Content-Type: multipart/mixed; boundary="-------------1109272144876" This is a multi-part message in MIME format. ---------------1109272144876 Content-Type: text/plain; name="11-137.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-137.keywords" Korteweg-de Vries equation, inverse scattering transform, Schr dinger operator, Hankel operator, Gevrey regularity. ---------------1109272144876 Content-Type: text/plain; name="bibryb.bib" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="bibryb.bib" @article {AblSatJMP78, AUTHOR = {Ablowitz, M. J. and Satsuma, J.}, TITLE = {Solitons and rational solutions of nonlinear evolution equations}, JOURNAL = {J. Math. 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An International Journal for Theory \& Applications}, VOLUME = {14}, YEAR = {2001}, NUMBER = {4}, PAGES = {493--512}, ISSN = {0893-4983}, MRCLASS = {34L40 (35C05 35Q53)}, MRNUMBER = {1799418 (2001k:34155)}, MRREVIEWER = {Vyacheslav N. Pivovarchik}, } @article {MR1856247, AUTHOR = {Rybkin, Alexei}, TITLE = {On the trace approach to the inverse scattering problem in dimension one}, JOURNAL = {SIAM J. Math. Anal.}, FJOURNAL = {SIAM Journal on Mathematical Analysis}, VOLUME = {32}, YEAR = {2001}, NUMBER = {6}, PAGES = {1248--1264 (electronic)}, ISSN = {0036-1410}, MRCLASS = {34L25 (34A55 34L40 47E05)}, MRNUMBER = {1856247 (2002f:34201)}, MRREVIEWER = {P. A. Mishnayevskiy}, DOI = {10.1137/S0036141000365620}, URL = {http://dx.doi.org/10.1137/S0036141000365620}, } @article {MR1855620, AUTHOR = {Rybkin, Alexei}, TITLE = {On a complete analysis of high-energy scattering matrix asymptotics for one dimensional {S}chr\"odinger operators with integrable potentials}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {130}, YEAR = {2002}, NUMBER = {1}, PAGES = {59--67 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {34L25 (34L40 81U05)}, MRNUMBER = {1855620 (2002g:34193)}, DOI = {10.1090/S0002-9939-01-06014-2}, URL = {http://dx.doi.org/10.1090/S0002-9939-01-06014-2}, } @article {MR1866429, AUTHOR = {Rybkin, Alexei}, TITLE = {Some new and old asymptotic representations of the {J}ost solution and the {W}eyl {$m$}-function for {S}chr\"odinger operators on the line}, JOURNAL = {Bull. London Math. Soc.}, FJOURNAL = {The Bulletin of the London Mathematical Society}, VOLUME = {34}, YEAR = {2002}, NUMBER = {1}, PAGES = {61--72}, ISSN = {0024-6093}, CODEN = {LMSBBT}, MRCLASS = {34L40 (34L25 81U05)}, MRNUMBER = {1866429 (2002j:34149)}, MRREVIEWER = {Christian Remling}, DOI = {10.1112/S0024609301008645}, URL = {http://dx.doi.org/10.1112/S0024609301008645}, } @article {MR1929041, AUTHOR = {Rybkin, Alexei}, TITLE = {Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional {S}chr\"odinger operators}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {131}, YEAR = {2003}, NUMBER = {1}, PAGES = {219--229 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {34L40 (34L25 47E05)}, MRNUMBER = {1929041 (2003g:34187)}, MRREVIEWER = {L. V. Kritskov}, DOI = {10.1090/S0002-9939-02-06555-3}, URL = {http://dx.doi.org/10.1090/S0002-9939-02-06555-3}, } @article {MR2043835, AUTHOR = {Rybkin, Alexei}, TITLE = {On the absolutely continuous and negative discrete spectra of {S}chr\"odinger operators on the line with locally integrable globally square summable potentials}, JOURNAL = {J. Math. Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {45}, YEAR = {2004}, NUMBER = {4}, PAGES = {1418--1425}, ISSN = {0022-2488}, CODEN = {JMAPAQ}, MRCLASS = {34L40 (47E05 81Q10)}, MRNUMBER = {2043835 (2005e:34261)}, MRREVIEWER = {Mikl{\'o}s Horv{\'a}th}, DOI = {10.1063/1.1650048}, URL = {http://dx.doi.org/10.1063/1.1650048}, } @incollection {MR2103376, AUTHOR = {Rybkin, Alexei}, TITLE = {On a transformation of the {S}turm-{L}iouville equation with slowly decaying potentials and the {T}itchmarsh-{W}eyl {$m$}-function}, BOOKTITLE = {Spectral methods for operators of mathematical physics}, SERIES = {Oper. Theory Adv. Appl.}, VOLUME = {154}, PAGES = {185--201}, PUBLISHER = {Birkh\"auser}, ADDRESS = {Basel}, YEAR = {2004}, MRCLASS = {34E10 (34B20 34E20 34L40 47E05)}, MRNUMBER = {2103376 (2005h:34147)}, MRREVIEWER = {Stanislav Kupin}, } @article {MR2194033, AUTHOR = {Rybkin, Alexei}, TITLE = {On the spectral {$L\sb 2$} conjecture, {$3/2$}-{L}ieb-{T}hirring inequality and distributional potentials}, JOURNAL = {J. Math. Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {46}, YEAR = {2005}, NUMBER = {12}, PAGES = {123505, 8}, ISSN = {0022-2488}, CODEN = {JMAPAQ}, MRCLASS = {81Q10 (34L40 47A55 47B25 47E05 47N50)}, MRNUMBER = {2194033 (2006k:81124)}, MRREVIEWER = {Jaouad Sahbani}, DOI = {10.1063/1.2142837}, URL = {http://dx.doi.org/10.1063/1.2142837}, } @article {MR2227810, AUTHOR = {Rybkin, Alexei}, TITLE = {The analytic structure of the reflection coefficient, a sum rule and a complete description of the {W}eyl {$m$}-function of half-line {S}chr\"odinger operators with {$L\sb 2$}-type potentials}, JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A}, FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics}, VOLUME = {136}, YEAR = {2006}, NUMBER = {3}, PAGES = {615--632}, ISSN = {0308-2105}, CODEN = {PEAMDU}, MRCLASS = {47E05 (34B20 34L25 34L40 81Q10)}, MRNUMBER = {2227810 (2007a:47053)}, MRREVIEWER = {Dmitry G. Shepelsky}, DOI = {10.1017/S0308210500005084}, URL = {http://dx.doi.org/10.1017/S0308210500005084}, } @incollection {MR2259114, AUTHOR = {Rybkin, Alexei}, TITLE = {Preservation of the absolutely continuous spectrum: some extensions of a result by {M}olchanov-{N}ovitskii-{V}ainberg}, BOOKTITLE = {Recent advances in differential equations and mathematical physics}, SERIES = {Contemp. Math.}, VOLUME = {412}, PAGES = {271--281}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2006}, MRCLASS = {47E05 (34B20 34L20 34L25 81Q10)}, MRNUMBER = {2259114 (2007g:47073)}, MRREVIEWER = {Alexander M. Gomilko}, } @article {MR2432031, AUTHOR = {Rybkin, Alexei}, TITLE = {On the evolution of a reflection coefficient under the {K}orteweg-de {V}ries flow}, JOURNAL = {J. Math. Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {49}, YEAR = {2008}, NUMBER = {7}, PAGES = {072701, 15}, ISSN = {0022-2488}, CODEN = {JMAPAQ}, MRCLASS = {35Q53 (34A55 34L25 35P25 35R30 37K15)}, MRNUMBER = {2432031 (2009h:35382)}, DOI = {10.1063/1.2951897}, URL = {http://dx.doi.org/10.1063/1.2951897}, } @article {MR2538578, AUTHOR = {Rybkin, Alexei and Vu Kim Tuan}, TITLE = {A new interpolation formula for the {T}itchmarsh-{W}eyl {$m$}-function}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {137}, YEAR = {2009}, NUMBER = {12}, PAGES = {4177--4185}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {65D05 (34B20 34L40 41A05 47E05)}, MRNUMBER = {2538578 (2010k:65023)}, MRREVIEWER = {Alexandru Ioan Mitrea}, DOI = {10.1090/S0002-9939-09-09983-3}, URL = {http://dx.doi.org/10.1090/S0002-9939-09-09983-3}, } @article {MR2540891, AUTHOR = {Rybkin, Alexei}, TITLE = {On the {M}archenko inverse scattering procedure with partial information on the potential}, JOURNAL = {Inverse Problems}, FJOURNAL = {Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data}, VOLUME = {25}, YEAR = {2009}, NUMBER = {9}, PAGES = {095011, 34}, ISSN = {0266-5611}, CODEN = {INPEEY}, MRCLASS = {35R30 (35P25 78A40)}, MRNUMBER = {2540891 (2011a:35581)}, DOI = {10.1088/0266-5611/25/9/095011}, URL = {http://dx.doi.org/10.1088/0266-5611/25/9/095011}, } @article {MR2558307, AUTHOR = {Rybkin, Alexei}, TITLE = {On the boundary control approach to inverse spectral and scattering theory for {S}chr\"odinger operators}, JOURNAL = {Inverse Probl. Imaging}, FJOURNAL = {Inverse Problems and Imaging}, VOLUME = {3}, YEAR = {2009}, NUMBER = {1}, PAGES = {139--149}, ISSN = {1930-8337}, MRCLASS = {35L25 (34A55 35L05 49N45 93C20)}, MRNUMBER = {2558307 (2011a:35313)}, MRREVIEWER = {Antonio C. G. Leit{\~a}o}, DOI = {10.3934/ipi.2009.3.139}, URL = {http://dx.doi.org/10.3934/ipi.2009.3.139}, } @article {Ryb10, AUTHOR = {Rybkin, Alexei}, TITLE = {Meromorphic solutions to the {K}d{V} equation with non-decaying initial data supported on a left half line}, JOURNAL = {Nonlinearity}, FJOURNAL = {Nonlinearity}, VOLUME = {23}, YEAR = {2010}, NUMBER = {5}, PAGES = {1143--1167}, ISSN = {0951-7715}, CODEN = {NONLE5}, MRCLASS = {35Q53 (34A55 34L25 37K15)}, MRNUMBER = {2630095 (2011i:35223)}, DOI = {10.1088/0951-7715/23/5/007}, URL = {http://dx.doi.org/10.1088/0951-7715/23/5/007}, } @incollection {MR2683250, AUTHOR = {Rybkin, Alexei}, TITLE = {Regularized perturbation determinants and {K}d{V} conservation laws for irregular initial profiles}, BOOKTITLE = {Topics in operator theory. {V}olume 2. {S}ystems and mathematical physics}, SERIES = {Oper. Theory Adv. Appl.