Content-Type: multipart/mixed; boundary="-------------0512011646771" This is a multi-part message in MIME format. ---------------0512011646771 Content-Type: text/plain; name="05-410.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-410.keywords" non-linear Schroedinger equation, potential, non-local non-linearity initial value problem, scattering, inverse scattering ---------------0512011646771 Content-Type: application/x-tex; name="sanwed.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sanwed.tex" \documentclass[12pt]{article} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{hyperref} \usepackage{amssymb,amsmath,amsthm} \usepackage{enumerate} %\usepackage{upref} %\usepackage{showkeys} %\setlength{\oddsidemargin}{0mm} %\setlength{\evensidemargin}{10mm} \setlength{\textwidth}{165mm} \setlength{\textheight}{218mm} \setlength{\topmargin}{6mm} %To get 1 inch margins use the following \setlength{\oddsidemargin}{-0mm} \setlength{\evensidemargin}{0mm} \setlength{\textwidth}{165mm} \setlength{\textheight}{228mm} \setlength{\topmargin}{-15mm} \newcommand{\sss}{\setcounter{equation}{0}} \newtheorem{theorem}{THEOREM}[section] \renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}} \newtheorem{lemma}[theorem]{LEMMA} \renewcommand{\thelemma}{\arabic{section}.\arabic{lemma}.} \newtheorem{corollary}[theorem]{COROLLARY} \renewcommand{\thecorollary}{\arabic{section}.\arabic{lemma}.} \newtheorem{prop}[theorem]{PROPOSITION} \renewcommand{\theprop}{\arabic{section}.\arabic{prop}.} \newtheorem{remark}[theorem]{REMARK} \renewcommand{\theremark}{\arabic{section}.\arabic{remark}.} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newtheorem{definition}[theorem]{DEFINITION} \renewcommand{\thedefinition}{\arabic{section}.\arabic{definition}.} \newtheorem{example}[theorem]{COUNTER-EXAMPLE} \renewcommand{\theexample}{\arabic{section}.\arabic{example}} %\newfont{\BBFONT}{msbm10 scaled 1200} %\newcommand{\ere}{\hbox{\BBFONT R}} %\newcommand{\ZETA}{\hbox{\BBFONT Z}} %\newcommand{\ese}{\hbox{\BBFONT S}} %\newcommand{\CE}{\hbox{\BBFONT C}} \newcommand{\ere}{ {\mathbb R}} \newcommand{\ZETA}{{\mathbb Z}} \newcommand{\ese}{{\mathbb S}} \newcommand{\CE}{{\mathbb C}} \newcommand{\ls}{L^2(\ese^{n-1})} %%%%macros%%%%%% \def\beq{\begin{equation}} \def\ene{\end{equation}} \def \ds {\displaystyle} \newcommand{\bull}{\hfill $\Box$} \def\qed{\ifhmode\unskip\nobreak\fi\ifmmode\ifinner \else\hskip5pt\fi\fi\hbox{\hskip5pt\vrule width4pt height6pt depth1.5pt\hskip1pt}} \def\1{L^{p+1}} \def\p{L^{ \frac{p}{p+1}}} \def\q{L^q} \def\P{L(P)} \def\m{\mathcal M} \def\var{\varphi} \newcommand{\Q}{{\mathbb H}_{Q}} \newcommand{\h}{{\mathbb H}^1} \begin{document} \baselineskip=20 pt \parskip 6 pt \title{The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schr\"odinger Equations with a Potential and a Non-Local Non-Linearity \thanks{ PACS classification scheme (2003): 03.65.Nk, 02.30.Zz, 02.30.Jr, 03.65.Db} \thanks{ Research partially supported by Universidad Nacional Aut\'onoma de M\'exico under Project PAPIIT-DGAPA IN 105799, and by CONACYT under Project P42553­F.}} \author{ Mar\'{\i}a de los \'Angeles Sandoval Romero and Ricardo Weder\thanks{ †Fellow Sistema Nacional de Investigadores.} \\Instituto de Investigaciones en Matem\'aticas Aplicadas y en Sistemas \\Universidad Nacional Aut\'onoma de M\'exico \\ Apartado Postal 20-726, M\'exico DF 01000 \\weder@servidor.unam.mx} \date{} \maketitle \begin{center} \begin{minipage}{5.75in} \centerline{{\bf Abstract}} \bigskip We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail the initial value problem for these equations. In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. We allow for a large class of unbounded potentials. We have no restriction on the growth at infinity of the positive part of the potential. We also construct the scattering operator in the case of potentials that go to zero at infinity. Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator. \end{minipage} \end{center} \newpage %%%%%%%%%%%%%%%% \section{Introduction} \sss In recent years there is a considerable interest in non-linear Schr\"odinger equations with a potential (NLSP) and a non-local non-linearity that is concentrated in a bounded region of space, for example, to model physical situations that appear in mesoscopic physics. In particular, in \cite{s1,s2,s3,s4,s5} the following equation was introduced to model a quantum capacitor (in \cite{s3} more general equations are discussed). \beq i\frac{\partial}{\partial t} u(x,t)= -\frac{d^2}{d x^2} u(x,t)+ V_0 \, u(x,t) + \lambda Q(u)\chi_{[b,c]}(x)\, u(x,t), \quad u(x,0)=\varphi(x), \label{1.1} \ene with, $x,t \in \ere$, and where we have set Planck's constant $\hbar$ equal to one and the mass equal to $1/2$. For any $O\in \ere, \chi_{O}$ is the characteristic function of $O$. The external potential, $V_0$, is a double barrier, $$ V_0(x)= \beta \left[\chi_{[a,b]}(x)+ \chi_{[c,d]}(x)\right],\qquad \beta > 0, $$ where, $ a < b 0$ there is a constant $K_\epsilon$ such that, \beq | (V_{0,2} \var,\var )| \leq \epsilon \| \var' \|_{L^2}^2+ K_\epsilon \|\var\|_{L^2}^2. \label{2.3b} \ene It follows that the quadratic form, \beq h(\varphi,\psi):=(\var',\psi')+(V_0\var,\psi), \,{\rm with \, domain},\,D(h):= \mathbb H^1 \cap D(Q), \label{2.5} \ene is closed and bounded from below. Let $H$ be the associated bounded-below, self-adjoint operator (see, \cite{ka1, rs2}). Then, \beq D(H)=\left\{\var \in D(h) :-\frac{d^2}{d x^2}\var + V \var \in L^2 \right\} \, {\rm and}\, H\var= -\frac{d^2}{d x^2}\var + V \var ,\, \var \in D(H). \label{2.6} \ene Let us take $N >0$ such that $H+N >1$ and let us denote $ D := \cap_{m=-\infty}^{\infty} D((H+N)^m)$. For any $s\in \ere$ let $\mathbb H_Q^s$ be the completion of $D$ in the norm, $\|(H+N)^{s/2}\|_{L^2}$. Notice that $ \Q^0=L^2, \mathbb H_Q^1 = D(\sqrt{H+N})=D(h)$ and that $ \mathbb H_Q^2= D(H)$. Observe that the following norm is equivalent to the norm of $\Q^1$, $$ \max\left[ \|\var\|_{\mathbb H^1},\,\|Q\var\|_{L^2}\right]. $$ Furthermore, if $V_{0,1}=0 $ the potential $V_0$ is a quadratic form bounded perturbation of $H_0$ with relative bound zero, and then, $\Q^1= \mathbb H^1$. In this case the $\Q^1$ solutions that we consider below are just $\mathbb H^1$ solutions. As $\Q^2=D(H)$, the following norm is equivalent to the norm of $\Q^2$, $$ \max \left[\|\var \|_{\Q^1}, \left\|\left(-\frac{d^2}{dx^2}+V_0\right)\var\right\|_{L^2} \right]. $$ The paper \cite{ka3} gives sufficient conditions in $V_{0} $ that assure that $ \Q^2 \subset \mathbb H^2$. Furthermore, if \beq \sup_{x\in \ere}\int_x^{x+1} |V_{0}(x)|^2\, dx < \infty, \label{2.19} \ene it follows from (\ref{2.3b}), and as $\|V_0 \var\|_{L^2}^2=(V_0^2\,\var, \var)$, and $(\var',\var') \leq \|H_0\var'\|^2+\|\var\|^2$, that $V_0$ is relatively bounded with respect to $H_0$ with relative bound zero, that is, for any $\epsilon >0$ there is a constant $K_\epsilon$ such that, \beq \| V_{0} \var \| \leq \epsilon \| H_0 \var \|_{L^2}+ K_\epsilon \|\var\|_{L^2}. \label{2.19b} \ene In this case (see \cite{ka1}, \cite{rs2}) $\Q^2 = \mathbb H^2$ and the $\Q^2$ solutions that we consider below are just $\mathbb H^2$ solutions. It follows from the functional calculus of self-adjoint operators that $H$ is bounded from $\Q^s$ to $\Q^{s-2}$ and that $e^{-itH}$ is a strongly-continuous unitary group on $\Q^s, s \in \ere$. Moreover, for any $\varphi \in \Q^s, s\in \ere, e^{-itH} \varphi \in C\left(\ere, \Q^s\right) \cap C^1\left(\ere, \Q^{s-2}\right)$ and, $$ i \frac{\partial}{\partial t} e^{-it H} \var = H e^{-it H} \var = e^{-it H} H \var, \quad \var \in \Q^s. $$ We introduce some further notation that we use below. Let $I=:[0,T],$ if $0 < T < \infty$ and $I= [0,\infty )$, if $T=\infty$. For any Banach space $\mathcal X$ we denote by $\mathcal X_R$ the closed ball in $\mathcal X$ with centre zero and radius $R$. If $ T < \infty$ we denote by $C(I,\mathcal X)$ the Banach space of continuous functions from $I$ into $\mathcal X$ and if $T=\infty$ we denote by $C_B(I, \mathcal X)$ the Banach space of continuous and bounded functions from $I$ into $\mathcal X$. For $T < \infty$, we define, $\mathcal N:= C(I,L^2)$ and $\mathcal N^j := C(I, \Q^j), j=1,2$. For functions $u(t,x)$ defined in $\ere^2$ we denote $u(t)$ for $u(t,\cdot)$. We study the initial value problem (\ref{2.1}) for $ t\geq 0$, but changing $t$ into $-t$ and taking the complex conjugate of the solution (time reversal) we also obtain the results for $t \leq 0$. By a $L^2$ solution on $I$ to (\ref{2.1}) we mean a function $u\in C(I,L^2)\cap C^1(I, \Q^{-2})$ that satisfies (\ref{2.1}). Multiplying both sides of (\ref{2.1}) (evaluated at $\tau$) by $e^{-i(t-\tau) H}$ and integrating in $\tau$ from zero to $t$ we obtain that, \beq u(t)=e^{-itH} \varphi + \frac{1}{i}\, G F(u),\, {\rm with}\, F(u):= (V_1 u,u)\, V_2 u \label{2.7} \ene and where, \beq G u := \int_0^t \, e^{-i(t-\tau)H} u(\tau)\, d\tau. \label{2.8} \ene Moreover, let $u \in C(I,L^2)$ be a solution to (\ref{2.7}). Then, it follows from (\ref{2.7}) that $ u \in C^1(I, \Q^{-2})$. We prove that $u$ solves (\ref{2.1}) taking the derivative of both sides of (\ref{2.7}). Hence, equations (\ref{2.1}) and (\ref{2.7}) are equivalent. We obtain our results below solving the integral equation (\ref{2.7}). In the next theorem we prove the existence of local solutions in $L^2$. \begin{theorem}\label{th2.1} Suppose that assumption A is satisfied and that $V_j \in L^\infty, j=1,2$. Then, for any $\varphi \in L^2$ there is $0 < T < \infty$, such that (\ref{2.1}) has a unique solution, $u \in C(I,L^2)$ with, $u(0)=\var$. $T$ depends only on $\|\varphi\|_{L^2}$. \end{theorem} \noindent {\it Proof:} We define, \beq \mathcal C(u): e^{-itH} \var +\frac{1}{i} G F(u). \label{2.9} \ene We will prove that we can take $R$ large enough, and $T$ so small, that $\mathcal C$ is a contraction on $\mathcal N_R$. It follows from Schwarz inequality that for any $u,v \in \mathcal N_R,$ \beq \|F(u)-F(v)\|_{\mathcal N} \leq C \,(\|u\|^2_{\mathcal N}+\|v\|^2_{\mathcal N})\, \| u-v\|_{\mathcal N} \leq C \,2\, R^2 \,\|u-v\|_{\mathcal N}. \label{2.10} \ene Then, as $e^{-itH}$ is unitary on $L^2$, \begin{gather} \|\mathcal C(u)\|_{\mathcal N}\leq \left[\|\var\|_{L^2} + C \, T \|u\|_{\mathcal N}^3 \right] \leq \left[\|\var\|_{L^2} + C \,T\, R^3 \right], \label{2.11} \\ \|\mathcal C(u)-\mathcal C(v)\|_{\mathcal N}\leq C T (\|u\|^2_{\mathcal N}+\|v\|^2_{\mathcal N})\, \| u-v \|_{\mathcal N} \leq C \,T\, 2\, R^2 \| u-v \|_{\mathcal N}. \label{2.12} \end{gather} Then, we can take $R,T$ such that $\|\var\|_{L^2} + C \,T\, R^3 \leq R$ and, $d:= C \,T\, 2\, R^2 < 1$, what makes $\mathcal C$ a contraction on $\mathcal N_R$. By the contraction mapping theorem \cite{rs1} $\mathcal C$ as a unique fixed point, $u$, in $ \mathcal N$ that is a solution to (\ref{2.7}). Suppose that there is another solution $v \in \mathcal N$. By the argument above, we have that $v(t)=u(t)$ for $ t \in [0,T_0]$ for some $ T_0 \leq T $. By iterating this argument we prove that $ v(t)=u(t), 0 \leq t \leq T$. \bull We now prove that the solution depends continuously on the initial data. \begin{theorem}\label{th2.2} Suppose that assumption A is satisfied and that $V_j \in L^\infty, j=1,2$. Then, the solution $u \in C([0,T],L^2),0< T < \infty,$ to (\ref{2.1}) with $u(0)=\var$, given by theorem \ref{2.1}, depends continuously on the initial value $\var$. In a precise way, let $\var_n \to \var$ strongly in $ L^2$. Then, for $n$ large enough, the solutions $u_n \in C([0,T],L^2)$ to (\ref{2.1}) with initial values $\var_n$ exist and $ u_n \to u $ in $C([0,T],L^2)$. \end{theorem} \noindent{\it Proof:} We first prove a local version of the theorem with $T$ replaced by a $T_0$ small enough . We define, \beq \mathcal C_n(u):= e^{-it H}\var_n+ \frac{1}{i} G F(u), \quad u \in \mathcal N(T_0) :=C([0,T_0],L^2). \label{2.13} \ene As $e^{-it H}\var_n \to e^{-it H}\var$ in $\mathcal N(T_0)$, for $n$ large enough $\mathcal C_n$ and $\mathcal C$ are contractions in $\mathcal N_R(T_0)$ with the same $R,T_0,d$. The unique fixed points, $u_n,u$ are solutions to (\ref{2.1}) in $C([0,T_0],L^2)$ with, respectively, $u_n(0)= \var_n, u(0)= \var$. Moreover, as $\mathcal C_n(u_n)-\mathcal C (u)= \mathcal C_n(u_n)-\mathcal C_n(u)+ \mathcal C_n(u)- \mathcal C(u)$, \beq \|u_n-u\|_{\mathcal N(T_0)}= \| \mathcal C_n(u_n)-\mathcal C(u)\|_{\mathcal N(T_0)} \leq d \| u_n-u \|_{\mathcal N(T_0)} + \|\var_n-\var \|_{L^2}, \label{2.14} \ene and as $ d < 1, u_n \to u$ in $\mathcal N(T_0)$. As the interval of existence of the solution given in theorem \ref{th2.1} depends only on the $L^2$ norm of the initial value, we can extend this argument, step by step, to the whole interval $[0,T]$. \bull \begin{remark}\label{rem2.3} {\rm let $T_m$ be the maximal time such that the solution $u$ given in theorem \ref{th2.1} can be extended to a solution $u \in C([0,T_m), L^2)$ with $u(0)=\var$. Then, if $T_m$ is finite we must have that $ \lim_{t \uparrow \infty}\|u(t)\|_{L^2}= \infty$. In other words, the solution exists for all times unless it blows up in the $L^2$ norm for some finite time. To prove this result suppose that $\|u(t)\|_{L^2}$ remains bounded as $t \uparrow T_m$. Then, by theorem \ref{th2.1} we can extend the solution continuously to $T_m+ \epsilon$ for some $\epsilon >0$, contradicting the definition of $T_m$. Another consequence of theorem \ref{th2.1} is that (\ref{2.1}) has at most one solution in $C(I, L^2)$ with $u(0)=\var$. Suppose, on the contrary, that there are two, $u_1,u_2$. Then, by theorem \ref{th2.1}, $u_1(t)=u_2(t), t \in [0,T_0]$ for some $0 0$, in contradiction with the definition of $T_m$. Then, $T_m=T$ and by continuity, $u_1(T)=u_2(T)$, completing the proof in the case $T < \infty$. A similar argument proves that if $ T=\infty$, $T_m$ can not be finite.