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Periodic Schr\"{o}dinger Operator, Density of States
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\documentstyle[12pt,mssymb]{article}
\begin{document}
\title{On the Density of States for the Periodic Schr\"{o}dinger Operator}
\author{Yu.\ E.\ Karpeshina \footnote{Research partially supported by USNSF Grant DMS-9803498.}}
\date{}
\maketitle
%%%%%%%%\nodate
\newtheorem{Lemma }{1}
\newtheorem{Theorem}{2}
\newtheorem{Corollary}{3}
\newtheorem{1}{Lemma }
\newtheorem{2}{Theorem}
\newtheorem{3}{Corollary}
\newtheorem{5}{Proposition}
\newtheorem{Proposition}{5}
\begin{quote}{\small ABSTRACT Asymptotic formula for the density of
states of the polyharmonic periodic operator $(-\Delta )^l+V $ in
$R^n$, $n\geq 2$, $l\geq 1$ is obtained. Special consideration is
given to the case of the Schr\"{o}dinger equation $n=3$, $l=1$,
where the second term of the asymptotic is found.}
\end{quote}

\section{Introduction.} We consider the
 operator
\begin{equation}
H=(-\Delta )^l+V  \label{1}
\end{equation}
in $L_2(R^n)$, where $V$ is the operator of multiplication by a real periodic
 potential,
 $n\geq 2$, $l\geq 1 $. Particular attention is paid to the case of
 the Schr\"{o}dinger operator  $l=1,n=3$.
For the sake of simplicity, we assume that the potential has
orthogonal periods $a_1,...,a_n$, however all the results are also
valid for nonorthogonal periods. Without the loss of generality we
consider that $\int _{Q}V(x)dx=0$, $Q=[0,a_1]\times ...\times
[0,a_n].$ We use the following representation of the potential:
\begin{equation}
V(x)=\sum _{m\in Z^3,m\neq 0}v_m\exp i(\vec{p}_m(0),x),  \label{6}
\end{equation}
where  $(\cdot ,\cdot )$ is the inner product in $R^n$ and
$\vec{p}_m(0)$ is a vector of the dual lattice:
\begin{equation}\vec{p}_m(0)=2\pi
(m_1a_1^{-1},m_2a_2^{-1},...,m_ma_n^{-1}). \label{p}
\end{equation}
The condition on the smoothness of the potential is:
\begin{equation}
\|V\|_*=\sum _{m\in Z^n,m\neq 0}|v_m||m|^{\nu } <\infty,
 \label{smooth1}
\end{equation}
$$\nu >(n-1)/2,\ \mbox{when}\ n\geq 3,\ \nu >2(l+1)/3,\
\mbox{when}\ n=2.$$
%%%%%%%%%The potential $V$ is real, so $v_m=\bar{v}_{-m}$.
%%%%%%%It may be assumed with no loss of generality that $v_0=0$.
It is well known (see f.e. \cite{RS})  that the
 spectral analysis of $H$ can be reduced
 to the study of a family of operators $H(t)$,
$t\in K$, where $K$ is the elementary cell of the dual lattice,
$$K=[0,2\pi a_1^{-1})\times [0,2\pi a_2^{-1})\times ...\times
[0,2\pi a_n^{-1}).$$ The vector $t$ is called the $quasimomentum$.
The operator $H(t)$, $t\in K$, acts in $L_2(Q)$.
%%%%%%, ($Q$ is the elementary cell of the periods of the potential,
%%%%%%%% $Q=[0,a_1)\times [0,a_2)...\times [0,a_n)$).
It is described by the
formula~(\ref{1}) and the quasiperiodic conditions:
\begin{equation}
u(x_1,...,x_{j-1},a_j, x_{j+1}...x_{n})= \exp
(it_ja_j)u(x_1,...,x_{j-1},0, x_{j+1},...,x_{n}), \label{quasi}
\end{equation}
$$j=1,2, ...,n.$$ The  derivatives with respect to $x_j$  must
also satisfy the analogous  conditions. The operator $H(t)$, $t\in
K$, has a discrete semibounded below spectrum $\Lambda (t)$:
$$\Lambda (t)=\cup _{n=1}^{\infty }\lambda _n(t),\ \lambda
_n(t)\to _{n\to \infty }\infty .$$ The spectrum $\Lambda $ of the
operator $H$ is the union of the spectra of
 the operators $H(t)$:
$\Lambda =\cup _{t\in K}\Lambda (t)=\cup _{n\in N,t\in K}\lambda _n(t).$
The functions $\lambda _n(t)$ are continuous, so $\Lambda $ has a band
structure:
$$\Lambda =\cup _{n=1}^{\infty }[q_n,Q_n],\ q_n=\min _{t\in K}\lambda _n(t),\
 Q_n=\max _{t\in K}\lambda _n(t).$$
The eigenfunctions of $H(t)$ and $H$ are simply related. Extending
all the eigenfunctions of the operators $H(t)$ quasiperiodically
(see (\ref{quasi})) to $R^n$, we obtain the complete system of
eigenfunctions of
 $H$. In the case of $V=0$, the eigenvalues and eigenfunctions of
 corresponding operators $H_0(t)$, $t\in K$, are  naturally
 indexed by  points of $Z^n$: $\lambda _m^0(t)=p_m^{2l}(t)$,
 $\psi _m^0(t,x)=\exp i(\vec{p}_m(t),x)$, $m\in Z^n$, here and below
 $\vec{p}_m(t)=\vec{p}_m(0)+t$, $\vec{p}_m(0)$ being given by (\ref{p}), and
 $p_m^{2l}(t)=|\vec{p}_m(t)|^{2l}$.


Let $N_V(\lambda ,t)$ be the counting function of eigenvalues,
i.e.,  $N_V(\lambda ,t)$ is the number of eigenvalues $\lambda
_n(t)$ of the operator $H(t)$, which are not greater than $\lambda
$:
\begin{equation} N_V(\lambda ,t)=\# \{n: \lambda _n(t)\leq \lambda \}=\sum
_{n\in N}\chi (\lambda -\lambda _n(t)), \label{N} \end{equation}
$\chi (\cdot )$ being the Heaviside function. In the case of
$V=0$, $ N_0(\lambda ,t)$ is the number of lattice points
$\vec{p}_m(t)$ in the sphere $\{x:|x|\leq \lambda
^{\frac{1}{2l}}\}$. Further we will use the notation $\lambda
=k^{2l}$. The asymptotic of the number of the lattice points in
the sphere $\{x:|x|\leq k\}$ when $k\to \infty $ depends
significantly on $t$ and properties of the lattice. Thus, even
$N_0(k^{2l},t)$ depends on $k$ and $t$ in a nontrivial way, while
perturbation $V$ makes this dependence even more sophisticated
(see f.e. \cite{B,C,Ch,DT,Sk1,Sk2,Y}). The proof of the estimate
$N_V(k^2,t)=\omega_nk^n+O_{t}(k^{n-1}), $ for the case $l=1$ one
can find in \cite{H4}, in the latter formula
$\omega_n=w_n|K|^{-1}$, $w_n$, $|K|$ being the volumes of the unit
sphere and of the cell $K$, correspondingly. The proof of the
better estimate $N_V(k^2,t)=\omega_nk^n+O_{t}(k^{n-1-\eta })$ is
given in \cite{V}.

The density of states $D_V(k^{2l})$ is the integral of
$N_V(k^{2l},t)$ over all $t$:
\begin{equation}
D_V(k^{2l})=\int _{K}N_V(k^{2l} ,t)dt. \label{1a}
\end{equation}
The physical sense of the density of states in the case of the
Schr\"{o}dinger operator (n=3, l=1) is the following: $D_V(\lambda
)$ is the limit number of the possible energy levels of a particle
in a body in the interval $(-\infty,\lambda  )$ divided by the
volume of the body as the volume expands to infinity (see f.e.
\cite{Z}).  M.A. Shubin \cite{Sh1,Sh2,RoShSo} gave the
mathematical justification of the passage to the limit for the
case of almost periodic (in particular, periodic) potentials. The
perturbation methods applied by M.A. Shubin provide the asymptotic
estimate for the density of states as $k\to \infty $:
 \footnote{Results for other nonperiodic cases
can be found f.e. in \cite{G,Ki,Sa}.}
\begin{equation}
D_V(k^{2l},t)=D_0(k^{2l},t)+O(k^{n-2l}),\label{1b*}
\end{equation}
where, as it is well known, $D_0(k^{2l})=\omega _nk^n.$
%%%%%%%5\begin{equation}D_0(k^{2l})=\omega _nk^n. \label{4a}
%%%%%%%%55\end{equation}
%%%The perturbation of this estimate by a periodic potentials in the case
%%%%%%%$l=1$ was considered by M.Shubin \cite{} and
%%%%%%%% B. Helffer, A. Mohamed \cite{}. M.A. Shubin proved that
%%%%%%%%\begin{equation}
%%%%%%%%D_V(k^2,t)=w_nk^n+O(k^{n-2}),\label{1b*}
%%%%%%%%\end{equation}
%%%%%using perturbation methods.\footnote{M.A. Shubin proved this
%%%%%%%%estimate for the more general case of almost periodic potentials.}
%%%%%%%%for almost periodic potentials.
 B. Helffer, A. Mohamed \cite{HM} improved this estimate for the case of periodic
$C^{\infty }$-potentials and $l=1$:
\begin{equation}
D_V(k^2,t)=\omega _nk^n+O(1)+O(k^{n-3+\varepsilon}),\ \
\varepsilon
>0. \label{2a}
\end{equation}
They used advanced methods of microlocal analysis.

This paper has two goals. The first goal  is to present a short
proof of  stronger estimates for $D_V(k^{2l},t)$ by using
perturbation methods. We consider the   general case $n\geq 2$,
$l\geq 1$, and $V$ satisfying (\ref{smooth1}). The following
estimate is proved in Section 2:
 \begin{equation}
D_V(k^{2l},t)=\omega _nk^n+O(k^{-\xi }\ln k),\ \ \xi
=4l-n-1.\label{3b*}
\end{equation}
Clearly (\ref{3b*}) is stronger than (\ref{1b*}): $\xi
>2l-n$. Comparing (\ref{2a}) and (\ref{3b*}) for $l=1$, we see that (\ref{3b*}) is stronger than (\ref{2a}), the
biggest difference is in the case $n=2$: $O(k^{-1}\ln k)$ instead
of $O(1)$.

