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Schrodinger operators, Dirac operators, Krein systems,  
singular part of the spectral measure.
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\begin{document}
\title[Singular spectrum and decay of potential]{On the singular spectrum of 
Schr\"odinger operators with decaying potential}
\author{S. Denisov}
\email{denissov@its.caltech.edu}
\author{S. Kupin}
\email{kupin@its.caltech.edu}
\thanks{\textit{Keywords:} Schr\"odinger operators, Dirac operators, Krein
systems, singular part of the spectral
measure.}
\date{January 19, 2002}
\address{Department of Mathematics, 253-37, Caltech, Pasadena CA 91125, USA.}

\begin{abstract}
The relation between Hausdorff dimension of the singular spectrum of a 
Schr\"odinger operator and the decay of its potential has been extensively
studied in many papers. In this work, we address similar questions from
a different point of view. Our approach relies on the
study of the so-called Krein systems. For Schr\"odinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling, are optimal in $L^p(\br_+)$ scale. 
\end{abstract}

\maketitle

\section*{Introduction}

\label{s0}

We consider a Schr\"odinger operator $L_qy=-y''+qy$ on the
positive half-line $\mathbb{R}_+$ with boundary condition $y'(0)+h y(0)=0$. Assume that $q\in L^\infty(\mathbb{R}_+)$ is a real-valued
function and $h \in \mathbb{R}\cup\{\infty\}$. Denote the spectral measure
of the operator $L_q$ by $\rho$.

Recently, Remling \cite{re1} proved the following theorem.

\begin{theorem}[\protect\cite{re1,re2}]
\label{t01} If $|q(x)|\le C(1+x)^{-\beta }$ with $1/2<\beta\leq 1$, then the
 support of the (possible) singular part of $\rho$ has Hausdorff dimension
less or equal to $2(1-\beta )$.
\end{theorem}

Actually, the stronger result was obtained, that is, the set of all positive
spectral parameters such that the transfer matrix is not bounded at 
infinity has Hausdorff dimension less or equal to $2(1-\beta )$. However,
the presence of a non-trivial singular continuous part of $\rho$ for some potentials
was only guessed. 
More attention to the subject was attracted when Simon \cite{si1} asked the
following question.

\begin{quest}
\label{conj1} Do there exist potentials $q$ on $\br_+$ so that
$|q(x)|\le C(1+x)^{-(1/2+\varepsilon)},\,$ $(\varepsilon>0)$ and the spectral
measure of $L_q$ has a non-trivial singular continuous part?
\end{quest}

The first example of a potential from $L^2(\br_+)$ with $\rho$ having a singular continuous component for some $h$ was given 
by Denisov \cite{de1}. Deift and Killip \cite{dk1} proved that for potentials from $L^2(\br_+)$, the essential support of the absolutely continuous part of the spectral measure is the whole positive half-line. Later, Kiselev \cite{ki1}
constructed Schr\"odinger operators with potentials decaying arbitrarily slower than $C(1+x)^{-1}$ and having an embedded singular continuous component.

In \cite{de1}, the construction was carried out in the opposite direction.
It was started with a specific spectral measure. Then the analysis of the corresponding
inverse spectral problem was used to establish the required 
properties of the potential.

In the present paper, we show that there exist potentials $q$ so that $\rho$ has a
singular component with 
Hausdorff dimension  exactly $%
2(1-\beta )$ and $q\in L^{p}(\mathbb{R}_{+})$ for any $p > p_0(\b)$ (see Section \ref{s32} for formulations). 
Our methods are essentially different from those of \cite{re1, ki1}. In writing this paper, we were motivated by certain well-known results from the theory of orthogonal polynomials \cite{ge1, sz1}.  
Therefore, we start with the continuous analogs of orthogonal polynomials, the solutions of the so-called Krein systems. For these systems, we study the above-mentioned questions. It turns out that
the problem can be reduced to the certain problem of minimization. We analyze this minimization problem using simple methods of complex analysis and approximation theory.   


The plan of the paper is as follows. In section 1, we introduce some
notation and discuss  well-known results that are necessary in the sequel. In section 2, we study the Krein systems case. Results obtained for Krein systems are applied to Dirac and Schr\"odinger operators in section 3. Certain theorems for
orthogonal polynomials are formulated in section 4.

We conclude the introduction with some notation. Given a measure $\s$ on $\br$, let $\s_s,\s_{ac}$ signify its singular and absolutely continuous components, respectively. The Lebesgue measure is denoted by $m$. The Hausdorff dimension of a measurable set $E\subset\br$ is denoted by $\hdim E$. As usual, $W^{2,m}(\br_+)$ and $H^p(\Omega)$ stand for standard Sobolev spaces on $\br_+$ and Hardy spaces on $\Omega$, $\Omega$ being a domain in $\bc$ with a smooth boundary. For  $f\in L^2(\br)$, $\hat f$ is its Fourier transform. $C$ is a constant changing from one relation to another.


\section{Preliminaries}\label{s1}
\subsection{}\label{s11} In this subsection, we briefly discuss some simple properties of
polynomials, orthogonal on the unit circle $\mathbb{T}=\{z:|z|=1\}$. A
detailed presentation of the subject can be found in \cite{ge1, sz1}.

Let $\sigma$ be a finite positive measure on $\mathbb{T}$. Let $\{\varphi_n\}
$ be polynomials, orthonormal with respect to $\sigma$, that is, $\int_\bt
\varphi_n\overline{\varphi_m}\,d\sigma=\d_{nm}$, $\d_{nm}$ being the
Kronecker symbol. We also consider monic orthogonal polynomials $\{\psi_n\}$%
, that is, 
\begin{equation*}
\int_\bt\psi_n(t)\overline{\psi_m}(t)\,d\sigma(t)=k_{n}\d_{nm}, 
\end{equation*}
where $\psi_n(z)=z^{n}+\ldots$, and $k_n=||\psi_n||^2_{2,\sigma}$. These
polynomials can be explicitly computed. Consider momenta of $\sigma$, given
by $c_k=\int_\bt e^{ikt}\,d\sigma(t)$. Define the matrix 
\begin{equation*}
M_n= \left[ 
\begin{array}{cccc}
c_0 & c_1 & \ldots & c_n \\ 
\overline{c_1} & c_0 & \ldots & c_{n-1} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
\overline{c_n} & \overline{c_{n-1}} & \ldots & c_0%
\end{array}
\right] 
\end{equation*}
and let $\Delta_n=\det M_n$. It is not difficult to see that 
\begin{equation}
\psi_n(z)=\sum_{k=0}^n (-1)^k\frac{\Delta_{n+1,n+1-k}}{\Delta_{n-1}}z^{n-k},
\label{e011}
\end{equation}
where $\Delta_{n+1,n+1-k}$ denotes the determinant of $M_n$ with dropped $(n+1)$-th
row and $(n+1-k)$-th column.
On the other hand, the sequence $\{\psi_n\}$ generates a sequence $\{a_n\}
$ of reflection (or Geronimus) coefficients by means of the relations 
\begin{equation}  \label{e021}
\left\{ 
\begin{array}{rcl}
\psi_{n+1}(z) & = & z\psi_n(z)-\overline{a_n}\psi^*_n(z), \\ 
\psi^*_{n+1}(z) & = & \psi^*_n(z)-a_nz\psi_n(z),%
\end{array}
\right.
\end{equation}
where $\psi_0(z)=\psi^*_0(z)=1$ and $\psi^*_n(z)=z^n\overline{%
\psi_n(1/\overline z)}$. Vice-versa, given a sequence $\{a_n\}$, $|a_n|<1 \,(n=0,1,\ldots)$, we can 
define orthogonal polynomials $\{\psi_n\}$ uniquely; see \cite{ge1} and \cite[ch. 5]%
{khr1} for details.

