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Schrodinger operator, spectrum, KdV equation
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\begin{document}
\*

\vskip 1,5cm
 {\bf \LARGE \noindent First KdV integrals and \\absolutely continuous 
   spectrum \\ for  1-D Schr\"{o}dinger  operator}
\vskip 0,5cm
 \noindent {\bf S.Molchanov$^{1}$, M.Novitskii$^{2}$, B.Vainberg$^{3}$ }
\vskip 0,5cm
{\small
\noindent $^{1}$ Mathematics, UNCC, Charlotte, NC 28223, USA,
 E-mail:molchanov@uncc.edu


\noindent $^{2}$ Mathematics, ILT,  Kharkov, 310164, Ukraine,
 E-mail: novitskii@ilt.kharkov.ua 


\noindent $^{3}$ Mathematics, UNCC, Charlotte, NC 28223, USA, E-mail: 
 brvainbe@uncc.edu } 
\vskip 0,5cm 

%\maketitle

\noindent {\bf Abstract:} 
We consider 1-D Schr\"odinger operators  on $L^{2}(R_{+})$ with slowly 
decaying potentials. Under some conditions  on the  potential,  related 
to  the first integrals  of the  KdV  equation, we prove that the 
  a.c. spectrum 
of the operator coincides  with the positive semiaxis  and the singular
spectrum is unstable.
Examples  show that for special classes of   sparse 
potentials  these results can not  be improved. 


\thispagestyle{empty}
\vskip1cm
\noindent {\bf Contents}

\noindent Introduction.

\noindent 1. Tests for the a.c. spectrum.

\noindent 2. Estimates for the first integrals  of the KdV equation.

\noindent 3. Theorem 1 and its generalization.

\noindent 4. Spectral bifurcation. Examples.

\noindent  References.

\thispagestyle{empty}
\vskip1cm


{\bf \large Introduction}
\par
  Let $H = H^{\alpha}$ be the  Hamiltonian on  $L^{2}(R_{+})$  given by 
   \begin{equation} H \psi = - \psi{''} + V(x) \psi,\quad  x\ge 0,
   \end{equation}
   \begin{equation} \label{boundZ}
      \psi(0) \cos \alpha - \psi' (0) \sin \alpha = 0  , \quad  
   \alpha \in S^{1} = [0,\pi ).
  \end{equation}
 It is assumed that the potential $V$ 
  vanishes  at 
 infinity  in some  sense (see (\ref {mnv})),  and from  this
 decay it follows that
  the  Birman condition:  
 \begin{equation}
  \lim_{x\to\infty} \int_{x}^{x+1  } |V(x)| d\,x = 0 \label{Birman}
 \end{equation}
  holds. Then (see the original paper
 \cite{B} or the monograph \cite{G})
   the operator  $H^{\alpha}$ 
 is   essentially self-adjoint,  and:
 \par
 a) the negative part of the spectrum of $H^{\alpha}$ is bounded from below 
 and discrete;
  \par
 b) the
 essential spectrum of $H^{\alpha}$  coincides with the positive semiaxis: 
 $$   \Sigma_{{ess}} =[0,\infty).
 $$
  Let $\mu^{\alpha}(d\,\lambda)$  be the spectral measure of $H^{\alpha},$
i.e. $H^{\alpha}$  is unitarily  equivalent to the operator of 
multiplication by $\lambda$  in the space $L^{2}(R, \mu^{\alpha}(d\,\lambda)).$
A   direct method of   construction of the measure will be recalled 
  in   the next section. 
\par
 The paper  concerns  the positive part of the 
   spectrum  of    the operator  $H^{\alpha}.$  Let $H^{p}, \quad p=0,1,2,..,$ 
   be the space of functions on $R_{+}$
  whose derivative of   order $p$ belongs  to $L^{2}.$
  The simplest form of our result is given by 
 \par
\begin{theorem}. \label{theorem1}\quad 
Let  $V \in  H^{p-1} \cap L^{p + 1}$ for some integer $p\ge 1, $ i.e 
\begin{equation} \label{mnv}
J_{p}(V) =
\int_{0}^{\infty} (|V^{(p-1)}(x)|^{2} + |V(x)|^{p + 1} )     d\,x < \infty.
\end{equation}
 Then:
  \par
  1)  for all $\alpha \in [0,\pi),$ the  essential support of the absolutely  
 continuous component of 
  the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$   coincides with
 $[0,\infty),$ i.e. $\mu^{\alpha}_{ac} (\Gamma ) > 0 $ for any  Borel set 
 $\Gamma \subset [0, \infty), |\Gamma | > 0;  $
 \par
2)  for a.e. $\alpha \in [0,\pi),$ the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$ is absolutely  
 continuous on $[0, \infty)$.
\end{theorem} 
 \par
For $p=1,$  the first    part of this theorem 
is identical  to the main result of the paper \cite{DK} 
 by P. Deift and R. Killip.   The paper \cite{DK} by itself is a  
  culmination of  a long sequence of 
publications  (see for 
instance \cite{CKR}, \cite{CK}, \cite{R})
 on spectral $L^{2}-$ conjecture. The central point of the proof in \cite{DK}
  was 
a relationship between the transmission coefficients of the operators with fast
decaying potentials and $L^{2}$- norm of their potentials.
The $L^{2}$  - norm  of a solution  is a first integral of the KdV  equation  
 $V_{t} = 6 V V_{x} - V_{xxx}. $ The idea  to use the high order 
  first  integrals:
 \begin{equation}
  I_{k} (V) = \int_{0}^{\infty} P_{k} (V, V', V'',..., ) d\,x 
  \end{equation}
 in a similar setting  is sitting on the surface.
Here $P_{k}$ are certain polynomials (see [4]).
  In particular,    $P_{1} =  V^{2},$
  $P_{2} = (V')^{2}/ 2 +  V^{3},$
 $ P_{ 3} = \frac12 V_{xx}^{2} - \frac{5}{2}V^{2} V_{xx} +\frac{5}{2}V^{4}.$ 
\par
Our  main  goal here is  an implementation  of this idea  and its
development in  several directions:

a) clarification of the proof  by separation of the measure  theory
and KdV
ideas;

b) simplification  of the result (instead of very  complicated expressions  for
  $ I_{k}, k=1,2,.. , $  we use very simple functionals  $ J_{k}, k=1,2,.. ); $
  
 c) construction of  several classes of examples where the spectral bifurcation
 $\Sigma_{ac} \Rightarrow  \Sigma_{pp} $  is related to the transition
 $$  J_{k} < \infty \quad \Rightarrow \quad   J_{k} = \infty ,\quad k=1,2,.. 
 \quad.$$ 
 \par
 The outline
 of the paper is the following. First we obtain a new test for the existence 
 of  the 
 a.c.  spectrum of the Schr\"{o}dinger operator based on the boundedness of the
 transfer operator in average (with respect to the spectral parameter).  Then 
 a formula is given for the norm of  the transfer operator in terms of 
 the transmission   coefficient.
  This  provides the existence of the a.c. spectrum if there is a sequence 
 $L_{n}\to \infty $ such that the transmission coefficients of the operators with
 potentials $V_{n}$ are bounded in average.  Here $V_{n}$ is the restriction  
 of  $V$  to the segment $[0,L_{n}].$  In section 2, the transmission coefficients
 are estimated in terms of the first integrals of the  KdV  equation.   Inequalities
 of the Kolmogorov  type allow us to estimate the first integrals 
 by  the
  functionals $J_{p}.$  A short section 3   contains the end of the proof
 of Theorem1. The last section
 provides examples showing the "exactness" of 
 Theorem1.
 \par
 In the spirit of   general functional analysis one could  expect that 
 the first term of the integrand in  condition (\ref{mnv}) dominates and 
 the second term can be omitted.
However, the   statement that 
$\int_{0}^{\infty} |V^{(p-1)}(x)|^{2}d\,x < \infty$ 
for some $p\ge 2$  implies the existence of the  a.c. spectrum
 on $[0,\infty),$ 
is obviously wrong.
If, for instance , $p=3,$    $V_{1}$ is an infinitely  smooth, 
compactly supported (or fast decaying) potential  and
$$ V(x) =  V_{0}(x) + V_{1}(x), \quad  
 V_{0}(x) = \alpha x,\quad \alpha> 0~,
$$
then $V'' \in L^{2}(R_{+}),
$  but the spectrum of 
$H = - d^{2}/ dx^{2} + V(x)$ is discrete for any $\alpha\in [0,\pi)$.
 Using piecewise smooth (and mainly linear)  functions  $V_{0}(x)$ 
 instead of 
$\alpha x,$  
one can  construct  examples  when   $ V'' \in L^{2}(R_{+})$, but
 the spectrum is 
a.c. above an arbitrary fixed level $ h_{0}= \sup  V_{0}(x) > 0 $,
  and is    singular below this  level.
\par
  We need some additional conditions on the
potential $ V(x)$ to guarantee that 
$\Sigma_{ac}= [0,\infty) $. We expect that the condition $|V(x)| \to 0$ 
 together with 
$\int_{0}^{\infty} |V^{(p-1)}(x)|^{2}   d\,x < \infty$ for some $p\ge 1$ is
 sufficient
for  $\Sigma_{ac}(H)= [0,\infty)$ in many interesting cases 
.  Even  the more general statement  is probably 
true:
\par
 If \, $||V||_{\infty} < \infty$ , \, $ h_{0} =\limsup_{x\to \infty} V(x)$ 
and 
 $V^{(p)}\in L^{2}(R_{+})$ for some $p\ge 1$, then the spectrum of 
$H = - d^{2}/ dx^{2} + V(x)$  is singular for $\lambda \le h_{0}$ and  contains 
the a.c. component which is  essentially supported on $[h_{0},\infty).$  
\par 
We shall  return  to the analysis  of this conjecture in a 
 different publication.

