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%%%%%%%%%%%%%%%%%%%%%%%%%%%% %MORE NEWCOMMANDS \newcommand{\de}{\delta} \newcommand{\PP}{\mathbb{P}} \newcommand{\OO}{\mathcal{O}} \newcommand{\HH}{\mathcal{H}} \newcommand{\GG}{\mathcal{G}} %NORME L^p grafo e tempo \newcommand{\LG}[1]{L^{#1}} \newcommand{\LT}[2]{L^{#1}_{[#2]}} \newcommand{\phisol}[1]{\phi_{#1}} %RENEWCOMMAND \renewcommand{\Im}{\operatorname{Im}\,} \renewcommand{\Re}{\operatorname{Re}\,} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{} \begin{document} %\input{copertina} \title{Fast solitons on star graphs} \author{Riccardo Adami${}^1$, Claudio Cacciapuoti${}^2$, Domenico Finco${}^3$, and Diego Noja${}^1$ %\author{Diego Noja}%${}^1$ %$\thanks{Partially supported %by a ``Borsa per giovani ricercatori GNFM, 2008.''} , %Andrea Sacchetti ${}^2$ \\ \\ ${}^1$Dipartimento di Matematica e Applicazioni, Universit\`a di Milano Bicocca \\ via R. Cozzi, 53, 20125 Milano, Italy\\ riccardo.adami@unimib.it, diego.noja@unimib.it \\ \\ ${}^2$Hausdorff Center for Mathematics, Institut f\"ur Angewandte Mathematik \\ Endenicher Allee, 60, 53115 Bonn, Germany\\ cacciapuoti@him.uni-bonn.de\\ \\ ${}^3$Dipartimento di Matematica, Universit\`a di Roma ``La Sapienza'' \\ P.le A. Moro, 2, 00185 Roma, Italy\\ finco@mat.uniroma1.it} \maketitle \begin{abstract} \par\noindent We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global strong and weak well-posedness and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff and the so called $\delta$ and $\delta'$. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper \cite {[HMZ07]} about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents a generalization of their work to graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph. \end{abstract} \section{Introduction} \setcounter{equation}{0} In the present paper we begin the study of the {\it nonlinear} wave propagation on graphs. While there exists an extensive literature on the behaviour of linear wave and Schr\"odinger equations on graphs (\cite {[Kuc04],[Kuc05],[KS99],[BCFK06],[BEH]}, and references therein) and a certain activity concerning the so called {\it discrete} nonlinear Schr\"odinger equation (DNLSE) in chains with edges and graphs inserted (``decorations", interpreted as defects in the chain), both from the physical or numerical point of view (see for example \cite {[KFTK],[BCSTV]}), as far as we known the subject of nonlinear Schr\"odinger evolution is new. \par The study of nonlinear propagation in ramified structures could be of relevance in several branches of pure and applied science, from condensed matter physics to nonlinear fiber optics, hydrodynamics and fluid transport (a non traditional example is blood flow in veins and arterias) and finally neural networks (see for example the study of {\it reaction diffusion type} Fitz-Nagumo-Rall equations on networks in \cite{[CMu]}, and references therein).\par Of course in all these examples there is a strong dependence on the modelization. The nonlinear Schr\"odinger equation with cubical nonlinearity is especially suitable to describe nonlinear electromagnetic pulse propagation in optical fibers and, under the name of Gross-Pitaevskii equation, the dynamics of Bose condensates. Better suited for other applications, for example the hydrodynamic flow, is the KdV equation or its relatives, which we do not treat here. \par A word has to be spent about the model of the ramified structure. Real world is not strictly one dimensional, and an abstract graph, which is just a set of copies of $\RE^+$ (``edges", or ``branches") with functions living on it satisfying certain boundary conditions at $0$ (see section $2$ for the rigorous notion) lacks some of the geometric meaningful characteristics of a real network, such as thickness and curvature of the branch or orientation between edges. On the other hand these problems are far from being well understood also for the linear propagation (see for example \cite {[ACF],[CE]}), and so we content ourselves with posing and analysing the nonlinear problem in the idealized and simplified framework of an abstract graph. \par We study in this paper the special case of a star graph with three edges. A generalization to a star graph with $n$ edges would be straightforward, but here our interest is in clarifying the main behaviours and techniques involved in the analysis. A preliminary point and not a trivial issue is the definition of the dynamics. Let us recall that for a star graph $\GG$, the linear Schr\"odinger dynamics is defined by giving a self-adjoint operator $H$ on the product of $n$ copies of $L^2(\RE^+)$ (briefly $L^2(\GG)$), with a domain $D(H)$ in which appears a linear condition involving the values at $0$ of the functions of the domain and of their derivatives. The admissible boundary conditions characterize the interaction at the vertex of the star graph. For the nonlinear problem, we establish the well-posedness of the dynamics in the case of a star graph with some distinguished boundary condition at the vertex, namely the free (or Kirchhoff), and the $\delta$ and $\delta'$ boundary conditions (see section $2$ for the relevant definitions). \par To be clear about the exact problem with we are faced to, the differential equation to be studied is of the form \begin{equation} \label{diffform} i \frac{d}{dt}\Psi_t \ = \ H \Psi_t - | \Psi_t |^2 \Psi_t \end{equation} where the function $\Psi$ is, for a three-edge graph, a column vector \begin{equation*} \Psi = \lf( \begin{array}{c} \psi_1 \\ \psi_2 \\ \psi_3 \end{array} \ri)\ , \end{equation*} that lies in the domain $D(H)$ of the linear Hamiltonian $H$ on the graph, so emboding the relevant boundary conditions. This is the abstract strong form of the equation, which, as it is apparent, is equivalent to a particular nonlinear coupled system of scalar equations. The coupling is not due to the nonlinearity, because of the definition \begin{equation*} |\Psi|^2\Psi\equiv \lf( \begin{array}{c} |\psi_1|^2\psi_1 \\ |\psi_2|^2\psi_2 \\ |\psi_3|^2\psi_3 \end{array} \ri)\ , \end{equation*} but to the boundary condition at the vertex. For example, in the simple case of a Kirchhoff boundary condition, the coupling between various components of the generic domain element is given by \begin{equation*} \Psi \in L^2(\GG) \text{ s.t. } \psi_i \in H^2 (\RE^+ ), \, \psi_1 (0) = \psi_2 (0) = \psi_3 (0), \, \psi_1' (0) + \psi_2' (0) + \psi_3'(0)=0 \ . \end{equation*} The $\delta$ or $\delta'$ boundary conditions allow a coupling between the values of the function $\Psi$ and values of their derivatives at the origin, but in principle the nature of the problem is unaltered.\par\noindent In the present paper, for several reasons, we prefer to write the dynamics in weak form, which is the following \begin{equation} \label{intform1} \Psi_t \ = \ e^{-iH t} \Psi_0 + i \int_0^t e^{-i H (t-s)}|\Psi_s|^2 \Psi_s \, ds\ ; \end{equation} here the $\Psi$ is in the form domain ${\mathcal D} ({\mathcal E}_{lin})$, where ${\mathcal E}_{lin}$ is the quadratic form of the linear Hamiltonian $H$. We interpret, according to the use, the form domain as the finite energy space for the problem. An adaptation of the methods in \cite{[AN09]} gives the local well-posedness of the equation \eqref{intform1} for every initial datum in ${\mathcal D} ({\mathcal E}_{lin})$. Moreover, charge and energy conservation laws hold true for such weak solutions, and as a consequence, the NLS on graph admits global solutions. Similar results could be shown without essential additional difficulties in the case of a general boundary condition at the vertex. These generalizations will be treated elsewhere.\par Apart from well-posedness, the main goal of this paper is to provide information on the interaction between a solitary wave and the boundary condition at the vertex. As it is well known, the NLS on the line admits a family of solitary non dispersive solutions, rigidly translating with a fixed velocity. The family of solitary solutions (``solitons") is given by the action of the Galilei group on the function \begin{equation*} \phi (x) \ = \ \sqrt 2 \cosh^{-1} x\ , \end{equation*} or explicitly, \begin{equation*} \phisol{x_0,v}(x,t) = e^{i \f v 2 x} {e^{- i t \f {v^2} 4}}e^{it} \phi ( x - x_0 - vt)\,. \end{equation*} \par We show that, after the collision of a similar solitary wave with the vertex there exists a timescale during which the dynamics can be described as the scattering of three split solitary waves, one reflected on the same branch where the originary soliton was running asymptotically in the past, and two transmitted solitary waves on the other branches. On the same timescale, the amplitudes of the reflected and transmitted solitary waves are given by the scattering matrix of the linear dynamics on the graph. The soliton-like character persists over time intervals that depend on the velocity of the impinging original soliton: the faster is the original soliton, the (logarithmically in the velocity) longer is the survival time of the solitary wave behaviour on every branch of the graph. The non-trivial point is that the persistence time of solitary behaviour after collision with the vertex is much longer, for fast solitons, than the time over which it is reasonable to approximate the nonlinear dynamics with the linear one. The same timescale of the order $\ln v$ of persistence of solitary behaviour appears in the paper \cite{[AbFS]} where the collision of two solitary waves with an underlying smooth potential is studied, and in the paper by Holmer, Marzuola and Zworski \cite{[HMZ07]} on the fast NLS-soliton scattering by a delta potential on the line, which is the main source of inspiration for the present paper. \par We give now an outline of our result and its proof.\p The initial datum is of the following form \begin{equation*} \Psi_0 (x) \ = \ \left( \begin{array}{c} \chi (x) e^{-i \f v 2 x} \phi (x - x_0) \\ 0 \\ 0 \end{array} \right) \end{equation*} where $\chi$ is a cut off function, that is $\chi \in C^\infty (\erre^+ )$, $\chi = 1$ in $(2, + \infty)$ and $\chi = 0$ in $(0,1)$. Apart from a small tail term truncated by the cutoff function, the first component is the initial datum of a free (i.e. without external potentials) NLS which on the line yields to a solitary wave running with velocity $v$; the center $x_0$ of the initial soliton is chosen far from the vertex; the precise condition, given in term of the velocity, is $x_0\geq v^{1-\delta}$ with $\delta \in (0,1)$. We are interested in the evolution $\Psi_t$ of this initial datum. \par The analysis of the dynamics can be divided in several phases. The pre-interaction phase, where the evolved initial datum is far from the vertex, and the undisturbed NLS evolution dominates. At the end of this phase, the solution enters the vertex zone, and differs (in $L^2$ norm) from the evolved solitary wave for an exponentially small error in the velocity $v$. The second phase is the interaction phase, in which a substantial fraction of the mass of the initial soliton has reached the vertex, and the linear dynamics dominates due to the shortness of the interaction time, leaving the system at the end of this phase with three scattered waves, the amplitudes of which are given by the action of the scattering matrix of the associated linear graph on the incoming solitary wave. The size of the corresponding error is (again in $L^2$ norm) a suitable negative inverse power ($v^{{-\frac{1}{2}}\delta}$) of the velocity. In both the phases one and two, the main technical tool consists in an accurate use of Strichartz estimates to control the deviations between the unperturbed NLS flow, the global NLS flow and the linear flow on the graph in the relevant time intervals. Finally there is the post-interaction phase, where the free NLS dynamics dominates again; however, now the initial data are not exact solitary waves, but waves with solitonlike profiles and ``wrong'' amplitudes (due to the scattering process in the interaction phase). \par The true evolution is confronted with a reference dynamics given by the superposition of the nonlinear evolution of the outgoing scattered profiles, and it turns out that the error is, in $L^2$ norm, of the order of an inverse power of velocity (which power depending on the size of the time interval of approximation). For a precise formulation one has to tackle the problem of representing the reference soliton dynamics to be confronted with the true dynamics. This problem arises because one would like to use crucial and known properties of NLS on the line (such as existence of infinite constants of motion), and various associated estimates, while on a star graph one has a NLS on halflines, jointly with boundary conditions. The problem occurs, of course, in each of the three phases in which the dynamics is decomposed. The details are in sections 4 and 5, and here to state the main result of the paper, we refer succinctly to the phase three only.\par Our choice of reference dynamics is the following. We associate to every edge of the star graph a companion edge chosen between the other two, in such a way to have three fictitious lines; then we glue the soliton on every single edge with the right tail on the companion edge, respecting the free nonlinear dynamics. One of the main technical point in the analysis of the true dynamics is to have a control in the errors brought by this schematization. More precisely, let us define \begin{equation*} \tilde\Phi^1_t(x_1 ,x_2, x_3)\equiv \begin{pmatrix} \tilde r e^{-i \f {v^2} 4 t}e^{i\f v 2 x_1} e^{it} \phi (x_1 + x_0 - vt)\\ \tilde r e^{-i \f {v^2} 4 t}e^{-i\f v 2 x_2} e^{it} \phi (x_2 - x_0 + vt)\\ 0 \end{pmatrix} \end{equation*} %%%%%% \begin{equation*} \tilde\Phi^2_{t}(x_1 ,x_2, x_3)\equiv \begin{pmatrix} 0\\ \tilde t e^{-i \f {v^2} 4 t}e^{i\f v 2 x_2} e^{it} \phi (x_2 + x_0 - vt)\\ \tilde t e^{-i \f {v^2} 4 t}e^{-i\f v 2 x_3} e^{it} \phi (x_3 - x_0 + vt) \end{pmatrix} \end{equation*} %%%%%% \begin{equation*} \tilde\Phi^3_{t}(x_1 ,x_2, x_3)\equiv \begin{pmatrix} \tilde te^{-i \f {v^2} 4 t}e^{-i\f v 2 x_1} e^{it} \phi (x_1 - x_0 + vt)\\ 0\\ \tilde te^{-i \f {v^2} 4 t}e^{i\f v 2 x_3} e^{it} \phi (x_3 + x_0 - vt) \end{pmatrix} \end{equation*} Each of these vectors represents a soliton on the fictitious line given by an edge and its companion, multiplied by the scattering coefficients of the linear dynamics considered, Kirchhoff, $\delta$ or $\delta'$ (and here unspecified). Up to small error, these functions represent outgoing waves at the end ($t=t_2$) of interaction phase, which is essentially a scattering process. Taking these as initial data for the free nonlinear dynamics on the pertinent fictitious line, we define their time evolution $\Phi_t^{j}$ as given by \begin{equation*} \Phi^j_{t}=e^{-i H_j (t-t_2)} \tilde\Phi^j_{t_2} + i \int_{t_2}^t ds \, e^{-i H_j (t-s)} | \Phi^j_s |^2 \Phi^j_s\qquad j=1,2,3\, \end{equation*} where the Hamiltonians $H_j$ are the linear Hamiltonians with Kirchhoff boundary conditions in which the $j$-branch is excluded, in fact a free dynamics (see subsection 2.3 for the precise definition of $H_j$ and remark 4.2 for a different characterization of the nonlinear evolution of $\Phi^j$). \p The $\{\Phi^j_{t}\}$ represent our reference dynamics (see complete definitions in section 4, where we adopt the slightly heavier notation $\Phi^j_{H,t}$, to take in account differences induced by the various vertex boundary conditions). \par With these premises, the main result of the paper is the following. \begin{theorem} There exist $\tau_* >0$ and $T_{\ast}>0 $ such that for $0< T 0$ and where $\tilde t^j$ is the scattering coefficient of the {\it linear} Hamiltonian which describes the vertex. So, in the limit of fast solitons, i.e. $v\to \infty$, one can assert that the ratio which defines the nonlinear scattering coefficient converges to the corresponding linear scattering coefficient. And analogously for the case of reflection coefficient $r(v)$, i.e. $j=1$, one has (for $t>t_2$, as before) $$ \lim_{v\to \infty}\frac{\|\Psi^1_t\|}{\|\Psi_t\|}=|\tilde r|\ . $$ \p This is true for every coupling between the ones considered, i.e. Kirchhoff, $\delta$ and $\delta'$.