}, VOLUME = {203}, PAGES = {427--444}, PUBLISHER = {Birkh\"auser Verlag}, ADDRESS = {Basel}, YEAR = {2010}, MRCLASS = {37K15 (34A55 34L40 35Q53 37K10)}, MRNUMBER = {2683250}, } ---------------1109272144876 Content-Type: application/x-tex; name="merom2011-10.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="merom2011-10.tex" \documentclass[reqno]{amsart} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.50.0.2953} %TCIDATA{} %TCIDATA{BibliographyScheme=BibTeX} %TCIDATA{LastRevised=Tuesday, September 27, 2011 18:35:31} %TCIDATA{} %TCIDATA{Language=American English} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}{Problem} \newtheorem{remark}[theorem]{Remark} \newtheorem{condition}[theorem]{Hypothesis} \numberwithin{equation}{section} \newcommand{\C}{\mathbb{C}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\abs}[1]{\left\vert #1 \right\vert} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\norm}[1]{\left\Vert #1 \right\Vert} \newcommand{\eps}{\varepsilon} \newcommand{\mynegspace}{\hspace{-0.12em}} \newcommand{\lvvvert}{\lvert\mynegspace\lvert\mynegspace\lvert} \newcommand{\rvvvert}{\rvert\mynegspace\rvert\mynegspace\rvert} \newcommand{\normtwo}[1]{\lvvvert #1 \rvvvert} \renewcommand{\Im}{\operatorname{Im}} \renewcommand{\Re}{\operatorname{Re}} \input{tcilatex} \begin{document} \title[Meromorphic solutions]{Spatial Analyticity of solutions to integrable systems. I. The KdV case} \author{Alexei Rybkin} \address{University of Alaska Fairbanks} \date{September, 2011} \address{Department of Mathematics and Statistics \\ University of Alaska Fairbanks\\ PO Box 756660\\ Fairbanks, AK 99775} \email{arybkin@alaska.edu} \thanks{Based on research supported in part by the NSF under grant DMS 1009673.} \subjclass[2010]{37K15, 47B35, 35B65} \keywords{Korteweg-de Vries equation, inverse scattering transform, Schr\"{o}% dinger operator, Hankel operator, Gevrey regularity. } \begin{abstract} We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles $q$'s which are, in a certain sense, essentially bounded from below and $q\left( x\right) =O\left( e^{-cx^{\varepsilon }}\right) ,x\rightarrow +\infty $, with some positive $c$ and $\varepsilon $. Using the inverse scattering transform, we show that the KdV flow turns such initial data into a function which is (1) meromorphic (in the space variable) on the whole complex plane if $\varepsilon >1/2$, (2) meromorphic on a strip around the real line if $\varepsilon =1/2$, and (3) Gevrey regular if $\varepsilon <1/2$. Note that $q$'s need not have any decay or pattern of behavior at $-\infty $. \end{abstract} \maketitle %\tableofcontents %-------------------------------------------Section 1: Introduction----------------------------------- \section{Introduction and statements of main results} The gain and persistence of regularity effects are important features of many dispersive (linear and nonlinear) partial differential equations (PDEs). The literature on the subject is truly enormous and we make no attempt to give a comprehensive review here. We only mention two recent relevant papers by Himonas et al \cite{Han_Himonas_2011}, \cite{Himonas_2011} where the interested reader can find further references on analytic and Gevrey regularity properties for KdV-type equations. In fact, we are interested in a much stronger effect of formation of meromorphic solution out of nonsmooth data. More specifically, in the current paper, we are concerned with the following problem. %-----------------------------------Problem 1-------------------------------------------------- \begin{problem} \label{pb1} Given the Cauchy problem for the KdV equation\footnote{% We use $z$ instead of $x$ for the spatial variable as it will frequently be complex.} \begin{equation} \begin{cases} \partial _{t}u-6u\partial _{z}u+\partial _{z}^{3}u=0 \\ u|_{t=0}=q% \end{cases}% , \label{eq1.1} \end{equation}% describe the largest possible class of (non-smooth) initial data $q$ which evolve into functions $u(z,t)$ meromorphic with respect to $z$ for any $t>0$. \end{problem} Meromorphic (or, more generally, analytic) solutions have of course been intensively studied since the boom around integrable systems started in the late 60s. A\ pure soliton (reflectionless) solution, historically the first explicit solution, is meromorphic on the whole complex plane having infinitely many double poles. This fact is of course a trivial observation immediately following from the explicit formula for multisoliton solutions. We emphasize that how those poles interact is not obvious at all. This question was raised back in earlier 70s by Kruskal and has been followed up by many. We refer the interested reader to a particularly influential 1977 paper \cite{Air77} by Airault-McKean-Moser and recent Bona-Weissler \cite% {Bona09} and the literature cited therein. More complicated examples of explicit solutions include algebraric, rational, meromorphic simply periodic, elliptic, etc. (see, e.g. \cite{AblSatJMP78}, \cite{Birnir87}, \cite{GUW06}, \cite{AktMee06} and the literature cited therein). All these examples are of course very specific and in addition those $q$'s are already meromorphic (i.e. smooth on the real line). Although Problem \ref{pb1} is not addressed in those papers but they demonstrate the importance of meromorphic solutions. Through the paper we deal with initial data subject to %--------------------------------------------Hypothesis 1.1------------------------------------ \begin{condition} \label{hyp1.1} $q$ is real and $L_{\limfunc{loc}}^{1}$ such that \begin{enumerate} \item (semiboundedness from below) \begin{equation} \label{Cond1} \inf\func{Spec}\left(-\partial_x^2+q(x)\right)=-h_0^2 \end{equation} with some $h_0\ge0$. \item (subexponential decay at $+\infty $) For $x$ large enough \begin{equation} \int_{x}^{\infty }\left\vert q\right\vert \leq C_{q}e^{-cx^{\varepsilon }} \label{Cond2} \end{equation}% with some positive $C_{q},c,\varepsilon $. \end{enumerate} \end{condition} We assume that the constants $c,\varepsilon $ in \eqref{Cond2} are chosen optimal. Note that the set of such functions is very large. Indeed, in terms of $q$ itself, Condition \eqref{Cond1} is satisfied if \begin{equation} \limfunc{Sup}\limits_{x}\int_{x-1}^{x}\max \left( -q,0\right) <\infty , \label{Cond_q} \end{equation}% i.e. $q$ is essentially bounded from below \cite{Glazman66}. The condition (% \ref{Cond_q}) cannot be improved since (\ref{Cond_q}) becomes