Content-Type: multipart/mixed; boundary="-------------0806270800551" This is a multi-part message in MIME format. ---------------0806270800551 Content-Type: text/plain; name="08-132.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-132.comments" 28 pages ---------------0806270800551 Content-Type: text/plain; name="08-132.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-132.keywords" QED, Mourre estimate, Fermi golden Rule ---------------0806270800551 Content-Type: application/x-tex; name="pf.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pf.tex" \documentclass[10pt]{amsart} %reqno \usepackage{amssymb,amsmath,amscd,verbatim, mathrsfs} %\usepackage[active]{srcltx} \usepackage[all]{xy} %\usepackage{showkeys} \usepackage{color} \newcommand\blue[1]{\textcolor{blue}{#1}} \newcommand\red[1]{\textcolor{red}{#1}} \newcommand\green[1]{\textcolor{green}{#1}} %\parindent0cm %\parskip=.5\baselineskip %\sloppy % Lettres rondes \font\tenrsf=rsfs10 at 11pt \font\sevenrsf=rsfs7 at 8pt \font\fiversf=rsfs5 at 6pt \newfam\rsffam \textfont\rsffam=\tenrsf \scriptfont\rsffam=\sevenrsf \scriptscriptfont\rsffam=\fiversf \def\rond#1{{\tenrsf\fam\rsffam#1}} % Counters \def\thesection{\arabic{section}} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{rem}[theorem]{Remark} % Letters \newcommand{\Ran}{{\rm Ran}} \newcommand{\Br}{\rond{B}} \newcommand{\Cr}{\rond{C}} \newcommand{\Dr}{\rond{D}} \newcommand{\Gr}{\rond{G}} \newcommand{\Hr}{\rond{H}} \newcommand{\hr}{\rond{h}} \newcommand{\Ir}{\rond{I}} 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\newcommand{\ess}{\mathrm{ess}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\id}{\mathrm{Id}} \newcommand{\un}{\mathrm{1}} \newcommand{\Krke}{\Kr_{k,\varepsilon}} \newcommand{\loc}{\mathrm{loc}} \newcommand{\oX}{\overline{X}} \newcommand{\px}{\partial_x} \newcommand{\rz}{\mathbb{R}} \newcommand{\supp}{\mathrm{supp}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\siges}{\sigma_\ess} \newcommand{\Spec}{\mathrm{Spec}} \newcommand{\ta}{\tilde{a}} \newcommand{\tal}{\tilde{\alpha}} \newcommand{\tih}{\tilde{h}} \newcommand{\tf}{\tilde{\varphi}} \newcommand{\tif}{\tilde{\phi}} \newcommand{\tit}{\tilde{\theta}} \newcommand{\tD}{\tilde{\Delta}} \newcommand{\tM}{\tilde{M}} \newcommand{\tmu}{\tilde{\mu}} \newcommand{\ucz}{\underline{\cz}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Xe}{X_\varepsilon} \newcommand{\zz}{\mathbb{Z}} \newcommand{\dG}{{\rm d}\Gamma} \newcommand{\rv}{{\rm v}} \newcommand{\slim}{\mathrm{s-}\!\lim} \newcommand{\wlim}{\mathrm{w-}\!\lim} % 2) Lettres gothiques: \font\teneuf=eufm10 at 12pt \font\seveneuf=eufm7 at 8pt \font\fiveeuf=eufm5 at 6pt \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\goth#1{{\teneuf\fam\euffam#1}} % \newcommand{\hg}{\goth{h}} \def\build#1_#2^#3{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}} \def\cchi{\raisebox{.45 ex}{$\chi$}}%an upper placed \chi \def\pp{<} \def\pg{>} \newcommand{\ind}{1} \begin{document} \title[On the Mourre estimate] {Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli-Fierz Hamiltonians. } \author{Sylvain Gol\'enia} \address{Mathematisches Institut der Universit\"at Erlangen-N\"urnberg Bismarckstr.\ 1 1/2 \\ 91054 Erlangen, Germany} \email{golenia@mi.uni-erlangen.de} %\subjclass[2000]{58J40, 58Z05} %\keywords{positive commutator estimates, threshold energy, Fermi % golden rule, quantum field theory} %\date{\today} \begin{abstract} We perform the spectral analysis of a zero temperature Pauli-Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy. We rely on a positive commutator approach introduced by Skibsted and pursued by Georgescu-G\'erard-M\o ller. We complete some results obtained so far by Derezi\'nski-Jak\u si\' c on one side and by Bach-Fr\"ohlich-Segal-Soffer on the other side. \end{abstract} \maketitle \begin{center} \emph{En hommage au 60\`eme anniversaire de Vladimir Georgescu.} \end{center} %\tableofcontents \section{Introduction} \setcounter{equation}{0} Pauli-Fierz operators are often used in quantum physics as generator of approximate dynamics of a (small) quantum system interacting with a free Bose gas. They describe typically a non-relativistic atom interacting with a field of massless scalar bosons. Pauli-Fierz operators appear also in solid state physics. They are used to describe the interaction of phonons with a quantum system with finitely many degrees of freedom. This paper is devoted to the justification of the second-order perturbation theory for a large class of perturbation. For positive temperature system, this property is related to the return to equilibrium, see for instance \cite{DJ2} and reference therein. This question has been studied in many places, see for instance \cite{BFS, BFS2, BFSS, DJ, FMS, FP, HHH} for zero temperature systems and \cite{DJ, JP, Merkli} for positive temperature. We mention also \cite{FGS, GGM, hsp, Ski} who studied certain spectral properties using positive commutator techniques. Here, we focus on the zero temperature setting. In \cite{BFS}, one initiates the analysis using analytic deformation techniques. In \cite{BFSS} and in \cite{DJ}, one introduces some kind of Mourre estimate approach. In the former, one enlarges the class of perturbation studied in \cite{BFS} and in the latter, one introduces another class. These two classes do not fully overlap. This is due to the choice of the conjugate operator. In this paper, we enlarge the class of perturbations used in \cite{DJ} for the question of the Virial theorem (one-commutator theory) and also for the limiting absorption principal (two-commutator theory). Now, we present the model. For the sake of simplicity and as in \cite{DJ}, we start with a $n$-level atom. It is described by a self-adjoint matrix $K$ acting on a finite dimensional Hilbert space $\Kr$. Let $(k_i)_{i=0,\ldots, n}$ be its eigenvalues, with $k_i0$. The aim of this paper is to give hypotheses on the form factor $\alpha$ to ensure that $H_\lambda$ has no eigenvalue in a neighborhood of $k_{i_0}$ for $\lambda$ small enough (and non-zero). First, we have to ensure that the perturbation given by the field operator will really couple the system at energy $k_{i_0}$; we have to avoid form factors like $\alpha(x)= \un \otimes b$ for all $x\in \Kr$ and some $b\in \hg$, see Section \ref{s:thre}. Here comes the second-order perturbation theory, namely the hypothesis of \emph{Fermi golden rule} for the couple $(H_0, \alpha)$ at energy $k_{i_0}$: \begin{eqnarray}\label{e:FGR0'} \wlim_{\varepsilon \rightarrow 0^+} P\phi(\alpha)\overline{P}\,\im(H_0-k +i\varepsilon)^{-1} \overline{P}\phi(\alpha) P >0, \mbox{ on } P\Hr, \end{eqnarray} where $P:=P_{k_{i_0}}\otimes P_\Omega$ and $\overline P:=1-P$. At first sight, this is pretty implicit. We make it explicit in Appendix \ref{s:level}. This condition involves the form factor, the eigenvalues of $H_0$ lower than $k_{i_0}$ and its eigenfunctions. Therein, we also explain why the ground state energy is tacitly excluded. In this paper, we are establishing an extended Mourre estimate, in the spirit of \cite{GGM2, Ski}; this is an extended version of the positive commutator technique initiated by E.\ Mourre, see \cite{ABG,M} and \cite{G2, GJe} for recent developments. Due to the method, we make further hypotheses on the form-factor. To formulate them, we shall take advantage of the \emph{polar coordinates} and of the unitary map: \begin{eqnarray}\label{e:T} \quad T:=\left\{\begin{array}{cll} L^2(\R^d, dk)&\longrightarrow& L^2(\R^+, dr)\otimes L^2(S^{d-1}, d\theta) :=\tilde \hg \\ u&\longmapsto& Tu:= (r, \theta)\mapsto r^{(d-1)/2}u(r\theta). \end{array}\right. \end{eqnarray} We identify $\hg$ and $\tilde \hg$ through this transformation. We write $\partial_r$ for $\partial_r\otimes \un$. We first give meaning to the commutator via: \begin{enumerate} \item[{\bf (I1a)}] $\alpha\in \Bc\big(\Kr, \Kr\!\otimes \dot\Hr^1(\R^+)\otimes L^2(S^{d-1})\big)$, $1\otimes \omega^{-1/2} \partial_r \alpha \in \Bc(\Kr, \Kr\!\otimes\hg)$. \end{enumerate} Here, the dot means the completion of $\Cc^\infty_c(\R^+)$ under the norm given by the space. We denote by $\|\cdot\|_2$ the $L^2$ norm. Recall the norm of $\Hr^1$ is given by $\|\cdot \|_2 + \|\partial_r\cdot\|_2$. We explain the method on a formal level. We start by choosing a conjugate operator so as to obtain some positivity of the commutator. We choose $A:=1_\Kr\!\otimes \dG(i\partial_r)$. Note this operator is not self-adjoint and only maximal symmetric. We set $N:=1_\Kr\!\otimes \dG(\id)$, the \emph{number operator}. Thanks to {\bf (I1a)}, one obtains \begin{align*} [H_\lambda, iA] = \underbrace{N + 1_\Kr\!\otimes P_\Omega}_{\geq 1} + \underbrace{\lambda\phi(\partial_r \alpha) - 1_\Kr\!\otimes P_\Omega}_{H_\lambda \mbox{-bounded }} =: M+ S. \end{align*} Consider a compact interval $\Jc$. Since $\dG(\omega)$ is non-negative, we have: \begin{eqnarray}\label{e:H0p} E_\Jc(H_0)= \sum_{0\leq i \leq \sup(\Jc)} P_{k_i}\otimes E_{\Jc-k_i}\big(\dG(\omega)\big). \end{eqnarray} We infer $(1_\Kr\!\otimes P_\Omega) E_\Jc(H_0)=0$ if and only if $\Jc$ contains no eigenvalues of $K$. We evaluate the commutator at an energy $\Jc$ which contains $k_{i_0}$ and no other $k_i$. Thus, \begin{align}\label{e:Ipositiv} M+ E_\Jc(H_0) SE_\Jc(H_0) \geq 1+ \big(-1+O(\lambda)\big)E_\Jc(H_0) \geq O(\lambda)E_\Jc(H_0), \end{align} since $\phi(i\partial_r \alpha)$ is $H_0$-bounded. We keep $M$ outside the spectral measure as it is not $H_\lambda$-bounded. Note we have no control on the sign of $O(\lambda)$ so far. We have not yet used the Fermi golden rule assumption. We follow an idea of \cite{BFSS} and set \begin{align*} B_\varepsilon := \im \big(\big((H_0-k_{i_0})^2+\varepsilon^2\big)^{-1}\overline{P} \phi(\alpha)P\big ) \end{align*} Observe that \eqref{e:FGR0'} implies there exists $c>0$ such that \begin{align*} P[H_\lambda, i \lambda B_\varepsilon]P= \frac{\lambda^2}{\varepsilon} P \phi (\alpha)\overline{P} \, \im (H_0-k_{i_0}+ i \varepsilon)^{-1}\, \overline{P} \phi (\alpha) P \geq \frac{c\lambda^2}{\varepsilon}P, \end{align*} holds true for $\varepsilon$ small enough. Let $\hat A:= A + \lambda B_\varepsilon$ and $\hat S := S + \lambda [H_\lambda,i B_\varepsilon]$. We have $[H_\lambda, i\hat A] = M + \hat S$. We go back to \eqref{e:Ipositiv} and infer: \begin{align}\label{e:Ipositiv2} M+ E_\Jc(H_0) \hat S E_\Jc(H_0) \geq \big( c\lambda^2/\varepsilon +O(\lambda)\big) E_\Jc(H_0) + \mbox{ error terms}. \end{align} By taking $\varepsilon:= \varepsilon(\lambda)$, one hopes to obtain the positivity of the constant in front of $E_\Jc(H_0)$, to control the errors terms and to replace the spectral measure by the one of $H_\lambda$. Using the Feshbach method and with a more involved choice of conjugate operator, we show in Section \ref{s:thre} that there are $\lambda_0, c', \eta>0$ so that \begin{align}\label{e:Ipositiv3} M+ E_\Jc(H_\lambda) \hat S E_\Jc(H_\lambda) \geq c'|\lambda|^{1+\eta} E_\Jc(H_\lambda), \mbox{ for all } |\lambda|\leq \lambda_0, \end{align} on the sense of forms on $\Dc(N^{1/2})$. One would like to deduce there is no eigenvalue in $\Jc$ from \eqref{e:Ipositiv3}. To apply a Virial theorem, one has at least to check that the eigenvalues of $H_\lambda$ are in the domain of $N^{1/2}$. One may proceed like in \cite{M}. In this article, we follow \cite{GGM, Ski} and construct a sequence of approximated conjugate operators $\hat A_n$ such that $[H_\lambda, i\hat A_n]$ is $H_\lambda$-bounded, converges to $[H_\lambda, i\hat A]$ and such that one may apply the Virial theorem with $A_n$. To justify these steps, we make a new assumption: \begin{enumerate} \item[{\bf (I1b)}] $\un_\Kr\! \otimes \omega^{-a}\alpha\in \Bc(\Kr, \Kr\!