} \end{remark} \bull We will use the uniqueness of $L^2$ solutions given by theorem \ref{2.1} and remark \ref{2.3} in the construction of the scattering operator in theorem \ref{th3.1} in section 3. We now study solutions in $\Q^1$. \begin{theorem}\label{th2.4} Suppose that assumption A holds, that $V_1$ satisfies (\ref{2.3}) and that $V_2 \in W_{1,\infty}$. Then, for any $\var \in \Q^1$ there is a $0 < T < \infty$ such that (\ref{2.1}) has a unique solution $ u \in C([0,T],\Q^1)$, with $u(0)= \var$. $T$ depends only on $\|\var\|_{\Q^1}$. \end{theorem} \noindent {\it Proof:} As $V_1$ satisfies (\ref{2.3}), it follows from (\ref{2.3b}) that, \beq |(V_1\var, \var)| \leq C \|\var\|_{\mathbb H^1}^2 ,\quad {\rm for \, all}\, \var \in \mathbb H^1. \label{2.15} \ene Then, defining $\mathcal C$ as in (\ref{2.9}) and as $V_2 \in W_{1,\infty}$, \begin{gather} \|\mathcal C (u)\|_{\mathcal N^1}\leq \|\var\|_{\Q^1}+ C \, T \|u\|_{\mathcal N^1}^3, \label{2.16} \\ \|\mathcal C (u) - \mathcal C (v) \|_{\mathcal N^1}\leq C\, T \left(\|u\|_{\mathcal N^1}^2+ \|v\|_{\mathcal N^1}^2\right)\, \|u-v\|_{\mathcal N^1}. \label{2.17} \end{gather} We take $R,T$ such that $\|\var\|_{\Q^1} + C \,T\, R^3 \leq R$ and, $d:= C \,T\, 2\, R^2 < 1$. By (\ref{2.16}) and (\ref{2.17}), with this choice $\mathcal C$ a contraction on $\mathcal N_R^1$ with contraction rate $d$. The unique fixed point, $u$, is a solution to (\ref{2.1}) with $ u(0)= \var$. We prove the uniqueness of the solution in $C([0,T],\Q^1)$ as in the proof of theorem \ref{th2.1}. \bull There is also continuous dependence of the solutions in $\Q^1$. \begin{theorem}\label{th2.5} Suppose that assumption A holds, that $V_1$ satisfies (\ref{2.3}) and that $V_2 \in W_{1,\infty}$. Then, the solution $u \in C([0,T],\Q^1),0 < T < \infty,$ to (\ref{2.1}) with $u(0)=\var$, given by theorem \ref{th2.4}, depends continuously on the initial value $\var$. In a precise way, let $\var_n \to \var,$ strongly in $ \Q^1$. Then, for $n$ large enough, the solutions $u_n \in C([0,T],\Q^1)$ to (\ref{2.1}) with initial values, $\var_n$ exist and $ u_n \to u $ in $C([0,T],\Q^1)$. \end{theorem} \noindent {\it Proof:} The theorem is proven as in the proof of theorem \ref{th2.2} replacing in the argument $\mathcal N(T_0)$ by $ C([0,T_0], \Q^1)$. \bull \begin{remark}\label{rem2.6}{\rm We prove as in remark \ref{rem2.3} that the solution to (\ref{2.1}) in $\Q^1$ exists for all $ t >0$ unless it blows up in the $\Q^1$ norm for some finite time and that theorem 2.4 implies that (\ref{2.1}) has at most one solution on $C(I,\Q^1)$.} \end{remark} \bull Let us now study solution in $\Q^2$. \begin{theorem}\label{th2.7} Suppose that assumption A holds, that $V_1$ satisfies (\ref{2.3}) and that $V_2 \in W_{2,\infty}$. Then, for any $\var \in \Q^2$ there is a $0 < T < \infty$ such that (\ref{2.1}) has a unique solution $ u \in C([0,T],\Q^2)$, with $u(0)= \var$. $T$ depends only on $\|\var\|_{\Q^2}$. \end{theorem} \noindent {\it Proof:} As $V_2 \in W_{2,\infty}$, for some constant $C$, \beq \|V_2 \var \|_{\Q^2} \leq C \|\var\|_{\Q^2} ,\quad {\rm for \, all}\, \var \in \Q^2. \label{2.20} \ene Then, with $\mathcal C$ defined as in (\ref{2.9}), \begin{gather} \|\mathcal C (u)\|_{\mathcal N^2}\leq \|\var\|_{\Q^2}+ C \, T \|u\|_{\mathcal N^2}^3, \label{2.21} \\ \|\mathcal C (u) - \mathcal C (v) \|_{\mathcal N^2}\leq C\, T \left(\|u\|_{\mathcal N^2}^2+ \|v\|_{\mathcal N^2}^2\right)\, \|u-v\|_{\mathcal N^2}. \label{2.22} \end{gather} Let $R,T$ be such that $\|\var\|_{\Q^2} + C \,T\, R^3 \leq R$ and, $d:= C \,T\, 2\, R^2 < 1$. Hence, it follows from (\ref{2.21}) and (\ref{2.22}) that $\mathcal C$ is a contraction on $\mathcal N_R^2$ with contraction rate $d$. The unique fixed point, $u$, is a solution to (\ref{2.1}) with $ u(0)= \var$. We prove the uniqueness of the solution in $C([0,T],\Q^2)$ as in the proof of theorem \ref{th2.1}. \bull \begin{theorem}\label{th2.8} Suppose that assumption A holds, that $V_1$ satisfies (\ref{2.3}) and that $V_2 \in W_{2,\infty}$. Then, the solution $u \in C([0,T],\Q^2), 0 < T < \infty,$ to (\ref{2.1}) with $u(0)=\var$, given by theorem \ref{th2.7}, depends continuously on the initial value $\var$. In a precise way, let $\var_n \to \var,$ strongly in $ \Q^2$. Then, for $n$ large enough, the solutions $u_n \in C([0,T],\Q^2)$ to (\ref{2.1}) with initial values $\var_n$ exist and $ u_n \to u $ in $C([0,T],\Q^2)$. \end{theorem} \noindent {\it Proof:} The theorem is proven as in the proof of theorem \ref{th2.2} replacing in the argument $\mathcal N(T_0)$ by $ C([0,T_0], \Q^2)$. \bull \begin{remark}\label{rem2.9}{\rm We prove as in remark \ref{rem2.3} that the solution to (\ref{2.1}) in $\Q^2$ exists for all $ t >0$ unless it blows up in the $\Q^2$ norm for some finite time and that theorem 2.7 implies that (\ref{2.1}) has at most one solution on $C(I,\Q^2)$.} \end{remark} \bull We now consider the problem of the regularity of solutions. Suppose that the conditions of theorems \ref{th2.1} and \ref{th2.4} are satisfied and that $\var \in \Q^1$. Then, by theorem \ref{th2.1}, (\ref{2.1}) has a unique $L^2$ solution and by theorem \ref{th2.4} a unique $\Q^1$ solution, both with initial value, $\var$. In the proposition below we prove that it is impossible that the $\Q^1$ solution blows-up before the $L^2$ solution. \begin{prop}\label{prop2.10} Suppose that assumption A holds that $V_1\in L^\infty$ and that $V_2 \in W_{1,\infty}$. Let $ u \in C([0,T],L^2),0 < T < \infty $ be a solution to (\ref{2.1}) with $u(0)=\var \in \Q^1$. Then, $u\in C([0,T],\Q^1)$. \end{prop} \noindent {\it Proof:} By theorem \ref{th2.4} there is a $ 0 < T_0 \leq T$ such that $u \in C([0,T_0],\Q^1)$. Let us denote, $ v:= \sqrt{H+N} u, 0\leq t \leq T_0$ . Multiplying both sides of (\ref{2.7}) by $\sqrt{H+N}$ we obtain that, \beq v(t)= e^{-it H}(\sqrt{H+N})\var + \frac{1}{i}\, G (V_1u, u)\left[(H+N)^{1/2} V_2 (H+N)^{-1/2}\right] v. \label{2.23} \ene Note that as $ V_2 \in W_{1,\infty}, \left[(H+N)^{1/2} V_2 (H+N)^{-1/2}\right]$ is a bounded operator in $L^2$. Equation (\ref{2.23}) is a linear equation for $v$, where $u$ is a fixed function in $\mathcal N$. Solving this equation in an interval $[T_0, T_0+ \Delta]$, with $\Delta $ small enough, we prove that $v(t) \in L^2$, for $T_0\leq t \leq T_0+\Delta$. Note that the length of $\Delta$ depends only on $\|u\|_{ \mathcal N}$. Repeating this argument, step by