\subsection{Estimates for entire functions.} An entire
function $f(z)$ is said to be of exponential type if there is a
constant $\b$ such that $|f(z)|\leq\const e^{\b |z|}$ everywhere.
The infimum of the set of $\b$ for which such inequality holds is
called the type of $f$.

\no {\bf Definition.}
{\it Let $\cE_\g, \g >0$ denote the set
 of exponential type  functions $f$, which satisfy
\[
\lb{D1} |f(k)|\ge 2|k|\qqq   \qqq \forall  \ k\in \R,
\]
\[
\lb{D2}
\begin{aligned}
&|f(k)-2ik+f_0|\le {Q^2\/|k|_1}e^{ \g (|\Im k|-\Im
k)+{Q\/|k|_1}},\qqq
\qq  \forall \ k\in \C,\\
& Q> 0,\qq  f_0\in \R,\qqq \qqq |f_0|\le Q,
\end{aligned}
\]
where the constants $Q=Q(f), f_0$ depend on $f$ and
$|k|_1=\max\{1,|k|\}$.}

In the proof of Theorem \ref{T1} we will need some properties of zeros of
$f\in \cE_\g $ in terms of the Carleson  measure. Recall that {\it a
positive Borel measure $M$ defined in $\C_-$ is called a
Carleson measure if there is
 a constant $C_M$ such that for all $(r,t)\in \R_+\ts\R$
\[
\lb{1.31}
   M(D_-(t,r))\leq C_Mr,\ \ \ {\rm where}\ \ \
   \ D_-(t,r)\ev\{z\in \C_-: |z-t|<r \},
\]
here $C_M$ is the Carleson  constant independent of $(t,r)$.}

For an entire function $f$ with zeroes $k_n, n\ge 1$
  we define an associated measure by
\[
\lb{MO}
 d\O (k,f)=\sum_{\Im k_n\le 0} \d (k-k_n+i)du dv,\qqq k=u+iv\in \C_-.
\]
We denote the number of zeros of function $f$ having modulus  $<r$ by $\cN (r,f)$,
each zero being counted according to its multiplicity.
In order to prove Theorem \ref{T1} we need


  \begin{theorem}
\lb{TEf}
 Let $f\in \cE_\g $ for some $\g >0$. Then

 i) The number of zeros of $f$ in the disk $\{|k|<r\},r>0, r_1=r+{1\/2}$
satisfies
\[
\lb{L1} \cN (r,f)\le {1\/\log 2}\rt({4r_1\g\/ \pi }+\log (1+{4r_1})+
{|f_0|+8Q\/1+4r_1}\rt).
\]

ii) $\O (\cdot,f)$ is a Carleson measure and for all
$(r,t)\in \R_+\ts \R $ satisfies
\[
\begin{aligned}
\lb{L2}
\O (D_-(t,r),f)\le\cN (r,f(t+\cdot))\le C(f)r ,\\
C(f)={12\/\log 4}\rt({\g \/\pi }+1+{|f_0| +3Q\/4}\rt).
\end{aligned}
\]
iii) Let $k_n, n\ge 1$ be all zeros of $f$. Then for each $p>1$ the following
estimate holds true:
\[
\lb{L3}
\sum_{\pm \Im k_n\le 0} {1\/|k_n-2i|^{p}}
\le   C Y_pC(f),
\]
where $C\le 2^5$ is the absolute  constant and  $Y_p$ is given by
\[
\lb{p1}
  Y_p=\int_\R
(1+x^2)^{-{p\/2}}dx=\sqrt \pi{\G({p-1\/2})\/\G({p\/2})} \qqq \forall \qq p>1.
\]