\section{ Tests for the a.c. spectrum }
%1.1
\subsection{Spectral measure }

 \par
  Let $\psi = \psi_{\lambda}(x)$  be the solution of the following initial 
  value problem on $R_{+}:$
   \begin{equation}
 H \psi = {\lambda}\psi,\quad \lambda >0, \quad\psi (0) = \sin \alpha, 
 \quad \psi'(0) = \cos \alpha,\quad \alpha \in [0,\pi). 
   \end{equation}
 The  spectral measure 
 $\mu^{\alpha}(d\,\lambda)$ of the operator  $H = H^{\alpha}$ 
  can be defined in terms 
 of the generalized Fourier
transform  and the Parseval equality  \cite{LS}: 
\begin{equation}
 L^{2}(R_{+}) \ni \varphi(x)  \rightarrow 
 \hat\varphi(\lambda) = \int_{0}^{\infty} \hat\varphi(x) 
 \psi_{\lambda}(x) d\,x,
\end {equation}
\begin{equation}
\varphi(x) = \int_{\Sigma(H^{\alpha})} \hat\varphi(\lambda) \psi_{\lambda}(x)
\mu^{\alpha}(d\,\lambda),
\end {equation}
\begin{equation}
(\varphi(x), \varphi(x))_{L^{2}(R_{+})} = 
(\hat\varphi(\lambda), \hat\varphi(\lambda))_{L^{2}
(\Sigma(H^{\alpha}),\mu^{\alpha} )}.
\end {equation}
 Here $\Sigma(H^{\alpha})$ is  the spectrum of $H^{\alpha}.$ We denote by 
 $T_{\lambda} (x_1, x_2)$  the  transfer   operator on the interval $[x_1, x_2]
 $: 
     \begin{equation}
  T_{\lambda}(x_1,x_2): (\psi_{\lambda}(x_1), \psi_{\lambda}^{'}(x_1)) 
  \rightarrow
  (\psi_{\lambda}(x_2), \psi_{\lambda}^{'}(x_2) ).
  \end {equation}
   The following  formula  (attributed   to  M.Krein's school, 
   see, for example, \cite{A})
   gives an  expression for the spectral measure,
    and  is very important  for our analysis:
 \begin{equation} \label{w1}
\mu^{\alpha}(d\,\lambda) = w-\lim_{L\to \infty}    \mu^{\alpha}_{L}(d\,\lambda),
\end {equation}  
 where
 \begin{equation}\label{w2}
 \mu^{\alpha}_{L}(d\,\lambda) =\frac{d\,\lambda}{\rho^{2}_{\lambda}(L)}
  \end {equation}
and 
 \begin{equation} \label{w3}
\rho^{2}_{\lambda}(x) = || T_{\lambda} (0 , x) e_{\alpha}||^{2}, \quad
e_{\alpha} = (\sin \alpha,  \cos \alpha     ).
\end {equation}
%1.2
\subsection {The a.c. spectrum  and the transfer operator}
Let $\Delta \subset(0,\infty) $  be  a  closed interval. 
It is well-known that the spectrum of $H^{\alpha}$ is  pure  a.c. on  $\Delta$
if  the transfer operator $T_{\lambda}(0,L)$ 
is   bounded uniformly in $L\in (0,\infty) $ and $\lambda \in \Delta$ 
 (see \cite{S2},  \cite{S3} and 
references there). 
A part of this  statement, namely, 
the existence of the a.c.  spectrum can be justified easily using 
(\ref{w1})  and (\ref{w2}). In fact, assume that 
\begin{equation}\label{bound}
|| T_{\lambda}(0,L_{n})|| \leq c_0 ,\quad   \lambda\in \Delta
 \end {equation}
 for  some  sequence $\{L_n\},  L_n\to \infty,$ and  a fixed interval 
$ \Delta\subset (0,\infty)$, $|\Delta|> 0. $ 
Then 
\begin{equation}
\frac{\mu^{\alpha}_{L}(d\,\lambda)}{d\,\lambda}\ge \frac{1}{c_{0}^{2}}, \quad 
\lambda \in \Delta,
 \end{equation}
and therefore, the same estimate is valid for the limit measure 
$\mu^{\alpha}(d\,\lambda).$
\par
One of  our central observations  is that the
uniform (\ref{bound}) or  even a  pointwise  estimate
can be replaced  by boundedness   in average.
 
\begin{theorem}.\label{mean} Let $\Delta \subset(0,\infty) $  be  a  closed 
interval.
  If  there is a  
 sequence $\{L_{n}\}, \quad L_{n}\to \infty, $ 
  such  that 
\begin{equation} \label{bb1}
 \int_{\Delta} \ln \,(|| T_{\lambda}(0,L_{n})||) \, d\,\lambda \le C(\Delta),
 \end {equation}
then  
\par
i)   for all  $\alpha \in [0,\pi),$ the  essential support of the measure 
$\mu_{ac}^{\alpha}(d\,\lambda)$ 
 contains $\Delta$;   
\par
ii)  for a.e. $\alpha \in [0,\pi),$ the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$ is absolutely  
 continuous on $\Delta$.
\end{theorem}
{\bf Remark.}  Theorem  \ref{mean}   remains valid if logarithm function in 
(\ref{bb1})  is replaced  by any monotonically increasing on  $[1,\infty)$
function  $ \Phi(\lambda)$.
\par
In order to prove  the theorem we need 
\begin{lemma}.\label{propos}
Let $\nu_n (d\,\lambda),  n=1,2,...,$  be a sequence
 of measures
on an interval $\Delta$,
   absolutely continuous with respect to  $d\,\lambda$ with  the
 densities
$$
\frac{\nu_n(d\,\lambda)}{d\,\lambda} =h I_{\Delta \setminus  \Gamma_n}
(\lambda ){d\,\lambda}, \quad  n=1,2,...
 $$
Here  $h$ is a constant, $I_{B}(\lambda )$  is the
 characteristic function of 
a Borel set $B,$  and  $$|\Gamma_n | \le \varepsilon .$$
 Then there is a subsequence $\nu_{n_{l}}(d\,\lambda)$ which converges weakly to a
 measure $\nu (d\,\lambda),$  and any such 
 weak limit $\nu (d\,\lambda)$  has the following properties:

 1) the  measure  $\nu (d\,\lambda)$ is absolutely continuous 
 with respect  to $d\,\lambda,$  and 
 $  {\nu(d\,\lambda)}/{d\,\lambda} \le h;$
 