step, we prove that $ v \in C([0,T], L^2)$ and, in consequence, that $u \in C([0,T],\Q^1)$. \bull In the following proposition we prove regularity between $\Q^1$ and $\Q^2$ solutions. \begin{prop}\label{prop2.11} Suppose that assumption A holds, that $V_1$ satisfies (\ref{2.19}) and that $V_2 \in W_{2,\infty}$. Let $ u \in C([0,T],\Q^1), 0 < T < \infty$, be a solution to (\ref{2.1}) with $u(0)=\var \in \Q^2$. Then, $u\in C([0,T],\Q^2)$. \end{prop} \noindent {\it Proof:} By theorem \ref{th2.7} there is a $ 0 < T_0 \leq T$ such that $u \in C([0,T_0],\Q^2)$ and then, by (\ref{2.1}) $v:=\frac{\partial}{\partial t} u(t)\in C_B([0,T_0],L^2)$. Moreover, taking the derivative in time of (\ref{2.7}) we obtain that, \beq i v = e^{-itH}[H\var + F(\var)]+ G \left( 2\,\{{\rm Re}\,(v, V_1 u)\}\, V_2 u+ (V_1u,u)\, V_2 v \right). \label{2.24} \ene Solving the real-linear equation (\ref{2.23})- where now $u$ is a fixed function in $\mathcal N^1$- in an interval $[T_0, T_0+ \Delta]$, with $\Delta $ small enough, we prove that $v(t) \in L^2$, for $T_0\leq t \leq T_0+\Delta $. Note that as $V_1$ satisfies (\ref{2.19}), it follows from (\ref{2.3b}) that $\|V_1 u(t)\|_{L^2}^2=(V_1^2 u(t),u(t)) \leq C \|u\|_{\mathcal N^1}^2$. In consequence, the length of $\Delta$ depends only on $\|u\|_{ \mathcal N^1}$. Repeating this argument, step by step, we prove that $ v \in C([0,T], L^2)$ and, in consequence, that $u \in C([0,T],\Q^2)$. \bull Let us now consider the existence of global $L^2$ solutions. For this purpose we prove that the $L^2$ norm is constant. \begin{lemma}\label{lem2.12} Suppose that assumption A holds, that $V_1\in L^\infty$, that $V_2 \in W_{1,\infty}$, and, furthermore, that $V_1,V_2$ are real valued. Then, the $L^2$ norm of the solution to (\ref{2.1}) given by theorem \ref{th2.1} and the $L^2$ norm of the $\Q^1$ solution given by theorem \ref{th2.4} are constant. \end{lemma} \noindent {\it Proof:} We first prove the theorem for the solution $u \in C([0,T],\Q^1)$. By (\ref{2.7}) $u \in C^1([0,T], \Q^{-1})$ and then, by (\ref{2.1}), \beq \frac{1}{2}\frac{d}{d t}\|u(t)\|_{L^2}= \,{\rm Re}\, \left(\frac{d}{d t}u(t),u(t)\right)= \,{\rm Re}\, \frac{1}{i} \left[( Hu,u)+(V_1u,u)\, (V_2u,u) \right]=0, \label{2.25} \ene and then, $\|u(t)\|_{L^2}= \|u(0)\|_{L^2}$. The result in the case of solutions $u \in C([0,T],L^2)$ follows approximating $u(0)=\var$ in the $L^2$ norm by $ \var_n \in \Q^1$ and applying theorem \ref{th2.2}. \bull \begin{theorem}\label{th2.13} Suppose that assumption A holds, that $V_1\in L^\infty$, that $V_2 \in W_{1,\infty}$, and, furthermore, that $V_1,V_2$ are real valued. Then, the $L^2$ solution, $u(t)$, given by theorem \ref{2.1} exists for all times, and $\|u(t)\|_{L^2}= \|u(0)\|_{L^2}, t \in [0,\infty)$. \end{theorem} \noindent {\it Proof:} the theorem follows from remark \ref{rem2.3} and lemma \ref{lem2.12}. \bull For $ u(t) \in C(I,\Q^1)$ we define the energy at time t as follows, \beq E(u(t)):=(u'(t),u'(t))+(V_0 u(t), u(t))+ \frac{1}{2}(V_1 u(t), u(t))\,(V_2u(t),u(t)). \label{2.26} \ene \begin{lemma}\label{lem2.14} Suppose that assumption A holds, that $V_1= \lambda V_2$, for some real $\lambda$, that $V_2$ is real valued, and that $V_2 \in W_{2,\infty}$. Then, the energy of the $\Q^1$ solution to (\ref{2.1}) given by theorem \ref{th2.4} and the energy of the $\Q^2$ solution to (\ref{2.1}) given by theorem \ref{th2.7} are constant in time. \end{lemma} \noindent {\it Proof:} We first prove the lemma for solutions $u \in C([0,T], \Q^2)$. If follows from (\ref{2.7}) that, $u \in C^1([0,T], L^2)$. As $ \Q^2 =D(H)$, we can write the energy as follows, $$ E(u(t)):=( u(t),H u(t))+ \frac{\lambda}{2}(V_2 u(t), u(t))^2. $$ It follows that, \begin{gather*} \frac{d}{dt} E(u(t))= 2 \,{\rm Re}\, \left[ (\dot{u}(t), H u(t))+ \lambda (V_2 u(t),u(t)) (V_2 \dot{u}(t),u(t))\right]= \\ = 2 \,{\rm Re}\, \frac{1}{i} \left[\|Hu(t)\|_{L^2}^2+ \lambda (V_2 u(t), u(t))\,\{ (V_2 u(t),Hu(t)) +(Hu(t), V_2 u(t))\}\right]=0. \end{gather*} The result in the case of solutions $u \in C([0,T],\Q^1)$ follows approximating $u(0)=\var$ in the $\Q^1$ norm by $ \var_n \in \Q^2$ and applying theorem \ref{th2.5}. \bull \begin{theorem}\label{th2.15} Suppose that assumption A holds, that $V_1= \lambda V_2$, for some real $\lambda$, that $V_2$ is real valued, and that $V_2 \in W_{2,\infty}$. Then, the $\Q^1$ solution to (\ref{2.1}) given by theorem \ref{th2.4} and the $\Q^2$ solution to (\ref{2.1}) given by theorem \ref{th2.7} exist for all times and the $L^2$ norm and the energy of the solutions are constant in time. \end{theorem} \noindent {\it Proof:} Let us first consider the solution $ u \in C(I, \Q^1)$. By lemmata \ref{lem2.12} and \ref{lem2.14} $$ (u'(t),u'(t))+(V_0 u(t), u(t))+N (u(t),u(t))\leq E(u(0))+ N\, \|u(0)\|^2_{L^2}+ \frac{|\lambda|}{2} \|V_2\|^2_{L^\infty} \|u(0)\|_{L^2}^4. $$ Then, by remark \ref{rem2.6} $u$ exists for all times, and the $L^2$ norm and the energy are constant in time. The theorem follows in the case of $\Q^2$ solutions by proposition \ref{prop2.11}. \section{Scattering}\sss In this section we construct the small amplitude scattering operator, $S,$ for equation (\ref{1.2}) and we give a method for the unique reconstruction of the potential $V_0$ and the coupling constant $\lambda$, from $S$. We first introduce some standard notations and some results that we need. For any $\gamma \in \ere$ let us denote by $L^2_{\gamma}$ the Banach space of all complex-valued measurable functions on $\ere$ such that, \beq \left\|\varphi \right\|_{\ds L^1_{\gamma}} := \int |\varphi(x)|\, (1+|x|)^\gamma\, dx < \infty. \label{3.1} \ene If $ V_0 \in L^1_1$ the differential expression $\tau:= -\frac{d^2}{dx^2}+V_0(x)$ is essentially self-adjoint on the domain, $$ D(\tau):= \left\{\varphi \in L^2_C: \varphi,\, {\rm and} \, \varphi' \, \hbox{\rm are absolutely continuous and} \, \tau \varphi \in L^2 \right\}, $$ where $L^2_C$ denotes the set of all functions in $L^2$ that have compact support. We denote by $H$ the unique self-adjoint realization of $\tau$. As is well known, \cite{dt}, \cite{wei}, $H$ has a finite number of negative eigenvalues, it has no positive of zero eigenvalues, it has no singular-continuous spectrum, and the absolutely-continuous spectrum is $[0,\infty)$. By $H_0$ we denote the unique self-adjoint realization of $-\frac{d^2}{d x^2}$ with domain $\mathbb H ^2$. The wave operators are defined as follows, $$ W_{\pm}:= {\rm s-} \lim_{t\rightarrow \pm \infty}\, e^{itH} \, e^{-it H_0}. $$ The limits above exist in the strong topology in $L^2$ and ${\rm Range}\, W_{\pm} = \mathcal H_{ac}$, where $\mathcal H_{ac}$ denotes the space of absolute continuity of $H$. Moreover, the intertwining relations hold, $H W_{\pm}= W_{\pm} H_0$. for these