We say that $\sigma$ is a Szeg\H{o} measure, if $\log\sigma^{\prime}_{ac}\in
L^1(\mathbb{T})$. The following theorem is classical.

\begin{theorem}[\protect\cite{ge1,sz1}]
\label{t02} The following assertions are equivalent:

\begin{itemize}
\item[\textit{i)}] $\sigma$ is a Szeg\H{o} measure.

\item[\textit{ii)}] The series $\sum_{k=0}^\infty |\varphi_n(z)|^2$
converges at least for one (and hence, for all) $z\in\mathbb{D}$.

\item[\textit{iii)}] There exists a subsequence $\{\varphi^*_{n_k}\}$
bounded at least for one (and hence, for all) $z\in\mathbb{D}$. 

\item[\textit{iv)}] The sequence $\{a_k\}$ lies in $l^2$.
\end{itemize}
In the above case, the limit $%
\pi(z)=\lim_{n\to\infty}\varphi^*_{n}(z),\ z\in\mathbb{D}$, exists, and $%
\pi^{-1}\in H^2(\mathbb{D})$.
\end{theorem}

Furthermore, if $\sigma$ is a Szeg\H{o} measure, we have that (see \cite[ch.
2]{ge1}) 
\begin{equation}  \label{e03}
\sum_{k=n}^\infty |a_k|^2\le C\inf_{p\in\mathcal{P}_n}||\pi_0-p||^2_{2,%
\sigma}.
\end{equation}
In this formula, $\mathcal{P}_n$ is the space of polynomials of degree less
or equal to $n$, and $\pi_0=\pi\chi_{\mathbb{T}\backslash E}$, where $E=%
\supp\, \sigma_s$.

\subsection{}\label{s12} 
In this subsection, we introduce the so-called Krein systems and briefly
discuss their properties. For more information on the topic, see %
\cite{kr1, ary, ry1}.

By a Krein system (a K-system), we mean the system of
differential equations 
\begin{equation}  \label{e04}
\left\{ 
\begin{array}{rcl}
P^{\prime}(r,\l) & = & i\l P(r,\l)-\overline{A(r)}P_*(r,\l), \\ 
P^{\prime}_*(r,\l) & = & -A(r)P(r,\l),%
\end{array}
\right.
\end{equation}
with boundary conditions $P(0,\l)=P_*(0,\l)=1$. We suppose that $A\in
L^1_{loc}(\mathbb{R}_+)$, $\l\in\mathbb{C},\ r\in\mathbb{R}_+$.

It turns out that the K-system defines the unique positive measure \, $\sigma$,\quad\, 
$\int_\br (1+\l^2)^{-1} d\sigma(\l) d\l<\infty$, with the following
 properties. Introduce the Fourier transform $\mathcal{F}: L^2(%
\mathbb{R}_+)\to L^2(\sigma)$ by the formula 
\begin{equation*}
(\mathcal{F} f)(\l)=\int^\infty_0 f(r)P(r,\l)\, dr. 
\end{equation*}
The inverse Fourier transform $\mathcal{F}%
^{-1}:L^2(\sigma)\to L^2(\mathbb{R}_+)$ is given by the relation 
\begin{equation*}
(\mathcal{F}^{-1} g)(r)=\int_\br g(\l)\overline{P(r,\l)}\,
d\sigma(\l), 
\end{equation*}
and we have  $\mathcal{F}^{-1}\mathcal{F} f=f$ for any $f\in
L^2(\mathbb{R}_+)$; see \cite{kr1}. The above integrals should be understood in the $L^2$%
-sense.

\begin{theorem}[\protect\cite{kr1}]
\label{t03} For any $f\in L^2(\mathbb{R}_+)$, the Parseval equality holds 
$$
\int_\br|\mathcal{F} f(\l)|^2\,d\sigma(\l)=\int^\infty_0 |f(s)|^2\, ds. 
$$
\end{theorem}

The parallels between K-systems and orthogonal polynomials may be extended much
further. We discuss the construction in a very particular case to be used in the sequel. The general construction can be found in \cite{kr1, ry1}. We let $\displaystyle \s_0=m/(2\pi)$ and assume that $\supp(\s-\s_0)$ is compact. Introduce
\begin{equation}
H(t)=\int_\br e^{i\l t}\, d(\s-\s_0)(\l).\label{e041}
\end{equation}
The function $H\in C^\infty(\br)$ gives rise to an integral 
equation for the ``resolvent'' kernel $\gg_r$  
\begin{equation}
\gg_r(t,\t)+\int^r_0 H(t-s)\gg_r(s,\t)\, ds=H(t-\t). \label{e042}
\end{equation}
Remarkably, it occurs that 
\begin{equation}\label{e05}
\begin{array}{rcl}  
P(r,\l)&=&\dsp e^{i\l r}\left( 1-\int^r_0 \gg_r(s,0)e^{-i\l s}\, ds\right), \\
P_*(r,\l)&=&\dsp 1-\int^r_0\gg_r(0,s)e^{i\l s}\, ds,   
\end{array}
\end{equation}
and $A(r)=\gg_r(0,r)$.
It is instructive to compare formulas \eqref{e04} and \eqref{e05} to %
\eqref{e021} and \eqref{e011}, respectively.

Because $H(t)\in C^\infty (\br_+)$, the Fredholm formula for the resolvents $\Gamma_r(t,\tau)$ yields $A\in C^\infty(\br_+)$. 
We also have a weakened version of Theorem \ref{t02}.
\begin{theorem}[\protect\cite{kr1, sa1, de2}]
\label{t04} The following assertions are equivalent:

\begin{itemize}
\item[\textit{i)}] $\sigma$ is a Szeg\H{o}-type measure on $\mathbb{R}$, that is $%
\displaystyle (1+\l^2)^{-1} \log \sigma^{\prime}_{ac}\in L^1(\mathbb{R})$.

\item[\textit{ii)}] The integral $\int^\infty_0 |P(r,\l)|^2\, dr$ converges
at least for one (and hence, for all) $\l\in\mathbb{C}_+$.

