. In this case, the operator $ \left. L \right|_{
\cN_e } $ is similar to a self-adjoint operator ($ A_0 $) which is
manifestly absolutely continuous in the sense of the self-adjoint
theory. Further motivations of the definition of the absolutely
continuous subspace can be found in \cite{Pav,Pavli,N,Nab}.
Let $ \omede ( k ) = \{ z \in \C_+: \; |z - k| < \delta \} $. A
point $ k \in \R $ is called {\it a proper point} of the operator
$ L $ if $\sup_{ z \in \omede ( k ) } \len S_e^{-1} ( z ) \rin $ $
< \infty $ for a $ \delta > 0 $. Following \cite{Pav} we call a
point $ k \in \R $ {\it a spectral singularity} if it is not
proper. In the case when the operator $ L $ has absolutely
continuous spectrum, and thus $ S = S_e $, this definition is
equivalent to the one described in Introduction, which is seen
from the fact \cite{Na} that for any dissipative operator $ L $
\be \la{RezS} \| S^{ -1 } ( z ) \| \asymp \im z \len \( L - z \)^{
-1 } \rin. \ee We refer to the papers \cite{NR,Pavli,Rybkin} and
references therein for a discussion of spectral projections for
operators with spectral singularities.
\bigskip
{\bf \S2}. We first establish a general assertion giving an upper
estimate for $ \len e^{ iLt } \rin $, $ t < 0 $, provided that we
have an upper estimate for the norm of the resolvent by a function
of $ \im z $. In the following theorem $ L $ is a maximal
dissipative operator with real spectrum satisfying $ \len \( L - z
\)^{ -1 } \rin \le \( \im z - \omega \)^{ -1 } $ for some $ \omega
> 0 $ and all $ z $ with $ \im z > \omega $. The latter condition
guarantees that the operator $ Z_t = e^{ i L t } $, $ t \ge 0 $,
has bounded inverse.
{\bf Theorem 1.} {\it Assume that for a locally bounded function $
\rho : [ t_0, +\infty ) \to \R_+ $, $ t_0 < \omega^{ -1 } $, the
estimate \be\la{assest} \len \( L - z \)^{ -1 } \rin \le \rho \(
\( \im z \)^{ -1 } \) \ee holds for all $ z \in \Pi $ where $ \Pi
= \left\{ z : \; 0 < \im z < t_0^{ -1 } \right\} $. Then for $ t
\ge t_0 $
\be
\la{th1} \len Z_t^{ -1 } \rin \le e + 2e \frac {\rho ( t )}t . \ee
Proof} of the theorem is based on the following inequality which
is valid for any maximal dissipative operator $ D $ \cite{N}:
\be\la{intest} \int_\R \len R_D ( k - i \von ) u \rin^2 dk \le
\frac \pi\von \len u \rin^2 , \; \von > 0 , \ee where $ R_D ( z )
= \( D - z \)^{ -1 } $. This inequality is deduced in \cite{N}
from the existence of a self-adjoint dilation of $ D $. An
elementary derivation of it is outlined in \cite{N1}.
By the Hille - Yosida theorem one has for all $ u $ from the
domain of $ L $ and $ t > 0 $ \be \la{sqzet} Z_t^{ -1 } u = -
\frac 1{ 2 \pi i } \mathop{\lim}_{ N \to \infty } \int_{ -N + i
\von }^{ N + i \von } e^{ - i z t } R_L ( z ) u \, dz \ee where $
\von > 0 $ is large enough. We first show that (\ref{sqzet}) holds
for any $ \von > 0 $. Indeed, the assumptions made about $ L $
imply that $ R_L (z) $ is uniformly bounded in each half plane $
\im z \ge \nu > 0 $. It then follows from the identity $$ R_L (z)u
= - \frac 1z u + \frac 1z R_L (z) Lu $$ that $ R_L (z) = O \( z^{
-1 } \) $ as $z$ ranges over any such a half plane. This allows us
to shift the contour of integration in (\ref{sqzet}), ensuring
that the integrals over vertical segments vanish in the limit $ N
\to \infty $. It also shows that $ \len R_L (z)u \rin $ is square
summable over any line $ \mathbb{R}+i\von $, $ \von > 0 $.
Notice that $$ \int_{ -N + i \von }^{ N + i \von } e^{ - i z t }
\llangle R_L ( \overline{z} ) u , v \rrangle dz = e^{ \von t }
\int_{ -N }^N e^{ - i k t } \llangle R_L ( k - i \von) u , v
\rrangle dk \mathop{\longrightarrow}_{ N \to \infty } 0 $$ for all
$ t > 0 $ by the Jordan lemma. We then have for $ t > 0 $
\begin{eqnarray} \la{aux}
\llangle Z_t^{ -1 } u , v \rrangle = - \frac 1{ 2 \pi i }
\mathop{\lim}_{ N \to \infty } \int_{ [-N, N ] + i \von } e^{ - i
z t } \llangle R_L ( z ) u , v \rrangle dz = \nonumber \\ - \frac
1{ 2 \pi i } \mathop{\lim}_{ N \to \infty } \int_{ [-N, N ] + i
\von } e^{ - i z t } \llangle ( R_L ( z ) - R_L ( \overline{z} ) )
u , v \rrangle dz = \nonumber \\ - \frac \von\pi \mathop{\lim}_{ N
\to \infty } \int_{ [-N, N ] + i \von } e^{ - i z t } \llangle R_L
( z ) u , R_L^* ( \overline{z} ) v \rrangle dz . \end{eqnarray}
The last integral admits the following estimate, \bequnan \left|
\int_{ [-N, N ] + i \von } \cdots \right| \le e^{ \von t } \(
\int_{ \R + i \von } \len R_L ( z ) u \rin^2 dz \)^{1/2} \( \int_{
\R + i \von } \len R_L^* ( \overline{z} ) v \rin^2 dz \)^{1/2} \le
\\ e^{ \von t } \( \int_{ \R + i \von } \len R_L ( z ) u \rin^2 dz
\)^{1/2} \sqrt{ \frac \pi\von } \len v \rin . \eequnan On the
last step we have applied the inequality (\ref{intest}) to the
dissipative operator $ -L^* $. To estimate the remaining integral
in the right hand side, we observe that from the resolvent
identity $$ R_L ( z ) = [ I + 2i \von R_L ( z ) ] R_L (
\overline{z} ), \; \im z = \von , $$ it follows that $$ \len R_L (
z ) u \rin \le \( 1 + 2 \von \rho \( \von^{ -1 } \) \) \len R_L (
\overline {z} ) u \rin $$ for $ z \in \Pi $. This gives for $ \von
< t_0^{ -1 } $ \bequnan \int_{ \R + i \von } \len R_L ( z ) u
\rin^2 dz \le \( 1 + 2 \von \rho \( \von^{ -1 } \) \)^2 \int_{ \R
+ i \von } \len R_L ( \overline {z} ) u \rin^2 dz \le \\ \frac
\pi\von \( 1 + 2 \von \rho \( \von^{ -1 } \) \)^2 \len u \rin^2 .
