Content-Type: multipart/mixed; boundary="-------------0803240430462" This is a multi-part message in MIME format. ---------------0803240430462 Content-Type: text/plain; name="08-59.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-59.keywords" nonlinear Schroedinger equation, ground states, asymptotic stability ---------------0803240430462 Content-Type: text/plain; name="NLS3Dstab.bbl" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="NLS3Dstab.bbl" \def\cprime{$'$} \begin{thebibliography}{10} \bibitem{bl:i} H.~Berestycki and P.-L. Lions. \newblock Nonlinear scalar field equations. {I}. {E}xistence of a ground state. \newblock {\em Arch. Rational Mech. Anal.}, 82(4):313--345, 1983. \bibitem{bp:asi} V.~S. Buslaev and G.~S. Perel{\cprime}man. \newblock Scattering for the nonlinear {S}chr\"odinger equation: states that are close to a soliton. \newblock {\em Algebra i Analiz}, 4(6):63--102, 1992. \bibitem{bp:asii} V.~S. Buslaev and G.~S. Perel{\cprime}man. \newblock On the stability of solitary waves for nonlinear {S}chr\"odinger equations. \newblock In {\em Nonlinear evolution equations}, volume 164 of {\em Amer. Math. Soc. Transl. Ser. 2}, pages 75--98. Amer. Math. Soc., Providence, RI, 1995. \bibitem{bs:as} Vladimir~S. Buslaev and Catherine Sulem. \newblock On asymptotic stability of solitary waves for nonlinear {S}chr\"odinger equations. \newblock {\em Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 20(3):419--475, 2003. \bibitem{caz:bk} Thierry Cazenave. \newblock {\em Semilinear {S}chr\"odinger equations}, volume~10 of {\em Courant Lecture Notes in Mathematics}. \newblock New York University Courant Institute of Mathematical Sciences, New York, 2003. \bibitem{sc:as} Scipio Cuccagna. \newblock Stabilization of solutions to nonlinear {S}chr\"odinger equations. \newblock {\em Comm. Pure Appl. Math.}, 54(9):1110--1145, 2001. \bibitem{dgps:bec} F.~Dalfovo, S.~Giorgini, L.P. Pitaevskii, and S.~Stringari. \newblock Theory of bose-einstein condensation in trapped gases. \newblock {\em Rev. Mod. Phys.}, 71(3):463--512, 1999. \bibitem{deiftsimon} P.~Deift, W.~Hunziker, B.~Simon, and E.~Vock. \newblock Pointwise bounds on eigenfunctions and wave packets in {$N$}-body quantum systems. {IV}. \newblock {\em Comm. Math. Phys.}, 64(1):1--34, 1978/79. \bibitem{htyau:gp} L{\'a}szl{\'o} Erd{\H{o}}s, Benjamin Schlein, and Horng-Tzer Yau. \newblock Derivation of the cubic non-linear {S}chr\"odinger equation from quantum dynamics of many-body systems. \newblock {\em Invent. Math.}, 167(3):515--614, 2007. \bibitem{gs:dis} M.~Goldberg and W.~Schlag. \newblock Dispersive estimates for {S}chr\"odinger operators in dimensions one and three. \newblock {\em Comm. Math. Phys.}, 251(1):157--178, 2004. \bibitem{gss:i} Manoussos Grillakis, Jalal Shatah, and Walter Strauss. \newblock Stability theory of solitary waves in the presence of symmetry. {I}. \newblock {\em J. Funct. Anal.}, 74(1):160--197, 1987. \bibitem{gss:ii} Manoussos Grillakis, Jalal Shatah, and Walter Strauss. \newblock Stability theory of solitary waves in the presence of symmetry. {II}. \newblock {\em J. Funct. Anal.}, 94(2):308--348, 1990. \bibitem{gnt:as} Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai. \newblock Asymptotic stability and completeness in the energy space for nonlinear {S}chr\"odinger equations with small solitary waves. \newblock {\em Int. Math. Res. Not.}, (66):3559--3584, 2004. \bibitem{kn:JSS} J.-L. Journ{\'e}, A.~Soffer, and C.~D. Sogge. \newblock Decay estimates for {S}chr\"odinger operators. \newblock {\em Comm. Pure Appl. Math.}, 44(5):573--604, 1991. \bibitem{kz:as2d} E.~Kirr and A.~Zarnescu. \newblock On the asymptotic stability of bound states in 2{D} cubic {S}chr\"odinger equation. \newblock {\em Comm. Math. Phys.}, 272(2):443--468, 2007. \bibitem{kz:as2d2} E~Kirr and A.~Zarnescu. \newblock Asymptotic stability of ground states in 2{D} nonlinear {S}chr\" odinger equation including subcritical cases. \newblock {\em In preparation}, 2008. \bibitem{lsy:3d} Elliott~H. Lieb, Robert Seiringer, and Jakob Yngvason. \newblock The ground state energy and density of interacting bosons in a trap. \newblock In {\em Quantum theory and symmetries (Goslar, 1999)}, pages 101--110. World Sci. Publ., River Edge, NJ, 2000. \bibitem{mm:ae} Minoru Murata. \newblock Asymptotic expansions in time for solutions of {S}chr\"odinger-type equations. \newblock {\em J. Funct. Anal.}, 49(1):10--56, 1982. \bibitem{ln:fa} Louis Nirenberg. \newblock {\em Topics in nonlinear functional analysis}, volume~6 of {\em Courant Lecture Notes in Mathematics}. \newblock New York University Courant Institute of Mathematical Sciences, New York, 2001. \newblock Chapter 6 by E. Zehnder, Notes by R. A. Artino, Revised reprint of the 1974 original. \bibitem{pw:cm} Claude-Alain Pillet and C.~Eugene Wayne. \newblock Invariant manifolds for a class of dispersive, {H}amiltonian, partial differential equations. \newblock {\em J. Differential Equations}, 141(2):310--326, 1997. \bibitem{rw:bs} Harvey~A. Rose and Michael~I. Weinstein. \newblock On the bound states of the nonlinear {S}chr\"odinger equation with a linear potential. \newblock {\em Phys. D}, 30(1-2):207--218, 1988. \bibitem{ss:ins} Jalal Shatah and Walter Strauss. \newblock Instability of nonlinear bound states. \newblock {\em Comm. Math. Phys.}, 100(2):173--190, 1985. \bibitem{sw:mc1} A.~Soffer and M.~I. Weinstein. \newblock Multichannel nonlinear scattering for nonintegrable equations. \newblock {\em Comm. Math. Phys.}, 133(1):119--146, 1990. \bibitem{sw:mc2} A.~Soffer and M.~I. Weinstein. \newblock Multichannel nonlinear scattering for nonintegrable equations. {II}. {T}he case of anisotropic potentials and data. \newblock {\em J. Differential Equations}, 98(2):376--390, 1992. \bibitem{sw:sgs} A.~Soffer and M.~I. Weinstein. \newblock Selection of the ground state for nonlinear {S}chr\"odinger equations. \newblock {\em Rev. Math. Phys.}, 16(8):977--1071, 2004. \bibitem{str:sw} Walter~A. Strauss. \newblock Existence of solitary waves in higher dimensions. \newblock {\em Comm. Math. Phys.}, 55(2):149--162, 1977. \bibitem{kn:Wed} Ricardo Weder. \newblock Center manifold for nonintegrable nonlinear {S}chr\"odinger equations on the line. \newblock {\em Comm. Math. Phys.}, 215(2):343--356, 2000. \bibitem{mw:ls} Michael~I. Weinstein. \newblock Lyapunov stability of ground states of nonlinear dispersive evolution equations. \newblock {\em Comm. Pure Appl. Math.}, 39(1):51--67, 1986. \end{thebibliography} ---------------0803240430462 Content-Type: application/x-tex; name="NLS3Dstab.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="NLS3Dstab.tex" \documentclass{article} \usepackage{amssymb,amsmath,amsfonts} %\usepackage{graphics,graphicx,color} %\usepackage[notref,notcite]{showkeys} %\usepackage[active]{srcltx} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9in} \setlength{\topmargin}{0in} \setlength{\headsep}{0in} \setlength{\headheight}{0in} \setlength{\oddsidemargin}{0in} \numberwithin{equation}{section} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{remark}{Remark}[section] \newcommand{\p}{\alpha} \newcommand{\pe}{\psi_E} \newcommand{\po}{\psi_0} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} %\newcommand{\V}[1]{\mathbf{#1}} %\renewcommand{\d}{\,{\mathrm d}} \renewcommand{\le}{\leqslant} \renewcommand{\ge}{\geqslant} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} \newcommand{\xs}{\langle x \rangle^{\sigma}} \newcommand{\xsn}{\langle x \rangle^{-\sigma}} \newcommand{\eht}{e^{-iH(t-\tau)}} \newcommand{\ehs}{e^{-iH(t-s)}} \newcommand{\ehts}{e^{-iH(\tau-s)}} \newcommand{\xst}{\langle x \rangle^{2\sigma}} \newcommand{\Ls}{L_{\sigma}^2} \newcommand{\Lsn}{L_{-\sigma}^2} \newcommand{\om}{\Omega} \newcommand{\ipt}{(\frac{1}{2}-\frac{1}{p_2})} \newcommand{\ipo}{(\frac{1}{2}-\frac{1}{p_1})} \newcommand{\ip}{(\frac{1}{2}-\frac{1}{p})} %\renewcommand{\labelenumi}%{($\mathrm{\alph{enumi}}$)} \begin{document} \title{Asymptotic stability of ground states in 3D nonlinear Schr\" odinger equation including subcritical cases} \author{E. Kirr\thanks{Department of Mathematics, University of Illinois at Urbana-Champaign}\ \ and \" O. M\i zrak\footnotemark[1]} \maketitle \begin{abstract} \noindent We consider a class of nonlinear Schr\"{o}dinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that ``shadows" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations. \end{abstract} \section{Introduction} In this paper we study the long time behavior of solutions of the nonlinear Schr\" odinger equation (NLS) with potential in three space dimensions (3-d): \begin{align} i\partial_t u(t,x)&=[-\Delta_x+V(x)]u+g(u), \quad t\in\R,\quad x\in\R^3 \label{u} \\ u(0,x)&=u_0(x) \label{ic} \end{align} where the local nonlinearity is constructed from the real valued, odd, $C^2$ function $g:\mathbb{R}\mapsto\mathbb{R}$ satisfying \begin{equation}\label{gest} |g''(s)|\leq C(|s|^{\alpha_1}+|s|^{\alpha_2}),\quad s\in\mathbb{R}\ 0<\alpha_1\le\alpha_2<3\end{equation} which is then extended to a complex function via the gauge symmetry: \begin{equation}\label{gsym} g(e^{i\theta}s)=e^{i\theta}g(s) \end{equation} The equation has important applications in statistical physics describing certain limiting behavior of Bose-Einstein condensates \cite{dgps:bec,lsy:3d,htyau:gp}. It is well known that this nonlinear equation admits periodic in time, localized in space solutions (bound states or solitary waves). They can be obtained via both variational techniques \cite{bl:i,str:sw,rw:bs} and bifurcation methods \cite{pw:cm,rw:bs,kz:as2d}, see also next section. Moreover the set of periodic solutions can be organized as a manifold (center manifold). Orbital stability of solitary waves, i.e. stability modulo the group of symmetries $u\mapsto e^{-i\theta}u,$ was first proved in \cite{rw:bs,mw:ls}, see also \cite{gss:i,gss:ii,ss:ins}. In this paper we show that solutions of \eqref{u}-\eqref{ic} with small initial data asymptotically converge to the orbit of a certain bound state, see Theorem \ref{mt}. Asymptotic stability studies of solitary waves were initiated in the work of A. Soffer and M. I. Weinstein \cite{sw:mc1,sw:mc2}, see also \cite{bp:asi,bp:asii,bs:as,sc:as,gnt:as}. Center manifold analysis was introduced in \cite{pw:cm}, see also \cite{kn:Wed}. The main contribution of our result is to allow for subcritical and critical ($L^2$) nonlinearities, $0<\alpha_1\le 1/3$ in \eqref{gest}. To accomplish this we develop an innovative technique in which linearization around a one parameter family of bound states is used to track the solution. Previously a fixed bound state has been used, see the papers cited in the previous paragraph. By continuously adapting the linearization to the actual evolution of the solution we are able to capture the correct effective potential induced by the nonlinearity $g$ into a time dependent linear operator. Once we have a good understanding of the semigroup of operators generated by the time dependent linearization, see Section \ref{se:lin}, we obtain sharper estimates for the nonlinear dynamics via Duhamel formula and contraction principles for integral equations, see Section \ref{se:main}. They allow us to treat a large spectrum of nonlinearities including, for the first time, the subcritical ones. The main challenge is to obtain good estimates for the semigroup of operators generated by the time dependent linearization that we use. This is accomplished in Section \ref{se:lin}. The technique is perturbative, and similar to the one developed by the first author and A. Zarnescu for 2-D Schr\" odinger type operators in \cite{kz:as2d}, see also \cite{kz:as2d2}. The main difference is that in 3-D one needs to remove the non-integrable singularity in time at zero of the free Schr\" odinger propagator: $$\|e^{i\Delta t}\|_{L^{1}\mapsto L^\infty}\sim |t|^{-3/2}.$$ We do this by generalizing a Fourier multiplier type estimate first introduced by Journ\' e, Soffer, and Sogge in \cite{kn:JSS} and by proving certain smoothness properties of the effective potential induced by the nonlinearity, see the Appendix. Since our methods rely on linearization around nonlinear bound states and estimates for integral operators we expect them to generalize to the case of large nonlinear ground states, see for example \cite{sc:as}, or the presence of multiple families of bound states, see for example \cite{sw:sgs}, where it should greatly reduce the restrictions on the nonlinearity. We are currently working on adapting the method to other spatial dimensions. The work in 2-D is almost complete, see \cite{kz:as2d,kz:as2d2}. \bigskip \noindent{\bf Notations:} $H=-\Delta+V;$ $L^p=\{f:\mathbb{R}^2\mapsto \mathbb{C}\ |\ f\ {\rm measurable\ and}\ \int_{\mathbb{R}^2}|f(x)|^pdx<\infty\},$ $\|f\|_p=\left(\int_{\mathbb{R}^2}|f(x)|^pdx\right)^{1/p}$ denotes the standard norm in these spaces; $=(1+|x|^2)^{1/2},$ and for $\sigma\in\mathbb{R},$ $L^2_\sigma$ denotes the $L^2$ space with weight $^{2\sigma},$ i.e. the space of functions $f(x)$ such that $^{\sigma}f(x)$ are square integrable endowed with the norm $\|f(x)\|_{L^2_\sigma}=\|^{\sigma}f(x)\|_2;$ $\langle f,g\rangle =\int_{\mathbb{R}^2}\overline f(x)g(x)dx$ is the scalar product in $L^2$ where $\overline z=$ the complex conjugate of the complex number $z;$ $P_c$ is the projection on the continuous spectrum of $H$ in $L^2;$ $H^n$ denote the Sobolev spaces of measurable functions having all distributional partial derivatives up to order $n$ in $L^2, \|\cdot\|_{H^n}$ denotes the standard norm in this spaces. \bigskip \noindent{\bf Acknowledgements:} The authors would like to thank Wilhelm Schlag and Dirk Hundertmark for useful discussions on this paper. Both authors acknowledge the partial support from the NSF grants DMS-0603722 and DMS-0707800. \section{Preliminaries. The center manifold.}\label{se:prelim} The center manifold is formed by the collection of periodic solutions for (\ref{u}): \begin{equation}\label{eq:per} u_E(t,x)=e^{-iEt}\psi_E(x) \end{equation} where $E\in\mathbb{R}$ and $0\not\equiv\psi_E\in H^2(\mathbb{R}^3)$ satisfy the time independent equation: \begin{equation}\label{eq:ev} [-\Delta+V]\psi_E+g(\psi_E)=E\psi_E \end{equation} Clearly the function constantly equal to zero is a solution of (\ref{eq:ev}) but (iii) in the following hypotheses on the potential $V$ allows for a bifurcation with a nontrivial, one parameter family of solutions: \bigskip \noindent{\bf (H1)} Assume that \begin{itemize} \item[(i)] There exists $C>0$ and $\rho >3$ such that: \begin{enumerate} \item $|V(x)|\le C^{-\rho},\quad {\rm for\ all}\ x\in\mathbb{R}^3;$ \item $\nabla V\in L^p(\mathbb{R}^3)$ for some $2\le p\le\infty$ and $|\nabla V(x)|\rightarrow 0$ as $|x|\rightarrow\infty ;$ \item the Fourier transform of $V$ is in $L^1.