 The second goal is to obtain the second term of the
asymptotic expansion  for the special case $n=3$, $l=1$, $V$
satisfying (\ref{smooth1}) with $\nu =900$ (Section 3). We prove
that
 \begin{equation}
D_V(k^2,t)=\omega _3k^3+d_V^0+O(k^{-\zeta }),\ \ \zeta
<1/130,\label{3a*}
\end{equation}
where $d_V^0$ is a constant which can be expressed as the  sum of
the integrals of
 the densities of states of some one-dimensional Schr\"{o}dinger
operators (Section 3.1). The number $1/130$ in the estimate
(\ref{3a*}) arises for technical reasons and could be improved by
somewhat longer considerations.
\section{The general case $n\geq 2$, $l\geq 1$.}
\begin{2}Let $n\geq 2$, $l\geq 1$ and $V$ satisfy the condition (\ref{smooth1}).
 Then, the density of states of the operator
$H$ satisfies the following
 asymptotic  as $k\to \infty $:
\begin{equation}
D_V(k^{2l})=\omega _nk^n+O(k ^{-\xi}\ln k), \ \ \xi =4l-n-1.\label{3}
\end{equation}
\end{2}
Let us describe the main steps in the proof of Theorem 1. First,
we represent operator $H(t)$ in the form
\begin{equation}
H(t)=H_0(t)+(I-P)V(I-P)+PV(I-P)+(I-P)VP+PVP, \label{t1}
\end{equation}
where $H_0(t)$ corresponds to $V=0$, $P=P(t)$ is a
finite-dimensional diagonal projection. We prove that
\begin{equation}
N_{(I-P)V(I-P)}(k^{2l},t)=N_0(k^{2l},t), \label{t2}
\end{equation}
i.e., the operators $H_0(t)$ and
%%%%%%%%%\begin{equation}
$\tilde{H}(t)=H_0(t)+(I-P)V(I-P)$
%%%%%%%%5 \label{*}\end{equation}
 have
the same number of eigenvalues which are not greater than
$k^{2l}$. The equality (\ref{t2}) holds due to the special choice
of the diagonal projection $P$. Indeed, $\tilde{H}(t)$ is a direct
sum of $PH_0$ and  $(I-P)H_0(t)+(I-P)V(I-P)$. Thus, $(I-P)V(I-P)$,
in fact, perturbs only $(I-P)H_0$. We construct $P$ in such a way
that $(I-P)H_0$ does not contain eigenvalues $p_m^{2l}(t)$ of
$H_0(t)$, which are ``in danger" to intersect the point $\lambda
=k^{2l}$ under the perturbation $(I-P)V(I-P)$. The simplest way
would be to chose $P$: $P_{mm}=1$, when $|p_m^{2l}(t)-k^{2l}| \leq
\|V\|$, however it would result, eventually, only in the estimate
(\ref{1b*}). So, we will choose $P$ in a somewhat more subtle way.
Since $(I-P)V(I-P)$ perturbs only $(I-P)H_0$, the relation
(\ref{t2}) holds.

 The second step: considering that
$PV(I-P)+(I-P)VP+PVP$ is a finite-dimensional operator, we get $$
|N_V(k^{2l},t)-N_{(I-P)V(I-P)}(k^{2l},t)|\leq 2\# (t),$$
 where $\# (t)$ is the dimension of
$P(t)$. Using (\ref{t2}), we obtain
\begin{equation}
|N_V(k^{2l},t)-N_{0}(k^{2l},t)|\leq 2\# (t). \label{14a'}
\end{equation}
In the third part of the proof we use counting arguments  to show
that
\begin{equation}
\int _{K}\#(t)dt<ck^{-\xi }\ln k,\ \ c=c(\|V\|_*,\nu ,
a_{\max},n,l), \label{4}
\end{equation}
here and below $a_{\max}=\max _{i}a_i$, $a_{\min }=\min _{i}a_i$.
Integrating both parts of (\ref{14a'}), we obtain (\ref{3}).

Let us introduce some notations: $$ \sigma
_{mq}(z,t)=\sqrt{|p_m^{2l}(t)-z||p_{m+q}^{2l}(t)- z|},\ \ m,q\in
Z^n,$$
\begin{equation}
\sigma _{m}(z,t)= \min _{q\in Z^n\setminus \{ 0 \}} |q|^{\nu
}\sigma _{mq}(z,t),
%%%%%55\sigma _{m}(k^{R_0},z,t)=
%%%%%%%%\min _{0<|q|<R_0}\sigma _{mq}(z,t),
%%%%%%%%%\ \ 0<\varepsilon <1.
\label{t5}
\end{equation}
$\nu $ being a parameter in the condition (\ref{smooth1}). We
define the diagonal projection $P$ as follows:
\begin{equation}
P_{mm}=\left\{ \begin{array}{ll}1,&\ \ \mbox{if}\
\sigma_{m}(k^{2l},t)<4\|V\|_*,\\ 0,&\ \
\mbox{otherwise;}\end{array}\right. \label{t6}
\end{equation}
$\|V\|_*$ being given by (\ref{smooth1}).
\begin{1} If $P$ is defined by (\ref{t6}), then
\begin{equation}
N_{(I-P)V(I-P)}(k^{2l},t)=N_0(k^{2l},t). \label{t2*}
\end{equation}
 \end{1} \underline{Proof.} We consider the following contour on
the complex plane:\\ \special{em:linewidth 0.4pt} \unitlength
1.00cm \linethickness{0.4pt}
\begin{picture}(1.0,2.0)
\put (0.0,0.0){\line(1,0){10.0}} \put (10,0){\vector(1,0){1}}
\put(2.0,0.0){\line(0,1){1.0}} \put (2.0,0.0){\line(0,-1){1.0}}
\put (2.0,1.0){\line(1,0){6.0}} \put (2.0,-1.0){\line(1,0){6.0}}
\put (8.0,1.0){\line(0,-1){1.0}} \put (8.0,-1.0){\line(0,1){1.0}}
\put (8.0,0.0){\circle*{0.1}} \put (3.0,0.0){\circle*{0.1}}
\put(8.3,-0.3){\makebox(0,0)[cc]{{\small $k^{2l}$}}}
\put(1.4,-0.3){\makebox(0,0)[cc]{{\small $-2\|V\|$}}}
\put(2.0,0.0){\circle*{0.1}}
\put(3.0,-0.3){\makebox(0,0)[cc]{{\small $0$}}}
\put(5.0,-2.0){\makebox(0,0)[cc]{{\small \ The contour $C_{0}$}}}
\end{picture}\\
\vspace{2cm}
\par
\noindent It is clear that $N_{(I-P)V(I-P)}(k^{2l},t)$ is given by
the integral: $$N_{(I-P)V(I-P)}(k^{2l},t)= \mbox{Tr}\oint
_{C_0}(\tilde{H}(t)-z)^{-1}dt, $$
 if there is no eigenvalue at
$k^{2l}$. Let us formally expand the resolvent in the perturbation
series with respect to  $H_0$:
\begin{equation}
(\tilde{H}(t)-z)^{-1}=(H_0(t)-z)^{-1}+\sum _{r=1}^{\infty }(H_0(t)-z)^{-1/2}\tilde{A}^r(H_0(t)-z)^{-1/2},
\label{5}
\end{equation}
$$\tilde{A}=(I-P)(H_0(t)-z)^{-1/2}V(H_0(t)-z)^{-1/2}(I-P).$$
%%%%%%%We prove that on the rich set $B_0(k)$ of $t$ this series converges and has
%%%%an asymptotic character.
 To prove the convergence of the series, it is enough to check that
\begin{equation}
\|\tilde{A}(z)\|<1/2,\ z\in C_0. \label{A*}
\end{equation}
 This estimate is obvious on the left side of the contour and on the horizontal parts. Let us consider the right side.
 It is easy to see that
$$\|\tilde{A}\|\leq  \max _{m: P_{mm}=0}\sum _{q\in Z^n\setminus
\{ 0 \}}\frac{|v_q|}{\sigma _{mq}}.$$
 Using the definition (\ref{t6}) of $P$, we get
$$\|\tilde{A}\|\leq \sum _{q\in Z^n\setminus \{ 0
\}}\frac{|v_q||q|^{\nu }}{4\|V\|_*}<1/4.$$ Thus,
 we obtain (\ref{A*}).
Note that
\begin{equation}
\|(I-P)(H_0-z)^{-1/2}\|= \left( \min _{m\in Z^n,\
m:P_{mm}=0}|p_m^{2l}(t)-z|\right) ^{-1/2}. \label{12}
\end{equation}
 If $\sigma _{m}>4 \|V\|_*$, then, obviously, $|p_m^{2l}(t)-k^{2l}|>0$. Thus,
$(I-P)(H_0-z)^{-1/2}$ is bounded. Estimate (\ref{A*}) provides
convergence of the series for the resolvent on the contour $C_0$.
Taking into account that $(H_0-z)^{-1}$ is a Sturm-Liouville
operator, we obtain that the series converges in the trace class.
Thus,
 $N_{\alpha (I-P)V(I-P)}(k^{2l},t)$ is a continuous
function of $\alpha V$, $0\leq \alpha \leq 1$. Considering that
$N_{(I-P)V(I-P)}(k^{2l},t)$ and $N_0(k^2,t)$ are integers, we get
(\ref{t2*}). $\Box $