\otimes \hg)$, \mbox{ for some } $a>1$. \end{enumerate} We now give our first result, based on the Virial theorem, see Proposition \ref{p:6.5}. \begin{theorem}\label{t:intro0} Let $\Ic$ be an open interval containing $k_{i_0}$ and no other $k_i$. Assume the Fermi golden rule hypothesis \eqref{e:FGR0'} at energy $k_{i_0}$. Suppose that {\bf(I0)}, {\bf(I1a)} and {\bf (I1b)} are satisfied. Then, there is $\lambda_0>0$ such that $H_\lambda$ has no eigenvalue in $\Ic$, for all $|\lambda|\in (0, \lambda_0)$. \end{theorem} We now give more information on the resolvent $R_\lambda(z):=(H_\lambda -z)^{-1}$ as the imaginary part of $z$ tends to $0$. We show it extends to an operator in some weighted spaces around the real axis. This is a standard result in the Mourre theory, when one supposes some $2$-commutators-like hypothesis, see \cite{ABG}. Here, as the commutator is not $H_\lambda$-bounded, one relies on an adapted theory. We use \cite{GGM} which is a refined version of \cite{Ski}. We check the hypotheses {\bf (M1)}--{\bf (M5)} given in Appendix \ref{s:ms} and deduce a limiting absorption principle, thanks to Theorem \ref{t:LAP}. Using again \eqref{e:T}, we state our class of form factors: \begin{enumerate} \item[{\bf (I2)}] $\alpha\in \Bc\big(\Kr, \Kr\!\otimes \dot\Br^{1,1}_2(\R^+)\otimes L^2(S^{d-1})\big)$. \end{enumerate} Recall that the dot denotes the completion of $\Cc^\infty_c$. One choice of norm for $\Br^{1,1}_2$ is: \begin{eqnarray*} \|f\|_{\Br^{1,1}_2(\R^+)}= \|f\|_2+ \int_0^1\big\|f(2t+\cdot)-2f(t+\cdot)+f(\cdot) \big\|_2\, \frac{dt}{t^2}. \end{eqnarray*} We refer to \cite{ABG, Tri} for Besov spaces and real interpolation. To express the weights, consider $\tilde b$ the square root of the Dirichlet Laplacian on $L^2(\R^+, dr)$. Using \eqref{e:T}, we define $b:=\un_\Kr\!\otimes T^{-1}\tilde bT$ in $\Hr$. Set $\rP_s :=\un_\Kr\!\otimes (\dG(b)+1)^{-s}(N+1)^{1/2}$. \begin{theorem}\label{t:intro1} Let $\Ic$ be an open interval containing $k_{i_0}$ and no other $k_i$. Assume the Fermi golden rule hypothesis \eqref{e:FGR0'} at energy $k_{i_0}$. Suppose that {\bf(I0)}, {\bf(I1a)} and {\bf (I2)} (and not necessarily {\bf (I1b)}), there is $\lambda_0>0$ such that $H_\lambda$ has no eigenvalue in $\Ic$, for all $|\lambda|\in (0, \lambda_0)$. Moreover, $H_\lambda$ has no singularly continuous spectrum in $\Ic$. For each compact interval $\Jc$ included in $\Ic$, and for all $s\in(1/2, 1]$, the limits \begin{align*} \rP_s^* R_\lambda(x \pm i0)\, \rP_s:=\lim_{y\rightarrow 0^+} \rP_s^*R_\lambda(x \pm iy) \rP_s \end{align*} exist in norm uniformly in $x\in\Jc$. Moreover the maps: \begin{align*} \Jc\ni x\mapsto \rP_s^* R_\lambda(x \pm i0) \rP_s \end{align*} are H\"older continuous of order $s-1/2$ for the norm topology of $\Bc(\Hr)$ \end{theorem} To our knowledge, the condition {\bf (I2)} is new, even for the question far from the thresholds. We believe it to be optimal for limiting absorption principle. We now compare our result with the literature. In \cite{BFSS}, they use a different conjugate operator, the second quantization of the generator of dilatation. With this choice they have $[H_0, iA]= \un_\Kr\!\otimes \dG(\omega)$. The commutator is $H_\lambda$-bounded. They modify the conjugate operator in the same way but the choice of parameters is more involved. The class of perturbations is thus different from ours. In \cite{DJ}[Theorem 6.3], one shows the absence of embedded eigenvalues by proving a limiting absorption principal with the weights $\un_\Kr\!\otimes (\dG(b)+1)^{-s}$, for $s>1/2$, without any contribution in $N$. They suppose essentially {\bf (I0)} and that $\alpha\in \Bc\big(\Kr, \Kr\!\otimes \dot\Hr^s(\R^+)\otimes L^2(S^{d-1})\big)$, for $s>1$. The class of perturbations is chosen in relation with the weights. Their strategy is to take advantage the Fermi golden rule at the level of the limiting absorption principle, with the help of the Feshbach method. The drawback is that they are limited by the relation weights/class of form-factors and they cannot give a Virial-type theorem. On the other hand, their method allows to cover some positive temperature systems and we do not deal with this question. Their method leads to fewer problems with domains questions. We mention that they do not suppose the second condition of {\bf (I1a)}. Therefore, concerning merely the disappearance of the eigenvalues, the conditions {\bf (I1a)} and {\bf (I1b)} do not imply $\alpha$ to be better than $\dot\Hr^1(\R^+)$, in the Sobolev scale. Hence, Theorem \ref{t:intro0} is a new result. We point out that the condition {\bf (I2)} is weaker than the one used in \cite{DJ}. The weights obtained in the limiting absorption principle are also better than the ones given in \cite{DJ}. We mention that one could improve them by using some Besov spaces, see \cite{GGM}. To simplify the presentation, we do not present them here. We believe they could hardly be reached by the method exposed in \cite{DJ} due to the interplay between weights and form-factors. In \cite{GGM2} and in \cite{Ski}, one cares about showing that the point spectrum is locally finite, i.e.\ without clusters and of finite multiplicity. Here, they use a Virial theorem. Between the eigenvalues, one shows a limiting absorption principle, and uses a hypothesis on the second commutator, something stronger than {\bf (I2)}, see Section \ref{s:2comm}. In our approach, we use the Virial theorem and the limiting absorption principle in an independent way. In particular, if one is interested only in the limiting absorption principle, one does not need to suppose the more restrictive condition {\bf (I1b)} but only {\bf (I0)}, {\bf (I1a)} and {\bf (I2)}. This is due to the fact that we are showing a strict Mourre estimate, i.e.\ without compact contribution. We now give the plan of the paper. In Section \ref{s:Pauli}, we recall some definitions and properties of Pauli-Fierz models. In Section \ref{s:conjop}, we construct the conjugate operators. In Section \ref{s:regu}, we prove the regularity properties so that one may apply the Mourre theory. The Virial theorem is discussed in Section \ref{s:virial}. In Section \ref{s:far}, we establish the extended Mourre estimate far from the thresholds for small coupling constants, we explain in Remark \ref{r:motiv} why the method should be improved to obtain the result above a threshold. In Section \ref{s:thre}, we settle the extended Mourre estimate above the thresholds under the hypotheses of a Fermi golden rule. In Appendix \ref{s:level}, we explain how to check the Fermi golden rule and why this hypothesis is compatible with the hypothesis {\bf (I0)}, {\bf (I1a)}, {\bf (I1b)} and {\bf (I2)}. In Appendix \ref{s:semiprop}, we gather some properties of $C_0$-semigroups and in Appendix \ref{s:ms} we recall the properties of the $\Cc^1$ class in this setting and the hypotheses so as to apply the extended Mourre theory. \subsection*{Notation:} Given a borelian set $\Jc$, we denote by $E_\Jc(A)$ the spectral measure associated to a self-adjoint operator $A$ at energy $\Jc$. Given Hilbert spaces $\Hr, \Kr$, we denote by $\Bc(\Hr, \Kr)$ the set of bounded operator from $\Hr$ to $\Kr$. We simply write $\Bc(\Hr)$, when $\Hr=\Kr$. We denote by $\sigma(H)$ the spectrum of $H$. We set $\langle x \rangle:=(1+x^2)^{1/2}$. We denote by $\|\cdot\|_\Hr$ and by $\langle \cdot, \cdot \rangle_\Hr$ the norm and the scalar product of $\Hr$, respectively. We omit the indices when no confusion arises. We denote by $\wlim$ and $\slim$ the weak and strong limit, respectively. A dot over a Besov or a Sobolev space denotes the closure of the set $\Cc_c^\infty$ of smooth functions with compact support, with respect to the norm of the space. \subsection*{Acknowledgments:} I express my gratitude to Jan Derezi\'nski who encouraged me in pursuing these ideas. I would also like to thank Volker Bach, Alain Joye, Christian G\'erard, Vladimir Georgescu, Wolfgang Spitzer, Claude-Alain Pillet and Zied Ammari for some useful discussions. This work was partially supported by the Postdoctoral Training Program HPRN-CT-2002-0277. \section{The Pauli-Fierz model}\label{s:Pauli} \setcounter{equation}{0} Pauli-Fierz operators are often used in quantum physics as generator of approximate dynamics of a (small) quantum system interacting with a free Bose gas. They describe typically a non-relativistic atom interacting with a field of massless scalar bosons. The quantum system is given by a (separable) complex Hilbert space $\Kr$. The Hamiltonian describing the system is denoted by a self-adjoint operator $K$, which is bounded from below. We will suppose that $K$ has some discrete spectrum. One may consider purely discrete spectrum, like \cite{GGM2}, or not, like in \cite{Ski}. To do not mutter the presentation, we will take $\Kr= {\rm Ran\, } E_{\Ic}(K)$, where $\Ic$ contains a finite number of eigenvalues and consider the restriction of $K$ to this space. Hence, we restrict the analysis to a self-adjoint matrix $K$ acting in a Hilbert space $\Kr$ of finite dimension. This corresponds to analyze $n$ level atoms. Doing so, we avoid some light problems of domains, which are already discussed in details in \cite{GGM2, Ski} and gain in clarity of presentation. \subsection{The bosonic field}\label{s:bosonic} We refer to \cite{BR, BSZ, RS} for a more thorough discussion of these matters. The bosonic field is described by the Hilbert space $\Gamma(\hg)$, where $\hg$ is a Hilbert space. We recall its construction. Set $\hg^{0\otimes}=\C$ and $\hg^{n\otimes}=\hg\otimes \dots \otimes \hg$. Given a closed operator $A$, we define the closed operator $A^{n\otimes}$ defined on $\hg^{n\otimes}$ by $A^{0\otimes}=1$ if $n=0$ and by $A\otimes \ldots \otimes A$ otherwise. Let $S_n$ be the group of permutation of $n$ elements. For each $\sigma\in S_n$, one defines the action on $\hg^{n\otimes}$ by $\sigma(f_{i_1}\otimes\ldots\otimes f_{i_n})= f_{\sigma^{-1}(i_1)}\otimes \ldots\otimes f_{\sigma^{-1}(i_n)}$, where $(f_i)$ is a basis of $\hg$. The action extends to $\hg^{n\otimes}$ by linearity to a unitary operator. The definition is independent of the choice of the basis. On $\hg^{n\otimes}$, we set \begin{eqnarray}\label{e:Pn} \Pi_n:=\frac{1}{n!}\sum_{\sigma\in S_n} \sigma \mbox{ and } \Gamma_n(\hg):= \Pi_n(\hg^{n\otimes}). \end{eqnarray} Note that $\Pi_n$ is an orthogonal projection. We call $\Gamma_n(\hg)$ the $n$-particle bosonic space. The bosonic space is defined by \begin{eqnarray*} \Gamma(\hg):=\bigoplus_{n=0}^{\infty} \Gamma_n(\hg). \end{eqnarray*} We denote by $\Omega$ the \emph{vacuum}, the element $(1,0,0,\ldots)$ and by $P_\Omega:=\Gamma(\hg)\rightarrow \Gamma_0(\hg)$ the projection associated to it. We define $\Gamma_{\rm fin}(\hg)$ the set of finite particle vectors, i.e.\ $\Psi=(\Psi_1, \Psi_2, \ldots)$ such that $\Psi_n=0$ for $n$ big enough. We now define the second quantized operators. We recall that a densely defined operator $A$ is closable if and only if its adjoint $A^*$ is densely defined. Given a closable operator $q$ in $\hg$. We define $\Gamma_{\rm fin}(q)$ acting from $\Gamma_{\rm fin}(\Dc(q))$ into $\Gamma_{\rm fin}(\hg)$ by \begin{eqnarray*} \Gamma_{\rm fin}(q)|_{\Pi_n( \Dc(q)^{n\otimes})}:= q\otimes \ldots \otimes q. \end{eqnarray*} Since $q$ is closable, $q^*$ is densely defined. Using that $\Gamma_{\rm fin}(q^*)\subset \Gamma_{\rm fin}(q)^*$, we see that $\Gamma_{\rm fin}(q)$ is closable and we denote by $\Gamma(q)$ its closure. Note that $\Gamma(q)$ is bounded if and only if $\|q\|\leq 1$. Let $b$ be a closable operator on $\hg$. We define $\dG_{\rm fin}(b):\Gamma_{\rm fin}\big(\Dc(b)\big)\longrightarrow \Gamma_{\rm fin}(\hg)$ by \begin{eqnarray*} \dG_{\rm fin}(b)|_{\Pi_n( \Dc(b)^{n\otimes})}:= \sum_{j=1}^{n} 1\otimes \ldots \otimes 1 \otimes \underbrace{b}_{j^{\rm th}} \otimes 1\otimes \ldots \otimes 1. \end{eqnarray*} As above, $\dG_{\rm fin}(b)$ is closable and $\dG(b)$ denotes also its closure. We link the objects. \begin{lemma}\label{l:semi} Let $\R^+\ni t\mapsto w_t\in\Bc(\hg)$ be a $C_0$-semigroup of contractions (resp.