\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline
{\em Electronic Journal of Differential Equations},
Conference 13, 2005, pp. 49--56.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{49}
\begin{document}
\title[\hfilneg EJDE/Conf/13 \hfil Friedrichs model operators]
{Friedrichs model operators of absolute type with one singular point}
\author[S. I. Iakovlev \hfil EJDE/Conf/13 \hfilneg]
{Serguei I. Iakovlev}
\address{Departamento de Matematicas\\
Universidad Simon Bolivar\\
Apartado Postal 89000, Caracas 1080-A, Venezuela\hfill\break
fax +58(212)-906-3278, tel. +58(212)-906-3287}
\email{iakovlev@mail.ru serguei@usb.ve}
\date{}
\thanks{Published May 30, 2005.}
\subjclass[2000]{47B06, 47B25}
\keywords{Analytic functions; eigenvalues; Friedrichs model;
linear system;\hfill\break\indent
modulus of continuity; selfadjoint operators; singular point}
\begin{abstract}
Problems of existence of the singular spectrum on the continuous
spectrum emerges in some mathematical aspects of quantum
scattering theory and quantum solid physics.
In the latter field, this phenomenon results from physical
effects such as the Anderson transitions in dielectrics.
In the study of this problem, selfadjoint Friedrichs model
operators play an important part and constitute
quite an apt model of real quantum Hamiltonians.
The Friedrichs model and the Schr\"{o}dinger operator are
related via the integral Fourier transformation.
Similarly, the relationship between the Friedrichs model and
the one dimensional discrete Schr\"odinger operator on
$\mathbb{Z}$ is established with the help of the Fourier series.
We consider a family of selfadjoint operators of the Friedrichs model.
These absolute type operators have one singular point $t=0$ of
positive order. We find conditions that guarantee the absence of
point spectrum and the singular continuous spectrum for such operators
near the origin. These conditions are actually necessary and sufficient.
They depend on the finiteness of the rank of a perturbation operator
and on the order of singularity. The sharpness of these conditions
is confirmed by counterexamples.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{proposition}
\section{Introduction}
Problem of existence of the singular spectrum on the continuous spectrum
emerges in some mathematical aspects of quantum scattering
theory and quantum solid physics. In the latter field
this phenomenon results from physical effects such as the Anderson
transitions in dielectrics. Note that we understand the singular
spectrum as the union of the point spectrum and the singular continuous
spectrum. In the study of this problem an important part is played
by the selfadjoint Friedrichs model operator
$S_1:=t\cdot\,+ V$ acting in $L_2(\mathbb{R})$
(where $t\cdot$ stands for the operator of multiplication by the
independent variable $t\in \mathbb{R}$, and $V$ is an integral
operator with a continuous Hermitian kernel). This operator
constitutes quite an apt model of real quantum Hamiltonians.
In \cite{f1} it was shown how the Friedrichs model can be used
for the study of the spectral properties of the Schr\"odinger
operator $(-\Delta+q)$. These operators are related via
the integral Fourier transformation.
A large body of literature is devoted to this model; we mention the papers
by Faddeev, Pavlov, Naboko, Iakovlev and others
\cite{d1, f1, p1, m1, n1, n2, n3, y1, y2}.
For the first time the fact that here the singular
spectrum may arise indeed was established by
Pavlov and Petras (1970) in \cite{p1}. Radically new conditions on $V$
guaranteeing the finiteness of the point spectrum of $S_1$ (the
singular continuous spectrum is missing) have been found in the
papers \cite{d1,n3}. Since, actually, these conditions are necessary and
sufficient in the context of the selfadjoint Friedrichs model the
problem in question was solved completely in \cite{d1}.