\eequnan Thus, $$ \left| \int_{ [-N, N ] + i \von } \cdots \right|
\le \frac \pi\von e^{ \von t } \( 1 + 2 \von \rho \( \von^{ -1 }
\) \) \len u \rin \len v \rin . $$ Inserting this into (\ref{aux})
we obtain $$ \left| \llangle Z_t^{ -1 } u , v \rrangle \right| \le
\frac \von\pi \frac \pi\von e^{ \von t } \( 1 + 2 \von \rho \(
\von^{ -1 } \) \) \len u \rin \len v \rin . $$ Now setting $ \von
= t^{ -1 } $ for $ t $ large enough one gets the result. \hfill
\endpr
Note that the above proof of the theorem does not hint on
sharpness of the assertion. Most of the results in this paper
refer to the important partial case when $ \rho $ is a power
function. For this choice of $ \rho $ theorem 1 was first proven
in \cite{NR}. From now on we assume that $ L $ is an operator of
the type considered in $ \S 1 $, that is, a maximal dissipative
operator with a bounded imaginary part and such that $ \sigma_{
ess } ( L ) \subset \R $. Introduce the notation $ I_\delta ( k )
= [ k - \delta, k + \delta ] $.
{\bf Corollary 2.} {\it Assume $ L $ has at most finitely many
spectral singularities such that for a certain real $ p > 0 $ the
estimate $$ \len S^{ -1 } ( k ) \rin \le C \left| k - k_j
\right|^{ -p } $$ holds for each singularity, $ k_j $, for a. e. $
k \in I_\delta ( k_j ) $, $ \delta > 0 $, and $ \mbox{ ess sup}_{
| k |
> M } \| S^{ -1 } ( k ) \| $ is finite for some $ M $. Then for
all $ t > 0 $ $$ \len \left. Z_t^{ -1 } \right|_{ \cN_e } \rin \le
C \( 1 + t^p \) $$ with some $ C
>0 $.
Proof.} Put $ \{ k_j \} $ be the set of spectral singularities,
and let $ m $ be an arbitrary scalar $ H^\infty $ - function in $
\C_+ $ such that $ | m ( z ) | \sim c_j \left| z - k_j \right|^p $
for $ z \in \omede ( k_j ) $ for all $ j $, for instance, $ m ( z
) = \prod_j \( \frac { z - k_j } { z+ i } \)^p $ will do. Then,
under the assumption of the corollary, the function $ \Omega = m
S_e^{ -1 } $ belongs to $ L^\infty ( \R, {\bf B} ( E ) ) $. Now, $
\Omega H_2^+ \subset H_2^+ $ since $ \Omega \cD \subset H_2^+ $
for $ \cD = S_e H_2^+ $, and $ \cD $ is dense in $ H_2^+ $. Thus,
$ \Omega $ is a bounded operator in $ L^2 ( \R, E ) $ commuting
with multiplication and preserving $ H_2^+ $. Therefore, \cite{Na}
there exists a function $ \Omega^\prime \in H^\infty ( {\bf B} (E)
) $ such that $ \Omega^\prime ( k ) = \Omega ( k ) $ for a.e. $ k
\in \R $. Multiplying this by $ S_e $ and applying the uniqueness
theorem one gets that $ \Omega^\prime ( z ) S_e ( z ) = m ( z ) I
$ for all $ z \in \C_+ $. Hence, $ m S_e^{ -1 } \in H^\infty (
{\bf B} (E) ) $, and the assumption of theorem 1 is fulfilled with
$ \rho ( t ) = C t^{ 1+p } $ in view of (\ref{RezS}). \hfill
\endpr
Let us remark that the condition of "absence of the spectral
singularity at infinity", that is, of finiteness of $ \mbox{ ess}
\sup_{ | k | > M } \| S^{ -1 } ( k ) \| $, in this corollary holds
true for the wide class of operators satisfying the condition $ S
( k ) \to I $ as $ k \to \infty $, often fulfilled in
applications. In fact, under the assumptions of the corollary $
L|_{\cN_e} $ is a linear sum of a bounded operator and an operator
similar to a self-adjoint one.
A situation covered by corollary 2 arises, for instance, if we
consider the Schr\"o\-din\-ger operator with exponentially
decreasing potential \cite{Lja,Nai}. Another example is given by
the Friedrichs model with analytic perturbation: consider the
Hilbert space $ H = L^2 ( I ) $ where $ I $ is a compact interval,
and define $ L $ to be $ L = A + i \langle \cdot , \varphi \rangle
\varphi $ where $ (A f ) (s) = s f (s) $ and $ \varphi \in L^2 ( I
) $ is a function analytic in a neighborhood of $ I $.
In both these examples the characteristic function is analytic on
the real axis, and $ S (k ) - I $ is compact, which implies that
the spectral singularities are exactly the real poles of $ S^{ -1
} $. The following definition provides the simplest natural
abstract generalization of this situation.
A point $ k_0 \in \R $ is said to be a {\it spectral singularity
of the order $ p > 0 $, $ p \in \R $, in the strong sense}, if
for some nonzero $ e_0 \in E $ \begin{eqnarray} \la{ordex} & \len
S ( k ) e_0 \rin_E \le C \left| k - k_0 \right|^p , &
\\ \la{deford}
& \len S^{ -1 } ( k ) \rin \le C \left| k - k_0 \right|^{ -p } &
\end{eqnarray} for a. e. $ k $ in a vicinity of $ k_0 $ on the
real axis.
{\bf Theorem 3.} {\it Assume $ L $ has a spectral singularity of
order $ p $ in the strong sense. Then for any sufficiently small $
\von > 0 $ there exists such a $ u \in \cN_e $ that
\be\la{exactness} \len Z_t^{ -1 } u \rin = t^{ p - \von } ( 1 + o
( 1 ) ), \hspace{.5cm} t \to + \infty . \ee}
{\it Proof.} We construct the required $ u $ for the functional
model of the operator. Without loss of generality one can assume
that the spectral singularity is located at the point $ 0 $.
Observe first that if for a sequence of functions $ \tg_\delta \in
H^2_+ (E) $ the sequence $ S \tg_\delta $ converges in $ L^2 ( \R
, E ) $ as $ \delta \to 0 $, then so does $ W \tg_\delta $ in $
\cN_e $. Indeed, then $ S \sqrt \Delta \tg_\delta $ converges in $
L^2 ( \R , E ) $, since for all $ \tv \in L^2 ( \R , E ) $ we have
$$ \len S \sqrt \Delta \tv \rin = \len \sqrt \Delta \sqrt { S^* S
} \tv \rin \le \len S \tv \rin , $$ and the fact follows from
(\ref{PK}) since $ P_- \Delta \tg_\delta = - P_- S^* S \tg_\delta
$ obviously converges.
Define the function $ h_\delta ( k ) = \( k + i \delta \)^{ - 1/2
- p + \von } $, $ \delta \ge 0 $, where the cut is drawn along the
negative imaginary axis, and let $ \tg_\delta = h_\delta e_0 $, $
\tg = \tg_0 $. Obviously, $ \tg_\delta \in H^2_+ ( E ) $ for all $
\delta > 0 $. One then has $$ \len S ( k ) \tg_\delta ( k ) \rin_E
\le C \frac {\left| k \right|^p } { \left| k + i \delta \right|^{
1/2 + p - \von } } \le C \left| k \right|^{ \von - 1/2 }, $$ and
therefore $ S \tg_\delta $ converge in $ L^2 ( \R, E ) $ as $
\delta \to 0 $. From this we infer that there exists the limit $ u
= \mathop{\lim} \limits_{ \delta \to 0 } W \tg_\delta $.