\par We add some other remarks on the result and further analysis and generalizations.\p To begin with, let us note that the real line with a point interaction at $0$ can be interpreted as a degenerate graph with two edges. The cited paper \cite{[HMZ07]} treat the special case of a $\delta$ interaction on the line, and our description shows how it could be possible to extend their results to other point interaction; the example here treated, apart $\delta$ and Kirchhoff conditions is (a version of) the $\delta'$ interaction, but in principle the analysis is applicable to other point interaction, also of more singular nature of the $\delta$.\par Concerning more general items, a sharper description of the post interaction phase can be achieved by an explicit characterization of the evolution of the modified solitary profiles, i.e. of the $ \Phi^j_t$ . This last part is somewhat delicate, and intersects with contemporary intense work on asymptotics for solitons in integrable and quasi integrable PDE, so we limit ourselves to the following remarks. In the case analysed in \cite{[HMZ07]}, the asymptotic behaviour of nonlinear Schr\"odinger evolution of solitary waveforms with modified amplitudes is given, and making use of inverse scattering theory it is shown (appendix A of the cited paper) that the evolution is close to a soliton up to times of order $\ln v$ and an error the order of which is an inverse power of velocity $v$. Borrowing from these results, it is possible to get in our case too, but we omit details, the asymptotics of the $ \Phi^j_t$, i.e. of the free NLS evolution of the modified solitary profiles outgoing from the phase two. It turns out that these outgoing wavefunctions can be approximated, on the same logarithmic timescale of Theorem 1.1, as new solitons with the same waveform of the unperturbed dynamics, modified amplitudes and phases, plus a dispersive (``radiation") contribution. The meaning of this statement is that the $L^{\infty}_x$ norm of the difference between the evolved modified solitary profiles $ \Phi^j_t$ and such final outgoing solitons has the usual dispersive behaviour, $|t-t_2|^{-\frac{1}{2}}$. Let us note that at the end of the phase three, there are two types of errors: errors due to the approximation procedure in phase one and two (${\mathcal{O}}_{L^2_x}$) ; and errors arising from neglecting dispersion in the reconstruction of the outgoing solitons (${\mathcal{O}}_{L^{\infty}}$) .\par An important question concerns the possibility of extending the timescale of validity of approximation by the solitary outgoing waves. As a quite generic remark, this possibility could be related to the asymptotic stability of the system, or of systems immediately related to it.\p More concretely, in a different type of model (scattering of two solitons on the line) in the already cited paper \cite{[AbFS]}, some considerations are given on obtaining longer timescales of quasiparticle approximation in dependence of the initial data and external potential, but it is unclear if similar considerations apply to the present case.\p %Let us mention, however, the recent results of G.Perelman on the asymptotics of two %colliding solitons for nonlinearity close to integrable on the line (\cite {[P]} and %results announced in ... ) Another issue is the nonlinearity. The fundamental asymptotics proved in \cite{[HMZ07]} and used in the present paper relies on the integrable nature of cubic NLS, and it is not immediate to extend these results to more general nonlinearities. One can conjecture that for nonlinearities close to integrable which admit solitary waves, the outgoing waves are close to solitons over suitable timescales. Let us mention, however, the recent results of G.Perelman on the asymptotics of colliding solitons for nonlinearity close to integrable or $L^2$ critical on the line (\cite {[P1]} and results announced in \cite{[P2]} ). \par A final, less difficult, problem is the extension of results of the present work to more general graphs. We believe that results similar to the ones of the present papers could be valid for more general boundary condition at the vertex of a star graphs, with the same proof, under the condition of absence of the eigenvalues of the linear Hamiltonian describing the graph. In the presence of eigenvalues, some Strichartz estimates weaken, and a more refined analysis is needed (see \cite{[DH]} for the analogous problem on the line with an attractive $\delta$ interaction). \p The extension to graphs of less trivial topology is an open problem.\p \subsection{Setting and notations} We consider a graph $\GG$ given by three infinite half lines attached to a common vertex. In order to study a quantum mechanical problem on $\GG$, the natural Hilbert space is then $L^2(\GG)\equiv L^2(\RE^+)\oplus L^2(\RE^+)\oplus L^2(\RE^+)$. \n We denote the elements of $L^2(\GG)$ by capital greek letters, while functions in $L^2(\RE^+)$ are denoted by lowercase greek letters. It is convenient to represent functions in $L^2(\GG)$ as column vectors of functions in $L^2(\RE^+)$, namely \begin{equation*} \Psi = \lf( \begin{array}{c} \psi_1 \\ \psi_2 \\ \psi_3 \end{array} \ri). \end{equation*} The norm of $L^2$-functions on $\GG$ is naturally defined by $$ \| \Psi \|_{L^2 (\GG)} : = \left( \sum_{j=1}^3 \| \psi_j \|^2_{L^2 (\erre^+)} \right)^{\f 1 2}. $$ Analogously, given $1 \leqslant p \leqslant \infty$, we define the space $L^r (\GG)$ as the set of functions on the graph whose components are elements of the space $L^r (\erre^+)$, and the norm is correspondingly defined by \begin{equation*} \big\|\Psi\big\|_{\LG{r} (\GG)}=\bigg(\sum_{j=1}^3\|\psi_j\|_{L^r(\RE^+)}^{r}\bigg)^{\f 1 r}. \end{equation*} When a functional norm is referred to a function defined on the graph, we omit the symbol $\GG$. Furthermore, from now on, when such a norm is $L^2$, we drop the subscript, and simply write $\| \cdot \|$. Accordingly, we denote by $(\cdot,\cdot)$ the scalar product in $L^2$. As it is standard when dealing with Strichartz's estimates, we make use of spaces of functions that are measurable as functions of both time (on the interval $[T_1,T_2]$) and space (on the graph). We denote such spaces by $L^p_{[T_1,T_2]}{\LG{r} (\GG)})$, and indices $1\leqslant r\leqslant \infty$, $1\leqslant p\leqslant \infty$; we endow them with the norm %%% \begin{equation*} %\big\|\Psi\big\|_{\LG{r}}=\bigg(\sum_{j=1}^3\|\psi_j\|_{L^r(\RE^+)}^{r}\bigg)^{1/r} %\;;\quad \big\|\Psi\big\|_{\LT{p}{T_1,T_2}\LG{r}{(\GG)}}= \bigg(\int_{T_1}^{T_2}\big\|\Psi_s\big\|_{\LG{r}}^p ds\bigg)^{1/p}\,. \end{equation*} %%% \n The extension to the case $p=\infty$ or $r=\infty$ is straightforward. %Among such spaces, we often use the one with $r = p = 6$, so, for %shorthand, we introduce the notation %$$ %X_{[T_1, T_2]} ([x_1, x_2]) \ = \ L^6_{[T_1, T_2]} L^6 ([x_1, x_2]). %$$ %When no index is written, we mean %the $L^2(\GG)$ norm, that is $\|\Psi\|\equiv %\big\|\Psi\big\|_{\LG{2}}$. \n Besides, we need to introduce the spaces $$H^1(\GG) \equiv H^1(\erre^+) \oplus H^1(\erre^+) \oplus H^1(\erre^+), \qquad H^2(\GG) \equiv H^2(\erre^+) \oplus H^2(\erre^+) \oplus H^2(\erre^+), $$ equipped with the norms $$ \| \Psi \|_{H^1(\GG)}^2 \ = \ \sum_{i=1}^3 \| \psi_i \|_{H^1(\erre^+)}^2, \qquad \| \Psi \|_{H^2(\GG)}^2 \ = \ \sum_{i=1}^3 \| \psi_i \|_{H^2(\erre^+)}^2. $$ %For a given time interval $[t_a,t_b]$ we shall denote by $X$ the space $X := L^{\infty}_{[t_a, t_b] } L^2 (\GG) \cap L^{6}_{[t_a, t_b] } L^6 ( \GG) $ and by $\|\cdot\|_X$ the corresponding norm. %\\ %{\bf Claudio: }{\it Va bene definire qui lo spazio $X$? \`E necessario esplicitare la dipendenza dall'intervallo temporale?} %\\ %Moreover we shall use $X(\RE^+)$ and $X(\RE)$ to denote the spaces $X(\RE^+) := L^{\infty}_{[t_a, t_b] } L^2 (\RE^+) \cap L^{6}_{[t_a, t_b] } L^6 (\RE^+)$ and $X(\RE) := L^{\infty}_{[t_a, t_b] } L^2 (\RE) \cap L^{6}_{[t_a, t_b] } L^6 (\RE)$, and $\|\cdot\|_{X(\RE^+)}$ and $\|\cdot\|_{X(\RE)}$ to denote the corresponding norms. %\n %We shall use the notation $\OO_{\LG{r}}(\eta)$ and $\OO_{\LT{p}{0,T}\LG{r}}(\eta)$. $\Psi=\Phi+\OO_{\LG{r}}(\eta)$ meaning that there exists $\eta_0>0$ such that for all $0<\eta<\eta_0$, $\big\|\Psi-\Phi\big\|_{\LG{r}}\leqslant C\eta$ where $C$ is a positive constant which does not depend on $\eta$. Similarly for $\OO_{\LT{p}{0,T}\LG{r}}(\eta)$. We also use the convention $\OO(\eta)\equiv\OO_{\LG{2}}(\eta)$. %{\bf Claudio: }{\it Forse usiamo solo $\OO_{\LG{2}}$. Nel caso semplifica la parte sopra.} \n The product of functions is defined componentwise, \begin{equation*} \Psi\Phi\equiv \lf( \begin{array}{c} \psi_1\phi_1 \\ \psi_2\phi_2 \\ \psi_3\phi_3 \end{array} \ri), \qquad \text{so that} \qquad |\Psi|^2\Psi\equiv \lf( \begin{array}{c} |\psi_1|^2\psi_1 \\ |\psi_2|^2\psi_2 \\ |\psi_3|^2\psi_3 \end{array} \ri). \end{equation*} \n We denote by $\mathbb I$ the $3\times 3$ identity matrix, while $\mathbb J$ is the $3\times 3$ matrix whose elements are all equal to one. \n When an element of $L^2 (\GG)$ evolves in time, we use in notation a subscript $t$: for instance, $\Psi_t$. Sometimes we shall write $\Psi(t)$ in order to highlight the dependence on time, or whenever such a notation is more understandable. \section{\label{sec:summary}Summary on linear dynamics on graphs} \subsection{Hamiltonians and quadratic forms} Standard references about Schr\"odinger equation on graphs are \cite{[BCFK06],[BEH],[Kuc04],[Kuc05],[KS99]}, to which we refer for complete treatment. Here we only give the definitions needed to have a self-contained exposition.\par We consider three Hamiltonian operators, denoted by $H_F$, $H_\delta^\alpha$, $H_{\delta^\prime}^\beta$, and called, respectively, the Kirchhoff, the Dirac's delta, and the delta-prime Hamiltonian. The three of those operators act as \begin{equation} \Psi \ \longmapsto \ \lf( \begin{array}{c} -\psi_1'' \\ -\psi_2'' \\ -\psi_3'' \end{array} \ri) \label{san} \end{equation} on some subspace of $H^2 (\mathcal G)$, to be defined by suitable boundary conditions at the vertex. Here and in the following subsection we collect some well-known facts (see \cite{[KS99]}, \cite{[Kuc04]},\cite{[Kuc05]} \cite{[BCFK06]}) on $H_F$, $H_{\delta}^\alpha$, and $H_{\delta^\prime}^\beta$. The Kirchhoff Hamiltonian $H_F$ acts on the domain \begin{equation} {\mathcal D} (H_F): = \{ \Psi \in H^2(\GG) \text{ s.t. } \, \psi_1 (0) = \psi_2 (0) = \psi_3 (0), \, \psi_1' (0) + \psi_2' (0) + \psi_3'(0)=0 \}. \label{fuji} \end{equation} It is well known, see \cite{[KS99]}, that \eqref{fuji} and \eqref{san} define a self-adjoint Hamiltonian on $L^2(\GG)$. Boundary conditions in \eqref{fuji} are usually called Kirchhoff boundary conditions. We use the index $F$ to remind that $H_F$ reduces to the free Hamiltonian on the line for a degenerate graph composed of two half lines. \n The quadratic form ${\mathcal E}_F$ associated to $H_F$ is defined on the subspace \begin{equation*} {\mathcal D} ({\mathcal E}_F) = \{ \Psi \in H^1(\GG) \text{ s.t. } \, \psi_1 (0) = \psi_2 (0) = \psi_3 (0) \} %\label{toukyou} \end{equation*} and reads \begin{equation*} {\mathcal E}_F [\Psi ] = \sum_{i=1}^3 \int_0^{+\infty} |\psi_i ' (x) |^2 \,dx\,. \end{equation*} The Dirac's delta Hamiltonian is defined on the domain \begin{equation*} {\mathcal D} (H_\delta^\alpha): = \{ \Psi \in H^2(\GG) \text{ s.t. } \, \psi_1 (0) = \psi_2 (0) = \psi_3 (0), \, \psi_1' (0) + \psi_2' (0) + \psi_3'(0)= \alpha \psi_1 (0) \} %\label{fujidelta} \end{equation*} Again, $H_\delta^\alpha$ is a self-adjoint opeator on $L^2(\GG)$ (\cite{[KS99]}). It appears that $H_\delta^\alpha$ generalizes the ordinary Dirac's delta interaction on the line. \n The quadratic form ${\mathcal E}_\delta^\alpha$ associated to $H_\delta^\alpha$ is defined on \begin{equation*} {\mathcal D} ({\mathcal E}_\delta^\alpha) = \{ \Psi \in H^1 (\GG) \text{ s.t. } \, \psi_1 (0) = \psi_2 (0) = \psi_3 (0) \} %\label{toukyoudelta} \end{equation*} and is given by \begin{equation*} {\mathcal E}_\delta^\alpha [\Psi ] = \sum_{i=1}^3 \int_0^{+\infty} |\psi_i ' (x) |^2 \,dx\, + \alpha | \psi_1 (0) |^2. \end{equation*} The delta-prime Hamiltonian is defined on the domain \begin{equation*} {\mathcal D} (H_{\delta^\prime}^\beta): = \{ \Psi \in H^2(\GG) \text{ s.t. } \, \psi_1^\prime (0) = \psi_2^\prime (0) = \psi_3^\prime (0), \, \psi_1 (0) + \psi_2 (0) + \psi_3(0)= \beta \psi_1^\prime (0) \} %\label{fujiprime} \end{equation*} Again, $H_{\delta^\prime}^\beta$ is a self-adjoint opeator on $L^2(\GG)$ (\cite{[KS99]}). \n The quadratic form ${\mathcal E}_{\delta^\prime}^\beta$ associated to $H_{\delta^\prime}^\beta$ is defined on $ {\mathcal D} ({\mathcal E}_{\delta^\prime}^\beta) \ = \ H^1 (\GG) %\text{ s.t. } \, %\psi_1 (0) = \psi_2 (0) = \psi_3 (0) \} %%\label{toukyouoprime} %\end{equation*} $ and is given by \begin{equation*} {\mathcal E}_{\delta^\prime}^\beta [\Psi ] = \sum_{i=1}^3 \int_0^{+\infty} |\psi_i ' (x) |^2 \,dx\, + \frac{1}{\beta} \left| \sum_{i=1}^3 \psi_i (0) \right|^2. \end{equation*} \p Notice that $H_{\delta^\prime}^\beta$ does not reduce to the standard $\delta'$ interaction on the line when it is restricted to a two edges graph. Here we are following the notation in \cite{[Kuc04]}, \cite{[Kuc05]}. The present $\delta'$ vertex is called sometimes $\delta'_s$ graph, where $s$ is for symmetric. A discussion of the correct extension of the usual $\delta'$ interaction is given in \cite{[Ex]} and \cite{[BEH]}. For completeness, we give the operator domain (the action is the same as the other cases). We use the denomination $\tilde\delta'$ to avoid confusion with the previously defined interaction. \begin{equation*} {\mathcal D} (H_{\tilde\delta^\prime}^{\tilde\beta}): = \{ \Psi \in H^2(\GG) \text{ s.t. } \, \psi_j(0) -\psi_k (0) =\frac{\tilde\beta}{n}(\psi_j^\prime(0) -\psi_k^\prime(0))\ ,\quad j,k= 1,2,...n\} \ . %\label{veraprime} \end{equation*} Throughout the paper we restrict to the case of repulsive delta and delta-prime ($\delta'$) interaction, i.e., $\alpha, \beta > 0$. It is easily proved, for example by inspection of the operator resolvents in the following subsection, that such a condition prevents the corresponding Hamiltonian operator from possessing bound states. \subsection{Resolvents, scattering, and propagators} First we consider the case of the Kirchhoff Hamiltonian. %For any $\lam$ such that $\Im \lam>0$ we denote by $G^\pm(\lam)$ the %bounded operators in $L^2(\RE^+)$ with