Content-Type: multipart/mixed; boundary="-------------0110142017457" This is a multi-part message in MIME format. ---------------0110142017457 Content-Type: text/plain; name="01-373.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-373.comments" Submitted on August 27, 2001 ---------------0110142017457 Content-Type: text/plain; name="01-373.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-373.keywords" relaxation, excited states, Schroedinger equations. ---------------0110142017457 Content-Type: application/x-tex; name="es-arXiv-08oct2001.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="es-arXiv-08oct2001.tex" % % es-submit-27aug2001.tex % AMS-LaTeX file % Tsai & Yau % \documentclass[12pt]{article} \usepackage{amsmath, amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textwidth 6in %%7in \oddsidemargin 0.25in %%-0.25in \evensidemargin \oddsidemargin \textheight 9.5in %9in \topmargin -0.85in %-0.5in \newcommand{\mybaselineskip}{\baselineskip 20pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % macro %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] %%[chapter] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \numberwithin{equation}{section} %%{chapter} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} %%\renewcommand{\theequation}{\arabic{equation}} \renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}} \renewcommand{\theenumi}{\Alph{enumi}} \newcommand{\R}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \renewcommand{\Re}{\mathop{\mathrm{Re}}} \renewcommand{\Im}{\mathop{\mathrm{Im}}} \newcommand{\eigen}{\mathtt{E}_1} %--- operators ------------------------------- \newcommand{\sign}{\mathop{\rm sign}} \newcommand{\supp}{\mathop{\mathrm{supp}}} \newcommand{\esssup}{\mathop{\rm ess{\,}sup}} \newcommand{\essinf}{\mathop{\rm ess{\,}inf}} \newcommand{\myspan}{\mathop{\rm span}} %% already defined: \liminf and \limsup %--------------------------------------------- \newcommand{\longto}{\longrightarrow} \newcommand{\norm}[1]{\left\Vert #1 \right\Vert} \newcommand{\length}[1]{\left| #1 \right|} \newcommand{\bkA}[1]{\left \langle #1 \right \rangle} \newcommand{\bke}[1]{\left( #1 \right)} \newcommand{\bkt}[1]{\left[ #1 \right]} \newcommand{\bket}[1]{\left\{ #1 \right\}} \newcommand{\e}{\varepsilon} \newcommand{\loc}{_{\mathrm{loc}}} \newcommand{\myremark}{\addtocounter{equation}{1}{{\bf Remark (\theequation)}}\quad} \newcommand{\myproof}{\noindent {\bf Proof.}\quad} \newcommand{\myendproof}{\hspace*{\fill}{{\bf \small Q.E.D.}} \vspace{10pt}} \newcommand{\al}{\alpha} \newcommand{\ka}{\kappa} \newcommand{\om}{\omega} \newcommand{\proj}{\big \lfloor} \renewcommand{\d}{\delta} \renewcommand{\L}{\mathcal{L}} \newcommand{\la}{\lambda} \newcommand{\Pc}{\, \mathbf{P}\! _\mathrm{c} \, \! } \newcommand{\PcA}{\, \mathbf{P}\! _\mathrm{c} \! ^A \,} \newcommand{\PcH}{\, \mathbf{P}\! _\mathrm{c} \! ^{H_0} \,} \newcommand{\PcL}{\, \mathbf{P}\! _\mathrm{c} \! ^\L \,} \newcommand{\Hc}{\, \mathtt{H} _\mathrm{c} \, \! } \newcommand{\Eigen}{\mathtt{E}} \newcommand{\Prg}{\mathtt{P}} \newcommand{\original}{^{(\mathrm{or})}} \newcommand{\diff}{{\pmb{\delta}}} \newcommand{\tl}[1]{\bket{#1}} \newcommand{\wt}{\widetilde} \newcommand{\conj}{\mathtt{C}} %\newcommand{\pfrac}[2]{{\dfrac{\partial #1}{\partial #2}}} \newcommand{\xii}{\xi_\infty} \newcommand{\pd}{\partial} \newcommand{\donothing}[1]{} \newcommand{\wbar}[1]{\overline{\rule{0pt}{2.4mm} {#1}}} \newcommand{\lbar}[1]{\underline{#1}} \newcommand{\n}{{n}} %{{- \! \! \! \! n}} %{\hbar} % \newcommand{\vect}[1]{\begin{bmatrix} #1 \end{bmatrix}} \newcommand{\leC}{\,\lesssim\,} %{\, \overset{<}{\sim}\, } \newcommand{\xm}{|x|} \newcommand{\dis}{\displaystyle} \newcommand{\xione}{\xi^{(3)}_1} \newcommand{\xitwo}{\xi^{(2)}} \newcommand{\xithree}{\xi^{(3)}} %_{2-5}} \newcommand{\uu}{\mu} %{{u}_\star} \newcommand{\vv}{\nu} %{{v}_\star} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % main document % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \mybaselineskip \title{Relaxation of Excited States in Nonlinear Schr\"odinger Equations} %\donothing{ \author{Tai-Peng Tsai\footnote{ttsai@cims.nyu.edu} \qquad Horng-Tzer Yau\footnote{Work partially supported by NSF grant DMS-0072098, yau@cims.nyu.edu} \\ \vspace*{-0.3cm} \\ Courant Institute, New York University} %, New York, NY 10012, USA. } \date{August 27, 2001} %} \donothing{ \author{Tai-Peng Tsai} \address{Courant Institute, New York University, New York, NY 10012, USA} %\curraddr{Institute for Advanced Study, Princeton, NJ 08540, USA} \email{ttsai@cims.nyu.edu} \author{Horng-Tzer Yau} \address{Courant Institute, New York University, New York, NY 10012, USA} \email{yau@cims.nyu.edu} \thanks{The work of the second author was partially supported by NSF grant DMS-0072098} \subjclass{Primary 35Q40, 35Q55} %\date{January 1, 1994 and, in revised form, June 22, 1994.} \date{January 27, 2001} \keywords{Relaxation, Excited states, Schr\"odinger equations} } \maketitle \footnotetext{{\it 2000 Mathematics Subject Classification.} Primary. 35Q40, 35Q55} \footnotetext{{\it Key words and phrases.} Relaxation, Excited states, Schr\"odinger equations.} \begin{abstract} We consider a nonlinear Schr\"odinger equation in $\R^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear \!{\it excited}\, state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schr\"odinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity. \end{abstract} %%\tableofcontents %\tableofcontents \section{Introduction} Consider the nonlinear Schr\"odinger equation % \begin{equation} \label{Sch} i \pd _t \psi = (-\Delta + V) \psi + \la |\psi|^2 \psi, \qquad \psi(t=0)= \psi_0 \end{equation} % where $V$ is a smooth localized potential, $\la $ is an order $1$ parameter and $\psi=\psi(t,x):\R\times \R^3 \longto \Complex$ is a wave function. The goal of this paper is to study the asymptotic dynamics of the solution for initial data $\psi_0$ near some {\it nonlinear excited state}. Recall that for any solution $\psi(t)\in H^1(\R^3)$ the $L^2$-norm and the Hamiltonian % \begin{equation} \label{1-2} {\mathcal H}[\psi] = \int \frac 12 |\nabla \psi|^2 + \frac 12 V |\psi|^2 + \frac 14 \la |\psi|^4 \, d x ~, \end{equation} are constants for all $t$. The global well-posedness for small solutions in $H^1(\R^3)$ can be proved using these conserved quantities and a continuity argument. We assume that the linear Hamiltonian $H_0 :=- \Delta + V$ has two simple eigenvalues $e_00$. It is proved in \cite{TY} that the evolution with initial data $\psi_0$ near some $Q_E$ will eventually settle down to some ground state $Q_{E_\infty}$ with $E_\infty$ close to $E$. See also \cite{BP} for the one dimensional case and \cite{SW2} for nonlinear Klain-Gorden equations. Denote by $L^2_{r}$ the weighted $L^2$ spaces ($r$ may be positive or negative) \begin{equation} \label{L2r.def} L^2_{r}\,(\R^3) \; \equiv \; \bket{\phi \in L^2(\R^3) \; : \; \bkA{x}^r \phi \in L^2(\R^3) } ~. \end{equation} % The space for initial data in \cite{TY} is \begin{equation} \label{Y.def} Y \equiv H^1(\R^3) \cap L^2_{r_0}\,(\R^3) ~, \qquad r_0>3 ~. \end{equation} We shall use $L^2\loc$ to denote $L^2 _{- r_0}$. The parameter $r_0>3$ is fixed and we can choose, say, $r_0=4$ for the rest of this paper. We now state the assumptions in \cite{TY} on the potential $V$. \noindent {\bf Assumption A0}: $- \Delta + V$ acting on $L^2(\R^3)$ has $2$ simple eigenvalues $e_0 |e_0|$ so that $2e_{01}$ is in the continuum spectrum of $H_1$. Let \begin{equation} \label{gamma0.def} \gamma_0 := \lim_{\sigma \to 0+} \Im \bke{\phi_0 \phi_1^2, \, \frac 1 {H_0 + e_0 - 2 e_1 -\sigma i} \Pc^{H_0}\phi_0 \phi_1^2} \end{equation} Since the expression is quadratic, we have $\gamma_0 \ge 0$. We assume, for some $s_0< C n_0^2$ small enough, \begin{equation} \label{A:gamma0} \inf_{|s| 0 ~. \end{equation} We shall use $0i$ to replace $\sigma i$ and the limit $\lim_{\sigma \to 0+}$ later on. \noindent {\bf Assumption A2}: For $\la Q_E^2$ sufficiently small, the bottom of the continuous spectrum to $-\Delta+ V+\la Q_E^2$, $0$, is not a generalized eigenvalue, i.e., not a resonance. Also, we assume that $V$ satisfies the assumption in \cite{Y} so that the $W^{k,p}$ estimates $k\le 2$ for the wave operator $W_H=\lim_{t\to \infty} e^{i t H}e^{it(\Delta+E)}$ hold for $k \le 2$, i.e., there is a small $\sigma>0$ such that, \[ |\nabla^\al V(x)| \le C \bkA{x}^{-5-\sigma}, \qquad \text{for } |\al|\le 2 ~. \] Also, the functions $(x\cdot \nabla)^k V$, for $k=0,1,2,3$, are $-\Delta$ bounded with a $-\Delta$-bound $<1$: \begin{equation*} \norm{(x\cdot \nabla)^k V\phi}_2 \le \sigma_0 \norm{-\Delta\phi}_2 + C\norm{\phi}_2, \qquad \sigma_0 < 1 , \quad k=0,1,2,3 ~. \end{equation*} \bigskip Assumption A2 contains some standard conditions to assure that most tools in linear Schr\"odinger operators apply. These conditions are certainly not optimal. The main assumption in A0-A2 is the condition $2e_{01} > |e_0|$ in assumption A1. The rest of assumption A1 are just generic assumptions. This condition states that the excited state energy is closer to the continuum spectrum than to the ground state energy. It guarantees that twice the excited state energy of $H_1$ (which one obtains from taking the square of the excited state component) becomes a resonance in the continuum spectrum (of $H_1$). This resonance produces the main relaxation mechanism. If this condition fails, the resonance occurs in higher order terms and a proof of relaxation will be much more complicated. Also, the rate of decay will be different. The main result in \cite{TY} concerning the relaxation of the ground states can be summarized in the following theorem. \medskip \noindent {\bf Theorem A} {\it Suppose that suitable assumptions on $V$ hold. Then there are small universal constants $\e_0, n_0>0$ such that, if the initial data $\psi_0$ satisfies $\norm{\psi_0-\n e^{i\Theta_0}\phi_0}_Y \le \e_0^{2} \n^2$ for some $\n\le \n_0$ and some $\Theta_0\in \R$, then there exists an $E_\infty$ and a function $\Theta(t)$ such that $\norm{Q_{E_\infty}}_Y - \n=O(\e_0^{2} \n)$, $\Theta(t) = -E_\infty t+ O(\log t)$ and} \begin{equation} \norm{\psi(t)-Q_{E_\infty} e^{ i \Theta(t)}}_{L^2 \loc} \le C(1+t)^{-1/2} ~. \end{equation} \medskip This theorem settles the question of asymptotic profile near ground states. Suppose that the initial data $\psi_0$ is now near some nonlinear excited state. From the physical ground, we expect that $\psi_t$ will eventually decay to some ground state unless the initial data $\psi_0$ is exactly a nonlinear excited state. We call this the "strong relaxation property". For comparison, we define a weaker property, the "generic relaxation property", as follows. Denote the space of initial data by $X$. Let $X_1$ ($X_0$ resp.) be the subspace of initial data such that the asymptotic profiles are given by some nonlinear excited (ground resp.) states. We shall say that the dynamics satisfy the generic relaxation property if $X_1$ has "measure zero". This concept depends on a notion of measure which should be specified in each context. With this definition, the strong relaxation property means that $X_1$ is exactly the space of excited states. In particular, $X_1$ is finite dimensional. We first note that the strong relaxation property is false. For any nonlinear excited state $Q_1$, define $X_{1, Q_1}$ to be the set of initial data converging to $Q_1$ asymptotically. It is proved in \cite{TY2} that for any given nonlinear excited state $Q_1$, $X_{1, Q_1}$ contains a finite co-dimensional set. Thus our goal is to establish some weaker statement