\end{theorem}
{\bf Proof.} i) Recall the Jensen formula (see p. 2 in \cite{Koo88})
for an entire function $F$:
\[
\lb{PJ}
\log |F(0)|+\int _0^r{\cN (t,F)\/ t}dt= {1\/ 2\pi }\int
_0^{2\pi}\log |F(re^{i\f})|d\f,
\]
 Rewrite $f$ in the form
 \[
 \lb{ef1}
 f=2ik-f_0+f_*, \qqq where \qq |f_*(k)|\le
{Q^2\/|k|_1}e^{{Q\/|k|_1}}e^{2\g v_-},\qqq v_-(k)=
{|\Im k|-\Im k\/2},
\]
for all $k\in \C$.
Consider the case $t\in \R\sm [-{1\/2}, {1\/2}]$. Let $z=t+k, k=re^{i\f}$.
Then \er{D2} and \er{ef1} imply
\[
\begin{aligned}
|f(t+k)|=|2it+2ik-f_0+f_*(z)|\le 2|t|\rt(1+{r\/|t|}\rt)\rt(1+{|f_0|+|f_*(z)|\/2(|t|+r)}\rt)\\
\le 2|t|\rt(1+{r\/|t|}\rt) e^{2\g v_-+{Q\/|z|_1}}\rt(1+{|f_0|+Q^2|z|_1^{-1}\/2(|t|+r)}\rt)\\
\le 2|t|\rt(1+{r\/|t|}\rt) e^{2\g v_-+{Q\/|z|_1}}
\rt(1+{|f_0|\/2(|t|+r)}\rt)  \rt(1+{Q^2\/2(|t|+r)|z|_1}\rt).
\end{aligned}
\]
Let $\cN (s)=\cN (s,f(t+\cdot))$.
Substituting this into \er{PJ} we obtain for the function $F(k)=f(t+k),
k=re^{i\f}, z=t+re^{i\f}$:
\[
\begin{aligned}
\lb{F1}
\log |f(t)|+\int_0^r{\cN (s)\/s}ds={1\/2\pi}\int_0^{2\pi}\log |f(z)|d\f\\
\le \log 2|t|+\log \rt(1+{r\/|t|}\rt)+\log \rt(1+{|f_0|\/2(|t|+r)}\rt)+{1\/2\pi}\int_0^{2\pi}2\g v_-d\f+X(t,r),\\
where \qq X(t,r)={1\/2\pi}\int_0^{2\pi}\rt[{Q\/|z|_1}+\log  \rt(1+{Q^2\/2(|t|+r)|z|_1}\rt) \rt]d\f,
\end{aligned}
\]
where $2v_-=r(|\sin \f| -\sin\f)$ and the simple integration yields
\[
\lb{F12}
\begin{aligned}
{1\/2\pi}\int_0^{2\pi}2\g v_-d\f=
{r\g\/2\pi }\int_0^{2\pi}\rt(|\sin \f|-\sin \f\rt)d\f={2r\g\/\pi}.
\end{aligned}
\]
Substituting the estimate ${|f(t)|\/2|t|}\ge 1$  into \er{F1}
together  with the simple one
$$
\int _0^r{\cN (s)ds\/s}\geq \cN ({r/2})\int _{r\/2}^r{ds\/s}=
\cN ({r/2})\log 2,
$$
we obtain
\[
\begin{aligned}
\lb{F3}
\cN ({r/2})\log 2\le {2r\g\/ \pi }+
\log \rt(1+{r\/|t|}\rt)+{|f_0|\/2(|t|+r)}+X(t,r).
\end{aligned}
\]

We show \er{L1}. Let $t={1\/2}$ and $r_1=r+{1\/2}$ for any $r>0$.
Then
$$
\{|k|<r\}\ss \{|k-{1\/2}|<r_1\},\qqq |z|_1=|(1/2)+k|_1\ge {1\/2}(1+|k|-(1/2))=
{1\/4}(1+2|k|)
$$
and \er{F3} give
\[
\lb{esr}
\begin{aligned}
\cN (r,f)\le \cN (r_1,f({1\/2}+\cdot))\le
{4r_1\g\/ \pi }+\log (1+4r_1)+{|f_0|\/(1+4r_1)}+X(t,r),\\
where \qq X(t,r)\le {1\/2\pi}\int_0^{2\pi}\rt[{4Q\/(1+4r_1)}+\log  \rt(1+{4Q^2\/(1+4r_1)^2}\rt) \rt]d\f\le {8Q\/(1+4r_1)},
\end{aligned}
\]
which yields  \er{L1}.



ii)
Let $r\le 1, t\in \R$. Then by the construction of $\O (\cdot,f)$, we obtain
$\O (D_-(t,r),f) =0$.

