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Spectral shift function, trace formula, Scrodinger operator, polyharmonic operator
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\begin{document}

\title{Estimates for the spectral shift function of the polyharmonic
operator}
\date{}

\author{Alexander Pushnitski
\thanks{School of Mathematical Sciences, 
University of Sussex, Falmer, 
East Sussex BN1 9QH, UK.
{\it E-mail:} a.pushnitski@sussex.ac.uk}
 }

\maketitle

\begin{abstract}
The Lifshits--Krein spectral shift function is considered for the pair of
operators $H_0=(-\laplace)^l$, $l>0$ and $H=H_0+V$ in $L_2(\R^d)$,
$d\geq1$; here $V$ is a multiplication operator.
The estimates for this spectral shift function $\xi(\l;H,H_0)$
are obtained in terms of the spectral parameter $\l>0$ and
the integral norms of $V$. These estimates are in a good agreement with
the ones predicted by the classical phase space volume considerations.
\end{abstract}


%\setcounter{section}{-1}
%=====================================================
\section{Introduction}
%=====================================================
\setcounter{equation}{0}


The main object of study of
this paper is the
I.~M.~Lifshits--M.~G.~Krein {\it spectral shift function}
(SSF). For an exposition of the SSF theory, see e.g.,
\cite{BYa} or \cite{Ya}.
For a general pair of selfadjoint operators $H_1$, $H_2$
in a Hilbert space $\H$,
satisfying some trace class condition (see \S2.3 below),
the SSF $\xi(\l;H_2,H_1)$ appears in connection
with the {\it trace formula} \cite{Lifshits,Krein}:
$$
\Tr
(\psi(H_2)-\psi(H_1))=\int_{-\infty}^\infty\xi(\l;H_2,H_1)\psi'(\l)d\l,
\quad\psi\in C_0^\infty(\R).
$$
Besides, the SSF is related to the scattering matrix $S(\l)$ for the pair
$H_1$, $H_2$ by the {\it Birman--Krein formula} \cite{BK}:
$$
\det S(\l)=e^{-2\pi i\xi(\l;H_2,H_1)},
$$
for almost every $\l$ on the absolutely continuous spectrum of $H_1$.
This formula allows one to interpret the SSF as the scattering phase.
Moreover, sometimes it is treated as the definition of the SSF.
(Note that in some concrete problems this formula in a slightly
different form appeared in \cite{Lifshits2,BuslaevFaddeev,Buslaev} before
the abstract paper \cite{BK}).


Let $H_0=(-\laplace)^l$, $l>0$ in $L_2(\R^d)$, $d\geq1$, and let $V=V(x)$,
$x\in\R^d$,
be a perturbation potential, which decays sufficiently fast as
$|x|\to\infty$.
Our aim is to obtain pointwise (in $\l>0$) bounds on
$\xi(\l;H_0+V,H_0)$ in terms of $\l$ and integral norms of $V$.
The most interesting case is $l=1$ (Schr\"odinger operator).
Nevertheless, the technique of this paper allows us to
obtain bounds on the SSF in a uniform way for all $l\in(0,\infty)$
(and all dimensions $d$), and thus we consider the problem in its natural
generality.


The SSF is monotonic with respect to a perturbation:
\[
\xi(\l;H_0-V_-,H_0)\leq
\xi(\l;H_0+V,H_0)\leq
\xi(\l;H_0+V_+,H_0),
\label{1.1}
\]
where $V_\pm=(|V|\pm V)/2$. The relations \er{1.1} reduce the problem
of estimating the SSF to the case of perturbations of a definite sign.
Thus, below we shall always assume that $V\geq0$ and consider the pair of
functions $\xi(\l;H_0\pm V,H_0)$.


Let us first consider a ``classical analogue'' of the SSF,
expressed in terms of the phase space volumes,
corresponding to the systems with the Hamiltonians
$h_0(p,x)=p^{2l}$ and $h_\pm(p,x)=p^{2l}\pm V(x)$.
Let
$$
\k=d/(2l)
$$
and let $\o_d$ be the volume of a unit ball in $\R^d$;
define
\bea
\xi^{cl}(\l;H_0+V,H_0)&=&
(2\pi)^{-d}\mbox{vol}\{ (p,x)\in\R^{2d}\mid
h_0(p,x)<\l<h_+(p,x)\}
\nonumber\\
&=&(2\pi)^{-d}
\omega_d\int_{\R^d}(\l^\k-(\l-V(x))_+^\k)dx,\quad \l>0,
\label{1.2}
\\
\xi^{cl}(\l;H_0-V,H_0)&=&-(2\pi)^{-d}
\mbox{vol}\{ (p,x)\in\R^{2d}\mid
h_-(p,x)<\l<h_0(p,x)\}
\nonumber\\
&=&-(2\pi)^{-d}
\omega_d\int_{\R^d}((\l+V(x))^\k-\l^\k)dx,\quad \l>0.
\label{1.3}
\eea
It is well known that $\xi^{cl}(\l;H_0\pm V,H_0)$ behaves in many
respects like $\xi(\l;H_0\pm V,H_0)$; for example, it has the same
asymptotics in most asymptotical regimes ---
see the review \cite{Robert} and references therein.
The integrands in \er{1.2}, \er{1.3} admit the following elementary
bounds:
\bea
\l^\k-(\l-V)_+^\k&\leq&\max\{\k,1\} \l^{\k-1}V,\quad \k>0;
\label{1.4}
\\
(\l+V)^\k-\l^\k&\leq&\k\l^{\k-1}V,\qquad\qquad\quad
\k\leq1;
\label{1.5}
\\
(\l+V)^\k-\l^\k&\leq&
C_1V^\k+C_2\l^{\k-1}V,\quad \k>1.
\label{1.6}
\eea
It is easy to give concrete explicit values for $C_1$, $C_2$ in
\er{1.6}; for example, $C_1=2^\k$, $C_2=2^\k-1$. Substituting
\er{1.4}--\er{1.6} into \er{1.2}, \er{1.3}, we get the following bounds
for $\xi^{cl}$:
\bea
\xi^{cl}(\l;H_0+V,H_0)&\leq&(2\pi)^{-d}
\o_d\max\{\k,1\}
\l^{\k-1}\|V\|_{L_1},\hskip 6.5em
\k>0;
\label{1.7}
\\
|\xi^{cl}(\l;H_0-V,H_0)|&\leq&(2\pi)^{-d}
\o_d\k\l^{\k-1}\|V\|_{L_1},
\hskip 10.5em \k\leq1;
\label{1.8}
\\
|\xi^{cl}(\l;H_0-V,H_0)|&\leq&C_1(2\pi)^{-d}
\o_d\|V\|^\k_{L_\k}
%\nonumber\\
+C_2(2\pi)^{-d}
\o_d\l^{\k-1}\|V\|_{L_1},
\quad\k>1.
\label{1.9}
\eea
Estimates \er{1.7}--\er{1.9} are in good agreement with the asymptotics of
$\xi^{cl}$ for high energy and large coupling constant:
\bea
\xi^{cl}(\l;H_0\pm V,H_0)\sim \pm
(2\pi)^{-d} \o_d\k\l^{\k-1} \int V(x)dx,\quad \l\to\infty,
\label{1.10}
%..................................................................(1.10)
\\
\xi^{cl}(\l;H_0-gV,H_0)\sim
-(2\pi)^{-d} \o_d g^\k \int V^\k(x)dx,\quad g\to\infty.
\label{1.11}
%..................................................................(1.11)
\eea