results see \cite{sch}. The linear scattering operator is defined as \beq S_L:= W_+^\ast \, W_-. \label{3.2} \ene For any pair, $\varphi, \psi,$ of solutions to the stationary Schr\"odinger equation \beq -\frac{d^2}{dx^2}\varphi +V_0 \varphi =k^2 \varphi, k \in \CE, \label{3.3} \ene let $[\varphi,\psi]$ denote the Wronskian of $\varphi$ and $\psi$, $$ [\varphi,\psi]:= \varphi' \psi-\varphi \psi'. $$ Let $f_j(x,k), j=1,2$ be the Jost solutions to (\ref{3.3}) that satisfy, $ f_1(x,k)\approx e^{ik x}, x \to \infty, f_2(x,k)\approx e^{-ikx}, x \to - \infty$, \cite{fa1,fa2,dt,cs}. The potential $V_0$ is said to be {\it generic} if $[f_1(x,0), f_2(x,0)]\neq 0$, and it is said to be {\it exceptional} if $[f_1(x,0), f_2(x,0)]= 0$. When $V_0$ is {\it exceptional} there is a bounded solution to (\ref{3.3}) with $k^2 =0$, that is called a half-bound state or a zero energy resonance. The trivial potential $V_0=0$ is exceptional. Below we will always assume that $V_0 \in L^1_{\gamma}$, where in the {\it generic case} $\gamma > 3/2$ and in the {\it exceptional case} $\gamma > 5/2$. In Theorem 1.1 of \cite{we1} it was proven that the operators $W_{\pm}$ and $W^{\ast}_{\pm}$ are bounded on $W_{j,p}, j=0,1, 1 < p < \infty$. By Theorem 3 in page 135 of \cite{st} \beq \|{\mathcal F}^{-1} (1+q^2)^{j/2}({\mathcal F} f)(q)\|_{L^p}, \label{3.4} \ene is a norm that is equivalent to the norm of $W_{j,p}, 1 < p < \infty$. In (\ref{3.4}) ${\mathcal F}$ denotes the Fourier transform. If $H$ has no eigenvalues the $W_{\pm}$ are unitary operators on $L^2$, and it follows from the intertwining relations that $$ (1+H)^{j/2}= W_{\pm}\, (1+H_0)^{j/2}\,W_{\pm}^{\ast}, $$ and then, by (\ref{3.4}), \beq \|(I+H)^{j/2}\,f \|_{L^p}, \label{3.5} \ene defines a norm that is equivalent to the norm of $W_{j,p}, j=0,1, 1 < p < \infty$. Below we use this equivalence without further comments. Furthermore, the following $L^p-L^{p'}$ estimate holds \beq \left\|e^{-it H}\right\|_{\ds{\mathcal B}\left( W_{1,p} ,W_{1,p'}\right)}\, \leq C\frac{1}{\ds |t|^{\frac{1}{p}-\frac{1}{2}}}, 1 \leq p \leq 2, \label{3.6} \ene where for any pair of Banach spaces $\mathcal X, \mathcal Y, \mathcal B(\mathcal X,\mathcal Y)$ denotes the Banach space of all bounded operators from $\mathcal X$ into $ \mathcal Y$. When $H$ has bound states estimate (\ref{3.6}) is proven in \cite{we2} for the restriction of $e^{-it H}$ to the subspace of continuity of $H$. The norm (\ref{3.5}) for the Sobolev spaces $W_{j,p}, 1 < p < \infty$, and the $L^p-L^{p'}$ estimate (\ref{3.6}) are the basic tools that we use in order to construct the scattering operator and to solve the inverse scattering problem. For any $1 < q < 3/2$ we denote, $p:= (q+1)/(q-1),$ $r:= (4q)/(2q-1)$. We designate, $P:=\left(1/r, 1/(p+1)\right)$ and we define, $$ L(P):= L^r\left(\ere, L^{p+1}\right). $$ Let $\mathcal M$ be the following Banach space, \beq \mathcal M:= C_B\left(\ere, L^{p+1}\right) \cap L(P), \label{3.7} \ene with norm, $$ \|\varphi\|_{\mathcal M}:= {\mathrm max }\left[\|\varphi\|_{ C_B\left(\ere,\, L^{p+1} \right)} , \|\varphi\|_{L(P)} \right]. $$ Recall that $C_B\left(\ere, L^{p+1}\right)$ denotes the Banach space of all bounded and continuous functions from $\ere$ into $L^{p+1}$. In the following theorem we construct the small amplitude non-linear scattering operator. \begin{theorem} \label{th3.1} Suppose that $V_0 \in L^1_{\gamma}$ where in the {\it generic case} $\gamma > 3/2$ and in the {\it exceptional case} $ \gamma > 5/2$ and that $H$ has no eigenvalues. Moreover, assume that $V_j \in L^q \cap L^\infty$ for some $1 < q < 3/2$. Then, there is a $\delta > 0$ such that for every $ \varphi_-\in \mathbb H^1$ with $\|\varphi_-\|_{\ds \mathbb H^1} < \delta$ there is a unique solution, $u,$ to (\ref{1.2}) such that $u\in \mathcal M \cap C_B(\ere,L^2)$ and \beq \lim_{t\to -\infty}\|u(t)-e^{-it H}\varphi_-\|_{\ds L^2}=0. \label{3.8} \ene Moreover, there is a unique $\varphi_+ \in L^2$ such that \beq \lim_{t\to \infty}\|u(t)-e^{-it H}\varphi_+\|_{\ds L^2}=0. \label{3.9} \ene Furthermore, $e^{-itH}\varphi_{\pm} \in \mathcal M$ and \beq \left\| u(t)- e^{-itH} \phi_{\pm}\right\|_{\mathcal M}\leq C \left\| e^{-it H} \varphi_{\pm} \right\|^{3}_{\mathcal M}, \label{3.10} \ene \beq \left\|\varphi_+ -\varphi_- \right\|_{\ds L^2} \leq C \left\|\varphi_-\right\|_{\ds \mathbb H^1}^3. \label{3.11} \ene The scattering operator $S_{ V_0}:\varphi_- \hookrightarrow \varphi_+$ is injective. \end{theorem} \noindent{\it Proof:} Observe that $u \in C_B(\ere, L^2)\cap\mathcal M$ is a solution to (\ref{1.2}) with $\lim_{t\to -\infty}\|u(t)-e^{-it H_0} \varphi_-\|_{\ds L^2}=0$ if and only if it is a solution to the following integral equation (this is proven as in the proof of the equivalence of (\ref{2.1}) and (\ref{2.7})) \beq u= e^{-itH} \varphi_- +\frac{1}{i} \int_{-\infty}^{t} e^{-i(t-\tau)\,H} F(u(\tau))\, d \tau, \label{3.12} \ene where, \beq F(u):= \lambda \left( V_1 u,u \right) \, V_2 \, u. \label{3.13} \ene We will prove that the integral in the right-hand side of (\ref{3.12}) is absolutely convergent in $\mathcal M$ and in $L^2$. For $u\in \mathcal M$ we define, \beq \mathcal P (u)(t):= \int_{-\infty}^t e^{-i(t-\tau)H}\, F(u(\tau))\, d \tau. \label{3.14} \ene By (\ref{3.6}) and H\"older's inequality, \beq \begin{array}{ll} \|( \mathcal Pu)(t)\|_{\1}\leq C \int_{-\infty}^t\, \frac{1}{|t-\tau|^d}\,\|F(u(\tau))\|_{\p} \\ \\ \leq C \,\int_{-\infty}^t\,\frac{1}{|t-\tau|^d}\, \|u(\tau)\|^3_{\1}\, d\tau, \\ \end{array} \label{3.15} \ene where, $d:=1/(2q)$. Then, \beq \|( \mathcal Pu)(t)\|_{\1} \leq C \, (I_1+I_2), \label{3.16} \ene where, $$ I_1:= \int_{-\infty}^{t-1}\,\frac{1}{|t-\tau|^d}\, \|u(\tau)\|^3_{\1}\, d\tau, $$ and $$ I_2:= \int_{t-1}^{t}\,\frac{1}{|t-\tau|^d}\, \|u(\tau)\|^3_{\1}\, d\tau. $$ Let us denote by $\chi_{(1,\infty)}$ the characteristic function of $(1,\infty)$. Then, by H\"older's inequality, $$ I_1 \leq \left\|\chi_{(1,\infty)}(\tau) \frac{1}{|\tau|^d}\right\|_{L^{\alpha}}\,\|u\|_{L(P)}^3 , $$ where, $\alpha:= r/(r-3)$. Note that as $d \alpha >1,\, \|\chi_{(1,\infty)}(\tau) \frac{1}{|\tau|^d}\|_{L^{\alpha}} < \infty$. Moreover, as $ d < 1$, $$ I_2 \leq C \|u\|_{C_B(\ere, \1 )}^3. $$ It follows that, \beq \|( \mathcal Pu)(t)\|_{\1} \leq C \,\|u\|_{\mathcal M}^3. \label{3.17} \ene We prove in a similar way that $( \mathcal Pu)(\cdot)$ is a continuous function on $\ere$ with values in $\1$. Furthermore, it follows from (\ref{3.15}) and the generalized Young inequality \cite{rs2}, that, \beq \|\mathcal P(u)\|_{\P}\leq C \|u\|_{\P}^3. \label{3.18} \ene Then, by (\ref{3.17}) and (\ref{3.18}), \beq \| \mathcal Pu\|_{\mathcal M} \leq C \,\|u\|_{\mathcal M}^3. \label{3.19} \ene In an analogous way we prove that, \beq \| \mathcal