 2) there exists  a set $\Gamma$ 
 such that $ |\Gamma| \le  2 \varepsilon$  and 
 $$\frac{\nu(d\,\lambda)}{d\,\lambda}\ge \frac{h}{2} \quad \mbox{on} \quad 
  \Delta\setminus\Gamma;
 $$ 
    \end{lemma}
{\it Proof. }  Let $F_{n}(\lambda) =\int_{-\infty}^{\lambda}\nu_n (d\,s) $ be 
the distribution function of the measure $\nu_{n}(d\,\lambda).$
 By
 the second Helly  theorem  there exists a subsequence 
 $\nu_{n_{l}}(d\,\lambda)$ which converges weakly to a
 measure $\nu (d\,\lambda)$, and  for any such subsequence
  the  distribution functions $F_{n_{l}}(\lambda)$ converge to
the  distribution function $F(\lambda)$ of the limit measure 
 at all  points $\lambda$  where   $F(\lambda)$  is continuous.
   The set where $F$ is discontinuous is at most countable. 
    Consider an interval $[\alpha,\beta) \subset \Delta $ with 
$ \alpha, \beta  $ outside  of this  set. Then 
$$  
\nu_{n_{l}} (\alpha,\beta)= F_{n_{l}}(\beta)-F_{n_{l}}(\alpha) = 
\int_{\alpha}^{\beta}\nu_{n_{l}} (d\,\lambda)\le 
\int_{\alpha}^{\beta} h d\,\lambda = h (\beta -  \alpha  ).
 $$
 Passing to the limit $l\to\infty,$ 
  we get 
 $$
\nu (\alpha,\beta)\le h(\beta-\alpha).
 $$ 
This estimate implies that the measure  $\nu (d\,\lambda)$ 
is absolutely continuous on $\Delta $  and 
 $  {\nu(d\,\lambda)}/{d\,\lambda} \le h;$
\par To prove the second statement of the lemma we note that 
 $$
 \nu_{n_{l}}(\Delta) \ge h(|\Delta| - \varepsilon) 
 $$   
and therefore,
\begin{equation} \label{in1}
\nu(\Delta) \ge h(|\Delta| - \varepsilon).
 \end{equation} 
 On the other hand, if 
$$
\Gamma = \{\lambda: \frac{\nu ( d\,\lambda)}{d\,\lambda}\le \frac{h}{2} \}
$$
then 
\begin{equation}\label{in2}
 \nu(\Delta) \le \frac{h}{2}|\Gamma| + h(|\Delta|  - |\Gamma|).
\end{equation}
 Inequalities (\ref{in1}) and (\ref{in2}) imply the second statement of Lemma.
\par
This completes the proof of the Lemma.
\par
{\it Proof of  Theorem  2}.  
We shall  prove Theorem  2 and the Remark simulteneously, i.e. we shall
assume that the logarithm function in (\ref{bb1}) is replaced  by a positive
monotonically increasing function $\Phi.$  Fix $\varepsilon > 0.$ 
 Let  $R_{\varepsilon}$  be  the solution of 
the equation 
$$  \Phi( R_{\varepsilon}) =\frac{C (\Delta)}{\varepsilon}
$$
where $ C (\Delta) $  is defined in (\ref{bb1}). Let 
$$
 \Gamma_{n}(\varepsilon) = 
 \{ \lambda : \lambda \in \Delta, \quad
 || T_{\lambda }(0,L_{n})|| \ge  R_{\varepsilon} \}.
 $$
 Since $ \Phi (t) > \frac{C (\Delta)}{\varepsilon} $  when 
 $t>  R_{\varepsilon}$, we have  
$$
\int_{\Delta} \Phi (|| T_{\lambda}(0,L_{n})||) d\,\lambda \ge
 \Phi (  R_{\varepsilon})| \Gamma_{n} | =\frac{C (\Delta)}{\varepsilon}
 | \Gamma_{n} |.
$$ 
 From here and (\ref{bb1}),  it immediately follows that 
 $ |\Gamma_{n}(\varepsilon)| \le \varepsilon.$ 
 \par
  Consider a  sequence of measures $\nu_{n, \varepsilon}(d\,\lambda)$  on 
  $\Delta $  defined by the  formula
  $$
  \nu_{n, \varepsilon}(d\,\lambda ) = 
  \frac{I_{ \Delta \setminus \Gamma_{n}(\varepsilon)} (\lambda)}
  { R_{\varepsilon}}
  d\, \lambda.
    $$ 
 Let   
$$\mu_{L_{n}}(d\,\lambda) = 
\frac{d\,\lambda}{||T_{\lambda}(0,L_n)e_{\alpha}||^{2}}$$
be the measure defined in (\ref{w2}) with $ L = L_{n}.$ Then
\begin{equation}  \label{inn3}
\mu_{L_{n}}(d\,\lambda) 
\ge \nu_{n, \varepsilon}(d\,\lambda), \quad n=1,2,.. \, , \quad \lambda \in \Delta. 
\end{equation}
Let $\nu_{ \varepsilon}(d\,\lambda)$ be a weak limit of some subsequence
$\nu_{{n_{l}}, \varepsilon}(d\,\lambda),\quad l\to \infty.$ 
Its existence follows from Lemma \ref{propos}.    
 From (\ref{w1})  and (\ref{inn3}) it follows that 
$$
\mu^{\alpha}(d\,\lambda )\ge \nu_{ \varepsilon}(d\,\lambda), \quad 
\lambda \in \Delta.
$$
Lemma 1 applied  to the sequence   $\nu_{n, \varepsilon}(d\,\lambda)$
leads to the  estimate
$$
\frac{\nu_{ \varepsilon}(d\,\lambda)}{d\,\lambda} \ge 
\frac{I_{\Delta \setminus \Gamma_{\varepsilon}}}{ R_{\varepsilon}}, \quad
 \lambda \in \Delta,
$$
with some $\Gamma_{\varepsilon}$  such that 
 $|\Gamma_{\varepsilon}| \le 2\varepsilon.$  
 Thus, the same estimate holds  for $\mu^{\alpha}(d\, \lambda ).$
  Since  $\varepsilon>0$ is an arbitrary small number,  
 the  essential support of 
  $\mu^{\alpha}_{ac}(d\,\lambda)$ contains  the interval $\Delta .$ 
 \par
 The  part i) of Theorem 2 is proved.
\par  Now we shall  prove  {\it  part ii) of Theorem 2.}
Let us consider first a specific  Hamiltonian $H^{\pi/2}$ 
(Neumann boundary conditions) and  let 
 $\mu^{\pi/2}(d\,\lambda) $ be its spectral measure.
  Under Birman's condition (3) on the  potential,
  \begin{equation} \label{tim}
\int_{\lambda_{0}}^{\infty}\frac{\mu^{\pi/2}(d\,\lambda)}{1 + |\lambda|} < \infty
  \end{equation} 
  where $\lambda_{0}$ is the bottom of the spectrum of $H^{\pi/2}.$
This estimates can be found in \cite{LS},  (Chapter 4) for much more general
potentials and for  boundary  condition (\ref{boundZ})
 with $\alpha \not= 0.$
Estimate (\ref{tim}) fails if $\alpha = 0.$ One could also prove a stronger
estimate for $\mu^{\alpha}(d\,\lambda)$:
\begin{equation} \label{tima}
  \int_{\lambda_{0}}^{\lambda}\mu^{\alpha}(d\,\lambda) 
 \le C_{\alpha} (1 +\sqrt{|\lambda|}), \quad  \lambda\ge \lambda_{0}(\alpha),
  \alpha \not= 0.
 \end{equation}  
 We shall not prove 
 (\ref{tima}) here, since (\ref{tim}) is enough for our purpose.
 \par
 Estimate (\ref{tim}) allows us to define 
  the  Borel transform $ m^{\pi/2}(z)$ of the spectral measure 
$\mu^{\pi/2}(d\,\lambda)$: 
$$
  m^{\pi/2}(z)= \int_{\lambda_{0}}^{\infty}\frac{\mu^{\pi/2}(d\,\lambda)}
  {z - \lambda},
   \quad \Im z > 0.
$$
Function $ m^{\pi/2}(z)$ is analytic in the half- plane $ \Im z  > 0.$
  Due to the general theory of Borel transforms (see \cite{S1}, Chapter I),
 $ m^{\pi/2}(z)$ has the following properties:
 
  a) for a.e. \quad $ \lambda \in R,$ 
   \begin{equation}
 0\le  \Im m^{\pi/2}(\lambda + i0) < \infty; 
 \quad |\Re m^{\pi/2}(\lambda + i0)| < \infty.
 \end{equation}
 
  b) the absolutely continuous  component of $\mu^{\pi/2}(d\,\lambda)$
    is  equal to   $$\pi^{-1}\Im m^{\pi/2}(\lambda +i 0)d\,\lambda;$$
  
  c) the  singular component of the measure $\mu^{\pi/2}(d\,\lambda)$ is 
  supported 
  on the set $ \{\lambda: \Im   m^{\pi/2}(\lambda +i 0) = \infty \}.$ 
  
