Content-Type: multipart/mixed; boundary="-------------0508220842830" This is a multi-part message in MIME format. ---------------0508220842830 Content-Type: text/plain; name="05-283.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-283.keywords" Pauli-Fierz enhanced binding ---------------0508220842830 Content-Type: application/x-tex; name="blv4.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="blv4.tex" \documentclass[12pt]{amsart} \usepackage{amssymb,amsfonts,latexsym,amscd} %\usepackage{showkeys} %\setlength\textwidth{6.5 in} \setlength\textheight{9 in} %\voffset=-0.6in \hoffset = -0.6in %\parindent = 0.4in %\pagestyle{plain} \def\1{{\bf 1}} \def\a{\alpha} \def\al{a_\lambda} \def\e{\epsilon} \def\d{{\rm d}} \def\mes{{\rm mes}} \def\cnj{{n\choose j}} \def\Re{{\mathcal Re}} \def\rc{\frac{1}{2}} \def\Rem{Rem} \def\nm{{|\!|\!|\,}} %triple norm \def\ua{\uparrow} %spin-up \def\da{\downarrow} %spindown \def\cw{C_W} \def\bra{\langle} \def\Bra{\Big\langle} \def\ket{\rangle} \def\Ket{\Big\rangle} \def\C{{\mathbb{C}}} % complex numbers \def\N{{\mathbb{N}}} %integers \def\Z{{\mathbb{Z}}} %ganze zahlen \def\R{{\mathbb{R}}} %real numbers \def\Exp{{\mathbb{E}}_\omega} %expectation probability \def\Tor{\mathbb{T}} %Torus \def\cA{{\mathcal A}} \def\cL{{\mathcal L}} \def\cT{{\mathcal T}} \def\cV{{\mathcal V}} \def\cE{{\mathcal E}} \def\cIm{{\mathcal Im}} \def\cB{{\mathcal B}} \def\gm{{\gamma}} \newcommand{\gH}{{\mathfrak H}} %Hilbert space \newcommand{\gF}{{\mathfrak F}} % Fock space \def\Dom{\mathfrak{Dom}} \def\alg{{\mathfrak A}} \def\vac{\Omega_f} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} %\def\Hspace{{\mathfrak H}} %\def\Mspace{{\mathfrak M}} %\def\Polyd{{\mathfrak V}} %\def\Wspace{{\mathfrak W}} %\def\Uspace{{\mathfrak U}} %\def\Tspace{{\mathfrak T}} %\def\Pl{{\mathcal P}} \def\eqnn{\begin{eqnarray*}} \def\eeqnn{\end{eqnarray*}} \def\eqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} \def\bal{\begin{align}} \def\eal{\end{align}} %\newtheorem{theorem}{Theorem}%[section] %\newtheorem{definition}{Definition}%[section] %\newtheorem{proposition}{Proposition}%[section] %\newtheorem{hypothesis}{Hypothesis}%[section] %\newtheorem{lemma}{Lemma}%[section] %\newtheorem{corollary}{Corollary}%[section] %\newtheorem{remark}{Remark}%[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{notation}[theorem]{Notation} %\def\prf{\noindent{\bf Proof.}$\;$} \begin{document} \title[Quantitative estimates on the enhanced binding] {Quantitative estimates on the enhanced binding for the Pauli-Fierz operator} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%% ACKNOWLEDGEMENTS %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \thanks{The authors gratefully acknowledge financial support from the following institutions: The European Union through the IHP network ``Analysis and Quantum'' HPRN-CT-2002-00277 (JMB and SV), the French Ministery of Research through the ACI ``jeunes chercheurs" (JMB), the Volkswagen Stiftung and DIPUC of the Pontific\'\i a Universidad Cat\'olica (HL), and the DFG grant WE 1964/2 (SV)} \author{Jean-Marie Barbaroux \and Helmut Linde \and Semjon Vugalter} \date{10 august 2005} \address{Centre de Physique Th\'eorique, Luminy Case 907, 13288 Marseille Cedex~9, France and D\'epartement de Math\'ematiques, Universit\'e du Sud-Toulon-Var, avenue de l'Universit\'e, 83957 La Garde Cedex, France, barbarou@univ-tln.fr} \address{Facultad de F\`isica, P. Universidad Cat\'olica de Chile, Casilla 306, Santiago 22, Chile } \address{Mathematisches Institut, Ludwig-Maximilians-Universit\"at M\"unchen, Theresienstrasse 39, 80333 M\"unchen and Institut f\"ur Analysis, Dynamik und Modelierung, Universit\"at Stuttgart, wugalter@mathematik.uni-muenchen.de} %%%%%%%%%%%%%%%%%%%%%%% abstract %%%%%%%%%%%%%%%%%%%%%%%% \maketitle \begin{abstract} For a quantum particle interacting with a short-range potential, we estimate from below the shift of its binding threshold, which is due to the particle interaction with a quantized radiation field. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Recently, the question of enhanced binding in nonrelativistic QED has been extensively studied in several publications \cite{HiroshimaSpohn2001, Hainzletal2003, Chenetal2003, CattoHainzl2004, BenguriaVugalter2004, Cattoetal2004}. Dressing a charged particle with photons increases the ability of a potential to confine it. %The free infraparticle binds a larger quantity of low-energetic %photons than the confined particle and thus possesses a larger %effective mass. In order for the particle to leave the potential %well, and additional energetic effort, proportional to the %difference of the two effective masses, is therefore necessitated, %relative to the situation with coupling to the quantized %electromagnetic field. For