P u- \mathcal P v\|_{\mathcal M} \leq C \, (\|u\|_{\mathcal M}^2+ \|v\|_{\mathcal M}^2)\, \|u-v\|_{\mathcal M}. \label{3.20} \ene Furthermore, \begin{gather*} \left\| (\mathcal P u)(t)\right\|_{L^2}^2= \int_{-\infty}^t\, d\tau \,\left( F(u(\tau)), \int_{-\infty}^t\, e^{-i (\tau-\tau')H}\, F(u(\tau'))\right)= \\ 2 {\mathrm Re}\int_{-\infty}^t\left( F(u(\tau)), \mathcal P(u)(\tau)\right)\, d\tau. \end{gather*} We define, $$ u_t(\tau):= \chi_{(-\infty, t)}(\tau)\, u(\tau), \qquad g_t(\tau):= \int_{-\infty}^{\tau} \frac{1}{|\tau -\tau'|^d} \|u_t(\tau')\|^3_{\1}\, d\tau. $$ Then, by (\ref{3.6}) and H\"older's inequality, \begin{gather} \left\| (\mathcal P u)(t)\right\|_{L^2}^2 \leq \int_{-\infty}^t \|u_t(\tau)\|_{L^{p+1}}^3\, g_t(\tau)\, d\tau, \nonumber\\ \leq C \, \left[\int \, \|u_t(\tau)\|^{3r/(r-1)}_{\1}\,d\tau\,\right]^{(r-1)/r} \, \|g_t\|_{L^r}. \label{3.21} \end{gather} By the generalized Young's inequality, $$ \|g_t\|_{L^r}\leq C \|u_t\|_{\P}^3. $$ Furthermore, $$ \left[\int \, \|u_t(\tau)\|^{3r/(r-1)}_{\1}\,d\tau\,\right]^{(r-1)/r} \leq \|u\|_{\m}^{4-r}\,\|u_t\|_{\P}^{r-1}. $$ Hence, by (\ref{3.21}) \beq \left\| (\mathcal P u)(t)\right\|_{L^2}^2 \leq \, \|u\|_\m^{4-r} \|u_t\|_{\m}^{r+2}. \label{3.22} \ene We prove in a similar way that the function $ t \in \ere \to (\mathcal P u)(t)$ with values in $L^2$ is continuous. We first prove that equation (\ref{3.12}) has at most one solution in $\m$ and then, we prove the existence of a solution for $\varphi$ small. Suppose that there are two solutions in $\m$ to (\ref{3.12}), $u,v,$ and denote, $u_T:= \chi_{(-\infty, T)} u,v_T:= \chi_{(-\infty,T)} v$. Then, \beq u_T(t)-v_T(t)= (\mathcal P u_T)(t)-(\mathcal P v_T)(t), \,{\mathrm for}\, t\leq T. \label{3.23} \ene Arguing as in the proof of (\ref{3.18}), and as $u_T(t)=v_T(t)=0$ for $t \geq T,$ we prove that, \beq \| u_T- v_T\|_{\P} \leq C \, (\|u_T\|_{\P}^2+ \|v_T\|_{\P}^2)\, \|u_T-v_T\|_{\P}. \label{3.24} \ene As $\lim_{T\to -\infty}\,(\|u_T\|_{\P}^2+ \|v\|_{\P}^2)=0$ we can take $T$ so negative that, $$ C \, (\|u_T\|_{\P}^2+ \|v\|_{\P}^2) < 1/2, $$ where $C$ is the constant in (\ref{3.24}). Then, for such $T$ equation (\ref{3.24}) implies that, $$ \| u_T- v_T\|_{\P} < \frac{1}{2}\, \| u_T- v_T\|_{\P}, $$ and then, $u_T(t)=v_T(t), t \leq T$ and by the uniqueness of the initial value problem at finite time (see theorem \ref{th2.1}), $u(t)=v(t), t \in \ere$. We now prove that $e^{-itH}\in {\mathcal B}\left(\mathbb H^1, \m \right)$. Since $e^{-itH}$ is a strongly continuous unitary group on $L^2$ that commutes with $(1+H)^{1/2}$ we have that, $ e^{-itH} \in {\mathcal B}\left(\mathbb H^1,C_B (\ere, \mathbb H^1 ) \right)$. Furthermore, as by Sobolev's theorem \cite{ad} and interpolation \cite{rs2} $\mathbb H^1$ is continuously imbedded in $\1$, it follows that, $e^{-itH} \in {\mathcal B}\left(\mathbb H^1, C_B(\ere, \1) \right).$ Moreover, by (\ref{3.6}) and Lemma 3.1 of \cite{ka2} it follows that (in \cite{ka2} the operator $e^{-itH_0}$ is considered, but the same proof applies in our case) $ e^{-itH} \in {\mathcal B}\left( L^2,L^r (\ere, L^{l} )\right)$ with, $l=2r/(r-2)$, and then, we have that, $e^{-itH}\in{\mathcal B}\left(\mathbb H^1,L^r (\ere, W_{1,l} )\right)$. Furthermore, as by Sobolev's theorem and interpolation \cite{rs2}, $W_{1,l}$ is continuously imbedded in $\1, $ we have that, $e^{-itH}\in{\mathcal B}\left(\mathbb H^1,L^r (\ere, \1 )\right).$ Then, \beq e^{-itH}\in {\mathcal B}\left(\mathbb H^1, \m \right). \label{3.25} \ene As in section 2, for $ R >0$ we denote, $$ \m_R:=\{ u \in \m : \| u\|_\m \leq R\}. $$ We now take $R$ so small that $ C \max [R^3, (2R)^2] < 1/2$, where $C$ is the biggest of the constants in (\ref{3.19}) and (\ref{3.20}), and $\delta$ so small that $$ \|e^{-itH}\varphi\|_\m \leq R/4 ,\, {\mathrm for}\, \|\varphi\|_{\mathbb H^1 }< \delta. $$ Then if $\|\varphi\|_{\mathbb H^1 }< \delta$ the operator, \beq \mathcal C(u):= e^{-itH}\varphi+ \mathcal P(u) \label{3.26} \ene is a contraction on $\m_R$. By the contraction mapping theorem \cite{rs1} $\mathcal C$ has a unique fixed point in $\m_R$ that is a solution to (\ref{3.12}), and moreover, \beq \|u\|_\m \leq \|e^{-itH}\varphi\|_\m + \frac{1}{2} \|u\|_\m, \label{3.27} \ene and hence, \beq \|u\|_\m \leq 2 \|e^{-itH}\varphi_-\|_\m. \label{3.28} \ene By (\ref{3.12}) and (\ref{3.22}), $u \in C_B(\ere, L^2)$ and (\ref{3.8}) holds. Equations (\ref{3.12}), (\ref{3.19}) and (\ref{3.28}) imply that (\ref{3.10}) holds for $\varphi_-$. We define, \beq \varphi_+ := \varphi_-+ \frac{1}{i}\, \int_{-\infty}^{\infty} e^{i\tau\,H} F(u(\tau))\, d \tau. \label{3.29} \ene Estimating as in the proof of (\ref{3.17}) we prove that $\varphi_+ \in \1$, and arguing as in the proof of (\ref{3.22}) it follows that $\varphi_+ \in L^2$ and that $$ \|\varphi_+-\varphi_-\|_{L^2}\leq C \|u\|_\m^3 \leq C \|e^{-itH}\varphi\|_\m^3. $$ Equation (\ref{3.11}) follows now from (\ref{3.25}). By (\ref{3.12}) and (\ref{3.29}), \beq u= e^{-itH} \varphi_+ -\frac{1}{i} \int_{t}^{\infty} e^{-i(t-\tau)\,H} F(u(\tau))\, d \tau. \label{3.30} \ene Equation (\ref{3.9}) follows from (\ref{3.30}) estimating as in the proof of (\ref{3.22}). Moreover, estimating as in the proof of (\ref{3.19}) we have that, \beq \left\| \int_{t}^{\infty} e^{-i(t-\tau)\,H} F(u(\tau))\, d \tau \right\|_\m \leq C \|u\|_\m^3. \label{3.31} \ene Multiplying both sides of (\ref{3.29}) by $e^{-itH}$ and estimating as in the proof of (\ref{3.19}) we prove that $e^{-itH}\varphi_+ \in \m$. By (\ref{3.30}), (\ref{3.31}) and arguing as in the proof of (\ref{3.28}) we obtain that, \beq \|u\|_\m \leq 2 \|e^{-itH}\varphi_+\|_\m. \label{3.32} \ene At this point, (\ref{3.10}) for $\varphi_+$ follows from (\ref{3.30})--(\ref{3.32}). Note that the uniqueness of $\varphi_+$ is immediate from the fact that $e^{-itH}$ is unitary on $L^2$. Finally, we prove that $S_{V_0}$ is injective. Suppose that $\varphi_+= S_{V_0}\varphi_-=0$. Then, by (\ref{3.30}) \beq u= -\frac{1}{i} \int_{t}^{\infty} e^{-i(t-\tau)\,H} F(u(\tau))\, d \tau. \label{3.33} \ene We prove that (\ref{3.33}) implies that $u=0$ arguing as in the proof of the uniqueness of the solution to equation (\ref{3.12}) and then, it follows from (\ref{3.8}) that $\varphi_-=0$. \bull We now define the scattering operator that relates asymptotic states that are solutions to the free Schr\"odinger equation, $$ i\frac{\partial}{\partial t}u= H_0 u, $$ given by, \beq S:= W_+^\ast\, S_{\ds V_0}\, W_-. \label{3.34} \ene In the following theorem we show that we can uniquely reconstruct the linear scattering operator from the small amplitude behaviour of $S$. \begin{theorem} \label{th3.2} Suppose that the assumptions of Theorem \ref{th3.1} are satisfied. Then, for every $\varphi \in \mathbb