\ of isometries), with generator $a$. Then $\R^+\ni t\mapsto \Gamma(w_t)\in\Bc\big(\Gamma(\hg)\big)$ is a $C_0$-semigroup of contractions (resp.\ of isometries) whose generator is $\dG(a)$. \end{lemma} \proof It is easy to see that $W_t:=\Gamma(w_t)$ is a $C_0$-semigroup of contractions (resp.\ of isometries). Let $A$ be its generator. Immediately, one gets $\dG_{\rm fin}(a)\subset A$. Since $\Gamma_{\rm fin}\big(\Dc(a)\big)$ is dense in $\dG(\hg)$ and invariant under $W_{t}$, the Nelson lemma gives that $\Gamma_{\rm fin}\big(\Dc(a)\big)$ is dense in $\Dc(A)$ for the graph norm and also that $\dG(a)=A$.\qed \subsection{The interacting system}\label{s:inter} Given a self-adjoint operator $\omega$ in $\hg$ and a finite dimensional Hilbert space $\Kr$. One defines the free Hamiltonian $H_0$ acting on the Hilbert space $\Hr:=\Kr\!\otimes \Gamma(\hg)$ by \begin{eqnarray}\label{e:H0} H_0:=K\otimes 1_{\Gamma(\hg)}+1_\Kr\! \otimes \dG(\omega). \end{eqnarray} We recall also the definition of the \emph{number operator} $N:=1_\Kr\!\otimes\dG(\id)$. We now define the interaction. Let $\alpha$ be an element $\Bc(\Kr, \Kr\!\otimes \hg)$. This is a \emph{form-factor}. We define $b(\alpha)$ on $\Hr$ by $b(\alpha):= \Kr\!\otimes h^{n\otimes}\rightarrow \Kr\!\otimes h^{(n-1)\otimes}$, where \begin{eqnarray*} b(\alpha)(\Psi\otimes \phi_1\otimes \ldots \phi_n):=\alpha^*(\Psi\otimes \phi_1)\otimes \phi_2\otimes \ldots \phi_n, \end{eqnarray*} for $n\geq 1$ and by $0$ otherwise. This operator is bounded and its norm is given by $\|\alpha\|_{\Bc(\Kr, \Kr\!\otimes \hg)}$. We define the \emph{annihilation operator} on $\Kr\!\otimes \Gamma(\hg)$ with domain $\Kr\!\otimes \Gamma_{\rm fin}(\hg)$ by \begin{eqnarray*} a(\alpha):=(N+1)^{1/2}b(\alpha)\big(1\otimes \Pi), \end{eqnarray*} where $\Pi:=\sum_n \Pi_n$, see \eqref{e:Pn}. As above, it is closable and its closure is denoted by $a(\alpha)$. Its adjoint is the \emph{creation operator}. It acts as $a^*(\alpha)= b^*(\alpha) (N+1)^{1/2}$ on $\Hr$. Note that $b^*(\alpha)(\psi\otimes\phi_1\otimes \ldots \otimes \phi_n)=(\alpha\phi)\otimes \phi_1\otimes \ldots \otimes \phi_n$. The \emph{(Segal) Field operator} is defined by \begin{eqnarray*} \phi(\alpha):=\frac{1}{\sqrt{2}}\big(a(\alpha)+a^*(\alpha)\big). \end{eqnarray*} We consider its closure on $\Kr\!\otimes \Dc(N^{1/2})$. We have the two elementary estimates: \begin{eqnarray}\label{e:bdN} \|(N+1)^{-1/2}a^{(*)}(\alpha)\|\leq \|\alpha\|, \quad \|(N+1)^{-1/2}\phi(\alpha)\|\leq \sqrt{2}\|\alpha\|. \end{eqnarray} An assertion containing $(*)$ holds with and without $*$. We give the following $N_\tau$-estimate and refer to \cite[Proposition 4.1]{DJ} for a proof of i). The point ii) is a direct consequence of the Kato-Rellich Lemma. This kind of estimates comes back to \cite{GJ}. See also \cite{BFS}. We refer to \cite{G}[Appendix A] and \cite{GGM2}[Proposition 3.7] for unbounded $K$. \begin{proposition}\label{p:ntau} Let $\omega$ be a non-negative, injective, self-adjoint operator on $\hg$. Let $\beta\in\Bc\big(\Kr, \Kr\!\otimes \Dc(\omega^{-1/2})\big)$. {\rm i)} Then $\phi(\beta)\in\Bc\big(\Kr\!\otimes\Dc(\dG(\omega)^{1/2}), \Hr\big)$ and for any $\Phi\in \Dc(\dG(\omega)^{1/2})$, \begin{align} \nonumber |\phi(\beta)\Phi\|^2\leq& \|\beta\|_{\Bc(\Kr, \Kr\!\otimes \hg)} \, \|\Phi\|^2 \\\label{e:ntau0} & + 2\|\omega^{-1/2}\beta\|_{\Bc(\Kr, \Kr\!\otimes \hg)} \, \langle \Phi, \un_\Kr\!\otimes\dG(\omega) \Phi \rangle. \end{align} {\rm ii)} The field operator $\phi(\alpha)$ is $H_0$-operator bounded with relative bound $\varepsilon $, for all $\varepsilon >0$. Hence, $H_\lambda:= H_0 +\lambda \phi(\alpha)$, for $\lambda\in\R$, defines a self-adjoint operator with domain $\Dc(H_\lambda)=\Kr\!\otimes\dG(\omega)$ and is essentially self-adjoint on any core of $H_0$. \end{proposition} \subsection{The zero-temperature Pauli-Fierz Model}\label{s:model} We now precise our model to the zero-temperature physical setting. The one particle space is given by $\hg:=L^2(\R^d, dk)$, where $k$ is the boson momentum. The \emph{one particle kinetic energy} is the operator of multiplication by $\omega(k):=|k|$. Consider a self-adjoint matrix $K$ on a finite dimensional Hilbert space $\Kr$ and denote by $(k_i)_{i=0, \ldots, n}$, with $k_i\pp k_{i+1}$ its eigenvalues. We denote by $P_{k_i}$ the projection onto the $i$-th eigenspace. The spectrum of $\dG(\omega)$ in $\Gamma(\hg)$ is $[0, \infty)$ and due the vacuum part, $0$ is the only eigenvalue. Its multiplicity is one. The spectrum of $H_0$ given by \eqref{e:H0} is $[k_0, \infty)$. The eigenvalues are given by $(k_i)_{i=0, \ldots, n}$ and have the same multiplicity as those of $\Kr$. The singularly continuous component of the spectrum is empty. Here, $(k_i)_{i=0, \ldots, n}$ play also the r\^ole of \emph{thresholds}. We consider a form-factor $\alpha$ satisfying hypothesis {\bf (I0)}. By applying Proposition \ref{p:ntau}, the operator $H_\lambda$, given by \eqref{e:H}, is self-adjoint and $\Dc(H_{\lambda})= \Kr\!\otimes \Dc\big(\dG(\omega)\big)$. Since we study form factors in $\Bc(\Kr,\Kr\!\otimes \hg)$, we forbid some eventual singularities of the form-factor from the very beginning. However, if the atomic part has a particular shape, one may use some gauge transformations and gains in singularity, see for instance \cite{GGM2}[Section 2.4] and \cite{DJ}[Section 1.6]. Nevertheless, it is an open question if there exists some gauge transformation that allows one to cover the physical form factor studied in \cite{BFS, BFSS}, from our conditions. Conversely, the classes of perturbations studied in the latter does not fully cover ours. \section{The conjugate operators}\label{s:conjop} \setcounter{equation}{0} In this paper, we analyze the spectrum of the Pauli-Fierz Hamiltonian $H_\lambda$ described in Section \ref{s:model} using some commutator techniques. We study the behavior of the embedded eigenvalues of $H_\lambda$ under small coupling constants and establish some refined spectral properties. To do so, we establish a version of the Mourre estimate, see Appendix \ref{s:LAPhyp}. Hence, we start by constructing the conjugate operator. We follow similar ideas as in \cite{GGM2, hsp, Ski}. Later, we modify it by a finite rank perturbation, in the spirit of \cite{BFSS}. Unlike in the standard Mourre theory, the conjugate operator is not self-adjoint and only maximal-symmetric. We refer to Appendix \ref{s:C1} for discussions about $1$-commutators properties in this setting. We point out that one may avoid to work with maximal-symmetric operator by symmetrizing the space and thus gluing non-physical free bosons, see \cite{DJ}[Section 5.2]. This trick leads to some problems of domains with our method and would be treated elsewhere. We point out that the real drawback of this choice of conjugate operator comes from the fact that the commutator is not $H_\lambda$ bounded, like in the standard Mourre theory and \cite{BFS, BFSS, FGS, FP}. Some difficulties appear to apply the Virial theorem. To overcome them, we follow ideas of \cite{Ski, GGM2} and construct a series of approximate conjugate operators. One may also proceed like in \cite{M}. \subsection{The semigroup on the $1$-particle space} Fix $\cchi\in\Cc^\infty_c\big(\R^+;[0,1]\big)$ decreasing such that $\cchi(x)=1$ for $x\leq 1$ and $0$ for $x\geq 2$. Set $\tilde \cchi := 1-\cchi$. We consider the following vector fields on $\R^+$: \begin{align}\label{e:sd} m_n(t):= \begin{cases} \tilde \cchi\left(n t\right), & \mbox{ for } n\in\N, \\ 1, & \mbox{ for } n=\infty, \end{cases} \mbox{ and } s_n(t)=\frac{m_n(t)}{t}. \end{align} Note that $m_n$ converges increasingly to $m_\infty$, almost everywhere, as $n$ goes to infinity. As in \cite{Ski} and in \cite{GGM2}, the r\^ole of $m_\infty$ would be to ensure the positivity of the commutator and the one of $m_n$ would be to guarantee of the Virial theorem. We define the associated vector fields in $\R^d$ as follows: \begin{eqnarray}\label{e:sdv} \overrightarrow{s_n}(k) :={s}_n \big(|k|\big)k, \mbox{ for } k\in\R^d \mbox{ and } n\in\N^*\cup\{\infty\}. \end{eqnarray} We shall construct the $C_0$-semi\-group of isometries associated to the vector fields $\overrightarrow{s_n}$ on $\hg=L^2(\R^d)$ and identify the generators. We define \begin{eqnarray}\label{e:sgp} a_n:=\displaystyle-\frac{1}{2}\Big(\overrightarrow{s_n}\cdot D_k + D_k \cdot\overrightarrow{s_n} \Big) \end{eqnarray} on $\Cc^\infty_c(\R^d\setminus \{0\})$ for all $n\in\N^*\cup\{\infty\}$ and where $D_k=i\nabla$. These operators are closable as the domains of their adjoints are dense. In the sequel, we denote by the same symbol their closure. We work in polar coordinates. We identify $\hg$ and $\tilde \hg$ through the transformation \eqref{e:T}. Given an operator $B$ in $\hg$, we denote by $\tilde B$ the corresponding operator acting in the $\tilde\hg$ and given by $\tilde B:= T B T^{-1}$. We have: % \begin{proposition}\label{p:ess} For $n$ finite, $a_n$ is essentially self-adjoint on $\Cc^\infty_c(\R^d\setminus \{0\})$ and $a_\infty$ is maximal symmetric with deficiency indices $(N, 0)$. Here, $N=\infty$ for $d\geq 2$ and $N=2$ for $d=1$. The operator $a_n$ generates a $C_0$-semigroup of isometries denoted by $\{w_{n,t}\}_{t\in\R^+}$. In polar coordinates, the domains are given by \begin{align*} \Dc(\tilde a_n)\supset \Dc(\tilde a_\infty)=&\, \dot\Hr^1(\R^+)\otimes L^2(S^{d-1}), \mbox{ for all } n\in \N^*, \\ \Dc(\tilde a_\infty^*)=&\, \Hr^1(\R^+)\otimes L^2(S^{d-1}), \end{align*} where $\dot\Hr^1(\R^+)$ is the closure of $\Cc^\infty_c(\R^{+})$ under the norm $\|\cdot \|+\|\partial_r \cdot\|$ and where $\Hr^1(\R^+)$ is the Sobolev space of first order. \end{proposition} See Section \ref{s:semiprop} for an overview on $C_0$-semigroups. For $n$ finite, the $C_0$-semigroup extends to a $C_0$-group since $a_n$ is self-adjoint. \proof When $n$ is finite, it is well known that $a_n$ is essentially self-adjoint on $\Cc^\infty_c(\R^d)$ and follows by studying $C_0$-group associated to the flow defined by the smooth vector field $\overrightarrow{s_n}$. The density follows by the Nelson lemma. See for instance \cite{ABG}[Proposition 4.2.3]. Hence, for $n$ finite, it remains to show that $\Cc^\infty_c(\R^d\setminus \{0\})$ is a core for $a_n$. Straightforwardly, for $n\in\N^*\cup\{\infty\}$, one gets \begin{eqnarray}\label{e:atilde} \quad \quad {\tilde a}_n:=Ta_n T^{-1}= i\big(m_n(\cdot)\partial_r +\frac{1}{2} (m_n)'(\cdot) \big)\otimes \un, \mbox{ where } m_n(r):=r s_n(r). \end{eqnarray} We extend $m_n$ on $\R$ by setting $m_n(-r):=m_n(r)$ for $r\pg 0$ and prolongate it by continuity in $0$. Let $\phi_{n,t}$ be the flow generated by the smooth vector field $m_n$ on $\R$. In other words, $\phi_{n,t}:=\phi_n(t,\cdot)$ is the unique solution of $(\partial_t\phi_n)(t,r)= m_n\big(\phi_n(t,r)\big)$, where $\phi_n(0,r)=r$. Since $m_n$ is globally Lipschitz, $\phi_{n,t}$ exists for all time $t$. Moreover, $\phi_{n,t}$ is a smooth diffeomorphism of $\R$ with inverse $\phi_{n,-t}$ for all $t\in\R$. Let $\tilde \phi_{n,t}$ be the restriction of $\phi_{n,t}$ from $\R^{+*}$ to $\R^{+*}$. Let $\Omega_{n,t}$ be the domain of this restriction, i.e.