the Pauli-Fierz operator which describes a nonrelativistic particle interacting with a radiation field, this effect was proved for small values of the fine structure constant $\alpha$, first under the simplifying assumption that the spin of the particle is absent \cite{Hainzletal2003}, and later generalized to the case of a particle with spin \cite{Chenetal2003, CattoHainzl2004}. In \cite{BenguriaVugalter2004}, it was shown that the effect of the enhanced binding is asymptotically small in $\alpha$ in the sense that the binding threshold for the Pauli-Fierz operator tends to the binding threshold for the corresponding Schr\"odinger operator as $\alpha$ tends to zero. Some quantitative estimates on this effect were obtained in \cite{Cattoetal2004} where it was proved that the difference between the binding threshold for the Schr\"odinger operator and the corresponding Pauli-Fierz operator with spin zero is at least of the order $\alpha$. In the work at hand, using a different method, we prove similar results for the more general case of a particle with spin zero or one half. Notice that studying the enhanced binding effect in the case of a particle with spin requires recovering one more term of the energy's expansion in powers of $\alpha$ than in the spinless case. The method of the proof is a further development of a method used in \cite{Hainzletal2003} and \cite{Chenetal2003}. We prove that the Pauli-Fierz operator has a ground state even for some value of the potential coupling constant that is smaller than the binding threshold for the corresponding Schr\"odinger operator. To do so, we construct a trial function for which the quadratic form of the Pauli-Fierz operator with this coupling constant takes a value strictly less than the self-energy. Then we apply \cite[Theorem~2.1]{Griesemeretal2001} which tells us that this implies the existence of a ground state. The trial function we use is similar to the one in \cite{Chenetal2003} with some modifications necessary to obtain quantitative estimates in the case with spin. It is constructed using the ground state of the self-energy operator with total momentum zero. As in all previous papers \cite{Hainzletal2003, Chenetal2003, CattoHainzl2004, BenguriaVugalter2004, Cattoetal2004}, our method is asymptotic in $\alpha$. Therefore, the problem of establishing the enhanced binding effect and estimating its strength for the physical value of $\alpha\approx 1/137$ still remains open. \section{Definitions and main result}\label{S1} The Pauli-Fierz Hamiltonian $H$ for a charged particle with or without spin in an external electrostatic potential and coupled to the quantized electromagnetic radiation field is defined by \begin{equation}\label{rpf} \begin{split} H \!\!=\!\! \left(- i\nabla_{x}\otimes I_f\! +\! \sqrt{\alpha} A(x)\right)^2\!\! +\! g\sqrt{\alpha}\sigma\!\!\cdot\!\! B(x)\!\! +\!\! \lambda W(x)\!\otimes\! I_f\! +\! I_{el}\!\otimes\! H_f\! -\!c_{\rm n.o.}\alpha. \end{split} \end{equation} The operator $H$ acts on the Hilbert space $\gH:=\gH^{el}\otimes \gF$. The Hilbert space $\gH^{el}$ of the nonrelativistic particle is $L^2(\R^3)\otimes \C^2$ in the case $g=1$ and $L^2(\R^3)$ in the case $g=0$. Here $\R^3$ is the configuration space of a single particle, while $\C^2$ accommodates its spin in the case $g=1$. We will describe the quantized electromagnetic field by use of the Coulomb gauge condition. Accordingly, the one-photon Hilbert space is given by $L^2(\R^3)\otimes \C^2$, where $\R^3$ denotes either the photon momentum or configuration space, and $\C^2$ accounts for the two independent transversal polarizations of the photon. The photon Fock space is then defined by $$ \gF = \bigoplus_{n=0}^\infty \gF_s^{(n)} , $$ where the n-photons space $\gF_s^{(n)} = \bigotimes_s^n\left(L^2(\R^3)\otimes\C^2\right)$ is the symmetric tensor product of $n$ copies of $L^2(\R^3)\otimes\C^2$. We use units such that $\hbar = c = 1$, and where the mass of the particle equals $m=1/2$. The particle charge is then given by $e=\sqrt{\alpha}$. As usual, we will consider $\alpha$ as a small parameter. The operator that couples a particle to the quantized vector potential is given by \begin{equation}\nonumber \begin{split} A(x) = & \sum_{\lambda = 1,2} \int_{\R^3} \frac{\zeta(|k|)}{2\pi|k|^{1/2}} \varepsilon_\lambda(k)\Big[ e^{ikx} \otimes a_\lambda(k) + e^{-ikx} \otimes a_\lambda^\ast (k) \Big] \d k \\ =: &D(x) + D^\ast (x), \end{split} \end{equation} where ${\rm div}A =0$ by the Coulomb gauge condition. The operators $a_\lambda$, $a_\lambda^*$ satisfy the usual commutation relations $$ [a_\nu(k), a^\ast_\lambda(k')] = \delta (k-k') \delta_{\lambda, \nu}, \quad [a_\nu(k), a_\lambda(k')] = 0 . $$ The vectors $\varepsilon_\lambda(k)\in\R^3$ are the two orthonormal polarization vectors perpendicular to $k$, \begin{equation}\label{def-eps} \varepsilon_1(k) = \frac{(k_2, -k_1, 0)}{\sqrt{k_1^2 + k_2^2}}\qquad {\rm and} \qquad \varepsilon_2(k) = \frac{k}{|k|}\wedge \varepsilon_1(k). \end{equation} The function $\zeta(|k|)$ describes the {\it ultraviolet cutoff} on the wavenumbers $k$. We assume $\zeta$ to be of class $C^1$ and to have compact support. The constant $c_{\rm n.o.