$ \end{enumerate} \item[(ii)] $0$ is a regular point\footnote{see \cite[Definition 6]{gs:dis} or $M_\mu=\{0\}$ in relation (3.1) in \cite{mm:ae}} of the spectrum of the linear operator $H=-\Delta+V$ acting on $L^2.$ \item [(iii)]$H$ acting on $L^2$ has exactly one negative eigenvalue $E_0<0$ with corresponding normalized eigenvector $\psi_0.$ It is well known that $\psi_0(x)$ is exponentially decaying as $|x|\rightarrow\infty,$ and can be chosen strictly positive. \end{itemize} \par\noindent Conditions (i)1. and (ii) guarantee the applicability of dispersive estimates of Murata \cite{mm:ae} and Goldberg-Schlag \cite{gs:dis} to the Schr\" odinger group $e^{-iHt}.$ Condition (i)2. implies certain regularity of the nonlinear bound states while (i)3. allow us to use commutator type estimates, see Theorem \ref{th:gjss}. All these are needed to obtain estimates for the semigroup of operators generated by our time dependent linearization, see Theorems \ref{th:lw} and \ref{th:lp} in section \ref{se:lin}. In particular (i)1. implies the local well posedness in $H^1$ of the initial value problem (\ref{u})-(\ref{ic}), see section \ref{se:main}. By the standard bifurcation argument in Banach spaces \cite{ln:fa} for (\ref{eq:ev}) at $E=E_0,$ condition (iii) guarantees existence of nontrivial solutions. Moreover, these solutions can be organized as a $C^1$ manifold (center manifold), see \cite[section 2]{kz:as2d}. Since our main result requires, we are going to show in what follows that the center manifold is $C^2.$ We note that for three and higher dimensions this has been sketched in \cite{gnt:as}, however they show smoothness by formal differentiation of certain equations without proof that at least one side has indeed derivatives. As in \cite{kz:as2d} we decompose the solution of (\ref{eq:ev}) in its projection onto the discrete and continuous part of the spectrum of $H:$ $$\psi_E=a\psi_0+h,\quad a=\langle \psi_0,\psi_E\rangle,\ h=P_c\psi_E.$$ Projecting now (\ref{eq:ev}) onto $\psi_0$ and its orthogonal complement $={\rm Range}\ P_c$ we get: {\setlength\arraycolsep{2pt} \begin{eqnarray} 0&=&h+(H-E)^{-1}P_cg(a\psi_0+h)\label{eq:evc}\\ 0&=&E-E_0- a^{-1}\langle\psi_0,g(a\psi_0+h)\rangle\label{eq:evp} \end{eqnarray}} Although we are using milder hypothesis on $V$ the argument in the Appendix of \cite{pw:cm} can be easily adapted to show that: $$\mathcal{F}(E,a,h)=h+(H-E)^{-1}P_cg(a\psi_0+h)$$ is a $C^2$ function from $(-\infty,0)\times\mathbb{C}\times \Ls\cap H^2$ to $L^2_\sigma\cap H^2$ and $\mathcal{F}(E_0,0,0)=0,$ $D_h\mathcal{F}(E_0,0,0)=I.$ Therefore the implicit function theorem applies to equation (\ref{eq:evc}) and leads to the existence of $\delta_1>0$ and the $C^2$ function $\tilde h(E,a)$ from $(E_0-\delta_1,E_0+\delta_1)\times \{a\in\mathbb{C}\ :\ |a|<\delta_1\} $ to $L^2_\sigma\cap H^2$ such that (\ref{eq:evc}) has a unique solution $h=\tilde h(E,a)$ for all $E\in (E_0-\delta_1,E_0+\delta_1)$ and $|a|<\delta_1.$ Note that, by gauge invariance, if $(a,h)$ solves (\ref{eq:evc}) then $(e^{i\theta}a,e^{i\theta}h),\ \theta\in[0,2\pi )$ is also a solution, hence by uniqueness we have: \begin{equation}\label{eq:hsym} \tilde h(E,a)=\frac{a}{|a|}\tilde h(E,|a|). \end{equation} Because $\psi_0$ is real valued, we could apply the implicit function theorem to (\ref{eq:evc}) under the restriction $a\in\mathbb{R}$ and $h$ in the subspace of real valued functions as it is actually done in \cite{pw:cm}. By uniqueness of the solution we deduce that $\tilde h(E,|a|)$ is a real valued function. Consider now the restriction of $\tilde h(E,a)$ to $a\in\mathbb{R},\ |a|<\delta_1.$ This is now a real valued $C^2$ function on $(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1)$ which, by \eqref{eq:hsym}, is odd in the second variable. We now differentiate \eqref{eq:evc} with $h=\tilde h(E,a),$ to obtain the following estimates for the first and second derivatives of $\tilde h$ on $(E,a)\in(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1):$ \begin{eqnarray} \frac{\partial\tilde h}{\partial a}(E,a)&=&-(D_h\mathcal{F})^{-1}(E,a,\tilde h(E,a))[(H-E)^{-1}P_c g'(a\psi_0+\tilde h(E,a))\psi_0]=\mathcal{O}(|a|^{1+\alpha_1})\nonumber\\ \frac{\partial\tilde h}{\partial E}(E,a)&=&(D_h\mathcal{F})^{-1}(E,a,\tilde h(E,a))[(H-E)^{-2}P_c g(a\psi_0+\tilde h(E,a))]=\mathcal{O}(|a|^{2+\alpha_1})\nonumber\\ \frac{\partial^2\tilde h}{\partial a^2}(E,a)&=& -(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h(E,a))\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)^2\right]=\mathcal{O}(|a|^{\alpha_1}) \nonumber \end{eqnarray} \begin{eqnarray} \frac{\partial^2\tilde h}{\partial E\partial a}(E,a)=\mathcal{O}(|a|^{1+\alpha_1})&=& (D_h\mathcal{F})^{-1}\left[(H-E)^{-2}P_c g'(a\psi_0+\tilde h)\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)\right]\nonumber\\ &-&(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h)\left(\psi_0+\frac{\partial\tilde h}{\partial a}\right)\frac{\partial\tilde h}{\partial E}\right]\nonumber\\ \frac{\partial^2\tilde h}{\partial E^2}(E,a)=\mathcal{O}(|a|^{2+\alpha_1})&=& +(D_h\mathcal{F})^{-1}\left[(H-E)^{-1}P_c g''(a\psi_0+\tilde h)\left(\frac{\partial\tilde h}{\partial E}\right)^2\right]\nonumber\\ &-&(D_h\mathcal{F})^{-1}\left[2(H-E)^{-3}P_c g(a\psi_0+\tilde h)-2(H-E)^{-2}P_c g'(a\psi_0+\tilde h)\frac{\partial\tilde h}{\partial E}\right]\nonumber \end{eqnarray} %scrie-le in forma h'=op^-{1}...=O(a^..) where we used $D_h\mathcal{F}(E,a,\tilde h(E,a))$ is invertible with bounded inverse and $D_h\mathcal{F}(E,0,0)=I,$ $(H-E)^{-1}$ is bounded and analytic operator in $E\in(E_0-\delta_1,E_0+\delta_1),$ and $g'(s)=\mathcal{O}(s^{1+\alpha_1}),$ $g''(s)=\mathcal{O}(s^{\alpha_1})$ as $s\rightarrow 0.$ Replacing now $h=\tilde h(E,a),\ (E,a)\in(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1) $ in (\ref{eq:evp}) we get: \begin{equation}\label{eq:evp1} E-E_0=a^{-1}\langle\psi_0,g(a\psi_0+\tilde h(E,a))\rangle. \end{equation} To this we can apply again the implicit function theorem by observing that $ G(E,a)=E-E_0-a^{-1}\langle\psi_0,g(a\psi_0+\tilde h(E,a))\rangle$ is a $C^1$ function from $(E_0-\delta_1,E_0+\delta_1)\times (-\delta_1,\delta_1)$ to $\mathbb{R}$ with the properties $G(E_0,0)=0,$ $\partial_E G(E_0,0)=1.$ We obtain the existence of $0<\delta\le\delta_1,$ and the $C^1$ even function $\tilde E:(-\delta , \delta)\mapsto (E_0-\delta,E_0+\delta)$ such that, for $|E-E_0|,|a|<\delta,$ the unique solution of (\ref{eq:evp}) with $h=\tilde h(E,a),$ is given by the $E=\tilde E(a).$ Note that $\tilde E$ is $C^2$ except at $a=0$ because $G$ is $C^2$ except at $a=0,$ and: \begin{eqnarray} \frac{d\tilde E}{da}(a)&=&-\frac{\partial_a G(E(a),a)}{\partial_E G(E(a),a)}=\mathcal{O}(|a|^{\alpha_1})\nonumber\\ \frac{d^2\tilde E}{da^2}(a)&=&\mathcal{O}(|a|^{\alpha_1-1})\qquad {\rm for}\ a\not=0,\ {\rm recall\ that\ }0<\alpha_1\le 1.\nonumber \end{eqnarray} If we now define the odd function: $$h(a)\equiv\tilde h(E(a),a),\qquad -\delta< a <\delta$$ we get a $C^2$ function because, for $a\not=0,$ based on the previous estimates on the derivatives of $\tilde h$ and $\tilde E,$ we have $$ \frac{d^2h}{da^2}(a)=\frac{\partial\tilde h}{\partial E}\frac{d^2\tilde E}{da^2}+\frac{\partial^2\tilde h}{\partial E^2}\left(\frac{d\tilde E}{da}\right)^2+2\frac{\partial^2\tilde h}{\partial E\partial a}\frac{d\tilde E}{da}+\frac{\partial^2\tilde h}{\partial a^2} =\mathcal{O}(|a|^{\alpha_1}), $$ hence, by L'Hospital $$\frac{d^2h}{da}(0)\stackrel{def}{=}\lim_{a\rightarrow 0}\frac{\frac{dh}{da}(a)-0}{a}=\lim_{a\rightarrow 0}\frac{d^2h}{da^2}(a)=0.$$ We now extend $h$ to complex values via the rotational symmetry \eqref{eq:hsym}: $$ h(a)=\frac{a}{|a|}\tilde h(E(|a|),|a|). $$ We have just proved: \begin{proposition}\label{pr:cm} There exist $\delta>0,$ the $C^2$ function $$h:\{a\in\mathbb{R}\times\mathbb{R}\ :\ |a|<\delta\}\mapsto L^2_\sigma\cap H^2 ,$$ and the $C^1$ function $E:(-\delta,\delta)\mapsto\mathbb{R}$ such that for $|E-E_0|<\delta$ and $|\langle\psi_0,\psi_E\rangle |<\delta$ the eigenvalue problem (\ref{eq:ev}) has a unique solution up to multiplication with $e^{i\theta},\ \theta\in [0,2\pi),$ which can be represented as a center manifold: \begin{equation}\label{eq:cm} \psi_E=a\psi_0+h(a),\ E=E(|a|), \quad \langle\psi_0,h(a)\rangle =0,\quad h(e^{i\theta}a)=e^{i\theta}h(a), |a|<\delta .\end{equation} Moreover $E(|a|)=\mathcal{O}(|a|^{1+\alpha_1})$, $h(a)=\mathcal{O}(|a|^{2+\alpha_1}),$ and for $a\in\mathbb{R},\ |a|<\delta,$ $h(a)$ is a real valued function with $\frac{d^2h}{da^2}(a)=\mathcal{O}(|a|^{\alpha_1})$ \end{proposition} Since $\psi_0(x)$ is exponentially decaying as $|x|\rightarrow\infty$ the proposition implies that $\psi_E\in L^2_\sigma .$ A regularity argument, see \cite{sw:mc1}, gives a stronger result: \begin{corollary}\label{co:decay} For any $\sigma\in\mathbb{R},$ there exists a finite constant $C_\sigma$ such that: $$\|^\sigma\psi_E\|_{H^2}\le C_\sigma\|\psi_E\|_{H^2}.$$ \end{corollary} \begin{remark}\label{rmk:reg} By standard regularity methods, see for example \cite[Theorem 8.1.1]{caz:bk}, one can show $\psi_E\in H^3.$ Hence by Sobolev imbeddings both $\psi_E$ and $\nabla\psi_E$ are continuous and converge to zero as $|x|\rightarrow\infty.$ \end{remark} \begin{remark}\label{rmk:pos} By standard variational methods, see for example \cite{rw:bs}, one can show that the real valued solutions of \eqref{eq:ev} do not change sign. Then Harnack inequality for $H^2\bigcap C(\mathbb{R}^3)$ solutions of \eqref{eq:ev} implies that these real solution cannot take the zero value. Hence $\psi_E$ given by \eqref{eq:cm} for $a\in\mathbb{R}$ is either strictly positive or strictly negative. \end{remark} In section \ref{se:lin} we also need some smoothness for the effective (linear) potential induced by the nonlinearity which modulo rotations of the complex plane is given by: $$Dg|_{\pe}[u+iv]=g'(\pe)u+i\frac{g(\pe)}{\pe}v,\qquad \psi_E\ge 0$$ namely: \bigskip \noindent{\bf (H2)} Assume that for the positive solution of \eqref{eq:ev} we have $\widehat{g'(\pe)},\ \widehat{\frac{g(\pe)}{\pe}}\in L^1(\mathbb{R}^3)$ where $\hat{f}$ stands for the Fourier transform of the function $f.$ \bigskip \noindent In concrete cases the hypothesis may be checked directly using the regularity of $\psi_E,$ the solution of an uniform elliptic e-value problem. In general we can prove the following result: \begin{proposition}\label{pr:dg} If the following holds \bigskip \noindent{\bf (H2')} $g$ restricted to reals has third derivative except at zero and $|g'''(s)|<\frac{C}{s^{1-\alpha_1}}+Cs^{\alpha_2 -1},$ $s>0,$ $0<\alpha_1\le\alpha_2;$ \bigskip \noindent then for the nonnegative solution of \eqref{eq:ev}, $\pe,$ we have $\widehat{g'(\pe)}\in L^1$ and $\widehat{\frac{g(\pe)}{\pe}}\in L^1$. \end{proposition} We will give the proof in the Appendix. We are going to decompose the solution of \eqref{u}-\eqref{ic} into a projection onto the center manifold and a correction. For orbital stability the projection which minimizes the $H^1$ norm of the correction is used, see for example \cite{mw:ls}, while for asymptotic stability one wants to remove periodic in time components of the correction. Currently there are two different ways to accomplish this. First and most used one is to keep the correction orthogonal to the discrete spectrum of a fixed linear Schr\" odinger operator ``close" to the dynamics, see \cite{kz:as2d,pw:cm}. For example in \cite{kz:as2d} the linear Schr\" odinger operator is $-\Delta + V$ and the correction is always orthogonal on its sole eigenvector $\psi_0,$ hence the decomposition becomes $$u=a\psi_0+h(a)+correction,\qquad {\rm where}\ a=\langle \psi_0,u\rangle.$$ Second technique is to use the invariant subspaces of the actual linearized dynamics at the projection, see for example \cite{gnt:as}. While more complicated the latter is the only one capable to render our main result. Since there are slight mistakes in the previous presentations of this decomposition we are going to describe it in what follows. Consider the linearization of \eqref{u} at function on the center manifold $\psi_E=a\psi_0+h(a),\ a=a_1+ia_2\in\mathbb{C},\ |a|<\delta:$ \begin{equation}\label{eq:ldE} \frac{\partial w}{\partial t}=-iL_{\psi_E}[w]-iEw \end{equation} where \begin{equation}\label{def:linop} L_{\psi_E}[w]=(-\Delta+V-E)w+Dg_{\psi_E}[w]=(-\Delta+V-E)w+\lim_{\varepsilon\in\mathbb{R},\ \varepsilon\rightarrow 0}\frac{g(\psi_E+\varepsilon w)-g(\psi_E)}{\varepsilon} \end{equation} {\bf Properties of the linearized operator}: \begin{enumerate} \item $L_{\psi_E}$ is real linear and symmetric with respect to the real scalar product $\Re\langle\cdot,\cdot\rangle,$ on $L^2(\mathbb{R}^3),$ with domain $H^2(\mathbb{R}^3).$ \item Zero is an e-value for $-iL_{\psi_E}$ and its generalized eigenspace includes $\left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\right\}$ \end{enumerate} The real linearity of $L_{\psi_E}$ follows from \eqref{def:linop}. For symmetry consider first the case of a real valued $\psi_E=a\psi_0+h(a),\ a\in (-\delta,\delta)\subset\mathbb{R}.$ Then for $w=u+iv\in H^2(\mathbb{R}^3),\ u,v$ real valued we have $$L_{\psi_E}[u+iv]=L_+[u]+iL_-[v]$$ with $L_+[u],$ $L_-[v]$ being real valued and symmetric: \begin{eqnarray} L_+[u]&=&(-\Delta+V-E)u+g'(\psi_E)u\nonumber\\ L_-[v]&=&(-\Delta+V-E)v+\frac{g(\psi_E)}{\psi_E}v.\nonumber \end{eqnarray} To determine the expression for $L_-$ we used the rotational symmetry \eqref{gsym}: $$ g(e^{i\theta}\psi_E)=e^{i\theta}g(\psi_E) $$ and we differentiate it with respect to $\theta$ at $\theta=0$ to get \begin{equation}\label{symgen} Dg_{\psi_E}[i\psi_E]=ig(\psi_E). \end{equation} Now, $$\Re\langle L_{\psi_E}[u+iv],u_1+iv_1\rangle=\Re\langle L_+[u],u_1\rangle +\Re\langle L_-[v],v_1\rangle=\Re\langle u,L_+[u_1]\rangle +\Re\langle v,L_-[v_1]\rangle=\Re\langle u+iv,L_{\psi_E}[u_1+iv_1]\rangle$$ hence $L_{\psi_E}$ is symmetric for real valued $\psi_E.$ For a complex valued function on the center manifold $\psi_E=a\psi_0+h(a),\ a\in\mathbb{C},\ |a|<\delta$ there exists $\theta\in [0,2\pi)$ such that $a=|a|e^{i\theta}$ and $$\psi_E=e^{i\theta}(|a|\psi_0+h(|a|))=e^{i\theta}\psi_E^{real}$$ where $\psi_E^{real}$ is real valued and on the center manifold. Using again the rotational symmetry of $g$ \eqref{gsym} we get: \begin{equation}\label{simlinop} L_{\psi_E}[w]=e^{i\theta}L_{\psi_E^{real}}[e^{-i\theta}w]. \end{equation} Since $e^{i\theta}$ is a unitary linear operator on the real Hilbert space $L^2(\mathbb{R}^3)$ and, due to the argument above, $L_{\psi_E^{real}}$ is symmetric we get that $L_{\psi_E}$ is symmetric. For the second property, we observe that substituting $w=i\psi_E$ in \eqref{def:linop} and using \eqref{symgen}, \eqref{eq:ev} we get $$L_{\psi_E}[i\psi_E]=i[(-\Delta+V-E)\psi_E+g(\psi_E)]=0.