%----------------------------------------------------------- % SPECTRAL SHIFT FUNCTION OF THE SCHRODINGER OPERATOR % IN THE LARGE COUPLING CONSTANT LIMIT % % A.B.Pushnitski % %----------------------------------------------------------- % LaTeX, 28 pages % \documentstyle[12pt]{article} \oddsidemargin 0in \topmargin -0.5in \textwidth 16truecm \textheight 22truecm \newcommand\<{\langle} \renewcommand\>{\rangle} \newcommand\qed{$\Box$} \def\a{{\alpha}} \def\b{{\beta}} \def\g{{\gamma}} \def\d{{\delta}} \def\e{{\varepsilon}} \def\k{{\kappa}} \def\l{{\lambda}} \def\m{{\mu}} \def\f{{\varphi}} \def\o{{\omega}} \def\p{{\pi}} \def\t{{\tau}} \def\s{{\sigma}} \def\r{{\rho}} \def\x{{\xi}} \def\z{{\zeta}} \def\D{{\Delta}} \def\L{{\Lambda}} \renewcommand\O{{\Omega}} \def\Np{{{\cal N}_+}} \def\Nm{{{\cal N}_-}} \def\Npm{{{\cal N}_\pm}} \def\Nmp{{{\cal N}_\mp}} \newcommand\laplace{{\bigtriangleup}} \newcommand\supp{\hbox{{\rm supp}}\,} \newcommand\mod{\hbox{{\rm mod}}\,} \newcommand\const{\hbox{{\rm const}}} \newcommand\sign{\hbox{{\rm sign}}\,} \newcommand\meas{\hbox{{\rm meas}}\,} \newcommand\rank{\hbox{{\rm rank}}\,} \newcommand\Ran{\hbox{{\rm Ran}}\,} \newcommand\Dom{\hbox{{\rm Dom}}\,} \newcommand\card{\hbox{{\rm card}}\,} \renewcommand\Im{\hbox{{\rm Im}}\,} \renewcommand\Re{\hbox{{\rm Re}}\,} \newcommand\slim{\hbox{{\rm s--}}\hskip-4pt\lim} \newcommand\vol{\hbox{{\rm vol}}\,} \newcommand\loc{\hbox{\rm\scriptsize loc}} \newcommand\Ker{\hbox{{\rm Ker}}\,} \newcommand\Tr{\hbox{{\rm Tr}}\,} \newcommand\R{{\bf R}} \newcommand\N{{\bf N}} \newcommand\Z{{\bf Z}} \newcommand\C{{\bf C}} \newcommand\Q{{\bf Q}} \newcommand\E{{\cal E}} \renewcommand\H{{\cal H}} \newcommand\K{{\cal K}} \newcommand\A{{\bf{A}}} \newcommand\B{{\bf B}} \newcommand\SS{{\bf S}} \newcommand\SSS{{\bf \Sigma}} \newcommand\V{\sqrt{V}} \newcommand\Vp{\sqrt{V_+}} \newcommand\Vm{\sqrt{V_-}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newcommand{\er}[1]{{\rm(\ref{#1})}} \newcommand{\erpm}[1]{{$(\ref{#1}\pm)$}} \newcommand{\erp}[1]{{$(\ref{#1}+)$}} \newcommand{\erm}[1]{{$(\ref{#1}-)$}} \renewcommand\[{\begin{equation}} \renewcommand\]{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\beao}{\begin{eqnarray*}} \newcommand{\eeao}{\end{eqnarray*}} \begin{document} {\sloppy \title{Spectral shift function of the Schr\"odinger operator in the large coupling constant limit} \author{A.~B.~Pushnitski} \date{} \maketitle \begin{abstract} We consider the spectral shift function $\xi(\l;H_0-\a V,H_0)$, where $H_0$ is a Schr\"odinger operator with a variable Riemannian metric and an electro-magnetic field and $V$ is a perturbation by a multiplication operator. We prove the Weyl type asymptotic formula for $\xi(\l;H_0-\a V;H_0)$ in the large coupling constant limit $\a\to\infty$. \end{abstract} %===================================================== \section*{0 Introduction} %===================================================== \renewcommand{\theequation}{0.\arabic{equation}} \setcounter{equation}{0} {\bf 0.1} Let $H_0=-\laplace$ in $L_2(\R^d)$, $d\geq1$ and let $V=V(x)$, $x\in\R^d$, be a perturbation potential which decays rapidly enough as $|x|\to\infty$. For a coupling constant $\a>0$ and a spectral parameter $\l<0$ denote \[ N(\l,\a)=\#\{n\in\N \mid \l_n(H_0-\a V)<\l\}, \label{0.1} %..............................................................(0.1) \] where $\l_n(H_0-\a V)$ are the eigenvalues of $H_0-\a V$, numbered with the multiplicities taken into account. Under certain restrictions on $V$ (which are different for $d=1$, $d=2$ and $d\geq3$), the following Weyl type asymptotic formula is valid (see, e.g., \cite{RS4,B2} and references therein): \[ \lim_{\a\to\infty}\a^{-d/2}N(\l,\a)=(2\pi)^{-d}\o_d\int V_+^{d/2}(x)dx,\quad \l<0, \label{0.2} %..............................................................(0.2) \] where $\o_d$ is the volume of a unit ball in $\R^d$ and $V_+(x)=\max\{0,V(x)\}$. Next, let $H_0$ be a Schr\"odinger operator of a more general type: \[ H_0=(-i\nabla-\A(x))^2+U(x), \label{0.3} %..............................................................(0.3) \] where $\A$ is a magnetic vector potential and $U$ is an electric potential, which satisfy some regularity conditions. Then $N(\l,\a)$ again obeys \er{0.2} for $\l<\inf\s(H_0)$. This fact is well known and has been proved in many particular cases; to the author's knowledge, the most general situation was considered in \cite{B} and \cite{BR} (see also the review \cite{B2} and references therein). The spectrum $\s(H_0)$ of the operator \er{0.3} may have gaps (apart from the semi-infinite gap $(-\infty,\inf\s(H_0))$). One can study the behaviour of the discrete spectrum of $H_0-\a V$ in these gaps for $\a\to\infty$. Let us discuss two generalizations of \er{0.2} known in this situation. {\bf 0.2} Let $V\geq0$; for $\a>0$ put\footnote{We write $\V$ in \er{0.4} in order to make our notation coherent with that of \protect\cite{B,Saf1}.} \[ N_\pm(\l;H_0,\V;\a):=\#\{t\in(0,\a)\mid \l_n(H_0\mp tV)=\l\}, \quad \l\in\R\setminus\s(H_0). \label{0.4} %..............................................................(0.4) \] In other words, $N_\pm(\l;H_0,\V;\a)$ is the number of eigenvalues (counting multiplicities) of $H_0\mp tV$, which pass the point $\l$ as $t$ grows monotonically from $0$ to $\a$. Since $V\geq0$, it follows that the eigenvalues of $H_0\mp\a V$ are monotone functions of $\a$. It is clear that \[ N(\l,\a)=N_+(\l;H_0,\V;\a), \quad \l<\inf\s(H_0). \label{0.5} %..............................................................