0.$ {\bf Remark 2. }In paper \cite{Den} it was considered the dependence of absence of singular component on certain interval on the local smoothness of Fourier transform. {\bf Remark 3. }From the corollary and method used in theorem 3 it follows that if $q(x)${\it \ is such that (A) and (B) are satisfied, }$\widehat{q(\omega )% }=\widehat{q(0)}+\widehat{\varphi (\omega )}${\it \ where }$\left| \widehat{% \varphi (\left| \omega \right| )}\right| \leq C|\omega |^{1/2+\varepsilon }\ ${\it \ in the vicinity of zero for some positive }$\varepsilon ${\it \ and }% $\frac{\widehat{\varphi (\left| \omega \right| )}}{|\omega |+1}\in L^2(R),$% {\it \ then the essential support of spectral measure of operator is }$% R^{+}. $\vspace{1cm} {\bf Section C.} In this section we will discuss the dependence of spectral measure $\sigma (\lambda )$ on the coefficient $A(x)$ of system (\ref{sys3}). In fact function $A(x)$ plays the role of sequence $a_n$ for polynomials orthogonal on the unit circle (see \cite{Ger} ). We will see that for the Dirac-type systems the situation is not so simple. The basic reason is the possible oscillation of $A(x)$. The following Lemma is true {\bf Lemma. }If measurable bounded function $A(x)$ is such that \[ \int\limits_x^\infty e^{-s}A(s)ds=\overline{o}(e^{-x}),\ A(x)e^x\int\limits_x^\infty A(s)e^{-s}ds\in L_1(R^{+}) \] then conditions (1)-(4) from section A are satisfied. Proof. Consider system (\ref{sys3}) with $\lambda =i$. If $P=e^{-x}Q$ then we have \begin{equation} \begin{array}{ccc} Q^{^{\prime }} &=&-Ae^xP_{*} \\ P_{*}^{^{\prime }} &=&-Ae^{-x}Q \end{array} \label{ff1} \end{equation} Consequently \begin{eqnarray*} P_{*}(x,i) &=&1-\int\limits_0^xA(s)e^{-s}ds+\int\limits_0^xA(s)e^sP_{*}(s,i)\int% \limits_s^\infty A(\xi )e^{-\xi }d\xi ds-\int\limits_x^\infty A(\xi )e^{-\xi }d\xi \int\limits_0^xA(s)e^sP_{*}(s,i)ds \\ && \end{eqnarray*} And now it suffices to use the standard argument. Let $M_n=\max_{x\in [0,n]}|P_{*}(x,i)|=|P_{*}(x_n,i)|$ So $M_n\leq 1+C+M_n\int\limits_0^\infty |A(s)|e^s\left| \int\limits_s^\infty A(\xi )e^{-\xi }d\xi \right| ds+\overline{o}(1)M_n$ If the whole integral in the last formula is less then $1$ then $M_n$ is bounded. Otherwise we should start to solve the equations (\ref{ff1}) not from zero but from some other point $x_0$ for which this condition is satisfied. {\bf Example. }$A(x)=(x^2+1)^{-\alpha }\sin (x^\beta )$ where $\alpha ,\beta >0.$ One can easily verify that conditions of Lemma are satisfied if $2\alpha +\beta /2>1$. Meanwhile $A(x)\in L_2(R^{+})$ if and only if $\alpha >1/4$. Nevertheless for nonpositive $A(x)$ with bounded derivative the condition $% A(x)\in L^2(R^{+})$ is necessary for (1)-(4) from Section A to be true. {\bf Proposition. }If one of the conditions (1)-(4) is true, $A(x)\leq 0$ and $A^{^{\prime }}(x)$ is bounded then $A(x)$ is from $L^2(R^{+}).$ Proof. Really, since $A(x)\leq 0$ both $P$ and $Q$ are not less then 1. Consequently if one of (1)-(4) holds then $P_{*}(x,i)$ is bounded and as it follows from (\ref{ff1}) $\int\limits_0^x|A(s)|e^s\int\limits_s^x\left| A(\xi )\right| e^{-\xi }d\xi ds$ is bounded as well. But we have the inequality \[ \int\limits_0^x|A(s)|e^s\int\limits_s^x\left| A(\xi )\right| e^{-\xi }d\xi ds\geq e^{-1}\int\limits_0^{x-1}|A(s)|\int\limits_s^{s+1}\left| A(\xi )\right| d\xi ds\geq C\int\limits_0^{x-1}\left| A(s)\right| ^2ds. \] which concludes the proof of proposition. The latter inequality follows from the boundedness of $A^{^{\prime }}(x).