integral kernels %%% %\be %G^-(\lam;x,y):=\f{i}{2\lam } e^{i \lam |x-y| }\;,\quad %G^+(\lam;x,y):=\f{i}{2\lam } e^{i \lam (x+y)}\,. %\end{equation*} %%% For any complex number $k$ with $\Im k>0$ we denote by $R_F (k)$ the resolvent of $H_F$, namely $R_F(k) := (H_F - k^2 )^{-1}$. The operator $R_F (k)$ is the integral operator on $L^2 (\GG)$ whose kernel reads \begin{equation*} R_F (k; x,y) =\f{i}{2k } e^{i k |x-y| } \mathbb{I} +\f{i}{2k } e^{i k (x+y)} \f13 \lf( \begin{array}{ccc} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array} \ri). %\label{kyoutou} \end{equation*} From the expression of the resolvent one immediately has the reflexion and transmission coefficient: \begin{equation} \label{scoefff} r (k) \ = \ - \f 1 3, \qquad t (k) \ = \ \f 2 3. \end{equation} \n We define the function $U_t$ as \begin{equation*} %\label{defu} U_t (x) \ : = \ \f {e^{i \f {x^2}{4t}}} {\sqrt{4 \pi i t}}\,. \end{equation*} In the following we shall use the same symbol $U_t$ to denote the operator in $L^2(\RE)$ defined by %%% \begin{equation*} \big[U_t\psi\big](x):=\int_{-\infty}^\infty U_t(x-y)\psi(y)dy\,. \end{equation*} %%% Moreover, we define two integral operators $U_t^\pm$ acting on $L^2 (\erre^+)$ \begin{equation*} %\label{u+-} U_t^\pm \ : \ L^2 (\erre^+) \rightarrow L^2 (\erre^+), \quad \big[U_t^\pm \psi\big](x) = \int_0^{+ \infty} U_t ( x \pm y ) \psi (y) \, dy\,. \end{equation*} We stress that, according to our definitions, the operators $U^-_t$ and $U_t$ have the same integral kernel, but act on different Hilbert spaces. With this notation the propagator $e^{- i H_F t}$ is given by \begin{equation} \label{freepro} e^{- i H_F t} = \ U_t^- \mathbb{I} + U_t^+ \f 1 3 \left( \begin{array}{ccc} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array} \right)\, = \, ( U_t^- - U_t^+ ) {\mathbb I} + \f 2 3 U_t^+ {\mathbb J} . \end{equation} Concerning the Hamiltonian $H_\delta^\alpha$, the related resolvent $R_\delta^\alpha (k) : = ( H_\delta^\alpha - k^2 )^{-1}$ reads \begin{equation*} R_{\delta}^\alpha (k; x, y) = \f{i}{2k } e^{i k |x-y| } {\mathbb I} - \f{i}{2k } \f {e^{i k(x+y) }} {\al-3ik} \lf( \begin{array}{ccc} {\al-ik} & {2ik} & {2ik} \\ {2ik} & {\al-ik} & {2ik} \\ {2ik} & {2ik} & {\al-ik} \end{array} \ri), \end{equation*} therefore the scattering coefficients read \begin{equation} \label{scoeffd} r_{H_\delta^\alpha} (k) \ = \ - \f {k + i \alpha} {3k + i \alpha}, \qquad t_{H_\delta^\alpha} (k) \ = \ \f {2k} {3k + i \alpha}\ . \end{equation} We use the standard formula %that links resolvent and propagator one has $$ U_{\delta,t}^\alpha (x, y) \ = \ \f 1 {\pi i} \int_{- \infty}^{+ \infty} e^{-ik^2 t}\ R_{\delta}^\alpha (k; x, y) \, k d k, $$ where $U_{\delta,t}^\alpha (x, y)$ denotes the integral kernel that represents the unitary propagator of the operator $e^{-i H_\delta^\alpha t}$, and then, owing to the identity \begin{equation*} \label{formul} \f 1 {2 \pi} \int_{- \infty}^{+ \infty} \f {e^{i k (x + y)}} {a - i k} \, e^{- i k^2 t} \, dk \ = \ \int_0^{+ \infty} e^{- a u} \, \f{e^{i \f{(x+y+u)^2}{4t}}}{\sqrt{4 \pi i t}} \, du, \end{equation*} we finally obtain \begin{equation} \begin{split} \label{propdelta} U_{\delta,t}^\alpha (x, y) \ = \ [ U_t (x-y) - U_t (x + y)] {\mathbb I} + \f 2 3 \left[ U_t (x + y) - \f \alpha 3 \int_0^{+ \infty} du \, e^{- \f \alpha 3 u} U_t (x + y + u) \right] {\mathbb J}. \end{split} \end{equation} For the delta-prime interaction one can proceed analogously: first, the resolvent $R_{\delta^\prime}^\beta (k) : = ( H_{\delta^\prime}^\beta - k^2 )^{-1}$ is known and coincides with the integral operator in $L^2 (\GG)$ whose kernel reads \begin{equation*} R_{\de '}^\beta (k ; x, y) = \f{i}{2k } e^{i k |x-y| } {\mathbb I} - \f{i}{2k } \f{e^{i k(x+y) }} {3-i\beta k} \lf( \begin{array}{ccc} -1+i\beta k & {2} & {2} \\ {2} & -1+i\beta k & {2} \\ {2} & {2} & -1+i\beta k \end{array} \ri). \end{equation*} \begin{equation} \label{scoeffdp} r_{H_{\delta^\prime}^\beta} (k) \ = \ \f {\beta k + i} {\beta k + 3 i }, \qquad t_{H_{\delta^\prime}^\beta} (k) \ = \ - \f {2i} {\beta k + 3 i }. \end{equation} \n Again, using \eqref{formul} we get the integral kernel $U_{\delta^\prime, t}^\beta (x,y)$ of the propagator $e^{-i H_{\delta^\prime}^\beta t}$, namely \begin{equation} \begin{split} \label{propdeltaprime} U_{\delta^\prime,t}^\beta (x, y) \ = \ [ U_t (x-y) + U_t (x + y)] {\mathbb I} - \f 2 \beta \int_0^{+ \infty} du \, e^{- \f 3 \beta u} U_t (x + y + u) {\mathbb J}. \end{split} \end{equation} Throughout the paper we'll need some auxiliary dynamics to be compared with the dynamics described by \eqref{diffform}, so, for later convenience, we introduce the two-edge Hamiltonians $H_j $ and the corresponding two-edge propagators $e^{- i H_j t} $, $j=1,2,3$. \n Let $H_j $ be defined by: %%% \begin{equation*} \begin{aligned} {\mathcal D} (H_j) := \{ \Psi \in H^2(\GG) \text{ s.t. } \, & \psi_j(0)=\psi_{j+1}(0)\,,\,\psi_j'(0)+\psi_{j+1}'(0)=0\, , \psi_k(0)=0\, ,\,k\neq j,j+1 \} \end{aligned} \label{honshu} \end{equation*} %%% \begin{equation*} H_j \Psi := \lf( \begin{array}{c} -\psi_1'' \\ -\psi_2'' \\ -\psi_3'' \end{array} \ri)\,, %\label{hokkaido} \end{equation*} where in equation \eqref{honshu} it is understood that $j=\{1,2,3\}$ modulo 3. \n The Hamiltonian $H_j$ couples the edges $j$ and $j+1$ with a Kirchhoff boundary condition and sets a Dirichlet boundary condition for the remaining edge, so that there is free propagation between the edges $j$ and $j+1$ and no propagation between them and the edge $j+2$. \n With a straightforward computation we have %%% \begin{equation} \label{twoedgepro-j} e^{- i H_j t} \ = \ U_t^- \mathbb{I} + U_t^+ {\mathbb T}_j\,, \end{equation} %%% where ${\mathbb T}_j$ are the matrices %%%% %\be %\big(T_j\big)_{kl}=\left\{ %\begin{aligned} %1&\textrm{ for } k=j, l=j+1\\ %1&\textrm{ for } k=j+1, l=j\\ %-1&\textrm{ for } k=l \textrm{ and } k\neq j, j+1;\; l\neq j,j+1\\ %0&\textrm{ otherwise} %\end{aligned} %\right. %\end{equation*} %%%% %%% \begin{equation*} {\mathbb T}_1=\begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&-1 \end{pmatrix}\;;\quad {\mathbb T}_2=\begin{pmatrix} -1&0&0\\ 0&0&1\\ 0&1&0 \end{pmatrix}\;;\quad {\mathbb T}_3=\begin{pmatrix} 0&0&1\\ 0&-1&0\\ 1&0&0 \end{pmatrix}\,. \end{equation*} %%% \subsection{Strichartz's estimates} A key tool in our method is the extension of the standard Strichartz's estimates (see e.g. \cite{[Caz]}) to the dynamics on $\GG$ described by the propagators $e^{-i H_F t}$, $e^{-i H_\delta^\alpha t}$, and $e^{-i H_{\delta^\prime}^\beta t}$. \n In this subsection we use the symbol $H$ to denote any of the three Hamiltonians of interest. As a preliminary step, we remark that from equations \eqref{freepro}, \eqref{propdelta}, \eqref{propdeltaprime}, the standard dispersive estimate immediately follows: %%% \begin{equation} \big\|e^{-iH t}\Psi\big\|_{\LG{\infty}}\leqslant \frac{c}{t^{1/2}}\big\|\Psi\big\|_{\LG{1}}\,. \label{osaka} \end{equation} %%% %%%%%%%%%% %PROPOSITION %%%%%%%%%% \begin{prop}[Strichartz Estimates for $e^{-iH t}$] \mbox{} \\ \label{prop:stric} \n Let $\Psi_0 \in L^2 (\GG)$, $\Gamma \in L^q_\erre \LG{k}$, with $1\leqslant q,k \leqslant 2, \frac{2}{q}+\frac{1}{k}=\frac{5}{2}$, and define \begin{equation*} \Psi(t) = e^{-i H t} \Psi_0 \qquad \Phi(t) = \int_0^t ds\, e^{-i H (t-s)} \Gamma(s)\,. \end{equation*} The following estimates hold true: \begin{equation} \label{stric1} \lf\| \Psi \ri\|_{L^{p}_\erre \LG{r} } \leqslant c \| \Psi_0 \| \end{equation} \begin{equation} \label{stric2} \lf\| \Gamma \ri\|_{L^p_\erre \LG{r} } \leqslant c \lf\| \Phi(\cdot) \ri\|_{L^q_\erre \LG{k} } \end{equation} %%% \n for any pair of indices $(r,p)$ satisfying %%% \begin{equation} \label{admissible} 2\leqslant p,r\leqslant \infty \;,\; \frac{2}{p}+\frac{1}{r}=\frac{1}{2}. \end{equation} %%% The constants $c$ in \eqref{stric1} and \eqref{stric2} are independent of $T$. \end{prop} %%%%%% %PROOF %%%%%% \begin{proof} The proof is standard due to the dispersive estimate \eqref{osaka}, see for instance \cite{[Caz]} and \cite{[KT]}. \end{proof} \begin{rem} {\em If $H = H^\alpha_\delta$ ($H^{\beta}_{\delta^\prime}$), the constants appearing in \eqref{osaka}, \eqref{stric1} and \eqref{stric2} are independent of $\alpha$ ($\beta$). Indeed, by the change of variable $u \to u \alpha$ ($u \to u / \beta$) the integral term in \eqref{propdelta} (\eqref{propdeltaprime}) can be easily estimated independently of $ \alpha$ ($\beta$), obtaining a dispersive estimate \eqref{osaka} independent of the parameters and therefore, by the standard Strichartz machinery, uniform inequalities \eqref{stric1} and \eqref{stric2}. } \end{rem} \section{\label{sec:wp}Well-posedness and conservation laws} Here we treat the problem of the well-posedness, i.e., the existence and uniqueness of the solution to equation \eqref{intform1} % IN EFFETTI L'EQUAZIONE IN FORMA INTEGRALE DOVREBBE GIA' ESSERE % STATA PRESENTATA NELL'INTRODUZIONE in the energy domain of the system. Such a domain turns out to coincide with the form domain of the linear part of equation \eqref{diffform}. Throughout this section, such a linear part is denoted by $H$, and, according to the particular case under consideration, it can be understood as the Hamiltonian operator $H_F$, $H_\delta^\alpha$, or $H_{\delta^\prime}^\beta$. Correspondingly, we denote the associated energy domain simply by ${\mathcal D} ({\mathcal E})$. All of the following formulas can be specialized to the particular cases ${\mathcal D} ({\mathcal E}_F)$, ${\mathcal D} ({\mathcal E}_\delta^\alpha)$, or ${\mathcal D} ({\mathcal E}_{\delta^\prime}^\beta)$. Let us stress that throughout the paper we do not approximate the dynamics in $H^1$, but rather in $L^2$. Furthermore, local well-posedness in $L^2$ is ensured by Strichartz estimates (proposition \ref{prop:stric}), as is easily seen following the line exposed in \cite{[Caz]}, chapters 2 and 3. Nonetheless, we prefer to deal with functions in the energy domain, since they are physically more meaningful.%, and moreover they are continuous, therefore allowing %estimates in $L^\infty$. We follow the traditional line of proving, first of all, local well-posedness, and then extending it to all times by means of a priori estimates provided by the conservation laws. %We give all proofs in a compact yet hopefully exhaustive %form. \n For a more extended treatment of the analogous problem for a two-edge vertex (namely, the real line with a point interaction at the origin), see \cite{[AN09]}. Let us preliminarily define a norm in the energy domain: $$ \| \Psi \|_{{\mathcal D} ({\mathcal E})}^2 \ : = \ \| \Psi \|^2 + \| \Psi^\prime \|^2\ , $$ where we used the notation $$ \Psi^\prime \ : = \ \left( \begin{array}{c} \psi_1^\prime \\ \psi_2^\prime \\ \psi_3^\prime \end{array} \right). $$ %Obviously, the ${\mathcal D} ({\mathcal E})$-norm is preserved by %the propagator $e^{-i H t}$. Furthermore, we denote ${\mathcal D} ({\mathcal E})^\star$ the dual of ${\mathcal D} ({\mathcal E})$, i.e., the set of the continuous linear functionals on ${\mathcal D} ({\mathcal E})$. We denote the dual product of $\Gamma \in {\mathcal D} ({\mathcal E})^\star$ and $\Psi \in {\mathcal D} ({\mathcal E})$ by $ \langle \Gamma, \Psi \rangle$. In such a bracket we'll often exchange the place of the factor in ${\mathcal D} ({\mathcal E})^\star$ with the place of the factor in ${\mathcal D} ({\mathcal E})$: indeed, the duality product follows the same algebraic rules of the standard scalar product. \n As usual, one can extend the action of $H$ to the space ${\mathcal D} ({\mathcal E})$, with values in ${\mathcal D} ({\mathcal E})^\star$, by $$ \langle H \Psi_1 , \Psi_2 \rangle \ : = \ ( H^{\f 1 2} \Psi_1, H^{\f 1 2} \Psi_2 ). $$ Furthermore, for any $\Psi \in {\mathcal D} ({\mathcal E})$ the identity \begin{equation} \label{extder} \f d {dt} e^{-i H t} \Psi \ = \ - i H e^{-i H t} \Psi \end{equation} holds in ${\mathcal D} ({\mathcal E})^\star$ too. To prove it, one can first test the functional $\f d {dt} e^{-i H t} \Psi$ on an element $\Xi$ in the operator domain ${\mathcal D} (H)$, obtaining \begin{equation*} \nonumber \left\langle \f d {dt} e^{-i H t} \Psi, \Xi \right\rangle \ = \ \lim_{h \to 0} \left( \Psi, \f {e^{i H (t+h)} \Xi - e^{i H t} \Xi} h \right)\ = \ ( \Psi, i H e^{i H t} \Xi) \ = \ \langle -i H e^{-i H t} \Psi, \Xi \rangle. \end{equation*} Then, the result can be extended to $\Xi \in {\mathcal D} ({\mathcal E})$ by a density argument. \n Besides, by \eqref{extder}, the differential version \eqref{diffform} of the Schr\"odinger equation holds in ${\mathcal D} ({\mathcal E})^\star$. In order to prove a well-posedness result we need to generalize standard one-dimensional Gagliardo-Nirenberg estimates to graphs, i.e. \begin{equation} \label{gajardo} \| \Psi \|_{L^p} \ \leqslant \ C \| \Psi^\prime \|^{\f 1 2 - \f 1 p}_{L^2} \| \Psi \|^{\f 1 2 + \f 1 p}_{L^2}, \end{equation} where the constant $C$ depends on the index $p$ only. The proof of \eqref{gajardo} follows immediately from the analogous estimates for functions of the real line, considering that any function in $H^1 (\erre^+)$ can be extended to an even function in $H^1 (\erre)$, and applying this reasoning to each component of $\Psi$. \begin{prop}[Local well-posedness in ${\mathcal D} ({\mathcal E})$] \label{loch2} For any $\Psi_0 \in {\mathcal D} ({\mathcal E})$, there exists $T > 0$ such that the equation \eqref{intform1} has a unique solution $\Psi \in C^0 ([0,T), {\mathcal D} ({\mathcal E}) ) \cap C^1 ([0,T), {\mathcal D} ({\mathcal E})^\star)$. \n Moreover, eq. \eqref{intform1} has a maximal solution $\Psi^{\rm{max}}$ defined on the interval $[0, T^\star)$, and the following ``blow-up alternative'' holds: either $T^\star = \infty$ or $$ \lim_{t \to T^\star} \| \Psi_t^{\rm{max}} \|_{{\mathcal D} ({\mathcal E})} \ = \ + \infty, $$ where we denoted by $\Psi_t^{\rm{max}}$ the function $\Psi^{\rm{max}}$ evaluated at time $t$. \end{prop} \begin{proof} We define the space $ {\mathcal X} : = L^\infty ([0,T), {\mathcal D} ({\mathcal E})),$ endowed with the norm $ \| \Psi \|_{\mathcal X} \ : = \ \sup_{t \in [0,T)} \| \Psi_t \|_{{\mathcal D} ({\mathcal E})}. $ Given $\Psi_0 \in {\mathcal D} ({\mathcal E})$, we define the map $G : {\mathcal X} \longrightarrow {\mathcal X}$ as $$ G \Phi : = e^{- i H \cdot} \Psi_0 + i \int_0^\cdot e^{- i H (\cdot - s)} | \Phi_s |^2 \Phi_s \, ds. $$ Notice that the nonlinearity preserves the space ${\mathcal D} ({\mathcal E})$. Indeed, since for any component $\psi_i$ of $\Psi$, $\psi_i^{\prime}$ belongs to $L^2 (\erre^+)$, then $ |\psi_i|^2 \psi_i$ belongs to $L^2 (\erre^+)$ too, and so the energy space for the delta-prime case is preserved. Furthermore, the product preserves the continuity at zero required by the Kirchhoff and the delta case. By estimates \eqref{gajardo} one obtains $$ \| | \Phi_s |^2 \Phi_s \|_{{\mathcal D} ({\mathcal E})} \ \leqslant \ C \| \Phi_s \|_{{\mathcal D} ({\mathcal E})}^3, $$ so \begin{equation} \label{contraz1} \begin{split} \| G \Phi \|_{\mathcal X} \ \leqslant \ & \| \Psi_0 \|_{{\mathcal D} ({\mathcal E})} + C \int_0^T \| \Phi_s \|_{{\mathcal D} ({\mathcal E})}^3 \, ds \ \leqslant \ \| \Psi_0 \|_{{\mathcal D} ({\mathcal E})} + C T \| \Phi \|_{\mathcal X}^3\,. \end{split} \end{equation} Analogously, given $\Phi, \Xi \in {\mathcal D} ({\mathcal E})$, \begin{equation} \label{contraz2} \begin{split} \| G \Phi - G \Xi \|_{\mathcal X} \ \leqslant \ & C T \left( \| \Phi \|_{\mathcal X}^2 + \| \Xi \|_{\mathcal X}^2 \right) \| \Phi - \Xi \|_{\mathcal X}\,. \end{split} \end{equation} We point out that the constant $C$ appearing in \eqref{contraz1} and \eqref{contraz2} is independent of $\Psi_0$, $\Phi$, and $\Xi$. Now let us restrict the map $G$ to elements $\Phi$ such that $\| \Phi \|_{\mathcal X} \leqslant 2 \| \Psi_0 \|_{{\mathcal D} ({\mathcal E})}$. From \eqref{contraz1} and \eqref{contraz2}, if $T$ is chosen to be strictly less than $(8C \| \Psi_0 \|_{{\mathcal D} ({\mathcal E})}^2)^{-1}$, then %$ \| G \Phi \|_{\mathcal % X} \leqslant 2 \| \Psi_0 \|_{{\mathcal D} ({\mathcal E}_F)}$, so the ball of radius %$ 2 \| \Psi_0 \|_{{\mathcal D} ({\mathcal E}_F)}$ is invariant under the action of $G$. If %furthermore one chooses $T < (16 C \| \Psi_0 \|_{{\mathcal D} ({\mathcal E}_F)}^3)^{-1}$, then $G$ is a contraction of the ball in ${\mathcal X}$ of radius $ 2 \| \Psi_0 \|_{{\mathcal D} ({\mathcal E})}$, and so, by the contraction lemma, there exists a unique solution to \eqref{intform1} in the time interval $[0, T)$. By a standard one-step boostrap argument one immediately has that the solution actually belongs to $C^0 ([0,T), {\mathcal D} ({\mathcal E}))$, and due to the validity of \eqref{diffform} in the space ${\mathcal D} ({\mathcal E})^\star$ we immediately have that the solution $\Psi$ actually belongs to $C^0 ([0,T), {\mathcal D} ({\mathcal E}))) \cap C^1 ([0,T),{\mathcal D} ({\mathcal E})^\star)$. The proof of the existence of a maximal solution is standard, while the blow-up alternative is a consequence of the fact that, whenever the ${\mathcal D} ({\mathcal E})$-norm of the solution is finite, it is possible to extend it for a further time by the same contraction argument. \end{proof} The next step consists in the proof of the conservation laws for our problem. \begin{prop} For any solution $\Psi \in C^0 ([0,T), {\mathcal D} ({\mathcal E})) \cap C^1 ([0,T), {\mathcal D} ({\mathcal E})^\star)$ to the problem \eqref{intform1}, the following conservation laws hold at any time $t$: \begin{equation*} %\label{conslaws} \| \Psi_t \| \ = \ \| \Psi_0 \|, \qquad {\mathcal E} ( \Psi_t ) \ = \ {\mathcal E} ( \Psi_0 ), \end{equation*} where the symbol $ {\mathcal E}$ denotes the {\em energy functional} \begin{equation*} %\label{energy} {\mathcal E} ( \Psi_t ) \ : = \ \f 1 2 {\mathcal E}_{lin} ( \Psi_t ) - \f 1 4 \| \Psi_t \|_{L^4}^4. \end{equation*} Here the functional ${\mathcal E}_{lin}$ coincides with ${\mathcal E}_{F}$, ${\mathcal E}_{\delta}^\alpha$ or ${\mathcal E}_{\delta^\prime}^\beta$, according to the case one considers. \end{prop} \begin{proof} First, one can directly prove that $e^{iHt} \Psi_t$ is differentiable as a function of $t$ with values in ${\mathcal D} ({\mathcal E})$, and \begin{equation} \label{stimella} \f d {dt} e^{iHt} \Psi_t \ = \ i e^{iHt} |\Psi_t |^2 \Psi_t. \end{equation} Then, according to proposition \ref{loch2}, given $\Xi \in {\mathcal D} ({\mathcal E})$ one has \begin{equation*} \nonumber \begin{split} \left\langle \f d {dt} \Psi_t, \Xi \right\rangle & \ : = \ \f d {dt} ( \Psi_t, \Xi ) \ = \ \f d {dt} ( e^{iHt} \Psi_t, e^{iHt} \Xi ) %\\ \ = \ ( i |\Psi_t |^2 \Psi_t, \Xi ) + \langle -i H \Psi_t, \Xi, \rangle \end{split} \end{equation*} where we used \eqref{stimella} and \eqref{extder}. Therefore, the conservation of the $L^2$-norm can be immediately obtained by $$ \f d {dt} \| \Psi_t \|^2 \ = \ 2 \, {\rm{Re}} \, \left\langle \Psi_t , \f d {dt} \Psi_t \right\rangle \ = \ 2 \, {\rm{Im}} \, \langle \Psi_t , H \Psi_t \rangle. $$ In order to prove the conservation of the energy, first we assume that $\langle \Psi_t, H \Psi_t \rangle$ is differentiable as a function of time and \begin{equation} \label{previous} \f d {dt} (\Psi_t , H \Psi_t)\ = \ 2 \, {\rm{Re}} \, \left\langle \f d {dt} \Psi_t , H \Psi_t \right\rangle \ = \ 2 \, {\rm{Im}} \, \langle | \Psi_t |^2 \Psi_t , H \Psi_t \rangle, \end{equation} where, in the last equality, we used \eqref{diffform} and the self-adjointness of $H$. Furthermore, \begin{equation} \label{naechst} \f d {dt} (\Psi_t , | \Psi_t |^2 \Psi_t) \ = \ \f d {dt} (\Psi_t^2 , \Psi_t^2) \ = \ 4 \, {\rm{Im}} \, \langle | \Psi_t |^2 \Psi_t , H \Psi_t \rangle. \end{equation} From \eqref{previous} and \eqref{naechst} one then obtains $$ \f d {dt} {\mathcal E} (\Psi_t) \ = \ \f 1 2 \f d {dt} \langle \Psi_t , H \Psi_t \rangle - \f 1 4 \f d {dt} (\Psi_t , | \Psi_t |^2 \Psi_t)_{L^2} \ = \ 0 $$ and the proposition is proved. %%%% LO METTIAMO DOPO %%%%% It remains to show \eqref{previous}. To this purpose, it suffices to notice that \begin{equation*} \begin{split} & \f 1 h \left[ \langle \Psi_{t+h}, H \Psi_{t+h} \rangle - \langle \Psi_{t}, H \Psi_{t} \rangle \right] %\\ \ = \ \left\langle \f{\Psi_{t+h} - \Psi_t} h, H \Psi_{t+h} \right\rangle + \left\langle H \Psi_{t} ,\f{\Psi_{t+h} - \Psi_t} h \right\rangle \end{split} \end{equation*} and then pass to the limit. \end{proof} \begin{cor} The solutions are globally defined in time. \end{cor} \begin{proof} By estimate \eqref{gajardo} with $p = \infty$ and conservation of the $L^2$-norm, there exists a constant $M$, that depends on $\Psi_0$ only, such that $$ {\mathcal E} (\Psi_0) \ = \ {\mathcal E} (\Psi_t) \ \geq \ \f 1 2 \| \Psi_t^\prime \|^2 - M \| \Psi_t^\prime \| $$ Therefore one obtains a uniform (in $t$) bound on $ \| \Psi_t^\prime \|^2$. As a consequence, one has that no blow-up in finite time can occur, and therefore, by the blow-up alternative, the solution is global in time. \end{proof} \section{Main Result} In this section we describe the asymptotic dynamics of a particular initial state, which resembles a soliton for the standard cubic NLS on the line.% as will be clear in the following. \n We use the notation \begin{equation*} %\label{fi} \phi (x) \ = \ \sqrt 2 \cosh^{-1} x\,, \end{equation*} and define \begin{equation} \label{fix0t} \phisol{x_0,v}(x,t) = e^{i \f v 2 x} {e^{- i t \f {v^2} 4}}e^{it} \phi ( x - x_0 - vt)\,. \end{equation} The function $\phisol{x_0,v}$ represents a soliton for the cubic NLS on the line centered at time $t=0$ in $x=x_0$ moving with speed $v$. Therefore, $\phisol{x_0,v}$ is the solution of the integral equation %%% \begin{equation} \label{eq:sol} \phisol{x_0,v}(x,t) = \big[U_t e^{i \f v 2 \cdot} \phi ( \cdot - x_0)\big](x) +i \int_0^t \big[U_{t-s}|\phisol{x_0,v}(\cdot,s)|^2\phisol{x_0,v}(\cdot,s)\big](x)ds \,. \end{equation} %%% \n We take as initial datum the following function \begin{equation*} %\label{init} \Psi_0 (x) \ = \ \left( \begin{array}{c} \chi (x) e^{-i \f v 2 x} \phi (x - x_0) \\ 0 \\ 0 \end{array} \right) \end{equation*} where $\chi$ is a cut off function, that is $\chi \in C^\infty (\erre^+ )$, $\chi = 1$ in $(2, + \infty)$ and $\chi = 0$ in $(0,1)$. For later use we define also $\chi_+ = \chi_{[0, +\infty)}$ and $\chi_- = \chi_{( -\infty,0]}$, where $\chi_{[a,b]}$ denotes the characteristic function of the interval $[a,b]$. \n Furthermore, let $x_0$ and $v$ be two positive constants and let the centre of the soliton be located at \begin{equation*} %\label{initcenter} x_0 \ \geqslant \ v^{1 - \delta} %x_0 \ \leqslant \ - v^{1 - \delta}. \end{equation*} with $0<\delta<1$. According to section \ref{sec:wp}, we use the symbol $H$ to generically denote the linear part of the evolution, regardless of the fact that we are considering the Kirchhoff, delta, or delta-prime boundary conditions. When necessary, we will distinguish between the three of them. Let $\Psi_{H,t}$ be the solution of the equation \begin{equation} \label{intform} \Psi_{t} \ = \ e^{-iH t} \Psi_0 + i \int_0^t ds \, e^{-i H (t-s)} | \Psi_s |^2 \Psi_s\,. \end{equation} Let us set $t_2:=x_0/v+v^{-\delta}$ and define the following functions: \begin{equation} \label{Phi1t2} \Phi^{1}_{H,t_2}(x_1 ,x_2, x_3)\equiv \begin{pmatrix} \tilde r_H e^{-i \f {v^2} 4 t_2}e^{i\f v 2 x_1} e^{it_2} \phi (x_1 + x_0 - vt_2)\\ \tilde r_H e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_2} e^{it_2} \phi (x_2 - x_0 + vt_2)\\ 0 \end{pmatrix} \end{equation} %%%%%% \begin{equation} \label{Phi2t2} \Phi^{2}_{H,t_2}(x_1 ,x_2, x_3)\equiv \begin{pmatrix} 0\\ \tilde t_H e^{-i \f {v^2} 4 t_2}e^{i\f v 2 x_2} e^{it_2} \phi (x_2 + x_0 - vt_2)\\ \tilde t_H e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_3} e^{it_2} \phi (x_3 - x_0 + vt_2) \end{pmatrix} \end{equation} %%%%%% \begin{equation} \label{Phi3t2} \Phi^{3}_{H,t_2}(x_1 ,x_2, x_3)\equiv \begin{pmatrix} \tilde t_H e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_1} e^{it_2} \phi (x_1 - x_0 + vt_2)\\ 0\\ \tilde t_H e^{-i \f {v^2} 4 t_2}e^{i\f v 2 x_3} e^{it_2} \phi (x_3 + x_0 - vt_2) \end{pmatrix} \end{equation} They represent solitons on the line multiplied by the scattering coefficients of the linear dynamics $\tilde r_H$ and $\tilde t_H$, %that will turn out to be equal to $\tilde r_F$, $\tilde %t_F$, %$\tilde r_\delta (\tilde \alpha)$, $\tilde t_\delta (\tilde \alpha)$, % $\tilde r_{\delta^\prime} (\tilde \alpha)$, $\tilde t_{\delta^\prime} % (\tilde \alpha)$, %according to the particular boundary condition we are assuming in the %vertex. The coefficient $\tilde r$ and $\tilde t$ that, in the particular regime we consider, are defined as follows: \begin{equation} \label{tildscoeff} \begin{split} & \tilde r_{H_F} \ = \ - 1/3, \qquad \tilde t_{H_F} \ = \ 2/3 \\ & \tilde r_{H_\delta^\alpha} \ = \ - \f {1 + 2 i \tilde \alpha} {3 + 2 i \tilde \alpha}, \qquad \tilde t_{H_\delta^\alpha} \ = \ \f 2 {3 + 2 i \tilde \alpha} \\ & \tilde r_{H_{\delta^\prime}^\beta} \ = \ \f{\tilde {\beta} + 2 i} {\tilde \beta + 6 i}, \qquad \tilde t_{H_{\delta^\prime}^\beta} \ = \ - \f {4 i} {\tilde \beta + 6 i } \end{split} \end{equation} where $\tilde \alpha$ and $\tilde \beta$ are the rescaled interaction parameters, namely $$ \tilde \alpha : = \alpha / v, \qquad \tilde \beta : = \beta v. $$ \begin{rem}{\em Notice that the coefficients $\tilde r_H$ and $\tilde t_H$ can be obtained by the scattering coefficients \eqref{scoeffd}, \eqref{scoeffdp}, identifying $k$ with $v/2$ and replacing $\alpha$ by $ \tilde \alpha v$ and $\beta$ by $\tilde \beta / v$. This is due to the fact that we implicitly considered a particle with mass equal to $1/2$, therefore the momentum $k$ is linked to the speed $v$ by $k = v/2$. %Inserting such a relationship into identities \eqref{scoeffd}, %and \eqref{scoeffdp}, %one obtains \eqref{tildscoeff}.} } \end{rem} For any $t>t_2$ we define the vectors $\Phi^j_{H,t}$ as the evolution of $\Phi^j_{H,t_2}$ with the nonlinear flows generated by $H_j$, i.e., they are solutions of the equation \begin{equation} \label{Phijt} \Phi^j_{H,t}=e^{-i H_j (t-t_2)} \Phi^j_{H,t_2} + i \int_{t_2}^t ds \, e^{-i H_j (t-s)} | \Phi^j_{H,s} |^2 \Phi^j_{H,s}\qquad j=1,2,3\,. \end{equation} %%%%%%% %REMARK %%%%%%% \begin{rem} {\em The vectors $\Phi^j_{H,t}$ can be represented by %%% \begin{equation} \label{Phi123t} \Phi^1_{H,t}(x_1 ,x_2, x_3)= \begin{pmatrix} e^{-i \f {v^2} 4 t_2}e^{it_2} \phi^{ref}_{t-t_2}(x_1 )\\ e^{-i \f {v^2} 4 t_2} e^{it_2} \phi^{ref}_{t-t_2}(-x_2)\\ 0 \end{pmatrix} \;;\quad \Phi^2_{H,t}(x_1 ,x_2, x_3)= \begin{pmatrix} 0\\ e^{-i \f {v^2} 4 t_2} e^{it_2} \phi^{tr}_{t-t_2} (x_2)\\ e^{-i \f {v^2} 4 t_2} e^{it_2} \phi^{tr}_{t-t_2} (-x_3) \end{pmatrix} \end{equation} \begin{equation} \label{Phi1234t} \Phi^3_{H,t}(x_1 ,x_2, x_3)= \begin{pmatrix} e^{-i \f {v^2} 4 t_2} e^{it_2} \phi^{tr}_{t-t_2} (-x_1)\\ 0\\ e^{-i \f {v^2} 4 t_2} e^{it_2} \phi^{tr}_{t-t_2} (x_3) \end{pmatrix}\, \end{equation} %%% where the functions $\phi^{ref}_t$ and $\phi^{tr}_t$ are the solutions to the following NLS on the line %%% \begin{equation} \label{phiref} \phi^{ref}_t(x)= \tilde r_H \int_{-\infty}^\infty U_{t}(x-y) e^{i\f v 2 y} \phi (y - v^{1-\de})dy +i\int_{0}^tds \int_{-\infty}^\infty U_{t-s}(x-y)|\phi^{ref}_s(y)|^2 \phi^{ref}_s(y)dy \end{equation} \begin{equation} \label{phitr} \phi^{tr}_t(x)= \tilde t_H\int_{-\infty}^\infty U_{t}(x-y) e^{i\f v 2 y} \phi (y - v^{1-\de})dy +i\int_{0}^tds \int_{-\infty}^\infty U_{t-s}(x-y)|\phi^{tr}_s(y)|^2 \phi^{tr}_s(y)dy\,. \end{equation}} \end{rem} Our main result is summarized in the following theorem: %%%%%%%% %THEOREM %%%%%%%% \begin{theorem} \label{mainth} There exists $\tau_* >0$ such that if we put $T_{\ast}= \tau_* \de /2$ and we take $0< T 0. \end{equation} Using the one-dimensional homogeneous Strichartz's estimates for $U_{t-t_a}$, namely, the analogous of \eqref{stric1} for functions of the half line, we can estimate the $X_{t_a, t_b}(\RE^+)$-norm of this term as $$ %\| U^-_t e^{i \f v 2 \cdot} \phi (\cdot + x_0) \|_{X(\erre^+)}\leqslant \|U_{t-t_a} \chi_+ e^{i \f v 2 \cdot} \phi (\cdot + x_0) \|_{X_{t_a, t_b}( \RE )} \leqslant C \| \chi_+ e^{i \f v 2 \cdot} \phi (\cdot + x_0) \| \ \leqslant C e^{-x_0},%%%%% \leqslant %\ C e^{-v^{1 -\delta}}, $$ where we used the notation $X_{t_a, t_b} (\erre) : = L^\infty_{[t_a, t_b]} L^2 (\erre) \cap L^6_{[t_a, t_b]} L^6 (\erre)$. \n The norm of the last term in the r.h.s. of equation \eqref{terms} can be estimated in a similar way by \begin{equation} \label{akak} \left\| \int_{t_a}^{\cdot} \big[ U_{\cdot-s} |\phi_{-x_0 , v} (s)|^2 \phi_{-x_0 , v} (s) \big]ds \right\|_{X_{t_a, t_b}(\RE^+)} \ \leqslant C \| \phi_{-x_0 , v}^3\|_{L^1_{[t_a,t_b]} L^2(\RE^+)} \leqslant C\f {1} {v} e^{-x_0 + vt_b}. \end{equation} Therefore, from \eqref{terms}, \eqref{kaka}, and \eqref{akak} we get \begin{equation*} \| K_1 \|_{X_{t_a, t_b}(\erre^+)} \ \leqslant \ C e^{-x_0 + vt_b}. \end{equation*} To estimate $K_2$, the first term in its definition \eqref{cappa} can be treated as in \eqref{kaka}, while the second is estimated following the line of \eqref{akak}. \end{proof} \begin{lemma} \label{blocco} Given $0 \ls t_a \ls t_b \ls t_1$, let $a$ and $b$ two positive numbers, with \begin{equation*} b \leqslant \f 1 {8 a^2 + 4 a}. \end{equation*} Moreover, let $y$ be a real, continuous function such that $0 \leqslant y (t_a) \leqslant a$, and \begin{equation} \label{constr} 0 \ls y (t) \ls a + b y^2 (t) + b y^3 (t), \qquad {\mbox{for any }} \ t \in [t_a, t_b]. \end{equation} Then, \begin{equation*} %\label{raddoppia} \max_{t \in [t_a, t_b]} y (t) \, \ls \, 2 a. \end{equation*} \end{lemma} \begin{proof} %Fixed $a > 0$, %notice that Consider the function $f_b(x) : = b x^3 + bx^2 - x + a$.