We consider \er{1.7}--\er{1.9} as the model estimates. Our goal is to
prove some analogues of \er{1.7}--\er{1.9} for the ``real'' SSF.
The estimates for the SSF which we are actually able to prove 
(see \er{2.12}, \er{2.15}, \er{2.16}, \er{2.16b}, \er{2.16d} below)
are slightly
worse than \er{1.7}--\er{1.9}, but they have a correct order in $\l$ and
$V$ (up to logarithmic terms). Our results extend the estimates of
\cite{Sob} and are closely related to the results of \cite{Pus2} --- see
discussion in the end of the next section.

Our technique is based on the new representation for the SSF obtained in
\cite{Pus1}.

%=====================================================
\section{Main results}
%=====================================================
\setcounter{equation}{0}

{\bf 2.1 Notation.}
For a closable linear operator $T$  in a Hilbert space $\H$, by
$\overline{T}$ we denote the closure of $T$.  For a selfadjoint operator
$A$, the symbols $\s(A)$, $\rho(A)$ denote its spectrum and resolvent set;
$R(z,A)=(A-zI)^{-1}$ (for  $z\in\rho(A)$)
and $E_A(\d)$ is the spectral projection
associated to
a Borel set $\d\subset\R$. By $\SS_\infty(\H)$ we denote the
space of all compact operators in $\H$.  For $T=T^*\in\SS_\infty(\H)$ we
introduce the counting functions of the spectrum by $n_\pm(s,T):=\rank
E_{\pm T}((s,+\infty))$, $s>0$.  Note that (see, e.g., \cite{BS-uchebnik})
\[
n_\pm(s_1+s_2,T_1+T_2)\leq n_\pm(s_1,T_1)+n_\pm(s_2,T_2),
\quad s_1,s_2>0.
\label{2.0}
%..................................................................(2.0)
\]
For $p\geq1$ the Neumann--Schatten classes $\SS_p$ are defined
in a usual way:
$$
T\in\SS_p\quad\mbox{if}\quad
\|T\|^p_{\SS_p}:=\sum_ns_n^p(T)<\infty,
$$
where $\{s_n(T)\}$ is a sequence of singular numbers of $T$.

Integral without the domain of
integration explicitly specified implies integration over $\R^d$.
Formulas and statements with double indices ($\pm$ and $\mp$) should be
read as pairs of statements, in one of which all the indices take upper
values and in another --- the lower ones.
By $C(d)$, $C(l)$, etc. (possibly with sub- and superscripts)
we denote various constants which depend only on
$d$, $l$, etc. and whose particular values are of no importance.
A constant which first appears
in formula $(i.j)$ is denoted by $C_{i.j}$.

Remind the definition of the lattice space
$l_1(L_r)\subset L_r(\R^d)$, $r\geq1$:
$$
u\in l_1(L_r)\quad\mbox{if}\quad
\|u\|_{l_1(L_r)}:=\sum_{j\in\Z^d}
\left(\int_{\Q^d+j}|u|^r dx\right)^{1/r}<\infty,
\quad  \Q^d=(0,1)^d\subset\R^d.
$$

Everywhere $\H=L_2(\R^d)$, $d\geq1$ and
$H_0=(-\laplace)^l$, $l>0$.
We shall need a notation for a logarithmic weight function.
Namely, for $x\in\R^d$ and $\a>0$ let
\[
F_\a(x)=1+(\log_+|x|)^\a.
\label{2.0a}
%..................................................................(2.0a)
\]



{\bf 2.2 Assumptions on $V$.}
Throughout the paper we assume that {\it the perturbation potential
$V=V(x)$ is nonnegative: $V\geq0$.}
Besides, in various places we shall use some of  the following assumptions
on $V$:
\bea
V\mbox{ is }H_0\mbox{--form compact;}
%..................................................................(2.1)
\label{2.1}\\
V\in L_1(\R^d);
%..................................................................(2.2)
\label{2.2}
\\
V\in L_\k(\R^d);
%..................................................................(2.3)
\label{2.3}\\
\int V(x)F_\a(x)dx<\infty,\quad \a>2.
%..................................................................(2.4)
\label{2.4}
\eea
Note that for $\k<1$ \er{2.1} follows from \er{2.2}, and for
for $\k>1$ --- from \er{2.3}.
The following assumption will appear only for $\k\geq2$:
\[
\mbox{if }\k\geq2,\mbox{ then }V\in l_1(L_2).
%..................................................................(2.5)
\label{2.5}
\]
{\bf 2.3 Existence of the SSF.}
Let $H_1$, $H_2$ be a pair of selfadjoint operators in a Hilbert space
$\H$ such that
\[
H_2-H_1\in\SS_1(\H).
%...................................................................(2.6)
\label{2.6}
\]
Then the SSF $\xi(\l;H_2,H_1)$ exists and
is given by the {\it Krein formula} \cite{Krein}
\[
\xi(\l;H_2,H_1)=\frac1\pi\lim_{\e\to+0}\arg\D_{H_2/H_1}(\l+i\e),
\mbox{  a.e. } \l\in\R,
%..................................................................(2.6a)
\label{2.6a}
\]
where
$\D_{H_2/H_1}(z)=\det(H_2-zI)(H_1-zI)^{-1}$ and the branch of the
argument is fixed by
$$
\arg\D_{H_2/H_1}(z)\to0 \quad\mbox{as}\quad
\Im z\to +\infty.
$$
In applications, instead of  \er{2.6}, it is usually
possible to check the inclusion $f(H_2)-f(H_1)\in\SS_1$, where $f$ is
some monotone smooth enough function. In this case one can first
take $h_1=f(H_1)$,  $h_2=f(H_2)$ and define
$\xi(\l;h_2,h_1)$ according to \er{2.6a} and then put
\[
\xi(\l;H_2,H_1):=\sign
f'\cdot\xi(f(\l);f(H_2),f(H_1)).
%..................................................................(2.7)
\label{2.7}
\]
Thus defined, $\xi(\l;H_2,H_1)$ still obeys the trace formula. See
\cite{BYa,Ya} for details.
%+++++++++++++++++++
\begin{proposition}\label{p2.1}
%+++++++++++++++++++
{\rm (i)} Let $\k<2$ and let $V$ obey \er{2.1}, \er{2.2}. Then, for any
$\l_0<\inf(\s(H_0)\cup\s(H_0\pm V))$, the relation
\[
R(\l_0,H_0)-R(\l_0,H_0\pm V)\in\SS_1
\label{2.8}
%..................................................................(2.8)
\]
holds.
Thus, the SSF $\xi(\l;H_0\pm V,H_0)$ is correctly defined by \er{2.7}
with $f(\l)=(\l-\l_0)^{-1}$.