}$ is $$ c_{\rm n.o.} = [D,D^*] = \frac{2}{\pi}\int_0^\infty r|\zeta(r)|^2 {\rm d}r, $$ and subtraction of the constant $c_{\rm n.o.}\alpha$ amounts to normal ordering of the operator $A^2$. The operator that couples a particle to the magnetic field $B = {\rm curl}A$ is given by \begin{equation}\nonumber \begin{split} B (x) = &\displaystyle\sum_{\lambda=1,2}\! \int_{\R^3}\! \frac{\zeta(|k|)}{2\pi|k|^{1/2}} k\times i\varepsilon_\lambda(k) \Big[ e^{ikx}\otimes a_\lambda(k) - e^{-ikx}\otimes a_\lambda^\ast(k)\Big] \d k \\ =: & K(x) + K^\ast (x). \end{split} \end{equation} In Equation~\eqref{rpf}, $\sigma = (\sigma_1, \sigma_2, \sigma_3)$ is the 3-component vector of Pauli matrices \begin{eqnarray*} \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\, , \ \ \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\, , \ \ \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\, . \end{eqnarray*} The photon field energy operator $H_f$ is given by \begin{equation}\nonumber H_f = \sum_{\lambda= 1,2} \int_{\R^3} |k| a_\lambda^\ast (k) a_\lambda (k) \d k. \end{equation} The multiplicative potential $W$ is assumed to be short range and in $L^4_{\rm loc}(\R^3)$, and $\lambda$ is a positive coupling constant. If the negative part of $W$ is nontrivial, then there exists a critical value $\lambda_0$ such that the Schr\"odinger operator $-\Delta + \lambda W$ has discrete spectrum for all $\lambda>\lambda_0$, but does not have any discrete spectrum for $0\leq \lambda<\lambda_0$. Analogously, the Pauli-Fierz operator also has a critical coupling constant $\lambda_1$, which depends on the fine structure constant $\alpha$. It is known \cite{BenguriaVugalter2004} that $\lambda_1$ converges to $\lambda_0$ from below as $\alpha$ goes to zero. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% main theorem %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Before stating our main result, let us introduce some notations. For $v$ a measurable function in $\R^3$, we define \begin{equation}\label{def-d1} d_v = \frac{1}{2\pi} \left(\int\frac{|v(x)||v(y|}{|x-y|^2}\mathrm{d}x \mathrm{d}y\right)^\frac12\, , \end{equation} if $v$ is not spherically symmetric and \begin{equation}\label{def-d2} d_v=\min\{\frac{1}{2\pi}\left(\int\frac{|v(x)||v(y|} {|x-y|^2}\mathrm{d}x \mathrm{d}y\right)^\frac12, \int_0^\infty t |v(t)| \mathrm{d} t\} \end{equation} if $v$ is spherically symmetric. Our main result is thus \begin{theorem}\label{mainthm} Assume that $W(x)$ satisfies the following conditions: $W\in L^4_{\rm loc}(\R^3)$ and there exists $a>0$, $c>0$ and $\delta>0$ such that for all $|x|>a$, $|W(x)|\leq c(1+|x|)^{-2-\delta}$. Then $$ \lambda_1 \leq \lambda_0 (1 - \alpha \eta^2 + \mathcal{O}(\alpha^\frac54)) $$ with $$ \eta^2 = \frac{1}{6\pi^2} \int_{\R^3} \frac{\zeta(|k|)}{|k|(k^2 + |k|+\cw)} \mathrm{d} k\, , $$ and $$ C_W = \lambda_0^2 (1+ \lambda_0 d_{W_+}) d_{W^2}\quad\mbox{and}\quad W_+ = (|W|+W)/2. $$ \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% PROOF %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of the main Theorem} In this section, we will prove the main theorem in the case of particle with spin $g=1$. The proof for $g=0$ can easily be deduced with several simplifications. We start with establishing some useful preliminary estimates. \subsection{Properties of the self-energy operator T(0) with zero total momentum} This section addresses the main properties of the self-energy operator $T(0)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% def. of self energy operator with total momentum 0 %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let us consider the case of a free particle coupled to the quantized electromagnetic field. The self-energy operator $T$ is given by $$ T = \left(- i\nabla_{x}\otimes I_f + \sqrt{\alpha} A(x)\right)^2 + g\sqrt\alpha\sigma \cdot B(x) + I_{el}\otimes H_f - c_{\rm n.o.}\alpha. $$ We note that this system is translationally invariant, that is, $T$ commutes with the operator of total momentum $$ P_{tot} = p_{el}\otimes I_f + I_{el}\otimes P_f , $$ where $p_{el}$ and $P_f = \sum_{\lambda =1,2} \int k a^\ast_\lambda(k) a_\lambda(k) \d k$ denote the particle and the photon momentum operators. Let $\gH_P\cong \C^2\otimes\gF$ denotes the fibre Hilbert space corresponding to conserved total momentum $P$. For any fixed value $P$ of the total momentum, the restriction of $T$ to the fibre space $\gH_P$ is given by (see e.g. \cite{Chen2001}) \begin{equation}T(P) = (P - P_f + \sqrt{\alpha} A(0))^2 + g \sqrt{\alpha}\sigma\cdot B(0) + H_f - c_{\rm n.o.