also necessary for (\ref{Cond1}) if $q$'s are negative. Therefore, any $q$ subject to Hypothesis \ref{hyp1.1} is essentially bounded from below, has subexponential decay at $+\infty $ and arbitrary otherwise. Such functions can grow (arbitrarily fast) at $-\infty $ or look like a stock market (Gaussian white noise on a left half line) but still satisfy our hypothesis as long as they exhibit rapid decay (\ref{Cond2}) at $+\infty $. In spectral terms (\ref{Cond2}) implies that $\left( 0,\infty \right) $ belongs to the absolutely continuous spectrum of $-\partial _{x}^{2}+q(x)$. We now state our main results. %-----------------------------------Theorem 1.2----------------------------------------- \begin{theorem} \label{thm1.2} Under Hypothesis \ref{hyp1.1} with $\varepsilon \geq 1/2$ on the initial data $q$ in \eqref{eq1.1} , the problem \eqref{eq1.1} has an analytic in $z$ solution $u(z,t)$ given by \begin{equation} u(z,t)=-2\partial _{z}^{2}\log \det \left( 1+\mathbb{M}(z,t)\right) , \label{det_form} \end{equation}% where $\mathbb{M}(z,t)$ is a trace class operator-valued function constructed in Proposition \ref{pr4.1} below for any $t>0$. Moreover, for any $t>0$ \begin{enumerate} \item If $\varepsilon>1/2$ then $u(z,t)$ is meromorphic on $\mathbb{C}$. \item If $\varepsilon =1/2$ then $u(z,t)$ is meromorphic in the strip \begin{equation} \left\vert \Im z\right\vert <\frac{9\sqrt{2}}{8}c\sqrt{t} \label{strip} \end{equation}% where $c$ is as in \eqref{Cond2}. \end{enumerate} \end{theorem} %-------------------------------Theorem 1.3--------------------------------- \begin{theorem} \label{thm1.3} Under Hypothesis \ref{hyp1.1} with $0<\varepsilon <1/2$ on the initial data $q$ in \eqref{eq1.1}, the operator-valued function $\mathbb{% M}(x,t)$ given in Proposition \ref{pr4.1} is trace class for any real $x$ and $t>0$ and \begin{equation*} \mathbb{M}(x,t)=\mathbb{M}^{\left( 1\right) }(x,t)+\mathbb{M}^{\left( 2\right) }(x,t), \end{equation*}% where $\mathbb{M}^{\left( 1\right) }(x,t)$ is meromorphic in $x$ and $% \mathbb{M}^{\left( 2\right) }(x,t)$ is Gevrey $G^{\frac{1}{2\varepsilon }-1}$ regular. If in addition $1+\mathbb{M}(x,t)$ is invertible for any real $x$ and $t>0$ then the problem \eqref{eq1.1} has a solution $u(x,t)$ given by \begin{equation} u(x,t)=-2\partial _{x}^{2}\log \det \left( 1+\mathbb{M}(x,t)\right) , \label{det_form1} \end{equation}% belonging to the Gevrey class $G_{\limfunc{loc}}^{\frac{1}{2\varepsilon }-1}$% . \end{theorem} Theorems \ref{thm1.2} and \ref{thm1.3} significantly improve our results in \cite{Ry11} which in turn improve Tarama \cite{Tarama04}. Theorems \ref% {thm1.2} and \ref{thm1.3} have also some important corollaries. We will come back to the relevant discussions in the last section when we have the necessary background. We only mention here that our approach is based on the Inverse Scattering Transform (IST)\ combined with pseudo-analytic continuation techniques developed by E.M. Dyn'kin (see e.g. \cite{Dyn76}, \cite{BorDyn93}) and we do not believe that any of the statements of Theorem % \ref{thm1.2} can be obtained by purely PDE techniques. The paper is organized as follows. In Section 2, for the reader's convenience we list our main notation and give the relevant preliminaries. In Section 3 we define a suitable reflection coefficient and investigate its properties which will play a central role in our consideration. The results of this section may have some independent interest. In Section 4 we give a brief review of the classical IST stated in terms of Hankel operators and further prepare to prove our main results in Section 5. Section 6, the last one, is devoted to discussions of our results and some corollaries which directly follow from them. It also contains some open problems. %--------------------------------Section 2: Notation and preliminaries--------------------------------- \section{Notation and Preliminaries} We adhere to standard terminology accepted in Analysis. Namely, $\mathbb{R}% _{\pm }\equiv[0,\pm \infty )$, $\mathbb{C}$ is the complex plane, \begin{equation*} \mathbb{C}_{\pm }=\left\{ z\in \mathbb{C}:\pm \Im z>0\right\} . \end{equation*}% Through the paper the subscript $\pm $ indicates objects (functions, operators, spaces, etc.) somehow related to $\mathbb{R}_{\pm }$ or $\mathbb{C% }_{\pm }$. The bar $\overline{z}$ denotes the complex conjugate of $z$. When appropriate, we write% \begin{equation*} y\eqsim x\text{ in place of \ }y=\limfunc{const}\cdot x \end{equation*}% and similarly whenever convenient \begin{equation*} y\lesssim _{a}x\text{ in place of }y\leq C_{a}x \end{equation*}% with some $C_{a}>0$ dependent on a parameter $a$ but independent of $x$. If $% C_{a}$ is an absolute constant we then write $y\lesssim x$.\ \ This will help us keep bulky formulas under control. We use $\left\Vert \cdot \right\Vert _{X}$ to denote the norm in a Banach (Hilbert) space $X$. We will need the Gevrey classes $G^{\alpha },\alpha >0,$ on $\mathbb{R}$ of all functions $f$: \begin{equation*} \left\Vert \partial _{x}^{n}f\right\Vert _{L^{\infty }}\lesssim _{f}Q_{f}^{n}\left( n!\right) ^{1+\alpha },n=0,1,2,... \end{equation*}% with some $Q_{f}>0.$ By \cite{BorDyn93}, Theorem 3, the statement $f\in G^{\alpha }$ is equivalent to the statement that $f$ admits a pseudo analytic extension to the whole complex plane such that \begin{equation} \left\vert \partial _{\overline{z}}f\right\vert \lesssim _{f}\exp \left\{ -Q\left\vert \Im z\right\vert ^{-\frac{1}{\alpha }}\right\} \label{lambda_bar} \end{equation}% with some $Q>0$. In a similar manner one introduces local Gevrey classes $G_{\limfunc{loc}% }^{\alpha }$. Next, $\mathfrak{S}_{2}$ denotes the Hilbert-Schmidt class% \begin{equation*} \mathfrak{S}_{2}=\left\{ A:\left\Vert A\right\Vert _{\mathfrak{S}% _{2}}^{2}\equiv\func{tr}\left( A^{\ast }A\right) <\infty \right\} \end{equation*}% and $\mathfrak{S}_{1}$ is the trace class: \begin{equation*} \mathfrak{S}_{1}=\left\{ A:\left\Vert A\right\Vert _{\mathfrak{S}_{1}}\equiv% \func{tr}\left( A^{\ast }A\right) ^{1/2}<\infty \right\} . \end{equation*} Note that $A\in\mathfrak{S}_1$ if and only if $A=A_1A_2$ with some $% A_1,A_2\in\mathfrak{S}_2$. Some other miscellaneous notation: $\chi _{S}\left( x\right) $ is the characteristic function of a set $S$, i.e. \begin{equation*} \chi _{S}\left( x\right) \equiv\left\{ \begin{array}{c} 1,x\in S \\ 0,x\notin S% \end{array}% \right. . \end{equation*}% In particular $\chi _{\pm }\equiv\chi _{_{\mathbb{R}_{\pm }}}$is the Heaviside function of $\mathbb{R}_{\pm }$. We also write \begin{equation*} \left. f\right\vert _{S}=\chi _{S}f. \end{equation*} The notation $H_{q}\equiv -\partial _{x}^{2}+q(x)$ for the Schr\"odinger operator on $L^{2}\left( \mathbb{R}\right) $ will be frequently used. %----------------------------------Section 3: The reflection coefficient and its analytic structure--------------------------------- \section{The reflection coefficient and its analytic structure} In this section we define a suitable reflection coefficient and investigate its properties which will play a central role in our consideration. The results of this section may have some independent interest. In the short-range scattering for the full line Schr\"{o}dinger operator, one typically introduces the right and left reflection coefficients $% R(\lambda ),L(\lambda )$ and the transmission coefficient $T(\lambda )$ as functions of the momentum $\lambda $ (see e.g. \cite{Deift79}). These quantities (also called transition coefficients) can also be properly defined