According to (\ref{PK}) and (\ref{interex}), for all $ \tv \in
\L2de $ we have $$ \len Z_t^{ -1 } W \tv \rin^2 = \len S \sqrt
\Delta \tv \rin^2 + \len P_- e^{ -ikt } \Delta \tv \rin^2, $$
hence \be \la{norde} \len Z_t^{ -1 } u \rin^2 = \len S \sqrt
\Delta \tg \rin^2 + \lim_{ \delta \to 0 } \len P_- e^{ -ikt }
\Delta \tg_\delta \rin^2 . \ee Since $ \tg_\delta - \Delta
\tg_\delta = S^* S \tg_\delta $ converges in $ L^2 (\R, E ) $, one
sees that (here the $ O $ - symbolic refers to the variable $ t
\in [ 0, + \infty ) $) $$ \lim_{ \delta \to 0 } \len P_- e^{ -ikt
} \Delta \tg_\delta \rin = \lim_{ \delta \to 0 } \len P_- e^{ -ikt
} \tg_\delta \rin + O ( 1 ) . $$ Assume first $ p \in ( 0 , 1 ]
$. Considering that $ P_- \tg_\delta = 0 $, one has the identity
$$ P_- e^{ -ikt } \tg_\delta = P_- \( e^{ -ikt } - 1 \) \tg_\delta
. $$ By majorized convergence theorem, one can now pass to the
limit $ \delta \to 0 $ in the right hand side to get $$ \lim_{
\delta \to 0 } \len P_- e^{ -ikt } \Delta \tg_\delta \rin = \len
P_- \( e^{ -ikt } - 1 \) \tg \rin + O ( 1 ) . $$
The projection $ P_- $ commutes with the dilations, $ \( P_- f
\)_a = P_- f_a $ where $ f_a ( x ) = f ( ax ) $, which gives $$
\len P_- \( e^{ -ikt } - 1 \) \tg \rin = \len \( P_- \left[ \( e^{
-ik } - 1 \) h_{ t^{ -1 } } e_0 \right] \)_t \rin = t^{ p - \von }
\len P_- \left[ \( e^{ -ik } - 1 \) h \, e_0 \right] \rin . $$ It
remains to notice that $ f = \( e^{ -ik } - 1 \) h \notin H^2_+ $
since $ f ( is ) \to \infty $ as $ s \to + \infty $, and therefore
the constant $ C_\von = \len P_- \left[ f e_0 \right] \rin \neq 0
$.
In the case of integer $ p > 1 $, we use a formula for $ P_- e^{
-ikt } \tg_\delta $ obtained by subtraction of the first $ p $
terms of the Taylor expansion for the function $ e^{ -ikt } $.
Since $ k^j \tg_\delta \in H^2_+ ( E ) $ for any nonnegaitve
integer $ j \le p -1 $, we have $$ P_- e^{ -ikt } \tg_\delta = P_-
\left[ \( e^{ -ikt } - \sum_{ j = 0 }^{ p-1 } \frac{ \( -ikt \)^j
} { j! } \) \tg_\delta \right]. $$ The function in square
brackets is $ O \( |k|^{ - 1/2 + \von } \) $ in a neighborhood of
$ 0 $ uniformly in $ \delta $, which makes it possible to pass to
the limit $ \delta \to 0 $ in the right hand side to obtain that
$$ \lim_{ \delta \to 0 } \len P_- e^{ -ikt } \tg_\delta \rin =
\len P_- \left[ \( e^{ -ikt } - \sum_{ j = 0 }^{ p-1 } \frac{ \(
-ikt \)^j } { j! } \) \tg \right] \rin . $$ The rest of the proof
coincides with that in the case $ p = 1 $. The case of non-integer
$ p > 1 $ is considered in a similar way by subtraction of the
first $ [p] + 1 $ terms of the Taylor expansion for the exponent.
\hfill \endpr
Another partial case of theorem 1 important in applications is
obtained if we take $ \rho ( t ) = \ln t $. This situation arises,
for instance, when studying the neutron transport operator for
slab geometry \cite{KNR,LW2}. The following assertion is analogous
to corollary 2.
{\bf Corollary 4.} {\it Assume $ L $ is bounded and has at most
finitely many spectral singularities such that the estimate $$
\len S^{ -1 } ( k ) \rin \le - C^\prime \ln | k - k_j | $$ holds
for each singularity, $ k_j $, for a. e. $ k \in I_\delta ( k_j )
$, $ \delta > 0 $. Then $$ \len \left. Z_t^{ -1 } \right|_{\cN_e}
\rin \le C \ln t $$ for all $ t \ge 2 $.
Proof.} This fact follows from theorem 1 with $ \rho ( t ) = \ln t
$. The proof can be taken verbatim from that of corollary 2, if we
set $ m ( z ) = \prod_j \frac 1{\ln ( z - k_j ) } $ where the cut
of the logarithm is chosen to be, for instance, the negative real
axis, and $ \ln 1 = 2 \pi i $. \hfill \endpr
{\it A logarithmic spectral singularity in the strong sense} is
defined in the way analogous to the power case, with (\ref{ordex})
replaced by $$ \len S ( k ) e_0 \rin_E \le \frac C{ | \ln | k -
k_0 | | } . $$
{\bf Theorem 5.}\la{exactlog} {\it Assume $ L $ has a logarithmic
spectral singularity in the strong sense. Then for any
sufficiently small $ \von > 0 $ there exists a $ u \in \cN_e $
such that $$ \len Z_t^{ -1 } u \rin = \( \ln t \)^{ 1 - \von } ( 1
+ o ( 1 ) ), \hspace{.5cm} t \to + \infty . $$
Proof} of this theorem is analogous to that of theorem 3. For the
function $ \tg $ we choose $ \tg ( k ) = \frac { \( \ln k \)^{ 1/2
- \von }} { ( k + i ) \sqrt k } e_0 $ where the cut of the square
root is chosen to be the negative imaginary axis, of the divisor
to be the positive real axis. Obviously, $ S \tg \in L^2 (\R; E )
$ since the function $ \frac 1{ k \left| \ln k \right|^{ 1 + 2
\von } } $ is integrable at zero. Also, with $ \tg_\delta $
defined as $ \tg_\delta = h_\delta e_0 $, $ h_\delta ( z ) = \frac
{\( \ln z \)^{ 1/2 - \von }} { ( z + i )\sqrt { z + i \delta } }
$, we see that $ S \tg_\delta $ converges in $ L^2 ( \mathbb{R}, E
) $ as $ \delta \downarrow 0 $. Since $ h_\delta \in H^2_+ $ for $
\delta > 0 $, it follows that there exists the limit $ u =
\mathop{\lim} \limits_{ \delta \to 0 } W \tg_\delta $.