  In order to show \er{L2} for thew case  $r> 1, t\in \R$ we need to consider two cases:

Firstly, let  $r>1, t\in \R\sm [-{1\/2}, {1\/2}]$. Then due to \er{F3} and \er{F1},
the measure $\O (\cdot,f)$ satisfies
\[
\begin{aligned}
\lb{F4}
& \O (D_-(t,r),f)\le\cN (r,f(t+\cdot))\le
{1\/\log 2}\rt(   {4r\g\/ \pi }+4r+|f_0| +Q+\log (1+Q)^2\rt)\le C_1r,\\
& \qqq \qqq  C_1={4\/\log 2}\rt({\g\/\pi }+1+{|f_0| +3Q\/4}\rt).
\end{aligned}
\]

Secondly, let  $r>1, t\in [0, {1\/2}]$. The proof for the case $t\in [-{1\/2},0]$
is similar. For two disks $D(t,r)$ and $D({1\/2},r_1), r_1=r+{1\/2}$ we have
\[
\begin{aligned}
\lb{F5}
D(t,r)\cap \{\Im k\le -1\}\ss D\big({1\/2},r_1\big)\cap \{\Im k\le -1\},
\\
r\le r_1=r+{1\/2}\le {3r\/2}.
\end{aligned}
\]
Then due to \er{L1}, the measure $\O (\cdot,f)$ satisfies
\[
\begin{aligned}
\lb{F4a}
\O (D_-(t,r),f)\le \O (D_-(1/2,r_1),f)\le \cN (r_1,f({\small{1\/2}}+\cdot))\le
C_1r_1\le {3\/2}C_1r.
\end{aligned}
\]
Thus $\O (\cdot,f)$ is a Carleson measure with the Carleson constant $C(f)={3\/2}C_1$, where $C_1$ is given by \er{F4}.

iii)  Consider the case $\Im k_n\le 0$. The proof for the case $\Im k_n> 0$
is similar, even simpler.
 In order to show \er{L3} we recall the Carleson result (see p.
63, Theorem 3.9, \cite{G81}):

Let $F$ be analytic on $\C_-$. For $0<p<\iy$  we say $F\in
\mH_p=\mH_p(\C_-)$ if
$$
\sup_{y<0}\int_\R|F(x+iy)|^pdx=\|F\|_{\mH_p}^p<\iy
$$
Note that the definition of the Hardy space $\mH_p$ involve all
$y<0$, instead of small only value of $y$, like say, $y\in (-1,0)$.
We define the Hardy space $\mH_p$ for the case $\C_-$, since below
we work with functions on $\C_-$.

{\it If $M$ is a Carleson measure and satisfies \er{1.31}, then the
following estimate holds:
\[
\lb{Ce}
  \int_{\C_-} |F|^pdM\leq C C_M \|F\|_{\mH_p}^p\qqq
 
  \forall \qq F\in \mH_p, \ p\in (0,\iy ),
\]
where  $C_M$ is the so-called Carleson constant from \er{1.31} and  $C\le 2^5$ is an
absolute constant.}

 In order to prove \er{L3} we  take the functions $F(k)={1\/k-i}$.
Then estimates \er{Ce}, \er{L2}  yield
\[
\lb{3.19} \sum_{\Im k_n\le 0} {1\/|k_n-2i|^p}=\int_{\C_-}
|F(\l)|^pd\O \leq C C(f) \|F\|_{\mH_p}^p, \qq p\in (1,\iy),
\]
$C$ is an absolute constant and $C(f)$ is defined in \er{L2}. Here
we have the simple identity
\[
\lb{3.19a} \|F\|_{\mH_p}^p=\int_\R {dt\/|t-i|^p}=\int_\R
{dt\/(t^2+1)^{p\/2}}=Y_p.
\]
The function $Y$ is studied in Lemma \ref{Tp}. Combine \er{3.19} and
\er{3.19a} we obtain \er{L3}.
\hfill  \BBox


