Content-Type: multipart/mixed; boundary="-------------0110302021819" This is a multi-part message in MIME format. ---------------0110302021819 Content-Type: text/plain; name="01-405.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-405.keywords" Excited states, Schr\"odinger equations, Stable direction ---------------0110302021819 Content-Type: application/x-tex; name="sd-cpde-26oct2001.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sd-cpde-26oct2001.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % VERSION October 9 2001 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amsmath, amssymb} %\usepackage{showkeys} \textheight 9.5in \topmargin -0.85in \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}{Definition}[section] \numberwithin{equation}{section} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newcommand{\R}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \renewcommand{\Re}{\mathop{\mathrm{Re}}} \renewcommand{\Im}{\mathop{\mathrm{Im}}} \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{\dist}{\mathop{\rm dist}} \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{\al}{\alpha} \newcommand{\ka}{\kappa} \newcommand{\om}{\omega} %\renewcommand{\d}{\delta} \newcommand{\la}{\lambda} \newcommand{\loc}{_{\mathrm{loc}}} \newcommand{\myremark}{{\bf Remark}\quad} \newcommand{\myproof}{\noindent {\bf Proof.}\quad} \newcommand{\myendproof}{\hspace*{\fill}{{\bf \small Q.E.D.}} \vspace{10pt}} \renewcommand{\L}{\mathcal{L}} \newcommand{\Pc}{\, \mathbf{P}\! _\mathrm{c} \, \! } \newcommand{\PcH}{\, \mathbf{P}\! _\mathrm{c} \! ^{H_0} \,} \newcommand{\PcL}{\, \mathbf{P}\! _\mathrm{c} \! ^{\L_1} \,} \newcommand{\Hc}{\, \mathtt{H} _\mathrm{c} \, \! } \newcommand{\eigen}{\mathtt{E}_1} \newcommand{\conj}{\mathtt{C}} \newcommand{\xii}{\xi_\infty} \newcommand{\pd}{\partial} \newcommand{\donothing}[1]{} \newcommand{\wbar}[1]{\overline{\rule{0pt}{2.4mm} {#1}}} \newcommand{\wt}{\widetilde} \newcommand{\vect}[1]{\begin{bmatrix} #1 \end{bmatrix}} \newcommand{\svect}[1] {\bkt{\begin{smallmatrix} #1\end{smallmatrix}}} \newcommand{\bQ}{\mathbf{Q}} \newcommand{\id}{\,\mathbf{id}\,} \newcommand{\Tmap}{{\bf \Omega}} \newcommand{\RE}{\mathop{\,\mathbf{RE}\,}} \newcommand{\evalue}{\omega_*} %{\theta} \newcommand{\evector}{\mathbf{\Phi}} \newcommand{\gs}{\wt \phi_0} \newcommand{\es}{\widehat Q_1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % main document % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip 20pt \vspace*{1in} \begin{center} {\bf \large STABLE DIRECTIONS FOR\\ EXCITED STATES OF\\ NONLINEAR SCHR\"ODINGER EQUATIONS} \bigskip {\bf Tai-Peng Tsai \ and \ Horng-Tzer Yau} \end{center} \bigskip \begin{abstract} We consider nonlinear Schr\"odinger equations in $\R^3$. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule. \end{abstract} \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 one 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}. For any solution $\psi(t)\in H^1(\R^3)$ the $L^2$-norm and the Hamiltonian % \begin{equation} \label{1-2} {\cal 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 $e_0 2e_1$. } \medskip The precise meaning of structural stability was given in \cite{G}. Roughly speaking, it means that the operator remains stable under small perturbations. Theorem A will not be directly used in this paper. As we will see later, the linearized operator around an excited state is a perturbation of $JH_1$. Thus, two different situations occurs: \medskip \hspace*{1.5cm}1. Non-resonant case: $e_0 > 2 e_1$. \quad ($e_{01}< |e_1|$). \medskip \hspace*{1.5cm}2. Resonant case: $e_0 < 2 e_1$. \quad ($e_{01}> |e_1|$). \medskip \noindent Here $e_{01} = e_1 -e_0 > 0$. In the resonant case, Theorem A says the linearized operator is in general unstable, which agrees with the physical picture. In the non-resonant case, however, the linearized operator becomes stable. The difference here is closely related to the fact that $ 2e_1-e_0$ lies in the continuum spectrum of $H_0$ only in the resonant case. In the resonant case, the unstable picture is confirmed for most data near excited states in our work \cite{TY2}. We prove that, as long as the ground state component in $\psi_0 - Q_1$ is larger than $\norm{\psi_0}^2$ times the size of the dispersive part corresponding the continuous spectrum, the solution will move away from the excited states and relax and stabilize to ground states locally. Since $\norm{\psi_0}^2$ is small, this assumption allows the dispersive part to be much larger than the ground state component. There is a small set of data where \cite{TY2} does not apply, namely, those data with ground state component in $\psi_0 - Q_1$ smaller than $\norm{\psi_0}^2$ times the size of the dispersive part. The aim of this paper is to show that this restriction is almost optimal: we will construct within this small set of initial data a ``hypersurface'' whose corresponding solutions converge to {\em excited states}. This does not contradict with the physical intuition since this hypersurface in certain sense has zero measure and can not be observed in experiments. These solutions, however, show that linear instability does not imply all solutions to be unstable. In the language of dynamical systems, {\it the excited states are one parameter family of hyperbolic fixed points and this hypersurface is contained in the stable manifold of the fixed points}. We believe that this surface is the whole stable manifold. We will also construct solutions converging to excited states in the non-resonant case, where it is expected since the linearized operator is stable. We now state our assumptions on the potential $V$: \noindent {\bf Assumption A0}: $H_0 := - \Delta + V$ acting on $L^2(\R^3)$ has two simple eigenvalues $e_00$ 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*} Assumption A1 contains some standard conditions to assure that most tools in linear Schr\"odinger operators apply. In particular, it satisfies the assumptions of \cite{Y} so that the wave operator $W_H=\lim_{t\to \infty} e^{i t H_0}e^{it\Delta}$ satisfies the $W^{k,p}$ estimates for $k\le 2$. These conditions are certainly not optimal. Let $e_{01}= e_1-e_0$ be the spectral gap of the ground state. In the resonant case $2e_{01} > |e_0|$ so that $ 2e_1-e_0$ lies in the continuum spectrum of $H_0$, we further assume \noindent {\bf Assumption A2}: For some $s_0>0$, %> C n_0^2$, % \begin{equation} \label{A:gamma0} \gamma_0 \equiv \inf_{|s| 0 . \end{equation} % Note that $\gamma_0 \ge 0$ since the expression above is quadratic. This assumption is generically true. %We shall use $0i$ to replace $\sigma i$ and the limit %$\lim_{\sigma \to 0+}$ later on. Let $Q_1=Q_{1,E_1}$ be a nonlinear excited state with $\norm{Q_{1,E_1}}_2$ small. Since $(Q_1,E_1)$ satisfies \eqref{Q.eq}, the function $\psi(t,x)=Q_1(x) e^{-iE_1t}$ is an exact solution of \eqref{Sch}. If we consider solutions $\psi(t,x)$ of \eqref{Sch} of the form \[ \psi(t,x) = \bkt{ Q_1(x) + h(t,x)} \, e^{-i E_1 t} \] with $h(t,x)$ small in a suitable sense, then $h(t,x)$ satisfies \[ \pd_t h = \L_1 h + \text{nonlinear terms} \] where $\L_1$, the {\it linearized operator around the nonlinear excited state solution} $Q_1(x) e^{-iE_1t}$, is defined by \begin{equation} \label{L1.def} \L_1 h = -i \bket{ (-\Delta + V -E_1 + 2 \la Q_1^2)\,h + \la Q_1^2 \,\wbar h\, }. \end{equation} % \begin{theorem} \label{th:1-2} Suppose $H_0 = - \Delta + V$ satisfies assumptions A0--A1. Suppose either (NR) $e_0> 2e_1$, or (R) $e_0< 2e_1$, and the assumption A2 for $\gamma_0$ holds. \noindent Then there are $n_0>0$ and $\e_0(n)>0$ defined for $n \in (0,n_0]$ such that the following holds. Let $Q_1: =Q_{1,E_1}$ be a nonlinear excited state with $\norm{Q_1}_{L^2}=n\le n_0$, and let $\L_1$ be the corresponding linearized operator. For any $\xii \in \Hc(\L_1)\cap (W^{2,1}\cap H^2)(\R^3)$ with $\norm{\xii}_{W^{2,1}\cap H^2}=\e$, $0 < \e \le \e_0(n)$, there is a solution $\psi(t,x)$ of \eqref{Sch} of the form % \[ \psi(t) = \bkt{ Q_1 + e^{t\L_1}\xi_\infty + g(t) }e^{-i E_1 t}, \qquad (t\ge 0), \] with $\norm{g(t)}_{H^2} \le C \e ^2 (1+t)^{-2}$. \end{theorem} To prove this theorem, a detailed spectral analysis of the linearized operator $\L_1$ is required. We shall classify the spectrum of $\L_1$ completely in both non-resonant and resonant cases, see Theorems \ref{th:2-1} and \ref{th:2-2}. It is well-known that the continuous spectrum $\Sigma_c$ of $\L_1$ is the same as that of $JH_1$, i.e., $\Sigma_c = \bket{si: s \in \R, |s| \ge |E_1|}$. The point spectrum of $\L_1$ is more subtle. By definition, $H_1 \phi_1 = -(E_1-e_1)\phi_1$ and $H_1 \phi_0 = -(E_1-e_0) \phi_0$, and thus the matrix operator $JH_1$ has 4 eigenvalues $\pm i(E_1-e_1)$ and $\pm i(E_1-e_0)$. In the non-resonant case, the eigenvalues of $\L_1$ are purely imaginary and are small perturbations of these eigenvalues. In the resonant case, the eigenvalues $\pm i(E_1-e_0)$ are embedded inside the continuum spectrum $\Sigma_c$. In general perturbation theory for embedded eigenvalues, they turn into resonances under self-adjoint perturbations. The operator $\L_1$ is however not a self-adjoint perturbation of $H_1$. In this case, we shall prove that {\it the embedded eigenvalues $\pm i(E_1-e_0)$ split into four