Then, $$ P_- e^{ -ikt } \Delta \tg_\delta = P_- e^{ -ikt } P_+
\Delta \tg_\delta + e^{ -ikt } P_- \Delta \tg_\delta . $$ Arguing
as in the proof of theorem 3 one concludes that $ \Delta $ in the
right hand side of this equality can be omitted with an accuracy
of $ O ( 1 ) $ in $ t $, and thus $$ \lim_{ \delta \to 0 } \len
P_- e^{ -ikt } \Delta \tg_\delta \rin = \lim_{ \delta \to 0 } \len
P_- e^{ -ikt } P_+ \tg_\delta \rin + O ( 1 ). $$ Let $
\varphi_\delta $ be the Fourier transform of the function $
h_\delta $. According to the Paley - Wiener theorem, $$ \len P_-
e^{ -ikt } P_+ \tg_\delta \rin^2 = \int_0^t \left| \varphi_\delta
( p ) \right|^2 dp \, \mathop{\longrightarrow} \limits_{\delta \to
0} \int_0^t \left| \varphi ( p ) \right|^2 dp $$ where $ \varphi $
is the Fourier transform of the function $ h \in L^1 ( \R ) $. The
passage to the limit $ \delta \to 0 $ in the right hand side is
justified by the obvious fact that $ h_\delta \stackrel{ L^1 ( \R
) } {\longrightarrow} h $, and therefore $ \varphi_\delta
\longrightarrow \varphi $ uniformly. Calculating the asymptotic of
$ \varphi ( p ) $ as $ p \to + \infty $ via the stationary phase
method, we find $$ \varphi ( p ) = \int \frac { e^{ -ikp } \( \ln
k \)^{ \frac 12 - \von }} { ( k + i ) \sqrt k } dk = C \frac {\(
\ln p \)^{ \frac 12 - \von }}{ \sqrt p } ( 1 + o ( 1 ) ) . $$ From
this we obtain, $$ \int_0^t \left| \varphi ( p ) \right|^2 dp = C
\( \ln t \)^{ 2 - 2 \von } ( 1 + o ( 1 ) ) , \ \ \ t \to + \infty
. $$ Therefore $$ \lim_{ \delta \to 0 } \len P_- e^{ -ikt } \Delta
\tg_\delta \rin = C \( \ln t \)^{ 1 - \von } ( 1 + o ( 1 ) ), $$
and the result follows. \hfill
\endpr
One should mention that the definition of a spectral singularity
of the order $ p $ we use is more restrictive than the requirement
of exactness of the estimate (\ref{deford}): the vector $ e_0 $ in
(\ref{ordex}) does not depend on $ k $. This circumstance is
reflected by the words "in the strong sense" inserted in the
definition. In fact, our definition does not even cover the case
of analytic $ S $, unless, for instance, we assume additionally
the relative compactness of the perturbation $ V $ with respect to
the real part of $ L $. The latter condition, however, is often
fulfilled in applications. Study of the general case calls for
further investigation.
We now proceed to study the local behaviour of the semigroup $ Z_t
$ with respect to the spectral representation of the real part of
the operator $ L $. This approach is natural when an operator can
be considered as a perturbation of its real part.
For an interval $ \omega \subset \R $ define $ P_\omega $ to be
the spectral projection of $ A = \re L $ for the $ \omega $.
{\bf Theorem 6.} {\it Let $ L $ be a dissipative operator as in
corollary 2 with $ p = n $ for a certain real $ n \ge 1 $. Then
for any closed interval $ \omega $ not containing the spectral
singularities} \be\la{estP} \len \left. P_\omega Z_{-t} \right|_{
\cN_e } \rin \le C_\omega \( 1 + t^{ n - 1/2 } \) , \hspace{.5cm}
t > 0 . \ee
We shall need the following abstract lemma. Let $ D $ be a
dissipative operator, $ S $ be the characteristic function of $ D
$, and $ \alpha = \sqrt{ 2 \ \im D } $.
{\bf Lemma 7.} {\it The following identity holds for all $ t \in
\R $, \be\la{exp} \alpha Z_t u = \frac i{\sqrt{ 2 \pi }} \int_\R
e^{ ikt } ( \Delta \tg ) ( k ) \, dk \ee for any smooth vector $ u
= W \tg $ such that $ \tg \in L^1 ( \R , E ) $.
Proof.} It is enough to proof the result for a compactly supported
$ \tg $. In view of the intertwining relation (\ref{interex}), it
suffices to proof the identity for $ t = 0 $. For any $ u = W \tg
\in \wt{\cN_e} $ we have from (\ref{cPK}) $$ u = P_\cK
\pmatrix{\tg \cr -S\tg \cr} = \pmatrix{\tg - P_+ \Delta \tg \cr
-S\tg \cr} \equiv \pmatrix{ g_1 \cr g_2 \cr}. $$ By \cite[Theorem
2]{N} we then have \be\la{F} \cases { g_1 + S^* g_2 = \cF_+ u \cr
S g_1 + g_2 = \cF_- u \cr} \Leftrightarrow \cases { P_- \Delta \tg
= \cF_+ u \cr - S P_+ \Delta \tg = \cF_- u \cr} \ee where \bequnan
\cF_+ u = - \frac 1{\sqrt { 2 \pi }} \alpha \( D - \cdot + i0 \)^{
-1 } u , \\ \cF_- u = - \frac 1{\sqrt { 2 \pi }} \alpha \( D^* -
\cdot - i0 \)^{ -1 } u . \eequnan Recall that $ \alpha \( D -
\cdot \)^{ -1 } v \in H^2_- ( E ) $ for any $ v \in H $
\cite[Theorem 4, Corollary 1]{N}, and therefore $ \cF_\pm u $
belong to $ H^2_\mp (E) $, respectively. Considering the second
equality in (\ref{F}) as an equality for elements of $ H^2_+ ( E )
$ we have for any $ z \in \C_+ $ $$ - S ( z ) ( P_+ \Delta \tg ) (
z ) = - \frac 1{ \sqrt{ 2 \pi } } \alpha \( D^* - z \)^{ -1 } u =
- \frac 1{ \sqrt{ 2 \pi } } S ( z ) \alpha \( D - z \)^{ -1 } u.
$$ Since $ \rho ( D ) \bigcap \C_+ $ is nonempty and $ S ( z ) $
is invertible for $ z \in \rho ( D ) \bigcap \C_+ $ one concludes
by analyticity that $ P_+ \Delta \tg = \frac 1{ \sqrt{ 2 \pi } }
\alpha \( D - \cdot \)^{ -1 } u $. We now get from (\ref{F}) $$
\Delta \tg = ( P_+ + P_- ) \Delta \tg = \frac 1{ \sqrt{ 2 \pi } }
\( \alpha \( D - \cdot + i0 \)^{ -1 } u - \alpha \( D - \cdot - i0
\)^{ -1 } u \), $$ the equality is being understood in the sense
of the space $ L^2 ( \R, E ) $. Let us integrate this equality
over an arbitrary open finite interval $ I $ such that $ \tg ( k )
= 0 $ for a.e. $ k \notin [ -A , A ] \subset I $. Since $ \( 2 \pi
i \)^{ -1 } \int_\gamma \( D - z \)^{ -1 } u = u $ for any
contour $ \gamma $ encircling $ [ -A , A ] $ which follows from
the intertwining relation (\ref{inter}), we obtain $$ \int_I (
\Delta \tg ) ( k ) \, dk = \frac 1{ \sqrt{ 2 \pi } } \lim_{ \von
\to 0 } \alpha \int_{ \gamma_\von } \( D - z \)^{ -1 } u \, dz = -
i \sqrt{ 2 \pi } \, \alpha u . $$ Here $ \gamma_\von $ is the
contour consisting of the intervals $ I \pm i \von $ oriented in
the opposite directions. \hfill \endpr
Applying now the Paley-Wiener theorem, we arrive at the following
{\bf Corollary.} {\it Let $ \cF $ be the Fourier transform in $
L^2 ( \R , E ) $. Then the following equality holds for any $u$
satisfying the assumption of lemma 7 for a.e. $ t \ge 0 $,
\be\la{expp} \alpha Z_{ -t } u = i \cF \left[ P_+ \Delta \tg
\right] (t) . \ee }
For $ u \in \wt{\cN_e} $ introduce the notation $ u^+ ( k ) $ for
the boundary values of the $ H^2_+ ( E ) $ - function from the
definition of the set $ \wt{\cN_e} $, $$ u^+ ( k ) = \frac 1{
\sqrt{ 2 \pi } } \, \alpha \( L - k - i0 \)^{ -1 } u . $$ Then
(\ref{expp}) can be rewritten as \be\la{expp1} \alpha Z_{ -t } u =
i \( \cF u^+ \) (t) . \ee This formula can, of course, easily be
derived directly from the Hille-Yosida theorem for all $ u \in
\wt{\cN_e} \cap \cD ( L ) $ without any use of the functional
model. We have used the argument above since we shall need lemma 7
anyway in what follows.