Note that the difference between $H(t)$ and $\tilde{H}(t)$ is a
finite dimensional operator:
$\tilde{H}(t)-H(t)=(I-P)VP+PV(I-P)+PVP.$ The dimension of this
operator is not greater than $2\# (t)$, where $\# (t)$ is the
dimension of $P$. According to the general perturbation theory
(see f.e. [Kato]):
\begin{equation}
|N_V(k^{2l},t)-N_{(I-P)V(I-P)}(k^{2l},t)|\leq 2\# (t).\label{16}
\end{equation}
\begin{1} The function $\# (t)$ satisfies the following estimate:
\begin{equation}
\int_{K}\#(t)dt<ck^{-\xi}\ln k, \ \ c=c(\|V\|_*,\nu
,a_{\max},l,n). \label{t8}
\end{equation}
\end{1}
\underline{Proof.} Let $\#(\Omega ,t)$ be the number of points
$\vec{p}_m(t)$ in a measurable set $\Omega \subset R^n$. It is
easy to see that $$\int_{K}\#(\Omega, t)dt=V(\Omega),$$ where $V$
is the volume.
%%%%%It is easy to see that inequality
%%%%%%%%%$$ \#(\Omega, t)>k^{s}V(\Omega}$$
%%%%%%%%can be satisfied only on the set of $t$ with measure less then $ck^{-s}$
Let us define $\Omega $ in $R^n$,  as follows:
\begin{equation}
\Omega =\cup _{q\in Z^n\setminus\{0\}}\Omega _{q},\ \ \Omega
_q=\left\{ x: ||x|^{2l}-k^{2l}|||x+\vec{p}_q(0)|^{2l}-k^{2l}|
<F(q)\right\},
\label{ah2}
\end{equation}
%%%%%%%\begin{equation}
$F(q)= \left( 4|q|^{-\nu }\|V\|_*\right)^2.$
%%%%%%%% \label{ah2} \end{equation}
By the definition of $P$, $P_{mm}=1$ if and only if $\vec{p}_m(t)$
is in $\Omega $. Hence, $\#(t)=\#(\Omega,t).$ Thus, it suffices to
prove the estimate:
\begin{equation}
V(\Omega )<ck^{-\xi }\ln k. \label{t13a}
\end{equation}
Let us represent $\Omega _q$ in the form $\Omega _q=\Omega
_q^>\cup \Omega _q^<$, where
 \begin{equation} \Omega _q^>= \{x\in
\Omega _q:||x+\vec{p}_q(0)|^{2l}-k^{2l}|\geq ||x|^{2l}-k^{2l}|\},
\label{ah1} \end{equation}
 and, correspondingly,
 $$\Omega _q^<=
\{x\in \Omega _q:||x+\vec{p}_q(0)|^{2l}-k^{2l}|\leq
||x|^{2l}-k^{2l}|\}.$$ It is easy to see that under the parallel
shift $x\rightarrow x+\vec{p}_q(0)$ the set $\Omega _q^>$ is
transformed into $\Omega _{-q}^<$. Thus, $V(\Omega _q^<)=V(\Omega
_{-q}^>)$. Considering (\ref{ah2}), we see that $$V(\Omega )\leq
2\sum _{q\in Z^{n}\setminus \{0\}}V(\Omega _q^>).$$ To prove
(\ref{t13a}), it suffices to obtain the estimate
\begin{equation}
V(\Omega _q^>)<c|q|^{-n-\delta }k^{-\xi }\ln k,\ \  \delta >0.
\label{t13m}
\end{equation}
for all $q\neq 0$. Let us prove (\ref{t13m}). Using (\ref{ah2})
and (\ref{ah1}), it is easy to see that  $\Omega _q^>$ lies in the
spherical shell
 $\hat{S}=
 \{x: ||x|^{2l}-k^{2l}|^2\leq \left( 4|q|^{-\nu }\|V\|_*\right) ^2\}=\{x: ||x|-k|\leq 4|q|^{-\nu }\|V\|_*k^{-2l+1}\} $.
 Let us split $\hat{S}$ into the spherical shells:
$$\hat{S}\subset \cup _{m=0}^{M}S_{m}(k),\ \  M=\left[ \ln \left(
12 |q|^{\nu+1}\|V\|_*k^{2l-1}\right) \right],$$ where
\begin{equation}
S_0=\{x: ||x|-k|<|q|^{-2\nu -1}k^{-n+1 -\xi }\}, \label{S0}
\end{equation}
$$S_{m}=\{x: |q|^{-2\nu -1}k^{-n+1-\xi }e^{(m-1)}\leq
||x|-k|<|q|^{-2\nu -1}k^{-n+1-\xi }e^{m}\},$$ $\ m=1,...,M.$
  Then,
$V(\Omega_q^>)\leq \sum _{m=0}^{M}V(\Omega_q^>\cap S_{m})$.
Clearly,
\begin{equation}
V(\Omega_q^> \cap S_{0})\leq V(S_{0})<c|q|^{-2\nu -1}k^{ -\xi
}<c|q|^{-n-\delta }k^{-\xi },\ \delta =2\nu +1-n>0. \label{t8a}
\end{equation}
Let us consider the intersection of a spherical shell $S_{m}$, $m>
0$, with the plane layer
%%%%%%  $T_{q, j\nu\xi }$:
$T_{q, m}=\left\{ x: ||x+\vec{p}_q(0)|^2-|x|^2|<|q|\left( 26
\|V\|_*\right) ^2ke^{-m}\right\} .$
 We prove that
\begin{equation}
\Omega _q^>\cap S_{m}\subset \Omega _q^>\cap S_{m}\cap T_{q,m}.
\label{t11}
\end{equation}
Suppose it is not so. Then there is $x\in \Omega  _q^>\cap S_{m}$,
such that $x\not \in T_{q,m}$. From the definitions of $S_{m}$ and
$T_{q,m}$, we see that
\begin{equation}
||x|^2-k^2|\geq |q|^{-2\nu -1}k^{-n+2 -\xi }e^{m-1}, \label{t9}
\end{equation}
\begin{equation}
||x|^2-k^2|< 2|q|^{-2\nu -1}k^{-n+2 -\xi }e^{m}, \label{t9a}
\end{equation}
\begin{equation}
||x+\vec{p}_q(0)|^2-|x|^2|\geq |q|\left( 26\|V\|_*\right) ^2
ke^{-m}. \label{t9b}
\end{equation}
Using (\ref{t9a}), (\ref{t9b}) and the inequality $m\leq M$, it is
easy to check that
\begin{equation}
||x+p_{q}(0)|^2-k^2|\geq  |q|\left( 13\|V\|_*\right)^2ke^{-m}.
\label{t10}
\end{equation}
 Multiplying (\ref{t9}) and (\ref{t10}), we get
$||x|^2-k^2|||x+p_{q}(0)|^2-k^2|>\left( 6|q|^{-\nu
}\|V\|_*k^{-2l+2}\right) ^2$. This contradicts to the fact that
$x\in \Omega _q$ (see (\ref{ah2})). Thus, (\ref{t11}) holds.
Therefore,
\begin{equation}
V(\Omega _q^>\cap S_{m})\leq V( S_{m}\cap T_{q,m}). \label{t11a}
\end{equation}
From the definitions of $\hat{S}$ and $T_{q,m}$ it follows that
$S_{m}\cap T_{q,m}=\emptyset$ if $|q|>c_1k$, $c_1=50(1+\|V\|_*)$,
hence it is enough to consider $|q|\leq c_1k$. Let us check that
\begin{equation} V( S_{m}\cap T_{q,m})<c|q|^{-n-\delta }k^{-\xi
},\ \delta >0 \label{***}.\end{equation}
 It is not difficult to estimate the volume of $S_{m}\cap T_{q,m}$, as the volume  of the intersection of
a spherical shell with a plane layer. \footnote{f.e. Lemma 2.3 in
\cite{K} gives the estimates for the measures of the intersections
of the sphere $|x|=k$ with  the spherical layers $T_{q,m}$,
depending on $q$ and the width of  $T_{q,m}$ . Integrating the
estimates over the radius of the sphere we get the estimates for $
V(S_{m}\cap T_{q,m})$.}  If $n\geq 3$, then
\begin{equation} V( S_{m}\cap T_{q,m})
<ca_mb_{mq}k^{n-2}, \label{ah}
\end{equation}
$a_m$ and $b_{mq}$ being the widths of $S_{m}$ and $T_{q,m}$,
correspondingly, $a_m=|q|^{-2\nu -1}k^{-n+1-\xi }e^{m}$,
$b_{mq}<8\pi a_{\max } \left( 26\|V\|_*\right)^2ke^{-m}$. Thus,
 $V( S_{m}\cap T_{q,m}))<c|q|^{-2\nu -1 }k^{-\xi }.$
 We see that (\ref{***}) holds for $\delta =2\nu +1 -n$, $n\geq 3$.
 If $n=2$ and  $0<|q|<k/2$, we can use the estimate (\ref{ah}).
 However, the estimate (\ref{ah}) is not valid when $|q|\approx k$.
 For  $2k >|q|\geq
 k/2$ we  use the estimate:
$V( S_{m}\cap T_{q,m})<ca_n\sqrt{b_{mq}k}\leq ck^{-\xi }
|q|^{-2\nu-1}e^{m/2}.$ Taking into account that $m\leq M$,
$M\approx (2l+\nu )\ln k$, we get \begin{equation} V( S_{m}\cap
T_{q,m})\leq ck^{-\xi -3\nu/2-1+l}\leq ck^{-\xi -2-\delta },\ \
\delta =3\nu /2-l-1. \label{41a} \end{equation}
 Using   (\ref{ah}) for $|q|\leq k/2$ and (\ref{41a})  for $|q|>k/2$, we get (\ref{***}) in the case
 $n=2$.
 Considering (\ref{t8a}), (\ref{t11a}), (\ref{***}) and summarizing the  estimates (\ref{***}) over $m$, we get
 $V(\Omega _q^>)\leq V(S_0)+\sum _{m=1}^M V( S_{m}\cap T_{q,m})\leq (M+1)|q|^{-n-\delta }k^{-\xi }
 \leq c|q|^{-n-\delta }k^{-\xi }\ln k.$
 Thus,  we have obtained (\ref{t13m}).
$\Box $