(0.5) \] For the function $N_+$, a relation similar to \er{0.2} is valid in the gaps of $\s(H_0)$ --- see \cite{H1,H2,ADH,B,BR,B2} and references therein: \[ \lim_{\a\to\infty}\a^{-d/2}N_+(\l;H_0,\V;\a)= (2\pi)^{-d}\o_d\int V^{d/2}(x)dx,\quad \l\in\R\setminus\s(H_0). \label{0.6} %..............................................................(0.6) \] Allowing some naive speculation, one can consider \er{0.6} as an expression of some intuitively attractive ``preservation law'' for the eigenvalues of $H_0-\a V$: as $\a$ grows, the eigenvalues which ``disappear'' at the left edge of a gap, eventually reappear at the right edge of the next gap (located to the left from the first one). Thus, we deal with a ``flow of eigenvalues'' leftwards, which sometimes ``disappears under the spectrum''. {\bf 0.3} Now suppose that the potential $V$ is not necessarily of a definite sign: $V=V_+-V_-$, $V_\pm\geq0$. For $\a>0$ put\footnote{Definition \protect\er{0.7} is correct due to the analyticity of $\l_n(H_0-\a V)$ in $\a$ --- see \cite{Saf1} for the details.} $$ N(\l;H_0,\Vp,\Vm;\a):=\#\{t\in(0,\a)\mid\l_n(H_0-tV)=\l, \frac{d}{ds}\l_n(H_0-sV)\mid_{s=t}<0\} $$ \[ -\#\{t\in(0,\a)\mid\l_n(H_0-tV)=\l, \frac{d}{ds}\l_n(H_0-sV)\mid_{s=t}>0\},\quad\l\in\R\setminus\s(H_0). \label{0.7} %..............................................................(0.7) \] In other words, $N(\l;H_0,\Vp,\Vm;\a)$ is a difference of the numbers of eigenvalues (counting multiplicities) of $H_0-tV$ which cross $\l$ leftwards and rightwards\footnote{The eigenvalues which ``turn'' at the point $\l$, do not enter the expression \protect\er{0.7} --- see \protect\cite{Saf1,Saf2}.} as $t$ grows from $0$ to $\a$. Obviously, for the perturbations of a definite sign, the counting function \er{0.7} coincides with $\pm N_\pm$: \bea N(\l;H_0,\Vp,0;\a)=N_+(\l;H_0,\Vp;\a),\quad \quad\l\in\R\setminus\s(H_0), \label{0.8} %..............................................................(0.8) \\ N(\l;H_0,0,\Vm;\a)=-N_-(\l;H_0,\Vm;\a),\quad \quad\l\in\R\setminus\s(H_0),\nonumber \eea and for $\l<\inf\s(H_0)$ --- with $N(\l,\a)$ of \er{0.1}: \[ N(\l;H_0,\Vp,\Vm;\a)=N(\l,\a),\quad \l<\inf\s(H_0). \label{0.9} %..............................................................(0.9) \] Note also that the function \er{0.7} admits the representation (see \cite{Saf1}) \[ N(\l;H_0,\Vp,\Vm;\a)=N_+(\l;H_0+\a V_-,\Vp;\a)- N_-(\l;H_0,\Vm;\a). \label{0.10} %..............................................................(0.10) \] For the function \er{0.7}, the following generalization of \er{0.2}, \er{0.6} is valid (see \cite{Saf1}): \[ \lim_{\a\to\infty}\a^{-d/2}N(\l;H_0,\Vp,\Vm;\a)= (2\pi)^{-d}\o_d\int V_+^{d/2}(x)dx, \quad \l\in\R\setminus\s(H_0). \label{0.11} %..............................................................(0.11) \] In this connection, see also \cite{ADH,Saf2} and references therein. {\bf 0.4} The aim of this paper is to present a certain analogue of \er{0.2}, \er{0.6}, \er{0.11} for $\l$ lying {\it on the spectrum} of $H_0$. In this case the generalization of the counting functions \er{0.1}, \er{0.4}, \er{0.7} is given by the I.~M.~Lifshits--M.~G.~Krein {\it spectral shift function} (SSF). Let, as above, $H_0$ be the Schr\"odinger operator \er{0.3} and $V=V(x)$ be a potential which decays sufficiently fast as $|x|\to\infty$. We prove the following asymptotic formula for the SSF: \[ \lim_{\a\to\infty}\a^{-d/2}\xi(\l;H_0-\a V,H_0)=- (2\pi)^{-d}\o_d\int V_+^{d/2}(x)dx,\quad\l\in\R \label{0.12} %..............................................................(0.12) \] (the precise statements are given in \S1.4). Note that \[ \xi(\l;H_0-\a V,H_0)=- N(\l;H_0,\Vp,\Vm;\a),\quad\l\in\R\setminus\s(H_0) \label{0.13} %..............................................................(0.13) \] (see Remark 6.4) and thus, by \er{0.8}, \er{0.9}, \bea \xi(\l;H_0-\a V,H_0)=-N_+(\l;H_0,\V;\a),& V\geq0,& \l\in\R\setminus\s(H_0);\nonumber\\ \xi(\l;H_0-\a V,H_0)=-N(\l,\a),&& \l<\inf\s(H_0). \label{0.14} %..............................................................(0.14) \eea We see that \er{0.12} can be considered as a natural generalization of \er{0.2}, \er{0.6}, \er{0.11}. Note, however, that the fact of existence of the SSF imposes strong restrictions on the rate of decay of $V$ at infinity. For this reason, \er{0.12} makes sense for a poorer class of $V$'s, than \er{0.2}, \er{0.6}, \er{0.11}. Formula \er{0.12} adds a new feature to our naive picture of the ``flow of eigenvalues'' of $H_0-\a V$ leftwards as $\a\to\infty$. It shows that this ``flow'' can be noticed and controlled not only in the gaps, but also on the spectrum of $H_0$, where it obeys the same asymptotic ``preservation law''. The relation \er{0.12} means that the family of functions in the l.h.s., which depend on the parameter $\a$, converges to a constant function in the r.h.s. as $\a\to\infty$. The type of convergence has to be specified. In fact, we prove two main results. Theorem 1.5 gives convergence in \er{0.12} for almost every $\l\in\R$ (see Remark 1.6). Theorem 1.7 gives convergence in the weighted space $L_1(\R;d\m(\l))$ with some power type weight $d\m(\l)$. The hypothesis of Theorem 1.5 includes the requirement $V\leq0$. On the other hand, Theorem 1.7 is valid only for $d\geq3$. Thus, neither of these theorems is exhaustive, but together they provide a fairly complete picture. As a by-product of the proof of Theorems 1.5, 1.7, we obtain some information on the asymptotic behaviour of $\xi(\l;H_0+\a V,H_0)$ as $\a\to\infty$ (for $V\geq0$) --- see Theorem 1.8. {\bf 0.5} The proof of Theorems 1.5 and 1.7 is based on some new integral representation for the SSF, which was found in \cite{Pus1}. Namely, let a nonnegative operator $V$ be factorized as $V=G^*G$. The above mentioned representation for the SSF reads (see Propositions 3.3, 3.4 below): \[ \xi(\l;H_0\pm\a V,H_0)=\pm\Nmp(\l;H_0,G;\a). \label{0.15} %..............................................................