such as the generic relaxation property. This is the first step toward a classification of asymptotic dynamics of the nonlinear Schr\"dinger equation. \donothing{ We can visualize these results in the following picture. Since the excited states and the ground states depend on the two parameters, the masses and the phases, they are two dimensional tubes. Each circle in these two tubes can be viewed as a fixed "point" of the dynamics in the sense that the evolution is just a rotation of the circle. On the other hand, each circle by itself is not stable since a small perturbation will in general change the mass. The family of the ground states taken as a whole is however stable under perturbation \cite{TY}. The family of excited states, even taken as a whole, is not stable, due to that $X_{1, Q_1}$ contains a finite co-dimensional set \cite{TY2}. } In this paper, we shall prove that for any excited state, there is a small neighborhood $\mathcal N$ so that $$ |{\mathcal N} \cap X_1 | \le C \|\psi_0\|^2 |{\mathcal N} \cap X_0 | $$ This estimates states that the ratio between $X_1$ and $X$ around excited states are bounded by the mass of the initial wave function $ \|\psi_0\|^2$. (Since we have not given a measure on the space of initial data, this statement is not well-defined and a precise statement will be given later on.) \donothing{ Recall our aim is to prove that $X_1$ has measure zero. Our result have shows only that $X_1$ is small relative to $X$ when the mass of the initial data is small. However, we have explicit estimates on the set $X\setminus X_1$. } In order to state the main result, we first decompose the wave function using the eigenspaces of the Hamiltonian $H_0$ as % \begin{equation} \label{psidec0} \psi = \lbar{x}\phi_0 + \lbar{y}\phi_1 +\lbar{\xi} , \quad \lbar{\xi} = \Pc^{H_0}\, \psi ~. \end{equation} % For initial data near excited states, this decomposition contains an error of order $y_0^3$ and it is difficult to read from \eqref{psidec0} whether the wave function is exactly an excited state. Thus we shall use the decomposition % \begin{equation} \label{psidec1} \psi = {x}\phi_0 + Q_1(y) +{\xi} ~. \end{equation} % where % \begin{equation} y=\lbar{y}, \quad x =\lbar{x} -(\phi_0, Q_1(y)) ,\quad \xi=\lbar{\xi}-\Pc Q_1(y) ~. \end{equation} % Here we have used the convention that $$ Q_1(y) :=Q_1(m) e^{i \Theta}, \qquad m= |y|, \; \; m e^{i \Theta} = y $$ % %\beq \label{psidec1} \psi = {x}\phi_0 + Q_1(y) +{\xi} ~. \eeq % %(Our trade back is the loss of some accuracy.) Here % We shall prove that for $\psi$ with sufficiently small $Y$ norm \eqref{Y.def}, such a decomposition exists and is unique in section 2. Thus we assume that $\psi_0= {x_0}\phi_0 + Q_1(y_0) +{\xi_0}$ is sufficiently small in $Y$. Let $n_0$ and $\e_0$ be the same small constants given in Theorem A. By choosing a smaller $n_0$, we may assume $n_0 \le \e_0^2/4$. We assume the initial data satisfies that % \begin{equation} \label{eq:1-10} \begin{split} &\norm{\psi_0}_{Y} =\n, \ 0 < n \le n_0, \qquad |y_0| \ge \tfrac 12 \n , \\ &|x_0| \ge 2 \n \, e^{-1/n} , \qquad |x_0| \ge \e_2^{-1} n^2 \norm{\xi_0}_Y ~, \end{split} \end{equation} % where $\e_2>0$ is a small universal constant to be fixed later in the proof. These conditions can be interpreted as follows: The excited state component, $y_0$, should account for at least half the mass of the initial data. Under this condition, if the ground state component, $x_0$ is not too small compared with the continuum component $\xi$, then the dynamics relaxes to some ground state. The condition $|x_0| \ge 2 \n \, e^{-1/n}$ is a very mild assumption to make sure that $x_0$ is not incredibly small. The following constant will be used in many places in this paper. Define % \begin{equation}\label{e.def} \e:= \min \bket{ \e_0/2, \, ( \log (2n/x_0))^{-1/2}}~. \end{equation} % Since $ n \le n_0\le \e_0^2/4$ and $|x_0| \ge 2 \n \, e^{-1/n}$, we have $n \le \e^2$. \begin{theorem} \label{th:1-1} Suppose the assumptions on $V$ given above hold. Let $\psi(t,x)$ be a solution of \eqref{Sch} with the initial data $\psi_0$ satisfying \eqref{eq:1-10}. Let % \begin{equation}\label{n1.def} n_1 = \bke{|x_0|^2 + \tfrac 12 \,|y_0|^2 }^{1/2} \sim n~. \end{equation} % Then, there exists an $E_\infty$ and a function $\Theta(t)$ such that $\norm{Q_{E_\infty}}_Y - n_1=O(\e \n)$, $\Theta(t) = -E_\infty t+ O(\log t)$ and % \begin{equation} C_1(1+t)^{-1/2} \le \norm{\psi(t)-Q_{E_\infty} e^{ i \Theta(t)}}_{L^2 \loc} \le C_2(1+t)^{-1/2} ~. \end{equation} for some constants $C_1$ and $C_2$. \end{theorem} %{\bf Moreover, we have Lower bound....} It is instructive to compare our result with the linear stability analysis of \cite{S} \cite{SS} \cite{G} \cite{G2} and \cite{SS2}. In our setup the main result in \cite{G2} states that the linearized operator around a nonlinear excited state is structural stable if $e_0 < 2 \, e_1$ and unstable if $e_0 > 2 \, e_1$. Hence the excited states considered in this article is unstable and is expected to decay under generic perturbations. The instability of the excited state contains in Theorem 1.1 is thus consistent with the linear analysis. Notice that Theorem 1.1 tracks the dynamics for all time including time regime when the dynamics are far away from the excited states. Furthermore, {\it for all initial data considered in Theorem 1.1, the relaxation rate to the asymptotic ground state is exactly of order $t^{-1/2}$}, a rate very different from the standard linear Schr\"odinger equations. In view of the linear analysis, the existence \cite{TY2} of (nonlinear) stable directions for excited states is a more surprising result. For the linear stable case, i.e., $e_0 > 2 \, e_1$, the only rigorous result is the existence \cite{TY2} of (nonlinear) stable directions in this case. Although the linear analysis states that all directions are linearly stable, on physics ground we still expect excited states remains generically unstable. We now explain the main idea of the proof for Theorem 1.1. The relaxation mechanism can be divided into three time regimes: 1. {\bf The initial layer}: The component of the wave function in the continuum spectrum direction gradually disperses away; the components in the bound states directions do not change much. 2. {\bf The transition regime}: Transition from the excited state to the ground state takes place in this interval. The component along the ground state grows in this regime; that along the excited state is slightly more complicated. We can further divide this time regime into two intervals. In part (i), the component along the excited state does not change much. In part (ii), it decreases steadily and eventually becomes smaller than the component along the ground state. 3. {\bf Stabilization}: The ground state dominates and is stable. Both the excited states and dispersive part gradually decay. In different time regimes, the dominant terms are different and we have to linearize the dynamics according to the dominant terms. In the first time region, $\psi(t)$ is near an excited state, and it is best to use operator linearized around the excited state. In the third time regimes, $\psi(t)$ is near a ground states, and it is best to use operator linearized around a ground state. For the transition regimes, the dynamics are far away from both excited and ground states and we can only use the linear Hamiltonian $H_0$. Besides technical problems associated with changing coordinate systems in different time intervals, there is an intrinsic difficulty related to the time reversibility of the Schrodinger equation. Imagine that we are now ready to show that our dynamics is in the third time regime and will stabilize around some nonlinear ground state. If we take the wave function $\psi_t$ at this time and time reverse the dynamics, then the dynamics will drive this wave function back to the initial state near some excited state. The time reversed state $\psi_t$ and the wave function $\psi_t$ itself will satisfy the same estimates in the usual Sobolev or $L_p$ senses. However, their dynamics are completely different: one stabilizes to a ground state; the other back to near an excited state. This suggests that $\psi_t$ carries information concerning the time direction and this information will not show if we measure it by the usual estimates. This time reversal difficulty manifest itself in the technical proofs as follows. We shall see that, when the third time regime begin, the dispersive part is not well-localized and its $L^2$-norm can be larger than that of the bound states---both violate conditions for approaching to ground states in \cite {TY}. To resolve this issue, we need to extract information which are time-direction sensitive so that even though the disperive part may be large, it is irrelevant since it is ``out-going''. Though the concept of ``out-going'' wave is known for linear Schr\"odinger equations, it is difficult to implement it for nonlinear Schr\"odinger equations. Our strategy is to identify the main terms of the dispersive part and calculate them explicitly. These terms carry sufficient information concerning the time direction. The rest are error terms and we can use various Sobolev or $L_p$ estimates. {\bf Resonance induced decay and growth} To illustrate the mechanism of resonance induced decay and growth, we consider the problem in the coordinates with respect to the linear Hamiltonian $H_0 = - \Delta + V$, \[ \psi(t) = {x(t)}\phi_0 + y(t) \phi_1 +{\xi(t)} ~, \quad, \xi(t) \in \PcH\psi (t) \] The nonlinear term $\psi^2 \bar \psi$, (assume $\la=1$) can be split into a sum of many terms using this decomposition. However, we claim that there is only one important nonlinear term in the equation for each component: % \begin{eqnarray} i\dot x &=& e_0 x + (\phi_0, (y \phi_1)^2 \bar \xi) + \cdots \label{model1} \\ i\dot y &=& e_1 y + (\phi_1, 2(x \phi_0)(\bar y \phi_1) \xi)+ \cdots \label{model2} \\ i\pd_t \xi &=& H_0 \xi + \PcH \bar x y^2 \phi_0\phi_1^2+ \cdots \label{model3} \end{eqnarray} % From \eqref{model1}, we know $ u(t) = e^{i e_0 t} x(t)$ has less oscillation of lower order than $x(t)$. Hence we say $x(t)$ has a phase factor $-e_0$. Similarly, $y(t)$ has a phase factor $-e_1$. The nonlinear term $\bar x y^2 \phi_0\phi_1^2$ has a phase factor $e_0 - 2e_1$, which, due to the assumption \eqref{evcon}, is the only term in $\psi^2 \bar \psi$ with a negative phase factor. %Therefore, it is the only relevent term %in the $\xi$ equation \eqref{model3}. It gives a term in $\xi$: \[ \xi(t) = \bar x y^2(t) \Phi + \cdots, \qquad \Phi=\frac 1{H +e_0 - 2e_1-0i} \, \phi_0\phi_1^2 \] Notice that $\Phi$ is complex. Substituting this term into \eqref{model1} and \eqref{model2}, we have % \begin{equation} \label{model4} \begin{aligned} i \dot x &= i \gamma_0 |y|^4 x + \cdots, \\ i \dot y &= -2 i \gamma_0 |x|^2 |y|^2 y + \cdots, \end{aligned} \end{equation} with $\gamma_0$ given in \eqref{gamma0.def}. In \eqref{model4} we have omitted two types of irrelevant terms: 1. Terms with same phase factors as $x$ or $y$: for example, $e_0 x$ and $|y|^2 x$ in \eqref{model1}. Since their coefficients are real, they disappear when we consider the equations for $|x|$ and $|y|$. 