\item[\textit{iii)}] The function $P_*(.,\l)$ is bounded at least for one
(and hence, for all) $\l\in\mathbb{C}_+$.
\end{itemize}
In the above cases, there exists a limit  $\Pi(\l)=\lim_{r_n\to\infty} P_*(r_n,\l),\ \l\in\mathbb{C}_+$, and ${[(\l+i)\Pi]}^{-1}\in H^2(\bc_+)$.
\end{theorem}
Under the assumptions of the theorem, $\displaystyle 2\pi \s'_{ac}(\lambda)=|\Pi(\lambda+i0)|^{-2}$; see \cite{sa1}.

\subsection{}
\label{s13} We will need one simple approximation argument. Complete
information on the subject can be found in \cite{akh}. We take  $f\in L^\infty(\br)$ and let
\begin{equation}
(f)_r(x)=r\int_\br K(r(x-s))f(s)\,ds,  \label{e0}
\end{equation}
where $\displaystyle K(x)=\frac{12}{\pi}\left\{\frac{\sin x/2}{x}\right\}^4$ and $r\ge 0$. For  $%
f\in L^2(\sigma)$, we define a continuity modulus
by the formula 
\begin{equation*}
\o^2_{2,\sigma}(\d,f)=\sup_{|h|\le\d} \int_\br |f(x+h)-f(x)|^2\,d\sigma(x). 
\end{equation*}
The following lemma is a modification of \cite[Lemma 3.1]{ge1},  and is
quoted here for the reader's convenience only.

\begin{lemma}
\label{l11} If $f\in L^2(\sigma)\cap L^\infty(\mathbb{R})$, then, for $r>0$, 
\begin{equation*}
||f-(f)_r||^2_{2,\sigma}\le C\o^2_{2,\sigma}\left(1/r,f\right). 
\end{equation*}
\end{lemma}

\begin{proof}
It is clear that 
\begin{equation*}
(f)_r(x)-f(x)=\frac{12}{\pi r^3}\int^\infty_0 \left\{\frac{\sin rs/2}{s}%
\right\}^4 A(s,x)\,ds, 
\end{equation*}
where $A(s,x)=f(x+s)+f(x-s)-2f(x)$. Applying  the H\"older inequality, we have
\begin{equation*}
|(f)_r(x)-f(x)|^2\le\frac{12}{\pi r^3}\int^\infty_0 \left\{\frac{\sin rs/2}{s}%
\right\}^4 |A(s,x)|^2\,ds. 
\end{equation*}
Integration in $x$ with respect to $%
\sigma$ yields 
\begin{equation*}
||(f)_r-f||^2_{2,\sigma}\le\frac{12}{\pi r^3}\int^\infty_0 \left\{\frac{\sin
rs/2}{s}\right\}^4 ||A(s,.)||^2_{2,\sigma}\,ds. 
\end{equation*}
Since $\o_{2,\sigma}(m\d,f)\le(m+1)\o_{2,\sigma}(\d,f)$, we have
\begin{equation*}
||A(s,.)||^2_{2,\d}\le \o^2_{2,\sigma}(|s|,f)\le
2(r|s|+1)^2\o^2_{2,\sigma}(1/r,f). 
\end{equation*}
Observing that 
\begin{equation*}
\frac{12}{\pi r}\int^\infty_0 \left\{\frac{\sin rs/2}{s}\right\}^4 s^2\,
ds\le C, 
\end{equation*}
with $C$ that does not depend on $r$, we finish the proof of the lemma.
\end{proof}

We denote the Paley-Wiener space of entire functions of exponential type $r$ by $\ccf_r$, that is
$$
\ccf_r=\lt\{F: F(\l)=\int^r_0 e^{i\l s}f(s)\,ds,\  f\in L^2[0,r]\rt\}.
$$
Observe that if $f\in H^2(\bc_+)$, then $(f)_r\in \ccf_{2r}$. Indeed, $\hat f$ is supported on $\br_+$, $\widehat{K(rx)}$ is supported on $[-2r,2r]$ (see \cite{akh}, Sect. 71), and, consequently, 
$\widehat{(f)_r}=\widehat{K(rx)}\hat f$ lives on $[0,2r]$.
  
 
\section{Krein systems}
\label{s2}

\subsection{}\label{s21} 
Suppose that the measure $\sigma$ of a K-system satisfies the Szeg\H{o}-type condition (see Theorem \ref{t04}). Our goal is to understand how the properties of the
singular and absolutely continuous parts of $\sigma$ and their mutual
location influence the properties of the coefficient $A$.   
We distinguish two different cases. In the first case,  $\s_{ac}$ and $\s_s$ are well-agreed. In the second, a ``good" $\s_{ac}$ and a singular component $\s_s$ are chosen more or less independently. By ``good" we mean that the density $\s'_{ac}$ is piecewise smooth, bounded above and bounded below from zero.

In the first case, we have the following theorem.

\begin{theorem}
\label{t1} There exists a K-system with the properties:

\begin{itemize}
\item[\textit{i)}] $\hdim\supp\,\sigma_s=2(1-\b)$, $1/2<\b<1
$.  

\item[\textit{ii)}] The corresponding coefficient $A$ satisfies the
inequality 
\begin{equation*}
\displaystyle \int^\infty_x |A(s)|^2\,ds\le \frac C{(1+x)^{2\b-1}}. 
\end{equation*}
\end{itemize}
\end{theorem}

Notice that the bound for the above integral matches perfectly the estimate
from \cite{re1} for Schr\"odinger operators.

As expected, the bound in the second case is worse.
 
\begin{theorem}
\label{t2} There exists a K-system with the properties:

\begin{itemize}
\item[\textit{i)}] $\hdim\supp \sigma_s=2(1-\b)$ with $%
1/2<\b<1$, and 
\begin{equation*}
\sigma^{\prime}_{ac}(x)=1/(2\pi)\left\{ 
\begin{array}{ll}
1/2, & |x|\le 1, \\ 
1, & |x|>1.%
\end{array}
\right. 
\end{equation*}

\item[\textit{ii)}] The coefficient $A$ lies in $L^p(\mathbb{R}_+)$ with any $%
p>4/(2+\gamma_0)$, where $\gamma_0=-\log(1-\b)/(\log2-\log(1-\b))$.
\end{itemize}
\end{theorem}

\begin{remark}
\label{rk1} If $\hdim\supp\,\sigma_s\to 0$, we get $A\in L^p(%
\mathbb{R}_+)$ with any $p>4/3$.
\end{remark}

\subsection{}
\label{s22} The proofs of  both theorems rely on several lemmas which are
proved in this subsection.