\ the set of $r\pg 0$ such that $\phi_{n,t}(r)\pg 0$. One has $\Omega_{n,t}=\R^{+*}$ for $t\geq 0$ as $m_n(r)$ is positive. For the same reason, $t\mapsto \Omega_{n,t}$ is increasing. Note also that we have $\Omega_{n,-t}=\phi_{n,t}(\R^{+*})$ for $t\geq 0$. For $u\in\tilde \hg$, we set: \begin{eqnarray}\label{e:wtilde} (\tilde w_{n,t} u)(r, \theta) := 1_{\Omega_{n,-t}}(r)\sqrt{\phi_{n,-t}'(r)} u(\phi_{n,-t}(r), \theta), \mbox{ for } t\geq 0. \end{eqnarray} A change of variable gives that $\tilde w_{n,t}$ is an isometry of $L^2(\R^+)$ with range $L^2(\Omega_{n,-t})$ for all $t\geq 0$. Since $\phi_{n,t}$ is a smooth flow, $\{\tilde w_{n,t}\}_{t\geq 0}$ is a $C_0$-semigroup of isometries. The adjoint $C_0$-semigroup is given by \begin{eqnarray}\label{e:wadj} (\tilde w_{n,t}^*u)(r, \theta) := 1_{\R^{+*}}(r) \sqrt{\phi_{n,t}'(r)} u(\phi_{n,t}(r), \theta), \mbox{ for } t\geq 0. \end{eqnarray} This is \emph{not} a semigroup of isometries when $n=\infty$. We compute the generator of the semigroup $\{\tilde w_{n,t}\}_{t\geq 0}$. Take $u\in \Cc^\infty_c\big(\tilde \hg\big)$. We have $\tilde w_{n,t} u\in \Cc^\infty_c(\Omega_{n,-t}\times S^{d-1})$. Let $r\in \Omega_{n,-t}$, we get \begin{eqnarray*} -\left(\frac{d}{dt} \tilde w_{n,t} u\right)(r, \theta)= \left(\tilde w_{n,t} \big(m_n(\cdot) \partial_r +\frac{1}{2} (m_n)'(\cdot)\big)u \right)(r,\theta). \end{eqnarray*} Denoting by the same symbol the closure of $\tilde a_n$ on $\Cc^\infty_c(\R^{+*}\times S^{d-1})$, we obtain \begin{eqnarray*} -i \frac{d}{dt} \tilde w_{n,t} u= \tilde w_{n,t} \tilde a_n u. \end{eqnarray*} The closed operator is \emph{a priori} only a restriction of the generator of the semigroup (in the sense of the inclusion of graph of operators). Now, since $\tilde w_{n,t}$ stabilizes $\Cc^\infty_c(\tilde \hg)$ for all $t\geq 0$, the Nelson lemma gives that this space is a core for generator of the $C_0$-semigroup $\{\tilde w_{n,t}\}_{t\geq 0}$. Since this one is an extension of $\tilde a_n$, we have shown that $\tilde a_n$ is really the generator. One may denote formally $\tilde w_{n,t}=e^{it\tilde a_n}$. The domain of $\tilde a_n$ contains $\dot\Hr^1(\R^+)\otimes L^2(S^{d-1})$. Easily, this is an equality for $n=\infty$. Considering the spectrum of $a_n$, we derive the deficiency indices of the closure of $a_n$ on $\Cc^\infty_c(\R^d\setminus\{0\})$ are of the form $(N,0)$. For $n$ finite these indices are equal, we infer the essential self-adjointness of $a_n$ on $\Cc^\infty_c(\R^d\setminus\{0\})$. At this point, one may feel the real difference between the case $n$ finite and infinite. On one hand $m_\infty\geq 1$ and on the other hand, for finite $n$, $m_n(r)$ tends to $0$ as $r$ tends to $0$. The domain of the adjoint of $\tilde a_\infty$ would be different. Indeed, \begin{eqnarray}\label{e:aadj} \big(\tilde a_\infty^*u\big)(r, \theta)= i\big(m_\infty(r)(\partial_r u)(r,\theta) +\frac{1}{2} (m_\infty)'(r)u(r,\theta) \big), \end{eqnarray} where $u\in\Dc(\tilde a_\infty^*)=\Hr^{1}(\R^+)\otimes L^2(S^{d-1})$. Moreover, when $n=\infty$, the deficiency indices are then $(\infty, 0)$, as the dimension of $L^2(S^{d-1})$ is infinite. \qed \subsection{The $C_0$-semigroup on the Fock space.} Thanks to Proposition \ref{p:ess} and Lemma \ref{l:semi}, we define the $C_0$-semigroups on the whole Hilbert space. We set: \begin{eqnarray}\label{e:semi} W_{n,t}:=\un_\Kr\!\otimes \Gamma(w_{n,t}) \mbox{ and } W^{*}_{n,t}=\un_\Kr\!\otimes \Gamma(w^{*}_t), \mbox{ for } t\geq 0. \end{eqnarray} Clearly, $\{W_{n,t}\}_{t\geq 0}$ is a $C_0$-semigroup of isometries. Let $A_\infty$ be its generator. In the same way, for $n$ finite, we set \begin{eqnarray}\label{e:semin} A_n:=\un_\Kr\!\otimes \dG(a_n). \end{eqnarray} This is the generator of the $C_0$-group $\un_\Kr\!\otimes\Gamma (e^{ita_n})$ by Lemma \ref{l:semi}. Recall the r\^ole of the $A_n$ is to ensure a Virial theorem, see Proposition \ref{p:6.5}. In Section \ref{s:far}, we see that the operator $A_\infty$ alone is not enough to deal with threshold energy as the system could be uncoupled. One needs to take in account the Fermi golden rule. One way is to follow \cite{DJ} and to take advantage of it in the limiting absorption principle. Another way is to modify the conjugate operator with a \emph{finite rank perturbation} so as to obtain more positivity above the thresholds, by letting appearing the Fermi golden rule in the commutator, see Section \ref{s:thre}. This idea comes from \cite{BFSS}. We follow it. Choose $k_{i_0}$ an eigenvalue of $K$ and assume that \eqref{e:FGR0} holds true at energy $k_{i_0}$ for the couple $(H_0, \alpha)$. Let $P$ be the projector $P_{k_{i_0}}\otimes P_\Omega$. For $\varepsilon<\varepsilon_0$, we define \begin{eqnarray*} \hat A_n:=A_n+\lambda \theta B_\varepsilon, \mbox{ for } n\in \N^*\cup\{\infty\}, \end{eqnarray*} where $B_\varepsilon :=\im (\overline{R_\varepsilon}^2\phi(\alpha)P)$, $R_\varepsilon:=\big((H_0-k_{i_0})^2+\varepsilon^2\big)^{-1/2}$ and $\overline{R_\varepsilon}:=\overline{P} R_\varepsilon$. Note that the conjugate operator depends on the two parameters $\lambda\in \R$ from the coupling constant, $\varepsilon>0$ from the Fermi golden rule hypothesis and on an extra technical $\theta>0$. For the sake of clarity, we do not write these extra dependences. Using Proposition \ref{p:kato} and the fact that $B_\varepsilon $ is bounded, one gets $\hat A_\infty$ is the generator of a $C_0$-semigroup. A bit more is true. \begin{lemma}\label{l:semiA} The operator $\hat A_\infty$ is maximal symmetric on $\Dc(A_\infty)$ and is the generator of $C_0$-semigroup of isometries, denoted by $\{\widehat W_{n,t}\}_{t\geq 0}$. For $n$ finite, the operator $\hat A_n$ is self-adjoint on the domain of $\Dc(A_n)$. \end{lemma} \proof The second point is obvious. We concentrate on the first one. By Proposition \ref{p:ess}, $A_\infty$ is maximal symmetric with deficiency indices $(N,0)$ for some $N\neq 0$. Since $B_\varepsilon$ is bounded, there is $c<0$ such that $\|B_\varepsilon(A_\infty -z)^{-1}\|<1$, for all $z\in \C$ where $\im(z)\leq c$. Since $(I+ B_\varepsilon(A_\infty -z)^{-1})(A_\infty -z)= A_\infty+B_\varepsilon-z$ on the domain of $A_\infty$, we get the spectrum of $\hat A_\infty$ is contained in an upper half plane $\R + i [c, \infty)$. Now, since $B_\varepsilon$ is symmetric, so is $\hat A_\infty$. If the indices of $\hat A_\infty$ would be both non-zero then its spectrum would be $\C$. Therefore, the deficiency indices of $\hat A_\infty$ are $(N',0)$ for some non-negative $N'$. Note that $N'\neq 0$ by the Kato-Rellich theorem applied on $\hat A_\infty$, since $B_\varepsilon$ is bounded. Hence, $\hat A_\infty$ is maximal symmetric on $\Dc(A_\infty)$ and its spectrum is $\R + i [0, \infty)$. It is automatically a $C_0$-semigroup of isometries. \qed %%%%%%%%%%% \section{Smoothness with respect to the $C_0$-semigroup}\label{s:regu} %%%%%%%%%%% \setcounter{equation}{0} In Section \ref{s:gene}, we recall a general result. In Section \ref{s:1comm}, we give some $1$-commutator properties for $A_n$. We check the hypothesis {\bf (M1)}--{\bf (M4)} of Appendix \ref{s:LAPhyp}. We identify the spaces and operators appearing therein in Lemma \ref{l:6.4}. In Section \ref{s:1comm'}, we extend these properties to $\hat A_n$, using Proposition \ref{p:kato} and Lemma \ref{l:finite}. The Virial theorem is discussed in Section \ref{s:virial}. At last, second commutator assumptions and the hypothesis {\bf (M5)} are discussed in Section \ref{s:2comm}. \subsection{A general result}\label{s:gene} In order to check the $C^1$ properties, the $b$-stability, see Definition \ref{d:bstable}, and to be able to deduce hypothesis {\bf (M1)-(M5)} of Appendix \ref{s:LAPhyp}, we recall \cite{GGM2}[Proposition 4.10]. We formulate it for bounded $K$. Set first a $C_0$-semigroup of isometries $\R^+\ni t \rightarrow v_t\in\Bc(\hg)$ with generator $a$. By Lemma \ref{l:semi}, $V_t:=\un_\Kr\!\otimes \Gamma(v_t)$ is a $C_0$-semigroup of isometries with generator $A=\un_\Kr\! \otimes\dG(a)$. Let $b\geq 0$ be a self-adjoint operator on $\hg$, and $K$ as in \eqref{e:H0}. Set \begin{eqnarray*} B:=K\otimes \un_{\Gamma(\hg)} + \un_\Kr\! \otimes \dG(b), \quad \Gr_B:=\Dc(B^{1/2})=\un_\Kr\! \otimes \Dc\big(\dG(b)^{1/2}\big). \end{eqnarray*} \begin{proposition}\label{p:4.10} Let $\omega$ and $b\geq 0$ acting in $\hg$. Then, {\rm i)} The space $\Gr_B$ is $b$-stable under $\{V_t\}_{t\in\R^+}$ {\rm (}resp.\ $\{V_t^*\}_{t\in\R^+}${\rm )}, if \begin{eqnarray}\label{e:4.19} v^*_t b v_t\leq C_t b, (\mbox{resp.\ } v_t bv^*_t \leq C_t b) \mbox{ with } \sup_{01/2$, dominated convergence proves it tends to $0$ in $\Bc(\Kr, \Kr\!\otimes \hg)$. The proof of the second point is the same but use the fact that $r^{-a}\alpha \in \Bc(\Kr, \Kr\!\otimes \hg)$ for the term in $m_n'$, for some $a>1$. \qed We point out that if one knows that $\omega^{-1}\alpha\in\Bc(\Kr, \Kr\!\otimes \Cc_0(\R^+)\otimes L^2(S^{d-1}))$, one may relax {\bf (I1b)} and take $a=1$. Here $\Cc_0(\R^+)$ denotes the continuous functions vanishing in $0$ and in $+\infty$. \begin{lemma}\label{l:6.3.0} Assume $n$ finite, {\bf (I0)} and {\bf (I1a)}. Then, $\{\widehat W_{n,t}\}_{t\in\R}$ $b$-stabilizes the form domain of $H_\lambda$. \end{lemma} \proof First we apply Proposition \ref{p:4.10} i) with $v_t=w_{n,t}$ and $b=w$. As we have a $C_0$-group, by taking $t$ negative we obtain the result for the adjoint. As in the proof of Proposition \ref{p:ess}, we denote by $\phi_{n,t}:\R^d\rightarrow \R^d$ the flow generated by the smooth vector field $\overrightarrow{s_n}$. Since $m_n(0)=0$, we have \begin{align} \nonumber |\phi_{n,t}(k)-k|&= |\phi_{n,t}(k)-\phi_{n,0}(k)|\leq \int_0^{|t|} \left|m_n(\phi_{n,s}(k))- m_n(0)\right|\, ds \\ \label{e:pourGron} &\leq \|\nabla m_n\|_{\infty} \int_0^{|t|} |\phi_{n,s}(k)| \, ds, \mbox{ for all } t\in \R. \end{align} By the Gronwall lemma, we infer there is $C$ such that $|\phi_{n,t}(k)|\leq C|k|$, for all $t\in[1,1]$. Plugging back into \eqref{e:pourGron}, we obtain $|\phi_{n,t}(k) - k|\leq C|tk|$, for all $t\in[1,1]$. Now using \eqref{e:wtilde} and \eqref{e:wadj}, we infer $e^{-ita_n} w e^{ita_n}= w\big(\phi_{n,t}(\cdot) \big)$. Since $m_n$ is globally Lipschitz, there is $C'$ such that \begin{eqnarray}\label{e:Gronw} |w(\phi_{n,t}(k))- w(k)|\leq C'|t|w(k), \mbox{ for all } t\in[1,1]. \end{eqnarray} Hence, we satisfy the hypothesis \eqref{e:4.19} and $\Dc(|H_\lambda|^{1/2})$ is $b$-stable under $\{W_{n,t}\}_{t\in\R}$. We now take care about $\{\widehat W_{n,t}\}_{t\in\R}$. Let $A_n'$ be the generator of $\{W_{n,t}\}_{t\in\R}$ in $\Dc(|H_\lambda|^{1/2})$. As in \eqref{e:Ahat}, set $\widehat{A_n}':= A_n'+ \lambda\theta B_\varepsilon$. By Lemma \ref{l:finite} ii) and the fact that $B_\varepsilon $ is with values in the $0$ and $1$ particles space, we get $B_\varepsilon$ bounded in $\Dc(|H_\lambda|^{1/2})$. Thanks to Proposition \ref{p:kato}, $\widehat{A_n}'$ is the generator of a $C_0$-group in $\Dc(|H_\lambda|^{1/2})$. We name it $\{\widehat W_{n,t}'\}_{t\in\R}$. By duality and interpolation, it extends to a $C_0$-group in $\Hr$. Comparing the generators, we obtain that $\{\widehat W_{n,t}'\}_{t\in\R}$ is really the restriction of $\{\widehat W_{n,t}\}_{t\in\R}$ and this gives the result.\qed \begin{lemma}\label{l:6.3.1} Assume $n$ finite, {\bf (I0)} and {\bf (I1a)}. Then $H_\lambda\in\Cc^1(\hat A_n)$. Moreover: \begin{eqnarray}\label{e:commun} \big[H_\lambda, i\hat A_n\big]= M_n+\hat S_n, \end{eqnarray} holds true in the sense of forms on $\Dc(|H_\lambda|^{1/2})$. \end{lemma} \proof Using again \eqref{e:Gronw}, we check \eqref{e:4.21}. We get $[H_0, iA_n]^\circ= \un_\Kr\!