{\rm (ii)} Let $\k\geq2$ and let
$V$ obey \er{2.1}, \er{2.5}. Then, for any
$\l_0<\inf(\s(H_0)\cup\s(H_0\pm V))$ with large enough absolute value, and
for any integer $k>\k-\frac12$, the relation
\[
R^k(\l_0,H_0)-R^k(\l_0,H_0\pm V)\in\SS_1
\label{2.9}
%...................................................................(2.9)
\]
holds.
Thus, the SSF $\xi(\l;H_0\pm V,H_0)$ is correctly defined by \er{2.7}
with $f(\l)=(\l-\l_0)^{-k}$.
\end{proposition}
Similar statements appeared in many different  versions in connection with
the trace class scattering theory; see, e.g., 
\cite{Yafaev2,ReedSimon,ReedSimon3}.
The proof of \prp{2.1} is based on \cite[Theorem XI.12]{ReedSimon3}
and is written down explicitly in  \cite{Pus2}.

Finally, note that, since $\s(H_0)=[0,\infty)$, one has
\[
\xi(-\l;H_0-V,H_0)=-\rank E_{H_0-V}((-\infty,-\l)),\quad \l>0.
\label{2.10}
%..................................................................(2.10)
\]



{\bf 2.4 Estimates for the SSF.}
First we state the estimates in their general form, which 
covers the case of all $\k>0$. Remind that $F_\a$ is defined by \er{2.0a}.
The main result of this paper is
%++++++++++++++
\begin{theorem}\label{t2.2}
%++++++++++++++
Under the assumptions \er{2.1}, \er{2.4}, \er{2.5}, the following estimates
hold for $\l>0$ and $\a>2$:
\[
|\xi(\l;H_0+V,H_0)|\leq C(d,l,\a)\l^{\k-1}
\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
\mbox{}+C(d,l)\l^{\k-1}(\log_+\l)\|V\|_{L_1},
%..................................................................(2.12)
\label{2.12}
\]
\bea
|\xi(\l;H_0-V,H_0)|&\leq&
C(d,l,\a)\l^{\k-1}
\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
+C(d,l)\l^{\k-1}(\log_+\l)\|V\|_{L_1}
\nonumber\\
&&\mbox{}+|\xi(-\l;H_0-6V,H_0)|.
%..................................................................(2.11)
\label{2.11}
\eea
\end{theorem}
The proof is given in \S7.
Clearly, \er{2.12} is in agreement with \er{1.7} up to the constants,
the logarithmic weight
$F_\a$ and the term $\log_+\l$.
In \er{2.11},
it remains to estimate the term $|\xi(-\l;H_0-6V,H_0)|$.
This will be done differently for
$\k>1$, $\k<1$ and $\k=1$.
The factor $6$ in \er{2.11} is chosen this way
in order to simplify the
constants appearing in the proof. Actually, this factor
can be replaced by any number greater than $1$; at the same time, the
constants  $C(d,l,\a)$ and $C(d,l)$ may have to be increased.
This affects only the constants in the final estimates \er{2.15},
\er{2.16}, \er{2.16b}, \er{2.16d} below.


{\bf 2.5 The case $\k\not=1$.}
If $\k>1$, the following estimate is well known:
\[
|\xi(-\l;H_0-V,H_0)|\leq C(d,l)\|V\|^\k_{L_\k}.
\label{2.13}
%...................................................................(2.13)
\]
For $l=1$, this is the famous Cwikel--Lieb--Rozenblum bound
\cite{Cwikel,Lieb,Roz}; note that the Cwikel's proof already
includes the case of
arbitrary $l\in(0,d/2)$ (and the Rozenblum's proof --- the case of
$l\in(0,d/2)\cap\N$).
Substituting \er{2.13} into \er{2.11}, we get
%+++++++++++++++++
\begin{corollary}\label{c2.3}
%+++++++++++++++++
Let $\k>1$.
Assume the hypothesis of \thm{2.2} and the inclusion \er{2.3}; then,
for any $\l>0$ and $\a>2$,
\bea
|\xi(\l;H_0-V,H_0)|&\leq&
C(d,l,\a)\l^{\k-1}
\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
+C(d,l)\l^{\k-1}(\log_+\l)\|V\|_{L_1}
\nonumber\\
&&\mbox{}+C(d,l)\|V\|^\k_{L_\k}.
\label{2.15}
%...................................................................(2.15)
\eea
\end{corollary}

For $\k<1$ let us use the Birman--Schwinger principle and
estimate $|\xi(-\l;H_0-V,H_0)|$ in the following
simple way:
\[
|\xi(-\l;H_0-V,H_0)|\leq\Tr(\sqrt{V}R(-\l,H_0)\sqrt{V})=
C(d,l)\l^{\k-1}\|V\|_{L_1}.
\label{2.14}
%...................................................................(2.14)
\]
Substituting \er{2.14} into \er{2.11} and taking into account 
the obvious estimate 
$\|V\|_{L_1}\leq\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}$,
we get the following
corollary.
%+++++++++++++++++
\begin{corollary}\label{c2.4}
%+++++++++++++++++
Let $\k<1$;
assume \er{2.4}.  Then,
for any $\l>0$ and $\a>2$
\[
|\xi(\l;H_0-V,H_0)|\leq
C(d,l,\a)\l^{\k-1}
\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
+C(d,l)\l^{\k-1}(\log_+\l)\|V\|_{L_1}.
\label{2.16}
%..................................................................(2.16)
\]
\end{corollary}


{\bf 2.6 The case $\k=1$.}
This degenerative case includes the important physical situation
$d=2$, $l=1$ and thus deserves proper attention. Now
$$
\xi^{cl}(-\l;H_0-V,H_0)=-(2\pi)^{-d}
\o_d\int V(x)dx,\quad
\forall \l\in\R.
$$
Nevertheless, as it is well known, the ``naive'' estimate
\[
|\xi(-\l;H_0-V,H_0)|\leq C\|V\|_{L_1}, \quad \l>0
\label{2.16a}
%..................................................................(2.16a)
\]
{\it is wrong}; see, e.g., \cite{BL} for the discussion.
Instead, there are numerous estimates of
$\xi(-\l;H_0-V,H_0)$, which are worse than \er{2.16a} by a logarithmic
term of some kind; see \cite{Sol,Weidl}. Such estimates are a bit
cumbersome as compared to \er{2.13}, \er{2.14}. Instead of discussing
them, we consider two rough but simple estimates.
The first one is (see, e.g., \cite[Proposition 5.5]{BS-vol7})
\[
|\xi(-\l;H_0-V,H_0)|\leq\|\sqrt{V}R(-\l,H_0)\sqrt{V}\|^q_{\SS_q}\leq
C(d,q)\l^{1-q}\|V\|^q_{L_q},
\quad q>1;
\label{2.16aa}
%..................................................................(2.16aa)
\]
it is of an ``almost correct'' order in $V$ and $\l$. Substituting
\er{2.16aa} into \er{2.11}, we get
%+++++++++++++++++
\begin{corollary}\label{c2.5}
%+++++++++++++++++
Let $\k=1$;
assume \er{2.1}, \er{2.4}
and let
$V\in L_q(\R^d)$ for some $q>1$.
Then, for any $\l>0$ and $\a>2$
\[
|\xi(\l;H_0-V,H_0)|\leq
C(d,\a)\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
+C(d)(\log_+\l)\|V\|_{L_1}+
C(d,q)\l^{1-q}\|V\|^q_{L_q}.
\label{2.16b}
%..................................................................(2.16b)
\]
\end{corollary}