H^1$, \beq \frac{d}{d\epsilon} S(\epsilon \varphi)= S_L\, \varphi, \label{3.35} \ene where the derivative exists in the strong convergence in $L^2$. \end{theorem} \noindent{\it Proof:} Since $S(0)=0$ and the wave operators $W_{\pm}$ are bounded on $\mathbb H^1$ \cite{we1} it is sufficient to prove that, \beq {\mathrm s-}\lim_{\epsilon \to 0}\frac{1}{\epsilon}\left[ S_{\ds V_0}(\epsilon \varphi)-\epsilon \varphi \right]=0. \label{3.36} \ene But (\ref{3.36}) follows from (\ref{3.11}) with $\varphi_-$ replaced by $\epsilon \varphi$. \bull \begin{corollary} \label{cor3.3} Suppose that the assumptions of Theorem \ref{th3.1} are satisfied. Then, $S$ uniquely determines the linear potential $V_0$. \end{corollary} \noindent{\it Proof:} By Theorem \ref{th3.2} we uniquely reconstruct $S_L$ from $S$. From $S_L$ we obtain the reflection coefficients for linear Schr\"odinger scattering on the line (see Section 9.7 of \cite{pe}). As $H$ has no bound states we uniquely reconstruct $V_0$ from one of the reflection coefficients using any of the standard methods. See, for example, \cite{fa1,fa2}, \cite{dt}, \cite{me}, \cite{mar}, \cite{cs}, \cite{ak}. \bull Note that our proof gives a constructive method to uniquely reconstruct $V_0$ from $S$. We first compute $S_L$ from the derivative in (\ref{3.35})and then, we obtain the reflection coefficients and we reconstruct $V_0$ from one of them. The following Theorem gives us a convergent expansion at low amplitude for $S_{\ds V_0}$ \begin{theorem} \label{th3.4} Suppose that the assumptions of Theorem \ref{th3.1} are satisfied. Then, for any $\varphi \in \mathbb H^1$, \begin{gather} i\left((S_{\ds V_0}-I)(\epsilon \varphi),\varphi\right)= \epsilon^3 \lambda \int_{-\infty}^{\infty}\, \left(V_1e^{-itH}\varphi,e^{-itH}\varphi\right)\, \left(V_2e^{-itH}\varphi,e^{-itH}\varphi\right)\, dt \nonumber \\ + O(\epsilon^5), \quad \epsilon \to 0. \label{3.40} \end{gather} \end{theorem} \noindent{\it Proof:} Suppose that $u_j \in \m , j=1,\cdots, 4$. Then, by H\"older's inequality, \begin{gather} \left| \int \,(V_1 u_1(t),u_2(t))\, (V_2 u_3(t),u_4(t))\, dt\right| \nonumber \\ \leq \|V_1\|_{L^q}\, \|V_2\|_{L^q}\, \int \|u_1(t)\|_{\1} \|u_2(t)\|_{\1} \|u_3(t)\|_{\1} \|u_4(t)\|_{\1}\, dt \nonumber \\ \leq \|V_1\|_{L^q}\, \|V_2\|_{L^q}\,\|u_1\|_{\P}\, \|u_2\|_{\P}\, \, \|u_3\|_{\P}\, \left[\int \, \|u_4(t)\|^{r/(r-3)}_{\1}\, dt\right]^{(r-3)/r}\nonumber \\ \leq \|V_1\|_{L^q}\, \|V_2\|_{L^q}\,\|u_1\|_{\P}\, \|u_2\|_{\P}\, \, \|u_3\|_{\P} \, \|u_4\|^{4-r}_{\m}\,\|u_4\|_{\P}^{r-3}\nonumber \\ \leq \|V_1\|_{L^q}\, \|V_2\|_{L^q} \Pi_{j=1}^4\, \|u_j\|_{\m}. \label{3.41} \end{gather} As $e^{-itH}\varphi \in \m$, (\ref{3.41}) with $u_j= e^{-itH}\varphi, j=1,2,3,4$, proves that the integral in the right-hand side of (\ref{3.40}) is absolutely convergent. By the contraction mapping theorem, the solution $u$ that satisfies (\ref{3.8}) with $ \epsilon \varphi$ instead of $\varphi_-$ is given by, \beq u(t)= \lim _{j \to \infty}{\mathcal C}^j (\epsilon \,e^{-itH}\varphi) =\epsilon \,e^{-itH}\varphi + v(t), \,{\rm where}\, v(t)= \sum_{j=1}^\infty {\mathcal P}^j(\epsilon e^{-itH}\varphi). \label{3.42} \ene Moreover, it follows from (\ref{3.19}) that if $\epsilon$ is small enough, \beq \|v\|_{\m} \leq C\, \epsilon^3. \label{3.43} \ene Then, (\ref{3.40}) follows by (\ref{3.29}) with $\epsilon \varphi$ instead of $\varphi_-$ and (\ref{3.41})-(\ref{3.43}). \bull \begin{corollary}\label{cor3.5} Suppose that the conditions of Theorem \ref{th3.1} are satisfied and that $V_1,V_2,$ are real-valued functions that are not identically zero. Moreover, assume either that $V_1= V_2$ or that $V_j, j=1,2$, do not change sign and $V_1\, V_2 \ne 0$ in a set of positive measure. Then, the scattering operator, $S$, and $V_j,j=1,2$, determine uniquely $\lambda$. \end{corollary} \noindent {\it Proof:} By (\ref{3.40}) \beq \lambda = \lim_{\epsilon \to 0}\frac{1}{\epsilon^3} \frac{i\left((S_{\ds V_0}-I)(\epsilon \varphi),\varphi\right)} {\int_{-\infty}^{\infty}\, \left(V_1e^{-itH}\varphi,e^{-itH}\varphi\right)\, \left(V_2e^{-itH}\varphi,e^{-itH}\varphi\right)\, dt}. \label{3.44} \ene By corollary \ref{cor3.3} $V_0$ is known, and then $H:= H_0+V_0$ is known. Then, $W_{\pm}$ are known, and $S$ uniquely determines $S_{\ds V_0}$ (see (\ref{3.34})). Hence, the right-hand side of (\ref{3.44}) is uniquely determined by our data. Moreover, under our conditions we can always find a $\varphi \in \mathbb H^1$ such that the denominator of the right-hand side of(\ref{3.44}) is not zero. \bull Note that (\ref{3.44}) gives us a formula for the reconstruction of $\lambda$. Let us now go back to the quantum capacitor (\ref{1.1}) where we take a slightly more general external potential $V_0$, namely, $$ V_0(x)= \left[\beta_1 \chi_{[a,b]}(x)+ \beta_2\chi_{[c,d]}(x)\right], $$ where $\beta_1,\beta_2 \in \ere$. By corollary \ref{cor3.3} we uniquely reconstruct $V_0$ from $S$. Then, $\beta_1,\beta_2, a,b,c$ and $d$ are uniquely reconstructed. Moreover, by corollary \ref{cor3.5} we uniquely reconstruct $\lambda$. Hence, from $S$ we uniquely reconstruct all the physical parameters of the quantum capacitor. %%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{nls6} Adami R~ and Teta A~ 2001 A class of nonlinear Schr\"odinger equations with concentrated nonlinearities, {\it J. Funct. Anal.} {\bf 180} 148--175 \bibitem {nls7} Adami R~ 2002 Blow-up for Schr\"odinger equations with pointwise nonlinearity, {\it Mathematical Methods in Quantum Mechanics }, Contemporary Mathematics {\bf 307}, ed Weder R, Exner P and Gr\'ebert B (Providence: Amer. Math. Soc.) pp 1--7 \bibitem{nls8} Adami R, Dell'Antonio G, Figari R and Teta A~ 2003 The Cauchy problem for the Schr\"odinger equation in dimension three with concentrated nonlinearity {\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire} {\bf 20} 477--500 \bibitem{nls9} Adami R, Dell'Antonio G, Figari R and Teta A~ 2004 Blow up solutions for the Schr\"odinger equation in dimension three with concentrated nonlinearity { \it Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire} {\bf 21} 121--137 \bibitem{ad} Adams R~ A~ 1975 {\it Sobolev Spaces} (New York: Academic) \bibitem{ak} Aktosun T and Klaus M~ 2002 Inverse Theory: Problem on the line, {\it Scattering}, vol. 1, ed Pike R~ and ~ Sabatier P (New York: Academic) pp 770--785 \bibitem{alw} Altshuler B~ L, Lee P~ A and Webb R~ A~ 1991 {\it Mesoscopic Phenomena in Solids} (New York: Elsevier) \bibitem{nls4} Bourgain J~ 1999 {\it Global Solutions of Nonlinear Schr\"odinger Equations}, Colloquiun Publications {\bf 46} (Providence: Amer. Math. Soc.) \bibitem{cs} Chadam K and Sabatier P~C ~ 1989 {\it Inverse Problems in Quantum Scattering Theory, Second Edition} (Berlin: Springer) \bibitem{m1} Davies E B~ 1979 Symmetry breaking for a nonlinear Schr\"odinger operator {\it Comm. Math. Phys.