(0.15) \] Here $\Npm(\l;H_0,G;\a)$ (see \er{3.7}) is an integral of a counting function of eigenvalues of a family of compact operators, related to $G$ and the resolvent of $H_0$. The functions $\Npm$ coincide with $N_\pm$ for $\l\in\R\setminus\s(H_0)$. The relation \er{0.15} is of an abstract nature. To a certain extent, it can be considered as a {\it Birman--Schwinger principle on the continuous spectrum}. The proof of Theorem 1.5 results from a straightforward analysis of the functions $\Np(\l;H_0,\V;\a)$ as $\a\to\infty$. The proof of Theorem 1.7 is based on the integral estimates on $\Npm$, which were obtained in \cite{Pus2}. These proofs are independent of each other. {\bf 0.6} The paper is organized as follows. In \S1 we give some necessary definitions and present the main results. We consider Schr\"odinger operators $H_0$ of a more general type than \er{0.3}; namely, we allow for a variable metric of the space (see \S1.3). The asymptotic formula \er{0.2} in this case has to be ``corrected'': the metric enters the r.h.s. The proof of this formula is given in \S2. In \S3 we discuss the representation \er{0.15} and formulate some necessary results of \cite{Pus1,Pus2}. In \S4, we prove some auxiliary estimates on the Schr\"odinger operator and discuss the existence of the SSF. Theorem 1.5 is proved in \S5, and Theorem 1.7 --- in \S6. {\bf 0.7 Acknowledgments.} The author is deeply grateful to M.~Sh.~Birman for suggesting the problem and for his constant attention to the work. The author is grateful to A.~Laptev and G.~Rozenblioum for useful discussions. Finally, the author expresses his gratitude to the Department of Mathematics of the Royal Institute of Technology, Stockholm, for the hospitality and to the ISF foundation for the financial support. %===================================================== \section{Main results} %===================================================== \renewcommand{\theequation}{\thesection.\arabic{equation}} \setcounter{equation}{0} {\bf 1.1 Notation} 1) The standard inner product in $\C^d$ is denoted by $\<\cdot,\cdot\>$; ${\bf 1}$ is a unit $d\times d$-matrix. Integral without the domain of integration explicitly specified implies integration over $\R^d$. By $\meas\d$ we denote the Lebesgue measure of a Borel set $\d\subset\R$. We denote $\o_d=\mbox{volume}\{x\in\R^d\mid |x|\leq1\}$. 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. A constant which appears in formula $(i.j)$ is denoted by $C_{i.j}$. 2) {\it Functions.} The spaces $L_p(\R^d)$ and $L_{p,\,\loc}(\R^d)$ are defined in a usual way. The space $l_\t(\Z^d;L_\s(\Q^d))$, $\t>0$, $\s\geq1$, consists of the functions $u\in L_{\s,\loc}(\R^d)$ such that the following functional is finite: $$ \|u\|_{l_\t(L_\s)}^\t:=\sum_{j\in\Z^d} \left(\int_{\Q^d+j}|u|^\s dx\right)^{\t/\s}, \quad \Q^d=(0,1)^d\subset\R^d. $$ For a real-valued function $F$ we put $F_\pm:=(|F|\pm F)/2$. For an open set $\O\subset\R^d$, $H^1(\O)$ is a Sobolev space and $H_0^1(\O)$ is the closure of $C_0^\infty(\O)$ in $H^1(\O)$. 3) {\it Operators.} Below $\H$, $\H_1$, $\H_2$ are separable Hilbert spaces. For a linear operator $A$, the notations $\Dom A$, $\Ran A$, $\Ker A$, $A^*$, $\s(A)$, $\r(A)$ are standard; $\overline{A}$ is the closure of $A$, $I$ is the identity operator. For a selfadjoint operator $A$ the symbol $E_A(\d)$ denotes the spectral measure of a Borel set $\d\subset\R$; $A_\pm:=(|A|\pm A)/2$. Resolvent of a selfadjoint operator $H_0$ is denoted by $R_0(z)=(H_0-zI)^{-1}$. By $\B(\H_1,\H_2)$ and $\SS_\infty(\H_1,\H_2)$ we denote respectively the spaces of bounded and compact operators acting from $\H_1$ into $\H_2$; $\B(\H):=\B(\H,\H)$, $\SS_\infty(\H):=\SS_\infty(\H,\H)$. For $T=T^*\in\SS_\infty(\H)$ and $s>0$ we denote $n_\pm(s,T):=\rank E_{T_\pm}((s,+\infty))$, and for $T\in\SS_\infty(\H_1,\H_2)$ put $n(s,T):=n_+(s^2,T^*T)$. 4) {\it Classes of compact operators.} For $0
0}s^pn(s,T)\right)^{1/p}.
$$
The functional $\|\cdot\|_{\SSS_p}$ is a quasinorm. The classes
$\SSS_p(\H_1,\H_2)$ are not separable
(if $\dim\H_1=\dim\H_2=\infty$);
a separable subspace
$\SSS_p^0\subset\SSS_p$ is defined by
$$
\SSS_p^0:=\{T\in\SSS_p\mid \lim_{s\to+0}s^pn(s,T)=0\}.
$$
Note that $\SS_p\subset\SSS_p^0$. For $T=T^*\in\SS_\infty$
the following functionals are introduced:
\bea
\D^{(\pm)}_p(T):=\limsup_{s\to\infty}s^pn_\pm(s,T),\\
\label{1.1}
%..............................................................(1.1)
\d^{(\pm)}_p(T):=\liminf_{s\to\infty}s^pn_\pm(s,T),
\label{1.2}
%..............................................................(1.2)
\eea
so that $0\leq\d_p^{(\pm)}(T)\leq\D_p^{(\pm)}(T)\leq\infty$. The
functionals $\D_p^{(\pm)}$, $\d_p^{(\pm)}$ are continuous in
$\SSS_p$ and do not change if their argument changes by an operator
of the class $\SSS_p^0$ (the last statement is essentially due to H.~Weyl).
{\bf 1.2 Spectral shift function}
Let $H_0$ and $H$ be selfadjoint operators in a Hilbert space $\H$
and let their difference, $V$, be a trace class operator:
\[
V:=H-H_0\in\SS_1(\H).
%..................................................................(1.3)
\label{1.3}
\]
Then the following {\it Lifshits---Krein trace formula}
\cite{Lifshits,Krein} holds:
\[
\Tr (\phi(H)-\phi(H_0))=\int_{-\infty}^\infty\xi(\l;H,H_0)\phi'(\l)d\l.
%..................................................................(1.4)
\label{1.4}
\]
Here $\phi$ is any function of some functional class and
$\xi(\l;H,H_0)$ is the SSF for the pair
$H_0$, $H$, which is given by the {\it Krein formula} \cite{Krein}
\[
\xi(\l;H,H_0)=\frac1\pi\lim_{\e\to+0}\arg\det(I+VR_0(\l+i\e)),
\mbox{ a. e. } \l\in\R.