$$ Hence zero is an e-value for $-iL_{\psi_E}$ and $i\frac{\psi_E}{|a|}$ for $a\not=0$ and $i\psi_0=\lim_{a\rightarrow 0}i\frac{\psi_E}{a}$ for $a=0$ are the corresponding eigenvectors. Moreover by differentiating \eqref{eq:ev} with respect to $a_1=\Re a\in\mathbb{R}$ or $a_2=\Im a\in\mathbb{R}$ we get $$-iL_{\psi_E}\left[\frac{\partial\psi_E}{\partial a_j}\right]=-\frac{\partial E}{\partial a_j}i\psi_E,\quad j=1,2.$$ Since $\frac{\partial E}{\partial a_j}=E'(|a|)\frac{\partial |a|}{\partial a_j}\in\mathbb{R}$ we deduce that $\frac{\partial\psi_E}{\partial a_j},\ j=1,2$ are in the generalized eigenspace of zero. Note that, by differentiating $h(e^{i\theta}a)=e^{i\theta}h(a)$ with respect to $\theta$ at $\theta=0$ we get $Dh|_a[ia]=ih(a)$ and, via \eqref{eq:cm}, $D\psi_E|_a[ia]=i\psi_E.$ Since the differential can be written with the help of the gradient: $$i\psi_E=D\psi_E|_a[ia]=\frac{\partial\psi_E}{\partial a_1}\Re [ia]+\frac{\partial\psi_E}{\partial a_2}\Im [ia],$$ we infer that $$i\psi_E\in {\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\right\}\quad {\rm or\ equivalently}\ \psi_E\in {\rm span} \left\{i\frac{\partial\psi_E}{\partial a_1},i\frac{\partial\psi_E}{\partial a_2}\right\}$$ where the span is taking over the reals\footnote{One can actually show that, for small $|a|,$ zero is the only e-value of $-iL_{\psi_E}$ and the corresponding eigenspace is spanned by $\frac{\partial\psi_E}{\partial a_j},\ j=1,2.$ However this is not needed in our argument.}. One can now decompose $L^2(\mathbb{R}^3)$ into invariant subspaces with respect to $-iL_{\psi_E}$: $$L^2(\mathbb{R}^3)={\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\right\}\oplus {\cal H}_a.$$ The standard choice is to use the projection along the dual basis: $${\cal H}_a=\{\phi_1,\phi_2\}^\perp$$ where the orthogonality is with respect to the real scalar product, and $\phi_1,\ \phi_2$ are in the generalized eigenspace of the adjoint of $-iL_{\psi_E}$ corresponding to the eigenvalue zero, and $\phi_1$ is orthogonal to $\frac{\partial\psi_E}{\partial a_2}$ but not to $\frac{\partial\psi_E}{\partial a_1}$ while $\phi_2$ is orthogonal to $\frac{\partial\psi_E}{\partial a_1}$ but not to $\frac{\partial\psi_E}{\partial a_2}.$ Since $L_{\psi_E}$ is symmetric we have $(-iL_{\psi_E})^*=L_{\psi_E}i$ and a direct calculations shows that one can choose $$\phi_1=-i\frac{\partial\psi_E}{\partial a_2},\qquad \phi_2=i\frac{\partial\psi_E}{\partial a_1}$$ as long as $\Re\langle i\frac{\partial \psi_E}{\partial a_1},\frac{\partial \psi_E}{\partial a_2}\rangle\not=0.$ But $$\Re\langle i\frac{\partial \psi_E}{\partial a_1},\frac{\partial \psi_E}{\partial a_2}\rangle=\Re\langle i\psi_0,i\psi_0\rangle=1,\qquad {\rm at}\ a=0 $$ and since $\psi_E$ is $C^2$ in $a_1,\ a_2$ we have: \begin{remark} By possible choosing $\delta>0$ smaller than the one in Proposition \ref{pr:cm} we get: \begin{equation}\label{jacobi:a0}\Re\langle i\frac{\partial \psi_E}{\partial a_1},\frac{\partial \psi_E}{\partial a_2}\rangle=\Re\langle i\psi_0,i\psi_0\rangle\ge\frac{1}{2}. \end{equation} \end{remark} Consequently, for $|a|<\delta,$ \begin{equation}\label{def:ha} {\cal H}_a=\left\{-i\frac{\partial\psi_E}{\partial a_2},i\frac{\partial\psi_E}{\partial a_1}\right\}^\perp,\qquad {\rm and}\ L^2(\mathbb{R}^3)={\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\right\}\oplus {\cal H}_a. \end{equation} Our goal is to decompose the solution of \eqref{u} at each time into: $$u=\psi_E+\eta=a\psi_0+h(a)+\eta,\qquad \eta\in{\cal H}_a$$ which insures that $\eta$ is not in the non-decaying directions (tangent space of the central manifold) ${\rm span} \left\{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\right\}$ of the linearized equation \eqref{eq:ldE} around $\psi_E$. The fact that this can be done in an unique manner is a consequence of the following lemma\footnote{This is an immediate consequence of the implicit function theorem but we find the proof in \cite{gnt:as} to be incomplete.}: \begin{lemma}\label{lem:decomp} There exists $\delta_1>0$ such that any $\phi\in L^2(\mathbb{R}^3)$ satisfying $\|\phi\|_{L^2}\le\delta_1$ can be uniquely decomposed: $$\phi =\psi_E+\eta=a\psi_0+h(a)+\eta$$ where $a=a_1+ia_2\in\mathbb{C},\ |a|<\delta,\ \eta\in {\cal H}_a.$ Moreover the maps $\phi\mapsto a$ and $\phi\mapsto \eta$ are $C^1$ and there exist constant $C$ independent on $\phi$ such that $$|a|\le 2\|\phi\|_{L^2},\qquad \|\eta\|_{L^2}\le C\|\phi\|_{L^2}.$$ \end{lemma} \smallskip\par{\bf Proof:} Consider the map $F:\{a=(a_1,a_2)\in\mathbb{R}^2\ :\ |a|<\delta\}\times L^2(\mathbb{R}^3)\mapsto\mathbb{R}\times\mathbb{R}:$ \begin{equation}\label{eq:jacobi} F(a_1,a_2,\phi)=\left(\Re\langle -i\frac{\partial \psi_E}{\partial a_2},\phi-\psi_E\rangle,\Re\langle i\frac{\partial \psi_E}{\partial a_1},\phi-\psi_E\rangle\right)\end{equation} where $\psi_E=a\psi_0+h(a),\ a=a_1+ia_2.$ Since $h(a)$ is $C^2,$ $F$ is a $C^1$ map and: \begin{eqnarray} F(0,0,0)&=&0\nonumber\\ \frac{\partial F}{\partial (a_1,a_2)}(0,0,0)&=&\mathbb{I}_{\mathbb{R}^2}\nonumber \end{eqnarray} where for the calculation of the Jacobi matrix we used \eqref{jacobi:a0}. The implicit function theorem implies that there exist $\delta_2\le \delta$ and a $C^1$ map: $$\tilde F=(\tilde F_1,\tilde F_2):B(0,\delta_2)\subset L^2(\mathbb{R}^3)\mapsto\mathbb{R}\times\mathbb{R}$$ such that the only solutions of $$F(a_1,a_2,\phi)=0$$ in $|a|=|a_1+ia_2|<\delta_2,\ \|\phi\|_{L^2}<\delta_2$ are given by $$(a_1=\tilde F_1(\phi),a_2=\tilde F_2(\phi),\phi).$$ Now, for an arbitrary $\phi\in B(0,\delta_2)\subset L^2(\mathbb{R}^3),$ since $$\phi =\psi_E+\eta=a\psi_0+h(a)+\eta$$ with $a=a_1+ia_2\in\mathbb{C},\ |a|<\delta_2\le\delta,\ \eta\in {\cal H}_a$ is equivalent to $F(a_1,a_2,\phi)=0$ we get that there is a unique choice: $$a_1=\tilde F_1(\phi),\quad a_2=\tilde F_2(\phi),\quad \eta=\phi-a\psi_0-h(a).$$ Moreover, by choosing $\delta_1\le\delta_2$ such that $$\|D\tilde F_\phi\|\le 2\qquad \forall\phi\in L^2(\mathbb{R}^3),\ \|\phi\|_{L^2}\le\delta_1$$ where the norm is the operator norm from $L^2(\mathbb{R}^3)$ into $\mathbb{R}\times\mathbb{R},$ we get, for all $\phi\in L^2(\mathbb{R}^3),\ \|\phi\|_{L^2}\le\delta_1:$ $$|a|=\sqrt{a_1^2+a_2^2}\le 2\|\phi\|_{L^2}$$ and $$\|\eta\|_{L^2}\le \|\phi\|_{L^2}+\|\psi_E\|_{L^2}\le \|\phi\|_{L^2}+|a|+\|h(a)\|_{L^2}\le C\|\phi\|_{L^2}$$ where $C\ge 3+2\sup_{a\in\mathbb{C},|a|\le\delta_2}\|Dh_a\|.$ Note that the existence of $\delta_1$ is insured by the continuity of $D\tilde F$ and, from the implicit function theorem: $$D\tilde F_0=D_\phi F|_{\phi=0}$$ and the latter has norm one being the projection operator onto $\psi_0.$ This finishes the proof of Lemma \ref{lem:decomp}. $\Box$ \begin{remark}\label{rmk:h-1} Both the decomposition \eqref{def:ha} and Lemma \ref{lem:decomp} can be extended without modifications to $H^{-1}(\mathbb{R}^3)$ the dual of $H^1$ because $\frac{\partial \psi_E}{\partial a_j}\in H^1,\ j=1,2.$ In this case $\langle u, \phi\rangle $ denotes the evaluation of the functional $\phi\in H^{-1} $ at $u\in H^1.$ \end{remark} We need one more technical result relating the spaces ${\cal H}_a$ and the space corresponding to the continuous spectrum of $-\Delta+V:$ \begin{lemma}\label{le:pcinv} There exists $\delta>\delta_2>0$ such that for any $a\in\mathbb{C},\ |a|\le\delta_2$ the linear map $P_c|_{{\cal H}_a}:{\cal H}_a\mapsto {\cal H}_0$ is invertible, and its inverse $R_a :{\cal H}_0\mapsto {\cal H}_a$ satisfies: \begin{eqnarray} \|R_a\zeta\|_{L^2_{-\sigma}}&\le &C_{-\sigma}\|\zeta\|_{L^2_{-\sigma}},\qquad \sigma\in\mathbb{R}\ {\rm and\ for\ all}\ \zeta\in {\cal H}_0\cap L^2_{-\sigma}\label{raest1}\\ \|R_a\zeta\|_{L^p}&\le &C_p\|\zeta\|_{L^p},\qquad 1\le p<\infty\ {\rm and\ for\ all}\ \zeta\in {\cal H}_0\cap L^p\label{raest2}\\ \overline{R_a\zeta}&=&R_a\overline\zeta\label{racc} \end{eqnarray} where the constants $C_{-\sigma},\ C_p>0$ are independent of $a\in\mathbb{C},\ |a|\le\delta_2.$ \end{lemma} \smallskip\par{\bf Proof:} Since $\psi_0$ is orthogonal to ${\cal H}_0,$ by continuity we can choose $\delta>\tilde\delta_2>0$ such that $\psi_0\notin {\cal H}_a$ for $|a|<\tilde\delta_2.$ Consequently $P_c|_{{\cal H}_a}$ is one to one, otherwise from $\phi\in {\cal H}_a,\ \phi\neq 0,\ P_c\phi=0$ we get $\phi=z\psi_0$ for some $z\in\mathbb{C},\ z\neq 0$ which contradicts $\psi_0\notin {\cal H}_a.$ Next, for $|a|<\tilde\delta_2$ we construct $R_a :{\cal H}_0\mapsto {\cal H}_a$ such that: \begin{equation}\label{pcinv} P_cR_a\zeta=\zeta,\qquad \forall\zeta\in {\cal H}_0. \end{equation} Since $P_c$ is the projection onto $\{\psi_0\}^\perp,$ condition \eqref{pcinv} is equivalent to \begin{equation}\label{radef} R_a\zeta=\zeta+z\psi_0\end{equation} for some $z\in\mathbb{C}.$ To insure that the range of $R_a$ is in ${\cal H}_a$ we impose \begin{equation}\label{zsys} \Re z\langle -i\frac{\partial\psi_E}{\partial a_2},\psi_0\rangle=-\Re\langle -i\frac{\partial\psi_E}{\partial a_2},\zeta\rangle,\qquad \Re z\langle i\frac{\partial\psi_E}{\partial a_1},\psi_0\rangle=-\Re\langle i\frac{\partial\psi_E}{\partial a_1},\zeta\rangle. \end{equation} This linear system of two equations with two unknowns, $\Re z$ and $\Im z,$ is uniquely solvable whenever $\psi_0\notin {\cal H}_a.$ Note that for $a=0$ the system becomes: $z=\langle\psi_0,\zeta\rangle.$ In \eqref{radef} we now choose $z$ to be the unique solution of \eqref{zsys} and obtain a well defined linear map $R_a :{\cal H}_0\mapsto {\cal H}_a$ satisfying \eqref{pcinv}. Consequently, $P_c|_{{\cal H}_a}$ is also onto, hence invertible and its inverse is $R_a.$ Moreover, by the continuity of the coefficients of \eqref{zsys} with respect to $a$ we can choose $\delta_2\le\tilde\delta_2$ such that, for all $|a|\le\delta_2:$ \begin{equation}\label{z:est}|z|\le 2\ \sqrt{(\Re\langle -i\frac{\partial\psi_E}{\partial a_2},\zeta\rangle)^2+(\Re\langle i\frac{\partial\psi_E}{\partial a_1},\zeta\rangle)^2}.\end{equation} Hence, via \eqref{radef} and H\" older inequality we get: $$\|R_a\zeta\|_Y\le\|\zeta\|_Y+2\|\psi_0\|_Y\|\zeta\|_Y\sqrt{\left\|\frac{\partial\psi_E}{\partial a_2}\right\|_{Y^*}^2+\left\|\frac{\partial\psi_E}{\partial a_1}\right\|_{Y^*}^2},$$ which, for the choice $Y=L^2_{-\sigma}(\mathbb{R}^3),\ Y^*=L^2_{-\sigma}(\mathbb{R}^3)$ respectively $Y=L^p(\mathbb{R}^3),\ Y^*=L^{p'}(\mathbb{R}^3),\ \frac{1}{p}+\frac{1}{p'}=1$ give \eqref{raest1}, respectively \eqref{raest2}. The constants are independent of $a$ due to the continuous dependence of $\frac{\partial \psi_E}{\partial a_j},\ j=1,2$ on $a\in\mathbb{C}$ in the compact $|a|\le\delta_2,$ and their exponential decay in time, see proposition \ref{pr:cm} and corollary \ref{co:decay}. Now, $P_c$ commutes with complex conjugation because it is the orthogonal projection onto ${\psi_0}^\perp$ and $\psi_0$ is real valued. Then \eqref{racc} follows from $R_a$ being the inverse of $P_c.$ The proof of Lemma \ref{le:pcinv} is now complete. $\Box$ We are now ready to prove our main result. \section{Main Result}\label{se:main} \begin{theorem}\label{mt} Assume that the nonlinear term in \eqref{u} satisfies \eqref{gest} and \eqref{gsym}. In addition assume that hypothesis (H1) and either (H2) or (H2') hold. Let $p_1=3+\alpha_1,\ p_2=3+\alpha_2$. Then there exists an $\varepsilon_0$ such that for all initial conditions $u_0(x)$ satisfying $$\max\{\|u_0\|_{L^{p_2'}},\|u_0\|_{H^1}\}\leq\varepsilon_0,\qquad \frac{1}{p_2'}+\frac{1}{p_2}=1$$ the initial value problem (\ref{u})-(\ref{ic}) is globally well-posed in $H^1$ and the solution decomposes into a radiative part and a part that asymptotically converges to a ground state. More precisely, there exist a $C^1$ function $a:\mathbb{R}\mapsto\mathbb{C}$ such that, for all $t\in\R$ we have: \begin{equation} u(t,x)=\underbrace{a(t)\psi_0(x)+h(a(t))}_{\psi_E(t)}+\eta(t,x) \label{dc} \end{equation} where $\psi_E(t)$ is on the central manifold (i.e it is a ground state) and $\eta(t,x)\in {\cal H}_{a(t)},$ see Proposition \ref{pr:cm} and Lemma \ref{lem:decomp}. Moreover there exists the ground states states $\psi_{E_{\pm\infty}}$ and the $C^1$ function $\theta:\R\mapsto \R$ such that $\lim_{|t|\rightarrow\infty}\theta(t)=0$ and: $$\lim_{t\rightarrow\pm\infty}\|\psi_E(t)-e^{-it(E_\pm-\theta(t))}\psi_{E_{\pm\infty}}\|_{H^2\bigcap L^2_\sigma}=0,$$ while $\eta$ satisfies the following decay estimates: \begin{eqnarray} \|\eta(t)\|_{L^2}&\leq &C_0(\alpha_1,\alpha_2)\varepsilon_0\nonumber\\ \|\eta(t)\|_{L^{p_1}}&\leq & C_1(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}},\quad p_1=3+\alpha_1\nonumber \end{eqnarray} and, for $p_2=3+\alpha_2:$ \begin{enumerate} \item [(i)] if $\alpha_1\geq \frac{1}{3}$ or $\frac{1}{3} >\alpha_1 > \frac{2\alpha_2}{3(3+\alpha_2)}$ then $$\|\eta(t)\|_{L^{p_2}}\leq C_2(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_2})}} $$ \item [(ii)] if $\alpha_1 = \frac{2\alpha_2}{3(3+\alpha_2)}$ then $$\|\eta(t)\|_{L^{p_2}}\leq C_2(\alpha_1,\alpha_2)\varepsilon_0\frac{\log(2+|t|)}{(1+|t|)^{3(\frac{1}{2}-\frac{1}{p_2})}} $$ \item [(iii)] if $\alpha_1 < \frac{2\alpha_2}{3(3+\alpha_2)}$ then $$\|\eta(t)\|_{L^{p_2}}\leq C_2(\alpha_1,\alpha_2)\frac{\varepsilon_0}{(1+|t|)^{\frac{1+3\alpha_1}{2}}} $$ \end{enumerate} where the constants $C_0,\ C_1$ and $C_2$ are independent of $\varepsilon_0$. \end{theorem} \begin{remark} Note that the critical and supercritical cases $\frac{1}{3}\leq\p_1< 3$ are contained in $(i)$. Our results for these cases are stronger than the ones in \cite{pw:cm,sw:mc1,sw:mc2} because we do not require the initial condition to be in $L^2_\sigma,\ \sigma>1.$ Compared to \cite{gnt:as} we have sharper estimates for the asymptotic decay to the ground state but we require the initial data to be in $L^{p_2'}.$ To the best of our knowledge the subcritical case $\alpha_1<1/3$ has not been treated previously. \end{remark} \begin{remark} One can obtain estimates for the radiative part $\eta$ in $L^p$, $2\leq p\leq p_1=3+\p_1$, or $p_1\leq p\leq p_2=3+\p_2$ by Riesz-Thorin interpolation between $L^2$ and $L^{p_1}$ respectively between $L^{p_1}$ and $L^{p_2}.