in much larger spectral situations through Wronskians and/or Titchmarsh-Weyl $m$-functions (see e.g. \cite{GNP97,GS97}). Such extensions need not be unique. However, in our setting of step-like potentials decaying at $+\infty $, there is a natural candidate for the right reflection coefficient $R(\lambda )$. %----------------------------------------Definition 3.1------------------------------------------- \begin{definition}[\protect\cite{Ry11}] \label{def3.1} Let $q(x)$ be real, locally integrable such that $q\in L^{1}\left( \mathbb{R}_{+}\right) $ and $-\partial _{x}^{2}+q(x)$ is in the limit point case at $-\infty $. Denoting by $R_{n}(\lambda )$ the right reflection coefficient (which is necessarily well defined) from the potential $q_{n}=q|_{(-n,\infty )}$, we call the weak limit (if it exists) \begin{equation} R(\lambda )\equiv \text{w-}\lim R_{n}(\lambda ),\;n\rightarrow \infty , \label{eq3.1} \end{equation}% the right reflection coefficient from the potential $q$. \end{definition} Note that one should not expect in \eqref{eq3.1}\ pointwise convergence as an explicit counterexample $q=$ $\chi _{-}$ readily shows. Uniform convergence in \eqref{eq3.1} is not available in general even in the short-range setting \cite{Deift79}. As shown in \cite{Ry11}, Lemma 5.4, the reflection coefficient introduced this way is well defined. The following statement will play a crucial role in our consideration. %------------------------------------------Proposition 3.2---------------------------------- \begin{proposition}[the analytic structure of the reflection coefficient] \label{pr3.2} Under Hypothesis \ref{hyp1.1}, the right reflection coefficient given by \eqref{eq3.1} exists and admits the representation \begin{equation} R(\lambda )=A(\lambda )+\frac{S(\lambda )G(\lambda )}{\lambda B(\lambda )} \label{Rep} \end{equation}% where functions $A,B,S,G$ have the properties \begin{enumerate} \item \label{it1} $A$ is an analytic on $\mathbb{C}^+\setminus[0,ih_0]$ function such that $\left\vert A \right\vert \le2$ on $\mathbb{R}$ and $% A(\lambda)=o\left(1/\lambda\right)$, $\lambda\to\infty$ along any ray in $% \mathbb{C}^+$ \item \label{it2} $B$ is the Blaschke product \begin{equation*} B(\lambda )=\prod_{k=1}^{N}\frac{\lambda -i\varkappa _{k}}{\lambda +i\varkappa _{k}} \end{equation*}% where real $\varkappa _{k}$'s are such that $\left\{ -\varkappa _{k}^{2}\right\} _{k=1}^{N}$ is the negative discrete spectrum of $H_{q_{+}}$% , $q_{+}\equiv q|_{\mathbb{R}_{+}}$ \item \label{it3} $\left\vert S(\lambda )\right\vert \leq 1$, $\lambda \in \mathbb{C}^{+}$ \item \label{it4} $G\in G^{\frac{1}{\varepsilon }-1}$ \item \label{it5} $\left\vert S(\lambda )G(\lambda )/\lambda \right\vert \leq 1$ a.e. on $\mathbb{R}$ \item \label{it6} If $R_{n}$ is as in Definition \ref{def3.1} then \begin{equation*} R_{n}(\lambda )=A_{n}(\lambda )+\frac{S(\lambda )G(\lambda )}{\lambda B(\lambda )} \end{equation*}% and \begin{equation*} A_{n}\rightarrow A,\;n\rightarrow \infty \end{equation*}% uniformly on any compact in $\mathbb{C}^{+}\setminus \lbrack 0,ih_{0}]$. \end{enumerate} \end{proposition} \begin{proof} Most of statements in Proposition \ref{pr3.2} (save \eqref{it4}) are proven in \cite{Ry11} and we restrict ourselves to some comments only. Note first that Condition 1 of Hypothesis \ref{hyp1.1} implies that $-\partial _{x}^{2}+q(x)$ is in the limit point case at $-\infty $ (see, e.g. \cite% {ClarkGeszt03} for complete results on this matter). Splitting \begin{equation} q=q_{-}+q_{+},\ \ q_{\pm }=q|_{\mathbb{R}_{\pm }} \label{split_q} \end{equation}% induces the representation \begin{equation*} R=\frac{T_{+}^{2}R_{-}}{1-R_{-}L_{+}}+R_{+} \end{equation*}% where $\pm $ label scattering quantities associated with $q_{\pm }$. The functions $T_{+},L_{+},R_{-}$ can be analytically continued into $\mathbb{C}% ^{+}$ and \begin{equation*} A\equiv \frac{T_{+}^{2}R_{-}}{1-R_{-}L_{+}} \end{equation*}% has properties \eqref{it1}, \eqref{it6}. For $R_{+}$, which is independent of $n$, we use the representation \cite{Deift79}, Theorem 2, \begin{equation*} R_{+}(\lambda )=\frac{T_{+}(\lambda )}{\lambda }G(\lambda ) \end{equation*}% where \begin{equation} G(\lambda )=\frac{1}{2i}\int_{-\infty }^{\infty }e^{-2i\lambda x}g(x)dx \label{eq6'.1} \end{equation}% with some $g$ obeying \begin{equation} \left\vert g(x)\right\vert \leq \left\vert q(x)\right\vert +\limfunc{const}% \int_{x}^{\infty }\left\vert q\right\vert . \label{Est_on_g} \end{equation}% Since $R_{+}(\lambda )$ is a reflection coefficient we have \eqref{it5}. Since $T_{+}$ is a transmission coefficient, \begin{equation*} T_{+}(\lambda )=\prod_{k=1}^{N}\frac{\lambda +i\varkappa _{k}}{\lambda -i\varkappa _{k}}\cdot S(\lambda )=B\left( \lambda \right) ^{-1}S(\lambda ) \end{equation*}% where $S$ is an outer function of $\mathbb{C}^{+}$: $\left\vert S(\lambda )\right\vert \leq 1$, $\lambda \in \mathbb{C}^{+}$. This proves \eqref{it2} and \eqref{it3}. The proposition is proven if we show \eqref{it4}. Due to \eqref{lambda_bar} we should demonstrate that $G$ admits a pseudo analytic extension the whole complex plane such that \begin{equation} \left\vert \partial _{\overline{\lambda }}G\right\vert \lesssim \exp \left\{ -Q\left\vert \Im \lambda \right\vert ^{-\frac{\varepsilon }{1-\varepsilon }% }\right\} \label{eq6''.1} \end{equation}% with some $Q>0$. There are a few explicit ways to construct pseudo analytic continuations (see e.g. \cite{Dyn76}, \cite{BorDyn93}, \cite{Tarama04}) producing different extensions. We modify the one used in \cite{Tarama04} to obtain a better $Q$ in \ref{eq6''.1}. Note that \begin{equation} G\left( \lambda \right) \eqsim \widehat{g}\left( 2\lambda \right) \label{Gg} \end{equation}% where $\widehat{g}$ is the Fourier transform of $g$ which due to % \eqref{Est_on_g} satisfies Condition 2 of Hypothesis \ref{hyp1.1} with some $% \widetilde{c}1$, $x_{n}=r^{n}$ and \begin{equation*} G_{n}(\lambda )=\int_{x_{n-1}}^{x_{n}}e^{-i\lambda x}g(x)dx. \end{equation*}% The formula \eqref{eq6''.1too} clearly defines an extension of $\widehat{g}% \left( \lambda \right) $ to complex $\lambda $. We next show that $% \widetilde{G}$ is uniformly bounded on $\mathbb{C}^{+}$. Bound $G_{n}$ first. By \eqref{est_g}% \begin{equation*} \left\vert G_{n}(\lambda )\right\vert \lesssim e^{\left\vert \Im \lambda \right\vert \cdot x_{n}}\int_{x_{n-1}}^{x_{n}}\left\vert g\right\vert \lesssim _{g}\exp \left\{ \left\vert \Im \lambda \right\vert \cdot x_{n}-% \widetilde{c}x_{n-1}^{\varepsilon }\right\} \end{equation*}% and one has \begin{equation} \left\vert \widetilde{G}(\lambda )\right\vert \lesssim _{g}\sum_{n\geq 1}\sum_{n\geq 1}\theta \left( r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{% \Im \lambda }{\widetilde{c}}\right) \exp \left\{ \left\vert \Im \lambda \right\vert \cdot x_{n}-\widetilde{c}x_{n-1}^{\varepsilon }\right\} . \label{eq6'''.1} \end{equation} In \eqref{eq6'''.1} many terms are in fact zero and nontrivial ones are subject to \begin{equation*} r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{\left\vert \Im \lambda \right\vert }{\widetilde{c}}\leq r. \end{equation*}% I.e. only nonzero terms in \eqref{eq6'''.1} are the ones obeying \begin{equation} x_{n}^{1-\varepsilon }\leq \frac{\widetilde{c}}{r^{\varepsilon +1}}\cdot \frac{1}{\left\vert \Im \lambda \right\vert }. \label{eq6'''.2} \end{equation} Under the condition \eqref{eq6'''.2}, for the argument of the exponential in % \eqref{eq6'''.1}, we have ($1/r<\delta <1$) \begin{align} \left\vert \Im \lambda \right\vert \cdot x_{n}-\widetilde{c}r^{-\varepsilon }x_{n}^{\varepsilon }& =\left( \left\vert \Im \lambda \right\vert \cdot x_{n}-\delta \widetilde{c}r^{-\varepsilon }x_{n}^{\varepsilon }\right) -(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon } \label{eq6'''.3} \\ & =\left\vert \Im \lambda \right\vert x_{n}^{\varepsilon }\left( x_{n}^{1-\varepsilon }-\delta \frac{\widetilde{c}}{\left\vert \Im \lambda \right\vert }\right) -(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon }. \notag \end{align}% By \eqref{eq6'''.2} the right hand side of \eqref{eq6'''.3} doesn't exceed \begin{eqnarray*} &&\left\vert \Im \lambda \right\vert x_{n}^{\varepsilon }\left( \frac{% \widetilde{c}}{\left\vert \Im \lambda \right\vert }\frac{1}{r^{\varepsilon +1}}-\frac{\widetilde{c}}{\left\vert \Im \lambda \right\vert }\frac{\delta }{% r^{\varepsilon }}\right) -(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon } \\ &=&-\widetilde{c}\left( \delta -\frac{1}{r}\right) x_{n-1}^{\varepsilon }-(1-\delta )x_{n-1}^{\varepsilon } \\ &<&-\limfunc{const}x_{n-1}^{\varepsilon }. \end{eqnarray*}% It follows now from this estimate and \eqref{eq6'''.1} that \begin{equation} \left\vert \widetilde{G}(\lambda )\right\vert \lesssim _{g}\sum_{n\geq 0}\exp \{-\limfunc{const}x_{n}^{\varepsilon }\}<\infty . \label{eq6iv.0} \end{equation}% Similarly one proves that all derivatives of $G$ are also bounded on $% \mathbb{C}^{+}$. It remains now to show \eqref{eq6''.1too}. One has \begin{align} \left\vert \partial _{\overline{\lambda }}\widetilde{G}\right\vert & \leq \sum_{n\geq 1}\theta ^{\prime }\left( r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{\left\vert \Im \lambda \right\vert }{\widetilde{c}}\right) \frac{% r^{\varepsilon +1}x_{n}^{1-\varepsilon }}{2\widetilde{c}}\left\vert G_{n}\right\vert \label{eq6iv.1} \\ & \lesssim _{g}\sum_{n\geq 1}x_{n}^{1-\varepsilon }\exp \{\left\vert \Im \lambda \right\vert \cdot x_{n}-\widetilde{c}r^{-\varepsilon }x_{n}^{\varepsilon }\}. \notag \end{align} Only terms subject to \begin{equation} \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert \Im \lambda \right\vert }% \leq x_{n}^{1-\varepsilon }\leq \frac{\widetilde{c}r^{-\varepsilon -1}}{% \left\vert \Im \lambda \right\vert } \label{eq6iv.2} \end{equation}% make a non trivial contribution to the series in \eqref{eq6iv.1}. The inequality \eqref{eq6iv.2} implies \begin{align} x_{n}& \geq \left( \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert \Im \lambda \right\vert }\right) ^{\frac{1}{1-\varepsilon }}, \notag \\ \intertext{or} x_{n}^{\varepsilon }& \geq \left( \frac{\widetilde{c}r^{-\varepsilon -2}}{% \left\vert \Im \lambda \right\vert }\right) ^{\frac{\varepsilon }{% 1-\varepsilon }}. \label{eq6iv.3} \end{align} Splitting the argument of the exponential in \eqref{eq6iv.1} same way as % \eqref{eq6'''.3} and using \eqref{eq6iv.3}, we have \begin{align*} \left\vert \Im \lambda \right\vert \cdot x_{n}^{\varepsilon }-\frac{% \widetilde{c}}{r^{\alpha }}x_{n}^{\alpha }& \leq \left\vert \Im \lambda \right\vert \cdot x_{n}^{\varepsilon }\left( \frac{\widetilde{c}% r^{-\varepsilon -1}}{\left\vert \Im \lambda \right\vert }-\frac{% r^{-\varepsilon }\delta \widetilde{c}}{\left\vert \Im \lambda \right\vert }% \right) -(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }% } \\ & =-x_{n}^{\varepsilon }\widetilde{c}r^{-\varepsilon -1}(r\delta -1)-(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }} \\ & \leq -\left( \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert \Im \lambda \right\vert }\right) ^{\frac{\varepsilon }{1+\varepsilon }}% \widetilde{c}r^{-\varepsilon -1}(r\delta -1)-(1-\delta )\frac{\widetilde{c}% x_{n}^{\varepsilon }}{r^{\varepsilon }} \\ & -\frac{\widetilde{c}^{\frac{1}{1-\varepsilon }}}{r^{\frac{2\varepsilon +1}{% 1-\varepsilon }}}\frac{r\delta -1}{\left\vert \Im \lambda \right\vert ^{% \frac{\varepsilon }{1-\varepsilon }}}-(1-\delta )\frac{\widetilde{c}% x_{n}^{\varepsilon }}{r^{\varepsilon }}. \end{align*} Inserting this into \eqref{eq6iv.1} we obtain \begin{align} \left\vert \partial _{\overline{\lambda }}\widetilde{G}\right\vert & \lesssim _{g}\left( \sum_{n\geq 0}x_{n}^{1-\varepsilon }\exp \{-\limfunc{% const}x_{n}^{\varepsilon }\}\right) \cdot \exp \left\{ -\widetilde{Q}% \left\vert \Im \lambda \right\vert ^{-\frac{\varepsilon }{1-\varepsilon }% }\right\} \label{eq6v.0} \\ \widetilde{Q}& \equiv(r\delta -1)\frac{\widetilde{c}^{\frac{1}{1-\varepsilon }}}{r^{\frac{2\varepsilon +1}{1-\varepsilon }}}<(r\delta -1)\frac{c^{\frac{1% }{1-\varepsilon }}}{r^{\frac{2\varepsilon +1}{1-\varepsilon }}}. \label{eq6v.1} \end{align}% The series in \eqref{eq6v.0} is convergent and $\widetilde{G}\left( \lambda \right) $ is an pseudo analytic extension of $\widehat{g}\left( \lambda \right) $ from the real line to the upper half plane. Due to \eqref{Gg} we have found a pseudo analytic extension of $G$ subject to \eqref{eq6''.1} with $Q=2\widetilde{Q}$. This completes our proof. \end{proof} %----------------------------------------------Remark3.3------------------ \begin{remark} The representation \eqref{Rep} is not unique. It depends on the reference point in the splitting of \eqref{split_q}. This flexibility will be used later. \end{remark} %-------------------------REamrk 3.4------------------------------ \begin{remark} We have also had some flexibility in choosing $r$ and $\delta $ in % \eqref{eq6v.1} subject to $r>1$, $1/r<\delta <1$. The range for $Q=2% \widetilde{Q}$ given by \eqref{eq6v.1} is \begin{equation*} 00$ under Hypothesis \ref{hyp1.1}. This means that $\det \left( 1+\mathbb{M}(z,t)\right) $ is an invariant, i.e. it produces the same value in any basis in $L^{2}\left( \mathbb{R}_{+}\right) $. In the setting of step-like potentials, the Marchenko operator has been intensively studied in the Kharkov mathematical school by Hruslov, Kotlyarov and their students\footnote{% Remark that this school has been greatly infuenced by Marchenko himself and he remains to be its part.} (see, e.g. \cite{Hruslov76}, \cite{KhrKot94}). We also refer to Cohen \ \cite{Cohen1984}, Kappeler \cite{Kappeler86}, Venakides \cite{Ven86} (and the literature cited therein), and recent Egorova-Teschl \cite{ET11}. In all the above papers save \cite{ET11}, $q$'s are assumed to have a specific type of behavior at $-\infty $ (approaching either a constant or a periodic function) and fall off at $+\infty$. In \cite% {ET11}, the interesting case of two finite gap potentials fused together is considered. We summarize important properties of the Marchenko operator in the following (see \cite{Ry11} for details). %--------------------------Proposition 4.1--------------------------------------------------------- \begin{proposition}[The structure of the Marchenko operator] \label{pr4.1} Assuming Hypothesis \ref{hyp1.1}, let $\mathbb{M}(z,t)$ be the Marchenko operator associated with $q$ and let $A$ be as in Proposition \ref% {pr3.2}. Then for any $z\in \mathbb{R}$, $t>0$, \begin{equation} \mathbb{M}(z,t)=\mathbb{M}_{+}(z,t)+\mathbb{A}(z,t), \label{eq4} \end{equation}% where $\mathbb{M}_{+}(z,t)$ is the Marchenko operator associated with $% q_{+}=q|_{\mathbb{R}_{+}}$ and $\mathbb{A}(z,t)$ is a Hankel integral operator with the kernel \begin{equation*} \frac{1}{2\pi }\int_{\mathbb{R}+ih}e^{2i\lambda (\cdot )}\zeta _{z,t}(\lambda )A(\lambda )d\lambda ,\quad h>h_{0}. \end{equation*}% Furthermore, $\mathbb{A}(z,t)$ is an entire operator-valued function of trace class for any complex $z$ and $t>0$, continuous with respect to $q$ in the following sense: If $q_{1},q_{2}$ are two functions subject to Hypothesis \ref{hyp1.1} then \begin{equation*} \left\Vert \mathbb{A}_{1}(z,t)-\mathbb{A}_{2}(z,t)\right\Vert _{\mathfrak{S}% _{1}}\leq \frac{1}{4\pi h}\left\Vert \zeta _{z,t}(A_{1}-A_{2})\right\Vert _{L^{1}(\mathbb{R}+ih)} \end{equation*}% for any $z\in \mathbb{C}$, $t>0$. \end{proposition} Note that $\mathbb{M}(z,t)$ depends on $(z,t)$ through $\zeta _{z,t}$. %------------------------------------Section 5: Proof of the main theorem and discussions --------------------------------------- \section{Proof of the main results} With all the preparations done in the previous sections, the actual proofs will be quite short. It is convenient to conduct both proofs at a time. Note first that, by a trivial shifting, we may assume without loss of generality that $H_{q_{+}}$ has at most one bound state $-\varkappa ^{2}$. Consider the problem \eqref{eq1.1} with \begin{equation*} q_{n}(x)=% \begin{cases} q(x)\quad & ,\quad x\geq -n \\ 0 & ,\quad x<-n% \end{cases}% . \end{equation*} It is well-known that for such initial profiles\footnote{% So far we only know that the determinant exists in the Fredholm sense.