Another simple estimate is valid (see, e.g.,
\cite[Proposition 5.4]{BS-vol7}) for $\l$ bounded away from zero:
\[
|\xi(-\l;H_0-V,H_0)|\leq C(d,r) \|V\|_{l_1(L_r)},
\quad r>1,\quad \l\geq1.
\label{2.16c}
%..................................................................(2.16c)
\]
Note that $V\in l_1(L_r)$, $r>1$, implies \er{2.1}. Thus,
substituting \er{2.16c} into \er{2.11}, we obtain
%+++++++++++++++++
\begin{corollary}\label{c2.6}
%+++++++++++++++++
Let $\k=1$;
assume \er{2.4}
and let
$V\in l_1(L_r)$ for some $r>1$.
Then,  for any $\l\geq1$ and $\g>2$
\[
|\xi(\l;H_0-V,H_0)|\leq
C(d,\g)\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}
+C(d)(\log_+\l)\|V\|_{L_1}+
C(d,r)\|V\|_{l_1(L_r)}.
\label{2.16d}
%..................................................................(2.16d)
\]
\end{corollary}
Substituting the estimates of \cite{Sol,Weidl} into \er{2.11},
one can obtain more precise statements.

{\bf 2.7 Comparison with the results of \cite{Sob}.}
In \cite{Sob}, the following result has been obtained. Let $d\geq2$, $l=1$
and let $V=V(x)\geq0$ satisfy the estimate
\[
V(x)\leq C_{\ref{2.17}}(1+|x|)^{-\rho},\quad\rho>d.
\label{2.17}
%..................................................................(2.17)
\]
For all $\l\geq C>0$ and all coupling constants $g>0$ the following bounds
have been established (see \cite[Theorem 4.2]{Sob}):
\bea
\xi(\l;H_0+gV,H_0)&\leq&
C_{\ref{2.19}}g\l^{\k-1}(|\log\l|+1).
\label{2.19}
%...................................................................(2.19)
\\
|\xi(\l;H_0-gV,H_0)|&\leq&
C_{\ref{2.18}}(g\l^{\k-1}(|\log\l|+1)+g^\k),
\label{2.18}
%..................................................................(2.18)
\eea
The constants $C_{\ref{2.19}}$,  $C_{\ref{2.18}}$ may depend on $d$ and
$C_{\ref{2.17}}$; this dependence has not been studied in \cite{Sob}.
Clearly, \er{2.19} follows from \er{2.12} and \er{2.18} --- from
\er{2.15}, \er{2.16d}.
The basic difference between our results and the ones of \cite{Sob}  
in the fact that the dependence on $V$ is explicit in
\er{2.12}, \er{2.15}, \er{2.16d} but not in \er{2.19},
\er{2.18}.  Besides, \thm{2.2} and Corollaries 
\ref{c2.3}--\ref{c2.6}
extend the results of \cite{Sob} in some other respects:

1)Theorem \ref{t2.2} and Corollary \ref{c2.4} deal with the case $\k<1$, 
which has not been considered in \cite{Sob};

2) the class of potentials $V$ in \thm{2.2} and
Corollaries \ref{c2.3}--\ref{c2.6} is broader
than the one given by \er{2.17};

3) \thm{2.2} and Corollaries \ref{c2.3}--\ref{c2.5} concern all $\l>0$
and in  \cite{Sob} $\l$ is assumed to be bounded away from zero.

{\bf 2.8 Integral estimates for the SSF.}
In \cite{Pus2}, the {\it integral} estimates for the SSF
$\xi(\l,H_0\pm V,H_0)$ were obtained (for the same operators $H_0$, $V$ as
in the present paper).
Before discussing them, let us write down the estimates for $\xi^{cl}$,
which easily follow from the definition \er{1.2}, \er{1.3}:
\bea
\int_0^R\xi^{cl}(\l;H_0+V,H_0)d\l&\leq&(2\pi)^{-d}\o_dR^\k\|V\|_{L_1},
\quad R>0,\, \k>0,
\label{2.20}
%..................................................................(2.20)
\\
\int_0^R|\xi^{cl}(\l;H_0-V,H_0)|d\l&\leq&(2\pi)^{-d}\o_dR^\k\|V\|_{L_1},
\quad R>0,\, \k\leq1,
\label{2.21}
%..................................................................(2.21)
\\
\int_0^R|\xi^{cl}(\l;H_0-V,H_0)|d\l&\leq&
C_1(d,l)R^\k\|V\|_{L_1}
+C_2(d,l)R\|V\|^\k_{L_\k},
\nonumber\\
&&\hskip 9.5em
R>0,\, \k>1.
\label{2.22}
%..................................................................(2.22)
\eea
It appears that
the estimates \er{2.20}, \er{2.22} can be carried over onto the
real SSF (see \cite{Pus2}, estimates (6.2) and (6.10) respectively);
it is interesting that even the constant in \cite[(6.2)]{Pus2} coincides
with the classical one, given by \er{2.20}.  For $\k<1$, the estimate
$$
\int_0^R|\xi(\l;H_0-V,H_0)|d\l\leq
C(d,l)R^\k\|V\|_{L_1},
\quad R>0,\, \k<1,
$$
(with the constant $C(d,l)$ greater than the classical one, given by
\er{2.21}) follows from \cite[(6.7)]{Pus2} and \er{2.14}.
For $\k=1$ the estimate \cite[(6.7)]{Pus2} together with  \er{2.16aa} implies
$$
\int_0^R|\xi(\l;H_0-V,H_0)|d\l\leq
C(d)R\|V\|_{L_1}+
C(d,q)R^{2-q}\|V\|_{L_q}^q,
\quad R>0,\, \k=1
$$
for any $q>1$, and together with \er{2.16c} it implies 
$$
\int_0^R|\xi(\l;H_0-V,H_0)|d\l\leq
C(d)R\|V\|_{L_1}+
C(d,r)R\|V\|_{l_1(L_r)},
\quad R>1,\, \k=1
$$
for any $r>1$.




{\bf 2.9 Remarks.}
1. Note that in \cite{Sob} some estimates for the SSF
have been found which have better order in $\l$ and $g$,
than \er{2.19}, \er{2.18},
depending on the exponent $\rho$ in \er{2.17}. These estimates, however,
are of a conditional character, since they depend on some hypothesis on
the boundary values of the resolvent of $H_0$, which has not been proved
yet.