eigenvalues $\pm \evalue$ and $\pm \bar \evalue$ with the real part given approximately by the Fermi golden rule} (see \cite{RS} Chap.XII.6): \[ n^4 \Im \bke{\la \phi_0 \phi_1^2, \, \frac 1{-\Delta + V +e_0 -2 e_1-0i} \,\Pc \la \phi_1^2 \phi_0 } . \] Here $n$ is the size of $Q_1$, see \eqref{gamma.main}. In particular, $e^{t \L_1}$ is {\em exponentially unstable} with the decay rate (or the blow-up rate) given approximately by the Fermi golden rule. In other words, {\it although self-adjoint perturbation turns embedded eigenvalues into resonances, the non-self adjoint perturbations given by $\L_1$ turns an embedded eigenvalue into two eigenvalues with the shifts in the real axis given to the leading order by the Fermi golden rule}. The dynamics of self-adjoint perturbation of embedded eigenvalues were studied in \cite{SW3}. In the appendix we will prove the existence of solutions vanishing locally as $t \to \infty$, independent of the number of bound states in $H_0$. Although some weaker versions of this proposition are expected, it has never been proved in current form and we include it for completeness. \begin{proposition} \label{th:1-1} Suppose $H_0 = - \Delta + V$ satisfies assumption A1. There is an $\e_0>0$ such that the following holds. For any $\xii \in \Hc(H_0)\cap (W^{2,1}\cap H^2)(\R^3)$ with $0<\norm{\xii}_{W^{2,1}\cap H^2}=\e \le \e_0$, there is a solution $\psi(t,x)$ of \eqref{Sch} of the form \[ \psi(t) = e^{-i tH_0} \xii + g(t), \qquad (t\ge 0), \] with $\norm{g(t)}_{H^2} \le C \e^2 (1+t)^{-2}$. \end{proposition} \section{Linear analysis for excited states} As mentioned in \S 1, there is a family $\bket{Q_{1,E_1}}_{E_1}$ of nonlinear excited states with the frequency $E_1$ as the parameter. They satisfy % \begin{equation} \label{Q1.eq} (-\Delta + V) Q_1 + \la |Q_1|^2 Q_1 = E_1Q_1 ~. \end{equation} % Let $Q_1=Q_{1,{E_1}}$ be a fixed nonlinear excited state with $n=\norm{Q_{1,{E_1}}}_2\le n_0 \ll 1$. The linearized operator around the nonlinear bound state solution $Q_1(x) e^{-iE_1t}$ is defined in \eqref{L1.def} \[ \L_1 h = -i \bket{ (-\Delta + V -E_1 + 2 \la Q_1^2)\,h + \la Q_1^2 \,\wbar h\, }. \] % We will study the spectral properties of $\L_1$ in this section. It properties are best understood in the complexification of $L^2(R^3,\Complex)$. \begin{definition} \label{CL2} Identify $\Complex$ with $\R^2$ and $L^2=L^2(\R^3, \Complex)$ with $L^2(\R^3, \R^2)$. Denote by $\Complex L^2=L^2(\R^3, \Complex^2)$ the complexification of $L^2(\R^3, \R^2)$. $\Complex L^2$ consists of 2-dimensional vectors whose components are in $L^2$. We have the natural embedding \[ f \in L^2 \longrightarrow \vect{ \Re f \\ \Im f} \in \Complex L^2. \] We equip $\Complex L^2$ with the natural inner product: For $f,g \in \Complex L^2$, $f=\svect{f_1\\f_2}$, $g=\svect{g_1\\g_2}$, we define % \begin{equation} \label{innerproduct} (f,g) = \int \bar f \cdot g \,dx = \int (\bar f_1 g_1 + \bar f_2 g_2)\,dx . \end{equation} % Denote by $\RE$ the operator taking the real part of functions in $\Complex L^2$. \end{definition} Recall the matrix operator $JH_1$ defined in Theorem A. Since $H_1 \phi_1 = -(E_1-e_1)\phi_1$ and $H_1 \phi_0 = -(E_1-e_0) \phi_0$, the matrix operator $JH_1$ has 4 eigenvalues $\pm i(E_1-e_1)$ and $\pm i(E_1-e_0)$ with corresponding eigenvectors % \begin{equation} \label{2-2A} \vect{\phi_1\\- i \phi_1}, \quad \vect{\phi_1\\ i \phi_1}, \quad \vect{\phi_0\\- i \phi_0}, \quad \vect{\phi_0\\ i \phi_0} . \end{equation} % Notice that % \begin{equation} \label{2-3} E_1-e_1 = O(n^2), \qquad E_1-e_0 = e_{01} + O(n^2). \end{equation} % The continuous spectrum of $JH_1$ is \begin{equation} \Sigma_c=\bket{ s i: \, s \in \R, |s| \ge |E_1|}, \end{equation} which consists of two rays on the imaginary axis. The operator $\L_1$ in its matrix form % \begin{equation} \label{3-3} \begin{bmatrix}0 & L_-\\ -L_+ &0\end{bmatrix} ,\qquad \text{with } \left\{ \begin{aligned} L_- &= - \Delta + V -E_1 + \la Q_1^2\\ L_+ &= - \Delta + V -E_1 + 3\la Q_1^2 \end{aligned} \right. \end{equation} % is a perturbation of $JH_1$. By Weyl's lemma, the continuous spectrum of $\L_1$ is also $\Sigma_c$. The eigenvalues are more complicated. In both cases ($e_{01}<|e_1|$ and $e_{01}>|e_1|$) they are near $0$ and $\pm i e_{01}$. As we shall see, in both cases $0$ is an eigenvalue of $\L_1$. The main difference between the two cases are the eigenvalues near $ ie_{01}$ and $-i e_{01}$. If $e_{01}<|e_1|$, then $ ie_{01}$ lies outside the continuous spectrum and $\L_1$ has an eigenvalue near $ie_{01}$ which is purely imaginary. On the other hand, if $e_{01}>|e_1|$, then $ ie_{01}$ lies inside the continuous spectrum. Generically it splits under perturbation and the eigenvalues of $\L_1$ near $\pm i e_{01}$ have non-zero real parts. We shall show that $L^2(\R^3,\Complex)$, as a real vector space, can be decomposed as the direct sum of three invariant subspaces % \begin{equation} \label{L2.dec} L^2(\R^3, \Complex) = S(\L_1)\oplus \eigen(\L_1) \oplus \Hc(\L_1) \end{equation} % Here $S(\L_1)$ is the generalized null space, $\eigen(\L_1)$ is a generalized eigenspaces and $\Hc(\L_1)$ corresponds to the continuous spectrum. Both $S(\L_1)$ and $\eigen(\L_1)$ are finite dimensional. Recall the Pauli matrices \[ \sigma_1 = \begin{bmatrix}0&1\\1&0\end{bmatrix},\quad \sigma_2 = \begin{bmatrix}0&-i\\i&0\end{bmatrix},\quad \sigma_3 = \begin{bmatrix}1&0\\0&-1\end{bmatrix}. \] They are self-adjoint and % \begin{equation}\sigma_1 \L_1 =\L_1^* \sigma_1, \qquad \sigma_3 \L_1 = - \L_1\sigma_3 , \end{equation} where $\L_1^* =\bkt{\begin{smallmatrix}0& - L_+\\L_-&0\end{smallmatrix}}$. Let $R_1 = \pd_{E_1} Q_{1,E_1}$. Direct differentiation of \eqref{Q1.eq} with respect to $E_1$ gives $L_+ R_1 = Q_1$. Since $L_-Q_1=0$ and $L_+ R_1 = Q_1$, we have $\L_1 \svect{0\\Q_1} = 0$ and $\L_1 \svect{R_1\\0} = -\svect{0\\Q_1}$. We will show dim$_\R S(\L_1)=2$, hence % \begin{equation} \label{S.def} S(\L_1)= \myspan_\R \bket{\vect{0\\ Q_1}, \vect{R_1\\0}} . \end{equation} % $\Hc(\L_1)$ can be characterized as % \begin{equation} \label{Hc.def} \Hc(\L_1) = \bket{ \psi \in \Complex L^2: (\sigma_1 \psi, f)=0 , \ \forall f \in S(\L_1)\oplus \eigen(\L_1) }. \end{equation} % We will use \eqref{Hc.def} as a working definition of $\Hc(\L_1)$. After we have proved the spectrum of $\L_1$ and the resolvent estimates, we will use the wave operator of $\L_1$ (see \cite{C,Y,Y2}) to show that \eqref{Hc.def} agrees with the usual definition of the continuous spectrum subspace. See \S 2.5. The space $\eigen(\L_1)$, however, has very different properties in the two cases, due to whether $\pm i (E_1-e_0)$ are embedded eigenvalues of $JH_1$. We will consider $\eigen = \eigen(\L_1)$ as a subspace of $L^2(\R^3, \R^2)$ and denote by $\Complex \eigen \subset \Complex L^2$ the complexification of $\eigen$. We will show that $\Complex \eigen$ is a direct sum of eigenspaces of $\L_1$ in $\Complex L^2$. We also have \begin{equation} \label{SEorth} (\sigma_1 f,g)=0 , \quad \forall f \in S(\L_1), \ \forall g \in \eigen(\L_1). \end{equation} We have the following two theorems for the two cases. {\bf Remark} The case $e_0 = 2e_1$: The spectral property of $\L_1$ is not clear. \begin{theorem} [Non-resonant case] \label{th:2-1} Suppose $e_0> 2e_1$, and the assumptions A0-A1 hold. Let $Q_1=Q_{1,E_1}$ be a nonlinear excited state with sufficiently small $L^2$-norm, and let $\L_1$ be defined as in \eqref{L1.def}. (1) The eigenvalues of $\L_1 $ are $0$ and $\pm \evalue $. The multiplicity of $0$ is two. The other eigenvalues are simple. Here $\evalue =i\ka$, $\ka$ is real, $\ka = e_{01} + O(n^2)$. There is no embedded eigenvalue. The bottoms of the continuous spectrum are not eigenvalue nor resonance. (2) The space $L^2= L^2(\R^3,\Complex)$, as a real vector space, can be decomposed as in \eqref{L2.dec}. Here $ S(\L_1)$ and $\Hc(\L_1)$ are given in \eqref{S.def} and \eqref{Hc.def}, respectively; $\eigen(\L_1)$ is the space corresponding to the perturbation of the eigenvalues $\pm i (E_1-e_0)$ of $JH_1$. We have the orthogonality relation \eqref{SEorth}. (3) Let $ \Complex \eigen$ denotes the complexification of $\eigen=\eigen(\L_1)$. $ \Complex \eigen$ is 2-complex-dimensional. $\eigen$ is 2-real-dimensional. We have % \begin{equation} \label{E1.def} \begin{aligned} \Complex \eigen&= \myspan_\Complex \bket{\evector,\bar \evector},\nonumber \\ \eigen&= \myspan_\R \bket{\vect{u\\0},\vect{0\\v} } \end{aligned} \end{equation} % Here $\evector = \svect{u\\ -i v}$ is an eigenfunction of $\L_1$ with eigenvalue $\evalue $. $u$ and $v$ are real-valued $L^2$-functions satisfying $L_+ u =-\ka v$, $L_- v =-\ka u$ and $(u,v)=1$. $u$ and $v$ are perturbations of $\phi_0$. $\bar \evector= \svect{u\\ i v}$ is another eigenfunction with eigenvalue $- \evalue $. We have $\L_1 \evector = \evalue \evector$, $\L_1 \bar \evector = - \evalue \bar \evector$. (4) For any function $\zeta \in \eigen(\L_1)$, there is a unique $\al \in \Complex$ so that \[ \zeta = \RE \al \evector, \] and we have $\L_1 \zeta = \RE \evalue \al \evector $, $e^{t\L_1} \zeta = \RE e^{t\evalue } \al \evector $. (5) We have the orthogonality relations in \eqref{Hc.def} and \eqref{SEorth}. Hence any $\psi \in L^2$ can be decomposed as (see \eqref{L2.dec}) % \begin{equation} \label{L2.dec2} \psi = a\vect{R_1 \\0} +b \vect{0\\ Q_1} + c\vect{ u \\ 0} + d \vect{0\\v} + \eta \end{equation} % with $\eta \in \Hc(\L_1)$, % \begin{equation} \label{3-10} \begin{gathered} a=(Q_1,R_1)^{-1}(Q_1,\Re \psi),\\ b=(Q_1,R_1)^{-1}(R_1,\Im \psi), \end{gathered} \qquad \begin{gathered} c=(u,v)^{-1}(v,\Re \psi),\\ d=(u,v)^{-1}(u,\Im \psi). \end{gathered} \end{equation} (6) Let $M_{1}\equiv \eigen(\L_1 ) \oplus \Hc(\L_1 )$. We have % \begin{equation} M_{1}\equiv \eigen(\L_1 ) \oplus \Hc(\L_1 ) = \begin{bmatrix} Q_1^\perp \\ R_1^\perp \end{bmatrix}. \end{equation} There is a constant $C_2>1$ such that, for all $\phi \in M_{1}$ and all $t\in \R$, we have % \begin{equation} \label{3-12} C_2^{-1}\norm{\phi}_{H^k} \le \norm{e^{t\L_1}\phi}_{H^k} \le C_2 \norm{\phi}_{H^k} ~, \qquad (k=1,2). \end{equation} (7) Decay estimates: For all $\eta \in \Hc(\L_1)$, for all $p \in [2,\infty]$, one has \[ \norm{e^{t\L_1}\eta}_{L^p} \le C |t|^{- 3(\frac 12 - \frac 1p)} \norm{\eta}_{L^{p'}}. \] \end{theorem} % \begin{theorem} [Resonant case] \label{th:2-2} Suppose $e_0< 2e_1$, and the assumptions A0-A2 hold. Let $Q_1=Q_{1,E_1}$ be a nonlinear excited state with sufficiently small $L^2$-norm, and let $\L_1$ be defined as in \eqref{L1.def}. (1) The eigenvalues of $\L_1$ are $0$, $\pm \evalue $ and $\pm \bar \evalue $. The multiplicity of $0$ is two. The other eigenvalues are simple. Here $\evalue = i \ka + \gamma$, $\ka, \gamma >0 $, $\ka = e_{01} + O(n^2)$, and $ \frac 34 \la^2 \gamma_0 n^4\le \gamma \le C n^4$. ($\gamma_0$ is given in \eqref{A:gamma0}). There is no embedded eigenvalue. The bottoms of the continuous spectrum are not eigenvalue nor resonance. There is an $\evalue$-eigenvector $\evector$, $\L_1 \evector = \evalue \evector$, which is of order one in $L^2$ and $\evector- \svect{\phi_0\\-i\phi_0}$ is locally small in the sense that % \begin{equation} \label{3-18} \length{(\phi , \evector- \svect{\phi_0\\-i\phi_0})} \le C n^2 \norm{\bkA{x}^{r} \phi}_{L^2}, \end{equation} % for any $r>3$. However, $\evector$ is not a perturbation of $\svect{\phi_0\\-i\phi_0}$ in $\Complex L^2$. In fact, $\evector = \svect{u \\ v}$ with $u- \phi_0$ and $v+ i\phi_0$ of order one in $L^2$, \[ u =\phi_0 - \frac { 1}{- \Delta + V- E_1 - \ka + \gamma i} \, \Pc(H_0) \la \phi_0 Q_1^2 + O(n^2) \quad \text{in }L^2, \] and $v = - L_+ u/\evalue$. Note $ -E_1 - \ka = e_0 - 2e_1 + O(n^2)$. (2) The space $L^2= L^2(\R^3,\Complex)$, as a real vector space, can be decomposed as in \eqref{L2.dec}. Here $ S(\L_1)$ and $\Hc(\L_1)$ are given in \eqref{S.def} and \eqref{Hc.def}, respectively; $\eigen(\L_1)$ is the space corresponding to the perturbation of the eigenvalues $\pm i (E_1-e_0)$ of $JH_1$. We have the orthogonality relation \eqref{SEorth}. (3) Let $ \Complex \eigen$ denotes the complexification of $\eigen=\eigen(\L_1)$. $ \Complex \eigen$ is 4-complex-dimensional. $\eigen$ is 4-real-dimensional. If we write $\evector=\vect{u_1+u_2i\\ v_1+ v_2 i}$ with $u_1,u_2, v_1, v_2$ real-valued $L^2$ functions, we have % \begin{equation} \label{3-16} \begin{aligned} \Complex \eigen&= \myspan_\Complex \bket{\evector,\bar \evector, \sigma_3 \evector, \sigma_3 \bar \evector},\\ \eigen&= \myspan_\R \bket{\vect{u_1\\0},\vect{u_2 \\0},\vect{0\\ v_1},\vect{0\\ v_2}}. \end{aligned} \end{equation} % Recall $\sigma_3 = \bkt{ \begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix}}$. The other eigenvectors are $\bar \evector$, $\sigma_3 \evector$ and $\sigma_3 \bar \evector$, % \begin{equation} \label{3-17} \L_1 \evector= \evalue \evector, \quad \L_1 \bar \evector= \bar\evalue \bar\evector, \quad \L_1 \sigma_3\evector= -\evalue (\sigma_3\evector), \quad \L_1 \sigma_3\bar\evector= -\bar \evalue (\sigma_3\bar \evector). \end{equation} (4) For any function $\zeta \in \eigen(\L_1)$, there is a unique pair $(\al,\beta) \in \Complex^2$ so that % \begin{equation} \label{zeta.dec2} \zeta = {\bf RE} \bket{\al \evector + \beta \sigma_3 \evector}, \end{equation} % and we have $\L_1 \zeta = \RE \bket{\evalue \al \evector - \evalue \beta \sigma_3 \evector} $, $e^{t\L_1} \zeta = \RE \bket{e^{t\evalue } \al \evector +e^{-t\evalue } \beta \sigma_3 \evector} $. (5) We have the orthogonality relations in \eqref{Hc.def} and \eqref{SEorth}. Moreover, $\sigma_1 \bar \evector \perp \bar \evector, \sigma_3 \evector, \sigma_3 \bar \evector$, $\sigma_1 \evector \perp \evector, \sigma_3 \evector, \sigma_3 \bar \evector$, and $\int \bar u v dx =0$, etc. % For any function $\psi \in \Complex L^2$, if we decompose % \begin{equation} \psi = a \vect{R_1\\0} + b \vect{0\\Q_1} + \al_1 \evector + \al_2 \bar \evector + \beta _1 \sigma_3 \evector + \beta_2 \sigma_3 \bar \evector + \eta \end {equation} % where $a,b,\al_1,\al_2,\beta_1,\beta_2 \in \Complex$ and $\eta \in \Hc(\L_1)$, then we have % \begin{gather} a= c_1 (\sigma_1 \svect{0\\Q_1},\psi), \qquad b= c_1 (\sigma_1 \svect{R_1\\0},\psi), \nonumber \\ \al_1 = c_2 (\sigma_1 \bar \evector,\psi), \qquad \al_2 = \bar c_2 (\sigma_1 \evector,\psi), \label{3-21} \\ \beta_1 = -c_2 (\sigma_1 \sigma_3\bar \evector,\psi), \qquad \beta_2 = - \bar c_2 (\sigma_1 \sigma_3\evector,\psi), \nonumber \end{gather} % where $c_1^{-1} = (Q_1,R_1)$ and $c_2^{-1}= (\sigma_1 \bar \evector, \evector) = \int 2 u v dx$. (Note $c_1 \la >0$.) % The statement that $\psi \in \eigen$ is equivalent to that $a,b \in \R$, $\al_1 =\al_2=\al/2$, $\beta_1 = \beta_2=\beta/2$ and $\eta$ is real. In this case, % \begin{equation} \psi = a \vect{R_1\\0} + b \vect{0\\Q_1} + {\bf RE} \bket{\al \evector + \beta \sigma_3 \evector} + \eta, \end{equation} with $a,b\in \R$, $\eta \in \Hc(\L_1)$ real, $\al, \beta \in \Complex$, and \begin{equation} \label{PalPbeta.def} \al = P_\al(\psi) \equiv 2 c_2 (\sigma_1 \bar \evector,\psi), \quad \beta = P_\beta (\psi)\equiv -2 c_2 (\sigma_1 \sigma_3\bar \evector,\psi). \end{equation} $P_\al$ and $P_\beta$ are maps from $L^2$ to $\Complex$. (6) There is a constant $C>1$ such that, for all $\eta \in \Hc(\L_{1})$ and all $t\in \R$, we have \[ C^{-1}\norm{\eta}_{H^k} \le \norm{e^{t\L_1}\eta}_{H^k} \le C \norm{\eta}_{H^k} ~, \qquad (k=1,2). \] (7) Decay estimates: For all $\eta \in \Hc(\L_1)$, for all $p \in [2,\infty]$, one has \[ \norm{e^{t\L_1}\eta}_{L^p} \le C |t|^{- 3(\frac 12 - \frac 1p)} \norm{\eta}_{L^{p'}}, \] where $C=C(n,p)$ depends on $n$. \end{theorem} \myremark (i). In (6), we restrict ourselves to $\Hc(\L_{1})$, not $M_1$ as in Theorem \ref{th:2-1}. (ii). In (3), $\evector$ is not a perturbaiton of $\svect{\phi_0\\-i \phi_0}$. Also, the $L^2$ functions $u_1$ and $u_2$ are independent of each other. So are $v_1$ and $v_2$. (iii) In (7) the constant depends on $n$ since there are eigenvalues which are very close to the continuous spectrum. Since the proof of Theorem \ref{th:2-1} is easier, we postpone it to the last subsection \S \ref{sec:pf21}. We will focus on proving Theorem \ref{th:2-2} in the following subsections. \subsection{Perturbation of embedded eigenvalues and their eigenvectors} In this subsection we study the eigenvalues of $\L_1$ near $ie_{01}$. By symmetry we get also the information near $-i e_{01}$. For our fixed nonlinear excited state $Q_1=Q_{1,E_1}$, let $H=-\Delta + V - E_1 + \la Q_1^2$. ($H$ is $L_1$ in \eqref{3-3}.) Let $ \gs$ denote a positive normalized ground state of $H$, with ground state energy $-\rho$ which is very close to $-e_{01}$. Hence the bottom of the continuous spectrum of $H$, which is close to $|e_1|$, is less than $\rho$. We have \[ HQ_1=0, \qquad H\gs = -\rho \gs. \] % \begin{equation} \label{5-1A} Q_1 = n \phi_1 + O(n^3), \qquad \gs = \phi_0 + O(n^2). \end{equation} We want to solve the eigenvalue problem $\L_1 \evector = \evalue \evector$ with $\evalue $ near $i e_{01}$. Write $\evector = \svect{u\\v}$. The problem has the form \[ \begin{bmatrix}0& H \\ -(H + 2\la Q_1^2) & 0 \end{bmatrix} \vect{u \\ v} = \evalue \vect{u \\ v} \] for some $\evalue$ near $i e_{01}$ and for some complex $L^2$-functions $u,v$. We have \[ H v = \evalue u, \qquad (H+2\la Q_1^2) u = - \evalue v. \] Thus $H (H+2\la Q_1^2) u = - \evalue^2 u$. Suppose $\evalue= i \ka + \gamma $ with $\gamma>0$. Since $\Im (- \evalue^2)<0$ and $H$ is real, it is more convenient to solve $H (H+2\la Q_1^2) \bar u = - \bar \evalue^2 \bar u$ instead. If we decompose $\bar u= a \gs + b Q_1 + h$ with $h\in \Hc(H)$, we find $b=0$ since $\bar u \in {\rm Image }\, H$. Since $a \not = 0$, we may assume $\bar u = \gs + h$. Let $A= H 2\la Q_1^2$ and $z= - \bar \evalue^2 \sim \rho^2 $. (A small $\Re \evalue >0$ corresponds to a small $\Im z > 0$.) We have \[ (H^2 + A) (\gs + h) = z (\gs + h), \] i.e., \begin{equation} \label{A-1} z \gs + z h =\rho^2\gs + A \gs + (H^2 + A) h. \end{equation} Taking projection $\Pc=\Pc(H)$, we get \[ z h=\Pc A \gs + (H^2 + \Pc A \Pc) h ~. \] If $\Im z \not = 0$, \begin{equation} \label{A-1.5} h = -(H^2 + \Pc A \Pc -z)^{-1}\Pc A \gs . \end{equation} % On the other hand, if $\Im z = 0$, then $h$ is generically not in $L^2$. We will assume $\Im z \not = 0$ in this subsection. The non-existence of eigenvalues with $\Im z = 0$ will be proved in next subsection. Taking inner product of \eqref{A-1} with $ \gs$, we get \[ z = \rho^2 + (\gs, A \gs) + \bke{\gs , A h} . \] Substituting \eqref{A-1.5}, we get \begin{equation} \label{A-2} z= \rho^2+ (\gs, A \gs) - \bke{\gs , A(H^2 + \Pc A \Pc -z)^{-1}\Pc A \gs } . \end{equation} If $A$ is self-adjoint, then the signs of the imaginary parts of the two sides of the equation are different. Thus $z$ is real and generically $h$ is not in $L^2$. In our case, $A= H 2 \la Q_1^2$ is not self-adjoint. Recall $H \gs = - \rho \gs$. Equation \eqref{A-2} becomes the following fixed point problem % \begin{equation} \label{5-5} z = f(z) \end{equation} % where \begin{multline} f(z) = \rho^2 -\rho (\gs 2\la Q_1^2 \gs) \\ + \rho \bke{\gs 2 \la Q_1^2, \, (H^2 + H \Pc 2 \la Q_1^2 \Pc -z)^{-1}H\Pc 2 \la Q_1^2 \gs }. \label{f.def} \end{multline} Let \begin{equation} \label{3-7} R(z) = (H^2 -z)^{-1}H = \frac 1{ 2(H - \sqrt z)} + \frac 1{ 2(H + \sqrt z)}, \end{equation} where $\sqrt z$ takes the branch $\Im \sqrt z >0$ if $\Im z > 0$. We can expand $f(z)$ as \begin{equation} \label{3-8} f(z) = \rho^2 -\rho(\gs 2\la Q_1^2 \gs) +\sum_{k=1}^\infty \rho 2 \la \bke{\gs Q_1, \bkt{2 \la Q_1\Pc R(z) \Pc Q_1}^k Q_1 \gs }. \end{equation} Let \begin{align*} z_0 &= \rho^2 -\rho(\gs 2 \la Q_1^2 \gs), \\ z_1 &= z_0 + 4 \rho \la^2\bke{\gs Q_1^2, \, R(z_0 + 0i)\Pc Q_1^2 \gs }. \end{align*} % We have $|z_1 - z_0|\le C n^4$ from its explicit form, (cf.