Consider  the  set
   \begin{equation}\label{ImRe}
 \Gamma = \{\lambda: 0< \Im m^{\pi/2}(\lambda  + i0) <\infty \},
  \end{equation}
and  let  $$\Omega = [0,\infty) \setminus \Gamma .$$
 \par
  Part i) of Theorem 2  and the properties  a) - c) above imply that  
  $|\Omega|=0$  and 
the  singular component  of $\mu^{\pi/2}(d\,\lambda)$  is supported 
  on $\Omega.$ We shall   show that  the  singular components of all
   measures $\mu^{\alpha}(d\,\lambda), \, \alpha \in (0,\pi),$  are supported 
  on the same set $\Omega.$
 Recall  (see \cite{LS}) that  Weyl function $ m^{\alpha}(z)$ 
  of the   operator $H^{\alpha}$ is defined by the relation 
 \begin{equation}\label{Weil}
\psi_{1}(x,z) + m^{\alpha}(z) \psi_{2}(x,z) \in L^{2}[0,\infty), \quad \Im z > 0,
\end{equation}
where $\psi_{1}(x,z)$ and $\psi_{2}(x,z)$  are solutions of the  equation
$ - \psi''  + V(x)\psi= z \psi $ 
 with initial  conditions
 $\psi_{1} (0) = \sin \alpha , \, \psi'_{1} (0) = \cos \alpha,$ \,
  and $\psi_{2} (0) = -\cos \alpha ,$\, $\psi'_{2} (0) = \sin \alpha.$
The following representation is  valid (see \cite{LS}): 
$$
  m^{\alpha}(z)=  \cot \alpha + \int_{\Sigma(H^{\alpha})}
  \frac{\mu^{\alpha}(d\,\lambda)}{z - \lambda}, \quad \Im z > 0,
  \quad \alpha \in (0,\pi).
$$
The  properties a) - c) of the Borel transform  of a measure are valid
  for $ m^{\alpha}(z).$
 Direct calculations based on (\ref{Weil}) 
  give  
\begin{equation}   \label{drob}
m^{\alpha}(z) = 
\frac{\cos\alpha + m^{\pi/2}(z)\sin \alpha}
{\sin \alpha - m^{\pi/2}(z)\cos \alpha }.
 \end{equation} 
Using (\ref{drob}), it is easy to check that 
  $$ 0< \Im m^{\alpha }(\lambda + i0) < \infty $$ 
at points $\lambda $ where $ 0<  \Im m^{\pi/2}(\lambda + i0) < \infty. $ 
 Hence, $\mu_{sing}^{\alpha}$
   is supported  on $\alpha -$  independent  set $\Omega$  of Lebesgue measure
   zero.
   \par
   The  spectral averaging of $\mu^{\alpha}(d\, \lambda )$  over
  $\alpha \in (0,\pi)$ 
    gives (see  \cite{A} or \cite{S1})
 $$
\frac{1}{\pi}\int_{0}^{\pi}\mu^{\alpha}(d\,\lambda)  = d\,\lambda.
$$
 This formula applied  to the set $\Omega$
leads to 
$$
 \frac{1}{\pi}\int_{0}^{\pi}  \mu^{\alpha}(\Omega)
  d\, \alpha=|\Omega|=0.
$$
Then $ \mu^{\alpha}(\Omega)= 0 $  for a.e. $\alpha$ , i.e. 
$ \mu_{sing}^{\alpha}(d\lambda)= 0$  for a.e. $\alpha$. 
\par
Theorem 2 is proved.
 
 \subsection {The a.c. spectrum and the  transmission coefficient }
First of all  we need a simple proposition which provides
 a connection
 between the transfer matrix and the scattering theory. 
 In this section, it is convenient to  replace spectral parameter
  $\lambda > 0$ by $k^{2}$.
Let  $\psi_{k}= \psi_{k^{2}} = \psi_{\lambda},$ and let 
a new transfer operator $P_{k}(x_1,x_2),\quad k = \sqrt{\lambda},$
    (Pruffer transformation) be  given by 
  \begin{equation}
  P_{k}(x_1,x_2): \left(\psi_{k}(x_1), \frac{\psi_{k}^{'}(x_1)}{k}\right) 
  \rightarrow
  \left(\psi_{k}(x_2), \frac{\psi_{k}^{'}(x_2)}{k}\right).
  \end {equation}
 It follows from  the definition of $P_{k}$ that,   for $\lambda = k^{2}$ 
 and    $D = \mbox{diag}(1,k),$
   \begin{equation} \label{diag}
 T_{\lambda}  = D^{-1}(k)  P_{k}D(k).
   \end {equation}
 \par
 \begin{lemma}.\label{a+b}  
  Let us  consider a
 compactly supported  on $(0,L)$ continuous  potential  $V(x)$,
  and let  $\psi(x,k)$ be the  Jost solution of the equation 
 $ \psi^{''}(x,k) + V(x)\psi(x,k) = k^{2}   \psi(x,k),  $  $x\in R,$ such that 
       \begin{equation} \label{a(k)}  \psi(x,k)   =    \begin{cases}
  \exp(-ikx), &   x\le 0,  \\ \notag
     a(k) \exp(-ikx) + b(k)\exp(ikx), &  x\geq  L.\\
         \end{cases} 
      \end{equation}
  Then  the norm of the   transfer operator  $P_{k}$ is given by 
  \begin{equation} \label{norm}
  || P_{k}(0,L) || = |a(k)| + |b(k)|.
 \end {equation}
\end{lemma}
{\bf Remark.} Everywhere below, we shall refer to  $a(k)$ 
 as the  transmission 
coefficient, although this term is more often
 used for ${1}/{a(k)}$, which is the 
corresponding coefficient  in  the asymptotic  expansion of the solution 
$ \psi / a $  of the  scattering problem.

{ \it Proof.} Denote  $a(k) = a_1 +i a_2; \quad b(k) = b_1 + i b_2 .    $
Vector $ (\Re  \psi(x,k), \Im  \psi(x,k))$ is equal to 
$(\sin k x, \cos k x)$  if $x\le 0,$ and it is equal to 
$$( (a_1 +b_1)\cos k x  + (a_2-b_2) \sin k x,
 (a_2 +b_2)\cos k x  + (b_1-a_1) \sin k x )
 $$
 if $x \ge L.$
Hence, the matrix of the  transfer  operator  $P_{k}(0, x), x\ge L $ 
is equal to 
\begin{equation}
  \left[
 \begin{array}{cc} 
               (a_1 +b_1)\cos k x  + (a_2-b_2) \sin k x &
                (a_2 +b_2)\cos k x  + (b_1-a_1) \sin k x \\                         
	   (a_2-b_2)\cos k x  -(a_1 +b_1) \sin k x &                                           
               (b_1-a_1)\cos k x  + (a_2 -b_2) \sin k x                         				 
\end{array} \right].
      \end{equation}Then  
\begin{equation}
\mbox{Tr} (P_{k}(0, x)P^{*}_{k}(0, x)) =2(|a|^2 + |b|^2 ).
\end{equation} 
The  norm  $ ||P_{k}|| $ is equal to  $\sqrt{\mu}$
 where $\mu$ is the larger  eigenvalue of 
 the matrix $P_{k}P_{k}^{*}.$  Since  $\det P_{k} = 1,$
 the characteristic equation for $\mu$ is  
\begin{equation}\label{charact}
\mu^2 - 2 (|a|^2 + |b|^2 ) \mu +1 =0.
\end{equation} 
The identity  $|a|^2 =  1 + |b|^2 $  holds  for coefficients $a$ and $b;$  
 and therefore, 
the roots of (\ref{charact}) are 
$\mu_{1,2}  = (|a| \pm |b| )^2.$
Then $$
||P_{k}(0, x)|| = \sqrt{\mu_1} =|a| + |b|.
$$
Lemma is proved. 
\par
 The  identity $|a|^2 =  1 + |b|^2 $  and  formulas (\ref{norm}), (\ref{diag}) 
 allow us to reformulate Theorem 2 as follows.
 Define by  $I_{[a,b]}(x)$  the characteristic function of the interval 
 $[a,b].$
   \begin{theorem}.\label{mean2} 
   Let $\Delta \subset(0,\infty) $  be  a  closed 
interval.  Assume that there exists  a sequence $ L_{n}\to \infty $ such that 
 the  transmission
  coefficients  $a_{n}(k)$  of the   operators 
 $$H= -d^{2}/dx^{2}  + V_{n}(x),\quad \hbox {where} \quad 
 V_{n}(x)= V(x) I_{[0,L_{n}]}(x),$$
   satisfy the condition
\begin{equation} \label{simple2}
 \int_{\Delta} \ln |a_{n}(k)| d\,k \le C(\Delta), \quad n =1,2,..\, .
 \end {equation}
Then  
\par
i) for all  $\alpha \in [0,\pi),$ the  essential support of the measure
$\mu_{ac}^{\alpha}(d\,\lambda)$ 
 contains $\Delta$;   
\par

ii)  for a.e. $\alpha \in [0,\pi),$ the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$ is absolutely  
 continuous  on $\Delta$.
\end{theorem} 
 