\otimes \dG([\omega, ia_n]^\circ)$ in the sense of form on $\Dc(|H_\lambda|^{1/2})$. By computing $[\omega, ia_n]^\circ$ on the core $\Cc^\infty_c(\R^d\setminus\{0\})$, we obtain $[\omega, ia_n]^\circ= m_n$. Now, by Lemma \ref{l:6.2}, we can use Proposition \ref{p:4.10} iii) and deduce $\big[H_\lambda, i A_n\big]= M_n+ S_n$ in the sense of forms on $\Dc(|H_\lambda|^{1/2})$. Finally, by Lemma \ref{l:commu}, $[H_\lambda, B_\varepsilon]$ is of finite rank, we also obtain \eqref{e:commun} on the same domain. Now, $H_\lambda\in\Cc^1\big(\hat A_n; \Dc\big(|H_\lambda|^{1/2}\big), \Dc\big( |H_\lambda|^{1/2} \big)^*\big)$ by Lemma \ref{l:6.3.0} and Proposition \ref{p:2comm}. We apply \cite{GGM2}[Lemma 6.3] to get $H_\lambda\in\Cc^1(\hat A_n)$. \qed Therefore, the Virial theorem holds true when $\hat A_n$ is the conjugate operator and when $n$ is finite. However, there is no Mourre estimate for $\hat A_n$ but only one for $\hat A_\infty$. To overcome this problem, we take advantage of the monotone convergence of $[H_0,i A_n]$ to $[H_0,i A_\infty]$ and of the uniformity given in Lemma \ref{l:unif} to prove: \begin{proposition}[Virial theorem]\label{p:6.5} Assume {\bf (I0)} and {\bf (I1)}. Let $u$ be an eigenfunction of $H_\lambda$ then $u\in \Dc( N^{1/2})$ and $\langle u, (M_\infty+ \hat S_\infty) u\rangle=0$, as a quadratic form on $\Dc(N^{1/2})\cap\Dc(H_\lambda)$. \end{proposition} \proof First, $M_n$ is a bounded form for $H_\lambda$. Note that $0\leq m_n\leq m$ implies $0\leq \dG(m_n)\leq \dG(m)$ for all $n$. Now, since $m_n$ is increasing and converges to $m$ as $n$ goes to infinity, monotone convergence gives \begin{eqnarray*} 0\leq \langle g, M_n g \rangle\leq \langle g, M_\infty g \rangle \mbox{ and } \langle g, M_n g \rangle \build{\longrightarrow}_{n\rightarrow \infty}^{} \langle g, M_\infty g \rangle, \end{eqnarray*} for all $g\in \Dc(M_\infty)\cap \Dc(H_\lambda)$. Using some Cauchy sequences, this holds true also in the sense of forms for $g\in \Dc(M_\infty^{1/2})\cap \Dc(H_\lambda)$. By authorizing the value $+\infty$ on the two r.h.s.\ when $g\notin\Dc(M_\infty^{1/2})$, one allows $g\in\Dc(H_\lambda)$. On the other hand, Lemma \ref{l:6.2} gives that $\hat S_n$ tends to $\hat S_\infty$ as a quadratic form on $\Dc(H)$. Let $\dot H$ be the closure of quadratic form $\langle u, \hat H'_\lambda u \rangle $ defined on $\Dc(M_\infty)\cap \Dc(H)$. It is given by the quadratic form $\langle u, (M_\infty + \hat S_\infty) u\rangle$ defined on $\Dc(M^{1/2}_\infty)\cap \Dc(H)$. Take now an eigenfunction $u$ of $H_\lambda$. By Lemma \ref{l:6.3.1} and the Virial theorem, see \cite{ABG}[Proposition 7.2.10], we get $\langle u, (M_n+\hat S_n) u \rangle=0$. By letting $n$ go to infinity and noticing that $\Dc(M_\infty^{1/2})=\Kr\!\otimes \Dc(N^{1/2})$, we get the result. \qed \subsection{Estimation on the second commutator}\label{s:2comm} In this section, we discuss the second commutator hypothesis {\bf (I2)} so as to obtain a limiting absorption principle through the Theorem \ref{t:LAP}. We stress we forgo the hypothesis {\bf (I1b)} in this section. We start with the important remark. \begin{lemma}\label{l:changeregu} We have $\Cc^2(A_\infty, \Gr, \Gr^*)= \Cc^2(\hat A_\infty, \Gr, \Gr^*)$. \end{lemma} \proof It is enough to show one inclusion. Using Proposition \ref{p:2comm} and the invariance of $\Gr$ and $\Gr^*$ given in Lemmata \ref{l:6.6.0} and \ref{l:6.6}, one may work directly with $A_\infty$ and $\hat A_\infty$. Let $H\in \Bc(\Gr, \Gr^*)$ be in $\Cc^2(A_\infty, \Gr, \Gr^*)$. One justifies the next expansion, by working in the form sense on $\Dc\big((A_\infty^*)^2|_{\Gr}\big)\times \Dc\big((A_\infty)^2|_{\Gr}\big)$. This is legal by using Lemma \ref{l:finite}. We have: \begin{align*} [[H, \hat A_\infty], \hat A_\infty]=& [[H, A_\infty], A_\infty] + [[H, A_\infty], \lambda\theta B_\varepsilon] \\ & [[H, \lambda \theta B_\varepsilon], A_\infty] + [[H, \lambda \theta B_\varepsilon] , \lambda \theta B_\varepsilon]. \end{align*} The first term is in $\Bc(\Gr, \Gr^*)$ by hypothesis. For the second one, note that $[H, A_\infty]\in\Bc(\Gr, \Gr^*)$ since $H$ is $\Cc^1(A_\infty, \Gr, \Gr^*)$. For the third one, we expand the commutator inside, use again that $H\in\Cc^1(A_\infty, \Gr, \Gr^*)$ and finish with Lemma \ref{l:finite} iii). For the last one, one expands it and use Lemma \ref{l:finite} ii). \qed We start by discussing the $\Cc^2$ theory used in \cite{GGM2, Ski} and check the point {\bf (M5')}. Through the isomorphism given by \eqref{e:T}, we suppose the stronger \begin{itemize} \item[{\bf (I2')}] $\alpha\in \Bc\big(\Kr, \Kr\!\otimes \dot\Hr^{2}(\R^+)\otimes L^2(S^{d-1})\big)$. \end{itemize} This hypothesis is stronger than $\alpha\in \Bc\big(\Kr, \Kr\!\otimes \dot\Hr^{s}(\R^+)\otimes L^2(S^{d-1})\big)$ for $s>1$, the one used in \cite{DJ}[Theorem 6.3]. \begin{lemma}\label{l:6.7} Assume {\bf (I0)}, {\bf (I1a)} and {\bf (I2')}. Then $H_\lambda\in\Cc^2(\hat A_\infty, \Gr, \Gr^*)$ and \begin{eqnarray*} [\hat H'_\lambda, i \hat A_\infty]= \lambda \phi\big(a^2_\infty \alpha\big) +\lambda\theta [[H_\lambda, B_\varepsilon ], iA] + \lambda^2\theta^2 [\hat H_\lambda', B_\varepsilon ]. \end{eqnarray*} Therefore, the hypothesis {\bf (M5')} is fulfilled. \end{lemma} \proof We use Proposition \ref{p:4.10} ii) and iii) for the operator $H:=N - \lambda \phi(i a_\infty \alpha)$. Point ii) is trivially satisfied. The hypothesis {\bf (I2)} and Proposition \ref{p:ntau} give \eqref{e:4.22}. We obtain $H\in \Cc^1(A_\infty; \Gr, \Gr^*)$. \qed We now work with the hypothesis {\bf (I2)} which is weaker than the one used in \cite{DJ}. Thanks to Lemma \ref{l:changeregu}, we have \begin{align*} \Cc^{1,1}(A_\infty, \Gr, \Gr^*) &:= \big(\Cc^{2}(A_\infty, \Gr, \Gr^*), \Bc(\Gr, \Gr^*)\big)_{1/2, 1} \\ &= \big(\Cc^{2}(\hat A_\infty, \Gr, \Gr^*), \Bc(\Gr, \Gr^*)\big)_{1/2, 1}=: \Cc^{1,1}(\hat A_\infty, \Gr, \Gr^*). \end{align*} We refer to \cite{ABG, Tri} for real interpolation. We obtain: \begin{lemma}\label{l:6.7'} Assume {\bf (I0)}, {\bf (I1a)} and {\bf (I2)}. Then $H_\lambda\in\Cc^{1,1}(\hat A_\infty, \Gr, \Gr^*)$ and the hypothesis {\bf (M5)} is fulfilled. \end{lemma} \proof By Lemma \ref{l:6.7}, we have $H_0\in \Cc^2(\hat A_\infty, \Gr, \Gr^*)$. It is enough to show that $\phi(\alpha)\in\Cc^{1,1}(A_\infty, \Gr, \Hr)$. By \cite{DJ}[Lemma 2.7], we have $W_{\infty, t} \phi(\alpha)= \phi(w_{\infty, t}\alpha) W_{\infty, t}$ for $t\geq 0$. By Proposition \ref{p:ntau} and $b\geq 1$ and since $\{W_{\infty, t}\}$ $b$-preserves $\Gr$, we get \begin{align*} \int_0^1\big\|[W_{\infty, t}[W_{\infty, t}, \phi(\alpha)]]\big\|_{\Bc(\Gr, \Hr)}\, \frac{dt}{t^2}& \leq \int_0^1\big\|\phi([w_{\infty, t}[w_{\infty, t}, \alpha]]) W_{\infty, 2t}\big\|_{\Bc(\Gr, \Hr)}\, \frac{dt}{t^2} \\ & \leq C \int_0^1\big\|[w_{\infty, t}[w_{\infty, t}, \alpha]] \big\|_{\Bc(\Kr, \Kr\!\otimes \hg)}\, \frac{dt}{t^2}. \end{align*} The latter is finite if and only if $\alpha$ belongs to $\big(\Bc\big(\Kr, \Dc(a_\infty^2)\big), \Bc(\Kr, \Kr\!\otimes \hg)\big)_{1/2, 1}$. On the other hand, using the isomorphism \eqref{e:T} and Proposition \ref{p:ess}, this space is the same as $\big( \Bc\big(\Kr, \Kr\!\otimes \dot\Hr^2(\R^+)\otimes L^2(S^{d-1})\big), \Bc(\Kr, \Kr\!\otimes \tilde\hg)\big)_{1/2, 1}$. Finally, using \cite{Tri}[Section 2.10.4], this is equivalent to the fact that $\alpha$ satisfies {\bf (I2)}. \qed \section{A Mourre estimate far from the thresholds}\label{s:far} \setcounter{equation}{0} \subsection{The result} The aim in this part is to show a Mourre estimate far from thresholds for small coupling constants. This is a well-known result, see \cite{BFS, DJ} for instance. For the sake of completeness, we give a proof of the estimate. Doing so, we point out, in Remark \ref{r:motiv}, where the lack of positivity occurs above the thresholds. We use the approach based on the theory described in Appendix \ref{s:ms}. To obtain information just above the thresholds and without supposing the Fermi golden rule, one should add a compact term in \eqref{e:ms}, see \cite{GGM, Ski}. \begin{theorem}\label{t:ms} Let $\Ic_0$ be a compact interval containing no element of $\sigma(K)$. Suppose also that {\bf(I0)} and {\bf(I1a)} are satisfied. Then, for all open interval $\Ic\subset \Ic_0$: {\rm i)} There are $M_\infty\geq 1$ and $S_\infty$ a $|H_\lambda|^{1/2}$-bounded operator such that $[H_\lambda, i A_\infty]= M_\infty+S_\infty$ holds in the sense of forms on $\Dc(N^{1/2})$. {\rm ii)} The conditions {\bf(M1)}--{\bf(M4)} are satisfied. {\rm iii)} There is $\lambda_0>0$ such that the following extended Mourre estimate \begin{eqnarray}\label{e:ms} M_\infty + S_\infty \geq a(\lambda) E_\Ic(H_\lambda) - b(\lambda) E_{\Ic^c}(H_\lambda)\langle H_\lambda \rangle. \end{eqnarray} holds true in the sense of forms on $\Dc(N^{1/2})$, for all $|\lambda| \leq \lambda_0$. Here, $a(\lambda)$ is positive and can be written as $\big(1+O(\lambda)\big)$. Besides, $b(\lambda)$ is also positive. {\rm iv)} If {\bf (I1b)} holds true, then $H_\lambda$ has no eigenvalue in $\Ic$, for all $|\lambda| \leq \lambda_0$. {\rm v)} If {\bf (I2)} holds true (and not necessarily {\bf (I1b)}), then $H_\lambda$ has no eigenvalue in the interior of $\Ic$, for all $|\lambda| \leq \lambda_0$. Moreover, one obtains the estimations of the resolvent given in Theorem \ref{t:intro1}. \end{theorem} \proof By Lemma \ref{l:6.4}, we have the first point. The point ii) is shown in Section \ref{s:1comm}. The point iii) follows from Proposition \ref{p:ms1}. Indeed, since $S_\infty$ is form bounded with respect to $H_\lambda$, we have that for all $\eta>0$ \begin{align}\nonumber E_\Ic(H_\lambda) S_\infty E_{\Ic^c}(H_\lambda) +E_{\Ic^c}(H_\lambda) S_\infty E_{\Ic}(H_\lambda) \geq & \\\label{e:eta} &\hspace{-5cm} -\eta E_\Ic(H_\lambda) S_\infty \langle H_\lambda \rangle^{-1} S_\infty E_\Ic(H_\lambda) - \eta^{-1} E_{\Ic^c}(H_\lambda) \langle H_\lambda \rangle. \end{align} The point iv) follows from the Virial Theorem, Proposition \ref{p:6.5}. Finally, Theorem \ref{t:LAP} gives point v), the space $\Gr$ appearing therein is identified in Lemma \ref{l:6.4}. \qed \subsection{The inequality} Here we establish the extended Mourre estimate away from the threshold. We use only {\bf (I0)} and {\bf (I1a)} and do not assume any Fermi golden rule assumption. \begin{proposition}\label{p:ms1} Let $\Ic_0$ be a compact interval such that $\sigma(K)\cap \Ic_0=\emptyset$. Let $\Ic$ be an open interval included in $\Ic_0$. Let $M_\infty:=N+ \un \otimes P_\Omega \geq 1$ and let $S_\infty:=-\un\otimes P_\Omega- \lambda\phi(ia_\infty \alpha)$. For $\lambda$ small enough, we get \begin{eqnarray}\label{e:ms1} M_\infty+E_\Ic(H_\lambda) S_\infty E_\Ic(H_\lambda) \geq \big(1+O(\lambda)\big)E_\Ic(H_\lambda), \end{eqnarray} holds true in the sense of forms on $\Dc(N^{1/2})$. \end{proposition} \proof Let $\Jc$ be a compact set containing $\Ic$ and contained in the interior of $\Ic_0$. Note that \eqref{e:H0p} gives $E_\Jc(H_0) \un_\Kr\! \otimes P_\Omega = 0$. By Proposition \ref{p:ntau}, we derive: \begin{eqnarray}\label{e:H0p'} E_\Jc(H_0) S_\infty E_\Jc(H_0) = \lambda E_\Jc(H_0) \phi(ia_\infty \alpha) E_\Jc(H_0) = O(\lambda)E_\Jc(H_0). \end{eqnarray} As $M_\infty\geq 1$, it remains to prove that $E_\Ic(H_\lambda) S_\infty E_\Ic(H_\lambda)=O(\lambda)E_\Ic(H_\lambda)$. We insert $E_\Jc(H_0)+E_{\Jc^c}(H_0)$ on the right and on the left of $S_\infty$. By \eqref{e:H0p'}, all the four terms are actually $O(\lambda)E_\Ic(H_\lambda)$. Indeed, Proposition \ref{p:ntau} gives for instance that \begin{align*} E_\Ic(H_\lambda)E_\Jc^c(H_0) S_\infty E_\Jc(H_0)E_\Ic(H_\lambda) = O(\lambda) E_\Ic(H_\lambda). \end{align*} For the right hand side, take $h\in\Cc^\infty_c(\Jc)$ so that $h|_{\Ic}=1$. We have \begin{align*} E_\Ic(H_\lambda)E_{\Jc^c}(H_0)= E_\Ic(H_\lambda)\big(h(H_\lambda)-h(H_0)\big) E_{\Jc^c}(H_0)=O(\lambda), \end{align*} by Lemma \ref{l:HS}. \qed \begin{rem}\label{r:motiv} This proof would not work over one of thresholds $\{k_i\}_{i=0,\ldots, n}$. Here, we use in a drastic way that $E_\Jc(H_0) \un \otimes P_\Omega=0$. However, when $\sigma(K)\cap \Ic= \{k_i \}$, this expression is never $0$ and is of norm $1$. A brutal estimation would give \begin{eqnarray} M+E_\Ic(H_\lambda) S_\infty E_\Ic(H_\lambda) \geq O(\lambda)E_\Ic(H_\lambda). \end{eqnarray} We have no control on the sign. This is no surprise as we know that one may uncouple the two parts of the system and an eigenvalue can remain, see Section \ref{s:thre}. To control the sign, one needs to gain some positivity just above $P_{k_i}\otimes P_\Omega$. This would be the r\^ole of the Fermi golden rule and of the operator $B_\varepsilon$. \end{rem} Here we have used the elementary: \begin{lemma}\label{l:HS} Let $h\in\Cc^\infty_c(\R)$ and $s\leq 1/2$. Let $V$ be symmetric operator being $H_0$-form bounded operator, with constant lower than $1$. Then, there is $C$ such that \begin{eqnarray*} \big\|\langle H_0\rangle^s\big(h(H_0) - h(H_0+\lambda V)\big)\big\|\leq C |\lambda|. \end{eqnarray*} \end{lemma} \section{A Mourre estimate at the thresholds}\label{s:thre} \setcounter{equation}{0} In this section we would like to study the absence of eigenvalue above one of the thresholds. From a physical point of view, as soon as the interaction is on, one expects the embedded eigenvalues to disappear into the complex plane and to turn into resonances. This is however not mathematically true as one may uncouple the Bosonic Field and the atom. Take for instance $\omega$ bounded, $\alpha\in\Bc(\Kr, \Kr\!