~\eqref{311-1} of Lemma \ref{th:A-2} below). We also have, by \eqref{5-1A} and \eqref{A:gamma0} \[ \Im z_1 = \Im 4\rho \la^2\bke{\gs Q_1^2, \, \frac 1{2(H - 0i)} \,\Pc Q_1^2 \gs } \ge \tfrac 74 \, e_{01} \la^2 n^4 \, \gamma_0 + O(n^6) > 0 . \] Let $r_0 = \frac 14( (e_{01})^2 - |e_1|^2)$ be a length of order 1. Denote the regions % \begin{equation} \label{G.def} G = \bket{x+iy:|x-\rho^2|< r_0, \ 03$. There is a constant $C_1>0$ such that, for all $z \in G$, \begin{equation} \label{311-1} \norm{\bkA{x}^{-r} \Pc R(z) \Pc \bkA{x}^{-r}}_{(L^2,L^2)}\le C_1 , \end{equation} % \begin{equation} \label{311-2} \norm{\bkA{x}^{-r} \Pc \frac {d}{dz} R(z) \Pc \bkA{x}^{-r}}_{(L^2,L^2)} \le C_1 (\Im z)^{-1/2}. \end{equation} % Here $\Pc = \Pc(H)$. Moreover, for $w_1,w_2\in G$, %with $w_1=a_1+ i b_1$, $w_2=a_2+ i b_2$, \begin{multline} \label{311-3} \norm{\bkA{x}^{-r} \Pc [R(w_1)-R(w_2)] \Pc \bkA{x}^{-r}}_{(L^2,L^2)} %\le C_1 |b_1^{1/2}- b_2^{1/2}| + C_1 (\max (b_1,b_2))^{-1/2}|a_1-a_2|. \\ \le C_1 (\max (\Im w_1, \Im w_2))^{-1/2}|w_1-w_2|. \end{multline} \end{lemma} \myproof We have \begin{equation} R(z) = (H^2 -z)^{-1}H = \frac 1{ 2(H - \sqrt z)} + \frac 1{ 2(H + \sqrt z)} \end{equation} % Since $\frac 1{ 2(H + \sqrt z)}$ is regular in a neighborhood of $\bar G$, it is sufficient to prove the lemma with $R(z)$ replaced by $R_1(z):=(H - \sqrt z)^{-1}$. That $\norm{\bkA{x}^{-r} \Pc R_1(z) \Pc \bkA{x}^{-r}} _{(L^2,L^2)}\le C_1$ is well-known, see e.g. \cite{Agmon}, \cite{JK}. The estimate \eqref{311-2} will follow from \eqref{311-3} by taking limit. We now show \eqref{311-3} for $R_1(z)$. For any $w_1,w_2 \in G$, we have $|\sqrt {w_1} -\sqrt {w_2}| \le |w_1 -w_2|$. Write $\sqrt {w_1} = a_1 + i b_1 $ and $\sqrt {w_2} = a_2 + i b_2 $. We may assume $0 < b_1 < b_2$. Let $w_3 \in G$ be the unique number such that $\sqrt {w_3} = a_1 + i b_2 $. For any $u,v \in L^2$ with $\norm{u}_2=\norm{v}_2=1$, let $u_1 = \Pc \bkA{x}^{-r} u$, $v_1 = \Pc \bkA{x}^{-r} v$. We have $u_1,v_1 \in L^1\cap L^2(\R^3)$ and % \begin{align*} &\length{\bke{u,\bkA{x}^{-r} \Pc \bkt{R_1(w_1)-R_1(w_3)} \Pc \bkA{x}^{-r}v}} \\ &=\length{\int_0^\infty (u_1, e^{-it(H- a_1)}v_1) \, (e^{-b_1 t} - e^{-b_2 t}) \, dt} \\ &\le \int_0^\infty (1+t)^{-3/2} (e^{-b_1 t} - e^{-b_2 t}) \, dt \le C ( b_2^{1/2} - b_1^{1/2}) \le b_2^{-1/2} (b_2 - b_1). \end{align*} % Here we have used the decay estimate for $e^{-itH}$ with $H=-\Delta + V -E_1 - \la Q_1^2$, namely, % \begin{equation} \label{decayest} \norm{ e^{-itH} \Pc \phi}_{L^\infty} \le C |t|^{- 3/2} \norm{\phi}_{L^1} \end{equation} % under our Assumption A1. See \cite{JK,JSS,Y}. We also have \begin{align*} &\length{\bke{u,\bkA{x}^{-r} \Pc \bkt{R_1(w_3)-R_1(w_2)} \Pc \bkA{x}^{-r}v}} \\ &=\length{\int_0^\infty (u_1, e^{-it(H- s_2 -i b_2)}v_1) \, (e^{i(a_1-a_2) t} - 1) \, dt} \\ &\le \int_0^\infty (1+t)^{-3/2} e^{-b_2 t}\, |e^{i(a_1-a_2) t} - 1|\, dt \le C b_2^{-1/2} |a_1-a_2|. \end{align*} % Since $|a_1-a_2| + |b_1-b_2| \sim |\sqrt {w_1} -\sqrt {w_2}| \le |w_1 -w_2|$, we conclude \[ \length{\bke{u,\bkA{x}^{-r} \Pc \bkt{R_1(w_1)-R_1(w_2)} \Pc \bkA{x}^{-r}v}} \le C b_2^{-1/2} |w_1 - w_2|. \] Hence we have \eqref{311-3}. \myendproof % \begin{lemma} \label{th:A-1} Recall the regions $G$ and $D$ are defined in \eqref{G.def}--\eqref{D.def}. (1) $f(z)$ defined by \eqref{f.def} is well-defined and analytic in $G$. (2) $|f'(z)|\le C n^4 (\Im z)^{-1/2}$ in $G$ and $|f'(z)|\le 1/2$ in $D$. (3) for $w_1,w_2 \in G$, \[ |f(w_1) - f(w_2)| \le C n^4 (\max (\Im w_1, \Im w_2))^{-1/2}|w_1-w_2|. \] (4) $f(z)$ maps $D$ into $D$. \end{lemma} \myproof By \eqref{311-1}, the expansion \eqref{3-8} can be bounded by \[ |f(z)| \le C + CC_1 n^4 + C C_1^2 n^6 + \cdots \] and thus converges. Since every term in \eqref{3-8} is analytic, $f(z)$ is well-defined and analytic. We also get the estimates in (2). To prove (3), let $b=\max (\Im w_1, \Im w_2)$. Then from \eqref{311-1}--\eqref{311-3}, \[ |f(w_1)-f(w_2)|\le \sum_{k=1}^\infty C kC_1^k n^{2k+2} b^{-1/2} |w_1-w_2| \le C n^4 b^{-1/2}|w_1-w_2|. \] It remains to show (4). We first estimate $|f(z_1)-z_1|$. Write $z_1 = z_0 + a + bi$. Recall that $|a|3$. We now show that $h_*$ is bounded in $L^2$ with a bound uniform in $n$. Recall $\sqrt{z_*} = \ka + i \gamma$ with $\ka \sim e_{01}$, $\gamma > \tfrac 12 \la^2 \gamma_0 n^4$. Since $Q_1 = n\phi_1 + O(n^3)$, by expansion and \eqref{5-1A} we have % \begin{align} h_* &= - (H^2 -z_*)^{-1}H \Pc(H) 2 \la \phi_0 Q_1^2 + O(n^2) \nonumber \\ &= - (H - \sqrt {z_*})^{-1} \Pc(H) \la \phi_0 Q_1^2 + O(n^2) \nonumber \\ &= - \frac { 1}{- \Delta + V + s - \gamma i} \Pc(H_0) \la \phi_0 Q_1^2 + O(n^2), \label{5-u.dec} \end{align} where $s= - E_1 - \ka = e_0 - 2e_1 + O(n^2)$. Here we have used the fact that \[ \Pc(H) \phi= \Pc(H_0) \phi + n^2 \sum_{k=1}^N (\psi_k^*,\phi) \psi_k \] for some local functions $\phi_k$, $\phi_k^*$ of order one. We will show that the leading term on the right of \eqref{5-u.dec} is of order one in $L^2$. It follows from the same proof that $O(n^2)$ on the right is also in $L^2$ sense. Observe that, for $f(p) \in L^2$ with $\norm{f}_2 \le 1$, \begin{align*} &\int \frac 1{p^2 - s + \gamma i} f(p) \cdot \frac 1{p^2 - s - \gamma i} \bar f(p) \, dp \\ &= \int |f(p)|^2 \frac 1{(p^2 - s)^2 + \gamma^2 } \, dp \\ &\le C + C\int _{\sqrt s/2 }^{3\sqrt s/2} \frac 1{(|r-\sqrt s| + \gamma)^2} \, dr \\ &= C + 2C \int _0^{\sqrt s/2} \frac 1{(r + \gamma)^2} \, dr \le C + C/\gamma. \end{align*} % Using wave operator for $-\Delta + V$, we have similar estimates if $p^2$ is replaced by $-\Delta + V$. Since $\la \phi_0 Q_1^2=O(n^2)$, \[ (h_*,h_*) \le C n^2\gamma^{-1} n^2 \le C, \] where $C$ is independent of $n$. Since $u = \gs + \bar h= \phi_0 + \bar h + O(n^2)$, we have obtained the $u$ part of the estimates $\norm{\evector}_{L^2} \le C$ and \eqref{3-18}. The corresponding estimate for $v$ can be proved using $v = (-L_+)u/\evalue$. \subsection{Resolvent estimates} In this subsection we study the resolvent $R(w) = (w - \L_1) ^{-1}$. Note that $R(w)$ had a different meaning in the previous subsection. We will prove resolvent estimates along the continuous spectrum $\Sigma_c$ and determine all eigenvalues outside of $\Sigma_c$. %As a result we show that there is no eigenvalue other %than $0$, $\pm \evalue$ and $\pm \bar \evalue$. Let $L^2_r$ denote the weighted $L^2$ spaces for $r \in \R$: \[ L^2_r = \bket{f: (1+x^2)^{r/2} f(x) \in L^2(\R^3)}. \] We will prove the following lemma. \begin{lemma} \label{th:resolvent} Let $R(w)= (w - \L_1)^{-1}$ be the resolvent of $\L_1$. Let ${\bf B}=B(L^2_r,L^2_{-r})$, the space of bounded operators from $L^2_r$ to $L^2_{-r}$ with $r>3$. Recall $\evalue = i \ka + \gamma$. For $\tau \ge |E_1|$ we have % \begin{equation} \norm{R(i \tau \pm 0)}_{\bf B}+ \norm{R(-i \tau \pm 0)}_{\bf B} \le C (1+\tau)^{-1/2}+ C (|\tau - \ka| + n^4)^{-1}. \end{equation} % The constant $C$ is independent of $n$. We also have % \begin{equation} \norm{R^{(k)}(i \tau \pm 0)}_{\bf B}+ \norm{R^{(k)}(-i \tau \pm 0)}_{\bf B} \le C (1+\tau)^{-(1+k)/2}+ C (|\tau - \ka| + n^4)^{-1}. \end{equation} for derivatives, where $k=1,2$. \end{lemma} We first consider $R_0(w)=(w-JH_1)^{-1}$. Recall $H_1 = - \Delta + V -E_1$. Since % \begin{align}\nonumber (w-JH_1)^{-1} &= \begin{bmatrix}w&-H_1\\H_1&w\end{bmatrix}^{-1} =\frac 1{H_1^2 + w^2}\begin{bmatrix} w&H_1\\-H_1 &w\end{bmatrix} \\ \label{JH-resol} &=\frac 12 \begin{bmatrix} -i&1\\-1 &-i\end{bmatrix}(H_1-i w)^{-1} + \frac 12 \begin{bmatrix} i&1\\-1 &i\end{bmatrix}(H_1+i w)^{-1}, \end{align} % the estimates of $R_0(w)$ can be derived from that of $(H_1-iw)^{-1}$ and $(H_1+iw)^{-1}$. By assumption, the bottom of the continuous spectrum of $H_1$, $-E_1$, is not an eigenvalue nor a resonance of $H_1$. Hence $(H_1-z)^{-1}$ is uniformly bounded in ${\bf B}$ for $z$ away from $e_0-E_1$ and $e_1-E_1$, see \cite{JK}. By \eqref{JH-resol} and \eqref{2-3}, $R_0(w)$ is uniformly bounded in ${\bf B}$ for $w$ with $\dist(w,\Sigma_p)\ge n$, where $\Sigma_p=\bket{0, i