{\it Proof of theorem 6.} Let $ u = W \tg $ be a smooth vector
satisfying the assumption of lemma 7. According to the Doimelle
formula one has \be\la{Doi} Z_{-t} u = \int_0^t e^{ i A ( t^\prime
- t ) } V Z_{ -t^\prime } u\ d t^\prime + e^{ -i A t } u . \ee Let
us substitute (\ref{expp1}) in this formula. We first notice that
\be\la{chi1} \alpha Z_{ -t } u = i \cF [ \chi_1 u^+ ] (t) + Q(t,u)
\ee where $ \chi_1 $ is the indicator of an arbitrary neighborhood
$ U = \bigcup_j I_d ( k_j ) $ of the set $ \{ k_j \} $ of spectral
singularities such that $ U \bigcap \omega = \emptyset $, and $ Q
$ satisfies $ \int_\R \len Q ( t , u ) \rin_E^2 dt \le C \| u \|^2
$. Indeed, letting $ \chi_2 = 1 - \chi_1 $, one has \bequnan
\int_\R \len Q ( t , u ) \rin_E^2 dt = \len \cF \left[ \chi_2 u^+
\right] \rin^2 = \len \chi_2 u^+ \rin^2
\le
\mathop{\hbox{ ess sup }}\limits_{ k \notin U } \len S^{ -1 } (k)
\rin^2 \len S u^+ \rin^2 \le \\ C \| u \|^2 . \eequnan Here we
have taken into account that $ \len S u^+ \rin \le \| u \| $,
which is implied\footnote{It also easily follows from (\ref{PK}).}
by the identity $$ S ( z ) u^+ ( z ) = \frac 1{ \sqrt{ 2 \pi } }
\, \alpha \( L^* - z \)^{ -1 } u , \; z \in \C_+ , $$ easily
verified by direct calculation, and the inequality (see \cite{N})
\be\la{noway} \int_\R \len \alpha \( L^* - k - i \von \)^{ -1 } u
\rin^2 dk \le \pi \len u \rin^2 . \ee It now follows that the
operator norm of the term corresponding to $ Q $ in (\ref{Doi}) is
estimated above by $ \sqrt t $, $$ \len \int_0^t e^{ i A (
t^\prime - t ) } \alpha Q ( t, u ) \ d t^\prime \rin \le \(
\int_0^t d t^\prime \)^{ 1/2 } \len \alpha \rin \( \int_0^t \len Q
( t , u ) \rin_E^2 d t^\prime \)^{ 1/2 } \le C \sqrt t \| u \| .
$$
Thus, it remains to estimate the contribution of the first term in
(\ref{chi1}) to (\ref{Doi}). We have, \bequnan \int_0^t e^{ i A (
t^\prime - t ) } \alpha \cF \left[ \chi_1 u^+ \right] (t^\prime )
\ d t^\prime = C e^{ - i A t } \int_0^t \int_U e^{ i ( A - k )
t^\prime } \alpha u^+ (k) dk \, d t^\prime = \\ C e^{ - i A t }
\int_U g_t ( A - k ) \alpha u^+ (k) \, dk , \eequnan where $ g_t
(k) = \( e^{ i k t } - 1 \)/k $. Since the projection $ P_\omega $
commutes with the functions of $ A $, the theorem will be proved
if we show that $$ \len \int_U g_t ( A - k ) P_\omega \alpha u^+
(k) \, dk \rin \le C \( 1 + t^{ n- 1/2 } \) \| u \| . $$
First, let us replace the integral over $ U $ by the integral over
the contour $ \Gamma_\delta = \bigcup \Gamma^j $, $ \Gamma^j =
\gamma^j_\delta \cup \{ k : \delta < | k - k_j | < d \} $, $
\gamma^j_\delta = \{ z \in \C_+ : \; | z - k_j | = \delta \} $,
where $ \delta > 0 $ will be chosen below. Then,
\begin{eqnarray} \la{gadel} \len \int_{\Gamma_\delta} g_t ( A - k
) P_\omega \alpha u^+ (k) \, dk \rin^2 \le \| V \| \( \int_{
\Gamma_\delta } \len g_t ( A - k ) P_\omega \rin^2 \len S_e^{ -1 }
( k ) \rin^2 |dk| \) \nonumber \\ \int_{ \Gamma_\delta } \len S_e
( k ) u^+ (k) \rin^2 |dk| .
\end{eqnarray} We shall now estimate separately the integrals over
$ \gamma_\delta^j $ and $ \varrho_\delta^j = \{ k: \; \delta < | k
-k_j | < d \} $ in the first factor. To this end, notice that it
follows from the proof of corollary 2 that $ \len S^{ -1 }_e ( z )
\rin = O \( | z - k_j |^{ - n } \) $ in the $d$-vicinity of the
point $ k_j $ in $ \overline{\C_+} $, provided that $ d $ is small
enough, since $ m S_e^{ -1 } $ is bounded in $ \C_+ $ for a scalar
function $ m $ such that $ m^{ -1 } $ satisfies this in the
vicinity of $ k_j $. This gives $$ \int_{ \gamma_\delta^j } \len
g_t ( A - k ) P_\omega \rin^2 \len S_e^{ -1 } ( k ) \rin^2 |dk|
\le C \delta^{ -2n } e^{ \delta t } \int_{ \gamma_\delta^j } |dk|
= C \pi \delta^{ 1-2n } e^{ \delta t } $$ since $ \len g_t ( A - k
) P_\omega \rin \le C e^{ \delta t } $ for $ k \in \gamma_\delta $
by the functional calculus, and
\begin{eqnarray} \la{except} \int_{ \varrho_\delta^j } \len g_t (
A - k ) P_\omega \rin^2 \len S_e^{ -1 } ( k ) \rin^2 dk \le \frac
4{ \( \mbox{dist} ( \omega , k_j ) - d \)^2 } \int_{ \delta < | k
-k_j | < d } \frac {dk}{\left| k - k_j \right|^{ 2n }} \le
\nonumber\\ C_1 \delta^{ 1 -2n } + C_2 .
\end{eqnarray} Combining these estimates and setting $ \delta = t^{ -1
} $, one obtains that the factor in parentheses in the right hand
side of (\ref{gadel}) is not greater than $ C \( 1 + t^{ 2n -1 }
\) $. Then, applying the Carleson embedding theorem, an
appropriate vector version of which can be found e. g. in
\cite{Pav}, one concludes that the norm of an $ H_+^2 $ - function
in $ L^2 \( \Gamma_\delta , |dk| \) $ is estimated\footnote{For
this very special case of the theorem the conclusion can, of
course, be verified in an elementary way.} from above by the norm
in $ H_+^2 $ with a constant $ C $ independent of $ \delta $,
hence $$ \int_{ \Gamma_\delta } \len S_e (k) u^+ (k) \rin^2 |dk|
\le C \len S_e u^+ \rin^2_{ H^2_+ (E) } \le C \len u \rin^2 . $$
It now remains to notice that the linear set described in lemma 7
is dense in $ \cN_e $. \hfill
\endpr
One should mention that one can eliminate completely the
functional model from the above proof of theorem 6, if we recall
that the fundamental inequality (\ref{noway}) can be obtained in
an elementary way (see \cite{N1} for an outline).
As we shall now show, an additional assumption of relative
smoothness of the perturbation $ V $ allows to establish a
generalization of the estimate (\ref{estP}) valid for any interval
$ \omega $ and all $ n \ge 1/2 $ (in the case $ n < 1 $ it only
follows from the proof of theorem 6 that the norm in the left hand
side of (\ref{estP}) is $ O \( \sqrt t \) $).