\underline{The proof of Theorem 1.} By definition,
$D_V(k^{2l})=\int _K N_V(k^{2l},t)dt $. Using Lemmas 1 and 2, and
estimate (\ref{16}), we obtain $$|D_V(k^{2l})-D_0(k^{2l})|\leq
2\int _{K}\# (t)dt \leq ck^{-\xi }\ln k.\ \  \Box $$
\section{The Schr\"{o}dinger Operator in $R^3$.}
\subsection{The main results.}
 First, we consider the case of a trigonometric polynomial:
\begin{equation}
V(x)=\sum _{m\in Z^3,\ 0<|m|<R_0}v_m\exp i(\vec{p}_m(0),x),\ \
R_0<\infty . \label{6*}
\end{equation}
 Before formulating the main theorems let us represent $V(x)$ as a
sum of potentials $V_q$, each changing only in one direction. To
do that,
 we start with the definition of a set $\Gamma (R_0)$: let us consider $m:m\in Z^3, 0<|m|<R_0$. In
this set some of the $m$ are scalar multipliers of others. Let us
keep from every family of the scalar multipliers only the minimal
representative, i.e., the representative having the minimal
length. We denote by $\Gamma (R_0)$ the union of these minimal
representatives. In other words, each $m:m\in Z^3, 0<|m|<R_0$ can
be uniquely represented in the form $m=rq$, where $r\in Z\setminus
\{ 0\}$, $q\in \Gamma (R_0)$. It is easy to see that
\begin{equation}
V=\sum _{q\in \Gamma (R_0)}V_q,    \label{23a}
\end{equation}
where $V_q$ depends only on $(x,\vec{p}_q(0))$, $$V_q=\sum _{r\in
Z\ ,0<|rq|<R_0}v_{rq}\exp ir(\vec{p}_q(0),x).$$ We consider the
operators $H_q=H_0+V_q$.
%%%%%%%% We will show in {\bf 3.2} that the spectral study of $H_q$ can be reduced to that of
%%%%%%55555a periodic Schr\"{o}dinger operator on the real line.
 Let $\delta D_{V_q}(k^2)=D_{V_q}(k^2)-D_{0}(k^2)$, and
 \begin{equation}
d_V(k^2)=\sum _{q\in \Gamma (R_0)}\delta D_{V_q}(k^2). \label{3b}
\end{equation}
\begin{2}If $V$ is a trigonometric polynomial (\ref{6*}), then the density of states of  operator $H$ satisfies
the following asymptotic formula as $k\to \infty $:
\begin{equation}D_V(k^2)=\omega _3k^3+d_V(k^2)+O(k ^{-\zeta }),\label{3a}
\end{equation}
where  $\zeta >1/105 $, and $d_V(k^2)$, given by (\ref{3b}),
satisfies the estimate:
\begin{equation}
|d_V(k^2)|<c\|V\|_*(1+\|V\|_*), \ \ c=c(a_{\max },a_{\min }).
\label{3b+}
\end{equation}
\end{2}
Estimate (\ref{3b+}) is uniform in $R_0$ when $\|V\|_*$ stays
finite, while $O(k ^{-\zeta })$ is not uniform (for details, see
the proof of the theorem in Section 3.3). The asymptotic formula
uniform in $R_0$ we will be given in Theorem 4.

 We
are going to show that $d_V$ is close to a constant $d_V^0$ and to
obtain the formula for $d_V^0$ . First, we consider $H_q=-\Delta
+V_q$. Let us direct $x_1$ along $\vec{p}_q(0)$ (the density of
states $D_{V_q}$ is invariant with respect to this operation
\cite{Sh1,Sh2,RoShSo}). Obviously, by separation of variables, the
spectral study of $H_q$ can be reduced to that of the
one-dimensional periodic Schr\"{o}dinger operator :
\begin{equation}
H_q^{(1)}=-\frac{d^2}{dx_1^2}+V_q^{(1)}(x_1), \label{one}
\end{equation}
$$V_q^{(1)}(x_1)=\sum _{r\in Z,0<|rq|<R_0}v_{rq}\exp \left(
irp_q(0)x_1\right) , \ \  {p}_q(0)=|\vec{p}_q(0)|.$$ Let $\rho
_{V_q}(k^2)$ be the integral density of states of $H_q^{(1)}$, and
$\delta \rho _{V_q}=\rho _{V_q}-\rho _{0}$. It is easy to show
(Section 3.3) that
 \begin{equation}
 \delta D_{V_q}=\pi \int _{-\infty }^{k^2}\delta \rho _{V_q}(y)dy, \label{gamma}
 \end{equation}
  $\delta \rho _{V_q}(y)$ satisfying the estimate: $$|\delta \rho _{V_q}(y)|<c_0(V_q)y^{-3/2},$$
   for $\sqrt{y}>10\|V_q\|p_q^{-1}(0)$, here and below
 \begin{equation}
 c_0(V_q)=c\|V_q\|_*(a^{-2}_{\min }+\|V_q\|_*).\ \ c\neq
 c(y,V_q,a_1,a_2,a_3), \label{47a}
 \end{equation}
Therefore,
 \begin{equation}
 \left| \delta D_{V_q}-\pi\int _{-\infty }^{\infty }\delta \rho
 _{V_q}(y)dy \right| <c_0(V_q)k^{-1},\ \ k>10\|V_q\|p_q^{-1}(0).\label{gamma1}
 \end{equation}
  Using the definition (\ref{3b}) of $d_V$, we will get
   \begin{equation}
 \left| d_V^0-d_V\right|<c_0(V)k^{-1},
  \label{d1} \end{equation}
  where
  \begin{equation}
  d_V^0=\sum _{q\in \Gamma (R_0)}\int _{-\infty }^{\infty }\delta \rho
  _{V_q}(y)dy.\label{d0}
  \end{equation}
  Note that $d_V^0$ is a constant with respect to $k^2$. We will show that
  \begin{equation}
  |d_V^0|<c\|V\|_*(1+\|V\|_*),\ \ c=c(a_{\max }, a_{\min }).\label{52a*}
  \end{equation}
   The next
  theorem follows from Theorem 2 and relations (\ref{d1}), (\ref{52a*}).
  \begin{2}The density of states of  operator $H$ satisfies
the following asymptotic formula as $k\to \infty $:
\begin{equation}
D_V(k^{2})=\omega _3k^3+d_V^0+O(k ^{-\zeta }),\label{3a**}
\end{equation}
where $d_V^0$ is a constant given by (\ref{d0}), $d_V^0$ satisfies
estimate (\ref{52a*}) and $\zeta >1/105$.
\end{2}
If $V$ satisfies (\ref{smooth1}) with $\nu =900$, then we can pass
to the limit in the definitions of  $d_V(k^2)$ and $d_V^0$ as
$R_0\to \infty $:
 $d_V(k^2)=\sum _{q\in \Gamma
(\infty)}\delta D_{V_q}(k^2)$ and $ d_V^0=\sum _{q\in \Gamma
(\infty )}\int _{-\infty }^{\infty }\delta \rho
  _{V_q}(y)dy $, where $\Gamma (\infty )=\lim _{R_0\to \infty }  \Gamma
  (R_0)=\cup _{R_0=1}^{\infty }\Gamma (R_0)$.
  \begin{2} If $V$ satisfies (\ref{smooth1}) with $\nu =900$, then relations (\ref{3a}), (\ref{3b+}) and
  (\ref{52a*}), (\ref{3a**} ) hold, where  $|O(k^{-\zeta
  })|<c_1k^{-\zeta}$, $\zeta >1/130$, $c_1=c_1(\|V\|_*,a_{\max
  },a_{\min })$.
   \end{2}
  Theorems 2 -- 4 are in correspondence with the result of G.Eskin,
J.Ralston, E.Trubowitz \cite{ERT1,ERT2,ERT3} that isospectral
potentials must have the same directional components $V_q$.



   We will start our considerations with the case of a
  trigonometric polynomial. We want to be able to extend proofs
  to the case of a smooth potential. For this reason everywhere
  below the only  estimate for $R_0$ which we  use is
  $R_0<k^{\delta /8}$, $\delta \leq 1/105 $.



\subsection{The spectra of the operators $H_q$.}
 The purpose of this
section is to show that eigenvalues $\lambda _m^q(t)$ of $H_q(t)$
can be  represented in the form: $\lambda
_m^q(t)=p_m^{2l}(t)+\Delta \Lambda _q(t)_{mm}$, where the shift
$\Delta \Lambda _q(t)_{mm}$ of $p_m^{2l}(t)$ under the
perturbation $V_q$ admits the estimate (\ref{3.3.26}).