2. The proof of \thm{2.2} borrows some elements of reasoning of
\cite{Sob}. But the operator theoretic part of our approach
(Propositions \ref{p3.2}, \ref{p3.3}) is completely different.

3. It is natural to compare the estimates for the SSF with its
asymptotics. Note that formula \er{1.10} for the real SSF and its various
extensions is well known (see, e.g., \cite{Robert} and references
therein). The relation \er{1.11} for the real SSF is a classical fact
for $\l<0$ (see, e.g., \cite{BS-vol7}) and has been proved in
\cite{Pus3} for $\l>0$ and $l=1$. Comparing the estimates  \er{2.12},
\er{2.15}, \er{2.16}, \er{2.16d} with the asymptotics \er{1.10},
\er{1.11}, we see that the estimates are of a correct order in the
coupling constant $g$ (as $g\to\infty$) and of an almost correct (up to
the logarithmic terms) order in $\l$ as $\l\to\infty$.


4. In the case $d=l=1$, a pointwise estimate on the SSF, which is somewhat
different from \er{2.16}, has been obtained in \cite{KostrSchrader}.
See the end of \S7 for the discussion of this estimate.


%=====================================================
\section{Representation for the SSF}
%=====================================================
\setcounter{equation}{0}

Assume \er{2.1}, \er{2.2} and
denote $W:=\sqrt{V}$. Assumption \er{2.1} means that
\[
W(H_0+I)^{-1/2}\in\SS_\infty.
%..................................................................(3.1)
\label{3.1}
\]
For $\Im z>0$ consider the ``sandwiched resolvent''
$$
T(z):=(W(H_0+I)^{-1/2})(H_0+I)R_0(z)(W(H_0+I)^{-1/2})^*=
\overline{WR_0(z)W};
$$
by \er{3.1}, $T(z)\in\SS_\infty$. Denote
$$
A(z)=\Re T(z),\quad K(z)=\Im T(z).
$$
By \er{2.2},
for any bounded interval $\d\subset\R$ one has
\[
WE_{H_0}(\d)\in\SS_2.
%...................................................................(3.2)
\label{3.2}
\]
{}From here follows
%+++++++++++++++++++++++++++++
\begin{proposition}\label{p3.1}
%+++++++++++++++++++++++++++++
Assume \er{2.1}, \er{2.2}. For a.e. $\l\in\R$,
\[
\exists\lim_{\e\to0+}T(\l+i\e)=:T(\l+i0)\in\SS_\infty
\quad\mbox{and}\quad K(\l+i0)\in\SS_1.
%..................................................................(3.3)
\label{3.3}
\]
\end{proposition}
{\it Proof }
For any $\d\subset\R$, denote
\[
\begin{array}{c}
T_\d(z):=(W(H_0+I)^{-1/2}E_{H_0}(\d))(H_0+I)R_0(z)
(W(H_0+I)^{-1/2}E_{H_0}(\d))^*,\\
A_\d(z)=\Re T_\d(z),\quad K_\d(z)=\Im T_\d(z).
\end{array}
%..................................................................(3.4)
\label{3.4}
\]
Now let $\d\subset\R$ be some open {\it bounded} interval. 
It is a classic
fact of the trace class scattering theory (see \cite{BE}
or \cite{Ya}) that \er{3.2} implies
\[
\exists\, T_\d(\l+i0)\in\SS_2
\quad\mbox{and}\quad K_\d(\l+i0)\in\SS_1,
\quad\mbox{a.e. }\l\in\R.
%...................................................................(3.4a)
\label{3.4a}
\]
On the other hand, the operator $T_{\R\setminus\d}(z)$ is analytically
extendable through $\d$; obviously,
\[
T_{\R\setminus\d}(\l)\in\SS_\infty
\quad\mbox{and}\quad K_{\R\setminus\d}(\l)=0,
\quad \l\in\d.
%..................................................................(3.4b)
\label{3.4b}
\]
Finally, writing $T(z)=T_\d(z)+T_{\R\setminus\d}(z)$
and taking into account \er{3.4a}, \er{3.4b},
we get \er{3.3} for a.e. $\l\in\d$.
Since $\d\subset\R$ is arbitrary, this implies the required statement.
\qed

It will follow from the reasoning of \S6 that under the additional
assumption \er{2.4}, the condition \er{3.3} actually holds for {\it all}
$\l\not=0$ and $T(\l+i0)$ depends continuously on $\l$ in the operator
norm and $K(\l+i0)$ --- in the trace norm.


Let $\l$ be such that \er{3.3} holds; denote
\[
\Npm(\l):=\frac1\pi\int_{-\infty}^\infty\frac{dt}{1+t^2}
n_\pm(1,A(\l+i0)+tK(\l+i0)).
%..................................................................(3.5)
\label{3.5}
\]
One easily checks
that \er{3.3} implies convergence of the integral in
\er{3.5}.
%++++++++++++++++++++++++++++++
\begin{proposition}\label{p3.2}
%++++++++++++++++++++++++++++++
{\rm\cite{Pus1,Pus2}}
Assume \er{2.1}, \er{2.2}, \er{2.5}; for a.e. $\l\in\R$
\[
\xi(\l;H_0\pm V,H_0)=\pm\Nmp(\l).
%...................................................................(3.6)
\label{3.6}
\]
\end{proposition}
Representation \er{3.5}, \er{3.6} for the SSF has been established
in \cite{Pus1} as an abstract
operator theoretic fact; application to the polyharmonic operator was
considered in \cite[Theorems 6.2, 6.3]{Pus2}.
Formula  \er{3.6} can be considered as the {\it Birman-Schwinger principle
on the continuous spectrum}; see \cite{Pus1} for the discussion.

A straightforward analysis of the r.h.s. of \er{3.5} gives the following
%++++++++++++++++++++++++++++++
\begin{proposition}\label{p3.3}
%++++++++++++++++++++++++++++++
{\rm\cite{Pus1}}
Let, for some $\l>0$, the condition \er{3.3} hold.
For any $\theta\in(0,1)$ the following estimate holds:
\[
\Npm(\l)\leq n_\pm(1-\theta,A(\l+i0))+\theta^{-1}\pi^{-1}
\|K(\l+i0)\|_{\SS_1}.
\label{3.7}
%..................................................................(3.7)
\]
\end{proposition}
In the proof of \thm{2.2}, we shall use \er{3.7} together with the
following decomposition:
\[
A(\l+i0)=A_\d(\l+i0)+A_{\R\setminus\d}(\l),\quad
\l>0,\quad\d=(0,2\l),
\label{3.8}
%..................................................................(3.8)
\]
where the operators in the r.h.s. are defined by \er{3.4}.
Substituting \er{3.8} into \er{3.7}, fixing $\theta=1/4$ and using
\er{2.0}, we get the following inequality:
\[
\Npm(\l)\leq n_\pm(1/2,A_{\R\setminus\d}(\l))+
n_\pm(1/4,A_\d(\l+i0))+4\pi^{-1}\|K(\l+i0)\|_{\SS_1}.
\label{3.9}
%..................................................................(3.9)
\]
The relation \er{3.9} plays the key role in the proof of \thm{2.2}.
In what follows we merely estimate each of
the three terms in the r.h.s. of \er{3.9}. Note that
$A_{\R\setminus\d}(\l)\geq0$; thus,
$n_-(1/2,A_{\R\setminus\d}(\l))=0$.
Certainly, there is much freedom in choosing the constants $\theta$ in
\er{3.7} and $s_{1,2}$ in \er{2.0}, but this choice affects only the
constants in the resulting formulas \er{2.12}, \er{2.15}, \er{2.16},
\er{2.16b}, \er{2.16d}.