$ \end{remark} \noindent\textbf{Proof of Theorem \ref{mt}} It is well known that under hypothesis (H1)(i) the initial value problem (1)-(2) is locally well posed in the energy space $H^1$ and its $L^2$ norm is conserved, see for example \cite[Cor. 4.3.3 at p. 92]{caz:bk}. Global well posedness follows via energy estimates from $\|u_0\|_{H^1}$ small, see \cite[Remark 6.1.3 at p. 165]{caz:bk}. We choose $\varepsilon_0\le \delta_1$ given by Lemma \ref{lem:decomp}. Then, for all times, $\|u(t)\|_{L^2}\le\delta_1$ and we can decompose the solution into a solitary wave and a dispersive component as in \eqref{dc}: $$u(t)=a(t)\psi_0+h(a(t))+\eta(t)=\psi_E(t)+\eta(t)$$ Moreover, by possible making $\varepsilon_0$ smaller we can insure that that $\|u(t)\|_{L^2}\le\varepsilon_0$ implies $|a(t)|\le\delta_2,\ t\in\mathbb{R}$ where $\delta_2$ is given by Lemma \ref{le:pcinv}. In addition, since $$u\in C(\mathbb{R},H^{1}(\mathbb{R}^3))\cap C^1(\mathbb{R},H^{-1}(\mathbb{R}^3)),$$ and $u\mapsto a$ respectively $u\mapsto \eta$ are $C^1,$ see Remark \ref{rmk:h-1}, we get that $a(t)$ is $C^1$ and $\eta\in C(\mathbb{R},H^{1})\cap C^1(\mathbb{R},H^{-1}).$ The solution is now described by the $C^1$ function $a:\mathbb{R}\in\C$ and $\eta(t)\in C(\R,H^1)\cap C^1(\R,H^{-1}).$ To obtain their equations we plug in (\ref{dc}) into (\ref{u}). Then we get \begin{align} \frac{\partial\eta}{\partial t}+D\pe|_a a'&=-i(L_{\pe}+E)\eta-Ei\pe-iF_2(\pe,\eta) \label{eq:udecomp} \end{align} where $L_{\pe}$ is defined by \eqref{def:linop} $$L_{\pe}\eta=(-\Delta+V-E)\eta-i\frac{d}{d\varepsilon}g(\pe+\varepsilon\eta)|_{\varepsilon=0}$$ and $F_2(\pe,\eta)$ denotes the nonlinear terms in $\eta$ \begin{equation} F_2(\psi_E,\eta)=g(\psi_E+\eta)-g(\psi_E)-\underbrace{\frac{d}{d\varepsilon}g(\pe+\varepsilon\eta)|_{\varepsilon=0}}_{F_1(\psi_E,\eta)} \label{g} \end{equation} Then projecting \eqref{eq:udecomp} onto the invariant subspaces of $-iL_{\psi_E},$ ${\cal H}_{a},$ see \eqref{def:ha} and the ${\rm span} \{\frac{\partial\psi_E}{\partial a_1},\frac{\partial\psi_E}{\partial a_2}\}$, we obtain the equations for $\eta(t)$ and $a(t):$ \begin{align} \frac{\partial\eta}{\partial t}&=-i(L_{\pe}+E)\eta-iF_2(\pe,\eta)-\tilde{F}_2(\pe,\eta) \label{eq:eta} \\ D\pe|_a a'&=-Ei\pe+\tilde{F}_2(\pe,\eta) \label{eq:a} \end{align} where \begin{equation} \tilde{F}_2(\pe,\eta)=\underbrace{\frac{\Re\<-i\frac{\partial\pe}{\partial a_2},-iF_2(\pe,\eta)\>}{\Re\<-i\frac{\partial\pe}{\partial a_2},\frac{\partial\pe}{\partial a_1}\>}}_{\beta_1(\pe,\eta)}\cdot\frac{\partial\pe}{\partial a_1}+\underbrace{\frac{\Re\}{\Re\}}_{\beta_2(\pe,\eta)}\cdot\frac{\partial\pe}{\partial a_2} \label{eq:f2t} \end{equation} \noindent In order to obtain the estimates for $\eta(t)$, we analyze \eqref{eq:eta}. The linear part of \eqref{eq:eta} is: \begin{align} \frac{\partial\zeta}{\partial t}&=-i(L_{\pe}+E)\zeta=(-\Delta+V)\zeta-i\frac{d}{d\varepsilon}g(\pe(t)+\varepsilon\zeta)|_{\varepsilon=0} \label{zeta} \\ \zeta(s)&=v \nonumber \end{align} Define $\Omega(t,s)v=\zeta(t)$. Then using Duhamel's principle \eqref{eq:eta} becomes \begin{equation} \eta(t)=\om(t,0)\eta(0)-\int_0^t \om(t,s)[iF_2(\pe,\eta)+\tilde{F}_2(\pe,\eta)]ds \label{eq:duheta} \end{equation} It is here where we differ from the approach \cite{sc:as,pw:cm,sw:mc1,sw:mc2}. The right-hand side of our equation contains only nonlinear terms in $\eta$. However the challenge is to obtain good dispersive estimates for the propagator $\om(t,s)$ of the linearization \eqref{zeta}, see Theorems \ref{th:lw} and \ref{th:lp}. In order to apply a contraction mapping argument for (\ref{eq:duheta}) we use the following Banach spaces. Let $p_1=3+\p_1$ and $p_2=3+\p_2$, $$Y_i=\{u\in L^2\cap L^{p_1}\cap L^{p_2}:\sup_t (1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}\|u\|_{L^{p_1}}<\infty, \sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\|u\|_{L^{p_2}}<\infty, \sup_t \|u\|_{L^2}<\infty \}$$ endowed with the norm $$ \|u\|_{Y_i}=\max\{\sup_t (1+|t|)^{3(\frac{1}{2}-\frac{1}{p_1})}\|u\|_{L^{p_1}}, \sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\|u\|_{L^{p_2}}, \sup_t \|u\|_{L^2} \}$$ for $i=1,2,3$, where $n_1=n_2=3(\frac{1}{2}-\frac{1}{p_2})$, $n_3=\frac{1+3\p_1}{2}$, $m_1=m_3=0$ and $m_2=1$. Consider the nonlinear operator in (\ref{eq:duheta}): $$N(u)=\int_0^t \om(t,s)[iF_2(\pe,u)+\tilde{F}_2(\pe,u)]ds$$ \begin{lemma}\label{lm} Consider the cases: $$ 1.\ \alpha_1\geq \frac{1}{3}\ {\rm or}\ \frac{1}{3} >\alpha_1 > \frac{2\alpha_2}{3(3+\alpha_2)}; \quad 2.\ \alpha_1 = \frac{2\alpha_2}{3(3+\alpha_2)}; \quad 3.\ \alpha_1 < \frac{2\alpha_2}{3(3+\alpha_2)}.$$ Then, for each case number i: $N : Y_i\rightarrow Y_i$ is well defined, and locally Lipschitz, i.e. there exists $\tilde{C_i}>0$, such that $$\|Nu_1 -Nu_2\|_{Y_i}\leq\tilde{C_i}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i}+\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1}+\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1 -u_2\|_{Y_i}. $$ \end{lemma} Note that the Lemma gives the estimates for $\eta $ in the Theorem \ref{mt}. Indeed, if we denote:$$v=\Omega(t,0)\eta(0),$$ then $$\|v\|_{Y_i}\leq C_0\|\eta(0)\|_{L^{p_2'}\cap H^1},$$ where $C_0=\max\{C,C_p\}$, see theorem \ref{th:lw}. We choose $\epsilon_0$ in the hypotheses of theorem \ref{mt}, such that $$C_0\epsilon_0\le\frac{1}{2}\Big(\sqrt{1+2/\tilde{C_i}}-1\Big)$$ Then by continuity there exists $0\le Lip\le 1$ such that: $$\|v\|_{Y_i}\leq\frac{2-Lip}{4}\Big(\sqrt{1+2Lip/\tilde{C_i}}-1\Big).$$ Let $R=L\|v\|_{Y_i}/(2-Lip)$ and $B(v,R)$ be the closed ball in $Y_i$ with center $v$ and radius $R$. A direct calculation shows that the right-hand side of (\ref{eq:duheta}): $$Ku=v+Nu$$ leaves $B(v,R)$ invariant, i.e. $K:B(v,R)\mapsto B(v,R)$, and it is a contraction with Lipschitz constant $Lip$ on $B(v,R)$. By the contraction mapping argument, (\ref{eq:duheta}) has a unique solution in $Y_i$. We now have two solutions of (\ref{eq:eta}), one in $C(\R,H^1)$ from classical well posedness theory and one in $C(\R,L^2\cap L^{p_1}\cap L^{p_2})$, $p_1=3+\p_1$, $p_2=3+\p_2$ from the above argument. Using uniqueness and the continuous embedding of $H^1$ in $L^2\cap L^{p_1}\cap L^{p_2}$, we infer that the solutions must coincide. Therefore, the time decaying estimates in the spaces $Y_{1-3}$ hold also for the $H^1$ solution. \noindent\textbf{Proof of Lemma \ref{lm}} Let $u_1,u_2$ be in one of the spaces $Y_i,\ i=1,2,3.$ Then at each $s\in\R$ we have: \begin{align} F_2(\pe(s),u_1(s))-&F_2(\pe(s),u_2(s))=g(\pe+u_1)-g(\pe+u_2)-F_1(\pe,u_1)+F_1(\pe,u_2) \nonumber \\ &=\int_0^1\Big[\frac{d}{d\tau}g(\pe+u_2+\tau(u_1-u_2))-\frac{d}{d\tau}g(\pe+\tau(u_1-u_2))|_{\tau=0}\Big]d\tau \nonumber \\ &=\int_0^1\int_0^1\frac{d}{ds}\cdot\frac{d}{d\tau}g\big(\pe+s(u_2+\tau(u_1-u_2))\big)ds\; d\tau \nonumber \end{align} Using the hypothesis \eqref{gest} we have $|g(u)|\leq C(|u|^{2+\p_1}+|u|^{2+\p_2})$, then taking the derivatives with respect to $\tau$ and $s$ and estimating the integral we get: \begin{align} |F_2(\pe,u_1)-F_2(\pe,u_2)|&\leq C\big[\underbrace{(|\pe|^{\p_1}+|\pe|^{\p_2})(|u_1|+|u_2|)|u_1-u_2|}_{A_1}\label{def:a13} \\ &+\underbrace{(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|}_{A_2}+\underbrace{(|u_1|^{1+\p_2}+|u_2|^{1+\p_2})|u_1-u_2|}_{A_3}\big].\nonumber \end{align} By \eqref{eq:f2t} and H\" older inequality, for any $1\le q\le \infty$ we have: \begin{eqnarray} \|\tilde{F}_2(\pe,u_1)-\tilde{F}_2(\pe,u_2)\|_{L^q}&\leq &\tilde C \Big(\Big\|\frac{\partial\pe}{\partial a_2}\Big\|_{L^{p_2}}\Big\|\frac{\partial\pe}{\partial a_1}\Big\|_{L^q}+\Big\|\frac{\partial\pe}{\partial a_1}\Big\|_{L^{p_2}}\Big\|\frac{\partial\pe}{\partial a_2}\Big\|_{L^q}\Big)(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})\nonumber \\ &+&\tilde C \Big(\Big\|\frac{\partial\pe}{\partial a_2}\Big\|_{L^{p_1}}\Big\|\frac{\partial\pe}{\partial a_1}\Big\|_{L^q}+\Big\|\frac{\partial\pe}{\partial a_1}\Big\|_{L^{p_1}}\Big\|\frac{\partial\pe}{\partial a_2}\Big\|_{L^q}\Big)\|A_2\|_{L^{p'_1}}\label{est:tildef}\\ &\leq &C(\|A_1\|_{L^{p'_2}}+\|A_2\|_{L^{p'_1}}+\|A_3\|_{L^{p'_2}}),\nonumber \end{eqnarray} where the uniform bounds on $\frac{\partial\pe}{\partial a_j}\in H^2(\R^3),\ j=1,2,$ follow from their continuous dependence on scalar $a,$ and $|a(t)|\le\delta_2,\ t\in\R.$ Now let us consider the difference $Nu_1-Nu_2$ \begin{equation}\label{eq:nu12} (Nu_1-Nu_2)(t)=\int_0^t \Omega(t,s) \big[iF_2(\pe (s),u_1(s))-iF_2(\pe (s),u_2(s))+\tilde{F}_2(\pe (s),u_1(s))-\tilde{F}_2(\pe (s),u_2(s))]ds \end{equation} \begin{itemize} \item \textbf{$L^{p_2}$ Estimate :} \begin{align} \|Nu_1-Nu_2\|_{L^{p_2}}&\leq\int_0^t \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}C\Big(2\|A_1\|_{L^{p'_2}}+\|A_2\|_{L^{p'_2}}+\|A_2\|_{L^{p'_1}}+2\|A_3\|_{L^{p'_2}}\Big)ds \nonumber \end{align} To estimate the term containing $A_1$, observe that $$\|(|\pe|^{\p_1}+|\pe|^{\p_2})(|u_1|+|u_2|)|u_1-u_2|\|_{L^{p'_2}}\leq\||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}(\|u_1\|_{L^{p_2}}+\|u_2\|_{L^{p_2}})\|u_1-u_2\|_{L^{p_2}}$$ with $\frac{1}{\beta}+\frac{2}{p_2}=\frac{1}{p'_2}$. Using Theorem \ref{th:lp} (see also Remark \ref{rmk:lp}), we have for each case number i: and $u_1,\ u_2\in Y_i:$ \begin{align} \int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_1\|_{L^{p'_2}}ds \nonumber \\ &\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}\frac{[\log(2+|s|)]^{2m_i}}{(1+|s|)^{2n_i}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} ds \nonumber \\ &\leq\frac{C(p_2)C_1 C_2}{(1+|t|)^{3\ipt}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where $C_2=\sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\int_0^t \frac{[\log(2+|s|)]^{2m_i}ds}{|t-s|^{3\ipt}(1+|s|)^{2n_i}}<\infty$ since $2n_i >1$ and $C_1=\sup_t \||\pe|^{\p_1}+|\pe|^{\p_2}\|_{L^{\beta}}.$ The uniform bounds in $t\in\R$ for $\|\pe\|_{L^{\alpha_j\beta}}^{\alpha_j},\ j=1,2$ follow from the continuous dependence of $\pe=a(t)\po+h(a(t))\in H^2(\R^3)$ on $a(t)$ and $|a(t)|\le \delta_2,\ t\in\R.$ To estimate the terms containing $A_2$, observe that $$\|(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|\|_{L^{p'_1}}\leq\big(\|u_1\|_{L^{p_1}}^{1+\p_1}+\|u_2\|_{L^{p_1}}^{1+\p_1}\big)\|u_1-u_2\|_{L^{p_1}}$$ since $\frac{1}{p'_1}=\frac{2+\p_1}{p_1}$ and \begin{eqnarray} \|(|u_1|^{1+\p_1}+|u_2|^{1+\p_1})|u_1-u_2|\|_{L^{p'_2}}&\leq &\left(\|u_1\|_{L^{p_1}}^{\theta(1+\p_1)}\|u_1\|_{L^{2}}^{(1-\theta)(1+\p_1)}+\|u_2\|_{L^{p_1}}^{\theta(1+\p_1)}\|u_2\|_{L^{2}}^{(1-\theta)(1+\p_1)}\right) \nonumber \\ &&\times\|u_1-u_2\|^\theta_{L^{p_1}}\|u_1-u_2\|^{1-\theta}_{L^{2}}\nonumber \end{eqnarray} where $\frac{1}{p'_2}=(2+\p_1)(\frac{1-\theta}{2}+\frac{\theta}{p_1}),\ 0\le\theta\le 1. $ Again using Theorem \ref{th:lp} (see also Remark \ref{rmk:lp}), we have \begin{align} \int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_2\|_{L^{p'_2}}ds \nonumber \\ &\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\cdot\frac{(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i}}{(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}} ds \nonumber \\ &\leq\frac{C(p_2)C_3[\log(2+|t|)]^{m_i}}{(1+|t|)^{n_i}}(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where the different decay rates $n_i$ depend on the case number in the hypotheses of this Lemma: \begin{enumerate} \item corresponds to $3(\frac{\p_1}{2}+\frac{1}{p_2})>1$, and $C_3=\sup_t (1+|t|)^{3\ipt}\int_0^t \frac{ds}{|t-s|^{3\ipt}(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}}<\infty;$ \item corresponds to $3(\frac{\p_1}{2}+\frac{1}{p_2})=1$, and $C_3=\sup_t \frac{(1+|t|)^{3\ipt}}{\log(2+|t|)}\int_0^t \frac{ds}{|t-s|^{3\ipt}(1+|s|)}<\infty;$ \item corresponds to $3(\frac{\p_1}{2}+\frac{1}{p_2})<1$, and $C_3=\sup_t (1+|t|)^{\frac{1+3\p_1}{2}}\int_0^t \frac{ds}{|t-s|^{3\ipt}(1+|s|)^{3(\frac{\p_1}{2}+\frac{1}{p_2})}}<\infty.$ \end{enumerate} To estimate the term containing $A_3$, observe that $$\|(|u_1|^{1+\p_2}+|u_2|^{1+\p_2})|u_1-u_2|\|_{L^{p'_2}}\leq\big(\|u_1\|_{L^{p_2}}^{1+\p_2}+\|u_2\|_{L^{p_2}}^{1+\p_2}\big)\|u_1-u_2\|_{L^{p_2}}$$ since $\frac{1}{p'_2}=\frac{2+\p_2}{p_2}$. Again using Theorem \ref{th:lp} (see also Remark \ref{rmk:lp}), we have \begin{align} \int_0^t & \|\om(t,s)\|_{L^{p'_2}\rightarrow L^{p_2}}\|A_3\|_{L^{p'_2}}ds \nonumber \\ &\leq\int_0^t \frac{C(p_2)}{|t-s|^{3\ipt}}\cdot\frac{[\log(2+|s|)]^{(2+\p_2)m_i}}{(1+|s|)^{(2+\p_2)n_i}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} ds \nonumber \\ &\leq\frac{C(p_2)C_4C_5[\log(2+|t|)]^{m_i}}{(1+|t|)^{n_i}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where $C_5=\sup_t \frac{(1+|t|)^{n_i}}{[\log(2+|t|)]^{m_i}}\int_0^t \frac{[\log(2+|s|)]^{(2+\p_2)m_i}ds}{|t-s|^{3\ipt}(1+|s|)^{(2+\p_2)n_i}}<\infty$ since $(2+\p_2)n_i >1$. \item \textbf{$L^{p_1}$ Estimate :} From \eqref{eq:nu12} we have \begin{eqnarray} \|Nu_1-Nu_2\|_{L^{p_1}}(t)&\le &\|\int_0^t \Omega(t,s) [iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s))]ds\|_{L^{p_1}}\nonumber\\ &+&C\int_0^t\|\Omega(t,s)\|_{L^{p'_1}\mapsto L^{p_1}}\|\tilde{F}_2(\pe(s),u_1(s))-\tilde{F}_2(\pe(s),u_2(s))\|_{L^{p'_1}}ds\nonumber \end{eqnarray} For the second integral we use \eqref{est:tildef} with $q=p'_1$ and the previous estimates on $A_1,\ A_2,\ A_3$ to obtain the required bound. For the first integral moving the norm inside the integration and applying $L^{p'_1}\mapsto L^{p_1}$ estimates for $\Omega(t,s)$ and \eqref{def:a13} for the nonlinear term would require the control of $A_3$ in $L^{p'_1}.$ The latter, unfortunately, can no longer be interpolated between $L^2$ and $L^{p_2}.$ To avoid this difficulty we separate and treat differently the part of the nonlinearity having an $A_3$ like behavior by decomposing $\R^3$ in two disjoints measurable sets related to the inequality \eqref{def:a13}: $$ V_1(s)=\{x\in\R^3\ |\ |F_2(\pe(s,x),u_2(s,x))-F_2(\pe(s,x),u_1(s,x))|\le C A_3(s,x)\},\qquad V_2(s)=\R^3\setminus V_1(s)$$ On $V_2(s),$ using polar representation of complex numbers, we further split the nonlinear term into: \begin{eqnarray} \lefteqn{iF_2(\pe(s,x),u_1(s,x))-iF_2(\pe(s,x),u_2(s,x))=e^{i\theta(s,x)}CA_3(s,x)}\nonumber\\ &&+\underbrace{e^{i\theta(s,x)}[|iF_2(\pe(s,x),u_1(s,x))-iF_2(\pe(s,x),u_2(s,x))|-CA_3(s,x)]}_{G(s,x)}\nonumber\end{eqnarray} where, due to inequality \eqref{def:a13}, $|G(s,x)|\le C(A_1(s,x)+A_2(s,x))$ on $V_2(s).$ Then we have: \begin{eqnarray}\lefteqn{\int_0^t \Omega(t,s) [iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s))]ds=\int_0^t \Omega(t,s) (1-\chi (s))G(s)ds}\nonumber\\ &&+\underbrace{\int_0^t \Omega(t,s) [\chi(s)(iF_2(\pe(s),u_1(s))-iF_2(\pe(s),u_2(s)))+(1-\chi(s))e^{i\theta(s)}CA_3(s)]ds}_{I(t)},\nonumber\end{eqnarray} where $\chi(s)$ is the characteristic function of $V_1(s).