In the following theorem $ L $ is a dissipative operator having at
most finitely many spectral singularities, $ k_j $, such that for
each $ k_j $ there exists a $ p_j > 0 $ such that the estimate $$
\len S^{ -1 } ( k ) \rin \le C \left| k - k_j \right|^{ -p_j } $$
holds for a. e. $ k \in I_\delta ( k_j ) $, $ \delta > 0 $, and $
\mbox{ ess sup}_{ | k |
> M } \| S^{ -1 } ( k ) \| $ is finite for some $ M $. Let $
\sigma_0 = \{ k_j \} $ be the set of spectral singularities of $ L
$, $ p = \max_j p_j $.
{\bf Theorem 6a.} {\it Assume that $ p \ne 1/2 $, and $ \sqrt V $
is $ \re L $ - smooth in the sense of Kato \cite{RS}. Then for any
closed interval $ \omega \subset \R $ we have \be\la{estPgen} \len
\left. P_\omega Z_{-t} \right|_{ \cN_e } \rin \le C_\omega \( 1 +
t^{ p - 1/2 } + \sum_{ j : k_j \in \omega } t^{ p_j } \) ,
\hspace{.5cm} t
> 0 . \ee
Proof.} Let $ U = \bigcup_{ j: k_j \notin \omega } U_j $ with the
intervals $ U_j = I_d (k_j) $ chosen to satisfy $ U_j \cap \omega
= \emptyset $, $ \chi_1 $ be the indicator of $ U $, and $ \chi_2
= 1 - \chi_1 $. Fix an arbitrary $ B > 0 $ such that $ U \cup
\omega \subset ( -B , B ) $, and define $ \chi_r $ to be the
indicator of $ \R \setminus [-B , B ] $, $ \chi_3 $ to be the
indicator of $ U^\prime = ( -B , B ) \setminus U $, so $ \chi_2 =
\chi_3 + \chi_r $. To take into account the contribution of the
singularities from $ \omega $ we observe that \be\la{essS}
\mathop{\sup}\limits_{ k \in \R \setminus U } \len S^{ -1 }_e ( k+
i\delta ) \rin \le C \max_{j: k_j \in \omega } \delta^{ - p_j }
\ee for $ \delta < 1 $, which also follows from the proof of
corollary 2.
The estimate of the contribution of the term in (\ref{Doi})
corresponding to $ \chi_1 $ holds in the situation under
consideration. Thus the theorem will be proved if we show that $$
\len \int_0^t e^{ i A t^\prime } \alpha \cF \left[ \chi_2 u^+
\right] (t^\prime) \, dt^\prime \rin \le C \( 1 + \sum_{ j : k_j
\in \omega } t^{ p_j } \) \| u \| . $$
We have for any $ \xi \in H $, \begin{eqnarray} \la{contF} \left|
\llangle \int_0^t \cdots , \xi \rrangle \right| = \left| \int_0^t
\llangle \cF \left[ \chi_2 u^+ \right] (t^\prime) , \alpha e^{ - i
A t^\prime } \xi \rrangle dt^\prime \right| \le \nonumber
\\ \( \int_0^t \len \cF \left[ \chi_2 u^+ \right] (t^\prime)
\rin^2 dt^\prime \)^{ 1/2 } \( \int_0^t \len \alpha e^{ - i A
t^\prime } \xi \rin^2 dt^\prime \)^{ 1/2 } . \end{eqnarray} We
first observe that \be \la{O1} \int_0^t \len \cF \left[ \chi_r u^+
\right] (t^\prime) \rin^2 dt^\prime \le \len \cF \left[ \chi_r u^+
\right] \rin^2 = \len \chi_r u^+ \rin^2 \le C \len S u^+ \rin^2
\le C \len u \rin^2 . \ee Then, transforming the contour of
integration, we obtain \bequnan \sqrt{ 2 \pi }\, \cF \left[ \chi_3
u^+ \right] (\tau) = \int_{ U^\prime } e^{ -ik\tau } u^+ (k) \, dk
= e^{ \delta \tau } \int_{ U^\prime } e^{ -ik\tau } u^+
(k+i\delta) \, dk + \\ \int_{ \rho_\delta } e^{ -ik\tau } u^+ (k)
\, dk , \eequnan where $ \rho_\delta $ is the contour consisting
of vertical segments of the length $ \delta = 1/t $ starting at $
\pm B $ and the ends of the intervals $ U_j $. For $ 0 \le \tau
\le t $ the second integral is estimated above by \bequnan e^{
\delta t } \int_0^\delta \len u^+ (k+is) \rin \, ds \le C \sqrt
\delta \( \int_0^\delta \len u^+ (k+is) \rin^2 \, ds \)^{ 1/2 }
\le
\\ C \sqrt \delta \( \mathop{\sup }\limits_{ k \in \rho_\delta } \len
S^{ -1 }_e (k) \rin \) \len S_e u^+ \rin_{ L^2 \( \rho_\delta ,
|dk| \) } \le C \sqrt \delta \| u \| . \eequnan Here we took into
account that the norm of $ S_e^{ -1 } $ is bounded outside an
arbitrary neighborhood of the set $ \sigma_0 $ (see the proof of
corollary 2). From this we have \bequnan \( \int_0^t \len \cF
\left[ \chi_3 u^+ \right] (t^\prime) \rin^2 dt^\prime \)^{ 1/2 }
\le C_1 e^{ \delta t } \len \cF \left[ \chi_3 u^+ (\cdot + i
\delta ) \right] \rin + C_2 \sqrt t \sqrt \delta \| u \| \le \\ C
\( \len \chi_3 u^+ (\cdot + i \delta ) \rin + \| u \| \) \le C \(
\max_{ j : k_j \in \omega } \( \delta^{ - p_j } \) \len \chi_3 (
S_e u^+ ) ( \cdot + i \delta ) \rin + \| u \| \) \le \\ C \(
\max_{ j : k_j \in \omega } \( \delta^{ - p_j } \) \len S_e u^+
\rin + \| u \| \) \le C \( 1 + \sum_{ j : k_j \in \omega } t^{ p_j
} \) \| u \| . \eequnan On the last but one step we have used
(\ref{essS}) here. Inserting this and (\ref{O1}) into
(\ref{contF}) and taking into account that $$ \int_0^t \len \alpha
e^{ - i A t^\prime } \xi \rin^2 dt^\prime \le C \| \xi \|^2 $$
under the imposed assumption about smoothness, we conclude the
proof. \hfill
\endpr
The results obtained approach the problem of detecting the
singularities formulated as follows. Assume we are given an
operator with finitely many spectral singularities, each of a
finite power order in the strong sense. How to calculate their
orders, $ p_j $, and locations, $ k_j $, from the asymptotic of $
Z_t $, $ t \to \pm\infty $ ? In this respect, corollary 2 and
theorem 3 answer the question of calculating the maximal order, $
p $, of spectral singularities. Once this is done, theorem 6a
provides a sufficient condition for the localization: if the
estimate (\ref{estP}) with $ n = p \ne 1/2 $ is not satisfied,
then $ \omega $ contains a singularity of an order greater than $
p - 1/2 $.
In the following theorem we give an asymptotic of the nonsmooth
vector constructed in the course of the proof of theorem 3.