%=====================================================
\section{Estimate for $A_{\R\setminus\d}$}
%=====================================================
\setcounter{equation}{0}

Assume \er{2.1};
as in the end of the previous section, let us fix some $\l>0$, denote
$\d=(0,2\l)$ and define the operator $A_{\R\setminus\d}$ according to
\er{3.4}.
%++++++++++++++++++++++++++++++
\begin{proposition}\label{p4.1}
%++++++++++++++++++++++++++++++
Under the above assumptions, for any $s>0$,
\[
n_+(s,A_{\R\setminus\d}(\l+i0))\leq
|\xi(-\l;H_0-3s^{-1}V,H_0)|.
\label{4.1}
%..................................................................(4.1)
\]
\end{proposition}
{\it Proof: }
A straightforward calculation shows that
$$
R(\l,H_0)E_{H_0}(\R\setminus\d)\leq3 R(-\l,H_0)
$$
in the quadratic form sense.
It follows that $A_{\R\setminus\d}(\l+i0)\leq3T(-\l)$. From here, using
the Birman-Schwinger principle, we get:
\beao
n_+(s,A_{\R\setminus\d}(\l+i0))&\leq&
n_+(s,3T(-\l))=\rank E_{H_0-3s^{-1}V}((-\infty,-\l))
\\
&=&|\xi(-\l;H_0-3s^{-1}V,H_0)|.
\quad\qed
\eeao

%=====================================================
\section{Estimate for $K$}
%=====================================================
\setcounter{equation}{0}

Assume \er{2.1}, \er{2.2}. For $\l>0$ consider the operator
\[
(WE((0,\l)))(WE((0,\l)))^*.
\label{5.00}
%..................................................................(5.00)
\]
By \er{3.2}, the operator \er{5.00} belongs to the trace class.
We will need a well known representation for the derivative of
\er{5.00} with respect to $\l$.
In order to write down
this representation, for every $t>0$ define the operator
$J(t):\,L_2(\R^d)\to L_2(\Sb^{d-1})$, which acts according to the formula
$$
J(t):\quad f(x)\mapsto\,(2\pi)^{-d/2}\int_{\R^d}
e^{it\<x,\nu\>}W(x)f(x)dx,\quad \nu\in\Sb^{d-1}.
$$
For $d=1$ by $\Sb^{d-1}$ we mean the set $\{-1,1\}$.
Clearly, $J(t)\in\SS_2$ and
\[
\|J(t)\|_{\SS_2}^2=d\o_d(2\pi)^{-d}\|V\|_{L_1}.
\label{5.00a}
%..................................................................(5.00a)
\]
A straightforward
calculation shows that the operator valued function \er{5.00}
is differentiable in $\l$ in the trace class and
\[
\F(\l):=
\frac{d}{d\l}(WE((0,\l)))(WE((0,\l)))^*
=(2l)^{-1}\l^{\k-1}J^*(\l^{1/(2l)})J(\l^{1/(2l)})
\geq0.
\label{5.0}
%..................................................................(5.0)
\]
If for some $\l>0$ the limit $T(\l+i0)$ exists, then, clearly,
\[
K(\l+i0)=\pi\F(\l).
\label{5.0a}
%..................................................................(5.0a)
\]
Thus,
\[
\|K(\l+i0)\|_{\SS_1}=\pi\Tr\F(\l)=
\pi(2l)^{-1}\l^{\k-1}\|J(\l^{1/(2l)})\|^2_{\SS_2}
%\nonumber\\
=\pi\k\o_d(2\pi)^{-d}\l^{\k-1}\|V\|_{L_1}.
\label{5.1}
%..................................................................(5.1)
\]

%=====================================================
\section{Estimate for $A_\d$}
%=====================================================
\setcounter{equation}{0}

{\bf 6.1 Preliminary estimates.}
Define the function $\f(t)$, $t>0$, by
\[
\f(t)=\left\{
\begin{array}{ll}
|e^{it}-1|^2,&t\leq \pi,\\
4,&t\geq\pi.
\end{array}
\right.
\label{6.0}
%..................................................................(6.0)
\]
%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.1}
%+++++++++++++++++++++++++++++++
Assume \er{2.2}.
For any $t_1>0$, $t_2>0$ the following estimate holds:
\[
\|J(t_1)-J(t_2)\|^2_{\SS_2}\leq
d\o_d(2\pi)^{-d}\int_{\R^d}V(x)\f(|x||t_1-t_2|)dx.
\label{6.1}
%..................................................................(6.1)
\]
\end{proposition}
{\it Proof: }
\beao
\|J(t_1)-J(t_2)\|^2_{\SS_2}&\leq&
(2\pi)^{-d}\int_{\R^d} V(x)\int_{\Sb^{d-1}}|e^{it_1\<\nu,x\>}-
e^{it_2\<\nu,x\>}|^2d\nu dx
\\
&\leq&(2\pi)^{-d}\int_{\R^d} V(x)\int_{\Sb^{d-1}}
\f(|x||t_1-t_2|)d\nu dx
\\
&=&(2\pi)^{-d}d\o_d\int_{\R^d} V(x)\f(|x||t_1-t_2|)dx. \quad\qed
\eeao
Remind that $\F(\l)$ is defined by \er{5.0}.
%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.2}
%+++++++++++++++++++++++++++++++
Assume \er{2.2}. For any $\l_1>0$, $\l_2>0$:
\bea
\|\F(\l_1)-\F(\l_2)\|_{\SS_1}&\leq&
\k\o_d(2\pi)^{-d}|\l_1^{\k-1}-\l_2^{\k-1}|\|V\|_{L_1}
+2\k\o_d(2\pi)^{-d}\l_2^{\k-1}\|V\|_{L_1}^{1/2}
\nonumber\\
&&\mbox{}\times
\left(\int V(x)\f(|x||\l_1^{1/(2l)}-\l_2^{1/(2l)}|)dx\right)^{1/2}.
\label{6.2}
%..................................................................(6.2)
\eea
\end{proposition}