$ Now $$\|\int_0^t \Omega(t,s) (1-\chi (s))G(s)ds\|_{L^{p_1}}\le \int_0^t\| \Omega(t,s)\|_{L^{p'_1}\mapsto L^{p_1}}C(\|A_1(s)\|_{L^{p'_1}}+\|A_2(s)\|_{L^{p'_1}})ds$$ and estimates as in the previous step for $A_1$ and $A_2$ give the required decay. For $I(t)$ we use interpolation: $$\|I(t)\|_{L^{p_1}}\leq\|I(t)\|_{L^2}^{1-\theta}\|I(t)\|_{L^{p_2}}^{\theta}\leq\|I(t)\|_{L^2}^{1-\theta}\left(\int_0^t \|\om(t,s)\|_{L^{p'_2}\mapsto L^{p_2}}\|A_3\|_{L^{p'_2}} ds\right)^{\theta}$$ where $\frac{1}{p_1}=\frac{1-\theta}{2}+\frac{\theta}{p_2}$. We know from previous step that the above integral decays as $(1+|t|)^{-3\ipt}$ and below we will show its $L^2$ norm will be bounded. Therefore $$\sup_t (1+|t|)^{3\ipo}\|I(t)\|_{L^{p_1}}<\infty$$ and the $L^{p_1}$ estimates are complete. \item \textbf{$L^2$ Estimate :} To estimate $L^2$ norm we cannot use $L^2\rightarrow L^2$ estimate for $\om(t,s)$ because that would force us to control $L^{2(\alpha_2+2)}$ which cannot pe interpolated between $L^2$ and $L^{p_2},\ p_2=\alpha_2+3.$ We avoid this by using the decomposition: $$T(t,s)v=[P_c\om(t,s)-\ehs P_c]v \quad \text{i.e.}\quad \om(t,s)=R_{a(t)} T(t,s)+R_{a(t)} \ehs P_c$$ For $T(t,s)$ we will use $L^{p'}\rightarrow L^2$ estimates, see Theorem \ref{th:lw}, while for $\ehs P_c$ we will use Stricharz estimates $L^\infty_tL_x^2$. We will also use a decomposition of the nonlinear term similar to the one for $L^{p_1}$ estimates that will allow us to estimate in a different manner this time the terms behaving like $A_2,$ see \eqref{def:a13}. All in all we have: \begin{align} \|Nu_1-Nu_2\|_{L^2}&\leq \int_0^t \|\om(t,s)\|_{L^2\rightarrow L^2}\|\tilde{F}_2(\pe,u_1)-\tilde{F}_2(\pe,u_2)\|_{L^2}ds \nonumber\\ &+\|R_{a(t)}\|_{L^2\mapsto L^2}\int_0^t \|T(t,s)\|_{L^{p'_2}\rightarrow L^2}C(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})ds \nonumber \\ &+\|R_{a(t)}\|_{L^2\mapsto L^2}\int_0^t \|T(t,s)\|_{L^{p'_1}\rightarrow L^2}C\|A_2\|_{L^{p'_1}}ds \nonumber \\ &+\|R_{a(t)}\|_{L^2\mapsto L^2}\|\int_0^t \ehs P_c (A_1(s)+A_3(s) ds\|_{L^2}+\|R_{a(t)}\|_{L^2\mapsto L^2}\|\int_0^t \ehs P_c A_2 ds\|_{L^2}\nonumber \end{align} For the first integral we use Theorem \ref{th:lp} part (i),\ \eqref{est:tildef} with $q=2$ and the estimates we have already obtained for $A_1,\ A_2$ and $A_3.$ We deduce that this integral is uniformly bounded in $t\in\R.$ Similarly we get uniform boundedness of the second and third integral by using Theorem \ref{th:lw} part (iv). For the fourth integral we use Stricharz estimate: $$\sup_{t\in\R}\|\int_0^t \ehs P_c A_1ds\|_{L^2}\leq C_s\left[\left(\int_\R \|A_1(s)\|_{L^{p'_2}}^{\gamma_2'}ds\right)^{\frac{1}{\gamma_2'}}+\left(\int_\R \|A_3(s)\|_{L^{p'_2}}^{\gamma_2'}ds\right)^{\frac{1}{\gamma_2'}}\right]$$ where $\frac{1}{\gamma_2'}+\frac{1}{\gamma_2}=1,$ and $\frac{2}{\gamma_2}=3\left(\frac{1}{2}-\frac{1}{p_2}\right).$ Using again the estimates we obtained before for $A_1$ and $A_3.$ we get: \begin{align} \| A_1\|_{L^{\gamma_2'}_sL^{{p'_2}}}&\leq C_{11}\Big[\int_\R \frac{(\log(2+|s|))^{2m_i\gamma_2'}}{(1+|s|)^{2n_i\gamma_2'}}ds\Big]^{\frac{1}{\gamma_2'}}(\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} \nonumber \\ &\leq C_{11} C_8 (\|u_1\|_{Y_i}+\|u_2\|_{Y_i})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where $C_8= \int_\R \frac{(\log(2+|s|))^{2m_i\gamma_1'}}{(1+|s|)^{2n_i\gamma_1'}}ds<\infty$ since $2n_i\gamma'>1$ and: \begin{align} \| A_3\|_{L^{\gamma_2'}_sL^{{p'_2}}}&\leq C_{12}\Big[\int_\R \frac{(\log(2+|s|))^{(2+\p_2)m_i\gamma_3'}}{(1+|s|)^{(2+\p_2)n_i\gamma_3'}}ds\Big]^{\frac{1}{\gamma_3'}}(\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} \nonumber \\ &\leq C_9 (\|u_1\|_{Y_i}^{1+\p_2}+\|u_2\|_{Y_i}^{1+\p_2})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where $C_9=\int_\R \frac{(\log(2+|s|))^{(2+\p_2)m_i\gamma_3'}}{(1+|s|)^{(2+\p_2)n_i\gamma_2'}}ds<\infty$ since $(2+\p_2)n_1\gamma_3'>1$. Similarly, for the fifth integral: $$\sup_{t\in\R}\|\int_0^t \ehs P_c A_2 ds\|_{L^2}\leq C_s\Big(\int_\R \|A_2\|_{L^{p_1'}}^{\gamma_1'}ds\Big)^{\frac{1}{\gamma_1'}}$$ where $\frac{1}{\gamma_1'}+\frac{1}{\gamma_1}=1,$ and $\frac{2}{\gamma_1}=3\left(\frac{1}{2}-\frac{1}{p_1}\right).$ Furthermore we have \begin{align} \| A_3\|_{L^{\gamma_1'}_sL^{{p'_1}}}&\leq C_{13}\Big[\int_\R \frac{ds}{(1+|s|)^{3(2+\p_1)\gamma_2'\ipo}}\Big]^{\frac{1}{\gamma_2'}}(\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i} \nonumber \\ &\leq C_{13}C_{10} (\|u_1\|_{Y_i}^{1+\p_1}+\|u_2\|_{Y_i}^{1+\p_1})\|u_1-u_2\|_{Y_i} \nonumber \end{align} where $C_{10}=\int_\R \frac{ds}{(1+|s|)^{3(2+\p_1)\gamma_2'\ipo}}ds<\infty$ since $3(2+\p_1)\gamma_2'\ipo>1$ . \end{itemize} The $L^2$ estimates are now complete and the proof of Lemma \ref{lm} is finished. $\Box$ We now finish the proof of Theorem \ref{mt} by analyzing the dynamics on the center manifold and showing it converges to a ground state. Using the fact that $$i\psi_E=D\psi_E|_a[ia]=\frac{\partial\psi_E}{\partial a_1}\Re [ia]+\frac{\partial\psi_E}{\partial a_2}\Im [ia]$$ equation \eqref{eq:a} becomes $$D\pe|_a(a'+iEa)=\frac{\partial\psi_E}{\partial a_1}\Re [a'+iEa]+\frac{\partial\psi_E}{\partial a_2}\Im [a'+iEa]=\tilde{F}_2(\pe,\eta)=\beta_1(\pe,\eta)\frac{\partial\psi_E}{\partial a_1}+\beta_2(\pe,\eta)\frac{\partial\psi_E}{\partial a_2}$$ Hence $$|a'+iEa|=\sqrt{\beta_1^2+\beta_2^2}=b(t)$$ and $$ \left|[a(t)e^{i\int_0^t E(s)ds}]'\right|=b(t) $$ Since $b(t)=\sqrt{\beta_1^2+\beta_2^2}$, and $$\beta_1\leq\left\|\frac{\partial\pe}{\partial a_2}\right\|_{L^{p_2}}(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})+\left\|\frac{\partial\pe}{\partial a_2}\right\|_{L^{p_1}}\|A_2\|_{L^{p'_1}}\leq C(\|\eta\|^{2}_{L^{p_2}}+\|\eta\|^{2+\p_2}_{L^{p_2}}+\|\eta\|^{2+\p_1}_{L^{p_1}})$$ $$\beta_2\leq\left\|\frac{\partial\pe}{\partial a_1}\right\|_{L^{p_2}}(\|A_1\|_{L^{p'_2}}+\|A_3\|_{L^{p'_2}})+\left\|\frac{\partial\pe}{\partial a_1}\right\|_{L^{p_1}}\|A_2\|_{L^{p'_1}}\leq C(\|\eta\|^{2}_{L^{p_2}}+\|\eta\|^{2+\p_2}_{L^{p_2}}+\|\eta\|^{2+\p_1}_{L^{p_1}})$$ we get $0\le b(t)\le C(1+|t|)^{1+\delta}$ for some $\delta>0,$ in each of the cases $(i)$, $(ii)$ and $(iii)$ in the Theorem \ref{mt}. Then, for any $\varepsilon>0$ we have \begin{equation} \left|a(t)e^{i\int_0^{t} E(s)ds}-a(t')e^{i\int_0^{t'} E(s)ds}\right|\le \int_{t'}^{t} b(s)ds<\varepsilon \end{equation} for $t,\ t'$ sufficiently large respectively sufficiently small. Therefore $a(t)e^{i\int_0^t E(s)ds}$ has a limit when $t\rightarrow\pm\infty$. This means $$e^{i\int_0^t E(s)ds}\pe=a(t)e^{i\int_0^t E(s)ds}\po+e^{i\int_0^t E(s)ds}h(a(t))=a(t)e^{i\int_0^t E(s)ds}\po+h(a(t)e^{i\int_0^t E(s)ds})\rightarrow\psi_{E_{\pm\infty}}$$ Above we used $h(e^{i\theta}a)=e^{i\theta}h(a)$, see Proposition \ref{pr:cm}. In addition $|a(t)|\rightarrow a_{\pm}$ as $t\rightarrow\pm$ at a rate $|t|^{-\delta}.$ Since $E(s)=E(|a(s)|$ is $C^1$ in $|a|$ on $|a|\le\delta_2,$ we deduce $|E(\pm s)-E_\pm|\le C(1+s)^{-\delta}$ for $s\ge 0$ and some constant $C>0.$ If we denote $$\theta(\pm t)=\frac{1}{\pm t}\int_0^{\pm t}E(s)-E_\pm ds,\qquad t\ge 0$$ then $\lim_{|t|\rightarrow\infty}\theta(t)=0$ and $$\lim_{t\rightarrow\pm}e^{it(E_\pm-\theta(t))}\psi_E(t)=\psi_{E_\pm}.$$ This finishes the proof of Theorem \ref{mt}. $\Box$ \section{Linear Estimates}\label{se:lin} Consider the linear Schr\" odinger equation with a potential in three space dimensions: \begin{align} i\frac{\partial u}{\partial t}&=(-\Delta+V(x))u \nonumber \\ u(0)&=u_0 \nonumber \end{align} It is known that if V satisfies hypothesis (H1) (i) and (ii) then the radiative part of the solution, i.e. its projection onto the continuous spectrum of $H=-\Delta+V$, satisfies the estimates: \begin{equation} \|e^{-iHt}P_c u_0\|_{\Lsn}\leq C_M\frac{1}{|t|^{\frac{3}{2}}}\|u_0\|_{\Ls} \label{eq:ls} \end{equation} for $\sigma>1$ and some constant $C_M>0$ independent of $u_0$ and $t\in\R$, and \begin{equation} \|e^{-iHt}P_c u_0\|_{L^p}\leq C_p\frac{1}{|t|^{3(\frac{1}{2}-\frac{1}{p})}}\|u_0\|_{L^{p'}} \label{eq:lp} \end{equation} for some constant $C_p>0$ depending only on $2\leq p$. The case $p=\infty$ in (\ref{eq:lp}) is proved by Goldberg and Schlag in \cite{gs:dis}. The conservation of the $L^2$ norm gives the $p=2$ case: $$\|e^{-iHt}P_c u_0\|_{L^2}=\|u_0\|_{L^2}$$ The general result (\ref{eq:lp}) follows from Riesz-Thorin interpolation. We would like to extend these estimates to the linearized dynamics around the center manifold. We consider the linear equation, with initial data at time $s$, \begin{align} i\frac{d\zeta}{dt}&=H\zeta+F_1(\psi_E,\zeta) \nonumber \\ \zeta(s)&=v \nonumber \end{align} where $F_1(\pe,\zeta)=\frac{d}{d\varepsilon}g(\pe+\varepsilon\zeta)|_{\varepsilon=0}=\frac{\partial}{\partial u}g(u)|_{u=\pe}\zeta+\frac{\partial}{\partial \bar{u}}g(u)|_{u=\pe}\overline{\zeta}$. For the sake of simpler notation, we will use $F_1(\zeta)$. By Duhamel's principle we have: \begin{equation} \zeta(t)=\ehs v(s)-i\int_s^t \eht F_1(\zeta)d\tau \label{soz} \end{equation} In the next theorems we will extend estimates of type (\ref{eq:ls})-(\ref{eq:lp}) to the operators $\om(t,s)$ and $T(t,s)$ considering the fact that $\pe(t)$ is small. Recall that $$T(t,s)=P_c\om(t,s)-\ehs P_c\quad\textrm{i.e.}\quad \om(t,s)=R_{a(t)} T(t,s)+R_{a(t)}\ehs P_c$$ \begin{theorem}\label{th:lw} There exists $\varepsilon_1>0$ such that for $\|\xs\pe\|_{H^2}<\varepsilon_1$ there exist constants $C$, $C_p >0$ with the property that for any t, s $\in\R$ the followings hold: \begin{align} (i)\quad &\|\om(t,s)\|_{\Ls\rightarrow\Lsn}\leq\frac{C}{(1+|t-s|)^{\frac{3}{2}}} \nonumber \\ (ii) \quad &\|T(t,s)\|_{L^1\rightarrow\Lsn}\leq\left\{ \begin{array}{ll} \frac{C}{|t-s|^{\frac{1}{2}}} & \textrm{for $s\leq t\leq s+1$} \\ \frac{C}{(1+|t-s|)^{\frac{3}{2}}} & \textrm{for $t>s+1$} \end{array} \right. \nonumber \\ (iii)\quad &T(t,s)\in L_t^2(\R,L^2\rightarrow\Lsn)\cap L_t^\infty(\R,L^2\rightarrow\Lsn) \nonumber \\ (iv)\quad&\|\om(t,s)\|_{L^{p'}\rightarrow\Lsn}\leq\frac{C}{|t-s|^{3(\frac{1}{2}-\frac{1}{p})}}\quad\textrm{for all $2\leq p\leq L^\infty$} \nonumber \\ &\|T(t,s)\|_{L^{p'}\rightarrow\Lsn}\leq\left\{ \begin{array}{ll} \frac{C}{|t-s|^{(\frac{1}{2}-\frac{1}{p})}} & \textrm{for $s\leq t\leq s+1$} \\ \frac{C}{(1+|t-s|)^{3(\frac{1}{2}-\frac{1}{p})}} & \textrm{for $t>s+1$} \end{array} \right. \nonumber \end{align} \end{theorem} \textbf{Proof of Theorem \ref{th:lw}} Fix $s\in\R$. $(i)$ By definition, we have $\om(t,s)v=\zeta(t)$ where $\zeta(t)$ satisfies equation \eqref{soz}. We project \eqref{soz} onto continuous spectrum of $H=-\Delta+V:$ \begin{equation} \xi(t)=\ehs P_c v-i\int_s^t \eht P_c F_1(R_a\xi)d\tau \label{sox} \end{equation} where $\xi=P_c\zeta$. We are going to prove the estimate for $P_c\Omega(t,s)$ by showing that the nonlinear equation \eqref{sox} can be solved via contraction principle argument in an appropriate functional space. To this extent let us consider the functional space $$X_1:=\{u\in C(\R,L_{-\sigma}^2(\R^3))|\sup_{t>s}(1+(t-s))^{\frac{3}{2}}\|u(t)\|_{L_{-\sigma}^2}<\infty\}$$ endowed with the norm $$\|u\|_{X_1}:=\sup_{t>s}\{(1+(t-s))^{\frac{3}{2}}\|u(t)\|_{L_{-\sigma}^2}\}<\infty$$ Note that the inhomogeneous term in (\ref{sox}) $\xi_0=\ehs P_c v$ satisfies $\xi_0\in X_1$ and \begin{equation} \|\xi_0\|_{X_1}\leq C_M\|v\|_{\Ls} \label{eq:z0} \end{equation} because of (\ref{eq:ls}). We collect the $\xi$ dependent part of the right hand side of (\ref{sox}) in a linear operator $L(s):X_1\rightarrow X_1$ \begin{equation} [L(s)\xi](t)=-i\int_s^t \eht P_c[F_1(R_a\xi)]d\tau \label{L} \end{equation} We will show that $L$ is a well defined bounded operator from $X_1$ to $X_1$ whose operator norm can be made less or equal to 1/2 by choosing $\varepsilon_1$ sufficiently small. Consequently $Id-L$ is invertible and the solution of the equation (\ref{sox}) can be written as $\xi=(Id-L)^{-1}\xi_0$. In particular $$\|\xi\|_{X_1}\leq(1-\|L\|)^{-1}\|\xi_0\|_{X_1}\leq 2\|\xi_0\|_{X_1}$$ which in combination with the definition of $\om$, the definition of the norm $X_1$ and the estimate (\ref{eq:z0}), finishes the proof of $(i)$. It remains to prove that $L$ is a well defined bounded operator from $X_1$ to $X_1$ whose operator norm can be made less than 1/2 by choosing $\varepsilon_1$ sufficiently small. \begin{align} \|L(s)\xi(t)\|_{L_{-\sigma}^2}\leq & \int_s^t \|\eht P_c\|_{L_{\sigma}^2\rightarrow L_{-\sigma}^2}\|F_1(R_a\xi)\|_{\Ls}d\tau \nonumber \\ \end{align} On the other hand $$\|F_1(R_a\xi)\|_{\Ls}\leq\|\xst(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\|_{L^{\infty}}\|R_a \xi\|_{\Lsn}\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2})\|\xi\|_{\Lsn}$$ and using the last three relations, as well as the estimate (\ref{eq:ls}) and the fact that $\xi\in X_1$ we obtain that \begin{align} \|L(s)\|_{X_1\rightarrow X_1}&\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \sup_{t>0}(1+|t-s|)^{\frac{3}{2}}\int_s^t \frac{1}{(1+|t-\tau|)^{\frac{3}{2}}} \cdot \frac{1}{(1+|\tau-s|)^{\frac{3}{2}}}{\ } d\tau \nonumber \\ &\leq(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \sup_{t>0}(1+|t-s|)^{\frac{3}{2}}\frac{1}{(1+|\frac{t-s}{2}|)^{\frac{3}{2}}}\leq C(\varepsilon_1^{1+\p_1}+\varepsilon_1^{1+\p_2}) \nonumber \end{align} Now choosing $\varepsilon_1$ small enough we get $$\|L\|_{X_1\rightarrow X_1}<\frac{1}{2}$$ Therefore $$\|P_c\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\leq\frac{\tilde C}{(1+|t-s|)^{\frac{3}{2}}}$$ and $$\|\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\leq \|R_{a(t)}\|_{\Lsn\rightarrow\Lsn}\|P_c\Omega(t,s)\|_{\Ls\rightarrow\Lsn}\le \frac{C}{(1+|t-s|)^{\frac{3}{2}}}$$ by Lemma \ref{le:pcinv}. $(ii)$ Recall that \begin{equation} P_c\om(t,s)v=T(t,s)v+\ehs P_c v \label{omt} \end{equation} Denote: \begin{equation} T(t,s)v=W(t) \label{w} \end{equation} then, by plugging in (\ref{sox}), $W(t)$ satisfies the following equation: \begin{equation} W(t)=\underbrace{-i\int_s^t \eht P_c [F_1(R_a\ehts P_c v)]d\tau}_{f(t)}+[L(s)W](t) \label{sow} \end{equation} By definition of $T(t,s)$ (\ref{w}) it is sufficient to prove that the solution of (\ref{sow}) satisfies $$\|W(t)\|_{\Lsn}\leq\left\{ \begin{array}{ll} \frac{C\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}} & \textrm{for $s\leq t\leq s+1$} \\ \frac{C\|v\|_{L^1}}{(1+|t-s|)^{\frac{3}{2}}} & \textrm{for $t>s+1$} \end{array} \right.$$ Let us also observe that it suffices to prove this estimate only for the forcing terms $f(t)$ because then we will be able to do the contraction principle in the functional space in which $f(t)$ will be, and thus obtain the same decay for $W$ as for $f(t)$. This time we will consider the functional space $$X_2=\{u\in C(\R,\Lsn{\R^3})| \sup_{|t-s|>1} (1+|t-s|)^\frac{3}{2}\|u\|_{\Lsn}<\infty, \sup_{|t-s|\leq1} |t-s|^\frac{1}{2}\|u\|_{\Lsn}<\infty \}$$ endowed with the norm $$\|u\|_{X_2}=\left\{ \begin{array}{ll} \sup_t |t-s|^\frac{1}{2}\|u\|_{\Lsn} & \textrm{for $|t-s|\leq 1$} \\ \sup_t (1+|t-s|)^\frac{3}{2}\|u\|_{\Lsn} & \textrm{for $|t-s|>1$} \end{array} \right.