\otimes \hg)$, given by $\alpha(x):= 1\otimes b$, for all $x\in \Kr$ and where $\omega b\in \hg$. After a dressing transformation, see for instance \cite{D}[Theorem 3.5], the operator $H_\lambda$ is unitarly equivalent to the free operator $K\otimes \un_{\Gamma(\hg)}+ \un_\Kr\!\otimes \dG(\tilde \omega_\lambda)$, for some $\tilde \omega_\lambda \in \Bc(\hg)$. Therefore, $H_\lambda$ has the same eigenvalues as $H_0$ for all $\lambda$. Note that this is no restriction to suppose that $\omega$ is bounded thanks to the exponential law, see for instance \cite{BSZ}[Section 3.2]. We couple the two systems through a Fermi golden rule assumption. \subsection{The Fermi golden rule hypothesis}\label{s:FGR} We choose one eigenvalue $k_{i_0}$ of $H_{\rm{el}}$ for $i_0\pg 0$. Let $P:=P_{k_{i_0}}\otimes P_{\Omega}$ and let $\overline{P}:=1-P$. Note that $P$ is of finite rank. We give an implicit hypothesis on $\alpha$ and explain how to check it in Appendix \ref{s:level}. \begin{definition} We say that the \emph{Fermi golden rule} holds true at energy $k$ for a couple $(H_0, \alpha)$ if there exist positive $\varepsilon_0$, $c_1$ and $c_2$ such that \begin{eqnarray}\label{e:FGR0} c_1 P\geq P\phi(\alpha)\overline{P}\,\im(H_0-k +i\varepsilon)^{-1} \overline{P}\phi(\alpha) P\geq c_2 P, \end{eqnarray} holds true in the sense of forms, for all $\varepsilon_0 >\varepsilon>0$. \end{definition} Due to the Fock space structure, one may omit $\overline P$ in \eqref{e:FGR0} but we keep it to emphasize the link between hypotheses of this type in other fields (like for Schr\"odinger operators). Since $P$ is of finite rank, this property follows from \eqref{e:FGR0'}. The upper and the lower bounds of \eqref{e:FGR0} would be crucial in our analysis. We shall keep track of the lower bound in the sequel so as to emphasis the gain of positivity it occurs. We set few notations. \begin{eqnarray}\label{e:Rbar} R_\varepsilon:=\big((H_0-k_{i_0})^2+\varepsilon^2\big)^{-1/2},\, \overline{R_\varepsilon}:=\overline{P} R_\varepsilon \mbox{ and } F_\varepsilon := \overline{R_\varepsilon}^2. \end{eqnarray} Note that $\varepsilon R_\varepsilon^2=\im(H_0-k_{i_0} +i\varepsilon)^{-1}$ and that $R_\varepsilon$ commutes with $P$. We get: \begin{eqnarray}\label{e:FGR} (c_1/\varepsilon) P\geq P\phi(\alpha)F_\varepsilon\phi(\alpha) P\geq (c_2/\varepsilon) P, \end{eqnarray} for $\varepsilon_0 > \varepsilon > 0$. It follows: \begin{eqnarray}\label{e:norm} \|R_\varepsilon\|= 1/\varepsilon \mbox{ and } \|P\phi(\alpha)\overline R_\varepsilon\|\leq c_1^{1/2}\varepsilon^{-1/2}. \end{eqnarray} As pointed out in Remark \ref{r:motiv}, we seek some more positivity for the commutator above the energy $P=P_{k_i}\otimes P_\Omega$. We proceed like in \cite{BFSS} and set \begin{eqnarray*} B_\varepsilon :=\im (\overline{R_\varepsilon}^2\phi(\alpha)P). \end{eqnarray*} It is a finite rank operator, see Lemma \ref{l:finite} for more properties. Observe now that we gain some positivity as soon as $\lambda\neq 0$: \begin{eqnarray}\label{e:motiv} P[H_\lambda, i \lambda B_\varepsilon]P= \lambda^2 P \phi (\alpha) F_\varepsilon \phi (\alpha) P \geq (c_2\lambda^2 /\varepsilon) P. \end{eqnarray} It is therefore natural to modify our conjugate operator. We set \begin{eqnarray}\label{e:Ahat} \hat A_n:=A_n+\lambda \theta B_\varepsilon, \mbox{ for } n\in \N^*\cup\{\infty\}. \end{eqnarray} It depends on the two parameters $\lambda\in \R$, $\varepsilon>0$ and on an extra technical $\theta>0$. For the sake of clarity, we do not write these extra dependences. Heuristically, the operator $A_\infty$ would give the positivity around the threshold and the $B_\varepsilon$ would complete it just above. We mention that $\hat A_\infty$ is maximal symmetric and generates a semigroup of isometries, see Lemma \ref{l:semiA}. \subsection{Main result}\label{s:main} We prove the extended Mourre estimate over the threshold $k_{i_0}$. This is the heart of the paper. The proof relies on the Feshbach method. We exploit the freedom we have so far on $\varepsilon$ and $\theta$: set $\varepsilon:= \varepsilon(\lambda)$ and $\theta=:\theta(\lambda)$ and suppose that $\lambda=o(\varepsilon)$, $\varepsilon = o(\theta)$ and $\theta=o(1)$ as $\lambda$ tends to $0$. We summarize this into: \begin{eqnarray}\label{e:size} |\lambda|\,\, \ll\,\, \varepsilon \,\,\ll\,\, \theta\,\,\ll\,\, 1, \mbox{ as } \lambda \mbox{ tends to } 0. \end{eqnarray} In \cite{BFSS}, this condition is more involved and the size of the interval comes into the play. We stress that the conjugate operator $\hat A_\infty$ depends on these three parameters. \begin{theorem}\label{t:msFGR} Let $\Ic_0$ be a compact interval containing $k_{i_0}$ and no other $k_i$. Assume the Fermi golden rule hypothesis \eqref{e:FGR0} and \eqref{e:size} hold true. Suppose also that {\bf(I0)} and {\bf(I1a)} are satisfied. Then, for all open interval $\Ic\subset \Ic_0$: {\rm i)} There are $M_\infty\geq 1$ and $\hat S_\infty$ a $|H_\lambda|^{1/2}$-bounded operator such that $[H_\lambda, i\hat A_\infty]= M_\infty+\hat S_\infty$ holds in the sense of forms on $\Dc(N^{1/2})$. {\rm ii)} There is $\lambda_0>0$ such that the following extended Mourre estimate \begin{eqnarray}\label{e:msFGR} M_\infty + \hat S_\infty \geq \, a(\lambda) E_\Ic(H_\lambda) -b(\lambda)E_{\Ic^c}(H_\lambda)\langle H_\lambda \rangle \end{eqnarray} holds true in the sense of forms on $\Dc(N^{1/2})$, for all $\lambda \in(0,\lambda_0)$. Here, one has $a(\lambda)= \lambda^2\theta c_2 / 5\varepsilon$ and $b(\lambda)>0$. {\rm iii)} If {\bf (I1b)} holds true, then $H_\lambda$ has no eigenvalue in $\Ic$. {\rm iv)} If {\bf (I2)} holds true (and not necessarily {\bf (I1b)}), then $H_\lambda$ has no eigenvalue in the interior of $\Ic$, for all $|\lambda| \leq \lambda_0$. Moreover, one obtains the estimation of the resolvents given in Theorem \ref{t:intro1}. \end{theorem} \begin{rem} By taking $\theta$ and $\varepsilon$ as power of $\lambda$, one may take $a(\lambda)= \lambda^{1+\eta}/5$, for some $\eta>0$. We do not reach the power $1$ as expected in Remark \ref{r:motiv}. This is due to the non-linearity in $\lambda$ of the conjugate operator. Note also, this is very small and then one does not expect a fast propagation of the state, i.e.\ the eigenvalue turns into a resonance. See for instance \cite{BFS2, HHH} for some lifetime estimates. \end{rem} The proof of this theorem needs few steps and is given in Section \ref{s:ext}. We first go into the Feshbach method and deal with unperturbed spectral measure in Proposition \ref{p:est3}. Next, in Proposition \ref{p:est4}, we change the spectral measure. \subsection{The infrared decomposition} As suggested by \eqref{e:motiv}, one expects to have to slip the space with the projector $P$ to take advantage of this positivity. To do so, we use the Feshbach method. As our result is local in energy, we fix a compact interval $\Jc$ which contains the selected eigenvalue $k_{i_0}$ and no others. We consider the Hilbert space $\Hr_{\Jc}:= E_{\Jc}(H_0)\Hr$. Let $\Hr^{\rm{v}}_{\Jc}:=P\Hr_{\Jc}$ and $\Hr_{\Jc}^{\overline{\rm{v}}}$ its orthogonal in $\Hr_{\Jc}$. The $\rm{v}$ subscript stands for vacuum. Given $H$ bounded in $\Hr_\Jc=\Hr^{\overline{\rm{v}}}_{\Jc}\oplus \Hr^{\rm{v}}_{\Jc}$, we write it following this decomposition in a matricial way: \begin{eqnarray}\label{e:dec} H= \left(\begin{array}{ll} H^{\overline{\rv}\overline{\rv}}& H^{\overline{\rv} \rv}\\ H^{\rv\overline{\rv}}& H^{\rv\rv} \end{array}\right). \end{eqnarray} We recall the Feshbach method, see \cite{BFS} and see also \cite{DJ}[Section 3.2] for more results of this kind. \begin{proposition}\label{p:fesh} Assume that $z\notin\sigma(H^{\overline{\rv}\overline{\rv}})$. We define \begin{eqnarray*} G_\rv(z)&:=& z \un^{\rv\rv} - H^{\rv\rv}- H^{\rv\overline{\rv}}\big(z\un^{\overline{\rv}\overline{\rv}}- H^{\overline{\rv}\overline{\rv}} \big)^{-1}H^{\overline{\rv} \rv}. \end{eqnarray*} Then, $z\in\sigma(H)$ if and only if $0\in \sigma\big(G_\rv(z)\big)$. \end{proposition} The reader should keep in mind that $\Jc$ would be chosen slightly bigger than the interval $\Ic$. This lost comes from the change of spectral measure from $H_0$ to $H_\lambda$. The aim of the section is to show the following proposition about $\hat S_\infty$, see \eqref{e:M&R'}. \begin{proposition}\label{p:est3} Let $\Jc$ be a compact interval containing $k$ and no other $k_i$. Suppose the Fermi golden rule \eqref{e:FGR0} and \eqref{e:size}, then one has \begin{align}\label{e:est3} E_\Jc(H_0)\hat S_\infty E_\Jc(H_0)\geq (c_2\lambda^2\theta \varepsilon^{-1}/3 -1) E_\Jc(H_0) \end{align} holds true in the sense of forms, for $\lambda$ small enough. \end{proposition} We go through a series of lemmata and give the proof at the end of the section. The $-1$ of the r.h.s.\ seems at first sight disturbing as we seek for some positivity. It would be balanced when we will add the operator $M_\infty\geq 1$, see Section \ref{s:ext}. In the first place, we estimate the parts of $\hat S_\infty$. \begin{lemma}\label{l:commu'} With respect to the decomposition \eqref{e:dec}, as $\lambda$ goes to $0$, we have \begin{eqnarray*} E_\Jc(H_0)\big(\lambda\phi(a_\infty\alpha) - P\big)E_\Jc(H_0)= \left(\begin{array}{cc} O(\lambda)& O(\lambda)\\ O(\lambda)& -1 \end{array}\right). \end{eqnarray*} \end{lemma} \proof The part in $P$ follows directly from \eqref{e:H0p}. The one in $\alpha$ results from Proposition \ref{p:ntau} and the fact that $P\phi(a_\infty\alpha)P=0$.