} {\bf 64} 191--210 \bibitem{m2} Davies E B~ 1995 Nonlinear Schr\"odinger operator and molecular structure {\it J. Phys. A: Math. and Gen.} {\bf 28} 4025--4041 \bibitem{dt} Deift P~ and Trubowitz E~ 1979 Inverse scattering on the line {\it Comm. Pure Appl. Math.} {\bf 32} 121--251 \bibitem{fa1} Faddeev L D~ 1964 Properties of the S matrix of the one--dimensional Schr\"{o}dinger equation, {\it Trudy Math. Inst. Steklov} {\bf 73}, 314--333 [ english translation, 1964, {\it Am. Math. Soc. Translation Series} 2 {\bf 65}, 139--166] \bibitem{fa2} Faddeev, L D~ 1974 Inverse problems of quantum scattering theory, II, {\it Itogi Nauki i Tekhniki Sovremennye Problemy Matematiki} {\bf 3}, 93--180 [ english translation, 1976 {\it J. Soviet Math.} {\bf 5}, 334--396] \bibitem{nls3} Ginibre J~ 1998 {\it Introduction aux \'Equations de Schr\"odinger non Lin\'eaires} (Paris: Onze \'Editions) \bibitem{m3} Grecchi V and Martinez A~ 1995 Non linear Stark effect and molecular localization {\it Comm. Math. Phys.} {\bf 166} 533--548 \bibitem{m5} Grecchi V, Martinez A and Sachetti A~ 2002 Destruction of the beating effect for a nonlinear Schr\"odinger equation {\it Comm. Math. Phys.}{\bf 227} 191--209 \bibitem{s2} Jona-Lasinio G, Presilla C and Capasso F~, 1992 Chaotic quantum phenomena without classical counterpart, {\it Phys. Rev. Lett.} {\bf 68} 2269--2272 \bibitem{s3} Jona-Lasinio G~ 1995 Stationary solutions and invariant tori for a class of non-linear non-local Schr\"odinger equations, {\it Advances in Dynamical Systems and Quantum Physics} (River Edge: World Scientific) pp 142--146 \bibitem{s4} Jona-Lasinio G, Presilla C and Sj\"ostrand J~ 1995 On Schr\"odinger equations with concentrated nonlinerities {\it Ann. Physics} {\bf 240} 1--21 \bibitem{ka1} Kato T~ 1976 {\it Perturbation Theory of Linear Operators, Second Edition} ( Berlin: Springer) \bibitem{ka2} Kato T~ 1989 Nonlinear Schr\"odinger equations, {\it Schr\"odinger operators}, Lecture Notes in Phys. {\bf 345} ed Holden H ~and Jensen A ( Berlin: Springer) pp 218--263 \bibitem{ka3} Kato T~ 1984 Remarks on holomorphic families of Schr\"odinger and Dirac operators, {\it Differential Equations} ed Knowles I W~ and Lewis R T~ (Amsterdam: Elsevier (North-Holland) pp 341--352 \bibitem{mar} Marchenko V A~ 1986 {\it Sturm-Liouville Operators and Applications} (Basel:Birkh\"auser) \bibitem{me} Melin A~ 1985 Operator methods for inverse scattering on the real line {\it Comm. in Partial Differential Equations} {\bf 10} 677--766 \bibitem{pe} Pearson D B~ 1988 {\it Quantum Mechanics and Spectral Theory} ( New York: Academic) \bibitem{s1} Presilla C~, Jona-Lasinio G~ and Capasso F~ 1991 Non-linear feedback oscillations in resonant tunneling through double barriers {\it Phys Rev. B} {\bf 43} 5200--5203 \bibitem{s5} Presilla C and Sj\"ostrand J~ 1996 Transport properties in resonant tunneling {\it J. Math. Phys.} {\bf 37} 4816--4844 \bibitem{nls2} Racke R~ 1992 {\it Lectures in Nonlinear Evolution Equations. Initial Value Problems}, Aspects of Mathematics {\bf E 19} (Braunschweig/Wiesbaden: Vieweg) \bibitem{rs1} Reed M and Simon B~ 1972 {\it Methods of Modern Mathematical Physics I Functional Analysis} (New York: Academic) \bibitem{rs2} Reed M and Simon B~ 1975 {\it Methods of Modern Mathematical Physics II Fourier Analysis, Self-Adjointness} (New York: Academic) \bibitem{m4}Sachetti A~ 2002 Tunneling destruction for a nonlinear Schr\"odinger equation, {\it Mathematical Results in Quantum Mechanics}, Contemporary Mathematics {\bf 307} ed Weder R~, Exner P~ and Gr\'ebert B (Providence: Amer. Math. Soc.) pp 275--279 \bibitem{m6}Sachetti A~ 2004 Nonlinear Time-dependent Schr\"odinger equation with double well potential, {\it Multiscale Methods in Quantum Mechanics} ed Blanchard P~ Dell'Antonio G~ (Basel: Birkh\"auser) \bibitem{m7} Sachetti A~ 2005 Nonlinear double well Schr\"odinger equation in the semiclassical limit {\it J. Statist. Phys.} {\bf 119} 1347--1381 \bibitem{sach} Sachetti A~ 2004 Nonlinear time -dependent one-dimensional Schr\"odinger equation with a double well potential {\it SIAM J. Math. Anal.} {\bf 35} 1160--1176 \bibitem{sch} Schechter M~ 1981 {\it Operator Methods in Quantum Mechanics} (New York: North Holland) \bibitem{st} Stein E M~ 1970 {\it Singular Integrals and Differentiability Properties of Functions} (Princeton: Princeton Univ. Press) \bibitem{nls1} Strauss W~ 1989 {\it Nonlinear Wave Equations}, CBMS-RCMS {\bf 73} (Providence: Amer. Math. Soc.) \bibitem{nls5} Sulem C~ and Sulem P-L~ 1999 {\it The Nonlinear Schr\"odinger Equation}, Appl. Math. Sciences {\bf 139} (New York: Springer) \bibitem{we1} Weder R~ 1999 The $W_{k,p}$ continuity of the Schr\"odinger wave operators on the line {\it Comm. Math. Phys.} {\bf 208} 507--520 \bibitem{we2} Weder R~ 2000 $L^p-L^{p'}$ estimates for the Schr\"odinger equation on the line and inverse scattering for the nonlinear Schr\"odinger equation with a potential, {\it J. Funct. Anal.} {\bf 170} 37--68 \bibitem{we4} Weder R~ 2000 Inverse scattering on the line for the nonlinear Klein-Gordon equation. Reconstruction of the potential and the nonlinearity {\it J. Math. Anal. Appl.} {\bf 252} 102-123 \bibitem{we10} Weder R~ 2000 Center manifolds for nonintegrable nonlinear Schr\"odinger equations on the line {\it Comm. Math. Phys. } {\bf 215} 343--356 \bibitem{we5} Weder R~ 2001 Inverse scattering for the nonlinear Schr\"odinger equation. Reconstruction of the potential and the nonlinearity {\it Math. Methods Appl. Sci.} {\bf 24} 245--254 \bibitem{we6} Weder R~ 2001 Inverse scattering for the nonlinear Schr\"odinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case {\it Proc. Amer. Math. Soc.} {\bf 129} 3637--3645 \bibitem{we7} Weder R~ 2002 Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential {\it J. Differential Equations} {\bf 184} 62-77 \bibitem{we8} Weder R~ 2003 The time dependent approach to inverse scattering {\it Advances in Differential Equations and Mathematical Physics}, Contemporary Mathematics {\bf 327} ed Karpeshina J, Stolz G, Weikard R, and Zeng Y~ (Providence: Amer. Math. Soc.) pp 359--377 \bibitem{we9} Weder R ~ 2005 Scattering for the forced non-linear Schr\"odinger equation with a potential on the half-line, {\it Math. Methods Appl. Sci.} {\bf 28} 1219--1236 \bibitem{we3} Weder R ~ 2005 The forced non-linear Schr\"odinger equation with a potential on the half-line, {\it Math. Methods Appl. Sci.} {\bf 28} 1237--1255 \bibitem{wei} Weidmann J~ {\it Spectral Theory of Ordinary Differential Operators}, Lecture Notes in Math. {\bf 1258} (Berlin: Springer) \end{thebibliography} \end{document} ---------------0512011646771--