%..................................................................(1.5)
\label{1.5}
\]
The branch of the argument in \er{1.5} is fixed by the condition
$\arg\det(I+VR_0(z))\to0$, $\Im z\to +\infty$.
For an exposition of the SSF theory, see \cite{BYa, Ya}.
The SSF is {\it monotonic} in $V$ \cite{Krein}:
\[
\pm V\geq0\quad \Rightarrow \quad \pm\xi(\l;H,H_0)\geq0.
%..................................................................(1.6)
\label{1.6}
\]
If $H_0$ is semibounded from below,
by choosing an appropriate $\phi$ in \er{1.4} one finds that
\[
\xi(\l;H,H_0)=-\rank E_H((-\infty,\l)),\quad \l<\inf\s(H_0),
%..................................................................(1.7)
\label{1.7}
\]
which coincides with \er{0.14}.
If the operators $H_0$, $H_1$, $H_2$ in $\H$ are such that $H_2-H_1$
and $H_1-H_0$ are trace class operators, then, clearly,
\[
\xi(\l;H_2,H_0)=\xi(\l;H_2,H_1)+\xi(\l;H_1,H_0).
%..................................................................(1.7a)
\label{1.7a}
\]
In applications, instead of \er{1.3}, it is usually possible to check
the inclusion
\[
f(H)-f(H_0)\in\SS_1(\H).
%..................................................................(1.8)
\label{1.8}
\]
Here $f:(a,b)\to\R$ is some smooth enough, monotone function and
$(a,b)\subseteq\R$ is an interval which contains $\s(H)\cup\s(H_0)$.
Then, the SSF for the pair $H_0$, $H$ is {\it defined} by a natural
formula
\[
\xi(\l;H,H_0):=\sign f'\cdot\xi(f(\l);f(H),f(H_0)).
%..................................................................(1.9)
\label{1.9}
\]
In this case the trace formula \er{1.4} is, of course, still valid
(only the class of admissible functions $\phi$ may have to be changed).
Thus, \er{1.7} is also valid. The relation \er{1.6} in this case has
been proved in \cite{Kopl} (see also \cite[\S8.10]{Ya}) for
$f(\l)=(\l-\l_0)^{-m}$ with any $m>0$ and $\l_0<\inf(\s(H)\cup\s(H_0))$.
Finally, note that if the operators $H_0$, $H_1$, $H_2$ in $\H$ are
such that $f(H_2)-f(H_1)$ and $f(H_1)-f(H_0)$ are trace class
operators, then
\er{1.7a} is still valid.
{\bf 1.3 Schr\"odinger operator}
In $\R^d$, $d\geq1$, we fix a real $d\times d$--matrix valued function
$g=g(x)$ such that
\[
g_-{\bf 1} \leq g(x)\leq g_+{\bf 1},
\quad 01\mbox{ --- any number}&\mbox{ if }q=1.
\end{array}
$$
{\bf Theorem 5.2}
{\it
Let the conditions \er{3.1}, \er{3.2}, \er{5.1} be satisfied.
Then, for a.e. $\l\in\R$:
\[
\lim_{\a\to\infty}\a^{-q_*}\Nm(\l;H_0,G;\a)=0.
\label{5.2}
%...............................................................(5.2)
\]
}
{\bf 5.2 Preliminary results.}
Below we use the notations \er{3.3}, \er{3.4}.
{\bf Proposition 5.3} \cite{BE,Naboko1}
{\it
Let $H_0$ be a selfadjoint operator in $\H$ and $G\in\SS_{2q}(\H,\K)$,
$q\leq1$. Then for a.e. $\l\in\R$ the operator $T(\l+i\e;H_0,G)$ has
limit values as $\e\to+0$ in $\SS_{q_*}(\K)$ and
$K(\l+i0;H_0,G)\in\SS_q(\K)$.
}
Proposition 5.2 for $q=1$, $q_*=2$ was proved in \cite{BE}; the
general case was studied in \cite{Naboko1}.
Further information can be found in \cite{Naboko2}.
{\bf Lemma 5.4}
{\it
Let the operators $A$, $K$ be as in \er{3.16}. Then, for any $p\geq
q$, the quantities \er{3.17} obey
\bea
\limsup_{s\to+0} s^p\eta_\pm(s;A,K)=\D_p^{(\pm)}(A),
\label{5.5}
%...............................................................(5.5)
\\
\liminf_{s\to+0}s^p\eta_\pm(s;A,K)=\d_p^{(\pm)}(A).
\label{5.6}
%...............................................................(5.6)
\eea
}
{\bf Proof}
Let us fix $0<\t<1$ and use \er{3.18}:
$$
s^p\eta_\pm(s;A,K)\leq s^p n_\pm(s(1-\t),A)+
s^{p-q}\t^{-q}\m_q(\t s,K)\|K\|_{\SS_q}.
$$
Passing to the upper limits as $\e\to+0$ and taking into account
\er{3.20}, we get
$$
\limsup_{s\to+0} s^p\eta_\pm(s;A,K)\leq(1-\t)^{-p}\D_p^{(\pm)}(A).
$$
Since $\t$ is arbitrary, this implies
$$
\limsup_{s\to+0} s^p\eta_\pm(s;A,K)\leq\D_p^{(\pm)}(A).
$$
Similarly, using \er{3.19} instead of \er{3.18}, we find
$$
\limsup_{s\to+0} s^p\eta_\pm(s;A,K)\geq\D_p^{(\pm)}(A),
$$
which gives \er{5.5}. In a similar way one proves \er{5.6}.
\qed
{\bf Lemma 5.5}
{\it
Conditions \er{5.3} and \er{5.1} for $q\leq\k$ imply
\[
GR_0(-1)\in\SSS^0_{2\k}(\H,\K).
\label{5.7}
%...............................................................(5.7)
\]
}
{\bf Proof}
By \er{5.1}, for any $R>0$ one has
$$
GR_0(-1)E_{H_0}([0,R))\in\SS_{2q}(\H,\K)\subset\SSS^0_{2\k}(\H,\K).
$$
Next,
\beao
\|GR_0(-1)-GR_0(-1)E_{H_0}([0,R))\|_{\SSS_{2\k}}=
\|GR_0(-1)E_{H_0}([R,\infty))\|_{\SSS_{2\k}}
\\
\leq
\|GR_0^{1/2}(-1)\|_{\SSS_{2\k}}(R+1)^{-1/2}\to0,\quad R\to\infty.
\eeao
Since $\SSS^0_{2\k}$ is closed in $\SSS_{2\k}$, this gives \er{5.7}.
\qed
{\bf Lemma 5.6}
{\it
Assume the hypothesis of Theorem 5.1. Then, for a.e. $\l\in\R$:
{\rm (i)}
$T(\l+i\e;H_0,G)$ has limit values as $\e\to+0$ in
$\SSS_\k(\K)$;
{\rm (ii)}
$K(\l+i0;H_0,G)\in\SS_q(\K)$.