}\alpha. \end{equation} We denote $\Sigma_0 := \inf\sigma( T(0))$. For the reader convenience, we first collect in the following theorem different known facts regarding the ground state of the operator $T(0)$, which will be used in the proof of the main theorem. From now on, we will denote by $\Pi_n$ the projection onto the subspace of $\C^2\otimes\gF$ corresponding to vectors which have all components zero except the $n$-photon components. We also define $\Pi_n^\geq = 1-\sum_{i=1}^{n-1}\Pi_n$. For vectors in $\C^2\otimes\gF$, the norm $\|.\|$ will refer to the standard norm in $\C^2\otimes\gF$. \begin{theorem}\label{Ogthm}\cite{Frohlich1974, Chen2001, Chenetal2003, Barbarouxetal2003} For $\a$ sufficiently small we have: \begin{itemize} \item $\Sigma_0$ is an eigenvalue bordering to continuous spectrum of $T(0)$ and $\Sigma_0=\inf\sigma(T)$. \item For any $\Omega_0\in {\rm Ker}(T(0)-\Sigma_0)$, its projection $\Pi_0\Omega_0$ onto the zero-photon sector of $\C^2\otimes\gF$ fulfils $\|\Pi_0\Omega_0\|\neq 0$. If $\Omega_0$ is normalized by $\|\Pi_0\Omega_0\|=1$, then the following inequalities are satisfied: $\|\Omega_0\| = 1+ \mathcal{O}(\alpha^{1/2})$, $\|D(0) \Omega_0 \| = \mathcal{O}(\alpha^{1/2})$, and $\|H_f^{1/2} \Omega_0 \| = \mathcal{O}(\alpha^{1/2})$. \item For the photon number operator $N_f:=\sum_{\lambda=1,2}\int \al^*(k)\al(k)\d k$, we have $\| N_f^{1/2} \Omega_0 \| = \mathcal{O}(\alpha^{1/2})$. \end{itemize} \end{theorem} \begin{corollary}\label{Ogcor} For any vector $\Omega_0\in {\rm Ker}(T(0)-\Sigma_0)$ normalized by $\|\Pi_0\Omega_0\|=1$, we have $\|\Omega_0\| = 1+ \mathcal{O}(\alpha)$, $\|D^*(0)\Pi_1^\geq\Omega_0\| = \mathcal{O}(\alpha^{1/2})$ and $\|\sigma\cdot K^*(0)\Pi_1^\geq\Omega_0\| = \mathcal{O}(\alpha^{1/2})$. \end{corollary} In the following, we consider two 4-vectors in $\C^2\otimes\left(L^2(\R^3)\otimes\C^2\right)$, of the form $\left(\xi(\ua,k,\lambda_1), \xi(\ua,k,\lambda_2), \xi(\da,k,\lambda_1), \xi(\da,k,\lambda_2)\right)$, where $\ua$ and $\da$ refer to the spin up and spin down of the particle, and $\lambda_1$, $\lambda_2$ refer to the two polarizations of the transverse photons. $$ \Gamma_{a,b}:= \left( \begin{array}{l} \Gamma(\ua,k,\lambda_1) \\ \Gamma(\ua,k,\lambda_2) \\ \Gamma(\da,k,\lambda_1) \\ \Gamma(\da,k,\lambda_2) \end{array} \right) := \left( \begin{array}{l} \frac{\zeta_\Lambda(k)}{|k|^\frac12} (- a \sqrt{k_1^2 + k_2^2} + b\, \frac{(k_1 - ik_2)k_3}{\sqrt{k_1^2 + k_2^2}}) \\ b \zeta_\Lambda(k) \frac{-k_2 - ik_1} {\sqrt{k_1^2 + k_2^2}} |k|^\frac12 \\ \frac{\zeta_\Lambda(k)}{|k|^\frac12} ( b \sqrt{k_1^2 + k_2^2} + a \frac{(k_1 + ik_2)k_3}{\sqrt{k_1^2 + k_2^2}}) \\ a\, \zeta_\Lambda(k)\frac{-k_2 + ik_1} {\sqrt{k_1^2 + k_2^2}} |k|^\frac12 \end{array} \right)\, . $$ Let \begin{equation}\label{eq:approx-min} \varphi_{a,b} = \sqrt{\alpha} \frac{i} {2\pi |k|(1 + |k|)}\Gamma_{a,b} . \end{equation} \begin{proposition}[Approximate ground state of the Pauli-Fierz operator]\label{main-prop} For $a$ and $b$ in $\C$ such that $|a|^2+|b|^2=1$, we consider the family of real-valued functionals $L_{a,b}$ defined on $\C^2\otimes L^2(\R^3)\otimes\C^2$ by $$ L_{a,b}(\xi) = \bra (k^2 + |k|)\xi,\xi\ket + 2\sqrt{\alpha}\Re \bra \xi, \Pi_1\sigma\cdot K^*(0) (\left(\begin{array}{l}a\\b\end{array}\right)0,0,\cdots)\ket, $$ where as before $B(0)=K(0) + K^*(0)$. Then we have $i)$ The vector $\varphi_{a,b}$ defined by \eqref{eq:approx-min} is the unique minimizer of $L_{a,b}$. $ii)$ $|\Sigma_0 - \inf L_{a,b}(\xi)| =\mathcal{O}(\alpha^{3/2})$. $iii)$ Let $\Omega_0\in {\rm Ker}(T(0)-\Sigma_0)$ be normalized by $\|\Pi_0\Omega_0\|=1$. Let us denote by $(a,b):= \Pi_0 \Omega_0$. We define the scalar product $\bra.,.\ket_1$ onto the one-photon sector $\Pi_1(\C^2\otimes\gF) = \C^2\otimes L^2(\R^3)\otimes\C^2$ by $\bra f,g\ket_1 = \bra (k^2 +|k|)f,g\ket_{\Pi_1(\C^2\otimes\gF)}$. Then for $\gamma\in\R$ and $R\in\C^2\otimes L^2(\R^3)\otimes\C^2$ such that $$ \Pi_1\Omega_0 = \gamma\varphi_{a,b} + R\ $$ and $\bra\varphi_{a,b}, R\ket_1 = 0$, we have \begin{equation}\label{eq:added2} \ \bra R,\, R\ket_1 = \mathcal{O}(\alpha^{3/2})\mbox{ and } \ |\gamma -1| = \mathcal{O}(\alpha^{3/4}) \end{equation} \end{proposition} \begin{remark} In the above proposition, and in the sequel, we use the same notation for $\Pi_1\Omega_0$ as a vector in $\C^2\otimes\gF$ which has all components zero except its one-photon component $(\Pi_1\Omega_0)^{(1)}$, as well as for the vector $(\Pi_1\Omega_0)^{(1)}$ in $\C^2\otimes L^2(\R^3)\otimes\C^2$. \end{remark} \begin{proof} In this proof, for the sake of simplicity of notations, we will drop the argument $0$ in the operators $A(0)$, $B(0)$, $D(0)$, $K(0)$ and their adjoint. We first prove i). Denoting $$ g_{a,b}:=\frac{1}{(k^2+|k|)}\Pi_1\sigma\cdot K^*((\begin{array}{l}a \\b\end{array}),0,0,\cdots)\, , $$ we have \begin{equation}\label{eq:lab-other} L_{a,b}(\xi) = \bra\xi,\xi\ket_1 + 2\sqrt{\alpha}\Re \bra \xi, g_{a,b}\ket_1 = \|\xi + \sqrt{\alpha} g_{a,b}\|_1^2 - \| \sqrt{\alpha} g_{a,b} \|_1^2 , \end{equation} where $\|.