{\bf Theorem 8.} {\it Assume $ L $ is a dissipative operator, and
$ \sqrt V $ is $ \re L $ - smooth. Then for any spectral
singularity $ k_0 \in \R $ of an order $ n > 1/2 $ and any $ \von
> 0 $ small enough there exists a $ u \in \cN_e $ satisfying (\ref{exactness})
and such that for any closed interval $ \omega $, $ k_0 \notin
\omega $, \be\la{nosmasab} P_\omega Z_{ -t } u = C e^{ -i k_0 t }
t^{ n - 1/2 - \von } \Psi_{k_0} + O \( t^{ \max\{ 0 , n - 3/2 -
\von \} } \) \ee where $ \Psi_{ k_0 } = \( A_\omega - k_0 \)^{ -1
} P_\omega \alpha e_0 $, $ A_\omega = A P_\omega $, $ e_0 $ is
defined in (\ref{ordex}), and the $ O $-symbol refers to the norm
in $ H $.}
In the above example of the Friedrichs model the smoothness
assumption is equivalent to the essential boundedness of the
function $ \varphi $. The asymptotic (\ref{nosmasab}) takes in
this case the form $$ Z_{ -t } u = C_0 e^{ - i k_0 t } t^{ n - 1/2
- \von } \Psi_{ k_0 } + r_t $$ where $$ \Psi_{ k_0 } ( s ) = \frac
{\varphi ( s )}{ s - k_0 }, $$ the function $ r_t $ satisfies $$
\( \int_{ | s - k_0 | > \delta } | r_t ( s ) |^2 ds \)^{ 1/2 } \le
C_\delta \( 1 + t^{ n - 3/2 - \von } \) $$ for any $ \delta > 0 $,
and the equality holds for each $ t > 0 $ for a. e. $ s \in I $.
The function $ \Psi_{ k_0 } $ makes sense of a regular part of the
generalized eigenfunction of $ L $ corresponding to the point $
k_0 $: any formal solution to $ L u = k_0 u $ coincides with $
\Psi_{ k_0 } $ for $ s \ne k_0 $. Thus, the asymptotic obtained
can be considered as a rigorous justification of the idea that
spectral singularity is a kind of resonance which has been drawn
into the continuous spectrum.
{\it Proof of theorem 8.} We define the $ u $ as in the proof of
theorem 3. Let us first show that \be\la{alphau} \alpha Z_{ -t } u
= C e^{ -i k_0 t } t^{ n - 1/2 - \von } e_0 + r(t) \ee with an $ r
\in L^2 ( \R , E ) $. In notation of the proof of theorem 3, let $
\Delta \tg_\delta = h_\delta e_0 + \rho_\delta $, then $
\rho_\delta $ converges in $ L^2 ( \R , E ) $. Let $ \rho = \lim_{
\delta \to 0 } \rho_\delta $. For a non-half-integer $ n $ we
define $ p_n ( s ) = \sum_{ j=0}^{ [n - 1/2 ]} \frac{ \( -is \)^j
}{j!} $. By lemma 7 we have for $ u_\delta = W \tg_\delta $ and
any $ t > 0 $
\begin{eqnarray}\la{subtrabstr} \alpha Z_{ -t } u_\delta = C \int_\R
e^{ -ikt } ( \Delta \tg_\delta ) ( k ) \, dk = C \( \int_\R e^{
-ikt } h_\delta ( k ) \, dk \) e_0 + C ( \cF \rho_\delta )( t )
\end{eqnarray}
Then, \bequnan \int_\R e^{ -ikt } h_\delta ( k ) \, dk = C e^{ -i
k_0 t } \int_\R \( e^{ -i (k - k_0 )t } - p_n ( ( k - k_0 ) t ) \)
h_\delta ( k ) \, dk + \\ e^{ -i k_0 t } \int_\R p_n ( ( k - k_0 )
t ) h_\delta ( k ) \, dk . \eequnan Blowing up the contour of
integration in the upper half plane, one verifies that the second
term in the right hand side is zero since $ n + 1/2 - [ n - 1/2 ]
- \von > 1 $. Obviously, in the first term one can pass to the
limit $ \delta \to 0 $ for any $ t > 0 $, which gives after
substitution $ k \mapsto ( k - k_0 )t $: $$ \int_\R e^{ -ikt }
h_\delta ( k ) \, dk \mathop{\longrightarrow} \limits_{\delta \to
0} C t^{ n - 1/2 - \von } e^{ -i k_0 t } , \ \ C = \int_\R \( e^{
-is } - p_n ( s ) \) h ( k ) \, dk . $$ Taking into account that $
\rho_\delta \mathop{\longrightarrow} \limits_{} \rho $ as $ \delta
\to 0 $ also in $ L^1 ( \R , E ) $, one concludes that both terms
in the right hand side of (\ref{subtrabstr}) converge in $ E $ as
$ \delta \to 0 $ to give (\ref{alphau}) for all $ t > 0 $. For $ n
$ half-integer the proof is the same, except that we have to take
$ n - 3/2 $ for the upper limit in the sum defining $ p_n $.
We now insert (\ref{alphau}) into the Doimelle formula
(\ref{Doi}). By the smoothness assumption, the term corresponding
to $ r ( t ) $ is bounded in $t$ (see the estimate of an analogous
term in the proof of theorem 6a, formulae
(\ref{contF}),(\ref{O1})). It now remains to consider the
contribution of the first term given by \be\la{inau} e^{ -i A t }
\int_0^t e^{ i ( A - k_0 ) \tau } \tau^{ n - 1/2 - \von } d \tau .
\ee Integrating by parts (once for $ n \in ( 1/2 , 3/2 ] $ and
twice for $ n > 3/2 $), one easily verifies that $$ \int_0^t e^{ i
(s - k_0) \tau } \tau^{ n - 1/2 - \von } d \tau = \frac 1{ i
(s-k_0) } \left[ e^{ i(s-k_0)t } t^{ n - 1/2 - \von } + O \( t^{
\max \{ 0 , n - 3/2 - \von \} } \) \right] $$ uniformly in $ s \in
\omega $. By the functional calculus, this implies the result.
\hfill \endpr
Substituting $ \tau \mapsto \tau/t $ in (\ref{inau}) we obtain
another asymptotic representation which describes the behaviour of
the solution in a vicinity of the spectral singularity.
\be\la{inside} Z_{ -t } u = e^{ -i A t } t^{ n + 1/2 - \von }
\theta ( ( A - k_0 ) t ) \alpha e_0 + O ( 1 ) \ee where $ \theta
$ is an entire function given by the formula\footnote{$ \theta $
is obviously bounded on the real axis, so the operator in the
right hand side defined by the functional calculus is bounded.} $
\theta ( s ) = \int_0^1 e^{ i s \tau } \tau^{ n - 1/2 - \von } d
\tau $.