For a.e. $\l,\,\m\in\R$:
{\rm (iii)}
$A(\l+i0;H_0,G)-A(\m+i0;H_0,G)\in\SSS_\k^0(\K)$.
}
{\bf Proof} Fix an open bounded interval $\d\in\R$. Below we check
(i), (ii) for a.e. $\l\in\d$ and (iii) for a.e. $\l,\m\in\d$.
1. One has
\[
T(z;H_0,G)=T(z;H_0,GE_{H_0}(\d))+
T(z;H_0,GE_{H_0}(\R\setminus\d)).
\label{5.7a}
%...............................................................(5.7a)
\]
By \er{5.1}, $GE_{H_0}(\d)\in\SS_{2q}(\H,\K)$. Hence, by Proposition 5.3, the
operator $T(\l+i\e;H_0,GE_{H_0}(\d))$ for a.e. $\l\in\R$ has limit
values as $\e\to+0$ in $\SS_{q_*}(\K)\subset\SSS_\k(\K)$ and
$K(\l+i0;H_0,GE_{H_0}(\d))\in\SS_q(\K)$. On the other hand, the limit
$T(\l+i0;H_0,GE_{H_0}(\R\setminus\d))$ exists in $\SSS_\k(\K)$
for all $\l\in\d$ and $K(\l+i0;H_0,GE_{H_0}(\R\setminus\d))=0$. This
proves (i), (ii).
2. Let us check (iii). For $\l,\,\m\in\d$ and $\e>0$ one has
\bea
A(\l+i\e;H_0,G)-A(\m+i\e;H_0,G)=
A(\l+i\e;H_0,GE_{H_0}(\d))
\nonumber\\
-A(\m+i\e;H_0,GE_{H_0}(\d))+(\l-\m)\Re(GR_0(\l+i\e)E_{H_0}(\R\setminus\d)
(GR_0(\m-i\e))^*).
\label{5.8}
%................................................................(5.8)
\eea
The first two terms in the right hand side of \er{5.8} for
a.e. $\l,\,\m\in\d$ have limits as $\e\to+0$ in
$\SS_{q_*}(\K)\subset\SSS_\k^0(\K)$ (by Proposition 5.3). The third
term for a.e. $\l,\,\m\in\d$ has a limit as $\e\to+0$, which can be
written as $(GR_0(-1))M(GR_0(-1))^*$
with some $M\in\B(\H)$. From here, taking into account \er{5.7}, we see
that the third term is in $\SSS_\k^0$, which gives (iii).
\qed
{\bf Lemma 5.7}
{\it
Assume the hypothesis of Theorem 5.2. Then, for a.e. $\l\in\R$:
{\rm (i)} $T(\l+i\e;H_0,G)$ has limit
values\footnote{In fact, this statement follows from Proposition 3.2}
as $\e\to+0$ in $\SS_\infty(\K)$;
{\rm (ii)} $K(\l+i0;H_0,G)\in\SS_q(\K)$;
{\rm (iii)}
$(A(\l+i0;H_0,G))_-\in\SS_{q_*}(\K)$.
}
{\bf Proof}
Fix arbitrary $R>0$ and an interval $\d=(-1,R)$.
It suffices to check (i)--(iii) for a.e. $\l\in\d$. As in the proof of
Lemma 5.6, we write the representation \er{5.7a} and see that for a.e.
$\l\in\R$ the first summand in the r.h.s.
has limit values in $\SS_{q_*}(\K)$ and for
all $\l\in\d$ the second summand has limit values in $\SS_\infty(\K)$,
which are nonnegative selfadjoint operators. This proves (i),
(iii). The statement (ii) follows from the inclusion
$K(\l+i0;H_0,GE_{H_0}(\d))\in\SS_q(\K)$, a.e. $\l\in\R$.
\qed
{\bf 5.3 Proof of Theorems 5.1, 5.2}
{\bf Proof of Theorem 5.1}
By Lemmas 5.4, 5.6(ii), for a.e. $\l\in\R$,
\beao
\limsup_{\a\to\infty}\a^{-\k}\Np(\l;H_0,G;\a)=
\D_{\k}^{(+)}(A(\l+i0)),
\\
\liminf_{\a\to\infty}\a^{-\k}\Np(\l;H_0,G;\a)=
\d_{\k}^{(+)}(A(\l+i0)).
\eeao
By Lemma 5.6(i), the quantities in the right hand sides are finite. By
Lemma 5.6(iii), they are independent of $\l$ (for the functionals
$\d_\k$, $\D_\k$ do not change if their argument changes by an
operator of the class $\SSS_\k^0$.)
\qed
{\bf Proof of Theorem 5.2}
By Lemmas 5.4, 5.7(ii), for a.e. $\l\in\R$:
$$
\limsup_{\a\to\infty}\a^{-q_*}\Nm(\l;H_0,G;\a)=
\D_{q_*}^{(-)}(A(\l+i0)).
$$
By Lemma 5.7(iii), $(A(\l+i0))_-\in\SS_{q_*}(\K)\subset\SSS_{q_*}^0(\K)$
and thus $\D_{q_*}^{(-)}(A(\l+i0))=0$.
\qed
{\bf 5.4 Proof of Theorems 1.5, 1.8}
{\bf Proof of Theorem 1.5}
1. Let us check the hypothesis of Theorem 5.1 for $\H=\K=L_2(\R^d)$,
$H_0=H_0(g,\A,U)$, $G=\V$, $\k=d/2$. We choose $q$ in \er{5.1} as
follows:
$q=1/2$ if $d=1$; $q=\max\{1/2,\t\}$ if $d=2$ (remind that $\t$ is an
exponent from \er{1.27}); $q=1$ if $d\geq3$.
Now, by Proposition 4.2,
\[
GR_0^m(-1)\in\SS_{2q}(\H,\K)
\label{5.11}
%...............................................................(5.11)
\]
for large enough $m$, where $q$ is as specified above.
Obviously, \er{5.11} implies \er{5.1}.
Finally, the inclusion \er{5.3} holds by \er{2.6}--\er{2.8}.
2. For $\l<0$, due to Proposition 1.3 and formula \er{3.11a}, the
quantities \er{5.4} coincide and are equal to $C_{\ref{1.22}}$. From here
and Theorem 5.1 we see that there exists a set $M_0\subset\R$,
$\meas(\R\setminus M_0)=0$, such that
\[
\lim_{\a\to\infty}\a^{-d/2}\Np(\l;H_0,\V;\a)=C_{\ref{1.22}},
\quad\forall\l\in M_0.
\label{5.12}
%...............................................................(5.12)
\]
3. Let a sequence $\a_n\to\infty$. By
\er{5.3}, \er{5.11} and Proposition 1.4,
for every $n\in\N$ the hypothesis of Proposition 3.4 holds for the
operators $H_0$, $H(\a_n)$. Thus, there exist such sets $M_n\subset
\R$, $\meas(\R\setminus M_n)=0$, that
\[
\xi(\l;H(\a_n),H_0)=-\Np(\l;H_0,\V;\a_n),
\quad\forall\l\in M_n,\, n\in\N.
