0$ and $\partial_x\psi(+0,z)$ are well-defined. As its regular counterpart, the Titchmarsh-Weyl $m$-function has the following properties: Properties of $m$. %(place before RC) \begin{enumerate} \item \label{pr1} $m$ is analytic and Herglotz. I.e. $m:\mathbb{C}^+\to% \mathbb{C}^+$. \item \label{pr2} Let $Q_n$ be a sequence of smooth $L^2_{loc}$ functions such that $\left\Vert Q-Q_n \right\Vert_{L^2_{loc}}\to0$, $n\to\infty$, and $% q=Q^{\prime }$ is limit point case at $-\infty$. Then ${m}_n\to m$ uniformly on compact subsets of $\mathbb{C}^+$. %\rightrightarrows m$ \end{enumerate} Define now the reflection coefficient $R$ from the right incident of a singular potential $q\in H^{-1}_{loc}(\mathbb{R})$ such that $q|_{\mathbb{R}% _+}=0$. Pick up a point $x_0>0$ and consider a solution to $L_qy=\lambda^2y$ which is proportional to the Weyl solution on $(-\infty,x_0)$ and is equal to $% e^{-i\lambda x}+re^{i\lambda x}$ on $(x_0,\infty)$. From the continuity of this solution and its derivative at $x_0$ one has \begin{equation*} r(\lambda,x_0)= e^{-2i\lambda x_0} \frac{i\lambda-\frac{\psi^{\prime }(x_0,\lambda^2)}{\psi(x_0,\lambda^2)}}{i\lambda+\frac{\psi^{\prime }(x_0,\lambda^2)}{\psi(x_0,\lambda^2)}}. \end{equation*} We define the right reflection coefficient by \begin{equation} \label{eq3.3} R(\lambda)=\lim_{x_0\to0^+}r(\lambda,x_0)= \frac{i\lambda-m(\lambda^2)}{% i\lambda+m(\lambda^2)}. \end{equation} \begin{example} \label{ex1} Let $q(x)=c \delta(x)$. The Weyl solution corresponding to $% -\infty$ can be explicitly computed by ($C\ne0$) \begin{equation*} \psi(x,\lambda^2)= C \begin{cases} e^{-i\lambda x} \quad & ,\quad x<0 \\ \frac{1}{2i\lambda} \left( ce^{i\lambda x}+(2i\lambda-c)e^{-i\lambda x}\right) \quad & , \quad x>0% \end{cases}% \end{equation*} and hence by \eqref{eq4.1} and \eqref{eq3.3} \begin{align*} m(\lambda^2) &= i\lambda-c, \\ R(\lambda) &= \frac{c}{2i\lambda-c}. \end{align*} \end{example} \section{Main result} In the last section we state and prove our main result. As customary, given self-adjoint operator $A$ we write $A\ge0$ if $A$ is positive. \begin{theorem} \label{thm1} Let $q$ in \eqref{eq1.1} supported on $\mathbb{R}_{-}$ be in $% H^{-1}$ and such that the Schr\"{o}dinger operator $L_{q}\geq 0$. Then there is a (unique) classical solution to \eqref{eq1.1} given by \begin{equation} u(x,t)=-2\partial _{x}^{2}\log \det \left( 1+\mathbb{H}_{x,t}\right) \label{eq2.1} \end{equation}% where $\mathbb{H}_{x,t}$ is the trace class Hankel operator on $L^{2}(% \mathbb{R}_{+})$ with the symbol \begin{equation*} \varphi _{x,t}(\lambda )=\frac{i\lambda -m(\lambda ^{2})}{i\lambda +m(\lambda ^{2})}e^{2i\lambda x+8i\lambda ^{3}t} \end{equation*}% where $m$ is the (Dirichlet) Titchmarsh-Weyl $m$-function of $L_{q}$ on $% L^{2}(\mathbb{R}_{-})$. The solution $u(x,t)$ is meromorphic in $\mathbb{C}^+$ for any $t>0$ except (double) poles none of which are real. \end{theorem} \begin{proof} It is proven in \cite{KapPerryTopalov2005} that \begin{equation*} L_{q}\geq 0\quad \Rightarrow \quad q\in B\left( L_{loc}^{2}\right) \subset H_{loc}^{-1} \end{equation*}% where $B(r)=r^{\prime }+r^{2}$ is the Miura map. Since compactly supported smooth functions are dense in $H^{-1}_{loc}$, we can approximate our $q$ by a sequence $\tilde{q}=\tilde{r}^{\prime }+\tilde{r% }^2$ where $\tilde{r}$'s are smooth and compactly supported. For each $\tilde{q}$ there exists the (classical) right reflection coefficient $\widetilde{R}$. The (classical) Marchenko operator $\widetilde{% \mathbb{H}}_{x,t}$ has no discrete component (since $L_{\tilde{q}} \ge0$) and hence it takes the form \begin{equation} \label{eq7.1} \left(\widetilde{\mathbb{H}}_{x,t}f\right)(\cdot)= \int_0^\infty \widetilde{H% }_{x,t}(\cdot+y)f(y)dy \end{equation} where \begin{equation} \label{eq7.2} \widetilde{H}_{x,t}(\cdot)= \frac{1}{2\pi} \int e^{2i\lambda x+8i\lambda^3t}e^{i\lambda(\cdot)}\widetilde{R}(\lambda)d\lambda. \end{equation} The reflection coefficient $\widetilde{R}$ can be computed by \begin{equation*} \widetilde{R}(\lambda)= \frac{i\lambda-\widetilde{m}(\lambda^2)}{i\lambda+% \widetilde{m}(\lambda^2)} \end{equation*} where $\tilde{m}$ is the Titchmarsh-Weyl $m$-function of $L^0_{\tilde{q}}$, the Dirichlet $-\partial_x^2+\tilde{q}(x)$ on $\mathbb{R}_-$. Since the function $\widetilde{R}(\lambda)$ is analytic in $\mathbb{C}^+$ and $% \widetilde{R}(\lambda)=O(1/\lambda) \; , \; \lambda\to\pm\infty$, and $% \left\vert \widetilde{R}(\lambda) \right\vert\le1\;,\; \lambda\in\mathbb{C}^+ $, one can obviously deform the contour of integration in \eqref{eq7.2} and % \eqref{eq7.2} reads \begin{equation} \label{eq8.1} \widetilde{H}_{x,t}(\cdot)= \frac{1}{2\pi} \int_{\Im\lambda=h} e^{2i\lambda x+8i\lambda^3t}e^{i\lambda(\cdot)}\widetilde{R}(\lambda)d\lambda. \end{equation} for any $h>0$. Since the integrand in \eqref{eq8.1} is clearly integrable along the line $\Im\lambda=h$, the operator $\widetilde{\mathbb{H}}_{x,t}$ is trace class (see \cite{Ryprep}) and the function \begin{equation} \label{eq8.2} \tilde{u}(x,t)=-2\partial_x^2\log\det\left(1+\widetilde{\mathbb{H}}% _{x,t}\right) \end{equation} is well-defined and solves \eqref{eq1.1} with initial data $\tilde{q}$. We now pass to the limit in \eqref{eq8.2} as $\tilde{r}\to r$ in $L^2_{loc}$% . By property \eqref{pr2} of the Titchmarsh-Weyl $m$-function, \begin{equation*} \widetilde{R}(\lambda)= \frac{i\lambda-\widetilde{m}(\lambda^2)}{i\lambda+% \widetilde{m}(\lambda^2)} \quad \longrightarrow \quad {R}(\lambda)= \frac{% i\lambda-{m}(\lambda^2)}{i\lambda+{m}(\lambda^2)} \end{equation*} on each compact set in $\mathbb{C}^+$. The oscillatory factor $e^{2i\lambda x+8i\lambda^3t}$ exhibits a superexponential decay on $\Im\lambda=h>0$. This means that (see \cite{Ryprep} for) \begin{equation*} \widetilde{\mathbb{H}}_{x,t} \; \longrightarrow \; \mathbb{H}_{x,t} \end{equation*} for any $x\in\mathbb{R}$, $t>0$ in trace class norm and hence \begin{equation*} \det\left(1+\widetilde{\mathbb{H}}_{x,t}\right) \; \longrightarrow \; \det\left(1+\mathbb{H}_{x,t}\right). \end{equation*} Note that $\widetilde{H}_{x,t}$ and \begin{equation*} {H}_{x,t}(\cdot)= \frac{1}{2\pi} \int_{\Im\lambda=h} e^{2i\lambda x+8i\lambda^3t}e^{i\lambda(\cdot)}{R}(\lambda)d\lambda \end{equation*} are clearly entire with respect to $x$, $\forall\;t>0$. It is quite easy to see that $\widetilde{\mathbb{H}}_{x,t}, \mathbb{H}_{x,t}$ are operator-valued functions entire with respect to $x$, $\forall\;t>0$. This means that the functions \begin{equation*} \tilde{u}(x,t)=-2\partial_x^2\log\det\left(1+\widetilde{\mathbb{H}}% _{x,t}\right) \end{equation*} are meromorphic in $x$ on the whole complex plane for any $t>0$ and converge to the meromorphic function \begin{equation*} {u}(x,t)=-2\partial_x^2\log\det\left(1+{\mathbb{H}}_{x,t}\right) \end{equation*} as $\tilde{r}\to r$ in $L^2_{loc}$. It remains to show that $\det(1+\mathbb{H}_{x,t})$ doesn't vanish on the real line for any $t>0$. Since $\mathbb{H}_{x,t}$ is trace class, this amounts to showing that $-1$ is not an eigenvalue of $\mathbb{H}_{x,t}$ for all $x\in\mathbb{R}\;,\;t>0$. We have two cases: $L_q$ has some a.c. spectrum, $L_q$ has no a.c. spectrum. The first case immediately follows from Lemma \ref{lem1}. The second case is a bit more involved. If the a.c. spectrum of $L_q$ is empty then the Titchmarsh-Weyl $m$-function is real a.e. on the real line and hence the reflection coefficient $\left\vert R(\lambda) \right\vert\le1$ in $\mathbb{C}^+$ and $\left\vert R(\lambda) \right\vert=1$ a.e. on $\mathbb{% R}$. I.e. $R$ is an inner function of the upper half plane. Lemma \ref{lem2} then applies. \end{proof} \begin{remark} \label{rem1} Theorem \ref{thm1} implies very strong WP of the KdV equation with eventually any steplike Miura initial data supported on $(-\infty,0)$. Each such solution $u(x,t)$ is smooth and hence solves the KdV equation in the classical sense. It also has a continuity property in the sense that if $% \{q_n\}$ is a sequence of smooth $H^{-1}_{loc}$ functions convergent in $% H^{-1}_{loc}$ to $q$ then the sequence of the corresponding solutions $% \{u_n(x,t)\}$ converges in $H^{-1}_{loc}$ to $u(x,t)$. This, in turn, implies uniqueness. The initial condition is satisfied in the sense that \begin{equation*} \left\Vert u(\cdot,t)-q \right\Vert_{H^{-1}_{loc}}\;\to\; 0 \quad,\quad t\to0. \end{equation*} \end{remark} \begin{remark} \label{rem2}It is unlikely that, under our conditions, $\mathbb{H}_{x,t}$ in % \eqref{eq2.1} is trace class for any $x$ if $t=0$. We conjecture however that if $Q$ is uniformly in $L_{loc}^{2}$, i.e. $\sup_{x\leq 0}\int_{x-1}^{x}\left\vert Q\right\vert ^{2}<\infty $, then $\mathbb{H}_{x,0} $ is also trace class for any real $x$. \end{remark} \begin{remark} \label{rem3} We assumed $q|_{\mathbb{R}_{+}}=0$ for simplicity and it can be replaced with a suitable decay condition but the consideration becomes much more involved due to serious technical circumstances. We plan to return to it elsewhere. \end{remark} \begin{thebibliography}{99} \bibitem{BGS2001} B\"{o}tcher, A.; Grudsky S.; Spitkovsky I. \emph{Toeplitz operators with frequency modulated semi-almost periodic symbols}. J. Fourier Anal. and Appl. 7 (2001), no. 5, 523--35. \bibitem{BotSil06} B\"{o}tcher, A.; Silbermann B. \emph{Analysis of Toeplitz operators}. Springer-Verlag, Berlin, 2002. 