Further elaboration of this topic seems to be of value. Namely, it is
of interest to investigate the singular spectrum of perturbations
of the operators of multiplication by a function $f(t)$ of the
independent variable (for example, $f(t)$ is equal to $\cos t$ or
$t^2$). Such operators naturally appear when various models of
the Schr\"odinger operator are considered in a momentum
representation. For example, the operator of multiplication by
$t^2$ is obtained if we write the Schr\"odinger operator in a
momentum representation. Similarly, the relationship between the
Friedrichs model and the one dimensional discrete Schr\"odinger
operator $S$ on $\mathbb{Z}$ is established with the help of the
Fourier series. The operator $S$ is equal to
$(U+U^{\ast})+q $ and is defined on the space
$l_2({\mathbb{Z}})$ of square summable complex
sequences $u=\{u_n\}_{n=-\infty}^{+\infty} $; here $U$
is the operator of right shift, $U^*$ is its adjoint, and
$q=\{ q_n\}_{-\infty}^{\infty}$, so that
$(Uu)_n=u_{n-1}$, $(U^*u)_n=u_{n+1}$, and
${(q\,u)}_n=q_n\cdot u_n $ \cite{n4,n5}. Under the isomorphism between
$l_2({\mathbb{Z}})$ and $ L_2(-\pi,\pi)$ given by the
map
\begin{equation}\label{1.1}
\Phi^{-1}:u\to \tilde u(t)=\sum_ {n=-\infty}^{+\infty}u_n\cdot
e^{\imath nt},
\end{equation}
the operator $S$ turns into $\tilde S$ acting by the formula
\begin{equation}\label{1.2}
\tilde S\tilde u(t)=2\cos (t)\cdot\tilde u(t)+
\int_{-\pi}^{\pi}\tilde q(t-x)\cdot\tilde u(x)\,dx ,
\end{equation}
where $\tilde q(t)=\sum_nq_n\cdot e^{\imath nt}$, $\tilde u\in
L_2(-\pi,\pi)$. Indeed,
\begin{equation}\label{1.3}
\begin{aligned}
(\tilde S\tilde u) = &\Phi^{-1} \left[ ( U+U^*) +
q\right] u =
\sum_{n=-\infty}^{+\infty}( u_{n-1}+u_{n+1}+q_nu_n)e^{\imath
nt}\\
=& \sum\nolimits_n u_n e^{\imath (n+1)t} + \sum\nolimits_n u_n
e^{\imath (n-1)t} + \sum\nolimits_nq_nu_n e^{\imath nt}\\
=& 2\cos (t) \sum\nolimits_n u_n e^{\imath nt} +
\int_{-\pi}^{\pi}\tilde q(t-x)\cdot\tilde u(x)\,dx \,.
\end{aligned}
\end{equation}
Obviously, that the change of variables $2\cos t=x$ would reduce
the study of $\sigma_{sing}({\tilde S})$, the singular spectrum of
the operator ${\tilde S}$, to that of
$\sigma_{sing}(S_1)$. However, since $(\cos t)'= -\sin
t\vert_{\pm\pi}=0$, this substitution is not smooth (that is, not
diffeomorphism) near the points $\pm\pi$ and, therefore, may lead
to a loss of subtle information concerning the structure of
$\sigma_{sing}({\tilde S})$. It is clear that, having an idea of
the structure of the set $\sigma_{sing}(S_1)$, we can deduce some
information about the set $\sigma_{sing}({\tilde S})$ (also near
the singular points $t=\pm\pi$) with the help of the change of
variables, but the results obtained in this way will be expressed
in inconvenient terms, and their sharpness near zero will be less
than satisfactory.
Thus, as a model in the theory of continuous spectrum
perturbations it seems reasonable to consider the perturbations
not of the operator of multiplication by the independent variable
$t$, but of the operator of multiplication by a function of $t$.
In this case the main attention must be paid to the singular spectrum
in a neighborhood of so called singular points next to which it is
impossible to introduce a smooth (locally) change of variables
reducing our problem to the standard Friedrichs model. It will be
shown that in a neighborhood of such points the behavior of the
singular spectrum acquires a quite different character.