\|_1$ is the norm associated to the scalar product $\bra.,.\ket_1$. Therefore, the minimizer of $L_{a,b}$ is $-\sqrt{\alpha}g_{a,b}$. A straightforward computation shows that $-\sqrt{\alpha}g_{a,b}= \varphi_{a,b}$. This implies that \begin{equation}\label{eq:added1} \inf L_{a,b} = L_{a,b}(\varphi_{a,b}) = - \|\varphi_{a,b}\|_1^2 . \end{equation} We now prove ii). We have \begin{equation}\label{eq:prop0} \begin{split} \bra T(0)\Omega_0,\Omega_0\ket = & \bra P_f^2 \Omega_0, \Omega_0\ket - \sqrt{\alpha}2\Re \bra P_f\Omega_0, A \Omega_0\ket + \alpha\bra A^2\Omega_0, \Omega_0\ket\\ & +\sqrt{\alpha} \bra\sigma\cdot B\Omega_0, \Omega_0\ket + \bra H_f\Omega_0, \Omega_0\ket -c_{\rm n.o.}\alpha \end{split} \end{equation} Let us estimate the terms in the above equality in order to identify those who are of order $\alpha^{3/2}$ and higher. \begin{equation}\label{eq:prop1} \begin{split} \bra P_f^2 \Omega_0, \Omega_0\ket = & \bra P_f^2 \Pi_0\Omega_0, \Pi_0\Omega_0\ket \!+ \! \bra P_f^2 \Pi_{1}\Omega_0, \Pi_{1}\Omega_0\ket \!+ \!\bra P_f^2 \Pi_2^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket \\ = & \bra P_f^2 \Pi_{1}\Omega_0, \Pi_{1}\Omega_0\ket \!+ \!\bra P_f^2 \Pi_2^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket . \end{split} \end{equation} \begin{equation}\label{eq:prop2} \begin{split} \bra H_f \Omega_0, \Omega_0\ket = & \bra H_f \Pi_0\Omega_0, \Pi_0\Omega_0\ket \!+ \! \bra H_f \Pi_{1}\Omega_0, \Pi_{1}\Omega_0\ket \!+ \!\bra H_f \Pi_2^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket \\ = & \bra H_f \Pi_{1}\Omega_0, \Pi_{1}\Omega_0\ket \!+ \!\bra H_f \Pi_2^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket . \end{split} \end{equation} Now, using the fact that $n$-photon sectors are invariant under $P_f$, $P_f\Pi_0\Omega_0 =0$, and $\bra P_f \Omega_0, A\Omega_0\ket = \bra A P_f \Pi_1^\geq\Omega_0, \Omega_0\ket = \bra A \Pi_1^\geq \Omega_0, P_f \Omega_0\ket = \bra A \Pi_1^\geq \Omega_0, P_f \Pi_1^\geq\Omega_0\ket$, we get \begin{eqnarray*} |\bra P_f \Omega_0, A\Omega_0\ket| & = & |\bra P_f \Pi_1^\geq\Omega_0, A \Pi_1^\geq\Omega_0\ket |\\ &\leq & |\bra P_f \Pi_1^\geq\Omega_0, D \Pi_2^\geq\Omega_0\ket | + |\bra P_f \Pi_2^\geq\Omega_0, D^* \Pi_1^\geq\Omega_0\ket | \\ & \leq & |\bra P_f\Pi_1\Omega_0, D\Pi_2\Omega_0\ket| + |\bra P_f \Pi_2^\geq \Omega_0, D \Pi_3^\geq \Omega_0\ket|\\ & & + |\bra P_f \Pi_2^\geq \Omega_0, D^* \Pi_1^\geq\Omega_0\ket| \\ & \leq & \|P_f\Pi_1\Omega_0\|\,\|D\Pi_2\Omega_0\| + \frac12 \|P_f\Pi_2^\geq\Omega_0\|^2 + \\ & & 2 \|D \Pi_3^\geq\Omega_0\|^2 + 2 \|D^*\Pi_1^\geq\Omega_0\|^2 . \end{eqnarray*} Using Theorem~\ref{Ogthm} and Corollary~\ref{Ogcor} and the fact that $\|P_f\Pi_1\Omega_0\| \leq c(\Lambda) \|\Pi_1\Omega_0\| = \mathcal{O}(\alpha^{1/2})$, where $c(\Lambda)$ depends only on the ultraviolet cutoff, yields \begin{equation}\label{eq:prop3} |\bra P_f \Omega_0, A\Omega_0\ket| \leq \frac12 \|P_f\Pi_2^\geq\Omega_0\|^2 + \mathcal{O}(\alpha) . \end{equation} We also have \begin{eqnarray*} \lefteqn{\bra A^2\Omega_0, \Omega_0\ket} & & \\ & = & \bra (D+D^*)^2 \Omega_0, \Omega_0\ket \\ & = & 2 \Re \bra DD \Omega_0, \Omega_0\ket + 2 \|D \Omega_0\|^2 + \|[D, D^*]\| \|\Omega_0\|^2 \\ & = & \mathcal{O}(\alpha^{1/2})+ \mathcal{O}(\alpha) + \|[D, D^*]\|(1+\mathcal{O}(\alpha))\, , \end{eqnarray*} where we used from Theorem~\ref{Ogthm} that $\|\Omega_0\| = 1+\mathcal{O}(\alpha)$ and $\|D\Omega_0\| = \mathcal{O}(\alpha^{1/2})$. Since the commutator $[D, D^*]$ equals $c_{\rm n.o.}$ we arrive at \begin{equation}\label{eq:prop4} \bra A^2\Omega_0, \Omega_0\ket = c_{\rm n.o.}+ \mathcal{O}(\alpha^{1/2}). \end{equation} Finally we have, writing $B = K+K^*$ \begin{equation}\nonumber \begin{split} \bra\sigma\cdot B \Omega_0, \Omega_0\ket = \bra\sigma\cdot K \Pi_1\Omega_0, \Pi_0\Omega_0\ket + \bra\sigma\cdot K\Pi_2^\geq\Omega_0, \Pi_1^\geq \Omega_0\ket \\ + \bra\sigma\cdot K^*\Pi_0\Omega_0, \Pi_1\Omega_0\ket + \bra\sigma\cdot K^* \Pi_1^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket\\ = 2\Re \bra\sigma\cdot K \Pi_1\Omega_0, \Pi_0\Omega_0\ket + 2 \Re \bra\sigma\cdot K^* \Pi_1^\geq \Omega_0, \Pi_2^\geq\Omega_0\ket . \end{split} \end{equation} using Theorem~\ref{Ogthm} and Corollary~\ref{Ogcor} we obtain \begin{equation}\label{eq:prop5} \bra\sigma\cdot B \Omega_0, \Omega_0\ket = 2\Re \bra\sigma\cdot K \Pi_1\Omega_0, \Pi_0\Omega_0\ket + \mathcal{O}(\alpha) . \end{equation} Collecting \eqref{eq:prop0}-\eqref{eq:prop5} and using $\bra H_f\Pi_2^\geq\Omega_0, \Pi_2^\geq\Omega_0\ket\geq 0$ we obtain \begin{equation} \begin{split} \bra T(0)\Omega_0, \Omega_0\ket \geq \bra P_f^2\Pi_1\Omega_0, \Pi_1\Omega_0\ket + \bra H_f\Pi_1\Omega_0, \Pi_1\Omega_0\ket \\+ 2\sqrt\alpha\Re \bra\sigma\cdot K \Pi_1\Omega_0, \Pi_0\Omega_0\ket + \mathcal{O}(\alpha^\frac32). \end{split} \end{equation} Since on the one-photon sector the operator $P_f^2$ reduces to multiplication by $k^2$, and the operator $H_f$ reduced to multiplication by $|k|$, we obtain \begin{equation}\label{eq:ineq1} \Sigma_0= \frac{\bra T(0)\Omega_0, \Omega_0\ket}{||\Omega_0||^2} \geq L_{a,b}(\Pi_1\Omega_0) + \mathcal{O}(\alpha^{3/2}) \geq \inf L_{a,b} + \mathcal{O}(\alpha^{3/2}). \end{equation} On the other hand, using i), and for $\psi_{a,b}=(\left(\begin{array}{l}a\\b\end{array}\right), \varphi_{a,b},0,0,\cdots)$ we have \begin{equation}\label{eq:ineq2} \begin{split} \inf L_{a,b} = & L_{a,b}(\varphi_{a,b}) = \bra T(0) \psi_{a,b}, \psi_{a,b}\ket \geq \Sigma_0 \|\psi_{a,b}\|^2 \\ = & \Sigma_0 (1 + \mathcal{O}(\alpha)) \geq \Sigma_0 + \mathcal{O}(\alpha^2) \end{split} \end{equation} Inequalities \eqref{eq:ineq1} and \eqref{eq:ineq2} conclude the proof of ii). Eventually, we prove iii). Due to the Inequalities \eqref{eq:ineq1} and \eqref{eq:ineq2}, we have $\inf L_{a,b} +\mathcal{O}(\alpha^{3/2}) = L_{a,b}(\Pi_1\Omega_0)$. Using \eqref{eq:lab-other}, the fact that $-\sqrt\alpha g_{a,b}=\varphi_{a,b}$, and \eqref{eq:added1}, we thus get \begin{equation} \begin{split} \inf L_{a,b} +\mathcal{O}(\alpha^{3/2}) = & L_{a,b}(\Pi_1\Omega_0) = \|\gamma\varphi_{a,b} + R -\varphi_{a,b}\|_1^2 - \|\varphi_{a,b}\|_1^2 \\ = & (\gamma-1)^2\|\varphi_{a,b}\|_1^2 + \|R\|_1^2 + \inf L_{a,b}\, , \end{split} \end{equation} which proves \eqref{eq:added2}. \end{proof} \subsection{Proof of Theorem~\ref{mainthm}} As it was mentioned in the introduction, we prove the theorem by constructing a trial function $\Psi$ for which the quadratic form of $H$ takes a value strictly smaller than $\Sigma_0 \|\Psi\|^2$. Let us start by proving an auxiliary result. For $\gm\in (0,1)$, we define $f_\gm\in L^2(\R^3)$ to be a normalized real valued eigenfunction, with associated eigenvalue $e_\gm$, of the Schr\"odinger operator $$ h_\gm:= -(1-\gm)\Delta + \lambda_0 W(x). $$ Here $\lambda_0$ is the critical coupling constant defined in Section~\ref{S1}. \begin{lemma}\label{lem:cw} Then for $\lambda\leq\lambda_0$, we have $$ \sum_i \!\bra (-\Delta + \lambda W) \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i} \ket \!\leq\! C_W \| \nabla f_\gm\|^2 + o_\gamma(1) \|\nabla f_\gm\|^2 , $$ with $C_W := \lambda_0^2 (1+ \lambda_0 d_{W_+}) d_{W^2}$, where $W_+ = (W + |W|)/2$ and $d_{W^2}$ and $d_{W_+}$ are defined by \eqref{def-d1}-\eqref{def-d2}. \end{lemma} \begin{proof} For a potential $V$ such that $V\in L^2_{\rm loc}$ and short range we have $$ |\bra V\psi,\psi\ket| \leq d_V \| \nabla \psi\|^2\, . $$ Moreover, we know that $f_\gm$ is an eigenfunction of $-(1-\gm)\Delta + \lambda_0 W$ and that the associated eigenvalue $e_\gm$ tends to zero as $\gamma$ tends to zero, since $\lambda_0$ is the critical coupling constant. Therefore we obtain the following sequence of inequalities \begin{equation}\label{eq:19bis} \begin{split} \sum_{i}\bra (-\Delta + \lambda W) \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i}\ket \leq \sum_{i}\bra (-\Delta + \lambda W_+) \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i}\ket\\ \leq \sum_{i}\bra (-\Delta + \lambda_0 W_+) \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i}\ket \leq \sum_i \bra -\Delta \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i}\ket (1+d_{\lambda_0 W_+}) \\ = (1+d_{\lambda_0 W_+}) \bra -\Delta f_\gm, -\Delta f_\gamma\ket = (1+d_{\lambda_0 W_+}) \bra \frac{e_\gm - \lambda_0 W } {1-\gamma}f_\gm, -\Delta f_\gm\ket \\ = \frac{e_\gm(1+d_{\lambda_0 W_+})}{1-\gamma} \bra f_\gm, -\Delta f_\gm\ket - \frac{\lambda_0(1+d_{\lambda_0 W_+})}{1-\gamma} \bra W f_\gm, -\Delta f_\gm\ket \end{split} \end{equation} We estimate the last term in the right hand side by \begin{equation}\label{eq:20bis} \begin{split} - \bra W f_\gm, -\Delta f_\gm\ket = \bra W f_\gm, \frac{\lambda_0 W - e_\gm} {1-\gamma} f_\gm\ket \\ \leq -\frac{e_\gm d_{W_+}}{1-\gamma} \|\nabla f_\gm\|^2 + \frac{\lambda_0}{1-\gamma} d_{W^2} \|\nabla f_\gm\|^2 \end{split} \end{equation} The Inequalities \eqref{eq:19bis} and \eqref{eq:20bis} imply for $\lambda\leq\lambda_0$ \begin{equation} \sum_{i}\bra (-\Delta + \lambda W) \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_i}\ket \leq \left(\lambda_0^2 (1+\lambda_0 d_{W_+}) d_{W^2} + o_{\gm}(1)\right) \|\nabla f_\gm\|^2 \end{equation} \end{proof} In the rest of this section, we will mainly work in the