\begin{lemma}
\label{l1} If $A\in W^{1,2}(\mathbb{R}_+)$ is a real-valued coefficient of a
K-system,  then 
\begin{equation}
\lim_{y\to+\infty} y^2\int^\infty_tP^2(\t,iy)\,d\t=\int^\infty_t A^2(\t)\,d\t
\label{e1}
\end{equation}
\end{lemma}
{\it for $t>0$.} 


\begin{proof}
Take the K-system \eqref{e04} with $\l=iy$ and introduce $Q(t)=e^{yt}P(t,iy)$.
Clearly, $Q$ is a solution to 
\begin{equation*}
\left\{ 
\begin{array}{rl}
Q^{\prime} & =-Ae^{yt}P_*, \\ 
P^{\prime}_* & =-Ae^{-yt}Q%
\end{array}%
\right. 
\end{equation*}
with boundary conditions $Q(0)=P_*(0)=1$.
We have 
\begin{eqnarray}
Q&=&1-\int^t_0 A(\t)e^{y\t}P_*(\t)\, d\t,  \label{e01} \\
P_*&=&1-\int^t_0 A(\t)e^{-y\t}\left(1-\int^\t_0 A(\xi)e^{y\xi}P_*(\xi)\,
d\xi\right)\, d\t.  \label{e02}
\end{eqnarray}
Plug relation \eqref{e02} in \eqref{e01} and express $P$ through $Q$.
This gives 
\begin{eqnarray}
P&=&\displaystyle e^{-yt}-e^{-yt}\int^t_0 A(\t)e^{y\t}\Bigl[1\Bigr.-\int^\t_0
A(\xi)e^{-y\xi}\,d\xi  \notag \\
&+&\int^\t_0 A(\xi)e^{-y\xi}\int^\xi_0 A(\eta)e^{y\eta}P_*(\eta)\, d\eta d\xi%
\Bigl]\Bigr.\,d\t=I_1-I_2+I_3-I_4.  \label{e2}
\end{eqnarray}
Now, we estimate integrals $\int^\infty_t |I_i|^2\,d\t$. It is clear that 
$$
\int^\infty_t |I_1|^2\, d\t\le C\frac{e^{-2yt}}{y}. 
$$
Furthermore, 
$$
I_2=\frac1y\left(A(t)-A(0)e^{-yt}-e^{-yt}\int^t_0 A^{\prime}(\t)e^{y\t}\,
d\t\right)=I^1_2+I^2_2+I^3_2, 
$$
and, obviously, \mbox{$||I^2_2||_2\le C/(y\sqrt y)$}. By the Young inequality for convolutions,  $||I^3_2||_2\le C/y^2$ if $y$ is big enough. For $I_3$ from \eqref{e2}, we obtain 
$$
|I_3|\le \frac{e^{-yt}}{\sqrt{y}}||A||_2\int^t_0|A(\t)|e^{y\t}\, d\t. 
$$
Consequently, $||I_3||_2\le C/(y\sqrt{y})$. Equality \eqref{e02} implies that $|P_*(x,iy)|\le C$ uniformly in $x$ and $y$; see \cite{de2}. Hence, we get  
 $||I_4||_2\le C/y^2$.

To compute the left-hand side of \eqref{e1}, we represent $|P(\t,iy)|^2$ with
the help of \eqref{e2}. The above estimates yield the claim of the lemma.
\end{proof}

In the lemma below, we use notations introduced in Section \ref{s13}. We also
let $k_{iy}(\l)=-(2\pi i)^{-1}(\l+iy)^{-1}$ and $\Pi_0=k_{iy}\Pi$.

\begin{lemma}
\label{l2} Let the measure $\sigma$ of a Krein system satisfy a Szeg\H{o}-type estimate.
Then 
\begin{equation}
\inf_{F\in\boldsymbol{\mathcal{F}}_r}||\Pi_0-F||^2_{2,\sigma}=\frac1{4\pi^2|%
\Pi(iy)|^2}\int^\infty_r |P(x,iy)|^2\, dx.  \label{e3}
\end{equation}
\end{lemma}

\begin{proof}
We begin with computation of $\cf^{-1}\Pi_0$. Notice that 
\begin{equation*}
\left(\cf^{-1}\Pi_0\right)(x)=\int_\br\Pi_0(\l)\ovl{P(x,\l)}\,d\s(\l)=
\frac1{2\pi}\ovl{\lb\frac1{2\pi
i}\int_\br\frac{P(x,\l)}{\Pi(\l)}\frac1{\l-iy}\, d\l\rb}. \label{e4} 
\end{equation*}
Taking into account the assumptions of the lemma and  relations \eqref{e05}, we see that the last integral
can be calculated by the Cauchy formula. Hence 
\begin{equation*}
(\cf^{-1}\Pi_0)(x)=\frac1{2\pi}\ovl{\lb\frac{P(x,iy)}{\Pi(iy)}\rb}. 
\end{equation*}
Using the simple identity \cite{kr1}
\begin{equation*}
|P_*(r,\lambda)|^2-|P(r,\lambda)|^2=2\im\l\int\limits_0^r |P(s,\l)|^2ds,
\end{equation*}
we can see that 
\begin{equation*}
\int_\br |\Pi_0(\l)|^2d\sigma(\l)=
\int^\infty_0 |\cf^{-1} \Pi_0|^2dx.
\end{equation*}
Therefore, $\Pi_0$ belongs to the range of $\cf$.



Pick a function $f_r\in L^2[0,r]$ and extend it to $[r,+\infty)$ by zero. It
follows that 
\begin{equation*}
\cf^{-1}(\Pi_0-\cf f_r)(x)=\frac1{2\pi\ovl{\Pi(iy)}}\lb \ovl{P(x,iy)}-2\pi%
\ovl{\Pi(iy)}f_r(x)\rb. 
\end{equation*}
Using the Parseval equality (see Theorem \ref{t03}) and observing that $\cf %
f_r\in \ccf_r$, we conclude the proof.
\end{proof}

% %\QTP{e KronecQ}
The next lemma is an elementary remark on the behavior of outer functions
of some special form. Let 
\begin{equation*}
g(z)=\exp\lt\{\frac1{2\pi i}\int_\br\frac{h_0(x)\,dx}{x-z}\rt\}, 
\end{equation*}
where $h_0\in L^1(\br)$ is a real-valued function, $f=g(.+h)/g$.

\begin{lemma}
\label{l3} Suppose that $h_0\in L^1(\br), g\in H^\infty(\bc_+)$, and $%
g^{-1}\in H^2(\bc_+)$. Then 
\begin{equation*}
\begin{array}{rcl}
i) &  & f(iy)=1+o(1/y),\ y\to+\infty, \\ 
ii) &  & \dsp \int_\br|k_{iy}|^2|f-1|^2\,dx\le\int_\br|k_{iy}|^2||f|^2-1|%
\,dx+o\lb 1/{y^2}\rb.%
\end{array}
\end{equation*}
\end{lemma}

\begin{proof}
Obviously, 
\begin{equation*}
f(z)=\exp\lt\{\frac1{2\pi i}\int_\br\frac{1}{x-z}(h_0(x+h)-h_0(x))\,dx\rt\}, 
\end{equation*}
and $k_{iy}f\in H^2(\bc_+)$. By dominated convergence theorem, 
$$%\begin{equation*}
\lim_{y\to+\infty} y\int_\br\frac{1}{x-iy}(h_0(x+h)-h_0(x))\,dx=i\int_%
\br(h_0(x+h)-h_0(x))\,dx=0, 
$$ %\end{equation*}
and the first claim of the lemma is proved. As for the second one, we see that 
$$%\begin{equation*}
||k_{iy}(f-1)||^2=||k_{iy}f||^2-\frac2{4\pi y}\re f(iy)+\frac1{4\pi
y}=||k_{iy}f||^2-\frac1{4\pi y} +o\lb 1/{y^2}\rb, 
$$%\end{equation*}
since $(k_{iy}f,k_{iy})=f(iy)/(4\pi y)$ and $(k_{iy},k_{iy})=1/(4\pi y)$ under the assumptions of the lemma.
\end{proof}