The following two functions $f_1(t)=|t|^m$ and $f_2(t)=\mathop{\rm
sgn}{t}\cdot |t|^m$ have one zero of order $m>0$ at the point
$t=0$. And near the origin $f_1$ and $f_2$ have a different
behavior. To these functions there correspond the selfadjoint
Friedrichs model operators $A_m,\, m>0,$ with one singular point
$t=0$
\begin{equation}\label{1.4}
A_m = |t|^m\cdot + V \qquad \textrm{(the absolute type
operators)\, ,}
\end{equation}
and the operators $S_m$
\begin{equation}\label{1.5}
S_m = \mathop{\rm sgn}{t}\cdot |t|^m\cdot + V \qquad \textrm{(the
symmetric type operators)}
\end{equation}
also with one singular point $t=0$ for $m\neq 1$. The operator
$S_1=t\cdot + V$ is the main operator of the Friedrichs model. It
has no singular points. In this paper we study the case of the
operators $A_m$. The operators $S_m\, ,m\ne 1$, were partially
considered in \cite{i1}.
\section{Statement of the problem and main result.}
In $L_2(\mathbb{R})$ we consider a family of selfadjoint operators
$A_m\, , m>0$, given by
\begin{equation}\label{2.1}
A_m = |t|^m\cdot + V\,.
\end{equation}
Here $|t|^m\cdot$ is the operator of multiplication by the
function $|t|^m$ of the independent variable $t\in\mathbb{R}$,and
$V$ (perturbation) is an integral operator with a continuous
Hermitian kernel $v(t,x)$. Thus, the action of the operator $A_m$
can be written as follows
\begin{equation}\label{2.2}
\bigl(A_m u\bigr)(t) = |t|^m\cdot u(t) +
\int_{\mathbb{R}}v(t,x)u(x)\>dx\,.
\end{equation}
We assume that $V$ is non-negative and belongs to the trace class
$\sigma_1$~:
\begin{equation}\label{2.3}
V\geq 0\,,\quad V\in\sigma_1 \,.
\end{equation}
Consequently, the operator $A_m$ is defined on the domain of
functions $u(t)\in L_2({\mathbb{R}})$ such that
$|t|^{m}u(t)\in L_2({\mathbb{R}})$. The kernel $v(t,x)$
is assumed to satisfy the following smoothness condition
\begin{equation}\label{2.4}
v(t+h,t+h)+v(t,t)-v(t+h,t)-v(t,t+h)\leq \omega^2(\vert
h\vert), \quad \vert h\vert \leq 1\, ,
\end{equation}
\noindent with the function $\omega (t)$ (the modulus of
continuity of $V$) monotone and satisfying a Dini condition:
\begin{equation}\label{2.5}
\omega (t)\downarrow 0 \quad \hbox{as } t\downarrow 0\,,
\quad\hbox{and}\quad \int_0^1 \frac{\omega (t)}{t}\, dt <\infty \,.
\end{equation}
Inequality~\eqref{2.4} may be regarded as a
smoothness condition for the kernel $v_{1/2}(t,x)$ of the integral
operator $V^{1/2}$, because, as shown in \cite{n2}, the expression on
the left in~\eqref{2.4} can be written as the integral $
\int_{\mathbb{R}}\vert v_{1/2}(t+h,x)-v_{1/2}(t,x)\vert^2\> dx \,
$ (and, therefore, is nonnegative). Together with~\eqref{2.4} the
fact that $V$ is of class $\sigma_1$ means that the kernel
$v(t,x)$satisfies a certain condition of decrease at infinity. The
requirement that the operator $V$ be of trace class $\sigma_1$ is
sufficient for the absolutely continuous spectrum of $A_m$ to
coincide with the real semi-axis $\mathbb{R_+}=[0\, ,+\infty)$
(see \cite{r1}).
Near the singular point $t=0$ we study the dependence of the
behavior of the point and singular continuous spectrum on the
smoothness of the kernel $v(t,x)$ . As noted above the structure
of the singular spectrum $\sigma_{sing}(S_1)$ of the operator
$S_1=t\cdot +V$ (the usual Friedrichs model operator) is pretty
well studied \cite{d1,f1,m1,n1,n2,n3,p1,y1,y2}. In particular, in
the papers \cite{n3,d1} it was shown that for this operator there
exists a sharp condition of finiteness of the singular spectrum.