%%%%% admits a unique local %minimum at the point %$$ %x_{min} : = - \f 1 3 + \f 1 3 \sqrt{1 + \f 3 b}. %$$ %Furthermore, i \ Denoted $\bar b := \f 1 {8 a^2 + 4 a}$, one has %the minimum of $f_{\bar b}$ is located at $\bar %x_{min} = 2a$ and $ f_{\bar b} (2a) \ = 0%\ - \f {a^2}{3a + 1} \ < \ 0 $. %\bar x_{min} : = \f 3 4 a + \sqrt{ \f 9 {16} a^2 + \f 5 8 a + \f 1 % {16}} - \f 1 4, %$$ %and $f_{\bar b} (\bar x_{min}) = 0$. \n If $b \ls \bar b$, then $f_b (x) \ls f_{\bar b} (x)$ for any $x > 0$. Besides, notice that $f_b (0) = a > 0$, then there must be a point $\tilde x \in (0, 2a]$ s.t. $f_b (\tilde x) = 0$. Finally, since the function $y$ is continuous, in order to satisfy the constraint \eqref{constr} one must have $$ y (t) \ls \tilde x \ls 2a. $$ \end{proof} %%%%%%%%%% %PROPOSITION %%%%%%%%%% \begin{prop} Let $\Psi_t$ be the solution of the equation \eqref{intform}, and $\Phi_t$ be the solution of equation \eqref{duhamsol}. There exists $C > 0$, independent of $t$ and $v$, such that \label{fase1} \begin{equation} \label{error1} \| \Psi_t - \Phi_t \| \ \leqslant \ C e^{-v^{1 - \delta}} \end{equation} for any $t \in [0, t_1]$. \end{prop} \begin{proof} \n Let us define $\Xi_t : = \Psi_t - \Phi_t$, and fix $t_a \in [0, t_1]$. Then, from equations \eqref{intform} and \eqref{duhamsol}, we have \begin{equation*} %\label{Xit} \begin{split} \Xi_t \ = & \ e^{-iH (t-t_a)} \Xi_0 + (e^{-iH (t-t_a)} - e^{-i H_1 (t-t_a)}) \Phi_0 + i \int_{t_a}^t (e^{-iH (t-s)} - e^{-i H_1 (t-s)}) | \Phi_s |^2 \Phi_s \, ds\\ & \ + i \int_{t_a}^t e^{-iH (t-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right]\\ = & \ e^{-iH (t-t_a)} \Xi_0 + F(t_a,t)\\ & + i \int_{t_a}^t e^{-iH (t-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right], \end{split} \end{equation*} where we defined \begin{equation*} %\label{effe} F (t_a, t) \ : = \ (e^{-iH (t - t_a)} - e^{-i H_1 (t - t_a)}) \Phi_0 + i \int_{t_a}^{t} (e^{-iH (t-s)} - e^{-i H_1 (t -s)}) | \Phi_s |^2 \Phi_s \, ds \end{equation*} Let us fix $t_b \in [t_a,t_1]$, and denote $X_{t_a, t_b} = L^{\infty}_{[t_a, t_b] } L^2 \cap L^{6}_{[t_a, t_b] } L^6$. Then \begin{equation} \begin{split} \label{split} \| \Xi \|_{X_{t_a, t_b}} \ \ls & \ \| e^{-iH (\cdot -t_a)} \Xi_{t_a} \|_{X_{t_a, t_b}} + \| F(t_a, \cdot) \|_{X_{t_a, t_b}} \\ &+ \left\| \int_{t_a}^\cdot e^{-iH (\cdot-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right] \right\|_{X_{t_a, t_b}} . \end{split} \end{equation} Using \eqref{stric1} the first term in the r.h.s can be estimated as $$ \| e^{-iH (\cdot -t_a)} \Xi_{t_a} \|_{X_{t_a, t_b}} \ \ls \ C \| \Xi_{t_a} \|. $$ \n We estimate the integral term in the r.h.s. of \eqref{split} also using Strichartz's estimates. Let us analyse in detail the cubic term. Since both pairs of indices $(\infty ,2 )$ and $(6,6)$ fulfil \eqref{admissible}, in \eqref{stric2} we can choose $q=k= 6/5$ and obtain \begin{equation} \lf\| \int_{t_a}^{\cdot} e^{-iH (\cdot-s)} | \Xi_s |^2 \Xi_s \,ds \ri\|_{X_{t_a, t_b}} \leqslant C \| | \Xi_{\cdot} |^2 \Xi_{\cdot} \|_{ \LT{6/5}{t_a,t_b} \LG{6/5} }. \label{yume} \end{equation} Moreover, by standard H\"older estimates, \begin{equation} \label{hoel1} \| | \Xi_{\cdot} |^2 \Xi_{\cdot} \|_{ \LT{6/5}{t_a,t_b} \LG{6/5} }\ \leqslant \ \left\| \| \Xi_\cdot \|^2_{L^6} \| \Xi_\cdot \|_{L^2} \right\|_{\LT{6/5}{t_a,t_b}} \ \ls \ \| \Xi \|^2_{ \LT{6}{t_a,t_b} \LG{6} } \| \Xi \|_{ \LT{2}{t_a,t_b} \LG{2} } \ \ls \ ( t_b - t_a)^{\f 1 2} \| \Xi \|^2_{ \LT{6}{t_a,t_b} \LG{6} } \| \Xi \|_{ \LT{\infty}{t_a,t_b} \LG{2} }. \end{equation} Then, by \eqref{yume} and \eqref{hoel1}, \begin{equation} \label{eq-dopo-hoel1} \lf\| \int_{t_a}^{\cdot} e^{-iH (\cdot-s)} | \Xi_s |^2 \Xi_s \,ds \ri\|_{X_{t_a, t_b}} \leqslant C (t_b-t_a )^{1/2} \|\Xi\|^3_{X_{t_a, t_b}}. \end{equation} Notice that the constant $C$ can be chosen independently of $t_a, t_b$, and of the boundary condition at the vertex. \n The other terms in the integral in the r.h.s. of \eqref{split} can be estimated analogously. One ends up with \begin{equation*} %\label{split2} \begin{split} \| \Xi \|_{X_{t_a, t_b}} \ls & \ C \ \| \Xi_{t_a} \| + \| F(t_a, \cdot) \|_{X_{t_a, t_b}} + C (t_b - t_a)^{\f 1 2} \| \Xi \|^3_{X_{t_a, t_b}} + C (t_b - t_a)^{\f 2 3} \| \Xi \|^2_{X_{t_a, t_b}} + C (t_b - t_a)^{\f 5 6} \| \Xi \|_{X_{t_a, t_b}}, \end{split} \end{equation*} where the arising norms of $\Phi$ were absorbed in the constant $C$. \n If $t_b$ and $t_a$ are sufficiently close, then $C (t_b - t_a)^{\f 5 6} < 1/2$. Furthermore, since the quantity $t_b - t_a$ is upper bounded, one can estimate $(t_b - t_a)^{\f 2 3}$ by $C (t_b - t_a)^{\f 1 2}$. So \begin{equation} \label{split3} \begin{split} \| \Xi \|_{X_{t_a, t_b}} \ls & \ \widetilde C \ \| \Xi_{t_a} \| + \widetilde C \ \| F(t_a, \cdot) \|_{X_{t_a, t_b}} + \widetilde C (t_b - t_a)^{\f 1 2} \left[ \| \Xi \|^3_{X_{t_a, t_b}} + \| \Xi \|^2_{X_{t_a, t_b}} \right]. \end{split} \end{equation} Applying lemma \ref{blocco} to the function $y(t) = \| \Xi \|_{X_{t_a, t}}$, which is continuous and monotone, one has that, if $$ t_b - t_a \ \ls \ \left( 8 \widetilde C^3 \ ( \ \| \Xi_{t_a} \| + \| F(t_a, \cdot) \|_{X_{t_a, t_b}})^2 + 4 \widetilde C^2 \ ( \| \Xi_{t_a} \| + \| F(t_a, \cdot) \|_{X_{t_a, t_b}} ) \right)^{-2}, $$ then $\| \Xi \|_{X_{t_a, t_b}} \ls \ 2 \widetilde C \ \| \Xi_{t_a} \| + 2 \wt C \ \| F(t_a, \cdot) \|_{X_{t_a, t_b}}$. From the immediate estimates \begin{equation*} %\label{easy} \| \Xi_{t_a} \| \ \ls \ 4, \qquad \| F(t_a, \cdot) \|_{X_{t_a, t_b}} \ \ls \ \| F(0, \cdot) \|_{X_{0, t_1}}, \end{equation*} if one denotes \begin{equation*} %\label{tauu} \tau \ : = \ \left( 8 \widetilde C^3 \ ( 4 + \| F(t_a, \cdot) \|_{X_{0, t_1}})^2 + 4 \widetilde C^2 \ ( 4 + \| F(t_a, \cdot) \|_{X_{0, t_1}} ) \right)^{-2}, \end{equation*} %the quantity $t_b - t_a$ can be chosen independently of $t_a$, so %there exists $\tau > 0$ such that, then for any $t \in [0, t_1)$ %one has $\| \Xi \|_{X_{t, t + \tau}} \ls \ 2 \widetilde C \ \| \Xi_{t} \| + 2 \wt C \ \| F(t, \cdot) \|_{X_{t, t+ \tau}}$. We divide the interval $[0, t_1]$ in $N + 1$ subintervals as follows $$ [0, t_1] \ = \ \left( \cup_{j=0}^{N-1} [j \tau, (j+1) \tau] \right) \cup [N \tau, t_1], $$ where $$ N : = \left[ \f {x_0 - v^{1 - \delta}}{v \tau} \right], \qquad [ \cdot ] = {\mbox{ integer part.}} $$ Making use of lemma \ref{blocco}, and noting that $\| \Xi_{(j+1) \tau} \| \ls \| \Xi \|_{X_{j\tau, (j+1) \tau}}$, one proves by induction that \begin{equation} \label{indu} \begin{split} \| \Xi \|_{X_{j\tau, (j+1) \tau}} \ & \ls \ (2 \widetilde C)^{j+1} \| \Xi_0 \| + \sum_{k = 0}^{j} (2 \widetilde C)^{j+1-k} \| F (k\tau, \cdot) \|_{X_{k\tau, (k+1) \tau}}, \qquad j = 0, \dots , N - 1, \\ \| \Xi \|_{X_{N\tau, t_1}} \ & \ls \ (2 \widetilde C)^{N+1} \| \Xi_0 \| + \sum_{k = 0}^{N-1} (2 \widetilde C)^{N+1-k} \| F (k\tau, \cdot) \|_{X_{k\tau, (k+1) \tau}} + 2 \wt C \ \| F (N\tau, \cdot) \|_{X_{N\tau, t_1 }} \end{split} \end{equation} where the last inequality comes from the fact that $t_1 - N \tau \ls \tau$, so lemma \ref{blocco} applies to this last step too. \n The norm of $\Xi$ as a function of the whole time interval $[0, t_1]$ can be estimated by \begin{equation} \label{almost} \begin{split} \| \Xi \|_{X_{0,t_1}} \ \ls & \ \sum_{j=0}^{N-1} \| \Xi \|_{X_{j\tau, (j+1) \tau}} + \| \Xi \|_{X_{N\tau, t_1}} \\ \ls & \ \sum_{j=0}^{N} (2 \widetilde C)^{j+1} \| \Xi_0 \| + \sum_{j=0}^{N} \sum_{k = 0}^{j} (2 \widetilde C)^{j+1-k} \| F (k\tau, \cdot) \|_{X_{k\tau, \min\{t_1, (k+1) \tau\}}} \,. \end{split} \end{equation} In order to prove the theorem using \eqref{almost}, we need more precise estimates for $\| \Xi_0 \|$ and $\| F (j\tau, \cdot) \|_{X_{j\tau, (j+1) \tau}}$. \n First, \begin{equation} \| \Xi_0 \|^2 \leqslant \ \int_0^{2} \phi^2 (x-x_0) \, dx + \int_0^\infty \phi^2 (x + x_0) = \ 2 (1 - {\rm{tanh}} (x_0 - 2)) \ \leqslant \ C e^{-2 x_0}. \label{tamago} \end{equation} To estimate $\| F (t_a, \cdot) \|_{X_{t_a, t_b}}$ we specialize $F$ to the three cases under analysis. From the explicit propagators \eqref{freepro}, \eqref{propdelta}, \eqref{propdeltaprime}, \begin{equation*} \begin{split} F_F(t_a, t)\ = & \ \f 1 3 \, U_{t-t_a}^+ \left( \begin{array} {ccc} -1 & -1 & 2 \\ -1 & -1 & 2 \\ 2 & 2 & 2 \end{array} \right) \Phi_0 + \f i 3 \int_{t_a}^t U_{t-s}^+ \left( \begin{array} {ccc} -1 & -1 & 2 \\ -1 & -1 & 2 \\ 2 & 2 & 2 \end{array} \right) | \Phi_s |^2 \Phi_s \, ds\, \\ F_\delta^\alpha (t_a,t)\ = & \ F_F(t_a,t) - \f 2 9 \alpha \int_0^{+\infty}du %\int_0^{+\infty} dy \, e^{-\f \alpha 3 u} \left(U_{t-t_a}^+ \, {\mathbb J} \, \Phi_0\right) (\cdot + u) - i \f 2 9 \alpha \int_{t_a}^t ds \int_0^{+\infty}du \, e^{-\f \alpha 3 u} \left( U_{t-s}^+ \, {\mathbb J} \, | \Phi_s |^2 \Phi_s \right) (\cdot + u) \\ F_{\delta^\prime}^\beta (t_a, t)\ = & U_{t-t_a}^+ \left( \begin{array} {ccc} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right) \Phi_0 + i \int_{t_a}^t U_{t-s}^+ \left( \begin{array} {ccc} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right) | \Phi_s |^2 \Phi_s \, ds\, \\ & - \f 2 \beta \int_0^{+\infty}du %\int_0^{+\infty} dy \, e^{-\f 3 \beta u} \left(U_{t-t_a}^+ \, {\mathbb J} \, \Phi_0\right) (\cdot + u) - i \f 2 \beta \int_{t_a}^t ds \int_0^{+\infty}du \, e^{-\f 3 \beta u} \left(U_{t-s}^+ \, {\mathbb J} \, | \Phi_s |^2 \Phi_s \right) (\cdot + u) \end{split} \end{equation*} It is immediately seen that $$ F_F (t_a, t,x ) \ = \ \f 1 3 \left( \begin{array}{c} - K_1 (x,t) - K_2 (x,t) \\ - K_1 (x,t) - K_2 (x,t) \\ 2 K_1 (x,t) + 2 K_2 (x,t), \end{array} \right) , $$ where $K_1$ and $K_2$ were defined in \eqref{cappa}. Lemma \ref{caudale} yields \begin{equation} \| F_F (t_a, \cdot) \|_{X_{t_a, t_b}} \ \leqslant \ C e^{-x_0 + vt_b}. \label{aka} \end{equation} \n Furthermore, since \begin{equation*} \begin{split} F_\delta^\alpha (t_a,t,x) \ = \ F_F (t_a,t,x ) - \f 2 9 \alpha \int_0^{+\infty} du \, e^{-\f \alpha 3 u} \left( \begin{array}{c} K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t)\end{array} \right), \end{split} \end{equation*} we conclude \begin{equation} \| F_\delta^\alpha (t_a, \cdot) \|_{X_{t_a, t_b}} \ \leqslant \ C e^{-x_0 + vt_b} \label{aka2} . \end{equation} \n Finally, \begin{equation*} \begin{split} F_{\delta^\prime}^\beta (t_a,t,x) \ = \ \left( \begin{array}{c} K_1 (x,t) - K_2 (x,t) \\ - K_1 (x,t) + K_2 (x,t) \\ 0 \end{array} \right) - \f 2 \beta \int_0^{+\infty} du \, e^{-\f 3 \beta u} \left( \begin{array}{c} K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t)\end{array} \right) \end{split} \end{equation*} yields \begin{equation} \| F_{\delta^\prime}^\beta (t_a, \cdot) \|_{X_{t_a, t_b}} \ \leqslant \ C e^{-x_0 + vt_b}. \label{aka3} \end{equation} Now we go back to estimate \eqref{almost}. Due to \eqref{tamago}, \eqref{aka}, \eqref{aka2}, and \eqref{aka3}, and estimating any geometric sum by the double of its largest term, we get \begin{equation} \label{ziemlich} \begin{split} \| \Xi \|_{X_{0,t_1}} \ \ls & \ 2 C \wt C e^{-x_0} %\f {(2 \wt C)^{N+1} % - 1} {2 \wt C - 1} \sum_{j = 0}^N ( 2 \wt C )^j + 2 C \wt C e^{-x_0 + v \tau} \sum_{j=0}^{N-1} (2 \widetilde C)^{j} \sum_{k = 0}^{j} \left( \f{e^{v\tau}} {2 \widetilde C}\right)^{k} \\ & \ + C e^{-x_0 + v \tau} (2 \widetilde C)^{N+1} \sum_{k = 0}^{N-1} \left( \f{e^{v\tau}} {2 \widetilde C}\right)^{k} + 2 C \wt C e^{-x_0 + v t_1} \\ \ \ls & \ 2 C (2 \wt C)^{N+1} e^{-x_0} + 4 C \wt C e^{-x_0 + v \tau} \sum_{j=0}^{N-1} e^{j v\tau} + 8 C \wt C^2 e^{-x_0 + N v \tau} + 2 C \wt C e^{-x_0 + v t_1} \\ \ \ls & \ 2 C (2 \wt C)^{N+1} e^{-x_0} + 8 C \wt C e^{-x_0 + N v \tau} + 8 C \wt C^2 e^{-x_0 + N v \tau} + 2 C \wt C e^{-x_0 + v t_1}. \end{split} \end{equation} Concerning the first term in the r.h.s. of \eqref{ziemlich}, we have \begin{equation} \label{zuerst} (2 \wt C)^{N} e^{-x_0} \ \ls \ \left( \f{{2 \wt C}^{\f 1 {v \tau}}} e \right)^{x_0 - v^{1 - \delta}} e^{ - v^{1 - \delta}} \ \ls \ C e^{ - v^{1 - \delta}}. \end{equation} From \eqref{ziemlich} and \eqref{zuerst} we get $\| \Xi \|_{X_{0, t_1}} \ls C e^{ - v^{1 - \delta}}$, so \eqref{error1} follows and the proof is concluded. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %SUBSECTION PHASE 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Phase 2} \subsection{Phase 2} We call ``phase 2'' the evolution of the system in the time interval $( t_1 , t_2 )$ with $t_2= \f {x_0} v + v^{- \delta}$. Let us define the vector %%% \begin{equation*} %\label{PhiS} \Phi^S_t:=\Phi^{S,in}_t+\Phi^{S,out}_t \end{equation*} %%% with %%% \begin{equation*} \Phi^{S,in}_t:= \begin{pmatrix} \phisol{x_0,-v}(t)\\ \\ 0\\ \\ 0 \end{pmatrix} \;,\qquad \Phi^{S,out}_t:= \begin{pmatrix} \tilde r\,\phisol{-x_0,v}(t)\\ \\ \tilde t\,\phisol{-x_0,v}(t)\\ \\ \tilde t\, \phisol{-x_0,v}(t) \end{pmatrix} \,, \end{equation*} where the function $\phi_{x_0,v}$ was defined in equation \eqref{fix0t} and the reflection and transmission coefficients, $\tilde r$ and $\tilde t$, must be chosen accordingly to the Hamiltonian $H$ taken in the equation \eqref{intform}. The explicit expressions of $\tilde r$ and $\tilde t$ in all the cases $H=H_F,\,H^\al_\de,\, H^\beta_{\de'}$ can be read in formula \eqref{tildscoeff}. %%%%%%%%%% %PROPOSITION %%%%%%%%%% \begin{prop} \label{lemma2} Let $t\in \left( t_1,t_2 \right)$ then there exists $v_0>0$ such that for all $v>v_0$ %%% \begin{equation} \label{prop2.1} \|\Psi_{t} - \Phi^S_t\|\leqslant C_1 v^{-\f\de2}\,, \end{equation} %%% moreover \begin{equation} \label{prop2.2} \|\Psi_{t_2}-\Phi^{S,out}_{t_2}\|\leqslant C_2v^{-\f\de 2}\,, \end{equation} where $C_1$ and $C_2$ are positive constants which do not depend on $t$ and $v$. \end{prop} %%%%%%%% %PROOF %%%%%%%% \begin{proof} From the definition of $\Psi_t$, see equation \eqref{intform}, %%% \begin{equation*} %\label{everything} \Psi_{t}= \ e^{-iH(t-t_1)} \Psi_{t_1} + i \int_{t_1}^{t} ds \, e^{-i H(t-s)} | \Psi_s |^2 \Psi_s\,. \end{equation*} %%% We start with the trivial estimate %%% \begin{equation} \label{ineq1} \begin{aligned} &\big\|\Psi_{t}-\Phi^S_t\big\|\\ &\leqslant \big\|\Psi_{t} - e^{-iH (t-t_1)} \Psi_{t_1}\big\|+\big\|e^{-iH(t-t_1)}\Psi_{t_1}-e^{-iH(t-t_1)}\Phi_{t_1}^{S,in}\big\|+\big\|e^{-iH(t-t_1)}\Phi_{t_1}^{S,in}-\Phi^S_t\big\| \end{aligned} \end{equation} %%% and estimate the r.h.s. term by term. The estimates involved in the analysis of the first term are similar to the ones used in the previous proposition then we omit the details. Similarly to what was done above we set $X_{t_1,t_2}= L^{\infty}_{[t_1, t_2] } L^2 \cap L^{6}_{[t_1, t_2] } L^6 $, then by Strichartz (see equation \eqref{eq-dopo-hoel1} and proposition \ref{prop:stric}) %%% \begin{equation} \label{john} \big\|\Psi - e^{-iH (\cdot-t_1)} \Psi_{t_1}\big\|_{X_{t_1,t_2}}\leqslant \bigg\| \int_{t_1}^{\cdot} ds \, e^{-i H (\cdot-s)} | \Psi_s |^2 \Psi_s\bigg\|_{X_{t_1,t_2}} \leqslant C(t_2-t_1)^{1/2}\big\|\Psi\big\|_{X_{t_1,t_2}}^3\,, \end{equation} %%% %%% \begin{equation*} \big\|e^{-iH(\cdot-t_1)}\Psi_{t_1}\big\|_{X_{t_1,t_2}}\leqslant C \big\|\Psi_{t_1}\big\|\,, \end{equation*} %%% which imply %%% \begin{equation*} \big\|\Psi\big\|_{X_{t_1,t_2}}\leqslant C\big\|\Psi_{t_1}\big\|+ C(t_2-t_1)^{1/2}\big\|\Psi\big\|_{X_{t_1,t_2}}^3\,. \end{equation*} %%% %By the continuity of the norm $\|\cdot\|_X$ in $t_2$, it follows that, if $C(t_2-t_1)^{1/2}\big[2C\big\|\Psi_{t_1}\big\|\big]^2\leqslant 1/2$, then $\|\Psi_{t}\|_X\leqslant 2C\|\Psi_{t_1}\|$; By lemma \ref{blocco} one has that if $(t_2-t_1)\leqslant\big[8 C^3 \|\Psi_{t_1}\|^2+4C^2 \|\Psi_{t_1}\|\big]^{-2}$, then $\|\Psi\|_{X_{t_1,t_2}}\leqslant 2C\|\Psi_{t_1}\|$; using this estimate in the inequality \eqref{john} we get %%% \begin{equation} \label{L2S1} \big\|\Psi_{t} - e^{-iH (t-t_1)} \Psi_{t_1}\big\|\leqslant \big\|\Psi - e^{-iH(\cdot-t_1)} \Psi_{t_1}\big\|_{X_{t_1,t_2}}\leqslant C(t_2-t_1)^{1/2} \big\|\Psi_{t_1}\big\|^3\leqslant Cv^{-\de/2} \end{equation} %%% where we used $t_2-t_1=2v^{-\de}$. We proceed now with the estimate of the second term in the r.h.s. of inequality \eqref{ineq1}. Let us set %%% \begin{equation*} \Psi_{t_1}=\Psi_{t_1}-\Phi_{t_1}+\Phi_{t_1}^{S,in}+\Phi_{t_1}^{tail}\,. \end{equation*} %%% Where the vector $\Phi_{t}$ was defined in equation \eqref{truesol}, and we introduced the vector %%% \begin{equation*} \Phi_{t_1}^{tail} := \begin{pmatrix} 0\\ \\ \phisol{-x_0,v}(t_1)\\ \\ 0 \end{pmatrix}\;. \end{equation*} We remark that $\Phi_{t_1}^{S,in}+\Phi_{t_1}^{tail}=\Phi_{t_1}$. The following trivial inequality holds