} \begin{equation} u_{n}(z,t)=-2\partial _{z}^{2}\log \det \left( 1+\mathbb{M}_{n}(z,t)\right) . \label{eq5.1} \end{equation}% By Proposition \ref{pr4.1} \begin{equation*} \mathbb{M}(z,t)=\mathbb{M}_{+}(z,t)+\mathbb{A}(z,t)+\delta \mathbb{A}(z,t) \end{equation*}% where $\delta \mathbb{A}\equiv\mathbb{A}_{n}-\mathbb{A}$ is meromorphic in $% z $ for any $t>0$ and small in the $\mathfrak{S}_{1}$-norm for $n$ large enough. I.e. \begin{equation} \left\Vert \mathbb{M}_{n}(z,t)-\mathbb{M}(z,t)\right\Vert _{\mathfrak{S}% _{1}}\rightarrow 0,\quad n\rightarrow \infty . \label{eq5.2.1} \end{equation}% Therefore, $\mathbb{M}(z,t)\in \mathfrak{S}_{1}$ is proven if we show that $% \mathbb{M}_{+}(z,t)\in \mathfrak{S}_{1}$. Split \begin{equation*} \mathbb{M}_{+}(z,t)=\mathbb{M}_{1}^{+}(z,t)+\mathbb{M}_{2}^{+}(z,t) \end{equation*}% where $\mathbb{M}_{1}^{+}(z,t),\mathbb{M}_{2}^{+}(z,t)$ are the Hankel operators with the kernels \begin{equation*} c_{0}^{2}\zeta _{z,t}(i\varkappa )e^{-\varkappa (x+y)}, \end{equation*}% and \begin{equation*} \frac{1}{2\pi }\int_{-\infty }^{\infty }e^{i\lambda (x+y)}\zeta _{z,t}(\lambda )R_{+}(\lambda )d\lambda \end{equation*}% respectively. Here $c_{0}$ stands for the norming constant associated with the bound state $-\varkappa ^{2}$. The operator $\mathbb{M}_{1}(z,t)$ is rank 1 and clearly entire in $z$. Thus we only need to properly control $\partial _{z}^{n}\mathbb{M}_{2}^{+}(z,t)$ in the $\mathfrak{S}_{1}$-norm. Evaluate (so far formally) the kernel of $% \partial _{z}^{n}\mathbb{M}_{2}^{+}(z,t),n=0,1,2,...,$ by the Green formula applied to the strip $\mathbb{R}\times (0,\varkappa /2)$ and by Proposition % \ref{pr3.2} ($\lambda =\alpha +i\beta ,\partial _{\overline{\lambda }}=\frac{% 1}{2}(\partial _{\alpha }+i\partial _{\beta })$) \begin{align} \frac{1}{2\pi }\int_{-\infty }^{\infty }e^{i\lambda (x+y)}(2i\lambda )^{n}\zeta _{z,t}(\lambda )R_{+}(\lambda )d\lambda & =\int_{\mathbb{R}% +i\varkappa /2}e^{i\lambda (x+y)}(2i\lambda )^{n}\left( B^{-1}SG\right) (\lambda )\frac{d\lambda }{2\pi } \notag \\ & \quad +2i\int_{0}^{\varkappa /2}d\beta \int d\alpha \;e^{i\lambda (x+y)}F(\alpha ,\beta ) \label{eq5.3.1} \\ & \equiv H_{1}(x+y)+H_{2}(x+y), \notag \end{align}% where \begin{equation*} F(\alpha ,\beta )\equiv \frac{1}{2\pi }\zeta _{z,t}(\lambda )(2i\lambda )^{n-1}\frac{S(\lambda )}{B(\lambda )}\partial _{\overline{\lambda }% }G(\alpha ,\beta ). \end{equation*} Due to the rapid decay of $e^{8i\lambda^3t}$ as $\lambda\to\infty$ along $% \mathbb{R}+ih$, the function $F(\alpha,\beta)$ is subject to the conditions of Proposition \ref{pr4.1} and hence the integral operator with kernel $H_1$ is trace class. Our analysis of the integral operator with kernel $H_2$ is based upon the following lemma. %-------------------------------------------------Lemma 5.1--------------------------------- \begin{lemma} \label{lem5.1} Let $F(\alpha ,\beta )$ be such that for some $h>0$ \begin{equation} \int_{0}^{h}\left( \int_{-\infty }^{\infty }\left\vert F\left( \alpha ,\beta \right) \right\vert d\alpha \right) \frac{d\beta }{\beta }<\infty . \label{star} \end{equation}% Then the integral Hankel operator $\mathbb{H}$ with the kernel ($\lambda =\alpha +i\beta $) \begin{equation*} H(x)=\int_{0}^{h}d\beta \int_{-\infty }^{\infty }\frac{d\alpha }{2\pi }% e^{i\lambda x}F(\alpha ,\beta ) \end{equation*}% is trace class and \begin{equation*} \left\Vert \mathbb{H}\right\Vert _{\mathfrak{S}_{1}}\leq \frac{1}{2}% \int_{0}^{h}\frac{d\beta }{\beta }\int_{-\infty }^{\infty }d\alpha \left\vert F(\alpha ,\beta )\right\vert . \end{equation*} \end{lemma} \begin{proof} We have \begin{align} H(x+y)& \eqsim \int_{0}^{h}d\beta e^{-\beta x}\int_{-\infty }^{\infty }d\alpha e^{i\alpha x}F(\alpha ,\beta ) \notag \\ & =\int_{0}^{h}e^{-\beta (x+y)}\widehat{F_{\beta }}(x+y)dx, \label{eq5.2.1too} \\ \widehat{F_{\beta }}(x+y)& \equiv\int_{-\infty }^{\infty }e^{i\alpha (x+y)}F(\alpha +\beta )d\alpha \notag \\ & \eqsim \widehat{F_{\beta }^{1/2}}\ast \widehat{F_{\beta }^{1/2}}(x+y) \notag \\ & =\int_{-\infty }^{\infty }F_{\beta }^{1/2}(x+s)F_{\beta }^{1/2}(y-s)ds. \label{eq5.2.2} \end{align} Here we have used the convolution theorem. Inserting \eqref{eq5.2.2} into % \eqref{eq5.2.1too} implies that \begin{equation} \mathbb{H}=\int_{0}^{h}\mathbb{H}_{\beta ,1}\mathbb{H}_{\beta ,2}d\beta, \label{eq5.2.3} \end{equation}% where $\mathbb{H}_{\beta ,1}$ and $\mathbb{H}_{\beta ,2}$ and integral (but not Hankel) operators on $L^{2}\left( \mathbb{R}\right) $ with the kernels \begin{align*} H_{\beta ,1}(x,s)& =\chi (x)e^{-\beta x}\widehat{F_{\beta }^{1/2}}(x+s), \\ H_{\beta ,2}(s,y)& =\chi (y)e^{-\beta y}\widehat{F_{\beta }^{1/2}}(y-s) \end{align*}% respectively. It follows from \eqref{eq5.2.3} that \begin{equation} \label{eq5.3.1too} \left\Vert \mathbb{H} \right\Vert_{\mathfrak{S}_1} \le \int_0^h \left\Vert \mathbb{H}_{\beta,1} \right\Vert_{\mathfrak{S}_2} \cdot\left\Vert \mathbb{H}% _{\beta,2} \right\Vert_{\mathfrak{S}_2}d\beta. \end{equation} Evaluate now the Hilbert-Schmidt norms of $\mathbb{H}_{\beta ,1}$ and $% \mathbb{H}_{\beta ,2}$. By the Plancherel equation we have \begin{align*} \left\Vert \mathbb{H}_{\beta ,1}\right\Vert _{\mathfrak{S_{2}}}^{2}& =\int_{0}^{\infty }\int_{-\infty }^{\infty }\left\vert H_{\beta ,1}(x,s)\right\vert ^{2}ds\;dx \\ & =\int_{0}^{\infty }dxe^{-2\beta x}\int_{-\infty }^{\infty }ds\left\vert \widehat{F_{\beta }^{1/2}}(x+s)\right\vert ^{2} \\ & =\frac{\left\Vert F_{\beta }^{1/2}\right\Vert _{L^{2}\left( \mathbb{R}% \right) }^{2}}{2\beta }=\frac{1}{2\beta }\left\Vert F_{\beta }\right\Vert _{L^{1}\left( \mathbb{R}\right) }. \end{align*}% That is \begin{equation*} \left\Vert H_{\beta ,1}\right\Vert _{\mathfrak{S}_{2}}\leq \frac{1}{\sqrt{% 2\beta }}\left\Vert F_{\beta }\right\Vert _{L^{1}\left( \mathbb{R}\right) }^{1/2}. \end{equation*}% Similarly, \begin{equation*} \left\Vert H_{\beta ,2}\right\Vert _{\mathfrak{S}_{2}}\leq \frac{1}{\sqrt{% 2\beta }}\left\Vert F_{\beta }\right\Vert _{L^{1}\left( \mathbb{R}\right) }^{1/2} \end{equation*}% and \eqref{eq5.3.1too} yields \begin{equation*} \left\Vert \mathbb{H}\right\Vert _{\mathfrak{S}_{1}}\leq \frac{1}{2}% \int_{0}^{h}\left\Vert F_{\beta }\right\Vert _{L^{1}}\frac{d\beta }{\beta }. \end{equation*}% The lemma is proven. \end{proof} Let us find suitable bounds on $\mathbb{R}\times \lbrack 0,\varkappa /2]$ for the functions involved in $F$: \begin{align*} \left\vert \zeta _{z,t}(\lambda )\right\vert & =\left\vert e^{8i\lambda ^{3}t+2i\lambda z}\right\vert =e^{8\beta ^{3}t-2\beta \Re z}\cdot e^{-24\beta t\alpha ^{2}-2\alpha \Im z} \\ & \leq e^{\varkappa (\varkappa ^{2}t+\left\vert z\right\vert )}\cdot \exp \left\{ -\left( \sqrt{24\beta t}\alpha +\frac{\Im z}{\sqrt{24\beta t}}% \right) ^{2}+\frac{\Im ^{2}z}{24\beta t}\right\} , \\ \left\vert \lambda ^{n-1}B^{-1}(\lambda )S(\lambda )\right\vert & \lesssim \left( \left\vert \alpha \right\vert +\beta \right) ^{n-1}, \\ \left\vert \partial _{\overline{\lambda }}G\right\vert & \lesssim _{q_{+}}\exp \left\{ -Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}\right\} . \end{align*} Thus \begin{equation} \left\vert F(\alpha ,\beta )\right\vert \lesssim _{q_{+}}e^{\varkappa \left( \varkappa ^{2}t+\left\vert z\right\vert \right) }\left( \left\vert \alpha \right\vert +\beta \right) ^{n-1}e^{-\left( \sqrt{24\beta t}\alpha +\frac{% \Im z}{\sqrt{24\beta t}}\right) ^{2}}\exp \left\{ \frac{\Im ^{2}z}{24t}\beta ^{-1}-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}\right\} . \label{eq5.4.0} \end{equation} To prove Theorem \ref{thm1.2} we only need to consider the case $n=1$. We have \begin{multline} \label{eq5.4.1} \int_{0}^{\varkappa /2}\frac{d\beta }{\beta }\int_{-\infty }^{\infty }\left\vert F\left( \alpha ,\beta \right) \right\vert d\alpha \\ \lesssim _{z,t,q_{+}}\int_{0}^{\varkappa /2}\beta ^{-3/2}\exp \left\{ \frac{% \Im ^{2}z}{24t}\beta ^{-1}-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }% }\right\} d\beta . \end{multline} So $F$ is subject to the condition of Lemma \ref{lem5.1} if the integral in % \eqref{eq5.4.1} converges, which depends on $\varepsilon$ and $\Im z$. \begin{itemize} \item[\protect\underline{Case 1}.] $\varepsilon >1/2$. Then\footnote{% We assume $\varepsilon <1$. If $\varepsilon \geq 1$ then Theorem \ref{thm1.2} is trivial.} $\frac{\varepsilon }{1-\varepsilon }>1$ and the right hand side of \eqref{eq5.4.1} is finite for any $z\in \mathbb{C}$. This means that $% \mathbb{M}_{+}(z,t)$ is an entire $\mathfrak{S}_{1}$-valued function for any $t>0$ and due to \eqref{eq5.2.1}, we can pass to the limit in \eqref{eq5.1} as $n\rightarrow \infty $ by standard properties of infinite determinants (see e.g. \cite{GGK00}). This proves \eqref{it1} in Theorem \ref{thm1.2}. \item[\protect\underline{Case 2}.] $\varepsilon =1/2$. Then $\frac{% \varepsilon }{1-\varepsilon }=1$ and the right hand side of \eqref{eq5.4.1} converges if and only if \begin{equation*} \frac{\Im ^{2}z}{24t}-Q<0 \end{equation*}% or when \begin{equation*} \left\vert \Im z\right\vert <\sqrt{12Q}\cdot \sqrt{t}. \end{equation*}% Choosing the maximum possible value of $Q$ in \eqref{eq7.1} we get \begin{equation*} \left\vert \Im z\right\vert <\frac{9\sqrt{2}}{8}c\sqrt{t} \end{equation*}% and \eqref{it2} of Theorem \ref{thm1.2} follows. Thus, Theorem \ref{thm1.2} is proven. \item[\protect\underline{Case 3}.] $0<\varepsilon <1/2$. Then $\frac{% \varepsilon }{1-\varepsilon }<1$ and \eqref{eq5.4.1} clearly diverges for any $\Im z\neq 0$ and our method fails to establish analyticity and we have to go back to \eqref{eq5.4.0} and analyze it for any natural $n$. Expanding $% \left( \left\vert \alpha \right\vert +\beta \right) ^{n-1}$ in % \eqref{eq5.4.0} by the binomial formula we have \begin{multline} \label{eq5.6.1} \int_{0}^{\varkappa /2}\frac{d\beta }{\beta }\int_{-\infty }^{\infty }F(\alpha ,\beta )d\alpha \\ \lesssim _{z,t,q_{+}}\sum_{k=0}^{n-1}\binom{n-1}{k}\int_{0}^{\varkappa /2}d\beta \beta ^{k-1}e^{-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }% }}\int_{0}^{\infty }\alpha ^{n-k-1}e^{-24\beta t\alpha ^{2}}d\alpha . \end{multline}% Reducing the inner integral in \eqref{eq5.6.1} to the gamma function% \footnote{% Recall $\Gamma (z)=\displaystyle\int_{0}^{\infty }\alpha ^{z-1}e^{-\alpha }d\alpha $.}, \begin{align} \eqref{eq5.6.1}& =\sum_{k=0}^{n-1}\binom{n-1}{k}\int_{0}^{\varkappa /2}d\beta \beta ^{k-1}e^{-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }% }}\cdot \frac{1}{(3\beta t)^{\frac{n-k}{2}}}\Gamma \left( \frac{n-k}{2}% \right) \notag \\ & \lesssim \sum_{k=0}^{n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}\Gamma \left( \frac{n-k}{2}\right) \int_{0}^{1}d\beta \beta ^{\frac{3k-n}{2}-1}e^{-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}}. \label{eq5.7.1} \end{align}% Introducing in the last integral the new variable $s=\beta ^{-\frac{% \varepsilon }{1-\varepsilon }}$ and setting $\gamma \equiv\frac{% 1-\varepsilon }{2\varepsilon }>1/2$ we get \begin{align} \int_{0}^{1}d\beta \beta ^{\frac{3k-n}{2}-1}e^{-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}}& =\frac{1-\varepsilon }{\varepsilon }\int_{1}^{\infty }s^{\frac{\varepsilon -1}{\varepsilon }\left( \frac{3k-n}{2}-1\right) -1}e^{-Qs}ds \notag \\ & \lesssim \int_{1}^{\infty }s^{\gamma (n-3k)-1}e^{-Qs}ds \notag \\ & \lesssim Q^{-\gamma (n-3k)+1}\int_{Q}^{\infty }s^{\gamma (n-3k)}e^{-s}ds. \label{eq5.7.2} \end{align}% The behavior of the last integral depends on the sign of $\omega _{k}\equiv\gamma (n-3k)$. If $\omega _{k}\geq 0$, i.e. $3k\leq n$, then \begin{align*} J_{k}& \equiv \int_{Q}^{\infty }s^{\omega _{k}}e^{-s}ds \\ & \leq \int_{0}^{\infty }s^{\omega _{k}-1}e^{-s}ds=\Gamma (\omega _{k}) \\ & =\Gamma (\gamma (n-3k)). \end{align*}% If $\omega _{k}<0$, i.e. $3k>n$, then \begin{equation*} J_{k}\leq Q^{\omega _{k}-1}\int_{Q}^{\infty }e^{-s}ds\leq Q^{\omega _{k}-1}. \end{equation*} Splitting the sum in \eqref{eq5.7.1} accordingly, we see that the right hand side of \eqref{eq5.7.1} is dominated by \begin{multline} \label{eq5.8.1} \sum_{0\leq 3k\leq n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}Q^{-\omega _{k}}\Gamma \left( \frac{n-k}{2}\right) \Gamma \left( \omega _{k}\right) +\sum_{n<3k\leq 3n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}\Gamma \left( \frac{n-k}{2}\right) \\ \equiv S_{1}+S_{2}. \end{multline} Analyze now $S_{1}$ and $S_{2}$. For $S_{1}$ we have \begin{align} S_{1}& \leq \Gamma \left( \frac{n}{2}\right) \Gamma (\gamma n)\sum_{0\leq 3k\leq n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}Q^{-\omega _{k}} \notag \\ & \leq \left( Q^{2\gamma }+\frac{1}{\sqrt{3t}Q^{\gamma }}\right) ^{n}\Gamma \left( \frac{n}{2}\right) \Gamma (\gamma n). \label{eq5.9.1} \end{align}% For $S_{2}$ we obtain \begin{equation*} S_{2}\leq \left( 1+\frac{1}{\sqrt{3t}}\right) ^{n}\Gamma \left( \frac{n}{3}% \right) \end{equation*}% and hence the contribution from $S_{2}$ to \eqref{eq5.7.2} produces a real analytic function. On the other hand, as it easily follows from % \eqref{eq5.9.1}, the contribution from $S_{1}$ produces a function from $% G^{\gamma -1/2}=G^{\frac{1}{2\varepsilon }-1}$. Thus we have proven that if $% 0<\varepsilon <1/2$ then \begin{equation*} \mathbb{M}_{+}(x,t)=\mathbb{M}_{+}^{\left( 1\right) }(x,t)+\mathbb{M}% _{+}^{\left( 2\right) }(x,t) \end{equation*}% where $\mathbb{M}_{+}^{\left( 1\right) }(x,t)$ is a real analytic $\mathfrak{% S}_{1}$-valued function and $\mathbb{M}_{+}^{\left( 2\right) }(x,t)$ is a $% \mathfrak{S}_{1}$-valued function from the Gevrey class $G^{\frac{1}{% 2\varepsilon }-1}$. Thus, we can pass to the limit as before. The limiting function is from the Gevrey class $G^{\frac{1}{2\varepsilon }-1}$ if $\det (1+\mathbb{M}(x,t))$ doesn't vanish for all $x\in \mathbb{R}$. The latter occurs if $1+\mathbb{M}(x,t)$ is invertible on $\mathbb{R}$ for any $t>0$. \end{itemize} Theorem \ref{thm1.3} is proven. \section{Discussions, corollaries, and open problems} \subsection{Discussions} %---------------------------Remark 6.1---------------------- \begin{remark} Theorem \ref{thm1.2} improves our main