\label{5.13}
%...............................................................(5.13)
\]
Combining \er{5.12} and \er{5.13}, we see that \er{1.29} holds true
for $M=M_0\cap(\cap_{n\in\N}M_n)$.
\qed
{\bf Proof of Theorem 1.8}
Let us check the hypothesis of Theorem 5.2 for $\H=\K=L_2(\R^d)$,
$H_0=H_0(g,\A,U)$, $G=\V$ and $q=\t$. Indeed, \er{3.1} and
\er{3.2} hold by hypothesis. By Proposition 4.2,
\[
GR_0^m(-1)\in\SS_{2\t}(\H,\K)
\label{5.14}
%................................................................(5.14)
\]
for large enough $m>0$; \er{5.14} implies \er{5.1}. Theorem 5.2 says
that for a.e. $\l\in\R$
$$
\lim_{\a\to\infty}\a^{-\r}\Nm(\l;H_0,\V;\a)=0,
$$
where $\rho$ is as specified in the statement of Theorem 1.8.
Finally, as in part 3 of
the proof of Theorem 1.5, taking into account
Proposition 3.4, we obtain \er{1.32}.
\qed
{\bf Remark 5.8}
Theorems 5.1 and 5.2 allow one to consider some other operators $H_0$,
as long as it is possible to establish some analogues of Propositions
2.1 and 4.2. In particular, one can take $H_0=(-\laplace)^{l}$,
$l>0$. In this case, the estimates of the type \er{2.1}--\er{2.3} and
\er{4.2} are well-known --- see, e.g., \cite{BS1,BS4,Simon4}. Here one can also
consider perturbations $V$ by differential operators
of an order lower than $l$ with decaying coefficients.
Another possibility is to consider the relativistic magnetic
Schr\"odinger operator
\[
\tilde H_0=(H_0({\bf 1},\A,0)+I)^{1/2}+U(x).
\label{5.15}
%................................................................(5.14)
\]
Here the bound similar to \er{2.1}--\er{2.3} can be obtained by using
the Heintz inequality. In order to get the bounds of the type
\er{4.2}, one can exploit the pointwise inequality
$$
|e^{-t\tilde H_0}\psi|\leq e^{-t(-\laplace+I)^{1/2}}|\psi|,\quad
t>0,\,\psi\in L_2(\R^d),
$$
in the spirit of \S4.1. In this connection, see also
\cite{RozSol1,BPus}.
%=====================================================
\section{Integral asymptotics}
%=====================================================
\setcounter{equation}{0}
The aim of this section is to prove Theorem 1.7. First we prove an
abstract result (Theorem 6.2) and then apply it to the Schr\"odinger
operator.
{\bf 6.1 Statement of an abstract result}
Let $H_0\geq0$ be a selfadjoint operator in a ``basic'' Hilbert space
$\H$. Next, let $\K_+$ and $\K_-$ be ``auxiliary'' Hilbert spaces,
$G_+:\H\to\K_+$ and $G_-:\H\to\K_-$ be closed operators and
\bea
G_-R_0^{1/2}(-1)\in\SS_\infty(\H,\K_-),
\label{6.1}
%...............................................................(6.1)
\\
G_+R_0^{1/2}(-1)\in\SSS_{2\k}(\H,\K_+),\quad
\k>0.
\label{6.2}
%...............................................................(6.2)
\eea
Let $V_+=G_+^*G_+$, $V_-=G_-^*G_-$ and define the operators \er{1.17}
as the form sums; denote
$$
R_+(\l;\a)=(H_+(\a)-\l I)^{-1};\quad
R_-(\l;\a)=(H_-(\a)-\l I)^{-1};\quad
R(\l;\a)=(H(\a)-\l I)^{-1}.
$$
Before stating the result on the SSF, we discuss the asymptotics of a
discrete spectrum below $\l=0$. Introduce the notation \er{1.18} and
for $\l<0$ denote
\[
\O_\k(\l):=\limsup_{\a\to\infty}\a^{-\k}N(\l;\a),
\quad
\o_\k(\l):=\liminf_{\a\to\infty}\a^{-\k}N(\l;\a).
\label{6.3}
%...............................................................(6.3)
\]
{\bf Proposition 6.1}
{\it
Let the conditions \er{3.1}, \er{6.1}, \er{6.2} hold and suppose that
for any $R>0$
\[
G_+E_{H_0}([0,R))\in\SSS^0_{2\k}(\H,\K_+).
\label{6.4}
%................................................................(6.4)
\]
Then the quantities $\O_\k$, $\o_\k$ in \er{6.3} do not depend on
$\l<0$.
}
The proof is given in the next subsection. Note that the condition
\er{6.1} in the hypothesis of Proposition 6.1 can be relaxed.
In \cite{Saf1},
in the framework of a similar abstract scheme but under slightly
different conditions on $H_0$, $G_+$, $G_-$, the
stability of the leading term of the asymptotics of
$N(\l;H_0,G_+,G_-;\a)$ (see \er{0.7}) has been established for
$\l\in\R\setminus\s(H_0)$.
In contrast to Proposition 6.1, the proof of the main result of
\cite{Saf1} requires the use of a fairly complicated technique.
The main result of this section (Theorem 6.2) deals with the SSF for
the pair $H_0$, $H(\a)$. Its proof uses also the SSF for the pairs
$H_0$, $H_+(\a)$ and $H_+(\a)$, $H(\a)$. In order to define these SSF,
suppose that for all $\a>0$ and some $k>0$,
$\l_0<\inf(\s(H_0)\cup\s(H(\a)))$, the following inclusions hold:
\bea
R_+^k(\l_0,\a)-R_0^k(\l_0)\in\SS_1(\H),
\label{6.5}
%...............................................................(6.5)
\\
R^k(\l_0,\a)-R_0^k(\l_0)\in\SS_1(\H).
\label{6.6}
%...............................................................(6.6)
\eea
The numbers $k$, $\l_0$ may depend on $\a$ but must be the same in
\er{6.5} and \er{6.6}. From \er{6.5} and \er{6.6} it follows that
\[
R_+^k(\l_0,\a)-R^k(\l_0,\a)\in\SS_1(\H).
\label{6.7}
%...............................................................(6.7)
\]
{\bf Theorem 6.2}
{\it
Let the conditions \er{3.1}, \er{6.1}, \er{6.2} for $\k>1$, and
\er{6.5}, \er{6.6} hold true. Next, suppose that for some $m>1/2$:
\bea
G_-R_0^m(-1)\in\SS_2(\H,\K_-),
\label{6.8}
%...............................................................(6.8)
\\
\sup_{\a>0}\|G_+R_+^m(-1;\a)\|_{\SS_2}
=:C_{6.9}<\infty.