e_{01}, -i e_{01} }$. Write \[ \L_1 = JH_1 + W, \qquad W=\begin{bmatrix} 0& \la Q_1^2 \\ -3\la Q_1^2 & 0 \end{bmatrix}. \] For $R(w)=(w - \L_1)^{-1}$ we have % \begin{equation} \label{3-40} R(w) = (1 - R_0(w)W)^{-1} R_0(w) = \sum_{k=0}^\infty [R_0(w)W)]^k R_0(w). \end{equation} % Since $R_0(w)$ is uniformly bounded in ${\bf B} $ for $w$ with $\dist(w,\Sigma_p)>n$, and $W$ is localized and small, \eqref{3-40} converges and $(w - \L_1)^{-1}$ is uniformly bounded in ${\bf B} $ for $w$ with $\dist(w,\Sigma_p)>n$ and we have % \begin{equation} \label{3-21N} \norm{R(w)}_B \le C \dist (w, \Sigma_p)^{-1}, \qquad (n\le \dist(w,\Sigma_p)\le 1). \end{equation} Recall $\Sigma_c=\bket{is: |s|\ge |E_1|}$ is the continuous spectrum of $JH_1$ and $\L_1$. For $w$ in the region % \begin{equation} \label{3-41} \bket{w: \dist (w,\Sigma_p)\ge n, w \not \in \Sigma_c}, \end{equation} % we have \[ \norm{R_0(w)}_{(L^2,L^2)} \le C\dist (w,\Sigma_c)^{-1}. \] By \eqref{3-40}, and because $W$ is localized and small, \begin{align*} &\norm{R(w)}_{(L^2,L^2)} \le \norm{R_0(w)}_{(L^2,L^2)} + \\ &\qquad +\sum_{k=1}^\infty C\norm{R_0(w)}_{(L^2,L^2)} \bket{C n^2 \norm{R_0(w)}_{\bf B}}^{k-1}\norm{R_0(w)}_{(L^2,L^2)} \\ & \le C\dist (w,\Sigma_c)^{-1}+ C\dist (w,\Sigma_c)^{-2}. \end{align*} % Hence $R(w)$ is uniformly bounded in $(L^2,L^2)$ in a neighborhood of $w$. In particular, there is no eigenvalue of $\L_1$ in the above region \eqref{3-41}. Note that this region includes a neighborhood of the bottom of the continuous spectrum $\Sigma_c$, $\pm i E_1$, except those in $\Sigma_c$. Hence the eigenvalues can occur only in $\bket{w: \dist (w,\Sigma_p)< n}$ or $\Sigma_c$. The circle $\bket{w:|w|=\sqrt n}$ is in the resolvent set of $\L_1$. By \cite{RS} Theorem XII.6, the Cauchy integral \[ P= \frac 1{2 \pi i} \oint_{|w|=\sqrt n} (w -\L_1)^{-1} \, d w \] gives the $L^2$-projection onto the generalized eigenspaces with eigenvalues inside the disk $\bket{w:|w|<\sqrt n}$. Moreover, the dimension of $P$ is an upper bound for the sum of the dimensions of those eigenspaces. However, since the projection $P_0=(2 \pi i)^{-1} \oint_{|w|= \sqrt n} R_0(w) \, dw$ has dimension $2$ (see \eqref{2-2A}--\eqref{2-3}), and \[ P -P_0 = \frac 1{2 \pi i} \oint_{|w|=\sqrt n} \sum_{k=1}^\infty \,[R_0(w)W]^k \, R_0(w) \, d w \] is convergent and bounded in $(L^2,L^2)$ by \begin{align*} &\le C \norm{R_0(w)}_{(L^2,L^2)} n^2 \sum_{k=0}^\infty \bke{ C n^2 \norm{R_0(w)}_{\bf B} }^k \norm{R_0(w)}_{(L^2,L^2)} \\ &\le C n^{-1/2} C n^2 n^{-1/2}= C n, \end{align*} (here we have used \eqref{3-21N}), the dimension of $P$ is also two. Since we already have two generalized eigenvectors $\svect{0\\Q_1}$ and $\svect{R_1\\0}$ with eigenvalue $0$, we have obtained all generalized eigenvectors with eigenvalues in the disk $|w|<\sqrt n$. Together with the results in \S 2.1, we have obtained all eigenvalues outside of $\Sigma_c$: $0$, $\pm \evalue$ and $\pm \wbar \evalue$. We next study $R(w) = (w - \L_1)^{-1}$ for $w$ near $\pm i e_{01}$: $|w- i e_{01}|0$, thus $- w^2$ lies in $G$ (defined in \eqref{G.def}). The other cases are similar. Let $\svect{f \\ g} \in \Complex L^2$. We want to solve the equation % \begin{equation} \label{2-48} (w - \L_1) \vect{ u \\ v} = \vect{ f \\ g}. \end{equation} We have \[ w u - H v = f, \qquad w v + (H + 2 \la Q_1^2) u = g. \] Cancelling $v$, we get (recall $A=H 2\la Q_1^2$) \[ w^2 u + (H^2 + A)u = F, \qquad F = w f + H g. \] Write $u = \al \gs + \beta \es + \eta$ with $\eta \in \Hc(H)$ and $\es =Q_1/\norm{Q_1}_2$. Also denote $\zeta =\al \gs + \beta \es$. We have \begin{align*} \bke{w^2 + H^2 + \Pc A} \eta &= \Pc F - \Pc A \zeta, \\ \bke{w^2 + H^2 + {\bf P}^\perp A} \zeta &= {\bf P}^\perp F - {\bf P}^\perp A \eta. \end{align*} Here $\Pc =\Pc(H)$ and ${\bf P}^\perp = 1 - \Pc$. Solving $\eta$ in terms of $\zeta$, we get % \begin{equation} \label{5-19} \eta =\bke{w^2 + H^2 + \Pc A \Pc}^{-1} (\Pc F - \Pc A \zeta). \end{equation} % Substituting the above into the $\zeta$ equation we get % \begin{equation} \label{5-20} \bke{w^2 + H^2 + {\bf P}^\perp A - {\bf P}^\perp A \bke{w^2 + H^2 + \Pc A \Pc }^{-1}\Pc A} \zeta = \wt F , \end{equation} % \[ \wt F= {\bf P}^\perp F - {\bf P}^\perp A \bke{w^2 + H^2 + \Pc A \Pc}^{-1} \Pc F. \] Using $\gs$ and $\es$ as basis, we can put \eqref{5-20} into matrix form % \begin{equation} \label{5-21} \begin{bmatrix} a & b \\ 0 & w^2 \end{bmatrix} \vect{\al \\ \beta} = \vect{(\gs, \wt F) \\ (\es, \wt F)}, \end{equation} % where (recall $H\gs = - \rho \gs$, $H \es = 0$) \[ a=w^2 + \rho^2 - \rho(\gs 2 \la Q_1^2 \gs) + \rho (\gs 2 \la Q^2, \bke{w^2 + H^2 + \Pc A \Pc }^{-1} H \Pc 2 \la Q_1^2 \gs), \] \[ b =- \rho(\gs 2 \la Q_1^2 \es) + \rho (\gs 2 \la Q^2, \bke{w^2 + H^2 + \Pc A \Pc }^{-1} H \Pc 2 \la Q_1^2 \es). \] Thus \begin{equation} \vect{\al \\ \beta} = \begin{bmatrix} 1/a & -b/(aw^2) \\ 0 & w^{-2} \end{bmatrix} \vect{(\gs, \wt F) \\ (\es, \wt F)}. \end{equation} \bigskip We now consider the case when $w$ is near the continuous spectrum $\Sigma_c$. We will assume $w= i\tau - \e$ with $|\tau - e_{01}|0$ much smaller. The case $w= i\tau + \e$ follows similarly. Let $z = - w^2$. It follows that $z\in G$ and $\Re z>0$ is small. The idea of what follows is to compare $z$ with $z_*$, the fixed point found in \S 2.1. We have $a=- z + f(z)= (z_* -z) + (-f(z_*) + f(z))$. Using Lemma \ref{th:A-1} (3) with $w_1=z$ and $w_2=z_*$, we have \[ |a|\ge |z-z_*| - |-f(z_*) + f(z)| \ge \tfrac 12 |z-z_*|= \tfrac 12 |w^2-\bar \evalue ^2| \ge C |w- \bar \evalue|. \] Recall $\evalue = i \ka + \gamma$ with $\gamma \sim n^4$. Hence $|a|\ge C (|\tau - \ka| + n^4)$. Thus \[ |\al|+ |\beta| \le C(1 + |a|^{-1}) \| \wt F\|_{L^2} \le C(|\tau - \ka| + n^4)^{-1} \bke{\norm{ f}_{L^2}+ \norm{g}_{L^2}}. \] By \eqref{5-19} and $F=wf + H g$, \[ \eta = \Omega w \Pc f + \Omega H \Pc g - \Omega w \Pc A \zeta, \] % where $\Omega =\bke{w^2 + H^2 + \Pc A \Pc}^{-1}$. Substituting the above into \eqref{5-19}, we can solve $\eta$ and we have \[ \norm{\eta}_{L^2} \le C \bke{\norm{ f}_{L^2}+ \norm{g}_{L^2}} + C n^2 (|\tau - \ka| + n^4)^{-1}\bke{\norm{ f}_{L^2}+ \norm{g}_{L^2}}. \] We conclude, for $u = \al \gs + \beta \es + \eta$, \[ \norm{u}_{L^2} \le \bke{ C + C (|\tau - \ka| + n^4)^{-1}} \bke{\norm{ f}_{L^2}+ \norm{g}_{L^2}}. \] We can estimate $v$ similarly. Thus, for $\tau \in (e_{01}-n, e_{01}+ n$), \[ %be \label{res-est} \norm{R(i \tau \pm 0)}_{\bf B} \le C + C (|\tau - \ka| + n^4)^{-1}. \] For $\tau > e_{01}+ n $ and $w=i\tau + 0$, using $R(w)= (1 + R_0(w ) W)^{-1} R_0(w)$ and the fact that $\norm{R_0(w)}_{\bf B} \le C (1+\tau)^{-1/2}$, (see \cite{JK} Theorem 9.2), we have $\norm{R(i\tau + 0)}_{\bf B} \le C \tau^{-1/2}$. For $\tau \in [|E_1|, e_{01}-n]$, by the same argument we have $\norm{R(i\tau + 0)}_{\bf B} \le C$. The derivative estimates for the resolvent is obtained by induction argument and by differentiating the relation $R(1+W R_0)=R_0$ and using the relations $(1+W R_0)^{-1} = 1- WR$ and $(1+ R_0 W)^{-1} = 1- RW$. See the proof of \cite{JK} Theorem 9.2. We have proved Lemma \ref{th:resolvent}. \subsection{Nonexistence of generalized $\evalue$-eigenvector} We now show that $\evalue$ is simple and $\evector$ is the only generalized $\evalue$-eigenvector, i.e., there is no vectors $\phi$ with $(\L_1 - \evalue) \phi\not =0$ but $(\L_1 - \evalue)^k \phi=0$ for some $k\ge 2$. Suppose the contrary, then we may find a vector $\vect{u\\v} $ with $(\evalue - \L_1) \vect{u\\v} =\vect{u_* \\ v_*}$. That is, $w=\evalue$ and $\vect{f\\g} = \vect{u_* \\ v_*}$ in the system \eqref{2-48}. We have $F= w u_* + H v_* = 2 \evalue u_*$. Since $u_* = \gs + \bar h_*$ with $\bar h_* \in \Hc(H)$, we have $(\es,\wt F)= (\es,F) = (\es,2\evalue u_*)=0$. Hence $\beta=0$. Also % \begin{align*} (\gs,\wt F)&=(\gs,F) - (\gs H 2\la Q_1^2 \bke{w^2 + H^2 + \Pc A \Pc}^{-1} \Pc F) \\ &= 2 \evalue + \rho (\gs 2 \la Q_1^2\bke{w^2 + H^2 + \Pc A \Pc}^{-1} 2 \evalue \bar h_*) \\ &= 2\evalue \bkt{ 1 +\rho (\Phi, \Omega \wbar \Omega H \Phi)}, \end{align*} % where $\Omega =\bke{w^2 + H^2 + \Pc A \Pc}^{-1}$ and $\Phi=\Pc \gs 2 \la Q_1^2$. Since the main term in $(\Phi, \Omega \wbar \Omega H \Phi)$, \[ (\Phi, (w^2 + H^2 )^{-1} (\bar w^2 + H^2)^{-1} H \Phi), \] is positive, $(\gs,\wt F)$ is not zero. On the other hand, $a= \evalue^2 + f(- \evalue^2) = -\bar z_* + f(\bar z_*)=0$. Hence there is no solution for $\al$. This shows $\evalue$ is simple (and so are $-\evalue, \pm \bar \evalue$). Once we have an eigenvector $\evector$ with $\L_1 \evector = \evalue \evector$ and $\evalue $ complex, then we have three other eigenvalues and eigenvectors as given in \eqref{3-17}. Hence we have found all eigenvalues and eigenvectors of $\L_1$. $\Complex \eigen$ is the combined eigenspace of $\pm \evalue$ and $\pm \bar \evalue$. It is easy to check that $\RE \Complex \eigen = \eigen$. We have proved parts (1)--(3) of Theorem \ref{th:2-2} in \S 2.1 to \S 2.3. \subsection{Nonexistence of embedded eigenvalues} \label{sec:embedded} In this subsection we prove that there is no embedded eigenvalue $i \tau$ with $|\tau|>|E_1|$. Suppose the contrary, we may assume $\tau > - E_1>0$ and $\L_1 \psi = i \tau \psi$ for some $\psi \in \Complex L^2$. We will derive a contradiction. Let $H_* = - \Delta - E_1$. We can