% %\QTP{e KronecQ}
\begin{remark}\label{rk2}
If $h_0$ is an even function, $g$ is  real for $z\in i\br_+$.
\end{remark}
% %\QTP{e KronecQ}

\subsection{}
\label{s23} In this section, for given $1/2<\b<1$, we construct a nonnegative function $w$
with certain special properties. This function gives rise to the 
measure $\s$. The measure, in  turn, generates the K-system from Theorem \ref{t1}.

% %\QTP{e KronecQ}
Let $E^0=[-1,1]$. In the first step, we set $E^1=E^0\bsl J_0$, where $J_0$
is the open middle interval of $E_0$, and $|J_0|=\b|E_0|$. In the $(n+1)$-th
step, we represent $E^n$ as $E^n=\cup^{2^n}_{k=1} I_{nk},\
|I_{nk}|=2((1-\b)/2)^n$. Similarly, we define $E^{n+1}=\cup^{2^n}_{k=1}
(I_{nk}\bsl J_{nk})$, where $J_{nk}$ are open middle intervals of $I_{nk}$,
and $|J_{nk}|=\b|I_{nk}|$, etc.

 %\QTP{e KronecQ}
Consider $E_\b=\cap^\infty_{k=0} E^k$. The Hausdorff dimension of $E_\b$ is
readily seen to be 
\begin{equation}
\hdim E_\b=\frac{\log 2}{\log 2-\log(1-\b)}.
\end{equation}
For $M>0$, we define the function $w=w_\b$ as follows: 
\begin{equation*}
w(x)=2\pi\left\{ 
\begin{array}{ll}
\min\{1, M|x+1|^\g\}, & x\le -1, \\ 
0, & x\in E_\b, \\ 
M\min\{|x-a|^\g, |x-b|^\g\}, & x\in J_{nk}=(a,b), \\ 
\min\{1, M|x-1|^\g\}, & x\ge 1,%
\end{array}%
\right. \label{e5} 
\end{equation*}
where 
\begin{equation}
\dsp 0<\g\le \g_0=\frac{-\log(1-\b)}{\log2-\log(1-\b)}.  \label{e5}
\end{equation}
In particular, $w$ is even. It lies in $\lip_\g (\br)$, and $w=0$ on $E_\b$.

Furthermore, for the chosen $\g$, we have $w^{-1}\in L^1[-1,1]
$, and, consequently, $\displaystyle \frac{\log w(x)}{1+x^2}\in L^1(\br)$. Indeed, denoting by $J_n$ an
arbitrary interval $J_{nk}$ (they are of the same length), we see that 
\begin{eqnarray}
\dsp\int^1_{-1} w^{-1}\,dx&\le&\frac CM \sum^\infty_{n=0} 2^n\int^{|J_n|/2}_0 x^{-\g}\,
dx  \notag \\
&\le&\frac C M \sum^\infty_{n=0} 2^n \lb\frac{1-\b}{2}\rb^{n(1-%
\g)}<\infty,  \label{e51}
\end{eqnarray}
since $2((1-\b)/2)^{1-\g}<1$ under condition \eqref{e5}. Moreover, we can fix $M$ big enough to ensure that 
$$
\int^2_{-2}w^{-1}\, dx<4/(2\pi).
$$


\medskip\nt \textit{Proof of Theorem \ref{t1}.}\quad Given $1/2<\b<1$, we
construct $E_\b$ and the function $w$ as described above. Consider $d\s=
w^{-1}dx+d\s_s$, where $\s_s$ is a finite singular measure on $%
E_\b$ so that
$$
\int^2_{-2}d\s(x)=\int^2_{-2}\lt(w^{-1}\,dx+d\s_s(x)\rt)=4/(2\pi).
$$
In particular, $\sigma_s$ can be singular continuous.
The measure $\sigma$ generates the Krein system with coefficient $A$; see Section \ref{s12}.
Since $\displaystyle \log [w/(2\pi)] \in L^1(\br)$, there exists an outer function $\Pi\in
H^\infty(\bc_+)$ such that $w=2\pi|\Pi|^2$ on $\br$. Consequently, $%
\Pi_0=k_{iy}\Pi\in H^2(\bc_+)$.