true %%% \begin{equation} \label{L2S2} \big\|e^{-iH(t-t_1)}\Psi_{t_1}-e^{-iH(t-t_1)}\Phi_{t_1}^{S,in}\big\|\leqslant \big\|\Psi_{t_1}-\Phi_{t_1}\big\|+ \big\|\Phi_{t_1}^{tail}\big\|\leqslant Ce^{-v^{1-\de}} \end{equation} %%% where in the last estimate we used proposition \ref{fase1} and the fact that $\|\Phi_{t_1}^{tail}\|\leqslant 2e^{-v^{1-\de}}$. Let us consider now the last term in the r.h.s. of inequality \eqref{ineq1}. We are going to prove that for all $t\in(t_1,t_2)$ and for $v$ big enough %%% \begin{equation} \label{seesee} \big\|e^{-iH(t-t_1)}\Phi_{t_1}^{S,in}-\Phi^{S}_t\big\|\leqslant Cv^{-\de}\,. \end{equation} %%% Let us introduce the functions %%% \begin{equation} \label{phi-} \phi^-_{t}(x):=\int_0^\infty U_{t-t_1}(x-y)\phisol{x_0,-v}(y,t_1) dy \end{equation} %%% and %%% \begin{equation} \label{phi+} \phi^+_{t}(x):=\int_0^\infty U_{t-t_1}(x+y)\phisol{x_0,-v}(y,t_1) dy\,. \end{equation} %%% First we prove a preliminary formula for the vector $e^{-iH(t-t_1)}\Phi^{S,in}_{t_1}$ (see equations \eqref{carnival} and \eqref{carnival2} below). For any constant $a$, not dependent on $v$ let us consider the term %%% \begin{equation*} \begin{aligned} &v a \int_0^{\infty}e^{-uva}\big[U^+_{t-t_1}\phi_{x_0,-v}(t_1)\big](u+x) du\\ =& v a \int_0^{\infty}du \int_0^\infty dy\, e^{-uv a} U_{t-t_1}(u+x+y)e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1) \end{aligned} \end{equation*} %%% where we set $\varphi(t):=- t \f {v^2} 4+t$. By integrating by parts we obtain the equality %%% \begin{equation} \begin{aligned} \label{misunderstood} &v a\int_0^{\infty}e^{-uv a}\big[U^+_{t-t_1}\phi_{x_0,-v}(t_1)\big](u+x) du\\ =& 2i a\int_0^{\infty}du \int_0^\infty dy\, e^{-uv a} U_{t-t_1}(u+x+y)e^{i\varphi(t_1)}\bigg[\frac{d}{dy}e^{-i\frac{v}{2}y}\bigg]\phi(y-x_0+vt_1)\\ %=&- %2ia\int_0^{\infty}du\, e^{-uva} U_{t-t_1}(u+x)e^{i\varphi(t_1)}\phi(-x_0+vt_1)\\ %&- %2ia\int_0^{\infty}du \int_0^\infty dy\, e^{-uv\tilde\al/3} \bigg[\frac{d}{dy}U_{t-t_1}(u+x+y)\bigg]e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1)\\ %&- %2i\tilde\alpha\int_0^{\infty}du \int_0^\infty dy\, e^{-uv\tilde\al/3} U_{t-t_1}(u+x+y)e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\bigg[\frac{d}{dy}\phi(y-x_0+vt_1)\bigg]\\ =&A_1(x,t)+A_2(x,t)+A_3(x,t) \end{aligned} \end{equation} %%% with %%% \begin{equation*} A_1(x,t) := - 2ia \int_0^{\infty}du\, e^{-uv a} U_{t-t_1}(u+x)e^{i\varphi(t_1)}\phi(-x_0+vt_1) \end{equation*} %%% %%% \begin{equation*} A_2(x,t) := - 2ia\int_0^{\infty}du \int_0^\infty dy\, e^{-uva} \bigg[\frac{d}{dy}U_{t-t_1}(u+x+y)\bigg]e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1)\\ \end{equation*} %%% %%% \begin{equation*} A_3(x,t) := - 2ia\int_0^{\infty}du \int_0^\infty dy\, e^{-uva} U_{t-t_1}(u+x+y)e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\bigg[\frac{d}{dy}\phi(y-x_0+vt_1)\bigg]\,. \end{equation*} %%% %%% We notice that \begin{equation*} %\begin{split} \bigg\| \int_0^{\infty}du\, e^{-uv a} U_{t-t_1}(u+\cdot)\bigg\|_{L^2(\RE^+)} % & %\ \leqslant %\ \bigg\| \int_0^{\infty}du\, e^{-uva} U_{t-t_1}(u+\cdot)\bigg\|_{L^2(\RE)} %\\ %& \ = \ \bigg\| %\int_{-\infty}^{+\infty}du\, \chi_+ (u)e^{-uva} %U_{t-t_1}(u+\cdot)\bigg\|_{L^2(\RE)} \\ %& \ = %\ \| \chi_+ e^{-va \cdot} \|_{L^2(\RE)} %\ = % \ \sqrt{ \frac{1}{2 v a}}. %\end{split} \end{equation*} %%% Then the following estimate for the term $A_1$ holds true %%% \begin{equation} \label{estA1} \|A_1(t)\|_{L^2(\RE^+)}\leqslant 2a \bigg\| \int_0^{\infty}du\, e^{-uva} U_{t-t_1}(u+\cdot)\bigg\|_{L^2(\RE^+)} |\phi(-x_0+vt_1)|\leqslant C\,\frac{e^{-v^{1-\de}}}{v^{1/2}}\,. \end{equation} %%% The term $A_3$ is estimated by %%% \begin{equation} \label{estA3} \begin{aligned} \|A_3(t)\|_{L^2(\RE^+)}\leqslant & 2 a\int_0^{\infty}du \,e^{-uva} \big\| \big[U_{t-t_1}\chi_+e^{-i\frac{v}{2}\cdot}\phi'(\cdot-x_0+vt_1)\big]\big(-(u+\cdot)\big)\big\|_{L^2(\RE^+)}\\ \leqslant & 2\big\|\phi'\big\|_{L^2(\RE)} a\int_0^{\infty}du \,e^{-uva } \leqslant \frac{C}{v}\,, \end{aligned} \end{equation} %%% where we used the equality $[U_t^+f](x)=[U_t \chi_+ f](-x)$. \noindent We compute finally the term $A_2$. By integration by parts %%% \begin{equation*} \begin{aligned} A_2(x,t)=&- 2ia\int_0^{\infty}du \int_0^\infty dy\, e^{-uva} \bigg[\frac{d}{du}U_{t-t_1}(u+x+y)\bigg]e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1)\\ %=& %2ia\int_0^\infty dyU_{t-t_1}(x+y)e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1)\\ %&-2iv a^2 %\int_0^{\infty}du \int_0^\infty dy\, e^{-uv a} U_{t-t_1}(u+x+y)e^{i\varphi(t_1)}e^{-i\frac{v}{2}y}\phi(y-x_0+vt_1)\\ =& 2ia \phi^+_{t}(x) -2iv a^2 \int_0^{\infty}e^{-uva}\big[U^+_{t-t_1}\phi_{x_0,-v}(t_1)\big](u+x) du\,, \end{aligned} \end{equation*} %%% where the function $ \phi^+_{t}$ was defined in equation \eqref{phi+}. Using the last equality in the equation \eqref{misunderstood} we get %%% \begin{equation} \label{night} va\int_0^{\infty}e^{-uva}\big[U^+_{t-t_1}\phi_{x_0,-v}(t_1)\big](u+x) du= \frac{2ia}{1+2ia} \phi^+_{t}(x)+ \frac{A_1(x,t)+A_3(x,t)}{1+2ia}\,. \end{equation} %%% From the definition of $\Phi^{S,in}_t$ and using the last equality with $a=\tilde\alpha/3$ in the formula for the integral kernel of $e^{- i H_\de^\al t}$, see equation \eqref{propdelta}, it follows that %%% \begin{equation} \label{carnival} e^{-iH_\de^{\alpha}(t-t_1)}\Phi_{t_1}^{S,in}= e^{-iH_\de^\alpha(t-t_1)} \begin{pmatrix} \phisol{x_0,-v}(t_1)\\ \\ 0\\ \\ 0 \end{pmatrix} = \begin{pmatrix} \phi^-_{t}-\frac{1+2i\tilde\al}{3+2i\tilde\al}\phi^+_t\\ \\ \frac{2}{3+2i\tilde\al}\phi^+_t\\ \\ \frac{2}{3+2i\tilde\al}\phi^+_t \end{pmatrix} -\frac{2}{3} \begin{pmatrix} A_{\tilde\alpha}(t)\\ \\ A_{\tilde\alpha}(t)\\ \\ A_{\tilde\alpha}(t) \end{pmatrix} \end{equation} %%% where the function $\phi_t^-$ was defined in equation \eqref{phi-} and we set $A_{\tilde\alpha}(x, t):=\big[(A_1(x,t)+A_3(x,t))/(1+2ia)\big]\big|_{a=\tilde\alpha/3}$. Similarly using equality \eqref{night} with $a=3/\tilde \beta$ in the formula for the integral kernel of the propagator $e^{- i H_{\de'}^\beta t}$, see equation \eqref{propdeltaprime}, we get %%% \begin{equation} \label{carnival2} e^{-iH_{\de'}^{\beta}(t-t_1)}\Phi_{t_1}^{S,in}= \begin{pmatrix} \phi^-_{t}+\frac{\tilde \beta+2i}{\tilde\beta+6i}\phi^+_t\\ \\ - \f {4i} {\tilde \beta + 6 i}\phi^+_t\\ \\ - \f {4i} {\tilde \beta + 6 i}\phi^+_t \end{pmatrix} -\frac{2}{3} \begin{pmatrix} A_{\tilde\beta}(t)\\ \\ A_{\tilde\beta}(t)\\ \\ A_{\tilde\beta}(t) \end{pmatrix} \end{equation} %%% where we introduced the notation $A_{\tilde\beta}(x, t):=\big[(A_1(x,t)+A_3(x,t))/(1+2ia)\big]\big|_{a=3/\tilde\beta}$. We notice that the analogous formula for $e^{-iH_F(t-t_1)}\Phi_{t_1}^{S,in}$ can be obtained from equation \eqref{carnival} by setting $\tilde\alpha=0$ and $A_{\tilde\alpha}=0$. To get the estimate \eqref{seesee} we show that, at the cost of an error of the order $(t_2-t_1)$, for $t\in(t_1,t_2)$, the functions $\phi^-_{t}(x)$ and $\phi^+_{t}(x)$ can be approximated with the solitons $\phisol{x_0,-v}(x,t)$ and $\phisol{-x_0,v}(x,t)$ respectively. We consider first the function $\phi^+_{t}$, by adding and subtracting a suitable term to the r.h.s. of equation \eqref{phi+} we get %%% \begin{equation} \label{pippo1} \begin{aligned} \phi^+_{t}(x)=& \int_{-\infty}^\infty U_{t-t_1}(x+y)\phisol{x_0,-v}(y,t_1) + i \int_{t_1}^{t} ds \int_{-\infty}^\infty U_{t-s} (x + y)| \phisol{x_0,-v}(y,s)|^2 \phisol{x_0,-v}(y,s) dy \\ &- \int_{-\infty}^0 U_{t-t_1}(x+y)\phisol{x_0,-v}(y,t_1) dy - i \int_{t_1}^{t} ds \int_{-\infty}^\infty U_{t-s} (x + y)|\phisol{x_0,-v}(y,s) |^2\phisol{x_0,-v}(y,s) \, dy\\ =&\phisol{-x_0,v}(x,t) +I(x,t)+II(x, t)\,, \end{aligned} \end{equation} %%% where we used the fact that $\phisol{x_0,-v}(-x,t)=\phisol{-x_0,v}(x,t)$ and we set %%% \begin{equation*} I(x,t):=- \int_{-\infty}^0 U_{t-t_1}(x+y)\phisol{x_0,-v}(y,t_1) dy \end{equation*} %%% and %%% \begin{equation*} II(x,t):=- i \int_{t_1}^{t} ds \int_{-\infty}^\infty U_{t-s} (x + y)|\phisol{x_0,-v}(y,s) |^2\phisol{x_0,-v}(y,s) \, dy\,. \end{equation*} %%% For the term $I$ we use the estimate %%% \begin{equation*} %\label{stimaI} \|I\|_{L^2(\RE^+)}\leqslant \|\phi (\cdot - x_0 + vt_1)\|_{L^2(\RE^-)} \leqslant 2e^{-v^{1-\de}}\,. \end{equation*} %%% The term $II$ is estimated by %%% \begin{equation*} \|II\|_{L^2(\RE^+)}\leqslant (t-t_1) \|\phi^3\|_{L^2(\RE)}\leqslant C v^{-\de}\,. \end{equation*} %%% \noindent Similarly, for the function $\phi^-_{t_2}$, we get %%% \begin{equation} \label{pippo2} \phi^-_{t}(x)=\phisol{x_0,-v}(x,t)+III(x,t)+IV(x,t) \end{equation} %%% where we set %%% \begin{equation*} III(x,t):= - \int_{-\infty}^0 U_{t-t_1}(x-y)\phisol{x_0,-v}(t_1,y) dy \end{equation*} %%% and %%% \begin{equation*} IV(x,t):= - i \int_{t_1}^{t} ds \int_{-\infty}^\infty U_{t-s} (x - y)|\phisol{x_0,-v}(s,y)|^2\phisol{x_0,-v}(s,y) \, dy\,. \end{equation*} %%% For $t\in(t_1,t_2)$ the estimates %%% \begin{equation*} \|III\|_{L^2(\RE^+)}\leqslant 2e^{-v^{1-\de}}\;;\quad \|IV\|_{L^2(\RE^+)}\leqslant C v^{-\de} \end{equation*} %%% are similar to the ones given above for the terms $I$ and $II$. Then from equations \eqref{pippo1} and \eqref{pippo2} we have %%% \begin{equation*} %\label{stimaph-} \|\phi^+_{t}-\phisol{-x_0,v}(t)\|_{L^2(\RE^+)}\leqslant C[e^{-v^{1-\de}}+v^{-\de}]\,,\quad \|\phi^-_{t}-\phisol{x_0,-v}(t)\|_{L^2(\RE^+)}\leqslant C[e^{-v^{1-\de}}+v^{-\de}] \end{equation*} for all $t\in(t_1,t_2)$. By using the last estimates and the estimates \eqref{estA1} and \eqref{estA3} in equations \eqref{carnival} and \eqref{carnival2} we get %%% \begin{equation*} %\label{L2S3} \big\|e^{-iH(t-t_1)}\Phi_{t_1}^{S,in}-\Phi^S_t\big\|\leqslant C\bigg[e^{-v^{1-\de}}+v^{-\de}+\frac{e^{-v^{1-\de}}}{v^{1/2}}+\frac{1}{v}\bigg] \end{equation*} %%% which in turn implies that for $v$ big enough the estimate \eqref{seesee} holds true. Using estimates \eqref{L2S1}, \eqref{L2S2} and \eqref{seesee} in the inequality \eqref{ineq1} we get the estimate \eqref{prop2.1}. The estimate \eqref{prop2.2} is a consequence of estimate \eqref{prop2.1} and of the fact that $\|\Phi^S_{t_2}-\Phi^{S,out}_{t_2}\|=\|\Phi^{S,in}_{t_2}\|\leqslant C e^{-v^{1-\de}}$. \end{proof} \begin{rem} Notice that estimate \eqref{prop2.1}, although non strictly necessary for the proof of theorem \ref{mainth}, enforces the picture that in the phase 2 a scattering event is occurring. The true wavefunction can be approximated by the superposition of an incoming and an outgoing wavefunction. At the end of this phase, only the outgoing wavefunction is not negligibile. \end{rem} \subsection{Phase 3} Let us put $t_3=t_2+T\ln v$. We call ``phase 3'' the evolution of the system in the time interval $(t_2,t_3)$. The approximation of $\Psi_t$ during this time interval is the content of theorem \ref{mainth}. \n We recall the following proposition (see \cite{[HMZ07]}). %%%%%%%%%%%% %PROPOSITION %%%%%%%%%%%% \begin{prop} \label{appAHMZ} Let $\phi^{tr}_t$ and $\phi^{ref}_t$ be defined as it was done in equations \eqref{phiref} and \eqref{phitr} above. Then $\forall k\in \NN$ there exist two constants $c(k)>0$ and $\si(k)>0$ such that %%% \begin{equation} \|\phi^\gamma_t\|_{L^2(\RE^-)}+\|\phi^\gamma_t\|_{L^\infty(\RE^-)}\leqslant \frac{c(k)(\ln v)^{\si(k)}}{v^{k(1-\de)}} \label{shodo} \end{equation} %%% for $\gamma = \{ \text{ref} , \text{tr} \}$, uniformly in $t\in[0,T\ln v]$. \end{prop} Notice that also the norms $\|\phi^\gamma_t\|_{L^p(\RE^-)}$, for $2 \leqslant p \leqslant \infty$, are estimated by the r.h.s. of \eqref{shodo}. %%%%%%%%%%%% %LEMMA %%%%%%%%%%%% Now we can prove theorem \ref{mainth} %%%%%%%%%%%%%%%%%%% %PROOF OF THE MAIN THEOREM %%%%%%%%%%%%%%%%%%% \begin{proof}[Proof of Theorem \ref{mainth}.] The strategy of the proof closely follows proposition \ref{fase1}. We will just sketch the common part of the proof while proving in details the different estimates. Let us define $ \Xi_t:=\Psi_t-\sum_{j=1}^3\Phi^j_t$ where the vectors $\Phi^j_t$ were given in equation \eqref{Phijt} and fix $t_a\in [t_2,t_3]$. From equations \eqref{intform} and \eqref{Phijt} it follows that the vector $\Xi_t$ satisfies the following integral equation \begin{equation} \label{amaranto} \begin{aligned} \Xi_t=& e^{-iH (t-t_a)} \Xi_{t_a} \\ &+\sum_{j=1}^3\bigg[\Big(e^{-iH (t-t_a)}-e^{-iH_j (t-t_a)}\Big) \Phi^j_{t_a} + i \int_{t_a}^t ds \, \Big( e^{-iH (t-s)}- e^{-i H_j (t-s)}\Big) | \Phi^j_s |^2 \Phi^j_s\bigg]\\ &+ i \int_{t_a}^t ds \, e^{-i H (t-s)} \sum_{j_1,j_2,j_3}(1-\de_{j_1j_2}\de_{j_2j_3})\overline{\Phi^{j_1}_s} \Phi^{j_2}_s \Phi^{j_3}_s\\ &+ i \int_{t_a}^t ds \, e^{-i H (t-s)} \bigg[\bigg| \Xi_s +\sum_{j=1}^3 \Phi^j_s\bigg|^2 \Xi_s +|\Xi_s|^2\sum_{j=1}^3 \Phi^j_s+2\Re\bigg[\overline{ \Xi_s}\sum_{j=1}^3 \Phi^j_s \bigg]\sum_{j=1}^3 \Phi^j_s\bigg]\\ =& e^{-iH (t-t_a)} \Xi_{t_a} + G(t_a,t) \\ &+ i \int_{t_a}^t ds \, e^{-i H (t-s)} \bigg[\bigg| \Xi_s +\sum_{j=1}^3 \Phi^j_s\bigg|^2 \Xi_s +|\Xi_s|^2\sum_{j=1}^3 \Phi^j_s+2\Re\bigg[\overline{ \Xi_s}\sum_{j=1}^3 \Phi^j_s \bigg]\sum_{j=1}^3 \Phi^j_s\bigg]\,. \end{aligned} \end{equation} \n Let us fix $t_b\in [t_a,t_3]$ and let $X_{t_a,t_b}= L^{\infty}_{[t_a, t_b] } L^2\cap L^{6}_{[t_a, t_b] }L^6 $. Using the Strichartz estimates as it was done in proposition \ref{fase1} (see equations \eqref{split} - \eqref{split3}), it is straightforward to prove that \begin{equation*} \| \Xi \|_{X_{t_a,t_b}} \leqslant \wt C\lf[ \| \Xi_{t_a} \| + \| G(t_a,\cdot)\|_{X_{t_a,t_b}} + (t_b-t_a)^{1/2} \lf( \| \Xi \|_{X_{t_a,t_b}}^2 + \| \Xi \|_{X_{t_a,t_b}}^3 \ri) \ri] \end{equation*} where $\wt C$ depends only on the constants appearing in the Strichartz estimates. Using lemma \ref{blocco} as it was done in the proof of proposition \ref{fase1} it follows that there exists $\tau>0$ such that, for any $t \in [t_2, t_3)$ one has $\| \Xi \|_{X_{t, t + \tau}} \ls \ 2 \wt C \ \| \Xi_{t} \| + 2 \wt C \ \| G(t, \cdot) \|_{X_{t, t+ \tau}}$. % %In particular there exists $\hat c>0$ such that if %\be % (\lf\| \Xi_{t_a} \| + \| G\|_X\ri)(t_b-t_a)^{1/2} \leqslant \hat c % %\label{conditer2} %\end{equation*} %then %\be %\| \Xi \|_X \leqslant \widehat{C} (\lf\| \Xi_{t_a} \| + \| G\|_X\ri)\,. %%\label{iter2} %\end{equation*} \n We divide the interval $[t_2,t_3]$ in $N+1$ subintervals: $[t_2+j\tau,t_2+(j+1)\tau)$, with $ j=0,...,N-1$; and $[t_2+N\tau, t_3)$. Where $N$ is the integer part of $(t_3-t_2)/\tau$. \n Then proceeding by induction as we did in the proof of proposition \ref{fase1}, see equations \eqref{indu} and \eqref{almost}, we get the inequality \begin{equation} \label{perf} \| \Xi \|_{X_{t_2,t_3}} \ \ls \sum_{j=0}^{N} (2\wt C)^{j+1} \| \Xi_{t_2} \| + \sum_{j=0}^{N} \sum_{k = 0}^{j} (2 \wt C)^{j+1-k} \| G (t_2+k\tau, \cdot) \|_{X_{t_2+k\tau, \min\{t_3,t_2+ (k+1) \tau\}}} \,. \end{equation} Now we estimate the initial datum $\| \Xi_{t_2}\|$ and the source term $\|G (t_a, \cdot) \|_{X_{t_a,t_b}}$ with $t_2\leqslant t_a\leqslant t_b\leqslant t_3$ and $t_b-t_a\leqslant\tau$. \n By proposition \ref{lemma2} (estimate \eqref{estA3}) and using the definitions \eqref{Phi1t2} - \eqref{Phi3t2} one has %%% \begin{equation*} \Xi_{t_2} = \Psi_{t_2}-\sum_{j=1}^3\Phi^j_{t_2}=\Psi_{t_2}-\Phi^{S,out}_{t_2} -\begin{pmatrix} \tilde t\,\phi_{x_0,-v}(t_2)\\ \\ \tilde r\,\phi_{x_0,-v}(t_2)\\ \\ \tilde t\,\phi_{x_0,-v}(t_2) \end{pmatrix} \,, \end{equation*} %%% %%%% %\be % \Xi_{t_2}(x_1, x_2 , x_3) = %\lf( \Psi_{t_2}-\sum_{j=1}^3\Phi^j_{t_2}\ri)(x_1, x_2 , x_3)= %\begin{pmatrix} %-\f23e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_1} e^{it_2} \phi (x_1 - x_0 + vt_2)\\ %\f13e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_2} e^{it_2} \phi (x_2 - x_0 + vt_2)\\ %-\f23e^{-i \f {v^2} 4 t_2}e^{-i\f v 2 x_3} e^{it_2} \phi (x_3 - x_0 + vt_2) %\end{pmatrix} %+ {\mathcal R} \,. %\end{equation*} %%%% with $\| \Psi_{t_2}-\Phi^{S,out}_{t_2}\|\leqslant C v^{-\de/2}$. Since %%% \begin{equation*} \int_0^\infty|\phi_{x_0,-v} (x,t_2)|^2dx=\int_0^\infty|\phi (x - x_0 + vt_2)|^2dx= 2\int_{v^{1-\de}}^\infty\sech(x)^2dx=\frac{4 e^{-2v^{1-\de}}}{1+e^{-2v^{1-\de}}}\leqslant 4 e^{-2v^{1-\de}} \end{equation*} %%% we have %%% \begin{equation} \label{ect} \|\Xi_{t_2} \|\leqslant C\big(v^{-\f \de 2}+e^{-2v^{1-\de}}\big)\,\leqslant C v^{-\f \de 2} \,. \end{equation} %%% Let us now consider the source term $G(t_a,t)$. We use the estimate $\|G(t_a,\cdot)\|_{X_{t_a,t_b}}\leqslant \|G(t_2,\cdot)\|_{X_{t_2,t_3}}$. To simplify the notation we set $G(t) \equiv G(t_2,t)$ and %%% \begin{equation*} G_1 (t) : =\sum_{j=1}^3\bigg[\Big(e^{-iH(t-t_2)}-e^{-iH_j (t-t_2)}\Big) \Phi^j_{t_2} + i \int_{t_2}^t ds \, \Big( e^{-iH(t-s)}- e^{-i H_j (t-s)}\Big) | \Phi^j_s |^2 \Phi^j_s\bigg] \end{equation*} %%% %%% \begin{equation*} G_2(t) : = i \int_{t_2}^t ds \, e^{-i H (t-s)} \sum_{j_1,j_2,j_3}(1-\de_{j_1j_2}\de_{j_2j_3})\overline{ \Phi^{j_1}_s} \Phi^{j_2}_s \Phi^{j_3}_s\,. \end{equation*} %%% By the definition of $G(t_a,t)$, see equation \eqref{amaranto} it follows that %%% \begin{equation*} G(t)=G_1(t)+G_2(t)\,; \end{equation*} %%% we estimate $G_1(t)$ and $G_2(t)$ separately. We proceed first with the estimate of the term $G_1$. From equations \eqref{freepro}, \eqref{propdelta}, \eqref{propdeltaprime} and \eqref{twoedgepro-j} one can see that for any (column) vector $F=(F_1,F_2,F_3)\in L^2$ %%% \begin{equation*} %\label{diff} \big[e^{-iH t}-e^{-iH_j t}\big]F= \mathbb{M}_j \begin{pmatrix} U^+_tF_1\\ \\ U^+_tF_2\\ \\ U^+_tF_3 \end{pmatrix} -\frac{2}{3}\mathbb{J} \begin{pmatrix} v a \int_0^\infty e^{-u v a} \big[U^+_tF_1\big](u+\cdot) du\\ \\ v a \int_0^\infty e^{-u v a} \big[U^+_tF_2\big](u+\cdot) du\\ \\ v a \int_0^\infty e^{-u v a} \big[U^+_tF_3\big](u+\cdot) du \end{pmatrix} \qquad j=1,2,3\,, \end{equation*} %%% where the constant $a$ and the matrices $ \mathbb{M}_j$ must be chosen accordingly to the Hamiltonian $H$:\\ for $H=H_\de^\alpha$, %%% \begin{equation*} a=\frac{\tilde \alpha}{3}\;,\qquad\mathbb{M}_j=-\mathbb{I}+\frac{2}{3}\mathbb{J}-\mathbb{T}_j\,; \end{equation*} %%% for $H=H_{\de'}^\beta$, %%% \begin{equation*} a=\frac{3}{\tilde \beta}\;,\qquad \mathbb{M}_j=\mathbb{I}-\mathbb{T}_j\,; \end{equation*} %%% and the formula for $H=H_F$ can be obtained by setting $\tilde\alpha=0$ in the formula for $H=H_\de^\alpha$. \n Then, denoting by $\big(G_1(x,t)\big)_l$, $l=1,2,3$, the $l$-th component of the vector $G_1$ one has %%% \begin{equation*} %\label{IIl} \big(G_1(x,t)\big)_l=\big(\widetilde{G}_1(x,t)\big)_l+\big(\widehat{G}_1(x,t)\big)_l\,, \end{equation*} %%% with %%% \begin{equation*} \big(\widetilde{G}_1(x,t)\big)_l:= \sum_{j,k=1}^3\big(\mathbb{M}_j\big)_{lk}\bigg[\big[U^+_{t-t_2}\big(\Phi^j_{t_2}\big)_k\big](x)+ i\int_{t_2}^tds\big[U^+_{t-s}\big|\big(\Phi^j_{s}\big)_k\big|^2\big(\Phi^j_{s}\big)_k\big](x)\bigg] \end{equation*} %%% %%% \begin{equation*} \big(\widehat{G}_1(x,t)\big)_l:= -\frac{2va}{3}\sum_{j,k=1}^3\int_0^\infty e^{-vua}\bigg[\big[U^+_{t-t_2}\big(\Phi^j_{t_2}\big)_k\big](u+x)+ i\int_{t_2}^tds\big[U^+_{t-s}\big|\big(\Phi^j_{s}\big)_k\big|^2\big(\Phi^j_{s}\big)_k\big](u+x)\bigg]du\,. \end{equation*} %%% %%%% %\be %%\label{IIl} %\begin{aligned} %\big(G_1(t)\big)_l=& %\sum_{j,k=1}^3\big(\mathbb{M}_j\big)_{lk}\bigg[\int_0^\infty U_{t-t_2}(x+y)\big(\Phi^j_{t_2}\big)_k(y)dy+ %i\int_{t_2}^tds\int_0^\infty U_{t-s}(x+y)\big|\big(\Phi^j_{s}\big)_k(y)\big|^2\big(\Phi^j_{s}\big)_k(y)dy\bigg]\\ %&-\frac{2va}{3}\sum_{j,k=1}^3\int_0^\infty e^{-vua}\bigg[\int_0^\infty U_{t-t_2}(u+x+y)\big(\Phi^j_{t_2}\big)_k(y)dy+ %i\int_{t_2}^tds\int_0^\infty U_{t-s}(u+x+y)\big|\big(\Phi^j_{s}\big)_k(y)\big|^2\big(\Phi^j_{s}\big)_k(y)dy\bigg]du\,. %\end{aligned} %\end{equation*} %%%% From the definition of the vectors $\Phi^j_{t}$, see equations \eqref{Phi123t} - \eqref{Phi1234t}, we see that for each $l=1,2,3$ the function $\big(\widetilde{G}_1(x,t)\big)_l$ is a linear combination of four functions, $f^{\gamma,+}_t$ and $f^{\gamma,-}_t$, with $\gamma $ being equal to $ref$ and $tr$, given by \begin{equation} \label{fal+} f^{\gamma,+}_t(x) := e^{-i \f {v^2} 4 t_2}e^{it_2}\bigg[\int_0^\infty U_{t-t_2}(x+y)\phi_0^\gamma(y)dy+i\int_{t_2}^tds\int_0^\infty U_{t-s}(x+y)|\phi^{\gamma}_{s-t_2}(y)|^2\phi^{\gamma}_{s-t_2}(y)dy\bigg] \end{equation} and \begin{equation*} %\label{fal-} f^{\gamma,-}_t(x) := e^{-i \f {v^2} 4 t_2}e^{it_2}\bigg[\int_0^\infty U_{t-t_2}(x+y) \phi_0^\gamma(-y)dy+i\int_{t_2}^tds\int_0^\infty U_{t-s}(x+y)|\phi^{\gamma}_{s-t_2}(-y)|^2\phi^{\gamma}_{s-t_2}(-y)dy\bigg] \end{equation*} where the functions $\phi^{ref}_t$ and $\phi^{ref}_t$ were defined in equations \eqref{phiref} and \eqref{phitr} respectively. Similarly one can see that for each $l=1,2,3$ the function $\big(\widehat{G}_1(x,t)\big)_l$ is a linear combination of %%% \begin{equation*} va \int_0^\infty e^{-vua} f^{ref,+}_t(u+x)du\;,\quad va \int_0^\infty e^{-vua} f^{tr,+}_t(u+x)du\;, \end{equation*} %%% %%% \begin{equation*} va \int_0^\infty e^{-vua} f^{ref,-}_t(u+x)du\;,\quad va \int_0^\infty e^{-vua} f^{tr,-}_t(u+x)du\;. \end{equation*} %%% First we study the function $f^{\gamma,+}_t$. We notice that, adding and subtracting a suitable term in equation \eqref{fal+} and using the definitions \eqref{phiref} and \eqref{phitr}, $f^{\gamma,+}_t(x)$ can be written as %%% \begin{equation*} %\label{phialpha+} f^{\gamma,+}_t(x) = I(x,t) + II(x,t) + III(x,t)\,, \end{equation*} %%% with %%% \begin{equation*} I(x,t):= e^{-i \f {v^2} 4 t_2}e^{it_2}\phi^{\gamma}_{t-t_2}(-x) \end{equation*} %%% %%% \begin{equation*} II(x,t):= -e^{-i \f {v^2} 4 t_2}e^{it_2}\int_0^{\infty} U_{t-t_2}(x-y)\phi_0^\al(-y)dy \end{equation*} %%% %%% \begin{equation*} III(x,t):=-ie^{-i \f {v^2} 4 t_2}e^{it_2}\int_{t_2}^{t}ds\int_{0}^\infty U_{t-s}(x-y)|\phi^{\gamma}_{s-t_2}(-y)|^2\phi^{\gamma}_{s-t_2}(-y)dy\,. \end{equation*} %%% Similarly to what was done above, % for any $t_2\leqslant t_a \leqslant t_b \leqslant t_3$ we set $X_{t_2,t_3}(\RE^\pm)= L^{\infty}_{[t_2, t_3] } L^2 (\RE^\pm) \cap L^{6}_{[t_2, t_3] } L^6 (\RE^\pm) $. By proposition \ref{appAHMZ} % and H\"older inequality , we have %%% \begin{equation} \label{glad0} \| I \|_{X_{t_2,t_3}(\RE^+)}=\|\phi^{\gamma}_{\cdot-t_2} \|_{X_{t_2,t_3}(\RE^-)} \leqslant \frac{c'(k)(\ln v)^{\si '(k)}}{v^{k(1-\de)}} \end{equation} %%% where $c'(k)$ and $\si '(k)$ are different constants from the one appearing in proposition \ref{appAHMZ}. For our purposes we do not need to compute them. \n Using the one dimensional Strichartz estimates for $U_t$, we have %%% \begin{equation} \label{glad1} \| II\|_{X_{t_2,t_3}(\RE^+)}\leqslant C %\lf( \int_{0}^\infty\big|\phi_0^\al(-y)\big|^2dy \ri)^{1/2} \| \chi_- \phi_0^\gamma\|_{L^2(\erre) } \leqslant C e^{-2v^{1-\de}}\,. \end{equation} %%% Finally, the term ${III}$ can be estimated using the inhomogeneous Strichartz estimate and proposition \ref{appAHMZ}. %%% \begin{equation} \label{glad2} \|III\|_{X_{t_2,t_3}(\RE^+)} \leqslant C \|( \chi_- \phi^{\gamma}_{\cdot-t_2} )^3 \|_{L^1_{[t_2, t_3] } L^2(\erre)} \leqslant C \bigg[ \frac{c(k)(\ln v)^{\si(k)'}}{v^{k(1-\de)}}\bigg]^3\,. \end{equation} Collecting the estimates \eqref{glad0}, \eqref{glad1} and \eqref{glad2}, it follows that %%% \begin{equation*} \|f^{\gamma,+}_\cdot\|_{X_{t_2,t_3}(\RE^+)}\leqslant C \bigg[ \frac{c(k)(\ln v)^{\si(k)'}}{v^{k(1-\de)}}\bigg]\,. \end{equation*} %%% The estimate of $f^{\gamma,-}_t$ is similar and we omit it. We have proved that for some $c'(k)$ and $\si'(k)$ possibly bigger than $c(k)$ and $\si(k)$ we have: \begin{equation*} \|\widetilde G_1\|_{X_{t_2,t_3}} \leqslant C \bigg[ \frac{c'(k)(\ln v)^{\si'(k)}}{v^{k(1-\de)}}\bigg] \end{equation*} where $\widetilde G_1$ is the vector in $L^2$ with components $\big(\widetilde{G}_1(x,t)\big)_l$, $l=1,2,3$. The estimate of $\widehat G_1(t)=\big(\big(\widehat{G}_1(t)\big)_1,\big(\widehat{G}_1(t)\big)_2,\big(\widehat{G}_1(t)\big)_3\big)$ is a trivial consequence of the fact that %%% \begin{equation*} \bigg\|va \int_0^\infty e^{-vua} f^{\gamma,\pm}_\cdot(u+\,\cdot\,)du\bigg\|_{X_{t_2,t_3}(\RE^+)} %\leqslant %va \int_0^\infty e^{-vua} \big\|f^{\gamma,\pm}_\cdot(u+\cdot)\big\|_{X(\RE^+)}du \leqslant \big\|f^{\gamma,\pm}_\cdot\big\|_{X_{t_2,t_3}(\RE^+)}\leqslant C \bigg[ \frac{c(k)(\ln v)^{\si(k)'}}{v^{k(1-\de)}}\bigg]\,, \end{equation*} %%% from which it follows that \begin{equation*} \|\widehat G_1\|_{X_{t_2,t_3}} \leqslant C \bigg[ \frac{c'(k)(\ln v)^{\si'(k)}}{v^{k(1-\de)}}\bigg]\,; \end{equation*} and \begin{equation} \label{estG1} \| G_1\|_{X_{t_2,t_3}} \leqslant C \bigg[ \frac{c'(k)(\ln v)^{\si'(k)}}{v^{k(1-\de)}}\bigg]\,. \end{equation} \n We analyse now the term $G_2$. Due to the presence of $(1-\de_{j_1j_2}\de_{j_2j_3})$ the components of the vector %%% \begin{equation*} (1-\de_{j_1j_2}\de_{j_2j_3})\overline{ \Phi^{j_1}_s}\Phi^{j_2}_s\Phi^{j_3}_s \end{equation*} %%% contains only terms (up to a phase) like %%% \begin{equation*} %\label{orange} \phi^{\gamma_1}_{t-t_2} (x) \phi^{\gamma_2}_{t-t_2} (x) \phi^{\gamma_3}_{t-t_2} (-x)\quad\textrm{or}\quad \phi^{\gamma_1}_{t-t_2} (x) \phi^{\gamma_2}_{t-t_2} (-x) \phi^{\gamma_3}_{t-t_2} (-x) \end{equation*} %%% where $\gamma_1$, $\gamma_2$ and $\gamma_3$ can be $ref$ or $tr$. This can be easily seen by using equations \eqref{Phi123t} - \eqref{Phi1234t}. By Strichartz, it is sufficient to estimate the $L^1_{[t_2, t_3] } L^2(\erre^+)$ norm of these terms. Then using H\"older inequality we have, for istance %%% \begin{equation*} \|\phi^{\gamma_1}_{t-t_2}\phi^{\gamma_2}_{t-t_2}\phi^{\gamma_3}_{t-t_2}(-\cdot)\|_{L^{2}(\RE^+)}\leqslant \|\phi^{\gamma_1}_{t-t_2}\|_{L^4(\RE^+)}\|\phi^{\gamma_2}_{t-t_2}\|_{L^4(\RE^+)}\|\phi^{\gamma_3}_{t-t_2}\|_{L^{\infty}(\RE^-)} \leqslant C \frac{c(k)(\ln v)^{\si(k)}}{v^{k(1-\de)}}\,. \end{equation*} %%% The second kind of terms can be estimated in the same way and we obtain \begin{equation*} \|G_2\|_{X_{t_2,t_3}} \leqslant C \bigg[ \frac{c'(k)(\ln v)^{\si'(k)}}{v^{k(1-\de)}}\bigg]\,, \end{equation*} which, together with the estimate \eqref{estG1}, gives \begin{equation} \|G\|_{X_{t_2,t_3}} \leqslant C \bigg[ \frac{c'(k)(\ln v)^{\si'(k)}}{v^{k(1-\de)}}\bigg]\,. \label{kagu} \end{equation} Fix $k$ such that $k(1-\de)>2$ then for $v$ sufficiently large \eqref{kagu} implies \begin{equation} \|G\|_{X_{t_2,t_3}} \leqslant \f 1 v\,. \label{unten} \end{equation} Using estimates \eqref{ect} and \eqref{unten} in the inequality \eqref{perf} we get \begin{equation*} \begin{aligned} \| \Xi \|_{X_{t_2,t_3}} \ \ls & Cv^{-\de/2} \sum_{j=0}^{N} (2\wt C)^{j+1} + v^{-1} \sum_{j=0}^{N} \sum_{k = 0}^{j} (2 \wt C)^{j+1-k}\\ \leqslant & 2 Cv^{-\de/2} (2\wt C)^{N+1} + v^{-1} \sum_{j=0}^{N} \big[ (2 \wt C)^{j+1}+(2 \wt C)^{j}+...+(2 \wt C)^{2}+(2 \wt C)\big] \\ \leqslant & 2 Cv^{-\de/2} (2\wt C)^{N+1} + 2 v^{-1} N (2 \wt C)^{N+1}\leqslant \widehat C N v^{-\de/2}(2 \wt C)^N\,. \end{aligned} \end{equation*} Since $N$ is the integer part of $(t_3-t_2)/\tau=\frac{T}{\tau}\ln v$ \begin{equation*} N v^{-\de/2}(2 \wt C)^N=\exp\Big[ \ln N-\frac{\de}{2}\ln v+N\ln (2 \wt C) \Big] \leqslant \exp\Big[\Big(-\frac{\de}{2}+\frac{T}{\tau}(\ln (2 \wt C)+1)\Big)\ln v \Big]\,. \end{equation*} We can finally set $\tau_*\equiv \tau/(\ln (2 \wt C)+1)$ and $T_*=\de \tau_*/2$ and obtain \begin{equation*} \| \Xi \|_{L^\infty_{[t_2,t_3]}L^2}\leqslant \| \Xi \|_{X_{t_2,t_3}} \leqslant \widehat C v^{ -\frac{T_*-T}{\tau_*}}\,. %\label{mokuteki} \end{equation*} which concludes the proof of theorem \eqref{mainth}. %Now we can finish the proof of the theorem by an iterative argument. By the previous estimates we can divide $(t_2, t_3)$ into intervals of constant width such that \eqref{conditer2} holds. %Let us assume for sake of simplicity that we can take intervals of length 1. %Let $l$ be the integer part of $T \ln v$. Iterating \eqref{iter2} $l+1$ times, we finally arrive at %\be %\| \Xi \|_{\LT{\infty}{t_2,t_3}\LG{2}} \leqslant C\, \widehat{C}^{T\ln v} v^{-\f \de 2} =C \, v^{ T \ln \widehat C -\f \de 2}\,. %%\label{mokuteki} %\end{equation*} %If we put $\tau^{-1} = \ln \widehat C$ and $T_{\ast} = \tau \de /2$ then we arrive at %\be %\| \Xi \|_{\LT{\infty}{t_2,t_3}\LG{2}} \leqslant C\, v^{-\f{T_{\ast} - T}{\tau} } %\end{equation*} %and the proof of theorem \ref{mainth} in concluded. \end{proof} \vspace{20pt} \n {\bf Acknowledgements.} \n The present research was partially supported by INDAM-GNFM research project ``Equazione di Schr\"odinger non lineare interagente con difetti sulla retta e su grafi''. \begin{thebibliography}{10} \bibitem{[AbFS]} W.~K. {Abu Salem}, J.~Fr{\"o}hlich, and I.~M. Sigal, \emph{Colliding solitons for the nonlinear {S}chr{\"o}dinger equation}, Comm. Math. 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Theor. % \textbf{42} (2009), no.~49, 495302, 19pp. % \bibitem{[ACF]} S.Albeverio, C.Cacciapuoti , D.Finco: Coupling in the singular limit of thin quantum waveguides, J.Math. Phys, {\bf 48}, 032103 (2007) % \bibitem{[BCFK06]} %G.~Berkolaiko, R.~Carlson, S.~Fulling, P.~Kuchment, Quantum graphs % and their applications, Contemporary Math., vol. 415, American Math. % Society, Providence, R.I., 2006. %\bibitem{[BEH]} J.Blank , P.Exner, M.Havlicek : {\em Hilbert spaces % operators in Quantum Physics}, Springer, New York (2008). %\bibitem{[BCSTV]} R.Burioni, D.Cassi, P.Sodano, A.Trombettoni, A.Vezzani : Soliton Propagation on chains with simple nonlocal defects, Physica D, {\bf 216}, 71-76, (2006) % \bibitem{[CE]} C.Cacciapuoti, P. Exner: Nontrivial edge coupling from a {D}irichlet network squeezing: the case of a bent waveguide, J. Phys A, Math.Theor. {\bf 40}, F511--F523 (2007) %\bibitem{[CM]} D.Cao Xiang , A.B.Malomed : Soliton defect collisions %in the nonlinear Schr\"odinger equation, {\it Phys. Lett. A} {\bf 206} 177-182 (1995) % \bibitem{[CMu]} S.Cardanobile, D.Mugnolo : Analysis of FitzHugh-Nagumo-Rall model of a neuronal network, Math.Meth.Appl.Sci., {\bf 30}, 2281-2308 (2007) %\bibitem{[Caz]} T. Cazenave : {\it Semilinear Schr\"odinger Equations}, vol. {\bf 10} Courant Lecture Notes in Mathematics {\it AMS}, Providence (2003) %\bibitem{[DH]} K.~Datchev, J.~Holmer: Fast soliton scattering by attractive delta impurities Communications in partial differential equations \textbf{34}, 1074-1113, (2009)\p %\bibitem{[Ex]} P. Exner : Contact interactions on graph superlattices, J.Phys A {\bf 29}, 87-102, (1996) %\bibitem{[GHW]} R.H. Goodman, P.J.Holmes, M.I.Weinstein: Strong NLS soliton-defect interactions {\it Physica D}, {\bf 192} (2004), 215-248 %\bibitem{[HMZ07]} %J.~Holmer, J.~Marzuola, M.~Zworski :Fast soliton scattering by delta % impurities, Commun. Math. Phys. \textbf{274} (2007), 187--216. % \bibitem{[HMZ2]} J.~Holmer, J.~Marzuola, M.~Zworski: Soliton splitting by delta impurities, J. Nonlinear. %Sci., {\bf 7}, 349-367, (2007) % \bibitem{[KT]} M.~Keel, T.~Tao: Endpoint Strichartz estimates, \emph{Amer.J.Math}, \textbf{120} 955,980, (1998)\p %\bibitem{[KFTK]} P. G. Kevrekidis, D. J. Frantzeskakis, G. Theocharis, I. G. Kevrekidis : Guidance of matter waves through Y-junctions Phys. Lett. A 317, 513 (2003) %\bibitem{[KS99]} %V.