\section{ The estimates for the first integrals of KdV  equation.}

  Our  next goal is to further   simplify  Theorem 2 by 
rewriting 
 the estimate 
 (\ref{simple2})  in  terms of the potential $V$ and its derivatives.
 The central  point here is the estimation of 
 $\int_{\Delta} \ln |a(k)| d\,k $  for  compactly supported or 
  fast decaying  potentials. 
 \par
    Let $V(t,x)$ be a
  solution  of the Cauchy problem for the  KdV- equation
   $$ V_{t} =  6 V V_{x} -  V_{x x x}, \quad V(0,x) = V(x),
   $$
 in the class  of the  fast decaying functions. 
  Let 
 $\lambda_{j} = - (\kappa_{j})^{2},  j=1,2,.., q,$   be   negative 
 eigenvalues   
  and  let $a(k)$  be the transmission  coefficient for the Hamiltonian  
  $H = - d^{2}/dx^{2} + V(t,x).$
  Then (as was discovered  by Gardner, Green, Kruskal and Miura,
 see details 
  in \cite{GGKM} or   \cite{DEGM}) 
  the  scattering data
 $ \{ \lambda_{j},  j=1,2,...q, \quad  a(k), k\in R \}$  
  do not depend on $t.$
   They are invariants of the KdV - dynamics. 
   \par
  It is also well known that the
 function  $a(k)$  admits analytic continuation 
  in the  half plane  $\Im z >0 $, and 
  $ i \ln a(z) $ has the asymptotic expansion 
\begin{equation}
     i\ln a(z) \sim  - \sum_{m=0}^{\infty}
 \frac{ I_m
 (u)}{z^{2m+1}}, \quad  \Im z >0 ,\quad  z\to \infty.  \label{asym}       
   \end{equation}
Coefficients $ I_m$  are invariants  of the KdV  equation also.
  They have two different  representations. First,
 they can 
be written in  the form 
 \begin{equation}\label{in}
I_m=  (-1)^{m+1}  \sum_{j=0}^q \frac{ 2 \kappa^{2m+1}_j}{2m+1}  +
\frac1{\pi}\int_{-\infty}^{+\infty}  
{k}^{2m} \ln|a(k)|) d\, k  ,  \quad  m= 0,1,2,...                               
\end{equation}
 This representation  can be found  in \cite{DEGM} and  can be 
  proved   in   the following  simple way.
 \par
  Let $B_q(z)$  be the  Blashke   product with
 the same zeros
$\{i\kappa_k\}_{k=1}^q $  as the zeros of the function $a(z)$ 
in the upper half plane:
\begin{equation}
B_q(z)=\prod_{j=1}^q \frac{z-i\kappa_j}{z+i\kappa_j}.
\end{equation}
 For real $z,$  we have $|B(z)|= 1.$
Hence, 
\begin{equation}
\Im [i\ln \frac{a(z)}{B_q(z)}] = \ln |a(z)|,  \quad  z\in R.
\end{equation}
Using the Gerglotz formula, we get
\begin{equation} \label{b1}   
i\ln a(z) = i\ln B_q(z) + 
\frac{1}{\pi } \int_{-\infty}^{+\infty} \frac{\ln |a(k)|}{k - z}
\,d \,k  .                                           
\end {equation}
It is  clear   that
\begin{equation}
i\ln \frac{z-i\kappa}{z+i\kappa} \sim 
\sum_{m=0}^{\infty}\frac{ 2 \kappa^{2m + 1 } }{2m+1}\frac{(-1)^{m}}{z^{2m+1}},
\quad  z\to \infty.  \label{b2}   
 \end{equation}
 Using  (\ref{b1}) and (\ref{b2}), we get (\ref{in}).
 \par
 Denote by $ r= (r_{0}, r_{1},....,r_{2m} )$ a vector with nonnegative integer
  coordinates, and let
  \begin{equation}
 d(r) = 2r_0+3r_1+\cdot\cdot \cdot + (2m + 2 ) r_{2m}
\quad .
\end {equation}  
 The second  representation  for $I_{m}$   is given by 
 \begin{lemma}.  
  The following    formulas  hold:
\begin{equation}\label{repr}
 I_{m} = \int_{-\infty}^{\infty} P_{m} ( V ,V' ,...)  d\,x ,\quad 
  n= 0,1,2,3,.. \, , 
\end {equation}
where $P_m$ 
 are the  polynomials in the function $ V $
and its derivatives:
\begin{equation} \label{pol}
P_m ( V ,V' ,...) = \sum _{ \{ \,r: \,d(r) =  2m + 2 \} } c_{m}(r)
 \prod_{l=0}^{2m}  {[V(x)^{(l)}]}^{r_l}. 
\end{equation}
\end{lemma}
{\bf Remark 1.}
The coefficients $c_{m}(r)$ can be calculated by the formula
$$
c_m(r) = \hbox{\bf E }\prod_{l=0}^{m} \frac{\xi_{l}^{r_l}} {r_l ! }.
$$
Here 
\begin{equation}
\xi_{l}(x)  =   \int_0^1  \omega^{l} (\tau )d\,\tau
\end{equation}
and  $w  ( \tau)  $  is  the "Wiener bridge",
 i.e.   conditional Wiener  process determined by the
condition $w  ( 0) =  w  ( 1 ) = 0   $
with zero means and the correlation function
$B ( s , t ) =  (s \wedge t) ( 1 - s \vee t ).  $

 \par 
{\bf Remark 2.} The structure (\ref{pol}) of the polynomials  $P_{m}$ is the consequence of the
 invariance of KdV equation under scaling $ V(t,x)  \to \varepsilon^{2}
 V(\varepsilon^{3} t,\varepsilon x).  $ This invariance implies  that 
 the polynomial
 $ P_m ( V ,V' ,...) $ must be homogeneous of order  $2m + 2$ under
  the transformation
 $ V(x)  \to \varepsilon^{2} V(\varepsilon x).  $  Specific values of the 
 coefficients  $c_{m}(r)$ in the "Minakshisundaram  type" expansion (\ref{pol})
 are not important for our analysis. However, the estimates for
  $c_{m}(r),$ 
 $m\to \infty,$  can be fundamental for some  applications.
\par
 Proof  of this lemma  uses  the  formula (\cite{DEGM}, p. 169):
\begin{equation} 
\mbox{tr}[(H-z^2I )^{-1}- (H_{0}-z^2I )^{-1}] = 
-\frac{1}{2z}\frac{d}{dz} \ln a(z) 
\label{trace}
\end{equation} 
 and the  Kac-Feynman  formula (for complete proof see \cite{Nov}).
 \par
The following inequalities are crucial.
\begin{theorem}. \label{KdV} 
The functional $I_{m}$  has  the  estimates:

a) 
\begin{equation}\label{twosides1}
I_{m } \le c_{m} 
	  \int_{0}^{\infty} (|V^{(m-1)}(x)|^{2} + |V(x)|^{m +1}) d\,x, 
	\qquad  m= 1,2,..\, ,                
\end{equation} 

b) 
\begin{equation}\label{twosides2}
\frac1{\pi}\int_{-\infty}^{+\infty} {k}^{4m+2}  \ln|a(k)| 
 d\, k \le I_{2m +1}, \qquad  m= 0,1, ,2,..\quad , 
\end{equation} 