2. Terms with different phase factors: for example, $\bar x y^2$ in \eqref{model1}. Since these terms have different phases, their contribution averaging over time will be small. This can be made precise by the Poincar\'e normal form. From \eqref{model4} we obtain the decay of $y$ and the growth of $x$ as well as the three time regimes mentioned previously. However, it should be warned that this set-up is only suitable when both $x$ and $y$ are of similar sizes. \section{The initial layer and the transition regimes: The set up} We now outline the basic strategy for the initial layer and the transition regimes. We first review the construction of the bound state families. \subsection{Nonlinear bound states} The basic properties of the ground state families can be summarized in the following lemma from \cite{TY}. \begin{lemma} \label{th:2-1} Suppose that $-\Delta+V$ satisfies the assumptions (A0) and (A2). Then there is an $\n_0$ sufficiently small such that for $E$ between $e_0$ and $e_0+ \la \n_0^2$ there is a nonlinear ground states $\bket{Q_{E}}_E$ solving \eqref{Q.eq}. The nonlinear ground state $Q_{E}$ is real, local, smooth, $ \la^{-1}(E-e_0)>0$, and % \[ Q_{E} = \n \,\phi_0 +O(\n ^{3 }) ~,\qquad \n \approx C \, [\la^{-1}(E-e_0)]^{1/2} \; , \quad C=(\int \phi_0^4 \, dx)^{-1/2}. \] Moreover, we have $R_E=\pd_E Q_{E} = O(\n^{-2} ) \, Q_{E} + O(\n )= O(\n^{-1} )$ and $\pd_E^2 Q_{E} = O(\n^{-3} )$. If we define $c_1\equiv (Q,R)^{-1}$, then $c_1=O(1)$ and $\la c_1 >0$. \end{lemma} This lemma can be proved using standard perturbation argument and similar conclusions hold for excited states as well. For the purpose of this paper, we prefer to use the value $m=(\phi_1, \, Q_1)$ as the parameter and refer to the family of excited states as $Q_1(m)$. It is straightforward to compute the leading corrections of $Q_1(m)$ via standard perturbation argument used in proving Lemma \ref{th:2-1}. Thus we can write $Q_1$ as % \begin{equation} Q_1(m)= m \phi_1 + \bke{m^3 q_3 + q^{(5)}(m)}: = m \phi_1 + q(m) \; , \quad q^{(5)}(m)=O(m^5), \quad q(m) \perp \phi_1 ~, \label{eq:2-2} \end{equation} where $q_3 = - \la (H_0 - e_1)^{-1} \pi \phi_1^3$ and $\pi$ is the projection % \begin{equation} \label{pi.def} \pi h = h - (\phi_1, h) \phi_1 . \end{equation} % Similarly, we can also expand $E_1(m)$ in $m$ as % \begin{equation} \label{E.dec} E_1(m)= e_1 + E_{1,2} m^2 + E_{1,4}m^4 + E_1^{(6)}(m), \qquad E_1^{(6)}(m)= O(m^6) ~. \end{equation} % Moreover, we can differentiate the relation of $Q_1(m)$ w.r.t. $m$ to have \begin{equation} Q_1'(m)= \frac d{d m}Q_1 =\phi_1 + q'(m) ~, \qquad q'(m)= \frac d{dm}q(m) = O(m^2), \quad q'(m) \perp \phi_1, \end{equation} \subsection{Equations} Thus in the first and second time regimes, we write % \begin{equation}\label{eq:2-4} \psi(t) = x(t)\phi_0 + Q_1(m(t))e^{i \Theta(t)} + \xi(t) ~, \end{equation} % where $\xi \in \Hc(H_0)$, see \eqref{psidec1}. If we write $\Theta(t)=\theta(t) - \int_0^t E_1(m(s)) \, ds$, we can write $y(t)$ as % \begin{equation} \label{y.def} y= m e^{i \Theta} = m \exp \bket{i\theta(t) - i\int_0^t E_1(m(s)) \, ds} ~. \end{equation} % Denote the part orthogonal to $\phi_1$ by $h ={x}\phi_0 +{\xi}$. From the Schr\"odinger equation, we have $h$ satisfies the equation \begin{align} i \pd_t h &= H_0 h + G + \Lambda, \nonumber \\ G &= \la |\psi|^2 \psi - \la Q_1^3 e^{i \Theta} \nonumber \\ &=\la Q_1^2(e^{i2\Theta} \bar h +2 h ) + \la Q_1(e^{i\Theta} 2h\bar h +e^{-i\Theta} h^2 ) + \la |h|^2 h \\ \Lambda &= \bke{\dot \theta Q_1 - i \dot m Q_1' } e^{i \Theta} ~. \end{align} % Since $m(t)$ and $\theta(t)$ are chosen so that \eqref{eq:2-4} holds, we have $0 = (\phi_1, i\pd_t h(t)) = \bke{ \phi_1, \, G + (\dot \theta Q_1 - i \dot m Q_1')e^{i \Theta} } $. Hence $m(t)$ and $\theta(t)$ satisfy % \begin{equation} \label{mt.eq} \dot m = \bke{ \phi_1, \Im G e^{-i \Theta} } \, , \qquad \dot \theta = -\frac 1m \, \bke{ \phi_1, \Re G e^{-i \Theta}} ~. \end{equation} % We also have the equation for $y$: \[ i \dot y = i \dot m e^{i \Theta} - ( \dot \theta- E_1(m) ) m e^{i \Theta} = E_1(m)y + e^{i \Theta} (i \dot m - m \dot \theta ) = E_1(m)y + (\phi_1 , G)~. \] Here we have used \eqref{mt.eq}. Denote $\Lambda_\pi= \pi \Lambda$. We can decompose the equation for $h$ into equations for $x$ and $\xi$. Summarizing, we have the original Schr\"odinger equation is equivalent to % \begin{equation} \label{xyxi.eq} \left \{ \begin{aligned} i\dot x &= e_0 \, x + \bke{ \phi_0, \, G + \Lambda_\pi } \\ i \dot y & = E_1(m)y + (\phi_1 , G) \\ i \pd_t \xi &= H_0 \, \xi + \Pc \bke{ G + \Lambda_\pi } \end{aligned} \right . \end{equation} Clearly, $x$ has an oscillation factor $e^{-ie_0t}$, and $y$ has a factor $e^{-ie_1t}$. Hence we define \begin{equation} \label{eq:uv} x= e^{-i e_0 t} u, \qquad y=e^{-i e_1 t} v ~. \end{equation} % Together with the integral form of the equation for $\xi$, we have % % \begin{align} \dot u &= -i e^{ie_0t} \bke{ \phi_0, \, G + \Lambda_\pi } ,\label{u.eq} \\ \dot v &= -i e^{ie_1 t} \bkt{(E_1(m)-e_1)y + (\phi_1 , G)}, \label{v.eq} \\ \xi(t) &= e^{-i H_0 t}\xi_0 +\int_0^t e^{-i H_0 (t-s)} \, \PcH G_\xi(s)\, d s ~,\qquad G_\xi = i^{-1}( G + \Lambda_\pi) ~.\label{xi.eq} \end{align} % This is the system we shall study. \subsection{Basic estimates and decompositions} It is useful to decompose various terms according to orders in $n$ so that we can identify their contributions. We now proceed to so this for $G$, $\Lambda_\pi$, $E(|y|)$ and $\xi(t)$. We expect that $x,y=O(n)$ and $\xi=O(n^3)$ locally. {\bf 1. $G$}\quad %We first decompose the nonlinear term $G$ %according to order in $n$. Recall that $G$ is given by \[ G= \la Q_1^2 (e^{i2\Theta} \bar h +2 h ) + \la Q_1 (e^{i\Theta} 2h\bar h +e^{-i\Theta} h^2 ) + \la |h|^2 h \] % with $h=x\phi_0 + \xi$ and $Q_1=Q_1(|y|)$. From the the decomposition \eqref{eq:2-2} of $Q_1=|y|\phi_1 + |y|^3 q_3 + q^{(5)}(|y|)$, we decompose $G$ as % \begin{align*} G&=\la (y^2 \phi_1^2 + 2 y^3 \bar y \phi_1 q_3) \, \bar h + \la (|y|^2 \phi_1^2 + 2 |y|^4 \phi_1 q_3) \, 2h \\ &\quad + \la (y \phi_1 + y^2 \bar y q_3 )\, 2 |h|^2 + \la (\bar y \phi_1 + y \bar y^2 q_3 ) \, h^2 + \la |h|^2 h + (*) \end{align*} % where $(*)=\la \bkt{2|y| \phi_1 q^{(5)} + (|y|^3 q_3 + q^{(5)})^2} \, (e^{i2\Theta} \bar h +2 h ) + \la q^{(5)}\, (e^{i\Theta} 2h\bar h +e^{-i\Theta} h^2 )$ with $q^{(5)}=q^{(5)}(|y|)$. We then substitute $h=x\phi_0 + \xi$ to obtain % \begin{equation} \label{G.dec} G= G_3 + G_5 + G_7, \end{equation} where % \begin{align} \label{G3.def} G_3 &= \la (y^2 \bar x + 2|y|^2 x) \phi_0 \phi_1^2 + \la (2|x|^2 y + x^2 \bar y) \phi_0^2 \phi_1 +\la |x|^2 x \phi_0^3 \\ \label{G5.def} G_5 &=\la (2 y^3 \bar y \bar x + 4 |y|^4 x ) \, \phi_0\phi_1 q_3 + \la (2|x|^2 y^2 \bar y + x^2 y \bar y^2 ) \, \phi_0^2 q_3 \\ \nonumber & \quad + \la (x \phi_0 + y \phi_1)^2 \wbar \xi + 2\la |(x \phi_0 + y \phi_1)|^2 \xi \end{align} % and \begin{align} \label{G7.def} G_7 &= \la \bkt{2|y| \phi_1 q^{(5)}(|y|) + (|y|^3 q_3 + q^{(5)}(|y|))^2} \, (e^{i2\Theta} \bar h +2 h ) \\ \nonumber &\quad + \la q^{(5)}(|y|)\, (e^{i\Theta} 2h\bar h +e^{-i\Theta} h^2 ) \\ \nonumber & \quad + \la \bke{ 2 y^3 \bar y \phi_1 q_3 \, \bar \xi + 4 |y|^4 \phi_1 q_3 \, \xi } + \la (y \phi_1 2 |\xi|^2 + \bar y \phi_1 \xi^2) \\ \nonumber &\quad +\la y^2 \bar y q_3 \, 2 ( x\phi_0 \bar \xi + \bar x \phi_0 \xi + |\xi|^2) + \la y \bar y^2 q_3 \, (2x \phi_0 \xi + \xi^2) \\ \nonumber &\quad + \la \phi_0(\bar x \xi^2 + 2 x |\xi|^2 ) + \la |\xi|^2 \xi ~. \end{align} % Note that $G_3=O(n^3)$, $G_5=O(n^5)$ and $G_7=O(n^7)$. If we use the convention that \[ f \leC g_1 + g_2 + \cdots \] for $\norm{f} \le C \norm{g_1}+\norm{g_2}+ \cdots$ for some suitable norms, we have % \begin{align} G &\leC n^2 x + n^2 \xi + \xi^3 \label{G.est0}\\ G_5 &\leC n^4 x + n^2 \xi \label{G5.est0}\\ G_7 &\leC n^6 x + n^4 \xi + n \xi^2 + \xi^3 \label{G7.est0} \end{align} It is crucial to observe that {\it no term in $G_3$ is of order $y^3$}. This is due to our setup emphasizing the role of nonlinear excited states. The price we pay is the introduction of terms involving $q_3$ and $q^{(5)}$. We now identify the main oscillation factors of various terms. For example, $y^2 \bar x= e^{i(-2e_1+e_0) t} v^2 \bar u$, and its factor is $-2e_1+e_0$. For terms in $G_3$ we have % \begin{equation} \label{eq:2-18} \begin{array}{c} y^2 \bar x \\ -2e_1 + e_0 \end{array} \quad \begin{array}{c} |y|^2 x \\ -e_0 \end{array} \quad \begin{array}{c} |x|^2 y\\ -e_1 \end{array} \quad \begin{array}{c} x^2 \bar y \\ -2 e_0 + e_1 \end{array} \quad \begin{array}{c} |x|^2 x \\-e_0 \end{array} \quad \end{equation} % % From the spectral assumption $|e_0| > 2|e_1|$, $-2e_1 + e_0$ is the only negative phase factor. Hence it is the only term of order $(n^3)$ that have resonance effect when we compute the main part of $\xi$. Also, since $|x|^2 y$ has the same phase as $y$, it will be resonant in the $y$-equation. Similarly, $|y|^2x$ and $|x|^2x$ have same phase as $x$ and will be resonant in $x$-equation. {\bf 2. $\Lambda_\pi$ and $E(m)$}\quad Recall $\Lambda_\pi= \pi (\dot \theta Q_1 - i \dot m Q'_1) e^{i \Theta}$. Since $\dot \theta = O(n^{-1}\norm{G}\loc)$ and $\dot m=O(\norm{G}\loc)$, % \begin{equation} \label{La.est} \norm{\Lambda_\pi(s)} = O(\dot \theta)\,O(\pi Q_1) + O(\dot m)\,O(\pi Q_1') \le C n^2 \norm{G}\loc ~, \end{equation} To find out the main part of $\Lambda_\pi$, we substitute equation \eqref{mt.eq} for $\dot m$ and $\dot \theta$ to obtain ( $m=|y|$), % \begin{align*} \Lambda_\pi &= \pi (\dot \theta Q_1 - i \dot m Q'_1 )\, e^{i \Theta} \\ &= -\bket{(\phi_1,\, G/2)m^{-1} \pi Q_1 + (\phi_1,\, \bar G/2) m^{-1} \pi Q_1 e^{2i\Theta}}\\ &\quad -i\bket{(\phi_1,\, G/2i) \pi Q'_1 + (\phi_1,\, \bar G/2i) \pi Q'_1 e^{2i\Theta}} \end{align*} % Since $G=G_3 + (G_5+G_7)$ and $\pi Q_1(m)=m^3 q_3 + q^{(5)}(m)$ by \eqref{eq:2-2}, we have $\pi Q'_1(m)=3 m^2 q_3 + O(m^4)$, and the main part of $\Lambda_\pi$ is (also recall $y=m e^{i \Theta}$) % \begin{align} \Lambda_{\pi,5} &= -\tfrac 12 \bket{(\phi_1,\, G_3)\,|y|^2 q_3 + (\phi_1,\, \bar G_3) \,y^2 q_3}\nonumber\\ &\quad -\tfrac 12 \bket{(\phi_1,\, G_3) \,3|y|^2 q_3 + (\phi_1,\, \bar G_3) \,3y^2 q_3}\nonumber \\ &=- 2 q_3 \, (\phi_1,\, G_3 |y|^2 + \bar G_3 y^2) \label{La5.def} \end{align} % Let $\Lambda_{\pi,7}=\Lambda_{\pi}-\Lambda_{\pi,5}$. We have % \begin{align} \Lambda_{\pi} &= \Lambda_{\pi,5}+\Lambda_{\pi,7} \label{La.dec}\\ \Lambda_{\pi,5} &\leC \norm{G_3}\loc |y|^2 \leC n^4 x \label{La5.est}\\ \Lambda_{\pi,7} &\leC \norm{G_5+G_7}\loc |y|^2 + \norm{G}\loc |y|^4 ~. \label{La7.est} \end{align} % {\bf 3. $\xi$}\quad Recall the equation for $\xi$ in \eqref{xi.eq}, % \[\xi(t) = e^{-i H_0 t}\xi_0 +\int_0^t e^{-i H_0 (t-s)} \, \PcH G_\xi(s)\, d s ~,\qquad G_\xi = i^{-1}( G + \Lambda_\pi) ~. \] % Since $\norm{\Lambda_\pi} \le C n^2 \norm{G}\loc $, the main terms in $G_\xi=i^{-1}(G+\Lambda_\pi)$ is $i^{-1}G_3$. We now compute the first term $\la y^2 \bar x \phi_0 \phi_1 ^2$ in $G_3$ using integration by parts: % \begin{align*} & - i \la \int_0^t e^{-i H_0 (t-s)} \, \Pc y^2 \bar x \phi_0 \phi_1 ^2 \, d s \\ &= - i \la e^{-i H_0 t} \int_0^t e^{i (H_0 -0i) s} \, e^{i (e_0-2e_1) s} \, v^2 \bar u \Pc \phi_0 \phi_1 ^2 \, d s \\ &=- i \la e^{-i H_0 t} \Bigg \{ \bkt{ \frac 1{ i (H_0 -0i +e_0- 2e_1)} e^{i H_0 s} \, e^{ i (e_0-2e_1) s} \, v^2 \bar u \Pc \phi_0\phi_1 ^2 }_0^t \\ & \qquad \qquad \qquad - \int_0^t \frac 1{ i (H_0 -0i +e_0- 2e_1)} e^{i H_0 s} \, e^{i (e_0-2e_1) s} \, \frac d{d s} \bke{ v^2 \bar u} \Pc \phi_0 \phi_1^2 \, d s \Bigg \} \\ &= y^2 \bar x \Phi_1 - e^{-i H_0 t} y^2 \bar x(0) \Phi_1 - \int_0^t e^{-i H_0 (t-s) } \, e^{i (e_0-2e_1) s} \frac d{d s} \bke{ v^2 \bar u} \Phi_1 \, d s ~, \end{align*} % where % \begin{equation}\label{Phi1.def} \Phi_1 = \frac {-\la} { H_0 -0i +e_0- 2e_1} \, \Pc \phi_0 \phi_1 ^2 ~. \end{equation} % This term, with the phase factor $e_0 - 2 e_1$, is the only one in $G_3$ having a negative phase factor (see \eqref{eq:2-4}). Since $-(e_0 - 2 e_1)$ is in the continuous spectrum of $H_0 $, $H_0 +e_0- 2e_1$ is not invertible, and needs a regularization $-0i$. We choose $-0i$, not $+0i$, so that the term $e^{-i H_0 t} y^2 \bar x(0) \Phi_1$ decays as $t \to \infty$. See Lemma \ref{th:2-2}. We can integrate all terms in $G_3$ and obtain the main terms of $\xi(t)$ as % \begin{equation} \xitwo(t)= y^2 \bar x \Phi_1 + |y|^2 x \Phi_2 + |x|^2 y \Phi_3 + x^2 \bar y \Phi_4 + |x|^2 x \Phi_5 \end{equation} % where \begin{alignat}{2} \label{Phij.def} \Phi_2 &= \frac {-2\la} { H_0 -e_0 } \, \Pc \phi_0 \phi_1 ^2 ~, \qquad &\Phi_3& = \frac {-2\la} { H_0 - e_1} \, \Pc \phi_0^2 \phi_1 ~, \\ \Phi_4 &= \frac {-\la} { H_0 -2 e_0 + e_1} \, \Pc \phi_0 ^2 \phi_1 ~, \qquad &\Phi_5 &= \frac {-\la} { H_0 - e_0} \, \Pc \phi_0 ^3 ~. \nonumber \end{alignat} The rest of $\xi(t)$ is % \begin{align*} \xithree(t)&= e^{-i H_0 t} \xi_0 -e^{-i H_0 t}\xitwo(0) -\int_0^t e^{-i H_0 (t-s)} \, \Pc \, G_4 \, d s \\ &\quad + \int_0^t e^{-i H_0 (t-s)} \, \Pc \bke{G_\xi-i^{-1}G_3-i^{-1} \la|\xi|^2\xi} \, d s \\ & \quad + \int_0^t e^{-i H_0 (t-s)} \, \Pc \bke{i^{-1}\la|\xi|^2\xi}\, d s \\ &\equiv \xithree_1(t)+ \xithree_2(t) +\xithree_3(t) +\xithree_4(t)+ \xithree_5(t) \end{align*} % where $\xithree_4(t)$ and $\xithree_5(t)$ are higher order terms in $G_\xi$ which we did not integrate and the integrand $G_4$ in $\xithree_3(t)$ consists of the remainders from the integration by parts: % \begin{align} G_4 &= e^{i(e_0 - 2 e_1) s} \frac d{d s} \bke{ v^2 \bar u} \Phi_1 + e^{i(-e_0 ) s} \frac d{d s} \bke{ |v|^2 u} \Phi_2 \label{G4.def} \\ & \quad + e^{i ( - e_1) s} \frac d{d s} \bke{ |u|^2 v} \Phi_3 + e^{i (-2 e_0 + e_1) s} \frac d{d s} \bke{ u^2 \bar v} \Phi_4 + e^{i (-e_0 ) s} \frac d{d s} \bke{ u^2 \bar u} \Phi_5 ~,\nonumber \end{align} Here we single out $\xithree_5(t)$ since $|\xi|^2\xi$ is a non-local term. Thus we have following decomposition for $\xi$: % \begin{equation}\label{xi.dec} \xi(t)= \xitwo(t)+\xithree(t) = \xitwo +\bke{\xithree_1 + \cdots + \xithree_5}~, \end{equation} % \subsection{Linear estimates} We now summarize known results concerning the linear analysis. The decay estimate was contained in \cite{JSS} and \cite{Y}; the estimate \eqref{eq:22-1B} was taken from \cite{SW2} and \cite{TY}; \begin{lemma}[decay estimates for $e^{-itH_0}$] \label{th:2-2} For $q \in [2, \infty]$ and $q'=q/(q-1)$, % \begin{equation} \label{eq:22-1A} \norm{ e^{-itH_0} \, \Pc^{H_0} \phi }_{L^q} \le C \,|t|^ {-3 \bke{\frac 12 - \frac 1q}} \norm{\phi}_{L^{q'}} ~. \end{equation} For smooth local functions $\phi$ and sufficiently large $r_0$, we have % \begin{equation} \label{eq:22-1B} \lim_{\sigma\to 0+} \norm{ \bkA{x}^{-r_0} \, e^{-it H_0} \, \frac 1{(H_0 + e_0 - 2 e_1- \sigma i)^k} \Pc^{H_0} \bkA{x}^{-r_0} \phi }_{L^2} \le C \bkA{t}^{-9/8} \end{equation} where $k=1,2$. %, and $0i$ means $\sigma i$ with $\lim_{\sigma \to 0+}$ %outside of the bracket. \end{lemma} Intuitively, we can write \[ e^{-i H_0 t} \Phi_1 = \lim_{ \e \to 0+ } e^{-i H_0 t} \int_0^\infty e^{-i (H_0 - \e i + e_0 - 2 e_1)s} \Pc \phi_0 \phi_1 ^2 \, ds ~. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The initial layer and the transition regimes: The estimates} In this section we wish to show the following picture for the solution $\psi(t)$: In the initial layer regime, the dispersive part gradually disperses away, while the sizes of the bound states do not change much. In the transition regime, the {\it original} dispersive part becomes negligible, while the $\phi_0$-components of $\psi(t)$ increases and the $\phi_1$-component decreases. Recall the orthogonal decomposition $\psi(t)=\lbar{x}\phi_0 + \lbar{y}\phi_1 + \lbar{\xi}$ \eqref{psidec0}. We have $ |\lbar{x}(t)|^2 + |\lbar{y}(t)|^2 + \norm{\lbar{\xi}(t)}_{L^2}^2 =\norm{\psi(t)}_{L^2}^2 \le \n^2 $. If we decompose $\psi$ via \eqref{psidec1}, i.e., % and \eqref{xi.dec} \begin{equation} \psi(t)=x\phi_0 + Q_1(y) + \xi ~, \qquad \xi= \xitwo + \xithree ~ , \end{equation} we have % % $\norm{\psi(t)}_{L^2} = %\norm{\psi_0}_{L^2}\le \norm{\psi_0}_{Y}\equiv n$. %For the components of the % $y=\lbar{y}$, $x=\lbar{x}+O(y^3)$ and $\xi=\lbar{\xi}+O(y^3)$. Thus \begin{equation} \label{eq:3-2} |x(t)|, \; |y(t)|, \; \norm{\xi(t)}_{L^2} \le \tfrac 54\n ~, \qquad \norm{\xi_0}_{Y} \le 4\n ~. \end{equation} For $p=1,2,4$, define the space \begin{equation}\label{eq:3-3} {L^p \loc } \equiv \bket{f: \; \bkA{x}^{-r_0}f(x) \in L^p(\R^3)} ~, \end{equation} % where $r_0>3$ is the exponent appeared in the linear estimate \eqref{eq:22-1B} in Lemma \ref{th:2-2}. Since $r_0>3$, we have $\norm{f}_{L^1 \loc} \le C\norm{f}_{L^p}$ for any $p$, The following proposition is the main result for the dynamics in the initial layer and the transition regimes. \begin{proposition} \label{th:3-1} Suppose that $V$ satisfies the assumptions given in \S 1. Let $\psi(t,x)$ be a solution of \eqref{Sch} with the initial data $\psi_0$ satisfying \eqref{eq:1-10}. Let $\e_3>0$ be a sufficiently small constant to be fixed later. Let $t_0=\e_3 n^{-4}$. Then there exists $t_1$ and $t_2$ such that for some constant $C \le 10000$ %and $ t_1 + 10100\; (\gamma_0 \n^4 \e^2)^{-1}$ we have \begin{equation} \label{eq:3-4} t_0\le t_1\le \frac {1.01}{\;\gamma_0 \n^4\;}\; \log \bke{\frac \n {|x_0|}} ~, \qquad t_1 + C \; (\n^4 \e^2)^{-1}\le t_2\le t_1 + 10100\; (\gamma_0 \n^4 \e^2)^{-1} ~, \end{equation} % and the following estimates hold: (i) For $0\le t\le 2\,t_2$, % \begin{equation} \label{eq:3-5} |x(t)|\ge \tfrac 34 \,\sup_{0\le s\le t} \, |x(s)| ~, \end{equation} % % \begin{align} \norm{\xi(t)}_{L^4} &\le C_2\, \n^2 t^{1/4}\,|x(t)| + C_2 \norm{\xi_0}_Y \bkA{t}^{-3/4} ~, \nonumber \\ \norm{\xi(t)}_{L^4\loc} &\le C_2 n^2 \,|x(t)| \quad + C_2 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~, \label{xi.est} \\ \norm{\xithree(t)}_{L^2 \loc} &\le \, C_2\, \n^{15/4} |x(t)| + C_2 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~. \nonumber \end{align} % where the constant $C_2$ will be specified in \eqref{eq:C2} of next subsection. (ii) (Initial layer) for $0\le t \le t_0$, % \begin{equation} \label{eq:3-7} \begin{array}{l} \tfrac 12 |x_0| \le |x(t)| \le \tfrac 32\, |x_0|, \\ 0.99\,|y_0| \le |y(t)|\le 1.01\,|y_0| \end{array} \end{equation} % (iii) For $n_1=\bke{|x_0|^2 + \tfrac 12 \, |y_0|^2}^{1/2}$ defined in \eqref{n1.def}, % \begin{equation} \label{eq:3-8} |x(t_1)| \ge 0.01 \n ~, \quad |x(t_2)| \ge 0.99 \n_1 ~, \quad \tfrac 12 \e n \le |y(t_2)| \le 2\e\n ~. \end{equation} \end{proposition} % \bigskip Notice that \eqref{eq:3-4} implies \begin{equation} \label{eq:3-4B} t_2 \le C_3 n^{-4} % \qquad C_3 = ... \end{equation} for some constant $C_3$. We will prove these estimates using \eqref{eq:1-10}, \eqref{eq:3-2} and a continuity argument. Hence we can assume the following weaker estimates: For $0\le t\le 2t_2$: % % % \begin{equation} \label{A:pf} \begin{aligned} |x(t)|&\ge \tfrac 12 \,\sup_{0\le s\le t} \, |x(s)| ~, \\ |x(t)|&\le 2 |x_0| \qquad \text{for } t< t_0 ~, \\ \norm{\xi(t)}_{L^4} &\le 2 C_2\, \n^2 t^{1/4}\,|x(t)| + 2 C_2 \norm{\xi_0}_Y \bkA{t}^{-3/4} ~, \nonumber \\ \norm{\xi(t)}_{L^4\loc} &\le 2 C_2 n^2 \,|x(t)| \quad + 2 C_2 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~, \\ \norm{\xithree(t)}_{L^2 \loc} &\le \, 2 C_2\, \n^{15/4} |x(t)| + 2 C_2 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~. \nonumber \end{aligned} \end{equation} % % By continuity, if we prove Proposition \ref{th:3-1} assuming these weaker estimates, we have proved the proposition itself. We shall see also estimates \eqref{A:pf} will be used only in estimating higher order terms. Recall from \eqref{La.est} that the local term $\Lambda_\pi$ satisfies $\norm{\Lambda_\pi}_r\le C n^2 \norm{G}_{L^1 \loc}$ for any $r$. Thus we have % \begin{equation} \label{uvdot.est} |\dot u(t)| \leC \norm{G}_{L^1 \loc}, \qquad |\dot v(t)| \leC \norm{G}_{L^1 \loc} + |y|^3 . \end{equation} The following lemma provides estimates for $G$ assuming the estimate \eqref{A:pf}. \begin{lemma} \label{th:3-2} Let $G$ be given by \eqref{G.dec}--\eqref{G7.def}. Suppose $n$ is sufficiently small and the estimate \eqref{A:pf} holds for $t\le C_3 n^{-4}$. % as long as \eqref{A:pf} remains Then we have the following estimates for $G$: % % \begin{equation} \label{G.est} \norm{G(t)}_{L^{4/3}\cap L^{8/7}} \le C_4 \n^{2}|x(t)| + C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8}~. \end{equation} % \begin{equation} \label{G57.est} \norm{(G-G_3)(t)}_{L^{4/3}\cap L^{8/7}} \le C_4 \n^{15/4}|x(t)|+ C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8}~. \end{equation} % \begin{equation} \label{G57.estB} \norm{(G-G_3)(t)}_{L^1 \loc} \le C_4 \n^{4}|x(t)|+ C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8}~. \end{equation} % % where $C_4$ is a constant independent of $C_2$ and $C(C_2)$ denotes constants depending on $C_2$. Moreover, \eqref{G.est} and \eqref{G57.est} remain true if we replace $G$ by $G_\xi$, and $(G-G_3)$ by $(G_\xi-i^{-1}G_3)$. Furthermore, we have % \begin{equation} \label{Gxi.L1est} \norm{G_\xi(t)}_{L^{1}}\le C_5 n^3 ~. \end{equation} \end{lemma} By the assumption \eqref{eq:1-10}, when $t>t_0$ the last term $C n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} $ is smaller and can be removed. The proof of this lemma is a straightforward application of the Holder and Schwarz inequalities. We will use \eqref{G.est} and \eqref{G57.est} for $\xi$, and \eqref{G57.estB} for $x$ and $y$. \myproof Recall \eqref{G.dec} that $G=G_3+G_5+G_7$. We first consider only the non-local term $\la |\xi|^2\xi$ in $G$, which belongs to $G_7$. Since $n\le \e^2$ and $t_2 \le C_3 \e^{-2} \n^{-4}$, by \eqref{A:pf} we have $\norm{\xi(s)}_{L^{4}}\le C n ^{3/4}|x(s)|+ C \norm{\xi_0}_Y \bkA{s}^{-3/4}$. Also using \eqref{eq:3-2} and the H\:older inequality we have % \begin{equation} \label{eq:3-21} \norm{|\xi|^2\xi(s)}_{L^{4/3}}\le C \norm{\xi(s)}_{L^{4}}^3 \le C \bke{n ^{3/4}|x(s)|}^3 + C \norm{\xi_0}_Y^3 \bkA{s}^{-9/4} ~. \end{equation} % % \begin{align} \norm{|\xi|^2\xi(s)}_{L^{8/7}}&\le C \norm{\xi(s)}_{L^{2}}^{1/2} \, \norm{\xi(s)}_{L^{4}}^{5/2} \nonumber \\ &\le C \n^{1/2}\,\bket{\bke{\n^{3/4} |x(s)|}^{5/2} + \norm{\xi_0}_Y^{5/2} \bkA{s}^{-15/8} }\label{eq:3-21A} \\ &\le C \n^{4 -1/8}|x(s)|+ C \n^{2}\,\norm{\xi_0}_Y \bkA{s}^{-15/8}~.\nonumber \end{align} % Hence this non-local term satisfies \eqref{G.est}--\eqref{G57.est}. Moreover, to prove \eqref{G57.estB}, we can bound $\norm{|\xi|^2\xi(s)}_{L^1 \loc}$ by $\norm{\xi(s)}_{L^{4}}^3$. For the local term $G -\la|\xi|^2\xi= G_3 + G_5 + (G_7-\la|\xi|^2\xi)$, all $L^p$-norms are equivalent. We can read from the explicit expressions of $G$ the following estimates: % \begin{align*} G_3 &\leC n^2 x \\ G_5 & \leC n^4 x + \n^2\,\xi \\ G_7-\la|\xi|^2\xi& \leC \n^6\,\xm+ \n^4\,\xi + \n\,\xi^2 \end{align*} % % To estimate $\xi$ in $G_\xi -\la|\xi|^2\xi$, we can use $\norm{\xi}_{L^{4}\loc}$. For example, \[ \norm{\bar y\phi_1\xi^2}_{L^{4/3}}\le C|y|\norm{\phi_1\bkA{x}^{2r_0}}_{L^{4}}\,\norm{\xi}_{L^{4}\loc}^2 \le C n \bke{(n^2 \xm)^2 +\norm{\xi_0}_Y^2 \bkA{s}^{-9/4}}~. \] Together with the explicit expressions of $G$ and $G_3$, similar arguments show \eqref{G.est}--\eqref{G57.estB}. Since $G_\xi= i^{-1}(G+\Lambda_\pi)$ and $\norm{\Lambda_\pi}\le Cn^2 \norm{G}\loc$ by \eqref{La.est}, \eqref{G.est} and \eqref{G57.estB} hold if we replace $G$ and $G-G_3$ by $G_\xi$ and $G_\xi- i^{-1}G_3$. obtain \eqref{Gxi.L1est}, we only need to check the non-local term $\la |\xi|^2\xi$. Since $\norm{\xi(s)}_{L^{4}}\le (2C_2C_3^{1/4}+8C_2)n$, we have % \begin{equation} \label{eq:3-20A} \norm{|\xi|^2\xi}_{L^{1}}\le C_{4,1} \norm{\xi(s)}_{L^{2}} \, \norm{\xi(s)}_{L^{4}}^{2} \le \frac 1{10}C_5 \n\, n^2 ~, \end{equation} % provided we choose $C_5 \ge 10(2C_2C_3^{1/4}+8C_2)C_{4,1}$. Thus the lemma is proved. \myendproof \subsection{Estimates of the dispersive part} We now prove the estimates for $\xi$ in Proposition \ref{th:3-1} by using \eqref{eq:3-2}, \eqref{A:pf}, Lemma \ref{th:2-2} and Lemma \ref{th:3-2}. %We first estimate the %$L^4$ and $L^4 \loc$-norms using %We then prove the $L^2 \loc$-estimates using the decomposition %\eqref{xi.dec}. \bigskip {\bf Step 1.