Namely, if $\omega(t)=O(\sqrt t)$ as $t\to 0 $, the singular
spectrum of $S_1$ consists of at most a finite number of
eigenvalues of finite multiplicity (the singular continuous
spectrum is absent). On the other hand, if
$\overline{\lim_{t\to0}}\omega (t)/\sqrt t=+\infty $, then
examples are constructed showing that even in the case when $V$ is
a rank 1 perturbation the eigenvalues of $S_1$ may have cluster
points. By using the simple change of variables $|t|^m=x$, we can
show that outside any neighborhood of the origin on the interval
$[0,+\infty)$ the structure of the spectrum $\sigma_{sing}(A_m)$
is locally identical with that of the operator $S_1$. This result
is explained by the smoothness of the above change of variables
outside any neighborhood of the origin, and also by the local
character of the main results of \cite{d1, f1, p1, m1, n1, n2, n3,
y1, y2} relating to the structure of $\sigma_{sing}(S_1)$ . Here
by locality we mean the following. Suppose that conditions
~(\ref{2.4}), (\ref{2.5}) are fulfilled only in some interval
$(c,d) \subset {\mathbb{R}}$, then the main results in \cite{d1,
f1, p1, m1, n1, n2, n3, y1, y2}
about the structure of $\sigma_{sing}(S_1)$
remain true in any closed subinterval $\Delta\subset (c,d)$.
However, as shown in this paper, in a neighborhood of the origin
the behavior of $\sigma_{sing}(A_m)$ is quite different. Here,
near zero, we can still use the change of variables $|t|^m=x$
mentioned above, but, since, e.g., $(|t|^m)^\prime_{\mid_{0}}=0$
for $m>1$, this change is not smooth (that is, not a
diffeomorphism) near zero. In this sense the zero point is a
singular point of the operators $A_m\, , m>0,$ so it needs a
special inspection. Observe that the origin is also a boundary
point of the continuous spectrum of $A_m$, which coincides with
the interval $[0,+\infty)$.
Naturally, there appears a problem of finding sharp, in a sense,
conditions on the kernel $v(t,x)$ that guarantee that the singular
spectrum is absent near the origin. In this paper it is shown
that such sufficient conditions are given in terms of
asymptotic behavior of the modulus of continuity $\omega(t)$ as
$t$ tends to zero. It appears that for $m\in (\, 1\, , 3]$ these conditions
also depend on a rank of the perturbation operator $V$. Namely,
if $\mathop{\rm rank} V <\infty$, then provided that
$\omega(t)=O(t^{(m-1)/2})$, $t\to 0,$ the spectrum near zero
is purely absolutely continuous. But if $\mathop{\rm rank} V=+\infty$, then
the structure of $\sigma_{sing}(A_m)$ depends on the value of a
constant $C$ in the condition $\omega(t)=Ct^{(m-1)/2}$.
The sharpness of these conditions is confirmed by
counterexamples. For $m\leq 1$ the spectrum is always purely
absolutely continuous in some neighborhood of the zero point
on the interval $[0,+\infty)$. At the same time for $m>3$
the singular spectrum may appear near zero for any modulus of
continuity $\omega(t)$. Hence, for $m>3$ near zero there is no
condition of the singular spectrum absence in terms of $\omega(t)$
as for $m\in (\, 1\, , 3]$.
In Sections 3, 4 the main results of the paper are formulated.
In Section 3 sufficient conditions on the perturbation $V$
are given ensuring the singular spectrum absence near the
origin. Counterexamples constructed in Section 4 show that
these conditions are sharp.
Note that some results of this
paper (for the case $m\in \mathbb{N}$) were announced in
\cite{y3}, and the case of $m=2$ has been in detail considered in
\cite{y4}.