 Let us consider the matrix of $H_q(t)$ in $l_2^3$, $l_2^3$ being the space of
square-summable sequences with indices in $Z^3$. We
associate with each $i$ in $Z^3$ the diagonal projection $P_i^q$ in $l_2^3$:
\begin{equation}
(P_i^q)_{mm}=\left\{ \begin{array}{ll}1,\ \ &
\mbox{if }\vec{p}_i(0)-\vec{p}_m(0)=l\vec{p}_q(0),\ l\in Z,\ \
\\0,\ \   &\mbox{otherwise}.\end{array}\right.
\label{3.3.16a}
\end{equation}
Obviously, $P_i^q=P_{j}^{q}$ if $\vec{p}_i(0)-\vec{p}_j(0)=l\vec{p}_q(0)$, $l\in Z$.
It is clear that there
exists a minimal subset $J_q^0$ of $Z^3$ such that
\begin{equation}
\sum _{i\in J_q^0}P_i^q=I. \label{3.3.17}
\end{equation}
Note that $(V_q)_{rn}=0$ if $\vec{p}_r(0)-\vec{p}_n(0)\neq
l\vec{p}_q(0)$, $l\in Z$, because $V_q$ depends only on
$(x,\vec{p}_q(0))$. This at once gives us that $P_i^qV_qP_i^q=
P_i^qV_q=V_qP_i^q$, and, therefore,
\begin{equation}
P_i^qH_q(t)P_i^q=P_i^qH_q(t)=H_q(t)P_i^q.   \label{3.3.18b}
\end{equation}
Considering relations~(\ref{3.3.17}) and~(\ref{3.3.18b}) we get
$H_q(t)=\sum _{i\in J_q^0}P_i^qH_q(t)P_i^q.$

We establish an isometric isomorphism between $P_i^ql_2^3$ and
$l_2^1$, $l_2^1$ being the space of square-summable sequences with
indices in $Z$. Denote by $\delta _m$, $m\in Z^3$, the element of
$l_2^3$ given by the formula $\{ \delta _m\} _n=\delta _{mn}$, and
by $\delta _l^1$, $l\in Z$,
 the analogous element of $l_2^1$ given by the formula $\{ \delta _l^1\} _r=
\delta _{lr}$, $\delta _{mn},\delta _{lr}$ being the Kronecker symbols. To construct an isometric isomorphism we represent $\vec{p}_i(t)
$, $i\in J_q^0$, in the form of a linear combination of $\vec{p}_q(0)$ and a vector in
the orthogonal complement of $\vec{p}_q(0)$: $\vec{p}_i(t)=
\tau _i(2\pi )^{-1}\vec{p}_q(0)+
\vec{d}^q_i$, $\tau _i\in R$, $\tau _i=\tau _i(t,q)$
, $(\vec{d}^q_i(t),\vec{p}_q(0))=0$. Thus, if $\delta _m\in P_i^ql_2^3$, then
$\vec{p}_m(t)$ is uniquely representable in the form
\begin{equation}
\vec{p}_m(t)=\vec{d}^q_i(t)+(\tau _i+2\pi l)(2\pi )^{-1}\vec{p}_q(0),\ \ l\in Z,
\label{3.3.19}
\end{equation}
where it can be assumed without loss of generality that $0\leq \tau _i<2\pi
$. From this,
\begin{equation}
(P_i^q)_{mm}=\left\{ \begin{array}{ll}1,\ \ &\mbox{if
}\vec{p}_m(t)= \vec{d}^q_i(t)+ (\tau _i+2\pi l)(2\pi
)^{-1}\vec{p}_q(0),\ \ l\in Z,\ \ \\0,\ \
&\mbox{otherwise.}\end{array}\right.
     \label{3.3.20}
\end{equation}
An isomorphism between $P_i^ql_2^3$ and $l_2^1$ is now established in the
natural way with the help of the formula~(\ref{3.3.19}): $\delta _m
\leftrightarrow \delta _l^1$,
 $\delta _m\in P_i^ql_2^3$,  $\delta _l^1\in l_2^1$, and it follows from~(\ref
{3.3.19}) that
\begin{equation}
l=[(\vec{p}_m(t) ,\vec{p}_q(0))p_q(0)^{-2}],   \label{3.3.21}
\end{equation}
\begin{equation}
\tau _i=2\pi(\vec{p}_m(t),\vec{p}_q(0))p_q(0)^{-2}-2\pi l.  \label{3.3.21a}
\end{equation}
It is easy to verify that the operator $P_i^qH_qP_i^q$ is
equivalent to the operator $H_1(\tau _i)+|(\vec{d}^q_i(t)|^2I$;
here the operator $H_1(\tau _i)$ is given by the matrix
\begin{equation}
H_1(\tau _i)_{lp}=(\tau _i+2\pi l)^2a_q^{-2}\delta _{lp}+
v_{(l-p)q},\ \ l,p\in Z, \ \ a_q=(2\pi )p_q(0)^{-1}. \label{3.3.22}
\end{equation}
This is the matrix representation of a periodic Schr\"{o}dinger
operator (\ref{one}) on the real axis.
 Each operator $H_1(\tau )$, $\tau \in [0,2\pi ),$ has a discrete spectrum.
It is well known that its eigenvalues $\lambda _l(\tau )$
 can be enumerated by integers in such a way that
\begin{equation}
|\lambda _l(\tau )-(2\pi l+\tau
)^2a_q^{-2}|<c_0(V_q)(lp_q(0))^{-2}, \label{3.3.15}
\end{equation}
when $lp_q(0)>10\|V_q\|p_q^{-1}(0)$, $c_0(V_q)$ being given by
(\ref{47a}). The functions $\lambda _l(\tau )$ are piecewise
continuous. Let $\Delta \lambda _l^q(\tau _i)$ be the shift of the
eigenvalue under the perturbation: $\Delta \lambda _l^q(\tau
_i)=\lambda _l^q(\tau _i)- (\tau _i+2\pi l)^2a_q^{-2}$. The
spectrum of the operator $P_i^qH_qP_i^q$ can be represented in the
form: $$\{ \lambda _l^q(\tau _i)+|(\vec{d}^q_i(t)|^2\} _{l\in
Z}=$$ $$=\{ \Delta \lambda _l^q(\tau _i)+(\tau _i+2\pi l)^2(2\pi
)^{-2}p_q(0)^2+ |(\vec{d}^q_i(t)|^2\} _{l\in Z}=$$ $$=\{ \Delta
\lambda _{l(m)}^q(\tau _i)+p_m^2(t)\} _{m:\delta _m\in
P_i^ql_2^3}= $$
\begin{equation}
=\{ \Delta \lambda _{l(m)}^q(\tau _m)+p_m^2(t)\} _{m:\delta _m\in P_i^ql_2^3};
\label{3.13.28af}
\end{equation}
here $\tau _m$ can be computed from the formula
\begin{equation}
\tau _m=2\pi(\vec{p}_m(t),\vec{p}_q(0))p_q(0)^{-2}-2\pi [(\vec{p}_m(t) ,\vec{p}_q(0))p_q(0)^{-2}],
  \label{3.3.24}
\end{equation}
and, as it is easily seen, $\tau _m$ coincides with $\tau _i$ for all $m:
\delta _m \in
P_i^ql_2^3$,
and $l(m)$ is given by the formula~(\ref{3.3.21}). The spectrum $\Lambda _q(t)$
of the operator $H_q$ is the union of the spectra of the operators
$P_i^qH_q(t)P_i^q$:
$$\Lambda _q(t)=\cup _{i\in J_q^0}
\{ \Delta \lambda _{l(m)}^q(\tau _m)+p_m^2(t)\} _{m:\delta _m\in P_i^ql_2^3}=
\{ \Delta \lambda _{l(m)}^q(\tau _m)+p_m^2(t)\} _{m\in Z^3}.$$
Introducing the diagonal matrix $\Delta \Lambda _q(t)$ of shifts of the
eigenvalues,
%%%%%\begin{equation}
$\Delta \Lambda _q(t)_{mm}=\Delta \lambda _{l(m)}^q(\tau _m)$,
%%%%%%5\label{3.3.25}
%%%%%%%%\end{equation}
we get that the diagonal matrix $\Lambda _q(t)$ of eigenvalues of the operator
$H_q(t)$ is $H_0(t)+\Delta \Lambda _q(t)$. Using (\ref{3.3.21}) and (\ref{3.3.15}), we obtain:
\begin{equation}
|\Delta \Lambda _q(t)_{mm}|< c_0(V_q)|(\vec{p}_m(t)
,\vec{p}_q(0))|^{-2},  \label{3.3.26}
\end{equation}
 when $|(\vec{p}_m(t) ,\vec{p}_q(0))|>10\|V_q\|$, where $ c_0(V_q)$ is given by (\ref{47a}).
\subsection{The densities of states of the operators $H_q$.}
\begin{1} The functions $\delta D_{V_q}(k^2)$ admit representation
(\ref{gamma}), estimates (\ref{gamma1}) and
\begin{equation}
|\delta D_{V_q}(k^2) |<c\|V_q\|_*(1+\|V_q\|_*),\ \  c=c(a_{\max },
a_{\min }). \label{dV0}
\end{equation}
\end{1}
\underline{Proof.} Let us calculate $D_{V_q}(k^2)$ in terms of the
 density of states of
  the corresponding one-dimensional Schr\"{o}dinger operator (\ref{one}). By rotating the
  coordinates, we direct $x_1$ along $\vec{p}_{q}(0)$. It  does not  change the density of states,
    \cite{Sh1,Sh2,RoShSo}. By separation of variables, we get
 $D_{V_q}(k^2)=$
  $$=\sum _{n_2,n_3\in Z, n_1\in N}\int_{K}\chi
  \left( k^2-(t_2+2\pi n_2)^2-(t_3+2\pi n_3)^2-\lambda
  _{n_1}(t_1)\right) dt_1$$
   $$= 2\pi \sum _{n_1\in N}\int _0^{\infty }\int _{0}^{2\pi }
  \chi
  (k^2-\rho^2-\lambda
  _{n_1}(t_1))\rho d\rho dt_1,$$
  where $\lambda _{n_1}(t_1)$ are Bloch eigenvalues of the
  operator (\ref{one}). Changing the
  variable $y=k^2-\rho ^2$, we get (\ref{gamma}).
  The regular perturbation theory arguments give:
  $$N_0^{(1)}(y - \|V_q\|)\leq N_{V_q}^{(1)}(y)\leq N_0^{(1)}(y +
  \|V_q\|),$$
  where $N_{V_q}^{(1)}$ is the counting function  (\ref{N}) of the  operator
  (\ref{one}), $\|V_q\|$ is the norm of $V_q$ in the class of bounded
  operators, $\|V_q\|<\|V_q\|_*$.
 Considering (\ref{3.3.15}) it is easy to get more precise
 estimate for   large $y$:
 $$N_0^{(1)}(y - c_0(V_q)y^{-1})\leq N_{V_q}^{(1)}(y)\leq N_0^{(1)}(y +
 c_0(V_q)y^{-1}),$$ when $\sqrt{y}>10\|V_q\|p_q^{-1}(0)$.
 Taking into account that $N_0(y)=\sqrt{y}$ and integrating the last estimates, we
 get
  $ \left|\delta \rho _{V_q}(y)\right|   < \|V_q\|(y+\|V_q\|)^{-1/2}$ and
  $\left|\delta \rho _{V_q}(y)\right| < c_0(V_q)y^{-3/2}$.
 {\footnote{The complete asymptotic expansion of $\rho _V$ for $V\in C^{\infty }$ is given in  \cite{ScSh}.}
Using the second estimate, we get (\ref{gamma1}).
 Using (\ref{gamma}) and combining both estimates, we get
 (\ref{dV0}).
 $\Box $

\begin{1}If $V$ is a trigonometric polynomial with $R_0<k^{\delta }$, $0<\delta <1/105$, then  the function $d_V(k^2)$,
 defined by (\ref{3b}), satisfies the estimates  (\ref{3b+}) and
\begin{equation}
|d_V-d_V^0|<c_0(V)k^{-1},\ \  c\neq c(V).  \label{d2}
\end{equation}
 $d_V^0$ being the constant given by (\ref{d0}).
 \end{1}
 {\bf Corollary.} Estimate (\ref{52a*}) holds.\\