decompose % \begin{equation} \label{L-Hstar} \L_1 = JH_* + A, \qquad A= \begin{bmatrix} 0 & V + \la Q_1^2 \\ - V - 3 \la Q_1^2 & 0 \end{bmatrix}. \end{equation} % Hence $(i \tau - JH_*)\psi = A \psi$. By the same computation of \eqref{JH-resol} we have % \[ (w -JH_*)^{-1}=(H_*-i w)^{-1}M_+ + (H_*+ i w)^{-1} M_-,\quad \] where \[ M_+=\frac 12 \begin{bmatrix} -i&1\\-1 &-i\end{bmatrix} , \qquad M_-= \frac 12 \begin{bmatrix} i&1\\-1 &i\end{bmatrix}. \] % Thus, with $w = i\tau$, we have % \begin{equation} \label{2-55} \psi =(i \tau - JH_*)^{-1} A \psi = (H_*+\tau)^{-1}\phi_+ + (H_*-\tau)^{-1} \phi_- , \end{equation} % where $\phi_+ =M_+A \psi $ and $\phi_- =M_- A \psi $. By Assumption A1 on the decay of $V$ and that $\psi \in L^2$, both $\phi_+,\phi_- \in L^2_{5+\sigma}$ with $\sigma>0$. Since $- \tau$ is outside the spectrum of $H_*$, we have $(H_*+\tau)^{-1}\phi_+ \in L^2_{5+\sigma}$. Let $s=E_1 + \tau>0$. We have $H_*-\tau = -\Delta - s$. By assumption $\psi \in \Complex L^2$, hence so is $(H_*-\tau)^{-1} \phi_- $. Therefore $(p^2 - s)^{-1} \widehat{\phi_-}(p)\in L^2$. Since $\phi_- \in L^2_{5+\sigma}$, $\phi_-$ is continuous and we can conclude % \begin{equation} \label{Restrict0} \widehat{\phi_-}(p)\big|_{|p|=\sqrt s} = 0. \end{equation} % We now recall \cite{RS2} page 82, Theorem IX.41: Suppose $f \in L^2 _r$ with $r>1/2$ and let $B_s f =\bke{(p^2 - s)^{-1} \widehat{f}} ^\vee $. Suppose $\hat f (p)\big|_{|p|=\sqrt s } = 0$. Then for any $\e>0$, one has $B_s f \in L^2_{r - 1 -2 \e}$ and $\norm{B_s f }_{ L^2_{r - 1 - 2 \e}}\le C_{r,\e,s} \norm{f}_{L^2_r}$ for some constant $C_{r,\e,s}$. In our case, we have $f=\phi_-$, $\e= \sigma/2$ and $r=5+\sigma$. We conclude $(H_*-\tau)^{-1} \phi_- =B_s f \in L^2_{4}$. Thus $\psi \in L^2_{4}$. However, since $(z-\L_1)\psi = (z-i\tau)\psi$, we have $R(z) \psi = (z-i\tau)^{-1} \psi$. Thus we have \[ \norm{(z-i\tau)^{-1} \psi}_{L^2_{-r}} \le C \norm{ \psi}_{L^2_{4}}, \] where the constant $C$ remains bounded as $z \to i \tau$ by Lemma \ref{th:resolvent}. This is clearly a contradiction. Thus $\psi$ does not exist. \subsection{Absence of eigenvector and resonance at bottom of continuous spectrum} In this subsection we show that there is no eigenvector and resonance at $\pm i E_1$. We want to show that, for $n=\norm{Q_{1,E_1}}_{L^2}$ sufficiently small, the null space of $\L_1 \pm i E_1$ in $X = L^2_{-r}$, $ r> 1/2$, is zero. Let us consider the case at $i|E_1|$. Suppose otherwise, we have a sequence $Q_{1,E_1(k)} \to 0$ and $\psi _k$ so that \[ \bke{\L_{1,E_1(k)} + i E_1(k)}\, \psi_k = 0, \quad \norm{\psi_k}_X =1. \] % As in the previous subsection, we write $\L_{1,E_1(k)} = JH_* + A_k$, where $H_*= -\Delta -E_1(k)$ and $A_k = JV + \begin{bmatrix} 0 & 1 \\ -3 & 0\end{bmatrix} \la Q_{1,E_1(k)}^2$. By \eqref{2-55} with $\tau = |E_1(k)|$ we have % \[ \psi_k =(i \tau - JH_*)^{-1} A_k \psi_k = (- \Delta + 2\tau)^{-1}M_+A_k \psi_k + (- \Delta)^{-1} M_- A_k \psi_k \] in $X$. Note that $(- \Delta + 2\tau)^{-1}M_+A_k$ and $(- \Delta) ^{-1} M_- A_k$ are compact operators in $X$, with a bound uniform in $k$. Since $X$ is a reflexive Banach space, we can find a subsequence, which we still denote by $\psi_k$, converging weakly to some $\psi_* \in X$. Thus $\tau \to |e_1|$, $(- \Delta + 2\tau)^{-1}M_+A_k \psi_k \to (- \Delta - 2e_1)^{-1}M_+JV \psi_*$ and $ (- \Delta)^{-1} M_- A_k \psi_k \to (- \Delta)^{-1}M_+JV \psi_*$ strongly in X. Thus % \[ \psi_* =(- \Delta - 2e_1)^{-1}M_+JV \psi_* + (- \Delta)^{-1}M_+JV \psi_* \] and $\psi_k \to \psi_*$ strongly. Hence $\norm{\psi_*}_X = \lim \norm{\psi_k}_X = 1$ and $(JH_1 + i e_1)\psi_*=0$ by \eqref{2-55} again. This contradiction to our assumption shows our claim. \bigskip {\bf Another proof:} We will use the resolvent estimates Lemma \ref{th:resolvent} to give a proof, without using that $\L_1$ is a perturbation of $JH_1$. Suppose the contrary that we have $\psi \in X$ which satisfies $\psi \not = 0$, $\L_1 \psi = i \tau \psi$, with $\tau = |E_1|$. Write $\L_1 = JH_*+A$ as before and let $R(z) = (z - \L_1)^{-1}$ and $R_0(z) = (z - JH_*)^{-1}$. We have $(i \tau - JH_*)\psi = A\psi$, hence % \begin{equation} \label{2-R0A} \psi = R_0(i \tau) A\psi \qquad \text{in } X. \end{equation} Let $w= \sigma_1 A \psi = A^* \sigma_1 \psi$. (Note $A^* = \sigma_1 A \sigma_1$). We have that $Vw \in L^2_{r}$. By Lemma \ref{th:resolvent} the ${L^2_{-r}}$-norm of $R(z) Vw $ is uniformly bounded as $z \to i \tau$. We will derive a contradiction. Recall the resolvent identity $R(z) A = - 1 - (1 - R_0(z)A)^{-1}$. Hence $ \rho(z) \equiv (1 - R_0(z)A)^{-1} w = R(z) A w + w$ is also uniformly bounded in $L^2_{-r}$ as $z \to i \tau$. Recall \cite{JK} Lemma 2.3 that $(- \Delta-z)^{-1} = (-\Delta)^{-1} + O(\sqrt{|z|})$ in ${\cal B} \bke{H^{-1}_{s}, H^1_{-s}}$ for $s>1$. Hence for $z$ near $i \tau$ we have by \eqref{2-55} \[ R_0(z) = R_0( i \tau) + O(\sqrt{|z-i\tau|}) \] in ${\cal B}\bke{H^{-1}_{s},H^1_{-s}}$. Therefore % \begin{align*} (w,w)&=(A^*\sigma_1 \psi,\ (1-R_0(z)A) \rho(z)) \\ &=(A^*\sigma_1 \psi, \ [1 - R_0(i \tau) A] \rho(z)) + (A^*\sigma_1 \psi, O(\sqrt{|z-i\tau|})A\rho(z)). \end{align*} % Since $\sigma_1 J = - J \sigma_1$, $\sigma_1 (z-JH_*) = (z +JH_*) \sigma_1 = (z -JH_*)^* \sigma_1$. Hence $\sigma_1 R_0(i \tau) = R_0(i \tau)^* \sigma_1$ and we have by \eqref{2-R0A} % \[ \sigma_1 \psi = \sigma_1 R_0(i \tau) A\psi= R_0(i \tau)^* \sigma_1 A\psi= R_0(i \tau)^* A^*\sigma_1\psi. \] Hence \[ (A^*\sigma_1 \psi, \ R_0(i \tau) A \rho(z)) = ( A^* R_0(i \tau)^* A^*\sigma_1 \psi, \ \rho(z)) = ( A^* \sigma_1 \psi, \ \rho(z)). \] Hence $(A^*\sigma_1 \psi, \ [1 - R_0(i \tau) A] \rho(z))=0$. Since $\rho(z)$ is uniformly bounded in $L^2_r$, we have \[ (w,w) =(A^*\sigma_1 \psi, O(\sqrt{|z-i\tau|})A\rho(z))= O(\sqrt{|z-i\tau|}) \] as $ z \to i \tau$. Thus $w=0$. Hence $(i \tau - JH_*)\psi =0$. If we write $\psi = \svect{u\\v}$, then \[ i \tau u - (-\Delta + \tau)v =0, \quad (-\Delta + \tau) u +i \tau v =0. \] One gets $\Delta (u -i v)=0$ immediately. Since $u,v\in X$, we conclude $u=iv$ and $(-\Delta + 2\tau)u =0$. Hence $u,v=0$. This finishes the proof. \subsection{Proof of Theorem \ref{th:2-2} (4)--(6)} We first show the orthogonality conditions. Recall $\sigma_1 = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$. It is self-adjoint in $\Complex L^2$. Let $\L_1^*$ be the adjoint of $\L_1$ in $\Complex L^2$. We have $\L_1 ^*= \begin{bmatrix}0 & -L_+ \\ L_- & 0\end{bmatrix}$ and $\L_1 ^*=\sigma_1 \L_1 \sigma_1$. Suppose $\L_1 f = \om _1 f$ and $\L_1 g = \om _2 g$ with $ \bar\om _1 \not = \om _2$. We have $\L_1^* \sigma_1 f =\sigma_1 \L_1 f= \om _1 \sigma_1 f$ and hence \[ \om _2 (\sigma_1 f, g) = (\sigma_1 f, \om _2 g) = (\sigma_1 f, \L_1 g) = (\L_1^*\sigma_1 f, g) = (\om _1 \sigma_1 f, g)= \bar \om _1( \sigma_1 f, g). \] Hence we must have $(\sigma_1 f, g) =0$. Therefore we have $\sigma_1 \bar \evector \perp \bar \evector, \sigma_3 \evector, \sigma_3 \bar \evector$, $\sigma_1 \evector \perp \evector, \sigma_3 \evector, \sigma_3 \bar \evector$, etc. If we write $u=u_1 + i u_2$, $v=v_1+i v_2$ and $\evector= \svect{u \\v}$, then we have % \begin{equation} \label{3-24} \int \bar u v \,d x = 0. \end{equation} % In other words, $(u_1,v_1)+(u_2,v_2)=0$ and $(u_1,v_2)= (u_2,v_1)$. If $f \in S(\L_1)$ and $\L_1 g = \om _2 g$ with $\om _2 \not = 0$. We have $(\L_1^*)^2\sigma_1 f = 0 $, hence \[ (\sigma_1 f, \om _2^2 g) = (\sigma_1 f, \L_1^2 g) = ((\L_1^*)^2\sigma_1 f, g) = (0, g). \] Hence $(\sigma_1 f, g)=0$. In terms of components, we get $(Q_1,u_1)=(Q_1,u_2)=0$, $(R_1,v_1)=(R_1,v_2)=0$. The above shows \eqref{3-21}. The rest of (4) and (5) follows directly. To prove (6), we first prove the following spectral gap % \begin{equation} \label{gap2} L_+|_{\bket{Q_1,v_1,v_2}^\perp} > \tfrac 12 |e_1| ~, \qquad L_-|_{\bket{R_1,u_1,u_2}^\perp} > \tfrac 12 |e_1| ~. \end{equation} % We will show the first assertion. Note that by \eqref{3-18} we have \[ v_1 = \Pc(L_-) v_1 + O(n^2), \qquad v_2 = - \phi_0 + \Pc(H_1) v_2 + O(n^2) \] in $L^2$. In particular $\norm{v_2}_{L^2} \ge 1/2$, and $ (v_1,L_- v_1)\ge ( v_1, L_- \Pc (L_-)v_1) - C n^2\ge - Cn^2$. By \eqref{3-24} \[ (v_1,L_- v_1) + (v_2,L_- v_2) =(v, L_-v )= (v,\om u) =0. \] % Hence $(v_2,L_+v_2) = (v_2, L_-v_2 ) + O(n^2) \le C n^2 $. We also have $(Q_1,L_+ Q_1)=(Q_1,L_- Q_1) + O(n^4) =0+ O(n^4)$. Let $Q_1'= Q_1 - (Q_1,v_2)v_2 /\norm{v_2}_2^2$. We have $Q_1' \perp v_j$ and $Q_1'= Q_1 + O(n^3)$ by \eqref{3-18} again. Hence $(Q_1',L_+ Q_1')=(Q_1,L_+ Q_1)+O(n^4)=O(n^4)\le C n^2 (Q_1',Q_1')$. We conclude that $L_+ |_{\myspan \bket{Q_1,v_2}} \le C n^2$. Since $L_+$ is a perturbation of $H_1$, it has exactly two eigenvalues below $\tfrac 12 |e_1|$. By minimax principle we have $L_+|_{\bket{Q_1,v_2}^\perp} > \tfrac 12 |e_1|$. This shows the first assertion of \eqref{gap2}. The second assertion is proved similarly. Let $\bQ(\psi)$ denote the quadratic form: (see e.g. \cite{W1,W2}) % \begin{equation} \label{bQ.def} \bQ(\psi)=( f, L_+ f) + ( g, L_- g)~, \qquad \text{if } \psi=f + i g ~. \end{equation} One can show for any $\psi \in L^2$ % \begin{equation} \label{3-13} \bQ(e^{t\L_1}\psi)=\mathbf{Q}(\psi)~, \qquad \text{for all } t, \end{equation} % by direct differentiation in $t$. By \eqref{gap2} one has % \[ \bQ(\eta) \sim \norm{\eta}_{H^1}^2 ~, \qquad \text{for any } \eta \in \Hc(\L_1 ) . \] % Thus \[ \norm{e^{t\L_1}\eta}_{H^1}^2 \sim \bQ(e^{t\L_1}\eta)= \bQ(\eta)\sim \norm{e^{t\L_1}\eta}_{H^1}^2. \] This shows (6). \subsection{Wave operator and decay estimate} It remains to prove the decay estimate (7). We will use the wave operator. We will compare $\L_1$ with $J H_*$, where $H_* =-\Delta - E_1$. Recall we write $\L_1 = JH_* + A$ in \S \ref{sec:embedded}, \eqref{L-Hstar}. % Keep in mind that $H_*$ has no bound states and $A$ is local. Define $W_+ = \lim_{t \to +\infty} e^{-t\L_1}e^{tJH_*}$. Let $R(z)=(z - \L_1)^{-1}$ and $R_*(z) = (z -JH_*)^{-1}$. We have % \begin{align*} &W_+ f -f \\ &= \lim_{\e \to 0+} \int_{|E_1|}^{+\infty} R(i\tau + \e) A \bkt{R_*(i\tau -\e)- R_*(i \tau + \e) } f \, d \tau %\label{3-30} \\ & - \lim_{\e \to 0+} \int_{|E_1|}^{+\infty} R(-i\tau + \e) A \bkt{R_*(-i\tau -\e)- R_*(-i \tau + \e) } f \, d \tau. %\label{3-31} \end{align*} Yajima \cite{Y,Y2} was the first to give a general method for proving the $(W^{k,p},W^{k,p})$ estimates for the wave operators of self-adjoint operators. This method was extended by Cuccagna \cite{C} to non-selfadjoint operators in the form we are considering. (He also used idea from Kato \cite{K}). One key ingredient in this approach is the resolvent estimates near the continuous spectrum, which in many cases can be obtained by the Jensen-Kato \cite{JK} method. (See \cite{Y} Lemmas 3.1 and 3.2 and \cite{C} Lemmas 3.9 and 3.10). % In our current setting, this estimate is provided by the Lemma \ref{th:resolvent}. We can thus follow the proof of \cite{C} to obtain that $W_+ $ is an operator from $\Complex L^2$ onto $\Hc(\L_1)$. Furthermore, $W_+$ and its inverse (restricted to $\Hc(\L_1)$) are bounded in $(L^p,L^p)$-norm for any $p\in [1,\infty]$. (Note this bound depends on $n$ since our bound on $R(w)$ depends on $n$.) By the intertwining property of the wave operator we have \[ e^{t\L_1} \Pc = W_+ e^{t JH_*} (W_+)^*\Pc. \] % The decay estimate in (7) follows from the decay estimate of $e^{t JH_*}$. % The proof of Theorem \ref{th:2-2} is complete. \donothing{ Since \eqref{3-30} and \eqref{3-31} are similar, we will only estimate \eqref{3-30}. Following \cite{Y}, we decompose \eqref{3-30} as $W_1f + W_2f$, where \begin{align*} W_1 f&=\lim_{\e \to 0+} \int_{|E_1|}^{+\infty} R_*(i\tau + \e) A \bkt{R_*(i\tau -\e)- R_*(i \tau + \e) } f \, d \tau, \\ W_2 f& =\lim_{\e \to 0+} \int_{|E_1|}^{+\infty} R_*(i\tau + \e) A R(i\tau + \e) A \bkt{R_*(i\tau -\e)- R_*(i \tau + \e) } f \, d \tau. \end{align*} The first term has an explicit form and can be estimated in $B(L^p,L^p)$ as in \cite{Y}. The second term can be estimated directly using our previous estimates of $R(w)$: \begin{align*} \norm{W_2}_{(L^p,L^p)} &\le \int _{|E_1|}^{+\infty} \norm{ R_*(i\tau + \e)}_{(L^p,L^p)} \norm{ R(i\tau + \e)}_{(L^2,L^2)} \\ &\qquad \qquad \qquad \cdot \norm{ R_*(i\tau -\e)- R_*(i \tau + \e) }_{(L^p,L^p)} \, d \tau \\ &\le \end{align*} } \subsection{Proof of Theorem \ref{th:2-1}} \label{sec:pf21} By the same Cauchy integral argument as in Subsection 2.2, the only eigenvalues of $\L_1$ are inside the disks $\bket{w:|w|< \sqrt n}$, $\bket{w:|w-ie_{01}|< \sqrt n}$ and $\bket{w:|w+ie_{01}|< \sqrt n}$. Moreover, their dimensions are $2$, $1$ and $1$, respectively, the same as that of $JH_1$. It counts the dimension of (generalized) eigenspaces of $\L_1$ in $\Complex L^2$. It also counts the dimensions of the restriction of these spaces in $L^2=L^2(\R^3,\R^2)$ as a real-valued vector space. By \eqref{S.def}, we already have two generalized eigenvectors near $0$. Hence we have everything near $0$. Since the dimension is 1 near $i e_{01}$, there is only a simple eigenvalue $\evalue$ near $i e_{01}$. We have $\evalue = i e_{01}+O(n^2)$ since the difference between $\L_1$ and $JH_1$ is of order $O(n^2)$. $\evalue$ has to be purely imaginary, otherwise $- \bar \evalue$ is another eigenvalue near $ie_{01}$, cf.~\eqref{3-17}, and the dimension can not be 1. (This also follows from the Theorem of Grillakis.) By the same arguments in \S 2.2-2.4 we can prove resolvent estimates and the non-existence of embedded eigenvalues. Also, the bottoms of the continuous spectrum are not eigenvalue nor resonance. Let $\evector$ be an eigenvector corresponding to $\evalue $. Since $\L_1 \evector = \evalue \evector$ and $\bar \evalue = -\evalue $, we have $\L_1 \bar \evector = -\evalue \bar \evector$. Hence the (unique) eigenvalue near $-ie_{01}$ is $-\evalue $ with eigenvector $\bar \evector$. Write $\evector=\svect{u\\ -iv}$. We may assume $u$ is real. Writing out $\L_1 \evector = i \ka \evector$ we get $L_- v = -\ka u$ and $L_+ u = -\ka v$. Hence $v$ is also real. We can normalize $u$ so that $(u,v)= 1$ or $-1$. Since $\evector$ is a perturbation of $\svect{\phi_0\\ -i\phi_0}$, we have $(u,v)= 1$. With this choice of $u,v$, let $\Complex \eigen$ and $\eigen$ be defined as in \eqref{E1.def}. $\Complex \eigen$ is the combined eigenspace corresponding to $\pm \evalue $. Clearly $\RE \Complex \eigen \subset \eigen$. Since \[ a\vect{u\\0}+b\vect{0\\v} = \RE \al \evector , \qquad \al=a +b i, \] we have $\RE \Complex \eigen =\eigen$. That the choice of $\al$ is unique can be checked directly. The statement that if $\zeta =\RE \al \evector $ then $\L_1 \zeta = \RE \evalue \al \evector $ and $e^{t\L_1} \zeta = \RE e^{t\evalue } \al \evector $ is clear. We have proved (3) and (4). Clearly, $S(\L_1)$, $\eigen(\L_1)$ and $\Hc(\L_1)$ defined as in \eqref{S.def}, \eqref{E1.def} and \eqref{Hc.def} are invariant subspaces of $L^2$ under $\L_1$, and we have the decomposition \eqref{L2.dec}. This is (2). For (5), note that \eqref{Hc.def} is by definition. For \eqref{SEorth}, we have \begin{gather*} (Q_1,u) = (Q_1, (-\ka)^{-1}L_- v) = (L_- Q_1, (-\ka)^{-1} v)=0,\\ (R_1,v) = (R_1, (-\ka)^{-1}L_+ u) = (-\ka)^{-1}(L_+ R_1, u)= (-\ka)^{-1} (Q_1, u) =0. \end{gather*} \eqref{3-10} comes from the orthogonal relations directly. The first statement of (6) is because of (5). For the rest of (6), We first prove the following spectral gap % \begin{equation} \label{gap1} L_+|_{\bket{Q_1,v}^\perp} > \tfrac 12 |e_1| ~, \qquad L_-|_{\bket{R_1,u}^\perp} > \tfrac 12 |e_1| ~. \end{equation} % Since $L_+$ is a perturbation of $H_1$, it has exactly two eigenvalues below $\tfrac 12 |e_1|$. Notice that $(Q_1, L_+Q_1) = (Q_1L_-Q_1)+O(n^4) = O(n^4)$ and $(v,L_+v) = (v,-\ka u) = - \ka$. Since $Q_1=n\phi_1 + O(n^3)$ and $v = \phi_0 + O(n^2)$, one has $(Q_1,v)=O(n^3)$. Thus one can show $L_+|_{\myspan \bket{Q_1,v}} \le C n^2$. If there is a $\phi \perp Q_1,v$ with $(\phi,L_+\phi) \le \tfrac 12 |e_1|(\phi,\phi)$, then we have $L_+|_{\myspan \bket{Q_1,v,\phi}} \le \tfrac 12 |e_1|$, which contradicts with the fact that $L_+$ has exactly two eigenvalues below $\tfrac 12 |e_1|$ by minimax principle. This shows the first part of \eqref{gap1}. The second part is proved similarly. Recall the quadratic form $\bQ(\psi)$ defined in \eqref{bQ.def} in \S 2.4. Also recall \eqref{3-13} that $\bQ(e^{t\L_1}\psi)= \mathbf{Q}(\psi)$ for all $t$ and all $\psi \in L^2$. By the spectral gap \eqref{gap1} one has % \begin{equation} \label{3-14} \bQ(\eta) \sim \norm{\eta}_{H^1}^2 ~, \qquad \text{for any } \eta \in \Hc(\L_1 ) ~. \end{equation} % For $\psi \in M_1$, we can write $\psi=\zeta +\eta $, where $\zeta =\RE \al \evector$, $\al\in \Complex$ and $\eta\in \Hc(\L_1 )$. Notice that, by orthogonality in \eqref{Hc.def}, \[ \bQ(\psi) = - |\al|^2 \ka(u,v) + \bQ(\eta) ~, \] which is not positive definite, (recall $(u,v)=1$). However, % \begin{equation} \label{3-15} \norm{\psi}_{H^1}^2 \sim |\al|^2 + \norm{\eta}_{H^1}^2~. \end{equation} % To see it, one first notes that $\norm{\psi}_{H^1}^2 $ is clearly bounded by the right side. Because of \eqref{3-10}, one has $|\al|^2\le C\norm{\psi}_{H^1}^2$. One also has $\norm{\eta}_{H^1}^2 \le C \norm{\phi}_{H^1}^2 + C|\al|^2$. Hence \eqref{3-15} is true. Therefore for $\psi=(\RE \al \evector) +\eta$ we have \begin{alignat*}{2} \norm{e^{t\L_1}\psi}_{H^1}^2 &\sim \norm{e^{t\L_1}\RE \al \evector}_{H^1}^2 + \norm{e^{t\L_1}\eta}_{H^1}^2 \qquad &&\text{(by } \eqref{3-15}) \\ &\sim |e^{-it\evalue }\al |^2 + \bQ(e^{t\L_1}\eta) &&\text{(by (4)}, \eqref{3-14}) \\ &\sim |\al|^2 + \bQ( \eta) &&\text{(by \eqref{3-13})} ~. \end{alignat*} Hence we have $\norm{e^{t\L_1}\psi}_{H^1}^2 \sim \norm{\psi}_{H^1}^2$ for all $t$. This shows (6). The decay estimate in (7) is obtained as in Theorem \ref{th:2-2} (7). The constant $C$, however, is independent of $n$ in the non-resonant case. The proof of Theorem \ref{th:2-1} is complete. \section{Solutions converging to excited states} In this section we prove Theorem \ref{th:1-2} using Theorems \ref{th:2-2} and \ref{th:2-2}. Fix $E_1$ and $Q_1= Q_{1,E_1}$. Let $\L_1$ be the corresponding linearized operator. For any $\xi_\infty \in \Hc(\L_1)$ with small $H^2 \cap W^{2,1}$ norm, we want to construct a solution $\psi(t)$ of the nonlinear Schr\"odinger equation \eqref{Sch} with the form \[ \psi(t) = \bkt{ Q_1 + h(t) }e^{-i E_1 t}~, \] where \begin{equation} h(t)=e^{t\L_1}\xi_\infty + g(t), \qquad g(t)= \text{error}. \end{equation} We decompose the error term $g(t)$ according to the decomposition \eqref{L2.dec} with respect to $\L_1$ and write \begin{equation} \label{g.dec} g(t)= a(t)R_1 + b(t) i Q_1 + \zeta(t) + \eta(t) \end{equation} % where $a(t)$ and $b(t)$ are real-valued functions, $\zeta(t) \in \eigen(\L_1)$ and $\eta(t) \in \Hc(\L_1)$. We have % \begin{align} \pd_t g &= \L_1 g + F(h), \nonumber \\ F(h) &= i^{-1}\la \bket{Q_1(2|h|^2+h^2)+|h|^2 h} \label{F.def} \\ g(t) &\to 0 \quad \text{as } t\to \infty . \nonumber \end{align} % The main term of $F(h(t))$ is $F(e^{t\L_1}\xi_\infty)$. Notice that, if $\norm{\xii}_{H^2 \cap W^{2,1}}\le \e \ll 1$, then $\xi(t) = e^{t\L_1}\xi_\infty $ satisfies % \begin{equation} \norm{\xi(t)}_{H^2} \le C_1 \e, \quad \norm{\xi(t)}_{W^{2,\infty}} \le C_1 \e |t|^{-3/2}, \quad \norm{|\xi|^2\xi(t)}_{H^2} \le C_1 \e^3 \bkA{t}^{-3}. \end{equation} % Here we have used the decay estimates in Theorem \ref{th:2-1} (7) and Theorem \ref{th:2-2} (7). Since our $Q_1$ is fixed, it does not matter that the constant depends on $n$ in the resonant case. Projecting the equation into components, we have % \begin{align} \label{a.eq} \dot a &= 0 + (Q_1,R_1)^{-1} (Q_1, \, \Re F(h)), \\ \label{b.eq} \dot b &= - a + (Q_1,R_1)^{-1} (R_1, \, \Im F(h)), \\ \pd_t \zeta &= \L_1 \zeta + {\bf P}_{\eigen} F(h) \\ \label{eta.eq} \pd_t \eta &= \L_1 \eta + \Pc^{\L_1} F(h) ~, \end{align} where ${\bf P}_{\eigen}$ and $\Pc^{\L_1}$ are corresponding projections with respect to $\L_{1}$. \subsection{Non-resonant case} For this case, we do not need to separate $\zeta$ and $\eta$. Hence we will write $u(t)=\zeta+\eta$, $u(t) \in {M_1}=M_1(E_1)$. Thus we have $\psi(t) = \bkt{ Q_1 +h(t) }e^{-i E_1 t}$, \[ h(t) = e^{t\L_1}\xi_\infty + a(t)R_1 + b(t) i Q_1 + u(t) \] and we have the equations for $u$: \begin{equation} \label{u.eq} \pd_t u = \L_1 u + P_{M_1} F(h) ~. \end{equation} Together with the equations \eqref{a.eq} and \eqref{b.eq} for $a$ and $b$, we get a complete system. We will define a contraction mapping $\Tmap$ in the following space % \begin{align*} {\cal A}=&\big \{ (a, b, u): [0, \infty) \to \R \times \R \times (M_1 \cap H^2), \\ &\quad |a(t)|,|b(t)|,\norm{u(t)}_{H^2} \le \e^2 (1+t)^{-2} \big \} \end{align*} % with the correponding norm \[ \norm{(a, b, u)}_{\cal A} := \sup _{t>0} \, (1+t)^2 \, \bke{ |a(t)|+ |b(t)|+\norm{u(t)}_{H^2} }. \] Our map $\Tmap$ is defined by \[ \Tmap : (a, b, u) \longto (a^\triangle, b^\triangle, u^\triangle) \] where, with $F(h)$ defined in \eqref{F.def}, \begin{align*} h(s) &=e^{s\L_1}\xi_\infty + a(s)R_1 + b(s) i Q_1 + u(s), \\ a^\triangle(t) &= \int_\infty^t (Q_1,R_1)^{-1} (Q_1, \, \Re F(h(s))) \,d s \\ b^\triangle(t) &= \int_\infty^t -a(s) + (Q_1,R_1)^{-1} (R_1, \, \Im F(h(s)))\,d s \\ u^\triangle(t) &=\int _\infty ^t e^{\L_1 (t-s)} P_{M_1} F(h(s)) \, d s \end{align*} % It is easy to check that the map $\Tmap$ is a contraction mapping if $\e$ is sufficiently small. Thus we have a fixed point in ${\cal A}$, which gives a solution to the system \eqref{a.eq}, \eqref{b.eq} and \eqref{u.eq}. Since it lies in ${\cal A}$, we also have desired estimates. We have proved Theorem \ref{th:1-2} under assumption (NR). \subsection{Resonant case} Recall from \eqref{zeta.dec2} that a function $\zeta(t) \in \eigen$ can be written as % \begin{equation}\label{4-9} \zeta = {\bf RE} \bket{\al \evector + \beta \sigma_3 \evector}, \quad \al,\beta \in \Complex . \end{equation} % Thus we write our solution $\psi(t)$ as $\psi(t) = \bkt{ Q_1 + h(t) }e^{-i E_1 t}$, \[ h(t)=e^{t\L_1}\xi_\infty + a(t)R_1 + b(t) i Q_1 + {\bf RE} \bket{\al(t) \evector + \beta(t) \sigma_3 \evector} + \eta(t). \] % $a$, $b$ and $\eta$ satisfy the equations \eqref{a.eq}, \eqref{b.eq} and \eqref{eta.eq}. For the equation of $\zeta$, because we have \[ \L_1 \zeta = {\bf RE} \bket{\la \evalue \evector -\evalue \beta \sigma_3 \evector}, \] hence % \begin{align} \dot \al &= \evalue \al + P_\al F, \\ \dot \beta &= -\evalue \beta+ P_\beta F, \end{align} where $P_\al$ and $P_\beta$ are defined in \eqref{PalPbeta.def}. Recall $\evalue = i \ka + \gamma$ with $\ka, \gamma >0$. The above two equations together with \eqref{a.eq}, \eqref{b.eq} and \eqref{eta.eq} form a complete system. As it will become clear, we have the freedom to choose $\xii$ and $\beta_0=\beta(0)$. We require that $\xii \in \Hc(\L_1)$ and \begin{equation} \norm{\xii}_{H^2 \cap W^{2,1}}\le \e , \qquad |\beta_0|\le \e^2. \end{equation} With given $\xii$ and $\beta_0$, we will define a contraction mapping $\Tmap$ in the following space % \begin{align*} {\cal A}=&\big \{ (a, b, \al,\beta, \eta): [0, \infty) \to \R \times \R \times \Complex \times \Complex \times (\Hc(\L_1) \cap H^2), \\ &\quad |a(t)|,|b(t)|, \norm{\eta(t)}_{H^2} \le \e^2 (1+t)^{-2} , \\ &\quad |\al(t)|,|\beta(t)|\le \e^2 (1+t)^{-2} \ \big \} \end{align*} % Our map $\Tmap$ is defined by \[ \Tmap : (a, b, \al, \beta, \eta) \longto (a^\triangle, b^\triangle, \al^\triangle, \beta^\triangle, \eta^\triangle) \] where, with $F(h)$ defined in \eqref{F.def}, % \begin{align*} h(s) &=e^{s\L_1}\xi_\infty + a(s)R_1 + b(s) i Q_1 + {\bf RE} \bket{\al(s) \evector + \beta(s) \sigma_3 \evector} + \eta(s) \\ a^\triangle(t) &= \int_\infty^t (Q_1,R_1)^{-1} (Q_1, \, \Re F(h(s))) \,d s \\ b^\triangle(t) &= \int_\infty^t -a(s) + (Q_1,R_1)^{-1} (R_1, \, \Im F(h(s)))\,d s \\ \al^\triangle(t) &= \int _\infty ^t e^{\evalue (t-s)} P_\al F(h(s)) \, d s \\ \beta^\triangle(t) &= e^{-\evalue t}\beta_0+\int _0 ^t e^{-\evalue (t-s)} P_\beta F(h(s)) \, d s \\ \eta^\triangle(t) &=\int _\infty ^t e^{\L_1 (t-s)} \PcL F(h(s)) \, d s . \end{align*} % Since $t-s<0$ in the integrand of $\al$, the integral converges. Similarly the $\beta$-integration converges. Observe that we have the freedom of choosing $\beta_0$ and $\xi_\infty$. Since $e^{-\evalue t}\beta_0$ decays exponentially, the main term of $\beta(t)$ when $t$ large is given by $F(e^{\L_1 t}\xi_\infty)$, not $e^{-\evalue t}\beta_0$. We obtain $\al(0)$, $a(0)$ and $b(0)$ as functions of $\xi_\infty$ and $\beta_0$. It is easy to check that the map $\Tmap$ is a contraction mapping if $\e$ is sufficiently small. Thus we have a fixed point in ${\cal A}$, which gives a solution to the system \eqref{a.eq}, \eqref{b.eq} and \eqref{u.eq}. Since it lies in ${\cal A}$, we also have the desired estimates. Direct estimates show that $\al(t)$ and $\beta(t)$ are of order $t^{-3}$; $a(t)$, $b(t)$ and $\eta(t)$ are of order $t^{-2}$ in $H^2(\R^3)$. We have proved Theorem \ref{th:1-2} under assumption (R). \section{Appendix: Vanishing solutions} In this appendix we prove Proposition \ref{th:1-1}. Recall $H_0=- \Delta +V$. The propagator $e^{-iH_0t}$ is bounded in $H^s$, $s\ge 0$, and satisfies the decay estimate of the form \eqref{decayest}: % \begin{equation} \label{decayest0} \norm{ e^{-itH_0} \Pc \phi}_{L^\infty} \le C |t|^{- 3/2} \norm{\phi}_{L^1} \end{equation} under our assumption A1. See \cite{JK,JSS,Y}. For any $\xii \in \Hc(H_0)$ with small $H^2\cap W^{2,1}$ norm, we want to construct a solution $\psi(t)$ of \eqref{Sch} with the form % \begin{equation} \label{2-1} \psi(t) = e^{-iH_0t} \xii + g(t), \qquad g(t)=\text{error}. \end{equation} % Let $\xi(t) = e^{-iH_0t} \xii$. Suppose $\norm{\xii}_{H^2\cap W^{2,1}} = \e$, $0< \e\le \e_0$, we have \[ \norm{\xi(t)}_{W^{2, \infty}} \le C_1 \e |t|^{-3/2}~, \quad \norm{|\xi|^2\xi(t)}_{H^2} \le C_1 \e^3 (1+t)^{-3} \] for some constant $C_1$. This uses the decay estimate \eqref{decayest0} for $e^{-it H_0} \PcH$. The error term $g(t)$ satisfies % \[ \pd_t g = -i H_0 g + F \] with $g(t) \to 0$ as $t \to \infty$ in certain sense, and % \begin{equation} \label{2-2} F(t)= -i \la |\psi|^2 \psi = -i \length{ \xi(t) + g(t)}^2 ( \xi(t) + g(t)) ~, \quad \xi(t)=e^{-iH_0t} \xii. \end{equation} % % We define a solution by \eqref{2-2} and % \begin{equation} g(t) = \int_\infty^t e^{-iH_0 (t-s)} F(s) \, d s ~. \end{equation} % Note that our $g(t)$ belongs to $L^2$ and is not restricted to the continuous spectrum component of $H_0$. Also note that the main term in $F$ is $|\xi|^2 \xi(t)$, which is of order $t^{-3}$ in $H^2$. Hence $g(t) \sim t^{-2}$. We define a contraction mapping in the following class \[ \mathcal{A} = \bket{ g(t):[0,\infty) \to H^2(\R^3), \, \norm{h(t)}_{H^2} \le C_1 \e^3(1+t)^{-2} } ~ . \] This class is not empty since it contains the zero function. We also define the norm \[ \norm{g}_{\mathcal A} := \sup _{t>0} (1+t)^2 \norm{g(t)}_{H^2}. \] For $g(t) \in \mathcal{A}$ we define \[ \Tmap: g(t) \to g^\triangle(t) =-i\la \int_\infty^t e^{-iH_0 (t-s)} \bke{|\xi + g|^2 (\xi + g)} (s) \, d s ~. \] It is easy to check that \begin{align*} \norm{g^\triangle(t)}_{H^2} &\le \int_t^\infty \norm{F(t)}_{H^2} \, d s \\ &\le \int_t^\infty C_1 \e^3 \bkA{s}^{-3} + C \e^5 {\bkA{s}^{-7/2}} \, d s \le C_1 \e^3\bkA{t}^{-2} ~, \end{align*} if $\e_0$ is sufficiently small. This shows that the map $\Tmap$ maps $\mathcal{A}$ into itself. Similarly one can show $\norm{\Tmap g_1 - \Tmap g_2}_{\mathcal A} \le \norm{ g_1 - g_2}_{\mathcal A}$, if $g_1, g_2 \in \mathcal{A}$. Therefore our map $\Tmap$ is a contraction mapping and we have a fixed point. Hence we have a solution $\psi(t)$ of the form \eqref{2-1} with $e^{-itH_0}\xii$ as the main profile. {\bf Remark.} The above existence result holds no matter how many bound states $H_0$ has. 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