c) 
\begin{equation} \label{twosides3}
\frac{1}{\pi}\int_{-\infty}^{+\infty} {k}^{4m }  \ln|a(k)| 
 d\, k \le I_{2m } +  d_{m} 
	  \int_{0}^{\infty} |V(x)|^{2m + 1}  d\,x,   \quad  m= 1, ,2,..\, .                
\end{equation} 
Here  $c_m$  and $d_m$  are  positive constants 
which  do not depend on $V$.
\end{theorem}
\begin{corollary}.
For any $m=1,2,..\, ,$ and  any closed interval $\Delta \in (0, \infty),$
$|\Delta| < \infty, $
\begin{equation}  \label{twosides4}
\int_{\Delta }\ln|a(k)|  d\, k  \le c_{m,\Delta } 
	  \int_{0}^{\infty} (| V^{(m-1)}(x)|^{2} + |V(x)|^{m +1})d\,x, 
	  \quad   m= 1,2,... \quad .               
\end {equation} 
  \end{corollary}  
 {\it Proof}. Inequalities b) and c)  follow from representation (\ref{in}).
  If $m$ is odd, then  both  terms
  in (\ref{in})
 are nonnegative  which implies b).   If $m$ is even,  we can use
  the one - dimensional 
 version  of  the Lieb-Thirring  estimate (see \cite{LT} 
 or \cite{RS}, p. 368):
  for any number  $\gamma > \frac12 $
 \begin{equation}\label{twosides5}
 \sum_{j=1}^{q} |\lambda_{j}|^{\gamma} \le C_{\gamma} 
 \int_{-\infty}^{+\infty}|V_{-}(x)|^{\frac12 + \gamma} d\,x.
 \end {equation}
 Here $V_{-}(x)$ is the negative part of the potential $V(x).$
 This estimate with   $\gamma = 2m + \frac12$  together with 
 (\ref{in}) implies c).
  \par
 To prove a) we use the second representation for $I_{m}$.
 Let us denote by $|| \cdot ||_{s}$  the norm in the space $L^{s}(R)$  and by 
 $A_{m} (r) $ the integral 
\begin{equation}
 A_{m} (r) = \int_{-\infty}^{\infty} \prod_{k=0}^{2m} 
  {[V^{(k)}(x)]}^{r_k} d\,x , \quad  r = (r_{0}, r_{1},....,r_{2m}).
\end {equation}
>From (\ref{repr} ) and  (\ref{pol} ) it follows that the estimate  
\begin{equation} \label{mon}
| A_{m} (r) | \le  C    ( {|| V||^{m + 1}_{m + 1}} +
 || {V^{(m-1)}||^{2}_{2}}), \quad  C = C(r,m),
\end{equation} 
implies a).  It is sufficient to prove (\ref{mon})   for integrals
 $ A_{m} (r) $  with  an additional condition 
\begin{equation}\label{con}
   r_{m} = r_{m + 1} = ...=r_{2m} = 0,
\end {equation}
since an arbitrary $ A_{m} (r) $ can be reduced via
integration by parts  to a linear combination  of  similar integrals for which
$d(r)$ is the same and condition (\ref{con}) holds.
Applying  generalized Holder inequality
\begin{equation}
|\int_{-\infty}^{\infty} \prod_{k=0}^{m-1} f_{k}(x) d\,x| \le
\prod_{k=0}^{m-1} \left(\int_{-\infty}^{\infty} |f_{k}(x)|^{\lambda_{k}}d\,x 
\right)^{1/\lambda_{k}},\quad \sum_{k=0}^{m - 1}1/\lambda_{k} = 1,
\end{equation}
with $\lambda_{k}= (2m + 2) / (k + 2)  r_{k}$   
 to  functions  $f_{k}= {[V^{(k)}(x)]}^{r_k}$       we  get 
\begin{equation} \label{A}
|A_{m} (r)|  \le  \prod_{k=0}^{m - 1} 
 {||V^{(k)}||}^{r_k}_{\frac{2m + 2}{k + 2}}. 
\end {equation}  
We also need a  Kolmogorov type  inequality 
( see, for example, \cite{Gab} ):
\begin{equation}
\label{kolm}
 || f^{(k)}||_{q} \le C(q,p,r, n,k) {|| f||_{p}}^{1-\alpha}
 || {f^{(n)}||_{r}}^{\alpha}
\end{equation}
which is valid  if 
\begin{equation} \label{kolmog}
 \frac{ 1}{q} =  \frac {n-k}{np} + \frac{k}{ nr},
  \qquad \alpha = \frac{k-1/q +1/p}{n-1/r + 1/p}, \qquad 0\le k\le n.
\end{equation}
Parameters  $q= (2m + 2) /(k + 2),$ \, $p=m + 1,$ \,$ r=2,$\, $n=m-1$
 satisfy  the first condition of 
(\ref{kolmog}). With these values of the parameters, $\alpha = k /(m-1),$ 
and (\ref{kolmog}) implies
\begin{equation} \label{B}
 || V^{(k)}||_{(2m + 2)/(k + 2)} \le C(m,k) || V||_{m+1 }^{\frac{m-k-1}{m-1}}
|| {V^{(m-1)}||_{2}}^{\frac{k}{m-1}}.
\end{equation}
  If 
\begin{equation}
   \kappa =  r_{0} +  r_{1} + .... + r_{m-1},
\end{equation}  
  then (\ref{A}) and (\ref{B}) imply 
\begin{equation}\label{kappa}
| A_{m} (r) | \le  C  
 {\left({|| V||^{m+1}_{m+1}}\right)}^{\frac{\kappa - 2}{m-1}}
 {\left(|| {V^{(m-1)}||^{2}_{2}}\right)}^{\frac{m-\kappa + 1}{m-1}}.
\end{equation}
 Holder inequality  $a b \le a^{p}/p + a^{q}/q $ 
with parameters $p= (m-1)/(\kappa-2)$ and $q= (m-1)/(m-\kappa + 1)$  applied to  
(\ref{kappa})  gives (\ref{mon}).
\par

Theorem  \ref{KdV} is  proved.

\section{Theorem 1 and its generalization}
\par
In order  to prove theorem 1 we need 
\begin{lemma}.\label{V_{B} + V_{R}}
 If  $V(x)\in H^{p-1}(R_{+}) \cap L^{p+1}(R_{+})$
  then it can 
 be represented in the form 
 \begin{equation} 
 V = V_{B} + V_{R}                                \label{decomp}
  \end{equation}
where :
\par
 a)  a "bump" potential $V_{B}$  has the form 
 
 \begin{equation}  V_{B} = \sum_{k=0}^{\infty} v_{k} .
 \end{equation}
 Here   potentials 
$v_{k}, k\ge 1, $  have compact supports  without intersection ,
and the support of $v_{k+1}$  is located to the right of the
support of  $v_{k}$; 
\par
 b) $V_{B}$ continued by zero for $x\le 0$ belongs to 
$ H^{p-1}(R) \cap L^{p + 1}(R) ;$
\par
 c)
 the remainder  potential $V_{R}$ is "small" in the following  sense:
  \begin{equation} 
 \int_{0}^{\infty}  | V_{R}(x)| d\, x < \infty. 
  \end{equation}
  \end{lemma} 
{\it Proof. }
The potential $V\in L^{p +1}$ and, as a result,   satisfies 
the Birman condition 
  (\ref{Birman}).
Hence, there exists a  sequence $\{L_{m}\}_{0}^{\infty}$ such that
  $ L_{0} = 0,$ 
$L_{m+1} - L_{m}\ge 2$ and 
 \begin{equation}\label{intr}
  \int_{L_{m}-1}^{L_{m}+1} |V(x)| d\,x < \frac{1}{2^{m}};\quad 
  \int_{L_{m}-1}^{L_{m}+1}( |V(x)|^{p+1} + 
  |V^{(p-1)}|^{2}) d\,x < \frac{1}{2^{m}},\quad m \ge 1. 
  \end{equation}
Consider the following  partition of unity $1=\sum_{k=0}^{\infty}\varphi_{k}$
 on $R.$
Let $\varphi\in C_{0}^{\infty}(R),$  $| \varphi|\le 1, $  
$\varphi =1 $ in a neighborhood of 
$x=0,$ and $\varphi = 0 $  when $|x| > 1.$  Then  $\varphi_{2k}(x)=
\varphi(x - L_{k})$,
 $\varphi_{2k+1}(x)= 1 - \varphi_{2k}(x) -\varphi_{2k+2}(x)$
if $x\in [L_{k}, L_{k+1}]$,  and $\varphi_{2k+1}(x) = 0$ if 
$x\notin [L_{k}, L_{k+1}].$
Obviously  $||\varphi_{k}^{(n)}(x)||_{\infty} < C,$
uniformly in
$k=1,2,..$  and $n=0,1,...,p.$
Define   
$$ 
v_{k}= V \varphi_{2k +1 },\quad
V_{R} (x) = V (x)\sum_{k=0}^{\infty} \varphi_{2k}(x).
$$
Then the statement a) is obvious.
According  to 
(\ref{intr}), the potential $V_{R} (x)$ belongs to $  L^{1}(0,\infty).$  
It remains to prove   that 
$$
  {|| V_{B}||^{p+1}_{p+1}} + || {V_{B}^{(p-1)}||^{2}_{2}} < \infty.
$$ 
Since  $V\in H^{p-1}(R_{+}) \cap L^{p+1}(R_{+}),$
it is sufficient to show   that 
   \begin{equation}\label{k-1}
  \int_{0}^{1}| V^{(k)} | d\,x \le C,\quad
   \int_{L_{m}-1}^{L_{m}+1}   |V ^{(k)}|^{2} d\,x < \frac{C}{2^{m}},\quad
   m\ge 1
  \end{equation}
  when $0\le k \le p-1.$
 Let $S=S_{m},  m\ge 0,$  be the interval of integration in (\ref{k-1}), 
 and let $||\cdot ||_{l,S}$ be the norm in $L^{l}(S).$ Then (\ref{k-1})
 follows from (\ref{intr}) and 
 the estimate  (see, for example \cite{Gab}) 
\begin{equation}
 || V^{(k)}||_{2,S} \le C_{1}|| V||_{2,S} + C_{2}  || {V^{(p-1)}||_{2,S}},
  \quad 0\le k\le p-1. 
\end{equation} 
Lemma \ref{V_{B} + V_{R}} is proved.
\par
{\it Proof of Theorem 1}. We  need to 
prove only the first statement of the theorem, since it implies the second one
(see the proof of Theorem 2).
 \par 
Since   the  essential support of the spectral measure is invariant 
under $L^{1}$- perturbations of the potentials (see \cite{K}),
  it is enough  to prove the first statement of the theorem only 
for $V= V_{B}.$  
\par
 Let $L_{k}$ be an arbitrary  point in the gap between supports of the potentials
$v_k$ and $v_{k+1}$ and let 
$$
V_{m} = \sum_{k=1}^{m}v_{k},\quad x>0; \quad V_{m} = 0, \quad x<0.
$$
Then  
  \begin{equation} \label{fin}
 J_{p}(V_{m})\le J_{p}(V)= C<\infty.
 \end {equation}
 According to Theorem 3, it is sufficient  to prove that 
for any closed interval $\Delta \subset (0,\infty)$ 
there is a constant $C(\Delta)$  such that 
\begin{equation} \label{ak}
 \int_{\Delta} \ln (|a_{m}(k)|) d\,k \le C(\Delta)
 \end {equation}  for any $m=1,2,..  \, .$
Here  $a_{m}(k)$ is the  transmission  coefficient  associated with operator 
 $H= -d^{2}/dx^{2}  + V_{m}(x).$
Inequality (\ref{ak}) follows immediately from (\ref{fin}),
 (\ref{twosides4}), 
and Theorem 3.
 \par
Theorem 1 is proved.
\par
The following statement can be proved similarly to Theorem 1.
   \begin{theorem}.  
Let \begin{equation} 
 V = V_{B} + V_{R},
  \end{equation} 
where   
 \begin{equation}
  V_{B} = \sum_{k=1}^{\infty} v_{k}, \quad 
  v_{k}\in  H^{p_{k}-1}(R_{+}) \cap L^{p_{k} + 1 }(R_{+}),
  \qquad V_{R}\in L^{1}(R_{+}),
   \end{equation} 
and the  supports of $v_{k}$ do not intersect each other.  Let $ p_{k} \le p, \,
 k=1,2,.. ,$  and  
   \begin{equation}
\sum_{k=1}^{\infty} {J_{p_{k}}(v_{k})}    < \infty.
\end{equation}
 Then:
 