\

\subsection{Estimates of resonances.} We consider Case 1 and without loss of
generality let $\supp q\ss [0,\g]$. The Schr\"odinger equation
\[
\lb{e1}
  -f''+q(x)f=k^2f,\ \ \  \ \ \ k\in \C \sm \{0\},
\]
has unique solutions $\p_{\pm}(x,k)$  such that $\p_+(x,k )=e^{ixk
},\ \ x\ge \g$ and $\p_-(x,k)=e^{-ik x},\ \ x\leq 0$. Outside of the
support of $q$ any solutions of \er{e1} have to be linear
combinations of $e^{\pm ik x}$. Functions $\p_\pm(x,\cdot),
\p_\pm'(x,\cdot),  x\in \R$ are entire. We define the functions $a,
w, s$ by
\[
\begin{aligned}
\lb{H1a}
w(k)=2ika(k)=\{\p_-(\cdot,k), \p_+(\cdot,k )\},\qqq
  \ s(k)=\{\p_+(\cdot,k ),\p_-(\cdot,-k )\},
\end{aligned}
\]
and $\{f, g\}=fg'-f'g$ denotes the Wronskian.  The functions $w(k),
s(k)$ are entire and the following asymptotic
estimates hold true:
\[
\lb{a1}
 a(k )=1+O(k ^{-1})\qqq {\rm as} \ \ |k |\to \iy ,\ \  k\in\ol\C_+.
\]
The scattering matrix for the operators $H, H_0=-{d^2\/dx^2}$ is given by
\[
S(k)=\ma a(k )^{-1}& r_-(k )\\ r_+(k )&a(k )^{-1}\am ,
\ \ \ \  r_{\pm}={s(\mp k)\/ w(k )}, \ \ \ \  k\in\R,
\lb{1.7}
\]
where $1/a$ is the transmission coefficient and $r_{\pm}$ are the reflection
coefficients. The matrix $S(k), k\in \R$ is unitary, which yields
\[
\lb{ws}
|w(k)|^2=4k^2+|s(k)|^2,\ \  \forall \ k\in \R.
\]
The solution $ \p_+$ of \er{e1}  satisfies the following equation
\[
\lb{3.1}
 \p_+(x,k)=e^{ixk}-\int _x^\g{\sin [k(x-t)]\/k}q(t)
\p_+(t,k)dt \qqq \forall\  (x,k)\in [0, \g]\ts \C.
\]
It is well known that equation \er{3.1} has a unique solution.
 Due to \er{3.1} the function $y(x,k)=e^{-ikx}\p_+(x,k)$ satisfies
the integral equation
\[
\lb{y1}
y(x,k)=1+\int _x^\g G(t-x,k)q(t)y(t,k)dt \qqq  G(t,k)={\sin kt\/k}e^{ikt},
\]
for all $(x,k)\in [0,\g]\ts \C$. We have the standard iterations
\[
\lb{y2}
y(x,k)=1+\sum_{n\ge 1}y_n(x,k),\qqq y_n(x,k)=
\int _x^\g G(t-x,k)q(t)y_{n-1}(t,k)dt,
\qq y_0=1.
\]
We need some properties of the functions introduced above.




\begin{lemma}
\label{Tw}
 Let $q\in L^1(\R)$ and  $\supp q\ss [0,\g]$. Then the functions
 $w, s$ and $\p _{+}(x,\cdot),\p _{+}'(x,\cdot)$, $x\in\R$ are entire and we have
\[
\lb{yn}
|y_n(x,k)|\le {h^n\/n!}e^{2(\g-x)v_-}, \qqq v_-=\ca 0,&  k\in \ol\C_+\\
|\Im k|,  &  k\in \C_-\ac,
\]
for any $n\ge 1, (x,k)\in [0,\g]\ts \C$, where
\[
\lb{v-}
 h={Q\/|k|_1}, \qq Q=\max\{\|q\|,\|q\|_1\},\qq
                                                |k|_1=\max\{1,|k|\};
\]
where $\|q\|=\int_0^\g |q(t)|dt$ and $\|q\|_1=\int_0^\g t|q(t)|dt$, and
\[
\begin{aligned}
\lb{y}
|y(x,k)|\le e^{2(\g-x)v_-+h},\\
|y(x,k)-1|\le h e^{2(\g-x)v_-+h},\\
|y(x,k)-1-y_1|\le {h^2\/2} e^{2(\g-x)v_-+h}.
\end{aligned}
\]
Moreover, $w\in \cE_\g$ and satisfies for any $k\in \C$:
\[
\lb{w1}
w(k)=i2k-q_0 +w_*(k), \qqq w_*(k)=-\int _0^\g q(t)\big(y(t,k)-1\big)dt,
\]
\[
\lb{w2}
|w_*(k)|\le \|q\| he^{h+2\g v_-},
\]
where $q_0=\int_0^\g q(t)dt$.