\section{Sufficient conditions for absolute
continuity of the spectrum on $[0,+\infty)$ near zero}
For $z\in {\mathbb{C}}\setminus [0,+\infty)$ we define
an analytic operator--valued function $T_m(z): E\to E$, where
$E:=\overline {R(V)}$ is the closure of the range of
$V$, as follows:
\begin{equation}\label{3.5}
T_m(z):=-\sqrt V\, (|t|^m-z)^{-1}\,\sqrt V\,.
\end{equation}
Here $(|t|^m -z)^{-1}$ denotes the
operator of multiplication by the corresponding function in
$L_2({\mathbb{R}})$. Obviously, that $\mathop{\rm Im} T_m(z)\geq
0$ if $\mathop{\rm Im} z >0$, and $T_m(z)\in \sigma_1$.
\begin{proposition}\label{pr3.1}
If $V$ satisfies conditions~\eqref{2.3}--\eqref{2.5}, then in the
complex plane cut along $[0,+\infty)$ the analytic
operator--valued function $T_m(z)$ admits a $\sigma_1$--norm
continuous extension to the upper and the lower parts of the cut
on the interval $(\,0\,;+\infty)$.
\end{proposition}
Let $T_m(\lambda):=T_m(\lambda+i0),\, \lambda>0$, denote
the corresponding boundary values of $T_m(z)$.
The set $N_m:=\{\lambda >0:
\exists g\in l_2 , g\ne 0 , T_m(\lambda)g=g\}
\equiv\{\lambda>0 : \ker (I-T_m(\lambda)) \neq \emptyset \}$ is
called a set of roots of the operator--function $T_m$. The vector
$g$ is called a root vector corresponding to the root $\lambda$.
\begin{proposition}\label{pr3.2}
If $V$ satisfies conditions~\eqref{2.3}--~\eqref{2.5}, then
$\sigma_{sing}(A_m)$, the singular spectrum of the operator $A_m =
|t|^m\cdot+ V\, ,m>0,$ embeds into the set $N_m$ supplemented
by the origin, i.e.,
\begin{equation}\label{3.12}
\sigma_{sing}({A_m}) =
\overline{\sigma_p({A_m})}\cup
\sigma_{s.c.}({A_m}) \subseteq N_m\cup\{0\}\, ,
\end{equation}
where $\sigma_{p}({A_m})$ is the point spectrum, and
$\sigma_{s.c.}({A_m})$ is the singular continuous
spectrum of the selfadjoint operator ${A_m}$.
\end{proposition}
From the Fredholm analytic alternative (see \cite[\S 8]{y0}) it follows
that the set $N_m\cup\{0\}\subset{\mathbb{R}}$ is a closed set of
Lebesgue measure zero. Also \cite[Theorem 3]{y4} says that under the
condition $V\geq 0$ the point $0$ is not an eigenvalue of the
operator $A_m = |t|^m\cdot+ V$. Below some conditions on the
modulus of continuity $\omega(t)$ of the perturbation operator $V$
are given guaranteeing the absolute continuity of the spectrum of
the operator $A_m$ on the interval $[0 ,+\infty)$ near zero.
For $m\in (1,3]$ these conditions depend on a rank of
the operator $V$.
\begin{theorem}\label{thm4.1}
Suppose that conditions~\eqref{2.3}--\eqref{2.5} are fulfilled.
Then for $m\in (0;1]$ the roots set $N_m$ is empty in some
neighborhood of the origin. And, hence, the spectrum of the
operator $A_m=|t|^m\cdot+ V$, defined by~\eqref{2.2}, is
purely absolutely continuous in some neighborhood of the origin on
the interval $[0,+\infty)$.
\end{theorem}
\begin{theorem}\label{thm4.2}
Suppose that the perturbation $V$ satisfies
conditions~\eqref{2.3}--~\eqref{2.5} with the function
$\omega(t)=C_{\omega}t^{\alpha}$, where $\alpha =(m-1)/2$, and
$m\in (1;3]$. If
\begin{equation}\label{4.6}
C_{\omega}< C_m:=( 2\int_0^1\frac{(1-x)^{m-1}}{1-x^m}dx)^{-1/2}\,,
\end{equation}
then the roots set $N_m$ is empty in some neighborhood of the
origin. Consequently, the spectrum of the operator
$A_m=|t|^m\cdot+ V$, defined by~\eqref{2.2}, is purely
absolutely continuous in some neighborhood of the origin on the
interval $[0,+\infty)$.