  1)  for all $\alpha \in [0,\pi),$  the essential support of the absolutely  
 continuous component of    the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$   coincides with
 $[0,\infty),$ i.e. $\mu^{\alpha}_{ac} (\Gamma ) > 0 $ for any  Borel set 
 $\Gamma \subset [0, \infty), |\Gamma | > 0.  $
  \par
2)  for a.e. $\alpha \in [0,\pi),$ the spectral measure 
 $\mu^{\alpha}(d\lambda)$ of the operator $H^{\alpha}$  is
  absolutely continuous on $[0, \infty)$.
\end{theorem} 

\section{ The spectral bifurcation. Examples.}  
\par
The examples below  show 
the  "exactness" of the statement of Theorem 1.  We shall construct 
a family of potentials  $V = V (x, \varepsilon; m),$  $m$ is integer,  
$m\ge 2,$  with the following properties:
\par
a) for each $m,$ the potential $V$ depends continuously on parameter $\varepsilon$ 
(in the uniform norm);
\par
b) $ J_{p}(V) = \infty $ for all $\varepsilon$ if $p \not= m,$ and therefore 
 neither Theorem  1 with $p \not= m$, nor the results  of
 \cite{DK} are 
 applicable to such potentials;
\par
c)
if $\varepsilon <0 $ then $J_{m}(V) < \infty, $   
and  Theorem 1 with $p=m$ can be applied. Hence,  the spectrum 
of $H^{\alpha}$  
on semiaxis $\lambda \ge 0$ is a.c.   for a.e.  $\alpha.$
If $\varepsilon \ge 0$ then $J_{m}(V) = \infty, $   
and the spectrum  of $H^{\alpha}$    is singular for a.e. $\alpha. $
\par
In order to make the idea of the examples transparent
 we shall discuss in detail the
case $m=2.$ Then, we shall breifly discribe the changes which are needed to
construct  $V$ for arbitrary $m.$
 \par 
 Let us consider  a sparse potential $V$ of the form
\begin{equation} \label{pott}
V = \sum_{n=1}^{\infty} h_{n}(\varepsilon) v_{n}(x-x_{n}),\quad x \in R,
\end{equation}  
where "bumps" $ v_{n}(x)$ have compact supports and $x_{n}\to \infty $ 
as $n\to \infty. $  
 We assume  that 
the functions $v_{n} = v_{n}(x)$ have the following properties: 
 $v_{n} \in C^{\infty}(R),$ 
$v_{n} $ are even (for simplicity), 
$ |v_{n}(x)|\le 1,  $   $ v_{n}(x) = 1$ 
  when $|x|\le L_n$, and  $ v_{n}(x) = 0  $  
when  $|x| \ge  L_n + 1. $ 
 Let  the behavior of all 
the functions $ v_{n}(x)$  be  the same  when   $ L_n \ge |x| \ge  L_n + 1,$ 
i.e. 
$ \omega (x) = v_{n}(L_n + \frac12 + x)$
does  not depend on $n$ when  $|x|\le \frac12$ .  Moreover, we assume  that 
$ \omega' (x)$  
 is even  when $|x|\le \frac12.$
 \par We choose
\begin{equation} \label{paramm}
L_n = n^{\beta}, \quad h_{n}= n^{-1/2 +\varepsilon}, \quad 
|\varepsilon|< \varepsilon_{0}    ,
 \quad x_n = \exp(c n \ln n), \quad c>1
\end{equation}  
with arbitrary $\beta \in (0,1/4)$ and  
$  \varepsilon_{0} = \min(\frac{1-4\beta}{6},\frac{\beta}{2}) .$
\par 
\begin{theorem}. \label{example1} 
1) Potential  $V$ with $v_{n}$  described above has the following properties:
 $$
 \int_{0}^{\infty} |V(x)|^{2} d\, x =\infty,
 $$
 \begin{equation}\label{no}
\int_{0}^{\infty} (|V^{(p-1)}(x)|^{2} + |V(x)|^{p+1} ) d\, x <\infty 
\quad  \hbox{if}\quad  \varepsilon <0, \quad p > 1, 
\end{equation} 
\begin{equation}\label{yes}
\int_{0}^{\infty} (|V^{(p-1)}(x)|^{2} + |V(x)|^{p+1} ) d\, x =\infty 
\quad  \hbox{if}\quad 0\le \varepsilon < \varepsilon_{0}, \quad p > 1.  
\end{equation} 
 \par
 2) For a.e. $\alpha,$ the spectrum of the  operator $H^{\alpha}$ with
  the potential $V$    is pure singular if $\varepsilon \ge 0 $  and is 
    a.c.  on the semiaxis $\lambda > 0$  if $\varepsilon < 0. $
 