By Lemma \ref{l2}, we have 
\begin{eqnarray}
\int^\infty_n |P(x,iy)|^2\,dx&=&C\inf_{F\in\ccf_n}||\Pi_0-F||^2_{2,\s}
\label{e52} \\
&\le&||\Pi_0-(\Pi_0)_{n/2}||^2_{2,\s} \le C\o^2_{2,\s}\lt(1/n, \Pi_0\rt). 
\notag
\end{eqnarray}
The remark given after the lemma explains why $(\Pi_0)_{n/2}$ lies in $\ccf_n$.
On the other hand, by the definition of $\o_{2,\s}$, 
$$
\o^2_{2,\s}\lt(1/n,\Pi_0\rt)=\sup_{|h|\le1/n}\left\{ \frac{1}{2\pi} \lt|\lt|\frac{%
\Pi_0(.+h)-\Pi_0}{\Pi}\rt|\rt|^2_2 +||\Pi_0(.+h)-\Pi_0||^2_{\s_s}\right\}. 
$$
These norms are not difficult to estimate. Recalling that $\supp\s_s=E_\b%
\subset[-1,1]$ and $w=2\pi|\Pi|^2=0$ on $E_\b$, we get  
\begin{equation*}
||\Pi_0(.+h)-\Pi_0||^2_{\s_s}=\int_\br |\Pi_0(x+h)|^2\,d\s_s(x)\le \frac
C{n^\g}\int^1_{-1} \frac{d\s_s(x)}{x^2+y^2}. 
\end{equation*}
Introduce the function $f(x)=\Pi(x+h)/\Pi(x)$.  Note that $k_{iy}f\in H^2(%
\bc_+)$. We have 
\begin{equation*}
\begin{array}{rcl}
\dsp\lt|\lt|\frac{\Pi_0(.+h)-\Pi_0}{\Pi}\rt|\rt|^2_2 & \le & \dsp C\Big\{ %
\int_\br |k_{iy}|^2|f-1|^2\, dx \\ 
&\quad + & \dsp\int_\br |k_{iy}(x+h)-k_{iy}(x)|^2 dx\Big\}.%
\end{array}
\end{equation*}
For the second integral, we readily get 
\begin{equation}
\int_\br |k_{iy}(x+h)-k_{iy}(x)|^2 dx\le \frac
1{n^2}\,O(1/y^3). \label{e521}
\end{equation}
The following representation for $\Pi$ holds \cite{kr1}
\[
\Pi(z)=\exp
\left\{ \frac{1}{2\pi i} \int_\br\frac{\ln(2\pi  \sigma^{\prime}_{ac})}{z-t}
\, dt\right\},
\]
where $z\in\bc_+$.
So, with the help of Lemma \ref{l3}, we obtain 
\begin{equation*}
\int_\br |k_{iy}|^2|f-1|^2\, dx 
\le\int_\br
|k_{iy}|^2|w(x+h)-w(x)|\frac{dx}{w(x)}+o\lb 1/{y^{2}}\rb. 
\end{equation*}
Since $w\in\lip_\g(\br)$ and $w(x+h)=w(x)$ outside the compact  $[-2,2]$, the last inequality yields 
\begin{equation*}
\int_\br |k_{iy}|^2|f-1|^2\, dx 
\le \frac C{n^\g}\int^2_{-2}\frac{1}{x^2+y^2}\frac{dx}{w(x)}+o\lb 1/{y^2}\rb. 
\end{equation*}
From \cite[Section 1]{de1},  we infer $A\in W^{1,2}(\br_+)$. Thus,
combining \eqref{e52} with Lemma \ref{l1}, we conclude that 
\begin{equation*}
\begin{array}{rcl}
\dsp\int_n^\infty A(s)^2\,ds & \le & \dsp\frac C{n^\g}
\lim_{y\to+\infty} y^2 \Big\{\int^2_{-2} \frac{d(\s_{ac}+\s_s)}{x^2+y^2}+o(y^{-2})\Big\}%
\le \frac C{n^\g}.%
\end{array}
\end{equation*}
The theorem is proved. \hfill $\Box$

 %\QTP{e KronecQ}
Notice that  $\g_0+\hdim E_\b=1$.

%\QTPQTP{e KronecQ}

\subsection{}
\label{s24} The proof of Theorem \ref{t2} is close in spirit to that of
Theorem \ref{t1}.  Once again, we begin with choosing parameters that define the measure of a
K-system. Suppose we have the set $E_\b,\ 1/2<\b<1$ (see Section \ref{s23}%
). Let $w_n=w_{\b,n}$ be 
\begin{equation*}
w_n(x)=\left\{ 
\begin{array}{ll}
\min\{1, r_n|x+1|^\g\}, & x\le-1, \\ 
0, & x\in E_\b, \\ 
\min\{1, r_n|x-a|^\g,r_n|x-b|^\g\}, & x\in J_{nk}=(a,b), \\ 
\min\{1,r_n|x-1|^\g\}, & x\ge 1,%
\end{array}%
\right. 
\end{equation*}
where $\{r_n\}$ is a sequence such that $r_n>0,\ r_n\nea+\infty$, $\g>0$.
The precise choice of $\{r_n\}$ and $\g$ will be made during the proof of the
theorem.

 %\QTP{e KronecQ}
Repeating, in essence, computations from \eqref{e51}, we deduce that 
\begin{equation*}
\int^2_{-2}w_n(x)^{-1}\, dx\le C, 
\end{equation*}
provided $0<\g\le\g_0$ (see \eqref{e5}).  Taking $\displaystyle j_n=\frac{\log
(C_0r_n)}{\g(\log 2-\log(1-\b))}$, we deduce 
\begin{equation*}
\begin{array}{rl}
\dsp\int^2_{-2}\log\frac1{w_n(x)}\,dx & 
\dsp\le C \sum_{k=1}^{j_{n}}2^{k} \int_0^{r_n^{-1/\g}}\log\frac1{r_nx^\g}\,dx\\ 
&\\ 
& \dsp \, +C\sum_{k=j_{n}+1}^{\infty } 2^k \int_0^{(1-\beta)^k2^{-k}}
\log\frac1{r_nx^\g}\,dx \le C\frac{\log
r_n}{r_n^{\g_0/\g}}.%
\end{array}
\end{equation*}

 %\QTP{e KronecQ}
\medskip\nt \textit{Proof of Theorem \ref{t2}.}\quad Take measure $\s$,
$d\s=d\s_{ac}+d\s_s$, where $\s_{ac}$ is given in the formulation of the
theorem and $\s_s$ is a finite singular measure supported on $E_\b$ so that
$$
\int^1_{-1}d\s=2/(2\pi).
$$
Again, $\sigma_s$ can be singular continuous. For any $n$, consider the outer functions $v_n\in H^\infty(\bc_+)$ with the
property $w_n=|v_n|^2$ and normalized as in Remark \ref{rk2}. We also take an outer function $\Pi\in H^\infty(\bc_+)$ so that ${\s'_{ac}}^{-1}=2\pi|\Pi|^2$. Moreover, let $%
v_{n0}=k_{iy}v_n\Pi$ and $\Pi_0=k_{iy}\chi_{\br\bsl E_\b}\Pi$.

In a trivial way, 
\begin{equation}
\inf_{F\in\ccf_N} ||\Pi_0-F||^2_{2,\s}\le C\left\{||\Pi_0-v_{n0}||^2_{2,\s%
}+\inf_{F\in\ccf_N}||v_{n0}-F||^2_{2,\s}\right\}.  \label{e53}
\end{equation}
We obtain a bound for the first summand in the last sum 
$$
\begin{array}{rl}
2\pi ||\Pi_0-v_{n0}||^2_{2,\sigma} & =||k_{iy}(v_n-1)||^2_2=\dsp\lb ||k_{iy}v_n||^2-%
\frac1{4\pi y}\rb \\ 
&  \\ 
& \dsp\,+\frac2{4\pi y}(1-\re v_n(iy))\le \frac2{4\pi y}(1-v_n(iy))%
\end{array}
$$
because $||v_n||_\infty=1$ and $v_n(iy)\in \br$ (see Remark \ref{rk2}). Furthermore, using the fact that $h_n=-\log w_n$ is supported on $%
[-2,2]$ and that $1-e^{-x}\le x$ for $x\in\br$, we proceed as follows
$$
\begin{array}{rl}
1-v_n(iy) & =\dsp1-\exp\lb-\frac1{2\pi}\int_\br\frac y{x^2+y^2}h_n(x)\, dx\rb
\\ 
&  \\ 
& \dsp\le\frac1{2\pi}\int^2_{-2}\frac{yh_n(x)}{x^2+y^2}\, dx.%
\end{array}
$$
Hence, we come to 
\begin{equation}  \label{e60}
||\Pi_0-v_{n0}||^2_2\le C\int^2_{-2}\frac{h_n(x)}{x^2+y^2}\, dx.
\end{equation}
We turn to the second summand at the right-hand side of \eqref{e53}. By  
\mbox{Lemma \ref{l11}}, we get 
\begin{eqnarray}
\dsp\inf_{F\in\ccf_N}\dsp ||v_{n0}-F||^2_{2,\s}&\le& C\o^2_{2,\s}\lt(1/N,
v_{n0}\rt)\le C\sup_{|h|\le1/N}\Big\{\int_\br |v_{n0}(x+h)-v_{n0}(x)|^2\, dx 
\notag \\
&+&\int_\br |v_{n0}(x+h)|^2\,d\s_s(x)\Big\}.  \label{e61}
\end{eqnarray}
For the first integral, we have
$$
\begin{array}{rcl}
&\dsp\int_\br&\dsp |v_{n0}(x+h)-v_{n0}(x)|^2\, dx\le C\Bigl\{\int_\br |k_{iy}(x+h)-k_{iy}(x)|^2\, |\Pi v_n|^2 dx\\
&&\\
&+&
\dsp\int_\br |k_{iy}|^2|v_{n}(x+h)-v_{n}(x)|^2\, dx+
\int_\br |k_{iy}|^2|\Pi(x+h)-\Pi(x)|^2\, dx
\Bigr\}.
\end{array}
$$
The first summand in the sum is bounded as in \eqref{e521}.