\end{lemma}
{\bf Proof.} The function $G(t,k)={\sin kt\/k}e^{ikt}$ satisfy
\[
\lb{y3}
|G(t,k)|\le {e^{2 v_-t}\/|k|_1}\ca 1 & if \ |k|\ge 1\\
                                   t & if \ |k|<1 \ac,
 \qqq t\ge 0, \ k\in \C,
\]
Consider the first case in \er{y3}: $|k|\ge 1$, the proof for the second case
$|k|<1$ is similar.
Substituting the estimate in \er{y3} for $|k|\ge 1$ into the identity
$$
y_n(x,k)=\int\limits_{x=t_0<t_1< t_2<...< t_n}
\lt(\prod\limits_{1\le j\le n}
G(t_{j}-t_{j-1},k)q(t_j)\rt)dt_1dt_2...dt_n,
$$
we obtain
\[
\begin{aligned}
|y_n(x,k)|\le {1\/|k|_1^n}\int\limits_{x=t_0<t_1< t_2<...< t_n}
\lt(\prod\limits_{1\le j\le n}e^{2v_-(t_{j}-t_{j-1})}
|q(t_j)|\rt)dt_1dt_2...dt_n\\
= {e^{-2v_- x}\/|k|_1^n}\int\limits_{x=t_0<t_1< t_2<...< t_n}
\lt(\prod\limits_{1\le j\le n}
|q(t_j)|\rt) e^{2v_-t_{n}}dt_1dt_2...dt_n\\
\le {e^{2(\g-x)v_-}\/|k|_1^n}\int\limits_{x=t_0<t_1< t_2<...< t_n}
|q(t_1)q(t_2)....q(t_n)| dt_1dt_2...dt_n\le e^{2(\g-x)v_-}{\|q\|^n\/n! |k|_1^n}.
\end{aligned}
\]


This shows that for each $x\ge 0$
the series \er{y2} converges uniformly on any bounded subset of $\C$.
Each term of this series is an entire function. Hence the sum is
an entire function. Summing the majorants we obtain estimates  \er{y}.
Thus the functions  $w, s$ and $\p _{+}(x,\cdot),\p _{+}'(x,\cdot),x\in\R$
are entire

The identity \er{3.1} gives
\[
\begin{aligned}
\lb{y5}
\p_+(0,k)'=ik-\int _0^\g\cos [kt] \ q(t)
\p_+(t,k)dt, \\
ik\p_+(0,k)=ik+i\int _0^\g\sin [kt] \ q(t)\p_+(t,k)dt,
\end{aligned}
\]
which yields
\[
\begin{aligned}
\lb{y6}
w(k)=\p_-(0,k )\p_+'(0,k )-\p_-'(0,k )\p_+(0,k )
=ik\p_+(0,k ) +\p_+'(0,k )\\
=2ik-q_0+w_*(k), \qqq w_*(k)=-\int _0^\g q(t)\big(y(t,k)-1\big)dt.
\end{aligned}
\]
Due to \er{y} we have the estimate
\[
\begin{aligned}
\lb{y7}
|w_*(k)|\le \int _0^\g|q(t)|h e^{2(\g-t)v_-+h}dt\le
h e^{2\g v_-+h}\int _0^\g|q(t)|dt=h \|q\|e^{2\g v_-+h},
\end{aligned}
\]
which is \er{w2}. Together with \er{ws} this implies that $w\in \cE_\g$.
\hfill \BBox


\

The identity \er{w1} and estimate \er{w2} give
 \[
\lb{w1x}
\begin{aligned}
w(k)=i2k-q_0 +w_*(k), \qq |w_*(k)|\le {Q^2\/|k|_1}e^{{Q\/|k|_1}+2\g v_-},\qq
 \forall \ k\in \C,\\
 where \ \ q_0=\int_0^\g q(t)dt,\qqq Q=\max\{\|q\|,\|q\|_1\}.
 \end{aligned}
\]