\end{theorem}
Note:~~Clearly that for the modulus of continuity
$\omega(t)=C_{\omega}t^{\alpha}$ the greatest possible value of
$\alpha$ is $1$. The value $\alpha=1$ exactly corresponds to $m=3$.
Observation:~~It is not difficult to obtain for the constant
$C_m$ a two--sided estimate. Indeed, since for $m>1$ and $x\in
[0,1]$
\begin{equation}\label{4.10}
1-x\leq 1-x^m\leq m(1-x)\, ,
\end{equation}
we have
\begin{equation}\label{4.11}
\frac{1}{m}\int_0^1\frac{(1-x)^{m-1}}{1-x}\, dx\leq
\int_0^1\frac{(1-x)^{m-1}}{1-x^m}\, dx\leq
\int_0^1\frac{(1-x)^{m-1}}{1-x}\, dx\,.
\end{equation}
Whence
\begin{equation}\label{4.12}
(\frac{m-1}{2})^{1/2}\; \leq\; C_m \leq
( \frac{m(m-1)}{2})^{1/2}\,.
\end{equation}
At the same time for $m=2$ and $m=3$ the integral
$\int_0^1{(1-t)^{m-1}}/{(1-t^m)}\, dt$ is evaluated exactly. For
$m=2$ we obtain $C_2=(1/\ln4)^{1/2}=0,849\dots$ that
coincides with the value of this constant from the paper \cite[Theorem 1]{y4}.
Likewise, we find that $C_3=(\pi/\sqrt{3}-\ln3)^{-1/2}=1,182\dots$.
Note also that from~\eqref{4.12} it follows immediately that
$C_m\to +0$ as $m\to 1^+$ (compare that with the assertion of
Theorem~\ref{thm4.1}).
%\end{proof}
If the perturbation $V$ is a finite rank operator, the result of
Theorem \ref{thm4.2} can be improved.
\begin{theorem}\label{thm4.3}
Suppose that the perturbation operator $V$ satisfies
conditions \eqref{2.3}--\eqref{2.5} and $\mathop{\rm rank} V<\infty$. If
$m\in (1;3]$ and $\omega(t)= O(t^{(m-1)/2})$ as $t\to
0^+$, then the origin is not a cluster point of the set of roots
$N_m$ of the operator-valued function $T_m$. Consequently, the
spectrum of the operator $A_m = |t|^m\cdot + V$, defined
by~\eqref{2.2}, is purely absolutely continuous in some
neighborhood of the origin on the interval $[0 ,+\infty)$.
\end{theorem}
\section{Sharpness of the absence conditions
for the singular spectrum: Counterexamples}
The following theorem states that for
an infinite rank perturbation ($\mathop{\rm rank} V=\infty$)
the absence condition for the singular spectrum of $A_m$,
$m\in (1;3]$, near the origin is indeed related to the constant
$C_\omega$ by $\omega(t)=C_\omega\cdot t^{(m-1)/2}$ as
$t\to 0$, (see Theorem~\ref{thm4.2}).
In particular, this means that the result
of Theorem~\ref{thm4.3} cannot be extended to perturbations of infinite rank.
Namely, Theorem~\ref{thm4.2}
is sharp in the following sense.
\begin{theorem}\label{thm5.1}
Let $m\in (1; 3]$. For any value of
$C_{\omega}\ge\Tilde{C}_m:=2^{(m-1)/2}m$ there exists an operator
$V$ with $\mathop{\rm rank} V=\infty$, such that $V$ satisfies
conditions~\eqref{2.3}--\eqref{2.5} with
$\omega(t)=C_{\omega}t^{(m-1)/2}$, and the origin is a cluster
point of the set of eigenvalues of the operator $A_m=|t|^m\cdot + V$, defined
by~\eqref{2.2}.