665 pp. \bibitem{ColKeStaTao03} Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. \emph{Sharp global well-posedness for KdV and modified KdV on }$% R $\emph{\ and }$T$\emph{. }J. Amer. Math. Soc. 16 (2003), no. 3, 705--49. MR1969209 (2004c:35352) \bibitem{DyGrud02} Dybin, V.; Grudsky S. \emph{Introduction to the theory of Toeplitz operators with infinite index}. Birkh\"{a}user Verlag, Basel, 2002. xii+299 pp. \bibitem{EGT09} Egorova, I.; Grunert K.; Teschl G. \emph{On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data I. Schwarz-type perturbations.} Nonlinearity 22 (2009), 1431--57. \bibitem{Gru01} Grudsky, S.M. \emph{Toeplitz operators and the modelling of oscillating discontinuities with the help of Blaschke products}. Operator theory: Advances and Applications, v. 121, Birkh\"{a}user Verlag, Basel, 2001, pp. 162-193. \bibitem{Guo09} Guo, Zihua \emph{Global Well-posedness of Korteweg-de Vries equation in }$H^{-3/4}\left( \mathbb{R}\right) $. J. Math. Pures Appl. (9) 91 (2009), no. 6, 583--97. \bibitem{Kap1986} Kappeler, T. \emph{Solutions to the Korteweg-de Vries equation with irregular initial data.} Comm. Partial Diff. Eq. 11 (1986), 927--45. %MR 87j:35326 \bibitem{KapTop06} Kappeler, T.; Topalov, P. \emph{Global wellposedness of KdV in $H^{-1}\left( \mathbb{T},\mathbb{R}\right)$.} Duke Math. J. Volume 135, Number 2 (2006), 327--60 \bibitem{KapPerryTopalov2005} Kappeler, T.; Perry, P.; Shubin, M.; Topalov, P. \emph{The Miura map on the line}. Int. Math. Res. Not. (2005), no. 50, 3091--133. MR2189502 (2006k:37191) \bibitem{Nik2002} Nikolski, N.K. \emph{Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel and Toeplitz.} Mathematical Surveys and Monographs, vol. 92, Amer. Math. Soc., Providence, 2002. 461 pp. \bibitem{Peller2003} Peller, Vladimir V. \emph{Hankel operators and their applications.} Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. xvi+784 pp. ISBN: 0-387-95548-8. \bibitem{Ryb10} Rybkin, Alexei \emph{Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line}. Nonlinearity 23 (2010), no. 5, 1143--67. %MR2630095 \bibitem{Ryprep} Rybkin, A. \emph{The Hirota $\tau $-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line}, preprint (2010). \bibitem{SS1999} Savchuk, A.M.; Shkalikov, A.A. \emph{Sturm-Liouville operators with distribution potentials.} Trans. Moscow Math. Soc. 64 (2003), 143--92. \bibitem{Tao06} Tao, Terence \emph{Nonlinear dispersive equations}. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp. ISBN: 0-8218-4143-2 \end{thebibliography} \end{document} ---------------1108102015343--