{\it Proof: } By  \er{5.0},
\beao
\|\F(\l_1)-\F(\l_2)\|_{\SS_1}&=&
(2l)^{-1}\|\l_1^{\k-1}J^*(\l_1^{1/(2l)})J(\l_1^{1/(2l)})-
\l_2^{\k-1}J^*(\l_2^{1/(2l)})J(\l_2^{1/(2l)})\|_{\SS_1}
\\
&\leq&(2l)^{-1}|\l_1^{\k-1}-\l_2^{\k-1}|
\|J^*(\l_1^{1/(2l)})J(\l_1^{1/(2l)})\|_{\SS_1}
\\
&&\mbox{}+(2l)^{-1}\l_2^{\k-1}
\|(J^*(\l_1^{1/(2l)})-J^*(\l_2^{1/(2l)}))J(\l_1^{1/(2l)})\|_{\SS_1}
\\
&&\mbox{}
+(2l)^{-1}\l_2^{\k-1}\|J^*(\l_2^{1/(2l)})
(J(\l_1^{1/(2l)})-J(\l_2^{1/(2l)}))\|_{\SS_1}.
\eeao
Substituting  \er{5.00a} and \er{6.1} into the r.h.s. of the last
estimate, we arrive at \er{6.2}. \qed


%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.3}
%+++++++++++++++++++++++++++++++
Assume \er{2.4}; then there exist such constants
$0<C^{(1)}_{\ref{6.3}}(\a)$ and
$0<C^{(2)}_{\ref{6.3}}(\a)<1$, that
\[
\int V(x)\f(|x|s)dx\leq C^{(1)}_{\ref{6.3}}(\a)
|\log s|^{-\a}\|VF_\a\|_{L_1}, \quad
\forall\,s\in(0,C^{(2)}_{\ref{6.3}}(\a)).
\label{6.3}
%..................................................................(6.3)
\]
\end{proposition}
{\it Proof:} One has
$$
\int V(x)\f(|x|s)dx\leq \sup_{x\in\R^d}\frac{\f(|x|s)}{F_\a(x)}
\int V(x)F_\a(x)dx.
$$
It is a straightforward calculation to check that
$$
\sup_{x\in\R^d}\frac{\f(|x|s)}{F_\a(x)}\leq
C^{(1)}_{\ref{6.3}}(\a)
|\log s|^{-\a}
$$
for some constant  $C^{(1)}_{\ref{6.3}}(\a)$ and for all small enough
$s>0$.
\qed

{\bf 6.2 Existence of $A_\d(\l+i0)$.}
%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.4}
%+++++++++++++++++++++++++++++++
Assume \er{2.4}, fix $\l>0$ and let $\d=(0,2\l)$. Then the limit
$A_\d(\l+i0)$ exists in the trace norm and
\[
A_\d(\l+i0)=\int_0^{2\l}\frac{\F(t)-\F(\l)}{t-\l}dt;
\label{6.4}
%..................................................................(6.4)
\]
the last integral is absolutely convergent in the trace norm.
\end{proposition}
{\it Proof }
We start from the obvious formula which is a consequence of the spectral
theorem:
$$
A_\d(\l+i\e)=\int_0^{2\l}\frac{t-\l}{(t-\l)^2+\e^2}
\F(t)dt,\quad \e>0.
$$
Clearly,
\[
\int_0^{2\l}\frac{t-\l}{(t-\l)^2+\e^2}
\F(t)dt=
\int_0^{2\l}\frac{t-\l}{(t-\l)^2+\e^2}
(\F(t)-\F(\l))dt.
\label{6.5}
%..................................................................(6.5)
\]
{}From Propositions \ref{p6.2}, \ref{p6.3} it follows that
\[
\|\F(\l+s)-\F(\l)\|_{\SS_1}\leq
C(\l,V)|\log|s||^{-\a/2}
\label{6.5a}
\]
for all small enough $s\in\R$. Thus, the integrands
both in \er{6.5} and in \er{6.4} are
dominated (in the trace norm) by an integrable function
$C(\l,V)|t-\l|^{-1}|\log|t-\l||^{-\a/2}$ in the neighbourhood of $t=\l$.
It follows that the r.h.s. of \er{6.5} converges to the r.h.s. of
\er{6.4}.
\qed


\begin{remark}
{\rm
Note that it follows from the estimate \er{6.5a}
that under the assumptions
\er{2.1}, \er{2.4} the condition \er{3.3} holds for all $\l>0$
and the operator
$A_\d(\l+i0)$ is continuous in $\l>0$ in the trace norm.
Thus,  $T(\l+i0)$ is continuous in $\l>0$ in the operator norm.
Next, if instead of \er{2.4} one assumes a stronger condition
$$
\int V(x)(1+|x|)^\g dx<\infty,\quad \g\in(0,2),
$$
then the same reasoning leads to the estimate
$$
\|\F(\l+s)-\F(\l)\|_{\SS_1}\leq C(\l,V)|s|^{\g/2}
$$
for all small enough $s\in\R$. This estimate implies  H\"older continuity
of $A_\d(\l+i0)$ in the trace norm with the exponent $\g/2$
and thus the H\"older continuity of $T(\l+i0)$ in the operator norm.
}
\end{remark}

{\bf 6.3 Estimates for $A_\d(\l+i0)$. }
First we consider the case $\l=1$.
%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.5}
%+++++++++++++++++++++++++++++++
Assume \er{2.4}; let $\d=(0,2)$. One has:
\[
\|A_\d(1+i0)\|_{\SS_1}\leq C(d,l)\|V\|_{L_1}+
C(d,l)\|V\|^{1/2}_{L_1}
\int_0^1\frac{ds}s\left(\int V(x)\f(|x|s)dx\right)^{1/2}.
\label{6.6}
%..................................................................(6.6)
\]
\end{proposition}
{\it Proof:}
Let us use \er{6.4} and \er{6.2}:
\bea
\|A_\d(1+i0)\|_{\SS_1}&\leq&
\int_0^{2}\frac{\|\F(\l)-\F(1)\|_{\SS_1}}{|\l-1|}d\l\leq\k\o_d(2\pi)^{-d}
\|V\|_{L_1}\int_0^2\frac{|\l^{\k-1}-1|}{|\l-1|}d\l
\nonumber\\
&+&
2\k\o_d(2\pi)^{-d}\|V\|_{L_1}^{1/2}\int_0^2\frac{d\l}{|\l-1|}
\left(\int V(x)\f(|x||\l^{1/(2l)}-1|)dx\right)^{1/2}
\label{6.6a}
%..................................................................(6.6a)
\eea
Changing variable in the last integral, we obtain:
\bea
\int_0^2\hskip-1em&&\hskip-1em\frac{1}{|\l-1|}
\left(\int V(x)\f(|x||\l^{1/(2l)}-1|)dx\right)^{1/2}d\l
\nonumber\\
&=&
\int\limits_{-1}^{2^{1/(2l)}-1}
\frac{2l(s+1)^{2l-1}}{|(s+1)^{2l}-1|}
\left(\int V(x)\f(|x|s)dx\right)^{1/2}ds
\nonumber\\
&\leq&
C(l)\int_{-1}^1\frac{ds}{|s|}\left(\int V(x)\f(|x|s)dx\right)^{1/2}
+
\int\limits_1^{2^{1/(2l)}-1}\frac{2l(s+1)^{2l-1}}{|(s+1)^{2l}-1|}
\left(\int V(x)\f(|x|s)dx\right)^{1/2}ds
\nonumber\\
&\leq&
C(l)
\int_{0}^1\frac{ds}s\left(\int V(x)\f(|x|s)dx\right)^{1/2}
+
C(l)\|V\|^{1/2}_{L_1}.
\label{6.6b}
%..................................................................(6.6b)
\eea
Note that the integral over $(1,2^{1/(2l)}-1)$
enters the last calculation only if $2^{1/(2l)}-1>1$, i.e., if
$l<1/2$. In order to estimate  this integral, we use the bound
$\f(t)\leq4$.  The estimates \er{6.6a} and \er{6.6b} together give
\er{6.6}.  \qed