\end{theorem}
Note: It is easy to verify that
$\Tilde{C}_m=2^{(m-1)/2}m\ge({m(m-1)/2})^{1/2}\ge C_m$ for $m>1$.
If $\mathop{\rm rank} V=1$, then $V=(\cdot,\varphi)\varphi$ with
$\varphi\in L_2(\mathbb{R})$. In this case the smoothness
condition~\eqref{2.4} is written in the form
\begin{equation}\label{6.1}
\vert \varphi (t+h)-\varphi (t)\vert\leq \omega\bigl
(\vert h\vert\bigr) ,\quad \vert h\vert\leq 1\,,
\end{equation}
with the function $\omega(t)$ satisfying condition \eqref{2.5}.
Note that for any function $\varphi(t)$ its actual modulus of
continuity $\widetilde{\omega}(h):=\sup\{ |\varphi (x)-\varphi (y)|:|x-y|1$. Suppose that $\omega(t)$, $t\ge 0$, is a monotone
nondecreasing function satisfying the condition
$\omega(0^+)=\omega(0)=0$ as well as the natural additional
condition of semiadditivity:
$\omega(t_1+t_2)\leq{\omega}(t_1)+{\omega}(t_2)$ for all
$t_1, t_2\geq 0$. If
$\limsup_{t\to 0} \omega(t)/t^{(m-1)/2}=+\infty$, then a compactly supported
function $\varphi:\mathbb{R}\to \mathbb{R}$ satisfying
condition~\eqref{6.1} is constructed and such that the operator
$A_m=|t|^m\cdot+ (\cdot,\varphi)\varphi$ has a sequence of positive
eigenvalues converging to zero.
\end{theorem}
\begin{corollary}\label{cor6.1}
It is not hard to show (see \cite[Lemma 2.2]{i1}) that if $\omega(t)$
is a nonnegative semiadditive function and $\omega(t)\downarrow 0$
as $t\downarrow 0$, then for any $a>0$ there exists a constant
$C>0$ such that $Ct\le \omega(t)$ for $t\in [0,a]$. Hence,
\[
\limsup_{t\to 0} \omega(t)/t^{(m-1)/2}=+\infty
\]
for all $m>3$. Therefore, it follows from Theorem~\ref{thm6.1} that for every
$m>3$ and for each monotone and semiadditive function
$\omega(t)$, $t\ge 0,$ satisfying the condition
$\omega(0^+)=\omega(0)=0$ (and thus nonnegative) a real--valued
compactly supported function $\varphi$ is constructed satisfying the
smoothness condition~\eqref{6.1} and such that the operator
$A_m=|t|^m\cdot + (\cdot,\varphi)\varphi$ has a sequence of positive
eigenvalues converging to zero. This means, in particular, that
for $m>3$ there is no condition guaranteeing the absence of the
singular spectrum of the operator $A_m=|t|^m\cdot + V$ near
the origin in terms of the modulus of continuity $\omega(t)$ of
the perturbation $V$.
\end{corollary}
\begin{corollary}\label{cor6.}
If $m\in(1,3]$, then, according to Theorem~\ref{thm6.1}, the
sufficient condition $\omega(t)=O(t^{(m-1)/2})$ as $t\to 0$
guaranteeing the absence of the singular spectrum of the operator
$A_m$ near the origin for the finite rank perturbation operator,
$\mathop{\rm rank} V< \infty$, (see Theorem~\ref{thm4.3}) is sharp. If this
condition is not fulfilled, that is,
$\limsup_{t\to 0}\omega(t)/t^{(m-1)/2}=+\infty$, then even
in the case when $V$ is a rank 1 perturbation there can exist
nontrivial singular spectrum near zero, and, in particular,
the operator $A_m$ can have a sequence of positive eigenvalues
converging to zero.
\end{corollary}
\subsection*{Acknowledgements}
The author thanks Professor S. N. Naboko for his
attention to this work.
\begin{thebibliography}{00}
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\end{document}