space representation for both particle and photons. Following \cite{Chenetal2003}, let us introduce, for given $x\in\R^3$, the shift operator on the photon space variables $\tau_x:\C^2\otimes\gF\rightarrow\C^2\otimes\gF$. For $\phi=(\phi_0, \phi_1, \ldots, \phi_n, \ldots )\in \C^2\otimes\mathcal{F}$, we have, writing by abuse of notation $\tau_x\phi = (\tau_x\phi_0 ,\tau_x\phi_1 ,\ldots)$, $$ \tau_x\phi_n (s; y_1, \ldots, y_n; \lambda_1, \ldots, \lambda_n) = \phi_n (s; y_1 - x, \ldots, y_n - x; \lambda_1, \ldots, \lambda_n) , $$ where $s$ is the spin of the particle and takes value in $\{ \uparrow,\, \downarrow\}$. We denote by $\Omega^x_0$ the ground state $\Omega_0$ written in space representation and shifted by $x$, i.e. $$ \Omega_0^x := \tau_x \mathcal{F}^{-1}\Omega_0, $$ where $\mathcal{F}$ stands for the Fourier transform. Recall that $D^*(0)$ is an operator valued vector with $3$ components which we denote by $D^*(0)_i$ ($i=1,2,3$). Then we consider the functions \begin{equation} \theta_{i} = (0,\theta_{i}^{(1)},0,\ldots) \in\C^2\otimes\gF \end{equation} with \begin{equation}\label{eq:def2-theta} \theta_{i}^{(1)} = (k^2+|k|+\cw)^{-1} \Pi_1 D^*(0)_i \left( (\begin{array}{l} a \\ b \end{array}),0,\ldots \right)\, \end{equation} and \begin{equation} \theta_{i}^x = \tau_x \mathcal{F}^{-1}\theta_{i}. \end{equation} We first state some properties of $\theta_{i}$. \begin{lemma}\label{lem-theta}$\ $ \noindent i) For $i\neq j$ we have $$ \bra \theta_{i}, \theta_{j} \ket = 0 \quad \mbox{and} \quad \bra \theta_{i}, \theta_{j} \ket_1 = 0 . $$ ii) For $i=1,2,3$ holds $$ \|\theta_{i}\sqrt{k^2 + |k|+\cw}\|^2 = \frac{1}{6\pi^2} \int_{\R^3} \frac{\zeta(|k|)}{|k|(k^2 + |k|+\cw)} \mathrm{d} k, $$ iii) For $i=1,2,3$, $\bra k_i \varphi_{a,b}, \Pi_1\theta_{i}\ket = 0$. \end{lemma} \begin{proof} To prove this Lemma we remind that $\theta_{i}$ has only a non zero component $\Pi_1 \theta_{i}$ in the one photon sector, and $$ \Pi_1 \theta_{i} = \left( \begin{array}{l} a \frac{\varepsilon_{1,i}(k) \zeta(|k|)} {|k|^\frac12 (k^2 + |k|+\cw)} \vspace{0.1cm}\\ a \frac{\varepsilon_{2,i}(k) \zeta(|k|)} {|k|^\frac12 (k^2 + |k|+\cw)} \vspace{0.1cm} \\ b \frac{\varepsilon_{1,i}(k) \zeta(|k|)} {|k|^\frac12 (k^2 + |k|+\cw)} \vspace{0.1cm}\\ b \frac{\varepsilon_{2,i}(k) \zeta(|k|)} {|k|^\frac12 (k^2 + |k|+\cw)}\ \end{array} \right), $$ where the two polarization vectors $\varepsilon_1(k)$ and $\varepsilon_2(k)$ are defined in \eqref{def-eps}. The properties stated in the Lemma follow straightforwardly from computations of the corresponding integrals. \end{proof} We consider the trial function $\Psi\in L^2(\R^3)\otimes\C^2\otimes\gF$: \begin{equation} \Psi := \Psi_1 + \Psi_2:= f_\gm(x) \Omega_0^x + i\sqrt{\alpha} \sum_{i=1}^3 \theta_{i}^x \frac{\partial f_\gm(x)}{\partial x_i}\, . \end{equation} Now we compute the expectation value of $H$ in the state $\Psi$. We have $$ \bra H\Psi,\Psi\ket = \bra H\Psi_1,\Psi_1\ket + \bra H\Psi_2,\Psi_2\ket + 2\Re \bra H\Psi_1,\Psi_2\ket. $$ As usual \cite{Chenetal2003}, due to the orthogonality $\bra f,\partial f/\partial x_i\ket =0$, we have \begin{equation}\label{eq:22} \bra H\Psi_1, \Psi_1\ket = \Sigma_0\|\Psi_1\|^2 + \bra (-\Delta + \lambda W(x))f_\gm,f_\gm\ket \|\Omega_0\|^2 \end{equation} Since $\Psi_2$ has only a non zero component in the one photon sector, in the quadratic form $\bra H \Psi_2,\Psi_2\ket$, all the terms involving $A(0)$ or $B(0)$ vanish. Moreover, using Lemma~\ref{lem-theta} and the orthogonalities $\bra \frac{\partial f_\gm}{\partial x_i}, \frac{\partial f_\gm}{\partial x_j}\ket =0$ and $\bra \frac{\partial f_\gm}{\partial x_i}, \frac{\partial^2 f_\gm}{\partial x_i\partial x_j}\ket =0$, for $i\neq j$, we arrive at \begin{equation}\label{eq:23} \begin{split} \bra H \Psi_2, \Psi_2\ket = & \alpha \sum_{l} \|\theta_{l}^x\|^2 \bra (-\Delta+\lambda W)\frac{\partial f_\gm}{\partial x_l}, \frac{\partial f_\gm}{\partial x_l}\ket \! + \! \mathcal{O}(\alpha^2) \|\nabla f_\gm\|^2 \\ & + \alpha \sum_{l}\|\frac{\partial f_\gm}{\partial x_l}\|^2 \bra (|k|+k^2) \Pi_1\theta_{l}^{x}, \Pi_1\theta_{l}^{x}\ket . \end{split} \end{equation} To compute the last term $\bra H\Psi_1,\Psi_2\ket$, we first note that \begin{equation}\nonumber\label{eq:cross-zero} \begin{split} \bra (-\Delta + \lambda W) f_\gm, \frac{\partial f_\gm}{\partial x_i}\ket =& \bra (-(1-\gm)\Delta + \lambda W) f_\gm, \frac{\partial f_\gm}{\partial x_i}\ket -\gm \bra \Delta f_\gm, \frac{\partial f_\gm}{\partial x_i}\ket \\ =& e_\gm \bra f_\gm, \frac{\partial f_\gm}{\partial x_i}\ket + \gamma \sum_j \bra \frac{\partial^2 f_\gm}{\partial x_j^2}, \frac{\partial f_\gm}{\partial x_i}\ket =0 . \end{split} \end{equation} The last equality holds since $f_\gm$ is a real function vanishing at infinity. Moreover, all other terms in the quadratic