%+++++++++++++++++++++++++++++++
\begin{proposition}\label{p6.6}
%+++++++++++++++++++++++++++++++
Assume \er{2.4}, fix $\l>0$ and let $\d=(0,2\l)$. The following estimate
holds:
\[
\|A_\d(\l+i0)\|_{\SS_1}\leq
C(d,l,\a)\l^{\k-1}\|V\|_{L_1}^{1/2}\|VF_\a\|_{L_1}^{1/2}+
C(d,l)\l^{\k-1}(\log_+\l)\|V\|_{L_1}.
\label{6.7}
%...................................................................(6.7)
\]
\end{proposition}
{\it Proof:}
We start from the standard dilatation argument. Namely, let $U_r$, $r>0$,
be the unitary dilatation operator
in $L_2(\R^d)$:
$(U_rf)(x)=r^{d/2}f(rx)$.
Then, obviously,
$$
U_rA(z)U_r^*=r^{2l}A^{(r)}(r^{2l}z),
$$
where $A^{(r)}$ corresponds to the
perturbation potential
$V^{(r)}(x)=V(xr)$. Thus, taking $r=\l^{-1/(2l)}$, using \er{6.6},
and changing variable in the resulting integrals, we obtain:
\beao
\|A(\l+i0)\|_{\SS_1}&=&\|U_rA(\l+i0)U_r^*\|_{\SS_1}=
\l^{-1}\|A^{(r)}(1+i0)\|_{\SS_1}
\leq
\l^{\k-1}C(d,l)\|V\|_{L_1}
\\
&&\mbox{}+C(d,l)\l^{\k-1}\|V\|^{1/2}_{L_1}
\int_0^{\l^{1/(2l)}}\frac{ds}{s}\left(\int V(x)\f(|x|s)dx\right)^{1/2}.
\eeao
It remains to estimate the last integral. First let $\l\geq1$. Using \er{6.3}
and the bound $\f(t)\leq4$, we get:
\beao
\int\limits_0^{\l^{1/(2l)}}
\frac{ds}s\left(\int V(x)\f(|x|s)dx\right)^{1/2}&\leq&
(C^{(1)}_{\ref{6.3}}(\a))^{1/2}\|VF_\a\|_{L_1}^{1/2}
\int\limits_0^{C^{(2)}_{\ref{6.3}}}
\frac{ds}s|\log s|^{-\a/2}
%\\&&\mbox{}
+
2\|V\|^{1/2}_{L_1}\int\limits_{C^{(2)}_{\ref{6.3}}}^{\l^{1/(2l)}}\frac{ds}s
\\
&\leq& C(\a) \|VF_\a\|_{L_1}^{1/2}+
\|V\|^{1/2}_{L_1}l^{-1}\log_+\l-2\|V\|^{1/2}_{L_1}\log C^{(2)}_{\ref{6.3}}
\\
&\leq&
(C(\a)-2\log C^{(2)}_{\ref{6.3}})\|VF_\a\|_{L_1}^{1/2}+
\|V\|^{1/2}_{L_1}l^{-1}\log_+\l
\\
&\leq&
C(\a)\|VF_\a\|_{L_1}^{1/2}+\|V\|^{1/2}_{L_1}l^{-1}\log_+\l.
\eeao
Finally, if $\l<1$, we merely replace the integration interval
$(0,\l^{1/(2l)})$ by
$(0,1)$, thus getting the upper bound.
\qed


%=====================================================
\section{Proof of \protect\thm{2.2}. Concluding remarks.}
%=====================================================
\setcounter{equation}{0}

{\bf 7.1 Proof of \thm{2.2}.}
Obviously,
$$
n_+(1/4,A_\d(\l+i0))\leq4\|A_\d(\l+i0)\|_{\SS_1}.
$$
It remains to substitute the last estimate together with \er{6.7},
\er{4.1}, \er{5.1}, into \er{3.9} and
take into account \prp{3.2}. \qed

{\bf 7.2 Remarks.}
1. Our \prp{6.4} is fairly close to the ``limiting absorption principle in
the trace class'' of \cite{Sob}, though these two statements are proved by
using very different technique.

2. One can exploit \er{3.9} by using some other (different from ours)
estimates for
$K(\l+i0)$, $A_\d(\l+i0)$, $A_{\R\setminus\d}(\l+i0)$, thus getting new
estimates for the SSF. Let us give an example.

In \cite{SobYaf}, the following estimate has been proved for $l=1$,
$d\geq2$:
\[
\|A_\d(\l+i0)\|_{\SS_2}^2\leq C(d)\l^{d-2-(q/2)} \int\int
\frac{V(x)V(x')}{|x-x'|^q}dxdx',\quad
q\in[0,d-1].
\label{7.1}
%...................................................................(7.1)
\]
For $d=1$ this estimate is obviously also true.
Observing that
\[
n_\pm(s,A_\d(\l+i0))\leq\frac12s^{-2}\|A_\d(\l+i0)\|_{\SS_2}^2
\label{7.2}
%..................................................................(7.2)
\]
and substituting \er{7.1}, \er{7.2} into \er{3.9}, one obtains the
following estimates for $l=1$,  $\l>0$, $q\in[0,d-1]$:
\beao
\xi(\l;H_0+V,H_0)&\leq&
C(d)\l^{d-2-(q/2)}\int\int
\frac{V(x)V(x')}{|x-x'|^q}dxdx'+
C(d)\l^{(d/2)-1}\|V\|_{L_1},
\\
|\xi(\l;H_0-V,H_0)|&\leq&
C(d)\l^{d-2-(q/2)}\int\int
\frac{V(x)V(x')}{|x-x'|^q}dxdx'
+C(d)\l^{(d/2)-1}\|V\|_{L_1}
\\
&&\mbox{}+|\xi(-\l;H_0-6V,H_0)|.
\eeao
One can combine the last estimate with \er{2.13}, \er{2.14}, \er{2.16aa},
\er{2.16c}
or similar bounds in an obvious way.
Note that the estimate (2.10) of \cite{KostrSchrader} is contained
in this series of estimates (for $d=1$).




\section*{Acknowledgments}
The author is grateful to M.~Sh.~Birman and A.~V.~Sobolev for useful
discussions.






























%\end{document}

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\end{document}



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