result from \cite{Ry11} where $% \mathbb{M}\left( x,t\right) \in \mathfrak{S}_{1}$ was not proven and only real analyticity of $u\left( x,t\right) $ was obtained. The main idea of \cite{Ry11} is to put together the analytic continuation arguments of \cite% {Ryb10} to treat initial data on $\mathbb{R}_{-}$ and Tarama's approach from \cite{Tarama04} to handle the data on $\mathbb{R}_{+}$. As far as we know the solution to Problem \ref{pb1} given in \cite{Tarama04} was best known back then. The main result of \cite{Tarama04} says that $u\left( x,t\right) $ is real analytic under the following conditions: $q$ is real and $L_{% \limfunc{loc}}^{2}$ such that \begin{equation*} \int_{-\infty }^{\infty }\left( 1+\left\vert x\right\vert \right) \left\vert q\left( x\right) \right\vert dx<\infty \end{equation*}% and for $x$ large enough there are positive $C_{q},c$ so that% \begin{equation*} \int_{x}^{\infty }\left\vert q\right\vert ^{2}\leq C_{q}e^{-cx^{1/2}}. \end{equation*}% Note that these conditions are much stronger than Hypothesis \ref{hyp1.1}. The techniques used in \cite{Tarama04} are also based upon the (classical) IST but his analysis relies on the properties of the Airy function as opposed to ours which is based on analytic and pseudo-analyitc continuations. The latter appears particularly well-suited for addressing Problem \ref{pb1} and consequently significantly less involved.\bigskip \end{remark} %------------------------------------Remark 6.2------------------- \begin{remark} It is proven in \cite{Deift79}, Theorem 7.2 that if $q$ is analytic in the strip $\left\vert \func{Im}z\right\vert 0$ in any strip around the real line accumulating only to infinity. By general theorems \cite{Steinberg69} on families of compact meromorphic operators these poles continuously depend on $t$ and hence may appear or disappear only on the boundary of analyticity of $u\left( x,t\right) $ (including infinity). \end{remark} \subsection{Corollaries} The following statement is a direct consequence of the analyticity of $% u\left( z,t\right) $ for $t>0$. \begin{corollary} \label{Corollary'}Under conditions of Theorem \ref{thm1.2} the solution $% u\left( z,t\right) $ can not vanish on an open set for any $t>0$ unless $q$ is identically zero. \end{corollary} This quickly recovers and improves a number of unique continuation results due to Zhang \cite{Zhang92}. E.g., one of the main results of \cite{Zhang92} says that $u\left( x,t\right) $ cannot have compact support at two different moments unless it vanishes identically. The techniques of \cite{Zhang92} rely upon the classical IST (coupled with some Hardy space arguments) and are valid under certain decay and regularity conditions on $q$. \begin{corollary} \label{Corol2} The class of (nonsmooth) initial data $q$ such that \begin{equation} \int_{-\infty }^{\infty }e^{c\left\vert x\right\vert ^{\varepsilon }}\left\vert q\left( x\right) \right\vert dx<\infty \text{ for some }% c,\varepsilon >0\text{ } \label{exp_decay} \end{equation}% is not preserved under the KdV flow. \end{corollary} \begin{proof} Assume that for some $t=t_{0}$ the function $u\left( x,t_{0}\right) $ is subject to (\ref{exp_decay}). Since the KdV equation is invariant under the transformation $\left( x,t\right) \rightarrow \left( -x,-t\right) $, the solution $u_{0}\left( x,t\right) $ to the problem (\ref{eq1.1}) with the initial data $q_{0}\left( x\right) =u\left( -x,t_{0}\right) $, by Theorems % \ref{thm1.2}, \ref{thm1.3}, will be at least smooth for any $t>0$. But $% u_{0}\left( x,t_{0}\right) =q\left( x\right) $ forcing original $q$ to be smooth too. \end{proof} Corollary \ref{Corol2}, in turn, implies that under the KdV flow neither an exponential decay at $-\infty $ nor smoothness persist in general. Note in this connection that issues related to persistence of regularity are also very important and have been extensively studied but we don't touch on this here. The explicit formula (\ref{det_form}), which was used to derive our analyticity results, does have some practical value. E.g. it implies that the large time asymptotic behavior of $u\left( x,t\right) $ is completely determined by the measure $\rho (\lambda )$ in (\ref{eq4.1}) alone. This fact is so far rigorously proven for $q$'s tending to a negative constant or a periodic function at $-\infty $ and was used to obtain explicit expressions for the so-called asymptotic solitons (see, e.g. \cite{Hruslov76}% , \cite{Ven86}, and \cite{KhrKot94}). We plan to return to this important issue elsewhere. \subsection{Open problems} \begin{enumerate} \item We believe that under Hypothesis \ref{hyp1.1} our solutions $u\left( x,t\right) $ have no singularities on the real line for any $t>0$. If this held then the problem \eqref{eq1.1} would be globally well-posed under Hypothesis \ref{hyp1.1} only and no blow-up solution could develop. That is to say that $1+\mathbb{M}(x,t)$ is automatically invertible for any real $x$ and $t>0$ under Hypothesis \ref{hyp1.1} alone. This fact is quite easy if in (\ref{eq4.1}) the support of $\rho \left( \lambda \right) $ is rich enough (a set of uniqueness of an analytic function) or $\left\vert R\left( \lambda \right) \right\vert <1$ on any set of positive Lebesgue measure (see \cite% {Ry11}). The situation is much less trivial if $R\left( \lambda \right) $ in (\ref{eq4.1}) is unimodular for a.e. real $\lambda $ (i.e. $q$ is completely reflecting). An affirmative answer is given in \cite{GR11} for the case of $% q $ such that $q|_{\mathbb{R}_{+}}=0$ and $H_{q}\geq 0$ (absence of negative spectrum). To address the problem as stated one needs to show that $1+% \mathbb{M}(x,t)$ is invertible in the case when in (\ref{eq4.1}) $\rho \left( \lambda \right) $ is supported on a set $\left\{ \lambda _{n}\right\} \subset \mathbb{R}_{+}$ such that $\dsum \lambda _{n}<\infty $ and $% \left\vert R\left( \lambda \right) \right\vert =1$ a.e. on the real line. In term of the Schrodinger operator $H_{q}$ itself this means that the absolutely continuous spectrum of $H_{q}$ is simple and supported on $% \mathbb{R}_{+}$ but there is a rich embedded positive singular spectrum. Physically relevant examples can be constructed from the Gaussian white noise, Pearson sparse blocks, Kotani potentials, etc. \item We do not know much about the Banach (or Hilbert) space of meromorphic function to which $u\left( z,t\right) $ from \ref{thm1.2} belongs. It would be very interesting to find such spaces as this would give, among others, important norm estimates for $u\left( z,t\right) $ which our paper lacks. \item We (cautiously) conjecture that in Theorem \ref{thm1.3} $u\left( x,t\right) $ could be represented for any $t>0$ as a meromorphic function plus a small Gevrey regular function. We can in fact show that the trace norm of $\mathbb{M}^{\left( 2\right) }(x,t)$ from Theorem \ref{thm1.3} can be made small but it is not clear if after taking the $\det $ and then $\log $ the analytic and small Gevrey parts will still be separated. Of course, this question will immediately have an affirmative answer if under conditions of Theorem \ref{thm1.3} the solution $u\left( x,t\right) $ happens to be real analytic. Our methods however fail to yield such results. \end{enumerate} \section*{Acknowledgement} We are grateful to Fritz Gesztesy for valuable discussions. %---------------------------------------Bibliography------------------------------------------------------ %{acm}%{ieeetr} %{unsrt} \bibliographystyle{abbrv} \bibliography{bibryb} \end{document} ---------------1109272144876--