\


We show the relations between the Cases 1, 2 and 3. In order to show
this we use a standard trick. For $q\in L^1(\R_+)$ with $\supp q\ss
[0,\g]$ we define the Schr\"odinger operator $\wt
H$ acting in $L^2(\R )$ with an even compactly
supported potential $\wt q$ given by
\[
\lb{wtq} \wt H=-{d^2\/dx^2}+\wt q,\qqq \qqq \wt q(x)=q(|x|), x\in \R.
\]
 The Schr\"odinger equation
$
  -f''+\wt q(x)f=k^2f,\ \ \  \ \ \ k\in \C \sm \{0\},
$ has unique solutions $\wt\p_{\pm}(x,k)$  such that $\wt\p_+(x,k
)=e^{ixk },\ \ x\ge \g$ and $\wt\p_-(x,k)=e^{-ik x},\ \ x\leq -\g$.
Note that the symmetry of the potential $\wt q$ yields
$$
\wt\p_+(x,k)=\p_+(x,k)=\wt\p_-(-x,k) \qqq \forall x\in [0,\g].
$$
This implies that the Wronskian $\wt w(k)$ for the potential $\wt q$
satisfies
\[
\lb{tw} \wt w(k)=\{\wt\p_+(x,k),
\wt\p_-(x,k)\}|_{x=0}=2\p_+(0,k)\p_+'(0,k).
\]
where $\p_+(0,k)$ and $\p_+'(0,k)$ are the Jost functions for the
Cases 2 and 3 respectively with the potential $q$. This identity is
very useful. For example, if we have estimate \er{3} for the Case 1,
then  \er{tw} gives the estimate \er{3} for the Case 2 and 3.

We obtain the estimate of the Wronskian $\wt w(k)$.
We need to define the potential
$q_1=\wt q(x-\g)$, which has the support $\supp q_1\ss [0,2\g]$.
Using Lemma \ref{Tw} and \er{w1}, \er{w2} we obtain
\[
\lb{wtw}
\begin{aligned}
\wt w(k)=i2k-2q_0 +\wt w_*(k), \\
|\wt w_*(k)|\le \|q_1\| {\wt Q^2\/|k|_1}e^{4\g v_-+{\wt Q\/|k|_1}},
\end{aligned}
\]
where the corresponding constant $\wt Q$ is given by
\[
\begin{aligned}
\lb{wtQ}
\wt Q=\max \{\|q_1\|, \|q_1\|_1\}=2\|q\|\max \{1, \g\},
\end{aligned}
\]
since
$$
\|q_1\|=\int_0^{2\g} |q_1(x)|dx=2\|q\|,\qqq \|q_1\|_1=\int_0^{2\g} x|\wt q(x-\g)|dx=
\int_{-\g}^{\g} (t+\g)|\wt q(t)|dt=2\g \|q\|.
$$



We describe few facts about the resonances in the disk $\{|k|<r\}$.

\begin{proposition}
\lb{Tr}
 Let $q\in L^1(\R)$ and  $\supp q\ss [0,\g]$ for some $\g>0$.

 i)  Let $r>0$ and  let $q$ satisfy
\[
\lb{q0}
\|q\|(1+he^{2\g r+h})<2r, \qqq {\rm where} \ h={\max \{\|q\|,\|q\|_1\}\/\max\{1,r\}}.
\]
Then the function $w$ has only one simple zero in the disk $\{|k|<r\}$.

ii) Let $1\le r\le {1\/2\g}$ and let $2\|q\|\le r$.
Then the function $w$ has only one simple zero in the disk $\{|k|<r\}$.


\end{proposition}
\no {\bf Proof}. i) Let $w_0=i2k$ and $|k|=r$.  Then the estimates \er{w1},
\er{w2} imply
\[
|w(k)-w_0(k)|\le \|q\| (1+he^{2\g+h})=|w_0(k)|C_0,\qqq
C_0={\|q\|\/2r}(1+he^{2\g r+h}).
\]
Hence, if $C_0<1$, then by Rouche's theorem, $w(k)$ has only one simple zero,
as $w_0=2ik$ in the disk $\{|k|<r\}$.


ii) If we assume that $f(h)=h+h^2e^{1+h}<2$, then \er{q0} holds true.

We have that the function $f$ is increasing. We get
the estimates $f({1\/2})<2$, since  $h={\|q\|\/r}\le {1\/2}$, which yield that $w$
has only one simple zero in the disk $\{|k|<r\}$.
\hfill\BBox

{\bf Remark.} 1) This proposition shows that for  any $\g, r>0$ and sufficiently
small $\|q\|$  the function $w$ has only one simple zero in the disk $\{|k|<r\}$.

2)  The number of zeros of $w$ in the disk with large $r$  depends on the diameter
$\g$. But it is very interesting to determine the asymptotics of $\cN(r,w)$ as
$r\to \iy$. This problem remains open for  long period.

3) We need to say that \er{q2} holds also true for the Case 2 and 3.
 Then \er{3} for Case 1 yields \er{3} for Cases 2 and 3.





\


 {\bf Proof of Theorem \ref{TrJS}}.
Lemma \ref{Tw} gives that the function $w\in \cE_\g$.