\label{6.9}
%...............................................................(6.9)
\eea
Then for any $E>0$:
\bea
\lim_{\a\to\infty}\int_{-E}^\infty(\a^{-\k}\xi(\l;H(\a),H_0)+\O_\k)_-
(\l+2E)^{-2m}d\l=0,
\label{6.10}
%...............................................................(6.10)
\\
\lim_{\a\to\infty}\int_{-E}^\infty(\a^{-\k}\xi(\l;H(\a),H_0)+\o_\k)_+
(\l+2E)^{-2m}d\l=0,
\label{6.11}
%...............................................................(6.11)
\eea
where $\O_\k$, $\o_\k$ are defined by \er{6.3} (and do not depend on
$\l$ by
Proposition 6.1
\footnote{Condition \protect\er{6.4} in the hypothesis of
Proposition 6.1 follows from \protect\er{6.9}:
$\protect\er{6.9}\,\Rightarrow\,
G_+R_0^m(-1)\in\SS_2\subset\SSS^0_{2\k}
\,\Rightarrow\,\er{6.4}$.}).
In particular, if $\O_\k=\o_\k$, then
\[
\lim_{\a\to\infty}\int_{-E}^\infty|\a^{-\k}\xi(\l;H(\a),H_0)+
\O_\k|(\l+2E)^{-2m}d\l=0.
\label{6.12}
%...............................................................(6.12)
\]
}
{\bf 6.2 Proof of Proposition 6.1}
1. Obviously, $N(\l_1,\a)\leq N(\l_2,\a)$ if $\l_1\leq\l_2<0$.
It follows that
$$
\O_\k(\l_1)\leq\O_\k(\l_2),\quad
\o_\k(\l_1)\leq\o_\k(\l_2),\quad
\l_1\leq\l_2<0.
$$
Thus, it suffices to check the opposite inequalities
$$
\O_\k(\l_1)\geq\O_\k(\l_2),\quad
\o_\k(\l_1)\geq\o_\k(\l_2),\quad
\l_1\leq\l_2<0.
$$
2. Denote
\[
X_\pm(\l):=G_\pm R_0^{1/2}(\l),\quad
Z_\pm(\l):=X_\pm^*(\l)X_\pm(\l),
\quad\l<0.
\label{6.14}
%...............................................................(6.14)
\]
Obviously,
$Z_\pm(\l_1)\leq Z_\pm(\l_2)$ for $\l_1\leq\l_2<0$.
Besides, below we will show that
\[
Z_+(\l_1)-Z_+(\l_2)=:M\in\SSS^0_\k(\H).
\label{6.15}
%...............................................................(6.15)
\]
{}From here by \er{2.5} it follows that
\beao
N(\l_1,\a)=n_+(\a^{-1},Z_+(\l_1)-Z_-(\l_1))\geq
n_+(\a^{-1},Z_+(\l_1)-Z_-(\l_2))
\\
=n_+(\a^{-1},Z_+(\l_2)-Z_-(\l_2)+M),
\\
\O_\k(\l_1)\geq\D^{(+)}_\k(Z_+(\l_2)-Z_-(\l_2)+M)=
\D^{(+)}_\k(Z_+(\l_2)-Z_-(\l_2))=
\O_\k(\l_2).
\eeao
Similarly, it follows that
$\o_\k(\l_1)\geq\o_\k(\l_2)$.
3. Now we need to check \er{6.15}. Clearly, \er{6.15} will follow if
we prove
\[
X_+(\l_1)-X_+(\l_2)\in\SSS^0_{2\k}(\H,\K_+).
\label{6.16}
%...............................................................(6.16)
\]
One can write the operator from \er{6.16} as
\[
X_+(\l_1)-X_+(\l_2)=G_+R_0(-1)B,\quad B\in\B(\H).
\label{6.16a}
%...............................................................(6.16a)
\]
As in Lemma 5.5, we find that \er{6.4} and \er{6.2} imply
$G_+R_0(-1)\in\SSS^0_{2\k}(\H,\K_+)$.
{}From here and \er{6.16a} follows \er{6.16}.
\qed
{\bf 6.3 Proof of Theorem 6.2}
{\bf Lemma 6.3}
{\it
Assume the hypothesis of Proposition 6.1. Denote
\[
L(\l,\a,a):=\a^{-\k}N_+(\l;H_+(\a),G_+;a\a),\quad
\l<0,\,
\a>0,\,
a>0.
\label{6.17}
%...............................................................(6.17)
\]
Then the quantities
\[
B(a):=\limsup_{\a\to\infty}L(\l,\a,a),\quad
b(a):=\liminf_{\a\to\infty}L(\l,\a,a)
\label{6.17a}
%...............................................................(6.17a)
\]
do not depend on $\l<0$ and depend continuously on $a>0$. In
particular,
\[
\lim_{a\to1}B(a)=\O_\k,\quad
\lim_{a\to1}b(a)=\o_\k.
\label{6.17b}
%...............................................................(6.17b)
\]
}
{\bf Proof}
For $\a>0$ and $\b>0$ define the operators
$H(\a,\b)=H_0+\a V_--\b V_+$ as the form sums. Clearly,
$$
N_+(\l;H_+(\a),G_+;\b)=\rank E_{H(\a,\b)}((-\infty,\l)),
\quad
\l<0.
$$
Introducing the notations \er{6.14} and using \er{2.5}, we find
$$
N_+(\l;H_+(\a),G_+;a\a)=n_+(\a^{-1};aZ_+(\l)-Z_-(\l)),
$$
and thus
$$
B(a)=\D_\k^{(+)}(aZ_+(\l)-Z_-(\l)),\quad
b(a)=\d_\k^{(+)}(aZ_+(\l)-Z_-(\l)).
$$
Now the continuity of $L(a)$, $l(a)$ follows from the continuity of
$\D_\k^{(+)}$, $\d_\k^{(+)}$ in $\SSS_\k$. The fact that $L(a)$,
$l(a)$ do not depend on $\l$, follows from Proposition 6.1 (after
the substitution $G_+\mapsto \sqrt{a}G_+$).
\qed
{\bf Proof of Theorem 6.2}
1. First note that, by \er{1.7a} and \er{6.5}--\er{6.7},
\[
\xi(\l;H(\a),H_0)=\xi(\l;H(\a),H_+(\a))+
\xi(\l;H_+(\a),H_0).
\label{6.18}
%...............................................................(6.18)
\]
Next, by Proposition 3.4,
\bea
\xi(\l;H(\a),H_+(\a))=-\Np(\l;H_+(\a),G_+;\a),
\label{6.19}
%...............................................................(6.19)
\\
\xi(\l;H_+(\a),H_0)=
\Nm(\l;H_0,G_-;\a).
\label{6.20}
%...............................................................(6.20)
\eea
Denote
\beao
F(\l,\a):=-\a^{-\k}\xi(\l;H(\a),H_0),
\\
Q(\l,\a):=\a^{-\k}\Np(\l;H_+(\a),G_+;\a),
\\
J(\l,\a):=\a^{-\k}\Nm(\l;H_0,G_-;\a).
\eeao
Write the equality \er{6.18} as
\[
F(\l,\a)=Q(\l,\a)-J(\l,\a).
\label{6.21}
%...............................................................(6.21)
\]
Finally, we denote for brevity $d\m(\l):=(\l+2E)^{-2m}d\l$ for
$\l\geq-E$ and $d\m(\l)=0$ for $\l<-E$.