\qed \begin{lemma}\label{l:commu} Suppose that the Fermi golden rule \eqref{e:FGR0} holds true. Then, the form $[H_\lambda, B_\varepsilon]$ defined on $\Dc(H_\lambda)\times \Dc(H_\lambda)$ extends to a finite rank operator on $\Hr$, still denoted by $[H_\lambda, B_\varepsilon]$. As $\lambda$ tends to $0$, we have \begin{eqnarray}\label{e:commua} \quad \big\|\,[H_\lambda, \lambda\theta B_\varepsilon]\,\big\|_{\Bc(\Hr)} \leq O(\lambda\theta\varepsilon^{-1/2}) + O(\lambda^2\theta\varepsilon^{-3/2}). \end{eqnarray} Besides, with respect to the decomposition \eqref{e:dec}, we have: \begin{eqnarray*} E_\Jc(H_0)[H_0, \lambda\theta B_\varepsilon ]E_\Jc(H_0)= \left(\begin{array}{cc} 0& O(\lambda\theta\varepsilon^{-1/2})\\ O(\lambda\theta\varepsilon^{-1/2}) & 0 \end{array}\right) \end{eqnarray*} and \begin{eqnarray*} E_\Jc(H_0)[\lambda \phi(\alpha), \lambda\theta B_\varepsilon]E_\Jc(H_0)= \left(\begin{array}{cc} O(\lambda^2\theta\varepsilon^{-3/2})& O(\lambda^2\theta\varepsilon^{-3/2})\\ O(\lambda^2\theta\varepsilon^{-3/2}) & \lambda^2\theta F_\varepsilon \end{array}\right). \end{eqnarray*} \end{lemma} \proof We give some estimates independent of $\Jc$. We expand the commutators, this could be justified by considering the commutator in the form sense on $\Dc(H_\lambda)$. \begin{align}\nonumber [\dG(\omega), \overline{R_\varepsilon}^2\phi(\alpha)P]&=[H_0-k_{i_0}, \overline{R_\varepsilon}^2\phi(\alpha)P] \\\label{e:commu1} &\hspace*{-2cm}=\overline{P}(H_0-k_{i_0})R_\varepsilon \overline{R_\varepsilon}\phi(\alpha)P + \overline{P}R_\varepsilon \overline{R_\varepsilon}\phi(\alpha)P(H_0-k_{i_0}) = \overline{P}O(\varepsilon^{-1/2})P +0. \end{align} Indeed, the first term derives from \eqref{e:norm} and $\|(H_0-k_{i_0})R_\varepsilon\|=O(1)$. For the second one, note that $(H_0-k_{i_0})P=0$. We turn to the second estimation and apply Proposition \ref{p:ntau}. We get $\phi(\alpha)R_\varepsilon=\phi(\alpha) R_1R_1^{-1}R_\varepsilon = O(\varepsilon^{-1})$. By \eqref{e:norm}, we have \begin{align}\nonumber [\phi(\alpha), \overline{R_\varepsilon}^2\phi(\alpha)P]&= P F_\varepsilon P + \overline{P} \phi(\alpha)R_\varepsilon \overline{R_\varepsilon}\phi(\alpha)P+\overline{P} R_\varepsilon\overline{R_\varepsilon}\phi(\alpha) P\phi(\alpha)(P +\overline{P}) \\ \label{e:commu2} &= P F_\varepsilon P + \overline{P} O(\varepsilon^{-3/2})P+ \overline{P} O(\varepsilon^{-3/2})\overline{P}. \end{align} Gathering lines \eqref{e:commu1} and \eqref{e:commu2}, we get \eqref{e:commua}. We finish by adding $E_\Jc(H_0)$. \qed We go into the Feshbach method and conclude. \proof[Proof of Proposition \ref{p:est3}] We set $C_\lambda:=E_\Jc(H_0)\hat S_\infty E_\Jc(H_0)$. First observe that for all $\mu\leq -3/4$, we get $C_\lambda^{\rm{\overline{vv}}}- \mu$ is invertible in $\Bc(\Hr^{\overline{\rm{vv}}})$ and $\|(C_\lambda^{\rm{\overline{vv}}}- \mu)^{-1}\|_{\Bc(\Hr^{\overline{\rm{vv}}})} \leq 2$. Indeed, from Lemma \ref{l:commu'} and \ref{l:commu}, we have that $C_\lambda^{\rm{\overline{vv}}}$ is bounded from below by $O(\lambda^2\theta \varepsilon^{-3/2})+O(\lambda)$. This is bigger than $-1/2$ by \eqref{e:size}, for $\lambda$ small enough. We now estimate from below the internal energy of $C_\lambda$, uniformly in $\mu\leq 3/4$. By Lemmata \ref{l:commu'} and \ref{l:commu}, the first part and the Fermi golden Rule \eqref{e:FGR}, we infer \begin{align*} C_\lambda^{\rv\rv} - C_\lambda^{\rm{v\overline{v}}} (C_\lambda^{\rm{\overline{vv}}}- \mu)^{-1} C_\lambda^{\rm{\overline{v}v}}+1 \geq & \\ & \hspace*{-5cm} \geq c_2\lambda^2 \theta\varepsilon^{-1}+ \big(O(\lambda\theta\varepsilon^{-1/2})+ O(\lambda^2\theta \varepsilon^{-3/2})+ O(\lambda)\big)^2 \\ & \hspace*{-5cm} = c_2\lambda^2 \theta\varepsilon^{-1} \big(O(\theta)+ O(\lambda \theta\varepsilon^{-1}) +O(\varepsilon^{1/2}) + O(\lambda^2 \theta\varepsilon^{-2}) + O(\lambda \varepsilon^{-1/2}) + O(\theta^{-1}\varepsilon )\big) \\ & \hspace*{-5cm} \geq c_2\lambda^2\varepsilon^{-1}/2, \mbox{ for } \lambda \mbox{ small enough.} \end{align*} We have used \eqref{e:size} for the last line. We are now able to conclude. Since $\Jc$ contains $k_{i_0}$ and no other $k_i$. We have $E_\Jc(H_0)P_\Omega=P$ by \eqref{e:H0p}. Let $\mu< c_2\lambda^2\theta \varepsilon^{-1}/2 -1$. Note that $\mu \leq -3/4$ for $\lambda$ small enough by \eqref{e:size}. Thanks to the previous lower bound, we can apply Proposition \ref{p:fesh} with respect to the decomposition \eqref{e:dec} for $C_\lambda$ and with $z=\mu$ to get the result. \qed \subsection{The extended Mourre estimate}\label{s:ext} At the end of the section, we establish the extended Mourre estimate. We start by enhancing Proposition \ref{p:est3}. \begin{proposition}\label{p:est4} Let $\Ic$ be a compact interval containing $k_{i_0}$ and not other $k_i$. Assume the Fermi golden rule \eqref{e:FGR0} and \eqref{e:size}. Then, \begin{eqnarray*} E_\Ic(H_\lambda)\hat S_\infty E_\Ic(H_\lambda)\geq c_2(\lambda^2\theta\varepsilon^{-1}/4 -1) E_\Ic(H_\lambda) \end{eqnarray*} holds true in the sense of forms for $\lambda$ small enough. \end{proposition} \proof Let $\Jc$ be a compact interval as in Proposition \ref{p:est3} such that $\Ic$ is included in its interior and contains $k_{i_0}$. By \eqref{e:size}, it is enough to prove \begin{eqnarray}\nonumber E_\Ic(H_\lambda)\big(\lambda\phi(a_\infty\alpha)+ [H_\lambda, i\lambda\theta B_\varepsilon]- P_\Omega\big)E_\Ic(H_\lambda)&\geq&\\ \label{e:est4'} &&\hspace*{-6cm} \big(c_2 \lambda^2\theta\varepsilon^{-1}/3+O(\lambda^2)+ O(\lambda^2\theta \varepsilon^{-1/2}) + O(\lambda^3\theta \varepsilon^{-3/2})-1\big) E_\Ic(H_\lambda). \end{eqnarray} We start with the left hand side of \eqref{e:est4'} and introduce $E_{\Jc}(H_0)+E_{\Jc^c}(H_0)$ on the right and on the left of $([H_\lambda, i\lambda\theta B_\varepsilon] +\lambda\phi(a_\infty\alpha)- P_\Omega)$. Note that both of spectral measures are bounded in $\Dc(H_0)$, endowed with the graph norm. We need to control the mixed term. Using Lemma \ref{l:HS} and \eqref{e:commua}, we get \begin{align*} E_\Ic(H_\lambda)E_{\Jc^c}(H_0)[H_\lambda, i\lambda\theta B_\varepsilon ] E_{\Jc}(H_0)E_\Ic(H_\lambda) =&\\ &\hspace*{-2cm} \big(O(\lambda^2\theta \varepsilon^{-1/2}) + O(\lambda^3\theta \varepsilon^{-3/2})\big) E_\Ic(H_\lambda), \end{align*} and a better term for $ E_\Ic(H_\lambda)E_{\Jc^c}(H_0)[H_\lambda, i\lambda\theta B_\varepsilon ] E_{\Jc^c}(H_0)E_\Ic(H_\lambda)$. Since the term $\phi(a_\infty\alpha)\langle H_0 \rangle^{-1/2}$ is bounded in $\Hr$ by Proposition \ref{p:ntau}, Lemma \ref{l:HS} gives \begin{eqnarray*} E_\Ic(H_\lambda)E_{\Jc^c}(H_0)\lambda\phi(a_\infty\alpha) E_{\Jc}(H_0)E_\Ic(H_\lambda) = O(\lambda^2)E_\Ic(H_\lambda), \end{eqnarray*} and a better term for the full-mixed term. As $H_0$ commute with $P_\Omega$, we infer $ E_\Ic(H_\lambda)E_{\Jc^c}(H_0) P_\Omega E_{\Jc}(H_0)E_\Ic(H_\lambda) = 0$. Now using Proposition \eqref{p:est3} we obtain \begin{eqnarray*} E_\Ic(H_\lambda)\big([H_\lambda, i\lambda\theta B_\varepsilon ] +\lambda\phi(a_\infty\alpha)- P_\Omega\big)E_\Ic(H_\lambda)&\geq&\\ &&\hspace*{-4cm} (c_2 \lambda^2\theta\varepsilon^{-1}/3 -1)E_\Ic(H_\lambda)E_\Jc(H_0)E_\Ic(H_\lambda)\\ &&\hspace*{-4cm}+\big(O(\lambda^2) + O(\lambda^2\theta \varepsilon^{-1/2}) + O(\lambda^3\theta \varepsilon^{-3/2})\big) E_\Ic(H_\lambda). \end{eqnarray*} Finally, the estimation \eqref{e:est4'} follows by noticing that $E_\Ic(H_\lambda)E_\Jc(H_0) E_\Ic(H_\lambda)$ is equal to $\big(1+O(\lambda^2)\big)E_\Ic(H_\lambda)$, again by Lemma \ref{l:HS}.\qed We are now able to prove the announced result. \proof[Proof of Theorem \ref{t:msFGR}] The operator $M_\infty$ and $\hat S_\infty$ are given in \eqref{e:M&R} and \eqref{e:M&R'}. Points i) and ii) are given in Section \ref{s:1comm'}. By Proposition \ref{p:est4} and since $M_\infty\geq 1$, \begin{eqnarray*} M_\infty + E_\Ic(H_\lambda)\hat S_\infty E_\Ic(H_\lambda)\geq c_2\lambda^2\theta\varepsilon^{-1}/4\, E_\Ic(H_\lambda) \end{eqnarray*} holds true in the form sense on $\Dc(N^{1/2})$. Then, \eqref{e:eta} gives iii). The point iv) follows from the Virial Theorem, Proposition \ref{p:6.5}. Finally Theorem \ref{t:LAP} gives point v). Indeed, the space $\Gr$ appearing therein is identified in Lemma \ref{l:6.4}. In remains to notice that the spaces \eqref{e:Gs} given for $\hat A_\infty$ and $A_\infty$ are the same. This follows from the fact that these operators have the same domain in $\Gr^*$, by Lemma \ref{l:finite} and that the spaces $\Gr^*_s$ are given by complex interpolation. \qed \appendix \section{Level shift operator}\label{s:level} \renewcommand{\theequation}{A.\arabic{equation}} \setcounter{equation}{0} In this paper, we never make the hypothesis that we analyse an eigenvalue which could be different than the ground state energy of $H_0$. The point is that it is well known that it is supposed to remain, even if the perturbation is switched on, see for instance \cite{AH, BFS, G0}. This leads to a contradiction to the hypothesis made on the Fermi golden rule. Therefore, in this section, we explain how one may check the Fermi golden rule assumption \eqref{e:FGR0}, why it is not fulfilled at ground state energy. This would also explain the compatibility with {\bf (I0)}--{\bf (I2)}. The computations we lead are standard, we keep it simple. See also \cite{BFS, DJ2, JP}. Let $e_i$ be an orthonormal basis of eigenvectors of $K$ relative to the eigenvalue $k_i$. To simplify the computation, say that $k_{i_0}$ is simple. Since $k_{i_0}$ is simple and since $\phi(\alpha)(e_{i_0}\otimes \Omega)= \alpha(e_{i_0})\in \Kr\!\otimes \hg$, \eqref{e:FGR0} is equivalent to: \begin{eqnarray*} c_1 \geq \big\langle \alpha(e_{i_0}), \im(H_0-k_{i_0} +i\varepsilon)^{-1}\alpha(e_{i_0}) \big\rangle \geq c_2>0, \mbox{ for } 0< \varepsilon \leq \varepsilon_0. \end{eqnarray*} We have $\alpha(e_{i_0})= \sum_{i=1,\ldots, n} e_i \otimes f_{i, i_0}\in \Kr\!\otimes \hg$, where $f_{i, i_0}=\big\langle e_i \otimes \un_{\hg}, \alpha(e_{i_0})\big\rangle$. As $\hg=L^2(\R^d,dk)$, we write $f_{i, i_0}$ as a function of $k$. We go into polar coordinates, see \eqref{e:T} and infer \begin{eqnarray*} c_1 \geq \sum_{i=1,\ldots, n} \int_0^\infty \int_{S^{d-1}} \varepsilon \frac{|f_{i, i_0}|^2(r\theta) r^{d-1}} {(r +\lambda_i-\lambda_{i_0} )^2+\varepsilon^2} \, d\sigma \, dr \geq c_2>0 \end{eqnarray*} Suppose now that $(r, \theta)\mapsto |f_{i, i_0}|^2(r\theta) r^{d-1}$ is continuous and in $L^1$. Then by dominated convergence, we let $\varepsilon$ go to zero and get: \begin{eqnarray}\label{e:FGR'} \quad\quad c_1 \geq \sum_{i=1,\ldots, i_0} c_i (\lambda_{i_0}-\lambda_i)^{d-1} \int_{S^{d-1}} \varepsilon |f_{i, i_0}|^2\big(\theta (\lambda_{i_0}-\lambda_i)\big) \, d\sigma \geq c_2>0 \end{eqnarray} Here note that, up to the constant $c_i$, $r\mapsto \varepsilon/\big((r +\lambda_i-\lambda_{i_0})^2+\varepsilon^2\big)$ is a Dirac sequence if and only if $\lambda_i\leq\lambda_{i_0}$. To satisfy the Fermi golden rule, it is enough to have a non-zero term in \eqref{e:FGR'}. When $d\geq 2$, we stress that the sum is taken till $i_0 -1$ and therefore is empty at ground state energy. When the $1$-particle space is over $\R$, it cannot be satisfied at this level of energy as well. Indeed, one would obtain a contradiction with the hypothesis {\bf{(I0)}} and the continuity of $(r, \theta)\mapsto |f_{i, i_0}|^2(r\theta)$. \section{Properties of $C_0$-semigroups}\label{s:semiprop} In this section, we gather various facts about $C_0$-semigroups we use along this article. Let $\Hr$ be a Hilbert space. Recall that $\wlim$ denotes the weak limit. \begin{definition}\label{d:semi} We say $\R^+\ni t\mapsto W_t$, with $W_t\in\Bc(\Hr)$ is a \emph{$C_0$-semigroup} if \begin{enumerate} \item $W_0=\id$ and $W_{s+t}=W_sW_t$, for all $s,t\geq 0$, \item $\wlim_{t\rightarrow 0^+} W_t=\id$. \end{enumerate} \end{definition} Automatically, this implies that $\R^+\supset t\mapsto W_t$ is strongly continuous, see \cite{HP}[Theorem 10.6.5]. We keep the convention of \cite{GGM2} and define the generator of $\{W_t\}_{t\geq 0}$ as being the closed densely defined operator $A$ defined on \begin{eqnarray*} \Dc(A):=\{u\in\Hr \mid \lim_{t\rightarrow 0^+}(it)^{-1}(W_t-\id)u \mbox{ exists} \}. \end{eqnarray*} We set $Au$ this limit. Formally, one reads $W_t=e^{itA}$. The map $\R^+\supset t\mapsto W_t^*$ being weakly continuous, $\{W_t^*\}_{t\geq 0}$ is also a $C_0$-semigroup. Its generator is $-A^*$. We recall the \emph{Nelson Lemma}, see for instance \cite{BR}[Corollary 3.1.7]. \begin{lemma}[Nelson Lemma]\label{l:Nelson} Let $\Dc$ be a dense subset of $\Hr$ and let $\{W_t\}_{t\geq 0}$ be a $C_0$-semigroup. If $W_t \Dc\subset \Dc$ then $\Dc$ is a core for the generator of $\{W_t\}_{t\geq 0}$. \end{lemma} Let $\Gr$ and $\Hr$ be two Hilbert spaces such that $\Gr\subset \Hr$ continuously and densely. Using the Riesz isomorphism, we identify $\Hr$ with $\Hr^*$, where the latter is the set of anti-linear forms acting on $\Hr$. We infer the following scale of spaces $\Gr\subset \Hr \simeq \Hr^* \subset \Gr^*$ with continuous and dense embeddings. In order to define the restriction of $W_t$ on $\Gr$, we set: \begin{definition}\label{d:bstable} Given a $C_0$-semigroup $\{W_t\}_{t\geq 0}$ on $\Hr$. We say that $\Gr$ is \emph{b-stable} (boundedly stable) under the action of $\{W_t\}_{t\geq 0}$ if {\rm i)} $W_t\Gr\subset \Gr$, for all $t\in \R^+$, {\rm ii)} $\sup_{t\in [0,1]}\|W_t u\|$ is bounded for all $u\in\Gr$. \end{definition} \begin{rem}\label{r:bstable} Note that unlike for $C_0$-groups, the second condition is required to ensure the continuity in $0$. These two conditions are equivalent to the fact that $\{W_t |_{\Gr}\}_{t\geq 0}$ is a $C_0$-semigroup on $\Gr$. \end{rem} Assuming that $\Gr$ is b-stable under the action of $\{W_t\}_{t\geq 0}$, we denote by $A_{\Gr}$ its generator. Thus, $A_\Gr$ is the restriction of $A$ and its domain is given by \begin{eqnarray*} \Dc(A_{\Gr})=\{u\in \Gr\cap \Dc(A) \mid Au\in\Gr \}. \end{eqnarray*} If $\Gr^*$ is also b-stable under $\{W_t^*\}_{t\geq 0}$, we denote by $A_{\Gr^*}$ the generator of $\{W_t\}_{t\geq 0}$ extended to $\Gr^*$. As above $A$ is a restriction of $A_{\Gr^*}$ and thanks to the Nelson lemma, we have that $A$ is the closure of $A_\Gr$ in $\Hr$ and that $A_{\Gr^*}$ is the closure of $A$ in $\Gr^*$. We would drop the subscript $\Gr$ when no confusion could arise. We recall the following result of perturbation, see \cite{kato}[Theorem IX.2.1]. \begin{proposition}\label{p:kato} Let $B$ be a bounded operator in a Hilbert space $\Hr$. Then $A$ is the generator a $C_0$-semigroup if and only if $A+B$ is also one. \end{proposition} \section{The Mourre method}\label{s:ms} \renewcommand{\theequation}{C.