$$ Now we will estimate $f(t)$. First we will investigate the short time behavior of this term. If $s\leq t\leq s+1$. Recall that $F_1(u)=\frac{d}{d\tau}g(\pe+\tau u)=\frac{\partial}{\partial u}g(u)|_{u=\pe}u+\frac{\partial}{\partial \bar{u}}g(u)|_{u=\pe}\overline{u}=g_{u}u+g_{\bar{u}}\overline{u}.$ $$\|f(t)\|_{\Lsn}\leq\|\xsn\int_s^t \eht P_c F_1(R_a\ehts P_c v) d\tau\|_{L^2}$$ For the term $g_u R_a\ehts P_c v$ we have \begin{align} &\|\xsn\int_s^t \|\eht P_c g_u R_a\ehts P_c v d\tau\|_{L^2} \nonumber\\ \leq&\|\xsn\|_{L^2}\int_s^t \|\ehs e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty} d\tau \nonumber\\ &\leq \int_s^t\frac{C}{|t-s|^{\frac{3}{2}}}\sup \|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}\sup\|\widehat{g_u}\|_{L^1} <\infty \nonumber \end{align} and for the term $g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v$ we have \begin{align} &\|\xsn\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v d\tau\|_{L^2} \nonumber\\ &\leq\|\xsn\|_{L^2}\int_s^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^\infty} d\tau \nonumber\\ &\quad+\int_{s+\frac{t-s}{4}}^t \|\eht P_c\|_{\Ls\rightarrow\Lsn}\|\xs g_{\bar{u}}\overline{R}_a \ehts P_c v\|_{L^2} d\tau \nonumber \\ &\leq\int_s^{s+\frac{t-s}{4}}\frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau+\int_{s+\frac{t-s}{4}}^t \frac{C}{(1+|t-\tau)^{3/2}}\|\xs g_{\bar{u}}\|_{L^2} \|\ehts P_c v\|_{L^\infty} d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}\sup (\|\widehat{g_{\bar{u}}}\|_{L^1}+\|\xs g_{\bar{u}}\|_{L^2})<\infty \nonumber \end{align} There we used J-S-S type estimate; see Appendix for $t=\tau-s$; $|\tau-s|\leq 1$. For the long time behavior of $f(t)$, we will split this integral into three parts to be estimated differently. For $t>s+1$, $$f(t)=\underbrace{\int_s^{s+\frac{1}{2}}\cdots}_{I_{1}}+\underbrace{\int_{s+\frac{1}{2}}^{\frac{t+s}{2}}\cdots}_{I_{2}}+\underbrace{\int_{\frac{t+s}{2}}^t \cdots}_{I_{3}}$$ Then for $t>s+1$, \begin{align} \|I_1\|_{\Lsn}&\leq\|\xsn\int_s^{s+\frac{1}{2}} \ehs P_c F_1(R_a \ehts P_c v) d\tau\|_{L^2} \nonumber \\ &\leq\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \|\ehs e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty} d\tau \nonumber \\ &\quad+\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^\infty} d\tau \nonumber \\ &\leq\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \frac{C}{|t-s|^{\frac{3}{2}}}\|e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^1} d\tau \nonumber \\ &\quad+\|\xsn\|_{L^2}\int_s^{s+\frac{1}{2}} \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c v\|_{L^1} d\tau \nonumber \\ &\leq C\|\xsn\|_{L^2}\Big(\frac{1}{|t-s|^{\frac{3}{2}}}+\frac{1}{|t-s-1|^{\frac{3}{2}}}\Big)\int_s^{s+\frac{1}{2}} \sup (\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\|v\|_{L^1}d\tau \nonumber \\ &\leq C\|\xsn\|_{L^2}\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\frac{1}{(1+|t-s|)^{\frac{3}{2}}}\|v\|_{L^1} \nonumber \end{align} For the second integral we have \begin{align} \|I_2\|_{\Lsn}&\leq\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \|\eht P_c\|_{\Ls\rightarrow\Lsn}\|F_1(R_a\ehts P_c v)\|_{\Ls} d\tau \nonumber \\ &\leq\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \frac{C}{(1+|t-\tau|)^{\frac{3}{2}}}\|\xs|\pe|^{1+\p}\|_{L^2}\|\ehts P_c v\|_{L^\infty}d\tau \nonumber \\ &\leq\frac{C\|v\|_{L^1}}{(1+|\frac{t-s}{2}|)^{\frac{3}{2}}}\int_{s+\frac{1}{2}}^{\frac{t+s}{2}} \frac{d\tau}{|\tau-s|^{\frac{3}{2}}}\leq\frac{C\|v\|_{L^1}}{(1+|t-s|)^{\frac{3}{2}}} \nonumber \end{align} $I_3$ is estimated similiar to $I_2$. $(iii)$ From \eqref{sow} we have \begin{align} \xsn W(t)=&\int_s^t \xsn\eht P_c[F_1(R_a\ehts P_c v)]d\tau+\int_s^t \xsn\eht P_c[F_1(R_a W(\tau))]d\tau \nonumber \end{align} Then \begin{align} \|\xsn W(t)\|_{L_t^2 L_x^2}&\leq\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}} \|\xs F_1(R_a\ehts P_c v)\|_{L^2}d\tau\Big\|_{L^2_t} \nonumber \\ &+\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}}(\|\xst g_u\|_{L^\infty}+\|\xst g_{\bar{u}}\|_{L^\infty})\|\xsn W(\tau)\|_{L^2_x}d\tau \Big\|_{L^2_t} \nonumber \\ &\leq C\|K\|_{L^1}\|v\|_{L^2}+\varepsilon_1 C\|K\|_{L^1}\|\xsn W\|_{L^2_x L^2_t} \nonumber\end{align} Where $K(t)=(1+|t|)^{-3/2}$. For the term $\xs F_1(R_a e^{-iHt} P_c v)=\xs (g_u R_a e^{-iHt} P_c v+g_{\bar{u}}\overline{R}_a e^{iHt} P_c v)$ we used $\|\xst g_u\|_{L^\infty}$ and $\|\xst g_{\bar{u}}\|_{L^\infty}$ is uniformly bounded in $t$ since $|g_u|=|g_{\bar{u}}|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})$ and the Kato smoothing estimate $\|\xsn e^{-iHt} P_c v\|_{L^2_t(\R,L^2_x)}\leq C\|v\|_{L^2_x}$. Choosing $\varepsilon_1$ small enough we get $\|\xsn W\|_{L^2_x L^2_t}<\infty$. In other words $T(t,s)\in L_t^2(\R, L^2\rightarrow\Lsn)$. And similarly \begin{align} \|\xsn W(t)\|_{L_x^2}&\leq\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}} \|\xs F_1(R_a\ehts P_c v)\|_{L^2}d\tau \nonumber \\ &+\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}}(\|\xst g_u\|_{L^\infty}+\|\xst g_{\bar{u}}\|_{L^\infty})\|\xsn W(\tau)\|_{L^2_x}d\tau \nonumber \\ &\leq C\|v\|_{L^2}+\varepsilon_1 C\|\xsn W\|_{L^2_x} \nonumber\end{align} This finishes the proof of $(iii)$, $T(t,s)\in L_t^2(\R, L^2\rightarrow\Lsn)\cap L_t^\infty(\R, L^2\rightarrow\Lsn).$ $(iv)$ By Riesz-Thorin interpolation between $(ii)$ and $(iii)$ (the $L_t^\infty$ part) we get the desired estimates. $\Box$ The next step is to obtain estimates for $\om(t,s)$ and $T(t,s)$ in unweighted $L^p$ spaces. \begin{theorem}{\label{th:lp}} Assume that $\|\xs\pe\|_{H^2} < \varepsilon_1$ (where $\varepsilon_1$ is the one used in Theorem \ref{th:lw}). Then there exist constants $C_2$, $C_2'$ and $C_\infty$ for all $t$, $s$ $\in$ $\R$ the following estimates hold: \begin{align} (i)\quad &\|\om(t,s)\|_{L^2\rightarrow L^2}\leq C_2,\quad \|T(t,s)\|_{L^2\rightarrow L^2}\leq C_2 \nonumber \\ (ii)\quad &\|\om(t,s)\|_{L^1\rightarrow L^\infty}\leq\frac{C_\infty}{|t-s|^{\frac{3}{2}}}, \quad \|T(t,s)\|_{L^1\rightarrow L^\infty}\leq\left\{ \begin{array}{ll} C_\infty|t-s|^{\frac{1}{2}} & \textrm{for $|t-s|\leq 1$} \\ \frac{C_\infty}{|t-s|^{\frac{3}{2}}} & \textrm{for $|t-s|>1$} \end{array} \right. \nonumber \\ %(iii)\quad&\|P_c\om(t,s)\|_{L^{p'}\rightarrow L^p}\leq\frac{C}{|t-s|^{3\ip}},\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\ %&\|T(t,s)\|_{L^{p'}\rightarrow L^p}\leq\left\{ \begin{array}{ll} %C|t-s|^{\ip} & \textrm{for $|t-s|\leq 1$} \\ %\frac{C}{|t-s|^{3\ip}} & \textrm{for $|t-s|>1$} \end{array} \right.,\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\ (iii)\quad &\|T(t,s)\|_{L^{p'}\rightarrow L^2}\leq C_2', \quad\textrm{for }\quad p=6 \nonumber\end{align} \end{theorem} \begin{remark}\label{rmk:lp} By Riesz-Thorin interpolation from $(i)$ and $(ii)$, and from $(i)$ and $(iii)$ we get \begin{eqnarray} \|\om(t,s)\|_{L^{p'}\rightarrow L^p}&\leq&\frac{C_p}{|t-s|^{3\ip}},\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\ \|T(t,s)\|_{L^{p'}\rightarrow L^p}&\leq&\left\{ \begin{array}{ll} C_p|t-s|^{\ip} & \textrm{for $|t-s|\leq 1$} \\ \frac{C}{|t-s|^{3\ip}} & \textrm{for $|t-s|>1$} \end{array} \right.,\quad\textrm{for all}\quad 2\leq p\leq\infty \nonumber \\ \|T(t,s)\|_{L^{p'}\rightarrow L^2}&\leq &C_p, \quad\textrm{for all}\quad 2\leq p\leq6 \nonumber \end{eqnarray} \end{remark} \noindent\textbf{Proof of Theorem \ref{th:lp}} Because of the estimate (\ref{eq:lp}) and relation $P_c\om=T+\ehs P_c$, It suffices to prove the theorem for $T(t,s)$. $(i)$ To estimate the $L^2$ norm we will use duality argument to make use of cancelations. \begin{align} &\|f(t)\|_{L^2}^2=\< f(t),f(t)\> \nonumber \\ &=\int_s^t\int_s^t \< \eht P_c F_1(R_a\ehts P_c v),e^{-iH(t-\tau')} P_c F_1(R_a e^{-iH(\tau' -s)} P_c v)\> d\tau' d\tau \nonumber \\ &=\int_s^t \int_s^t \< F_1(R_a\ehts P_c v),e^{-iH(\tau-\tau')} P_c F_1(R_a e^{-iH(\tau'-s)} P_c v)\> d\tau' d\tau \nonumber \\ &=\int_s^t \int_s^t \< \xs F_1(R_a\ehts P_c v), \xsn e^{-iH(\tau-\tau')} P_c F_1(R_ae^{-iH(\tau'-s)} P_c v) \> d\tau' d\tau \nonumber \\ &\leq\int_s^t \int_s^t \|F_1(R_a\ehts P_c v)\|_{\Ls}\|e^{-iH(\tau-\tau')} P_c F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{\Lsn} d\tau' d\tau \nonumber \\ &\leq\int_s^t \|\xs F_1(R_a\ehts P_c v)\|_{L^2}\int_s^t\frac{C}{(1+|\tau-\tau'|)^{3/2}}\|\xs F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{L^2} d\tau' d\tau\nonumber \\ &\leq C\|\xs F_1(R_a\ehts P_c v)\|_{L_{\tau}^2 L_x^2} \Big\|\int_s^t\frac{C}{(1+|\tau-\tau'|)^{3/2}} \|\xs F_1(R_a e^{-iH(\tau'-s)} P_c v)\|_{L_x^2} d\tau\Big\|_{L^2_{\tau}}\nonumber \\ &\leq C\|K\|_{L^1}\|\xs F_1(R_a e^{-iHt} P_c v)\|_{L_t^2 L^2_x}^2\leq C\|v\|^2_{L^2}<\infty \nonumber \end{align} At the last line, $K(t)=(1+|t|)^{-3/2}$ and we used convolution estimate. For the term $\xs F_1(R_a e^{-iHt} P_c v)=\xs (g_u R_a e^{-iHt} P_c v+g_{\bar{u}}\overline{R}_a e^{iHt} P_c v)$ we used the Kato smoothing estimate $\|\xsn e^{-iHt} P_c v\|_{L^2_t(\R,L^2_x)}\leq C\|v\|_{L^2_x}$. We will estimate $L^2$ norm of $L$ similiar to $f$. %First we will show $W$ satisfies $\|\xsn W\|_{L^2_x}\in L^2_t$, which is similiar to Kato's smoothing estimate. %\begin{align} \xsn W(t)=&\int_s^t \xsn\eht P_c[F_1(\ehts P_c v)]d\tau \nonumber \\ %&+\int_s^t \xsn\eht P_c[F_1(W(\tau))]d\tau \nonumber \end{align} %Then %\begin{align} &\|\xsn W(t)\|_{L_x^2 L_t^2} \nonumber \\ %&\leq\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}} \|\xs F_1(\ehts P_c v)\|_{L^2}d\tau\Big\|_{L^2_t} \nonumber \\ %&+\Big\|\int_s^t \frac{C}{(1+|t-\tau|)^{3/2}}(\|\xst g_u\|_{L^\infty}+\|\xst g_{\bar{u}}\|_{L^\infty})\|\xsn W(\tau)\|_{L^2_x}d\tau \Big\|_{L^2_t} \nonumber \\ %&\leq C\|K\|_{L^1}\|v\|_{L^2}+\varepsilon_1 C\|K\|_{L^1}\|\xsn W\|_{L^2_x L^2_t} \nonumber\end{align} %Where $K(t)=(1+|t|)^{-3/2}$. Choosing $\varepsilon_1$ small enough we get $\|\xsn W\|_{L^2_x L^2_t}<\infty$. Now we will use this estimate to show that, in fact, $L^2$ norm of $L(s)W$ is finite. \begin{align} &\|L(s)W\|_{L^2}^2=\ \nonumber \\ &=\<\int_s^t \eht P_c F_1(W(\tau))d\tau,\int_s^t e^{-iH(t-\tau')} P_c F_1(W(\tau'))d\tau'\> \nonumber \\ &=\int_s^t \int_s^t \< F_1(W(\tau)),e^{-iH(\tau-\tau')} P_c F_1(W(\tau'))\>] d\tau' d\tau \nonumber \\ &\leq\int_s^t (\|\xs g_u\|_{L^\infty}+\|\xs g_{\bar{u}}\|_{L^\infty})\|\xsn W\|_{L^2} \nonumber \\ &\quad\quad\quad\quad\quad\quad\times\int_s^t C K(\tau-\tau')(\|\xs g_u\|_{L^\infty}+\|\xs g_{\bar{u}}\|_{L^\infty})\|\xsn W\|_{L^2} d\tau' d\tau\nonumber \\ &\leq C\|\xsn W\|_{L_x^2 L_\tau^2} \Big\|\int_s^t CK(\tau-\tau') \|\xsn W\|_{L_x^2} d\tau'\Big\|_{L^2_{\tau}}\nonumber \\ &\leq C\|K\|_{L^1}\|\xsn W\|_{L_\tau^2 L^2_x}<\infty \nonumber \end{align} By Theorem \ref{th:lw} $(iii)$, $\|\xsn W\|_{L_\tau^2 L^2_x}<\infty$. \noindent Therefore we conclude $\|T(s,t)\|_{L^2\rightarrow L^2}\leq C$ and $\|\om(s,t)\|_{L^2\rightarrow L^2}\leq C$ $(ii)$ Let us first investigate the short time behavior of the forcing term $f(t)$. We will assume $s\leq t\leq s+1$, \begin{align} \|f(t)\|_{L^\infty}&=\|\int_s^t \eht P_c F_1(R_a \ehts P_c v)d\tau\|_{L^\infty} \nonumber \\ &\leq\int_s^t \|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty}\|e^{iH(\tau-s)}P_c g_u R_a\ehts P_c v\|_{L^1}d\tau \nonumber \\ &+\int_s^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\ &+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}} \|e^{-iH(t-\tau)}P_c\|_{L^1\rightarrow L^infty}\|g_{\bar{u}} \overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\ &+\int_{t-\frac{t-s}{4}}^t \|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\ &\leq\int_s^t \frac{C}{|t-s|^{\frac{3}{2}}}\sup\|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau+\int_s^{s+\frac{t-s}{4}} \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau \nonumber \\ &+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\|v\|_{L^1}d\tau+\int_{t-\frac{t-s}{4}}^t \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\sup \|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{1}{2}}}(\|g_{\bar{u}}\|_{L^1}+\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})) \nonumber \end{align} Now let us investigate the long time bevaviour of the forcing term $f(t)$. We will assume $t>s+1$ and seperate $f(t)$ into four parts as follows, $$f(t)=\underbrace{\int_s^{s+\frac{1}{4}}\cdots}_{I_1}+\underbrace{\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\cdots}_{I_2}+\underbrace{\int_{t-\frac{1}{4}}^t\cdots}_{I_3}$$ We will start with $I_2$ for which we are away from the singularities around $\tau=s$ and $\tau=t$. Then for $I_1$ and $I_3$ we will use J-S-S type estimate to remove the singularities. \begin{align} \|I_2\|_{L^\infty}&\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \|\eht P_c (g_u R_a\ehts P_c v+g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v})\|_{L^\infty}d\tau \nonumber \\ &\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}} (\|g_u R_a\ehts P_c v\|_{L^1}+\|g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1})d\tau \nonumber \\ &\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}( \|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1})\|\ehts P_c v\|_{L^\infty}d\tau \nonumber \\ &\leq\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}( \|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1}) \frac{\|v\|_{L^1}}{|\tau-s|^{\frac{3}{2}}}d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}}(\|g_u\|_{L^1}+\|g_{\bar{u}}\|_{L^1}) \nonumber \end{align} %For $I_1$ and $I_3$ we need to remove the singularity around $\tau=s$ and $\tau=t$. \begin{align} \|I_1\|_{L^\infty}&\leq\int_s^{s+\frac{1}{4}} \|\ehs P_c e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^\infty}d\tau \nonumber \\ &\quad+\int_s^{s+\frac{1}{4}} \|e^{-iH(t+s-2\tau)} \ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^\infty}d\tau \nonumber \\ &\leq\int_s^{s+\frac{1}{4}}\frac{C}{|t-s|^{\frac{3}{2}}}\|e^{iH(\tau-s)} P_c g_u R_a\ehts P_c v\|_{L^1}d\tau \nonumber \\ &\quad+\int_s^{s+\frac{1}{4}}\frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\ehts P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^1}d\tau \nonumber \\ &\leq\int_s^{s+\frac{1}{4}} C \Big(\frac{1}{|t-s|^{\frac{3}{2}}}+\frac{1}{|t-s-1|^{\frac{3}{2}}}\Big)\sup(\|\widehat{g_u}\|_{L^1}+\|\widehat{g_{\bar{u}}}\|_{L^1})\|v\|_{L^1}d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \begin{align} \|I_3\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)} P_c e^{-iH(t-s)} P_cv\|_{L^\infty}d\tau \nonumber \\ &\quad+\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)} P_c e^{iH(t+s-2\tau)} P_c \bar{v}\|_{L^\infty}d\tau \nonumber \\ &\leq C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} Now it remains to show that $L(s)W$ is bounded in $L^\infty$. Again to remove the singularities we will split the integral in different parts. Let us consider $s\leq t\leq s+1$, %$$L(s)W=\underbrace{\int_s^{t-\frac{t-s}{4}} \cdots}_{L_1}+\underbrace{\int_{t-\frac{t-s}{4}}^t \cdots}_{L_2}$$ %First we will show $L_1$ is bounded in $L^\infty$. %\begin{align} %\|L_1\|_{L^\infty}&\leq\int_s^{t-\frac{t-s}{4}} \|\eht P_c F_1(W(\tau))\|_{L^\infty} d\tau \nonumber \\ %&\leq\int_s^{t-\frac{t-s}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\| F_1(W(\tau))\|_{L^\infty} d\tau \nonumber \\ %&\leq\int_s^{t-\frac{t-s}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}}\|F_1(W)\|_{L^1} d\tau \nonumber \\ %&\leq\int_s^{t-\frac{t-s}{4}} \frac{C}{|t-s|^{\frac{3}{2}}}(\|\xs g_u\|_{L^2}+\|\xs g_{\bar{u}}\|_{L^2})\|W\|_{\Lsn} d\tau \nonumber \\ %&\leq\frac{C}{|t-s|^{\frac{3}{2}}}\int_s^{t-\frac{t-s}{4}} \frac{\|v\|_{L^1}}{(1+|\tau-s|)^{\frac{1}{2}}} d\tau \nonumber \\ %&\leq\frac{C}{|t-s|^{\frac{1}{2}}}\|v\|_{L^1} \nonumber \end{align} %To estimate $L_2$, first we will plug in (\ref{sow}) in the this integral once more. This will help us to get some cancellations and we will get rid of the singularity around $\tau=t$. \begin{align} &L(s)W=\int_s^t \eht P_c F_1(R_a W(\tau)) d\tau\nonumber \\ &=\int_s^t \eht P_c g_u R_a\Big[\int_s^\tau e^{-iH(\tau-\tau')} P_c [F_1(R_a e^{-iH(\tau'-s)}P_c v)+F_1(R_a W(\tau'))]d\tau'\Big]d\tau \nonumber \\ &\quad+\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a\Big[\int_s^\tau e^{iH(\tau-\tau')}P_c[\overline{ F_1(R_a e^{-iH(\tau'-s)}P_c v)}+\overline{ F_1(R_a W(\tau'))}]d\tau'\Big]d\tau \nonumber \end{align} All the terms will be either of the following forms $$L_1=\int_s^t \eht P_c g_u R_a\int_{s}^\tau e^{-iH(\tau-\tau')} P_c X(\tau')d\tau'd\tau$$ $$L_2=\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a\int_{s}^\tau e^{iH(\tau-\tau')} P_c \overline{X(\tau')}d\tau'd\tau$$ where $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v, g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}, g_u R_a W(\tau'), g_{\bar{u}}\overline{R_a W(\tau')}$ In what follows we will add $e^{iH(t-\tau)}$ and $\eht$ terms after $g_u R_a$ and $g_{\bar{u}}\overline{R}_a$ then we will estimate the terms in a similiar way as we estimated $I_1$ and $I_3$. \begin{equation} L_1=\int_s^t \eht P_c g_u R_a e^{iH(t-\tau)}\int_{s}^\tau e^{-iH(t-\tau')} P_c X(\tau')d\tau'd\tau \label{l1} \end{equation} \begin{equation}L_2=\int_s^t \eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\int_{s}^\tau e^{-iH(t-2\tau+\tau')} P_c \overline{X(\tau')}d\tau'd\tau \label{l2} \end{equation} \begin{itemize} \item For $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v$ we have \begin{align} \|L_1\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau'd\tau \nonumber \\ &\leq\int_s^t \|\widehat{g_u}\|_{L^1}\int_s^\tau \frac{C}{|t-s|^{\frac{3}{2}}}\|\widehat{g_u}\|_{L^1}\|v\|_{L^1}d\tau'd\tau\leq C\sqrt{t-s}\|v\|_{L^1}\leq C\|v\|_{L^1}\quad\textrm{for }s\leq t\leq s+1 \nonumber \end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_s^{t-\frac{t-s}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a\int_s^{\tau}e^{-iH(\tau-s)}P_c e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'd\tau \nonumber \\ &+\int_{t-\frac{t-s}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty} \nonumber \\ &\quad\quad\quad\quad\times \int_s^{\tau}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'd\tau \nonumber \\ &\leq\int_s^{t-\frac{t-s}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}} \|g_{\bar{u}}\|_{L^1}\int_s^\tau \frac{C}{|\tau-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau'd\tau \nonumber \\ &+\int_{t-\frac{t-s}{4}}^t \|\widehat{g_{\bar{u}}}\|_{L^1}\int_s^\tau \frac{C}{|t+s-2\tau|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}\|v\|_{L^1}d\tau'd\tau\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align} \item For $X(\tau')=g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}$ we have \begin{align} \|L_1\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\ &\quad\quad\quad\times\Big[\int_{s}^{s+\frac{t-s}{4}} \|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\nonumber \\ &\quad\quad\quad+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}}\|e^{-iH(t-\tau')}\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\ &\quad\quad\quad+\int_{t-\frac{t-s}{4}}^{\tau}\|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\ &\leq\int_s^t \|\widehat{g_u}\|_{L^1}\Big[\int_s^{s+\frac{t-s}{4}}\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'+\int_{s+\frac{t-s}{4}}^{t-\frac{t-s}{4}}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\frac{C}{|\tau-s|^{\frac{3}{2}}}d\tau'\nonumber \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{t-\frac{t-s}{4}}^\tau\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'\Big]\|v\|_{L^1}d\tau\nonumber \\ &\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber\end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\Big[\int_{s}^{s+\frac{t-s}{4}}\| e^{-iH(t-2\tau+2\tau'-s)}e^{iH(\tau'-s)} P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\ &\quad\quad\quad+\int_{s+\frac{t-s}{4}}^\tau \|e^{-iH(t-2\tau+\tau')} P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \Big]d\tau \nonumber \\ &\leq\int_s^t \|\widehat{g_{\bar{u}}}\|_{L^1}\Big[\int_s^{s+\frac{t-s}{4}} \frac{C}{|t-2\tau+2\tau'-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1} \|v\|_{L^1}d\tau'+\int_{s+\frac{t-s}{4}}^\tau \frac{C}{|t-2\tau+\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1} \frac{\|v\|_{L^1}}{|\tau'-s|^{\frac{3}{2}}}d\tau'\Big] d\tau \nonumber \\ &\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align} %\begin{align} %\|L_2\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_{\bar{u}} e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\|\int_{s}^\tau e^{-iH(t-2\tau+2\tau'-s)}e^{iH(\tau'-s)} P_c g_u e^{-iH(\tau'-s)}P_c vd\tau'\|_{L^\infty}d\tau \nonumber \\ %&\leq\int_s^t \|\widehat{g_{\bar{u}}}\|_{L^1}\int_s^\tau \frac{C}{|t-2\tau+2\tau'-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1} \|v\|_{L^1}d\tau'd\tau\leq C\|v\|_{L^1} \nonumber \end{align} %The last inequality is due to $-(t-s)\leq t-2\tau+2\tau'-s\leq -\frac{(t-s)}{2}$. \item For $X(\tau')=g_u R_a W(\tau')$ and $g_{\bar{u}} \overline{R_a W(\tau')}$ we will change the order of integration, \begin{align} \|L_1\|_{L^\infty}&\leq\int_s^t\int_{\tau'}^t\|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\|e^{-iH(t-\tau')}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\ &\leq\int_s^t\int_{\tau'}^t \|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\|W\|_{\Lsn}d\tau d\tau' \nonumber \\ &\leq\int_s^t\|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{1}{2}}} d\tau' \nonumber \\ &\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\|g_u R_a e^{-iH(\tau-\tau')} P_c g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\ &+\int_s^t\int_{t-\frac{t-\tau'}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\| e^{-iH(t+\tau'-2\tau)} P_c g_u R_a W(\tau')\|_{L^\infty}d\tau d\tau' \nonumber \\ &\leq\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}}\frac{C}{|t-\tau|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^2}\|e^{-iH(\tau-\tau')}P_c\|_{L^2\rightarrow L^2}\|g_u R_a W(\tau')\|_{L^2}d\tau d\tau' \nonumber \\ &+\int_s^t\int_{\tau'}^{t-\frac{t-\tau'}{4}} \|\widehat{g_u}\|_{L^1}\frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\|g_u R_a W(\tau')\|_{L^1}d\tau d\tau' \nonumber \\ &\leq\int_s^t\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|g_{\bar{u}}\|_{L^2}\|\xs g_u\|_{L^\infty}\|W(\tau')\|_{\Lsn}d\tau' \nonumber \\ &+\int_s^t \|\widehat{g_u}\|_{L^1}\frac{C}{|t-\tau'|^{\frac{1}{2}}}\|\xs g_u\|_{L^2}\|W(\tau')\|_{\Lsn}d\tau' \nonumber \\ &\leq C\sqrt{|t-s|}\|v\|_{L^1} \nonumber \end{align} \end{itemize} Similarly we will investigate the long time behavior of the operator $L(s)$ for $t>s+1$. $$L(s)W(t)=\underbrace{\int_s^{t-\frac{1}{4}}\cdots}_{L_3}+\underbrace{\int_{t-\frac{1}{4}}^t\cdots}_{L_4}$$ \begin{align} \|L_3\|_{L^\infty}&\leq\int_s^{t-\frac{1}{4}} \|\eht P_c F_1(R_a W(\tau))\|_{L^\infty} d\tau \nonumber \\ &\leq\int_s^{t-\frac{1}{4}} \|\eht P_c\|_{L^1\rightarrow L^\infty}\| F_1(R_a W(\tau))\|_{L^1} d\tau \nonumber \\ %&\leq\int_s^{t-\frac{1}{4}} \frac{C}{|t-\tau|^{\frac{3}{2}}}\|F_1(W)\|_{L^1} d\tau \nonumber \\ &\leq\int_s^{t-\frac{1}{4}} \frac{C}{|t-s|^{\frac{3}{2}}}(\|\xs g_u\|_{L^2}+\|\xs g_{\bar{u}}\|_{L^2})\|W\|_{\Lsn} d\tau \nonumber \\ &\leq\frac{C}{|t-s|^{\frac{3}{2}}}\int_s^{t-\frac{1}{4}} \frac{\|v\|_{L^1}}{(1+|\tau-s|)^{\frac{3}{2}}} d\tau\leq\frac{C}{|t-s|^{\frac{3}{2}}}\|v\|_{L^1} \nonumber \end{align} In $L_4$ we will plug in \eqref{sow} once more: \begin{align} L_4&=\int_{t-\frac{1}{4}}^t \eht P_c F_1(R_a W(\tau)) d\tau\nonumber \\ &=\int_{t-\frac{1}{4}}^t \eht P_c g_u R_a\Big[\int_s^\tau e^{-iH(\tau-\tau')} P_c [F_1(R_a e^{-iH(\tau'-s)}P_c v)+ F_1(R_a W(\tau'))]d\tau'\Big]d\tau \nonumber \\ &\quad+\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a\Big[\int_s^{\tau} e^{iH(\tau-\tau')}P_c[\overline{F_1(R_a e^{-iH(\tau'-s)}P_c v)}+\overline{F_1(R_a W(\tau'))}]d\tau'\Big]d\tau \nonumber \end{align} Again we will add $e^{iH(t-\tau)}$ and $\eht$ terms after $g_u R_a$ and $g_{\bar{u}}\overline{R}_a$. Then all the terms will be similar to $L_1$, $L_2$, $(\ref{l1})-(\ref{l2})$ respectively. After seperating the the inside integrals into pieces, we will estimate short time step integrals exactly the same way we did short time behavior by using JSS estimate, and the other integrals will be estimated using the usual norms. \begin{itemize} \item For $X(\tau')=g_u R_a e^{-iH(\tau'-s)}P_c v$ we have \begin{align} \|L_1\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\ &\quad\quad\Big[\int_s^{s+\frac{1}{4}}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \nonumber \\ &\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}} \|e^{-iH(t-\tau')} P_c\|_{L^1\rightarrow L^\infty}\|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\ &\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t-s)}P_c\|_{L^1\rightarrow L^\infty} \|e^{iH(\tau'-s)}P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau'\Big]d\tau \nonumber \\ &\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty} \nonumber \\ &\quad\quad\quad\quad\times\Big[ \int_s^{s+\frac{1}{4}}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\ &\quad\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\|e^{-iH(t-2\tau+\tau')}P_c\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}} \overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\ &\quad\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t+s-2\tau)}P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\ &\leq\frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \item For $X(\tau')=g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}$ we have \begin{align} \|L_1\|_{L^\infty}\leq&\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\nonumber \\ &\times\Big[\int_{s}^{s+\frac{1}{4}} \|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\nonumber \\ &\quad\quad\quad+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\|e^{-iH(t-\tau')}\|_{L^1\rightarrow L^\infty}\|g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau' \nonumber \\ &\quad\quad\quad+\int_{t-\frac{1}{4}}^{\tau}\|e^{-iH(t+s-2\tau')} P_c\|_{L^1\rightarrow L^\infty}\|e^{-iH(\tau'-s)}P_c g_{\bar{u}}\overline{R}_a e^{iH(\tau'-s)}P_c \bar{v}\|_{L^1}d\tau'\Big]d\tau \nonumber \\ \leq&\int_{t-\frac{1}{4}}^t \|\widehat{g_u}\|_{L^1}\Big[\int_s^{s+\frac{1}{4}}\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'+\int_{s+\frac{1}{4}}^{t-\frac{1}{4}}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1}\frac{C}{|\tau-s|^{\frac{3}{2}}}d\tau'\nonumber \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{t-\frac{1}{4}}^\tau\frac{C}{|t+s-2\tau'|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1}d\tau'\Big]\|v\|_{L^1}d\tau\nonumber \\ \leq& C\frac{\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber\end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_s^t \|\eht P_c g_{\bar{u}}\overline{R}_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\Big[\int_{s}^{s+\frac{1}{4}}\| e^{-iH(t-2\tau+2\tau'-s)}e^{iH(\tau'-s)} P_c g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^\infty}d\tau' \nonumber \\ &\quad\quad\quad+\int_{s+\frac{1}{4}}^\tau \|e^{-iH(t-2\tau+\tau')} P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a e^{-iH(\tau'-s)}P_c v\|_{L^1}d\tau' \Big]d\tau \nonumber \\ &\leq\int_s^t \|\widehat{g_{\bar{u}}}\|_{L^1}\Big[\int_s^{s+\frac{1}{4}} \frac{C}{|t-2\tau+2\tau'-s|^{\frac{3}{2}}}\|\widehat{g_{\bar{u}}}\|_{L^1} \|v\|_{L^1}d\tau'+\int_{s+\frac{1}{4}}^\tau \frac{C}{|t-2\tau+\tau'|^{\frac{3}{2}}}\|g_{\bar{u}}\|_{L^1} \frac{\|v\|_{L^1}}{|\tau'-s|^{\frac{3}{2}}}d\tau'\Big] d\tau \nonumber \\ &\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \item $L_1$, $L_2$ terms corresponding to $X(\tau')=g_u R_a W(\tau')$ and $g_{\bar{u}} \overline{R_a W(\tau')}$ \begin{align} \|L_1\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_u R_a e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t-\tau')}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau'd\tau \nonumber \\ &\leq\int_{t-\frac{1}{4}}^t \|\widehat{g_u}\|_{L^1}\int_s^{\tau}\frac{C}{|t-\tau'|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau' d\tau \nonumber \\ &\leq\int_{t-\frac{1}{4}}^t\Bigg[\int_s^{\frac{t+s}{2}} \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau +\int_{\frac{t+s}{2}}^\tau \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau\Bigg] d\tau' \nonumber \\ &\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \begin{align} \|L_2\|_{L^\infty}&\leq\int_{t-\frac{1}{4}}^t \|\eht P_c g_{\bar{u}}\overline{R_a} e^{iH(t-\tau)}\|_{L^\infty\rightarrow L^\infty}\int_s^{\tau}\|e^{-iH(t+\tau'-2\tau)}P_c\|_{L^1\rightarrow L^\infty} \|g_u R_a W(\tau')\|_{L^1}d\tau'd\tau \nonumber \\ &\leq\int_{t-\frac{1}{4}}^t \|\widehat{g_{\bar{u}}}\|_{L^1}\int_s^{\tau}\frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\|\xs g_u\|_{L^2}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau' d\tau \nonumber \\ &\leq\int_{t-\frac{1}{4}}^t\Bigg[\int_s^{\frac{\tau+s}{2}} \frac{C}{|t+\tau'-2\tau|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau +\int_{\frac{\tau+s}{2}}^\tau \frac{C}{|t-\tau'|^{\frac{3}{2}}}\frac{C\|v\|_{L^1}}{(1+|\tau'-s|)^{\frac{3}{2}}}d\tau\Bigg] d\tau' \nonumber \\ &\leq \frac{C\|v\|_{L^1}}{|t-s|^{\frac{3}{2}}} \nonumber \end{align} \end{itemize} \noindent Now combining all the above estimates we get $$\|W(t)\|_{L^\infty}\leq\left\{ \begin{array}{ll} C |t-s|^{\frac{1}{2}} & \textrm{for $|t-s|\leq 1$} \\ \frac{C}{|t-s|^{\frac{3}{2}}} & \textrm{for $|t-s|>1$} \end{array} \right.$$ This finishes the proof of $(ii)$. %$(iii)$ From part $(i)$ and $(ii)$ we have $$\|T(t,s)\|_{L^2\rightarrow L^2}\leq C$$ and $$\|T(t,s)\|_{L^1\rightarrow L^\infty}\leq\left\{ \begin{array}{ll} %C \sqrt{|t-s|} & \textrm{for $|t-s|\leq 1$} \\ %\frac{C}{|t-s|^{\frac{3}{2}}} & \textrm{for $|t-s|>1$} \end{array} \right.