We set $f_n=v_n(.+h)/v_n$ and obtain a bound for the second summand with the help of Lemma \ref{l3} 
\begin{eqnarray}  \label{e7}
\int_\br|k_{iy}|^2|f_n-1|^2|v_n|^2\, dx&\le& \int_\br
|k_{iy}|^2||f_n|^2-1|\, dx+o\lb1/{y^2}\rb  \notag \\
&\le& \int_\br |k_{iy}|^2\lt||v_n(x+h)|^2-|v_n(x)|^2\rt|\frac{dx}{|v_n(x)|^2}+o%
\lb1/{y^2}\rb  \notag \\
&\le&\frac{Cr_n}{N^\g}\int^2_{-2}\frac1{x^2+y^2}\frac{dx}{|v_n(x)|^2}+o\lb1/%
{y^2}\rb,
\end{eqnarray}
that is because $w_n=|v_n|^2$, and the function $%
|v_n(.+h)|^2-|v_n|^2$ is supported on $[-2,2]$.

Let $f=\Pi(x+h)/\Pi(x)$. Applying Lemma \ref{l3} once again, we get

\[
\int_\br |\Pi(x+h)-\Pi(x)|^2|k_{iy}|^2 dx\leq 
C \left( \int_\br |k_{iy}|^2 ||f|^2-1|dx+o(1/y^2)\right).
\]
Clearly,
\[
\int_\br |k_{iy}|^2 ||f|^2-1|dx=\int_\br |k_{iy}|^2 |{\sigma_{ac}^{\prime}}^{-1}(x+h)-{\sigma_{ac}^{\prime}}^{-1}(x)|dx\leq Ch\int\limits_{-2}^{2} \frac{dx}{x^2+y^2}.
\]


For the second integral in \eqref{e61}, we have 
\begin{equation}  \label{e8}
\int_\br |v_{n0}(x+h)|^2\, d\s_s(x)\le\frac{Cr_n}{N^{\g}}\int^1_{-1}\frac{%
d\s_s(x)}{x^2+y^2}.
\end{equation}
Take $N=n$. Hence, \eqref{e53}, \eqref{e60}, %
\eqref{e7}, \eqref{e8}, and Lemma \ref{l1} yield 
$$
\begin{array}{rl}
\dsp\int^\infty_n A(s)^2\, ds & \dsp\le C\lim_{y\to+\infty} y^2\Big\{%
\int^2_{-2}\frac{\log w_n(x)^{-1}\,dx}{x^2+y^2} \\ 
&  \\ 
& \dsp\,+\frac{r_n}{n^\g}\int^2_{-2}\frac1{x^2+y^2}\frac{dx}{w_n(x)}+o\lb 1/{%
y^2}\rb \dsp+\frac{1}{n y^2}+\frac{r_n}{n^{\g}}\int^1_{-1}\frac{d\s_s(x)}{x^2+y^2}\Big\}
\\ 
&  \\ 
& \,\, \le\dsp C\lt\{\frac{\log r_n}{r_n^{\g_0/\g}}%
+\frac{r_n}{n^{\g}}\rt\}.%
\end{array}
$$
To optimize the result,  pick $\g=\g_0-\ep,\ r_n=n^\a$ with $\a=\g_0/2$, and arbitrarily small $\ep>0$. 
The bound becomes 
$$
\int^\infty_n A(s)^2\, ds\le\frac C{n^{\g_0/2-\ep}}. 
$$
From \cite{de1}, we have $A\in L^\infty (\br_+)$. Thus, $A\in L^p(\br_+)$ with any $p>p_0=4/(2+\g_0)$.   

\hfill$\Box$


\section{Schr\"odinger and Dirac operators}\label{s3}

\subsection{}\label{s31}
In this section, we  apply results obtained for Krein systems to Dirac
and one-dimensional Schr\"odinger operators. 

Assume that we have a K-system \eqref{e04} with coefficient $A$.
We define the functions $\phi(x,\l)=\re e^{-i\l x}P(2x,\l)$ and $\psi(x,\l)=\im e^{-i\l x}P(2x,\l)$. An easy computation shows that the functions $\phi, \psi$ are solutions of the following Dirac system
\begin{equation}
\left\{ 
\begin{array}{ccc}
\phi^\prime & = & -\lambda\psi-a_1\phi+a_2\psi, \\ 
\psi^\prime & = & \lambda\phi+a_2\phi+a_1\psi, 
\end{array}
\right.  \label{dir}
\end{equation}
with boundary conditions $\phi(0)=1,\psi(0)=0$. Above, $a_1(x)=2\re A(2x)$ and $a_2(x)=2\im A(2x)$.
This allows us to say %
(see \cite{kr1}) that $\rho_{Dir}=2\sigma$, where 
$\rho_{Dir}$ is the spectral measure of the Dirac system (\ref{dir}).