In these notations, we need to prove that
\bea
\lim_{\a\to\infty}\int(F(\l,\a)-\O_\k)_+d\m(\l)=0,
\label{6.21a}
%................................................................(6.21a)
\\
\lim_{\a\to\infty}\int(F(\l,\a)-\o_\k)_-d\m(\l)=0.
\label{6.21b}
%...............................................................(6.21b)
\eea
2. Let us prove \er{6.21a}. In \er{3.21}, take $H_+(\a)$ for $H_0$ and
$G_+$ for $G$. Using the notation \er{6.17}, we obtain:
\beao
\int(Q(\l,\a)-(1-\theta)^{-2\k} L(-E,\a,(1-\theta)^{-2}))_+d\m(\l)
\leq\a^{1-\k}\theta^{-1-m}\|G_+R_+^m(-E,\a)\|^2_{\SS_2}
\\
\leq \max\{1,E^{-2m}\}C_{6.9}
\theta^{-1-m}\a^{1-\k}.
\eeao
Next, taking into account \er{6.21} and the inequality $J(\l,\a)\geq0$,
\beao
(Q(\l,\a)-(1-\theta)^{-2\k}L(-E,\a,(1-\theta)^{-2}))_+
\geq(F(\l,\a)-(1-\theta)^{-2\k}L(-E,\a,(1-\theta)^{-2}))_+
\\
\geq
(F(\l,\a)-\O_\k)_+-
((1-\theta)^{-2\k} L(-E,\a,(1-\theta)^{-2})-\O_\k)_+.
\eeao
{}From here we find
\beao
\int(F(\l,\a)-\O_\k)_+d\m(\l)\leq \max\{1,E^{-2m}\}C_{6.9}
\theta^{-1-m}\a^{1-\k}
\\
+((1-\theta)^{-2\k}L(-E,\a,(1-\theta)^{-2})-
\O_\k)_+\int d\m(\l).
\eeao
Passing to the limit as $\a\to\infty$, we find
(using the notations \er{6.17a}):
$$
\limsup_{\a\to\infty}\int(F(\l,\a)-\O_\k)_+d\m(\l)\leq
((1-\theta)^{-2\k}B((1-\theta)^{-2})-\O_\k)_+\int d\m(\l).
$$
By \er{6.17b}, the right hand side of the last inequality goes to 0 as
$\theta\to+0$, and we arrive at \er{6.21a}.
3.
Let us prove \er{6.21b}.
In \er{3.22}, take $H_+(\a)$ for $H_0$ and $G_+$ for $G$. Using the
notation \er{6.17}, we obtain:
\beao
\int(Q(\l,\a)-(1+\theta)^{-\k}L(-2E,\a,(1+\theta)^{-1}))_-d\m(\l)
\leq
\a^{1-\k}\theta^{-1}(2/(2m-1)+4)
\\
\times\|G_+R_+^m(-2E,\a)\|^2_{\SS_2}\leq
(2/(2m-1)+4)\max\{1,(2E)^{-2m}\}C_{6.9}
\theta^{-1}\a^{1-\k}.
\eeao
Next, taking into account \er{6.21},
\beao
(Q(\l,\a)-(1+\theta)^{-\k}L(-2E,\a,(1+\theta)^{-1}))_-
\\
=(F(\l,\a)-\o_\k+
J(\l,\a)+(\o_\k-(1+\theta)^{-\k} L(-2E,\a,(1+\theta)^{-1})))_-
\\
\geq
(F(\l,\a)-\o_\k)_--
J(\l,\a)-(\o_\k-(1+\theta)^{-\k} L(-2E,\a,(1+\theta)^{-1}))_+.
\eeao
{}From here we find:
$$
\int(F(\l,\a)-\o_\k)_-d\m(\l)\leq
(2/(2m-1)+4)\max\{1,(2E)^{-2m}\}C_{6.9}
\theta^{-1}\a^{1-\k}
+\int J(\l,\a)d\m(\l)
$$
\[
+(\o_\k-(1+\theta)^{-\k} L(-2E,\a,(1+\theta)^{-1}))_+\int d\m(\l).
\label{6.22}
%...............................................................(6.22)
\]
By \er{3.23},
$$
\limsup_{\a\to\infty}\int J(\l,\a)d\m(\l)\leq
\limsup_{\a\to\infty}\a^{1-\k}\|G_-R_0^m(-2E)\|^2_{\SS_2}=0.
$$
Passing to the limit as $\a\to\infty$ in \er{6.22} and taking into
account the last relation, we obtain:
$$
\limsup_{\a\to\infty}(F(\l,\a)-\o_\k)_-d\m(\l)\leq
(\o_\k-(1+\theta)^{-\k} b((1+\theta)^{-1}))_+\int d\m(\l).
$$
By \er{6.17b}, the right hand side of the last inequality goes to $0$
as $\theta\to+0$, and we obtain \er{6.21b}.
\qed
{\bf Remark 6.4}
Under the hypothesis of Theorem 6.2, formula \er{0.13} is valid.
Indeed, using \er{0.10}, \er{6.18}--\er{6.20}, for
$\l\in\R\setminus\s(H_0)$ we obtain:
\beao
\xi(\l;H(\a),H_0)=-\Np(\l;H_+(\a),G_+;\a)+
\Nm(\l;H_0,G_-;\a)
\\
=-N_+(\l;H_+(\a),G_+;\a)+N_-(\l;H_0,G_-;\a)=
-N(\l;H_0,G_+,G_-;\a).
\eeao
{\bf 6.4 Proof of Theorem 1.7}
Let us check the hypothesis of Theorem 6.2 for $H_0=H_0(g,\A,U)$,
$G_\pm=\sqrt{V_\pm}$, $\H=\K_+=\K_-=L_2(\R^d)$ and
$\k=d/2$.
Inclusions \er{6.1}, \er{6.2} follow from \er{2.6}. Conditions
\er{6.8}, \er{6.9} follow from \er{4.2} (for $q=1$);
while checking \er{6.9}, we include the term $+\a V_-$ into the
background potential $U$ and use the fact that the constant $C_{4.2}$
does not depend on $U\geq0$.
Conditions \er{6.5}, \er{6.6} follow from Proposition 1.4.
Next, by Proposition 1.3, $\O_\k=\o_\k=C_{\ref{1.22}}$.
Thus, \er{6.12} gives \er{1.30} with $p=2m$.
\qed
{\bf Remark 6.5}
Let us comment on the possibility of applying Theorem 6.2 to other
differential operators $H_0$. Here the main difficulty is in the check
of the condition \er{6.9}, which in our case required the use of
rather specific technique of pointwise estimates for the Gaussian
kernels. This technique is applicable to the elliptic differential
operators of the order $l\leq2$. Thus, one can apply Theorem 6.2 to
the relativistic magnetic Schr\"odinger operator \er{5.15}.
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\vskip 1cm
\noindent
Department of Mathematical Physics\\
Faculty of Physics\\
St-Petersburg State University\\
198904, St.Petersburg, Ul'yanovskaya, 1\\
RUSSIA\\
{\it E-mail:} pushnit@mph.phys.spbu.ru
\end{document}