$$ %We can now use Riesz-Thorin interpolation between the spaces $L^2$ and $L^1$ as starting spaces and between $L^2$ and $L^\infty$ as arrival spaces to get the claimed estimate $$\|T(t,s)\|_{L^{p'}\rightarrow L^p}\leq\left\{ \begin{array}{ll} %C|t-s|^{\ip} & \textrm{for $|t-s|\leq 1$} \\ %\frac{C}{|t-s|^{3\ip}} & \textrm{for $|t-s|>1$} \end{array} \right.,\quad\textrm{for all}\quad 2\leq p\leq\infty $$ $(iii)$ We split $f$ given by \eqref{sow}: $$f=\underbrace{\int_s^{s+1}\cdots}_{I_1}+\underbrace{\int_{s+1}^t\cdots}_{I_2}$$ %\begin{equation} %\|I_1\|_{L^2}\leq C\int_s^{s+1}\|g_u\ehts P_c v+g_{\bar{u}}e^{iH(\tau-s)}P_c \bar{v}\|_{L^2}d\tau \nonumber \end{equation} For $I_1$ integral, it suffices to show that $\|g_u(\tau)R_a\ehts P_c v\|_{L^2}\in L^1_\tau[s,s+1]$. Since $g_u(\tau)$ has bounded derivatives we have $\|g_u(\tau)-g_u(s)\|_{L^3}\leq C|\tau-s|$, then by H\"{o}lder inequality in space, $$\|(g_u(\tau)-g_u(s))R_a\ehts P_c v\|_{L^2}\leq\|g_u(\tau)-g_u(s)\|_{L^3}\|\ehts P_c v\|_{L^6} \in L^1_\tau$$ Now it suffices to show $\|g_u(s)R_a\ehts P_c v\|_{L^2}\in L^1_\tau$. For any $\tilde{v}\in L^2$ we have $$\|g_u(s)R_a\ehts P_c v\|_{L^2}=\<\tilde{v},g_u(s)R_a\ehts P_c v\>=\\leq\underbrace{\|e^{iH(\tau-s)}P_c R_a g_u(s)\tilde{v}\|_{L^6}}_{\in L^2_\tau}\|v\|_{L^{6/5}}$$ Since $L^2_\tau[s,s+1]\hookrightarrow L^1_\tau[s,s+1]$, $\|e^{iH(\tau-s)}P_c R_a g_u(s)\tilde{v}\|_{L^6}\in L^1_\tau$. \begin{align} \|I_2\|_{L^2}&\leq C\Big(\int_{s+1}^t \|g_u R_a\ehts P_c v\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}+C\Big(\int_{s+1}^t \|g_{\bar{u}}\overline{R}_a e^{iH(\tau-s)} P_c \bar{v}\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\ &\leq C\Big(\int_{s+1}^t \|\xs g_u\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|\ehts P_c v\|_{L_{-\sigma}^2}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}+C\Big(\int_{s+1}^t \|\xs g_{\bar{u}}\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|e^{iH(\tau-s)} P_c \bar{v}\|_{L_{-\sigma}^2}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\ &\leq C\Big(\int_{s+1}^t \frac{d\tau}{|\tau-s|^{3(\frac{1}{2}-\frac{1}{p})\gamma'}}\Big)^{\frac{1}{\gamma'}}<\infty \nonumber \end{align} At the first inequality we used Strichartz estimate with $(\gamma,\rho)$ with $\gamma>2$ and the last inequality holds since $3(\frac{1}{2}-\frac{1}{p})\gamma'>1$ for $p=6$ and $\gamma>2$. Similarly we will estimate $L(s)W$. \begin{align} \|L(s)W(t)\|_{L^2}&\leq C\Big(\int_s^t \|g_u R_a W+g_{\bar{u}}\overline{R_aW}\|_{L^{\rho'}}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}} \nonumber \\ &\leq C\Big(\int_s^t \|\xs (g_u+g_{\bar{u}})\|_{L^{\frac{3\gamma}{2}}}^{\gamma'}\|W\|_{\Lsn}^{\gamma'}d\tau\Big)^{\frac{1}{\gamma'}}\nonumber \\ &\leq C\Big(\int_s^t \frac{1}{(1+|\tau-s|)^{3(\frac{1}{2}-\frac{1}{p})\gamma'}}\Big)^{\frac{1}{\gamma'}}<\infty \nonumber \end{align} Hence $T(t,s):L^{p'}\rightarrow L^2$ is bounded for $p=6$. %Now using the estimate obtained in part $(i)$, above estimate for $p=6$ and Riesz-Thorin interpolation we obtain the desired estimate for $T(t,s)$. This finishes the proof of part $(iii)$ and the theorem. $\Box$ %Because of the estimate (\ref{eq:lp}) and relation $\om=R_a T+R_a \ehs P_c$, we have the following corollary. %\begin{corollary}{\label{cor}} %Assume that $\|\xs\pe\|_{H^2} < \varepsilon_1$ (where $\varepsilon_1$ is the one used in Theorem \ref{th:lw} and \ref{th:lp}). %Then for all $t$, $s$ $\in$ $\R$ the following estimates hold: %\begin{align} %&(i)\quad \|\om(t,s)\|_{L^2\rightarrow L^2}\leq C \nonumber \\ %&(ii)\quad \|\om(t,s)\|_{L^1\rightarrow L^\infty}\leq\frac{C}{|t-s|^{\frac{3}{2}}} \nonumber \\ %&(iii)\quad\|\om(t,s)\|_{L^{p'}\rightarrow L^p}\leq \frac{C}{|t-s|^{3\ip}},\quad\textrm{ for all}\quad 2\leq p\leq\infty \nonumber \end{align} %\end{corollary} \section{Appendix}\label{se:ap} \subsection{J-S-S type estimates} In \cite{kn:JSS} the authors obtain the following estimate\footnote{Their theorem is stated differently but the proof can be easily adapted to obtain the advertised estimate}: \begin{theorem}\label{th:jss} If $W:\mathbb{R}^n\mapsto\mathbb{C}$ has Fourier transform $\widehat{W}\in L^1(\mathbb{R}^n)$ then for any $t\in\mathbb{R}$ and any $1\le p\le\infty$ we have: $$\|e^{-i\Delta t}We^{i\Delta t}\|_{L^p\mapsto L^p}\le \|\widehat{W}\|_{L^1}$$ \end{theorem} In what follows we are going to generalize the estimate to the semigroup of operators generated by $-\Delta+V:$ \begin{theorem}\label{th:gjss} Assume $V:\mathbb{R}^n\mapsto\mathbb{R}$ and $W:\mathbb{R}^n\mapsto\mathbb{C}$ have Fourier transforms in $L^1(\mathbb{R}^n).$ Then for any $T>0$ there exist a constant $C_T$ independent of $W$ such that for any $-T\le t\le T$ and any $1\le p\le\infty$ we have: $$\|e^{-i(-\Delta +V) t}We^{i(-\Delta +V)t}\|_{L^p\mapsto L^p}\le C_T\|\widehat{W}\|_{L^1}.$$ One can choose $C_T=\exp(2\|\widehat{V}\|_{L^1}T).$ \end{theorem} The proof relies on existence of finite time wave operators: \begin{lemma}\label{lem:waveop} If $V:\mathbb{R}^n\mapsto\mathbb{R}$ has Fourier transform in $L^1(\mathbb{R}^n)$ then for any $T>0$ there exist a constant $C_T$ such that for any $-T\le t\le T$ and any $1\le p\le\infty$ we have: $$\|e^{-i(-\Delta +V) t}e^{-i\Delta t}\|_{L^p\mapsto L^p}\le C_T\|\widehat{W}\|_{L^1},\quad \|e^{i\Delta t}e^{i(-\Delta +V) t}\|_{L^p\mapsto L^p}\le C_T\|\widehat{W}\|_{L^1}.$$ One can choose $C_T=\exp(\|\widehat{V}\|_{L^1}T).$ \end{lemma} {\bf Proof of Lemma:} Let $$H=-\Delta+V$$ then $H$ is a self adjoint operator on $L^2$ with domain $H^2,$ (note that $V\in L^\infty,$) hence it generates a group of isometric operators: $$e^{-iHt}:L^2\mapsto L^2,\qquad t\in\mathbb{R}.$$ Consequently: \begin{equation}\label{def:qt}Q(t)=e^{-iHt}e^{-i\Delta t}:L^2\mapsto L^2,\qquad t\in\mathbb{R}, \end{equation} is also a family of isometric operators. Their infinitesimal generators are: $$\frac{dQ}{dt}=-ie^{-iHt}Ve^{-i\Delta t}=-i\underbrace{e^{-iHt}e^{-i\Delta t}}_{Q(t)}\ \underbrace{e^{i\Delta t}Ve^{-i\Delta t}}_{Q_0(t)}$$ Hence \begin{equation}\label{eq:qt} Q(t)=\mathbb{I}_d-i\int_0^tQ(s)Q_0(s)ds \end{equation} where $$Q_0(t)=e^{i\Delta t}Ve^{-i\Delta t}:L^p\mapsto L^p,\qquad 1\le p\le \infty$$ is bounded uniformly by $\|\widehat V\|_{L^1},$ see Theorem \ref{th:jss}. The contraction principle shows that for any $T>0$ and any $1\le p\le\infty$ the linear equation \eqref{eq:qt} has a unique solution in the Banach space $C([-T,T],B(L^p,L^p)).$ Since on $L^2\bigcap L^p$ the solution is given by \eqref{def:qt} and $L^2\bigcap L^p$ is dense in $L^p$ we obtain that for any $-T\le t\le T$ and any $1\le p\le\infty,$ $e^{-iHt}e^{-i\Delta t}$ has a unique extension to a bounded operator on $L^p.$ Applying the $L^p$ norm in \eqref{eq:qt} we get: $$\|Q(t)\|_{L^p}\le 1+\int_0^t\|Q(s)\|_{L^p} \|\widehat V\|_{L^1}ds$$ and by Gronwall inequality: $$\|Q(t)\|_{L^p}\le e^{\|\widehat V\|_{L^1} |t|}\le e^{\|\widehat V\|_{L^1} T},\quad {\rm for} -T\le t\le T.$$ A similar argument can be made for $Q^*(t)=e^{i\Delta t}e^{iHt}.$ The Lemma is now completely proven. \bigskip \noindent {\bf Proof of Theorem \ref{th:gjss}:} For $H,$ $Q$ and $Q^*$ as in the proof of the previous Lemma we have: $$ e^{-iHt}WE^{iHt}=\underbrace{e^{-iHt}e^{-i\Delta t}}_{Q(t)}\underbrace{e^{i\Delta t}We^{-i\Delta t}}_{L^p\mapsto L^p\ {\rm bounded}}\underbrace{e^{i\Delta t}e^{iHt}}_{Q^*(t)}.$$ Hence using Theorem \ref{th:jss} and Lemma \ref{lem:waveop} we get for any $1\le p\le\infty:$ $$\|e^{-iHt}WE^{iHt}\|_{L^p\mapsto L^p}\le e^{2\|\widehat V\|_{L^1} T}\|\widehat W\|_{L^1},\qquad {\rm for}\ -T\le t\le T. $$ The theorem is now completely proven. \begin{remark}\label{jss} To obtain the linear estimates in Section \ref{se:lin} we used Theorem \ref{th:gjss} in the form: $$\|e^{iHt}WR_ae^{-iHt}\|_{L^p\mapsto L^p}\le C\|\widehat W\|_{L^1},\qquad {\rm for}\ 0\le t\le 1$$ where $W$ is the effective potential induced by the nonlinearity, see next subsection, while $R_a$ is the linear operator defined in Lemma \ref{le:pcinv}. \end{remark} To see why the above estimate holds consider $f\in L^p\bigcap L^2\bigcap {\cal H}_0.$ Then by \eqref{radef} we have for a certain $z=z(f)\in\mathbb{C}:$ $$e^{iHt}WR_ae^{-iHt}f=e^{iHt}We^{-iHt}f+ze^{iHt}W\psi_0.$$ Theorem \ref{th:gjss} applies directly to the first term on the right hand side, while for the second term we use, see \eqref{z:est}: $$|z|\le 2\|f\|_{L^p}\sqrt{\left\|\frac{\partial \psi_E}{\partial a_2}\right\|_{L^{p'}}+\left\| \frac{\partial \psi_E}{\partial a_1}\right\|_{L^{p'}}},\qquad \frac{1}{p}+\frac{1}{p'}=1$$ and the fact that $\psi_0$ is an e-vector of $H$ with e-value $E_0<0$ hence $$\|e^{iHt}W\psi_0\|_{L^p}=\|e^{iHt}We^{-iHt}e^{iE_0t}\psi_0\|_{L^p}\le C\|\widehat W\|_{L^1}\|\psi_0\|_{L^p},$$ where again we used Theorem \ref{th:gjss}. \subsection{Smoothness of the effective potential} In this section we will prove Proposition \ref{pr:dg} i.e. $\widehat{g'(\pe)}$ and $\widehat{(\frac{g(\pe)}{\pe})}$ \bigskip \noindent From by Corollary \ref{co:decay}, we have $\pe\in H^2$ which implies $\pe\in L^p$ for $2\leq p\leq\infty$. Also from \eqref{gest}, by integrating, we get $|g'(s)|\leq C(|s|^{1+\p_1}+|s|^{1+\p_2})$. Hence $|g'(\pe)|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\in L^2$ and $|g''(\pe)|\leq C(|\pe|^{\p_1}+|\pe|^{\p_2})\in L^\infty$. Now we have \begin{align} \|\widehat{g'(\pe)}\|_{L^1}&=\|\frac{1}{1+|\xi|^2}(1+|\xi|^2)\widehat{g'(\pe)}\|_{L^1}\nonumber \\ &\leq\|\frac{1}{1+|\xi|^2}\|_{L^2}\|(1+|\xi|^2)\widehat{g'(\pe)}\|_{L^2}\nonumber \\ &\leq C(\|\widehat{g'(\pe)}\|_{L^2}+\|\widehat{\Delta g'(\pe)}\|_{L^2}) \nonumber \\ &\leq C(\|\underbrace{g'(\pe)}_{\in L^2}\|_{L^2}+\|\Delta g'(\pe)\|_{L^2}) \nonumber\end{align} So it suffices to show that $\Delta g'(\pe)\in L^2$. Similarly it is enough to show that $\Delta (\frac{g(\pe)}{\pe})\in L^2$. \begin{equation} \Delta g'(\pe)=g'''(\pe)|\nabla \pe|^2+\underbrace{g''(\pe)}_{\in L^\infty}\underbrace{\Delta\pe}_{\in L^2}\label{eq:dgpe} \end{equation} and \begin{equation} \Delta (\frac{g(\pe)}{\pe})=(\frac{g''(\pe)}{\pe}-2\frac{g'(\pe)}{\pe^2}+2\frac{g(\pe)}{\pe^3})|\nabla \pe|^2+(\underbrace{\frac{g'(\pe)}{\pe}-\frac{g(\pe)}{\pe^2}}_{\in L^\infty})\underbrace{\Delta\pe}_{\in L^2}\label{eq:gpe} \end{equation} We will use the following comparison theorem proved in \cite[Theorem 2.1]{deiftsimon} to get the upper bound for the $\nabla\pe$ and lower bound for $\pe$: \begin{theorem}\label{th:comp} Let $\varphi\geq0$ be continuous on $\overline{\R^3\setminus K}$ and $A\geq B\geq0$ for some closed set $K$. Suppose that on $\overline{\R^3\setminus K}$, in the distributional sense, $$\Delta |\psi|\geq A|\psi|;\quad\quad \Delta \varphi\leq B\varphi$$ and that $|\psi|\leq\varphi$ on $\partial K$ and $\psi$, $\varphi\rightarrow0$ as $x\rightarrow\infty$. Then $|\psi|\leq\varphi$ on all of $\R^3\setminus K$. \end{theorem} \noindent Note that $\frac{\partial \pe}{\partial x_1}$ and $\pe$ are continuous and $\frac{\partial \pe}{\partial x_1},\ \pe\rightarrow0$ as $|x|\rightarrow\infty$. Hence $\left|\frac{g(\pe)}{\pe}\right|\leq C(|\pe|^{1+\p_1}+|\pe|^{1+\p_2})\rightarrow0$ as $x\rightarrow\infty$. \bigskip \noindent First we need the standard upper bound for $\pe\geq0$. For any $A<-E$, there exists $C_A$ depending on $A$ such that $\pe\leq C_A e^{-\sqrt{A}|x|}$. Indeed if $R$ is sufficiently large, on $\overline{\R^3\setminus B(0,R)}$ we have $$\Delta\pe=[-E+V(x)+\frac{g(\pe)}{\pe}]\pe\geq A\pe,\quad\text{and}\quad \Delta\varphi=A\varphi-\frac{2\sqrt{A}}{|x|}\varphi\leq A\varphi$$ and on $\partial B(0,R)$ we have $\pe\leq C_A e^{-\sqrt{A}|x|}$ for $C_A$ big enough. Then by Theorem \ref{th:comp} we have $\pe\leq C_A e^{-\sqrt{A}|x|}$ on $\R^3\setminus B(0,R)$. \noindent To get the lower bound for $\pe$ we will choose $\varphi=\pe$ and $\psi=C e^{-\sqrt{A_2}|x|}$ in Theorem \ref{th:comp}. On $\overline{\R^3\setminus B(0,R)}$, fix $\varepsilon>0$, $A_2\geq-E+2\varepsilon$ and choose $R$ large enough such that $\frac{2\sqrt{A_2}}{|x|}\leq\varepsilon$ for $|x|\geq R$. Then from \eqref{eq:ev} we have $$\Delta \pe=[-E+V(x)]\pe+g(\pe) \leq[-E+V+\frac{g(\pe)}{\pe}]\pe\leq (-E+\varepsilon)\pe$$ and for $A_2\geq -E+2\varepsilon$ we have $$\Delta \psi=A_2\psi-\frac{2\sqrt{A_2}}{|x|}\psi\geq(-E+\varepsilon)\psi$$ Choose $C$ such that $C e^{-\sqrt{A_2}|x|}\leq\pe$ on $\partial B(0,R)$. Then by theorem \ref{th:comp}, we have $C e^{-\sqrt{A_2}|x|}\leq\pe$ for $|x|>R$. \noindent We will show that for $\psi=\frac{\partial \pe}{\partial x_1}$ and $\varphi=C e^{-\sqrt{A_1}|x|}$ where $A_1<-E$ hypothesis of the theorem \ref{th:comp} is satisfied. Differentiating the eigenvalue equation \eqref{eq:ev} with respect to $x_1$ we get $$\Delta \frac{\partial \pe}{\partial x_1}=[-E+V(x)]\frac{\partial \pe}{\partial x_1}+g'(\pe)\frac{\partial \pe}{\partial x_1}+\frac{\partial V}{\partial x_1}\pe$$ Let $$f^{\pm}=\max\{ 0,\pm f\}\quad\text{and}\quad S_{\leq}=\{x\in\R^3|\Big|\frac{\partial \pe}{\partial x_1}\Big|\leq\pe\}\quad\text{and}\quad S_{\geq}=\{x\in\R^3|\Big|\frac{\partial \pe}{\partial x_1}\Big|\geq\pe\}$$ Fix $A_1<-E$, choose $R$ large enough such that $-E+V(x)+g'(\pe)-\Big|\frac{\partial V}{\partial x_1}\Big|\geq A_1$ on $|x|\geq R$. Let $S=S_{\leq}\cup B(0,R)$, then on $\overline{\R\setminus S}$ we have $$\Delta \Big|\frac{\partial \pe}{\partial x_1}\Big|\geq A_1\Big|\frac{\partial \pe}{\partial x_1}\Big|$$ Now, by continuity of $\frac{\partial \pe}{\partial x_1}$ there exists $C_1$ such that $\Big|\frac{\partial \pe}{\partial x_1}\Big|e^{\sqrt{A_1}|x|}\leq C_1$ on $|x|=R$. Since both on $\frac{\partial \pe}{\partial x_1}$ and $\pe$ are continuous we have $\Big|\frac{\partial \pe}{\partial x_1}\Big|=\pe\leq C_2 e^{\sqrt{A_1}|x|}$ on $\partial S_\leq$. So on $\partial S$, we have $\Big|\frac{\partial \pe}{\partial x_1}\Big|\leq\max\{C_1,C_2\} e^{\sqrt{A_1}|x|}$ \noindent Therefore by theorem \ref{th:comp}, we have $|\nabla \pe|\leq C e^{-\sqrt{A_1}|x|}$ Now we can prove Proposition \ref{pr:dg} \textbf{Proof of Proposition \ref{pr:dg}} By {\bf (H2')} we have $|g'''(s)|<\frac{C}{s^{1-\alpha_1}}+Cs^{\alpha_2 -1},$ $s>0,$ $0<\alpha_1\le\alpha_2;$ then $$|g'''(\pe)|\nabla \pe|^2|\leq\frac{C}{\pe^{1-\p_1}}|\nabla \pe|^2$$ and $$|(\frac{g''(\pe)}{\pe}-2\frac{g'(\pe)}{\pe^2}+2\frac{g(\pe)}{\pe^3})|\nabla \pe|^2|\leq\frac{C}{\pe^{1-\p_1}}|\nabla \pe|^2$$ Using the estimates for $|\nabla \pe|$ and $\pe$ and choosing $2\sqrt{A_1}>\sqrt{A_2}$, we get that $\Delta g'(\pe),\Delta (\frac{g(\pe)}{\pe})\in L^2$. Hence we get the desired estimates for $\widehat{g'(\pe)}$ and $\widehat{\frac{g(\pe)}{\pe}}$. \bibliographystyle{plain} \bibliography{ref} \end{document} ---------------0803240430462--