\end{theorem}
{\bf Remark.} Theorem 1 with any $p>1$  can be applied to potential (\ref{pott})
if $\varepsilon < 0.$ 
Hence, this potential provides an example when Theorem 1 garantees the existence
of the a.c. spectrum in spite of the fact that $ V \notin L^{2}.$ Then, 
$V(x,\varepsilon,m)$ with $m=2$ can be realized as
$$
\tilde{ V} = V + V_{1}
$$
where $V$ is given by (\ref{pott}), $V_{1}$ is a compactly supported function such
that  $| V'_{1}| < C,$  $V''_{1} \notin L^{2}.$  
Obviously, $J_{p}(\tilde{ V}) < \infty$  only if $p=2, $ $\varepsilon < 0.$
On the other hand, the a.c. spectrum of operators $ H^{\alpha}$ with potentials 
$V$ and $\tilde{ V}$  coincide, since the
 perturbation by $V_{1}$ is of the trace class.
 \par
{\it Proof. } First of all note that 
$$
 \int_{-\infty}^{\infty} |V(x)|^{2} d\, x \ge \sum_{n=1}^{\infty}
  h_{n}^{2} L_{n} = \infty.
 $$
   Further, 
\begin{equation}
\int_{-\infty}^{\infty} |V(x)|^{p+1}  d\, x = 
2 \sum_{n=1}^{\infty} h_{n}^{p+1}( L_{n} + a  ),\quad \hbox{where }\quad
a = \int_{0}^{1} |V( L_{n} + x)|^{p+1}  d\, x.
\end{equation} 
Thus , if $p>1$, then 
\begin{equation}\label{V^{4}}
\int_{-\infty}^{\infty} |V (x)|^{p+1} d\, x < \infty 
\end{equation} 
Similarly, if $p>1$, then 
\begin{equation}
\int_{-\infty}^{\infty} |V^{(p-1)}(x)|^{2}  d\, x = 
b \sum_{n=1}^{\infty} h_{n}^{2},\quad \hbox{where }\quad
b = 2 \int_{0}^{1} |V^{(p-1)}( L_{n} + x)|^{2}  d\, x.
\end{equation} 
Together with (\ref{paramm}) and (\ref{V^{4}})  this implies 
(\ref{no}) and (\ref{yes}). 
\par
 From (\ref{no}) 
 and Theorem \ref{theorem1} it 
follows  that the spectrum  of $ H^{\alpha}$ on the
 semiaxis $\lambda > 0 $  is a.c.  
for a.e. $\alpha $  if $\varepsilon  < 0.$ Since $V$ vanishes at infinity, the
spectrum of $ H^{\alpha}$ on the semiaxis $\lambda > 0 $  is discrete.
It  only remains  to show  that $\Sigma ( H^{\alpha})$ on the semiaxis 
$\lambda > 0 $
 is  pure singular   for  a.e.  $\alpha$ if 
 $ 0\le \varepsilon < \varepsilon_{0}$.
 \par
 Let  $\psi$ be the solution of the problem 
\begin{equation}\label{ equation}
-\psi''(x) + h_{n}v_{n}\psi(x) =  k^{2}\psi(x),\quad x\in R,
\quad \psi(x)=e^{-ikx},\quad  x < -L_{n} - 1. 
\end{equation} 
 Then 
\begin{equation}
 \psi(x)=  a_{n}e^{-ikx} +   b_{n}e^{+ikx},  \quad  x > L_{n} + 1. 
\end{equation} 
   The
 function $ \psi(x)/ a_{n}$ describes the scattering of the plane wave, coming 
    from $x= \infty$,
 on the  "one  bump" potential $v_{n}$ .
  The spectrum of  one-dimensional
 Schr\"{o}dinger operators with sparse potentials was  studied in 
  \cite{M1},  \cite{M2}, where  it was proved 
    that under an assumption  on 
 $x_{n}$
 \begin{equation}
  \sum_{n=1}^{\infty} \frac{x_{n}}{x_{n+1}} <\infty
\end{equation} 
 (which is called super
 lacunarity ), the spectrum of  $H^{\alpha}$ 
 with a sparse potential
 is singular  on any   interval $\Delta \in (0,\infty)$ 
 for  a.e.  ${\alpha}$ 
  if the series
 \begin{equation} \label{singular}
  \sum_{n=1}^{\infty} |b_{n}|^{2} 
\end{equation} 
diverges  for almost all $  k^{2}  \in \Delta $ ( and the spectrum is  a.c. 
if the  series
 converges  for $  k^{2}  \in \Delta $). 
  Thus, in order  to complete the proof of Theorem \ref{example1},
  it is only left to show  the divergence of the series (\ref{singular}) 
    when $\varepsilon> 0.$
  \par
  We fix  a closed interval $\Delta \in (0,\infty)$. We will show  that
  \begin{equation}\label{ B_{n}}
  b_{n} (k) = h_{n}\frac{\alpha (k)}{k} \sin (2k (L_{n}+ 1/2))  + O(h^{2}_{n}L^{2}_{n}), \quad 
  n\to \infty,
\end{equation} 
   where $\alpha (k) $ is an entire function and the remainder decays uniformly 
  in $  k^{2}  \in \Delta, $ 
   i.e. 
   \begin{equation}
 | O(h^{2}_{n}L^{2}_{n})| \le C(\Delta )h^{2}_{n}L^{2}_{n}, \quad  n=1,2,3,.. \,. 
 \end{equation} 
 Here $C(\Delta )$ does not depend on $n$
 and $  k^{2}  \in \Delta $. 
   Since   $\sum_{n=1}^{\infty} h_{n}^{3} L_{n}^{2} < \infty ,$
  $\sum_{n=1}^{\infty} h_{n}^{4} L_{n}^{4} < \infty ,$ and $\alpha (k) = 0 $ only at a
   discrete set of points,
  the divergence of (\ref{singular})  when 
   $ 0\le \varepsilon < \varepsilon_{0}$ 
   follows 
  from the divergence   of the series
  \begin{equation}
  \sum_{n=1}^{\infty} h_{n}^{2}|\sin (2k (L_{n}+ 1/2))|^{2} =
  \sum_{n=1}^{\infty} h_{n}^{2}[ 1 - \cos (4k (L_{n}+ 1/2))],\quad 
   0\le \varepsilon < \varepsilon_{0}.
  \end{equation}  
  The divergence of the last series is obvious, since
   $\sum_{n=1}^{\infty} h_{n}^{2}= \infty$
  if $\varepsilon> 0$  and 
    \begin{equation}
  \sum_{n=1}^{\infty} n^{-1 + 2\varepsilon}\cos (4k ({n}^{\beta}+ 1/2)) < \infty, \quad 
  2\varepsilon < \beta, \quad  k\not= 0.
    \end{equation} 
      Finally, it is  only left  to justify (\ref{ B_{n}}).
  \par
  The solution 
  $\psi $ of (\ref{ equation}) satisfies the integral equation
  \begin{equation} \label{intequation}
  \psi(x) = e^{-ikx} + \int_{-L_{n}- 1}^{x} \frac{\sin k(x-\xi)}{k}
   h_{n} v_{n}(\xi) \psi(\xi) d\,\xi.
    \end{equation}  
  The norm of the integral operator   in the right-hand side of (\ref{intequation}) 
  in the space $C(R)$ does not exceed 
  $$ h_{n}(L_{n} + 2 ) /|k|  \le C_{1}(\Delta)h_{n} L_{n}.
  $$
  It is  less than 1 if $n$ is large enough. Thus,
 \begin{equation}
  \psi(x) = e^{-ikx} + \int_{-L_{n}- 1}^{x} \frac{\sin k(x-\xi)}{k}
   h_{n} v_{n}(\xi) e^{-ik\xi} d\,\xi  + O(h^{2}_{n}L^{2}_{n}),
    \end{equation}  
  and therefore,
 \begin{equation}\label{integra}
  b_{n}(k) = \frac{h_{n}}{2k} \int_{-\infty}^{\infty} 
   v_{n}(\xi) e^{-2ik\xi} d\,\xi  + O(h^{2}_{n}L^{2}_{n}),
   \quad  k^{2}  \in \Delta, \quad n \to \infty.
    \end{equation}  
   Since the function $v'_{n}$ is supported  on the set
   $\{\xi : L_{n} \le |\xi|\le L_{n} + 1  \}$
      and  is an odd function,
   after integration by  parts, (\ref{integra})   takes the form
$$
  b_{n}(k) = \frac{-i h_{n}}{(2k)^{2}} \int_{-\infty}^{\infty} 
   v'_{n}(\xi) e^{-2ik\xi} d\,\xi  + O(h^{2}_{n}L^{2}_{n})=
   $$
   $$
 - \frac{ h_{n}}{2k^{2}} \int_{L_{n} }^{L_{n} + 1} 
   v'_{n}(\xi) \sin{2ik\xi} d\,\xi  + O(h^{2}_{n}L^{2}_{n}) = 
 $$
 $$ 
    - \frac{ h_{n}}{2k^{2}} \int_{- 1/2 }^{1/2} 
   \omega'(\xi) \sin[2k( \xi + L_{n}+\frac{1}{2})] d\,\xi  + O(h^{2}_{n}L^{2}_{n}) = 
  $$
    \begin{equation}\label{end}
    - \frac{ h_{n}}{2k^{2}} \int_{- 1/2 }^{1/2} 
   \omega'(\xi) \cos2k \xi d\,\xi  \sin2k(  L_{n}+\frac{1}{2}) + 
    O(h^{2}_{n}L^{2}_{n}). 
       \end{equation}  
   The last equality is the consequence of the fact  that $\omega'(\xi)$ is
   even on the interval $|\xi|< 1/2 .$  Equality (\ref{end}) implies
    (\ref{ B_{n}}). 
\par The proof  of Theorem  \ref{example1}    is complete.
\par
In order to construct potentials $V(x,\varepsilon;m)$    for $m>2$
one can consider first the family of potentials:    
$$
W = \sum_{n=1}^{\infty} h_{n}(\varepsilon) v_{n}(x - x_{n})
 \varphi(x - x_{n}, n), \quad x\in R,
$$    
where $h_{n},$    $v_{n}$ are the same  as in (\ref{pott}), and  $\varphi$ 
is a $C^{\infty}$ -  function such that $|\varphi^{(j)}| < C_{j}$  if $j\ge m,$
and
$$
\varphi(x - x_{n}, n) = (L_{n} - |x - x_{n}|)^{m-2}, \quad  |x - x_{n}| \ge 1.
$$    
We assume that (\ref{paramm}) holds with small enough  $\beta > 0$ and 
$ \varepsilon_{0} = \varepsilon_{0}(\beta) > 0.$  Then one can easily check 
that 
$$
J_{p}(W) = \infty ,\quad \hbox{if}\quad p < m \quad\hbox{or}\quad p \ge m, \quad
\varepsilon \ge 0; 
$$
$$
J_{p}(W) < \infty ,\quad \hbox{if}\quad p \ge m, \quad \varepsilon < 0. 
$$
Thus, Theorem 1 with  any $ p \ge m$ can be applied to the operator $H^{\alpha}$
with potential $W$  when  $\varepsilon < 0.$  Arguments similar to those used in the
proof of Theorem 6 can be applied  to prove that operator  $H^{\alpha}$
with potential $W,$  has singular spectrum for a.e $\alpha $ if $\varepsilon \ge 0.$
Then, $V(x,\varepsilon; m),$  $m>2,$  can be realized  as
 $V = W + V_{1}$  where $V_{1}$ is a compactly supported function 
such that  $|V_{1}^{(m-1)}| < C, $ $V_{1}^{(m)} \notin  L^{2} $  (compare with  
the Remark following Theorem6).

\vskip1cm    

\noindent{\small {\it Acknowledgments}. We would like to thank V.Jaksic
(University of Ottawa)  and B.Simon (Caltech) for 
useful discussions. 
M.N.  would like to thank the  Department of Mathematics 
 of the University of  North Carolina for the hospitality.
The  work of the first and third authors  was 
supported in part  by NSF Grant $\#$ DMS -  9971592.

  

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