} $L^4$ and $L^4\loc$ norms, $0\le t \le 2t_2$ Recall the equation \eqref{xi.eq} for $\xi$: % \begin{equation} \label{E3-21} \xi(t) = e^{-i H_0 t}\xi_0 +\int_0^t e^{-i H_0 (t-s)} \, \PcH G_\xi(s)\, d s ~,\qquad G_\xi = i^{-1}( G + \Lambda_\pi) ~. \end{equation} %We first estimate $\norm{\xi(t)}_{L^4}$ by estimating the right %side of \eqref{E3-21}. By \eqref{A:pf}, Lemma \ref{th:2-2} and Lemma \ref{th:3-2}, we have % \begin{align*} \norm{\xi(t)}_{L^4} &\le \norm{ e^{-i H_0 t}\xi_0 }_{L^4} + \int_0 ^{t} C \length{t-s}^{-3/4} \norm{G_\xi(s)}_{4/3} \, d s \\ &\le C \norm{\xi_0}_Y \bkA{t}^{-3/4} + \int_0 ^{t} C \length{t-s}^{-3/4} \bke{C_4\n^2 |x|(s) + C(C_2) n^2 \norm{\xi_0}_Y \bkA{s}^{-9/8} } \, d s \\ &\le C_{2,1} \norm{\xi_0}_Y \bkA{t}^{-3/4} + C_{2,1} \n^2 \,t^{1/4}\, |x(t)| \, + \, C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-4/3}, \end{align*} where $C_{2,1}$ is some explicit constant. %Since the main terms in $ \norm{\xi(t)}_{L^4} $ is from %$e^{-itH_0} \xi_0$ and terms of order $n^2 x$ in the integrand, %$C_{2,1}$ can be chosen independent of $C_2$. %In the computation we have used %\[ %\int_0^t |t-s|^{-3/4} \, \bkA{s}^{-9/8}\, d s \le C \bkA{t}^{-3/4}. %\] We now estimate $\norm{\xi(t)}_{L^4 \loc}$. If $t\le 1$, we can bound $L^4 \loc$-norm by $L^4$-norm. Hence we may assume $t>1$. For , We divide the time integral in \eqref{E3-21} into $s\in [0,t-1]$ and $s\in [t-1,t]$. In the first interval, we estimate $L^4 \loc$ by $L^8$ norm; in the second we estimate $L^4 \loc$ simply by $L^4$. Using similar arguments in the previous estimate of the $L_4$ norm, we have % \begin{align*} \norm{\xi(t)}_{L^4 \loc} &\le C \norm{\xi_0}_Y \bkA{t}^{-9/8} + \int_0^{t-1} \frac C{|t-s|^{9/8}} \norm{G_\xi}_{L^{8/7}} \, d s + \int_{t-1}^t \frac C{|t-s|^{3/4}} \norm{G_\xi}_{L^{4/3}} \, d s \\ &\le C \norm{\xi_0}_Y \bkA{t}^{-9/8} + \int_0^t \frac C {\bkA{t-s}^{9/8}} \bke{C_4\n^2 |x|(s) + C(C_2) n^2 \norm{\xi_0}_Y \bkA{s}^{-9/8} } \, d s \\ &\qquad + \sup_{t-1 \le s \le t} \norm{G_\xi(s)}_{L^{4/3}} \\ &\le C_{2,2} \norm{\xi_0}_Y \bkA{t}^{-9/8} + C_{2,2} n^2 |x(t)| \, + \, C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \end{align*} % where $C_{2,2}$ is some explicit constant. \bigskip {\bf Step 2.} $L^2 \loc $-norm, $0\le t \le 2t_2$ % Recall the decomposition \eqref{xi.dec}: $\xi=\xitwo+\xithree= \xitwo +(\xithree_1 + \cdots + \xithree_5)$. We will estimate the $L^2 \loc $-norm of each term. {\bf 0}. $\xitwo$. Since $\Phi_1 \in L^2 \loc$, and $\Phi_j\in L^2$, $(j>1)$, we have \[ \norm{\xitwo(t)}_{L^2 \loc} \le C_{2,3} C\n^2 \, |x(t)| ~, \] for some explicit constant $C_{2,3}$. {\bf 1}. $\xithree_1$. We have \[ \norm{\xi^{(3)}_1(t)}_{L^2 \loc}\le C_{2,4} \norm{\xi_0}_Y \bkA{t}^{-9/8} ~, \] for some explicit constant $C_{2,4}$ by the $L^{p',p}$ estimate of $e^{-itH_0}$ in Lemma \ref{th:2-2}. {\bf 2}. $\xithree_2$. By the linear estimate \eqref{eq:22-1B} in Lemma \ref{th:2-2} we have, for some constant $C_{2,5}$, \[ \norm{\xithree_2(t)}_{L^2 \loc}\le C_{2,5} \n^2 |x_0| \bkA{t}^{-9/8} ~. \] {\bf 3}. $\xithree_3$. To estimate $\xithree_3(t)=-\int_0^t e^{-iH_0(t-s)} \Pc G_4 \, ds$ with $G_4$ defined in \eqref{G4.def}, we need estimates \eqref{uvdot.est} for $\dot u$ and $\dot v$ and the linear estimate \eqref{eq:22-1B} in Lemma \ref{th:3-2}. Hence % \begin{align*} \norm{\xithree_3(t)}_{L^2 \loc}& \le \int_0^t \norm{e^{-iH_0(t-s)} \Pc G_4}_{L^2 \loc} \, ds \\ &\overset{\eqref{eq:22-1B}} \le C\int_0^t\bkA{t-s}^{-9/8} \bke{n^2 |\dot u| + n |x \dot v|}\, ds \\ &\overset{\eqref{uvdot.est}} \le C \int_0^t \bkA{t-s}^{-9/8} \bke{n^2 \norm{G}_{L^{4/3}} + n^4 |x|}\, ds \\ %%&\overset{\rm Lemma \ref{th:3-2}} &\overset{\eqref{G.est}} \le C \int_0^t \bkA{t-s}^{-9/8} \bke{ n^4 |x| + C(C_2)n^4 \norm{\xi_0}_Y \bkA{s}^{-9/8}}\, ds \\ & \le C_{2,6}n^4 |x(t)| + C (C_2)n^4 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~, \end{align*} {\bf 4}. $\xithree_4+\xithree_5$. We write $\xithree_4+\xithree_5=\int_0 ^{t} e^{-iH_0(t-s)} \Pc G_{\xi,5}(s) \, d s$, where $G_{\xi,5}(s):=(G_\xi-i^{-1}G_3)(s)$. By Lemma \ref{th:3-2} and Lemma \ref{th:2-2}, we have for $t>1$, % \begin{align*} &\norm{(\xithree_4+\xithree_5)(t)}_{L^2 \loc} \\ &\le \int_0 ^{t-1} \norm{e^{-iH_0(t-s)} \Pc G_{\xi,5}(s) }_{L^8} \, d s + \int_{t-1} ^{t} \norm{e^{-iH_0(t-s)} \Pc G_{\xi,5}(s)}_{L^4} \, d s \\ &\le C \int_0 ^{t-1} C \length{t-s}^{-9/8} \norm{G_{\xi,5}(s)}_{8/7} \, d s + \int_{t-1} ^{t} C \length{t-s}^{-3/4} \norm{G_{\xi,5}(s)}_{4/3} \, d s \\ &\le C \bke{ \int_0 ^{t-1} \length{t-s}^{-9/8} +\int_{t-1} ^{t} \length{t-s}^{-3/4} } \bke{ (C_4 n^{15/4} |x(s)|+ C(C_2) n^2\norm{\xi_0} \bkA{s}^{-9/8}} \, d s \\ &\le C_{2,7}\n^{15/4}\, |x(t)| + C(C_2) n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8}~, \end{align*} for some explicit constant $C_{2,7}$. If $t<1$, we can bound the $L^2 \loc$-norm by the $L^4$-norm. Hence the last estimate for $t<1$ follows from the estimate in Step 1. We have obtained estimates on $\xi$ involving explicit constants $C_{2,1},\ldots, C_{2,7}$ and $C(C_2)$. We now define the constant $C_2$ in \eqref{xi.est} to be: % \begin{equation}\label{eq:C2} C_2 \equiv C_{2,1}\, +\cdots + \, C_{2,7} ~. \end{equation} Since all terms involving $C(C_2)$ have some extra $n$ factor, $\xi$ satisfies the estimates in \eqref{xi.est} provided that $n$ is sufficiently small. %Note that \eqref{A:pf} are only used to %estimate higher order terms. Since the constants $C_{2,j}$ do %not depend on $C_2$, we do not have the danger of recurrent %definition. Summarizing, we have proved the following lemma: \begin{lemma} If $n$ is sufficiently small, there is an explicit constant $C_2$ such that, if \eqref{A:pf} holds in $[0, t]$ for some $t\le C_3 n^{-4}$, then the estimates \eqref{xi.est} in Proposition \ref{th:3-1} also hold in $[0,t]$. \end{lemma} \myremark In the proof, we only used \eqref{eq:3-2}, \eqref{A:pf}, Lemma \ref{th:2-2} and Lemma \ref{th:3-2}. The information we need on the size of bound states is in \eqref{eq:3-2} and the first estimate of \eqref{A:pf}. Since \eqref{eq:3-2} is always true, we only need to ensure that the first estimate in \eqref{A:pf} holds. %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \subsection{Normal form for equations of bound states} We now compute the Poincar\'e normal form for the bound states. This normal form will be used to estimate the bound states components $x$ and $y$ in next subsection. Recall that we write \[ x(t)= e^{-ie_0t} u(t) , \qquad y(t)= e^{-ie_1t} v(t) \] and the equations \eqref{u.eq} and \eqref{v.eq} for $u$ and $v$, \[ \dot u = -i e^{ie_0t} \bke{ \phi_0, \, G + \Lambda_\pi } ,\qquad \dot v = -i e^{ie_1 t} \bkt{(E_1(m)-e_1)y + (\phi_1 , G)}. \] % Using the decompositions \eqref{La.dec} for $\Lambda_\pi$ and \eqref{E.dec} for $E_1(m)$, we can decompose the equations for $u$ and $v$ according to orders in $n$: % \begin{align} \dot u &=-i e^{ie_0t} (\phi_0, G_3) -i e^{ie_0t} (\phi_0,G_5+\Lambda_{\pi,5}) -i e^{ie_0t} (\phi_0,G_7+\Lambda_{\pi,7}) \nonumber \\ &\equiv R_{u,3} + R_{u,5}+ R_{u,7} ~,\label{u.eq1} \\ \dot v&=-i e^{ie_1 t} [(\phi_1, G_3) + E_{1,2}|y|^2 y ] -i e^{ie_1 t} [(\phi_1, G_5) + E_{1,4} |y|^4 y]\nonumber\\ &\quad -i e^{ie_1 t} [(\phi_1, G_7) + E_1^{(6)}(|y|) \,y] \nonumber\\ &\equiv R_{v,3} + R_{v,5}+ R_{v,7}\label{v.eq1} ~. \end{align} % Using \eqref{La5.est}, \eqref{La7.est} and Lemma \ref{th:3-2}, which assume \eqref{A:pf}, we have % \begin{align} |R_{u,5}| &\leC \norm{G_5}_{L^1\loc} + \norm{\Lambda_{\pi,5}}_{L^1} \leC n^4 |x| + n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{Ru5.est} \\ |R_{u,7}| &\leC \norm{G_7}_{L^1\loc} + \norm{\Lambda_{\pi,7}}_{L^1} \leC n^6 |x| + n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{Ru7.est}\\ |R_{v,5}| &\leC \norm{G_5}_{L^1\loc} + |y|^5 \leC n^5 + n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{Rv5.est}\\ |R_{v,7}| &\leC \norm{G_7}_{L^1\loc} + |y|^7 \leC n^7 + n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{Rv7.est} \end{align} % We shall first integrate $R_{u,3}$ and $R_{v,3}$ in step 1, and then integrate $R_{u,5}$ and $R_{v,5}$ in step 2. % \noindent{\bf Step 1} \quad Integration of terms of order $n^3$ In the equation of $u$, \eqref{u.eq1}, the terms of order $n^3$ are contained in $R_{u,3}=-i e^{ie_0t} (\phi_0, G_3)$. The resonant terms from $G_{3}$ are $|y|^2 x$ and $|x|^2x$, whose phase cancels the factor $e^{ie_0t}$. The other four terms of order $n^3$ in $G_3$ have different frequencies and can be exploited using integration by parts. By \eqref{G3.def} we have % \begin{align*} \dot u &=-i e^{ie_0t} (\phi_0, G_3) + R_{u,5} + R_{u,7} \\ &= c_1 \, |u|^2 u +c_2 \, |v|^2 u + \frac d{dt} (u_1^-) + g_{u,1} + R_{u,5} + R_{u,7} \end{align*} % where % \begin{equation*} c_1=- i \la (\phi_0^2 ,\phi_0^2) , \qquad c_2=- i 2\la (\phi_0^2 ,\phi_1^2) \end{equation*} % \[ u_1^- = - \bke{ \la \phi_0, \frac { e^{i(2e_0-2e_1)t} v^2 \bar u }{2e_0-2e_1} \phi_0 \phi_1^2 + \frac {e^{i(e_0-e_1)t} 2|u|^2 v} {e_0-e_1}\phi_0 ^2\phi_1 + \frac {e^{i(-e_0+e_1)t} u^2 \bar v} {-e_0+e_1} \phi_0^2 \phi_1 } \] and \begin{equation*} g_{u,1} = \Bigg( \la \phi_0, \frac { e^{i(2e_0-2e_1)t} \frac d{dt}(v^2 \bar u) }{2e_0-2e_1} \phi_0 \phi_1^2 + \frac {e^{i(e_0-e_1)t} \frac d{dt}(2|u|^2 v)} {e_0-e_1}\phi_0 ^2\phi_1 + \frac {e^{i(-e_0+e_1)t} \frac d{dt}(u^2 \bar v)} {-e_0+e_1} \phi_0^2 \phi_1 \Bigg). \end{equation*} In the equation of $v$, \eqref{v.eq1}, the terms of order $n^3$ are in $R_{v,3}= -i e^{ie_1 t} [(\phi_1, G_3) + E_{1,2}|y|^2 y ]$. There is only one resonant term in $G_3$, namely, $|x|^2 y$. Another resonant term of order $n^3$ is from the term $E_{1,2}|y|^2 y$. The other four terms of order $n^3$ in $G_3$ have different frequencies and can be integrated. We thus have % \begin{align*} \dot v &= -i e^{ie_1 t} [(\phi_1, G_3) + E_{1,2}|y|^2 y ] +R_{v,5} + R_{v,7} \\ &= c_6 |u|^2 v +c_7 |v|^2v + \frac d{d t} ( v_1^-) + g_{v,1} +R_{v,5} + R_{v,7} \end{align*} where % \begin{equation*} c_6=- i 2\la (\phi_0^2, \phi_1^2), \qquad c_7=- i E_{1,2} \end{equation*} % \begin{align*} v_1^- &= - \Bigg( \la \phi_1, \frac { e^{i(-e_1+e_0)t} v^2 \bar u }{-e_1+e_0} \phi_0 \phi_1^2 + \frac {e^{i(e_1-e_0)t} 2|v|^2 u} {e_1-e_0}\phi_0 \phi_1^2 \\ &\qquad \qquad \qquad + \frac {e^{i(2e_1-2e_0)t} u^2 \bar v} {2e_1-2e_0} \phi_0^2 \phi_1 + \frac {e^{i(e_1-e_0)t} |u|^2 u} {e_1-e_0}\phi_0^3 \Bigg) \end{align*} and \begin{align*} g_{v,1} = \Bigg( \la \phi_1, &\frac { e^{i(-e_1+e_0)t} \frac d{dt}( v^2 \bar u) }{-e_1+e_0} \phi_0 \phi_1^2 + \frac {e^{i(e_1-e_0)t} \frac d{dt}(2|v|^2 u)} {e_1-e_0}\phi_0 \phi_1^2 \\ &\qquad + \frac {e^{i(2e_1-2e_0)t} \frac d{dt}(u^2 \bar v)} {2e_1-2e_0} \phi_0^2 \phi_1 + \frac {e^{i(e_1-e_0)t} \frac d{dt}(|u|^2 u)} {e_1-e_0}\phi_0^3 \Bigg) \end{align*} We now define \begin{equation} %\label{gv1.est} u_1 = u - u_1^- ~, \qquad v_1 = v - v_1^- ~. \end{equation} % The equations for $u_1$ and $v_1$ are \begin{align*} \dot u_1 &= c_1 |u|^2 u + c_2 |v|^2 u + g_{u,1} +R_{u,5} + R_{u,7} \nonumber \\ &= c_1 |u_1|^2 u_1 + c_2 |v_1|^2 u_1 + g_{u,2} + g_{u,1} +R_{u,5} + R_{u,7} %\label{u1.eq} \end{align*} \begin{equation*} %\label{gu2.est} g_{u,2} = c_1 (|u|^2 u-|u_1|^2 u_1) + c_2 (|v|^2 u-|v_1|^2 u_1) \end{equation*} and \begin{align*} \dot v_1 &= c_6 |u|^2 v + c_7 |v|^2 v + g_{v,1} +R_{v,5} + R_{v,7} \nonumber \\ &= c_6 |u_1|^2 v_1 + c_7 |v_1|^2 v_1 + g_{v,2} + g_{v,1} +R_{v,5} + R_{v,7} %\label{v1.eq} \end{align*} \[ g_{v,2} = c_6 (|u|^2 v-|u_1|^2 v_1) + c_7 (|v|^2 v-|v_1|^2 v_1) ~. \] We have finished the integration of order $n^3$ terms. Note that both $u_1^-$ and $v_1^-$ enter the equations of $u_1$ and $v_1$. This is the reason we compute their normal form together. %We shall re-decompose the $O(n^5)$ terms and give some simple estimates. Observe that % \begin{equation} \label{uv1.est} |u^-_1|\leC n^2 |u|~, \qquad |v^-_1|\leC n^2 |v| ~. \end{equation} % We now decompose $g_{u,1}$, $g_{v,1}$, $g_{u,2}$ and $g_{v,2}$ according to their orders in $n$. We want to write them as sum of order $n^5$ and order $n^7$ terms. We first claim that $g_{u,1}$ and $g_{v,1}$ are of the forms \[ g_{u,1}= e^{ie_0t} g_{u,1,5} + g_{u,1,7} ~, \qquad g_{v,1}= e^{ie_1t} g_{v,1,5} + g_{v,1,7} ~, \] where $g_{u,1,7}$ and $g_{v,1,7}$ are order $n^7$ terms, and $g_{u,1,5}$ and $g_{v,1,5}$ are explicit homogeneous polynomials of degree 5 in $x,\bar x, y, \bar y$ with { purely imaginary} coefficients. Moreover, every term in $g_{u,1,5}$ has a factor $x$ or $\bar x$. For example, the first term in $g_{u,1}$ is % % \begin{align*} &C e^{i(2e_0-2e_1)t} \frac d{dt}(v^2 \bar u) \\ &=C e^{i(2e_0-2e_1)t} (2 \bar u v \dot v + v^2 \bar u) \\ &=C e^{ie_0 t} \bke{ 2 \bar x y e^{-ie_1 t} \dot v + y^2e^{ie_0 t} \wbar{\dot u} } \\ &=C e^{ie_0 t} \bke{ 2 \bar x y e^{-ie_1 t} [R_{v,3}+ R_{v,5} + R_{v,7}] + y^2e^{ie_0 t} [ \wbar {R_{u,3}}+ \wbar{R_{u,5}} + \wbar{R_{u,7}} ] } \end{align*} % where $C=(\la \phi_0 , (2e_0 - 2e_1)^{-1} \phi_0 \phi_1^2)$ is real. Repeating this calculations for all terms in $g_{u,1}$ and collecting terms of order $n^5$, we obtain $g_{u,1,5}$. The rest belongs to $g_{u,1,7}$. There are two terms of order $n^5$ in the last expression: the terms of the form $ 2 \bar x y R_{v,3}$ and $y^2 \wbar {R_{u,3}}$. By definitions of $R_{u,3}$ and $R_{v,3}$, they are explicit polynomials of degree 5 in $x,\bar x, y, \bar y$ with purely imaginary coefficients. From the estimates of \eqref{Ru5.est}--\eqref{Rv7.est}, we can bound $g_{u,1,7}$ by \begin{equation} \label{guv17.est} \begin{aligned} |g_{u,1,7}(t)| &\leC n^6 |u| + n^4 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~, \\ |g_{v,1,7}(t)| &\leC n^6 |v| + n^4 \norm{\xi_0}_Y \bkA{t}^{-9/8} ~. \end{aligned} \end{equation} Similarly, we can write $g_{u,2}$ and $g_{v,2}$ as \[ g_{u,2}= e^{ie_0t} g_{u,2,5} + g_{u,2,7} ~, \qquad g_{v,2}= e^{ie_1t} g_{v,2,5} + g_{v,2,7} ~, \] where $g_{u,1,5}$ and $g_{v,1,5}$ are explicit homogeneous polynomials of degree 5 in $x,\bar x, y, \bar y$ with {purely imaginary} coefficients and $g_{u,2,7}$ and $g_{v,2,7}$ are order $n^7$ terms satisfying \begin{equation} \label{guv27.est} |g_{u,2,7}(t)| \leC n^6 |u| ~,\qquad |g_{v,2,7}(t)| \leC n^6 |v| ~. \end{equation} % Here we have used \eqref{uv1.est} in last estimate. Moreover, every term in $g_{u,2,5}$ has a factor $x$ or $\bar x$. We shall not perform calculations and estimates in details as they are similar to the previous step. To gain some idea, we shall do one example and show it is of the right form. By using $u-u_1 = u_1^-$, the first term in $g_{u,2}$ can be written as % \begin{align*} |u|^2 u -|u_1|^2 u_1 &= u^2 \bar u - (u - u_1^-)^2 (\bar u - \wbar{u_1^-}) = u^2 \wbar{(u_1^-)} + 2 |u|^2 u_1^- + O(u |u_1^-|^2) \\ &= x^2 \wbar{(u_1^-)} + 2 |x|^2 u_1^- + O(u |u_1^-|^2) \end{align*} % The first two terms, $x^2 \wbar{(u_1^-)} + 2 |x|^2 u_1^-$, contributes to $g_{u,2,5}$. Since $u_1^-$ equals to $e^{ie_0t}$ times a polynomial of degree $3$ in $x,\bar x, y, \bar y$ with real coefficients and $c_1$ in $g_{u,2}$ is purely imaginary, they are of the desired form. Summarizing, we can write \[ g_{u,1} + g_{u,2} = e^{i e_0 t} \wt R_{u,5} + g_{u,3} \] \[ g_{v,1} + g_{v,2} = e^{i e_1 t} \wt R_{v,5} + g_{v,3} \] where $\wt R_{u,5}= g_{u,1,5} + g_{u,2,5}$ and $\wt R_{v,5}=g_{u,1,5} + g_{u,2,5}$ are explicit homogeneous polynomials of degree 5 in $x, \bar x, y$ and $\bar y$ with {\bf purely imaginary} coefficients. Moreover, every term in $\wt R_{u,5}$ has a factor $x$ or $\bar x$. Also, $g_{u,3}= g_{u,1,7} + g_{u,2,7}$ and $g_{v,3}=g_{v,1,5} + g_{v,2,5}$. From the assumption \eqref{A:pf}, we have % \begin{equation} \label{Guv5B.est} |\wt R_{u,5}| \leC n^4|x| \, ,\quad |\wt R_{v,5}| \leC n^5. \end{equation} \begin{equation} \label{guv3.est} |g_{u,3}| \leC n^6 |x| + n^4 \norm{\xi_0}_Y \bkA{t}^{-9/8} \, ,\quad |g_{v,3}| \leC n^7 + n^4 \norm{\xi_0}_Y \bkA{t}^{-9/8}~. \end{equation} The final equations for $u_1$ and $v_1$ are % \begin{align} \dot u_1 &= c_1 |u_1|^2 u_1 + c_2 |v_1|^2 u_1 + (R_{u,5} + e^{i e_0 t} \wt R_{u,5}) + (R_{u,7} + g_{u,3} ) \label{u1.eq} \\ \dot v_1 &= c_6 |u_1|^2 v_1 + c_7 |v_1|^2 v_1 + (R_{v,5} + e^{i e_1 t} \wt R_{v,5}) + ( R_{v,7} + g_{v,3} ) \label{v1.eq} \end{align} \bigskip \noindent{\bf Step 2} \quad Integration of terms of order $n^5$ We now integrate terms of order $n^5$. In $u_1$-equation \eqref{u1.eq} we have $R_{u,5} + e^{ie_0t}\wt R_{u,5}$, where $R_{u,5}$ is from the decomposition of original equation \eqref{u.eq1} and $\wt R_{u,5}$ is from the error terms $g_{u,1} + g_{u,2}$. Similarly, terms of order $n^5$ in $v_1$-equation \eqref{v1.eq} is $R_{v,5} + e^{ie_1t}\wt R_{v,5}$. Observe that they are either of the form $x^\al y^\beta$ with $|\al|+|\beta|=5$, or of the form $xy\xi$. Also note that there are two sources in $R_{u,5} $: $G_5$ and $\Lambda_{\pi,5}$. Among all these terms the main term is $G_5$. We have already studied $\wt R_{u,5}$ and $\wt R_{v,5}$. They are explicit homogeneous polynomials of degree 5 in $x, \bar x, y$ and $\bar y$ with {\bf purely imaginary} coefficients. Moreover, every term in $\wt R_{u,5}$ has a factor $x$ or $\bar x$. We next look at $\Lambda_\pi$. Recall \eqref{La.dec} that $\Lambda_\pi = \Lambda_{\pi,5} + \Lambda_{\pi,7}$ and $\Lambda_{\pi,5}=- 2 q_3 \, (\phi_1,\, G_3 |y|^2 + \bar G_3 y^2)$ \eqref{La5.def}. Thus $\Lambda_{\pi,5}$ is a homogeneous polynomial in $x,\bar x, y$ and $y$ of degree $5$ with purely real functions as coefficients. Therefore the term $-i e^{ie_0t} (\phi_0,\Lambda_{\pi,5})$ in $\dot u$ equation \eqref{u.eq1} gives only polynomials with purely imaginary coefficients and a phase $e^{ie_0t}$. Recall $G_5$ is given by % \begin{align*} G_5 &=\la (2 y^3 \bar y \bar x + 4 |y|^4 x ) \, \phi_0\phi_1 q_3 + \la (2|x|^2 y^2 \bar y + x^2 y \bar y^2 ) \, \phi_0^2 q_3 \\ & \quad + \la (x \phi_0 + y \phi_1)^2 \wbar \xi + 2\la |(x \phi_0 + y \phi_1)|^2 \xi \end{align*} % Recall the decomposition $\xi=\xitwo + \xithree$, where \[ \xitwo(t)= y^2 \bar x \Phi_1 + |y|^2 x \Phi_2 + |x|^2 y \Phi_3 + x^2 \bar y \Phi_4 + |x|^2 x \Phi_5 \] with $\Phi_1$ the only function with nonzero imaginary part \eqref{Phi1.def}, \eqref{Phij.def}. Let % \begin{equation}%\label{} \Phi_1=\Phi_{1,R} + i \Phi_{1,I} \end{equation} % with both $\Phi_{1,R}$ and $ \Phi_{1,I} $ real. Denote the real part of $\xitwo(t)$ by % \begin{equation}%\label{} \xitwo_R(t)= y^2 \bar x \Phi_{1,R} + |y|^2 x \Phi_2 + |x|^2 y \Phi_3 + x^2 \bar y \Phi_4 + |x|^2 x \Phi_5. \end{equation} % We can write $\xi= y^2 \bar x i \Phi_I +\xitwo_R +\xithree$. Thus we can further decompose $G_5$ as % \begin{equation} G_5 =G_{5,1} + G_{5,2} + G_{5,3} \end{equation} where % \begin{align*} G_{5,1} &= (x \phi_0 + y \phi_1)^2 \bar y^2 x (-i) \Phi_{1,I} + 2|(x \phi_0 + y \phi_1)|^2 y^2 \bar x i \Phi_{1,I} \\ G_{5,2} &=\la (2 y^3 \bar y \bar x + 4 |y|^4 x ) \, \phi_0\phi_1 q_3 + \la (2|x|^2 y^2 \bar y + x^2 y \bar y^2 ) \, \phi_0^2 q_3 \\ & \quad + \la (x \phi_0 + y \phi_1)^2 \wbar {\xitwo_R} + 2\la |(x \phi_0 + y \phi_1)|^2 \xitwo_R \\ G_{5,3} &=\la (x \phi_0 + y \phi_1)^2 \wbar \xithree + 2\la |(x \phi_0 + y \phi_1)|^2 \xithree \end{align*} % The term $G_{5,3}$ will be shown to be smaller than $G_{5,1}$ and $G_{5,2}$. Although $G_{5,1}$ and $G_{5,2}$ are of the same size, $G_{5,2}$ consists of monomials in $x$, $\bar x$, $y$ and $\bar y$ with {\it real} functions as coefficients, while $G_{5,1}$ with purely imaginary coefficients. The reason that $G_{5,1}$ has purely imaginary coefficients is due to the resonance of some linear combination of eigenvalues with the continuum spectrum of $H_0$ appearing in the form $(H_0 - 0i - 2e_1 + e_0)^{-1}$. The only resonant term in $u$-equation from $G_{5,1}$ is $|y|^4x$ (from $y^2 \bar \xi$): \[ -i e^{i e_0 t} (\phi_0,(y \phi_1)^2 \bar y^2 x (-i) \Phi_{1,i}) = - (\phi_0 \phi_1^2,\Phi_{1,i}) |v|^4 u ~, \] and the only resonant term in $v$-equation from $G_{5,1}$ is $|x|^2|y|^2y$ (from $x \bar y \xi$): \[ -i e^{i e_1 t} (\phi_1, 2 (x\phi_0)( \bar y \phi_1) y^2 \bar x i \Phi_{1,i}) = 2 (\phi_0 \phi_1^2,\Phi_{1,i}) |u|^2 |v|^2 v ~. \] Note their coefficients only differ by a factor $-2$. We recall % \begin{equation} \gamma_0 =-(\phi_0 \phi_1^2,\Phi_{1,i}) = -\Im \bke{ \la \phi_0 \phi_1^2 \, , \, \frac {-\la} {H_0 - 0i - 2e_1 + e_0} \, \Pc \phi_0 \phi_1^2 } >0 \end{equation} Together with the definitions of $R_{u,5}$ and $ R_{v,5}$ in \eqref{u.eq1}, we can rewrite % \begin{align*} &R_{u,5} + e^{i e_0 t} \wt R_{u,5} = e^{i e_0 t} \bkt{\wt R_{u,5} -i (\phi_0,\, G_{5,1}+G_{5,2} +\Lambda_{\pi,5}) } -i e^{i e_0 t} (\phi_0,\, G_{5,3}) \\ & R_{v,5} + e^{i e_1 t} \wt R_{v,5} = e^{i e_1 t} \bkt{\wt R_{v,5} -i (\phi_1,\, G_{5,1}+G_{5,2} +E_{1,4}|y|^4y)} -i e^{i e_1 t} (\phi_1,\, G_{5,3}) \end{align*} As in Step 1, we now integrate by parts the non-resonant terms inside the square brackets. The resonant terms can't be integrated and we shall only collect them. This procedure is the same as in Step 1 and we only summarize the conclusion: there exists constants $c_3, c_4, c_5, c_8, c_9,c_{10}$, $u_2^-=O(u^5 + u v^4)$ and $v_2^-=O(u^5 + u v^4)$ two homogeneous polynomials in $u$ and $v$ of degree $5$, and $g_{u,4}$ and $g_{v,4}$ the integration remainders such that % \begin{align*} &R_{u,5} + e^{i e_0 t} \wt R_{u,5} = \bke{c_3 |u|^4 + c_4 |u|^2 |v|^2 + c_5|v|^4} u \\ &\qquad + \frac d{dt}\bke{u_2^-} + g_{u,4} -i e^{i e_0 t} (\phi_0,\, G_{5,3}) ~, \\ & R_{v,5} + e^{i e_1 t} \wt R_{v,5} = \bke{c_8 |u|^4 + c_9 |u|^2 |v|^2 + c_{10}|v|^4} v \\ &\qquad + \frac d{dt}\bke{v_2^-} + g_{v,4} -i e^{i e_1 t} (\phi_1,\, G_{5,3}) \end{align*} % Furthermore, except $c_3$ and $c_9$, all other constants are purely imaginary. The real parts of $c_3$ and $c_9$ are from $G_{5,1}$ and they are given explicitly by % \begin{equation} \label{3-46} \Re c_3 = \gamma_0 , \qquad \Re c_9 = -2\gamma_0 ~. \end{equation} The explicit forms of $u_2$ or $v_2$ are not important. We only need to know their sizes. We can now write the equations for $u$ and $v$ as \begin{align*} \dot u_1 &= c_1 \, |u_1|^2 u_1 + c_2\, |v_1|^2 u_1 + \bke{c_3 |u|^4 + c_4 |u|^2 |v|^2 + c_5|v|^4} u \\ &\qquad + \frac d{dt}\bke{u_2^-} + g_{u,4} -i e^{i e_0 t} (\phi_0,\, G_{5,3}) + g_{u,3} + R_{u,7} ~, \\ \dot v_1 &= c_6 \, |u_1|^2 v_1 + c_7\, |v_1|^2 v_1 + \bke{c_8 |u|^4 + c_9 |u|^2 |v|^2 + c_{10}|v|^4} v \\ &\qquad + \frac d{dt}\bke{v_2^-} + g_{v,4} -i e^{i e_1 t} (\phi_1,\, G_{5,3}) + g_{v,3} + R_{v,7} ~, \end{align*} % We now define % \begin{align} \uu &\equiv u_1 - u_2^- =u - u_1^- - u_2^- \label{uu.def} \\ \vv &\equiv v_1 - v_2^- = v - v_1^- - v_2^- \label{vv.def} \end{align} % We have % \begin{align*} \dot \uu &= c_1 \, |u_1|^2 u_1 + c_2\, |v_1|^2 u_1 + \bke{c_3 |u|^4 + c_4 |u|^2 |v|^2 + c_5|v|^4} u \\ & \qquad + g_{u,4} -i e^{i e_0 t} (\phi_0,\, G_{5,3}) + g_{u,3}+ R_{u,7} \\ &=c_1 \, |\uu|^2 \uu + c_2\, |\vv|^2 \uu + \bke{c_3 |\uu|^4 + c_4 |\uu|^2 |\vv|^2 + c_5|\vv|^4} \uu + g_u \end{align*} % and \begin{align*} \dot \vv &= c_6 \, |u_1|^2 v_1 + c_7\, |v_1|^2 v_1 + \bke{c_8 |u|^4 + c_9 |u|^2 |v|^2 + c_{10}|v|^4}v \\ &\qquad + g_{v,4} -i e^{i e_1 t} (\phi_1,\, G_{5,3}) + g_{v,3} + R_{v,7} \\ &=c_6 \, |\uu|^2 \vv + c_7\, |\vv|^2 \vv + \bke{c_8 |\uu|^4 + c_9 |\uu|^2 |\vv|^2 + c_{10}|\vv|^4} \vv + g_v \end{align*} % with \begin{align} g_u &= g_{u,4} + g_{u,5} + g_{u,3} + R_{u,7} -i e^{i e_0 t} (\phi_0,\, G_{5,3}) \label{gu.def} \\ g_v&= g_{v,4} +g_{v,5} + g_{v,3} + R_{v,7} -i e^{i e_1 t} (\phi_1,\, G_{5,3}) \label{gv.def} \end{align} % and % \begin{align*} g_{u,5}&=c_1 \, (|u_1|^2 u_1-|\uu|^2 \uu) + c_2\, (|v_1|^2 u_1-|\vv|^2 \uu) \nonumber \\ &\quad + c_3 (|u|^4u-|\uu|^4 \uu) + c_4 (|u|^2 |v|^2 u-|\uu|^2 |\vv|^2 \uu) + c_5(|v|^4 u-|\vv|^4 \uu) \\ g_{v,5}&=c_6 \, (|u_1|^2 v_1-|\uu|^2 \vv) + c_7\, (|v_1|^2 v_1-|\vv|^2 \vv) \nonumber \\ &\quad + c_8 (|u|^4v-|\uu|^4 \vv) + c_9 (|u|^2 |v|^2 v-|\uu|^2 |\vv|^2 \vv) + c_{10}(|v|^4 v-|\vv|^4 \vv) ~. \end{align*} Observe that the error terms $g_u$ and $g_v$ are of the form \begin{equation*} g_u, g_v \sim (x^7 + x y^6) + (x^2 + y^2)\, \xithree + (x^4 + y^4) \xi + \bke{\phi,\xi^3} + \cdots ~, \end{equation*} % where $\phi$ denotes some local function. For $g_v$ we should add $|y|^7$: Since $g_v$ has a term $-i e^{ie_1t}\, (\phi_1, E^{(6)}(|y|)y)$ from $R_{v,7}$. This term has no factor in $x$ and is of order $|y|^7$. Under the assumption of \eqref{A:pf}, the error terms $g_{u,4}$ and $g_{v,4}$ can be estimated similarly as for $g_{u,1}$ and $g_{v,1}$. Also, $g_{u,5}$ and $g_{v,5}$ can be estimated similarly as for $g_{u,2}$ and $g_{v,2}$. We also have, for $j=1,2$, \[ | e^{i e_j t} (\phi_j,\, G_{5,3}) | \le C n^2 \norm{\xi^{(3)}}_{L^2 \loc} \overset{\eqref{A:pf}} \le C n^{6-1/4} |x| + C n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8}~. \] % Together with the estimates \eqref{Ru7.est}, \eqref{Rv7.est} and \eqref{guv3.est}, we conclude % \begin{align*} &|g_u(t)| \le C_6 \n^{6-1/4} |x(t)| + C_6 n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \\ &|g_v(t)| \le C_6 \n^{7-1/4} \qquad + C_6 n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \end{align*} % Summarizing our effort, we have obtained the following lemma. \begin{lemma} \label{th:3-4} Let $\uu$ and $\vv$ be defined as in \eqref{uu.def}--\eqref{vv.def}. They satisfy % \begin{equation} \label{eq:NF}\begin{split} \dot \uu &= (c_1 |\uu|^2 + c_2|\vv |^2) \uu + (c_3 |\uu|^4 + c_4 |\uu |^2|\vv|^2 + c_{5} |\vv |^4)\uu + g_u \\ \dot \vv &= (c_6 |\uu|^2 + c_7 |\vv |^2) \vv + (c_8 |\uu|^4 + c_9 |\uu |^2|\vv|^2 + c_{10}|\vv |^4)\vv + g_v \end{split}\end{equation} % All coefficients $c_1, \cdots, c_{10}$ except $c_3$ and $c_9$ are purely imaginary and we have \eqref{3-46}, i.e., \begin{equation} \Re c_3 = \gamma_0 ~, \qquad \Re c_9 = -2\gamma_0~, \end{equation} where $\gamma_0>0$ is defined in \eqref{gamma0.def}. Moreover, assuming \eqref{A:pf} and using the estimates \eqref{eq:3-2} and Lemma \ref{th:3-2}, we have % \begin{align} &|u(t)-\uu(t)|\le C_6\,\n^2|x(t)| ~,\quad |v(t)-\vv(t)|\le C_6 \,\n^3 \label{eq:33-1} \\ &|g_u(t)| \le C_6 \n^{6-1/4} |x(t)| + C_6 n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{gu.est}\\ &|g_v(t)| \le C_6 \n^{7-1/4} \qquad + C_6 n^2 \norm{\xi_0}_Y \bkA{t}^{-9/8} \label{gv.est} \end{align} % for some explicit constant $C_6$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Estimates for bound states} In this subsection we will conclude the estimates for $x$ and $y$ stated in Proposition \ref{th:3-1}. %That is, for some %$t_1, t_2$, $00$ and we have that $f(t) \ge 5 \cdot 10^{-5} \,\n^2$ for $t_1 \le t \le t_2'$. Hence \[ \dot g \le -(1.99)\, \gamma_0 f g^2 \le - 9.95 \cdot 10^{-5}\,\gamma_0 \n^2 g^2 ~. \] Hence \[ g(t) \le [g(t_1)^{-1} + 9.95 \cdot 10^{-5}\,\gamma_0 \n^2 (t-t_1)]^{-1} ~, \quad (t_1 \le t \le t_2'). \] and $ g(t_2) < (\e \n)^2$. This contradiction shows the existence of $t_2$ satisfying \eqref{eq:33-8}. Since $\dot g \ge - (2.01) f g^2$, similar argument shows $t_2 \ge C\e^{-2} \n^{-4}$ if $|y(0)|> 2 \e n$. % and we also have $g(t) \in [\e %\n /2,\, 2 \e \n]$ for $t\in [t_2,\, 2 t_2]$. Combining with estimate \eqref{3-53} for $f+g$, we have estimates for $f(t_2)$. From \eqref{3-50}, these estimates of $f$ and $g$ can be translated into estimates of $x(t_2)$ and $y(t_2)$ stated in Proposition \ref{th:3-1}. We have proved \eqref{eq:3-4}, \eqref{eq:3-5}, \eqref{eq:3-7} and \eqref{eq:3-8} in Proposition \ref{th:3-1}, using the estimates \eqref{eq:1-10}, \eqref{eq:3-2} and the assumption \eqref{A:pf}. Since \eqref{A:pf} holds for $t=0$, by continuity it holds for all $t \le t_2$. From Lemmas \ref{th:3-2}--\ref{th:3-4} and the above estimates, Proposition \ref{th:3-1} is proved. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Stabilization regime} In this section we study the solution $\psi(t)$ in the third time regime, after the solution has become near some nonlinear ground state. In this regime, it is natural to use the decomposition \eqref{psidec2} for the solution $\psi(t)$ which emphasizes nonlinear ground states. A key issue here is to pass the information from the coordinates system \eqref{psidec1} to the \eqref{psidec2}. As emphasized in the introduction, it is not sufficient to use only the estimates of $\psi$ at $t=t_2$. We will also use the explicit form of the main terms in the dispersive part of $\psi(t)$ to ensure that they do not come back to affect the local dynamics, i.e., the part of the wave represented by these terms is ``out-going'. The set-up and proof here are similar to those in \cite{TY} except the big terms of the dispersive part just mentioned. We shall first show that certain local estimates used in \cite{TY} are still small (see Lemma \ref{th:4-2}). \subsection{Preliminaries} In the time regime $t\ge t_2$, we will use the set-up in \cite{TY}, which we now briefly recall. For the proofs we refer the reader to \cite{TY}. For all nonlinear ground state $Q_E$ with frequency $E$, Let $\L_E$ be the linearized operator around $Q_E$: \begin{equation} \label{L.def} \L h = -i \bket{ (-\Delta + V -E + 2 \la Q_E^2)\,h + \la Q_E^2 \,\wbar h\,} \end{equation} With respect to $\L_E$, we can decompose $L^2(\R^3,\Complex)$, as a real vector space, as the direct sum of three invariant subspaces: \begin{equation} \label{L2.dec} L^2(\R^3,\Complex) = S(\L_E) \oplus \eigen(\L_E) \oplus \Hc(\L_E) \end{equation} where $S(\L_E)$ and $\eigen(\L_E)$ are generalized eigenspaces, obtained from perturbation of $\phi_0$ and $\phi_1$ respectively and $\Hc(\L_E)$ corresponds to the continuous spectrum of $\L_E$. Notice that this decomposition is not orthogonal. Also, $ S(\L_E)= \myspan_\R(iQ_E, R_E)$, where $R_E = \pd_E Q_E$. For each $\psi$ sufficiently close to $Q_E$, we can decompose $\psi$ as % \begin{equation} \label{psidec2} \psi = \bkt{Q_E + a_E R_E + \zeta_E + \eta_E}\, e^{ i \Theta_E} ~. \end{equation} % Here $a_E, \Theta_E \in \R$ and $\zeta_E \in \eigen(\L)$ and $\eta_E\in \Hc(\L)$. The direction $iQ_E$ is implicitly given in $Q_E (e^{ i \Theta} -1)$. Moreover, for this $\psi$ there is a unique frequency $E'$ such that in the decomposition \eqref{psidec2} the coefficient $a$ vanishes. In some sense it means that $Q_{E'}$ is the closest nonlinear ground states to $\psi$. \donothing{ Thus one can define $E(t)$ for each $t\ge t_2$ as the unique $E$ such that the coefficient $a$ vanishes. For each time $T\ge t_2$, let $Q_E= Q_{E(T)}$ and $R_E=Q_{E(T)}$. For all $t\in [t_2,T]$, we can decompose the solution $\psi(t)$ according to \eqref{psidec2} with $\L=\L_{E(T)}$: % \begin{equation} \label{eq:4-4} \psi(t) = \bkt{Q_E + a_E(t) R_E + \zeta_E(t) + {\eta_E} (t)}\, e^{-E(T)t + i \Theta(t)} ~, \end{equation} % for $(t_2 \le t \le T)$. Here $\zeta \in \eigen(\L_E)$, $\eta\in \Hc(\L_E)$. } \subsection{Estimates} %At this point we come back to the issue raised in the introduction. Our aim is to show that $\psi(t_2)$ satisfies the conditions of Theorem 3 (resonance dominated solutions) in \cite{TY}. By Proposition \ref{th:3-1} one can show that $\psi(t_2)$ is close to a nonlinear ground state $Q_{E_0}e^{i \Theta_0}$ in $L^2\loc$-norm, i.e., $\norm{Q_{E_0}}_{L^2}=n_{t_2}\le n_0$, $\norm{\psi(t_2)-Q_{E_0}e^{i \Theta_0}}_{L^2\loc} \le \e_0 n_{t_2}$. From the same Proposition, we have $n_{t_2} \sim n_1 \sim n$ where $n_1$ is defined in \eqref{n1.def} and $n$ is defined in \eqref{eq:1-10}. Thus for the purpose of order of magnitude, we are free to interchange $n_{t_2}$ with $n$. We now state the conditions of Theorem 3 (resonance dominated solutions) in \cite{TY}: Suppose that the initial data $\psi (t_2)$ is decomposed as in \eqref{psidec2} with the frequency $E=E_0$ chosen so that the coefficient $a$ vanishes. Then the excited state component $\zeta$ satisfies \begin{equation} \label{ty22} %\norm{\psi_0} = \n <\n_0 ~, 0 < \norm{\zeta} \le \e_0 \n \end{equation} and the dispersive part satisfies \begin{equation} \label{ty23} \norm{ {\eta_E} (t_2)}_Y \le C\norm{\zeta}^2 \end{equation} for all $E$ close to $E_0$ with $|E -E_0 |\le \norm{\zeta}^2$. Here the $Y$ norm is defined in \eqref{Y.def}. We shall see that the condition \eqref{ty22} is easy to verify by Proposition \ref{th:3-1}. The dispersive part, however, is no longer localized and there is no hope to satisfy \eqref{ty23}. In fact, even its $L^2$-norm is not small enough. Recall we know from the previous sections that $|x(t_2)|^2 - |x_0|^2$ is roughly one half of $|y_0|^2 - |y(t_2)|^2$. Thus, by conservation of $L^2$-norm, the dispersive part gains the other half of $|y_0|^2 - |y(t_2)|^2$ and $\norm{\xi(t_2)} \approx n$. One believes on physics ground that most mass of the dispersive part is far away and it has little influence to the local dynamics. The local part of the dispersive component, on the other hand, is generated by changes of the bound states and is small. Thus the results in \cite{TY} should still hold. This idea, however, requires to clarify the concept of ``out-going waves'' for nonlinear equations. Instead of directly approach this problem, we examine the condition \eqref{ty23} in the proof in \cite{TY}. It is used only to guarantee that for all $s \ge 0$ %(\e n)^{-2}+ \gamma_0 n^2 (t-t_2) \sim n^{2} t \begin{equation} \label{eta31A} \begin{aligned} \norm{e^{s\L} \eta_E(t_2) }_{L^4} &\le \tfrac 1{20}\, \bke{(\e n)^{-2}+ \gamma_0 n^2 s} ^{-3/4+1/100} ~, \\ \norm{e^{s\L} \eta_E(t_2) }_{L^2 \loc} &\le C \bke{(\e n)^{-2}+ \gamma_0 n^2 s } ^{-1} , \end{aligned} \end{equation} % % These estimates are used to estimate the local contributions of $e^{s\L} {\eta_E} (t_2) $ in (3.19) (see the following remark) and in the proofs for the $L^4$ and $L^2\loc$ estimates for $ {\eta_E} (t)$ in Lemmas 5.2 and 5.3. of \cite{TY}. {\it Remark:} In \cite{TY}, \eqref{eta31A} is used to estimate $\tilde \eta^{(3)}_1 (t) =e^{-iA (t-t_2)} \tilde \eta(t_2) $, with $\tilde \eta(t_2) = e^{i \Theta(t_2)} U {\eta_E} (t_2)$, and $A$ being a self-adjoint perturbation of $-\Delta+V$, $\L = U^{-1} (-iA) U$, $U$ a bounded operator. Since $e^{s\L} = U^{-1} e^{-isA} U$, \[ \tilde \eta^{(3)}_1 (t)=e^{-iA (t-t_2)} \, e^{i \Theta(t_2)} U {\eta_E} (t_2)\ =e^{i \Theta(t_2)} U e^{(t-t_2)\L} {\eta_E} (t_2) \] Thus we can choose freely to estimate either $A$ or $\L$. Since $e^{(t-t_2)\L} {\eta_E} (t_2)$ is easier to estimate than $e^{-iA (t-t_2)} \tilde \eta(t_2) $ by using the Duhamel's expansion, we