\underline{Proof.}
 Summarizing (\ref{dV0}) and (\ref{gamma1}) over all $q$ and taking into account that
  \begin{equation}
  \sum _{q}\|V_q\|_*^j\leq \|V\|_*^j, \ \ j=1,2, \label{66b}
  \end{equation}
   we obtain  (\ref{3b+}) and (\ref{d2}). $\Box $

Let us introduce  the notations:
\begin{equation}
\delta \chi _q(\vec{p}_m(t),k)=
\chi \left( k^2-p_m^2(t)-\Delta \Lambda _q(t)_{mm}\right) -\chi \left( k^2-p_m^2(t)\right) ,\label{33}
\end{equation}
\begin{equation}
P(V_q,k^2,M)=\int _{K}\sum _{m\in
Z^3:|(\vec{p}_m(0),\vec{p}_q(0))|<M,\ |p_m^2(t)-k^2|\leq \|V_q\|}
\delta \chi _q(\vec{p}_m(t),k)dt. \label{21}
\end{equation}
It turns out that $P(V_q,k^2,M)$ is a good approximation for
$\delta D_{V_q}(k^2)$ as $M\to \infty $:
\begin{1} If $M>8\pi a_{\min }^{-1}p_q(0)+10\|V_q\|$, then the following estimate holds:
\begin{equation}
\left| \delta D_{V_q}(k^2)-P(V_q,k^2,M)\right|
<c_0(V_q)p_q^{-1}(0)M^{-1}.\label{20}
\end{equation}
\end{1}
\underline{Proof.} By definition, $\delta D_{V_q}(k^2)=\int _{K}
\delta N_{V_q}(k^2,t)dt$, where $\delta N_{V_q}(k^2,t)=\sum _{m\in
Z^3}\delta \chi _q(\vec{p}_m(t),k).$ Let us break the area of
summation and, correspondingly, the sum into the two: $\delta
N_{V_q}(k^2,t)=\Sigma _1+\Sigma _2$, $$\Sigma _1(M,k,t)= \sum
_{m\in Z^3:|(\vec{p}_m(0),\vec{p}_q(0))|<M}\delta \chi
_q(\vec{p}_m(t),k),$$ $$ \Sigma _2(M,k,t)= \sum _{m\in
Z^3:|(\vec{p}_m(0),\vec{p}_q(0))|\geq M}\delta \chi
_q(\vec{p}_m(t),k).$$ Thus,
\begin{equation}
 \delta D_{V_q}(k^2)=\int _{K}\Sigma _1dt+\int _{K}\Sigma _2dt.
 \label{70a}
 \end{equation}
 We consider $\Sigma _1$. Note  that
$|\Delta \Lambda _q(t)_{mm}|\leq \|V_q\|$ and, therefore, taking
into account (\ref{33}), we see that
\begin{equation}
\delta \chi (\vec{p}_m(t),k)=0,\ \mbox{when}\
|k^2-p_m^2(t)|>\|V_q\|. \label{52a}
\end{equation}
Using notation (\ref{21}), we easily obtain:
\begin{equation}
\int _{K}\Sigma _1(M,k,t)dt=P(V_q,k^2,M).\label{56a}
\end{equation}
%%%%%%%%%$$\Sigma _{01}=\sum _{m\in Z^3:|(\vec{p}_m(t),\vec{p}_q(0))|<M}\chi (k^2-p_m^2(t)).$$
Let us consider $\Sigma _2$.
 Taking into account (\ref{3.3.26}), we get
\begin{equation}
\delta \chi (\vec{p}_m(t),k)=0,\ \mbox{when}\ |k^2-p_m^2(t)|>c_0
(V_q)|(\vec{p}_m(t),\vec{p}_q(0))|^{-2},
 \label{52att}
\end{equation}
 and, therefore,
$$\Sigma _2= \sum _{m\in Z^3:|(\vec{p}_m(0),\vec{p}_q(0))|\geq M,
\ |k^2-p_m^2(t)|\leq
c_0(V_q)|(\vec{p}_m(t),\vec{p}_q(0))|^{-2}}\delta \chi
_q(\vec{p}_m(t),k).$$ Considering that $|\delta \chi _q|\leq 1$
and $|(t,\vec{p}_q(0))|\leq M/2$, we obtain: $\left| \Sigma
_{2}(M,k,t)\right| \leq \# (\Omega _{1q},t)$, where $\Omega
_{1q}\subset R^3$,
 $$\Omega _{1q}(M)=\{ x:||x|^2-k^2|\leq
c_0(V_q)|(x, \vec{p}_q(0))|^{-2},\ |(x, \vec{p}_q(0))|\geq
M/2\}.$$ It is easy to show that
%%%%%%%\begin{equation}
$V(\Omega _{1q}(M))<c_0(V_q)p_q^{-1}(0)M^{-1}$.
 Thus,
\begin{equation}
\int _{K}\left| \Sigma _{2}(M,k,t)\right| dt\leq \int _{K}\#
(\Omega _{1q},t)dt=V(\Omega _{1q}(M))<c_0(V_q)p_q^{-1}(0)M^{-1}.
\label{55b}
\end{equation}
Using (\ref{70a}), (\ref{56a}) and (\ref{55b}), we  obtain
 (\ref{20}). $\Box $
\subsection{Auxiliary Operator $\hat{H}(t)$.}
Our considerations are based on the expansion of the resolvent in
a perturbation series. In [K] it was proved that in the case of
the Schr\"{o}dinger operator in $R^3$ the perturbation series
converges not with respect to the free operator $H_0(t)$, but with
respect to some auxiliary operator $\hat{H}(t)$. Let us describe
this operator
 for a trigonometric polynomial (\ref{6*}) with $R_0<k^{\delta /8}$, $0<\delta <1/105$.

We consider the following sets in $Z^3$: $$\Pi(k^{1/5})=\cup _{q
\in \Gamma (R_0)}\Pi _q(k^{1/5}),$$ $$\Pi _q(k^{1/5})=\{ m:\mid
(\vec{p}_m(0),\vec{p}_q(0)) \mid <k^{1/5}\} ,$$ $$T(k)= \cup
_{q,q'\in \Gamma (R_0), q\neq q'}T_{qq'},$$
\begin{equation}
T_{qq'}=\Pi _q(k^{1/5})\cap \Pi _{q'}(k^{3/5})=
\label{29a}
\end{equation}
$$=\{ m:
|(\vec{p}_m(0),\vec{p}_q(0))|<k^{1/5} ,
|(\vec{p}_m(0),\vec{p}_{q'}(0))|
<k^{3/5 }\} .    $$
We define the diagonal projection $P_q$ and the auxiliary operator $\hat{H}$
as follows:
\begin{equation}
(P_q)_{mm} =
\left\{ \begin{array}{ll}1,\mbox{  if }m\in \Pi _q(k^{1/5})
\setminus T(k),\\0,\mbox{  otherwise;} \end{array}\right. \label{30}
\end{equation}
\begin{equation}
\hat{H}(t)=H_0(t)+\sum _{q\in \Gamma (R_0)}P_qV_qP_q.    \label{31}
\end{equation}
It is clear that $\hat{H}(t)$ has a block structure. Each block
$H_0(t)P_q+P_qV_qP_q$ is the ``piece" of $H_q(t)$, corresponding
to $m\in \Pi _q(k^{1/5}) \setminus T(k)$. From the definition of
$T(k)$ it follows that the blocks do not intersect. Because each
block is big enough, it is easy to show, that its eigenvalues
$\hat{\lambda }_m(t)$ are close to corresponding ones of $H_q$,
namely, (see f.e. \cite{K}):
\begin{equation}
\hat{\lambda }_m(t)= p_m^2(t)+\Delta \Lambda _q(t)_{mm}+O(
c_0(V_q)k^{-2/5}),\ \label{57a*}
\end{equation}
$$\mbox{if } m\in \Pi _q(k^{1/5}-M_q)\setminus T(k),$$
\begin{equation}
\hat{\lambda }_m(t)=p_m^2(t)+\Delta \Lambda _q(t)_{mm}+O(
c_0(V_q)k^{-1/5})\ \label{57b*}
\end{equation}
$$\mbox{if } m\in \Pi _q(k^{1/5}) \setminus\left( \Pi
_q(k^{1/5}-M_q)\cup T(k)\right),$$
 $M_q\approx 10R_0p_q(0)$. Thus,
the eigenvalues $\hat{\lambda }_m(t)$ of $\hat{H}(t)$ can be
enumerated by indexes $m\in Z^3$, $\hat{\lambda }_m(t)$ satisfies
(\ref{57a*}), (\ref{57b*}) if $ m\in \Pi _q(k^{1/5})\setminus
T(k)$ for some $q$, and $\hat{\lambda }_m(t)=p_m^2(t)$ otherwise.