form $\bra H\Psi_1,\Psi_2\ket$ which contain $\bra f_\gm, \frac{\partial f_\gm}{\partial x_i}\ket$ vanish also. So we arrive at \begin{equation}\label{eq:24} \begin{split} 2\Re \bra H\Psi_1,\Psi_2\ket = & - 2\Re \bra P\cdot (P_f-\sqrt{\alpha}A(0))\Psi_1, \Psi_2\ket \\ = & 2 \sqrt{\alpha} \sum_i \|\frac{\partial f_\gm} {\partial x_i}\|^2 \Re \bra (P_f- \sqrt{\alpha} A(0))_i\Omega_0, \theta_{i}\ket \end{split} \end{equation} The term with $P_f$ on the right hand side is estimated as follows \begin{equation}\label{eq:above} \begin{split} \Re \bra (P_f)_i \Omega_0, \theta_{i}\ket_{\C^2\otimes\gF} = & \Re \bra \Pi_1(P_f)_i(\gamma\varphi_{a,b} + R), \Pi_1\theta_{i}\ket_{\C^2\otimes L^2(\R^3)\otimes\C^2}\\ = & \Re \bra k_i(\gamma\varphi_{a,b} + R), \Pi_1\theta_{i} \ket_{\C^2\otimes L^2(\R^3)\otimes\C^2}\\ \leq & \|R\|_1\||k|^\frac12 \Pi_1\theta_{i}\| + \gamma \Re \bra k_i\varphi_{a,b}, \Pi_1\theta_{i} \ket_{\C^2\otimes L^2(\R^3)\otimes\C^2} \end{split} \end{equation} Using Proposition~\ref{main-prop} yields the following bound for the first term in the right hand side of \eqref{eq:above} \begin{equation}\label{eq:26} \|R\|_1\||k|^\frac12 \Pi_1\theta_{i}\| = \mathcal{O}(\alpha^{\frac34}) \end{equation} According to Lemma~\ref{lem-theta}~iii), the second term in the right hand side of \eqref{eq:above} equals zero. Therefore, collecting \eqref{eq:24}-\eqref{eq:26}, we arrive at \begin{equation}\label{eq:27} 2\Re\bra H \Psi_1, \Psi_2\ket = - 2 \alpha \sum_i \|\frac{\partial f_\gm} {\partial x_i}\|^2 \Re \bra A(0)_i\Omega_0, \theta_{i}\ket + \mathcal{O}(\alpha^{\frac54}) \|\nabla f_\gm\|^2 \end{equation} Now we have, using Theorem~\ref{Ogthm} and the fact that $D(0)$ restricted to the 2-photon sector is a bounded operator \begin{equation}\label{eq:28} \begin{split} \Re \bra A(0)_i\Omega_0, \theta_{i}\ket = &\Re \bra D(0)_i\Pi_2\Omega_0, \theta_{i}\ket + \Re \bra D^*(0)_i\Pi_0\Omega_0, \theta_{i}\ket\\ & = \mathcal{O}(\alpha^{\frac12}) + \Re \bra D^*(0)_i\Pi_0\Omega_0, \theta_{i}\ket . \end{split} \end{equation} Due to the definition~\eqref{eq:def2-theta} of $\theta_{i}$, the second term on the right hand side of \eqref{eq:28} is $\| \theta_{i} \sqrt{k^2 +|k| + C_W}\|^2$. Therefore, collecting the Equalities~\eqref{eq:22}, \eqref{eq:23}, \eqref{eq:27} and \eqref{eq:28} we obtain \begin{equation} \begin{split} \bra H\Psi, \Psi\ket = & \Sigma_0\|\Psi_1\|^2 + \|\Omega_0\|^2 \bra (-\Delta + \lambda W)f_\gm,f_\gm\ket \\ & + \alpha \sum_{l} \|\theta_{l}\|^2 \bra (-\Delta+\lambda W)\frac{\partial f_\gm}{\partial x_l}, \frac{\partial f_\gm}{\partial x_l}\ket \\ & - 2\alpha \sum_{l}\|\frac{\partial f_\gm}{\partial x_l}\|^2 \| \theta_{i} \sqrt{k^2 +|k| + C_W}\|^2 + \mathcal{O}(\alpha^{\frac54}) \|\nabla f_\gm\|^2 \\ & + \alpha \sum_{l}\|\frac{\partial f_\gm}{\partial x_l}\|^2 \|\theta_{l}\|_1^2 . \end{split} \end{equation} From Lemma~\ref{lem-theta}, we know that $\|\theta_l \sqrt{k^2 +|k| + C_W}\|^2$ is independent of $l$. We denote this constant by $\eta^2$. With Lemma~\ref{lem:cw} we thus arrive at \begin{equation}\label{eq:last1} \begin{split} \bra H\Psi, \Psi\ket \leq \Sigma_0\|\Psi\|^2 -\Sigma_0\|\Psi_2\|^2 + \|\Omega_0\|^2\left( \|\nabla f_\gm\|^2 + \bra \lambda W f_\gm,f_\gm\ket\right)\\ - \alpha \sum_l \|\frac{\partial f_\gm}{\partial x_l}\|^2 \eta^2 + \alpha o_\gamma(1)\|\nabla f_\gm\|^2 + \mathcal{O}(\alpha^{\frac54}) \|\nabla f_\gm\|^2\, . \end{split} \end{equation} Note that $\Sigma_0 \|\psi_2\|^2 = \mathcal{O}(\alpha^2) \|\nabla f_\gm\|^2$. We thus obtain \begin{equation}\label{eq:last2} \begin{split} \bra H\Psi, \Psi\ket - \Sigma_0\|\Psi\|^2 \leq \\ \|\Omega_0\|^2 \Big((1-\frac{\alpha}{\|\Omega_0\|^2} \eta^2 + \frac{\alpha}{\|\Omega_0\|^2} (o_\gm(1)+ \mathcal{O}(\alpha^\frac14)))\|\nabla f_\gm\|^2 + \bra \lambda W f_\gm,f_\gm\ket \Big) \end{split} \end{equation} Using from Corollary~\ref{Ogcor} that $\|\Omega_0\|^2 = 1+\mathcal{O}(\alpha)$, we obtain \begin{equation}\label{eq:last3} \begin{split} \bra H\Psi, \Psi\ket - \Sigma_0\|\Psi\|^2 \leq \\ \|\Omega_0\|^2 \Big( (1-\alpha \eta^2 + \alpha o_\gm(1)+ \mathcal{O}(\alpha^\frac54))\|\nabla f_\gm\|^2 + \bra \lambda W f_\gm,f_\gm\ket \Big)\, . \end{split} \end{equation} Therefore \begin{equation}\label{eq:last4} \begin{split} \bra H\Psi, \Psi\ket - \Sigma_0\|\Psi\|^2 \leq \|\Omega_0\|^2 (1-\alpha \eta^2 + \alpha o_\gm(1)+ \mathcal{O}(\alpha^\frac54)) \\ \times \Big( \|\nabla f_\gm\|^2 + (1+\alpha \eta^2 + \alpha o_\gm(1)+ \mathcal{O}(\alpha^\frac54))^{-1} \bra \lambda W f_\gm,f_\gm\ket \Big)\, . \end{split} \end{equation} If $\lambda > \lambda_0 (1 - \alpha \eta^2 + \mathcal{O}(\alpha^\frac54))$, choosing $\gamma$ (depending on $\alpha$) small enough, we arrive at $\bra H\Psi,\Psi\ket - \Sigma_0\|\Psi\|^2 <0$. 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