~Kostrykin, R.~Schrader : {K}irchhoff's rule for quantum wires, J. % Phys. A: Math. Gen. \textbf{32} (1999), no.~4, 595--630 %\bibitem{[Kuc04]} P. Kuchment: %Quantum graphs: I. Some basic structures, % Waves in Random and Complex Media, 14, {\bf 1}, %(2003), S107-S128. %\bibitem{[Kuc05]} P.Kuchment : Quantum graphs II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A {\bf 38} 4887-4900, (2005). %\bibitem{[P1]} G.Perelman : Two soliton collision for nonlinear Schr\"odinger equation in dimension one, International Conference on Spectral Theory, St. Petersbourg, (2009) %\bibitem{[P2]} G.Perelman : A remark on soliton-potential interaction for nonlinear Schr\"odinger equations Math. Res. Lett. 16 (2009), no. 3, 477Ð486 %\end{thebibliography} %%BIBLIOGRAFIA SUL MIO POWERBOOK DA FILE SPECIFICO %\bibliographystyle{/Users/claudio/works/myamsplain} %\bibliography{adami-cacciapuoti-finco-noja10} %nome file .bib nella stessa cartella %%BIBLIOGRAFIA SUL MIO POWERBOOK %\bibliographystyle{/Users/claudio/works/myamsalpha} %\bibliography{/Users/claudio/works/mywwb10} %%BIBLIOGRAFIA SUL MAC DELL'UNIVERSITA' %\bibliographystyle{/private/Network/Servers/him-addc-01.him.uni-bonn.de/home/cacciapuoti/works/myamsalpha} %\bibliography{/private/Network/Servers/him-addc-01.him.uni-bonn.de/home/cacciapuoti/works/mywwb10} %\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} %\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. %\providecommand{\MRhref}[2]{% % \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} %} %\providecommand{\href}[2]{#2} %\begin{thebibliography} %\bibitem{[AbFS]} W.K. Abu Salem, J.~Fr\"ohlich, I.M.Sigal, Colliding solitons for the nonlinear Schr\"odinger equation, Comm.Math.Phys, \textbf{291}, 151-176 (2009)\p %\bibitem{[AN09]} %R.~Adami and D.~Noja, \emph{Existence of dynamics for a 1{D} {NLS} equation % perturbed with a generalized point defect}, J. Phys. A: Math. Theor. % \textbf{42} (2009), no.~49, 495302, 19pp.\p %\bibitem{[BCFK06]} %G.~Berkolaiko, R.~Carlson, S.~Fulling, and P.~Kuchment, \emph{Quantum graphs % and their applications}, Contemporary Math., vol. 415, American Math. % Society, Providence, R.I., 2006.\p %\bibitem{[Caz]} T.~Cazenave, {\it Semilinear Schr\"odinger equations} Lecture Notes of the Courant Institute, AMS, (2000)\p %\bibitem{[HMZ07]} %J.~Holmer, J.~Marzuola, and M.~Zworski, \emph{Fast soliton scattering by delta % impurities}, Commun. Math. Phys. \textbf{274}, 187--216, (2007)\p %\bibitem{[DH]} K.~Datchev, J.~Holmer \emph{Fast soliton scattering by attractive delta impurities} Communications in partial differential equations \textbf{34}, 1074-1113, (2009)\p %\bibitem{[KT]} M.~Keel, T.~Tao, Endpoint Strichartz estimates, \emph{Amer.J.Math}, \textbf{120} 955,980, (1998)\p %\bibitem{[KS99]} %V.~Kostrykin and R.~Schrader, \emph{{K}irchhoff's rule for quantum wires}, J. % Phys. A: Math. Gen. \textbf{32} (1999), no.~4, 595--630.\p %\bibitem{[Kuc04]} %P.~Kuchment, \emph{Quantum graphs. {I}. {S}ome basic structures}, Waves Random % Media \textbf{14} (2004), no.~1, S107--S128.\p %\end{thebibliography} \end{document} @article{albeverio-cacciapuoti-finco:07, author = {Sergio Albeverio and Claudio Cacciapuoti and Domenico Finco}, title = "Coupling in the singular limit of thin quantum waveguides", journal ={J. Math. Phys.}, volume = {48}, year = {2007}, number = {}, pages = {032103} } @article{cacciapuoti-exner:07, author = {Claudio Cacciapuoti and Pavel Exner}, title = "Nontrivial edge coupling from a {D}irichlet network squeezing: the case of a bent waveguide", journal ={J. Phys. A: Math. Theor.}, volume = {40}, year = {2007}, number = {26}, pages = {F511--F523}, note = {} } Squeezing di guide d'onda con condizioni al bordo generiche @article{cacciapuoti-finco:08pp, author = {Claudio Cacciapuoti and Domenico Finco}, title = "Graph-like models for thin waveguides with Robin boundary conditions", journal ={arxiv:0803.4314 [math-ph]}, volume = {}, year = {2008}, number = {}, pages = {}, note = {} } \subsection{Phase 1} We call ``phase 1'' the dynamics in the time interval $\left(0, t_1 \right)$ with $t_1= \f {x_0} v - v^{-\delta}$. In this interval we approximate the solution by the soliton \eqref{truesol}. Before estimating the error, we prove the following lemma. \begin{lemma} \label{caudale} For the functions \begin{equation} \begin{split} \label{cappa} K_1 (t,x)& \ : = \ \int_0^\infty U_t (x + y) e^{-i \f v 2 y} \phi ( y - x_0) \, dy + i \int_0^t ds \int_0^\infty U_{t-s} (x + y) e^{-i \f v 2 y} {e^{- i s \f {v^2} 4}} e^{is} \phi^3 ( y - x_0 + vs) \, dy \\ K_2 (t,x) & \ : = \ \int_0^\infty U_t (x + y) e^{i \f v 2 y} \phi ( y + x_0) \, dy + i \int_0^t ds \int_0^\infty U_{t-s} (x + y) e^{i \f v 2 y} {e^{- i s \f {v^2} 4}} e^{is} \phi^3 ( y + x_0 - vs) \, dy \end{split} \end{equation} the following estimate holds: \be %\label{stimacaudale} \| K_i \|_{X (\erre^+)} \leqslant \ e^{-x_0 + v T}, \qquad i = 1,2, \end{equation*} where $X (\erre^+) : = L^\infty_{[0,T]} L^2 (\erre^+) \cap L^6_{[0,T]} L^6 (\erre^+)$, if $v$ is large enough. \end{lemma} \begin{proof} Let us start with $K_1$. Adding and subtracting a contribution on the negative $y$ one can write \be \begin{split} K_1 (t,x) & \ = \ \int_{-\infty}^\infty U_t (x + y) e^{-i \f v 2 y} \phi ( y - x_0) \, dy + i \int_0^t ds \int_{-\infty}^\infty U_{t-s} (x + y) e^{-i \f v 2 y} {e^{- i s \f {v^2} 4}} e^{is} \phi^3 ( y - x_0 + vs) \, dy \\ & \ - \int_{-\infty}^0 U_t (x + y) e^{-i \f v 2 y} \phi ( y - x_0) \, dy - i \int_0^t ds \int_{-\infty}^0 U_{t-s} (x + y) e^{-i \f v 2 y} {e^{- i s \f {v^2} 4}} e^{is} \phi^3 ( y - x_0 + vs) \, dy \\ = & \ e^{i \f v 2 x} {e^{- i t \f {v^2} 4}}e^{it} \phi ( x + x_0 - vt) \\ & \ - \int_{-\infty}^0 U_t (x + y) e^{-i \f v 2 y} \phi ( y - x_0) \, dy - i \int_0^t ds \int_{-\infty}^0 U_{t-s} (x + y) e^{-i \f v 2 y} {e^{- i s \f {v^2} 4}} e^{is} \phi^3 ( y - x_0 + vs) \, dy \end{split} %\label{terms} \end{equation*} where we used the integral equation \eqref{eq:sol}. Since all the components have the same structure, we reduce ourselves to estimate functions on the real axis instead of on $\GG$. %Therefore let us put $X(\erre^+)\equiv L^{\infty}_{[t_a, t_b] } %L^2 (\erre^+) \cap L^{6}_{[t_a, t_b] } L^6 ( \erre^+) $ and %$X(\erre)\equiv L^{\infty}_{[t_a, t_b] } L^2 (\erre) \cap L^{6}_{[t_a, t_b] } L^6 ( \erre) $. Then, the $X(\erre^+)$-norm of the first term can be estimated by a straightforward computation by $ C e^{ -(x_0 - vt_b) } \leqslant C e^{-v^{1 -\delta}}. $ To evaluate the size of the second term, let us write it as follows: \be %\label{kaka} \int_0^\infty U_t (x-y) e^{i \f v 2 y} \phi (y + x_0)dy \ =[ U^-_t e^{i \f v 2 \cdot} \phi (\cdot + x_0)] (x)= \ [U_t \chi_+ e^{i \f v 2 \cdot} \phi (\cdot + x_0)] (x), \quad x > 0. \end{equation*} Therefore, using now the one dimensional (homogeneous) Strichartz estimates for $U_t$, we can estimate the $X(\RE^+)$ norm of this term as $$ \| U^-_t e^{i \f v 2 \cdot} \phi (\cdot + x_0) \|_{X(\erre^+)}\leqslant \|U_t \chi_+ e^{i \f v 2 \cdot} \phi (\cdot + x_0) \|_{X(\RE)} \leqslant C \| \chi_+ e^{i \f v 2 \cdot} \phi (\cdot + x_0) \| \ \leqslant C e^{-x_0} \leqslant \ C e^{-v^{1 -\delta}}, $$ where we used the notation $X (\erre) : = L^\infty_{[0,T]} L^2 (\erre) \cap L^6_{[0,T]} L^6 (\erre)$. The norm of the last term in the r.h.s. of equation \eqref{terms} can be estimated in a similar way by \be %\label{akak} \left\| \int_0^{\cdot} \big[ U_{\cdot-s} |\phi_{x_0 , v} (s)|^2 \phi_{x_0 , v} (s) \big]ds \right\|_{X(\RE^+)} \ \leqslant C \| \phi_{x_0 , v}^3\|_{L^1_{[t_a,t_b]} L^2(\RE^+)} \leqslant C\f {1} {v} e^{-x_0 + vt_b}\,. \end{equation*} Therefore, \be \| K_1 \|_{X(\erre^+)} \ \leqslant \ C \f {1} {v} e^{-x_0 + vt_b} \leqslant e^{-x_0 + vt_b} %\label{aka} \end{equation*} and in the last inequality we have assumed that $v \geqslant C$. To estimate $K_2$, the first term can be treated as in \eqref{kaka}, while the second is estimated following the line of \eqref{akak}. \end{proof} %%%%%%%%%% %PROPOSITION %%%%%%%%%% \begin{prop} Let $t\in \left(0, t_1 \right)$ then there exists $v_0>0$ such that for all $v>v_0$ %\label{fase1} \be %\label{error1} \| \Psi_t - \Phi_t \| \ \leqslant \ c e^{-v^{1 - \delta}} \end{equation*} where $C$ is a positive constant which independent of $t$ and $v$. \end{prop} \begin{proof} \n Let us define $\Xi_t : = \Psi_t - \Phi_t$. Then, from equations \eqref{intform} and \eqref{duhamsol}, we have \be %\label{Xit} \begin{split} \Xi_t \ = & \ e^{-iH t} \Xi_0 + (e^{-iH t} - e^{-i H_1 t}) \Phi_0 + i \int_0^t (e^{-iH (t-s)} - e^{-i H_1 (t-s)}) | \Phi_s |^2 \Phi_s \, ds\\ & \ + i \int_0^t e^{-iH (t-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right]\\ = & \ e^{-iH t} \Xi_0 + F(t)\\ & + i \int_0^t e^{-iH (t-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right]\,. \end{split} \end{equation*} \n Let us fix $0\leqslant t_a \leqslant t_b \leqslant t_1$ and let $X= L^{\infty}_{[t_a, t_b] } L^2 (\GG) \cap L^{6}_{[t_a, t_b] } L^6 ( \GG) $. If we consider the equation \eqref{Xit} on the time interval $[t_a,t_b]$, we have \be \begin{split} \Xi_t \ =& \ e^{-iH (t-t_a)} \Xi_{t_a} + F(t) \\ &+ i \int_{t_a}^t e^{-iH (t-s)} \left[ | \Xi_s |^2 \Xi_s + | \Xi_s |^2 \Phi_s + | \Phi_s |^2 \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Xi_s + 2 {\mbox{Re}} ( \overline{\Xi_s} \Phi_s) \Phi_s \right]\,. \end{split} \end{equation*} \n Now we want to derive an estimate of $\| \Xi \|_X$ w.r.t. to $ \Xi_{t_a}$ and $F$. We shall estimate the nonlinear term using Strichartz estimates. Let us analyze in details the cubic term. Since both $(\infty ,2 )$ and $(6,6)$ are admissibile indices in proposition \ref{prop:stric} we can choose $q=k= 6/5$ and the following estimates holds: \be \lf\| \int_{t_a}^{\cdot} e^{-iH (\cdot-s)} | \Xi_s |^2 \Xi_s \,ds \ri\|_X \leqslant C \| | \Xi_{\cdot} |^2 \Xi_{\cdot} \|_{ \LT{6/5}{t_a,t_b} \LG{6/5} }\,. %\label{yume} \end{equation*} Using H\"older estimates $ \| \Xi \|_{\LG{6/5}} \leqslant \| \Xi \|_{\LG{3}} \| \Xi \|_{\LG{2}} $ and $\|f \|_{\LT{1}{t_a,t_b}} \leqslant \|f \|_{\LT{5/2}{t_a,t_b}} \|f \|_{\LT{5/3}{t_a,t_b} } $ in the r.h.s. of \eqref{yume} we arrive at \be \lf\| \int_{t_a}^{\cdot} e^{-iH_F (\cdot-s)} | \Xi_s |^2 \Xi_s \,ds \ri\|_X \leqslant C (t_b-t_a )^{1/2} \|\Xi\|_X^3\,. \end{equation*} The other terms can be treated in the same way, see \cite[Lemma 3.1]{[HMZ07]} for details, so we get: \be \| \Xi \|_X \leqslant c\lf[ \| \Xi_{t_a} \| + \| F\|_X + (t_b-t_a)^{1/2} \lf( \| \Xi \|_X^2 + \| \Xi \|_X^3 \ri) \ri], %\label{kagaku} \end{equation*} where $C$ depends only on the constants appearing in the Strichartz estimates. If $t_b-t_a$ is small enough, then \eqref{kagaku} provides the required estimates. In fact there exists $\tilde c >0$ such that if \be (\lf\| \Xi_{t_a} \| + \| F\|_X\ri)(t_b-t_a)^{1/2} \leqslant \tilde c %\label{conditer} \end{equation*} then \be \| \Xi \|_X \leqslant \widetilde C (\lf\| \Xi_{t_a} \| + \| F\|_X\ri)\,. %\label{iter} \end{equation*} Before using \eqref{iter} for an iterative argument, we have to estimate the size of the initial datum and of the forcing term in \eqref{Xit}. \begin{equation} \| \Xi_0 \|^2 \leqslant \ \int_0^{2} \phi^2 (x-x_0) \, dx + \int_0^\infty \phi^2 (x + x_0) = \ 2 (1 - {\rm{tanh}} (x_0 - 2)) \ \leqslant \ C e^{-2 x_0}. \label{tamago} \end{equation} It remains to estimate the source term $F(t)$ defined in \eqref{Xit} by \be F(t) \ : = \ (e^{-iHt} - e^{-iH_1t} ) \Phi_0 + \int_0^t (e^{-iH(t-s)} - e^{-iH_1(t-s)}) | \Phi_s |^2 \Phi_s \, ds. \end{equation*} Let us specialize $F(t)$ to the three cases under analysis. We have: \be \begin{split} F_F(t)\ = & \ \f 1 3 \, U_t^+ \left( \begin{array} {ccc} -1 & -1 & 2 \\ -1 & -1 & 2 \\ 2 & 2 & 2 \end{array} \right) \Phi_0 + \f i 3 \int_0^t U_{t-s}^+ \left( \begin{array} {ccc} -1 & -1 & 2 \\ -1 & -1 & 2 \\ 2 & 2 & 2 \end{array} \right) | \Phi_s |^2 \Phi_s \, ds\, \\ F_\delta^\alpha (t)\ = & \ F_F(t) - \f 2 9 \alpha \int_0^{+\infty}du %\int_0^{+\infty} dy \, e^{-\f \alpha 3 u} [U_t^+ \, {\mathbb J} \, \Phi_0] (\cdot + u) - i \f 2 9 \alpha \int_0^t ds \int_0^{+\infty}du \, e^{-\f \alpha 3 u} [U_{t-s}^+ \, {\mathbb J} \, | \Phi_s |^2 \Phi_s] (\cdot + u) \\ F_{\delta^\prime}^\beta (t)\ = & U_t^+ \left( \begin{array} {ccc} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right) \Phi_0 + i \int_0^t U_{t-s}^+ \left( \begin{array} {ccc} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right) | \Phi_s |^2 \Phi_s \, ds\, \\ & - \f 2 \beta \int_0^{+\infty}du %\int_0^{+\infty} dy \, e^{-\f 3 \beta u} [U_t^+ \, {\mathbb J} \, \Phi_0] (\cdot + u) - i \f 2 \beta \int_0^t ds \int_0^{+\infty}du \, e^{-\f 3 \beta u} [U_{t-s}^+ \, {\mathbb J} \, | \Phi_s |^2 \Phi_s] (\cdot + u) \end{split} \end{equation*} It is immediately seen that $$ F_F (t,x ) \ = \ \f 1 3 \left( \begin{array}{c} - K_1 (x,t) - K_2 (x,t) \\ - K_1 (x,t) - K_2 (x,t) \\ 2 K_1 (x,t) + 2 K_2 (x,t), \end{array} \right) $$ so lemma \ref{caudale} yields \be \| F_F \|_X \ \leqslant \ e^{-x_0 + vt_b} %\label{aka} \end{equation*} for a sufficiently large $v$. Furthermore, since \be \begin{split} F_\delta^\alpha (t,x) \ = \ F_F (t,x ) - \f 2 9 \alpha \int_0^{+\infty} du \, e^{-\f \alpha 3 u} \left( \begin{array}{c} K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t)\end{array} \right) \end{split} \end{equation*} we conclude \be \| F_\delta^\alpha \|_X \ \leqslant \ e^{-x_0 + vt_b} %\label{aka} \end{equation*} for a sufficiently large $v$. Finally, \be \begin{split} F_{\delta^\prime}^\beta (t,x) \ = \ \left( \begin{array}{c} K_1 (x,t) - K_2 (x,t) \\ - K_1 (x,t) + K_2 (x,t) \\ 0 \end{array} \right) - \f 2 \beta \int_0^{+\infty} du \, e^{-\f 3 \beta u} \left( \begin{array}{c} K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t) \\ K_1 (x+u,t) + K_2 (x+u,t)\end{array} \right) \end{split} \end{equation*} yields \be \| F_{\delta^\prime}^\beta \|_X \ \leqslant \ e^{-x_0 + vt_b} %\label{aka} \end{equation*} for a sufficiently large $v$. Now we can start the iteration. First notice that due to the previous estimates, if $v$ is sufficiently large, we can divide $(0,t_1)$ into subintervals of constant length where condition \eqref{conditer} holds. For sake of simplicity let us assume that we can take subintervals of length 1. Then \eqref{tamago} and \eqref{aka} with \eqref{iter} lead to \begin{align} &\| \Xi_{t=1}\| \leqslant \widetilde C e^{-x_0 } + e^{-x_0 + v} \leqslant 2\widetilde C e^{-x_0 + v} \nonumber \\ & \|\Xi_{t=2}\| \leqslant \widetilde C\lf( 2C e^{-x_0 + v} + e^{-x_0 + 2v} \ri) = \widetilde C e^{-x_0 + v} \lf(2C + e^v \ri) \leqslant 2\widetilde C e^{-x_0 + 2v} \nonumber \\ & \|\Xi_{t=3}\| \leqslant \widetilde C \lf( 2\widetilde C e^{-x_0 + 2v} + e^{-x_0 + 3v} \ri) = \widetilde C e^{-x_0 +2 v} \lf(2 \widetilde C + e^v \ri) \leqslant 2\widetilde C e^{-x_0 + 3v} \nonumber \\ &\ldots & \nonumber \\ &\| \Xi_{t=k} \| \leqslant 2\widetilde C e^{-x_0 + kv} %\label{kuro} \end{align} and we have assumed in \eqref{kuro} that $2\widetilde C \leqslant e^v$. \n Let $\overline k$ be the integer part of $t_1$, applying one last time \eqref{iter} to $[ \overline k , t_1 ]$ we obtain \begin{equation*} \| \Xi _{t_1} \| \leqslant2 \widetilde C e^{-x_0 + v t_1} \leqslant 2\widetilde C e^{- v^{1-\de} } . \end{equation*} \end{proof} ---------------1004141506738--