Let $\hat{D}_V(k^2)$ be the density of states for the operator $\hat{H}(t)$.
\begin{1} When $k^{1/5}>8\pi a_{\min }^{-1}p_q(0)+10\|V_q\|$,
the following relations hold:
\begin{equation}
\hat{D}_V(k^2)=D_0(k^2)+d_V(k^2)+f(k), \label{25}
\end{equation}
$$ |f(k)|<cR_0^3\|V\|_*(1+\|V\|_*)k^{-1/5},\ \
c=c(a_{\max},a_{\min }).$$
\end{1}
\underline{Proof.} By definition,
$\hat{D}_V(k^2)=\int _{K}\hat{N}_V(k^2,t)dt,$ where
$\hat{N}_V(k^2,t)=
\sum _{m\in Z^3}\chi (k^2-\hat{\lambda }_m(t)).$
 Considering (\ref{33}) it is easy to see that
$\hat{N}_V(k^2,t)=N_0(k^2,t)+\sigma _1+\sigma _2+\sigma _3,$ where
$$\sigma _1=\sum _{q\in \Gamma (R_0)}\sigma _{1q},\ \ \sigma
_{1q}=\sum _{m \in \Pi _q(k^{1/5})\setminus T(k)}\delta \chi
_q(\vec{p}_m(t),k),$$ $$\sigma _2=\sum _{q\in \Gamma (R_0)}\sum
_{m \in \Pi _q(k^{1/5}-M_q)\setminus T(k)}\delta \chi
'_q(\vec{p}_m(t),k),$$ $$\sigma _3=\sum _{q\in \Gamma (R_0)}\sum
_{m \in \Pi _q(k^{1/5})\setminus \left( \Pi _q(k^{1/5}-M_q)\cup
T(k)\right) }\delta \chi '_q(\vec{p}_m(t),k),$$
 $$\chi
'_q(\vec{p}_m(t),k)=\chi \left( k^2-\hat{\lambda }_m(t)\right)
-\chi \left( k^2-p_m^2(t)-\Delta \Lambda _q(t)_{mm}\right) .$$
%%%%%%%$$\chi ''(\vec{p}_m(t),k)=\chi (k^2-p_m^2(t))-\chi (k^2-\hat{\lambda }_m(t)).$$
We represent $\sigma _{1q}(t)$ in the form $\sigma _{1q}=\sigma
_{1q}'-\sigma _{1q}''$, where $$\sigma _{1q}'=\sum _{m \in \Pi
_q(k^{1/5})}\delta \chi _q(\vec{p}_m(t),k),\ \ \sigma _{1q}''=\sum
_{m \in \Pi _q(k^{1/5})\cap T(k)}\delta \chi _q(\vec{p}_m(t),k).$$
Considering  (\ref{21}) and (\ref{52a}), we see that $\int
_K\sigma _{1q}'(t)dt=P(V_q, k^2, k^{1/5})$. To estimate $\sigma
_{1q}''$, let us note that
 $\Pi _q(k^{1/5})\cap T(k)\subset \cup _{q'\in \Gamma (R_0),q'\neq q}T_{qq'}$, $T_{qq'}$ being given by
(\ref{29a}). Again, taking into account (\ref{52a}) and the
inequalities $|\delta \chi _q |\leq 1$,
$|(t,\vec{p}_q(0))|<k^{1/5}$, we get $|\sigma _{1q}''|\leq
\#(\Omega_{3q},t)$, where $\Omega_{3q}= \cup _{q'\in \Gamma
(R_0),q'\neq q}\Omega _{qq'},$
 $\Omega _{qq'}=
\{x:|k^2-|x|^2|\leq \|V_q\|,\ |(x,\vec{p}_{q}(0))|\leq 2k^{1/5},\
|(x,\vec{p}_{q'}(0))|\leq 2k^{3/5}\}.$ Thus, $\left| \int _K\sigma
_{1q}''(t)dt\right| \leq V(\Omega
_{3q})<cR_0^3\|V_q\|p_q(0)^{-1}k^{-1/5}$, and,  hence, $$\left|
\int _K\sigma _{1q}(t)dt - P(V_q,k^2,k^{1/5})\right|
<cR_0^3\|V_q\|k^{-1/5}.$$ Summarizing this estimate over $q$
  and using (\ref{20}), (\ref{3b}) and (\ref{66b}), we obtain
\begin{equation}
\left| \int _K\sigma _1dt - d_V(k^2)\right| <
cR_0^3\|V\|_*(1+\|V\|_*)k^{-1/5}, c=c(a_{\max },a_{\min
}).\label{bb}
\end{equation}
%%%%%%55555The estimate (\ref{64a}) follows from (\ref{20}) and (\ref{22}).
 Considering  estimate (\ref{57a*}) and
the definition of $\chi'_q$,
%%%%%%%for $\hat{\lambda }_m(t)$, $m\in \Pi _q(k^{1/5}-C_q)\setminus T(k)$,
 we get
$$|\sigma _2|\leq \# (\Omega _4,t), \ \ \Omega _4=\cup _{q\in
\Gamma (R_0)}\Omega _{4q},$$ $$ \Omega _{4q}(k^2,
c_0(V_q)k^{-2/5},2k^{1/5})=$$ $$ \{x\in R^3:|k^2-|x|^2-\Delta
_q(x)|<c_0(V_q)k^{-2/5}, |(x, \vec{p}_q(0))|<2k^{1/5}\},$$ the
function $\Delta _q(x)$ being defined by two relations:
$x=\vec{p}_m(t)$ (a unique representation) and $\Delta
_q(x)=\Delta \Lambda _q(t)_{mm}$. In the next lemma we will prove
that $V(\Omega _{4q})<c_0(V_q)p_q(0)^{-1}k^{-1/5}$. Therefore,
$V(\Omega _4)\leq c_0(V)R_0^3k^{-1/5}$. This gives us $\left| \int
_K\sigma _2(t)dt\right| \leq \int _K\# (\Omega _4,t)=V(\Omega
_4)\leq c_0(V)R_0^3k^{-1/5}$. Similarly, considering (\ref{57b*}),
we prove that $\left| \int _K\sigma _3dt\right|
<c_0(V)R_0^3k^{-1/5}$.
 Using the formulae for the integrals of $\sigma _i$, $i=1,2,3$,
 we get (\ref{25}).
  $\Box $
\begin{1} If $L_1, L_2=o(k)$, then the volume of the set $\Omega _{4q}(k^2,L_1,L_2)$:
\begin{equation}
\Omega _{4q}(k^2,L_1,L_2)=\{ x\in R^3:|k^2-|x|^2-\Delta _q(x)|<L_1, |(x,\vec{p}_{q}(0))|\leq L_2\}
 \label{64a*}
\end{equation}
satisfies the estimate
\begin{equation}
V(\Omega _{4q})<cp_q(0)^{-1}L_1L_2,\ \ c\neq c(k,L_1,L_2). \label{35}
\end{equation}
\end{1}
\underline{Proof.} Let us introduce a cylindrical coordinates $(z,
\rho ,\vartheta  )$, $z$ being directed along $\vec{p}_q(0)$.
Using the fact that $\Delta \Lambda _q(x)$ depends only on $z$,
$\Delta \Lambda _q(x)\equiv \varphi (z)$, we get the following
representation for $\Omega _{4q}$: $\Omega _{4q}=\{(z, \rho
,\vartheta ):|\rho ^2-\rho _0^2|<cL_1, |z|\leq L_2p_q(0)^{-1}\},$
where $\rho _0^2=k^2-z^2-\varphi (z)$. From this we easily obtain
(\ref{35}). $\Box $
\subsection{Proof of the Main Result.}
First, we prove that there is a rich subset $B_1(k)$ of $K$, such
that $N_V(k^2,t)=\hat{N}_V(k^2,t)$ for all $t$ in $B_1(k)$ and $k$
big enough (Lemma 9). The set $K\setminus B_1(k)$ has an
asymptotically zero measure as $k\to \infty $, and, if $t\in
K\setminus B_1(k)$, then $|N_V(k^2,t)-\hat{N}_V(k^2,t)|<\# (\Omega
_0, t)$, where the step-function $\# (\Omega_0, t)$ satisfies the
estimate $\int _{K\setminus B_1}\# (\Omega _0,
t)dt<cR_0^2k^{-\delta }$, $\delta
<1/105$. (Lemma 10). Combining Lemmas 6, 9 and 10, we get Theorem
2. Theorem 3 follows from Theorem 2 and Lemma 4. Passing to the
limit as $R_0\to \infty $, we obtain Theorem 4. Lemma 8 describes
the properties of the set $B_1(k)$.

 Let us consider the set
$\Omega _0=\Omega _1\cup \Omega '\cup M \cup T_1$, where $\Omega
_1$ is the spherical shell $\Omega _1=\{ ||x|^2-k^2|\leq
k^{-1-\delta }\} $, $\Omega '$ is  defined by (\ref{ah2}) with
$F_q=k^{-4\delta }$,
 $M=\cup _{q\in \Gamma (R_0)}\Omega _{4q}(k^2,
k^{-1/5-\delta }, k^{1/5})$, $T_1=\cup _{q,q'\in \Gamma
(R_0)}\tilde{\Omega }_{qq'},$ $\tilde{\Omega }_{qq'}=\{
x:||x|^2-k^2|<k^{1/5-10\delta }, |(x,\vec{p}_q(0))|<k^{1/5},
|(x,\vec{p}_{q'}(0))|<k^{3/5}\},$ $\Omega _{4q}$ being given by
formula (\ref{64a*}). Obviously, $V(\Omega _1)<ck^{-\delta }$.
Considering like in Lemma 2, we get $V(\Omega ')<R_0^3k^{-4\delta
}\ln k $. Using Lemma 7, we get $V(M)<R_0^2k^{-\delta }$. It is
easy to show (see f.e. \cite{K}) that $V(\Omega
_{qq'})<cR_0p_q(0)^{-1}p_{q'}(0)^{-1}k^{-10\delta }$, and,
therefore,
$
V(T_1)\leq V(T_1)<cR_0^5k^{-10\delta }.$ Adding above estimates
and taking into account that $R_0<k^{\delta /8}$, we get $V(\Omega
_0 )<cR_0^2k^{-\delta }$ and, therefore,
\begin{equation}
\int _K \# (\Omega _0,t)
<cR_0^2k^{-\delta },\ \ c=c(a_{\max }, a_{\min }). \label{86a}
\end{equation}
This estimate means that $\# (\Omega _0,t)$ differs from zero only
on the set with a  volume of order $R_0^2k^{-\delta }$.