Using this  simple relation between the measures of Krein and Dirac systems, we immediately obtain the following corollaries of Theorem \ref{t1} and Theorem \ref{t2}.

\begin{theorem}\label{t3} 
There exists a Dirac system (\ref{dir}) with the properties:
\begin{itemize}
\item[\textit{i)}] The coefficient $a_2(x)=0$ for all $x>0$.
\item[\textit{ii)}] $\hdim\supp\,\sigma_{Dir,s}=2(1-\b)$, $1/2<\b<1
$. %%and $\supp\sigma_{Dir,s}=\{ x: \sigma_{Dir, ac}^\prime(x)=+\infty\}$,
\item[\textit{iii)}] The coefficient $a_1$ satisfies the
inequality 
\begin{equation*}
\displaystyle \int^\infty_x |a_1(s)|^2\,ds\le \frac C{(1+x)^{2\b-1}}. 
\end{equation*}
\end{itemize}
\end{theorem}

\begin{theorem}\label{t4}
There exists a Dirac system (\ref{dir}) with the properties:
\begin{itemize}
\item[\textit{i)}] The coefficient $a_2(x)=0$ for all $x>0$.
\item[\textit{ii)}] $\hdim\supp\sigma_{Dir,s}=2(1-\b)$ with $%
1/2<\b<1$, and 
\begin{equation*}
\sigma^{\prime}_{Dir,ac}(x)=1/(2\pi)\left\{ 
\begin{array}{ll}
1/2, & |x|\le 1, \\ 
1, & |x|>1.%
\end{array}
\right. 
\end{equation*}
\item[\textit{iii)}] The coefficient $a_1$ is in $L^p(\mathbb{R}_+)$ for any $%
p>4/(2+\gamma_0)$, $\gamma_0$ is given by \eqref{e5}.
\end{itemize}
\end{theorem}

\subsection{}\label{s32}
When the coefficient $a_1$ is absolutely continuous and $a_2=0$, we deduce from \eqref{dir} that
\begin{equation}
\begin{array}{ccc}
\psi^{\prime\prime}-q\psi+\lambda^2\psi&=&0, \\ 
\phi^{\prime\prime}-q_1\phi+\lambda^2\phi&=&0, 
\end{array} 
\end{equation}
where $q=a_1^2+a_1^\prime,\ q_1=a_1^2-a_1^\prime$. 
The corresponding boundary conditions are
$$
\begin{array}{ll}
\psi(0)=0, & \psi^\prime(0)=\lambda,\\ 
\phi(0)=1, & \phi^\prime(0)+a_1(0)\phi(0)=0.
\end{array}
$$
Therefore, the spectral measure $\rho_{Sch}$ of the Schr\"odinger
operator 
\begin{equation}
L_q y=-y^{\prime\prime}+qy \label{e9}
\end{equation} 
with boundary condition $y(0)=0$ is related to $\sigma$ by the formula 
\begin{equation}
\rho_{Sch}(\lambda)=4\int_0^{\lambda^{1/2}}\xi^2d\sigma(\xi),\ \label{liza}
\end{equation}
where $\lambda>0$.

Let a real-valued coefficient $A$ be
as in Theorems \ref{t1} or \ref{t2} and $a_1(x)=2A(2x)$.
The arguments from \cite[Section 2]{de1} show that $q$ lies not only in $L^2(\br_+)$, but also in $W^{m,2}(\br_+)$ for any integer $m$. Indeed, if the first derivatives of $A$ at $x=0$ vanish, then the first trace formulas for the K-system and trace identities for the Schr\"odinger operator coincide. In the general case, the trace formulas for K-systems involve  
derivatives of $A$ at $x=0$. Still, the approximation argument used in \cite{de1} works and we have that $q(x)\in W^{m,2}(\br_+)$ for any $m$. A simple interpolation argument yields the following statement:

\begin{theorem}\label{t5}
There exists a potential $q$ with the properties:
\begin{itemize}
\item[\textit{i)}] The spectral measure of $L_q$ (see \eqref{e9}) has a singular continuous component $\rho_{Sch,s}$ such that $\hdim\supp\rho_{Sch,s}=2(1-\beta)$ with $1/2<\beta< 1$. 

\item[\textit{ii)}] The following estimate holds
\begin{equation*}
\int_x^\infty |q(s)|^2 ds\leq \frac{C}{(1+x)^\gamma},
\end{equation*}
where $0\le\g<2\beta -1$.
\end{itemize}
\end{theorem}

\medskip\nt
{\it Sketch of the proof.}\quad Take $A$ as in Theorem \ref{t1}. By \cite{de1}, the coefficient $A\in W^{1,2}(\br_+)$ and is therefore bounded. Since $q=a_1^\prime+a_1^2$, it suffices to show that 
\begin{equation}
\int_x^\infty |a_1^\prime(s)|^2 ds\leq \frac{C}{(1+x)^\gamma}.\label{anna}
\end{equation}
Since $q\in W^{m,2}(\br_+)$, we can easily prove that $a_1\in W^{m,2}(\br_+)$ for any integer $m$ as well. 
It suffices to use an inequality from \cite{Gab}
\[
\|f^\prime\|_2\leq C_m \|f\|_2^{1-1/m}\|f^{(m)}\|_2^{1/m} 
\]
to obtain \eqref{anna}. The required properties of $\rho_{Sch}$ now follow from Theorem \ref{t1} and \eqref{liza}. 
\hfill $\Box$ 

\medskip
Similarly, the following corollary of Theorem \ref{t5} can be proved.

\begin{theorem}\label{t6} 
Fix any $p_0>4/3$. There exists a potential $q$ such that 
\begin{itemize}
\item[\textit{i)}] The spectral measure $\rho_{Sch}$ of the Schr\"odinger operator \eqref{e9}
has a nontrivial singular continuous component for some boundary condition at $x=0$. 

\item[\textit{ii)}] The positive density $\rho'_{Sch,ac}$ is piecewise constant.

\item[\textit{iii)}] $q\in L^p(\br_+)$ for any $p>p_0$.
\end{itemize}
\end{theorem}


\label{s3}

\section{Orthogonal polynomials}

\label{s4} The following theorems are counterparts of Theorems \ref{t1} and %
\ref{t2} for orthogonal polynomials on $\bt$. Their proofs follow
word-by-word the proofs of the results for Krein systems. The only
difference is that we need to use inequality \eqref{e03} instead of Lemmas %
\ref{l1} and \ref{l2}. That is why the arguments below are omitted. 

\begin{theorem}
\label{t7} There exists a measure $\s$ with the properties:

\begin{itemize}
\item[\textit{i)}] $\hdim\supp\s_s=2(1-\b)$, $1/2<\b<1$.

\item[\textit{ii)}] The sequence $\{a_n\}$ of reflection coefficients is such
that 
\begin{equation*}
\dsp \sum^\infty_{k=n}|a_k|^2\le \frac C{n^{2\b-1}}. 
\end{equation*}
\end{itemize}
\end{theorem}

In fact, some ingredients of the proof of the above theorem can be found in \cite{ge1}.

\begin{theorem}
\label{t8} There exists a singular continuous measure $\s_s$ so that

\begin{itemize}
\item[\textit{i)}] $\hdim\supp\s_s=2(1-\b)$ with $1/2<\b<1$.

\item[\textit{ii)}] The sequence $\{a_n\}$ associated to $\displaystyle d\s=dm/2+d\s_s$ lies
in $l^p$ for any \mbox{$p>4/(2+\g_0)$}, where $\g_0$ is given by \eqref{e5}.
\end{itemize}
\end{theorem}

 %\QTP{e KronecQ

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\end{document}
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