This theorem shows, in particular, that the contribution of
arbitrary small neighborhood of the spectral singularity in the
asymptotic of $ Z_{ -t } u $ is of the maximal order, $ t^{ n -
\von } $, in norm. For $ n > 1 $ this is immediate, and for $ n
\in [ 1/2, 1 ] $ this follows from
{\bf Theorem 8a.} {\it Assume $ L $ is an operator as in theorem
8. Then for any spectral singularity $ k_0 \in \R $ of an order $
n^\prime \le 1/2 $ and any $ \von > 0 $ small enough there exists
a $ u \in \cN_e $ satisfying (\ref{exactness}) and such that for
any closed interval $ \omega $, $ k_0 \notin \omega $, $$ P_\omega
Z_{ -t } u = O ( 1 ) . $$
Proof.} The representation (\ref{alphau}) holds in the situation
under consideration if understood as an equality a. e. in $ t > 0
$. To verify this, it is enough to notice that for any $ t > 0 $
$$ \int_\R e^{ -ikt } h_\delta ( k ) \, dk
\mathop{\longrightarrow} \limits_{\delta \to 0} C t^{ n^\prime -
1/2 - \von } e^{ -i k_0 t } , \ \ C = \int_\R e^{ -ik } k^{ -1/2 -
n^\prime + \von } \, dk , $$ and this limit is (locally) uniform
in $ t $, and apply an approximation argument to (\ref{exp}). The
only difference is that $ r_\delta $, although converges in $ L^2
$, does not belong to $ L^1 $. Then, the integral $ \int_0^t e^{
is \tau } \tau^{ n^\prime - 1/2 - \von } d \tau $ is obviously
uniformly bounded in $ s $ and $ t $ for $ t \ge 1 $ and $ s \ge
s_0 > 0 $. By the functional calculus, the result follows. \hfill
\endpr
Another corollary of theorem 8 is the fact that the estimate
(\ref{estP}) with $ n = p \ge 1/2 $ cannot be improved in the
power scale {\it for any operator $ L $} of the class under
consideration. Indeed, otherwise $ \Psi_{ k_0 } = 0 $, for a $ k_0
$, a singularity of the order $ p $, and thus $ P_\omega \alpha
e_0 = 0 $ for any $ \omega $, $ \omega \bigcap \sigma_0 =
\emptyset $. This implies that $ \alpha e_0 \in \Ran P_{ \{
\sigma_0 \} } $, that is, $ \alpha e_0 = \sum c_j u_j $, $ u_j \in
\ker ( A - k_j ) $. By (\ref{inside}) we obtain that $$ Z_{ -t } u
= t^{ p + 1/2 - \von } \sum_j c_j \theta ( ( k_j - k_0 )t ) e^{
-i k_j t } u_j + O ( 1 ) . $$ Since $ \theta ( a t ) = O \( t^{ -1
} \) $ for $ a \ne 0 $, the leading term at large $t$ is given by
the term with $ j = 0 $, which is $ t^{ p + 1/2 - \von } c_0
\theta ( 0 ) e^{ -i k_0 t } u_0 $. But then the upper estimate for
the norm of $ Z_{ -t } $ implies that $ c_0 = 0 $, for $ \theta (
0 ) \ne 0 $. Thus, we have obtained a contradiction because the
terms corresponding to $ j \ne 0 $ are of the order $ t^{ p - 1/2
- \von } $ which is incompatible with the asymptotic
(\ref{exactness}).
Combining the results obtained, let us formulate separately the
solution to the localization problem for spectral singularities of
higher orders in terms of the local norms $ \| P_\omega Z_{ -t }
\| $. First, we determine using corollary 2 and theorem 3 the
maximal order, $ p $. Assume first that $ p \ne 1/2 $. Consider a
closed interval $ \omega \subset \R $. If $ \len \left. P_\omega
Z_{-t} \right|_{ \cN_e } \rin \le C \( 1 + t^{p^\prime } \) $ for
a $ p^\prime \ge 0 $, then the interior of $ \omega $ does not
contain a singularity of an order greater than $ p^\prime $. In
the case when $ \len \left. P_\omega Z_{-t} \right|_{ \cN_e } \rin
\le C_\omega $ ($ p^\prime = 0 $) this means that the interior of
$ \omega $ does not contain singularities at all. Then, theorem 6a
shows that if the estimate above holds with a positive $ p^\prime
> p - 1/2 $, and this order cannot be made smaller, then $ \omega
$ contains a singularity of the order $ p^\prime $. In the case $
p = 1/2 $ the only difference is that (\ref{estPgen}) holds with $
1 $ changed by $ \ln t $ in the parentheses (see (\ref{except})).
We thus have
{\bf Corollary 9.} {\it Let a dissipative operator $ L = A + iV $,
$ V \ge 0 $, have at most finitely many spectral singularities,
each of a finite power order in the strong sense, and $ p $ be the
maximal order. Suppose that $ \sqrt V $ is $ A $ - smooth. For a
closed interval $ \omega \subset \R $ define the number $ p (
\omega ) = \inf_{ p^\prime \ge 0 } \{ p^\prime : \len \left.
P_\omega Z_{-t} \right|_{ \cN_e } \rin \le C (p^\prime ) \( 1 +
t^{p^\prime } \) \} $. Then
(i) $ \omega $ contains a singularity of the order $ p ( \omega )
$ if $ p ( \omega ) > p - 1/2 $, provided that $ p ( \omega ) \ne
0 $,
(ii) The interior of $ \omega $ does not contain a singularity of
an order greater than $ p ( \omega ) $.}
This result comprises an algorithmic solution to the problem of
localization of spectral singularities of orders greater than $
\max\{ 0, p - 1/2 \} $. The separation of singularities of lower
orders cannot be achieved in this way since, as it is seen from
(\ref{nosmasab}), the norm estimate for an interval containing
such singularities is determined by the contribution of
singularities outside the interval. The appearance of spectral
projections for the real part of operator in the procedure for
localization, however, suggests that one can localize the
singularities of lower orders in terms of time asymptotic if we
increase the smoothness of the perturbation. This problem will be
considered in the forthcoming papers of the series.
\bigskip
{\bf \S3}. As an example we consider an operator arising from the
one - speed transport theory\footnote{The detailed exposition of
the results listed below can be found in the paper \cite{KNR}.}.
Let $ d \Omega_p $ be the Lebesgue measure on the unit sphere $
{\mathbb S}^2 $. Given a nonnegative compactly supported function
$ c \in L^\infty \( \R^3 \) $, define the operator, $T$, in the
Hilbert space $ H = L^2 ( \R^3 \times {\mathbb S}^2 ) $ by the
formula ($ q \in \R^3, \hp \in {\mathbb S}^2 $) $$ T = i \hp \cdot
\nabla_q + i c(q) K, \hspace{.4cm} K = \frac 1{4 \pi}
\int_{{\mathbb S}^2} \cdot \ d \Omega_p $$ on the natural domain
of its real part. Then $ T $ is a maximal dissipative operator,
and the essential spectrum of $ T $ coincides with the real axis.
The characteristic function, $ S $, of the operator $ T $ is
analytic on the real axis, and $ S (z) - I $ is a compact
operator. Let $ H_{ess} $ be the invariant subspace of $ T $
corresponding to $ \sigma_{ess} ( T )$. Define the set $ \cE
\subset L^\infty \( \R^3\) $ as $\cE = \{ c \ : \ \ker ( I + Q (0)
)\neq 0 \} $ where $ Q ( 0 ) $ is the integral operator in $ L^2 (
\R^3 ) $ with the kernel $ - \frac 1{4\pi } \sqrt{c(q)} \frac
1{\left| q^\prime -q\right|^2 } \sqrt{c( q^\prime)} $.
{\bf Theorem.} {\it \cite{KNR} The completely nonself-adjoint part
of the restriction, $ T_{ess} $, of $ T $ to $ H_{ess} $ has
purely absolutely continuous spectrum. The function $ S $
satisfies the condition of absence of a spectral singularity at
infinity from corollary 2. If $ c \notin \cE $ then the operator $
T_{ ess } $ is similar to a self-adjoint operator. If $ c \in \cE
$ then the operator $ T $ has the unique spectral singularity of
the order $ 1 $ located at the point $ 0 $.}
Applying corollary 2 and theorem 3 one obtains the following
result announced in \cite{KNR}.
{\bf Corollary.} {\it Let $ Z_t^e = \left. \exp ( - i T t )
\right|_{ H_{ess} } $, $ t \ge 0 $. If $ c \in \cE $ then $$ \|
Z_t^e \| \le C ( 1 + t ) , $$ and this estimate is exact in the
sense given by theorem 3.}
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\end{document}
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