state all conditions in terms of $\L$. %easier to expand $\L$ than to expand $A$ using . Because we only need \eqref{eta31A}, the same proof in \cite{TY} actually gives the following stronger result. \begin{theorem} \label{th:4-1} Suppose that $\psi(t_2)$ is close to a nonlinear ground state $Q_{E_0}e^{i \Theta_0}$ in $L^2\loc$-norm, and suppose that in the decomposition \eqref{psidec2} with $E=E_0$ one has $\norm{Q_{E_0}}_{L^2}=n_{t_2} \sim n \le n_0$, % $\norm{\psi(t_2)-Q_{E_0}e^{i \Theta_0}}_{L^2\loc} \le \e_0 ^2 \n^2$. $\norm{\zeta_{E_0}} \le \e_0 n$, $|a|+\norm{\eta_{E_0}}_{L^2 \loc} \le \e_0 ^2 \n^2$. If for all $E$ close to $E_0$ with $|E -E_0 |\le \e_0 ^2 n^2$ the dispersive part $ {\eta_E} (t_2)$ in the decomposition \eqref{psidec2} satisfies the estimates \eqref{eta31A}, then the conclusion and the proof of Theorem 1 in \cite{TY} remain valid. In particular, there is a frequency $E_\infty$ with $|E_\infty -E_0|\le \e_0 ^2 \n^2$ and a function $\Theta(t)= - E_\infty t + O(\log(t))$ for $t\in [t_2,\infty)$ such that \[ \norm{\psi(t) - Q_{E_\infty}e^{i \Theta(t)} }_{L^2 \loc} \le C_2 \bke{(\e n)^{-2}+ \gamma_0 n^2 (t-t_2)} ^{-1/2} \] for some constant $C_2$. Suppose that the initial data $\psi (t_2)$ is decomposed as in \eqref{psidec2} with the frequency $E$ chosen so that the coefficient $a$ vanishes. Suppose that, in addition to the previous assumption that \eqref{eta31A} holds for all frequency $E$ with $|E -E_0 |\le \e_0 ^2 n^2$, the excited state component $\zeta$ satisfies \eqref{ty22}--\eqref{ty23}. Then the lower bound \[ C_1 \bke{(\e n)^{-2}+ \gamma_0 n^2 (t-t_2)} ^{-1/2} \le \norm{\psi(t) - Q_{E_\infty}e^{i \Theta(t)} }_{L^2 \loc} \] holds as well. \end{theorem} The merit of this modification is that we do not need the initial data to be localized. We only need to know that its dispersive part is ``outgoing'' in certain sense. Notice that the condition on the size of the excited component $\zeta$ is a simple consequence of the estimates \eqref{xi.est}, \eqref{eq:3-8} and \eqref{eq:3-4}. To see this, we first pretend that the size of $\zeta$ is given by $y(t_2)$ and the size of the ground state component is given by $x(t_2)$. Then the condition \eqref{ty22} is just a simple consequence of \eqref{eq:3-8}. Since the difference between the decompositions \eqref{psidec1} and \eqref{psidec2} are higher order terms, the condition \eqref{ty22} is easy to check. Therefore, Theorem \ref{th:1-1} follows from the following Lemma: \begin{lemma} \label{th:4-2} Let $\psi(t)$ be the solution of \eqref{Sch} in Theorem \ref{th:1-1} and $t_2$ be the time in Proposition \ref{th:3-1}. Let $E_0=E(t_2)$ be the unique energy such that in the decomposition \eqref{psidec2} the coefficient $a$ vanishes. Then for all $E$ close to $E_0$, i.e., $|E-E_0|\le C \e_0^2 n^2$, we have for all $t \ge t_2$ % \begin{equation} \label{eta31B} \norm{e^{(t-t_2)\L} {\eta_E} (t_2) }_{L^4} \le n %^{1/100} (n^2 t)^{-3/4+1/100} ~, \quad \norm{e^{(t-t_2)\L} {\eta_E} (t_2)}_{L^2 \loc} \le n %^{1/100} (n^2 t) ^{-1} ~. \end{equation} % \end{lemma} Notice that, since $t_2 \sim \e^{-2} n^{-4}$ by Proposition \ref{th:3-1}, we have $(\e n)^{-2}+ \gamma_0 n^2 (t-t_2) \sim n^{2} t$ % for all $t\ge t_2$, no matter $t>2 t_2$ or $t < 2 t_2$. Hence \eqref{eta31B} implies \eqref{eta31A} with a big margin \noindent{\bf Proof of Lemma \ref{th:4-2}} We have the two decompositions at $t=t_2$ % \begin{equation} %\label{eq:4-7} \begin{split} \psi(t) & = x(t)\phi_0 +Q_1(y(t))+\xi(t) \\ & = \bkt{Q_E + a_E(t) R_E + \zeta_E(t)+ {\eta_E} (t)}\, e^{i\Theta_E(t) } \end{split} \end{equation} % Since $E$ will be fixed for the rest of this proof, we shall drop all subscripts $E$. Hence \begin{equation} \label{eq:4-12a} \zeta (t)+ {\eta } (t) =\bkt{x(t)\phi_0 \, e^{-i \Theta (t)} -Q_{T}} +\bkt{ Q_1(y(t))+\xi(t)} \, e^{-i \Theta(t)} -a_E(t) R_E~. \end{equation} % Thus we have \[ {\eta } (t_2)=\PcL \bket{\bkt{x(t_2)\phi_0 \, e^{-i \Theta (t_2)} -Q_{T}} + \bkt{ Q_1(y(t_2))+\xi(t_2)} \, e^{-i \Theta(t_2)} } =\eta_{0,1} + \eta_{0,2} ~, \] where \begin{align*} \eta_{0,1}&=\PcL \bket{\xi(t_2)e^{-i \Theta(t_2)} } ~, \\ \eta_{0,2}&=\PcL \bket{ \bke{x(t_2)\phi_0\,e^{-i \Theta(t_2)} - Q_E} +Q_1(y(t_2))\, e^{-i \Theta(t_2)} } ~. \end{align*} % Note that $\eta_{0,2}$ is a local $H^1$ function and is bounded by $O(\n^3)$, i.e., $\norm{\eta_{0,2}}_Y \le C n^3$. Therefore we have \begin{equation} \label{eq:4-12} \norm{e^{(t-t_2)\L}\eta_{0,2} }_{L^4} \le C \n^3 \; \bkA{t-t_2}^{-3/4} ~, \qquad \norm{e^{(t-t_2)\L}\eta_{0,2} }_{L^2 \loc} \le C \n^3 \; \bkA{t-t_2}^{-3/2} ~. \end{equation} Hence $e^{t\L}\eta_{0,2} $ satisfies the desired estimates with a big margin. We now focus on the non-local term $\eta_{0,1}=\PcL \bket{\xi(t_2) e^{-i \Theta(t_2)} }$. Recall $\xi(t_2)$ is bounded by $2 n$ in $L^2$ \eqref{eq:3-2} and by $4C_2 \n^{5-1/4}$ in $L^2 \loc$ from Proposition \ref{th:3-1}. In particular, we have $\norm{\eta_{0,1}}_{L^2} \le C \norm{\xi(t_2)}_{L^2} \le C\n$. Thus, by Lemmas 2.6 and 2.9 of \cite{TY}, %\label{eq:4-27a} we have $ \norm{e^{(t-t_2)\L} \eta_{0,1}}_{L^2} \le C\n $. For convenience of notation, we write \[ \L = -i H_0 + iV_1 + iV_2 \conj ~, \] where $V_1=2\la Q_E^2$, $V_2=\la Q_E^2$ and $\conj$ denotes the conjugation operator. By Duhamel's principle, % \begin{align*} e^{(t-t_2)\L}\eta_{0,1} &= \PcL e^{(t-t_2)\L} \bket{\xi(t_2) e^{-i \Theta(t_2)} } \\ &=\PcL e^{-i(t-t_2) H_0} \xi(t_2) e^{-i \Theta(t_2)} \\ &\quad + \int_{t_2}^t e^{(t-s)\L} \PcL i(V_1 + V_2 \conj) e^{-i(s-t_2)H_0} \xi(t_2) e^{-i \Theta(t_2)} \, d s ~. \end{align*} We now substitute \eqref{xi.eq} with $t=t_2$, i.e., % \begin{equation} \label{eq:4-13} \xi(t_2) =e^{-it_2H_0} \xi_0 + \int_0^{t_2} e^{-i(t_2-\tau)H_0} \PcH G_\xi(\tau) \, d \tau , \end{equation} % into the above equation. We have % \[ e^{(t-t_2)\L}\eta_{0,1} =E_1 + E_2 + E_3 + E_4 \] % where \begin{align*} \Omega_1 &= \PcL e^{-it H_0} \,e^{-i \Theta(t_2)}\xi_0 \\ \Omega_2&= \int_{t_2}^t e^{(t-s)\L} \PcL i(V_1 + V_2 \conj) e^{-is H_0} \,e^{-i \Theta(t_2)} \xi_0 \, d s \\ \Omega_3&=\int_0^{t_2} \PcL e^{-i(t -\tau) H_0} \PcH \, e^{-i \Theta(t_2)} \, G_\xi(\tau) \, d \tau \\ \Omega_4&= \int_{t_2}^t\int_0^{t_2} e^{(t-s)\L} \PcL i(V_1 + V_2 \conj) e^{-i(s-\tau)H_0} \PcH \,e^{-i \Theta(t_2)} \, G_\xi(\tau) \, d \tau \,d s \end{align*} The only estimates we need here from \S 3 are: (cf.\eqref{Gxi.L1est}) \begin{equation} \norm{\xi_0}_Y \le 4n , \qquad \norm{G_\xi(\tau)}_{L^1} \le Cn^3 , \quad (0\le \tau \le t_2),\qquad t_2 \sim \e^{-2} \n^4 ~. \end{equation} From the linear estimate Lemma \ref{th:2-2} we can estimate $\Omega_1$ by \begin{align*} &\norm{ \Omega_1 }_{L^4} \le C t ^{-3/4}\norm{ \xi_0}_{L^{4/3}} \\ &\norm{ \Omega_1 }_{L^2 \loc} \le C \norm{ \Omega_1 }_{L^8 } \le C t ^{-9/8} \norm{ \xi_0}_{L^{8/7}} \end{align*} For $\Omega_2$, since $t_2>1$, we have \[ \norm{ \Omega_2 }_{L^4} \le C \int_{t_2}^{t} |t-s|^{-3/4} \, |s|^{-9/8} \norm{ \xi_0}_{L^{8/7}} \, d s \le C t^{-3/4} \norm{ \xi_0}_{L^{8/7}} ~. \] If $t>t_2+1$, we bound its $L^2\loc$-norm by $L^8$ for $s\in [t_2,t-1]$ and by $L^4$ for $s\in [t-1,t]$. Thus we have \begin{align*} \norm{ \Omega_2 }_{L^2 \loc} &\le C \bket{ \int_{t_2}^{t-1} |t-s|^{-9/8} + \int_{t-1}^t |t-s|^{-3/4} } |s|^{-9/8}\norm{ \xi_0}_{L^{8/7}} \, d s \\ &\le C t^{-9/8} \norm{\xi_0}_{L^{8/7}} ~. \end{align*} % If $t \le t_2 + 1$, we use only $L^4$ norm for the whole interval $[t_2,t]$ and the same estimate holds. For $\Omega_3$, notice that $\Omega_3 = \PcL e^{-i(t-t_2)H_0} J$ where $J$ is the integral in \eqref{eq:4-13}, \[ J= \int_0^{t_2} e^{-i(t_2-\tau)H_0} \PcH G_\xi(\tau) \, d \tau = \xi(t_2) - e^{-i t_2H_0} \xi_0 ~. \] Since it is the difference of two $L^2$ functions of order $n$, $\norm{J}_{L^2} \le C \n$ and $\norm{ \Omega_3 (t)}_{L^2} \le C \n$ for all $t$. Suppose that $t\ge 2t_2$. Since $\norm{ G_\xi}_{L^{1}}\le C \n^3$ by \eqref{Gxi.L1est}, we have % \[ \norm{ \Omega_3 }_{L^\infty} \le C \int_0^{t_2} |t-\tau|^{-3/2}\norm{ G_\xi(\tau)}_{L^{1}} \, d \tau \le C \int_0^{t_2} t^{-3/2} \n^3 \, d \tau \le C \n^3 t_2 t^{-3/2} \] Interpolating, we have \[ \norm{ \Omega_3 }_{L^4}\le C \norm{ \Omega_3 }_{L^2}^{1/2}\, \norm{ \Omega_3 }_{L^\infty}^{1/2} \le Cn^2 t_2^{1/2} t^{-3/4} ~. \] Since $L^2 \loc$-norm is bounded by $L^\infty$-norm, we conclude that % \[ \norm{ \Omega_3 }_{L^2 \loc} \le C \n^3 t_2 t^{-3/2} \le C \n^3 t_2^{5/8} t^{-9/8}~. \] Suppose now that $t<2t_2$. From similar arguments we have \[ \norm{ \Omega_3 }_{L^4} \le C\int_0^{t_2} |t-\tau|^{-3/4} \n^3 \, d \tau \le C\int_0^{t_2} |t_2-\tau|^{-3/4} \n^3 \, d \tau = C t_2^{1/4} \n^3 \] \begin{align*} \norm{ \Omega_3 }_{L^2 \loc} &\le C\int_0^{t_2-1} \norm{\cdot}_{L^8} +C\int_{t_2-1}^{t_2} \norm{\cdot}_{L^4} \\ &\le C\int_0^{t_2-1} |t_2-\tau|^{-9/8} n^3 \,d \tau + C\int_{t_2-1}^{t_2} |t_2-\tau|^{-3/4} \n^3 \, d \tau \le C \n^3 + C n^3 \end{align*} We now estimate $\Omega_4$. Since $\norm{ G_\xi}_{L^{1}}\le C \n^3$ by \eqref{Gxi.L1est}, we have, \begin{align*} \norm{ \Omega_4 }_{L^4} &\le C\int_{t_2}^t |t-s|^{-3/4} \int_0^{t_2} |s-\tau|^{-3/2}\norm{ G_\xi(\tau)}_{L^{1}} \, d \tau \,d s \\ &\le C\int_{t_2}^t |t-s|^{-3/4} t_2 s^{-3/2} \n^3 \,d s \\ &\le C t_2 \n^3 \int_{t_2}^t |t-s|^{-3/4} s^{-9/8}t_2^{-3/8} \,d s \\ & \le C \n^3 t_2 ^{5/8} \int_{0}^t |t-s|^{-3/4} \bkA{s}^{-9/8} \,d s \\ & \le C \n^3 t_2 ^{5/8} t^{-3/4} \end{align*} % We can bound the $L^2\loc$-norm by $L^8$ for $s\in [t_2,t-1]$, and by $L^4$ for $s\in [t-1,t]$ (if $t\le t_2 + 1$, we use only $L^4$ norm for the whole interval $[t_2,t]$). Thus we have the bound \begin{align} \norm{ \Omega_4 }&_{L^2 \loc} \le \bke{C\int_{t_2} ^{t-1} |t-s|^{-9/8} + C\int_{t-1}^t |t-s|^{-3/4}} \bke{\int_0^{t_2} |s-\tau|^{-3/2}\norm{ G_\xi(\tau)}_{L^{1}} \, d \tau} \,d s \nonumber \\ &\le C\int_{t_2} ^{t-1} |t-s|^{-9/8} (t_2 s^{-3/2} n^3) \,d s + C\int_{t-1}^t |t-s|^{-3/4} (t_2 s^{-3/2} n^3) \,d s \label{eq:4-28A} \end{align} % The second integral in \eqref{eq:4-28A} is bounded by $C\n^3 t_2 t^{-3/2}$. The first integral, when $t \ge 2 t_2$, is bounded by \begin{align} \label{eq:4-29} &\le C\n^3 t_2 \int_{t_2}^{t/2} |t-s|^{-9/8} s^{-3/2} \, d s + C\n^3 t_2 \int_{t/2}^{t-1} |t-s|^{-9/8} s^{-3/2} \, d s \\ \nonumber &\le C\n^3 t_2 t^{-9/8} \int_{t_2}^{t/2} s^{-3/2} \, d s + C\n^3 t_2 t^{-3/2} \int_{t/2}^{t-1} |t-s|^{-9/8} \, d s \\ \nonumber &\le C \n^3 t_2 t^{-9/8} t_2^{-1/2} + C \n^3 t_2 t^{-3/2} \end{align} % On the other hand, if $t_2+1\le t \le 2 t_2$, then the first integral in \eqref{eq:4-28A} is bounded by \[ C\n^3 t_2 \int_{t/2}^{t-1} |t-s|^{-9/8} s^{-3/2} \, d s \] which is the second integral in \eqref{eq:4-29} and is bounded by $C \n^3 t_2 t^{-3/2}$. Thus, for all $t \in [t_2,T]$, \[ \norm{ \Omega_4 }_{L^2 \loc} \le C \n^3 t_2^{1/2} t^{-9/8} + C \n^3 t_2 t^{-3/2} \] Summarizing, we have \[ \norm{e^{(t-t_2)\L} \eta_{0,1}}_{L^4} \le \sum_{j=1}^4 \norm{\Omega_j}_{L^4} \le C\bket{\norm{\xi_0}_{L^{4/3}\cap L^{8/7}} + n^2 t_2^{1/2} +n^3 t_2^{5/8}}t^{-3/4} \] \[ \norm{e^{(t-t_2)\L} \eta_{0,1}}_{L^2 \loc} \le \sum_{j=1}^4 \norm{\Omega_j}_{L^2 \loc} \le C\bket{\norm{\xi_0}_{L^{4/3}\cap L^{8/7}} +n^3 t_2^{5/8}} t^{-9/8} \] % Since $t_2 \le \e^{-2}n ^{-4}$, equation \eqref{eta31B} holds if $n$ is sufficiently small. We have proved Lemma \ref{th:4-2}. \begin{thebibliography}{XX} \bibitem{BP} V.S.~Buslaev and G.S.~Perel'man, On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, 75--98, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995. \bibitem{G} M.~Grillakis: Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Comm. Pure Appl. 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