$ $\Box$ Function $A(x)$ from the example above with $\alpha =1/4,\ 1<\beta \leq 3/2\ $ satisfies the conditions of Lemma (consequently (1)-(4) holds ), has bounded derivative but is not from $L^2(R^{+}).$ The explanation is that this function is not nonpositive. We would like to conclude the paper with two open problems the first of which is much more difficult then the second one. {\bf Open problems}. 1. Prove that Theorem 2 holds for $W\in L^2(R^{+})$. 2. Prove that the presence of a.c. component on the half-line pertains to those potentials which Fourier transform is from $L^{2}$ near the zero. Specifically, if $q$ admits the Fourier transform $\widehat{q}$ such that $% \widehat{q}\in L^{2}_{loc}(R)$ and $\frac{\widehat{q}}{|\omega |+1}$$\in L^2(R), $ then the a.c. part of the spectrum fills the whole positive half-line. This conjecture seems reasonable at least with some additional constraints since we can represent $\widehat{q}=\widehat{q_1}+\widehat{q_{2}}$. Where the $\widehat{q_1}$ is localized near the zero and is from $L^2$ so the methods of paper \cite {Den} works. The other function $\widehat{q_{2}}$ is such that Theorem 3 can be applied.% \vspace{1cm} Acknowledgment. Author is grateful to C.Remling for attention to this work. e-mail address: saden@cs.msu.su \begin{thebibliography}{99} \bibitem{Kr} M.G.Krein Continuous analogues of propositions on polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR, {\bf 105}, 637-640, (1955). \bibitem{LS} B.M.Levitan, I.S.Sargsjan Introduction to the Spectral Theory, Trans. Math. Monographs, 39, Amer. Math. Soc., Providence, RI, 1976. \bibitem{Atk} F.V.Atkinson Discrete and continuous boundary problems, Academic Press, New York, London, 1964. \bibitem{Kur} T.Kato Perturbation theory for linear operators, Reprint of the 1980 edition, Berlin, Springer, 1995. \bibitem{Rem1} C.Remling Relationships between the $m$-function and subordinate solutions of second order differential operators. J.\ Math.\ Anal.\ Appl.\ {\bf 206}, 352--363, (1997). \bibitem{JL} S.Jitomirskaya, Y.Last Dimensional Hausdorff properties of singular continuous spectra. Phys.\ Rev.\ Letters {\bf 76}, 1765--1769, (1996). \bibitem{Rem2} C.Remling The absolutely continuous spectrum of one-dimensional Schr\"odinger operator with decaying potentials Commun. Math. Physics, {\bf 193}, 151-170, (1998). \bibitem{St} G.Stolz Bounded solutions and absolute continuity of Sturm-Liouville operators. J.\ Math.\ Anal.\ Appl.\ {\bf 169}, 210--228, (1992). \bibitem{Behn1} H.Behncke Absolute continuity of Hamiltonians with von Neumann Wigner potentials. Proc.\ Amer.\ Math.\ Soc.\ {\bf 111}, 373--384, (1991). \bibitem{Behn2} H.Behncke Absolute continuity of Hamiltonians with von Neumann Wigner potentials II. Manuscr.\ Math.\ {\bf 71}, 163--181, (1991). \bibitem{vNW} J. von Neumann, E. Wigner Uber merkw\"urdige diskrete Eigenwerte. Z.\ Phys.\ {\bf 30}, 465--467, (1929). \bibitem{RS} M.Reed, B.Simon Methods of modern mathematical physics, Scattering theory, 3, Academic Press, 1979. \bibitem{Mench} D.Menchoff Sur les series de fonctions orthogonales. Fund. Math. {\bf 10 }, 375-420, (1927). \bibitem{CK} M.Christ, A.Kiselev WKB asymptotics of generalized eigenfunctions of one-dimensional Schr\"odinger operators (preprint ). \bibitem{Den} S.A.Denisov Absolutely continuous transform of Schr\"odinger operators and Fourier transform of the potential (submitted to Russian Journal of Mathematical Physics ). \bibitem{Ger} Y.L.Geronimus Polinomials orthogonal on the segment and unit circle (in Russian ), 1958. \end{thebibliography} \end{document} ---------------0007210912660--