Let $B_1(k,\delta ,V)\subset K$ is the set of $t$ such that  $\#
(\Omega _0,t)=0$.
  \begin{1} If $t\in B_1(k,\delta ,V)$, then
\begin{equation}
|p_m^{2}(t)-k^2|>k^{-1-\delta   }\ \mbox{for all } m\in Z^n,
\label{6a1}
\end{equation}
\begin{equation}
\min _{q\in Z^3 0<|q|<R_0} |p_m^{2}(t)-k^2||p_{m+q}^{2}(t)-k^2|
>k^{-4\delta }, \ \mbox{for all }
m\in Z^n,\label{6*1}
\end{equation}
\begin{equation}
|k^2-p_m^2(t)-\Delta \Lambda _q(t)_{mm}|>k^{-1/5-\delta }\
\mbox{for all } m \in \Pi _q(k^{1/5})  \mbox{and } q\in \Gamma
(R_0)\label{36}
\end{equation}
\begin{equation}
|k^2-p_m^2(t)|>k^{1/5-10\delta }\ \mbox{for all } m \in T(k).
\label{37}
\end{equation}
\end{1}
\underline{Proof.} Let us prove (\ref{6a1}). Suppose it does not
hold. This means that $\vec{p}_m(t)\in \Omega _1\subset \Omega
_0$, therefore $\# (\Omega _0,t)\geq 1$, i.e., $t \not \in B_1$,
what contradicts to the hypothesis of the lemma. Conditions
(\ref{6*1})
 -- (\ref{37}) can be proved
in the analogous way.  $\Box $
\begin{1} If
$t\in B_1(k, \delta ,V)$, then for sufficiently large $k$,
$k^{\delta /8}>R_0+\max _m |v_m|+c(a_{\max}, a_{\min }),$
\begin{equation}
N_V(k^2,t)=\hat{N}_V(k^2,t).  \label{9a}
\end{equation}
\end{1}
\underline{Proof.} To construct the convergent series for the resolvent we take the
unperturbed operator to be not $H_0(t)$ but
$\hat{H}(t)$ defined by
formulae~(\ref{30}) and (\ref{31}).
%%%%%55To construct the perturbation theory series
It is convenient to reduce $\hat{H}(t)$ to diagonal form,
$\hat{H}(t)\to U\hat{H}(t)U^*\equiv H_0+\Delta \hat{\Lambda }(t)$,
$U=U(t)$, and then to consider the operator $H(t)$ in this
representation: $UH(t)U^*=H_0+\Delta \hat{\Lambda }(t)+B$, where
$B=UWU^*$,
 $W=V-\sum _{q\in \Gamma (R_0)}P_qV_qP_q$.
We expand the resolvent of $UH(t)U^*$  in the formal series:
\begin{equation}
U(H(t)-z)^{-1}U^*= (H_0+\Delta \hat{\Lambda }(t)-z)^{-1}+\label{3.5.81}
\end{equation}
$$ \sum _{r=1}^{\infty }(H_0++\Delta \hat{\Lambda
}(t)-z)^{-1/2}A_1^r(H_0++\Delta \hat{\Lambda }(t)-z)^{-1/2},$$ $$
A_1(z,t)= (H_0(t)+\Delta \hat{\Lambda }(t)-z)^{-1/2}B(t)
(H_0(t)+\Delta \hat{\Lambda }(t)-z)^{-1/2}. $$ Lemma 4.14 in [K]
proves that under conditions (\ref{6a1}) -- (\ref{37}) and
$R_0<k^{\delta /8}$, $\max _{m} |v_m| <k^{\delta /8} $, $k^{\delta
/8}>c(a_{\max }, a_{\min })$, the following estimates hold:
\begin{equation}
\|A_1(k^2,t)\|<k^{4\delta }, \ \ \
\|A_1^3(k^2,t)\|<k^{-1/5+21\delta }. \label{3.5.84}
\end{equation}
The estimates (\ref{3.5.84})  provides the convergence of the
series (\ref{3.5.81}) for $z\in C_0$. The series converges in the
trace class since $(H_0+\Delta \hat{\Lambda }(t)-z)^{-1}$ is a
Sturm-Liouville operator. This means that  the number of
eigenvalues inside the contour is a continuous function of $\alpha
B$, $0\leq \alpha \leq 1$, i.e., the relation (\ref{9a}) holds.
$\Box $
\begin{1}If $t\in K\setminus B_1$, then for sufficiently large
$k$, $k^{\delta /8}>R_0+\max _m |v_m|+c(a_{\max }, a_{\min })$,
the following estimate holds:
\begin{equation}
|N_V(k^{2l},t)-\hat{N}_V(k^{2l},t)|\leq 2\# (\Omega _0,t).
\label{14a}
\end{equation}
\end{1}
\underline{Proof.} Let $t\in K\setminus B_1$. Then there is a
positive number of different indexes $m$ such that
$\vec{p}_m(t)\in \Omega _0$. Let us introduce the diagonal
projection $P'$ with elements equal to 1 for such $m$:
\begin{equation}
P'_{mm}=\left\{\begin{array}{ll}1,&\mbox{if }\vec{p}_m(t)\in
\Omega _0;\\ 0,&\mbox{otherwise.}\end{array}\right. \label{15a}
\end{equation}
Let us consider the operator $H'(t)=H_0(t)+\Delta \hat{\Lambda
}(t)+(I-P')B(I-P')$. This means that in the matrix of the operator
$UHU^*=H_0(t)+\Delta \hat{\Lambda }(t)+B$ we replace all columns
and rows, corresponding to $m:\vec{p}_m(t)\in \Omega _0$, by zeros
with the exception of the diagonal elements, i.e., we delete from
$B$ all columns and rows, which can break convergence of the
perturbation series for the resolvent.
 Let $N'_V(K^2,t)$ be the number of eigenvalues
for $H'(t)$. Considering as in Lemma 4.14 \cite{K}, we show that
the perturbation series converges and
\begin{equation}
\hat{N}_V(k^{2},t)=N'_{V}(k^{2},t). \label{13a**}
\end{equation}
Note that the difference between $UH(t)U^*$ and $H'(t)$ is a
finite dimensional operator:
$UH(t)U^*-H'(t)=(I-P')BP'+P'B(I-P')+P'BP'.$ The dimension of this
operator is not greater than $2\# (\Omega  _0,t)$. According to
the general perturbation theory (see f.e. [Kato]):
$|N_V(k^{2},t)-N'_V(k^{2},t)|\leq 2\# (\Omega  _0,t).$ Combining
the last estimate with (\ref{13a**}), we get (\ref{14a}). $\Box$

\underline{The proof of Theorem 2.} Clearly, $D_V(k^{2})=\int
_{B_1(k)}N_V(k^{2},t)dt +\int _{K\setminus B_1}N_V(k^{2},t)dt $.
Using Lemmas 9 and 10, and estimate (\ref{86a}), we obtain
$|D_V(k^{2})-\hat{D}_V(k^{2})|\leq 2\int _{K\setminus B_1}\#
(\Omega _0,t)dt=2V(\Omega  _0)\leq cR_0^2k^{-\delta }.$
Considering (\ref{25}), we get
\begin{equation}
\left| D_V(k^2)-D_0(k^2)+d_V(k^2)\right| <cR_0^2k^{-\delta}+
cR_0^3\|V\|_*(1+\|V\|_*)k^{-1/5},\label{97a}
\end{equation}
$c= c(a_{\max }, a_{\min })$. Note that the only restriction on
$\delta $ is $1/5-21\delta
>0$ (see (\ref{3.5.84})), therefore (\ref{3a}) holds. The estimate
(\ref{3b+}) is proved in Lemma 4. $\Box
$

\underline{The proof of Theorem 3.} It follows immediately from
Theorem 2 and estimate (\ref{d2}). $\Box $

\underline{The proof of Theorem 4.} Suppose $V$ satisfies
(\ref{smooth1}) with $\nu =900$. Let $V_0$ be a trigonometric
polynomial with $R_0=k^{\delta /8}$, $\delta \leq 1/105$, which is
the truncation of the Fourier series of $V$. Theorems 2 and 3 hold
for $V_0$. Clearly, $\|V-V_0\|<\|V\|_*R_0^{-\nu }$. Taking into
account  (\ref{dV0}), it is not difficult to show that
$|d_V-d_{V_0}|<cR_0^{-\nu}\|V\|_*(1+\|V\|_*)$. Using this estimate
and (\ref{3b+}) for $d_{V_0}$, we get (\ref{3b+}) for  $d_V$.

Obviously, $N_{V_0}(k^2-\|V-V_0\|,t)\leq N_{V}(k^2,t)\leq
N_{V_0}(k^2+\|V-V_0\|,t).$ Integrating the last inequalities and
taking into account the above estimate for $\|V-V_0\|$, we get
$|D_{V}- D_{V_0}|<12\pi^2\|V\|_*R_0^{-\nu }k$. Using (\ref{97a})
for $V_0$ we get $|D_{V}-D_{0} -d_{V_0}|<12\pi ^2 \|V\|_*R_0^{-\nu
}k+cR_0^2k^{-\delta }+cR_0^3\|V\|_*(1+\|V\|_*)k^{-1/5}$.  From
this estimate and above estimate for $|d_V-d_{V_0}|$, we obtain:
$|D_{V}-D_{0} -d_{V}|<c_1k^{-3\delta /4}$, $\delta
<1/105 $, hence $|D_{V}-D_{0} -d_{V}|<c_1k^{-\zeta }$, $\zeta >1/130$,
$c_1=c_1(\|V\|_*, a_{\max }, a_{\min })$. Similarly, we get
$|D_{V}-D_{0} -d_{V}^0|<c_1k^{-\zeta }$.
 $\Box $

%%%%%%\centerline{\bf References.}
%%%%%%%\bigskip
%%%%%%%\par
%%%%%%%\noindent
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