We apply the estimate \er{L1} to the function $w$. Thus  due to the estimate \er{w1x} and the simple fact $|q_0|\le Q$,
we obtain \er{q2}.

We show \er{q3}.
The identity \er{tw} gives $\cN (r,\p_+(0,\cdot))+\cN
(r,\p_+'(0,\cdot))=\cN (r,\wt w)$, where $\wt w$ is the Wronskian
for the potential $\wt q$ given by \er{wtq} and $\wt w$ satisfies \er{wtw}, \er{wtQ}. Then these arguments and  the estimate \er{q2} implies
 \er{q3}. \hfill $\BBox$


 \medskip


{\bf Proof of Theorem \ref{T1}.} We consider Case 1 and let $\supp
q\ss [0,\g]$. By Lemma \ref{Tw}, the function $w$ belongs to
$\cE_\g$ with $Q=\max\{\|q\|,\|q\|_1\}$ and satisfies \er{w1x}.
 Thus  the estimate \er{L3}, \er{L2}  from  Theorem
\ref{TEf} give \er{3} for  Case 1.


Consider Cases 2 and 3.  Then we have the identity $\wt w(k)=2\p_+(0,k)\p_+'(0,k)$,
where $\wt w$ is the Wronskian for the potential $\wt q$ given by \er{wtq}
(see  \er{tw}). Then  \er{3} for $\wt q$ (Case 1) together with  \er{wtQ} implies
\er{3} for Cases 2 and 3.
\hfill $\BBox$


\begin{lemma}
\lb{Tp} The function $Y_p=\int_\R {(1+x^2)^{-{p\/2}}}dx, p>1$ is
strongly monotonic and convex on $(1,\iy)$, since $Y_p'<0, Y_p''>0$,
and  satisfies \er{Yp}  and the identity \er{p1}, i.e., $Y_p=\sqrt \pi{\G({p-1\/2})/\G({p\/2})}$ for all $p>1$.
\end{lemma}
\no {\bf Proof.} We rewrite $Y_p$ in the form $Y_p=\int_\R e^{-p
f(x)}dx$, where $f(x)={1\/2}\log (1+x^2)$. This yields
$Y_p'=-\int_\R f(x)e^{-p f(x)}dx<0$ and $Y_p''=\int_\R f^2(x)e^{-p
f(x)}dx>0$. We rewrite $Y_p$ in another form
$$
Y_p=2\int_0^\iy {(1+x^2)^{-{p\/2}}}dx=\int_0^\iy
{t^{-{1\/2}}(1+t)^{-{p\/2}}}dt=\sqrt \pi{ \G(z)\/\G(z+{1\/2})}, \qq
z={p-1\/2},
$$
where $\G$ is the Gamma function, see identities (2), (5) in Section
1.5 in \cite{EMOT53}.

Asymptotics  ${ \G(z)\/\G(z+{1\/2})}=z^{-{1\/2}}(1+O({1\/z}))$ as
$z\to \iy$, see  (5) in Sect. 1.18 in \cite{EMOT53} give \er{Yp} as
$p\to \iy$. Recall that $\G$ is meromorphic in $\C$ with simple
poles $0,-1,-2,..$. Then the identities $\G(1+z)=z\G(z)$ and
$\G(1)=1, \G(1/2)=\sqrt \pi$ yield  \er{Yp} as $p\to 1$:
$$
Y_p=\sqrt \pi{ \G(z)\/\G(z+{1\/2})}={\sqrt\pi \G(1+z)\/
z\G(z+{1\/2})}={1+O(z)\/z}\qq \as \  \ z={p-1\/2}\to 0,
$$
 \hfill $\BBox$


 \medskip

\no {\bf Acknowledgments.} \footnotesize I am grateful to  Michael
Loss, Atlanta, Grigori Rozenblum, G\"oteborgs, and Sergey Morozov,
Munich, for useful comments about Lieb-Thirring inequalities and
Elliot Lieb, Princeton, for the reference \cite{DFLP06}. Moreover, I
am also grateful to  Alexei Alexandrov  (St. Petersburg) for
stimulating discussions about the Carleson Theorem and about the
absolute constant $C$ in \er{Ce}.

This work was supported by the Ministry of education and science of
the Russian Federation, state contract 14.740.11.0581 and the RFFI
grant "Spectral and asymptotic methods for studying of the
differential operators" No 11-01-00458.