\arabic{equation}} \setcounter{equation}{0} \subsection{The $C^1$ class.}\label{s:C1} Given a self-adjoint operator $A$, the so-called $C^1(A)$ class of regularity is a key notion within the Mourre's theory, see \cite{ABG} and \cite{GG}. This guarantees some properties of domains and that the commutator of an operator $H$ with $A$ would be $H$-bounded. In this paper, we have to deal with maximal symmetric conjugate operators and thus have to extend the standard class exposed in details in \cite{ABG}[Section 6.2]. As some refinements appear, we present an overview of the properties and refer to \cite{GGM}[Section 2] for proofs. Within this section, we consider a closed densely defined operator $A$ acting in a Hilbert space $\Hr$. Note this implies that $\Dc(A^*)$ is dense in $\Hr$. We first defined the class of bounded operators belonging to $\Cc^1(A)$. Let $S\in\Bc(\Hr)$. We denote by $[S,A]$ the sesquilinear form defined on $\Dc(A^*)\times\Dc(A)$ by \begin{eqnarray*} \langle u, [S,A] v\rangle := \langle A^* u, S v\rangle - \langle S^*u, Av\rangle, \mbox{ for } u\in\Dc(A^*), v\in\Dc(A). \end{eqnarray*} \begin{definition}\label{d:C1bd} An operator $S\in\Bc(\Hr)$ belongs to $\Cc^1(A)$ if the sesquilinear form $[S,A]$ is continuous for the topology of $\Hr\times\Hr$. We denote by $[S,A]^\circ$ the unique bounded operator in $\Hr$ extending this form. \end{definition} We now extend the definition to unbounded operator by asking the resolvent $R(z):=(S-z)^{-1}$ to be $\Cc^1(A)$. We precise the statement. We first recall that given $S$ a closed densely defined operator on $\Hr$, the \emph{$A$-regular resolvent set} of $S$ is the set $\rho(S,A)\subset \C\setminus\sigma(S)$ such that $R(z)$ is of class $\Cc^1(A)$. \begin{definition} Let $S$ be a closed and densely defined operator on $\Hr$. We say that \emph{$S$ is of class $\Cc^1(A)$} if there are a constant $C$ and a sequence of complex numbers $z_\nu\in\rho(S,A)$ such that $|z_\nu|\rightarrow \infty$ and $\|R(z_\nu)\|\leq C|z_\nu|^{-1}$. If $S$ is of class $\Cc^1(A)$ and $\rho(S,A)=\C\setminus \sigma(S)$ then we say that $S$ is of \emph{full class $\Cc^1(A)$}. \end{definition} In many cases these two definitions coincide. Indeed, given $S\in\Cc^1(A)$, one shows that if $A$ is regular or if $S$ is self-adjoint with a spectral gap then $S$ is in the full class $\Cc^1(A)$. We recall that a closed densely defined operator $B$ is \emph{regular} if there is a constant $C$ and $\alpha_n\in\C\setminus \sigma(B)$ such that $\|(B-\alpha_n)\|\leq C|\alpha_n|^{-1}$ and such that $|\alpha_n|\rightarrow \infty$. The generators of $C_0$-semigroups are regular for instance. \begin{definition} Let $A$ and $S$ be two closed and densely defined operators in $\Hr$. We define $[A,S]$ as the sesquilinear form acting on $\big(\Dc(A^*)\cap \Dc(S^*)\big) \times \big(\Dc(A)\cap \Dc(S)\big)$ and given by $\langle u, [S,A] v\rangle := \langle A^* u, Sv\rangle - \langle S^*u, A v\rangle.$ \end{definition} \begin{proposition}\label{p:ext} Let $S\in\Cc^1(A)$. Then $\Dc(A^*)\cap \Dc(S^*)$ and $\Dc(A)\cap \Dc(S)$ are cores for $S$ and $S^*$ respectively and the form $[A,S]$ has a unique extension to a continuous sesquilinear form denoted by $[A,S]^\circ$ on $\Dc(S^*)\cap \Dc(S)$. Moreover, \begin{eqnarray*} [A, R(z)]^\circ=-R(z)[A,S]^\circ R(s), \mbox{ for all } z\in\rho(S,A), \end{eqnarray*} where on the right hand side, $[A,S]^\circ$ is considered as an element of $\Bc\big(\Dc(S), \Dc(S^*)\big)$. \end{proposition} We stress the fact that $[A,S]$ extends to an element of $\Bc\big(\Dc(S), \Dc(S^*)\big)$ is not enough to ensure $S\in\Cc^1(A)$, see \cite{GG}. Some conditions of compatibilities are to be added, see \cite{GGM}[Proposition 2.21]. This could also be bypassed by knowing some invariance under a $C_0$-semigroup generated by $A$. \begin{definition} Let $\{W_{1,t}\}_{t\in\R^+}, \{W_{2,t}\}_{t\in\R^+}$ be two $C_0$-semigroups on the Hilbert spaces $\Hr_1$ and $\Hr_2$ with generator $A_1$ and $A_2$. We say that $B\in\Bc(\Hr_1, \Hr_2)$ is of \emph{class $\Cc^1(A_1, A_2)$} if: \begin{eqnarray*} \|W_{2,t}S- SW_{1,t} \|_{\Bc(\Hr_1, \Hr_2)}\leq ct, \, 0\leq t\leq 1. \end{eqnarray*} \end{definition} If $\Gr\subset \Hr$ are two Hilbert spaces continuously and densely embedded and if a $C_0$-semigroup $\{W_t\}_{t\in\R^+}$, with generator $A$ on $\Hr$, $b$-stabilizes $\Gr$ and $\Gr^*$, we denote the class $\Cc^1(A_\Gr, A_{\Gr^*})$ by $\Cc^1(A;\Gr, \Gr^*)$. We have the following result. \begin{proposition}\label{p:2comm} $S\in\Cc^1(A_1, A_2)$ if and only if the sesquilinear form ${}_2[S,A]_1$ on $\Dc(A_2^*)\times \Dc(A_1)$ defined by $\langle u_2, {}_2[S,A]_1 u_1\rangle:= \langle S^* u_2, A_1 u_1\rangle- \langle A^*_2u_2, S u_1\rangle $ is bounded for the topology of $\Hr_2\times \Hr_1$. Let ${}_2[S,A]_1^\circ$ be the closure of this form in $\Bc(\Hr_1, \Hr_2)$. We have: \begin{eqnarray*} {}_2[S,A]_1^\circ=\slim_{t\rightarrow 0^+}(S W_{1,t}- W_{2,t}S). \end{eqnarray*} \end{proposition} Note that for $S\in \Bc(\Hr)$, with $\Hr_i=\Hr$ and $W_{i,t}=W_t$, one has $S\in \Cc^1(A_1, A_2)$ if and only if $B\in\Cc^1(A)$. \subsection{Regularity assumptions for the limiting absorption principle}\label{s:LAPhyp} In this part, we recall a set of assumptions presented in \cite{GGM} so as to ensure a limiting absorption principle, see Theorem \ref{t:LAP}. Consider $H$ a self-adjoint operator, $H'$ symmetric closed and densely defined and $A$ closed and densely defined. These operators are linked by $H'=[H,iA]$ in a sense defined lower. Denote also $\Dr:=\Dc(H)\cap \Dc(H')$ endowed with the intersection topology, namely the topology associated to the norm $\|\cdot\|+ \|H \cdot\|+\|H'\cdot\|$. We start by some assumptions on $H$ and on $H'$. \begin{itemize} \item[{\bf (M1)}] \emph{$H$ is of full class $\Cc^1(H')$, $\Dr=\Dc(H)\cap \Dc(H'^*)$ and this is a core for $H'$.} \item[{\bf (M2)}] \emph{There are $\Ic\subset\R$ open and bounded and $a,b\pg 0$ such that} \begin{equation}\label{e:mstrict} H'\geq \big(a \un_\Ic(H) -b \un_{\Ic^c}(H) \big)\langle H\rangle \end{equation} \emph{holds true in the sense of forms on $\Dr$.} \end{itemize} The last one is the \emph{strict Mourre estimate}. In order to check the first hypothesis, we rely on \cite{GGM}[Lemma 2.26], see also \cite{Ski}[Lemma 2.6]: \begin{lemma}\label{l:avoid} Let $H,M$ be self-adjoint operators such that $H\in\Cc^1(M)$ and that $\Dc(H)\cap \Dc(M)$ is a core of $M$. Let $R$ be a symmetric operator such that $\Dc(R)\supset \Dc(H)$. Set $H'$ the closure of $M+R$ defined on $\Dc(R)\cap\Dc(M)$. Then $H$ is of full class $\Cc^1(H')$ and $\Dc(H)\cap\Dc(H')$ is a core for $H'$ and $\Dc(H)\cap \Dc(H')= \Dc(H)\cap \Dc(H'^*)=\Dc(H)\cap \Dc(M)$. \end{lemma} Assuming {\bf (M2)}, one chooses $c\pg 0$ such that $H'+c \langle H\rangle \geq \langle H\rangle$ (take for instance $c=b+1$). Since $H'+c \langle H\rangle$ is symmetric and positive, it possesses a Friedrichs extension $G\geq \langle H\rangle$. We name the form domain of $G$: \begin{eqnarray}\label{e:G} \Gr:=\Dc(G^{1/2}), \mbox{ endowed with the graph norm } \|\cdot\|_\Gr. \end{eqnarray} Note that $\Gr$ is also obtained by completing the space $\Dr$ with the help of the norm $\|u\|_\Gr= \sqrt{\langle u, (H'+c \langle H\rangle)u \rangle}$. We identify these spaces in Lemma \ref{l:6.4}. We now recall the dual norm $\|\cdot\|_{\Gr^*}$ of $\Gr$. Given $u\in\Hr$, we set \begin{eqnarray}\label{e:dual} \|u\|_{\Gr^*}:=\sup_{v\in\Dr, \, \|v\|_{\Gr}\leq 1} |\langle u, v\rangle|= \|G^{-1/2}u\|. \end{eqnarray} Using the Riesz isomorphism, we identify $\Hr$ with $\Hr^*$ the space of anti-linear forms on $\Hr$. The space $\Gr^*$ is given by the completion of $\Hr$ with respect to the norm $\|\cdot\|_{\Gr^*}$. We get the following scale space: \begin{eqnarray*} \Dr\, \subset \, \Gr \, \subset \, \Hr\simeq \Hr^*\, \subset \, \Gr^* \, \subset \, \Dr^*, \end{eqnarray*} with dense and continuous embeddings. We turn to the assumptions concerning the conjugate operator $A$ and higher commutators. Suppose $A$ to be the generator of $\{W_t\}_{t\in\R^+}$ \begin{itemize} \item[{\bf (M3)}] \emph{The $C_0$-semigroup $\{W_t\}_{t\in\R^+}$ is of isometries and $b$-stabilizes $\Gr$ and $\Gr^*$,} \item[{\bf (M4)}] \emph{$H\in\Cc^1(A; \Gr, \Gr^*)$,} \item[{\bf (M5)}] \emph{$H\in\Cc^{1,1}(A; \Gr, \Gr^*)$.} \end{itemize} The hypothesis {\bf (M4)} implies that \begin{eqnarray*} \lim_{t\rightarrow 0^+}\big(\langle u, W_t Hu\rangle - \langle Hu, W_t u\rangle\big)= \langle u, H'u\rangle, \mbox{ for all } u\in\Dr. \end{eqnarray*} The hypothesis {\bf (M5)} means that $H\in \Bc(\Gr, \Gr^*)$ and that \begin{eqnarray*} \int_0^1 \big\|\, [W_t, [W_t, H]]\, \big\|_{\Bc(\Gr, \Gr^*)}\, \frac{dt}{t^2} <\infty. \end{eqnarray*} This is equivalent to the fact that $H$ belongs to $\big(\Cc^{2}(A; \Gr, \Gr^*), \Bc(\Gr, \Gr^*)\big)_{1/2, 1}$. We refer to \cite{ABG, Tri} for real interpolation. One may also consider the stronger $H'\in\Cc^1(A; \Gr, \Gr^*)$, i.e.\ \begin{itemize} \item[{\bf (M5')}] \emph{$H\in\Cc^{2}(A; \Gr, \Gr^*)$.} \end{itemize} We now give the result. Let $A_{\Gr^*}$ be the generator of $\{W_t\}_{t\in\R^+}$ generator in $\Gr^*$. For $s\in(0,1)$, we set: \begin{eqnarray}\label{e:Gs} \Gr^*_s:=\Dc\big(|A_{\Gr^*}|^s\big) \mbox{ and } \Gr_{-s}:=(\Gr^*_s)^*. \end{eqnarray} Here, the absolute value is taken with respect to the Hilbert structure of $\Gr^*$. Given $\Jc$ an interval, we define $J_0^{\pm}:=\{\lambda\pm i \mu, \lambda\in \Jc \mbox{ and } \mu >0\}$. Finally, set $R(z):=(H-z)^{-1}$. From \cite{GGM}, we obtain: \begin{theorem}\label{t:LAP} Assume that {\bf(M1)}--{\bf(M5)} hold true. Let $\Jc$ be a compact interval included in $\Ic$. Then if $z\in \Jc_0^\pm$, $R(z)$ induces a bounded operator in $\Bc(\Gr^*_s, \Gr_{-s})$, for all $s\in (1/2, 1]$. Moreover the limit $R(\lambda\pm i0)= \lim_{\mu\rightarrow \pm 0} R(\lambda+i\mu)$ exists in the norm topology of $\Bc(\Gr^*_s, \Gr_{-s})$, locally uniformly in $\lambda\in \Jc$ and the maps $\lambda\mapsto R(\lambda \pm i0)\in\Bc(\Gr^*_s, \Gr_{-s}) $ are H\"older continuous of order $s-1/2$. \end{theorem} This theorem can be improved by considering weights in some Besov spaces related to the conjugate operator. We refer to \cite{GGM} for more details. 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